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Journal of the Physical Society of Japan

Fluctuations in Amplification of Quanta with Application to Maser Amplifiers

Kōichi Shimoda, Hidetosi Takahasi, Charles Hard Townes

Published Jun 1, 1957• 315 citations

Abstract

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 .

DOI: 10.1143/jpsj.12.686

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In this paper we consider further the quantum-statistical properties of radiation in nondegenerate parametric-type amplifiers. In particular we find moment-generating functions and probability distribution functions for the field variables at the output of an amplifier for various input conditions. We find that a classical description of the input fields and of the amplification process is completely valid provided we take correctly into account the response of the amplifier to the input zero-point noise fields. This result is valid for inputs of arbitrarily small power.

Non-relativistic quantum mechanics is treated using Feynman's space-time approach and Wigner's probablity density of position and momentum. The mechanics of a system is described by the probability of each path of the coordinates in space-time. The influence of one system on another is described by the probability of a force function. It is shown that linear systems follow only classical paths. The quantum theory merely puts uncertainty into their initial conditions, just like thermal noise. The formalism apparently does not simplify intrinsically non-linear problems. However, linear systems are sometimes coupled to non-linear quantum systems which operate only on a linear portion of their charactersitcs (like a vacuum tube in a class A amplifier). Then the reaction on the linear systems can be reduced to an equivalent impedence and noise power spectrum, so that the linear problem can be solved by classical methods. Solid state and beam type masers are treated as examples. Masers employ excited non-linear systems to acheive linear amplification of the electromagnetic signal. The modes of the waveguides and cavities are the linear systems. To them the amplifying systems appear as nagative resistance, reactance, and noise.

Non-relativistic quantum mechanics is treated using Feynman's space-time approach and Wigner's probablity density of position and momentum. The mechanics of a system is described by the probability of each path of the coordinates in space-time. The influence of one system on another is described by the probability of a force function. It is shown that linear systems follow only classical paths. The quantum theory merely puts uncertainty into their initial conditions, just like thermal noise. The formalism apparently does not simplify intrinsically non-linear problems. However, linear systems are sometimes coupled to non-linear quantum systems which operate only on a linear portion of their charactersitcs (like a vacuum tube in a class A amplifier). Then the reaction on the linear systems can be reduced to an equivalent impedence and noise power spectrum, so that the linear problem can be solved by classical methods. Solid state and beam type masers are treated as examples. Masers employ excited non-linear systems to acheive linear amplification of the electromagnetic signal. The modes of the waveguides and cavities are the linear systems. To them the amplifying systems appear as nagative resistance, reactance, and noise.

We investigate amplification of optical images by means of a traveling-wave optical parametric amplifier. As shown recently by Kolobov and Lugiato [Phys. Rev. A 52, 4930 (1995)] for a cavity-based geometry, such a scheme can amplify images, without deteriorating their sign-to-noise ratio, thus working as a noiseless amplifier. Here we consider a configuration without cavity, which is more realistic for a possible experimental realization. We investigate in detail the quantum fluctuations of the amplifier and formulate criteria for its noiseless performance. The spatial resolution power, which guarantees noiseless amplification is estimated. We demonstrate how the optimum phase matching of a phase-sensitive wavefront of the image can improve the noise performance of the amplifier and bring it to the ultimate value achievable under given physical conditions. We discuss the effect of improvement of the signal-to-noise ratio in the case of observation of both the input and the amplified image, with an ideal photodetector of small area (i.e., much less than the coherence area of the amplifier). This improvement is due to the fact that the signal-to-noise ratio in the input is deteriorated by the observation with the photodetector of small area.