Observation of Massless and Massive Collective Excitations with Faraday Patterns in a Two-Component Superfluid

We report on the experimental measurement of the dispersion relation of the density and spin collective excitation modes in an elongated two-component superfluid of ultracold bosonic atoms. Our parametric spectroscopic technique is based on the external modulation of the transverse confinement frequency, leading to the formation of density and spin Faraday waves. We show that the application of a coherent coupling between the two components reduces the phase symmetry and gives a finite mass to the spin modes.

Figure 1

Generation mechanism of excitation pairs in the density and spin branches of a two-component system. The external excitation at ${\omega }_M$ is converted into two excitations with opposite wave vectors and half the energy. In the absence of coupling between the two components, ${\mathrm{\Omega }}_R=0$ (left), two symmetries are preserved and both modes have a linear behavior, with density speed of sound $c_d$ and spin speed of sound $c_s$. When a coherent coupling is present, ${\mathrm{\Omega }}_R\ne 0$ (right), one of the two symmetries is broken, introducing a curvature in the spin dispersion relation, which makes the excitation acquire a mass.

Figure 2

Density and spin Faraday patterns. (a) Sketch of the experimental configuration. The transverse trapping frequency is modulated in time at frequency ${\omega }_M$, periodically compressing the elongated condensate. (b) The parametrically generated excitations appear as a spatial pattern with a well-defined wavelength along the axis of the condensate. (c) Density (red) and spin (blue) 2D experimental patterns and corresponding integrated 1D profiles for ${\omega }_M/2\pi =400\mathrm{Hz}$ (left) and ${\omega }_M/2\pi =200\mathrm{Hz}$ (right).

Figure 3

PSD of density (a) and spin (b) excitations as a function of the modulation frequency. The thick lines indicate theoretical predictions [Eqs. (1) and (2)] for the dispersion relations (dark) and subharmonics (light) (see text), with ${\mathrm{\Omega }}_R=0$ and no fitting parameters. The line thickness corresponds to one standard deviation confidence interval originating from the uncertainty in the atomic density. Insets show the modulation amplitude in time and the corresponding fringe visibility for ${\omega }_M/2\pi =200\mathrm{Hz}$. The dashed line in panel (a) indicates the position of the spin branch, where a spurious signal is present due to the crosstalk between spin and density modes.

Figure 4

PSD of the spin excitations in the presence of a coherent coupling. In (a), ${\omega }_p/2\pi =120\mathrm{Hz}$, ${\mathrm{\Omega }}_R/2\pi =33\mathrm{Hz}$, and ${\mu }_s/h=225\mathrm{Hz}$; in (b), ${\omega }_p/2\pi =175\mathrm{Hz}$, ${\mathrm{\Omega }}_R/2\pi =80\mathrm{Hz}$, and ${\mu }_s/h=150\mathrm{Hz}$. The thick lines indicate the theoretical predictions [Eqs. (1) and (2)] for the dispersion relations (dark) and subharmonics (light) (see text). The line thickness corresponds to one standard deviation confidence interval originating from the uncertainty in the atomic density. Insets show the corresponding PSD of the density channel (unaffected by the coupling). The dashed line in panel (b) indicates the position of the density branch, where a spurious signal is present due to the crosstalk between spin and density modes.