In-memory multilevel control of generic SO(m) holonomy in photonics

Domain: physics
Published: 2026-02-07
Editorial outcome: minor_revision

Abstract
Non-Abelian geometric phases offer fault-tolerant unitary operations that are essential for holonomic quantum computation. While the “all-geometric” approach has successfully enabled robust control of atoms, ions, and electrons in various quantum systems, its photonic implementations have remained limited by a lack of tunability—an essential feature for modern classical or quantum optical information processing. Here, we demonstrate non-volatile multilevel tunable generic special-orthogonal (SO) geometric phases on a multilayer silicon photonic platform incorporating the phase-change material (PCM) of Sb₂Se₃. By toggling Sb₂Se₃ between crystalline and amorphous states, we dynamically control the number of degenerate states involved in the holonomy, enabling switching between different holonomic paths and SO(m) geometric transformations. Our results bridge the key gap between robust geometric-phase control and in-memory reconfigurable photonics, establishing a paradigm for fault-tolerant, high-dimensional operations in integrated optical systems.

Introduction
The non-Abelian geometric phase is well known for its inherent robustness in realizing unitary transformations. Building upon the foundational work on the Berry phase1, Wilczek and Zee pointed out that a matrix-valued geometric phase can emerge from the adiabatic evolution of degenerate quantum states2. Unlike the dynamic phase, which depends on the system’s energy and evolution time, the geometric phase is determined purely by the path traced in the Hilbert space. This unique property underpins the ‘all-geometric’ approach3,4, which serves as the foundation for fault-tolerant holonomic quantum computing5–14. Recently, classical wave systems—such as light and sound—with their rich and versatile degrees of freedom, have emerged as promising platforms for exploring non-Abelian geometric phases15–24. In this context, the ‘all-geometric-phase’ approach offers advantages including broad operational bandwidth and enhanced fabrication tolerance in optical computing applications24. However, the robustness that makes geometric phases appealing but also introduces a fundamental challenge: it inherently limits their tunability. The reported scheme for high-dimensional special-orthogonal (SO) holonomy24 is predetermined and fixed after fabrication, which cannot fulfill the requirements of programmable photonic circuits. Although the reconfigurable non-Abelian geometric phase has been demonstrated using the thermo-optic effect25, this implementation is limited to the braid group operations. While this scheme has potential for specialized tasks26, it falls short of enabling general-purpose photonic matrix computation. Therefore, realizing tunable non-Abelian holonomy over the full SO(m) group—combining the dynamic reconfigurability and the robustness of holonomy—is essential to support in-situ training27 and diverse computing tasks.

To realize the tunable non-Abelian geometric phase, a large refractive index modulation in the waveguide is necessary to manipulate the degenerate subspace. Conventional tuning mechanisms—such as the thermo-optic effect, free carrier dispersion effect, or Pockels effect28—typically induce only small changes in the refractive index. In contrast, chalcogenide phase change materials (PCMs) exhibit dramatic refractive index change during phase transitions, exceeding the tuning strength of conventional methods by 1 to 2 orders of magnitude29. Crucially, the optical states of PCMs are non-volatile, enabling lower energy consumption and in-memory storage of tuning states. The phase transition of PCMs is thermally driven and can be triggered by a laser within the PCM’s absorption band. A variety of applications have been demonstrated, including nonvolatile optical switching30–35, photonic memory36,37, analog-to-digital converter38, and in-memory photonic computing39–45. The low-loss PCMs, exemplified by Sb₂Se₃46, have enabled functionalities beyond memories47, such as post-fabrication tuning48, non-volatile tunable filtering49, and power splitting50,51. Combined with the simplicity of laser-driven reconfiguration, low-loss PCMs enable a powerful platform for investigating advanced physical mechanisms with flexible reconfigurability, including the tuning of non-Abelian geometric phases. The large refractive index contrast of PCMs enables complete and stable switching of holonomic evolution paths. Moreover, unlike thermally tuned approaches, PCM-based tuning is free of thermal crosstalk, allowing for independent control of individual waveguides and effectively avoiding degeneracy breaking caused by thermal-crosstalk induced refractive index perturbations.

In this work, we demonstrate tunable generic SO holonomy based on a three-layer photonic platform incorporating PCM waveguides, specifically utilizing Sb₂Se₃52. Some waveguides are made of Sb₂Se₃, while others are made of silicon. By switching the Sb₂Se₃ waveguide between its crystalline and amorphous states, the dimensionality of the degenerate subspace alternates between two and three, resulting in different holonomic paths and the enclosed solid angles. As a result, we achieve reconfigurable geometric operations of generic SO(2) transformations. Furthermore, multi-level tunability is available through the cascading of a series of basic SO(2) transformation components. For higher-dimensional matrices, an 8-level reconfigurable SO(3) transformation is synthesized by Givens rotations. Moreover, the proposed architecture supports high-dimensional non-Abelian braiding, which can be harnessed to construct scalable optical switch meshes. Figure 1a shows the conceptual illustration of tunable generic SO holonomies on a multilayer, heterogeneous photonic integration platform incorporating PCM, along with its potential applications in classical neuromorphic computing and holonomic quantum computing. Our work provides a promising pathway toward practical implementations of non-Abelian geometric phase in integrated photonics.

Results
Tunable SO(2) holonomy
We implement the tunable non-Abelian holonomy in a three-layer heterogeneous integrated photonic platform incorporating PCM. As shown in Fig. 1b, all waveguides share an identical cross-section. The top waveguide is made of Sb₂Se₃, while the remaining waveguides are made of amorphous silicon. The inset in Fig. 1b shows the focused ion beam (FIB) cross-sectional image of the fabricated device. The top Sb₂Se₃ waveguide is patterned by inductively coupled plasma reactive ion etching using a Cl₂/Ar gas mixture. We use silica as both a protective layer and a hard mask for patterning. Figure 1c,d show the FIB images of a single Sb₂Se₃ waveguide fabricated on the substrate, confirming the quality of the Sb₂Se₃ waveguide. For the two-layer silicon waveguides, the scanning electron microscopy (SEM) images are provided in the Supplementary Figs. 8–10. The detailed fabrication process can be found in Methods and Supplementary Fig. 1.

The Hamiltonian of this five-waveguide system can be written as:

H(z) = \begin{pmatrix} \beta & \kappa_{AX}(z) & \kappa_{BX}(z) & \kappa_{CX}(z) & \kappa_{DX}(z) \\ \kappa_{AX}(z) & \beta & 0 & 0 & 0 \\ \kappa_{BX}(z) & 0 & \beta & 0 & 0 \\ \kappa_{CX}(z) & 0 & 0 & \beta & 0 \\ \kappa_{DX}(z) & 0 & 0 & 0 & \beta_P \end{pmatrix} (1)

Where β is the effective index of the individual silicon waveguides A, B, C, and X, where β_P is the effective index of the Sb₂Se₃ waveguide D. κ_{iX} (i=A, B, C, and D) represents the coupling coefficient between the surrounding waveguide i and the central waveguide X. Among these, κ_{AX} and κ_{DX} are fixed throughout the evolution, while κ_{BX} and κ_{CX} evolve adiabatically and are modulated by the gap distance. The number of degenerate states supported by this Hamiltonian can be actively controlled by switching the phase state of the Sb₂Se₃ waveguide. When Sb₂Se₃ is in the crystalline phase (Phase 1 in Fig. 1b), this Hamiltonian supports two degenerate states, as a result of the significant refractive index mismatch between crystalline Sb₂Se₃ and amorphous silicon (Supplementary Fig. 5, resulting in β ≠ β_P). As derived in Eqs. S2–S7, these two degenerate states (|D_1〉 and |D_2〉) can be utilized to construct an SO(2) holonomy20,21,24. When light is injected from waveguides B and C, the state lying in the degenerate subspace |ϕ(z)〉 = a_1(z)|D_1〉 + a_2(z)|D_2〉 is excited, and evolves under an SO(2) transformation e^{iθσ_y} (σ_y is the Pauli matrix53). θ is the solid angle enclosed by the holonomic path in the parameter space spanned by κ_{AX}, κ_{BX}, and κ_{CX}. In contrast, when Sb₂Se₃ transitions to its amorphous phase (Phase 2 in Fig. 1b), three degenerate states are preserved by chiral symmetry54 due to the refractive index matching between amorphous Sb₂Se₃ and amorphous silicon (Supplementary Fig. 5, resulting in β = β_P). As derived in Eqs. S11-S15, three degenerate states (|D_1〉, |D_2〉, and |D_3〉) form an SO(3) holonomy, with the Berry-Wilczek-Zee (BWZ) connection elements A_{3i} = A_{i3} = 0 (i=1, 2, 3) due to the fixed κ_{AX} and κ_{DX} throughout the evolution. When light is injected from waveguides B and C, only |D_1〉 and |D_2〉 are excited. The resulting evolution in the degenerate subspace spanned by |D_1〉 and |D_2〉 corresponds to an SO(2) rotation generated by σ_y which is a subgroup of the full SO(3) holonomy. But the accumulated geometric phase α differs from the phase θ observed in Phase 1 since it involves the participation of κ_{DX} (detailed in η and η’ of Eq. S7 and Eq. S14), and the control manifold is redefined as κ = \sqrt{κ_{AX}^2 + κ_{DX}^2}, κ_{BX}, κ_{CX}. Thus, the value of κ_{DX} and κ_{AX} determines the contrast of θ and α of this two-level tunable geometric phase.

Supplementary Fig. 3 illustrates an example of the two-level tunable SO(2) holonomy. Supplementary Fig. 3d simulates the optical evolution in Phase 1, where light remains confined to waveguides A, B, and C throughout the process, while waveguide D is decoupled from the degenerate space. In contrast, in Phase 2 (Supplementary Fig. 3e), waveguide D actively participates in the evolution of the degenerate space and significantly influence the final results of the SO(2) transformation compared to Phase 1 (Supplementary Fig. 3d). More information about the numerical simulation methods is provided in Methods.

To demonstrate the wide-range tunability of the system, we further propose a reconfigurable braiding protocol in which the geometric phase can be modulated between π/2 and 0. As illustrated in Fig. 1e, the structure incorporates three stimulated Raman adiabatic passage (STIRAP)55–59 processes, with the configuration parameters set as κ_{DX} ≫ κ_{AX}, κ_{BX} and κ_{CX}, enabling a wide-range tunability. When waveguide D is in Phase 1 (refractive index mismatched), it becomes decoupled from the degenerate space, enabling two-mode braiding16,18,23. In this scenario, the simulation results of the fundamental building blocks—namely, the lateral and vertical STIRAP processes—are shown in Supplementary Fig. 2a and 2c, respectively. As expected, when the input light excites the zero mode, the output exhibits a complete power swap. In contrast, when waveguide D transitions to Phase 2, where the refractive indices across all waveguides are matched, the configuration (κ_{DX} ≫ κ_{AX}, κ_{BX}, and κ_{CX}) leads to a complete power swap, as displayed in Supplementary Fig. 2b and 2d. The corresponding simulation confirms that the light remains localized in the input waveguide throughout the evolution, indicating the absence of any nontrivial SO(2) geometric phase.

In practical device models, certain factors may affect the degeneracy, such as the next-nearest-neighbor coupling coefficients60. Supplementary Fig. 14b-c presents the simulated next-nearest-neighbor coupling coefficients, which are approximately an order of magnitude smaller than the nearest-neighbor couplings. Supplementary Fig. 14f-g compares the optical evolution of a holonomy with (dashed lines) and without (solid lines) next-nearest-neighbor coupling. The agreement between the two cases indicates that next-nearest-neighbor coupling has a minimal impact on the holonomy process. To further account for all potential factors that may affect the degeneracy, we performed full-vectorial electromagnetic eigenmode simulations of the five-waveguide system using the commercial software Ansys Lumerical. Supplementary Fig. 7a-b illustrates the calculated effective indices of the five eigenmodes as a function of waveguide gap distance for both amorphous and crystalline PCM states. Threefold and twofold degeneracy are well preserved across most gap distances, except for a slight splitting observed at small gap distances. Therefore, the designed holonomic path should avoid these narrow-gap regimes. More details are provided in Supplementary Note 2.

Figure 2c presents the experimental results of a two-level tunable SO(2) holonomy. The light input/output from waveguides B and C excites the state within twofold degenerate subspace, which undergoes an SO(2) transformation. The measured intensities correspond to the absolute square of the matrix elements of the SO(2) transformation. The gap distances and the corresponding coupling coefficients between waveguides throughout the evolution are detailed in Supplementary Fig. 4a. By switching the phase state of the Sb₂Se₃ waveguide, the holonomic path traverses distinct degenerate subspaces, enabling the realization of different SO(2) holonomies. The holonomic paths in parameter space, along with the enclosed solid angles of the two phase states, are shown in Supplementary Fig. 4b-c. The conformity between theoretical prediction and experimental results is evaluated as fidelity; the fidelity of Phase 1 is 0.9991, and the fidelity of Phase 2 is 0.9993. The fidelity is defined as F=(1/N)|Tr(U_t^† U_{exp})|61–64 based on the Frobenius inner product, Tr(A) represents the trace of A, N is the dimension of the unitary matrix, U_t, U_{exp} are the theoretical and experimental unitary matrices, respectively. The phase transition of the Sb₂Se₃ waveguide between its crystalline and amorphous states can be achieved via laser direct-writing techniques, as the Sb₂Se₃ waveguide is located on the top layer of the device.

