Non-cascade random walks in solid-state high harmonic generation

Domain: physics
Published: 2026-02-18
Editorial outcome: minor_revision

Abstract
Random walks—both classical and quantum—unlocked new possibilities in search algorithms and information processing. Although linear photonic systems, with flexible tunability and multiple degrees of freedom, have served as efficient carriers for random walks, they typically require cascaded implementations, presenting a potential limitation on realizing integrated photonic circuits. In this work, we demonstrate a non-cascade, high-dimensional random walk in the orbital angular momentum (OAM) space of light using solid-state high-harmonic spectroscopy. The crystal nonlinearity enables the simultaneous conversion of multiple photons into a series of harmonics with distinct colors and whose OAM distributions are determined by the symmetry of the crystal. This approach reveals the dynamics of photonic degrees of freedom in high-harmonic generation can be naturally framed as an ultrafast, high-dimensional random walk, paving the way for compact, highly stable photonic platforms tailored for solid-state information processing.

Introduction
The random walk, a cornerstone of stochastic process modeling, traces its origin to 1827 when Robert Brown observed the erratic motion of pollen grains suspended in water1. Its mathematical formalism was later proposed by Karl Pearson2 and subsequently solidified by Albert Einstein through molecular kinetic theory3. In its simplest form, a discrete one-dimensional random walk describes propagation via sequential random steps, where a walker’s position evolves probabilistically—akin to coin tosses—ultimately converging to a Gaussian distribution after many iterations. This framework has found broad applications across disciplines, from solid-state physics4 and astronomy5 to polymer chemistry6, biology7, computer science8, and finance9.

The quantum walk, a quantum-mechanical counterpart, is distinguished by its reliance on quantum superposition and the dispersed distributions. Experimental realizations have been demonstrated in diverse platforms, including superconducting systems10–12, nuclear magnetic resonance13,14, cold atoms15, optical lattices16,17, free electrons18,19, and linear photonic systems20–32. Among these, the photonic system with flexible tunability and rich degrees of freedom33–35 has been developed into an ideal carrier for quantum computation and precision measurements. However, most photonic random walks rely on cascaded interferometric networks (Fig. 1a), where the physical footprint scales rapidly with step number36. This undoubtedly gives rise to inherent challenges in integration and scalability, as accumulated phase noise and environmental perturbations degrade performance.

Here, we introduce a non-cascade, high-dimensional random walk in orbital angular momentum (OAM) space of photons, enabled by solid-state high-harmonic generation (HHG)—a nonlinear up-conversion process that coherently generates radiation with multiples of the incident laser frequency37–39. The simultaneously emitted harmonics exhibit structured spatial and polarization profiles, where the harmonic order serves as an intuitive walking step. Unlike conventional cascaded photonic walks, our approach leverages the frequency-resolved, simultaneous emission of HHG (Fig. 1b), where the ultrafast electron dynamics in solids intrinsically encode the amplitude and phase for all walking steps at the same time. By exploiting the macroscopic symmetry of the crystal, we unveil a walking mode based on solid-state HHG, distinct from conventional paradigms. The combination of high-dimensional OAM space and non-cascade operation of this approach establishes a route toward compact, ultrafast photonic systems for high-dimensional information processing.

Results
Construction of initial state for random walks
To implement the non-cascade random walks in photonic systems, we first identify two fundamental physical quantities of photons as vital subspaces: two-dimensional spin angular momentum (SAM) of |s⟩ = |L⟩ or |R⟩ as the coin space Hc and infinite-dimensional OAM of |‘⟩ = |−1⟩!|+1⟩ as the walker space Hw 22,23. Compared to conventional cascaded random walks where a sequence of walking units U^ is requested to propagate the photonic system in the Hilbert space H = Hc ⊗ Hw (Fig. 1a), simultaneous and individual different-order up-conversion processes with the absorption of multiple fundamental photons enable us to realize the non-cascade high-harmonic random walks (Fig. 1b). The absence of subsequent optical elements in non-cascade implementation necessitates pre-configuring the entire propagation operator in the input state. In this respect, the incident fundamental wave should carry the basic walking unit U^, including the conditional displacement of the walker and the coin-flip operation, which is implemented in experiments via a q-plate (QP) and a quarter-wave plate (QWP), respectively:

U^ = 1/sqrt(2) * [sum_{‘=-1}^{1} (|R⟩ + i|L⟩)⟨R| ⊗ |‘+2q⟩⟨‘| + sum_{‘=-1}^{1} (i|R⟩ + |L⟩)⟨L| ⊗ |‘-2q⟩⟨‘|] (1)

