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Dynamics of quantum observables in entangled states
Physics Letters A 373 (2009) 2814–2819
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Physics Letters A www.elsevier.com/locate/pla
Dynamics of quantum observables in entangled states C. Sudheesh a , S. Lakshmibala b,∗ , V. Balakrishnan b a b
Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel Department of Physics, Indian Institute of Technology Madras, Chennai 600 036, India
a r t i c l e
i n f o
Article history: Received 22 April 2009 Accepted 4 June 2009 Available online 6 June 2009 Communicated by P.R. Holland
a b s t r a c t We examine the dynamics of a radiation field propagating through a nonlinear medium. A time series analysis of the mean photon number illustrates how an open quantum system interacting with a quantum environment can exhibit remarkably diverse ergodicity properties, both nonlinearity and departure from coherence playing a crucial role. © 2009 Elsevier B.V. All rights reserved.
PACS: 42.50.-p 03.67.Mn 42.50.Dv 42.50.Md Keywords: Open quantum systems Photon-added coherent states Wave packet dynamics Bipartite entanglement Time series analysis Recurrence time statistics
1. Introduction and statement of the problem Quantum wave packets spread while propagating through nonlinear media, and lose their original form. Under appropriate conditions, however, they could subsequently display interesting nonclassical effects such as revivals, i.e., the return of the initial wave packet to its original form, apart from an overall phase. Wave packet revivals occur at specific instants of time, which are non-zero integer multiples of a fundamental time T rev . (That is, |ψ(0)|ψ(t )|2 = 1 at t = nT rev .) It is evident that at these instants all expectation values return to their initial values. Revivals arise due to specific quantum interference between the basis states that constitute the wave packet. (See for instance, [1].) In general, it has been established that even in those systems where exact revivals are absent, the initial state returns approximately to its original form infinitely often during its temporal evolution, if the governing Hamiltonian is time-periodic with a discrete quasi-energy spectrum [2]. (This phenomenon is the analogue of Poincaré recurrences in classical systems.) As a result, chaotic dynamics does not occur in isolated, bounded quantum systems with discrete spectra.
*
Corresponding author. Tel.: +91 44 22574869; fax: +91 44 22570509. E-mail address: [email protected] (S. Lakshmibala).
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.06.010
The dynamics is always recurrent, although the recurrence time could be extremely long, perhaps comparable to the age of the universe, even in certain simple quantum systems [3]. However, a subsystem of the isolated quantum system is effectively an open quantum system, which interacts with its quantum ‘environment’. Its density matrix (obtained by tracing out the environment variables from the full density matrix) is not governed, in general, by Hamiltonian dynamics. It evolves non-unitarily according to a dynamical map, which describes the state change over a given time interval. As a consequence, the dynamics of expectation values of the subsystem operators could, in principle, display a wide spectrum of properties ranging from mere ergodicity to ‘chaos’. In this Letter we examine the temporal evolution of an open quantum system with a view to understanding the precise manner in which the nature of the initial state affects the dynamics of expectation values. For this purpose, we consider a bipartite system comprising of a radiation field interacting with a nonlinear atomic medium. The subsystem of interest is the field, and we investigate the dynamics of the field wave packet and the manner in which it spreads by examining the ergodicity properties of appropriate field observables, as the initial state unitarily evolves to very different superpositions of the basis states in the Hilbert space considered. In general, the total wavefunction |ψ(t )
C. Sudheesh et al. / Physics Letters A 373 (2009) 2814–2819
is highly entangled even if |ψ(0) is an unentangled direct product state. Earlier work [4–6] revealed that clean signatures of nonclassical effects such as wave packet revivals are manifested in the temporal behaviour of quantum expectation values, even in the dynamics of a single system. In the present work we establish that, in fact, the dynamics of expectation values reveals much more, such as a plethora of ergodicity properties corresponding to the subsystem dynamics of a bipartite entangled system which at best exhibits only near-revivals/recurrences. (A near-revival of an initial state |ψ(0) at t = T rev means that |ψ(0)|ψ( T rev )|2 lies in some -neighbourhood of unity.) Now, there exists a strong analogy between the dynamics of a quantum wave packet which is initially a Gaussian, and the behaviour (in classical phase space) of a classical ensemble of particles whose Liouville density function is initially a Gaussian. (A similar analogy could perhaps be extended to non-Gaussians as well.) However, a smooth cross-over from the quantum to the classical problem is in general not straightforward, as the classical spreading would vanish if the h¯ → 0 limit is taken naively. For precisely this reason, the dynamics of quantum observables is not easily captured through classical analogs, and so is often regarded [7] as a purely quantum phenomenon, with no classical counterparts. It is in this same spirit that we refer, in this paper, to the rich ergodicity properties exhibited by quantum observables as a quantum mechanical effect. We emphasise that this approach is in marked contrast to more common approaches that exist in the literature to examine the ergodicity properties of the quantum counterparts of generic classically chaotic systems [8]. For our purpose, we need a system in which revival phenomena can either occur or be suppressed, depending on the values of the parameters in the Hamiltonian H . An uncomplicated but nontrivial H for our purposes is the one that describes the interaction of a single-mode field of frequency ω with the atoms of the (Kerr-like) nonlinear medium through which it propagates. The medium is modelled [9] by an anharmonic oscillator with frequency ω0 and nonlinearity parameter γ . The Hamiltonian of the total system is given by
H = h¯
ωa†a + ω0 b† b + γ b†2 b2 + g a† b + b†a .
