Mathematical and numerical studies on ground states of trapped unitary Fermi gases

TL;DR

The paper studies ground states of unitary Fermi gases described by a nonlinear Schrödinger equation with quantum pressure and rotation. It develops dimensionless and reduced models for common trap geometries, proves existence and (where possible) uniqueness of ground states, and introduces a regularized gradient-flow method to compute ground states in nonrotating and rotating settings. It demonstrates that quantum pressure qualitatively changes vortex lattice structures, particularly at weak interactions, and validates dimension-reduction and Thomas-Fermi regimes. The framework provides a robust approach to explore vortex patterns and lays groundwork for future dynamics studies in unitary Fermi gases.

Abstract

We mathematically and numerically study the ground states of unitary Fermi gases. Starting from the three-dimensional nonlinear Schrödinger equation that contains a quantum pressure term and an angular momentum rotation term, we first nondimensionalize the equation and then obtain its one-dimensional and two-dimensional counterparts in some limit regimes of the external potentials. Existence and uniqueness of the ground states of the unitary Fermi gases are studied with/without the angular momentum rotation term. We present a regularized normalized gradient flow method to compute the ground states of trapped unitary Fermi gases. Our numerical results show that the quantum pressure term has a significant effect on the ground state properties. Specifically, with the presence of the quantum pressure term, the vortex lattices are very different from those obtained in conventional Bose-Einstein condensation.

Figures (8)

• Figure 1: Ground states in the one dimensional harmonic potential $V_1(x) = \frac{1}{2}x^2$.
• Figure 2: Ground states in the one dimensional optical lattice potential $V_2(x) = \frac{1}{2}x^2+5\sqrt{2}\sin^2\left(\frac{\pi x}{2}\right)$.
• Figure 3: Numerical verification of the dimension reduction approximations for a cigar-shaped condensate under the harmonic potential $V({\bf x} ) = (\gamma_x^2 x^2 + \gamma_y^2 y^2 + \gamma_z^2 z^2)/2$. Here we fix $\gamma_x = 1$ and set $\gamma_y = \gamma_z = 50$, 100, and 200, respectively. The first row compares the normalized axial profile $\phi_{3D}(x) := \iint \phi(x,y,z) \, dydz / \| \iint \phi(x,y,z) \, dydz \|$, where $\phi(x,y,z)$ is the numerical ground state computed from the full 3D model, with the corresponding normalized solution $\phi_{1D}(x)$ obtained from the effective one-dimensional model \ref{['gpe1d']}, and with the Thomas–Fermi approximation $\phi_{1D}^{TF}(x)$\ref{['TFgs']}. The second row compares the normalized transverse profile $\phi_{\perp}(y,z=0) := \int \phi(x,y,z=0) \, dx / \| \int \phi(x,y,z) \, dx \|$ with the Gaussian ansatz \ref{['0gs']}.
• Figure 4: Numerical verification of the dimension reduction approximations for a disk-shaped condensate under the harmonic potential $V({\bf x} ) = (\gamma_x^2 x^2 + \gamma_y^2 y^2 + \gamma_z^2 z^2)/2$. Here we fix $\gamma_x = \gamma_y = 1$ and choose $\gamma_z = 50$, 100 and 200, respectively. The first row compares the normalized solution $\phi_{3D}(x, y=0) := \int \phi(x,y=0,z) \, dz / \| \int \phi(x,y,z) \, dz \|$, where $\phi(x,y,z)$ is the numerical solution computed from the 3D model, with the normalized numerical solution $\phi_{2D}(x, y=0)$, computed from the effective 2D model \ref{['gpe2d']}, and with the Thomas-Fermi approximation $\phi_{2D}^{TF}(x, y=0)$\ref{['TFgs']}. The second row compares the normalized axial profile $\phi_{\perp}(z) := \int\phi(x,y,z) \, dxdy / \| \iint \phi(x,y,z) \, dx dy \|$ with the Gaussian ansatz \ref{['0gs']}.
• Figure 5: Ground states (top) and central vortex ground states with index $m = 1$ (bottom) in harmonic potential $V(x,y) = \frac{1}{2}(x^2+y^2)$.

Theorems & Definitions (6)

• Lemma 3.1
• Lemma 3.2
• Theorem 3.1
• Theorem 3.2
• Remark 4.1
• Remark 4.2