Dipole and monopole modes in the Bose-Hubbard model in a trap

The lowest-lying collective modes of a trapped Bose gas in an optical lattice are studied in the Bose-Hubbard model. An exact diagonalization of the Hamiltonian is performed in a one-dimensional five-particle system in order to find the lowest few eigenstates. The dipole and breathing character of the eigenstates is confirmed in the limit where the tunneling dominates the dynamics, but under Mott-like conditions the excitations do not correspond to oscillatory modes.

Figure 1

Ground-state density profile for different values of on-site interaction $U$ in a shallow trap with $\omega =0.3$. $r$ is the dimensionless site index.

Figure 2

Quantum fluctuations in the ground-state density, $⟨\delta n^2⟩=⟨n^2⟩-{⟨n⟩}^2$, for different values of $U$ in the shallow trap, with $\omega =0.3$.

Figure 3

Lowest excitation frequencies in units of the trap frequency $\omega$ in a shallow trap with $\omega =0.3$. Full lines represent the results for the three lowest excited states obtained by exact diagonalization. The dashed line is the outcome of the Bogoliubov approximation.

Figure 4

Time dependence of the density profile for a system with $U=0.1$ and $\omega =0.3$ (weak trapping and weak interactions). The different sets of bars show the density at successive time instances for a system prepared in a superposition of the ground state and an excited state (left panel, first excited state; right panel, second excited state) with the amplitudes 0.92 and 0.4, respectively. The curves in the lower two panels show the variation of the center-of-mass position and mean squared radius with time, showing the dipole and breathing character, respectively, of the oscillations. To enhance visibility, the curve for the mean squared radius has been shifted down by an amount equal to its time average, defining the plotted quantity as $⟨\delta r^2⟩=⟨\psi \left(t\right)\mid r^2\mid \psi \left(t\right)⟩-\overline{r^2}$ where $\overline{r^2}$ is the time average.

Figure 5

Same as Fig. 4, but here the coupling is $U=100$.

Figure 6

Ground-state density in the case of a tight trap, $\omega =4.0$.

Figure 7

Quantum fluctuations in the ground-state density in a tight trap, $\omega =4.0$.

Figure 8

Lowest few excitation frequencies in a tight trap, $\omega =4.0$. Dashed lines represent the Bogoliubov approximation. The pointed feature in the topmost curve for $U∕J\approx {10}^1$ is indicative of a crossing with the energy level above it, not shown in the picture.

Figure 9

Visualization of the two lowest modes for a tight trap, $\omega =4.0$, and weak coupling, $U=0.1$. Panels are as in Fig. 4.

Figure 10

Same as Fig. 4, but here the trapping frequency is $\omega =4.0$ and the coupling is $U=40$.

Figure 11

Same as Fig. 4, but here the trapping frequency is $\omega =4.0$ and the coupling is $U=100$.