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Laser control of population dynamics with bichromatic fields: direct search through field amplitude, frequency, and phase space
2 August 1996
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical PhysicsLetters 257 (1996) 658-664
Laser control of population dynamics with bichromatic fields: direct search through field amplitude, frequency, and phase space Ashish K. Gupta a, Peter Gross b, Deepa B. Bairagi a,l, Manoj K. Mishra a a Department of Chemistry, Indian Institute of Technology, Powai, Bombay 400 076, India b Tara Institute o f Fundamental Research, Horai Bhabha Road, Bombay 400 005, India
Received 11 July 1995; in final form 17 April 1996
Abstract
A general technique for integrating through parameter space in order to determine how the eigenvalues and eigenfunctions of the time-dependent or time-independent Schr'6dinger equation change with parameter values is outlined, and its application to laser field amplitude, frequency, and phase (in a bichmmatic field) are presented. The enhanced efficiency of this method provides 3-D plots of the objective versus pairs of laser parameters (frequency and phase, amplitude and phase) thereby permitting a visual determination of suitable field parameters which will help achieve desired populations in a multi-level system using a bichromatic field. Applications charting the field parameters for selective vibrational excitation of hydrogen fluoride are presented.
1. Introduction
In a recent communication [ 1] we have proposed a new method to obtain quantum dynamics in the presence of strong (nonperturbative) periodic fields of the form A cos(tot) over a wide range of field amplitudes and frequencies by solving the time-dependent Schr&linger equation (TDSE) only once. The method utilizes Floquet theory [2] and integrates through frequency and amplitude space using parametric equations of motion (PEM) for quasienergies and Floquet eigenvectors. Computational savings achieved by the PEM approach to the solution of TDSE in some cases is even greater than 100 times as compared to the brute force way of solving these problems repeatedly for different field parameters. l CSIR Senior Research Fellow.
Although the results presented were based on eigenstate population, any observable can be calculated using PEM since one obtains the wavefunction directly. These results, however, were limited to single color lasers. Bichromatic control of chemical reactions by manipulating the phase between the two colors has received considerable attention [3-5] and the dominant role of phase in achieving two-color coherent control of photodissociation of H~ [6] and D~" [7] has been demonstrated recently. The calculations presented in these works are, however, limited only to a few discrete values of the phase parameter since solution of the TDSE for a large number of phase values is computationally demanding. It is our purpose in this Letter to extend our earlier formulation of the amplitude and frequency PEM [1] for the TDSE to allow an efficient treatment
0009-2614/96/$12.00 Copyright© 1996 Elsevier Science B.V. All rights reserved PII S0009-2614(96)00571-4
659
A.K. Gupta et a L / Chemical Physics Letters 257 (1996) 658-664
of photodynamics as a function of phase difference between two colors of a bichromatic laser, once again, without having to repeatedly solve the TDSE for each value of the phase difference. The large computational savings permit us to obtain 3-D plots of population dynamics as a simultaneous function of frequency and phase or amplitude and phase thereby permitting global insights into the field induced dynamics which can be utilized to determine appropriate field parameters for the given photodynamical task. In particular, extreme sensitivity of photolysis branching ratios to initial vibrational state of the molecule is well known [8-14] and a scheme for efficient search of appropriate field parameters to prepare the optimal initial state is an important task. As will be seen, the PEM-based method presented here allows considerable flexibility in the choice of field parameters for a given level population target, the field forms, however, are restricted to two cw pulses and therefore other field forms may exist which provide better population dynamics, i.e. dynamics which more closely match the desired ones. For example, chirped pulses have been shown to effectively populate high lying states [15,16] and dissociate. In our formulation, the amplitudes and the frequencies of the two cw pulses are by definition time-independent, unlike a chirped pulse. Also, the flexibility provided in allowing for time variability in amplitude functions of the cw components have been shown to be necessary for obtaining the optimal field from optimal control theory [17]. Thus, the restrictive nature of our field form (time-independent amplitude and frequency) cannot allow for the best possible population results. However, it does allow for computationally efficient search of the field parameters using the method described below which do provide adequate results, and the fact that the same result may be obtained through many different choices adds flexibility to experimental attainment of these goals. The balance of this Letter is as follows. In Section 2 we outline the general formalism of the PEM and their application to the TDSE for evaluating quantum dynamics over a range of field parameters. Earlier formulation [1], was strictly valid for the situation where the Hamiltonian depends linearly on the field parameters, and here we extend the PEM formalism
to phase in bichromatic fields which is a nonlinear parameter. Application of the phase related PEM method to a discrete-level system (vibrational eigenstates of the HF molecule) is presented. Eigenstate populations as a function of field parameters (amplitudes and phase, frequency and phase) are shown in 3-D plots and their relevance to control of population dynamics via suitable choice of field parameters is discussed in Section 3. Some concluding remarks and other possible applications of this method are collected in Section 4.
