Pattern formation of quantum Kelvin-Helmholtz instability in binary superfluids

We study theoretically the nonlinear dynamics induced by shear-flow instability in segregated two-component Bose-Einstein condensates in terms of the Weber number, which is defined by extending the past theory on the Kelvin-Helmholtz instability in classical fluids. Numerical simulations of the Gross-Pitaevskii equations demonstrate that dynamics of pattern formation is well characterized by the Weber number We, clarifying the microscopic aspects unique to the quantum fluid system. For $\text{We}\lesssim 1$, the Kelvin-Helmholtz instability induces flutter-finger patterns of the interface and quantized vortices are generated at the tip of the fingers. The associated nonlinear dynamics exhibits a universal behavior with respect to We. When $\text{We}\gtrsim 1$ in which the interface thickness is larger than the wavelength of the interface mode, the nonlinear dynamics is effectively initiated by the counter-superflow instability. In a strongly segregated regime and a large relative velocity, the instability causes transient zipper pattern formation instead of generating vortices due to the lack of circulation to form a quantized vortex per a finger. In a weakly segregating regime and a small relative velocity, the instability leads to the sealskin pattern in the overlapping region, in which the frictional relaxation of the superflow cannot be explained only by the homogeneous counter-superflow instability. We discuss the details of the linear and nonlinear characteristics of this dynamical crossover from small to large Weber numbers, where microscopic properties of the interface become important for the large Weber number.

Figure 1

The density profiles of the stationary solution of Eq. (2) are shown in the left panel of panel (a) for $\mathrm{\Delta }=1, {10}^{-1}, {10}^{-2}$, and ${10}^{-3}$ from the start to the end of the arrow. Here, the red solid and blue dashed curves correspond to $n_1={\left|{\varphi }_1\right|}^2$ and $n_2={\left|{\varphi }_2\right|}^2$, respectively, and their profiles are symmetric with respect to $y=0$. The right panel shows the enlarged view of the left one around $y=0$. Panels (b) and (c) show the surface tension $\tilde{\alpha }=\alpha /\left(2P_0\xi \right)$ and the thickness of the interface $l_{\text{d}}/\xi$ as a function of $\mathrm{\Delta }=(g_{12}/g)-1$, respectively, by the (red) solid curves. In panel (b), we also plot the analytic formula of ${\alpha }_{\text{weak}}$ [Eq. (12)] and ${\alpha }_{\text{strong}}$ [Eq. (13)] by the thin dashed line and the thin dotted curve, respectively. In panel (c), we draw the fitting line $l_{\text{d}}/\xi =0.914{\mathrm{\Delta }}^{-1/2}$. The panel (d) shows the distribution of the surface tension density divided by $\alpha$ for $\mathrm{\Delta }={10}^{-1}$ (red solid curve), ${10}^{-2}$ (blue dashed curve), and ${10}^{-3}$ (green dotted curve).

Figure 2

The expected phase diagram of the dynamical pattern formation by the shear flow instability in the $\mathrm{\Delta }-V_R^2$ plane. The resulting pattern is schematically depicted in each region, where the gray region represents the interface between the binary superfluids. We also show the contours of the typical Weber number We by the dashed curves. The red solid curve represents the relation $k_0{l}_d=1$, which divide roughly the two characteristic regions of the instability. For $k_0{l}_d\lesssim 1(\text{We}\lesssim 1)$ corresponding to the right side of the red curve, the instability of the thin interface is well described by the classical KHI. In the left side with $k_0{l}_d\gtrsim 1(\text{We}\gtrsim 1)$, the interface becomes thick so that the KH theory is not directly applicable. The green dashed-dotted curve shows the relation ${\nu }_{\kappa }\equiv {\lambda }_0{V}_R/2\kappa =1$; see the discussion in Sec. 4c. The orange dashed double-dotted line represents $\text{Ma}=1$ with the Mach number Ma of the bulk flow, which is supersonic above this line. The blue dashed curve shows the relation $k_{\text{CSI}}{l}_{\text{overlap}}=1$; see the discussion in Sec. 6. The parameter points given in the previous papers are shown by the symbols: $▾$ Fig. 1 in Ref. [7], $▴$ Fig. 8 in Ref. [29], $\blacksquare$ Fig. 2(a) in Ref. [8], and $\times$ Fig. 2(b) in Ref. [8]. The black dot and the dashed arrows represent the parameter range along which we show the numerical results in the following sections.

Figure 3

Snapshots of the unstable dynamics for $V_R/V=\sqrt{0.45}$ and $\mathrm{\Delta }=1.0$, where the corresponding Weber number is $\text{We}=0.291$. In panel (a), the density difference $\mathrm{\Delta }n=(n_1-n_2)/n_0$ is depicted, in which the red (blue) region corresponds to the area where the density $n_1$ ($n_2$) is located. In panel (b), the profile of the phase ${\theta }_1$ of the first component is shown only in the region $\mathrm{\Delta }n>0$; there is only noisy phase fluctuations in the region $\mathrm{\Delta }n<0$ since the amplitude $n_1$ is almost zero there. A vortex is located at the endpoint of a branch cut (jump from ${\theta }_1=-\pi$ to $\pi$). The distributions of the vorticity $\omega$ in the $x-y$ plane is shown in panel (c). The spatial region of the plot is $-150\xi \le x,y\le 150\xi$. The time is represented by the dimensionless value $g{n}_{0}t/\hslash \equiv \tilde{t}$.

