Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2509.07215

Construction of the Jaynes-Cummings interaction over the finite two-dimensional oscillator

Alejandro R. Urzúa

Published Sep 8, 2025• 0 citations
arXivHTMLPDF

Abstract

The interaction between a two-level atom and the finite two-dimensional oscillator in the Cartesian coordinate system is addressed. The construction of the coupling between the degenerate energy states of the finite oscillator and the two levels of the atom allows the identification of an excitation-conserved eigenbasis. A dynamical approach to the evolution of mean values of the fields mediated by the atom is shown in the interaction picture. As an example, the eigenenergies states and $\mathfrak{su}(2)$ coherent states are studied. Perspectives and prospectives are given to settle a path in the study of atomic systems coupled to finite versions of the harmonic oscillator.

DOI
10.48550/arxiv.2509.07215

Related papers

Construction of the Jaynes-Cummings interaction over the finite two-dimensional oscillator

Alejandro R. Urzúa

The interaction between a two-level atom and the finite two-dimensional oscillator in the Cartesian coordinate system is addressed. The construction of the coupling between the degenerate energy states of the finite oscillator and the two levels of the atom allows the identification of an excitation-conserved eigenbasis. A dynamical approach to the evolution of mean values of the fields mediated by the atom is shown in the interaction picture. Furthermore, a quantitative characterization of the atom–field entanglement induced by the finite dynamics is presented. As an example, the eigenenergies states and [Formula: see text] coherent states are studied. Perspectives and prospects are presented to chart a path in the study of atomic systems coupled to finite versions of the harmonic oscillator.

International Journal of Modern Physics B•Mar 20, 2026•0 citations

Decoherence and entanglement in quantum physics

Jorge Medina Hernández

Throughout this project, the case of a composite system formed by a harmonic oscillator and a two-state atom will be studied; considering both affect each other as \na consequence of a Jaynes-Cummings interaction, and being both coupled to different reservoirs that are assumed to be independent from each other. The effects \nof the decoherence in the evolution of the populations and coherences will be analyzed; specifically the energy dissipation and the coherence loss, as well as the stationary states and the relevance of the Jaynes-Cummings interaction. Finally, the \nsolution of stochastic equations will be analyzed, particularly the evolution of the \nexpectation value of the position operator for the harmonic oscilator; for the case of \nsimultaneous measurements in both subsystems, as well as for a measurement in \nthe harmonic oscillator when the two-state atom undergoes a decoherence process.

Scientia Insularum Revista de Ciencias Naturales en islas•Jan 1, 2019•0 citations

Atom and quantum oscillator coupled by the vacuum field: Radiation pattern, emission spectrum, and decay dynamics

P. R. Berman, C. H. Raymond Ooi

We study the dynamics of a system consisting of a two-level atom and a quantum oscillator coupled by the vacuum radiation field. The oscillator and the atom are assumed to have the same resonance frequency. In contrast to the two-atom or two-oscillator case, the atom-oscillator system leads to an unclosed hierarchy of equations for the operators that determine the radiation pattern emitted by the atom-oscillator system. Instead of using the operator equations, we formulate the problem in terms of density-matrix equations which can be solved once the initial conditions are specified. As an example, we looked at the radiation pattern, the decay dynamics, and the spectrum for an initial condition in which the oscillator is in its $n=1$ state and the atom is in its excited state. Dressed states (dressed by the static interaction between the atom and oscillator) prove to be especially useful for interpreting the results and comparing them with the two-atom and two-oscillator systems. All calculations are carried out using the resonance or rotating-wave approximation. The relationship of our work to damped Jaynes-Cummings models is noted.

Physical Review A•Jan 4, 2016•2 citations

Excitation of coherent correlated states in the Jaynes-Cummings model by external deterministic forces: II

É. M. Verlan, Margarita A. Razumova

In terms of excitation creation and annihilation operators of the Jaynes-Cummings model, acting in the representation of dressed states, the Hamiltonian is written which describes the character of the spectrum of excitations of two modes, representing a quantum analog of the classical behavior of two interacting one-dimensional anharmonic oscillators, namely, the field and atomic oscillators. The anharmonicity is caused by the nonlinearity of the oscillator interaction and manifests itself in the dependence of the frequencies of both modes on the number of excitations, i.e., on the energy. It is shown that an external deterministic force, acting on the system during a certain time t 0, transfers it from a vacuum state to a coherent state or from one of the coherent states to another coherent state. The probability of the transition from the vacuum state to the coherent state with a given number of excitations represents the Poissonian distribution for the number of excitations formed in the (atom + field) system by the end of action of the external force. It was found to be proportional to the excitation time t 0.

Optics and Spectroscopy•Nov 1, 2001•2 citations

Coherent Interactions

Stephen Mark Barnett, PAUL M. RADMORE

Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.

•Nov 14, 2002•0 citations

Echoes in the coherent-state Jaynes–Cummings model

P. K. Aravind, Joseph O. Hirschfelder

The Jaynes–Cummings model (JCM) is used to give a detailed analysis of the long-time behavior of a two-level atom interacting with a (Glauber) coherent-state field. This work is based on the earlier work of J. H. Eberly and collaborators, who drew attention to the collapse and revival of the Rabi oscillations of the atomic inversion. We use the three components S1, S2, and S3 of the pseudospin vector S to analyze the dynamics of the atom within the rotating-wave approximation and exact resonance. We show that, at the times when the revivals peak, the variables σ = |S| and η = S3/σ become independent of (1) the transition dipole moment of the atom, (2) the intensity of the coherent-state field, and (3) the (common) frequency of the atom and the field. Thus these special values of σ and η are a universal feature of the JCM that should be observed in experiments with any two-level atom in an arbitrarily intense (but resonant) coherent-state field. The derivation of the above results was facilitated by transforming into a rotating coordinate frame in which S1 = 0 so that S2 and S3 are the only dynamical variables. The two-dimensional nature of S in this frame makes it easy to visualize the dynamical changes in the atom. The dynamics of the JCM is clarified by comparison with the closely related Buck–Sukumar model. The relationship of the phenomena discussed here to photon echoes and to quench echoes (in condensed media) is pointed out.

