Quantum Advantage in the Charging Process of Sachdev-Ye-Kitaev Batteries

The exactly solvable Sachdev-Ye-Kitaev (SYK) model has recently received considerable attention in both condensed matter and high energy physics because it describes quantum matter without quasiparticles, while being at the same time the holographic dual of a quantum black hole. In this Letter, we examine SYK-based charging protocols of quantum batteries with $N$ quantum cells. Extensive numerical calculations based on exact diagonalization for $N$ up to 16 strongly suggest that the optimal charging power of our SYK quantum batteries displays a superextensive scaling with $N$ that stems from genuine quantum mechanical effects. While the complexity of the nonequilibrium SYK problem involved in the charging dynamics prevents us from an analytical proof, we believe that this Letter offers the first (to the best of our knowledge) strong numerical evidence of a quantum advantage occurring due to the maximally entangling underlying quantum dynamics.

Figure 1

The charging protocol of a QB made of $N$ spin-$1/2$ units, described by the ${\widehat{\mathcal{H}}}_0$ in Eq. (1). At time $t<0$, the battery is fully discharged. In the time interval $0<t<\tau$, the interacting charging Hamiltonian ${\widehat{\mathcal{H}}}_1$ is switched on, and energy is injected via the quench. Finally, at time $\tau$, interactions are switched off and ${\widehat{\mathcal{H}}}_0$ is switched back on, so that the stored energy $E_N(\tau )$ is conserved thereafter.

Figure 2

Dynamics of the dimensionless population $p_k(\tau )$ of the QB energy levels as a function of time $\tau$ (in units of $1/J$) and the level index $k$ for three different charging protocols: c-SYK (a), b-SYK with $\overline{J}=J$ (b), and parallel with $K=J$ (c). Data in (a) and (b) correspond to a single realization of disorder in the couplings $J_{i,j,k,l}$ and ${\overline{J}}_{i,j,k,l}$.

Figure 3

(a) The relevant quantities for the bound (12), evaluated at the optimal time ${\tau }^{*}$, and averaged over disorder: time-averaged variances $⟨⟨{\mathrm{\Delta }}_{{\tau }^{*}}{\widehat{\mathcal{H}}}_0^2⟩⟩$ (blue triangles, in units of ${\omega }_0^2$), $⟨⟨{\mathrm{\Delta }}_{{\tau }^{*}}{\widehat{\mathcal{H}}}_1^2⟩⟩$ (green squares, in units of $J^2$), $⟨⟨{\mathrm{\Delta }}_{{\tau }^{*}}^{\mathrm{ent}}{\widehat{\mathcal{H}}}_0^2⟩⟩$ (black circles, in units of ${\omega }_0^2$), as functions of $N$. Dashed curves denote linear (green) and quadratic (blue, black) fits to the numerical results. The four data points corresponding to the smallest $N$ have been discarded from the fits. (b) The optimal power (red) $⟨⟨P_N({\tau }^{*})⟩⟩$ and the quantity in the right-hand side of Eq. (12) (blue) are plotted as functions of $N$, in a log-log scale and in units of ${\omega }_{0}J$. Dashed lines correspond to power laws $\sim N^{1+k}$ ($k=0.5$, red; $k=0$, orange) and are plotted as guides to the eye. Data in this figure refer to the c-SYK QB model, and have been obtained after averaging over $N_{\mathrm{dis}}={10}^3$ (for $N=4,\dots ,10$), $5\times {10}^2$ (for $N=11$, 12), and ${10}^2$ (for $N=13,\dots ,16$) instances of disorder in the couplings $\{J_{i,j,k,l}\}$.