A Number Of Quantum States Of Non-classical Nature Of Its Wigner Function

In recent years, people extensively know that in quantum mechanics the principle of superposition is the source of all kinds of non-classical effects of quantum states. So some quantum states exhibited remarkable non-classical effects are constructed on the basis of the principle. It is a fundamental way to constructing new quantum states. For example, the superposed states of two coherent states have squeezing and anti-bunching effect, etc. However the single coherent state cannot exhibit them. Another way to constructing new quantum state is employing the associated operators on the referenced states, where the states are arbitrary states in principle, usually the vacuum state and coherent state are looked on as the referenced states. For example, the photon-added or photon-depleted coherent is constructed by employing the creation operator or annihilation operator on the coherent state by many times. On the other hand, some new quantum states are constructed by virtue of eigenvalue equation of quantum operators. For example, Glauber's coherent state is defined as the eigenstate of the annihilation operator a . It is as soon as possible to construct some allowable quantum states in quantum mechanics, then one shall study their quantum statistical properties and then find some new non-classical properties and the relations among all kinds of non-classical effects as soon as possible. This a very important and effective approach to studying quantum states. Therefore, in theory it is important practical meaning to constructing some new quantum states and investigating their non-classical properties.We know that the Wigner function of a quantum state possesses all informations of the quantum state in the whole evolutionary process, so the variations of quantum state can be described by its Wigner function. However, usually the Wigner functions cannot be measured like the quantum state, so people want to reconstruct Wigner functions for all kinds of quantum states and measure these Wigner functions by using of some observable quantities, then realize the indirect measurement of the corresponding quantum states. On the other hand, the tomogram function of a quantum state is positive quasi-probability function which can be measured directly and be used in the description of evolutionary process of the quantum state. Therefore, it is higher theoretical and practical interest to study the evolutionary process of quantum states by using their Wigner functions and tomogram functions.In this paper we introduce some research progress in new even and odd nonlinear coherent states and Wigner functions for some associated quantum states. We chiefly discuss the following contents: in theory a type of new even and odd nonlinear coherent states is constructed and their non-classical properties are investigated. On the other hand, using the Wigner operator in the coherent representation and entangled representation and the technique of integration within an ordered product(IWOP) of operators, we reconstruct and obtain Wigner functions for some quantum states. Then in term of the variations of the Wigner functions with respect to complex variables, we discuss their non-classical properties in detail. Finally, using the Radon transformations between Wigner operator and the projector of intermediate coordinate-momentum state or the entangled state, the tomogram functions for these states are obtained. The main work and results are summarized as follows:1. New even and odd nonlinear coherent state and their non-classical propertiesNew even and odd nonlinear coherent states(EONLCSs) are constructed and their squeezing, amplitude-squared squeezing, anti-bunching, phase probability distribution and number-phase squeezing are studied. The results show that new EONLCSs have rather different non-classical properties from those of usual even and odd coherent states and even and odd nonlinear coherent states. On the other hand, we obtain Husimi functions and Wigner functions for new EONLCSs, then from the view point of phase space their non-classical properties are discussed.2. Wigner functions for the excited squeezed vacuum state and pair coherent stateUsing two different ways the excited squeezed vacuum state(ESVS) is normalized and then new form of Legendre polynomials is obtained. Using Wigner operator in the coherent state (or in the entangled state) representation and the IWOP technique, the Wigner function for the ESVS (or pair coherent state) is obtained. In term of the variations of the Wigner function with the complex variables, we find that, for different parameters r and m , the ESVSs show different squeezing effects and quantum interference features. On the other hand, pair coherent state(PCS) always exhibits non-classical properties and PCS is more pronounced non-classical properties when q is taken a odd number. However, for different values of q , the state shows different quantum interference properties. Finally, using the intermediate coordinate-momentum state representation and the entangled state representation, we obtain the tomogram functions of the ESVS and PCS.3. The two-mode excited squeezed vacuum state and its propertiesThe two-mode excited squeezed vacuum state(TESVS) is normalized and the new form of Jacobi polynomials is given using the normalization constant. Then the photon number distribution of the TESVS is calculated. Finally, using the Wigner operator in the entangled state representation, we construct the Wigner function for the TESVS. On the basis of the variance of the Wigner function against the parameters m,n and r in the phase space, we find that, for different values of m and n , the TESVS shows different quantum interference effects and the squeezing effect of the TESVS only has relations with the squeezd parameter r .4. Phase properties and Wigner functions for the photon-added even and odd coherent statesUsing the Pegg-Barnett phase operator formalism and the numerical computation method, the phase probability distributions and the squeezed properties of number operator and phase operator for the photon-added even and odd coherent states(PAEOCSs) are investigated. The results show that the phase probability distributions exhibit different quantum phase information and interference features. We also find that the PAEOCSs exhibit squeezed effects in the directions of the number operator and phase operator in different ranges ofα. On the other hand, using the coherent state representation of the Wigner operator, the Wigner functions for the PAEOCSs are obtained. In terms of the variations of the Wigner function withαin the phase space, the non-classical properties of the PAEOCSs are discussed. It is found that the PAEOCSs always exhibit non-classical properties, and the photon-added even (or odd) coherent state exhibits the non-classical properties more easily when m is odd (or even). Using the marginal distributions of the Wigner functions for the PAEOCSs, we illuminate the physical meaning of the Wigner functions. Finally, based on the intermediate coordinate-momentum representation the quantum tomograms of the PAEOCSs are obtained.5. Non-classical properties of the finite dimensional Roy-type EONLCS and the finite dimensional even and odd pair coherent stateThe finite dimensional Roy-type EONLCSs and the finite dimensional even and odd pair coherent states(EOPCSs) are constructed and their orthonormalized property, completeness relations, amplitude-squared squeezing, anti-bunching and phase probability distribution are discussed. The results show that, for the two states in finite dimensional space, there are always normalized property and completeness relations. The finite dimensional Roy-type EONLCSs exist amplitude-squared squeezing for keeping the phase angleθfixed and the relations between the conditions of squeezing and the parameters s,r and the function f ( n ) are given. However, we find that, for any value of q , in different ranges of |ξ| the 5-dimensional EOPCSs for the modes 1 and 2 exhibit anti-bunching effect and the photons between the two modes are always correlated. The different peak structures of the phase probability distributions for the 7-dimensional EOPCSs show remarkably different quantum interference properties.