arX

iv:c

hao-

dyn/

9803

038v

1 2

6 M

ar 1

998

Preprint HKBU-CNS-9714

August 1997

Squeezed state dynamics of kicked quantum systems

Bambi Hu[1,2], Baowen Li[1], Jie Liu[1,3], and Ji-Lin Zhou[1,4]

[1]Department of Physics and Centre for Nonlinear Studies, Hong Kong Baptist University, Hong

Kong, China[2] Department of Physics, University of Houston, Houston TX 77204

[3] Institute of Applied Physics and Computational Mathematics, P.O.Box 8009, 100088 Beijing,

China[4] Department of Astronomy, Nanjing University, 210093 Nanjing, China

Abstract

In this paper, at first, we apply squeezed state to quantize harmonic oscilla-

tor. It is shown that this approach produces not only energy levels, but also

eigenfunctions, exactly. At second, by using the squeezed state, we demon-

strate that quantum fluctuations can enhance chaos as well as suppress chaos.

Finally, energy diffusion of kicked quantum systems is studied in the squeezed

state approximation. Both numerical and analytical results show that the en-

ergy diffusion may behave either as t in analogy to the classical diffusion,

or behave as t2 due to the effect of the quantum fluctuations, or, in special

case, be strongly localized. We conjecture that these are general three typical

quantum diffusion behaviors.

PACS numbers: 05.45.+b,03.65.sq,05.40.+j

Typeset using REVTEX

1

I. INTRODUCTION

Squeezed state is a generalized coherent state, which has a wide applications in manybranches of physics such as quantum optics and high energy physics etc. In recent years,there are growing applications of squeezed state to study the chaotic dynamical systems[1–12]. We call this squeezed state approach. Sometimes, people in this field also called thisapproach semiquantum approach, thus the time evolution of the expectation values and thefluctuations of the squeezed state is called semiquantum dynamics.

The main purpose of this approach is to study how the quantum fluctuations manifestthemselves on the classical trajectories. It starts directly from the quantum systems withno reference to the classical limit. In fact, it has been shown [5] that the squeezed statedynamics exits even for the systems without a well-defined classical dynamics. In generalcase, the squeezed state approach simplifies the quantum version and gives us a way to gobeyond the semiclassical approximation [1–5,7,8]. It provides a very useful tool to study theclassical-quantum correspondence problems.

The squeezed state approach has proven to be very successful in studying the dynamicalsystems ranging from integrable to many-body non-integrable systems. Among many others,let’s give just a few examples here. In calculating the ground state energy for the quantumsystem with potential V (q) = −V0/cosh

2αq, Tsui [8] discovered that the ground state energyobtained by the squeezed state approach is much closer to the exact ground state energy thanthat from the usual WKB method. More recently, Pattanayak and Schieve [11] applied thisapproach to a classically chaotic system, for which the WKB method completely fails, theyhave been successfully calculated low-lying eigenenergies which agree within a few percentwith the pure quantum (numerical) results. In addition to the ground state and/or the lowerexcited state energies, the squeezed state approach can provide a way to obtain correcteigenfunctions. For instance, in applying the squeezed state approach to the harmonicoscillator we can produce not only the eigenenergies but also the eigenfunctions (includingthe ground state) exactly as we shall see in Section III, whereas the semiclassical methodcannot yield the correct eigenfunctions. As a further example for many-body systems, wewould like to mention that we have used the squeezed state approach to the one dimensionalquantum Frenkel-Kontoroval model [13], which is a nontrivial many-body non-integrablesystem. Our results show that the squeezed state approach captures very nicely the featureof the quantum effect, namely, the standard map which determining the coordinates ofclassical ground state is renormalized to an effective sawtooth map in the quantum case.The squeezed state results agree well with that of quantum Monte Carlo method [13].

In this paper we would like to apply the squeezed state to study the generic behaviors ofquantum kicked systems. As is well know that, in the development of quantum chaos, thekicked quantum systems play a very important role. The prototype of these kicked systemsare kicked rotator and kicked harmonic oscillator (KHO). They represent two different classesof dynamical systems. On the one hand, the kicked rotator obeys the Kolmogorov-Arnold-Moser (KAM) theorem. Classically, as the kick strength increases, the invariant curvesgradually breaks up. When the strength of kick exceeds a critical value of Kc = 0.9716..., thelast invariant curve disappears and the bounded chaos turns into global chaos, characterizingunbounded diffusion in the momentum direction [14]. Quantum mechanically, the diffusionfollows classical one only up to a certain time, after which it is completely suppressed thus

2

leads to the dynamical localization [15]. The dynamical localization has been connected tothe Anderson localization [16], and has been confirmed experimentally [17].

On the other hand, since the harmonic oscillator is a degenerate system, the KHO modelis thus out of the framework of the KAM theorem, namely diffusion can occur along thestochastic webs for any small perturbation strength. The KHO model stems from a realphysical system. It describes a charged particle moving in a magnetic field, and underthe disturbance of a wave packet [18]. It has 11

2degrees of freedom. In the classical case,

depends on the ratio between frequency of harmonic oscillator and that of external kicks (seeEq.(47)), this system displays abundant structures in phase space such as the crystal, quasi-crystal and stochastic webs etc [18,19]. Since the phase space of the KHO is unbounded andcannot be reduced to a cylinder as in the case of the kicked rotator model, the numericalinvestigation of quantum KHO is much more difficult than that of kicked rotator model.Therefore, in contrary to the kicked rotator, only few attempts have been made in thequantum KHO [20–24]. A general picture about the quantum behavior of the KHO is stilllacking.

Using the squeezed state approach, as we shall see later, we are able to obtain not onlynumerically but also analytically the general diffusion behaviour of this model.

The paper is organized as follows. In Sec. II we give a brief introduction of the squeezedstate approximation for the purpose of self-contained. In Sec. III we shall apply thisapproach to quantize the harmonic oscillator which yields exactly the eigenenergies as wellas the eigenfunctions. In Section IV, by using the KHO model, we shall demonstrate that thequantum fluctuations will enhance chaos at small perturbation regime, while it suppresseschaotic diffusion at large perturbation regime. In Section V we shall study numerically andanalytically the diffusion and localization phenomena. In Sec. VI some comparisons betweenthe squeezed state results and those very few available quantum results will be given. Weshall conclude our paper by discussions and remarks in Sec. VII.

II. SQUEEZED STATE APPROACH

The squeezed state approach starts from the time-dependent variational principle(TDVP) formulation,

δ∫

dt〈Ψ(t)|ih ∂

∂t− H|Ψ(t)〉 = 0. (1)

Variation w.r.t. 〈Ψ(t)| and |Ψ(t)〉 gives rise to the Schrodinger equation and its complexconjugate (see e.g. [25]), respectively. The true solution may be approximated by restrictingthe choice of states to a subspace of the full Hilbert space and finding the path along whichthe above equation is satisfied within this subspace. In the squeezed state approach, thesqueezed state is chosen as |Ψ(t)〉. In this manner, as we shall see later, in addition to thedynamics of the centroid of the wave packet, we will also have the equations for the motionof the fluctuations, the spread of wave packet. Therefore, this approach enables us to studythe effects of the quantum fluctuations on the dynamical behavior.

