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Nonperturbative methods for the study of multiphoton dynamics in intense periodic and multi-color laser fields
482
Computer Physics Communications 63 (1991) 482—493 North-Holland
Nonperturbative methods for the study of multiphoton dynamics in intense periodic and multi-color laser fields Shih-I Chu Department of Chemisizy, University of Kansas, Lawrence, KS 66045, USA
and Tsin-Fu Jiang Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, 30050 Received 2 April 1990
We present a brief account of some recently developed generalized Floquet methods (both time-dependent and time-independent) for the nonperturbative treatment of the time-development of wavefunctions or the density matrix operator of quantum systems driven by intense periodic or multi-color (multi-frequency) laser fields. Both the Schrodinger and Liouville equations are considered. We also explore the feasibility of the fast Fourier transformation (FFT) propagation method for the study of Rydberg atom dynamics. We then present a FF1T case study of the time propagation of the wavefunction and coarse-grained Wigner density distribution for the problem of microwave-driven multiphoton excitation and ionization of Rydberg hydrogen atoms.
1. Introduction The study of the nonlinear response and timedevelopment of multiphoton dynamics of atomic and molecular systems in intense laser fields is a subject of both fundamental and practical interest in modem science and technology *, The theory of multiphoton processes can be formulated in a fully quantum mechanical or semiclassical fashion. In the former approach, both the system and the field are treated quantum mechanically, while in the semiclassical approach, the system is described by a time-dependent schrodinger equation in which the effect of the radiation field is represented by an effective Hamiltonian consistent with Maxwell’s equations. The fully quantized methods are generally difficult to apply as here one is
*
Recent reviews can be found in ref. [11.
0O1O-4655/91/$03.50 © 1991
—
dealing with a large number of time-independent coupled second-order differential equations subject to appropriate asymptotic boundary conditions. In contrast, only first-order time-dependent initial-value problems are involved in semiclassical methods. The semiclassical approach usually leads to Hamiltomans periodic in time through an assumed sinusoidal time variation of the electromagnetic field. The use of the Floquet theorem [21can further greatly simplify the dynamical study. In this article, we shall address the problems and difficulties encountered by the conventional Floquet methods and discuss some possible generalized Floquet formalisms and solutions for nonperturbative treatment of multiphoton and nonlinear optical processes in intense periodic and quasi-periodic fields. Before that, let us first review briefly how the Floquet theorem may be implemented in the semiclassical treatment of multiphoton processes in periodic fields.
Elsevier Science Publishers B.V. (North-Holland)
S. -I. Chu T -F. Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
2. Implementation of the Floquet theorem in the
483
and U(t, t0) is the time-evolution operator defined as
study of multiphoton processes in periodic fields
~P (t)
There are two ways to implement the Floquet theorem into numerical calculations. The first method is the time-dependent propagation technique [5aJ in which one numerically intergrates the Schrodinger equation (h 1)
t~t~)~P (t0),
=
with
(7( t0,
t0)
=
1. (2)
The Floquet theorem asserts that
=
U(t, i a,L7(t,
t~)= ñ(t)U(t,
(1)
ta).
A+2w1
B
B [HF]=
Where
=
n=O n=—1 n=—2
A+wI
0
0
0
n’=2
B
0
0
n’=l
B
0
n’=O
0
B
A
o o
0
B
0
0
A—wi B n’=—l B A—2w1 n’=—2
v=O E00
v=1
0
0
0
0
0
(0) E01
~
0
0
0
(0) E10
o
o
A=
~‘•~
-
I
I
~(0)
I ~(O) 12
and
v=O o
—
B=
v’=O
v’=i
I
v=1
b000, o o
(3)
=
=
n=1
exp[—1Q(t—to)],
where Q is the time-independent quasi-energy operator, and P(t, t0) P(t + T, t0) is periodic. Since U(t0 + nT, t0) [U(t0 + T, t0)J’~, one sees
Here the Hamiltoman is periodic in time, namely, + T) .U(t) with T= 2.rr/w being the period, n=2
t~)=P(t, t0)
b0102:
o b001 1 b0110 a
o
a ....
b0211
a b0112i
~‘
=
a
v’
=
1
0
____________
a
b101~ a a
b11121
i
Fig. 1. Structure of Floquet Hamiltonian HF for multiphoton excitation of molecules. ~ are unperturbed vibrational—rotational energies and ~ are electric dipole coupling matrix elements.
S-I. Chu, T -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
484
that the time-evolution operator over one period, 0(10 + T, ta), provides essentially all the information we ever need about the long-time behavior of the system. Further, in the truncated basis, U( t0 + T, t0) may be diagonalized by some unitary transformation S, S~O(t~4- T, t0)S
=
(4)
c_ID,
where D is a diagonal matrix. Thus O(t0 + T, t0) S e_WS± =
(5)
whereupon
0(t~+ nT,
t0)
S eDS+.
=
(6)
The second way to implement the Floquet theorem is via themethod so-called time-independent Floquet Hamiltonian [3—5].In this approach, one transforms the periodically time-dependent Hamiltonian 1-2(t) into an equivalent time-independent infinite-dimensional Floquet matrix eigenvalue problem. As an example, consider the multiphoton excitation of vibrational—rotational states of a diatomic molecule under the periodic perturbation V(t)
=
—~t(r) E 0 cos(wt
+
~)
(7)
where ~a is the dipole moment operator of the molecule, E0, ~o and t~ are the field amplitude, frequency, and phase, respectively. The corresponding time-independent Floquet matrix HF possesses a block tridiagonal form as shown in fig. 1. The determination of the vibration—rotational quasi-energies and quasi-energy states (QES) thus reduces to the solution of a time-independent Floquet matrix eigenproblem. Figure 1 shows that HF possesses a periodic structure with only the number of w’s in the diagonal elements varying from block to block. This structure endows the quasi-energy eigenvalues and eigenvectors of HF with important periodic properties. Detailed discussions of various Floquet studies of multiphoton excitation of finite-level systems in periodic fields can be found in the recent review articles [4,5]. In the following we shall discuss several major problems encountered by these conventional Floquet techniques when applied to realistic systems.
