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Journal of Shandong Institute of Light Industry

The phase characteristic of the non-classical optical field with negative P representation

Yingying Tuo

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

The phase characteristic and phase variance of the non-classical optical field with negative P representation according to P-B quantum phase theory was studied in this paper.It is shown that the non-classical optical field with negative P representation is a totally stochastically phase state.The fluctuation of the phase operators is a constant(π2/3).

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The phase characteristic and phase variance of the non-classical optical field with negative P representation according to P-B quantum phase theory was studied in this paper.It is shown that the non-classical optical field with negative P representation is a totally stochastically phase state.The fluctuation of the phase operators is a constant(π2/3).
The Wehrl′s Entropy and Shannon′s entropy for the non-classical optical fields with negative P representation are studied in this paper. The results show that the Wehrl′s entropy of the states decreases along with the growth of the state parameter and the average photon numbers and tend to be constant. While the Shannon′s entropy increases along with the increase of state parameter and the average photon numbers. It is also found that the Wehrl′s entropy of the non-classical optical fields with negative P representation decreases while the total noise increases and the Shannon′s entropy decreases along with the decrease of observable fluctuation.
Quantum-statistical properties of the phase of electromagnetic field in the case of phase superpositions of coherent states of the type Nα,β,ξneiϕα+eiξneiϕβ have been studied. Formulas for the average values and fluctuations (variances) of Hermitian phase operator ϕˆθ of the electromagnetic field (so-called Pegg–Barnett optical phase operator) in the case of general superposition of coherent states are obtained and analysed. The states of phase superpositions of the ‘Schrödinger cat’ type Nα,−α,ξ(|α⟩+eiξ|−α⟩),ϕβ=ϕα+π, and superpositions of coherent states of the type Nα,α∗,ξ(|α⟩+eiξ|α∗⟩),ϕβ=−ϕα, are considered separately.
In our paper we consider the non-classical behaviour of both the Hermitian (observable) Stokes parameters of light and the phase difference of two modes that describe the quantum polarization states of optical field. To characterize the degree of polarization of light we introduce a new quantity taking into account the quantum properties of different quantum states of two orthogonally polarized modes. The problem of determination of the phase difference in two modes of optical field for the quantum polarization states of light is discussed. To describe in general such a quantum field we introduce two pairs of the phase operators: the phase angles for the Stokes parameters of light in a three-dimensional picture of the Poincaré sphere. We also consider a special type of the eight-port polarization interferometer (polarimeter) for simultaneous homodyne detection of both the Stokes parameters of light and the polarization phase operators and their fluctuations as well. Using an anisotropic (spatioperiodic) Kerr-like nonlinear medium associated with the polarization interferometer we could generate and also observe the polarization-squeezed phase states of light. The fluctuations in the phase difference between two orthogonally polarized modes for these non-classical states are less than the fluctuations for light in coherent state.
We extend the theory of the Hermitian optical phase operator to analyze the quantum phase properties of pairs of electromagnetic field modes. The operators representing the sum and difference of the two single-mode phases are simply the sum and difference of the two single-mode phase operators. The eigenvalue spectra of the sum and difference operators have widths of 4\ensuremath{\pi}, but phases differing by 2\ensuremath{\pi} are physically indistinguishable. This means that the phase sum and difference probability distributions must be cast into a 2\ensuremath{\pi} range. We obtain mod(2\ensuremath{\pi}) probability distributions for the phase sum and difference that unambiguously reveal the signatures of randomness, phase correlations, and phase locking. We use our approach to investigate the phase sum and difference properties for uncorrelated modes in random and partial phase states and the phase-locked properties of the two-mode squeezed vacuum states. We reveal the fundamental property of two-mode squeezed states that the phase sum is locked to the argument of the squeezing parameter. The variance of the phase sum depends dilogarithmically on 1+tanhr, where r is the magnitude of the squeezing parameter, vanishing in the large squeezing limit.
Applying the approach of modelling physical quantities in terms of measurable maps to the phase of a light field, we distinguish a multiple-valued phase from its representation on the unit circle and the single-valued phases in the case of one random phase variable. We study two multiple-valued random phase variables, whose multiple value forms a displaced lattice, and we compare them with their representation on the torus. We relate the phase expectations and variances to single-valued phases, whereas we associate the preferred phase and the phase dispersion with the representation on the circle. We propose alternatives for the covariance of two single-valued random phase variables. Starting from the concept of a characteristic sequence, we introduce the phase characteristics of higher order. As an application to quantum optics, we reformulate the study of phase properties of the states of one- and two-mode light fields.
The comparison between the classical and quantum theories of optical coherence is presented by using an idea used in the problems of modulation of light beams of classical fields. In such problems it is assumed that it is possible to vary the mean light intensity of a beam without changing its statistical properties defined by a set of coherence functions. We generalize this idea and introduce precisely the class of optical fields which are consistent for modulation. It is shown that all the quantum fields of this class have a positive PPP representation and are strictly equivalent to classical fields. Moreover, when the field is assumed to be stationary, an interpretation of this condition is given which in particular makes precise the relations between photon-counting and light-intensity measurements. Finally it is shown that all the quantum fields without PPP representation cannot be consistent for modulation, and the condition of consistency for modulation appears as a characteristic property of fields strictly equivalent to classical ones.