Formation of Spatial Shell Structure in the Superfluid to Mott Insulator Transition

We report on the direct observation of the transition from a compressible superfluid to an incompressible Mott insulator by recording the in-trap density distribution of a Bosonic quantum gas in an optical lattice. Using spatially selective microwave transitions and spin-changing collisions, we are able to locally modify the spin state of the trapped quantum gas and record the spatial distribution of lattice sites with different filling factors. As the system evolves from a superfluid to a Mott insulator, we observe the formation of a distinct shell structure, in good agreement with theory.

Figure 1

Integrated distribution of a superfluid (a) and a Mott insulating state (b) calculated for a lattice with harmonic confinement. Gray solid lines denote the total density profiles, blue (red) lines the density profiles from singly (doubly) occupied sites. A vertical magnetic field gradient is applied which creates almost horizontal surfaces of equal Zeeman shift over the cloud (dashed lines in insets). A slice of atoms can be transferred to a different hyperfine state by using microwave radiation only resonant on one specific surface (colored areas in insets). Spin-changing collisions can then be used to separate singly (blue) and doubly (red) occupied sites in that plane into different hyperfine states.

Figure 2

Integrated in-trap density profiles of the atom cloud for different lattice depths and atom numbers: (a) $1.0\times {10}^5$ atoms in the superfluid regime ($V_0=3E_{\mathrm{r}}$); (b) $1.0\times {10}^5$ atoms in the Mott regime ($V_0=22E_{\mathrm{r}}$); (c) $2.0\times {10}^5$ atoms; (d) $3.5\times {10}^5$ atoms. The gray data points denote the total density distribution and the red points the distribution of doubly occupied sites. The blue points show the distribution of sites with occupations other than $n=2$. The solid lines are fits to an integrated Thomas-Fermi distribution in (a) and an integrated shell distribution for (b)–(d). The $n=2$ data points are offset vertically for clarity.

Figure 3

Fitted radii of the Mott shells for different atom numbers and a lattice depth of $22E_{\mathrm{r}}$. The solid, gray, and open data points denote the radii of the shells with single, double, and triple occupation, respectively. The solid, dashed, and dash-dotted lines are calculations of these first three Mott shell radii for zero temperature and zero tunneling.

Figure 4

Evolution of the shell structure through the superfluid to Mott insulator transition for $N_{\mathrm{tot}}=(1.0\pm 0.1)\times {10}^5$ atoms. Shown is the relative size of the area in which doubly occupied lattice sites are found compared to the total cloud size. The dashed line marks the lattice depth at which the SF to MI transition is expected for a site occupation $n=1$.