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Rigorous derivations of the approach of individual elements of large isolatedsystems to a state of thermal equilibrium, starting from arbitrary initialstates, are exceedingly rare. This is particularly true for quantum mechanicalsystems. We demonstrate here how, through a mechanism of repeated scattering,an approach to equilibrium of this type actually occurs in a specific quantumsystem, one that can be viewed as a natural quantum analog of severalpreviously studied classical models. In particular, we consider an optical modepassing through a reservoir composed of a large number ofsequentially-encountered modes of the same frequency, each of which itinteracts with through a beam splitter. We then analyze the dependence of theasymptotic state of this mode on the assumed stationary common initial state$\sigma$ of the reservoir modes and on the transmittance $\tau=\cos\lambda$ ofthe 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 notthemselves initially thermalized. We show in addition that, when the initialstates are pure, the asymptotic state of the optical mode is maximallyentangled with the reservoir and exhibits less nonclassicality than the stateof the reservoir modes.

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