From Light-Cone to Supersonic Propagation of Correlations by Competing Short- and Long-Range Couplings

TL;DR

The paper investigates how correlations spread in quantum many-body systems with competing short-range and global-range couplings, revealing a crossover from light-cone-like propagation to supersonic, distance-independent spreading. Using a Bose-Hubbard model coupled to a cavity field, the authors combine numerically exact time-dependent matrix product state methods with analytical and semiclassical approaches to identify the essential role of global-range fluctuations in driving rapid, system-wide correlations. They show that the supersonic spreading originates from cavity-field fluctuations and persists across one and two dimensions, even in the presence of dissipation, while a mean-field cavity description fails to capture this behavior. A minimal analytic model predicts a characteristic $\mathcal{C}_{nn}(d,t) \propto J^4 \Omega^4 t^8$ scaling governed by cavity fluctuations, highlighting the mechanism behind the nonlocal propagation. The findings offer guidance for experiments with ultra-cold atoms in optical cavities and extend understanding of information scrambling in systems with mixed interaction scales.

Abstract

We investigate the dynamical spreading of correlations in many-body quantum systems with competing short- and global-range couplings. We monitor the non-equilibrium dynamics of the correlations following a quench, showing that for strong short-range couplings the propagation of correlations is dominated at short and intermediate distances by a causal, light-cone, dynamics, resembling the purely short-range quantum systems. However, the interplay of short- and global-range couplings leads to a crossover between space-time regions in which the light-cone persists to regions where a supersonic, distance-independent, spreading of the correlations occurs. We identify the important ingredients needed for capturing the supersonic spreading and demonstrate our findings in systems of interacting bosonic atoms, in which the global range coupling is realized by a coupling to a cavity light field, or atomic long-range interactions, respectively. We show that our results hold in both one and two dimensions and in the presence of dissipation. Furthermore, we characterize the short time power-law scaling of the distance-independent growth of the density-density correlations.

Figures (7)

