dorsal/arxiv
View SchemaQuantum-Classical Correspondence for the Equilibrium Distributions of Two Interacting Spins
| Authors | J. Emerson, L. E. Ballentine |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0103050 |
| URL | https://arxiv.org/abs/quant-ph/0103050 |
| DOI | 10.1103/PhysRevE.64.026217 |
| Journal | Phys. Rev. E 64, 026217 (2001). |
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 $(\hbar/{\mathcal J})^{1/2}$, where ${\mathcal J}$ 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 (${\mathcal J}/\hbar \to \infty$) of a non-dissipative model endowed with only a few degrees of freedom.
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"abstract": "We consider the quantum and classical Liouville dynamics of a non-integrable\nmodel of two coupled spins. Initially localised quantum states spread\nexponentially to the system dimension when the classical dynamics are chaotic.\nThe long-time behaviour of the quantum probability distributions and, in\nparticular, the parameter-dependent rates of relaxation to the equilibrium\nstate are surprisingly well approximated by the classical Liouville mechanics\neven for small quantum numbers. As the accessible classical phase space becomes\npredominantly chaotic, the classical and quantum probability equilibrium\nconfigurations approach the microcanonical distribution, although the quantum\nequilibrium distributions exhibit characteristic `minimum\u0027 fluctuations away\nfrom the microcanonical state. The magnitudes of the quantum-classical\ndifferences arising from the equilibrium quantum fluctuations are studied for\nboth pure and mixed (dynamically entangled) quantum states. In both cases the\nstandard deviation of these fluctuations decreases as $(\\hbar/{\\mathcal\nJ})^{1/2}$, where ${\\mathcal J}$ is a measure of the system size. In\nconclusion, under a variety of conditions the differences between quantum and\nclassical Liouville mechanics are shown to become vanishingly small in the\nclassical limit (${\\mathcal J}/\\hbar \\to \\infty$) of a non-dissipative model\nendowed with only a few degrees of freedom.",
"arxiv_id": "quant-ph/0103050",
"authors": [
"J. Emerson",
"L. E. Ballentine"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevE.64.026217",
"journal_ref": "Phys. Rev. E 64, 026217 (2001).",
"title": "Quantum-Classical Correspondence for the Equilibrium Distributions of Two Interacting Spins",
"url": "https://arxiv.org/abs/quant-ph/0103050"
},
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