Figure 2d shows the experimental implementation of tunable two-mode braiding. The phase state of the Sb₂Se₃ waveguide determines whether the transformation corresponds to a ‘swap’ or ‘non-swap’ operation in the resulting matrix. The gap distances, coupling coefficients, and holonomic paths can be found in Supplementary Fig. 6. To demonstrate multi-level discrete reconfigurability, Fig. 2e illustrates the cascading of two independently tunable SO(2) building blocks. The detailed information is shown in Supplementary Fig. 3a–c. By adjusting the phase states of each unit, a composite transformation with multi-level tunability is achieved. As a proof of concept, we demonstrate three discrete geometric phase levels by cascading two identical building blocks. By employing non-identical units, the number of achievable levels can be expanded. In general, cascading n distinct units enables 2^n (n bits) discrete phase levels. Here, we propose a scalable scheme for generating equidistant geometric phases, as shown in Supplementary Fig. 11.

For the modulation of the phase states of Sb₂Se₃, Fig. 2a displays the microscope images of the reversible switching of the Sb₂Se₃ thin film, as well as the laser direct writing setup. We started with a crystalline Sb₂Se₃ film sandwiched between μm-thick SiO₂ layers on the Si substrate, and the crystallization state was obtained by thermally annealing the as-deposited amorphous film. A rectangular area was amorphized using 408-nm laser pulses from a commercial laser writing system. The grain texture of the pristine substrate transitioned to a smooth morphology in the written region, which suggested that the Sb₂Se₃ was laser-induced melted and rapidly quenched into a disordered glassy state. Simultaneously, the refractive index change of Sb₂Se₃ caused a distinct color change in the written area. The amorphization process was confirmed by the Raman spectra in Fig. 2b. The black curve shows the Raman spectrum of the crystalline Sb₂Se₃ thin film, where three characteristic peaks at 190 cm⁻¹ (A_g mode of Sb–Se bonds), 150 cm⁻¹ (B_g mode of Sb–Sb bonds), and 120 cm⁻¹ (A_g mode of Se–Se bonds)46,65–67 could be observed. The blue curve shows the Raman spectrum of the laser-switched area, where a broad peak centered around 190 cm⁻¹ stands for the amorphous feature65. Amorphous patterns on the crystalline substrate can be erased via annealing or laser pulses, which induce the recrystallization of amorphous Sb₂Se₃. During annealing, grains tend to start growing from the boundary between the amorphous and crystalline regions. Recrystallization of amorphous Sb₂Se₃ can also be achieved using our home-built 639-nm laser direct-writing system, where the laser scanning direction can form larger grains and guide the grain growth orientation to some extent. This creates granular contrasts between the recrystallized region and the original crystalline substrate, as seen in Fig. 2a. The Raman spectroscopy confirmed the laser-induced crystallization of an amorphous Sb₂Se₃ film, as indicated by the orange curve in Fig. 2b (the Raman intensity has been normalized), and the micro-area X-ray Diffraction (Micro-XRD) pattern further confirmed the lattice structure (as shown in Supplementary Fig. 12). More details are provided in Methods. Furthermore, to verify that our tuning method can achieve complete phase transition across the full thickness of the 100 nm-thick PCM layer, we conducted transmission electron microscopy (TEM, Talos F200X G2, Thermo Fisher Scientific) tests, which are discussed in Supplementary Note 3 and illustrated in Supplementary Fig. 13. The TEM and FFT analyses indicate that the 100 nm thin film exhibits no significant inhomogeneity or morphological differences between the upper and lower sections after laser-controlled phase transition switching. This suggests that the phase switching occurs throughout the entire thickness direction.

Tunable high-dimensional holonomy
To demonstrate the scalability of the architecture, a high-dimensional reconfigurable mesh is presented in Fig. 3. A 3-bit (8-level) tunable SO(3) transformation is implemented using the framework of Givens rotations. By selectively switching the phase state of each Sb₂Se₃ waveguide—either through laser direct-writing to induce the amorphous state or erasing to restore the crystalline state—we systematically measure all eight levels of the SO(3) holonomy. Figure 3a–c presents microscope images of laser direct writing on Sb₂Se₃ waveguides. Distinct textural differences are evident in the Sb₂Se₃ waveguides when comparing the crystalline and amorphous phases (e.g., Fig. 3a). Besides, due to the wider laser-written patterns compared to the Sb₂Se₃ waveguide width, the color contrast between the crystalline and amorphous regions can be clearly observed in the surrounding film areas. The corresponding mathematical models, experimentally measured matrix elements, and transformation fidelity for the 8-level SO(3) holonomy are also summarized in Fig. 3d–k. The presented magnitude corresponds to the absolute value of the matrix elements of the SO(3) transformation. The relatively high fidelity (>0.96) demonstrates the excellent repeatability of the cyclic erasing and writing process.

Figure 4 presents the experimental realization of reconfigurable three-mode braiding. Building upon the tunable two-mode braiding demonstrated in Fig. 2d, high-dimensional braiding can be constructed by assembling a scalable mesh through parallel and cascaded integration of the basic two-mode braiding unit. We experimentally demonstrate four representative three-mode braiding scenarios through three braiding devices, with corresponding theoretical models and measured intensity distributions shown in the figures. This approach offers a promising route toward large-scale, programmable optical switching and topological photonic circuits.

Methods
Fabrication process
The fabrication process of the device is compatible with CMOS technology. The devices were fabricated on a 500-μm-thick silicon substrate wafer with a 3-μm-thick buried oxide layer. For the first layer, a 100-nm-thick amorphous silicon thin film was deposited by plasma-enhanced chemical vapor deposition (PECVD, Oxford Instruments). The PECVD gas included SiH_4 and Ar. Then, photoresist (AR-P 6200.09) was spun at a speed of 4000 r/min to have a thickness of 200 nm on the chip, and baked at a temperature of 180 °C for 2 min. The designed patterns were defined by electronic beam lithography (EBL, Vistec EBPG-5200), and the developing used methyl isobutyl ketone (MIBK, developing for 75 s) and isopropanol (IPA, developing for 60 s). Then it was fully etched through an inductively coupled plasma deep reactive ion etching system for silicon (ICP Deep RIE, SPTS Technologies). After that, the chip was coated with silica as the inter-layer dielectric using PECVD, and the chemical–mechanical polishing (CMP) was employed to planarize the silica. For the second layer, similar processes were repeated as the first layer, such as depositing a 100-nm-thick amorphous silicon film, defining patterns through EBL, fully ICP Deep RIE, PECVD, etc. For the patterning of the third-layer Sb₂Se₃ waveguides, we first deposited a 100-nm-thick Sb₂Se₃ film using magnetron sputtering (Denton). Subsequently, a 200-nm-thick silica protective layer was deposited using inductively coupled plasma chemical vapor deposition (ICPCVD, Oxford Instruments). Next, EBL and development were performed to define the waveguide pattern. This was followed by ICP etching of the silica protective layer, and finally, the Sb₂Se₃ film was etched by ICP reactive ion etching (Sentech) using a Cl₂/Ar gas mixture. The scanning electron microscope (SEM) figures and focused ion beam (FIB) images of the fabricated Sb₂Se₃ waveguides are provided in Fig. 1c,d. Finally, a 1-μm-thick silica cladding was deposited through ICPCVD. The fabrication process is summarized in Supplementary Fig. 1.

Measurement setup
The light is emitted from a wideband source (Wuhan Leicheng Photonics Technology Co., Ltd.), which covers a wavelength range of 1150–1680 nm. Then it propagates through a polarization controller and is coupled in/out to the integrated photonic chip. Edge couplers are utilized to interface the light between a single-mode fiber and a silicon waveguide. The light output is monitored by a power meter (JW3216, Shanghai Joinwit), and the spectra are analyzed through OSA (AQ6370C, YOKOGAWA).

Phase transition control of Sb₂Se₃
The transition from the crystalline to the amorphous phase is achieved via laser direct writing using a DWL 66+ system (Heidelberg Instruments) with a nominal laser power of 15 mW and a writing speed of 3 mm²/min. The transition from amorphous state to crystalline state can be realized through annealing (whole chip erasing, 200 °C, 5 min) or through laser-diode-writting (locally erasing) at a wavelength of 639 nm. Figure 2a illustrates the setup. Laser-diode-induced crystallization can be achieved using two distinct modes: (1) a continuous scanning mode with a laser power of 40 mW, a scanning speed of 0.01 mm/s, and an energy density of ~80 μJ/μm² on the sample surface; and (2) a single-pulse mode with a laser power of 6.95 mW, a pulse duration of 300 ms, and the same energy density of ~89 μJ/μm². To calibrate the time and energy needed for laser-controlled amorphization, amorphization was also demonstrated on this system using a single pulse with a power of 44 mW, a duration of 200 ns, and an energy density of ~5.7 nJ/μm².

Numerical simulation
We employ multiple simulation approaches for cross-validation. For the numerical calculation, one method involves computing the Wilczek–Zee connection integral using MATLAB. The other method is based on solving the Schrödinger-like equation i∂|φ(z)〉/∂z = Ĥ(z)|φ(z)〉 via the transfer matrix method, also implemented in MATLAB. These two methods fit well with each other. For the actual device simulation, a 3D finite-difference time-domain (3D-FDTD) full-wave simulation for the electromagnetic field is utilized through commercial software Lumerical. The coupling coefficients between two identical waveguides are calculated through Mode solutions that κ = (n_s - n_a)/2 75, where n_s, n_a are effective index of symmetric and anti-symmetric supermodes, respectively. The results from the numerical calculation and actual device simulation are also in close agreement.