2. Formulation For a time-independent Hamiltonian H ( k ) = H 0 + v(k) where H 0 does not depend on k, H o e . ffi Ent~n with (t~= 10n) ffi 8=n, the evolution of energy level E n and eigenfunctions t~n as a function of the linear perturbation (v(k) = kV, d H / d k ffi H' ffi V) are given by [ 18,19] den
dk -- H~n ffi Vnn' dH" n
[ H',, [ 2
y'.
dk
m ./. n
=
E,
d'~-
= E m~'n
(2)
En -E.' n',n;.
ItE-E+ 1) -
+
s
d0,,
(1)
H~,, ~E, m
e. '
C3)
(4)
where H%=
660
A.K. Gupta et aL / Chemical Physics Letters 257 (1996) 658-664
also been applied to many interesting quantum chemistry problems [20]. The TDSE in the semi-classical dipole length approximation is:
and they belong to a composite R $ T configuration and temporal Hilbert space introduced by Sambe [26] and further discussed in Refs. [25,2]. The eigenvectors satisfy the orthonormality condition
a ~ ( at r , t) ffi - -hi [ H° ^ - ~( r) A c o s ( t o t ) ] ~ ( r , t),
((+m I ,l,.)) ffi -~-f0 dt(,l,,, I ,I,.) -- 8~,.
.8
(5) where H0 is the field-free Hamiltonian. Expanding • (r, t) in terms of its field-free eigenstates and performing the standard manipulations, we obtain the equations of motion for the vector an(t) containing the time-dependent eigenstate amplitudes: d
i
-~a, ffi - - ~ [ e - p.A cos(tot)]an(t),
(6)
1
T
(9)
If k = A, then using this def'mition of the inner product, we have extended the PEM approach to the time-dependent problem in an earlier communication [1] where the matrix elements H" n of the amplitude PEM are
((+.
dH(t)
= - ((+. I
+.)) (r) cos(tot) I
(10)
where the subscript n on vector an(t) indicates a particular initial condition (see below). Here, E is the diagonal energy matrix and p. is the dipole coupling matrix [ 1]. An appealing and economic method for solution of the TDSE utilizes the Flequet approach when the perturbing field is periodic. The Floquet approach has been used extensively in studies ranging from multiphoton ionization [22] to state-selective vibrational excitation [23]; a good summary of the method is provided in Ref. [24]. Following Ref. [25], the vector an(t) is propagated over one optical cycle (0 ~ "r, where -rffi 2"tr/to) N times (n = 1, 2,. • -, N), once for each linearly independent initial condition (1.0, 0.0, 0.0, • .- ), (0.0, 1.0, 0.0, • • • ), etc. The N x N propagator matrix U(~, 0) defined by a(x)ffi U('r, 0)a(0) (where a(0) is the identity matrix here) is constructed and diagonalized to obtain the quasienergies En and eigenvectors at time t - - 0 , i.e. d~.(0). The propagator at any time L-r where L is an integer can then be constructed using
Thus the energies E n in Eqs. (1)-(4) are replaced with the quasienergies c n, the eigenvectors ~. are replaced by the Floquet vectors ~bn(0), and the matrix elements H" n are replaced by their Sambe counterparts in Eq. (10). Propagation of Eqs. (1)-(4) for k (here amplitude A) then leads to the variation of the quasienergies and Fioquet vectors with respect to A from which the propagator matrix in Eq. (7) is constructed to determine the wavefnnction at any time L-r. Similarly, if k ffi to, then with the transformarion tot ~--~ [1] the Sambe matrix elements become a H ' n ffi ((~bm I -ih~--~ I & . ) ) (11)
U( L,, 0) -- +(0) exp(-iEL,)Op*(0).
H(t)t~n(t ) -- ( H 0 - ~ [ A I cos(tort)
(7)
These Flequet vectors are eigenvectors of the
time-dependent SchrSdinger equation whose eigenvalues are the quasienergies en. They are related by the eigenvalue equation a ( H 0 - p.A c o s ( t o t ) - i h ~ t ) d ~ . ( t ) = ~ . * . ( t ) ,
(8)
and wavefunction dynamics for any range of frequencies may be determined using F_,qs. (1)-(4) with the new definition for H" n. For bichromatic light the Floquet eigenvector and eigenvaluc equation is given by
+a2 cos(to2t + x+.(t)
°)
- igg
(12)
Unlike either the amplitude case (k---A) or the frequency case (k ffi to) (after the tot = ? transformation), the Hamiltonian is not linearly dependent on the variable field parameter in the case of phase
A.K. Gupta et aL / Chemical Physics Letters 257 (1996) 658-664
(1~ -- 8). Therefore the four equations in Eqs. (1)-(4) will not form a closed set. However, taking into account the fact that dH(t)ldSffi-d3H(t)/d83, we can form a closed set of equations for the phase: dEn(k) dk -- Hnn'
(13)
d
H,. dX
dn.,, dk
I,I,,(x)),
= E
(14)
1
H.~n;'~
v"
=/~ E.(k)-Ei(k)
H.IH~.
