Effective interactions between two impurities in quasi-two-dimensional dipolar Bose【-逻*辑*与-】ndash;Ei
本站小编 Free考研考试/2022-01-02
Fu-Lin Deng1,2, Tao Shi1,3, Su Yi,1,2,3,41CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100049, China
First author contact:4Author to whom any correspondence should be addressed. Received:2020-03-17Revised:2020-04-9Accepted:2020-04-9Online:2020-06-04
Abstract We investigate the effective interaction between two heavy impurities immersed in a quasi-two-dimensional dipolar Bose–Einstein condensate via a variation approach. We show that the mediated interaction is highly tunable via the contact and the dipole–dipole interactions between the background gas atoms. Interestingly, the mediated interaction potential may become an oscillating function of inter-impurity distance when roton excitation emerges under sufficiently strong dipolar interaction. Our system therefore provides an efficient way for tuning the mediated interaction between impurities. Keywords：dipolar gas;impurity;effective interaction;roton excitation
PDF (498KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Fu-Lin Deng, Tao Shi, Su Yi. Effective interactions between two impurities in quasi-two-dimensional dipolar Bose–Einstein condensates. Communications in Theoretical Physics, 2020, 72(7): 075501- doi:10.1088/1572-9494/ab8a2a
1. Introduction
When a quantum impurity immerses in and interacts with a macroscopic background, it forms a polaron with background particles. Dressed by the surroundings, polaron may have properties significantly different from the bare impurity particle. One typical example is an electron moving in lattice fields and coupled to vibrations of the underlying lattice [1, 2]. The concept is proved useful to understand a variety of physical systems, such as semiconductors [3, 4], magnetic materials [5, 6], and strongly correlated systems [7].
Ultracold atomic gases, with their outstanding controllability on inter-atom interactions [8] and on external potentials [9], have attracted tremendous interests in studying the polaron physics. Both fermion polarons [10–13] and boson polarons [14–16] have been realized experimentally. The large majority of theoretical studies of the boson polarons concentrate on properties of a single impurity, such as its ground-state properties [17–22], dynamics [23, 24], and Efimov effect [25]. Moreover, understanding the interaction between polarons is also interesting and essential. For instance, Bardeen studied the effective interaction between He3 in superfluid He4 via the exchange of virtual phonons [26]. In the context of cold atomic gases, the medium-mediated interactions between two impurities have been studied via the Born–Oppenheimer approximation [27, 28], in which the motion of two heavy impurities is decoupled from the light background atoms. We note that in those studies background atoms always interact with each other via short-range s-wave interaction.
In this work, we study the effective interaction between two heavy impurities immersed in quasi-two-dimensional (quasi-2D) dipolar Bose–Einstein condensates (BECs) by using the Chevy variation ansatz [27–30]. The interaction potentials between the two impurities mediated by the background condensate is computed in the Born–Oppenheimer limit. We show that by tuning the strength of the contact and the dipole–dipole interaction (DDI), one may dramatically alter the effective interaction between impurities. In particular, at the strong DDI limit, the effective interaction potential exhibits a oscillatory tail, indicating that the mediated interaction can be either attractive or repulsive depending on the inter-impurity distance. We also analytically demonstrate that the oscillation is attributed to roton excitation of the dipolar gas [31–34], which leads to singularities on the density of states at certain frequencies. The behavior has the same origin as the non-Markovian dephasing dynamics of a quantum two-level impurity coupled to a quasi-2D dipolar gas [35]. Finally, we show that when the DDI interaction between gas atoms is anisotropic on the 2D plane, the mediated interaction between impurities is also anisotropic.
This paper is organized as follows. In section 2, we introduce our model and derive the equation for the mediated interaction potential. Section 3 devotes to present our result of the mediated interactions under various circumstances. Finally, we summarize in 4.
2. Formulation
We consider two impurity atoms immersed into a ultracold gas of N polarized dipolar bosons (figure 1(a)). The atoms in the gas interact via the short-range collisional and the long-range dipolar interactions such that the two-body interaction potential is$ \begin{eqnarray}{V}_{G}^{(3{\rm{D}})}({\boldsymbol{r}})={g}_{0}\delta ({\boldsymbol{r}})+\displaystyle \frac{3{g}_{d}}{4\pi }\displaystyle \frac{1-3{\left(\hat{{\boldsymbol{d}}}\cdot \hat{{\boldsymbol{e}}}\right)}^{2}}{{r}^{3}},\end{eqnarray}$where ${g}_{0}=4\pi {{\hslash }}^{2}{a}_{0}/{m}_{G}$ with a0 being the s-wave scattering length and mG the mass of the gas atoms, gd represents the strength of the dipolar interaction, $\hat{{\boldsymbol{e}}}={\boldsymbol{r}}/r$, and $\hat{{\boldsymbol{d}}}$ is the unit vector along the polarized dipole moments of the atoms. As to the impurity atoms, we assume that they do not interact with each other directly, but both interact with gas atoms through s-wave collision, i.e.$ \begin{eqnarray}{V}_{{IG}}({\boldsymbol{r}})={g}_{{IG}}\delta ({\boldsymbol{r}}),\end{eqnarray}$where ${g}_{{IG}}=2\pi {{\hslash }}^{2}{a}_{3{\rm{D}}}/{m}_{{IG}}$ with a3D being the scattering length between an impurity and a gas atom, ${m}_{{IG}}\,={m}_{I}{m}_{G}/({m}_{I}+{m}_{G})$ being the reduced mass, and mI the mass of the impurity atoms.