(1)
(a, a† ) are the field annihilation and creation operators, (b, b† ) are the corresponding atomic oscillator operators, and g quantifies the coupling between the field and atom modes. Importantly, although the total number operator N tot = (a† a + b† b) commutes with H (for all values of the parameters in H ), the photon number operator N = a† a does not do so for any g = 0. Thus, while H can be cast in block-diagonal form in a direct-product basis of field and atom Fock states, the model is not trivial. It is pertinent to follow the time evolution of the expectation values of operators corresponding to physical observables, because these quantities alone are the ones that are measurable experimentally, and hence available in practice. Now, if we simply follow the unitary evolution of an (exactly specified) initial pure state according to the Schrödinger equation with the Hamiltonian H of Eq. (1), compute the expectation values of the full, infinite set of independent observables pertaining to the system as functions of time, and examine whether any chaotic behaviour occurs by calculating the Liapunov spectrum, the exponents will all be zero. However, we must note that the naive Ehrenfest relation no longer governs the evolution of expectation values when the Hamiltonian is more than quadratic, as is the case in the present instance. The generalised Ehrenfest equations (that is, the evolution equations for the full set of expectation values in a system) are generically nonlinear, and pertain to an infinite set of coupled equations, except in cases when the algebra of observables closes. It does not do so in our case, as may be seen by re-writing the Hamiltonian in terms of the angular momentum operators J + =
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a† b, J − = ab† , J z = (a† a − b† b)/2. We then have
H = h¯
1 2
(ω + ω0 + γ ) N tot + (ω − ω0 + γ ) J z
2 1 + γ N tot − 2 J z + 2g J x . 4
(2)
The problem is similar to that of an Ising spin in a transverse magnetic field. It is straightforward to check that the system of equations of motion of the angular momentum operators does not close. In this sense the effective phase space (for an arbitrary initial state) is not finite dimensional, nor is the system ‘linear in a dynamical sense’, even though the Hamiltonian is blockdiagonal. In practice, of course, it is not possible either to solve the infinite set of Ehrenfest equations for the expectation values, or to obtain all of them experimentally. The question then arises as to what extent the state of the system can be reconstructed from a knowledge of a finite number of expectation values. In particular, we are interested in reconstructing the dynamics using information on a subsystem—for instance, the field mode in our model. The expectation value N (t ), or the mean energy of the field mode, serves as a very convenient variable in this regard. N (t ) varies with time because of the coupling between the two modes, and it deviates from periodicity because of the nonlinearity in H . We now ask: suppose we have a time series on a single physical quantity, namely, the subsystem variable N (t ). What can we deduce regarding the underlying dynamics? For this purpose, we have taken the computed values of N (t ) at discrete time intervals as our time series. Using this time series as the input information, we reconstruct the ‘effective’ phase space, estimate its embedding dimensions, and examine the extent of the sensitivity to initial conditions that is displayed in the subsequent dynamics. Revivals of the initial state, i.e., the periodic return of the quantum state to its initial unentangled form, would imply the periodic return of a† a (and all other relevant observables) to their initial values, at the instant of revival. In the space of observables therefore, their dynamics would merely correspond to closed orbits. In order to explore the hierarchy of randomness in the dynamics, we report here only on those cases (or parameter regimes) in which revivals are absent. We use the value of the subsystem von Neumann entropy as an indicator of revivals (its initial value being zero for the unentangled state), and consider only those cases for which this entropy does not vanish at any t > 0. It is important to note that even small changes in the initial value of a† a are consistent with, and may correspond to, states that in general can be quite far removed (in the sense of the corresponding probability distributions) from the initial distribution. It turns out that the temporal behaviour of N (t ) in the reconstructed phase space can be quite diverse, ranging from quasiperiodicity to exponential sensitivity to initial conditions, depending strongly on the initial state and the parameter regime—in particular, on the degree of coherence of the initial state, and on the ratio γ / g of the strengths of the nonlinearity and the field-atom coupling. 2. Dynamics of the field-mode: Time series analysis and power spectra We consider initial states that are initially unentangled direct products of the field and atomic oscillator states: specifically, states with the field in a coherent state |α (CS) or an m-photon-added coherent state |α , m (PACS), while the atomic oscillator is in its ground state |0. Recall that the CS |α (α ∈ C) satisfies a|α = α |α , and is a minimum uncertainty state. The normalized PACS is defined [10] as |α , m = (a† )m |α /[m! L m (−ν )]1/2 where m is a
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C. Sudheesh et al. / Physics Letters A 373 (2009) 2814–2819
Fig. 1. Power spectrum of the mean photon number vs. the frequency (in units of g) for the initial states (a) |α ; 0 and (b) |(α , 5); 0 with
positive integer, ν = |α |2 , and L m is the Laguerre polynomial of order m. A PACS possesses the useful properties of quantifiable and tunable degrees of departure from perfect coherence and Poissonian photon statistics. With the recent experimental production and characterization of a 1-photon-added coherent state by quantum state tomography [11], it may be expected that photon-added coherent states will play an increasingly significant role in future investigations. For brevity of notation, we write |α ⊗ |0 = |α ; 0 and |α , m ⊗ |0 = |(α , m); 0 for the initial states considered. N (0) = ν and [(m + 1) Lm+1 (−ν )/ Lm (−ν )] − 1, respectively, in the two cases. We have carried out a detailed analysis of the time series (using a time step δt ranging from 10−2 for small γ / g to 10−1 for large γ / g) generated by the values of the mean photon number N computed over long intervals of time (106 time steps), including phase space reconstruction, estimation of the minimum embedding dimension demb , calculation of the power spectrum [12–14], and recurrence-time statistics (using very long time series of 107 steps when necessary). We use a robust algorithm developed by Rosenstein et al. [15] and Kantz [16] for the estimation of the maximal Liapunov exponent λmax from data sets represented by time series. The phase-space reconstruction procedure (including the extraction of demb ) has been carried out carefully, and it has been checked that any further increase in demb does not alter the inferences made regarding the exponential instability, if any, in N . We have also repeated our calculations using the procedure by Wolf et al. [17] for time series analysis and verified that our results stand. Finally, as a consistency check, we have also repeated the entire procedure independently for b† b, and verified that N tot indeed remains constant in time, as it should. Our results can be summarized as follows: For small values ( 1) of γ / g, near-revivals and fractional revivals of the initial state occur. A near-revival of the energy exchange is signaled by rapid oscillations of the mean photon number. For an initial coherent state of the field mode, the near-revival time [9] is approximately equal to 4π /γ . (In general, the revival time is inversely proportional to the coefficient of the term in H that is quadratic in the quantum numbers concerned [18].) The occurrence of revival phenomena is also mirrored in the behaviour of the extent of entanglement of the bipartite system. The latter can be characterized, for instance, by the subsystem von Neumann entropy (we could also use the subsystem linear entropy). Concomitant with the revival phenomena, these entropies of entanglement undergo well-marked oscillations in time [19]. (At the instants of near-revivals of the initial non-entangled
γ / g = 10−2 and ν = |α |2 = 1.