E
dH" n
'
(15)
H~IH~%
dk
E~,(X) - E l ( k ) - H i "
H'iH;n -.I-
E E,(l~'~--~-iCk),
(16)
l
where
Hm.= (~.(k) [
dH(t)(k)
dk
I~n(l~))
(17)
and H,',, ffi ( , I , , ( k ) i
compute the dipole matrix elements were computed via the Fourier grid Hamiltonian (FGH) method [28]. Numerically, both the TDSE (Eq. (6)) and the PEM equations (Eqs. (1)-(4)) were integrated with a fifth-order Runge-Kutta integrator. In all results presented below we present the time-averaged populations in accordance with actual laboratory conditions. In other words, at each value of k ffi 8 where the population dynamics is to be evaluated, we compute (see Ref. [29]) the time-averaged population of the nth state:
P~ = ~ I~bnm(O)12 ] ~bt,(0)
i
d2H(t)(k) dk 2 i ~I',(k)).
661
] 2,
(19)
n--I
where m is the total number of states included in the calculation (here 14 states were found to be sufficient). The molecules are initially assumed to be in the ground vibrational state. The dependence of the HF vibrational population dynamics as a function of field amplitude and frequency has been discussed in some detail earlier [1]. In the present work we examine the population dynamics as a function of two variables simultaneously. The effect of phase and frequency variation on the population in the first excited level is presented in Fig. 1. In this case, we set ~oI = 0.01814 au and to 2 ffi ¢ o l / 2 . T h e frequency co I is c h o s e n t o b e nearly
(18)
Eqs. (13)-(16) furnish a closed set of equations for the phase parameter and have been solved for discrete vibrational levels of the I-IF molecule using a fifth-order Runge-Kutta numerical integrator. These results are discussed in Section 3.
3. Results and discussion Due to its large dipole moment, hydrogen fluoride has been a prototypical molecule for testing different methods for solving the TDSE. Our previous application [1] of the frequency and amplitude PEM were also for this molecule and this is the system of our choice for the initial application of the phase PEM as well. We consider vibrational excitation of rotationless hydrogen fluoride in its ground electronic state. Potential and dipole functions are from Ref. [27]. Bound state energies and wavefunctions used to
,Sin --Aapo-
Fig. 1. Plot of population lransferred to the first excited vibrational level by the bichromatic field E(t)ffi A, cos(tot)+ A 2 co~21--tot-{-8) as a function of to and 8. The critical role of resonance is manifested through the peak at to ffi 0.1814 au for all values of the phase difference 8.
662
A.g. Gupta et al. / Chemical Physics Letters 257 (1996) 658-664
(a)
i~'1~
~.~
l
~'~(b)
i!' todiedifferentcxcit~lvibrationallevelsbythebichmmatifield c ~(t)
~)],
Fig. 2. Plot of population I~l~sf~n~l = A[co~t) + c~m2t + ¢0~ = 0.01820 au, co2 = 0.95to,, as a function of A and 8. The large variation with respect m 8 affirms the sensitivity to phase seen in earlier investigations [6] and offers the possibility of phase based control. (a) o = 1, (b) u = 2 and (c) v = 7. The identification of a narrow range of amplitude and phase parameters responsible for large population transfer in (c) can be a significant input for design of an appropriate laser field.
