Adiabatic control of molecular excitation and tunneling by short laser

J . Phys. Chem. 1993,97, 12634-12643

12634

Adiabatic Control of Molecular Excitation and Tunneling by Short Laser Pulses Heinz-Peter Breuer Fakultiit fur Physik der Universitiit, Hermann- Herder-Strasse 3. 79104 Freiburg i. Br., Germany

Martin Holthaus’ Department of Physics, Center for Nonlinear Sciences, and Center for Free- Electron Laser Studies, University of California. Santa Barbara, California 931 06 Received: April 27, 1993; In Final Form: July 2, 1993’

W e discuss a principle to control both the selective excitation of molecular vibrations and the tunneling process in a double-well potential by intense, short laser pulses with systematically shaped, smooth envelopes. In both cases, the theory of adiabatic response of Floquet states yields functionals of the pulse shape which allow an accurate prediction of the parameters of optimal pulses. The example of selective excitation leads to a farreaching correspondence between classical and quantum mechanics; nonlinear resonances in the classical phase space play an important role.

1. Introduction Controlling a molecular reaction by intense laser pulses is an outstanding topic in chemical physics.’ It is also a difficult one: On the experimental side, one of the many problems is an accurate determination of all interaction parameters including the precise variation of the field strength during the pulses; the theorist, on theother hand, has to facea time-dependent Schrijdinger equation with a Hamiltonian that couples several degrees of freedom. It has been theoretically realized2J and experimentally demonstrated4.s that quantum mechanical interference phenomena between independent excitation pathways open up important possibilities to “guide the evolution of a quantum system”.6 Moreover, a significant recent development is the consistent application of optimal control to molecular dynamics under the influence of strong laser fields. Given the enormous progress in pulse-shaping technologies, as well as the development of new computational tools to solve the quantum equations of motion, there is a realistic hope that this approach will lead to laboratory applications in the near future.’ As an illustrative example of how quantum mechanical optimal control theory works, it has been demonstrated that it is possible to predict electric fields which result in the selective excitation of eigenstates of a Morse oscillator.’O However, those fields turned out to be rather complex, and it may be hard to get an intuitive understanding of the excitation mechanism. That should not be too surprising: to achieve an efficient control, the strength of forces associated with the external laser field should be comparable to intramolecular forces, which means that the mechanisms will be nonperturbative in character; and to prevent unintended damage from the molecule, the pulses should be as short as possible, which implies that pulse shape effects will play a substantial role. The purpose of the present paper is to demonstrate that, nevertheless, there are situations where quantum dynamics can be controlled with very simple, smooth pulses and where the mechanism is fairly transparent, even if the fields are so strong that they substantially modify the level structure of the unperturbed system. More important than the particular numerical results will be the language in which they are discussed; the key point will be the adiabatic response to laser pulses with a “slowly” varying amplitude.

The theoretical framework used in the following discussion is the Floquet picture of quantum mechanics. This picture fulfills two basic requirements: it incorporates the external laser field in a nonperturbative manner, and it is particularly well suited for an analysis of pulse shape effects. In addition, it also allows us to investigate the classicalqtantum correspondence in the case of laser-molecule interaction. It will be shown that the search for rules for laser-assisted molecular control can lead to concepts that differ significantly from traditional ones: selective excitation will be described in terms of constructive interference and pulseshape-controlled tunneling in terms of an adiabatically induced phase shift. The material is organized as follows: The next two sections analyze model calculations for the controlled excitation of molecular vibrations.11#12First, it is demonstrated in section 2 that the theory of adiabatic response of Floquet states13-15leads to a functional of the laser pulse shape that makes possible an accurate prediction of the parameters of optimal pulses.I6 Section 3 then discusses how the association of Floquet states with classical vortex tubes17 can be exploited to gain some additional insights into thedynamical processes from a semiclassical point of view.lG20 Section 4 addresses the seemingly different problem of pulseshape-controlled tunneling in a laser field.2’ It will be shown that there is actually a close conceptual link to the case of selective excitation; adiabatic response is again the salient feature. Finally, section 5 contains a concluding discussion.

2. Excitation by Interference The considerationsin this section are motivated by an important question: Can one selectively excite a particular vibrational state of a molecular bond within less than 1 ps? This is the typical time scale for intramolecular vibrational energy redistribution, and a process aiming a t molecular control should be faster than that. Pioneering model calculations22and subsequent numerical studies’lJ2 have indicated that there may be a possibility to reach this goal. The aim here is not to reproduce these results but to clarify the underlying quantum mechanical principles. A model for the dynamics of molecular stretching modes in the presence of a laser field is provided by a forced Morse oscillator:

~

* Author to whom correspondence should be addressed. Address after

Oct. 1, 1993: Universitit Marburg, Fachbereich Physik, Renthof 6, 35032 Marburg, Germany. Abstract published in Adounce ACS Abstracts, November 1 , 1993.

0022-3654/93/2091- 12634$04.00/0

with

0 1993 American Chemical Society

The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12635

Excitation and Tunneling by Short Laser Pulses

m b) w = (€6

and

Hi,,(x,t) = dx F ( t ) cos(wt)

(2.3)

For the parameters (all data in atomic units)

m = 1744.805, D = 0.2251,

= 1.174, d = 0.3099 (2.4) this model yields an approximate description of vibrational states of a H F molecule interacting with laser radiation of frequency w;23,24 the function F ( t ) describes the envelope of the electrical field during the pulse. These parameters have been chosen only to give one out of many possible examples and their precise values will not be important, but it is important that the reduced mass m of the hydrogen and the fluorine atom is rather large; the value of m is almost that of the proton mass. To formulate the problem as clear as possible, let us first consider the result of a "numerical experiment": For pulses of length t,, let us fix the shape

I[

F ( t ) = F,,, sin2(rt/t,) (2.5) and let us assume that the Morse oscillator is in its ground state at t = 0. We then solve the SchrMinger equation ia,+(x,t) = H@,x,t) +(xA

