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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

The quantum mechanics of signals and noise in attenuators and maser amplifiers

F.N.H. Robinson

Published Aug 3, 1965• 2 citations
Publisher

Abstract

Abstract A method used by Gordon, Walker & Louisell (1963) to treat the temporal behaviour of signals and noise is applied to obtain their spatial or steady-state behaviour in a lossy line or attenuator and a maser amplifier. The results relate the expectation values of the output voltage and its power spectrum to a classical signal input and the thermal state of the system. They differ in principle but not in practice from the results obtained by classical arguments.

DOI
10.1098/rspa.1965.0162

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Journal of the Physical Society of Japan•Jun 1, 1957•315 citations
Abstract A method used by Gordon, Walker & Louisell (1963) to treat the temporal behaviour of signals and noise is applied to obtain their spatial or steady-state behaviour in a lossy line or attenuator and a maser amplifier. The results relate the expectation values of the output voltage and its power spectrum to a classical signal input and the thermal state of the system. They differ in principle but not in practice from the results obtained by classical arguments.
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.
A rigorous basis for the quantum analysis of the steady state of linear distributed systems is established. The analysis of a distributed system of finite length requires, for self-consistency, that excitations be stated at the boundaries of the system even in the absence of externally applied excitations. The commutators of the amplitudes at the boundaries are stated and a useful analogy with thermal noise of classical systems is established. The use of these boundary conditions enables one to formulate the theory of the steady state for distributed quantum systems. When the system under consideration is coupled to a dissipation mechanism, operator-noise sources have to be assigned to the dissipative elements. The commutation relations that must be obeyed by these noise sources are derived. This formalism enables one to analyze the steady-state operation of an attenuator and of a maser amplifier. Finally, properties of multiterminal-pair networks are discussed using the steady-state quantum approach.
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.
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 .