Eur. Phys. J. B 84, 699-705 (2011)
https://doi.org/10.1140/epjb/e2011-20725-5

Regular Article

Nonlinear interference in a mean-field quantum model

¹ Université de Nice – Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France
² Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland

^a e-mail: vidar@hi.is

Received: 1 September 2011
Received in final form: 19 October 2011
Published online: 23 November 2011

Abstract

Using similar nonlinear stationary mean-field models for both a 2D axisymmetrical Bose-Einstein Condensate and an electron pair in a parabolic trap, we propose to describe the original many-particle ground state as a one-particle mixed state (in contrast to a pure state), i.e. as a statistical ensemble of several one-particle quantum states. These quantum states are the eigenfunctions of the corresponding stationary nonlinear Schrödinger equation (hence called “nonlinear eigenstates”). Due to their nonlinearity, they are not orthogonal. Therefore, taking the simple example of a two-level system, we show that each of these two nonlinear eigenstates |i⟩  and  |j⟩  occurs with a probability (or statistical weight) that is defined by their non-orthogonality  ⟨i|j⟩ 0. We give the corresponding density matrix. We search for physical grounds in the interpretation of our two main results, namely, a quantum-classical nonlinear transition and the interference between two “nonlinear eigenstates”.