Relaxation of A Thermally Bathed Harmonic Oscillator: A Study Based on the Group-theoretical Formalism

TL;DR

The paper investigates the open quantum dynamics of a thermally bathed harmonic oscillator by applying the group-theoretical characteristic function (GCF) to the quantum-optical master equation, recasting it as first-order PDEs on the Heisenberg-Weyl group and solving analytically for Gaussian and Fock initial states. It reveals nonmonotonic entropy relaxation in the quantum regime, with temperature- and state-dependent phase transitions: for Gaussian states, a critical occupation $N_c= ext{sinh}^2 r$ (equivalently a critical temperature) separates monotonic and hump-like relaxation; for Fock states, an additional phase with two extrema emerges, characterized by thresholds $N_{eta}=n$ and $N_{eta}=N_c(n)$. A classical comparison shows monotonic entropy growth in the coarse-grained sense and distinct energy-variance behavior, underscoring genuine quantum relaxation features. The methodology and results offer analytic insight for experimental tests (e.g., photon-number distributions) and potential quantum-control applications in dissipative settings, while illustrating the utility of the GCF framework in open quantum systems.

Abstract

Quantum dynamics of a damped harmonic oscillator has been extensively studied since the sixties 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. By taking advantage of the GCF, the master equation can be transformed into a first-order linear partial differential equation that allows 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 nonmonotonic relaxation characteristics. In particular, when the initial state is a Gaussian state (i.e., a squeezed coherent state), it is found that 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. For the temperature higher than the critical value, the entropy will converge to its equilibrium value from below monotonically. However, when the initial state is a Fock state, it is found that there is a new phase additional to the previous case, 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.

Figures (9)

• Figure 1: The entropy as a function of time, $S(t)$, for an initial Gaussian state. The parameter $N_{\beta}$ is fixed to be $N_{\beta}=1$. From bottom to top, the red curve is for $r=0$ and $N_c=0$, the orange curve is for $r=1$ and $N_{c}\approx 1.38$, and the blue curve is for $r=2$ and $N_{c}\approx 13.15$. Note that $N_{c}=\sinh^{2}r$.
• Figure 2: Time function $V_{2}$ for an initial Gaussian state with $\gamma =2$ (a) and $\gamma = 0.2$ (b). In both panels, the red line is for $r=0$ and $\gamma_c=0$, the orange line is for $r=1$ and $\gamma_c\approx 3.626$, and the blue line if for $r=2$ and $\gamma_c\approx 27.29$.
• Figure 3: Entropy $S_{1}(t)$ for the initial Fock state $|1\rangle$. From top to bottom, $N_{\beta}=1.0$ (red), $N_{\beta}=1.2$ (orange), $N_{\beta}=N_c(1)\approx1.366$ (green), and $N_{\beta}=1.5$ (blue). Note that the red and the green are for the two critical cases, in between $S(t)$ curve features two extreme points (e.g., the orange curve).
• Figure 4: The same as Fig. \ref{['St-Fock']} but for $R_1 (t)$ instead.
• Figure 5: Energy variance $\langle(\Delta E)^{2}\rangle$ as a function of time for $N_{\beta}=1.5$. The three curves, from bottom to top, are for $n=1$ (red), $n=2$ (orange), and $n=5$ (blue), respectively.