PPHYSICAL
REVIEW E VOLUME
NUMBER FEBRUARY
Orbital instability and the loss of quantum coherence Allan
Tameshtit and J. E. Sipe
Department of Physics and
Ontario
Laser and
Lightmave
Research
Centre,
University of Toronto, Toronto,
Ontario,
Canada
MSSXA7 (Received April revised manuscript received November We compare quantum decoherence in generic regular and chaotic systems that interact with a thermal reservoir via a dipole coupling. Using a time-dependent, self-consistent approximation in the spirit of Hartree, we derive in the high temperature limit an expression for the ofF-diagonal elements of the system density operator that initially corresponds to a coherent superposition of two adjacent wave packets. We relate the decoherence rate to the Lyapunov exponent in the Ehrenfest regime. In this regime, the greater the instability of the system the faster the loss of coherence occurs. PACS number(s): +b, — w I. INTRODUCTION
Classical behavior is ubiquitous in a universe that is fundamentally quantum mechanical. From one point of view, this observation is rationalized by noting the similarity between the quantum and classical equations of Ehrenfest and
Newton.
But recent developments in the theory of chaos complicate this explanation. Realis- tic
Hamiltonians give rise to significant regions of phase space that are classically chaotic. In such regions the Ehrenfest regime, in which quantum expectation values behave as do classical trajectories, is much more fleet- ing than in regular regions [1, From this point of view, then, the simultaneous pervasiveness in nature of classically chaotic Hamiltonians and classical behavior is clearly perplexing. Prom a di8'erent viewpoint, however, it is the loss of coherence that occurs when a system interacts with its environment that permits a classicaldescription of phenomena [3, This line of reasoning, together with the main result of this paper, circumvents the difBculty raised above; we show that coherence de- cays faster in systems that are chaotic rather than regu- lar. Hence, &om this second viewpoint, the emergence of classical behavior should occur sooner in chaotic systems than in their regular counterparts. Most investigations of decoherence and the emergence of classical behavior have relied on the simple model of the coupling of a system, desc. , ibed by Hamiltonian H, to a thermal environment of harmonic oscillators. The total
Hamiltonian is taken as HT = H + ) Rusbtb, + hAR, where B = g. [K(urz)bs + r*(ur )b s], twith K being a cou- pling parameter, and A is a system operator. We wish to compare here the corresponding decoherence when H is either classically regular or chaotic; only making use of generic properties of the two classes, we need not spec- ify H for our analysis. In the secular regime, we have previously calculated [5] the chaotic decay rates of coher- ences in the energy representation [6]. And, in the case of A = H, we found [7, that coherence, as measured by the purity Trp, decays faster in chaotic systems than in their regular counterparts. Here we generalize some of these results and examine a more physical and much studied coupling, A = q, where q is the position opera- tor of the system. Recently [9], an analysis of a Brown- ian equation describing an unstable harmonic oscillator (potential= — [k~q /2) revealed that, at long times, the entropy of a Gaussian state increases linearly at a rate determined by k. In contrast, we use a diferent approach to look at generic regular and chaotic systems and exam- ine not the long time entropy growth of a single Gaussian, but rather the decoherence of a superposition of two adja- cent localized states. The perspective we adopt has been proposed previously [4, and may be summarized by the following excerpt [4]: "The destruction of interference terms is often considered as caused in a classical way by an 'uncontrollable influence' of the environment on the system of interest. . . . But the opposite is true: The sys- tem disturbs the environment, thereby dislocalizing the phases. " We find that in some regimes chaotic systems disturb the environment more profoundly, resulting in faster decoherence. It is instructive to first re-examine the case A = H Rom this perspective; here the interaction with the envi- ronment can be thought of as performing quantum non- demolition measurements on the system's energy. Denote by ~E„) an energy eigenstate of the system and ~PR) an arbitrary reservoir state. An initial total operator (~E„)(E ~) I3 (~P~)(P~~) evolves exactly as e "" '(IE-)(E-I) UR(t
E-)(l&~)(4RI)UR(t
E-) (2) where cu = (E — E )/h and UR(t,
E„) = exp ~ — — HR+ hs„) (~, b, + '. bt) The
American
Physical
Society BRIEF
REPORTS We see that the reservoir bra and ket are driven by difer- ent amplitudes if n g m. One would expect the reduced system operator [obtained by tracing expression (2) with respect to the reservoir variables] to decay faster, the greater the difference IE„— E I. Since it is known [12] that nearest neighbor energy differences are greater in chaotic systems (the so called "energy repulsion" ) than in regular systems, our previous result [7, is further elucidated. Taking instead the dipole coupling A = q, we show below that a similar process in the Ehrenfest regime leads to faster decay of coherence in chaotic sys- tems than in regular systems. What there takes the place of energy repulsion is "trajectory repulsion:" that is, the well-known exponentiation of chaotic trajectories. II.
