Tunneling of quantum particle in parametrically driven double-well system

Chemical Physics 309 (2005) 95–101 www.elsevier.com/locate/chemphys

Tunneling of quantum particle in parametrically driven double-well system Akira Igarashi a, Hiroaki Yamada a

b,*

Graduate School of Science and Technology, Niigata University, Ikarashi 2-Nochou 8050, Niigata 950-2181, Japan b Aoyama 5-7-14-205, Niigata 950-2002, Japan Received 25 June 2004; accepted 26 August 2004 Available online 21 September 2004

Abstract We numerically study a wavepacket tunneling in one-dimensional symmetric double-well potential with a coherent perturbation. In the classical dynamics, such a system shows chaotic behavior by the classical driving force. In the present paper, we show how the tunneling dynamics depends on the parameters such as the amplitude and frequency of the perturbation in comparison with the classical counterpart. The resonance and chaos strongly affect on the tunneling dynamics. It is shown that the classical chaos enhances tunneling rate and the quantum fluctuation in both resonant and non-resonant tunneling processes. We also discuss a relation between our model and decoherence in the quantum system.
2004 Elsevier B.V. All rights reserved. Keywords: Tunneling; Double-well; Coherent perturbation; Chaos; Resonance

1. Introduction The study on the dynamics of periodically perturbed one-dimensional systems such as the kicked quantum pendulum, driven double-well oscillators have shown that the classical stochasticity gets suppressed in the fully quantum dynamics due to the quantum interferences [1,2]. On the other hand, relation between chaotic behavior and tunneling phenomenon in classically chaotic systems are also interesting and important subject in study of quantum physics [3,4]. Recently, the semiclassical description for the tunneling phenomena in the chaotic system have been developed by several groups [5–7]. Furthermore, in the experimental realization the controllability of the quantum tunneling by external pertur*

Corresponding author. Tel.: +81252671941. E-mail addresses: [email protected] (A. Igarashi), [email protected] (H. Yamada). 0301-0104/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.08.016

bation based on optimal control theory [8,9] is an important topic. This controllability of the tunneling is strongly related to quantum measurement problem such a quantum Zeno effect [10]. Lin and Ballentine studied interplay between the tunneling and classical chaos for a particle in a double-well potential with oscillatory driving force [11]. The system is a quantum driven Duffing oscillator without dissipation. They have found that coherent tunneling takes place between small isolated classical stable regions of phase space bounded by Kolmogorov–Arnold–Moser (KAM) surfaces, which are much smaller than the volume of a single potential well. With increasing nonlinearity, regular tori successively dissolve in adjacent chaotic layers which grow in size and merge until the whole phase space is uniformly diffusive. Ha¨nggi and the coworkers studied the chaos-suppressed tunneling in the driven double-well model in terms of the Floquet formalism [12,13]. They found a one-dimensional manifold in the parameter space, where

