Dissipative dynamics of quantum states in the fiber channel

In this paper we consider the Liouville equation that describes the quantum nonunitary dynamics of quantum states in optical fiber. We consider a particular case of thermalization aiming to applications related to various quantum information protocols; however, the model can be generalized in various ways taking into account more features (external pump, nonlinear interactions, continuous spectra, free space propagation, etc.). In order to obtain the appropriate evolution models for states in the channel we use the $\text{su}(1,1)$ algebra formalism in the Liouville representation. The developed model is applied in cases of two different initial states as an example. The first is single- and multifrequency-mode (e.g., wavelength-division-multiplexed) weak coherent states in quantum notation as a rather simple but useful example. This particular example is of interest since weak coherent states are a commonly used tool in various fields of optics, e.g., optical communication, quantum key distribution, etc. Results for coherent states are well agreed upon with classical theories; however, in order to highlight quantum features of developed theory we also consider the case of nonclassical light, in particular, Fock states. Considered implementation of the model takes into account dichroism, retardance, thermalization, dispersion, and decoherence in the polarization domain. We derive expressions of evolved states, mean photon number, and estimate the Stokes parameters as well as the degree of polarization. Considered examples explicitly demonstrate all the features and effects of the developed approach. The described approach allows one to connect the information properties of quantum channels with its physical ones. In order to illustrate this statement we consider BB84 quantum key distribution protocol and investigate the behavior of the quantum bit error rate affected by considered physical phenomena in an optical fiber.