Data availability
The source data generated in this study are available in the Figshare repository https://doi.org/10.6084/m9.figshare.30876017. Source data are provided with this paper.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The control of holonomy has recently emerged as a highly active topic, attracting broad interest across various physical disciplines. In this context, the manuscript entitled “In-memory multilevel control of generic SO(m) holonomy in photonics” by Chen et al. presents a novel approach to implementing nonvolatile and tunable holonomic control by manipulating degenerate photonic states. Specifically, the authors cleverly introduce a five-waveguide architecture incorporating a phase-change material (PCM) waveguide, which enables the realization of two- and three-fold degeneracies. This in turn allows for the demonstration of fundamental SO(2) holonomic transformations, as well as the prospect of extending to in-memory, multilevel control of higher-dimensional SO(m) holonomies through cascaded designs. The proposed scheme is original and timely, offering a potentially significant contribution to both the fundamental study of non-Abelian photonics and the development of reconfigurable photonic devices. Since this work represents a big advance that is likely to stimulate further research in this area, I would support the publication of the manuscript after the authors address my following comments.\n1. Please clarify whether the targeted degeneracies are well preserved in both the amorphous and crystalline states of the five-waveguide system. Since the presence of multiple coupled waveguides can sometimes introduce unwanted perturbations that affect the robustness of degeneracy (see, for example, Phys. Rev. A 105, 013507 (2021)), a more detailed analysis would strengthen the work. In particular, simulations of the effective indices of the five modes as a function of gap distance in the actual device model could help demonstrate that the two- or three-fold degeneracies remain well preserved across both material states. Besides, the detailed parameters of reconfigurable two-mode braiding, including the evolution of gap distances, coupling coefficients, and holonomic path, should be provided.\n2. Regarding fabrication, Supplementary Fig. 1 and the Methods section do not mention any planarization steps. Since multilayer structures, particularly those involving braiding, typically require good planarity, it would be useful to clarify whether a planarization process (e.g., chemical–mechanical polishing) was employed. For the PCM integration, the manuscript presents SEM images of Sb₂Se₃ waveguides, but not of the silicon waveguides used in the holonomy structures. Providing SEM figures of the silicon waveguides would improve the completeness of the work. In addition, while PCM layers in photonic devices are often on the order of tens of nanometers, the PCM waveguide here is significantly thicker. Given the limited penetration depth of visible laser light in PCM, this could lead to non-uniform phase transitions and refractive index gradients along the thickness direction after laser writing. A discussion on how such effects may influence the reversibility of laser switching and the stability of reconfigurable geometric phases would further strengthen the work.\n3. For the in-memory reconfiguration of SO(m) holonomy, two laser direct writing systems operating at 408 nm and 639 nm were employed for amorphization and crystallization of the PCM, respectively. Why are two distinct laser systems used? Also, the parameters of the specific control pulse for the phase switching process should be provided in detail. Further, the laser-triggered phase transition in Sb2Se3 is characterized by grain morphology under microscopy and Raman spectroscopy. While X-ray diffraction analysis would offer more direct observation of the lattice structure changes through diffraction peaks, I would suggest the authors to add XRD results as Supplementary data.\n4. The demonstration of two-level SO(2) holonomy relies on the binary switching between the amorphous and crystalline states of the PCM. It would be interesting to discuss how the system behaves when the PCM is in intermediate states (see, e.g., Nat. Commun. 15, 2786 (2024)), as this may introduce additional levels of control. Furthermore, the current design assumes a uniform phase change across the entire waveguide. Considering the case where only a portion of the waveguide undergoes a phase transition may open the possibility of generating multiple geometric-phase levels within a single device. A discussion along these lines would strengthen the manuscript by highlighting the broader tunability and versatility of the proposed approach.\n5. For the multilevel control of SO(2) holonomy, the authors demonstrate three discrete levels by cascading two identical building blocks. While this provides a useful proof of concept, the scheme could potentially be extended further for a more insightful discussion. In particular, cascading non-identical building blocks may enable a richer set of discrete levels and more versatile control.\nSome small remarks for the authors’ consideration:\n1. Some typos should be double-checked carefully, i.e. in the abstract the word \"reconfiguable\" should be “reconfigurable”.\n2. In Fig. 1b, the annotation “Two degenerate subspace” is recommended to be corrected as “Two degenerate states” or “Two-dimensional degenerate subspace”. Similar descriptions, such as “Three degenerate subspace” in the annotation of Fig. 1b and “the number of degenerate spaces” in the caption of Fig. 1, are also recommended to be revised.\n3. In the sentence “This creates slight color contrasts between the recrystallized region and the original crystalline substrate, as seen in Supplementary Fig. 2a,” it appears that the referenced image should be Fig. 2a in the main text rather than the supplementary material. Please verify and correct this reference for clarity and consistency.\n4. The unit of the scale bar in Fig. 1d should be provided.\n5. The symbol “μ” in Supplementary Eq. (S3) should be well defined.", "reviewer_2": "This work demonstrates in-memory multilevel control of generic SO(m) holonomy using a multilayer silicon photonic platform integrated with the phase-change material Sb₂Se₃. By optically switching the Sb₂Se₃ waveguides between amorphous and crystalline states, the authors dynamically modulate the number of degenerate modes in the system, enabling reconfigurable non-Abelian geometric phases without volatile external controls. They experimentally achieve high-fidelity, nonvolatile SO(2) and SO(3) transformations, along with tunable two- and three-mode braiding operations, and showcase multi-level programmability through cascaded SO(2) units and Givens rotations. This approach merges the inherent robustness of geometric phases with the nonvolatile tunability of PCMs, offering a promising pathway toward fault-tolerant photonic quantum processing, reconfigurable optical switching networks, and high-dimensional classical neuromorphic computing.\nIn my opinions, this work demonstrates the first experimental realization of reconfigurable control over non-Abelian holonomy, which exhibits notable innovation. However, several issues should be addressed prior to publication:\n1. First and foremost, regarding novelty, this work combines the concepts of \"reconfigurability\" and \"non-Abelian holonomy,\" which brings to mind two related works also published in Nature Communications: Nat. Commun. 16, 7089 (2025) and Nat. Commun. 16, 3650 (2025). Compared to the former, this study extends braiding operations to more general holonomic scenarios and introduces a new refractive index tuning method; compared to the latter, it achieves dynamic control over the SO(m) group. The authors should provide more detailed and specific comparisons with these relevant works, and include necessary discussions in the manuscript to better highlight the originality and significance of their contribution.\n2. As a follow-up to the point above, different to Nat. Commun. 16, 7089 (2025), which employs the thermo-optic effect for active tuning, this work utilizes the phase-change material, claiming advantages in lower energy consumption and improved stability. The authors are encouraged to provide a detailed comparison of the advantages and disadvantages of different active tuning methods to underscore the technical and systemic innovation of their approach.\n3. While the active tuning strategy presented is appealing, it remains limited to switching between SO(2) and SO(3) based on the presence or absence of onsite index mismatch. It would be valuable if the authors could discuss whether their method can be extended to design more complex or interesting functionalities.\n4. The caption of Figure 2e is overly brief. A more detailed explanation of how the SO(2) transformation is realized in parameter space should be provided to assist readers unfamiliar with the process of non-Abelian holonomy.\n5. The multilayer quasi-3D waveguide platform constructed using deposited SiO₂ is a promising architecture. However, a valid concern is whether next-nearest-neighbor coupling coefficients (such as κCD) can be neglected in such multilayer waveguides, and whether they might disrupt the holonomy process. Supplementary data on coupling coefficients should be provided in the Supplementary Information of this work to facilitate reader understanding.\n6. Several minor issues require attention: the parameter space vectors on lines 91 and 136 contain a square symbol on the left-hand side of the equation, which may be a display error. The matrix alignment in Eq. (1) is also irregular. The authors should correct these and carefully check for other similar issues throughout the text."}, "author_response": "Dear reviewers,\nWe are very grateful for your insightful comments and for your time and efforts in assessing our work. In this response letter, we provide a detailed revision checklist and point-by-point response to your comments. We express our sincere gratitude once again for your further efforts in assessing our revised manuscript.\nSincerely yours,\nXuhan Guo on behalf of all authors\nRevision checklist\nReviewer #1\nComments Main manuscript revision Supplementary Information\nrevision\n(Page 7, lines 178-185): (Supplementary Note 2):\nSimulate the effective indices of Add a new section to analyze\nthe five modes to demonstrate that the degeneracy of the\nthe three-fold and two-fold five-waveguide system.\ndegeneracy are well preserved.\n(Supplementary Fig. 7):\n(Page 9, lines 230-231): Add the simulated effective\nAdd the parameters of the indices of five modes to\nreconfigurable two-mode braiding. demonstrate that the degeneracy\n1 is well preserved.\n(Supplementary Fig. 6):\nAdd the detailed parameters\nof the reconfigurable two-mode\nbraiding in Supplementary Fig.\n6.\n(Page 15, lines 389-390): (Supplementary Fig. 1):\nAdd the chemical-mechanical Add the CMP step into the\npolishing (CMP) process to the fabrication process flow\nfabrication process in Methods. diagram.\n(Page 5, lines 119-120): (Supplementary Figs. 8-10):\nAdd the SEM figures of the Add the SEM figures of the\nsilicon waveguides. silicon waveguides.\n(Page 10, lines 264-271): (Supplementary Note 3):\nAdd a discussion on the Added a discussion on the\n2\ncomplete phase switching through verification of complete phase\nTEM analysis. transition through TEM analysis.\n(Page 14, lines 339-343): (Supplementary Fig. 13):\nAdd a discussion on the effects Added the TEM analysis to\nof subtle deviations in PCM Supplementary Fig. 13.\nswitching.\n(Pages 15-16, lines 412-423):\nAdd the reason for using two\nlaser systems, and provide the\nparameters of laser pulses.\n3 (Page 10, lines 260-264): (Supplementary Fig. 12):\nAdded a discussion of the XRD Added the XRD results.\nresults.\n(Page 14, lines 339-343): (Supplementary Fig. 15):\nAdded a discussion on the Add the behavior of the\nintermediate states. system at intermediate states.\n(Page 14, lines 358-365):\nscenario where only a portion of\nthe waveguides undergoes phase\ntransition.\n(Page 9-10, lines 235-239): (Supplementary Fig. 11) :\nDiscuss the case of cascading Added the scheme of\nnon-identical building blocks for cascading non-identical building\n5\ngenerating more geometric phase blocks for generating more\nlevels. geometric phase levels.\n1. (Page 1, lines 23-26): Correct 5. (Supplementary page 2, lines\nthe spelling of “reconfigurable”. 55-57): Add the definition of . \n2. (Page 4, Fig. 1b, e): Correct to 𝜇\n“Two degenerate states” and\n“Three degenerate states”;\n6 (Page 5, lines 97-99): Correct to\n(Small “the number of degenerate states”\nremarks)\n3. (Page 10, lines 259-260):\nCorrect the referenced image to\nFig. 2a.\n4. (Page 4, Fig. 1d): Add the unit\n“nm”.\nReviewer #2\nComments Main manuscript revision Supplementary Information\nrevision\n(Page 2, lines 42-50):\nAdd the detailed comparisons\nwith [Nat. Commun. 16, 7089\n1 (2025)] and [Nat. Commun. 16,\n3650 (2025)].\n(Page 2, lines 66-69):\nAdd a detailed comparison of\nthe different tuning methods.\n2\n(Page 14, lines 366-371):\nAdd a discussion on new\nfunctionalities beyond switching\n3\nbetween SO(2) and SO(3).\n(Page 9, lines 200-209): (Supplementary Fig. 16):\nAdd a detailed caption for Add a detailed device structure\nin parameter space. parameter space.\n(Page 7, lines 172-178): (Supplementary Fig. 14):\nAdd the discussion on the Add the data on\ninfluence of next-nearest-neighbor next-nearest-neighbor coupling\n5\ncoupling coefficients. coefficients, and simulate the\nevolution of a holonomy.\n(Page 5, lines 99-102): Correct the\ndisplay error of κ.\n6 (Page 6, lines 145-148): Correct\n(Small the display error of κ.\nremarks)\n(Page 5, Eq. (1)): Correct the\nmatrix alignment.