A. Igarashi, H. Yamada / Chemical Physics 309 (2005) 95–101

the tunneling completely suppressed by the coherent driving. The quasienergies of chaotic singlets frequently intersect the tunnel doublets which are supported by regular tori. It should be noted that the Duffing-type system is used as an effective model in many physics fields, such as proton transfer in the biomolecules [14], isomerization reactions under external laser fields [15], atomic tunneling between ATM tip and surface in terms of lattice vibration [16] and so on [17,18]. Furthermore, oscillation of the atomic populations in the macroscopic quantum tunneling due to overlaps between nonlinear resonance have been studied in weakly coupled Bose– Einstein condensation (BEC) with a periodically timevarying atomic scattering length [19]. In the present paper, we investigate influence of chaos on quantum tunneling in a parametrically driven double-well system. The system parameters are set suitable values that the quantum tunneling take places in the energetically and/or dynamically forbidden region. The equation of motion that neglected the nonlinear term shows parametric instability. Note that Lin et al. dealt with a double-well system driven by forced oscillator, therefore, the asymmetry of the potential play a role in the chaotic behavior and tunneling transition between the quasienergy states. However, in our model the potential is remained symmetric during the time evolution process and different mechanism from the forced oscillation makes the classical chaotic behavior. Based on the model, we have carried out a numerical analysis of quantum mechanical tunneling between the energetically forbidden wells. We present some numerical results concerning to a classical and quantum description of the field-induced barrier tunneling and analyses how classical chaos makes its presence in a regime of the parameters space spanned by the amplitude and frequency region of the oscillatory driving force. For the resonance frequency to the intrinsic one, the initially localized wavepacket rapidly spreads, and the quantum tunneling take place even for the relatively small perturbation. It is shown that in the non-resonant cases with small perturbation the wave packet shows quasiperiodic motion. Moreover, classical chaos significantly enhances the quantum fluctuation as well as the quantum barrier tunneling between the right and left well. We discuss relation between our model and a bistable system coupled with a heat bath, which is used as a simple model for some biological systems and chemical reactions. Outline of the present paper is as follows. In the next section, the model we investigated is introduced with the necessary background of the system in the driven tunneling. In Section 3, the numerical results for tunneling probability and the chaotic behavior are given. The last section contains summary and discussion.

2. Models The model studied in the present paper is a symmetric quartic double-well system with a parametric oscillatory driving force. The Hamiltonian is H ðtÞ ¼

p 2 q4 q2 þ
AðtÞ ; 2 4 2

ð1Þ

AðtÞ ¼ a

sin Xt;

ð2Þ

where q and p represent the generalized coordinate and the conjugate momentum, respectively. In the concrete, the parameters are set as a = 5,
= 0.1–1.0, to emphasize the tunneling effect in the energetically forbidden region during the time evolution. In the classical dynamics, such a system shows chaotic behavior by the driving force A(t). In Fig. 1 we show the sketch of the double-well potential V(q,t) and some energy levels for
= 0. In the present paper, we show how the tunneling dynamics depends on the parameters (X,
), in comparison with the classical counterpart. Then, we use Gaussian wavepacket as the initial state ( ) ðq
q0 Þ2 wðq; t ¼ 0Þ ¼ N exp
; ð3Þ 2r p where q0 = a at a bottom of the right well. N and r are normalization factor and spread of the initial packet, respectively. The Gaussian wavepacket can be approximately generated by the linear combination of the ground state doublet as wðq; t ¼ 0Þ  p12 ðu0 ðqÞþ u1 ðqÞÞ, where u0 and u1 denote the ground state doublet. We set r = 1/3.4(0.3) throughout this paper (see Fig. 1.). Indeed, the ammonia molecule is well described

15

t=0 t=π/2Ω t=3π/2Ω

10

5 V(q,t)

96

0

-5

-10 -3

-2

-1

0 q

1

2

3

Fig. 1. Sketch of the driven double-well potential described by the time-dependent Hamiltonian at different time (t = 0, p/2X, 3p/2X). The parameters of the oscillatory force are set at a = 5,
= 1.0, X = 2.8. The horizontal lines denote some eigen energy levels in the absent of the oscillation (
= 0). The lines below the barrier come in doublets.

A. Igarashi, H. Yamada / Chemical Physics 309 (2005) 95–101

by this barrier height with two number of doublet with energies below the top of the barrier, and the tunneling dynamics is well approximated by several levels from the bottom of the well [17]. We used second order unitary integration in the numerical simulation for the time evolution. h = 1.0 and time mesh dt = 5 · 10
5–1.3 · 10
3. 7 The spatial discretization is chosen so p p that 2 (=128) points cover the interval [
2.5 a, 2.5 a]. Before going on the quantum tunneling, we briefly sketch the classical property of the nonlinear parametric oscillator. The NewtonÕs equation of the motion is d2 q
ða

