Stochastic Optical Skyrmions

• Stochastic optical skyrmions are S²-valued fields with integer topological charge preserved under perturbations up to a 60° tilt.
• They are realized via optical anisotropy in liquid-crystal retarder arrays, enabling programmable, topologically protected information storage.
• Femtosecond pulses stochastically switch the helicity in magnetic skyrmions, bridging optical control and spintronic behavior.

Stochastic optical skyrmions are skyrmionic topological textures in optical or optically driven systems for which stochasticity enters either as robustness of an $S^{2}$-valued optical field under random perturbations or as random light-induced switching of a skyrmionic substructure. In one formulation, a spatially varying optical anisotropy field is reduced to a distinguished axis geometry field $n(x,y)\in S^{2}$, so that skyrmionic charge can be encoded directly in structured matter and remain invariant under artificially introduced stochastic perturbations, with the physically realized $S^{2}$-valued field allowed to differ from the designed field everywhere by a large margin of error, up to $60^\circ$, without affecting the underlying topological charge. In another formulation, femtosecond optical pulses induce stochastic switching of the Néel-cap helicity of dipolar skyrmions in Fe$_3$Sn$_2$ at room temperature, while the internal Bloch component remains unchanged over a wide fluence range (Zhang et al., 22 Aug 2025, Weber et al., 7 Mar 2025).

1. Conceptual scope and topological setting

Skyrmions are presented as universal features of $S^{2}$-valued fields, and their occurrence across diverse physical domains is taken to reflect the ubiquity of topology in the study of the natural world. The central technical requirement is that the physically relevant field be representable as a map from a two-dimensional domain into the target sphere $S^{2}$. Within that setting, the skyrmion charge is the degree of the map and is therefore quantized as an integer (Zhang et al., 22 Aug 2025).

The optical-anisotropy construction broadens this framework to fields taking values in manifolds of dimension greater than $2$. The relevant move is parameter-space dimensionality reduction: one begins with a smooth field $f:U\to M$ with $n(x,y)\in S^{2}$0 and selects a surjective projection $n(x,y)\in S^{2}$1 that extracts a physically relevant subcomponent. The resulting field $n(x,y)\in S^{2}$2 inherits skyrmion charge. In the cited optical example, this is not a director-field skyrmion in liquid crystals; it is explicitly described as fundamentally different from the more commonly known skyrmions formed by director fields in liquid crystals, because the skyrmion is encoded in optical anisotropy through a distinguished axis (Zhang et al., 22 Aug 2025).

In the Fe$n(x,y)\in S^{2}$3Sn$n(x,y)\in S^{2}$4 work, the relevant skyrmions are magnetic rather than optical-material skyrmions, but the stochastic optical element arises from all-optical control. There, femtosecond light pulses do not create an $n(x,y)\in S^{2}$5 field by projection from optical anisotropy; instead, they drive stochastic transitions among Néel-cap configurations of pre-existing dipolar skyrmions. The two works therefore use the phrase in different but related senses: one emphasizes topological encoding in an optical medium under stochastic perturbations, and the other emphasizes stochastic optical manipulation of magnetic skyrmion textures (Weber et al., 7 Mar 2025).

2. Optical anisotropy as an $n(x,y)\in S^{2}$6-valued skyrmion field

The optical-anisotropy formulation starts from a spatially varying pure retarder whose $n(x,y)\in S^{2}$7 Mueller matrix is written as

$n(x,y)\in S^{2}$8

where $n(x,y)\in S^{2}$9 is the lower-right $S^{2}$0 rotation submatrix acting on the Stokes vector. Every $S^{2}$1 is parametrized by an axis-angle pair consisting of a unit axis $S^{2}$2 and a retardance angle $S^{2}$3. The axis geometry field is then defined by

$S^{2}$4

and the skyrmion field is chosen as

$S^{2}$5

This choice identifies the physically distinguished anisotropy axis as the topological field (Zhang et al., 22 Aug 2025).

For an order-$S^{2}$6 Néel-type skyrmion, one convenient design in polar coordinates $S^{2}$7 is specified by

$S^{2}$8

with

$S^{2}$9

Equivalently, in Cartesian coordinates with $60^\circ$0 and $60^\circ$1,

$60^\circ$2

which satisfies $60^\circ$3. The topological charge is

$60^\circ$4

By direct substitution of the designed field, the paper states that $60^\circ$5, so the integer charge coincides with the winding number (Zhang et al., 22 Aug 2025).

