Smorodinsky-Winternitz potentials: Coherent state approach

In this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional harmonic oscillator problem and the Kepler-Coulomb problem. In the first case we find the nonspreading wave packets by transforming the system into four oscillators in Cartesian, and also polar, coordinates. In the second case, the coherent states are constructed in Cartesian coordinates by transforming the system into three nonisotropic harmonic oscillators. All of these states evolve in physical-time. In the third case, the system is transformed into four oscillators and the parametric-time coherent states are constructed in two coordinate frames. In the fourth case, the system is transformed into two oscillators with the reflection symmetry and the parametrictime coherent states are constructed in two coordinate frames.     

D-dimensional Smorodinsky-Winternitz potential: Coherent state approach

In this study, we construct the coherent states for a particle in the D-dimensional maximally superintegrable Smorodinsky-Winternitz potential. We, first, map the system into 2D harmonic oscillators, second, construct the coherent states of them by evaluating the transition amplitudes. Third, in the Cartesian and the hyperspherical coordinates, we find the coherent states and the stationary states of the original sytem by reduction.     

Many fields interaction: Beam splitters and waveguide arrays

We study the interaction of many fields. We obtain an effective Hamiltonian for this system by using a method recently introduced that produces a small rotation to the Hamiltonian that allows to neglect some terms in the rotated Hamiltonian. We show that coherent states remain coherent under the action of a quadratic Hamiltonian and by solving the eigenvalue and eigenvector problem for tridiagonal matrices we also show that a system of n interacting harmonic oscillators, initially in coherent states, remain coherent during the interaction.     

Coherent states with elliptical polarization

Coherent states of the two-dimensional harmonic oscillator are constructed as superpositions of energy and angular momentum eigenstates. It is shown that these states are Gaussian wave-packets moving along a classical trajectory, with a well-defined elliptical polarization. They are coherent correlated states with respect to the usual Cartesian position and momentum operators. A set of creation and annihilation operators is defined in polar coordinates, and it is shown that these same states are precisely coherent states with respect to such operators.     

Pseudoharmonic oscillator and their associated Gazeau–Klauder coherent states

In the paper we have constructed and examined the properties of the Gazeau–Klauder coherent states (GK-CSs) for the pseudoharmonic oscillator (PHO), one of three possible kinds in order to define the coherent states for this oscillator potential. In the second part, we have examined some nonclassical properties of these states. Our attention has been concentrated on the mixed states (thermal states). The diagonal P-representation of the corresponding density operator and some thermal expectations for the quantum canonical ideal gas of pseudoharmonic oscillators have also been examined. Like the CSs for the harmonic oscillator (HO), the GK-CSs for the PHO can be useful in the quantum information theory (QIT).     

Spin Number Coherent States and the Problem of Two Coupled Oscillators

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.     

Spin Number Coherent States and the Problem of Two Coupled Oscillators

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.     

Overlaps of deformed and non-deformed harmonic oscillator basis states

A systematic approach for expanding non-deformed harmonic oscillator basis states in terms of deformed ones, and vice versa, is presented. The objective is to provide analytical results for calculating these overlaps (transformation brackets) between deformed and non-deformed basis states in spherical, cylindrical, and Cartesian coordinates. These overlaps can be used for reducing the complexity of different research problems that employ three-dimensional harmonic oscillator basis states, for example as used in coherent state theory and the nuclear shell-model, especially within the context of ab initio symmetry-adapted no-core shell model.     

Classical Interpretation of a Deformed Quantum Oscillator

Following the same procedure that allowed Shcrödinger to construct the (canonical) coherent states in the first place, we investigate on a possible classical interpretation of the deformed harmonic oscillator. We find that, these oscillator, also called q-oscillators, can be interpreted as quantum versions of classical forced oscillators with a modified q-dependant frequency.     

Factorizing coupled quantum systems and damped nonlinear Schrödinger equations

We show that the wave function of a coupled quantum system may factorize for certain coupling operators, resulting in wave functions and effective nonlinear Hamiltonians for the subsystems. Systems of coupled harmonic oscillators with discrete or continuous spectra are considered, where all degrees of freedom move in time-dependent coherent Glauber states.We present the general formalism and study two examples in detail. The problem of radiation damping results under drastic assumptions in exponentially damped harmonic motion, obeying a nonlinear Schrödinger equation. In the second example, a different type of coupling is studied which yields inverse power law damping.     

On the exact solution of the harmonic quadrupole collective hamiltonian

Explicit expressions for the eigenfunctions of the harmonic quadrupole collective Hamiltonian both in the lab and intrinsic systems of references are given. Two alternative approaches, the technique of projective coherent states and the theory of harmonic polynomials in collective coordinates, are used. Symmetry properties and recursive formulae for the internally labelled wave functions are established. Applications to the yrast states as well as to the low angular momentum states in the case of the asymmetric rotor Hamiltonian are also presented.     

LANDAU LEVELS IN A TWO-DIMENSIONAL NONCOMMUTATIVE SPACE: MATRIX AND QUATERNIONIC VECTOR COHERENT STATES

The behavior of an electron in an external uniform electromagnetic background coupled to an harmonic potential, with noncommuting space coordinates, is considered in this work. The thermodynamics of the system is studied. Matrix vector coherent states (MVCS) as well as quaternionic vector coherent states (QVCS), satisfying required properties, are also constructed and discussed.     

