Emergent phenomena in an interacting Bose gas

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In this thesis, I intend to study the quantum fluctuation dynamics in a Bose gas on a torus Λ = (L𝕋)³ that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a Bose-Einstein condensate (BEC) with density N surrounded by thermal fluctuations with density 1, we assume that the system is described by a mean-field Hamiltonian. We extract a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics. Using a Fock-space approach, we provide explicit error bounds. It is known that the BEC and the HFB fluctuations both evolve at microscopic time scales t ∼ 1. Given a quasifree initial state, we determine the time evolution of the centered correlation functions ⟨a⟩, ⟨aa⟩ − ⟨a⟩², ⟨a⁺a⟩ − |⟨a⟩|² at mesoscopic time scales t ∼ λ⁻², where 0 < λ ≪ 1 denotes the size of the HFB interaction. For large but finite N, we consider both the case of fixed system size L ∼ 1, and the case L ∼ λ⁻²⁻. In the case L ∼ 1, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in ℚ; this phenomenon is reminiscent of the Talbot effect. For the case L ∼ λ⁻²⁻, we prove that the collision operator is well approximated by the expression predicted in the literature. In either of those cases, we have λ ∼ (log log N / log N)[superscript α], for different values of α > 0.



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