Central limit theorem for Bose-Einstein condensates

Simone Rademacher, Benjamin Schlein

Published Jul 1, 2019• 14 citations

Abstract

We consider a Bose gas trapped in the unit torus in the Gross-Pitaevskii regime. In the ground state, we prove that fluctuations of bounded one-particle observables satisfy a central limit theorem

DOI: 10.5167/uzh-174530

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Central limit theorem for Bose-Einstein condensates

Simone Rademacher, Benjamin Schlein

Published Jul 1, 2019• 14 citations

Publisher

Abstract

We consider a Bose gas trapped in the unit torus in the Gross-Pitaevskii regime. In the ground state, we prove that fluctuations of bounded one-particle observables satisfy a central limit theorem

DOI: 10.5167/uzh-174530

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Central limit theorem for Bose-Einstein condensates

Simone Rademacher, Benjamin Schlein

We consider a Bose gas trapped in the unit torus in the Gross-Pitaevskii regime. In the ground state, we prove that fluctuations of bounded one-particle observables satisfy a central limit theorem.

arXiv:1903.00365Journal of Mathematical PhysicsJul 1, 201915 citations

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The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.

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We consider the Bose-Einstein condensation of atoms in a trap where the density of particles is so high that the low-density approach of Gross [J. Math. Phys. (N.Y.) 4, 195 (1963)] and Pitaevskii $($ V. L. Ginsburg and L. P. Pitaevskii, Zh. \'Eksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 1858 (1958)] $)$ will not be applicable. For this purpose we use the slave boson representation which is valid for hard-core bosons at any density. This description leads to the same results as the Gross-Pitaevskii approach in the low-density limit, but for higher densities, it predicts the depletion of the condensate in the regions where the density of the atomic cloud is high.We consider the Bose-Einstein condensation of atoms in a trap where the density of particles is so high that the low-density approach of Gross [J. Math. Phys. (N.Y.) 4, 195 (1963)] and Pitaevskii $($V. L. Ginsburg and L. P. Pitaevskii, Zh. \'Eksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 1858 (1958)]$)$ will not be applicable. For this purpose we use the slave boson representation which is valid for hard-core bosons at any density. This description leads to the same results as the Gross-Pitaevskii approach in the low-density limit, but for higher densities, it predicts the depletion of the condensate in the regions where the density of the atomic cloud is high.

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N

goes to infinity with

Na

fixed, where

a

is the scattering length of the interparticle interaction, measured in units of the trap size.

Abstract We consider a system ofNbosons in the limit

N \rightarrow \infty

N→∞ , interacting through singular potentials. For initial data exhibiting Bose–Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic nonlinear Schrödinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem.

We consider the Bose-Einstein condensation of atoms in a trap where the density of particles is so high that the low-density approach of Gross [J. Math. Phys. (N.Y.) 4, 195 (1963)] and Pitaevskii

(

V. L. Ginsburg and L. P. Pitaevskii, Zh. \'Eksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 1858 (1958)]

)

will not be applicable. For this purpose we use the slave boson representation which is valid for hard-core bosons at any density. This description leads to the same results as the Gross-Pitaevskii approach in the low-density limit, but for higher densities, it predicts the depletion of the condensate in the regions where the density of the atomic cloud is high.