Two-dimensional vortex quantum droplets

It was recently found that the Lee-Huang-Yang (LHY) correction to the mean-field Hamiltonian of binary atomic boson condensates suppresses the collapse and creates stable localized modes (two-component quantum droplets, QDs) in two and three dimensions (2D and 3D). In particular, the LHY effect modifies the effective Gross-Pitaevskii equation (GPE) in 2D by adding a logarithmic factor to the usual cubic term. In the framework of the accordingly modified two-component GPE system, we construct 2D self-trapped modes in the form of QDs with vorticity $S$ embedded into each component. Because of the effect of the logarithmic factor, the QDs feature a flat-top shape, which expands with the increase of $S$ and norm $N$. An essential finding, produced by a systematic numerical investigation and analytical estimates, is that the vortical QDs are stable (which is a critical issue for vortex solitons in nonlinear models) up to $S=5$, for $N$ exceeding a certain threshold value, which is predicted to scale as $N_{\mathrm{th}}\sim S^4$ for large $S$ (for three-dimensional QDs, the scaling is $N_{\mathrm{th}}\sim S^6$). The prediction is corroborated by numerical findings. Pivots of QDs with $S\ge 2$ are subject to structural instability, as specially selected perturbations can split the single pivot in a set of $S$ or $S+2$ pivots corresponding to unitary vortices; however, the structural instability remains virtually invisible, as it occurs in a broad central “hole” of the vortex soliton, where values of fields are very small, and it does not cause any dynamical instability. In the condensate of ${}^{39}\mathrm{K}$ atoms, in which QDs with $S=0$ and a quasi-2D shape were created recently, the vortical droplets may have radial size $\lesssim 30\mu \mathrm{m}$, with the number of atoms in the range of ${10}^4–{10}^5$. The role of three-body losses is considered too, demonstrating that they do not prevent the creation of the vortex droplets but may produce a noteworthy effect, leading to sudden splitting of “light” droplets. In addition, hidden-vorticity (HV) states in QDs, with topological charges $S_{+}=-S_{-}=1$ in their components, which are prone to strong instability in other settings, have their stability region too. Unstable HV states tend to spontaneously merge into zero-vorticity solitons. Collisions of QDs, which may lead to their merger, and dynamics of elliptically deformed QDs (which form rotating elongated patterns or ones with oscillations of the eccentricity) are briefly considered too.

Figure 1

(a) Cross sections of density patterns ${\left|\varphi \left(\mathbf{r}\right)\right|}^2$ for the QDs of the flat-top type with $S=0$ (the system's ground state) and norms $N=100$ (solid curve), 400 (dotted curve), and 800 (dashed curve). (b) Cross sections of density patterns ${\left|\varphi \left(\mathbf{r}\right)\right|}^2$ for the QDs of the Gaussian type with $S=0$ (the system's ground state) and norms $N=10$ (dashed curve), 25 (dotted curve), and 48 (solid curve). (c) The effective area (10) for QDs with embedded vorticities $S=0$ (solid line), 1 (short-dashed line), 2 (dotted line), 3 (dash-dotted line) vs $N$. The long-dashed line represents the TF prediction, $A=N/n_{\mathrm{p}}^{\left(\mathrm{TF}\right)}$ for $S=0$; see Eq. (11). [(d1), (d2)] The largest (peak) local density, $|{\varphi }_{\mathrm{p}}{|}^2\equiv n_{\mathrm{p}}$ [dash-dotted curve in panel (d1)], and chemical potential $\mu$ [dash-dotted curve in penal (d2)] vs $N$. Solid lines in these two panels show the TF limit values, $n_{\mathrm{p}}^{\left(\mathrm{TF}\right)}=1/\sqrt{e}$ and ${\mu }_{\mathrm{TF}}=-1/\left(2\sqrt{e}\right)$; see Eqs. (13) and (14), respectively. The rhombus in panel (d1) designates the border between the QD profiles of the Gaussian and flat-top types.

Figure 2

Panels (a1)–(a4) display density patterns of vortex QDs with $S=1,2$, 3, 4 and norm $N=1000$. The first three QDs are stable, while the one corresponding to $S=4$ is not; see Fig. 3. (b) Cross sections of the density patterns from panels (a1)–(a4) (dotted, dashed, dash-dotted, and solid curves represent $S=1$, 2, 3, and 4, respectively). (c) The same as in panel (b) for $S=1$ and different values of $N$ (solid, dotted, and dashed curves represent $N=300$, 700, and 1000, respectively). All vortex QDs shown in panel (c) are stable.