Journal of the Optical Society of America B•May 1, 1985•6 citations
The interaction between a two-level atom and the finite two-dimensional oscillator in the Cartesian coordinate system is addressed. The construction of the coupling between the degenerate energy states of the finite oscillator and the two levels of the atom allows the identification of an excitation-conserved eigenbasis. A dynamical approach to the evolution of mean values of the fields mediated by the atom is shown in the interaction picture. As an example, the eigenenergies states and su(2)\mathfrak{su}(2)su(2) coherent states are studied. Perspectives and prospectives are given to settle a path in the study of atomic systems coupled to finite versions of the harmonic oscillator.
The interaction between a two-level atom and the finite two-dimensional oscillator in the Cartesian coordinate system is addressed. The construction of the coupling between the degenerate energy states of the finite oscillator and the two levels of the atom allows the identification of an excitation-conserved eigenbasis. A dynamical approach to the evolution of mean values of the fields mediated by the atom is shown in the interaction picture. Furthermore, a quantitative characterization of the atom–field entanglement induced by the finite dynamics is presented. As an example, the eigenenergies states and [Formula: see text] coherent states are studied. Perspectives and prospects are presented to chart a path in the study of atomic systems coupled to finite versions of the harmonic oscillator.
Throughout this project, the case of a composite system formed by a harmonic oscillator and a two-state atom will be studied; considering both affect each other as \na consequence of a Jaynes-Cummings interaction, and being both coupled to different reservoirs that are assumed to be independent from each other. The effects \nof the decoherence in the evolution of the populations and coherences will be analyzed; specifically the energy dissipation and the coherence loss, as well as the stationary states and the relevance of the Jaynes-Cummings interaction. Finally, the \nsolution of stochastic equations will be analyzed, particularly the evolution of the \nexpectation value of the position operator for the harmonic oscilator; for the case of \nsimultaneous measurements in both subsystems, as well as for a measurement in \nthe harmonic oscillator when the two-state atom undergoes a decoherence process.
We study the dynamics of a system consisting of a two-level atom and a quantum oscillator coupled by the vacuum radiation field. The oscillator and the atom are assumed to have the same resonance frequency. In contrast to the two-atom or two-oscillator case, the atom-oscillator system leads to an unclosed hierarchy of equations for the operators that determine the radiation pattern emitted by the atom-oscillator system. Instead of using the operator equations, we formulate the problem in terms of density-matrix equations which can be solved once the initial conditions are specified. As an example, we looked at the radiation pattern, the decay dynamics, and the spectrum for an initial condition in which the oscillator is in its n=1n=1n=1 state and the atom is in its excited state. Dressed states (dressed by the static interaction between the atom and oscillator) prove to be especially useful for interpreting the results and comparing them with the two-atom and two-oscillator systems. All calculations are carried out using the resonance or rotating-wave approximation. The relationship of our work to damped Jaynes-Cummings models is noted.
In terms of excitation creation and annihilation operators of the Jaynes-Cummings model, acting in the representation of dressed states, the Hamiltonian is written which describes the character of the spectrum of excitations of two modes, representing a quantum analog of the classical behavior of two interacting one-dimensional anharmonic oscillators, namely, the field and atomic oscillators. The anharmonicity is caused by the nonlinearity of the oscillator interaction and manifests itself in the dependence of the frequencies of both modes on the number of excitations, i.e., on the energy. It is shown that an external deterministic force, acting on the system during a certain time t 0, transfers it from a vacuum state to a coherent state or from one of the coherent states to another coherent state. The probability of the transition from the vacuum state to the coherent state with a given number of excitations represents the Poissonian distribution for the number of excitations formed in the (atom + field) system by the end of action of the external force. It was found to be proportional to the excitation time t 0.
Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.
The Jaynes–Cummings model (JCM) is used to give a detailed analysis of the long-time behavior of a two-level atom interacting with a (Glauber) coherent-state field. This work is based on the earlier work of J. H. Eberly and collaborators, who drew attention to the collapse and revival of the Rabi oscillations of the atomic inversion. We use the three components S1, S2, and S3 of the pseudospin vector S to analyze the dynamics of the atom within the rotating-wave approximation and exact resonance. We show that, at the times when the revivals peak, the variables σ = |S| and η = S3/σ become independent of (1) the transition dipole moment of the atom, (2) the intensity of the coherent-state field, and (3) the (common) frequency of the atom and the field. Thus these special values of σ and η are a universal feature of the JCM that should be observed in experiments with any two-level atom in an arbitrarily intense (but resonant) coherent-state field. The derivation of the above results was facilitated by transforming into a rotating coordinate frame in which S1 = 0 so that S2 and S3 are the only dynamical variables. The two-dimensional nature of S in this frame makes it easy to visualize the dynamical changes in the atom. The dynamics of the JCM is clarified by comparison with the closely related Buck–Sukumar model. The relationship of the phenomena discussed here to photon echoes and to quench echoes (in condensed media) is pointed out.