Structure and stability of Mott-insulator shells of bosons trapped in an optical lattice

We consider the feasibility of creating a phase of neutral bosonic atoms in which multiple Mott-insulating states coexist in a shell structure and propose an experiment to spatially resolve such a structure. This spatially inhomogeneous phase of bosons, arising from the interplay between the confining potential and the short-ranged repulsion, has been previously predicted. While the Mott-insulator phase has been observed in an atomic gas, the spatial structure of this phase in the presence of an inhomogeneous potential has not yet been directly probed. In this paper, we give a simple recipe for creating a structure with any desired number of shells, and explore the stability of the structure under typical experimental conditions. The stability analysis gives some constraints on how successfully these states can be employed for quantum information experiments. The experimental probe we propose for observing this phase exploits transitions between two species of bosons, induced by applying a frequency-swept, oscillatory magnetic field. We present the expected experimental signatures of this probe, and show that they reflect the underlying Mott configuration for large lattice potential depth.

Figure 1

The mean-field phase diagram for the Bose-Hubbard Hamiltonian at first perturbative order in $w∕U$. Here, $\tilde{\mu }\equiv \mu -V\left(r\right)$, as described in the text, and $Z$ is the coordination number of the lattice ($Z=6$ for the three-dimensional cubic lattice under consideration here). The shaded regions are Mott insulating with the indicated number of bosons per site $n$. The white regions are superfluid. Notice that the superfluid regions at integer values of $\tilde{\mu }$ extend all the way to zero tunneling $(w∕U=0)$. While the exact location of the boundaries between superfluid and Mott regions at finite tunneling is different at higher orders in perturbation theory, the qualitative features of the phase diagram persist [19].

Figure 2

Illustration of the bounding values of $r$ for an spherically symmetric confining potential centered at $r=0$ which supports shells of Mott-insulator phases.

Figure 3

Spatial profile of occupation number for six different temperatures (see legend, color available online). A Metropolis algorithm is used to calculate the distribution of roughly $3\times {10}^4$ atoms among lattice sites with a harmonic external potential characterized by $\alpha =h\times 113\mathrm{Hz}∕\mu {\mathrm{m}}^2$ and the experimental parameters detailed in the Introduction. The average number of atoms per site $⟨n⟩$ is shown at radii $r$ which are multiples of the lattice spacing. The radius is measured from the minimum of the external potential.

Figure 4

Probability distribution of occupation number for six different temperatures: (a) $U∕\left(100k_B\right)$, (b) $U∕\left(30k_B\right)$, (c) $U∕\left(10k_B\right)$, (d) $U∕\left(3k_B\right)$, (e) $U∕k_B$, and (f) $2U∕k_B$. The probability $P_n$ to find a site occupied by $n$ atoms is shown for radii $r$ which are multiples of the lattice spacing. The data are extracted from the Metropolis algorithm used in Fig. 3 and are displayed as a stacked column plot (see legend, color available online).

Figure 5

The energy spectrum of ${\widehat{H}}_{ab}$ for lattice sites with $n=1,2,3$ (a) and the Rabi frequencies for transitions between states that are coupled by the interaction Hamiltonian (b). In (a), the energy difference between states and the shift of the ground state are shown.

Figure 6

The spectrum for transitions between $\mid a⟩$ and $\mid b⟩$ for $n=1,2,3$. The transition frequency between states with different $n_a$ and $n_b$ is shown by the thick line. The dashed lines indicate the boundaries between sites with different $n$ in the lattice, which occur at radii $R_0$, $R_1$, and $R_2$. Sites with $n=1,2,3$ are indicated with green, blue, and red coloring, respectively (color available online). The interaction energy shifts are indicated, and the figure is drawn assuming $\beta >1$.

Figure 7

Simulation of rapid adiabatic passage experiment for roughly $3\times {10}^4$ atoms at six different temperatures (see the legend, where $k_B$ is Boltzmann’s constant). The number of atoms $N_b$ in state $\mid b⟩$ is shown as the frequency ${\omega }_{\mathrm{rf}}$ of an oscillating magnetic field is swept (color available online).