3. Problems and difficulties associated with conventional Floquet techniques and the developments of generalized Floquet methods for the solutions of these problems The first difficulty is the large atom (molecule)—field basis set problem. Consider the dimensionality of the Floquet matrix such as fig. 1 for example. In practical calculations, one truncates the Floquet matrix into N by N in dimension, where N NFNVNJ, where NF is the number of Floquet photon blocks, N~the number of vibrational levels included in one Floquet block, and N~the number of rotational states included in one vibrational level. As N increases rapidly with the size of the molecule and with the order of multiphoton transitions, and as the computational cx3, the full Floquet analysis can become prohibitively costly even for simple molepense grows as N cules. While the use of a supercomputer can provide solutions to some of these problems, the development of new approximation techniques capable of providing accurate results yet involving =
much smaller Floquet matrix manipulation is in great demand. The most probable path approach (MPPA) [6], using artificial intelligence algorithms (see for example ref. [7]) to select the important Floquet basis states, is one such strategy. The method allows orders-of-magnitude reduction of the dimensionality of the Floquet matrix and yet maintains good accuracy. It has been applied successfully to the study of infrared multiphoton excitation dynamics in SO2 [6] as well as the multiphoton ionization of microwave-driven Rydberg hydrogenic atoms [8]. Other alternative strategies for reducing the dimensionality of the Floquet matrix can be found in the review article [5a]. The second general limitation of conventional Floquet techniques is of more fundamental nature, namely they are applicable only to monochromatic (i.e. one-laser) field problems. However, many recent experiments involve the use of more than one laser field, namely, multi-color or multifrequency excitation (with frequencies w~incommensurate). In such cases, the Hamiltonians are no longer periodic in time and the Floquet theorem is simply not valid. Such limitation has been
5.-I. Chu, T. -F Jiang
/ Nonperturbative methodsfor the study of multiphoton dynamics
recently removed with the development of the so-called many-mode Floquet theorem (MMFT) [5,9—11].The theorem allows the exact transformation of the polychromatic time-dependent
A+2w
21
[Hr]=
B
8
Schrodinger equation into an equivalent time-independent infinite-dimensional eigenvalue prob1cm. Consider for example the interaction of an N-
0
A +
0
-
A
-
0
W21
A
2w21
WHERE C+2w11 X
A1
=
E
C
=
C+w11
0
0
0
X
0
0
x
a
x
c
o
0
X
o
0
0
0 E~
a
B
X
0
x X
:~ 0
v(2) v(2) By
C
-
2w11
0 0
0
I
=
a
0
485
v(2)
v(2) ay
o 0
Fig. 2. Structure of two-mode Floquet Hanültonian. It can be extended to N-frequency problems.
S-I. Chu, T -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
486
level quantum system with two linearly polarized
Performing the long-time average over
monochromatic fields with frequencies to~ and W2 (w0/w2 irrational number). According to MMFT, the time-evolution operator (in matrix form) can be written as
the time-averaged transition probability
t
—
t
0
gives
=
~
=
1112
k1k,
L2~
I<$k~k2I x~1112)(x~111, I aOO) 12
Y
52(t,t0)
(13)
U(t,
t0)
The MMFT has been applied to the study of
72)
~ (y~n1n~ Iexp[
=
nI
—iIF(t
—
t0)]
17200)
two-color multiple quantum NMR transitions in spin systems [10], to the study of SU(N) dynami-
~2
xexp[i(n1co1
+
n2w2)t].
(8)
Here HF is the time-independent two-mode Floquet Hamiltonian defined in the generalized Floquet state basis,
I in~n2)
Ii) x
n1)
x In2)
and satisfies the infinite-dimensional eigenvalue equation, (y1n1n~I HF I -y2k1k2)
~ y2k1k2
=X(yinin2IX).
(9)
Figure 2 depicts the structure of the two-mode Floquet Hamiltonian. The components are ordered in such a way that y runs over unperturbed atomic or molecular states (denoted by Greek letters) of H0 before each change in n1 and n1, in turn, runs over before n2. The quasi-energy eigenvalues (X~1,~2 }and their corresponding eigenvectors (I Xyn1n2)) of ‘F have the following useful bichromatic properties, namely X~flfl
+
=
~1~1
+
n2~2,
(10)
and (Yi, =
m1
+
q1, m2
+
q2
Xyzni±qin2+q2)
(11)
The transition probabilty averaged over the initial time t0 while keeping the elapsed time t t0 fixed is given by —
=
(
—
~
t0)
new measure of the nonlinear optical property of the quantum systems in multi-color laser fields [14]. The third general problem arises from the fact that conventional Floquet techniques are based on the use of the time-dependent Schrodinger equation and classical fields. They cannot therefore be applied to the study of nonlinear optical processes where spontaneous decays and collisional dampings play an essential role. The correct starting point is to consider the Liouville equation for the time development of the density matrix operator of atoms and molecules, allowing for a more natural introduction of the damping mechanisms. The Liouville equations for the time evolution
processes is (see for example ref. [15]) (h i 0,~3(t) [12(t), ,3(t)] =
+
i[k, p3(t)].
=
1) (14)
Here p is the density matrix of the system, reduced by an averaging over all irrelevant degrees 2(t) 2(t). acting i2~ is as thea thermal unperturbed of freedom bath, Hamiltonian and 1 ‘o + J with eigenvalues (E~) and eigenvectors (la)}, a 0, 1, 2 N 1, and V(t) is the interaction =
(/3k 1k~ exp[ —iHF(t
k
been further found that polychromatic quasi-energy eigenfunctions exhibit striking fractal behavior in temporal Fourier space [13] and that the corresponding fractal dimensions provide a unique
of a set of N-level quantum systems, interacting with several coherent linearly polarized monochromatic fields, undergoing relaxation by Markovian
/3
cal evolution and symmetry breaking in N-level systems driven by polychromatic fields [11], and to the study of quasi-periodic or chaotic behavior of quantum systems under the perturbation of strong multi-color fields [12]. Most recently, it has
2
—
to)] I a00)~
1k2
(12)
=
—
5.-I. Chu, T. -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
Hamiltoman between the system and the M monochromatic laser fields given by
dent infinite-dimensional non-Hermitian supereigenvalue equation [5,16], namely
(15)
~~(aTh(m}ILFIoT;(k})
[.k,p(t)]
The relaxation term
consists of T1 (population damping) and T2 (coherent damping) mechanisms which are due to the coupling of the system to the thermal bath by radiative decays and coffisional relaxations, etc. Equation (14) is solved usually by perturbation methods [15] or by direct numerical integration. It has recently been shown, however, that the polychromatic time-dcpendent Liouville equation, eq. (14), can be cxactly reformulated as an equivalent time-indepen-
A+2w21
LF
WHERE
-
•.~
B
B 0 ________
o o
_______
X C*w11
o o
B*
}
I ~p;(n))
{m }
fJ{)(a$
I~
(16)
where L F is the time-independent many-mode Floquet—Liouville superoperator defined in terms of the generalized tetradic-Floquet basis I { m })~Ia>($ I X I (m }>, with (m } m0, m2,..., m~.The structure of the Floquet—Liouvile super-matrix LF is illustrated in fig. 3 for the simplest two-level two-mode case. For N-level sys=
0
B A ________ B* 0
0
0 B ________ Aw21
B~
0 X
0 0 0
0
0
o
Y
000
0 0
0 0
Y 0
0 Y
0 0
0
0
0
0
Y
Y
=
0 0 ________ B A-2w21
_______
C+2w11 X~ A~
x
0
A~2I B* ________ 0 0
487
0
0
0 0
0 0
c
x
o
X* 0
C-w11 X~
c-2~1r
I(yab+yba)
‘Yob
X
0
0
0
‘Yba
0
0
C~ 0
0
0
0
0
wbo~IFba
0
WbaIFba
AND 0
a -v~
_______
_____-
0
-V~
V~
o v~
vbo
ob
a
0
-v~
0
0
______
0
1~ -v111
o .~.
