Topological robustness of classical and quantum optical skyrmions in atmospheric turbulence

Domain: neuroscience
Published: 2026-01-27
Editorial outcome: minor_revision

Abstract
The degradation of classical and quantum structured light induced by complex media, such as atmospheric turbulence, constitutes a critical barrier to its practical implementation in communication, energy transport, imaging and sensing. Here we construct both classical and quantum optical skyrmions and transmit them through experimentally simulated atmospheric turbulence, revealing the embedded topological resilience of their structure. For nonlocal quantum skyrmions, we show that although photon entanglement rapidly degrades, the topological characteristics of the states remain stable. Similarly, while the vectorial structure of local classical skyrmions is strongly distorted by the medium, their topology is preserved across a wide range of turbulence strengths. Our experimental results are supported by rigorous analytical and numerical modelling, validating that the quantum-classical equivalence of the topological behaviour is due to the non-separability of the states and the one-sided nature of the channel. This equivalence enables robust information transport in noisy environments, opening pathways for resilient terrestrial and satellite-to-ground communication.

Introduction
Skyrmions are topologically protected quasiparticles that have found broad relevance across disciplines, from particle physics1 and condensed matter physics2–5 to photonics6. For instance, magnetic skyrmions7, which can be manipulated at the nanoscale by electric currents, have demonstrated strong potential for applications in the development of silicon-based high-density data memory devices8–10. Moreover, in contrast to these stationary implementations, skyrmions can be constructed as propagating light waves in free space. This has been demonstrated using structured light11 in both classical and quantum regimes, opening opportunities for topologically protected information transfer. However, under realistic conditions, maintaining such topological protection becomes challenging in complex propagation environments. In particular, understanding how optical skyrmions preserve their topology during propagation through atmospheric turbulence remains a key open question.

By leveraging the unique degrees of freedom (DoF) intrinsic to photonic systems, researchers have successfully achieved both controlled generation and dynamic manipulation of skyrmions across multidimensional parameter spaces, encompassing spin angular momentum12–14, Stokes parameters15–17, electric or magnetic field vectors18–20, energy flows21,22, and momentum-space pseudospin vectors23–26. In contrast, previous studies have primarily treated skyrmions as localized field structures or particles. Nonlocal quantum skyrmions have been experimentally realized by entangling the orbital angular momentum (OAM) and polarization of biphoton states27. This approach enables a global encoding of topological information through correlations between spatial and polarization DoF, marking a fundamental shift from conventional single-particle descriptions and allowing topological information to remain robust against isotropic noise28.

Here we demonstrate the robustness of nonlocal quantum skyrmions under realistic conditions by transmitting them through experimentally simulated atmospheric turbulence. The experiment is grounded in a correlation framework that connects perturbations in complex media to variations in quantum states, while coordinate transformation analysis systematically elucidates the fundamental principles governing topological stability. In parallel, we create a local analogue of the quantum state, a classical optical skyrmion, and send it through the same channel, again showing robustness. As illustrated in Fig. 1, both quantum and classical skyrmions exhibit remarkable robustness under turbulence, with their topological charges remaining invariant. The equivalent topological resilience observed for classical and quantum skyrmions arises from the nature of atmospheric turbulence, which acts effectively as a one-sided quantum channel. Together, the theoretical and experimental results confirm that both classical and quantum skyrmions preserve their topological characteristics when propagating through atmospheric turbulence.

Results
Classical-quantum equivalence of topologies
Spin-textured structured light can be compactly written as |Ψ⟩ = M1A e1B + M2A e2B, where the polarization DoF is expressed as any pair of orthonormal states {e1, e2} and the spatial mode DoF is given by the orthonormal basis states {M1, M2}. Normalisation is omitted for simplicity. In the classical picture, the subscripts A and B denote two DoF, whereas in the quantum picture they correspond to two photons. The non-separability is local in the classical case and nonlocal in the quantum case29–31.

Such states can be endowed with a topology, and without loss of generality we consider
|Ψ⟩ = λ1LGℓ1(r, ϕ) |H⟩A |H⟩B + λ2LGℓ2(r, ϕ) |V⟩A |V⟩B, (1)
where λi are the normalized complex coefficients, |H⟩ and |V⟩ denote horizontal and vertical polarization states, respectively, and |ℓ⟩ represents the spatial mode state |LGℓ(r, ϕ)⟩, where LGℓ(r, ϕ) denotes the Laguerre-Gaussian mode of light32, and (r, ϕ) are the radial and azimuthal coordinates in the transverse plane.

A skyrmionic topology emerges only when |ℓ1| ≠ |ℓ2|, where the field establishes a nontrivial 2D mapping from the transverse plane R2 to the Poincaré sphere S2. In contrast, for |ℓ1| = |ℓ2|, the mapping becomes topologically trivial if ℓ1 = ℓ2, or topologically nontrivial in a lower dimension (mapping a closed loop in the transverse plane S1 to a closed loop S1 in the Poincaré sphere) for ℓ1 = −ℓ2. The topological invariant N, referred to as the skyrmion number or wrapping number, quantifies how many times the mapping covers the Poincaré sphere under stereographic projection and can be calculated as
N = 1/4π ∫R2 ϵijk Si ∂x Sj ∂y Sk dxdy, (2)
where ϵijk is the Levi-Civita symbol and Si with i=1,2,3 are the locally normalized Stokes parameters such that Σ3 i=1 Si2 = 1. This normalization ensures that the mapping onto S2 always maps onto a unit sphere. For the above states, Eq. (2) simplifies to (see Methods for detailed derivation)
N = n|ℓ1 − ℓ2|, (3)
where n = ± 1 denotes the skyrmion orientation determined by the relative ordering of ℓ1 and ℓ2.

We now consider the evolution of such a state under a one-sided channel, where one DoF is affected and the other remains unaffected. Atmospheric turbulence serves as such a channel since the spatial mode (DoFA) is distorted, while the polarization (DoFB) is not. We can frame a general equivalence between the classical and quantum states, thus enabling an agnostic analysis, where pure topological quantum states are equated to their classical counterparts by virtue of non-separability33. Each realization of a one-sided channel can be modeled as a unitary transformation acting on the input pure state, since any physical process can be represented by a completely positive trace-preserving map that preserves both trace and positivity, thereby ensuring the purity of the output. The Choi-Jamiolkowski isomorphism34 establishes a correspondence between the channel operator, T, and a quantum state, so that a measurement on one returns the other. The state after the channel, both classical and quantum, is |Ψout⟩ = (TA ⊗ 1B) |Ψin⟩, where 1B is the identity on the unaffected DoF. This shows that the spin texture of the field will in general change, but that the distorted spatial modes at the output will still be orthogonal, thus maintaining the non-separability (inhomogeneous nature of the spin texture) of the state35. The open question which we seek to address here is whether this implies that the topology, as measured by N, changes or remains intact?

Topological robustness of nonlocal skyrmions in turbulence
We address this question in the framework of quantum light, following a developed formulation of nonlocal quantum skyrmions36. Photons, as flying qubits, play a pivotal role in quantum information processing. However, environmental disturbances such as atmospheric turbulence severely degrade their coherence, posing a fundamental obstacle to the realization of practical quantum networks. To overcome this challenge, considerable efforts have been made, including the introduction of turbulence eigenmodes37,38 and experimental demonstrations of the turbulence-resilient nature of vector structured light35. Nevertheless, an effective strategy for directly mitigating turbulence effects on quantum states remains elusive. The advent of nonlocal quantum skyrmions provides a promising route for robust transmission of topological information through turbulence. Their globally defined topological mapping ensures the invariance of the skyrmion number, demonstrating exceptional resilience under turbulent conditions.

When the photon propagates through a single realization of turbulence, the disturbance can be modeled as a unitary operator UA acting on photon A39. This operator coherently redistributes the OAM components of the photon while preserving the overall purity of the two-photon state. Its action on an OAM eigenstate can be expressed as
|ℓ⟩A → UA |ℓ⟩ = Σℓ0 αℓ0 |ℓ0⟩A, (4)
where αℓ0 denotes the complex amplitude coefficient. This process shows that turbulence redistributes the initial OAM eigenstate into a superposition of multiple modes. The two-photon state after turbulence can therefore be written as
|Ψ0⟩ = λ01 |ℓ1⟩A |H⟩B + λ02 |ℓ2⟩A |V⟩B + λ03 |ℓ1⟩A |H⟩B + λ04 |ℓ2⟩A |V⟩B, (5)
where λ0i represents the normalized complex coefficients. By performing a position-projection measurement on photon A, the spatial-polarization correlation is mapped onto photon B, which can be represented by a spatially dependent Stokes vector. The topological structure of this normalized Stokes field defines the nonlocal skyrmion configuration, and the invariance of its skyrmion number serves as a direct measure of the system’s robustness under turbulence. More generally, the topological properties of a quantum system are determined by the global structure of its wavefunction, which ensures their insensitivity to local perturbations.

