Coherent versus incoherent amplification: from parametric resonance to the kinetic regime in stimulated axion decay

Technical Summary: Coherent versus Incoherent Amplification in Stimulated Axion Decay

Problem Statement
The paper addresses a theoretical discrepancy regarding the scaling of the growth rate ($s$) for stimulated axion decay into two photons ($a \to \gamma\gamma$). Two distinct frameworks yield conflicting predictions for this rate:

1. Coherent Field Description: Treating the axion as a classical, coherent background field leads to modified Maxwell equations. For a monochromatic axion background, these equations map to a Mathieu equation, predicting a parametric resonance with a growth exponent scaling linearly with the axion-photon coupling: $s \propto g_{a\gamma}$.
2. Kinetic Description: Treating the process via Boltzmann equations for phase-space distributions relies on the squared matrix element of the interaction. This naturally predicts a rate scaling quadratically with the coupling: $s \propto g_{a\gamma}^2$.

The central question is determining the conditions under which stimulated axion decay transitions between these two regimes and how the coherence properties of the axion field govern this behavior.

Methodology
The authors adopt a classical field framework, treating both axion and electromagnetic fields as classical waves. The analysis proceeds in three stages:

1. Coherent Baseline: The authors first review the standard parametric resonance for a coherent, zero-momentum axion background, deriving the Mathieu instability and the linear scaling $s \propto g_{a\gamma}$. They then generalize this to monochromatic axions with non-zero momentum, showing that while the resonance condition shifts (via pair detuning), the maximum growth rate remains linear in $g_{a\gamma}$.
2. Discrete Superposition: To model a realistic axion field with finite momentum dispersion, the authors consider a discrete superposition of axion momentum modes. They derive coupled equations for the signal and idler photon modes. By integrating out the idler modes, they obtain a linear Volterra integro-differential equation for the signal amplitude, where the memory of the axion momentum distribution is encoded in a kernel $K(\tau)$.
3. Continuum Limit: The discrete sum is taken to the continuum limit, assuming a Maxwell-Boltzmann velocity distribution representative of virialized dark matter. This allows for the analytical evaluation of the kernel and the derivation of an implicit equation for the complex growth exponent $s_k$.

Key Contributions and Results
The primary contribution is the derivation of a unified expression for the photon growth exponent that smoothly interpolates between the coherent and incoherent regimes.

• The Kernel and Dephasing: The authors demonstrate that the transition is driven by the dephasing of different axion momentum modes. The kernel $K(\tau)$, which encodes the cumulative effect of these modes, decays over a characteristic timescale $\tau_{coh}$ (parametrically equal to the photon crossing time of an axion coherence domain).
• The Dimensionless Parameter: The transition is controlled by the dimensionless parameter $\tilde{\Theta} = \kappa / \sigma_\Delta$, where $\kappa$ is the coherent on-resonance growth rate and $\sigma_\Delta$ characterizes the width of the axion momentum dispersion.
• Asymptotic Regimes:
  • Coherent Regime ($\tilde{\Theta} \gg 1$): When the amplification timescale is shorter than the dephasing time, the axion field appears coherent to the photon. The implicit equation recovers the standard Mathieu result, yielding $s \propto g_{a\gamma}$.
  • Incoherent Regime ($\tilde{\Theta} \ll 1$): When amplification is slow compared to dephasing, the photon samples multiple uncorrelated coherence patches. The growth exponent becomes proportional to the integral of the kernel, recovering the kinetic scaling $s \propto g_{a\gamma}^2$.
• Implicit Equation: The central result is Equation (4.18), an implicit equation for the dimensionless growth exponent $x_k$:
  $$x_k = \sqrt{\frac{\pi}{2}} \tilde{\Theta} \exp\left[ \frac{(\tilde{\Theta}x_k - i d_k)^2}{2} \right] \text{erfc}\left( \frac{\tilde{\Theta}x_k - i d_k}{\sqrt{2}} \right)$$
  where $d_k$ is the normalized momentum mismatch. This equation provides an approximate description of the crossover between the two limits.

Significance and Claims
The paper claims to resolve the apparent contradiction between the parametric resonance and kinetic descriptions by showing they are distinct limits of the same underlying classical dynamics, distinguished solely by the coherence properties of the axion field.

• Unified Framework: The work provides the first derivation within a unified framework that connects the $g_{a\gamma}$ and $g_{a\gamma}^2$ scalings.
• Physical Mechanism: It identifies the loss of phase coherence among momentum modes (dephasing) as the mechanism that suppresses the linear growth term, leaving a weaker, quadratic growth term.
• Robustness: The authors argue that while the detailed shape of the crossover depends on the specific axion momentum distribution, the asymptotic scaling behaviors ($s \propto g_{a\gamma}$ vs. $s \propto g_{a\gamma}^2$) are largely insensitive to the specific distribution assumed.
• Astrophysical Context: The analysis suggests that for standard virialized galactic dark matter, the system lies deep in the incoherent regime ($\tilde{\Theta} \ll 1$). Entering the coherent regime would require extreme densities (e.g., in axion stars or black hole superradiant clouds) or significantly different velocity distributions, though the paper notes that applying the virialized Maxwell-Boltzmann assumption to such bound objects requires caution.

The authors explicitly state that their treatment of the crossover is approximate in the intermediate regime due to neglected mode couplings in their simplified ansatz, and they propose numerical simulations and non-equilibrium quantum field theory extensions as necessary future work to refine the transition details.