Properties of a static dipolar impurity in a 2D dipolar BEC

Dipolar BECs have long been of interest in cold atom physics due to the large variety of physics they display Lahaye et al. (2009); Chomaz et al. (2022). The first such system was a condensate of ⁵²Cr atoms discovered in 2004 Griesmaier et al. (2005) utilizing its magnetic dipole moment, but BECs using electric dipole moments have recently been realized Bigagli et al. (2024). While the condensates themselves are of interest, experiments with impurity species implanted in them may become of great interest in the coming years. Boson impurities in non-dipolar condensates have been well-studied both experimentally Hu et al. (2016); Jørgensen et al. (2016) and theoretically Scazza et al. (2022); Mistakidis et al. (2023); Grusdt et al. (2024).

Given the high degree of experimental control and precision, cold atomic systems are a great sandbox for impurity physics. When an impurity is introduced into a cold atomic system, it becomes dressed by the medium particles. These interactions alter the properties of the impurity from its bare form and also allow the impurity to act as a local probe of its environment  Volosniev et al. (2019); Mehboudi et al. (2019); Bouton et al. (2020). The impurity may also alter properties of the bath local to the impurity, as it will change the bath density around it. The extent of its effects depend on both the strength and range of the impurity’s interactions and the properties of the medium. These effects have been studied experimentally and have also been studied in the case of two-dimensional (2D) systems Zhang et al. (2012); Ong et al. (2015); Koschorreck et al. (2012).

There has been other work with impurities in dipolar media Kain and Ling (2014); Ardila and Pohl (2018); Volosniev et al. (2023); Shukla et al. (2024). Most of these works Kain and Ling (2014); Ardila and Pohl (2018); Volosniev et al. (2023) have been in momentum space, while our work is in coordinate space. This gives us access to important quantities such as the density. Our aim is to further the exploration of impurities in cold dipolar gases with a study in which the impurity is also a dipole. Furthermore, the gas is confined to two spatial dimensions, which has been less explored than 3D. This is in contrast to our previous work, Shukla et al. (2024), where we worked in 3D and with a Gaussian impurity which was strongly interacting and had other tuneable properties. In our system, the dipolar BEC is confined in a 2D harmonic trap with a dipolar impurity implanted inside the trap. The dipoles are polarized in the $z$-direction, and we have studied both of the extreme orientations, where the trap contains the dipoles in the plane perpendicular to the polarization direction or parallel to it.

This paper is structured as follows: after the introduction, we will discuss the methods used in our calculations, we will then report our results for the two geometries, discuss them, and then conclude.

Our dipolar BEC consists of $N$ dipoles in a 2D harmonic trap with a single impurity dipole implanted inside. The Hamiltonian of our system is:

	$\displaystyle H$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{N}\left(\frac{p_{k}^{2}}{2m}+V_{trap}(\mathbf{r}_{k})+\beta V_{dip}(\mathbf{r_{k}}-\mathbf{r})\right)+\frac{p^{2}}{2m_{imp}}$		(1)
			$\displaystyle+\sum_{i<j}^{N}V_{c}(\mathbf{r_{i}}-\mathbf{r_{j}})+\sum_{i<j}^{N}V_{dip}(\mathbf{r_{i}}-\mathbf{r_{j}}),$

where $m$ is the mass of a dipolar boson, $m_{imp}$ and $\mathbf{r}$ are the mass and position of the impurity dipole, respectively, $V_{c}$ is the contact interaction, $V_{dip}$ is the dipole-dipole interaction, which will be detailed subsequently, and $\beta$ is a scaling factor for the strength of the dipole-impurity interaction.

