3590
J . Phys. Chem. 1986, 90, 3590-3594
reflect the photon absorption cross sections not the dissociation dynamics. These processes are schematically represented in Figure 9. When the Kr.S02+ ground state absorbs a photon one of two things happen. First, a repulsive state might be accessed and direct dissociation (DD) to Kr+/S02 products occurs. Second, a bound state could be accessed followed by direct vibrational predissociation or internal conversion (IC) and vibrational predissociation (VP) to SO,+/Kr products. For reasons discussed above we feel the SO2+ products are primarily electronically excited. In Figure 9, the repulsive state pictured directly correlates to Kr+/S02 products. This is a reasonable assumption. We also pictured the bound state initially accessed as also correlating to the same Kr+/S02products. This need not be necessarily so. This assumption is reasonable in that such a Kr+-SO2"charge-transfer" bound state almost certainly exists in this energy region and we
have evidence in the Ar.C02+ system the analogous Ar+.C02 state is accessed.8 However, we have no direct evidence this is the state; any bound excited state might be involved. For example, it is possible a bound state correlating to a new SO2+excited state at 1 .5 eV may be formed.
-
Acknowledgment. The support of the Air force Office of Scientific Research under Grant AFOSR-86-0059 and the National Science Foundation under Grant CHESS12711 is gratefully acknowledged. Finally, M.T.B. expresses his deep appreciation to Rudy Marcus for the professional lifetime of intellectual inspiration he has provided. I am pleased to count him as both mentor and friend. Registry No. Kr, 7439-90-9; SO2+, 12439-77-9; Kr+, 16915-28-9; SO*, 7446-09-5.
Barriers to Chaotic Classical Motion and Quantum Mechanical Localization in Multiphoton Dissociation Robert C. Brown+and Robert E. Wyatt* Institute of Theoretical Chemistry and Department of Chemistry, University of Texas, Austin, Texas 78712- 1167 (Received: January 15, 1986)
Recent work (MacKay, R. S.; Meiss, J. D.; Percival, I. C. Physica D 1984, 13, 55. Bensimon, D.; Kadanoff, L. E. Physica D 1984, 13, 82.) on locating and calculating the flux across global bottlenecks to classical diffusion in strongly chaotic regions is used to study IR multiphoton dissociation for a model diatomic molecule. Particular attention is given to the correspondence between the classical barriers and quantum mechanical dissociation rates. It is found that quantum mechanical localization can arise in regions of x-p phase space which are strongly stochastic when the classical flux is smaller than h .
I. Introduction The interaction between molecular systems and intense laser fields has received a great deal of attention over the past several years. Motivated, in part, by the possibility of using lasers to selectively excite specific nuclear motions, the theoretical work in this area has primarily focused on dynamical effects which tend to promote or hinder mode-selective excitation and dissociation.' One consequence of this work has been the recognition that, under intense radiation, there can exist bottlenecks to energy transfer between the laser field and the various molecular degrees of freedom. While the bottlenecks are often associated with either anharmonicity or a weak dipole coupling, they can also result from nonlinear interactions with the field which are not easily anticipated from either the field-free molecular energy level spacings or the dipole couplings.2 Multiphoton dissociation (MPD) is one area for which this is particularly true. The difficulty in this case stems from two sources. First, from the perspective of classical mechanics, dissociation can only occur for those trajectories which are initially in stochastic regions (regular orbits being contrained to an invariant torus). However, it has been difficult to readily analyze chaotic regions due to the apparent lack of structure in the system's phase space. Thus, while it has been recognized that classical orbits can exhibit nonrandom effects even in strongly stochastic r e g i ~ n s , ~it- lhas ~ only been recently4-sJ2 that theories have been developed which support any quantitative analysis of the phase space features which induce this nonrandom behavior. Second, theoretical studies on model systems perturbed by an external field (e.g. the periodically kick rotorI3,l4)have indicated strong discrepancies between classical and quantum dynamics in stochastic regions. This is, in part, another manifestation of a general 'Present address: Aerodyne Research, Inc., 45 Manning Road, Billerica, MA 01821.
