  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. The validity of this model can break down,
however, in systems such as dense, ordered atomic arrays, where strong
interference in emission can give rise to phenomena such as subradiance and
"selective" radiance. Here, we develop a general formalism that finds the
maximum storage efficiency for a collection of atoms with discrete, known
positions, and a given spatial mode in which an optical field is sent. 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 $N_\mathrm{a}$ like $\sim (\log N_\mathrm{a})^2/N_\mathrm{a}^2$,
and that, remarkably, an array of just $4 \times 4$ atoms in principle allows
for an efficiency comparable to a disordered ensemble with optical depth of
around 600.

Quantum Physics