Many-Body Interference at the Onset of Chaos

We unveil the signature of many-body interference across dynamical regimes of the Bose-Hubbard model. Increasing the particles’ indistinguishability enhances the temporal fluctuations of few-body observables, with a dramatic amplification at the onset of quantum chaos. By resolving the exchange symmetries of partially distinguishable particles, we explain this amplification as the fingerprint of the initial state’s coherences in the eigenbasis.

Figure 1

Fluctuations of two-particle observables in the Bose-Hubbard model (1) for $N=L=6$. Top: average temporal variance $v$ [Eq. (3)] versus $J/U$, for 100 initial states with one particle per site and variable particle indistinguishability $\gamma$ (color bar). The black curve highlights the case analyzed in Fig. 2. The gray-shaded area indicates the quantum-chaotic region of (1) [also see Figs. 2 and 2]. Insets show the plateau value $v_{\infty }$ (large $J/U$) and the enhancement $q=v_{\max }/v_{\infty }$ of the maximal fluctuation $v_{\max }$, as functions of $\gamma$. Bottom: time series of $⟨N_2(t)N_3(t)⟩$ for four combinations of $J/U$ and $\gamma$ (identified by correspondingly colored crosses in the top panel). For visibility, the curves for $J/U=4.1$ are shifted upward by 0.8.

Figure 2

Ingredients determining the averaged temporal variance $v$ shown in Fig. 1, as functions of $J/U$ and of the particles’ indistinguishability $\gamma$. The legend color codes the symmetry sectors $\lambda$ and indicates their corresponding block dimensions $d_{\lambda }$. (a) Weights $p_{\lambda }$ [see Eq. (4)] of the initial states for varying $\gamma$ (vertical black line marks $\gamma =0.15$). (b) Bottom: comparison of $v_{\lambda }$ (solid), Eq. (4), to the factorized approximation $R_{\lambda }{A}_{\lambda }$ (dashed), Eq. (5), versus $J/U$, for $\gamma \simeq 0.15$. Top: qualitative illustration of the competition between the delocalization $d_{\lambda }^{\mathrm{eff}}$ of the initial state’s component ${\rho }^{\lambda }$ and the bandwidth $W_{\lambda }$ of the averaged observable $\widehat{|O_{\alpha \beta }^{\lambda }{|}^2}$, in the eigenbasis of the symmetry sector $\lambda$, for three values of $J/U$. (c) Evolution of $W_{\lambda }$ (dashed) and $d_{\lambda }^{\mathrm{eff}}$ (solid) with $J/U$ (horizontal lines highlight block dimensions $d_{\lambda }$). Inset: delocalization of the 60 eigenstates closest in energy to the initial state, quantified by the mean value (solid) and variance (dashed) of their fractal dimensions with respect to the Young basis, identifying the chaotic phase [shaded regions in the inset and in the main panels of (b) and (c)]. (d) Distribution of the sector-specific enhancement $q_{\lambda }$ for initial states with variable ${\gamma }_{\lambda }$. For clarity, (b)–(d) show data for the largest six sectors only.