Decoherence produces coherent states: an explicit proof for harmonic chains

Abstract

We study the behavior of infinite systems of coupled harmonic oscillators as t->infinity, and generalize the Central Limit Theorem (CLT) to show that their reduced Wigner distributions become Gaussian under quite general conditions. This shows that generalized coherent states tend to be produced naturally. A sufficient condition for this to happen is shown to be that the spectral function is analytic and nonlinear. For a rectangular lattice of coupled oscillators, the nonlinearity requirement means that waves must be dispersive, so that localized wave-packets become suppressed. Virtually all harmonic heat-bath models in the literature satisfy this constraint, and we have good reason to believe that coherent states and their generalizations are not merely a useful analytical tool, but that nature is indeed full of them. Standard proofs of the CLT rely heavily on the fact that probability densities are non-negative. Although the CLT generally fails if the probability densities are allowed to take negative values, we show that a CLT does indeed hold for a special class of such functions. We find that, intriguingly, nature has arranged things so that all Wigner functions belong to this class.

Final published version. 17 pages, Plain TeX, no figures. Online at http://astro.berkeley.edu/~max/gaussians.html (faster from the US), from http://www.mpa-garching.mpg.de/~max/gaussians.html (faster from Europe) or from [email protected]

References in corpus (2)

• Apparent wave function collapse caused by scattering
• Steady states of harmonic oscillator chains and shortcomings of harmonic heat baths

Cited by in corpus (30)

• Decoherence, einselection, and the quantum origins of the classical
• The importance of quantum decoherence in brain processes
• Quantum Darwinism
• Einselection and Decoherence from an Information Theory Perspective
• Quantum limit of decoherence: Environment induced superselection of energy eigenstates
• Decoherence, Einselection, and the Existential Interpretation (the Rough Guide)
• Consciousness as a State of Matter
• Decoherent Histories and Hydrodynamic Equations
• Generalized Uncertainty Relations and Long Time Limits for Quantum Brownian Motion Models
• Decoherence, irreversibility and the selection by decoherence of quantum states with definite probabilities
• Quantum Theory of the Classical: Einselection, Envariance, Quantum Darwinism and Extantons
• Decoherence and thermalization dynamics of a quantum oscillator
• Pointer states via Decoherence in a Quantum Measurement
• Predictability sieve, pointer states, and the classicality of quantum trajectories
• Decoherence of Histories and Hydrodynamic Equations for a Linear Oscillator Chain
• Emergence of Quantum-Classical Dynamics in an Open Quantum Environment
• Quantum Prediction Algorithms
• Generalized Coherent States as Preferred States of Open Quantum Systems
• Measurable signatures of quantum mechanics in a classical spacetime
• From Davydov solitons to decoherence-free subspaces: self-consistent propagation of coherent-product states
• Equilibration and thermalization of the dissipative quantum harmonic oscillator in a non-thermal environment
• Information Geometry of Quantum Entangled Gaussian Wave-Packets
• Approximate and Exact Consistency of Histories
• Measurement-induced decoherence and information in double-slit interference
• Adapted Caldeira-Leggett Model
• The Minimal Modal Interpretation of Quantum Theory
• Quantum Uncertainties in the Schmidt Basis Given by Decoherence
• On the Spatial Density Matrix for the Centre of Mass of a one dimensional Perfect Gas
• The Copenhagen Interpretation as an Emergent Phenomenon
• Decoherence-Free Subspaces in Supersymmetric Oscillator Networks