-v~
0
0
-V~
vt2~
V~ —v121
_______
ba
ab
v~
0
0
v~2l -v~2~
0
Fig. 3. Structure of the Floquet—Liouville supermatrix LF for the case of two-level systems (with level spacing w 50) in linearly polarized bichromatic fields. w1 and a2 are the two radiation frequencies, J’~)(i = 1, 2) are the electric dipole couplings, and ye,,, Yb~,and ~b* (i~b+ y~~)/2 are relaxation parameters.
488
S-I. Chu, T -F Jiang / Nonperturbative methods for the stu4v of multipholon dynamics
tems and near-resonance processes, the infinite-dimentional super-matrix LF can be further reduced to a N2 by N2 effective non-Hermitian Hamiltonian [16] via the use of generalized van Vleck nearly degenerate perturbation theory [16,17], allowing analytical treatment of intensity-dependent nonlinear optical properties. The development of the Floquet—Liouville super-matrix (FLSM) formalism thus facilitates the study of intense-field nonlinear optical processes in terms of a few complex super-eigenvalues and eigenvectors. This yields a numerically stable and computationally efficient approach for the unified treatment of nonresonant and resonant, one- and multiple-photon, steady-state, and transient phenomena in nonlinear optical processes, much beyond the conventional perturbative and rotating wave approximation (RWA) approaches [15]. The FLSM method has been applied to the study of intense field multiphoton resonance fluorescence spectra [16], intensity-dependent nonlinear optical susceptibilities [18], and multiple-wave mixings [Sb], etc. Much remains to be explored in this direction.
We use the split-operator algorithm [19] to propagate the wavefunction: (h 1) =
~~
+
t)
exp[ _i(P2/4m) ~t] exp[—il2i~t] xexp[ —i(P2/4m)
~t1~~(t)
+
(17) where the effect of applying the operator cx F—il P2/4m /
is evaluated in terms of discrete Fourier transforms and the FFT algorithm. The Hamiltoman under consideration is the one-dimensional (1D) atomic hydrogen (stretched in the field direction) in the presence of ac and dc fields (in atomic units): H(t) H~+ V(t), where
(18)
=
H
2/dx2
0 and V(t) =
—
1/x,
(19)
d
—
~
=
—Fax sin cot—Fdx.
(20)
4. Microwave-driven multiphoton excitation dy-
F
namics in Rydberg atoms: fast Fourier fransformalion propagation method
0 and Fd are, respectively, the electric field strength of the ac and dc fields, and w is the microwave frequency. Note the singularity of the Coulomb potential at the origin (x 0). This is =
The fast Fourier transformation (FFT) method [19,20] is an attractive and increasingly popular technique for numerically integrating the time-dcpendent Schrodinger equation without the use of basis-set expansion. It also allows the treatment of the continuum exactly. By combining the FFT algorithm along with the Floquet theorem, we have a powerful numerical method for solving the time propagation of the wavefunction m penodic fields. In this section, we shall first look into the feasibility and reliability of the FFT method for treating the Rydberg atom problem. Then we shall apply the method for the study of the microwavedriven multiphoton excitation and ionization of Rydberg hydrogen atoms, a subject of considerable current interest in atomic physics and chaotic dynamics [21].
the source of major problem and error for timedependent propagation method for atomic problems, particularly when the imtial state starts from the ground state (is) orbital. For the Rydberg atom dynamics the singularity problem is cxpected to be less severe. To see this, we first study the time propagation of the unperturbed system a, ~p (t) ñ0~p(t). (21) =
The initial state is chosen to be at n0 72. We use 1024 grid points for the discretization of the coordinate space from x x0 to 25 000 a0, where x0 is the starting point of integration, typically 5 to 10 a0. The result is not sensitive to the choice of x0. Figure 4 shows the percentage error in energy [m (
=
—
=
=
S. -I. Chu, T. -F Jiang / Nonperturbative methods for the study of multiphoton dynamics
489
2.~
1.~.
—1.~
—
—
I
0
I 10000
5000
I 15000
20000
Time (e.u.)
Fig. 4. Percentage error of the energy (E(t)) of atomic hydrogen at n = 72, using the FFT propagation method for the unperturbed Hamiltonian. 0.0008 _________________________________________ = 13.0GHz
00006
00004
00004
A
~
0.0000
=
0.0
T
c.~ =
0.0000
\~.
_________________________________________ t
=
13.0GHz 46.0 r
~
~
0 0008 _______________________________________________
0 0008 _______________________________________________
0.0004
00004
t
=
1 0
T
~
0.0000
o,oooo
01 0.0008 _________________________________________ * * 0.0004
t=2.0-r
~
o.oooo
t .
=
47,0 r
.
~ 00008 _________________________________________ * 8OT * — 0.0004 ~t=4
~
o.oooo
....._..~..=
0 0008 _______________________________________________
0 0008
_____________________________________________
A
00004
~=
0.0004 0.0000
~
~
t = 3.0
T
\ç\..,. ~~‘\_
0.0000
0.0008 _______________________________________________
0.0000
5625
~
Y~.
,~
/~_-,_~
t=500-r 0.0004
~ 0
.
