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AIP conference proceedings

Spectra of Photon Down Conversion

Jan Smotlacha, Goce Chadzitaskos, C. Daskaloyannis, Piotr Kielanowski, S. Twareque Ali, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov

Published Jan 1, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.3275589

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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.
We use the methods of constructions of and deformed coherent states in order to construct the coherent states for down conversion processes. The down conversion process can be understood as a quasi-exactly solvable model of quantum mechanics. After the reduction of the Hamiltonian, we use the Turbiner polynomials approach, and the eigenvalues of the Hamiltonian for low number of photons are calculated and the approximation formula is found out. After the discussion on the time evolution and the entanglement, the coherent states are constructed as the eigenstates of the reduced annihilation operator.
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.
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.
Without using the assumption of the Bethe ansatz, we have obtained exact explicit analytical expressions for the energy spectrum and eigenstates in terms of a parameter λ for a model of optical second harmonic generation. We have shown that the parameter λ is determined by the roots of a polynomial of order [( N +1)/4]. Here [ y ] denotes y ' s integer part and N = n a +2 n b with n a and n b being photon numbers of the fundamental and the second harmonic modes respectively. We also obtain explicit analytical expressions for λ for , and explicit analytical expressions for all the eigenstates without showing any interactions between the two modes.