Lieb-Robinson Bounds for Spin-Boson Lattice Models and Trapped Ions

Abstract

We derive a Lieb-Robinson bound for the propagation of spin correlations in a model of spins interacting through a bosonic lattice field, which satisfies itself a Lieb-Robinson bound in the absence of spin-boson couplings. We apply these bounds to a system of trapped ions, and find that the propagation of spin correlations, as mediated by the phonons of the ion crystal, can be faster than the regimes currently explored in experiments. We propose a scheme to test the bounds by measuring retarded correlation functions via the crystal fluorescence.PACS numbers: 03.65. Ud, 03.67.Ac, 37.10.Ty, 03.67.Mn The possibility of designing or simulating many-body systems in quantum-optical setups, such as ultracold atoms [1][2][3][4] or large ion crystals [5], is stimulating considerable progress in our understanding of non-equilibrium quantum many-body phenomena. However, the interpretation of these experiments demands powerful theoretical tools, including the LiebRobinson bounds (LRBs) [6] developed in this work. On the surface, LRBs show that non-relativistic quantum many-body systems, under certain conditions, display a causal structure analogous to relativistic quantum field theories. More deeply, LRBs are essential to prove fundamental quantum many-body properties, such as the exponential decay of correlations in the ground-state of gapped local Hamiltonians -the so-called "clustering of correlations" [7]-, scaling laws for entanglement entropy [8] -the "area laws" [9] -, or the robustness of topological order under local perturbations [10].Causality limits how local measurements and perturbations, described by an operator O X in region X, affect later measurements of another operator O Y in a separate region Y [11]. In analogy to Heisenberg's principle [12], this uncertainty is quantified by a commutator C Y,X (t) = [O Y (t), O X (0)] . Lorentz invariance and the mathematical structure of relativistic theories guarantee causality [11]. Thus, C Y,X (t) = 0 when the distance d XY > ct places both regions outside the light cone defined by the speed of light c. In non-relativistic quantum mechanics, causality is violated at the few particle level [11]. Remarkably, in the many-body regime, an approximate light cone emerges, outside of which such correlations are vanishingly small. This phenomenon, first demonstrated by Lieb and Robinson [6] for a lattice of locally-interacting spins, has been generalized to finite-dimensional models, anharmonic oscillators and master equations [7,[13][14][15][16].In this work, we address the role of bosons as mediators of interactions between particles in the light of LRBs. This is done for a general model of finite dimensional systems interacting through a bosonic field that satisfies a LRB itself. New bounds are derived, which are then applied to a crystal of trapped atomic ions, where the spins and the bosons map to the ions' internal states and the crystal's phonons, respectively. These LRBs work for all spin-boson lattice models of any dimensionality and geometry realized with sta...

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