Simultaneous dipole and quadrupole moment contribution in the Bogoliubov spectrum: Application of the non-integral Gross–Pitaevskii equation

Abstract

We present the Gross-Pitaevskii equation for Bose-Einstein condensates (BECs) possessing the electric dipole and the electric quadrupole moments in a non-integral form. These equations are coupled with the Maxwell equations. The model under consideration includes the dipole-dipole, the dipole-quadrupole, and the quadrupole-quadrupole interactions in terms of the electric field created by the dipoles and quadrupoles. We apply this model to obtain the Bogoliubov spectrum for three dimensional BECs with a repulsive short-range interaction. We obtain an extra term in the Bogoliubov spectrum in compare with the dipolar BECs. We show that the quadrupole-quadrupole interaction gives a positive contribution in the Bogoliubov spectrum. Hence this spectrum is stable.

Low-energy scatterings and pseudopotential of polarized quadrupoles

“…Another interesting platform for exploring the novel properties of interactions is the ultracold gases consisting of particles possessing large quadrupole moment, such as alkaline-earth and rare-earth atoms in the metastable 3 P 2 states [12][13][14][15][16][17][18][19][20] and homonuclear diatomic molecules [21][22][23][24]. Following the approach for the dipolar interaction, a widely used pesudopotential for the quadrupolequadrupole interaction (QQI) between two polarized (along the z axis) bosonic quadrupoles is [25][26][27][28] V qq (r) = 4π 2 a 00 M δ(r)…”

“…Lahrz et al explored the exotic roton excitations in a two-dimensional quadrupolar Bose-Einstein condensates (BECs) [26]. Andreev studied the Bogoliubov spectrum of a BEC in the presence of both dipolar and quadrupolar interactions [27]. Wang and Yi investigated the groundstate properties and stability of quadrupolar condensates by numerically solving the Gross-Pitaevskii equation [28].…”

Low-energy scatterings and pseudopotential of polarized quadrupoles

“…Another interesting platform for exploring the novel properties of interactions is the ultracold gases consisting of particles possessing large quadrupole moment, such as alkaline-earth and rare-earth atoms in the metastable 3 P 2 states [12][13][14][15][16][17][18][19][20] and homonuclear diatomic molecules [21][22][23][24]. Following the approach for the dipolar interaction, a widely used pseudopotential for the quadrupole-quadrupole interaction (QQI) between two polarized (along the z axis) bosonic quadrupoles is [25][26][27][28] V qq (r) = 4πℏ 2 a 00 M δ(r)…”

“…Lahrz et al explored the exotic roton excitations in a two-dimensional quadrupolar Bose-Einstein condensate (BEC) [26]. Andreev studied the Bogoliubov spectrum of a BEC in the presence of both dipolar and quadrupolar interactions [27]. Wang and Yi investigated the ground-state properties and stability of quadrupolar condensates by numerically solving the Gross-Pitaevskii equation [28].…”

Trapped Bose–Einstein condensates with quadrupole–quadrupole interactions

“…[17] Recently, a new quantum simulation platform based on atoms or molecules with electric quadrupole-quadrupole interaction (QQI) was theoretically proposed. [18][19][20][21][22][23][24] The quantum phases of quadrupolar Fermi gases in a two-dimensional (2D) optical lattice [18] and in two coupled one-dimensional (1D) pipes [20] were studied. Lahrz et al proposed to detect quadrupolar interactions in ultracold Fermi gases via the interaction-induced mean-field shift.…”