OO22-3654/86/2090-3590$01.50/0
tendency for quantum mechanics to suppress classical c h a o ~ ' ~ - ~ I which has made it difficult to extrapolate between classical and quantum mechanical descriptions in stochastic regions. Significant progress has been recently made by MacKay, Meiss, and Percival' and Bensimon and Kadanofa in the analysis of the ~~
~~
~~
(1) Wyatt, R. E.; Hose, G.; Taylor, H. S. Phys. Reu. A 1983, 28, 815. (2) Brown, R. C.; Wyatt, R. E. J. Chem. Phys. 1982, 82, 4777. (3) Chirikov, B. V. Plasma Phys. 1960, I , 253. Chirikov, B. V. Phys. Rep. 1979, 52C, 265. (4) Percival, I. C. In Nonlinear Dynamics and Beam-Beam Interactions,
Month, M., Herrera, J. C. Eds.; American Institute of Physics: New York, 1979; AIP Conf. Proc. No. 57, p 302. ( 5 ) Channan, S. R.; Lebowitz, J. L. In Nonlinear Dynamics; New York Academy of Sciences: New York, 1980; p 108. (6) Bartlett, J. H. Cell. Mech. 1982, 28, 295. (7) Mackay, R. S.; Meiss, J. D.; Percival, I. C. Phys. Reu. Lett. 1984, 52, 697. Mackay, R. S.;Meiss, J. D.; Percival, I. C. Physica D 1984, 13, 55. (8) Bensimon, D.; Kadanoff, L. E. Physica D 1984, 13, 82. (9) Chirikov, B. V. In Dynamical Systems and Ckaos, Garrito, L., Ed.; Lecture Notes in Physics, Vol. 179; Springer: New York, 1983; p 29. (10) Reinhardt, W. P. J. Phys. Chem. 1982, 86, 2158. Shirts, R. B.; Reinhardt, W. P.J. Chem. Phys. 1982, 77, 5204. Jaffe, C.; Reinhardt, W. P. Ibid. 1982, 77, 5191. (11) Davis, M. Chem. Phys. Lett. 1984, 110, 491. (12) Davis, M. J. Chem. Phys. 1985,83, 1016. Davis, M.; Gray, S . K. J . Chem. Phys., submitted for publication. (13) Casati, G.; Chirikov, B. V.; Izraelev, F. M.; Ford, J. In Stochastic Behaoior in Classical and Quantum Systems, Casati, G., Ford, J., Eds.; Springer Lecture Notes in Physics No. 93; Springer: New York, 1979; p 334. (14) Iwailev, F. M,; Shepelyanskii, D. L. Sou. Phys. Dokl. 1979,24,996. (15) Blumel, R.; Smilansky, U.Phys. Rev. Lett. 1984, 52, 137. ( I 6) Heller, E. U.Phys. Rev. Lett. 1984, 53, 15 15. (17) Petrosky, T. Y.; Schieve, W. C. Phys. Reu. A 1985, 31, 3097. (18) Fishman, S.; Grempel, D. R.; Prange, R. E. Phys. Reu. Lett. 1982, 49, 509. (19) Dorizzi, B.; Grammaticos, B.; Pomeau, Y. J. Stat. Phys. 1984, 37, 93. (20) Casati, G.; Guarneri, I. Commun. Math. Phys. 1984, 80, 1 (21) Chang, S.-J.; Shi, K.-J. Phys. Rev. Lett. 1985, 55, 269
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 3591
Quantum Mechanical Localization in MPD classical structure in chaotic regions associated with the remnants of KAM curves having irrational winding numbers. These workers have developed a theory for diffusion through chaotic regions which (1) suggests the location of the strongest bamers to diffusion and (2) provides a methodology for constructing phase space representations of the barriers and calculating the fluxes and rates for transport across them. These ideas were first adapted to a realistic molecular model by Davis, who used the theory to characterize the global bottlenecks to intramolecular vibrational relaxation in collinear 0 C S . l 2 In that work, the theoretical relaxation rates were found to be in good agreement with rates calculated by numerically integrating classical distributions. In this paper, we numerically examine the response of quantum mechanics to the classical structure in stochastic regions. Our primary purpose is to present numerical evidence that quantum mechanics may sense this underlying structure, and that its response is manifested by stronger localization and slower diffusion rates than predicted by classical dynamics. These