Figure 3

(a1) The numerically found radius of the vortex's inner hole vs $S$ [each point corresponds to $N_{\mathrm{th}}\left(S\right)$ in panel (a2)] and the dashed line showing the prediction given by the TF approximation, according to Eq. (18). (a2) Minimum norms, $N_{min}$, necessary for the existence of the vortex QDs (circles) and threshold values of the norm necessary for their stability, $N_{\mathrm{th}}$ (triangles), vs $S$. The dashed curve shows the fit to analytically predicted scaling (24), $N_{\mathrm{th}}=6S^4$. Numerically exact values of $N_{min}\left(S\right)$ and $N_{\mathrm{th}}\left(S\right)$ are additionally produced in Table 1. [(b1), (b2)] The peak density and chemical potential of QDs with different vorticities vs $N$ (dash-dotted, dashed, and solid curves represent $S=1$, 3, and 5, respectively). Horizontal dotted lines show the TF-predicted limit values $n_{\mathrm{p}}^{\left(\mathrm{TF}\right)}$ and ${\mu }_{\mathrm{TF}}$; see Eqs. (13) and (14), respectively. (c) The evolution of an unstable QD with $(N,S)=(30,1)$. The inset shows the respective spectrum of stability eigenvalues $\mathrm{\Lambda }$. The above panels pertain to $L_3=0$ (no three-body loss). (d) The evolution of the density pattern of the QD with $S=1$ and initial norm $N=1000$, produced by simulations of Eq. (3) with $L_3=0.01$, starting from $t=0$ (d1). Other plots pertain to $t=464$ (d2), 1584 (d3), 3008 (d4), and 3120 (d5).

Figure 4

Panels (a1)–(a4) display density patterns of vortex QDs with $S=1$, 2, 3, and 4 and norms $N=60$, 200, 510, and 1380, respectively, which are selected from the stability boundary of the QDs; see Fig. 3. (b) Cross sections of the density patterns from panels (a1)–(a4).

Figure 5

(a) An enlargement (in domain $\left|x,y\right|\le 8$) of the density pattern, ${\left|\psi \left(x,y\right)\right|}^2$, for a QD with $(S,N)=(3,1000)$, which was initially perturbed as per Eq. (26), with $s=+1$ and $\varepsilon =0.0013+0.0023i$. The pattern is produced by the simulation of Eq. (3) up to $t=5000$. (b) The corresponding phase pattern clearly identifies three unitary vortices, whose pivots are located at zeros of the local amplitude. [(c), (d)] The same as in panels (a) and (b), but with the initial perturbation carrying vorticity $s=-1$, as per Eq. (28). In this case, the amplitude and phase patterns demonstrate splitting into a set of five pivots, one with winding number $-1$ located in the middle, surrounded by four satellites with winding numbers $+1$. Note the extremely small scale of the local amplitude, $\left|\psi \left(x,y\right)\right|\sim {10}^{-7}$, in panels (a) and (c). On the normal scale, such as one in Figs. 2 and 4, these splitting patterns remain invisible.

Figure 6

(a) The stability area of HV modes, defined as per Eq. (30), in the plane of $(N,4\pi /g)$, in the model based on Eq. (1) with $L_3=0$. The indigo (medium gray) area is bounded by the lower and upper limit values, see Eq. (32), with nearly constant values of $N_{\mathrm{th}}^{\left(\mathrm{low}\right)}$, as seen in the inset. (b) Density and phase plots showing opposite vorticities of the two components of a stable HV mode, with $(N,4\pi /g)=(8000,0.11)$, which corresponds to point A in panel (a), as a result of the evolution simulated up to $t=10000$.

Figure 7

(a1)–(a4) Merger of two zero-vorticity QDs into a single droplet with strongly oscillating eccentricity, initiated by $\psi (\mathbf{r},t=0)=\varphi (x-x_0,y)+\varphi (x+x_0,y)$, with $x_0=12$ and norm $N=100$ of each QD. (b1)–(b4) Merger of two transversely kicked zero-vorticity QDs, with $S=0$, into a rotating elongated droplet, initiated by $\psi (\mathbf{r},t=0)=\varphi (x-x_0,y)e^{-iky}+\varphi (x+x_0,y)e^{iky}$, with $x_0=12$, $N=100$, and kick $k=0.015$. (c1)–(c4) Steady rotation of an oval-shaped vortex QD with vorticity $S=1$ and $N=200$. These results are produced by simulations of Eq. (3) with $L_3=0$.