Quantum-classical correspondence for the equilibrium distributions of two interacting spins

Abstract

We consider the quantum and classical Liouville dynamics of a non-integrable model of two coupled spins. Initially localised quantum states spread exponentially to the system dimension when the classical dynamics are chaotic. The long-time behaviour 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 → ∞) of a non-dissipative model endowed with only a few degrees of freedom.05.45. MT,03.65.TA,03.65.Yz

Tsallis’ deformation parameter quantifies the classical–quantum transition

“…[24,25] and the references therein). In particular, the investigation of ''quantum'' chaotic motion is considered important in this limit.…”

Is the Quantum State Real? An Extended Review of ψ-ontology Theorems

“…If nothing else, this strongly impacts how we are to understand the classical limit of quantum theory (e.g. see [68][69][70][71]). So, whilst the ontic/epistemic question may at first sight seem abstract and philosophical, it does in fact have concrete implications for physics.…”

Information-Theoretic Equilibration: The Appearance of Irreversibility under Complex Quantum Dynamics

“…We estimate the equilibration time using the location of the inflection points of the curves in Fig. 2, and find it scales as O(log(D)), which is characteristic of quantum chaotic systems [22,27,28]. Fig.…”

“…Pure-state fluctuations satisfying the scaling of Theorem 1 were observed already in the two-body, classically chaotic quantum system studied in Refs. [27,28], which motivated the question: was the behaviour of that complex system exceptional, or was it evidence of a universal equilibration behaviour for closed chaotic systems? If the latter, does this effect carry over from chaotic quantum systems to sufficiently complex many-body quantum systems?…”