Figure 4

Maximum length ${\mathrm{\Lambda }}_{\text{max}}$ of the growing fingers before the vortex emission. Panel (a) shows a single shot data of the time development of the finger length $\mathrm{\Lambda }$ as well as that of the time derivative of the intercomponent interaction energy ${\dot{E}}_{12}=d{E}_{12}/dt$ for $V_R/V=\sqrt{0.45}$ and $\mathrm{\Delta }=0.6$. The length ${\mathrm{\Lambda }}_{\text{max}}$ is extracted at the moment when ${\dot{E}}_{12}$ starts to make a rapid chaotic oscillation, denoted by the downward arrow. Panel (b) shows ${\overline{\mathrm{\Lambda }}}_{\text{max}}$, an average with five different initial conditions for a single plot, as a function of $\mathrm{\Delta }$ for $V_R/V=\sqrt{0.18}$ (red circles) and $\sqrt{0.45}$ (blue triangles), where the error bars represent the standard deviation. The bottom panel (c) shows ${\overline{\mathrm{\Lambda }}}_{max}$ as a function of the Weber number We, where ${\overline{\mathrm{\Lambda }}}_{\text{max}}$ is scaled by $l_d$. The green dashed line serves as a guide to the eye for the ${\text{We}}^{-1}$ dependence.

Figure 5

Typical dynamics of the density difference $\mathrm{\Delta }n$ in the case of strongly phase-separated condensate with $\mathrm{\Delta }=1.0$ and the relative velocity (a) $V_R/V=2$ ($\text{We}=2.58$) and (b) $V_R/V=\sqrt{20}$ ($\text{We}=12.92$). The red (blue) region corresponds to the area where the density $n_1$ ($n_2$) is located. The profile of the phase ${\theta }_1$ in the region with $\mathrm{\Delta }n>0$, corresponding to panel (b), is shown in panel (c). The spatial region of the plot is $-25\xi \le x,y\le 25\xi$.

Figure 6

The imaginary part of the excitation spectrum, $\text{Im}\left[\omega \right]$, of the BdG equations for $\mathrm{\Delta }=1.0$ and (a) $V_R/V=1$, (b) 2, and (c) $\sqrt{20}$, shown by the red dots. The blue dashed curve represents Eq. (6) obtained by the KH theory in Sec. 2b. The green dotted curve represents Eq. (24) in the theory of the CSI in Ref. [34], where $n=0.318n_0$ [taken from Fig. 1], $g_{12}=2g$, and $k_{\perp }=0$.

Figure 7

Dynamics of the density difference $\mathrm{\Delta }n$ for $V_R/V=\sqrt{0.45}$ and (a) $\mathrm{\Delta }={10}^{-1}$ ($\text{We}=2.12$), (b) $\mathrm{\Delta }={10}^{-2}$ ($\text{We}=20.6$), and (c) $\mathrm{\Delta }={10}^{-3}$ ($\text{We}=205.6$). The red (blue) region corresponds to the area where the density $n_1$ ($n_2$) is located. In panel (d), the profile of the phase ${\theta }_1$ corresponding to panel (c) is shown. The spatial region of the plot is $-150\xi \le x,y\le 150\xi$.

Figure 8

(a) A semilog plot of $\text{Im}\left[\omega \right]$ of the dispersion relations for $V_R/V=\sqrt{0.45}$ and $\mathrm{\Delta }=0.001$. The numerical results obtained by the BdG equation are plotted by red points. The blue dashed curve represents the dispersion relation (6) of the KHI. Also, we plot $\mathrm{Im}\left[\omega \right]$ obtained from the dispersion relation (24) of the CSI with $k_{\perp }=0$ by the green dashed-dotted curve and $k_{\perp }=k_y^{\text{fit}}$ extracted from the eigenfunction in panel (b) by the gray dashed-dotted curve, where the fitting function $u^{\text{fit}}=\left|u_j\right|e^{i{k}_y^{\text{fit}}(y-y^{\text{fit}})}$ with the parameters $k_y^{\text{fit}}$ and $y^{\text{fit}}$ is employed; we here get $k_{\perp }\xi =0.155$. In panels (b), we plot the eigenfunctions $(u_1,v_1,u_2,v_2)$ corresponding to the largest value of $\text{Im}\left[\omega \right]$ in panel (a).

Figure 9

The schematic illustration of our initial setup for the strongly segregating BECs with a shear flow. (a) The condensates are phase separated into $y<0$ (${\psi }_1$ component) and $y>0$ (${\psi }_2$ component) regions with the relative velocity $V_R$. The interface is located along the $x$ axis, the vorticity being distributed in alignment with the $x$ axis. (b) The electrostatic analogy of the configuration of panel (a). The vorticity corresponds to the positive charge, while the electric field is parallel to the contour lines of the phase ${\theta }_j$.

Figure 10

The imaginary part of the eigenvalue of Eq. (B5) in the $\mathit{K}$ space for $g_{12}=g, n=n_0/2$, and several values of $V_R$: (a) $V_R=\sqrt{0.4}$, (b) $\sqrt{5}$, (c) $\sqrt{10}$, and (d) $\sqrt{20}$. The plot range is determined as $0<k_{\parallel }\xi <V_R/\left(2V\right)$ and $0<k_{\perp }\xi <V_R/\left(4V\right)$. The eigenvalue is scaled by $\mu =g{n}_0$.