Charging a quantum battery via nonequilibrium heat current

When a quantum system is subject to a thermal gradient it may sustain a steady nonequilibrium heat current by entering into a so-called nonequilibrium steady state (NESS). Here we show that NESS constitute a thermodynamic resource that can be exploited to charge a quantum battery. This adds to the list of recently reported sources available at the nanoscale, such as coherence, entanglement, and quantum measurements. We elucidate this concept by showing analytic and numerical studies of a two-qubit quantum battery that is alternatively charged by an incoherent heat flow and discharged by application of a properly chosen unitary gate. The presence of a NESS for the charging step guarantees steady operation with positive power output. Decreasing the duration of the charging step results in a time-periodic steady state accompanied by increased efficiency and output power. The device is amenable to implementation with different nanotechnology platforms.

Figure 1

Two stroke NESS-based quantum heat engine scheme: First, the quantum working fluid playing the role of a quantum battery is charged, as it is traversed by an incoherent heat flux and reaches a steady state; then energy is extracted in the form of work by suitably coupling the battery to a work source that evolves it unitarily.

Figure 2

Steady-state ergotropy of the two coupled qubits setup as a function of inverse effective temperatures. ${\beta }_S\simeq 2.75$ is equivalent to ${\mathrm{\Gamma }}_S^{+}={\mathrm{\Gamma }}_A^{+}=3.5\times {10}^{-4}{\omega }_0$, ${\mathrm{\Gamma }}_S^{-}=5.5\times {10}^{-3}{\omega }_0$, and ${\mathrm{\Gamma }}_A^{-}=5\times {10}^{-4}{\omega }_0$. These values were used in other simulations reported below. The inset shows the configuration of the energy levels with the transitions induced by external pumping and dissipation channels. The thickness of the levels represents the typical population distribution required to operate the thermal engine via the $|A\rangle \leftrightarrow |S\rangle$ exchange.

Figure 3

Full cycle for the two-qubit thermal machine with instantaneous work extraction happening after a short idle phase and complete recharging to the NESS, shown in terms of the level populations as a function of time. Here parameters are chosen as $\lambda ={10}^{-2}{\omega }_0$, ${\mathrm{\Gamma }}_k^{+}={\mathrm{\Gamma }}_0{\overline{n}}_k$, ${\mathrm{\Gamma }}_k^{-}={\mathrm{\Gamma }}_0({\overline{n}}_k+1)$, with ${\overline{n}}_k=1/(e^{{\omega }_0/{\mathcal{T}}_k}-1)$, ${\mathcal{T}}_A=2{\omega }_0$, ${\mathcal{T}}_S=0.1{\omega }_0$, and ${\mathrm{\Gamma }}_0={10}^{-3}{\omega }_0$.

Figure 4

(a) Efficiency and power for the thermal machine operated at ${\rho }_{\mathrm{OSS}}$ as a function of $\tau$. (b) Ergotropy extracted per step for different choices of $\tau$ starting from ${\rho }_{\mathrm{NESS}}$ and converging to ${\rho }_{\mathrm{OSS}}$; (c) dynamics of the system reaching the operational steady state for $\tau =150$. In all panels we set $\lambda ={10}^{-2}{\omega }_0$, ${\mathrm{\Gamma }}_k^{+}={\mathrm{\Gamma }}_0{\overline{n}}_k$, ${\mathrm{\Gamma }}_k^{-}={\mathrm{\Gamma }}_0({\overline{n}}_k+1)$, with ${\overline{n}}_k=1/(e^{{\omega }_0/{\mathcal{T}}_k}-1)$, ${\mathcal{T}}_A=2{\omega }_0$, ${\mathcal{T}}_S=0.1{\omega }_0$, and ${\mathrm{\Gamma }}_0={10}^{-3}{\omega }_0$.