Hamiltonian of photons in a single-mode optical fiber for quantum communications protocols

A phenomenological Hamiltonian of photons in a single-mode stochastic inhomogeneous optical fiber (OF) is derived. Quantization of radiation is performed in the basis of an ideal OF with proper calibration that ensures transversality of the electric-field-displacement vector. Stochastic parameters of the Hamiltonian are determined by using the reciprocal tensor of the dielectric permittivity averaged over the OF segment volume. The Hamiltonian is parametrized by three phenomenological parameters and preserves the number of photons. It is assumed that the segment of the OF is divided into random subsegments with optical parameters defined by the Wiener process with respect to the longitudinal coordinate. The temporal dynamics of the single-photon density matrix is analyzed in the basis of states with orthogonal polarizations. The relative quantum beat error rate in the sifted quantum key distributed according to the BB84 protocol with polarization coding of information averaged over the scatter of the OF parameters is calculated.