Treelike Interactions and Fast Scrambling with Cold Atoms

We propose an experimentally realizable quantum spin model that exhibits fast scrambling, based on nonlocal interactions that couple sites whose separation is a power of 2. By controlling the relative strengths of deterministic, nonrandom couplings, we can continuously tune from the linear geometry of a nearest-neighbor spin chain to an ultrametric geometry in which the effective distance between spins is governed by their positions on a tree graph. The transition in geometry can be observed in quench dynamics, and is furthermore manifest in calculations of the entanglement entropy. Between the linear and treelike regimes, we find a peak in entanglement and exponentially fast spreading of quantum information across the system. Our proposed implementation, harnessing photon-mediated interactions among cold atoms in an optical cavity, offers a test case for experimentally observing the emergent geometry of a quantum many-body system.

Figure 1

Nonlocal spin models with cold atoms. (a) Linear light cone vs fast scrambling. (b) Spins (green) coupled locally along a chain of length $N=8$ or nonlocally according to a tree of depth $n={\log }_2(N)$. Green cube shows full pattern of couplings specified in Eq. (2). (c) Scheme for controlling the graph of interactions mediated by a cavity via the spectrum of a drive field and the gradient of a magnetic field $\mathbf{B}$. Depending whether short-range (blue) or long-range (orange) couplings dominate, the graph resembles either the linear chain (i) or the tree (ii).

Figure 2

Dynamics in the single-magnon sector. (a) Occupation $⟨n_j(t)⟩$ for $N=128$ sites initialized with a single excitation on site $N/2$ with (i) $s=-2$, (ii) $s=0$, or (iii) $s=2$; dashed red lines show fits to slowest-growing occupations. (iv) Occupation $⟨n_j(t)⟩$ at $s=0$ for $N=1024$, vs graph distance $r_{ij}$ from initial site $i$. (v) Reemergence of light cone at $s=2$ after rearranging $N=128$ sites according to Monna map $\mathcal{M}$. (b) Magnon dispersion relation for $N=128$ sites arranged according to either (i) physical location or (ii) Monna-mapped order for $s=-2$ (blue), $s=0$ (black), and $s=2$ (red). (c) Breakdown of polynomial light cone: polynomial exponent $b$ (red) and logarithmic exponent $c^{\prime }$ (blue) vs $s$.

Figure 3

Growth of entanglement after a quench, for an initial product state of $N=16$ sites with $S=1/2$ oriented along $\widehat{\mathbf{x}}$. (a) Entanglement entropy $S_A$ vs partition size $L$ and time $t$, for Archimedean (top) or 2-adic (bottom) partitions. (b) Minimum entanglement vs $s$ over all possible partitions of size $L=1$, 2, 4, 8 (light to dark solid curves), compared with entanglement for Archimedean (blue dashed) and 2-adic (red dot-dashed) partitions.

Figure 4

Chaos and fast scrambling at $s=0$. (a) Average semiclassical sensitivity $C_{\mathrm{cl}}(t)$ for $N=4096$. (i) $C_{\mathrm{cl}}$ vs time for graph distances $r_{ij}=0,1,\dots ,6$ (blue to red). (ii) Fixed-time contours of $C_{\mathrm{cl}}$ vs $r_{ij}$, showing exponential decay of $C_{\mathrm{cl}}$ vs $r_{ij}$ at early times. Contours are for $0.2\le J_{0}t\le 4$ in increments of 0.2. (b) Lyapunov exponent $\lambda /J_0$ (i) and scrambling time $\lambda t_{*}$ (ii) vs $N$ (blue circles) with fit $\lambda t_{*}=\alpha \log (N)+\beta$ (red dashed). (c) MPS calculations for $N=64$, $S=1/2$, and open boundary conditions.