On thermalization of two-level quantum systems

Providing the microscopic behavior of a thermalization process has always been an intriguing issue. There are several models of thermalization, which often requires interaction of the system under consideration with the microscopic constituents of the macroscopic heat bath. With an aim to simulate such a thermalization process, here we look at the thermalization of a two-level quantum system under the action of a Markovian master equation corresponding to memory-less action of a heat bath, kept at a certain temperature, using a single-qubit ancilla. A two-qubit interaction Hamiltonian ($H_{th}$, say) is then designed -- with a single-qubit thermal state as the initial state of the ancilla -- which gives rise to thermalization of the system qubit in the infinite time limit. Further, we study the general form of Hamiltonian, of which ours is a special case, and look for the conditions for thermalization to occur. We also derive a Lindblad-like non-Markovian master equation for the system dynamics under the general form of system-ancilla Hamiltonian.