effects are interpreted as resulting from the quantum uncertainty which masks any breakup in the classical phase space on a scale smaller than h. This is not meant to imply that the classical stochasticity is localized to a volume of phase space which is less than the quantum uncertainty, but that the breakup of regular KAM tori and resonance islands occurs on a scale smaller than h. In this case, the quantum dynamics may appear more regular than would be predicted from the classical dynamics even in stochastic regions which are large enough to support several quantum states. For this analysis, we will treat multiphoton dissociation in a model for an anharmonic oscillator (the diatomic molecule HF). The results should be qualitatively indicative, however, of the types of effects that may occur in other dynamical processes.’2 The paper has been organized as follows. The classical description of the system is presented in section 11. This includes an analysis of the classical barriers to energy transfer between the IR field and the diatomic molecule. In section 111, Floquet theory is used to study the quantum dynamics for the system, which is then correlated with classical barriers defined in section 11. The results are summarized in section IV. 11. Barriers to Classical Diffusion The classical Hamiltonian for a nonrotating diatomic interacting with a laser field can be writtenZ2 H c = HM HF XHI = E (1)
+
+
where HF and H M are Hamiltonians for the radiation field and the H F (Morse oscillator) diatomic, respectively, and H , is the field-molecule, nonlinear, dipole interaction H~ = p 2 / ( 2 p ) D ~ [ I e(-a(x-xo)l]2 (2a)
+
HF = (1 /2) [PF2
+ oF2xF2]
HI = -D(x)XF
xo, au P , au HF dipole moment A, au @, au radiation field parameters 1, W/cml Eo,Vlao wF,cm-I optical potential v, au Vo,au
~
(22)Miller, W. H.J . Chem. Phys. 1978, 69, 2188. (23) Davis, M. J.; Wyatt, R. E. Chem. Phys. Lett. 1982, 86, 235. (24)Gray, S. K.Chem. Phys. 1983, 75, 67. (25)Shirts, R. B.;Davis, T. F. J. Phys. Chem. 1984, 88, 4665.
0.35 0.02
1
U
N I
M
0
3 : 2 RESONANCE ZONE
0 (D r
0 0 0 0 0 (0 l-
I
REGULAR R E G I O N
0 0
’ *’, 8 I
N M
I
:&w.,
I
l!00
2125
0
3150
I
4175
6.00
x
0
b
N 1 M
0 0
v
W r
4
0 0 0
0 0 W
= 2 + Y
r
I
M
where c is the speed of light, eo is the permittivity of free space, and E is the conserved total energy of the laser/oscillator system ( E = 4.17 X lo7 au). Values for the parameters in eq 1-3 are listed in Table I. This system has been previously used in several classical studies of the IR multiphoton excitation of HF.23-25 The Hamiltonian in eq 1 represents a coupled, two-dimensional, nonlinear oscillator with conserved total energy. Such systems are amendable to a surface-to-section analysis in which orbits in the four-dimensional (x,p,XF,PF) phase space are projected onto
2.90 x 1013 0.7821 3922
0
(2c)
(3)
0.4541 0.0064
0
9
N
I
= W F E O / ( ~ E )=” WF[I/CE0]1/2 ~
0.225(6.124eV) 1.174 1.7329 1744.59
Do,au a,au
0 0
D(x) = Ax exp(-@x4] (24 The coupling parameter is related to the radiation field strength Eo and intensity I by
~~
HF diatomic Hamiltonian
(2b)
with
~
TABLE I: Values of Parameters Used in Calculations
I!00
2125
3!50
I
4.75
I 6.00
x Figure 1. The x-p surface of section for the Hamiltonian describing the interaction between an anharmonic oscillator and an intense radiation field. (b) Phase space representation of the cantori resulting from the breakup of KAM surfaces with winding numbers equal to 1 and 2 plus the golden mean. Also shown are the fixed points for a stable (0) and unstable ( X ) 2:l resonance.
