Collective generation of quantum states of light by entangled atoms

We present a theoretical framework to describe the collective emission of light by entangled atomic states. Our theory applies to the low-excitation regime, where most of the atoms are initially in the ground state, and relies on a bosonic description of the atomic excitations. In this way, the problem of light emission by an ensemble of atoms can be solved exactly, including dipole-dipole interactions and multiple light scattering. Explicit expressions for the emitted photonic states are obtained in several situations, such as those of atoms in regular lattices and atomic vapors. We determine the directionality of the photonic beam, the purity of the photonic state, and the renormalization of the emission rates. We also show how to observe collective phenomena with ultracold atoms in optical lattices and how to use these ideas to generate photonic states that are useful in the context of quantum information.

Figure 1

$\Lambda$ scheme.

Figure 2

Photon emission by a spin wave with linear momentum ${\mathbf{K}}_n$ in an atomic chain. The maxima in the photon distribution are at angles $\theta$ such that momentum along the chain is conserved up to reciprocal lattice wave vectors.

Figure 3

Angular photon distribution of the photonic mode emitted by the completly symmetric spin wave in a chain of $N=20$ atoms. We have used the exact egenvalues of the density matrix and followed the method described in the previous section.

Figure 4

Angular photon distribution of the photonic mode emitted by the completly symmetric spin wave from a chain of $N=20$ atoms, $\lambda =5d_0$, calculated assuming periodic boundary conditions (solid line) and performing an exact diagonalization of the master equation (dashed line). The distribution in the exact case shows a departure from the ${\mathrm{sinc}}^2$ shape due to collective effects.

Figure 5

Calculation of the collective rates of a chain with $N=20$ atoms. Solid squares: result predicted by the plane-wave approximation (50). Open circles: eigenvalues of the matrix $J_{ij}$ defined by (14). In the last case, to each eigenvalue of $J_{ij}$ we assign a wave number $n$, which corresponds to the maximum Fourier component of the corresponding eigenvector.

Figure 6

(i) Creation of a spin wave with linear momentum ${\mathbf{k}}_{\mathrm{L},\mathrm{A}}+{\mathbf{k}}_{\mathrm{L},\mathrm{B}}$ in an optical lattice. (ii) Mapping from the spin wave to a photonic mode.

Figure 7

Double-$\Lambda$ scheme for the generation of photons entangled in polarization.

Figure 8

Angular photon distribution of the photonic mode emitted by the completely symmetric spin wave in an ion chain with $N=10$ ions.

Figure 9

Scheme for the generation of entangled spin waves by using the Rydberg blockade.

Figure 10

Level scheme for the implementation of the atom-photon mapping, avoiding incoherent processes.