Stable and Efficient Charging of Superconducting Capacitively Shunted Flux Quantum Batteries

TL;DR

This work addresses fast, stable charging of three-level quantum batteries implemented on a capacitively-shunted flux qubit. It compares STIRAP and counterdiabatic (CD) STIRAP under a total-Hamiltonian-norm constraint, and introduces a unifying performance metric $\mathcal{S}=1/(\tau_c\xi)$ that blends charging speed with stability. The authors demonstrate near-quantum-speed-limit charging, achievable by isolating the $|g\rangle\leftrightarrow|f\rangle$ path and applying a protective $\Omega_{gf}$ drive, assisted by a fast $Z$-pulse, and quantify stability via ergotropy oscillations post-peak. The results establish the capacitively shunted flux qubit as a strong platform for three-level quantum batteries and offer practical routes for optimizing energy transfer under experimental constraints, with implications for quantum thermodynamics and energy-efficient quantum information processing.

Abstract

Quantum batteries, as miniature energy storage devices, have sparked significant research interest in recent years. However, achieving rapid and stable energy transfer in quantum batteries while obeying quantum speed limits remains a critical challenge. In this work, we experimentally optimize the charging process by leveraging the unique energy level structure of a superconducting capacitively-shunted flux qubit, using counterdiabatic pulses in the stimulated Raman adiabatic passage. Compared to previous studies, we impose two different norm constraints on the driving Hamiltonian, achieving optimal charging without exceeding the overall driving strength. Furthermore, we experimentally demonstrate a charging process that achieves the quantum speed limit. In addition, we introduce a dimensionless parameter $\mathcal{S}$ to unify charging speed and stability, offering a universal metric for performance optimization. In contrast to metrics such as charging power and thermodynamic efficiency, the $\mathcal{S}$ criterion quantitatively captures the stability of ergentropy while also considering the charging speed. Our results highlight the potential of the capacitively-shunted qubit platform as an ideal candidate for realizing three-level quantum batteries and deliver novel strategies for optimizing energy transfer protocols.

Figures (10)

• Figure 1: Schematic representation of the charging method and the C-shunt flux qubit device. (a) Illustration of the STIRAP and CD-STIRAP methods in a three-level system. (b) Steady-state charging curve of the ergotropy, where $\tau_c$ denotes the charging time when the maximum value is first reached. The standard deviation of the ergotropy for $\tau > \tau_c$ is denoted as $\xi$, which is used to assess the stability of the charging process after it completes. The enlarged grey region corresponds to the part of $\mathcal{E}$ with $\tau > \tau_c$. The green dashed line indicates the average value $\overline{\mathcal{E}}$ in this region. The maximum ergotropy is denote as $\mathcal{E}_\text{max}$, which is equal to the energy of the highest energy level in the quantum system. The blue curve is obtained from numerical simulations based on the STIRAP method, while the red curve corresponds to simulations based on the quantum speed limit (QSL). (c) Schematic diagram of the C-shunt flux qubit. (d) 3D schematic representation of the C-shunt flux qubit, which shows the actual structure of the qubit.
• Figure 2: Properties of the C-shunt flux qubit near $0.5\Phi_0$ point. (a) The lower plot shows the energy spectrum of the qubit's $|g\rangle \leftrightarrow |e\rangle$ transition, while the upper plot shows the energy spectrum of the $|g\rangle \leftrightarrow |f\rangle$ transition. Transition frequencies of the qubit at sweet point are $\omega_{ge}/2\pi = 2.6612~\text{GHz}, \omega_{gf}/2\pi = 6.1703~\text{GHz}$ and the qubit anharmonicity: $(\omega_{ef} - \omega_{ge})/2\pi = 0.8479~\text{GHz}$. Fitting the energy spectrum yields the qubit parameters: $C_{\text{J}} =9~\text{fF}$, $C_{\text{sh}}= 45~\text{fF}$, $\alpha =0.471$, and $I_c =88~\text{nA}$. (b) The Rabi oscillation frequency between the $|g\rangle$ and $|f\rangle$ states at different external magnetic flux. Using these results, the transition probability as a function of the external flux can be calibrated as summarizing in (c), where $A$ is a factor related to the coupling between the XY control lines and the qubit. (d) Population decay with time when $\Phi_{\text{ext}} = 0.5\Phi_0$. The left panel shows the qubit initially in the $|e\rangle$ state, while the right panel shows the qubit initially in the $|f\rangle$ state. From the data, we get $\Gamma_{fg} = 0.2~\text{kHz}$. (e) Population decay with time when $\Phi_{\text{ext}} = 0.496\Phi_0$. From the data, we get $\Gamma_{fg} = 20.0~\text{kHz}$.
• Figure 3: Charging optimization of CD-STIRAP method under different constraint conditions. (a) Numerical simulation of the charging process for different values of $\eta$ under the constraint $\Omega^2_{ge}(t) + \Omega^2_{ef}(t) + \Omega^2_{gf}(t) = \Omega^2_{\text{max}}$. The obtained values of $\tau_c$, $\xi$, and $\mathcal{S}$ for different $\eta$ values are normalized, and the results are shown in (b). The maximum value of $S$ is obtained for $\eta = 0.12$. (c) Comparison of the charging curves for $\eta = 0.12$ and $\eta = 0$, with the dashed line representing the numerical simulation results. The error bars indicate standard error (SE) of the data. All curves are within the QSL limit. The inset shows the pulse envelope of $\Omega^{\text{tri}}_{ge}(t)$ (blue) and $\Omega^{\text{tri}}_{ef}(t)$ (yellow). (d) Numerical simulation of the charging process for different values of $\eta$ under the constraint $\Omega_{ge}(t) + \Omega_{ef}(t) + \Omega_{gf}(t) = \Omega_{\text{max}}$. The maximum value of $S$ is obtained for $\eta = 0.14$ as shown in (f). (e) Comparison of the charging curves for $\eta = 0.14$ and $\eta = 0$, with the dashed line representing the numerical simulation results. The inset shows the Pulse envelope of $\Omega^{\text{cyc}}_{ge}(t)$ (blue) and $\Omega^{\text{cyc}}_{ef}(t)$ (yellow).
• Figure 4: A charging process achieving QSL. (a) Blue points represent experimental data, and the dashed line represents the numerical simulation results. (b) Schematic of the pulse sequence used in this method. The detuning between the drive frequency and the $g$–$f$ transition frequency is denoted by $\Delta$, where $\Delta = (E_f - E_g) - \omega_{gf}$. The protective pulse leads to a resonant cutoff when $\Delta = 0$, and a detuned cutoff when $\Delta / 2\pi = 48~\mathrm{MHz}$.
• Figure 5: Photograph of the device. (i) False-color micrograph of the device. The circuit elements are color-coded as follows: readout line (green), readout resonator (blue), shunt capacitor (yellow), XY control line (orange), Z control line (pink) and airbridges (purple). (ii) Localized micrograph of the shunt capacitor, with the qubit loop outlined in red. (iii)–(iv) Scanning electron micrographs: (iii) qubit loop, (iv) small Josephson junction, (v) Large Josephson junction.