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Physical Review A

Quantum model of decoherence in the polarization domain for the fiber channel

Козубов Антон Владимирович, Andrei Gaidash, George P. Miroshnichenko

Published May 28, 2019• 16 citations
Publisher

Abstract

In this work we consider the Liouville equation; it describes the dynamics of the photon density matrix in the Schr\"odinger representation based on the Markov approximation in the channel without dispersion. The equation contains a relaxation superoperator dependent on the phenomenological parameters of the optical fiber. These parameters allow one to take into account the phenomena of birefringence and optical activity, isotropic absorption, and dichroism. We also present in our work that these parameters affect not only the polarization of the states but the length of the Stokes vector. Hence the developed technique describes the decoherence process in the polarization domain in the quantum case and allows one to analyze the dynamics of single-photon states in the quantum (depolarizing) channel more properly. We also present a visual illustration of polarization states' evolution in the polarization-coded quantum key distribution BB84 protocol as an example. We estimate quantum bit error rates' dependence on channel length. Also we examine maximal allowed channel length, dependent on various configurations of channel parameters.

DOI
10.1103/physreva.99.053842

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In this work we consider the Liouville equation; it describes the dynamics of the photon density matrix in the Schr"odinger representation based on the Markov approximation in the channel without dispersion. The equation contains a relaxation superoperator dependent on the phenomenological parameters of the optical fiber. These parameters allow one to take into account the phenomena of birefringence and optical activity, isotropic absorption, and dichroism. We also present in our work that these parameters affect not only the polarization of the states but the length of the Stokes vector. Hence the developed technique describes the decoherence process in the polarization domain in the quantum case and allows one to analyze the dynamics of single-photon states in the quantum (depolarizing) channel more properly. We also present a visual illustration of polarization states' evolution in the polarization-coded quantum key distribution BB84 protocol as an example. We estimate quantum bit error rates' dependence on channel length. Also we examine maximal allowed channel length, dependent on various configurations of channel parameters.
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 su(1,1)\text{su}(1,1)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.
A phenomenological Hamiltonian of photons in single-mode stochastic fiber depending on the vector of random parameters is proposed. The time dynamics of single-photon density matrix in the basis of states with orthogonal polarizations is considered. The fiber-parameter-averaged quantum bit error rate (QBER) in a sifted quantum key distributed over the BB84 protocol using entangled polarization states of biphotons is found. It is shown that QBER can be significantly reduced even at large dispersions of random fiber parameters. To this end, identically fabricated fibers must be used for quantum channels A and B. The choice of pairs of fiber segments must be correlated, with a correlation coefficient close to unity. This approach is based on a remarkable property of the singlet biphoton state, which is “free of collective decoherence.” A correlated choice of fiber segments for channels A and B reduces significantly QBER, making its value below critical (i.e., equal to 0.11, a level below which a distributed key is accepted for cryptographic purposes).
In this paper, we consider the influence of a divergence of polarization of a quantum signal transmitted through an optical fiber channel on the quantum bit error rate of the subcarrier wave quantum key distribution protocol. Firstly, we investigate the dependence of the optical power of the signal on the modulation indices' difference after the second phase modulation of the signal. Then we consider the Liouville equation with regard to relaxation in order to develop expressions of the dynamics of the Stokes parameters. As a result, we propose a model that describes quantum bit error rate for the subcarrier wave quantum key distribution depending on the characteristics of the optical fiber. Finally, we propose several methods for minimizing quantum bit error rate.
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.
We investigate the polarization state dynamics of single photon pulse for optical fiber quantum communication channels. On the basis of a birefringence vector model in which amplitude and direction are both stochastic variables, Jones vector is obtained by solving the frequency domain wave equation. The fidelity of output quantum state and degree of polarization of the pulse are also obtained from the density operators. It is shown that the fidelity of quantum state decreases quickly and tends to a stable value along optical fiber, and increases for larger mean fluctuation magnitude of the stochastic fiber birefringence. Degree of polarization is nearly constant for small mean fluctuation magnitude of the birefringence. The fidelity and degree of polarization vary in the same way for Gaussian and rectangular frequency spectrum envelope, while the value of Lorentzian spectrum is smaller.