Under a single realization of turbulence, although the spatial distribution of the state in Eq. (5) differs from that of the initial state, the normalized Stokes field after turbulence can be obtained from the initial one through a smooth, orientation-preserving coordinate transformation (x,y) → (x0,y0). This invariance can be quantitatively characterized by the skyrmion number, defined as
N0 = 1/4π ∫R2 ϵijk Si ∂x0 Sj ∂y0 Sk dx0dy0 = 1/4π ∫R2 ϵijk Si ∂x Sj ∂y Sk dxdy = N. (6)
Here (x0,y0) denote the transformed coordinates corresponding to the smooth, orientation-preserving mapping introduced above. It should be noted that the skyrmion number remains invariant only under such a mapping; discontinuous or orientation-reversing transformations would alter the degree of the mapping and hence the topology. A detailed proof of this requirement is provided in Supplementary Section IA. Under these conditions, the correspondence between the spatial coordinates and the Poincaré sphere remains continuous and bijective, thereby preserving the degree of the mapping and ensuring the invariance of the skyrmion number.

The above analysis corresponds to a single realization of turbulence. In realistic conditions, two additional effects arise. First, turbulence scatters part of the initial state into unmeasured OAM modes, causing entanglement decay due to amplitude leakage. Second, ensemble averaging over multiple random realizations introduces statistical mixing, rendering the state effectively mixed. Nevertheless, as long as the effective operation remains smooth and orientation-preserving, the mapping from the transverse plane to the Poincaré sphere remains continuous, thereby ensuring conservation of the skyrmion number. Detailed proofs are provided in Supplementary Sections IB and IC.

The experimental procedure consists of three stages: the preparation of nonlocal skyrmions, the introduction of turbulence, and the measurement of quantum states. The experimental setup is depicted in Fig. 2a. Specifically, a 405 nm continuous-wave laser serves as the pump source, and spatial filtering via a single-mode fiber is employed to produce a Gaussian spatial mode. This configuration allows precise control over the spiral spectrum bandwidth of the entangled photon pairs, thereby enabling direct manipulation of the skyrmions’ topological structures. The filtered pump beam is then directed onto a 3 mm-thick β-barium borate nonlinear crystal, where type-I spontaneous parametric down-conversion generates entangled photon pairs exhibiting OAM correlations. The photons pass through the 810 ± 10 nm band-pass filter, and the quantum state at this stage can be expressed as:
|Ψ⟩ = 1/Σ cℓ |ℓ⟩A |−ℓ⟩B, (7)
where cℓ denotes the complex spiral-bandwidth coefficients40. To generate nonlocal skyrmions, the initial OAM-entangled state must be transformed into a hybrid-entangled state. In the experiment, a beam splitter spatially separates the photon pairs: photon A retains its OAM DoF to carry spatial information, while photon B undergoes a fully digital spatial-polarization conversion process, coupling its OAM DoF to a specific polarization state. The conversion process41 implemented by a spatial light modulator (SLM) is based on its polarization-selective operating characteristics. The procedure is as follows: first, a half-wave plate (HWP) transforms the incident photons into diagonally polarized light, introducing a horizontal polarization component that can be modulated by the SLM. Upon reflection, the initially unmodulated vertical component is converted into an SLM-responsive polarization state by an optical system consisting of a quarter-wave plate (QWP) and a mirror, thereby enabling full modulation of both orthogonal polarization components. After this polarization conversion, the nonlocal skyrmion was experimentally realized.

Photons carrying OAM first pass through an SLM followed by a coupling-lens system before being coupled into a single-mode fiber. Notably, the SLM is configured with both holographic gratings for quantum state tomography and additional phase-modulation gratings designed to simulate atmospheric turbulence (see Supplementary Material). Meanwhile, polarization-encoded photons are directed through a Stokes parameter measurement system comprising a QWP, HWP, and linear polarizer, after which they are coupled into another fiber via a lens assembly. Coincidence counts are recorded by single-photon detectors placed at both output channels, providing the experimental data for analysis.

Quantum state tomography was performed by conducting mutually unbiased basis measurements on both photons’ OAM and polarization DoF (see Supplementary Materials). First, we performed quantum state tomography on nonlocal skyrmion state |Ψ⟩ = 1/√2 (|0⟩A |H⟩B + |1⟩A |V⟩B) in the absence of turbulence, with the joint measurement data shown in Fig. 2b. Based on this dataset, the quantum state’s density matrix was reconstructed using maximum likelihood estimation, as shown in Fig. 2c. The Gaussian profile of the pump beam naturally imposes a finite spiral-bandwidth distribution on the OAM modes, resulting in nonuniform weighting where the |H, 0⟩ component typically dominates40. Analysis of the density matrix further allowed us to extract the spin texture characteristics of the nonlocal skyrmion. Notably, under turbulence-free conditions, the measured spin texture (Fig. 2d) exhibits a clearly defined Neel-type topological structure. Subsequently, we performed state tomography on nonlocal skyrmions under varying turbulence intensities to investigate their topological and entanglement properties. The experimentally obtained state tomography data, reconstructed density matrices, and corresponding spin-textured fields under turbulence strength Ω = 2ω/r0 = 1.2 are shown in Fig. 2e–g, where ω denotes the second-moment beam waist and r0 is the Fried coherence length characterizing the transverse coherence scale of atmospheric turbulence (see Supplementary Materials, Section III). In the experiment, we quantified the entanglement strength of the two-qubit system using concurrence C, a standard entanglement measure for biphoton systems that ranges from 0 for separable states to 1 for maximally entangled states42. A higher concurrence indicates stronger nonlocal correlations between the two photons (see Supplementary Material). Other common entanglement measures, such as the entanglement entropy or tangle, are monotonically related to the concurrence for two-qubit states. As shown in Fig. 2h, even as increasing turbulence deforms the quantum texture and reduces the concurrence, the topological charge of the nonlocal skyrmion remains rigorously preserved. This robust topological protection has been confirmed through both experiment and simulation, following the mechanism described in Eq. (6): nonlocal correlations transform turbulence-induced perturbations into smooth coordinate mappings, thereby ensuring the conservation of topological properties.

Topological robustness of classical skyrmions in turbulence
Investigating optical skyrmions in the classical regime allows us to easily assess their robustness in a basis independent manner. In the quantum regime, spatial projective measurements are made only onto the OAM subspace of choice. In the classical regime, we can make use of polarization optics and a camera to directly obtain the spatially resolved Stokes parameters. In this way, we may observe the effects of inter-modal coupling across all basis modes with four simple intensity measurements, and not just coupling within a specific subspace. This provides a complementary analysis of the topological robustness, as different scenarios and applications will require and may benefit from different measurement approaches. The close analogy between classical vector beams and quantum hybrid entangled states implies that the results in this work are directly applicable and easily adaptable from one regime to another31,43.

Classical optical skyrmions were generated using an SLM hologram with a complex amplitude modulation scheme44 in combination with a Sagnac interferometer. An example SLM hologram is shown in Fig. 3a and the experimental setup is shown in Fig. 3b with further details provided in the Methods section. Figure 3c shows a summary of results in the near-field (i.e. in the image plane of the SLM) where each data-point represents the mean measured N over 100 phase screen realizations and the error bars denote the standard deviation. In this regime, we expect the added turbulent phase screen to act only as a phase perturbation and thus have no effect on the skyrmion number45. We tested 5 distinct skyrmion topologies from N=1 to N=5 across 5 different turbulence strengths. The turbulence strengths were quantified by the unitless parameters Ω. We see over the range tested, N remains almost perfectly unchanged with only slight deviations evident at Ω=10 where the correlation length of the medium is an order of magnitude smaller than the transverse beam size and constitutes a very strong perturbation. A summary of the far-field results is shown in Fig. 3d. We see that for low and moderate strengths Ω=0.5, 1 and 2, the skyrmion number remains almost perfectly unchanged for all 5 of the tested topologies. As the strength increases past these, we see that there is some noticeable variation in the skyrmion number up until Ω=5, where the error bars begin to overlap, indicating a regime where one can no longer confidently identify the original topological number. The far-field represents the extreme limit of propagation. In order to investigate a regime that lies between the near field and the far-field, we also tested the robustness of optical skyrmions through an experimentally simulated 100 m channel using a 1 m propagation distance in the lab. These results were performed in such a manner such that they can be generalized to channels of arbitrary lengths but of same Rytov variance σ2R according to Fresnel scaling procedure described in ref. 46. The Rytov variance σ2R for a horizontal path with uniform perturbation strength is given by47
σ2R = 5.32 k7/6 L5/6 r0−5/3, (8)
where k is the wavenumber and L is the length of the channel. The use of this metric is paramount in quantifying the turbulence strengths as it accounts not only for the strength of the phase perturbation r0, but also scintillation due to propagation. A summary of these experimental results using a single phase screen is shown in Fig. 3e. For all topologies except N=5, there is an excellent agreement between the encoded and measured skyrmion number. The error bars are also barely noticeable, indicating almost negligible variation over multiple realizations and demonstrating remarkable stability of the topology. In Fig. 3d and e, N = 5 demonstrates a small but noticeable decay as the aberration strength increases. This can be attributed to the fact that higher order topologies exhibit polarization structures that change rapidly across the transverse plane. When these structures are distorted and no longer symmetrical, it can be challenging to reliably identify all of the polarization singularities needed to accurately determine the skyrmion number. This can possibly be improved by more intelligently choosing the component scalar modes to allow for a more spread out singularity distribution. Such state engineering approaches have already shown an increase in the reliability of measuring other topological structures such as optical knots48.

