Propagation of light in cold emitter ensembles with quantum position correlations due to static long-range dipolar interactions

We analyze the scattering of light from dipolar emitters whose disordered positions exhibit correlations induced by static, long-range dipole-dipole interactions. The quantum-mechanical position correlations are calculated for zero-temperature bosonic atoms or molecules using variational and diffusion quantum Monte Carlo methods. For stationary atoms in dense ensembles in the limit of low light intensity, the simulations yield solutions for the optical responses to all orders of position correlation functions that involve electronic ground and excited states. We calculate how coherent and incoherent scattering, collective linewidths, line shifts, and eigenmodes, and disorder-induced excitation localization are influenced by the static interactions and the density. We find that dominantly repulsive static interactions in strongly confined oblate and prolate traps introduce short-range ordering among the dipoles, which curtails large fluctuations in the light-mediated resonant dipole-dipole interactions. This typically results in an increase in coherent reflection and optical depth, accompanied by reduced incoherent scattering. The presence of static dipolar interactions permits the highly selective excitation of subradiant eigenmodes in dense clouds. This effect becomes even more pronounced in a prolate trap, where the resonances narrow below the natural linewidth. When the static dipolar interactions affect the optical transition frequencies, the ensemble exhibits inhomogeneous broadening due to the nonuniformly experienced static dipolar interactions that suppress cooperative effects, but we argue that, e.g., for Dy atoms such inhomogeneous broadening is negligible.

Figure 1

Schematic illustration of the trapping geometries illuminated by coherent light. (a) An oblate (pancake-shaped) trap in the $xy$ plane (${\ell }_z\ll {\ell }_x={\ell }_y$), with the static atomic dipoles aligned perpendicular to the plane along the light propagation direction. An elongated prolate (cigar-shaped) trap (${\ell }_z\gg {\ell }_x={\ell }_y$) with the static dipoles aligned perpendicular to the long axis. The incident light in (b) propagates along either the long or short axis of the trap.

Figure 2

Transmission and reflection of $x$-polarized Gaussian beam from 200 atoms in an oblate trap ${\ell }_x/{\ell }_z=18$, with ${\ell }_z=1/k$, as a function of the detuning for different atom numbers. The peak densities for different $N$ are ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 0.01,0.05,0.1,0.5,1$. (a) Coherent optical depth ${\mathrm{OD}}_{\mathrm{coh}}$, (b) incoherent transmission $T_{\mathrm{inc}}$, (c) coherent reflection $R_{\mathrm{coh}}$, (d) incoherent reflection $R_{\mathrm{inc}}$ [Eqs. (20, 21, 22, 23)]. Coherent transmission resonance (e) HWHM and (f) shift. The lens NA 0.2 and 0.8 for coherent and incoherent light, respectively.

Figure 3

Dipolar potential between two dipoles perpendicular to the separation and the 2D density profile as a function of the radius for 100 atoms for different interaction lengths $R_{\mathrm{dip}}$ [Eq. (4)] for ${\ell }_x/{\ell }_z=100$.

Figure 4

Transmission and reflection from 200 two-level atoms in an oblate trap at peak densities ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 0.1$ (top row) and 1 (bottom row), illuminated with a Gaussian beam, as a function of the laser detuning for different interaction strengths. [(a),(e)] Coherent optical depth ${\mathrm{OD}}_{\mathrm{coh}}$; [(b),(f)] incoherent transmission $T_{\mathrm{inc}}$; [(c),(g)] coherent reflection $R_{\mathrm{coh}}$; [(d),(h)] incoherent reflection $R_{\mathrm{inc}}$ [Eqs. (20, 21, 22, 23)]. The lens NA 0.2 (top row) 0.25 (bottom row) and 0.8 for coherent and incoherent light, respectively.

Figure 5

Change in the resonance width, shift, and the intensity extrema of coherently reflected and transmitted light for 200 atoms at the peak densities [(a),(b)] ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 0.1$ and [(c),(d)] 1 extracted from Fig. 4. [(a),(c)] Coherent transmission and reflection resonance widths $T_{\mathrm{coh}},R_{\mathrm{coh}}$; [(b),(d)] maximum $R_{\mathrm{coh}}$ and minimum $T_{\mathrm{coh}}$.

Figure 6

Atom separations for the system of 200 atoms in an oblate trap at density ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 0.1$ with the resonant wavelength and the trap parameters from Fig. 4. Probability distributions of [(a),(c)] nearest-neighbor pair $d_{\mathrm{nn}}$ and [(b),(d)] all-pair separations $d_{\mathrm{aa}}$. The short-range distributions are shown on a linear scale while the long-range plots are shown on a logarithmic scale.

