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.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:2411.14204Journal of Physics A Mathematical and TheoreticalMar 6, 20252 citations

Quantum Nonlinear Optics

Eiichi Hanamura, Yutaka Kawabe, Akio Yamanaka

Springer eBooksJan 1, 200732 citations

Realization of the nonlinear optical gates in the $\Lambda$ -type medium

Andrei V. Rybin, Ilya P Vadeiko

We propose a method to control optical transparency of a Bose-Einstein condensate with working energy levels of the $\\Lambda$ -type. The reported effects are essentially nonlinear and are considered in the framework of an exactly solvable model describing interaction of light with a $\\Lambda$ -type medium. We show how the complicated nonlinear interplay between fast and slow solitons in the $\\Lambda$ -type medium points to a possibility to create optical gates as well as to a possibility to storage optical information.We propose a method to control optical transparency of a Bose-Einstein condensate with working energy levels of the $\\Lambda$-type. The reported effects are essentially nonlinear and are considered in the framework of an exactly solvable model describing interaction of light with a $\\Lambda$-type medium. We show how the complicated nonlinear interplay between fast and slow solitons in the $\\Lambda$-type medium points to a possibility to create optical gates as well as to a possibility to storage optical information.

Jul 19, 20040 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.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/0504072Il Nuovo Cimento BApr 24, 20051 citation

Quantum theory of nonlinear optics

Y. R. Shen

Dec 1, 19676 citations

We propose a method to control optical transparency of a Bose-Einstein condensate with working energy levels of the

\\Lambda

-type. The reported effects are essentially nonlinear and are considered in the framework of an exactly solvable model describing interaction of light with a

\\Lambda

-type medium. We show how the complicated nonlinear interplay between fast and slow solitons in the

\\Lambda

-type medium points to a possibility to create optical gates as well as to a possibility to storage optical information.