a two-dimensional plane. For example, an x-p surface of section is generated by plotting x and p , whenever XF = 0 a n d PF > 0, for several different trajectories. Figure l a shows the Poincare x-p surface of section for the diatomic-field Hamiltonian Hc with X ( X / w F= 1.00 X au) chosen to correspond to a field intensity of I = 2.90 X lOI3 W/cm2. The field frequency uFwas 3922 cm-’ (0.001 78 au), which is slightly red-shifted from the H F fundamental of 4138 cm-’. Initial conditions for the trajectories are given by x(t=O)
= xg
(4a)
3592
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986
where E ( n ) is the energy of the nth field-free Morse oscillator eigenstate and E is the total conserved energy for the system. At this field intensity, the phase space is characterized by a regular region, corresponding to orbits with n < 1 1 and n = 11 with p(t=O) positive, and a chaotic region, corresponding to orbits with n = 11 and p(t=O) negative, and n > 11. In the regular region, each orbit is quasiperiodic and constrained to an invariant two-dimensional torus in the four-dimensional phase space. There are two fundamental frequencies for the motion on each torus. In the present application, these correspond to the unperturbed field frequency wF and a dynamical frequency wM for the diatomic that depends upon the x-p initial conditions. When the ratio between these two frequencies, Le., the winding number a = wF/oM, is irrational, the orbit’s projection on the surface of section is a single, smooth closed curve. If a is rational, a = wF/wM = M / N , the x-p surface of section will exhibit a set of M closed curves. The latter condition indicates a region of resonance between the molecular and field degrees of freedom. In the chaotic region, the trajectories are not constrained to a torus and can sample all energetically accessible regions of phase space. This is indicated in the surface of section as an apparently random splattering of points extending from the regular region described above out toward large positive values of x. Thus these orbits can diffuse through the chaotic region and eventually dissociate. However, as they dissociate, the trajectories may exhibit very nonrandom behavior as they temporarily get trapped by dynamical barriers in the stochastic region. There barriers correspond to the remnants of the regular tori that have broken up (in the sense that they are no longer invariant) under the nonlinear coupling. From the theory developed in ref 7 and 8, it is possible to locate the strongest barriers to diffusion through the chaotic region. The basic ingredients of the theory can be summarized as follows: (1) As regular, invariant tori break up under the nonlinearity, they are replaced by structures, ~ a n t o r i which, ,~ although not invariant, can still act as partial barriers to diffusion. (2) The strongest barriers (in the sense that the corresponding flux is a local minimum) arise from cantori associated with the breakup of tori for which the winding number a is most irrational. The most irrational number, i.e. the most difficult to represent by rational approximations, is the golden mean vg = [5(1/2)- 1]/2 = 0.618 .... Hence, cantori, for which a = n vg, where n is a nonnegative integer, tend to define the strongest dynamical barriers. (3) Finally, it is possible to (a) construct a representation of a given cantorus on the surface of section, and (b) calculate the associated flux by approximating the true cantorus ( a irrational) with the successively higher order periodic (rational a) convergents obtained by truncating the continued fraction representation of a . Explicit details for the method can be found in ref 7, 8, and 12. Figure l b shows the x-p representation of cantori that correspond to the breakup of invariant tori with a equal to one and two plus the golden mean. The flux across a = 1 + vg is 0.1 au; the flux across a = 2 + vg is 1.1 au. The cantori and fluxes for a = n + vg, with n > 2 could not be obtained because the phase space was too unstable. Also shown are the fixed points for the stable and unstable periodic orbits for a 2:l resonance that lies between these two cantori. Note, in Figure la, there is very little indication of either the 2.1 resonance or any barrier associated with the 2 + vg cantorus. The classical structures illustrated in Figure l b partition the phase space into several distinct regions. First, there is the region interior to the 1 vg cantorus. This region is predominately regular and, consequently, most of the classical orbits do not dissociate. Quantum mechanically, however, the system can dissociate via tunneling through the invariant tori. Second, there