HARTREE
ANALYSIS We start our analysis with a two dimensional version of (1) with A being replaced by the position (q, q„): , r, dI"") = &-)-n. bt b +(~lq*l~)m (&lqwl&) q~ ~ + (@I (6) We call the solution of these last two equations a Hartree (total) ket and denote it by ~QT (t)). The error in this scheme is given by IAQT (t)) = l@F(t)) — li//T (t)), where l@T" (t)), is the exact solution of the Schrodinger equation with total
Hamiltonian (3) and initial condition (4). By writing down the evolution equation that Ib, @T(t)) satis- fies, switching to an interaction picture, and performing a Born expansion, a general expression for the error may be obtained to first order in the coupling parameter [ll]; this we seek to characterize the initial Hartree kets li//;„) that lead to the most accurate description of early time behavior. Assuming a continuum of oscillator &equencies with density of modes g(w), we adopt perhaps the most common model: HT —— ~+ ) ) ~qbj~„bq, ~ + &q~&i + Qy&2~ (8) g(~)IK(~)l' = ~e(~)e(~-- — ~) (7) where Ri ~2l = p. [K(~~ ) b~ i ~2l + /(, " (cu~) b. ~ l]. Obtaining the exact evolution corresponding to HT for arbitrary H is out of the question. We resort to an ap- proximation in the spirit of a time-dependent, Hartree analysis [13].
Such an approach has been employed be- fore to treat Hamiltonians similar to HT [14, Graham and
Hohnerbach [15] (see also [16] for related work) used this approximation to study a two level, chaotic system, although there a single mode treated classically (as op- posed to a quantum reservoir) was coupled to the system; moreover, the chaos there was induced by strong coupling to the mode. This should be contrasted to our work be- low, where H is intrinsically chaotic and the coupling to the reservoir is weak. Suppose we take the following product initial state: I@'-) l@R, '-) (4) (@Iil&
I&~) (@~IWR) Rl&ji) „(41(4 l(q*& +q. &. )l@ ) l@)~l (Welf a) &@IV) where lg;„) and lvPR;„) are, respectively, system and reservoir kets. We first find the best product state which approximates the evolution of (4) over a time long enough to capture the relevant decoherence; we are interested in the usual situation where this time is much shorter than that during which significant en- ergy relaxation occurs. By "best" we mean that to- tal state l@T(t))— : l@(t)) (8) IQR(t)), with initial condi- tion (4), that results from the variational equation [17] (biz l(ihd//dt — IIT)l@T ) = By considering variations in (@ I and (@R I, the following coupled equations describ- ing the evolution of l@T) may be obtained: where C is a coupling constant, is the unit step func- tion, and w „ is some high &equency cutofF. Noting that
Eq. (6) evolves reservoir coherent states, denoted by lcxR) or IP&), into other reservoir coherent states, we choose them for the lg~;„); they form a convenient basis with which to expand the reservoir canonical operator that we consider below [see state (9)], particularly so be- cause they yield an asymptotic expression for the error that is independent of reservoir operators. To wit, if the coherent states are chosen for the li//R;„), we find [18] that the error, to first order in the coupling, is then (&@T(t) l&@T(t)) - [(&q*)'(t) + (&q )'(t)1 (8) For arbitrary H, it is a nontrivial task to determine which class of initial states li/'/;„) minimizes the error. If an eigenstate of q is chosen as the initial system ket, the uncertainty in the position, although initially zero, will quickly increase significantly because of the infinite mo- mentum dispersion. For a nonzero time interval, rela- tion (8) suggests that the smallest error results when the iiutial Hartree system ket lg;„) is a compromise be- tween being completely localized in position or momen- tum. Only this qualitative feature of the li//;„), and its independence of the specific reservoir coherent state cho- sen for l@~;„) under the approximation employed below, is indicated for the next step of our analysis. As a simple example of a more . general initial total state, in what follows we take the reservoir in thermal as „= cu t — + oo for a fixed time t, and where Lq ~„~ is the uncertainty of the position q ~„~ in the Hartree system ket l@(t)); the bar denotes the time aver-age t I (Aq ) (w) + (Aq„) (w) uri(w)dw with weight — cos(B (1 — c/t)) — cos(A s/0) )„ — It ~/t BRIEF
REPORTS where the Hartree system ket
Ized) is centered about the four dimensional phase space point z~; p~, ~„ is the reser- voir operator corresponding to a canonical ensemble at temperature T. Now define y(t) to be the trace, over the reservoir variables, of the operator that results by propa- gating pR, „ Izi)(z2I for a time t; the system operator y(t) reflects the coherence that subsists between the two states Izi) and
Iz2).
