Coherence-Driven Charger

• The coherence-driven charger is a quantum system that transfers energy via coherent pumping rather than thermal energy injection, ensuring work extractability.
• It employs a Markovian master equation framework to balance coherent energy transfer with dissipative thermalization using Lindblad operators and exchange Hamiltonians.
• Optimizing the interplay between coherent driving and dissipation directly enhances the battery’s ergotropy, allowing energy to be stored in a fully extractable form.

A coherence-driven charger in quantum battery research refers to an ancillary quantum system that mediates energy transfer from an external supply to a battery, enabling (and optimizing) the transfer of energy in a form that is directly useful for work extraction—namely, via coherently pumped quantum excitations. This concept differs fundamentally from protocols limited to thermal energy/entropy injection, by focusing on the generation and preservation of quantum coherence during the charging process. The detailed open-system framework, formulated in terms of Markovian master equations, reveals the competition and interplay between coherent energy transfer and thermalization and establishes the unique role of coherence as a resource for both the magnitude and extractability of stored energy.

1. Theoretical Model and Master Equation Formalism

The coherence-driven charging model is structured around three main elements: an external energy source (E), a quantum charger (A), and the battery (B). The charger A acts as a transducer, receiving energy from E through two distinct channels:

• Coherent Channel: Represented by a time-dependent modulation term $$\Delta H_A(t) = F \left[e^{-i\omega_0 t} a^\dagger + e^{i\omega_0 t} a\right]$$, where $F$ is the driving field amplitude, $\omega_0$ is the resonance frequency, and $a$, $a^\dagger$ are the charger's lowering and raising operators.
• Thermal Channel: Modeled by a Lindblad dissipator $\mathcal{D}_A^{(T)}$ that brings A into equilibrium with a thermal bath at temperature $T$.

The closed charger–battery system evolves according to the master equation (in an appropriate interaction picture): $$\frac{d}{dt} \tilde{\rho}_{AB}(t) = -i\left[\Delta H_A + H_{AB}^{(1)}, \tilde{\rho}_{AB}(t)\right] + \mathcal{D}_A^{(T)}[\tilde{\rho}_{AB}(t)]$$ where $H_{AB}^{(1)} = g(a b^\dagger + a^\dagger b)$ is the exchange Hamiltonian (respecting conservation of excitation number) with coupling strength $g$, and $b$, $b^\dagger$ are the battery's bosonic operators.

The dissipator for a thermal bath is explicitly

$$\mathcal{D}_A^{(T)}[\rho] = \gamma(N_b(T)+1)\mathcal{D}_A^{[a]}[\rho] + \gamma N_b(T)\mathcal{D}_A^{[a^\dagger]}[\rho]$$

with $$\mathcal{D}_A^{[x]}[\rho] = x \rho x^\dagger - \frac{1}{2}\{x^\dagger x, \rho\}$$ and $N_b(T) = [\exp(\hbar\omega_0/k_B T) - 1]^{-1}$.

2. Energy Storage, Ergotropy, and the Role of Coherence

Two central performance metrics emerge:

• Stored Energy: $E_B(\tau) = \mathrm{Tr}(H_B \rho_B(\tau))$ (with $\rho_B(\tau)$ the reduced battery density matrix at time $\tau$).
• Ergotropy: $$\mathcal{E}_B(\tau) = E_B(\tau) - \min_U \mathrm{Tr}[H_B U \rho_B(\tau) U^\dagger]$$, the maximum extractable work via local unitary operations.

Key results demonstrate an energy decomposition: $$E_B(\tau)|_{F,T} = E_B(\tau)|_{F=0,T} + E_B(\tau)|_{F,T=0}$$ However,

$$\mathcal{E}_B(\tau)|_{F=0,T} = 0, \qquad \mathcal{E}_B(\tau)|_{F,T} = E_B(\tau)|_{F,T=0}$$

Thermal energy raises the battery’s internal energy but yields a passive (thermal) final state from which no work is extractable. Only coherent pumping (finite $F$, $T=0$) contributes to ergotropy. This principle is seen both in analytic derivations and explicit models. For instance, for two harmonic oscillators with a coherent drive, the battery can remain in a (quasi-)coherent state whose energy is fully available as work, while thermal driving leads to fast relaxation towards a passive (thermal) state and nullifies the ergotropy in the steady state.

