Research On Non-Gaussian Quantum States And Their Nonclassicality

Nonclassicality of optical fields has been a topic of great interest in quantum optics andquantum information processing. Usually, the nonclassicality manifests itself in specificproperties of quantum statistics, such as the antibunching, sub-Poissonian photon statistics,squeezing in one of the quadratures of the field, partial negative Wigner distribution, etc.Among them, the Wigner function is the quasi-probability distribution function, whose valuecan be positive or negative. For the quasi-classical state (such as a typical coherent state), itsWigner function is always non-negative. Thus, the partial negativity of Wigner function isindeed a good indication of the highly nonclassical character of quantum states. In order toobtain new and non-classical quantum states, many researchers have proposed a variety ofschemes. The basic approach for constructing non-classical quantum states is making use ofthe principle of superposition in quantum mechanics, for example, some superposition statesof Fock states, coherent states, squeezed states, displaced Fock states and so on. Anotherapproach is that acting an operator on a reference state, for example, the usual squeezedstates can be generated by operating the squeezing operator on the coherent states. In recentyears, it is found that subtracting photons from or adding photons to quantum states canobtain some non-classical states such as photon-added coherent states. In fact, these statesabove are not involved in the Gaussian quantum states. The so-called Gaussian quantumstate is defined as state with a Gaussian Wigner function. However, with the developmentof experimental techniques, experimental and theoretical physicists are trying to use non-Gaussian state as the source of information. The most notable example is certainly their usefor an optical quantum computer, alongside their employment for improving teleportation,cloning, and storage. Therefore, considerable attention has been paid to a class of non-Gaussian quantum states. In this thesis, some new non-Gaussian quantum states, exhibitingtheir non-classical features, can be obtained from a number of Gaussian quantum states afteroperating non-Gaussian operation. This thesis is organized as follows:1. We brie?y introduce some basis theories of quantum optics. To begin with, thetechnique of integration within ordered product of operators (IWOP), first proposed by Prof. Fan, is brie?y reviewed, and based on this technique fundamental representations in quantummechanics are derived from new point of view. Especially we give the IWOP technique andskills to easily deduce several distribution functions in the phase space, such as the Wignerfunction, Husimi function and Tomogram function and so on. At the same time, two newphoton counting formulas are presented. It also lists other methods describing non-classicalproperties of quantum states.2. We employ a new way, the partial trace method and the IWOP technique, to derivegeneralized thermal vacuum states corresponding to some complex systems, namely, obtain-ing the corresponding pure state of a mixed state in the extended Hilbert space. So for acomplex system, the ensemble average of turns into calculating the pure state's expectation,this is not only convenient for calculations, but also develops and enriches the theory of ther-mal field dynamics. In addition, it is very convenient for solving the internal energy and itsdistribution in the complex system system by using its corresponding thermal vacuum state.3. We obtain some non-Gaussian states associated with thermal field after non-Gaussianoperation, including photon-added, photon-subtracted, and photon-modulated, and studytheir non-classical properties. After deriving normalization coefficients on these non-Gaussian states, we obtain their explicit expressions of the Wigner functions, by adoptingthe IWOP technique and the Weyl ordered operators'invariance under similar transforma-tions. Next, we focus on investigating the decoherence of statistical properties for photon-added thermal field in the thermal environment, such as photon number distribution and theWigner function with time evolution. It shows that the partial negativity of Wigner func-tion is gradually disappear with the increase of time. We discuss the ?uctuation problemsof these non-Gaussian states in quantum mesoscopic RLC circuit. Secondly, we also studythe photon-modulation thermal state, generated by operating the linear combination of thecreation and annihilation operators on the thermal state, and discuss its non-classical prop-erties such as fidelity, quasi-probability distribution function, photon counting distribution,and Tomogram function.4. We study non-Gaussian states evolved from single-mode squeezed states and theirnon-classical properties. At first, we introduce the generalized squeezing operator in thecoherent state representation and the corresponding squeezed Fock state, analyze its non-classical characteristics. Its analytical expression of the Wigner function with time evolu-tion is obtained in the dissipative channel. Then, we discuss photon-added-then-subtractedsqueezed vacuum state, which is actually the superposition of two kind squeezed Fock states,evaluate its non-Gaussianity by using Hilbert-Schmidt distance measure and present the gen- eration scheme through the cavity QED. Finally, we propose photon-added and photon-subtracted squeezed vacuum states, deduce their normalization coefficients and Wignerfunctions, and deduce the analytical expressions of the fidelity between photon subtracted(or added) squeezed vacuum state and squeezed cat state. For the same number photon-subtraction as photon-addition, a squeezed cat state with a lower fidelity yet higher amplitudecan be generated by the case of photon-addition. In this sense, although photons addition op-erator are more difficult than that of photons subtraction in experiment, photons addition canalso be a powerful tool to generate a cat state with large amplitude.5. We study non-Gaussian states evolved from two-mode squeezed states and their non-classical properties. Firstly we discuss the single- and two-mode successive squeezed oper-ator and its squeezed states. Some statistical properties such as squeezing effect, correlationfunction, anti-bunching effect, and photon number distribution are investigated. In partic-ular, we analytically derive its Wigner function by virtue of the Weyl ordered operators'invariance under similar transformations. Moreover, we focus on investigating the statisticalproperties of photon subtractions from a two-mode squeezed vacuum state, which is also anon-Gaussian state, and its non-classical properties with the help of the IWOP technique. Itis found that the normalization of this state is the Jacobi polynomial of the squeezing pa-rameter, A compact expression for the Wigner function is related to two-variable Hermitepolynomial.6. We investigate nonclassical properties of the field states generated by subtracting(adding) any number photon from the squeezed thermal state. It is found that the normaliza-tion factor of photon-subtracted squeezed thermal state is a Legendre polynomial of squeez-ing parameter and average photon number of thermal state. Their expressions of severalquasi-probability distributions are derived analytically. Additionally, we devote to calcu-lating the fidelity between the photon-subtracted squeezed thermal state and the squeezedthermal state. It is shown that the fidelity decreases monotonously with the increment ofboth photon-subtraction number and the squeezing parameter. Finally, we also discuss theirdecoherence.