49.0
0.0008 _______________________________________________
t=4.Oi
(1
0.0004
—
t
11250
x
16875
22500
0.0000 __________________________________________________ 0 5625 11250 16875 22500
x
Fig. 5. Time development of the probability density I t~(x)12 versus time. The physical parameters used are w = 13.0 GHz, Fd = 6.5 V/cm, and F 0 = O.O18/N~(au), where N0 = 72.
490
S. -I. Chu, T. -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
in the initial period. After that the system is stabilized to about 0.25% error level. This is rather encouraging considering the large spatial distance covered. Higher accuracy may be achieved by increasing the number of spatial mesh points. We now consider the multiphoton excitation dynamics in strong microwave and dc fields. The
times at t/T = 0, 1, 2, 3, 4, and at some larger times with t/T = 46—50, where T = 2’rr/w is one period of microwave field oscillation (w = 13.0 GHz). Typically, we used about 26000 time stems (t~t)per field oscillation (T) in the time propagation. In this calculation, the spatial coordinate x spans from x0 to 25000 a0. To prevent unphysi-
Schrodinger equation is solved using the split-operator and FFT algorithms with the Hamiltonian given by eq. (18), and the initial state at n0 = 72. The dc field strength is chosen to be F,~,= 6.5 V/cm as used in the recent experiment [22]. We have performed the calculations for several microwave frequencies. Some of the results are presented below, Figure 5 shows the probability density I %~(x) I 2 versus the spatial coordinate x for several initial
cal reflection from the outer boundary, we have adopted the absorbing boundary method (see for example ref. [23]) and place a slowly vanishing filter function (b = 0.004) —
f ( x)
=
{
1
+
exp [ b ( x
—
x1)] }
(22)
centered at X~= 22500 a0. The result is found to be insensitive to the choice of b and X~, provided X~ is chosen large enough and b small enough.
a
011
b
T0~
—011
‘
0
12500
25000
Fig. 6. (a) 3D coarse-grained Wigner density distribution for the unperturbed (1D) atomic hydrogen at n = 72. (b) The corresponding 2D contour plot (dotted lines from Outer to inner region: probabity density = iO~,~ and 102, respectively. Solid lines from outer to inner region: probability density = ~o 1.5 10 ~, and 10 05, respectively.).
S-I. Chu, T. -F. Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
491
b
a
= 0
12500
25000
c~
=~ 0
0
12500
25000
d
L
=~ 12500
25000
0
Fig. 7. The time development of the CGW density contour plot for (a) t = 2T, (b) Parameters used same as fig. 5.
Figure 5 reveals that the electronic probability flux spreads and recollects (in x) as time advances with occasional leaking to the continuum. At this stage, it is instructive to construct the coarse-grained Wigner (COW) density distribution (see for example ref. [24]), providing a vivid visualization of the time-development of the quantum system in phase space (x, p). Figure 6a shows the 3D CGW density plot for the unperturbed (1D) atomic hydrogen at n = 72. The corresponding 2D contour diagram is depicted in fig. 6b. It is seen that the largest probability density is initially located nearby x 11 000 a0 and p 0. Figures 7a—d show the time-development of the COW density contour diagram (for t = 2, 5, 10
12500
t =
5r, (c)
t =
lOr, and (d)
25000
t =
50r, respectively.
and 50 T), after the microwave and dc fields have been turned on. We have also performed the calculation of the population of the bound states by projecting ( t) onto unperturbed hydrogenic states I n). A timeaveraged result is shown in fig. 8 (dots joined by solid lines). Also shown here are the recent experimental results by Bayfield and Sokol [22] (solid curve). In this preliminary calculation, we have not yet taken into account the time-varying envelope of the microwave fields and there is some ambiguity regarding how to take the time average ~/i
appropriately. Nevertheless, the overall agreement in shape and magnitude appears reasonably good. As far as we know, this is the first calculation of
492
S-I. Chu, T -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
—I 60
70
I
I
I
80
90
100
110
State Quantum Number
Fig. 8. Population versus state quantum number n. Solid curve: experimental results; dots joined by solid lines: FF1’ results. Parameters used same as fig. 5.
microwave-driven Rydberg atom dynamics, taking into account the effect of continuum exactly * This preliminary result shows that the FF1’ propagation method is a feasible and powerful numerical technique for probing strong-field Rydberg atom quantum dynamics. More refined calculations are being performed and will be discussed elsewhere [26].
References [11M.S.
Feld and V.S. Letokhov, eds., Coherent Nonlinear
Optics (Springer, New York, 1980). J. Jortner, R.D. Levine and S.A. Rice, eds., Photoselective
Chemistry, Advances in Chemical Physics, vol. 47 (Wiley, New York, 1981). S.H. Lin, ed., Advances in Mutliphoton Processes and
This work was partially supported by US Department of Energy (Division of Chemical Sciences), and by the University of Kansas General Research Funds.
Spectroscopy, vols. 1—3 (World Scientific, Singapore, 1984, 1985, 1987). J.O. Hirschfelder, R.E. Wyatt and R.D. Coalson, eds., Lasers, Molecules and Methods, Advances in Chemical Physics, vol. 73 (Wiley, New York, 1989). [2] G. Floquet, Ann. de l’Ecole Norm. Sup. 12 (1983) 47. S.H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703. [3] J.H. Shirley, Phys. Rev. 138 (1965) B979. [4] D.R. (1976)Dion 265. and J.O. Hirschfelder, Adv. Chem. Phys. 35 [5] (a) SI. Chu, Adv. At. Mol. Phys. 31 (1985) 197.
Former quantal studies of this problem have used primarily
(b) SI. Chu, Adv. Chem. Phys. 73 (1989) 739. [6] J.V. Tietz and SI. Chu, Chem. Phys. Lett. 101 (1983) 446. [7] P.H. Winston, Artificial Intelligence (Addison—Wesley,
basis set expansion methods: either hydrogemc basis (which 2 disignores the continuum spectrum) or Surmian basis (L cretization of the continuum). See for example ref. [25].For detailed references in this field see also references in ref. [211.
Reading, 1987). [81 K. Wang MA, and S.!. Chu, Phys. Rev. A 39 (1989) 1800. [9] T.S. Ho, S.!. Chu, and J.V. Tietz, Chem. Phys. Lett. 96 (1983) 464. [101 T.S. Ho and SI. Chu, J. Phys. B 17 (1984) 2101.
