Free-space photonic quantum link and chiral quantum optics

We present the design of a chiral photonic quantum link, where distant atoms interact by exchanging photons propagating in a single direction in free space. This is achieved by coupling each atom in a laser-assisted process to an atomic array acting as a quantum phased-array antenna. This provides a basic building block for quantum networks in free space, i.e., without requiring cavities or nanostructures, which we illustrate with high-fidelity quantum state transfer protocols. Our setup can be implemented with neutral atoms using Rydberg-dressed interactions.

Figure 1

Free-space chiral quantum optics. (a) A master atom (qubit) is coupled to a bilayer atomic array of $N_{\mathrm{a}}=N_{\perp }\times N_{\perp }\times 2$ atoms as “quantum antenna” achieving unidirectional photon emission. (b) Spatial distribution of the emitted field $|\vec{\phi }\left(\vec{r}\right)|$ (see text) for two master atoms as qubits interacting via a paraxial mode, with $2z_0=90{\lambda }_0,N_{\perp }=17,{\delta }_{\perp }=0.7{\lambda }_0$. (c) Basic model of the antenna: level schemes of the qubit off-resonantly coupled to a two-level antenna atom. (d) Chiral coupling in waveguide QED, with two atomic qubits coupling to right-propagating modes with rate ${\gamma }_R$ and to nonguided modes with rate ${\gamma }^{\prime }$. (e) Fidelity for quantum state transfer between atomic qubits, with interatomic spacing ${\delta }_{\perp }=0.8{\lambda }_0$ and various $N_{\perp }$. The red curve corresponds to $N_{\perp }\to \infty$ (see text). Inset: corresponding infidelity for $z_0\to 0$, with $50/N_{\perp }^4$ (dashed black line) for reference.

Figure 2

Few-atom quantum antenna. (a) Spatial distribution of the emitted field $|\vec{\phi }\left(\vec{r}\right)|$ (see text) for antenna configurations as bilayer $2\times 2,3\times 3$, and $8\times 8$ regular arrays. The field can be interfaced with optical lenses. (b) Emission to a superposition of two focusing modes, and mode matching to an optical fiber, with bilayer $17\times 17$ atomic arrays as antennas.

Figure 3

Performance of a “few-atom” antenna. (a) Effective optical depth and corresponding Purcell factor $\beta$ [inset (a)] for an optimal Gaussian mode $n_0$, for an antenna consisting of $N_a$ atoms. Regular arrays with $N_z=2,4,8$ are shown in blue (dark gray), orange (gray), and green (light gray) solid lines, respectively (transversal sizes of $N_{\perp }=2,3,4$ are highlighted by different markers as shown in the legend). A lattice imperfection modeled as classical randomization of atomic positions with normal distribution with dispersion ${\sigma }_{\mathrm{th}}=0.01{\lambda }_0$ (${\sigma }_{\mathrm{th}}=0.02{\lambda }_0$) is shown in dashed (dot-dashed) lines with the corresponding color code. Results for randomly distributed atoms with the atomic density of the regular arrays $n_a=2/{\lambda }_0^3$ are shown with a black dotted line. (b) Statistics of optimal Gaussian mode waists $w_0$ (blue dots) vs $L_{\perp }$ for all configurations of $N_{\perp }$ and $N_z$ up to $16\times 16\times 8$. Fitted linear dependence (blue dashed line) of $w_0$ and the corresponding opening angle $\theta$ of the Gaussian beam (red solid line). (c) Statistics of optimal interatomic distances ${\delta }_{\perp }$ vs $N_{\perp }$. (d)–(f) Imperfections for arrays with $N_z=2,4,8$ and $N_{\perp }=10$. Purcell factor as a function of (d) percentage of defects, (e) Gaussian dispersion ${\sigma }_{\mathrm{th}}$ of randomized atomic positions, and (f) detuning from resonance for two-level atoms.

Figure 4

Free-space quantum link. (a) Purcell factor $\beta$ as a function of the distance $z_0$ from each antenna to the focusing point of the Gaussian mode for various $N_{\perp }$, with ${\delta }_{\perp }=0.75{\lambda }_0$. Inset: corresponding optimal waists. The dot-dashed curves have analytical expressions (see text). (b),(c) State transfer fidelity, for $N_{\perp }=10$ and (b) ${\delta }_z=0.75{\lambda }_0$, (c) ${\delta }_{\perp }=0.8{\lambda }_0$.

Figure 5

Rydberg implementation. (a) Atomic level schemes, with the relevant states for our implementation in blue. (b),(c) Probability of emission in a Gaussian mode, (b) as a function of the distance $z_m$ between master atom and antenna [dashed line: value for two-level model; see Fig. 3], and (c) as a function of the number of atoms for the optimal $z_m$, using dressing laser with a LG profile [blue (dark gray)] or with an optimized spatial distribution of ${\mathrm{\Omega }}_d\left({\vec{r}}_i\right)$ [orange (light gray)]. We consider an ensemble of $10\times 10\times 2$ atoms with ${\delta }_{\perp }=0.7{\lambda }_0,{\delta }_z=1.25{\lambda }_0,{\mathrm{\Omega }}_c=2\pi \times 2.5\text{MHz}, |{\mathrm{\Omega }}_d/{\mathrm{\Delta }}_d|\le 0.02$.