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arXiv:2410.19118 · Advanced Quantum Technologies

Reproducing the Effects of Quantum Deformation in the Undeformed Jaynes‐Cummings Model

Thiago Takaji Tsutsui, Antonio S. M. de Castro, Fabiano M. Andrade

Published Apr 29, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract In the Jaynes‐Cummings (JC) model, the time dependence in the coupling parameter allows changes in the forms of the Rabi oscillations. In the inverse problem approach (IPA), the time‐dependent coupling parameter is obtained from the resulting population inversion. In this work, the IPA is employed to obtain a time‐dependent coupling that reproduces the effects of ‐deformation in the population inversion of an undeformed JC. This is relevant because it may pave the way for simulating quantum deformation in the JC model and possibly enables an experimental verification in a setting where the coupling can be precisely controlled.

DOI
10.1002/qute.202400559

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Abstract In the Jaynes‐Cummings (JC) model, the time dependence in the coupling parameter allows changes in the forms of the Rabi oscillations. In the inverse problem approach (IPA), the time‐dependent coupling parameter is obtained from the resulting population inversion. In this work, the IPA is employed to obtain a time‐dependent coupling that reproduces the effects of ‐deformation in the population inversion of an undeformed JC. This is relevant because it may pave the way for simulating quantum deformation in the JC model and possibly enables an experimental verification in a setting where the coupling can be precisely controlled.
Jaynes–Cummings (JC) models are by far the most widely used to study the dynamics of two-level systems, such as atoms or molecules under suitable conditions, coupled to quantized electromagnetic fields. Nowadays, analytical solutions of JC models are more useful than ever for understanding and controlling radiation-matter interactions relevant to quantum information processing, quantum computing, and other quantum technologies. This study presents the analysis and generalization of a method for the solution of the dynamical problem in the time-independent JC model (Carbonaro et al., 1979) that enables an exact description of the atom-field dynamics in the resonant time-dependent JC model and in a nontrivial class of off-resonance time-dependent JC models. Our analysis produces the ready-to-use forms of the evolution operator in the resonant and non-resonant cases, which are then used to derive the evolution of the atomic population inversion for some types of atom-field coupling. The central idea behind the proposed approach offers opportunities for its extension to other time-dependent cases and its generalization to JC-related models.
It is shown that various variants of the deformed Jaynes-Cummings model (JCM) correspond to the JCM with an intensity-dependent coupling characterized by two additional phenomenological parameters (p,q). The standard JCM is obtained for p=q=1. The quantum collapse and revival effects and the squeezing properties of a particular variant of the (p,q)-deformed Jaynes-Cummings model are studied numerically. The model is based on the q-deformed oscillator algebra ${\mathit{AA}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$-${\mathit{qA}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$A=1 that interpolates between Fermi-Dirac and Bose-Einstein statistics. If the cavity field is prepared initially in a q-deformed coherent state, the quantum collapse and revival effects are observed only for q\ensuremath{\approxeq}1. For q>1, the atomic inversion 〈${\mathrm{\ensuremath{\sigma}}}_{3}$(t)〉 exhibits chaoticlike behavior, which is a feature observed also in other q-deformed JCM's. Strong squeezing is observed only for small positive and small negative q values. If q\ensuremath{\approxeq}1, the squeezing is very weak. In the limit q=1\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon} with \ensuremath{\varepsilon}\ensuremath{\ll}1, the algebra can be interpreted as describing a small violation of Bose-Einstein statistics in the JCM.
An extension of the standard one-photon as well as the two-photon Jaynes-Cummings (JC) model has been done to include transient effects due to a modulation of the atom-field coupling coefficient. An explicit time dependence is given for the case of a linear sweep of the coupling parameter. We have shown the effects of such a sweep on the dynamical evolution of a two-level atom undergoing a one- or two-photon process in a single nondecaying mode of the field in a coherent state. The statistical aspects of the field, such as intensity-intensity correlation and squeezing, are also investigated and are found to be both qualitatively as well as quantitatively different from the standard JC models.