Quantum Battery-Charging Process

• Quantum battery-charging processes are protocols that use quantum systems, such as qubits and harmonic oscillators, to store and transfer energy by leveraging non-Gaussian states.
• Key methodologies include Hamiltonian models like the Jaynes–Cummings interaction, state preparation of Fock and squeezed states, and full counting statistics to quantify energy fluctuations.
• Applications span high-precision quantum processors, metrology, and thermodynamics studies, demonstrating superior scaling and robustness against noise.

Quantum battery-charging processes refer to protocols employing quantum systems—typically two-level systems ("qubits") or harmonic oscillators—as energy storage elements, with the explicit aim of exploiting quantum coherence and non-classicality to enhance energy transfer metrics beyond what is achievable classically. Such protocols, central to the emerging field of quantum thermodynamics, are distinguished by their focus on both the mean work (or energy) deposited and the quantum statistical fluctuations during transfer, with significant research interest in identifying genuine quantum advantages in power, efficiency, and precision over Gaussian (classical or quasi-classical) energy sources.

1. Theoretical Framework and Hamiltonian Models

Quantum battery-charging protocols are commonly modeled as light–matter interactions, primarily via the number-conserving Jaynes–Cummings (JC) Hamiltonian. The fundamental system consists of a single-mode bosonic “charger” (an optical or microwave cavity, or a mechanical oscillator) at frequency $\omega_{\rm cav}$, coupled to a qubit (the “battery”) at frequency $\omega_{\rm qub}$:

$$H_{\rm JC} = \hbar \omega_{\rm qub} \frac{\sigma_z}{2} + \hbar \omega_{\rm cav} a^\dagger a + \hbar g(t) (\sigma_+ a + \sigma_- a^\dagger)$$

where $a$, $a^\dagger$ are bosonic annihilation and creation operators, $\sigma_\pm$ and $\sigma_z$ are the standard Pauli operators, and $g(t)$ is a tunable, typically time-windowed, interaction strength. The rotating-wave approximation (RWA) ($g \ll \omega_0$, $$\omega_{\rm qub} \approx \omega_{\rm cav} \approx \omega_0$$) is employed so that counter-rotating terms are neglected (Rinaldi et al., 2024, Rinaldi et al., 17 Jan 2026).

Extensions to multi-qubit batteries involve either sequential or parallel interaction schemes. In the sequential protocol, each of $M$ qubits interacts serially with the charger through:

$$H_{\rm int}(t) = \sum_{j=1}^M A_j(t) g (\sigma_+^{(j)} a + \sigma_-^{(j)} a^\dagger)$$

with $A_j(t)$ activating only for the $j$th interaction window (Rinaldi et al., 17 Jan 2026).

Analogous frameworks exist in optomechanical settings, where the interaction Hamiltonian in pulsed regimes reads:

$$H(t) = \Delta\, a^\dagger a + \frac{p_m^2}{2m} + \frac{1}{2}m\omega_m^2 x_m^2 - G x_m (a + a^\dagger)$$

where $x_m$, $p_m$ refer to the position and momentum of a levitated nanoparticle (Bemani et al., 20 Nov 2025). Linearization of this interaction allows for "beam-splitter" (phonon-subtraction) and "parametric amplification" (phonon-addition) processes essential for state preparation.

2. Classification and Preparation of Charger States

The performance of a quantum battery-charging process is fundamentally contingent on the quantum state of the charger. Key classes include:

• Coherent states $|\alpha\rangle$ (Gaussian; Poissonian photon statistics; Wigner function positive-definite).
• Squeezed (vacuum/coherent) states $| \zeta, \alpha\rangle$ (Gaussian; quadrature uncertainties asymmetrically distributed).
• Fock states $|N\rangle$ (number states; non-Gaussian; Wigner function exhibits negativity, oscillatory structure).