sin XtÞq þ q3 ¼ 0: dt2

ð4Þ

When q is small (1) and the third nonlinear term q3 can be negligible, the variable q exponentially grows in terms of the parametric instability in our parameter range A(t) > 0. For larger value of q (>1) the nonlinearity begin to affect the motion. In our initial condition (q = q0, p = 0) the forced oscillator term works to make the initial deviation from the initial position. Fig. 2 shows surface of section of classical phase space for four different combinations of
and X. There are regular trajectories p confined in the stable island centered at q = ± 5, p = 0 in the case with X = 1.2,
= 0.1. The regular islands occur in symmetry-related pairs due to H(q,p,t) = H(
q,
p,t + 2p/X). For the quantum system it appears as parity of the states. As
increases, the chaotic region surrounding the central island increases in size. In Fig. 2(c) we can see lots of tori caused by the higher order nonlinear resonances. In the next section, we mainly investigate the four cases with the parameters, X = 1.2, 2.8 and
= 0.1, 0.9. The value of the frequency X = 2.8 is near some intrinsic frequency in the quantum energy levels. In this case the homoclinic tangle of stable and unstable manifolds replaces the separ-

Fig. 2. Stroboscopic plots of the classical trajectories in phase space (q,p). The parameters are: (a) X = 1.2,
= 0.1; (b) X = 1.2,
= 0.9; (c) X = 2.8,
= 0.1 and (d) X = 2.8,
= 0.9.

97

atrix. In the case of Fig. 2(d), the dynamical barriers are almost broken. The details of the chaotic behavior based on the Melnikov method has been reported in [20–22].

3. Numerical results In this section, we give numerical results of the driven tunneling phenomena for some combinations of the parameters (
,X). We define the quantum probability function PL(t) Z 0 P L ðtÞ  jwðq; tÞj2 dq; ð5Þ
1

which roughly gives the probability of a packet to go through the barrier from the right well to the left one. This can also be interpreted by a survival probability PR(t) of the wavepacket in the initially prepared right well by PR(t) = 1
PL(t). The tunneling rate is enhanced in the adiabatically slow case, X < DE01/ h, comparing with the unperturbed case, where DE01 (=6.74726 · 10
4) is an energy difference between the tunneling doublet of the ground state. On the other hand, for X  x0 the dynamics of the double-well system behaves independently of the driving force, where p x0(= 2A(t)) denotes eigen frequency near bottom of the each well. The frequencies we used in this paper is in a region X 2 [DE01/h,x0]. 3.1. Tunneling Fig. 3 shows the quantum probability PL(t) as a function of time for two typical perturbation frequencies in a relatively short time region. For the small
(60.4) at X = 1.2 the time dependence of the tunneling probability is similar to the unperturbed case (
= 0) and is well suppressed by the wall. For the larger
(P0.6) the tunneling probability increases as the time elapses even in this time scale. On the other hand, at X = 2.8 in Fig. 3(b) the tunneling probability rapidly increases and oscillates around 0.5 except for some cases with small
(=0.1, 0.2). It is apparently chaotic oscillation. The resonant energy absorption and tunneling is due to some intrinsic frequencies between the quantum levels around X = 2.8. We roughly see the resonant structure between the coherent driving force and double-well system for relatively short time t = 120. Fig. 4 shows
and X dependence of the tunneling rate at t = 120. We can see resonance structure for comparatively larger X P 2.5 in Fig. 4(b). The resonance peaks correspond to the difference between the quantum energy levels of the double-well system. According to the numerical result, let us divide the tunneling phenomena into two kinds of the tunneling process, i.e., resonant tunneling as in

98

A. Igarashi, H. Yamada / Chemical Physics 309 (2005) 95–101

1 0.8 0.6

0.8 0.6

0.4

0.4

0.2

0.2

0 0

(a)

100

200

0

300

t 1

1

0.2

0.4

ε

0.6

0.8

ε=0.0 ε=0.1 ε=0.3

0.8

0.6

0.6 PL

PL(t)

0

(a)

ε=0.1 ε=0.4 ε=0.9

0.8

0.4

0.4

0.2

0.2

0 0

40

80

0

120

t

Fig. 3. Quantum tunneling probability PL(t) as a function of time for various
at (a) X = 1.2 and (b) X = 2.8.

