Beyond the Floquet theorem: generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser fields

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Physics Reports 390 (2004) 1 – 131 www.elsevier.com/locate/physrep

Beyond the Floquet theorem: generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser 'elds Shih-I Chua; b;∗ , Dmitry A. Telnovc a

Department of Chemistry, University of Kansas, 2010 Malott Hall, 1251 Wescoe Hall Dr., Lawrence, KS 66045-7582, USA b Kansas Center for Advanced Scienti(c Computing, Lawrence, KS 66045, USA c Department of Physics, St. Petersburg State University, St. Petersburg 198504, Russia Accepted 7 October 2003 editor: J. Eichler

Abstract The advancement of high-power and short-pulse laser technology in the past two decades has generated considerable interest in the study of multiphoton and very high-order nonlinear optical processes of atomic and molecular systems in intense and superintense laser 'elds, leading to the discovery of a host of novel strong-'eld phenomena which cannot be understood by the conventional perturbation theory. The Floquet theorem and the time-independent Floquet Hamiltonian method are powerful theoretical framework for the study of bound–bound multiphoton transitions driven by periodically time-dependent 'elds. However, there are a number of signi'cant strong-'eld processes cannot be directly treated by the conventional Floquet methods. In this review article, we discuss several recent developments of generalized Floquet theorems, formalisms, and quasienergy methods, beyond the conventional Floquet theorem, for accurate nonperturbative treatment of a broad range of strong-'eld atomic and molecular processes and phenomena of current interests. Topics covered include (a) arti'cial intelligence (AI)—most-probable-path approach (MPPA) for e9ective treatment of ultralarge Floquet matrix problem; (b) non-Hermitian Floquet formalisms and complex quasienergy methods for nonperturbative treatment of bound–free and free–free processes such as multiphoton ionization (MPI) and above-threshold ionization (ATI) of atoms and molecules, multiphoton dissociation (MPD) and above-threshold dissociation (ATD) of molecules, chemical bond softening and hardening, chargeresonance enhanced ionization (CREI) of molecular ions, and multiple high-order harmonic generation (HHG), etc.; (c) many-mode Floquet theorem (MMFT) for exact treatment of multiphoton processes in multi-color laser 'elds with nonperiodic time-dependent Hamiltonian; (d) Floquet–Liouville supermatrix (FLSM) formalism for exact nonperturbative treatment of time-dependent Liouville equation (allowing for relaxations and dephasing mechanisms) and high-order nonlinear optical processes (such as intensity-dependent nonlinear optical susceptibilities and multiphoton resonance >uorescence, etc.); (e) generalized Floquet approaches for ∗

Corresponding author. Fax: +1-785-864-5396. E-mail addresses: [email protected] (S.-I Chu), [email protected] (D.A. Telnov). c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter  doi:10.1016/j.physrep.2003.10.001

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the treatment of nonadiabatic and complex geometric phases involving multiphoton transitions; (f) generalized Floquet techniques for the treatment of multiphoton processes in intense laser pulse 'elds with nonperiodic time-dependent Hamiltonians; (g) Floquet formulations of time-dependent density functional theory (DFT) and time-dependent current DFT for nonperturbative treatment of multiphoton processes of many-electron quantum systems in periodic or polychromatic (quasiperiodic) laser 'elds. For each generalized Floquet approach, we present also the corresponding development of new computational techniques for facilitating the study of strong-'eld processes and phenomena. The advancement of these generalized Floquet formalisms and quasienergy methods provides powerful new theoretical frameworks and accurate computational methods for nonperturbative and ab initio treatment of a wide range of interesting and challenging laser-induced chemical and physical processes and insightful exploration of strong-'eld atomic and molecular physics. c 2003 Elsevier B.V. All rights reserved.
PACS: 32.80.−t; 33.80.−b; 42.65.−k

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Floquet theorem and general properties of quasienergy states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The Floquet theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. General properties of quasienergy states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Time-independent Floquet Hamiltonian method: stationary treatment of periodically time-dependent SchrGodinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Quasienergy diagram and multiphoton excitation of molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Beyond the Floquet theorem: generalized Floquet formalisms and quasienergy methods . . . . . . . . . . . . . . . . . . . . . . . 5. Arti'cial intelligence in multiphoton dynamics: most-probable-path approach for ultralarge Floquet matrix problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Non-Hermitian Floquet formalisms and complex quasienergy methods for multiphoton ionization and dissociation in monochromatic 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Non-Hermitian Floquet formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Non-Hermitian Floquet calculations by L2 basis set expansion methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Non-Hermitian Floquet calculations by complex-scaling generalized pseudospectral methods . . . . . . . . . . . . . . 6.3.1. Uniform complex scaling—generalized pseudospectral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2. Exterior complex scaling—generalized pseudospectral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Complex-scaling generalized pseudospectral method for two-center systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Applications of non-Hermitian Floquet methods: atomic multiphoton processes in strong 'elds . . . . . . . . . . . . . . . . 7.1. Multiphoton and above-threshold ionization of atomic hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Intensity-dependent threshold shift and ionization potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. AC Stark shifts of Rydberg states in strong 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Intensity- and frequency-dependent multiphoton detachment of H − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1. Intensity-dependent multiphoton detachment rates and AC Stark shifts of H − . . . . . . . . . . . . . . . . . . . 7.4.2. Averaged multiphoton detachment rates: comparison of theoretical and experimental results . . . . . . . 7.5. Above-threshold multiphoton detachment of negative ions: angular distributions and partial widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. General expressions for the photoelectron energy distributions and partial rates . . . . . . . . . . . . . . . . . . 7.5.2. Multiphoton detachment of H − near one-photon threshold: exterior complex-scaling calculations . . . 7.6. Precision calculation of two-photon detachment of H − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 6 6 8 8 12 13 14 15 15 16 18 19 21 23 26 27 30 33 34 35 36 38 39 40 43

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131 8. Applications of non-Hermitian Floquet methods: molecular multiphoton processes in strong 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Multiphoton and above-threshold dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Nature of chemical bond in strong 'elds: laser induced chemical bond softening and hardening . . . . . . . . . . 8.3. Charge resonance enhanced multiphoton ionization of molecular ions in intense low-frequency laser 'elds . 9. Many-mode Floquet theorem for nonperturbative treatment of multiphoton processes in multi-color or quasi-periodic laser 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Many-mode Floquet theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. SU(N ) dynamical symmetries and nonlinear coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Fractal character of quasienergy states in multi-color or quasi-periodic 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Multiphoton above-threshold ionization in two-color laser 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1. Incommensurate frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Commensurate frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3. Multiphoton detachment of H − in two-color laser 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Chemical bond hardening and molecular stabilization in two-color laser 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6. High-order harmonic generation in two-color laser 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1. Non-Hermitian Floquet treatment of HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2. Two-color phase control of HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Floquet–Liouville supermatrix formalism for nonlinear optical processes in intense laser 'elds . . . . . . . . . . . . . . . . 10.1. The FLSM formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Intensity-dependent generalized nonlinear optical susceptibilities and multiple wave mixings . . . . . . . . . . . . . . 10.2.1. Exact FLSM nonperturbative treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2. High-order nearly degenerate perturbative treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Multiphoton resonance >uorescence in intense laser 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Floquet study of nonadiabatic and complex geometric phases in multiphoton transitions . . . . . . . . . . . . . . . . . . . . . . 11.1. Cyclic quantum evolution and nonadiabatic geometric phases for spin-j systems driven by periodic 'elds . . 11.2. Biorthogonal density matrix formulation of complex geometric phases for dissipative systems and nonlinear optical processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1. Geometric representation of non-Hermitian SchrGodinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2. Complex geometric phase in dissipative two-level systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3. Complex geometric phase for multiphoton transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Generalized Floquet approaches for multiphoton processes in intense laser pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Nonadiabatic coupled dressed-states formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. Multiphoton adiabatic inversion of multilevel systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3. A stationary formulation of time-dependent Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4. Adiabatic Floquet approach to multiphoton detachment of negative ions by intense laser pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1. General expressions for electron energy distributions in multiphoton above-threshold detachment . . . 12.4.2. Adiabatic approximation for smooth laser pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Generalized Floquet formulation of time-dependent density functional theory for many-electron quantum systems in intense laser 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. Generalized Floquet formulation of time-dependent density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1. Periodic 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2. Multi-color or quasiperiodic 'elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2. Generalized Floquet formulation of time-dependent current-density-functional theory . . . . . . . . . . . . . . . . . . . . . 13.3. Non-Hermitian Floquet formulation of TDDFT and TDCDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4. Exact relations of quasienergy functional in the Floquet formulation of TDDFT . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1. Time derivatives of kinetic, potential, and exchange-correlation energies . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2. Virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5. Applications of Floquet-TDDFT formalism to multiphoton ionization problems . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1. Multiphoton ionization of He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

44 44 47 51 56 56 58 61 65 65 67 67 68 70 71 74 76 76 80 80 82 85 89 89 92 93 95 96 98 99 101 103 105 105 107 108 110 110 113 113 116 117 117 119 120 120

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13.5.2. Multiphoton detachment of Li− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 123 124 124

1. Introduction The advancement of high-power and short-pulse laser technology in the last two decades has greatly facilitated the exploration of atomic and molecular multiphoton and very high-order nonlinear optical processes, leading to the discovery of a number of novel strong-'eld phenomena, such as multiphoton and above-threshold ionization (MPI/ATI) of atoms, multiphoton and above-threshold dissociation (MPD/ATD) of molecules, multiple high-order harmonic generation (HHG), chemical bond softening and hardening, nonsequential double ionization, Coulomb explosion, and coherent control of chemical and physical processes, etc. [1–14]. These experimental advancements have stimulated considerable e9orts in the development of new theoretical and computational methods for nonperturbative investigation of the electronic structure and quantum dynamics of atomic and molecular systems in the presence of intense and super intense laser 'elds. There are two general nonperturbative approaches currently widely used in the study of strong-'eld atomic and molecular physics. The 'rst is the stationary treatment of the time-dependent SchrGodinger equation. In particular, the development of generalized Floquet formalisms allows the reduction of the periodical or quasiperiodical time-dependent SchrGodinger equation into a set of time-independent coupled equations or Floquet matrix eigenvalue problem. The Floquet methods have been applied to a wide range of atomic and molecular multiphoton processes in the last two decades. Some of these works can be found in the earlier Floquet review articles [15–20]. The second general approach is to solve numerically the time-dependent SchrGodinger equation directly in space and time. The advantage of the time-dependent approach is that it can be applied directly to the problems of multiphoton excitation with arbitrary laser pulse shape and duration. However, generalized Floquet formalisms have now been also developed, allowing stationary treatment of laser pulse excitation problems [21,22]. In this article, we shall con'ne our discussion to the time-independent generalized Floquet formalisms and associated computational methods. For time-dependent treatment of intense-'eld processes of one-electron or single-active-electron systems, the volume [23] contains a good collection of di9erent time-dependent techniques up to 1991. The direct numerical solution of time-dependent SchrGodinger equation is currently feasible only for one- and two-electron systems in strong 'elds [23,24]. Even for two-electron systems, involving 6D time-dependent partial di9erential equations, converged calculations are rather dif'cult to achieve within the current computer technology. However, the recent developments of self-interaction-free time-dependent density functional theory (TDDFT) [25–27] and time-dependent generalized pseudospectral techniques [28,29] allow comprehensive nonperturbative treatment of multiphoton processes of many-electron atomic [26,30,31] and molecular [27,32] systems under arbitrary laser pulse 'elds. The literature on the subject related to Floquet theories is expanding rather rapidly in recent years. In this review article, we can only survey a subset of the literature with primary emphasis on the

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Fig. 1. Overview of the scope of recent generalized Floquet developments and various atomic and molecular processes amenable to these new Floquet treatments.

recent advancement of generalized Floquet formalisms and their applications to multiphoton and high-order nonlinear optical processes of atomic and molecular systems in intense laser 'elds. Other aspects of Floquet developments for intense 'eld problems not covered in this review such as the R-matrix Floquet theory [33,34] and high-frequency Floquet theory [35,36], etc., can be found in other recent reviews or articles. The outline of this article is as follows. In Section 2, we start form the conventional Floquet theorem and discuss the general properties of Floquet quasienergy states. In Section 3, the time-independent Floquet Hamiltonian method is introduced. The method allows the transformation of the periodically time-dependent SchrGodinger equation into an equivalent in'nite dimensional time-independent Floquet matrix eigenvalue problem. This (Hermitian) Floquet Hamiltonian technique has been extensively applied to the nonperturbative studies of bound–bound transitions such as multiphoton excitation (MPE) of two-level [37,38] and multi-level atomic and molecular systems [15–20], and multiple quantum (MQ) NMR transitions in the spin-systems [39], etc. In Section 4, we discuss the limitations of the Floquet Hamiltonian methods for the treatment of various other important multiphoton processes such as the nonperiodic time-dependent processes, bound–free transitions, and nonlinear optical processes, etc. We outline a list of various generalized Floquet formalisms, beyond the conventional Floquet theorem, that have been developed in the last two decades for overcoming the major diNculties encountered by the conventional Floquet Hamiltonian techniques. Fig. 1 shows an overview of the scope of recent generalized Floquet developments and chemical and physical processes amenable to these new Floquet treatments. Not all the subjects

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listed here will be treated in this article due to the space limitation. Some of the topics not covered here can be found in earlier Floquet reviews [15–20]. In Section 5, we discuss the methodology for the e9ective treatment of the large-scale Floquet matrix problem often encountered in the study of molecular multiphoton excitation (MPE) processes. A most-probable-path approach (MPPA) [40,41] using arti(cial intelligence algorithms to preselect the most important Floquet-state paths in multiphoton processes is introduced. The MPPA allows the reduction of the in'nite-dimensional Floquet matrix to a manageable scale and yet maintains high precision. The MPPA–AI method has been applied successfully to the study of MPE of molecules and Rydberg atoms as well as to electron transfer in biological systems. In Section 6, we discuss the extension of the Hermitian Floquet matrix method to include the complete set of both discrete and continuum states. The incorporation of the complex-scaling transformation [42–46] allows analytical continuation of the Hermitian Floquet Hamiltonian onto the complex energy planes in higher Riemann sheets. The complex poles of the resulting non-Hermitian Floquet Hamiltonian [47,48] provide the solutions of the decaying quasienergy states. These quasienergy states possess complex quasienergies (ER ; −=2): the real parts (ER ) give rise to the ac Stark shifts of the perturbed atomic (or molecular) energy levels, whereas the imaginary parts () provide the total multiphoton ionization (or dissociation) rates (widths). The non-Hermitian Floquet matrix can be constructed in two di9erent fashions: (a) the use of square integrable (L2 ) atomic (molecular) basis set expansion [15–20,47,48]; and (b) the discretization of the Floquet Hamiltonian by means of the generalized pseudospectral techniques [29,49] more recently developed. Then in Section 7, we discuss several applications of the non-Hermitian Floquet methods for the studies of atomic multiphoton processes in strong 'elds, including MPI/ATI of neutral atoms, multiphoton detachment of negative ions, and ac Stark shifts of Rydberg states, etc. Section 8 is devoted to the application of the non-Hermitian Floquet methods to the study of molecular multiphoton processes in strong 'elds. The novel behavior of chemical bond in intense laser 'elds, namely, the laser-induced chemical bond softening and hardening [50] phenomena, are discussed at length. The Floquet matrix methods described above are suitable only for problems involving periodically time-dependent Hamiltonians. In Section 9, we introduce the many-mode Floquet theorem (MMFT) [51,52] which is the generalization of the conventional Floquet theorem to allow for nonperturbative treatment of multi-color or multi-frequency multiphoton processes, involving nonperiodic time-dependent Hamiltonians. The MMFT allows exact transformation of the incommensurate multi-frequency or polychromatic (quasiperiodic) time-dependent SchrGodinger equation into an equivalent time-independent in'nite dimensional many-mode Floquet matrix eigenvalue problem. Further, for N -level resonant or near resonant multiphoton processes, an e9ective Hamiltonian (of order N × N ) can be constructed from the in'nite-dimensional many-mode Floquet Hamiltonian using appropriate nearly degenerate perturbation theoretical techniques. This allows analytical treatment of the generalized Bloch–Siegert shift, power broadening, and spectral line shapes, etc., in multi-frequency 'elds well beyond the traditional rotating wave approximation (RWA). Following this, in Section 9, we discuss several applications of the MMFT to physical and chemical processes in multi-color strong 'elds: (a) SU(N ) dynamical symmetry and symmetry breaking in the time evolution of N -level quantum systems; (b) quantum fractal character of multi-mode quasienergy eigenstates; (c) multiphoton and above-threshold detachment of H− in two-color 'elds: angular distributions and partial widths; (d) chemical bond hardening and molecular stabilization of diatomic molecules in two-color laser 'elds, and (e) high-order harmonic generation in two-color laser 'elds, etc.

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The generalized Floquet formalisms described so far are all based on the framework of the time-dependent SchrGodinger equation. However, for nonlinear optical processes involving radiative decays and collisional dampings, etc., we need to go beyond the SchrGodinger equation and invoke the use of density matrix formulation. In Section 10, we present the extension the many-mode Floquet theorem to the density matrix framework and introduce the Floquet–Liouville Supermatrix (FLSM) formalism [53,54]. The FLSM formalism allows the exact transformation of the (periodically or quasiperiodically) time-dependent Liouville equation into an equivalent time-independent non-Hermitian Floquet–Liouvillian (superoperator) eigenvalue problem. This provides a powerful nonperturbative approach for the study of a wide range of high-intensity nonlinear optical processes beyond the traditional perturbative and rotating wave approximation. Applications of the FLSM formalism to the study of the intensity-dependent multiphoton resonance >uorescence and nonlinear optical susceptibilities in strong laser 'elds are discussed in Section 10. In Section 11, we discuss the extension of the generalized Floquet formalisms to the study of geometric phases in multiphoton transitions [55,56]. Two speci'c topics are treated in this context: (a) cyclic quantum evolution and nonadiabatic geometric phase for spin j-systems driven by periodic 'elds, and (b) biorthogonal density matrix formulation of complex geometric phase in dissipative systems and nonlinear optical processes. In Section 12, we introduce several generalized Floquet approaches for the stationary treatment of multiphoton processes in intense laser pulse 'elds [21,22]. These approaches can be extended to the nonperturbative study of atomic and molecular multiphoton or scattering processes driven by short laser pulses with nonperiodic time-varying and/or chirped laser 'elds. All the generalized Floquet formalisms discussed up to this point have been largely applied to the nonperturbative investigation of multiphoton and nonlinear optical processes of one- or two-electron or 'nite-level atomic and molecular systems. Similar to the time-dependent approaches, the ab initio Floquet wave function treatment of many-electron quantum systems in time-dependent 'elds is well beyond the capability of current computer technology. In Section 13, we introduce the latest development in this 'eld, the generalized Floquet formulation of time-dependent density-functional theory (TDDFT) [57,58] for overcoming this grand challenge. The generalized Floquet–TDDFT formulation allows the extension of various Floquet formalisms for nonperturbative treatment of a broad range of multiphoton processes of many-electron quantum systems (atoms, molecules, and clusters). Several recent applications of this new development are presented in Section 13. Much remains to be explored along this direction in the future. Finally, in Section 14, we conclude the review and present some future outlook.

2. The Floquet theorem and general properties of quasienergy states When the perturbing laser 'eld is suNciently strong, it is useful to introduce the notion of quasienergy which can be considered to be a (time-averaged) characteristic energy of the combined system (namely, atoms/molecules) and electromagnetic (EM) 'elds together. The description of the response of atoms and molecules to monochromatic laser 'elds can be greatly facilitated by the use of the Floquet theorem [59]. We start the discussion in this section from the conventional Floquet theorem for monochromatic or periodic 'elds.

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2.1. The Floquet theorem The solutions of linear di9erential equations with periodic coeNcients were 'rst considered by Floquet [59] and PoincarOe [60] about a century ago. The Floquet theorem was later used by Autler and Townes [61] to obtain wave functions for the two-level system in terms of in'nite continued fractions. Application of Floquet theory to quantum systems began to grow only after the mid-1960s. In particular, Shirley [37] reformulated the time-dependent problem of the interaction of a two-level quantum system with a strong oscillating classical 'eld as an equivalent time-independent in'nite-dimensional Floquet matrix. While this is a semi-classical theory (namely, the system is treated quantum mechanically, whereas the EM 'eld is treated classically by Maxwell equations) without explicit 'eld quantization, Shirley showed that the Floquet states can be interpreted physically as quantum 'eld states. In fact, the Floquet quasienergy diagram is identical to the (fully quantized) dressed-atom picture introduced by Cohen-Tannoudji and Haroche [62]. A comprehensive survey of di9erent Floquet techniques for two-level systems published before 1976 is given by Dion and Hirschfelder [38]. Generalization of the Floquet theory for nonperturbative treatment of in'nite-level systems, including both bound and continuum states, was 'rst introduced by Chu and Reinhardt [47] in 1977. Let us now consider the properties of the wave functions of a quantum system driven by a periodic external 'eld with period (and fundamental frequency ! = 2 = ). The SchrGodinger equation for the system may be written as (} = 1) ˆ t)(r; t) = 0 ; H(r;

(2.1)

ˆ t) ≡ Hˆ (r; t) − i9=9t : H(r;

(2.2)

where Hˆ (r; t) is the total Hamiltonian given by Hˆ (r; t) = Hˆ 0 (r) + Vˆ (r; t) ;

(2.3)

where Vˆ (r; t) is the periodic perturbation due to the interaction between the system and the monochromatic 'eld, Vˆ (r; t + ) = Vˆ (r; t) ;

(2.4)

and the unperturbed Hamiltonian Hˆ 0 (r) has a complete orthonormal set of eigenfunctions: Hˆ 0 (r)|(r) = E0 |(r);

(r)|(r) =  :

(2.5)

The wavefunction , called the quasienergy-state (QES), can be written, according to the Floquet theorem [59], in the following form: (r; t) = e−it (r; t) ;

(2.6)

where (r; t) is periodic in time, i.e., (r; t + ) = (r; t) ;

(2.7)

and  is a real parameter called the Floquet characteristic exponent or the quasienergy. The term quasienergy re>ects the formal analogy of the states, Eq. (2.6), with the Bloch eigenstates in a solid

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with the quasimomentum k. Substituting Eq. (2.6) into Eq. (2.1), we obtain an eigenvalue equation for the quasienergy, ˆ t) (r; t) =   (r; t) ; H(r;

(2.8)

subject to the periodicity condition (2.7). Note that the following transformation  =  + m! ;  (r; t) = exp(im!t) (r; t) ;

(2.9) (2.10)

m being an arbitrary integer number, converts any eigenstate in Eq. (2.8) into another eigenstate. The wave function (r; t) in Eq. (2.1), however, remains unchanged upon this transformation. That means, the Floquet states are physically equivalent if their quasienergies di9er by m!. ˆ t), one can introduce the composite Hilbert space S For the Hermitian operator H(r; [15,17,37,38,63] which contains time-periodic wave functions. The spatial part is spanned by any orthonormal basis set of square-integrable (L2 ) functions in the con'guration space:
(r)|(r) ≡ ∗ (r)(r) dr =  : (2.11) The temporal part is spanned by the complete orthonormal set of functions {exp(im!t)}, where m = 0; ±1; ±2; : : : is the Fourier index, and
1

exp[i(n − m)!t] dt = nm : (2.12)

0 The inner product in the composite Hilbert space S is de'ned as follows:
1

 T (r; t)|(r; t)U = dt (r; t)|(r; t) :

0

(2.13)

ˆ satisfy the orthonormality condition The eigenvectors of H T |
U = 
;

and form a complete set in S:  | UT | = I :

(2.14) (2.15)



2.2. General properties of quasienergy states The use of the quasienergy-state (QES) framework is signi'cant as it plays a similar role in studying quantum systems in time-periodic 'elds as the bound states do for the time-independent Hamiltonian. The QES with di9erent quasienergies  are mutually orthogonal for each moment of time and form a complete orthonormal set, as indicated in Eqs. (2.14)–(2.15). The quasienergy eigenvalue equation (2.8) has the form of the “time-independent” SchrGodinger equation in the composite Hilbert space S. It can be readily shown that all the general quantummechanical theorems for the time-independent SchrGodinger equation, such as the variational principle,

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the Hellman–Feynman, hypervirial, Wigner–Neumann and other theorems, can be extended also to the QES in periodic 'elds. Thus, for example, the variational form of Eq. (2.8) can be written as ˆ U T|H| [] = 0; [] ≡ : (2.16) T|U While the energy of the system is not a conserved quantity for an explicitly time-dependent Hamiltonian, it is possible to determine the “mean energy” Hˆ  of the system in the QES  (r; t):
1

 (r; t)|Hˆ (r; t)| (r; t) dt Hˆ  =

0 9 | (r; t)U : 9t Using the Hellmann–Feynman theorem, it can be shown that =  + T (r; t)|i

(2.17)

Hˆ  =  − !9=9! :

(2.18)

Other properties of quasienergy states can be found in Ref. [15]. 3. Time-independent Floquet Hamiltonian method: stationary treatment of periodically time-dependent Schr"odinger equation Exact analytical solution of the time-dependent SchrGodinger equation with periodic Hamiltonian is generally not possible even for a simple two-level system. Thus it is necessary to develop approximate methods for the treatment of multiphoton excitation (MPE) of atoms and molecules. In this section we discuss a time-independent Floquet Hamiltonian method [15–20,37,38] for the nonperturbative treatment of multiphoton bound–bound transitions in atoms and molecules. The quasienergy state (QES) function  (r; t), Eq. (2.6), can be expanded in a Fourier series, ∞   (r; t) = exp(−i t) A(n) (r) exp(−in!t) : (3.1) n=−∞

Thus a QES can be considered as a superposition of stationary states with energies equal to ( +n!). The functions A(n) (r) of (3.1) can be further expanded in terms of the orthonormal set of unperturbed eigenfunctions of Hˆ 0 (r), namely, {|(r)},  (n)  |(r) : (3.2) A(n) (r) = 

Substituting Eqs. (3.1) and (3.2) into Eq. (2.1), we obtain the following system of coupled equations  (n) [|Hˆ (m−n) | − ( + m!)mn  ] =0 ; (3.3) n

where




1 ˆ (n) ˆ H (r; t) exp(in!t) dt : (3.4) H (r) ≡

0 As an example, consider the interaction of a quantum system with a linearly polarized monochromatic 'eld. In this case, V ∼ cos !t, and only the matrix elements |Hˆ (n) | with n = 0, ±1 are

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nonvanishing. The system of Eq. (3.3) is similar to the system for a constant (i.e. time-independent) perturbation. It is expedient to introduce at this stage the Floquet-state nomenclature |n ≡ | ⊗ |n, where  is the system index, and |n are the Fourier vectors (n = 0; ±1; ±2; : : :) such that t|n = exp(in!t). The system of Eq. (3.3) can be recast into the form of a matrix eigenvalue equation:  (k) (n) n|Hˆ F |k =   ; (3.5) 

k

where Hˆ F is the time-independent Floquet Hamiltonian whose matrix elements are de'ned by (n−m) + n! nm : n|Hˆ F |m = Hˆ 

(3.6)

It follows from Eq. (3.5) that the quasienergies are eigenvalues of the secular equation det|Hˆ F − I | = 0 :

(3.7)

As an example, consider the multiphoton excitation (MPE) of the vibrational–rotational states of a diatomic molecule with dipole moment (r) by a monochromatic 'eld with amplitude E0 , frequency !, and phase ’, respectively. In the electric dipole approximation, the interaction potential energy between the quantum system and the classical EM 'eld is given by V (r; t) = −((r) · E0 ) cos(!t + ’) :

(3.8)

The Floquet matrix Hˆ F possesses a block tridiagonal form as shown in Fig. 2. The determination of the vibrational–rotational quasienergies n and quasienergy state |n  thus reduces to the solution of a time-independent Floquet matrix eigenproblem. Fig. 2 shows that Hˆ F has a periodic structure with only the number of !’s in the diagonal elements varying form block to block. This structure endows the quasienergy eigenvalues and eigenvectors of Hˆ F with the following general periodic properties: n = 0 + n! ; ; n + p|; m+p  = ; n|m  :

(3.9) (3.10)

Consider now the transition probability from an initial quantum state | to a 'nal quantum state |. The time-evolution operator Uˆ (t; t0 ), in matrix form, can be expressed as Uˆ  (t; t0 ) ≡ |Uˆ (t; t0 )|  n| exp[ − iHˆ F (t − t0 )]|0 exp(in!t) : =

(3.11)

n

Eq. (3.11) shows that Uˆ  (t; t0 ) can be interpreted as the amplitude that a system initially in the Floquet State |0 at time t0 evolve to the Floquet State |n by time t according to the timeindependent Floquet Hamiltonian Hˆ F , summed over n with weighting factors exp(in!t). The latter interpretation enables one to solve problems involving Hamiltonians periodic in time by methods applicable to time-independent Hamiltonians. The transition probability going from the initial quantum state | and a coherent 'eld state to the 'nal quantum state |, summed over all 'nal 'eld states,

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Fig. 2. Structure of the time-independent Floquet Hamiltonian Hˆ F in the Floquet state basis (|vj; n). The Hamiltonian is composed of the diagonal Floquet blocks of type A and the o9-diagonal blocks of type B. Evj(0) are unperturbed vibrational–rotational energies and bvj; v
j
are electric dipole coupling matrix elements.

can now be written as P→ (t; t0 ) = |Uˆ  (t; t0 )|2 =

 k

k| exp[ − iHˆ F (t − t0 )]|0

m

×exp(im!t0 ) m| exp[ − iHˆ F (t − t0 )]|k : The quantity of experimental interest, however, is the transition probability averaged over initial times t0 (or equivalently averaged over the initial phases of the 'eld seen by the system), keeping

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Fig. 3. (a) Quasienergy plots and (b) time-average MPE transition probabilities P00→vj for the HF molecule subjected to both laser (Eac = 1:0 TW=cm2 ) and dc electric 'elds (Edc = 10−4 a:u:) simultaneously. Dash-dotted line, one-photon peaks; dashed line, two-photon peaks; and solid line, three-photon peaks. Nonlinear e9ects such as power broadening, dynamical Stark shift, Autler–Townes multiplet splitting, hole burning, and S-hump behaviors, etc., are observed and can be correlated with the avoided crossing pattern of the quasienergy levels (adapted from Ref. [64]).

the elapsed time t − t0 'xed. This yields  |k| exp[ − iHˆ F (t; t0 )]|0|2 : P→ (t − t0 ) =

(3.12)

k

Finally, averaging over t − t0 , one obtains the long-time average transition probability  |k|l  l |0|2 : PU → (t − t0 ) = k

(3.13)

l

3.1. Quasienergy diagram and multiphoton excitation of molecules Much information can be obtained from the plot of the quasienergy eigenvalues (or the characteristic exponents) of the Floquet Hamiltonian [15–20,39]. The main feature of the quasienergy plot is illustrated in Fig. 3 for the case of the HF molecule subject to both the ac and dc 'elds [64]. Nonlinear e9ects such as power broadening, hole burning, S-hump behaviors are observed and may

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be correlated with the avoided crossing patterns in the quasienergy diagram (Fig. 3). The addition of a dc electric 'eld, spoils the restriction of the rotational dipole selection rule and induces signi'cant intermixing of the bare molecular rovibrational states. Due to the greater number of strongly coupled nearby states in the dc 'eld, nonlinear optical e9ects such as those mentioned above appear at a much lower ac 'eld strength than they would be in the absence of the dc 'eld [64]. The introduction of an additional external dc 'eld, therefore, strongly enhances the MPE probabilities and results in a much richer spectrum, in accord with the experimental observations [65]. 4. Beyond the Floquet theorem: generalized Floquet formalisms and quasienergy methods The Floquet Hamiltonian method described in the last section provide a powerful nonperturbative technique for the treatment of bound–bound multiphoton transitions for simple 'nite-level quantum systems. However, for more complex systems and/or processes, the capability of the Floquet Hamiltonian method is limited. Some of the major limitations are listed below: (i) For complex atomic and molecular systems involving a large number of energy levels and photons, the dimensionality of the Floquet matrix can become prohibitively large and intractable. (ii) Conventional Floquet methods with Hermitian Hamiltonians can only treat bound–bound multiphoton excitation (MPE) transitions, but not bound–free and free–free transitions such as multiphoton ionization (MPI), above-threshold ionization (ATI), multiphoton dissociation (MPD), and above-threshold dissociation (ATD) processes, etc. (iii) For “multi-color” (multi-frequency, polychromatic) laser excitations in which the laser frequencies !i (i = 1; 2; : : :) are incommensurate, the Hamiltonian becomes nonperiodic in time, and the Floquet theorem is no longer valid. (iv) Conventional Floquet methods (based on the SchrGodinger equation) cannot deal with nonlinear optical processes with (radiative, collisional, and phase) relaxations. (v) For laser 'elds with arbitrary pulse shape, the Hamiltonians are nonperiodic in time and the Floquet theorem is again not valid. (vi) As the quantum system being considered includes more than two electrons, the dimensionality of the Floquet matrix becomes prohibitively large and intractable. These are the common bottlenecks encountered by the conventional Floquet Hamiltonian techniques. To overcome these diNculties, new theoretical developments beyond the Floquet theorem and Floquet Hamiltonian method are necessary. In the following we list several generalized Floquet formalisms and associated computational techniques that have been developed in the past two decades for the nonperturbative treatment of strong-'eld atomic and molecular physics: (i) Non-Hermitian Floquet matrix formalisms and complex quasienergy methods for nonperturbative treatment of bound–free and free–free MPI/ATI of atoms and MPI/MPD/ATD of molecules. (ii) Many-mode Floquet theorem (MMFT) for exact treatment of multi-color (multi-frequency) laser excitations, where the Hamiltonian is nonperiodic (polychromatic, quasiperiodic) in time. (iii) Most probably path approach (MPPA) using arti(cial intelligence (AI) algorithms for selecting the most important multiphoton excitation Floquet-state pathways, allowing many orders of magnitude reduction in the size of Floquet matrix yet maintaining good accuracy.

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(iv) Floquet–Liouville supermatrix (FLSM) formalism (i.e., non-Hermitian MMFT treatment of density matrix operator) for exact nonperturbative treatment of intensity- and frequencydependent nonlinear optical processes allowing for relaxation and dissipative mechanisms. (v) Generalized Floquet treatments of multiphoton processes in intense (arbitrarily shaped) laser pulse 'elds. (vi) Floquet formulation of time-dependent density functional theory (TDDFT) for the treatment of multiphoton processes of many-electron quantum systems in intense laser 'elds. Fig. 1 shows an overview of the scope of generalized Floquet formalisms and various strong-'eld processes amenable to theses new Floquet treatments developed at the University of Kansas. Not all the topics listed in Fig. 1 will be treated in this article. In the following section (Sections 5–13), several of the main topics of current signi'cance in strong-'eld atomic and molecular physics will be presented in depth.

5. Arti'cial intelligence in multiphoton dynamics: most-probable-path approach for ultralarge Floquet matrix problems The time-independent Floquet Hamiltonian formalism outlined in Section 3 is a general nonperturbative approach and is applicable in principle to the study of multiphoton excitation of atoms and molecules in intense monochromatic 'elds. However, as the number of quantum states and the number of photons absorbed or emitted involved become very large, the dimensionality of the Floquet matrix will increase rather rapidly and can become prohibitively large. In this section we discuss a procedure, the so-called most-probable-path approach (MPPA), to tackle such a large Floquet matrix challenge. The MPPA was 'rst introduced by Tietz and Chu in 1983 in an ab initio study of high-order nonlinear multiphoton excitation (MPE) of SO2 molecule [40]. A brute-force attempt to calculate polyatomic MPE would soon become computationally very intensive due to the large size of Floquet matrix needed for convergence. For typical 15-photon calculation of SO2 , for example, a Floquet matrix on the order of 10; 000 × 10; 000 would have to be diagonalized at each frequency and 'eld strength. In most exact Floquet calculations, however, the majority of the molecule-'eld Floquet states are unimportant due to either large detuning or very small coupling matrix elements. The MPPA is a practical strategy introduced to determine which Floquet states are, in fact, important at each step of the multiphoton processes. The procedure is derived from algorithms which utilize arti'cial intelligence (AI) to prune the number of choices at each node (photon order) of a decision tree [66]. Similar to the minimax game playing programs, the MPPA examines the possible paths to take at each photon order iteration with the static evaluation function given by the N th order perturbation theory (this is a breadth-'rst search). If all paths were followed exhaustively, the problem would be beyond practical solutions. In game theory, one answer uses a breadth-limiting heuristic technique and discards any paths for which the N th order coupling is small (with respect to other N th order terms). The MPPA begins by calculating all possible second-order perturbative terms. The Np largest couplings (where Np is the number of paths to keep at each step) are chosen as the most probable paths through second order. The initial state and intermediate states of the chosen paths are marked

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as important and are used in the 'nal calculations. At each iterative step, the method calculates all possible (N + 1)st-order couplings (paths) using only the Np N th-order paths saved in the last iteration. The (N + 1)st-order couplings are then examined and the largest Np are saved for further traversal. The N th-order states which have now become intermediate to a large (N + 1)st-order path are “important” and are marked for later use. By iterating long enough, one can traverse the entire molecular-'eld Floquet basis space, saving only those states that are important to various nth-order processes. The reduction of the Floquet basis set and the dimensionality of the Floquet matrix is considerable, leading to many-order-of-magnitude savings in computer time and yet maintaining good accuracy (≈ 95%) in most cases. Using MPPA, Tietz and Chu have studied the collisionless MPE spectra of SO2 in intense IR laser 'elds [40]. The MPPA predicted results are in good agreement with the experimental data of Simpson and Bloembergen [67]. The selection of important multiphoton pathways by the MPPA/AI algorithms is later further exploited by Chang and Wyatt in the study of MPE of a spherical top molecule [68] and by Wang and Chu in the study of MPE and quantum di9usion phenomena of 3D Rydberg atoms driven by strong microwave 'elds [41]. In the latter case, the MPPA allows the reduction of an ultralarge Floquet matrix (on the order of several million) to an e9ective matrix of a manageable size (on the order of several thousands) [41]. More recent applications of the AI search algorithms include, for example, the study of intramolecular dynamics in large polyatomic molecules [69,70] and the search of electron transfer pathways by pruning the protein [71], etc.

