Construction Of Coherent States Research Articles

Perturbative construction of coherent states

Perturbative construction of coherent states

Ladder operators with no vacuum, their coherent states, and an application to graphene

In literature ladder operators of different nature exist. The most famous are those obeying canonical (anti-) commutation relations, but they are not the only ones. In our knowledge, all ladder operators have a common feature: the lowering operators annihilate a non zero vector, the vacuum. This is connected to the fact that operators of these kind are often used in factorizing some positive operators, or some operators which are bounded from below. This is the case, of course, of the harmonic oscillator, but not only. In this paper we discuss what happens when considering lowering operators with no vacua. In particular, after a general analysis of this situation, we propose a possible construction of coherent states, and we apply our construction to graphene.

The Dunkl oscillator in the momentum representation and coherent states

We discuss quantum mechanical systems with Dunkl derivatives by constructing the Dunkl-Heisenberg relation in the momentum representation by means of the reflection operator for momentum and we obtain the corresponding position quantum eigenfunction. We examine the one-dimensional Dunkl oscillator in the momentum space in terms of ν-deformed Hermite polynomials. We obtain the energy levels as well as the ground-state and excited wave functions in terms of the ν-deformed Hermite polynomials. We also describe some properties of the ν-deformed Hermite polynomials. We apply the method to the construction of coherent states.

De Sitter space as a BRST invariant coherent state of gravitons

The $S$-matrix formulation indicates that a consistent embedding of the de Sitter state in quantum gravity is possible exclusively as an excited quantum state constructed on top of a valid $S$-matrix vacuum such as the Minkowski vacuum. In the present paper we offer such a construction of the de Sitter state in the form of a coherent state of gravitons. Unlike previous realizations of this idea, we focus on BRST invariance as the guiding principle for physicality. In order to establish the universal rules of gauge consistency, we study the BRST-invariant construction of coherent states created by classical and quantum sources in QED and in linearized gravity. Introduction of $N$ copies of scalar matter coupled to gravity allows us to take a special double scaling limit, a so-called species limit, in which our construction of the de Sitter state becomes exact. In this limit, the irrelevant quantum gravitational effects vanish, whereas the collective phenomena, such as Gibbons-Hawking radiation, are calculable.

Multi-dimensional q-deformed bosonic Newton oscillators and the related q-calculus, q-coherent states, and Hermite q-polynomials

The multi-dimensional q-deformed bosonic Newton oscillator algebra with SU(d)-symmetry is considered. In this framework, we first introduce some new properties concerning the q-deformed calculus related to the algebra, and we then discuss possible consequences of applying these deformed oscillators in some quantum optical issues such as in the construction of coherent states and their effects on the photon statistics. Second, we investigate the role of q-deformation on both the energy levels and the wave functions of the bosonic Newton oscillators by constructing the q-deformed Hermite polynomials. The results obtained in this work might have some implications for studies on quantum information based technologies such as in photonic quantum computing.

Graph coherent states for loop quantum gravity

In this article we further investigate the construction of graph coherent states, first introduced in [1], in the context of loop quantum gravity. We specifically investigate the possibility of defining a family of graph coherent states adapted to the canonical loop quantum gravity Hamiltonian. After discussing various aspects of the general framework and the choice of operators, we use the Euclidean part of the Hamiltonian operator to propose a generator of the generalized canonical structure, necessary to define the coherent states. We then apply the construction procedure, leading to a new family of graph coherent states partially adapted to the gravity Hamiltonian in loop quantum gravity.

Generalized su(1,1) algebra and the construction of nonlinear coherent states for Pöschl-Teller potential

We introduce a generalization structure of the su(1,1) algebra which depends on a function of one generator of the algebra, f(H). Following the same ideas developed to the generalized Heisenberg algebra (GHA) and to the generalized su(2), we show that a symmetry is present in the sequence of eigenvalues of one generator of the algebra. Then, we construct the Barut-Girardello coherent states associated with the generalized su(1,1) algebra for a particle in a Pöschl-Teller potential. Furthermore, we compare the time evolution of the uncertainty relation of the constructed coherent states with that of GHA coherent states. The generalized su(1,1) coherent states are very localized.

