Engineering a heat engine purely driven by quantum coherence

The question of whether quantum coherence is a resource beneficial or detrimental to the performance of quantum heat engines has been thoroughly studied but remains undecided. To isolate the contribution of coherence, we analyze the performance of a purely coherence-driven quantum heat engine, a device that does not include any heat flow during the thermodynamic cycle. The engine is powered by the coherence of a multiqubit system, where each qubit is charged via interaction with a coherence bath using the Jaynes-Cummings model. We demonstrate that optimal coherence charging and hence extractable work is achieved when the coherence bath has an intermediate degree of coherence. In our model, the extractable work is maximized when four copies of the charged qubits are used. Meanwhile, the efficiency of the engine, given by the extractable work per input coherence flow, is optimized by avoiding the coherence being stored in the system-bath correlations that is inaccessible to work. We numerically find that the highest efficiency is obtained for slightly lower temperatures and weaker system-bath coupling than those for optimal coherence charging.

Figure 1

Schematic showing a coherence-driven heat engine protocol using the sequential coherence charging of individual qubits at uniform inverse temperature $\beta$. Each qubit is charged sequentially to a state ${\rho }_S$ with coherence by input coherence flow $Q_{\mathrm{coh}}^{\left(\mathrm{in}\right)}={\sum }_k{Q}_{\mathrm{coh},k}^{\left(\mathrm{in}\right)}$, where $Q_{\mathrm{coh},k}^{\left(\mathrm{in}\right)}=-k_{B}T\mathrm{\Delta }{C}_{\mathrm{ext}}\left({\rho }_B\right)$ corresponds to the change of coherence of the coherence bath $|{\gamma }_B\rangle$ used for the $k\mathrm{th}$ qubit weighted by its temperature (see the main text for details). The coherence flow is partially stored in the total system via internal coherence and external coherence but is also stored in the system-bath correlations. While only the internal coherence $C_{\mathrm{int}}\left({\rho }_S^{\otimes N}\right)$ can be extracted as coherent work $W_{\mathrm{coh}}=k_{B}T{C}_{\mathrm{int}}\left({\rho }_S^{\otimes N}\right)$, the external coherence $C_{\mathrm{ext}}\left({\rho }_S^{\otimes N}\right)$ is locked and erased from the system during work extraction while the coherence stored in the system-bath correlations is allowed to flow as $Q_{\mathrm{coh}}^{\left(\mathrm{out}\right)}$ into an incoherent thermal bath ${\gamma }_B$.

Figure 2

The coherence of (a) the system $C_{\mathrm{ext}}({\rho }_S\left(t\right))$ and (b) the coherence change of the bath $\mathrm{\Delta }{C}_{\mathrm{ext}}({\rho }_B\left(t\right))$ as a function of inverse temperature $\beta {\omega }_0$ and coupling strength $gt$ for individual qubit charging via the Jaynes-Cummings interaction. As both the system and bath have nondegenerate energy levels, only the external coherence contributes to the total amount of coherence after the interaction. Higher temperatures increase the coherence change of the bath, which we can interpret as coherence being locked in the system-bath correlations (see Eq. (13)).

Figure 3

The internal coherence per qubit $C_{\mathrm{int}}\left({\rho }_S^{\otimes N}\right)/N$ is shown as a function of the number of qubits $N$. The top panel shows the internal coherence per qubit at different inverse temperatures $\beta$ for the maximally coherent state $|{\gamma }_S\rangle \langle {\gamma }_S{|}^{\otimes N}$, where $|{\gamma }_S\rangle =\left(\right|0\rangle +e^{-\beta {\omega }_0/2}|1\rangle )/\sqrt{Z_S}$. The lower panel shows the internal coherence per qubit given the parameter values that yield the optimal efficiency ${\eta }_C$ in our protocol for each system size. In both cases, the internal coherence per qubit is maximized for $N=4$ qubits. While the internal coherence per qubit is maximized for $\beta =0$ for the maximally coherent state, the optimal efficiency is obtained for intermediate temperatures.

Figure 4

The efficiency of the single-copy coherence charging protocol ${\eta }_C$ is shown as a function of inverse temperature $\beta {\omega }_0$ and coupling strength $gt$ for various $N$. It is shown that efficiency is maximized for $N=4$ qubits at intermediate temperatures.

Figure 5

The efficiency ${\eta }_C$ of the single-copy coherence charging protocol is shown as a function of the number of qubits $N$. The inverse temperatures $\beta {\omega }_0$ and coupling strengths $gt$ were chosen such that ${\eta }_C$ is optimized for each $N$. The maximum efficiency is reached for $N=4$ qubits.

Figure 6

The ground-state population of the system ${\rho }_{00}\left(t\right)$ after time $t$ with an initial state ${\rho }_S\left(0\right)=|0\rangle \langle 0|$. The system parameters are taken to be $\beta {\omega }_0=1$ and $g=1$.

Figure 7

The ground-state population of the system ${\rho }_{00}\left(t\right)$ after time $t$ with an initial state ${\rho }_S\left(0\right)={\gamma }_S+i\kappa \left(\right|0\rangle \langle 1|-|1\rangle \langle 0\left|\right)$. The system parameters are taken to be $\kappa =0.3, \beta {\omega }_0=1$, and $g=1$.