• Figure 1: (a), (c), (e) The propagation of $\mathcal{C}_{nn}(d,t)$, computed with $\hat{H}$, Eq. \ref{['eq:Hamiltonian']}, for the atoms-cavity coupling strength $\hbar\Omega/J\in\{0.03,0.42,0.99\}$. The blue line is a guide to the eye, approximating the front of the light-cone propagation for $\Omega=0$. (b), (d), (f) $\mathcal{C}_{nn}(d,t)$ for several distances and $\hbar\Omega/J\in\{0.03,0.42,0.99\}$. The curves depicted with shades of orange to brown correspond to $\hat{H}$, Eq. \ref{['eq:Hamiltonian']}, the dashed curves with shades of green to $\hat{H}_{\text{atom-only}}$, Eq. \ref{['eq:Hamiltonian_atom_only']}, and the dashed blue curves to the $\Omega=0$ case. The magenta lines show the algebraically scaling $\propto t^\alpha$, with $\alpha\in\{2,4,8\}$. The parameters used are $N=10$ particles, $L=20$ sites, $U/J=2$, $\hbar\delta/J=2$, $\hbar\Gamma/J=0$.
• Figure 2: (a)-(b) The dependence of the correlations $\mathcal{C}_{nn}$ as a function of rescaled time $t\sqrt{\Omega J/\hbar}$ for (a) $d=5$ and (b) $d=16$, obtained for the atom-cavity Hamiltonian $\hat{H}$, Eq. \ref{['eq:Hamiltonian']}. (c)-(d) The dependence of the correlations $\mathcal{C}_{nn}$ as a function of rescaled time $t\Omega^{2/3} (J/\hbar)^{1/3}$ for (c) $d=5$ and (d) $d=16$, obtained for the atom-only Hamiltonian $\hat{H}_{\text{atom-only}}$, Eq. \ref{['eq:Hamiltonian_atom_only']}. The magenta lines represent the algebraic scaling (a)-(b) $\propto t^8 \Omega^4$ and (c)-(d) $\propto t^6 \Omega^4$. The values of the coupling are $0.03\leq\hbar\Omega/J\leq1.02$, and the same parameters as in Fig. \ref{['fig:lightcone']}.
• Figure 3: (a) The propagation of $\mathcal{C}_{nn}(d,t)$, in the presence of dissipation, Eqs. (\ref{['eq:Lindblad']})-(\ref{['eq:Hamiltonian']}), for $\hbar\Omega/J=0.99$, $\hbar\Gamma/J=10$, $N=10$ particles, $L=20$ sites, $U/J=2$, $\hbar\delta/J=2$. The blue line is a guide to the eye, approximating the front of the light-cone propagation for $\Omega=0$ and $\Gamma=0$. (b) The propagation of correlations when cavity field is replaced with a classical field supp, for the same parameters as in (a). Correlations $\mathcal{C}_{nn}(d,t)$ calculated from TWA in (c) 1D for a lattice with L=21 and in (d) 2D for lattice with $L\times L=21\times21$. For the TWA simulations we have used $U/J=0.1$, $\hbar\delta/J=2$, $\hbar\Gamma/J=1$, $\sqrt{N}\hbar\Omega/J=4$ in (c), and $\sqrt{N}\hbar\Omega/J=8$ in (d). We initialize the atoms with alternating densities of $n=10$ and $n=11$ bosons corresponding to the total atom numbers (c) $N=220$ and (d) $N=4630$.
• Figure C1: (a), (c), (e) The space-time propagation of the correlations $\mathcal{C}_{nn}(d,t)$, in the presence of dissipation, and (b), (d), (f) the time-dependence of $\mathcal{C}_{nn}(d,t)$ for several distances. In panels (a), (b) we show the full quantum dynamics of the atoms-cavity model, Eqs. (1)-(2) in the main text, in panels (c), (d) the cavity dynamics has been replaced with a classical field, Eq. (\ref{['eq:Hamiltonian_MF']}) and Eq. (\ref{['eq:mf_photon_field']}), and in panels (e), (f) stochastic noise has been added to the classical field evolution, Eq. (\ref{['eq:Hamiltonian_MF']}) and Eq. (\ref{['eq:mf_photon_field_noise']}). The blue lines are a guide to the eye and approximates the front of the light-cone propagation for $\Omega=0$ and $\Gamma=0$. The magenta lines represent algebraically increasing curves $\propto t^\alpha$, with $\alpha\in\{2,4,8\}$. The parameters used are $\hbar\Omega/J=0.99$$\hbar\Gamma/J=10$, $N=10$ particles, $L=20$ sites, $U/J=2$, $\hbar\delta/J=2$. Panels (a) and (c) correspond to Fig. 3(a) and Fig. 3(b) in the main text, we reproduce them here for completeness.
• Figure C2: The space-time propagation of the correlations $\text{Log}[\mathcal{C}_{nn}(d,t)]$, in the presence of dissipation, where in (a) the cavity dynamics has been replaced with a classical field, Eq. (\ref{['eq:Hamiltonian_MF']}) and Eq. (\ref{['eq:mf_photon_field']}), and in panel (b) stochastic noise has been added to the classical field evolution, Eq. (\ref{['eq:Hamiltonian_MF']}) and Eq. (\ref{['eq:mf_photon_field_noise']}). The panels show the same data as in Fig. \ref{['fig:dissipation_supp']}(c) and Fig. \ref{['fig:dissipation_supp']}(e), but in a logarithmic scale. The blue lines are a guide to the eye and approximates the front of the light-cone propagation for $\Omega=0$ and $\Gamma=0$. In the white regions the value of the correlations is smaller than $10^{-4}$. The parameters used are $\hbar\Omega/J=0.99$$\hbar\Gamma/J=10$, $N=10$ particles, $L=20$ sites, $U/J=2$, $\hbar\delta/J=2$.