We study collective 'free-space' radiation properties of two distant single-layer arrays of quantum emitters as two-level atoms. We show that this system can support a long-lived Bell superposition state of atomic excitations exhibiting strong subradiance, which corresponds to a non-local excitation of the two arrays. We describe the preparation of these states and their application in quantum information as resource of non-local entanglement, including deterministic quantum state transfer with high fidelity between the arrays representing quantum memories. We discuss experimental realizations using cold atoms in optical trap arrays with subwavelength spacing, and analyze the role of imperfections.Introduction. -Recent advances in preparing regular arrays of atoms with optical traps [1-4] offer new opportunities to engineer strong collective coupling between atoms and light, with applications in quantum information science. In particular, a single layer of atoms loaded into a regular 2D array with sub-wavelength spacing has been proposed as an atomic mirror with high reflectivity [5][6][7][8][9], as quantum memory with efficient storage and retrieval [10], and to implement topological quantum optics [11,12]; in addition, emission of single photons from bilayer atomic arrays can be engineered to be highly directional in free-space [13]. Moreover, single-layered atomic arrays have been shown to support subradiant collective excitations [14][15][16], which consist of excited superposition states of atoms decaying much slower than a single isolated excited atom, due to interference in spontaneous emission [17][18][19][20][21][22][23].Here we show that the composite quantum system consisting of two distant single-layered arrays of atoms [cf. Figs. 1(a-c)] can support an atomic Bell superposition state exhibiting strong subradiance. Remarkably, this non-radiating 'dark' state is a non-local entangled state, i.e. a superposition state of a collective excitation living in the first or second array, where the two arrays can be separated by a distance L much larger than the transverse size L ⊥ of each individual array. This phenomenon relies on two ingredients. First, spontaneous emission from a collective atomic excitation in a single layer can be directional, with a proper phasing of the atomic dipoles, corresponding to light emission in both directions perpendicular to the atomic array, as in Fig. 1(a) [5]. Second, radiation from two distant atomic arrays can -provided the separation length L is commensurate with half the optical wavelength [upper panel in Fig. 1(c)] -lead to destructive interference of light emitted to the left and to the right of the two arrays, corresponding to a subradiant state, i.e. this 'dark' state will show strongly suppressed radiative loss to the outside world. In contrast, the lower panel in Fig. 1(c) displays a 'bright' (i.e., radiating) state due to constructive interference.Below we will show that these non-local subradiant

We experimentally demonstrate that a nonclassical state prepared in an atomic memory can be efficiently transferred to a single mode of free-propagating light. By retrieving on demand a single excitation from a cold atomic gas, we realize an efficient source of single photons prepared in a pure, fully controlled quantum state. We characterize this source using two detection methods, one based on photon-counting analysis and the second using homodyne tomography to reconstruct the density matrix and Wigner function of the state. The latter technique allows us to completely determine the mode of the retrieved photon in its fine phase and amplitude details and demonstrate its nonclassical field statistics by observing a negative Wigner function. We measure a photon retrieval efficiency up to 82% and an atomic memory coherence time of 900  ns. This setup is very well suited to study interactions between atomic excitations and use them in order to create and manipulate more sophisticated quantum states of light with a high degree of experimental control.

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. arXiv:1802.05592v2 [quant-ph]