The macroscopic description of a quantum oscillator with linear passive dissipation is formulated in terms of a master equation for the reduced density matrix. The procedure used is based on the asymptotic methods of nonlinear dynamics, which enables one to obtain an expression for the general term in the weak coupling expansion. For the special example of a charged oscillator interacting with the electromagnetic field, an explicit form of the master equation through third order in this expansion is obtained. This form differs from that generally obtained using the rotating wave approximation in that there is no electromagnetic (Lamb) shift and that an explicit expression is given for the decay rate.
1996 Academic Press, Inc.
I. INTRODUCTION
In the field of quantum optics, a central topic is the subject of master equations. It is discussed in all the modern textbooks [1 5] and review articles [6] in the area, with special emphasis being placed on the coupling of the radiation field to either the two-level atom or the harmonic oscillator. In every case the discussion involves the so-called rotating-wave Hamiltonian. But this approximate Hamiltonian has serious defects [7], the most serious being that as an operator it has no lower bound on its spectrum. This has the consequence that, strictly speaking, there is no equilibrium state and, therefore, no fluctuation-dissipation theorem. On the other hand, to the level of approximation generally used (weak coupling and only through second order) this defect does not lead to unphysical results. However, the approximation does, as we shall see, lead to an incorrect formula for the level shifts.
The master equation is a macroscopic equation for the density matrix of an atomic system (in our case the oscillator) coupled to a thermal reservoir. In the physically interesting case of coupling to the electromagnetic field, the reservoir is the radiation in a blackbody cavity. The master equation corresponds to a reduced description of the approach to equilibrium, in the sense that it does not involve the article no. 0137 362 0003-4916Γ96 18.00