Determining the evolution of y(t) entails that we solve Eqs. (5) and (6) self-consistently. A common technique to accomplish this is by iteration. One may start the iterative algorithm by assuming the evolution of the Hartree system ket to be unaffected by the reservoir so that Iz. (t)), j = obeys (1) . dlz, " (t)) dt (io) where the superscript of lzI l(t)) refers to the nth it- erative step. Next, the expectation value (q )(t, zz) (z. (t)lq Iz (t)) and that for the y coordinate are in- serted into Eq. (6) and one then calculates
IPR (t)).
This is the end of one iteration and we may then form equilibrium and the system in a pure state consisting of a superposition of two such Hartree kets.
That is, we consider the following initial condition: p~, -- ). c'c, *lz')(z'I (9) taking z1 — — z and Z2 —— z + bz (13) f t gg Bz — g~ z q- jqz (z, z q- qz) — q ( zz)j z) dz O(ts), regular
O(e2~i+ l~ ) chaotic (14) as t ~ oo. In view of the corresponding expression in Eq. (12), for initial conditions (13) it is clear that co- herence decays faster in the Ehrenfest regime for chaotic systems than for regular systems.
The source of this faster decoherence is the unstable orbits. By imprint- ing their exponentiation on the reservoir states correlated with them, the coherence between them is lost at a cor- respondingly [Eqs. (12) and (14)] faster rate than it is for orbits in a regular system. so that Iz2) is now formally a state centered about a point in phase space only inf1nitesimally displaced from z1. Consider now a system in the Ehrenfest regime [21], where the the expectation values (q)(t) behave as would the classical coordinates q(t). We can then obtain univer- sal behavior by appealing to the different rates of diver- gence of initially adjacent orbits. Generically [22], this divergence exhibits average linear [23, and exponen- tial [25, growth; denoting by A ) the maximum Lyapunov exponent [24] and by e any positive number, we have x R t n„ t (3 z, t z, t (ii) where these last integrals run over the two polarizations and the infinite number of reservoir modes. For the sec- ond iteration, one would insert (P&~llb~lP&~l) (j = as well as the last mentioned system expectation val- ues, into Eq. (5) and solve to obtain Iz l(t)). Contin- uing in this manner, one could generate g~ l (t) and the higher iterates, the hope being that convergence to y(t) occurs suKciently rapidly. Even with this approximate scheme, calculating the y~ & (t) becomes increasingly pro-hibitive. Our simple goal here is to compute y~ & (t), leav- ing aside for now the diKcult and important mathemati- cal problem of determining the convergence properties of th. &~"l(t) [19]. For a axed time t & we may calculate the controlling factor of the modulus of (zi~ l(t) ly~ l(t) lz2~ l(t)). We find l- ( "(t)l~" (t) I "(t)) t g~ 'T) Z2 Q~ ) Z + j(qz)(z zz) (qz)(z z~)l ) ~z (qq) as Ru „/kT ~ first, followed by „-+ oo [20]. Henceforth, we shall restrict the system initial state by III.
DISCUSSION
For real, quantum systems, broaching this issue be- yond the
Ehrenfest regime is complicated by the fact that the "break time" tg (i. e. , the time after which the quantum expectation values are no longer approximated by the classical trajectories) is much shorter in chaotic than in regular systems. Although this matter is not completely resolved, some theoretical work [1, indicates that for chaotic and regular systems ts is O(in') and
O(h), respectively (see also [26,
Nevertheless, nu- merical analyses [28] indicate that, for short times, nar- row wave packets propagated quantum mechanically can exhibit large, classical-like sensitivities to initial condi- tions before spreading quickly dampens this instability. Moreover, because the differences of the expectation val- ues in expression (12) enter as exponents, coherence de- cay will be greatly affected by any disparity in the diver- gence rates of neighboring wave packets — whether expo- nentially sensitive to initial conditions or not in chaotic and regular systems. As alluded to in the Introduction and discussed in [1], &om one point of view the shorter break time in clas- sically chaotic systems leads one to conclude that the greater the instability, the more a system resembles a BRIEF
REPORTS quantum one.
However, greater instability is precisely what gives rise to more effective decoherence when, unlike the previous point of view, the surroundings are taken into account. And when coherence is lost, the hallmark of quantum mechanics the interference effects that arise from the quantum prescription of adding amplitudes, and not probabilities, of alternative events — is quashed. By demonstrating that orbital instability leads to faster de- coherence, we have thus established an important link between chaos and the appearance of classical behavior. [1] G. M. Zaslavsky, in Quantum
Dynamics of Chaotic Sys- tem8, edited by J. -M. Yuan, D. H. Feng, and G. M. Za- slavsky (Gordon and
Breach,
Langhorne,
PA, [2] M. V. Berry and N. L. Balazs, J. Phys. A (1979). [3] W. H. Zurek,
Phys.
