Collective effects on the performance and stability of quantum heat engines

Recent predictions for quantum-mechanical enhancements in the operation of small heat engines have raised renewed interest in their study both from a fundamental perspective and in view of applications. One essential question is whether collective effects may help to carry enhancements over larger scales, when increasing the number of systems composing the working substance of the engine. Such enhancements may consider not only power and efficiency, that is, its performance, but, additionally, its constancy, that is, the stability of the engine with respect to unavoidable environmental fluctuations. We explore this issue by introducing a many-body quantum heat engine model composed by spin pairs working in continuous operation. We study how power, efficiency, and constancy scale with the number of spins composing the engine and introduce a well-defined macroscopic limit where analytical expressions are obtained. Our results predict power enhancements, in both finite-size and macroscopic cases, for a broad range of system parameters and temperatures, without compromising the engine efficiency, accompanied by coherence-enhanced constancy for finite sizes. We discuss these quantities in connection to thermodynamic uncertainty relations.

Figure 1

Schematic representation of the heat engine composed of two ensembles of spins (blue and red spins) with different energy spacing, collectively dissipating in respective thermal baths at different temperatures ${\beta }_1$ and ${\beta }_2$, and subjected to a collective coherent drive $V\left(t\right)$ mediated by a classical field. A heat current from hot to cold baths allows extracting work in the external field (which acts as an ideal battery). Analogously the heat current can be inverted, refrigerating the cold bath, by consuming external work. We explore the power, efficiency, and constancy of the engine with the number of spins and their enhancements with respect to an equivalent number of two-spin engines working in parallel

Figure 2

(a) Power output for a single spin-pair thermal machine, ${\mathcal{P}}_1$ as a function of energy spacing difference $\mathrm{\Delta }E=E_2-E_1$ and average temperature $\overline{T}\equiv (T_1+T_2)/2$. The black thick curve, $\mathrm{\Delta }E=\mathrm{\Delta }{E}_{*}$, corresponds to zero power where all average currents vanish and highlights the boundary between heat engine (reddish area) and refrigerator (bluish area) modes of operation. (b) Example of fittings of the power ratio ${\mathcal{P}}_N/{\mathcal{P}}_1\propto N^{\alpha }$ as a power law with respect to the number $N$ of spin pairs in the machine, for sufficiently large $N\simeq 20$. Colored circles correspond to simulations for different values of $k_B\overline{T}$ (see legend) and $\mathrm{\Delta }E=1.25E_1$, while red dashed lines are the result of a linear regression. (c) Results for the fitted power-law exponent $\alpha$ as a function of $k_B\overline{T}$ and $\mathrm{\Delta }E$. Values of $\alpha >1$ in extensive regions of the parameter space indicate a superlinear scaling of the collective power ${\mathcal{P}}_N$ (see color legend on the right). The white thick line corresponds again to $\mathrm{\Delta }E=\mathrm{\Delta }{E}_{*}$ in Eq. (23). In all plots energetic quantities are given in units of $E_1$ and the power output in units of ${\mathrm{\Gamma }}_0$. Other parameters are $E_1=100{\omega }_0,{\mathrm{\Gamma }}_0={\omega }_0,k_B\mathrm{\Delta }T=T_2-T_1=100{\omega }_0$.

Figure 3

Collective engine constancy ${\mathcal{C}}_N$ as a function of the number of spin pairs $N$ for different values of the average temperature in the range $k_B\overline{T}=[10E_1,13E_1]$ from bottom (blue) to top (gray) curves and TUR bound (black dotted line). Inset: Augmentation of the region close to the TUR bound. Other parameters are ${\omega }_0=0.006E_1$, ${\mathrm{\Gamma }}_0=0.001E_1$, $\mathrm{\Delta }E=9E_1$, and $\mathrm{\Delta }T=10E_1$. The constancy of the engine ${\mathcal{C}}_N$ decreases with the number of spin pairs $N$, which indicates that the increasing in the power output with the size is accompanied by an even faster growing of the power fluctuations.

Figure 4

Purely dissipative case (${\omega }_0=0$). (a) We show the steady-state magnetization computed analytically from Eq. (33) and those obtained from the dynamical equations of motion (38) and (39). Both methods agree with great precision. (b) We show the Lindbladian gap for finite system sizes, as well the Jacobian gap in the macroscopic limit. (c) We show the steady-state magnetization for varying bath temperatures and system sizes. (d) We show the transient system size $N^{*}$ obtained from the Lindbladian gap and from the steady-state magnetization, using $\epsilon ={10}^{-10}$ for their computation. For sufficient small system size, $N\ll \beta E$, the Lindbladian gap and steady-state magnetization are practically indistinguishable from the zero temperature case until they reach the transient system size $N^{*}$.

