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arXiv:0712.0355 · Physical Review A

Bright and dark solitary waves in a one-dimensional spin-polarized gas of fermionic atoms with p -wave interactions in a hard-wall trap

Marvin D. Girardeau, Ewan M. Wright

Published Apr 9, 2008• 4 citations
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Abstract

We elucidate the physics underlying the fact that both bright and dark solitary waves can arise in a one-dimensional spin-polarized gas of fermionic atoms with attractive three-dimensional $p$-wave interactions in a hard-wall trap. This is possible since the one-dimensional fermion system can be mapped to a system of bosons described by the Lieb-Linger model with either repulsive or attractive $\ensuremath{\delta}$-function interactions which can support solitary waves.

DOI
10.1103/physreva.77.043612

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We elucidate the physics underlying the fact that both bright and dark solitary waves can arise in a one-dimensional spin-polarized gas of fermionic atoms with attractive three-dimensional ppp-wave interactions in a hard-wall trap. This is possible since the one-dimensional fermion system can be mapped to a system of bosons described by the Lieb-Linger model with either repulsive or attractive \ensuremathδ\ensuremath{\delta}\ensuremathδ-function interactions which can support solitary waves.
We investigate the ground state of the one-dimensional fermionic system enclosed in a hard-wall trap with attractive contact ppp-wave interactions. Based on the Bethe ansatz method, the explicit wave function is derived by numerically solving the Bethe ansatz equations for the full physical regimes $(\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{c}_{F}\ensuremath{\le}0)$. With the exact wave function some quantities which are important in many-body physics are obtained, including the one-body density matrix and the momentum distribution of the ground state for finite system. It is shown that the shell structure of the density profiles disappears with the increase of the interaction and in the fermionic Tonks-Girardeau limit the density distribution shows the same behavior as that of an ideal Bose gas. However, the one-body density matrix and the momentum distribution exhibit completely different structures compared with their bosonic counterparts.
We use a time-dependent dynamical mean-field-hydrodynamic model to study the formation of fermionic bright solitons in a trapped degenerate Fermi gas mixed with a Bose-Einstein condensate in a quasi-one-dimensional cigar-shaped geometry. Due to a strong Pauli-blocking repulsion among spin-polarized fermions at short distances there cannot be bright fermionic solitons in the case of repulsive boson-fermion interactions. However, we demonstrate that stable bright fermionic solitons can be formed for a sufficiently attractive boson-fermion interaction in a boson-fermion mixture. We also consider the formation of fermionic solitons in the presence of a periodic axial optical-lattice potential. These solitons can be formed and studied in the laboratory with present technology.
Bright solitons are non-dispersive wave solutions, arising in a diverse range of nonlinear, one-dimensional systems, including atomic Bose-Einstein condensates with attractive interactions. In reality, cold-atom experiments can only approach the idealized one-dimensional limit necessary for the realization of true solitons. Nevertheless, it remains possible to create bright solitary waves, the three-dimensional analogue of solitons, which maintain many of the key properties of their one-dimensional counterparts. Such solitary waves offer many potential applications and provide a rich testing ground for theoretical treatments of many-body quantum systems. Here we report the controlled formation of a bright solitary matter-wave from a Bose-Einstein condensate of (85)Rb, which is observed to propagate over a distance of ∼1.1 mm in 150 ms with no observable dispersion. We demonstrate the reflection of a solitary wave from a repulsive Gaussian barrier and contrast this to the case of a repulsive condensate, in both cases finding excellent agreement with theoretical simulations using the three-dimensional Gross-Pitaevskii equation.
Using the asymptotic Bethe ansatz, we study the stabilization problem of the one-dimensional spin-polarized Fermi gas confined in a hard-wall potential with a tunable ppp-wave scattering length and finite effective range. We find that the interplay of two factors, i.e., the finite interaction range and the hard-wall potential, will stabilize the system near resonance. The stabilization occurs even on the positive scattering length side, where the system undergoes a many-body collapse if any of the factors is absent. At ppp-wave resonance, the fermion system is found to feature ``quasiparticle condensation'' for any value of effective range, which is stabilized if the range is above twice the mean particle distance. Slightly away from resonance, the correction to the stability condition linearly depends on the inverse scattering length. Finally, a global picture is presented for the energetics and stability properties of fermions from the weakly attractive to deep bound-state regime. Our results raise the possibility of achieving stable ppp-wave superfluidity in quasi-one-dimensional (1D) atomic systems, and meanwhile shed light on the intriguing sss- and ppp-wave physics in 1D that violate the Bose-Fermi duality.
We consider the formation of bright solitons in a mixture of Bose and Fermi degenerate gases confined in a three-dimensional elongated harmonic trap. The Bose and Fermi atoms are assumed to effectively attract each other whereas bosonic atoms repel each other. Strong enough attraction between bosonic and fermionic components can change the character of the interaction within the bosonic cloud from repulsive to attractive making thus possible the generation of bright solitons in the mixture. On the other hand, such structures might be in danger due to the collapse phenomenon existing in attractive gases. We show, however, that under some conditions (defined by the strength of the Bose-Fermi components attraction) the structures which neither spread nor collapse can be generated. For elongated enough traps the formation of solitons is possible even at the ``natural'' value of the mutual Bose-Fermi ($^{87}\mathrm{Rb}\text{\ensuremath{-}}^{40}\mathrm{K}$ in our case) scattering length.
We consider the interplay of repulsive short-range and same-sign long-range interactions in the dynamics of dark solitons, as prototypical coherent nonlinear excitations in a trapped one-dimensional Bose gas. First, the form of the ground state is examined, and then both the existence of the solitary waves and their stability properties are explored, and corroborated by direct numerical simulations. We find that single- and multiple-dark-soliton states can exist and are generically robust in the presence of long-range interactions. We analyze the modes of vibration of such excitations and find that their respective frequencies are significantly upshifted as the strength of the long-range interactions is increased. Indeed, we find that a prefactor of the long-range interactions considered comparable to the trap strength may upshift the dark soliton oscillation frequency by an order of magnitude, in comparison to the well established one of $\mathrm{\ensuremath{\Omega}}/\sqrt{2}$ in a trap of frequency $\mathrm{\ensuremath{\Omega}}$.