Most of the continuous variable quantum cryptography protocols are constructed on the basis of standard coherent and squeezed states. However, other Gaussian quantum states can be also utilized in these protocols. In this thesis, some nonlinear coherent states are used in squeezed state cryptography protocols. Shanon mutual information is used to investigate the security of communication and its variations contingent on several parameters, including the noise factor and coherency parameter are studied. It is obsereved that harmonious, superposed, SU(1,1) and four photon nonlinear coherent states provide more security than standard coherent states, whereas photon added coherent state have less security.

A generalized form of coherent states are displaced number states. It has been shown that these states and their superpositions have non-classical properties; for example they show anti-bunching effect and may have sub-poissonian photon statistics. In this thesis, using two-mode displaced number states, we construct quasi Bell states. Then, we investigate their non-classical properties such as anti-bunching effect, sub-poissonian photon statistics and squeezing. We calculate Mandel parameter and second order coherence function to investigate sub-poissonian photon statistics and anti-bunching effect, respectively. We observed that all the states show sub-poissonian photon statistics in some ranges of the parameters involved while of them show anti-bunching effect and also one of them show squeezing.

Displaced number states are obtained by operating displaced operator on excited number states of harmonic oscillator. These states are the generalized form of coherent states and a type of nonorthogonal states. In this thesis, using the superposed two-mode displaced number states with and out of phase, we construct 13 quasi Bell states. Then, we study the entanglement properties of these states. To this end, first, they are written in terms of orthogonal states. Then, using concurrence for pure bipartite states, we obtain entanglement values for all the ranges of the parameters.
It is observed that the two of the thirteen states are maximally entangled for all the ranges of the parameters while the other states are maximally entangled or seperable depending on the parameters
 

In quautum mechanics, position dependent potentials are devided to the following two groups: polynomial and non-polynomial (fractional). Fractional potentials are also devided to singular and non-singular potentials. Spiked and isotonic potentials are among singular potentials. In addition, CPRS and the second and fourth order fractional potentials are included in the non-singular potentials class. In this thesis second and fourth order fractional potentials are studied. To study these potentials, we employ a matrix method. First, we find the Hamiltonian matrix representation in the harmonic oscillator eignstates basis. To this end, we employ a mathematical method to solve integrals including the multiplication of the Hermite functions. Then, by the use of mathematica, a computer programme, we obtain energy levels and wave functions. Finally, the energy spectrum of the second and fourth order fractional oscillators are analyzed as a function of the potential parameters.

First, the formulation of supersymmetry quantum mechanics is introduced based on concepts such as partner potentials, superpotential, supersymmetric and supercharge operators. Then, expressing the shape invariance condition of partner potentials, we examine the application of this method to obtain eigenfunctions and eigenvalues of Morse potential. Next, we will review how to categorize the shape invariant potentials and related operator method. Then, by expressing the generalization of the supersymmetry quantum mechanics to second order parasupersymmetric quantum mechanics, we study the interaction of three-level atoms with the shape invariant potentials in the framework of parasupersymmetry. By the use of second order parasupersymmetry, the three configuration of the three-level atom Hamiltonians are written in terms of scale operators and partner Hamiltonians. Considering the scale operator for three configuration of the three-level atom and solveing the eigenvalue equation for them, we obtain eigenvalues and eigenfunctions. Finally, we examine the obtained results for several special potentials.

Squeezed states are an important class of quantum states. The eminent specification of these states is the reduction of fluctuations of a component of the field less than the values of similar quantity of a coherent state. Due to numerous theoretical and experimental applications, these states have attracted much attention, in recent years. Hence, these states have been constructed for various physical system. In the mean while, quadratic Hamiltonians are studied in various aspects in recent years. Two harmonic oscillators coupled with λxy interaction term is a sample of the system with quadratic Hamiltonians. The aim of this research is the construction of the vacuum squeezed states for two dimensional harmonic oscillator with λxy interaction term. This work is performed by generalizing squeezing operator. To this end, first using a matrix method, we obtain ladder operators for a general quadratic Hamiltonian. Then, we construct squeezing operator for each one of the oscillation modes in term of ladder operators. In the following the total squeezing operator is obtained as a product of squeezing operator of all the modes and we introduce the functional form of the squeezing operator in term of position and momentum operators. Then, we obtain the total squeezing operator by multiplying the quadratic exponential terms. By applying this operator on the wave function of a vacuum state, we obtain the wave function of the squeezed vacuum state. Finally, we study time evolution of the wave function of the squeezed vacuum state