A.K. Gupta et aL / Chemical Physics Letters 257 (1996) 658-664
resonant with the first fundamental (v -- 0 -o 1) transition. Note that there is very little population transfer for frequency values which do not satisfy the resonance condition and the same feature is observed for population transfer to other levels as well. This 3-D plot therefore establishes the critical role of the resonance condition in photoexcitation. Similar resuits were observed in our earlier study [1] where an increase in intensity only leads to some power broadening of the excitation spectrum. Otherwise the maximum population transfer was indeed seen only for the resonant frequency. We next examine the case where to~ ffi 0.01820 and to 2 --0.95a h. Here, the to I field component is nearly resonant with the v ffi 0 -o 1 transition and to 2 is nearly resonant with v ffi 1 ~ 2. Note that in this case there will be greater transition probability to states above v ffi 1. Computationally, we note that the period -t in this case is 20 x 2"it/tel; in this way an integral number of optical cycles of both the to t and to 2 components fit within the FIoquet period "r. Although multiple cycles of both field components are incorporated into % the Floquet formalism is still valid since we define -~ as the Floquet period: a similar situation occurred in Ref. [30] where population dynamics with a Gaussian pulse train were studied. From both Figs. 2a and 2b it is clear that even for low field amplitudes, a suitable choice of phase can induce substantial population transfer which would otherwise have required much more intense fields. In particular, Fig. 2c offers a welcome surprise where a narrow range of field amplitude and phase difference 9.01E--4 •
g.ooC- 4 .
8.ggE-',
8.gBE-4 '
8.97E-4
B.gS£-4
8.gsE-4
663
is seen to transfer sizable population to a very highly excited state. Such results would be difficult without the 3-D plots presented here, which in turn would be much too demanding to generate without recourse to the PEM-based approach to the solution of TDSE advocated here. It may also be noted that the variation of population dynamics with respect to phase is not due to constructive interference between the two cw components whereby for some 8 values the intensity increases significantly; this was tested by plotting maximum intensity versus phase for B = 0 -o q¢ (see Fig. 3). Thus from this we can infer that the change of populations with phase is a coherent phenomenon and transition are occurring via interfering paths.
4. Concluding remarks In this Letter we have formulated phase related parametric equations of motion for an efficient investigation of quantum dynamics in the presence of bichromatic lasers. The resulting equations offer computational savings of up to a factor of 100 over the brute force approach to the solution of the TDSE for each new value of the phase difference. These PEM-based savings have made possible 3-D plots of population in various levels as a simultaneous function of phase and frequency or phase and amplitude permitting a global insight which could be tapped advantageously for selecting suitable field parameters for a given photodynamical task. In particular, appropriately populated initial vibrational levels play a critical role in initial state based coherent control of photodissociation [3,4,14] and the tools developed here could be quite useful in this important chemical task. Finally, though our initial applications have focussed on vibrational population dynamics the technique presented here is general and could be used for the investigation and control of photodynamics of other discrete optical and spin levels in atoms and molecules.
J Phase
(Rod|ansi
Fig. 3. Maximum laser intensity as a function o f p h a s e f o r the bichromatic field E ( t ) ffi A[coc,(tolt) + c o s ( t o 2 t + 8)], to t ffi 0 . 0 1 8 2 0 au, to2 ffi 0-95tot, At ffi 0.01 a u a n d A 2 ffi 0 . 0 2 au.
Acknowledgement DBB acknowledges support from CSIR Junior Research Fellowship no. 9/87(143) 92/EMR-I.
664
A.lg. Gupta et a L / Chemical Physics Letters 2.$7 (1996) 658-664
MKM acknowledges support from the Board of Research in Nuclear Sciences of the Department of Atomic Energy, India.
References [1] P. Gross, A.K. Gupta, D.B. Bairagi and M.K. Mishra, Chem. Phys. Letters 236 (1995) 8. [2] S.I. Chu, Advan. At. Mol. Phys. 21 (1985) 197. [3] P. Brumer and M. Shapiro, Ann. Rev. Phys. Chem. 43 (1992) 257. [4] P. Bmmer and M. Shapiro, Accounts Chem. Res. 22 (1989) 407. [5] W.S. Warren, H.A. Rabitz and M. Dahleh, Science 259 (1993) 1581. [6] E. Charmn, A. Giusti-Suzor and F.H. Mies. Phys. Rev. Letters 71 (1993) 692. [7] J. Wang and S.L Chu, Chem. Phys. Letters 227 (1994) 663. [8] E.F. Dishoeck, M.C. Van Hermet and A. Dalgamo, J. Chem. Phys. 77 (1982) 3693. [9] S.C. Givertz and G.G. Balint-Kurti, J. Chem. Soc. Faraday Trans. II 82 (1986) 1231. [1(3] M.H. Alexander, B. Pouilly and T. Duhoo, J. Chem. Phys. 99 (1993) 1752. [I 1] I.H. Gersonde, S. Hennig and H. Gabriel, J. Chem. Phys. 101 (1994) 9558. [12] I. Levy and M. Shapiro, J. Chem. Phys. 89 (1988) 2900.
