Analysis of the operation of gradient echo memories using a quantum input-output model

The gradient echo memory (GEM) technique is a promising candidate for real devices due to its demonstrated performance, but to date high performance experiments can only be described numerically. In this paper we derive a model for GEM as a cascade of infinite interconnected harmonic oscillators. We take a quantum input-output approach to analyse this system, describing the read and write processes of GEM each as a linear-time-invariant process. We provide an analytical solution to the problem in terms of transfer functions which describe the memory behaviour for arbitrary inputs and operating regimes. This allows us to go beyond previous works and analyse the storage quality in the regimes of high optical depth and memory-bandwidth comparable to input bandwidth, exactly the regime of high-efficiency experiments.