For simplicity, we shall focus on 2D systems. To this end, we assume that the gas is tightly confined along the z axis by a harmonic potential$ \begin{eqnarray}{U}_{G}(z)=\displaystyle \frac{1}{2}{m}_{G}{\omega }_{z}^{2}{z}^{2},\end{eqnarray}$where ωz is trapping frequency. By further assume that ${\hslash }{\omega }_{z}$ is much larger than the interaction energies, the motion of the condensate atoms along the z axis is frozen to the ground state of the harmonic potential, i.e.$ \begin{eqnarray}{\phi }_{G}(z)=\displaystyle \frac{1}{{\left(\pi {{\ell }}^{2}\right)}^{1/4}}{{\rm{e}}}^{-{z}^{2}/(2{{\ell }}^{2})},\end{eqnarray}$where ${\ell }=\sqrt{{\hslash }/({m}_{G}{\omega }_{z})}$ is the harmonic oscillator width. While on the xy plane, the Bose gas is homogeneous on the xy plane with area density n. For simplicity, we assume that impurity atoms also possess the same wave function along the z axis.
Now, after integrating out the z variable, the system Hamiltonian, in the momentum space, takes the form$ \begin{eqnarray}\begin{array}{rcl}H & = & \displaystyle \sum _{{\boldsymbol{k}}}{\varepsilon }_{G}({\boldsymbol{k}}){b}_{{\boldsymbol{k}}}^{\dagger }{b}_{{\boldsymbol{k}}}+\displaystyle \frac{1}{2S}\displaystyle \sum _{{\boldsymbol{k}},{\boldsymbol{k}}^{\prime} ,{\boldsymbol{p}}}{\widetilde{V}}_{G}({\boldsymbol{p}}){b}_{{\boldsymbol{k}}^{\prime} -{\boldsymbol{p}}}^{\dagger }{b}_{{\boldsymbol{k}}+{\boldsymbol{p}}}^{\dagger }{b}_{{\boldsymbol{k}}}{b}_{{\boldsymbol{k}}^{\prime} }\\ & & +\displaystyle \sum _{{\boldsymbol{k}}}{\varepsilon }_{I}({\boldsymbol{k}}){c}_{{\boldsymbol{k}}}^{\dagger }{c}_{{\boldsymbol{k}}}+\displaystyle \frac{g}{S}\displaystyle \sum _{{\boldsymbol{k}},{\boldsymbol{k}}^{\prime} ,{\boldsymbol{p}}}{b}_{{\boldsymbol{k}}^{\prime} -{\boldsymbol{p}}}^{\dagger }{c}_{{\boldsymbol{k}}+{\boldsymbol{p}}}^{\dagger }{c}_{{\boldsymbol{k}}}{b}_{{\boldsymbol{k}}^{\prime} ,}\end{array}\end{eqnarray}$where ${\varepsilon }_{G,I}({\boldsymbol{k}})={{\hslash }}^{2}{k}^{2}/(2{m}_{G,I})$ are kinetic energy terms for condensate and impurity atoms with ${\boldsymbol{k}}\equiv ({k}_{x},{k}_{y})$ being the 2D momentum, S is the area of the quasi-2D condensate, and ${b}_{{\boldsymbol{k}}}$ and ${c}_{{\boldsymbol{k}}}$ are annihilation operators of the condensate and the impurity atoms, respectively. Moreover, $g={g}_{{IG}}/(\sqrt{2\pi }{\ell })$ and$ \begin{eqnarray}{\widetilde{V}}_{G}({\boldsymbol{k}})=\displaystyle \frac{1}{2\pi }\int {\widetilde{V}}_{G}^{(3{\rm{D}})}({\boldsymbol{q}}){{\rm{e}}}^{-{{\ell }}^{2}{q}_{z}^{2}/2}{\rm{d}}{q}_{z},\end{eqnarray}$where ${\boldsymbol{q}}=({k}_{x},{k}_{y},{q}_{z})$ and ${\widetilde{V}}_{G}^{(3{\rm{D}})}({\boldsymbol{q}})={g}_{0}-{g}_{d}[1-3{\left(\hat{{\boldsymbol{d}}}\cdot \hat{{\boldsymbol{q}}}\right)}^{2}]$ is the Fourier transform of the interaction potential ${V}_{G}^{(3{\rm{D}})}$. To proceed further, we assume, without loss of generality, that the dipole moment ${\boldsymbol{d}}$ always lies on the xz plane and makes an angle ϑ to the z axis. It is then be shown that$ \begin{eqnarray}{\widetilde{V}}_{G}({\boldsymbol{k}})=2\sqrt{2\pi }{\hslash }{\omega }_{z}{\ell }{a}_{0}\left[1+\chi {\tilde{v}}_{d}({\ell }{k}_{x},{\ell }{k}_{y})\right],\end{eqnarray}$where $\chi ={g}_{d}/{g}_{0}$ represents the relative DDI interaction strength and$ \begin{eqnarray}\begin{array}{rcl}{\tilde{v}}_{d}(x,y) & = & 3{\cos }^{2}\vartheta -1-3\sqrt{\displaystyle \frac{\pi }{2}}\\ & & \times \left(\rho {\cos }^{2}\vartheta -\displaystyle \frac{{x}^{2}}{\rho }{\sin }^{2}\vartheta \right){{\rm{e}}}^{{\rho }^{2}/2}\mathrm{erfc}\left(\displaystyle \frac{\rho }{\sqrt{2}}\right)\end{array}\end{eqnarray}$with $\rho =\sqrt{{x}^{2}+{y}^{2}}$ and $\mathrm{erfc}(\cdot )$ being the complementary error function.
At ultracold temperature, the gas is dominated by condensed atoms such that the Bose gas Hamiltonian, HG (the first line of equation (5)), can be approximately expressed in the quadratic form. Then, utilizing the Bogoliubov transformation, the uncondensed atoms are described by the quasiparticle Hamiltonian$ \begin{eqnarray}{H}_{B}=\sum _{{\boldsymbol{k}}\ne 0}{E}_{{\boldsymbol{k}}}{\beta }_{{\boldsymbol{k}}}^{\dagger }{\beta }_{{\boldsymbol{k}}},\end{eqnarray}$where the quasiparticle energy is [33]$ \begin{eqnarray}{E}_{{\boldsymbol{k}}}=\displaystyle \frac{1}{2}{\hslash }{\omega }_{z}\sqrt{{\left({\ell }k\right)}^{4}+P{\left({\ell }k\right)}^{2}\left[1+\chi {\tilde{v}}_{d}({\ell }{k}_{x},{\ell }{k}_{y})\right]}\end{eqnarray}$with $P=8\sqrt{2\pi }{\ell }{a}_{0}n$ being a dimensionless parameter measuring the contact interaction strength and ${\beta }_{{\boldsymbol{k}}}={u}_{{\boldsymbol{k}}}{b}_{{\boldsymbol{k}}}\,+{v}_{{\boldsymbol{k}}}{b}_{-{\boldsymbol{k}}}^{\dagger }$ is the quasiparticle annihilation operator with transformation coefficients ${u}_{{\boldsymbol{k}}}={\left[\left({\varepsilon }_{G}({\boldsymbol{k}})+n{\widetilde{V}}_{G}({\boldsymbol{k}}\right)/{E}_{{\boldsymbol{k}}}+1\right]}^{1/2}/\sqrt{2}$ and ${v}_{{\boldsymbol{k}}}={\left[\left({\varepsilon }_{G}({\boldsymbol{k}})+n{\widetilde{V}}_{G}({\boldsymbol{k}}\right)/{E}_{{\boldsymbol{k}}}-1\right]}^{1/2}/\sqrt{2}$. Interestingly, if the dipole moments are polarized along the z axis, i.e. $\vartheta =0$, the quasiparticle spectrum is isotropic on the xy plane. For sufficiently large χ, roton minimum appears in the excitation spectrum.
