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arXiv:quant-ph/0503170 · Physical Review A

Quantum mechanics of Hyperion

Nathan Wiebe, L. E. Ballentine

Published Aug 19, 2005• 42 citations
arXivHTMLPDF

Abstract

This paper is motivated by the suggestion [W. Zurek, Phys. Scri. T, 76, 186 (1998)] that the chaotic tumbling of the satellite Hyperion would become nonclassical within $20\phantom{\rule{0.3em}{0ex}}\text{years}$, but for the effects of environmental decoherence. The dynamics of quantum and classical probability distributions are compared for a satellite rotating perpendicular to its orbital plane, driven by the gravitational gradient. The model is studied with and without environmental decoherence. Without decoherence, the maximum quantum-classical (QC) differences in its average angular momentum scale as ${\ensuremath{\hbar}}^{2∕3}$ for chaotic states, and as ${\ensuremath{\hbar}}^{2}$ for nonchaotic states, leading to negligible QC differences for a macroscopic object like Hyperion. The quantum probability distributions do not approach their classical limit smoothly, having an extremely fine oscillatory structure superimposed on the smooth classical background. For a macroscopic object, this oscillatory structure is too fine to be resolved by any realistic measurement. Either a small amount of smoothing (due to the finite resolution of the apparatus) or a very small amount of environmental decoherence is sufficient to ensure the classical limit. Under decoherence, the QC differences in the probability distributions scale as ${({\ensuremath{\hbar}}^{2}∕D)}^{1∕6}$, where $D$ is the momentum diffusion parameter. We conclude that decoherence is not essential to explain the classical behavior of macroscopic bodies.

DOI
10.1103/physreva.72.022109

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This paper is motivated by the suggestion [W. Zurek, Phys. Scri. T, 76, 186 (1998)] that the chaotic tumbling of the satellite Hyperion would become nonclassical within 20years20\phantom{\rule{0.3em}{0ex}}\text{years}20years, but for the effects of environmental decoherence. The dynamics of quantum and classical probability distributions are compared for a satellite rotating perpendicular to its orbital plane, driven by the gravitational gradient. The model is studied with and without environmental decoherence. Without decoherence, the maximum quantum-classical (QC) differences in its average angular momentum scale as \ensuremathℏ2∕3{\ensuremath{\hbar}}^{2∕3}\ensuremathℏ2∕3 for chaotic states, and as \ensuremathℏ2{\ensuremath{\hbar}}^{2}\ensuremathℏ2 for nonchaotic states, leading to negligible QC differences for a macroscopic object like Hyperion. The quantum probability distributions do not approach their classical limit smoothly, having an extremely fine oscillatory structure superimposed on the smooth classical background. For a macroscopic object, this oscillatory structure is too fine to be resolved by any realistic measurement. Either a small amount of smoothing (due to the finite resolution of the apparatus) or a very small amount of environmental decoherence is sufficient to ensure the classical limit. Under decoherence, the QC differences in the probability distributions scale as (\ensuremathℏ2∕D)1∕6{({\ensuremath{\hbar}}^{2}∕D)}^{1∕6}(\ensuremathℏ2∕D)1∕6, where DDD is the momentum diffusion parameter. We conclude that decoherence is not essential to explain the classical behavior of macroscopic bodies.
Violation of correspondence principle may occur for very macroscopic byt isolated quantum systems on rather short timescales as illustrated by the case of Hyperion, the chaotically tumbling moon of Saturn, for which quantum and classical predictions are expected to diverge on a timescale of approximately 20 years. Motivated by Hyperion, we review salient features of ``quantum chaos'' and show that decoherence is the essential ingredient of the classical limit, as it enables one to solve the apparent paradox caused by the breakdown of the correspondence principle for classically chaotic systems.
The environment – external or internal degrees of freedom coupled to the system – can, in effect, monitor some of its observables. As a result, the eigenstates of these observables decohere and behave like classical states: Continuous destruction of superpositions leads to environment-induced superselection (einselection). Here I investigate it in the context of quantum chaos (i.e., quantum dynamics of systems which are classically chaotic). I show that the evolution of a chaotic macroscopic (but, ultimately, quantum) system is not just difficult to predict (requiring an accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg indeterminacy (which limits the resolution available in the initial conditions). This happens after a time t ℏ which is only logarithmic in the Planck constant. A definitely macroscopic (if somewhat outrageous) example is afforded by various components of the solar system which are chaotic, with the Lyapunov timescales ranging from a bit more than a month (Hyperion) to millions of years (planetary system as a whole). On the timescale t ℏ the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of radii of their orbits into “Schrödinger cat – like” states, and the concept of a trajectory would cease to apply. In reality, such paradoxical states are eliminated by decoherence which helps restore quantum-classical correspondence. The price for the recovery of classicality is the loss of predictability: In the classical limit (associated with effective decoherence, and not just with the smallness of ℏ) the rate of increase of the von Neumann entropy of the decohering system is independent of the strength of the coupling to the environment, and equal to the sum of the positive Lyapunov exponents. Algorithmic aspects of entropy production are briefly explored to illustrate the effect of decoherence from the point of view of the observer. We show that “decoherence strikes twice”, introducing unpredictability into the system and extracting quantum coherence from the observers memory, where it enters as a price for the classicality of his records.
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.
This work formulates classical physics as a stable residual-rigid macroscopic phase of quantum dynamics. The central thesis is not that classical mechanics is an independent fundamental theory, nor that decoherence alone solves the measurement problem, but rather that classical behavior arises conditionally when quantum states belong to a protected small-residual basin around a regular classical stratum. Let ρ denote a quantum state and let Mcl denote a regular classical stratum in the quantum state space. The non-classicality residual is defined as the multi-component map Rcl(ρ) = (Roff(ρ), Rloc(ρ), REhr(ρ), Rdyn(ρ), Rred(ρ)), where the components measure, respectively, off-diagonal coherence in a pointer decomposition, failure of semiclassical localization, Ehrenfest defect, transverse dynamical instability, and lack of redundant environmental records. The associated residual energy is ηcl(ρ) = woff ‖Roff‖² + wloc ‖Rloc‖² + wEhr ‖REhr‖² + wdyn ‖Rdyn‖² + wred ‖Rred‖², with positive weights determined by the macroscopic resolution scale of the physical model. The mathematical structure is residual-quadratic. If R(u*) = 0, η(u) = ‖R(u)‖², and L = DR(u*), then the canonical second variation satisfies D²η(u*)[h,h] = 2 ‖Lh‖², and the associated stability operator is A = L∗L. Applied to the classicality residual, this gives the transverse stability operator Acl = DRcl∗ DRcl. The principal rigidity statement is conditional: if the linearized residual has a coercive transverse spectral gap and if small residual energy cannot occur far from the classical stratum, then ηcl(ρ) ≤ ε* implies dist(ρ,Mcl)² ≤ C ηcl(ρ). Thus classical physics is interpreted as a protected low-residual basin in quantum state space: a stable macroscopic regime in which decoherence, semiclassical localization, Ehrenfest control, dynamical invariance, and environmental redundancy jointly enforce rigidity of classical behavior. The framework also identifies the main failure modes of classicality: closing of the transverse spectral gap, absence of decoherence, loss of localization, failure of redundant records, tunneling, revivals, mesoscopic superpositions, and concentration-like escape channels. Consequently, the classical regime is not asserted as a universal limit of all quantum states, but as a gapped residual-rigid phase valid under explicit stability and anti-concentration hypotheses.
In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.
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.