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Physical Review A

Quadrature squeezing in the nondegenerate parametric amplifier

Mark J. Fernée, Paul Kinsler, P. D. Drummond

Published Jan 1, 1995• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.51.864

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•Nov 21, 2007•0 citations
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.
We calculate the squeezing or quadrature spectrum of a nondegenerate parametric amplifier in an interferometer. Squeezing is found to occur below and above threshold. The new feature in the above threshold behavior is the existence of spontaneous phase diffusion in the transmitted signal, even with a stable pump. A novel technique is developed to calculate the quadrature fluctuations in the presence of phase diffusion. We find that squeezing occurs but is restricted to relatively high frequencies due to the phase diffusion. Another application of our theory is in the case of a subharmonic oscillator with a phase-diffusing pump. Squeezing is maintained here also, except at low frequencies. The presence of phase diffusion in quadrature phase sensitive detection can be expected to be universal. In all cases it tends to reduce or destroy the low-frequency squeezing and thereby increases low-frequency noise in a phase sensitive measurement. Our new stochastic technique, therefore, has a wide range of uses.
We investigate the time evolution of squeezing in the signal mode of the undamped parametric amplifier. Both the degenerate and nondegenerate cases are considered. These are the simplest models that give limits to squeezing caused by both the intrinsic pump noise and pump depletion. Conflicting analytic expressions have been reported. We resolve this by the use of exact numerical methods, employing both number state and stochastic techniques. Our central result is that in both cases we find that the minimum in the squeezed quadrature variance scales as N${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}1/2}$, where N\ifmmode\bar\else\textasciimacron\fi{} is the initial coherent pump photon number. This bound on the amount of squeezing obtainable, coupled with the loss in energy due to the conversion efficiency, serves to limit the phase information in the squeezed channel to no more than that of the coherent pump. We believe this to be a fundamental limit for using the squeezed radiation generated in coherently driven parametric amplifiers.
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
In this article we explore the quantum properties of a degenerate optical parametric oscillator when it is tuned to the first family of transverse modes at the down-converted frequency. Recently we found [C. Navarrete-Benlloch et al., Phys. Rev. Lett. 100, 203601 (2008)] that above threshold a TEM10{}_{10}10​ mode following a random rotation in the transverse plane emerges in this system (we denote it as the bright mode), breaking thus its rotational invariance. Then, owing to the mode orientation being undetermined, we showed that the phase quadrature of the transverse mode orthogonal to this one (denoted as the dark mode) is perfectly squeezed at any pump level and without an increase in the fluctuations on its amplitude quadrature (which seems to contradict the uncertainty principle). In this article we go further in the study of this system and analyze some important features not considered previously. First we show that the apparent violation of the uncertainty principle is just that---``apparent''---as the conjugate pair of the squeezed quadrature is not another quadrature but the orientation of the bright mode (which is completely undetermined in the long term). We also study a homodyne scheme in which the local oscillator is not perfectly matched to the dark mode, as this could be impossible in real experiments due to the random rotation of the mode, showing that even in this case large levels of noise reduction can be obtained (also including the experimentally unavoidable phase fluctuations). Finally, we show that neither the adiabatic elimination of the pump variables nor the linearization of the quantum equations are responsible for the remarkable properties of the dark mode (which we prove analytically and through numerical simulations, respectively), which were simplifying assumptions used in Navarrete-Benlloch et al. [Phys. Rev. Lett. 100, 203601 (2008)]. These studies show that the production of noncritically squeezed light through spontaneous rotational symmetry breaking is a robust phenomenon.
This thesis is primarily a theoretical study of degenerate parametric amplification as a means of generating squeezed-state light. i) A wideband traveling-wave formalism is developed for analyzing quantum mechanically a degenerate parametric amplifier. The formalism is based on spatial differential equations — spatial Langevin equations — that propagate temporal Fourier components of the field through the nonlinear medium. In addition to the parametric nonlinearity, the Langevin equations include absorption and associated fluctuations, dispersion, and pump quantum fluctuations. The dominant effects of dispersion and pump quantum fluctuations on the squeezing produced by a degenerate parametric amplifier are analyzed. ii) The wideband formalism of i) is used to carry out a more detailed analysis of the effects of phase mismatching. With the assumption of a lossless medium and a classical pump, we find that parametric amplification is capable of generating squeezed-state light over a wide band if materials with large χ(2) nonlinearities can be found, and that the squeezing bandwidth can be enhanced by phase mismatching away from degeneracy. iii) We consider again the effect of pump quantum fluctuations on the squeezing produced by parametric amplification. We perform discrete-mode calculations for a parametric amplifier with a quantum pump, and discuss some of the limitations of calculations of this sort in quantum optics. We derive stochastic differential equations (SDEs) for one- and two-mode parametric amplifiers, and from them obtain an iterative solution showing that pump quantum fluctuations impose a limitation on the degree of squeezing obtainable from a parametric amplifier. iv) A possible application of squeezing is considered; in particular, we study the effects of squeezing the intracavity noise in a laser oscillator. We solve the classical noise problem of a realistic laser model by making a bold — and possibly unrealizable — assumption, that the in-phase and quadrature Langevin sources which are responsible for the "noisiness" of the laser can be squeezed. We show that the effect of squeezing the in-phase quadrature is to reduce the phase noise, including the linewidth, of the laser but, due to amplitude-phase coupling, not to eliminate them altogether.