Proposal for an autonomous quantum heat engine

TL;DR

The paper addresses the challenge of realizing an autonomous quantum heat engine powered solely by heat flow in a superconducting circuit. It develops a non-Markovian, quasiclassical framework and a Green's-function reduction to a two-mode dynamics, showing that coherent microwave generation can emerge from the internal heat flow and nonlinear coupling. It provides quantitative analyses of power output, efficiency (η < 1%), and parameter dependencies, and demonstrates an Otto-cycle–like, cosine-modulated operation of the working body within a realistic circuit. The results establish a concrete, experimentally feasible path toward the first autonomous QHE in circuit QED, with potential applications as a cryogenic coherent microwave source powered by thermal gradients and opportunities for optimization via reservoir engineering.

Abstract

We propose and theoretically analyse a superconducting electric circuit which can be used to experimentally realize an autonomous quantum heat engine. Using a quasiclassical, non-Markovian theoretical model, we demonstrate that coherent microwave power generation can emerge solely from the heat flow through the circuit determined by non-linear circuit quantum electrodynamics. The predicted energy generation rate is sufficiently high for experimental observation with contemporary techniques, rendering this work a significant step toward the first experimental realization of an autonomous quantum heat engine based on Otto cycles.

Figures (9)

• Figure 1: Schematics illustrating the dynamics of the proposed quantum heat engine. (a) Idealised description of the quantum Otto cycle. The working-body mode is depicted as a quadratic potential traversing between frequency values $\omega_\mathrm{c}$ and $\omega_\mathrm{h}$. In the photon number representation, the average photon numbers after interacting with the heat reservoirs are given by $n_\mathrm{c}$ and $n_\mathrm{h}$, respectively. The work done by the oscillator $W_\mathrm{out}$ exceeds the work done on the oscillator $W_\mathrm{in}$. Consistently, the heat obtained from the hot reservoir by the system $Q_\mathrm{h}$ exceeds the heat released from the system to the cold reservoir $Q_\mathrm{c}$. (b) Feasible Lorentzian line shapes of the cold (blue colour) and hot (red colour) reservoirs with indicated centre angular frequencies $\omega_\mathrm{c}$ and $\omega_\mathrm{h}$, respectively, along with the frequency-modulated angular frequency $\omega_\mathrm{a}'(t)$, shown by the sinusoidal purple line. The shaded areas depict regions in time and frequency where the effective resonator interacts with the heat reservoirs, with darker colours depicting stronger coupling.
• Figure 2: Lumped-element circuit model of the quantum heat engine. The heat engine circuit consists of four LC resonators denoted by A, B, C, and H as marked by the dotted line boxes. The main capacitances and inductances of the resonators are denoted by $C_i$ and $L_i$, respectively, with $i=\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{h}$. The bare angular frequencies of the resonators are given by $\omega_i=1/\sqrt{L_iC_i}$. The capacitances $C_\mathrm{ca}$ and $C_\mathrm{ha}$ couple resonator A to resonators C and H, respectively. Resonators C and H contain dissipative elements, indicated by the blue and red boxes, maintained at temperatures $T_\mathrm{c}$ and $T_\mathrm{h}$, respectively. The Josephson junctions in the symmetric SQUID loop both have a Josephson energy of $E_\mathrm{J}$. In addition, a small piece of an inductor directly connected to resonator B with the inductance $L_\mathrm{g}$ forms a part of the SQUID loop, through which resonator B is also grounded. We indicate the circuit nodes with green dots, and assign a flux variable $\varphi_i$, with $i=\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{h},\mathrm{s},\mathrm{g}$, for each node. The flux-tunable effective resonator acting as the working body of the quantum heat engine is denoted by the blue colour in the centre, with the time-dependent and flux-tunable frequency $\omega_\mathrm{a}'(t)$. The purple arrows illustrate the total magnetic flux $\Phi(t)=\Phi_\mathrm{ext}+\varphi_\mathrm{g}(t)$ of the SQUID loop, with $\Phi_\mathrm{ext}$ as the static external magnetic flux. The turquoise quarter wavelength cosines illustrate the field mode $\varphi_\mathrm{b}$ in resonator B.
• Figure 3: Absolute value of the imaginary part of the time-independent Green's function $G_0(\omega)$ of the SQUID mode as a function of angular frequency at different time-independent values of $\phi_\mathrm{b}$. The red and blue vertical dashed lines represent the bare frequencies of hot and cold filters, respectively, while the black dashed line shows the flux tunable angular frequency $\omega_\mathrm{a}'$, defined in the main text. We demonstrate the dependence of the Green's function on $\phi_\mathrm{b}$ by using the values $\phi_\mathrm{b}\in\{0,\pm 0.27\}$. The parameters used for these results are listed in Table \ref{['tab:table1']}.
• Figure 4: (a) Total dissipation rate $\Gamma_\mathrm{tot}(A_\mathrm{b},\theta_\mathrm{b})$ as a function of field amplitude at different intrinsic quality factors $Q_\mathrm{b}$. The stable and unstable stationary points are marked by the cyan and magenta diamonds, respectively. The dashed line segment on the $Q_\mathrm{b}\rightarrow\infty$ curve denotes the region where stable points essentially occur in the primary dissipation valley for some $Q_\mathrm{b}$. (b) Output power as a function of the intrinsic quality factor, corresponding to the dashed line segment in panel (a). The star marks the location of the maximum value of output power. The parameters used for computing the results are found in Table \ref{['tab:table1']}.
• Figure 5: (a) Total dissipation rate $\Gamma_\mathrm{tot}(A_\mathrm{b},\theta_\mathrm{b})$ as a function of the field amplitude $A_\mathrm{b}$ at various indicated hot-bath temperatures $T_\mathrm{h}$ and for an intrinsic dissipation rate of $\gamma_\mathrm{b}=0$. (b) Output power as a function of $Q_\mathrm{b}$, determined from the stable points of the primary dissipation valley. The black stars mark the maximum values for each temperature. Note that the three lowest-temperature curves are cut out because we limit $Q_\mathrm{b}$ to $100000$. The data is presented on a logarithmic horizontal axis. (c) Maximum attainable output power as a function of $T_\mathrm{h}$. (d) Efficiency of the heat engine corresponding to the data in panel (c). (e) Lowest intrinsic quality factor of resonator B required for the heat engine to start spontaneously ($Q_\mathrm{init}$ marked by the plus signs) and the lowest possible $Q_\mathrm{b}$ enabling self-sustained oscillations ($Q_\mathrm{stop}$ marked by the crosses) as functions of temperature. The dotted vertical lines in panels (c), (d), and (e) indicate the value of $T_\mathrm{h}$ given in Table \ref{['tab:table1']}. In panels (a) and (b), we only show every other curve, as compared with the markers in panels (c) and (d), for the sake of clarity. The values in the legend are in units of kelvin, and the parameters used for computing these results, not given here, are found in Table \ref{['tab:table1']}.