Effective algorithms for calculation of quasiprobability distributions of bright "banana'' states

Non-Gaussian quantum states, described by negative valued Wigner functions, are important both for fundamental tests of quantum physics and for emerging quantum information technologies. One of the promising ways of generation of the non-Gaussian states is the use of the cubic (Kerr) optical non-linearity, which produces the characteristic banana-like shape of the resulting quantum states. However, the Kerr effect in highly transparent optical materials is weak. Therefore, big number of the photons in the optical mode ($n\gtrsim10^6$) is necessary to generate an observable non-Gaussianity. In this case, the direct approach to calculation of the Wigner function becomes extremely computationally expensive. In this work, we develop quick algorithms for computing the Husimi and Wigner quasiprobability functions of these non-Gaussin states by means of the Kerr nonlinearity. This algorithm can be used for any realistic values of the photons number and the non-linearity.Comment: 24 pages, 10 figure