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arXiv:1207.1738 · Journal of Physics B Atomic Molecular and Optical Physics

One- and two-dimensional reductions of the mean-field description of degenerate Fermi gases

Pablo Díaz, D. Laroze, Ivan A. Schmidt, Boris A. Malomed

Published Jul 4, 2012• 11 citations
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Abstract

We study collective behaviour of Fermi gases trapped in various external potentials, including optical lattices (OLs), in the framework of the mean-field (hydrodynamic) description. Using the variational method, we derive effective dynamical equations for the one- and two-dimensional (1D and 2D) settings from the general 3D mean-field equation. The respective confinement is provided by trapping potentials with the cylindrical and planar symmetries, respectively. The resulting equations are nonpolynomial Schrödinger equations coupled to equations for the local transverse size of the trapped states. Numerical simulations demonstrate close agreement of results produced by the underlying 3D equation and the effective low-dimensional ones. We consider the ground state in these settings. In particular, analytical solutions are obtained for the effectively 2D non-interacting Fermi gas. Differences between the 1D and 2D configurations are highlighted. Finally, we analyse the dependence of the 1D and 2D density patterns of the trapped gas, in the presence of the OL, on the strengths of the confining and OL potentials, and on the scattering length which determines the strength of interactions between non-identical fermions.

DOI
10.1088/0953-4075/45/14/145304

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We study collective behaviour of Fermi gases trapped in various external potentials, including optical lattices (OLs), in the framework of the mean-field (hydrodynamic) description. Using the variational method, we derive effective dynamical equations for the one- and two-dimensional (1D and 2D) settings from the general 3D mean-field equation. The respective confinement is provided by trapping potentials with the cylindrical and planar symmetries, respectively. The resulting equations are nonpolynomial Schrödinger equations coupled to equations for the local transverse size of the trapped states. Numerical simulations demonstrate close agreement of results produced by the underlying 3D equation and the effective low-dimensional ones. We consider the ground state in these settings. In particular, analytical solutions are obtained for the effectively 2D non-interacting Fermi gas. Differences between the 1D and 2D configurations are highlighted. Finally, we analyse the dependence of the 1D and 2D density patterns of the trapped gas, in the presence of the OL, on the strengths of the confining and OL potentials, and on the scattering length which determines the strength of interactions between non-identical fermions.
We study collective behavior of Fermi gases trapped in various external potentials, including optical lattices (OLs), in the framework of the mean-field (hydrodynamic) description. Using the variational method, we derive effective dynamical equations for the one- and two-dimensional (1D and 2D) settings from the general 3D mean-field equation. The respective confinement is provided by trapping potentials with the cylindrical and planar symmetry, respectively. The resulting equations are nonpolynomial Schr% \"{o}dinger equations (NPSEs) coupled to equations for the local transverse size of the trapped states. Numerical simulations demonstrate close agreement of results produced by the underlying 3D equation and the effective low-dimensional ones. We consider the ground state in these settings. In particular, analytical solutions are obtained for the effectively 2D non-interacting Fermi gas. Differences between the 1D and 2D configurations are highlighted. Finally, we analyze the dependence of the 1D and 2D density patterns of the trapped gas, in the presence of the OL, on the strengths of the confining and OL potentials, and on the scattering length which determines the strength of interactions between non-identical fermions.
We present a summary of some recent theoretical results for matter-wave patterns in Fermi and Bose–Fermi degenerate gases, obtained in the framework of the quasi-mean-field approximation. We perform a dimensional reduction from the three-dimensional (3D) equations of motion to 2D and 1D effective equations. In both cases, comparison of the low-dimensional reductions to the full model is performed, showing very good agreement for ground-state solutions. Some complex dynamical regimes are reported too for the corresponding 1D systems.
Within the framework of the mean-field hydrodynamic model of a degenerate Fermi gas (DFG), we study, by means of numerical methods and variational approximation (VA), the formation of fundamental gap solitons (FGSs) in a DFG (or in a BCS superfluid generated by weak interaction between spin-up and spin-down fermions), which is trapped in a periodic optical-lattice (OL) potential. An effectively one-dimensional (1D) configuration is considered, assuming strong transverse confinement; in parallel, a proper 1D model of the DFG (which amounts to the known quintic equation for the Tonks-Girardeau gas in the OL) is considered too. The FGSs found in the first two bandgaps of the OL-induced spectrum (unless they are very close to edges of the gaps) feature a ( tightly bound ) shape, being essentially confined to a single cell of the OL. In the second bandgap, we also find antisymmetric tightly bound subfundamental solitons (SFSs), with zero at the midpoint. The SFSs are also confined to a single cell of the OL, but, unlike the FGSs, they are unstable. The predicted solitons, consisting of ∼10 4 –10 5 atoms, can be created by available experimental techniques in the DFG of 6 Li atoms.
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 use numerical simulation to study ultracold, quantum degenerate, atomic gases. In the first part of the thesis we study the effects of disorder, introduced via a bichromatic optical lattice, in one and two dimensional systems. We employ the Aubry-Andr\'{e} model and use time-dependent numerical simulations to investigate the disorder dependent transition to strong localisation present in the model. Weak s-wave interactions are added to the model and we observe the interaction between localisation and interaction induced self-trapping. We then add a tilted lattice potential to the model. In the homogeneous model this induces Bloch oscillations. While one might expect that a strong enough force will break the strong localisation or self-trapping, within the bounds of the single-band model, the trapping effect of the Bloch oscillations reinforces both of the other effects leading to increased confinement, albeit lacking the clear single frequency oscillation signature of pure Bloch oscillations. Along with the two dimensional bichromatic optical lattice we add a term to the Hamiltonian equivalent to that of a uniform external magnetic field on charged particles. Since the experimental realisation of this model would employ neutral atoms, the magnetic field is synthetic, the equivalent effect being produced by an appropriate set of lasers and magnetic fields. We show that in the ballistic regime (weak bichromatic disorder) the system displays positive magnetoresistance. Conversely in the strong localisation regime the system exhibits negative magnetoresistance. In the latter part of the thesis we use density functional theory to calculate the ground-state density of a harmonically trapped dipolar Fermi gas. We then use these to calculate the lowest energy collective mode oscillation frequencies under the hydrodynamic approximation. We find that increasing the strength of the dipoles has the effect of increasing the mode frequencies. The increase saturates for large dipole strengths. We verify this analytically and show that such is due to the local nature of the two dimensional energy functional and not dependent on the specific equation of state. We employ an average density approximation to construct an energy functional for the inhomogeneous, 2D degenerate Fermi gas. The ground-state densities for a cylindrically symmetric harmonic trap are compared to the Kohn-Sham results, showing extremely good agreement in the tail region and good agreement with the exact ground-state energy. We then do the same for higher order polynomial traps and obtain improved agreement for higher degree.
We study the properties of a single-component (spin polarized) degenerate dipolar Fermi gas of 161 Dy atoms using a hydrodynamic description. Under axially-symmetric trapping we suggest reduced one- (1D) and two-dimensional (2D) descriptions for the cigar and disc shapes, respectively. In addition to a complete numerical solution of the hydrodynamic model we also consider a variational approximation. For a trapped system under appropriate conditions, the variational approximation as well as the reduced 1D and 2D models are found to yield results for the shape, size and chemical potential of the system in agreement with the full numerical solution of the three-dimensional (3D) model. For the uniform system we consider anisotropic sound propagation in 3D. An analytical result for anisotropic sound propagation in a uniform dipolar degenerate Fermi gas is found to be in agreement with the results of numerical simulation in 3D.