state, the entropies are very close to zero.) For a given (small) value of γ / g, signatures of the revival phenomena in general become less pronounced as the degree of coherence of the initial field state is decreased. The entropy of entanglement does not return to the neighbourhood of its initial value of zero, either. Correspondingly, we find that the dynamics of observables in the phase space reconstructed from the single time series for N (t ) ranges from periodicity through quasiperiodicity to ergodicity, but is not exponentially sensitive to initial conditions. As a case representative of weak nonlinearity, we have chosen the parameter values γ = 1, g = 100. Fig. 1(a), a log–linear plot of the power spectrum S ( f ) (computed from the time series of N (t )) is indicative of quasiperiodic behaviour. With increasing lack of coherence of the initial field state, the number of frequencies in S ( f ) increases, as seen in Fig. 1(b). In contrast to the case of weak nonlinearity, the nature of the reconstructed dynamics changes drastically when γ / g 1. Wellmarked revival phenomena are no longer evident. The entropy of entanglement increases rapidly from its initial value of zero to a saturation value in each case, and shows small, irregular fluctuations about this value thereafter. As representative values for this nonlinearity-dominated regime, we have set γ = 5, g = 1. We first examine the case corresponding to ν = 1. For an initial field CS, both the time series and S ( f ) confirm that the subsystem dynamics is not exponentially sensitive to initial conditions. In contrast to this, an initial PACS leads to a broadband form for S ( f ), for sufficiently large values of m. This is supported by an estimation of the extent of sensitivity to initial conditions displayed in the reconstructed phase space of observables from the time series for N (t ). The initial set of separations between the j th pair of nearest neighbours in this phase space evolves to the set {d j (k)} after k time steps. λmax , the quantifier that determines the extent of sensitivity to ‘errors’ in the initial prescription of observables, is the slope of the plot of ln d j (k) (the average is over all values of j) against t in the linear region lying in between the initial transient and final saturation regions. Figs. 2(a) and 2(b) depict two typical cases in which λmax > 0. We find that λmax increases systematically with m for an initial PACS. We have verified that the results of Fig. 2(b) are reaffirmed using the algorithm of Wolf et al. [17]. Table 1 summarises our conclusions for various initial states and parameter values. The entropy of entanglement provides independent corroboration [19] of the dynamical behaviour as deduced from S ( f ) and λmax .
C. Sudheesh et al. / Physics Letters A 373 (2009) 2814–2819
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Fig. 2. ln d j (k) vs. t (= kδt), using the algorithm of Rosenstein et al., for the initial states (a) |(α , 5); 0 with γ / g = 5 and solid line corresponds to an embedding dimension demb = 5, and the dotted lines to values of demb from 6 to 10.
Table 1 Qualitative dynamical behaviour of the mean photon number of a single-mode electromagnetic field interacting with a nonlinear medium. “Regular” ⇒ λmax = 0. Increasing departure from coherence → Initial state
|a; 0 Increasing nonlinearity
↓
|(a, 1); 0
|(a, 5); 0
γ / g = 10−2 regular |α |2 = v = 1 γ / g = 10−2 regular
regular
regular
regular
regular
regular
regular
λmax ≈ 0.5
λmax ≈ 0.6
λmax ≈ 0.7
λmax ≈ 0.9
λmax ≈ 0.7
λmax ≈ 0.8
λmax ≈ 1.0
v =5 γ /g = 5 v =1 γ /g = 5 v =5 γ /g = 5 v = 10
ν = 1 and (b) |α ; 0 with γ / g = 5 and ν = 5. The
portantly, F (τ ) is very well fitted by the exponential distribution μe −μτ . This is precisely the distribution expected in a hyperbolic dynamical system, for a sufficiently small cell size [23,24]. Using even longer time series (107 steps), we have further confirmed that two successive recurrences to a cell are uncorrelated, with a distribution μ2 τ e −μτ , the next term in the Poisson distribution [25,26]. 4. The classical system
3. Recurrence statistics Our conclusions are reinforced by a detailed analysis of another important characterizer of dynamical behaviour: recurrence statistics of the coarse-grained dynamics of N (t ) as represented by its time series. For the range of parameter values we use, the scatter in N (t ) is typically 1. We use a cell size ∼ 10−2 and very long time series (105 –106 time steps). This enables us to numerically construct the invariant density ρ (and hence the stationary measure μ for any cell C ), as well as the distribution F (τ ) of the time τ (in units of the time step δt) of first recurrence or return to C . The mean recurrence time τ can then be calculated, and compared with the result τ = μ−1 that follows from the Poincaré recurrence theorem [20]. As the latter is derived from the requirement of ergodicity alone, an agreement between the two values confirms that the dynamics is indeed ergodic in all the cases studied. We present here just two representative cases, both of which are also included in Table 1, for ready reference. The first corresponds to weak nonlinearity (γ / g = 10−2 ) and |ψ(0) = |(α , 1); 0, ν = 1. According to Table 1, this case is regular. Fig. 3(a) shows the invariant density, while (b) shows the actual recurrence time distribution. The discrete nature of the latter is a clear indication that the dynamics is actually quasiperiodic [21,22]. In marked contrast, consider a case of strong nonlinearity, γ / g = 5 and |ψ(0) = |(α , 1); 0, ν = 10. According to Table 1, this case has λmax = 0.80. Fig. 4(a) shows that the invariant density is in fact well-approximated by a Gaussian in this case. More im-
The quantum case considered by us allows for more than one classical limit. Only some of these correspond to classically chaotic systems. Such inequivalent classical limits are possible because the quantum Hamiltonian considered here can be recast, as shown earlier, in terms of spin operators, and spin has no well-defined classical counterpart. This feature is illustrated even in a simpler model where a spin 1/2 operator interacts with a Hermitian amplitude operator proportional to (b + b† ), where b and b† are Bose annihilation and creation operators. The Hamiltonian in this case is given by
H = ωb† b + ω0 S z + g b + b† ( S + + S − ).
(3)
Here, ω , ω0 and g are constants, and S + , S − and S z are spin operators. A straightforward semi-classical approximation which treats the boson amplitude as a classical variable driven by the expectation values of S + and S − gives a set of equations which correspond to a non-integrable classical system [27]. In this same spirit, our system Hamiltonian (which, as shown earlier, can be recast in terms of angular momentum operators) can have a chaotic classical limit. Now consider another classical counterpart of the Hamiltonian in Eq. (1). Let the linear oscillator associated with (a, a† ) have a mass m, position x and momentum p x , and let that associated with (b, b† ) have a mass M, position y and momentum p y . Putting in all the constant factors (including h¯ ) in the definitions of the raising and lowering operators in Eq. (1), we get H (x, p x , y , p y ). When h¯ → 0, the only consistent way to obtain a non-trivial, finite expression for the classical Hamiltonian H cl is to let the nonlinearity parameter γ → 0 simultaneously, such that the ratio γ /¯h → λ = a finite number. Then, with H 1 = p 2x /(2m) + mω2 x2 /2 and H 2 = p 2y /(2M ) + M ω02 y 2 /2, we find
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C. Sudheesh et al. / Physics Letters A 373 (2009) 2814–2819
Fig. 3. (a) Invariant density and (b) first-recurrence-time distribution for the cell C from the time series of N (t ), for weak nonlinearity.