The squeezed state is defined by the ordinary harmonic oscillator displacement operatorD(α) acting on a squeezed vacuum state S(β)|0〉:

3

|αβ〉 = D(α)S(β)|0〉D(α) = exp

(

αa+ − α∗a)

, (2)

S(β) = exp(

1

2(βa+2 − β∗a2)

)

a+ and a are boson creation and annihilation operators which satisfy the canonical commu-tation relation: [a, a+] = 1. The coherent state is just operator D acting on the vacuumstate |0〉,

|α〉 = D(α)|0〉. (3)

In terms of number eigenstate |n〉, the coherent state can be written as

|α〉 = exp(

−1

2|α|2

) ∞∑

n=0

αn√n!|n〉. (4)

α is the eigenvalue of the creation operator, i.e.

a|α〉 = α|α〉. (5)

Denote,

α = |α|eiη, (6)

and taking the integral over the angle η from 0 to 2π on both sides of Eq.(4), we obtain thenumber eigenstate expressed in terms of the coherent state,

|n〉 =1

2πexp

(

1

2|α|2

)

|α|−n√

n!∫ 2π

0dηe−inη|α〉. (7)

The coordinate and momentum operators are defined as:

p = i

√

h

2(a+ − a),

q =

√

h

2(a+ + a). (8)

Thus we have the expectation values and the variances,

p ≡ 〈Ψ(t)|p|Ψ(t)〉 = i

√

h

2(α∗ − α),

q ≡ 〈Ψ(t)|q|Ψ(t)|〉 =

√

h

2(α∗ + α). (9)

∆q2 ≡ 〈Ψ(t)|(q − q)2|Ψ(t)|〉 = hG,

∆p2 ≡ 〈Ψ(t)|(p − p)2|Ψ(t)|〉 = h(1

4G+ 4Π2G). (10)

4

The canonical coordinate (G, Π) was introduced by Jakiw and Kerman [26] for the quantumfluctuations, and its relation with β in Eq.(3) is [5,8]:

G ≡ 1

2| cosh |β| + β

|β| sinh |β||2,

Π ≡ i

2

β∗ − β

|β|sinh |β| cosh |β|

|cosh|β| + β|β|

sinh |β||2. (11)

The Heisenberg uncertainty relation becomes:

∆q∆p =h

2

√1 + 16G2Π2 ≥ h

2. (12)

In fact, the squeezed state |αβ〉 is equivalent to the following Gaussian-type state [7]

|Ψ(t)〉 ≡ 1

(2G)1/4exp

(

i

h(pq − qp)

)

exp(

1

2hΩq2

)

|0〉

= e−iψ|αβ〉, (13)

where

Ω = 1 − 1

2G+ 2iΠ,

e−2iψ =1√G

(

cosh|β| + β

|β|sinh|β|)

. (14)

From the TDVP, we can obtain the dynamical equations for the expectation value and thequantum fluctuations,

q =∂H

∂p, p = −∂H

∂q, (15)

hG =∂H

∂Π, hΠ = −∂H

∂G, (16)

where the dot denotes the time derivative and the H is defined by

H ≡ 〈Ψ(t)|H|Ψ(t)〉. (17)

These equations give us a very simple and very clear picture about the motion of the expec-tation values as well as the evolution of the quantum fluctuations, which are responsible forthe quantum diffusion. If the Hamiltonian consists of separate kinetic and potential termssuch as,

H =1

2p2 + V (q), (18)

then the Hamiltonian function can be written as

5

H =1

2p2 + V (q) + h

(

1

8G+ 2GΠ2

)

+

exp

h

2G

(

∂

∂q

)2

− 1

V (q). (19)

Initial Conditions

In order to solve the equations of motion Eqs.(15) and (16), appropriate initial conditionsfor the variables (q, p) and (G, Π) should be posed. In principle, the initial condition mustbe physically meaningful. Thus the following two conditions are selected.

(1) Minimal uncertainty

The initial state |Ψ(t0)〉 should satisfy this condition. Because G0 is always larger thanzero, from uncertainty principle Eq.(12), we have

Π(t0) = 0. (20)

(2) Least quantum effect

We need to determine the initial value of G. This can be achieved by requesting the mini-mization of H with respect to G, i.e.

∂H

∂G= 0,

∂2H

∂G2> 0. (21)

For instance for the harmonic oscillator we have: G(t0) = 12ω0

.

III. QUANTIZATION OF THE HARMONIC OSCILLATOR BY THE SQUEEZED

STATE APPROACH

Harmonic oscillator is a simple but very important model in quantum mechanics. It is anintegrable system, its eigenenergies as well as eigenfunctions can be obtained analytically.Therefore, this model is very suitable for testing the approximated methods such as theWKB method and so on.

As is well know that the WKB approximation can give us the exact eigenenergies for thisintegrable system. However, it can not yields the exact eigenfunctions, in particular for thelow-lying eigenstates. It gives only the envelope of the wavefunction at the semiclassical limith → 0. In this section, we shall demonstrate that when applying the squeezed state approachto the harmonic oscillator model, one can not only obtain the energy levels precisely butalso the eigenfunctions as well.

The harmonic oscillator has the Hamiltonian operator

H =p2

2+

ω20 q

2

2. (22)

6

Applying the squeezed state approach to this system, one can easily obtain,

H =p2

2+

ω20q

2

2+

h

2

(

1

4G+ 4Π2G + ω2

0G)

. (23)

This Hamiltonian can be expressed in form of action-angle variables,

H = ωoI + 2ω0(J + h1

4), (24)

where,

I =1

2π

∮

pdq, J =1

2π

∮

Πd(hG). (25)

The transformations between (q, p), (G, Π) and (I, φ), (J, θ) has the following form

q =√

2I/ωo sin φ,

p =√

2Iωo cos φ,

G =1

ω0

(

(2J +1

2) −

√

2J(2J + 1) cos θ)

, (26)

Π =ω0

2

√

2J(2J + 1) sin θ

(2J + 12) −

√

2J(2J + 1) cos θ.

From Eq. (27) one can clear see that, the motion of both the two degrees of freedom (DOF)

are degenerate, namely, ∂Ho

∂Iis independent of I and

∂Hf

∂Jindependent of J . Furthermore,

(G, Π) are decoupled from (q, p). Thus the centroid of the wavepacket goes exactly along theclassical trajectory. While the fluctuations in momentum and position are time-independent,i.e. the width of the wavepacket keeps constant. From the minimal uncertainty principle forthe initial condition mentioned in previous section, we have G = 1

2ω0, Π = 0. Therefore, the

wavepacket along the periodic orbit always keeps its form as a coherent state.The time evolution of both (q, p) and (G, Π) are periodic with period T0(= 2π/ω0) and

T0/2, respectively. So we can apply the EBK quantization to the extended phase space,

I = nh, J = mh, n, m = 0, 1, 2, · · · , (27)

Substituting I and J into Eq.(24) and keeping in mind that G = 12ω0

and Π = 0 thus m=0,we obtain the squeezed state eigenenergy,

En = hω0(n +1

2). (28)

This is exactly the eigenenergy of the harmonic oscillator. Here, the zero point energy 12hω0

comes into the formula in a very natural and straight way. This is quite different fromthe WKB method. In WKB method, the 1

2hω0 comes from the Maslov phase correction

which is necessary because of the singularity of the wavefunction. In the squeezed stateapproach, since we have not any singularities, the Maslov-Morse correction is incorporated

7

by the extended variables G and Π. Furthermore, the energy of the system is in the form ofthe expectation value of the underlying Hamiltonian operator, whereas in the usual WKBmethod, the energy is taken to be the classical form, i.e E = Hcl. This is one of the reasonwhy the people call this approach semiquantum approach.

