Localization and delocalization of ultracold bosonic atoms in finite optical lattices

We study bosonic atoms in small optical lattices by exact diagonalization and observe a striking similarity to the superfluid to Mott insulator transition in macroscopic systems. The momentum distribution, the formation of an energy gap, and the pair correlation function show only a weak size dependence. For noncommensurate filling we reveal in deep lattices a mixture of localized and delocalized particles, which is sensitive to lattice imperfections. Breaking the lattice symmetry causes a Bose-glass-like behavior. We discuss the nature of excited states and orbital effects by using an exact diagonalization technique that includes higher bands.

Figure 1

(Color online) The lowest coefficients $c_n$ for two $(\circ )$ and eight atoms $(+)$ in a double well potential in dependency on the lattice depth $V_0$ calculated by exact diagonalization. The lines without markers are obtained using the LBA. The inset shows a symmetric and an antisymmetric basis wave function.

Figure 2

(Color online) The momentum distribution of six bosons in a quasi-one-dimensional chain with six sites. The crossover from a delocalized wave function $(V_0\lesssim 5E_R)$ to a localized wave function $(V_0\gtrsim 10E_R)$ can be observed.

Figure 3

(Color online) The pair correlation function of six bosons in a chain with six sites shows the localization of particles for $V_0\gtrsim 10E_R$. Due to the low filling factor $\nu =1$, the deviation from results obtained using the LBA (dotted lines) are only noticeable for $V_0\lesssim 5E_R$.

Figure 4

(Color online) The integral over the local correlation function for chains with $N_s$ sites and small two-dimensional lattices (6, 9, and 12 sites) with filling factor $\nu =1$ (in LBA). Inset (a) is a logarithmic plot in units of $V_0∕E_R$ and inset (b) is a double logarithmic plot in units of $U∕J$ for a chain with ten sites.

Figure 5

The energy spectrum of six bosons in a chain with six sites relative to the energy $E_0$ of the noninteracting system. The spectrum shows the ground state and the two lowest bands.

Figure 6

(Color online) The momentum distribution (normalized to 1) of a two-dimensional $3\times 3$ lattice with nine atoms for various lattice depth $V_0$ using the LBA.

Figure 7

(Color online) The momentum distribution of $N=5$ to 10 atoms in a chain with five sites at $V_0=40E_R$. For a perfectly symmetric potential the localization of all particles can only be observed at integer filling (dotted lines).

Figure 8

The energy spectrum for six particles on five sites shows the formation of a lowest many-particle band.

Figure 9

(Color online) The momentum distribution (normalized to 1) of the lowest band states of a two-dimensional $3\times 3$ lattice with ten atoms at $V_0=30E_R$ (in LBA). The seventh excited state, which corresponds to the sixth excited state (rotated by $\pi ∕2$), is not shown here [29].

Figure 10

(Color online) The momentum distribution of six atoms in a chain with five sites at $V_0=20E_R$ for a laser beam with a finite Gauss width $w_0$. The inset shows the corresponding site offsets ${ϵ}_1$ and ${ϵ}_2$.