Fig. 4. (a) Invariant density and (b) first-recurrence-time distribution for the cell C from the time series of N (t ), for strong nonlinearity.
H cl = H 1 + H 2 +
+√
g
ωω0
λ
ω02
H 22
√ √ ( mM ωω0 xy + p x p y / mM ).
(4)
tot The counterpart of N tot is N cl = H 1 /ω + H 2 /ω0 , which is in involution with H cl . Hence the 2-freedom classical system is Liouville– Arnold integrable. Further, although H cl has cross terms that could tot change sign, N cl = constant is a hyperellipsoid in the 4-dimensional phase space, which guarantees that the motion is bounded for any set of initial conditions. All four Liapunov exponents vanish, and the classical motion is always regular. This behaviour is indeed very different from the much more diverse one found for the quantum expectation value N (t ) in the reconstructed phase space. Our results bring out an important aspect: quantum expectation values in a nonlinear, entangled system could exhibit an interesting range of ergodic behaviour depending on the extent of coherence of the initial state.
5. Concluding remarks We note that a comparison of the quantum and classical cases is not always straightforward [7], and the case at hand is one such instance. An outcome of this feature is that the Liapunov expo-
nents that characterize the dynamical behaviour of classical and quantum expectation values of the same observable can indeed be very different from each other [28]. The interpretation of the dynamics of quantum observables in our case rests on the observation by Peres [3] that there are infinitely many states that can lead to the same, or infinitesimally separated, values for the expectation values of one or more observables (Peres has focused on the energy). In an entangled bipartite system, the subsystem density matrix is obtained by tracing out the variables corresponding to the quantum environment from the full pure state density matrix. This reduced density matrix corresponds to a mixed state, in general, as is clear by considering even simple examples such as the Bell states. To rephrase it differently: In our procedure, the effective reconstructed space of observables is built from the knowledge of the initial state we consider and its subsequent evolution, and therefore its dimensions are determined by our reference initial state. However, several states exist in the neighbourhood of a given initial state with very different coherence properties, although the mean photon number in these states may be very close numerically to that in our reference initial state. Consequently, the time series of the mean photon number in those cases will be very different from the one sampled by us, and in general even the dimensionality of the corresponding reconstructed space will be different. In this sense,
C. Sudheesh et al. / Physics Letters A 373 (2009) 2814–2819
small changes in the initial values of the observables can lead to very different final values of observables. The chaotic behaviour reported here is ultimately a reflection of the nature of the spread of the initial wave packet under suitable conditions, i.e., the manner in which different states are accessed by the system, between two successive recurrences of the full system. Our results also show that the degree of coherence of the initial state (which is a Gaussian wave packet for an ideal coherent state) is an important factor in this regard. In this connection we must note that, even in cases when the effective dimensionality of the Hilbert space of states is relatively small owing to the existence of other constants of the motion (as is the case in the model at hand), the revival times can be exceedingly large if the spectrum is not linear in the quantum number, precisely because of the anharmonic term in H . The states and parameter values for which we see exponential sensitivity in the dynamics in the reconstructed phase space are indeed those that have exceedingly large near-revival times. The issues we have discussed are of significance, inasmuch as (a) quantum mechanical systems are not chaotic in the usual sense, and (b) measured data are generically the time series of an incomplete set of observables. As we have shown, an open quantum system can exhibit remarkably diverse ergodicity properties even when the environment with which it interacts is itself fully quantum mechanical. References [1] R.W. Robinett, Phys. Rep. 392 (2004) 1.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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T. Hogg, B.A. Huberman, Phys. Rev. Lett. 48 (1982) 711. A. Peres, Phys. Rev. Lett. 49 (1982) 1118. C. Sudheesh, S. Lakshmibala, V. Balakrishnan, Phys. Lett. A 329 (2004) 14. C. Sudheesh, S. Lakshmibala, V. Balakrishnan, Europhys. Lett. 71 (2005) 744. C. Sudheesh, S. Lakshmibala, V. Balakrishnan, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S728. R.G. Littlejohn, Phys. Rep. 138 (1986) 193. F. Haake, Quantum Signatures of Chaos, Springer-Verlag, Berlin, 1991. G.S. Agarwal, R.R. Puri, Phys. Rev. A 39 (1989) 2969. G.S. Agarwal, K. Tara, Phys. Rev. A 43 (1991) 492. A. Zavatta, S. Viciani, M. Bellini, Science 306 (2004) 660. H.D.I. Abarbanel, Analysis of Observed Chaotic Data, Springer-Verlag, New York, 1995. P. Grassberger, I. Procaccia, Phys. Rev. Lett. 50 (1983) 346. A.M. Fraser, H.L. Swinney, Phys. Rev. A 33 (1986) 1134. M.T. Rosenstein, J.J. Collins, C.J. De Luca, Physica D 65 (1993) 117. H. Kantz, Phys. Lett. A 185 (1994) 77. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16 (1985) 285. S. Seshadri, S. Lakshmibala, V. Balakrishnan, J. Stat. Phys. 101 (2000) 213. C. Sudheesh, S. Lakshmibala, V. Balakrishnan, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 3345. M. Kac, Probability and Related Topics in Physical Sciences, Interscience, New York, 1959. M. Theunissen, C. Nicolis, G. Nicolis, J. Stat. Phys. 94 (1999) 437. S. Seshadri, S. Lakshmibala, V. Balakrishnan, Phys. Lett. A 256 (1999) 15. M. Hirata, Ergod. Th. Dyn. Systems 13 (1993) 533. V. Balakrishnan, M. Theunissen, Stoch. Dyn. 1 (2001) 339. M. Hirata, Dyn. Syst. Chaos 1 (1995) 87. V. Balakrishnan, C. Nicolis, G. Nicolis, Stoch. Dyn. 1 (2001) 345. R. Graham, M. Höhnerbach, Z. Phys. B: Condens. Matter 57 (1984) 233. L.E. Ballentine, Y. Yang, J.P. Zibin, Phys. Rev. A 50 (1994) 2854.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Dynamics of quantum observables in entangled states C. Sudheesh a , S. Lakshmibala b,∗ , V. Balakrishnan b a b
Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel Department of Physics, Indian Institute of Technology Madras, Chennai 600 036, India
a r t i c l e
i n f o
Article history: Received 22 April 2009 Accepted 4 June 2009 Available online 6 June 2009 Communicated by P.R. Holland
a b s t r a c t We examine the dynamics of a radiation field propagating through a nonlinear medium. A time series analysis of the mean photon number illustrates how an open quantum system interacting with a quantum environment can exhibit remarkably diverse ergodicity properties, both nonlinearity and departure from coherence playing a crucial role. © 2009 Elsevier B.V. All rights reserved.