Construction of coherent states for Morse potential: A su(2)-like approach

We propose a generalized su(2) algebra that perfectly describes the discrete energy part of the Morse potential. Then, we examine particular examples and the approach can be applied to any Morse oscillator and to practically any physical system whose spectrum is finite. Further, we construct the Klauder coherent state for Morse potential satisfying the resolution of identity with a positive measure, obtained through the solution of truncated Stieltjes moment problem. The time evolution of the uncertainty relation of the constructed coherent states is analyzed. The uncertainty relation is more localized for small values of radius of convergence.

Pseudo-fermionic coherent states with time-dependent metric

In this paper, we construct time-dependent pseudo-fermionic coherent states for non-Hermitian Hamiltonian systems. Our construction of pseudo-fermionic coherent states is based on an introduction of time-dependent pseudo-fermionic creation and annihilation operators subjected to time-dependent metrics such that the latter are integrals of motion. As an illustration, we study a time-dependent non-Hermitian two-level system.

Nonlinear coherent states of the para-Bose oscillator and their non-classical features

The construction of nonlinear coherent states via a unitary displacement operator is possible only for a few quantum-mechanical systems. In this paper, we define two non-unitary and a unitary displacement operators with the help of corresponding f -deformed bosonic annihilation and creation operators. While the action of the non-unitary displacement type operators on the vacuum state of field results in two new families of nonlinear coherent states (NLCSs), the unitary displacement operator reproduces Wigner-Heisenberg coherent states of the Gilmore-Perelomov type. We prove that the introduced NLCSs satisfy the resolution of the identities through positive definite measures. We also examine the non-classical properties of the obtained NLCSs by evaluating Klyshko’s criterion, Mandel’s parameter, quadrature squeezing and Wigner quasi-probability distribution function, in detail. Finally, we propose a simple scheme for the physical generation of the introduced NLCSs of the Gilmore-Perelomov type.

Coherent states of quantum linear rotator

A system of coherent states is constructed for quantum linear rotator — a model for rotation of a diatomic molecule. The rotator in such a state possesses definite averages of orientation and angular momentum. Construction of coherent states is done in the framework of generalized Jordan–Schwinger approach to quantum angular momentum. The final definition of the coherent states involves operators which span the Lie algebra of metaplectic group Mp(4,R).

A construction of quarter BPS coherent states and Brauer algebras

BPS coherent states closely resemble semiclassical states and they have gravity dual descriptions in terms of semiclassical geometries. The half BPS coherent states have been well studied, however less is known about quarter BPS coherent states. Here we provide a construction of quarter BPS coherent states. They are coherent states built with two matrix fields, generalizing the half BPS case. These states are both the eigenstates of annihilation operators and in the kernel of dilatation operator. Another useful labeling of quarter BPS states is by representations of Brauer algebras and their projection onto a subalgebra $\mathbb{C}[S_n\times S_m]$. Here, the Schur-Weyl duality for the Walled Brauer algebra plays an important role in organizing the operators. One interesting subclass of these Brauer states are labeled by representations involving two Young tableaux. We obtain the overlap between quarter BPS Brauer states and quarter BPS coherent states, where the Schur polynomials are used. We also derive superposition formulas transforming a truncated version of quarter BPS coherent states to quarter BPS Brauer states. The entanglement entropy of Brauer states as well as the overlap between Brauer states and squeezed states are also computed.

Generalized Coherent States for Position-Dependent Effective Mass Systems

A generalized scheme for the construction of coherent states in the context of position-dependent effective mass systems has been presented. This formalism is based on the ladder operators and associated algebra of the system which are obtained using the concepts of supersymmetric quantum mechanics and the property of shape invariance. In order to exemplify the general results and to analyze the properties of the coherent states, several examples have been considered.

