Quantum Battery Dynamics

• Quantum battery is a finite quantum system that utilizes collective many-body interactions and correlations for enhanced energy storage and ergotropy.
• The system employs interacting spin-chain models with coherent and thermal charging protocols, demonstrating distinct energy deposition and extraction dynamics.
• Disorder and quantum phase transitions critically influence the battery's performance by boosting stored energy while potentially reducing extractable work.

A quantum battery is a finite quantum system designed for temporary energy storage, in which quantum effects—especially collective many-body interactions and correlations—play a pivotal role in the charging, storing, and work-extraction cycle. The paper of quantum batteries aims to uncover mechanisms leading to enhanced energy storage, higher ergotropy (maximal extractable work), and nontrivial dynamical phenomena compared to classical analogs. Interacting spin-chain models represent a paradigmatic platform for exploring quantum battery functionality, particularly in the presence of environmental noise, rich many-body dynamics, and varying charging protocols.

1. Model Architecture: Interacting Spin Chain in a Cavity

The core system consists of an $N$-site chain of two-level systems (spins), each with energy spacing $\omega_a$. These spins interact via a nearest-neighbor hopping (exchange) term of strength $J$, and collectively couple to a single mode of a cavity field with frequency $\omega_c$ and coupling constant $g$. The total system Hamiltonian is: $H_S = H_A + H_B + H_I$ with

$H_A = \omega_c c^\dagger c,$

$$H_B = \omega_a \sum_{i=1}^{N} \sigma_+^{(i)}\sigma_-^{(i)} + J \sum_{i=1}^{N-1} (\sigma_+^{(i)} \sigma_-^{(i+1)} + \text{h.c.}),$$

$$H_I = \sum_{i=1}^{N} g(\sigma_+^{(i)} c + \text{h.c.}),$$

where $c$ ($c^\dagger$) are cavity photon annihilation (creation) operators, and $\sigma_\pm^{(i)}$ are the raising/lowering operators for the $i$th spin.

The hopping term $J$ implements band splitting within the spin chain and is critical for generating collective quantum phenomena—the key to enhanced storage and ergotropy.

2. Charging Protocols: Coherent Versus Thermal

Charging is enacted via the cavity mode, using either a classical coherent drive or thermal bath coupling:

• Coherent Charging:

A driving field with amplitude $f$ couples to the cavity as $H_d = f(c^\dagger + c)$. In the rotating frame and under resonant drive ($\omega_d = \omega_c$), the dynamics (including cavity decay with rate $\kappa$) are described by the Lindblad master equation:

$$\frac{d\rho_S}{dt} = -i[H_S + H_d, \rho_S] + \kappa L_c[\rho_S],$$

where $$L_c[\rho_S] = c \rho_S c^\dagger - \frac{1}{2}\{c^\dagger c, \rho_S\}$$.

• Thermal Charging:

Thermalization involves connecting the cavity to a heat bath at temperature $T$. The evolution is governed by:

$$\frac{d\rho_S}{dt} = -i[H_S, \rho_S] + \kappa(n_B+1)L_c[\rho_S] + \kappa n_B L_{c^\dagger}[\rho_S],$$

where $n_B = 1/(\exp(\omega_c/(k_BT)) - 1)$.

The population and coherence transfer pathways depend crucially on these protocols, with thermal driving enabling nonzero ergotropy only for $N\geq 2$ due to collective effects.

3. Quantifying Energy Storage and Ergotropy

The energy stored in the battery subsystem at time $t$ is: $E_B(t) = \operatorname{tr}[H_B \rho_B(t)],$ where $\rho_B(t)$ is the reduced density matrix of the spin chain (after tracing out the cavity mode). The net charged energy, measured with respect to the ground state $E_G$, is: $\Delta E(t) = E_B(t) - E_G.$ The fraction of stored energy that can be extracted as work—i.e., the ergotropy—obeys

$$\epsilon_B(t) = E_B(t) - \min_U \operatorname{tr}[H_B U \rho_B(t) U^\dagger],$$

where the minimization is over all unitaries $U$ and, for a diagonalized $H_B$ ($e_n$ ascending) and $\rho_B$ ($r_n$ descending), $$\min_U \operatorname{tr}[H_B U \rho_B(t) U^\dagger] = \sum_n r_n e_n$$.

