Non-Markovian Collective Emission from Macroscopically Separated Emitters

We study the collective radiative decay of a system of two two-level emitters coupled to a one-dimensional waveguide in a regime where their separation is comparable to the coherence length of a spontaneously emitted photon. The electromagnetic field propagating in the cavity-like geometry formed by the emitters exerts a retarded backaction on the system leading to strongly non-Markovian dynamics. The collective spontaneous emission rate of the emitters exhibits an enhancement or inhibition beyond the usual Dicke superradiance and subradiance due to self-consistent coherent time-delayed feedback.

Figure 1

Two two-level emitters prepared in a collective state coupled to an optical waveguide. The emitters are located at positions $x_{1,2}=\pm d/2$, with $d$ comparable to the coherence length $\sim v_g/\gamma$. The rates ${\gamma }_{3D}$ and ${\gamma }_{1D}$ refer to the emitter spontaneous emission rates into free space and guided modes, respectively. The mode operators $\widehat{a}(\omega )$ and $\widehat{b}(\omega )$ refer to annihilation operators for the right- and left-propagating waveguide modes, respectively.

Figure 2

Emitters excitation probabilities as a function of time and for different separations $\eta$ for initial (a) superradiant and (b) subradiant states, assuming perfect coupling efficiency ($\beta =1$). Solid and dotted curves depict the dynamics for $\eta =0$ and $\eta =\infty$, respectively. For intermediate emitter separations, the emitters decay independently with a rate $\gamma$ until $\gamma t=\eta$ (indicated by the dashed vertical lines), and collectively afterwards. For the critical separation $\eta \approx {\eta }_c\approx 0.56$, we observe an instantaneous superradiant spontaneous emission rate of ${\gamma }_{\mathrm{inst}}\approx 4.59\gamma$. The $1/e$ value of the initial emitters excitation probability is reached first for coincident emitters (depicted by the gray horizontal dashed-dotted line). In the subradiant case the emitter excitation probability reaches the asymptotic value $(1+\eta /2{)}^{-2}$, shown by the horizontal dashed-dotted lines.

Figure 3

Instantaneous decay rate ${\gamma }_{\mathrm{inst}}^{\max }$ (blue solid line) and associated critical emitter separation ${\eta }_c$ (red dashed-dotted line) as a function of the emitter-waveguide coupling efficiency $\beta$ for an initial superradiant state. The horizontal dashed line depicts an instantaneous collective emission rate of $2\gamma$, which corresponds to a coupling efficiency of $\beta \approx 0.13$. This illustrates that the collective emission rate of $2\gamma$ of usual Dicke superradiance can be exceeded for sufficiently large emitter-waveguide coupling efficiency and appropriate emitter separations.

Figure 4

Normalized field intensity as a function of position and time for $\eta =0.5$ and (a) superradiant and (b) subradiant initial state. The positions of emitters at $\gamma x_{1,2}/(2v_g)\approx \pm 0.25$ are depicted by the vertical dashed line. As the field from one of the emitters reaches the other at $\gamma t=\eta$ (dashed-dotted line), their interference results in either a collective radiation burst or reflection of the field into the cavity formed by the emitters.