Here, the topological charge q=1/2 denotes the net displacement of 2q=1 in the OAM space after each step, ‘ denotes the OAM of the incident photon, and L or R denote the SAM of the incident photon. This walking unit points to that after each step of random walk, the walker still carries a new coin in hand and has a uniform probability to decide the direction of the next step, which definitely constitutes the whole random walk process. Thus, the constructed fundamental wave is derived via the action of the walking unit on a linearly polarized Gaussian beam represented as |ψ0⟩ = (|L,0⟩ + |R,0⟩)/sqrt(2) (the notation |s,‘⟩ represents a direct product |s⟩ ⊗ |‘⟩ of a certain SAM and OAM state for convenience), as shown in Fig. 1b. The experimental spatial profile of the fundamental wave with a central singularity (first row of Fig. 2a) is prepared in the spanned 2⊗2 Hilbert space as follows:

|ψ1⟩ = U^|ψ0⟩ = i/2 |L,+1⟩ + 1/2 |L,-1⟩ + 1/2 |R,+1⟩ + i/2 |R,-1⟩ (2)

Despite the net OAM of the generated laser pulse being zero, it provides four eigenstates in both SAM and OAM subspaces with values of |s|=|‘|=1. Here, the relative phase among different angular momentum components is determined by the rotation angles of the QP and QWP. Although it introduces a global rotation in the far-field intensity profile (see Supplementary Note 1 for more details on the effect of relative phase), the probability distribution across OAM modes remains invariant, as it depends only on the square amplitude of each mode. These four fundamental photons with distinct SAMs and OAMs can be absorbed during the up-converted HHG process (see Methods for more details on the construction of incident vector beam). Although a coherent laser source rather than single-photon state is employed in experiments, the quantum characteristics of the random walk are preserved and fully captured through quantum interference among all possible pathways from a wave perspective23,29,40–43.

Demonstration of non-cascade random walks
By focusing the fundamental wave |ψ1⟩ into a bulk crystal of single-crystalline α-quartz oriented along the <0001> direction, we generate different-order harmonics as various steps of high-harmonic random walks. The inversion-symmetry-broken property of α-quartz enables one to observe both odd- and even-order harmonics44–47. The spatial and polarization profiles of different-order harmonics are crucial for characterizing the corresponding SAM and OAM states through the high-harmonic random walks. Here, polarization-resolved measurements are performed to identify six distinct polarization states of the harmonics. The spatial profiles of fundamental wave (H1), second harmonic (H2), third harmonic (H3), and fourth harmonic (H4) are illustrated in Figs. 2a, 2c, 2e, and 2g, respectively, where the first row displays the overall profile without any polarization selection, and the other six display the polarization-resolved spatial profiles of |ψH⟩, |ψV⟩, |ψD⟩, |ψA⟩, |ψL⟩, and |ψR⟩. In contrast to the pumping beam with a single singularity, various structured profiles are generated from distinct walking steps, which are determined by the intrinsic symmetry of the α-quartz. Taking the H2 emission as an example, the spatial profile with a central spot and four nodes around signifies the interference of the ‘=0,±2 components from the second step of high-harmonic random walks.

In order to gain insight into the characteristics of non-cascade high-harmonic random walks, it is essential to understand the nonlinear HHG process in the photon scenario with the principle of conservation of energy, momentum, and parity48–53. The incident vector laser pulse acting as a symmetric initial state in 2⊗2 angular momentum space can be divided as four fundamental waves with the same frequency and individual angular momentum components (j=1!4). Then, the harmonic frequency is described as the net absorption of the photons from the driving fields: Ωn = sum_{j} m_j ω_j with m_j and ω_j the absorbed photon number and frequency of the j-th driving photon, respectively, and n = sum_{j} m_j. The OAM of the harmonics is proved to be the summation of the OAM carried by each involved photon, which reads ‘_n = sum_{j} m_j ‘_j 54–57. However, the SAM of the harmonics is more complicated due to the spin restriction of the emitted harmonic photon. The selection rule for SAM of the harmonics considering the R-fold rotational symmetry of solids is expanded as s_n = sum_{j} s_j + kR = ±1 with k an arbitrary integer58,59. The selection rule derived theoretically from both classical tensor calculation60 and quantum Floquet theory61 shows that the extra angular momenta of photons are transferred to the electron wave packets with certain macroscopic symmetry via the coupling between the light field and materials. Thus, the yield of the n-order harmonic illumination of random walk can be described as

