arXiv subjects

Simone Rademacher

Publications and source records attributed to Simone Rademacher.

At least 19 recordsLinked to original sources

arXiv2026

Probabilistic representation and limit theorems for particle numbers of quasi-free states

We study the particle number distribution of locally interacting bosonic quasi- free states, which arise in various areas of mathematical physics. We show that the particle number decomposes into an infinite sum of independent geometrically distributed random variables, confirming predictions from the physics literature. This representation yields exponential tail bounds for the particle number, a law of small numbers together with a large deviation principle at logarithmic speed when the lattice spacing diverges, and a central limit theorem when it vanishes. As an application, we obtain a detailed description of the statistics of the quantum depletion in Bose--Einstein condensates, with the constants explicit in terms of the scattering length of the interaction potential.

math-ph↗

arXiv2026

Exponential concentration of fluctuations in mean-field boson dynamics

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.

math-ph↗

arXiv2026

Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian

Mean-field Hartree theory is a central tool for reducing interacting many-body dynamics to an effective nonlinear one-particle evolution. This approximation has been employed also when the Hamiltonian that governs the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail. We analytically solve a model of $N$ bosonic qubits with two-body interactions generated by a purely anti-Hermitian Hamiltonian, determine an analytic expression for the limit for $N\to\infty$ of the one-particle marginal state and show that such a limit does not agree with the solution of the non-Hermitian Hartree evolution equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.

quant-ph↗

arXiv2025

Dispersive estimates and long-time validity for Bogoliubov dynamics of interacting Bose gases

We consider the Bogoliubov approximation for the many-body quantum dynamics of weakly interacting Bose gases and establish a uniform-in-time validity of the Bogoliubov theory. The proof relies on a detailed analysis of the dispersive behavior of the symplectic Bogoliubov dynamics, which allows for a rigorous derivation of the Bogoliubov theory as an effective description of quantum fluctuations around the Bose-Einstein condensate on all time scales.

math-ph↗

arXiv2025

On Bose-Einstein condensates in disordered media

We consider the quantum dynamics of interacting bosons in the mean-field regime when they are subjected to a disordered potential, which is either random or quasi-periodic. Starting from a spatially localized Bose-Einstein condensate, we prove that fluctuations around the condensate propagate with a small velocity due to the disorder. This provides an example of a disordered many-body system with provably slow transport behavior in any spatial dimension. The main technical novelty is an interaction picture analysis relative to the Anderson-localized one-body dynamics.

math-ph↗

arXiv2025

Exponential Control of Excitations for Trapped BEC in the Gross-Pitaevskii Regime

We consider trapped Bose gases in three dimensions in the Gross-Pitaevskii regime whose low energy states are well known to exhibit Bose-Einstein condensation. That is, the majority of the particles occupies the same condensate state. We prove exponential control of the number of particles orthogonal to the condensate state, generalizing recent results for translation invariant systems.

math-ph↗

arXiv2024

Local enhancement of the mean-field approximation for bosons

We study the quantum many-body dynamics of a Bose-Einstein condensate (BEC) on the lattice in the mean-field regime. We derive a local enhancement of the mean-field approximation: At positive distance $\rho>0$ from the initial BEC, the mean-field approximation error at time $t\leq \rho/v$ is bounded as $\rho^{-n}$, for arbitrarily large $n\geq 1$. This is a consequence of new ballistic propagation bounds on the fluctuations around the condensate. To prove this, we develop a variant of the ASTLO (adiabatic spacetime localization observable) method for the particle non-conserving generator of the fluctuation dynamics around Hartree states.

math-ph↗

arXiv2024

Generating function for quantum depletion of Bose-Einstein condensates

We consider a Bose gas on the unit torus at zero temperature in the Gross-Pitaevskii regime, known to perform Bose-Einstein condensation: a macroscopic fraction of the bosons occupy the same quantum state, called condensate. We study the Bose gas' quantum depletion, that is the number of bosons outside the condensate, and derive an explicit asymptotic formula of its generating function. Moreover, we prove an upper bound for the tails of the quantum depletion.

math-ph↗

arXiv2023

Out-of-time-ordered correlators of mean-field bosons via Bogoliubov theory

Quantum many-body chaos concerns the scrambling of quantum information among large numbers of degrees of freedom. It rests on the prediction that out-of-time-ordered correlators (OTOCs) of the form $\langle [A(t),B]^2\rangle$ can be connected to classical symplectic dynamics. We rigorously prove a variant of this correspondence principle for mean-field bosons. We show that the $N\to\infty$ limit of the OTOC $\langle [A(t),B]^2\rangle$ is explicitly given by a suitable symplectic Bogoliubov dynamics. In practical terms, we describe the dynamical build-up of many-body entanglement between a particle and the whole system by an explicit nonlinear PDE on $L^2(\mathbb{R}^3) \oplus L^2(\mathbb{R}^3)$. For higher-order correlators, we obtain an out-of-time-ordered analog of the Wick rule. The proof uses Bogoliubov theory. Our finding spotlights a new problem in nonlinear dispersive PDE with implications for quantum many-body chaos.

math-ph↗

arXiv2023

Limit theorems for empirical measures of interacting quantum systems in Wasserstein space

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, trapped Bose gase 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 through an appropriate central limit theorem.

math-ph↗

arXiv2023

Ground state of Bose gases interacting through singular potentials

We consider a system of $N$ bosons on the three-dimensional unit torus. The particles interact through repulsive pair interactions of the form $N^{3\beta-1} v (N^\beta x)$ for $\beta \in (0,1)$. We prove the next order correction to Bogoliubov theory for the ground state and the ground state energy.

math-ph↗

arXiv2023

Exponential bounds of the condensation for dilute Bose gases

We consider N bosons on the unit torus $\Lambda = [0,1]^3$ in the Gross-Pitaevski regime where the interaction potential scales as $N^2 V (N(x -y))$. We prove that the thermal equilibrium at low temperatures exhibits the Bose-Einstein condensation in a strong sense, namely the probability of having $n$ particles outside of the condensation decays exponentially in $n$.

math-ph↗

arXiv2023

Large deviations for the ground state of weakly interacting Bose gases

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 $N \rightarrow \infty$ 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.

math-ph↗

arXiv2022

Traveling waves and effective mass for the regularized Landau-Pekar equations

We consider the regularized Landau-Pekar equations with positive speed of sound and prove the existence of subsonic traveling waves. We provide a definition of the effective mass for the regularized Landau-Pekar equations based on the energy-velocity expansion of subsonic traveling waves. Moreover we show that this definition of the effective mass agrees with the definition based on an energy-momentum expansion of low energy states.

math-ph↗

arXiv2021

A note on dependent random variables in quantum dynamics

We consider the many-body time evolution of weakly interacting bosons in the mean field regime for initial coherent states. We show that bounded k-particle operators, corresponding to dependent random variables, satisfy both, a law of large numbers and a central limit theorem.

math-ph↗

arXiv2021

Large deviation estimates for weakly interacting bosons

We study the many-body dynamics of an initially factorized bosonic wave function in the mean-field regime. We prove large deviation estimates for the fluctuations around the condensate. We derive an upper bound extending a recent result to more general interactions. Furthermore, we derive a new lower bound which agrees with the upper bound in leading order.

math-ph↗

arXiv2021

The effective mass problem for the Landau-Pekar equations

We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar [10].

math-ph↗

arXiv2020

A large deviation principle in many-body quantum dynamics

We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates, that are consistent with central limit theorems that have been established in the last years.

math-ph↗