Autonomous rotor heat engine

The triumph of heat engines is their ability to convert the disordered energy of thermal sources into useful mechanical motion. In recent years, much effort has been devoted to generalizing thermodynamic notions to the quantum regime, partly motivated by the promise of surpassing classical heat engines. Here, we instead adopt a bottom-up approach: we propose a realistic autonomous heat engine that can serve as a test bed for quantum effects in the context of thermodynamics. Our model draws inspiration from actual piston engines and is built from closed-system Hamiltonians and weak bath coupling terms. We analytically derive the performance of the engine in the classical regime via a set of nonlinear Langevin equations. In the quantum case, we perform numerical simulations of the master equation. Finally, we perform a dynamic and thermodynamic analysis of the engine's behavior for several parameter regimes in both the classical and quantum case and find that the latter exhibits a consistently lower efficiency due to additional noise.

Figure 1

Autonomous rotor heat engine. A harmonic mode is pushing down a piston attached to a rotor through radiation pressure. Concurrently, the angular position $\phi$ of the rotor (defined relative to the upper turning point) modulates the coupling of the mode to baths at respective occupations ${\overline{n}}_H$ and ${\overline{n}}_C$. This leads to a preferred clockwise motion of the rotor. Note that the specific implementation of the valves depicted here realizes exactly the modulation functions Eq. (5). The terms $\kappa , g$, and $I$ denote the bath thermalization rate, the torque per excitation in the mode, and the moment of inertia, respectively.

Figure 2

Classical simulation of the engine dynamics. Panels (a) and (b) depict, respectively, the angular momentum and the unwrapped angle of the rotor for two exemplary cases of fast ($\kappa =100g$) and slow ($\kappa =g$) thermalization. The lines represent the mean values for a numerical sample over ${10}^5$ trajectories; the shaded areas cover two standard deviations. Red and blue areas on the top indicate the sign of $sin〈\phi \left(t\right)〉$ for $\kappa =100g$, i.e., where the working mode is mostly in contact with the hot and the cold bath. Panel (c) shows the angular two-time correlation function $S_{\phi \phi }(t_1,t_2)$ for $\kappa =100g$. $S_{\phi \phi }(t_1,t_2)=1$ implies that the angular position at time $t_2$ can be predicted from its value at time $t_1$ (and vice versa), while $S_{\phi \phi }(t_1,t_2)=0$ corresponds to uncorrelated angular random variables. All simulations start with the rotor at rest ($L_z\left(0\right)=0, \phi \left(0\right)=\pi /2, I=\hslash /g$), and reservoirs at $({\overline{n}}_H,{\overline{n}}_C)=(1,0)$.

Figure 3

Rate of increase of (a) the average angular momentum $〈L_z〉$ and (b) its variance $\mathrm{\Delta }{L}_z^2$ for the parameters of Fig. 2. The green solid line corresponds to the direct computation of the derivative from the simulated dynamics. The dotted and dashed lines are obtained from the analytical rates given in Eq. (12), respectively, before and after taking the limit of free rotation. As expected, the latter description is only valid once the engine has started and the gain in angular momentum within each cycle is sufficiently small.

Figure 4

Engine cycle in a pressure-volume diagram, where the mode volume grows linearly with the piston position $x$. The solid line represents the ideal cycle running clockwise and producing ${\mathcal{W}}_{\mathrm{cyc}}$, where the cavity has time to thermalize ${\left|a\right|}^2\to \overline{n}\left(\phi \right)$. The color indicates the sign of $sin\varphi$, i.e., the dominant bath coupling. The markers show the mean radiation pressure in 100 bins of $\phi mod2\pi$, sampled from ${10}^6$ trajectories evolved to the time $gt=30$, with the parameters of the fast (outer cycle) and slow (inner cycle) cases chosen as in Fig. 2.

Figure 5

Efficiency of the engine in units of $2g/{\omega }_0$ for the parameters of Fig. 2. The green solid line corresponds to the direct computation of the efficiency from the simulated dynamics, using Eqs. (14) and (17) before the adiabatic elimination and the free-rotation limit. The black line is obtained from the analytical result given in Eq. (18) and is plotted in the regime of operation (solid) and in the saturated regime $〈L_z〉/I\kappa \ge {10}^{-1}$ (dotted). At short times, the efficiency oscillates as the free-rotation limit is not valid, while in the long-time limit the efficiency deviates from Eq. (18) due to the mode not thermalizing fast enough.

Figure 6

Simulation of the engine dynamics for $Ig/\hslash =100k$ (first row) and $Ig/\hslash =k$ (second row). All the results obtained from the quantum model (in yellow) are compared to the classical backaction-free case (in green). Panels (a) and (d) depict the angular momentum while (b) and (c) show the evolution of the angular distribution. Panels (c) and (f) show the angular two-time correlation function $S_{\phi \phi }(t_1,t_2)$, which we symmetrize in the quantum regime to obtain a real quantity (note that the classical version is always real and symmetric). The gray area in (d) shows the uncertainty predicted by a classical model including back-action noise. The engine parameters are ($k=10, \mu =\pi /2, \kappa ={10}^{3/2}\sqrt{\hslash g/I}$).

Figure 7

Performance of the engine in the regime $Ig/\hslash =k$ of Fig. 6 where quantum effects play a role. The quantum model (in yellow) is compared to the classical results averaged over $5\times {10}^6$ trajectories with (black dashed) and without backaction noise (green). (a) Efficiency in units of $2g/{\omega }_0$. (b) Heat input power (top three lines) and work output power (bottom three lines).

Figure 8

(a) Entropies of the quantum engine in the regime $Ig/\hslash =k$ of Fig. 6, with ${\overline{n}}_C={10}^{-3}$. The von Neumann entropy of the engine state in units of $k_B$ (black) is compared to the entropies of the reduced work mode and rotor states (yellow and green) and to the sum of both (black dotted). (b) Rate of entropy change (black dotted) versus entropy flows in and out of the engine (red and blue) in units of $k_B\kappa$ at short times. The black solid line shows the intrinsic entropy production rate Eq. (A5). It is always positive, with a minimum value of $7.7\times {10}^{-4}{k}_B\kappa$ at $t=0.1\sqrt{I/\hslash g}$.