Hearing the light: stray-field noise from the emergent photon in quantum spin ice

TL;DR

This work addresses the challenge of directly observing the emergent photons in the U(1) quantum spin liquid (Coulomb) phase of quantum spin ice (QSI). It proposes stray-field magnetometry as a direct probe, exploiting that emergent photons are transverse magnetization waves and behave like cavity- or waveguide-like modes whose spectra and spatial patterns depend on boundary conditions. By deriving insulating and superconducting long-wavelength boundary conditions and analyzing both bulk-cavity and thin-film geometries, the authors predict sharp, mode-specific signatures in the stray-field noise that should be detectable with current solid-state defect magnetometry (e.g., NV centers) at millikelvin temperatures. The results offer a concrete, experimentally accessible route to confirm Coulomb-phase physics in QSI and to characterize the emergent electrodynamics through its boundary-sensitive photon spectra.

Abstract

Decisive experimental confirmation of the $U(1)$ quantum spin liquid phase in quantum spin ice remains an outstanding challenge. In this work, we propose stray-field magnetometry as a direct probe of the emergent photons -- the gapless excitation of the emergent electrodynamics in quantum spin ice. The emergent photons are transverse magnetization waves, which, in a finite sample, form discrete modes governed by one of two sets of natural boundary conditions: ``insulating'' or ``superconducting''. Considering cavity and thin film geometries, we find that the spectrum and spatial structure of the stray magnetic noise provide a sharp qualitative signature of the underlying electrodynamics. The predicted stray-field noise power lies comfortably within the detection range of present-day solid-state defect magnetometry.

Figures (4)

• Figure 1: Stray-field magnetic noise from quantum spin ice. a) We consider a finite sample of QSI, and probe the stray-field magnetic noise with a scanning magnetometer above it. b) Structure of QSI on the pyrochlore lattice. The strong ZZ interactions force the spins to be in 2-in-2-out "ice" configurations udagawaSpinIce2021. Flipping spins aligned along hexagonal rings (green) connect the ice configurations.
• Figure 2: Magnetic noise generated by a finite quantum spin ice sample with "superconducting" boundary conditions at temperature 100 mK. a) $T_2$ decoherence time of an NV center in the magnetic stray-field outside the sample (geometry in inset) for an XY8-$4$ dynamical decoupling protocol, plotted versus the pulse spacing $\tau_{\rm XY8}$. The $T_2$ time is computed by convolving the longitudinal component of the discrete magnetic noise spectrum with the XY8-$N$ filter function degenQuantumSensing2017rovnyNanoscaleDiamondQuantum2024SM. The curves show the $T_2$ times at two different points above the sample, as indicated in panel (b). b) Spatially resolved magnetic noise magnitude for a selection of low-frequency modes labeled $(n_x, n_y, n_z)$, with the longitudinal component ($|B^z|^2$) on the left and the transverse component ($|B^\perp|^2=|B^x|^2+|B^y|^2$) on the right, and mapped out in a plane 0.85 $\mu$m above the sample surface (see inset in (a)). Note that for the "insulating" boundary conditions, the stray-field noise is exactly zero everywhere outside the sample. Converted to experimentally relevant units using $v=10$ m/s, $\alpha'=0.1$, and $\mu_0^2 g^2 = 10^{-38}\ {\rm T^2 m^{-2} s^2}$.
• Figure 3: $T_1$ decoherence time of an NV center placed at different distances, $d$, from a thin film QSI with "superconducting" boundaries. The decoherence time is directly determined from the stray-field magnetic noise: $T_1^{-1}=\frac{3}{2}\gamma_e^2(\mathcal{C}^{xx}+\mathcal{C}^{yy})$. The steps at frequencies $\omega = n \pi v/L_z$, appear as each additional longitudinal mode $n$ becomes available. Converted to experimentally relevant units using $\gamma_e=2\pi \times 28$ GHz/T, $v=10$ m/s, $\alpha'=0.1$, and $\mu_0^2 g^2 = 10^{-38}\ {\rm T^2 m^{-2} s^2}$, and assuming a thermal photon population at temperature 100 mK.
• Figure 4: Termination of pyrochlore QSI with (a) $(111)$ plane and (b) $(110)$ plane. The broken tetrahedra host $e$-charges, which can hop along broken hexagons on the boundary (green paths). For these example terminations, the hopping terms are generated at second order in a Schrieffer-Wolff expansion and each involves four spins in the boundary layer (marked by red spheres). Note that first-order, two-spin terms would also appear at the boundaries if the termination planes were shifted to leave cut tetrahedra containing three spins.