Tunable geometries from a sparse quantum spin network

摘要

Nonlocal light-mediated interactions between cold atoms coupled to the mode of an optical cavity present uniqueprospects for simulating the quantum dynamics of strongly-interacting many-body systems. In a recent publication,we introduced a tunable, nonlocal sparse spin network that can be engineered in near-term single-modecavity QED platforms.1 In this companion paper, we study this spin network in detail and pedagogically review itsbasic dynamical properties, providing theoretical details and calculations that expand on the statements made inour original publication. We show that the network exhibits two distinct notions of emergent geometry { linear andtreelike { that can be accessed using a single tunable parameter. In either of these two extreme limits, we nd asuccinct description of the resulting dynamics in terms of two distinct metrics on the network, encoding a notion ofeither linear or treelike distance between spins. We also show that the network can be mapped in these two extremelimits onto exactly solvable models: a linear Heisenberg spin chain in one limit, and a Dyson hierarchical model inthe other. These observations highlight the essential role played by the geometry of the interaction structure indetermining a system's dynamics, and raise prospects for novel studies of nonlocal and highly chaotic quantumdynamics in near-term experiments.