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Chemical Physics 327 (2006) 351–360

Quantum dynamics of an electron moving in coupled quarticand coupled double-well oscillators under intense laser fields

Neetu Gupta a, B.M. Deb b,c,*

a Theoretical Chemistry Group, Department of Chemistry, Panjab University, Chandigarh 160 014, Indiab S.N. Bose National Centre for Basic Sciences, JD Block, Sector-III, Salt Lake, Kolkata 700 098, India

c Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560 064, India

Received 19 April 2006; accepted 9 May 2006Available online 13 May 2006

Abstract

The phenomenon of quantum chaos is investigated by employing the model of an electron moving in coupled nonlinear two-dimen-sional oscillators, namely, coupled quartic and coupled double-well oscillators, under intense laser fields. The unperturbed ground-statewavefunctions of the oscillators, obtained by solving the time-dependent Schrodinger equation (TDSE) in imaginary time, are evolved inreal time by numerically solving the TDSE. Various signatures like autocorrelation function, distance function, quantum ‘‘phase space’’volume, ‘‘phase space’’ trajectories and overlap integral (similar to quantum fidelity or Loschmidt echo) have been studied to diagnosequantum chaos in terms of sensitivity towards an initial state characterized by a mixture of quantum states (wavepacket), brought aboutby small changes in the Hamiltonian, rather than towards a ‘‘pure’’ quantum state (i.e., a single eigenstate). Other quantum dynamicalaspects such as time-dependent probability density distributions as well as power spectra and high harmonic generation (HHG) spectraat different laser intensities have also been examined. In case of the coupled double-well oscillator, suppression of quantum chaos hasbeen observed.� 2006 Elsevier B.V. All rights reserved.

PACS: 05.45.Mt; 05.45.Xt; 42.50.Hz; 42.65.Ky

Keywords: Coupled quartic oscillator; Coupled double-well oscillator; Quantum chaos; High-order harmonics generation

1. Introduction

The study of quantum chaos constitutes a fascinatingand active branch of present-day physics, chemistry, andmathematics. In the past few years, considerable effortshave been made to study the connection between classicaland quantum dynamics in situations where classical chaos[1,2] dominates. It is generally agreed that classical chaosarises from the nonlinearity of classical equations ofmotion, whereas there is a debate on whether and how itmanifests itself in the corresponding quantum mechanicalsystems as quantum mechanics is a linear theory in itstime-independent as well as time-dependent (TD) form.

0301-0104/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2006.05.007

* Corresponding author.E-mail address: bmdeb@yahoo.co.in (B.M. Deb).

In a classically chaotic system, the exponential sensitivityto initial position in the phase space is the confirmatory sig-nature of chaos but to decide whether a quantum system ischaotic or not, a similar unambiguous answer is difficult toobtain since the uncertainty principle in quantum mechan-ics rules out the deterministic concept of classical mechan-ics. However, it is clear by now that quantum signatures ofclassical chaos do exist and thus the study of quantumchaos is a field of intense ongoing research. The presentpaper studies the sensitivity to initial conditions (SIC) fora quantum system. If the initial state is a ‘‘pure’’ quantumstate (i.e., a single eigenstate), then due to the unitarity ofthe time-evolution operator SIC does not exist. Hence, fol-lowing Ballentine et al. [3], we have taken the initial state asa mixture of quantum states (or wavepacket), resultingfrom the action of a perturbed Hamiltonian on a single

352 N. Gupta, B.M. Deb / Chemical Physics 327 (2006) 351–360

eigenstate; the initial wavepacket can be changed byslightly changing the Hamiltonian.

Chaotic phenomena in quantum systems have beenidentified in few-body atomic and molecular systems [4].The case of one electron in an external magnetic field isthe simplest example of a chaotic quantum system [5].The microwave ionization of highly excited atomic hydro-gen provides another example of a classically chaotic quan-tum system [6]. Footprints of chaos have also beenidentified in spin systems, e.g., in the spectrum of latticespin systems [7] and spin glass clusters [8]. It has beenshown recently [9] that even nonrelativistic linear harmonicoscillator (LHO) behaves chaotically if it is driven by anexternal force that varies in both time and space. The rela-tivistic LHO also exhibits chaos when it is driven by a forcethat is periodic in time, because the frequency is no longerconstant in the relativistic regime.

