Cavity-QED Simulation of a Maser beyond the Mean-Field Approximation

TL;DR

This paper advances maser modeling by moving beyond the mean-field approximation through a geometry-informed discretization of emitter–mode couplings into $J$ bins, enabling a Tavis–Cummings–type Hamiltonian weighted over these bins. Using a second-order cumulant expansion implemented in QuantumCumulants.jl, the authors solve the resulting dynamics on a laptop and demonstrate markedly improved agreement with a room-temperature, optically pumped pentacene-doped para-terphenyl maser, achieving $R^2$ ≈ 0.774 compared with MFA’s 0.265. The work highlights how inhomogeneous coupling and heating can influence threshold behavior and detuning, offering a quantitative framework to optimize maser designs. Overall, the methodology combines geometry-driven coupling distributions, efficient quantum cumulant dynamics, and accessible computation to enable practical, beyond-MFA maser engineering.

Abstract

Based on the well-known Tavis-Cummings (TC) model of cavity quantum electrodynamics (QED), we introduce a method for quantum-mechanically simulating the dynamics of experimental masers beyond the mean-field approximation (MFA) that takes into account the spatial variation of the a.c. magnetic field of the maser's amplified microwave mode across its gain medium. The distribution in the coupling between the amplified mode and the medium's very large number (typically $10^{17}$) of spatially distributed quantum emitters can be determined straightforwardly for a given geometry and composition using an electromagnetic-field solver. Upon discretising this distribution as a histogram over a small finite number of bins, we assign -- as an approximation -- the same coupling to all emitters that fall within the same bin, where the value of this coupling equals the center value of the bin's range. With our approximate Hamiltonian arranged as a weighted sum over these bins, we generate expressions for expectation values of operators in the Heisenberg picture to second order in cumulant expansion, using the publicly available QuantumCumulants.jl package in Julia. For ten evenly spaced bins, our model, which can be run on a laptop computer, is used to simulate the recorded output from an experimental maser with a pentacene-doped para-terphenyl gain medium. We find that it replicates the quantum-mechanical features of the measured maser's dynamics, in particular its damped collective Rabi oscillations, more closely than the standard TC model under the MFA can, with an $R^2$ value of 0.774, as opposed to 0.265. Our model should thus aid the quantitative engineering of improved, optimised maser designs.

Figures (4)

• Figure 1: Pentacene molecules are driven from their singlet ground states, $S_0$ into their first excited singlet states, $S_1$, by their absorption of yellow pump light (photons) at rate $\xi$. From there, the molecules undergo spin-selective intersystem crossing into the triplet manifold $T_2$pentacenepentacene2, before decaying to $T_1$ through internal conversion, which preserves the ISC-induced population inversion across the X and Z sub-levels. The frequency difference is 1.4495 GHz, corresponding to an L-band microwave photon.
• Figure 2: (a) Rendered image of the maser cavity with magnetic vector field (white arrows) and magnetic energy density map with magnetic field vectors on a plane cut through the middle of gain medium, obtained using COMSOL's solver of Maxwell's equations. (b) A 10-bin histogram of coupling strengths obtained from calculated magnetic energy densities.
• Figure 3: (a) Maser output power at 1.4495 GHz as a function of time for the experiment, recorded using a digital storage oscilloscope at a resolution bandwidth of 100 MHz, overlaid with the results of an MFA simulation and our 10-bin simulation (b) Output for the first 0.30 ms highlighting collective Rabi oscillations.
• Figure 4: Maximum maser output power at 1.4495 GHz in 0.5 ms of optical pumping, against pump power, for Gaussian distributions of emitter-photon coupling strength, with $\bar{g}=0.18$ and a $\sigma (g_j)$ of 0, 0.02, 0.04, 0.06, respectively.