Charging a quantum battery from the Bloch sphere

TL;DR

This work analyzes a two-qubit quantum battery in which the charger qubit can start in any pure state on the Bloch sphere, parameterized by the polar angle $\theta$. Solving the exact dynamics under a resonant coupling $J$ yields closed-form expressions for stored energy $E$, ergotropy $\mathcal{E}$, and capacity $\mathcal{K}$ as functions of $\theta$, highlighting how coherences and population inversions drive extractable work and how a quantum area theorem–like relationship emerges. The study identifies the optimal energetic time $t_E=\pi/(2J)$ (independent of $\theta$) but shows that ergotropy and ergotropic power have $\theta$-dependent optimal times, while a dissipative Lindblad model provides quantitative corrections for realistic setups. The analysis clarifies the roles of coherence and population inversion in quantum thermodynamics, offering guidance for experiments with coupled two-level systems and a springboard for exploring many-body quantum energy devices.

Abstract

We reconsider the quantum energetics and quantum thermodynamics of the charging process of a simple, two-component quantum battery model made up of a charger qubit and a single--cell battery qubit. We allow for the initial quantum state of the charger to lie anywhere on the surface of the Bloch sphere, and find the generalized analytical expressions describing the stored energy, ergotropy and capacity of the battery, all of which depend upon the initial Bloch sphere polar angle in a manner evocative of the quantum area theorem. The origin of the ergotropy produced, as well as the genesis of the battery capacity, can be readily traced back to the quantum coherences and population inversions generated (and the balance between these two mechanisms is contingent upon the starting Bloch polar angle). Importantly, the ergotropic charging power and its associated optimal charging time display notable deviations from standard results which disregard thermodynamic considerations. Our theoretical groundwork may be useful for guiding forthcoming experiments in quantum energy science based upon coupled two-level systems.

Figures (4)

