Thermalization of quantum systems by finite baths

– We consider a discrete quantum system coupled to a finite bath, which may consist of only one particle, in contrast to the standard baths which usually consist of continua of oscillators, spins, etc. We find that such finite baths may nevertheless equilibrate the system though not necessarily in the way predicted by standard open system techniques. This behavior results regardless of the initial state being correlated or not. Due to the linearity of the Schrödinger equation concepts like ergodicity or mixing are strictly speaking absent in quantum mechanics. Hence the tendency towards equilibrium is not easy to explain. However, except for some ideas [1, 2] the approaches to thermalization in the quantum domain seem to be centered around the idea of a thermostat, i.e., some environmental quantum system (bath, reservoir), enforcing equilibrium upon the considered system. Usually it is assumed that this bath’s classical analogon contains an infinite number of decoupled degrees of freedom. Theories addressing such scenarios are the projection operator techniques (time-convolutionless, Nakajima Zwanzig), the Born approximation (BA) [3] and the path integral technique (Feynman Vernon [4]). The projection operator techniques are exact if all orders of the systembath interaction strength are taken into account which is practically unfeasible. However, assuming weak interactions and accordingly truncating at leading order in the interaction strength (BA) produces an exponential relaxation behavior (c.f. [5, 6]) whenever the bath consists of an continuum of oscillators, spins, etc. The origin of statistical dynamics is routinely based on this scheme, if it breaks down no exponential thermalization can a priori be expected. We find that this scheme breaks down (i.e. the BA produces wrong results) if the bath features a special spectral structure which cannot arise from an uncoupled multitude of subsystems or modes (see below). We refer to this type of bath as finite bath. This holds true even and especially in the limit of weak coupling and arbitrarily dense bath spectra. Nevertheless a statistical relaxation behavior can be induced by finite baths. It simply is not the behavior predicted by the BA. Thus the principles of statistical mechanics in some sense apply below the infinite particle number limit and beyond the BA.