6. Non-Hermitian Floquet formalisms and complex quasienergy methods for multiphoton ionization and dissociation in monochromatic 'elds 6.1. Non-Hermitian Floquet formalism The Floquet matrix methods described in previous sections, involving time-independent Hermitian Floquet Hamiltonians, provide powerful nonperturbative techniques for the treatment of bound–bound multiphoton transitions. These methods cannot, however, be applied to bound–free or free–free transitions such as multiphoton ionization (MPI) of atoms or multiphoton dissociation (MPD) of molecules. In this section we discuss a non-Hermitian Floquet formalism and complex quasienergy method, 'rst introduced by Chu and Reinhardt [47], for nonperturbative treatment of MPI processes in linearly polarized monochromatic 'elds. Applying the uniform complex scaling transformation [72,73], r → r exp(i), to the SchrGodinger equation, we obtain from Eqs. (2.1)–(2.2), i

9(r exp(i); t) = Hˆ (r exp(i); t)(r exp(i); t) ; 9t

(6.1)

where Hˆ (r exp(i); t) is now a non-Hermitian periodic Hamiltonian. The wavefunction (r exp(i); t), can be written, according to the Floquet theorem, (r exp(i); t) = (r exp(i); t) exp(−it) ;

(6.2)

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where  is the complex quasienergy, and the periodic function (r exp(i); t) = (r exp(i); t + ) satis'es the eigenvalue equation ˆ exp(i); t)(r exp(i); t) = (r exp(i); t) ; H(r (6.3) ˆ where H is given in Eq. (2.2). Following the procedure described in Section 3, Eq. (6.3) can be transformed into a time-independent Floquet matrix eigenvalue equation, namely, Hˆ F (r exp(i))(r exp(i)) = (r exp(i)) ; (6.4) where Hˆ F (r exp(i)) ≡ Hˆ F () is now an analytically continued, time-independent non-Hermitian Floquet Hamiltonian, and  is the complex quasienergy. The complex scaling transformation distorts the continuous spectrum away from the real axis, exposing the quasienergy resonances in appropriate higher Riemann sheets, and also allowing the use of 'nite variational expansions employing L2 basis function chosen from a complete discrete basis. The use of a complete L2 basis obviates the necessity of explicit introduction of exact atomic or molecular bound and continuum states, thus reducing all computations to those involving 'nite-dimensional non-Hermitian matrices. The use of complex coordinates not only allows direct calculation of eigenvalue parameters associated with complex dressed states, but completely avoids numerical problems arising from strong coupling between overlapping atomic or molecular continua. The real part of the complex quasienergy (ER ) provides the ac-Stark shifted energy, whereas the imaginary part (=2) determines the total MPI or MPD rate. In the following we discuss several numerical techniques for the implementation of Eq. (6.4) and their applications to atomic and molecular multiphoton processes. 6.2. Non-Hermitian Floquet calculations by L2 basis set expansion methods As discussed earlier, a conventional method for the solution of the non-Hermitian Floquet Hamiltonian, Eq. (6.4), is to use the L2 -basis expansion technique [47,48]. Consider, for example, the problem of MPI of atomic hydrogen, a signi'cant prototype strong 'eld process. Corresponding to the periodically time-dependent Hamiltonian, 1 1 (6.5) Hˆ (r; t) = − ∇2 − + Fz cos(!t) ; 2 r describing the interaction of atomic hydrogen with a monochromatic, linearly polarized, coherent 'eld of frequency ! and peak 'eld strength F, an equivalent time-independent Hamiltonian Hˆ F (r) may be obtained by an extension of the Floquet Hamiltonian method described in Section 3. The structure of Hˆ F is shown in Fig. 4. The Floquet Hamiltonian Hˆ F shows a tridiagonal block structure, consisting of the diagonal A ± n!I (n = 0; ±2; ±4; : : :) blocks and the o9-diagonal B blocks. Each diagonal block is composed of angular momentum blocks S, P, D, ... representing the projection of the atomic electronic Hamiltonian onto states of l = 0; 1; 2; : : :, and Vl; l
s are electric dipole coupling matrix elements. Thus, for example, in the case of atomic hydrogen, the S block consists of the 1s, 2s, 3s, ... ns, ... bound states and the entire ks Coulomb continuum. The Hamiltonian of Fig. 4 has no discrete spectrum, and the time evolution is dominated by poles of the resolvent (E − Hˆ F )−1 near the real axis but on higher Riemann sheets. These complex poles, which correspond to decaying complex quasienergy states (QES), maybe found directly from the analytically continued Floquet Hamiltonian, Hˆ F (), obtained by the uniform complex scaling transformation r → r exp(i). This transformation e9ects an analytical continuation of (E − Hˆ F )−1 into the lower half-plane on an

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Fig. 4. Structure of the Floquet Hamiltonian for atomic H in linearly polarized 'elds (adapted from Ref. [47]).

appropriate higher Riemann sheet, allowing the complex QES to be determined by solution of a non-Hermitian eigenproblem. In practice, the atomic blocks can be made discrete by use of a 'nite subset of the complete L2 basis such as Laguerre or Sturmian functions. In practice, the convergence of complex quasienergy (ER ; −=2) calculations may achieve arbitrary precision by systematically increasing the basis size and the number of angular momentum blocks. The complex quasienergy eigenvalues and eigenfunctions of the non-Hermitian Floquet Hamiltonian Hˆ F () can be eNciently determined by means of an inverse iteration technique developed in [47,74]. Since the Floquet matrix possesses a periodic block structure, eNcient numerical algorithms can be developed so that only the matrix information in one A and one B block (Fig. 4) is all needed.

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19

Fig. 5. Intensity-dependent generalized cross section, )ˆ2 , for ionization of bare 1s state of H in the frequency region dominated by resonant two-photon ionization (adapted from Ref. [47]).

Using this procedure, the 'rst converged nonperturbative intensity-dependent “generalized” cross sections, )N = =I N , where I is the laser intensity, for an atomic H atom subjected to intense monochromatic laser 'eld were obtained [47]. See Fig. 5 for an example. Numerous extensions and applications of the non-Hermitian Floquet formalism have been performed in the last two decades. For reviews on non-Hermitian Floquet methods using L2 basis functions, we refer to [15,17,18] for details. 6.3. Non-Hermitian Floquet calculations by complex-scaling generalized pseudospectral methods In this section, we describe an alternative and more recent approach, the complex-scaling generalized pseudospectral (CSGPS) method, 'rst introduced by Yao and Chu [49] for accurate and eNcient treatment of atomic and molecular resonances, including multiphoton quasienergy resonances. The method does not require the computation of potential matrix elements (which is usually the most time-consuming part of atomic and molecular structure calculations using the conventional L2 basis set expansion-variational method), is simple to implement, and provides the values of the wavefunctions directly at the spatial grid points. As shown elsewhere [49,75], the generalized pseudospectral (GPS) method is computationally more eNcient and accurate than the 'nite di9erence method. It is

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S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

also computationally more advantageous than the L2 basis set expansion-variational method, particularly for highly excited resonance states. While the pseudospectral method has been extensively studied in mathematics in the last decade and applied to >uid dynamics [76] (such as aerodynamics, meteorology, and oceanology), little attention, however, has been paid to the usefulness of the method (at least in its most updated form) in the study of atomic and molecular structure and resonances. As such, we discuss below the essence of the pseudospectral method and its several generalizations for the treatment of bound and resonance states, as well as complex quasienergy resonances. 6.3.1. Uniform complex scaling—generalized pseudospectral method Consider the eigenvalue problem for the radial SchrGodinger equation de'ned on the semi-in'nite axis [0; ∞] with the Dirichlet boundary conditions: Hˆ (r) (r) = E (r);

(0) = (∞) = 0 ;

(6.6)

where 1 d2 Hˆ (r) = − + U (r) : 2 dr 2

(6.7)

For atomic structure calculations involving Coulomb potential, one typical problem with the grid methods is the singularity at r =0 and the long-range nature of the potential. Generally, one truncates the semi-in'nite domain into 'nite domain [rmin ; rmax ] to avoid the problems of both the singularity at the origin and the in'nite domain. For this purpose, rmin must be chosen to be suNciently small and rmax suNciently large. This results in the need of a large number of grid points, in addition to possible truncation errors. To overcome this problem, one can map the semi-in'nite domain [0; ∞] into the 'nite domain [−1; 1] using the mapping r =r(x), and then apply the Legendre or Chebyshev pseudospectral technique. A useful algebraic mapping for the Coulomb problem is r = r(x) = Rm

1+x ; 1−x

(6.8)

where Rm is a mapping parameter. However, the introduction of nonlinear mapping can lead to either an asymmetric or a generalized eigenvalue problem. Such undesirable features can be avoided by the following symmetrization procedure [49,75]. Thus by introducing  (r(x)) = r  (x)f(x) ; (6.9) we obtain the following transformed Hamiltonian, leading to a symmetric eigenvalue problem [75,77] (in atomic units): 1 1 d2 1 Hˆ (x) = −  + U (r(x)) + Um (x) ; 2 r (x) d x2 r  (x)

(6.10)

where Um (x) =

3(r  )2 − 2r  r  ; 8(r  )4

(6.11)

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

21

and primes are used to denote the derivatives of r(x) with respect to x: r  (x) =

dr ; dx

r  (x) =

d2 r ; d x2

r  (x) =

d3 r : d x3

Note that for the special mapping, Eq. (6.8), Um (x) = 0. Discretizing the Hamiltonian operator, Eq. (6.10), by the Legendre pseudospectral method, leads to the following set of coupled linear equations:  N   1 (2)

− Dj
j + j j U (r(xj )) + j j Um (r(xj )) Aj = EAj
; j  = 1 : : : N − 1 : (6.12) 2 j=0 Here the coeNcients Aj are related to the wavefunction values at the collocation points as Aj = r  (xj )f(xj )[PN (xj )]−1 = [r  (xj )]1=2 (r(xj ))[PN (xj )]−1 ;

(6.13)

where PN (x) is the Legendre polynomial, and the matrix Dj(2)
j , representing the second derivative with respect to r, is given by  −1 (2)  dj
j [r (xj )]−1 : Dj(2)
j = [r (xj
)]

(6.14)

The matrix d(2) j
j is related to the second derivative of the Legendre cardinal function gj (x) with respect to x: gj (xj
) = d(2) j
j d(2) j
j = −

2 (xj
− xj )2

(2) d(2) 0N = dN 0 =

d(2) jj = −

PN (xj
) ; PN (xj )

N (N + 1) − 2 ; 4

N (N + 1) 3(1 − xj2 )

(2) d(2) 00 = dNN =

(j  = j; (j  j) = (0N ); (j  j) = (N 0)) ;

(j = 0; j = N ) ;

N (N + 1)[N (N + 1) − 2] : 24

(6.15a) (6.15b) (6.15c) (6.15d) (6.15e)

The pseudospectral approximation for the 'rst derivative of the wavefunction (r) with respect to r, calculated at the points r(xj
), can be expressed through the coeNcients Aj :  N  d (r)   − 1=2 = PN (xj
)[r (xj
)] Dj(1) j = 0 : : : N ; (6.16)
j Aj ;  dr r(xj
) j=0 with the matrix Dj(1)
j given by:  −1=2 (1)  dj
j [r (xj )]−1=2 ; Dj(1)
j = [r (xj
)]

(6.17)

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S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

where d(1) j
j is related to the 'rst derivative of the cardinal function gj (x) with respect to x: PN (xj
) ; (6.18a) gj (xj
) = d(1) j
j PN (xj ) 1 d(1) (j  = j); d(1) (j = 0; j = N ) ; (6.18b) jj = 0 j
j = xj
− x j N (N + 1) N (N + 1) ; d(1) : (6.18c) d(1) NN = 00 = − 4 4 The GPS method described above can now be extended to the resonance state problems by means of the uniform complex scaling method [49,75] or by the exterior complex scaling method to be described below. For the uniform complex scaling [72,73], r → r exp(i), we have the following mapping transformation [49,75]: 1+x r = Rm exp(i) : (6.19) 1−x Here both parameters Rm and  are real; Rm is the mapping parameter which determines the density of the grid points while  is the complex rotation angle. Under this transformation, the semi-axis r ∈ [0; ∞] is rotated in the complex plane by the angle  and then mapped to the interval x ∈ [−1; 1]. Note that for transformation (6.19), the additional potential Um (x) vanishes, so the Hamiltonian matrix in Eq. (6.12) takes the following simple form: 1 Hj
j = − Dj(2) j ; j = 1 : : : N − 1 ; (6.20)
j + j
j U (r(xj )); 2 with the Dirichlet boundary conditions taken into account. 6.3.2. Exterior complex scaling—generalized pseudospectral method The exterior complex scaling transformation method was 'rst described in detail in 1979 by Simon [78] for the treatment of molecular resonances in the Born–Oppenheimer approximation. The idea of regularization of the normalization integral through evaluating it along the exterior complex scaling contour in the complex plane was also discussed in 1978 by Nicolaides and Beck [79]. The method has been subsequently extended to the study of atomic and molecular resonances, particularly, for potentials which behave nonanalytically (or de'ned only numerically or piecewise analytically) in the interior region of the coordinates. For such nonanalytical potentials, although the uniform complex scaling is still possible by means of certain transformation techniques developed by Datta and Chu [80,81], the exterior complex scaling (ECS) provides a direct and alternative procedure. The principal idea of ECS is to perform the analytical continuation (complex scaling) of the coordinates beyond some distance Rb only. Thus for the one-particle system, the contour R(r) in the complex plane of the coordinate can be de'ned as follows: R(r) = r;

0 6 r 6 Rb ;

R(r) = Rb + (r − Rb ) exp(i);

r ¿ Rb :

(6.21)

Here r is assumed to be real-valued while R(r) becomes complex-valued beyond the radius Rb . For many-body systems, the same transformation is performed for each interparticle coordinate. A number of applications of the exterior complex scaling procedure has been developed in the time-independent

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

23

calculations of atomic and molecular resonances [82–86], cross sections in electron-atom collisions [87], as well as in time-dependent calculations [88]. Various numerical techniques were used to solve the second-order di9erential equation along the contour de'ned by Eq. (6.21): propagation and matching methods [82,85,86], global basis-set expansions [83], and 'nite-element basis-set expansions [84,88], etc. The function R(r) is not analytical at the point Rb , and some care should be taken when solving the equation along contour (6.21). The boundary conditions at the point Rb can be inserted in the Hamiltonian leading to appearance of an additional zero-range potential [83,89]. The singular potential does not appear if the transition between the interior (unscaled) and exterior (complex scaled) regions of the coordinate is performed with an analytical function R(r) [86,90]. However, nontrivial mapping functions also complicate the problem producing additional terms in the Hamiltonian. In this section, we discuss a new implementation of the exterior complex scaling method by means of the generalized pseudospectral (GPS) technique 'rst introduced by Telnov and Chu [77], providing a simple yet highly accurate and eNcient procedure. For the exterior scaling, the whole range of the coordinate is split into two domains, the pseudospectral discretization being performed separately in each domain. The complex scaling is applied in the exterior domain only. The boundary conditions at the boundary point Rb can be incorporated in the discretized Hamiltonian, modifying the matrix elements. The matrix elements also have simple explicit expressions, and the calculation of the Hamiltonian matrix in the GPS method with the exterior complex scaling is as simple as with the uniform complex scaling. For the exterior domain, one can use the mapping transformation rex (x), slightly di9erent from (6.19): 1+x exp(i) ; (6.22) rex = Rb + Rm 1−x while in the interior domain the linear map rin (x), 1 (6.23) rin = Rb (1 + x) ; 2 serves the purpose. The boundary point Rb , as well as Rm and  are the parameters of the transformations. Both the maps (6.22) and (6.23) do not generate the additional potential Vm (x), and the sets of linear equations for the coeNcients Aj in the interior and exterior domains read as: Nin 

in Hjin
j Ain j = EAj
;

j  = 1 : : : Nin − 1 ;

(6.24)

j  = 1 : : : Nex − 1 ;

(6.25)

j=0 Nex 

ex Hjex
j Aex j = EAj
;

j=0

Nin and Nex being the numbers of collocation points in the interior and exterior domains, respectively. The Hamiltonian matrices Hjin
j and Hjex
j in the interior and exterior domains have the simple form of Eq. (6.20): 1 in Hjin
j = − Dj(2); + j
j V (rin (xjin )); j  ; j = 1 : : : Nin − 1 ; (6.26)
j 2 1 ex Hjex
j = − Dj(2); + j
j V (rex (xjex )); j  ; j = 1 : : : Nex − 1 : (6.27)
j 2

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S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

Here xjin and xjex are the collocation points for the interior and exterior domains, respectively, and in ex the second derivative matrices Dj(2); and Dj(2); are de'ned according to Eq. (6.14):
j
j in  −1 = [rin (xjin
)]−1 d(2) ; Dj(2);
j j
j [rin (xj )]

(6.28)

ex   −1 = [rex (xjex
)]−1 d(2) : Dj(2);
j j
j [rex (xj )]

(6.29)

The Dirichlet boundary conditions at r = 0 and r = ∞ imply that ex Ain 0 = ANex = 0 :

(6.30)

ex However, there is no Dirichlet condition at the point r = Rb which corresponds to Ain Nin and A0 . One has to impose the continuity condition of the wave function and its 'rst derivative instead. This condition can be incorporated into the Hamiltonian matrix elements, leading to the following matrix eigenvalue problem:  N N ex −1 in −1  1 in in Nex 2 in H Hjin
j − Hjin
; Nin DN(1); A + (−1) D0;(1);j ex Aex j j
; Nin j in ; j 1 1 j=1 j=1

=EAin j
; N ex −1 

Hjex
j

j=1

=EAex j
;

j  = 1 : : : Nin − 1 ;  Nin −1 1 ex  1 ex (1); ex ex in in Hj
; 0 Aj − (−1)Nex + Hj
; 0 D0; j DN(1); Aj in ; j 1 21 j=1 j  = 1 : : : Nex − 1 ;

where the constants 2 and 1 are de'ned as follows:      rin (1) 1=2 1 Nin (Nin + 1) Nex (Nex + 1) + : 2=  ; 1=  (−1) rex (−1) 4 rin (1) rex

(6.31)

(6.32)

The total matrix of the eigenvalue problem (6.31) has the dimensions (Nin +Nex −2) by (Nin +Nex −2). ex The diagonalization of this matrix yields the eigenvalues and the eigenvectors {Ain j } and {Aj } inside the interior and exterior domains, respectively. Extension of the ECS–GPS method has been recently implemented into the non-Hermitian Floquet formalism for the study of ATI of negative ions H− and Li− in strong 'elds [77,91,92]. 6.4. Complex-scaling generalized pseudospectral method for two-center systems In this section, we discuss the recent extension of the generalized pseudospectral (GPS) method for nonuniform spatial grid discretization and high-precision electronic structure calculations of two-center diatomic molecular systems [29]. Without loss of generality, consider the 'eld-free electronic Hamiltonian of H2+ , 1 1 1 − ; Hˆ = − ∇2 − 2 |r − R1 | |r − R2 |

(6.33)

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

25

where r is the electronic coordinate, and R1 = (0; 0; a) and R2 = (0; 0; −a) are the coordinates of the two nuclei in Cartesian coordinates. The internuclear separation R is equal to 2a. Now consider the bare electronic Hamiltonian in the prolate spheroidal coordinates (2; 1; ’), 0 6 2 ¡ ∞, 0 6 1 6 , 0 6 ’ 6 2 , where x = a sinh 2 sin 1 cos ’, y = a sinh 2 sin 1 sin ’, z = a cosh 2 cos 1. Eq. (6.33) can be recasted into the following form:   1 9 1 9 ˆ H =− 2 sinh 2 2a 92 (sinh2 2 + sin2 1)sinh2 92  1 9 9 + sin 1 2 2 91 91 (sinh 2 + sin 1) sin 1

1 2 cosh 2 92 + − : (6.34) 2 2 2 9’ sinh 2 sin 1 a(cosh2 2 − cos2 1) Due to the axial symmetry of the system, the solutions of the static SchrGodinger equation, Hˆ = E , take the form, m (r)

= eim’ (2; 1)

(m = 0; ±1; ±2; : : :) :

(6.35)

In the two-center GPS method [29], one expands (2; 1) by N2 ; N1 (2; 1), the polynomials of order N2 and N1 in 2 and 1, respectively, N2 ; N1

(2; 1)  N2 ; N1 (2; 1) =



6(2i ; 1j )gi (x(2))gj (y(1)) ;

(6.36)

i=0; j=0

and further require the approximation to be exact, i.e., N2 ; N1 (2i ; 1j ) = 6(2i ; 1j ) ≡ 6ij , where {x(2i )} and {y(1j )} are the two sets of collocation points to be described below. In Eq. (6.36), gi (x) and gj (y) are the cardinal functions [29,76] de'ned as gi (x) = −

(1 − x2 )PN 2 (x) 1 ; N2 (N2 + 1)PN2 (xi ) x − xi

(6.37)

(1 − y2 )PN 1 (y) 1 : N1 (N1 + 1)PN1 (yj ) y − yj

(6.38)

gj (y) = −

In the case of the Legendre pseudospectral method, the boundary points are x0 = y0 = −1 and xN2 = yN1 = 1. xi (i = 1; : : : ; N2 − 1) and yj (j = 1; : : : ; N1 − 1) are the collocation points determined, respectively, by the roots of the 'rst derivative of the Legendre polynomial PN2 (x) with respect to x and the 'rst derivative of PN1 (y) with respect to y. It follows that the cardinal functions possess the following unique properties: gi (xi
) = i; i
;

gj (yj
) = j; j
:

(6.39)

The following mapping relations between 2 and x and between 1 and y are found to be convenient for two-center systems [29]: 2=L

1+x ; 1−x

1=

(1 + y) ; 2

(6.40)

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S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

where x ∈ [ − 1; 1], y ∈ [ − 1; 1], 2 ∈ [0; ∞], 1 ∈ [0; ], and L is the mapping parameter. Having constructed the mesh structure, one can now de'ne a set of discrete weights wix (i = 0; : : : ; N2 ), wjy (j = 0; : : : ; N1 ), and a pair of discrete matrices dx and dy , which generate approximate integrals and partial derivatives on the mesh. Direct pseudospectral discretization of the Hamiltonian in Eq. (6.34) leads to an asymmetric eigenvalue problem. To symmetrize the Hamiltonian discretization, consider the following alternative but equivalent variational form of the SchrGodinger equation [29]   d 3 r ∗ (Hˆ − E) =0 : (6.41)  ∗ For the case of H2+ in the prolate spheroidal coordinates, the integral in Eq. (6.43) has the explicit form
Fs ≡ d 3 r ∗ (Hˆ − E)   1 9 2 9 2 d r + 92 91 sinh2 2 + sin2 1  

m2 2 2 cosh 2 3 3 + 2 d r − d r + E 2 ; 2a sinh2 2 sin2 1 a(cosh2 2 − cos2 1)

1 = 2 2a




3

(6.42)

where d 3 r = a3 (sinh2 2 + sin2 1) sinh 2 sin 1 d2 d1 d’ : Discretizing Fs under the polynomial approximation, Eq. (6.36), leads to     6i
j 6ij qkj d2ki d2ki
+ 6ij
6ij qil d1lj d1lj
Fs  i; i
; j

+ m2

i; j; j


k



62ij wij −

i; j



62ij wij

i; j



l

2 cosh 2i +E a(cosh2 2i − cos2 1j )

 ;

(6.43)

where 6ij stands for 6(2i ; 1j ), and q, w, and w are the weights [29]   d2 d1 x y ; qij = 2 awi wj sinh 2i sin 1j d x x=xi dy y=yj wij = wij =sinh2 2i sin2 1j

wij = qij (sinh2 2i + sin2 1j ); and d2ij

 =

dx d2

x=xi



dgj (x) dx

x=xi

;

d1ij

 =

dy d1

x=xi



dgj (y) dy

y=yi

:

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

It is now straightforward to perform the variation of Fs with respect to two-dimensional grid. This leads to a symmetric eigenvalue problem [29]: qkj 1 1 qil :i
j √ d2ki d2ki
+ :ij
√ d1 d1
2
wi
j wi; j 2
wij wij
lj lj k; i

27

, Eq. (6.41), on the

j ;l

 wij 2 2 cosh 2i :ij = E:ij ; m − (6.44) wij a(cosh2 2i − cos2 1j ) √ where :ij = wij 6ij . Eq. (6.44) is the 'nal working equation for the GPS discretization of the H2+ Hamiltonian in the prolate spheroidal coordinates. It has the form of a standard eigenvalue problem of a sparse real symmetric matrix. As a measure of the accuracy and usefulness of the two-center GPS procedure, the ground-state energy of H2+ is calculated [29] with the help of Eq. (6.44). Using a modest number of grid points (12 points in 2 and 10 points in 1), the result is E = −1:1026342144949 a.u., in complete agreement with the exact value of −1:1026342144949 a.u. [93]. Similar procedure has been extended to the precision calculation of the electronic structure of more complex diatomic molecules [27,32]. The GPS method presented above for bound-state eigenvalue problems can be extended to the resonance-state complex eigenvalue problems by means of the complex-scaling transformation. In the prolate spheroidal coordinates, however, only the coordinate 2 needs to be complex rotated, namely, 2 → 2 exp(i), where  is the rotation angle. Consider, for example, the dc 'eld ionization of H2+ . In the presence of a static electric 'eld parallel to the molecular (z) axis, the Hamiltonian of H2+ becomes 

+

1 2 cosh 2 Hˆ = − ∇2 − + Fa cosh 2 cos 1 ; 2 a(cosh2 2 − cos2 1)

(6.45)

where F is the electric 'eld amplitude. Following the complex-scaling GPS procedure as discussed before, one arrives at the following sparse complex symmetric matrix eigenvalue problem from which the 'eld ionization rates can be accurately performed [29]: qkj 1 1 qil :i
j √ d2ki d2ki
+ :ij
√ d1 d 1
2
2
wij wij
lj lj wi
j wi; j k; i

+



j ;l

2 cosh 2i Fa cosh 2i cos 1j − a(cosh2 2i − cos2 1j )

:ij = E:ij :

(6.46)

Extension of the complex-scaling GPS method to the study of charge resonance enhanced multiphoton ionization of molecular ions in intense laser 'elds will be discussed in Section 8.3. 7. Applications of non-Hermitian Floquet methods: atomic multiphoton processes in strong 'elds In this section, we present several applications of non-Hermitian Floquet formalisms and complex quasienergy methods for nonperturbative studies of atomic MPI/ATI in intense monochromatic laser 'elds.

28

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

7.1. Multiphoton and above-threshold ionization of atomic hydrogen The 'rst nonperturbative calculations of intensity-dependent “generalized” MPI cross sections of atoms were performed in 1977 [47] for atomic hydrogen driven by intense linearly polarized monochromatic 'elds. This work introduced the 'rst non-Hermitian Floquet formalism and employed the use of uniform complex-scaling transformation and Laguerre basis. The work was extended to the case of circularly polarized 'eld in 1978 [48]. In the latter case, the periodically time-dependent SchrGodinger equation, 9 (r; t) = [Hˆ 0 (r) − F(x cos !t + ;y sin !t)](r; t) ; (7.1) 9t can be transformed to a system of coordinates rotating with the frequency of the 'eld, yielding the time-independent eigenvalue equation, i

ˆ ˆ ˆ Q(r)  (r) = [H 0 (r) − ;!lz − Fx] (r) =  (r) ;

(7.2)

where Qˆ is the quasienergy operator,  is the quasienergy, lˆz is the z-component of the orbital angular momentum, and ; = +1(−1) corresponds to the left (right) circularly polarized light. Similar to the Floquet Hamiltonian Hˆ F in the linearly polarized case, the quasienergy operator Qˆ has no discrete ˆ −1 are located near the real axis but on higher spectrum. The complex poles of the resolvent (E − Q) Riemann sheet. These complex quasienergy states may be determined by analytical continuation of ˆ Q(r) to the complex plane using the uniform complex scaling transformation: ˆ ˆ exp(i)) = Hˆ 0 (r exp(i)) − ;!lˆz − Fx exp(i) : Q(r) → Q(r

(7.3)

The complex quasienergy eigenvalue equation, ˆ exp(i)) (r exp(i)) =  (r exp(i)) ; Q(r 2

(7.4)

can be solved by either L basis set expansion or generalized pseudospectral discretization technique. Fig. 6 shows the Floquet Hamiltonian for H atom in circularly polarized light [48]. Let us now return to the linearly polarization case. Although the non-Hermitian Floquet formalism is nonperturbative, the Floquet results should reproduce the perturbative results in the weak 'eld limit. Maquet et al. have exploited in detail the relationship between the complex poles of the resolvent of the Floquet Hamiltonian and continued-fraction perturbation theory [74]. Connection is also made to the diagrammatic representations of the in'nite-order perturbation summations [74]. For resonant two-, three-, and four-photon processes, Holt et al. have made connection of the non-Hermitian Floquet theory with the standard two-level model [94]. Later Chu and Cooper [95] presented benchmark calculations of the intensity- and frequency-dependent complex quasienergies (ER ; −=2), threshold shifts, and above-threshold-ionization (ATI) branching ratios for the perturbed ground state of atomic-hydrogen in linearly polarized laser 'elds. Table 1 shows representative intensity-dependent complex quasienergies of H(1s) for the two- and three-photon resonance frequency regimes [74]. This work is further extended to low-lying excited states in strong linearly polarized laser 'elds [96]. Table 2 shows an example of the intensity-dependent complex quasienergies of the perturbed low-lying excited states (2s, 2p, 3s, 3p) of atomic hydrogen at = = 530 nm. For atomic hydrogen, Shakeshaft and his collaborators have performed a series calculations using the non-Hermitian Floquet technique and Sturmian basis functions [97–99]. Their works

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

29

Fig. 6. Structure of the non-Hermitian quasienergy Hamiltonian for MPI of an H atom in circularly polarized monochromatic 'elds. The Hamiltonian consists of diagonal angular momentum blocks Lm (where L = S; P; D; : : : ; etc., and m are the magnetic quantum numbers) and o9-diagonal dipole coupling blocks of types X and Y (adapted from Ref. [48]). Table 1 Intensity-dependent complex quasienergies (ER ; −=2) in atomic units of the perturbed ground state of the H atom nearby Nm = 2 (! ¿ 0:25 a.u.) and Nm = 3 (! ∼ 0:2 a.u.) regions !

ER

−=2

Frms = 0:01 0.30 0.28 0.27 0.26 0.25 0.22 0.20 0.19 0.18

−0:5005167 −0:5004353 −0:5004059 −0:5003816 −0:5003613 −0:5003353 −0:5002932 −0:5002452 −0:5002898

−0:3769(−5) −0:4513(−5) −0:5021(−5) −0:5616(−5) −0:6286(−5) −0:1799(−6) −0:1055(−6) −0:7147(−5) −0:7911(−6)

Frms = 0:025 0.30 0.28 0.27 0.26 0.25 0.22 0.20 0.19 0.18

−0:502899 −0:502474 −0:502320 −0:502197 −0:502100 −0:502192 −0:501585 −0:502396 −0:501968

−0:1314(−3) −0:1608(−3) −0:1800(−3) −0:2021(−3) −0:2259(−3) −0:1888(−4) −0:4667(−4) −0:4458(−3) −0:9484(−4)

!

ER

−=2

Frms = 0:075 0.30 0.28 0.27 0.26 0.22 0.20 0.19 0.18

−0:51301 −0:51215 −0:51203 −0:51243 −0:51944 −0:51892 −0:51733 −0:51484

−0:639(−2) −0:815(−2) −0:920(−2) −0:110(−1) −0:173(−1) −0:683(−2) −0:437(−2) −0:376(−2)

Frms = 0:10 0.30 0.28 0.27 0.26 0.22 0.20 0.19 0.18

−0:5158 −0:5178 −0:5250 −0:5315 −0:5355 −0:5250 −0:5208 −0:5188

−0:167(−1) −0:228(−1) −0:252(−1) −0:103(−1) −0:163(−1) −0:821(−2) −0:961(−2) −0:132(−1)

Frms is the root-mean-square electric 'eld strength, and the numbers in parentheses indicate the power of 10 (adapted from Ref. [95]).

30

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Table 2 Intensity-dependent complex quasienergies (ER ; −=2) in atomic units of the perturbed low-lying excited states of the H atom at = = 530 nm Frms

ER

−=2

Frms

ER

−=2

2s 1:0(−4) 5:0(−4) 1:0(−3) 2:0(−3) 3:0(−3) 4:0(−3) 5:0(−3) 7:5(−3) 1:0(−2)

−0:12499970 −0:1249926 −0:124970 −0:124876 −0:124709 −0:124455 −0:124102 −0:12277 −0:12125

−0:1383(−7) −0:1853(−7) −0:8769(−7) −0:1124(−5) −0:5125(−5) −0:1432(−4) −0:3041(−4) −0:1092(−3) −0:5898(−3)

2p 1:0(−4) 5:0(−4) 1:0(−3) 2:0(−3) 3:0(−3) 4:0(−3) 5:0(−3) 7:5(−3) 1:0(−2)

−0:12499905 −0:1249762 −0:124905 −0:124626 −0:124177 −0:123581 −0:122876 −0:12136 −0:11898

−0:7705(−9) −0:5371(−8) −0:7432(−7) −0:1159(−5) −0:5693(−5) −0:1726(−4) −0:4047(−4) −0:6558(−3) −0:4256(−2)

3s 1:0(−4) 5:0(−4) 1:0(−3) 2:0(−3) 3:0(−3) 4:0(−3) 5:0(−3) 6:0(−3) 7:5(−3)

−0:05555488 −0:0555383 −0:055486 −0:055279 −0:054933 −0:054448 −0:053823 −0:05306 −0:05165

−0:1846(−6) −0:5219(−5) −0:2096(−4) −0:8397(−4) −0:1892(−3) −0:3370(−3) −0:5278(−3) −0:7624(−3) −0:1197(−2)

3p 1:0(−4) 5:0(−4) 1:0(−3) 2:0(−3) 3:0(−3) 4:0(−3) 5:0(−3) 6:0(−3)

−0:05555525 −0:0555478 −0:055525 −0:055432 −0:055280 −0:055062 −0:05479 −0:05445

−0:2443(−6) −0:6329(−5) −0:2540(−4) −0:1025(−3) −0:2341(−3) −0:4248(−3) −0:6807(−3) −0:1010(−2)

The numbers in parentheses indicate the power of 10 (adapted from Ref. [96]).

were reviewed in [18]. Fig. 7 shows the total ionization rates of H(1s) by linearly polarized 616 nm light calculated by DGorr et al. [97]. By averaging the Floquet results over the spatial and temporal pro'les of the laser pulse, DGorr et al. [97] were able to obtain the photoelectron spectra of atomic H at 608 nm which are in fair agreement with the experimental data by Rottke et al. [100]. No absolute measurements, however, were achieved in these earlier experiments. Kyrala and Nichols [102] have performed the 'rst absolute rate measurement for MPI of atomic hydrogen at 248 nm with subpicosecond pulses between 1012 and 1014 W=cm2 . Their results were compared with perturbative and nonperturbative theories. The agreement between experimental and theoretical results is fair but can be improved if the laser pro'le is taken into account in the calculations. So far we have discussed only the MPI of atomic hydrogen. In the presence of strong enough laser 'elds, however, above-threshold ionization (ATI) can occur: the emitted electron can absorb (Nm + S) electrons, where Nm is the minimum number of photons required to ionize the atoms and S = 0; 1; 2; : : : . Thus the electron energy spectrum consists of a series of peaks occurring approximately at (Nm + S)}! − Eg , where Eg is the ionization potential of the ground state. When the external 'eld strengths further increases, the electron peaks broaden and shift and the slowest electron peaks eventually disappear, a typical ATI “peak switching” phenomenon in long pulse experiments particularly for rare gas atoms [103]. For atomic hydrogen, usually only a few ATI

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Fig. 7. Total rate for ionization of H(1s) by 616 nm light. (b) and (c) are magni'cations of the structure inside the boxes (b) and (c) of (a) (adapted from Ref. [97]).

peaks can be detected [101,102] even at high laser intensity (see Fig. 8 for an example.) The ATI is essentially negligible for intensity less than 1014 W=cm2 . For a survey on experimental and theoretical works on MPI/ATI, the readers are referred to the review [104]. 7.2. Intensity-dependent threshold shift and ionization potential The non-Hermitian Floquet matrix method discussed in previous sections provides a nonperturbative technique for the exploration of the MPI/ATI phenomena. It is found that the ionization potential is frequency and intensity dependent and is determined by both the ac Stark shift of the ground state and the continuum threshold upshift. The disappearance of the lowest electron energy peaks in the ATI can be accounted for by the shift of the ionization threshold in intense 'elds [103]. The ionization potential in intense 'elds may be de'ned as [95] th (F) = Uosc + |ER (F)| ;

(7.5)

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Fig. 8. Energy spectrum of the photoelectrons from MPI of atomic hydrogen. The laser parameters are wavelength 608 nm, peak intensity ∼ 1014 W=cm2 , and pulse duration ∼ 0:4 ps. The smooth line below 1 eV is the signal averaged over the statistical >uctuations in this energy range (adapted from Ref. [101]).

where F is the (peak) 'eld strength, ER (F) (¡ 0) is the 'eld-dependent perturbed ground-state energy obtained from the complex quasienergy calculation, and Uosc = e2 F 2 =4m!2 ;

(7.6)

is the average quiver kinetic energy (also known as the ponderomotive potential) picked up by an electron of mass m and charge e driven sinusoidally by the 'eld. Since in the limit of high quantum numbers, a Rydberg electron becomes a free electron, the continuum threshold is shifted up by the amount equal to Uosc . Electrons traversing a laser beam scatter elastically from regions of high light intensity by the ponderomotive potential. Thus an electron with energy el (F) less than Uosc cannot escape from the Coulomb potential and is trapped. From Eq. (7.5), we can de'ne the threshold shift as Yth (F) = th (F) − th (F = 0) ;

(7.7)

where th (F = 0) is the 'eld-free ionization threshold. The total energy of the emitted electron in the 'eld can be written as el (F) = N }! + ER (F) = e2 F 2 =4m!2 + PT2 =2m ;

(7.8)

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Fig. 9. Intensity-dependent threshold shift Yth (F) = YER + YEC for ! = 0:5 a.u. (Nm = 1) (adapted from Ref. [105]).

