Topological optical skyrmion transfer to matter Open Access

Structured light¹ of spatially varying polarization patterns, as a field of topological photonics, has recently garnered significant interest. In paraxial optics, the polarization of a light beam is commonly described by Stokes vectors. A monotonically varying Stokes vector, covering all orientations over the beam’s cross-sectional area, can be represented as a two-dimensional skyrmion texture.^2–4 These “baby-skyrmions” are non-singular, localized topological objects that take a uniform asymptotic value sufficiently far from the center of the structure. They are lower-dimensional analogs of three-dimensional skyrmions, originally introduced in nuclear physics^5–8 and later studied in superfluids,^9–15 elementary particle physics, and cosmology.¹⁶ Owing to their easier experimental accessibility and recognized applications, two-dimensional skyrmions are generally better known. In superfluid liquid ³He, they are also known as Anderson–Toulouse–Chechetkin^17,18 and Mermin–Ho¹⁹ vortices. In magnetic materials,^20,21 they hold great promise for data storage and spintronics, and they have been prepared in exciton–polariton structures.^4,22–24 Analogous skyrmions have also been theoretically studied^25–28 and experimentally observed^29–31 in atomic Bose–Einstein condensates.

Although skyrmions in optical fields² are a more recent subject, they have already been intensively studied theoretically^3,32–39 and demonstrated in laboratory experiments, both in evanescent fields^40–44 and in free space.^33,45–47 Skyrmions are valuable for exploring fundamental aspects of topological photonics and structured light and hold potential applications in metrology and imaging techniques,^41,42 as well as in optical communication and information storage.^2,48 A key element for many of these applications is the high-fidelity mapping of skyrmion topology from optical fields to matter, its storage, processing, and retrieval. However, experimental studies addressing these processes have been notably absent. In fact, there has been an almost complete lack of even reports on the interaction of optical skyrmions with matter, with the notable exception of a recent study addressing the imprinting of surface relief structures of polarization patterns on an azopolymer film.⁴⁹ This method, however, only imprints rippled marks of the polarization structures on the material surface without transferring the topological properties of the skyrmion.

In this work, we prepare a topological skyrmion texture in a laser beam and measure its topological charge. This optical skyrmion is then made to interact with atomic matter. By leveraging the well-established technology of imprinting optical vortices in Laguerre–Gaussian (LG) laser beams onto atomic vortices in atom clouds,^50–53 we demonstrate a high-fidelity mapping of skyrmion topology onto matter, enabling it to be detected in its new form of an atomic pseudospin excitation. Transfer of skyrmions and hopfions from light to atoms was recently analyzed theoretically in Ref. 35 via internal level transitions. In our approach, we take an alternative route, achieving state transfer through adiabatic passage in a dark state. This mechanism is analogous to electromagnetically-induced transparency, which has been successfully employed to slow and store light.^54–56 The topological charge density within the matter is determined by detecting the atomic dark state population and projecting onto the diabatic state. By integrating the measured charge density, we obtain a topological charge Q ≃ 0.84 in the atomic case. This value is slightly reduced compared to the original Q ≃ 0.91 observed in the optical skyrmion, primarily due to the chosen laser beam width being significantly larger than the atom cloud. Within the spatial overlap region, the integrated topological charge is effectively transferred, while the different coupling strengths associated with the light polarization lead to a slight spatial stretching of the skyrmion texture.

In Fig. 1, we present the experimentally measured pseudospin profile of the Stokes vectors for the propagating light field along the beam’s transverse direction before interaction with the atomic sample. Since our focus is on transferring the topological properties of the skyrmion to matter rather than on skyrmion light propagation, we did not attempt to demonstrate the stability of solution (2) during propagation.² The optical skyrmion topology is depicted by illustrating the Stokes vectors both on a 2D plane at a cross section of the laser field and on a sphere. Compactification on a sphere is achieved by defining a closed path on the plane along which the Stokes vector maintains a constant angle between the radial and vertical (z) axes. From the experimental data, we determine the largest such path that allows integration of the skyrmion topological charge and find that it corresponds to an angle of 0.88π [see the solid-black curve in Fig. 1(a)]. The region outlined by this path then serves to determine a compactification region for the skyrmion topology and is projected onto the sphere, with the points along the path representing the north pole. The Stokes vector wraps over the Poincaré sphere almost entirely once, giving a topological charge Q ≃ 0.91 in Eq. (1).

