Experimental investigation of a quantum battery using star-topology NMR spin systems

Theoretical explorations have revealed that quantum batteries can exploit quantum correlations to achieve faster charging, thus promising exciting applications in future technologies. Using NMR architecture, here we experimentally investigate various aspects of quantum batteries with the help of nuclear spin systems in a star-topology configuration. We first carry out numerical analysis to study how charging a quantum battery depends on the relative purity factors of charger and battery spins. By experimentally characterizing the state of the battery spin undergoing charging, we estimate the battery energy as well as the ergotropy, the maximum amount of work that is unitarily available for extraction. The experimental results thus obtained establish the quantum advantage in charging the quantum battery. We propose using the quantum advantage, gained via quantum correlations among chargers and the battery, as a measure for estimating the size of the correlated cluster. We develop a simple iterative method to realize asymptotic charging that avoids the oscillatory behavior of charging and discharging. Finally, we introduce a load spin and realize a charger-battery-load circuit and experimentally demonstrate battery energy consumption after varying the duration of battery storage, for up to 2 min.

Figure 1

A single spin-1/2 particle in an external magnetic field $B_0$ as a quantum battery. The ground state (a) and the excited state (b) correspond respectively to uncharged and charged states of the battery.

Figure 2

Two charging schemes: (a) parallel charging scheme where a single battery is charged by an individual charger and (b) the collective charging scheme where a single battery is charged by multiple chargers.

Figure 3

(a) Star-topology configuration showing the central battery spin symmetrically surrounded by charger spins. (b)–(f) The star-topology nuclear spin systems studied in this work. The strength $J$ of the battery-charger interaction for each system is shown with the molecular structure, while other details are tabulated in panel (g). Note that all the nuclei considered here (B and C) are spin-1/2 nuclei.

Figure 4

(a) Battery energy ${\mathtt{e}}_B$ versus charging phase $\theta =2\pi J\tau$ for different numbers $N$ of charger spins in pure (solid lines) and mixed (dashed lines; $\epsilon ={10}^{-5}$) state cases. (b) Quantum advantage $\mathrm{\Gamma }$ versus $N$ for different purity values $\epsilon$.

Figure 5

(a) The NMR pulse sequence for charging the quantum battery and measuring its energy. The wide and the narrow rectangular pulses correspond to $\pi$ and $\pi /2$ pulses, respectively. The shaped pulse in the lowest row corresponds to the pulsed field gradient (PFG) which dephases the coherences and retains populations. (b) The symbols correspond to the experimentally measured battery energy values ${\mathtt{e}}_B$ versus the normalized charging duration $\tau /{\overline{\tau }}_N$ for the five spin-systems shown in Fig. 3. Here the dotted lines are spline fits to guide the eye. (c) The quantum advantage $\mathrm{\Gamma }$ versus the number $N$ of charger spins showing $\sqrt{N}$ dependence.

Figure 6

The dots represent the experimentally estimated ergotropy of the battery spin versus the normalized charging duration $\tau /{\overline{\tau }}_N$ for all five spin-systems. Here the ergotropy is scaled by $\epsilon \hslash {\omega }_B{\mathtt{e}}_B^{\max }$ [see Eq. (13)], where ${\mathtt{e}}_B^{\max }$ is taken from Fig. 5 indicated by solid squares. The solid lines in small spin-systems represent the theoretical fits accounting also for the experimental nonidealities.

Figure 7

Numerically calculated battery energy (with pure and mixed states), entanglement entropy (for pure state; $\epsilon =1, \gamma =1$), and quantum discord (for mixed state; $\epsilon ={10}^{-5}, \gamma =1$) versus the normalized charging duration $\tau /{\overline{\tau }}_9$ for the $N=9$ star system involving a single battery spin and nine charger spins.

Figure 8

(a) The NMR pulse sequence for asymptotic charging of a quantum battery. (b) Battery energy ${\mathtt{e}}_B$ versus charging duration $n\mathrm{\Delta }$ for three values of delay $\mathrm{\Delta }$. Here the dashed lines represent the fits to asymptotic charging functions as described in the text. The charging time constants for these three cases are plotted in the inset. (c) Battery energy at saturation ${\mathtt{e}}_B\left(20\mathrm{\Delta }\right)$ (after $n=20$ iterations) versus the delay $\mathrm{\Delta }$ showing the optimal delay range from 7.5 to 10 s. Here the dashed line is a spline curve fit to guide the eye.

Figure 9

(a) The QCBL circuit and its implementation in the 38-spin star-topology system (left) and the NMR pulse sequence for the QCBL circuit (right). Here the dashed lines are spline curve fits to guide the eye. (b) The energy of the battery (${\mathtt{e}}_B$) and the load (${\mathtt{e}}_L$) versus the discharging parameter $J_{BL}{\tau }^{\prime }$. (c) The energy of the load (${\mathtt{e}}_L$) extracted from the battery after a storage time ${\tau }_s$. The dashed line is an exponential fit as discussed in the text.