Multivariate quantum memory as controllable delayed multiport beamsplitter

Abstract: The addressability of parallel spatially multimode quantum memory for light allows one to control independent collective spin waves within the same cold atomic ensemble. Generally speaking, there are transverse and longitudinal degrees of freedom of the memory that one can address by a proper choice of the pump (control) field spatial pattern. Here we concentrate on the mutual evolution and transformation of quantum states of the longitudinal modes of collective spin coherence in the cavity-based memory scheme. We assume that these modes are coherently controlled by the pump waves of the on--demand transverse profile, that is, by the superpositions of waves propagating in the directions, close to orthogonal to the cavity axis. By the write-in, this allows one to couple a time sequence of the incoming quantized signals to a given set of superpositions of orthogonal spin waves. By the readout, one can retrieve quantum states of the collective spin waves that are controllable superpositions of the initial ones and are on demand coupled to the output signal sequence. In general case, the memory is able to operate as a controllable delayed multi-port beamsplitter, capable of transformation of the delays, the durations and time shapes of signals in the sequence. We elaborate the theory of such light--matter interface for the spatially multivariate cavity-based off--resonant Raman--type quantum memory. Since in order to speed up the manipulation of complex signals in multivariate memories it might be of interest to store relatively short light pulses of a given time shape, we also address some issues of the cavity-based memory operation beyond the bad cavity limit.