So far, we have only investigated the effect of turbulence as acting at a single plane with the single phase screen approximation. However, to investigate the resilience of optical skyrmions through thick/extended media such as long path atmospheric turbulence, one must make use of a multiple phase screen approach. Due to the inefficiencies of polarization insensitive SLMs such as digital micro-mirror devices, we opted to make use of numerical wave-optics simulations, consisting of 5 phase screens evenly spaced out over an L=100 m channel as seen in Fig. 4a. Following the approach outlined in ref. 46, we tested strengths ranging from σ2R=0.5 to σ2R=3.0, with full simulation parameters and details in the Supplementary Materials. We show example intensity distributions of a Laguerre-Gaussian beam with ℓ=2 after propagating through the simulated channel in Fig. 4b for distortion strengths σ2R=1.5, σ2R=2.0, σ2R=2.5 and σ2R=3.0, demonstrating the significant distortion induced on the input beam at all strengths. We show the measured skyrmion number of the beams after propagating through the channel in Fig. 4c, where each data point represents the average over 100 independent realizations of the turbulent channel and the error bars give the standard deviation. We see that overall turbulence strengths and for all encoded skyrmion numbers, the topology of the beam is well maintained at the output of the channel, with very little deviation, even at considerably high distortion strengths. Zoomed-in views of the results for encoded skyrmion numbers N=4 and N=1 are shown in Fig. 4d and e, respectively. We see that the topology only begins to noticeably vary in both cases at a distortion strength of σ2R=2 and greater. However, the deviation is still significantly small when compared to the value of the skyrmion number and, as a consequence, the topologies are still easily distinguished from each other over the entire range of tested strengths.

The robustness of optical skyrmions in extended propagation through complex channels has seen very little study, primarily due to the complex dynamics of spatially varying polarization fields as they diffract. In general, it cannot be assumed that optical skyrmions will remain robust when propagating through arbitrary complex media, even if that medium is unitary. A simple example of this would be a cylindrical mode converter where an input Laguerre-Gaussian beam is converted to a Hermite-Gaussian beam either at a plane49 or permanently50. In such simple optical systems, the mode conversion will lead to a decay and eventual loss of the skyrmionic topology. It is therefore surprising to see that optical skyrmions are robust through atmospheric turbulence, which are often considered to be an extreme example of a complex channel. This is even more surprising when one considers that the Noll covariance matrix of atmospheric turbulence weights astigmatism (the key aberration in mode converters) as its most significant component aberration51,52. Mode conversion typically manifests only in the far-field53, explaining why the propagation results in Fig. 3e and Fig. 4c exhibit almost perfect robustness when compared to the far-field results in Fig. 3d. Even so, we only see the measured N deviating from the encoded value for higher order topologies N=5. These results then provide strong evidence for the robustness of optical skyrmions for a larger range of turbulent conditions and for vast majority of real-world scenarios.

Methods
Theory of nonlocal quantum skyrmions
Nonlocal quantum skyrmions arise from a nonlocal mapping between spatial coordinates and Stokes parameters, mediated by the entanglement of OAM and polarization in biphoton systems, forming topologically structured fields. The nonlocal skyrmion state can be defined as Eq. (1). The polarization states are selected for experimental convenience, where a simple wave plate operation enables conversion between bimeron and skyrmion55. To describe the spatial-polarization correlation in position space, we transform photon A from the OAM basis to the position basis rA = (r, ϕ), using the overlap relation ⟨rA|ℓ⟩ = LGℓ(r, ϕ), with LGℓ(r, ϕ) = LGℓ(r, ϕ). By performing the position-projection measurement on photon A, we obtain the conditional state of photon B:
|Ψ⟩B|A = 1/√1+|μ|2 (|H⟩B + μ(r,ϕ) |V⟩B), (9)
with
μ(r,ϕ) = λ2/λ1 (LGℓ2(r)/LGℓ1(r)) ei(ℓ2−ℓ1)ϕ = f(r)eiΦ(ϕ), (10)
where f(r)=|μ(r,ϕ)|, Φ(ϕ)=arg[μ(r,ϕ)]. The complex parameter μ(r,ϕ) characterizes the spatially varying coupling between the OAM and polarization DoF, giving rise to a position-dependent polarization distribution. To quantitatively analyze this structure, the quantum Stokes parameters are extracted from the reconstructed two-photon density matrix using the following procedure:
Sj(r) = ⟨Ψ|σB,j|Ψ⟩ = ⟨rA|⟨Ψ|σB,j|Ψ⟩|rA⟩, (11)
where σB,j denotes the usual Pauli matrices. By substituting Eq. (9) into Eq. (11), the Stokes parameters of the nonlocal skyrmion can be obtained as follows:
Sx(r) = 2f(r)cosΦ / 1+f(r)2
Sy(r) = 2f(r)sinΦ / 1+f(r)2
Sz(r) = 1−f(r)2 / 1+f(r)2. (12)
Because the Stokes parameters vary spatially in the position representation, the skyrmion number is obtained from the following expression15:
N = 1/4π ∫ ∫ (1/r ∂S/∂r × 1/r ∂S/∂ϕ) rdrdϕ. (13)
Using the properties of Laguerre-Gaussian functions, the explicit form of the topological charge can be derived as
N = 1/2 (ℓ2 − ℓ1) (1/1+f2(0) − 1/1+f2(∞)). (14)
When |ℓ1| ≠ |ℓ2|, a nontrivial topological configuration emerges, and the topological charge of the nonlocal quantum skyrmion becomes N = ±(ℓ2 − ℓ1).

Generation of classical optical skyrmions
We experimentally investigated the robustness of optical Stokes skyrmions in a classical turbulent channel with the setup shown in Fig. 3. Figure 3a shows an example of the digital phase hologram H(x,y) used to spatially structure the beam using a complex amplitude modulation scheme44,
H(x,y) = J−1(A(x,y)) sin[ϕ(x,y) + 2π(Gx x + Gy y)], (15)
where A(x,y) and ϕ(x,y) is the amplitude and phase of the desired field respectively, J−1 is the inverse Bessel function of the first kind and Gx(y) are the grating frequencies in the horizontal and vertical directions. We encoded a hologram for each of the component scalar modes onto each half of the SLM allowing us to shape each component of the vector beam separately and independently apply and dynamically change the turbulence phase mask on the beam as described in ref. 46.

The experimental setup is shown in Fig. 3b, where a horizontally polarized Gaussian beam from a HeNe laser (λ = 633 nm) was expanded using a 10× objective lens L1 and collimated using a f=250 mm lens L2 before impinging on the screen of a liquid crystal on silicon SLM. The SLM was encoded with holograms for a Gaussian beam of waist w0 = 0.5 mm on the one half and an LGℓ beam of variable OAM ℓ and a constant embedded Gaussian beam waist of w0 = 0.5 mm on the other. The employed complex amplitude modulation scheme generates the desired optical field in the first diffraction order of the digital hologram. This was isolated with the use of a spatial filtering iris and a 4f imaging system consisting of two lenses L3 and L4, both of focal length f=300 mm. To create a vectorial superposition, both beams were passed through a HWP to convert them to diagonal polarization and then through a Sagnac interferometer to isolate and coaxially combine the orthogonal polarization components, forming a vector beam of the form |Ψ⟩ = (|0⟩|V⟩ + |ℓ⟩|H⟩)/√2. The exact geometry of the Sagnac interferometer is detailed in the supplementary information. The encoded skyrmion/wrapping number of the beam is subsequently given by the OAM of the horizontally polarized spatial mode, i.e. N=ℓ.

