Superconducting-like Heat Current: Effective Cancellation of Current-Dissipation Trade-Off by Quantum Coherence

Quantum coherence is a useful resource for increasing the speed and decreasing the irreversibility of quantum dynamics. Because of this feature, coherence is used to enhance the performance of various quantum information processing devices beyond the limitations set by classical mechanics. However, when we consider thermodynamic processes, such as energy conversion in nanoscale devices, it is still unclear whether coherence provides similar advantages. Here we establish a universal framework, clarifying how coherence affects the speed and irreversibility in thermodynamic processes described by the Lindblad master equation, and give general rules for when coherence enhances or reduces the performance of thermodynamic devices. Our results show that a proper use of coherence enhances the heat current without increasing dissipation; i.e., coherence can reduce friction. In particular, if the amount of coherence is large enough, this friction becomes virtually zero, realizing a superconducting-like “dissipation-less” heat current. Since our framework clarifies a general relation among coherence, energy flow, and dissipation, it can be applied to many branches of science from quantum information theory to biology. As an application to energy science, we construct a quantum heat engine cycle that exceeds the power-efficiency trade-off bound on classical engines and effectively attains the Carnot efficiency with finite power in fast cycles.

Figure 1

Schematic diagram of the $2N$-state model. (a) No coherence ($\rho ={\rho }_{\mathrm{sd}}$). In this case, $A_{\mathrm{cl}}=O(N)$ and $A_{\mathrm{qm}}=0$ hold for arbitrary $\rho$ (see Supplemental Material, Sec. II B [33]). Therefore, in order to obtain $O(N)$ heat current, dissipation inevitably scales as $O(N)$. (b) With $O(N)$ coherence (e.g., ${\rho }^{+}$). In this case, correlated decays and excitations occur and $A_{\mathrm{cl}}=O(N)$ and $A_{\mathrm{qm}}=O(N^2)$ hold. As a result, an $O(N)$ heat current with a constant-order dissipation is realized.

Figure 2

Schematic diagram of the fast cycle attaining Carnot efficiency with finite power. The cycle consists of four steps. Step 1: the $2N$-state system is connected to the hot bath, absorbing the dissipation-less heat $Q_{\mathrm{H}}$ for a time duration ${\tau }_{\mathrm{H}}$. Step 2: the interaction between the system and the hot bath is turned off, and the energy gap of the system is changed from $\hslash {\omega }_H$ to $\hslash {\omega }_C$ [37]. Step 3: the system is connected to the cold bath, releasing the dissipation-less heat $Q_{\mathrm{C}}$ for a time duration ${\tau }_{\mathrm{C}}$. Step 4: the interaction between the system and the cold bath is turned off, and the energy gap of the system is changed from $\hslash {\omega }_C$ to $\hslash {\omega }_H$.

Figure 3

Numerical check of the current-dissipation trade-off inequalities (3) and (4) as a function of time $t$ during step 1 (interaction with the hot bath) of the 2-qubit heat engine cycle. Red and orange solid curves are the current-dissipation ratio $J_H^2/\dot{\sigma }$ for the states $\rho$ and ${\rho }_{\mathrm{sd}}$, respectively. Black dashed curve is the quantum bound $(A_{\mathrm{cl}}+A_{\mathrm{qm}})/2$ and the blue dashed curve is the classical bound $A_{\mathrm{cl}}/2$. Note that the ratio $J_H^2(\rho )/\dot{\sigma }(\rho )$ exceeds the classical bound, demonstrating the coherence-induced quantum lubrication effect. The parameters are ${\omega }_H=1.9$, ${\omega }_C=1$, ${\beta }_H=1.1$, ${\beta }_C=2.1$, ${\tau }_H=0.2$, ${\mathrm{\Gamma }}_0=1$ (Figs. 3 and 4) and ${\tau }_C=0.4$ (Fig. 3).

Figure 4

Numerical calculation of the power-efficiency trade-off relation (8) by varying the time duration ${\tau }_C$ of the heat engine cycle, step 3. Black dashed curve is the quantum bound $c({\overline{A}}_{\mathrm{cl}}+{\overline{A}}_{\mathrm{qm}})$ and the blue dashed curve is the classical bound $c{\overline{A}}_{\mathrm{cl}}$. Red solid curve shows the power-efficiency ratio $(W/\tau )\times ({\eta }_{\text{Car}}-\eta {)}^{-1}$, which exceeds the classical bound for some parameter range.