Decoherence and Nonclassicality of Quantum States

Abstract

The phase space representation of quantum mechanics is a well-known powerful tool for studying the correspondence between the density operator and classical distributions in phase space. This representation, known as the third formulation of quantum mechanics, is given in terms of the joint probability distribution (or more precisely the quasi-probability), and is independent of the conventional Hilbert space or the path integral formulations. In this representation one needs not choosing − coordinate or momentum − it works in the full phase space, accommodating the uncertainty principle, and offering a unique insight into the classical limit of quantum theory [1]. A variety of these representations exist, including Wigner [2], Husimi [3], P [4], Huwi [5], and are distinct from one another by the way they highlight classical structures against a background of quantum interferences. The study of decoherence is a long sought after goal in quantum optics. The question very often underpinning is how one can measure this decoherence or alternatively to what extent a system is quantum (with respect of course to the reference classical system, the coherent state). Recently an indicator of nonclassicality (quantumness) of a given system has been proposed [6] and has been successfully tested in a large number of quantum states of infinite dimensional Hilbert space. This indicator is based on the relative volume of the negative part of the Wigner function and is a quantitative measure of the degree to which a system is quantum. Attempts have been trying to link the negativity of the Wigner function with the entanglement of the analysed state on a composed Hilbert space [7]. The fundamentals of quantum mechanics in phase space were reviewed in this work, focusing mainly on the role of different distribution functions (quasi-distribution functions) with respect to the study of intriguing quantum phenomena. In particular, Wigner Distribution [2] and Non Classicality Indicator [6] were used to explore few quantum states [8], including the superposition of two Cat states, Squeezed Fork state and the Compass state [9] (a superposition of two pairs of Gaussians) for which the sub-Planck structures have been well demonstrated. Confidently, the resulting outputs will help to better understand and predict the quantumness of these quantum states as a function of their sizes or when they are exposed to perturbations.