where N = (Nm + S) is the total number of photons absorbed by the electron near the atom. Since a free electron cannot absorb or emit photons after leaving the Coulomb 'eld, the electron has an energy el which is the same in the laser 'eld as it is at the detector. Thus the ponderomotive potential acts to alter the kinetic energy (PT2 =2m) of the electron from its value outside the laser 'eld to a lower value inside the laser. Figs. 9 and 10 show typical examples of intensity- and frequency-dependent threshold shifts of atomic hydrogen in intense laser 'elds, where YER = ER (F = 0) − ER (F) is the ac Stark shift of the ground state, YEC = e2 F 2 =4m!2 is the continuum threshold upshift due to ponderomotive potential, and Yth (F) is the net threshold shift de'ned by Eq. (7.5) and is equivalent to the sum of YER and YEC . Fig. 9 shows the threshold shifts typical to the one-photon (Nm = 1) dominant process (! ¿ 0:5 a.u.) while Fig. 10 shows the typical phenomena for multiphoton (Nm = 3 in this case) dominant process (! ¡ 0:5 a.u.). Note the marked di9erence between the two cases. For Nm = 1 (Fig. 9), both the ground state (ER (F) ¿ ER (0)) and the continua are upshifted with the ac Stark shift |YER | being greater than YEC . The resulting net threshold shift Yth (F) becomes more negative as the 'eld strength increases. Hence the ionization potential decreases with increasing F. On the other hand, for Nm ¿ 2 such as the case ! = 0:2 a.u. shown in Fig. 10, the ground state energy shifted downward (ER (F) ¡ ER (0)) while YEC shifts the continuum threshold upward. The result is a large positive net threshold shift and the ionization potential increases rapidly with increasing 'eld strength F. As a general rule, the ponderomotive potential YEC becomes more and more important than the ac Stark shift |YER | as Nm increases or ! decreases. The consequence is that the ionization potential

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Fig. 10. Intensity-dependent threshold shift Yth (F) = YER + YEC for ! = 0:2 a.u. (Nm = 3) (adapted from Ref. [105]).

increases rather rapidly with both F and Nm . The disappearance of the lowest energy electrons in the MPI/ATI experiment of xenon [103], for example, can be attributed to this threshold shift e9ect. 7.3. AC Stark shifts of Rydberg states in strong (elds Giant ac Stark shifts of high-lying atomic states in strong 'elds have been observed experimentally [106]. While perturbation calculation can be performed to arbitrary excited states, it is valid only for weak 'elds. The behavior of ac Stark Shift of Rydberg atoms in strong 'elds is still not well understood particularly for resonant excitation cases. In this section, we discuss some theoretical results obtained from a generalized Floquet technique, using the Sturmian basis. The method allows nonperturbative treatment of ac Stark shifts of arbitrary excited states [96]. Fig. 11 shows the ac Stark shifts of n = 12 atomic states of atomic H for l = 0; 1; 2; 3. Several essential energy-shift behaviors of excited states are noted: (a) All the excited levels shown are shifted upward and closely follow the shift caused by the ponderomotive potential Uosc (shown by dotted curves) in the weaker-'eld region. This e9ective potential has its origin in the A2 term (where A is the vector potential) and has been shown to be equal to the average quiver kinetic energy picked up by an electron of mass m and charge e driven sinusoidally by the 'elds. The results lend further support to the view that all Rydberg states and the continuum are upshifted by the same amount, described by Uosc . However, this description appears valid only in the weak-'eld regime where no strong mixings exist among atomic states. (b) Above some critical 'eld strengths (Fc ), the atomic energy levels (for a given n

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35

Fig. 11. Field-dependent ac Stark shift of n = 12; l = 0; 1; 2; 3 states of atomic hydrogen. The dotted curve is the ponderomotive potential quadratic shift (adapted from Ref. [96]).

but di9erent l) split, and signi'cant deviation from the A2 curve occurs. The critical 'eld strength Fc depends on n and decreases rather rapidly as n increases. One should therefore use the A2 ponderomotive shift law with caution in the interpretation of energy-level shifts in high-intensity MPI/ATI experiments. (c) For F ¿ Fc , strong mixings exist among nearby atomic states, and the level identities usually cannot be discerned. Fig. 12 shows the intensity-dependent energy-level shift pattern for highly excited states (n = 49; 50; 51; 52). (For clarity, only the even-parity (s, d, g) states are shown.) The level-shift pattern is similar to that shown in Fig. 11, except that the critical 'eld strengths Fc are now considerably lower. For F ¿ Fc , a large departure from the A2 shift occurs, and strong inter-n mixings take place. This behavior is expected to prevail for all Rydberg levels. 7.4. Intensity- and frequency-dependent multiphoton detachment of H − The study of multiphoton detachment of H− is a subject of considerable interest, stimulated mainly by the experimental work at LAMPF in Los Alamos [107,108] in the last decade. The experimental setup uses the relativistic Doppler e9ect, allowing the continuous tuning of the laser frequency (in the atom frame) over a wide range of photon energies. For moderately strong laser intensities (109 –1011 W=cm2 ) used in the experiments, doubly excited states are far above the ionization threshold of the ground state and can thus be safely ignored. Since H− has only one weakly bound state, one can construct accurate one-electron model potential to describe H− [109] which reproduces precisely the known H− detachment energy and the low-energy e-H(1s) elastic-scattering phase shifts. The model potential so constructed reproduces exactly the one-photon detachment cross sections [110,111] obtained from ab initio two-electron

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Fig. 12. Field-dependent energy-level shift pattern for even-parity (s,d,g) states of Rydberg hydrogen atoms (n = 49–52). Ponderomotive potential shifts are shown in dotted lines (adapted from Ref. [96]).

correlated calculations. This model potential has been used to perform several H− multiphoton detachment studies in the last decade, providing useful insights regarding intensity- and frequencydependent detachment rates and electron angular distributions. 7.4.1. Intensity-dependent multiphoton detachment rates and AC Stark shifts of H − In this section, we discuss some typical intensity- and frequency-dependent behavior of the complex quasienergy (ER ; −=2) of H− in strong laser 'elds by means of the non-Hermitian Floquet Hamiltonian formalism and the complex-scaling generalized pseudospectral technique [75]. The structure of the Floquet Hamiltonian Hˆ F (r) for H− is similar to that of atomic hydrogen, as H− is treated by the e9ective one-electron model potential described above. The laser frequencies range 0.20 –0:42 eV and laser intensities from 1 to 40 GW=cm2 . The laser frequency range covers both two- and three-photon dominant detachment processes. Figs. 13a and b show, respectively, the energies (ER ’s) and the multiphoton detachment rates (’s) of H− for laser intensity I = 4; 8; 12; 16, and 20 GW=cm2 and laser frequency from ! = 0:20 to 0:42 eV. Figs. 14a and b show the same complex quasienergies for higher laser intensity I = 20; 30, and 40 GW=cm2 . Several distinct behaviors are noticed: (i) The H− ground-state energy shows signi'cant intensity-dependent ac Stark shifts. The larger the laser intensity, the larger the ac Stark shift. The energy of H− ground state (at a given laser intensity) generally shows smooth dependence on the laser frequency !, except nearby the onset of multiphoton ionization thresholds where ER ’s show dips. The positions of the dips are blue shifted and the dips are more pronounced as the laser intensity increases. (ii) The multiphoton detachment rates () are strongly intensity dependent. For each laser intensity, the photodetachment rate shows rapid change with photon frequency ! nearby the onset of each multiphoton ionization threshold. The lower the laser intensity, the sharper the threshold

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37

Fig. 13. The frequency- and intensity-dependent complex quasienergies (ER ; −=2) of H− for I = 4; 8; 12; 16, and 20 GW=cm2 and ! = 0:20–0:40 eV: (a) ER (real energies), showing the ac Stark shifts of the H− ground state, and (b)  (imaginary energies), showing the multiphoton detachment rates (adapted from Ref. [75]).

behavior. Similar to the behavior of ER , the positions of the photodetachment threshold-jumps are blue shifted as the laser intensity increases. 7.4.2. Averaged multiphoton detachment rates: comparison of theoretical and experimental results In the previous Section 7.4.1, we discuss the multiphoton detachment rates of H− driven by monochromatic laser 'elds. To compare with the experimental measurement [108], it is necessary to perform the simulations of intensity-averaged multiphoton detachment rates [75]. The experiment is simulated by a uniform H− beam (diameter 3 mm) and a Gaussian distribution of laser

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Fig. 14. Same as Fig. 13 except for higher laser intensities I = 20; 30 and 40 GW=cm2 (adapted from Ref. [75]).

intensity in space:  W0 2 exp(−2D2 =w2 (z)) : I (D; z) = I0 W (z)

(7.9)

I0 is the laser peak intensity in the atom frame at the spot center (D = z = 0), W (z) is the spot size given by W (z) = W0 (1 + z 2 =zR2 )1=2 ;

(7.10)

where W0 (=110 m) is the waist of the focus beam, zR = W02 == is the associated Rayleigh range, = = 10:6 m is the laser frequency in the laboratory frame, and z is the distance from the waist. The laser pulse used in the experiment is linearly polarized and temporarily smooth and had a duration of 136 ns (FWHM), and a 1-s long tail. Because of the long duration of the laser pulse, it is a good approximation to treat it as a monochromatic laser 'eld.

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It is useful to recall the experimental setup for the H− experiment [108]. The 800-MeV H− target ions travel at a speed ≈ 2:53 × 108 m=s, corresponding to  = v=c = 0:842 and  = 1:85, where  and  are the usual relativistic parameters. The photon energy in the laboratory frame (Elab ) was Doppler shifted to Ec:m: in the center-of-mass (c.m.) (or equivalently, the atom frame of the H− ion) according to Ec:m: = (1 +  cos )Elab ;

(7.11)

where  is the angle of intersection ( = 0 when head on) of the H− and laser beams. The laser intensity in the c.m. frame (Ic:m: ) transforms from the laboratory value Ilab according to Ic:m: = 2 (1 +  cos )2 Ilab :

(7.12)

In the Los Alamos experiment [108], the laboratory laser wavelength is 'xed at =lab = 10:6 m. By adjusting the intersection angle , one can generate di9erent laser wavelength in the atom-frame via Eq. (7.11). Further according to Eqs. (7.11) and (7.12), Ic:m: =Ilab = (!c:m: =!lab )2 ;

(7.13)

where !c:m: is the laser frequency in the atom-frame, and !lab = 2 c==lab . Eq. (7.13) shows that for di9erent !c:m: , the H− ions are exposed to di9erent laser peak intensity Ic:m: even the laboratory peak intensity Ilab is 'xed. Further, for larger !c:m: , the larger the laser peak intensity Ic:m: needs to be considered. The averaged multiphoton detachment rates can thus be determined via the expression
Ipeak U c:m: ) = W (I )(!c:m: ; I ) dI ; (7.14) (! 0

where I is the laser intensity in the c.m. frame, Ipeak = Ic:m: is the peak intensity in the c.m. frame given in Eq. (7.12), and W (I ) is a weighting factor. In principle, W (I ) depends also upon !c:m: , since for di9erent !c:m: , Ipeak is di9erent. Fig. 15 shows the averaged multiphoton detachment rates determined from the above-mentioned simulation procedure corresponding to the case of Ilab = 4 GW=cm2 , and =lab = 10:6 m. Also shown in Fig. 15 are the experimental data for comparison. The overall agreement appears quite satisfactory, well within the estimated experimental uncertainty of a factor of 5. A similar simulation was also performed using the generalized cross sections from perturbative calculations [109]. The results there were less satisfactory as the predicted onset of n-photon thresholds is much sharper than the experimental data and the Floquet results. These studies indicate that nonperturbative treatment is required and the non-Hermitian Floquet formalism is capable of describing satisfactorily the process of multiphoton detachment of H− in the Los Alamos experiments. 7.5. Above-threshold multiphoton detachment of negative ions: angular distributions and partial widths In this section, we discuss a general Floquet procedure for calculating the angular distributions and partial rates associated with the above-threshold multiphoton detachment of negative ions.

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Fig. 15. Comparison of the intensity-averaged photodetachment rates for the case Ilab =4 GW=cm2 : black circles, theoretical simulation, open circles, experimental data (adapted from Ref. [75]).

7.5.1. General expressions for the photoelectron energy distributions and partial rates We make use of the Floquet solution (r; t) of the time-dependent SchrGodinger equation,   9 1 2 i (r; t) = − ∇ + U (r) + (F · r) cos !t (r; t) ; (7.15) 9t 2 where (r; t) is the quasienergy wave function in the length gauge; U (r) is the atomic potential; F and ! are, respectively, the laser 'eld strength and frequency (linear polarization of the laser 'eld is assumed in Eq. (7.15)). Following the Floquet theorem, the wave function (r; t) can be represented as (r; t) = exp(−it) (r; t) ;  being the quasienergy. The periodically time-dependent wave function Fourier series:  (r; t) = m (r) exp(−im!t) :

(r; t) can be expanded in (7.16)

m

In the F → 0 limit, the Fourier component m = 0 corresponds to the unperturbed wave function. The expression for the electron angular distributions after absorption of n linearly-polarized photons can be written as [112,113] dn = (2 )−2 kn |An |2 : (7.17) dF Here, kn is the electron drift momentum:  (7.18) kn = 2(Re En − UP ) ; En is the electron energy after the absorption of n photons, and UP is the ponderomotive potential: En =  + n!;

UP = F 2 =4!2 :

(7.19)

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The detachment is possible only if Re En ¿ UP ; from this inequality one can extract nmin —the minimal number of photons required for detachment. With the increase of the laser 'eld strength F, the minimal number of photons nmin also increases due to the increase of the positive ponderomotive energy shift of the detachment threshold. The n-photon detachment amplitude An is de'ned as follows [112,113]:  
1

F2 (kn · F) An = dt exp in!t − i sin(2!t) + i cos(!t)

0 (2!)3 !2  
(r · F) 3 sin(!t) U (r) (r; t) ; (7.20) × d r exp −i(kn · r) + i ! where =2 =! is the period. Eqs. (7.17) and (7.20) assume that the wave function (r; t) is properly normalized. This expression is extracted from the exact integral equation for the decay wave function (r; t), the latter equation being obtained with the help of the Green function for the motion in the uniform ac 'eld [114]. For spherical symmetric binding potential U (r) and the initial state with de'nite angular momentum and its z-projection, the electron distribution does not depend on the azimuthal angle ’. Expression (7.20) is suitable for practical computations since the integration over the angles in the spatial integral can be performed analytically, and the integral over the variable can be computed e9ectively using the fast Fourier transform routines. The quantity dn =dF represents the number of electrons per unit time detached with absorption of n photons and emitted within the unit solid angle under direction of the vector kn . Integration of the angular distributions (7.17) with respect to the angles specifying the direction kn gives the partial rates n :
dn : (7.21) n = dF dF The sum of all partial rates with n ¿ nmin , where nmin is the minimum number of photons required for detachment, is equal to the total rate : =

∞ 

n :

(7.22)

n=nmin

7.5.2. Multiphoton detachment of H − near one-photon threshold: exterior complex-scaling calculations In this section, we discuss an application of the non-Hermitian Floquet formalism to abovethreshold detachment of H− , using the exterior complex-scaling—generalized pseudospectral (ECS–GPS) method [77] for the discretization and solution of non-Hermitian Floquet Hamiltonian. One can expand dn =dF (Eq. (7.17)) as a function of the angle G between the detection kn and 'eld F directions on the basis of the Legendre polynomials. Due to parity restrictions, only even Legendre polynomials are present in the expansion:   ∞  dn n (n) 1+ 2l P2l (cos G) : (7.23) = dF 4 l=1

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(n) The coeNcients 2l are the anisotropy parameters since they determine the deviation of the real (n) angular distribution from the isotropic one. When analyzing the behavior of the coeNcients 2l for weak and medium-strong external 'elds, a comparison with the results of the lowest-order perturbation theory (LOPT) is valuable. For the one-photon detachment, the prediction of the perturbation (1) theory is 2(1) =2; 2l =0 (l ¿ 1). The situation is more complicated if the number of absorbed photons n=2. According to LOPT, the emitted electrons in this case may possess the angular momentum (2) 0 or 2. For the emitted electron in the pure d-state, we have 2(2) = 10=7; 4(2) = 18=7; 2l = 0 (l ¿ 2) (2) whereas for the pure s-state the distribution is isotropic, i.e. all 2l = 0. In reality, however, s- and (2) d-waves are mixed in the wave function of the emitted electron, so the coeNcients 2l cannot be calculated with pure angular algebra even for the lowest intensities. According to LOPT, the detachment amplitude should behave as   1 5  (7.24) P0 (cos G) + P2 (cos G) ; 2 2   i.e. it contains contributions from s- and d-partial waves. The factors 1=2 and 5=2 are added as normalization coeNcients for the Legendre polynomials. The mixing coeNcient  can be calculated within LOPT: in general it depends not only on the angular algebra, but also on the radial wave functions. Squaring the absolute value of the amplitude written above and expanding it over the (2) even-order Legendre polynomials, one obtains for the coeNcients 2l [77,112]: √ 10 + 14 Re  5 18 ; 4(2) = ; (7.25) 2(2) = 2 7(1 + || ) 7(1 + ||2 )

other coeNcients being zero within LOPT. Given the mixing parameter , one can calculate the anisotropy parameters 2(2) and 4(2) . For example, if one puts  = 0 (pure d-wave in the 'nal state), the results 10/7 and 18/7 mentioned above are obtained. On the other hand, if we take the 2(2) and 4(2) coeNcients from our calculations, we can 'nd the complex mixing parameter :  92(2) =4(2) − 5 18 √ Re  = ; Im  = − 1 − (Re )2 : (7.26) 7 5 74(2) The coeNcient  calculated in this way is intensity-dependent. In the limit of the weak external 'eld this result should converge to the intensity-independent value which can be determined within LOPT. An example of recent H− Floquet study [77] is discussed below in connection with the recent experiments on the electron angular distribution [115]. The non-Hermitian Floquet calculations were performed with the help of ECS–GPS technique for the laser 'eld intensities in the range 109 W=cm2 –1012 W=cm2 and the wavelengths 1.640 and 1:908 m. For the wavelength 1:640 m, the photon energy (! = 0:756 eV) is very close to the one-photon detachment threshold (0:754 eV). The one-photon channel is open only for the weak intensity 109 W=cm2 ; for the higher intensities it is closed due to ac Stark shift of the detachment threshold. In the 1:908 m (! = 0:650 eV) 'eld, a minimum of two photons is required for detachment for all the intensities used in the calculations. Tables 3 and 4 contain the (above-threshold) partial and total rates for the detachment of H− by 1.640 and 1:908 m laser 'eld, respectively. The results show that the detachment rates for the same intensity are generally larger for the wavelength 1:908 m. The exception is made by the intensity

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Table 3 Partial and total rates for the detachment of H− by 1:640 m radiation Partial rates (a.u.) Laser intensity W=cm2 1.0(09) 1.0(10) 1.0(11) 2.0(11) 4.0(11) 8.0(11) 1.0(12)

Total rates (a.u.)

Number of photons absorbed 1

2

3

4

5

3:316(−09)

3:010(−09) 2:978(−07) 2:717(−05) 9:938(−05) 3:353(−04) 9:482(−04) 1:231(−03)

7:155(−13) 7:099(−10) 6:753(−07) 5:134(−06) 3:695(−05) 2:316(−04) 3:926(−04)

1:553(−16) 2:087(−12) 1:336(−08) 2:109(−07) 3:185(−06) 4:202(−05) 9:064(−05)

9:706(−15) 2:643(−10) 8:655(−09) 2:708(−07) 7:339(−06) 1:981(−05)

6:327(−09) 2:985(−07) 2:786(−05) 1:047(−04) 3:757(−04) 1:229(−03) 1:734(−03)

The numbers in parentheses indicate the powers of 10 (adapted from Ref. [77]).

Table 4 Partial and total rates for the detachment of H− by 1:908 m radiation Partial rates (a.u.)

Total rates (a.u.)

Laser intensity W=cm2

Number of photons absorbed 2

3

4

5

1.0(09) 1.0(10) 1.0(11) 2.0(11) 4.0(11) 8.0(11) 1.0(12)

4:846(−09) 4:778(−07) 4:165(−05) 1:434(−04) 4:231(−04) 9:195(−04) 1:126(−03)

2:211(−12) 2:191(−09) 1:996(−06) 1:433(−05) 9:070(−05) 4:137(−04) 6:353(−04)

8:278(−16) 8:223(−12) 7:650(−08) 1:118(−06) 1:453(−05) 1:416(−04) 3:008(−04)

3:054(−19) 3:037(−14) 2:844(−09) 8:346(−08) 2:162(−06) 4:155(−05) 1:138(−04)

4:848(−09) 4:800(−07) 4:373(−05) 1:589(−04) 5:304(−04) 1:516(−03) 2:176(−03)

The numbers in parentheses indicate the powers of 10 (adapted from Ref. [77]).

109 W=cm2 , where the total detachment rate at the wavelength 1:640 m is larger than that at the wavelength 1:908 m. The di9erence is due to the one-photon contribution which is present for the 1:640 m 'eld and not for the 1:908 m 'eld. The above-threshold contribution is found to be not very signi'cant for the intensities up to 1011 W=cm2 . For the higher intensities, the contribution of the above-threshold channels to the total rate (i.e. deviation from the LOPT predictions) becomes very important. As it can be expected from the general theory, at the same intensity, the breakdown of the perturbation theory is more pronounced for the larger wavelength 1:908 m. This observation is con'rmed by the analysis of the angular distributions by means of the anisotropy parameters 2l for the two- and three-photon detachment as well as the mixing parameters  for the two-photon detachment. Further it is found that for laser intensity less than 1011 W=cm2 ,

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Fig. 16. Angular distributions for two-photon detachment of H− . The wavelength of the laser 'eld is 2:15 m, and the intensity is 6:5 × 1011 W=cm2 . The curve represents the Floquet results of Ref. [91] while the black dots represent the experimental data [116] (adapted from Ref. [91]).

the two-photon detachment processes are dominated by the d-wave electrons, in accord with the recent experimental observations by Praestegaard et al. [115]. More recently, the non-Hermitian Floquet formalism along with the ECS–GPS technique has been extended to the study of the laser-frequency and intensity e9ects on the shape of the electron angular distribution from two-photon detachment of H− near ionization threshold [91]. The external 'eld parameters are chosen to correspond to the recent experiment by Reichle et al. [116]. It is found that the angular distribution pattern can be interpreted in terms of the interference of the s and d partial waves in the 'nal state and the Wigner threshold law. Fig. 16 shows the comparison of the angular distribution patterns obtained from the experiment [116] and the Floquet prediction [91] for the case of laser wavelength 2:15 m and laser intensity 6:5 × 1011 W=cm2 . 7.6. Precision calculation of two-photon detachment of H − In the last several subsections, we have discussed several recent non-Hermitian Floquet studies of multiphoton detachment of H− for energy range well below the doubly excited resonance states. There are also various other theoretical treatments of multiphoton detachment of H− [117–119]. In this subsection, we consider two-photon detachment of H− near the doubly excited 1 S and 1 D resonance states. Several (perturbative or nonperturbative) theoretical methods have been used to study the multiphoton detachment cross section of H− in this higher energy region [120–124]. There has been also a number of two-photon detachment cross section (TPDCS), )2 , experiments for H− , both above and below the single-photon detachment threshold [115,125–128] mainly in the weaker 'eld or perturbative regime. In particular, a prominent 1 D resonance structure above this threshold has been observed in the experiment [125] and predicted by theory [120–124]. The latest measurement [128] yields a peak )2 of 3:2(+1:8; −1:2) × 10−49 cm4 s or 420(+240; −160) (=I 2 ) a.u., whereas the theoretical predictions are 703 [122,123] and 710 [120,121] (=I 2 ) a.u.  is the width and I is the radiation intensity. Hence, despite the rather large error bars quoted in the experiment, the

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45

existing theoretical data still fall outside of the experimental uncertainty. Recently, Chung and Chu [129] have performed a correlated two-electron precision calculation for TPDCS of H− by means of the non-Hermitian Floquet method. A highly accurate initial state wave function is used along with fully correlated saddle-point wave functions for the intermediate and 'nal states. The cross section is investigated for energies below the n = 2 and 3 thresholds. The peak cross section predicted for the 1 D resonance is 3:10 × 10−49 cm4 s at 10:8732 eV. It represents the 'rst ab initio theoretical prediction in complete agreement with the experimental result of 3:2(+1:8; −1:2) × 10−49 cm4 s at 10:8732(27) eV [128].

8. Applications of non-Hermitian Floquet methods: molecular multiphoton processes in strong 'elds It is well known that multiphoton dissociation (MPD) of polyatomic molecules is a rather eNcient process and can occur in relatively weak infrared laser 'elds [130,131]. On the other hand, MPD of small molecules such as diatomic and triatomic molecules is a slow and ineNcient process, due to the low density and anharmonicity of vibrational states. Indeed, MPD of diatomic molecules from the ground vibrational states of diatomic molecule has never been observed experimentally until 1986. The only exception, as far as diatomic molecules are concerned, is the experimental observation of two-photon dissociation from highly excited vibrational states of HD+ in weak 'elds [132]. Floquet studies [133,134] have shown that MPD from the weakly bound high vibrational levels is usually considerably more eNcient than from those more tightly bound low-lying levels, when the laser intensity is weak. Stimulated by the discoveries of a number of novel nonlinear phenomena in the response of atoms to strong laser 'elds, there has been a growing new interest in the study of nonlinear multiphoton dynamics of diatomic molecules in the last decade. Nonlinear optical phenomena such as MPD, above-threshold dissociation (ATD), dissociative ionization, Coulomb explosion, charge-resonanceenhanced ionization (CREI), chemical bond softening and hardening, high-order harmonic generation (HHG), etc. have been extensively explored. For a topical review on the dynamics of H2+ in intense laser 'elds, see [135]. In this section we discuss the development of generalized non-Hermitian Floquet formalisms and complex quasienergy methods for the study of MPD/ATD/CREI/HHG processes of diatomic molecules in strong 'elds. As will be shown below, the presence of additional interatomic degrees of freedom in molecules enriches greatly the problem of the nonlinear interaction of molecules with intense laser 'elds. 8.1. Multiphoton and above-threshold dissociation The development of the non-Hermitian Floquet formalism and complex vibrational quasienergy (VQE) method for the study of molecular MPD processes was 'rst introduced in 1981 [136], using L2 basis functions and uniform complex scaling procedure. Alternative procedures for the calculation of complex VQE’s using the complex-scaling Fourier grid Hamiltonian (CSFGH) method [137] (with uniform grid spacing) and complex-scaling generalized pseudospectral (CSGPS) technique [138] (with nonuniform grid spacing) will be discussed later in this section.

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Fig. 17. Potential energy curves for the ground 1s)g and 'rst excited 2p)u electronic states of H2+ as a function of internuclear separation R. Also displayed is a schematic diagram showing the absorption of one, two, and three photons Z from the ground vibrational level of the 1s)g state (adapted from Ref. [137]). of wavelength 2660 A

Consider the response of the prototype diatomic molecular ion, H2+ , to intense monochromatic laser 'elds. Fig. 17 shows the potential energy curves of the ground (1s)g ) and 'rst excited (2p)u ) electronic states of H2+ as a function of internuclear separation R. Also displaced is a schematic diagram showing the multiphoton and above-threshold dissociation (MPD/ATD) processes from the ground vibrational level of the 1s)g state. In the presence of external laser 'elds, all the vibrational levels of H2+ molecules in the ground electronic state (1s)g ) are coupled to the dissociative continuum of the upper (repulsive) electronic state 2p)u , and thus become (shifted and broadened) vibrational quasienergy (VQE) resonances. Each VQE resonance possesses an intensity- and frequency-dependent complex energy eigenvalue (ER ; −=2), the real part of which is related to the ac Stark shift and the imaginary part (width) provides the total MPD/ATD rate. Fig. 18 shows the dressed-state (electronic-'eld potential energy curves) picture (sold lines: diabatic curves; dotted lines: adiabatic curves). Each curve corresponds to Ui (R) + n}!, which Ui (R) are the electronic potential energy for 1s)g or 2p)u state, and n = 0; −1; −2; −3 are the Fourier photon indices. Formally, the photodissociation or multiphoton dissociation between a bound and a repulsive electronic states is a half-collision process and can be regarded as a (diabatic) curve-crossing or an (adiabatic) avoided-crossing predissociation problem. The Hamiltonian for the perturbed molecular system is (with r and R being, respectively, the electronic and the internuclear coordinates), Hˆ (r; R; t) = Tˆ R + Hˆ el (r; R) + (r; R) · E0 f(t) sin !t ;

(8.1)

where Tˆ R is the nuclear kinetic energy operator, Hˆ el (r; R) is the electronic Hamiltonian, (r; R) is the dipole moment operator, E0 is the electric 'eld amplitude of the laser pulse with pulse shape f(t).

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Fig. 18. Electronic-'eld potential-energy curves of the two electronic states of H2+ dressed by n = 0; −1; −2; −3 photons Z Solid lines: diabatic curves. Dotted lines: adiabatic curves (adapted from Ref. [137]). of wavelength 2660 A.

The SchrGodinger equation under the Born–Oppenheimer approximation can be reduced to (assuming E0 is parallel to R) 9 ˆ ˆ i (8.2) g (R; t) = [T R + U 1 (R)] g + 2(R)E0 f(t) sin !t u ; 9t 9 ˆ ˆ i (8.3) u (R; t) = [T R + U 2 (R)] u + 2(R)E0 f(t) sin !t g ; 9t where g (R; t) and u (R; t) are the probability amplitudes at internuclear distance R with the electron being in the 1s)g and 2p)u states, respectively, Uˆ 1 (R) and Uˆ 2 (R) are the corresponding internuclear potentials, and 2(R) is the transition dipole moment between the two electronic states. In the case of f(t) = 1, i.e., when the perturbation is periodic in time, Eqs. (8.2) and (8.3) can be transformed into an equivalent time-independent in'nite-dimensional Floquet Hamiltonian (Hˆ F ) eigenvalue problem [136,137]. To determine the complex vibrational quasienergy states, the uniform complex-scaling transformation, R → R exp(iG), can be made which leads to a non-Hermitian Floquet Hamiltonian, Hˆ F (R exp(iG)). Instead of expanding the nuclear wavefunctions in terms of a set of basis functions (L2 basis set expansion method) as was done in earlier works [136], one can discretize the Floquet Hamiltonian using the complex-scaling Fourier-grid Hamiltonian (CSFGH) method [137] with equal grid spacing. The matrix elements in the Floquet basis |; n ≡ |⊗|n (where | denotes the electronic states and n is the photon Fourier index ranging from −∞ to +∞) can be written as Ri |[Hˆ F (R exp(iG))]n; m |Rj  = {exp(−2iG)Ri |Tˆ R |Rj YR + [U (Ri exp(iG)) + n}!]ij } nm +

1 2(Ri exp(iG))E0 ij n; m±1 (1 −  ) ; 2

(8.4)

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10

-2

2

Γ/Frms (a.u.)

10

-3

10

-4

10

-5

10

1010

1011

1012

1013

1014

2

I (W/cm ) 2 Fig. 19. Reduced widths (=Frms ) vs. intensity I , for the ground vibrational level (v = 0) of the H2+ (1s)g ) state at Z At weaker 'elds, the photodissociation is a dominant one-photon process. Above some critical 'eld intensity = = 2660 A. (≈ 1012 W=cm2 ), MPD/ATD become signi'cant and dominant in the photodissociation process (adapted from Ref. [137]).

where Ri |Tˆ R |Rj  =

1 N YR




L

l=−L

kl2 exp[ikl (Ri − Rj )] ; 2

(8.5)

with N being the number of grid points, Rj = jYR (j = 1; 2; : : : ; N ), kl = lYk = 2 l=N YR, and L = (N − 1)=2. The desired complex vibrational quasienergies (VQEs) can be then identi'ed by the stationary points of the G trajectories of the complex quasienergies of the non-Hermitian Floquet Hamiltonian Hˆ F (G) [137]. The real parts of the complex VQEs correspond to the energies of the shifted vibrational states in the laser 'elds and the imaginary parts to the (half) widths (MPD/ATD rates) of the vibrational resonances. Fig. 19 shows the intensity-dependent MPD half-widths (=2), obtained from the CSFGH method, as a function of the laser intensity I for the ground vibrational level (v=0) of the H2+ (1s)g ) electronic 2 state. At weaker 'elds, =Frms (proportional to =I ) is seen to be independent of the laser intensity I , and the photodissociation is dominantly a one-photon process. Above some critical 'eld intensity (I  1012 W=cm2 ), ATD sets in, and the process becomes highly nonlinear [137]. 8.2. Nature of chemical bond in strong (elds: laser induced chemical bond softening and hardening We now discuss the intriguing behavior of vibrational quasienergy resonances in intense laser 'elds as well as the chemical bond hardening phenomenon as 'rst revealed from the non-Hermitian Floquet studies by Yao and Chu [50,139].

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Fig. 20. (a) Vibrational quasienergy level structure and dressed adiabatic potentials of H2+ , (b) photodissociation halfwidths (=2) of vibrational quasienergy states (labeled by v of H2+ ), at = = 775 nm and I = 1011 W=cm2 (weaker 'eld case (adapted from Ref. [50])).

Figs. 20–22 show the real (ER ) and imaginary (=2) parts of the complex vibrational quasienergies of H2+ at 775 nm for laser intensity at I = 1011 W=cm2 (weaker 'eld case), 5 × 1012 W=cm2 (medium strong 'eld case), and 5 × 1013 W=cm2 (strong 'eld case), respectively. In Figs. 20(a) –22(a), the 'eld-modi'ed adiabatic potentials are displayed and labeled by Floquet-state basis index |g; n (or |u; n) in the asymptotic (R) region. The horizontal line segments at the left side column represent the converged (real parts of the) energies of VQE resonances. The line segments at the right side column(s) denote the energy positions of bound vibrational levels or shape resonances supported by the corresponding adiabatic potential well(s). The imaginary (half) widths (=2) of VQE resonances are shown in Figs. 20(b) –22(b) in ascending order (labeled by v ) according to the magnitude of their corresponding ER ’s displayed in Figs. 20(a) –22(a). Figs. 20(a) and (b) show the expected weak-'eld behavior. At this intensity (I = 1011 W=cm2 ), the (perturbed) VQE resonance positions shown in Fig. 20(a) are very close to those of ('eld-free)

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Fig. 21. Same as Fig. 20 except for the intensity I = 5 × 1012 W=cm2 (medium strong 'eld case) (adapted from Ref. [50]).

vibrational states supported by the ground 1s)g potential curve. The behavior of the photodissociation widths (Fig. 20(b)) of these resonance states is also expected: low-lying states generally have smaller dissociation widths and longer (photodissociation) lifetimes than those of the high-lying levels. In fact, the photodissociation rates of high-lying resonances (v ¿ 8) can be nine to ten orders of magnitude larger than the photodissociation rate of the tightly bound ground vibrational level. The situation becomes more dedicated when the laser intensity increases. For example, at the medium strong intensity I = 5 × 1012 W=cm2 , the gap of one-photon avoided crossing (R  5 a.u.) already becomes suNciently large and the structure of VQE resonances signi'cantly distorted. In fact, the VQE resonances now break into two groups (Fig. 21(a)): lower-lying resonance group (v = 0–10) and higher-lying resonance group (v ¿ 11), widely separated in energy. The widths of the upper-group resonances (v ¿ 11) are consistently smaller than those of the higher members of the lower-lying group resonances (e.g. v = 7–10), see Fig. 21(b). As compared with the weaker 'eld case (Fig. 20(b)), all the VQEs in the lower-lying group are now broadened substantially, i.e.

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Fig. 22. Same as Fig. 20 except for the intensity I = 5 × 1013 W=cm2 (strong 'eld case) (adapted from Ref. [50]).

molecules become more unstable in stronger 'elds, a phenomenon known as “bond softening” [140]. What is more intriguing here is the unexpected behavior of the upper group resonances. A comparison of Figs. 20(b) and 21(b) reveals that the photodissociation rates of these high-lying VQE resonances actually decrease with increasing laser intensity! That is, molecules become more stable at stronger 'elds, a novel phenomenon which was termed as “chemical bond hardening” [50,139]. These bond hardened states arise from the trapping of molecular vibrational wave functions at longer (R  5 a.u.) internuclear separation by the (one-photon) adiabatic potential well. These trapped states are in fact not bound states but slowly leaking quasi-bound resonance states due to the nonadiabatic couplings to other Floquet-state channels. As the laser intensity increases, the one-photon gap becomes larger, leading to weaker nonadiabatic couplings and therefore smaller photodissociation rates (widths). As the laser intensity further increases to strong 'eld regime, multiphoton avoided crossings now play signi'cant role and the VQE resonance structure undergoes dramatic changes. Fig. 22 shows a strong 'eld case at I = 5 × 1013 W=cm2 , in which the VQE resonances now break into several

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di9erent groups. The topmost resonance-group states (v = 16–19) are well separated in energy from the lower-lying groups. These highest-lying resonances exhibit the following distinct features: (a) They are supported by the (now very shallow) adiabatic potential well near the one-photon avoided cross region (Re  6:4 a.u.). (b) Their photodissociation rates are extremely small, smaller than that of any lower-lying group resonance states. Molecules associated with these VQE states are therefore very stable against photodissociation even their binding energies are very small, a full manifestation of the bond hardening phenomenon. The phenomenon of such a laser-induced stabilization of molecules has been observed experimentally [141] and chemical bond hardening is a topic of much current interest in strong-'eld molecular physics [142]. Non-Hermitian Floquet study of MPD/ATD and laser-induced VQE resonance states of D+ 2 molecular ions in intense one- and two-color (fundamental plus its third harmonic) laser 'elds was also pursued [143] following the initial H2+ work [50]. Stabilization and chemical bond hardening for high-lying VQE resonances are observed for both one- and two-color excitation processes. It is found that by tuning the relative phase between the fundamental and the third harmonic laser 'elds, the electronic-'eld potential surface and the MPD/ATD rates can be modi'ed signi'cantly, suggesting some degree of “coherent control” of multiphoton dynamics may be feasible [143]. The kinetic energy spectra of diatomic molecules undergoing MPD/ATD can be performed by means of an extension of the non-Hermitian Floquet formalism and an integral equation approach for the partial rates [138]. By discretizing the non-Hermitian Floquet Hamiltonian by the complex-scaling generalized pseudospectral (CSGPS) technique with nonuniform grid spacing, and using a “back rotation” procedure to extract the partial widths from the total resonance wavefunctions, Telnov and Chu were able to obtain the ATD energy spectra from individual vibrational level of H2+ molecular ions in intense 775 nm laser 'elds [138]. 8.3. Charge resonance enhanced multiphoton ionization of molecular ions in intense low-frequency laser (elds The study of dissociative ionization of diatomic molecules in intense laser 'elds is a subject of considerable current interest both experimentally [5,144–146] and theoretically [29,147–149]. For the prototype molecular ion system, H2+ , both time-dependent wavepacket method [147] and Floquet approach (using complex basis functions) [148,149] have been used for the study of the multiphoton ionization in strong 'elds. Experimentally it has been found that linear molecules tend to align along the linear polarization of the laser 'elds. Further, the kinetic energy of the dissociated fragment ions appears to be independent of the laser pulse and ionization fraction, and is only a fraction of the Coulomb energy of the ions at the equilibrium separation Re [146]. A possible interpretation of the latter observation is that “Coulomb explosion” does not take place at the equilibrium internuclear distance Re but at a larger “critical” distance (Rc ) at which the ionization rate peaks [150]. The charge resonance enhanced ionization (CREI) at some larger internuclear distance has been observed experimentally [145] and con'rmed by theoretical consideration [29,147–149]. In the case of low-frequency ac 'elds, the theoretical analysis of the enhanced ionization process is generally proceeded by means of the over-the-barrier breakup mechanism in the static (dc) 'eld limit at 'xed internuclear distance R [147–149]. However, there is still some disagreement on the actual detailed mechanisms responsible for the CREI phenomenon.

