Quantum Statistical Enhancement of the Collective Performance of Multiple Bosonic Engines

We consider an ensemble of indistinguishable quantum machines and show that quantum statistical effects can give rise to a genuine quantum enhancement of the collective thermodynamic performance. When multiple indistinguishable bosonic work resources are coupled to an external system, the internal energy change of the external system exhibits an enhancement arising from permutation symmetry in the ensemble, which is absent when the latter consists of distinguishable work resources.

Figure 1

Schematic picture of the setup. Multiple work resources collectively denoted by $R$ deliver energy $\mathrm{\Delta }U$ to an external system $S$ through the coupling Hamiltonian $H_C$. If the work resources are $N$ quantum heat engines $E_1,\dots ,E_N$, all the engines and the heat baths collectively denoted by $E$ and $B$, respectively, are included in $R$.

Figure 2

Quantum-enhanced performance of multiple identical engines in the perturbative impulse coupling. (a) Ratio $\mathcal{E}\equiv {⟨\mathrm{\Delta }U⟩}_N^{\text{indist}}/{⟨\mathrm{\Delta }U⟩}_N^{\mathrm{dist}}$ for an outcoupling with an impulse of strength $g=0.01$ to an external system. The atomic energy gap is driven with a linear speed of $v=0.1\mathrm{\Omega }(0{)}^2$ over a total protocol time of $T=20/\mathrm{\Omega }(0)$. The impulse kick occurs at $t_1=0.35T/2$, and ${\theta }_{t_1}$ is tuned by changing $\mathrm{\Delta }=\{4.2\mathrm{\Omega }(0),1.4\mathrm{\Omega }(0),0\}$ (different curves). The bath temperatures are given by ${\beta }_c{E}_0=2$ and ${\beta }_h{E}_{T/2}=1/4$. Solid lines are from analytical expressions derived in the perturbative limit, and dots are from a numerical calculation with the external system given by a HO of frequency $\omega =2\pi \times 0.05/T$. (b) ${⟨\mathrm{\Delta }U⟩}_N^{\text{indist}}$ produced by indistinguishable particles when $\mathrm{\Delta }=0$ and ${\beta }_h{E}_{T/2}=1/4$ as a function of $N$ and ${\beta }_c{E}_0$ for impulse-type coupling and other parameters as in (a). Regions with linear scaling in $N$ of the contour gradients correspond to quadratic scaling of ${⟨\mathrm{\Delta }U⟩}_N$ with $N$.

Figure 3

Quantum performance of multiple identical engines with continuous nonperturbative coupling. (a) ${⟨\mathrm{\Delta }U⟩}_N^{\text{indist}}$ done by $N$ indistinguishable engines outcoupled via a continuous nonperturbative coupling. A quadratic scaling is shown for small $N$. (b) Scaling of the enhancement $\mathcal{E}$. Engine parameters are $v=0.1\mathrm{\Omega }(0{)}^2$, $T=20/\mathrm{\Omega }(0)$, $\mathrm{\Delta }=0$, and $\mathrm{\Omega }(0)=1$. The coupling $g_C(t)$, to the external HO system of frequency $\omega =2\pi \times 0.05/T$, is chosen with $g=0.5$, ${\delta }_t=0.9$, $\alpha =2142/T$, $t_{\mathrm{on}}=(1-{\delta }_t)T/4$, and $t_{\mathrm{off}}=t_{\mathrm{on}}+{\delta }_{t}T/2$.