Low-energy scatterings and pseudopotential of polarized quadrupoles

Interatomic interactions in ultracold atomic gases are of fundamental importance in determining the properties of the systems. Under a ultralow temperature, the van der Waals force between two neutral atoms can be described by a contact potential characterized by a single $s$-wave scattering length. Such a simplification has gained great success in cold atomic physics (see, e.g., Refs. Stringari1999 ; Bloch2008 ). In addition to the isotropic contact interaction, the long-range and anisotropic dipole-dipole interaction may become dominant and lead to the dipolar quantum gases for particles with large magnetic moments  Baranov2002 ; Baranov2008 ; Lahaye2009 ; Baranov2012 ; Bottcher2021 ; Guo2021 . For the mean-field treatment of the dipolar gases, an important step is to identify that the interaction between two identical bosonic dipoles can be modeled by a pseudopotential consisting of a $s$-wave contact potential and a bare dipole-dipole interaction potential whose scattering amplitude reproduces that of the real potential away from scattering resonance You1998 ; DDI-pseu1 ; DDI-pseu2 .

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 Nagel2003 ; Loftus2001 ; Takahashi2007 ; Schreck2013 ; Takahashi2011 ; Schreck2010 ; Sterr2009 ; Schreck2009 ; Killian2009 and homonuclear diatomic molecules Stellmer2012 ; Reinaudi2012 ; Gould2015 ; Schreck2017 . Following the approach for the dipolar interaction, a widely used pesudopotential for the quadrupole-quadrupole interaction (QQI) between two polarized (along the $z$ axis) bosonic quadrupoles is Li2013 ; Lahrz2015 ; Andreev2017 ; Wang2018

where $M$ is the mass of the bosons, $a_{00}$ is the $s$-wave scattering length, $g_{Q}=\Theta^{2}/(\sqrt{\pi}\varepsilon_{0})$ is the QQI strength with $\Theta$ being the electric quadrupole moment and $\varepsilon_{0}$ the vacuum permittivity, $r=|{\mathbf{r}}|$, and $\hat{\mathbf{r}}={\mathbf{r}}/r$. In a relevant study, Pikovski considered the general form for the anisotropic interaction Pikovski2014 .

Armed with pseudopotential Eq. (1), various properties of quadrupolar gases were investigated for ultracold quadrupolar Bose gases. In particular, Li et al. studied the shapes, stability, mobility, and collisions of solitons of quadrupolar gases in two-dimensional lattices Li2013 . Lahrz et al. explored the exotic roton excitations in a two-dimensional quadrupolar Bose-Einstein condensates (BECs) Lahrz2015 . Andreev studied the Bogoliubov spectrum of a BEC in the presence of both dipolar and quadrupolar interactions Andreev2017 . Wang and Yi investigated the ground-state properties and stability of quadrupolar condensates by numerically solving the Gross-Pitaevskii equation Wang2018 .

Furthermore, Bhongale et al. showed that QQI might lead to the unconventional Bardeen-Cooper-Schrieffer and charge density wave phases for quadrupolar Fermi gases trapped in a 2D square optical lattice Bhongale2013 . Huang et al. found that quadrupolar Fermi gases in coupled one-dimensional tubes supported the triplet superfluid and spin-density wave phases Huang2014 . More recently, using the diffusion Monte Carlo technique, Astrakharchik et al. predicted a quantum phase transition from a gas to a solid phase in a two-dimensional Bose system with quadrupolar interactions Astrakharchik2021 .

For the experimental detection of the quadrupolar effects, Lahrz et al. proposed to measure the mean-field induced frequency shift in a two-dimensional optical square lattice of Yb or Sr atoms in the ${}^{3}P_{2}$ state Lahrz2014 . Interestingly, Han et al. experimentally observed the quadrupolar blockade in a gas of Rb atoms Han2022 .

