Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited. 1 1
Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited. 1 1
C λ -extended oscillator algebras, generalizing the Calogero-Vasiliev algebra, where C λ is the cyclic group oforder λ , have recently proved very useful in the context of supersymmetric quantum mechanics and some of its variants. Here, we determine the spectrum generating algebra of the C λ -extended oscillator. We then construct its coherent states, study their nonclassical properties, and compare the latter with those of standard λ -photon coherent states, which are obtained as a special case. Finally, we briefly review some other types of coherent states associated with the C λ -extended oscillator.
The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
In the present series of papers, the coherent states of Sp(2d,R), corresponding to the positive discrete series irreducible representations 〈λd+n/2,..,λ1+n/2〉, encountered in physical applications, are analyzed in detail with special emphasis on those of Sp(4,R) and Sp(6,R). The present paper discusses the annihilation-operator coherent states, i.e., the eigenstates of the noncompact lowering generators corresponding to complex eigenvalues. These states generalize the coherent states introduced by Barut and Girardello for Sp(2,R), and later on extended by Deenen and Quesne to the Sp(2d,R) irreducible representations of the type 〈(λ+n/2)d〉. When λ1,...,λd are not all equal, it was shown by Deenen and Quesne that the eigenvalues do not completely specify the eigenstates of the noncompact lowering generators. In the present work, their characterization is completed by a set of continuous labels parametrizing the (unitary-operator) coherent states of the maximal compact subgroup U(d). The resulting coherent states are therefore of mixed type, being annihilation-operator coherent states only as regards the noncompact generators. A realization in a subspace of a Bargmann space of analytic functions shows that such coherent states satisfy a unity resolution relation in the representation space of 〈λd +n/2,...,λ1+n/2〉, and therefore may be used as a continuous basis in such space. The analytic functions and the differential operators representing the representation space discrete bases and the Sp(2d,R) generators, respectively, are found in explicit form. It is concluded that the annihilation-operator coherent state representation provides the mathematical foundation for the use of differentiation operators with respect to the noncompact raising generators in symbolic expressions of the Sp(2d,R) generators. This is to be compared with the habit of replacing a boson annihilation operator by a symbolic differentiation with respect to the corresponding creation operator, accounted for by the Bargmann representation of such operators.
Recently we have obtained, on the basis of a group approach to quantization, a Bargmann-Fock-like realization of the Relativistic Harmonic Oscillator as well as a generalized Bargmann transform relating fock wave functions and a set of relativistic Hermite polynomials. Nevertheless, the relativistic creation and annihilation operators satisfy typical relativistic commutation relations of the Lie product (vector-z, vector-z(sup dagger)) approximately equals Energy (an SL(2,R) algebra). Here we find higher-order polarization operators on the SL(2,R) group, providing canonical creation and annihilation operators satisfying the Lie product (vector-a, vector-a(sup dagger)) = identity vector 1, the eigenstates of which are 'true' coherent states.
Recently an f -deformed Fock space which is spanned by | n 〉 λ was introduced. These bases are the eigenstates of a deformed non-Hermitian Hamiltonian. In this contribution, we will use rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by | n 〉 λ bases, to a new one, spanned by extended two-parameters bases | n 〉 λ 1 , λ 2 . These bases are now the eigenstates of a non-Hermitian Hamiltonian H λ 1 ,λ 2 = a † λ 1 ,λ 2 a +½, where a † λ 1 , λ 2 and a are, respectively, the deformed creation and ordinary bosonic annihilation operators. The bases | n 〉 λ 1 , λ 2 are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states in our two-parameter Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states numerically.
The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su (1, 1) algebra (for κ > 0), the su (2) algebra (for κ < 0) and the h 4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k -fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k -fermionic) variables. Some of the results are applied to su (2), su (1, 1) and the harmonic oscillator (in a truncated or not truncated form). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.