Core-bound waves on a Gross-Pitaevskii vortex

Abstract

We find the dispersion relations of two elusive families of core-bound excitations of the Gross-Pitaevskii (GP) vortex, varicose (axisymmetric) and fluting (quadrupole) waves. For wavelengths of order the healing length, these two families -- and the well-known Kelvin wave -- possess an infinite sequence of core-bound, vortex-specific branches whose energies lie below the Bogoliubov dispersion relation. In the short-wavelength limit, these excitations can be interpreted as particles radially bound to the vortex, which acts as a waveguide. In the long-wavelength limit, the fluting waves unbind from the core, the varicose waves reduce to phonons propagating along the vortex, and the fundamental Kelvin wave is the only core-bound vortex-specific excitation. Finally, we propose a realistic spectroscopic protocol for creating and detecting the varicose wave, which we test by direct numerical simulations of the GP equation.

Figures (10)

• Figure 1: Excitations of a Gross-Pitaevskii vortex (a) (top) The wavefunction of the condensate with the vortex $\psi_0$ and a 3D density contour. (bottom) Radial excitation wavefunctions $u$ (solid) and $v$ (dashed)---see text---with the density contours for the corresponding excited vortex. (b) The dispersion relations of the fundamental branches of these excitations $\epsilon^{(n=0)}_{m}(k)$. We show the Bogoliubov dispersion relation $\epsilon_{\mathrm B} = \sqrt{k^2(k^2+2)}$ (black line), the scattered phonons (green region), the varicose branch $m=0$ (blue line), and the fluting branch $m=-2$ (purple line). The Kelvin branch $m=-1$ is shown as an orange line and its long-wavelength approximation $\epsilon_{\mathrm P} = -k^2\ln{k}$ as the dot-dashed orange line. Inset: zoomed-in dispersion relations.
• Figure 2: Spectrum of the GP vortex excitations (a) Binding energies $\Delta \epsilon \equiv \epsilon_{\mathrm B}-\epsilon$ of varicose waves (blue; $m=0$), fluting waves (purple; $m=-2$) and Kelvin waves (orange; $m=-1$); the fundamental branches are shown as solid lines and the higher branches are dashed, with shorter dashes indicating higher $n$. The two double arrows show ratios of $\mathrm{e}^{-2\pi}$ and $\mathrm{e}^{-\sqrt{2}\pi}$ (see text). Note that despite appearances, in the limit $k\rightarrow\infty$, $\Delta\epsilon_{-1}^{(2)}\neq\Delta\epsilon_0^{(1)}$. (b) The spatial profiles $u(r)$ of the first three $m=0$ (resp. $m=-1$) modes in the limit $k\rightarrow \infty$ are shown on the left (resp. on the right); in this limit $v \rightarrow 0$. The color code and line styling follow the same convention as in (a). Insets: Zoomed-out views of the same plots. The normalization is arbitrary.
• Figure 3: Numerical spectroscopy of varicose waves. (a) Typical response curves for the injected energy $E_\mathrm{i}$ versus drive frequency $\Delta\epsilon_\mathrm{d}$ are plotted for $k_\mathrm{d}=1$ (red) and $k_\mathrm{d}=3$ (blue), with (solid) and without (dashed) a vortex. Each spectrum is driven with $r_\mathrm{d}=0.25$ for $T=150$ (the Fourier-limited FWHM is $\approx 0.04$); here $N_r=200$, $N_z=12$, and $N_t=2\times10^5$. (b) The spectroscopically extracted dispersion relation is shown as solid circles; the linear analysis calculation is the dash-dotted line. Inset: the varicose peak position versus radial grid points $1/N_r$ for $k_\mathrm{d}=3$, together with a linear fit for the $N_r\to\infty$ extrapolation. Each resonance was driven with $r_\mathrm{d}=5$ for $T=100$ (the Fourier-limited FWHM is $\approx 0.06$). Throughout this panel, $N_z=12$ and $N_t=3.5\times 10^5$. The gray band is the $95\%$ confidence interval on the fitted parameters; the resulting error on the main panel is smaller than point size. (c) The spatial profile $\delta \tilde{\psi}(r,k_\mathrm{d},\epsilon_\mathrm{p})$ is plotted for $k_\mathrm{d}=1$ (red) and $k_\mathrm{d}=3$ (blue) for resonant drive; the infinite-system expectation is shown as a dash-dotted line. Here $r_\mathrm{d}=5$, $T=150$, $N_r=200$, and $N_t=2\times 10^5$.
• Figure 4: Finite-size effects on the m$\boldsymbol{ ~=0}$ modes (top) Varicose wave's spatial profile $u(r)$ and localization length $\ell$ (dashed line, see text) at fixed system size $R$ (left) and wavenumber $k$ (right). (bottom) Spectrum of excitations at fixed radius ($R=50$, left) and fixed wavenumber ($k=1$, right). The color indicates $\ell/R$.
• Figure 5: Finite-size effects on the varicose modes. System radius $R_\mathrm{c}$ at which $\Delta\epsilon_0^{(0)} = 0$ (solid line) and $\Delta\epsilon_0^{(1)} = 0$ (dashed line) as a function of $k$.