(2.6) for a laser pulse (2.5) and thus calculate the occupation probabilities of the unperturbed Morse eigenstates at t = t,, when the interaction is over. Repeating this procedure for different values of the peak field strength F,, we obtain a good description of the responseof the Morseoscillator to the laser pulses. Detailed investigations of this kind can be found in ref 12. Figure l a 4 shows the result of such a model calculation. Starting in the ground state n = 0 with energy EO,the aim was to populate the sixth excited state n = 6 with energy E6. Different resonant frequencies w were chosen: w = (E6 - Eo)/m with m = 1, ..., 7. In addition, all pulses have a length of 50 laser cycles: t p = SOT, with T = 2 ~ / w . Considering the approximate selection rule An = 1, valid for weakly anharmonic systems like the Morse oscillator, one might guess that a 'six-photon-resonance", w = (E6 - E0)/6,could open a path for "sequential ladder climbing". Indeed, Figure ICshows that this frequency is the only one that leads to a strong response; pulses with F,,, = 0.07 au yield a high degree of excitation of the target state n = 6. Note that in this case the interaction time t , = SOT is only 0.472 ps. For even stronger pulses with F,,, = 0.09 au the final excitation probability is rather low again; a major part of the probability returns to the ground state at the end of the pulses. This excitation pattern cannot be explained by the naive ladder climbing mechanism: why should the possibility to reach the top of the ladder depend so strongly on the peak field strength? The selection rule An = *l is a perturbative result, and it is not obvious that it should survive in the strong field regime. In addition, it does not say anything about the effect of the pulse shape. In all the other cases summarized in Figure 1 there is some excitation during the pulses, but the system returns almost completely to the initial state at t = tp' These results, as well as similar ones,11J2lead to the following questions: (i) Is there another way to characterize the frequency used in Figure IC, rather than stating that it leads to a "sixphoton resonance"? (ii) What determines in the case of Figure ICthe field strength FmXthat leads to maximal excitation? Can one reach an excitation probability of loo%? Both problems will be dealt with in reverse order. The rest of this section follows ref 16 to answer the questions posed in (ii); the following section addresses the issue raised in (i). Let us start by reviewing some theoretical concepts.25-28 The Hamiltonian (2.1) contains the slowly varying pulse shape function

*

-Eo)/5

F w e 1. Occupation probability of the ground state n = 0 and of the sixthexcitedstaten= 6 aftertheMorscoscillator(2.2), (2.4) hasinteractcd with pulses (2.3), (2.5). The system was initially in its ground state; the frequencies were w = (E6 - Eo)/m,with m = 1, 2, ..., 7. The length of each pulse was 50 laser cycles, Le., tp = 50 X 2 r / w .

F ( t ) . If we freeze the field strength at some instantaneous value Fwith 0 I F I F,,,, we obtain a Hamiltonian HF that is periodic in time:

HF(p,x,t) = H0(p,x) + dx F cos(d)

(2.7) For such a time-periodicquantum system,there exist quasienergies €:and Floquet states u:(x,t). These quantities are defined by the eigenvalue equation

(HF(p,x,t)- ia,)u;(x,t) = E;u;(x,t) with periodic boundary conditions:

(2.8)

u;(x,t) = uf(x,t+T) (2.9) Due to the periodicity in time, quasienergies are defined up to an integer multiple of the photon energy w , and they can be arranged in Brillouin zones. The Floquet states, on the other hand, share many properties of stationary states. In the present context, the most important observation is that they obey an adiabatic principle. Let us assume that at t = tothe field strength is F(to)= FOand that the system is in an arbitrary Floquet state:

rL(x,t,) = &,to)

(2.10)

If the field strength then varies "slowly" compared to the time T = 2 ~ / wof one laser cycle (and if the initial state is nonresonant), the wave function will evolve in time as #(x,t) = u;(')(x,r) exp(-iS'dr e;()' 10

(2.1 1)

Le., it remains in the "adiabatically connected" Floquet state and acquires a dynamical phase that is determined by the integral over the instantaneous quasienergies. Expressing the same thing in a more pictorial manner, one may speak of quasienergy surfaces which emerge when the instantaneous quasienergies e: are regarded as functions of laser parameters like F and 0;when these parameters are changing slowly, the wave function evolves on these quasienergy surfaces in a Born-Oppenheimer like manner.

Breuer and Holthaus

12636 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993

Fmaz (a,..)

0

.04

.02

Figure 2. Occupation probability of the fifth excited state n = 5 (solid line) and dissociation probability (broken line) after the HF Morse oscillator has interacted with pulses (2.5) of frequency w = (E5 - & ) / 5 = 0.016 489 au and length tp = 50 X 2 r / w = 0.461 ps.

Before proceeding, a word of caution is in order. For vanishing field strength, that is for F = 0, the Floquet states are determined by the eigenstates of Ho.Since the energy spectrum of HOis continuous above the dissociation border E = D, and the quasienergy spectrum has a Brillouin zone structure, even the quasienergy spectrum at F = 0 consists of the whole real axis, with embedded discrete eigenvaluesdue to the bound states. But for F > 0, these embedded eigenvalues turn into poles of the resolvent of the quasienergy 0perator.2~ In other words, the quasienergies obtain negative imaginary parts, which expresses the fact that the Floquet states are metastable in the presence of the continuum and, therefore, “decay”. Of course, this fact is intuitively clear since the laser can dissociate the molecule at any field strength. The question then is whether the time scale t, of the laser pulse can be chosen such that, on the one hand, it is long enough to guarantee essentially adiabatic behavior; on the other, 1, should be short compared to the lifetimes of the Floquet states involved. Again, numerical model calculations indicate that such a choice is possible. Figure 2 shows the response of a H F Morse oscillator to pulses with frequency w = (E5 - E0)/5; the pulse length again is t, = SOT. The full line denotes the excitation probability of the state n = 5 after pulses (2.5) with peak strength Fmax; the broken line is thedissociation probability, i.e., the excitationof thecontinuum. Obviously, continuum effects are important for FmX > 0.1 au and can be neglected for smaller amplitudes;the excitationpattern is very similar to that observed in Figure IC. Since we are not interested indissociation-theaim is toexcite themolecule without destroying it-we will restrict the following analysis to pulses with Fmx I 0.1 au. In this regime, the imaginary parts of the relevant quasienergies are small, and they can be neglected if the pulses are short. The expression (2.1 1) already suggests an explanation for the negative results displayed in Figure 1: When the laser field strength F(t) rises “slowly”, the wave function of the initial vibrational state n = 0 is shifted adiabatically into the “connected” Floquet state; later, when F(t) decreases, it is shifted back into the original state. But the results shown in Figures ICand 2 can only be explained if adiabaticity is at least partially violated. The clue can be found by inspecting the quasienergy spectrum. Figure 3a shows one Brillouin zone of quasienergiesfor the driven H F Morse oscillator, plotted versus the instantaneous field strength F. The frequency is w = (E5 - E0)/5, as in Figure 2. Because of this choice, the quasienergies :e and e t that originate from the energies EOand E5 are degenerate (moduloo)at F = 0. The important observation is that this degeneracy is significantly removed at Fc = 0.03 au; both levels respond differently

.06

F

.8 3

.08

(a,..)

