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:1303.1802

Mirror-field-atom interaction: Hamiltonian diagonalization

D. Rodríguez-Méndez, Omar Aguilar Loreto, H. M. Moya-Cessa

Published Mar 7, 2013• 1 citation
arXivHTMLPDF

Abstract

We show that the interaction between a movable mirror with a quantized field that interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operators (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a series of small rotations, into an effective Hamiltonian that may be solved. We would like to stress that the transformation in terms of SGO already simplifies enough the Hamiltonian in the sense that, in an exact way, it "eliminates" one of the three-subsystems, namely the quantized field.

DOI
10.48550/arxiv.1303.1802

Related papers

Equivalence between mirror-field-atom and ion-laser interactions

Raúl Juárez-Amaro, Arturo Zúñiga-Segundo, Francisco Soto +1 more

We show that the interaction between a movable mirror with a quantized field tha t interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operator s (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a unitary transformation, into a Hamiltonian that is equivalent to the ion-laser Hamiltonian. We would like to stress that the transformation in terms of SGO already simplifies e nough the Hamiltonian in the sense that, in an exact way, it eliminates one of the three-subsystems, namely the quantized fie ld.

Applied Mathematics & Information Sciences•Apr 4, 2013•2 citations

Operator approach to quantum optomechanics

C. Ventura-Velázquez, B. M. Rodríguez-Lara, H. M. Moya-Cessa

We study the mirror-field interaction in several frameworks: when it is driven, when it is affected by an environment, and when a two-level atom is introduced in the cavity. By using operator techniques, we show either how these problems may be solved or how the Hamiltonians involved, via sets of unitary transformations, may be taken to known Hamiltonians for which there exist approximate solutions.

arXiv:1411.7303•Physica Scripta•May 13, 2015•15 citations

AN ALGEBRAIC APPROACH TO THE INTERACTION OF A THREE-LEVEL ATOM INTERACTING WITH QUANTIZED RADIATION FIELD

Xu‐Bo Zou, Jing‐Bo Xu, Xiao-Chun Gao +1 more

The system of a three-level atom in the Ξ configuration coupled to two quantized field modes with arbitrary detuning and density-dependent multiphoton interaction is studied by dynamical algebraic method. With the help of an su(3) algebraic structure, we diagonalize the Hamiltonian by making use of unitary transformations and obtain the eigenvalues, eigenstates and time evolution operator for the system. Based on this su(3) structure, we also show that the system of a three-level atom in the Ξ configuration can be exactly transformed to an effective two-level Hamiltonian by an unitary transformation. Finally, we show that there exist an su (N) algebraic structure in the system of a N-level atom interacting with N-1 field modes.

International Journal of Modern Physics B•Jun 10, 2000•0 citations

Intrinsic decoherence in the interaction of two fields with a two-level atom

R. Juárez-Amaro, J.L. Escudero-Jiménez, H. M. Moya-Cessa

We study the interaction of a two-level atom and two fields, one of them classical. We obtain an effective Hamiltonian for this system by using a method recently introduced that produces a small rotation to the Hamiltonian that allows to neglect some terms in the rotated Hamiltonian. Then we solve a variation of the Schrödinger equation that models decoherence as the system evolves through intrinsic mechanisms beyond conventional quantum mechanics rather than dissipative interaction with an environment.

arXiv:1309.4382•Annalen der Physik•Jun 10, 2009•3 citations

The exact solution of generalized Dicke models via Susskind–Glogower operators

B. M. Rodríguez-Lara, H. M. Moya-Cessa

Abstract. We show a right unitary transformation approach based on Susskind– Glogower operators that diagonalizes a generalized Dicke Hamiltonian in the field basis and delivers a tridiagonal Hamiltonian in the Dicke basis. This tridiagonal Hamiltonian is diagonalized by a set of orthogonal polynomials satisfying a three-term recurrence relation. Our result is used to deliver a closed form, analytic time evolution for the case of a Jaynes–Cummings–Kerr model and to study the time evolution of the population inversion, reduced field entropy, and Husimi’s Q-function of the field for ensembles of interacting two-level systems under a Dicke–Kerr model. PACS numbers: 02.30.Ik,03.65.Fd,42.50.Pq

•Aug 16, 2016•7 citations

Simple Prescription for Quantizing Fields Applied to Interacting Spins in an Electromagnetic Field and to Other Systems