The same paper places this construction in a wider abstract class. If $60^\circ$6 is, for example, $60^\circ$7 or a subset of $60^\circ$8 representing a full Mueller matrix field, then a suitable projection to $60^\circ$9 can still define a skyrmion-bearing field. Examples mentioned include the Stokes normalization map and the Hopf map from $_3$0. This suggests that the optical-anisotropy realization is intended as one instance of a broader topological procedure rather than an isolated optical device concept (Zhang et al., 22 Aug 2025).

3. Stochastic perturbations and charge invariance

The stochastic aspect of the optical-anisotropy skyrmion is formulated by replacing the ideal axis field $_3$1 with a perturbed field

$_3$2

where $_3$3 is a local random vector and $_3$4 projects back to unit length. The associated homotopy is written as

$_3$5

A homotopy argument shows that if $_3$6, equivalently at most a $_3$7 tilt everywhere, then the deformation remains well defined for all $_3$8, and the topological charge is unchanged:

$_3$9

Under this criterion, the skyrmion charge is protected even under stochastic Gaussian noise (Zhang et al., 22 Aug 2025).

The experimental noise model is implemented at the level of the liquid-crystal control variables. Independent Gaussian noise is added to each SLM phase,

$_2$0

so that the resulting perturbed anisotropy field $_2$1 is a complicated but well-defined random field. The noise strength is swept from $_2$2 to $_2$3, and for each $_2$4 the study generates $_2$5 noise realizations, measures the corresponding $_2$6, and computes $_2$7 from the skyrmion-charge integral. Both simulation and experiment exhibit three regimes: low noise with $_2$8, where $_2$9 with negligible spread; intermediate noise with $S^{2}$0, where boundary distortions appear and $S^{2}$1 begins to fluctuate; and high noise with $S^{2}$2, where $S^{2}$3 collapses to zero on average as homotopy breaks down (Zhang et al., 22 Aug 2025).

A common misconception would be to treat the observed robustness as immunity to arbitrary disorder. The stated protection criterion is conditional: it depends on the perturbation remaining within the homotopy-preserving bound. The paper therefore frames stochastic robustness not as unlimited fault tolerance but as topological invariance within a quantitatively specified error envelope (Zhang et al., 22 Aug 2025).

4. Reconfigurable realization in a liquid-crystal retarder array

The proof-of-concept platform consists of three cascaded phase-only LC-SLMs that form a pixel-addressable synthetic elliptical retarder. Each SLM imparts a fast-axis rotation $S^{2}$4 plus half-wave retardance $S^{2}$5, and the net Mueller matrix is that of a pure retarder with fast axis $S^{2}$6 and retardance $S^{2}$7. The required per-pixel phase patterns $S^{2}$8 are computed from the desired $S^{2}$9 prescription, so that when the three SLMs are driven with these patterns, the combined action is exactly the designed $S^{2}$0 (Zhang et al., 22 Aug 2025).

Read-out is performed by full Mueller-matrix polarimetry at each pixel, using four input analyzer states and four output measurements. The lower-right $S^{2}$1 submatrix is decomposed to extract the rotation axis $S^{2}$2, yielding the measured field $S^{2}$3. This is the experimental bridge between the abstract topological construction and the material implementation: the topological object is not inferred indirectly from a scalar image but reconstructed from the anisotropy tensorial response encoded in the Mueller formalism (Zhang et al., 22 Aug 2025).

This realization is explicitly reconfigurable. Because writing is accomplished by updating the SLM phase patterns and read-out by polarimetric decomposition, the system combines programmable synthesis with full optical read/write. A plausible implication is that the experiment is intended less as a fixed skyrmion medium than as a tunable topological platform for studying how skyrmionic charge behaves under controlled anisotropy perturbations (Zhang et al., 22 Aug 2025).

5. Topological encoding and protected information storage

The information-storage demonstration uses skyrmion bags. A bag consists of an outer skyrmion of charge $S^{2}$4 whose chirality sets the read order, together with four enclosed skyrmions carrying individual charges $S^{2}$5. These enclosed charges map to two $S^{2}$6-bit words via a balanced ternary scheme. In the reported experiment, the six-letter string “SKYRME” was written, the enclosed charges were measured under real-world noise, and every character was recovered correctly (Zhang et al., 22 Aug 2025).