Figure 3c shows results for the measured N in the near-field. These were obtained by using a second 4f imaging system consisting of lenses L5 and L6 (both of focal length f=300 mm) to image the beam from image plane of the SLM just after the output of the Sagnac interferometer to a polarization sensitive camera capable of measuring all 4 linear polarization projection simultaneously. A QWP was used to measure the circular polarization projections. These projections allowed for the spatially varying Stokes parameters of the beam to be measured and the skyrmion number was calculated according using the method described in ref. 56. The far-field results presented in Fig. 3d were measured with use of lens L7 of focal length f=1000 mm, which Fourier transformed the image plane after the Sagnac interferometer onto the polarization sensitive camera for full stokes polarimetry measurements. The propagation results shown in Fig. 3e require propagation in two stages. First, the desired modes were numerically propagated using angular spectrum propagation57 (0.5 m) of the desired channel length. The propagated version of the field and the turbulence phase screen were then combined before being used to generate the hologram. The modulated light then propagated through the same 4f imaging system, spatial filter and Sagnac interferometer as before, and then propagated the remaining channel length (0.5 m) to the polarization camera.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the manuscript of ``Topological robustness of classical and quantum optical skyrmions in atmospheric turbulence’’, the authors propose to utilize classical and quantum optical skyrmions through channels with atmospheric turbulence and show the robustness of topological resilience of the structure. In quantum experiment, they utilize a bi-photon system to show although the entanglement is degraded, the topological metric, skyrmion number, remains nearly unaffected. Similar results are also identified in experiments of classical regime. The manuscript is overall well organized with solid theory and experiment support. However, we still have several questions and concerns.\n\nMajor questions and concerns\n1. In Fig. 2 c, the quantum state tomography of the skyrmion state in the absence of turbulence shows an uneven distribution in the four components. Is it due to the imperfect state preparation in experiment?\n2. In the paragraph following Eq. (6), the authors states that the effect of turbulence on quantum states can be characterized through coordinate transformation thus the quantum skyrmions are ruboust. We suggest to provide more high-level explanation on the how the turbulence affects the coordinate transformation as this is the key point for the fundamental theory of this manuscript.\n3. In Fig. 2, the authors utilize concurrence to characterize the entanglement. What is the reason for using this metric, and how about other entanglement measures?\n4. In Fig.4, the numerical simulations show the robustness of skyrmions against thick turbulence. Does the Rytov variance depend on the length of channel, spacing separation between turbulence phase screens, number of turbulence phase screens? An added clarification here can help readers understand the thickness of turbulence to be studied.\n5. For a curiosity question, provided the robustness of topological measure against turbulence, is it possible to encode information utilizing this approach? We expect to see some future discussions or extensions.\n\nMinor concerns and questions\n1. Following Eq. (1), the abbreviation LG is used without definition or explanation.\n2. In Fig.2 b and c, the heatmap and z-axis are not well explained. Is it amplitude or probability? Similar questions also appear in e and f.\n3. What are the insets at right top corner of Fig.2 c and f? They are not explained in the caption.\n4. In Fig. 2h, the y-axis of only N is not appropriate since both concurrence and skyrmion number are plotted here.\n5. In Eq. (5), we suggest to consider other notions for the coefficients to distinguish from the ones in Eq. (1).\n6. The turbulence strength Ω = 2𝜔/𝑟 is used without explanation on the physical meaning of each parameter.\n7. The authors utilize concurrence to characterize the entanglement. We suggest to add a brief introduction of its definition and physical meaning in the main text and related references, which would be friendly to a wider range of readers.\n8. In the second paragraph of Classical Skyrmions in Turbulence, we notice a typo of “1!m”. Please check it and other possible typos as well.\n9. In the same paragraph, it is a little awkward to say ``We see that for all topologies except N=5, we see excellent agreement between encoded and measured skyrmion number’’. Please revise this sentence.\n10. In the abstract, the authors conclude that their work ``blurs’’ the classical-quantum divide. We would suggest to revise the wording though we do see similar behavior in both classical and quantum experiments.\n\nWith all the above questions and concerns well addressed, we would like to recommend it to publish in Nature Communications.", "reviewer_2": "This manuscript presents a significant experimental advancement in the topological robustness of optical skyrmions in the classical and quantum domains. This work demonstrates the topological resilience of both quantum and classical optical skyrmions in simulating atmospheric turbulence. The results are well supported by rigorous theory and high-quality experiments. The manuscript is clearly written, methodical, and addresses an important gap in non-local skyrmions as topologically resilient in atmospheric turbulence. Hence, I recommend acceptance after revisions if some definitions and theories are clarified and the claims are contextualized.\nMy comments are listed below：\n1. The definition of quantum skyrmion. The author's definition is given as Eq.8, which is a two photon entanglement state (if all coefficients are not equal to 0).\na) Must all quantum skyrmions be written in terms of two photons? Can the entangled state between different degrees of freedom of a single photon be regarded as a quantum skyrmion?\nb) Are all two-photon entangled states satisfying this form quantum skyrmions? Are some restrictions required for the coefficients or OAM values? If two OAMs in Eq.8 have the same absolute value but opposite signs, the quantum skyrmion number N=0.\nc) The author theoretically demonstrates that due to coordinate transformations, the quantum skyrmion number in Eq. S1 can be calculated. Experimentally, it is shown that N remains unchanged. Therefore, can Eq. S1 also be used to represent quantum skyrmion?\nd) Eq. S1 can be regarded as a superposition state of two quantum skyrmions. Does the superposition of any two quantum skyrmions still result in a quantum skyrmion? Eq. S1 can be used to represent any two-photon pure state. Is it the case that quantum skyrmions should generally be represented by two-photon pure states due to the invariance of N?\n2. When considering the effects of atmospheric turbulence, the authors claim that only qubits are taken into account, making Eq. 5 valid. However, failing to account for other higher-order OAM modes formed by atmospheric turbulence will lead to a brightness reduction of the quantum skyrmion. Given the presence of noise, this may reduce the experimental value of N. The author should evaluate the impact of reduced brightness.\n3. In the discussion and conclusion section of the main text, the author mentions that this discovery unveils a profound shared characteristic of quantum and classical skyrmions—the unitary nature of any single-sided quantum channel acting on input pure states, which is somewhat confusing. The author appears to suggest that the invariance of quantum skyrmions results from the local transformation's unitarity. It is well-known that entanglement remains unchanged under local unitary transformations. However, the author's experiments demonstrate that N remains preserved even during entanglement decay. So is the physical basis for the robust of quantum skyrmion that entanglement remains invariant under local unitary transformations? If so, why does N remain unchanged during entanglement degenerates? If not, the separated states might also exhibit skyrmion properties—does this mean skyrmions do not depend on non-separability?