Environment-mediated charging process of quantum batteries

We study the charging process of open quantum batteries mediated by a common dissipative environment in two different scenarios. In the first case, we consider a quantum charger-battery model in the presence of a non-Markovian environment, where the battery can be properly charged in a strong coupling regime, without any external power and any direct interaction with the charger, i.e., a wireless-like charging happens. The environment plays a major role in the charging of the battery, while this does not happen in a weak coupling regime. In the second scenario, we show the effect of individual and collective spontaneous emission rates on the charging process of quantum batteries by considering a two-qubit system in the presence of Markovian dynamics. Our results demonstrate that open batteries can be satisfactorily charged in Markovian dynamics by employing an underdamped regime and/or strong external fields. We also present a robust battery by taking into account subradiant states and an intermediate regime. Moreover, we propose an experimental setup to explore the ergotropy in the first scenario.

Figure 1

Schematic representation of the model considered in Sec. 2.

Figure 2

Dynamics of $\left|\mathrm{\Delta }E\right|/{\omega }_0$ as a function of the dimensionless quantity $\lambda \tau$. Dotted red line depicts $|\mathrm{\Delta }{E}_A|/{\omega }_0$, and solid blue line shows $\mathrm{\Delta }{E}_B/{\omega }_0$. The initial state is $|\mathrm{\Phi }\left(0\right)\rangle ={|e\rangle }_A{|g\rangle }_B\otimes {|0\rangle }_{\mathcal{E}}$ and $c_1=1/\sqrt{2}$ with (a) $\mathcal{R}=500$, (b) $\mathcal{R}=30$, and (c) $\mathcal{R}=0.3$.

Figure 3

Dynamics of $\mathcal{W}/{\mathcal{W}}_{\max }$ as a function of the dimensionless quantity $\lambda \tau$. (a) Dotted red line depicts $\mathcal{R}=30$, and solid blue line $\mathcal{R}=0.3$. The initial state is $|\mathrm{\Phi }\left(0\right)\rangle ={|e\rangle }_A{|g\rangle }_B\otimes {|0\rangle }_{\mathcal{E}}$. (b) Here the entangled initial state of the qubits is assumed as Eq. (13) and $\mathcal{R}=30$ is considered. Solid darker blue line shows $(c_1={\alpha }_{-}=\sqrt{3}/2)$, dot-dot-dashed black line $(c_1={\alpha }_{-}=1/\sqrt{2})$, dotted red line $(c_1=1/\sqrt{2},{\alpha }_{-}=0.92)$, dashed green line $(c_1=\sqrt{3}/2,{\alpha }_{-}=0.5)$, and dot-dashed magenta line $(c_1=1/\sqrt{2},{\alpha }_{-}=0.2)$.

Figure 4

Single-cell QB. Time evolution of $\mathcal{W}/{\mathcal{W}}_{\max }$ as a function of the dimensionless quantity $\mathrm{\Omega }\tau$ for $\gamma =0.9\mathrm{\Gamma }$, and $l_1=l_2=0$. Dot-dashed-dashed purple line exhibits $(\mathrm{\Gamma }=0.5\mathrm{\Omega })$, solid red line $(\mathrm{\Gamma }=0.1\mathrm{\Omega })$, and dot-dashed black line $(\mathrm{\Gamma }=0.01\mathrm{\Omega })$. (a) The initial state is assumed $|\phi \rangle ={|e\rangle }_{\text{A}}{|g\rangle }_{\text{B}}$. (b) The initial entangled state is chosen with ${\alpha }_{-}=c_1=\sqrt{3}/2$.

Figure 5

Two-cell QB. Dynamics of $\mathcal{W}/{\mathcal{W}}_{\max }$ as a function of the dimensionless quantity $l\tau$ by considering $l_1=l_2=l$, and ${\omega }_0={\omega }_L=\omega$. Solid darker blue line shows $(\mathrm{\Gamma }=\gamma =\mathrm{\Omega }=0)$, dot-dot-dashed-dashed darker orange line $(\mathrm{\Gamma }=\gamma =\mathrm{\Omega }=l)$, dot-dashed red line $(\mathrm{\Gamma }=0.1l,\gamma =\mathrm{\Omega }=0)$, dotted black line $(\mathrm{\Gamma }=0.1l,\mathrm{\Omega }=0.1\mathrm{\Gamma },\gamma =0.9\mathrm{\Gamma })$, dot-dashed-dashed magenta line $(\mathrm{\Gamma }=0.1l,\mathrm{\Omega }=5\mathrm{\Gamma },\gamma =0.9\mathrm{\Gamma })$, dot-dot-dashed green line $(\mathrm{\Gamma }=5l,\mathrm{\Omega }=0.1\mathrm{\Gamma },\gamma =0.9\mathrm{\Gamma })$, and dot-dot-dot-dashed purple line $(\mathrm{\Gamma }=5l,\mathrm{\Omega }=5\mathrm{\Gamma },\gamma =0.9\mathrm{\Gamma })$. The battery initial state is ${|\phi \rangle }_{\text{B}}=|gg\rangle$ (a) and ${|\phi \rangle }_B=|{\phi }_{-}\rangle$ with $c_1=1/\sqrt{2}$ (b).

Figure 6

Schematic representation of $N$-cell stable quantum battery by considering $2N$ qubits where every two qubits interact with a common environment in the presence of driving fields as well as the intermediated regime.

Figure 7

Schematic diagram of optical setup, taken from Ref. [29]: SP (S-wave plate), PBS (polarizing beam splitter), SF (spatial filter), HWP (half wave plate), DP (Dove prism), MZIM (Mach-Zehnder interferometer with an additional mirror), PZT (piezoelectric ceramic), DMZIM (double input/output MZIM), BS (beam splitter), CNOT (controlled-not gate), and CCD camera (charge coupled device camera).