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arXiv:2303.04883

A Comment on "Algebraic approach to the Tavis-Cummings model with three modes of oscillation" [J. Math. Phys. 59, 073506 (2018)]

Viani S. Morales-Guzman, Jorge G. Hirsch

Published Mar 8, 2023• 0 citations
arXivHTMLPDF

Abstract

Choreño et al. [J. Math. Phys. 59, 073506 (2018)] reported analytic solutions to the resonant case of the Tavis-Cummings model, obtained by mapping it to a Hamiltonian with three bosons and applying a Bogoliubov transformation. This comment points out that the Bogoliubov transformation employed is not unitary, cannot be inverted, and cannot enforce the symmetries of the model.

DOI
10.48550/arxiv.2303.04883

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Choreño et al. [J. Math. Phys. 59, 073506 (2018)] reported analytic solutions to the resonant case of the Tavis-Cummings model, obtained by mapping it to a Hamiltonian with three bosons and applying a Bogoliubov transformation. This comment points out that the Bogoliubov transformation employed is not unitary, cannot be inverted, and cannot enforce the symmetries of the model.
We study the Tavis-Cummings model with three modes of oscillation by using four different algebraic methods: the Bogoliubov transformation, the normal-mode operators, and the tilting transformation of the SU(1, 1) and SU(2) groups. The algebraic method based on the Bogoliubov transformation and the normal-mode operators lets us obtain the energy spectrum and eigenfunctions of a particular case of the Tavis-Cummings model, while with the tilting transformation we are able to solve the most general case of this Hamiltonian. Finally, we compute some expectation values of this problem by means of the SU(1, 1) and SU(2) group theory. The authors of the article agree to the retraction of the article effective 21 September 2023
We discuss a recently proposed method [R. J\'auregui and J. R\'ecamier, Phys. Rev. A 46, 2240 (1992)] based on the application of iterative Bogoliubov transformations to anharmonic oscillators and show that, if the algorithm converges, one easily obtains the final result directly in one step. We prove that the Bogoliubov transformation can be written in terms of scaling and translation parameters and present exact results for the coefficients of the Bogoliubov transformation for some selected examples.
The Jaynes-Cummings (JC) Hamiltonian is a paradigmatic model of light-matter interaction and, more generally, qubit-boson interactions, widely used across atomic, optical, and superconducting qubit platforms. In the multi-qubit setting, where n qubits are identically coupled to a single boson mode, this interaction is known as the Tavis-Cummings (TC) Hamiltonian. The structure of the TC model is usually understood in terms of two standard symmetries: permutation invariance of the qubits and a U(1) symmetry associated with conservation of the total excitation number. Here we identify an additional, independent "accidental" symmetry of the TC Hamiltonian and construct the corresponding conserved observable. We show that, for n>2 qubits, this symmetry imposes strong constraints on the realizable unitary transformations. These constraints persist in the presence of the global JzJ_zJz​ Hamiltonian, but are removed by adding Jz2J_z^2Jz2​, even though Jz2J_z^2Jz2​ preserves both permutation invariance and the U(1) symmetry. Finally, we explain the origin of this previously unnoticed symmetry using Schwinger's boson representation of angular momentum. These restrictions have important implications for controllability of the TC system and for its applications to quantum computing, which are investigated further in a companion paper.
Abstract Using the quantum group technique we construct a one-parametric family of integrable modifications of the n -level, n − 1 mode Tavis–Cummings Hamiltonian possessing an additional Stark-type term. We show that in the ‘quasiclassical’ limit the constructed Hamiltonian transforms into the integrable Hamiltonian of the quantum n -level, n − 1 mode Tavis–Cummings model with the equal interaction strengths considered in Skrypnyk (2008 J. Phys. A: Math. Theor . 41 475202, 2009 J. Math. Phys . 50 103523). We diagonalize the constructed ‘modified’ Tavis–Cummings Hamiltonian and its second order integrals of motion using the nested Bethe ansatz.