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Physical Review

Noise in a Molecular Amplifier

Marcel W. Muller

Published Apr 1, 1957• 50 citations
Publisher

Abstract

The quantum theory of noise in lossy electrical circuits is extended to systems which are not in thermal equilibrium. The theory is applied to microwave molecular amplifiers (masers) and predicts a correction to the classically calculated noise figure which can be identified with spontaneous emission from the molecules of the active medium. The correction is small at ordinary temperatures but becomes significant at very low temperatures.

DOI
10.1103/physrev.106.8

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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]

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Physical Review Letters•Mar 14, 1988•33 citations
The quantum theory of noise in lossy electrical circuits is extended to systems which are not in thermal equilibrium. The theory is applied to microwave molecular amplifiers (masers) and predicts a correction to the classically calculated noise figure which can be identified with spontaneous emission from the molecules of the active medium. The correction is small at ordinary temperatures but becomes significant at very low temperatures.
Fluctuations in the amplification and absorption of waves by quantum processes are considered. Assuming for each quantum the probability (per unit time) a of producing another quantum, probability b of being absorbed, and assuming a probability c that a new quantum is introduced, a set of differential equations is obtained. By solving these equations, a complete expression for the probability of distribution of quanta is obtained, as well as expressions for the average values and fractional fluctuation. The expressions developed are applied in particular to maser-type amplifiers, and certain fluctuations in the amplification of electromagnetic waves are pointed out which are important when their quantum character becomes significant. This condition can occur in maser-type amplifiers, where thermal and extraneous noises may be very small. For such an amplifier, a is proportional to the number of excited molecules, whereas b consists of a term proportional to the number of molecules in the ground state plus terms due to certain other losses. The noise temperature of a maser-type traveling-wave amplifier is the "effective temperature" a ( a - b ) -1 h ν/ k . In superregenerative and regenerative amplifiers using resonant cavities, the noise temperature is rather similar, if losses through the input coupling hole are excluded from b . In any case, the limiting noise for an ideal amplifier corresponds to a classical noise temperature of h ν/ k .
The contribution of the saturation field to the noise figure of a three-level maser is calculated. It is shown that the effect is very small, under ordinary circumstances.Certain aspects of spontaneous-emission noise are considered. It is shown that the spontaneous-emission equivalent temperature for a free-electron vacuum-tube amplifier is one-half that of a maser.The possibility of eliminating spontaneous-emission noise is considered and it is concluded that more general quantum-mechanical amplifiers can in principle be constructed which will not have spontaneous-emission noise.
We derive the fundamental noise properties of a parametric amplifier (paramp) which is driven by a monochromatic pump but which has many lossy signal and idler modes. We use a quantum-mechanical approach similar to that employed by Louisell and co-workers in their treatment of paramps and lasers with lossless modes. Our results are not directly deducible from the lossless mode analysis, but they do give the same limiting noise temperature as that analysis. This minimum noise temperature is the same as that of an ideal laser. In fact, we find that one can always design a laser whose field correlation functions would be the same as those of any given paramp. Many results for laser noise in various configurations can be carried over directly and applied to parametric amplifiers by using correspondence substitutions which we set down. For example, in analogy to Pound's description of the maser (in terms of a Nyquist theorem extended to negative temperatures), the calculation of paramp noise is found to be equivalent to applying Nyquist's theorem to both the active and passive elements in the (classically computed) equivalent circuit of a signal mode of interest. From this one obtains the spectra of effective noise generators which, when amplified in the circuit, duplicate the "quantum" as well as thermal noise that is present in the output. The Nyquist theorem is extended to apply to each active element by simply taking the element temperatures to be minus the real temperature of the corresponding idler mode (whose parametric coupling into the signal circuit gives rise to that element) multiplied by the ratio of the pump minus the idler to the idler frequencies.
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]