Although the pseudopotential Eq. (1) is widely used, its validity has not been strictly checked through scatterings calculations DDI-pseu1 ; DDI-pseu2 ; You1998 . In this work, we study the low-energy scattering of two identical particles interacting via a simple model potential consisting of a hard core and a bare QQI. We show that, similar to the dipolar scatterings You1998 ; You2001 ; Blume2008 , a series of broad resonances appear on the $s$-wave scattering length as the quadrupolar interaction strength increases. Interestingly, sequences of broad resonances also appear on the generalized scattering lengths of the $p$ wave for fermions and of the $s$ and $d$ wave coupling for bosons, which is in striking contrast to dipolar scatterings. We further show that the scattering amplitudes for higher partial waves with incoming ($l$) and outgoing ($l^{\prime}$) channels satisfying $l+l^{\prime}\geq 4$ are determined by the first Born approximation. These observations inspire us to propose new pseudopotentials for quadrupolar interaction which incorporates the anisotropic short-range contributions due to the $p$-wave scattering for fermions and the $s$ and $d$ waves coupling for bosons. Finally, to further understand the scattering resonances, we also study the bound-state properties of two particles in both free space and harmonic traps.

The rest of this paper is organized as follows. In Sec. II, we introduce the model potential and give a brief analysis to the threshold behavior of the quadrupolar scattering. In Sec. III, we present results on the generalized scattering lengths and the bound-state properties in the vicinity of collision resonances. We also propose a new pseudopotential for the quadrupolar interaction between identical bosons or fermions. Finally, we conclude in Sec. IV.

We consider the collisions between two identical polarized quadrupoles. The interaction potential is modeled as

where we have introduced a short-distance truncation $r_{c}$ such that the interaction potential is a hard sphere for $r<r_{c}$ and a pure quadrupolar interaction for $r>r_{c}$. Apparently, the QQI is anisotropic as it depends on the polar angle $\theta$ of $\hat{\mathbf{r}}$. In particular, the QQI is repulsive along $\theta=0^{\circ}$ and $90^{\circ}$ and is most attractive along $\theta=49.1^{\circ}$. We point out that the quadrupolar interactions were also involved in the studies of the cold collisions between the metastable alkaline-earth atoms QQI-scattering1 ; Kokoouline2003 ; QQI-scattering2 .

The quadrupolar scatterings are described by the Schrödinger equation

where $\mu=M/2$ is the reduce mass of the colliding atoms and $E=\hbar^{2}k^{2}/(2\mu)$ is the incident energy with $k=|{\mathbf{k}}|$ being the incident momentum. To proceed, we expand the scattering wave function in terms of the partial waves, i.e., $\psi({\mathbf{r}})=\sum_{lm}r^{-1}\phi_{lm}(r)Y_{lm}(\hat{\mathbf{r}})$, where $l$ is the orbital angular momentum quantum number and $m$ is projection quantum number. After substituting the partial wave expansion into Eq. (5), we obtain, for $r>r_{c}$, the coupled equations

where $\bar{g}_{Q}=g_{Q}\mu/\hbar^{2}$ characterizes the quadrupolar interaction strength and is of the dimension of volume and

Apparently, because the model potential $V_{\rm model}$ conserves the $z$ component of the orbital angular momentum, $\zeta_{lm}^{l^{\prime}m^{\prime}}=0$ if $m\neq m^{\prime}$. Without loss of generality, we shall restrict ourselves to the $m=0$ case as scattering channels with different $m$ quantum numbers can be treated separately. Moreover, the $3j$ symbols in Eq. (II) imply that $\zeta_{lm}^{l^{\prime}m^{\prime}}$ is nonzero only when $|l-l^{\prime}|\leq 4\leq l+l^{\prime}$ and $l+l^{\prime}$ is even. As a result, QQI does not directly contribute to the lowest partial waves due to $\zeta_{00}^{00}=\zeta_{00}^{20}=0$ for bosons and $\zeta_{10}^{10}=0$ for fermions.

