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arXiv:1010.4033 · Physical Review A

Spectroscopy of the soliton lattice formation in quasi-one-dimensional fermionic superfluids with population imbalance

Roman M. Lutchyn, Maxim Dzero, Victor M. Yakovenko

Published Sep 6, 2011• 35 citations
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Abstract

Motivated by recent experiments in low-dimensional trapped fermionic superfluids, we study a quasi-one-dimensional (quasi-1D) superfluid with a population imbalance between two hyperfine states using an exact mean-field solution for the order parameter. When an effective ``magnetic field'' exceeds a critical value, the superfluid order parameter develops spatial inhomogeneity in the form of a soliton lattice. The soliton lattice generates a band of quasiparticle states inside the energy gap, which originate from the Andreev bound states localized at the solitons. Emergence of the soliton lattice is accompanied by formation of a spin-density wave, with the majority fermions residing at the points in space where the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) order parameter vanishes. We discuss possibilities for experimental detection of the quasi-1D FFLO state using elastic and inelastic optical Bragg scattering and radiofrequency spectroscopy. We show that these measurements can provide necessary information for unambiguous identification of the spatially inhomogeneous quasi-1D FFLO state and the soliton lattice formation.

DOI
10.1103/physreva.84.033609

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Motivated by recent experiments in low-dimensional trapped fermionic superfluids, we study a quasi-one-dimensional (quasi-1D) superfluid with a population imbalance between two hyperfine states using an exact mean-field solution for the order parameter. When an effective ``magnetic field'' exceeds a critical value, the superfluid order parameter develops spatial inhomogeneity in the form of a soliton lattice. The soliton lattice generates a band of quasiparticle states inside the energy gap, which originate from the Andreev bound states localized at the solitons. Emergence of the soliton lattice is accompanied by formation of a spin-density wave, with the majority fermions residing at the points in space where the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) order parameter vanishes. We discuss possibilities for experimental detection of the quasi-1D FFLO state using elastic and inelastic optical Bragg scattering and radiofrequency spectroscopy. We show that these measurements can provide necessary information for unambiguous identification of the spatially inhomogeneous quasi-1D FFLO state and the soliton lattice formation.
Superfluidity with imbalanced populations of harmonically trapped ultracold fermions in a two-dimensional optical lattice is investigated by solving the Bogoliubov-de Gennes equations. The intriguing richness of patterns between competing ferromagnetic and superfluid phases for different filling densities has been revealed. At low density, the main experimental features could be reproduced, especially the bimodal distribution of ferromagnetism. The superfluid-pairing gap oscillates radially, demonstrating the realization of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. In the high-density regime, another type of FFLO state with gap oscillation along the angular direction shows up and the accompanying ferromagnetic order modulates accordingly. The square lattice like FFLO state may appear at the intermediate densities. Moreover, we suggest other experimental proposals of observing the unique FFLO features straightforwardly.
We propose and analyze a variational wave function for a population-imbalanced one-dimensional Fermi gas that allows for Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type pairing correlations among the two fermion species, while also accounting for the harmonic confining potential. In the strongly interacting regime, we find large spatial oscillations of the order parameter, indicative of an FFLO state. The obtained density profiles versus imbalance are consistent with recent experimental results as well as with theoretical calculations based on combining Bethe ansatz with the local density approximation. Our variational wave function displays no signature of the FFLO state in the densities of the two fermion species. Nonetheless, the oscillations of the order parameter appear in density-density correlations, both in situ and after free expansion. Furthermore, above a critical polarization, the value of which depends on the interaction, we find the unpaired Fermi-gas state to be energetically more favorable.
In a very interesting recent Letter\cite{machida}, the authors suggested the possibility of realizing the spatially modulated, or Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid state in trapped atomic fermion systems. The authors this Letter used a 1D mean field solution as guidance to estimate the parameter range for the existence of the FFLO phase, and also discussed the possibility of its detection by imaging the atomic density of the system. In this comment I wish to make two points. (i) In 1D there exists an exact solution based on bosonization, which fully takes into account the important quantum fluctuation effects\cite{yang01}; the exact solution suggests a wider parameter range for the FFLO state than that obtained from the mean-field solution of the present Letter. (ii) One can detect the FFLO pairing (in which Cooper pairs carry finite momenta) more directly by extending the methods used to detect BCS pairing\cite{regal,altman,greiner}.
We characterize the collective modes of a soliton train in a quasi-one-dimensional Fermi superfluid, using a mean-field formalism. In addition to the expected Goldstone and Higgs modes, we find novel long-lived gapped modes associated with oscillations of the soliton cores. The soliton train has an instability that depends strongly on the interaction strength and the spacing of solitons. It can be stabilized by filling each soliton with an unpaired fermion, thus forming a commensurate Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We find that such a state is always dynamically stable, which paves the way for realizing long-lived FFLO states in experiments via phase imprinting.
We calculate the zero-temperature (T=0) phase diagram of a polarized two-component Fermi gas in an array of weakly coupled parallel one-dimensional (1D) "tubes" produced by a two-dimensional optical lattice. Increasing the lattice strength drives a crossover from three-dimensional (3D) to 1D behavior, stabilizing the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) modulated superfluid phase. We argue that the most promising regime for observing the FFLO phase is in the quasi-1D regime, where the atomic motion is largely 1D but there is weak tunneling in the other directions that stabilizes long-range order. In the FFLO phase, we describe a phase transition where the quasiparticle spectrum changes from gapless near the 3D regime to gapped in quasi-1D.
Ground-state Hartree-Fock-Bogoliubov (HFB) theory is applied to imbalanced spin-1/21/21/2 one-dimensional (1D) Fermi systems that are spatially confined by either a harmonic or a hard-wall trapping potential. It has been hoped that such systems, which can be realized using ultracold atomic gases, would exhibit the long-sought-after Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase. The HFB formalism generalizes the standard Bogoliubov quasiparticle transformation by allowing for Cooper pairing to exist between all possible single-particle states and accounts for the effects of the inhomogeneous trapping potential as well as the mean-field Hartree potential. This provides an unbiased framework to describe inhomogeneous densities and pairing correlations in the FFLO state of a confined 1D gas. In a harmonic trap, numerical minimization of the HFB ground-state energy yields a spatially oscillating order parameter reminiscent of the FFLO state. However, we find that this state has almost no imprint in the local fermion densities (consistent with experiments that found no evidence of the FFLO phase). In contrast, for a hard-wall geometry, we find a strong signature of the spatial oscillations of the FFLO pairing amplitude reflected in the local in situ densities. In the hard-wall case, the excess spins are strongly localized near regions where there is a node in the pairing amplitude, creating an unmistakable crystalline modulation of the density.