Among the nonlinear oscillators, the Henon-Heiles[10,11] and coupled quartic oscillators [12–15] have beenextensively studied to elucidate the role of underlyingclassical chaos in quantum mechanics. Since the studyof nonlinear dynamical systems leads to a betterunderstanding of the complexity of chaos in quantummechanics, this paper studies the quantum dynamics oftwo two-dimensional nonlinearly coupled oscillators,namely, coupled quartic (CQ) and coupled double-well(DW) oscillators which have relevance to some of theatomic systems mentioned above under intense laserfields. The importance of this problem lies in its theoret-ical relevance as well as different physical applications,such as the interaction of atoms with electromagneticfields. The CQ oscillator has been employed in the studyof Rydberg states of atomic hydrogen and DW poten-tials play an important role in accounting for manyinteresting phenomena in solid state and molecularphysics, e.g., for describing the elements responsible forstructural phase transitions (such as protons in hydrogenbonds), defect structures in nanocrystals, ligand migra-tion of biomolecules [16–18], etc.

Although there is an extensive literature on the calcu-lation of energy eigenvalues of DW oscillators [19,20]and the classically chaotic CQ oscillator is one of themost widely studied models of quantum chaos [12–15],all the previous studies were time-independent andrequired the calculation of accurate eigenenergies andeigenfunctions of highly excited states. A TD quantummechanical study on these systems does not seem to bereported so far. The present work employs the modelof an electron moving in nonlinearly CQ and coupledDW oscillator under intense laser fields, to study thepossibility of quantum chaos as well as to compare thequantum dynamics of the two systems. Since quantumchaos is known to manifest itself in the highly excitedstates, our objective is to create a situation in whichthe electron could get excited to the continuum/‘‘pseudo-continuum’’ under a laser field of high intensity and thusdynamical ‘‘signatures’’ of quantum chaos may be

observed. A ‘‘pseudocontinuum’’ [21] may be defined asthe semiclassical limit of the unperturbed quantum sys-tem, in accordance with the Correspondence Principle.Note that our classical system of reference is the unper-turbed nonlinear oscillator and not the classical oscilla-tor + laser system.

The accuracy of our numerical algorithm was estab-lished by calculating the ground, first and second excitedstates of the unperturbed oscillators by numerically solvingthe time-dependent Schrodinger equation (TDSE) in imag-inary time. The unperturbed ground-state is the input attime t = 0 for the real time Schrodinger equation incorpo-rating the TD potential which is taken as arising from anintense laser field. Although the initial input is theground-state for both the Hamiltonians, after the first timestep the initial input is a mixture of quantum states and isslightly different for two Hamiltonians involving slightlydifferent laser intensities. Several dynamical indicators suchas autocorrelation function, quantum ‘‘phase space’’ vol-ume, quantum ‘‘phase space’’ trajectory, distance function,decay of overlap integral (similar to quantum fidelity orLoschmidt echo [22–25]), etc., have been studied in orderto examine quantum chaos through sensitivity towardsan initial state brought about by small changes in the Ham-iltonian. Such a combination of different signatures ofquantum chaos under an intense perturbation may nothave been studied before.

