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Journal of the Optical Society of America B

Quantum analysis of a traveling-wave electro-optic phase modulator in the presence of the phase noise from a radio frequency oscillator and a laser

Rana Pratap, Harishankar Ramachandran

Published Aug 10, 2020• 3 citations
Publisher

Abstract

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.

DOI
10.1364/josab.383773

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IEEE Journal of Quantum Electronics•Jan 1, 2004•39 citations
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.
Abstract A model of photon statistics and quantum phase distribution is analyzed for coherent optical communication links in the presence of quantum phase noise, dispersion and phase noise of the radio-frequency (RF) oscillator using an in-line optical amplifier. The quantum analysis of a dual drive Mach–Zehnder modulator and the quantum transfer function for an optical field mode across an in-line amplifier is performed by taking into account both the quantum and the classical phase noises. The resulting photon statistics are compared with the semiclassical model. We analyze the quantum phase distribution by considering the Wigner quasiprobability distribution for a coherent optical link. Here, we investigate the broadening of the quantum phase distribution due to quantum phase noise of the laser, the dispersive optical fiber, the phase noise of the RF oscillator and the inevitable noise coefficient of the optical amplifier for arbitrary amplified spontaneous emission (ASE) field modes. We analytically investigate the quantum analog to the bit error rate (BER) as a function of the laser’s linewidth, the linewidth of the RF oscillator and chromatic dispersion. The results of quantum models are compared with the semiclassical model. Quantum treatment of both the BER and variance is found to agree with prior results for a single-mode ASE field in the absence of phase noise but deviate when the effect of phase noise is considered. Including phase noise effects in the amplitude in the semiclassical treatment of the BER yields a new semiclassical expression that agrees with quantum expressions of machine precision for single-mode ASE fields.
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.
The phase noise affecting the beat photocurrent produced by coherent detection of a single-frequency laser field with a time-delayed and frequency-shifted image of itself in a heterodyne two-wave interferometric system using a single-mode optical fiber is investigated as a function of source spectrum and phase-decorrelation time under the assumptions of quantum laser phase noise and high-local-oscillation level. The influence on the photocurrent spectrum of partial correlation between detected fields is pointed out. These calculations are an extension of a previously reported high-resolution spectroscopic method and account for the phase-noise limitation in unbalanced heterodyne fiber sensors and heterodyne-transmission receivers with an optical reference wave generated from the received signal. An experimental study is reported consisting of a Mach–Zender-type interferometer employing a single-frequency amplitude-stabilized laser, an acousto-optical single-side-band modulator for frequency shifting one of the arms, and several lengths of single-mode optical fiber to decorrelate the detected fields partially. Results are found to be in good agreement with theory.
The basic equations governing noise phenomena are derived from first principles and applied to examples in optical communications. Quantum noise arises from two sources, the momentum fluctuations of electrons at optical frequencies and the uncertainty-related fluctuations of the electromagnetic field. Shot noise results from the beating of the noise sources with the signal field. In high-gain amplifiers, the spontaneous-emission noise dominates shot noise and results in a noise figure of at least 3 dB. It is shown explicitly how, at high power, both the laser field and the laser noise source become classical. The increase in noise in lasers with open cavities is not due to enhanced spontaneous emission as once thought, but to single-pass amplification. The noise fields and spontaneous currents have Gaussian distributions, while nonlasing modes have exponential photon-number distributions. Low-frequency intensity fluctuations arise from the electric current driving the laser and can be sub-Poissonian, in contrast to shot noise, which has a Poissonian distribution. The calculational tools are a wave equation for the field operator and a rate equation for the carrier-number operator, each containing spontaneous current noise sources. The correlation functions of these sources are determined by the fluctuation-dissipation theorem. [S0034-6861(96)00503-X]
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.
A quantum theory for the noise of optical pulses in actively mode-locked lasers is presented. In the presence of phase modulation and/or group velocity dispersion, the linear operator that governs the time evolution of the pulse fluctuations inside the laser cavity is not Hermitian (or normal) and the eigenmodes of this operator are not orthogonal. As a result, the eigenmodes have excess noise and the noise in different eigenmodes is highly correlated. We construct quantum operators for the pulse photon number, phase, timing, and frequency fluctuations. The nonorthogonality of the eigenmodes results in excess noise in the pulse photon number, phase, timing, and frequency. The excess noise depends on the frequency chirp of the pulse and is present only at low frequencies in the spectral densities of the pulse noise operators.