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Physical Review A

Ground-state phase transitions of the degenerate parametric amplifier

Christopher C. Gerry

Published May 1, 1988• 4 citations
Publisher

Abstract

Ground-state phase transitions of the degenerate parametric amplifier assuming the pump mode in an ordinary coherent state and the signal mode in an SU(1,1) coherent state are discussed. A second-order phase transition with the coupling constant playing the role of a control parameter is found. The lower-energy state can be a squeezed state in the signal mode. An additional anharmonic term related to optical bistability is shown to enhance the effect.

DOI
10.1103/physreva.37.3619

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Annual Meeting Optical Society of America•Jan 1, 1989•0 citations
Ground-state phase transitions of the degenerate parametric amplifier assuming the pump mode in an ordinary coherent state and the signal mode in an SU(1,1) coherent state are discussed. A second-order phase transition with the coupling constant playing the role of a control parameter is found. The lower-energy state can be a squeezed state in the signal mode. An additional anharmonic term related to optical bistability is shown to enhance the effect.
We study the effect of non-coherent-state pumps on the behavior of the degenerate parametric amplifier. This is done both by developing some exact results and by extending the parametric approximation. Relations are found between the number fluctuations in the pump and those in the signal. The effect of initial phase uncertainty in the pump on the squeezing in the signal is also determined. In addition, we find that rotational symmetry in phase space of the signal-mode state is twice that of the pump-mode state. This implies, for example, that a pump in the squeezed vacuum state will produce a signal-mode state with a four-pronged structure. The effect of a squeezed pump with nonzero amplitude is studied by using a more general form of the parametric approximation. It is shown that in the time interval during which the parametric approximation is valid the effect of this pump is the same as one which is an incoherent superposition of coherent states. We find that while the squeezing of the pump degrades the squeezing of the signal it enhances its amplitude-squared squeezing.
The time evolution of squeezed vacuum states, taken to be SU(1,1) coherent states, interacting with a two-photon medium is examined using numerical methods. The non-photon-conserving medium is modeled by a Hamiltonian which consists of the free-field term, a non-photon-conserving term in the form of a degenerate parametric amplifier, and an anharmonic term which gives rise to optical bistability. This system has previously been shown to produce enhanced squeezing when interacting with ordinary coherent light. In the present work we find that the squeezing property is generally revoked by the anharmonic term and increased by the nonconserving term. The combined effects are not additive, however, and a quantum interference effect is observed in the calculations.
The parametric amplifier is usually studied under the assumption that the pump mode is in a large-amplitude coherent state. We consider what happens when this is no longer the case. We first derive some exact relations between the number fluctuations of the pump and signal, the pump and the idler, and also the signal and the idler. It is then determined how much phase noise can be tolerated in the initial pump state if two-mode squeezing is to be produced in the signal-idler field. The relation between the rotational symmetry in phase space of the initial pump state and that of the signal-idler subsystem is determined. We find that the sum phase distribution of the signal and idler has twice the rotational symmetry as that of the initial pump state. Finally, we generalize the parametric approximation to treat pump states which are not coherent states. We examine the effect of a phase-squeezed pump state and find that the pump squeezing can actually decrease the two-mode squeezing of the signal-idler field while enhancing a form of higher-order squeezing, sum squeezing, for these fields.
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.
The degenerate parametric oscillator is an important source of squeezed light. If the pump mode is adiabatically eliminated, its operation is governed by two dimensionless parameters: the pump parameter λ and the coupling constant g. The pump parameter controls the threshold behavior predicted by semiclassical theory (threshold is reached for λ = 1). The coupling constant g determines the strength of the quantum fluctuations. The optical parametric oscillators used to generate squeezed light have g ~ 10–4 ≪ 1.1 Therefore, in these systems the quantum fluctuations only produce very small perturbations about semiclassical states. We have recently shown how the steady state of the degenerate parameter oscillator changes as a function of coupling strength.2 In the strong coupling limit, major departures from semiclassical states occur; for g2 ≫ 1 and 2λ2/g2 ≪ 1 the steady state approches the coherent state superposition | ψ + 〉 = | λ / g 〉 + | − λ / g .