Rigorous results on approach to thermal equilibrium, entanglement, and nonclassicality of an optical quantum field mode scattering from the elements of a non-equilibrium quantum reservoir

Stephan De Bievre1, Marco Merkli2, and Paul E. Parris3

1Univ. Lille, CNRS, Inria, UMR 8524, Laboratoire P. Painlevé, F-59000 Lille, France
2Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7, Canada
3Missouri University of Science and Technology, Rolla, Missouri, 65409, USA

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Abstract

Rigorous derivations of the approach of individual elements of large isolated systems to a state of thermal equilibrium, starting from arbitrary initial states, are exceedingly rare. This is particularly true for quantum mechanical systems. We demonstrate here how, through a mechanism of repeated scattering, an approach to equilibrium of this type actually occurs in a specific quantum system, one that can be viewed as a natural quantum analog of several previously studied classical models. In particular, we consider an optical mode passing through a reservoir composed of a large number of sequentially-encountered modes of the same frequency, each of which it interacts with through a beam splitter. We then analyze the dependence of the asymptotic state of this mode on the assumed stationary common initial state $\sigma$ of the reservoir modes and on the transmittance $\tau=\cos\lambda$ of the beam splitters. These results allow us to establish that at small $\lambda$ such a mode will, starting from an arbitrary initial system state $\rho$, approach a state of thermal equilibrium even when the reservoir modes are not themselves initially thermalized. We show in addition that, when the initial states are pure, the asymptotic state of the optical mode is maximally entangled with the reservoir and exhibits less nonclassicality than the state of the reservoir modes.

It is a tenet of thermodynamics that a macroscopic system, starting from almost any initial state, will inevitably evolve towards a well-defined state of thermodynamic equilibrium. Perhaps surprisingly, rigorous mathematical derivations of how, exactly, this approach to equilibrium comes about as a consequence of the underlying dynamics of the individual elements making up the system are exceedingly rare. This rarity is particularly acute for large quantum mechanical systems. In this paper, the authors demonstrate through a rigourous mathematical analysis, how an approach to equilibrium of this type actually occurs in a specific quantum system that can be experimentally realized using an array of simple optical elements. In particular, they consider an optical (e.g. laser) mode of fixed frequency (the system mode) that sequentially passes through a linear sequence of beam splitters all with the same transmittance. Each beam splitter is assumed to be illuminated through a second input port by identically prepared “reservoir” modes of the same frequency. The beam splitters thus allow the system mode to interact and exchange energy with the reservoir modes. The authors’ analysis shows that when the transmittance of the beam splitters is high, the initial optical system mode will, after passing through many beam splitters, approach a state of thermal equilibrium even when the reservoir modes are not themselves initially thermalized.

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