Abstract
A major application for atomic ensembles consists of a quantum memory for light, in which an optical state can be reversibly converted to a collective atomic excitation on demand. There exists a well-known fundamental bound on the storage error, when the ensemble is describable by a continuous medium governed by the Maxwell–Bloch equations. However, these equations are semi-phenomenological, as they treat emission of the atoms into other directions other than the mode of interest as being independent. On the other hand, in systems such as dense, ordered atomic arrays, atoms interact with each other strongly and spatial interference of the emitted light might be exploited to suppress emission into unwanted directions, thereby enabling improved error bounds. Here, we develop a general formalism that fully accounts for spatial interference, and which finds the maximum storage efficiency for a single photon with known spatial input mode into a collection of atoms with discrete, known positions. As an example, we apply this technique to study a finite two-dimensional square array of atoms. We show that such a system enables a storage error that scales with atom number Na like , and that, remarkably, an array of just 4 × 4 atoms in principle allows for an error of less than 1%, which is comparable to a disordered ensemble with an optical depth of around 600.
Highlights
2 a for storage from a Gaussian-like mode, and remarkably, that a 4 × 4 array in principle already enables an error below 1%
A major application for atomic ensembles consists of a quantum memory for light, in which an optical state can be reversibly converted to a collective atomic excitation on demand
We have introduced a prescription to calculate the maximum storage and retrieval efficiency of a quantum memory, which fully accounts for re-scattering and interference of light emission in all directions
Summary
The full dynamics of light emission and re-scattering of an arbitrary collection of atoms in free space, specified only by their discrete, fixed positions rj, can be related to an effective model containing only the atomic degrees of freedom and the incident field [36,37,38,39,40,41]. Given the evolution of the atomic state under Heff, any observables associated with the total field operator Eout(r) can be derived from the input–output relation [37,38,39,40,41] This equation states that the total field is a superposition of the incoming field and the fields emitted by the atoms, whose spatial pattern is contained in the Green’s function. The first condition is obviously satisfied for atoms in free space, as the vacuum’s Green’s function has a frequency spectrum that is much broader than the atomic linewidth Neglecting retardation in both the photon-mediated interactions between atoms and the field produced by the atoms requires the characteristic length L of the atomic system to be much smaller than that of a spontaneously-emitted photon, which is ~c/Γ0 ⩽ 1 m [45,46,47,48], where Γ0 = μ0ω2egde2g ∕ 3πħc is the single-atom spontaneous emission rate in vacuum. The spin model presented above has a natural extension to the multi-excitation case (e.g., studying the storage of multiple photons and their subsequent nonlinear interaction [52,53,54]), whereas exact solutions are only available in a limited number of cases [47, 55, 56]
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