The storage characteristics are stated explicitly. One bag encodes two $S^{2}$7-bit numbers, and a $S^{2}$8 bag array yields $S^{2}$9 bits. Writing is performed through the SLM patterns $2$0, while reading is performed by Mueller polarimetry and decomposition. The error tolerance is tied directly to the stochastic-topological analysis: the charge is protected up to $2$1 noise, corresponding to $2$2, and the reported bit-error rate is $2$3 in the trials described (Zhang et al., 22 Aug 2025).

The significance of this demonstration lies in the fact that the encoded variable is topological charge rather than a purely geometric local state. That does not eliminate implementation error, but it relocates the information-bearing degree of freedom from pointwise anisotropy accuracy to homotopy class. The paper’s formulation therefore presents topological encoding not merely as a descriptive label but as the operative mechanism behind the stated perturbation tolerance (Zhang et al., 22 Aug 2025).

6. Femtosecond optical switching of dipolar skyrmions in Fe$2$4Sn$2$5

A second line of work relevant to stochastic optical skyrmions concerns all-optical stochastic switching in the kagome magnet Fe$2$6Sn$2$7. Fe$2$8Sn$2$9 is described as a layered kagome magnet with rhombohedral stacking of Fe–Sn bilayers and strong uniaxial anisotropy, with the easy axis parallel to $f:U\to M$0, competing against shape anisotropy in thin lamellae. In zero field at remanence, the film shows stripe domains; applying $f:U\to M$1 generates isolated, circular domain particles identified as dipolar skyrmions (Weber et al., 7 Mar 2025).

These dipolar skyrmions have a hybrid Bloch–Néel topology. In the mid-plane of the film, the magnetization winds in a Bloch-type circular wall carrying an integer topological charge $f:U\to M$2 for type I bubbles. Near each surface, the spins twist into Néel-type caps that may co-rotate or counter-rotate. Co-rotating Néel caps produce a bright or dark central dot in a Fresnel image, whereas counter-rotating caps yield neutral central contrast because the surface contributions cancel. Type II bubbles, by contrast, are described as particles with two internal Bloch-wall segments and without net topological charge, $f:U\to M$3 (Weber et al., 7 Mar 2025).

The optical control protocol uses single femtosecond pulses with wavelength $f:U\to M$4, pulse duration $f:U\to M$5, and a $f:U\to M$6 laser from which single pulses are selected via a Pockels cell. The pulse energy in focus on the sample is varied from $f:U\to M$7 to $f:U\to M$8, corresponding to approximately $f:U\to M$9–$n(x,y)\in S^{2}$00 fluence depending on spot size. A single pulse is absorbed and rapidly heats the electronic system; via electron–spin coupling the spin temperature rises on the $n(x,y)\in S^{2}$01–$n(x,y)\in S^{2}$02 timescale. The system then enters a transient hot-spin state in which the energy barriers between different Néel-cap configurations are substantially lowered. Upon cooling, each surface Néel cap relaxes into one of its two degenerate helicities, and because the two surfaces are effectively decoupled by the thick Bloch core, the final helicities are chosen essentially at random, leading to a stochastic $n(x,y)\in S^{2}$03 cw, $n(x,y)\in S^{2}$04 mixed, $n(x,y)\in S^{2}$05 ccw distribution. Over a wide fluence range, the Bloch-wall rotation in the midplane remains unchanged (Weber et al., 7 Mar 2025).

The measurements are carried out by in situ Lorentz TEM using a modified JEOL JEM-F200 UTEM at $n(x,y)\in S^{2}$06 in continuous beam mode, with the objective lens off and defocus $n(x,y)\in S^{2}$07–$n(x,y)\in S^{2}$08. Lorentz micrographs with $n(x,y)\in S^{2}$09 exposure are acquired immediately after each single light pulse. A typical field of view spans several $n(x,y)\in S^{2}$10, permitting the tracking of $n(x,y)\in S^{2}$11–$n(x,y)\in S^{2}$12 dipolar skyrmions over $n(x,y)\in S^{2}$13 successive pulses. Pixel-intensity histograms exhibit three well-separated peaks, enabling automated classification of the Néel-cap state after each pulse. At $n(x,y)\in S^{2}$14 per pulse, the average probabilities are reported as $n(x,y)\in S^{2}$15, $n(x,y)\in S^{2}$16, and $n(x,y)\in S^{2}$17, with minor variations between individual skyrmions and standard deviation $n(x,y)\in S^{2}$18. Below approximately $n(x,y)\in S^{2}$19 the switching rate drops sharply, while above approximately $n(x,y)\in S^{2}$20 permanent damage occurs (Weber et al., 7 Mar 2025).