-

-

-

1

-

-

2 . 6-

.4L

.2 -

-

--t-

I

l

l

I

l

l

I

l

l

I t 1

I

l

l

at higher field strength. This fact has consequences for the dynamics of the pulsed system:l6 When the field strength rises, the initial state first evolves adiabatically according to (2.1 1). When the rising laser amplitude starts to remove the quasienergy degeneracy, Le., when F = Fc at t = t,, the wave function splits into a superposition of the two Floquet states u:(x,t) and uc(x,t) with almost equal weights. When the field strength rises further, both components evolve separately on their own quasienergy surfaces, and each acquires its own dynamical phase. When the field strength decreases during the second half of the pulse, they move back adiabatically until the critical value Fcis reached again at t = t1. Then both parts of the wave function interact. This interaction is a quantum mechanical interferenceeffect which determines the new occupation probabilities of uc(x,t) and uf(x,t). The final part of the pulse, during which F(t) decreases from Fc back to zero, can again be described in terms of adiabatic motion. Translating this simple physical picture-adiabatic evolution, splitting with equal weight at t = t,, adiabatic evolution of two components, interference at t = 21, adiabatic evolution-into a quantum mechanical two-Floquet-state calculation, one obtains the followingexpres~ion’~ for the excitation probability Pi+ after


Excitation and Tunneling by Short Laser Pulses the interaction:

Pi,f = sin2(Ai,/2)

(2.12)

with (2.13) (The subscript i denotes the initial and f the final state.) These formulas clearly reflect the physical mechanism. Equation 2.12 is an interference pattern, and whether the laser pulses lead to maximal or minimal excitation is determined by the difference of the dynamical phases (2.13) which both components have acquired between splitting and interference. Since the quasienergies tr and e: of the two participating states are almost degenerate for F < FE,we can extend the integration over the whole pulse:

0

.02

Fmax

(2.14) The idealization behind this line of reasoning should be clear. We are dealing with wave mechanics, and hence, the “rough spot“ at F = F,must result in a smooth reorganizationof the wave function; both splitting and interference will require a finite amount of time rather than happening at a moment t s or t1. Nevertheless, the simple result (2.12), (2.14) does a very good job: The full line in Figure 4a shows the quantity sin2(&5/2), calculated for pulses (2.5) with w = (E5 - &)/S and t , = lOOT (that is, tp = 0.922 ps) as a function of F,,,; little boxes denote occupation probabilities of the fifth excited state n = 5 which were obtained by a numerical solution of the Schriidinger equation. The agreement is striking. There are, of course, many alternative theoretical approaches to the problem of molecules interacting with strong laser fields, each of which has its own particular strengths and weaknesses. Results like that shown in Figure 4a demonstrate a major strength of the Floquet picture: Excitation probabilities after the interaction with short laser pulses can be calculated from the instantaneous quasienergies and the pulse shape F(t). In fact, the phase integral (2.14) can be regarded as a functional of the pulse shape

and once the quasienergies have been computed, this functional can be evaluated for arbitrary pulse shapes. If we are in the adiabatic regime, the precise form of the pulse shape does not matter; there is nothing special about the pulses (2.5) used in the numerical examples. An “optimal pulse” is a pulse which leads to maximal excitation, i.e., to constructive interference. These pulses are simply characterized by

bf[F(t)] = ~ ( 2 + k l), k = 0, f l , f 2 , and k = 0 (or k = -1) yields the first maximum.

...

.04

0

.02

.04

.06

.08

(a,..)

.06

.08

Fmax (a,..)

1 .8 x

.t: .6

3

3

8 .4 .2

(2.16)

Now that the physics underlying Figures 1c and 2 is understood, the problem can be turned around. Using the condition 2.16, one can “design” those pulse shapes F(t) which lead to maximal excitation. The functional Air is not injective, so that there can be a great variety of pulses which achieve selective excitation in a given situation. It is now also possible to find the reason for the negative results shown in Figure 1. For example, Figure 3b shows the quasienergy spectrum for w = (Ea - Eo)/& corresponding to one of the cases summarized in Figure la. The two relevant quasienergies eiand e: remain almost degenerate in the whole range 0 5 F I0.1 au, which means that there is no possibility of acquiring a relative dynamical phase during a short pulse. Up to this point, the theoretical considerations rely mainly on the adiabatic response of Floquet states. But adiabaticity cannot

0 0

.02

.04

.06

.08

Fmaz (u.u.)

Figure 4. Numerically computed excitation probabilities of the fifth excited state (boxes) and sin2(&5/2) (full line) for pulses with frequency w = (E5 - E 0 ) / 5 and length tp = 100 X 2 r / w (a), 1, = 50 X 2 r / w (b), and tp = 30 X 2 r / w (c). Note the gradual breakdown of adiabaticity.

be taken for granted for very short laser pulses, a point which leads to a further natural question: What is the length t, of the shortest pulses for which the adiabatic picture is still applicable? This question is difficultto answer in general;some partial answers are contained in the Figure 4b,c. As in Figure 4a, the full lines represent sin2(&5/2); the boxes are numerically computed excitation probabilities for t, = 50T = 0.461 ps (b) and tp = 30T = 0.277 ps (c). The agreement is still good in Figure 4b and, although the maximal excitation probability drops below 0.8,


holds at least qualitatively for the very short pulses of Figure 4c. Adiabaticity is destroyed only gradually. Let us finish this section with a mathematical remark. If one writes the adiabatic response of Floquet states as in (2.1 l ) , one tacitly assumes that an important detail is properly taken care of. The eigenvalue equation (2.8) does not specify the instantaneous Floquet states u:(x,t) completely, but only up to a phase factor. On the other hand, the wave function (2.1 1) is a solution of the Schriidinger equation, and given the initial condition (2. lo), there is no freedom left in the choice of the phases. This observation immediately leads to the theory of Berry’s phase.3’J2 Let us assume that not only the field strength F but also the laser frequency w varies during the interaction with the molecule. We then havea two-dimensional parameter space with elements (F,w), and for each set of instantaneous parameters there is a Hilbert space H(F~w) spanned by the instantaneous Floquet states. A variation of the parameters corresponds to a curve ( F ( t ) , w ( t ) ) in parameter space, and the adiabatic principle yields a way of transporting an initial Floquet state uhF*F’”)(x,t) along such a curve,33 Le., a “ c ~ n n e x i o n ” .The ~ ~ repeatedly used phrase “adiabatically connected Floquet state” was meant in this differential geometric sense. To be more precise, one can start with an arbitrary single-valued assignment (F,w) uhF,W)(x,t)and write

-

uhF*w’(x,t) = exp(iy)uT)(x,t)

(2.17)

