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arXiv:2101.02462

Photon-added BarutGirardello like coherent states of time-dependent\n Landau problem

Latévi M. Lawson, Komi Sodoga, Gabriel Y. H. Avossevou

Published Jan 7, 2021• 1 citation
arXivHTMLPDF

Abstract

Recently, we have determined the spectrum and the wave functions of the Hamiltonian of a Landau particle with time-dependent mass and frequency undergoing the influence of a uniform time-dependent electric field[J. Math. Phys. 56, 072104 (2018)]. In the present paper we extend the study of this model that we name the time-dependent Landau problem into the context of coherent states. By means of the traditional factorization method of the eigenfunctions of this system expressed in terms of the generalized Laguerre polynomials, we derive the generators of the su(1,1) Lie algebra and we construct the coherent states a la Barut-Girardello. These states are shown to satisfy the Klauder's mathematical requirement to build coherent states and some of their statistical properties are calculated and analyzed. We find that these states are sub-Poissonian in nature. We show that, addition of photons from these coherent states, increases the statistical properties and changes the mathematical properties of these states.

DOI
10.48550/arxiv.2101.02462

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Recently, we have determined the spectrum and the wave functions of the Hamiltonian of a Landau particle with time-dependent mass and frequency undergoing the influence of a uniform time-dependent electric field[J. Math. Phys. 56, 072104 (2018)]. In the present paper we extend the study of this model that we name the time-dependent Landau problem into the context of coherent states. By means of the traditional factorization method of the eigenfunctions of this system expressed in terms of the generalized Laguerre polynomials, we derive the generators of the su(1,1) Lie algebra and we construct the coherent states a la Barut-Girardello. These states are shown to satisfy the Klauder's mathematical requirement to build coherent states and some of their statistical properties are calculated and analyzed. We find that these states are sub-Poissonian in nature. We show that, addition of photons from these coherent states, increases the statistical properties and changes the mathematical properties of these states.
The ordinary Landau problem consists of describing a charged particle in time-independent magnetic field. In the present case the problem is generalized onto time-dependent uniform electric fields with time-dependent mass and harmonic frequency [1].The spectrum of a Hamiltonian describing this system is obtained. The configuration space wave functions of the model is expressed in terms of the generalised Laguerre polynomials. To diagonalize the time-dependent Hamiltonian we employ the Lewis-Riesenfeld method of invariants. To this end, we introduce an unitary transformation in the framework of the algebraic formalism to construct the invariant operator of the system and then to obtain the exact solution of the Hamiltonian. We recover the solutions of the ordinary Landau problem in the absence of the electric and harmonic fields, for a constant particle mass. The quantization of this system exhibits many symmetries such as U(1),SU(2),SU(1,1)U(1), SU(2), SU(1,1)U(1),SU(2),SU(1,1). We therefore construct the coresponding coherent states and the associated photon added and nonlinear coherent states.
The spectrum of a Hamiltonian describing the dynamics of a Landau particle with time-dependent mass and frequency undergoing the influence of a uniform time-dependent electric field is obtained. The configuration space wave function of the model is expressed in terms of the generalised Laguerre polynomials. To diagonalize the time-dependent Hamiltonian, we employ the Lewis-Riesenfeld method of invariants. To this end, we introduce a unitary transformation in the framework of the algebraic formalism to construct the invariant operator of the system and then to obtain the exact solution of the Hamiltonian. We recover the solutions of the ordinary Landau problem in the absence of the electric and harmonic fields for a constant particle mass.
We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting three different methods which are due to (i)(i)(i) Barut-Girardello, (ii)(ii)(ii) Gazeau-Klauder and (iii)(iii)(iii) Perelomov associated to Lie algebra su(1,1)su(1,1)su(1,1). Resolution of unity and overlapping properties are examined in each method.
Photon-added BarutndashndashndashGirardello coherent states of the pseudoharmonic oscillator
In a previous paper (Popov D 2001 J. Phys. A: Math. Gen. 34 5283) we have constructed the coherent states (CSs) of the pseudoharmonic oscillator (PHO). These states are of the Barut–Girardello type. In the present paper, we have constructed and investigated some properties of the photon-added Barut–Girardello coherent states (PA-BGCSs) of the PHO. These states are not orthogonal but are normalized and satisfy a unity resolution relation (the completeness relation). We have found the analytical form for the positive weight function in the resolution of unity. By using these states, we have calculated some expectation values (the first two powers of the number operator N ), which have evinced some non-classical properties. In order to examine the statistical properties of a PHO gas, which obeys the quantum canonical distribution, the diagonal P -representation of the density operator ρ ( m ) is constructed and the thermal expectation values are calculated. All these quantities are expressed in terms of Meijer G -functions, and so the PA-BGCSs are a new field of application for these functions. Also, the time dependence of the PA-BGCSs is examined. If in the obtained results we put m = 0, we recover all the results from our previous paper.
We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting the Gazeau–Klauder approach, where resolution of unity and overlapping properties are examined. Coherent states are found to be similar to those found for a particle trapped in a Pöschl–Teller potential of the trigonometric type. Some comparisons with Barut–Girardello and Klauder–Perelomov methods are noted.