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Bosons in a double well: two-mode approximation and fluctuations

Alessandro Olgiati, Nicolas Rougerie and Dominique Spehner

Vol. 16 (2023), No. 8, 1885–1954

We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity. Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose–Hubbard Hamiltonian. We prove an energy expansion, including the two-mode Bose–Hubbard energy and two independent Bogoliubov corrections (one for each potential well), and a variance bound for the number of particles falling inside each potential well. The latter is a signature of a correlated ground state in that it violates the central limit theorem.

many-body quantum mechanics, Bose–Einstein condensation, Bose–Hubbard model, Bogoliubov theory
Mathematical Subject Classification
Primary: 35Q40, 81V73, 82B10
Secondary: 47A75, 49S05
Received: 31 May 2021
Revised: 28 January 2022
Accepted: 16 March 2022
Published: 16 October 2023
Alessandro Olgiati
Institut fur Mathematik
Universitat Zurich
Nicolas Rougerie
École Normale Supérieure de Lyon & CNRS, UMPA (UMR 5669)
Dominique Spehner
Departamento de Ingeniería Matemática
Universidad de Concepción
Université Grenoble Alpes & CNRS
Institut Fourier & LPMMC

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