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Physical Review

Quantum Noise in Linear Amplifiers

H. A. Haus, James A. Mullen

Published Dec 1, 1962• 318 citations
Publisher

Abstract

The classical definition of noise figure, based on signal-to-noise ratio, is adapted to the case when quantum noise is predominant. The noise figure is normalized to "uncertainty noise." General quantum mechanical equations for linear amplifiers are set up using the condition of linearity and the requirement that the commutator brackets of the pertinent operators are conserved in the amplification. These equations include as special cases the maser, the parametric amplifier, and the parametric up-converter. Using these equations the noise figure of a general amplifier is derived. The minimum value of this noise figure is equal to 2. The significance of the result with regard to a simultaneous phase and amplitude measurement is explored.

DOI
10.1103/physrev.128.2407

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The classical definition of noise figure, based on signal-to-noise ratio, is adapted to the case when quantum noise is predominant. The noise figure is normalized to "uncertainty noise." General quantum mechanical equations for linear amplifiers are set up using the condition of linearity and the requirement that the commutator brackets of the pertinent operators are conserved in the amplification. These equations include as special cases the maser, the parametric amplifier, and the parametric up-converter. Using these equations the noise figure of a general amplifier is derived. The minimum value of this noise figure is equal to 2. The significance of the result with regard to a simultaneous phase and amplitude measurement is explored.
If the uncertainty principle of quantum mechanics is applied to the process of signal measurement, two theorems relating to amplifier noise performance can be deduced. First, it can be shown that it is impossible to construct a linear noiseless amplifier. Second, if the amplifier is characterized as having additive white Gaussian noise, it can be shown that the minimum possible noise temperature of any linear amplifier is T.= In 2 -IIG -hv 1-1/ IG k In the limit of high gain G this expression reduces to that previously derived for the ideal maser and parametric amplifier. It is shown that the minimum noise amplifier does not degrade the signal but rather allows the use of an inaccurate detector to make measurements on an incoming signal to the greatest accuracy consistent with the uncertainty principle.
The principal problem of light amplifiers is to find a means to amplify a coherent amplitude without decreasing the SNR. Consider a signal with amplitude squeezed quantum fluctuations, that is, less noise in the amplitude quadrature than the phase quadrature. The objective is to find a phase sensitive amplifier, which, while amplifying the coherent amplitude, adds quantum noise preferentially to the phase quadrature. We take as a concrete model for the amplifier an intracavity parametric amplifier. We go beyond the usual linear analysis and include the nonlinear regime where pump depletion is important. We find that in the linear region the quantum noise added by the amplification process may be reduced by squeezing the idler mode, while in the nonlinear regime the quantum fluctuations from the pump become more important. These results are of direct application to practical light amplifiers.
The quantum-limited noise figure (NF) for a general combination of linear optical elements is obtained in a compact form. The entries in this expression are obtained from the classical input–output transmission coefficient of the signal, and of the various transmission coefficients from the noise inputs to the output. The same result is also obtained by means of a semiclassical derivation. The linear elements can be: population-inversion amplifiers, parametric amplifiers and/or wavelength converters, beam splitters, lossy fibers, etc. They form a network with a simple directed graph (no loops). We identify a class of networks for which Friis’ formula for concatenated elements holds. We verify the general formula for a variety of well-known cases. We also verify that the NF for an optical communication link periodically amplified by nondegenerate phase-sensitive parametric amplifiers (PSAs) is only about half of that for a similar system using degenerate PSAs.
The purpose of a phase-preserving linear amplifier is to make a small signal larger, regardless of its phase, so that it can be perceived by instruments incapable of resolving the original signal, while sacrificing as little as possible in signal-to-noise ratio. Quantum mechanics limits how well this can be done: A high-gain linear amplifier must degrade the signal-to-noise ratio; the noise added by the amplifier, when referred to the input, must be at least half a quantum at the operating frequency. This well-known quantum limit only constrains the second moments of the added noise. Here we derive the quantum constraints on the entire distribution of added noise: We show that any phase-preserving linear amplifier is equivalent to a parametric amplifier with a physical state for the ancillary mode; the noise added to the amplified field mode is distributed according to the Wigner function of the ancilla state.
The concept of noise figure F and noise measure M applicable to radio frequency and microwave amplifiers is reviewed and extended to cover optical amplifiers. Two noise figures are defined in the optical regime: one gives the ASE output, while the other gives the ratio of signal-to-noise ratios at the input and output. In the quantum regime, the lower limit on these optimum values is set by a generalization of Heisenberg's uncertainty relation. A Raman back-pumped fiber link with post-amplification with an erbium-doped fiber amplifier is used to illustrate the use of noise measure.
This chapter begins by specializing the general definition of optical amplifiers to a special class of linear amplifiers that includes all those in common use. Amplifiers are divided into the categories of phase insensitive or phase sensitive, and each of these categories is subdivided into the categories of traveling-wave or regenerative. Illustrative examples are given for each of the four types. This is followed by a general description of single-input, single-output linear amplifiers that includes the input-output equation, the conditions for phase insensitivity, and the constraints imposed by unitarity. This general theory is used to determine the noise limits for linear amplifiers. The general amplifier uncertainty principle obtained in this way shows that there are noise free amplifiers.