PACS: 42.50.-p 03.67.Mn 42.50.Dv 42.50.Md Keywords: Open quantum systems Photon-added coherent states Wave packet dynamics Bipartite entanglement Time series analysis Recurrence time statistics
1. Introduction and statement of the problem Quantum wave packets spread while propagating through nonlinear media, and lose their original form. Under appropriate conditions, however, they could subsequently display interesting nonclassical effects such as revivals, i.e., the return of the initial wave packet to its original form, apart from an overall phase. Wave packet revivals occur at specific instants of time, which are non-zero integer multiples of a fundamental time T rev . (That is, |ψ(0)|ψ(t )|2 = 1 at t = nT rev .) It is evident that at these instants all expectation values return to their initial values. Revivals arise due to specific quantum interference between the basis states that constitute the wave packet. (See for instance, [1].) In general, it has been established that even in those systems where exact revivals are absent, the initial state returns approximately to its original form infinitely often during its temporal evolution, if the governing Hamiltonian is time-periodic with a discrete quasi-energy spectrum [2]. (This phenomenon is the analogue of Poincaré recurrences in classical systems.) As a result, chaotic dynamics does not occur in isolated, bounded quantum systems with discrete spectra.
*
Corresponding author. Tel.: +91 44 22574869; fax: +91 44 22570509. E-mail address: [email protected] (S. Lakshmibala).
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.06.010
The dynamics is always recurrent, although the recurrence time could be extremely long, perhaps comparable to the age of the universe, even in certain simple quantum systems [3]. However, a subsystem of the isolated quantum system is effectively an open quantum system, which interacts with its quantum ‘environment’. Its density matrix (obtained by tracing out the environment variables from the full density matrix) is not governed, in general, by Hamiltonian dynamics. It evolves non-unitarily according to a dynamical map, which describes the state change over a given time interval. As a consequence, the dynamics of expectation values of the subsystem operators could, in principle, display a wide spectrum of properties ranging from mere ergodicity to ‘chaos’. In this Letter we examine the temporal evolution of an open quantum system with a view to understanding the precise manner in which the nature of the initial state affects the dynamics of expectation values. For this purpose, we consider a bipartite system comprising of a radiation field interacting with a nonlinear atomic medium. The subsystem of interest is the field, and we investigate the dynamics of the field wave packet and the manner in which it spreads by examining the ergodicity properties of appropriate field observables, as the initial state unitarily evolves to very different superpositions of the basis states in the Hilbert space considered. In general, the total wavefunction |ψ(t )
C. Sudheesh et al. / Physics Letters A 373 (2009) 2814–2819
is highly entangled even if |ψ(0) is an unentangled direct product state. Earlier work [4–6] revealed that clean signatures of nonclassical effects such as wave packet revivals are manifested in the temporal behaviour of quantum expectation values, even in the dynamics of a single system. In the present work we establish that, in fact, the dynamics of expectation values reveals much more, such as a plethora of ergodicity properties corresponding to the subsystem dynamics of a bipartite entangled system which at best exhibits only near-revivals/recurrences. (A near-revival of an initial state |ψ(0) at t = T rev means that |ψ(0)|ψ( T rev )|2 lies in some -neighbourhood of unity.) Now, there exists a strong analogy between the dynamics of a quantum wave packet which is initially a Gaussian, and the behaviour (in classical phase space) of a classical ensemble of particles whose Liouville density function is initially a Gaussian. (A similar analogy could perhaps be extended to non-Gaussians as well.) However, a smooth cross-over from the quantum to the classical problem is in general not straightforward, as the classical spreading would vanish if the h¯ → 0 limit is taken naively. For precisely this reason, the dynamics of quantum observables is not easily captured through classical analogs, and so is often regarded [7] as a purely quantum phenomenon, with no classical counterparts. It is in this same spirit that we refer, in this paper, to the rich ergodicity properties exhibited by quantum observables as a quantum mechanical effect. We emphasise that this approach is in marked contrast to more common approaches that exist in the literature to examine the ergodicity properties of the quantum counterparts of generic classically chaotic systems [8]. For our purpose, we need a system in which revival phenomena can either occur or be suppressed, depending on the values of the parameters in the Hamiltonian H . An uncomplicated but nontrivial H for our purposes is the one that describes the interaction of a single-mode field of frequency ω with the atoms of the (Kerr-like) nonlinear medium through which it propagates. The medium is modelled [9] by an anharmonic oscillator with frequency ω0 and nonlinearity parameter γ . The Hamiltonian of the total system is given by
H = h¯
ωa†a + ω0 b† b + γ b†2 b2 + g a† b + b†a .
