On thermalization of two-level quantum systems

Sagnik Chakraborty, Prathik Cherian J, Sibasish Ghosh

Published 2016-04-18Version 1

It has always been a difficult issue in Statistical Mechanics to provide a generic interaction Hamiltonian among the microscopic constituents of a macroscopic system which would give rise to equilibration of the system. One tries to evade this problem by incorporating the so-called $H-theorem$, according to which, the (macroscopic) system arrives at equilibrium when its entropy becomes maximum over all the accessible micro states. This approach has become quite useful for thermodynamic calculations using the (thermodynamic) equilibrium states of the system. Nevertheless, the original problem has still not been resolved. In the context of resolving this problem it is important to check the validity of thermodynamic concepts -- known to be valid for macroscopic systems -- in the microscopic world. Quantum thermodynamics is an effort in that direction. As a toy model towards this effort, we look here at the process of 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 certain temperature. A two-qubit interaction Hamiltonian ($H_{th}$, say) is then designed -- with a single-qubit mixed state as the initial state of the bath -- which gives rise to thermalization of the system qubit in the infinite time limit. We then look at the question of equilibration by taking the simplest case of a two-qubit system $A + B$, under some interaction Hamiltonian $H_{int}$ (which is of the form of $H_{th}$) with the individual qubits being under the action of separate heat baths of temperatures $T_A$, and $T_B$. Different equilibrium phases of the two-qubit system are shown to appear -- both the qubits or one of them get cooled down.