Momentum distributions and numerical methods for strongly interacting one-dimensional spinor gases

F. Deuretzbacher, D. Becker, and L. Santos

Phys. Rev. A 94, 023606 – Published 4 August 2016

Abstract

One-dimensional spinor gases with strong $δ$ interaction fermionize and form a spin chain. The spatial degrees of freedom of this atom chain can be described by a mapping to spinless noninteracting fermions and the spin degrees of freedom are described by a spin-chain model with nearest-neighbor interactions. Here, we compute momentum and occupation-number distributions of up to 16 strongly interacting spinor fermions and bosons as a function of their spin imbalance, the strength of an externally applied magnetic field gradient, the length of their spin, and for different excited states of the multiplet. We show that the ground-state momentum distributions resemble those of the corresponding noninteracting systems, apart from flat background distributions, which extend to high momenta. Moreover, we show that the spin order of the spin chain—in particular antiferromagnetic spin order—may be deduced from the momentum and occupation-number distributions of the system. Finally, we present efficient numerical methods for the calculation of the single-particle densities and one-body density matrix elements and of the local exchange coefficients of the spin chain for large systems containing more than 20 strongly interacting particles in arbitrary confining potentials.

Received 8 March 2016

DOI:https://doi.org/10.1103/PhysRevA.94.023606

Physics Subject Headings (PhySH)

Cold atoms & matter waves Fermi gases

Atomic, Molecular & Optical

Authors & Affiliations

F. Deuretzbacher^1,*, D. Becker², and L. Santos¹

¹Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, DE-30167 Hannover, Germany
²I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstrasse 9, DE-20355 Hamburg, Germany

^*frank.deuretzbacher@itp.uni-hannover.de

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Issue

Vol. 94, Iss. 2 — August 2016

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Images

Figure 1
Single-particle densities of particles 1 (left) to 25 (right) of a spin chain in a harmonic trap consisting of 50 particles. The densities 26–50 are obtained by mirroring at the vertical axis through the origin. $l$ is the harmonic-oscillator length.
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Figure 2
One-body density matrix elements $ρ^{(i, j)} (z, z^{'})$ of 20 harmonically trapped particles. $l$ is the harmonic-oscillator length.
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Figure 3
Effect of population imbalance. Left column: Momentum distributions $ρ_{↑} (k)$ (solid blue line) and $ρ_{↓} (k)$ (dashed red line) of 16 spin-1/2 fermions with $(N_{↑}, N_{↓}) = (8, 8)$ (top) to $(15, 1)$ (bottom). Right column: Occupation-number distributions $ρ_{↑} (n)$ (solid blue line) and $ρ_{↓} (n)$ (dashed red line). $l, n$ : length scale and quantum number of the harmonic oscillator.
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Figure 4
Effect of magnetic field gradient. Top left: Momentum distributions $ρ (k) = ρ_{↑} (k) + ρ_{↓} (k)$ of 16 spin-balanced spin-1/2 fermions for $G / J = 0$ (dashed red line), 1 (dash-dotted blue line), and 10 (solid black line). Top right: Occupation-number distributions $ρ (n) = ρ_{↑} (n) + ρ_{↓} (n)$ for $G / J = 0$ (dashed red line), 1.5 [light blue (gray) line], 3 (dash-dotted blue line), and 10 (solid black line). Thin dashed black lines: 16 spinless noninteracting fermions. Bottom left: Momentum distributions of 16 spin-balanced spin-1/2 bosons for $G / J = 0$ (dashed red line), 0.1 [light blue (gray) line], and 10 (solid black line). Bottom right: Occupation-number distributions for $G / J = 0$ (dashed red line) and 10 (solid black line). $G$ : strength of $B$ -field gradient; $J = \sum_{i = 1}^{N - 1} J_{i} / (N - 1)$ : mean value of local exchange coefficients; $l, n$ : length scale and quantum number of the harmonic oscillator.
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Figure 5
Effect of increasing excitations. Top left: Momentum distributions $ρ (k) = ρ_{↑} (k) + ρ_{↓} (k)$ of 16 spin-balanced spin-1/2 fermions in the ground state (dashed red line), the 100th excited state [light blue (gray) line], the 2000th excited state (dash-dotted blue line), the 10 000th excited state (solid black line), and the highest-excited state (thin dashed black line). Top right: Occupation-number distributions $ρ (n) = ρ_{↑} (n) + ρ_{↓} (n)$ of the ground state (dashed red line), the 1000th excited state [light blue (gray) line], the 8000th excited state (dash-dotted blue line), the 12 000th excited state (solid black line), and the highest-excited state (thin dashed black line). Bottom left: Momentum distributions of 16 spin-balanced spin-1/2 bosons in the ground state (dashed red line), the 30th excited state [light blue (gray) line], the 700th excited state (dotted gray line), the 10 000th excited state (dash-dotted blue line), and the highest-excited state (solid black line). Bottom right: Occupation-number distributions of the ground state (dashed red line), the 10 000th excited state (dash-dotted blue line), and the highest-excited state (solid black line). Thin dashed black lines (in all subfigures): 16 spinless noninteracting fermions. $l, n$ : length scale and quantum number of the harmonic oscillator.
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Figure 6
Effect of increasing particle spin. Left: Momentum distributions $ρ (k) = \sum_{m} ρ_{m} (k)$ of eight spin-balanced fermions in the ground state and a particle spin of 1/2 (dashed red line), 3/2 (dash-dotted blue line), and 7/2 (solid black line). Right: The same for eight spin-balanced bosons in the highest-excited state. $l$ is the harmonic-oscillator length.
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