2. Methodology

The two-dimensional TDSE is (atomic units employedunless mentioned otherwise)

HWðx; y; tÞ ¼ ioWðx; y; tÞ

otð1Þ

Our model system of an electron moving under two-dimensional nonlinear oscillator potentials in the presenceof a laser field in the x-direction is defined by theHamiltonian

H ¼�ð1=2Þo2=ox2�ð1=2Þo2=oy2þ V ðx;yÞ� x�0f ðtÞ sinðxLtÞð2Þ

where xL is the laser frequency, f(t) is the ramp function(see later), e0 is laser peak electric field given by (8pI/c)1/2,c is the velocity of light and V(x,y) is the oscillator poten-tial. For CQ oscillator, V(x,y) is given by

V ðx; yÞ ¼ x4 þ y4 þ ax2y2 ð3Þwhere a is a coupling constant and is taken here as 45. Onthe other hand, the coupled DW potential is given by

V ðx; yÞ ¼ �5x2 � 5y2 þ 0:5x4 þ 0:5y4 þ x2y2 ð4ÞThe method of obtaining the ground-state wavefunction

W(x,y, t = 0) of the unperturbed potentials given by Eqs.(3) and (4) is based on replacing t in Eq. (1) by imaginarytime s and putting s = �it, thereby transforming the TDSEinto an equation that resembles a diffusion quantum Monte

N. Gupta, B.M. Deb / Chemical Physics 327 (2006) 351–360 353

Carlo equation [26]. Successive higher energies arecalculated by the same imaginary time evolution butadditionally maintaining the orthogonality constraintbetween all the states. This diffusion quantum Monte Carloapproach has successfully been employed earlier to obtainenergies of atomic and molecular systems [27,28] as well asthose of double-well [29], multiple-well [30], and self-inter-acting [31] nonlinear oscillators. The numerical methodreported earlier for one-dimensional oscillators [29] hasbeen adopted for two-dimensional oscillators. Our calcu-lated ground – and first excited state energy eigenvaluesare in excellent agreement with literature values (given inparentheses): For the CQ oscillator [32]; E0 = 2.56982182(2.5698121), E1 = 8.93389282 (8.9337805) while for theDW oscillator [19]; E0 = �10.346278667 (�10.3462789),E1 = �10.2287095 (�10.2287098). The same algorithm isthen employed to evolve W(x,y, t = 0) under the CQ andcoupled DW potentials in the intense laser fields, by numer-ically solving the TDSE in real time. For the present calcu-lations, the laser wavelength kL = 1064 nm, frequencyxL = 0.0428228. In order to identify the threshold intensityat which the electron moving in CQ/DW potential getsexcited to the ‘‘pseudocontinuum’’, both the model systemshave been evolved under different laser intensities from5 · 1013–5 · 1022 W cm�2. The space grid in both x and y

directions is taken as �40 6 x,y 6 40, Dx = Dy = 0.1, withDt = 0.0716432. The computations were carried out from0 6 t 6 2934.51 (71 fs), i.e., 20 optical cycles. The linearramp f(t) = t/t0 up to five optical cycles and unitythereafter.

Since chaos is characterized by exponential SIC and ourearlier work on the Henon–Heiles potential [33] indicatedgreater sensitivity of quantum motion to the Hamiltonianthan to the initial ‘‘pure’’ quantum state (single eigenstate),we studied SIC for the dynamics of an electron moving innonlinearly coupled oscillators under intense laser fields, bygenerating two slightly different initial inputs as follows:The ground-state wavefunction W(x,y, t = 0) of CQ/DWpotential obtained by numerically solving the TDSE inimaginary time is evolved in laser fields of slightly differentintensities, i.e., 5 · 1013 and 5.001 · 1013 W cm�2, 5 · 1018

and 5.001 · 1018 W cm�2, 5 · 1022 and 5.001 · 1022

W cm�2. When W(x,y, t = 0) is evolved in time under aslightly different laser intensity, it is designated as W 0(x,y, t

= 0). Thus, keeping in view the unitarity of the time-evolu-tion operator, the initial input (the unperturbed ground-state) becomes a mixture of quantum states for both theHamiltonians after the first time step (i.e., after time0.0716432) and the same input state is slightly differentfor the two Hamiltonians involving slightly different laserintensities.

With respect to different initial inputs, the TDSE wasnumerically solved in real time for lasers of varying inten-sities to let the system generate its own wave packet.Finally, we examine a combination of several dynamicalsignatures of quantum chaos which have emerged overthe years.

