Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons

Abstract

We consider the dynamics of $N$ interacting bosons initially exhibiting Bose-Einstein condensation. Due to an external trapping potential, the bosons are strongly confined in two spatial directions, with the transverse extension of the trap being of order $\epsilon$. The non-negative interaction potential is scaled such that its scattering length is positive and of order $(N/{\epsilon }^2{)}^{-1}$, the range of the interaction scales as $(N/{\epsilon }^2{)}^{-\beta }$ for $\beta \in (0,1)$. We prove that in the simultaneous limit $N\to \infty$ and $\epsilon \to 0$, the condensation is preserved by the dynamics and the time evolution is asymptotically described by a cubic defocusing nonlinear Schrödinger equation in one dimension, where the strength of the nonlinearity depends on the interaction and on the confining potential. This is the first derivation of a lower-dimensional effective evolution equation for singular potentials scaling with $\beta \ge \frac{1}{2}$ and lays the foundations for the derivation of the physically relevant one-dimensional Gross-Pitaevskii equation ($\beta =1$). For our analysis, we adapt an approach by Pickl to the problem with strong confinement.

References in corpus (20)

• Realization of Bose-Einstein condensates in lower dimensions
• Experimental demonstration of painting arbitrary and dynamic potentials for Bose-Einstein condensates
• Derivation of the Cubic Non-linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems
• Observations of density fluctuations in an elongated Bose gas: ideal gas and quasi-condensate regimes
• Mean-Field Dynamics: Singular Potentials and Rate of Convergence
• Bloch oscillations in the absence of a lattice
• Quantitative Derivation of the Gross-Pitaevskii Equation
• Gross-Pitaevskii Dynamics for Bose-Einstein Condensates
• Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation
• Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction
• One-Dimensional Behavior of Dilute, Trapped Bose Gases
• On the rigorous derivation of the 2D cubic nonlinear Schrödinger equation from 3D quantum many-body dynamics
• Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions
• Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D
• Well-posedness of non-autonomous linear evolution equations in uniformly convex spaces
• Derivation of the Time Dependent Two Dimensional Focusing NLS Equation
• Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials
• Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons
• The NLS limit for bosons in a quantum waveguide
• On the Time Dependent Gross Pitaevskii- and Hartree Equation

Cited by in corpus (5)

• Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions
• Higher order corrections to the mean-field description of the dynamics of interacting bosons
• Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the 2d Gross-Pitaevskii equation for strongly confined 3d bosons
• On the nonlinear Schrödinger equation with a toroidal trap in the strong confinement regime