+
+
Brown and Wyatt is the stochastic region between the two cantori. Classically the flux across 2 + vg is large, and trajectories with initial conditions in this region tend to dissociate quickly. However, this flux, quantum mechanically, is still on the order of h and the breakup of this region of phase space can still be masked by the quantum uncertainty. Finally, there is the region exterior to 2 + vg. In the following section we examine the extent to which quantum mechanics senses this underlying classical structure. 111. Quantum Dynamics in Stochastic Regions A. Quantum Hamiltonian and Numerical Methods. It was found advantageous to use a time-dependent semiclassical Hamiltonian of the form:
HQ = HM - Eo COS
(WFf)D(X)
(5)
to describe the quantum dynamics. In eq 5, HMand D(x) are quantum mechanical operators for the molecular Hamiltonian and dipole moment defined in eq 2. The moleculefield interaction is then incorporated as a periodically time varying, dipole coupling between field-free Morse oscillator eigenstates I X k ) , The parameters Eo and wF (Table I) represent the radiation field strength and frequency, respectively. The Hamiltonian HQhas been shown to be equivalent to Hc in the limit of high-intensity fields, when the nonlinear coupling parameter X is related to the field strength and intensity Z by eq 3. Dissociation is treated by (1) discretizing (constructing an infinite wall at x = x* = 16.75 a,) the continuum energy levels, and (2) associating a decay width yk with each discretized state I X k ) whose energy & is greater than the dissociation energy Do, by defining the complex energy
The decay widths are defined by diagonal matrix elements of the optical potential26 YK
= (xKIVO~IXK)
(7)
where Vo,(x) = Vo[1
+ exp(-(x
- x*)/q]]-l
(8)
The optical potential has been shownz6to give good agreement with dissociation probabilities obtained by numerically integrating the time-dependent Schrodinger equation over a grid in two variables, x and t . There are two advantages in formulating the quantum system as described above. First, because the nonlinear interaction is periodic in time, Floquet theoryz7 can be used to provide a straightforward, nonperturbative method for following the time evolution of the molecular wave function, \k,(x,t), for any given initial condition, \k,(x,t=O) = x,(x). From \k,(x,t), time-dependent state-testate transition probabilities PFl.(t) = I(x,l\k,(t))Iz,survival probabilities, P d ( t ) = P,,(t),and dissociation probabilities, P D f ( t ) = 1 - Cp1P,,(t) with N Bequal to the number of bound states, are easily generated. Second, since \k, depends only on the diatomic internuclear distance and time, the Wigner t r a n s f o r ~ n ~ ~ - ~ ~
can be unambiguously displayed on the x-p surface of section. Thus, Hc will first be used in describing the phase space and locating the strongest barriers to classical dissociation. Then, using (26) Leforestier, C.; Wyatt, R. E. J. Chem. Phys. 1983, 78, 2334. (27) Leasure, S . C.; Wyatt, R. E. Opt. Eng. 1980, 19, 46. Leasure, S . C.; Milfeld, K. F.;Wyatt, R. E. J . Chem. Phys. 1981, 74, 6197. Milfeld, K. F.; Wyatt, R. E. Phys. Rev. A 1982, 27, 12. (28) Wigner, E. P. Phys. Rev. 1932, 40, 49. (29) Hutchinson, J. S.; Wyatt, R. E. Phys. Lett. 1980, 72, 378. Phys. Rev. A 1981, 23, 1567. (30) Korsch, H. J.; Berry, M. V. Physica D 1981, 3, 627. (31) For example, see: Marcus, R. A. In Chaotic Behavior in Quantum Systems, Casati, G . ;Plenum: New York, 1985; p 295.
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 3593
Quantum Mechanical Localization in MPD 0.23
DISSOCIATION ENERGY 10
t 1‘’
15
0 , 0 1 4 8 0.%---6?28
E
( 0 . u . ) 0.17
I t
= 5
t
t
‘10
13
0’13
O“‘
1
t
7
f
15
8 20.00
40.00
60.00
80.00
100.00
FIELD PERIODS
8
n
r0
m
0
a
a0 -I
46
> n > E
2*
cnN 0
0 0
0.00
20.00
40.00
t
= 4 5
,
I
0 , 0 1 5 6 0 , 0 1 3 4 0.0/71
-1
O0.00
= 3 5
I
I
0
;I
t
11
Figure 2. Energy level diagram for field-freeMorse oscillator eigenstates which provide a resonant pathway to dissociation.
