Quantitative Derivation of the Gross‐Pitaevskii Equation

By Niels Benedikter, Gustavo Alves de Oliveira, B. Schlein. 2012

152 citations recorded by the source corpus.

Starting from first‐principle many‐body quantum dynamics, we show that the dynamics of Bose‐Einstein condensates can be approximated by the time‐dependent nonlinear Gross‐Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one‐particle reduced density, the form of the initial data is preserved by the many‐body evolution, up to a small error that vanishes as N−1/2 in the limit of large N.© 2015 Wiley Periodicals, Inc.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/ee63cf086b30752d02d096b312001aee18f31670

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Effective Evolution Equations from Quantum Dynamics (2015, background): “he bound (5.49) becomes a simple exponential. Quantitative convergence towards the Gross-Pitaevskii equation. From (5.36) and Proposition 5.3 we obtain the following theorem, which has been proved in [12]. 39 Theorem 5.4. Let V ≥ 0, spherically symmetric and short range. Let ϕ∈ H4(R3) and ξN ∈ F suc”
Mean–Field Evolution of Fermionic Systems (2013, methodology): “In particular, bosonic Bogoliubov transformations have been used in [5], to describe fluctuations around the condensate, and in [6], to implement the short scale correlation structure produced by the singular potential.”