3. Results and discussion

Although a large grid was employed to avoid reflection/transmission of the wavefunction at/through the gridboundaries, still the TD norm N(t) increased/decreasedbeyond unity under high intensity laser fields. This is char-acteristic of systems undergoing transitions to highlyexcited states, including the continuum or ‘‘pseudocontin-uum’’, such that the probability density spreads over alarge domain of space and cannot be restricted within afinite computation grid. Therefore, for such intensitiesN(t) was renormalized to unity at every time step for boththe systems (note that N(t) remained unity under a laserfield of relatively low intensity, i.e., 5 · 1013 W cm�2).

Among the TD quantities examined by us to study thequantum dynamical motion of an electron moving in theCQ/DW potential under intense laser fields, is the autocor-relation function C(t) which also helps characterizequantum chaos. C(t) measures the correlation of thetime-evolved state of the system with its initial state [10]and is defined as

CðtÞ ¼ W1ðx; y; t ¼ 0Þ W1ðx; y; tÞjh ij j2 ð5ÞIn the case of the CQ oscillator, C(t) maintains the

steady initial value of unity at I = 5 · 1013 W cm�2

(Fig. 1.1(a)). It oscillates but keeps returning to the startingvalue at I = 5 · 1018 W cm�2 (Fig. 1.1(b)), whereas itdecays sharply to zero at I = 5 · 1022 W cm�2 and doesnot return to the starting value at all (Fig. 1.1(c)). In thecase of coupled DW potential, C(t) oscillates at5 · 1013 W cm�2 but keeps returning to the starting value(Fig. 1.2(a)). But, at 5 · 1018 W cm�2, it does not returnto the initial value (Fig. 1.2(b)) and at 5 · 1022 W cm�2 itdecays sharply to zero (Fig. 1.2(c)). Thus, it is obvious thatin both the systems the electron loses correlation with itsinitial state under a high intensity laser field. This may beregarded as an indication of quantum chaos.

The corresponding power spectra are obtained in termsof the following fast fourier transform (FFT) of C(t) for thelast eight optical cycles, with integration limits t1 =1173.730 and t2 = 2347.532

AðxÞ ¼Z t2

t1

CðtÞe�ixtdt

��������2

; �1 6 x 6 þ1;CðtÞ ¼ Cð�tÞ

ð6Þ

The power spectra of CQ and DW potentials are given inFig. 2. A sharp peak corresponding to the ground-state(E0 = 2.56) of CQ oscillator is seen in the spectrum atI = 5 · 1013 W cm�2 (Fig. 2.1(a)). At I = 5 · 1018 W cm�2,other peaks also appear (Fig. 2.1(b)) and at 5 · 1022 W cm�2

the spectrum is very rich with high energy peaks appearingwith considerable intensity (Fig. 2.1(c)). In the case of DWpotential, the spectrum shows a few lines at I =5 · 1013 W cm�2 (Fig. 2.2(a)). At I = 5 · 1018 (Fig. 2.2(b))and 5 · 1022 W cm�2 it has many peaks but the lower energypeaks are more intense than the higher energy ones

Fig. 1. Plots of correlation function C(t) against time t in a.u. for (1.1) coupled quartic oscillator, and (1.2) coupled double-well oscillator at:(a) I = 5 · 1013 W cm�2, (b) I = 5 · 1018 W cm�2, and (c) I = 5 · 1022 W cm�2, respectively.

Fig. 2. Power spectra plotted against the photon energy (x), in a.u. for (2.1) coupled quartic oscillator, and (2.2) coupled double-well oscillator at: (a)I = 5 · 1013 W cm�2, (b) I = 5 · 1018 W cm�2, and (c) I = 5 · 1022 W cm�2, respectively.