I_HHG(Ω_n) ∝ |sum_{sum m_j=n} σ_n prod_{j} e^{iφ_j} e^{i‘_jϕ} E_{m_j}^s_j |^2 = |sum_{sum m_j=n} σ_n e^{iφ_n} e^{i‘_nϕ} E_{m_L}^1+m_2 E_{m_R}^3+m_4 |^2 (3)

where σ_n = σ(n; m_1, m_2, m_3, m_4) is the ratio of the multiphoton absorption pathways introduced from the number of permutation, φ_j is the phase factor of each incident photon, φ_n is the overall phase factor of a certain multiphoton absorption pathway expressed as φ_n = sum_{j} m_j φ_j, ϕ is the azimuthal angle, and E_{m_j}^s_j is the electric field of the j-th photon component.

During the multiphoton absorption process, photons carrying angular momentum superposition state are absorbed indiscriminately by the electron in the crystal, which can be regarded as an analog to random walks in photonic systems (Detailed derivation and discussion about random-walk behavior during successive photon absorption are elaborated in Supplementary Note 2). The SAM of each involved photon acts as a two-sided coin to decide the walking direction, and the OAM of the harmonic acts as a walker in the one-dimensional space. Compared to conventional random walks, the step number in our work is clearly represented by an additional dimension—the harmonic order. Here we take the fundamental wave, second harmonic, and third harmonic as examples, as shown in Fig. 3. The fundamental wave labeled H1 is the standard product of the step operator on the Gaussian mode at the zero point, resulting in four individual positions in the spanned s⊗‘ Hilbert space shown in Fig. 3c. The measured polarization-resolved spatial profiles shown in Fig. 2a are in great accordance with the theoretically simulated ones shown in Fig. 2b. The second harmonic labeled H2 illustrated in Fig. 3d demonstrates the two-photon up-converted process as a result of sequential evolution of two steps in the s⊗‘ space. Although there are nine possible pathways in the Hilbert space, the pathways of absorbing three identical SAM photons are forbidden considering the selection rule s_n = ±1 discussed above. Meanwhile, the SAM of the second harmonic that absorbs two fundamental photons with the same SAM is deduced to be opposite. Thus, the two-step second-harmonic emission is written as

|ψH2⟩ = (-| -2⟩ + 2i|0⟩ + |+2⟩)|L⟩ + (| -2⟩ + 2i|0⟩ - |+2⟩)|R⟩ (4)

We present the theoretically simulated spatial profiles in Fig. 2d as a direct comparison with the experimental measurements. In the OAM space, the walker moves from the zero point towards (-1,+1) at the first step and (-2,0,+2) at the second step, and the walker at each OAM position has a two-sided coin at hand, which is actually the superimposed SAM states. This undoubtedly proves that the harmonics generated from a bulk crystal meet the fundamental criteria of the random walk and even expands the realization toward the frequency domain.

While for the third harmonic labeled H3 shown in Fig. 3e, the three-photon process is complex, and the populated Hilbert space becomes larger. Upon the absorption of three photons, there are sixteen possible pathways in the Hilbert space. However, the pathways of absorbing three identical SAM photons are forbidden considering the three-fold rotational symmetry of the α-quartz. Therefore, the third-harmonic emission can be derived as

|ψH3⟩ = (i|-3⟩ - |-1⟩ + i|+1⟩ - |+3⟩)|L⟩ + (-|-3⟩ + i|-1⟩ - |+1⟩ + i|+3⟩)|R⟩ (5)

the simulated results from which are illustrated in Fig. 2f. The great agreement between the experiments and simulations proves that the walker randomly moves in the OAM space with a two-sided coin at hand, providing an extra knob to steer the high-harmonic random walks. The OAM space is expanded into (-3, -1, +1, +3) after the three-step walk for the H3 emission. However, one can find that the walker still undergoes the complete second step during the H3 generation although it cannot reach the position of 0⊗‘ in H2 generation restricted by the selection rule. This, to our surprise, implies an advantageous non-cascade characteristic of the high-harmonic random walks in solids. All these random walks with different steps originate from the same initial state and will not interfere with the previous step. In contrast to the cascaded random walks in linear photonic systems, where sequential iterations are required, different-order high-harmonic random walks can be simultaneously generated.

