Roton-Maxon Excitation Spectrum of Bose Condensates in a Shaken Optical Lattice

We present experimental evidence showing that an interacting Bose condensate in a shaken optical lattice develops a roton-maxon excitation spectrum, a feature normally associated with superfluid helium. The roton-maxon feature originates from the double-well dispersion in the shaken lattice, and can be controlled by both the atomic interaction and the lattice modulation amplitude. We determine the excitation spectrum using Bragg spectroscopy and measure the critical velocity by dragging a weak speckle potential through the condensate—both techniques are based on a digital micromirror device. Our dispersion measurements are in good agreement with a modified Bogoliubov model.

Figure 1

Generation of roton-maxon dispersion in a shaken lattice. (a) For a single atom, the lattice modulation creates a double-well structure above a critical modulation amplitude (top three lines) [21]. In our experiment, the atoms are prepared at the minimum with zero or negative momentum ($q^{*}\le 0$, red dot); see text. (b) With atomic interactions, a roton minimum (circle) and a maxon maximum (square) in the excitation spectrum can form. The dashed line indicates the critical velocity limited by the roton minimum according to the Landau criterion for superfluidity. Dispersions are upward offset with increasing modulation amplitude for clarity. The lattice reciprocal momentum is $\hslash k_L=h/\lambda$ where $\lambda$ is the wavelength of the lattice beams and $h=2\pi \hslash$ is the Planck constant.

Figure 2

Excitation spectra. (a) We measure the excitation spectra with $N_0=30000$ atoms in a harmonic trap (square) and in a stationary lattice (circle) with DMD-based Bragg spectroscopy. The inset illustrates the moving optical potential with velocity $v$ and periodicity $d$ created by the DMD on the BEC (tilted ellipse); see text. The solid lines correspond to the Bogoliubov model with chemical potentials equal to the trap-averaged values. (b) For a BEC with $N_0=9000$ atoms loaded in a shaken optical lattice, we measure the excitation spectrum along the lattice direction. The modulation amplitude (peak to peak) is $\mathrm{\Delta }x=33\mathrm{nm}$. The solid line is the best fit based on Eq. (1). The inset shows a typical atom loss spectrum taken at $k=-0.38$ $k_L$. In both panels, the scattering length is $a=47$ $a_0$.

Figure 3

Roton or maxon energy vs scattering length. (a) We measure the excitation spectra at different scattering lengths $a/a_0=5$ (circle), 13 (triangle), 24 (square), 40 (diamond), 55 (pentagon), and 70 (star). The condensate number is $N_0=9000$. Solid curves are fits based on Eq. (1). A global optimization procedure gives $V=6.7(2)$ $E_{\mathrm{R}}$ and $\mathrm{\Delta }x=43(3)\mathrm{nm}$. (b) Roton energies (circle) and maxon energies (square) extracted from the fits in panel (a) are shown at different scattering lengths. Solid curves are fits based on ${\mathrm{\Delta }}_r=A(a/a_0{)}^{2/5}$ and ${\mathrm{\Delta }}_m=B+C(a/a_0{)}^{2/5}$, from which we obtain $A=h\times 9(1)\mathrm{Hz}$, $B=h\times 37(9)\mathrm{Hz}$, and $C=h\times 8(1)\mathrm{Hz}$.

Figure 4

Superfluid critical velocity. (a) We measure the residual condensate number fraction after dragging a speckle pattern through the center of the cloud at different velocities $v$ along the roton direction ($p>0$, solid dots) and the nonroton direction ($p<0$, solid squares). The solid lines are fits used to determine the critical velocity. The inset illustrates the experimental scheme; see text. (b) Critical velocities as a function of modulation amplitude are shown. Above the critical modulation amplitude, $\mathrm{\Delta }x>12\mathrm{nm}$, the critical velocity is significantly lower in the roton direction. Our measurement is compared with the critical velocity calculated from Eq. (1) using the Landau criterion (dashed lines). In both panels, the scattering length is $a=47$ $a_0$ and the initial condensate number is $N_0=9000$.