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Physical Review A

Quantum-to-classical limit in a Hamiltonian system

L. E. Ballentine

Published Sep 28, 2004• 17 citations
Publisher

Abstract

The dynamics of quantum and classical probability distributions are computed for a model of two coupled rotors (or pendulums), with emphasis on the $\ensuremath{\hbar}$ scaling of the quantum-classical (QC) differences. This scaling is not the same for coarse-grained and fine-grained quantities. The QC differences in the averages of observables scale as ${\ensuremath{\hbar}}^{2∕3}$ for chaotic states, but as ${\ensuremath{\hbar}}^{2}$ for regular states. The QC differences in probability distributions scale as ${\ensuremath{\hbar}}^{1∕3}$ for chaotic states. No simple scaling is found for those differences in regular states, although their overall magnitudes are similar to those in chaotic states. QC differences arise first in the short-wavelength regime, and subsequently spread to all wavelengths. A satisfactory classical limit is obtained without invoking environmental decoherence, which in any case would be effective in suppressing only short-wavelength QC differences.

DOI
10.1103/physreva.70.032111

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The dynamics of quantum and classical probability distributions are computed for a model of two coupled rotors (or pendulums), with emphasis on the \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ scaling of the quantum-classical (QC) differences. This scaling is not the same for coarse-grained and fine-grained quantities. The QC differences in the averages of observables scale as \ensuremathℏ2∕3{\ensuremath{\hbar}}^{2∕3}\ensuremathℏ2∕3 for chaotic states, but as \ensuremathℏ2{\ensuremath{\hbar}}^{2}\ensuremathℏ2 for regular states. The QC differences in probability distributions scale as \ensuremathℏ1∕3{\ensuremath{\hbar}}^{1∕3}\ensuremathℏ1∕3 for chaotic states. No simple scaling is found for those differences in regular states, although their overall magnitudes are similar to those in chaotic states. QC differences arise first in the short-wavelength regime, and subsequently spread to all wavelengths. A satisfactory classical limit is obtained without invoking environmental decoherence, which in any case would be effective in suppressing only short-wavelength QC differences.
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.
The equations of motion for phase-space moments and correlations are derived systematically for quantum and classical dynamics, and are solved numerically for chaotic and regular motions of the H\'enon-Heiles model. For very narrow probability distributions, Ehrenfest's theorem implies that the centroid of the quantum state will approximately follow a classical trajectory. But the error in Ehrenfest's theorem does not scale with $\ensuremath{\Elzxh},$ and is found to be governed essentially by classical quantities. The difference between the centroids of the quantum and classical probability distributions, and the difference between the variances of those distributions, scale as ${\ensuremath{\Elzxh}}^{2},$ and so are the true measures of quantum effects. For chaotic motions, these differences between quantum and classical motions grow exponentially, with a larger exponent than does the variance of the distributions. For regular motions, the variance of the distributions grows as t2,{t}^{2},t2, whereas the differences between the quantum and classical motions grow as t3.{t}^{3}.t3.
In this thesis, a comparison between quantum and classical mechanics was made for two systems whose classical behaviour can be chaotic.The general correspondence between classical mechanics and quantum mechanics in the classical regime has always been assumed but never proved.Recent research in the field of quantum chaos has brought this assumption into question.For some systems, such as the kicked rotor in the classically chaotic regime, quantum mechanics does not display the classical behaviour of unbounded momentum.It has been suggested that quantum mechanics will fail to produce classical chaos for a broad range of simple systems.A comparison of quantum and classical mechanics was made using numerical simulations, without recourse to taking limits such as 7i going to zero.The two systems studied were a driven pendulum and a pair of pendula joined by a spring.The quantum and classical expectation values for a number of observables were computed, as were the probability distributions in phase space.For periodic motion of the driven pendulum it was found that the theories agreed well for times up to 150 driving periods.Similar agreement was found for initial conditions whose classical behaviour was chaotic, but only for times up to a few tens of driving periods.For longer times, it was found that the quantum system displayed a boundedness in position and/or momentum that was absent in the classical system.A possible explanation was given in terms of the quasienergy expansion.In the coupled-pendula system no such disagreements between the theories were found.I would like to thank Dr. Ballentine for his continued support during this project and his help in editing and revising this thesis.I would like to thank the members of my committee, Drs.Bechhoefer, Boa1 and Loss for their many suggestions.For many helpful discussions, I would like to thank Dominic Mimnagh and Jim Zibin.
We consider the quantum and classical Liouville dynamics of a nonintegrable model of two coupled spins. Initially localized quantum states spread exponentially to the system size when the classical dynamics are chaotic. The long-time behavior of the quantum probability distributions and, in particular, the parameter-dependent rates of relaxation to the equilibrium state are surprisingly well approximated by the classical Liouville mechanics even for small quantum numbers. As the accessible classical phase space becomes predominantly chaotic, the classical and quantum probability equilibrium configurations approach the microcanonical distribution, although the quantum equilibrium distributions exhibit characteristic "minimum" fluctuations away from the microcanonical state. The magnitudes of the quantum-classical differences arising from the equilibrium quantum fluctuations are studied for both pure and mixed (dynamically entangled) quantum states. In both cases the standard deviation of these fluctuations decreases as (h/J)(1/2), where J is a measure of the system size. In conclusion, under a variety of conditions the differences between quantum and classical Liouville mechanics are shown to become vanishingly small in the classical limit (J/h-->infinity) of a nondissipative model endowed with only a few degrees of freedom.
Understanding how classical physics emerges from quantum mechanics remains a central problem in the foundations of physics. Here we derive a classical limit from finite-resolution measurements, modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared with Planck's constant, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable and the induced probability density is positive. We derive the exact evolution equation for this density and show that, in the strong coarse-graining regime, its non-Liouville corrections are suppressed up to an Ehrenfest time, resulting in classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics, thereby closing the quantization--classical-limit loop. Repeated finite-resolution measurements then generate stochastic records confined, with high probability, to tubes around classical trajectories. Our results provide a unified operational framework for the quantum-to-classical transition in microscopic and macroscopic systems, and establish the consistency of the quantization--classical-limit cycle.
Making use of appropriate quantum-classical correspondence we have examined the differential behavior of regular and chaotic trajectories in terms of quantum decoherence, the evolution of average quantities, and variances and entropy in a driven double-well oscillator in contact with the surroundings. Our numerical analysis has shown that even on a short time scale the decay of the average quantum coherence is multiexponential. While the onset of decoherence is much faster for a regular trajectory, the decay is faster asymptotically for a chaotic trajectory compared to a regular one. The coupling of the system to the environment turns chaotic in a regular evolution. The environment also affects the evolution of quantum variances for a regular trajectory almost from the beginning, while it has an insignificant effect on chaotic evolution up to a time after which, for both trajectories, noise in the quantum variances exhibits remarkable suppression, although fluctuations of the reservoir modes, in general, tend to increase the level of variances. We identify three stages of quantum evolution; a short decoherence regime followed by a Liouville flow, the latter regime being dominated by the classical curvature of the potential. This is the regime at which growth of entropy or quantum variances is exponential, the rate being determined by the classical largest Lyapunov exponent. The last stage is the irreversible flow dominated by diffusion which suppresses noises in the quantum variances. \textcopyright{} 1996 The American Physical Society.