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Investigation on Statistical Properties of Nonclassical States Derived from Coherent States

Author: WangZhen
Tutor: FanHongYi
School: Shanghai Jiaotong University
Course: Physics
Keywords: coherent states the technique of integration within an ordered product of opera-tors sub-Poissonian distribution Wigner function decoherence
CLC: O431.2
Type: PhD thesis
Year: 2013
Downloads: 48
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Abstract


The nonclassicality of optical fields has been a hot topic in the development of quantum op-tics and quantum information processing. Usually, the nonclassicality manifests itself in specificproperties of quantum statistics, such as sub-Poissonian photon statistics, antibunching, squeez-ing in one of the quadratures, and partial negative Wigner function, etc. In recent years，peopleextensively know that it is the principle of superposition in quantum mechanics that produce var-ious kinds of non-classical effects of quantum states. Therefore, people construct many quantumstates exhibiting remarkable nonclassical effects on the basis of the principle. For example, thesuperposed states of two coherent states have squeezing and anti-bunching effect. Another wayto constructing new quantum states is employing the associated operators on the original states，where the original states can be arbitrary states in principle, e.g., vacuum states, coherent states,and thermal states, etc. For example, the squeezed states can be generated by operating the squeez-ing operator on vacuum states or coherent states; the photon-added coherent states is constructedby employing the photon creation operator a on coherent states repeatedly, etc. An importantand effective approach to studying nonclassical states is to constructing quantum states as manyas possible in the framework of quantum mechanics, and then find out new nonclassical effects bystudying their quantum statistical properties. Therefore, it is of practical meaning to constructingnew quantum states theoretically and investigating their nonclassical properties. We know that theWigner function of a quantum state possesses all information of the quantum state in the wholephase space, so the evolution of quantum states can be described by Wigner function. However,Wigner function is not probability distribution function but quasi-probability distribution function,whose value can be positive or negative. For the classical or quasi-classical states (e.g., coherentstates), their Wigner function is always non-negative. Thus the partial negativity of Wigner func-tion is indeed a good indication of the highly nonclassical character of quantum states. In this paperwe mainly introduce the research progress in the manipulation of coherent states and squeezed co-herent states. Using the Wigner operator in coherent states representation and the technique ofintegration within an ordered product of operators, we reconstruct and obtain the Wigner functionsfor these quantum states. And then in term of the variations of the Wigner function with respectto complex variables in phase space, we discuss their nonclassical properties in detail. Finally, by virtue of the evolution formula of Wigner function in amplitude damping channel and thermalchannel derived by the theory of thermal field dynamics and thermal entanglement states represen-tation, we discuss the decoherence effect of these quantum states. The main works are summarizedas five parts:1. The construction and investigation of photon manipulation squeezed vacuum states thatintegrate photon-added and photon-subtracted squeezed vacuum states. Firstly, by applying theoperation of the coherent superposition of photon addition operator a and subtraction operator aon squeezed vacuum states, we obtain the so-called photon manipulation squeezed vacuum states.And then, we derive the normalization constant by virtue of the technique of integration within anordered product and the newly found expression of Legendre polynomials. And also derive thephoton-number distribution, the Wigner function, and the fidelity between squeezed Schro¨dingercat states.2. The investigation on statistical properties of photon-added coherent states in thermal chan-nel. We obtain the evolution of density operator in thermal channel obtained by thermal entan-glement states representation and the evolution of Wigner function in thermal channel by usingthe Wigner operator in coherent states representation. Additionally, based on the coordinate—momentum intermediate representation, we obtain the tomogram function of photon-added coher-ent states in thermal channel. Indicated by the analysis and investigation is that the density operator,Wigner function, and tomogram function of photon-added coherent states in thermal channel shallreduce to the density operator, Wigner function, and tomogram function of thermal states when theevolution time tends to infinity.3. The study on statistical properties of photon-added-then-subtracted coherent states. Bytaking advantage of the operator ordering theory in quantum optics and the technique of inte-gration within an ordered product, the analytical expressions of normalization constant, photon-number distribution, Mandel’s Q-parameter, P-function, Q-function, Wigner function, and fidelitybetween photon-added-then-subtracted coherent states and initial coherent states are derived, andits nonclassicality is discussed in detail accordingly. In addition, using the time evolution of Wignerfunction in amplitude damping channel obtained by thermal entanglement states representation, thedecoherence effect of photon-added-then-subtracted coherent states in amplitude damping channelis studied.4. The investigation on statistical properties of photon-subtracted and photon-added squeezedcoherent states. Employing the invariance of Weyl ordered operator under similar transformationand the technique of integration within an ordered product, we derive the normally ordered formof density operator for squeezed coherent states. On the basis of the normally ordered densityoperator, by repeatedly subtracting from and adding photons to squeezed coherent states, we the-oretically construct so-called photon-subtracted and photon-added squeezed coherent states. By analytical calculating Mandel’s Q-parameter, quadratures squeezing, photon-number distribution,and Wigner function, etc. we study the nonclassicality of photon-subtracted and photon-addedsqueezed coherent states. On the other hand, the decoherence process of photon-subtracted andphoton-added squeezed coherent states in thermal channel is studied through the time evolution ofdensity operator and Wigner function, respectively.5. The photon-added squeezing enhanced thermal states is introduced theoretically by re-peatedly applying photon creation operator on the squeezing enhanced thermal states. And thenby virtue of the normally ordered form density operator for enhanced squeezing thermal states,we investigate the statistical properties on account of the analytical expressions of normalizationconstant, Mandel’s Q-parameter, photon-number distribution, Wigner function, and fidelity be-tween photon-added squeezing enhanced thermal states and squeezing enhanced thermal states. Inaddition, the decoherence process of photon-added squeezing enhanced thermal states in thermalchannel is also included.The structure of this dissertation is arranged as follows:In Chap.1, we briefly introduce the basic theories of nonclassical quantum states and coherentstates, including some way that can generate nonclassical states and some effective criteria thatcan measure nonclassicality, and the definition and properties of coherent states is also reviewed.In Chap.2, we introduce the technique of integration within an ordered product. And based onthis technique some fundamental representations in quantum mechanics, Wigner function, timeevolution of density operator and Wigner function in several channels are derived. And it is thefoundation for further investigation in the following chapters. In Chap.3, we study a kind of statesthat integrate photon-added and photon-subtracted squeezed vacuum states—photon modulationsqueezed vacuum states. In Chap.4, we study the statistical properties of photon-added coherentstates and photon-added-then-subtracted coherent states. In Chap.5, we obtain the nonclassicaland non-Gaussian states associated with squeezed coherent states after nonclassical non-Gaussianoperation, including photon addition and photon subtraction. In Chap.6, we obtain the nonclassicaland non-Gaussian states associated with squeezing enhanced thermal states. We believe that theinvestigation of these quantum states, to some extent, can further enrich the quantum states ma-nipulation and quantum states engineering theory, possessing high academic value and practicalsignificance.

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CLC: > Mathematical sciences and chemical > Physics > Optics > The light nature of the theory > Quantum optics
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