At sufficiently large $r$, the scattering wave functions satisfy the asymptotical boundary conditions

where $j_{l}(x)$ and $n_{l}(x)$ are the spherical Bessel and spherical Neumann functions, respectively, and $K^{l^{\prime}0}_{l0}$ is the element of the $K$ matrix corresponding to the incoming channel $(l0)$ and the outgoing channels $(l^{\prime}0)$. Physically, the first term at the right-hand-side of Eq. (8) is the free spherical wave solution and the second one accounts the contributions of the interaction. For a spherical potential, the $K$ matrix elements reduce to the familiar form $K^{l^{\prime}0}_{l0}=\delta_{ll^{\prime}}\tan\delta_{l}(k)$ with $\delta_{l}(k)$ being the phase shift. While for an $1/r^{n}$-type anisotropic interaction, $K_{l0}^{l^{\prime}0}$ behaves in the low-energy limit ($k\to 0$) as $k^{l+l^{\prime}+1}$ if $l+l^{\prime}<n-3$ and as $k^{n-2}$ otherwise TYWu . Consequently, in the low-energy limit, the generalized scattering lengths are then defined as

where $a_{00}$ has dimension of length and all other $a_{ll^{\prime}}$’s are of dimension of volume. To find the generalized scatterings, we numerically integrate Eq. (6) from $r=r_{c}$ up to $10^{4}r_{c}$ using Johnson’s log-derivative propagator method log-Johnson . The $K$ matrix elements can then be obtained by matching the scattering wave function with the asymptotic boundary conditions Eq. (8), which subsequently leads to the generalized scattering lengths.

Before we present the results on the generalized scattering lengths and the bound states, let us first specify the interaction parameter covered in this work. The typical quadrupole moment of alkaline-earth atoms and homonuclear molecules is about $10-40$ a.u. Dere2001 ; Loftus2002 ; Santra2003 ; Santra2004 ; Buch2011 ; Byrd2011 and, without loss of generality, we choose $r_{c}=100$ a.u.. Then the dimensionless quadrupolar interaction strength $\bar{g}_{Q}/r_{c}^{3}$ can be as large as 1000 for Yb atom in ${}^{3}P_{2}$ state ($\Theta=30$ a.u. Buch2011 ), which, as shall be shown, is sufficiently large for the experimental observations of the quadrupolar scattering effects. It can be estimated that the magnetic dipole-dipole interaction between Yb atoms is much smaller than the QQI within the interatomic distance of a few hundreds Bohr radii, a range in which the atomic collision takes place Kokoouline2003 . We therefore neglect the magnetic dipole-dipole interaction in our calculations.

Numerically, we solve Eq. (5) with the incident energies $E/E_{r_{c}}=4\times 10^{-3}$, $4\times 10^{-4}$, and $4\times 10^{-6}$, where $E_{r_{c}}=\hbar^{2}/(\mu r_{c}^{2})$ is a characteristic energy associated with $r_{c}$. For Yb atoms, these incident energies correspond to temperatures $8\times 10^{-7}$, $8\times 10^{-8}$, and $8\times 10^{-10}\,{\rm K}$, respectively. It is found numerically that the generalized scattering lengths quickly converge as the collision energy $E$ is lowered. Finally, for practical purpose, we introduce a truncation $l_{\rm cut}$ for $l$ in numerical calculations. It turns out that, for all results presented in this work, $l_{\rm cut}=34$ for bosons and $35$ for fermions are sufficient to ensure the convergence of the scattering wave functions.

In summary, we have studied the low-energy scattering and the bound-state properties of two identical particles interacting via QQI. We numerically compute the generalized scattering lengths $a_{ll^{\prime}}$ as functions of the quadrupolar interaction strength. It has been shown that the short-range part of the interaction potential gives rise to $a_{00}$ and $a_{02}$ for bosons and $a_{1m}$ for fermions. These generalized scattering lengths exhibit a series of scattering resonances as the quadrupolar interaction strength grows. On the other hand, the long-range part of the interaction, i.e., the bare QQI, contributes to the generalized scattering lengths through the first Born approximation. Consequently, we propose new pseudopotentials that correctly take into account the contributions of $a_{02}$ and $a_{1m}$ for bosonic and fermionic quadrupoles, respectively. These pseudopotentials should pave the way for studying the many-body physics of the quadrupolar quantum gases. Finally, for a better understanding the scattering resonances, we have also presented a detailed analysis of the bound-state properties in the vicinity of the resonances.