Photon-induced atom recoil in collectively interacting planar arrays

The recoil of atoms in arrays due to the emission or absorption of photons is studied for subwavelength interatomic spacing. The atoms in the array interact with each other through collective dipole-dipole interactions and with the incident laser field in the low-intensity limit. Shining uniform light on the array gives rise to patterns of excitation and recoil in the array. They arise due to the interference of different eigenmodes of excitation. The relation between the recoil and the decay dynamics is studied when the array is in its excitation eigenstates. The recoil experienced by a subradiant collective decay is substantially larger than that from independent atom decay. A method to calculate the rate of recoil when steady state has been achieved with a constant influx of photons is also described.

Figure 1

A schematic of the system considered. The atoms are the blue spheres and are placed in an ordered array in the $x-y$ plane with interatomic separation $d$. The incoming light (red wavy arrow) is in the $\widehat{\mathbf{z}}$ direction with wave vector ${\mathbf{k}}_0$.

Figure 2

The net kinetic-energy kick and the linear fit for an $8\times 8$ array, initially in an eigenstate, with respect to the inverse of the decay rate of the eigenvalue. Blue dots denote the calculated plot points, and the red line denotes the linear fit of the data. (a) denotes the kick in the out-of-plane direction, which is proportional to the lifetime with a slope of 2/5, as expected from Eq. (16). (b) denotes the net kick in the plane of the array.

Figure 3

The recoil energy deposited in the $z$ direction on one array of a cavity with two $11\times 11$ curved arrays with $d=0.75\lambda$ and $L=19.75\lambda$ corresponding to the system in Sec. 3b1. Each cell denotes one atom in the array. Note the center atom has a kick of $\sim 900E_r$.

Figure 4

The excitation and recoil distribution pattern on an $11\times 11$ atom array with $d=0.68\lambda$ when excited by a laser pulse with peak Rabi frequency ${\mathrm{\Omega }}_0=0.02\mathrm{\Gamma }$ and pulse width $t_w=16/\mathrm{\Gamma }$ at zero detuning. Each cell denotes one atom in the array. (a) shows the integral of the excitation probability in time for each atom in arbitrary units, and (b) shows $\mathrm{\Delta }{K}_z/E_r$ deposited on each atom of the array per photon incident on the atom.

Figure 5

The effects of an atom missing from an array of $11\times 11$ atoms with $d=0.68\lambda$ and pulsed light illumination with no detuning. The plots show the total integrated probability of excitation of each atom in arbitrary units. Compare with Fig. 4, which is the case where no atoms are missing. (a) has the atom at coordinate (2,2) missing. It has a larger impact and affects up to second-nearest-neighbor atoms. (b) has the atom at coordinate (3,3) missing. It has a smaller impact and affects only the immediate neighbors.

Figure 6

The decomposition of the state into its eigenmodes when excited by uniform light for a $5\times 5$ array with $d=0.4\lambda$ versus the detuning $\delta$. The thick lines show the probability of each eigenmode in rescaled arbitrary units while varying the detuning. The thin vertical lines show the line shift ${\mathrm{\Delta }}_{\alpha }$ of the eigenmode with the corresponding color. Only the five states that have a significant contribution are shown.