To investigate the mediated interaction between two impurities, we adopt the Born–Oppenheimer approximation [27, 28] by assuming that ${m}_{I}/{m}_{G}\to \infty $, which implies that the position of the impurity atoms are fixed. Then without loss of generality, we assume that one of the impurities lies at the origin and the other locates at the position ${\boldsymbol{r}}=(x,y)$. Under these condition, the Hamiltonian of the system reduces to$ \begin{eqnarray}\begin{array}{rcl}H({\boldsymbol{r}}) & = & \displaystyle \sum _{{\boldsymbol{k}}\ne 0}{E}_{{\boldsymbol{k}}}{\beta }_{{\boldsymbol{k}}}^{\dagger }{\beta }_{{\boldsymbol{k}}}+2{ng}\\ & & +\displaystyle \frac{g\sqrt{N}}{S}\displaystyle \sum _{{\boldsymbol{k}}\ne 0}({u}_{{\boldsymbol{k}}}-{\upsilon }_{{\boldsymbol{k}}})({\beta }_{{\boldsymbol{k}}}^{\dagger }+{\beta }_{-{\boldsymbol{k}}})(1+{{\rm{e}}}^{-{\rm{i}}{\boldsymbol{k}}\cdot {\boldsymbol{r}}})\\ & & +\displaystyle \frac{g}{S}\displaystyle \sum _{{\boldsymbol{k}},{\boldsymbol{k}}^{\prime} \ne 0}({u}_{{\boldsymbol{k}}}{u}_{{\boldsymbol{k}}^{\prime} }+{v}_{{\boldsymbol{k}}}{v}_{{\boldsymbol{k}}^{\prime} }){\beta }_{{\boldsymbol{k}}}^{\dagger }{\beta }_{{\boldsymbol{k}}^{\prime} }\left(1+{{\rm{e}}}^{-{\rm{i}}({\boldsymbol{k}}-{\boldsymbol{k}}^{\prime} )\cdot {\boldsymbol{r}}}\right),\end{array}\end{eqnarray}$with ${\boldsymbol{r}}$ being a parameter. We shall solve for the ground-state energy ${ \mathcal E }({\boldsymbol{r}})$ of the Hamiltonian(11) which is a function of ${\boldsymbol{r}}$, representing the effective interaction potential between two impurities. To this end, we introduce the trial wave functions [27, 30]$ \begin{eqnarray}| \psi \rangle ={\alpha }_{0}| 0\rangle +\sum _{{\boldsymbol{q}}\ne 0}{\alpha }_{{\boldsymbol{q}}}{\beta }_{{\boldsymbol{q}}}^{\dagger }| 0\rangle ,\end{eqnarray}$where $| 0\rangle $ represents BEC vacuum with two impurities. The ansatz(12) describes a variational state truncated to one Bogoliubov excitation.
Applying the variational principal $\langle \delta \psi | H({\boldsymbol{r}})-{ \mathcal E }| \psi \rangle =0$ with the effective Hamiltonian and the trial wave function, one obtains a set of two coupled equations$ \begin{eqnarray}\begin{array}{l}(2{ng}-{ \mathcal E }){\alpha }_{0}+\displaystyle \frac{g\sqrt{N}}{S}\displaystyle \sum _{{\boldsymbol{k}}}({u}_{{\boldsymbol{k}}}-{\upsilon }_{{\boldsymbol{k}}})\left({{\rm{e}}}^{-{\rm{i}}{\boldsymbol{k}}\cdot {\boldsymbol{r}}/2}+{{\rm{e}}}^{{\rm{i}}{\boldsymbol{k}}\cdot {\boldsymbol{r}}/2}\right){\alpha }_{{\boldsymbol{k}}}\\ \quad =\,0,\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}({E}_{{\boldsymbol{k}}}-{ \mathcal E }){\alpha }_{{\boldsymbol{k}}}+\displaystyle \frac{g\sqrt{N}}{S}({u}_{{\boldsymbol{k}}}-{\upsilon }_{{\boldsymbol{k}}})\left({{\rm{e}}}^{{\rm{i}}{\boldsymbol{k}}\cdot {\boldsymbol{r}}/2}+{{\rm{e}}}^{-{\rm{i}}{\boldsymbol{k}}\cdot {\boldsymbol{r}}/2}\right){\alpha }_{0}\\ +\,\displaystyle \frac{g}{S}\displaystyle \sum _{{\boldsymbol{k}}^{\prime} }({u}_{{\boldsymbol{k}}}{u}_{{\boldsymbol{k}}^{\prime} }+{v}_{{\boldsymbol{k}}}{v}_{{\boldsymbol{k}}^{\prime} })\left({{\rm{e}}}^{{\rm{i}}({\boldsymbol{k}}-{\boldsymbol{k}}^{\prime} )\cdot {\boldsymbol{r}}/2}+{{\rm{e}}}^{-{\rm{i}}({\boldsymbol{k}}-{\boldsymbol{k}}^{\prime} )\cdot {\boldsymbol{r}}/2}\right){\alpha }_{{\boldsymbol{k}}^{\prime} }=0.\end{array}\end{eqnarray}$Here the bare coupling strength, g, has an ultraviolet divergence which can be regularized according to the following procedure [36]$ \begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{1}{g} & = & \mathop{\mathrm{lim}}\limits_{{{\boldsymbol{k}}}_{0}\to 0}\left[-\displaystyle \frac{{m}_{G}}{\pi {{\hslash }}^{2}}\mathrm{ln}\left(\displaystyle \frac{{{k}}_{0}{a}_{2D}{{\rm{e}}}^{\gamma }}{2}\right)\right.