Figure 5

Jacobian gap ${\lambda }_{\mathrm{gap}}^J$ for different energy splittings $\mathrm{\Delta }E$ and varying average temperatures $\overline{T}$, for a system with coupling parameters ${\omega }_0=0.006,{\mathrm{\Gamma }}_0=0.001,E_1=1,\mathrm{\Delta }T=T_2-T_1=20$. The dashed vertical lines highlight the boundary between heat engine (left part) and refrigerator (right part) modes of operation in the system, for the different fixed energy splitting difference. The relaxation to the steady state is faster for higher average temperatures seeing that the Jacobian gap increases linearly with the average temperature, $\mathrm{Re}\left({\lambda }_1^J\right)\sim c\overline{T}$. In the heat engine mode of operation the slope of the growth is independent of the energy spacing difference, which is not true for the refrigerator mode of operation.

Figure 6

Power, collective gain, and constancy of the collective heat engine in the macroscopic limit, with fixed parameters ${\omega }_0=0.006,{\mathrm{\Gamma }}_0=0.001,E_1=1,\mathrm{\Delta }T=T_2-T_1=20$, as a function of the energy splitting difference $\mathrm{\Delta }E$ and average temperature $\overline{T}=(T_1+T_2)/2$. (a) Output power in units of ${\mathrm{\Gamma }}_0$ for the collective engine, ${\lim }_{N\to \infty }({\mathcal{P}}_N/N{\mathrm{\Gamma }}_0)$ in the steady state. The black curve highlights the boundary between heat engine (reddish area) and refrigerator (bluish area) modes of operation, similar to the single spin-pair case. (b) Log-ratio of the collective power output and $N$ independent single-pair engines, ${\lim }_{N\to \infty }{log}_{10}({\mathcal{P}}_N/N{\mathcal{P}}_1)$. The black dashed curve separates the region with collective gain (${\mathcal{P}}_N/N{\mathcal{P}}_1>1$) from the one with collective loss (${\mathcal{P}}_N/N{\mathcal{P}}_1<1$). (c) Constancy ${\mathcal{C}}_N$ of the collective engine working in the steady state, along the same set of system parameters. While the constancy lies below the classical TUR limit, ${\mathcal{C}}_N\le 1$, we observe a high-stability region (${\mathcal{C}}_N\simeq 0.7$) overlapping with high gains in the collective power, ${\lim }_{N\to \infty }({\mathcal{P}}_N/N{\mathcal{P}}_1)\simeq {10}^3$.

Figure 7

Constancy ${\mathcal{C}}_N$ for increasing system sizes and in the macroscopic limit (inset panel). We set system parameters $E_1=1.0,E_2=10.0,{\mathrm{\Gamma }}_0=0.001,T_1=2$, and $T_2=25$, for varying ${\omega }_0$ and considering the system in the high-temperature regime. There are regions of parameters with quantum enhancements of the constancy, ${\mathcal{C}}_N\ge 1$, for finite sizes. These enhancements are lost in the macroscopic limit, where the constancy is always below the classical limit.

Figure 8

Mutual information in the macroscopic limit. We show the mutual information $I_N$ between two spins, each belonging to a different collective spin, for fixed system parameters ${\omega }_0=0.006,{\mathrm{\Gamma }}_0=0.001,E_1=1,\mathrm{\Delta }T=T_2-T_1=20$. For this range of parameter, the mutual information between the both spins is qualitatively associated with the power output of the system.

Figure 9

We show in the upper panels the expectation values for the magnetization $\langle {\widehat{m}}^z\rangle$ and its quadrature $\langle {\left({\widehat{m}}^z\right)}^2\rangle$, for different system sizes and temperatures. In the bottom panels we show the exact dynamics for finite system sizes and those from the effective dynamical equations of motion (38) and (39). We show the dynamics of the magnetization $\langle {\widehat{m}}^z\rangle$ for a fixed temperature (bottom-left panel) $\beta E=0.1$ and (bottom-right panel) $\beta E=10$. The results show an agreement regarding the macroscopic limit and the finite size exact diagonalization trends for the system.

Figure 10

We show the dependence of the power output enhancements with the energy splinting $\mathrm{\Delta }E$, for fixed parameters ${\omega }_0=0.006,{\mathrm{\Gamma }}_0=0.001,E_1=1,{\beta }_1=50,{\beta }_2={10}^{-2}$, both for finite system sizes and in the macroscopic limit. The collective gain of the power output has an exponential growth with the energy spacing difference.