The time-convolutionless master equation for driven open system derived in a previous paper, is applied to a simple system of a driven quantal oscillator. This system is assumed to be interacting weakly with its surroundings (heat bath). The operator form of equation is transcribed into the c -number one by the phase-space method. The equations of motion for the expectation values of the boson operator and boson number operator are derived from the c -number equations and are solved for the case of a non-adiabatic oscillator-bath interaction. The results are compared with the ones in the conventional Markoffian approximation for the time-convolution equation in the narrowing limit.
Time-convolutionless forms of the quantal master equations for driven open system are derived from the Liouville equation for the total system under an arbitrary initial condition for the two cases of weak external driving fields and of arbitrary ones. They have forms convenient for the perturbational expansions, and are respectively compared with the time-convolution forms of equations in the lowest Born approximation for the interaction H SB of the system with its surroundings. It is shown that the time-convolutionless forms of equations prevail over the time-convolution ones, and that the conventional Markoffian approximation for the latter in the narrowing limit leads to the incorrect conclusion. The time-convolutionless form of master equation is expanded up to n -th order in powers of the external driving fields in the lowest Born approximation for H SB .