Theory of Scalable Spin Squeezing with Disordered Quantum Dipoles

TL;DR

The paper develops a theory of scalable spin squeezing in disordered two-dimensional dipolar spin ensembles described by the dipolar XXZ Hamiltonian $H_{\mathrm{XXZ}}=-\sum_{i<j} J_{ij}(s_i^x s_j^x+s_i^y s_j^y+\Delta s_i^z s_j^z)$ with $J_{ij}=J/r_{ij}^3$, where positional disorder is encoded by the filling fraction $f$. Using path-integral quantum Monte Carlo, the authors map the finite-temperature XY order phase diagram as a function of $f$ and $\Delta$, deriving the critical temperature $T_c$ and translating it into a phase diagram for scalable squeezing via the criterion that the initial energy density $E_x$ lies below the critical energy density $E_c$. They show that increasing disorder compresses the scalable-squeezing region toward the Heisenberg point $\Delta=1$ due to rare, strongly coupled dimers that heat the bath upon quench; a dimer-augmented mean-field theory captures the observed shift of the peak in $T_c$ and reproduces the phase boundary. For NV color centers, shelving a fraction of spins with large local fields $J_i$ can remove dimers from dynamics, restoring scalable squeezing as demonstrated by QMC and DTWA simulations, highlighting a practical route toward disorder-resilient quantum sensing.

Abstract

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.

Figures (3)

• Figure 1: Scalable squeezing with disordered dipoles in 2D. (a) Depicts an example of a disordered spin configuration in two dimensions. For strong disorder (here $f=0.1$), near-neighbor “dimers” (red glow) are strongly excited when prepared in an $x$-polarized product state. For such a product state, the Wigner distribution of the collective spin on the Bloch sphere (red) is concentrated around the $+x$ direction with $O(\sqrt{N})$ projection noise. (b) Under evolution with $H_\mathrm{XXZ}$, dimers relax toward their local ground states (black arrows), releasing energy that can heat the system above the XY ordering temperature $T_\mathrm{c}$. (c) Time evolution with $H_\mathrm{XXZ}$ yields scalable spin squeezing when the energy density of the initial $x$-state ($E_x$) is below the energy density of the critical state ($E_\mathrm{c}$), i.e. when the excess energy density $(E_x-E_\mathrm{c})/|E_x| < 0$ (density plot). The phase boundary is obtained from QMC (black points, red dashed guide for the eye) and for $f \to 0$, from analytic arguments (red solid). As $f \to 0$, the region of scalable spin squeezing is localized to a regime near the Heisenberg point ($\Delta=1$). From the non-squeezing phase, scalable squeezing can be restored by (d) increasing the filling toward the uniform case $f=1$, which reduces the hierarchy of energy scales between strong dimer bonds and typical interactions; or (e) increasing $\Delta$ toward the Heisenberg point, which lowers the dimer excitation energy. Optimal squeezing is believed to occur at timescales $t \sim N^{2/5}$block_scalable_2024.
• Figure 2: Thermodynamic phase diagram of XY order. (a) Critical temperature $T_\mathrm{c}$ from QMC (solid lines) versus Ising anisotropy $\Delta$ for several filling fractions $f$ (legend shared with panel (b)). Temperature is rescaled by $f^{-3/2}$ to account for the change in the typical nearest-neighbor dipolar interaction strength as one dilutes the system. Note that $T_\mathrm{c}$ extends further below the numerically accessible points shown. For the three largest fillings, the first-order mean-field cluster expansion is also shown (faint lines). (b) The analogous $T_\mathrm{c}$ curves for the dimer mean-field ansatz [Eq. \ref{['eq:MF_dimers']}], which captures full intra-dimer correlations. (c) Depicts the anisotropy $\Delta_{\rm peak}$ which maximizes $T_\mathrm{c}$ as extracted from QMC for different filling fractions $f$ (colored circles), and from the mean-field theory with dimer corrections (dashed black). In contrast, the cluster expansion yields $\Delta_{\rm peak} = -0.5$ for all $f$ (dotted red). (d) The critical temperature $T_\mathrm{c}$ is larger when the dimers are more susceptible to XY ordering; this is quantified for a given dimer (with internal coupling $J_{ij}$) by its magnetic susceptibility $\chi$ to a transverse field from the other spins [Eq. \ref{['eq:chi']}], shown here at various anisotropies $\Delta$ and temperatures $T$. At $T\!\approx\!E_{\mathrm{gap}}$ the susceptibility crosses over from Curie behavior $\chi\sim 1/T$ to $\chi\sim 1/E_{\mathrm{gap}}$. (inset) Shows the energy spectrum of a dimer as a function of the anisotropy.
• Figure 3: Scalable squeezing via the shelving of dimers. (a) Rescaled distribution $p(J_i)$ of local fields $J_i := \sum_{j\ne i} J_{ij}$ for $f=0.05$ (solid blue) and $f \to 0$ (dashed black) SM. The markers on the $x$ axis label peaks (to their right) in the $f=0.05$ distribution, each corresponding to the fields indicated by the local motifs in the insets (top right). By shelving all spins with $J_i$ to the right of a given marker, (b) one lowers the excess energy density of the initial $x$-state $(E_x-E_\mathrm{c})/|E_x|$ by an amount roughly proportional to the fraction of spins shelved, as demonstrated by QMC for $\Delta = 0$. (c) Dynamical simulations using cDTWA confirm the squeezing phase diagram derived from QMC in (b). Increasing the fraction of spins shelved from 1.5% (red) to 35% (blue) leads to an optimal squeezing parameter $\xi^2$ below $1$ and decreasing with post-shelving system size $N$. We have filtered out fast oscillations in $\xi^2(t)$ due to dimers SM.