The adiabatic transport then fixes the phase y along the curve. An interesting question3’ arises when the curve in parameter space is a closed loop 6: What is the value of the phase y(6) after the parameters have returned to their original values? The pulse (2.5) is an example of such a loop: F = F ( t ) , w = constant and F(0) = F ( t p ) = 0. This loop does not enclose a finite area in parameter space, and in such a case the phase y(S) vanishes (which entitled us to neglect it in the preceding discussion). But if the loop actually encloses a finite area, i.e., if in addition to the field strength Falso the frequency w varies during the interaction and if w ( 0 ) = w ( t p ) ,then y(6) can be different from zero. It can be shown that this phase has a purely geometrical origin.32 To conclude: In addition to the dynamical phase factors exp(-iJdt )e: there is also a geometrical phase which has to be taken into account explicitly if more than one parameter is varied.31932 In the context of laser-assisted molecular control, a possible application occurs when both field strength and frequency are changing cyclically during the interaction. Some questions concerningan adiabatic frequency variation have been addressed in ref 35.

3. Splitting by Separatix Crossing The picture outlined in the preceding section is almost closed. Is rests on well-established quantum mechanical principles, and it has predictive power. Nevertheless, one question is left unanswered: Why is Figure I C so vastly different from the other figures in that series? What is so special about the frequency w = (E6 - Eo)/6 for a transition to n = 6, and w = (E5 - Eo)/5 for n = 5? It is at this point that the large reduced mass m (2.4) becomes important, because it pushes us far into the semiclassicalregime. Introducing the dimensionless coordinateP with

Q, = w(m/2D/3’)’’’

(3.2)

the classical action functional

(3.3)

Breuer and Holthaus

Px i------

e-

0 2x Figure5 (a) Classical trajectoriesof a particle moving in an unperturbed Morse potential are confined to a “circle”E = constant. (b) Following all trajectorieswith a specified energy E in time, a vortex tube is obtained. (c) In action angle variables (I,@ of the unperturbed Morse oscillator, a vortex tube is characterized by its action I. A Poincar6 section for F = 0 simplyshowsthe sections of the vortex tubes with a plane ? = constant. Note that the angles 0 = 0 and 0 = 2r have to be identified. (d) For F > 0, resonances appear in the classical phase space. At the resonant actionI,, an ’elliptic island”emergcs which is surroundedby a ‘stochastic layer”. Note that vortex tubes ‘outside” an elliptic island cannot be deformed continuously to tubes “inside” the resonance.

can be written as

with the scaled coupling strength X = dF/2D/3 (3.5) The important observation is that the action scales with a factor a = (2mDlp’)’’’ (3.6) This factor drops out of the scaled classical equations of motion, but it remains in the quantum problem: it scales Planck’s constant h to the “effective value” h , =~ h/u.17 Since a is proportional to the square root of m, it is quite large in our case: u = 23.87h. Hence, the characteristic action her is comparatively small, a fact which indicates that semiclassical methods will be very powerful. A semiclassicalanalysis of periodicallydrivenquantum systems relies on the fact that Floquet states are associated with classical invariant vortex tubes in the same way as usual stationary states are associated with invariant tori.’’ The classical trajectories of the unperturbed Morse oscillator (2.2) are confined to manifolds E = constant of constant energy. For bound motion, these manifolds are, topologically speaking, circles C in the (p,x) phase space (see Figure Sa). However, in the periodically driven case it is advantageous to consider the dynamics in the three-dimensional extended phase space (p,x,?) spanned by momentum, position, and time coordinate. If we choose every point @O,XO) of a particular circle C as initial condition for Hamilton’s equations ( F = 0), the resulting trajectories fill out a cylinder or “vortex tube”” in the extended


Excitation and Tunneling by Short Laser Pulses (p,x,r) phase space, as shown in Figure Sb. In action angle variables of the unperturbed system, the initial manifolds Ccanberepresentedby straight lines (Figure 5c). It isunderstood that points at 0 = 0 and 0 = 2 r have to be identified. The unperturbed Morse oscillator is integrable: the phase space is completely stratified into invariant manifolds. Integrability is lost when the system is driven by a periodic force. The Kolmogorov-Arnold-Moser the0rem3~implies that some of the vortex are merely deformed and become periodically time dependent; others are destroyed completely. One way of visualizing the dynamics is to compute a PoincarCsection. Starting with a representative set of initial values, one solves the equations of motion numerically and plots the coordinatesafter every period. In our context, the crucial point is the emergence of resonances. If the ratio of the driving frequency w and the frequency Q(1)of oscillations in the Morse potential is equal to a ratio of small integers,stable (elliptic) and unstable (hyperbolic) periodic orbits appear. In a PoincarC section, periodic orbits correspond to fixed points, and close to the elliptic fixed points the dynamics look like that of a pendulum:39 Mainly regular motion within the "elliptic island" is bounded by a separatrix. Actually, instead of a smooth separatix curve, there is a 'stochastic layer" (very schematically drawn in Figure 5d). The size of these resonances depends on the integers which form the ratio w:Q; the largest is the principal resonance with o:Q = 1:l. What has all this to do with our problem? In terms of the action variable (3.7)

the classical Hamiltonian HOof the unperturbed Morse oscillator can be expressed as

Ho(l) = wd - w:12/40

(3.8)

where wo is the frequency of small oscillations, wo = (20B2/m)'/2

(3.9)

By BohrSommerfeld quantization, the bound quantum states are associated with discrete values I,,= (n I/*) of the action:

+

are the correct energy eigenvalues; n is an integer ranging from nmin= 0 to nmax= [a- l/2]. The classical angular frequency is given by (3.1 1)

In the example of the 1:l resonance, the condition (3.12)

determines the resonant action Ira,Le., the center of the most important elliptic island (cf. Figure 5d). Motivated by the quantum mechanical problem, let us now choosea driving frequencyw which, using conventionallanguage, corresponds to a "An photon transition" between the quantum states with energy E,, and E,,+&: 0=

(l/An)(En+&-EJ

(3.13)