Acknowledgement
*
S. -I. Chu, T -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics [11] T.S. Ho and SI. Chu, Phys. Rev. A 31 (1985) 659; A 32 (1985) 377. [12] SI. Chu, Advances in Laser Science — II, AlP Conf. Proc. 160 (1987) 282. [13] K. Wang and S.I. Chu, Chem. Phys. Lett. 153 (1988) 87. [14] S.!. Chu, Conunents At. Mol. Phys. (1990). [15] YR. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984). [16] T.S. Ho, K. Wang and SI. Chu, Phys. Rev. A 33 (1986) 1798. [17] K. Aravind and J.O. Hirschfelder, J. Phys. Chem. 88 (1984) 4788, and references therein. [18] K. Wang and S.!. Chu, J. Chem. Phys. 86 (1987) 3225. [19] M.D. Feit, J.A. Fleck, Jr., and A. Steiger, J. Comput. Phys. 47 (1982) 412.
493
[20] R. Kosloff and D. Kosloff, J. Chem. Phys. 79 (1983) 1823. [21] G. Casati, By. Chirikov, DL. Shepelyansky and I. Guarneri, Phys. Rep. 154 (1987) 77. [22] J.E. Bayfield and D.W. Sokol, Phys. Rev. Lett. 61(1988) 2007. [23]R. Kosloff and D. Kosloff, J. Comput. Phys. 63 (1986) 363. [24] S.J. Chang and K.J. Shi, Phys. Rev. Lett. 55 (1985) 269. [25] J.N. Bardsley, B. Sundaram, L.A. Pinnaduwage and J.E. Bayfield, Phys. Rev. Lett. 56 (1986) 1007. SM. Susskind and R.V. Jensen, Phys. Rev. A 38 (1988) 711. K. Wang and SI. Chu, Phys. Rev. A 39 (1989) 1800. [26] J.F. Jiang and S.!. Chu, to be published.
Computer Physics Communications 63 (1991) 482—493 North-Holland
Nonperturbative methods for the study of multiphoton dynamics in intense periodic and multi-color laser fields Shih-I Chu Department of Chemisizy, University of Kansas, Lawrence, KS 66045, USA
and Tsin-Fu Jiang Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, 30050 Received 2 April 1990
We present a brief account of some recently developed generalized Floquet methods (both time-dependent and time-independent) for the nonperturbative treatment of the time-development of wavefunctions or the density matrix operator of quantum systems driven by intense periodic or multi-color (multi-frequency) laser fields. Both the Schrodinger and Liouville equations are considered. We also explore the feasibility of the fast Fourier transformation (FFT) propagation method for the study of Rydberg atom dynamics. We then present a FF1T case study of the time propagation of the wavefunction and coarse-grained Wigner density distribution for the problem of microwave-driven multiphoton excitation and ionization of Rydberg hydrogen atoms.
1. Introduction The study of the nonlinear response and timedevelopment of multiphoton dynamics of atomic and molecular systems in intense laser fields is a subject of both fundamental and practical interest in modem science and technology *, The theory of multiphoton processes can be formulated in a fully quantum mechanical or semiclassical fashion. In the former approach, both the system and the field are treated quantum mechanically, while in the semiclassical approach, the system is described by a time-dependent schrodinger equation in which the effect of the radiation field is represented by an effective Hamiltonian consistent with Maxwell’s equations. The fully quantized methods are generally difficult to apply as here one is
*
Recent reviews can be found in ref. [11.
0O1O-4655/91/$03.50 © 1991
—
dealing with a large number of time-independent coupled second-order differential equations subject to appropriate asymptotic boundary conditions. In contrast, only first-order time-dependent initial-value problems are involved in semiclassical methods. The semiclassical approach usually leads to Hamiltomans periodic in time through an assumed sinusoidal time variation of the electromagnetic field. The use of the Floquet theorem [21can further greatly simplify the dynamical study. In this article, we shall address the problems and difficulties encountered by the conventional Floquet methods and discuss some possible generalized Floquet formalisms and solutions for nonperturbative treatment of multiphoton and nonlinear optical processes in intense periodic and quasi-periodic fields. Before that, let us first review briefly how the Floquet theorem may be implemented in the semiclassical treatment of multiphoton processes in periodic fields.
Elsevier Science Publishers B.V. (North-Holland)
S. -I. Chu T -F. Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
2. Implementation of the Floquet theorem in the
483
and U(t, t0) is the time-evolution operator defined as
study of multiphoton processes in periodic fields
~P (t)
There are two ways to implement the Floquet theorem into numerical calculations. The first method is the time-dependent propagation technique [5aJ in which one numerically intergrates the Schrodinger equation (h 1)
t~t~)~P (t0),
=
with
(7( t0,
t0)
=
1. (2)
The Floquet theorem asserts that
=
U(t, i a,L7(t,
t~)= ñ(t)U(t,
(1)
ta).
A+2w1
B
B [HF]=
Where
=
n=O n=—1 n=—2
A+wI
0
0
0
n’=2
B
0
0
n’=l
B
0
n’=O
0
B
A
o o
0
B
0
0
A—wi B n’=—l B A—2w1 n’=—2
v=O E00
v=1
0
0
0
0
0
(0) E01
~
0
0
0
(0) E10
o
o
A=
~‘•~
-
I
I
~(0)
I ~(O) 12
and
v=O o
—
B=
v’=O
v’=i
I
v=1
b000, o o
(3)
=
=
n=1
exp[—1Q(t—to)],
where Q is the time-independent quasi-energy operator, and P(t, t0) P(t + T, t0) is periodic. Since U(t0 + nT, t0) [U(t0 + T, t0)J’~, one sees
Here the Hamiltoman is periodic in time, namely, + T) .U(t) with T= 2.rr/w being the period, n=2
t~)=P(t, t0)
b0102:
o b001 1 b0110 a
o
a ....
b0211
a b0112i
~‘
=
a
v’
=
1
0
____________
a
b101~ a a
b11121
i
Fig. 1. Structure of Floquet Hamiltonian HF for multiphoton excitation of molecules. ~ are unperturbed vibrational—rotational energies and ~ are electric dipole coupling matrix elements.
S-I. Chu, T -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
484
that the time-evolution operator over one period, 0(10 + T, ta), provides essentially all the information we ever need about the long-time behavior of the system. Further, in the truncated basis, U( t0 + T, t0) may be diagonalized by some unitary transformation S, S~O(t~4- T, t0)S
=
(4)
c_ID,
where D is a diagonal matrix. Thus O(t0 + T, t0) S e_WS± =
(5)
whereupon
0(t~+ nT,
t0)
S eDS+.
