Algebraic Coherent States and Squeezing

2012

It is the aim of this paper to show how to constructà la Perelomov andà la Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction ofà la Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.

Journal of Mathematical Physics, 2002

States which minimize the Schrödinger-Robertson uncertainty relation are constructed as eigenstates of an operator which is a element of the h(1) ⊕ su(2) algebra. The relations with supercoherent and supersqueezed states of the supersymmetric harmonic oscillator are given. Moreover, we are able to compute gneneral Hamiltonians which behave like the harmonic oscillator Hamiltonian or are related to the Jaynes-Cummings Hamiltonian.

1999

We present a general unified approach for finding the coherent states of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the non-compact cases, to obtain the annihilation operator eigenstates, by finding the canonical conjugates of these operators. Generalized coherent states, in the Perelomov sense, also follow from this construction. This allows us to explicitly construct coherent states associated with various quantum optical systems. *

Journal of Nonlinear Mathematical Physics, 2021

We study some properties of the SU (1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K 0 of the su(1, 1) Lie algebra. Analogous results for the SU (2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.

Journal of Physics A: Mathematical and General, 1996

We consider two analytic representations of the SU (1, 1) Lie group: the representation in the unit disc based on the SU (1, 1) Perelomov coherent states and the Barut-Girardello representation based on the eigenstates of the SU (1, 1) lowering generator. We show that these representations are related through a Laplace transform. A 'weak' resolution of the identity in terms of the Perelomov SU (1, 1) coherent states is presented which is valid even when the Bargmann index k is smaller than 1 2. Various applications of these results in the context of the two-photon realization of SU (1, 1) in quantum optics are also discussed.

Journal of Mathematical Physics, 2021

Susskind-Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coherent state quantization map. In this regard, the modified Susskind-Glogower coherent states have been introduced as an alternative family of states that resolve the identity resolution. In the present manuscript, the quantization map related to the modified Susskind-Glogower coherent states is exploited, which naturally leads to a particular representation of the su(1, 1) Lie algebra in its discrete series. The latter provides evidence about further generalizations of coherent states, built from the Susskind-Glogower ones by extending the indexes of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind. In this form, the new families of Susskind-Glogower-I and Susskind-Glogower-II coherent states are introduced. The corresponding quantization maps are constructed so that they lead to general representations of elements of the su(1, 1) and su(2) Lie algebras as generators of the SU(1, 1) and SU(2) unitary irreducible representations respectively. For completeness, the optical properties related to the new families of coherent states are explored and compared with respect to some well-known optical states. Contents 1. Introduction 2. Modified Susskind-Glogower coherent states 3. Quantization map related to the modified Susskind-Glogower coherent states 4. Susskind-Glogower-I coherent states 5. Quantization map with SGI CS 6. One-photon SU(1, 1) coherent states 7. Susskind-Glogower-II coherent states 8. Quantization map with SGII CS 9. Boson realization and contraction of algebras 10. Photon statistics and nonclassical properties of the SGI and SGII coherent states 11. Conclusions Appendix A. Normalization constant N κ (r)

Physical Review A, 2005

We construct a general state which is an eigenvector of the annihilation operator of the generalized Heisenberg algebra. We show, for several systems characterized by different energy spectra, that this general state satisfies the minimal set of conditions required to obtain Klauder's minimal coherent states.

2012

In a previous paper [{\it J. Phys. A: Math. Theor.} {\bf 40} (2007) 11105], we constructed a class of coherent states for a polynomially deformed $su(2)$ algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed $su(1,1)$ algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su(1,1) algebra which contains the Barut-Girardello set and the Perelomov set of the SU(1,1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.

Journal of Physics A: Mathematical and Theoretical, 2009

In a previous paper [J. Phys. A: Math. Theor. 40 (2007) 11105], we constructed a class of coherent states for a polynomially deformed su(2) algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed su(1, 1) algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su(1, 1) algebra which contains the Barut-Girardello set and the Perelomov set of the SU (1, 1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.

International Journal of Theoretical Physics

We generalise the notion of coherent states to arbitrary Lie algebras by making an analogy with the GNS construction in $C^*$-algebras. The method is illustrated with examples of semisimple and non-semisimple finite dimensional Lie algebras as well as loop and Kac-Moody algebras. A deformed addition on the parameter space is also introduced simplifying some expressions and some applications to conformal field theory is pointed out, e.g. are differential operator and free field realisations found. PACS: 02.20.S, 03.65.F, 11.25.H Keywords: coherent states, Lie and Kac-Moody algebras, realisations.

Canadian Journal of Physics, 2004

By introducing a generalization of the (p, q)-deformed boson oscillator algebra, we establish a two-parameter deformed oscillator algebra in an infinite-dimensional subspace of the Hilbert space of a harmonic oscillator without first finite Fock states. We construct the associated coherent states, which can be interpreted as photon-added deformed states. In addition to the mathematical characteristics, the quantum statistical properties of these states are discussed in detail analytically and numerically in the context of conventional as well as deformed quantum optics. Particularly, we find that for conventional (nondeformed) photons the states may be quadrature squeezed in both cases Q = pq 1 and their photon number statistics exhibits a transition from sub-Poissonian to super-Poissonian for Q 1 they are always sub-Poissonian. On the other hand, for deformed photons, the states are sub-Poissonian for Q > 1 and no quadrature squeezing occurs while for Q

1999

We provide a unified approach for finding the coherent states of various deformed algebras, including quadratic, Higgs and q-deformed algebras, which are relevant for many physical problems. For the non-compact cases, coherent states, which are the eigenstates of the respective annihilation operators, are constructed by finding the canonical conjugates of these operators. We give a general procedure to map these

Communications in Physics, 2013

In recent years, one of the new applications of the coherent state method was to construct representation of superalgebras and quantum superalgebras. Following this stream, we had a contribution to working out explicit representation of Uq[gl(2|1)]. Up to now, Uq[gl(2|1) is still the biggest quantum superalgebra representations in coherent state basis of which can be built. In this article, we will show some detailed techniques used in our previous work but useful for our further investigations. The newest results on building representations in a coherent state basis of Uq[osp(2|2)], which has the same rank as Uq[gl(2|1)], are also briefly exposed.

Journal of Mathematical Physics, 2002

We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2 N -1 -1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labelled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.

Symmetries and Groups in Contemporary Physics, 2013

A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is presented. A natural approach to building coherent states associated to deformed algebras is then deduced.