Complexity in parametric Bose-Hubbard Hamiltonians and structural analysis of eigenstates

We consider a family of chaotic Bose-Hubbard Hamiltonians parametrized by the coupling strength $k$ between neighboring sites. As $k$ increases the eigenstates undergo changes, reflected in the structure of the local density of states. We analyze these changes, both numerically and analytically, using perturbative and semiclassical methods. Although our focus is on the quantum trimer, the presented methodology is applicable for the analysis of longer lattices as well.

Figure 1

(Color online) The kernel $P(n\mid m)$ of the BHH plotted as a function of the perturbed energies $E_n$ (LDOS representation) and for various perturbation strengths $\delta k>\delta k_{\mathrm{qm}}$. In the lower plane we report the parametric evolution of the energy levels within the bandwidth ${\Delta }_b\approx 230$. The energy width $\Gamma$ is shown (bold line) as a function of $\delta k$. The averaged shape of eigenfunctions is given by the same kernel $P(n\mid m)$ and is obtained by just inverting the energy axis. Here, $N=70$ and ${\lambda }_0=0.053$.

Figure 2

(Color online) Upper panel: Comparison of the quantum band profile $⟨{\mid B_{\mathrm{nm}}\mid }^2⟩(2\pi \tilde{U}∕N^2\Delta )$ with the classical power spectrum $\tilde{C}\left(\omega \right)$. The number of particles is $N=230$ and ${\lambda }_0=0.053$. The inset shows a snapshot of the matrix $\mathbf{B}$. Lower panel: The kernel $P(n\mid m)$ of the BHH [see Eq. (1)] for $\delta k=0.05$ and for $\delta k=0.3$ compared to the corresponding theoretical expressions $P_{\mathrm{FOPT}}$ and $P_{\mathrm{prt}}$. Here $\delta k_{\mathrm{qm}}=0.09$ and $\delta k_{\mathrm{prt}}=1.02$.

Figure 3

(Color online) The parameters (a) $\delta k_{\mathrm{qm}}$ and (b) $\delta k_{\mathrm{prt}}$ for various $\tilde{U}$, $N$ and for ${\lambda }_0=0.053$. A nice scaling in accordance with Eq. (2) is observed. In the inset of (b) we report the scaling of the energy width $\Gamma$ with $\delta k$. The dashed line has slope 2.

Figure 4

(Color online) The kernel $P(n\mid m)$ (LDOS representation) for the BHH in the nonperturbative regime $(\delta k=10)$ for $N=230$ and ${\lambda }_0=0.053$. In the upper panel we plot a time series $E\left(t\right)$ which leads to the classical profile $P_{\mathrm{cl}}\left(E\right)$ (see text for details).