Superradiant Peak Emission Rate and Time in Quantum Emitter Arrays

Abstract

Determining the peak photon emission time and rate for an ensemble of $N$ quantum systems undergoing collective superradiant decay typically requires tracking the time evolution of the density operator. Generally, the dimension of the density operator grows exponentially ($\sim \! 2^N$) with the number of emitters, in the absence of any symmetries such as in Dicke superradiance with full or partial permutational symmetry. We present a detailed study of the superradiant peak emission rate and time for initially fully excited quantum emitter ensembles, for one-, two- and three-dimensional arrays in free-space and emitter chains coupled to waveguide reservoirs. For few emitters ($N\lesssim 14$) we utilize the full quantum master equation, and for mesoscopic emitter numbers ($N\lesssim 400$) we use a second- and third-order cumulant expansion of the operator averages to track the time evolution of the system.

Figures (9)

• Figure 1: Superradiant peak emission occurs in free-space atomic emitter ensembles, when the nearest-neighbor atomic distance $a$ is less than or comparable to the emitter resonance wavelength. In this case, the initial emission rate $R(t)$ of $N$ two-level emitters rises to a peak emission rate $R_\mathrm{peak}>N\Gamma$ at a later time. In this work we will study ensembles of two-level systems with spontaneous decay rate $\Gamma$ from the excited state ($|e\rangle$) to the ground state ($|g\rangle$) in 1D, 2D and 3D arrays embedded in a 3D photonic environment, as well as equidistant chains coupled to a waveguide (1D photonic environment).
• Figure 2: The peak emission rate and time in Dicke superradiance based on exact numerics using the quantum master equation. The numerics are performed in the symmetric Dicke subspace were computational complexity scales linear with emitter number. Plots of the peak emission rate and time using exact numerics based on the quantum master equation (continuous lines) as a function of $N$. The exact value of the peak emission rate converges to $\Gamma N^2/5$ at larger $N$. The peak emission time shows excellent agreement with $\ln(N \!- \!1)/[\Gamma(N+1)]$, and also shown is the literature prediction $\tau_d = \ln N / (\Gamma N)$gross_haroche (gray dashed line).
• Figure 3: One-dimensional chain. (a) The normalized peak emission rate as a function of the emitter number $N$ for various emitter spacings $a$. Shown are circular ($\circ$) and linear ($\square$) polarized dipoles, and both converge to a linear scaling with $N$. (b) The peak emission time as a function of emitter number shows a saturation as opposed to the literature result for Dicke superradiance, $t_\mathrm{peak}\Gamma = \ln(N)/(\Gamma N)$. The values are obtained with a third-order cumulant expansion.
• Figure 4: Two-dimensional square array. (a) The normalized peak emission rate as a function of the emitter number $N$ for various emitter spacings $a$. Shown are circular ($\circ$) and linear ($\square$) polarized dipoles, and both converge to a superlinear scaling with $N$. (b) The peak emission time as a function of emitter number. All values are obtained with a third-order cumulant expansion.
• Figure 5: Three-dimensional cubic array. (a) The normalized peak emission rate as a function of the emitter number $N$ for various emitter spacings $a$. Shown are circular ($\circ$) and linear ($\square$) polarized dipoles, and as in the case of square arrays, both converge to a superlinear scaling with $N$. (b) The peak emission time as a function of emitter number. All values are obtained with a third-order cumulant expansion.