Ergotropy from coherences in an open quantum system

We show that it is possible to have nonzero ergotropy in the steady states of an open quantum system consisting of qubits that are collectively coupled to a thermal bath at a finite temperature. The dynamics of our model leads the qubits into a steady state that has coherences in the energy eigenbasis when the system consists of more than a single qubit. We observe that even though the system does not have inverted populations, it is possible to extract work from the coherences and we analytically show that in the high-temperature limit, ergotropy per unit energy is equal to the $l_1$ norm of coherence for the two-qubit case. Further, we analyze the scaling of coherence and ergotropy as a function of the number of qubits in the system for different initial states. Our results demonstrate that one can design a quantum battery that is charged by a dissipative thermal bath in the weak-coupling regime.

Figure 1

(a) Coherence as measured by $C_{l_1}$ and (b) ergotropy calculated for different environment temperatures $\beta =0.01$ (solid), $\beta =1$ (dot-dashed), and $\beta =10$ (dashed) with $\omega =1$.

Figure 2

Scaling of coherence (blue circles) and ergotropy (orange squares) as a function of the number of qubits initiated in their ground states in the high-temperature limit, $\beta \to \infty$, and $\omega =1$.

Figure 3

(a) Scaling of mean coherence (blue circles) and ergotropy (orange squares) as a function of the number of qubits initiated in ${10}^5$ random initial states in the high-temperature limit, $\beta \to \infty$, and $\omega =1$. Error bars denote the standard deviation and if not visible they remain inside the data points. (b) Ergotropy vs coherence scatter plot for all ${10}^5$ random initial states for each two- (blue), three- (brown), four- (red), five- (gray), six- (pink), and seven- (green) qubit systems. In gray scale stacks of points corresponding to $N=2$ to $N=7$ is displayed from dark to light, respectively.

Figure 4

Concurrence and quantum discord in the two-qubit steady states of our model in the collective coupling limit for different temperatures of the bath. It is informative to compare these plots with that presented in Fig. 1.