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arXiv:cond-mat/0106003 · Physical Review A

Hydrodynamic excitations in a spin-polarized Fermi gas under harmonic confinement in one dimension

Anna Minguzzi, Patrizia Vignolo, M. L. Chiofalo, M. P. Tosi

Published Aug 13, 2001• 53 citations
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Abstract

We consider a time-dependent nonlinear Schr\"odinger equation in one dimension (1D) with a fifth-order interaction term and external harmonic confinement, as a model for both (i) a Bose gas with hard-core contact interactions in the local-density approximation, and (ii) a spin-polarized Fermi gas in the collisional regime. We evaluate analytically in the Thomas-Fermi limit the density fluctuation profiles and the collective excitation frequencies, and compare the results for the low-lying modes with those obtained from numerical solution of the Schr\"odinger equation. We find that the excitation frequencies are multiples of the harmonic-trap frequency even in the strong-coupling Thomas-Fermi regime. This result shows that the hydrodynamic and the collisionless collective spectra coincide in the harmonically confined 1D Fermi gas, as they do for sound waves in its homogeneous analog. It also shows that in this case the local-density theory reproduces the exact collective spectrum of the hard-core Bose gas under harmonic confinement.

DOI
10.1103/physreva.64.033605

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We consider a time-dependent nonlinear Schr"odinger equation in one dimension (1D) with a fifth-order interaction term and external harmonic confinement, as a model for both (i) a Bose gas with hard-core contact interactions in the local-density approximation, and (ii) a spin-polarized Fermi gas in the collisional regime. We evaluate analytically in the Thomas-Fermi limit the density fluctuation profiles and the collective excitation frequencies, and compare the results for the low-lying modes with those obtained from numerical solution of the Schr"odinger equation. We find that the excitation frequencies are multiples of the harmonic-trap frequency even in the strong-coupling Thomas-Fermi regime. This result shows that the hydrodynamic and the collisionless collective spectra coincide in the harmonically confined 1D Fermi gas, as they do for sound waves in its homogeneous analog. It also shows that in this case the local-density theory reproduces the exact collective spectrum of the hard-core Bose gas under harmonic confinement.
The collective excitations of a zero-temperature, spin-polarized, harmonically trapped, two-dimensional dipolar Fermi gas are examined within the Thomas-Fermi--von Weizs\"acker hydrodynamic theory. We focus on repulsive interactions and investigate the dependence of the excitation frequencies on the strength of the dipolar interaction and particle number. We find that the mode spectrum can be classified according to bulk modes, whose frequencies are shifted upward as the interaction strength is increased, and an infinite ladder of surface modes, whose frequencies are independent of the interactions in the large particle limit. We argue quite generally that it is the local character of the two-dimensional energy density that is responsible for the insensitivity of surface excitations to the dipolar interaction strength and not the precise form of the equation of state. This property will not be found for the collective excitations of harmonically trapped, dipolar Fermi gases in one and three dimensions, where the energy density is manifestly nonlocal.
We present a comprehensive study of the discretized modes of an atomic gas in different conditions of confinement. Starting from the equations of hydrodynamics we derive a closed equation for the velocity field, depending on the adiabatic and isothermal compressibilities and applicable to different dimensions and quantum statistics. At zero temperature the equation reproduces the irrotational behavior of superfluid hydrodynamics. It is also applicable above the critical temperature in the collisional regime, where the appearance of rotational components in the velocity field is caused by the external potential. In the presence of harmonic trapping, a general class of analytic solutions is obtained for systems exhibiting a polytropic equation of state, characterized by a power law isoentropic dependence of the pressure on the density. Explicit results for the compressional modes are derived for both Bose and Fermi gases in the pancake and cigar as well as in the deep two- and one-dimensional regimes. Our results agree with the analytical predictions available in the literature in some limiting cases. They are particularly relevant in one-dimensional configurations, where the study of the collective frequencies could provide a unique test of the achievement of the collisional regime at finite temperature.
We consider a system of one-dimensional non-interacting fermions in external harmonic confinement. Using an efficient Green's function method we evaluate the exact profiles and the pair correlation function, showing a direct signature of the Fermi statistics and of the single quantum-level occupancy. We also study the dynamical properties of the gas, obtaining the spectrum both in the collisionless and in the collisional regime. Our results apply as well to describe a one-dimensional Bose gas with point-like hard-core interactions.
By using a sum-rule approach we investigate the transition between the hydrodynamic and the collisionless regime of the collective modes in a one-dimensional (1D) harmonically trapped Bose gas. Both the weakly interacting gas and the Tonks-Girardeau limits are considered. We predict that the excitation of the dipole compression mode is characterized in the high-temperature collisionless regime by a beating signal of two different frequencies (${\ensuremath{\omega}}_{z}$ and $3{\ensuremath{\omega}}_{z}$), while in the high-temperature collisional regime, the excitation consists of a single frequency ($\sqrt{7}{\ensuremath{\omega}}_{z}$). This behavior differs from the case of the lowest breathing mode whose excitation consists of a single frequency ($2{\ensuremath{\omega}}_{z}$) in both regimes. Our predictions for the dipole compression mode open promising perspectives for the experimental investigation of collisional effects in 1D configurations.
The collective oscillations of one-dimensional (1D) repulsive Bose gas with external harmonic confinement in two different regimes are studied. The first regime is the mean-field regime when the density is high. The second regime is the Tonks-Girardeau regime when the density is low. We investigate the resonances under periodic modulations of the trap potential and the effective nonlinearity. Modulations of the effective nonlinear coefficient result from modulations of the atomic scattering length by the Feshbach resonance method or variations of the transverse trap frequency. In the mean-field regime we predict bistability in the nonlinear oscillations of the condensate. In the Tonks-Girardeau regime the resonance has the character of a linear parametric resonance. In the case of rapid strong modulations of the nonlinear coefficient we find analytical expressions for the nonlinearity managed soliton width and the frequency of the slow secondary oscillations near the fixed point. We confirm the analytical predictions by direct numerical simulations of the 1D Gross-Pitaevskii equation and the effective nonlinear Schr\"odinger equation with quintic nonlinearity and trap potential.
Using a time-dependent modified nonlinear Schrödinger equation (MNLSE)-where the conventional chemical potential proportional to the density is replaced by the one inferred from Lieb-Liniger's exact solution-we study frequencies of the collective monopole excitations of a one-dimensional Bose gas. We find that our method accurately reproduces the results of a recent experimental study [E. Haller et al., Science 325, 1224 (2009)] in the full spectrum of interaction regimes from the ideal gas, through the mean-field regime, through the mean-field Thomas-Fermi regime, all the way to the Tonks-Giradeau gas. While the former two are accessible by the standard time-dependent NLSE and inaccessible by the time-dependent local density approximation, the situation reverses in the latter case. However, the MNLSE is shown to treat all these regimes within a single numerical method.