(1)
(a, a† ) are the field annihilation and creation operators, (b, b† ) are the corresponding atomic oscillator operators, and g quantifies the coupling between the field and atom modes. Importantly, although the total number operator N tot = (a† a + b† b) commutes with H (for all values of the parameters in H ), the photon number operator N = a† a does not do so for any g = 0. Thus, while H can be cast in block-diagonal form in a direct-product basis of field and atom Fock states, the model is not trivial. It is pertinent to follow the time evolution of the expectation values of operators corresponding to physical observables, because these quantities alone are the ones that are measurable experimentally, and hence available in practice. Now, if we simply follow the unitary evolution of an (exactly specified) initial pure state according to the Schrödinger equation with the Hamiltonian H of Eq. (1), compute the expectation values of the full, infinite set of independent observables pertaining to the system as functions of time, and examine whether any chaotic behaviour occurs by calculating the Liapunov spectrum, the exponents will all be zero. However, we must note that the naive Ehrenfest relation no longer governs the evolution of expectation values when the Hamiltonian is more than quadratic, as is the case in the present instance. The generalised Ehrenfest equations (that is, the evolution equations for the full set of expectation values in a system) are generically nonlinear, and pertain to an infinite set of coupled equations, except in cases when the algebra of observables closes. It does not do so in our case, as may be seen by re-writing the Hamiltonian in terms of the angular momentum operators J + =
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a† b, J − = ab† , J z = (a† a − b† b)/2. We then have
H = h¯
1 2
(ω + ω0 + γ ) N tot + (ω − ω0 + γ ) J z
2 1 + γ N tot − 2 J z + 2g J x . 4
(2)
The problem is similar to that of an Ising spin in a transverse magnetic field. It is straightforward to check that the system of equations of motion of the angular momentum operators does not close. In this sense the effective phase space (for an arbitrary initial state) is not finite dimensional, nor is the system ‘linear in a dynamical sense’, even though the Hamiltonian is blockdiagonal. In practice, of course, it is not possible either to solve the infinite set of Ehrenfest equations for the expectation values, or to obtain all of them experimentally. The question then arises as to what extent the state of the system can be reconstructed from a knowledge of a finite number of expectation values. In particular, we are interested in reconstructing the dynamics using information on a subsystem—for instance, the field mode in our model. The expectation value N (t ), or the mean energy of the field mode, serves as a very convenient variable in this regard. N (t ) varies with time because of the coupling between the two modes, and it deviates from periodicity because of the nonlinearity in H . We now ask: suppose we have a time series on a single physical quantity, namely, the subsystem variable N (t ). What can we deduce regarding the underlying dynamics? For this purpose, we have taken the computed values of N (t ) at discrete time intervals as our time series. Using this time series as the input information, we reconstruct the ‘effective’ phase space, estimate its embedding dimensions, and examine the extent of the sensitivity to initial conditions that is displayed in the subsequent dynamics. Revivals of the initial state, i.e., the periodic return of the quantum state to its initial unentangled form, would imply the periodic return of a† a (and all other relevant observables) to their initial values, at the instant of revival. In the space of observables therefore, their dynamics would merely correspond to closed orbits. In order to explore the hierarchy of randomness in the dynamics, we report here only on those cases (or parameter regimes) in which revivals are absent. We use the value of the subsystem von Neumann entropy as an indicator of revivals (its initial value being zero for the unentangled state), and consider only those cases for which this entropy does not vanish at any t > 0. It is important to note that even small changes in the initial value of a† a are consistent with, and may correspond to, states that in general can be quite far removed (in the sense of the corresponding probability distributions) from the initial distribution. It turns out that the temporal behaviour of N (t ) in the reconstructed phase space can be quite diverse, ranging from quasiperiodicity to exponential sensitivity to initial conditions, depending strongly on the initial state and the parameter regime—in particular, on the degree of coherence of the initial state, and on the ratio γ / g of the strengths of the nonlinearity and the field-atom coupling. 2. Dynamics of the field-mode: Time series analysis and power spectra We consider initial states that are initially unentangled direct products of the field and atomic oscillator states: specifically, states with the field in a coherent state |α (CS) or an m-photon-added coherent state |α , m (PACS), while the atomic oscillator is in its ground state |0. Recall that the CS |α (α ∈ C) satisfies a|α = α |α , and is a minimum uncertainty state. The normalized PACS is defined [10] as |α , m = (a† )m |α /[m! L m (−ν )]1/2 where m is a
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Fig. 1. Power spectrum of the mean photon number vs. the frequency (in units of g) for the initial states (a) |α ; 0 and (b) |(α , 5); 0 with
positive integer, ν = |α |2 , and L m is the Laguerre polynomial of order m. A PACS possesses the useful properties of quantifiable and tunable degrees of departure from perfect coherence and Poissonian photon statistics. With the recent experimental production and characterization of a 1-photon-added coherent state by quantum state tomography [11], it may be expected that photon-added coherent states will play an increasingly significant role in future investigations. For brevity of notation, we write |α ⊗ |0 = |α ; 0 and |α , m ⊗ |0 = |(α , m); 0 for the initial states considered. N (0) = ν and [(m + 1) Lm+1 (−ν )/ Lm (−ν )] − 1, respectively, in the two cases. We have carried out a detailed analysis of the time series (using a time step δt ranging from 10−2 for small γ / g to 10−1 for large γ / g) generated by the values of the mean photon number N computed over long intervals of time (106 time steps), including phase space reconstruction, estimation of the minimum embedding dimension demb , calculation of the power spectrum [12–14], and recurrence-time statistics (using very long time series of 107 steps when necessary). We use a robust algorithm developed by Rosenstein et al. [15] and Kantz [16] for the estimation of the maximal Liapunov exponent λmax from data sets represented by time series. The phase-space reconstruction procedure (including the extraction of demb ) has been carried out carefully, and it has been checked that any further increase in demb does not alter the inferences made regarding the exponential instability, if any, in N . We have also repeated our calculations using the procedure by Wolf et al. [17] for time series analysis and verified that our results stand. Finally, as a consistency check, we have also repeated the entire procedure independently for b† b, and verified that N tot indeed remains constant in time, as it should. Our results can be summarized as follows: For small values ( 1) of γ / g, near-revivals and fractional revivals of the initial state occur. A near-revival of the energy exchange is signaled by rapid oscillations of the mean photon number. For an initial coherent state of the field mode, the near-revival time [9] is approximately equal to 4π /γ . (In general, the revival time is inversely proportional to the coefficient of the term in H that is quadratic in the quantum numbers concerned [18].) The occurrence of revival phenomena is also mirrored in the behaviour of the extent of entanglement of the bipartite system. The latter can be characterized, for instance, by the subsystem von Neumann entropy (we could also use the subsystem linear entropy). Concomitant with the revival phenomena, these entropies of entanglement undergo well-marked oscillations in time [19]. (At the instants of near-revivals of the initial non-entangled
γ / g = 10−2 and ν = |α |2 = 1.