Let’s construct the eigenfunction by the squeezed state approach now. We see that whenthe trial wave function |Ψ(t)〉 is transformed to

|Ψ(t)〉 = exp

(

iλ(t)

h

)

|Ψ(t)〉, (29)

the derived variational equations of motion remain invariant. Substituting |Ψ(t)〉 into theSchrodinger equation, we can determine the equation of λ(t),

λ(t) =∫ t

0dt′〈Ψ(t′)|ih ∂

∂t′− H|Ψ(t′)〉 = λG + λD. (30)

The second part of the integral corresponds to the dynamical phase. We denote it λD. Thefirst term is the geometrical phase noted as λG, which is

λG =1

2

∫ t

0(pq − qp)dt + h

∫ t

0ΠGdt. (31)

Since the motion of (p, q) and (G, Π) are periodic, this geometrical phase is the Aharanov-Anandan form of Berry’s phase. During the evolution, each points along the periodic orbitacquires a phase factor. However, the dynamical phase does not change during the evolution,only the geometrical phase matters. So, the eigenfunction is a weighted sum over the pointsof the commensurate periodic orbit [10]. The weight factor at each point is an appropriategeometrical phase. Furthermore, as mentioned before, according to the requirement of theinitial condition, the initial wavepacket is a coherent wavepacket, and it does not changeits form when cycling along the periodic orbit. Substituting the expressions of p and q inEq.(27) into Eq.(31) and keeping in mind that Π = 0, we can easily evaluate the integraland obtain the geometrical phase at time t which is

λG(t) = nhφ. (32)

where φ = π/2− η. Thus, the eigenfunction for the bound state having the eigenenergy En

in Eq.(28) is:

c∫ 2π

0ei

λGh |α〉dφ = C

∫ 2π

0e−inη|α〉dη, (33)

where, |α〉 is the coherent state. φ is angle in the (ω0q, p) plane as is shown in Fig. 1. Cthe normalization constant. This is nothing but just the number eigenstate |n〉 given inEq.(7) except the prefactor. This constant C can be easily calculated by the normalization.Therefore, in the coordinate representation, the wavefunction is:

Ψn(q) = C∫ 2π

0dηe−inη〈q|α〉 (34)

This is the exact wavefunction of the harmonic oscillator.

8

IV. ENHANCEMENT AND SUPPRESSION OF CHAOS

In this section, we would like to discuss effect of the quantum fluctuations on classicalchaos. It is commonly argued that the quantum fluctuations suppress classical chaos dueto interference, while it enhances chaos due to tunneling. This is a rather qualitative argu-ment. Using the squeezed state approach it is very readily to do quantitative analysis of thequantum fluctuations.

There are only a few works on this issue. In a certain model, the suppression of chaoscomes about, while in other model the enhancement takes place. Zhang and his coworkershave observed the suppression of the chaos in the kicked spin system and the kicked rotatorsystem [1,2,4]. Using a one-dimension problem with a Duffing potential without any externalperturbation, thus the classical as well as the quantum behavior are regular, Pattanayak andSchieve [9] have demonstrated that the squeezed state behavior is chaotic, and concludedthat the quantum fluctuations induce chaos. The effect of enhancement was also confirmedin the kicked double well [6] model. As we shall see later that, the KHO model provides aprototype for studying these two effects. The enhancement and suppression can be observedby changing the external potential.

The Hamiltonian of the quantum KHO model can be written as [18],

H =p2

2+

ω20

2q2 + V (q)δT , (35)

where

δT =∞∑

n=−∞

δ(t − nT ). (36)

Using the squeezed state as the trial wavefunction of the Hamiltonian(35), one can readilyobtain

H = Ho + Hf + Hp, (37)

where

Ho =p2

2+

ω20q

2

2, (38)

is the Hamiltonian of the harmonic oscillator;

Hf =∆p2

2+

ω20∆q2

2, (39)

denotes the contribution from quantum fluctuations; and

Hp = exp

∆q2

2

(

∂

∂q

)2

V (q)δT , (40)

from the external perturbative potential. The external potential can be even or odd. With-out loss of generality, we denote it as

9

V (q) = KΘ(q), Θ(q) =

sin(k0q), for odd

cos(k0q), for even(41)

From Eq.(15) and Eq.(16) we have:

q = p,

p = −ω20q − KeffΘ

′(q)δT ,

G = 4ΠG, (42)

Π =1

8G2− 2Π2 − ω2

0

2+

k20

2KeffΘ(q)δT ,

where,

Keff = K exp

(

− hk20G

2

)

, (43)

is called effective potential whose physical meaning will be discussed later. Θ′ is the firstderivative of Θ with respect to q. It is clear to see from Eq.(43) that, in the time intervalnT < t < (n + 1)T , the harmonic oscillator takes free motion governed by

q = p,

p = −ω20q,

G = 4ΠG, (44)

Π =1

8G2− 2Π2 − ω2

0

2.

while at the time t = (n + 1)T , it is kicked by the external potential. Therefore, before andafter the kick the momentum and its fluctuation undergo an jump:

pn+1(T+) = pn(T

−) − Ke−hGn(T−)

2 Θ′(qn(T−)),

Πn+1(T+) = Πn(T

−) +k2

0

2Ke−

hGn(T−)2 Θ(qn(T

−)). (45)

Eq.(45) and Eq.(45) are coupled differential equations. From these equations, we can readilyfind out that the squeezed state dynamics differs from the classical motion in two things.(1) There are two additional dimensions for the squeezed state motion, namely the particlemoves in a 4 dimensional extended phase space. (2) The classical quantity (q, p) are coupledwith the quantum fluctuations, which make the semiquantal motion complicated. On theone hand, because of these two additional dimensions in phase space, we expect that theinvariant curves in the classical phase space would not be able to prevent the trajectoryfrom penetrating or acrossing them semiquantally. On the other hand, since the quantumfluctuations are always positive, they equal to zero only at the limit case of h = 0, theeffective potential strength Keff is always less than K. The reduction of the effectivepotential acting on the wavepacket leads to the suppression of chaos. These two mechanismsco-exist in the semiquantal system. They compete with each other and determine thedynamical behavior of the underlying system. So, we expect that the quantum fluctuation

10

may enhance the chaos in a certain case, and may suppress the classical chaos as well. Thisargument is nicely illustrated numerically in the following.

In this section we restrict our calculations on the potential:

V (q) = −K sin(q). (46)

But we should point out that the main conclusions given in this section does not dependneither on the parity of the potential nor the sign of K.

