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Exponential concentration of fluctuations in mean-field boson dynamics

Matias Gabriel Ginzburg, Simone Rademacher, Giacomo De Palma

Abstract

We study the mean-field dynamics of a system of $N$ interacting bosons starting from an initially condensated state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having $n$ particles outside the condensate decays exponentially in $n$ for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having $n$ excitations.

Exponential concentration of fluctuations in mean-field boson dynamics

Abstract

We study the mean-field dynamics of a system of interacting bosons starting from an initially condensated state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having particles outside the condensate decays exponentially in for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having excitations.
Paper Structure (16 sections, 2 theorems, 91 equations)

This paper contains 16 sections, 2 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Exponential condensation
  3. Mean-field models with bounded potentials
  4. Mean-field models with unbounded potentials
  5. Comparison to the literature
  6. Probabilistic picture
  7. Singular scaling regimes
  8. Idea of the proofs
  9. Excitation map
  10. Proof of the theorems
  11. Step 1
  12. Step 2
  13. Mean-field models of type (i)
  14. Mean-field models of type (ii)
  15. Step 3
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $w$ be an arbitrary bounded two-body interaction, let $\Psi_N(t)\in\mathrm{Sym}^N\mathfrak{h}$ denote the solution to the Schrödinger equation eq:SchrodingerEq with initial data eq:initialCondensate satisfying the condensate condition eq:initial_condensate_condition. Furthermore, let $\phi (t)$ for all $t\geq 0$ and $0\leq \beta < \beta_c(t) = -\ln\tanh(3\|w\|\,t)$, where

Theorems & Definitions (4)

  • Theorem 2.1: Exponential Condensation for bounded potentials
  • Remark 2.2
  • Remark 2.3: Probabilisitc picture
  • Theorem 2.4: Exponential condensation for unbounded interaction potentials