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Due to the large number of Floquet photon blocks involved in the low-frequency 'elds, “exact” Floquet calculations are diNcult to achieve and previous studies of this problem have used approximations such as the cycle averaged rate (i.e., an average of the ionization rates at di9erent dc 'elds over one optical cycle) [148,149] or the asymptotic expansion of the complex quasienergy to the !2 order [149]. In this section, we discuss a recent development of non-Hermitian Floquet theory and computational method for two-center systems, allowing accurate and converged calculations of multiphoton ionization of H2+ (at 'xed R) in the low-frequency (1064 nm) regime and the detailed exploration of the origin of CREI phenomenon [29]. The periodically time-dependent Hamiltonian describing the electronic interaction of H2+ with a monochromatic, linearly polarized laser 'eld with frequency ! and electric 'eld F along the internuclear axis z, is given by, in prolate spheroidal coordinates (2; 1; ’) [29], 1 2 cosh 2 Hˆ (r; t) = − ∇2 − + aF cosh 2 cos 1 cos !t ; (8.6) 2 a(cosh2 2 − cos2 1) where the two nuclear positions are set at (0; 0; a) and (0; 0; −a) along the z-axis in Cartesian coordinates, and the internuclear separation R is equal to 2a. Corresponding to Eq. (8.6), an equivalent time-independent Floquet Hamiltonian Hˆ F can be constructed, 1 [Hˆ 0 − j − n!]n + Fa cosh 2 cos 1[n−1 + n+1 ] = 0; (n = 0; ±1; ±2; : : :) ; (8.7) 2 where Hˆ 0 is the unperturbed electronic Hamiltonian and the quasienergy-state Fourier component n (r) is time independent. Performing the complex scaling transformation, 2 → 2 exp(i), and using the variational formulation for the Floquet Hamiltonian [29], it leads to the minimization of the functional  2  2
2 1 9 9 n n Fs = 2 d 3 r + R 92 91 sinh2 2 + sin2 1  
2 cosh 2 3 + n! + j (n )2 − d r 2 2 a(cosh 2 − cos 1)
+Fa d 3 r cosh 2 cos 1(n−1 + n+1 )n (n = 0; ±1; ±2; : : :) : (8.8) Discretizing Eq. (8.8) by means of the complex-scaling (CS) generalized pseudospectral (GPS) method for the two-center system in the prolate spheroidal coordinates (see Section 6.4) and performing the minimization, one obtains [29] qkj 1 n qil 1 n :i
j √ d2ki d2ki
+ :ij
√ d1 d1
2
wi
j wi; j 2
wij wij
lj lj k; i

j ;l

+Fa cosh 2i cos 1j (:ijn−1 + :ijn+1 ) − (n = 0; ±1; ±2; : : :)

2 cosh 2i :ijn = j:ijn ; a(cosh2 2i − cos2 1j ) (8.9)

where qij and wij are the weights used in the GPS procedure. Eq. (8.9) leads to an in'nite-dimensional complex symmetric matrix, whose complex eigenvalues (j = (ER ; −=2)) are related to the positions and widths of the shifted and broadened complex quasienergy states of H2+ .

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Fig. 23. Electronic energy levels of I electronic states of 'eld-free H2+ molecular ion vs. internuclear distance R. The proton–proton Coulomb repulsion energy is not included. The n)g states are in solid lines, while the n)u states are in dotted lines (adapted from Ref. [29]).

Fig. 23 shows the 'eld-free electron energy levels of H2+ as a function of internuclear distance R, obtained by the GPS method. The two lowest electronic states, 1)g and 1)u , become nearly degenerate at larger R. In the presence of the external 'elds, the electric dipole coupling of 1)g and 1)u is linearly proportional to R and becomes very signi'cant. This phenomenon, known as the “charge resonance” (CR) e9ect [151], occurs only in the odd-charged molecular ion systems. As to be shown later in this section, the combined e9ect of CR and the multiphoton transitions to excited electronic states is the main mechanism responsible for the CREI phenomenon observed for the molecular ion systems [29]. Figs. 24 and 25 show, respectively, the R-dependent real and imaginary parts of the complex quasienergies of H2+ in the presence of the linearly polarized (LP) 1064 nm monochromatic laser 'eld with peak intensity 1014 W=cm2 . The number of grid points used are 46 in the 2 coordinate and 34 in the 1 coordinate. Due to the symmetry of the system, only half of the 1 grid points are actually needed. Up to 121 Floquet photon blocks were used to achieve fully converged results. The largest dimension of the non-Hermitian Floquet matrix considered in this study is 87; 120. The complex quasienergy eigenvalues can be determined accurately and eNciently by means of the implicitly restarted Arnoldi algorithm for sparse complex matrix [29]. Due to the large number of the electronic and Floquet blocks involved, the resulting Floquet energy level structure is rather complicated. In Fig. 24, only those quasienergy states whose major components are the 'eld-free 1)g and 1)u states are shown. In fact, the information regarding enhanced ionization can be extracted from these complex quasienergy states alone. Fig. 24 reveals several intriguing behaviors of the real parts of

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Fig. 24. The real parts of the complex quasienergies vs. R. Only those quasienergy states whose dominant components are from 1)g and 1)u states are included. Two di9erent groups of quasienergy states can be identi'ed. The two solid lines indicate one representative quasienergy level from each group (adapted from Ref. [29]).

Fig. 25. The imaginary parts of the complex quasienergies vs. R from the lower and upper groups of quasienergy levels (adapted from Ref. [29]).

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the quasienergy levels. Two di9erent groups, called the “lower” and “upper” groups of quasienergy states can be identi'ed. Two of the representative quasienergy levels (one from each group) are labeled by solid lines in Fig. 24. An important consequence of the Floquet symmetry is that all of these quasienergy states in the lower (upper) group, separated by 2m! (m integer) in energy, are in fact physically indistinguishable and contain the same information regarding multiphoton dynamics. Thus, for example, while the (real) energy is separated by 2m!, each quasienergy state in the lower (or upper) group has the identical imaginary energy, Im(j), since the latter is related to the total ionization rate of the physical state. Thus the dynamical information contained in the two quasienergy levels (denoted by solid lines in Fig. 24) can be used to explore the multiphoton dynamics and the mechanisms responsible for the CREI phenomenon. Fig. 25 shows the R-dependent imaginary parts (ionization widths) of the complex quasienergies in the lower (solid line) and upper (dotted line) groups, respectively. It is interesting to note that both curves exhibit two major peaks in the ionization rate at certain larger distances R. For the “lower” group, the largest ionization enhancement occurs at R  9a0 and a second enhancement occurs around R  6:2a0 . For the “upper” group, the major enhancement occurs around R  8a0 and the second enhancement at R  5a0 . The ionization enhancement phenomenon at some larger internuclear distance R has been reported in other recent theoretical studies [147–149] and interpreted in terms of the over-the-barrier ionization picture in the dc-'eld limit. In the dc 'eld, the ionization width of the upper “2p” level (in the 'eld-free united-atom language) is much larger than that of the lower “1s” state [149]. In a low frequency laser 'eld, certain amount of the electron population is excited to the 2p level due to charge resonance and/or multiphoton absorption. Thus in the dc-'eld picture, the 2p level is considered to be the major state responsible for the observed ionization enhancement [147–149]. Zuo et al. [147] argued that the major ionization peak arises from over-the-barrier ionization of the 2p level out of the higher electronic-'eld potential well. Mulyukov et al. [149], on the other hand, suggested that the ionization enhancement is due to the mixing of the 2p state, which is localized in the higher well of the double-well electronic-'eld potential, with energetically nearby highly excited states that are localized in the lower potential well. Over-the-barrier ionization from the lower well can proceed without the impediment of back scattering of the electron from the hump between the wells. These authors also performed perturbative corrections, through order !2 , of the shifts of 1s and 2p levels in the low-frequency ac 'eld [149]. However, the perturbative corrections break down near ionization peak positions. Also the dc-'eld predicted peak positions are somewhat di9erent from those of the quasienergy calculations. Finally, the Floquet results in Fig. 25 indicate that the quasienergy states in both “upper” and “lower” groups both show double-peak enhancement features. Qualitatively, this latter observation may be attributed to the fact that the latest Floquet calculation [29] is a genuine ac-'eld study and the “upper” and “lower” quasienergy states are the dynamical combination of both the “1s” and “2p” levels in the sense that the majority of the electron population is transferring back and forth between the 'eld-free 1)g and 1)u states. A more detailed analysis of the nature and dynamical behavior of these quasienergy states reveals that the ionization enhancement is mainly due to the e9ect of charge resonance enhanced multiphoton resonances of the 1)g and 1)u states with excited electronic states at some particular internuclear distances [29]. These “critical” distances depend on the details of molecular electronic structure and the laser frequency and intensity used in the study. Further, if the laser 'eld is turned on adiabatically, only the “lower” quasienergy level (solid line in Fig. 25) contributes to the enhanced ionization at intermediate and larger R [29].

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9. Many-mode Floquet theorem for nonperturbative treatment of multiphoton processes in multi-color or quasi-periodic laser 'elds 9.1. Many-mode Floquet theorem One of the major limitations of conventional Floquet techniques described in previous sections is that they are applicable only to monochromatic (i.e. one-color laser-'eld) problem where the Hamiltonian is periodic in time. However, many recent experiments involve the use of more than one laser 'eld, namely, multi-color or multi-frequency laser 'elds (with frequencies !i incommensurate). In such cases, the Hamiltonians are no longer periodic in time and the Floquet theorem is simply not valid. Such a bottleneck has been circumvented with the development of the so-called many-mode Floquet theorem (MMFT) by Ho et al. in 1983 [51,52,152–154]. The MMFT allows the exact transformation of any polychromatic or quasi-periodic time-dependent SchrGodinger equation into an equivalent time-independent in'nite-dimensional eigenvalue problem. Without loss of generality, let us consider the interaction of an arbitrary N -level system with two incommensurate monochromatic radiation 'elds. Extension to arbitrary number of radiation 'elds is straightforward. In the electric dipole approximation, the SchrGodinger equation can be written as 9 i (r; t) = Hˆ (r; t)(r; t) ; (9.1) 9t where the Hamiltonian Hˆ (r; t) is bichromatic in time, Hˆ (r; t) = Hˆ 0 (r) − (r) · [E1 (t) + E2 (t)] :

(9.2)

Hˆ 0 and  are respectively the unperturbed Hamiltonian and the dipole moment of the system, and E1 and E2 are classical 'elds given by (9.3) Ei (t) = Re[Ei Jˆi e−i!i t ] ; ˆ where Ei ; i and !i are respectively the electric 'eld amplitude, the polarization vector, and the frequency associated with the ith 'eld. Note that here we assume the two 'eld frequencies !1 and !2 are incommensurate so that the total Hamiltonian, Eq. (9.2), is nonperiodic in time. The MMFT [51,52] states that the exact solution of the time-dependent SchrGodinger equation, with the Hamiltonian Eq. (9.2), has the following explicit form: (r; t) = exp(−it)(r; t) ;

(9.4)

where  is the generalized (two-mode) quasienergy, and (t) is bichromatic in time. Thus the concept of quasienergy is preserved even when the Hamiltonian is no longer periodic in time! This greatly facilitates the exploration of multiphoton dynamics in multi-color laser 'elds. The MMFT also allows the exact transformation of the bichromatic time-dependent problem, Eq. (9.1), into an equivalent time-independent in'nite-dimensional matrix eigenvalue problem [51,52]:  1 n1 n2 |Hˆ F |2 k1 k2 2 k1 k2 |= = =1 n1 n2 |= ; (9.5) 2

k1

k2

where 1 n1 n2 |Hˆ F |2 k1 k2  = H[n1 12−k1 ; n2 −k2 ] + (n1 !1 + n2 !2 )1 2 n1 k1 n2 k2 ;

(9.6)

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with H[n1 12; n2 ] = E1 1 2 n1 0 n2 0 +

2 

V(i) (ni ;1 + ni ;−1 ) ; 1 2

(9.7)

i=1

and 1 = − Ei 1 | · i |2  : (9.8) V(i) 1 2 2 Here Hˆ F is the (time-independent) two-mode Floquet Hamiltonian de'ned in terms of the generalized Floquet basis state |n1 n2  = | ⊗ |n1  ⊗ |n2 , with  being an atomic (or molecular) state of H0 , and the integer index ni (=0; ±1; ±2; : : :) is a Fourier component of the ith 'eld. Fig. 26 depicts the structure of the two-mode Floquet Hamiltonian for the linear polarization case. The components are ordered in such a way that  runs over unperturbed states (denoted by Greek letters) of H0 before each change in n1 , and n1 , in turn, runs over before n2 . The quasienergy eigenvalues {=n1 n2 } and their corresponding eigenvectors {|=n1 n2 } of Hˆ F have the following useful bichromatic forms, namely, =n1 n2 = =00 + n1 !1 + n2 !2 ;

(9.9)

and 1 ; n1 + q1 ; n2 + q2 |=2 ;n1 +q1 ;n2 +q2  = 1 n1 n2 |=2 n1 n2  :

(9.10)

The time evolution operator Uˆ (t; t0 ) can be expressed in the following matrix form: Uˆ  (t; t0 ) ≡ |Uˆ (t; t0 )| =

∞ 

∞ 

n1 n2 | exp[ − iHˆ F (t − t0 )]|00exp[i(n1 !1 + n2 !2 )t] :

(9.11)

n1 =−∞ n2 =−∞

The transition probability averaged over the initial time t0 while keeping the elapsed time t − t0 'xed is given by  |k1 k2 | exp[ − iHˆ F (t − t0 )]|00|2 : (9.12) P→ (t − t0 ) = k1 k 2

Performing the long time average over t − t0 gives the time averaged transition probability  |k1 k2 |=l1 l2 =l1 l2 |00|2 : (9.13) PU → = k1 k2 l1 l2

We note that while the many-mode Floquet Hamiltonian shown in Fig. 26 is of in'nite dimensional, an e9ective Hamiltonian (of the order N × N ) can be constructed for important near-resonant or resonant multiphoton processes (for N -level systems) by means of appropriate nearly degenerate perturbation techniques [51,52,153,155–158]. This allows analytic treatment of the e9ective multi-mode Hamiltonian, yielding useful analytical expressions for the generalized Bloch–Siegert shift (ac Stark shift), power broadening, and spectral line shape etc. in multi-color 'elds. In the following subsections, we shall discuss the application of the MMFT to the study of several intense-'eld multiphoton processes of current interest.

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59

Fig. 26. Generalized two-mode Floquet Hamiltonian Hˆ F for two linearly polarized radiation 'eld problems (adapted from Ref. [51]).

9.2. SU(N ) dynamical symmetries and nonlinear coherence It was 'rst shown by Feynman et al. [159] in 1957 that for two-level systems, the description of magnetic and optical resonance phenomena can be simpli'ed by the use of the Bloch spin or pseudospin vector. The extension of the vector description to N -level (N ¿ 3) systems was made later by Elgin [160] and Hioe and Eberly [161] in 1980 –81. It is found that the dynamical evolution of N -level nondissipative systems can be expressed in terms of the generalized rotation of an (N 2 − 1)-dimensional coherence vector S whose property can be analyzed by appealing to SU(N ) group symmetry. For example, the time-evolution of three-level systems can be described by a coherence vector of constant length rotating in an eight-dimensional space.

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The study of SU(N) dynamical evolution of the coherent vector S and the symmetry-breaking e9ects embodied in N -level systems subjected to an arbitrary number of monochromatic 'elds can be greatly facilitated by the use of MMFT [52,153]. Consider, for example, the SU(3) case corresponding to the dynamical evolution of a three-level system driven by a bichromatic 'eld. We adopt the standard form of the SU(3) generators used by Gell-Mann [162], namely, sˆ ≡ {sˆi |i = 1; : : : ; 8} where



0

1

 sˆ1 =  1

0



0

 0 ;

0

0

0

0  sˆ4 =  0 1  0  sˆ7 =  0 0

0

1



0 0 0 0 i





0

 sˆ2 =  i 

0

−i

0





0

 0 ;

0

0

1

 sˆ3 =  0





0

0 0 −i 0        0  ; sˆ5 =  0 0 0  ; sˆ6 =  0 0 i 0 0 0    0 1 0 0    1 0 1 0  : √ −i  = ; s ˆ 8   3 0 0 0 −2

0

0

 0  ;

−1 0 0



0

0 

0

 1  ;

1

0 (9.14)

In terms of the eight 3 × 3 matrices in Eq. (9.14) and the 3 × 3 identity matrix Iˆ, the Hamiltonian for any three-level system in a bichromatic 'eld can be expressed as  3  8  1 1 ˆ ˆ E I + j (t)sˆj ; (9.15) H (t) = 3 =1 2 j=1 and the density matrix of the system can be written as 8 1 1 D(t) ˆ = Iˆ + Sj (t)sˆj ; 3 2 j=1

(9.16)

where j (t) = Tr[Hˆ (t)sˆj ] ;

(9.17)

Sj (t) = Tr[D(t) ˆ sˆj ] :

(9.18)

and Substituting Eqs. (9.15) and (9.16) into the Liouville equation 9D(t) ˆ i = [Hˆ (t); D(t)] ˆ (9.19) 9t results in an equation of motion (i.e. the generalized Bloch equation) for the coherence vector S(t), namely,  d fjkl k (t)Sl ; (9.20) Sj = dt kl

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61

where fjkl ’s are the structure constants associated with the Gell-Mann type generators, i.e., Eqs. (9.14) of the group SU(3). The length of the coherent vector S(t) is a constant of motion, and is given by

 1 2 : K ≡ |S| = 2 Tr(DˆD) ˆ − 3 At the exact two-photon resonance condition (i.e. L1 + L2 = 0, where L1 and L2 are the detunings which are given by, for a cascade system (E1 ¡ E2 ¡ E3 ), L1 =(E2 −E1 )−!1 and L2 =(E3 −E2 )−!2 ), within the RWA, S(t) can be factored into three subvectors, namely, S(t) = A(t) + B(t) + C (t) ;

(9.21)

where A(t); B(t), and C (t), of dimensions three, four, and one, rotate independently, and their respective lengths are preserved in the course of time. In the more general cases, deviations from either the RWA limit, or the two-photon resonance condition, will modify the trajectory of the S(t) described by Eq. (9.11), and thus break the dynamic symmetries embodied in the independence of the subvectors A; B, and C in the course of the time. The study of this symmetry breaking e9ect can be facilitated by means of Eq. (9.18) and the relation D(t) ˆ = U (t; t0 )D(t ˆ 0 )U † (t; t0 ) :

(9.22)

Here D(t ˆ 0 ) is the density matrix at the initial time t0 (initial conditions) and the time-evolution operator U (t; t0 ) can be determined by the method of MMFT, Eq. (9.11), and expressed in terms of a few time-independent quasi-energy eigenvalues and eigenvectors. Furthermore, the generalized Van Vleck (GVV) nearly degenerate perturbation theory [52,156–158] can be extended to the analytical treatment of the time-independent many-mode Floquet Hamiltonian. The general idea behind the MMFT-GVV technique [52] is to block-diagonalize the time-independent Floquet Hamiltonian Hˆ F so that the coupling between the model space (consisting of nearly degenerate and strongly coupled Floquet states of interest) and the remainder of the con'guration space (called the external space) diminishes to a desired order. One important feature of the MMFT-GVV approach is that if the perturbed model space wave functions are exact to the nth order, the corresponding quasi-energy eigenvalues in the model space will be accurate to the (2n+1)th order. In that regard, it is interesting to note that the RWA is merely the lowest order (i.e., n = 0) limit, namely, model space wave functions correct only to the zeroth order and eigenvalues accurate to the 'rst order. Furthermore, while the RWA can only deal with sequential one-photon processes, the MMFT-GVV approach is capable of treating both one-photon and multiphoton processes on an equal footing. Thus the MMFT-GVV approach provides a natural and powerful extension beyond the conventional RWA limit for nonperturbative treatment of multiphoton processes in intense polychromatic 'elds. Figs. 27(a) and (b) show an example of the time evolution patterns, obtained by the MMFT-GVV [Fig. 27(a)] and RWA [Fig. 27(b)] method respectively, of the projection of the subvector B(t) trajectory onto the B1 –B4 plane [52]. It is seen that the MMFT-GVV trajectory displays >uctuation around the RWA trajectory caused by the anti-rotating and higher order terms. The MMFT has been also extended to the study of coherent population trapping and SU(3) dynamical evolution of dissipative 3-level systems in intense bichromatic 'elds [153]. It is found that the dynamical evolution of the dissipative SU(3) eight-dimensional coherent vector S(t) evolves

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Fig. 27. Projection of the trajectory of the subvector B(t) on the B1 –B4 plane. (a) GVV(3; 1) results and (b) RWA results (adapted from Ref. [52]).

predominantly to a one-dimensional scalar C(t) = S8 (t) at the two-photon or multiphoton resonant quasi-trapping conditions. Fig. 28 shows the comparison of the time evolution of the projection of the coherent vector S(t) onto the S5 –S6 plane for the dissipative [Fig. 28(a)] and the nondissipative [Fig. 28(b)] three-level systems [153]. 9.3. Fractal character of quasienergy states in multi-color or quasi-periodic (elds The discovery that chaotic behavior is universal and inherent in all classical nonlinear physical systems has stimulated the search for analogous phenomena in quantum mechanics [163]. In this section, we shall discuss the dynamical behavior of quantum systems under the in>uence of

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63

Fig. 28. Projection of the trajectory of the coherence vector on the S5 –S6 plane for (a) dissipative three-level system and (b) nondissipative three-level system (adapted from Ref. [153]).

multi-frequency (quasiperiodic) time-dependent perturbations, a subject of fundamental interest in atomic and molecular spectroscopy and nonlinear optics. It is known that any discrete bound quantum system driven by a periodic (monochromatic) 'eld exhibits quasiperiodic (i.e., nonchaotic) behavior and reassembles itself in'nitely often in the course of time [164]. The behavior of the corresponding quantum system driven by bichromatic or polychromatic (nonperiodic) 'elds is less clear and is a subject of controversy [165,166]. Using the MMFT, however, it can be shown rigorously that the quantum motion is in fact quasiperiodic rather than chaotic. This topic has been reviewed in [17,167]. In this section we discuss a new prospect of the quantum dynamical behavior in multi-color 'elds: the quantum fractal behavior of quasienergy state wavefunctions in temporal Fourier space

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Fig. 29. Modulus of a quasienergy state eigenfunction component |Xn1 n2 |(≡ | n1 n2 |= vs. the Floquet Fourier indices n1 and n2 (adapted from Ref. [168]).

Fig. 30. Cross sections of ln|Xn1 n2 | as a function of n2 for several 'xed n1 values. Note the self-similarity of the wave function components (adapted from Ref. [168]).

[167,168]. This was borne out from a MMFT study of the character of quasienergy eigenfunctions of a two-level (spin −1=2) system in intense bichromatic 'elds. Fig. 29 shows an example of the mountainous quasienergy eigenfunction in the two-dimensional Fourier space |n1  ⊗ |n2 . As an example of the subtleties of the wave function behavior, Fig. 30 displays the cross sections of the logarithm of the modulus of the same eigenfunction as a function

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65

Fig. 31. Plot of ln A(L) vs. ln(2L + 1) for several di9erent 'eld strengths b for quasienergies = in the reduced zone. (a) b = 0:05, (b) b = 0:1, (c) b = 0:2 (adapted from Ref. [168]).

of n2 for several di9erent ('xed) values of n1 . The self-similarity of the recurrence pattern in the n2 direction is clearly evident. Similar behavior is observed in the n1 direction when n2 is held 'xed. The self-similar and fragmented character of the quasienergy eigenfunctions suggests that they may be fractal objects. For a more quantitative measure, let us de'ne a density correlation function [167,168] A(L) for the quasienergy eigenfunction |=n1 n2  in the two-dimensional Fourier |n1  ⊗ |n2  discrete space as ∞ ∞    A(L) = | n1 n2 |=m1 m2 |2 n1 =−∞ n2 =−∞ 
=;

×

L L   

k1 =−L k2 =−L

| ; n1 + k1 ; n2 + k2 |=m1 m2 |2 :

(9.23)



=;

Fractal character is obtained if A(L) ˙ (2L + 1)Df , where L is now the (discrete) integer scaling length in the Fourier space, and Df is the fractal dimension. Fig. 31 shows the ln A(L) vs. ln(2L + 1) plot for several di9erent 'eld strengths for quasi-energy states in the reduced zone. It is seen that ln A(L) vs. ln(2L + 1) follows a straight line quite well and then bends. The point of bending is related to the localization length of the quasi-energy eigenfunction in the (n1 ; n2 ) Fourier space. Beyond the localization length, the eigenfunction decays rapidly. As expected, the localization length extends when the 'eld strength increases. The fractal dimension Df can be obtained from a least squares 'tting of the slope of the linear portion of the graphs in Fig. 31. Further, it is found that all quasi-energy eigenfunctions |=m1 m2  (for given 'eld frequencies and intensities) possess the same fractal dimension (independent of ; m1 and m2 ). Thus Df is a unique new nonlinear optical property of the 'eld-driven quantum system [167,168].

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Let us now explore the physical mechanism leading to this type of quantum fractal behavior in the multi-frequency problem [167,168]. The key point is that the polychromatic time-dependent SchrGodinger equation can be mapped into a (time-independent) tight-binding equation similar to the Anderson-model type [169,170] equation in disordered crystalline systems:  Tn u n + Wr un+r = =un ; (9.24) r

where n = (; n1 ; n2 ; : : : ; nM ),  is the system index (=; ; : : :), (n1 ; n2 ; : : : ; nM ) are the Fourier indices for the M incommensurate frequencies of the driving 'elds, Tn = E + n1 !1 + n2 !2 + · · · + nM !M , and Wr are the electric dipole couplings. [Eq. (9.24) can be obtained directly from Eq. (9.5).] Based on what we have learned from the wave function behavior in the crystalline disordered systems [171], we can interpret the results (Figs. 29–31) as a sort of manifestation of the Anderson localization e9ect in the Floquet state space n = (; n1 ; n2 ). The source of “randomness” can be traced to the diagonal “disorder”. Elements in diagonal directions, Tn = E + n1 !1 + n2 !2 exhibit a sort of “pseudo-randomness” as n1 and n2 can take any positive or negative integer values (0; ±1; ±2; : : :) and !1 =!2 is an irrational number. In particular, when !1 ∼ = !2 ∼ = |E − E |, the diagonal Tn elements contain a set (band) of nearly degenerate but “pseudo-random” diagonal values. There is, however, no randomness in the o9-diagonal Wr terms as there are only two di9erent couplings b(1) and b(2) . The diagonal “disorder” tends to localize the quasienergy states, whereas the o9-diagonal couplings tend to delocalize the wavefunction. The fractal dimension Df and the localization length thus provide useful new measure and characterization of this delicate balance of nonlinear optical interactions. 9.4. Multiphoton above-threshold ionization in two-color laser (elds In this section we discuss the extension of MMFT th the study of multiphoton above-threshold ionization (ATI) of atoms or negative ions in two-color laser 'elds. Consider the following timedependent SchrGodinger equation for the electron bound in the atomic potential U (r) and subject to the in>uence of the external two-color laser 'elds:   9 1 2 i (r; t) = − ∇ + U (r) + [F1 cos !1 t + F2 cos (!2 t + )] · r (r; t) : (9.25) 2 9t Eq. (9.25) implies linear polarization for both the 'elds with the frequencies !1 and !2 , however the orientation of the 'eld vectors F1 and F2 can be arbitrary.  is the phase di9erence between the two laser 'elds for t=0. 9.4.1. Incommensurate frequencies Consider 'rst the case of two incommensurate frequencies !1 and !2 . In this case the combined external 'eld is not periodic in time, so the conventional Floquet solutions of Eq. (9.25) do not exist. However, according to MMFT, one can look for the wave function (r; t) in the form given by Eq. (9.4), where (r; t) can be expanded in a double Fourier series:  (r; t) = 6m1 m2 (r) exp[ − i(m1 !1 t + m2 !2 t)] : (9.26) m1 ; m2

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The wave function (r; t) satis'es the following equation:   1 9 = − ∇2 + U (r) + [F1 cos !1 t + F2 cos (!2 t + )] · r −   ; i 9t 2

67

(9.27)

which is equivalent to the in'nite set of time-independent equations for the Fourier components 6m1 m2 (r):   1 2 1 − ∇ + U (r) − Em1 m2 6m1 m2 + {(F1 · r) [6m1 −1;m2 + 6m1 +1;m2 ] 2 2 + (F2 · r) [e−i 6m1 ;m2 −1 + ei 6m1 ;m2 +1 ]} = 0 ;

(9.28)

where E m 1 m 2 =  + m 1 ! 1 + m 2 !2 :

(9.29)

The 'nal expression for the electron energy and angular distributions has the following form [172]: 1 dn1 n2 = kn n |An n |2 ; dF (2 )2 1 2 1 2 where k n1 n 2

  = 2 −

(9.30)

F12 F22 − + n 1 !1 + n 2 !2 (2!1 )2 (2!2 )2

;

(9.31)

and An1 n2 is the transition amplitude (vector kn1 n2 points at the direction where the electrons are detected):

1  An 1 n 2 = exp [i(m2 − n2 )] d 1 d 2 (2 )2 m ; m − − 1

2

 ×exp i(n1 − m1 ) 1 + i(n2 − m2 ) 2 − i

F12 F22 sin 2

− i sin 2 2 1 (2!1 )3 (2!2 )3

(kn1 n2 · F1 ) (kn1 n2 · F2 ) (F1 · F2 ) cos 1 + i cos 2 − i (2!1 !2 ) !12 !22   1 1 sin( 1 + 2 ) − sin( 1 − 2 ) × (!1 + !2 ) (!1 − !2 )  
(r · F1 ) (r · F2 ) 3 sin 1 + i sin 2 × d r exp −i(kn1 n2 · r) + i !1 !2 +i

×U (r)6m1 m2 (r) :

(9.32)

Analysis of Eq. (9.32) shows that in the case of incommensurate frequencies, the electron distributions do not actually depend on the relative phase shift .

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9.4.2. Commensurate frequencies Now we turn to the case of commensurate frequencies and consider the case when !2 is a harmonic of !1 : !1 = !;

!2 = p! ;

(9.33)

where p is a positive integer number. We can use the conventional Floquet theorem since the Hamiltonian is now periodic in time. The wave function (r; t) is expanded in the Fourier series with one fundamental frequency !:  (r; t) = 6m (r) exp(−im!t) : (9.34) m

For the di9erential electron >ux dn =dF, the following result is obtained [172]: 1 dn = kn |An |2 ; dF (2 )2 where   F12 F22 − + n! ; kn = 2  − (2!)2 (2p!)2 and

(9.35)

(9.36)

 F12 F22 d exp i(n − m) − i sin 2

− i sin (2p + 2) (2!)3 (2p!)3 −  (kn · F2 ) (F1 · F2 ) (kn · F1 ) cos + i cos (p + ) − i + i 2 2 ! (p!) 2p!2   1 1 sin((p + 1) + ) − × sin((p − 1) + ) × (p + 1)! (p − 1)!  
(r · F2 ) (r · F1 ) 3 sin + i sin(p + )) × d r exp −i(kn · r) + i ! p!

1  An = 2 m






×U (r)6m (r) :

(9.37)

In Eq. (9.37), vector kn points at the direction of the electron ejection. Expression (9.37) is general and contains three-dimensional space and one-dimensional time integration. We note that in the commensurate frequency case, the electron angular and energy distributions depend upon the relative phase . 9.4.3. Multiphoton detachment of H − in two-color laser (elds Let us consider an application of the theory discussed in the previous Section 9.4.2 for the multiphoton above-threshold detachment of H− subject to the fundamental 'eld of CO2 laser (wavelength 10:6 m) and its third harmonic [172]. In what follows we consider the special case when the two 'elds are polarized in the same direction. The calculations are performed for the fundamental 'eld intensity 1010 W=cm2 and harmonic 'eld intensities 109 and 108 W=cm2 . The results show the following novel features. First, the total and partial rates for the two-color detachment, when the

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harmonic 'eld is relatively strong (such as 10 times smaller than that of fundamental 'eld), are generally much larger than the rates for the one-color detachment by the fundamental or harmonic 'eld alone. However, the opposite situation is also possible if the harmonic 'eld is weaker (say, 100 times smaller than that of fundamental 'eld intensity), and relative phase  is close to . Second, the total and partial rates manifest a strong dependence on the relative phase between the two 'elds. The total rate is the largest for the phase  = 0, and the smallest for  = . Such a dependence on the relative phase is also valid for the 'rst few ATI peaks. However, for the subsequent ATI peaks, the picture is quite di9erent. The energy spectrum for the case of  = is broader and the peak heights decrease more slowly compared to the case of = 0. The strong phase dependence is also manifested in the angular distributions of the ejected electrons for this commensurate frequency case. Figs. 32(a) – (c) show an example of the phase-dependent ATI spectrum of H− in two-color laser 'elds [172]. 9.5. Chemical bond hardening and molecular stabilization in two-color laser (elds In the previous Section 9.4, we have shown that multiphoton above-threshold detachment rates of negative ions in two-color commensurate 'elds can be signi'cantly modi'ed by the phase-dependent interactions. In this section we address the problem of two-color multiphoton above-threshold dissociation (ATD) of diatomic molecular ions. Consider the case of D+ 2 ions driven by a fundamental laser 'eld and its third harmonic [143]. The Hamiltonian for the system under consideration is given by Hˆ (r; R; t) = Tˆ R + Hˆ el (r; R) + (r; R) · F(t) ;

(9.38)

F(t) = F1 sin !t + F3 sin(3!t + ) ;

(9.39)

where r and R are, respectively, the electronic and internuclear coordinates. Tˆ R is the nuclear kinetic-energy operator, Hˆ el (r; R) is the electronic Hamiltonian, (r; R) is the dipole moment operator, ! is the fundamental frequency, F1 and F3 are, respectively, the fundamental and the third-harmonic 'eld amplitude, and  is the relative phase between the two laser 'elds at t = 0. The electronic potential curves for 1s)g and 2p)u and the transition dipole moment for D+ 2 are the + same as those we for H2 . The velocity gauge can be adopted to facilitate fast convergence in strong 'elds. The complex eigenvalues of the VQE resonances can be eNciently and accurately determined by means of the complex-scaling Fourier grid Hamiltonian (CSFGH) method [137] to discretize the non-Hermitian Floquet Hamiltonian as discussed in previous section for H2+ . Figs. 33–35 show the results of VQE resonances of D+ 2 driven by fundamental laser 'eld at 775 nm and its third harmonic both with intensity at I = 2:5 × 1013 W=cm2 , and the phase  = 0, =2, and , respectively. As the laser phase  is varied in the two-color excitation, one sees signi'cant modi'cation of both adiabatic electronic-'eld potential avoided crossing patterns and MPD/ATD rates. The behavior for the case of  = (Fig. 35) is the most instructive one here. It is shown here that the total number of VQE resonances is the smallest one among all the cases considered, indicating its correspondence to the “strongest” excitation situation. Further the VQE resonances can be rather clearly distinguished into four well separate groups, the three upper groups are supported by the three separate electronic-'eld potential wells shown in Fig. 35(a). The topmost group resonances (v = 13–20) have the longest photodissociation lifetimes (bond hardening) and even the second

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Fig. 32. Electron energy spectrum after multiphoton above-threshold detachment of H− by 10:6 m radiation (intensity 1010 W=cm2 ) and its third harmonic (intensity 108 W=cm2 ). The heights of the bars correspond to the partial rates after absorption of n fundamental frequency photons, starting with n = 8. The relative phase between the fundamental and harmonic 'elds is  = 0 (a),  = (b), and  = =2 (c) (adapted from Ref. [172]).

highest group resonances (v =10–12) now also begin to show substantial stabilization. In comparison, the  = =2 case (Fig. 34) shows overlap in energy span between two middle resonance groups, and the  = 0 case (Fig. 33) shows the “weakest” excitation situation, namely, only two upper resonance groups are formed and the avoided crossing between the |u; −1 and |g; −2 adiabatic curves are accidentally very narrow. In summary, by tuning the relative phase  between the fundamental and the third harmonic laser 'elds, one can achieve certain degree of “coherent control” of multiphoton dynamics, particularly the electronic-'eld avoided crossing patterns, the trapping in di9erent potential wells and the chemical bond hardening phenomenon [143].

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Fig. 33. (a) Vibrational quasienergy level diagram and dressed adiabatic electronic-'eld potentials of D+ 2 driven by a fundamental laser 'eld at 775 nm and its third harmonic, both with intensity at 2:5 × 1013 W=cm2 . The relative phase between the two laser 'elds is  = 0. (b) The MPD/ATD halfwidths (=2) of the corresponding vibrational quasienergy resonance states (labeled by v ) (adapted from Ref. [143]).

