Coherent correlation imaging for resolving fluctuating states of matter

The study addresses a long-standing dilemma in imaging stochastic nanoscale dynamics: achieving high spatial resolution requires distributing limited signal over many pixels, which typically forces extensive temporal averaging and blurs time-varying phenomena. Constraints from source brightness, optics, detectors, and sample perturbations further limit signal per frame, enforcing a trade-off between spatial and temporal resolution. The research question is whether one can recover both high spatial and temporal resolution for irregular, non-repeating dynamics without increasing probe intensity to perturbative levels. The authors propose coherent correlation imaging (CCI), which classifies low-signal coherent scattering snapshots by their underlying real-space state before averaging, thereby decoupling temporal resolution (set by single-frame acquisition and classification accuracy) from spatial resolution (set by captured q-range and informed averaging). They demonstrate CCI on thermally driven fluctuations of a highly degenerate magnetic maze domain state in a low-pinning perpendicular ferromagnetic multilayer at 310 K, aiming to reveal discrete states, transition pathways, and the microscopic pinning energy landscape governing domain-wall dynamics.

Conventional full-field and scanning imaging approaches are constrained by signal-limited trade-offs: higher spatial resolution demands greater signal, but practical and fundamental limits (including sample damage) necessitate temporal averaging that blurs dynamics. Mode decomposition approaches can enhance the signal-to-noise of repeating modes but fail for irregular temporal signals. Mixed-state ptychography can reconstruct static images of the most visited states over an averaging interval but does not recover the actual sequence of events. Prior coherent X-ray and electron scattering studies have probed many fluctuating systems in reciprocal space; however, high-resolution real-space time-resolved imaging has been largely inaccessible due to insufficient per-frame signal at relevant timescales. Related developments in photon-correlation spectroscopy, nanoparticle tomography, and gene-expression data clustering provide tools for similarity assessment and hierarchical clustering that CCI adapts to classify coherent scattering frames at low photon counts. The work positions CCI as a bridge between reciprocal-space sensitivity (speckle fingerprints) and real-space imaging via phase retrieval, overcoming limitations of prior techniques for non-repetitive, stochastic dynamics.

• System: Thermally evolving magnetic maze (stripe) domain state in a low-pinning perpendicular ferromagnetic multilayer [Pt (2.7 nm)/Co0.6Fe0.4B20 (0.8 nm)/MgO (1.5 nm)] at a constant temperature of 310 K, observed in a fixed field of view.
• Data acquisition: Record a nearly continuous sequence of coherent scattering snapshots I(q,t) in Fourier (momentum) space using resonant X-ray scattering in forward geometry. A holographic mask enables robust reconstruction of real-space magnetization images mz(x,t) by holographically aided iterative phase retrieval.
• CCI principle: Each low-photon-count scattering frame contains a speckle pattern serving as a fingerprint of the underlying real-space domain configuration. CCI classifies frames into states based on similarity before any averaging.
• Classification pipeline (three phases):
  1. Exhaustive configuration discovery: Compute normalized pair correlations between frames to derive a low-noise similarity distance (e.g., Pearson distance). Apply a modified, iterative agglomerative hierarchical clustering algorithm to identify natural divisions (using inconsistencies between linkage step sizes and lower-level variability) and obtain clusters representing spatially resolved domain configurations (“internal modes”).
  2. Misclassification assessment: Quantify expected error rates for assigning single frames to the discovered configurations as a function of inter-configuration similarity.
  3. State grouping: Define an acceptable misclassification threshold (here <1%). Group similar configurations (internal modes) until the frame-to-group assignment error falls below threshold. The resulting groups are termed “states” (resolved in space and time), while members are “internal modes” (resolved in space but not in time).
• Image reconstruction: For each state or internal mode, average the assigned frames in Fourier space and reconstruct high-resolution real-space images via holographically aided phase retrieval (e.g., RAAR / Gerchberg–Saxton style algorithms as referenced). Spatial resolution is determined by collected q-range and assessed via Fourier ring/shell correlation criteria.
• Temporal resolution: Determined by single-frame acquisition time and timestamp precision; independent of spatial resolution. High temporal fidelity arises from very low misclassification rates achieved by correlation-based similarity metrics and hierarchical clustering.
• Dataset scale: 28,800 frames processed. 99.5% of frames assigned to states. Resulting in 32 temporally resolved states composed of 72 internal modes.
• Dynamic network analysis: Construct a transition network where node distances reflect pairwise similarity (Pearson distance) and edge widths indicate observed transition counts. Identify agglomerates by combining similarity distances and transition frequencies, revealing clusters of frequently interconverting states and linker states mediating transitions.
• Pinning landscape inference: From spatial distributions of internal modes, reconstruct likely domain-wall pathways, identify attractive pinning sites (convergent points where many domain-wall paths merge), and detect repulsive pinning regions (areas never occupied by quasistatic domain walls, high switching counts needed to traverse). Use spatiotemporal occurrence maps to distinguish attractive (point-like) vs repulsive (area-like) pinning, informed by the topological constraint that domain walls cannot terminate within the film interior.

