Charger-mediated energy transfer in exactly solvable models for quantum batteries

We present a systematic analysis and classification of several models of quantum batteries involving different combinations of two-level systems and quantum harmonic oscillators. In particular, we study energy-transfer processes from a given quantum system, termed a “charger,” to another one, i.e., the proper “battery.” In this setting, we analyze different figures of merit, including the charging time, the maximum energy transfer, and the average charging power. The role of coupling Hamiltonians which do not preserve the number of local excitations in the charger-battery system is clarified by properly accounting for them in the global energy balance of the model.

Figure 1

(a) shows the time-dependent interaction protocol that allows energy flow between the charger, described by the Hamiltonian ${\mathcal{H}}_{\mathrm{A}}$, and the battery, described by the Hamiltonian ${\mathcal{H}}_{\mathrm{B}}$. At time $t<\tau$ the two systems A and B do not interact and cannot exchange energy, their dynamics being governed by the Hamiltonian ${\mathcal{H}}_0={\mathcal{H}}_{\mathrm{A}}+{\mathcal{H}}_{\mathrm{B}}$. In the time interval $0<t<\tau$ the Hamiltonian ${\mathcal{H}}_1$ is switched on and the two systems interact. Finally, the interaction is switched off at time $\tau$, and the energy $E_{\mathrm{B}}\left(\tau \right)$ stored in the battery B is a conserved quantity. (b) illustrates cartoons of the three charger-battery toy models introduced and studied in this paper. Subpanel (1): Energy transfer is studied between two qubits. Subpanel (2): Energy transfer is studied between a quantum harmonic oscillator and a qubit. Subpanel (3): Energy transfer is studied between two quantum harmonic oscillators.

Figure 2

(a) displays the stored energy $E_{\mathrm{s}}\left(\tau \right)$ (in units of ${\omega }_0$) as a function of $g\tau$, for the case of two coupled qubits. (b) shows the average charging power $P_{\mathrm{s}}\left(\tau \right)$ (in units of $g{\omega }_0$) as a function of $g\tau$. We clearly see that the quantum battery is charged by the charging qubit via Rabi oscillations.

Figure 3

(a) displays the stored energy $E_{\mathrm{s}}\left(\tau \right)$ (in units of ${\omega }_0$) as a function of $\sqrt{K}g\tau$, for the case of a qubit charged by a QHO. The initial number of excitation is $K=3$. Different curves refer to results obtained for three different choices of the initial state of the charger: Fock state (blue solid line), coherent state (red dashed line), and Gibbs state (dashed-dotted green line). (b) shows the average charging power $P_{\mathrm{s}}\left(\tau \right)$ (in units of $\sqrt{K}g{\omega }_0$) as a function of $\sqrt{K}g\tau$. The initial number of excitation is $K=3$. Color coding as in (a). (c) Same as in (a) but for $K=20$. The black dotted line represents the energy of a Gibbs state of the qubit, with temperature equal to that of the initial Gibbs state of the QHO. Note that, for long times (not shown), the system has revivals, due the unitarity of the time evolution. (d) shows the average charging power corresponding to (c). This figure clearly shows the “optimality” of the Fock state, which is the best choice for maximizing the energy and power. Note that, for $K\gg 1$ as in (c) and (d), the coherent state well approximates the quantities $E_{\mathrm{s}}\left(\tau \right)$ and $P_{\mathrm{s}}\left(\tau \right)$ calculated for an initial Fock state.

Figure 4

Figures of merit for the detuning protocol described in Sec. 3a. (a) shows the stored energy $E_{\mathrm{s}}\left(\tau \right)$ (blue solid line) and the switching energy $\delta E_{\mathrm{sw}}\left(\tau \right)$ (red dashed line), in units of $K{\omega }_0$, and as functions of $g\tau$. Results in this panel have been obtained by setting $\delta \omega ={\omega }_0/2$ and $g={\omega }_0/10$. Since $\delta \omega >0$, there is no difference between the stored energy and the transferred energy, i.e., $E_{\mathrm{t}}\left(\tau \right)=E_{\mathrm{s}}\left(\tau \right)$. (b) shows the stored energy $E_{\mathrm{s}}\left(\tau \right)$ (blue solid line), the transferred energy $E_{\mathrm{t}}\left(\tau \right)$ (black dashed-dotted line), and the switching energy $\delta E_{\mathrm{sw}}\left(\tau \right)$ (red dashed line). Results in this panel have been obtained by setting $\delta \omega =-{\omega }_0/2$ and $g={\omega }_0/10$. Since $\delta \omega <0$, some energy is injected into the system and $E_{\mathrm{t}}\left(\tau \right)\le E_{\mathrm{s}}\left(\tau \right)$.

Figure 5

(a) displays the stored energy $E_{\mathrm{s}}\left(\tau \right)$ (in units of $K{\omega }_0$) as a function of $g\tau$, for the case of two coupled QHOs, evaluated by setting $g=0.35{\omega }_0$. Different curves refer to results obtained for three different choices of the initial state of the charger: Stored energy for an initial Fock or a thermal state evaluated by setting $K=3$ (blue solid line); stored energy for an initial Fock or a thermal state evaluated by setting $K=100$ (dark blue solid line); stored energy for an initial coherent state evaluated by setting $K=3$ (red dashed-dotted line); stored energy for an initial coherent state evaluated by setting $K=100$ (dark red dashed-dotted line). The same color code is used for all other panels. The stored energy shows oscillations similar to the RWA case (see Fig. 2), with counter-rotating terms causing only quantitative corrections. (c) displays the switching energy $\delta E_{\mathrm{sw}}\left(\tau \right)$ (in units of $K{\omega }_0$) as a function of $g\tau$, evaluated for $g=0.35{\omega }_0$. (e) displays the transferred energy $E_{\mathrm{t}}\left(\tau \right)$ (in units of $K{\omega }_0$) as a function of $g\tau$, evaluated for $g=0.35{\omega }_0$. (b) displays the stored energy $E_{\mathrm{s}}\left(\tau \right)$ (in units of $K{\omega }_0$) as a function of ${\omega }_{+}\tau$ and evaluated for $g\to {\omega }_0/2$. Due to the vicinity to the critical point, the stored energy increases as a power law. (d) displays the switching energy $\delta E_{\mathrm{sw}}\left(\tau \right)$ (in units of $K{\omega }_0$) as a function of ${\omega }_{-}\tau$, evaluated for $g\to {\omega }_0/2$. This quantity measures the energy that is externally injected. The power-law increase of this quantity is clear. (f) displays the transferred energy $E_{\mathrm{t}}\left(\tau \right)$ (in units of $K{\omega }_0$) as a function of ${\omega }_{-}\tau$, evaluated for $g\to {\omega }_0/2$. Only a small amount of the corresponding stored energy seen in (b) can be counted as transferred energy, while the majority of the energy is externally injected.