A. Enhancement of chaos

In solving Eqs.(45) and (45), we used the seventh and eighth order Runge-Kutta formulawith adaptive stepsize control. The permissible error is fixed at 10−12. In Fig. 2a we plotthe classical phase space (qcl, pcl) for a trajectory starting from (0,0) and evolving 104 kicks.The parameter K = 0.8 and σ = 1/π, where

σ =ω0

ωT, (47)

is the ratio between the angular frequency of the kick ωT (ωT = 2π/T , T is the periodof kicks) and the angular frequency of the harmonic oscillator ω. In our calculations, weput ω0 = 1. It is obvious that, in the classical phase space, the regular region (stableislands) and the chaotic region coexists. Fig. 2b shows the semiquantal phase space, thewave packet starts from (q0, p0, G0, Π0) = (0, 0, 0.5, 0), with h = 0.1. As was shown that ifthere was no kicks, the wave packet starting from this point will evolve exactly along theclassical particle’s trajectories forever. The fluctuations both in momentum and coordinatekeep constant, and are independent of time. In this case squeezed state dynamics describesexactly the classical one. Now if we switch on the kick, the situation becomes quite different.As is shown in Fig. 2a, the initial point just lies in the stochastic sea, thus it is evident thatin the classical case the trajectory will never enter the stable islands due to the existenceof the invariant curves. However, as we predicted that the invariant curve could not beable to prevent the trajectory from acrossing it via other dimensions semiquantally. This isdemonstrated by Fig. 2b, where all stable islands in the classical phase space are ”visited”by the semiquantal trajectory.

As a quantitative verification, we have numerically calculated the maximal Lyapunovexponent,

λ = limn→∞

λn, (48)

for the trajectories in both cases. The time behavior of λn is shown in Fig. 3. This figureexhibits the coexistence of the enhancement and suppression. At the initial stage the en-hancement mechanism dominant. While after a certain time the squeezed state λn becomeslarger than that its classical counterpart, which means that the enhancement mechanismbecomes dominant, and consequently results in the enhancement of chaos. The chaotic mo-tion in the extended phase space is characterized by the two positive Lyapunov exponentsin the four dimensional phase space (q, p, hG, Π).

11

B. Suppression of chaos

We would like go to another limit, namely at very large perturbation, to investigate thesuppression of the chaos. Classically, when K increases, the classical motion becomes moreand more chaotic. For a sufficient large K such as K = 6 the classical motion is completelychaotic as is shown in the phase space of Fig. 4a, where σ = 1/π. Like Fig. 2, Fig. 4 isfor the trajectory starts from the origin and evolves 104 kicks. The classical chaotic anddiffusive process is easily seen from the evolution of this phase plot. To demonstrate thesuppression of the chaos (or the diffusion process), we start a wave packet from (0, 0, 0.5, 0)in the 4-D squeezed state phase space. The evolution is shown in Fig. 4b. ComparingFig. 4a and Fig. 4b, it is obvious that in the classical case, the phase space is chaotic anddiffusive, whereas in the semiquantal case the diffusion process is largely slowed down andsuppressed. Furthermore, there are invariant-curve-like structures appear in the semiquantalphase space. These structures seems to form a barrier for the diffusion and thus suppresschaos. The suppression of chaos is quantitatively demonstrated by the large decrement ofthe λn as is shown in Fig. 6, where the suppression mechanism is most important.

To see the suppression more clear, we plot the variation of Keff with time (in unit of thekick) in Fig. 6. This plot indeed demonstrates that the effective perturbation strength aremuch less than its classical counterpart in most of the time during the evolution. This is themain reason of the suppression. In fact, the deduction of the effective potential acting onthe wavepacket have a clear physical picture. The width of the wavepacket centred at (q, p)in coordinate space is ∆q =

√hG, while the external potential has the wavelength of 2π/k0.

Therefore, there are in fact m(=√

hGk0/2π) periods of the external potential acting on thewavepacket simultaneously. This is quite different from the classical model, there only onekick acting on the particle at one time. Since the external potential is negative in someplaces and positive in other places, the wider the wavepacket, the larger the number of m,thus in total, the smaller the effective potential acting on the harmonic oscillator. Whereas,if the wavepacket ∆q is so small that it is smaller than the period of the external potential,then the effective potential is big. As a matter of fact, the effective potential Keff in Eq.(43)can be written as:

Keff = K exp

(

− m2

2π2

)

. (49)

As a further evidence of the suppression, it is convenient to calculate the energy diffusionwith time n (in the unit of kicks) for an ensemble of trajectories. The diffusion is defined by〈En〉 subtracts the initial averaging energy. For the classical case, En = 1

2(p2n + q2

n)cl wherethe 〈· · ·〉 means the ensemble average over many trajectories. In our calculations we havetaken such an ensemble averaging over 104 initial points which are uniformly distributedinside a disk area centred at the origin of the phase space. As for the the semiquantaldynamics, 〈En〉 is defined by,

En =1

2

⟨

Ψ|p2n + ω2

0 q2n|Ψ

⟩

=1

2(p2n + ω2

0q2n) +

1

2h(

1

4Gn+ 4Π2

nGn + ω20Gn

)

(50)

12

In Fig. 7 we show the energy diffusion of K = 6 and σ = 1/π for the classical and thesemiquantal cases. The suppression of the classical diffusion is very obvious.

C. Transition from enhancement to suppression

We have seen so far the enhancement at small K regime and the suppression at large Kregime. Now, we would like to discuss the transition from enhancement to suppression bychanging the strength of external potential for fixed quantum fluctuations. We show herethat there exists a threshold value of Kc distinguishing the enhancement from suppression.

To do this, we need to take an appropriate ensemble average over many trajectories inthe phase space. However, since the classical phase space of the KHO model is unbounded,it is impossible to do such an average over the whole phase space. This makes the numericalworks very difficult. After many numerical experiments, we find a compromise, namely, wetake the average over a disk centered at origin with radius of π. We spread 15 × 15 initialpoints uniformly distributed inside this area. In classical case, we calculate the Lyapunovexponent for each trajectory after 104 kicks. And plotted the averaged value denoted as 〈λcl〉in Fig. 8. This averaged value is analogy to the Kolmogorov entropy in a bounded system.But strictly speaking, this quantity cannot be called Kolmogorov entropy. Nevertheless,this parameter captures more or less the chaoticity of the underlying system. In the case ofsemiquantal dynamics, since we have 4D extended phase space, we have always two positiveLyapunov exponents. We add these two values, and denote it as 〈λsq〉. It is plotted in Fig.8 in comparison with the classical one.

From Fig. 8 we can draw the following conclusions: (1) There exists a certain thresholdvalue of Kc. Before this point, 〈λsq〉 > 〈λcl〉, which means that the degree of chaos isenhanced; while after this point 〈λsq〉 < 〈λcl〉, the chaos is suppressed. (2) At the region ofK ≫ Kc, the 〈λsq〉 fluctuates around a certain value. It does not change with increasing ofK. (3) The enhancement and suppression depends largely on h.

The results discussed in this section are restricted to the irrational frequency ratio. Onemight ask, does our conclusion applies also to the rational frequency ratio. It is well knowthat KHO model is a degenerate system out of KAM theorem. In classical phase space,there exists slow diffusion along the stochastic web for any small value of perturbation. Ournumerical results show also the enhancement and suppression. We give one example withσ = 1/4 and K = 6 in Fig. 9 for suppression. The corresponding Lyapunov exponent is alsoshown in Fig. 10.

Finally, we would like to say a few words about the initial conditions and parity of theexternal potential. We have performed a wide range of numerical investigations and foundthat the above discussed qualitative and quantitative conclusions does not depend on theselection of initial condition and the parity of the external potential. However, the selectionof initial condition must be physically meaningful as we discussed in Sec. II.