[13] C. Kalyanaraman and N. Sathyamurthy, Chem. Phys. Letters 2O9 (1993) 52. [14] P. Gross, A. Gupta, D.B. Bairagi and M.K. Mishra, J. Chem. Phys., in press. [15] S. Chelkowski, A.D. Bandrauk and P.B. Corkora, Phys. Rev. Letters 65 (1990) 2355. [16] S. Chelkowski and A.D. Bandrank, J. Chem. Phys. 99 (1993) 4279. [17] H. Shen, J.P. Dussautt and A.D. Bandrauk, Chem. Phys. Letters 221 (1994) 498. [18] P. Pechukas, Phys. Rev. Letters 51 (1983) 943. [19] T. Yukawa, Phys. Letters A 116 (1986) 227. [20] D.A. Mazzioti, M.K. Mishra and H. Rabitz, J. Phys. Chem. 99 (1995) 112. [21] X. Yang and J. BurgdSrf, Phys. Rev. A 48 0993) 83. [22] J.N. Bardsley, A. SzSke and M.J. Coraella, J. Phys. B 21 (1988) 3899. [23] H.P. Brener, K. Dietz and M. Holthans, J. Phys. B 24 (1991) 1343. [24] F.H.M. Faisal, Theory of raultiphoton processes (Plenum Press, New York, 1987). [25] K.F. Milfeld and R.E. Wyatt, Phys. Rev. A 27 (1983) 72. [26] H. Samhe, Phys. Rev. A 7 (1973) 2203. [27] A. Guldebrand and G.D. Billing, Chem. Phys. Letters 186 0991) 229. [28] C.C. Marston and G.G. Balint-Kurti, J. Chem. Phys. 91 (1989) 3571. [29] S.C. Leasure, K.F. Milfeld and R.E. Wyatt, J. Chem. Phys. 74 (1981) 6197. [30] B.G. Dibble and R.B. Shirts, J. Chem. Phys. 94 (1991) 3451.
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical PhysicsLetters 257 (1996) 658-664
Laser control of population dynamics with bichromatic fields: direct search through field amplitude, frequency, and phase space Ashish K. Gupta a, Peter Gross b, Deepa B. Bairagi a,l, Manoj K. Mishra a a Department of Chemistry, Indian Institute of Technology, Powai, Bombay 400 076, India b Tara Institute o f Fundamental Research, Horai Bhabha Road, Bombay 400 005, India
Received 11 July 1995; in final form 17 April 1996
Abstract
A general technique for integrating through parameter space in order to determine how the eigenvalues and eigenfunctions of the time-dependent or time-independent Schr'6dinger equation change with parameter values is outlined, and its application to laser field amplitude, frequency, and phase (in a bichmmatic field) are presented. The enhanced efficiency of this method provides 3-D plots of the objective versus pairs of laser parameters (frequency and phase, amplitude and phase) thereby permitting a visual determination of suitable field parameters which will help achieve desired populations in a multi-level system using a bichromatic field. Applications charting the field parameters for selective vibrational excitation of hydrogen fluoride are presented.
1. Introduction
In a recent communication [ 1] we have proposed a new method to obtain quantum dynamics in the presence of strong (nonperturbative) periodic fields of the form A cos(tot) over a wide range of field amplitudes and frequencies by solving the time-dependent Schr&linger equation (TDSE) only once. The method utilizes Floquet theory [2] and integrates through frequency and amplitude space using parametric equations of motion (PEM) for quasienergies and Floquet eigenvectors. Computational savings achieved by the PEM approach to the solution of TDSE in some cases is even greater than 100 times as compared to the brute force way of solving these problems repeatedly for different field parameters. l CSIR Senior Research Fellow.
