Dynamics of the Density Matrix in Contact with a Thermal Bath and the Quantum Master Equation

We study the structure of the time evolution of the density matrix in contact with a thermal bath in a standard projection operator scheme. In general, the equation of motions of the density matrix with dissipative effects tends to lead any initial state into a steady state. The reduced density matrix of the system in the steady state is obtained by tracing out the degree of freedom of the thermal bath from the density matrix for the equilibrium state of the total system. This reduced matrix is modified by the interaction, and is different from that of the equilibrium of the system alone. In a commonly used equation of motion of the reduced density matrix, we have three terms, i.e., a term of the quantum mechanical evolution of the system density matrix, a term of the non-Markov evolution due to memory effects, and a term depending on the initial total density matrix. We make clear roles of the three terms by explicit calculations of the contributions of the terms to the steady state density matrix. By making use of the role of each term, the properties of the commonly used quantum master equation are examined. For example, if we do not include the imaginary part of the second term, the quantum master equation leads the reduced density matrix into that of the equilibrium of the system without modification. On the other hand, if we include the imaginary part, the steady state density matrix of the system satisfies the master equation up to the second order of the interaction. This property indicates that, in the representation that diagonalizes the system Hamiltonian, the leading order (the second order) terms of the off-diagonal elements of the steady state solution of the master equation agree with those of the density matrix of the equilibrium state.