\nReply to Reviewer 1\nComments from Reviewer 1:\nReviewer #1 (Remarks to the Author):\nThe control of holonomy has recently emerged as a highly active topic, attracting\nbroad interest across various physical disciplines. In this context, the manuscript\nentitled “In-memory multilevel control of generic SO(m) holonomy in photonics” by\nChen et al. presents a novel approach to implementing nonvolatile and tunable\nholonomic control by manipulating degenerate photonic states. Specifically, the\nauthors cleverly introduce a five-waveguide architecture incorporating a phase-change\nmaterial (PCM) waveguide, which enables the realization of two- and three-fold\ndegeneracies. This in turn allows for the demonstration of fundamental SO(2)\nholonomic transformations, as well as the prospect of extending to in-memory,\nmultilevel control of higher-dimensional SO(m) holonomies through cascaded designs.\nThe proposed scheme is original and timely, offering a potentially significant\ncontribution to both the fundamental study of non-Abelian photonics and the\ndevelopment of reconfigurable photonic devices. Since this work represents a big\nadvance that is likely to stimulate further research in this area, I would support the\npublication of the manuscript after the authors address my following comments.\nResponse:\nWe sincerely thank the reviewer for the careful work and positive comments.\nThese instructive comments have significantly improved the quality of our manuscript.\nWe have carefully addressed all the raised points and revised the manuscript\naccordingly.\nComments from Reviewer 1:\n1.Please clarify whether the targeted degeneracies are well preserved in both the\namorphous and crystalline states of the five-waveguide system. Since the presence of\nmultiple coupled waveguides can sometimes introduce unwanted perturbations that\naffect the robustness of degeneracy (see, for example, Phys. Rev. A 105, 013507\n(2021)), a more detailed analysis would strengthen the work. In particular, simulations\nof the effective indices of the five modes as a function of gap distance in the actual\ndevice model could help demonstrate that the two- or three-fold degeneracies remain\nwell preserved across both material states.\nResponse:\nWe sincerely thank the reviewer for the instructive comments on the degeneracy.\nIndeed, in practical device models, certain factors may affect degeneracy, such as\nself-coupling and next-nearest-neighbor coupling effects, which have been thoroughly\ndiscussed in the literature referenced by the reviewer [Phys. Rev. A 105, 013507\n(2022)]. To fully consider all these factors, we perform finite-difference eigenmode\n(FDE) analysis of the electromagnetic eigenmodes in the multi-waveguide system\nusing the commercial software Ansys Lumerical. Fig. R1c-d shows the simulated\neffective indices of the five eigenmodes in the five-waveguide system with different\ngap distances. The system exhibits three-fold degeneracy when the phase-change\nmaterial waveguide is in the amorphous state and two-fold degeneracy in the\ncrystalline state. In most cases, the degeneracy is well preserved, except for a slight\nsplitting of the effective refractive indices of the degenerate modes observed at small\ngap distances. Therefore, the design of the holonomic path could bypass regions of\nsignificant degeneracy splitting. Fig. R1e-f shows the splitting of the effective\nrefractive indices of the degenerate modes with different gap distances, where δ is\ndefined as | - | for the three-fold degeneracy and | - | for the two-fold\ndegeneracy,\n𝑛\nre𝐷s1pe\n𝑛\nc𝐷ti3vely. In addition to designing the holon\n𝑛\no𝐷m1i\n𝑛\nc 𝐷p2ath, optimizing the\ndevice's propagation length also contributes to high-fidelity transformations ([Phys.\nRev. A 105, 013507 (2022)], [Nat. Commun. 16, 3650 (2025)]). By appropriately\nshortening the device length in regions with severe degeneracy splitting—while\nmaintaining adiabaticity—the adverse effects of imperfect degeneracy can be\nmitigated, thereby enabling high-fidelity transformations.\nFig. R1 | Degeneracy analysis of the five-waveguide system. a-b, The structure of\nthe five-waveguide system. a, The phase-change material waveguide D is in its\namorphous state. b, The phase-change material waveguide D is in its crystalline state.\nc-d, Simulated effective indices of the five eigenmodes with different gap distances\nfor the structure shown in (a-b). c, The phase-change material waveguide is in its\namorphous state, and the system supports three-fold degeneracy. d, The phase-change\nmaterial waveguide is in its crystalline state, and the system supports two-fold\ndegeneracy. e-f, Splitting of the effective indices of the degenerate modes, and the\ndesign of the holonomic path should bypass regions of significant degeneracy\nsplitting.\nRevision:\nMain text:\n(Page 7, lines 178-185) To further account for all potential factors that may affect the\ndegeneracy, we performed full-vectorial electromagnetic eigenmode simulations of\nthe five-waveguide system using the commercial software Ansys Lumerical.\nSupplementary Fig. 7a-b illustrates the calculated effective indices of the five\neigenmodes as a function of waveguide gap distance for both amorphous and\ncrystalline PCM states. Three-fold and two-fold degeneracy are well preserved across\nmost gap distances, except for a slight splitting observed at small gap distances.\nTherefore, the designed holonomic path should avoid these narrow-gap regimes. More\ndetails are provided in Supplementary Note 2.\nSupplementary:\n(Supplementary Note 2) We have added this part of the degeneracy analysis to\nSupplementary Note 2.\n(Supplementary Fig. 7) We have added the content of Fig. R1 to Supplementary Fig.\n7.\nComments from Reviewer 1:\nBesides, the detailed parameters of reconfigurable two-mode braiding, including the\nevolution of gap distances, coupling coefficients, and holonomic path, should be\nprovided.\nResponse:\nWe sincerely thank the reviewer for the instructive comments. We now provide\ndetailed parameters of reconfigurable two-mode braiding. Fig. R2a shows the\nevolution of gap distances and coupling coefficients. Among the four coupling\ncoefficients, κ remains large throughout the entire evolution, satisfying the\nDX\ncondition κ κ , κ , and κ . The evolution of the other three coupling\nDX AX BX CX\ncoefficients can be divided into the following three stages: At Step 1, waveguide B is\n≫\nisolated, maintaining a large separation from the other waveguides. The gap between\nwaveguide X and C gradually narrows, while the separation between waveguide X\nand A increases. Consequently, κ increases, κ decreases, and κ remains\nCX AX BX\nnegligible. At Step 2, waveguide A is isolated, with negligible κ . The gap between\nAX\nwaveguides X and B gradually decreases, enhancing their coupling strength (κ ),\nBX\nwhile the separation between X and C increases, leading to a reduction in κ . At Step\nCX\n3, waveguide C is decoupled from other waveguides, waveguide X moves to A from\nB, leading to an increase in κ while a decrease in κ .\nAX BX\nFig. R2b shows the holonomic paths of two states. When waveguide D is in\nPhase 1 (refractive index mismatched). The path in parameter space follows the\ngeodesic arc along the boundary of the spherical first octant, leading to a standard\ntwo-mode braiding. In contrast, when waveguide D transitions to Phase 2 (refractive\nindex matched), the configuration (κ κ , κ , and κ ) leads to a collapse of the\nDX AX BX CX\ngeometric phase to zero.\n≫\nFig. R2 | Detailed parameters of reconfigurable two-mode braiding. a, The\nevolution of gap distances and coupling coefficients at three steps: At Step 1,\nwaveguide B is isolated. The gap between waveguide X and C gradually narrows,\nwhile the separation between waveguide X and A increases. Consequently, κ\nCX\nincreases, κ decreases, and κ remains negligible. At Step 2, waveguide A is\nAX BX\nisolated, with negligible κ . The gap between waveguides X and B gradually\nAX\ndecreases, enhancing their coupling strength (κ ), while the separation between X\nBX\nand C increases, leading to a reduction in κ . At Step 3, waveguide C is decoupled\nCX\nfrom other waveguides, waveguide X moves to A from B, leading to an increase in\nκ while a decrease in κ . b, The holonomic paths of two states. When waveguide\nAX BX\nD is in Phase 1 (refractive index mismatched). The path in parameter space follows\nthe geodesic arc along the boundary of the spherical first octant, leading to a standard\ntwo-mode braiding. In contrast, when waveguide D transitions to Phase 2 (refractive\nindex matched), the configuration (κ κ , κ , and κ ) leads to a collapse of the\nDX AX BX CX\ngeometric phase to zero.\n≫\nRevision:\nMain text:\n(Page 9, lines 230-231) The gap distances, coupling coefficients, and holonomic path\ncan be found in Supplementary Fig. 6.\nSupplementary:\n(Supplementary Fig. 6) We provide the detailed parameters of reconfigurable\ntwo-mode braiding in Figs. R2 toSupplementary Fig. 6.\nComments from Reviewer 1:\n2.Regarding fabrication, Supplementary Fig. 1 and the Methods section do not\nmention any planarization steps. Since multilayer structures, particularly those\ninvolving braiding, typically require good planarity, it would be useful to clarify\nwhether a planarization process (e.g., chemical–mechanical polishing) was employed.\nResponse:\nWe sincerely thank the reviewer for the detailed comments on the planarization\nprocess. Chemical–mechanical polishing (CMP) is necessary for the fabrication of\nbraiding. As Fig. R3 shows, we employed CMP to planarize the silica layer on the\nsilicon waveguide of the first layer. After planarization, the second-layer waveguide\ncan be fabricated with high quality, free from distortion or deformation caused by the\ntopography of the silica layer.\nFig. R4 shows the FIB (focused ion beam) images of the cross-section of three\nelements in tunable two-mode braiding. The images clearly show that the silica layer\non the first-layer silicon waveguide is highly planarized after the CMP process,\nenabling high-quality fabrication of the second-layer silicon waveguides with intact\nmorphology without distortion.\nFig. R3 | CMP technique for planarizing the silica layer. Due to the presence of the\nfirst-layer silicon waveguides, the surface of the silica deposited by PECVD exhibits\ntopographical variations. We employed CMP to polish the silica layer, achieving a flat\nand planar surface.\nFig. R4 | FIB images of three elements in tunable two-mode braiding. The silica\nlayer on the first-layer silicon waveguide is highly planarized after the CMP process,\nenabling high-quality fabrication of the second-layer silicon waveguides.\nRevision:\nMain text:\n(Page 15, lines 388-389) After that, the chip was coated with silica as the inter-layer\ndielectric using PECVD, and the chemical-mechanical polishing (CMP) was\nemployed to planarize the silica.\n(Page 4, Fig. 1e) We add a FIB image of two-mode braiding shown in Fig. R4 to Fig.\n1e, which demonstrates the good planarity of two-mode braiding.\nSupplementary:\n(Supplementary Fig. 1) The CMP step has been incorporated into the modified\nprocess flow diagram inSupplementary Fig. 1.\nComments from Reviewer 1:\nFor the PCM integration, the manuscript presents SEM images of Sb₂Se₃ waveguides,\nbut not of the silicon waveguides used in the holonomy structures. Providing SEM\nfigures of the silicon waveguides would improve the completeness of the work.\nResponse:\nWe would like to thank the reviewer for the instructive comments on the SEM\nfigures. Here, we will provide detailed SEM figures of the silicon waveguides on\nlayer 1 and layer 2. Fig. R5 shows the SEM figures of two-layer silicon waveguides\nfor the tunable two-mode braiding, where the dashed boxes represent the imaging\nposition. Fig. R6 shows the SEM figures of silicon waveguides for the tunable\nSO(2) holonomy. Besides, Fig. R7 provides the silicon waveguide SEM figures of tunable\nSO(3) holonomy. We now provide these figures in the Supplementary Information.\nFig. R5 | SEM figures of two-layer silicon waveguides for the tunable two-mode\nbraiding. The dashed boxes represent the imaging position.\nFig. R6 | SEM figures of two-layer silicon waveguides for the tunable SO(2)\nholonomy. The dashed boxes represent the imaging position.\nFig. R7 | SEM figures of two-layer silicon waveguides for the tunable SO(3)\nholonomy. The dashed boxes represent the imaging position.\nRevision:\nMain text:\n(Page 5, lines 119-120) For the two-layer silicon waveguides, the scanning electron\nmicroscopy (SEM) images are provided in the Supplementary Figs. 8-10.\nSupplementary:\n(Supplementary Figs. 8-10) We provide detailed SEM figures in Figs. R5-R7 to\nSupplementary Figs. 8-10.\nComments from Reviewer 1:\nIn addition, while PCM layers in photonic devices are often on the order of tens of\nnanometers, the PCM waveguide here is significantly thicker. Given the limited\npenetration depth of visible laser light in PCM, this could lead to non-uniform phase\ntransitions and refractive index gradients along the thickness direction after laser\nwriting. A discussion on how such effects may influence the reversibility of laser\nswitching and the stability of reconfigurable geometric phases would further\nstrengthen the work.\nResponse:\nWe would like to thank the reviewer for this instructive question. Indeed, in\nconventional PCM-integrated photonic devices, a thin PCM layer (typically tens of\nnanometers) is usually covered on top of the photonic waveguide to form a hybrid\nstructure, interacting with light via the evanescent field, resulting in relatively weak\nrefractive index modulation. In contrast, our work directly uses PCMs as the\nwaveguide material itself, enabling significantly stronger modulation of the effective\nrefractive index. This necessitates a relatively thicker PCM layer. To ensure effective\nrefractive index matching between the Si waveguide and the PCM waveguide, our\nPCM waveguide is designed with a thickness of 100 nm. Low-loss PCMs with slow\ncrystallization kinetics typically exhibit relatively good glass-forming ability, which\nallows them to achieve reversible switching at larger thicknesses. Laser switching of\nPCMs is a complex process involving light absorption, heat generation, and phase\ntransition. Therefore, to verify whether the cross-section of the laser-switched PCM\nexhibits amorphous/crystalline stratification or obvious nonuniformity related to the\npenetration depth of the laser, transmission electron microscopy (TEM, Talos F200X\nG2, Thermo Fisher Scientific) tests were conducted. The Sb₂Se₃ film is sandwiched\nbetween SiO₂ layers, and except for the absence of waveguide patterning, the\nstructure is identical to that of the device in the manuscript.