  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.
