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arXiv:nlin/0111042 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Ehrenfest times for classically chaotic systems

P. G. Silvestrov, C. W. J. Beenakker

Published Mar 7, 2002• 38 citations
arXivHTMLPDF

Abstract

We describe the quantum-mechanical spreading of a Gaussian wave packet by means of the semiclassical WKB approximation of Berry and Balazs [J. Phys. A $2,$ 625 (1979)]. We find that the time scale $\ensuremath{\tau}$ on which this approximation breaks down in a chaotic system is larger than the Ehrenfest times considered previously. In one dimension $\ensuremath{\tau}=\frac{7}{6}{\ensuremath{\lambda}}^{\ensuremath{-}1}\mathrm{ln}(A/\ensuremath{\Elzxh}),$ with $\ensuremath{\lambda}$ the Lyapunov exponent and A a typical classical action.

DOI
10.1103/physreve.65.035208

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We describe the quantum-mechanical spreading of a Gaussian wave packet by means of the semiclassical WKB approximation of Berry and Balazs [J. Phys. A 2,2,2, 625 (1979)]. We find that the time scale \ensuremathτ\ensuremath{\tau}\ensuremathτ on which this approximation breaks down in a chaotic system is larger than the Ehrenfest times considered previously. In one dimension \ensuremathτ=76\ensuremathλ\ensuremath−1ln(A/\ensuremath\Elzxh),\ensuremath{\tau}=\frac{7}{6}{\ensuremath{\lambda}}^{\ensuremath{-}1}\mathrm{ln}(A/\ensuremath{\Elzxh}),\ensuremathτ=67​\ensuremathλ\ensuremath−1ln(A/\ensuremath\Elzxh), with \ensuremathλ\ensuremath{\lambda}\ensuremathλ the Lyapunov exponent and A a typical classical action.
Two approaches to quantum-classical correspondence are distinguished according to the classical dynamical theory with which quantum theory is compared. The first of these, Ehrenfest correspondence, defines a dynamical regime in which the quantum expectation values follow approximately a classical trajectory. The second of these, Liouville correspondence, applies when the quantum probability distributions remain well approximated by a density in the classical phase space. The former applies only for narrow states, whereas the latter may remain valid even for quantum states that have spread to the system size. A spin model is adopted for this correspondence study because the quantum state is discrete and finite- dimensional, and thus no artificial truncation of the Hilbert space is required. The quantum time-evolution is given by a discrete unitary mapping. The corresponding classical model is volume-preserving (non-dissipative) and the time-evolution is given by a symplectic map. In classically chaotic regimes, the widths of initially narrow quantum states grow, on average, exponentially with time, until saturation at the system size. This initial spreading rate is well approximated by the classical Lyapunov exponent when the accessible classical phase space is predominantly chaotic. Because of the exponential growth rate of the quantum variance, the Ehrenfest regime is delimited by a break-time that grows logarithmically with increasing quantum numbers. The small differences between quantum expectation values and corresponding Liouville averages of dynamical variables also grow exponentially, initially, if the classical behaviour is chaotic. This exponential rate is independent of h and consistently larger than the classical Lyapunov exponent by at least a factor of two. Interestingly, this exponential growth rate does not continue until these differences approach magnitudes of order the system size, but crosses over to power-law growth (or simply saturates), when the differences have reached magnitudes no larger than order ħ. Because of the exponential growth of these quantum-classical differences, the Liouville regime is also delimited by a break-time that grows logarithmically with increasing quantum numbers. However, due to the early saturation of the exponential growth of the differences, the Liouville log break-time applies only in a restricted domain. After spreading to the system size, the quantum states relax to an approximately time-independent state. This is characterised by equilibrium distributions that are subject to rapidly oscillating fluctuations about the coarse-grained classical steady-state. If the accessible classical phase space is predominantly chaotic, then the equilibrium quantum probability distributions are nearly microcanonical. Otherwise the steady-state quantum probability distributions accurately reflect the details of the KAM surfaces in the classical phase space. The equilibrium regime quantum fluctuations, which are quantum-classical differences, are shown to approach zero as a negative power of the quantum numbers. Since the differences between quantum expectation values and individual classical trajectories grow exponentially in time to a magnitude that scales with the system size, the Ehrenfest regime is delimited by an unrestricted log break-time. The log break-time may be short compared to experimental time-scales for some chaotic macroscopic bodies, and, as a result, an observable breakdown of Ehrenfest correspondence may arise for some macroscopic systems. Although the Liouville regime is also delimited by a log break-time, an observable breakdown of correspondence between quantum mechanics and Liouville mechanics for macroscopic bodies is not likely since their differences, expressed relative to the dimensions of the observable, approach zero in the limit of large quantum numbers.
We analyze the semiclassical evolution of Gaussian wave packets in chaotic systems. We show that after some short time a Gaussian wave packet becomes a primitive WKB state. From then on, the state can be propagated using the standard time-dependent WKB scheme. Complex trajectories are not necessary to account for the long-time propagation. The Wigner function of the evolving state develops the structure of a classical filament plus quantum oscillations, with phase and amplitude being determined by geometric properties of a classical manifold.
The quantum correspondence of one particular signature of classical chaos--the exponential instability of motion--can be characterized by the initial exponential growth rate of the spreading of the propagating quantum wave packet. The growth rate is approximately twice the classical maximum Lyapunov exponent of the system. In the regular case, the dispersion of the wave packet is only due to the usual quantum effect that should vanish in the classical limit. In contrast, in the chaotic case, the evolution behavior of the wave packet is due to the dynamical effect associated with the nonlinearity and persists as long as the spatial extension of the initial wave packet is kept finite.
The initial quantum dynamics of an initially well-localized, minimum-uncertainty Gaussian wave packet is studied analytically for the periodically kicked pendulum. The general results presented here include our earlier analytic result [Phys. Rev. A 41, 2952 (1990)] as a special case. We find that as long as the packet remains localized, it remains approximately Gaussian, while its center and widths evolve in time approximately according to, respectively, the classical map and a variance map which utilize the instantaneous values of the classical Jacobi matrix. When the classical map is very strongly chaotic, we predict that the average kinetic energy of the well-localized wave packet does not initially grow diffusively (linearly with time). We show that, instead, in the presence of chaos, the deviations initially grow, on the average, exponentially at a rate equal to the corresponding classical Liapunov exponent under the appropriate conditions. The time scale (Ehrenfest time) on which the Ehrenfest-Hamilton correspondence holds in the presence of chaos is estimated and is slightly different from a previous estimate made by Chirikov, Israilev, and Shepelyansky [Sov. Sci. Rev. C 2, 209 (1981)]. Numerical evidence in support of these analytical results is presented.