Perelomov Coherent States Research Articles

Spin Number Coherent States and the Problem of Two Coupled Oscillators* *Supported by SNI-México, COFAA-IPN, EDD-IPN, EDI-IPN, SIP-IPN Project No. 20150935

From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.

SU(1,1)coherent states for Dirac–Kepler–Coulomb problem inD+1dimensions with scalar and vector potentials

We decouple the Dirac's radial equations in $D+1$ dimensions with Coulomb-type scalar and vector potentials through appropriate transformations. We study each of these uncoupled second-order equations in an algebraic way by using an $su(1,1)$ algebra realization. Based on the theory of irreducible representations, we find the energy spectrum and the radial eigenfunctions. We construct the Perelomov coherent states for the Sturmian basis, which is the basis for the unitary irreducible representation of the $su(1,1)$ Lie algebra. The physical radial coherent states for our problem are obtained by applying the inverse original transformations to the Sturmian coherent states.

Semiclassical description of quantum rotator in terms of SU(2) coherent states

We introduce coordinates of the rigid body (rotator) using mutual positions between body-fixed and space-fixed reference frames. Wave functions that depend on such coordinates can be treated as scalar functions of the group SU(2). Irreducible representations of the group SU(2) × SU(2) in the space of such functions describe their possible transformations under independent rotations of the both reference frames. We construct sets of the corresponding group SU(2) × SU(2) Perelomov coherent states (CS) with a fixed angular momentum j of the rotator as special orbits of the latter group. Minimization of different uncertainty relations is discussed. The classical limit corresponds to the limit j → ∞. Considering Hamiltonians of rotators with different characteristics, we study the time evolution of the constructed CS. In some cases, the CS time evolution is completely or partially reduced to their parameter time evolution. If these parameters are chosen as Euler angles, then they obey the Euler equations in the classical limit. Quantum corrections to the motion of the quantum rotator can be found from exact equations on the CS parameters.

Dynamics of a three level atom interacting with a detuned SU (1,1) quantum system

Starting from the interaction between a 3-level atom in the cascade configuration with an SU(1, 1) quantum system, an effective Hamiltonian for a 2-level atom with the presence of a Stark shift structure is obtained under large detuning and adiabatic elimination. The evolution operator for the new system is obtained and expectation values for different dynamical observables are obtained taking the initial state of the SU(1, 1) system to be a Perelomov coherent state and a superposition state for the 2-level atom. Atomic inversion, atomic variable squeezing and degree of entanglement are investigated for various values of the involved parameters. The effect of the Stark shift parameters is displayed in these phenomena.

Barut—Girardello and Gilmore—Perelomov coherent states for pseudoharmonic oscillators and their nonclassical properties: Factorization method

In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrödinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU (1, 1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut—Girardello and Gilmore—Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.

SU(1,1) Lie algebraic approach for the evolution of the quantum inflationary universe

Abstract Quantum behavior of scalar fields and vacuum energy density in the inflationary universe are investigated using SU(1,1) Lie algebraic approach. Wave functions describing the evolution of scalar fields that have been thought to have driven cosmic inflation are identified in several possible quantum states at the early stage of the universe, such as the Fock state, the Glauber coherent state, and the SU(1,1) coherent states. In particular, we focus in this research on two important classes of the SU(1,1) coherent states, which are the so-called even and odd coherent states and the Perelomov coherent state. It is shown in the spatially flat universe driven by a single scalar field that the probability densities in all these states have converged to the origin (ϕ = 0, where ϕ is the scalar field) as time goes by. This outcome implies that the vacuum energy density characterized by the scalar field dissipates with time. The probability density in the matter-dominated era converged more rapidly than that in the radiation-dominated era. Hence, we can confirm that the progress of dissipation for the vacuum energy density became faster as the matter era began after the end of the early dominance of radiation. This consequence is, indeed, in agreement with the results of our previous researches in cosmology (for example, see [Chin. Phys. C 35 (2011) 233] and references there in).

SU(1,1) Coherent States for Position-Dependent Mass Singular Oscillators

The Schroedinger equation for position-dependent mass singular oscillators is solved by means of the factorization method and point transformations. These systems share their spectrum with the conventional singular oscillator. Ladder operators are constructed to close the su(1,1) Lie algebra and the involved point transformations are shown to preserve the structure of the Barut-Girardello and Perelomov coherent states.

Aspects of coherent states of nonlinear algebras

Various aspects of coherent states of nonlinear su(2) and su(1, 1) algebras are studied. It is shown that the nonlinear su(1, 1) Barut–Girardello and Perelomov coherent states are related by a Laplace transform. We then concentrate on the derivation and analysis of the statistical and geometrical properties of these states. The Berry's phase for the nonlinear coherent states is also derived.

Entanglement generation from deformed spin coherent states using a beam splitter

Using the linear entropy as a measure of entanglement, we investigate the effect of a beam splitter on the Perelomov coherent states for the q-deformed Uq(su(2)) algebra. We distinguish two cases: in the classical q → 1 limit, we find that the states become Glauber coherent states as the spin tends to infinity; whereas for q ≠ 1, the states, contrary to the earlier case, become entangled as they pass through a beam splitter. The entanglement strongly depends on the q-deformation parameter and the amplitude Z of the state.