We enable the interaction between the optical skyrmion and matter through an adiabatic passage into a dark state.⁵⁶ This method is based on resonant interactions, unlike off-resonance imprinting of LG beams,^50,52 and can be employed in storing vortices on atomic ensembles.⁶⁰ The experimental study of off-resonance imprinting⁵² was extended in Ref. 61 to report the creation of an atomic skyrmion. However, due to a misinterpretation of the F = 2 spin manifold, the prepared object did not represent a skyrmion⁶² but rather a singular spin-2 vortex.⁶³ 

A schematic of the experiment is shown in Fig. 2(b). To generate the optical skyrmion, a Gaussian beam is split into two orthogonal polarization components, with one component transformed into an LG beam using a spatial light modulator (see  Appendix B). The two polarization components are then recombined to create a beam exhibiting the desired transverse skyrmion pattern. This beam is directed into the cold-atom cloud (1/e²-waist ρ_c ≃ 107(3) μm) and resonantly coupled to the atomic transitions. The atomic skyrmion properties are extracted from the state populations measured after various interaction sequences with optical beams (see  Appendix B).

In the dark state adiabatic passage of the optical skyrmion to matter, the atoms are initially optically pumped into the |1⟩ state. The Gaussian beam on the |2⟩ → |3⟩ transition is subsequently turned on, followed by an adiabatic ramp of the LG beam on the |1⟩ → |3⟩ transition. This process adiabatically connects the initial |1⟩ state to the dark state of Eq. (6). By assuming that the atoms are in the dark state of Eq. (6), we then determine from the measurement data the unknown parameters θ and ϕ, subsequently referred to as θ_a and ϕ_a (see  Appendix B). The resulting profiles are shown in Figs. 3(a) and 3(b).

The pseudospin texture of the transferred skyrmion is shown in Fig. 3(c) on the xy plane, along with its compactified version on a sphere [Fig. 3(d)]. We also display the Poincaré sphere representation, illustrating how the Stokes vectors cover most of the order-parameter space [Fig. 3(e)], defined by the angles (0 ≤ θ_a ≤ π, 0 ≤ ϕ_a ≤ 2π) in Eq. (6). Integrating the topological charge from the experimental data according to Eq. (1) yields Q ≃ 0.84.

In Fig. 4, we show radially averaged relative phases and azimuthally averaged mixing angles, representing the Poincaré sphere spherical angles for the different scenarios of Eq. (6), when the optical and matter skyrmions are determined within the same spatial region. Given the imbalance in the atom–light coupling strengths between the Gaussian and LG beams (see  Appendix A), an ideal transfer of topology would yield ϕ_a = ϕ_o and $θa=θoc$⁠, as explained previously. The former equality is experimentally well-verified, while some mismatches are observed in the latter, likely due to residual aberrations in our imaging system.

We demonstrated that topological optical skyrmion textures can be effectively mapped onto matter by leveraging well-established methods for imprinting optical vortices of LG beams onto atomic vortices.^50–53 Due to the smaller spatial region of the atom cloud, the spin vector orientation between the radial and vertical axes at the edge of the cloud maintains a smaller value of 0.8π than in the optical skyrmion. When integrating over the same spatial area, the optical skyrmion’s topological charge is 0.84, which decreases to 0.80 when accounting for the imbalance in atom–light coupling strength. This value is in reasonable agreement with the atomic skyrmion’s topological charge of 0.84, with the remaining discrepancy likely due to systematic errors, such as aberrations in our imaging system.

Mapping skyrmion topology onto matter and detecting it through atomic measurements open up possibilities for topological data encoding and storage. Future work could extend this approach to skyrmion laser pulses, leveraging existing technologies for storing and releasing entire pulses within an atomic cloud.^54,60 In addition, our approach offers improved detection and analysis of more complex topologies in structured light. This is particularly relevant because the traditional description of paraxial light waves in terms of the Stokes vectors on a Poincaré sphere is incomplete; it discards the variation in the total electromagnetic phase of vibration.^33,35 At optical frequencies, the spatial profile of this phase is challenging to measure due to rapid oscillations³³ and is typically overlooked in the topological analyses. However, this phase variation leads to an optical hypersphere description and the emergence of genuinely 3D topologies. The most dramatic examples are full 3D particle-like skyrmions and hopfions,^33,35 analogous to knotted solitons of the Skyrme–Faddeev model in high-energy physics.^6,64 Such constructions based on links and knots are related to fundamental ideas proposed by Kelvin, who suggested them as building blocks of atomic particles.⁶⁵ In the context of electromagnetic fields, they have been suggested even as models of ball lightning.^66,67 Once transferred to atomic media, the detection of these structures no longer faces similar challenges as in the optical case.¹⁵ 

This work was supported by the CQT/MoE (Grant No. R-710-002-016-271), the Singapore Ministry of Education Academic Research Fund Tier 2 (Grant No. MOE-T2EP50220-0008), and EPSRC UK (Grant No. EP/S002952/1).

The authors have no conflicts to disclose.

