Bounds on the capacity and power of quantum batteries

Quantum batteries, composed of quantum cells, are expected to outperform their classical analogs. The origin of such advantages lies in the role of quantum correlations, which may arise during the charging and discharging processes performed on the battery. In this theoretical work, we introduce a systematic characterization of the relevant quantities of quantum batteries, i.e., the capacity and the power, in relation to such correlations. For these quantities, we derive upper bounds for batteries that are a collection of noninteracting quantum cells with fixed Hamiltonians. The capacity, that is, a bound on the stored or extractable energy, is derived with the help of the energy-entropy diagram, and this bound is respected as long as the charging and discharging processes are entropy preserving. While studying power, we consider a geometric approach for the evolution of the battery state in the energy eigenspace of the battery Hamiltonian. Then, an upper bound for power is derived for arbitrary charging process, in terms of the Fisher information and the energy variance of the battery. The former quantifies the speed of evolution, and the latter encodes the nonlocal character of the battery state. Indeed, due to the fact that the energy variance is bounded by the multipartite entanglement properties of batteries composed of qubits, we establish a fundamental bound on power imposed by quantum entanglement. We also discuss paradigmatic models for batteries that saturate the bounds both for the stored energy and power. Several experimentally realizable quantum batteries, based on integrable spin chains, the Lipkin-Meshkov-Glick and the Dicke models, are also studied in the light of these newly introduced bounds.

Figure 1

Schematic of a charging (discharging) process in the quantum state space. The trajectories represent two different charging processes $A$ (blue) and $B$ (red) in which the initial and final states ($|{\mathrm{\Psi }}_i\rangle$ and $|{\mathrm{\Psi }}_f\rangle$ respectively) coincide. The process $A$ undergoes a path with path length $L_A$ at a speed $I_E^{\left(A\right)}$. Similarly, the process $B$ traverses the path length $L_B$ and the speed $I_E^{\left(B\right)}$. The time required ($t_A$ and $t_B$ for the two processes) to reach to the final state, from the initial one, depends both on the path length and the speed. For instance, the charging process requires shorter time, i.e., $t_A<t_B$, although the speed of the process $B$ is larger, $I_E^{\left(A\right)}<I_E^{\left(B\right)}$. This is because path length of $B$ is also larger, $L_A<L_B$.

Figure 2

Energy-entropy diagram. Any quantum state $\rho$ is represented in the diagram as a point with coordinates $x_{\rho }:=(E\left(\rho \right),S\left(\rho \right))$. The point in the thermal boundary with slope $\beta$ corresponds to the thermal state with inverse temperature $\beta$. The trajectory of a charging (discharging) process is represented with a dotted (dashed) line. They connect, respectively, the initial state ${\rho }_0$, with entropy $S_0$, and the final state ${\rho }_s$ (${\rho }_e$). The capacity $C$ corresponds to the energetic amplitude at a given entropy.

Figure 3

A schematic representation of speed of quantum evolution in Hilbert space ($I_Q$) and the speed of evolution in the energy eigenspace of the battery Hamiltonian ($I_E$). Notice that, in general, $I_Q⩾I_E$.

Figure 4

Dynamics of energy levels $p_k$, according to their definition (21), during a parallel (a) and a global (b) charging process. Here one can see the contribution of each energy level ($y$ axis) to the total energy, as a function of time. One can see absolute nonlocality in energy space for the global case, where there is a coexistence in time of the highest and lowest energy level, and locality of the parallel one, where at a given time only levels that are close contribute to the total energy.

Figure 5

Dynamics of the stored energy for different spin integrable models described by the Hamiltonian Eq. (A1). The legend indicates to which model corresponds each line. We have studied the XY (XX) model that correspond to ${\lambda }_m=0$ (${\lambda }_m={\gamma }_m$). For each of these models, we have studied the nearest-neighbor (NN) case, where ${\gamma }_1=1$ and ${\gamma }_{m\ne 1}=0$, and also the power law (pow) case, where ${\gamma }_m=m^{-2}$. We have fixed the system size to $N=20$ spins and normalized the stored energy (Y axis) accordingly, such that it is bounded to 1. We observe that for all the models the maximum stored energy is about $50\%$ of the total due to the desynchronization between the different modes that contribute to this quantity.

Figure 6

Over the grey dashed line, we plot the value of $cos{\theta }_P:=P/\sqrt{{\langle \mathrm{\Delta }{H}_B^2\rangle }_{\mathrm{\Delta }t}{\langle I_E\rangle }_{\mathrm{\Delta }t}}$ as a function of the number of spins $N$ for the models presented in Fig. 5. We observe a fast saturation to an approximate 0.8 value. Below the grey dashed line, we plot the same quantity but substituting ${\langle I_E\rangle }_{\mathrm{\Delta }t}\to 4{\mathrm{\Delta }}^2{H}_{\mathrm{JW}}$, and it saturates to an approximate 0.6 value. This shows how lower is our bound obtained using the Fisher information in energy eigenspace instead of the speed in Hilbert space for these particular models.

Figure 7

Dynamics of the energy levels during the charging process based on the LMG model for $\lambda =5$. Due to the structure of $H_{\mathrm{LMG}}$, only every second energy level starting from the initial state is occupied during the evolution.

Figure 8

The figures consider time-averaged quantities of the LMG model, which are relevant in the study of power, as a function of the number of spins $N$. The final time has been chosen as the one for which the capacity is maximized. Blue color is used for $\lambda =20$, whereas red color is used for $\lambda =5$. Quantities that carry units of time (i.e., power and Fisher information) have been renormalized with the coupling strength $\lambda$. The legends indicate the scaling with $N$ of the different curves.

Figure 9

Capacity properties of the LMG battery. In (a) and (b), one can see the expected scaling of the capacity and the final relative energy variance as a function of the system size. The blue color is used for $\lambda =20$, whereas red color is used for $\lambda =5$, and the legends indicate the scaling law. In (c), we plot the dependence with the anisotropy parameter $\gamma$ of both the capacity and power. For this plot, we used a system of 50 spins and set $\lambda =5$.

Figure 10

Dynamics of the energy levels $p_k$ during the charging process based on Dicke model for $\lambda =0.01$ (a) and 0.5 (b). In (a), we see that for the weak-coupling regime the behavior is similar to the parallel charging case. On the other hand, (b) shows an intermediate behavior between parallel and global charging.

Figure 11

The figures consider time-averaged quantities of the Dicke model, which are relevant in the study of power, as a function of the number of spins $N$ inside the cavity. The final time has been chosen as the one for which the capacity is maximized. The blue color (circles) is used for the weak-coupling regime ($\lambda =0.01$), whereas the red color (squares) is used for the strong-coupling regime ($\lambda =0.5$). Quantities that carry units of time (i.e., power and Fisher information) have been renormalized with the coupling strength $\lambda$. The legends indicate the scaling with $N$ of the different curves.

Figure 12

The figures show the relevant quantities for the capacity of the Dicke model evaluated at the final time $t_f$, as a function of the number of spins $N$ inside the cavity. Blue color (circles) is used for the weak-coupling regime ($\lambda =0.01$), whereas red color (squares) is used for the strong-coupling regime ($\lambda =0.5$). In (c), we plot the entanglement entropy $S^B:=-\mathrm{Tr}\left({\rho }_B{log}_2{\rho }_B\right)$ of the battery reduced density matrix, defined as ${\rho }_B={\mathrm{Tr}}_c\left(\rho \right)$, where the trace is performed over the cavity degree of freedom. We normalize this quantity by its maximum value $S_{\max }^B={log}_2(N+1)$.