9.6. High-order harmonic generation in two-color laser (elds Recently a great deal of attention has been devoted to the study of multiple high-order harmonic generation (HHG) processes in intense short laser pulses [173,174]. Besides its fundamental interest for strong-'eld atomic and molecular physics, the HHG provides a potential tunable coherent light source in the extreme ultraviolet (xuv) region, a so-called “tabletop synchrotron”. Moreover, the HHG may lead to a promising way of generating sub-femtosecond (attosecond) ultrashort pulses of radiation of high frequency. In the presence of intense one-color laser 'elds, several nonperturbative methods have been used in the theoretical studies of HHG processes: time-independent Floquet formalism [175,176], numerical integration of the time-dependent SchrGodinger equation [5,23,28,177,178], as well as purely classical approach [96,179]. Such theoretical treatments usually consider the interaction of a single atom with strong classical electromagnetic 'elds. In this section we discuss the extension of non-Hermitian Floquet approach to the study of HHG processes in atomic systems driven by intense two-color laser 'elds. It is shown that by varying both

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Fig. 34. The same as Fig. 33 except for laser phase angle  = =2 (adapted from Ref. [143]).

the relative phase and the relative intensity of the fundamental to the harmonic 'eld, it is possible to coherently control the enhancement or the decrement of the HHG yields. 9.6.1. Non-Hermitian Floquet treatment of HHG According to the classical theory of 'elds [180], the intensity of radiation produced by an accelerated charge per unit solid angle and summed over all possible polarizations is given by the following expression: dI |a(t)|2 = sin2 # ; dF 4 c3

(9.40)

where a(t) is the acceleration of the charge, c is the velocity of the light, and # is the angle (with respect to a) under which the radiation is detected. In the case of periodic motion, the acceleration a(t) can be expanded in Fourier series: a(t) =

∞  n=−∞

an exp(−in!t) ;

(9.41)

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73

Fig. 35. The same as Fig. 33 except for laser phase angle  = (adapted from Ref. [143]).

where the frequency ! = 2 = , being the period. The total (angle-integrated) intensity of the nth harmonic in the acceleration form is given by In =

4|an |2 : 3c3

(9.42)

Alternatively, the intensity of the nth harmonic can be rewritten in the length form: In =

4n4 !4 |dn |2 : 3c3

(9.43)

In the present case, the charge particle is represented by an electron moving under the in>uence of the core and external laser 'eld. Let the atomic potential U (r) be spherically-symmetrical, and the external 'eld linearly polarized in the z direction. Then the Hamiltonian Hˆ for single-active-electron systems reads as (in atomic units) 1 Hˆ = − ∇2 + U (r) + zF(t) ; 2

(9.44)

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where F(t) is the laser 'eld strength. For the two-color 'elds, one has the following expression for F(t): F(t) = F1 cos(!1 t) + F2 cos(!2 t + ) ;

(9.45)

with F1 and F2 being the 'eld strengths for the 'rst and the second 'elds, respectively, and  being the relative phase between the two 'elds. In the present formulation, the atomic system will be treated quantum mechanically. Thus the acceleration a(t) and its Fourier components should be replaced with the corresponding quantum expectation values. If the initial state of the atom is spherically symmetrical, then only the z-component of the mean acceleration does not vanish. According to the Ehrenfest theorem [181], d2 i z = |[Hˆ ; pˆ z ]| ; (9.46) 2 dt m where (r; t) is a quasienergy wave function describing the electron subject to the in>uence of the core as well as the external 'eld, and pˆz is the momentum operator. Consider 'rst the case of commensurate frequencies !1 and !2 . Here the conventional Floquet theory applies, and the wave function (r; t) can be expanded in the Fourier series with a single fundamental frequency ! ( is the quasienergy): (r; t) = exp(−it)

∞ 

m (r) exp(−im!t)

:

(9.47)

m=−∞

If the second laser 'eld is a harmonic of the 'rst one, namely, !2 = N!1 , N being integer, and the fundamental frequency ! is equal to the frequency of the 'rst 'eld, ! = !1 , then the expression for the squared absolute value of the Fourier component |an |2 can be written as [176]  ∞        z dW   2  m + 1 F1 [n; 1 + n; −1 ]  |an | =  m− n   r dr  2 m=−∞

2  1 + F2 [n; N exp(−i) + n; −N exp(i)] : 2

(9.48)

In analogy, the squared absolute value of the Fourier component |dn |2 is expressed as follows:  ∞ 2       m−n |z| m  : (9.49) |dn |2 =    m=−∞

The value In given by Eqs. (9.42) and (9.43) has the meaning of the energy radiated per unit time through the mode with the frequency n!. The corresponding photon emission rate n (the number of photons of frequency n! emitted per unit time) is obtained by 4|an |2 4n3 !3 |dn |2 = : (9.50) 3n!c3 3c3 Both forms (acceleration and length) are equivalent in the Floquet theory. Now consider brie>y the general case of incommensurate frequencies !1 and !2 . According to the many-mode Floquet theory, the wave function (r; t) can be expanded in a double Fourier series n =

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75

with two fundamental frequencies, !1 and !2 :  (r; t) = exp(−it) m1 m2 (r) exp(−i(m1 !1 + m2 !2 )t) :

(9.51)

m1 ; m2

The long-time average of the squared acceleration is equal to
T ∞  1 |a(t)|2 dt = |an1 n2 |2 ; lim T →∞ T 0 n ; n =−∞ 1

(9.52)

2

with the squared absolute value of the Fourier components |an1 n2 |2 de'ned as   z dW 1  |an1 n2 |2 =   m1 −n1 ;m2 −n2 | | m1 m2  + F1 (n1 ;1 + n1 ;−1 )n2 ;0  r dr 2 m1 ; m2

2  1 + F2 [n2 ;1 exp(−i) + n2 ;−1 exp(i)]n1 ;0  : 2

(9.53)

Note that although the relative phase  is present in Eq. (9.53), it does not a9ect the result. When driven in the two-color 'elds with incommensurate frequencies !1 and !2 , the electron can emit radiation with the frequencies |n1 !1 +n2 !2 |, where n1 and n2 are integers. The corresponding photon emission rate can be determined by [176]  n1 n 2 =

4|an1 n2 |2 : 3|n1 !1 + n2 !2 |c3

(9.54)

9.6.2. Two-color phase control of HHG In this section we consider HHG by the hydrogen atom driven by the fundamental frequency laser 'eld and its third harmonic: F(t) = F1 cos(!t) + F2 cos(3!t + ) :

(9.55)

The quasienergy wave function (r; t) can be expanded in the double series over the time Fourier components and Legendre polynomials Pl (cos #), # being the angle between r and the 'eld direction (z-axis): (r; t) = exp(−it)

∞ 

exp(−im!t)

m=−∞

∞   l=0

l + 1=2

1 r

ml (r)Pl (cos #)

:

(9.56)

The complex-scaling generalized pseudospectral (CSGPS) technique [75] can be extended to the discretization and solution of the non-Hermitian Floquet Hamiltonian Hˆ F (r exp(i)). To facilitate the convergence in strong 'eld calculations, the velocity-gauge Hamiltonian may be adopted instead of the length Hamiltonian: 1 d ; Hˆ = − ∇2 + W (r) − iA(t) 2 dz d F(t) = − A(t) : dt

(9.57) (9.58)

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Table 5 Harmonic generation rates by the hydrogen atom for the fundamental 'eld 532 nm, 5×1013 W=cm2 and its third harmonic; n is the harmonic order n

Harmonic generation rates (a.u.) One-color fundamental 'eld

Two-color, harmonic 'eld 5 × 1011 W=cm2

3 5 7 9 11 13 15 17 19 21 23

2:18(−13) 1:64(−12) 1:12(−13) 7:28(−15) 3:03(−16) 3:07(−18) 1:10(−20) 1:87(−23) 1:81(−26) 1:10(−29) 4:62(−33)

5 × 109 W=cm2

=0

=

=0

=

5:14(−12) 4:54(−12) 5:90(−13) 1:19(−13) 2:68(−15) 1:45(−17) 6:72(−20) 2:03(−22) 4:98(−25) 9:89(−28) 1:67(−30)

1:77(−12) 2:43(−12) 1:71(−12) 8:45(−14) 1:41(−15) 1:35(−17) 8:49(−20) 3:94(−22) 1:46(−24) 4:50(−27) 1:20(−29)

4:19(−13) 1:75(−12) 9:07(−14) 1:02(−14) 4:17(−16) 3:68(−18) 1:10(−20) 1:45(−23) 9:94(−27) 3:82(−30) 8:35(−34)

8:20(−14) 1:57(−12) 1:55(−13) 6:25(−15) 2:20(−16) 2:51(−18) 1:07(−20) 2:34(−23) 3:13(−26) 2:86(−29) 1:94(−32)

The number in parentheses indicates the power of 10 (adapted from Ref. [176]).

The integrations over the angles # and ’ (in the spherical coordinate system with the polar axis along the 'eld direction) with the function (9.56) in Eqs. (9.48), (9.49) can be performed analytically, giving rise to the following expressions for the squared Fourier components of the induced acceleration and displacement, respectively [176]:      l+1 1 1  2  |an | = −  (m−n); l | 2 | m; (l+1)  +  (m−n); (l+1) | 2 | m; l   r r (2l + 1)(2l + 3) m;l

2  1 1 1 1 + F1 n; 1 + F1 n; −1 + F2 exp(−i)n; 3 + F2 exp(i)n; −3  ; 2 2 2 2  2   l + 1   2  |dn | =  [ (m−n); l |r| m; (l+1)  +  (m−n); (l+1) |r| m; l ] :   (2l + 1)(2l + 3) m;l

(9.59) (9.60)

As it was mentioned above, the acceleration and length forms of the expression for HHG rates are equivalent in the Floquet theory, if the wavefunction is converged and if an appropriate “regularization” of the length form integral is performed. However, the acceleration form is generally expected to be more stable and reliable, especially for high order harmonics, because it makes use of the coordinate r range which is not far from the nucleus, where the wave functions calculated are generally more accurate. Table 5 shows an example of the HHG calculation of atomic H under the irradiation of two-color laser 'elds, with fundamental wavelength 532 nm and its third harmonic [176]. We see that for stronger harmonic 'eld (5 × 1011 W=cm2 ) the HHG rates are generally signi'cantly enhanced,

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77

compared with the one-color case, for both the phase  values used in the calculations. However, for the weaker harmonic intensity (5 × 109 W=cm2 ) and  = , the rates for the 'rst few harmonics are smaller than that for the one-color HHG. We see that a small admixture of the harmonic 'eld can lead to a dramatic change in the HHG rates. Thus by tuning the relative phase and the relative intensity of the fundamental to the harmonic 'eld, one can control the enhancement or decrement of the HHG yield. The dependence of the HHG rates on the relative phase  is the same for both strong and weak 3rd harmonic 'elds. The general observation is that the HHG spectrum is broader for  = than for  = 0, the rates decrease slower in the tail portion of the spectrum whereas in the top portion (the 'rst few harmonics) they have the magnitudes smaller than that for  = 0. These results are analogous to that for above-threshold detachment by two-color laser 'elds from H− negative ions [172] discussed in Section 9.4.3. 10. Floquet–Liouville supermatrix formalism for nonlinear optical processes in intense laser 'elds The subject of nonlinear optical processes such as multiphoton dissociation of molecules, resonance >uorescence, Raman scattering, and multiple wave mixings, etc. is a very active 'eld in science and technology in the past 3 decades [182–185]. At lower 'elds, perturbative and diagrammatic methods [182–185] are often used for nonresonant phenomena, whereas the rotating wave approximation (RWA) is commonly adopted for near resonant processes [182–188]. The generalized Floquet formalisms based on the SchrGodinger equation discussed in previous section while providing powerful nonperturbative techniques for the studies of multiphoton ionization, excitation, and dissociation processes, etc., cannot be adopted directly to processes undergoing relaxations (due to radiative decays and collision dampings, etc). In this section we discuss a general nonperturbative approach for exact treatment of Liouville equation (allowing for relaxation mechanisms) and density matrix operator of atomic or molecular systems exposed to intense monochromatic or polychromatic 'elds. By extending the many-mode Floquet theorem (MMFT) [51,52], the time-dependent Liouville equation can be transformed into an equivalent time-independent Floquet–Liouville super-matrix (FLSM) eigenproblem [53,54,189]. As will be shown below, the FLSM approach provides a powerful theoretical framework for nonperturbative and uni'ed treatment of nonresonant and resonant, one-photon and multiphoton, steady-state and transient phenomena, well beyond the RWA and traditional perturbative methods. 10.1. The FLSM formalism The Liouville equation for the time evolution of a set of N -level quantum systems, interacting with several coherent linearly polarized monochromatic 'elds, undergoing relaxation by Markovian processes, is 9 ˆ D(t)] D(t) ˆ = [Hˆ (t); D(t)] ˆ + i[R; ˆ : (10.1) 9t Here Dˆ is the density matrix of the system, reduced by an averaging over all irrelevant degrees of freedom acting as a thermal bath, and Hˆ (t) = Hˆ 0 + Vˆ (t). Hˆ 0 is the unperturbed atomic Hamiltonian with eigenvalues {E } and eigenvectors {|}  = 0; 1; 2; : : : ; N − 1; and Vˆ (t) is the interaction i

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Hamiltonian between the system and the M -mode classical 'elds given by M  ˆ  · i cos(!i t + ’i ) ; V (t) = −

(10.2)

i=1

where  is the atomic dipole moment, i the 'eld amplitude, !i the frequency and ’i the phase ˆ D(t)] of the ith 'eld. The relaxation term [R; ˆ consists of T1 (population damping) and T2 (coherent damping) mechanisms which are due to the coupling of the atomic system to the thermal bath by radiative decays and collisions, etc. More explicitly [53,183,188]   D (T1 ) ; (10.3) (Rˆ D) ˆ  = − D + =

(Rˆ D) ˆ  = − D

( = )

(T2 ) ;

(10.4)

where the phenomenological damping parameter  describes the population decay,  the phase relaxation and  the feeding. In the following we shall con'ne our discussion to closed systems, ˆ D] namely, Tr[R; ˆ = 0. Extension to open systems is straightforward. In the tetradic or Liouville space [190] spanned by ||, where ;  = 0; 1; : : : ; N − 1, Eq. (10.1) ˆˆ ˆ + ifˆ, or in matrix form, can be rewritten as i9=9t ˆ = L(t)  9 Lˆˆ; 21 (t)D21 (t) + if ; (10.5) i D (t) = 9t 21 ˆˆ where L(t) is the superoperator or Liouvillian which is nonsingular, whose matrix elements are, assuming |0 is the ground level, Lˆˆ00;21 (t) = Hˆ 02 (t)01 − Hˆ 10 (t)02 − i(00 + 0 )20 10    −i (1 − 2 )0  21 (1 − 02 ) ;

(10.6)

=0

Lˆˆ;21 (t) = Hˆ 2 (t)1 − Hˆ 1 (t)2 − i(21 2 1 − 2  21 ) ( = 0;  = 0) ;



(10.7)

and fˆ is the source term, f21 = 0 20 0 with 0 = =0 0 . The homogeneous solution of Eq. (10.5) can be solved expediently by invoking the many-mode Floquet theorem (MMFT) [51,52], analogous to solving the SchrGodinger equation with Hamiltonian having the same time dependence as that in Eq. (10.1). The MMFT renders the time-dependent Liouville equation into an equivalent time-independent in'nite-dimensional super-eigenvalue equation [53,54,189],  ; {m}|LˆˆF |) ; {k}) ; {k}|F21;{n}  = F21;{n} ; {m}|F21;{n}  ; (10.8) )

{k }

where LˆˆF is the time-independent many-mode Floquet–Liouville superoperator de'ned in terms of the generalized tetradic-Floquet basis |; {m} ≡ || ⊗ |{m}, with {m} = m1 ; m2 ; : : : ; mM .

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79

Fig. 36. Structure of the Floquet–Liouville supermatrix LˆˆF for the case of two-level system (with level spacing !ba ) in i linearly polarized bichromatic 'elds. !1 and !2 are the two radiation frequencies, Vab (i = 1; 2) are the electric dipole couplings, and ab , ba , and ba = (ab + ba )=2 are relaxation parameters (adapted from Ref. [189]).

The structure of the Floquet–Liouville super-matrix LˆˆF is illustrated in Fig. 36 for the two-level ˆˆ two-mode case. The super-eigenvalues and eigenvectors Mof LF posses the following important properties: (i) Im(F21;{n} ) ¡ 0, (ii) F21;{n+k } = F21;{n} + i=1 ki !i , and (iii) ; {m + k}|F21;{n+k }  = ; {m}|F21;{n} . Further, it can be shown that in the limit of  =  = 0 (i.e. no relaxations), the super-eigenvalues F and eigenvectors |F of LˆˆF are related to the quasi-energy eigenvalues = and eigenvectors |= of Hˆ F , where Hˆ F is the Floquet Hamiltonian for the nondamping case, by the following relations: F;{m} = =; {0} − =; {0} + 21; {k}|F;{0}  =



M 

m i !i ;

(10.9)

i=1

2; {n}|=; {0} =; {0} |1; {n − k} :

(10.10)

{ n}

Thus the super-eigenvalues F have the physical interpretation as the “di9erence spectrum” of the quasienergies. Fig. 37 shows an example of the super-eigenvalues (real parts only) as a function of

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Fig. 37. Supereigenvalues (real parts), Re F, are shown as a function of !2 − !1 , for the case of a closed two-level system driven by two intense linearly polarized monochromatic 'elds of frequencies !1 and !2 . Parameters used are !ba = !1 = 100, ba = 0:1, ab = 0:0, ba = (1=2)ba , || = 10ba , || = 20ba and 61 = 62 = 0:0. The tetradic-Floquet indices |21; k1 k2  are shown on the right-hand side. Note that Faa;k1 k2 and Fbb;k1 k2 are almost degenerate and cannot be distinguished in the 'gure (adapted from Ref. [54]).

!1 −!2 for a closed two-level systems with purely radiative relaxation and driven by two linearly polarized laser 'elds of frequencies !1 and !2 , respectively. The 'rst frequency !1 is 'xed at resonance with the level spacing (i.e., !ba = !1 = 100:0 arbitrary units). The tetradic-Floquet indices are shown on the right-hand side of the 'gure. Each avoided crossing between the super-eigenvalues |a; b; 1; 0 and |ba; −1; 0 corresponds to a multiphoton (subharmonic) resonance transition. For example, the positions of avoided crossings from the right-hand side to the line center correspond, respectively, to the subharmonics (2!1 − !2 ), (3!1 − 2!2 ), (4!1 − 3!2 ), and so on, while those from the left-hand side to the center correspond, respectively, to (2!2 − !1 ); (3!2 − 2!1 ); (4!2 − 3!1 ); : : : ; subharmonic multiphoton resonant transitions. The central part of the diagram contains in'nite number of higher-order processes which are not shown. Several unique features of the super-eigenvalue plot worth mentioning: (i) The super-eigenvalue pattern is not exactly symmetrical with respect to the line center. (ii) There is one-to-one correspondence between the avoided crossing pattern and the multiphoton resonance absorption line shape (such as power broadening, ac Stark shift, etc.), similar to the well known quasienergy plot (see, for example, Fig. 3) for the nondamping case. In terms of the eigenvalues and eigenvectors of the superoperator LˆˆF , the reduced density matrix (t) ˆ can be expressed as (t) ˆ = Uˆˆ (t; t0 )(t ˆ 0 ), where Uˆˆ is the super-evolution-operator given by, in matrix form, Uˆˆ ;21 (t; t0 ) =

 { m}

; {m}| exp[ − iLˆˆF (t − t0 )]|21; {0}

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

+ 0 21

 )

81

∗ ; {m}|F) ;{k }  F) ; {k } |00; {0}

{k }





× {1 − exp[ − iF) ;{k } (t − t0 )]}=iF) ;{k } expi

M 

 mj !j t  :

(10.11)

j=1

Furthermore, since Im F ¡ 0 for all F, the reduced density matrix has a simple form at large times t → ∞,  ∗ (; {m}|F) ;{k }  F) ; D (t) → 0 {k } |00; {0}=iF) ;{k } ) t →∞

{m} D {k }



×expi

M 



mj !j t  ;

(10.12)

j=1

which is oscillatory rather than completely stationary as would be the case in the RWA limit. 10.2. Intensity-dependent generalized nonlinear optical susceptibilities and multiple wave mixings The determination of nonlinear optical susceptibilities represents a signi'cant area of both experimental and theoretical research in nonlinear optics [182–185]. Calculations of nonlinear optical susceptibilities in a medium with discrete quantum levels are usually performed by means of perturbative methods. The perturbative treatment is adequate when both the pump and the probe 'elds are weak and the corresponding nonlinear optical susceptibilities are independent of 'eld strengths. However, most recent experimental works were carried out under the conditions that both the pump and the probe 'elds are strong. Distinct new features such as subradiative structures, multiphoton absorption peaks, and very high-order nonlinear wave mixings, etc. have been observed. In most of these nonlinear optical processes, when the 'elds are intense enough to saturate the transitions, nonlinear optical susceptibilities become intensity dependent. Nonperturbative response functions are required to explain these intense-'eld e9ects and the rotating wave approximation (RWA) is commonly used for the approximate treatment of exact or near resonant processes [182–188]. However, important strong-'eld e9ects (such as ac Stark shifts) and o9-resonant processes cannot be properly treated by these conventional RWA or perturbative techniques. In the following subsection we discuss an exact nonperturbative method for the treatment of intensity-dependent nonlinear optical susceptibilities in polychromatic 'elds valid for arbitrary laser intensities, detunings, and relaxation, based on the extension of the Floquet–Liouville supermatrix (FLSM) formalism [53,54]. 10.2.1. Exact FLSM nonperturbative treatment The nonlinear response of an ensemble of systems to the incident polychromatic 'elds takes the form of a dielectric polarization density P(t) which acts as a source term in Maxwell’s wave equations. The polarization density is related to the expectation value of the dipole moment operator ˆ and can be calculated from the density matrix D(t) ˆ P(t) = N0 2 = N0 Tr(2ˆD(t)) ˆ ; (10.13) where N0 is the number density in the ensemble and D(t) ˆ can be determined by the FLSM method.

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Without loss of generality, let us consider the response of two-level systems driven by intense M -mode polychromatic 'elds. The polarization density now takes the form P(t) = N0 [2ba Dab (t) + 2ab Dba (t)] ;

(10.14)

where 2ab is the transition dipole matrix element between the unperturbed atomic states |a and |b (assumed to be of opposite parity and Ea ¡ Eb ). In the steady state (t → ∞), the polarization density may be expanded as a Fourier series in the incident frequencies (as shown by Eq. (10.12)),  Pm1 m2 ···mM exp[ − i(m1 !1 + m2 !2 + · · · mM !M )t] ; (10.15) P(t) = m1 m2 ···mM

where Pm1 m2 ···mM ≡ P{m} (!) is the Fourier component at frequency ! = m1 !1 + m2 !2 + · · · mM !M . As an example, consider the two-mode (M = 2) case with !1 the pump frequency and !2 the probe frequency. We have from Eq. (10.15) P(t) = P1; 0 (!1 )e−i!1 t + P0; 1 (!2 )e−i!2 t + P2; −1 (2!1 − !2 )e−i(2!1 −!2 )t + · · · :

(10.16)

The physical meaning of these terms is as follows: P1; 0 (!1 ) and P0; 1 (!2 ) give rise to absorption (or ampli'cation) of the pump and probe waves, respectively, while the mixing response P2; −1 (2!1 −!2 ) is responsible for the generation of an optical wave with frequency ! = 2!1 − !2 , and so on. Note that P{m} (!) is a nonperturbative result. If expanded in terms of a power series of incident 'elds, P{m} (!) can be related to the conventional perturbative nonlinear susceptibilities (to in'nite order, in principle). For example, in the case of bichromatic 'elds (M = 2), P2; −1 (2!1 − !2 ) = :(3) (−2!1 + !2 ; !1 ; !1 ; −!2 )12 (!1 )2∗ (!2 ) + :(5) (−2!1 + !2 ; !1 ; !1 ; !1 ; −!1 ; −!2 )13 (!1 )1∗ (!1 )2∗ (!2 ) + :(5) (−2!1 + !2 ; !1 ; !1 ; !2 ; −!2 ; −!2 )12 (!1 )|2 (!2 )|2 2∗ (!2 ) +···; where

 i (ni !i ) =

etc:

i (!i )ni ;

ni ¿ 0 ;

i∗ (!i )|ni | ;

ni ¡ 0

is the Fourier transform of the ith optical 'eld at !i and :(q) is the conventional (intensityindependent) perturbative qth order optical susceptibility [182–185]. At weak incident 'elds, the lowest (nonvanishing) order susceptibility dominates and the conventional perturbative approach for :(q) is adequate. Thus if both the pump and probe 'elds are weak, the generation of a coherent signal at 2!1 − !2 (four-wave mixing), for example, is described by the third-order (q = 3) nonlinear susceptibility :(3) (−2!1 + !2 ; !1 ; !1 ; −!2 ). However, for strong saturating 'elds, higher-order nonlinear susceptibilities can contribute signi'cantly. This leads to the concept of intensity-dependent generalized nonlinear optical susceptibility de'ned by [54] :{m} (!) = P{m} (!)=[1 (m1 !1 )2 (m2 !2 ) · · · M (!M )] ;

(10.17)

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where ! = m1 !1 + m2 !2 + · · · + mM !M . In the limit of weak 'elds, :{m} (!) reduces to the lowest nonvanishing order (intensity-independent) :(q) , as it should be. Using the results of Eqs. (10.12), (10.16), and (10.17), one obtains the following nonperturbative expression for generalized nonlinear optical susceptibility (for the two-level M-mode case) in terms of the supereigenvalues and eigenvectors of the Floquet–Liouvillian LˆˆF [54]: :{m} (!) = m1 !1 + m2 !2 + · · · + mM !M )   [ba; {m}|F) ;{k } 2ab + ab; {m}|F) ;{k } 2ba ] = N0 ba  )

{k }

∗ −1 × F) ; {k } |aa; {0} [iF) ;{k } ]

  

=[1 (m1 !1 )2 (m2 !2 ) · · · M (mM !M )] :

(10.18)

10.2.2. High-order nearly degenerate perturbative treatment To exploit analytical properties of nonlinear optical processes and to make a connection with commonly used perturbative and RWA approaches, we shall discuss now the extension of Salwen’s almost degenerate perturbation theory [155] to the analytical treatment of the Floquet–Liouvillian LˆˆF . Consider the important class of a system of dipole-allowed two-level atoms (molecules) undergoing (2|m| + 1)-photon [!ba ∼ = (m + 1)!1 − m!2 ] near-resonant transitions in the presence of two intense linearly polarized laser 'elds characterized by the frequencies (!1 ; !2 ), amplitudes (1 ; 2 ) and initial phases (’1 ; ’2 ) respectively. The two-level |a and |b (Ea ¡ Eb ) are assumed to be of opposite parity. In a proper rotating frame (not the RWA) de'ned by the unitary transformation   1 0 R(t) = ; (10.19) 0 exp(i[(m + 1)!1 − m!2 ]t) the density-matrix super-operator (t) ˆ satis'es approximately the Salwen–Liouville equation, namely, i

9(t) ˆ ˆ + ifˆS ; = LˆˆS (t) 9t

where fˆS is the source supervector given by   ba    0    : ˆ fS =  0  

(10.20)

(10.21)

0 When the resonance condition !ba ∼ = (m+1)!1 −m!2 , m arbitrary integer, is satis'ed, the unperturbed tetradic-Floquet states |aa; 00; |bb; 00; |ab; m+1; −m, and |ba; −(m+1); m, form a four-dimensional almost degenerate set and span the Salwen’s “model space.” In terms of this model space, the

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e9ective Salwen–Liouvillian  −i(ab + ba )  iab  LˆˆS =  ∗  −u a 

LˆˆS has the following matrix form: 0

−ua∗

ua

−iba

−ub∗

ub

−u b

−(L + ) − iba

0

ub

0

(L + ) − iba

∗

ua

    ;  

(10.22)

where ab , ba and ba (≡ ab ) are the damping constants due to spontaneous emission and collisions etc., L is the detuning de'ned by L=!ba −[(m+1)!1 −m!2 ], and  (bichromatic Block–Siegert resonance shift) and u’s (power broadening or resonance width parameters) represent intensity-dependent higher-order perturbation corrections for the rest of the supermatrix LˆˆF (called the “external” space). The steady-state solutions (d (t)=dt ˆ = 0) for the density matrix in Eq. (10.20) can be solved readily to give the coherence (i.e. o9-diagonal) density matrix elements, ∗

∗

DUba = −ba [(L +  + iba )(ba ua + ab ub ) + ub∗ (u b ua − ub u a )]= DU ;

(10.23a)

DUab = DU∗ba ;

(10.23b)

and where ∗ 2 ](ab + ba ) + 2 Re(z) − 4 Im(ua u b )Im(ua ub∗ ) ; DU = ba [(L + )2 + ba

and z = [ba − i(L + )][(ab + ba )ub ub + ba ua ua + ab ua ub ] : From Eqs. (10.17) and (10.23), the following general analytical expression for intensity-dependent nonlinear optical susceptibility is obtained [54]: :m+1; −m [! = (m + 1)!1 − m!2 ] = − N0 2ab ba {[L +  + iba ] · [ba ua + ab ub ]: ∗

∗

+ ub∗ [u b ua − ub u a ]}=[DU · j1 ((m + 1)!1 ) · j2 (−m!2 )] :

(10.24)

Note that Eqs. (10.23) and (10.24) possess the following two distinct features and advantages over other conventional perturbative or RWA approaches: (i) the intensity-dependent nature of D and :(!) is clearly determined by the two physical parameters  and u only; and (ii) D and :(!) have the same general functional form as shown by Eqs. (10.23) and (10.24), respectively, regardless of the order (2|m| + 1) of multiphoton processes.  and u, of course, depend on m and can be determined via the nearly degenerate perturbative treatment. Analytical expressions for  and u for various cases can be found in Ref. [54]. Figs. 38(a) and (b) show the intensity-dependent four-wave mixing nonlinear responses :(! = 2!1 − !2 ), as functions of !2 − !1 , subject to both radiative and collisional dampings [54]. The pumping frequency !1 is 'xed at resonance with the level spacing !ba (!1 = !ba = 100:0 (arbitrary units)), the probe 'eld strength || is 'xed at a low value (|| = 0:01), while the pump 'eld strength || varies from weak to medium strong. First note that in the limit of a weak pump

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Fig. 38. Intensity-dependent nonlinear optical susceptibilities corresponding to the four-wave mixing process, !=2!1 −!2 , as a function of !2 − !1 . The pump frequency !1 is 'xed at the resonance frequency (!1 = !ba = 100:0), and the damping parameters used are ba = 0:1, ba = 5ba (arbitrary units). The probe 'eld strength || is 'xed at 0.01, while the pump 'eld strength || varies. Curves labeled a, b, and c correspond, respectively, to ||=ba = 0:1; 0:5; 1:0. The dotted curves are third-order perturbative results which are intensity independent (adapted from Ref. [54]).

'eld, :(2!1 − !2 ) approaches the third order perturbative result (dotted curves) which are intensity independent. However, as the pump 'eld strength || increases, signi'cant changes in line shapes can be seen. In particular, an extra absorption peak (hole) appears at the line center (!1 = !2 = !ba ) at stronger pump 'eld (Fig. 38(b)). This can be attributed to the contribution from the 5th-order perturbation terms. When both the pump and the probe 'eld intensities are further increased, various higher-order contributions will eventually set in, leading to pronounced subharmonic multi-peak structures in the generalized nonlinear optical susceptibility, particularly around the line center region (Figs. 39(a) and (b)). Using the FLSM approach [54], Kavanaugh and Silbey derived the intensity-dependent expressions for susceptibility of a three-level system interacting with one monochromatic laser 'eld [191], and the nonlinear response of two- and three-level systems in the presence of pulsed laser 'elds [192]. A model three-level system that mimics the excited electronic states of typical nonlinear optical

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Fig. 39. Generalized nonlinear optical susceptibilities :(!=2!1 −!2 ) at intense bichromatic 'elds are shown as a function of !2 − !1 . The dispersive responses are shown in (a) and the absorptive responses in (b). The solid curves are the results for the pure radiative damping case (ba = (1=2)ba ), and the dotted curves include the e9ects of collisional relaxation (ba = 2ba ). The multiphoton subradiative structures are labeled as (n1 ; n2 ) corresponding to the n1 !1 + n2 !2 processes, where n1 and n2 are (positive or negative) integers (adapted from Ref. [54]).

polymers was used. They found that the total susceptibility obtained from the FLSM approach correctly accounts for many observed nonlinear optical phenomena such as the intensity-dependent behavior of a conjugated system near resonance as well as the presence of extra resonances when collisional relaxations are incorporated, etc. 10.3. Multiphoton resonance Auorescence in intense laser (elds The FLSM formalism has been also applied to the study of multiphoton-induced resonance >uorescence and light scatterings of N -level systems illuminated by strong polychromatic 'elds [189]. Resonance >uorescence scattering by atoms and molecules in the presence of strong laser 'elds is a delicate nonlinear process in two aspects: (i) it is a cascade process via an in'nite number of dressed atomic or molecular states, and (ii) it requires strong resonance mixings, either by one photon or by several photons, between unperturbed atomic levels. The strong mixings of levels produce sidebands due to the ac Stark e9ect, in addition to those corresponding to the natural transition frequencies.

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Fig. 40. Schematic cascade >uorescence processes of two-level atoms driven by a monochromatic 'eld of frequency !L . The splitting u of the doublets in each column is the splitting of the adjacent quasienergy levels and is caused by the ac Stark e9ect and possible detuning L = !ba − (2n + 1)!L at nearly resonant conditions. Each column is a collection of quasienergy levels of like parity; quasienergy levels belonging to di9erent columns are of opposite parity. Arrows indicate parts of cascade >uorescence down the in'nite number of quasienergy levels (adapted from Ref. [189]).

The resonance >uorescence processes of two-level systems driven by a monochromatic laser 'eld of frequency !L are schematically depicted in Fig. 40, where each doublet is characterized by a splitting u between a pair of nearly degenerate quasi-energy levels, and arrows indicate >uorescence cascade patterns. At each near-resonance condition !ba ∼ (2n + 1)!L , n = 0; 1; 2; : : : ; the most intense >uorescence light comes around ! ∼ (2n + 1)!L and shows a triplet pattern. There are intimate relationships between the super-eigenvalues and the long-time-averaged population [Figs. 41(a) and (b)]. Fig. 41(a) shows the strong mixed regions of the two levels caused by one-, three- and 've-photon resonance transitions, respectively. The splittings and stretches of the avoided crossing regions re>ects the widths of the corresponding lineshapes of the long-time-averaged excitation spectrum DUbb as a function of !L , depicted in Fig. 41(b). Fig. 42 shows the >uorescence power spectra corresponding to the shifted three-photon resonance condition (!ba  3!L ). Note the strong triplet >uorescence spectra nearby !  !L and 3!L [Figs. 42(a) and (b)] as well as

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Fig. 41. (a) Supereigenvalues (real parts) and (b) long-time-averaged population DUbb for a closed system of two-level (1) atoms driven by an intense monochromatic 'eld of frequency !L . Parameters: !ba =100:0, ba =1:0, ab =0:0, |Vab |=25:0, (1) 6 =0:0 (arbitrary units). The one-, three-, and 've-photon resonances [solid curves in (b)] occur at !L =106:335; 41:295, and 24:525, respectively. Also shown are results of the corresponding nondamping (ba = 0:0) case [dotted curves in (b)] for comparison (adapted from Ref. [189]).

a much weaker triplet around !  5!L [Fig. 42(c)]. Particularly interesting is the strongly asymmetric three-peak structure near !  !L . This asymmetry can be largely attributed to the strong mixings not only among the resonant, or nearly resonant, unperturbed tetradic Floquet states (e.g., |aa; 0, |bb; 0, |ab; +3), and |ba; −3 but also of the nonresonant states (such as |ab; +1, |ba; −1, etc.). At a much weaker 'eld, only those nearly resonant states are mixed; thus intense >uorescence light can only be observed at 3!L and possesses a symmetric triplet-peak appearance, the well-known Mollow spectrum [193]. The intense >uorescence light and its asymmetric outlook at !  !L in this three-photon resonance case are genuine strong 'eld e9ects [189]. Moreover, in addition to the

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Fig. 42. Fluorescence spectra IU(!) near (a) !  !L , (b) !!L , and (c) !  5!L for a system of two-level atoms driven by a monochromatic 'eld; !L tuned at the shifted three-photon resonance !L = 41:295. Parameters same as in Fig. 41. The inset in each 'gure shows the schematic cascade diagram (not to scale) (adapted from Ref. [189]).

time-averaged power spectrum, another dynamical quantity of interest is the time-dependent physical spectrum which can be also calculated by means of the FLSM formalism. For a detailed discussion of this topic, we refer the readers to [189].