\\ & & \left.+{ \mathcal P }{\displaystyle \int }_{{k}\lt {\rm{\Lambda }}}\displaystyle \frac{{{\rm{d}}}^{2}{k}}{{\left(2\pi \right)}^{2}}\displaystyle \frac{1}{{\varepsilon }_{G}({{\boldsymbol{k}}}_{0})-{\varepsilon }_{G}({\boldsymbol{k}})}\right],\end{array}\end{eqnarray}$where $\gamma =0.577\,216\cdots $ is Euler’s constant, ${ \mathcal P }$ stands for the principal value integral, and Λ is a cutoff of the momentum which will be taken to be $\infty $ at the end of the calculation. Moreover, a2D is the quasi-2D scattering length between impurity and gas atoms which relates to the 3D scattering length a through the relation ${a}_{2{\rm{D}}}{e}^{\gamma }/(2\ell )=\sqrt{\pi /B}{{\rm{e}}}^{-\sqrt{\pi /2}\ell /{a}_{3{\rm{D}}}}$ with B=0.915 [37]. For simplicity, we shall treat a2D as control parameter. Combining equations (13)–(15), one gets the self-consistent equation for ${ \mathcal E }$, i.e.$ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{2n}{{ \mathcal E }}=\mathop{\mathrm{lim}}\limits_{{{k}}_{0}\to 0}\left[-\displaystyle \frac{1}{\pi {{\ell }}^{2}{\hslash }{\omega }_{z}}\mathrm{ln}\left(\displaystyle \frac{{{k}}_{0}{a}_{2{\rm{D}}}{{\rm{e}}}^{\gamma }}{2}\right)\right.\\ \left.+\,{ \mathcal P }\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{2}{k}}{{\left(2\pi \right)}^{2}}\left(\displaystyle \frac{1}{{\varepsilon }_{G}({{\boldsymbol{k}}}_{0})-{\varepsilon }_{G}({\boldsymbol{k}})}-\displaystyle \frac{{u}_{{\boldsymbol{k}}}^{2}(1+\cos ({k}\cdot {\boldsymbol{r}}))}{{ \mathcal E }-E({\boldsymbol{k}})}\right)\right].\end{array}\end{eqnarray}$We note that this equation is the same as that derived in [28], though we consider a quasi-$2{\rm{D}}$ system here. Of particular interest, when the interaction between gas atoms is isotropic on the xy plane, we may integrate out the angle of ${\boldsymbol{k}}$ and find that$ \begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{2n}{{ \mathcal E }} & = & \mathop{\mathrm{lim}}\limits_{{k}_{0}\to 0}\left[-\displaystyle \frac{1}{\pi {{\ell }}^{2}{\hslash }{\omega }_{z}}\mathrm{ln}\left(\displaystyle \frac{{k}_{0}{a}_{2{\rm{D}}}{{\rm{e}}}^{\gamma }}{2}\right)\right.\\ & & +\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{0}^{\infty }k{\rm{d}}k\left(\displaystyle \frac{{u}_{k}^{2}}{E(k)-{ \mathcal E }}-\displaystyle \frac{1}{{\varepsilon }_{G}(k)-{\varepsilon }_{G}({k}_{0})}\right)\\ & & \left.+\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{0}^{\infty }k{\rm{d}}k\displaystyle \frac{{u}_{k}^{2}{{J}}_{0}({kr})}{E(k)-{ \mathcal E }}\right],\end{array}\end{eqnarray}$where ${{J}}_{n}(x)$ is the Bessel functions of the first kind. Solving equation (16) or (17) will give rise to the ground-state energy ${ \mathcal E }(r)$ which determines the behavior of the mediated interaction between impurities.
3. Results
To present our result, we consider a concrete example by assuming a typical harmonic oscillator length ${\ell }=2.5\,\times {10}^{-5}\,\mathrm{cm}$. We further assume a typical peak gas density ${10}^{14}\,{\mathrm{cm}}^{-3}$ along the z axis which leads to the area density $n=2.77{{\ell }}^{-2}$. If we take the scattering length for gas atoms to be a0=5.9 nm [38, 39], we have P=1.4. Further, we shall treat the relative DDI strength χ, the scattering length between impurity and gas atom ${a}_{2{\rm{D}}}$, and the orientation angle of the dipole moments ϑ as control parameters. In addition, to simplify the calculations, we always use ℓ and ${\hslash }{\omega }_{z}$ as the length and the energy units, respectively.
Figure 1.
