Calculation of collective modes for the Bose-Hubbard model with confinement

The collective excitations in the Bose-Hubbard model in a trap are studied by means of numerical diagonalization in one dimension. The strength function is calculated for monopole and dipole perturbations, and moments of the strength function are utilized in order to obtain information about the collective behavior under external forces. In the superfluid regime, the spectrum is found to be exhausted by one single frequency, while in systems that contain a Mott insulating plateau, several frequencies are excited. An explanation of recent experimental findings in terms of a Mott plateau is suggested.

Figure 1

Strength function for a monopole excitation of a weakly interacting system, with ${\omega }_t=0.3$, $U=0.1$. The height of the stems represent the strength $S_{k0}$ for each excited state $\mid k⟩$, and their horizontal position indicates the frequency of the excited state, $\omega =E_k$. The crosses mark the magnitude of three moment-related frequencies, in increasing order, ${\omega }_{10}$, ${\omega }_{20}$, and ${\omega }_{21}$. (However, the positions of the three crosses coincide completely in this figure.) The inset shows the density profile as circles and the quantum fluctuations in the local density as asterisks. All quantities are plotted in the dimensionless units employed throughout the paper.

Figure 2

Same as Fig. 1, but here the perturbation has dipole symmetry.

Figure 3

Same as Fig. 1, but here the trap strength is ${\omega }_t=1$ and the coupling is intermediate, $U=20$. In the inset, the asterisks represent the quantum fluctuations in the density magnified by a factor 10.

Figure 4

Same as Fig. 1, but here the trap and the coupling are stronger, ${\omega }_t=4$ and $U=100$. In the inset, the asterisks represent the quantum fluctuations in the density magnified by a factor 100.

Figure 5

Properties of the system as functions of the coupling $U$ and the trap strength ${\omega }_t$. The latter is swept from ${\omega }_t=0.3$ to ${\omega }_t=4$, and $U$ varies as $U=20{\omega }_t$ through the sweep. The topmost panel shows the moment-related frequencies (from top to bottom ${\omega }_{10}$, ${\omega }_{20}$, and ${\omega }_{21}$) in units of the lowest monopole frequency $E_2$. The middle panel shows the magnitude of the zeroth moment $m_0$. The bottom panel plots the quantum fluctuations in local density, $\delta n^2=⟨n^2⟩-{⟨n⟩}^2$, at the point $r=0$ (solid line) and $r=2$ (dashed). The units are the dimensionless units employed throughout the paper.