Two-Dimensional Quantum Droplets in Binary Dipolar Bose-Bose Mixture

Abstract

We study two-dimensional (2D) isotropic quantum droplets (QDs) in dipolar binary Bose–Einstein condensates (BECs). The QDs are supported by the competition between the 2D form of the Lee-Huang-Yang (LHY) term and the isotropic dipole-dipole interactions (DDIs). Moreover, the DDIs in the 2D plane can be tuned to be either repulsive or attractive. Before that, QDs in dipolar BECs were often explored in three-dimensional (3D) systems, with competition between the attractive DDIs and the repulsive LHY term. Unlike the 3D system, the LHY term of the 2D binary system behaves in a logarithmic form, which can feature both attraction and repulsion. In this case, the QDs can be produced regardless of the interactions (attraction, repulsion, or zero) that the mean-field effect represents. In this paper, we model the aforementioned QDs via the 2D binary dipolar BECs with the competition between isotropic DDIs and the logarithmic LHY term. Their characteristic parameters (the peak density, $I_{\mathrm{P}}$, chemical potential, $\mu$, and effective area, $A_{\mathrm{eff}}$) using both numerical and theoretical methods are discussed. The centripetal collision and oblique collision between moving QDs are also studied.

1. Introduction

According to mean-field (MF) theory, a condensed bosonic mixture collapses when the interspecies attraction becomes stronger than the geometrical average of the intraspecies repulsions [1,2], $g_{12}^2>g_{11}{g}_{22}$. Perhaps the most effective solution to tackling the collapses has recently been theoretically presented [3] and experimentally implemented [4,5,6,7,8,9,10,11,12,13,14,15,16]. It is based on the LHY correction provided by quantum fluctuations to the MF dynamics of BECs [17,18,19,20]. In binary BECs, the interaction between atoms is nearly eliminated by modifying the repulsion within the component and the attraction between the components using the Feshbach resonance technique [21]. These soliton-like states, called Quantum droplets, are formed as a result of the equilibrium between the MF dynamics and beyond the mean-field (BMF) effect, which is effectively represented by the famous LHY term, i.e., the local quartic self-repulsive term in the 3D Gross–Pitaevskii equations (GPEs).

Subsequently, the theoretical scheme was extended from 3D systems to low-dimensional systems, and the specific expressions for the LHY terms in two- and one-dimensional systems were re-derived and modified [22]. However, in the two-dimensional (2D) case, the LHY term is very intriguing because it is proportional to $n[ln(n/n_0)-1]$ ($n_0$ is the equilibrium density). Due to the nature of the logarithmic function, this LHY correction can show both attraction and repulsion. This characteristic indicates that whether the MF interaction is attractive or repulsive, it could be used to create a QD with LHY corrections in 2D binary BECs.

The MF attraction can be provided not only by cubic contact (local) inter-component attraction in binary BEC [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46], but also by long-range (nonlocal) dipole-dipole interactions in a single-component condensate of atoms carrying permanent magnetic moments [12,47,48,49,50]. The situation of a single-component dipolar BECs system has drawn much attention [51,52,53,54]. However, in this system framework, the conflict between the higher-order repulsive LHY term and attractive DDIs has been mentioned frequently. Regarding the scenario involving DDIs in a low-dimensional two-component system, some relevant reports consider this proposal [55,56,57,58]. When we discuss low-dimensional binary dipolar BECs, especially in the 2D case, unlike the initial stereotypical repulsive effect, the logarithmic LHY term at a 2D dynamic will expand the system’s options.

It is worth noting that in prior 2D dipolar BECs studies, the polarization direction of the dipoles is required to be parallel to the 2D plane to ensure the generally attractive DDIs, which can form QDs in equilibrium with the repulsive LHY term [59,60]. If the dipole is perpendicular to the 2D plane, QDs cannot be formed unless the rotation field effect is introduced to tune the effective dipole coupling from the positive value to negative value through the rapid rotation of the dipole direction [61,62,63]. Whether repulsive DDIs and beyond mean-field effects can form QDs in a 2D dipolar BECs system is still debatable. Yet, this problem has an entirely new possible solution since the LHY term in the 2D binary system is in logarithmic form.

We consider DDIs in 2D binary Bose-Bose mixtures with all the dipoles oriented perpendicular to the 2D plane and theoretically construct the QDs supported by the combination of the logarithmic LHY correction and MF interaction. In the current formula, not only attractive DDIs, but also repulsive DDIs can form QDs because the LHY correction can be adaptively changed into attraction or repulsion. We also discuss how QDs behave in this physical environment and how their collision dynamics perform.