The examples shown in Figures IC and 2 are special cases of such a choice with n = 0 and An = 6 and An = 5 , respectively. Using

eq 3.10 it follows that w: w = w --(An

40

+ 2n + 1) (3.14)

and a comparison with eq 3.11 yields

Ire = '/2(~,,+& + 1,) (3.15) Thechoice(3.13)ofthefrequencyo has theeffectthat theelliptic island of the 1:l resonance develops right in the middle between the action I,,corresponding to the initial state and the action I,,+& of the target state. Collecting all these pieces, we can now discuss the interaction of a Morse oscillator with a pulsed periodic force on a purely classical level.19 Let us start with a classical 'initial state" consisting of a set of initial values (po,xo), all of them confined to the manifold I = I,,of the unperturbed oscillator, (2.2) and let us fix the frequency (3.13). When the interaction is turned on, Le., when F(r) slowly rises, the initial manifold is deformed; but, by the principle of adiabatic invariance, the trajectories stay close to it. At the same time, the resonant elliptic island emerges at I = Ira;its width grows approximately with the square root of F.38 At a certain critical field strength F,, it becomes so large that its separatrix touches the deformed initial manifold. At this moment, the adiabatic invarianceof theaction breaksdown: The trajectories have to cross the separatrix, and separatrix crossing leads to a "jump" of the action.40-42 Because the Morse potential is only weakly anharmonic, the width of the resonance increases almost symmetrically to higher and to lower values of the action. Thus, the separatrix touches both the deformed manifold with action I,, and that with action I,,+b at the same time. Both manifolds combine to a new object when "captured" inside the elliptic island, and, again by virtue of the adiabatic principle, the trajectories remain close to this object when the field strength rises further and, later, starts to decrease. But when the field strength reaches Fcagain, separatrix crossing takes place a second time and the trajectories are distributed over the manifolds with the actions I , and I,,+&, to which they remain confined until the end of the pulse. As a result of this process-adiabatic motion, separatrix crossing, adiabatic motion, separatrix crossing, adiabatic motion-a "classical transition" from I,,to I,,+& has taken place. A numerical study of this scenario, as well as further discussions, can be found in ref 19. Ascending from the classical to the semiclassicallevel, we can no longer refer to individual trajectories, but rather have to focus on the quantized vortex tubes (i.e., those manifolds to which the trajectories are confined). A quantized vortex tube is a semiclassical Floquet state.17 The deformation of the vortex tubes with increasing field strength F can be regarded as a continuous process as long as F(t) is smaller than the critical value Fc.But because of the existence of the separatrix, low-field vortex "outside" the resonance cannot be deformed continuouslyto vortex tubes "inside" the resonance. In fact, if one applies the semiclassical quantization rules,17 the assignment of quantum numbers changes when Floquet states are captured by resonances.18 Of course, this change of quantum numbers is a reflection of the classical "jump of the action". On the quantum level, the classical discontinuity is not immediately visible. The quasienergies e: appear to vary smoothly with F, and as a consequence, the functional Aif[F(t)] depends smoothly on the pulse shape. Nevertheless, classical separatrix crossing leaves its traces in quantum mechanics. The change of quantum numbers indicates a strong change of the nodal structure of the instantaneous Floquet states, and that is indeed what is found numerically.16J8 Thus, we arrive at the following, refined picture of the excitation process:


When the field strength F ( t ) rises, the initial state is adiabatically shifted into the connected Floquet state; this Floquet state is associated with a classicalvortex tube. When the classical separatrix crossing occurs at Fc,the vortex tubes have to reorganize themselves from "outside" to "inside" the resonance. This reorganization process leads to a strong change of structure of the instantaneous Floquet states uf(x,t) in a small interval of field strength close to F,; this change of structure is too strong for the wave function to follow adiabatically. Instead, it is torn into a superposition of two Floquet states; these two states are those which are associated with the two actions that are coupled by the classical resonance. After that, both components of the wave function evolve adiabatically and acquire a relative dynamical phase which is determined by the difference of the instantaneousquasienergies. When the second separatrix crossing occurs, quantum mechanics dominates the scene: quantum mechanical interference determines the excitation probabilities of the final states. As a result of all these considerations, we can answer the question posed in the beginning of this section. The frequencies chosen in Figures IC and 2 are special because they lead to a coupling of the initial and the target state via a classical 1:l resonance. An argument that goes back to Berman and Zaslavsky43 shows how a classical 1:1 resonance affects the quantum mechanical quasienergy spectrum. Let us consider an arbitrary onedimensional quantum system with Hamiltonian Ho, and let this system interact with an external periodic force of strength A:

H = H,+ AX cos(wt)

(3.16)

The existence of a 1:l resonance then implies that close to the "resonant" eigenstate cp,of HOwith eigenvalueE, the level spacing is approximately equal to the photon energy w , so we also assume

E:=w

(3.17)

(The prime denotes differentiation with respect to quantum number.) Applying the techniques outlined in ref 43, it is possible to derive an approximate analytical expression for the quasienergies ck of near-resonant states in terms of characteristic values of the Mathieu equation9 c k ( 4 ) = E, + '/*E':ak(q) mod 0

(3.18)

where the Mathieu parameter q is a scaled coupling strength between neighboring states close to the resonance

and ak(q) is one of those characteristic values that are associated with (even or odd) *-periodic solutions of the Mathieu equation; these characteristic values are usually denoted44 by ao, a2, a4, ... (for even Mathieu functions) or by b2, b4, bb, ... (for odd functions). Thus, the similarity of the quasienergy spectrum close to a resonance (Figure 3a) and the stability diagram of the Mathieu equationU is no coincidence. (The quasienergiesappear inverted, because E': is negative.) Since both aznand bZnapproach (2n)* for vanishing coupling strength q (or A), one easily sees that expression 3.18 becomes, in that limit, the second-order Taylor expansion of the unperturbed energy eigenvalues around E,, with the "mod w" as linear term (see (3.17)). For the particular example of a Morse oscillator, where the energy eigenvalues depend only quadratically on the quantum number, the approximation (3.18) is quite accurate. In fact, if the resonance couples the initial ground state with a final state fwith even quantum number, eq 3.17 can be fulfilled exactly. For instance, for f = 4 the resonant state is r = 2. In this case the removal of the degeneracy of the quasienergies e: and ,e: which eventually determines the relative phase between the interfering

Breuer and Holthaus Floquet states, is expressed by the difference between the characteristic values44

a4(q)= 16 +

5* + & 433 + ...

(3.20)

and b4(4)= 1 6 + & - & +317

...

