Experimental Extraction of Coherent Ergotropy and Its Energetic Cost in a Superconducting Qubit

TL;DR

The paper investigates how initial-state coherence affects the ergotropy that can be extracted from a superconducting transmon qubit and the thermodynamic cost of extraction. It advances by implementing three work-extraction protocols—dephasing extraction, direct extraction, and sequential extraction—to separate coherence-independent ergotropy from coherence-consuming ergotropy, while preparing pure states with a tunable Rabi angle $\theta$ on a two-level system ${\rho_S = |\psi(\theta)\rangle\langle\psi(\theta)|}$ where ${|\psi(\theta)\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle}$. The study quantifies coherent ergotropy $\mathcal{E}_c$ and incoherent ergotropy $\mathcal{E}_i$, demonstrates how $\mathcal{E}_c$ tracks quantum coherence $C(\rho_S)$, and identifies the optimal initial states under different decoherence channels (energy relaxation vs dephasing) to maximize charging efficiency. By incorporating thermodynamic costs via gate operation energies and defining efficiency $\eta = \Delta E /(\Delta E + \Sigma)$ with $\Sigma = \frac{1}{\tau}\int_0^{\tau} \|H_d(t)\|dt$, the results reveal a trade-off between coherence-assisted work and energetic cost, highlighting a practical route to coherence-controlled quantum energy storage and informing design principles for scalable quantum batteries. The work further shows that, for superconducting platforms where dephasing dominates, incoherent storage (tuning $\theta$ toward $\pi$) can maximize efficiency, while coherent storage (around $\theta \approx \pi/2$) becomes favorable when dephasing is suppressed or when dynamical decoupling extends $T_2$, offering guidelines for optimizing initial-state design in quantum thermodynamic devices.

Abstract

Quantum coherence, encoded in the off-diagonal elements of a system's density matrix, is a key resource in quantum thermodynamics, fundamentally limiting the maximum extractable work known as ergotropy. While previous experiments have isolated coherence-related contributions to work extraction, it remains unclear how coherence can be harnessed in a controllable and energy-efficient manner. Here, we experimentally investigate the role of initial-state coherence in work extraction from a superconducting transmon qubit. By preparing a variety of pure states and implementing three complementary extraction protocols, we reveal how coherence governs the partitioning of ergotropy. We find that the choice of initial state depends on the dominant decoherence channel-energy relaxation or dephasing. By further accounting for thermodynamic costs, we identify optimal initial states that maximize the efficiency. Our results demonstrate that the initial-state design provides a scalable approach to coherence control and advances the development of efficient quantum thermodynamic devices.

Figures (4)

• Figure 1: Coherent and incoherent ergotropy in a superconducting qubit. (a) Bloch sphere representation of ergotropy extraction from a pure state $\rho_S = \ket{\psi(\theta)}\bra{\psi(\theta)}$ (black arrow). The ergotropy $\mathcal{E}(\rho_S)$ corresponds to the green arc to the ground state (passive state). Dephasing $\Delta$ yields the mixed state $\delta_{\rho_S}$ (upward orange arrow), whose ergotropy is the orange arc. The blue arc denotes incoherent ergotropy—the maximal work extractable without consuming coherence—ending in $\sigma_{\rho_S}$. The remainder, coherent ergotropy, is extractable via a unitary $U_c$. (b) Coherent ergotropy as a function of coherence and Rabi angle $\theta$. It increases with coherence at fixed $\theta$, and depends on $\theta$ due to gate-induced coherence. (c) Experimental implementation using a transmon qubit. $\Phi_0 = h/2e$ is the flux quantum, where $e$ is the elementary charge and $h$ is Planck’s constant. The flux dependence of the qubit frequency $\omega_q$ shows a sweet spot (purple triangle) with long dephasing time $T_2 = 32.7\mu\mathrm{s}$ and short $T_1 = 25.7\mu\mathrm{s}$, and a detuned point (red star) with $T_2 = 2.2\mu\mathrm{s}$ and $T_1 = 64.5\mu\mathrm{s}$, enabling preparation of $\delta_{\rho_S}$.
• Figure 2: Experimental comparison of three work extraction protocols from a superconducting qubit. (a) Bloch sphere representation of quantum states under dephasing extraction (orange), direct extraction (green), and sequential extraction (blue). Dots indicate individual state preparations; arrows show average Bloch vectors. (b) Energy–coherence diagram for the extracted states. Each trajectory illustrates how energy and coherence are reduced under different protocols. The error bars represent the standard error. Gray diamonds represent numeric simulation results accounting for all decoherence effects.
• Figure 3: Variation of ergotropy and coherence in the sequential protocol as a function of the Rabi angle $\theta$. (a) Change in $\mathcal{E}$ and coherence $\Delta C$ of the prepared state $\rho_\theta$ relative to the ground state. The blue bars represent experimental data, and the brown bars represent theoretical calculations. (b) Coherent ergotropy $\mathcal{E}_c$ versus $\theta$; the inset shows the coherence drop after coherent extraction during $U_c$, indicating full consumption. (c) Incoherent ergotropy $\mathcal{E}_i$ as a function of $\theta$, becoming dominant near $\theta = \pi$. The inset shows coherence remaining unchanged during $V_\pi$. (d) Initial coherence $C(\rho_\theta)$ plotted against $\mathcal{E}_c$ and $\mathcal{E}_i$, revealing their respective positive and negative correlations with coherence. The black solid line represents the theoretical calculation. All error bars represent the standard error.
• Figure 4: Thermodynamic efficiencies $\eta$ of state preparation $R_Y(\theta)$, coherent extraction $U_c$, incoherent extraction $V_\pi$, and total extraction versus coherence $C(\rho_\theta)$. The inset marks a local maximum preparation efficiency at $\theta_m \approx 0.742\pi$ with $C_m \approx 0.432$ and equal extraction efficiencies at $\theta_e \approx 0.722\pi$ with $C_e \approx 0.469$. The error bars represent the standard error.