The rest of the paper is structured as follows. The model is introduced in Section 2. Basic numerical and theoretical results of QDs are presented in Section 3. Then, the collisions between the moving QDs are reported in Section 4. The work is concluded in Section 5.

2. Model

We consider isotropic QDs in Petrov’s proposal [22] trapped in a 2D binary dipolar BEC system with all the dipoles oriented perpendicular to the 2D plane (as shown in Figure 1). The GPE-LHY of this system is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill i\frac{\partial {\psi }_{+}}{\partial t}=-\frac{1}{2}{\nabla }^2{\psi }_{+}+\kappa {\psi }_{+}\left(\mathbf{r}\right)\int \int d{\mathbf{r}}^{\prime }R\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)\left[{\left|{\psi }_{+}\left({\mathbf{r}}^{\prime }\right)\right|}^2+{\left|{\psi }_{-}\left({\mathbf{r}}^{\prime }\right)\right|}^2\right]\\ \hfill +\gamma {\psi }_{+}\left[{\left|{\psi }_{+}\left({\mathbf{r}}^{\prime }\right)\right|}^2+{\left|{\psi }_{-}\left({\mathbf{r}}^{\prime }\right)\right|}^2\right]ln\left[{\left|{\psi }_{+}\left({\mathbf{r}}^{\prime }\right)\right|}^2+{\left|{\psi }_{-}\left({\mathbf{r}}^{\prime }\right)\right|}^2\right],\end{array}\end{array}$$

(1)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill i\frac{\partial {\psi }_{-}}{\partial t}=-\frac{1}{2}{\nabla }^2{\psi }_{-}+\kappa {\psi }_{-}\left(\mathbf{r}\right)\int \int d{\mathbf{r}}^{\prime }R\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)\left[{\left|{\psi }_{-}\left({\mathbf{r}}^{\prime }\right)\right|}^2+{\left|{\psi }_{+}\left({\mathbf{r}}^{\prime }\right)\right|}^2\right]\\ \hfill +\gamma {\psi }_{-}\left[{\left|{\psi }_{-}\left({\mathbf{r}}^{\prime }\right)\right|}^2+{\left|{\psi }_{+}\left({\mathbf{r}}^{\prime }\right)\right|}^2\right]ln\left[{\left|{\psi }_{-}\left({\mathbf{r}}^{\prime }\right)\right|}^2+{\left|{\psi }_{+}\left({\mathbf{r}}^{\prime }\right)\right|}^2\right].\end{array}\end{array}$$

(2)

If we consider a symmetry form (here, we assumed that ${\psi }_{+}={\psi }_{-}=\psi /\sqrt{2}$), the GPE-LHY can be expressed as

$$\begin{array}{c}\hfill i\frac{\partial }{\partial t}\psi =-\frac{1}{2}{\nabla }^2\psi +\kappa \psi \int R(\mathbf{r}-{\mathbf{r}}^{\prime })|\psi \left({\mathbf{r}}^{\prime }\right){|}^{2}d{\mathbf{r}}^{\prime }+{\gamma |\psi |}^2\psi ln{|\psi |}^2,\end{array}$$

(3)

where $\mathbf{r}$ = (x,y) is the set of the coordinates, $\psi$ is the wave function, and $\kappa$ is the strength of the DDI. Notably, $\kappa$ is an adjustable quantity, i.e., its sign can be positive or negative, and the total nonlocal interaction can be adjusted to become repulsive or attractive. $\gamma$ is the coefficient of the beyond-MF effect in 2D binary BECs. The isotropic kernel corresponding to the particles’ dipolar moments polarized perpendicular to the (x,y) plane is defined as

$$\begin{array}{c}\hfill R(\mathbf{r}-{\mathbf{r}}^{\prime })=\frac{1-3{cos}^2\mathrm{\Theta }}{{\left[b^2+{(\mathbf{r}-{\mathbf{r}}^{\prime })}^2\right]}^{3/2}},\end{array}$$