=
(6)
The second way to implement the Floquet theorem is via themethod so-called time-independent Floquet Hamiltonian [3—5].In this approach, one transforms the periodically time-dependent Hamiltonian 1-2(t) into an equivalent time-independent infinite-dimensional Floquet matrix eigenvalue problem. As an example, consider the multiphoton excitation of vibrational—rotational states of a diatomic molecule under the periodic perturbation V(t)
=
—~t(r) E 0 cos(wt
+
~)
(7)
where ~a is the dipole moment operator of the molecule, E0, ~o and t~ are the field amplitude, frequency, and phase, respectively. The corresponding time-independent Floquet matrix HF possesses a block tridiagonal form as shown in fig. 1. The determination of the vibration—rotational quasi-energies and quasi-energy states (QES) thus reduces to the solution of a time-independent Floquet matrix eigenproblem. Figure 1 shows that HF possesses a periodic structure with only the number of w’s in the diagonal elements varying from block to block. This structure endows the quasi-energy eigenvalues and eigenvectors of HF with important periodic properties. Detailed discussions of various Floquet studies of multiphoton excitation of finite-level systems in periodic fields can be found in the recent review articles [4,5]. In the following we shall discuss several major problems encountered by these conventional Floquet techniques when applied to realistic systems.
3. Problems and difficulties associated with conventional Floquet techniques and the developments of generalized Floquet methods for the solutions of these problems The first difficulty is the large atom (molecule)—field basis set problem. Consider the dimensionality of the Floquet matrix such as fig. 1 for example. In practical calculations, one truncates the Floquet matrix into N by N in dimension, where N NFNVNJ, where NF is the number of Floquet photon blocks, N~the number of vibrational levels included in one Floquet block, and N~the number of rotational states included in one vibrational level. As N increases rapidly with the size of the molecule and with the order of multiphoton transitions, and as the computational cx3, the full Floquet analysis can become prohibitively costly even for simple molepense grows as N cules. While the use of a supercomputer can provide solutions to some of these problems, the development of new approximation techniques capable of providing accurate results yet involving =
much smaller Floquet matrix manipulation is in great demand. The most probable path approach (MPPA) [6], using artificial intelligence algorithms (see for example ref. [7]) to select the important Floquet basis states, is one such strategy. The method allows orders-of-magnitude reduction of the dimensionality of the Floquet matrix and yet maintains good accuracy. It has been applied successfully to the study of infrared multiphoton excitation dynamics in SO2 [6] as well as the multiphoton ionization of microwave-driven Rydberg hydrogenic atoms [8]. Other alternative strategies for reducing the dimensionality of the Floquet matrix can be found in the review article [5a]. The second general limitation of conventional Floquet techniques is of more fundamental nature, namely they are applicable only to monochromatic (i.e. one-laser) field problems. However, many recent experiments involve the use of more than one laser field, namely, multi-color or multifrequency excitation (with frequencies w~incommensurate). In such cases, the Hamiltonians are no longer periodic in time and the Floquet theorem is simply not valid. Such limitation has been
5.-I. Chu, T. -F Jiang
/ Nonperturbative methodsfor the study of multiphoton dynamics
recently removed with the development of the so-called many-mode Floquet theorem (MMFT) [5,9—11].The theorem allows the exact transformation of the polychromatic time-dependent
A+2w
21
[Hr]=
B
8
Schrodinger equation into an equivalent time-independent infinite-dimensional eigenvalue prob1cm. Consider for example the interaction of an N-
0
A +
0
-
A
-
0
W21
A
2w21
WHERE C+2w11 X
A1
=
E
C
=
C+w11
0
0
0
X
0
0
x
a
x
c
o
0
X
o
0
0
0 E~
a
B
X
0
x X
:~ 0
v(2) v(2) By
C
-
2w11
0 0
0
I
=
a
0
485
v(2)
v(2) ay
o 0
Fig. 2. Structure of two-mode Floquet Hanültonian. It can be extended to N-frequency problems.
S-I. Chu, T -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
486
level quantum system with two linearly polarized
Performing the long-time average over
monochromatic fields with frequencies to~ and W2 (w0/w2 irrational number). According to MMFT, the time-evolution operator (in matrix form) can be written as
the time-averaged transition probability
t
—
t
0
gives
=
~
=
1112
k1k,
L2~
I<$k~k2I x~1112)(x~111, I aOO) 12
Y
52(t,t0)
(13)
U(t,
t0)
The MMFT has been applied to the study of
72)
~ (y~n1n~ Iexp[
=
nI
—iIF(t
—
t0)]
17200)
two-color multiple quantum NMR transitions in spin systems [10], to the study of SU(N) dynami-
~2
xexp[i(n1co1
+
n2w2)t].
(8)
Here HF is the time-independent two-mode Floquet Hamiltonian defined in the generalized Floquet state basis,
I in~n2)
Ii) x
n1)
x In2)
and satisfies the infinite-dimensional eigenvalue equation, (y1n1n~I HF I -y2k1k2)
~ y2k1k2
=X(yinin2IX).
(9)
Figure 2 depicts the structure of the two-mode Floquet Hamiltonian. The components are ordered in such a way that y runs over unperturbed atomic or molecular states (denoted by Greek letters) of H0 before each change in n1 and n1, in turn, runs over before n2. The quasi-energy eigenvalues (X~1,~2 }and their corresponding eigenvectors (I Xyn1n2)) of ‘F have the following useful bichromatic properties, namely X~flfl
+
=
~1~1
+
n2~2,
(10)
and (Yi, =
m1
+
q1, m2
+
q2
Xyzni±qin2+q2)
(11)
The transition probabilty averaged over the initial time t0 while keeping the elapsed time t t0 fixed is given by —
=
(
—
~
t0)
new measure of the nonlinear optical property of the quantum systems in multi-color laser fields [14]. The third general problem arises from the fact that conventional Floquet techniques are based on the use of the time-dependent Schrodinger equation and classical fields. They cannot therefore be applied to the study of nonlinear optical processes where spontaneous decays and collisional dampings play an essential role. The correct starting point is to consider the Liouville equation for the time development of the density matrix operator of atoms and molecules, allowing for a more natural introduction of the damping mechanisms. The Liouville equations for the time evolution
processes is (see for example ref. [15]) (h i 0,~3(t) [12(t), ,3(t)] =
+
i[k, p3(t)].
