Nonlinear Two-Photon Quantum Battery

• The paper demonstrates that nonlinear two-photon interactions enable ultrafast charging dynamics and enhanced energy storage compared to traditional linear protocols.
• The methodology relies on engineered quadratic couplings in platforms like circuit-QED and photonic cavities, which induce squeezed states and collective Rabi oscillations.
• Key results reveal exponential energy scaling and maximal ergotropy from pure Gaussian states, paving the way for advanced hybrid quantum technologies.

A nonlinear two-photon driving quantum battery model utilizes nonlinear photon-photon interactions to implement energy transfer and storage protocols with substantial advantages over single-photon counterparts. These models leverage engineered quadratic (two-photon) couplings in cavity-QED, circuit-QED, and related bosonic systems, as well as in ensembles of two-level systems (TLS) coupled to a resonator. Central features include ultrafast charging dynamics, generation of nonclassical (squeezed) states, and enhanced scaling of stored energy and available work (ergotropy). The following sections synthesize definitions, formalism, dynamical analysis, performance metrics, and implementation considerations, with direct reference to the established literature (Crescente et al., 2020, Downing et al., 2023, Downing et al., 9 Jul 2024, Downing et al., 24 Oct 2025, Xu et al., 15 Nov 2025).

1. Theoretical Framework and Model Hamiltonians

Nonlinear two-photon quantum battery models generalize first-order (dipolar, “linear”) charging by incorporating quadratic coupling—driving or interaction terms proportional to $a^2 + a^{\dagger 2}$ (bosonic systems) or $[(a^\dagger)^2 + a^2]$ coupled to collective spin operators (Dicke-type models). Two principal classes arise:

(a) Collective Two-Photon Dicke Model

The system consists of $N$ two-level systems (TLS), described by collective spin operators $$J_\alpha = \frac{1}{2} \sum_{i=1}^N \sigma_\alpha^{(i)}$$, coupled to a single quantized mode (resonator) of frequency $\omega_c$: $$H = \omega_c a^\dagger a + \omega_a J_z + g_1 J_x (a^\dagger + a) + g_2 J_x \left[ (a^\dagger)^2 + a^2 \right]$$ Here, $g_1$ and $g_2$ parameterize single-photon and two-photon coupling strengths. Suppressing $g_1$ yields the pure two-photon Dicke Hamiltonian: $$H_{2ph} = \omega_c a^\dagger a + \omega_a J_z + g_2 J_x \left[ (a^\dagger)^2 + a^2 \right]$$ Typically, $g_2$ is engineered via circuit-QED architectures (e.g., symmetric SQUIDs, bichromatic driving) to maximize nonlinear exchange (Crescente et al., 2020).

(b) Bosonic Two-Mode Quadratic Driven Model

A bosonic "charger" ($a$) is coupled (either linearly or nonlinearly) to a "battery" ($b$), both of frequency $\omega$, with a time-dependent or static quadratic drive: $$H(t) = \omega a^\dagger a + \omega b^\dagger b + g (a b^\dagger + a^\dagger b) + \lambda(t) (a^2 + a^{\dagger 2})$$ $\lambda(t)$ realizes a two-photon (parametric) drive; $g$ enables energy transfer between charger and battery (Downing et al., 2023, Downing et al., 24 Oct 2025). In some cases, direct quadratic ("downconversion") coupling $J(a^\dagger b^2 + b^{\dagger 2} a)$ between charger and battery is considered (Downing et al., 24 Oct 2025).

2. Dynamical Evolution, Squeezing, and Charging Dynamics

Squeezing Generation and State Evolution

The quadratic drive executes a squeezing operation on the recipient mode (battery), with the propagator

$$U_I(t) = \exp\left[ \frac{1}{2} r(t) \left( a^{\dagger 2} - a^2 \right) \right], \qquad r(t) = \int_0^t \Omega(t')\,dt'$$

where $\Omega(t)$ is the two-photon drive amplitude (Downing et al., 9 Jul 2024). For a bosonic battery initially in vacuum, squeezing populates higher Fock states, leading to

$$\langle n(t) \rangle = \sinh^2[r(t)], \qquad E(t) = \hbar\omega\,\sinh^2[r(t)]$$

i.e., the stored energy grows exponentially with the integrated pulse area $r(t)$. For time-dependent protocols (e.g., Gaussian pulse envelopes), both the time-maximum stored energy and maximum charging power display exponential scaling in the drive parameters.

Spin-Boson Collective Dynamics

In the two-photon Dicke model, with all TLSs initially in their ground state and the cavity loaded with $2N$ photons, the system undergoes collective Rabi oscillations at the frequency $G_2 = g_2 N$. The average stored energy evolves as

$E(t) = N \omega_a \sin^2(G_2 t)$

with fluctuations $\Delta E(t) = \omega_a \frac{N}{2} |\sin(2G_2 t)|$ (Crescente et al., 2020).

3. Performance Metrics: Energy, Power, and Ergotropy

Key figures of merit for quantum batteries include the maximum stored energy ($E_{\max}$), maximum and average charging power ($P_{\max}$, $\overline{P}$), and ergotropy (extractable work, $\mathcal E$).

