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Physical Review A

Sound waves and dynamics of superfluid Fermi gases in optical lattices

Ai‐Xia Zhang, Ju-Kui Xue

Published Oct 20, 2009• 21 citations
Publisher

Abstract

The sound waves, the stability of Bloch waves, the Bloch oscillation, and the self-trapping phenomenon in interacting two-component Fermi gases throughout the BEC-BCS crossover in one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) optical lattices are discussed in detail. Within the hydrodynamical theory and by using the perturbative and tight-binding approximation, sound speed in both weak and tight 1D, 2D, 3D optical lattices, and the criteria for occurrences of instability of Bloch waves and self-trapping of Fermi gases along the whole BEC-BCS crossover in tight 1D, 2D, 3D optical lattices are obtained analytically. The results show that the sound speed, the criteria for occurrences of instability of Bloch waves and self-trapping, and the destruction of Bloch oscillation are modified dramatically by the lattice parameters (lattice dimension and lattice strength), the atom density or atom number, and the atom interaction.

DOI
10.1103/physreva.80.043617

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The sound waves, the stability of Bloch waves, the Bloch oscillation, and the self-trapping phenomenon in interacting two-component Fermi gases throughout the BEC-BCS crossover in one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) optical lattices are discussed in detail. Within the hydrodynamical theory and by using the perturbative and tight-binding approximation, sound speed in both weak and tight 1D, 2D, 3D optical lattices, and the criteria for occurrences of instability of Bloch waves and self-trapping of Fermi gases along the whole BEC-BCS crossover in tight 1D, 2D, 3D optical lattices are obtained analytically. The results show that the sound speed, the criteria for occurrences of instability of Bloch waves and self-trapping, and the destruction of Bloch oscillation are modified dramatically by the lattice parameters (lattice dimension and lattice strength), the atom density or atom number, and the atom interaction.
Bloch band and stabilities of Bloch waves of superfluid Fermi gases in one-dimensional (1D) periodic optical lattices are discussed. Within the hydrodynamical theory and the two-mode approximation, the Bloch band structure, the energetic and dynamical instabilities of Bloch waves at the first Brilliouin zone are presented. The results show that, when the atom density is beyond a critical value, a loop structure in the Bloch band at the zone edge is developed along the BEC-BCS crossover. The Bloch band structure and the stabilities of Bloch waves are modified dramatically when the system crosses from the BEC side to the BCS side, and they can be adjusted to the required characteristics by changing the atom's interaction (with the Feshbach resonance technique), the atom density, and the lattice parameters. The analytical expressions of the critical atom density for exciting the loop structure and maintaining the stabilities of Bloch waves are obtained.
We predict the existence of self-trapping, stable, moving solitons and breathers of Fermi wave packets along the Bose-Einstein condensation (BEC)-BCS crossover in one dimension (1D), 2D, and 3D optical lattices. The dynamical phase diagrams for self-trapping, solitons, and breathers of the Fermi matter waves along the BEC-BCS crossover are presented analytically and verified numerically by directly solving a discrete nonlinear Schrödinger equation. We find that the phase diagrams vary greatly along the BEC-BCS crossover; the dynamics of Fermi wave packet are different from that of Bose wave packet.
The nonlinear Landau-Zener tunneling and the nonlinear Rabi oscillations of superfluid Fermi gases between Bloch bands in an accelerating optical lattice are discussed. Within the hydrodynamic theory and a two-level model, the tunneling probability of superfluid Fermi gases between Bloch bands is obtained. We find that, as the system crosses from the Bose-Einstein condensation (BEC) side to the BCS side, the tunneling rate is closely related to the particle density: when the density is smaller (larger) than a critical value, the tunneling rate at unitarity is larger (smaller) than that in the BEC limit. This is well explained in terms of an effective interaction and an effective potential. Furthermore, the nonlinear Rabi oscillations of superfluid Fermi gases between the bands are discussed by imposing a periodic modulation on the level bias and the strength of the lattice. Analytical expressions of the critical density for suppressing or enhancing the Rabi oscillations are obtained. It is shown that, as the system crosses from the BEC side to the BCS side, the critical density strongly depends on the modulation parameters (i.e., the modulation amplitude and the modulation frequency). For a fixed density, a high-frequency or low-frequency modulation can suppress or enhance the Rabi oscillations both at unitarity and in the BEC limit. For an intermediate modulation frequency, the Rabi oscillations are chaotic along the entire BEC-BCS crossover, especially, on the BCS side. Interestingly, we find that the modulation of the lattice strength only with an intermediate modulation frequency has significant effect on the Rabi oscillations both in the BEC limit and at unitarity; that is, an intermediate-frequency modulation can enhance the Rabi oscillations, especially on the BCS side.
The superfluidity and pairing phenomena in ultracold atomic Fermi gases have been of great interest in recent years, with multiple tunable parameters. Here we study the BCS-BEC crossover behavior of balanced two-component Fermi gases in a one-dimensional optical lattice, which is distinct from the simple three-dimensional (3D) continuum and a fully 3D lattice often found in a condensed matter system. We use a pairing fluctuation theory which includes self-consistent feedback effects at finite temperatures and find widespread pseudogap phenomena beyond the BCS regime. As a consequence of the lattice periodicity, the superfluid transition temperature Tc{T}_{\mathrm{c}}Tc​ decreases with pairing strength in the BEC regime, where it approaches asymptotically ${T}_{\mathrm{c}}=\ensuremath{\pi}an/2m$, with aaa being the sss-wave scattering length, and nnn (mmm) the fermion density (mass). In addition, the quasi-two dimensionality leads to fast-growing (absolute value of the) fermionic chemical potential $\ensuremath{\mu}$ and pairing gap $\mathrm{\ensuremath{\Delta}}$, which depends exponentially on the ratio d/ad/ad/a. Importantly, Tc{T}_{\mathrm{c}}Tc​ at unitarity increases with the lattice constant ddd and hopping integral ttt. The effect of the van Hove singularity on Tc{T}_{\mathrm{c}}Tc​ is identified. The superfluid density exhibits T3/2{T}^{3/2}T3/2 power laws at low TTT, away from the extreme BCS limit. These predictions can be tested in future experiments.
We discuss the Bloch-state solutions of the stationary Gross-Pitaevskii equation and of the Bogoliubov equations for a Bose-Einstein condensate in the presence of a one-dimensional optical lattice. The results for the compressibility, effective mass and velocity of sound are analysed as a function of the lattice depth and of the strength of the two-body interaction. The band structure of the spectrum of elementary excitations is compared with the one exhibited by the stationary solutions (Bloch bands). Moreover, the numerical calculations are compared with the analytic predictions of the tight binding approximation. We also discuss the role of quantum fluctuations and show that the condensate exhibits 3D, 2D or 1D features depending on the lattice depth and on the number of particles occupying each potential well. We finally show how, using a local density approximation, our results can be applied to study the behaviour of the gas in the presence of harmonic trapping.
The higher-order interactions of Bose-Einstein condensate in multi-dimensional optical lattices are discussed both analytically and numerically. It is demonstrated that the effects of the higher-order atomic interactions on the sound speed and the stabilities of Bloch waves strongly depend on the lattice strength. In the presence of higher-order effects, tighter and high-dimensional lattices are confirmed to be two positive factors for maintaining the system's energetic stability, and the dynamical instability of Bloch waves can take place simultaneously with the energetic instability. In addition, we find that the higher-order interactions exhibit a long-range behavior and the long-lived coherent Bloch oscillations in a tilted optical lattice exist. Our results provide an effective way to probe the higher-order interactions in optical lattices.