• Figure 1: Charging a quantum battery from the Bloch sphere. Panel (a): a geometrical representation of the space of possible quantum states of the qubit modelling the quantum mechanical charger [cf. Eq. \ref{['eq:zdfdzdfdzf']}]. Panel (b): the quantum battery is built from a pair of coupled two-level systems, where the charger is associated with the label $a$ and the battery with $b$. The qubits are each characterized by the transition frequency $\omega_b$ [cf. Eq. \ref{['eq:dsfsdfsdf']}] and they exchange energy at the coupling rate $J$ [cf. Eq. \ref{['eq:dsfsdfsdffhfhff']}] during the charging phase.
• Figure 2: Energetics of the quantum battery. Panel (a): the energy $E$ stored in the quantum battery, in units of the transition frequency $\omega_b$, as a function of the time $t$ which has elapsed (in units of the inverse coupling rate $1/J$) since the coupling was switched on [cf. Eq. \ref{['eq:sfddddddffsdfsf']}]. Dashed line: the optimal time $t_{E}$ [cf. Eq. \ref{['eq:dvddeddfg']}]. Panel (b): the charging power $P$, in units of $\omega_b J$, as a function of time $t$ [cf. Eq. \ref{['eq:sfgfgfgfgfgdsdfsf']}]. Dashed line: the optimal time $t_{P}$ [cf. Eq. \ref{['eq:dvdxfgxffdvdsgfg']}]. Coloured lines in panels (a, b): the various values of the Bloch sphere polar angle $\theta$ considered [cf. Eq. \ref{['eq:zdfdzdfdzf']}]. Panel (c): the optimal stored energy $E(t_{E})$ (purple line) and optimal charging power $P(t_{P})$ (yellow line), in units of $\omega_b$ and $B \omega_b J$ respectively, as a function of the angle $\theta$ [cf. Eq. \ref{['eq:dfgdgdfgbv']} and Eq. \ref{['eq:dfxgfxgfggdg']}]. Thin green line: the energetic standard deviation $\sigma_{E} (t_{E})$ [cf. Eq. \ref{['eq:sdfsdfssdfsdfsdsf']}].
• Figure 3: Thermodynamics of the quantum battery. Panel (a): the energy $E$ (pink line), passive state energy $\textsf{E}$ (cyan line), and ergotropy $\mathcal{E}$ (red line) in the quantum battery, all in units of the transition frequency $\omega_b$, as a function of the time $t$ which has elapsed (in units of the inverse coupling rate $1/J$) since the coupling was switched on [cf. Eq. \ref{['eq:sfddddddffsdfsf']} and Eq. \ref{['eq:fsdfs']}]. Vertical pink and red lines: the optimal times $t_{E}$ and $t_{\mathcal{E}}$ respectively [cf. Eq. \ref{['eq:dvddeddfg']} and Eq. \ref{['eq:dvdfg']}]. Panel (b): the charging power $P$ (pink line), passive state charging power $\textsf{P}$ (cyan line), and ergotropic charging power $\mathcal{P}$ (red line), all in units of $\omega_b J$, as a function of time $t$ [cf. Eq. \ref{['eq:sfgfgfgfgfgdsdfsf']}]. Vertical pink and red lines: the optimal times $t_{P}$ and $t_{\mathcal{P}}$ respectively [cf. Eq. \ref{['eq:dvdxfgxffdvdsgfg']} and Eq. \ref{['eq:dgsfsdf']}]. In this column, the Bloch sphere polar angle $\theta = \pi/2$. Panels (c, d): as for the first column, but where $\theta = \pi$. Panel (e): the optimal charging times for energy and ergotropy $t_{E}$ and $t_{\mathcal{E}}$ (purple line), for charging power $t_P$ (pink line) and for ergotropic charging power $t_{\mathcal{P}}$ (yellow line), all in units of $1/J$, as a function of $\theta$ [cf. Eq. \ref{['eq:dvddeddfg']}, Eq. \ref{['eq:dvdxfgxffdvdsgfg']} and Eq. \ref{['eq:dvdfg']}]. Dashed orange line: the analytic approximation of Eq. \ref{['eq:dgsfsdf']}. Panel (f): the optimal quantities associated with the charging times of panel (e), including the peak energy $E(t_{E})$ and peak ergotropy $\mathcal{E} (t_{\mathcal{E}})$ (purple line), both in units of $\omega_b$, and the peak ergotropic charging power $\mathcal{P} (t_{\mathcal{P}})$ (yellow line) and peak charging power $P(t_{P})$ (pink line), both in units of $\omega_b J$, all as a function of $\theta$ [cf. Eq. \ref{['eq:dfgdgdfgbv']}, Eq. \ref{['eq:dfxgfxgfggdg']} and Eq. \ref{['eq:dfgdg']}]. Dashed orange line: the analytic approximation of Eq. \ref{['eq:sdfsdf']}.
• Figure 4: The origin of the ergotropy and the genesis of the battery capacity. Panel (a): the ergotropy $\mathcal{E}$ (red line) and capacity $\mathcal{K}$ (dashed cyan line), both in units of the transition frequency $\omega_b$, as a function of the time $t$ which has elapsed (in units of the inverse coupling rate $1/J$) since the coupling was switched on [cf. Eq. \ref{['eq:dgffgfg']} and Eq. \ref{['eq:dgffgfg22222']}]. Green line: the population inversion parameter $\mathcal{I}$ [cf. Eq. \ref{['eq:sfdsgdgfdfg']}]. Yellow line: the coherence parameter $\mathcal{C}$ [cf. Eq. \ref{['eq:sfdsgdgfdfg222']}]. In this panel the Bloch sphere polar angle $\theta = \pi/2$. Panel (b): as for panel (a) but where $\theta = \pi$. Panel (c): the dynamical concurrence $C$ generated in the two--qubit system [cf. Eq. \ref{['eq:conccc']}]. Coloured lines: the various values of the Bloch sphere polar angle $\theta$ considered [cf. Eq. \ref{['eq:zdfdzdfdzf']}].