Charging Quantum Batteries via Indefinite Causal Order: Theory and Experiment

In the standard quantum theory, the causal order of occurrence between events is prescribed, and must be definite. This has been maintained in all conventional scenarios of operation for quantum batteries. In this study we take a step further to allow the charging of quantum batteries in an indefinite causal order (ICO). We propose a nonunitary dynamics-based charging protocol and experimentally investigate this using a photonic quantum switch. Our results demonstrate that both the amount of energy charged and the thermal efficiency can be boosted simultaneously. Moreover, we reveal a counterintuitive effect that a relatively less powerful charger guarantees a charged battery with more energy at a higher efficiency. Through investigation of different charger configurations, we find that ICO protocol can outperform the conventional protocols and gives rise to the anomalous inverse interaction effect. Our findings highlight a fundamental difference between the novelties arising from ICO and other coherently controlled processes, providing new insights into ICO and its potential applications.

Figure 1

Illustrations of the three scenarios. (a) DCO scenarios, where two chargers are sequentially arranged, resulting in either ${\mathcal{C}}_1\circ {\mathcal{C}}_2$ (upper) or ${\mathcal{C}}_2\circ {\mathcal{C}}_1$ (lower) configurations. (b) ICO charging realized by a QS that entangles two causal orders, allowing the causal order of operations on a quantum battery to be in a quantum superposition by preparing the order system in a superposition state. (c) NUCC protocol based on coherently controlling the path a battery takes, thereby achieving the effect of performing charging with ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ simultaneously.

Figure 2

Experimental setup, illustrating single photons produced through the type-I spontaneous parametric down-conversion (SPDC) process. A Mach-Zehnder interferometer structure realizes a QS, enabling the ICO charging process. The projecting measurements are carried out via a combination of QWP-HWP-PBS. Beam splitter (BS), half-wave plate (HWP), quarter-wave plate (QWP), beam displacer (BD), polarization beam splitter (PBS), $\beta$-barium borate (BBO), interference filter (IF), avalanche photodiode (APD).

Figure 3

Coupling strength dependence of performance for ICO charging protocol, its DCO, and NUCC counterparts. (a) Logarithm for population ratio ${\log }_{(1-p)/p}\mathcal{R}$ versus $\kappa /\omega$. ${\mathcal{R}}_{*}^{-}$, ${\mathcal{R}}_{\max }^{-}$, ${\mathcal{R}}_{\min }^{-}$, ${\mathcal{R}}_{\max }^{\mathrm{D}}$, and ${\mathcal{R}}_{*}^{\mathrm{N},-}$ are shown by (blue) solid, dashed, dotted, red, and orange curves, respectively. The corresponding experimental results are shown by symbols. (b) Thermal efficiency $\eta$ versus $\kappa /\omega$, where the meaning of curves(dots) with distinct colors are similar to those in (a).

Figure 4

Performance of protocol with fixed coupling strength ($\kappa /\omega =1$). (a) Population ratio $\mathcal{R}$ versus parameter $p$ of initial battery states. (b) Thermal efficiency $\eta$ versus $p$. Experimental results and theoretical predictions are shown by dots and curves, respectively. Blue and red curves correspond to ${\mathcal{R}}_{*}^{-}$ (${\eta }_{*}^{-}$) and ${\mathcal{R}}^{\mathrm{D}}$ (${\eta }^{\mathrm{D}}$), while gray curve represents ${\mathcal{R}}^{\mathrm{D}}$ (${\eta }^{\mathrm{D}}$) with coupling strength set to infinity.