354 N. Gupta, B.M. Deb / Chemical Physics 327 (2006) 351–360

N. Gupta, B.M. Deb / Chemical Physics 327 (2006) 351–360 355

(Fig. 2.2(c)). Note that at I = 5 · 1022 W cm�2, the appar-ently faint low-energy peaks (Fig. 2.2(c)) are numerous andcomparable in intensity to the high-energy peaks atI = 5 · 1018 W cm�2 (Fig. 2.2(b)).

Although the electron cannot escape over the infinitebarriers of both the potentials, a semiclassical limit is pre-sumably reached for very high vibrational numbers and thehigh-lying levels crowd together thereby approximating a‘‘continuum’’. As mentioned in Section 1, this may beregarded as a ‘‘pseudocontinuum’’ [21]. The presence ofhigh energy peaks at high intensity in both the potentialssuggests that the electron has reached the ‘‘pseudocontin-uum’’ and quantum chaos might be manifested in thisregion.

The variations of the laser electric field e(t), the distancefunction D(t) and quantum ‘‘phase space’’ volume (withinthe Ehrenfest theorem) V(t) for both CQ and coupledDW oscillators at I = 5 · 1022 W cm�2 are depicted in Figs.

Fig. 3. Plots of laser electric field e(t), distance function D(t) and quantum‘‘phase space’’ volume V(t) against time t in a.u. for the coupled quarticoscillator at I = 5 · 1022 W cm�2.

3 and 4, respectively. D(t), obtained from the same initialwavefunction that is evolved under slightly different Ham-iltonians, is given by [34,35]

DðtÞ ¼ x1ðtÞh i � x2ðtÞh ið Þ2 þ px1ðtÞ

� �� px2

ðtÞ� �� �2

n o12 ð7Þ

where hx1(t)i, hpx1(t)i and hx2(t)i and hpx2

(t)i are expecta-tion values at time t of position and momentum operatorsin x-direction with respect to W(x,y, t) and W 0(x,y, t),respectively. Note that W(x,y, t = 0) = W 0(x,y, t = 0). Hereexpectation values in y-direction, i.e., hyi and hpyi are notconsidered as the potential is symmetric in y-directionand therefore these values vanish, as verified computation-ally to check the numerical accuracy.

The quantum ‘‘phase space’’ volume is defined as theuncertainty product [10]

Fig. 4. Plots of laser electric field e(t), distance function D(t) and quantum‘‘phase space’’ volume V(t) against time t in a.u. for the coupled double-well oscillator at I = 5 · 1022 W cm�2.

356 N. Gupta, B.M. Deb / Chemical Physics 327 (2006) 351–360

V ðtÞ ¼ x2� �� xh i2

� �p2

x

� �� pxh i

2� �

y2� �

p2y

D En o12 ð8Þ

where all the expectation values are with respect toW(x,y, t). In the laser field of intensity 5 · 1013 and 5 ·1018 W cm�2, V(t) and D(t) remain almost constant forboth the oscillators. At 5 · 1022 W cm�2, D(t) increasesfor the CQ oscillator (Fig. 3). For the DW oscillator,although it shows a rapid increase to nearly the CQ valueat t � 400 a.u., it later oscillates to a steady value whichis approximately half the largest value (Fig. 4). Thus, thedistance corresponding to two initially identical wavefunc-tions that are evolving in quantum ‘‘phase space’’ underslightly different laser intensities increases for both theoscillators. This behavior is analogous to the extreme sen-sitivity of a classically chaotic system to initial conditions.However, for the DW oscillator, a partial suppression ofquantum chaos at later stages is indicated.

In a similar manner, V(t) increases for the CQ potentialand reaches up to 5500 (Fig. 3) but for the DW potential, itreaches only up to 1500 at t � 800 a.u. and does notincrease thereafter (Fig. 4). A large increase in V(t) forthe CQ oscillator at 5 · 1022 W cm�2 is an indicator ofquantum chaos as the large ‘‘phase space’’ spanned bythe electron implies greater uncertainty in the informationabout the wavepacket being generated by the electron.Note that both D(t) and V(t) show similar behavior forthe DW oscillator, i.e., a sharp increase followed by an

Fig. 5. Quantum ‘‘phase space’’ trajectories, in a.u., of the coupled double-(c) I = 5 · 1022 W cm�2, as well as of the coupled quartic oscillator at: (d) I =

almost steady value, indicative of a partial suppression ofquantum chaos.