Before we conclude this section, we would like to discuss the connection of suppressionto the dynamicla localization. In fact, it is a challenge for squeezed state approach for thissubtile phenomenon. We argued that the dynamical localization observed in kicked rotatorare due to the quantum suppression of chaos discussed above. In fact, in limiting caseof ω0 = 0, the KHO model (35) is reduced to the famous kicked rotator model, in which

13

chaotic diffusion would be completely suppressed by the quantum fluctuations and results indynamical localization, a well established fact observed by Casati et al [15] almost 20 yearsago numerically and confirmed recently by experiment [17]. This has been nicely illustratedby Zhang and Lee [4] with the squeezed state approach.

V. DIFFUSION AND LOCALIZATION

In preceding section, we have presented how the quantum fluctuations enhance and sup-press chaos. This fact will definitely affect the diffusion behavior. For instance, in thelimiting case, when the suppression becomes dominant the localization is expected to hap-pen. Thus in this section we shall give a detail study on this. In particular, we concentrateon the large K regime. Because this is the most difficult region in pure quantum computa-tion. As we shall see quickly that the squeezed state approach not only provide an easy wayto do numerical calculations but also make it possible to do some analytical estimations.

The energy En of the kicked harmonic oscillator in the squeezed state approximationEq.(50) can be written as two parts,

En = Ecn + Ef

n . (51)

Ecn contains the first two terms in Eq.(50), which is due to the motion of the centroid of

a wave packet relating to the p and q equations in Eq.(45). They mimic the effects ofclassical diffusion (ECD). Ef

n includes the last three terms in Eq(50), is attributed to theeffects of quantum fluctuations (EQF), which are governed by G and Π equations in Eq.(45).These two kinds of effects are the main reasons of the diffusion process in the squeezed statedynamics.

The ratio σ is an important quantity, as we shall see later. We take σ the golden meanσg = (

√5−1)/2... and its continued-fraction expansion r/s: 2/3, 3/5, 5/8,... as examples. r, s

are generated by the Fibonacci sequence defined as: F0 = 1, F1 = 1, and Fn = Fn−2 + Fn−1

for n > 1. Without loss of generality, in all the calculations , we keep the parameters h = 1,K = 6, k0 = 1 and ω0 = 1, and the initial point is chosen as (0, 0, 0.5, 0), which correspondsto the ground state of the unperturbed quantum harmonic oscillator.

A. Numerical results

Figs. 11-12 and Figs. 13-14 show the numerical results of energy diffusion of

V (q) =

K cos(k0q)

K sin(k0q)(52)

for even and odd parity, respectively. Now we discuss these two cases separately.Even potential: In this case, for all rational frequencies, the energy will finally goes

quadratically with time i.e. En ∼ n2. It is shown in Fig. 11, the slope is equals 2 asymp-totically in double logarithm plot. However, for the case σ = r/s with relatively largerr, s, the diffusion only starts after a certain time. Before this time the energy diffusion is

14

localized. The transient time depends on the frequency ratios, and approximately is of orderof (σ − σg)

−1. We call this transient region a transient dynamical localization region.For the irrational case, the dynamical localization occurs as is clearly seen in Fig.12. This

significant phenomenon has been observed and investigated in various quantum systems inthe past years.

It is worth pointing out that for the two trivial cases, i.e. σ = 1/1, 1/2, our squeezedstate results agree completely with the quantum analytical results of Berman et al [20] whichis the only existing analytical results of the quantum diffusion of this model up to now. Thisdemonstrates the validity of the squeezed state approach. Moreover, by using the squeezedstate approach, we have also recovered the quantum results obtained numerically by Bor-gonovi et al [21]. See more in detail in the next section.

Odd potential: In this case, the quadratic law is observed only in the case of the ra-tional frequency ratios σ = r/s with odd s. For other situations the energy diffuses linearlywith time approximately, see Fig. 13 and Fig. 14.

B. Analytical estimation

The above numerical results can be understood very well by analyzing the the evolutionequations Eqs.(43). In fact, we can derive analytically the energy diffusion by studying thesystem (43). Starting from Eqs.(43), we find that when the external perturbation is absent,the two degrees of freedom (DOF) (p, q) and (G, Π) are decoupled, and each undergoes thefree motion. In terms of action-angle variable, the Hamiltonian of the free motion can beexpressed as Eq.(27). From this formula, we have already seen that, both of the free motionof the two DOF are degenerate. It is due to this degeneracy which makes the resonancebetween the two frequency possible in the whole phase space. Consequently, the squeezedstate dynamical behavior of the kicked harmonic oscillator is quite different from that ofthe kicked rotator [4]. That is, the motions of the centroid and the fluctuations of the wavepacket behave like an oscillator with fixed frequencies ω0 and 2ω0, respectively.

However, when the kicks are added, the two degree of freedom becomes coupled, theenergy may start diffusing. It is convenient to express the evolution of system in terms ofthe action-angle variables. From Eqs. (25) and (27) one can readily obtain a four dimensionalmap

In+1 = In − KeffΘ′(

k0

√

2In/ω0 sin φn

)

k0

√

2In/ω0 cos φn,

φn+1 = φn + ω0T + KeffΘ′(

k0

√

2In/ω0 sin φn

)

k0/√

2Inω0 sin φn,

Jn+1 = Jn + Keffh

4k2

0

√

(4Jn + 1)2 − 1 sin θnΘ(

k0

√

2In/ω0 sin φn

)

, (53)

θn+1 = θn + 2ω0T − Keffk2h

1 − 4Jn + 1√

(4Jn + 1)2 − 1cos θn

Θ(

k0

√

2In/ω0 sin φn

)

.

With this 4D map, we are able to do analytical estimation of the energy diffusion. We shalltreat it at two different limit cases.

15

Classical diffusion effect (h = 0)

In the classical limit case h = 0, Efn = 0, and the effect of classical diffusion becomes

dominant. Therefore, the change of energy during one kick is

∆Ecn = k0KΘ′(k0qn) (pn cos ω0T − qnω0 sin ω0T ) +

1

2k2

0K2Θ′2(k0qn). (54)

For K ≫ 1, the orbit can be supposed to be approximately ergodic. After ensemble averagingover the variables p, q, the first two terms vanishes approximately, thus we obtain the linearenergy diffusion, namely,

Ecn ∼ 〈∆Ec

n〉n ≈ 1

4k2

0K2n. (55)

Note that the average of the first two terms of ∆Ecn, though is much smaller than the last

term for large K, is nevertheless not exactly zero. This results in some oscillations of En

around the linearity (see Fig. 13-14).