\n1)Annealed crystalline state: Figure R8a presents the TEM image of crystalline\nSb Se (Cry-Sb Se ) obtained through annealing, where the long-range ordered\natomic arrangement is distinctly visible. A Fast Fourier Transform (FFT) analysis\n₂ ₃ ₂ ₃\nreveals the orthorhombic lattice structure (Fig. R8b), and multiple sets of reflexes are\nobserved due to the polycrystalline nature of the material with varying orientations. 2)\nLaser amorphization: The amorphization is conducted using the commercial laser\ndirect writing system, as mentioned in the manuscript. Fig. R8c and Fig. R8d show\nthe TEM images of the top and bottom areas of the amorphized Sb Se (am-Sb Se )\nthin film. The ordered atomic arrangement is no longer observed, and no obvious\n₂ ₃ ₂ ₃\ndifference appears in the top and bottom regions of the film. The FFT of Fig. R8d\nshown in Fig. R8e displays typical amorphous diffraction rings. 3) Laser\ncrystallization: Laser crystallization of an amorphous Sb Se thin film was achieved\nusing our home-built system. Fig. R8g and Fig. R8h present the TEM images of the\n₂ ₃\ntop and bottom regions of the laser-crystallized Sb₂Se₃, revealing lattice fringes with\nno significant differences observed between the upper and lower sections. The FFT of\nFig. R8h shown in Fig. R8f also indicates distinct crystallographic signals.\nFig. R8 | TEM analysis of the laser-controlled phase switching of Sb Se thin\n2 3\nfilms. TEM image (a) and local FFT (b) of annealed-crystalline Sb Se ;\n2 3\nTEM image\nof the top (c, g) and bottom (d, h) areas of the laser-amorphized (c, d) and\nlaser-crystallized (g, h) Sb Se thin films, respectively. e, The local FFT of (d)\n2 3\nconfirms laser-induced amorphization of the bottom part of the Sb Se ; f, the local\n2 3\nFFT of (h) confirms the laser-induced crystallization of the bottom part of the Sb Se\n2 3\nthin film.\nThe TEM and FFT analyses indicate that, under our device structure and using\nthe same laser parameters as in our manuscript, the 100 nm thin film exhibits no\nobvious inhomogeneity or morphological differences between the upper and lower\nsections after laser-controlled phase transition. This suggests that the phase switching\noccurs throughout the entire thickness direction. The distinct refractive index gradient\nin PCM primarily arises from the coexistence of amorphous and crystalline phases\nwith different volume fractions. As the material transitions toward either a fully\namorphous or fully crystalline state, the refractive index variation significantly\ndiminishes, in accordance with the Arrhenius law. Since our 100 nm-thick PCM is\nvery close to complete phase-transition, it can be considered free of notable\nnonuniformity or resulting gradient, and this does not impede its reversible switching.\nWhen certain lower energy levels are employed, it is indeed possible to create\nintermediate states with a mixed state of amorphous and crystalline phases. However,\nthe performance of the photonic devices primarily depends on the effective refractive\nindex of the entire waveguide. Even with minor variations in the refractive index\nconfiguration due to random crystallization, the non-Abelian geometric phase remains\ninherently unaffected due to its intrinsic tolerance to parameter fluctuations and\nrobustness against perturbations. This ensures stable and reliable performance despite\nminor variations, making such devices highly practical for advanced applications in\nphotonics and quantum technology.\nRevision:\nMain text:\n(Page 10, lines 264-271) Furthermore, to verify that our tuning method can achieve\ncomplete phase transition across the full thickness of the 100 nm-thick PCM layer, we\nconducted transmission electron microscopy (TEM, Talos F200X G2, Thermo Fisher\nScientific) tests, which are discussed in Supplementary Note 3 and illustrated in\nSupplementary Fig. 13. The TEM and FFT analyses indicate that the 100 nm thin film\nexhibits no significant inhomogeneity or morphological differences between the upper\nand lower sections after laser-controlled phase transition switching. This suggests that\nthe phase switching occurs throughout the entire thickness direction.\n(Page 14, lines 339-343) It is worth noting that the stochastic nucleation\ncharacteristics of PCMs usually lead to subtle deviations in conventional dynamic\nphase modulation devices, but these deviations are not an issue in reconfigurable\ngeometric phase devices. It is worth noting that the stochastic nucleation\ncharacteristics of PCMs usually lead to subtle deviations in conventional dynamic\nphase modulation devices, but these deviations are not an issue in reconfigurable\ngeometric phase devices.\nSupplementary:\n(Supplementary Note 3) We have added the discussion on the verification of complete\nphase transition via TEM analysis in Supplementary Note 3.\n(Supplementary Fig. 13) We have added the TEM analysis shown in Fig. R8 to\nSupplementary Fig. 13 in the revised manuscript.\nComments from Reviewer 1:\n3. For the in-memory reconfiguration of SO(m) holonomy, two laser direct writing\nsystems operating at 408 nm and 639 nm were employed for amorphization and\ncrystallization of the PCM, respectively. Why are two distinct laser systems used?\nAlso, the parameters of the specific control pulse for the phase switching process\nshould be provided in detail.\nResponse:\nWe thank the reviewer for the questions. We used a commercial laser direct\nwriting system (equipped with a 408-nm laser) to amorphize the PCM. The\nadvantages lie in its high-precision stage, convenient computer control system, and\nhigh throughput, which are highly beneficial for scalable networks like reconfigurable\nSO(m) holonomy. A limitation is that the commercial equipment in our institute does\nnot provide users with full flexibility to adjust parameters, which poses challenges in\nachieving crystallization. In contrast, our home-built laser switching system (using a\n639 nm wavelength laser) can operate in both continuous scanning mode and pulsed\ndot matrix mode, with flexible editing of pulse width, power, or pulse train, enabling\nboth amorphization and crystallization of Sb Se . Therefore, we employed the\n2 3\nhome-built laser switching system here to complement the functionalities of the\ncommercial system, and ultimately successfully achieved the reversible switching of\nSb Se .\n2 3\nThe amorphization of Sb Se was realized on the DWL 66+ system (Heidelberg\n2 3\nInstruments) by adopting a nominal laser power of 15 mW and a writing speed of 3\nmm2/min. Laser-diode-induced crystallization can be achieved using two distinct\nmodes: (1) a continuous scanning mode with a laser power of 40 mW, a scanning\nspeed of 0.01 mm/s, and an energy density of ~80 μJ/μm² on the sample surface; and\n(2) a single-pulse mode with a laser power of 6.95 mW, a pulse duration of 300 ms,\nand the same energy density of ~89 μJ/μm². To calibrate the time and energy needed\nfor laser-controlled amorphization, amorphization was also demonstrated on this\nsystem using a single pulse with a power of 44 mW, a duration of 200 ns, and an\nenergy density of ~5.7 nJ/μm². The laser switching parameters have been added to the\nmain text for clarity.\nRevision:\nMain text:\n (Page 16, lines 412-423) The transition from the crystalline to the amorphous phase is\nachieved via laser direct writing using a DWL 66+ system (Heidelberg Instruments)\nwith a nominal laser power of 15 mW and a writing speed of 3 mm2/min. The\ntransition from amorphous state to crystalline state can be realized through annealing\n(whole chip erasing, 200°C, 5 mins) or through laser-diode-writting (locally erasing)\nat a wavelength of 639 nm. Figure 2a illustrates the setup. Laser-diode-induced\ncrystallization can be achieved using two distinct modes: (1) a continuous scanning\nmode with a laser power of 40 mW, a scanning speed of 0.01 mm/s, and an energy\ndensity of ~80 μJ/μm² on the sample surface; and (2) a single-pulse mode with a laser\npower of 6.95 mW, a pulse duration of 300 ms, and the same energy density of ~89\nμJ/μm². To calibrate the time and energy needed for laser-controlled amorphization,\namorphization was also demonstrated on this system using a single pulse with a\npower of 44 mW, a duration of 200 ns, and an energy density of ~5.7 nJ/μm².\nComments from Reviewer 1:\nFurther, the laser-triggered phase transition in Sb Se is characterized by grain\n2 3\nmorphology under microscopy and Raman spectroscopy. While X-ray diffraction\nanalysis would offer more direct observation of the lattice structure changes through\ndiffraction peaks, I would suggest the authors to add XRD results as Supplementary\ndata.\nResponse:\nWe thank the reviewer for the suggestion. To confirm the lattice structure of the\nlaser-switched Sb Se , we performed micro-area X-ray diffraction (Micro-XRD)\n2 3\nmeasurement on the laser-switched region of an Sb Se thin film using a Bruker D8\n2 3\nDiscover instrument. To enhance the weak signal intensity and mitigate interference\nfrom surrounding areas, we employed the LynxEye 1D mode in Micro-XRD\nmeasurement with prolonged measurement time. As shown in Fig. R9, the\nlaser-crystallized Sb Se displays sharp diffraction peaks corresponding to specific\n2 3\nlattice planes, which are absent in the amorphous region. These results align with\nTEM and FFT analyses, confirming the laser-triggered phase transition in Sb Se . The\n2 3\nXRD results are added to the supplementary materials.\nFig. R9 | Micro-XRD patterns of the laser-switched Sb Se . The laser-crystallized\n2 3\nSb Se displays sharp diffraction peaks corresponding to specific lattice planes, which\n2 3\nare absent in the amorphous region.\nRevision:\nMain text:\n(Page 10, lines 260-264) The Raman spectroscopy confirmed the laser-induced\ncrystallization of an amorphous Sb₂Se₃ film, as indicated by the orange curve in Fig.\n2b (the Raman intensity has been normalized), and the micro-area X-ray Diffraction\n(Micro-XRD) pattern further confirmed the lattice structure (as shown in\nSupplementary Fig. 12).\nSupplementary:\n(Supplementary Fig. 12) We have added the micro-XRD patterns shown in Fig. R9 to\nSupplementary Fig. 12 in the revised manuscript.\nComments from Reviewer 1:\n4. The demonstration of two-level SO(2) holonomy relies on the binary switching\nbetween the amorphous and crystalline states of the PCM. It would be interesting to\ndiscuss how the system behaves when the PCM is in intermediate states (see, e.g., Nat.\nCommun. 15, 2786 (2024)), as this may introduce additional levels of control.\nResponse:\nWe are grateful to the reviewer’s insightful comments regarding the behaviors of\nintermediate states. Since the fully amorphous state is designated to achieve the onsite\nindex matching, any deviation into an intermediate state introduces an index\nmismatch. Most such states behave like the fully crystalline phase due to strong\nmismatch, but a few with weak mismatch cause population leakage into\nnon-degenerate subspaces, resulting in non-unitary dynamics and the absence of a\nwell-defined geometric phase. We will use reconfigurable braiding as an example to\ndemonstrate this point.\nThe intermediate states are composed of mixtures of amorphous and crystalline\nphases in varying ratios. Fig. R10a shows the evolution results of reconfigurable\nbraiding as a function of the crystalline fraction. The similarity between the matrices\nof intermediate states and the target unitary operation is defined as S =\n|Tr(U †U )|/ Tr U U )Tr(U U )1. Where Tr(A) represents the trace of a\ntheo inter theo theo inter inter\n† †\nmatrix A, U c √ orre(sponds to the target unitary operation or , U\ntheo inter\nrepresents the transfer matrix of the system when the PC0M w1aveguid1e i0s in the\n[ ] [ ]\n−1 0 0 1\nintermediate states. The behavior of intermediate states can be divided into three\nregimes: (i) When the PCM slightly deviates from the amorphous state, the three-fold\ndegeneracy is only weakly broken, and the system can still operate with high fidelity.\n(ii) As the index mismatch increases, as illustrated in Fig. R10c-d, the effective index\nof gradually detaches from the zero mode and . Since remains\nclose to the zero-mode plane, the evolution of degenerate states easily leaks into\n|𝐵3⟩ |𝐷1⟩ |𝐷2⟩ |𝐵3⟩\nnon-degenerate states, leading to a sharp drop in fidelity and non-unitary behaviors of\ndegenerate states. (iii) With further increase in index mismatch, fully separates\nfrom the degenerate plane (Fig. R10e), restoring a robust two-fold degeneracy that is\n|𝐵3⟩\nresilient to fluctuations in the index mismatch. This circumstance occupies the\nmajority of intermediate states, and the evolution is equivalent to the fully crystalline\ncase. Therefore, this two-level control is robust against mild partial switching and\nintermediate mismatch fluctuations, enabling high-fidelity operation even in the\npresence of modest tuning variations such as fluctuations in laser-writing penetration\ndepth.\nFig. R10 | Behavior of the system at different intermediate states. a, The similarity\nbetween the system’s transfer matrix (when the PCM waveguide is in intermediate\nstates) and the target unitary operation ( or ), as a function of the\ncrystalline fraction. The behavior of interm0edia1te state1s ca0n be divided into three\n[ ] [ ]\n−1 0 0 1\nregimes. b-e, Eigenvalues of the five modes in a five-waveguide system at different\ncrystalline fractions. b, Fully amorphous state, three-fold degeneracy can be observed.