Fig. 5 shows the quantum ‘‘phase space’’ trajectoriesfor an electron moving in coupled DW and CQ poten-tials under different laser intensities. In case of the DWoscillator at I = 5 · 1013 W cm�2, the trajectory starts asa spiral pattern and the electron flips back and forth inthe two minima (Fig. 5(a)). At a higher intensity, the tra-jectory gets more complicated while the electron stillkeeps flipping between the two minima (Fig. 5(b) and(c)). The trajectory starts as a spiral pattern for the CQoscillator at I = 5 · 1018 W cm�2 (Fig. 5(d)). As the laserintensity increases, the expectation values increase andthe trajectory gets more complicated (Fig. 5(e)). Thedivergence of ‘‘phase space’’ trajectories for the CQpotential is shown in Fig. 6 where hpxi and hxi are plot-ted at the end of each optical cycle for the wavefunctionsevolved under 5 · 1022 and 5.001 · 1022 W cm�2 laserintensities, respectively. The trajectories diverge afterthree optical cycles thereby displaying quantum chaos.Note that, in accordance with the behavior of D(t) andV(t) for the DW oscillator which maintained a steadyvalue after an initial increase, the quantum ‘‘phase space’’trajectories corresponding to initial wavefunctions thatare evolved under slightly different intensities, do notdiverge and superimpose on each other at this laserintensity.

well oscillator at: (a) I = 5 · 1013 W cm�2, (b) I = 5 · 1018 W cm�2, and5 · 1018 W cm�2 and (e) I = 5 · 1022 W cm�2, respectively.

Fig. 6. The expectation values (a.u.) hpx(t)i of the momentum operator inthe x-direction and the expectation values of the position operator hx(t)i atthe end of each optical cycle plotted against optical cycle for the coupledquartic oscillator at I = 5 · 1022 W cm�2. The dotted line corresponds toI = 5.001 · 1022 W cm�2.

Fig. 7. Overlap integral I(t) plotted against optical cycle for (7.1) coupledquartic oscillator and (7.2) coupled double-well oscillator at: (a)I = 5 · 1018 W cm�2 and (b) I = 5 · 1022 W cm�2, respectively.

N. Gupta, B.M. Deb / Chemical Physics 327 (2006) 351–360 357

Another criterion for identifying chaos is the decay ofthe overlap integral I(t) between the time-evolved states,i.e., W(x,y, t) and W 0(x,y, t). I(t) is given by [22]

IðtÞ ¼ Wðx; y; tÞ W0ðx; y; tÞjh ij j ð9ÞI2(t) may be identified as the quantum fidelity or Loschmidtecho [22–25] except that both W(x,y, t) and W 0(x,y, t) de-note perturbed states. However, if one regards W(x,y, t)as corresponding to an ‘‘unperturbed’’ Hamiltonian, whichincludes the original oscillator potential plus the laser po-tential, and W 0(x,y, t) as corresponding to a ‘‘perturbed’’Hamiltonian, where the laser intensity is slightly changed,then I2(t) indeed becomes the fidelity (note that since ourclassical system of reference is the oscillator without a laserfield, we do not adopt such a viewpoint here). It is knownthat fidelity decay can be taken as a reliable indicator ofquantum chaos in the unperturbed system provided the ap-plied perturbation commutes with a classical coordinate[23]. It is clear from Eq. (2) that this condition is satisfiedin the present case. For systems manifesting classical chaos,fidelity decay has been found to be Gaussian or exponen-tial, including a Lyapunov regime [25]. Since this was notconsidered fully satisfactory, Wang et al. [25] have pro-vided a general semiclassical treatment for fidelity decayin the limit of large perturbations. In the present case, wewish to see how I(t) in Eq. (9) decays from unity in courseof time. Fig. 7.1(a) and 7.2(a) show that I(t) remains almostunity for the CQ oscillator and DW oscillator, respectively,under the laser intensities 5 · 1013 and 5 · 1018 W cm�2,thereby showing no quantum chaos. However, at I =5 · 1022 W cm�2, although I(t) for the DW oscillator de-cays sharply to about 0.3 at the initial stage, indicating