For quantum advantage, emphasis is on genuinely non-Gaussian charger states, operationally certified through preparation and contrast with Gaussian benchmarks. In cavity QED or circuit QED architectures, Fock state synthesis is well-established, while in optomechanical systems, heralded protocols using photon-counting projective measurements enable single- and multi-phonon Fock-state creation with fidelities exceeding 0.9 under optimized conditions (low mechanical heating $\gamma_m n̄ \ll \kappa$, ground-state occupation $n_0 \lesssim 0.1$) (Bemani et al., 20 Nov 2025).

Detecting quantum non-Gaussianity can proceed via thresholds for Fock-state occupation probabilities $Q_n > Q_n^G$ (with $Q_n^G$ the Gaussian-state maximum for occupation $n$), or via Wigner function negativity (Bemani et al., 20 Nov 2025, Rinaldi et al., 2024).

3. Quantification of Charging Performance

A rigorous evaluation of the charging process incorporates not only mean energy transfer but also fluctuations and reliability. The central figures of merit are:

• Average stored energy: $E(\tau) = \langle \Delta U_\tau \rangle$
• Average charging power: $P(\tau) = E(\tau)/\tau$
• Stochastic fluctuation (variance): $\Delta E^2(\tau) = \text{Var}(\Delta U_\tau)$

The full counting statistics (FCS) framework is employed to characterize not just mean values but the entire probabilistic distribution of $\Delta U_\tau$. Introducing a counting field $\chi$, the moment-generating function $G(\chi, \tau)$ allows extraction of all energy cumulants:

$$G(\chi, \tau) = \mathrm{Tr} \left\{ U_{\chi/2}(\tau)[\rho_{\rm qub}(0) \otimes \rho_{\rm cav}(0)] U_{-\chi/2}^\dagger(\tau)\right\}$$

The $k$th cumulant follows as $$C_k = (-i\partial/\partial\chi_1)^k \ln G_\tau |_{\chi_1=0}$$ (Rinaldi et al., 2024, Rinaldi et al., 17 Jan 2026).

The charging efficiency (or reliability) is defined as

$$\eta(\tau) = \frac{E(\tau)}{E(\tau) + \Delta E(\tau)}$$

while signal-to-noise ratio (SNR) is

$\text{SNR} = \frac{E(\tau)^2}{\Delta E^2(\tau)}$

These express the tradeoff between deposited energy and its fluctuations, critical for precise energy delivery (Rinaldi et al., 2024, Rinaldi et al., 17 Jan 2026).

4. Quantum Advantage Through Non-Gaussianity

A defining result is the identification of true quantum advantage when the charger is prepared in a Fock state. Analytically, for a single-qubit battery $|g\rangle$ and a Fock-state charger $|N\rangle$, at optimal interaction time $\tau_{\rm opt} = \pi/(2g\sqrt{N})$ (under resonance $\Delta\omega=0$):

• Full energy transfer: $E(\tau_{\rm opt}) = g^2 N \hbar \omega_{\rm qub}$
• Zero variance: $\Delta E^2(\tau_{\rm opt}) = 0$, hence SNR $\rightarrow \infty$
• Maximal power scaling: $$P_{\rm max} = (2 \hbar \omega_{\rm qub}/\pi) g \sqrt{N}$$

This scaling $P_{\rm max} \propto \sqrt{N}$ contrasts sharply with Gaussian chargers, whose maximal SNR remains finite, power growth is merely linear or saturates, and energy-exchange fluctuations persist (Rinaldi et al., 2024, Rinaldi et al., 17 Jan 2026). For multi-qubit systems, a sequential protocol with $N=M$ (number of qubits) enables perfect swaps with zero variance for each qubit, maintaining unit fidelity and maximum precision per round (Rinaldi et al., 17 Jan 2026).

In parallel protocols (Tavis–Cummings model with simultaneous multi-qubit interaction), the Fock-state advantage persists but SNR does not diverge and fidelity is less than unity. Scaling behaviors converge for large $M$.

