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arXiv:2304.05461 · Physica Scripta

Supersymmetric quantum mechanics, multiphoton algebras and coherent states

Juan D. García-Muñoz, David J. Fernández C., F Vergara-Méndez

Published Sep 6, 2023• 2 citations
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Abstract

Abstract The multiphoton algebras for one-dimensional Hamiltonians with infinite discrete spectrum, and for their associated kth-order SUSY partners are studied. In both cases, such an algebra is generated by the multiphoton annihilation and creation operators, as well as by Hamiltonians which are functions of an appropriate number operator. The algebras obtained turn out to be polynomial deformations of the corresponding single-photon algebra previously studied in literature. The Barut-Girardello coherent states, which are eigenstates of the annihilation operator, are obtained and their uncertainty relations are explored by means of the associated quadratures.

DOI
10.1088/1402-4896/acf743

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Abstract The multiphoton algebras for one-dimensional Hamiltonians with infinite discrete spectrum, and for their associated kth-order SUSY partners are studied. In both cases, such an algebra is generated by the multiphoton annihilation and creation operators, as well as by Hamiltonians which are functions of an appropriate number operator. The algebras obtained turn out to be polynomial deformations of the corresponding single-photon algebra previously studied in literature. The Barut-Girardello coherent states, which are eigenstates of the annihilation operator, are obtained and their uncertainty relations are explored by means of the associated quadratures.
Abstract In this paper we will realize the polynomial Heisenberg algebras through the harmonic oscillator. We are going to construct then the Barut–Girardello coherent states, which coincide with the so-called multiphoton coherent states, and we will analyze the corresponding Heisenberg uncertainty relation and Wigner distribution function for some particular cases. We will show that these states are intrinsically quantum and cyclic, with a period being a fraction of the oscillator period. The associated geometric phases will be as well evaluated.
We investigate some aspects of q Heisenberg algebra. We show how su(2) and su(1,1) generators can be constructed in terms of the q creation and annihilation operators. We also construct the coherent states for the q oscillator and show that they can be obtained by the action of a displacement operator on the vacuum. For the multimode case with q=0, corresponding to the infinite statistics of Greenberg [Phys. Rev. Lett. 64, 705 (1990)], we generalize our single-mode construction to obtain the corresponding coherent states. These states, which are eigenstates of the annihilation operators, interestingly, turn out to be degenerate owing to the noncommutativity of the displacement operators.
We present a general unified approach for finding the coherent states (CSs) 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 noncompact cases, to obtain the annihilation operator eigenstates, by finding the canonical conjugates of these operators. Generalized CSs, in the Perelomov sense, also follow from this construction. This allows us to explicitly construct CSs associated with various quantum optical systems.
We review the higher‐order supersymmetric quantum mechanics (H‐SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first‐order SUSY transformations are used to derive in a simple way the higher‐order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H‐SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.
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 coherent states, 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.