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Journal of Physics Conference Series

Richardson—Gaudin form of Bethe Ansatz solutions for an atomic-molecular Bose-Einstein condensate model

Yibing Shen, Jon Links

Published Apr 13, 2015• 1 citation
Publisher

Abstract

An integrable model describing the interconversion between Bose-Einstein condensates of atomic and molecular degrees of freedom is studied. An exact solution for this model has been previously derived via the algebraic Bethe Ansatz formalism. Here we derive an alternative solution of the Richardson-Gaudin form, and highlight some of the implications of the result.

DOI
10.1088/1742-6596/597/1/012068

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An integrable model describing the interconversion between Bose-Einstein condensates of atomic and molecular degrees of freedom is studied. An exact solution for this model has been previously derived via the algebraic Bethe Ansatz formalism. Here we derive an alternative solution of the Richardson-Gaudin form, and highlight some of the implications of the result.
We study the ground-state properties of an atomic-molecular Bose–Einstein condensate model through an exact Bethe Ansatz solution. For a certain range of parameter choices, we prove that the ground-state Bethe roots lie on the positive real-axis. We then use a continuum limit approach to obtain a singular integral equation characterising the distribution of these Bethe roots. Solving this equation leads to an analytic expression for the ground-state energy. The form of the expression is consistent with the existence of a line of quantum phase transitions, which has been identified in earlier studies. This line demarcates a molecular phase from a mixed phase. Certain correlation functions, which characterise these phases, are then obtained through the Hellmann–Feynman theorem.
The Bose-Hubbard dimer Hamiltonian is a simple yet effective model for describing tunneling phenomena of Bose-Einstein condensates.One of the significant mathematical properties of the model is that it can be exactly solved by Bethe ansatz methods.Here we review the known exact solutions, highlighting the contributions of V.B. Kuznetsov to this field.Two of the exact solutions arise in the context of the Quantum Inverse Scattering Method, while the third solution uses a differential operator realisation of the su(2) Lie algebra.
Integrable systems take a special niche in quantum many-body systems. Due to the underlying algebraic structure, it is possible to capture all the quantum correlations exactly within a Bethe Ansatz wavefunction at a favourably scaling computational cost. Richardson [1] and Gaudin [2] have shown that the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian of superconductivity [3] is integrable and that the Bethe Ansatz state corresponds to a product wave function of generalized Cooper pairs [4]. In this formulation, the Bethe ansatz state is equivalent to a non-orthogonal geminal, which is widely used in quantum chemistry to describe electron pair correlations in atoms and molecules [5]. Inspired by Richardson-Gaudin integrable systems, it is possible to construct classes of non-orthogonal geminals, allowing for an accurate description of realistic and non-integrable strongly correlated atoms and molecules, while retaining the favorable computational scaling [6, 7]. In this contribution, I will discuss Richardson-Gaudin integrability and why it is useful for strongly correlated systems.