state, the entropies are very close to zero.) For a given (small) value of γ / g, signatures of the revival phenomena in general become less pronounced as the degree of coherence of the initial field state is decreased. The entropy of entanglement does not return to the neighbourhood of its initial value of zero, either. Correspondingly, we find that the dynamics of observables in the phase space reconstructed from the single time series for N (t ) ranges from periodicity through quasiperiodicity to ergodicity, but is not exponentially sensitive to initial conditions. As a case representative of weak nonlinearity, we have chosen the parameter values γ = 1, g = 100. Fig. 1(a), a log–linear plot of the power spectrum S ( f ) (computed from the time series of N (t )) is indicative of quasiperiodic behaviour. With increasing lack of coherence of the initial field state, the number of frequencies in S ( f ) increases, as seen in Fig. 1(b). In contrast to the case of weak nonlinearity, the nature of the reconstructed dynamics changes drastically when γ / g 1. Wellmarked revival phenomena are no longer evident. The entropy of entanglement increases rapidly from its initial value of zero to a saturation value in each case, and shows small, irregular fluctuations about this value thereafter. As representative values for this nonlinearity-dominated regime, we have set γ = 5, g = 1. We first examine the case corresponding to ν = 1. For an initial field CS, both the time series and S ( f ) confirm that the subsystem dynamics is not exponentially sensitive to initial conditions. In contrast to this, an initial PACS leads to a broadband form for S ( f ), for sufficiently large values of m. This is supported by an estimation of the extent of sensitivity to initial conditions displayed in the reconstructed phase space of observables from the time series for N (t ). The initial set of separations between the j th pair of nearest neighbours in this phase space evolves to the set {d j (k)} after k time steps. λmax , the quantifier that determines the extent of sensitivity to ‘errors’ in the initial prescription of observables, is the slope of the plot of ln d j (k) (the average is over all values of j) against t in the linear region lying in between the initial transient and final saturation regions. Figs. 2(a) and 2(b) depict two typical cases in which λmax > 0. We find that λmax increases systematically with m for an initial PACS. We have verified that the results of Fig. 2(b) are reaffirmed using the algorithm of Wolf et al. [17]. Table 1 summarises our conclusions for various initial states and parameter values. The entropy of entanglement provides independent corroboration [19] of the dynamical behaviour as deduced from S ( f ) and λmax .
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Fig. 2. ln d j (k) vs. t (= kδt), using the algorithm of Rosenstein et al., for the initial states (a) |(α , 5); 0 with γ / g = 5 and solid line corresponds to an embedding dimension demb = 5, and the dotted lines to values of demb from 6 to 10.
Table 1 Qualitative dynamical behaviour of the mean photon number of a single-mode electromagnetic field interacting with a nonlinear medium. “Regular” ⇒ λmax = 0. Increasing departure from coherence → Initial state
|a; 0 Increasing nonlinearity
↓
|(a, 1); 0
|(a, 5); 0
γ / g = 10−2 regular |α |2 = v = 1 γ / g = 10−2 regular
regular
regular
regular
regular
regular
regular
λmax ≈ 0.5
λmax ≈ 0.6
λmax ≈ 0.7
λmax ≈ 0.9
λmax ≈ 0.7
λmax ≈ 0.8
λmax ≈ 1.0
v =5 γ /g = 5 v =1 γ /g = 5 v =5 γ /g = 5 v = 10
ν = 1 and (b) |α ; 0 with γ / g = 5 and ν = 5. The
portantly, F (τ ) is very well fitted by the exponential distribution μe −μτ . This is precisely the distribution expected in a hyperbolic dynamical system, for a sufficiently small cell size [23,24]. Using even longer time series (107 steps), we have further confirmed that two successive recurrences to a cell are uncorrelated, with a distribution μ2 τ e −μτ , the next term in the Poisson distribution [25,26]. 4. The classical system
3. Recurrence statistics Our conclusions are reinforced by a detailed analysis of another important characterizer of dynamical behaviour: recurrence statistics of the coarse-grained dynamics of N (t ) as represented by its time series. For the range of parameter values we use, the scatter in N (t ) is typically 1. We use a cell size ∼ 10−2 and very long time series (105 –106 time steps). This enables us to numerically construct the invariant density ρ (and hence the stationary measure μ for any cell C ), as well as the distribution F (τ ) of the time τ (in units of the time step δt) of first recurrence or return to C . The mean recurrence time τ can then be calculated, and compared with the result τ = μ−1 that follows from the Poincaré recurrence theorem [20]. As the latter is derived from the requirement of ergodicity alone, an agreement between the two values confirms that the dynamics is indeed ergodic in all the cases studied. We present here just two representative cases, both of which are also included in Table 1, for ready reference. The first corresponds to weak nonlinearity (γ / g = 10−2 ) and |ψ(0) = |(α , 1); 0, ν = 1. According to Table 1, this case is regular. Fig. 3(a) shows the invariant density, while (b) shows the actual recurrence time distribution. The discrete nature of the latter is a clear indication that the dynamics is actually quasiperiodic [21,22]. In marked contrast, consider a case of strong nonlinearity, γ / g = 5 and |ψ(0) = |(α , 1); 0, ν = 10. According to Table 1, this case has λmax = 0.80. Fig. 4(a) shows that the invariant density is in fact well-approximated by a Gaussian in this case. More im-
The quantum case considered by us allows for more than one classical limit. Only some of these correspond to classically chaotic systems. Such inequivalent classical limits are possible because the quantum Hamiltonian considered here can be recast, as shown earlier, in terms of spin operators, and spin has no well-defined classical counterpart. This feature is illustrated even in a simpler model where a spin 1/2 operator interacts with a Hermitian amplitude operator proportional to (b + b† ), where b and b† are Bose annihilation and creation operators. The Hamiltonian in this case is given by
H = ωb† b + ω0 S z + g b + b† ( S + + S − ).
(3)
Here, ω , ω0 and g are constants, and S + , S − and S z are spin operators. A straightforward semi-classical approximation which treats the boson amplitude as a classical variable driven by the expectation values of S + and S − gives a set of equations which correspond to a non-integrable classical system [27]. In this same spirit, our system Hamiltonian (which, as shown earlier, can be recast in terms of angular momentum operators) can have a chaotic classical limit. Now consider another classical counterpart of the Hamiltonian in Eq. (1). Let the linear oscillator associated with (a, a† ) have a mass m, position x and momentum p x , and let that associated with (b, b† ) have a mass M, position y and momentum p y . Putting in all the constant factors (including h¯ ) in the definitions of the raising and lowering operators in Eq. (1), we get H (x, p x , y , p y ). When h¯ → 0, the only consistent way to obtain a non-trivial, finite expression for the classical Hamiltonian H cl is to let the nonlinearity parameter γ → 0 simultaneously, such that the ratio γ /¯h → λ = a finite number. Then, with H 1 = p 2x /(2m) + mω2 x2 /2 and H 2 = p 2y /(2M ) + M ω02 y 2 /2, we find
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Fig. 3. (a) Invariant density and (b) first-recurrence-time distribution for the cell C from the time series of N (t ), for weak nonlinearity.
