Light scattering from dense cold atomic media

We theoretically study the propagation of light through a cold atomic medium, where the effects of motion, laser intensity, atomic density, and polarization can all modify the properties of the scattered light. We present two different microscopic models: the “coherent dipole model” and the “random-walk model”, both suitable for modeling recent experimental work done in large atomic arrays in the low-light-intensity regime. We use them to compute relevant observables such as the linewidth, peak intensity, and line center of the emitted light. We further develop generalized models that explicitly take into account atomic motion. Those are relevant for hotter atoms and beyond the low-intensity regime. We show that atomic motion can lead to drastic dephasing and to a reduction of collective effects, together with a distortion of the line shape. Our results are applicable to model a full gamut of quantum systems that rely on atom-light interactions, including atomic clocks, quantum simulators, and nanophotonic systems.

Figure 1

Scheme. (a) Microscopic models: random-walk model (left) and coherent dipole model (right). In the random-walk model, a photon is randomly scattered by the atoms. Scattering events are characterized by the incident and outgoing wave vectors of the photons and their corresponding polarizations. In the coherent dipole model, atoms are coupled by dipole-dipole interactions $G_{ab}$, and all atoms contribute to the fluorescence. (b) Experimental setup for measuring the fluorescence from a cloud of atoms. An incident laser drives an atomic transition with a spontaneous emission rate $\mathrm{\Gamma }$. Atoms absorb and emit light. The detectors collect scattered photons at an angle $\theta$ measured from the incident beam direction.

Figure 2

CD model: on-resonance intensity. (a) Angular distribution: due to the constructive interference along the forward direction ($\theta =0$), the intensity is drastically enhanced within a small angular region. (b) and (c) The intensities for three different directions ($\theta =0,\pi /10,\pi /2$) are shown as a function of atom number, each normalized to the value at $N=50$. All are calculated for a spherical cloud of fixed size. The right vertical axes label the intensity for $\theta =\pi /2,\pi /10$. In (b) the OD and the density are relatively low (when $N=500, \tau =2, n_0{k}_0^{-3}=0.0015$). The transverse intensity increases $\sim N$, while the forward intensity increases $\sim N^2$, showing a collective enhancement for a small window of $\theta$ around zero. Outside this narrow angular window the enhancement disappears, and the intensity becomes almost $\theta$ independent, as indicated by the identical behavior observed for two different angles, $\theta =\pi /10$ and $\pi /2$. In (c) the OD and density are relatively large (when $N=500, \tau =10, n_0{k}_0^{-3}=0.017$). The rate at which the intensity increases with $N$ slows down in both the forward and transverse directions, with $I\sim N^1$ for $\theta =0$ and $I\sim N^{0.5}$ for $\theta =\pi /2$, respectively. (d) On-resonance intensity as a function of OD: it is highly suppressed at large OD. Here, the intensity is normalized to the corresponding value at $\tau =25$ for each direction.

Figure 3

CD model: collective broadening. (a) FWHM linewidth as a function of OD. At small OD, the FWHM increases linearly with OD, but as OD increases, density effects set in, multiple scattering events become relevant, and the linewidth dependence on OD is no longer linear. Two different atom numbers are used for the blue solid ($N=1000$) and black dashed ($N=200$) lines, and the OD is varied by changing the cloud size. For the blue solid line, $n_0{k}_0^{-3}=0.14$ at $\tau =25$ and 0.003 at $\tau =2$. For the black dashed line, the density at the same OD is doubled. With smaller density, the linewidth increases to a larger value in the large-OD regime. (b) The line shape at small OD values ($\tau =2, n_0{k}_0^{-3}=0.002$) for different angles $\theta [\theta$ is defined in Fig. 1] is mainly Lorentzian. Here, the intensity is normalized to the on-resonance intensity for each $\theta$. (c) At large OD ($\tau =20, n_0{k}_0^{-3}=0.06$), the fluorescence line shape significantly broadens and stops being Lorentzian. The brown dots for $\theta =0$ show the Lorentzian fit, which fails to describe the line shape. At intermediate $\theta$, a double-peak structure shows up. For all panels, the cloud shape is spherical.

Figure 4

CD model: frequency shift. For small density, the shift calculated from the coherent dipole model (blue solid line) increases linearly with density as predicted by the mean-field theory. However, when density is large, there is a significant deviation from the mean-field result. When motional effects are taken into account (red dotted line, Doppler width of $5\mathrm{\Gamma }$, see Sec. 4a), the nonlinear suppression of the frequency shift with density is less severe.