Effect of Quantum Fluctuations

As for the second limit case, suppose a wave packet starts from (q, p, G, Π) = (0, 0, 0.5, 0)with an even potential, the center of the wave packet keeps fixed, thus Ec

n ≡ 0. The energydiffusion is purely caused by the effects of the quantum fluctuations. In this case we shallanalyze the diffusion process for two different frequency ratios, i.e. rational and irrational.For the case of rational ratio, let’s take the simple case of σ = 2/3 as an example. Duringthe time of 3T, there are 3 kicks acting on the harmonic oscillator. Since the frequencyof fluctuation is 2ω0, (G, Π) evolves 4 periods. Note that the effective amplitude of kickacting on the wave packet is Keff rather than K. Among these three kicks, only that oneat a relative small G affect the free motion of the oscillator significantly. We call this kickthe effective kick, the effects from other two kicks can be neglected due to the very largeG, and thus very small Keff . At the time when the next effective kick is in action, G isapproximately the same because of the resonance, see Fig. 15. Therefore, the increment ofΠ is almost constant, which means that Π3n ≈ n. Thus, from Eq.(50) we obtain

En ≈ n2, (56)

which gives rise to the quadratic law observed in Figs. 11-12.If σ is an irrational number, a very interesting thing can arise. From Eq. (24) we know

that the angular variable when a kick is added is

θn = 2πnσ + θ0(mod2π). (57)

This is nothing but a pseudo-random number generator, indicating that the jump of Π mayhappen in the upper part (Π > 0) and lower part (Π < 0) with the same probability, andthus the increment of the energy in the upper part will be canceled out by the decrease in the

16

lower part. This leads to the localization phenomenon observed in Fig. 12. This localiza-tion mechanism, resorting to a pseudo-random number generator, reminds us that happensin the kicked rotator, where the localization is related to the Anderson’s localization for aquantum particle in a one-dimensional lattice in the presence of a static-random potential[16]. Our results imply that it might also be possible to construct a connection between thekicked harmonic oscillator and the Anderson’s problem in the framework of the squeezedstate approximation. The mechanism discussed can also explain the transient dynamical lo-calization occurred in the case of the rational frequency ratio, as is shown in Fig. 11. Sinceduring the time t ≤ (σ−σg)

−1, a rational number behaves just like a pseudo-irrational, thusoccurs the transient phenomenon.

General case

As to the general case of system (43), both ECD and EQF may coexist. To illustratethis, let’s consider the case of σ = r/s, where r, s are co-primed integer. Suppose s is odd.As we have explained above, between two effective kicks, the (q, p), and (G, Π) evolves freely.Thus, the angle variables of (q, p) at the two successive effective kicks are φ and φ + 2πr,respectively, and that of (G, Π) are θ and θ+4πr. From Eq. (27) and Eq. (43), we find thatin this case, the increment of ∆p and ∆Π have the same sign, which means that both ECDand EQF are excited, which is independent of the potential parity. Because the diffusiondue to EQF is ∼ t2 which is much more faster than that of ECD (∼ t), thus asymptoti-cally t2 diffusion shows up as is shown in Fig. 13. While if s is even, there are additionalone effective kick between above mentioned two, namely the s/2th kick, at which the anglevariable of (G, Π) is θ + 2πr, while that of (q, p) is φ + πr. Thus, for the even potentialcase, the change of Π due to the two consecutive effective kicks have the same sign, whichimplies that the EQF is excited and t2 diffusion will show up. For the odd potential case,the change of Π due to the two consecutive effective kicks have opposite sign, the EQF isthus suppressed, and we obtain the linear diffusion as is seen in Fig. 14.

For the irrational σ case, the EQF is suppressed and the ECD becomes dominant. If(p, q) happens to be the fixed points in (p, q) plane as is the case of Fig. 12, the localizationoccurs. This is the reason for the different diffusion behavior of the irrational σ case inFig. 12 and Fig. 14 for even and odd external potential, respectively. Please note that(p, q) = (0, 0) are the expectation values of all the eigenstates of the harmonic oscillator,thus the localization we observed in Fig. 12 is not only restricted to the case of ground stateas we discussed up to now, but it is very general.

Based on the numerical results and analytical estimate of the squeezed state dynamics,we conjecture that the quantum energy diffusion of the kicked harmonic oscillator, canbehave either as E ∼ t in analogy to the classical diffusion, or behave as E ∼ t2 due tothe effects of the quantum fluctuations, or be strongly localized. We believe that thesethree classes of diffusion might be not only limited to the models we studied here but arather general conclusion valids for a general quantum system. This is because in a generalquantum systems the diffusion process are governed by the two effects: effects of classicaldiffusion and the effect of quantum fluctuations. The coexistence and competition of thesetwo effects lead to different diffusion behavior.

17

C. Transition from localization to diffusion

The results discussed above are focused on the diffusion at very large perturbation, inthis case the phase space is completely chaotic. As a further example we would like todemonstrate a very interesting and important phenomenon in quantum mechanics, i.e. thetunneling effect. We start a wavepacket from point (0,7.5) in the classical phase space(q, p). Here we have σ = 1/5 and V (q) = K cos q with K = 0.5. This point lies inside astable island. The wavepacket has parameter G0 = 1/2, Π0 = 0 according to the minimaluncertainty principle. It is a Gaussian wavepacket. The classical phase space is shown inFig. 16a. As we see that this point is inside a stable island. Classically, the trajectorystarts from this point will never be able to go out. However, in quantum case, the situationbecomes very different. We expect that if the width of the wavepacket is much smaller thanthe size of this stable island, the wavepacket will be confined by this stable island, thusleads to the localization. However, if the wavepacket becoming wider than the size of thestable island, it will spread out. Here we demonstrate this quantum phenomenon by thesqueezed state approach. In Figs. 16(b)-16(f) we plotted the energy evolution for differentPlanck constant h, which corresponds to different width of initial wavepacket. At h = 1, 2we observed localization phenomenon. Like other quantum system. The energy oscillatesaround a certain value. When we increase the h further to h = 5. The transition fromlocalization to delocalization happens, which is shown in Fig. 16.

VI. COMPARISON WITH QUANTUM RESULTS

To give the reader a clear picture about the accuracy of the squeezed state approach, wewould like to compare our results obtained from squeezed state approach as that obtainedfrom quantum one. However, as mentioned before since the diffusion occurs in the wholeunbounded phase space and cannot be reduced to a motion on a cylinder like that case ofkicked rotator, the pure quantum (numerical) investigation is very difficult, in particularat large K regime. Nevertheless, with a large amount of CPU time, one would be able toobtain some results at small K regime for short time evolution. This is why there are onlya few available quantum results in this regime. We will compare them with our squeezedstate results here.

A. Numerical results

At first, let us look at the results obtained by Berman et al [20]. As already shown by Fig.11. As for the case of 1/1 and 1/2, our squeezed state results agree completely with Berman’squantum analysis. In this case both squeezed state and pure quantal analysis predict the t2

growth of the energy. Another important remark. Berman et al [20] conjectured that for thegeneral case of 1/q, the energy growth should less than t2. From our squeezed state analysiswe reached that there are only three different diffusion: t2, t and localization. Therefore,our squeezed state analysis agrees also with their prediction.

Now we turn to the results obtained by Borgonovi and Rebuzzini [21]. Since the timeunit given in their pictures is not clear, we cannot do the quantitative comparison with our

18

squeezed state results. Therefore, we performed the quantum calculation by using our ownprogram (see next subsection). All the system parameters are kept the same as Borgonoviand Rebuzzini used. The results are given in Figs. 17 and 18. These pictures correspondsto different diffusion behaviors. In Fig. 17 a t2 diffusion is obtained (t is in unit of kick),the squeezed state approach gives rise also to the t2 diffusive behavior, although there isdifference in one constant. In Fig. 18 both the quantum and squeezed state results showlocalization. Please compare these two pictures to Borgonovi and Rebuzzini’s Fig. 1 andFig. 8, respectively.