Although the results presented were based on eigenstate population, any observable can be calculated using PEM since one obtains the wavefunction directly. These results, however, were limited to single color lasers. Bichromatic control of chemical reactions by manipulating the phase between the two colors has received considerable attention [3-5] and the dominant role of phase in achieving two-color coherent control of photodissociation of H~ [6] and D~" [7] has been demonstrated recently. The calculations presented in these works are, however, limited only to a few discrete values of the phase parameter since solution of the TDSE for a large number of phase values is computationally demanding. It is our purpose in this Letter to extend our earlier formulation of the amplitude and frequency PEM [1] for the TDSE to allow an efficient treatment
0009-2614/96/$12.00 Copyright© 1996 Elsevier Science B.V. All rights reserved PII S0009-2614(96)00571-4
659
A.K. Gupta et a L / Chemical Physics Letters 257 (1996) 658-664
of photodynamics as a function of phase difference between two colors of a bichromatic laser, once again, without having to repeatedly solve the TDSE for each value of the phase difference. The large computational savings permit us to obtain 3-D plots of population dynamics as a simultaneous function of frequency and phase or amplitude and phase thereby permitting global insights into the field induced dynamics which can be utilized to determine appropriate field parameters for the given photodynamical task. In particular, extreme sensitivity of photolysis branching ratios to initial vibrational state of the molecule is well known [8-14] and a scheme for efficient search of appropriate field parameters to prepare the optimal initial state is an important task. As will be seen, the PEM-based method presented here allows considerable flexibility in the choice of field parameters for a given level population target, the field forms, however, are restricted to two cw pulses and therefore other field forms may exist which provide better population dynamics, i.e. dynamics which more closely match the desired ones. For example, chirped pulses have been shown to effectively populate high lying states [15,16] and dissociate. In our formulation, the amplitudes and the frequencies of the two cw pulses are by definition time-independent, unlike a chirped pulse. Also, the flexibility provided in allowing for time variability in amplitude functions of the cw components have been shown to be necessary for obtaining the optimal field from optimal control theory [17]. Thus, the restrictive nature of our field form (time-independent amplitude and frequency) cannot allow for the best possible population results. However, it does allow for computationally efficient search of the field parameters using the method described below which do provide adequate results, and the fact that the same result may be obtained through many different choices adds flexibility to experimental attainment of these goals. The balance of this Letter is as follows. In Section 2 we outline the general formalism of the PEM and their application to the TDSE for evaluating quantum dynamics over a range of field parameters. Earlier formulation [1], was strictly valid for the situation where the Hamiltonian depends linearly on the field parameters, and here we extend the PEM formalism
to phase in bichromatic fields which is a nonlinear parameter. Application of the phase related PEM method to a discrete-level system (vibrational eigenstates of the HF molecule) is presented. Eigenstate populations as a function of field parameters (amplitudes and phase, frequency and phase) are shown in 3-D plots and their relevance to control of population dynamics via suitable choice of field parameters is discussed in Section 3. Some concluding remarks and other possible applications of this method are collected in Section 4.
2. Formulation For a time-independent Hamiltonian H ( k ) = H 0 + v(k) where H 0 does not depend on k, H o e . ffi Ent~n with (t~= 10n) ffi 8=n, the evolution of energy level E n and eigenfunctions t~n as a function of the linear perturbation (v(k) = kV, d H / d k ffi H' ffi V) are given by [ 18,19] den
dk -- H~n ffi Vnn' dH" n
[ H',, [ 2
y'.
dk
m ./. n
=
E,
d'~-
= E m~'n
(2)
En -E.' n',n;.
ItE-E+ 1) -
+
s
d0,,
(1)
H~,, ~E, m
e. '
C3)
(4)
where H%=
660
A.K. Gupta et aL / Chemical Physics Letters 257 (1996) 658-664
also been applied to many interesting quantum chemistry problems [20]. The TDSE in the semi-classical dipole length approximation is:
and they belong to a composite R $ T configuration and temporal Hilbert space introduced by Sambe [26] and further discussed in Refs. [25,2]. The eigenvectors satisfy the orthonormality condition
a ~ ( at r , t) ffi - -hi [ H° ^ - ~( r) A c o s ( t o t ) ] ~ ( r , t),
((+m I ,l,.)) ffi -~-f0 dt(,l,,, I ,I,.) -- 8~,.
.8
(5) where H0 is the field-free Hamiltonian. Expanding • (r, t) in terms of its field-free eigenstates and performing the standard manipulations, we obtain the equations of motion for the vector an(t) containing the time-dependent eigenstate amplitudes: d
i
-~a, ffi - - ~ [ e - p.A cos(tot)]an(t),
(6)
1
T
(9)
If k = A, then using this def'mition of the inner product, we have extended the PEM approach to the time-dependent problem in an earlier communication [1] where the matrix elements H" n of the amplitude PEM are
((+.
