We investigate the time evolution of the momentum of an impurity atom injected into a degenerate TonksGirardeau gas. We establish that given an initial momentum p 0 the impurity relaxes to a steady state with a nonvanishing momentum p ∞ . The nature of the steady state is found to depend drastically on whether the masses of the impurity and the host are equal. This is due to multiple coherent scattering processes leading to a resonant interaction between the impurity and the host in the case of equal masses. The dependence of p ∞ on p 0 remains nontrivial even in the limit of vanishing interaction between the impurity and host particles. In this limit p ∞ (p 0 ) is found explicitly.

The adiabatic theorem is a fundamental result in quantum mechanics, which states that a system can be kept arbitrarily close to the instantaneous ground state of its Hamiltonian if the latter varies in time slowly enough. The theorem has an impressive record of applications ranging from foundations of quantum field theory to computational molecular dynamics. In light of this success it is remarkable that a practicable quantitative understanding of what "slowly enough" means is limited to a modest set of systems mostly having a small Hilbert space. Here we show how this gap can be bridged for a broad natural class of physical systems, namely, many-body systems where a small move in the parameter space induces an orthogonality catastrophe. In this class, the conditions for adiabaticity are derived from the scaling properties of the parameter-dependent ground state without a reference to the excitation spectrum. This finding constitutes a major simplification of a complex problem, which otherwise requires solving nonautonomous time evolution in a large Hilbert space. DOI: 10.1103/PhysRevLett.119.200401 The adiabatic theorem (AT) is a profound statement that applies universally to all quantum systems having slowly varying parameters. It was originally conjectured by Born in 1926 [1], and its complete proof was given in a joint paper by Fock and Born two years later [2]. A number of refinements have been proposed over the years, see Ref.[3] and references therein. The theorem addresses the time evolution of a generic quantum system having a HamiltonianĤ λ , which is a continuous function of a dimensionless timedependent parameter λ ¼ Γt, where t is time and Γ is called the driving rate. For each λ one defines an instantaneous ground state, which is the lowest eigenvalue solution to Schrödinger's stationary equationFor simplicity, we assume that Φ λ is unique for each λ.Imagine that at t ¼ 0 the system is prepared in the Hamiltonian's instantaneous ground state Φ 0 . Then, as the parameter λ changes with time, the wave function of the system, Ψ λ , evolves according to Schrödinger's equationIt is natural to expect that as time goes by, the physical state Ψ λ will depart from the instantaneous ground state Φ λ ; in other words, the quantum fidelitywill decrease from its initial value of unity. The adiabatic theorem states that this departure can be made arbitrarily small provided that the driving is slow enough. In more rigorous terms, for any λ and for any small positive ϵ there exists small enough Γ such that 1 − F ðλÞ < ϵ. A process in which the fidelity (3) remains within a prescribed vicinity of unity is called adiabatic. The AT is a powerful tool in quantum physics, with applications ranging from the foundations of perturbative quantum field theory [4,5] to computational recipes in atomic and solid state physics [6]. The recent upsurge of interest in the AT has been driven by the ongoing developments in the theory of quantum topological order [7] and quantum information processing [8]. The universal applicabi...