Spatial Phase Control of Energy and Ergotropy in Quantum Batteries

The gist The relative geometry between the charger and the quantum battery in a structured environment acts as a fundamental control resource for engineering quantum battery performance, governing energy transfer, charging dynamics, and extractable work through collective interference effects

Thermodynamic Framework

The thermodynamic framework employs the Hamiltonian HB to describe the quantum battery system, with its state represented by the density operator ρB (Page 2). The energy stored in the battery is given by EB = ⟨HB⟩ρB, which represents the total energy content of the system (Page 2). To quantify extractable work, ergotropy W is introduced as W = tr (ρBHB) − tr (σρHB), where σρ is a passive state from which no further work can be extracted through unitary operations (Page 2). The charging power P is defined as the energy stored in the battery over a given charging interval divided by the duration of the energy-storage process, quantified as P = EB/t (Page 2).

Physical Model and Geometry Encoding

The physical model considers two identical qubits located at x1 = −l and x2 = +l inside a mirror-terminated one-dimensional waveguide of length L (Page 3). The total Hamiltonian H is written as H = HS + HB + HI, where HS describes the free Hamiltonian of the two qubits, HB represents the discrete bosonic modes of the structured environment, and HI accounts for the interaction between the qubits and environmental modes (Page 3). The effects of geometry are encoded in coupling amplitudes ξ(j)k = ξkeikxj, with xj showing the location of the j-th qubit (Page 3). Specifically, for qubit 1 acting as the charger and qubit 2 as the battery, the position-dependent coupling coefficients can be expressed as ξ(1)k ≃ ξke + ik0l and ξ(2)k ≃ ξke - ik0l (Page 3).

Collective Channels and Geometric Control

The system dynamics are simplified by introducing collective symmetric and antisymmetric basis states, denoted by +⟩ = (ecgB⟩ + gceB⟩)/√2 and -⟩ = (ecgB⟩−gceB⟩/√2, respectively (Page 3). In this basis, the system–environment interaction Hamiltonian can be compactly expressed as HSB = ∑k √2 [Γs Ξ+s + Γa Ξ+a] ak + H.c., where Γs = ξk cos(k0l), Γa = iξk sin(k0l), and the collective raising operators are defined as Ξ+s = (σ+1 +σ+2)/√2 and Ξ+a = (σ+1 − σ+2)/√2 (Page 3). This form of the interaction Hamiltonian explicitly reveals how the geometric dependence enters the dynamics through the factors cos(k0l) and sin(k0l), which determine the effective coupling strength of the symmetric and antisymmetric collective channels to the structured reservoir (Page 3).

Performance Metrics and Optimal Operation

The charging performance is characterized by several optimal energetic figures of merit: ∆EmaxB = max τ [∆EB(τ)], Wmax = max τ [W(τ)], and Pmax = max τ [P(t)] (Page 4). The analysis reveals that optimal charging occurs at separations where collective bright-state coupling is maximized, whereas dark-state configurations suppress energy transfer despite finite system–environment interaction (Page 4). Furthermore, the results demonstrate that in a structured environment the spatial geometry of the system acts as an effective control parameter for tuning the charging efficiency, dynamical stability, and the interplay between coherent exchange and dissipation (Page 7). Optimal quantum battery performance is achieved in the combined regime of strong collective interference (appropriate geometry) and long reservoir memory (small λ/γ) (Page 6).

Role of Non-Markovian Dynamics

In the presence of environmental memory, non-Markovian effects can give rise to distinct regions in parameter space where the revival of extractable work occurs (Page 2). The appearance of revival-like features signals partial information and energy backflow from the reservoir, which is a characteristic signature of non-Markovian dynamics (Page 2). This behavior indicates that geometry functions as a fundamental control parameter for the thermodynamic activity of the quantum battery, determining not only how energy is stored but whether and when it becomes operationally useful (Page 7). The combined analysis shows that constructive collective interference is essential for maximizing extractable work, while geometries inducing dark-state formation limit accessible energy and ergotropy (Page 7).

Conclusion on Geometry Control

Spatial geometry constitutes a fundamental control parameter for the operation of a quantum battery embedded in a structured environment (Page 7). By engineering the relative separation between the charger and the battery, collective interference between symmetric and antisymmetric channels can be selectively enhanced or suppressed, directly modulating the effective system–reservoir interaction (Page 7). The study establishes geometry as a powerful and experimentally accessible resource for engineering quantum battery performance (Page 7). Optimal charging and maximal work extraction are achieved when both collective pathways contribute coherently to the dynamics (Page 4). The ability to tune charging dynamics through spatial phase control opens new avenues for designing interference-based quantum energy devices in structured photonic environments (Page 7). The combined analysis of internal energy, charging power, and ergotropy jointly determine the operational performance of the battery (Page 7).