• Scaling Laws:
  • In the two-photon Dicke model:
  • $E_{\max}(N) \propto N$
  • $P_{\max}(N) \propto N^2$
  • Charging time $t_E \propto 1/(g_2 N)$
  • For linear models, $P_{\max} \propto N^{3/2}$ and $t_E \propto 1/(g_1 \sqrt{N})$ (Crescente et al., 2020).
• Ergotropy and Minimum-Uncertainty States:
  • In continuous-variable two-photon models, squeezing induces pure minimum-uncertainty (Gaussian) states of the battery mode, saturating Heisenberg's relation, and thus all stored energy is extractable:

  $\mathcal{E}(t) = E(t)$ - The determinant of the covariance matrix $\mathcal{D} = 1$ indicates absence of passive (non-extractable) energy (Downing et al., 24 Oct 2025). - For general Gaussian states, ergotropy is

  $$W_{\rm erg} = E_b - E_b^\beta, \qquad E_b^\beta = \hbar\omega_0\,\frac{\sqrt{\mathcal{D}}-1}{2}$$

  (Downing et al., 2023).

• Nonreciprocal and Dissipative Effects:

  • Optimizing dissipation asymmetry (suppressing battery loss) and balancing collective dissipation channels enhances both $E$ and $\mathcal{E}$ (Xu et al., 15 Nov 2025).

4. Comparison with Single-Photon Driving and Other Charging Mechanisms

Direct benchmarking against linear (single-photon) protocols reveals qualitative and quantitative superiority of two-photon models in several metrics:

• The two-photon Dicke protocol delivers an additional $\sqrt{N}$ speedup in charging and $N^{1/2}$ enhancement in power, on top of the usual Dicke collective scaling (Crescente et al., 2020).
• In bosonic Gaussian batteries, both stored energy and ergotropy can become parametrically larger near critical points of the driven-dissipative system, while some energy becomes passive and unextractable near but not at minimum-uncertainty (Downing et al., 2023).
• Under strong two-photon driving in nonreciprocal models, the absolute energy storage and ergotropy advantages persist, though conversion efficiency per unit supplied energy may be lower than in linear cases (Xu et al., 15 Nov 2025).

5. Open-System and Nonequilibrium Considerations

Open quantum system dynamics, encompassing Markovian loss, engineered dissipation, and environmental coupling, define available performance regimes and physical limits:

• Master equations of Lindblad type describe decay (rates: $\kappa, \gamma$), cross-dissipation (rate: $\Gamma$), and engineered nonreciprocal energy flow (Xu et al., 15 Nov 2025).
• Dynamical equilibration times, steady-state existence, and critical points are set by interplay of two-photon drive strength and decay rates. Stability requires $G < \Lambda/4$, where $\Lambda$ is the relevant total loss rate (Xu et al., 15 Nov 2025).
• At strong driving and near dissipative exceptional points, parametrically large energy storage is feasible, while real devices require technical regularization to avoid unphysical divergences (Downing et al., 2023).

6. Experimental Implementation and Platform-Specific Parameters

Realization of nonlinear two-photon quantum batteries requires platforms supporting significant two-photon (parametric) interactions:

• Circuit QED: Superconducting circuits with flux qubits, symmetric SQUIDs, or Josephson parametric amplifiers; frequencies $\omega/2\pi \sim 5$–$10$ GHz; $g_2/\omega_a$ up to $0.1$–$0.5$; drive amplitudes up to $100$ MHz; coherence times $\sim\mu$s–ms (Crescente et al., 2020, Downing et al., 2023, Xu et al., 15 Nov 2025).
• Photonic Microcavities: Second-order nonlinear ($\chi^{(2)}$) crystals for optical frequency implementations; parametric downconversion for two-photon driving (Downing et al., 9 Jul 2024).
• Trapped Ions and Magnonic Resonators: Bichromatic driving and engineered couplings enable implementation of the quadratic terms (Crescente et al., 2020, Xu et al., 15 Nov 2025).
• Optomechanical/Nanomechanical: Two-phonon modulation mechanisms support equivalent quadratic interactions (Downing et al., 9 Jul 2024).

A table summarizing representative parameter regimes:

Platform	Two-Photon Drive (G, $\Omega$)	Loss Rate ($\kappa$, $\gamma$)	Coupling ($g$, $J$)
Circuit QED	$1$–$100$ MHz	$0.1$–$1$ MHz	$1$–$10$ MHz
Photonic Cavity	$1$–$10$ GHz	$0.1$–$10$ MHz	platform dependent
Magnonic Resonators	$5$–$50$ MHz	$0.01$–$0.1$ MHz	$0.25$ $\omega$
Optomechanics	variable	$\ll$ resonance frequency	system specific

7. Outlook and Implications

Nonlinear two-photon quantum battery models provide a platform for ultrafast, high-capacity quantum energy storage, with superextensive scaling properties and the intrinsic generation of nonclassical states (notably, squeezed minimum-uncertainty Gaussian states). These features yield maximal extractable work and open prospects for hybrid quantum technologies leveraging stored energy and quantum correlations for sensing or computation. Fine control of parametric driving, environmental couplings, and two-photon interactions is central to optimizing device performance and realizing experimental prototypes consistent with theoretical predictions (Crescente et al., 2020, Downing et al., 2023, Downing et al., 24 Oct 2025, Xu et al., 15 Nov 2025, Downing et al., 9 Jul 2024).