Charging Dicke and Tavis--Cummings quantum batteries with a $\mathcal{PT}$-symmetric lossy--gain cavity pair

Technical Summary: Charging Dicke and Tavis–Cummings Quantum Batteries with a PT-Symmetric Lossy–Gain Cavity Pair

Problem Statement
The paper investigates the charging dynamics of quantum batteries modeled as collections of $N$ two-level atoms (Dicke or Tavis–Cummings models) coupled to a photonic environment. While quantum charging advantages via collective effects are well-established in closed systems, the behavior of these batteries in open, non-Hermitian environments remains less explored. Specifically, the authors address how a Parity-Time ($\mathcal{PT}$)-symmetric photonic environment—consisting of a balanced pair of lossy and gain cavities—affects the charging of quantum batteries. The study aims to determine if the unique spectral properties of $\mathcal{PT}$-symmetry, particularly the exceptional point (EP), can enhance energy storage, power, and extractable work (ergotropy) compared to isolated batteries or those coupled to standard dissipative environments.

Methodology
The authors employ a combination of analytical derivations and numerical simulations:

1. Analytical Models:
   • Isolated Batteries: Short-time perturbation theory is used to derive the scaling of excitation numbers for both Tavis–Cummings (TC) and Dicke models. An exact bound on stored energy is derived for the TC model based on excitation number conservation.
   • PT-Symmetric Cavity Pair: An exact closed system of moment equations is derived for the empty cavity pair to locate the exceptional point and characterize the growth rates in the broken and unbroken phases.
   • Loaded System: In the low-excitation limit, the system is linearized using the Holstein–Primakoff mapping to solve for the eigenvalues of the full four-mode network (two cavities + two batteries). For the Dicke model, a Floquet analysis is performed on the Bogoliubov system to account for counter-rotating terms.
2. Numerical Simulations:
   • The full Lindblad master equation is solved using a batched Monte-Carlo wave-function method with $K=300$ trajectories.
   • Simulations cover the Tavis–Cummings model (rotating-wave approximation) and the Dicke model (including counter-rotating terms) across a range of inter-cavity couplings ($J$) relative to the loss/gain rate ($\kappa = \gamma/2$).
   • Key observables include stored energy ($E$), charging power ($P$), and ergotropy ($W$), calculated for both the gain-side and lossy-side batteries.

Key Contributions and Results

1. Isolated Battery Dynamics:

   • For atoms prepared in the collective ground state, the excitation number grows quadratically with time ($t^2$). The coefficient is $\lambda^2 \bar{n}_0$ for the TC model and $\lambda^2(4\bar{n}_0 + 1)$ for the Dicke model, where $\bar{n}_0$ is the initial mean photon number.
   • An exact upper bound on stored energy for the TC model is established: $E_{\text{max}} \leq \omega_0 \min(\bar{n}_0, N)$. This bound is saturated numerically to within 2%.
   • The Dicke model violates this bound at ultrastrong coupling due to counter-rotating terms creating real photons from virtual ones, exceeding the bound by a few percent.

2. PT-Symmetric Charging Mechanism:

   • The system consists of two identical batteries, one in a lossy cavity and one in a gain cavity, maintaining strict $\mathcal{PT}$ symmetry of the generator.
   • Broken Phase ($J < \kappa$): The gain cavity accumulates photons while the lossy cavity remains nearly empty. Consequently, the gain-side battery charges significantly, while the lossy-side battery remains uncharged.
   • Unbroken Phase ($J > \kappa$): Strong coherent coupling shuttles photons to the lossy cavity. The lossy-side battery intercepts these photons before they are dissipated, eventually overtaking the gain-side battery in stored energy.
   • Exceptional Point: The transition between these regimes occurs at $J = \kappa$.

3. Performance Metrics:

   • Optimal Regime: The best charging performance is found in the broken phase, close to but not exactly at the exceptional point.
   • Quantitative Gains: Compared to an isolated battery, the gain-side battery in the broken phase ($J/\kappa = 0.25$) gains approximately 39% in stored energy, 26% in power, and 28% in ergotropy.
   • Deep Unbroken Phase: Conversely, deep in the unbroken phase ($J/\kappa = 8$), the gain-side battery loses roughly 65% of its potential energy and work capacity compared to the isolated case, while the lossy-side battery recovers a significant portion of the remaining energy.

4. Impact of Counter-Rotating Terms (Dicke vs. TC):

   • Restoring counter-rotating terms (switching from TC to Dicke) does not alter the qualitative charging behavior or the envelope of stored energy and ergotropy within statistical error.
   • The primary effect is the addition of a reactive oscillation at $2\omega_0$. This oscillation has zero net work over a cycle but nearly doubles the peak instantaneous power.
   • However, this extra energy is of lower quality; the recoverable fraction (ergotropy) drops from ~98% (isolated) to ~90% in the broken phase, indicating that the gain channel introduces entropy.

5. Exceptional Point Shift:

   • For the TC model, the exceptional point remains exactly at $J = \kappa$ in the low-excitation limit, as the batteries load the cavities equally without dissipation. The growth rate in the broken phase is halved compared to the empty cavity pair.
   • For the Dicke model, the exceptional point shifts slightly ($J \approx 0.996\kappa$) due to the violation of excitation number conservation by counter-rotating terms. The shift is of order $(\lambda/2\omega_0)^2$.
   • Symmetry Breaking: If the batteries are not identical (unequal coupling), the symmetric phase is destroyed entirely, and the system exhibits directional behavior regardless of $J$.

Significance and Claims
The paper claims that a $\mathcal{PT}$-broken environment acts as an effective charger for quantum batteries, delivering more work than an isolated system, albeit with a trade-off in the quality of the stored energy (increased entropy). The directionality of the charging is dictated by the initial state (photons placed in the gain cavity) rather than an asymmetry in the Liouvillian itself.

The authors emphasize that while the Dicke model introduces reactive power oscillations that inflate the instantaneous power peak, the fundamental charging capacity and work extraction remain consistent with the Tavis–Cummings description at the coupling strengths studied. The work provides a quantitative framework for understanding how non-Hermitian, gain-loss balanced environments can be engineered to enhance quantum energy storage, identifying the broken phase near the exceptional point as the optimal operating regime. The study concludes that the "extra" energy provided by the gain channel is largely recoverable as work, though the process is "slightly dirty" due to the classical disorder introduced by the gain mechanism.