Nonequilibrium Many-Body Quantum Engine Driven by Time-Translation Symmetry Breaking

Quantum many-body systems out of equilibrium can host intriguing phenomena such as transitions to exotic dynamical states. Although this emergent behaviour can be observed in experiments, its potential for technological applications is largely unexplored. Here, we investigate the impact of collective effects on quantum engines that extract mechanical work from a many-body system. Using an optomechanical cavity setup with an interacting atomic gas as a working fluid, we demonstrate theoretically that such engines produce work under periodic driving. The stationary cycle of the working fluid features nonequilibrium phase transitions, resulting in abrupt changes of the work output. Remarkably, we find that our many-body quantum engine operates even without periodic driving. This phenomenon occurs when its working fluid enters a phase that breaks continuous time-translation symmetry: The emergent time-crystalline phase can sustain the motion of a load generating mechanical work. Our findings pave the way for designing novel nonequilibrium quantum machines.

Figure 1

Cavity-atom quantum engine. The atomic working fluid is held in an optical cavity with one movable mirror (mass $m$) attached to a spring (characteristic frequency $\omega$). The cavity length $L=L_0+x$ decomposes into an equilibrium length $L_0$ and a small deviation $x$. The motion of the mirror is damped by mechanical friction (proportional to the coefficient $\gamma$) and driven by thermal fluctuations and the radiation pressure inside the cavity. Each atom is described as a two-level system with ground state $|g⟩$, excited state $|e⟩$, and energy splitting ${\omega }_{\text{at}}$. Excitations are generated and destroyed through interactions with a light mode in the cavity and with the driving laser (Rabi frequency $\mathrm{\Omega }$ and detuning $\mathrm{\Delta }$). The cavity loss rate is $\kappa$.

Figure 2

Periodic driving. (a) Periodic motion of the mirror driven by the many-body engine. The average velocity of the mirror, ${\overline{v}}_t$, is given in units of ${\text{v}}_0=({\omega }_{\mathrm{cav}}^0/\omega )(\hslash D_0/m)$, the average position ${\overline{x}}_t$ in units of ${\text{v}}_0/\omega$, and the force in units of $\hslash {\omega }_{\mathrm{cav}}^0{D}_0$. Time is given in units of ${\omega }^{-1}$. This representative cycle is obtained for ${\omega }_{\text{at}}-{\omega }_{\mathrm{cav}}^0=0.1\omega$, ${\mathrm{\Delta }}_{\max }=2\omega$, and ${\mathrm{\Delta }}_{\min }=\kappa =g=\omega$. The velocity ${\overline{v}}_t$, averaged over noise realizations, is not always zero during the cycle, proving that the engine constantly delivers energy to the mirror through the force $f_t$. (b) Power output, in units of $m{\text{v}}_0^2\omega$, as a function of $\mathrm{\Omega }/\omega$ and ${\mathrm{\Delta }}_{\min }/\omega$ with fixed ${\mathrm{\Delta }}_{\max }-{\mathrm{\Delta }}_{\min }=\omega$. In the insets, we show the mean velocity and the force for ${\mathrm{\Delta }}_{\min }=0.2\omega$ and two slightly different values of the Rabi frequency, $\mathrm{\Omega }/\omega =1.55$ and 1.56, for which the power output differs substantially. The force profile changes abruptly from an oscillatory pattern to a two-plateau-like shape indicating a nonequilibrium phase transition. Numerical results have been obtained by simulating the dynamics of the mirror for sufficiently long times, such that the system has converged to its asymptotic cycle.

Figure 3

Time-crystal quantum engine. (a) Mean velocity of the mirror in units of ${\text{v}}_0=({\omega }_{\mathrm{cav}}^0/\omega )(\hslash D_0/m)$ as a function of time (in units of ${\omega }^{-1}$) for two different photon-loss rates. For $\kappa /\omega =0.5$, the mirror comes to rest, while sustained oscillations emerge for $\kappa /\omega =1.5$. For this plot, we have set $\mathrm{\Delta }=0$, ${\omega }_{\text{at}}-{\omega }_{\mathrm{cav}}^0=0.1\omega$, and $\mathrm{\Omega }=g=\omega$. (b) Average power output, in units of $m{\text{v}}_0^2\omega$, as a function of the photon-loss rate $\kappa$ and the Rabi frequency $\mathrm{\Omega }$, both in units of $\omega$. The remaining parameters are the same as before. Along the dashed line $\kappa \mathrm{\Omega }\sim 1$, a phase transition occurs, where the average power jumps to a finite value as the working fluid spontaneously forms a time crystal. The scale has been truncated at 0.2, but significantly larger values for the work output ($P_{\mathrm{av}}>2$) are found. The maximum value, for the chosen parameters, is given by $P_{\mathrm{av}}\sim 2.8$.