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arXiv:2309.05368

Scalable spin squeezing in two-dimensional arrays of dipolar large-$S$ spins

Youssef Trifa, Tommaso Roscilde

Published Sep 11, 2023• 0 citations
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Abstract

Controlling the quantum many-body state of arrays of qudits, possessing a large local Hilbert space, opens the path to a broad range of possibilities for many-particle entanglement, interesting both for fundamental quantum science, as well as for potential metrological applications. In this work we theoretically show that the spin-spin interactions realized in two-dimensional Mott insulators of large-spin magnetic atoms (such as Cr, Er or Dy) lead to scalable spin squeezing along the non-equilibrium unitary evolution initialized in a coherent spin state. An experimentally relevant perturbation to the collective squeezing dynamics is offered by a quadratic Zeeman shift, which leads instead to squeezing of individual spins. Making use of a truncated cumulant expansion for the quantum fluctuations of the spin array, we show that, for sufficiently small quadratic shifts, the spin squeezing dynamics is akin to that produced by the paradigmatic one-axis-twisting (OAT) model -- as expected from an effective separation between collective spin and spin-wave variables. Spin squeezing with OAT-like scaling is shown to be protected by the robustness of long-range ferromagnetic order to quadratic shifts in the equilibrium phase diagram of the system, that we reconstruct via quantum Monte Carlo and mean-field theory.

