Collection efficiency of optical photons generated from microwave excitations of a Bose-Einstein condensate

Stimulated Raman scattering on $\mathrm{\Lambda }$ atoms is a promising tool for generating optical photons from a microwave excitation in the ground-state hyperfine manifold. We consider an atomic Bose-Einstein condensate coupled to a microwave field that scatters photons into the guided modes of a nearby optical fiber. Due to momentum transfer to the condensate, stimulated photon scattering can occur outside of the phase-matched direction, which can be used to separate the converted photons from the strong Raman readout pulse. Conversely, in the phase-matched direction, superradiant scattering due to bosonic enhancement leads to an increased collection efficiency in the guided modes of the fiber, for which we determine optimal conditions.

Figure 1

Level scheme depicting the many-body states available to the system starting from $|G\rangle \equiv |0,0,0\rangle$. The other states are defined as $|S\rangle \equiv |0,1,0\rangle , |E_{\mathbf{k}}\rangle \equiv |0,0,1_{\mathbf{k}}\rangle$, and $|G_{\mathbf{k}}\rangle \equiv |1_{\mathbf{k}},0,0\rangle$. Black lines with arrows show the transitions induced by the interaction with the microwave cavity, the drive, and the free electromagnetic fields.

Figure 2

The geometry of the system of atoms, paraxial optics, and the fiber.

Figure 3

Angular dependence of the geometrical factor $|{\xi }_0{\left(\theta \right)|}^2$ (curves) corresponding to the superradiance and the value of the geometrical factor $\xi /N$ (horizontal lines) for the isotropic radiation for a fixed number of atoms $N=10$ for various values of (a) the transverse size ${\overline{\sigma }}_z$ and (b) the longitudinal size $\overline{\sigma }$. $\theta$ is measured in radians.

Figure 4

The dependence of the critical angle ${\theta }^{*}$ (the two channels have the same contribution) on the atom number $N$ for fixed geometry $\overline{\sigma }=50, {\overline{\sigma }}_z=100$. ${\theta }^{*}$ is measured in radians.