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17th Congress of the International Commission for Optics: Optics for Science and New Technology

Preferred values and dispersions of phase of any order for l-photon Kerr states

Vlasta Peřinová

Published Sep 1, 1996• 0 citations
Publisher

Abstract

It is well known that during the ℓth-subharmonic generation general ℓ-photon states evolve from an arbitrary state of the pump mode. For the second-subharmonic generation with an initial coherent state the assumption of pump nondepletion, the so called parametric approximation, is very useful for obtaining closed formulas. The resulting state then is an ordinary squeezed vacuum. The parametric approximation is ineffective for ℓ > 2 not only because closed formulas are not obtained, but also because it is proven that no quantum state is defined for nonzero higher-order squeezing [1]. This finding motivated a definition of the ℓ-photon coherent state [2],

DOI
10.1117/12.2316110

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It is well known that during the ℓth-subharmonic generation general ℓ-photon states evolve from an arbitrary state of the pump mode. For the second-subharmonic generation with an initial coherent state the assumption of pump nondepletion, the so called parametric approximation, is very useful for obtaining closed formulas. The resulting state then is an ordinary squeezed vacuum. The parametric approximation is ineffective for ℓ > 2 not only because closed formulas are not obtained, but also because it is proven that no quantum state is defined for nonzero higher-order squeezing [1]. This finding motivated a definition of the ℓ-photon coherent state [2],
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 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 study a single mode of the radiation field subjected to ideal k-photon parametric amplification (photons created or destroyed k at a time). In an attempt to generalize ordinary squeezing (k = 2), Fisher et al.1 considered the case k > 2. They concluded that there is something seriously wrong with the evolution operator after discovering that for k > 2 its matrix elements in the number-state basis have divergent Taylor series expansions in time. We show that this divergence is due to a branch cut along the negative time axis, and we obtain useful information by treating these Taylor series as asymptotic expansions. We investigate the states generated by a k-photon paramp by calculating the evolution of the quantum O-function. We watch the vacuum evolve into a state whose O-function has k-fold symmetry. The classical behavior which corresponds to a k-photon parametric interaction is described by phase-space trajectories near an unstable fixed point. We compare the classical and quantum behavior and find that the quantum corrections smear out classical phase-space features which are smaller than allowed by the uncertainty principle. Finally, we consider the evolution of an initial coherent state which has large amplitude, and we find that a three-photon paramp generates squeezing at a rate proportional to the amplitude of the initial coherent state.