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arXiv:2210.07628 · Physical Review Applied

Divergence of the Variance of the Optical Phase in Gain-Switched Semiconductor Lasers Described by Stochastic Rate Equations

Angel Valle A

Published May 2, 2023• 7 citations
arXivHTMLPDF

Abstract

We report a theoretical study of the phase diffusion in a gain-switched single-mode semiconductor laser. We use stochastic rate equations for the electric field to analyze the phase statistics of the gain-switched laser. Their use avoids the instabilities obtained with rate equations for the photon number and optical phase when the photon number is small. However, we show that a new problem appears when the field equations are integrated: the variance of the optical phase becomes divergent. This divergence cannot be observed with the numerical integration of the commonly used equations for the photon number and optical phase because of the previous instabilities. The divergence of the phase variance means that this quantity does not reach a fixed value as the integration time step is decreased. We find that the phase variance increases as the integration time step decreases, with no sign of saturation behavior even for tiny steps. We explain the divergence by making the analogy of our problem with two-dimensional Brownian motion. The fact that the divergence appears is not surprising because in 1940 Paul L\`evy demonstrated that the variance of the polar angle in two-dimensional Brownian motion is a divergent quantity. Our results show that stochastic rate equations for the photon number and phase are not appropriate for describing the phase statistics when the photon number is small. Simulation of the stochastic rate equations for the electric field are consistent with L\`evy's results but gives unphysical results since an infinite value is obtained for a quantity that can be measured.

DOI
10.1103/physrevapplied.19.054005

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Quantum random number generation (QRNG) has become a topic of growing interest in recent years due to important applications in cryptography and simulations. Interferometric detection of phase diffusion in gain-switched single-mode semiconductor lasers is one of the main generation techniques. In this paper, we study experimentally and theoretically the phase diffusion in gain-switched discrete mode laser diodes. We derive a stochastic rate equations model for the laser electric field that avoids numerical instabilities that appear when simulating amplitude and phase equations. Measurements are performed in order to extract the parameters of our semiconductor laser. Spontaneous emission rate coupled into the lasing mode is measured as a function of the carrier number for bias currents below threshold. A quadratic dependence is obtained that permits us to evaluate the validity of the linear approximations that have been used to describe laser phase diffusion in QRNG. The good agreement between experiments and theory permits us to give a realistic quantitative description of the dynamical evolution of the phase statistics in this type of QRNGs.
We performed theoretical analysis of the phase randomness in a gain-switched semiconductor laser in the context of its application as a quantum entropy source. Numerical simulations demonstrate that phase diffusion r.m.s. exhibits non-linear dependence on the bias current, which could be of significant practical importance, particularly, in application to high-speed optical quantum random number generators. It is shown that phase diffusion between laser pulses cannot always be assumed to exhibit required efficiency, particularly, at high pulse repetition rates. It was also revealed that the gain saturation significantly affects the r.m.s. value of the phase diffusion and, in essence, determines the degree of non-linearity of its dependence on the pump current.
The statistics of the optical phase of the light emitted by a semiconductor laser diode when subject to periodic modulation of the applied bias current are theoretically analyzed. Numerical simulations of the stochastic rate equations describing the previous system are performed to describe the temporal dependence of the phase statistics. These simulations are performed by considering two cases corresponding to random and deterministic initial conditions. In contrast to the Gaussian character of the phase that has been assumed in previous works, we show that the phase is not distributed as a Gaussian during the initial stages of evolution. We characterize the time it takes the phase to become Gaussian by calculating the dynamical evolution of the kurtosis coefficient of the phase. We show that, under the typical gain-switching with square-wave modulation used for quantum random number generation, quantity is in the ns time scale; that corresponds to the time it takes the system to lose the memory of the distribution of the initial conditions. We compare the standard deviation of the phase obtained with random and deterministic initial conditions to show that their differences become more important as the modulation speed is increased.
The quantum phase noise in a pulsed semiconductor laser is studied thoroughly in the context of its utilization as a quantum entropy source in a quantum random number generator (QRNG) device. We performed a numerical analysis of the phase diffusion process for a semiconductor laser in the continuous-wave operation mode and gain-switched (GS) mode. The result demonstrates the amplification of randomness in the GS mode, which is gauged physically by the variance VarΔϕ. The variance value, which is mathematically related to the temporal distance between the laser pulses used in the experimental setup, also determines the stability of the setup. Furthermore, we show how the QRNG probability distribution is influenced by several experimental factors such as the quality of the interference process and the noise in the detection system.
In this paper, we report a theoretical and experimental study of the phase diffusion in a gain-switched single-mode semiconductor laser. Our modelling includes nonlinear dependencies of the carrier recombination rate and the semiconductor material gain on the carrier number. Also a comparison with the results obtained with linear models is performed. We focus on the below threshold operation, that is, the regime where most of the phase diffusion occurs in quantum random number generators based on the above mentioned system. Using the extracted parameters for our laser and simple analytical expressions for the laser linewidth, we show that logarithmic material gain and cubic carrier recombination dependence on the carrier number must be considered for a good agreement between experiments and theory. In this way we show that consideration of these nonlinearities is essential for obtaining a good quantitative description of the phase diffusion when using rate equations modelling.
The stochastic dynamics of the quantum state of a laser conditioned on continuous intensity and phase-sensitive measurements on its output are examined. We first develop the essential model for a Poissonian laser, and then generalize it for regularly pumped lasers. We show that the rate of phase diffusion (which gives the laser linewidth) is not affected by the regularity of the pump. In both cases, heterodyne detection causes the phase variance in the conditioned laser state to become very small, yet remain significantly above (at least three times) that of a coherent state. The phase diffusion is manifest by a random walk undertaken by the mean phase, with the stochasticity arising from the local-oscillator shot noise. In contrast, intensity measurements have no effect on a Poissonian laser. For a perfectly regularly pumped laser, coarse-grained intensity measurements (for which an approximate theory is developed here) collapse the state to one with an arbitrarily small photon-number variance-to-mean ratio.