(4)

where b is the cutoff of the kernel, which is determined by the confinement length $a_{\perp }$ in the transverse dimension [64,65,66]. In our isotropic 2D system, the angle $\mathrm{\Theta }$ between the dipole polarization direction and its relative distance is $\frac{\pi }{2}$, i.e., ${cos}^2\mathrm{\Theta }=0$. Furthermore, if we apply a notation

$$\begin{array}{c}\hfill t\leftarrow \gamma t,(x,y)\leftarrow \sqrt{\gamma }(x,y),\kappa \leftarrow \frac{\kappa }{\gamma },\end{array}$$

(5)

Eqution (3) can be simplified as

$$\begin{array}{c}\hfill i\frac{\partial }{\partial t}\psi =-\frac{1}{2}{\nabla }^2\psi +\kappa \psi \int R(\mathbf{r}-{\mathbf{r}}^{\prime })|\psi \left({\mathbf{r}}^{\prime }\right){|}^{2}d{\mathbf{r}}^{\prime }+{|\psi |}^2\psi ln{|\psi |}^2,\end{array}$$

(6)

the total energy of the system is

$$\begin{array}{c}\hfill E=\frac{1}{2}\int d\mathbf{r}\left[{|\nabla \psi |}^2+{\kappa |\psi |}^2\int R(\mathbf{r}-{\mathbf{r}}^{\prime })|\psi \left({\mathbf{r}}^{\prime }\right){|}^{2}d{\mathbf{r}}^{\prime }+{|\psi |}^{4}ln\left(\frac{{|\psi |}^2}{\sqrt{e}}\right)\right],\end{array}$$

(7)

and the total number of particles is defined as

$$\begin{array}{c}\hfill N={\int |\psi |}^{2}d\mathbf{r}.\end{array}$$

(8)

While units are not required in the figures shown below, because all the quantities in Equations (6)–(8) are written in scaled form, it is relevant to summarise here estimates for the pertinent quantities in physical units. To achieve this, we need to convert the results into physical units that correspond to the experimental realization of QDs in Bose-Bose mixtures, using the values of the magnetic moment for ${}^{164}$Dy atoms. The typical size of confinement length in underlying physical units is $a_{\perp }\sim 3$ $\mathsf{\mu }\mathrm{m}$ [67,68,69,70]. We thus conclude that the stable 2D binary dipolar QDs may be created with the number of atoms in the range from $N_{atom}\sim {10}^3$ to $N_{atom}\sim {10}^4$, and physical lateral sizes $l_{phys}\sim 1$ $\mathsf{\mu }\mathrm{m}$. The corresponding relation between the physical quantities and scaled ones is

$$\begin{array}{c}\hfill N_{atom}\sim 10N,{(x,y)}_{phys}\sim (x,y)\times 1\mathsf{\mu }\mathrm{m}.\end{array}$$

(9)

The stationary solution is obtained as a usual form with the wavefunction $\varphi \left(\mathbf{r}\right)$ and real chemical potential $\mu$, i.e.,

$$\begin{array}{c}\hfill \psi (\mathbf{r},t)=\varphi \left(\mathbf{r}\right)e^{-i\mu t}.\end{array}$$

(10)

A dynamical invariant of the system is the total norm, which is proportional to the total number of atoms in the dipolar bosonic mixtures. The free control parameters in the system are the strength of the DDI, $\kappa$, and the total norm of the QDs, N.

3. Stationary Solution of the Quantum Droplets

The isotropic QDs in this system are numerically found using the imaginary-time method (ITM) [71,72] by inputting a special ansatz

$${\varphi }^{\left(0\right)}(x,y)=A\tilde{r}exp\left(-\alpha {\tilde{r}}^2\right).$$

(11)

where $A>0$ and $\alpha >0$ are real constants. $\tilde{r}=\sqrt{x^2+y^2}$.

Typical examples of stable fundamental QDs are shown in Figure 2a,b. The size of the QDs created by the attractive DDI with $\kappa =-0.05$ is smaller than another with $\kappa =0.05$. The stability of the fundamental QDs is demonstrated by direct simulation with 1% random noise, as plotted in Figure 2c,d. This result demonstrates that both attracting and repelling DDIs can create stable self-localized states in 2D dipolar binary BECs. At the equilibrium condition, the QD is a flat-top QD; hence, we can neglect the contribution from the kinetic energy term by applying the Thomas–Fermi (TF) approximation. In this case, the density distribution of the QDs satisfies $n\left(\mathbf{r}\right)={|\psi \left(\mathbf{r}\right)|}^2=\mathrm{const}.$ Therefore, the total energy (7) is

$$\begin{array}{c}\hfill E=\frac{1}{2}\left(\kappa \epsilon n^2+n^{2}ln\left(\frac{n}{\sqrt{e}}\right)\right)A_S,\end{array}$$