• CCI performance and output: 99.5% of 28,800 frames were assigned to one of 32 temporally resolved states composed of 72 internal modes, yielding nanometre-scale real-space images with single-frame temporal resolution.
• Dynamics: The system exhibits discrete, recurring states with non-cyclic, non-evolving behavior over the observed timescales. States are revisited in irregular sequences with intermittent calm periods and bursts of rapid hopping; the same state can persist for tens of minutes or switch on subsecond timescales. This is incompatible with simple Arrhenius activation over a fixed energy barrier, suggesting a dynamic, configuration-dependent energy landscape.
• Transition network: Nodes (states) arranged by similarity (Pearson distance) and connected by observed transitions reveal agglomerates where intra-agglomerate transitions are much more frequent than inter-agglomerate ones. Several agglomerates center on linker states (for example, states 1, 2, 7) that mediate many transitions.
• Pinning landscape mapping: Real-space internal modes reveal domain-wall pathways, enabling identification of attractive pinning sites (point-like locations where many paths converge) and repulsive pinning regions (extended areas avoided by domain walls). Inter-agglomerate transitions predominantly require traversing repulsive pinning regions, while most intra-agglomerate transitions involve only attractive pinning sites.
• Mechanistic insight: Due to topological constraints (domain walls cannot terminate inside the material), attractive pinning acts point-like (local energy minima), while repulsive pinning acts area-like (extended energy penalties). Repulsive regions increase the energetic cost for entire wall segments and suppress zipper-like one-by-one reordering between multiple attractive sites, leading to higher activation barriers for inter-agglomerate transitions and explaining their lower frequency. The fragmentation by repulsive areas also accounts for the relatively small number of observed states out of many possible connections among attractive sites.
• Broader impact: CCI decouples spatial from temporal resolution, enabling real-space imaging of stochastic, non-repeating dynamics that were previously only accessible in reciprocal space or via mode analysis. It opens pathways to study fluctuating order parameters and pinning/topological effects in a wide range of materials.

The findings demonstrate that coherent speckle patterns retain sufficient information to classify the underlying real-space state at low photon counts, allowing temporal resolution down to single-frame acquisition times without sacrificing spatial resolution. By accurately assigning frames to states with extremely low misclassification rates, CCI reconstructs high-fidelity real-space images through informed averaging and recovers the full temporal sequence of stochastic dynamics, addressing the core challenge of signal-limited spatiotemporal imaging. The observed non-Arrhenius, non-cyclic dynamics in the maze domain system are explained by a configuration-dependent energy landscape shaped by both attractive and repulsive pinning sites and by long-range stray-field interactions. The separation of transitions into agglomerates and the role of linker states reflect how repulsive, area-like pinning imposes higher activation costs across entire wall segments, while attractive, point-like pinning facilitates lower-barrier rearrangements. This mechanistic understanding clarifies why intra-agglomerate transitions dominate and why only a limited subset of states is explored. The approach and insights generalize to many fluctuating systems where order parameters vary in space and time, enabling real-space investigation of morphology, connectivity, and topology of ordered regions and their relation to material inhomogeneities, which were previously inaccessible due to temporal averaging constraints.

Coherent correlation imaging (CCI) resolves the long-standing trade-off between spatial and temporal resolution in imaging stochastic nanoscale dynamics by classifying coherent scattering frames prior to averaging. Applied to thermally fluctuating magnetic maze domains, CCI assigned 99.5% of 28,800 frames to 32 discrete states (72 internal modes), revealing a complex transition network with agglomeration in similarity space and irregular temporal behavior. The spatiotemporal reconstructions enabled mapping of attractive and repulsive pinning sites and yielded a general mechanism for the dynamics based on topological constraints and configuration-dependent pinning energies. CCI is broadly compatible with existing coherent imaging setups, sources from visible to X-rays/electrons, and can be combined with mode decomposition to further analyze dynamics or mitigate timestamp uncertainties. With emerging high-repetition-rate X-ray free-electron lasers and high-harmonic sources, CCI promises real-time, nonperturbative imaging from submicrosecond to femtosecond timescales at nanometre resolution. Future work includes extending CCI to diverse fluctuating systems (antiferromagnets, spin/charge order in high-Tc superconductors, frustrated magnets, structural transitions), integrating few-photon correlation techniques and machine learning for single-pulse state identification, and quantitative energy-landscape reconstruction over larger fields of view.

• Applicability requires that, during the experiment and within the field of view, the system revisits the same configurations or close approximations so that frames can be binned into a discrete set of states.
• Classification fidelity depends on state separability (speckle contrast and inter-state similarity), photon statistics per frame (exposure time), and precise timestamps; performance may degrade for extremely low contrast or highly overlapping states.
• Field of view and reciprocal-space coverage constrain spatial resolution and the completeness of reconstructed domain-wall pathways and pinning maps.
• The method reconstructs internal modes spatially but not temporally within a state; very fast intra-state dynamics below single-frame timescales remain unresolved.