Coupling of quantum fluctuations in a two-component condensate

We model frozen light stored via electromagnetically induced transparency quantum-memory techniques in a Bose-Einstein condensate. The joint evolution of the condensate and the frozen light is typically modeled using coupled Gross-Pitaevskii equations for the two atomic fields, but these equations are only valid in the mean-field limit. Even when the mean-field limit holds individually for each atomic-field component, coupling between the neglected fluctuations of the two components could lead to a breakdown of the mean-field approximation even if it is a good approximation for each species individually. We solve and test the effect of coupled quantum fluctuations on coherent nonlinear evolution of the frozen light pulse to see whether this two-species condensate could enable nonlinear quantum optical phenomena. Our analysis commences with a full second-quantized Hamiltonian for a two-component condensate. The field operators are broken up into a mean-field and a quantum fluctuation component. The quantum fluctuations are truncated to lowest non-vanishing order. The transformation diagonalizing the second-quantized approximate Hamiltonian is described by coupled differential equations that are solved with a power series expansion. We compare the consequent dynamics with the mean-field evolution given by the two-component GrossPitaevskii equation.