An architecture for quantum networking of neutral atom processors

Abstract

Development of a network for remote entanglement of quantum processors is an outstanding challenge in quantum information science. We propose and analyze a two-species architecture for remote entanglement of neutral atom quantum computers based on integration of optically trapped atomic qubit arrays with fast optics for photon collection. One of the atomic species is used for atom–photon entanglement, and the other species provides local processing. We compare the achievable rates of remote entanglement generation for two optical approaches: free space photon collection with a lens and a near-concentric, long working distance resonant cavity. Laser cooling and trapping within the cavity remove the need for mechanical transport of atoms from a source region, which allows for a fast repetition rate. Using optimized values of the cavity finesse, remote entanglement generation rates $> 10^3~\mathrm{s}^{-1}$ are predicted for experimentally feasible parameters.

Appendix

1.1 Fiber-coupling efficiency of a high-NA lens

To calculate the coupling efficiency of the scattered photons into a single-mode fiber, we assume that the fiber supports a Gaussian mode. In other words, we assume that a Gaussian mode can be coupled into the fiber with 100% efficiency. Thus, we find the field distribution of the radiating dipole after the collection lens, i.e., between the two lenses shown in Fig. 2, and calculate its overlap with a collimated Gaussian mode.

Referring to the geometry shown in the inset of Fig. 2, the three components of the electric field of a radiating dipole are given by

$$\begin{aligned} {\mathbf {E}}_{\sigma _{\pm }}(\theta ,\phi )= & {} \sqrt{\frac{3}{16\pi }} \frac{i e^{\imath (kr \pm \phi )}}{r} \left( \pm \cos {\theta } {\hat{\theta }} + i {\hat{\phi }}\right) \end{aligned}$$

(19)

$$\begin{aligned} {\mathbf {E}}_{\pi }(\theta ,\phi )= & {} \sqrt{\frac{3}{16\pi }} \frac{i e^{\imath kr}}{r} \sin {\theta } {\hat{\theta }}, \end{aligned}$$

(20)

where $\phi$ is the azimuthal angle. After passing through the collection lens, the spherical coordinates are transformed in the following way:

$$\begin{aligned} {\hat{\theta }} \rightarrow {\hat{\rho }} \\ {\hat{r}} \rightarrow {\hat{z}}\\ {\hat{\phi }} \rightarrow {\hat{\phi }}. \end{aligned}$$

Converting Eqs. (19) and (20) to cylindrical coordinates gives the following expressions for the field amplitudes after the collection lens:

$$\begin{aligned}&{{\mathbf {E}}}_{\sigma \pm }(\rho ,\phi ) \nonumber \\&\quad =\sqrt{\frac{3}{16\pi }} \frac{i e^{\pm \imath \phi }f^{1/2}}{(f^2+\rho ^2)^{3/4}} \left[ \frac{\pm f}{(f^2+\rho ^2)^{1/2}} {\hat{\rho }} + i {\hat{\phi }}\right] , \end{aligned}$$

(21)

$$\begin{aligned}&{{\mathbf {E}}}_{\pi }(\rho ,\phi ) = \sqrt{\frac{3}{16\pi }} \frac{i e^{\pm \imath \phi }f^{1/2}}{(f^2+\rho ^2)^{3/4}} \frac{\rho }{(f^2+\rho ^2)^{1/2}}{\hat{\rho }}, \end{aligned}$$

(22)

where f is the focal length of the lens, $\rho$ is the radial coordinate from the axis of the lens, and the factor of $f^{1/2}/(f^2+\rho ^2)^{1/4}$($\sqrt{\cos {\theta }}$ in spherical coordinates before the lens) accounts for the projection of the field onto the lens plane. Note that in both coordinate systems, the integral of the dipole field $\mathbf{E}_D=\mathbf{E}_{\sigma \pm }$ or $\mathbf{E}_D=\mathbf{E}_\pi$ over half of the full solid angle gives half of the total energy emitted by the dipole

$$\begin{aligned} \int _0^{\frac{\pi }{2}} d\theta \sin {\theta } \int _{0}^{2\pi }d\phi \, \left| {\mathbf {E}}_{D}(\theta ,\phi ) \right| ^2= & {} \frac{1}{2}, \end{aligned}$$

(23)

$$\begin{aligned} \int _{0}^{\infty } d\rho \rho \, \int _{0}^{2\pi }d\phi \, \left| {\mathbf {E}}_{D}(\rho ,\phi )\right| ^2= & {} \frac{1}{2}. \end{aligned}$$

(24)

The field of a circularly polarized Gaussian laser mode with waist w is given by

$$\begin{aligned}&{\mathbf {E}}_{G} = \frac{1}{\sqrt{\pi }w} e^\frac{-\rho ^2}{w^2} \left[ \left( \cos \phi \pm i \sin \phi \right) {\hat{\rho }} - \left( \sin \phi \mp i \cos \phi \right) {\hat{\phi }} \right] . \end{aligned}$$

(25)

We then calculate the overlap integral of the collimated dipole field, Eqs. (21) and (22), with the Gaussian mode of Eq.(25)

$$\begin{aligned} O =\bigg |\int _{0}^{\rho _{\mathrm{NA}}}d\rho \rho \, \int _{0}^{2\pi }d\phi \, {\mathbf {E}}_G \cdot {\mathbf {E}}_D^* \bigg |^2, \end{aligned}$$

(26)

where $\rho _{\mathrm{NA}} = \frac{f \mathrm NA}{\sqrt{1-(\mathrm NA)^2}}$. From this integral, we see that the overlap of the $\pi$-polarized light emitted by the atom and the Gaussian mode is zero. Thus, the $\pi$-polarized photons do not couple into the single-mode fiber. We also calculate the overlap of the circularly polarized light emitted by the atom with the Gaussian mode which is shown in Fig. 2. As the NA of the lens changes, the beam waist w in Eq. (25) has to be adjusted to maximize the overlap integral.

We note that the transverse profile of the photons collected and collimated by the lens is not Gaussian. Also, the polarization of the photons changes gradually from circular at the center ($\rho =0$) to elliptical as $\rho$ increases. Both of these effects contribute to decreasing the overlap integral in Eq. (26).