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arXiv:1312.6107 · Physical Review A

Spectroscopy for a few atoms harmonically trapped in one dimension

Nathan L. Harshman

Published Mar 28, 2014• 43 citations
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Abstract

Spectroscopic labels for a few particles with spin that are harmonically trapped in one dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the noninteracting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multicomponent fermions and bosons transform under adiabatic evolution from noninteracting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases $N=3--5$, are provided.

DOI
10.1103/physreva.89.033633

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Spectroscopic labels for a few particles with spin that are harmonically trapped in one dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the noninteracting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multicomponent fermions and bosons transform under adiabatic evolution from noninteracting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases N=3−−5N=3--5N=3−−5, are provided.
This article investigates the properties of a few interacting particles trapped in a few wells and how these properties change under adiabatic tuning of interaction strength and interwell tunneling. While some system properties are dependent on the specific shapes of the traps and the interactions, this article applies symmetry analysis to identify generic features in the spectrum of stationary states of few-particle, few-well systems. Extended attention is given to a simple but flexible three-parameter model of two particles in two wells in one dimension. A key insight is that two limiting cases, hard-core repulsion and no interwell tunneling, can both be treated as emergent symmetries of the few-particle Hamiltonian. These symmetries are the mathematical consequences of infinite barriers in configuration space. They are necessary to explain the pattern of degeneracies in the energy spectrum, to understand how degeneracies are broken for models away from limiting cases, and to explain separability and integrability. These symmetry methods are extendable to more complicated models and the results have practical consequences for stable state control in few-particle, few-well systems with ultracold atoms in optical traps.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a hard-wall trap or a harmonic oscillator trap with a tunable $\ensuremath{\delta}$-function barrier at the trap center. The mixture consists of a 1D Bose gas with repulsive interactions and of a 1D noninteracting spin-aligned Fermi gas, with both species interacting through hard-core interactions. Using a generalized Bose-Fermi mapping, we calculate the reduced single-particle density matrix and the momentum distribution of the gas as a function of barrier strength and the parity of particle number. The secondary peaks in the momentum distribution show remarkable correlation between particles on the two sides of the split.
Recent developments in experimental techniques have made it possible to use magnetic fields to tune interactions between trapped cold atoms with different spin components allowing for detailed experimental investigations of the properties of quantum mechanical few-body systems. In this thesis we investigate the properties of two-component cold atoms trapped in a harmonic oscillator potential with a zero range interaction of arbitrary strength between the different species. In the limit of infinite interaction the atoms will tend to avoid each other. This is reminiscent of the Pauli principle and we will address differences and similarities to a system of identical fermions. Exact diagonalization of the Hamiltonian in a harmonic oscillator basis is used to obtain the eigenvectors and eigenvalues of the system and we also employ numerical methods borrowed from nuclear physics to generate effective interactions using unitary transformations. This method proves to be very effective for improving the convergence and the computation time is significantly decreased even for very strongly interacting systems.
We present highly accurate energy spectra and eigenfunctions of small 1D harmonically trapped two-component Fermi gases with interspecies δ-function interactions, and analyze the correlations of the so-called upper branch (i.e., the branch that describes a repulsive Fermi gas consisting of atoms but no molecules) for positive and negative coupling constants. Changes of the two-body correlations as a function of the interspecies coupling strength reflect the competition of the interspecies interaction and the effective repulsion due to the Pauli exclusion principle, and are interpreted as a few-body analog of a transition from a nonmagnetic to a magnetic phase. Moreover, we show that the eigenstate ψadia of the infinitely strongly interacting system with |n1+n2|>2 and |n1-n2|<n (n1 and n2 denote the number of fermions of components 1 and 2, respectively), which is reached experimentally by adiabatically changing the system parameters, does not, as previously proposed, coincide with the wave function ψG obtained by applying a generalized Fermi-Fermi mapping function to the eigenfunction of the noninteracting single-component Fermi gas.
This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for NNN identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the NNN particle case.
We explore a few-fermion mixture consisting of two components that are repulsively interacting and confined in a one-dimensional harmonic trap. Different scenarios of population imbalance ranging from the completely imbalanced case where the physics of a single impurity in the Fermi sea is discussed to the partially imbalanced and equal population configurations are investigated. For the numerical calculations the multiconfigurational time-dependent Hartree method is employed, extending its application to few-fermion systems. Apart from numerical calculations we generalize our ansatz for a correlated pair wave function proposed recently [I. Brouzos and P. Schmelcher, Phys. Rev. Lett. 108, 045301 (2012)] for bosons to mixtures of fermions. From weak to strong coupling between the components the energies, the densities and the correlation properties of one-dimensional systems change vastly with an upper limit set by fermionization where for infinite repulsion all fermions can be mapped to identical ones. The numerical and analytical treatments are in good agreement with respect to the description of this crossover. We show that for equal populations each pair of different component atoms splits into two single peaks in the density while for partial imbalance additional peaks and plateaus arise for very strong interaction strengths. The case of a single-impurity atom shows rich behavior of the energy and density as we approach fermionization and is directly connected to recent experiments [G. Z\"urn et al., Phys. Rev. Lett. 108, 075303 (2012)].