(3.21)

Because, according to (3.19). q is proportionalto the field strength

F,one finds that for such an example of a "four-photontransition" the quasienergies split with the fourth power of the field strength, as expected from perturbation theory. All these considerationsshow how the correspondenceprinciple works if there is a 1:l resonance in the classical phase space. What about the other resonances? For instance, if w = (E6 Eo)/3, there is a classical resonance w:O(I,) = 2:l at I,, = (16 + 10)/2. With slight modifications,the preceding discussionalso applies to this case. But at equal values of the field strength F, the width of the 2:l resonance (which leads to two elliptic islands in a Poincart section) is much smaller than that of the "principal" 1:l resonance.38 This fact, in turn, implies that the field strength Fcnecessary for separatrix crossingor, correspondingly,to achieve an appreciable quasienergy splitting, is much higher. Figure l a shows no effect of the resonance even for F , = 0.1 au; up to this field strength there is (mainly) deformation of the initial vortex tube or, quantum mechanically, adiabatic motion. Going to even higher field strengths does not seem to be sensible, because F = 0.1 au = 5.142 X 1O8 V/cm is already a very strong field; such a strong field is likely to have several undesired effects on a real molecule.4s Thus, if we define an "optimal laser pulse" as a pulse which leads to (almost) complete excitation of a target state with the smallest possible field strength, we have to conclude that such a pulse is linked to that classical resonance which encloses the largest area in phase space, Le., the 1:l resonance. Of course, this does not mean that other resonances are of no importance at all. One can enhance the "classical character" of the driven Morse oscillator by choosing an even larger value of the parameter CY (see eq 3.6). The effective Planck constant h,ff = h/a determines how many quantized vortex tubes fit into a classicalresonance," and if a is large enough, even "small" higher resonances can become important. The Iz+ molecule,46 for instance, yields a = 180 h. In such a case, the classicalquantum correspondencewill be even more pronouncedthan in our example. In addition, there is a simple rule: The larger a,i.e., the smaller the anharmonicity, the larger the "quantum jump" An that can be induced by the mechanism under discussion. 4. Laser-Assisted Tunneling

This section investigates the problem of pulse-shape-controlled tunneling in a laser field.21 Whereas the mechanism of pulseshape-controlled selectiveexcitationas discussed in section 2 could partially be linked to classical mechanics, the tunneling process is a genuinely quantum mechanical phenomenon. Nevertheless, it will be demonstrated that the Floquet picture leads to a very similar description of both cases; in fact, no further theoretical developments will be necessary. Let us choose the model Hamiltonian47 H(p,x,t) = p2/2

+ VDW(x)+ F ( ~ ) xsin(ot)

(4.1)

with the quartic double-well potential (4.2) For F = 0, the value of D determines the number of energy eigenstates with negativeenergy. Let us fix D = 2.5 so that three pairs of states fall "below" the barrier; very similar dynamical conditions govern the inversion of an NH, molecule.4*

The Journal of Physical Chemistry, Vol. 97,No. 48, 1993 12641

Excitation and Tunneling by Short Laser Pulses Let cpl(x)and *(x) be the lowest two double-well eigenstates with energies El and E2, respectively. Because m(x) has even and * ( x ) has odd parity, the linear combinations cp*(x) = ( l / a c c l ( x ) f cpz(4) (4.3) are "localized" in the "right" or the "left" well. Without the laser field, an initial wave function +(x,t=O)

= cp+(x)

(4.4)

evolves in time as

= (1/v'WE1'(cpl(x) + cp2(x)e-i(ErE1)') (4.5) and, due to the tunnel splitting AE = (E2 - E l ) ,the relative phase +(x,t)

between both components grows linearly with time. In particular, after the 'bare tunneling time" (4.6)

-5L2

-6 0

there is a relative factor of (-1), which means that (up io an overall phase) cp+(x) has been shifted to cp-(x); the particle has tunneled from one well to the other. Thus, tunneling in a double well is a matter of quantum phases, and the following question arises:21 Is it possible to control the relative phase by an external laser pulse such that the tunneling process is accelerated? Can one, again, design "optimal laser pulses" which lead to a fast, controlled population transfer from one well to the other? The results of section 2 immediately suggest a possible mechanism: When the laser field is turnedon smoothly,the initial double-well eigenstates p,(x) evolve, according to eq 2.1 1, into the "adiabatically connected" Floquet states uf(x,t), and the initial wave function (4.4) is shifted into

uf(')(x,t) exp(-iK$'?

.2

.4

F

.e

.a

1

Figure 6. Logarithm of the tunnel splitting el:- c g / w for the periodically driven quartic double well (4.2), plotted as function of the strength Fof the driving force (D = 2.5, w = 1.5). 0 .2 .4 .6 6 5 ...........................................

1

dt')) (4.7)

Let us consider a smooth pulse F ( t ) which starts at t = 0, reaches a maximal field strength FmaX, and decreases back to F = 0 after a total pulse time t,. Exactly as in the case of selective excitation, the resulting relative phase AT is the integral over the difference of the instantaneous quasienergies: (4.8) Thus, it is possible to influence the tunneling process by a laser pulse. In particular, onecan manipulate the tunneling time: The relative phase factor becomes (-1) if the conditions are such that

AT(?,) = ~ ( 2 -t k I), k = 0,fl,f2,... (4.9) The tunneling time T is therefore no longer given by eq 4.6, but rather determined by AT(7) = f7r

(4.10)

The magnitude of T depends on the behavior of the quasienergies. If the absolute value of the difference (et- e:) becomes smaller than the original tunnel splitting ( E 2- El), the tunneling process is slowed down; if it becomes larger, the particle can tunnel faster. Let us illustrate this scenario with a numerical example.2l The value D = 2.5 leads to (E2- E l ) = 1 SO7 X 10-5; the bare tunneling time (4.6) is T b = 2.085 x 105. Measuring time in units of cycles T = 2u/w for a driving frequency w = 1.5, we have Tb = 49766T. Figure 6 shows the logarithm of the absolute value of (e: cf)/w as function of the field strength F. Both quasienergies approach each other and cross at FE 0.05, but then the difference increases by several orders of magnitude. In the presence of a driving force with a constant amplitude close to Fc, the tunneling

0

.2

.4

.6

1

Fmax

Figure 7. Tunneling functional AT in units of ?I,evaluated for pulses (2.5) with w = 1.5 (see Figure 6) and tp = 500 X 2?1/w (above), and numerically computed tunneling probabilities (below). time will be larger than ~ b ; it~will ~ be , shorter ~ ~ than Tb for larger amplitudes. The crossingof the two "lowest" quasienergiesshould be no surprise; it can be deduced from a simple two-level approximation.23 Since we are interested in fast, controlled population transfer, we will need strong pulses. For convenience, let us choose again the shape function (2.5) and fix the pulse length t, = 500T. which is 2 orders of magnitude shorter than T b . The upper half of Figure 7 shows the phase AT as a function of the maximal amplitude Fmpx,calculated from eq 4.8. On the other hand, we can integrate SchrBdinger's equation numerically for the initial condition (4.4) and pulses (2S);inthis way, thetunnelingprobabilitiesI(&(rP))l2 are calculated by brute force. The lower half of Figure 7 shows the result. Exactly as predicted by eq 4.9, a practically complete

Breuer and Holthaus


population transfer is achieved if AT is equal to an odd integer multiple of ?r. Now we can proceed in complete analogy to the case of selective excitation: AT is a functional of the pulse shape, AT = &[F(t)], and if the instantaneous quasienergies are known, the condition (4.9) characterizes the possible “optimal laser pulses” in the adiabatic regime. Using the tunneling functional, it is possible to “design” such pulses without solving the Schrijdinger equation. Compared to the mechanism leading to the results shown in Figures IC and 2, the present case is much simpler. There is no need to split the initial wave function, because the functions localized in one of the wells already consist of two components; there is no interference to determine the final occupation probabilities. Nonetheless, there is a strong resemblance: In both cases, adiabatic evolution of two components and the emergence of a relative dynamical phase determine the result. Obviously, the present discussion relies in an essential way on the symmetryof thedouble-well potential V,W(X). But the essence is clear: Quantum phasescan be controlled by smooth laser pulses, a fact which also allows us to manipulate the tunneling process.