dH(t)
= - ((+. I
+.)) (r) cos(tot) I
(10)
where the subscript n on vector an(t) indicates a particular initial condition (see below). Here, E is the diagonal energy matrix and p. is the dipole coupling matrix [ 1]. An appealing and economic method for solution of the TDSE utilizes the Flequet approach when the perturbing field is periodic. The Floquet approach has been used extensively in studies ranging from multiphoton ionization [22] to state-selective vibrational excitation [23]; a good summary of the method is provided in Ref. [24]. Following Ref. [25], the vector an(t) is propagated over one optical cycle (0 ~ "r, where -rffi 2"tr/to) N times (n = 1, 2,. • -, N), once for each linearly independent initial condition (1.0, 0.0, 0.0, • .- ), (0.0, 1.0, 0.0, • • • ), etc. The N x N propagator matrix U(~, 0) defined by a(x)ffi U('r, 0)a(0) (where a(0) is the identity matrix here) is constructed and diagonalized to obtain the quasienergies En and eigenvectors at time t - - 0 , i.e. d~.(0). The propagator at any time L-r where L is an integer can then be constructed using
Thus the energies E n in Eqs. (1)-(4) are replaced with the quasienergies c n, the eigenvectors ~. are replaced by the Floquet vectors ~bn(0), and the matrix elements H" n are replaced by their Sambe counterparts in Eq. (10). Propagation of Eqs. (1)-(4) for k (here amplitude A) then leads to the variation of the quasienergies and Fioquet vectors with respect to A from which the propagator matrix in Eq. (7) is constructed to determine the wavefnnction at any time L-r. Similarly, if k ffi to, then with the transformarion tot ~--~ [1] the Sambe matrix elements become a H ' n ffi ((~bm I -ih~--~ I & . ) ) (11)
U( L,, 0) -- +(0) exp(-iEL,)Op*(0).
H(t)t~n(t ) -- ( H 0 - ~ [ A I cos(tort)
(7)
These Flequet vectors are eigenvectors of the
time-dependent SchrSdinger equation whose eigenvalues are the quasienergies en. They are related by the eigenvalue equation a ( H 0 - p.A c o s ( t o t ) - i h ~ t ) d ~ . ( t ) = ~ . * . ( t ) ,
(8)
and wavefunction dynamics for any range of frequencies may be determined using F_,qs. (1)-(4) with the new definition for H" n. For bichromatic light the Floquet eigenvector and eigenvaluc equation is given by
+a2 cos(to2t + x+.(t)
°)
- igg
(12)
Unlike either the amplitude case (k---A) or the frequency case (k ffi to) (after the tot = ? transformation), the Hamiltonian is not linearly dependent on the variable field parameter in the case of phase
A.K. Gupta et aL / Chemical Physics Letters 257 (1996) 658-664
(1~ -- 8). Therefore the four equations in Eqs. (1)-(4) will not form a closed set. However, taking into account the fact that dH(t)ldSffi-d3H(t)/d83, we can form a closed set of equations for the phase: dEn(k) dk -- Hnn'
(13)
d
H,. dX
dn.,, dk
I,I,,(x)),
= E
(14)
1
H.~n;'~
v"
=/~ E.(k)-Ei(k)
H.IH~.
E
dH" n
'
(15)
H~IH~%
dk
E~,(X) - E l ( k ) - H i "
H'iH;n -.I-
E E,(l~'~--~-iCk),
(16)
l
where
Hm.= (~.(k) [
dH(t)(k)
dk
I~n(l~))
(17)
and H,',, ffi ( , I , , ( k ) i
compute the dipole matrix elements were computed via the Fourier grid Hamiltonian (FGH) method [28]. Numerically, both the TDSE (Eq. (6)) and the PEM equations (Eqs. (1)-(4)) were integrated with a fifth-order Runge-Kutta integrator. In all results presented below we present the time-averaged populations in accordance with actual laboratory conditions. In other words, at each value of k ffi 8 where the population dynamics is to be evaluated, we compute (see Ref. [29]) the time-averaged population of the nth state:
P~ = ~ I~bnm(O)12 ] ~bt,(0)
i
d2H(t)(k) dk 2 i ~I',(k)).
661
] 2,
(19)
n--I
where m is the total number of states included in the calculation (here 14 states were found to be sufficient). The molecules are initially assumed to be in the ground vibrational state. The dependence of the HF vibrational population dynamics as a function of field amplitude and frequency has been discussed in some detail earlier [1]. In the present work we examine the population dynamics as a function of two variables simultaneously. The effect of phase and frequency variation on the population in the first excited level is presented in Fig. 1. In this case, we set ~oI = 0.01814 au and to 2 ffi ¢ o l / 2 . T h e frequency co I is c h o s e n t o b e nearly
(18)
Eqs. (13)-(16) furnish a closed set of equations for the phase parameter and have been solved for discrete vibrational levels of the I-IF molecule using a fifth-order Runge-Kutta numerical integrator. These results are discussed in Section 3.
3. Results and discussion Due to its large dipole moment, hydrogen fluoride has been a prototypical molecule for testing different methods for solving the TDSE. Our previous application [1] of the frequency and amplitude PEM were also for this molecule and this is the system of our choice for the initial application of the phase PEM as well. We consider vibrational excitation of rotationless hydrogen fluoride in its ground electronic state. Potential and dipole functions are from Ref. [27]. Bound state energies and wavefunctions used to
,Sin --Aapo-
Fig. 1. Plot of population lransferred to the first excited vibrational level by the bichromatic field E(t)ffi A, cos(tot)+ A 2 co~21--tot-{-8) as a function of to and 8. The critical role of resonance is manifested through the peak at to ffi 0.1814 au for all values of the phase difference 8.
