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arXiv:1612.07805

Entanglement and spin-squeezing without infinite-range interactions

Michael Foss‐Feig, Zhexuan Gong, Alexey Vyacheslavovich Gorshkov, Charles W. Clark

Published Dec 22, 2016• 10 citations
arXivHTMLPDF

Abstract

Infinite-range interactions are known to facilitate the production of highly entangled states with applications in quantum information and metrology. However, many experimental systems have interactions that decay with distance, and the achievable benefits in this context are much less clear. Combining recent exact solutions with a controlled expansion in the system size, we analyze quench dynamics in Ising models with power-law ($1/r^α$) interactions in $D$ dimensions, thereby expanding the understanding of spin squeezing into a broad and experimentally relevant context. In spatially homogeneous systems, we show that for small $α$ the scaling of squeezing with system size is identical to the infinite-range ($α=0$) case. This indifference to the interaction range persists up to a critical value $α=2D/3$, above which squeezing degrades continuously. Boundary-induced inhomogeneities present in most experimental systems modify this picture, but it nevertheless remains qualitatively correct for finite-sized systems.

DOI
10.48550/arxiv.1612.07805

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Infinite-range interactions are known to facilitate the production of highly entangled states with applications in quantum information and metrology. However, many experimental systems have interactions that decay with distance, and the achievable benefits in this context are much less clear. Combining recent exact solutions with a controlled expansion in the system size, we analyze quench dynamics in Ising models with power-law (1/rα1/r^α1/rα) interactions in DDD dimensions, thereby expanding the understanding of spin squeezing into a broad and experimentally relevant context. In spatially homogeneous systems, we show that for small ααα the scaling of squeezing with system size is identical to the infinite-range (α=0α=0α=0) case. This indifference to the interaction range persists up to a critical value α=2D/3α=2D/3α=2D/3, above which squeezing degrades continuously. Boundary-induced inhomogeneities present in most experimental systems modify this picture, but it nevertheless remains qualitatively correct for finite-sized systems.
Long-range spin-spin interactions are known to generate nonequilibrium dynamics that can squeeze the collective spin of a quantum spin ensemble in a scalable manner, leading to states whose metrologically useful entanglement grows with system size. Here, we show theoretically that scalable squeezing can be produced in 2D U(1)-symmetric systems even by short-range interactions, i.e., interactions that at equilibrium do not lead to long-range order at finite temperatures, but rather to an extended Berezinskii-Kosterlitz-Thouless critical phase. If the initial state is a coherent spin state in the easy plane of interactions, whose energy corresponds to a thermal state in the critical Berezinskii-Kosterlitz-Thouless phase, the nonequilibrium dynamics exhibits critical slowing down, corresponding to a power-law decay of the collective magnetization in time. This slow decay protects scalable squeezing, whose scaling reveals in turn the decay exponent of the magnetization. Our results open the path to realizing massive entangled states of potential metrological interest in many relevant platforms of quantum simulation and information processing-such as Mott insulators of ultracold atoms, or superconducting circuits-characterized by short-range interactions in planar geometries.
We investigate many-body spin squeezing dynamics in an XXZ model with interactions that fall off with distance r as 1/r^{α} in D=2 and 3 spatial dimensions. In stark contrast to the Ising model, we find a broad parameter regime where spin squeezing comparable to the infinite-range α=0 limit is achievable even when interactions are short ranged, α>D. A region of "collective" behavior in which optimal squeezing grows with system size extends all the way to the α→∞ limit of nearest-neighbor interactions. Our predictions, made using the discrete truncated Wigner approximation, are testable in a variety of experimental cold atomic, molecular, and optical platforms.
Long-range interacting spin systems can be used to generate metrologically useful entanglement in the form of spin squeezing resulting in quantum enhanced sensitivity. This work considers power-law spin exchange interactions to generate two-mode squeezing in a bilayer geometry. It shows how spatial control over the geometry allows for achieving the same sensitivity with power-law interactions as with infinite-range interactions, and how spatially Floquet-engineered interactions allow for achieving the ultimate Heisenberg scaling of sensitivity.
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Spin squeezed entanglement enables metrological precision beyond the classical limit. Understood through the lens of continuous symmetry breaking, dipolar spin systems exhibit the remarkable ability to generate spin squeezing via their intrinsic quench dynamics. To date, this understanding has primarily focused on lattice spin systems; in practice however, dipolar spin systems$\unicode{x2014}$ranging from ultracold molecules to nuclear spin ensembles and solid-state color centers$\unicode{x2014}$often exhibit significant amounts of positional disorder. Here, we develop a theory for scalable spin squeezing in a two-dimensional randomly diluted lattice of quantum dipoles, which naturally realize a dipolar XXZ model. Via extensive quantum Monte Carlo simulations, we map out the phase diagram for finite-temperature XY order, and by extension scalable spin squeezing, as a function of both disorder and Ising anisotropy. As the disorder increases, we find that scalable spin squeezing survives only near the Heisenberg point. We show that this behavior is due to the presence of rare tightly-coupled dimers, which effectively heat the system post-quench. In the case of strongly-interacting nitrogen-vacancy centers in diamond, we demonstrate that an experimentally feasible strategy to decouple the problematic dimers from the dynamics is sufficient to enable scalable spin squeezing.
Recent work has identified a dynamical squeezing phase transition in power-law interacting bilayer XXZ spin models, separating a fully collective phase with Heisenberg-limited squeezing from a partially-collective phase with universal critical scaling. Here we test and establish the universality of this transition along two qualitatively different microscopic axes: lattice geometry, by studying square, triangular, and honeycomb 2D2\mathrm{D}2D bilayers as well as 1D1\mathrm{D}1D ladders, and a symmetry-preserving rescaling λλλ of the interlayer couplings relative to the intralayer ones. Combining a Bogoliubov instability analysis with discrete truncated Wigner simulations, we find that the transition persists across all four lattice geometries and over a wide range of λλλ with critical exponents consistent within error, providing strong evidence for a genuine non-equilibrium universality class. The Bogoliubov theory recovers the previously identified scaling aZ∗∝La_Z^* \propto LaZ∗​∝L in the long-range interacting regime α<d+2α< d+2α<d+2, and yields an analytical scaling aZ∗∝L2/(α−d)a_Z^* \propto L^{2/(α-d)}aZ∗​∝L2/(α−d) for the critical aspect ratio with system size for α>d+2α>d+2α>d+2, with ααα the power-law exponent in dimension ddd. This uncovers a previously unrecognized sub-linear regime for short-range interactions. By tuning λλλ we vary the interlayer coupling strength at fixed layer spacing, demonstrating that the dynamical transition can be driven purely through interaction engineering without modifying the underlying geometry. These findings provide a versatile route toward controlling entanglement generation in Rydberg-array, polar molecule, and trapped-ion platforms with applications in quantum sensing and simulation.