Pseudopotential treatment of two aligned dipoles under external harmonic confinement

Dipolar Bose and Fermi gases, which are currently being studied extensively experimentally and theoretically, interact through anisotropic, long-range potentials. Here, we replace the long-range potential by a zero-range pseudopotential that simplifies the theoretical treatment of two dipolar particles in a harmonic trap. Our zero-range pseudopotential description reproduces the energy spectrum of two dipoles interacting through a shape-dependent potential under external confinement very well, provided that sufficiently many partial waves are included, and readily leads to a classification scheme of the energy spectrum in terms of approximate angular momentum quantum numbers. The results may be directly relevant to the physics of dipolar gases loaded into optical lattices.

Figure 1

Relative eigenenergies $E$ for (a) two identical bosonic dipoles and (b) two identical fermionic dipoles interacting through $V_{\mathit{pp},\mathit{reg}}$ [using $a_{00}=0$ in (a)] under spherical harmonic confinement as a function of $D_{*}∕a_{ho}$. The line style indicates the approximate quantum number of the corresponding eigenstates. In (a), a solid line refers to $l\approx 0$, a dashed line to $l\approx 2$, and a dotted line to $l\approx 4$; in (b), a solid line refers to $l\approx 1$, a dashed line to $l\approx 3$, and a dotted line to $l\approx 5$.

Figure 2

(Color online) Panel (a) shows the relative energies $E$ for two aligned identical bosonic dipoles under external spherical harmonic confinement as a function of $D_{*}∕a_{ho}$. Solid lines show the numerically determined energies obtained using $V_{\mathit{\text{model}}}$ with $b=0.0097a_{ho}$. Crosses show the energies obtained using $V_{\mathit{pp},\mathit{reg}}$ with essentially infinitely many terms, and $a_{00}$ calculated for $V_{\mathit{\text{model}}}$. Panel (b) shows a blow-up of the energy region around $E\approx 5.5\hslash \omega$. Note that the horizontal axis in (a) and (b) are identical.

Figure 3

(Color online) Crosses show the relative eigenenergies $E$ as a function of $D_{*}∕a_{ho}$ for two identical bosons with $a_{00}=0$ interacting through $V_{\mathit{\text{model}}}$ with $b=0.0031a_{ho}$ in three different energy regions. Lines show $E$ for two identical bosons with $a_{00}=0$ interacting through $V_{\mathit{ps},\mathit{reg}}$ with $a_{l{l}^{\prime }}$ given by Eqs. (9, 10). As in Fig. 1a, solid, dashed, and dotted lines show the energies of levels characterized by approximate quantum numbers $l\approx 0$, 2, and 4. The agreement between the crosses and the lines is good at small $D_{*}∕a_{ho}$ but less good at larger $D_{*}∕a_{ho}$. Squares show the eigenenergies obtained for the energy-dependent pseudopotential at $D_{*}=0.242a_{ho}$; the agreement between the squares and the crosses is excellent, illustrating that usage of the energy-dependent $K$ matrix greatly enhances the applicability regime of $V_{\mathit{pp},\mathit{reg}}$.

Figure 4

Energy-dependent scattering lengths $a_{00}\left(k\right)$ (solid line), $a_{20}\left(k\right)$ (dashed line), and $a_{22}\left(k\right)$ (dash-dotted line) for the model potential $V_{\mathit{\text{model}}}$ with $D_{*}=78.9b$ as a function of the relative energy $E$. In oscillator units, $V_{\mathit{\text{model}}}$ is characterized by $b=0.0031a_{ho}$ and $D_{*}=0.242a_{ho}$. For comparison, horizontal dotted lines show the energy-independent scattering lengths $a_{22}$, Eq. (9), and $a_{20}$, Eq. (10), calculated in the first Born approximation.