662
A.g. Gupta et al. / Chemical Physics Letters 257 (1996) 658-664
(a)
i~'1~
~.~
l
~'~(b)
i!' todiedifferentcxcit~lvibrationallevelsbythebichmmatifield c ~(t)
~)],
Fig. 2. Plot of population I~l~sf~n~l = A[co~t) + c~m2t + ¢0~ = 0.01820 au, co2 = 0.95to,, as a function of A and 8. The large variation with respect m 8 affirms the sensitivity to phase seen in earlier investigations [6] and offers the possibility of phase based control. (a) o = 1, (b) u = 2 and (c) v = 7. The identification of a narrow range of amplitude and phase parameters responsible for large population transfer in (c) can be a significant input for design of an appropriate laser field.
A.K. Gupta et aL / Chemical Physics Letters 257 (1996) 658-664
resonant with the first fundamental (v -- 0 -o 1) transition. Note that there is very little population transfer for frequency values which do not satisfy the resonance condition and the same feature is observed for population transfer to other levels as well. This 3-D plot therefore establishes the critical role of the resonance condition in photoexcitation. Similar resuits were observed in our earlier study [1] where an increase in intensity only leads to some power broadening of the excitation spectrum. Otherwise the maximum population transfer was indeed seen only for the resonant frequency. We next examine the case where to~ ffi 0.01820 and to 2 --0.95a h. Here, the to I field component is nearly resonant with the v ffi 0 -o 1 transition and to 2 is nearly resonant with v ffi 1 ~ 2. Note that in this case there will be greater transition probability to states above v ffi 1. Computationally, we note that the period -t in this case is 20 x 2"it/tel; in this way an integral number of optical cycles of both the to t and to 2 components fit within the FIoquet period "r. Although multiple cycles of both field components are incorporated into % the Floquet formalism is still valid since we define -~ as the Floquet period: a similar situation occurred in Ref. [30] where population dynamics with a Gaussian pulse train were studied. From both Figs. 2a and 2b it is clear that even for low field amplitudes, a suitable choice of phase can induce substantial population transfer which would otherwise have required much more intense fields. In particular, Fig. 2c offers a welcome surprise where a narrow range of field amplitude and phase difference 9.01E--4 •
g.ooC- 4 .
8.ggE-',
8.gBE-4 '
8.97E-4
B.gS£-4
8.gsE-4
663
is seen to transfer sizable population to a very highly excited state. Such results would be difficult without the 3-D plots presented here, which in turn would be much too demanding to generate without recourse to the PEM-based approach to the solution of TDSE advocated here. It may also be noted that the variation of population dynamics with respect to phase is not due to constructive interference between the two cw components whereby for some 8 values the intensity increases significantly; this was tested by plotting maximum intensity versus phase for B = 0 -o q¢ (see Fig. 3). Thus from this we can infer that the change of populations with phase is a coherent phenomenon and transition are occurring via interfering paths.
4. Concluding remarks In this Letter we have formulated phase related parametric equations of motion for an efficient investigation of quantum dynamics in the presence of bichromatic lasers. The resulting equations offer computational savings of up to a factor of 100 over the brute force approach to the solution of the TDSE for each new value of the phase difference. These PEM-based savings have made possible 3-D plots of population in various levels as a simultaneous function of phase and frequency or phase and amplitude permitting a global insight which could be tapped advantageously for selecting suitable field parameters for a given photodynamical task. In particular, appropriately populated initial vibrational levels play a critical role in initial state based coherent control of photodissociation [3,4,14] and the tools developed here could be quite useful in this important chemical task. Finally, though our initial applications have focussed on vibrational population dynamics the technique presented here is general and could be used for the investigation and control of photodynamics of other discrete optical and spin levels in atoms and molecules.
J Phase
(Rod|ansi
Fig. 3. Maximum laser intensity as a function o f p h a s e f o r the bichromatic field E ( t ) ffi A[coc,(tolt) + c o s ( t o 2 t + 8)], to t ffi 0 . 0 1 8 2 0 au, to2 ffi 0-95tot, At ffi 0.01 a u a n d A 2 ffi 0 . 0 2 au.
Acknowledgement DBB acknowledges support from CSIR Junior Research Fellowship no. 9/87(143) 92/EMR-I.
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A.lg. Gupta et a L / Chemical Physics Letters 2.$7 (1996) 658-664
MKM acknowledges support from the Board of Research in Nuclear Sciences of the Department of Atomic Energy, India.
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