Journal of Nonlinear Mathematical Physics

Volume 24, Issue 3, June 2017, Pages 426 - 464

Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates

Authors
Alessandro Michelangeli
SISSA – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy, alemiche@sissa.it
Alessandro Olgiati
SISSA – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy, aolgiati@sissa.it
Received 2 April 2017, Accepted 19 May 2017, Available Online 6 January 2021.
DOI
https://doi.org/10.1080/14029251.2017.1346348How to use a DOI?
Keywords
effective non-linear evolution equations, many-body quantum dynamics, pseudo-spinor Bose-Einstein condensates, partial trace, reduced density matrix, Gross-Pitaevskii scaling, cubic NLS, coupled nonlinear Schrödinger system
Abstract

We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl’s projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.

Copyright
© 2017 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
24 - 3
Pages
426 - 464
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2017.1346348How to use a DOI?
Copyright
© 2017 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Alessandro Michelangeli
AU  - Alessandro Olgiati
PY  - 2021
DA  - 2021/01
TI  - Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates
JO  - Journal of Nonlinear Mathematical Physics
SP  - 426
EP  - 464
VL  - 24
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2017.1346348
DO  - https://doi.org/10.1080/14029251.2017.1346348
ID  - Michelangeli2021
ER  -