Fluctuations are ubiquitous in nanometer-scale systems, spanning orders of magnitude in space and time. Real-space access to fluctuating states is impeded by a dilemma
between spatial and temporal resolution. Averaging over an extended period of time (or repetitions) is key for the majority of high-resolution imaging experiments, especially in weak
contrast systems. If, by lack of better knowledge, averaging is indiscriminate, it leads to a loss of temporal resolution and to motion-blurred images. We present coherent correlation imaging (CCI) – a high-resolution, full-field imaging technique that realizes multi-shot, time-resolved imaging of stochastic processes. The key of CCI is
the classification of camera frames that correspond to the same physical state (Fig. 1) even at low photon count, where imaging is not possible. CCI combines a correlation-based similarity
metric with powerful classification algorithm developed for genome research [1] realizing informed, non-sequential signal averaging while maintaining single frame temporal resolution. We apply CCI to study previously inaccessible magnetic fluctuations in a highly degenerate magnetic stripe domain state. Our material is a Co-based chiral ferromagnetic multilayer with
magnetic pinning low enough to exhibit stochastically recurring dynamics that resemble thermally-induced Barkhausen jumps near room temperature. CCI reconstructs sharp, high-contrast
images of all domain states by phase retrieval [2] and, unlike previous approaches, also tracks the time when these states occur. The spatiotemporal imaging reveals an intrinsic transition
network between the states and unprecedented details of the magnetic pinning landscape (Fig. 2). 
**References** 
[1] G. Sherlock, et. al., Current Opinion in Immunology 12, 201-205 (2000)
[2] Flewett, S. et al., Optics Express 20, 29210–29216, 2012 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/575255bbb0f7b8891a22722531593fa6.png) 
Principle of time-resolved coherent correlation imaging. Top: Sequence of camera frames showing Fourier-space coherent scattering patterns. Coherent correlation imaging classifies
scattering frames by their underlying domain state, as indicated by the colors. Bottom: Real-space images reconstructed from an informed average of same-state frames. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/c06609f11a71267f63b9f1a35eff95c7.png) 
Map of attractive (blue dots) and repulsive (red areas) pinning sites. The background shows the position of the domain walls and their relative occurrence observed in the experiment.

Coherent Correlation Imaging: Resolving fluctuating states of matter

Fluctuations are ubiquitous in magnetically and charge ordered systems. Spanning multiple orders of magnitude in space and time, they often lead to a wealth of fascinating physical effects. Recent examples of such dynamics are fluctuations of spin- and charge density waves in antiferromagnets [1], diffusion-like motion of magnetic skyrmions in low pinning magnetic multilayer or spontaneous fluctuations between ordered stripe and skyrmion lattices phases [2]. High-resolution real-space imaging can resolve the spatial information. However, due to insufficient signal on a timescale faster than the relevant dynamics, averaging over an extended period of time (or repetitions) is key for the majority of high-resolution imaging experiments. So far, averaging has been performed blindly and continuously, leading to a trade-off between extracting either spatial or temporal information, especially in weak contrast systems. Here, we present coherent correlation imaging (CCI) – a high-resolution, full-field imaging technique that realizes multi-shot, time-resolved imaging of stochastic processes. The key of CCI is the classification of camera frames that correspond to that same physical state by combining a correlation-based similarity metric with powerful classification algorithm developed for nanoparticle tomography and genome research (see Fig. 1) [3]. CCI extends these algorithms to the temporal domain, realizing informed, non-sequential signal averaging while maintaining single frame temporal resolution. CCI does not require prior knowledge of the imaged system, rendering it the first time-resolved x-ray imaging technique capable of accessing systems with entirely unknown dynamics. To develop and demonstrate CCI, we study thermal hopping of magnetic domain walls – a prototypical example of stochastic nanometer-scale dynamics – on timescales ranging from sub-seconds to hours under equilibrium conditions. Our material is [Pt (2.7 nm)/Co60Fe20B20 (0.8 nm)/MgO (1.5 nm)]15, a chiral ferromagnet with magnetic pinning low enough to exhibit stochastically recurring dynamics that resemble thermally induced Barkhausen jumps near room temperature [4]. Imaging is performed through Fourier-transform x-ray holography with contrast provided by the x-ray magnetic circular dichroism [5]. CCI reconstructs sharp, high-contrast images of all domain states and, unlike previous approaches, also tracks the time when these states occur (see Fig. 2). This uniquely reveals not only the occupation probabilities of all states, but further, the complex network of stochastic transition trajectories between states. The full potential of this technique will manifest at the next generation of pulsed high-coherence x-ray sources, where it promises to resolve a wealth of previously hidden dynamic processes, such as the nucleation dynamics of externally-induced phase transitions and the intrinsic fluctuations of weakly pinned textures in spin and charge ordered systems.

**References:**

[1] - Shpyrko, O. G. et al. Direct measurement of antiferromagnetic domain fluctuations. Nature 447, 68–71 (2007) [2] - Seaberg, M. H. et al. Nanosecond X-Ray Photon Correlation Spectroscopy on Magnetic Skyrmions. Physical Review Letters 119, 067403 (2017) [3] - Eisen, M. B. et al. Cluster analysis and display of genome-wide expression patterns. Proceedings of the National Academy of Sciences 95, 14863–14868 (1998) [4] - Lemesh, I. et al. Current-Induced Skyrmion Generation through Morphological ThermalTransitions in Chiral Ferromagnetic Heterostructures. Advanced Materials 30, 1805461 (2018) [5] - Eisebitt, S. et al. Lensless imaging of magnetic nanostructures by X-ray spectro-holoraphy. Nature 432, 885–888 (2004)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/02c68b10457f8b0b09df682b3321c147.jpg)
Figure 1| Single-frame correlation and single-frame imaging. a, CCI two-time correlation 
analysis showing the similarity between the states recorded at different times. b, Cross-section of the two-time correlation function at t2 = 28 s (red line in (a)). The colored shades define frames that correspond to the same physical state of the system. c, Image corresponding to the blind average of all frames. High-contrast regions indicate stationary behavior of the local out-of-plane magnetization mz, while seemingly semi-transparent features correspond to a superposition of two different magnetic states. d,e, Images obtained by selectively averaging the blue and yellow shaded regions in (b), respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/d7e9df838294c2b83f25a91cbc1552ab.jpg)
Figure 2| CCI-resolved dynamics in our sample. a, Real-space images of all domain states. Field of view is 720 nm in diameter. b, Temporal evolution of the states. The horizontal axis indicates the measurement time. The vertical axis shows which state has existed at the given time.

Coherent Correlation Imaging: High-resolution imaging of stochastic dynamics INVITED

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