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Controlling the quantum many-body state of arrays of qudits, possessing alarge local Hilbert space, opens the path to a broad range of possibilities formany-particle entanglement, interesting both for fundamental quantum science,as well as for potential metrological applications. In this work wetheoretically show that the spin-spin interactions realized in two-dimensionalMott insulators of large-spin magnetic atoms (such as Cr, Er or Dy) lead toscalable spin squeezing along the non-equilibrium unitary evolution initializedin a coherent spin state. An experimentally relevant perturbation to thecollective squeezing dynamics is offered by a quadratic Zeeman shift, whichleads instead to squeezing of individual spins. Making use of a truncatedcumulant expansion for the quantum fluctuations of the spin array, we showthat, for sufficiently small quadratic shifts, the spin squeezing dynamics isakin to that produced by the paradigmatic one-axis-twisting (OAT) model -- asexpected from an effective separation between collective spin and spin-wavevariables. Spin squeezing with OAT-like scaling is shown to be protected by therobustness of long-range ferromagnetic order to quadratic shifts in theequilibrium phase diagram of the system, that we reconstruct via quantum MonteCarlo and mean-field theory.

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