(12)

where $A_S=N/n$ is the total area of the QDs and $\epsilon =\int d\mathbf{r}R\left(\mathbf{r}\right)\approx 6.2$ is the totally nonlocal effect. Then, we can obtain the density value of the QD by solving $dE/dn=0$, which yields the

$$\begin{array}{c}\hfill n_e=\frac{1}{e^{1/2}{e}^{\kappa \epsilon }},\end{array}$$

(13)

and the chemical potential is

$$\begin{array}{c}\hfill {\mu }_e=\kappa \epsilon n_e+n_{e}ln{n}_e.\end{array}$$

(14)

If we replace $n_e=N/A_S$, substitute it into Equation (12), and then solve $dE/d{A}_S=0$, one can obtain the total area for the equilibrium state satisfying the equation

$$\begin{array}{c}\hfill ln{A}_S=\kappa \epsilon +lnN+\frac{1}{2}.\end{array}$$

(15)

To study the characteristics of the QDs, we define the effective area for the QDs as

$$A_{\mathrm{eff}}=\frac{{\left({\int |\varphi |}^{2}d\mathbf{r}\right)}^2}{{\int |\varphi |}^{4}d\mathbf{r}}.$$

(16)

The peak density ($I_{\mathrm{P}}$), chemical potential ($\mu$), and effective area ($A_{\mathrm{eff}}$) of the fundamental QDs as functions of N for repulsive and attractive DDIs are displayed in Figure 3(a1–b3), respectively. When the MF interaction and LHY term compete, a superfluid, incompressible state is maintained, whose density (at extremely low values) cannot rise above a particular maximum. Thus, this quantum macroscopic state is referred to as a fluid, and localized states filled by it are referred to as “droplets” [4]. In Figure 3(a1,b1), the peak values saturate at $I_{\mathrm{P}}\approx 0.472$ and $I_{\mathrm{P}}\approx 0.837$, respectively, if N is sufficiently large, as may be expected for the QDs. According to Equation (13), their theoretical prediction equilibrium densities are $n_e\approx 0.445$ and $n_e\approx 0.827$, which are also close to the numerical results [see the red dashed line in Figure 3(a1,b1)].

Figure 3(a2,b2) shows that the chemical potential decreases with an increasing N, i.e., $\frac{d\mu }{dN}<0$, which indicates that it satisfies the Vakhitov–Kolokolov (VK) criterion, which is the well-known necessary stability condition for self-trapped modes [73]. This criterion has a straightforward physical interpretation: If we introduce one particle into the system and the energy of this bound state falls, the system is stable. They saturate at $\mu \approx$$-0.219$ and $\mu \approx$ $-0.397$ when N is sufficiently large. The theoretical predictions of the chemical potential are ${\mu }_e\approx$ $-0.222$ and ${\mu }_e\approx$ $-0.414$, which are also close to the numerical results [see the red dashed line in Figure 3(a2,b2)].

When $\kappa$ is positive, the dipole interaction shows repulsion, and in this case, $I_{\mathrm{P}}$ significantly increase with an increasing N at the beginning [at approximately $N<25$], then $I_{\mathrm{P}}$ gradually decreases with an increasing N until it reaches equilibrium. The changing trend of the effective area in Figure 3(a3) reveals the cause of this phenomenon. In this case, QDs were not formed as expected because when N is very small (just a few particles), its size is enormous and comparable to the gas state. At this point, the system will be encouraged by the increase in the number of atoms to produce droplets of condensate, which causes the effective area $A_{\mathrm{eff}}$ to decrease rapidly and the corresponding peak density $I_{\mathrm{P}}$ to increase rapidly. The peak density of the QD remains static once it enters equilibrium due to its incompressibility; hence, the effective area $A_{\mathrm{eff}}$ grows linearly as N increases.

When $\kappa$ is negative, the dipole interaction exhibits attraction, and the situation is similar to that mentioned above. The peak density of the negative $\kappa$ increases continuously, while the other peak density increases first and then falls because the effective area of the latter increases more quickly than the former.

The $I_{\mathrm{P}}$, $\mu$, and $A_{\mathrm{eff}}$ of the fundamental QDs as functions of the strength of $\kappa$ with $N=500$ are also shown in Figure 3(c1–c3). The numerical results are consistent with those of the theoretical analysis.

