Stable adiabatic quantum batteries

With the advent of quantum technologies comes the requirement of building quantum components able to store energy to be used whenever necessary, i.e., quantum batteries. In this paper we exploit an adiabatic protocol to ensure a stable charged state of a three-level quantum battery which allows one to avoid the spontaneous discharging regime. We study the effects of the most relevant sources of noise on the charging process, and, as an experimental proposal, we discuss superconducting transmon qubits. In addition we study the self-discharging of our quantum battery where it is shown that spectrum engineering can be used to delay such phenomena.

Figure 1

(a) Schematic representation of the stored charge in classical and quantum batteries. While classical batteries present a charge stability after the charging time ${\tau }_{\text{c}}$, quantum batteries typically have oscillatory behavior. (b) Pictorial representation of a three-level quantum battery and its charger based on resonant fields. Each energy level represents a charging step in which the energy gap between them is $\mathrm{\Delta }{\varepsilon }_{nm}={\varepsilon }_n-{\varepsilon }_m$. In real physical settings, relaxation and dephasing could induce some nonunitary discharging by losses described by rates $\mathrm{\Gamma }$ and $\gamma$, respectively.

Figure 2

(a) Ergotropy and (b) power for a linear ramp including the effects of both relaxation and dephasing. We chose the energy spectrum of our three-level system as ${\omega }_1=0$, ${\omega }_2=1$, and ${\omega }_3=1.95$ to account for the anharmonicity of the energy gaps in transmon qubits, resulting in ${\mathcal{C}}_{\text{max}}=1.95\hslash$. We set the rates characterizing the noise as ${\mathrm{\Gamma }}_{32}=2{\mathrm{\Gamma }}_{21}$, ${\gamma }_2={\mathrm{\Gamma }}_{21}$, and ${\gamma }_3={\mathrm{\Gamma }}_{32}=2{\mathrm{\Gamma }}_{21}$, again to match the state-of-the-art parameters measured for transmon qubits [51].

Figure 3

Ergotropy during a self-discharging process of our battery as function of ${\mathrm{\Gamma }}_{21}t$ for different choices of the energy gaps, and the rest of the parameters remain as in Fig. 2. Note that the black dash-dotted line with $\delta {\varepsilon }_{32}/\delta {\varepsilon }_{21}=0.95$ corresponds to the energy gap configuration considered in Fig. 2 and represents the case most relevant to a transmon qubit implementation.

Figure 4

(a) The sketch of a superconducting transmon qubit circuit, where a Josephson junction of capacitance $C_{\text{J}}$ is shunted by a large capacitance $C_{\text{B}}$. (b) Energy level structure of the transmon qutrit, where the maximum stored energy is given by means of the energies $E_{\text{J}}$ and $E_{\text{C}}$ as ${\mathcal{C}}_{\text{max}}={\varepsilon }_3-{\varepsilon }_1=\mathrm{\Delta }{\varepsilon }_{32}+\mathrm{\Delta }{\varepsilon }_{21}=4\sqrt{2E_{\text{J}}{E}_{\text{C}}}-E_{\text{C}}$.