Quantum model for traveling-wave electro-optical phase modulator

Abstract

We develop a quantum model of a traveling-wave electro-optical phase modulator by considering a parametric interaction of the optical and radiofrequency waves in the modulator medium by means of the electro-optic effect. The model includes a possible dispersion of the modulator medium in the optical domain. In the case of negligible dispersion, we derive a simple analytic expression for the evolution operator of the optical field, which has been proposed previously in the literature on the basis of an analogy to classical field transformation. We study by means of a perturbative approach the influence of small dispersion on the field evolution operator and formulate conditions of negligibility of modulator dispersion. We demonstrate the signature of dispersion in phase modulation: breaking the symmetry of sidebands with respect to the mean photon number. We obtain the law of transformation of a multimode coherent state by a phase modulator, which is important for understanding the structure of signal states in quantum cryptographic schemes utilizing phase modulation of coherent optical beams.