Fig. 3(b) and non-resonant tunneling as in Fig. 3(a), as seen in a biased weakly coupled superlattice [23,24]. The resonant tunneling occurs by the rapid absorption of the energy of the driving force by the resonance as seen in Fig. 3(b). In the off-resonant frequency region with small
, the transition between the wells can be described by the non-resonant tunneling. In the present paper, we mainly consider a relatively small time scale 2p/X < t   h/DE01. The transfer of the wavepacket takes place for at least a time scale, t  h/DE01, due to the quantum fluctuation. Fig. 5 shows the long time behavior of PL(t) in the case with X = 2.8,
= 0.1. The complete tunneling from the left well to the right well occurs, and it shows quasiperiodic oscillation with much larger frequency than the unperturbed case with the recursion time  h/DE01  2 · 104. Moreover, population dynamics can also give the significant information in the tunneling phenomena. The details for the population dynamics of several levels will be given elsewhere [25]. 3.2. Chaotic behavior In this subsection we consider the relation between the tunneling probability and the chaotic behavior in the quantum dynamics. First, we illustrate in Fig. 6 the average energy hHi as a function of the average position hqi upto a relatively short-time, t 6 50 2p , for four X

1

(b)

1.5

2 Ω

2.5

3

Fig. 4. (a) Quantum tunneling probability PL(t) versus
for various X at t = 120. (b) Quantum tunneling probability PL(t) versus X for various
at t = 120.

1

0.8

0.6 PL(t)

(b)

Ω=1.0 Ω=1.3 Ω=2.1

PL

PL(t)

1

ε=0.0 ε=0.1 ε=0.4 ε=0.7 ε=0.9

0.4

0.2

0 0

5000

10000 t

15000

20000

Fig. 5. Long-time behavior (upto t ¼ 104 2p ) of Quantum tunneling X probability PL(t) as a function of time for
= 0.1, X = 2.8.

cases in Fig. 2. In the non-resonant case with small perturbation strength as in Fig. 6(a), the wavepacket is well localized in the right well in this time scale. We can see that the particle tunnels the barrier in Fig. 6(b), (c) and (d). In the cases, the energy of the wavepacket increases and the packet go through the wall around the top of the barrier. In the cases of small perturbation strength, quasiperiodic like motion appear regardless of the frequency

A. Igarashi, H. Yamada / Chemical Physics 309 (2005) 95–101 -4

-6

0

0

-2

-2

-6

-6 1

1.5

2 〈q〉

2.5

3

-3

(b)

-2

-1

0 1 〈q〉

2

3

-6 -3

(a)

-4

-2

-1

0 1 〈q〉

2

3

(b)

0

2 -2

-2

-1

0 1 〈q〉

2

3

(d)

-4 -6

-6 -3

-1

0 1 〈q〉

2

3

-3

-2

-1

0 1 〈q〉

2

3

0 -2 -4

-4 -6

〈H〉

〈H〉

〈H〉

-4

-2

2

-2

0

-3

4

4

-2

(c)

-5

〈H〉

-4

0

〈H〉

〈H〉

-5

〈H〉

〈H〉

-4

(a)

99

-3

-2

-1

0 1 〈q〉

2

3

Fig. 6. Plots of the average energy hHi versus the average position hqi for various cases. Short-time behavior by the data upto t ¼ 50 2p are X plotted on the figure to show the trajectory clearly. The parameters are: (a) X = 1.2,
= 0.1; (b) X = 1.2,
= 0.9; (c) X = 2.8,
= 0.1 and (d) X = 2.8,
= 0.9. The unperturbed double-well potential is shown by the broken line as a guide for eyes.

-6 -3

(c)

-2

-1

0 1 〈q〉

2

3

(d)

Fig. 7. Long-time behavior (upto t ¼ 104 2p ) of the stroboscopic plots X of the average energy hHi and the average position hqi for various cases. The parameters are: (a) X = 1.2,
= 0.1; (b) X = 1.2,
= 0.9; (c) X = 2.8,
= 0.1 and (d) X = 2.8,
= 0.9. The unperturbed doublewell potential is shown by the broken line as a guides for eyes.