\n4. In Section topologically protected quantum skrimings resist turbulence of the supplementary materials, the author demonstrates the robust of skyrmions through coordinate transformations. When the quantum state evolves to Eq.S1, the projection for the OAM component can undergo a corresponding coordinate transformation; however, the projection result for the photon B also changes accordingly, which no longer takes form of Eq.9. Proving that the projections of the photon A can be achieved through coordinate transformations does not seem to fully demonstrate the invariance of the quantum skyrmion number.\nThe author should clarify how the calculation was performed—whether the previous formula was applied again (Eq.9, Eq.12) or the projection result of photon B was recalculated.", "reviewer_3": "The work investigates the robustness of skyrmion characteristics, such as their topological charge. The authors demonstrate that this charge remains invariant under changes in spatial coordinates. This property could have significant implications for optical skyrmions, which are generated from superpositions of the internal degrees of freedom of the electromagnetic field, such as polarization and orbital angular momentum.\nThe atmospheric turbulence, which acts only on the spatial modes of the field, induces distortions and decoherence that currently hinder the use of skyrmions in both classical and quantum communications. A key finding of this paper is that the skyrmion number remains unchanged under those turbulence that can be modeled as a coordinate transformation. This intuition is validated through two experiments in the quantum and classical regimes.\nIn the quantum experiment, a skyrmionic state of two entangled photons is generated via spontaneous parametric down-conversion. Atmospheric turbulence is simulated using phase holograms on a Spatial Light Modulator (SLM). The classical experiment is performed with coherent light, analyzing the effects of similarly simulated turbulence in both the near- and far-field. Both experiments rigorously confirm the invariance of the skyrmion number under such turbulent conditions.\nThe work is well-written, accessible, and methodologically rigorous. While the phenomenon is of fundamental interest, I do not find that the potential impact of the study meets the high standards required for publication in Nature Communications. The work certainly merits publication in a more specialized journal.\nBelow are specific comments and questions:\n1) The study does not address whether this invariance is unique to skyrmions or can also be observed in other topologically non-trivial states of light. For instance, are more generic vector vortex beams, of which optical skyrmions are a specific subset, also endowed with similar robustness?\n2) The analysis focuses primarily on atmospheric turbulence, whose effects can be modeled as a coordinate transformation, thus leading to the observed topological invariance. Can all practically relevant turbulence effects be captured by this model? The effect described in Eq. (5) appears to be a very specific case, where turbulence populates other OAM modes without compromising the state's purity. What would be the impact of other effects, such as genuine decoherence? It would be insightful to consider other types of disturbances, for instance, in channels like underwater links or generic scattering media. This would provide a clearer and more comprehensive picture of the practical applicability of this invariance.\n3) The paper lacks an extensive and convincing discussion on how this property can be leveraged in the proposed application fields for skyrmions. A more detailed exploration of potential applications would strengthen the manuscript.\nMinor Comments:\n- Equation 6 is a repetition of Equation 2. It would be more useful to include further details on the proof of invariance from the Supplementary Information, as this constitutes a central result of the paper.\n- In Figure 3, the laser beam reflections in the Sagnac interferometer part are somewhat confusing. I suggest reducing the beam size and using a lighter color for the reflections to improve clarity.", "reviewer_4": "In the manuscript of ``Topological robustness of classical and quantum optical skyrmions in atmospheric turbulence’’, the authors propose to utilize classical and quantum optical skyrmions through channels with atmospheric turbulence and show the robustness of topological resilience of the structure. In quantum experiment, they utilize a bi-photon system to show although the entanglement is degraded, the topological metric, skyrmion number, remains nearly unaffected. Similar results are also identified in experiments of classical regime. The manuscript is overall well organized with solid theory and experiment support. However, we still have several questions and concerns.\n\nMajor questions and concerns\n1. In Fig. 2 c, the quantum state tomography of the skyrmion state in the absence of turbulence shows an uneven distribution in the four components. Is it due to the imperfect state preparation in experiment?\n2. In the paragraph following Eq. (6), the authors states that the effect of turbulence on quantum states can be characterized through coordinate transformation thus the quantum skyrmions are ruboust. We suggest to provide more high-level explanation on the how the turbulence affects the coordinate transformation as this is the key point for the fundamental theory of this manuscript.\n3. In Fig. 2, the authors utilize concurrence to characterize the entanglement. What is the reason for using this metric, and how about other entanglement measures?\n4. In Fig.4, the numerical simulations show the robustness of skyrmions against thick turbulence. Does the Rytov variance depend on the length of channel, spacing separation between turbulence phase screens, number of turbulence phase screens? An added clarification here can help readers understand the thickness of turbulence to be studied.\n5. For a curiosity question, provided the robustness of topological measure against turbulence, is it possible to encode information utilizing this approach? We expect to see some future discussions or extensions.\n\nMinor concerns and questions\n1. Following Eq. (1), the abbreviation LG is used without definition or explanation.\n2. In Fig.2 b and c, the heatmap and z-axis are not well explained. Is it amplitude or probability? Similar questions also appear in e and f.\n3. What are the insets at right top corner of Fig.2 c and f? They are not explained in the caption.\n4. In Fig. 2h, the y-axis of only N is not appropriate since both concurrence and skyrmion number are plotted here.\n5. In Eq. (5), we suggest to consider other notions for the coefficients to distinguish from the ones in Eq. (1).\n6. The turbulence strength Ω = 2𝜔/𝑟 is used without explanation on the physical meaning of each parameter.\n7. The authors utilize concurrence to characterize the entanglement. We suggest to add a brief introduction of its definition and physical meaning in the main text and related references, which would be friendly to a wider range of readers.\n8. In the second paragraph of Classical Skyrmions in Turbulence, we notice a typo of “1!m”. Please check it and other possible typos as well.\n9. In the same paragraph, it is a little awkward to say ``We see that for all topologies except N=5, we see excellent agreement between encoded and measured skyrmion number’’. Please revise this sentence.\n10. In the abstract, the authors conclude that their work ``blurs’’ the classical-quantum divide. We would suggest to revise the wording though we do see similar behavior in both classical and quantum experiments.\n\nWith all the above questions and concerns well addressed, we would like to recommend it to publish in Nature Communications."}, "author_response": "Response Letter to Reviewers\nTopological robustness of classical and quantum op- tical skyrmions in atmospheric turbulence\nZhenyuGuo,CadePeters,NiloMata-Cervera,AntonN.Vetlugin,RuixiangGuo,PeiZhang,AndrewForbes, YijieShen\nNatureCommunications,Manuscript ID NCOMMS-25-70646\n1. ResponsestotheCommentsofReviewer1andReviewer2\nComment: Thismanuscriptpresentsasignificantexperimentaladvancementinthetopologicalrobustnessofoptical skyrmionsintheclassicalandquantumdomains. Thisworkdemonstratesthetopologicalresilienceof bothquantumandclassicalopticalskyrmionsinsimulatingatmosphericturbulence. Theresultsarewell supportedbyrigoroustheoryandhigh-qualityexperiments. Themanuscriptisclearlywritten,methodical, andaddressesanimportantgapinnon-localskyrmionsastopologicallyresilientinatmosphericturbulence. Hence,Irecommendacceptanceafterrevisionsifsomedefinitionsandtheoriesareclarifiedandtheclaims arecontextualized. Mycommentsarelistedbelow.