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11. Floquet study of nonadiabatic and complex geometric phases in multiphoton transitions In recent years, there is considerable interest generated by the discovery by Berry [194] regarding the geometrical phase factor associated with the adiabatic transport of a quantum system around a closed circuit in some parameter space [195]. In addition to the “normal” dynamical phase,
i  (t)|Hˆ (t)| (t) dt ; (11.1) D = − } a geometric phase factor G (C), now known as the Berry’s phase, evolves during the cyclic evolution, which depends only upon the geometry of the circuit C. Several experiments have been reported that demonstrate the e9ects of this phase. These include observations on nuclear magnetic resonance (NMR) [196], photons [197,198], neutrons [199], nuclear spins [200], molecular energy levels [201], electrons [202], and spin echos [203], etc. Aharonov and Anandan [204] (AA) have further introduced a new cyclic quantum phase that is a gauge-invariant generalization of the Berry phase without recourse to adiabaticity. The AA phase is a more general concept and is associated with the evolution of any cyclic state, i.e., a quantum state | (t) which returns to itself, apart from a phase factor, after some time T : | (T ) = exp(i)| (0), where (total phase) = D (dynamical phase) + G (geometric phase)y. The AA geometric phase is related to an holonomy [195] associated with parallel transport around the circuit in projective Hilbert space. The importance of the AA formulation is that it applies whether or not the Hamiltonian Hˆ (t) is cyclic or adiabatic. The AA geometric phase only depends upon the cyclic evolution of the system. This establishes a simple connection of the geometric phase to the Aharonov–Bohm e9ect [205,206] which does not invoke the adiabaticity of the circuit. The AA phase has also been detected experimentally by means of NMR interferometry [207]. More recently, it has been suggested that the geometric phase may be used to realize controlled NOT gate operations in spin 1/2 systems [208] and in superconducting nanocrystals [209,210], etc. This cyclic evolution has the potential to form universal operations of quantum bits. In the following subsections, we discuss several generalizations of the AA phase formulation to multiphoton and nonlinear optical processes in strong laser 'elds by means of the Floquet theory 'rst presented by Wu et al. [55,56]. Application of the Floquet theory to the study of geometric phases in multiphoton ionization of atomic H is discussed in [211]. 11.1. Cyclic quantum evolution and nonadiabatic geometric phases for spin-j systems driven by periodic (elds In this section we consider the Aharonov–Anandan geometric phases for the cyclic quantum evolution of any spin-j driven by periodic 'elds. In general, for any N -level system [subject to SU(N ) dynamical symmetry], the cyclic evolution of the quantum state (t) is diNcult to realize if N ¿ 3 since there are N 2 − 1 degrees of freedom in the Hilbert space. However, we shall show below that for the spin-j system, it is feasible to satisfy the cyclic condition [55] and that the AA phase can be measured experimentally. The quantum evolution of a particle with spin j and magnetic moment 2 in a variable magnetic 'eld B(t) = (Bx (t); By (t); Bz (t)) is governed by the time-dependent SchrGodinger equation d (11.2) i | (t) = Hˆ (t)| (t) : dt

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91

Here Hˆ (t) is the Hamiltonian Hˆ (t) = − · B(t) = −(2=j)J · B ; (11.3) and J = (Jˆx ; Jˆy ; Jˆz ) is the spin angular momentum operator with Jˆi corresponding to the operator of in'nitesimal rotation around the axis (i = x; y; z). It is known that the spin motion can be described by the SU(2) group [212], as the Hamiltonian, Eq. (11.3), contains only the generators of the SU(2) group Hˆ (t) = i[a(t)Jˆ+ − a∗ (t)Jˆ− − ib(t)Jˆ0 ] ; (11.4) where a(t) = i(2=2j)[Bx (t) − iBy (t)] ;

(11.5)

b(t) = −(2=j)Bz (t) ;

(11.6)

and the in'nitesimal operators Jˆ± = Jˆx ± iJˆy , Jˆ0 = Jˆz obey the commutation relations [Jˆ0 ; Jˆ± ] = ±Jˆ± ; [Jˆ− ; Jˆ+ ] = −2Jˆ0 :

(11.7)

The quantum evolution of the spin-j system, Eq. (11.2), can be studied either by the determination of the time-evolution propagator Uˆ (t; t0 ) or by resorting to the use of spin coherent states (CS) concept [55,212]. The time-propagator method is more general and can also be applied to nonspin systems but the CS approach provides an elegant SU(N ) group theoretical approach for the spin systems. In the following, we discuss the Floquet approach for the study of AA phases in spin systems driven by periodically time-dependent 'elds [55]. First consider the motion of a spin-j system under the in>uence of a static magnetic 'eld (z) ˆ and a linearly polarized magnetic 'eld in the x direction, as in typical NMR experiments. Thus B(t) = (Bx0 cos !t; 0; Bz0 ) : The Hamiltonian, Eq. (11.3), now reads Hˆ (t) = !0 Jˆz + 4!⊥ Jˆx cos !t ;

(11.8) (11.9)

where !0 = −2Bz0 =j;

!⊥ = −2Bx0 =4j : (11.10) As the Hamiltonian Hˆ (t)= Hˆ (t +2 =!) is periodic in time, the time-dependent SchrGodinger equation, Eq. (11.2), can be transformed into an equivalent time-independent in'nite-dimensional Floquet Hamiltonian eigenvalue analysis. We 'rst introduce the Floquet state basis |jm; 2=|jm⊗|2, where |jm are the (unperturbed) spin eigenfunctions (m = −j; −j + 1; : : : ; j) and |2 are the Fourier vectors (2 = 0; ±1; ±2; : : :) such that t|2 = exp(i2!t). In terms of the basis |jm; 2, the time-independent Floquet Hamiltonian Hˆ F has the following matrix form: jm; 2|Hˆ F |jn; 1 = (n!0 + 1!)mn 21 + 2!⊥ jm; 2|Jˆx |jn; 1(2; 1+1 + 2; 1−1 ) ;

(11.11)

where 2 and 1 are Fourier indices (−∞ to +∞). The problem involved here is to solve the time-independent Floquet matrix eigenproblem for the spin-j system: ( j) ( j) ( j) Hˆ F |m1  = m1 |m1  : (11.12)

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( j) ( j) Here m1 and |m1  are, respectively, the quasienergy eigenvalues and eigenvectors. The time propagator can be constructed from

Uˆ (t; t0 ) =

j ∞   m=−j 1=−∞

( j) ( j) ( j) |m1 exp[ − im1 (t − t0 )]m1 |:

(11.13)

In general, there is no closed-form solution for quasienergy eigenvalues and eigenvectors. However, in the rotating-wave approximation (RWA), an exact analytic solution can be obtained for the spin-j problem. This approximation is equivalent to replacing the linearly polarized magnetic 'eld by a rotating 'eld. The RWA is generally valid in NMR conditions where |L=!0 |1 and |!⊥ =!0 |1. (Analytical results can still be obtained beyond the RWA limit using high-order nearly degenerate perturbation techniques. The more accurate treatment of Floquet eigenvectors will not, however, a9ect the geometric phase formulation in general.) In the RWA, one drops all energy-nonconserving terms, and the in'nite-dimensional Floquet Hamiltonian reduces to block-diagonal RWA Hamiltonians Hˆ RWA = −j!Iˆ + YJˆ0 + 2!⊥ Jˆx ; (11.14) where L = !0 − !, and Iˆ is the identity operator. Notice Hˆ RWA possesses the common eigenfunctions as the operator hˆ = YJˆ0 + 2!⊥ Jˆx , Hˆ RWA |m  = m |m  ;

(11.15)

ˆ m  = =m |m  ; h|

(11.16)

with m = =m − j!. ˆ a rotation in the xz plane is necessary. Thus To diagonalize h, ˆ ; 0)−1 hˆR(0; ˆ ; 0) R(0; =(Y cos  + 2!⊥ sin )Jˆ0 + (−Y sin  + 2!⊥ cos )Jˆx :

(11.17)

To put the right-hand side of Eq. (11.17) into diagonalized form (in the |jm basis), the coeNcient of the Jˆx , operator must vanish −Y sin  + 2!⊥ cos  = 0 : This leads to tan = 2!⊥ =L

(11.18a)

2 1=2 cos  = L=(L2 + 4!⊥ ) = L=F :

(11.18b)

and ˆ in the |jm basis, can now be obtained readily form Eq. (11.17), The eigenvalues of h, =m = m(Y cos  + 2!⊥ sin ) = mF ; from which the quasienergy eigenvalues of Hˆ RWA are determined, m = −j! + mF;

(m = −j; −j + 1; : : : ; j) :

(11.19)

The quasienergy eigenfunctions, in the rotating frame of coordinates, are ˆ ; 0)|jm : |m  = R(0;

(11.20)

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The total wave function for spin-j systems, in rotating frames of coordinates and in RWA, can now be written in closed-form solution | R (t) = e

ij!t

j 

e−imFt |m m | (0) :

(11.21)

m=−j

Eq. (11.21) shows that after a period T = 2 =F, the system returns to | R (0) with an extra total overall phase : | R (T ) = ei | R (0) ;

(11.22)

 = j!T − 2 j :

(11.23)

and The dynamical phase D is determined by
T  R (t)|Hˆ RWA | R (t) dt mod (2 ) D = − 0

= j!T − 2

j 

m| (0)|m |2

mod (2 ) :

(11.24)

m=−j

Finally, from Eqs. (11.23) and (11.24), we arrive at the AA geometric phase G (= − D ),   j  G = −2 j − (11.25) m| (0)|m |2 : m=−j

This expression is general in that there is no restriction on the initial wave function | (0). In the event that | (0) = |j; −j, Eq. (11.25) reduces to G = −2 j(1 + cos ) = −2 j(1 + L=F) :

(11.26)

On the other hand, if | (0) = |j; j, the result is G = −2 j(1 − cos ) = −2 j(1 − L=F) :

(11.27)

These expressions are in agreement with the results obtained by means of the method of SU(N ) spin coherent states (CS) [55]. The CS study further reveals that the AA geometric phase is equivalent to the solid angle enclosed by the generalized Bloch vector’s closed circuit times j. Fig. 43 shows the generalized Bloch sphere for the spin-j systems [55]. Extension of the nonadiabatic multiphoton AA phase to include the 'eld modulation e9ects is discussed in [213]. 11.2. Biorthogonal density matrix formulation of complex geometric phases for dissipative systems and nonlinear optical processes Most of the studies of geometric phases have been con'ned to the evolution of unitary operators for Hermitian Hamiltonians. In this section, we present a density matrix formulation of complex geometric phases for dissipative quantum systems involving non-Hermitian Hamiltonians [56]. Since both | (t) and the conventional density matrix de'ned by D(t) ˆ = | (t) (t)| are decaying in time,

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Fig. 43. Generalized Bloch sphere model for any spin j. Each point nˆ = (sin G cos 6; sin G sin 6; cos G) on the (unit radius) Bloch sphere represents a spin-coherent state. The unit vector nˆ0 (south pole) represents the fundamental vector | 0  = |j; −j. Also shown here is the mapping of a three-dimensional unit vector nˆ to a point R (complex parameter) in the xy plane (adapted from Ref. [55]).

a cyclic state cannot be de'ned. In [56], a generalized density matrix formulation is introduced to avoid this diNculty. In addition, it is shown that the celebrated Feynman–Vernon–Hellwarth geometric representation [159] of the Hermitian SchrGodinger equation can be extended to the case of non-Hermitian SchrGodinger equation. This provides a natural framework for the description of complex geometric phases in dissipative systems. We then present a theorem which relates the complex geometric phase to a complex solid angle. Finally the formalism is applied to the study of multiphoton Rabi >oppings in dissipative two-level systems to obtain general analytic formulas for complex geometric phases. 11.2.1. Geometric representation of non-Hermitian SchrDodinger equation Consider the following time-dependent SchrGodinger equation: i

d ˆ | (t) = H(t)| (t) ; dt

(11.28)

where ˆ H(t) = Hˆ (t) − iGˆ

(11.29)

Hˆ (t) = Hˆ 0 + Vˆ (t) :

(11.30)

and Hˆ 0 is the unperturbed Hamiltonian of the two-level system with eigenstates | and | and eigenvalues E and E , and Vˆ (t) is the time-dependent perturbation. In Eq. (11.29), Gˆ is the diagonal

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95

damping operator with eigenvalues g and g : ˆ = g |; G|

( =  or ) :

(11.31)

g can represent, for example, the spontaneous decay rate of level |, etc. To construct the density matrix, the conventional way is to adopt Dˆ (t) = | (t) (t)| ;

(11.32)

where | (t) is the solution of Eq. (11.28). This leads to the Liouville equation of the following form: d ˆ Dˆ (t)}+ ; (11.33) i Dˆ (t) = [Hˆ (t); Dˆ (t)]− − i{G; dt where ˆ B] ˆ − = Aˆ Bˆ − Bˆ Aˆ [A;

and

ˆ B} ˆ + = Aˆ Bˆ + Bˆ Aˆ : {A;

Due to the dissipative { ; }+ term, the density matrix Dˆ described by Eq. (11.33) does not have a conserved norm and its trace, Tr Dˆ (t), is decreasing in time. This causes diNculty in the description of the AA geometric phase as the density matrix is required to return to its initial value after a cyclic evolution of the system [204]. To overcome the diNculty, consider the following generalized density matrix [56]: D(t) ˆ = | (t):(t)| ;

(11.34)

de'ned by the biorthonormal Hilbert space [15,17]. Here |:(t) is the solution of the SchrGodinger ˆ † (t) equation with the adjoint Hamiltonian H d ˆ † (t)|:(t) : i |:(t) = H (11.35) dt Eq. (11.34) leads to the following generalized Liouville equation: d ˆ ˆ = [H(t); D(t)] ˆ − ; i D(t) (11.36) dt the form of which is identical to the ordinary Liouville equation without dissipation. Further, in the biorthonormal Hilbert space, one has Tr D(t) ˆ = :(t)| (t) = :(0)| (0) = 1 ;

(11.37)

and the norm of the three-vector (to be de'ned below) is conserved in time even as the system is dissipating. The solutions of Eqs. (11.28) and (11.35) can be written, for a two-level system, in the general form | (t) = a(t)| + b(t)|;

U :(t)| = a(t)| U + b(t)| :

In terms of column and row vectors, we have   a(t) U | (t) = ; :(t)| = (a(t); U b(t)) ; b(t)

(11.38)

(11.39)

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subject to the biorthogonal relation, U U a(t)a(t) U + b(t)b(t) = a(0)a(0) U + b(0)b(0) =1 :

(11.40)

Following Feynman et al. [159], one can construct the three-vector r as ˆ )ˆx ) = abU + baU ; r1 (t) ≡ u(t) = Tr(D(t)

(11.41a)

r2 (t) ≡ v(t) = Tr(D(t) ˆ )ˆy ) = i(abU − ba) U ;

(11.41b)

r3 (t) ≡ !(t) = Tr(D(t) ˆ )ˆz ) = aaU − bbU ;

(11.41c)

where the )’s ˆ are the Pauli spin matrices, and ri (t) are complex quantities. (For non-dissipative ˆ systems, G = 0, aU → a∗ , bU → b∗ , and ri (t) become real quantities.) From Eqs. (11.41), one can readily show that the norm of the complex Bloch vector S(t) = (u; v; w) is conserved in time, r12 (t) + r22 (t) + r32 (t) = u2 (t) + v2 (t) + w2 (t) = 1 :

(11.42)

Thus S(t) = (u; v; w) is a complex three-vector with unit norm and traces out a trajectory in the complex three-space. Further it can be shown that the di9erential equation for r is dr =×r ; dt

(11.43)

where  is also a three-vector in “r” space de'ned by ˆ )ˆx ); F1 = Tr(H(t)

ˆ )ˆy ); F2 = Tr(H(t)

ˆ )ˆz ) : F3 = Tr(H(t)

Eqs. (11.34) – (11.43) are the generalization of the Feynman–Vernon–Hellwarth geometric representation to the non-Hermitian SchrGodinger equation 'rst introduced by Chu et al. in 1989 [56]. 11.2.2. Complex geometric phase in dissipative two-level systems Under a cyclic quantum evolution, D(t ˆ + T ) = D(t); ˆ

S(t + T ) = S(t) ;

(11.44)

and | (t + T ) = exp(i)| (t);

|:(t + T ) = exp(i∗ )|:(t) ;

(11.45)

where  (∗ ) is the total (complex) phase associated with the cyclic evolution of | (t) (|:(t)). To evaluate , we follow the procedure of Aharonov and Anandan [204] by introducing a modi'ed adjoint pair | ˜ (t) = exp[ − if(t)]| (t);

|:(t) ˜ = exp[ − if∗ (t)]|:(t) :

(11.46)

Here f(t) is an arbitrary function satisfying f(t + T ) − f(t) = , and | ˜ (t + T ) = | ˜ (t);

|:(t ˜ + T ) = |:(t) ˜ ;

(11.47)

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are periodic in time with period T . Using Eqs. (11.28) and (11.35), we get 9 ˜ df ˆ ˜ ; = :|i ˜ |  − :| ˜ H| dt 9t and thus  = D + G ; with

(11.48)

(11.49)


D = dynamical phase =

97

0

T

ˆ  dt ; :|H|

and


G = AA geometric phase =

0

T

:|i ˜

9 ˜ |  dt : 9t

(11.50)

(11.51)

We shall now introduce the following theorem [56]. Theorem. The complex geometric phase G de(ned in Eq. (11.51) is equal to one-half of the complex solid angle F(C) enclosed by the complex trajectory S(t) = (u; v; w). Here the solid angle F(C) enclosed by a closed curve C is de'ned as
T [1 − cos G(t)]6˙ dt ; F(C) = 0

(11.52)

and the theorem implies that G = F(C)=2. Assuming (u; v; w) form an orthogonal three-vector, one can de'ne U + abU U tan 6 = u = ab cos G = w = aaU − bb; ; (11.53) v i(abU − ba) U where G and 6 are complex spherical angles. 11.2.3. Complex geometric phase for multiphoton transitions In this subsection, we present an example of complex geometric phase for dissipative two-level systems undergoing multiphoton Rabi >oppings [56]. Consider the time evolution of the SchrGodinger equation, Eq. (11.28), for a two-level dissipative system driven by an intense periodic 'eld. The perturbation Vˆ (t) in Eq. (11.30) is now given by the electric dipole interaction, Vˆ (t) = − · 0 cos(!t + ’) ;

(11.54)

where  is the electric dipole moment of the system and 0 , !, and ’ are, respectively, the 'eld amplitude, frequency, and phase. In terms of the unperturbed bases {|; |} of the two-level system, ˆ the total Hamiltonian H(t), Eq. (11.29), has the following matrix form:

E − ig V (t) ˆ H(t) = ; (11.55) E − ig V (t)

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where V (t) = |V (t)|, and g (g ) is the damping constant. The non-Hermitian time-dependent ˆ + 2 =!) = H(t), ˆ SchrGodinger equation, Eq. (11.28), with periodic Hamiltonian H(t given in Eq. (11.55), can be transformed into an equivalent in'nite-dimensional time-independent nonHermitian Floquet matrix (Aˆ F ) eigenvalue problem, Aˆ F |=n  = =n |=n  ;

(11.56)

where =n and |=n  are the complex quasi-energy eigenvalues and eigenvectors, respectively, with  =  or  and n (Fourier index) = − ∞ to +∞. For nearly resonant multiphoton processes, E − E ≡ !0 ≈ (2n + 1)!; n = 1; 2; : : :, the in'nitedimensional Floquet matrix Aˆ F can be further reduced to a two-by-two e9ective Hamiltonian by means of appropriate nearly degenerate high-order perturbation theory [56],

u E + (c) ; (11.57) Aˆ e9 = u E − (2n + 1)! + (c) where (c) , ((c) ) and u , (u ) are, respectively, the (complex) ac Stark shifts and e9ective couplings. For (2n + 1)-photon transition, the leading terms in  and u can be derived analytically using the (2n + 1)-order nearly degenerate perturbation method [213]. The e9ective Hamiltonian Aˆ e9 possesses two complex eigenvalues =± and eigenvectors |=± , Aˆ e9 |=±  = =± |=±  ;

(11.58)

=± = K ± q ;

(11.59)

where

K=

$ 1 1 L2 + 4u u ; Tr(Aˆ e9 ); q = 2 2 and L is the detuning parameter,

(11.60)

L = E − [E − (2n + 1)!] + (c) − (c) :

(11.61)

The quasienergy eigenvectors |=±  are

(u =u )1=4 cos(G=2) −(u =u )1=4 sin(G=2) |=+  = ; |=−  = ; (u =u )1=4 sin(G=2) (u =u )1=4 cos(G=2) where G is a complex angle de'ned by √ tan G = 2 u u =L :

(11.62)

(11.63)

∗ and eigenvectors |  of the adjoint Hamiltonian Similarly, the complex eigenvalues ± = =± ±

Aˆ + e9 |±  = =± |±  ;

(11.64)

can be obtained. The wavefunctions | (t) and :(t)| can now be approximated as | (t) ≈ exp[ − iAˆ e9 (t − t0 )]| (t0 ) ;

(11.65)

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99

and :(t)| ≈ :(t0 )|exp[iAˆ e9 (t − t0 )] :

(11.66)

The complex Bloch vector S(t) = (u; v; w) can be constructed according to Eq. (11.41). Following the procedure described in the previous section, one arrives at the following general formula for the complex AA geometric phase for multiphoton Rabi >oppings with period T = q, G(C) = [1 + :(0)|=+ + | (0) − :(0)|=− − | (0)] ;

(11.67)

when the generalized density matrix D(t) ˆ = | (t):(t)| returns to its initial value, D(T ˆ ) = D(0). ˆ The corresponding formula for nondissipative systems has the following simpler form, G = [1 + |=+ | (0)|2 − |=− | (0)|2 ] ;

(11.68)

where G is real. As a simple application, consider a two-level system initially prepared in the ground state, | (0)= |, and the laser phase is ’ = 0. In this case u = u , and the complex nonadiabatic geometric phase has the following simple form: G(C) = (1 + cos G) ;

(11.69)

where L

cos G = $

2 L2 + 4u

:

(11.70)

In the special case of one-photon transition, n = 0, and in the limit of rotating wave approximation, we have L = (! − !0 ) − i(g − g ), and u = b = − 12 || · 0 . The complex geometric phase, Eq. (11.69), now becomes   (! − !0 ) − i(g − g )  : G(C) = 1 + $ (11.71) 2 2 [(! − !0 ) − i(g − g )] + 4b Similar expression for Eq. (11.71) has been also obtained by a di9erent procedure [214]. 12. Generalized Floquet approaches for multiphoton processes in intense laser pulses Recently there has been much interest in the study of multiphoton and nonlinear optical processes using intense short-pulse laser 'elds. In general, compared to a 'eld with constant amplitude, a pulse 'eld has more degrees of freedom. Recent advances in ultrafast optics have made available laser pulses as short as a few femtoseconds [215–217]. A substantial e9ort is also underway to develop single attosecond optical pulses [218] or trains of attosecond pulses [219]. In this section, we discuss several generalizations of the Floquet theory for nonperturbative treatment of multiphoton processes in the presence of pulsed laser 'elds, where the Hamiltonian is no longer a periodic function of time.

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12.1. Nonadiabatic coupled dressed-states formalism In this section, we present a general nonadiabatic coupled dressed-state formalism for nonperturbative treatment of multiphoton processes driven by pulsed laser 'elds with time-varying 'eld amplitude, frequency, and phase [21]. The time evolution of a nondegenerate N -level system irradiated by a linearly polarized laser pulse 'eld can be described within the electric dipole approximation by the SchrGodinger equation d| (t) = {Hˆ 0 −  · E0 (t) cos[!(t)t + 6(t)]}| (t) ; i (12.1) dt where Hˆ 0 is the unperturbed Hamiltonian with eigenvalues E(0) and eigenvectors |,  = 1; 2; : : : ; N ; 2 is the electric dipole moment of the system; and E0 (t), !(t), and 6(t) are, respectively, the amplitude, frequency, and phase of the applied laser 'eld. If E0 (t); !(t); and 6(t) change only slowly in any time interval of length 2 =!(t), then the pulse or pulse sequence can be viewed in such a way that E0 (t) is the amplitude modulation and 6(t) is the phase modulation on some carrier wave at frequency !(t). U The discussion that follows will focus on this speci'c, albeit important, type of pulse excitation. Collectively, the set of parameters E0 , !, and 6 characterizing the laser 'eld shall be abbreviated as X = {E0 ; !; 6}. Thus, the total time derivative in Eq. (12.1) can be written simply as  9 d 9 + X˙ · = ; (12.2) dt 9t X 9X where 9 9 9 9 ≡ E˙ 0 · + 6˙ : (12.3) + !˙ X˙ · 9X 9E0 9! 96 Because X = {E0 ; !; 6} vary only very slowly in time, one can momentarily freeze the parameter X at a certain value, analogous to Born–Oppenheimer approximation, and establish the parameterized SchrGodinger equation,  9 i |(t; X ) = Hˆ (t; X )|(t; X ) ; (12.4) 9t X where the Hamiltonian Hˆ (t; X ) is given in Eq. (12.1) with X 'xed at a certain value. The periodic nature of the Hamiltonian Hˆ (t; X ), that is, Hˆ (t+ ; X )= Hˆ (t; X ) with =2 =!, guarantees a complete set of orthonormal quasi-energy state basis {| (t; X );  = 1; 2; : : : ; N } that possesses the following product form (Floquet theorem): | (t; X ) = exp[ − i= (X )t]|˜  (t; X ) ;

(12.5)

where the quasi-energies = ’s are real numbers, and the functions | (t; X ) are periodic, that is, |˜  (t + ; X ) = |˜  (t; X ) ;

(12.6)

and satisfy the orthonormal condition ˜  (t; X )|˜  (t; X ) =  (t; X )| (t; X ) =  :

(12.7)

The quasi-energies = (X ) and quasi-energy states | (t; X ) of Eq. (12.4) can be readily solved by the time-independent Floquet Hamiltonian method described in earlier sections.

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101

In general, one can expand the total wavefunction | (t) of Eq. (12.1) in terms of {| (t; X );  = 1; 2; : : : ; N } at each 'xed value of X , namely, | (t) =

N 

a (X )| (t; X ) :

(12.8)

=1

Substituting Eq. (12.8) into Eq. (12.1) yields a set of coupled equations, N

i

 9 da (X (t)) = −i | (t; X ) : a (X (t)) (t; X )|X˙ · dt 9X

(12.9)

=1

Noting that the changes in a (X (t)) and da (X (t))=dt are negligible within a period , Eq. (12.9) may be further reduced by averaging it over a duration to obtain N

i

 9 da (X (t)) = −i | (t; X )U : a (X (t))T (t; X )|X˙ · dt 9X

(12.10)

=1

Here the outer bracket represents a time average, that is,
1

T···U ≡ dt· · · ;

0 so that the coupling matrix elements have no explicit time dependence. Eq. (12.10) shows that transitions between various adiabatic quasienergy states {| (t; X )} are caused by the nonadiabatic coupling matrix elements T (t; X )|X˙ ·9=9X | (t; X )U due to the variation of the 'eld quantities E, !, and 6 in time. The coupling matrix elements can be analytically evaluated using the Hellmann– Feynman theorem and expressed in terms of quasienergy eigenvalues and eigenvectors. The coupled dressed-states formalism presented here provides an exact nonperturbative approach for the treatment of multiphoton excitation of quantum systems with arbitrary pulse shapes [21]. The method has been applied successfully to the study of multiphoton adiabatic inversion of multilevel molecular systems [21]. We note in passing that similar generalized Floquet methods have also been developed for the treatment of the following di9erent classes of laser-induced atomic and molecular processes is intense 'elds. (i) Nonadiabatic theory for high-resolution molecular multiphoton absorption (MPA) spectra [220,221]. The approach is based on the adiabatic separation of fast vibrational motion from slow rotational motion, incorporating the fact that the infrared laser frequency is close to the frequencies of adjacent vibrational transitions. One can thus 'rst solve the vibrational quasienergy (VQE) states (or, equivalently, the dressed vibrational states) with molecular orientation (xed. This reduces the computationally often formidable (vibrational–rotational) Floquet matrix analysis to a manageable scale and, in addition, provides useful physical insights for understanding the nonlinear MPA dynamics. The VQE levels are found to be grouped into distinct energy bands characterized by the infrared frequency, with each band providing an e9ective potential for molecular rotation. While the interband couplings are totally negligible, the intraband nonadiabatic angular couplings are the main driving mechanisms for inducing resonant vibrational–rotational multiphoton transitions [220,221].

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(ii) Multicharged ion–atom collisions in intense laser (elds: coupled dressed-quasi-molecular-states (DQMS) approach [16,222,223]. Here we are concerned with nonperturbative treatment of charge transfer processes at low collision velocities and strong laser intensities. These processes are important in determining the particle densities in Tokamak plasma [224] and have potential usefulness in the development of X-ray lasers [225,226]. More recent studies in this direction are focused on the exploration of new excitation mechanisms in energetic ion–atom collisions embedded in short laser pulses [227] or the control of the interaction pathways in ion–atom collisions [228], i.e., to enhance the population of favored 'nal states and to suppress the production of undesired ones by suitable laser parameters. This could be useful for various applications such as laser-driven fusion or plasma heating. When the laser frequency of interest is in the range of quasimolecular (A · · · B)+Z electronic energy separations, the laser 'eld oscillates much faster than the motion of the nuclei. It is legitimate to 'rst construct the solutions of the (A − B)+Z + 'eld system, namely, the dressed quasimolecular electronic states with the internuclear separation R 'xed. The DQMS constructed in this way are adiabatic, and their associated quasienergies (depending parametrically on R) exhibit regions of avoided crossings, where the electronic transition can be induced by the nonadiabatic radial couplings due to the nuclear movement. By further transforming the adiabatic DQMS into an appropriate diabatic DQMS representation, de'ned via the vanishing of the radial couplings, one obtains a new set of coupled (diabatic) equations that o9er substantial computational advantages. Application of the DQMS approach has been made for the study of laser assisted ion–atom collisions such as He++ + H(1s) [222] and Li+++ + H(1s) [223] systems, etc. 12.2. Multiphoton adiabatic inversion of multilevel systems As an application of the nonadiabatic coupled dressed-states formalism, consider below the problem of population inversion among the low-lying vibrational levels of a diatomic molecule induced by a strong laser 'eld of 'xed amplitude (E0 ) and phase (6) but slowly varying frequency !(t) [21]. Rotational motion is not considered but can be implemented if needed. Figs. 44 and 45 show the results of the population inversion of the 12 C16 O Morse oscillator from v=0 to 3 by slowly sweeping the laser frequency ! from 2300 to 2000 cm−1 . A linear scanning rate of the form !(t)=!(t =0)−Tt is assumed, where T is the sweeping rate. The quasienergies =v; 0 (!), v = 0; 1; 2; 3, assuming initially the system is at its unperturbed ground vibrational level (v = 0), is depicted in Fig. 44. Thus starting from the adiabatic quasienergy level connected to v = 0 in the extreme right (left) and proceeding to the left (right), one encounters a series of avoided crossings. Population redistribution among the quasienergy levels can occur at these avoided-crossing points and is induced by the nonadiabatic coupling terms in Eq. (12.10). Since the narrower are the avoided crossing regions, the larger will be the nonadiabatic couplings, it is clearly far more favorable to sweep the laser frequency from 2300 to 2000 cm−1 than to sweep from the opposite direction. To avoid population redistribution and to preserve adiabaticity (i.e., staying on a single quasienergy level), at these anticrossing points, the sweeping rate has to be suNciently slow (but faster than the relaxation rates). Fig. 45 shows that by sweeping the laser frequency (from right to left) slow enough, one can in fact overcome the bottleneck and achieve nearly 100% population transfer from level 0 (far right) to level 3 (far left). For more complicated systems, there could be many pathways

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103

Fig. 44. Quasienergies =v; 0 (!) as functions of the laser frequency ! for a four-level system (v = 0; 1; 2; 3 of the CO Morse oscillator) at the laser intensity 50 GW=cm2 and the laser phase 6 = 0:0 (adapted from Ref. [21]).

Fig. 45. Time-dependent population Pv
→v (!(t)), here v = 0, for the four-level system (same as Fig. 44) as functions of 2 the laser frequency ! swept at the rate T = !˙ equal to 0:1F01 , where F01 is the Rabi frequency. The laser intensity is 2 'xed at 50 W=cm and the laser pulse phase 6 is 'xed at zero. The arrow, bottom right, indicates the direction of the frequency sweeping (adapted from Ref. [21]).

leading from an initial state to a desired 'nal excited state. A Floquet quasienergy diagram such as Fig. 44 is therefore very useful as a guide toward the choice of optimum pathway. In addition, Massey’s adiabatic criterion [229] for collision process can be extended here to establish a simple adiabaticity condition for the rate of change of the laser parameter X .

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12.3. A stationary formulation of time-dependent Hamiltonian systems In this section we describe a general stationary nonperturbative treatment of time-dependent Hamiltonian systems, driven by arbitrary pulse 'elds, by means of the two-mode Floquet theorem and a feature of initial value problem of the SchrGodinger equation [22]. The SchrGodinger equation i

9 |(t) = Hˆ (t)|(t) 9t

(12.11)

is an initial value problem because of the 'rst derivative of time, t, involved in the equation. The wavefunction, |(t1 ), at time, t1 , is uniquely determined by the following equation, |(t1 ) = Uˆ (t1 ; t0 )|(t0 ) :

(12.12)

Here |(t0 ) is the wavefunction at initial time, t0 , and Uˆ (t1 ; t0 ) is the time-evolution operator and can be written formally in the time-ordering form, 
Uˆ (t1 ; t0 ) = Tˆ exp −i

t0

t1

Hˆ (t) dt

 ;

(12.13)

where Tˆ is the time-ordering operator. The time-dependent Hamiltonian Hˆ (t) can be written as a sum of a time-independent part and time-dependent potential, i.e. Hˆ (t) = Hˆ 0 + Vˆ (t) :

(12.14)

From Eq. (12.13), it is obvious that Uˆ (t1 ; t0 ) is uniquely determined by Hˆ 0 , the time-independent part of the Hamiltonian, and Vˆ (t), the time dependent potential during the time t0 and t1 (t1 ¿ t0 ). Consequently, the time-dependent potential, Vˆ (t), before time t0 and after time t1 has no e9ect on the solution of the wavefunction |(t1 ) if the initial condition |(t0 ) is 'xed. This physical signi'cance of the initial value problem leads us to the following stationary approach of solving the time-dependent SchrGodinger equation, Eq. (12.11). The time-dependent potential, Vˆ (t), is assumed to start at time t0 and end at time t0 + . The wavefunction |(t0 ) at time t0 is assumed to be given either by preparation or by means of time-independent methods. Justi'ed by the feature of the initial value problem of the time-dependent SchrGodinger equation, we impose a quasiperiodic condition to the time-dependent potential Vˆ (t). By the quasiperiodicity, we mean that the envelope of the time-dependent potential, Vˆ (t), is reproduced with the period in the time domain while the other time variations of Vˆ (t) continue. In the case of pulsed laser 'elds, the envelope of Vˆ (t) is related to the pulse shape and the other time variations correspond to the oscillations with laser frequencies. The wavefunction, |(t), during t0 and t0 +

is independent to the arti'cial periodicity of Vˆ (t) provided that |(t0 ) is 'xed. Most importantly, this introduction of the quasiperiodicity of Vˆ (t) and Hˆ (t) suNces for one to cast the time-dependent SchrGodinger equation in a stationary form using the many-mode Floquet formalism (MMFT) [51,52].

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105

The MMFT [51,52] states that there are Floquet wavefunctions, |= (t), for a quasiperiodic Hamiltonian system with incommensurate frequencies, which are the solutions of  9 ˆ ˆ ˆ |= (t) = 0 ; (12.15) H|= (t) = H 0 + V (t) − i 9t and |= (t) can be written as |= (t) = exp(−iq= t)|q= (t) ;

(12.16)

where |q= (t) is quasiperiodic in time t. The time-evolution operator, Uˆ (t; t0 ), can be represented, instead of the time-ordering form, by the Floquet states, |= ,  Uˆ (t; t0 ) = |= (t)= (t0 )| : (12.17) =

Using Eq. (12.16) and the completeness relationship stationary form,  ˆ − t0 )]q= (t0 )| Uˆ (t; t0 ) = |q= (t)exp[ − iH(t

 =

|q= UTq= | = 1, Uˆ (t; t0 ) can be cast into a

=

 ˆ − t0 )]|t0  = t| (|q= UTq= |) exp[ − iH(t =

ˆ − t0 )]|t0  : = t| exp[ − iH(t

(12.18)

We should point out that the term (t − t0 ) in Eq. (12.18) commutes with the extended Hamiltonian ˆ Hˆ (t) − i9=9t), and we can rewrite Eq. (12.18) as H(= ˆ − t0 )]|t0 | =t ; Uˆ (t; t0 ) = t| exp[ − iH(

(12.19)

ˆ which is the projection of the super operator exp(−iH( −t 0 )) in the extended space onto the normal physical space at time t0 and t [22]. Physically, the time evolution operator between time t0 and t ˆ − t0 ))| =t from the supersurface at in the physical space is re>ected by the propagation exp(−iH(

t0 to the supersurface at t in the extended space, which bears resemblance to a physical propagation in the physical space for time-independent Hamiltonian. Similar expression for the time evolution operator Uˆ (t; t0 ) has been also obtained in a di9erent context [230,231] using Howland’s stationary scattering theory [232]. In particular, Peskin and Moiseyev [231] have advanced the (t; t  ) formalism for the solution of time-dependent SchrGodinger equation. By regarding the time as an extra coordinate in the extended Hilbert space, it avoids the needs to introduce the time ordering operator when the time-dependent SchrGodinger equation is integrated. Thus a time-independent expression for state-to-state transition probabilities can be derived by using the analytical time dependence of the time evolution operator in the extended Hilbert space. As a matter of fact, the derivation of the expression of the time-evolution operator, Eq. (12.19), is justi'ed for any time t, since t0 + can be arbitrary. In a practical calculation, however, an extended dimension means more work. To be eNcient and optimal, it is ideal to do the calculation with the extra dimension during the application of the time-dependent potential only. In addition, ˆ Hˆ (t) − i9=9t), is always an unbound it should be pointed out that the extended Hamiltonian, H(= Hamiltonian in the extended Hilbert space, and a quasiperiodic boundary condition therefore con'ne

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the calculation performed in the feature of L2 wavefunctions in a 'nite time domain. Eq. (12.19) can be solved expediently by the Faber polynomial expansion [233], a general polynomial expansion for Hermitian and non-Hermitian Hamiltonians. For an explicit treatment of the e9ects of laser pulses in multiphoton processes using the stationary formulation, we refer to Ref. [22] for details. 12.4. Adiabatic Floquet approach to multiphoton detachment of negative ions by intense laser pulses Recent advances in short-pulse high-power laser technology have generated considerable interest in the study of multiphoton above-threshold ionization (ATI) of atoms and above-threshold dissociation of molecules. Multiphoton dynamics in such short subpicosecond lasers are qualitatively di9erent from those in longer pulses which consist of nanoseconds and hundreds of picosecond of 'eld oscillations. From the theoretical point of view, it is a good approximation to treat the long-pulse 'elds as continuous waves. For short pulse 'elds, however, it is usually necessary to take into account the temporal distributions of the laser beam for a close comparison of the experimental and theoretical results. Unlike the earlier long pulse experimental observation [103], the measured ATI electron spectra for subpicosecond pulses show satellite lines to the main peaks [104]. These subpeak structures may be attributed to the resonances with intermediate states and/or to the electron interference between the rising and falling edges of the pulses. In this section we present an adiabatic Floquet approach for the treatment of the laser pulse shape e9ects on multiphoton ATI of atoms [234]. Firstly, we shall present a general integral equation formulation for the transition amplitude for multiphoton ATI processes driven by short pulsed laser 'elds. Secondly, we introduce a general adiabatic theory for the description of interference phenomena in the ATI spectra. Although the laser pulses under consideration are short enough, they can still contain tens to hundreds of cycles of radiation frequency. Under these conditions, one can extend an adiabatic approach for the description of interference phenomena in the electron spectra [235]. It is based on the concept of adiabatic quasienergy states and allows to predict the oscillatory behavior of the electron energy spectra. We shall show that analytical insights on the ATI subpeak structure can be obtained from this extended adiabatic Floquet approach. Finally we present an application the adiabatic Floquet approach to the study of multiphoton above-threshold detachment of H− in intense short pulsed laser 'elds. 12.4.1. General expressions for electron energy distributions in multiphoton above-threshold detachment Since the detachment process is under consideration, the distribution of the ejected electrons with respect to their momenta is of great importance. It is described by the following transition amplitude from the initial bound state i (r; t) to the 'nal state k (r; t) with the de'nite momentum k: Tik = k (r; t)|Uˆ (t; t0 )|i (r; t0 );

t→∞ :

(12.20)

Here Uˆ (t; t0 ) is the full time evolution operator taking into account the interaction with the atomic core as well as with the laser 'eld. Eq. (12.20) assumes that in the remote future (t → ∞) both the interactions are turned o9, so that k (r; t) is a plane wave. To make use of Eq. (12.20), note that |(r; t) = Uˆ (t; t0 )|i (r; t0 )

(12.21)

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is the exact wave function which describes the detachment process. It satis'es the time-dependent SchrGodinger equation for the electron bound in the atomic potential U (r) and subject to the in>uence of the pulsed laser 'elds:   9 1 i (r; t) = − ∇2 + U (r) + (F · r)f(t) cos !t (r; t) : (12.22) 9t 2 Here U (r) is the atomic potential, f(t) is the pulse envelope, ! and F are, respectively, the laser frequency and 'eld strength (linear polarization is assumed). Introducing
t
t  F dt and b = a dt  ; (12.23) a=− and after making several unitary transformations, Eq. (12.22) can be recast into the following integral form 

1 t 2   d 3 k exp[ − iE(k) · (t − t0 )] (r; t) = exp i(a · r) − i a (t ) dt 2 t0  

1 t 2   a (t ) dt ×k (r − b) d 3 r  k∗ (r  )i (r  ; t0 ) − i exp i(a · r) − i 2 t0


×

d3 r 

d 3 k k (r − b)

1 + i 2


t0

t


t

t0

dt  exp −iE(k) · (t − t  ) − i(a · r  )

a2 (t  ) dt  k∗ (r  − b )U (r  )(r  ; t  ) :

(12.24)

In deriving Eq. (12.24), use has been made of the spectral expansion of the 'eld-free evolution operator exp[ − iHˆ 0 (t − t  )]:
(12.25) exp[ − iHˆ 0 (t − t  )] = d 3 k exp[ − iE(k) · (t − t  )]|k k | ; where Hˆ 0 = − 12 ∇2 , and k (r) are the plane waves with the wave vector k: k (r) = (2 )−3=2 exp[i(k · r)] ;

(12.26)

normalized in the following way: k
|k  = (3) (k − k ) :

(12.27)

The energy E(k) depends on k as usual: E(k) = k2 =2 :

(12.28)

Note that in Eq. (12.25) the values a, b, are taken at the time t, and the values a , b at the time t  . Now we can use Eqs. (12.20) and (12.21) to calculate the transition amplitude. When calculating the transition amplitude, one has to take the limits t → ∞, t0 → −∞. In the remote past/future the laser 'eld vanishes (a; b → 0), and the transition amplitude Tik to the state k within the approximation

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described above is given by [234]


t

∞ 1 dt  exp iE(k)t  + i a2 (t  ) dt  Tik = −i(2 )−3=2 2 −∞ −∞
× d3 r  exp[ − i(a · r  ) + i(k · b ) − i(k · r  )]U (r  )(r  ; t  ) :

(12.29)

Eq. (12.25) gives the transition amplitude after the pulse is over, t → ∞. And the di9erential probability per unit energy and solid angle is shown to be dP(k) = k|Tik |2 ; dE dF √ where k = 2E and dF is the solid angle around the k direction.