\nc-d, As the index mismatch increases, the effective index of gradually detaches\nfrom the zero mode and , but still remains close to the zero-mode plane.\n|𝐵3⟩\ne,\nFully crystalline state, fully separates from the degenerate plane, restoring a\n|𝐷1⟩ |𝐷2⟩\nrobust two-fold degeneracy.\n|𝐵3⟩\nRevision:\nMain text:\n(Page 14, lines 341-343) Slight deviations from the fully crystalline or amorphous\nstates do not significantly affect the two-level geometric phase exhibited by the device\n(Supplementary Fig. 15).\nSupplementary:\n(Supplementary Fig. 15) We have added the behavior of the system at different\nintermediate states shown in Fig. R10 to Supplementary Fig. 15.\nComments from Reviewer 1:\nFurthermore, the current design assumes a uniform phase change across the entire\nwaveguide. Considering the case where only a portion of the waveguide undergoes a\nphase transition may open the possibility of generating multiple geometric-phase\nlevels within a single device. A discussion along these lines would strengthen the\nmanuscript by highlighting the broader tunability and versatility of the proposed\napproach.\nResponse:\nWe sincerely appreciate the reviewer’s insightful comments. When only a portion\nof the waveguides undergoes a phase transition, a mapping between the three-fold and\ntwo-fold degenerate subspace is induced. This mapping can lead to population\nleakage out of the degenerate manifold during dimensionality reduction if the\ninterface parameters do not satisfy κ →0 and κ →0. Therefore, partial phase\nBX CX\ntransition of the PCM waveguide does not create new levels between the two-state\ngeometric phases, but instead leads to a non-unitary transformation. A detailed\nderivation is provided in the following:\nWhen the PCM waveguide is in its crystalline state (onsite index mismatch), the\nHamiltonian supports two degenerate states, which we define as:\n(R1)\n|𝐷1⟩=sin 𝜑|B⟩−cos 𝜑|C⟩\nWhere, =arctan(κ|𝐷/2κ⟩=), −c=oasrc𝜂t|aAn⟩(κ+s/in 𝜂(cos 𝜑|)B ⟩+sin 𝜑|C⟩)\nCX BX AX\n2 2\nWh𝜑en the PCM is i𝜂n its amorph√o𝜅uBsX +sta𝜅tCeX (with onsite index matching), three\ndegenerate states are supported by the Hamiltonian, and we define the three states as:\n(R2)\n|𝐷1′⟩ =sin 𝜑|B⟩−cos 𝜑|C⟩\n|𝐷2′⟩=−cos 𝜂′(cos 𝜎|A⟩+sin 𝜎|D⟩) + sin 𝜂′(cos 𝜑|B⟩+sin 𝜑|C⟩)\nWhere, =arctan(κ /κ ), =|𝐷a3rc′⟩ta=n(sin 𝜎|A⟩−c/os 𝜎|D⟩ ), =arctan(κ /κ )\nCX BX DX AX\n2 2 2 2\nAs 𝜑illustrated in Fig. 𝜂R′11a, at √th𝜅eA Xph+as𝜅eD Xtr√an𝜅sBiXti+on𝜅 CinXter𝜎face, the system undergoes\na mapping between the three degenerate states ( , , ) and two degenerate\nstates ( , ). Based on Eq. R1-2, we can get:\n|𝐷1′⟩ |𝐷2′⟩ |𝐷3′⟩\n|𝐷1⟩ |𝐷2⟩ (R3)\n|𝐷1⟩=|𝐷1′⟩\n|𝐷2⟩=(cos 𝜂cos 𝜂′cos 𝜎+sin 𝜂sin 𝜂′)|𝐷2′⟩ −cos 𝜂sin 𝜎|𝐷3′⟩\nEspecially, if the transition is at position of κ →0 and κ →0, since →π/2,\nBX CX\n'→π/2, we can get:\n𝜂\n𝜂 (R4)\n|𝐷1⟩=|𝐷1′⟩\nTherefore, there is no population loss from the degenerate subspace if the transition\n|𝐷2⟩=|𝐷2′⟩\noccurs exclusively at positions where κ →0 and κ →0. This corresponds to the\nBX CX\nmulti-level tunable SO(2) scheme in the main text.\nOtherwise, if the transition position does not satisfy κ →0 and κ →0, the\nBX CX\ndimensionality-reduction projection in Eq. R3 may result in population loss of the\ndegenerate subspace, leading to a non-unitary transformation of the degenerate states.\nAlthough this cannot generate additional levels of SO(2) geometric phase, it may\nenable potentially new functionalities. The system effectively operates as a singular\nvalue decomposition (SVD)-like architecture, as illustrated in Fig. R11b. Therefore, it\nenables a potential functionality: the programmable realization of general 2×2\nmatrices via an SVD-based architecture, considering that different transition positions\ncorrespond to distinct matrices.\nFig. R11 | The dynamics of the system when a portion of the PCM waveguide\nundergoes a phase transition. a, At the phase transition interface, a mapping is\nestablished between the three-fold degenerate states ( , , ) and two-fold\ndegenerate states ( , ). b, Since the mapping may result in population loss of\n|𝐷1′⟩ |𝐷2′⟩ |𝐷3′⟩\nthe degenerate subspace, the system effectively implements a singular value\n|𝐷1⟩ |𝐷2⟩\ndecomposition (SVD)-like architecture, which supports a potential functionality of\nprogrammable general 2×2 matrices.\nRevision:\nMain text:\n(Page 14, lines 358-365) Based on the architecture demonstrated in this work,\nadditional functionalities can be further developed. For example, the basic two-level\ntunable SO(2) geometric phase discussed in the manuscript is achieved by fully\nswitching the entire PCM waveguide. However, if only a portion of the waveguides\nundergoes phase transition, the mapping between the three-fold and two-fold\ndegenerate subspaces may lead to population leakage from the degenerate subspace.\nThis effect enables the system to effectively realize a singular value decomposition\n(SVD)-like architecture, thereby supporting the programmable implementation of\ngeneral 2×2 matrices.\nComments from Reviewer 1:\n5. For the multilevel control of SO(2) holonomy, the authors demonstrate three\ndiscrete levels by cascading two identical building blocks. While this provides a\nuseful proof of concept, the scheme could potentially be extended further for a more\ninsightful discussion. In particular, cascading non-identical building blocks may\nenable a richer set of discrete levels and more versatile control.\nResponse:\nWe would like to thank the reviewer for this insightful suggestion. Cascading\nnon-identical building blocks can indeed generate a larger number of discrete SO(2)\ngeometric phases. In general, cascading n distinct units can produce (n bits) \ndistinct levels. Here, we exemplify a scheme for generating equidistant g𝑛eometric\n2\nphase levels. As illustrated in Fig. R12a, a 4-level equidistant tuning is achieved by\ncombining two distinct building blocks. The first unit is designed to achieve\ngeometric phases of －θ and －2θ, and the second unit provides geometric phases of\n3θ and 5θ. The combination of these states can generate equidistant geometric phases\nranging from θ to 4θ. Fig. R12b illustrates an 8-level equidistant tuning scheme. The\nfirst two units traverse the inverse holonomic path of the device shown in Fig. R12a,\ngenerating geometric phases ranging from －θ to －4θ. Cascading an additional unit\nthat provides geometric phases of 5θ and 9θ, the overall device achieves eight\nequidistant phase levels, uniformly distributed from θ to 8θ.\nFig. R12 | Scheme for generating (n bits) equidistant levels SO(2) geometric\nphase by cascading n distinct build𝒏ing blocks. a, A 4-level equidistant geometric\n𝟐\nphase scheme based on the combination of two distinct building blocks. The first unit\nis designed to achieve geometric phases of －θ and －2θ, and the second unit\nprovides geometric phases of 3θ and 5θ. The combination of these states can generate\nequidistant geometric phases ranging from θ to 4θ. b, An 8-level equidistant tuning\nscheme. The first two units traverse the inverse holonomic path of the device shown\nin (a), generating geometric phases ranging from －θ to －4θ. Cascading an\nadditional unit that provides geometric phases of 5θ or 9θ, the overall device achieves\neight equidistant phase levels, uniformly distributed from θ to 8θ.\nRevision:\nMain text:\n(Page 9-10, lines 235-239) As a proof of concept, we demonstrate three discrete\ngeometric phase levels by cascading two identical building blocks. By employing\nnon-identical units, the number of achievable levels can be expanded. In general,\ncascading n distinct units enables 2n (n bits) discrete phase levels. Here, we propose a\nscalable scheme for generating equidistant geometric phases, as shown in\nSupplementary Fig. 11.\nSupplementary:\n(Supplementary Fig. 11) We have added a new illustration (Fig. R12) as\nSupplementary Fig. 11 to clarify the scheme for generating more geometric phase\nlevels.\nComments from Reviewer 1:\nSome small remarks for the authors’ consideration:\n1. Some typos should be double-checked carefully, i.e. in the abstract the word\n\"reconfiguable\" should be “reconfigurable”.\n2. In Fig. 1b, the annotation “Two degenerate subspace” is recommended to be\ncorrected as “Two degenerate states” or “Two-dimensional degenerate subspace”.\nSimilar descriptions, such as “Three degenerate subspace” in the annotation of Fig. 1b\nand “the number of degenerate spaces” in the caption of Fig. 1, are also recommended\nto be revised.\n3. In the sentence “This creates slight color contrasts between the recrystallized region\nand the original crystalline substrate, as seen in Supplementary Fig. 2a,” it appears\nthat the referenced image should be Fig. 2a in the main text rather than the\nsupplementary material. Please verify and correct this reference for clarity and\nconsistency.\n4. The unit of the scale bar in Fig. 1d should be provided.\n5. The symbol “μ” in Supplementary Eq. (S3) should be well defined.\nResponse:\nWe sincerely appreciate the reviewer’s very detailed inspection, we have made\ncorrections according to the comments in the following:\nRevision:\nMain text:\n2. (Page 1, lines 23-26) Our results bridge the key gap between robust\ngeometric-phase control and in-memory reconfigurable photonics, establishing a new\nparadigm for fault-resilient, high-dimensional operations in integrated optical\nsystems.\n2. (Page 4, Fig. 1b, e) Correct to“Two degenerate states”; “Three degenerate states”;\n(Page 5, lines 97-99) The crystalline and amorphous states of Sb Se can be tuned by\n2 3\nlaser direct writing, which modulates the number of degenerate states, further\nconstructing different SO(2) holonomies.\n3. (Page 10, lines 259-260) This creates granular contrasts between the recrystallized\nregion and the original crystalline substrate, as seen in Fig. 2a.\n4. (Page 4,Fig. 1d) Add the unit “nm” for the SEM figures.\nSupplementary:\n5. (Supplementary page 2, lines 55-57) Where is the coordinates of κ space. For\nthe SO(2) holonomy here, since κ space isomorphic to a unit 2-sphere, can be\n𝜇\nchosen as and (a pair of surface coordinates), is the path-ordering operator\n𝜇\nand is the non-Abelian connection, which reads:\n𝜑 𝜂 𝑃\n𝐴\n(S4)\n(𝐴 𝜑,𝜂)𝑖𝑗 = i⟨𝐷𝑖|∂𝜑,𝜂|𝐷𝑗⟩\nReply to Reviewer 2\nComments from Reviewer 2:\nReviewer #2 (Remarks to the Author):\nThis work demonstrates in-memory multilevel control of generic SO(m) holonomy\nusing a multilayer silicon photonic platform integrated with the phase-change material\nSb₂Se₃. By optically switching the Sb₂Se₃ waveguides between amorphous and\ncrystalline states, the authors dynamically modulate the number of degenerate modes\nin the system, enabling reconfigurable non-Abelian geometric phases without volatile\nexternal controls. They experimentally achieve high-fidelity, nonvolatile SO(2) and\nSO(3) transformations, along with tunable two- and three-mode braiding operations,\nand showcase multi-level programmability through cascaded SO(2) units and Givens\nrotations. This approach merges the inherent robustness of geometric phases with the\nnonvolatile tunability of PCMs, offering a promising pathway toward fault-tolerant\nphotonic quantum processing, reconfigurable optical switching networks, and\nhigh-dimensional classical neuromorphic computing.\nIn my opinions, this work demonstrates the first experimental realization of\nreconfigurable control over non-Abelian holonomy, which exhibits notable innovation.\nHowever, several issues should be addressed prior to publication:\nResponse:\nWe sincerely thank the reviewer for the hard work and positive comments. These\ninstructive comments have significantly improved the quality of our manuscript. We\nwill fully address these issues.\nComments from Reviewer 2:\n1. First and foremost, regarding novelty, this work combines the concepts of\n\"reconfigurability\" and \"non-Abelian holonomy,\" which brings to mind two related\nworks also published in Nature Communications: Nat. Commun. 16, 7089 (2025) and\nNat. Commun. 16, 3650 (2025). Compared to the former, this study extends braiding\noperations to more general holonomic scenarios and introduces a new refractive index\ntuning method; compared to the latter, it achieves dynamic control over the SO(m)\ngroup. The authors should provide more detailed and specific comparisons with these\nrelevant works, and include necessary discussions in the manuscript to better highlight\nthe originality and significance of their contribution.\nResponse:\nWe sincerely appreciate the reviewer’s instructive comments and detailed\noverview of the recent related works. We now provide detailed comparisons and\nhighlight our originality.\n(i) Compared with the reconfigurable braiding [Nat. Commun. 16, 7089 (2025)].\nThe first distinction lies in the theoretical mechanism. As illustrated in Fig. R12,\n[Nat. Commun. 16, 7089 (2025)] employ a four-waveguide structure that supports\ntwo degenerate modes for braiding in the absence of heating. Upon applying heat to\nmodulate the refractive index of waveguides, chiral symmetry of the Hamiltonian is\nbroken, causing the eigenmodes to localize at the input waveguides of each STIRAP\nchannel, and it’s notable that the degeneracy is lifted off in the meantime. In contrast,\nour scheme is based on directly manipulating the dimensionality of degenerate\nsubspaces. The five-waveguide structure supports either a two-fold or three-fold\ndegenerate states, enabling a switchable holonomic path in different degenerate\nsubspaces. Thus, both states are based on geometric phase.\nThe second distinction lies in the tuning range. [Nat. Commun. 16, 7089 (2025)]\nachieves reconfigurability within braid groups. In contrast, our system enables control\nover the SO(2) group, which is further expanded to the control of SO(m) by Givens\nrotations.\nThe third distinction lies in the tuning methods, [Nat. Commun. 