1

2

〈p〉

〈p〉

1 0

0 -1

-1 1.8

(a)

-2 1.9

2

2.1 〈q〉

2.2

2.3

(b)

2

2

1

1 〈p〉

〈p〉

X. On the other hand, in the cases with relatively large
as in Fig. 6(b) and (d) it seems that the wavepackets show the chaotic behaviors around the top of the wall. The instability around the homoclinic fixed point is suppressed by the quantum effect in a sense that the quantum wavepacket stays for much larger time around the top of the wall when comparing with the classical cases. The chaos-induced weak localization can be also observed in quantum barrier crossing problem when the energy of the initial state is higher than height of the potential wall. Next, for the sake of confirming the long-time behaviors, in Fig. 7 we show the stroboscopic plots of the average energy hHi versus the average position hqi for various cases in Fig. 6. In the nearly integrable case as seen in Fig. 7(a), the simple path between the wells with small fluctuation by the perturbation is dominate in the tunneling phenomenon. It follows that in Fig. 7(c) the motion is apparently not fully chaotic but almost quasiperiodic or weakly chaotic when
is small. As
increases the motion of the wave packet approaches the chaotic behavior. In particular, the fully chaotic case as seen in Fig. 7(d), the wave packet clearly shows activated behavior by crossing the top of the barrier. Fig. 8 is the plots of the average momentum hpi versus position hqi upto t ¼ 50 2p , corresponding to the clasX sical cases in Fig. 2. The dynamical behaviors in the phase space well corresponds to Fig. 6. For X = 1.2 with
= 0.1 the wavepacket is confined in the initially localized well for this time scale and the chaotic behavior is very weak. Even in the case with X = 2.8,
= 0.1, the quantum wavepacket shows quasiperiodic or weak chaotic motion for the larger time scale. On the other hand, for X = 2.8,
= 0.9 the quantum trajectory by the

0 -1

-1

0

-2

-1

0

〈q〉

1

2

3

1

2

3

0 -1 -2

-2

(c)

-2

-3 -2

-1

0

〈q〉

1

2

3

(d)

〈q〉

Fig. 8. Plots of the expectation values (hqi,hpi). The parameters are: (a) X = 1.2,
= 0.1; (b) X = 1.2,
= 0.9; (c) X = 2.8,
= 0.1 and (d) X = 2.8,
= 0.9. The data upto t ¼ 50 2p are plotted on the figure. X

expectation values is chaotic and moves between the wells. It is worth noting that in the dynamical process the complete separation between the chaotic behavior and the resonance effect is impossible. 3.3. Quantum fluctuation We consider the relation between chaos and the quantum fluctuation as a generic quantum effect. First, Fig. 9 shows p the time dependence of pthe standard deviations, Dq  h(q
hqi)2i and Dp  h(p
hpi)2i for various cases. It seems that the fluctuations, Dq and Dp, grow as
increases. In the resonant cases corresponding to fully chaotic one, the wave packet does not return back to the nearly initial state once it spreads over the

100

A. Igarashi, H. Yamada / Chemical Physics 309 (2005) 95–101 3

ε=0.1 ε=0.9

7

2

6

∆q 1

1

0

0

(a) 0

100

200

300

t

3

(b) 0

80

3

∆p

∆p 1 0 200

300

t

1 ε=0.1 ε=0.9

(d)

0

40

80

0 120

0

0.2

t

0.4

0.6

0.8

ε

Fig. 9. Plots of the standard deviations, Dq and Dp, as a function of time for
= 0.1 and 0.9 in non-resonant and resonant cases. The values of the frequency are X = 1.2 in the panels (a), (c) and X = 2.8 in the panels (b), (d), respectively.

space. Additionally, we have confirmed that throughout the time-evolution the fluctuation of the energy is less than the barrier height between the wells of the potential except for the fully chaotic case. Next, Fig. 10 presents the uncertainly product, i.e. phase space volume, as a function of time for various

5

∆ q∆ p

3 2

0 100

4

2 1

0

120

t

4

ε=0.1 ε=0.9

2

(c)

