Heat, work, and fluctuations in a driven quantum resonator

TL;DR

This study analyzes the thermodynamics of a driven open quantum harmonic oscillator serving as the working fluid in a nanoscale heat engine. By modeling the system with a time-dependent frequency and a Lindblad bath, it characterizes both average energetics (work and heat) and the full fluctuations through photon-counting statistics, including the cumulants and the complete photon-transfer distribution. In linear response, explicit relations connect changes in temperature, power, and heat to the drive, while beyond linear response the work-heat interplay and non-Gaussian fluctuations become prominent, captured by the first few cumulants and the full distribution. The work provides quantitative guidance for designing quantum heat engines and refrigerators based on resonator platforms, with potential extensions to Otto/Stirling cycles and multi-reservoir configurations in various quantum technologies.

Abstract

A central building block of a heat engine is the working fluid, which mediates the conversion of heat into work. In nanoscale heat engines, the working fluid can be a quantum system whose behavior and dynamics are non-classical. A particularly versatile realization is a quantum resonator, which allows for precise control and coupling to thermal reservoirs, making it an ideal platform for exploring quantum thermodynamic processes. Here, we investigate the thermodynamic properties of a driven quantum resonator whose temperature is controlled by modulating its natural frequency. We evaluate the work performed by the external drive and the resulting heat flow between the resonator and its environment, both within linear response and beyond. To further elucidate these processes, we determine the full distribution of photon exchanges between the resonator and its environment, characterized by its first few cumulants. Our results provide quantitative insights into the interplay between heat, work, and fluctuations, and may help in designing future heat engines.

Figures (7)

• Figure 1: Driven quantum resonator. (a) An external drive performs work on the resonator by changing its natural frequency, $\omega_0(t)$, which leads to a changing temperature, $T(t)$. The temperature of the resonator also changes because heat is exchanged with a thermal reservoir at the temperature $T_e$. The coupling to the reservoir is denoted by $\gamma$. (b) Square-wave drive of the resonator frequency. (c) Corresponding temperature of the resonator. Parameters are $k_B T_e = 1.5 \hbar \bar{\omega}_0$ in terms of the resonator frequency without the drive. The period and amplitude of the drive are $\tau = 2 \pi /(0.1 \bar{\omega}_0)$ and $\Delta\omega_0=0.1\bar{\omega}_0$.
• Figure 2: External drive and time-dependent temperature. (a,b,c) Square-wave, sawtooth, and harmonic drive of the resonator frequency. (d,e,f) Time-dependent temperature of the resonator for three different couplings to the environment. For the smallest coupling, the temperature follows the drive according to Eq. (\ref{['eq:Wtemp']}). The reservoir temperature is $k_B T_e = 1.5 \hbar \bar{\omega}_0$ in terms of the resonator frequency without the drive. The period and amplitude of the drive are $\tau = 2 \pi /(0.1 \bar{\omega}_0)$ and $\Delta\omega_0=0.7\bar{\omega}_0$.
• Figure 3: Power and heat. (a,b,c) Power for the threes drives in Fig. \ref{['fig:fig2']}. (d,e,f) Heat currents for the three drives. The value of the reservoir coupling is $\gamma=0.05\bar{\omega}_0$, while the other parameters are the same as in Fig. \ref{['fig:fig2']}. We have also defined $P_0=J_0=\hbar\bar{\omega}_0/\tau$.
• Figure 4: Linear response. (a,b,c) Temperature, power, and heat for the harmonic drive $\omega_0(t)=\bar{\omega}_0+\Delta\omega_0\sin(\Omega t)$ with $\Omega = 0.1\bar{\omega}_0$ and $\Delta \omega_0 = 0.1 \bar{\omega}_0, 0.5 \bar{\omega}_0$. The other parameters are $\gamma = 0.1 \bar{\omega}_0$ and $k_B T_e = 1.5 \hbar \bar{\omega}_0$. Solid lines show numerical results, while dashed lines are the linear-response expressions in Eqs. (\ref{['eq:TLR']}), (\ref{['eq:PLR']}), and (\ref{['eq:JLR']}). (d,e,f) Same quantities in a parametric plot.
• Figure 5: Second, third, and fourth cumulants. (a,b,c) The solid lines show numerical results for a harmonic drive, while the dashed lines are the cumulants without the drive. The drive reads $\omega_0(t)=\bar{\omega}_0+\Delta\omega_0\sin(\Omega t)$ with $\Omega = 0.1\bar{\omega}_0$ and $\Delta \omega_0 = 0.6 \bar{\omega}_0$. The other parameters are $\gamma = 0.1 \omega_0$ and $k_B T_e = 4 \hbar \bar{\omega}_0$. The vertical lines in panel (a) correspond to the distributions in Fig. \ref{['fig:FCS']}.