(12.30)

12.4.2. Adiabatic approximation for smooth laser pulses To calculate the di9erential probability, one has to solve Eq. (12.23) for the function (r; t). If the pulse envelope f(t) is smooth enough, the wave function (r; t) originating from the initial state function exp(−iEi t)i (r) represents an adiabatic quasienergy state [235]:  
t   (r; t) = exp −i (t ) dt (12.31) m (r; t) exp(−im!t) ; m

where (t) and m (r; t) are the adiabatic quasienergy and Fourier components of the wave function de'ned for the laser 'eld peak value at the moment t. The adiabatic quasienergy (t) contains an imaginary part, so the decay during the pulse is taken into account. Approximation (12.29) requires a smooth laser pulse (with |(df=dt)|!) and assumes the quasienergy levels not to cross as the 'eld amplitude varies. In other words, the process should be nonresonant. If the resonance between two states does exist then the wave function (r; t) will contain the superposition of two adiabatic quasienergy states [235]. We consider here only the nonresonant process which is appropriate for negative ions such as H− which has only one bound state. For the smooth laser pulse the following approximate relations hold: a = −!−1 Ff(t) sin !t; b = −!−2 Ff(t) cos !t ;

t 1 t 2   −2 2 a (t ) dt = (2!) F f2 (t  ) dt  + (2!)−3 F 2 f2 (t) sin 2!t : 2 −∞ −∞

(12.32)

When relations (12.32) are used in Eq. (12.29), one arrives at the following expression for the transition amplitude Tik : 
t 
∞ −3=2 −2 2 Tik = −i(2 ) dt exp iE(k)t + i(2!) F f2 (t  ) dt 
− i

n t

−∞

−∞



(t  ) dt  − in!t An (kn ; t) ;

(12.33)

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where An (k; t) is the n-photon detachment amplitude for the monochromatic laser 'eld with the peak value Ff(t):

1   An (k; t) = dt exp(in!t ) d 3 r exp[i!−1 f(t)(r · F) sin !t  + i!−2 f(t)(k · F) cos !t 

0 − i(k · r) − i(2!)−3 F 2 f2 (t) sin 2!t  ]W (r) (r; t; t  ) ;

(12.34)

(notation stands for the period of the monochromatic 'eld: = 2 =!). The di9erential transition probability can now be expressed as follows [234]:  

t k  ∞ dP(k) −2 2 = dt exp iE(k)t + i(2!) F f2 (t  ) dt   dE dF (2 )3  −∞ −∞ n


−i

t

−∞





Y(t ) dt − i(E

(0)

2  + n!)t An (k; t) ; 

(12.35)

where E (0) is the unperturbed energy of the initial state and Y(t) is a complex value with the real part providing the ac Stark shift YES (t) and the imaginary part equal to minus halfwidth of the state (total detachment rate) (t)=2: (t) (t) = E (0) + Y(t); Y(t) = YES (t) − i : 2 Eq. (12.35) is the main result of the adiabatic Floquet approach to the laser pulse detachment. According to Eq. (12.35), the multiphoton detachment probability for the pulse can be expressed via the photodetachment amplitudes for the monochromatic laser 'elds. The more slow varies the envelope f(t), the more accurate is the approximation. The adiabatic Floquet approach has been applied to the investigation of the pulse shape e9ects on the multiphoton above-threshold detachment of H− by the radiation of CO2 laser (wavelength 10:6 m) used in Los Alamos experiment [107]. Both Gaussian pulse and a square pulse with smooth edge are considered. The angle-resolved and angle-integrated ATI electron energy distributions are analyzed. It is found that they contain oscillatory satellite structures to the main peaks due to the interference of the electrons detached on the rising and falling edges of the pulse [234]. Fig. 46 shows typical pattern of the angle-integrated electron energy distribution for the peak laser 'eld intensity 5 × 1010 W=cm2 . The laser pulse shape used is Gaussian, exp[ − (2 t=T )2 ], and the pulse width (parameter T ) is 5, 10, and 20 ps. 13. Generalized Floquet formulation of time-dependent density functional theory for many-electron quantum systems in intense laser 'elds The various generalized Floquet formalisms and quasienergy methods presented in the previous sections provide powerful nonperturbative theoretical frameworks and practical computational techniques for the exploration of multiphoton dynamics of quantum systems in strong 'elds. In actual computations, the systems that have been investigated so far are mainly one-or two-electron atomic or molecular systems, due to large dimensionality of Floquet matrix involved. Similar challenge exists in the time-dependent methods for direct numerical solution of time-dependent SchrGodinger equation in space and time.

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Fig. 46. Angle-integrated energy spectrum dP=dE for the peak intensity 5 × 1010 W=cm2 and several pulse widths T (Gaussian pulse). (a) T = 5 ps, (b) T = 10 ps, (c) T = 20 ps (adapted from Ref. [234]).

In the last several years, a new generation of nonperturbative theoretical and computational techniques has been initiated aiming for the investigation of the quantum dynamics of many-electron quantum systems in strong 'elds. These approaches are based on the extension and generalization of the time-dependent density functional theory (TDDFT) to strong 'elds. While much progress has been made in the steady-state DFT [236–241] since the fundamental works of Hohenberg and Kohn [242] and Kohn and Sham [243], the development of TDDFT is relatively recent [244,245] and is primarily limited to the treatment weak-'eld processes. The central theme of modern TDDFT [246] is a set of time-dependent Kohn–Sham (TDKS) equations which are structurally similar to the time-dependent Hartree–Fock equations but include in principle exactly all the many-body e9ects through a local time-dependent exchange-correlation potential. We 'rst note that the conventional

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weak-(eld TDDFT [244–251] (for the calculation of excitation energies and frequency-dependent polarizabilities) based on perturbation theory cannot be applied to strong 'eld processes. Additional theoretical and numerical advancements are required for the extension of TDDFT to the study of strong 'eld phenomena. The 'rst strong-(eld TDDFT approach involves the direct numerical solution of self-interactionfree time-dependent Kohn–Sham (TDKS) equation in space and time [25–27,31,32,252]. More accurate numerical techniques such as time-dependent generalized pseudospectral methods [27,28] have been recently developed for nonuniform spatial grid discretization (essential for accurate treatment of problems involving the Coulomb interactions) and high-precision numerical solution of TDKS equations for both (one-center) atomic [26,31] and (two-or multi-center) molecular [27,32] systems. In the second strong-(eld TDDFT approach, the generalized Floquet formulations of TDDFT have been developed [57,58,253,254], allowing exact transformation of the periodically (one-color) or quasiperiodically (multi-color) time-dependent Kohn–Sham equation into an equivalent time-independent in'nite-dimensional Floquet matrix eigenvalue problem. Moreover, for bound–free processes such as multiphoton ionization or dissociation, uniform [49,75] and exterior [77] complex scaling techniques have been implemented into the Floquet-TDDFT formulation, allowing analytical continuation of the Floquet Hamiltonian into higher Riemann sheets in the complex energy plane and direct determination of the complex quasienergy (dressed) states of many-electron systems by the solution of a non-Hermitian Floquet Hamiltonian eigenvalue problem. The time-independent Floquet-TDDFT formalism, when applicable, has several advantages over the time-dependent approaches. First, the Floquet approach is numerically more accurate and stable since it involves only the solution of a (Hermitian or non-Hermitian) time-independent Floquet matrix eigenvalue problem and there are no time propagation errors involved as in the time-dependent methods. Second, the Floquet-state or the dressed-state picture of many-electron systems is conceptually appealing and provides useful physical insights regarding multiphoton dynamics in terms of the avoided crossing pattern of (time-independent) quasienergy levels. Third, for near-resonant or resonant multiphoton processes, nearly-degenerate (high-order) perturbation methods can be applied to the Floquet Hamiltonian, leading to analytical expressions valuable for both theoretical and experimental investigation of multiphoton and high-order nonlinear optical phenomena. In this section, we shall focus our discussion on the recent developments and applications of generalized Floquet formulations of TDDFT and time-dependent current density functional theory (TDCDFT). The Floquet-TDDFT (TDCDFT) formalism allows the extension of various Floquet wavefunction approaches discussed in previous sections to many-electron quantum systems, providing powerful new theoretical frameworks for the exploration of a broad range of strong-'eld phenomena in more complex atomic, molecular, cluster, and condensed matter systems in the future. 13.1. Generalized Floquet formulation of time-dependent density functional theory 13.1.1. Periodic (elds Consider the quasienergy eigenvalue equation (2.8) where (r; t) is a wave function of a manyelectron system (notation r stands for all coordinates of the system under consideration). We can de'ne a quasienergy functional (see Section 2): ˆ U : F = T|H|

(13.1)

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Variation of the functional, Eq. (13.1), under the normalization condition T|U = 1

(13.2)

leads to Eq. (2.8) as an equation for the function (r; t) which brings the stationary point to functional (13.1). Eq. (2.8) resembles the conventional eigenvalue problem, but unlike the unperturbed Hamiltoˆ in the composite Hilbert space S: the nian Hˆ 0 , there is no ground state for the Hamiltonian H quasienergies span the whole range [ − ∞; ∞] due to property (2.9). That is why the traditional Hohenberg–Kohn theorem [242] is not applicable to this case. However, there exist extensions of the density functional theory beyond the Hohenberg–Kohn theorem [255]. The theory of Ref. [255] treats all the states of the system on the same footing and represents a rigorous basis for the analysis of ˆ Hˆ (t), excited states within the density functional theory. Thus it can be rigorously justi'ed that H; (r; t) and the quasienergy  are all unique functionals of the electron density (spin-densities in the spin-polarized theory) provided a particular Floquet eigenstate is selected. In what follows we shall consider the speci'c Floquet state (r; t), Eq. (2.6), which evolves from the ground state of the unperturbed system upon adiabatic switch on the external 'eld. Taking into account that discussed above, the quasienergy functional in Eq. (13.1) can be expressed as a functional of the density. One can now extend the Kohn–Sham formalism [243] and introduce the Kohn–Sham spin-orbitals 6)k (r; t) (the notation ) stands for both possible spin functions; we shall use the notations  and  when we need to distinguish di9erent spin projections). As (r; t) in Eq. (13.1) is periodic in time, the Kohn–Sham spin-orbitals are also periodic in time. The spatial parts of the spin-orbitals are orthonormal to each other according to 6)k (r; t)|6)i (r; t) = ki : The total electron density D(r; t) can be written as a sum of spin-densities D) (r; t):  D(r; t) = D) (r; t) ;

(13.3)

(13.4)

)

D) (r; t) =



|6)k (r; t)|2 :

(13.5)

k

The summation with respect to k in Eq. (13.5) is performed over all occupied spatial orbitals; for the closed-shell atoms the number of spatial orbitals is equal to N=2, N being the total number of electrons. The quasienergy functional (13.1) can be rewritten in the following form:
1

F= dt[Ts (t) + J (t) + U (t) + V (t) + Ds (t) + Exc (t)] ; (13.6)

0 where = 2 =! is the period. The time-dependent quantities under integral (13.6) are de'ned as follows:  1 Ts (t) = 6)k (r; t)| − ∇2 |6)k (r; t) ; (13.7) 2 k;)

1 D(r; t)D(r  ; t) 3 d r d3 r  ; (13.8) J (t) = 2 |r − r  |

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113




d 3 r D(r; t)u(r) ; 
V (t) = d 3 r D) (r; t)v) (r; t) ; U (t) =

)

Ds (t) =



6)k (r; t)| − i

k;)

9 ) |6 (r; t) : 9t k

(13.9) (13.10) (13.11)

Here Ts (t) is a noninteracting kinetic energy, J (t) is a classical electron–electron repulsion (Hartree) energy, U (t) is an expectation value of a single-particle potential u(r) (interaction with the nucleus), V (t) is the expectation value of the external electromagnetic 'eld, Exc (t) is the exchange-correlation energy. The latter contains the di9erence between the exact kinetic energy and Ts (t) as well as nonclassical part of electron–electron interaction. For linearly polarized monochromatic laser 'elds, within the electric dipole approximation, the external 'eld potential v) (r; t) has the following spin-independent form: v (r; t) = v (r; t) = (F · r) cos !t ;

(13.12)

F being the electric 'eld strength. The Kohn–Sham equations for the time periodical orbitals 6)k (r; t) can be obtained from the stationary principle for the quasienergy functional (13.6) under the constraints of Eq. (13.3). Using the functional di9erentiation of Eqs. (13.3) and (13.6) with respect to the orbitals 6)k (r; t), one arrives at the following set of time-dependent Kohn–Sham (TDKS) equations:   9 ) 1 2 ) ) ) i 6k (r; t) = − ∇ + u(r) + v (r; t) + vs (r; t) − k 6)k (r; t) : (13.13) 9t 2 The single-particle potential vs) (r; t) includes the classical electron–electron repulsion (the Hartree potential) as well as the exchange-correlation interaction: ) vs) (r; t) = vH (r; t) + vxc (r; t) ;
Exc D(r  ; t) ) ; vxc vH (r; t) = d 3 r  (r; t) = :  |r − r | D)

(13.14) (13.15)

The Lagrange multipliers k) play the role of orbital quasienergies. The set of Eqs. (13.23) is solved self-consistently producing the Kohn–Sham orbitals 6)k (r; t) and orbital quasienergies k) . Then the total quasienergy  of the N -electron system can be determined according to Eq. (13.6) as a stationary point of the quasienergy functional:



 1 d 3 r vs) (r; t)D) (r; t) : k) + dt J (t) + Exc (t) − (13.16) =

0 ) k;)

Expressions (13.13) and (13.16) are the main equations of the Floquet formulation of TDDFT [57,58]. One can expand the periodic functions 6)k (r; t), D) (r; t) and vs) (r; t) in the Fourier series: ∞  exp(−im!t)6)km (r) ; (13.17) 6)k (r; t) = m=−∞

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D) (r; t) =

m=−∞

vs) (r; t)

∞ 

=

m=−∞

exp(−im!t)D)m (r) ;

(13.18)

exp(−im!t)(vs) )m (r) :

(13.19)

Here the Fourier components D)m (r) are related to the Fourier components of the Kohn–Sham orbitals as follows: ∞   6)k; m+n (r)(6)kn (r))∗ : (13.20) D)m (r) = k

n=−∞

Substituting Eqs. (13.17)–(13.19) into Eq. (13.13) results in the following set of in'nite-dimensional time-independent coupled equations (form (13.12) is used for the external 'eld potential) [57,58]:   1 2 1 − ∇ + u(r) − m! 6)km (r) + (F · r)[6)k; m−1 (r) + 6)k; m+1 (r)] 2 2 +

∞  n=−∞

(vs) )m−n (r)6)kn (r) = k) 6)km (r) :

(13.21)

Eqs. (13.21) are the working equations of the Floquet-TDDFT formalism for periodic 'elds. Their solution is signi'cantly facilitated compared with that of the time-dependent equations, Eq. (13.13). They can also be rewritten in the form of time-independent Floquet matrix eigenvalue problem: ˜) ˜ ) = ) 6 Hˆ )F (r)6 k k k ;

(13.22)

˜ ) is the vector consisting of the where Hˆ )F is the Floquet Hamiltonian de'ned by Eq. (13.21) and 6 k ˜ ) are to be solved components 6)km (r). The orbital quasienergy eigenvalues k) and eigenfunctions 6 k self-consistently. 13.1.2. Multi-color or quasiperiodic (elds For the general case of many-electron systems in polychromatic or multi-color laser 'elds with incommensurate frequencies, the Hamiltonian Hˆ (t) is not a periodic function of time and the conventional Floquet theory is not applicable. In this case, we can extend the many-mode Floquet theorem [51,52,152,153] which allows the transformation of the multi-color or quasiperiodically time-dependent Kohn–Sham equations into an equivalent time-independent many-mode Floquet matrix eigenvalue problem. We refer readers to Ref. [253] for detailed presentation of the many-mode Floquet-TDDFT formalism. 13.2. Generalized Floquet formulation of time-dependent current-density-functional theory For problems where the magnetic properties are of interest, the conventional DFT is not suNcient, even in the time-independent case, and it is necessary to use the current density functional theory (CDFT) [256–258]. In this theory, the electron current density is included as an additional variable and the energy is minimized with respect to variations in the paramagnetic current as well as in the density. To study the more interesting dynamical properties, one needs a time-dependent CDFT

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(TDCDFT). Several attempts have been made recently to develop TDCDFT [250,259,260]. The central result of TDCDFT is a set of TDKS equations which follow from the principle that the action is stationary with respect to variations in paramagnetic current density as well as the density itself. These equations include in principle exactly all many-body e9ects through a local time-dependent exchange-correlation potential. These earlier TDCDFT treatments, however, have been limited to the study of weak-'eld processes only. In this section, we present the Floquet formulation of TDCDFT for nonperturbative treatment of multiphoton processes in the presence of intense electric and magnetic 'elds [58]. This is an extension of the Floquet formulation of TDDFT (Section 13.1), and is particularly relevant to processes where the magnetic 'eld plays a signi'cant role. Examples are multiphoton processes in the presence of both laser and static magnetic 'elds, processes involving open shell atoms or molecules, and multiphoton processes in the presence of superintense laser 'elds (where both time-dependent electric and magnetic 'elds make important contributions), etc., to mention only a few. We show that the time-dependent problems in TDCDFT can be exactly transformed into an equivalent time-independent Floquet Hamiltonian matrix eigenvalue problems [58]. The time-periodic Hamiltonian operator Hˆ (r; t) in Eq. (2.3) is now the Pauli Hamiltonian taking into account the external 'eld coupling to the electron spins: 2  N  N 1 1 ˆ H (r; t) = −i∇ + A(rj ; t) − ’(ri ; t) 2 j=1 c i=1 N

N

 1 1 1 : (B(ri ; t) · sˆi ) + u(ri ) + + c i=1 2 |ri − rj | i=1

(13.23)

i=j

In Eq. (13.23), c is the velocity of light, the vector operator sˆi is the spin operator of the ith electron; the potential u(ri ) describes the Coulomb interaction of the ith electron with the nucleus; ’(r; t) and A(r; t) are the scalar and vector potentials of the external 'eld related to the electric and magnetic 'eld strengths E and B: E(r; t) = −∇’(r; t) − B(r; t) = ∇ × A(r; t) :

1 9A(r; t) ; c 9t (13.24)

The electron spin-densities D) (r; t) and the total density D(r; t) are calculated according to Eqs. (13.4) and (13.5), and the paramagnetic current spin-densities jp) (r; t) are de'ned as 1  [(6)k (r; t))∗ ∇6)k (r; t) − 6)k (r; t)(∇6)k (r; t))∗ ] : (13.25) jp) = 2i k

The summation with respect to k in Eq. (13.25) is performed over all occupied spatial orbitals. The quasienergy functional can be recast in the familiar form of Eq. (13.6), however the exchange-correlation energy within CDFT and TDCDFT is a functional of the spin-densities as well as the current spin-densities (see below). Regarding the contribution of the external electromagnetic 'eld, we shall assume that the magnetic 'eld has a constant direction in space (along the quantization axis for the spins). In this case one can account for the spin coupling to the external

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magnetic 'eld through the spin-dependent scalar potential [256,257]. Introducing the spin-dependent scalar potential v) (r; t) such as 1 1  B(r; t) ; (13.26) [v (r; t) − v (r; t)] = 2c 2 1  (13.27) [v (r; t) + v (r; t)] = −’(r; t) ; 2 we can express V (t) (see Eq. (13.10)) as follows: 
1 d 3 r[v) (r; t) + 2 A2 (r; t)]D) (r; t) V (t) = 2c ) −i

1  ) 6k (r; t)|(A · ∇) + (∇ · A)|6)k (r; t) : 2c

(13.28)

k;)

A more general theory which allows magnetic 'elds with variable directions is required to employ spin-density and spin-current matrices [256,257]. The Kohn–Sham equations for the time-periodic orbitals 6)k (r; t) are obtained from the stationary principle for the quasienergy functional (13.16) with respect to variation of the spin-densities and current spin-densities:  1 1 1 9 ) [(A · ∇) + (∇ · A)] i 6k (r; t) = − ∇2 + u(r) + v) (r; t) + vs) (r; t) + 2 A2 (r; t) − i 9t 2 2c 2c  1 ) ) ) − i [(Axc · ∇) + (∇ · Axc )] − k 6)k (r; t) ; (13.29) 2c where the exchange-correlation vector potential A)xc (r; t) is de'ned as a functional derivative of the exchange-correlation functional Exc with respect to the paramagnetic current jp) (that means, the projections of A)xc (r; t) are the functional derivatives of Exc with respect to the corresponding projections of jp) ): Exc [D ; D ; jp ; jp ] 1 ) Axc (r; t) = : (13.30) jp) c The single-particle potential vs) (r; t) has the same de'nition as in Section 13.1 (see Eqs. (13.14), (13.15)). The total quasienergy  of the N -electron system now can be calculated as
T 
 1 ) d 3 r vs) (r; t)D) (r; t) k + dt J (t) + Exc (t) − = T 0 ) k;)

1  ) ) ) ) 6k (r; t)|(Axc · ∇) + (∇ · Axc )|6k (r; t) : + 2c

(13.31)

k;)

Since the paramagnetic currents jp) are gauge-dependent (the physical current j ) = jp) + 1c D) A is gauge-invariant), it may seem that the exchange-correlation functional also depends on the gauge. However, it can be shown [256,257] that the functional actually depends on the vorticity jp) ) = ∇ × ) (13.32) D

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which is gauge-invariant. Speci'c forms of this functional are available within the adiabatic local density approximation (ALDA) [258]. Like the Floquet-TDDFT formalism (Section 13.1), one can recast the set of TDKS equations (13.29) into a set of time-independent Floquet matrix equations by means of the Fourier expansion of the time-periodic functions 6)k (r; t), D) (r; t), v) (r; t), A(r; t), A)xc (r; t), and vs) (r; t). Detailed expressions can be found in Ref. [58]. 13.3. Non-Hermitian Floquet formulation of TDDFT and TDCDFT In the presence of intense external electromagnetic 'elds, atoms (molecules) can be ionized (dissociated) by the absorption of multiple photons, and all the bound states become shifted and broadened metastable resonance states possessing complex quasienergies  = r − i=2. The real parts of the complex quasienergies, r , provide the ac Stark shifted energy levels, while  are equal to the total ionization (dissociation) rates of the corresponding atomic (molecular) states. To determine these complex quasienergy states of many-electron systems, the non-Hermitian Floquet Hamiltonian formalisms discussed in Sections 6 and 7 can be extended to TDDFT and TDCDFT as described below. The use of the complex scaling transformation, r → r exp(i), allows the analytical continuation of the Hermitian Floquet Hamiltonian Hˆ )F (r), Eq. (13.22), into a non-Hermitian Floquet Hamiltonian Hˆ )F (r exp(i)), reducing the problem of the determination of the complex quasienergy ˜ ) (r exp(i)) to the solution of a non-Hermitian matrix eigenvalue eigenvalues k) and eigenvectors 6 k problem. In the non-Hermitian Floquet formulation of TDDFT and TDCDFT, all the quantities given in the quasienergy functional, Eq. (13.16), as well as the spin-densities and current spin-densities themselves become complex quantities. A delicate task is to perform the analytical continuation of the exchange-correlation scalar and vector potentials in (13.13) and (13.29), which depend on the spin-densities D) and paramagnetic current spin-densities j)p , to the complex plane. We introduce the following de'nition of the complex spin-density [58]:  D) (r; t) = (6)k (r ∗ ; t))∗ 6)k (r; t) : (13.33) k

Eq. (13.33) represents explicitly analytically continuable quantity (the notation r∗ stands for the vector with the complex-conjugated radial coordinate r), and for real r reduces exactly to the conventional de'nition (13.5). That means, the spin-density de'ned by Eq. (13.33) is always real and nonnegative on the real r axis. In the same manner, we can de'ne the complex current spin-density: 1  [(6)k (r ∗ ; t))∗ ∇6)k (r; t) − 6)k (r; t)∇(6)k (r ∗ ; t))∗ ] : (13.34) jp) (r; t) = 2i k

For real r, Eq. (13.34) reduces to the conventional form Eq. (13.25). Using Eqs. (13.33) and (13.34) for analytical continuation of the spin-densities and current spin-densities, one can also analytically continue the vorticities ) in Eq. (13.32) to the complex plane of the radial coordinate. The analytical continuation of the potentials in Eqs. (13.13) and (13.29) is as follows. The potentials u(r), v) (r; t), and A(r; t) are explicit functions of r; their calculation for complex r is straightforward. For the potentials vs) (r; t) and A)xc (r; t), which are functionals of D) and ) , Eqs. (13.33), (13.34) and (13.32) can be applied to obtain those quantities for complex r.

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Now turn to the calculation of the ionization or multiphoton ionization rates. There is a well-known relation between the total ionization rate  and the imaginary part of the total quasienergy :  = −2 Im  :

(13.35)

The total quasienergy can be calculated according to Eq. (13.31). All the terms in the right-hand side of (13.31) are real except the sum of the orbital quasienergies k) . This is because the spin-densities and current spin-densities are real on the real axis of the radial coordinate r. Thus one arrives at the following result:  Im  = Im k) : (13.36) k;)

In addition to the total (multiphoton) ionization (dissociation) rates of atoms (molecules) in external electromagnetic 'elds, it is also important to determine the partial ionization (dissociation) rate from each individual electronic orbital [58]. It can be shown by means of the equation of continuity that the imaginary parts of the spin-orbital quasienergies j)k have the usual physical meaning, namely: k) = −2 Im j)k ;

(13.37)

where k) is the ionization rate from the particular Kohn–Sham spin-orbital with the indexes k and ). Summing Eq. (13.37) over all occupied spin-orbitals and taking into account Eqs. (13.31) and (13.36) one obtains:  = k) : (13.38) k;)

Thus the total ionization rate can be expressed as a sum of spin-orbital ionization rates [58], and the individual spin-orbital ionization rates can be determined by means of the non-Hermitian Floquet formulation of TDDFT or TDCDFT. 13.4. Exact relations of quasienergy functional in the Floquet formulation of TDDFT In this section, we present several exact relations regarding the quasienergy functional, Eq. (13.3), in the Floquet formulation of TDDFT. The relations involving the exchange-correlation energy and potential are of the primary importance since the exact time-dependent exchange-correlation energy functional is unknown and largely unexplored. The exact relations presented below can serve as additional constraints in search for better time-dependent exchange-correlation functionals in the future. Exact relations and theorems for the time-dependent quantities described above may be established in the framework of the Floquet formulation of TDDFT [254], like they hold in the general TDDFT [261]. Since the quasienergy in the Floquet formulation of TDDFT is a time-independent quantity, some additional constraints do exist which are not available in the general TDDFT. 13.4.1. Time derivatives of kinetic, potential, and exchange-correlation energies Consider TDKS for the time-periodic spin-orbital 6)k (r; t), Eq. (13.13). In general, the following normalization condition holds for the Kohn–Sham spin-orbitals: 6)k (r; t) | 6)k (r; t) = 1 :

(13.39)

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Multiplying Eq. (13.13) by [6)k (r; t)]∗ and taking the integral with respect to the coordinate r, one obtains: 1 9 6)k | − ∇2 + u(r) + vs) (r; t) + v) (r; t) | 6)k  + 6)k | − i | 6)k  = j)k : (13.40) 2 9t Performing summation of Eq. (13.40) over all spin-orbitals results in the following relation which must be satis'ed at arbitrary time moment t: 
 ) Ts (t) + 2J (t) + U (t) + Vext (t) + Ds (t) + d 3 r D) (r; t)vxc (r; t) = j)k : (13.41) )

k;)

Di9erentiating Eq. (13.41) with respect to time, one obtains: 
d 9D) (r; t) ) d Ts (t) + Ds (t) + [vs (r; t) + v) (r; t) + u(r)] d3 r dt dt 9t )   )
 9vs (r; t) 9v) (r; t) 3 ) + =0 : d r D (r; t) + 9t 9t )

(13.42)

On the other hand, di9erentiating Eq. (13.40) with respect to time and performing summation over all spin-orbitals, one arrives at the following expression:   ) 
d 9vs (r; t) 9v) (r; t) 3 ) Ds (t) + + =0 : (13.43) d r D (r; t) dt 9t 9t ) Substituting Eq. (13.43) into Eq. (13.42), we obtain the following expression: 
d 9D) (r; t) d3 r Ts (t) = − [u(r) + vs) (r; t) + v) (r; t)] : dt 9t )

(13.44)

Eqs. (13.43) and (13.44) are exact relations which express the dependence of the single-particle kinetic energy and time-derivative operator expectation values on time through the density and potentials, including the exchange-correlation potential. Now we consider the most important relations involving the time-dependent exchange-correlation energy and potential. Taking into account the de'nition of the quasienergy functional Eq. (13.1) and that of the exchange-correlation energy, the relation Eq. (13.41) can be re-written in the following form:  
) Exc (t) =  − d 3 r D) (r; t)vxc j)k + J (t) + (r; t) : (13.45) k;)

)

The exchange-correlation energy Exc (t) itself can be expressed through the expectation value of the exchange-correlation potential, Hartree energy J (t), total quasienergy  and Kohn–Sham spin-orbital quasienergies j)k . Di9erentiating Eq. (13.45) with respect to time, one obtains the equation which does not contain quasienergies and relates directly the time dependence of the exchange-correlation energy to that of the exchange-correlation potential expectation value and Hartree energy:
d d d ) d 3 r D) (r; t)vxc (r; t) + J (t) : Exc (t) = (13.46) dt dt ) dt Eqs. (13.45) and (13.46) are the most important results of this analysis. Eq. (13.46) can serve as a constraint in search for approximate time-dependent exchange-correlation functionals since it

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establishes a relation between the time derivatives of the exchange-correlation energy Exc (t) and the ) exchange-correlation potential vxc (r; t). 13.4.2. Virial theorem The virial theorem in the Floquet formulation of TDDFT can be obtained by a generalization of that in the traditional quantum mechanics. It follows from TDKS Eq. (13.13) that the expectation value of the operator i A = − [(r · ∇) + (∇ · r)] 2 satis'es the following equation:

(13.47)

d ) 1 6k |A|6)k  = 26)k | − ∇2 |6)k  − 6)k |(r · ∇u(r))|6)k  dt 2 − 6)k |(r · ∇v) (r; t))|6)k  − 6)k |(r · ∇vs) (r; t))|6)k  :

(13.48)

Taking into account the relation d ) 1 d2 ) 2 ) 6k |A|6)k  = 6 |r |6k  ; (13.49) dt 2 dt 2 k and making use of the homogeneity properties of the Hartree potential and performing summation over all spin-orbitals, one arrives at the following relation:  1 d2  ) 2 ) 6k |r |6k  = 2Ts (t) − 6)k |(r · ∇u(r))|6)k  + J (t) 2 2 dt k;)

−



k;)

) 6)k |(r · ∇vxc (r; t))|6)k  −

k;)



6)k |(r · ∇v) (r; t))|6)k  :

(13.50)

k;)

Like the Kohn–Sham system of noninteracting particles, an analogous derivation can be performed for the original system with interacting particles. Since the spin-densities for the both systems are the same, the following relation can be obtained: 
) d 3 r D) (r; t)(r · ∇vxc Exc (t) + Tc (t) − Dc (t) = − (r; t)) ; (13.51) )

where the correlation kinetic energy Tc (t) and correlation time derivative Dc (t) are de'ned as differences between the corresponding quantities of the original interacting system (T (t) and D(t)) and the Kohn–Sham noninteracting system (Ts (t) and Ds (t)): Tc (t) = T (t) − Ts (t) ;

(13.52)

Dc (t) = D(t) − Ds (t) :

(13.53)

Eq. (13.51) is the main result from the virial theorem analysis. It may serve as an additional constraint when searching for approximate forms of the time-dependent exchange-correlation energy functional and potential in the future formalisms.

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Fig. 47. Photoionization cross sections for the ground state of He atom. Solid line: calculated results [57]; dashed line, experimental data [262].

13.5. Applications of Floquet-TDDFT formalism to multiphoton ionization problems To demonstrate the procedure and usefulness of the Floquet–TDDFT formalism, we present in this section several initial case studies regarding multiphoton ionization (MPI) of neutral atoms and multiphoton detachment of negative ions in intense laser 'elds. Much remains to be explored in the extension and application of the Floquet-TDDFT formalism to strong-'eld atomic and molecular physics in the future. 13.5.1. Multiphoton ionization of He First we consider MPI of the ground state of He atoms. To compare with the experimental results [262], we have 'rst computed [57] weak-'eld one-photon ionization rates in the photon energy range of 25 –50 eV. Fig. 47 shows the comparison of the Floquet-TDDFT results with the experimental data. The deviation from the experimental curve is well within 5% which is surprisingly good, for the calculations [57] use the Hartree–Fock quasienergy functional, and have not yet taken into account the correlation energy functional. Besides the one-photon cross sections, we present here the calculations of MPI rates of He in two-color laser 'elds with the fundamental wavelength 248 nm and its third harmonic [253]. We applied the Hartree–Fock quasienergy functional (Exc = − 12 J [D]) which exactly eliminates the self-interaction. The Kohn–Sham orbital function 61 and quasienergy j1 (the same for both spin projections) were obtained by solving the non-Hermitian Floquet matrix eigenvalue problem Eq. (13.22), and the total complex quasienergy  was calculated according to Eq. (13.16). Table 6 shows the dependence of the total complex quasienergies on the intensity of the harmonic 'eld and the relative phase  between the fundamental and its third harmonic 'elds. Several novel nonlinear features in two-color 'elds were observed. First, MPI rates for the relative phase  = 0 case are always larger than those for  = case. Second, by mixing the strong fundamental 'eld with a weaker third harmonic 'eld, the rates can be signi'cantly enhanced. For example, for the case of harmonic 'eld intensity 1 × 1013 W=cm2 , there is an enhancement by nearly two orders of magnitude in the multiphoton ionization rates in the two-color 'elds. The only exception is the weakest

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Table 6 Total complex quasienergies for the multiphoton ionization of the ground state of the helium atom by the fundamental 'eld (248 nm) with the intensity IL = 1014 W=cm2 and its third harmonic 'eld with several di9erent intensities IH IH (W=cm2 )

Total quasienergies j (a.u.) =0

0 1 × 1013 1 × 1012 1 × 1010

= −8

−2:86266–i7:482 × 10 −2:86283–i5:628 × 10−6 −2:86268–i8:332 × 10−7 −2:86266–i1:112 × 10−7

−2:86266–i7:482 × 10−8 −2:86282–i3:356 × 10−6 −2:86268–i1:816 × 10−7 −2:86266–i4:703 × 10−8

The total multiphoton ionization rates are equal to 2|Im(j)|.  is the relative phase of the two laser 'elds (adapted from Ref. [253]).

harmonic 'eld (1 × 1010 W=cm2 ) case: due to the destructive interference of di9erent pathways to the continuum when  = , the MPI rate is smaller than that for the fundamental 'eld alone at the same intensity. Finally, we note that the ac Stark shift of the ground state of the He atom increases as the harmonic 'eld intensity increases. 13.5.2. Multiphoton detachment of Li− As another example of recent application of the Floquet-TDDFT formalism, we present here some recent results of the study of the multiphoton detachment of Li− [92]. We make use of the (spin-polarized) Becke exchange [263,264] and Lee–Yang–Parr correlation [265] functionals (BLYP exchange-correlation). For the self-interaction correction, we extend the Krieger–Li–Iafrate (KLI) procedure [266,267] with the implementation of an explicit self-interaction-correction (SIC) term of Tong and Chu [31]. The combination of BLYP exchange-correlation potential and KLI/SIC self-interaction correction (BLYP–KLI/SIC) has proved its accuracy in recent extensive atomic structure calculations across the periodic table [270,271]. The electron aNnity of Li calculated by this procedure is 0.02294 a.u., in good agreement with the experimental value of 0.02271 a.u. [272]. In Fig. 48 we show the results for the one-photon detachment cross section obtained from the weak-'eld calculations. Also shown here are the results of multichannel R-matrix calculation [268] and experimental data [269] for comparison. It is seen that the Floquet–TDDFT results are in fair agreement with the more sophisticated multichannel calculations [268] and with the experiment [269]. In Table 7 we present the partial and total multiphoton above-threshold detachment rates of Li− by the linearly polarized infrared laser 'eld [92]. The laser frequency range corresponds to the two-photon dominant process. Also shown are the real parts of the 2s orbital quasienergy, and the ac Stark shift can be readily obtained. The calculations were performed for the laser 'eld intensities 1×109 , 1×1010 , and 1×1011 W=cm2 . For the highest intensity used, 1×1011 W=cm2 , the two-photon detachment channel is closed at the frequencies 0.012 and 0:014 a:u: because of large ac Stark and ponderomotive shifts. One can see that for higher intensities, 1 × 1010 and 1 × 1011 W=cm2 , the above-threshold detachment makes a signi'cant contribution to the total rate. We have also studied the electron angular distributions for the two-photon detachment of Li− . The results are shown in Fig. 49 for the laser 'eld intensity 1 × 1010 W=cm2 and several frequencies. Detailed analysis of the electron angular distributions reveals dramatic interference of s- and

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Fig. 48. Cross section of one-photon detachment of Li− . Full curve, Floquet–TDDFT calculation [92]; dashed curve, multichannel R-matrix calculation [268]; diamonds, experiment [269] (adapted from Ref. [92]).

