An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive. Recently, meaningful experimental studies of the problem have become possible, stimulating theoretical interest. In generic isolated systems, non-equilibrium dynamics is expected to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics. However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible. For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete. Some recent studies even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems. Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription. Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch and Srednicki. A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages-any eigenstate in the microcanonical energy window will do, because they all give the same result.

We calculate, within the pseudopotential approximation, a one-dimensional scattering amplitude and effective onedimensional interaction potential for atoms confined transversally by an atom waveguide or highly elongated "cigar"-shaped atomic trap. We show that in the low-energy scattering regime, the scattering process degenerates to a total reflection suggesting an experimental realization of a famous model in theoretical physics -a one-dimensional gas of impenetrable bosons ("Tonks" gas). We give an estimate for suitable experimental parameters for alkali atoms confined in waveguides.Rapid progress in producing Bose condensates of alkali atoms [1] opened up new areas of ultra-low energy collisional physics. Concurrently, work has progressed on confinement of atoms in the light-induced [2] and magnetic-field-induced [3] atom waveguides.We develop a theory for the binary atomic collisions in presence of a transverse external confinement. Using the pseudopotential approximation, we derive an expression for an effective one-dimensional scattering amplitude and show that the interparticle interaction can be approximated by an effective one-dimensional δ-potential of a known strength. In the case of a dilute atomic gas, when the three body collisions are negligible, our effective potential can also be used to describe quasi-one-dimensional many-body systems: a project on the experimental realization of the one-dimensional Bose condensate is already presented in the literature [4]. Results of our paper will allow one to properly take into account the trap-induced corrections to the strength of the atomic mean-field potential.Furthermore, we show that in the low-energy scattering limit (k z |a 1D | ≪ 1), the effective one-dimensional scattering degenerates to a total reflection (hk z is the longitudinal component of the atomic momentum and a 1D is the one-dimensional scattering length defined below). This conclusion allows us to suggest an experimental realization of another famous model in theoretical physics -a one-dimensional gas of impenetrable bosons [5] also referred to as a "Tonks" gas. Such a system provides us with an unusual example of a boson-fermion duality: the elementary excitations of such a bosonic system obey Fermi statistics [6] (which is possible only in one dimension [7]). The Tonks gas is, therefore, a system complementary to the one-dimensional Bose condensate: the latter requires the high-energy scattering regime (k z |a 1D | ≫ 1) [8] and its excitation spectrum is represented by the Bogoliubov bosons.To describe the binary collisions between cold atoms confined in a waveguide, we suggest the following model: where g = 2πh 2 a/µ is the potential strength, a is the s-wave scattering length for the "true" interaction potential, µ = m/2 is the reduced mass, m is the mass of the atoms. The regularization operator ∂ ∂r (r · ), that removes the 1/r divergence from the scattered wave (see [9]), plays an important role in the derivation below;(d) Atomic motion (both transverse and longitudinal component...

In this Letter we pose the question of whether a many-body quantum system with a full set of conserved quantities can relax to an equilibrium state, and, if it can, what the properties of such state are. We confirm the relaxation hypothesis through a thorough ab initio numerical investigation of the dynamics of hard-core bosons on a one-dimensional lattice. Further, a natural extension of the Gibbs ensemble to integrable systems results in a theory that is able to predict the mean values of physical observables after relaxation. Finally, we show that our generalized equilibrium carries more memory of the initial conditions than the usual thermodynamic one. This effect may have many experimental consequences, some of which having already been observed in the recent experiment on the non-equilibrium dynamics of one-dimensional hard-core bosons in a harmonic potential [T. Kinoshita, T. Wenger, D. S. Weiss, Nature (London) 440, 900 (2006)].

M. Olshanii [Phys. Rev. Lett. 81, 938 (1998)] recently solved the atom-atom scattering problem with a pseudopotential interaction in the presence of transverse harmonic confinement, i.e. within an 'atom waveguide', deriving an effective one-dimensional coupling constant that diverged at a "confinement induced resonance" (CIR). Here, we report numerical results for finite range potentials that corroborate this resonance. In addition, we now present a physical interpretation of this effect as a novel type of Feshbach resonance in which the transverse modes of the waveguide assume the roles of 'open' and 'closed' scattering channels. [1,2,3,4,5,6,7,8]. One goal of such experiments is to reach the 'single-mode' or quasi-1D regime, where only the ground state of transverse motion is significantly populated at thermal equilibrium. This regime is of great practical interest due to the potential for ultra-sensitive rotation and gravitational gradient detection with guided single-mode atom interferometers. In addition to such applications, reaching the quasi-1D regime is of significant theoretical interest as the 1D delta-interacting boson gas represents one of the few known fully integrable quantum field theories. In a finite system with infinitely strong (hard-core) deltafunction interactions, boson many-body states in 1D have been shown to correspond via a one-to-one mapping with the highly-correlated states of the corresponding noninteracting Fermi gas [9]. The properties of this TonksGirardeau gas have been a topic of significant current theoretical interest [9,10,11,12,13,14,15] in anticipation of future atom-waveguide experiments. Additionally, the homogeneous 1D Bose with arbitrary-strength delta-function interactions, known as the Lieb Liniger model, is also a fully integrable system [16].To make the connection between experiments in tightly confining waveguides and theoretical models in 1D, it is necessary to know the relationship between the effective 1D coupling constant, g 1D , and the 3D scattering length, a. This problem was first addressed rigorously in [22], where it was predicted that a 'confinement induced resonance' (CIR) modifies the effective interaction, resulting in an effective 1D coupling strength which can be tuned from −∞ to +∞ by varying the transverse width of the * Electronic address: thbergeman@notes.cc.sunysb.edu † Electronic address: mmoore@cfa.harvard.edu ‡ Electronic address: olshanii@phys4adm.usc.edu waveguide, a ⊥ over a small range in the vicinity of the resonance at a ⊥ = Ca, where C = 1.4603 . . .. Hence this resonance clearly has significant implications for atomwaveguide experiments. Until now, however, there has been no convincing physical explanation for the effect, thus raising questions concerning its appearance in systems with finite-range interactions.The primary goal of this Letter, therefore, is to present numerical calculations of scattering in the presence of a cylindrical harmonic potential using finite-range atomatom potentials, confirming the existence of the CIR. ...