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Nanosystems Physics Chemistry Mathematics

Parameterization of an interaction operator of optical modes in a single-mode optical fiber

George P. Miroshnichenko

Published Dec 21, 2015• 2 citations
Publisher

Abstract

The phenomenological parameters of the Hamiltonian for the photons produced in earlier studies [4] are associated with the parameters of the deformed optical fiber (OF).This Hamiltonian is necessary for the correct description of the propagation of photons through the quantum channel in a quantum communication protocols.Models of a compressing strain of the OF profile and a twisting deformation are considered.As a consequence, the phenomenological parameters of the Hamiltonian expressed in terms of such strains characteristics, as a relative compression of the profile, OF radius, the orientation angle of the deformed profile, rotation angle per unit length, elasto-optical tensor, and refraction coefficient.

DOI
10.17586/2220-8054-2015-6-6-857-865

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The phenomenological parameters of the Hamiltonian for the photons produced in earlier studies [4] are associated with the parameters of the deformed optical fiber (OF).This Hamiltonian is necessary for the correct description of the propagation of photons through the quantum channel in a quantum communication protocols.Models of a compressing strain of the OF profile and a twisting deformation are considered.As a consequence, the phenomenological parameters of the Hamiltonian expressed in terms of such strains characteristics, as a relative compression of the profile, OF radius, the orientation angle of the deformed profile, rotation angle per unit length, elasto-optical tensor, and refraction coefficient.
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.
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).
The implementation of quantum communication protocols in long optical fibers is limited by several decoherence mechanisms. In this contribution we review this mechanisms and analysis their effect on the quantum states. Because of asymmetry in the real fibers, the two orthogonal polarization modes are propagated at different phase and group velocities. The difference in group velocities results in Polarization Mode Dispersion (PMD). It is created by random fluctuations of the residual birefringence in optical fibers, such that the State of Polarization (SOP) of an optical signal will turn randomly over time, in an unpredictable way. In optical quantum communication, quantum state measurement is necessary. The bottleneck for communication between far apart nodes is the increasing of the error probability with the length of the channel connecting the nodes. For an optical fiber, the value of absorption and depolarization of a photon (i.e., the qubit) increases exponentially with the length of the fiber.
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.
The problem of quantitative description of the total effect of electromagnetic fields of lightning discharges on the polarization state (SOP) of light in fiber-optic communication lines (FOCL) of the ITU-T G. 652 standard is considered. The cumulative angle parameter is proposed, which is defined as the integral of the norm of the local Stokes rotation vector along the line length, taking into account random birefringence, magnetooptical Faraday effect and electro-optical Kerr effect. Using the Jones matrix formalism, analytical expressions for cumulative polarization rotation angle are obtained in terms of the Verde constant, the induction of the lightning magnetic field, and the correlation length of the polarization-mode dispersion (PMD). It is established that the cumulative angle is related to the dynamic differential group delay (DGD), as well as to the quantum error coefficient QBER for the BB84 quantum key distribution (QKD) protocol. Based on the obtained dependences, a classification of exposure modes (I-IV) is formed according to the lightning current amplitude and the distance to the discharge channel. Numerical estimates are derived for SSMF (G. 652), DSF (G. 653), and NZ-DSF (G. 655) fibers with line lengths up to 50 km and lightning currents of10−105 kA\mathbf{1 0 - 1 0 5 ~ k A}10−105 kA. It is shown that at Category I currents (over 105 kA) at a distance of 100 m, the rate of SOP change reaches17.5Mrad/s17.5 \text{Mrad} / \mathrm{s}17.5Mrad/s, which leads to a critical increase in QBER and termination the QKD session. The results allow us to use the cumulative angle as a single integral criterion for lightning resistance of high-speed and quantum fiber-optic lines in the design of routes and justification of lightning protection parameters.