Photon-echo quantum memory in solid state systems

Many applications of quantum communication crucially depend on reversible transfer of quantum states between light and matter. Motivated by rapid recent developments in theory and experiment, we review research related to quantum memory based on a photon-echo approach in solid state material with emphasis on use in a quantum repeater. After introducing quantum communication, the quantum repeater concept, and properties of a quantum memory required to be useful in a quantum repeater, we describe the historical development from spin echoes, discovered in 1950, to photon-echo quantum memory. We present a simple theoretical description of the ideal protocol, and comment on the impact of a non-ideal realization on its quantum nature. We extensively discuss rare-earth-ion doped crystals and glasses as material candidates, elaborate on traditional photon-echo experiments as a test-bed for quantum state storage, and describe the current state-of-the-art of photon-echo quantum memory. Finally, we give a brief outlook on current research.