\nResponse: Wesincerelythankthereviewersfortheirtimeandeffortdevotedtocarefullyreadingandevaluatingour manuscript. Wegreatlyappreciatetheconstructiveandinsightfulcomments. Wehavecarefullyaddressed allthecommentspointbypointandrevisedthemanuscriptaccordingly. Wehopethatourresponsesbelow satisfactorilyresolvethereviewers’concerns.\nComment: 1. Thedefinitionofquantumskyrmion. Theauthor’sdefinitionisgivenasEq.8,whichisatwophoton entanglementstate(ifallcoefficientsarenotequalto0).\n(a)Mustallquantumskyrmionsbewrittenintermsoftwophotons? Cantheentangledstatebetween differentdegreesoffreedomofasinglephotonberegardedasaquantumskyrmion?\n(b)Arealltwo-photonentangledstatessatisfyingthisformquantumskyrmions? Aresomerestrictions required for the coefficients or OAM values? If two OAMs in Eq.8 have the same absolute value but oppositesigns,thequantumskyrmionnumberN=0.\n(c)Theauthortheoreticallydemonstratesthatduetocoordinatetransformations,thequantumskyrmion numberinEq. S1canbecalculated. Experimentally,itisshownthatNremainsunchanged. Therefore, canEq. S1alsobeusedtorepresentquantumskyrmion?\n(d)Eq. S1canberegardedasasuperpositionstateoftwoquantumskyrmions. Doesthesuperposition of any two quantum skyrmions still result in a quantum skyrmion? Eq. S1 can be used to represent any two-photon pure state. Is it the case that quantum skyrmions should generally be represented by two-photonpurestatesduetotheinvarianceofN?\nResponse: (a)Indeed,theconceptofaquantumskyrmionisnotlimitedtotwo-photonsystems. Inourmanuscript,the quantumskyrmionwediscussedactuallycorrespondstoanonlocalquantumskyrmion,anotionintroduced inNat. Photon. 18,258–266(2024),wherespatialandpolarizationinformationaretopologicallymapped throughhybridentanglementbetweentwophotons. Inthiscase,thetopologicalmappingisnaturallydefined 1 inabipartiteHilbertspace.\nInprinciple,aquantumskyrmioncanalsobeconstructedwithinasinglephotonpossessingmultipleinternal degreesoffreedom,suchaspolarization–OAMentanglement. Theessentialrequirementistheexistence ofanontrivialmappingbetweenthejointstatespaceandthePoincarésphere,whichsupportsaquantized topologicalinvariant. Indeed,inarecentwork(Nat. Phys. 21,1462–1468(2025)),Maetal. experimentally realizedasingle-photonskyrmionbyexploitingtheentanglementbetweenpolarizationandOAMdegrees offreedominaquantum-dot–semiconductor-cavityplatform.\nToavoidambiguity,wehaverevisedtheterminologyinourmanuscript,replacing“quantumskyrmion”with “nonlocalquantumskyrmion”(orsimply“nonlocalskyrmion”whereappropriate). Thismodificationhas beenhighlightedintherevisedmanuscript.\n(b)Notallbiphotonhybrid-entangledstatescanberegardedasnonlocalquantumskyrmions. Additional constraintsmustbeimposedsothattheOAMindicesshouldensureacontinuousanddifferentiablemapping thatpreservesawell-definedtopologicalcharge. When|ℓ |=|ℓ |,themappingbecomestopologicallytrivial 1 2 ifℓ =ℓ ,ortopologicallynontrivialinalowerdimension(mappingaclosedloopinthetransverseplane 1 2 S1 toaclosedloopS1 inthePoincarésphere)forℓ = −ℓ . Therefore,askyrmionictopologyemerges 1 2 onlywhen|ℓ |̸=|ℓ |,wherethefieldestablishesanontrivial2DmappingfromthetransverseplaneR2to 1 2 thePoincarésphereS2. WehaverevisedtheparagraphbelowEq. (1)inthemaintexttoemphasizethis clarification.\nAskyrmionictopologyemergesonlywhen|ℓ | ̸= |ℓ |,wherethefieldestablishesanontrivial2D 1 2 mapping from the transverse plane R2 to the Poincaré sphere S2. In contrast, for |ℓ | = |ℓ |, the 1 2 mappingbecomestopologicallytrivialifℓ = ℓ ,ortopologicallynontrivialinalowerdimension 1 2 (mappingaclosedloopinthetransverseplaneS1 toaclosedloopS1 inthePoincarésphere)for ℓ =−ℓ .\n1 2\n(c)WewouldliketoclarifythatEq. (S1)inthepreviousversionwasincorrectlywrittenasatwo-photon state,whichmayhavecausedconfusion. IthasnowbeencorrectedintherevisedSupplementarySectionIA torepresenttheconditionalstateofphotonBafterprojectingphotonAontoposition⃗r :\nA\n⟨⃗r |Ψ′⟩=|Ψ′ ⟩=λ′LG |H⟩ +λ′LG |V⟩ +λ′LG |H⟩ +λ′LG |V⟩ , (1)\nA B|A 1 ℓ1 B 2 ℓ1 B 3 ℓ2 B 4 ℓ2 B\nwhichcanbenormalizedas:\n|H⟩ +µ′(r,ϕ)|V⟩\n|Ψ′ ⟩= B B,\nB|A (cid:112) 1+|µ′(r,ϕ)|2\n(2)\nµ′(r,ϕ)=\nλ′\n2\n|LG\nℓ1\n(r)|+λ′\n4\n|LG\nℓ2\n(r)|ei(ℓ2−ℓ1)ϕ\n.\nλ′\n1\n|LG\nℓ1\n(r)|+λ′\n3\n|LG\nℓ2\n(r)|ei(ℓ2−ℓ1)ϕ\nTheactualstateaffectedbyturbulenceis\n|Ψ′⟩=λ′|ℓ ⟩ |H⟩ +λ′|ℓ ⟩ |V⟩ +λ′|ℓ ⟩ |H⟩ +λ′|ℓ ⟩ |V⟩ . (3)\n1 1 A B 2 1 A B 3 2 A B 4 2 A B\nwhichdescribesageneraltwo-photonpurestateintheencodedOAM–polarizationsubspace. Asshowninthe\nrevisedSupplementarySectionIA,thisstatedoesnotautomaticallyqualifyasaquantumskyrmion. Topolog- icalprotectionrequirestwonecessaryconditions:(i)nonzeroentanglementand(ii)agloballynonvanishing\ndenominatorensuringacontinuous,orientation-preservingcoordinatemapping. Onlywhenthesecriteriaare 2 metdoestheconditionalStokesfieldpreservetheskyrmionnumber. Therefore,|Ψ′⟩ingeneralrepresentsa\nsuperpositionofcomponentsthatmayretaintheskyrmiontopologyundertheaboveconditions,butitisnot\nitselfaquantumskyrmionasdefinedbyEq. (1)inthemaintext.\n(d) As clarifiedabove, Eq. (S1)has been corrected to denote theconditional state of photon Bas Eq. 1.\nAlthoughthisfieldcanbeexpressedasasuperpositionofskyrmioniccomponents,suchasuperpositiondoes\nnotnecessarilyyieldanotherskyrmionunlessthecoefficientssatisfytheentanglementandcontinuitycriteria\nstatedabove. AsrigorouslydemonstratedinSupplementarySectionIA,onlywhentheseconditionsaremet\ndoestheStokesfieldpreserveacontinuousmappingandaquantizedskyrmionnumber. Therefore, |Ψ′⟩\nrepresentsageneraltwo-photonpurestatewhoseconditionaltopologymayremainprotectedunderthese\ncriteria,butthequantumskyrmionitselfisuniquelydefinedbyEq. (1)inthemaintext,ratherthanbyan\narbitrarysuperposition.\nInourexperiment,theturbulentperturbationwasrealizedusingasinglespatiallightmodulatortoimpose\ncontrolledrandomphasefluctuations. Thisconfigurationcorrespondstotheweakscintillationconditions\n(Physical Review A, 2014, 90(5): 052115), where the dominant field components (λ′,λ′) remain much\n1 4\nstrongerthanthecross-modecouplings(λ′,λ′).Undersuchconditions,bothtopologicalprotectioncriteria\n2 3\narefulfilled,ensuringthatthemappingremainscontinuousandfreeofsingularities. Consequently,despite\nminorlocaldistortions,theskyrmionnumberisexperimentallyobservedtoremaininvariant.\nComment: 2. Whenconsideringtheeffectsofatmosphericturbulence,theauthorsclaimthatonlyqubitsaretaken\nintoaccount,makingEq. 5valid. However,failingtoaccountforotherhigher-orderOAMmodesformed\nbyatmosphericturbulencewillleadtoabrightnessreductionofthequantumskyrmion.Giventhepresence\nofnoise,thismayreducetheexperimentalvalueofN.Theauthorshouldevaluatetheimpactofreduced\nbrightness.\nResponse: Inthequantumcase,thedetectedstateisalwaysrenormalized,sorestrictingtheanalysistoasmallerencoded\nsubspacedoesnotchangethetraceofthedensitymatrix. Likewise,inboththequantumandclassicalregimes,\nthecalculationoftheskyrmionnumberreliesonlocalnormalizationoftheStokesvectors. Consequently,the\nabsoluteopticalpowerorbrightness(anditsquantumequivalent)isfactoredoutofthetopologicalobservable.\nThiscontrastswithotherquantitiessuchasentanglement,whichindeeddecreaseswhenpartofthestate\nisprojectedout. Thisdistinctionillustratesafundamentaladvantageoftopologicalobservables—theyare\ninsensitivetooverallsignalloss.\nAsdetailedintherevisedSupplementarySectionIC,turbulenceinducesaneffectivenon-unitaryprojection\noperatorT =PU P,whichredistributespartoftheopticalamplitudeintohigher-orderOAMmodes.\nA\nThis apparentnon-unitarityarisessolelyfromprojectingthegloballyunitaryevolutionontoafinitedetection\nsubspace. Itreducesthetotaldetectionprobabilityp=Trρ˜. Afternormalization,ρ =ρ˜/p,thebrightness\neff\nreductionmerelylowersthecoincidencerateandeffectiveconcurrencebutdoesnotalterthenormalized\nStokesfieldwithinthemeasuredsubspace. Therefore,whilethenon-unitaritymanifestsasreducedconcur-\nrence,theunderlyingmappingoftheStokesfieldcontinuestofollowthesmooth,orientation-preserving\ncoordinatetransformationproveninthemaintext. Consequently,theexperimentallyevaluatedskyrmion\nnumber remains invariant, confirming that the topological structure is preserved even under brightness\nreduction.\nFurthermore,althoughquantumstatetomographyinherentlylimitsobservationstotwo-dimensionalprojec-\ntionswithintheencodedsubspace,ourclassicalskyrmionexperimentsprovideacomplementarymeansto\nverifythefullspatialtopology. Byemployingpolarisationopticsandacamera,wedirectlyobtainspatially\nresolvedStokesparameters. Throughonlyfoursimpleintensitymeasurements,wecanobserveinter-modal\ncoupling across all basis modes, not just within a specific subspace. This classical approach offers an\n3 independentandbasis-freevalidationoftopologicalrobustness,experimentallyandnumericallyconfirming\nthestrongprotectionoftheskyrmionstructureunderturbulence.