5. Discussion The previous model calculations have shown under which conditions the adiabatic principle can allow an efficient control of quantum mechanical dynamics. The case of pulse-shape-controlled selective excitation of molecular vibrational states is somewhat involved: If the pulse length lies in the adiabatic regime, Le., if (apart from those moments when the wave function splits or interferes) the time evolution can be described as adiabatic motion on quasienergy surfaces, and if the classical 1:1 resonance is large enough at peak field strength F,,, so that the quantized vortex tubes associated with the initial and the target states fit inside the main elliptic island, and if the peak field strength is small enough so that dissociation remains negligible, then there are two basic rules: (i) The laser frequency w should be chosen such that a classical 1:l resonance couples the vortex tubes of the initial and the target state state by means of the separatrix crossing mechanism. In the exampleof a Morse oscillator the Hamiltonian dependsquadratically on the action, which implies that the optimal frequency is close to (3.13). (ii) The smooth pulse shape should be chosen such that the interference of the two Floquet states participating in the dynamics is constructive. The first interference maximum corresponds to optimal pulses with the smallest field strength; it is determined by Aif[F(t)] = f?r (see eq 2.16). The excitation mechanism can be described in terms of splitting and interference of the quantum mechanical wave function. From a practical point of view, it is important that the whole process can be induced by pulses with smooth shapes, which can be realized without much effort. In addition, condition 2.16 leaves considerable freedom. One can choose a suitable pulse length t , and use smooth, but otherwise arbitrary, envelopes. The mechanism itself is an interesting example of the classicalquantum correspondence for time-dependent processes. On the classical level, the periodically forced Morse oscillator is a system with “soft chaos”;SO apart from the stochastic layers close to the separatrices the dynamics are mainly regular. What has been observed in a number of different cases is confirmed again: It appears “as if quantum mechanics were capable of riding smoothly over the rough spots in classical mechanics” (see ref 50, p 207); a discontinuous rearrangement of classical vortex tubes results in a smooth phase functional Air. One may ask whether there are similar possibilitiesof control over a quantum system if its classical counterpart exhibits “hard chaos”. The example of pulse-shape-controlled tunneling in a symmetric double well appears comparatively simple: If the pulse length lies in the adiabatic regime, then the tunneling process can be manipulated systematically by laser pulses with

smooth shapes. The tunneling process can be slowed down or accelerated, and if the difference of the two relevant quasienergies becomes large, a fast, controlled population transfer from one well to the other is possible. The condition which characterizes optimal laser pulses is AT[F(~)]= * A (see eq 4.9); the mechanism is adiabatic motion of two separate components. All in all, the technique of “adiabatic control” is certainly not as ambitious or general in scope as the systematic application of the theory of optimal control, but it does yield a lot of physical insight. Even in the context of our simple models, many questions remain. For example, what about the nonadiabatic regime? Can one find further rules which allow the controlled application of extremely short pulses? If yes, to which extent could such rules by universal? The problem of selective excitation leads to further questions. Numerical studies of a periodically driven Morse oscillator with parameters of an 0-H bondllJ2 have shown that smooth pulses can induce the transition (n = 0) (n = 5 ) with high efficiency, but not the transition (n = 0) (n = 10). From the semiclassical point of view, this result is easy to understand. The principal 1:1 resonance can be large enough to couple the vortex tubes that originate from the states with n = 0 and n = 5 , but not large enough to couple the more “distant” states n = 0 and n = 10. Hence, the situation is qualitatively very different in both cases. But there is a simple possibility to circumvent this problem:s1 One can employ two resonances and achieve the transition in two steps, e.g., (n = 0) (n = 5 ) (n = lo), where each single step follows the now familiar scheme. But the limiting quantity remains the phase space area covered by the principal resonance(s). It is an important question whether there are alternative mechanisms which are not subject to such limitations and which can be exploited to achieve transitions between distant states in a single step. Another class of problems concerns the many degrees of freedom of a real molecule. Is it possible to control them individually? Can one, for instance, use the scheme of selective excitation to manipulate one particular bond in a large molecule? A major difficulty in actual experiments is due to the spatial variation of the intensity in a laser focus. Different molecules will, in general, “see” different pulse profiles, and an interference pattern like that shown in Figure I C or 2 might be averaged out. It is, therefore, encouraging to know that signatures for the mechanisms discussed in this contribution have already been observed: Experimentson Rydberg atoms interacting with pulses of “strong” microwave radiation can be carried out under extremely well-defined conditions, so that even subtle questions concerning the classicalquantum correspondence can be investigated.52 There are experimental data53 which show clear indications for separatrix cro~sing;’~ very recent experimentss4sSs have demonstrated for the first time that adiabatically evolving, interfering Floquet states lead to an oscillating transition probability. Thus, one is no longer restricted to “numerical experiments”; real experiments with Rydberg atoms can help to understand the mechanisms which, in a different parameter regime, are candidates for the coherent control of molecular dynamics. In summary, the analysis of simple models has revealed some general principles which determine the response of quantum systems to short, strong pulses of a periodic force; particularly important is the possibility to use the pulse shape to manipulate relative dynamical phases and thereby to control excitation and tunneling processes. Whether these principles can actually be exploited in the laboratory to achieve the laser-assisted control of molecular systems remains to be seen.

