$N$ scaling of large-sample collective decay in inhomogeneous ensembles

We experimentally study collective decay of an extended disordered ensemble of $N$ atoms inside a hollow-core fiber. We observe up to 300-fold enhanced decay rates, strong optical bursts, and a coherent ringing. Due to inhomogeneities limiting the synchronization of atoms, the data does not show the typical scaling with $N$. We show that an effective number of collective emitters can be determined to recover the $N$ scaling known to homogeneous ensembles over a large parameter range. This provides physical insight into the limits of collective decay and allows for its optimization in extended ensembles as used, e.g., in quantum optics, precision time-keeping, or waveguide QED.

Figure 1

(a) Simplified three-level system with $|1\rangle \widehat{=}{}^2{S}_{1/2}F=1, |2\rangle \widehat{=}{}^2{S}_{1/2}F=2$, and $|3\rangle \widehat{=}{}^2{P}_{1/2}{F}^{\prime }=2$. The excited state decay rate is $\mathrm{\Gamma }=2\pi \times 5.75\mathrm{MHz}$ [52]. (b) Corresponding effective TLS comprised of the ground states $|1\rangle , |2\rangle$. (c) Simplified experimental setup.

Figure 2

(a) Exemplary traces of the transmitted pump power without atoms (blue, 16 averages, scaled down by factor 0.5) and the Stokes power with atoms loaded into the HCPBGF (single-shot data). The peak pump Rabi frequency is ${\mathrm{\Omega }}_p^{\left(0\right)}=6.4\mathrm{\Gamma }, {\mathrm{\Delta }}_p=18.4\mathrm{\Gamma }$ with $N=83\times {10}^3$ (orange) and $N=220\times {10}^3$ (green). $t=0$ corresponds to the moment the pump power reaches $50\%$ of its mean maximum value. The extracted delay $t_D$ of the orange signal is shown by the grey shaded area. (b) Measured mean delays of the first emitted SF burst versus $N_{\mu }$ for ${\mathrm{\Delta }}_p=18.4\mathrm{\Gamma }$, with error bars representing the standard deviation. The expected theoretical dependence [Eq. (2)] is shown considering $N_{\mu }$ (dashed green line) as well as the MCN $N_{\text{mc}}$ (solid orange line) for $\beta =0.07$.

Figure 3

(a), (b) Experimentally determined initial relative collective decay rate ${\mathrm{\Gamma }}_N/{\langle {\mathrm{\Gamma }}_R\rangle }_r$ versus $N_{\mu }=\mu N$ (a) and MCN $N_{\text{mc}}=\eta \mu N$ (b) for $2\mathrm{\Gamma }\le {\mathrm{\Delta }}_p\le 26.4\mathrm{\Gamma }$ at ${\mathrm{\Omega }}_p^{\left(0\right)}=6.4\mathrm{\Gamma }$. The symbol size is proportional to $N$. The solid blue lines show the dependence ${\mathrm{\Gamma }}_N/{\langle {\mathrm{\Gamma }}_R\rangle }_r=N_{\mu }$ (a) and ${\mathrm{\Gamma }}_N/{\langle {\mathrm{\Gamma }}_R\rangle }_r=N_{\text{mc}}$ (b) for homogeneous conditions. (c, d) Measured mean peak power of the first SF burst versus $N_{\mu }^2$ (c) and $N_{\text{mc}}^2$ (d) for ${\mathrm{\Delta }}_p=18.4\mathrm{\Gamma }$. The green (orange) dashed line in panel (c) represents a linear (quadratic) scaling with $N_{\mu }$ for reference. The orange solid line in panel (d) represents a least-squares linear fit to the data, excluding the last data point (see text).

Figure 4

Relative MCN $N_{\text{mc}}({\mathrm{\Delta }}_p,N)/N_{\mu }$ versus $N_{\mu }$ and detuning ${\mathrm{\Delta }}_p$. The black line shows where $\tilde{\alpha }({\mathrm{\Delta }}_p,N)=1$ and the white line where ${\eta }_{\text{inh}}$ is as large as ${\eta }_s(\tilde{\alpha }=1)$.

Figure 5

Same as Fig. 3 but using the widths of the first bursts instead of the mean delays to determine ${\mathrm{\Gamma }}_N$.

Figure 6

Initial relative collective decay rate ${\mathrm{\Gamma }}_N/{\langle {\mathrm{\Gamma }}_R\rangle }_r$ versus corresponding MCN when considering only inhomogeneous broadening (a), inhomogeneous broadening and pump absorption (without conversion) (b), pump attenuation (absorption and conversion) without inhomogeneous broadening (c). Compare to Figs. 3 and 3.

Figure 7

Scaling factor $\beta \left({\mathrm{\Delta }}_p\right)$ as determined from manual fits to the measured $\langle t_D\left(N\right)\rangle$. The solid line shows a linear fit for ${\mathrm{\Delta }}_p>6\mathrm{\Gamma }$ with $\beta \left({\mathrm{\Delta }}_p\right)=0.182-6.1\times {10}^{-3}{\mathrm{\Delta }}_p/\mathrm{\Gamma }$.