Table 7 Partial and total rates for the multiphoton above-threshold detachment of Li− Laser 'eld frequency (a.u.)

Partial rates (a.u.) Number of photons absorbed 2 3

Total rates (a.u.)

Re j (a.u.)

7.94(−8) 9.67(−8) 8.68(−8) 6.17(−8) 3.80(−8)

−2.296132(−2) −2.296269(−2) −2.296554(−2) −2.297061(−2) −2.298074(−2)

4

Laser (eld intensity 1 × 109 W=cm2 0.012 7.81(−8) 0.014 9.62(−8) 0.016 8.66(−8) 0.018 6.17(−8) 0.020 3.80(−8)

1.34(−09) 5.54(−10) 2.12(−10) 7.26(−11) 2.41(−11)

Laser (eld intensity 1 × 1010 W=cm2 0.012 4.88(−6) 0.014 8.16(−6) 0.016 7.63(−6) 0.018 5.75(−6) 0.020 3.72(−6)

1.59(−6) 4.29(−7) 1.84(−7) 6.22(−8) 2.01(−8)

8.77(−8) 1.37(−8) 2.80(−9) 8.14(−10) 3.34(−10)

6.56(−6) 8.60(−6) 7.82(−6) 5.81(−6) 3.74(−6)

−2.341102(−2) −2.321540(−2) −2.320871(−2) −2.323095(−2) −2.329926(−2)

Laser (eld intensity 1 × 1011 W=cm2 0.012 0.014 0.016 1.60(−4) 0.018 2.82(−4) 0.020 2.81(−4)

1.42(−4) 7.59(−5) 9.17(−5) 5.26(−5) 2.27(−5)

4.13(−5) 2.97(−5) 1.76(−5) 6.94(−6) 2.67(−6)

1.99(−4) 1.12(−4) 2.73(−4) 3.42(−4) 3.06(−4)

−2.600547(−2) −2.658997(−2) −2.768378(−2) −2.690298(−2) −2.649207(−2)

The numbers in parentheses indicate the powers of 10 (adapted from Ref. [92]).

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Fig. 49. Angular distributions for 2-photon detachment of Li− . The laser 'eld intensity is 1 × 1010 W=cm2 . The laser 'eld frequency is: (a) 0.012 a.u., (b) 0.014 a.u., (c) 0.016 a.u., and (d) 0.020 a.u. (adapted from Ref. [92]).

d-waves in the detachment amplitude [92]. For higher frequencies (0:016 and 0:020 a.u.), the d-wave dominates the amplitude, and the angular distributions show the strongly anisotropic pattern as in Fig. 49(c) or (d), with the maximum pointing at the 'eld direction. For smaller frequencies (0:012 and 0:014 a.u.), the quasienergy level of the ground state is brought closer to the threshold, and the relative weight of the s-wave increases, in accordance with the Wigner threshold law [273]. For the frequency 0:012 a.u. the electron angular distribution is nearly isotropic (Fig. 49(a)). 14. Conclusion In this article, we have reviewed the recent advancement of generalized Floquet formalisms and the associated computational methods and their applications to the study of a broad range of intense-'eld multiphoton and nonlinear optical processes of current interest as listed in Fig. 1. The advantages of generalized Floquet methods may be summarized as follows: (i) They provide a transparent and insightful physical picture for intensity- and frequency-dependent multiphoton and nonlinear optical phenomena in terms of the avoided crossings of a few (real or complex) quasienergy (or dressed) states. (ii) They take into account self-consistently all the intermediate-level shifts and broadenings and multiple coupled continua. (iii) In the case of bound-free or free-free MPI/ATI/MPD/ATD transitions, only L2 basis functions or generalized pseudospectral spatial discretizations are required, and no asymptotic boundary conditions need to be enforced. (iv) The generalized Floquet procedures are computationally more accurate and eNcient than the alternative time-dependent methods, since the former involves only the solution of a (Hermitian or non-Hermitian) matrix or supermatrix eigenvalue problem. The time-dependent methods, on the other hand, are subject to short-time approximation and time propagation errors, typically,

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O((Yt)3 ). Moreover, for resonant or nearly resonant multiphoton processes, one can extend the nearly degenerate perturbation techniques to the analysis of the Floquet Hamiltonian (or Floquet–Liouville Supermatrix). This allows the reduction of the in'nite-dimensional Floquet matrix (or supermatrix) problem into an N by N (or N 2 by N 2 ) e9ective Hamiltonian (for N -level systems), from which insightful analytical results of multiphoton transitions or nonlinear optical properties can be obtained. (v) The generalized Floquet approaches are nonperturbative in nature, applicable to arbitrary 'eld strengths. They provide general and practical procedures for comprehensive investigation of single- and multi-photon, resonant and nonresonant, steady-state and time-dependent phenomena in a uni'ed fashion, well beyond the conventional high-order perturbation theory and rotating wave approximation. Much remains to be explored in strong-'eld atomic and molecular physics in the future using the generalized Floquet techniques. In particular, the latest development of Floquet formulations of TDDFT and TDCDFT provide powerful new ab initio theoretical frameworks for quantitative and comprehensive exploration of the multiphoton dynamics and high-order nonlinear optical phenomena of many-electron quantum systems in strong 'elds, a subject of considerable current challenges in atomic and molecular and optical physics. Research along this direction will be reported elsewhere. Acknowledgements The developments of generalized Floquet formalisms and associated computational techniques in the past two decades have been contributed by many former postdoctors and students at the University of Kansas, particularly, Dr. T.S. Ho, Dr. J.V. Tietz, Dr. K.K. Datta, Dr. S. Bhattacharya, Dr. K. Wang, Dr. E. Layton, Dr. G. Yao, Dr. Y. Huang, Dr. J. Wang, and Dr. X. Chu. SIC acknowledges invaluable former collaborations and discussions with Professor William Reinhardt, Professor Jinx Cooper, Dr. Cecil Laughlin, and late Professor Joseph Hirschfelder. This work was supported by the Chemical Sciences, Geosciences, and Biosciences Division, ONce of Basic Energy Sciences, ONce of Science, U.S. Department of Energy, and by the National Science Foundation under grant number PHY-0098106. We acknowledge Kansas Center for Advanced Scienti'c Computing for the use of Origin2400 supercomputer facilities sponsored by NSF-MRI program DMS-9977352 for some of our recent investigations. References [1] [2] [3] [4] [5]

M. Gavrila (Ed.), Atoms in Intense Laser 'elds, Academic Press, New York, 1992. M.H. Mittleman, Introduction to the Theory of Laser–Atom Interactions, Plenum Press, New York, 1993. F.H.M. Faisal (Ed.), Theory of Multiphoton Processes, Plenum Press, New York, 1987. L.F. DiMauro, P. Agostini, Adv. At. Mol. Opt. Phys. 35 (1995) 79. L.S. Cederbaum, K.C. Kulander, N.H. March (Eds.), Atoms and Molecules in Intense Fields, Springer, New York, 1997. [6] P. Lambropoulos, H. Walther (Eds.), Multiphoton Processes, Institute of Physics Pub., Bristol and Philadelphia, 1997. [7] P.H. Bucksbaum, Nature (London) 396 (1998) 217.

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[8] L.F. DiMauro, R.R. Freeman, K.C. Kulander (Eds.), Multiphoton Processes, AIP Conference Proceedings, Vol. 525, American Institute of Physics, New York, 2000. [9] N.B. Delone, V.P. Krainov, Multiphoton Processes in Atoms, 2nd Edition, Springer, New York, 2000. [10] S.A. Rice, M. Zhao, Optical Control of Molecular Dynamics, Wiley, New York, 2000. [11] H. Rabitz, R.d. Vivie-Riedle, M. Motzkusand, K. Kompa, Science 288 (2000) 824. [12] D. Batani, C.J. Joachain, S. Martellucci, A.N. Chester (Eds.), Atoms, Solids, and Plasmas in Super-Intense Laser Fields, Academic/Plenum Pub., New York, 2001. [13] A.D. Bandrauk, Y. Fujimura, R.J. Gordon, Laser Control and Manipulation of Molecules, in: ACS Symposium Series, Vol. 821, Oxford University Press, Oxford, 2002. [14] P. Brumer, M. Shapiro, Coherent Control of Atomic and Molecular Processes, Wiley, New York, 2003. [15] S.I. Chu, Adv. At. Mol. Phys. 21 (1985) 197. [16] S.I. Chu, in: S.H. Lin (Ed.), Advances in Multiphoton Processes and Spectroscopy, Vol. 2, World Scienti'c, Singapore, 1986, p. 175. [17] S.I. Chu, Adv. Chem. Phys. 73 (1989) 739. [18] R.M. Potvliege, R. Shakeshaft, in: M. Gavrila (Ed.), Atoms in Intense Laser Fields, Academic Press, New York, 1992, p. 373. [19] S.I. Chu, in: C.D. Lin (Ed.), Review of Fundamental Processes and Applications of Atoms and Ions, World Scienti'c, Singapore, 1993, p. 403. [20] S.I. Chu, in: D. Truhlar, B. Simon (Eds.), Multiparticle Quantum Scattering with Applications to Nuclear, Atomic, and Molecular Physics, Springer, New York, 1997, p. 343. [21] T.S. Ho, S.I. Chu, Chem. Phys. Lett. 141 (1987) 315. [22] Y. Huang, S.I. Chu, Chem. Phys. Lett. 225 (1994) 46. [23] K.C. Kulander (Ed.), Time-dependent Methods for Quantum Dynamics, in: Computer Physics Communications, Vol. 63, North-Holland, Amsterdam, 1991. [24] J.S. Parker, E.S. Smyth, K.T. Taylor, J. Phys. B 31 (1998) L571. [25] C.A. Ullrich, U.J. Gossmann, E.K.U. Gross, Phys. Rev. Lett. 74 (1995) 872. [26] X.M. Tong, S.I. Chu, Phys. Rev. A 57 (1998) 452. [27] X. Chu, S.I. Chu, Phys. Rev. A 63 (2001) 023411. [28] X.M. Tong, S.I. Chu, Chem. Phys. 217 (1997) 119. [29] X. Chu, S.I. Chu, Phys. Rev. A 63 (2001) 013414. [30] X.M. Tong, S.I. Chu, Phys. Rev. A 58 (1998) R2656. [31] X.M. Tong, S.I. Chu, Phys. Rev. A 64 (2001) 013417. [32] X. Chu, S.I. Chu, Phys. Rev. A 64 (2001) 063404. [33] P.G. Burke, P. Francken, C.J. Joachain, J. Phys. B 24 (1991) 751. [34] P.G. Burke, J. Colgan, D.H. Glass, K. Higgins, J. Phys. B 33 (2000) 143. [35] M. Gavrila, in: M. Gavrila (Ed.), Atoms in Intense Laser Fields, Academic Press, New York, 1992, p. 435. [36] M. Gavrila, J. Phys. B 35 (2002) R147. [37] J.H. Shirley, Phys. Rev. 138 (1965) B979. [38] D.R. Dion, J.O. Hirschfelder, Adv. Chem. Phys. 35 (1976) 265. [39] K. Wang, Nonperturbative treatments of the interaction of light and matter, Ph.D. Thesis, University of Kansas, 1988. [40] J.V. Tietz, S.I. Chu, Chem. Phys. Lett. 101 (1983) 446. [41] K. Wang, S.I. Chu, Phys. Rev. A 39 (1989) 1800. [42] W.P. Reinhardt, Ann. Rev. Phys. Chem. 33 (1982) 223. [43] B.R. Junker, Adv. At. Mol. Phys. 18 (1982) 208. [44] Y.K. Ho, Phys. Rep. 99 (1983) 1. [45] S.I. Chu, Int. J. Quantum Chem. Quantum Chem. Symp. 20 (1986) 129. [46] N. Moiseyev, Phys. Rep. 302 (1998) 211. [47] S.I. Chu, W.P. Reinhardt, Phys. Rev. Lett. 39 (1977) 1195. [48] S.I. Chu, Chem. Phys. Lett. 54 (1978) 367. [49] G. Yao, S.I. Chu, Chem. Phys. Lett. 204 (1993) 381. [50] G. Yao, S.I. Chu, Chem. Phys. Lett. 197 (1992) 413.

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131 [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100]

127

T.S. Ho, S.I. Chu, J.V. Tietz, Chem. Phys. Lett. 96 (1983) 464. T.S. Ho, S.I. Chu, Phys. Rev. A 31 (1985) 659. T.S. Ho, S.I. Chu, Chem. Phys. Lett. 122 (1985) 327. K. Wang, S.I. Chu, J. Chem. Phys. 86 (1987) 3225. E. Layton, Y. Huang, S.I. Chu, Phys. Rev. A 41 (1990) 42. S.I. Chu, Z.C. Wu, E. Layton, Chem. Phys. Lett. 157 (1989) 151. D.A. Telnov, S.I. Chu, Chem. Phys. Lett. 264 (1997) 466. D.A. Telnov, S.I. Chu, Phys. Rev. A 58 (1998) 4749. G. Floquet, Ann. l’Ecol. Norm. Sup. 12 (1883) 47. J.H. PoincarOe, Les Methodes Nouvelles de la Mechanique Celeste, Vol. I, II, III, IV, Paris, 1892, 1893, 1899. S.H. Autler, C.H. Townes, Phys. Rev. 100 (1955) 703. C. Cohen-Tannoudji, S. Haroche, J. Phys. (Paris) 30 (1969) 153. H. Sambe, Phys. Rev. A 7 (1973) 2203. S.I. Chu, J.V. Tietz, K.K. Datta, J. Chem. Phys. 77 (1982) 2968. R. Duperrx, H. van den Bergh, J. Chem. Phys. 73 (1980) 585. P.H. Winston, Arti'cial Intelligence, Addison-Wesley, Reading, MA, 1979. T.B. Simpson, N. Bloembergen, Chem. Phys. Lett. 100 (1983) 325. J. Chang, R.E. Wyatt, J. Chem. Phys. 85 (1986) 1826. Y. Zhang, S.J. Klippenstein, R.A. Marcus, J. Chem. Phys. 94 (1991) 7319. Y. Zhang, R.A. Marcus, J. Chem. Phys. 96 (1992) 6073. J.N. Gehlen, I. Daizadeh, A.A. Stuchebrukhov, R.A. Marcus, Inorg. Chim. Acta 243 (1996) 271. A. Aguilar, J.M. Combes, Commun. Math. Phys. 22 (1971) 265. E. Balslev, J.M. Combes, Commun. Math. Phys. 22 (1971) 280. A. Maquet, S.I. Chu, W.P. Reinhardt, Phys. Rev. A 27 (1983) 2946. J. Wang, S.I. Chu, C. Laughlin, Phys. Rev. A 50 (1994) 3208. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988. D.A. Telnov, S.I. Chu, Phys. Rev. A 59 (1999) 2864. B. Simon, Phys. Lett. A 71 (1979) 211. C.A. Nicolaides, D.R. Beck, Phys. Lett. A 65 (1978) 11. K.K. Datta, S.I. Chu, Chem. Phys. Lett. 87 (1982) 357. S.I. Chu, K.K. Datta, J. Chem. Phys. 76 (1982) 5307. J. Turner, C.W. McCurdy, Chem. Phys. 71 (1982) 127. N. Lipkin, N. Moiseyev, E. BrGandas, Phys. Rev. A 40 (1989) 549. A. Scrinzi, N. Elander, J. Chem. Phys. 98 (1993) 3866. C.A. Nicolaides, H.J. Gotsis, M. Chrysos, Y. Komninos, Chem. Phys. Lett. 168 (1990) 570. N. Rom, N. Moiseyev, R. Levebvre, J. Chem. Phys. 95 (1991) 3562. T.N. Rescigno, M. Baertschy, D. Byrum, C.W. McCurdy, Phys. Rev. A 55 (1997) 4253. C.W. McCurdy, C.K. Stroud, M.K. Wisinski, Phys. Rev. A 43 (1991) 5980. P.B. Kurasov, A. Scrinzi, N. Elander, Phys. Rev. A 49 (1993) 5095. N. Rom, E. Engdahl, N. Moiseyev, J. Chem. Phys. 93 (1990) 3413. D.A. Telnov, S.I. Chu, Phys. Rev. A 66 (2002) 063409. D.A. Telnov, S.I. Chu, Phys. Rev. A 66 (2002) 043417. D.E. Ramaker, J.M. Peek, At. Data 5 (1973) 167. C.R. Holt, M.G. Raymer, W.P. Reinhardt, Phys. Rev. A 27 (1983) 2971. S.I. Chu, J. Cooper, Phys. Rev. A 32 (1985) 2769. S.I. Chu, K. Wang, E. Layton, J. Opt. Soc. Am. B 7 (1990) 425. M. DGorr, R.M. Potvliege, R. Shakeshaft, Phys. Rev. A 41 (1990) 558. R.M. Potvliege, R. Shakeshaft, Phys. Rev. A 38 (1988) 1098. R.M. Potvliege, R. Shakeshaft, Phys. Rev. A 38 (1988) 4597. H. Rottke, B. Wol9, M. Brickwedde, D. Feldmann, K.H. Welge, in: G. Mainfray, P. Agostini (Eds.), Multiphoton Processes, Commissariat a` l’Energie Atomique, Paris, 1991. [101] H. Rottke, B. Wol9, M. Brickwedde, D. Feldmann, K.H. Welge, Phys. Rev. Lett. 64 (1990) 404.

128

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

[102] G.A. Kyrala, T.D. Nichols, Phys. Rev. A 44 (1991) R1450. [103] P. Kruit, J. Kimman, H.G. Muller, M.J. van der Wiel, Phys. Rev. A 28 (1983) 248. [104] R.R. Freeman, P.H. Buksbaum, W.E. Cooke, G. Gibson, T.J. McIlrath, L.D. van Woerkom, Adv. At. Mol. Opt. Phys. Suppl. 1 (1992) 43. [105] S.I. Chu, R.Y. Yin, J. Opt. Soc. Am. B 4 (1987) 720. [106] P. Agostini, P. Breger, A. L’Huillier, H.G. Muller, G. Petite, Phys. Rev. Lett. 63 (1989) 2208. [107] C.Y. Tang, P.G. Harris, A.H. Mohagheghi, H.C. Bryant, C.R. Quick, J.B. Donahue, R.A. Reeder, S. Cohen, W.W. Smith, J.E. Stewart, Phys. Rev. A 39 (1989) 6068. [108] C.Y. Tang, P.G. Harris, H.C. Bryant, A.H. Mohagheghi, R.A. Reeder, H. Shari'an, H. Toutounchi, C.R. Quick, J.B. Donahue, S. Cohen, W.W. Smith, Phys. Rev. Lett. 66 (1991) 3124. [109] C. Laughlin, S.I. Chu, Phys. Rev. A 48 (1993) 4654. [110] A.L. Stewart, J. Phys. B 11 (1978) 3851. [111] A.W. Wishart, J. Phys. B 12 (1979) 3511. [112] D.A. Telnov, S.I. Chu, J. Phys. B 29 (1996) 4401. [113] D.A. Telnov, J. Phys. B 24 (1991) 2967. [114] I.J. Berson, J. Phys. B 8 (1975) 3078. [115] L. Prastegaard, T. Andersen, P. Balling, Phys. Rev. A 59 (1999) R3154. [116] R. Reichle, H. Helm, I.Y. Kiyan, Phys. Rev. Lett. 87 (2001) 243001. [117] S. Geltman, Phys. Rev. A 43 (1991) 4930. [118] C.R. Liu, B. Gao, A.F. Starace, Phys. Rev. A 46 (1992) 5985. [119] L.A.A. Nikolopoulos, P. Lambropoulos, Phys. Rev. Lett. 82 (1999) 3771. [120] D. Proulx, M. Pont, R. Shakeshaft, Phys. Rev. A 49 (1994) 1208. [121] D. Proulx, R. Shakeshaft, Phys. Rev. A 46 (1992) R2221. [122] I. SOanchez, H. Bachau, F. MartObn, J. Phys. B 30 (1997) 2417. [123] I. SOanchez, F. MartObn, H. Bachau, J. Phys. B 28 (1995) 2863. [124] J. Purvis, M. DGorr, M. Terao-Dunseath, C.J. Joachain, P.G. Burke, C.J. Noble, Phys. Rev. Lett. 71 (1993) 3943. [125] A. Stintz, X.M. Zhao, C.E.M. Strauss, W.B. Ingalls, G.A. Kyrala, D.J. Funk, H.C. Bryant, Phys. Rev. Lett. 75 (1995) 2924. [126] X.M. Zhao, M.S. Gulley, H.C. Bryant, C.E.M. Strauss, D.J. Funk, A. Stintz, D.C. Rislove, G.A. Kyrala, W.B. Ingalls, W.A. Miller, Phys. Rev. Lett. 78 (1997) 1656. [127] D.C. Rislove, C.E.M. Strauss, H.C. Bryant, M.S. Gulley, D.J. Funk, X.M. Zhao, W.A. Miller, Phys. Rev. A 58 (1998) 1889. [128] D.C. Rislove, C.E.M. Strauss, H.C. Bryant, M.S. Gulley, D.J. Funk, X.M. Zhao, W.A. Miller, Phys. Rev. A 59 (1999) 906. [129] K.T. Chung, S.I. Chu, Phys. Rev. A 61 (2000) 060702(R). [130] N. Bloembergen, E. Yablonovitch, Phys. Today 31 (1978) 23. [131] P.A. Schulz, A.S. Sudbo, D.J. Krajnovich, H.S. Kwok, Y.R. Shen, Y.T. Lee, Ann. Rev. Phys. Chem. 30 (1979) 311. [132] A. Carrington, J. Buttenshaw, Mol. Phys. 44 (1981) 267. [133] S.I. Chu, C. Laughlin, K.K. Datta, Chem. Phys. Lett. 98 (1983) 476. [134] C. Laughlin, K.K. Datta, S.I. Chu, J. Chem. Phys. 85 (1986) 1403. [135] A. Giusti-Suzor, F.H. Mies, L.F. DiMauro, E. Charron, B. Yang, J. Phys. B 28 (1995) 309. [136] S.I. Chu, J. Chem. Phys. 75 (1981) 2215. [137] S.I. Chu, J. Chem. Phys. 94 (1991) 7901. [138] D.A. Telnov, S.I. Chu, Chem. Phys. Lett. 255 (1996) 223. [139] G. Yao, S.I. Chu, Phys. Rev. A 48 (1993) 485. [140] P.H. Bucksbaum, A. Zavriyev, H.G. Muller, D.W. Schumacher, Phys. Rev. Lett. 64 (1990) 1883. [141] A. Zavriyev, P.H. Bucksbaum, J. Squier, F. Saline, Phys. Rev. Lett. 70 (1993) 1077. [142] L.J. Frasinski, J.H. Posthumus, J. Plumridge, K. Codling, Phys. Rev. Lett. 83 (1999) 3625. [143] J. Wang, S.I. Chu, Chem. Phys. Lett. 227 (1994) 663. [144] E. Constant, H. Stapelfeldt, P.B. Corkum, Phys. Rev. Lett. 76 (1996) 4140. [145] G.N. Gibson, M. Li, C. Guo, J. Neira, Phys. Rev. Lett. 79 (1997) 2022.

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131 [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190] [191] [192] [193] [194]

129

K. Codling, L.J. Frasinski, J. Phys. B 26 (1993) 783. J. Zuo, A.D. Bandrauk, Phys. Rev. A 52 (1995) 2511. M. Plummer, J.F. McCann, J. Phys. B 29 (1996) 4625. Z. Mulyukov, M. Pont, R. Shakeshaft, Phys. Rev. A 54 (1996) 4299. T. Seideman, M.Y. Ivanov, P.B. Corkum, Phys. Rev. Lett. 75 (1995) 2819. R.S. Mulliken, J. Chem. Phys. 7 (1939) 20. T.S. Ho, S.I. Chu, J. Phys. B 17 (1984) 2101. T.S. Ho, S.I. Chu, Phys. Rev. A 32 (1985) 377. K. Wang, T.S. Ho, S.I. Chu, J. Phys. B 18 (1985) 4539. H. Salwen, Phys. Rev. 99 (1955) 1274. P. Aravind, J.O. Hirschfelder, J. Phys. Chem. 88 (1984) 4788. P.R. Certain, J.O. Hirschfelder, J. Chem. Phys. 52 (1970) 5977. B. Kirtman, J. Chem. Phys. 49 (1968) 3890. R.P. Feynman, F.L. Vernon Jr., R.W. Hellwarth, J. Appl. Phys. 28 (1957) 49. J.N. Elgin, Phys. Lett. A 80 (1980) 140. F.T. Hioe, J.H. Eberly, Phys. Rev. Lett. 47 (1981) 838. M. Gell-Man, Y. Neeman, The Eight-Fold Way, Benjamin, New York, 1964. B. Eckhardt, Phys. Rep. 163 (1988) 205. T. Hogg, B.A. Huberman, Phys. Rev. Lett. 48 (1982) 711. Y. Pomeau, B. Dorizzi, B. Grammaticos, Phys. Rev. Lett. 56 (1986) 681. R. Badii, P.F. Meier, Phys. Rev. Lett. 58 (1987) 1045. S.I. Chu, in: J.-C. Gay (Ed.), Irregular Atomic Systems and Quantum Chaos, Gordon and Breach, London, 1992, p. 109. K. Wang, S.I. Chu, Chem. Phys. Lett. 153 (1988) 87. P.W. Anderson, Rev. Mod. Phys. 50 (1978) 191. P.A. Lee, T.V. Ramakrishnam, Rev. Mod. Phys. 57 (1985) 287. C.M. Soukoulis, E.N. Economou, Phys. Rev. Lett. 52 (1984) 565. D.A. Telnov, J. Wang, S.I. Chu, Phys. Rev. A 51 (1995) 4797. Z. Chang, A. Rundquist, H. Wang, M.M. Murnane, H.C. Kapteyn, Phys. Rev. Lett. 79 (1997) 2967. C. Spielmann, H. Burnett, R. Sartania, R. Koppitsch, M. Schnurer, C. Kan, M. Lenzner, P. Wobrauschek, F. Krausz, Science 278 (1997) 661. R.M. Potvliege, R. Shakeshaft, Phys. Rev. A 40 (1989) 3061. D.A. Telnov, J. Wang, S.I. Chu, Phys. Rev. A 52 (1995) 3988. J.L. Krause, K.J. Schafer, K.C. Kulander, Phys. Rev. A 45 (1992) 4998. T.F. Jiang, S.I. Chu, Phys. Rev. A 46 (1992) 7322. G. Bandarage, A. Maquet, J. Cooper, Phys. Rev. A 41 (1990) 1744. L.D. Landau, E.M. Lifshitz, Classical Theory of Fields, Pergamon, New York, 1975. A. Messiah, Quantum Mechanics, North-Holland, Amsterdam, 1965. N. Bloembergen, Nonlinear Optics, Benjamin, New York, 1965. Y.R. Shen, The Principle of Nonlinear Optics, Wiley, New York, 1984. M.D. Levenson, Introduction to Nonlinear Laser Spectroscopy, Academic, New York, 1982. P.N. Prasad, D.J. Williams, Introduction to Nonlinear Optical E9ects in Molecules and Polymers, Wiley, New York, 1991. L. Allen, J.H. Eberly, Optical Resonance and Two-Level Atoms, Wiley, New York, 1975. M. Sargent, M.O. Scully, W.E. Lamb Jr., Laser Physics, Addison-Wesley, Reading, MA, 1974. B. Dick, R.M. Hochstrasser, Chem. Phys. 75 (1983) 133. T.S. Ho, K. Wang, S.I. Chu, Phys. Rev. A 33 (1986) 1798. U. Fano, Phys. Rev. 131 (1963) 259. T.C. Kavanaugh, R.J. Silbey, J. Chem. Phys. 96 (1992) 6443. T.C. Kavanaugh, R.J. Silbey, J. Chem. Phys. 98 (1993) 9444. B.R. Mollow, Phys. Rev. 188 (1969) 1969. M.V. Berry, Proc. R. Soc. London, Ser. A 392 (1984) 45.

130 [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239] [240]

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131 B. Simon, Phys. Rev. Lett. 51 (1983) 2167. D. Suter, G. Chingas, R.A. Harris, A. Pines, Mol. Phys. 61 (1987) 1327. A. Tomita, R.Y. Chiao, Phys. Rev. Lett. 57 (1986) 937. R. Simon, H.J. Kimble, E.C.G. Sudarshan, Phys. Rev. Lett. 61 (1988) 19. T. Bitter, D. Dubbers, Phys. Rev. Lett. 59 (1987) 251. R. Tycko, Phys. Rev. Lett. 58 (1987) 2281. R.S. Ham, Phys. Rev. Lett. 58 (1987) 725. D.M. Bird, A.R. Preston, Phys. Rev. Lett. 61 (1988) 2863. M. Tian, R.R. Reibel, Z.W. Barber, J.A. Fischer, W.R. Rabbitt, Phys. Rev. A 67 (2003) 011403. Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58 (1987) 1593. Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. C.A. Mead, D.G. Truhlar, J. Chem. Phys. 70 (1979) 2284. D. Suter, K.T. Mueller, A. Pines, Phys. Rev. Lett. 60 (1988) 1218. X. Wang, K. Matsumoto, J. Phys. A 34 (2001) L631. G. Falci, R. Fazio, G.M. Palma, J. Slewert, V. Vedral, Nature (London) 407 (2000) 355. M.D. Lukin, P.R. Hemmer, Phys. Rev. Lett. 84 (2000) 2818. M. Pont, R.M. Potvliege, R. Shakeshaft, P.H.G. Smith, Phys. Rev. A 46 (1992) 555. A.M. Perelomov, Sov. Phys. USP. 20 (1977) 703. E. Layton, S.I. Chu, in: Radiation E9ects and Defects in Solids, Vol. 122–123, Gordon and Breach, London, 1991, p. 73. J.C. Garrison, E.M. Wright, Phys. Lett. A 128 (1988) 177. G. Steinmeyer, D.H. Sutter, L. Gallmann, N. Matuschek, U. Keller, Science 286 (1999) 1507. T. Brabec, F. Krausz, Rev. Mod. Phys. 72 (2000) 545. A. Zewail, J. Phys. Chem. A 104 (2000) 5660. M. Hentschel, R. Kienberger, C. Spielmann, G.A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, F. Krausz, Nature 414 (2001) 509. P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H.G. Muller, P. Agostini, Science 292 (2001) 1689. S.I. Chu, T.S. Ho, J.V. Tietz, Chem. Phys. Lett. 99 (1983) 422. T.S. Ho, S.I. Chu, J. Chem. Phys. 79 (1983) 4708. T.S. Ho, S.I. Chu, C. Laughlin, J. Chem. Phys. 81 (1984) 788. T.S. Ho, C. Laughlin, S.I. Chu, Phys. Rev. A 32 (1985) 122. R.C. Isler, E.C. Crume, Phys. Rev. Lett. 41 (1978) 1296. R.H. Dixon, J.F. Seely, R.C. Elton, Phys. Rev. Lett. 40 (1978) 122. W.R. Green, M.D. Wright, J.F. Young, S.E. Harris, Phys. Rev. Lett. 43 (1979) 120. L.B. Madsen, J.P. Hansen, L. Kocbach, Phys. Rev. Lett. 89 (2002) 093202. T. Kirchner, Phys. Rev. Lett. 89 (2002) 093203. H.S.W. Massey, Rep. Prog. Phys. 12 (1949) 248. P. Pfeifer, R.D. Levine, J. Chem. Phys. 79 (1983) 5512. U. Peskin, N. Moiseyev, J. Chem. Phys. 99 (1993) 4590. J.S. Howland, Math. Ann. 207 (1974) 315. Y. Huang, D.J. Kouri, D.K. Ho9man, Chem. Phys. Lett. 225 (1994) 37. D.A. Telnov, S.I. Chu, J. Phys. B 28 (1995) 2407. A.K. Kazansky, D.A. Telnov, Zh. Eksp. Teor. Fiz. 94 (1988) 73 [Sov. Phys.-JETP 67 (1988) 253]. R.G. Parr, W. Yang, Density–Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, 1989. R.M. Dreizler, E.K.U. Gross, Density Functional Theory, An Approach to the Quantum Many-Body Problem, Springer, Berlin, 1990. E.K.U. Gross, R.M. Dreizler (Eds.), in: NATO Advanced Studies Institute, Series B: Physics, Vol. 337, Plenum, New York, 1995. N.H. March, Electron Density Theory of Atoms and Molecules, Academic, San Diego, 1992. J.K. Labanowski, J.W. Andzelm (Eds.), Density Functional Methods in Chemistry, Springer, Berlin, 1991.

S.-I Chu, D.A. Telnov / Physics Reports 390 (2004) 1 – 131

131

[241] J.F. Dobson, G. Vignale, M.P. Das (Eds.), Electron Density Functional Theory: Recent Progress and New Directions, Plenum Press, New York, 1998. [242] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [243] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1113. [244] A. Zangwill, P. Soven, Phys. Rev. A 21 (1980) 1561. [245] E. Runge, E.K.U. Gross, Phys. Rev. Lett. 52 (1984) 997. [246] E.K.U. Gross, J.F. Dobson, M. Petersilka, in: R.F. Nalewajski (Ed.), Density Functional Theory, Topics in Current Chemistry, Vol. 181, Springer, Berlin, 1996, p. 81. [247] E.K.U. Gross, W. Kohn, Phys. Rev. Lett. 55 (1985) 2850. [248] S. Hirata, M. Head-Gordon, Chem. Phys. Lett. 302 (1999) 375. [249] G.D. Mahan, K.R. Subbaswamy, Local Density Theory of Polarizability, Plenum Press, New York, 1990. [250] S.M. Colwell, N.C. Handy, A.M. Lee, Phys. Rev. A 53 (1996) 1316. [251] S.J.A. van Gisbergen, J. Snijders, E.J. Baerends, J. Chem. Phys. 109 (1998) 10657. [252] C.A. Ullrich, E.K.U. Gross, Comm. At. Mol. Phys. 33 (1997) 211. [253] D.A. Telnov, S.I. Chu, Int. J. Quant. Chem. 69 (1998) 305. [254] D.A. Telnov, S.I. Chu, Phys. Rev. A 63 (2001) 012514. [255] A. GGorling, Phys. Rev. A 59 (1999) 3359. [256] G. Vignale, M. Rasolt, Phys. Rev. Lett. 59 (1987) 2360. [257] G. Vignale, M. Rasolt, Phys. Rev. B. 37 (1988) 10685. [258] G. Vignale, M. Rasolt, D.J.W. Geldart, Adv. Quant. Chem. 21 (1990) 235. [259] A.K. Rajagopal, Phys. Rev. A 50 (1994) 3759. [260] S.K. Ghosh, A.K. Dhara, Phys. Rev. A 38 (1988) 1149. [261] P. Hessler, J. Park, K. Burke, Phys. Rev. Lett. 82 (1999) 378. [262] J.A.R. Samson, Z.X. He, L. Yin, G.N. Haddad, J. Phys. B 27 (1994) 887. [263] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [264] A.D. Becke, J. Chem. Phys. 96 (1992) 2155. [265] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [266] J.B. Krieger, Y. Li, G.F. Iafrate, Phys. Rev. A 45 (1992) 101. [267] Y. Li, J.B. Krieger, G.F. Iafrate, Phys. Rev. A 47 (1993) 165. [268] C.A. Ramsbottom, K.L. Bell, K.A. Berrington, J. Phys. B 27 (1994) 2905. [269] H.J. Kaiser, E. Heinicke, R. Rackwitz, D. Feldmann, Z. Phys. 270 (1974) 259. [270] X.M. Tong, S.I. Chu, Phys. Rev. A 55 (1997) 3406. [271] X.M. Tong, S.I. Chu, Phys. Rev. A 57 (1998) 855. [272] G. Haecer, D. Hanstorp, I. Kiyan, A.E. KlinkmGuller, U. Ljungbad, D.J. Pegg, Phys. Rev. A 53 (1996) 4127. [273] E.P. Wigner, Phys. Rev. 73 (1948) 1002.