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

Physical Review A

Explicitly solvable models of a two-mode coupler in Kerr media

Agnieszka Tereszkiewicz, Anatol Odzijewicz, Maciej Horowski, Igor Jex, Goce Chadzitaskos

Published Jun 19, 2007• 7 citations
Publisher

Abstract

We present exact solutions for two nonlinear models each of which describes parametric down conversion of photons as well as the Kerr effect. The Hamiltonians for these models are related to the dual Hahn finite orthogonal polynomials. Explicit expressions are obtained for the spectra and for the eigenvectors of the Hamiltonians. A discussion of the physical characteristics of the systems is presented.

DOI
10.1103/physreva.75.063817

Related papers

Systems with intensity-dependent conversion integrable by finite orthogonal polynomials

Maciej Horowski, Goce Chadzitaskos, Anatol Odzijewicz +1 more

We present exact solutions of a class of the nonlinear models which describe the parametric conversion of photons. Hamiltonians of these models are related to the classes of finite orthogonal polynomials. The spectra and exact expressions for eigenvectors of these Hamiltonians are obtained.

arXiv:math-ph/0310058•Journal of Physics A Mathematical and General•May 25, 2004•8 citations

Exact solution for a class of quantum models of interacting bosons

V. S. Shchesnovich

Abstract Quantum models of interacting bosons have a wide range of applications, including the propagation of optical modes in nonlinear media, such as the k-photon down-conversion. Many of these models are related to nonlinear deformations of finite group algebras and, in this sense, are exactly solvable. While advanced group-theoretic methods were developed to study the eigenvalue spectrum, in quantum optics, the primary focus is not on the spectrum of the Hamiltonian but rather on the evolution of an initial state—such as the generation of optical signal modes by a strong pump mode propagating through a nonlinear medium. I propose a simple and general method to solve the state evolution problem, applicable to a broad class of quantum models of interacting bosons. For the k-photon down-conversion model and its generalizations, the solution to the state evolution problem is expressed as an infinite series expansion in powers of the propagation time, with coefficients determined by a recursion relation involving only a single polynomial function. This polynomial function is unique to each nonlinear model. As an application, I compare the exact solution of the parametric down-conversion process with the semiclassical parametric approximation.

arXiv:2411.14204•Journal of Physics A Mathematical and Theoretical•Mar 6, 2025•2 citations

Quantum theory of two-mode nonlinear directional couplers

Anthony Chefles, Stephen Mark Barnett

The quantum two-mode nonlinear directional coupler exhibits intriguing variations on the self-trapping effect seen in the corresponding classical model. We study and compare the quantum counterparts of this phenomenon for number and coherent state inputs, and discuss their relation to the nature of the eigenvalues and eigenvectors of the Hamiltonian.

Journal of Modern Optics•Apr 1, 1996•59 citations

Spectra of Photon Down Conversion

Jan Smotlacha, Goce Chadzitaskos, C. Daskaloyannis +5 more

We demonstrate that quasi‐exactly solvable models of quantum mechanics can be used in nonlinear optical processes for a down conversion or second‐harmonic generation processes. After the reduction we use the Turbiner and Bender ‐Dunne polynomial approach. The eigenvalues of Hamiltonian for low number of photons are calculated and the approximation formula is found out.

AIP conference proceedings•Jan 1, 2009•0 citations

Quasi-exactly solvable models in nonlinear optics

Gabriel Aacute lvarez, Federico Finkel, A Gonz lez-L pez +1 more

We study a large class of models with an arbitrary (finite) number of degrees of freedom, described by Hamiltonians which are polynomial in bosonic creation and annihilation operators, and including as particular cases n th harmonic generation and photon cascades. For each model, we construct a complete set of commuting integrals of motion of the Hamiltonian, fully characterize the common eigenspaces of the integrals of motion and show that the action of the Hamiltonian on these common eigenspaces can be represented by a quasi-exactly solvable reduced Hamiltonian, whose expression in terms of the usual generators of 𝔰𝔩 2 is computed explicitly.

arXiv:math-ph/0205043•Journal of Physics A Mathematical and General•Oct 2, 2002•12 citations

Exact solution of a nonlinear model of two-wave mixing in Kerr media

Pochi Yeh

The coupling of two electromagnetic waves in Kerr media is considered. An exact solution of the coupled-mode equations is derived in the most general case when both pump depletion and material absorption are considered.

Journal of the Optical Society of America B•May 1, 1986•63 citations
We present exact solutions for two nonlinear models each of which describes parametric down conversion of photons as well as the Kerr effect. The Hamiltonians for these models are related to the dual Hahn finite orthogonal polynomials. Explicit expressions are obtained for the spectra and for the eigenvectors of the Hamiltonians. A discussion of the physical characteristics of the systems is presented.
Abstract Quantum models of interacting bosons have a wide range of applications, including the propagation of optical modes in nonlinear media, such as the k-photon down-conversion. Many of these models are related to nonlinear deformations of finite group algebras and, in this sense, are exactly solvable. While advanced group-theoretic methods were developed to study the eigenvalue spectrum, in quantum optics, the primary focus is not on the spectrum of the Hamiltonian but rather on the evolution of an initial state—such as the generation of optical signal modes by a strong pump mode propagating through a nonlinear medium. I propose a simple and general method to solve the state evolution problem, applicable to a broad class of quantum models of interacting bosons. For the k-photon down-conversion model and its generalizations, the solution to the state evolution problem is expressed as an infinite series expansion in powers of the propagation time, with coefficients determined by a recursion relation involving only a single polynomial function. This polynomial function is unique to each nonlinear model. As an application, I compare the exact solution of the parametric down-conversion process with the semiclassical parametric approximation.
We study a large class of models with an arbitrary (finite) number of degrees of freedom, described by Hamiltonians which are polynomial in bosonic creation and annihilation operators, and including as particular cases n th harmonic generation and photon cascades. For each model, we construct a complete set of commuting integrals of motion of the Hamiltonian, fully characterize the common eigenspaces of the integrals of motion and show that the action of the Hamiltonian on these common eigenspaces can be represented by a quasi-exactly solvable reduced Hamiltonian, whose expression in terms of the usual generators of 𝔰𝔩 2 is computed explicitly.