Asymptotic evaluation of bosonic probability amplitudes in linear unitary networks in the case of large number of bosons

An asymptotic analytical approach is proposed for bosonic probability amplitudes in unitary linear networks, such as the optical multiport devices for photons. The asymptotic approach applies for large number of bosons $N\gg M$ in the $M$-mode network, where $M$ is finite. The probability amplitudes of $N$ bosons unitarily transformed from the input modes to the output modes of a unitary network are expressed through a multidimensional integral with the integrand containing a large parameter (N) in the exponent. The integral representation allows an asymptotic estimate of bosonic probability amplitudes up to a multiplicative error of order 1/N by the saddle point method. The estimate depends on solution of the scaling problem for the $M\times M$-dimensional unitary network matrix: to find the left and right diagonal matrices which scale the unitary matrix to a matrix which has specified row and column sums (equal, respectively, to the distributions of bosons in the input and output modes). The scaled matrices give the saddle points of the integral. For simple saddle points, an explicit formula giving the asymptotic estimate of bosonic probability amplitudes is derived. Performance of the approximation and the scaling of the relative error with N are studied for two-mode network (the beam-splitter), where the saddle-points are roots of a quadratic and an exact analytical formula for the probability amplitudes is available, and for three-mode network (the tritter).