the onset of quantum chaos, it oscillates later with decreas-ing amplitude and returns to almost unity, indicating thesuppression of quantum chaos (Fig. 7.2(b)). In otherwords, the DW oscillator undergoes a transition from reg-ularity to quantum chaos and then back to regularity. Forthe CQ oscillator, however, I(t) initially decays rapidly toabout 0.45, then fluctuates around a average value of 0.7,but never returns to the starting value of unity. This indi-cates an initial rapid onset of quantum chaos, followedby the latter’s partial suppression. One may thus concludethat the CQ oscillator exhibits greater sensitivity to slightchanges in the Hamiltonian, compared to the DWoscillator.

Fig. 8 shows the potential energy surface (PES) and theprobability density plots for the CQ oscillator. At t = 0 (theground-state), the PES has a four-pole symmetry(Fig. 8(a)) and the probability density is given by a single

Fig. 8. The coupled quartic oscillator potential energy surface: (a) at t = 0, (b) at the crest (e(t) = e0, t = 2824.4603), and (c) at the trough (e(t) = �e0,t = 2897.8229 a.u.) of the 20th optical cycle for I = 5 · 1022 W cm�2. The corresponding probability densities are given in d, e, and f, respectively.

358 N. Gupta, B.M. Deb / Chemical Physics 327 (2006) 351–360

centrosymmetric peak (Fig. 8(d)) corresponding to theminima of the PES. It was observed that after 20 opticalcycles, the density spreads over a large grid for both theoscillators at I = 5 · 1022 W cm�2. As mentioned earlier,the change in norm N(t) at this intensity was taken careof by renormalizing N(t) to unity. The spread of probabil-ity density suggests that the electron gets excited to high-

lying states, i.e., a semiclassical domain, and reaches a‘‘pseudocontinuum’’. The leakage of probability densityinto the ‘‘pseudocontinuum’’ is also supported by the pres-ence of higher energy peaks in the power spectrum.Fig. 8(b) and (c) show that the minima in the PES andthe probability density lie in the positive x-direction atthe crest (e0 = 1193.6013, t = 2824.4603) while they shift

N. Gupta, B.M. Deb / Chemical Physics 327 (2006) 351–360 359

to the negative x-direction at the trough (e0 = �1193.6013,t = 2897.8229; Fig. 8(e) and (f)) of the 20th optical cycle.Note that the four-pole symmetry of the potential getsdestroyed when the laser field is applied. This indicates thatselection rules for spectral transitions, which are derivedfor static perturbations, need reexamination for such largetime-dependent perturbations.

The high-harmonic generation (HHG) spectrum H(x)for the last eight optical cycles, with integration limits givenbelow, is obtained in terms of the FFT of the time-varyingdipole moment, l(t) = hW1(x,y, t)jxjW1(x,y, t)i,

HðxÞ ¼Z t2

t1

lðtÞe�ixtdt

��������2

; �1 6 x 6 þ1; lðtÞ ¼ lð�tÞ

ð10Þ

with t1 = 1173.730 and t2 = 2347.532. When H(x) is plottedagainst the harmonic order at I = 5 · 1013 W cm�2, no sig-nificant spectrum is observed for both the systems (Fig. 9(a)and (d)). At I = 5 · 1018 and 5 · 1022 W cm�2, the spectrashow several plateaus like a staircase (Fig. 9(b) and (c))for the DW potential. This can be explained in terms ofthe low-lying energy levels of DW oscillator (E0 =�10.346278667, E1 = � 10.2287095). Interestingly, thespectra for the DW oscillator show only odd harmonics(see inset of Fig. 9(c)). At I = 5 · 1018 and 5 · 1022 W cm�2,significantly rich spectra with both even and odd harmonics

