State diagram for continuous quasi-one-dimensional systems in optical lattices

We studied the appearance of Mott insulator domains of hard sphere bosons on quasi-one-dimensional optical lattices when a harmonic trap was superimposed along the main axis of the system. Instead of the standard approximation represented by the Bose-Hubbard model, we described those arrangements by continuous Hamiltonians that depended on the same parameters as the experimental setups. We found that for a given trap the optical potential depth, $V_0$, needed to create a single connected Mott domain decreased with the number of atoms loaded on the lattice. If the confinement was large enough, it reached a minimum when, in the absence of any optical lattice, the atom density at the center of the trap was the equivalent of one particle per optical well. For larger densities, the creation of that single domain proceeded via an intermediate shell structure in which Mott domains alternated with superfluid ones.

Figure 1

$\varphi \left(z\right)$ for ${\omega }_z=2\pi \times 415$ Hz.

Figure 2

Particle density, $\rho \left(z\right)$ (dashed line), and number of particles per potential well, $n_i$ (solid line), for a set of $N$= 15 particles, ${\omega }_z=2\pi \times 415$ Hz, and $V_0$ = 7.6$E_R$. No Mott domain is observed. An arrow indicates the central well, in whose center the longitudinal harmonic potential equals 0.

Figure 3

$m{\omega }_z^2{\kappa }_i$ (dashed line and triangles) and ${\Delta }_i$ (solid line and circles) for the same system as in Fig. 2.

Figure 4

Same as in Fig. 2 but for $V_0$ = 15.2$E_R$.

Figure 5

Same as in Fig. 3 but for $V_0$ = 15.2$E_R$.

Figure 6

Same as in Fig. 2 but for $N$ = 31 and $V_0=6.3E_R$.

Figure 7

Critical values of ${(V_0/E_R)}_C$ for the apparition of a Mott domain in the center of cluster, in terms of the number of particles $\left(N\right)$ for $N=5$–49. Squares, ${\omega }_z=2\pi \times 4.15$ Hz; circles, ${\omega }_z=2\pi \times 60$ Hz. The dotted line is the value for the homogeneous case with the same optical lattice parameters, taken from Ref. [19].

Figure 8

Same as in Fig. 7, but for ${\omega }_z=2\pi \times$ 415 Hz and $N=5$–37.

Figure 9

Number of particles at the central site, $i=0$, as a function of $V_0/E_R$ for different values of $N$ and ${\omega }_z=2\pi \times$ 415 Hz.