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arXiv:2606.08361

A century of coherent states

Dusan Popov

quant-phSubmitted Jun 6, 2026• Announced Jun 9, 2026• 0 citations
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Abstract

During the century of existence of the notion of coherent states, either linear or nonlinear, several schemes for their construction, theoretical or experimental, have been developed. Generally, the mathematical structure of coherent states depends on the choice of ladder operators, and consequently on the structure constants. In this paper, we propose a way to construct generalized coherent states for anharmonic oscillators that is based on a diagonal operator ordering technique (DOOT) applied to generalized hypergeometric functions, that is, on some of the most general special functions. These states are generated by the action of a pair of ladder operators, the creation and the annihilation, whose ordered normal product is equal to the dimensionless Hamiltonian of the quantum system. In addition, the action of these operators is easy to find if the expression for the dimensionless energy eigenvalues is known.

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43 pages, No figures

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In the paper we developed a procedure for constructing generalized coherent states with shifted argument, as a result of the action of the generalized displacement operator. This was based on the action of a pair of nonlinear ladder operators, which generate nonlinear coherent states. To examine the properties of coherent states with shifted argument, the rules of the normal operator ordering technique (DOOT) were used. The results obtained were verified for a series of particular cases, obtaining expressions consistent with those in the literature. The expressions obtained will be, among others, useful in quantum optics, e.g. for calculating the Wigner operator in the representation of coherent states.
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The unknown ladder operators for the Wei Hua potential have been derived within the algebraic approach. The method is extended to include the rotating oscillator. The annihilation and creation operators have been obtained with the use of the factorization method. The coherent states for the Wei Hua anharmonic oscillator, which are eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed in the framework of the supersymmetric quantum mechanics. The constructed ladder operators can be applied to compute the matrix elements in theoretical spectroscopy due to the fact that this potential is widely used to a proper description the large-amplitude vibrations of diatomic molecules.
In this work we make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) As eigenstates of a deformed annihilation operator and ii) by application of a deformed displacement operator to the vacuum state. We also construct the coherent states for the same systems using the ladder operators obtained by traditional methods with the knowledge of the eigenfunctions and eigenvalues of the corresponding Schrödinger equation. We show that both methods yield coherent states with identical algebraic structure.
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