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THEORETICAL STUDY OF HIGHER ORDER NONCLASSICALITY IN INTERMEDIATE STATES
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THEORETICAL STUDY OF HIGHER ORDER NONCLASSICALITY IN INTERMEDIATE STATES



1 Introduction

A state of light is called nonelassieal, if its Glauber Sudarshan P function is negative or more singular than a delta function [1]. In these situations quasi probability distribution P is not accepted as classical probability and thus one can not obtain an analogous classical state. For example, squeezed state and antibunched state are well known nonelassieal states. These two lowest order nonelassieal states have been studied since long but the interest in higher order nonelassieal states is relatively new. Possibilities of observing higher order nonclassicalities in different physical systems have been investigated in recent past [2]-[16]. For example, i) higher order squeezed state of Hong Mandel type [2]-[4], ii) higher order squeezed state of Hillery type [5], [6], ii) higher order subpoissonian photon state [7]-[9] and iv) higher order antibunched state [10]-[16] are recently studied in different physical systems. All these interesting recent studies and their potential applications in quantum optics, quantum information, quantum computing and other fields motivated us to study the higher order nonelassieal states of light in detail.
Among the different class of states which shows nonclassicality. Intermediate states form an interesting group. An intermediate state is a quantum state which reduces to two or more distinguishably different states (normally, distinguishable in terms of photon number distribution) in different limits. In 1985, such a state was first time introduced by Stoler et al. [17]. To be precise, they introduced Binomial state (BS) as a state which is interme diate between the most nonelassieal number state \n) and the most classical coherent state \a). Since the introduction of BS, the intermediate states have attracted considerable at tention of physicists. Consequently, different properties of binomial states have been studied [18]-[22]. In these studies it has been observed that the nonelassieal phenomena (such as, antibunching, squeezing and higher order squeezing) can be seen in BS. This trend of search for nonclassicality in Binomial state, continued in nineties. In one hand, several versions of generalized BS have been proposed [19]-[27] and in the other hand, people went beyond binomial states and proposed several other form of intermediate states (such as, excited bi nomial state (EBS) [21], odd excited binomial state (OEBS) [22], negative hypergeometrie state (NHS) [23], reciprocal binomial state (RBS) [24], photon added coherent state (PACS) [25] etc.) and hypergeometrie state (HS) [27] . The studies in the last century were mainly limited to theoretical predictions of lower order nonclassicalities but the recent developments in the experimental techniques made it possible to experimentally verify some of those the oretical predictions. For example, we can note that, as early as in 1991 Agarwal and Tara [25] had introduced PACS but the experimental generation of the state has happened only in recent past when /avalla. Viciani and Bellini [26] succeed to produce it in 2004. This exper imental observation, several existing reports of lower order nonelassiealities in intermediate states and the intrinsic nonelassieal character of the intermediate states have motivated us to study the possibilities of observing higher order nonelassiealities in intermediate states. Present work aims to provide a clear understanding of higher order nonelassieal states in general with specific attention towards intermediate states and many wave mixing processes [8], [12]-[16], [28]-[32]. Chronological development of the subject is provided in Table 1. Our achievements appear in the lower part of the Table 1. The results of the present work, which are reported in [8],[12],[28]-[30], will be discussed in detail in the proposed thesis.
In this synopsis we try to provide a brief sketch of the proposed doctoral thesis. To do so we have categorized this synopsis in 6 sections including introduction. Each section contain brief summary of individual chapters of the thesis. To be precise, first chapter of the proposed thesis will provide an introduction to the subject and major content of that chapter is accepted for publication in our review article [7]. Section 2, briefly describes the content of chapter 2 of the proposed thesis. Here we describe the possibility of observing higher order antibunching in intermediate states and in some simple nonlinear optical phenomena. The results reported in this chapter are published in [12],[29]-[30]. In section 3 we report that a generalized notion of higher order nonclassicality (in terms of higher order moments) can be introduced. Under this generalized framework of higher order nonclassicality, conditions of higher order squeezing and higher order subpoissonian photon statistics are derived. These simplified criteria play the central role in the present work as these criteria are used in next chapter to study the presence of higher order nonelassiealities in different intermediate states. The results summarized in section 3 are published in [8] and they will be described in detail in the chapter 3 of the proposed thesis. The section 4 provides a glimpse of the work to be reported in detail in the chapter 4 of the proposed thesis. In this section we briefly describe our work on the possibilities of observing higher order squeezing and higher order subpoissonian photon statistics in different intermediate states. Corresponding results are partly published [7],[8] and partly communicated. Section 5 shows reduction of quantum phase fluctuations (U parameter), which is a stronger criterion of nonclassicality (compared to the lowest order antibunching), is possible in intermediate states. Corresponding results are published in [28] and will be described in detail in chapter 5 of the thesis. Section 6 briefly summarizes the content of the concluding chapter of the proposed thesis. The conclusions are derived from the earlier chapters of the thesis (i.e. from the works reported in [7],[8],[12],[28]-[30]). Limitations of the present work & scope of future works are also described here. Definition of different intermediate states is given in Appendix A, mathematical criteria of different kind of nonclassicality are shown in Appendix B and the results related to the study of higher order nonelassiealities in binomial state are provided in Appendix C as an example.
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