van der Waals dephasing for Dicke subradiance in cold atomic clouds

We investigate numerically the role of near-field dipole-dipole interactions on the late-emission dynamics of large disordered cold atomic samples driven by a weak field. Previous experimental and numerical studies of subradiance in macroscopic samples have focused on low-density samples of pure two-level atoms, without internal structure, which corresponds to a scalar representation of the light. The cooperative nature of the late emission of light is then governed by the resonant optical depth. Here, by considering the vectorial nature of the light, we show the detrimental role of the near-field terms on cooperativity in higher-density samples. The observed reduction in the subradiant lifetimes is interpreted as a signature of the inhomogeneous broadening due to the near-field contributions, in analogy with the van der Waals dephasing phenomenon for superradiance.

Figure 1

Temporal dynamics of the scattered light after the switch-off of the driving field at $t=0$, for a given optical depth $b_0=14$ and for several densities $\rho {\lambda }^3$. As the density is increased, the decay at late times becomes slightly faster. The total scattered light (all polarizations together) is computed from the vectorial CDE model at $\theta ={45}^{\circ }$ from the laser propagation axis, after the system has been driven to steady state with a laser detuned by $\mathrm{\Delta }=-15{\mathrm{\Gamma }}_0$ and with circular polarization ${\sigma }^{-}$. The exclusion volume is $r_{min}={\rho }^{-1/3}/\pi$.

Figure 2

Subradiant lifetime ${\tau }_{\mathrm{sub}}$ as a function of (a) the on-resonance optical depth $b_0$ for several densities of the sample $\rho {\lambda }^3$ and (b) the density $\rho {\lambda }^3$ for several values of $b_0$. The insets are close-ups of the two lowest $b_0$ data sets. The lifetimes ${\tau }_{\mathrm{sub}}$ are obtained from an exponential fit of the total scattered light intensity (vectorial model) collected at $\theta ={45}^{\circ }$ in the fit range $I/I_0=[{10}^{-6},5\times {10}^{-6}]$. The parameters of the simulations are the same as in Fig. 1.

Figure 3

Subradiant lifetimes as a function of the resonant optical depth $b_0$ for different densities obtained using the scalar (diamonds) and vectorial models (filled circles). The fit interval is $I\left(t\right)/I\left(0\right)=[2\times {10}^{-7},{10}^{-6}]$. The parameters of the simulation are the same as in Fig. 1.

Figure 4

Eigenvalue distribution for two different densities, (a), (c)$\rho {\lambda }^3=6$ and (b), (d) $\rho {\lambda }^3=16$ obtained using (a), (b) scalar and (c), (d) vectorial models. The resonant optical depth is $b_0=20$ for both and the number of realizations is (a) $N_{\mathrm{r}}=36$, (b) $N_{\mathrm{r}}=255$, (c) $N_{\mathrm{r}}=40$, and (d) $N_{\mathrm{r}}=285$ such that the number of eigenvalues is the same in all cases. Here there is no exclusion volume and the detuning of the driving laser is $\mathrm{\Delta }=-15\mathrm{\Gamma }$ (for the sake of consistency with the CDEs, which are written in the laser-rotating frame, the eigenenergies are also shifted by $\mathrm{\Delta }$). The color code corresponds to the logarithm of the eigenstate density.

Figure 5

Vectorial model. (a), (b) Eigenvalue distribution of the collective coupled-dipole modes computed without any exclusion volume and with (a) a red-detuned driving laser $\mathrm{\Delta }=-15{\mathrm{\Gamma }}_0$ and (b) a blue-detuned laser $\mathrm{\Delta }=+15{\mathrm{\Gamma }}_0$. The two subradiant pair branches (with $\text{IPR}\approx 0.5$) are asymmetric: for a given interatomic distance they have different frequencies and lifetimes. The extent of the branches is very long and, depending on the sign of the detuning, one of the branches crosses the resonance ${\mathrm{\Delta }}_k=0$ [45]. The black lines correspond to the analytical expressions with a cutoff corresponding to the exclusion volume $r_{min}={\rho }^{-1/3}/\pi$. Here $b_0=30,\rho {\lambda }^3=20$. (c) Decay of the scattered light collected at $\theta ={45}^{\circ }$ for opposite sign detunings and for different exclusion volumes. Without any exclusion volume the entire long-lived dynamics is influenced by pairs, with a noticeable difference between detunings of opposite signs (green lines). On the contrary, for $r_{min}={\rho }^{-1/3}/\pi$, the decay dynamics is independent of the sign of the detuning. With $r_{min}=0.5/k_0$, only the intermediate dynamics is independent of the sign of the detuning and for very late times the decay for $\mathrm{\Delta }=+15{\mathrm{\Gamma }}_0$ starts to be slower, showing the influence of pairs. (d) Decay of the light scattered at $\theta ={90}^{\circ }$ decomposed into two polarization channels: parallel ($\parallel$) and orthogonal ($\perp$) to the scattering plane, which is defined by the wave vector of the incoming laser beam and the observation direction. Without any exclusion volume, both superradiant and subradiant parts are partially polarized. With the exclusion volume $r_{min}={\rho }^{-1/3}/\pi$, the superradiant part is polarized while the subradiant part is depolarized. Inset shows two polarization channels of light scattered at $\theta ={90}^{\circ }$ for the exclusion volume $r_{min}=0.5/k_0$. For red detuning, the entire subradiant part is depolarized, whereas for blue detuning, light is depolarized at intermediate times and at late times, when there is an influence of pairs, it becomes polarized. For panels (c) and (d), $b_0=16,\rho {\lambda }^3=10$.

Figure 6

Vectorial model. (a) Comparison of subradiant lifetimes obtained with $r_{min}={\rho }^{-1/3}/\pi$ (full circles) and $r_{min}=0.5/k$ (diamonds) with a red-detuned laser $\mathrm{\Delta }=-15{\mathrm{\Gamma }}_0$. (b) Subradiant lifetimes as a function of the exclusion volume $r_{min}$ for $b_0=16, \rho {\lambda }^3=10$ and a red-detuned laser $\mathrm{\Delta }=-15\mathrm{\Gamma }$. Lifetimes were obtained in the fit range $I/I_0=[{10}^{-6},5\times {10}^{-6}]$.