16, 7089 (2025)]\nutilizes the thermo-optics effect to modulate the effective index of waveguides. Our\nwork uses phase-change material (PCM) as the active in-memory manipulation\nmethod. PCM has the advantages of large-index-contrast, zero static-power\nconsumption, and zero thermal crosstalk, which we will discuss in detail in the next\nresponse.\nFig. R12 | Comparison between this work and the reconfigurable braiding work.\na, [Nat. Commun. 16, 7089 (2025)] achieved reconfigurability within braid groups.\nUpon applying heat to break the chiral symmetry of the Hamiltonian, the degenerate\nband is split into non-degenerate bands, thereby enabling the reconfigurability of\nbraiding. b, The summary of our work. The system enables control over the SO(m)\ngroup based on manipulating the dimensionality of degenerate subspaces. The active\ntuning is achieved using a phase-change material (PCM).\n(ii) Compared with the static SO(m) holonomy [Nat. Commun. 16, 3650 (2025)].\nThe key distinction lies in the introduction of dynamic control, along with the\ntheoretical control framework and a heterogeneous integration platform. As Fig. R13a\nshows, [Nat. Commun. 16, 3650 (2025)] uses a four-waveguide system to construct a\nbasic SO(2) holonomy in a silicon nitride platform, further expanding to construct\nSO(6) holonomy through Givens rotation. The scheme is static and non-tunable, and\nthe generated geometric phase is predetermined and remains fixed after device\nfabrication. To realize active and in-memory tunability, this work employs a\nfive-waveguide structure on a multilayer amorphous silicon heterogeneous integration\nplatform integrated with a phase-change material (Fig. R13b). By controlling the\ncrystalline-to-amorphous transition of the phase-change material waveguide, the\ndimensionality of the degenerate subspace supported by the Hamiltonian can be\ndynamically tuned, transforming the holonomy from static to reconfigurable with\nnon-volatility. Therefore, our scheme realizes generic control over SO(2) holonomy,\nwhich can be further expanded to the control of SO(m) by Givens rotations. This\nnon-volatile tunability combines the robustness of geometric phases with the high\ndynamic reconfigurability, offering promising prospects for low-power, in-memory\nprogrammable photonic computing.\nFig. R13 | Comparison between this work and the SO(m) holonomy work. a, [Nat.\nCommun. 16, 3650 (2025)] uses a four-waveguide system to construct a basic SO(2)\nholonomy, further expanding to construct SO(6) holonomy through Givens rotation.\nThe generated geometric phase is predetermined and remains fixed after device\nfabrication. b, The summary of our work. This work realizes active and in-memory\ntunability of SO(m) holonomy. By controlling the state of the PCM waveguide, the\ndimensionality of the degenerate subspace supported by the Hamiltonian can be\ndynamically tuned, transforming the holonomy from static to reconfigurable with\nnon-volatility.\nRevision:\nMain text:\n(Page 2, lines 42-50) The reported scheme for high-dimensional SO(m) holonomy24 is\npredetermined and fixed after fabrication, which can not fulfill the requirements of\nprogrammable photonic circuits. Although the reconfigurable non-Abelian geometric\nphase has been demonstrated using the thermo-optic effect25, this implementation is\nlimited to the braid group operations. While such scheme has potential for specialized\ntasks like the Ising problem solving26, it falls short of enabling general-purpose\nphotonic matrix computation. Therefore, realizing tunable non-Abelian holonomy\nover the full SO(m) group—combining the dynamic reconfigurability and the\nrobustness of holonomy, is essential to support in-situ training27 and diverse\ncomputing tasks.\nComments from Reviewer 2:\n2. As a follow-up to the point above, different to Nat. Commun. 16, 7089 (2025),\nwhich employs the thermo-optic effect for active tuning, this work utilizes the\nphase-change material, claiming advantages in lower energy consumption and\nimproved stability. The authors are encouraged to provide a detailed comparison of\nthe advantages and disadvantages of different active tuning methods to underscore the\ntechnical and systemic innovation of their approach.\nResponse:\nWe are very thankful for the reviewer’s instructive comments. Indeed, both\nthermal turning and non-volatile tuning are important active tuning approaches.\nHowever, we believe that phase-change materials (PCMs) offer some advantages over\nconventional thermo-optic (TO) modulation in integrated photonic devices,\nparticularly in terms of refractive index contrast, non-volatility, and no thermal\ncrosstalk, as will be discussed in detail below.\n(i) Large refractive index contrast. PCMs exhibit drastic changes in optical\nproperties between amorphous and crystalline phases, with refractive index\ndifferences (Δn) exceeding 0.7 in the near-infrared regime. In contrast, TO effects\ninduce only small index changes (typically ~10-2). The large refractive index contrast\nfacilitates significant onsite index matching or mismatching, enabling the complete\nand stable switching from a three-fold degeneracy to a two-fold degeneracy, and\nthereby supporting reconfigurable holonomic evolution paths.\n(ii) Ultra-low power consumption and non-volatility. A key advantage of\nPCMs is their non-volatile nature: once switched, the state is retained without power.\nEnergy is consumed only during the transient switching event (fJ to pJ per bit). In\ncontrast, TO devices require continuous power to maintain a state, resulting in high\nstatic power consumption and thermal crosstalk.\n(iii) Absence of thermal crosstalk. Thermal crosstalk is a notable problem of\nthermo-optic modulation in high-density programmable integrated photonics. The\npersistent heat diffuses laterally through the substrate and dielectric layers, altering\nthe refractive index of neighboring waveguides. Especially for the non-Abelian\nholonomy with densely packed waveguides, thermal crosstalk makes independent\ncontrol of nearest-neighboring waveguides challenging. Evident thermal crosstalk\nmay influence the degeneracy of tunable SO(2) holonomy. As Fig. R14a shows, [Nat.\nCommun. 16, 7089 (2025)] also simulates the influence of thermal crosstalk in\nreconfigurable braiding, and the propagation constant of neighboring waveguide X\nslightly varies as the distance to the electrode. Fig. R14b shows the temperature\ndistribution observed in an integrated thermo-optic modulation device, captured using\na thermal imaging camera (FOTRIC 280+). Evident thermal crosstalk can be\nobserved in the spot point (Sp1). In contrast, PCM is non-volatile with no static heat,\nand its switching heat is transient and highly localized, enabling crosstalk-free,\nindependent tuning of individual waveguides within the holonomy, and preventing\ndegradation of the degeneracy caused by thermal crosstalk.\nFig. R14 | Thermal crosstalk of thermo-optic modulation. a, Temperature\ndistribution simulated in [Nat. Commun. 16, 7089 (2025)], and the propagation\nconstant of neighboring waveguide X is slightly affected. b, Temperature distribution\nobserved in an integrated thermo-optic modulation device, we capture it using a\nthermal imaging camera to demonstrate the significant thermal crosstalk of\nthermo-optic modulation. Evident thermal crosstalk can be observed in the spot point\n(Sp1).\nAlthough phase-change material offers the aforementioned advantages, its\nswitching speed is slower than that of thermal tuning, as crystallization of the PCM\nrequires relatively long pulses (on the order of milliseconds to hundreds of\nmilliseconds), whereas thermal tuning achieves significantly faster switching times.\nTherefore, the thermo-optic modulation effect exhibits a higher switching speed.\nTable R1 summarizes some important performance metrics of phase-change\nmaterials and thermo-optic tuning. Fig. R15 provides a visual comparison between the\noptically controlled PCM approach demonstrated in this work and reported\nthermo-optic tuning schemes, offering clearer insight into their advantages and\ndisadvantages.\nRefractive\nTypical Static power Thermal\nindex Switching time Volatility\nperformance consumption crosstalk\ncontrast\nThermo-optic ~0.01 ~10 mW 2~236 μs\nY Y\n(Si)2-5\nThermo-optic\n(Polymer)6 ~0.003 120 mW 1 ms Y Y\nPCM 200 ns (A)\n(Sb Se )(O, ~0.8 0 300 ms (C) N N\n2 3\nThis work)\nPCM 500 ns (A)\n~0.8 0 N N\n(Sb Se )(E)7 100ms(C)\n2 3\nTable R1 | The performance comparison of thermo-optic modulation and phase\nchange materials. A, amorphization; C, crystallization. O, optically reversible\nswitching. E, electrically reconfigurable switching.\nFig. R15 | The performance comparison of thermo-optic modulation and phase\nchange materials.\nRevision:\nMain text:\n(Page 2, lines 52-73) Conventional tuning mechanisms—such as the thermo-optic\neffect, carrier dispersion effect, or Pockels effect28—typically induce only small\nchanges in the refractive index. In contrast, chalcogenide PCMs exhibit dramatic\nrefractive index change during phase transitions, exceeding the tuning strength of\nconventional methods by 1 to 2 orders of magnitude29. Crucially, the optical states of\nPCMs are nonvolatile, enabling lower energy consumption and in-memory storage of\ntuning states. ... low-loss PCMs enable a powerful platform for investigating\nadvanced physical mechanisms with flexible reconfigurability, including the tuning of\nnon-Abelian geometric phases. The large refractive index contrast of PCMs enables\ncomplete and stable switching from three-fold to two-fold degeneracy, thereby\nsupporting reconfigurable holonomic evolution paths. Moreover, PCM-based tuning\nis inherently free of thermal crosstalk, allowing for independent control of individual\nwaveguides. In contrast, thermally tuned systems suffer from thermal crosstalk that\nperturbs the refractive indices of neighboring waveguides, easily degrading the\ndegeneracy of the SO(2) holonomy. Thus, the PCM-based tuning approach effectively\navoids degradation of the degeneracy caused by thermal crosstalk.\nComments from Reviewer 2:\n3. While the active tuning strategy presented is appealing, it remains limited to\nswitching between SO(2) and SO(3) based on the presence or absence of onsite index\nmismatch. It would be valuable if the authors could discuss whether their method can\nbe extended to design more complex or interesting functionalities.\nResponse:\nWe sincerely appreciate the insightful comments. Here, we will provide some\ndiscussions on two potential functionalities of this scheme beyond SO(2) and SO(3).\n(i) Potential tunable generic U(2) holonomy\nThe tunable geometric phases in this work are confined to the SO(m) group due\nto the positive real coupling coefficients. To realize the generic U(m) geometric\nphases, complex-valued coupling coefficients are required8-19. In photonics, effective\ncomplex-valued couplings between lossless optical waveguides can be realized\nthrough artificial gauge fields20-22, and this scheme has been incorporated to achieve\ngeneric U(m) holonomy23. Based on these reported complex-valued coupling\nconfigurations, the active tuning method of this work has the potential to reach\ntunable generic U(2) holonomies.\nHere, we start with introducing the reconfigurable Berry phase to one degenerate\nstate. Similar to the holonomy with complex couplings discussed in [Phys. Rev. A,\n101, 062314 (2020)], our configuration is shown in Fig. R16b-c: κ =0 throughout\nBX\nthe holonomy, κ , κ are positive real numbers, and κ |κ eiφCX is a\nAX DX CX CX\ncomplex number. Therefore, the Berry connection for in Fig. R=16b is|nontrivial,\nwhich can be expressed as A =i D ∂ D =-sin2(|𝐷 θ 2),⟩ and acquires a Berry\nφ 2 φ\nCX CX 2 2\nphase = -sin2( θ )dφ througho ⟨ ut | the holo ⟩ nomic evolution |𝐷 , 2⟩ where we define\nα ∮\nθ\n=arctan(|κ |/|κ |). This holonomy generates a U(2) transformation for D and\n1 0 | ⟩\nD : . In contrast, when the PCM waveguide switches to its amorphous state\n2 0 e-i\n|(ons⟩ite\n[\n-indexα\n]\nmatching) in Fig. R16c, the Berry connection of in Fig. R16c\nbecomes A =i D ∂ D =-sin2( ), where =arctan(|κ |/ |𝐷|κ2⟩ 2 |κ 2).\nφ CX 2 φ CX 2 𝜃 2 ′ 𝜃′ CX AX DX\nThus, the Berry ⟨ pha | se acqu ⟩ ired by D is 2= -sin2( )dφ √ , an | d + the U | (2)\n𝜃′\n| ⟩ β1 0 ∮ 2\ntransformation for D and D becomes: . Fig. R16d-e illustrates the\n1 2 0 e-i\nsimulated Berry pha|se a⟩cquired| by⟩ D\n2\nin two\n[\nstateβs.\n]\nBy tuning the PCMwaveguide\nin its amorphous or crystalline, D can acquires different Berry phases throughout\n2\n| ⟩\nthe holonomy. The calculation demonstrates the potential functionality in\n| ⟩\nreconfigurable phase gates.\nThrough cascading this tunable phase gate, we can further construct tunable\nunitary matrices , . Combined the tunable SO(2) geometric phase ( ) in\nthis work, we ca𝑖nα 𝜎𝑧imp𝑖lIαement the potential functionality: tunable generi𝑖cα 𝜎𝑦U(2)\nholonomy24. e e e\nFig. R16 | Potential functionality: tunable U(2) holonomy a, The structure of the\ntunable phase gates based on complex-valued couplings configuration. The\ncomplex-valued couplings can be induced by artificial gauge fields through periodic\nbending waveguides20-22. b-c, Configuration of coupling coefficients, which is\ndesigned as: κ =0; κ , κ are positive real numbers, and κ |κ eiφCX is a\nBX AX DX CX CX\ncomplex number. By tuning the PCMwaveguide in its amorphous or c\n=\nrystall\n|\nine, D\n2\ncan acquires different Berry phases throughout the holonomy. d-e, Simulated Berry\n| ⟩\nphase acquired by D in two states, demonstrating the tunable phase gates. d, The\n| ⟩\nin its amorphous state, corresponding to onsite index matching.\n(ii) Injection of multi-photon Fock states enable higher-dimensional tunability.\nWe now turn to the second potential functionality. Injecting multiple photons\nsignificantly expands the dimensionality of the degenerate subspace [Nat. Phys. 19,\n30-34 (2023), Phys. Rev. A, 101, 062314 (2020)]. This multiphoton injection scheme\nis compatible with our tunable platform, enabling reconfigurable higher-dimensional\nquantum holonomy of the basic five-waveguide system. For example, as Fig. R17b\nillustrates, two indistinguishable photons occupying two degenerate modes give rise\nto three orthogonal two-photon basis states, thereby supporting three-dimensional\nquantum holonomy, which is similar to the scheme in [Nat. Phys. 19, 30-34 (2023)].