Marshall S. Sparks

A simple prescription for quantizing fields is developed and applied to the sytsem of interacting spins in an electromagnetic field. The classical equations of motion are written in the standard form $\ensuremath{-}i(\frac{d}{\mathrm{dt}})|V〉+\mathfrak{M}|V〉=0$, where the state vector $|V〉$ has the state variables, such as the generalized coordinates and momenta, as its components, and the time development operator $\mathfrak{M}$ contains derivative, integral, and matrix operators in general. By diagonalizing $\mathfrak{M}$ and replacing the resulting classical variables with operators obeying boson commutation relations, the system is quantized, and the Hamiltonian and Heisenberg equations of motion are automatically cast into the harmonic-oscillator form. Thus the quantization is reduced to the simple problem of diagonalizing a matrix. This avoids the laborious process of finding a Lagrangian density (by trial and error in general), determining the canonical variables, forming the Hamiltonian density, and inventing transformations to creation and annihilation operators. Problems with zero-frequency modes, which can be annoying in the standard treatment, are automatically eliminated by the prescription. The bookkeeping of forward and backward waves, different branches of dispersion curves, etc., is simplified. In addition to affording a straightforward and simple method of quantizing fields, the method also gives the necessary and sufficient conditions for a Hamiltonian to be transformable into harmonic-oscillator form, gives a direct relation between the classical equations of motion and the Heisenberg quantum-mechanical equations, and gives a simple method of determining the constants of the motion.

Physical Review•May 25, 1969•0 citations
We show that the interaction between a movable mirror with a quantized field that interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operators (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a series of small rotations, into an effective Hamiltonian that may be solved. We would like to stress that the transformation in terms of SGO already simplifies enough the Hamiltonian in the sense that, in an exact way, it "eliminates" one of the three-subsystems, namely the quantized field.
We show that the interaction between a movable mirror with a quantized field tha t interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operator s (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a unitary transformation, into a Hamiltonian that is equivalent to the ion-laser Hamiltonian. We would like to stress that the transformation in terms of SGO already simplifies e nough the Hamiltonian in the sense that, in an exact way, it eliminates one of the three-subsystems, namely the quantized fie ld.
The system of a three-level atom in the Ξ configuration coupled to two quantized field modes with arbitrary detuning and density-dependent multiphoton interaction is studied by dynamical algebraic method. With the help of an su(3) algebraic structure, we diagonalize the Hamiltonian by making use of unitary transformations and obtain the eigenvalues, eigenstates and time evolution operator for the system. Based on this su(3) structure, we also show that the system of a three-level atom in the Ξ configuration can be exactly transformed to an effective two-level Hamiltonian by an unitary transformation. Finally, we show that there exist an su (N) algebraic structure in the system of a N-level atom interacting with N-1 field modes.
We study the interaction of a two-level atom and two fields, one of them classical. We obtain an effective Hamiltonian for this system by using a method recently introduced that produces a small rotation to the Hamiltonian that allows to neglect some terms in the rotated Hamiltonian. Then we solve a variation of the Schrödinger equation that models decoherence as the system evolves through intrinsic mechanisms beyond conventional quantum mechanics rather than dissipative interaction with an environment.
Abstract. We show a right unitary transformation approach based on Susskind– Glogower operators that diagonalizes a generalized Dicke Hamiltonian in the field basis and delivers a tridiagonal Hamiltonian in the Dicke basis. This tridiagonal Hamiltonian is diagonalized by a set of orthogonal polynomials satisfying a three-term recurrence relation. Our result is used to deliver a closed form, analytic time evolution for the case of a Jaynes–Cummings–Kerr model and to study the time evolution of the population inversion, reduced field entropy, and Husimi’s Q-function of the field for ensembles of interacting two-level systems under a Dicke–Kerr model. PACS numbers: 02.30.Ik,03.65.Fd,42.50.Pq
A simple prescription for quantizing fields is developed and applied to the sytsem of interacting spins in an electromagnetic field. The classical equations of motion are written in the standard form $\ensuremath{-}i(\frac{d}{\mathrm{dt}})|V〉+\mathfrak{M}|V〉=0$, where the state vector ∣V〉|V〉∣V〉 has the state variables, such as the generalized coordinates and momenta, as its components, and the time development operator M\mathfrak{M}M contains derivative, integral, and matrix operators in general. By diagonalizing M\mathfrak{M}M and replacing the resulting classical variables with operators obeying boson commutation relations, the system is quantized, and the Hamiltonian and Heisenberg equations of motion are automatically cast into the harmonic-oscillator form. Thus the quantization is reduced to the simple problem of diagonalizing a matrix. This avoids the laborious process of finding a Lagrangian density (by trial and error in general), determining the canonical variables, forming the Hamiltonian density, and inventing transformations to creation and annihilation operators. Problems with zero-frequency modes, which can be annoying in the standard treatment, are automatically eliminated by the prescription. The bookkeeping of forward and backward waves, different branches of dispersion curves, etc., is simplified. In addition to affording a straightforward and simple method of quantizing fields, the method also gives the necessary and sufficient conditions for a Hamiltonian to be transformable into harmonic-oscillator form, gives a direct relation between the classical equations of motion and the Heisenberg quantum-mechanical equations, and gives a simple method of determining the constants of the motion.