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Quantum Optics Journal of the European Optical Society Part B

Discrete superpositions of coherent states and phase properties of the m-photon anharmonic oscillator

Małgorzata Paprzycka, R. Tanaś

Published Oct 1, 1992• 31 citations
Publisher

Abstract

The formation of discrete superpositions of coherent states via the unitary evolution of the m-photon anharmonic oscillator is studied. Exact analytical formulae for the superposition coefficients are obtained. It is shown that, in contrast to the two-photon anharmonic oscillator, for m>2 the superposition components enter the superposition with different amplitudes. The Pegg-Barnett phase formalism is used to calculate the phase distributions for the resulting states and to visualize their symmetry.

DOI
10.1088/0954-8998/4/5/008

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The formation of discrete superpositions of coherent states via the unitary evolution of the m-photon anharmonic oscillator is studied. Exact analytical formulae for the superposition coefficients are obtained. It is shown that, in contrast to the two-photon anharmonic oscillator, for m>2 the superposition components enter the superposition with different amplitudes. The Pegg-Barnett phase formalism is used to calculate the phase distributions for the resulting states and to visualize their symmetry.
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The problem of generating discrete superpositions of coherent states in the process of light propagation through a nonlinear Kerr medium, which is modelled by the anharmonic oscillator, is discussed. It is shown that under an appropriate choice of the length (time) of the medium the superpositions with both even and odd numbers of coherent states can appear. Analytical formulae for such superpositions with a few components are given explicitly. General rules governing the process of generating discrete superpositions of coherent states are also given. The maximum number of well distinguished states that can be obtained for a given number of initial photons is estimated. The quasiprobability distribution Q( alpha , alpha *,t) representing the superposition states is illustrated graphically, showing regular structures when the component states are well separated.
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The generation of discrete superpositions of coherent states in the anharmonic oscillator model is discussed from the point of view of their quasi-probability distribution Q(α, α*) and phase probability distribution P(θ). It is shown that for the superposition with well-distinguishable states both distributions show the same rotational symmetry. The maximum number of well-distinguishable states is estimated. The two functions are illustrated graphically to show explicitly their symmetry and the influence of the interference terms. The similarity between the Q function integrated over the amplitude and the phase distribution P(θ) is shown to exist for the anharmonic oscillator states.
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A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.