arXiv:1903.00365 · Journal of Mathematical Physics

Central limit theorem for Bose-Einstein condensates

Simone Rademacher, Benjamin Schlein

Published Jul 1, 2019• 15 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: 10.1063/1.5094348

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arXiv:1903.00365 · Journal of Mathematical Physics

Central limit theorem for Bose-Einstein condensates

Simone Rademacher, Benjamin Schlein

Published Jul 1, 2019• 15 citations

arXiv HTML PDF

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: 10.1063/1.5094348

Related papers

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

Jul 1, 201914 citations

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We prove fundamental properties of empirical measures induced by measurements performed on quantum N -body systems. More precisely, we consider measurements performed on the ground state of an interacting Bose gas on the torus in the Gross–Pitaevskii regime, known to exhibit Bose–Einstein condensation. For the corresponding empirical measure, we prove a weak law of large numbers with limit induced by the condensate wave function and characterize the fluctuations around it through an appropriate central limit theorem.We prove fundamental properties of empirical measures induced by measurements performed on quantum N -body systems. More precisely, we consider measurements performed on the ground state of an interacting Bose gas on the torus in the Gross–Pitaevskii regime, known to exhibit Bose–Einstein condensation. For the corresponding empirical measure, we prove a weak law of large numbers with limit induced by the condensate wave function and characterize the fluctuations around it through an appropriate central limit theorem.

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Abstract We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose–Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables due to the bosons’ correlation. We prove that in the limit $N \rightarrow \infty$ N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.Abstract We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose–Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables due to the bosons’ correlation. We prove that in the limit $$N \rightarrow \infty $$ N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.

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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.

N \rightarrow \infty

N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.