40

Ω=1.2 Ω=2.8

5

ε=0.1 ε=0.9

∆ q∆ p

∆q

2

3

ε=0.0 4 ε=0.1 ε=0.4 ε=0.9 3 2 1

Fig. 11. Plots of the uncertainty product versus
for various cases X = 1.2, 2.8 at t = 200. The error bars are estimated by the data in an interval t 2 [150,250] in Fig. 10.

cases, which is defined by DqDp. The uncertainty product can be used as a measure of quantum fluctuation. The initial value is DqDp = h/2(=0.5) for Gaussian wavepacket in Eq. (3). In Fig. 10(a), it is found that the increase of the perturbation strength enhances the increment of the quantum fluctuation for X = 1.2. For the larger time scale, it oscillates around the corresponding certain level. In the resonance case in Fig. 10(b) the quantum fluctuation rapidly increases and fluctuates around a level (4.0) except for a small
(=0.1). As we can expect, the structure of the time dependence well corresponds to ones of the tunneling rate PL(t) in Fig. 3. Fig. 11 shows the uncertainly product versus the perturbation parameter
. As a result, in the both cases, X = 1.2 and 2.8, the classical chaos enhances the quantum fluctuation. The similar result can be confirmed by the phase space dynamic of Wigner function [25].

0 0

(a)

100

200

300

t

4. Summary and discussion

8

∆ q∆ p

6

4

2

ε=0.1 ε=0.4 ε=0.9

0 (b)

0

40

80

120

t

Fig. 10. Plots of the uncertainty product DqDp versus time for various
, (a) X = 1.2 and (b) X = 2.8.

We numerically investigated a wavepacket tunneling in one-dimensional symmetric double-well potential with coherent perturbation. We showed the frequency dependence of the tunneling probability in the parametrically perturbed cases in comparison to the classical dynamics. We have confirmed that two types of the tunneling process, i.e., resonant and non-resonant tunneling, exists depending on the characteristic frequency of the perturbation. In the resonant tunneling, the wavepacket rapidly spreads in the system and fluctuates even for comparatively small
. On the other hand, in the non-resonant tunneling the tunneling takes place very slowly and the behavior is similar to the unperturbed

A. Igarashi, H. Yamada / Chemical Physics 309 (2005) 95–101

case. However, the oscillation is weakly chaotic even for the long time that is different from the unperturbed case. As a result, in both cases, the increase of the perturbation strength enhances the quantum barrier tunneling between the right and left well as well as quantum fluctuation. We can extend the monochromatically perturbed double-well model used in this paper in order to investigate pre-dissipative property which is organized in quantum systems. It is accomplished by changing the time-dependent perturbation to the following oscillation: M
X AðtÞ ¼ a
p cosðXi t þ hi Þ; M i¼1

ð6Þ

where Xi and hi are the mutually incommensurate frequency components and the initial phases [26]. We can expect that in the long time regime the tunneling probability PL(t) is enhanced as the number M of the frequency components increases up to some extent because of the increasing of the stochasticity in the total system [28]. As we mentioned in the introduction, while the mutual influence of quantum coherence and classical chaos has been under investigation since many years ago, the additional effects caused by coupling the chaotic system to the other degrees of freedom or an environment, namely decoherence and dissipation, have been studied only rarely [27] as well as the tunneling phenomena in the chaotic system. It is expected that the tunneling is extremely sensitive to any disruption of coherence as it occurs due to the unavoidable coupling to the environment. Indeed, the tunneling transition dominates over thermally activated barrier crossing below crossover temperature for the quantum stochastic resonance. Then the temperature dependence of the signal-to-noise ratio is influenced by the quantum fluctuation [29–31]. If the time dependence of the potential comes up with the stochastic fluctuation as h(A(t1)
a)(A(t2)
a)i / Td(t2
t1), where T denotes the temperature, the stochastic perturbation models a heat bath coupled with the system. The detailed investigation of the stochastic resonance and decoherence with tunneling in the parametrically driven symmetric double-well system will be given elsewhere [25].

101

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