Detecting the Amplitude Mode of Strongly Interacting Lattice Bosons by Bragg Scattering

We report the first detection of the Higgs-type amplitude mode using Bragg spectroscopy in a strongly interacting condensate of ultracold atoms in an optical lattice. By the comparison of our experimental data with a spatially resolved, time-dependent bosonic Gutzwiller calculation, we obtain good quantitative agreement. This allows for a clear identification of the amplitude mode, showing that it can be detected with full momentum resolution by going beyond the linear response regime. A systematic shift of the sound and amplitude modes’ resonance frequencies due to the finite Bragg beam intensity is observed.

Figure 1

Comparison of the experimental visibility (a) and theoretically predicted energy absorption (b) at $|{\mathbf{p}}_B|=\pi /a$ using a Blackman-Harris pulse of 10 ms in an optical lattice with $s=13$ in a 3D trap. A maximum intensity $V=0.27E_r$ and the experimentally determined total particle number $N_{\mathrm{tot}}=5\times {10}^4\pm 33\%$ and $\omega =2\pi \times (26,26,21)\mathrm{Hz}$ trapping frequency were used for (b), leading to a maximum central density $n=1.05$. The lower peak is the sound mode, mainly broadened by the high intensity of the Bragg beam to lower frequencies. The upper peak at 650 Hz is the gapped amplitude mode, broadened mainly by the trap. Figures (c),(d),(e) show the theoretically predicted trap broadened logarithmic quasimomentum distributions in the first Brillouin zone at the frequencies marked by the green lines in (b).

Figure 2

(a),(b) Theoretical dispersion relations for the homogeneous system in the linear response limit at $n=1$ (a) and density-dependence at $|\mathbf{k}|=\pi /a$ (b) in the SF for weak ($s=5$, black circles) and strong ($s=13$, blue squares) interactions. Corresponding Bogoliubov results [27] are shown as black dashed lines in (a). (c),(d) Illustration of the order parameter for a coherent excitation of the phase (sound) mode (c) and the amplitude mode (d) in a homogeneous condensate at $\mathbf{k}=(0.8/a,0,0)$ and $s=13$. The projection of all ${\psi }_{\mathbf{i}}$’s in the complex plane is shown by the black ellipses: for the sound mode (c), the oscillation is almost exclusively in the tangential, for the amplitude mode (d) mainly in the radial (i.e., in the amplitude) direction.

Figure 3

Energy absorption (a) and quasimomentum density (b),(c) spectra for a square pulse at high intensity $V=0.1E_r$ (insets: weak intensity $V=0.005E_r$) in the intermediate interaction $s=9$ regime and for Bragg momentum $|{\mathbf{p}}_B|=\pi /a$. The resonance frequencies predicted from the maxima of the high intensity energy absorption spectra [plotted as gray squares in (d)] contain a systematic uncertainty quantified by the FWHM of the pulse after $10\mathrm{ms}\approx 3.26/J$ indicated by the error bars and shaded region in (d). The comparison with the true quasiparticle energies [dashed white lines in (a)–(c), black circles in (d)] reveals significant discrepancies. For comparison in (d): The blue dotted line is the Bogoliubov result, the green dashed lines are the results from Ref. 21 for the amplitude and sound modes ($\omega (\mathbf{k})={\psi }_0\sqrt{2Un{ϵ}_{\mathbf{k}}}$ with ${\psi }_0$ determined by $GW$).