P,
‘ 1 5
11
0.15
1
0 , 0 1 4 6 0 . 0 1 3 1 0 , 0 1 4 3
13
60.00
80.00
100.00
FIELD PERIODS
Figure 3. The (a) probability of dissociation and (b) survival probability as a function of the field period (IT = 0.054 ps) for the diatomic oscillator initially in the 8-th, 11-th, 13-th, 15-th, and 18-th Morse eigenstates.
%, the time development of the Wigner transform of a nonstationary state can be viewed relative to the topology of the classical barriers. In our analysis, we will focus on the sets of states indicated by the energy level diagram in Figure 2. These states were selected because for wF = 3922 cm-’ the dipole couplings and field-free energy level spacings indicate they should provide a strong path to dissociation. B. State-testate Transition and Dissociation Probabilities. Figure 3a shows the probability of dissociation as a function of the number of field periods ( I T = 0.054 ps) for the diatomic initially in each of the Morse oscillator states displayed in Figure
(t)
(E 0 4 . 0 2 8 7 ,
0!000!31 Wo-
0.%--6?61
0.0179)
Figure 4. State-to-statetransition probabilities for the diatomic initially in the (a) 11-th and (b) 13-th Morse oscillator states. Time is in units of field periods.
2. When the system is initially in the 18-th oscillator state, PDI8(t) (specifically, 1 - PDI8(t))shows the exponential decay that is indicative of states which are directly coupled to the continuum.26 At the other extreme, when the diatomic is initially in the 8-th state, PDs(t)increases approximately linearly during the first 100 field periods (5.4 ps). This behavior corresponds to the fact that the system initially has its greatest “density” in the regular region interior to the 1 + vg cantorus and, consequently, dissociation is a tunneling process. In contrast to these two extreme types of behavior, when the diatomic is initially in the region (in oscillator states 11, 13, or 15) between the two classical barriers, PD’(t)shows neither the exponential behavior characteristic of direct coupling to the continuum nor the slow, linear growth indicative of tunneling. The field-free 13-th state is a little off resonance and a depressed dissociation probability is not unexpected. However, the field-free 1 5 t h state is in resonance with the continuum via both a single two-photon transition and two one-photon transitions through the 18-th state. Yet the probability of dissociation is still suppressed. Also, state-to-state transition probabilities indicate that the bottleneck to dissociation is associated with the transition to the 18-th state, with most the amplitude that reaches this state dissociating. Similar trends are seen in plots of the survival probability P,’(t) vs. time shown in Figure 3b. In particular, the following features should be noted: (1) PsI8(t)shows a sharp falloff (approximately 1/8 of its initial value in 20 field periods) with very little recurrence at later times. (2) P,’(t) for i = 8 and 13 indicates a slow (approximately) linear decay, with Ps13(t)showing the greatest falloff. Neither curve falls to zero during the first 100 periods, indicating that the molecular motion is to a large extent trapped in the initial regions where the Wigner transform has significant “density”. PsI3(t) also exhibits a fast oscillatory structure with a period of approximately 6.3 field periods (0.34 ps). (3) The time dependence of Pi(?)for i = 11 and 15 are almost identical. This includes a fast initial decay with Ps’(t) falling to about 1/2 of its initial value within the first 5 periods (0.27 ps) and a broad set of recurrences occurring about every 35 field periods. The recurrences are characterized by a secondary fast oscillatory structure having the same period as observed for PsI3(t). To understand the time dependence of Ps’(t) for n = 11, 13, and 15 it is useful to consider the state-to-state transition probabilities shown in Figure 4. These diagrams are representative of the behavior that we have observed throughout the first 100 field periods. In Figure 4a, the system is initially in the 1 1-th Morse oscillator eigenstate. The subsequent time development can be characterized as a Rabi cycling between states 11, 13, and 15, with the dominant contribution arising from states 11 and 15. In Figure 4b, the system was initially in the 13-th oscillator state. The subsequent time development indicates that the system is to
3594
Brown and Wyatt
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986
It = o
a
t
t = o
b
= 5
Q
t = 5
I I” I
M
~
It
b
C
t
=
d
15
t
= 10
C
t
= 15
d
t
= 20
e
t
= 25’
f
= l o
0
0
t
e
= 20
t
9
f
= 25
l
n
n
1
i
t
X
X (r
a
a b
N‘
N
0
0
9 c-32.00
-~
-10.67
10.67
00
r
2.00
-‘10.67
lb.67
* 00
P Figure 5. Contour plots (positive contours only) for the Wigner transform of a nonstationary state for an anharmonic oscillator evolving in response to the radiation field after (a) 0, (b) 5, (c) 10, (d) 15, (e) 20, and (f) 25 field periods (Is = 0.054 ps). The cantori shown in Figure I b are also shown in each of the diagrams a-f. The oscillator was initially, Figure Sa, in the 1 5 t h Morse eigenstate.
