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arXiv:0806.0990 · New Journal of Physics

Excitations of attractive 1D bosons: binding versus fermionization

Emmerich Tempfli, Sascha Zöllner, Peter Schmelcher

Published Oct 16, 2008• 35 citations
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Abstract

The stationary states of a few bosons in a one-dimensional harmonic trap are investigated throughout the crossover from weak to strongly attractive interactions. For sufficient attraction, three different classes of states emerge: (i) N -body bound states, (ii) bound states of smaller fragments and (iii) gas-like states that fermionize, that is, map to ideal fermions in the limit of infinite attraction. The two-body correlations and momentum spectra characteristic of the three classes are discussed, and the results are illustrated using the soluble two-particle model.

DOI
10.1088/1367-2630/10/10/103021

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The stationary states of a few bosons in a one-dimensional harmonic trap are investigated throughout the crossover from weak to strongly attractive interactions. For sufficient attraction, three different classes of states emerge: (i) N -body bound states, (ii) bound states of smaller fragments and (iii) gas-like states that fermionize, that is, map to ideal fermions in the limit of infinite attraction. The two-body correlations and momentum spectra characteristic of the three classes are discussed, and the results are illustrated using the soluble two-particle model.
This thesis studies the one-dimensional Bose gas in harmonic and double-well traps from a few-body perspective. The main emphasis is on the crossover from weak interactions to the fermionization limit of infinite repulsion, where the system maps to an ideal Fermi gas. To explore the structure as well as the quantum dynamics throughout that crossover, we both develop an exact-diagonalization approach and resort to a multi-configurational time-dependent method (MCTDH). The basic mechanism of the fermionization crossover for the ground state is shown to consist in the formation of a correlation hole in the two-body density, which culminates in a localization of the individual particles for strong repulsion. This is accompanied by a reduction of coherence. We demonstrate how the concrete pathway depends on the trap geometry, on the shape of the interaction, as well as on the atom number. By extension, we also investigate the lowest excitations, whose understanding is a base for studying the impact of the fermionization crossover on the tunneling dynamics in a double well. In symmetric wells, a pathway from single-particle to fragmented-pair tunneling shows up. By energetically offsetting the two wells, tunnel resonances become accessible, which may be used to extract single atoms.
We study a one-dimensional system of two-component fermions in the limit of strong attractive particle-particle interactions. First, we analyze scattering in the corresponding few-body problem, which is analytically solvable via Bethe ansatz. This allows us to engineer effective interactions between the system's effective degrees of freedom: fermions and bosonic dimers (tightly bound pairs of fermions). We argue that, although these interactions are strong, the resulting effective problem can be mapped onto a weakly interacting one, paving the way for the use of perturbation theory. This finding simplifies studies of many-fermion systems under confinement that are beyond reach of state-of-the-art numerical methods. We illustrate this statement by considering an impurity atom in a Fermi gas.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions, predict the existence of two three-body bound states when both interactions are attractive, and determine their binding energies as universal functions of the two-body and three-body scattering lengths. We also show that an infinitesimal three-body attraction induces an excited bound state only for 3, 39, or more bosons. Our findings herein have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
Crossover of the pair excitation spectrum in the ground state of a dilute two-dimensional Fermi gas is studied as a function of the attraction strength $\frac{{\ensuremath{\varepsilon}}_{0}}{{\ensuremath{\varepsilon}}_{F}}$, where ${\ensuremath{\varepsilon}}_{0}$ is the two-fermion binding energy and ${\ensuremath{\varepsilon}}_{F}$ is the Fermi energy. It is shown that at $\frac{{\ensuremath{\varepsilon}}_{0}}{{\ensuremath{\varepsilon}}_{F}}\ensuremath{\simeq}\frac{1}{4}$ the spectrum changes its character from the continuous one in the strong-coupling, or "Bose-gas," limit ($\frac{{\ensuremath{\varepsilon}}_{0}}{{\ensuremath{\varepsilon}}_{F}}\ensuremath{\gg}1$) to that which consists of two different types of excitations in the weak-coupling ($\frac{{\ensuremath{\varepsilon}}_{0}}{{\ensuremath{\varepsilon}}_{F}}\ensuremath{\ll}1$) regime: (i) long-wavelength soundlike mode with the cutoff at momentum ${q}_{c}\ensuremath{\sim}\frac{1}{{\ensuremath{\xi}}_{0}}$, and (ii) pair excitations with $q\ensuremath{\ge}\sqrt{8m\ensuremath{\mu}}$.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.