Fig. 4. (a) Invariant density and (b) first-recurrence-time distribution for the cell C from the time series of N (t ), for strong nonlinearity.
H cl = H 1 + H 2 +
+√
g
ωω0
λ
ω02
H 22
√ √ ( mM ωω0 xy + p x p y / mM ).
(4)
tot The counterpart of N tot is N cl = H 1 /ω + H 2 /ω0 , which is in involution with H cl . Hence the 2-freedom classical system is Liouville– Arnold integrable. Further, although H cl has cross terms that could tot change sign, N cl = constant is a hyperellipsoid in the 4-dimensional phase space, which guarantees that the motion is bounded for any set of initial conditions. All four Liapunov exponents vanish, and the classical motion is always regular. This behaviour is indeed very different from the much more diverse one found for the quantum expectation value N (t ) in the reconstructed phase space. Our results bring out an important aspect: quantum expectation values in a nonlinear, entangled system could exhibit an interesting range of ergodic behaviour depending on the extent of coherence of the initial state.
5. Concluding remarks We note that a comparison of the quantum and classical cases is not always straightforward [7], and the case at hand is one such instance. An outcome of this feature is that the Liapunov expo-
nents that characterize the dynamical behaviour of classical and quantum expectation values of the same observable can indeed be very different from each other [28]. The interpretation of the dynamics of quantum observables in our case rests on the observation by Peres [3] that there are infinitely many states that can lead to the same, or infinitesimally separated, values for the expectation values of one or more observables (Peres has focused on the energy). In an entangled bipartite system, the subsystem density matrix is obtained by tracing out the variables corresponding to the quantum environment from the full pure state density matrix. This reduced density matrix corresponds to a mixed state, in general, as is clear by considering even simple examples such as the Bell states. To rephrase it differently: In our procedure, the effective reconstructed space of observables is built from the knowledge of the initial state we consider and its subsequent evolution, and therefore its dimensions are determined by our reference initial state. However, several states exist in the neighbourhood of a given initial state with very different coherence properties, although the mean photon number in these states may be very close numerically to that in our reference initial state. Consequently, the time series of the mean photon number in those cases will be very different from the one sampled by us, and in general even the dimensionality of the corresponding reconstructed space will be different. In this sense,
C. Sudheesh et al. / Physics Letters A 373 (2009) 2814–2819
small changes in the initial values of the observables can lead to very different final values of observables. The chaotic behaviour reported here is ultimately a reflection of the nature of the spread of the initial wave packet under suitable conditions, i.e., the manner in which different states are accessed by the system, between two successive recurrences of the full system. Our results also show that the degree of coherence of the initial state (which is a Gaussian wave packet for an ideal coherent state) is an important factor in this regard. In this connection we must note that, even in cases when the effective dimensionality of the Hilbert space of states is relatively small owing to the existence of other constants of the motion (as is the case in the model at hand), the revival times can be exceedingly large if the spectrum is not linear in the quantum number, precisely because of the anharmonic term in H . The states and parameter values for which we see exponential sensitivity in the dynamics in the reconstructed phase space are indeed those that have exceedingly large near-revival times. The issues we have discussed are of significance, inasmuch as (a) quantum mechanical systems are not chaotic in the usual sense, and (b) measured data are generically the time series of an incomplete set of observables. As we have shown, an open quantum system can exhibit remarkably diverse ergodicity properties even when the environment with which it interacts is itself fully quantum mechanical. References [1] R.W. Robinett, Phys. Rep. 392 (2004) 1.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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T. Hogg, B.A. Huberman, Phys. Rev. Lett. 48 (1982) 711. A. Peres, Phys. Rev. Lett. 49 (1982) 1118. C. Sudheesh, S. Lakshmibala, V. Balakrishnan, Phys. Lett. A 329 (2004) 14. C. Sudheesh, S. Lakshmibala, V. Balakrishnan, Europhys. Lett. 71 (2005) 744. C. Sudheesh, S. Lakshmibala, V. Balakrishnan, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S728. R.G. Littlejohn, Phys. Rep. 138 (1986) 193. F. Haake, Quantum Signatures of Chaos, Springer-Verlag, Berlin, 1991. G.S. Agarwal, R.R. Puri, Phys. Rev. A 39 (1989) 2969. G.S. Agarwal, K. Tara, Phys. Rev. A 43 (1991) 492. A. Zavatta, S. Viciani, M. Bellini, Science 306 (2004) 660. H.D.I. Abarbanel, Analysis of Observed Chaotic Data, Springer-Verlag, New York, 1995. P. Grassberger, I. Procaccia, Phys. Rev. Lett. 50 (1983) 346. A.M. Fraser, H.L. Swinney, Phys. Rev. A 33 (1986) 1134. M.T. Rosenstein, J.J. Collins, C.J. De Luca, Physica D 65 (1993) 117. H. Kantz, Phys. Lett. A 185 (1994) 77. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16 (1985) 285. S. Seshadri, S. Lakshmibala, V. Balakrishnan, J. Stat. Phys. 101 (2000) 213. C. Sudheesh, S. Lakshmibala, V. Balakrishnan, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 3345. M. Kac, Probability and Related Topics in Physical Sciences, Interscience, New York, 1959. M. Theunissen, C. Nicolis, G. Nicolis, J. Stat. Phys. 94 (1999) 437. S. Seshadri, S. Lakshmibala, V. Balakrishnan, Phys. Lett. A 256 (1999) 15. M. Hirata, Ergod. Th. Dyn. Systems 13 (1993) 533. V. Balakrishnan, M. Theunissen, Stoch. Dyn. 1 (2001) 339. M. Hirata, Dyn. Syst. Chaos 1 (1995) 87. V. Balakrishnan, C. Nicolis, G. Nicolis, Stoch. Dyn. 1 (2001) 345. R. Graham, M. Höhnerbach, Z. Phys. B: Condens. Matter 57 (1984) 233. L.E. Ballentine, Y. Yang, J.P. Zibin, Phys. Rev. A 50 (1994) 2854.