a large extent trapped in that state. C. Quantum Mechanical Localization in Strongly Stochastic Regions. The time-dependent state-to-state transition, survival, and dissociation probabilities shown in Figures 3 and 4 indicate that the quantum dynamics exhibits a strong localization in the region between the two classical barriers. This localization exists even though the classical phase space is strongly chaotic, and the classical flux across the 2 vg cantorus is relatively large. As additional evidence for the correspondence between the classical barriers and the quantum localization, Figures 5 and 6, show contour plots of the Wigner transform rw’(x,p,t) of the evolving wave function \k,(x,t)at selected times during the first 25 field periods. For clarity, only positive contours are shown. The contour values were taken in equal increments ranging from to approximately l/JWmax. The the largest value of rWmax, diatomic was initially in the 15-th, Figure 5, and 13-th, Figure 6, Morse eigenstates. In Figure 5, the diatomic is initially in the 15th Morse oscillator eigenstate. The Wigner transform of this state (Figure 5a) has its greatest “density” in the region near the 2 + vg cantorus. During its subsequent evolution, the quantum state oscillates in the classically chaotic region localized between the two contori. Dissociation results from leakage of probability from the small extensions in the Wigner transform across the 2 + vg cantorus. When the diatomic is initially in the 13-th Morse eigenstate (Figure 6), the localization is even stronger. In this case, the quantum state initially (Figure 6a) has its greatest “density” on the stable and unstable 2:l fned points which lie between the two cantori. Parts b-f of Figure 6 indicate that the subsequent evolution of this state may be strongly influenced by these resonances, even though the resonance islands have been completely broken
+
Figure 6. Same as Figure 5, except the diatomic was initially, Figure 6a: in the 13-th Morse eigenstate.
up under the nonlinear coupling. Other oscillator states which initially lie between the cantori (e.g., the 12-th and 14-th) show a similar behavior. However, since these states are not resonantly coupled to the continuum, the probability which leaks across the 2 + vg cantorus tends to buildup outside, eventually recrossing back into the region between the two cantori.
IV. Discussion We have numerically studied the quantum mechanical implications of dynamical barriers in stochastic regions corresponding to the breakup of invariant tori. The analysis focused on laser/diatomic interactions, but the results are expected to extend to other dynamical processes, including intramolecular energy relaxation and unimolecular dissociation.I2 Three. main qualitative features of the laser induced of diatomics are suggested by the results in this paper: (1) The quantum bottlenecks to dissociation appear to occur in the same regions of phase space as the dynamical barriers to classical diffusion. (2) A quantum wave function can remain localized in a stochastic region of phase space with an area greater than h,if the region is bounded by classical barriers whose flux is less than or approximately equal to h. (3) When the system is strongly quantum mechanical (large effective h ) ,the classical flux across the barriers can become very large and still remain less than h. In this case, the quantum bottlenecks and localization can persist even though there are no apparent classical barriers. Finally, we acknowledge the enormous impact that the work of Professor R. A. Marcus has had upon the field of nonlinear dynamics, particularly intramolecular dynamic^.^' Acknowledgment. This work was supported, in part, by research grants from the Robert A. Welch Foundation and the National Science Foundation.