\nComment: 3. In the discussion and conclusion section of the main text, the author mentions that this discovery\nunveilsaprofoundsharedcharacteristicofquantumandclassicalskyrmions—theunitarynatureofany\nsingle-sidedquantumchannelactingoninputpurestates, whichissomewhatconfusing. Theauthor\nappears to suggest that the invariance of quantum skyrmions results from the local transformation’s\nunitarity. Itiswell-knownthatentanglementremainsunchangedunderlocalunitarytransformations. However,theauthor’sexperimentsdemonstratethatNremainspreservedevenduringentanglementdecay.\nSoisthephysicalbasisfortherobustofquantumskyrmionthatentanglementremainsinvariantunder\nlocalunitarytransformations? Ifso,whydoesNremainunchangedduringentanglementdegenerates? If\nnot,theseparatedstatesmightalsoexhibitskyrmionproperties—doesthismeanskyrmionsdonotdepend\nonnon-separability?\nResponse: Indeed,free-spacepropagationthroughatmosphericturbulenceisaunitaryprocessinthefullHilbertspace\n[APL123,021103(2023)]. Thisglobalunitarity,however,doesnotcontradicttheeffectivenon-unitarity\nobservedinexperiments. Theunitarity(andentanglement)ispreservedglobally,asscatteringredistributes\namplitudesintomodesoutsidetheoriginallyencodedsubspace. Whentheevolutionisprojectedontoafinite\ndetectionsubspace,partoftheamplitudeistracedout,sotheeffectivetransformationappearsnon-unitary.\nInpractice,(1)entanglementquantificationinarbitrarilylargedimensionsisextremelychallenging,and(2)\nmanyquantum-informationprotocols(e.g.,quantumcommunicationandQKD)operatewithinafixed,finite\nbasis. Consequently,theapparententanglementdecayandbrightnessreductionaredirectmanifestations\nofthisprojection-inducednon-unitarity,notofalossofglobalunitarity. Importantly,theevolutionwithin\nthemeasuredsubspacestillfollowsasmooth,orientation-preservingcoordinatetransformation,ensuring\ntheconservationoftopologicalcharge. Asdiscussedinthemanuscript,thisfollowsfromthefactthatthe\nturbulentchannelismap-preservingratherthanamplitude-preserving.\nTheseconclusionsarerigorouslyverifiedintherevisedSupplementarySectionI.SectionIAprovesthatnon-\nseparabilityisthenecessaryconditionfortheexistenceofanontrivialskyrmionictopology,whileSectionIB\ndemonstratesthatevenafterensembleaveragingleadstoamixedstate,thenormalizedStokesfieldundergoes\nonlyasmooth,orientation-preservingdeformationoftheoriginalfield. Therefore,theskyrmionnumber\nremainsinvariant,confirmingthattopologicalprotectionpersistsfromthepuretothemixed-stateregime.\nInaddition,wehaveincludednewdataanalysisinSectionIB(seeFig. S1),wheretheensemble-averaged\nmixed-statebehaviorisquantitativelyexamined. Furthermore,inSectionIC,weprovethatthiseffective\nnon-unitarityleadstoobservableentanglementdecaywhileleavingthetopologicalchargeinvariant. This\nadditionalanalysisfurthersupportsourconclusionthat,despiteentanglementdecay,thetopologicalstructure\nremainsprotectedunderturbulence.\nComment: 4. InSectiontopologicallyprotectedquantumskrimingsresistturbulenceofthesupplementarymaterials,\ntheauthordemonstratestherobustofskyrmionsthroughcoordinatetransformations. Whenthequantum\nstateevolvestoEq.S1,theprojectionfortheOAMcomponentcanundergoacorrespondingcoordinate\ntransformation;however,theprojectionresultforthephotonBalsochangesaccordingly,whichnolonger\ntakes form of Eq.9. Proving that the projections of the photon A can be achieved through coordinate\ntransformationsdoesnotseemtofullydemonstratetheinvarianceofthequantumskyrmionnumber. The\nauthor should clarify how thecalculation was performed—whether the previousformulawasapplied\nagain(Eq.9,Eq.12)ortheprojectionresultofphotonBwasrecalculated.\nResponse: Indeed, the earlier expression was incorrectly written as a two-photon entangled state. It has now been\ncorrectedintherevisedSupplementarySectionItorepresenttheconditionalstateofphotonBafterprojecting\n4 photon A onto position⃗r . The corrected expressions of |Ψ′ ⟩ and µ′(r,ϕ) are given in Eq. 2. With\nA B|A\nthis correction, the calculation of the Stokes field for photon B is performed exactly as in the main text,\nreplacingµ(r,ϕ)byµ′(r,ϕ). Becausethecoordinatetransformationissmooth,non-singular,andorientation-\npreserving,itinducesacontinuoushomotopyoftheStokesfieldwithoutchangingitstopologicaldegree. The\nskyrmionnumberisthereforeinvariant,confirmingtherobustnessclaimedinthemaintext.\nFurthermore,asfullydetailedintherevisedSupplementarySectionI,wehavesystematicallyanalyzedtheim-\npactofatmosphericturbulence—includingpuritydegradation,modediffusion,andbrightnessreduction—and\nprovedthatallsucheffectsmerelydeformthefieldcontinuouslywithoutalteringthetopologicalinvariant.\nTheserevisionsprovidearigorousandself-containedverificationofthetopologicalprotectionofquantum\nskyrmionsunderturbulence.\n2. ResponsestotheCommentsofReviewer3\nComment: Theworkinvestigatestherobustnessofskyrmioncharacteristics,suchastheirtopologicalcharge. The\nauthorsdemonstratethatthischargeremainsinvariantunderchangesinspatialcoordinates.Thisproperty\ncouldhavesignificantimplicationsforopticalskyrmions,whicharegeneratedfromsuperpositionsof\nthe internal degrees of freedom of the electromagnetic field, such as polarization and orbital angular\nmomentum. The atmospheric turbulence, which acts only on the spatial modes of the field, induces\ndistortionsanddecoherencethatcurrentlyhindertheuseofskyrmionsinbothclassicalandquantum\ncommunications. Akeyfindingofthispaperisthattheskyrmionnumberremainsunchangedunderthose\nturbulencethatcanbemodeledasacoordinatetransformation. Thisintuitionisvalidatedthroughtwo\nexperimentsinthequantumandclassicalregimes. Inthequantumexperiment,askyrmionicstateoftwo\nentangledphotonsisgeneratedviaspontaneousparametricdown-conversion. Atmosphericturbulence\nissimulatedusingphasehologramsonaSpatialLightModulator(SLM).Theclassicalexperimentis\nperformedwithcoherentlight,analyzingtheeffectsofsimilarlysimulatedturbulenceinboththenear-\nandfar-field. Bothexperimentsrigorouslyconfirmtheinvarianceoftheskyrmionnumberundersuch\nturbulent conditions. The work is well-written, accessible, and methodologically rigorous. While the\nphenomenonisoffundamentalinterest,Idonotfindthatthepotentialimpactofthestudymeetsthehigh\nstandardsrequiredforpublicationinNatureCommunications. Theworkcertainlymeritspublicationina\nmorespecializedjournal. Belowarespecificcommentsandquestions.\nResponse: Wesincerelythankthereviewerforthepositiveevaluationofourmanuscriptandforrecognizingtheclarity,\nmethodologicalrigor,andaccessibilityofourwork. Wealsoappreciatethecommentregardingthepotential\nimpactofthestudyandwouldliketofurtherclarifyitssignificanceinthebroadercontextofphotonicsand\nquantuminformationscience.\nThecentralquestionaddressedinourwork—howthetopologyofopticalskyrmionswithstandsatmospheric\nturbulence—representsalong-standingbottleneckthathashinderedthetransitionoftopologicallightfrom\nlaboratory demonstrations to real-world systems. By resolving this challenge, our study establishes a\ncrucialfoundationforpushingopticalskyrmionstowardpracticalimplementationsinbothclassicaland\nquantuminformationnetworks. Opticalskyrmions,beingstructuredlightfieldswithquantizedtopological\ncharge,provideapowerfulframeworkforencodinginformationinmultidimensionalspaces. However,their\nreal-worlddeploymenthasbeenlimitedbytheuncertaintyofhowturbulence-induceddistortionsaffecttheir\ntopologicalintegrity.\nOurresultsdemonstrate,forthefirsttime,thattheskyrmionnumberremainsinvariantevenunder\nrealisticturbulence,confirmingtheinherenttopologicalresilienceofskyrmionicstates. Thisfinding\n5 establishesthetopologicalresilienceofskyrmionicstatesasafundamentalphysicalproperty—bridgingthe\ngapbetweentheoreticaltopologyandexperimentalphotonics. Thisresilienceisessentialforimplementing\nturbulence-immuneopticalandquantumcommunicationchannels,wheretheinformationisencodednotin\nfragileamplitudesorphases,butinrobusttopologicalinvariants.\nWethereforebelievethattheimpactofthisworkextendsbeyondaspecializedstudyofopticalfields. It\naddressesacorefoundationalquestionthatthebroadercommunitymustresolvebeforepracticalimplementa-\ntionsoftopologicalinformationtransfercanberealized. Inthissense,theresultsreportedhererepresenta\nconceptualmilestone,directlyalignedwithNatureCommunications’aimtopublishsignificantadvancesof\nbroadrelevancewithineachdiscipline.