Finite-temperature analysis of a quasi-two-dimensional dipolar gas

We present a finite-temperature analysis of a quasi-two-dimensional (Q2D) dipolar gas. To do this, we use the Hartree-Fock-Bogoliubov method within the Popov approximation. This formalism is a set of nonlocal equations containing the dipole-dipole interaction and the condensate and thermal correlation functions, which are solved self-consistently. We detail the numerical method used to implement the scheme. We present density profiles for a finite-temperature dipolar gas in Q2D and compare these results to those for a gas with zero-range interactions. Additionally, we analyze the excitation spectrum and study the impact of the thermal exchange.

Figure 1

$N_0/N$ as a function of temperature for the Cr dipolar gas (thick black line with circles), the contact gas (green dash-dotted line), and for 2000 ideal trapped particles (solid black line). Additionally, a gas of 300 Dy particles (dashed red with $+$) and one of 300 ideal particles (red open squares) are shown.

Figure 2

The chemical potential for three systems: dipolar (solid black), contact (dashed red), and dipolar without exchange (dash-dotted blue) and the interaction terms for each system in the Hamiltonian as a function of temperature. The contributing terms are the direct condensate ($\langle {\phi }_0\left|D_0\right|{\phi }_0\rangle /\mu$, triangles), the direct thermal ($\langle {\phi }_0\left|\tilde{D}\right|{\phi }_0\rangle /\mu$, circles), and the thermal exchange ($\langle {\phi }_0|\tilde{X}/|{\phi }_0\rangle /\mu$, squares) interactions.

Figure 3

The total and condensate density of (a) a Cr dipolar gas and (b) a contact gas, and the comparison of the total densities (c). The total density is in blue (black) for the dipolar (contact) gas, and the condensate density is in red. The temperatures from top to bottom are $T/T_{c0}=$ 0.25 (dashed), 0.55 (dotted), 0.75 (dash-dotted), and 0.90 (solid). (c) The dipolar gas is more dense in the center of the trap at all temperatures.

Figure 4

Comparison of densities with and without thermal exchange. Both the total and condensate densities are shown at four temperatures; from top to bottom, $T/T_{c0}=$ 0.25 (dashed), 0.55 (dotted), 0.75 (dash-dotted), and 0.90 (solid). (a) A Cr gas with (blue) and without (black) the thermal exchange; the corresponding condensate density with (red) and without (green) thermal exchange is shown. (b) For a Dy gas, the total density with (blue) and without (black) the thermal exchange; the corresponding condensate density with (red) and without (green) thermal exchange is shown. Both the condensate and total densities without thermal exchange have a lower central density. (c) A comparison of Dy (black) and Cr (blue) with a large number of atoms (3700) when the chemical potentials are the same at zero temperature.

Figure 5

The excitation spectra of the dipolar gas (blue circles), dipolar gas without exchange (red triangles), and contact gas (black pluses) as functions of temperature for $N=2000$. The azimuthal symmetry of the excitations is shown next to each curve.

Figure 6

The excitation spectrum as a function of $N$ for the dipolar gas (blue circles) at $T/T_{c0}=0.05$. The plots on the right are contour plots of the quasiparticle, $u_{\alpha }$, modes for $N=5000$ dipolar atoms. The contour lines are in 0.25 increments of the maximum value of $u_{\alpha }$ (i.e., 0.75,0.5,$...,-0.75$). Additionally, the negative regions are shaded blue. The condensate density contours (dashed black) are on the same scale.