Pzz is a real but non-positive one.Nevertheless, the field state description with the Glauber-Sudarshan P-distribution remains the most demonstrative and consumable.For example, a proposed definition of non-classical states of electromagnetic field (Bogolyubov (Jr.) et al., 1988) uses the expression (1) for the statistical operator.A state is referred to as non-classical one if one of two requirements is obeyed: either average number of photons in a mode is less than 1, or P-function is not positively determined or has singularity that is higher than the -function.www.intechopen.com
Pzz is a real but non-positive one.Nevertheless, the field state description with the Glauber-Sudarshan P-distribution remains the most demonstrative and consumable.For example, a proposed definition of non-classical states of electromagnetic field (Bogolyubov (Jr.) et al., 1988) uses the expression (1) for the statistical operator.A state is referred to as non-classical one if one of two requirements is obeyed: either average number of photons in a mode is less than 1, or P-function is not positively determined or has singularity that is higher than the -function.www.intechopen.com
In 1963 Glauber /Phys. Rev. 130, 2529 and 131, 2766/ published a basic theoretical study that revealed the broad variety of possible photon states. Later on a quantum-statistical analysis showed that states with classical analog (this light Is termed classical light) as well as states without classical analog (so-called nonclassical light) can occur. Following the definition given by Teich, Saleh, and Perina /JOSAB2 (1985) 275) nonclassical light exhibits one or more of the attributes squeezed, antibunched, sub-Poissonian, what is connected with a negative value of the field-fluctuation excess, the bunching excess, or the Poisson excess. These negative values mean quantum fluctuations below the classical standard quantum limits of ideal laser light; thus, nonclassical light may yield an essential improvement of the optical measurement accuracy, in particular of the signal-to-noise ratio. There exists a common property of the three attributes concerning the Glauber-Sudarshan representation P(c ), which is the real weight function belonging to the incoherent mixture (density operators) of Glauber states with different complex amplitudes. In the case of classical light P(oç ) is a positive definite function, whereas nonclassical light requires a nonpositive definite function P(o.), which cannot be interpreted as a proper probability distribution - as it is possible in the case of light with a classical analog /M. Schubert, Ann. d. Phys. 4k (1987) 53/. In addition to its possible important applications nonclassical light yields a deep insight into the quantum nature of the radiation field.
The existence of positive well-defined quasiprobabilities is known to be related to the measure of nonclassicality for a given field state. We test the coincidence of the loss of well-known nonclassical properties with the appearance of the positive well-defined quasiprobabilities for various fields influenced by phase-insensitive reservoirs. For Gaussian fields the appearance of a positive Glauber-Sudarshan P function coincides with the loss of nonclassical properties in quadrature fluctuations and in photon statistics. We find that when an initial Fock state dissipates or is amplified the initial field loses its nonclassicality in photon statistics before the P function becomes positive, which is different from the Gaussian field. The Wigner function may be positive even when the decaying Fock state is sub-Poissonian. For the amplification of the Fock state, however, the Wigner function is positive only when the field is super-Poissonian. For the dissipation and amplification of the Fock state the point when a certain quasiprobability function becomes positive definite depends only on the average number of photons N of the heat bath. \textcopyright{} 1996 The American Physical Society.
Modern electromagnetic theory based on quantum conceptions supposes turn to averages of operator dynamic variables for describing the field state and analyzing experimental results with computational methods. Theoretical analysis of such non-equilibrium phenomenon as Dicke superfluorescence leads to an algebra of variables which are necessary for building a set of evolution equations. Rather different correlation functions can be determined in measurements according to the coincidence scheme using Glauber's quantum detectors. A natural way of connecting various correlation functions is expressing them via the photon density matrix and Wigner function. The problem of comparing the theoretical and experimental approaches is analyzed in the present paper.
As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons. First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution. In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P -representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P -representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q -representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.
One of the most relevant problems in the quantum theory is the question whether the appropriate state of the system can be described within a classical theory. The famous examples of nonclassical states are Schrodinger-cats states or entanglement states. In spite of many practical applications of entanglement as a special kind of nonclassicality (like quantum teleportation [1], dense coding [2], or implementation of super-fast [3] and fast algorithms [4]), we still have difficulties with describing and specifying it. One cannot find such an operational optimal method, which would allow us to maximally use the technological and scientific potential arising from the adaptation properties of quantum correlations. Thus, my thesis is focused on giving a proposal of a theoretical recipe for constructing experimentally achievable procedures for the study of quantum correlations. In my thesis I have focused on quantum correlations described in terms of nonclassicality (quantumness) and specific kind of it,i.e., entanglement. The nonclassicality definition, used by me in the thesis, is based on the Glauber-Sudarshan function (P -function). One can assume, that a state is nonclassical, if its P -function is negative or more singular than Dirac’s delta [5]. As follows, the state is nonclassical, when its P -function cannot be treated as a “real” probabilistic distribution. Based on specific properties of the P -function, Agarwal and Tara [6], as well as, Shchukin, Richter and Vogel [7, 8] proposed nonclassicality and entanglement [9] criteria based on matrices of moments of annihilation and creation operators. The operational procedures for analyzing nonclassicality and an efficient method for measuring such moments developed by Shchukin and Vogel [10] creates basis for my thesis. It can be seen that the nonclassicality criteria based on matrices of moments offer an effective way of deriving specific inequalities, which might be useful in the verification of nonclassicality of particular states generated in experiments. Therefore, criteria constructed based on the above definition of nonclassicality can be used to find practical and effective methods of generating and testing nonclassicality and, therefore, also quantum entanglement of optical fields. The results presented in my thesis can be divided into three main groups: 1. Finding operational and practical criteria to classify states in terms of nonclassicality and entanglement based on fundamental classical inequalities like Cauchy-Schwarz inequality. It was also shown how some known entanglement inequalities can be derived as nonclassicality inequalities or as sums of more than one inequality [Bartkowiak2010a]. 2. Describing properties and behaviour of quantum correlations for different optical fields (e.g. for multiand single-mode systems, for interacting and noninteracting modes). In particular, general occurrence of sudden vanishing of nonclassicality, which can be observed not only for twoor multimode but also for single-mode fields, was proven [Bartkowiak2011]. 3. Finding the methods of generating and testing of quantum correlations, which would be practical and easy to implement with available resources in both linear and nonlinear optics.