Ergoregion instabilities in rotating two-dimensional Bose-Einstein condensates: Perspectives on the stability of quantized vortices

We investigate the stability of vortices in two-dimensional Bose–Einstein condensates. In analogy with rotating space-times and with a careful account of boundary conditions, we show that the dynamical instability of multiply quantized vortices in trapped condensates persists in untrapped, spatially homogeneous geometries and has an ergoregion nature with some modification due to the peculiar dispersion of Bogoliubov sound. Our results open perspectives to the physics of vortices in trapped condensates, where multiply quantized vortices can be stabilized by interference effects and singly charged vortices can become unstable in suitably designed trap potentials. We show how superradiant scattering can be observed also in the short-time dynamics of dynamically unstable systems, providing an alternative point of view on dynamical (in)stability phenomena in spatially finite systems.

Figure 1

Real (a) and imaginary (b) parts of the BdG eigenfrequencies for modes of azimuthal number $m=2$ on a charge $\ell =2$ vortex in a harmonic trap for different values of the interparticle interaction energy. Here $n_0=N/l_0^2$ with $N$ number of atoms and $l_0=\sqrt{\hslash /\left(2M{\omega }_0\right)}$. Black (solid), red (dotted), and green (thicker) lines correspond to positive-, negative-, and zero-norm modes. In the inset (c), a plot of the energetically unstable mode for $g{n}_0/\hslash {\omega }_0=500$: the black (thinner) and red (thicker) lines show the modulus of the $u$ and $v$ components of the BdG spinor, the dashed line is a rescaled plot of the condensate density.

Figure 2

Real (a) and imaginary (b) parts of the Bogoliubov eigenfrequencies for modes of azimuthal number $m=2$ on a charge $\ell =2$ vortex in a BEC of size $R$. Black (solid), red (dotted), green (thicker) lines corresponds to positive-, negative-, and zero-norm modes. A wider view of the imaginary part is given in (d). Here, the $1/\sqrt{R}$ dashed line envelopes the instability maxima up to moderate $R$. The horizontal line indicates the instability rate extracted from the time-dependent simulation with absorbing boundary conditions shown in Fig. 3. (c) Spatial shape of the dynamically unstable core mode for the case with $R=60\xi$. The black (thin) and red (thick) lines, respectively, show the moduli of the $u_{\varphi }$ and $v_{\varphi }$ components of the Bogoliubov spinor. The dashed line shows the (rescaled) density profile of the vortex. In the inset, the mode on a shorter amplitude range is plotted to highlight the structure at large $r$.

Figure 3

Snapshots of the time evolution of a $m=2$ perturbation scattering on a $\ell =2$ vortex. Panels (a)–(d) correspond to the times $t=0, t=40\hslash /\mu , t=200\hslash /\mu$, and $t=1000\hslash /\mu$. Black (thin) and red (thick) lines, respectively, show the moduli of the $u_{\varphi }$ and $v_{\varphi }$ components of the Bogoliubov spinor (arbitrary units). Outgoing boundary conditions are imposed by including a wide and smooth imaginary potential of Gaussian spatial shape centered at the edge of the integration box $r=700\xi$, of variance $120\xi$ and amplitude $0.15\mu$, so to effectively absorb the perturbation spinor $|\varphi \rangle$ and suppress the reflected waves.

Figure 4

$R$ dependence of the Bogoliubov spectrum of a $\ell =4$ vortex for different azimuthal $m$. Panels (a)–(h) correspond to azimuthal numbers between $m=1$ and $m=8$. Black (solid), red (dotted), and green (thicker) lines correspond to positive-, negative-, and zero-norm modes.

Figure 5

Imaginary (black circles) and real (red squares) parts of the $m=2$ unstable mode frequency for growing vortex charge $\ell$. The meaning of the error bars on the black circles is explained in the text.

Figure 6

Left column: BdG spectra of a $\ell =1$ vortex in a harmonically trapped condensate as a function of the interparticle interaction energy (a). Spatial profile of the energetically unstable mode for $g{n}_0=1000\hslash {\omega }_0$ (thin black and thick red lines correspond to the moduli of the $u_{\varphi }$ and $v_{\varphi }$ components of the Bogoliubov spinor in arbitrary units) along with (dashed line) a rescaled version of the condensate density profile (c). Right column: BdG spectra of a $\ell =1$ vortex in the presence of an attractive Gaussian potential of the form Eq. (15) of strength $A=2g{n}_{\infty }$ and spatial size $\sigma =5\xi$ (with $n_{\infty }$ asymptotic density and $\xi$ the associated healing length) as a function of the total radius $R$ of the condensate (b). Spatial profile of the dynamically unstable mode for $R=100\xi$ along with (dashed line) the condensate density profile (d).

Figure 7

(a)–(d): Snapshots at times $t=0, 350\hslash /\mu , 750\hslash /\mu , 1100\hslash /\mu$ of the evolution of a Gaussian Bogoliubov wave packet of $m=2$ incident waves hitting a charge $\ell =4$ vortex. The frequency of the incident wave packet is centered around $\omega =0.2\hslash /\mu$, namely, the frequency of the core mode visible in the spectra shown in Fig. 4. Dirichlet boundary conditions are imposed to the Bogoliubov modes at $R=380\xi$. The arrows indicate the direction of the radial group velocity. Panel (e) shows the time dependence of the core mode amplitude (red solid line), as measured by the $r=2\xi$ value of the $v_{\varphi }$ component of the BdG spinor (arbitrary units). The black dashed line is the time dependence of the same core mode amplitude for a slightly different size $R=388.4\xi$, for which the system is dynamically unstable. Around $t=900\mu /\hslash$, the two curves show clear signature of the destructive vs constructive interference between the reflected wave packet and the core mode.