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:quant-ph/0410068 · TURKISH JOURNAL OF PHYSICS

Realizations of the osp(2,1) Superalgebra and Related Physical Systems

Hayriye Tütüncüler, Ramazan Koç

Published Oct 10, 2004• 0 citations
arXivHTMLPDF

Abstract

Eigenvalues and eigenfunction of two-boson 2x2 Hamiltonians in the framework of the superalgebra osp(2,1) are determined by presenting a similarity transformation. The Hamiltonians include two bosons and one fermion have been transformed in the form of the one variable differential equations and the conditions for its solvability have been discussed. It is observed that the Hamiltonians of the various physical systems can be written in terms of the generators of the osp(2,1) superalgebra and under some certain conditions their eigenstates can exactly be obtained. In particular, the procedure given here is useful in determining eigenstates of the Jaynes-Cummings Hamiltonians.

DOI
10.48550/arxiv.quant-ph/0410068

Related papers

Exactly and Quasi-Exactly Solvable two-mode Bosonic Hamiltonians

Ramazan Koç, Hayriye Tütüncüler, Eser Olğar

We develop a method to determine the eigenvalues and eigenfunctions of two-boson Hamiltonians include a wide class of quantum optical models. The quantum Hamiltonians have been transformed in the form of the one variable differential equation and the conditions for its solvability have been discussed. Applicability of the method is demonstrated on some simple physical systems.

arXiv:quant-ph/0410089•Chinese Journal of Physics•Oct 12, 2004•1 citation

Supercanonical transformations and time-dependent Hamiltonians

A. Anzaldo‐Meneses

Abstract Starting from general self-adjoint linear combinations of generators of the superalgebra a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp . The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients.

Europhysics Letters (EPL)•Jan 1, 2021•1 citation

Dynamical Superalgebra of the "Dressed" Jaynes-Cummings Model

Carla Buzano, Mario Rasetti, Maria Luisa Rastello

We show how, by resorting to a one-fermion realization of the Pauli operators, the Hamiltonian of the Jaynes-Cummings (JC) model can be identified as an element of the superalgebra u(1|1), which plays the role of a dynamical algebra. The extension of this notion to osp(2|2) allows adding both virtual and real two-photon processes to the JC Hamiltonian. The exact diagonalization problem is tackled here in the special case when the coupling constants of the fermionic terms of the "dressed" JC Hamiltonian are assumed to nilpotent Grassman-Banach numbers.

Physical Review Letters•Jan 9, 1989•66 citations

Two-level Jaynes-Cummings Hamiltonian as a supersymmetric quantum-mechanical system

H. A. Schmitt, A. Mufti

Using a realization of the noncompact orthosymplectic superalgebra $\mathrm{Osp}(\frac{2}{2}, R)$, the two-level Jaynes-Cummings Hamiltonian is shown to be a supersymmetric quantum-mechanical system in the sense of Witten. The existence of the supersymmetry is shown to place a restriction on the physical parameters of the Hamiltonian.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1991•8 citations

Solution of two-mode bosonic Hamiltonians and related physical systems

Ramazan Koç, Hayriye Tütüncüler, Mehmet Koca

We have constructed the quasi-exactly-solvable two-mode bosonic realizations of su(2) and su(1, 1) algebra. We derive the relations leading to the conditions for quasi-exact solvability of two-boson Hamiltonians by determining a general procedure which maps the Schwinger representations of the su(2) and su(1, 1) algebras to the Gelfand-Dyson representations, respectively. This mapping allows us to study nonlinear quantum-optical systems in the framework of quasi-exact solvability. Our approach also leads to a simple construction of special functions of two variables which are the most appropriate functions to study quasi-probabilities in quantum optics.

arXiv:math-ph/0504072•Il Nuovo Cimento B•Apr 24, 2005•1 citation

On the similarity transformations in second quantized fermion-boson interacting hamiltonian and the BCS theory

César Augusto Dartora, Fillipi Klos Rodrigues de Campos

A similarity transformation is an equivalence relation between square matrices which preserves determinant, trace and eigenvalues, playing a key role in quantum mechanics in simplifying complex hamiltonian systems and improving analytical results attainable from the use of perturbation theory. As a prototypical example, the conventional BCS theory of superconductivity is usually derived from a similarity transformation of the original electron-phonon hamiltonian, written in second quantized version. Here we discuss the general method for writing the similarity transformation operator in second quantized form, allowing one to recast a hamiltonian describing an interacting fermion-boson system into an effective theory in which only the desired degrees of freedom are kept after the transformation.

Revista Brasileira de Ensino de Física•Jun 4, 2016•0 citations
Eigenvalues and eigenfunction of two-boson 2x2 Hamiltonians in the framework of the superalgebra osp(2,1) are determined by presenting a similarity transformation. The Hamiltonians include two bosons and one fermion have been transformed in the form of the one variable differential equations and the conditions for its solvability have been discussed. It is observed that the Hamiltonians of the various physical systems can be written in terms of the generators of the osp(2,1) superalgebra and under some certain conditions their eigenstates can exactly be obtained. In particular, the procedure given here is useful in determining eigenstates of the Jaynes-Cummings Hamiltonians.
Abstract Starting from general self-adjoint linear combinations of generators of the superalgebra a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp . The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients.
We show how, by resorting to a one-fermion realization of the Pauli operators, the Hamiltonian of the Jaynes-Cummings (JC) model can be identified as an element of the superalgebra u(1|1), which plays the role of a dynamical algebra. The extension of this notion to osp(2|2) allows adding both virtual and real two-photon processes to the JC Hamiltonian. The exact diagonalization problem is tackled here in the special case when the coupling constants of the fermionic terms of the "dressed" JC Hamiltonian are assumed to nilpotent Grassman-Banach numbers.
Using a realization of the noncompact orthosymplectic superalgebra Osp(22,R)\mathrm{Osp}(\frac{2}{2}, R)Osp(22​,R), the two-level Jaynes-Cummings Hamiltonian is shown to be a supersymmetric quantum-mechanical system in the sense of Witten. The existence of the supersymmetry is shown to place a restriction on the physical parameters of the Hamiltonian.
We have constructed the quasi-exactly-solvable two-mode bosonic realizations of su(2) and su(1, 1) algebra. We derive the relations leading to the conditions for quasi-exact solvability of two-boson Hamiltonians by determining a general procedure which maps the Schwinger representations of the su(2) and su(1, 1) algebras to the Gelfand-Dyson representations, respectively. This mapping allows us to study nonlinear quantum-optical systems in the framework of quasi-exact solvability. Our approach also leads to a simple construction of special functions of two variables which are the most appropriate functions to study quasi-probabilities in quantum optics.
A similarity transformation is an equivalence relation between square matrices which preserves determinant, trace and eigenvalues, playing a key role in quantum mechanics in simplifying complex hamiltonian systems and improving analytical results attainable from the use of perturbation theory. As a prototypical example, the conventional BCS theory of superconductivity is usually derived from a similarity transformation of the original electron-phonon hamiltonian, written in second quantized version. Here we discuss the general method for writing the similarity transformation operator in second quantized form, allowing one to recast a hamiltonian describing an interacting fermion-boson system into an effective theory in which only the desired degrees of freedom are kept after the transformation.