Abstract:
Polya states of single mode radiation field are proposed and their algebraic characterization and nonclassical properties are investigated. They degenerate to the binomial (atomic coherent) and negative binomial (Perelomov's su(1,1) coherent) states in two different limits and further to the number, the ordinary coherent and Susskind-Glogower phase states. The algebra involved turn out to be a two-parameter deformation of both su(2) and su(1,1). Nonclassical properties are investigated in detail.

Abstract:
We introduce the negative binomial states with negative binomial distribution as their photon number distribution. They reduce to the ordinary coherent states and Susskind-Glogower phase states in different limits. The ladder and displacement operator formalisms are found and they are essentially the Perelomov's su(1,1) coherent states via its Holstein-Primakoff realisation. These states exhibit strong squeezing effect and they obey the super-Poissonian statistics. A method to generate these states is proposed.

Abstract:
The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite dimensional cyclic representations. As a realistic physical system of GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically. The new creation and annihilation operators of the Pegg-Barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit q->1.

Abstract:
A class of squeezed states for the su(1,1) algebra is found and expressed by the exponential and Laguerre-polynomial operators acting on the vacuum states. As a special case it is proved that the Perelomov's coherent state is a ladder-operator squeezed state and therefore a minimum uncertainty state. The theory is applied to the two-particle Calogero-Sutherland model. We find some new squeezed states and compared them with the classical trajectories. The connection with some su(1,1) quantum optical systems (amplitude-squared realization, Holstein-Primakoff realization, the two mode realization and a four mode realization) is also discussed.

Abstract:
We show that the binomial states (BS) of Stoler {\it et al.} admit the ladder and displacement operator formalism. By generalizing the ladder operator formalism we propose an eigenvalue equation which possesses the number and the squeezed states as its limiting solutions. The explicit forms of the solutions, to be referred to as the {\it generalized binomial states} (GBS), are given. Corresponding to the wide range of the eigenvalue spectrum these GBS have as widely different properties. Their limits to number and {\it squeezed} states are investigated in detail. The time evolution of BS is obtained as a special case of the approach.

Abstract:
We propose a displacement-operator approach to some aspects of squeezed states for general multiphoton systems. The explicit displacement-operators of the squeezed vacuum and the coherent states are achieved and expresses as the ordinary exponential form. As a byproduct the coherent states of the $q$-oscillator are obtained by the {\it usual exponential} displacement-operator.

Abstract:
Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the $su(2)$ coherent states, we propose ``interpretation'' of $su(1,1)$ and $su(r,1)$ coherent states ``in terms of probability theory''. They will be called the ``negative binomial'' (``multinomial'') ``states'' which correspond to the ``negative'' binomial (multinomial) distribution, the non-compact counterpart of the well known binomial (multinomial) distribution. Explicit forms of the negative binomial (multinomial) states are given in terms of various boson representations which are naturally related to the probability theory interpretation. Here we show fruitful interplay of probability theory, group theory and quantum theory.

Abstract:
`Hypergeometric states', which are a one-parameter generalization of binomial states of the single-mode quantized radiation field, are introduced and their nonclassical properties are investigated. Their limits to the binomial states and to the coherent and number states are studied. The ladder operator formulation of the hypergeometric states is found and the algebra involved turns out to be a one-parameter deformation of $su(2)$ algebra. These states exhibit highly nonclassical properties, like sub-Poissonian character, antibunching and squeezing effects. The quasiprobability distributions in phase space, namely the $Q$ and the Wigner functions are studied in detail. These remarkable properties seem to suggest that the hypergeometric states deserve further attention from theoretical and applicational sides of quantum optics.

Abstract:
We have systematically studied the hard-core Bose-Hubbard model with correlated hopping on a triangular lattice using density-matrix renormalization group method. A rich ground state phase diagram is determined. In this phase diagram there is a supersolid phase and a pair superfluid phase due to the interplay between the ordinary frustrated boson hopping and an unusual correlated hopping. In particular, we find that the quantum phase transition between the supersolid phase and the pair superfluid phase is continuous.

Abstract:
We introduce new kinds of states of quantized radiation fields, which are the superpositions of negative binomial states. They exhibit remarkable non-classical properties and reduce to Schr\"odinger cat states in a certain limit. The algebras involved in the even and odd negative binomial states turn out to be generally deformed oscillator algebras. It is found that the even and odd negative binomial states satisfy a same eigenvalue equation with a same eigenvalue and they can be viewed as two-photon nonlinear coherent states. Two methods of generating such states are proposed.