Charger-mediated energy transfer for quantum batteries: An open-system approach

The energy charging of a quantum battery is analyzed in an open quantum setting, where the interaction between the battery element and the external power source is mediated by an ancilla system (the quantum charger) that acts as a controllable switch. Different implementations are analyzed putting emphasis on the interplay between coherent energy pumping mechanisms and thermalization.

Figure 1

Pictorial representation of the model analyzed in this work. Here, energy from the external world E flows into the ancillary system A, which acts as a classical-to-quantum transducer for B (the quantum battery). The subsystems A and B interact via a time-dependent coupling, which is switched on during the charging interval $[0,\tau ]$ only. In our model, the EA coupling may either occur via the interaction with a thermal source (represented by the yellow lamp) or coherently via the modulation of the local Hamiltonian of A (represented by the green laser), or both.

Figure 2

Local energy $E_{\mathrm{B}}\left(\tau \right)$ and ergotropy ${\mathcal{E}}_{\mathrm{B}}\left(\tau \right)$ of the battery B (both in units of ${\omega }_0)$ as functions of $g\tau$, for the two-harmonic-oscillator model. (a) The black dashed-dotted, red dashed, and magenta dotted curves represent $E_{\mathrm{B}}\left(\tau \right)$ for $N_b\left(T\right)=1$ and $F=0.1{\omega }_0,N_b\left(T\right)=1$, and $F=0$ (no coherent driving), and $N_b\left(T\right)=0$ and $F=0.1{\omega }_0$ (no thermal driving), respectively. The blue solid curve represents the ergotropy ${\mathcal{E}}_{\mathrm{B}}\left(\tau \right)$ for $N_b\left(T\right)=1$ and $F=0.1{\omega }_0$. Note that this curve is superimposed to the magenta one: this is because, as emphasized in Eq. (25), ${\left.{\mathcal{E}}_{\mathrm{B}}\left(\tau \right)\right|}_{F,T}={\left.E_{\mathrm{B}}\left(\tau \right)\right|}_{F,T=0}$. All numerical results in (a) have been obtained by setting $g=0.2{\omega }_0$ and $\gamma =0.05{\omega }_0$ (underdamped regime). (b) Same as in (a) but for $\gamma ={\omega }_0$ (overdamped regime).

Figure 3

The ratio (7), which measures the fraction of energy stored in the battery which can be extracted as work, as a function of $g\tau$ and for the two-harmonic-oscillator model. (a) Different curves correspond to different values of the loss parameter $\gamma$. Blue solid line: $\gamma =0.05{\omega }_0$ (underdamped regime); $\gamma ={\omega }_0$ (overdamped regime): red dashed line. The other parameters have been set as follows: $F=0.2{\omega }_0,g=0.2{\omega }_0,N_b\left(T\right)=0.2$. (b) Same as in (a) but for $N_b\left(T\right)=1$. $R_{\mathrm{B}}$ decreases for increasing temperature. In both panels, all curves approach the asymptotic value (28) for $\tau \gg 1/g$.

Figure 4

(a) The local energy $E_{\mathrm{A}}\left(\tau \right)$ of the ancilla A [in units of $N_b\left(T\right){\omega }_0]$ [Eq. (31)] as a function of $g\tau$ for the two-harmonic-oscillator model. Different curves correspond to different values of the ratio $\gamma /g$. Red dashed line: $\gamma /g=1/2$; green solid line: $\gamma /g=4$; blue dashed-dotted line: $\gamma /g=25$. (b) Same as in (a) but for the energy $E_{\mathrm{B}}\left(\tau \right)$ stored in the battery B [Eq. (30)]. (c) The maximum average storing power ${\tilde{P}}_{\mathrm{B}}$ [in units of $g{\omega }_0{N}_b\left(T\right)]$ [Eq. (32)] as a function of $\gamma /g$. All results in (a)–(c) have been obtained for the purely energy supply regime (i.e., $F=0)$.

Figure 5

(a) $E_{\mathrm{B}}\left(\tau \right)$ (in units of ${\omega }_0{F}^2/g^2)$ [Eq. (37)] as a function of $g\tau$ for the two-harmonic-oscillator model. Different curves correspond to different values of the loss parameter $\gamma$. Black dotted line: $\gamma =0$; red dashed line: $\gamma =0.1{\omega }_0$; blue dashed-dotted line: $\gamma =0.4{\omega }_0$; black solid line: $\gamma =0.8{\omega }_0$. Results have been obtained by setting $g=0.2{\omega }_0$. (b) Same as in (a) but for $F=F_0\sqrt{\gamma }$. Different curves correspond to different values of the loss parameter $\gamma$. Red dashed line: $\gamma =0.1{\omega }_0$; black solid line: $\gamma =0.8{\omega }_0$; green dashed-dotted line: $\gamma =2{\omega }_0$; magenta dotted line: $\gamma =5{\omega }_0$. We notice that in this case $\gamma$ needs to be tuned with $g$ in order to get high power in a short time. All results in this figure refer to the case of the purely coherent energy supply regime, i.e. $N_b\left(T\right)=0$.

