BEYOND BOGOLIUBOV DYNAMICS - Boßmann, Lea; Petrat, Sören; Pickl, Peter; Soffer, Avy (Pure and Applied Analysis, 2021)
We consider a system of N interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to arbitrary precision. The N-independent corrections are given in terms of the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick’s theorem. We determine the n-point correlation functions of the excitations around the condensate, as well as the reduced densities of the N-body system, to arbitrary accuracy, given only the knowledge of the two-point correlation functions of a quasifree state and the solution of the Hartree equation. In this way, the complex problem of computing all n-point correlation functions for an interacting N-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs for the Bogoliubov two-point correlation functions.
CRIS-Team Scopus Importer:2024-10-02
Keywords: asymptotic expansion; Bogoliubov equation; Bose–Einstein condensate; effective equations; perturbation theory; quantum many-body dynamics

Derivation of the 1d Gross–Pitaevskii Equation from the 3d Quantum Many-Body Dynamics of Strongly Confined Bosons - Boßmann, Lea; Teufel, Stefan (Annales Henri Poincaré, 2019)
We consider the dynamics of N interacting bosons initially forming a Bose–Einstein condensate. Due to an external trapping potential, the bosons are strongly confined in two dimensions, where the transverse extension of the trap is of order ε. The non-negative interaction potential is scaled such that its range and its scattering length are both of order (N/ε2)-1, corresponding to the Gross–Pitaevskii scaling of a dilute Bose gas. We show that in the simultaneous limit N→ ∞ and ε→ 0 , the dynamics preserve condensation and the time evolution is asymptotically described by a Gross–Pitaevskii equation in one dimension. The strength of the nonlinearity is given by the scattering length of the unscaled interaction, multiplied with a factor depending on the shape of the confining potential. For our analysis, we adapt a method by Pickl (Rev Math Phys 27(01):1550003, 2015) to the problem with dimensional reduction and rely on the derivation of the one-dimensional NLS equation for interactions with softer scaling behaviour in Boßmann (Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons. arXiv preprint, 2018. arXiv:1803.11011).
CRIS-Team Scopus Importer:2024-10-02

Derivation of the 2d Gross–Pitaevskii Equation for Strongly Confined 3d Bosons - Boßmann, Lea (Archive for Rational Mechanics and Analysis, 2020)
We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order ε. The interaction is non-negative and scaled in such a way that its scattering length is of order ε/ N, while its range is proportional to (ε/ N) β with scaling parameter β∈ (0 , 1]. We consider the simultaneous limit (N, ε) → (∞, 0) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter β. For β∈ (0 , 1) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice β= 1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.
CRIS-Team Scopus Importer:2024-10-02

Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons - Boßmann, Lea; Pavlović, Nataša; Pickl, Peter; Soffer, Avy (Journal of Statistical Physics, 2020)
In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of Nd-dimensional bosons for large N. The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form (N- 1) - 1Nd βv(Nβ·) for β∈[0,14d). We derive a sequence of N-body functions which approximate the true many-body dynamics in L2(RdN) -norm to arbitrary precision in powers of N- 1. The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.
CRIS-Team Scopus Importer:2024-10-02
Keywords: Bose-Einstein condensation; Many-body quantum dynamics; Mean-field scaling; Norm approximation

On the dipole approximation with error estimates - Boßmann, Lea; Grummt, Robert; Kolb, Martin (Letters in Mathematical Physics, 2018)
The dipole approximation is employed to describe interactions between atoms and radiation. It essentially consists of neglecting the spatial variation of the external field over the atom. Heuristically, this is justified by arguing that the wavelength is considerably larger than the atomic length scale, which holds under usual experimental conditions. We prove the dipole approximation in the limit of infinite wavelengths compared to the atomic length scale and estimate the rate of convergence. Our results include N-body Coulomb potentials and experimentally relevant electromagnetic fields such as plane waves and laser pulses.
CRIS-Team Scopus Importer:2024-10-02
Keywords: Dipole approximation; Ionization; Laser; N-body; Schrödinger operator