The non-Gaussianity—manifest in high-order statistical behavior and negativity in quasi-probability distributions—directly suppresses energy-transfer fluctuations below Gaussian/state-of-art classical limits. This advantage is robust with respect to moderate levels of cavity thermal noise, Fock-state attenuation, detunings, and preparation imperfections (Rinaldi et al., 17 Jan 2026).

5. Implementation in Optomechanics and State Preparation Protocols

Heralded quantum non-Gaussian state generation in levitated optomechanics constitutes a practical method for charging mesoscopic quantum batteries (Bemani et al., 20 Nov 2025). Pulsed optomechanical interactions under the Hamiltonian $H_{\rm PA} = -g(a^\dagger b^\dagger + ab)$ (blue detuned) enable addition of single or multiple phonons to a mechanical oscillator via photon-count "heralding" events. Cascading $j$ such pulses followed by single-photon detection prepares mechanical $j$-phonon Fock states:

$$\rho_B^{(j)} \approx \frac{b^{\dagger j} \rho_B(0) b^j}{\text{Tr}[b^j \rho_B(0) b^{\dagger j}]}$$

Quantitative quantum non-Gaussianity is verified when occupation probabilities $Q_n = \langle n | \rho_B | n \rangle$ exceed maximal Gaussian thresholds $Q_n^G$ (Bemani et al., 20 Nov 2025). Experimental parameters favor resolved-sideband regime, weak-coupling ($g\lesssim0.1\kappa$), low heating rates, and enable preparation of single- and two-phonon states with $>90\%$ fidelity (for $n_0<0.05$, $\gamma_m n̄ < 0.01\kappa$).

6. Applications, Robustness, and Scaling Laws

Quantum batteries with non-Gaussian charging resources offer compelling applications for high-precision quantum technology and quantum thermodynamics. Notable use-cases include:

• Fault-tolerant, high-precision energy delivery: Maximal SNR protocols are suited for quantum processors and nanoscale devices requiring minimized energy-exchange uncertainty.
• Quantum metrology: Multi-phonon Fock-state probes in optomechanics outperform standard quantum limits in phase-randomized displacement sensing (Bemani et al., 20 Nov 2025).
• Thermodynamic cycle studies: Non-Gaussian energy charging cycles provide testbeds for fluctuation theorems, work extraction, and resource-theoretic aspects of quantum thermodynamics.
• Robustness: Fock-state-based protocols preserve their precision advantage under non-idealities, maintaining $D_{x,N} > 0$ (resource-normalized SNR difference between Fock and Gaussian chargers) for moderate thermal, attenuation, detuning, and preparation noise (Rinaldi et al., 17 Jan 2026).

Scaling laws govern protocol speed, precision, and resource requirements. Sequential Fock-state protocols achieve zero variance per qubit with interaction time $\tau_j \propto 1/\sqrt{N_j}$, while parallel schemes exhibit saturating precision with increasing $M$.

7. Summary Table: Chargers and Performance Regimes

Initial Cavity State	Non-Gaussian?	Max. SNR at Optimum	Max. Power Scaling
Fock state $|N\rangle$	Yes (canonical)	$\infty$ (sequential protocol)	$P_{\max} \propto \sqrt{N}$
Coherent $|\alpha\rangle$	No (Gaussian)	Finite (bounded, $O(1)$)	Linear/saturating in $N$
Squeezed vacuum	No (nonclassical, Gaussian)	Finite (bounded)	Linear/saturating
Thermal	No (classical, Gaussian)	Low (increases noise)	Linear/saturating

Fock state chargers enable perfect, noiseless swaps in sequential protocols, while all Gaussian chargers are limited by residual photon (or phonon) number fluctuations (Rinaldi et al., 2024, Rinaldi et al., 17 Jan 2026, Bemani et al., 20 Nov 2025).

In summary, contemporary research demonstrates that quantum non-Gaussian resource states, operationalized as Fock states in cavity or vibrational modes, are uniquely capable of delivering high-speed, robust, and maximally precise charging of quantum batteries, establishing a genuine quantum advantage in both theory and feasible experimental practice.