New window|Download| PPT slide Figure 1.Schematic diagrams of (a) two impurities immersed in a ultracold quasi-2D dipolar gas and (b) the typical excitation spectrum of dipolar BECs under different χ's. Roton excitation emerges under sufficiently large DDI strength.
Let us first consider the simplest situation by assuming that the Bose gas is noninteracting, i.e. ${\widetilde{V}}_{G}=0$. Then by noting that ${u}_{{\boldsymbol{k}}}=1$, ${v}_{{\boldsymbol{k}}}=0$, and ${E}_{{\boldsymbol{k}}}={\varepsilon }_{G}(k)$, all the integrals can be worked out analytically to yield$ \begin{eqnarray}\displaystyle \frac{2n}{{ \mathcal E }}=\displaystyle \frac{2}{2\pi }\left[-\mathrm{ln}\left(\displaystyle \frac{\sqrt{-2{ \mathcal E }}{a}_{2{\rm{D}}}{{\rm{e}}}^{\gamma }}{2}\right)+{K}_{0}\left(\sqrt{-2{ \mathcal E }}r\right)\right],\end{eqnarray}$where Kn(x) is nth order modified Bessel function of the second kind. To gain more insight into equation (18), we examine the asymptotic behavior of ${ \mathcal E }(r)$. By noting that ${K}_{0}(x\gg 1)\simeq \sqrt{\tfrac{\pi }{2x}}{{\rm{e}}}^{-x}$, one obtains$ \begin{eqnarray}{ \mathcal E }(r)\mathop{\longrightarrow }\limits^{r\gg 1}{{ \mathcal E }}_{\infty }-\displaystyle \frac{{{ \mathcal E }}_{\infty }^{2}}{2\pi n}\sqrt{\displaystyle \frac{\pi }{2\sqrt{-2{{ \mathcal E }}_{\infty }}}}\displaystyle \frac{{{\rm{e}}}^{-\sqrt{-2{{ \mathcal E }}_{\infty }}r}}{\sqrt{r}},\end{eqnarray}$where ${{ \mathcal E }}_{\infty }\equiv {{ \mathcal E }}_{r\to \infty }$ and satisfies the equation ${{ \mathcal E }}_{\infty }\,=\,-2\exp (-4\pi n/{{ \mathcal E }}_{\infty })/({a}_{2{\rm{D}}}^{2}{{\rm{e}}}^{2\gamma })$. In figure 2, we compare the numerical solution of equation (16) with the asymptotic result equation (19), which exhibits very good agreement at large r. Of particular interest, for sufficiently strong interaction (small a2D), since the exponential part ${{\rm{e}}}^{-4\pi n/{{ \mathcal E }}_{\infty }}$ tends to 1, we find ${{ \mathcal E }}_{\infty }\approx -2{a}_{2{\rm{D}}}^{-2}{{\rm{e}}}^{-2\gamma }$. Similar to the 3D case [27, 28], this result suggests that one boson forms a bound state with one impurity. However, different from 3D systems [28] where ${ \mathcal E }(r)$ decays as a Yukawa potential, equation (19) suggests that the 2D effective potential decays slower.
Figure 2.
New window|Download| PPT slide Figure 2.Ground-state energy versus inter-impurity distance for various P's with χ=0 and ${a}_{2{\rm{D}}}=10{\ell }$. Dashed lines show the asymptotic ${ \mathcal E }(r)$'s for P=0 (equation (19)) and P=2 (equation (20)), respectively.
For interacting gases, we shall first turn on the contact interaction between gas atoms; while the dipolar interaction strength χ remains to be zero. Figure 2 plots the typical behavior of ${ \mathcal E }(r)$ under different contact interaction strengths. As can be seen, ${ \mathcal E }(r)$ still increases monotonously with inter-impurity distance, indicating that the effective interaction between impurity is attractive. In addition, the asymptotic value ${{ \mathcal E }}_{\infty }$ increases with the repulsive interaction strength P, which is a natural consequence as the effective interaction is mediated by the gas atoms. It should also be noted that, compared to the noninteracting case, ${ \mathcal E }(r)$ now contains a 1/r tail at large distance, which, as shall be shown, can be attributed to low energy phonon excitation. To see this, let us focus on the small k limit [28], where one can apply the approximation ${u}_{k}^{2}/[E(k)-{ \mathcal E }]\,\approx -\sqrt{P}/(4{ \mathcal E }k)$. Then the asymptotic expression for the ground-state energy becomes$ \begin{eqnarray}{ \mathcal E }(r)\mathop{\longrightarrow }\limits^{r\to \infty }{{ \mathcal E }}_{\infty }+\displaystyle \frac{\sqrt{P}{{ \mathcal E }}_{\infty }}{16\pi n}\displaystyle \frac{1}{r}.\end{eqnarray}$As can be seen, the ${ \mathcal E }(r)$ exhibits a 1/r decay at large r, in analog to the Coulomb potential. As an example, we compare the asymptotic result equation (20) with the numerical solutions of equation (16) in figure 2 for P=2, which shows very good agreement at large r.
