Deterministic Free-Space Source of Single Photons Using Rydberg Atoms

We propose an efficient free-space scheme to create single photons in a well-defined spatiotemporal mode. To that end, we first prepare a single source atom in an excited Rydberg state. The source atom interacts with a large ensemble of ground-state atoms via a laser-mediated dipole-dipole exchange interaction. Using an adiabatic passage with a chirped laser pulse, we produce a spatially extended spin wave of a single Rydberg excitation in the ensemble, accompanied by the transition of the source atom to another Rydberg state. The collective atomic excitation can then be converted to a propagating optical photon via a coherent coupling field. In contrast to previous approaches, our single-photon source does not rely on the strong coupling of a single emitter to a resonant cavity, nor does it require the heralding of collective excitation or complete Rydberg blockade of multiple excitations in the atomic ensemble.

Figure 1

Schematics of the system. A single source atom is initially prepared in the Rydberg state $|u⟩$ (top left). The transition $|u⟩\to |d⟩$ of the source atom is coupled nonresonantly to the Rydberg transition $|i⟩\to |s⟩$ of the medium atoms (top right) with the dipole-dipole exchange interaction $D$. All medium atoms are initially in the ground state $|g⟩$. A laser pulse couples $|g⟩$ to the intermediate Rydberg state $|i⟩$ with Rabi frequency $\mathrm{\Omega }$ and detuning $\mathrm{\Delta }\gg |\mathrm{\Omega }|,D$. Together with the dipole-dipole exchange $|i⟩|u⟩\to |s⟩|d⟩$, this leads to the transition $|g⟩\to |s⟩$ of the medium atoms detuned by $\delta$. The resulting single Rydberg excitation of the medium atoms has the spatial amplitude profile $S(\mathit{r})\propto D(\mathit{r}-{\mathit{r}}_s)$. The stored excitation can be converted to a propagating photon $\mathcal{E}$ by applying the control field ${\mathrm{\Omega }}_c$ on the transition from $|s⟩$ to the electronically excited state $|e⟩$.

Figure 2

Preparation of a single collective Rydberg excitation of an atomic ensemble. (a) Time dependence of Rabi frequency $\mathrm{\Omega }$ (left vertical axis) and two-photon detuning $\delta$ (right vertical axis) of the driving laser. (b) Dynamics of populations $P_G$ and $P_S$ of the collective ground and single Rydberg excitation states of the atoms. (c) Final spatial distribution of the Rydberg excitation in an elongated volume containing the atoms: The main panel shows the density of excitation $p_s(z)$ along the longer $z$ axis of the volume (integrated over the transverse $x$ and $y$ directions), while the left and right insets show the densities $p_s(x)$ and $p_s(y)$ along $x$ and $y$. We average over 2000 independent realizations of the ensemble of $N=1000$ atoms with the peak density ${\rho }_{\max }\simeq 10\mu {\mathrm{m}}^{-3}$ and Gaussian distribution with ${\sigma }_{x,y}=1\mu \mathrm{m}$ and ${\sigma }_z=6\mu \mathrm{m}$. The source atom is placed at $x_{\mathrm{s}}=7\mu \mathrm{m}$, $y_{\mathrm{s}}$, $z_{\mathrm{s}}=0$. In the simulations, we take the peak Rabi frequency ${\mathrm{\Omega }}_{\max }=2\pi \times 10\mathrm{MHz}$, the Rydberg state decay ${\mathrm{\Gamma }}_s=10\mathrm{kHz}$, and dephasing ${\gamma }_{sg}=10\mathrm{kHz}$, and we use the atomic parameters corresponding to a Cs source atom with $|u⟩\equiv 70P_{3/2}$ and $|d⟩\equiv 70S_{1/2}$ and Rb medium atoms with $|i⟩\equiv 58P_{3/2}$ and $|s⟩\equiv 57D_{5/2}$ [37]. With the intermediate state detuning $\mathrm{\Delta }=9.4{\mathrm{\Omega }}_{\max }$ and the dipole-dipole interaction coefficient $C_3=11.7\mathrm{GHz}\mu {\mathrm{m}}^3$, the medium atoms near the cloud center experience the strongest interaction $D\simeq 2\pi \times 5.4\mathrm{MHz}$. The corresponding second-order transition rate is ${\tilde{D}}_{\max }\simeq 2\pi \times 0.6\mathrm{MHz}$, while the collective coupling rate is $\overline{D}\simeq 2\pi \times 13\mathrm{MHz}$.

Figure 3

Angular probability distribution of the photon emitted by the atomic medium. In the upper polar plot, the polar angle ${\theta }_x$ is varied in the $x\text{−}z$ plane (azimuth $\phi =0$, $\pi$); in the lower plot, the polar angle ${\theta }_y$ is varied in the $y\text{−}z$ plane (azimuth $\phi =\pi /2$, $3\pi /2$). The red solid line corresponds to the control field wave vector ${\mathit{k}}_c\parallel {\mathit{k}}_0$ and the blue dashed line to a small inclination ${\mathit{k}}_c\angle {\mathit{k}}_0=0.04\pi$. In the collinear geometry (red solid line), the resulting angular width (FWHM) of the emitted radiation is $\mathrm{\Delta }{\theta }_x\simeq 0.07\pi$ and $\mathrm{\Delta }{\theta }_y\simeq 0.068\pi$, and the total probability of radiation emitted into the phase-matched direction $z$ within the solid angle $\mathrm{\Delta }\mathrm{\Omega }=2\pi (1-\cos \mathrm{\Delta }\theta )$, with $\mathrm{\Delta }\theta \simeq 0.07\pi$, is $\sim 0.74$. The plots are obtained from a single realization of random positions of Rb atoms with the parameters in Fig. 2, but the quantum state of the emitted radiation $|{\psi }_{\mathrm{ph}}⟩$ is highly reproducible for different realizations ($m$, $m^{\prime }$) of the atomic ensemble, $|⟨{\psi }_{\mathrm{ph}}^{(m)}|{\psi }_{\mathrm{ph}}^{(m^{\prime })}⟩|\gtrsim 0.96P_S$.