Exact time dependence of solutions to the time-dependent Schrödinger equation

Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.

Comment on “Barut–Girardello and Klauder–Perelomov coherent states for the Kravchuk functions” [J. Math. Phys. 48, 112106 (2007)

We call attention to the misconstructions in a paper recently published in this journal [A. Chenaghlou and O. Faizy, J. Math. Phys. 48, 112106 (2007)]. It is shown that the constructed Barut–Girardello coherent states are problematic from the view points of the definition and the measure. The claimed coherencies for the Kravchuk functions cannot actually exist.

SU(1,1) Coherent States for the Generalized Two-Mode Time-Dependent Quadratic Hamiltonian System

The SU(1,1) coherent states, so-called Barut-Girardello coherent state and Perelomov coherent state, for the generalized two-mode time-dependent quadratic Hamiltonian system are investigated through SU(1,1) Lie algebraic formulation. Two-mode Schrodinger cat states defined as an eigenstate of \(\hat{K}_{-}^{2}\) are also studied. We applied our development to two-mode Caldirola-Kanai oscillator which is a typical example of the time-dependent quadratic Hamiltonian system. The time evolution of the quadrature distribution for the probability density in the coherent states are analyzed for the two-mode Caldirola-Kanai oscillator by plotting relevant figures.

Coherent States for Hopf Algebras

Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras possessing a Haar integral. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. A noncommutative resolution of identity formula is proved in that setup. Examples come from quantum groups.

SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System

Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola–Kanai oscillator. The probability density of these coherent states for the Caldirola–Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed.

Analytic Representations Based on SU(1,1) Lie Algebra Coherent States for Squeezed Displaced Fock States

The coherent states (CSs) of the SU(1,1) group can be divided into two broad categories: (a) the Barut-Girardello coherent states (BGCSs) and (b) the Perelomov coherent states (PCSs). Some definitions for the squeezed displaced Fock states (SDFSs) are given. The hyperbolic analytic representation in the complex plane is considered. An analytic representation of the SU(1,1) Lie group is given and the representation in the unit disk based on the SU(1,1) PCSs for SDFSs is considered.

SU(1,1) LIE ALGEBRA APPLIED TO THE TIME-DEPENDENT QUADRATIC HAMILTONIAN SYSTEM PERTURBED BY A SINGULARITY

We realized SU (1,1) Lie algebra in terms of the appropriate SU (1,1) generators for the time-dependent quadratic Hamiltonian system perturbed by a singularity. Exact quantum states of the system are investigated using SU (1,1) Lie algebra. Various expectation values in two kinds of the generalized SU (1,1) coherent states, that is, BG coherent states and Perelomov coherent states are derived. We applied our study to the CKOPS (Caldirola–Kanai oscillator perturbed by a singularity). Due to the damping constant γ, the probability density of the SU (1,1) coherent states for the CKOPS converged to the center with time. The time evolution of the probability density in SU (1,1) coherent states for the CKOPS are very similar to the classical trajectory.

Darboux transformations of coherent states of the time-dependent singular oscillator

Darboux transformation of both Barut-Girardello and Perelomov coherent states for the time-dependent singular oscillator is studied. In both cases the measure that realizes the resolution of the identity operator in terms of coherent states is found and corresponding holomorphic representation is constructed. For the particular case of a free particle moving with a fixed value of the angular momentum equal to two it is shown that Barut-Giriardello coherent states are more localized at the initial time moment while the Perelomov coherent states are more stable with respect to time evolution. It is also illustrated that Darboux transformation may keep unchanged this different time behavior.

Gazeau-Klauder and Klauder-Perelomov coherert states for the anharmonic oscillator

Using the Kinani-Daoud method we have constructed the Gazeau-Klauder(GK) and KlauderPerelomov(KP) coherent states for the anharmonic oscillator. It is shown that the two types of coherent states have totally different forms in their expressions. We have also discussed the resolution of unity of the two coherent states and the Hilbert space of each state. The study of Mandel Q pa rameter of the states shows that GK coherent state coincides with the sub-Poissonian and KP coherent state coincides with the super-Poissonian statistical distributions.

Squeezing evolution with non-dissipativeSU(1,1) systems

We investigate the squeezed regions in the phase plane for non-dissipative dynamical systems controlled by SU(1,1) Lie algebra. We analyze such study for the two SU(1,1) generalized coherent states, namely, the Perelomov coherent state (PCS) and the Barut-Girardello Coherent state (BGCS).

A realization of the dynamical group for the square-well potential and its coherent states

A realization of the ladder operators for the infinitely deep well potential is presented. Both the asymmetric and symmetric wells are discussed. It is shown that these operators satisfy the commutation relations for the SU(1, 1) group. Closed analytical expressions for the matrix elements of some relevant functions are obtained. Perelomov coherent states for this system are explicitly constructed. The behaviour of the dispersion relation and the time evolution for such states are examined.