Both $\Delta E$ and $\epsilon_B$ exhibit protocol-dependent dynamics and encode the thermodynamic usefulness of the charged state.

4. Role of the Hopping Interaction and Quantum Phase Transition

The hopping term $J$ fundamentally alters the battery spectrum. For $J = 0$, spins are independent and ergodic effects are negligible; the ground state is trivial ($E_G=0$, fully polarized). Introducing $J\neq 0$ splits energy bands and hybridizes excitations, raising the capacity for energy storage and extractable work.

A ground-state quantum phase transition (QPT) emerges as a non-analyticity in $E_G$ at critical $J_c$ (e.g., $J_c \sim 1/\sqrt{N}$ or $J_c \sim \sqrt{N}$, depending on $N$). This transition manifests as discontinuities or cusps in both $\Delta E$ and $\epsilon_B$ at $J_c$, and is reflected in order parameters: $$M_z = \langle S_z \rangle / N, \quad \xi_z = \langle S_z^2 \rangle / N^2,$$ with $S_z = \sum_{i=1}^{N} \sigma_z^{(i)}$.

The quantum phase transition sharply demarcates regimes of low and high battery performance, with the post-QPT phase exhibiting considerable enhancement of both stored energy and ergotropy due to the lowering of $E_G$.

5. Dynamics and Efficiency of Charging Protocols

• Coherent Driving: Generates oscillations in both $\Delta E(t)$ and $\epsilon_B(t)$, arising from Rabi-like energy exchange between the cavity and the spin chain. Increasing the drive strength $f$ boosts both the total stored energy and ergotropy, but the ratio $R_B(t) = \epsilon_B(t)/\Delta E(t)$ (the efficiency) can show non-monotonic dependence on $f$ due to these oscillations.
• Thermal Driving: For $N=1$, thermal charging produces a passive (zero-ergotropy) state in the battery. For $N \geq 2$, many-body effects generate nonzero ergotropy, distinguishing this regime from both the trivial non-interacting case and prior single-spin results. Thus, incoherent heat sources can indeed “charge” a quantum battery provided that collective degrees of freedom are available.

6. Impact of Onsite Disorder

Onsite disorder is introduced as random shifts $\delta_i$ in the energy spacing for each spin, modifying the Hamiltonian term: $$H_B = \omega_a \sum_{i=1}^{N} (1 + \delta_i)\sigma_+^{(i)} \sigma_-^{(i)} + J \sum_{i=1}^{N-1} (\sigma_+^{(i)} \sigma_-^{(i+1)} + \text{h.c.}),$$ where $\delta_i \in [-W/2, W/2]$ describes the disorder amplitude.

Key effects are:

• Energy enhancement: Disorder increases the overall stored energy in the battery, attributable to disorder-assisted (noise-assisted) energy transfer processes.
• Ergotropy stability: While more energy can be charged in the presence of disorder, the fraction of extractable work (ergotropy) becomes less robust: strong disorder can substantially reduce $\epsilon_B/\Delta E$.

This interplay highlights a trade-off in using disorder to boost energy deposition at the potential cost of extractable work.

7. Synthesis and Physical Implications

This spin-chain quantum battery model underscores how many-body effects, controlled interactions ($J$), and environmental features (disorder, dissipation) fundamentally shape both achievable storage and work-extraction capacities. The key phenomena include:

• Band splitting and QPT-induced performance enhancement at distinct interaction strengths.
• Nontrivial ergotropy generation under thermal driving via collective many-body states, beyond single-spin limitations.
• Disorder-induced energy enhancement (with potential ergotropy reduction), pointing to regimes where noise can be converted into useful battery performance.
• Protocol-dependent efficiency with oscillatory behavior and nontrivial dependence on external drive strength.

The outlined model is experimentally relevant for quantum platforms such as spin chains engineered in cavity QED, circuit QED, or similar architectures, with clear prescriptions for optimizing battery performance via tuning hopping interaction $J$, cavity driving strength $f$, and disorder amplitude $W$.

The comprehensive analysis provides both a quantitative and qualitative foundation for designing quantum batteries that leverage many-body physics and open-systems effects to achieve energy storage performance unattainable with classical or non-interacting models.