Fig. 9. HHG spectra in a.u. plotted against harmonic order (x/xL) foI = 5 · 1018 W cm�2, and (c) I = 5 · 1022 W cm�2, respectively. The inset of (c)oscillator are shown at: (d) I = 5 · 1013 W cm�2, (e) I = 5 · 1018 W cm�2, and

are obtained for the CQ oscillator (Fig. 9(e) and (f)). Thespectra for both the oscillators show the characteristicHHG features for atoms and molecules, viz., a rapid de-crease in signal intensity followed by a long plateau of har-monics, but here it is not followed by a sharp fall in intensitybut rather shows a dramatic cut off. The HHG spectra alsoconstitute an evidence for the existence of a ‘‘pseudocontin-uum’’ for these oscillators.

4. Conclusion

The present approach characterizes quantum chaos assensitivity to the initial quantum state, in the same spiritas in classical (deterministic) chaos. However, the sensitiv-ity is not towards an initial ‘‘pure’’ quantum state (i.e., asingle eigenstate) but towards an initial state characterizedby a mixture of quantum states. This mixture or wavepac-ket is created by the time-dependent perturbation itselfacting on a single eigenstate at t = 0. Two slightly differ-ent initial mixtures of quantum states have been createdat the end of first time step (� 0.072 a.u.) by smallchanges in the laser intensity, i.e., the Hamiltonian.Although sensitivity to small changes in the Hamiltonianhas been observed before, this was in the absence of astrong external perturbation. To the best of our knowl-edge, the results and insights presented in this work bybringing a strong, external time-dependent perturbation

r the coupled double-well oscillator at: (a) I = 5 · 1013 W cm�2, (b)shows only the odd harmonics. The HHG spectra for the coupled quartic(f) I = 5 · 1022 W cm�2, respectively.

360 N. Gupta, B.M. Deb / Chemical Physics 327 (2006) 351–360

and the quantum ‘‘pseudocontinuum’’ as well as a combi-nation of signatures of quantum chaos including a quan-tity akin to fidelity (the overlap integral I(t)) intoconsideration, have not been reported before.

The present work identifies the threshold intensity(5 · 1022 W cm�2 in the present case) required to excitethe electron to the ‘‘pseudocontinuum’’ in both the sys-tems. The meandering of the electron density into the‘‘pseudocontinuum’’ is demonstrated through the powerspectra, HHG spectra and the probability density plotsthat show the density spreading over a large space grid.At the threshold intensity, quantum chaos has been eluci-dated in the CQ oscillator through various dynamical(time-dependent) quantities such as the: (i) decay of auto-correlation function, (ii) increase in distance function, (iii)large increase in quantum ‘‘phase space’’ volume, (iv)decay of overlap integral, and (v) divergence of quantum‘‘phase space’’ trajectories. However, in contrast to theCQ oscillator, the coupled DW oscillator goes througha transition from regularity to chaotic behavior and backto regularity as both D(t) and V(t) maintain a steadyvalue after an initial increase, quantum ‘‘phase space’’ tra-jectories obtained at slightly different laser intensities donot diverge and the overlap integral I(t), related to quan-tum fidelity or Loschmidt echo, returns to near unity afteran initial sharp decay.

In view of the relevance of CQ and coupled DW oscilla-tors as models for biophysicochemical systems, the presentstudy encourages the idea of employing the existing signa-tures of quantum chaos to a variety of problems fromnuclear physics to atomic/molecular physics to quantumbiophysics.

Acknowledgements

We thank Prof. Ramesh Kapoor for creating thecomputation facilities. N.G. also thanks the C.S.I.R.,New Delhi, for financial support.

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