=
1) (14)
Here p is the density matrix of the system, reduced by an averaging over all irrelevant degrees 2(t) 2(t). acting i2~ is as thea thermal unperturbed of freedom bath, Hamiltonian and 1 ‘o + J with eigenvalues (E~) and eigenvectors (la)}, a 0, 1, 2 N 1, and V(t) is the interaction =
(/3k 1k~ exp[ —iHF(t
k
been further found that polychromatic quasi-energy eigenfunctions exhibit striking fractal behavior in temporal Fourier space [13] and that the corresponding fractal dimensions provide a unique
of a set of N-level quantum systems, interacting with several coherent linearly polarized monochromatic fields, undergoing relaxation by Markovian
/3
cal evolution and symmetry breaking in N-level systems driven by polychromatic fields [11], and to the study of quasi-periodic or chaotic behavior of quantum systems under the perturbation of strong multi-color fields [12]. Most recently, it has
2
—
to)] I a00)~
1k2
(12)
=
—
5.-I. Chu, T. -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
Hamiltoman between the system and the M monochromatic laser fields given by
dent infinite-dimensional non-Hermitian supereigenvalue equation [5,16], namely
(15)
~~(aTh(m}ILFIoT;(k})
[.k,p(t)]
The relaxation term
consists of T1 (population damping) and T2 (coherent damping) mechanisms which are due to the coupling of the system to the thermal bath by radiative decays and coffisional relaxations, etc. Equation (14) is solved usually by perturbation methods [15] or by direct numerical integration. It has recently been shown, however, that the polychromatic time-dcpendent Liouville equation, eq. (14), can be cxactly reformulated as an equivalent time-indepen-
A+2w21
LF
WHERE
-
•.~
B
B 0 ________
o o
_______
X C*w11
o o
B*
}
I ~p;(n))
{m }
fJ{)(a$
I~
(16)
where L F is the time-independent many-mode Floquet—Liouville superoperator defined in terms of the generalized tetradic-Floquet basis I { m })~Ia>($ I X I (m }>, with (m } m0, m2,..., m~.The structure of the Floquet—Liouvile super-matrix LF is illustrated in fig. 3 for the simplest two-level two-mode case. For N-level sys=
0
B A ________ B* 0
0
0 B ________ Aw21
B~
0 X
0 0 0
0
0
o
Y
000
0 0
0 0
Y 0
0 Y
0 0
0
0
0
0
Y
Y
=
0 0 ________ B A-2w21
_______
C+2w11 X~ A~
x
0
A~2I B* ________ 0 0
487
0
0
0 0
0 0
c
x
o
X* 0
C-w11 X~
c-2~1r
I(yab+yba)
‘Yob
X
0
0
0
‘Yba
0
0
C~ 0
0
0
0
0
wbo~IFba
0
WbaIFba
AND 0
a -v~
_______
_____-
0
-V~
V~
o v~
vbo
ob
a
0
-v~
0
0
______
0
1~ -v111
o .~.
-v~
0
0
-V~
vt2~
V~ —v121
_______
ba
ab
v~
0
0
v~2l -v~2~
0
Fig. 3. Structure of the Floquet—Liouville supermatrix LF for the case of two-level systems (with level spacing w 50) in linearly polarized bichromatic fields. w1 and a2 are the two radiation frequencies, J’~)(i = 1, 2) are the electric dipole couplings, and ye,,, Yb~,and ~b* (i~b+ y~~)/2 are relaxation parameters.
488
S-I. Chu, T -F Jiang / Nonperturbative methods for the stu4v of multipholon dynamics
tems and near-resonance processes, the infinite-dimentional super-matrix LF can be further reduced to a N2 by N2 effective non-Hermitian Hamiltonian [16] via the use of generalized van Vleck nearly degenerate perturbation theory [16,17], allowing analytical treatment of intensity-dependent nonlinear optical properties. The development of the Floquet—Liouville super-matrix (FLSM) formalism thus facilitates the study of intense-field nonlinear optical processes in terms of a few complex super-eigenvalues and eigenvectors. This yields a numerically stable and computationally efficient approach for the unified treatment of nonresonant and resonant, one- and multiple-photon, steady-state, and transient phenomena in nonlinear optical processes, much beyond the conventional perturbative and rotating wave approximation (RWA) approaches [15]. The FLSM method has been applied to the study of intense field multiphoton resonance fluorescence spectra [16], intensity-dependent nonlinear optical susceptibilities [18], and multiple-wave mixings [Sb], etc. Much remains to be explored in this direction.
We use the split-operator algorithm [19] to propagate the wavefunction: (h 1) =
~~
+
t)
exp[ _i(P2/4m) ~t] exp[—il2i~t] xexp[ —i(P2/4m)
~t1~~(t)
+
(17) where the effect of applying the operator cx F—il P2/4m /
is evaluated in terms of discrete Fourier transforms and the FFT algorithm. The Hamiltoman under consideration is the one-dimensional (1D) atomic hydrogen (stretched in the field direction) in the presence of ac and dc fields (in atomic units): H(t) H~+ V(t), where
(18)
=
H
2/dx2
0 and V(t) =
—
1/x,
(19)
d
—
~
=
—Fax sin cot—Fdx.
(20)
4. Microwave-driven multiphoton excitation dy-
F
namics in Rydberg atoms: fast Fourier fransformalion propagation method
0 and Fd are, respectively, the electric field strength of the ac and dc fields, and w is the microwave frequency. Note the singularity of the Coulomb potential at the origin (x 0). This is =
The fast Fourier transformation (FFT) method [19,20] is an attractive and increasingly popular technique for numerically integrating the time-dcpendent Schrodinger equation without the use of basis-set expansion. It also allows the treatment of the continuum exactly. By combining the FFT algorithm along with the Floquet theorem, we have a powerful numerical method for solving the time propagation of the wavefunction m penodic fields. In this section, we shall first look into the feasibility and reliability of the FFT method for treating the Rydberg atom problem. Then we shall apply the method for the study of the microwavedriven multiphoton excitation and ionization of Rydberg hydrogen atoms, a subject of considerable current interest in atomic physics and chaotic dynamics [21].
the source of major problem and error for timedependent propagation method for atomic problems, particularly when the imtial state starts from the ground state (is) orbital. For the Rydberg atom dynamics the singularity problem is cxpected to be less severe. To see this, we first study the time propagation of the unperturbed system a, ~p (t) ñ0~p(t). (21) =
The initial state is chosen to be at n0 72. We use 1024 grid points for the discretization of the coordinate space from x x0 to 25 000 a0, where x0 is the starting point of integration, typically 5 to 10 a0. The result is not sensitive to the choice of x0. Figure 4 shows the percentage error in energy [m (
=
—
=
=
S. -I. Chu, T. -F Jiang / Nonperturbative methods for the study of multiphoton dynamics
489
2.~
1.~.