3. Physical Dynamics and Operational Protocol

The coherence-driven charging process occurs over a finite interval $[0,\tau]$, with the charger–battery–environment couplings “switched on” during this stage. The energy transfer workflow can be summarized:

1. External supply (E) injects energy into A either via a coherent field (amplitude $F$) or a thermal bath (temperature $T$).
2. Charger (A) interacts with battery (B) via $H_{AB}^{(1)}$, facilitating exchange of quantum excitations.
3. The charging interval $\tau$ is optimized to maximize ergotropy (not just stored energy), and following this interval, A and B are isolated from their respective environments.
4. Potential trade-offs arise: tuning of the dissipation rate $\gamma$ and coupling $g$ is necessary to balance fast absorption from E, sufficient transfer to B, and resilience against unwanted thermalization.

Performance depends not only on the drive amplitude $F$ and thermal bath temperature $T$ but critically on the detailed balance between coherent pumping and dissipative loss. The process is most efficient when the coherent drive predominates, while thermalization has a double-edged role—facilitating rapid energy capture by A, but at the expense of potentially filling the battery with non-extractable (passive) energy if coherence is lost.

4. Interplay and Additivity of Injection Channels

Results indicate an additive structure in expectation values and energies: $$\langle x \rangle = \langle x \rangle_{F=0,T} + \langle x \rangle_{F,T=0}$$

$E_B(\tau) = E_B(\tau)_{F=0,T} + E_B(\tau)_{F,T=0}$

This property signifies that coherent and thermal channels act independently at the level of energy accumulation. In contrast, for ergotropy, only the coherent channel is non-zero: $\mathcal{E}_B(\tau)|_{F,T} = E_B(\tau)|_{F,T=0}$ At long times and with only thermal injection, the system settles to

$$E_B(\infty) = \omega_0 N_b(T), \qquad \mathcal{E}_B(\infty) = 0$$

In practical application, the timing of the charging interval, as well as the ratio of $F$ to $g$ and $\gamma$, are adjusted to maximize ergotropy. Excessive dissipation or thermal driving can reduce extractable work despite increasing stored energy, underlining the necessity of coherence in the drive.

5. Implications for Quantum Battery Design and Performance Optimization

The detailed analysis leads to several design principles for quantum batteries:

• Coherence is a unique and necessary resource for creating non-passive, high-ergotropy battery states.
• Thermalization is double-edged: while it can quickly “pump” energy into the charger under the right conditions, its ultimate effect is to “passivize” stored energy.
• Regime balancing: In qubit or hybrid models, interplay between coherent field, temperature, and dissipation is nontrivial—moderate thermal backgrounds can enhance transfer in some parameter regimes (noise-assisted transfer, reminiscent of quantum biology effects).
• Timing and parameter regime: The interval $\tau$ and system-environment parameters ($\gamma$, $g$) are not arbitrary—precision control can be necessary to avoid over-thermalization or incomplete charging.

This underlines that a high-performance “coherence-driven charger” must prioritize coherent energy pumping, optimize for rapid energy absorption (by tuning $\gamma$ or introducing engineered loss), and keep the overall protocol duration sufficiently short to avoid full relaxation to passive equilibria. Only under these conditions can the quantum battery realize its maximal operational advantage—namely, the rapid storage of energy in a form that is entirely available as work output.

6. Generalization to Other Physical Realizations

While the principal calculations employ harmonic oscillators for both charger and battery, the framework is adaptable to qubit models (with corresponding operator substitutions) or more complex hybrid systems. In each case, the essential physics remains: the charger must mediate the conversion of external classical energy into quantum excitations that maintain their coherence until transferred to the battery.

The analysis provides a foundation for future studies on many-body batteries, collective effects, and engineered environments in quantum battery technology.

Summary Table: Key Mechanistic Contrasts

Energy Supply	Stored Energy $E_B$	Extractable Work (Ergotropy) $\mathcal{E}_B$
Coherent drive ($F$)	Non-passive, additive	High (all excess energy is extractable)
Thermal drive ($T$)	Nonzero, additive	Zero (passive, no work extractable)
Both	Additive contributions	Only coherent part is useful

In conclusion, the coherence-driven charger paradigm establishes that achieving high, extractable energy storage in quantum batteries is contingent on engineering protocols and environments that maximize and preserve quantum coherence during energy transfer, while managing (rather than eliminating) thermalization and dissipation to maintain operational efficiency and robustness (Farina et al., 2018).