Fermionic Atoms in a Three Dimensional Optical Lattice: Observing Fermi Surfaces, Dynamics, and Interactions

We have studied interacting and noninteracting quantum degenerate Fermi gases in a three-dimensional optical lattice. We directly image the Fermi surface of the atoms in the lattice by turning off the optical lattice adiabatically. Because of the confining potential, gradual filling of the lattice transforms the system from a normal state into a band insulator. The dynamics of the transition from a band insulator to a normal state is studied, and the time scale is measured to be an order of magnitude larger than the tunneling time in the lattice. Using a Feshbach resonance, we increase the interaction between atoms in two different spin states and dynamically induce a coupling between the lowest energy bands. We observe a shift of this coupling with respect to the Feshbach resonance in free space which is anticipated for strongly confined atoms.

Figure 1

Observing the Fermi surface. Time of flight images obtained after adiabatically ramping down the optical lattice. The characteristic density increases from left to right. (a) Image of 3500 atoms per spin state and a potential depth of the optical lattice of $5E_r$. Images (b)–(e) were obtained with 15 000 atoms per spin state. The potential depths of the optical lattices were (b) $5E_r$, (c) $6E_r$, (d) $8E_r$, and (e) $12E_r$. The images show the optical density (OD) integrated along the vertically oriented $z$ axis after 9 ms of ballistic expansion.

Figure 2

Analysis of the density distributions. The dots are cuts through the measured density distribution for quasimomentum $q_y=0$ after adiabatically ramping down the optical lattice. (a) Normal state with ${\rho }_c=14.5$, (b) band insulator with ${\rho }_c=137$, and (c) band insulator with ${\rho }_c=2500$. We have numerically calculated the momentum distribution function of fermions in the lowest band of a three-dimensional lattice with $20\times 20\times 20$ sites and characteristic lengths ${\zeta }_x/d=3.2$, ${\zeta }_y/d=2.6$, ${\zeta }_z/d=2.5$ [(a),(b)] and ${\zeta }_x/d=1$, ${\zeta }_y/d=0.8$, ${\zeta }_z/d=0.8$ (c), assuming zero temperature (solid lines). Experimental data of (c) are averaged over five images. Imperfect adiabaticity during the switch-off of the optical lattice may cause the rounding off of the experimental data at the edge of the Brillouin zone in (b) and (c). The calculated momentum distribution function is scaled to match the experimental data using identical scale factors for all graphs.

Figure 3

Restoring phase coherence. (a) Control sequence for the depth of the optical lattice. (b) Pseudocolor image of the momentum distribution after releasing the atoms from the initial optical lattice of $5E_r$ and 6 ms ballistic expansion. It reveals the central momentum peak and the matter wave interference peaks at $\pm 2\hslash k$. The data are averaged over five repetitive measurements. (c) Width of the central momentum peak obtained from Gaussian fits to the atomic density distribution. The initial width is determined by the momentum spread of an atom localized in the vibrational ground state of a lattice well. The $10\%$ difference in this size comes from slightly different magnifications of the imaging system in the two orthogonal directions. The difference in the asymptotic values of the width can most likely be attributed to the loading sequence of the lattice and to the asymmetry of the confining potentials due to the different beam waists. The error bars show the statistical error of four repetitive measurements.

Figure 4

Interaction-induced transition between Bloch bands. (a) Transferring fermions into higher bands using a sweep across the Feshbach resonance (filled symbols). The inverse magnetic field sweep rate is $12\mu \mathrm{s}/\mathrm{G}$. The line shows a sigmoidal fit to the data. The open symbols show a repetition of the experiment with the atoms prepared in the spin states $|F=9/2,m_F=-9/2⟩$ and $|F=9/2,m_F=-7/2⟩$ where the scattering length is not sensitive to the magnetic field. The magnetic field is calibrated by rf spectroscopy between Zeeman levels. Because of the rapid ramp, the field lags behind its asymptotic value and the horizontal error bars represent this deviation. (b) Fraction of atoms in higher bands for a final magnetic field of 233 G for different magnetic field sweep rates. The vertical error bars show the statistical error of four repetitive measurements. (c) Momentum distribution for a final magnetic field of $B=233\mathrm{G}$ and a $12\mu \mathrm{s}/\mathrm{G}$ sweep rate. Arrows indicate the atoms in the higher bands.