—1.~
—
—
I
0
I 10000
5000
I 15000
20000
Time (e.u.)
Fig. 4. Percentage error of the energy (E(t)) of atomic hydrogen at n = 72, using the FFT propagation method for the unperturbed Hamiltonian. 0.0008 _________________________________________ = 13.0GHz
00006
00004
00004
A
~
0.0000
=
0.0
T
c.~ =
0.0000
\~.
_________________________________________ t
=
13.0GHz 46.0 r
~
~
0 0008 _______________________________________________
0 0008 _______________________________________________
0.0004
00004
t
=
1 0
T
~
0.0000
o,oooo
01 0.0008 _________________________________________ * * 0.0004
t=2.0-r
~
o.oooo
t .
=
47,0 r
.
~ 00008 _________________________________________ * 8OT * — 0.0004 ~t=4
~
o.oooo
....._..~..=
0 0008 _______________________________________________
0 0008
_____________________________________________
A
00004
~=
0.0004 0.0000
~
~
t = 3.0
T
\ç\..,. ~~‘\_
0.0000
0.0008 _______________________________________________
0.0000
5625
~
Y~.
,~
/~_-,_~
t=500-r 0.0004
~ 0
.
49.0
0.0008 _______________________________________________
t=4.Oi
(1
0.0004
—
t
11250
x
16875
22500
0.0000 __________________________________________________ 0 5625 11250 16875 22500
x
Fig. 5. Time development of the probability density I t~(x)12 versus time. The physical parameters used are w = 13.0 GHz, Fd = 6.5 V/cm, and F 0 = O.O18/N~(au), where N0 = 72.
490
S. -I. Chu, T. -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
in the initial period. After that the system is stabilized to about 0.25% error level. This is rather encouraging considering the large spatial distance covered. Higher accuracy may be achieved by increasing the number of spatial mesh points. We now consider the multiphoton excitation dynamics in strong microwave and dc fields. The
times at t/T = 0, 1, 2, 3, 4, and at some larger times with t/T = 46—50, where T = 2’rr/w is one period of microwave field oscillation (w = 13.0 GHz). Typically, we used about 26000 time stems (t~t)per field oscillation (T) in the time propagation. In this calculation, the spatial coordinate x spans from x0 to 25000 a0. To prevent unphysi-
Schrodinger equation is solved using the split-operator and FFT algorithms with the Hamiltonian given by eq. (18), and the initial state at n0 = 72. The dc field strength is chosen to be F,~,= 6.5 V/cm as used in the recent experiment [22]. We have performed the calculations for several microwave frequencies. Some of the results are presented below, Figure 5 shows the probability density I %~(x) I 2 versus the spatial coordinate x for several initial
cal reflection from the outer boundary, we have adopted the absorbing boundary method (see for example ref. [23]) and place a slowly vanishing filter function (b = 0.004) —
f ( x)
=
{
1
+
exp [ b ( x
—
x1)] }
(22)
centered at X~= 22500 a0. The result is found to be insensitive to the choice of b and X~, provided X~ is chosen large enough and b small enough.
a
011
b
T0~
—011
‘
0
12500
25000
Fig. 6. (a) 3D coarse-grained Wigner density distribution for the unperturbed (1D) atomic hydrogen at n = 72. (b) The corresponding 2D contour plot (dotted lines from Outer to inner region: probabity density = iO~,~ and 102, respectively. Solid lines from outer to inner region: probability density = ~o 1.5 10 ~, and 10 05, respectively.).
S-I. Chu, T. -F. Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
491
b
a
= 0
12500
25000
c~
=~ 0
0
12500
25000
d
L
=~ 12500
25000
0
Fig. 7. The time development of the CGW density contour plot for (a) t = 2T, (b) Parameters used same as fig. 5.
Figure 5 reveals that the electronic probability flux spreads and recollects (in x) as time advances with occasional leaking to the continuum. At this stage, it is instructive to construct the coarse-grained Wigner (COW) density distribution (see for example ref. [24]), providing a vivid visualization of the time-development of the quantum system in phase space (x, p). Figure 6a shows the 3D CGW density plot for the unperturbed (1D) atomic hydrogen at n = 72. The corresponding 2D contour diagram is depicted in fig. 6b. It is seen that the largest probability density is initially located nearby x 11 000 a0 and p 0. Figures 7a—d show the time-development of the COW density contour diagram (for t = 2, 5, 10
12500
t =
5r, (c)
t =
lOr, and (d)
25000
t =
50r, respectively.
and 50 T), after the microwave and dc fields have been turned on. We have also performed the calculation of the population of the bound states by projecting ( t) onto unperturbed hydrogenic states I n). A timeaveraged result is shown in fig. 8 (dots joined by solid lines). Also shown here are the recent experimental results by Bayfield and Sokol [22] (solid curve). In this preliminary calculation, we have not yet taken into account the time-varying envelope of the microwave fields and there is some ambiguity regarding how to take the time average ~/i
appropriately. Nevertheless, the overall agreement in shape and magnitude appears reasonably good. As far as we know, this is the first calculation of
492
S-I. Chu, T -F Jiang / Nonperturbative methodsfor the study of multiphoton dynamics
—I 60
70
I
I
I
80
90
100
110
State Quantum Number
Fig. 8. Population versus state quantum number n. Solid curve: experimental results; dots joined by solid lines: FF1’ results. Parameters used same as fig. 5.
microwave-driven Rydberg atom dynamics, taking into account the effect of continuum exactly * This preliminary result shows that the FF1’ propagation method is a feasible and powerful numerical technique for probing strong-field Rydberg atom quantum dynamics. More refined calculations are being performed and will be discussed elsewhere [26].
References [11M.S.
Feld and V.S. Letokhov, eds., Coherent Nonlinear
Optics (Springer, New York, 1980). J. Jortner, R.D. Levine and S.A. Rice, eds., Photoselective
Chemistry, Advances in Chemical Physics, vol. 47 (Wiley, New York, 1981). S.H. Lin, ed., Advances in Mutliphoton Processes and
This work was partially supported by US Department of Energy (Division of Chemical Sciences), and by the University of Kansas General Research Funds.
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Former quantal studies of this problem have used primarily
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Acknowledgement
*
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