Rev. D (1981); (1982); Phys.
Today (10), (1991); C. M. Savage and D. F. Walls,
Phys.
Rev. A (1985); J. E. Sipe and N. Arkani-Hamed, ibid. (1992). [4] E. Joos and H. D. Zeh, Z. Phys. B (1985). [5] A. Tameshtit and J. E. Sipe,
Phys.
Rev. A (1994). [6] For a temporally periodic perturbation applied to a Hamiltonian system (typically having one degree of free- dom) in the absence of any thermal effects, Berman and
Zaslavsky [29] found semiclassically that the off- diagonal elements of the density operator (in the en- ergy representation of the unperturbed Hamiltonian) de- cay only if the perturbation induces chaos. Again, ne- glecting thermal noise, but including dissipation,
Dittrich and
Graham [30] examined the stroboscopically kicked (and hence nonconservative) rotor only within the rotat- ing wave approximation; restricting this speci6c model to the chaotic diffusive regime and using order of mag- nitude estimates, they found that the phase coherence lifetime of the quasienergies was inversely proportional to the square of the kicking strength. See also
Cohen [31], for further work on the kicked rotor or particle, and Blumel et al. [32]. [7] A. Tameshtit and J. E. Sipe, in Quantum
Dynamics of Chaotic
Systems [1]. [8] A. Tameshtit and J. E. Sipe,
Phys.
Rev. A (1993). [9] W. H. Zurek and J. P. Paz,
Phys.
Rev.
Lett. (1994). [10] A Stern Y. Aharonov, and Y. Imry,
Phys.
Rev. A (1990) [11] N. Arkani-Hamed and J. E. Sipe (unpublished). [12] F. Haake,
Quantum
Chaos (Springer-Verlag,
Berlin, and references cited therein. [13] D. R. Hartree,
The
Calculation of Atomic
Structures (Wi- ley,
New
York, [14] H.
Primas, in Sixty
Two
Years of Unc-ertainty, edited by A. I. Miller (Plenum
Press,
New
York, p. 259. [15] R. Graham and M. Hohnerbach, Z. Phys. B (1984). [16] [»] [19] [20] [21] [22] [23] [24] [26] [27] [28] [29] [30] [»] [32] [33] [34] L. L. Bonilla and F. Guinea,
Phys.
Lett. B (1991); Phys.
Rev. A (1992). J. I. Frenkel,
Wave
Mechanics,
Part II (Oxford University
Press,
Oxford,
Some technical assumptions were used to derive (8) from the order expression; for example, we assumed (v/r(t)]q [g(t)) and the analogous expression for y to be a thrice continuously differentiable function of time. Note also that these position expectation values depend on the cutoff frequency u We have not investigated to what extent this Hartree approximation captures the initial slip or jolt [33] previ- ously found in analyses of a system harmonic oscillator coupled to a bath of harmonic oscillators. Inasmuch as the jolt is partly due to unphysical initial conditions, and appears to have no signi6cant effect on decoherence when the coupling is weak [34], this is probably not a serious drawback. See also [10] for related expressions involving the inter- ference of two given electron paths (that start and end together) in a metal. For a discussion of the validity of the Ehrenfest ap- proximation see, e. g. , A. Messiah,
Quantum
Mechanics (North-Holland,
Amsterdam,
Vol. Chap.
VI.
Harmonic oscillators and periodic orbits are excluded. G. Casati, B. V. Chirikov, and J. Ford,
Phys.
Lett. (1980). H. -D. Meyer, J. Chem.
Phys. (1986). V. I. Oseledec,
Trans.
Moscow
Math.
Soc. (1968). R. G. Littlejohn,
Phys.
Rep. (1986). S. Tomsovic and E. J. Heller,
Phys.
Rev. E (1993). R. S. Judson, S. Shi, and H. Rabitz, J. Chem.
Phys. (1989). G. P. Berman and G. M. Zaslavsky,
Physica (1979); G. M. Zaslavsky,
Chaos in Dynamic
Systems (Harwood,
Chur,
Switzerland,
Chap. T. Dittrich and R. Graham,
Ann.
Phys. (1990). D. Cohen,
Phys.
Rev. A (1991); (1991). R. Blumel, A. Buchleitner, R. Graham, L. Sirko, U. Smi- lansky, and H. Walther,
Phys.
Rev. A (1991). F. Haake and R. Reibold,
Phys.
Rev. A (1985). J. P. Paz, S. Habib, and W. H. Zurek,
Phys.
Rev. D488