Thermodynamics of trajectories of a quantum harmonic oscillator coupled to $N$ baths

We undertake a thorough analysis of the thermodynamics of the trajectories followed by a quantum harmonic oscillator coupled to $N$ dissipative baths by using an approach to large-deviation theory inspired by phase-space quantum optics. As an illustrative example, we study the archetypal case of a harmonic oscillator coupled to two thermal baths, allowing for a comparison with the analogous classical result. In the low-temperature limit, we find a significant quantum suppression in the rate of work exchanged between the system and each bath. We further show how the presented method is capable of giving analytical results even for the case of a driven harmonic oscillator. Based on that result, we analyze the laser cooling of the motion of a trapped ion or optomechanical system, illustrating how the emission statistics can be controllably altered by the driving force.

Figure 1

Large-deviation function $\theta \left(s\right)$ for different temperatures of the second thermal bath. From top to bottom $\left(0<s\lesssim 0.1\right)$, we have $n_2=20$ (red), $n_2=30$ (green), and $n_2=60$ (blue). The vertical dashed lines highlight the branch points for each curve. $(n_1=10,{\gamma }_1={\gamma }_2/2=0.01$.)

Figure 2

Mean rate of excitations exchanged [i.e., mean net activity $k\left(0\right)]$ between the harmonic oscillator and its reference bath as a function of the bath temperature $T_1$. The classical case is illustrated by the red solid curve and the quantum case by the green dashed curve. $(T_2=2T_1,{\gamma }_1={\gamma }_2/2=0.01$.)

Figure 3

Left: The steady-state number of phonons, $n_{\mathrm{st}}$, in the vibrational state of a trapped ion against the detuning $\mathrm{\Delta }$ between the ion's internal transition and the laser field; note that $n_{\mathrm{st}}$ is independent of the driving. Right: $lnk\left(0\right)$ for the same situation, and for three different driving scenarios; from top to bottom we have (i) periodic driving with $\mathcal{F}=10$ (blue curve), (ii) constant driving with $\mathcal{F}=10$ (green), and (iii) no driving $(\mathcal{F}=0$; red). We have used the parameters $\delta =0,\mathrm{\Omega }/\omega =0.5,\gamma /\omega =1,\phi =\pi /4$.