How two spins can thermalize a third spin

We consider thermalization of a microscopic quantum system by interaction with a thermal bath. Our interest is the minimal size the bath can have while still being able to thermalize the system. Within a specific thermalization scheme, we show that a single spin 1/2 can be fully thermalized by interaction with a bath that consists of just two other spins 1/2. The two bath spins are initially in a pure entangled state and the thermalizing interaction is a Heisenberg exchange interaction of the system spin with one of the bath spins. The time-dependent coupling strength has to obey a single integral constraint. We also present a simple generalization of this minimal model in which the bath consists of an arbitrary number of spin-1/2 pairs.

Figure 1

A system spin $A$ in an arbitrary initial state $\rho$ interacts with two bath spins $B$ in a pure, entangled initial state $\psi$. By the interaction the entanglement is swapped to the upper spin and the reduced state of the upper spin becomes a thermal state $\tau$.