Analytic treatment of controlled reversible inhomogeneous broadening quantum memories for light using two-level atoms

It has recently been discovered that the optical analog of a gradient echo, in an optically thick material, could form the basis of an optical memory that is both completely efficient and noise-free. Here we present analytical calculations showing that this is the case. There is close analogy between the operation of the memory and an optical system with two beam splitters. We can use this analogy to calculate efficiencies as a function of optical depth for a number of quantum memory schemes based on controlled inhomogeneous broadening. In particular, we show that multiple switching leads to a net 100% retrieval efficiency for the optical gradient echo even in the optically thin case.

Figure 1

The domain over which the Maxwell-Bloch equations are solved. Each horizontal slice represents one position in the sample of atoms as a function of time. The light enters the sample at $z=-z_0$ and leaves at $z=+z_0$. The dotted line represents the point in time at which the detunings of the ions are flipped. In region $A$ the dynamics are described by Eqs. (1, 2), in region $B$ by Eqs. (3, 4).

Figure 2

Optical gradient echo memory represented as a pair of beam splitters. The bold arrows represent the flow of energy in the optically thick case. The spatial and temporal modes that correspond to each port of the beam splitter are discussed in the text.

Figure 3

Network of analog beam splitters relevant to a transverse CRIB memory modeled as a large number of optically thin gradient echo memories.

Figure 4

Network of analog beam splitters relevant to an optical gradient echo memory where the field is switched a number of times.

Figure 5

(Color online) Echo efficiency for an optical gradient echo as a function of $\beta$, showing the effect of multiple switching. The ratio of the energies of the input and transmitted pulse is given by $\exp (-2\pi \beta )$.

Figure 6

(Color online) Echo efficiency as a function of optical depth for transverse broadening CRIB echo, showing the effects of multiple switching. As is shown in the figure, efficiencies significantly greater than 54% can be achieved with multiple switching, but 100% efficiency is approached relatively slowly as the number of echoes increases. Approximately 100 echoes are needed to achieve greater than 90% efficiency.