Charging a quantum battery from the Bloch sphere

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.

• The paper formulates a two-qubit quantum battery model using an arbitrarily prepared charger state on the Bloch sphere to drive energy transfer.
• The paper derives closed-form expressions for optimal charging times, showing that stored energy and ergotropy coincide at peak performance.
• The paper decomposes energy contributions into population inversion and quantum coherence, while also detailing the impact of dissipation.

Analytical Framework for Charging a Quantum Battery from the Bloch Sphere

Introduction and Motivation

This paper formulates and analyzes a two-qubit quantum battery, in which one qubit functions as a charger and the other as the battery. It generalizes the standard quantum battery models by permitting the initial state of the charger qubit to be an arbitrary pure state on the Bloch sphere, parameterized by polar and azimuthal angles $(\theta,\phi)$. The central questions addressed involve quantifying how the quantum coherences and populations inherent in the charger's initial state affect the battery's quantum energetic and thermodynamic properties, particularly focusing on the stored energy, ergotropy, and capacity, as well as their respective charging rates and optimal charging times.

The model is relevant for experimental implementations in solid-state and quantum optical systems where precise state preparation and strong, coherent inter-qubit couplings are feasible. The analysis targets foundational thermodynamic quantities and their origins, distinguishing the contributions from population inversion and off-diagonal coherence.

Model Specification

The quantum battery system consists of two interacting qubits with matched transition frequencies $\omega_b$, coupled via a coherent Hamiltonian with strength $J$. The charger qubit $a$ is initialized in a general pure state on the Bloch sphere, while the battery qubit $b$ starts in its ground state. The system is driven by the following Hamiltonian:

$$\hat{H} = \omega_b (\sigma_a^\dagger \sigma_a + \sigma_b^\dagger \sigma_b) + J (\sigma_a^\dagger \sigma_b + \sigma_b^\dagger \sigma_a)$$

where $\sigma_s$ are standard qubit ladder operators. Charging occurs during the interval $0 < t < T$, after which the coupling is turned off, and the battery enters the storage phase.

The initial charger state on the Bloch sphere is

$$|{\alpha}\rangle = \cos(\tfrac{\theta}{2})|0\rangle_a + e^{i\phi}\sin(\tfrac{\theta}{2})|1\rangle_a$$

with $0\leq \theta \leq \pi$, $0\leq \phi < 2\pi$.

Quantitative Measures: Energetics and Thermodynamics

Key performance measures are established for both energetics and thermodynamics:

• Energy $E(t)$: population in the excited state of the battery qubit.
• Power $P(t)=E(t)/t$: charging rate as a function of time.
• Variance $\sigma_E^2$: fluctuations in stored energy during charging.
• Ergotropy $\mathcal{E}$: extractable work defined as $E-\textsf{E}$, where $\textsf{E}$ is the passive state energy.
• Ergotropic Power $\mathcal{P}(t) = \mathcal{E}(t)/t$ and its maximization.
• Capacity $\mathcal{K}$: difference between active and passive state energies, quantifying the maximal possible energy storage for cyclic unitary evolution.

For all of these metrics, both their optimal values and the times at which these optima occur are derived analytically as functions of $\theta$.

Analytical Solutions and Key Results

The quantum dynamics are exactly solvable under the chosen Hamiltonian for dissipationless evolution. The relevant second and first moments needed for energetics and ergotropy, respectively, are computed explicitly. Main analytical expressions include:

• Stored energy:

$$E_\theta (t) = \omega_b \sin^2(\tfrac{\theta}{2})\sin^2(J t)$$

with the optimal charging time $t_E = \pi/2J$. The dependence on the initial Bloch polar angle captures a quantum area theorem-like structure.

• Charging power peaks at $t_P = A/J$, where $A$ solves $\tan(A)=2A$, independent of $\theta$, but the magnitude scales as $\propto \sin^2(\tfrac{\theta}{2})$.

For the thermodynamic qualities, the ergotropy is given by

$$\mathcal{E}_\theta = \omega_b \left[\sin^2(\tfrac{\theta}{2})\sin^2(J t) - \tfrac{1}{2}(1 - \sqrt{1 - \sin^4(\tfrac{\theta}{2})\sin^2(2Jt)})\right]$$

with the key result that at the optimal time $t_\mathcal{E}=\pi/2J$, the ergotropy attains

$$\mathcal{E}_\theta(t_\mathcal{E}) = \omega_b \sin^2(\tfrac{\theta}{2}) = E_\theta(t_E)$$

i.e., the ergotropy and stored energy coincide at the optimal points for this two-qubit system.

However, contrary to purely energetic considerations, the optimal time for maximum ergotropic charging power $\mathcal{P}$ depends sensitively on $\theta$ and is generally later than the maximum for the standard energetic power. Explicit and numerically accurate approximations are presented for these dependencies.

Regarding the origins of ergotropy and capacity, the paper rigorously decomposes these into contributions from population inversion $\mathcal{I}$ and coherence $\mathcal{C}$. When $\theta = \pi$ (the "south pole"), ergotropy is exclusively due to population inversion; on the equator ($\theta = \pi/2$) at optimal charging, it stems purely from coherence.

Effect of Dissipation

The inclusion of environmental dissipation for the charger (modeled via Lindblad dynamics with rate $\gamma$) produces a renormalization of Rabi frequencies and an exponential decay of all performance metrics. Analytical corrections are provided in terms of the small parameter $\gamma/J$, demonstrating that in the strong-coupling regime ($J\gg \gamma$), the energy and ergotropy optima are both reduced and occur at earlier times, but the qualitative scaling with $\theta$ persists.

Entanglement and Physical Interpretations

Time-resolved concurrence quantifies the charger-battery entanglement throughout charging. At the optimal energy storage time, the two-qubit state is separable, while maximal entanglement occurs at intermediate times associated with Bell-type states. Entanglement plays a secondary role for single-cell batteries, but may become crucial in many-body or collective settings.

Implications and Outlook

This study provides a rigorous and compact analytical foundation for two-level system-based quantum batteries with arbitrary charger initializations. Strong claims substantiated by closed-form results include the precise equivalence between optimum ergotropy and stored energy, and the explicit decomposition of ergotropy/capacity into population and coherence terms, as functions of the initial Bloch polar angle. The analysis demonstrates that quantum coherence can alone suffice to create ergotropy in the absence of population inversion.

Practically, these results offer guidance for state preparation protocols in experiments targeting optimal power and work extraction. The theoretical framework also facilitates the benchmarking of more complex quantum battery architectures, including multipartite or many-body variants.

Future research directions could focus on:

• Scaling to multiple charger and battery cells to assess entanglement-enhanced charging or collective quantum advantages,
• Extension to non-Markovian and strong dissipation regimes,
• Optimal control of charger trajectories on the Bloch sphere for maximal thermodynamic efficiency,
• Real-time adaptive charging and feedback protocols.

Conclusion

This work delivers a complete analytical characterization of the quantum dynamics and thermodynamics of a fundamental two-qubit quantum battery, where the initial charger state is arbitrary on the Bloch sphere. It elucidates the precise roles of population inversion and coherence in generating extractable work and battery capacity, provides optimal charging protocols dependent on the quantum geometric starting point, and incorporates dissipation effects relevant for experimentation. The formalism and key results directly inform both foundational quantum thermodynamics and applied design of quantum energy devices.