Minimal velocity estimates and soft mode bounds for the massless spin-boson model

TL;DR

This paper establishes bounds on the number of bosons in the massless spin-boson model, demonstrating that unbound bosons travel at a minimal speed and providing insights into the asymptotic behavior of the system.

Contribution

It extends previous work by providing detailed bounds on boson counts and minimal velocities, advancing understanding of the model's long-term dynamics and asymptotic completeness.

Findings

Mean number of bosons tends to a ground state bound asymptotically.

Unbound bosons travel at speeds not lower than the propagation speed.

Bounds on low-momentum (soft) emitted bosons are established.

Abstract

We consider generalised versions of the spin-boson model at small coupling. We assume the spin (or atom) to sit at the origin $0 \in \mathbb{R}^d$ and the propagation speed $v_p$ of free bosons to be constant, i.e.\ independent of momentum. In particular, the bosons are massless. We prove detailed bounds on the mean number of bosons contained in the ball $\{|x| \leq v_p t \}$. In particular, we prove that, as $t \to \infty$, this number tends to an asymptotic value that can be naturally identified as the mean number of bosons bound to the atom in the ground state. Physically, this means that bosons that are not bound to the atom, are travelling outwards at a speed that is not lower than $v_p$, hence the term 'minimal velocity estimate'. Additionally, we prove bounds on the number of emitted bosons with low momentum (soft mode bounds). This paper is an extension of our earlier work in 'Approach to ground state and time-independent photon bound for massless spin-boson models' (arXiv:1109.5582, Ann. H. Poincare, 2012). Together with the results in that paper, the bounds of the present paper suffice to prove asymptotic completeness, as we describe in 'Asymptotic completeness for the massless spin-boson model'(arXiv:1301.2357).