Quick charging of a quantum battery with superposed trajectories

We propose charging protocols for quantum batteries based on quantum superpositions of trajectories. Specifically, we consider that a qubit (the battery) interacts with multiple cavities or a single cavity at various positions, where the cavities act as chargers. Further, we introduce a quantum control prepared in a quantum superposition state, allowing the battery to be simultaneously charged by multiple cavities (the multiple-charger protocol) or a single cavity with different entry positions (the single-charger protocol). To assess the battery's performance, we evaluate the maximum extractable work, referred to as ergotropy. The primary discovery lies in the quick charging effect, wherein we prove that the increase in ergotropy stems from the quantum coherence initially present in the quantum control. Moreover, the induced “Dicke-type interference effect” in the single-charger protocol can further lead to a “perfect charging phenomenon”, enabling a complete conversion of the stored energy into extractable work across the entire charging process, with just two entry positions in superposition. Furthermore, we propose circuit models for these charging protocols and conduct proof-of-principle demonstrations on IBMQ and IonQ quantum processors. The results validate our theoretical predictions, demonstrating a clear enhancement in ergotropy.

Figure 1

The quantum battery $Q$ is first sent into a multiport beam splitter (MPBS1), which allows the quantum battery to travel along $N$ different trajectories (denoted by ${|j\rangle }_D$ with $j=1\cdots N$) in a manner of quantum superposition. We consider two charging processes: (a) the trajectories each lead to a charger (cavities) $\left\{C_j\right\}$, causing the quantum battery to interact with all chargers simultaneously, (b) the trajectories lead to a single charger $C$ but at different positions $\left\{r_j\right\}$, causing the quantum battery to interact with the charger with various coupling strengths. Once the charging process is completed, a second multiport beam splitter (MPBS2) is used to perform measurement on the trajectories' degree of freedom $D$. This measurement captures the quantum interference effect between different trajectories and results in $N$ possible reduced states ${\rho }_j$. We then extract work from each ${\rho }_j$, where the maximum amount of extractable work is called the ergotropy. The work extraction operations are described by unitary operators $U_j$, which transform each of the batteries to a passive state ${\phi }_j$.

Figure 2

The stored energy $E$ and average ergotropy $\overline{W}$ (both in units of $\hslash {\omega }_c$) on time $\tau$ (in units of $1/{\omega }_c$) for $\lambda =0.05$. The black solid curve plots the stored energy while the dashed curves plot the average ergotropy. From bottom to top, the blue, red, and green dashed curves show the results for $N=1$, 2, and 4, respectively.

Figure 3

(a) The stored energy $E$ and average ergotropy $\overline{W}$ (both in units of $\hslash {\omega }_c$) as functions of time $\tau$ (in units of $1/{\omega }_c$) for $\lambda =0.5$. (b) The stored energy $E$ and the average ergotropy contributed by states of $k=1,{\overline{W}}_{k=1}$. Two brown vertical lines at $\tau =0.63,1.00$ indicate the inversion points $T_N$ for $N=1$ (blue) and $N=2$ (red). Here, ${\overline{W}}_{k=1}$ is 0 when $N=7$ (green) and $N=\infty$ (magenta). (c) The stored energy $E$ and the average ergotropy contributed by states of $k\ne 1,{\overline{W}}_{k\ne 1}$. Except for $N=1$ (blue), these states contribute average ergotropy when $\tau >0$. (d) The dashed curves plot the change in average purity $\mathcal{P}$ against time $\tau$. The cutoff photon number is set to 9 in the above results.

Figure 4

(a) Maximum average ergotropy ${\overline{W}}_{\text{max}}$ defined in Eq. (23) with respect to different coupling strength $\lambda$. The blue, red, green, and magenta dashed lines represent the results for $N=1$, 10, 100, and $\infty$, respectively. (b) The average purity $\mathcal{P}$ on dimensionless coupling strength $\lambda$. The blue, red, green, and magenta dashed lines represent the results for $N=1$, 10, 100, and $\infty$, respectively. The cutoff photon number is set to 9 in the above results.

Figure 5

Under the multiple-charger protocol, the numerical simulations for the von Neumann entropy of the premeasurement state $\mathcal{S}\left(\rho \right)$ and quantum-classical mutual information $I$ with (a) $\lambda =0.05$ and (c) $\lambda =0.5$, and the ergotropy gain $\mathrm{\Delta }\overline{W}$ with (b) $\lambda =0.05$ and (d) $\lambda =0.5$.

Figure 6

The stored energy $E$ and average ergotropy $\overline{W}$ for different numbers ($M$) of QBs under two superposed trajectories ($N=2$) but different charging protocols. (a) Multiple-charger protocol with the coupling strength $\lambda =0.05$. (b) Multiple-charger protocol with the coupling strength $\lambda =0.5$. (c) Single-charger protocol with $r_1=0.1l, r_2=0.9l$, and the coupling strength $\lambda =0.05$. (d) Single-charger protocol with $r_1=0.1l, r_2=0.9l$, and the coupling strength $\lambda =0.5$. Note that the rotating-wave approximation is considered for (a) and (c).

Figure 7

(a) Quantum circuit for multiple-chargers protocol. Here, $D,Q,C_1,C_2$ represent the control qubit, battery qubit, first charger, and second charger, respectively. “Tr” is trace out. (b) Decomposition of a controlled unitary in (a). (c) Qubit configuration used on $\mathrm{ibmq}\_\mathrm{algiers}$. (d) Decomposition of a $XX$ gate into CNOT gates.

Figure 8

Charging process of the single-charger setup, where the coupling strengths of the two superposed trajectories have the same magnitude but are out of phase.

Figure 9

The stored energy $E$ and average ergotropy $\overline{W}$ (both in units of $\hslash {\omega }_c$) on time $\tau$ (in units of $1/{\omega }_c$) for $\lambda =0.05,N=2$. The black solid and dashed curves represent the stored energy and average ergotorpy predicted by numerical simulations. The green circles and blue “$x$”s represent experimental results performed on $\mathrm{ibmq}\_\mathrm{algiers}$. The red triangles and magenta diamonds represent experimental results performed on IonQ Aria 1. Each data point is obtained after averaging 1000 experimental repetitions.

Figure 10

The stored energy $E$ and average ergotropy $\overline{W}$ (both in units of $\hslash {\omega }_c$) on time $\tau$ (in units of $1/{\omega }_c$) with $r_1=0.1l$ and $r_2=0.9l$. The black curve represents the stored energy and average ergotropy predicted by numerical simulations. The green circle and blue “$x$”s represent the experimental results performed on $\mathrm{ibmq}\_\mathrm{algiers}$. The red triangles and magenta diamonds represent experimental results performed on IonQ Aria 1. Each data point is obtained after averaging 1000 experimental repetitions.