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arXiv:quant-ph/0610106

Semi-classical theory of quiet lasers. I: Principles

Jacques Arnaud, Laurent Chusseau, Fabrice Philippe

Published Oct 13, 2006• 0 citations
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Abstract

When light originating from a laser diode driven by non-fluctuating electrical currents is incident on a photo-detector, the photo-current does not fluctuate much. Precisely, this means that the variance of the number of photo-electrons counted over a large time interval is much smaller that the average number of photo-electrons. At non-zero Fourier frequency $Ω$ the photo-current power spectrum is of the form $Ω^2/(1+Ω^2)$ and thus vanishes as $Ω\to 0$, a conclusion equivalent to the one given above. The purpose of this paper is to show that results such as the one just cited may be derived from a (semi-classical) theory in which neither the optical field nor the electron wave-function are quantized. We first observe that almost any medium may be described by a circuit and distinguish (possibly non-linear) conservative elements such as pure capacitances, and conductances that represent the atom-field coupling. The theory rests on the non-relativistic approximation. Nyquist noise sources (in which the Planck term $\hbarω/2$ is being restored) are associated with positive or negative conductances, and the law of average-energy conservation is enforced. We consider mainly second-order correlations in stationary linearized regimes.

DOI
10.48550/arxiv.quant-ph/0610106

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•Jan 1, 2003•1 citation
When light originating from a laser diode driven by non-fluctuating electrical currents is incident on a photo-detector, the photo-current does not fluctuate much. Precisely, this means that the variance of the number of photo-electrons counted over a large time interval is much smaller that the average number of photo-electrons. At non-zero Fourier frequency ΩΩΩ the photo-current power spectrum is of the form Ω2/(1+Ω2)Ω^2/(1+Ω^2)Ω2/(1+Ω2) and thus vanishes as Ω→0Ω\to 0Ω→0, a conclusion equivalent to the one given above. The purpose of this paper is to show that results such as the one just cited may be derived from a (semi-classical) theory in which neither the optical field nor the electron wave-function are quantized. We first observe that almost any medium may be described by a circuit and distinguish (possibly non-linear) conservative elements such as pure capacitances, and conductances that represent the atom-field coupling. The theory rests on the non-relativistic approximation. Nyquist noise sources (in which the Planck term ℏω/2\hbarω/2ℏω/2 is being restored) are associated with positive or negative conductances, and the law of average-energy conservation is enforced. We consider mainly second-order correlations in stationary linearized regimes.
We call quiet a stationary laser that generates in detectors regular photo-electrons (sub-Poisson statistics). It follows from the law of conservation of energy that this is so when the laser power supply does not fluctuate. Various configurations are analyzed on the basis of the Planck (1907) semi-classical concept: am not seeking the meaning of light quanta in the vacuum but rather in places where emission and absorption occur, and I assume that what happens in the vacuum is rigorously described by Maxwell's equations. Exact agreement with Quantum Optics results is noted. Comments welcome!
This article is the shorten version of quant-phys/0610106 with a supplemented theory and new results concerning a single-electron laser driven by a constant-potentiel battery. "Quiet (or sub-Poissonian) oscillators generate a number of dissipation events whose variance is less than the mean. It was shown in 1984 by Golubev and Sokolov that lasers driven by regular pumps are quiet in that sense. We consider in the present paper two oscillators that should exhibit in principle the same property. First, a reflex klystron, a vacuum tube operating in the microwave range of frequency. Second a laser involving a single electron permanently interacting with the field. It is unnecessary to quantize the optical field, that is, the theory is semi-classical, yet exact. As an example, the battery-driven one-electron laser delivers a detected noise of 7/8 of the shot-noise level, and is therefore sub-Poissonian. Our calculations are related to resonance-fluorescence treatments but with a different physical interpretation. Previous theories considering excited-state atoms regularly-injected in low-loss resonators, on the other hand, do require light quantization. The theory presented here is restricted to above-threshold stationary single-mode oscillators. The paper is written in such a way that readers should be able to follow it without having to refer to quantum-optics texts."
Quiet (or sub-Poissonian) oscillators generate a number of dissipation events whose variance is less than the mean. It was shown in 1984 by Golubev and Sokolov that lasers driven by regular pumps are quiet in that sense. The purpose of this paper is to show that, as long as the laser-detector system is strictly stationary, quantization of the optical field is not required to explain such phenomena. The theory presented here is semi-classical, yet exact. Previous theories considering excited-state atoms regularly-injected in resonators, on the other hand, do require in principle light quantization. Specifically, we consider a laser involving a single electron permanently interacting with the field and driven by a constant-potential battery, and point out a similarity with reflex klystrons. The detected noise is found to be only 7/8 of the shot-noise level. It is therefore sub-Poissonian. Our calculations are related to resonance-fluorescence treatments but with different physical interpretations.
A previously reported expression [Opt. Quantum Electron.27, 63 (1995)] for the variance of photon number m in a laser diode driven by nonfluctuating currents is generalized to account for spontaneous carrier recombination. The semiclassical theory used here yields Q≡〈Δm2〉/m-1=b+(n/m)/(1+b)-1/2, where n is the average carrier number and b is the spontaneous-to-stimulated rate ratio. This expression is in exact agreement with a recently reported quantum-theory result [J. Opt. Soc. Am. B13, 761 (1996)]. At high power, b≈0 and m≫n imply that 〈Δm2〉=〈m〉/2, i.e. sub-Poissonian internal-field statistics. The Q factor that relates to carrier number is given by the same expression, except that the first b term is missing. Practical means of measuring the photon-number variance and the role of spectral-hole burning are briefly discussed.
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]
'l*he noise is the was a saying of Rolf Landauer, l one of the founding fathers of mesoscopic physics. What he meant was that fluctuations in time of a measurement can be a source of Information that is not present in the time-averaged value. As figure l reminds us, some types of noise are more interesting than others. A physicist with access to sensitive ways of distinguishing the granularity in a signal may delight in the noise. Noise plays a uniquely informative role in the particle-wave duality. In 1909, Albert Einstein realized that electromagnetic fluctuations vary depending on whether the energy is carried by waves or by particles. The magnitude of energy fluctuations scales linearly with the mean energy for classical waves, but it scales with the square root of the mean energy for classical particles. Since a photon is neither a classical wave nor a classical particle, the linear and square-root contributions coexist. Typically, the square-root (particle) contribution dominates at optical frequencies and the linear (wave) contribution takes over at radio frequencies. If Isaac Newton could have measured noise as the time-dependent fluctuations from the mean intensity, he would have been able to settle his dispute with Christiaan Huygens on the corpuscular nature of light—without actually needing to observe an individual photon. Such is the power of noise. The diagnostic power of photon noise was further developed in the 1960s, when it was discovered that one can teil the difference between radiation from a laser and that from a black body on the basis of their fluctuating Signals: The wave contribution to the fluctuations is entirely absent for a laser; it is merely small for a black body. Noise measurements are now a routine technique in quantum optics, and Roy Glauber's quantum mechanical theory of photon statistics is textbook material. Because electrons share the particle—wave duality with photons, one might expect fluctuations in the electrical current to play a similar diagnostic role. Current fluctuations due to the discreteness of the electrical charge are known as shot noise. Although the first observations of shot noise date from work on vacuum tubes in the 1920s,