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Spatial Phase Control of Energy and Ergotropy in Quantum Batteries

Maryam Hadipour 1 and Soroush Haseli 1, ∗

1Faculty of Physics, Urmia University of Technology, Urmia, Iran

(Dated: March 2, 2026)

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I. INTRODUCTION

In recent years the thermodynamic effects and consequences arising from quantum evolution have emerged as a fundamental framework for the development of novel architectures for high-efficiency energy storage devices (Page 1). These advances also provide the basis for the formulation and optimization of precise protocols for the control, conversion, and management of work and heat, approaches that open perspectives beyond the intrinsic limitations of classical thermodynamic processes [1–4] (Page 1). Within this framework, quantum energy storage devices, commonly referred to as Quantum Batteries (QBs), have emerged in recent years as a central subject of both theoretical and experimental research (Page 1). Owing to their ability to exploit uniquely quantum mechanical features, such as superposition and entanglement, these systems offer novel prospects for efficient energy storage and transfer (Page 1). In realistic scenarios, any physical quantum system inevitably interacts with its surrounding environment (Page 1). Consequently, investigating quantum batteries within the framework of open quantum systems is of fundamental importance (Page 1). Environmental interactions lead to dissipation, decoherence, and memory effects that significantly influence charging dynamics, energy retention, and extractable work (Page 1). Therefore, studying quantum batteries in the context of open quantum systems is essential for assessing their practical performance and operational stability [5] (Page 1).

Consequently, they have been extensively investigated and systematically analyzed in the contemporary literature [4, 6–20] (Page 1). Quantum batteries have been theoretically and experimentally investigated across a broad range of physical realizations (Page 1). Proposed schemes include spinchain models [21–24], atomic ensembles coupled to optical cavities [25–29], and superconducting systems [30–32] (Page 1). These diverse approaches rely on different interaction mechanisms and control techniques, thereby enabling the exploration of multiple strategies for microscopic energy storage and controlled energy transfer within quantum regimes (Page 1).

Although the enhancement of charging power in quantum batteries, together with its favorable scaling with system size [33–35], presents an encouraging prospect, practical implementation remains significantly challenged by the phenomenon of self-discharge induced by environmental interactions (Page 1). Energy dissipation and the degradation of quantum coherence, arising from unavoidable coupling to the surroundings, can substantially constrain the long-term performance and stability of these devices (Page 1).

To mitigate these limitations, several sophisticated strategies have been proposed. These include the deliberate exploitation of quantum coherence in conjunction with disordered local fields [36], the storage of energy in decoherence-resistant dark states [37, 38], and the investigation of non-Markovian dynamical effects as a means of alleviating dissipative losses [39–42] (Page 1). Collectively, these approaches seek to extend the operational lifetime over which quantum batteries can reliably retain stored energy without significant performance degradation (Page 1). Furthermore, considerable research efforts have been devoted to elucidating the possible connection between quantum correlations and the charging efficiency of quantum batteries [43] (Page 1). Despite these extensive investigations, a comprehensive and universally accepted relationship has yet to be established (Page 1).

In this work, we investigate the role of the relative geometry between the charger and the quantum battery in a structured environment (Page 1). In particular, we examine how the spatial seperation between the two qubits and the associated geometric phase influence the energy transfer process, charging dynamics and the amount of extractable work of the battery through their control over collective interference and the spectral structure of the environment (Page 1). Within this framework, we demonstrate that the system geometry not only regulates the effective coupling strength between the charger and the environment, but also, through the engineering of bright and dark collective channels, can suppress or enhance decay processes and energy backflow (Page 1). As a result, the thermodynamic performance of the battery, including charging time, charging power, and maximum ergotropy, becomes directly governed by the geometric distance parameter (Page 1).

1. The AI system can optimize quantum battery performance by treating spatial geometry as a fundamental control resource, specifically tuning the relative separation between the charger and the battery to achieve maximal work extraction or charging power.

2. The improved system will be capable of predicting energy storage and work extraction dynamics based on geometric configurations, as the paper demonstrates that optimal charging occurs at separations where collective bright-state coupling is maximized.

3. The AI can design interference-based quantum energy devices in structured photonic environment by leveraging the findings that constructive interference enhances energy transfer at specific separations, while destructive interference suppresses it, giving rise to geometry-induced dark configurations.

4. The system can dynamically assess the operational utility of stored energy by calculating ergotropy, as the analysis shows that extractable work is highly selective in both time and geometry, distinguishing between total stored energy and the extractable portion of the stored energy.

5. The AI will be able to diagnose dynamic stability issues by identifying conditions where dark-state protection occurs or where performance is limited, as the study shows that geometries inducing dark states limit the accessible energy and ergotropy.

We investigate the role of spatial geometry in controlling energy storage and work extraction in a non-Markovian quantum battery. The system consists of two identical two-level systems coupled to a structured waveguide reservoir, where one qubit serves as the charger and the other as the battery. The spatial separation between the qubits introduces a geometry-dependent phase that governs collective interference, directly modulating the decay channels and energy transfer dynamics. We show that tuning the interqubit distance allows for the engineering of bright and dark collective pathways, which, in conjunction with environmental memory, significantly enhances the charging power, maximal stored energy, and ergotropy revivals.