In the presence of the DDI, the behavior of mediated interaction become more interesting. We plot, in figure 3, the ground state energy as a function of inter-impurity distance under various DDI strengths and with the parameters P=2 and ${a}_{2{\rm{D}}}=10{\ell }$. Here, for simplicity, we have also assumed that the dipole moments are polarized along the z axis such that the DDI is isotropic in the xy plane. An immediately observation is that the asymptotic value of ${{ \mathcal E }}_{\infty }$ decreases with χ, indicating that the DDI is of attractive nature. This may seem strange at first sight as, for a quasi-2D system, the DDI is supposed to be repulsive when the dipole moments are polarized along the z axis. To understand this, we note that ${{ \mathcal E }}_{\infty }$ is mainly determined by the short-distance behavior of the ground-state energy as a small discrepancy in ${ \mathcal E }(r)$ at small r may lead to a large difference in ${{ \mathcal E }}_{\infty }$. While at small r, our quasi-2D BEC is in fact a 3D system, for which the DDI interaction can certainly be attractive. Consequently, it is seen that ${{ \mathcal E }}_{\infty }$ decrease with growing χ.
Figure 3.
New window|Download| PPT slide Figure 3.Ground-state energy versus inter-impurity distance for various χ's with P=2, ${a}_{2{\rm{D}}}=10{\ell }$, and ϑ=0. The dashed line shows the result computed using equation (23) with χ=5.6.
As to the behavior of ${ \mathcal E }(r)$, we note that for small χ, ${ \mathcal E }(r)$ appears to be a monotonically increasing function of r, in analog to those in figure 2. However, for sufficiently large χ, the value of ${ \mathcal E }(r)$ oscillates with r, which indicates that, depending on the inter-impurity distance, the effective interaction between the impurities can be either attractive or repulsive. To understand this behavior, we recall that a particular interesting features for the excitation spectrum of a quasi-2D dipolar BEC is that, as schematically shown in figure 1(b), roton excitation emerges at sufficiently large χ. For example, simple calculation using equation (10) shows that, with P=2, the roton excitation sets in when $\chi \gt {\chi }^{* }\simeq 4.23$ and the condensate becomes unstable due to the softening of the roton mode for $\chi \gt {\chi }^{* * }\simeq 5.67$. Figure 3 clearly implies that the oscillatory behavior of ${ \mathcal E }(r)$ is closely related to the roton spectrum of the condensate.
To gain more insight into the underlying physics, we follow the technique used in [35], in which the momentum-domain integral in equation (17) was transferred to the frequency domain according to$ \begin{eqnarray}{\int }_{0}^{\infty }{\rm{d}}{k}(\cdots )=\sum _{i}{\int }_{{l}_{i}}\displaystyle \frac{{\rm{d}}{E}({k})}{{E}^{\prime} ({k})}(\cdots ),\end{eqnarray}$where we have denoted $E(k)={E}_{{\boldsymbol{k}}}$ due to the axial symmetry and li represents the ith monotonic interval of E(k). As analyzed in [35], in the presence of the roton excitation, the most important contributions to the integral over the frequency domain in equation (21) come from the vicinity of the van Hove singularities where $E^{\prime} (k)=0$. For convenience, let us denote the local maximum and minimum of E(k) as EM and Em, respectively. We can then expand E(k) around EM and Em to obtain $E^{\prime} (k)=\sqrt{2| {E}_{M}^{{\prime\prime} }| [{E}_{M}-E(k)]}$ and $E^{\prime} (k)=\sqrt{2{E}_{m}^{{\prime\prime} }[E(k)-{E}_{m}]}$, respectively. Now, we have approximately$ \begin{eqnarray}\begin{array}{rcl}{\displaystyle \int }_{0}^{\infty }k{\rm{d}}k\displaystyle \frac{{u}_{k}^{2}{{\rm{J}}}_{0}({kr})}{{E}_{k}-{ \mathcal E }} & \approx & {\displaystyle \int }_{0}^{{E}_{M}}\displaystyle \frac{{k}_{M}{u}_{M}^{2}}{{E}_{M}-{ \mathcal E }}\displaystyle \frac{2{{\rm{J}}}_{0}({kr}){\rm{d}}{E}_{k}}{\sqrt{2| {E}_{M}^{{\prime\prime} }| ({E}_{M}-{E}_{k})}}\\ & & +{\displaystyle \int }_{{E}_{m}}^{\infty }\displaystyle \frac{{k}_{m}{u}_{m}^{2}}{{E}_{m}-{ \mathcal E }}\displaystyle \frac{2{{\rm{J}}}_{0}({kr}){\rm{d}}{E}_{k}}{\sqrt{2{E}_{m}^{{\prime\prime} }({E}_{k}-{E}_{m})}},\end{array}\end{eqnarray}$which can be worked out analytically and yields$ \begin{eqnarray}\widetilde{{ \mathcal E }}(r)\mathop{\longrightarrow }\limits^{r\to \infty }{{ \mathcal E }}_{\infty }-\displaystyle \frac{{{ \mathcal E }}_{\infty }^{2}}{4\pi n}\left[\displaystyle \frac{{k}_{M}^{2}{u}_{M}^{2}{F}_{1}({k}_{M}r)}{{E}_{M}-{{ \mathcal E }}_{\infty }}-\displaystyle \frac{{k}_{m}^{2}{u}_{m}^{2}{F}_{2}({k}_{m}r)}{{E}_{m}-{{ \mathcal E }}_{\infty }}\right],\end{eqnarray}$where ${k}_{M,m}$ are momenta correspond to ${E}_{M,m}$, ${u}_{M,m}={u}_{{k}_{M},{k}_{m}}$, ${F}_{1}(x)=\pi {{\rm{J}}}_{1}(x){{\rm{H}}}_{0}(x)+{{\rm{J}}}_{0}(x)[2-\pi {{\rm{H}}}_{1}(x)]$ and ${F}_{2}(x)={F}_{1}(x)-2/x$ with ${{\rm{H}}}_{n}(x)$ being the Struve functions. Since both Jn and Hn are oscillating functions, it is natural to find that $\widetilde{{ \mathcal E }}(r)$ is also a oscillating function. In figure 3, we compare the analytical $\widetilde{{ \mathcal E }}(r)$ with the numerical ${ \mathcal E }(r)$ for χ=5.6, here we have used numerically obtained ${{ \mathcal E }}_{\infty }$ in equation (23). As can be seen, these two results agree qualitatively. It is now clear that the oscillatory behavior of ${ \mathcal E }(r)$ is due to the roton spectrum such that the density of state $\rho (E)\propto 1/E^{\prime} (k)$ diverges at Em and EM. As a result, these two energy scales give rise to the most important contributions to ${ \mathcal E }(r)$, which leads to its oscillating behavior. On the contrary, in the absence of the roton excitation, all energy components contribute to ${ \mathcal E }(r)$ nearly equally, which washes out the oscillation on ${ \mathcal E }(r)$.
Finally, we assume that the DDI is anisotropic on the xy plane as the tilting angle of the dipole moment ϑ is nonzero. In figure 4(a), we plot the excitation spectrum equation (10) on the ${k}_{x}{k}_{y}$ plane for P=2, χ=3.03, and ϑ=π/4. As expected, ${E}_{{\boldsymbol{k}}}$ is anisotropic. Of particular interest, it exhibits a roton spectrum in the vicinity of the ky axis. In figure 4(b), we present the corresponding ground-state energy on the xy plane. As can be seen, ${ \mathcal E }({\boldsymbol{r}})$ now depends on the relative position of the impurities. In addition, ${ \mathcal E }({\boldsymbol{r}})$ is oscillatory along the direction where the excitation spectrum of the dipolar BEC is of the roton nature.
Figure 4.
New window|Download| PPT slide Figure 4.Contour plot of (a) the excitation spectrum ${E}_{{\bf{k}}}$ and (b) ground-state energy ${ \mathcal E }({\bf{r}})$, when the dipole moments are not polarized along the z axis with P=2, χ=3.03, a2D=10ℓ, and ϑ=π/4.
4. Conclusion
In summary, we have studied the mediated interaction potential of two heavy impurities embedded in a quasi-2D ultracold dipolar gas. With a variational ansatz truncated to one excitation, we show that the mediated interaction ${ \mathcal E }(r)$ between two impurities is strongly influenced by the structure of excitation spectrum of BEC. In the absence of the DDI, ${ \mathcal E }(r)$ is monotonically increasing function, which leads to an attractive potential. While under sufficiently strong DDI where the roton excitation emerges in dipolar gas, ${ \mathcal E }(r)$ is no longer momotonic, instead, it becomes oscillatory. Moreover, when the dipolar interaction is anisotropic on the xy plane, ${ \mathcal E }({\boldsymbol{r}})$ also exhibits anisotropy.
Acknowledgments
Fulin Deng is grateful to Dr Lijuan Jia for her help on numerical code debugging. This work was supported by the NSFC (Grants No. 11 674 334, No. 11 974 363, and No. 11 947 302), and by the Strategic Priority Research Program of CAS (Grant No. XDB28000000). TS acknowledges the Thousand-Youth-Talent Program of China.
,1,2,3,41CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, 
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