The contact interaction is $V_{c}(\mathbf{r})=4\pi\hbar^{2}a\delta(\mathbf{r})/m$, where $a$ is the scattering length. The dipole-dipole interaction is long-range and anisotropic. In general form, it reads for magnetic dipoles

$$V_{dip}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\frac{d^{2}-3(\mathbf{d}\cdot\mathbf{\hat{r}})^{2}}{|\mathbf{r}|^{3}},$$

(2)

where, $\mathbf{d}$ is the dipole moment, $\mu_{0}$ is the permeability of free space, and $\mathbf{\hat{r}}$ is the unit vector along $\mathbf{r}$. In our system, the dipoles are polarized in the $z$-direction by an external field. In this case, the dipole-dipole interaction can be written as

$$V_{dip}(\mathbf{r})=\frac{C_{dd}}{4\pi}\frac{1-3\cos^{2}(\theta_{d})}{r^{3}},$$

(3)

where, $\theta_{d}$ is the angle between $\mathbf{r}$ and $\mathbf{d}$. We have used the standard notation $C_{dd}=\mu_{0}d^{2}$, allowing us to introduce a useful length scale $a_{dd}=C_{dd}m/(12\pi\hbar^{2})$, known as the ’dipolar length’Lahaye et al. (2009). The dipolar length allows us to easily compare the relative strengths of interactions in the problem, by comparing it with the scattering length of the contact interaction. The scaling factor that appears in Eq. (1) allows us to treat an impurity with a different dipole moment than the background dipolar gas. In such a situation, Eqs. (2) and (3) would need two different dipole moments, $\mathbf{d_{g}}$ and $\mathbf{d_{i}}$, instead of the $d^{2}$ terms. We have chosen to handle this by using a scaling factor $\beta=|\mathbf{d_{i}}|/|\mathbf{d_{g}}|$, where $\mathbf{d_{g}}$ is the dipole moment of a bulk gas atom and $\mathbf{d_{i}}$ is the dipole moment of the impurity. In terms of the dipolar lengths $a_{dd,g}$ and $a_{dd,i}$ (the dipolar lengths of the gas and impurity, respectively), $\beta=\sqrt{\frac{a_{dd,i}m_{g}}{a_{dd,g}m_{i}}}$, where $m_{g}$ and $m_{i}$ are the masses of a dipolar gas atom and impurity atom, respectively. Note that the dipole-dipole interaction is singular at the origin. In order to treat this difficulty, at the origin we have taken the potential to be a constant value equal to the average of the potential value at the four nearest grid points to the origin in our numerical solution. The short-range behavior of the dipole-dipole interaction is beyond the scope of this paper and not vital to our findings.

In order to solve Eq. (1), we assume that our wave function $\psi$, can be written as a product of one-particle wave functions, i.e., $\psi(\mathbf{r_{1}},\mathbf{r_{2}},\dots,\mathbf{r_{N}})=\psi(\mathbf{r_{1}})\psi(\mathbf{r_{2}})\dots\psi(\mathbf{r_{n}})$. If there was no trap, we can transform the resulting Schrödinger equation ($H\psi=E\psi$) into the frame of the impurity and obtain Gross (1962); Volosniev and Hammer (2017); Hryhorchak et al. (2020); Jager et al. (2020); Drescher et al. (2020); Guenther et al. (2021):

	$\displaystyle-\frac{1}{2}\left(\frac{1}{m}+\frac{1}{{m_{imp}}}\right)\nabla^{2}\psi(\mathbf{r})+\beta V_{dip}(\mathbf{r})\psi(\mathbf{r})+\psi(\mathbf{r})\int\left[V_{c}(\mathbf{r^{\prime}}-\mathbf{r})+V_{dip}(\mathbf{r^{\prime}}-\mathbf{r})\right]|\psi(\mathbf{r^{\prime}})|^{2}d\mathbf{r^{\prime}}$
	$\displaystyle=E\psi(\mathbf{r}).$		(4)

This equation is a Gross-Pitaevskii equation (GPE). For numerical convenience as well as a recognition of experimental realtities, we will solve the GPE in a trap. If the trap is large, the properties of the impurity are insensitive to it. It also makes our results somewhat relevant for mobile impurities. Alternatively, one can use an external laser to pin the impurity at a certain location, such as the center of the trap Catani et al. (2012), which results in a similar GPE. We will solve the GPE numerically by adapting a publicly available code Kumar et al. (2015) to our problem. The code solves the GPE using the split-step Crank-Nicolson method.

