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arXiv:quant-ph/0105152

An implementation of the polynomial Lie algebra methods for solving a class of nonlinear models in quantum optics

V. P. Karassiov, Alexander A. Gusev, Sergue Il'ich Vinitsky

Published May 31, 2001• 0 citations
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Abstract

We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras $su_{pd}(2)$. These schemes, based on "diagonal" representations of model evolution operators (via diagonalizing Hamiltonians with the help of the $su_{pd}(2)$ defining relations), are implemented in the form adapted for numerical calculations. Their efficiency is demonstrated on the example of the second-harmonic-generation model.

DOI
10.48550/arxiv.quant-ph/0105152

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We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras supd(2)su_{pd}(2)supd​(2). These schemes, based on "diagonal" representations of model evolution operators (via diagonalizing Hamiltonians with the help of the supd(2)su_{pd}(2)supd​(2) defining relations), are implemented in the form adapted for numerical calculations. Their efficiency is demonstrated on the example of the second-harmonic-generation model.
We compare exact and SU(2)-cluster approximate calculation schemes to determine dynamics of the second-harmonic generation model using its reformulation in terms of the polynomial Lie algebra supd(2). These schemes, based on diagonal representations of the model evolution operator (due to diagonalizing the model Hamiltonian) and realized in the algorithmic forms, enabled us to implement computer experiments exhibiting a satisfactory accuracy of the cluster approximations in a large range of characteristic model parameters.
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 reelaborate on a general method for diagonalizing a wide class of nonlinear Hamiltonians describing different quantum optical models. This method makes use of a nonlinear deformation of the usual su(2) algebra and when some physical parameter, dictated by the particular model under consideration, becomes small, it gives a diagonal effective Hamiltonian that describes correctly the dynamics for arbitrary states and long times. We extend the technique to NNN-level atomic systems interacting with quantum fields, finding the corresponding effective Hamiltonians when the condition of kkk-photon resonance is fulfilled.
In this body of work, we will examine two classes of Lie algebra deformations and their appli- cation to the exact solutions of various quantum integrable systems. The first class of Lie algebra deformations are polynomial deformations of the A1 Lie algebra. We construct unitary representations of these algebras in terms of bosonic and spin operators as well as obtain their single-variable differential operator realizations. Using these results, we obtain the exact Bethe ansatz solutions of a variety of quantum integrable systems such as the Bose-Hubbard and Tavis-Cummings model. The second class of Lie algebra deformations are the so-called quasi-Gaudin algebra. This algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation this algebra, we obtain partial solutions for a class of bosonic models which do not preserve U(1) symmetry