Figure 6

(a) Two-dimensional color plot of ${\mathcal{E}}_{\mathrm{B}}\left(\infty \right)$ (in units of ${\omega }_0)$ [Eq. (45)] as a function of $F$ (in units of ${\omega }_0)$ and $N_b\left(T\right)$ for the two-qubit model. Notice that ${\mathcal{E}}_{\mathrm{B}}\left(\infty \right)$ reaches its maximum value (53) for $N_b\left(T\right)=0$ (zero temperature) and $F\simeq 1.09g$. For large enough $F$ we notice that ${\mathcal{E}}_{\mathrm{B}}\left(\infty \right)$ is not monotonically decreasing in $N_b\left(T\right)$. (b) ${\mathcal{E}}_{\mathrm{B}}\left(\infty \right)$ (in units of ${\omega }_0)$ as a function of $N_b\left(T\right)$. Different curves correspond to different values of $F$. Magenta solid line: $F=0$ [which yields ${\mathcal{E}}_{\mathrm{B}}\left(\infty \right)=0]$; green dashed-dotted line: $F=0.05{\omega }_0$; blue dashed line: $F=0.1{\omega }_0$; red dotted line: $F=0.5{\omega }_0$. The nonmonotonic behavior as a function of $N_b\left(T\right)$ is clearly evident for $F=0.5{\omega }_0$. (c) Same as in (a) but for the asymptotic value $E_{\mathrm{B}}\left(\infty \right)$ of the energy stored in B [Eq. (44)]. (d) Same as in (a) and (c) but for the ratio $R_{\mathrm{B}}\left(\infty \right)$ [Eq. (7) in the $\tau \to \infty$ limit]. $R_{\mathrm{B}}\left(\infty \right)$ reaches its maximum value for $N_b\left(T\right)=0$ and in the $F\to 0$ limit. All results in this figure have been obtained by setting $g=0.1{\omega }_0$ and $\gamma ={\omega }_0$.

Figure 7

(a) $E_{\mathrm{B}}\left(\infty \right)/{\omega }_0$ (red dashed line) [Eq. (51)] and ${\mathcal{E}}_{\mathrm{B}}\left(\infty \right)/{\omega }_0$ (blue solid line) [Eq. (50)] as functions of $2F/g$, for the two-qubit model. Results in this panel have been obtained by setting $\gamma \gg F,g$. (b) The ratio $R_{\mathrm{B}}\left(\infty \right)$ [Eq. (7) in the $\tau \to \infty$ limit] is plotted as a function of $2F/g$. Different curves correspond to different values of $\gamma$. Blue solid line: $\gamma \gg F,g$; red dashed-dotted line: $\gamma /g=5$; magenta dashed line: $\gamma /g=1$. Both panels refer to the purely coherent energy supply regime, i.e., $N_b\left(T\right)=0$.

Figure 8

(a) $E_{\mathrm{B}}\left(\tau \right)$ (in units of ${\omega }_0)$ as a function of $g\tau$, for the two-qubit model. Different curves refer to different values of $F$ (in units of ${\omega }_0)$. Blue dashed line: $F=0.05{\omega }_0$; red solid line: $F=0.2{\omega }_0$; black dashed-dotted line: $F={\omega }_0$. (b) Same as in (a) but for ${\mathcal{E}}_{\mathrm{B}}\left(\tau \right)$. Numerical results in (a) and (b) have been obtained by setting $g=0.2{\omega }_0$ and $\gamma =0.05{\omega }_0$. (c), (d) Same as in (a) and (b) but for $\gamma ={\omega }_0$. All results in this figure refer to the purely coherent energy supply regime, i.e., $N_b\left(T\right)=0$.

Figure 9

${\mathcal{E}}_{\mathrm{B}}$ (in units of ${\omega }_0)$ as a function of $g\tau$, for the two-qubit model. Different curves refer to different values of $N_b\left(T\right)$. Red solid line: $N_b\left(T\right)=0$; blue dashed line: $N_b\left(T\right)=0.1$; magenta dotted line: $N_b\left(T\right)=0.5$; cyan dashed-dotted line: $N_b\left(T\right)=1$. Numerical results in this plot have been obtained by setting $g=F={\omega }_0/5$ and $\gamma ={\omega }_0$. Notice that, in a finite range of values of $g\tau$, the result for $N_b\left(T\right)=0.1$ (blue dashed line) lies above the result for $N_b\left(T\right)=0$ (red solid line).

Figure 10

(a) $E_{\mathrm{B}}\left(\tau \right)$ (in units of ${\omega }_0)$ as a function of $g\tau$, for the hybrid model. Different curves refer to different values of $F$ (in units of ${\omega }_0)$. Green solid line: $F=0.1{\omega }_0$; red dashed line: $F=0.5{\omega }_0$; blue dashed-dotted line: $F=1.5{\omega }_0$. (b) Same as in (a) but for ${\mathcal{E}}_{\mathrm{B}}\left(\tau \right)$. Numerical results in (a) and (b) have been obtained by setting $g=0.1{\omega }_0,\gamma ={\omega }_0$, and $N_b\left(T\right)=0$. (c), (d) Same as in (a) and (b) but for $N_b\left(T\right)=1$.

Figure 11

The timescale $\overline{\tau }$ (in units of $1/g)$ at which $E_{\mathrm{B}}\left(\tau \right)$ reaches its maximum value (blue circles) is plotted as a function of $F/g$. Red squares denote the same quantity but for the case of ${\mathcal{E}}_{\mathrm{B}}\left(\tau \right)$. Both results refer to the hybrid model. Numerical results in this figure have been obtained by setting $g=0.2{\omega }_0,\gamma ={\omega }_0$, and refer to the purely coherent energy supply regime, i.e., $N_b\left(T\right)=0$.