Relaxation of a thermally bathed harmonic oscillator: A study based on the quantum group-theoretical formalism

The quantum dynamics of a damped harmonic oscillator has been extensively studied since the 1960s of the last century. Here, with a distinct tool termed the “group-theoretical characteristic function (GCF)”, we investigate analytically how a harmonic oscillator immersed in a thermal environment would relax to its equilibrium state. We assume that the oscillator is at a pure state initially and its evolution is governed by a well-known quantum-optical master equation. Taking advantage of the GCF, the master equation can be transformed into a first-order linear partial differential equation, allowing us to write down its solution explicitly. Based on the solution, it is found that, in clear contrast with the monotonic relaxation process of its classical counterpart, the quantum oscillator may demonstrate some intriguing non-monotonic relaxation characteristics. In particular, when the initial state is a Gaussian state (i.e., a squeezed coherent state), there is a critical value of the environmental temperature below which the entropy will first increase to reach its maximum value, then turn down and converge to its equilibrium value from above. Conversely, when the temperature exceeds the critical value, the entropy converges monotonically to its equilibrium value from below. In contrast, for an initial Fock state, there are two critical temperatures instead and, in between, a new additional phase emerges, where the time curve of entropy features two extreme points. Namely, the entropy will increase to reach its maximum first, then turn down to reach its minimum, from where it begins to increase and converges to the equilibrium value eventually. Other related issues are discussed as well.