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Physical Review

Quantum Fluctuations and Noise in Parametric Processes. I.

W. H. Louisell, Avigail Yariv, Anthony E.. Siegman

Published Dec 15, 1961• 716 citations
Publisher

Abstract

A quantum mechanical model for parametric interactions is used to evaluate the effect of the measuring (amplifying) process on the statistical properties of radiation. Parametric amplification is shown to be ideal in the sense that it allows a simultaneous determination of the phase and number of quanta of an electromagnetic wave with an accuracy which is limited only by the uncertainty principle. Frequency conversion via parametric processes is shown to be free of zero-point fluctuations.

DOI
10.1103/physrev.124.1646

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A quantum mechanical model for parametric interactions is used to evaluate the effect of the measuring (amplifying) process on the statistical properties of radiation. Parametric amplification is shown to be ideal in the sense that it allows a simultaneous determination of the phase and number of quanta of an electromagnetic wave with an accuracy which is limited only by the uncertainty principle. Frequency conversion via parametric processes is shown to be free of zero-point fluctuations.
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.
In this paper the quantum noise properties of phase-insensitive and phase-sensitive parametric processes are studied. Formulas for the field-quadrature and photon-number means and variances are derived, for processes that involve arbitrary numbers of modes. These quantities determine the signal-to-noise ratios associated with the direct and homo-dyne detection of optical signals. The consequences of the aforementioned formulas are described for frequency conversion, amplification, monitoring, and transmission through sequences of attenuators and amplifiers.
A two-step nonlinear optical interaction consisting of (i) a spontaneous frequency down-conversion process coupled with (ii) a frequency up-conversion process is treated quantum mechanically in the experimentally observed case of simultaneous collinear phase matching of both processes. The temporal behavior of the output signals is found to depend on which process, (i) or (ii), predominates. The nature of the photon statistics for each of the output modes is found to be "super-Poisson," regardless of which process, (i) or (ii), is dominant, and the zero-point fluctuations characteristic of parametric amplification are exhibited.
We develop for the first time a quantum theory of simultaneous amplification and frequency up-conversion of optical images in two coupled three-wave processes. The theory developed takes into account the diffraction effect and the difference of the group velocities of the interacting waves. Due to the degenerate parametric process involved in the nonlinear optical interactions, the up-conversion also possesses phase sensitivity. As a result, the statistical characteristics of the amplified image and the converted one depend on the phase relation between the signal and the pump waves. It is shown that at the output of the nonlinear crystal the signal-to-noise ratios for the amplified image and the converted one may be same as those for the initial image, i.e. transformation of the optical image may be 'noiseless'.
Quantum properties of parametric amplification and frequency conversion of an optical image in coupled parametric interactions are analyzed for the case of a remote object. The coupled-wave interaction consists of a traditional process of parametric amplification in the high-frequency pump field accompanied by the frequency mixing of generated and pump waves. The photon number density and signal-to-noise ratio at the interacting frequencies are studied in the image plane at the output of the nonlinear frequency converter in the fixed pump field approximation. The degree of entanglement of optical images at frequencies above and below the pump frequency is investigated.
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 .