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Canadian Journal of Physics

Calculation of the squeezing spectrum using a Feynman-diagram technique

Haider R. Zaidi

Published Feb 1, 1988• 4 citations
Publisher

Abstract

The squeezing spectrum of a parametric amplifier, with a quantized pump field, is calculated using the Feynman-diagram technique. The results of the existing theories are recovered. The connection with the theory of superconductivity becomes transparent in this approach.
DOI
10.1139/p88-024

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The quantum theory of the degenerate parametric amplifier is usually treated in the parametric approximation where the pump field is treated classically. In this paper we present a fully quantized theory of this nonlinear optical device using a path-integral approach. A perturbation series, the first term of which corresponds to the parametric approximation, is employed to evaluate explicitly the coherent-state propagator. The question of the validity of the parametric approximation is considered and the conditions under which this approximation is justified are elucidated. Finally, certain correlation functions for the signal-mode operators are calculated that are needed to study squeezed states. It is shown that the quantum nature of the pump field tends to decrease the squeezing.
We perform discrete-mode calculations for a parametric amplifier with a quantum pump and discuss some of the limitations on calculations of this sort in quantum optics. We calculate corrections to the squeezing due to pump quantum fluctuations to order N${\ifmmode\bar\else\textasciimacron\fi{}}^{\mathrm{\ensuremath{-}}2}$, for a pump initially in a coherent state with average photon number N\ifmmode\bar\else\textasciimacron\fi{}. We find that the limit to the variance of the squeezed quadrature due to the quantum nature of the pump goes as N${\ifmmode\bar\else\textasciimacron\fi{}}^{\mathrm{\ensuremath{-}}1/2}$. .AE
The squeezing generated in a lossless nondegenerate parametric amplifier with pump quantization has been numerically simulated in a previous paper [G. Drobn\'y and I. Jex, Phys. Rev. A 46, 499 (1992)]. The report squeezing in a quadrature of up to 90%. We give analytic and numeric results that show that the quadratures studied by Drobn\'y and Jex do not exhibit squeezing if the standard technique of homodyne detection is employed. This is simply shown by a calculation that reveals that the variance in both quadratures is equal for all times. On the other hand, if homodyne detection is not used, the variance would oscillate at the optical carrier frequency---hence, squeezing would be unobservable. We believe that this illustrates the importance of using analytical and physical checks of computer-generated results.
This work proposes a graphical impedance-matching method for superconducting parametric amplifiers with quantum-limited noise performance using a modified Smith chart. It enables one to quickly apply the impedance-matching methods in microwave engineering to the design of parametric amplifiers. When the matching networks are composed of resonators and their equivalents, the matching of a parametric amplifier can be reduced to that of a negative-resistance amplifier, which is generally invalid due to the connection between the signal-frequency and idler-frequency networks introduced by the parametrically modulated component. Several examples are presented to demonstrate the effective and efficient matching of parametric amplifiers using this method, circumventing the need for complex calculations. This method also offers intuitive insight into how variations in pump strength and bias affect the amplification behavior of the circuit, which is useful in characterizing the parametric amplifier and finding the optimal operating state.