--

- -

Acknowledgment. The suggestion to employ the Floquet picture to analyze the previous results contained in refs 11 and 12 was made by J. Manz, whom we also thank for inspiring discussions

Excitation and Tunneling by Short Laser Pulses and helpful comments. M.H.gratefully acknowledgesa Feodor Lynen research grant from the Alexander von Humboldt-Stiftung, as well as support from the Office of Naval Research (N0001492-5-1452),

References and Notes (1) For a recent review, see: Warren, W. S.;Rabitz, H.; Dahleh, M. Science 1993, 259, 1581. (2) Tannor, D. J.; Kosloff, R.; Rice, S . A. J. Chem.Phys. 1%. 85,5805. (3) Brumer, P.; Shapiro, M. Acc. Chem. Res. 1989, 22, 407. (4) Lu. S.-P.; Park, S.M.; Xie, Y.; Gordon, R. J. J . Chem. Phys. 1992, 96, 6613. ( 5 ) Scherer, N. F.; Carlson, R. J.; Matro, A.; Du,M.; Ruggiero, A. J.; Romero-Rochin, V.; Cina, J. A.; Fleming, G . R.; Rice, S . A. J. Chem. Phys. 1991, 95, 1487. (6) Rice, S . A. Science 1992, 258, 412. (7) Rerce, A.; Dahleh, M.; Rabitz, H. Phys. Reo. A 1988, 37, 4950. (8) Armstrup, B.; Carlson, R. J.; Matro, A,; Rice, S.A. J . Phys. Chem. 1991, 95, 8019. (9) Judson, R. S.; Rabitz, H. Phys. Rev. Lett. 1992, 68, 1500. (IO) Shi, S.;Rabitz, H. J. Chem. Phys. 1990, 92, 364. (11) Jakubetz, W.; Just, B.; Manz, J.; Schreier, H.-J. J. Phys. Chem. 1990,94,2294. (12) Just, B. DiplomaThesis,Institut filr PhysikaliheChcmie, Universitat Wilrzburg, 1991. (13) Kuz'min, M. V.;Sazonov, V.N. Zh. Eksp. Teor. Fiz. 1980,79,1759 [Sou.Phys. JETP 1981,52, 8891. (14) Ho, T. S.;Chu, S . 4 . Chem. Phys. Lett. 1987, 141, 315. (15) Breuer, H. P.; Holthaus, M. Z. Phys. D 1989, 11, 1. (16) Breuer, H. P.; Dietz, K.;Holthaus, M. J . Phys. B 1991, 24, 1343. (17) Breuer, H. P.; Holthaus, M. Ann. Phys. (N.Y.) 1991, 211,249. (18) Henkel, J.; Holthaus, M. Phys. Reo. A 1992,45, 1978. (19) Dietz, K.; Henkel, J.; Holthaus, M. Phys. Rev. A 1992, 45, 4960. (20) Connaade, J. P.; Conen, K.; Dietz, K.; Henkel, J. J . Phys. B 1992, 25, 3771. (21) Holthaus, M. Phys. Rev. Lett. 1992, 69, 1596. (22) Paramanov, G . K.; Sawa, V. A. Phys. Lett. A 1983, 97, 340. (23) Walker, R. B.; R a t o n , R. K. J. Chem. Phys. 1977, 67, 2017. (24) Goggin, M. E.; Milonni, P. W. Phys. Reo. A 1988, 37, 796. (25) Shirley, J. H. Phys. Reu. 1965, 138, B979. (26) Zel'dovich, Ya. B. Zh. Eksp. Teor. Fir. 1966,51, 1492 [Sou. Phys. JETP 1967, 24, 1006). (27) Ritus, V. I. Zh. Eksp. Teor. Fiz. 1966,51, 1544 [Sw. Phys. JETP 1967, 24, 10411.

The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12643 (28) For a review, see: Chu, S . 4 . Ado. Chem. Phys. 1989, 73, 739. (29) Yajima, K. Commun. Math. Phys. 1982.87, 331. (30) Note that the definition of 4fdiffers by a factor of I / * from that given in ref 16, so that here & denotes the actual phase difference. (31) Berry, M. V. Proc. R. Soc. London, A 1984,392,45. (32) Simon, B. Phys. R N . Lett. 1983, 51, 2167. (33) Breuer. H. P.; Dietz, K.; Holthaus, M. Radiat. Efl. Def. Solids 1991, 122-1 23.9 1. (34) Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. Modern

Geometry-Methods and Applications. Part II. The Geometryand Topology of Manlfolds;Springer: New York, 1985. (35) Breuer, H. P.; Holthaus, M. Phys. Lett. A 1989, 140, 507. (36) Graham, R.; Hbhnerbach, M. Phys. Reu. A 1991, 43, 3966. (37) Arnold, V. I. Mathematical Methods of Classical Mechanics; Springer: New York, 1978. (38) Lichtenberg, A. J.; Liebeman, M. A. RegularandSt~harricMotion; Springer: New York, 1983. (39) Chirikov, B. V. Phys. Rep. 1979.52, 263. (40) Tennyson, J. L.; Cary, J. R.; Escande, D. F. Phys. Rev. Lett. 1986, 56, 2117. (41) Cary, J. R.; Escande, D. F.; Tennison, J. L. Phys. Reu. A 1986.34, 4256. (42) Dana, I.; Reinhardt, W. P. Physica D 1987, 28, 115. (43) Berman, G. P.; Zaslavsky, G . M. Phys. Lett A 1977, 61, 295. (44) Abramowitz, M., Stegun, I. A., Eds. Handbook of Mathematical Functions; Dover: New York, 1972. (45) Chelkowski, S.;Bandrauk, A. D.; Corkum, P. B. Phys. R w . Lett. 1990,65, 2355. (46) Tanner, J. J.; Maricq, M. M. Phys. Reo. A 1989,40,4054. (47) Grossmann, F.; Dittrich, T.; Jung, P.; Hlnggi, P. Phys. Reu. Lett. 1991, 67, 516. (48) Rauk, A.; Allen, L. C.; Clementi, E . J. Chem. Phys. 1970,52,4133. (49) Grossmann, F.; Jung, P.; Dittrich, T.; Hlnggi, P. Z. Phys. B 1991, 84, 315. (50) Gutnviller, M. C. Chaos in Classical and Quantum Mechanics; Springer: New York, 1990. (51) Just, B.; Manz, J.; Paramonov, G . K.; Chem. Phys. Lett. 1992,193, 429. (52) Sauer, B. E.; Bellermann, M. R. W.; Koch, P. M. Phys. Reu. Lett. 1992, 68, 1633. (53) Bayfield, J. E.; Sokol, D. W. Phys. Rev. Lett. 1988, 61, 2007. (54) Baruch, M. C.; Gallagher, T. F. Phys. Rev. Lett. 1992, 68, 3515. (55) Yoakum, S.;Sirko, L.; Koch, P. M. Phys. Reo. Lett. 1992,69,1919.

Adiabatic control of molecular excitation and tunneling by short laser

Recommend Documents