Metastability, chaos and spectrum tomography for Bose-Hubbard rings and chains

Abstract

We analyze the metastability of Bose-Hubbard condensates for finite-size one-dimensional ring lattices and open chains, using a semiclassical tomographic perspective that emphasizes the relation of the many-body spectrum to the underlying classical phase-space structures. This constitutes an arena for inspection of quantum ergodicity and localization, in far-from-equilibrium scenarios of experimental interest. Both local aspects (via Bogoliubov analysis) and global aspects (by inspecting the mixed regular-chaotic dynamics) are addressed. We also clarify how chaos is diminished in the limit of the Gross-Pitaevskii equation.

Figures (11)

• Figure 1: Stability regimes for 5-site ring. (a) The $(\phi,u)$ regime diagram. White regions indicate Landau energetic stability (all Bogoliubov frequencies are real and positive). Yellow region indicates linear dynamical stability (Bogoliubov frequencies are real, but some become negative). The gray region indicates instability (Bogoliubov frequency becomes complex). (b) Bogoliubov frequencies along the indicated ${u=1}$ section. In the region where they become complex, the real part is plotted in magenta, while the imaginary part is plotted as a red dashed line.
• Figure 4: Bogoliubov freq versus $u$, complex frequencies. (a) The imaginary part $\hbox{Im}[\omega_q]$ of the Bogoliubov frequency is plotted versus $u$ for a chain with 3 (blue), 5 (orange), and 7 (yellow) sites. We inspect the SP that supports condensation in the first excited orbital. (b) The critical value $u_c$ above which complex frequencies appear as a function of $L_s$. Note that for 3 sites ${u_c=0}$. For ${u_L > u_cL}$, the SP becomes unstable; hence, in the GPE limit, the instability regime gets excluded, and chaos is globally diminished.
• Figure 5: The structure of the $\tilde{\bm{P}}$ matrix. Image of the $\tilde{\bm{P}}$ matrix (multiplied by $L=L_s{+}1$) in the orbital representation for the $m_o{=}4$ stationary state of an $L_s{=}51$ chain. Left and right panels are for $u{=}0$ and $u{=}8$. Exact zero values are colored in gray, while blue color indicates very small non-zero numerical values.
• Figure 6: Representative classical trajectories for 3 site chain. The trajectories are launched at $n_o=1$ (blue) and at $n_o=0.5$ (red). Plot of $n_o$ versus time, from left to right: $u=0.5$ (quasi-regular phase-space that contains an unstable SP); $u=1.5$ (chaotic phase-space that contains an unstable SP); $u=3.5$ (separated chaotic sea and stable SP island).
• Figure 10: Tomographic spectra for a 5 site Ring. The columns are for: (a) Classical ${(n_{o},E)}$ energy-landscape. The plots display a uniform distribution of phase-space points $(E, n_{o})$. Some of the points are used as initial conditions for a long trajectory of duration $t{=}2500$, and color-coded according to the temporal average $\left\langle n_{o} \right\rangle$. (b) Classical $(\left\langle n_{o} \right\rangle ,E)$ spectrum, color-coded by inverse classical purity $1/\mathcal{S}$. (c) Quantum $(\left\langle n_{o} \right\rangle ,E)$ spectrum for $N{=}30$ particles, color coded by inverse purity $1/\mathcal{S}$. Only $P{=}0$ eigenstates are included. The rows from top to bottom are for: ${(u{=}4, \phi{=}1.1 \pi)}$ demonstrating ES of the SP; ${(u{=}1, \phi{=}2.1 \pi)}$ demonstrating DS of the SP; ${(u{=}1, \phi{=}2.7 \pi)}$ where the SP becomes unstable; ${(u{=}4, \phi{=}2.7 \pi)}$ where, due to the increased chaos, we get ergodicity, as seen in the upper part of the first-row spectrum.