B. Procedure for quantum computation

Now we describe the procedure of our quantum computation. Since the Hamiltonian isperiodic in time, the Floquet theory can be applied. Thus the time evolution can be reducedto the evolution of the eigenstate over one driving period,

|Ψ(t + T )〉 = U(T )|Ψ(t)〉, (58)

where

U(T ) = UfreeUkick = exp

(

−iH0T

h

)

exp

(

−iV (q)

h

)

(59)

is the Floquet operator.To simulate the quantum diffusion in this degenerate system (35), the Fourier spectral

method is employed. The time interval of free propagation is divided into many slices, eachhas width of ∆. For each slice the evolution operator is factored into a product of kinetic andpotential propagator arranged in a symmetric way so that a full potential step is sandwichedbetween two half kinetic steps, namely,

exp

(

−iH0∆

h

)

= exp

(

−ip2∆

4h

)

exp

(

−iω2

0 q2∆

2h

)

exp

(

−ip2∆

4h

)

+ O(∆3) (60)

This technique of symmetrically splitting the kinetic propagator reduces the error introducedby neglecting the commutator between the kinetic and potential operators. The error isreduced to O(∆3) from O(∆2) in a nonsymmetric splitting. The kinetic propagation iscarried out in momentum space, since in this space the time evolution is simplified asmultiplication. The potential step in performed in coordinate space for the same reason.The kick step, performed once per period, is also done in the coordinate space. A FastFourier Transform (FFT) routine is used to transform the wave function between these twospaces.

For the KHO model is a degenerate system, a wave packet may diffuse rapidly, even toinfinity in both coordinate and momentum space. The average energy of the wave packetmay reach rather high value during the diffusion. This amplifies the error caused by the ap-proximation made in Eq.(60). Therefore, the self-adaptative procedure is used to adjust thetime slice in Eq.(60) in each period to make sure that the width of time slice is much smaller

19

than the inverse energy. Second, both coordinate and momentum spaces should be largeenough. So a large number (32768) of Fourier components are used in our computations.

Our technique described above is different from that used by Berman et al [20]. Forself-consisten test, we have used the same parameter as that of Fig. 2 in Ref. [20], andcomputed the energy diffusion with our method. We found that our results completelyagree with theirs even in every details.

As already emphasized, because of the unbounded phase space, the quantum computa-tion is very time-consuming, even for small perturbation. For instance, about 10 days CPUtime (IBM RISC System/6000 42T, with 192 MB RAM) has been spent for Fig. 17, while20 days CPU time for Fig. 18.

VII. CONCLUSIONS AND DISCUSSIONS

In this paper, by using the harmonic oscillator, we have demonstrated that, the squeezedstate approach, or semiquantal approach, is a very powerful method. In this approach theMaslov phase enters into the zero energy as a geometrical phase. This approach gives usnot only the exact eigenenergies but also the exact wavefunction in a very convenient way.It provides an easy way to exceed the semiclassical method.

Applying the squeezed state approach to the kicked harmonic oscillator, we illustrate howthe quantum fluctuation affect the classical dynamics. We have shown that the chaoticitycan be enhanced as well as suppressed by the quantum fluctuations. A transition fromenhancement to suppression is observed when we change the strength of the kicks.

Moreover, with this squeezed state approach, we are able to investigate the energy dif-fusion numerically and analytically. Based on the squeezed state analysis we come to theconclusion that the energy diffusion for a general quantum system can be categorized intothree classes: (1) E ∼ t2, this is due to the quantum fluctuations; (2) E ∼ t, which minicsthe classical diffusion; (3)E ∼ t0 which is strongly localized. In this aspect, for those casesfor which the quantum investigation is extremely difficult and sometimes even impossible,the squeezed state approach provides a substitute way.

Finally, we would like to remark on the quantization of quantum system whose classicalcounterpart is chaotic. As is well know that this is a tough problem which has attractedtremendous attentions in last two decades. Among many others, Gutzwiller’s trace formulamight be the very plausible one [27]. However, this approach encounters the difficulty ofthe divergence, although many important contributions have been made to overcome thisdifficulty. We believe that the squeezed state approach might be an alternative way and cancontribute to this. Recent successful application of this method to calculate the eigenenergiesof a chaotic system by Pattanayak and Schieve [11] sheds light on this direction.

Acknowledgement. The work was supported in part by grants from the Hong KongResearch Grants Council (RGC) and the Hong Kong Baptist University Faculty ResearchGrant (FRG).

20

REFERENCES

[1] W. M. Zhang, J. M. Yuan, D. H. Feng, R. Qi and J. Tjon, Phys. Rev. A 42, 3615(1990).[2] W. M. Zhang and D. H. Feng, and R. Gilmore, Rev. Mod. Phys. 62, 867 (1990).[3] W. M. Zhang and D. H. Feng, Int. J. Mod. Phys. A 8 1417(1993).[4] W. M. Zhang and M. T. Lee, Preprint, IP-ASTP-26-94.[5] W. M. Zhang and D. H. Feng, Phys. Rep. 252, 1(1995).[6] S.T. Mo, Master Thesis, Taiwan, China, 1995.[7] Y. Tsui and Y. Fujiwara, Prog. Theor. Phys. 86, 443, 469(1991).[8] Y.Tsui, Prog. Theor. Phys. 88, 911 (1992).[9] A. K. Pattanayak and W. C. Schieve, Phys. Rev. Lett. 72, 2855(1994).

[10] A. K. Pattanayak and W. C. Schieve, Phys. Rev. E 50, 3601 (1994).[11] A. K. Pattanayak and W. C. Schieve, Phys. Rev. E 56, 278 (1997).[12] W. V. Liu and W. C. Schieve, Phys. Rev. Lett. 78, 3278 (1997).[13] B. Hu, B. Li, and W.-M. Zhang, Preprint CNS-9702.[14] B. V. Chirikov, Phys. Rep. 52, 263 (1979).[15] G. Casati, B. Chirikov, J. Ford and F. M. Izrailev, in Stochastic behavior in Classical

and Quantum Hamiltonian Systems, Lecture Notes in Physics, edited by G. Casati andJ. Ford, (Springer-Verlag, Berlin 1979), Vol. 93, P.334

[16] D. R. Grempel, R. E. Prange, and S. Fishman, Phys. Rev. A 29, 1639 (1984).[17] F.L. Moore, J. C. Robinson, C. F. Bharucha, B. Sundaram and M. G. Raizen, Phys.

Rev. Lett. 75, 4598 (1995).[18] G. M. Zaslavsky, R. Z. Sagdeev, D. A.Usikov and A. A. Chernikov, ”Weak chaos and

quasi-regular patterns”, Cambridge University Press, 1992.[19] A. A Chernikov, R. Z. Sagdeev, and G. M. Zaslavsky, Physica D 33, 65 (1988).[20] G. P. Berman, V. Yu. Rubaev and G. M. Zaslavsky, Nonlinearity 4, 543 (1991).[21] F. Borgonovi and L. Rebuzzini, Phys. Rev. E 52, 2302 (1995).[22] D. Shepelyansky and C. Sire, Europhys. Lett. 20, 95 (1992).[23] I. Dana, Phys. Rev. Lett. 73, 1609 (1994).[24] M. Frasca, Phys. Lett. A 231, 344 (1997).[25] P. Kramer and M Saraceno, ”Geometry of the Time-Dependent Variational Principle

in Quantum Mechanics, Springer-Verlag Berlin Heidelberg 1981.[26] R. Jackiw and A. Kerman, Phys. Lett. A 71, 158 (1979).[27] M. C. Gutzwiller, ”Chaos in Classical and Quantum Mechanics”, Springer-Verlag 1990,

New York Inc.

