Exact solution of gradient echo memory and analytical treatment of gradient frequency comb

Gradient echo memory (GEM) stores and retrieves photon wave packet in forward direction with high efficiency and fidelity using photon-echo mechanism. It is an important technique for quantum memory applications. By breaking the continuity of the gradient absorption structure, the scheme becomes gradient frequency comb (GFC), which is a hybrid of GEM and atomic frequency comb (AFC). To elucidate the non-trivial field-atom evolution of gradient echo, we derive its exact analytical solution in a medium of arbitrary optical thickness subjecting to any linear gradient of transition frequency, and discuss its physical processes. Based on this solution, we further suggest and analysis two types of GFC, one with stepwise gradient and the other with discontinuous gradient.