Séminaire: Problèmes Spectraux en Physique Mathématique


Année 2021-2022



Pour tout renseignement complémentaire, veuillez contacter les organisateurs, Hakim Boumaza, Mathieu Lewin ou Stéphane Nonnenmacher.

Prochain séminaire le lundi 9 mai 2022 après-midi à l'Institut Henri Poincaré, en salle Pierre Grisvard (ex-314, 3e étage)


14h - 15h Alessandro Olgiati (University of Zurich)
Bosons in a double well: two-mode approximation and fluctuations

Abstract:
I will discuss the ground state properties of a system of bosonic particles trapped by a double-well potential, in a joint limit of large inter-well separation and high potential barrier. The bosons mutually interact via a two-body potential in the mean-field regime. The leading-order physics of the model is governed by a Bose-Hubbard Hamiltonian coupling two low-energy modes, each supported in the bottom of one well. Fluctuations beyond these two modes are ruled by two independent Bogoliubov Hamiltonians, one for each well. Our main result is that, when the system is in the ground state, the variance of the number of particles occupying the low-energy modes is suppressed. This is a violation of the central limit theorem that holds in the occurrence of Bose-Einstein condensation, and therefore a signature of the emergence of strong correlations in the ground state.
We achieve the result by proving a precise ground state energy expansion in terms of Bose-Hubbard and Bogoliubov energies.
Joint work with Nicolas Rougerie and Dominique Spehner.

 15h15 - 16h15 Cécilia Lancien (Grenoble) Spectrum of random quantum channels 

Abstract:
The main question that we will investigate in this talk is the following: what does the spectrum of a quantum channel typically look like? We will see that various natural models of random quantum channels generically exhibit a large spectral gap, between their first and second largest eigenvalues. This is in tight analogy with what is observed for the spectral gap of transition matrices associated to random graphs. In both the classical and the quantum settings, establishing results of this kind is interesting as it has important consequences regarding the speed of convergence to equilibrium of the corresponding dynamics. We will also present implications of the quantum result in terms of typical decay of correlations in so-called matrix product states (which are used to describe the states of 1D many-body quantum systems with local interactions, appearing for instance in quantum condensed matter physics).



Prochaines séances :

13 juin 2022: Amirali Hannani

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