21

FIGURES

FIG. 1. Phase space of a periodic motion of harmonic oscillator.

FIG. 2. Comparison between the classical phase space (a) and the semiquantal one (b) at

K = 0.8 for an irrational frequency ratio σ = 1/π. One classical trajectory starts from (0, 0), while

the squeezed state wave packet starts from the same point having initial fluctuation parameter

(G0,Π0) = (0.5, 0), and h = 0.1.

FIG. 3. The time (in the unit of the kick) behavior of λn for the trajectories shown in Fig.2.

The increment of the λn after a certain time in the semiquantal case indicates the enhancement of

the chaos.

FIG. 4. The same as Fig.2 but for K = 6 with an irrational frequency ratio σ = 1/π for

classical (a) and squeezed state (b) case (h = 1). The semiquantal phase space displays an obvious

suppression of the classical diffusion.

FIG. 5. The time behavior of the λn for the trajectory shown in Fig. 4. The large decrement

of the λ in the semiquantal case demonstrates the strong suppression of the classical chaos.

FIG. 6. The time evolution of the effective external potential Keff for the orbit shown in

Fig.4b.

FIG. 7. The diffusion at K = 6 for an irrational frequency ratio σ = 1/π, for classical and

semiquantal case. As is defined, the averaging energy shown in this plot has been subtracted the

initial averaging energy. The ensemble averaging is taken over 104 initial points which are uniformly

distributed in an area of disk centered at (0,0) with radius of π in the phase space. The diffusion

coefficient in the semiquantal case is obviously much smaller than that of the classical case, which

indicates strong suppression of the classical chaos.

FIG. 8. The transition from enhancement to suppression. Averaged Lyanpunov exponent

versus the external potential K for classical case and semiquantal case with h = 0.1 and 1. The

average is taken over 15×15 points uniformly distributed inside a disk of radius π centred at (0, 0).

Here σ = 1/π.

FIG. 9. Suppression for the trajectory for rational σ = 1/4, and K = 6. (a) classical phase

space. (b) semiquantal (qsq, psq) with h = 1.

FIG. 10. The Lyapunov exponent for the trajectories shown in Fig. 9.

22

FIG. 11. Evolution of the energy with time (in the unit of kicks) for the case of even potential

with rational frequency ratio. The numbers indicating the frequency ratio. The case of 1/1

coincides with that of 1/2. Not that the curves have slope 2 asymptotically, and a transient

dynamical localization phenomenon shows up.

FIG. 12. Same as Fig. 11 but for an irrational frequency ratio σ = (√

5−1)/2. The dynamical

localization occurs is clearly seen.

FIG. 13. The same as Fig. 12 but for the case of odd potential. The dashed line with slope 1

is drawn for guiding the eyes.

FIG. 14. The same as Fig.13 but for the case of odd potential. The dashed line with slope 1

is drawn for guiding the eyes.

FIG. 15. The (G,Π) plot for K = 6, σ = 2/3.

FIG. 16. Transition from localization to delocalization of a wavepacket driving by the quantum

fluctuation. The wavepacket starting from a stable island (q0, p0) = (0, 7.5). σ = 1/5, ω0 = 1 and

K = 0.5. (a) Classical phase space. (b-f) semiquantal energy diffusion. (b) h = 1, (c) h = 2, (d)

h = 5.

FIG. 17. Comparison of semiquantal (dashed line) and quantum (solid line) diffusion behaviors

for σ = 1/4, ω0 = 2/π, ω1 = 8π, K = 0.5, and h = 1. The trajectory starts from (q0, p0) = (3.15, 0).

The semiquantal trajectory has the same (q0, p0) and (G0,Π0) = ( 12ω0

, 0). Please compare it with

Fig.1 in Ref.[8].

FIG. 18. Comparison of semiquantal (top) and quantum (bottom) diffusion behaviors for

σ = (√

5 − 1)/2, ω0 = 1, K = 1, and h = 1. The trajectory starts from (q0, p0) = (15, 0).

The semiquantal trajectory has the same (q0, p0) and (G0,Π0) = ( 12ω0

, 0). Please compare it with

Fig.8 in Ref.[8].

23

Hu et al Fig. 1

ω0q

p

φ

-4 -2 0 2 4

-4

-2

0

2

4

Hu et al Fig.2

p cl

qcl

p sq

qsq

-4 -2 0 2 4

-4

-2

0

2

4

0 2000 4000 6000 8000 100000.025

0.030

0.035

0.040

0.045

0.050

Hu et al Fig. 3

Semiquantal

Classical

λ n

n

-60 -40 -20 0 20 40 60-60

-40

-20

0

20

40

60

Hu et al Fig.4

p cl

qcl

p sq

qsq

-300 -200 -100 0 100 200 300-300

-200

-100

0

100

200

300

0 2000 4000 6000 8000 100000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Hu et al Fig. 5

Semiquantal

Classicalλ n

n

0 2000 4000 6000 8000 100000

1

2

3

4

5

6

7

Hu et al Fig. 6

Kef

f

n

0 200 400 600 800 10000

200

400

600

800

1000

1200

1400

Classical

Semiquantal

Hu et al Fig. 7

<E

n> -

<E

0>

n

0.6 0.8 1.0 1.2 1.40.00

0.02

0.04

0.06

0.08

0.10

0.12

Classical h=1 h=0.1

K

<λ>

Hu et al Fig. 8

-150 -100 -50 0 50 100 150

-150

-100

-50

0

50

100

150

Hu et al Fig.9

p cl

qcl

p sq

qsq

-400 -200 0 200 400

-400

-200

0

200

400

0 2000 4000 6000 8000 100000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Hu et al Fig. 10

Semiquantal

Classical

λ n

n

100

101

102

103

104

105

106

10-1

100

101

102

103

104

105

106

21/343/5

2/3

1/2

En

Hu et al Fig. 11

n

100

101

102

103

104

105

106

10-1

100

101

102

103

104

En

Hu et al Fig. 12

n

100

101

102

103

104

105

106

100

101

102

103

104

105

106

21/34

3/5

2/3

En

Hu et al Fig. 13

n

100

101

102

103

104

105

106

100

101

102

103

104

105

En

Hu et al Fig. 14

n

| |__

G

Hu et al Fig 15

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5

-20 -10 0 10 20

-20

-10

0

10

20(a)

q cl

qcl

En

0 200 400 600 800 1000

26

28

30

32(b)

nn

En

n

0 1000 2000 3000 4000 500022

24

26

28

30

32

(c)

0 200 400 600 800 10000

500

1000

1500

2000

Hu et al. Fig. 16

(d)

100

101

102

100

101

102

103

En

Hu et al Fig. 17

n

0 200 400 600 800 1000

60

80

100

120

140

n

Hu et al Fig.18

n

E

0 200 400 600 800 1000

60

80

100

120

140

E