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Journal of Physics Conference Series

Phase-space representation of non-classical behaviour of scalar wave-fields

Gustavo Cañas, Román Castañeda, Herbert Vinck-Posada

Published Jan 1, 2011• 0 citations
Publisher

Abstract

The modelling of optical fields by using radiant and virtual point sources for the spatial coherence wavelets in the phase-space representation evidences some effects, conventionally attributed to non-classical correlations of light, although such type of correlations are not explicitly included in the model. Specifically, a light state is produced that has similar morphology to the Wigner Distribution Function of the well-known quantum Schrödinger cat and squeezed states.

DOI
10.1088/1742-6596/274/1/012141

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The modelling of optical fields by using radiant and virtual point sources for the spatial coherence wavelets in the phase-space representation evidences some effects, conventionally attributed to non-classical correlations of light, although such type of correlations are not explicitly included in the model. Specifically, a light state is produced that has similar morphology to the Wigner Distribution Function of the well-known quantum Schrödinger cat and squeezed states.
In the framework of the spatial coherence wavelets, different features of the first-order spatial coherence (Young's interference) are analysed by calculating the corresponding marginal power spectrum, a close related quantity to the classical Wigner Distribution Function (WDF) of the optical field. The consideration of radiant and virtual point sources evidences some effects, conventionally attributed to non-classical correlations of light, although such type of correlations are not explicitly included in the model. Specifically, a light state is produced that has similar morphology to the WDF of the well-known quantum Schrödinger cat state.
The border between the descriptions of the classical optical fields in any state of spatial coherence and the quantum coherence state of light is revisited in the framework of the phase-space representation. Although it is established that such descriptions are not completely equivalent, the exact calculation of the marginal power spectrum leads to new analogies that suggest that some features exclusively attributed to quantum states of light can be also shared by classical optical fields due to their spatial coherence state.
Approximating a continuous superposition by a discrete one leads to an understanding of the role of quantum interference in generating squeezed coherent states, and in its quantum properties. It was shown that discrete superposition with a large enough number of coherent states with appropriate coefficients can have practically the same features as the squeezed coherent state. Dependent on the parameters W, r and 8, different numbers of coherent states are required for the interference fringe to form a Wigner function resembling that of the squeezed vacuum state and to reduce the deviation below a certain value. For the case of rotating the direction of squeezing, about 20 states were required to lower the deviation to 0.05%, while 640 states would be necessary when a unit displacement along the imaginary axis is included. When only a few coherent states are superimposed, the resulting state changes chaotically. In the case when the approximated squeezed coherent state was displaced along the imaginary axis an interesting Schrodinger cat-like state was found with about 20 coherent states on the real axis. At this point a significant peak occurs in the variances of the quadratures. It turned out that non-classical states can be effectively produced by a discrete superposition of coherent states with the coefficients derived from the one-dimensional representation of the given state.
The elementary quantum theory of quasi-classical states for mechanical harmonic oscillators is applied to the radiation oscillators describing light, and thus used to define quasi-classical states of light. These states are called “coherent”, because the expectation value of any normally ordered product of creation and annihilation operators factorizes for them. A theoretical discussion of possible sources for coherent states is followed by experimental evidence for their existence. The mathematical properties of coherent states, e.g., nonorthogonality and overcompleteness, are explained in order to use them to represent states and operators. These representations provide a suitable replacement for the classical phase space description of light, which is forbidden by the uncertainty principle. The quantum phase space approach leads to the definition of the Wigner distribution, the Q-function, the P-function, and Gaussian mixed states.