7. Micromagnetic framework, interconversion, and interpretive issues

The Fe$n(x,y)\in S^{2}$21Sn$n(x,y)\in S^{2}$22 study supplements the microscopy with micromagnetic modeling based on the energy density functional

$n(x,y)\in S^{2}$23

with parameters $n(x,y)\in S^{2}$24, $n(x,y)\in S^{2}$25, $n(x,y)\in S^{2}$26, $n(x,y)\in S^{2}$27. The dynamics are evolved with the Landau–Lifshitz–Gilbert equation,

$n(x,y)\in S^{2}$28

where $n(x,y)\in S^{2}$29 and $n(x,y)\in S^{2}$30. Ultrafast heating is represented by a thermal field

$n(x,y)\in S^{2}$31

with $n(x,y)\in S^{2}$32 a random unit-vector field updated every timestep $n(x,y)\in S^{2}$33 (Weber et al., 7 Mar 2025).

The simulations use a $n(x,y)\in S^{2}$34 slab with cell size $n(x,y)\in S^{2}$35, and soft-magnetic surface layers with $n(x,y)\in S^{2}$36 to mimic damaged or oxidized TEM specimen surfaces. The temperature pulse ramps from $n(x,y)\in S^{2}$37 to $n(x,y)\in S^{2}$38 within $n(x,y)\in S^{2}$39 and relaxes back to $n(x,y)\in S^{2}$40 over approximately $n(x,y)\in S^{2}$41, with best fit at $n(x,y)\in S^{2}$42. The thermal stage is integrated with timestep $n(x,y)\in S^{2}$43, and static equilibria are obtained with $n(x,y)\in S^{2}$44 integration. Electron phase-shift maps are then computed from the vector potential, showing that the outer Fresnel ring arises predominantly from the Bloch midplane whereas the central dot contrast is dominated by the two Néel caps. The post-pulse simulations reproduce stochastic reversal of only the surface caps while the ring remains unchanged, in quantitative agreement with experiment (Weber et al., 7 Mar 2025).

The same work reports interconversion between type I and type II bubbles. At zero applied field, light pulses switch only the Néel caps of existing type I dipolar skyrmions. Under a tilted-field protocol with $n(x,y)\in S^{2}$45 during field-on and subsequent ramp to remanence off-axis, the film is populated by approximately $n(x,y)\in S^{2}$46 type II bubbles. Subsequent femtosecond pulses of $n(x,y)\in S^{2}$47 convert all type II bubbles to type I dipolar skyrmions within approximately $n(x,y)\in S^{2}$48–$n(x,y)\in S^{2}$49 pulses, with random resulting Bloch-wall helicities; at moderate out-of-plane field of approximately $n(x,y)\in S^{2}$50, rare conversions back from type I to type II are observed, which the paper interprets as implying that type II has higher free energy and only forms when transient barriers are overcome (Weber et al., 7 Mar 2025).

An important interpretive issue concerns the coupling mechanism. The study favors a transient heating model: femtosecond absorption heats the electron bath, ultrafast electron–spin coupling raises the spin temperature in less than $n(x,y)\in S^{2}$51, the elevated spin temperature flattens the free-energy landscape of surface twists, and subsequent cooling quenches the system into one of the local minima. By contrast, all experiments were performed with $n(x,y)\in S^{2}$52-polarized light without helicity control, and no net magnetic field from the inverse Faraday effect was observed. The paper therefore treats inverse-Faraday-driven switching not as the operative explanation for the reported data, while noting that future experiments with circular polarization could address possible inverse Faraday effect contributions or ultrafast spin–orbit torques (Weber et al., 7 Mar 2025).

Taken together, the two cited works delineate two technically distinct meanings of stochastic optical skyrmions. One is an $n(x,y)\in S^{2}$53-reduced optical-anisotropy field whose integer charge is preserved under bounded stochastic perturbations and exploited for topologically protected encoding. The other is a magnetic skyrmion texture whose internal helicity substructure is switched stochastically by ultrafast light pulses. The shared theme is not a single material platform but the use of optical control or optical encoding to access stochastic topological behavior in skyrmionic systems (Zhang et al., 22 Aug 2025, Weber et al., 7 Mar 2025).