Figure 5

CD model: finite-size scaling. (a) The linewidth is calculated for different numbers of atoms at the same OD ($\tau =4$ for $N=3000$) by varying the density (blue line with circles). From $N=1000$ to 5000 the linewidth is not obviously changed. In contrast, by keeping the same density ($n_0{k}_0^{-3}=0.0037$ for $N=3000$) while varying the OD (magenta line with triangles), the linewidth keeps changing. (b) The frequency shift is calculated for different numbers of atoms. By keeping constant density (magenta line with triangles), the frequency shift remains almost constant, while for constant OD (blue line with circles), there is a significant variation of the frequency shift with $N$. Here, the cloud aspect ratio is $R_x:R_y:R_z=2:2:1$ [53].

Figure 6

CD model: effect of laser polarization. (a) Intensity distribution in the plane perpendicular to the laser propagation direction, i.e., $\theta =\pi /2$ for all $\varphi$'s (inset shows the geometry). $\varphi =0$ is the $z$ direction. Here, the incident laser is polarized along $z$. The intensity detected along the laser polarization is suppressed compared to other directions. (b) and (c) show the line shape and linewidth of light detected at $\theta =\pi /2,\varphi =0$. (b) At small OD, the FWHM linewidth is below the natural linewidth. (c) As OD increases, the linewidth is collectively broadened due to multiple-scattering processes [Fig. 3]. For all panels, degeneracy in excited levels has been assumed.

Figure 7

RW: transformation of Stokes vectors. In the random-walk model, a scattering event is determined by two consecutive transformations of local coordinates: $\{{\mathbf{e}}_3,{\mathbf{e}}_{\mathrm{r}},{\mathbf{e}}_{\mathrm{l}}\}\to \{{\mathbf{e}}_3^{\prime },{\mathbf{e}}_{\mathrm{r}}^{\prime },{\mathbf{e}}_{\mathrm{l}}^{\prime }\}$ via rotation $\varphi$ and $\{{\mathbf{e}}_3^{\prime },{\mathbf{e}}_{\mathrm{r}}^{\prime },{\mathbf{e}}_{\mathrm{l}}^{\prime }\}\to \{{\mathbf{e}}_3^{\prime \prime },{\mathbf{e}}_{\mathrm{r}}^{\prime \prime },{\mathbf{e}}_{\mathrm{l}}^{\prime \prime }\}$ via rotation $\theta$ [68].

Figure 8

RW model. (a) Distribution of on-resonance intensity in the plane $\theta =\pi /2$. Along the direction where single-scattering events are forbidden ($\varphi =0$), the light intensity is suppressed. (b) The FWHM linewidth increases linearly with ${\tau }_p$ in a dilute medium. The linewidth along the direction parallel to the laser polarization can drop below $\mathrm{\Gamma }$ for small ${\tau }_p$ (purple squares). (c) Line shapes for different ${\tau }_p$'s and angles of observation. The line shape is Lorentzian for low ${\tau }_p$ but gets distorted and develops a double-peak structure as the ${\tau }_p$ increases. Here, for all angles, the signal is collected within a small angular region $\delta \theta =5^{\circ }$. The intensity is normalized to the value at zero detuning for $\theta =0$ and ${\tau }_p=0.3$, and the width of laser beam is 5 times the Gaussian width of the atomic cloud. Here, all ${\tau }_p$ are labeled as the value at $\mathrm{\Delta }=0$.

Figure 9

Modified frozen dipole model: peak intensity. (a) Collective forward enhancement: forward intensity normalized by the transverse intensity as a function of Doppler width. (b) Intensity detected along different directions at Doppler width ${\mathrm{\Delta }}_{\mathrm{D}}=6\mathrm{\Gamma }$ as a function of atom number for fixed cloud size (when $N=500, \tau =2, n_0{k}_0^{-3}=0.0015$). $I\sim N^{1.6}$ for $\theta =0$, and $I\sim N^{0.7}$ for $\theta =\pi /2$. The right vertical axis labels the intensity for $\theta =\pi /2,\pi /10$.

Figure 10

Semiclassical model: single-atom line shape. The excitation line shape is calculated for three different driving strengths, each normalized by ${\mathrm{\Omega }}^2$. The atomic motion is allowed to be one-dimensional and parallel to the laser propagating direction. Here, we used ${\omega }_r=0.6\mathrm{\Gamma }$ (the value of the ${}^{1}S_0\to {}^{3}P_1$ transition of Sr [69]).

Figure 11

Semiclassical model: effect of motion on frequency shift. Two atoms are driven by a pair of counterpropagating lasers, with $\mathrm{\Omega }=0.1\mathrm{\Gamma }, {\omega }_r=0.6\mathrm{\Gamma }$. The red dashed line shows the result of the semiclassical model where motion is allowed in three dimensions. The blue solid line shows the result from the modified CD. The center of the excitation line shape is calculated at $t=5/\mathrm{\Gamma }$ for different Doppler widths for both models.