Our system is in a harmonic trap, which we have initially chosen to be radially symmetric:

$$V_{trap}(\mathbf{r})=\frac{1}{2}m\left(\omega^{2}r^{2}+\omega^{2}_{3D}x_{3}^{2}\right),$$

(5)

where $r^{2}=x^{2}+y^{2}$ or $r^{2}=x^{2}+z^{2}$, depending on the confinement geometry of the system, $x_{3}=z$ or $y$ and $\omega_{3D}$ is the confinement frequency in the third dimension, and we will assume $\omega_{3D}\gg\omega$. In our calculations, we have chosen the oscillator length, $\ell=\sqrt{\hbar/m\omega}$ to be 1 $\mu$m, which is within the range of typical experimental values. Our dipolar length is 130$a_{0}$, which is the value for ¹⁶²Dy, which has a very large magnetic moment and is of great current experimental interest Chomaz et al. (2022). We have used $a=150a_{0}$ for the contact interaction scattering length. While this is close to the background value for Dy, this is also something that is tunable experimentally via Feschbach-Fano resonances Chin et al. (2010). It is also important for the stability of dipolar systems that $a>a_{dd}$ Lahaye et al. (2009).

The reader will have noticed that the Hamiltonian, Eq. (1), is a three-dimensional equation, whereas we are interested in studying 2D systems. As hinted at above, we work in quasi-2D, where one of the confining frequencies in the harmonic trap is large and we assume that the particles are in the ground state in that direction, which is a Gaussian wave function. That direction is then integrated out and the resulting equation is then in 2D Kumar et al. (2015).

In this section we will present our results starting with static properties calculated in the two different geometries mentioned in the introduction: confining the particles to the $xy$-plane (perpendicular to the dipole polarization) and the $xz$-plane (parallel to the dipole polarization). One of the static results we will show in both subsections is the self-energy of the impurity. The self-energy is defined as

$$E_{self}=E[\beta]-E[\beta=0],$$

(6)

where $E[\beta]$ is the energy of the system with an impurity of strength $\beta$. This quantity is an important characteristic for an impurity system as it indicates how the system’s energy is affected by the introduction of an impurity. Several different impurities were chosen with different strengths: ¹⁶²Dy, ⁵²Cr, and ¹⁶⁸Er. While at first glance, it may seem that ¹⁶²Dy cannot be an impurity in a gas of ¹⁶²Dy atoms, the impurity atom can be kept in a different hyperfine state and thus distinct from the background gas.

In this paper we have shown density and self-energy results for a dipolar impurity immersed in a 2D dipolar BEC for two different geometries. We have found that the impurity distorts the gas in its immediate environment while mostly leaving the gas far away unchanged. Our dynamic results show, however, that when an impurity is first introduced it can cause density fluctuations at a much greater distance. We have primarily used impurity strengths that are experimentally relevant and hope that our results provide some impetus for experimental exploration of these systems.

Some related systems that could be studied further include adding an additional impurity or impurities. Not only would they add further distortion to the BEC, but the impurities themselves would interact via the exchange of elementary excitations of the mediumParedes et al. (2024). One could also imagine tilting the confinement to orientations between the $xy$- and $xz$-plane extremes. One might expect interesting behavior at the so-called ’magic angle’$=\cos^{-1}(1/\sqrt{3})\approx 54.74^{\circ}$ where the dipolar interaction vanishes in one direction, but the interaction in the perpendicular direction would remain dipolar.

The authors acknowledge that this material is based upon work supported by the National Science Foundation/EPSCoR RII Track-1: Emergent Quantum Materials and Technologies (EQUATE), Award OIA-2044049.