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arXiv:1505.01517

Universal Scaling Properties of Cold Atom Scattering Dynamics in Confined Low Dimensional Geometries

Jeff Maki, Fei Zhou

Published May 6, 2015• 0 citations
arXivHTMLPDF

Abstract

Intermediate energy scale physics plays a very important role in non-equilibrium dynamics of quasi-low dimensional cold atom systems. In this article we obtain the universal scaling relations for the generalized reflection coefficient, i.e., the fraction of atoms scattered out of the initial state, at intermediate energies scales. These intermediate energies are larger than the trap frequency, but necessarily smaller than the effective range of the inter-atomic potential. For harmonically confined one and two dimensional geometries, whenever the energy of the system is commensurate with a transverse energy level, it is shown that the system becomes non-interacting. For energies close to a transverse energy level, the reflection coefficient and scattering dynamics take on universal scaling forms, only depending on the difference between the energy of the particles and the transverse energy level. In this article, the universal power law behaviors of these scattering observables are obtained.

DOI
10.48550/arxiv.1505.01517

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Intermediate energy scale physics plays a very important role in non-equilibrium dynamics of quasi-low dimensional cold atom systems. In this article we obtain the universal scaling relations for the generalized reflection coefficient, i.e., the fraction of atoms scattered out of the initial state, at intermediate energies scales. These intermediate energies are larger than the trap frequency, but necessarily smaller than the effective range of the inter-atomic potential. For harmonically confined one and two dimensional geometries, whenever the energy of the system is commensurate with a transverse energy level, it is shown that the system becomes non-interacting. For energies close to a transverse energy level, the reflection coefficient and scattering dynamics take on universal scaling forms, only depending on the difference between the energy of the particles and the transverse energy level. In this article, the universal power law behaviors of these scattering observables are obtained.
In quasi-one- or quasi-two-dimensional traps with strong transverse confinements, quantum gases behave like strictly one- or two-dimensional systems at large length scales. However, at short distance, the two-body scattering intrinsically has three-dimensional characteristics such that an exact description of any universal thermodynamic relation requires three-dimensional contacts since the range of interaction (a few nm) is orders of magnitude smaller than the harmonic-oscillator length of the transverse confinement ($\ensuremath{\sim}{10}^{2}\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$ for a 100 kHz trap). A fundamental question arises as to whether one- or two-dimensional contacts, which were originally defined for strictly one or two dimensions, are capable of describing quantum gases in quasi-one- or quasi-two-dimensional traps. Here, we point out an exact relation between the three- and low-dimensional contacts in these highly anisotropic traps. Such relation allows us to directly connect physical quantities at different length scales and to characterize the quasi-one- or quasi-two-dimensional traps using universal thermodynamic relations that were derived for strict one or two dimensions.
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In this dissertation, we study the few-body ultracold Bose-Fermi mixture and quasi-1D scattering at ultracold temperature. Degenerate quantum gases have attracted enormous attentions during the past two decades. They have opened new platforms of quantum simulation, precision measurement and quantum chemistry. The scattering properties of degenerate quantum gases are the first things to study in order to gain insight into various novel phenomena at ultracold temperatures.\nTo address the role of quantum statistics at ultracold temperature, we study the spectrum and dynamics of a few-body Bose-Fermi mixtures. In particular, we focus the dynamical evolution of a few-body Bose-Fermi mixture and concentrate on its universal behavior at large inter-particle scattering length. We predict the molecule formation efficiency in many-body Bose-Fermi mixtures by mapping this critical observable in few-body calculations. We also propose that a the quantum beat experiment could be used to measure the energy of the lowest Efimov trimer at unitarity.\nTo address the role of dimensionality, we study confinement induced resonances and similar phenomena in general transverse confining potentials. Well-separated energy scales in different dimensions allow the creation of reduced dimensional systems at ultracold temperatures. We develop a general framework to regularize the low-energy quasi-1D scattering phase shift associated with a zero range interaction.\nLastly, we discuss future prospects for the study of an ultracold Bose-Fermi mixture, and scattering in reduced dimensional systems of a more general topology.
Many physical systems can be described by one-dimensional (1D) models. It is the case of ultra-cold atoms: under certain circumstances their dynamics occurs only in one dimension. During my PhD I studied some aspects of 1D integrable systems. First, I present a study on the ground state of a system of 2-component repulsive fermions in 1D under harmonic confinement. I use the Bethe ansatz solution to calculate the phase diagram of the system in the homogeneous case. Adding a harmonic confinement I show that the atoms are distributed in a two-shell structure: the partially polarised phase in the inner shell and the fully polarised phase at the edges of the trap. Next I study the finite size effects for the gap of the quasiparticle excitation spectrum in the 1D Hubbard model. Two type of corrections to the result of the thermodynamic limit are obtained: a power law correction inversely proportional to the size of the system L, due to gapless excitations, and an exponential correction on L related to the existence of gapped excitations. In the weakly interacting regime this last correction can become important. Finally I study the response of a highly excited 1D gas to a periodic modulation of the coupling constant. I consider the Lieb-Liniger model and the non-integrable model of a single mobile impurity in a Fermi gas. I show that the non-integrable system is sensitive to excitations with frequencies as low as the mean level spacing, whereas the threshold frequency in the integrable case is much larger. This effect can be used as a probe of integrability for mesoscopic 1D systems, and can be observed experimentally by measuring the heating rate of a parametrically excited gas.
Phase transitions are ubiquitous in our three-dimensional world. By contrast, most conventional transitions do not occur in infinite uniform low-dimensional systems because of the increased role of thermal fluctuations. The crossover between these situations constitutes an important issue, dramatically illustrated by Bose-Einstein condensation: a gas strongly confined along one direction of space may condense along this direction without exhibiting true long-range order in the perpendicular plane. Here we explore transverse condensation for an atomic gas confined in a novel trapping geometry, with a flat in-plane bottom, and we relate it to the onset of an extended (yet of finite-range) in-plane coherence. By quench crossing the transition, we observe topological defects with a mean number satisfying the universal scaling law predicted by Kibble-Zurek mechanism. The approach described can be extended to investigate the topological phase transitions that take place in planar quantum fluids. Interacting quantum many-body systems in low dimensions is an active research area in ultra-cold gases. Here, Chomaz et al.study the dimensional crossover of Bose–Einstein condensation and observe the emergence of phase coherence in an ultra-cold quasi-2D Bose gas confined to a flat-bottom trapping potential.
We study some aspects of equilibrium and off-equilibrium quantum dynamics of dilute bosonic gases in the presence of a trapping potential. We consider systems with a fixed number of particles and study their scaling behavior with increasing the trap size. We focus on one-dimensional bosonic systems, such as gases described by the Lieb-Liniger model and its Tonks-Girardeau limit of impenetrable bosons, and gases constrained in optical lattices as described by the Bose-Hubbard model. We study their quantum (zero-temperature) behavior at equilibrium and off equilibrium during the unitary time evolution arising from changes of the trapping potential, which may be instantaneous or described by a power-law time dependence, starting from the equilibrium ground state for an initial trap size. Renormalization-group scaling arguments and analytical and numerical calculations show that the trap-size dependence of the equilibrium and off-equilibrium dynamics can be cast in the form of a trap-size scaling in the low-density regime, characterized by universal power laws of the trap size, in dilute gases with repulsive contact interactions and lattice systems described by the Bose-Hubbard model. The scaling functions corresponding to several physically interesting observables are computed. Our results are of experimental relevance for systems of cold atomic gases trapped by tunable confining potentials.