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Modern Physics Letters B

TWO-MODE PHASE-STATE REPRESENTATIONS IN QUANTUM OPTICS

Hong-Yi Fan, Xian-Ting Liang, Zhihu Sun

Published Oct 30, 2001• 0 citations
Publisher

Abstract

A phase state presentation |ξ= reiθ> for a two-mode light field is constructed in the context of quantum optics. This phase space formalism is directly connected to phase-related operators. The generalized quasi-probability distribution function Q(ξ) and measurable phase probability distribution function P(θ) are constructed and are useful to obtain information about phase properties of light fields.

DOI
10.1142/s0217984901002804

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A phase state presentation |ξ= reiθ> for a two-mode light field is constructed in the context of quantum optics. This phase space formalism is directly connected to phase-related operators. The generalized quasi-probability distribution function Q(ξ) and measurable phase probability distribution function P(θ) are constructed and are useful to obtain information about phase properties of light fields.
A phase space for the single-mode light field is constructed by using optical phase variables. This phase-space formalism is directly connected to optical-phase-related operators. The phase-state basis \ensuremath{\Vert}\ensuremath{\theta}〉 is used to represent various quantum operators and states. Properties of the density operator \ensuremath{\rho}^ phase-state matrix elements P(\ensuremath{\theta},\ensuremath{\theta}')=〈\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}\ensuremath{\theta}'〉 and their relations to phase-related quasiprobability distributions are discussed. By using the phase representation, different types of number-phase intelligent states are analyzed.
Various quasiprobability distributions have been developed in the past using the Hilbert space of the single-mode light field. The development of a quasiprobability distribution associated with a phase operator has previously been impossible because of the absence of a unique Hermitian phase operator defined on the Hilbert space. Recently, however, Pegg and Barnett [Europhys. Lett. 6, 483 (1988); Phys. Rev. A 39, 1665 (1989)] and Barnett and Pegg [J. Mod. Optics 36, 7 (1989)] introduced a new formalism that does allow the construction of a Hermitian phase operator and associated phase eigenstates. In this paper we develop a quasiprobability distribution associated with the number and phase operators of the single-mode light field in the new formalism. The new distribution, which we call the number-phase Wigner function, has properties analogous to the Wigner function. We also derive the number-phase Wigner representation of number states, phase states, general physical states, coherent states, and the squeezed vacuum. We find this new representation has features that are related to the number and phase properties of states. For example, the number-phase Wigner representation of a number state is nonzero only on a circle, while the representation of a phase state is only nonzero along a radial line.
The probability distribution for finding a quantum state of the light field with a particular phase can be described by a number of theoretical methods. One of the most frequently used in quantum optics is the Pegg–Barnett phase distribution which identifies the phase distribution as the (necessarily positive) modulus square of the overlap of the quantum state with the relevant phase state. Other phase distributions have been proposed which employ the phase-sensitivity of various quantum quasi-probabilities such as the Wigner function. In this paper we discuss the connections between these approaches and illustrate their similarities and important differences for a number of pertinent quantum states of light. For some of these states, quantum interference can generate significant regions of phase space for which the Wigner quasi-probability is negative, and as we show, this may render the quasi-probability phase distribution negative. We discuss the general significance of this phenomenon.
The language of statistical moments and covariance matrices we introduced in the previous chapter offers a compact and efficient formalism to deal with Gaussian states. A much more general approach may be taken to describe any quantum state in a setting which is reminiscent of classical phase space. This is the framework of quantum characteristic functions and quasi-probability distributions, which goes back to seminal work by Wigner on quantum corrections to classical statistical mechanics, and bloomed in the sixties with the rise of theoretical quantum optics and the emergence of a general unifying picture. Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which we shall prove in the form of the Fourier–Weyl relation between density matrices and characteristic functions. This relationship will constitute the bridge between phase space and Hilbert space descriptions, which will be useful in several applications in the context of quantum information. It will be convenient to handle most of the proofs and mathematical arguments concerning characteristic functions and quasi-probability distributions on a single mode of the system. Because displacement operators of multimode systems are just tensor products of local displacement operators, the extension of the formalism to systems with many degrees of freedom will be straightforward. Nevertheless, we shall always take care of explicitly linking the single-mode formulae which will appear in this chapter to the general multimode description adopted in the previous one.
The Hilbert space of a single mode quantum system is spanned by a countable set of orthonormal basis states denoted |n〉 for n = 0, 1, 2, … Such a basis can be used to define quasi-probability distributions over a two-dimensional quantum phase plane. Multi-mode quantum systems are spanned by orthonormal product states |n1,…nk〉 = |n1〉 ⊗ … ⊗ |nk〉 which can be used to define quasiprobability distributions over a 2k-dimensional phase volume. However, this multi-mode basis set is also countable and can be uniquely specified by a single integer label. This labeling allows us to define new collective operators and generalized coherent states to construct probability distributions over a new two-dimensional phase plane. These distributions represent the system's 2k-dimensional phase volume with perfect fidelity, and allow us to apply phase space interference (PSI) approaches to multi-mode quantum systems.