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Journal of Optics

Photon statistics and quantum phase distribution for coherent optical links in the presence of phase noise from both the laser source and the RF oscillator, and dispersion

Rana Pratap, Harishankar Ramachandran

Published Jun 16, 2023• 3 citations
Publisher

Abstract

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.

DOI
10.1088/2040-8986/acdefd

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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.
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.
Adopting a new classical approach, the additive noise power generated in optical amplifiers is calculated in terms of power spectral density. The classical formalism used combines a coruscular approach to a phase-amplitude description of the optical field. The noise contributions of the input field fluctuations, including zero-point fluctuations, and of the electron momentum fluctuations at optical frequency linked to the amplifier itself, are clearly identified. The excess of noise associated to coupling or built-in losses is determinated. The well-known result of the Amplified Spontaneous Emission (ASE) is obtained for the laser amplifiers. This description is applied to linear phase-insensitive amplifiers and to inhomogeneous, nonlinear phase-insensitive Raman amplifier, pointing out the effects of gain compression and gain distribution. This new approach makes possible the treatment of the squeezed-state of light and the quadrature reduced noise amplifications.
The fundamental source of noise in laser and optical amplifier is spontaneous emission. However, it is possible to reduce this noise by preparing the gain medium (atoms) in coherent superposition of states. There are essentially two schemes for generating atomic coherence in optical amplifiers. \n \nFirst scheme involves the preparation of atoms in a coherent superposition of states before injecting them inside the cavity. In the other scheme atoms (initially pumped incoherently) are continuously driven by an external pump field inside the cavity. Almost all the studies concerning the quenching of spontaneous emission noise in optical amplifiers, based on the notion of atomic coherent superposition, ignore the finite bandwidth of the field that is responsible for generating coherence. In this dissertation, we have examined the effect of the finite bandwidth associated with the phase fluctuations of the driving field on a number of noise quenching schemes. \n \nIt is known that coherent preparation of atoms can, under certain conditions, lead to noise-free amplification and almost 50 percent phase squeezing in a two-photon correlated emission laser. We study the effect of phase fluctuations in the injected atoms on the phase noise of the field in these systems. We discuss the conditions under which squeezing is preserved in the two-photon linear amplifiers. We also show that, even for very large phase fluctuations in the injected atoms, almost 50 percent phase squeezing can be achieved. The corresponding gain is however significantly reduced. \n \nWe also study a system of phase-sensitive amplifier where atoms are prepared in a coherent superposition of states by a strong classical field. We study the role of pump phase fluctuations associated with the finite linewidth on the quantum statistical properties of the radiation field. A study of output field characteristics reveals that the presence of a weak pump field gives rise to equal variances in both the field quadratures which are independent of linewidth. On the other hand, the presence of strong pump field alters the output field characteristics with the generation of squeezing which is later on destroyed due to the presence of phase fluctuations
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]
This article presents an application of a recently published probablistic model for traveling wave optical amplification. Both optical signal and amplified spontaneous emission (ASE) noise are modeled as bunches of photons traveling along the amplifying region with size varying according to simple random rules. The results of the theory enable one to obtain a complete and accurate characterization of the signal and noise counting at the amplifier output. On the basis of the above statistics, we find a realistic evaluation of the limits imposed by both quantum and ASE noise on the performances of a receiver including an optical amplifier at a preamplifier stage.
In a fiber-optic link, the continuous semiclassical and discrete photon statistics models are widely used to characterize the output probability distribution of an optical amplifier. The semiclassical model describes the output as a chi-square distribution while the discrete photon statistics model describes it through the noncentral negative binomial (NNB) distribution. In this paper, we present a way of understanding the discrete photon statistical distribution at the output of the amplifier and the convergence of the same to its semiclassical counter part. We highlight this convergence through the relation between the NNB and chi-square distribution demonstrated through deviation analysis and simulations. The penalty incurred in bit error rate analysis due to semiclassical approximation over NNB is also discussed.