4. Collision between the Moving Quantum Droplets

Stable 2D dipolar binary BEC mixtures can be set in motion by opposite kicks $\pm \eta$ applied along the x or y-direction. Accordingly, it is possible to simulate collisions between two mixtures moving in opposite directions. Collisions in the x and y directions behave similarly due to the isotropic properties. Without loss of generality, we discuss centripetal collisions that occur in the x direction. Generally, the initial states for both moving QDs can be constructed as follows:

$$\begin{array}{c}\hfill \varphi (x,y)=\varphi (x-x_0,y)e^{-i\eta x}+\varphi (x+x_0,y)e^{i\eta x}.\end{array}$$

(17)

Several typical collision results are shown in Figure 4. We investigate the binary collisions between QDs created in 2D dipolar binary BECs and distinguish between the separation and merging as the two main outcomes, i.e., quasi-elastic collisions and completely inelastic collisions. The quasi-elastic collisions will merge at the beginning, but when compressed to the limit, they will split along the positive and negative directions of the y axis; that is, they will deflect with an angle of 90 degrees relative to the motion before the collision, as shown in Figure 4(a1–a6). Similar results have been reported in Ref. [74]. The completely inelastic collisions are displayed in Figure 4(b1–b6). Two QDs merge into a quadrupolar breather, repeatedly elongating and oscillating along the y and x directions.

In the present setting, the simulations show that the values of $\kappa$ will affect the results of collisions. When $\kappa >0$, for instance, quasi-elastic collision is more likely to occur than when $\kappa <0$. This might result from the repulsive interaction of DDIs, which produces a low surface tension that cannot absorb the collision pair’s kinetic energy [75].

Depending on the pair’s collision velocity, which is proportional to $\eta$, the increase in $\eta$ leads to a transition from completely inelastic collisions between slow QDs to quasi-elastic outcomes for fast QDs. When $\eta$ becomes too large, the collision will split the initial binary mixture into several pieces, as shown in Figure 4(c1–c6). In addition, the values of N mainly affect the size of droplets, which will cause the critical velocity that discriminates between the completely inelastic and quasi-elastic cases to display a different dependence on the norm N for small and large droplets [8].

We also discuss the collision dynamics between two QDs whose centers of mass are not at the same level, i.e., off-center collisions. In our simulations, we move the two mixtures by a distance of $y_0$ in the positive and negative directions of the y axis and then apply appropriate opposite kicks $\pm \eta$. The initial states for these moving QDs can be constructed as follows:

$$\begin{array}{c}\hfill \varphi (x,y)=\varphi (x-x_0,y-y_0)e^{-i\eta x}+\varphi (x+x_0,y+y_0)e^{i\eta x},\end{array}$$

(18)

and a typical result is shown in Figure 4(d1–d6). We find that the formation of a rotating quadrupolar breathing body is the result of an oblique collision. Some characteristics of the collision result, such as the rotational speed and the duration of the breathing motion, are affected by the values of $\kappa$, N, $y_0$, and $\eta$. Of course, an excessive $\eta$ will also cause the mixture to separate.

5. Conclusions

Based on the adaptive balance mechanism of the LHY term in the 2D binary BECs that represents the correction of the beyond mean-field effects, we prove that both repulsive and attractive DDIs can construct 2D isotropic QDs in a binary dipolar BEC system with all the dipoles oriented perpendicular to the 2D plane. We discuss the characteristic parameters of the QDs formed by the combination of attracting or repelling DDIs and the logarithmic LHY term, and the numerical results are consistent with the theoretical predictions. In addition, we also discussed the collision dynamics under this physical setting from the two cases of centripetal collision and non-centripetal collision. The numerical simulation results show that when the collision velocity is appropriate, there will be completely inelastic collisions and quasi-elastic collisions, while a large collision velocity will destroy the mixtures and split them into multiple components.

The present analysis can be extended further. First, the QDs outlined above are fundamental modes, and it is natural to expect that the vortex (alias spinning) modes may offer an opportunity to study more sophisticated properties of the QD state of matter. Furthermore, anisotropic QD embedded vorticity has been constructed in a 2D binary dipolar system [60]. Another relevant possibility is constructing anisotropic QD modes with vorticity in a 3D binary dipolar BEC system, which will be intriguing and challenging.