Kelvin-Helmholtz instability in two-component Bose gases on a lattice

We explore the stability of the interface between two phase-separated Bose gases in relative motion on a lattice. Gross-Pitaevskii-Bogoliubov theory and the Gutzwiller ansatz are employed to study the short- and long-time stability properties. The underlying lattice introduces effects of discreteness, broken spatial symmetry, and strong correlations, all three of which are seen to have considerable qualitative effects on the Kelvin-Helmholtz instability. Discreteness is found to stabilize low flow velocities because of the finite energy associated with displacing the interface. Broken spatial symmetry introduces a dependence not only on the relative flow velocity but also on the absolute velocities. Strong correlations close to a Mott transition will stop the Kelvin-Helmholtz instability from affecting the bulk density and creating turbulence; instead, the instability will excite vortices with Mott-insulator-filled cores.

Figure 1

Growth rate of the dynamically most unstable mode (i.e., the imaginary part of the mode with the largest imaginary frequency) as a function of flow wave vector $k$ and and mode wave vector $k_e$ (note that a shift $k_e-k/2$ has been applied to the $y$ axis to make the symmetry around $k_e=k/2$ more transparent). Here one component is in motion with wave vector $k$ and the other is stationary. The lattice has 256 sites, the parameter values are $J/U=1$, $U_{12}/U=1.1$, and the asymptotic density is $\overline{n}=1$.

Figure 2

Time development of the instability for the case $k=5\pi /64$, with the remaining parameter values as in Fig. 1. (a), (c), (e), and (g) The density of the component 1, where red indicates high density and blue is zero density; (b), (d), (f), and (h) the corresponding phase. Snapshots are taken at times $t=44$ for (a) and (b), $t=176$ for (c) and (d), $t=265$ for (e) and (f), and $t=400$ for (g) and (h). The field of view is $64\times 64$ lattice sites.

Figure 3

Largest imaginary part of the excitation frequency as in Fig. 1, but with counterflowing condensates having wave vectors $(k/2,-k/2)$. The lattice has 256 sites, parameter values are $J/U=1$, $U_{12}/U=1.1$, and the asymptotic density is $\overline{n}=1$.

Figure 4

Comparison of the largest imaginary part of the excitation frequency as a function of the excitation wave number $k_e$ in the cases of one flowing component, $(k,0)$ (dashed red line), and counterflow, $(k/2,-k/2)$ (solid blue line). Parameter values are as in Fig. 1, but the wave vector is fixed at $k=1.3$ in (a) and $k=1.0$ in (b). For the case of one flowing component, the $k_e$ axis is shifted by $k_e\to k_e-k/2$. The lattice has 256 sites, parameter values are $J/U=1$, $U_{12}/U=1.1$, and the asymptotic density is $\overline{n}=1$.

Figure 5

Growth rate of the dynamically most unstable mode (i.e., the imaginary part of the mode with the largest imaginary frequency) as a function of flow wave vector $k$ and and mode wave vector $k_e$. Here both components are in motion with counterdirected wave vectors $(k/2,-k/2)$. The parameters are $J/U=1$, $U_{12}/U=1.5$, and the asymptotic density is $\overline{n}=4$.

Figure 6

A few Bogoliubov eigenvalues for a fixed excitation wave vector $k_e=0.3$ as functions of the flow wave vector $k$. Parameters are as in Fig. 5. (top) Real parts and (bottom) imaginary parts.

Figure 7

Bogoliubov eigenstates for the lowest-lying surface modes, functions $u_j\left(z\right)$ (blue solid lines) and $v_j\left(z\right)$ (red dashed lines) at $k_e=0.3$ and $k=1.25$, which is below the instability threshold. (a) and (b) Quasiparticle mode; (c) and (d) quasihole mode. (left) Mode functions $u_1\left(z\right),v_1\left(z\right)$ associated with component 1 and (right) mode functions $u_2\left(z\right),v_2\left(z\right)$. The lattice has 256 sites, parameter values are $J/U=1$, $U_{12}/U=1.5$, and the asymptotic density is $\overline{n}=4$.

Figure 8

Phase diagram for a mixture of two identical Bose gases, calculated in the Gutzwiller approximation. The chemical potential is taken to be $\mu =0.7U$. Points $A$ and $B$ mark the parameters for which the two simulations described in the text are done.

Figure 9

Cross sections of the density profile of two phase-segregated Bose gases at points (top) $A$ and (bottom) $B$ in the phase diagram. Squares denote the density for component 1 $n_1$, crosses denote $n_{c1}$, asterisks $n_2$, and circles denote $n_{c2}$. Lines are to guide the eye.

Figure 10

Time development in the Gutzwiller approximation of a system with $J=0.045U$, $U_{12}=0.85U$, and $\mu =0.7U$, with wave vectors $(k/2,-k/2)$, where $k=7\pi /16$. Snapshots displayed in the different rows are taken at times (from top to bottom) $100/U$, $200/U$, $300/U$, and $4000/U$. (left) Condensate density $n_c$, (middle) total density $n$, and (right) the phase $\varphi$ of bosons of component 1. The corresponding quantities for component 2 are not shown but are similar to the ones shown mirrored in the $x$ axis. The field of view is $64\times 64$ lattice sites.

Figure 11

Time development in the Gutzwiller approximation of a system with $J=0.035U$, $U_{12}=0.85U$, and $\mu =0.7U$, with wave vectors $(k/2,-k/2)$, where $k=7\pi /16$. Snapshots displayed in the different rows are taken at times (from top to bottom) $100/U$, $200/U$, $300/U$, and $4000/U$. (left) Condensate density $n_c$, (middle) total density $n$, and (right) the phase $\varphi$ of bosons of component 1. The corresponding quantities for component 2 are not shown but are similar to the ones shown mirrored in the $x$ axis. The field of view is $64\times 64$ lattice sites.