Figure 7

Atom separations for the system of 200 atoms in an oblate trap at density ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 1$ with the resonant wavelength and the trap parameters from Fig. 4. Probability distributions of (a) nearest-neighbor pair $d_{\mathrm{nn}}$ and (b) all-pair separations $d_{\mathrm{aa}}$.

Figure 8

Distribution of eigenmodes as a function of the collective resonance linewidth and line shift for two cases of Fig. 4 with [(a),(b)] ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 0.1$ and [(c),(d)] 1 for [(a),(c)] independent atoms $R_{\mathrm{dip}}=0$ and [(b),(d)] static DD interactions with $R_{\mathrm{dip}}\sqrt{{\overline{\rho }}_{2\mathrm{D}}}\simeq 1.6$. The bin size $[\mathrm{\Delta }{log}_{10}(\nu /\gamma )\times \mathrm{\Delta }\delta /\gamma ]=[0.017\times 0.04]$.

Figure 9

Occupation of collective excitation eigenmodes in the steady-state response of Fig. 4 at peak densities ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 0.1$ (top row) and 1 (bottom row), at detunings $\mathrm{\Delta }/\gamma =0.4$ and $-0.6$, respectively, as a function of the collective linewidth $\nu$ and line shift $\delta$. [(a),(c)] Independent atoms; [(b),(d)] $R_{\mathrm{dip}}\sqrt{{\overline{\rho }}_{2\mathrm{D}}}\simeq 1.6$ [Eq. (4)].

Figure 10

(a) Coherent optical depth ${\mathrm{OD}}_{\mathrm{coh}}$, (b) incoherent transmission $T_{\mathrm{inc}}$ as in Fig. 4 at ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 1$, but for an isotropic $J=0\to J^{\prime }=1$ transition. The expectation values of the magnitudes of the average excitation $\langle |\mathcal{P}{|}^{\mathrm{av}}\rangle$, and its in-plane $\langle |{\mathcal{P}}_{\parallel }{|}^{\mathrm{av}}\rangle$, and out-of-plane $\langle |{\mathcal{P}}_{\perp }{|}^{\mathrm{av}}\rangle$ components for $R_{\mathrm{dip}}\sqrt{{\overline{\rho }}_{2\mathrm{D}}}\simeq 0.15$ for (c) ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 0.1$ and (d) 1.

Figure 11

Inhomogeneously broadened transition frequencies induced by static dipolar interactions for two-level atoms in an oblate trap for $N=200, {\ell }_{x}k=6, {\ell }_{z}k=0.15, R_{\mathrm{dip}}k=0.15$ at ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 1$. (a) Coherent optical depth ${\mathrm{OD}}_{\mathrm{coh}}$, (b) incoherent transmission $T_{\mathrm{inc}}$, (c) coherent reflection $R_{\mathrm{coh}}$, and (d) incoherent reflection $R_{\mathrm{inc}}$ [Eqs. (20, 21, 22, 23)] with coupling strength $D$ [Eq. (25)] in terms of detuning $\zeta$ from the most likely atomic transition frequency $\overline{\delta }$ in each case.

Figure 12

Coherently and incoherently forward-scattered power (in units of incident light intensity/$k^2$) from 10 atoms in a prolate trap (${\ell }_z/{\ell }_x=25$) for different static dipole interaction strengths at the peak densities [(a),(b)] ${\overline{\rho }}_{1\mathrm{D}}/k\simeq 0.1$ and [(c),(d)] ${\overline{\rho }}_{1\mathrm{D}}/k\simeq 1$, and [(e),(f)] the atomic pair distributions. The light propagates parallel to the long trap axis. Normalized (e) nearest-neighbor and (f) all-pair distributions between the atoms. The lens NA 0.8.

Figure 13

Scattered light resonance (a) widths and (b) shifts of maxima from Fig. 12 for ${\overline{\rho }}_{1\mathrm{D}}/k\simeq 1$. The HWHM is calculated by averaging the contributions of the left and right sides of the peak.

Figure 14

Eigenmode occupations at the steady-state response in Fig. 12 at the detuning [(a)–(d)] $\mathrm{\Delta }=0$ and [(e),(f)] $-0.9\gamma$ for (a) and (b) ${\overline{\rho }}_{1\mathrm{D}}/k\simeq 0.1$ and [(c)–(f)] 1. The static interaction [(a), (c), (e)] $R_{\mathrm{dip}}=0$ and [(b), (d), (f)] $R_{\mathrm{dip}}{\overline{\rho }}_{1D}\simeq 0.47$.