Time-dependent pseudofermionic systems and coherent states

We show, by means of similarity transformations, that the time-dependent fermionic systems are associated to the time-dependent pseudofermionic systems. A general construction of time dependent fermionic coherent states (FCSs) describing the two-level dissipative system driven by a periodic electromagnetic field is developed, and a strict parallelism between FCS and the time dependent pseudofermionic coherent states (PFCSs) is established and examined. We discuss properties of the constructed FCS and PFCS.

Orthogonal polynomials, Laguerre Fock space, and quasi-classical asymptotics

Continuing our earlier investigation of the Hermite case [S. T. Ali and M. Engliš, J. Math. Phys. 55, 042102 (2014)], we study an unorthodox variant of the Berezin-Toeplitz quantization scheme associated with Laguerre polynomials. In particular, we describe a “Laguerre analogue” of the classical Fock (Segal-Bargmann) space and the relevant semi-classical asymptotics of its Toeplitz operators; the former actually turns out to coincide with the Hilbert space appearing in the construction of the well-known Barut-Girardello coherent states. Further extension to the case of Legendre polynomials is likewise discussed.

Semi-classical trace asymptotics for magnetic Schrödinger operators with Robin condition

We compute the sum and number of eigenvalues for a certain class of magnetic Schrödinger operators in a domain with boundary. Functions in the domain of the operator satisfy a (magnetic) Robin condition. The calculations are valid in the semi-classical asymptotic limit and the eigenvalues concerned correspond to eigenstates localized near the boundary of the domain. The formulas we derive display the influence of the boundary and the boundary condition and are valid under a weak regularity assumption of the boundary function. Our approach relies on three main points: reduction to the boundary, construction of boundary coherent states, and handling the boundary term as a surface electric potential and controlling the errors by various Lieb-Thirring inequalities.

Construction of photon-added spin coherent states and their statistical properties

In the present work, we construct and investigate some properties of the photon-added spin coherent states (PA-SCSs). The Klauder’s minimal set of conditions required to obtain coherent states are discussed. We give the analytical form for the positive weight function in the resolution of unity. Finally, we examine the statistical properties of the PA-SCSs in terms of different parameters using the Mandel’s Q-parameter. All these quantities are expressed in terms of hypergeometric and Meijer G-functions, and so, the PA-SCSs are a new field of application for these functions.

Supersymmetric infinite wells and coherent states

Gaussian Klauder coherent states are discussed in the context of the infinite well quantum model, otherwise known as the particle in a box. A supersymmetric partner system is also presented, as well as a construction of coherent states in this new system. We show that these states can be chosen, in both systems to have many properties usually expected for coherent states. In particular, they yield highly localised wave packets for a short period of time, which evolve in a quasi-classical manner and which saturate approximately Heisenberg uncertainty relation. These studies are elaborated in one- and two-dimensional contexts. Finally, some relations are established between the Gaussian states being mostly used here and the generalised coherent states, which are more standardly found in the literature.

Complexifier Method for Generation of Coherent States of Nonlinear Harmonic Oscillator

In this work we present a construction of coherent states based on ”complexifier” method for a special type of one dimensional nonlinear harmonic oscillator presented by Mathews and Lakshmanan (Q Appl Math 32:215, 1974). We will show the state quantization by using coherent states, or to build the Hilbert space according to a classical phase space, is equivalent to departure from real coordinates to complex ones.

Pöschl-Teller Hamiltonian: Gazeau-Klauder type coherent states, related statistics, and geometry

This work mainly addresses a construction of Gazeau-Klauder type coherent states (CSs) for a Pöschl-Teller model. Relevant characteristics are investigated. Induced geometry and statistics are studied. Then, the Berezin-Klauder-Toeplitz quantization of the classical phase space observables is presented.