DOI
10.48550/arxiv.2309.05368

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Scalable spin squeezing in two-dimensional arrays of dipolar large-SSS spins
Controlling the quantum many-body state of arrays of qudits, possessing a large local Hilbert space, opens the path to a broad range of possibilities for many-particle entanglement, interesting both for fundamental quantum science, as well as for potential metrological applications. In this work we theoretically show that the spin-spin interactions realized in two-dimensional Mott insulators of large-spin magnetic atoms (such as Cr, Er or Dy) lead to scalable spin squeezing along the non-equilibrium unitary evolution initialized in a coherent spin state. An experimentally relevant perturbation to the collective squeezing dynamics is offered by a quadratic Zeeman shift, which leads instead to squeezing of individual spins. Making use of a truncated cumulant expansion for the quantum fluctuations of the spin array, we show that, for sufficiently small quadratic shifts, the spin squeezing dynamics is akin to that produced by the paradigmatic one-axis-twisting (OAT) model -- as expected from an effective separation between collective spin and spin-wave variables. Spin squeezing with OAT-like scaling is shown to be protected by the robustness of long-range ferromagnetic order to quadratic shifts in the equilibrium phase diagram of the system, that we reconstruct via quantum Monte Carlo and mean-field theory.
We theoretically show that the spin-spin interactions realized in two-dimensional Mott insulators of large-spin magnetic atoms (such as Cr, Er, or Dy) lead to scalable spin squeezing along the nonequilibrium unitary evolution initialized in a coherent spin state. An experimentally relevant perturbation to the collective squeezing dynamics is offered by a quadratic Zeeman shift, which leads instead to squeezing of individual spins. Making use of a truncated cumulant expansion for the quantum fluctuations of the spin array, we show that, for sufficiently small quadratic shifts, the spin squeezing dynamics is akin to that produced by the paradigmatic one-axis-twisting model-as expected from an effective separation between collective-spin and spin-wave variables. Scalable spin squeezing is shown to be protected by the robustness of long-range ferromagnetic order to quadratic shifts in the equilibrium phase diagram of the system that we reconstruct via quantum Monte Carlo and mean-field theory.
The production of large-scale entangled states is one of the main goals of next-generation quantum technologies, with an immediate potential for applications in the context of entanglement-assisted quantum sensing. A very promising platform to achieve this goal is offered by ultracold spinor gases, made of atoms with a large internal spin sensitive to magnetic fields. Here we show that the native spin-changing collisions in a spinor Bose-Einstein condensate, combined with an arbitrary quadratic Zeeman shift, can generate scalable spin squeezing in the collective spin of the ensemble, following the universal paradigm of the celebrated one-axis-twisting model. Squeezing dynamics is driven by the quadratic Zeeman shift when this shift is small; and by the spin-changing collisions for large shifts, in the form of stroboscopic squeezing. Turning off the Zeeman shift freezes out the collective-spin dynamics, so that the ensuing collective spin dynamics can be uniquely governed by an external field to be sensed. Our theoretical results pave the way for the use of spinor Bose gases with a large spin in fundamental studies of entanglement, as well as in advanced metrological applications.
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 study the generation of metrologically useful entanglement in a three-level (spin-1) system naturally realized in arrays of dipole-interacting Rydberg atoms confined in optical tweezers. In the spin-quadrupolar operator basis, the interaction Hamiltonian decomposes into effective SU(2) subspaces, within which quench dynamics from product initial states generate scalable spin-nematic squeezing. For symmetric interactions, we identify a mapping to effective one-axis twisting within bright and dark manifolds and demonstrate that the squeezing parameter scales as ξ2∝N−2/3ξ^{2}\propto N^{-2/3}ξ2∝N−2/3 (ξ2∝N−0.5ξ^{2}\propto N^{-0.5}ξ2∝N−0.5) with system size for all-to-all (two-dimensional dipolar) couplings. In both cases the quantum Fisher information reaches FQ∝N2F_Q\propto N^{2}FQ​∝N2. For antisymmetric interactions supplemented by a microwave drive we find a distinct two-axis countertwisting mechanism. This results in squeezing ξ2∝N−0.7ξ^{2}\propto N^{-0.7}ξ2∝N−0.7 for all-to-all interactions and moderate squeezing for dipolar interactions in 2D. Our results constitute a first theoretical step beyond the well-studied qubit setting toward scalable entanglement generation in qudit systems with dipolar interactions, directly relevant to current Rydberg tweezer experiments.
Spin squeezing is both a fundamental witness of quantum entanglement and a key resource for quantum-enhanced metrology. While all-to-all interactions provide a paradigmatic route to generate spin squeezing, many programmable quantum simulators naturally realize finite-range interactions that induce coupling between collective spin dynamics and finite-momentum spin-wave modes, thereby degrading the achievable squeezing. Here we develop an optimization-based global-control strategy for enhancing spin squeezing in a two-dimensional XX model with dipolar interactions. By combining rotor-spin-wave theory with an extension to open boundary conditions, we enable efficient optimization of control protocols for large finite-range interacting systems. We show that a single collective transverse field generates squeezing beyond the conventional two-axis-twisting benchmark across system sizes accessible to numerical benchmarking. The optimized dynamics exploits squeezing-axis reorientation while suppressing spin-wave excitations, thereby stabilizing strong squeezing over extended times. We further implement the optimized protocol in a two-dimensional Rydberg-atom array and experimentally observe enhanced, long-lived spin squeezing in good agreement with theory.
Spin squeezing serves as both a fundamental witness of quantum entanglement and a critical resource for quantum-enhanced metrology. While generating substantial spin squeezing in finite-range interacting systems remains challenging, such capability is important for advancing quantum technologies. In this work, we develop an optimal control strategy for achieving enhanced spin squeezing in a two-dimensional XX model with dipolar interactions. Leveraging rotor-spin-wave theory for periodic boundary conditions, we circumvent computational bottlenecks to explore control strategies at unprecedented scales. Remarkably, optimizing a single collective transverse field is sufficient to achieve substantial squeezing enhancement, exceeding the two-axis-twisting benchmark. The optimized control field achieves this breakthrough by dynamically suppressing inter-subspace mixing induced by the finite-range interactions, thereby confining the system evolution predominantly within the maximal spin subspace. We further extend rotor-spin-wave theory to open boundary conditions and incorporate dephasing noise, providing a scalable framework for realistic systems. Under these conditions, the optimized protocol remains effective, highlighting its robustness and suitability for experimental implementation.