Directional emission of single photons from small atomic samples

We provide a formalism to describe deterministic emission of single photons with tailored spatial and temporal profiles from a regular array of multilevel atoms. We assume that a single collective excitation is initially shared by all the atoms in a metastable atomic state and that this state is coupled by a classical laser field to an optically excited state which rapidly decays to the ground atomic state. Our model accounts for the different field polarization components via reabsorption and emission of light by the Zeeman manifold of optically excited states.

Figure 1

Atoms with three excited substates, $e^{-1}$, $e^0$, and $e^{+1}$, a ground state $g$, and a long-lived state $f$ are arranged in a regular lattice. State $f$ is coupled to state $e^{+1}$ by a classical laser field. The excited states decay to the ground state, emitting a photon with ${\sigma }^{-}$, $\pi$, and ${\sigma }^{+}$ polarized photons, respectively. The direction of the emission is determined by the wave number of the atomic $g$-$f$ coherence and the wave number of the out-coupling field, ${\vec{k}}_{em}=-{\vec{k}}_{gf}-{\vec{k}}_L$.

Figure 2

A snapshot of the total intensity emitted by a cloud of $14\times 14\times 10$ (blue) and $3\times 3\times 10$ (green) atoms with the lattice spacing $d=0.25{\lambda }_0$ and a single atom (red) using a square out-coupling pulse indicated by the dotted line. The larger atomic ensembles experience the faster decay and the sharpest emission peaks. Parameters used for this simulation are ${\Omega }_L=8.2\Gamma$, a vanishing single atom detuning of the the ${\Omega }_L$ light field, and a coupling pulse length $t_w=0.2{\Gamma }^{-1}$.

Figure 3

Emission of a single photon from a small atomic sample of $3\times 3\times 8$ atoms arranged in a lattice with the spacing $d=0.60{\lambda }_0$. (a) Integrated intensity $I={\sum }_{\epsilon }\int d\Omega r^2{I}_{\epsilon }(\vec{r},t)$ as a function of the distance from the atomic sample (solid line) and the out-coupling pulse shape (dotted line). (b) Spatial distribution of intensity of left-handed polarized light $I_{\Omega ,l}=r^2{I}_l(\vec{r},t)$ for a given time $t=200{\lambda }_0/c$. The spatial coordinates are given in units of $c/\Gamma$. (c) Same as (b), but for the opposite handedness of the light (right handed). The intensity scale is given in the same arbitrary units in (b) and (c). (d) Polar plot of the angular intensity distributions of light at $R=199c/\Gamma$ corresponding to (b) at the moment of the maximum intensity. More than $95\%$ of the intensity is emitted in the positive $z$ direction within a cone with a half opening angle $\Delta \theta =0.33$ rad. (e) The corresponding intensity distribution for the right-handed polarization. The same intensity units are applied for the two polarizations in (d) and (e) (note the values differ by an order of magnitude). The parameters used for this simulation are ${\Omega }_L=2\Gamma$, and the single-atom detuning of the ${\Omega }_L$ field is $10\Gamma$.

Figure 4

Polar plot of emitted light for a lattice of $3\times 3\times 8$ atoms for the cases (a) $d=0.50{\lambda }_0$ ($\Delta \theta =0.38$ rad), (b) $d=0.53{\lambda }_0$ ($\Delta \theta =0.36$ rad), (c) $d=0.60{\lambda }_0$ ($\Delta \theta =0.33$ rad), (d) $d=0.90{\lambda }_0$ ($\Delta \theta =0.22$ rad), (e) $d=1.00{\lambda }_0$ ($\Delta \theta =0.20$ rad), and (f) $d=1.10{\lambda }_0$ ($\Delta \theta =0.18$ rad). The total angular distribution for lattices (g) $6\times 6\times 8$ ($\Delta \theta =0.21$ rad) and (h) $8\times 8\times 8$ ($\Delta \theta =0.18$ rad) with $d=0.60{\lambda }_0$. The coupling laser parameters are the same as in Fig. 3.

Figure 5

Temporal shaping of the photon emission from an atomic array of $3\times 3\times 8$ atoms with $d=0.60{\lambda }_0$. (a) Output intensity (thick blue dashed line) using a constant pulse (thin red dashed line) with ${\Omega }_{L0}=10.5\Gamma$ and $\delta =120\Gamma$. With a specially designed out-coupling pulse with peak Rabi frequency ${\Omega }_{L0}=42.0\Gamma$ and the temporal shape $f\left(t\right)$ (thin red solid line), we obtain a Gaussian temporal shape at $R=260c/\Gamma$ of the emitted light intensity (thick blue solid line). (b) Integrated intensities $n^0\left(t\right)$ (dashed line) and $n\left(t\right)$ (thick solid curve) and a graphical solution (red arrows) of the equation $n\left(t\right)=n^0\left(\tau \left(t\right)\right)$. (c) Out-coupling of a double-peaked pulse (thick blue line) using ${\Omega }_{L0}=38.85\Gamma$ and $f\left(t\right)$ (thin red line).

Figure 6

Frustrated emission from a sample of $6\times 6\times 12$ atoms with the lattice spacing $d=0.60{\lambda }_0$. (a) A snapshot of the angularly integrated intensity of the emitted light after excitation with a short pulse (red dashed line) with the same parameters as in Fig. 2. The case where all $|e^{-1}\rangle$, $|e^0\rangle$, and $|e^{+1}\rangle$ levels are included is shown with the blue (dark gray) curve, and the case restricted to a single excited state $|e^0\rangle$ is shown with the green (light gray) curve. Angular distribution at $R=96.6c/\Gamma$ (b) for the case of the full manifold of the excited state and (c) for the case restricted to $|e^0\rangle$.