Quasi-exact solvability of Dirac-Pauli equation and generalized Dirac oscillators

In this paper we demonstrate that neutral Dirac particles in external electric fields, which are equivalent to generalized Dirac oscillators, are physical examples of quasi-exactly solvable systems. Electric field configurations permitting quasi-exact solvability of the system based on the sl(2) symmetry are discussed separately in the spherical, cylindrical, and Cartesian coordinates. Some exactly solvable field configurations are also exhibited.     

Modelling free energy of a rigid protein in solid water: Comparison between rigid-body motions and harmonic oscillators

We test two methods to estimate a partition function of a system consisting of multi-atom molecules with intermolecular interaction. Our test case is a protein in water. The first method is based on rigid-body motions. The space in which the protein (bovine pancreatic trypsin inhibitor) would be moving is limited, so by making a correction to a partition function of a rigid body, we can obtain the function. The function depends on temperature and space. The second method is based on harmonic oscillators. Under the potential field produced by surrounding water molecules, a protein behaves like a set of harmonic oscillators. We obtain the partition function for the oscillators within a harmonic approximation. The function also depends on temperature and the strength of potential energy between the protein and waters. Comparison of the two methods indicates that the second method is better for estimating a partition function for a protein in water.     

Time-dependent harmonic oscillator in a magnetic field and an electric field on the non-commutative plane

In the non-commutative space, wave functions and geometric phases are derived for the time-dependent harmonic oscillator in external time-dependent magnetic and electric field. Explicit forms of the coherent states are also given, which are not the minimum uncertainty states for the coordinates and momenta.     

Radial coherent states—From the harmonic oscillator to the hydrogen atom

We construct spectrum generating algebras of SO(2, 1) ～ SU(1, 1) in arbitrary dimension for the isotropic harmonic oscillator and the Sturm-Coulomb problem in radial coordinates. Using these algebras, we construct the associated radial Barut-Girardello coherent states for the isotropic harmonic oscillator (in arbitrary dimension). We map these states into the Sturm-Coulomb radial coherent states and show that they evolve in a fictitious time parameter without dispersing.     

Coherent States of Position-Dependent Mass Oscillator

In this paper, we study Gazeau-Klauder and displacement-type coherent states of two-dimensional position-dependent mass oscillators, which is called Λ-dependent oscillators and Λ can be interpreted as the curvatures of the spherical and the hyperbolic spaces, on which oscillators are constrained. In addition, we consider the effect of Λ parameter on the physical properties of these coherent states, including minimized Heisenberg uncertainty relation and Mandel’s Q parameter. We also elaborate the relation between the curvature of the physical space and the curvature of the Λ-dependent coherent state manifold.     

Two-mode Gaussian states as resource of secure quantum teleportation in open systems

For secure quantum teleportation (SQT) of coherent states two conditions are necessary to be fulfilled: Gaussian-state resources with two-way steering and teleportation fidelity higher than 2/3. We investigate and compare squeezed thermal states and squeezed vacuum states as initial resource states for SQT in an open quantum system, consisting of two uncoupled harmonic oscillators interacting with a thermal environment. The evolution of the open system is obtained in terms of the covariance matrix, by using the Gorini-Kossakowski-Lindblad-Sudarshan master equation. The SQT conditions are satisfied in a longer period of time in the case of initial squeezed vacuum states, therefore these states are better resource states for SQT than squeezed thermal states. We show that the admissible time for SQT decreases by increasing temperature, dissipation coefficient and average number of thermal photons, while for greater values of the squeezing parameter, SQT conditions are satisfied in a longer period of time.     

Interbasis expansions for isotropic harmonic oscillator

The exact solutions of the isotropic harmonic oscillator are reviewed in Cartesian, cylindrical polar and spherical coordinates. The problem of interbasis expansions of the eigenfunctions is solved completely. The explicit expansion coefficients of the basis for given coordinates in terms of other two coordinates are presented for lower excited states. Such a property is occurred only for those degenerated states for given principal quantum number n.     

Coherent States and Modified de Broglie-Bohm Complex Quantum Trajectories

This paper examines the nature of classical correspondence in the case of coherent states at the level of quantum trajectories. We first show that for a harmonic oscillator, the coherent state complex quantum trajectories and the complex classical trajectories are identical to each other. This congruence in the complex plane, not restricted to high quantum numbers alone, illustrates that the harmonic oscillator in a coherent state executes classical motion. The quantum trajectories we consider are those conceived in a modified de Broglie-Bohm scheme. Though quantum trajectory representations are widely discussed in recent years, identical classical and quantum trajectories for coherent states are obtained only in the present approach. We may note that this result for standard harmonic oscillator coherent states is not totally unexpected because of their holomorphic nature. The study is extended to coherent states of a particle in an infinite potential well and that in a symmetric Poschl-Teller potential by solving for the trajectories numerically. For the Gazeau-Klauder coherent state of the infinite potential well, almost identical classical and quantum trajectories are obtained whereas for the Poschl-Teller potential, though classical trajectories are not regained, a periodic motion results as t→∞. Similar features were found for the SUSY quantum mechanics-based coherent states of the Poschl-Teller potential too, but this time the pattern of complex trajectories is quite different from that of the previous case. Thus we find that the method is a potential tool in analyzing the properties of generalized coherent states.