Cooperative Resonances in Light Scattering from Two-Dimensional Atomic Arrays

Ephraim Shahmoon, Dominik S. Wild, Mikhail D. Lukin, and Susanne F. Yelin

Phys. Rev. Lett. 118, 113601 – Published 14 March 2017

Abstract

We consider light scattering off a two-dimensional (2D) dipolar array and show how it can be tailored by properly choosing the lattice constant of the order of the incident wavelength. In particular, we demonstrate that such arrays can operate as a nearly perfect mirror for a wide range of incident angles and frequencies, and shape the emission pattern from an individual quantum emitter into a well-defined, collimated beam. These results can be understood in terms of the cooperative resonances of the surface modes supported by the 2D array. Experimental realizations are discussed, using ultracold arrays of trapped atoms and excitons in 2D semiconductor materials, as well as potential applications ranging from atomically thin metasurfaces to single photon nonlinear optics and nanomechanics.

Received 1 October 2016

DOI:https://doi.org/10.1103/PhysRevLett.118.113601

Physics Subject Headings (PhySH)

Light propagation, transmission & absorption Light-matter interaction Quantum optics Superradiance & subradiance

Atomic systems

Atomic, Molecular & Optical

Authors & Affiliations

Ephraim Shahmoon¹, Dominik S. Wild¹, Mikhail D. Lukin¹, and Susanne F. Yelin^1,2

¹Department of Physics, Harvard University, Cambridge Massachusetts 02138, USA
²Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA

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Vol. 118, Iss. 11 — 17 March 2017

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Images

Figure 1
(a) 2D array of atoms spanning the $x y$ plane at $z = 0$ , with interatomic spacing $a$ on the order of the resonant wavelength of the atoms, $λ$ . For resonant light, the individual atomic cross section is of order $λ^{2}$ (dashed circles). (b) Light scattering off the array in the single diffraction order regime: The incident field (yellow arrow) produces a forward scattered field at $z > 0$ and a reflected field at $z < 0$ . (c) Intensity transmission coefficient $T$ and reflection coefficient $R$ for a square lattice at normal incident and resonant light ( $δ = 0$ ) as a function of the lattice constant $a$ . Strong scattering is observed with perfect reflection occurring at $a / λ \approx 0.2$ , 0.8.
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Figure 2
(a) The cooperative shift $Δ$ , Eq. (4), as a function of the lattice constant $a$ (normal incidence). This plot is central in the design of the scattering since the shift determines the collective resonances of the array according to Eq. (3). Perfect reflection occurs when the cooperative shift equals the incident detuning, $δ = Δ$ . For example, $Δ = 0$ at $a / λ \approx 0.2$ , 0.8 explains the resonances in Fig. 1. (b) Intensity reflection coefficient $R$ as a function of lattice constant $a$ and detuning $δ$ . We note that the emerging contour of perfect reflection (bright yellow) coincides with the cooperative resonance plotted in (a) (marked here by the dashed black curve).
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Figure 3
Scattering at a general angle of incidence for a lattice constant $a = 0.2 λ$ . (a) Intensity reflection coefficient $R_{s s}$ for $s$ -polarized incident and scattered fields at zero detuning from the bare atomic resonance ( $δ = 0$ ) as a function of the in plane components $k_{x, y}$ of the incident wave vector. (b) $s s$ component of the cooperative shift matrix $\bar{\bar{Δ}}$ . The variation of the energy shift around the resonance $Δ_{s s} = δ = 0$ is small compared to an atomic linewidth, which explains the high reflection $R_{s s}$ at all angles.
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Figure 4
(a) Band structure of the collective surface modes of the atom array for $a = 0.2 λ$ . The three bands in solid lines correspond to the three eigenvalues of the cooperative shift $\bar{\bar{Δ}} (k_{∥})$ , whereas the single dashed band is that of array atoms with a single circularly polarized transition, $e_{0} = (1, i, 0) / \sqrt{2}$ . The inset shows the location of the special points $Γ$ , $X$ , $M$ in the first Brillouin zone. The dotted vertical lines (main figure) and the circle (inset) indicate the light cone $| k_{∥} | = 2 π / λ$ . (b) An impurity atom (red) is placed at the center of the array. The black and white colors of the array atoms represent the two sublattices which form the surface mode $M$ . (c) Decay rate to surface modes inside (dashed) and outside (solid) the light cone, of the impurity atom with polarization $e_{I} = (1, - i, 0) / \sqrt{2}$ as a function its transition frequency $ω_{I}$ . (d) Intensity of the electric field produced by an impurity (red dot), driven at detuning $(ω - ω_{a}) / γ = 1.1$ and resonant with the surface modes near the corner of the Brillouin zone ( $M$ ). An alternating periodic potential, $δ ω / γ = \pm 0.1$ , is applied on an array of $30 \times 30$ atoms (white circles), with all other parameters the same as in (b). The bright yellow region near $z = 0$ corresponds to the excited surface modes, which are then coupled into the collimated beam by the periodic potential. (e) Emitted power integrated over a cone of half angle $θ$ for the situation described in (d).
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