Motion-induced directionality of collective emission in a non-chiral waveguide

Abstract

We report on the observation of motion-induced directionality in the collective emission of atoms confined within a hollow-core waveguide. Unlike in chiral waveguides, the atom-field coupling is here isotropic in the forward and backward direction. However, Raman-induced effective two-level emitters with spatially oscillating phases of the transition dipole enable thermally induced, but controllable directionality of the collective emission. By tuning the characteristic rate of collective decay we achieve a directionality of up to 0.89(1). We furthermore study the correlations of the emitted light close to and well above the threshold to collective emission, showing a buildup of coherence in the superfluorescent bursts while exhibiting thermal statistics below the threshold. To understand the underlying mechanism we employ numerical simulations based on the Truncated Wigner Approximation for spins and find good agreement. Additionally we present a simple model capable of reproducing the observed directionality via location blurring induced by the thermal motion of the atoms during collective emission. Our results will enable studies of collective, nonreciprocal interactions in non-chiral systems.

Figures (7)

• Figure 1: Simplified three level system of $^{87}$Rb (a) with $|1\rangle,|2\rangle,|3\rangle\, \widehat{=}\, \mathrm{^2S_{1/2}\,F=1}, \, \mathrm{^2S_{1/2}\,F=2},\, \mathrm{^2P_{1/2}\,F'=2}$ that corresponds to an effective two-level system (b) for pump detuning $\Delta_p\gg\Gamma'$ of decay rate $\Gamma = \Gamma_R \propto \Omega_p^2$ with pump Rabi frequency $\Omega_p$. (c) Experimental setup. (d,e,f) Experimental examples for superfluorescent bursts measured in $(+)$ (blue) and $(-)$ direction (orange) for $(\sigma_\textrm{v} ; N_\textrm{mc})$ of $(1.5;160)$ (d), $(1.5;106)$ (e), and $(5;132)$ (f).
• Figure 2: Temporal evolution of the pulse power and correlation functions above (a,b) and close to the SF threshold (c,d) for N$_\textrm{mc} = 248(7)$ and N$_\textrm{mc} = 55(6)$, respectively. Two-time auto-correlation functions $g^{(2)}(t_1,t_2)$ are shown in (a,c) with the corresponding average count rates (blue) during the bursts along with the equal-time $g^{(2)}(t,t)$. Intervals used for averaging $g^{(2)}(t,t)$ are marked by blue dotted lines in (a,c). Measured at $\Gamma = 2\pi\times 18.1(6)~$kHz.
• Figure 3: (a) Mean temporal burst power (blue) and equal-time $g^{(2)}(t,t)$ (red) vs $N_\mathrm{mc}$ in (+) direction for $N=71(6)\times10^{3}$ (corresponding to $N_\textrm{mc}=280$) and $\Gamma = 2\pi\times 33(2)~$kHz. Dashed lines show TWA simulations, while data points show measurement results. Horizontal lines indicate thermal (orange) and coherent (black) correlations. (b) Experimental SF burst widths (blue symbols) vs $N_\textrm{mc}$ for $\Gamma = 2\pi\times 33(2)~$kHz. The lines show a least-squares fit of type $\tau_\mathrm{FWHM} =6.0(2)/(N_{\textrm{mc}}\Gamma)$ to the experimental data (light blue) and the results of full numerical calculations averaged over 5$\times 10^4$ trajectories (purple).
• Figure 4: (a) Measured directionality $\kappa$ (symbols) of the SF bursts amplitudes in $(\pm)$ direction vs $N_\mathrm{mc}$. The pump power was set to three different values at $\Delta_p = 26.4\Gamma_{D_1}$ corresponding to $\Gamma_R / (2\pi) = 20(2)~$kHz, $33(2)$ kHz, $67(2)~$kHz. This results in $\sigma_\mathrm{v}$ of 5.0(2)($\bullet$), 3.0(1)($\bullet$) and 1.50(2)($\bullet$) respectively. The vertical lines indicate the SF thresholds, respectively. (b) Corresponding $\kappa$ as obtained from the TWA simulations including motion (lines). The results of the static effective model for $\sigma_\mathrm{v} = 3$ are shown by symbols, assuming constant $\tau$ ($\circ$) as well as $\tau(N_\textrm{mc}) = (N_\textrm{mc}\Gamma/2)^{-1}$ ($\bullet$). Each data point was averaged over $5\times 10^4$ trajectories, resulting in uncertainties smaller than the symbol size.
• Figure 5: Timescales $\tau^*(N_\textrm{mc})$ (crosses) and $\tau(N_\textrm{mc})$ (full circles) in units of the spontaneous emission rate $\Gamma$ vs $N_\mathrm{mc}$ used to map the dynamic onto the static model for $\sigma_\mathrm{v} = 3$. For $\tau^*$ we achieve the best match between static and dynamic model in Fig. \ref{['fig:fwd_bwd_data']}(b) while $\tau(N_\textrm{mc}) = (N_\textrm{mc} \Gamma/2)^{-1}$ is the characteristic superradiant time of the system.