We consider the evolution of a gas of N bosons in the three-dimensional Gross–Pitaevskii regime (in which particles are initially trapped in a volume of order one and interact through a repulsive potential with scattering length of order 1/N ). We construct a quasi-free approximation of the many-body dynamics, whose distance to the solution of the Schrödinger equation converges to zero, as N \to \infty , in the L^{2} (\mathbb{R}^{3N}) -norm. To achieve this goal, we let the Bose–Einstein condensate evolve according to a time-dependent Gross–Pitaevskii equation. After factoring out the microscopic correlation structure, the evolution of the orthogonal excitations of the condensate is governed instead by a Bogoliubov dynamics, with a time-dependent generator quadratic in creation and annihilation operators. As an application, we show a central limit theorem for fluctuations of bounded observables around their expectation with respect to the Gross–Pitaevskii dynamics.
Consider a system of
N bosons in three dimensions interacting via a repulsive short range pair potential
N2V(N(xi−xj)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let
HN denote the Hamiltonian of the system and let
ψN,t be the solution to the Schrödinger equation. Suppose that the initial data
ψN,0 satisfies the energy condition
<ψN,0,HNkψN,0>≤CkNk for $k=1,2,... $. We also assume that the
k-particle density matrices of the initial state are asymptotically factorized as
N→∞. We prove that the
k-particle density matrices of
ψN,t are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential
V. We also prove the same conclusion if the energy condition holds only for
k=1 but the factorization of
ψN,0 is assumed in a stronger sense.
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.
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.
Abstract 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 ε . The interaction is non-negative and scaled in such a way that its scattering length is of order
ε/N ε / N , while its range is proportional to
(ε/N)β ( ε / N ) β with scaling parameter
β∈(0,1] β ∈ ( 0 , 1 ] . We consider the simultaneous limit
(N,ε)→(∞,0) ( 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 β . For
β∈(0,1) β ∈ ( 0 , 1 ) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice
β=1 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.