Chirantan Mitra: Data curation (equal); Formal analysis (lead); Investigation (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Chetan Sriram Madasu: Data curation (equal); Formal analysis (equal); Investigation (equal); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal). Lucas Gabardos: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal). Chang Chi Kwong: Investigation (supporting); Project administration (supporting); Writing – review & editing (equal). Yijie Shen: Conceptualization (supporting); Methodology (supporting); Writing – review & editing (equal). Janne Ruostekoski: Conceptualization (equal); Methodology (lead); Writing – original draft (lead); Writing – review & editing (lead). David Wilkowski: Conceptualization (equal); Methodology (lead); Project administration (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are openly available in Dataverse at https://doi.org/10.21979/N9/EAVRTG, Ref. 68.

The optical skyrmion is coupled to a cold 87-strontium gas at a temperature of 6.9(3) μK, using the |¹S₀, F = 9/2⟩ → |³P₁, F = 9/2⟩ hyperfine transition of the intercombination line at 689 nm.⁶⁹ The gas contains 4.2(3) × 10⁶ atoms, within a 1/e²-width ρ_c = 107(3) μm of a quasi-Gaussian shape. A Λ-scheme connects the Zeeman ground states ψ₁ ≡|1⟩ ≡|m = 9/2⟩ and ψ₂ ≡|2⟩ ≡|m = 5/2⟩ to the excited state |3⟩ ≡|m = 7/2⟩. The LG and Gaussian beam polarizations and frequencies are set to couple to the |1⟩ → |3⟩ and |2⟩ → |3⟩ transitions, respectively. The Clebsch–Gordan coefficients $C1=−18/99$ and $C2=32/99$⁠, associated with the atom–light coupling strengths of these transitions, differ, resulting in unbalanced dipole matrix elements μ_i,3 = C_iμ, where μ denotes the reduced dipole moment. This imbalance affects the radial mixing angle θ_o, yielding the corrected value $θoc$⁠, such that $tan(θoc/2)/tan(θo/2)=(18/32)1/2$ in Fig. 4(b).

The skyrmion lifetime in this experiment is mainly limited by the thermal motion of the atoms, which scrambles the topological structure. The lifetime can be estimated to be on the order of $τ≃d/v̄$⁠, where $v̄≃2.5cms−1$ is the thermal speed and d ≃ 5 μm is the typical size of the topological structure, leading to τ ≃ 200 μs. For applications with long storage times, significantly longer skyrmion lifetimes can be achieved with atomic Bose–Einstein condensates (in rotation, only limited by the condensate lifetime of up to tens of seconds) or in optical lattice potentials that freeze the atomic motion.

An optical skyrmion is represented by a Poincaré beam,⁵⁸ generated by a superposition of a Gaussian and an LG beam with orthogonal polarizations.^3,4 To produce the LG beam, a Gaussian beam is first split into two orthogonal polarization components. One component is transformed into an LG beam using a phase mask imprinted on a spatial light modulator, employing a digital-hologram method.⁷⁰ To measure the relative phase profile of the Gaussian and the LG beams, we use a polarizer to project the polarization of each beam. The resulting interference pattern is detected on a CCD camera.

To characterize the atomic skyrmion, the populations of states |1⟩ and |2⟩ are detected using spin-sensitive shadow imaging techniques.⁷¹ More precisely, the |1⟩ population profile is measured with a circularly polarized probe beam resonantly tuned to the |¹S₀, F = 9/2, m = 9/2⟩ → |³P₁, F = 11/2, m = 11/2⟩ transition. Subsequently, the |2⟩ population profile is measured by tuning the probe beam to the |¹S₀, F = 9/2, m = 5/2⟩ → |³P₁, F = 11/2, m = 7/2⟩ transition. The θ_a mixing angle profile is extracted from population measurements following the adiabatic transfer of the optical skyrmion to the atomic ensemble. The relative phase ϕ_a is extracted after the transfer by switching off the LG beam and simultaneously turning on an equally polarized co-propagating plane wave that addresses the same transition. If the relative local phase between the two beams is zero with equal field amplitude, the atoms remain in the dark state defined by Eq. (6). Otherwise, a fraction of the |1⟩ and |2⟩ populations will be excited to state |3⟩, leading to an asymmetric profile from which ϕ_a can be extracted.

As beams follow different optical paths before being recombined, their relative phase is not stable, resulting in shot-to-shot random variations. However, the relative phase, with a coherence time of about 10 ms, does not significantly vary during the short interrogation time of about 100 μs (duty cycle $∼0.01%$⁠). If an image, obtained after each run, contains phase information, it is fitted and properly reoriented to avoid phase blurring during the summation of the images. The latter is used to improve the signal-to-noise ratio.