Algebraic power scaling in a slowly-quenched bosonic quantum battery

Technical Summary: Algebraic Power Scaling in a Slowly-Quenched Bosonic Quantum Battery

Problem Statement
Quantum batteries aim to leverage coherence and entanglement to achieve charging speeds and power outputs superior to classical counterparts. While spin-based quantum batteries can exhibit superextensive power scaling ($N^\alpha$ with $\alpha > 1$) due to collective interactions, they are fundamentally limited by bounded energy spectra. Conversely, bosonic oscillator batteries offer unbounded energy spectra, yet existing models utilizing coherent driving suffer from a critical limitation: the stored energy is capped by coherent oscillations (Rabi flopping) between the charger and the battery. In standard constant-coupling protocols, only a few low-lying excited states are accessible regardless of the charging duration, preventing the exploitation of the oscillator's extensive capacity. Furthermore, optimizing driving sources often requires complex time-dependent controls.

Methodology
The authors propose a charging protocol for a bosonic quantum battery consisting of a charger (mode $A$) and a battery (mode $B$), both initially in the ground state. The system is driven by an external coherent field $F$ applied solely to the charger, while the interaction between the charger and battery, $g(t)$, is subjected to a continuous "slow quench."

• Hamiltonian: The system is modeled in a frame rotating with the drive frequency. The Hamiltonian is given by $H(t) = g(t)(ab^\dagger + a^\dagger b) + F(a + a^\dagger)$, where $a$ and $b$ are bosonic annihilation operators.
• Quench Profile: The coupling strength follows a power-law ramp $g(t) = g_f (t/\tau_Q)^r$ for $0 \le t \le \tau_Q$, where $\tau_Q$ is the quench duration and $r \ge 0$ is the ramp exponent. For $t > \tau_Q$, the coupling remains at its maximum $g_f$.
• Dynamics: The authors analyze the system using the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation to include charger dissipation ($\gamma$). However, for the primary derivation of scaling laws, they utilize an effective non-Hermitian Hamiltonian approach, which is equivalent for quadratic systems and reduces computational complexity.
• Analysis: The study focuses on the first moments of the operators ($\langle a \rangle, \langle b \rangle$) to derive the equations of motion. The authors solve these equations analytically for the closed system and numerically for open systems to determine the maximum stored energy ($E_{B,m}$) and maximum average power ($P_{B,m}$). They also calculate the ergotropy (extractable work) to ensure the stored energy is usable.

Key Contributions and Results

1. Algebraic Scaling Laws:
   The central finding is that introducing a slow quench ($r > 0$) allows the maximum stored energy and maximum charging power to scale algebraically with the quench duration $\tau_Q$, rather than being bounded as in constant-coupling cases.

   • Maximum Stored Energy: $E_{B,m} \propto \tau_Q^{2\alpha}$
   • Maximum Power: $P_{B,m} \propto \tau_Q^{\alpha}$
   • Where the exponent is defined as $\alpha = r/(r+1)$. Since $r \in [0, \infty)$, the scaling exponent is bounded by $\alpha \in [0, 1]$.
   • Counterintuitive Result: Slower quenches (larger $\tau_Q$) lead to higher charging power. This occurs because the slow ramp suppresses the coherent back-and-forth energy oscillations between the charger and battery, allowing the coherent drive to effectively pump energy into the battery over a longer duration.

2. Temporal Extensivity:
   The authors define this phenomenon as "temporal extensivity." Unlike spatial extensivity, where power scales with the number of battery units $N$, this protocol achieves scaling with time. In the limit of infinite $\tau_Q$ (for a closed system), arbitrarily high excited states become accessible.

3. Ergotropy and Extractability:
   In the ideal closed system, the battery state remains a pure coherent state throughout the process. Consequently, the ergotropy (maximum extractable work) is equal to the total stored energy ($W_B = E_B$). This implies that 100% of the accumulated energy is extractable, provided the coupling is switched off at the first maximum of the energy curve.

4. Impact of Dissipation:
   When charger dissipation ($\gamma \neq 0$) is included, the monotonic algebraic scaling is modified. The charger energy saturates at long times due to the competition between driving and loss.

   • This results in a non-monotonic dependence of power on $\tau_Q$.
   • There exists an optimal quench duration $\tau_Q^{\text{opt}} \approx 2.513 \gamma^{-1}$ that maximizes power.
   • For $\tau_Q < \tau_Q^{\text{opt}}$, the algebraic scaling holds. For $\tau_Q > \tau_Q^{\text{opt}}$, the power decays as $\tau_Q^{-1}$.
   • This establishes a finite-time scaling window for experimental realization, bounded by the effective decoherence time.

5. Mapping to Tavis-Cummings Model:
   The bosonic model is mapped to a coherently driven Tavis-Cummings (or Dicke) model in the low-excitation limit via the Holstein-Primakoff transformation. This suggests the temporal algebraic enhancement is not unique to single-mode oscillators but may apply to collective-spin quantum batteries, provided the excitation number remains much smaller than the total spin ($n_B \ll 2s$).

Significance and Claims
The paper claims to demonstrate a mechanism to overcome the energy limitations inherent in standard coherent charging of bosonic batteries. By utilizing a slow quench, the protocol suppresses coherent oscillations that typically limit energy storage, enabling an "unbounded" increase in power relative to the quench duration in the ideal limit.

The authors emphasize that this is not a purely closed-system artifact; the algebraic enhancement persists in open systems but is bounded by dissipation, leading to a practical, finite optimal charging time. The work connects the physics of slow quenches in quantum batteries to experimental platforms such as superconducting circuit QED (using tunable beam-splitter interactions) and organic microcavities (collective light-matter coupling). The study concludes that while higher ramp exponents $r$ yield better scaling, they require precise timing and are subject to larger energy fluctuations, and the presence of dissipation necessitates an optimization of the quench duration to maximize power.