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IEEE Journal of Quantum Electronics

Evolution of Quantum States in an Electro-Optic Phase Modulator

Pradeep K Kumar, Anil Prabhakar

Published Dec 17, 2008• 22 citations
Publisher

Abstract

We present a quantum mechanical model to evaluate the time evolution of quantum states of light in an electro-optic phase modulator (EOPM). The phase modulator is analogous to a multilevel atomic system with equally spaced energy levels, each of which corresponds to the sidebands of the phase modulated optical field. Using perturbation theory, we solve the equations governing the interaction of the optical field with a modulating field both with and without phase velocity mismatch. A unitary operator that evolves any given quantum state allows us to evaluate the response of EOPM for single photon, coherent and squeezed states. We apply these results to the analysis of frequency coded quantum key distribution scheme.

DOI
10.1109/jqe.2008.2002673

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We present a quantum mechanical model to evaluate the time evolution of quantum states of light in an electro-optic phase modulator (EOPM). The phase modulator is analogous to a multilevel atomic system with equally spaced energy levels, each of which corresponds to the sidebands of the phase modulated optical field. Using perturbation theory, we solve the equations governing the interaction of the optical field with a modulating field both with and without phase velocity mismatch. A unitary operator that evolves any given quantum state allows us to evaluate the response of EOPM for single photon, coherent and squeezed states. We apply these results to the analysis of frequency coded quantum key distribution scheme.
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.
Quantum analysis has been done for the traveling-wave (TW) electro-optic phase modulator in the presence of the phase noise of a radio frequency (RF) oscillator and a laser source, in which the Hamiltonian for an active section of the phase modulator system is derived. The effect of RF phase noise from the oscillator and a phase velocity mismatch (PVM) between optical photons and the RF field are included in the Hamiltonian. The time evolution of the field operator of the laser in the TW electro-optic phase modulator is derived by the Heisenberg equation of motion. The action of a TW electro-optic phase modulator in the presence of the two distinct weak noises—classical phase noise because of RF and quantum phase noise caused by a laser—has been evaluated and explained in terms of the second-order temporal correlation function and power spectrum. The power spectrum has been written as a function of the semiconductor laser linewidth, an enhancement factor in the laser linewidth, and the RF oscillator linewidth. The results are applied to analyze the basic model of a frequency-coded quantum key distribution (FC-QKD) system. The quantum bit error rate (QBER), the first-order optical intensity correlation function, and the spectrum of intensity fluctuation have been calculated for the partially coherent regime to evaluate the performance of the FC-QKD system.
We investigate evolution of multimode quantum light in electro-optical modulator, using semiclassical model of phase modulation process. For the case of monochromatic incoming signal, we consider transformation of its spectrum and calculate statistical properties of each frequency mode. Particularly, we find that for the case of initial coherent state our result is covered by the well-known classical model. While for the case of initially squeezed state we observe single- and multi-mode squeezing in frequency spectrum of modulated light.
Phase modulation of light plays a key role in modern-day optical communication and processing systems. For this purpose, electro-optic phase modulators are indispensable because of their unique operational flexibility in manipulating the output light signal and providing a large modulation bandwidth. We propose a characterization of the performance of a non-resonant electro-optic phase modulator using what we believe to be a novel quantum-interference-based method. The evaluation is done to study (i) sideband generation, (ii) V π voltage, (iii) modulation parameter, and (iv) capability to introduce controlled phase noise in the system. Electromagnetically induced absorption, a product of constructive quantum interference, is used to study the above-mentioned aspects. These results are compared with those obtained from scanning Fabry–Perot Etalon fringes. In the case of a non-resonant modulator, the efficiency of analysis based on electromagnetically induced absorption is proved to be better than that of Etalon fringe analysis. The result of the noise through the phase modulator in the atom-field system is cross-verified with the resulting heterodyne beat signal obtained between two similar external cavity lasers (free-running, passive stabilization using an extra layer of thermal and vibrational isolation), where one of them is quieter than the other and experiences the introduction of noise through the electro-optic phase modulator.
We present a detailed analysis of the quantum description of electro-optical phase modulation. The results define a black-box type model for this device, which may be especially useful in the engineering steps leading to the design of complex quantum information systems incorporating one or more of these devices. By constructing an explicit representation of the phase modulation scattering operator, it is shown that an approach based entirely on its classical description leads to unphysical modes associated to non-positive frequencies. After modifying this operator, phase modulation is described, for the first time to the best of our knowledge, in terms of a unitary scattering operator Ŝ defined over positive-frequency modes. The modifications introduced by Ŝ in the process of sideband generation by phase modulation are shown to be insignificant when the radiation belongs to optical bands, thus being consistent with the classical description. Finally, the model is employed to characterize the important case of multitone radio- frequency modulation of an optical signal.
We theoretically study electro-optic light modulation based on a quantum model where the linear electro-optic effect and the externally applied microwave field result in the interaction between optical cavity modes. The model assumes that the number of interacting modes is finite, and effects of the mode overlapping coefficient on the strength of the intermode interaction can be taken into account through dependence of the coupling coefficient on the mode characteristics. We show that, under certain conditions, the model is exactly solvable and can be analyzed using the technique of the Jordan mappings for the su(2) Lie algebra. Analytical results are applied to study effects of light modulation on the frequency dependence of the photon counting rate. In contrast to the limiting case of an infinitely large number of interacting modes, when the number of interacting modes is finite, the sideband intensities reveal strongly nonmonotonic behavior supplemented with asymmetry of the intensity distribution provided the pumped mode is not central. We also analyze different regimes of two-modulator transmission and establish the conditions of validity of the semiclassical approximation by applying the methods of polynomially deformed Lie algebras for analysis of the model with quantized microwave field.