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Fortschritte der Physik

Nonclassicality of States in Quantum Optics

Alfred Wünsche, V. V. Dodonov, Olga V. Man’ko, and V.I. Man'ko

Published Oct 1, 2001• 26 citations
Publisher

Abstract

The nearest Hilbert-Schmidt distance of (pure) states to coherent states is discussed as a quantitative measure of nonclassicality of states in quantum optics or quantum mechanics of harmonic oscillators. Fock states, squeezed coherent states and finite superpositions of coherent states (Schrödinger cat states) are considered as examples. Some other approaches to nonclassicality of states in quantum optics are shortly discussed.

DOI
10.1002/1521-3978(200110)49:10/11<1117::aid-prop1117>3.0.co;2-4

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The nearest Hilbert-Schmidt distance of (pure) states to coherent states is discussed as a quantitative measure of nonclassicality of states in quantum optics or quantum mechanics of harmonic oscillators. Fock states, squeezed coherent states and finite superpositions of coherent states (Schrödinger cat states) are considered as examples. Some other approaches to nonclassicality of states in quantum optics are shortly discussed.
The Hilbert—Schmidt distance between two arbitrary normalizable states is discussed as a measure of the similarity of the states. Unitary transformations of both states with the same unitary operator (e.g. the displacement of both states in the phase plane by the same displacement vector and squeezing of both states) do not change this distance. The nearest distance of a given state to the whole set of coherent states is proposed as a quantitative measure of non-classicality of the state which is identical when considering the coherent states as the most classical ones among pure states and the deviations from them as non-classicality. The connection to other definitions of the non-classicality of states is discussed. The notion of distance can also be used for the definition of a neighbourhood of considered states. Inequalities for the distance of states to Fock states are derived. For given neighbourhoods, they restrict common characteristics of the state as the dispersion of the number operator and the squared deviation of the mean values of the number operator for the considered state and the Fock state. Possible modifications in the definition of non-classicality for mixed states with dependence on the impurity parameter and by including the displaced thermal states as the most classical reference states are discussed.
The degree of nonclassicality of states of a field mode is analysed considering both phase-space and distance-type measures of nonclassicality. By working out some general examples, it is shown explicitly that the phase-space measure is rather sensitive to superposition of states, with finite superpositions possessing maximum nonclassical depth (the highest degree of nonclassicality) irrespective of the nature of the component states. Mixed states are also discussed and examples with nonclassical depth varying between the minimum and the maximum allowed values are exhibited. For pure Gaussian states, it is demonstrated that distance-type measures based on the Hilbert–Schmidt metric are equivalent to the phase-space measure. Analyzing some examples, it is shown that distance-type measures are efficient to quantify the degree of nonclassicality of non-Gaussian pure states.
Abstract Superposition of two coherent states, the Schrodinger’s cat state, can exhibit different nonclassical properties having foundational applications in quantum information processing. We consider the ‘superposition of Schrodinger’s cat state with the vacuum state (SCVS)’ of the optical field. We discuss different witness of nonclassicality properties such as lower- and higher-order squeezing (viz., squeezing, Hong & Mandel’s fourth-order squeezing, amplitude-squared squeezing) and sub-Poissonian photon statistics. Further, we discuss the negativity of the Wigner function of SCVS indicating the nonclassicality of the state under investigation. We find that the vacuum state contribution in SCVS exhibits different nonclassicalities under some conditions stronger where the nonclassicalities exhibited by the state without vacuum state contribution are weaker.
Homodyne measurements are a widely used quantum measurement. Using a coherent state of large amplitude as the local oscillator, it can be shown that the quantum homodyne measurement limits to a field quadrature measurement. In this work, we give an example of a general idea: injecting nonclassical states as a local oscillator can lead to nonclassical measurements. Specifically, we consider injecting a superposition of coherent states, a Schr\"odinger cat state, as a local oscillator. We derive the Kraus operators and the positive operator-valued measure in this situation.
Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.
This chapter presents two essentially different constructions of harmonic oscillator states in a finite-dimensional (FD) Hilbert space. The authors propose some new definitions of the states and find their explicit forms in the Fock representation. For the convenience of the reader, they bring together several known FD quantum-optical states, including FD coherent states, FD phase coherent states, FD displaced number states, FD Schrödinger cats, and FD squeezed vacuums. The discrete Wigner function is applied in order to demonstrate intriguing properties of the various states.