\nComment: 1. Thestudydoesnotaddresswhetherthisinvarianceisuniquetoskyrmionsorcanalsobeobservedin\nothertopologicallynon-trivialstatesoflight. Forinstance,aremoregenericvectorvortexbeams,ofwhich\nopticalskyrmionsareaspecificsubset,alsoendowedwithsimilarrobustness?\nResponse: Intheclassicalcase,theinhomitted for brevity... [See full text in PDF]\n(Note: The response continues for several pages addressing all reviewer comments point-by-point.)"}, {"round": 2, "reviewer_comments": {"reviewer_1": "This paper “Topological robustness of classical and quantum optical skyrmions in atmospheric turbulence” investigates the topological robustness of non-local quantum skyrmions against atmospheric turbulence. I am pleased to report that your revisions have convincingly addressed all concerns raised. The other Referees made a number of observations that I consider to be very important, and the authors have answered them adequately.\nI believe this subject to be of sufficient interest of the readership to warrant publication in Nature Communications. The revised manuscript provides a rigorous proof of the relationship between topological stability theory and unitary evolution. Relevant definitions are well-established, theoretical explanations are logically structured, and conclusions are clearly stated. The results seem sound and are well presented. A number of important details were also included and/or clarified. I have no new observations concerning the new version of the manuscript, and I recommend publication in Nature Communications.", "reviewer_3": "The authors have revised the manuscript convincingly and have exhaustively addressed many of my previous comments. I appreciate the materials added to the Supplementary Information and the main text, which clarify the modelling of various turbulence effects on skyrmions. The Discussion section has also been improved and now provides a clear framework on the applicability of skyrmion topological invariance in imaging and communication protocols. I remain somewhat skeptical, however, regarding the use of skyrmion topological invariance for encoding quantum information and its use in quantum protocols, as these require the transmission of superposition states. The quantum experiment indeed shows a degradation of concurrence, which is expected for a decoherent channel such as turbulence. Then, although the skyrmion number N remains invariant and robust, it is not clear to me how this topological invariant can be used to encode robust entangled states in this degree of freedom.\nConcerning the manuscript's suitability for publication in Nature Communications, I still have some concerns about whether it meets the broad audience and high-impact criteria, or if it might be better suited for a more technical and specialized journal. However, I do not have a strong objection to its publication here, as it is a high-quality work that has undergone significant improvement following revision.\nMinor comments:\n- Is there a specific reason why only the case N=1 has been investigated in the quantum experiment? Is there a maximum N value that remains robust against turbulence, particularly in the presence of entangled skyrmions?\n- In Eq. (3), is the parameter n necessary given the presence of the modulus in the formula?", "reviewer_4": "We appreciate the authors' work in revision the manuscript to address our questions and concerns. All of our questions and concerns are well addressed in the current manuscript, and we would like to recommend it for publication in Nature Communications."}, "author_response": "Response Letter to Reviewers\nTopological robustness of classical and quantum op- tical skyrmions in atmospheric turbulence\nZhenyuGuo,CadePeters,NiloMata-Cervera,AntonN.Vetlugin,RuixiangGuo,PeiZhang,AndrewForbes, YijieShen\nNatureCommunications,Manuscript ID NCOMMS-25-70646A\n1. ResponsestotheCommentsofReviewer1andReviewer2\nComment: Thispaper“Topologicalrobustnessofclassicalandquantumopticalskyrmionsinatmosphericturbulence” investigatesthetopologicalrobustnessofnon-localquantumskyrmionsagainstatmosphericturbulence. Iampleasedtoreportthatyourrevisionshaveconvincinglyaddressedallconcernsraised. Theother RefereesmadeanumberofobservationsthatIconsidertobeveryimportant,andtheauthorshaveanswered themadequatey. Ibelievethissubjecttobeofsufficientinterestofthereadershiptowarrantpublication inNatureCommunications. Therevisedmanuscriptprovidesarigorousproofoftherelationshipbetween topologicalstabilitytheoryandunitaryevolution. Relevantdefinitionsarewell-established,theoretical explanationsarelogicallystructured, andconclusionsareclearlystated. Theresultsseemsoundand arewellpresented. Anumberofimportantdetailswerealsoincludedand/orclarified. Ihavenonew observations concerning the new version of the manuscript, and I recommend publication in Nature Communications.\nResponse: WethanktheReviewerforthepositiveevaluationofourrevisedmanuscript. Weappreciatetherecognition thattherevisionshaveadequatelyaddressedallconcernsandthatthemanuscriptnowprovidesaclearand rigorousanalysiswithwell-structuredtheoryandsoundresults. WearepleasedthattheReviewerhasno furthercomments.\n2. ResponsestotheCommentsofReviewer3\nComment: The authors have revised the manuscript convincingly and have exhaustively addressed many of my previouscomments. IappreciatethematerialsaddedtotheSupplementaryInformationandthemain text,whichclarifythemodellingofvariousturbulenceeffectsonskyrmions. TheDiscussionsectionhas also been improved and now provides a clear framework on the applicability of skyrmion topological invarianceinimagingandcommunicationprotocols. Iremainsomewhatskeptical,however,regarding theuseofskyrmiontopologicalinvarianceforencodingquantuminformationanditsuseinquantum protocols, as these require the transmission of superposition states. The quantum experiment indeed shows a degradation of concurrence, which is expected for a decoherent channel such as turbulence. Then, although the skyrmion number N remains invariant and robust, it is not clear to me how this topologicalinvariantcanbeusedtoencoderobustentangledstatesinthisdegreeoffreedom. Concerning themanuscript’ssuitabilityforpublicationinNatureCommunications,Istillhavesomeconcernsabout whetheritmeetsthebroadaudienceandhigh-impactcriteria,orifitmightbebettersuitedforamore technicalandspecializedjournal. However,Idonothaveastrongobjectiontoitspublicationhere,asitis ahigh-qualityworkthathasundergonesignificantimprovementfollowingrevision.\nResponse: WethanktheReviewerforthecarefulreassessmentofourmanuscriptandforacknowledgingthesubstantial improvements. Weagreethatturbulenceleadstodecoherenceandadegradationofconcurrence,andwedo notclaimrobustnessofquantumentanglementitself. Ourworkemphasizestherobustnessoftheskyrmion numberasastructuraltopologicaldescriptor,whichwehaveclarifiedintherevisedmanuscript.\nComment: MinorComments:\nIsthereaspecificreasonwhyonlythecaseN=1hasbeeninvestigatedinthequantumexperiment? Isthere amaximumNvaluethatremainsrobustagainstturbulence,particularlyinthepresenceofentangled skyrmions?\nResponse: Inthequantumexperiment, wefocusonthecaseN = 1primarilyduetoexperimentalconstraints. The entangledspiralspectrumgeneratedbynonlinearcrystalsisdominatedbythe|0⟩|0⟩component,whilehigher- orderOAMmodeshaveprogressivelylowerweights,leadingtosignificantlyreducedcoincidencecountrates. Asaresult,currentexperimentalstudiesofquantumskyrmionsaretypicallylimitedtolowskyrmionnumbers (generallyN ≤3),asalsoreportedinpreviousworks[Nat. Photon. 18,258–266(2024);NatCommun16, 2934(2025).]\nWeemphasizethatthislimitationisexperimentalinnatureanddoesnotimplyareductionintheintrinsic robustnessofskyrmionswithhigherN. Thegenerationandinvestigationofhigher-orderquantumskyrmions constituteanimportantdirectionforfutureresearch.\nComment: InEq. (3),istheparameternnecessarygiventhepresenceofthemodulusintheformula?\nResponse: InEq. (3),N isdefinedasasignedskyrmionnumber. Themodulus|ℓ −ℓ |specifiesitsmagnitude,while 1 2 thefactorn=±1encodestheskyrmionorientation. Wethereforeconsiderthecurrentformoftheequation appropriate.\n3. ResponsestotheCommentsofReviewer4\nComment: Weappreciatetheauthors’workinrevisionthemanuscripttoaddressourquestionsandconcerns. Allof ourquestionsandconcernssarewelladdressedinthecurrentmanuscript,andwewouldliketorecommend itforpublicationinNatureCommunications.\nResponse: WethanktheReviewerforthepositiveassessmentofourrevisedmanuscriptandforconfirmingthatall questionsandconcernshavebeensatisfactorilyaddressed."}]

Decision letter