Figure 15

(a) Coherently and (b) incoherently forward-scattered power (in units of incident light intensity/$k^2$) from 10 atoms in a prolate trap for different static dipole interaction strengths at the peak density ${\overline{\rho }}_{1\mathrm{D}}/k\simeq 1$ and ${\ell }_z/{\ell }_x=25$. The light propagates perpendicular to the long trap axis. The lens NA 0.25 and 0.8 for coherent and incoherent light, respectively.

Figure 16

Eigenmode occupations in Fig. 15 at the detuning [(a),(b)] $\mathrm{\Delta }=0$; (c) $-10\gamma$; and (d) $4\gamma$. (a) $R_{\mathrm{dip}}k=0$ [(b)–(d)] $R_{\mathrm{dip}}k\simeq 1$. The histogram bins $[\mathrm{\Delta }{log}_{10}(\nu /\gamma ),\mathrm{\Delta }\delta /\gamma ]=[0.016,0.04]$, [0.016,0.04], [0.03,0.12], [0.03, 0.07], respectively.

Figure 17

Localization of excitations $\left|\mathcal{P}\right|$ [in units of $\mathcal{D}\mathcal{E}/\left(\hslash \gamma \right)]$ as a function of the distance from the most excited atom $|\mathit{\varrho }-{\mathit{\varrho }}_{\max }|$ for 200 atoms in an oblate trap at the peak density ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 1$ for (a) all data, for (b) 2% of the most localized cases (1000 realizations). The corresponding profile in an individual stochastic realization with $R_{\mathrm{dip}}\sqrt{{\overline{\rho }}_{2\mathrm{D}}}\simeq 0.15$ for (c) ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 0.04$, (d) 1.

Figure 18

[(a),(b)] Peak localized excitations $\langle \max \left(\right|\mathcal{P}\left|\right)\rangle$ [in units of $\mathcal{D}\mathcal{E}/\left(\hslash \gamma \right)]$ and [(c),(d)] localization length $\langle {\varrho }_0\rangle$ as a function of interaction strength $R_{\mathrm{dip}}\sqrt{{\overline{\rho }}_{2\mathrm{D}}}$ [Eq. (4)] for 200 atoms in an oblate trap. [(e),(f)] The localization length for the $2\%$ most localized cases. The values of (d) $\langle {\rho }_0\rangle k\simeq 1.28,1.69,1.68,1.77$ and (f) $\langle {\rho }_0\rangle k\simeq 0.27,0.15,0.16,0.199$ for the density ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 1$

Figure 19

Transmission and reflection from 200 two-level atoms in an oblate trap as a function of the laser detuning for different peak densities ${\overline{\rho }}_{2\mathrm{D}}/k^2$ with ${\ell }_x/{\ell }_z=25$, for which $R_{\mathrm{dip}}\sqrt{{\overline{\rho }}_{2\mathrm{D}}}\simeq 0.12,0.18, {\ell }_{z}k\simeq 0.27,0.18,{\ell }_{x}k\simeq 6.7,4.4$ (solid curves) $R_{\mathrm{dip}}=0$ (dashed curves). (a) Coherent optical depth ${\mathrm{OD}}_{\mathrm{coh}}$, (b) incoherent transmission $T_{\mathrm{inc}}$, (c) coherent reflection $R_{\mathrm{coh}}$, and (d) incoherent reflection $R_{\mathrm{inc}}$ [Eqs. (20, 21, 22, 23)]. The lens NA 0.45 and 0.8 for coherent and incoherent light, respectively.

Figure 20

Variation of scattered light intensity extrema, resonance width and shift with peak density ${\overline{\rho }}_{2\mathrm{D}}/k^2$, and eigenmode occupations for the system shown in Fig. 19. (a) Peak coherent transmission $\min \left({\mathrm{T}}_{\mathrm{coh}}\right)$ and reflection $max\left(R_{\mathrm{coh}}\right)$, (c) HWHM, and (d) shift. The independent-atom case (the static DD case) is represented by dashed (solid) lines. Eigenmode occupations for ${\overline{\rho }}_{2\mathrm{D}}/k^2\simeq 3.3$ ($R_{\mathrm{dip}}\sqrt{{\overline{\rho }}_{2\mathrm{D}}}\simeq 0.18$) for (d) $\mathrm{\Delta }=0$, and (e) $3\gamma$. The bin size $[\mathrm{\Delta }{log}_{10}(\nu /\gamma ),\mathrm{\Delta }\delta /\gamma ]=[0.016,0.08]$.