Figure 1

(Color online) (a) Velocity field amplitude $\alpha$ of the stationary solutions of Eq. (5) versus rotation frequency $\Omega$ for $ϵ=0$ (solid line), 0.02 (dotted line), and 0.1 (dashed line). Regions of dynamical instability for $ϵ=0.02$ and 0.1 are shown (circles). (b) Phase diagram of $ϵ$ versus $\Omega$. Plotted are the bifurcation point ${\Omega }_{\mathrm{bif}}\left(ϵ\right)$ (solid line) from Eq. (5), the onset of dynamical instability ${\Omega }_{\mathrm{ins}}\left(ϵ\right)$ from Eq. (6) (dashed line), and experimental data of Hodby et al. [3] (circles). Simulations of Eqs. (1, 2) (with $\mu =10\hslash {\omega }_{\perp }$) show the critical ellipticities beyond which the condensate deviates from an elliptical shape ${ϵ}_{\mathrm{cr}}^{\mathrm{dev}}$ (crosses) and beyond which lattice formation ultimately occurs ${ϵ}_{\mathrm{cr}}^{\mathrm{inst}}$ (points with error bars). The bifurcation point for $ϵ=0$ (dotted line) and the crossing frequency ${\Omega }_{\mathrm{X}}$ (dot-dashed line) at which ${\Omega }_{\mathrm{bif}}\left(ϵ\right)={\Omega }_{\mathrm{ins}}\left(ϵ\right)$ are indicated.Reuse & Permissions

Figure 2

(Color online) Dynamics under a continuous increase of $ϵ$ (at a rate of $dϵ∕dt={10}^{-4}{\omega }_{\perp }$) for (I) ripple instability $(\Omega ∕{\omega }_{\perp }=0.7)$, (II) interbranch instability $(\Omega ∕{\omega }_{\perp }=0.75)$, and (III) catastrophic instability $(\Omega ∕{\omega }_{\perp }=0.8)$. (a) Velocity field amplitude $\alpha$ versus $ϵ$ according to Eq. (5) [(black) solid lines] and TD simulations of Eqs. (1, 2) [(green) dots]. To the right of the dashed line, Eq. (3) is no longer a valid fit to the simulated velocity field (the standard deviation of the fit becomes of the order of $\alpha$). The regions of dynamical instability are indicated [(red) circles]. Density snapshots showing the (b) onset of instability, (c) disrupted state seeded by instability, and (d) vortex lattice. Dark (light) regions represent high (low) density. Each box represents a region $12\times 12\mathrm{\mu }\mathrm{m}=(20\times 20)a_{\perp }$, where $a_{\perp }=(\hslash ∕m{\omega }_{\perp }{)}^{1∕2}$ is the mean harmonic oscillator length (the simulation box is much greater than this). In (Ib) and (IIb) the density scale is limited to $0.1\%n_0$ to highlight low-density features.Reuse & Permissions