\nCombined with our active tuning scheme, where single-photon evolution is governed\nby a tunable SO(2) transformation (Fig. R17a), the two injected indistinguishable\nbosons collectively undergo a reconfigurable SO(3) transformation (Fig. R17b). This\nscheme offers a compatible alternative for realizing higher-dimensional\nreconfigurable holonomic quantum computation, thereby extending the tunable\nholonomy beyond the Givens-rotation-based approach presented in the main text.\n[FIGURE REDACTED]\nFig. R17 | Injection of multi-photon Fock states. a, Injection of single photons to\nthe basic five-waveguide system, and the evolution is governed by a tunable SO(2)\ntransformation. b, Injection of two indistinguishable photons, which is similar to the\nscheme in [Nat. Phys.19, 30-34 (2023)]. The five-waveguide system is governed by a\nreconfigurable SO(3) transformation.\nRevision:\nMain text:\n(Page 14, lines 365-371) Besides, this work demonstrates tunable SO(m) holonomy.\nBy incorporating complex-valued coupling coefficients, the active tuning method of\nthis work has the potential to reach tunable U(2) holonomy. For higher-dimensional\nholonomic control, multiphoton Fock-state injection20,73 can provide a compatible and\ncomplementary route to reconfigurable higher-dimensional holonomic quantum\ncomputation beyond the Givens-rotation-based approach presented in this work.\nComments from Reviewer 2:\n4. The caption of Figure 2e is overly brief. A more detailed explanation of how the\nSO(2) transformation is realized in parameter space should be provided to assist\nreaders unfamiliar with the process of non-Abelian holonomy.\nResponse:\nWe are very thankful for the reviewer’s instructive comments. We will provide a\ndetailed description of the evolution of holonomic paths in parameter space. Figure 2e\nof the main text cascades two identical tunable SO(2) holonomies to realize multilevel\nprogrammability. As Fig. R18 shows, each building block can be divided into three\nsteps. The parameter space spanned by coupling coefficients is isomorphic to a\n2-sphere.The start/end points are located at the north pole (the star points) since κ ,\nBX\nκ → 0 at the input/output port. At Step 1, waveguide C moves towards X while B\nCX\nremains far from X, consequently κ increases and κ remains zero. Thus, the\nCX BX\npath in parameter space follows the geodesic arc along the boundary of the spherical\nfirst octant. Then, at step 2, waveguide B moves to X while C departs from X, leading\nto an increase in κ while a decrease of κ to zero. At Step 3, waveguide B\nX X\nmoves away from X, while waveguide C remains distant from X, with both couplings\nB C\nfinally vanishing. The evolution moves along the geodesic arc along the edge of the\nspherical first octant at Step 3, completing a closed holonomic path in parameter\nspace. Through the holonomy, the two degenerate states follow an SO(2)\ntransformation θ y, where θ is the solid angle enclosed by the holonomic path. The\nsecond cascaded id𝜎evice follows a similar evolution, the difference is the parameter\ne\nspace switch to 2 ＋ 2 ,resulting in the different enclosed solid angle α.\nAX DX\n[√𝜅 𝜅 ,𝜅BX,𝜅CX]\nFig. R18 | Illustration of the holonomic evolution in parameter space and device\nstructure for multilevel tunable SO(2) holonomy. The device consists of two\nidentical tunable SO(2) geometric phase units, and both units can be divided into three\nsteps. At Step 1, waveguide C moves towards X while B remains far from X, and the\npath in parameter space follows the boundary of the spherical octant. Then at step 2,\nwaveguide B moves to X while C departs from X, leading to an increase in κ while\nBX\na decrease of κ to zero. At Step 3, waveguide B moves away from X, while\nCX\nwaveguide C remains distant from X. The evolution moves along the geodesic arc\nalong the edge of the spherical first octant at Step 3, completing a closed holonomic\npath in parameter space.\nRevision:\nMain text:\n(Page 9, lines 200-209) Fig. 2 | Experimental results of multi-level tunable SO(2)\nholonomy. ... e,Three-level reconfigurable SO(2) holonomy. Upper: Schematic of the\nholonomic evolution in parameter space, and the corresponding device structure is\nshown in Supplementary Fig. 16. The overall process consists of two identical tunable\nSO(2) holonomies, and each unit can be divided into three steps.The evolution begins\nand ends at the north pole (star markers), where κ , κ → 0. At Step 1, κ\nBX CX CX\nincreases and κ remains zero, and the path in parameter space follows the\nBX\nboundary of the spherical octant. Then at step 2, κ increases as κ decreases to\nX X\nzero. At Step 3, κ decreases and κ remains zero, with both couplings vanishing\nX CX\nB C\nfinally. The evolution returns to the north pole. Through the holonomy, the two\nB\ndegenerate states follow an SO(2) transformation θ y, where θ is the solid angle\nenclosed by the holonomic path. Lower: measured inite𝜎nsity of the three states of the\ne\nthree-level reconfigurable SO(2) holonomy transformation.\nSupplementary:\n(Supplementary Fig. 16) We have added the detailed description of three-level tunable\nSO(2) holonomy in Fig. R18 to Supplementary Fig. 16.\nComments from Reviewer 2:\n5. The multilayer quasi-3D waveguide platform constructed using deposited SiO₂ is\na promising architecture. However, a valid concern is whether next-nearest-neighbor\ncoupling coefficients (such as κ ) can be neglected in such multilayer waveguides,\nCD\nand whether they might disrupt the holonomy process. Supplementary data on\ncoupling coefficients should be provided in the Supplementary Information of this\nwork to facilitate reader understanding.\nResponse:\nWe sincerely appreciate the reviewer’s insightful comments. As Fig. R19b shows,\nwe simulate next-nearest-neighbor coupling coefficients (κ , κ , κ , κ , dashed\nCD BD BA CA\nlines) as a function of the gap distance (g or g ), and we also make a comparison\nBX CX\nwith the nearest-neighbor coupling (κ , κ , the solid line). The\nBX CX\nnext-nearest-neighbor coupling coefficients are approximately an order of magnitude\nsmaller than the nearest-neighbor couplings. Fig. R19c shows the evolution of\ncoupling coefficients in a holonomy. Compared to the nearest-neighbor couplings\n(solid lines), the next-nearest-neighbor couplings (dashed lines) are significantly\nweaker.\nTo further analyze the influence on the holonomy process, we calculate the\neigenvalues of the Hamiltonian in Eq. R5 as a function of gap distance to analyze the\ndegeneracy. Figure R19d-e illustrates the eigenvalues of five modes for the case of\nonsite-index match (d=0 in Eq. R5, amorphous PCM) and onsite-index mismatch\n(d=0.53 in Eq. R5, crystalline PCM), respectively. Three-fold and two-fold\ndegeneracy are observed in two states, respectively. The degeneracy is well preserved\nin most gap distances, except for a slight splitting of the eigenvalues of degenerate\nmodes observed at small gap distances. Therefore, the design of the holonomic path\nshould bypass regions of significant degeneracy splitting. Fig. R19f-g shows the\nsimulated optical evolution with different injections in two states, and we compare the\nideal evolution (solid lines) with the evolution under the influence of\nnext-nearest-neighbor coupling (dashed lines). The agreement of the two evolution\ntrajectories indicates that next-nearest-neighbor coupling has a minimal impact on the\nholonomy process.\n0 𝜅AX(𝑧) 𝜅BX(𝑧) 𝜅CX(𝑧) 𝜅DX(𝑧)\n(R5)\n⎡ ⎤\n𝜅AX(𝑧) 0 𝜅BA(𝑧) 𝜅CA(𝑧) 0\n⎢ ⎥\n𝐻 (𝑧)= 𝜅BX(𝑧) 𝜅BA(𝑧) 0 0 𝜅BD(𝑧)\n⎢ ⎥\n⎢𝜅CX(𝑧) 𝜅CA(𝑧) 0 0 𝜅CD(𝑧)⎥\n⎣𝜅DX(𝑧) 0 𝜅BD(𝑧) 𝜅CD(𝑧) 𝑑 ⎦\nFig. R19 | Analysis of next-nearest-neighbor coupling coefficients. a, Illustration of\nthe gap distance, nearest-neighbor coupling coefficients, and next-nearest-neighbor\ncoupling coefficients for the five-waveguide system. b, Simulated\nnext-nearest-neighbor coupling coefficients (dashed lines) and the nearest-neighbor\ncoupling coefficients (the solid line) as a function of the gap distance. c, Evolution of\nthe nearest-neighbor and next-nearest-neighbor coupling coefficients in a holonomy.\nd-e, Eigenvalues of Eq. R5 for the case of onsite-index match (d=0 for amorphous\nPCM) and onsite-index mismatch (d=0.53 for crystalline PCM), respectively.\nThree-fold and two-fold degeneracy are observed in two states, respectively. f-g, The\nsimulated optical evolution with different injections for two states. The solid lines\nrepresent the ideal evolution, while dashed lines show the evolution in the presence of\nnext-nearest-neighbor couplings. Upper panel: injecting into waveguide B. Lower\npanel: injecting into waveguide C. f, Amorphous state. g, Crystalline state.\nRevision:\nMain text:\n(Page 7, lines 172-178) In practical device models, certain factors may affect the\ndegeneracy, such as the next-nearest-neighbor coupling coefficients. Supplementary\nFig. 14b-c presents the simulated next-nearest-neighbor coupling coefficients, which\nare approximately an order of magnitude smaller than the nearest-neighbor couplings.\nSupplementary Fig. 14f-g compares the optical evolution of a holonomy with (dashed\nlines) and without (solid lines) next-nearest-neighbor coupling. The agreement\nbetween the two cases indicates that next-nearest-neighbor coupling has a minimal\nimpact on the holonomy process.\nSupplementary:\n(Supplementary Fig. 14) We have added the data on next-nearest-neighbor coupling\ncoefficients from Fig. R19 to Supplementary Fig. 14.\nComments from Reviewer 2-5:\n6. Several minor issues require attention: the parameter space vectors on lines 91\nand 136 contain a square symbol on the left-hand side of the equation, which may be\na display error. The matrix alignment in Eq. (1) is also irregular. The authors should\ncorrect these and carefully check for other similar issues throughout the text.\nResponse:\nWe are very thankful for the reviewer’s instructive comments. We sincerely\napologize for these display errors, and we have now addressed these issues carefully,\nwhich are summarized as follows:\nRevision:\nMain text:\n(Page 5, lines 100-102): The left inset shows the FIB cross-sectional image of the\ndevice. The right insets show the parameter spaces, where the control manifold\nκ= κ ,κ , κ or κ= κ2 +κ2 ,κ , κ is isomorphic to a unit 2-sphere, and the\nAX BX CX AX DX BX CX\nsta[r represents ]the start/[e√nd point of a holo]nomy.\n(Page 6, lines 145-148): But the accumulated geometric phase α differs from the\nphase θ observed in Phase 1 since it involves the participation of κ (detailed in η\nDX\nand η′ of Eq. S7 and Eq. S14), and the control manifold is redefined as κ\n=\nκ2 κ2 κ κ .\nAX DX BX CX\n[√ + , , ]\n(Page 5, Eq. (1)): The Hamiltonian of this five-waveguide system can be written as:\nβ\nβ\n𝜅AX(𝑧) 𝜅BX(𝑧) 𝜅CX(𝑧) 𝜅DX(𝑧)\nβ (1)\n⎡ ⎤\n𝜅AX(𝑧) 0 0 0\n⎢ β ⎥\n𝐻 (𝑧)=⎢𝜅BX(𝑧) 0 0 β0 ⎥\n⎢ 𝜅CX(𝑧) 0 0 0 P ⎥\n⎢ ⎥\nReferences ⎣𝜅DX(𝑧) 0 0 0 ⎦\nunitaries. Optics Express25, 28236-28245 (2017).\nplatform: state-of-the-art and a review. Frontiers of Optoelectronics 15, 9\n(2022).\noperating at 2 μm. Optics Express29, 23508-23516 (2021).\n733-735 (2013).\nin silicon. Optics express22, 10487-10493 (2014).\nCommunications16, 7089 (2025).\noptical neural networks with nanoseconds in situ training capability. Advanced\nPhotonics5, 046004 (2023).\nstructures for holonomic computation. Physical Review A101, 062314 (2020).\nOperations in Classical Mechanics by Nonadiabatic Evolutions. Physical\nReview Letters135, 027201 (2025).\nSynthetic magnetic fields for ultracold neutral atoms. Nature 462, 628-632,\ndoi:10.1038/nature08609 (2009).\ngauge potentials for neutral atoms. Reviews of Modern Physics 83, 1523-1543\n(2011).\nstrontium gas. Nature communications9, 3580 (2018).\nnon-adiabatic geometric gates. Nature496, 482-485 (2013).\nand Quantum Computation with Josephson Junctions. Physical Review Letters\n90, 028301, doi:10.1103/PhysRevLett.90.028301 (2003).\ngeometric gates. Physical review letters122, 080501 (2019).\ngeometric two ion-qubit phase gate. Nature422, 412-415 (2003).\nholonomic single quantum gates with a geometric spin under a zero field.\nNature photonics11, 309-314 (2017).\nquantum gates over geometric spin qubits with polarised microwaves. Nature\ncommunications9, 3227 (2018).\ntemperature high-fidelity holonomic single-qubit gate on a solid-state spin.\nNature communications5, 4870 (2014).\nArtificial Gauge Field. Physical Review Letters 129, 053901,\ndoi:10.1103/PhysRevLett.129.053901 (2022).\ncouplers. Physical Review A—Atomic, Molecular, and Optical Physics 71,\n065801 (2005).\nperiodically curved waveguides. Opt. Lett.30, 2137-2139 (2005).\nphotonics. arXiv preprint arXiv:2507.01394 (2025).\nJournal of Mathematical Physics15, 1677-1679 (1974)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed all my comments by adding new simulations, new experimental tests and new discussions. I am happy to recommend the publication of the manuscript in its present form.", "reviewer_2": "After reviewing the revised manuscript and the authors’ detailed responses, I am satisfied that my previous concerns have been adequately addressed. The authors now provide clearer comparisons with related work and convincingly highlight the novelty of PCM-based reconfigurability for controlling SO(m) holonomy. The added discussions on tuning mechanisms, mode degeneracy, and the robustness of the multilayer platform significantly strengthen the presentation. The clarifications in figures and the correction of minor issues also improve readability. Overall, the revision enhances both the technical depth and clarity of the manuscript, and I recommend its publication in Nature Communications."}, "author_response": ""}]

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