OAM distributions of high-harmonic random walks
Due to the non-cascade property of the high-harmonic random walks, an important feature of the implementation occurs that the distribution probability of this type of random walk shown in Fig. 4a is quite different from the ones of previous classical and quantum walks, acting as a potential choice of the walking algorithm. To validate the proposed high-harmonic random walks, we have performed experimental measurements to resolve the OAM components of individual harmonic signals based on spin-orbit tomography62. The measured OAM distributions for high harmonics in Fig. 4b shows a good agreement with the simulated results in Fig. 4a (see “Methods” and Supplementary Note 4 for more details on the spin-orbit tomography of high harmonics). One can observe a Gaussian distribution of classical walks in OAM space with the increment of walking step, as shown in Figs. 4c, d, while the distribution of quantum walks shows a clear diffused characteristic in Fig. 4d. For the non-cascade random walks in solid-state HHG proposed here, the distribution demonstrates a complicated structure such as a flat-topped distribution for H3 in Fig. 4c and a diffused distribution for H4, the same as the four-step quantum walks, as shown in Fig. 4d. This can be understood as the synergy of the classical ratio stemming from multiphoton absorption pathways and the quantum selection rule depending on the macroscopic symmetry of the crystal.

Furthermore, our high-harmonic random walk exhibits a pronounced degree of programmability, achieved through active waveform manipulation of the driving laser field. For instance, by introducing a second-harmonic field with a specific polarization state, additional photon absorption pathways are activated, thereby modifying the output probability distribution. A theoretical demonstration of this programmability—using a two-color field to reconfigure the output states—is detailed in Supplementary Note 5. The adoption of an unbalanced coin with only a left-handed second-harmonic field leads to asymmetric walk distributions. Such control, extendable to other laser parameters like intensity and phase, underscore the potential of this non-cascade system for implementing diverse random walks and applications in parallel optical information processing.

Methods
Experimental technique
A linearly polarized infrared laser pulse centered at 1310 nm with a repetition rate of 10 kHz and a pulse duration of 45 fs (full width at half maximum) is produced from a commercial TOPAS-Prime in connection to a multipass amplified Ti-sapphire laser system. The linearly polarized Gaussian beam can be converted into a typical radially polarized beam with zero intensity at its center via passing through a QP with topological charge q=1/2, which has been widely applied in the realm of quantum information and high-harmonic spectroscopy. The generated vector beam is a superposition of two standard Laguerre-Gaussian modes of ‘=±1 with opposite circular polarizations s=∓1. However, the radially polarized beam only has two components in the spanned 2⊗2 Hilbert space such as |R, -1⟩ and |L, +1⟩, which is asymmetric for the random walk in OAM space. To maintain the basic symmetry in the Hilbert space, we construct a complex vector beam using an additional QWP after the propagation through the QP. The QWP with 0° rotational angle acts as a Hadamard gate operator to perform a rotation operation in SAM subspace, giving birth to a superposition state of both right and left circular polarization. The spatial and polarization profiles of the generated vector beam are demonstrated in Fig. 3a. The polarization state evolves from a right-hand circular one to a linear one, and back to a left-handed circular one when the azimuthal angle rotates from 0° to 90° in the first quadrant. The polarization states of the other three quadrants hold mirror symmetry compared to the adjacent quadrants and a two-fold rotational symmetry in the whole polarization plane.

The spatially structured laser beam is then focused with 25-cm focus length on an 80 μm α-quartz with the cut of <0001>, resulting in a peak intensity of ~5×10^13 W cm^-2 in the interaction region. The spatial structure of the crystal, belonging to the space group of P321, is depicted in Fig. 3b. The band gap of the α-quartz is ~9.5 eV and the in-plane rotational symmetry is three-fold (R=3). The inversion symmetry of the α-quartz is broken and thus both odd- and even-harmonics are observed. The generated harmonics are propagated through bandpass filters and imaged by a CMOS camera. The polarization states of these harmonics are identified by a combination of a broadband QWP and a Glan prism.

To determine the OAM components of the generated harmonics, we perform modal decomposition using a spatial light modulator (SLM). This technique compensates for the OAM of the collimated harmonics by applying a helical phase factor exp(iΔ‘ϕ), which transforms the initial state |s,‘⟩ into a modified state |s,‘+Δ‘⟩. The reflected harmonic beam is then Fourier-transformed by a lens and imaged onto a CMOS camera. The schematic of the experimental setup is shown in Supplementary Fig. 2. A distinct central spot is observed when the compensating topological charge Δ‘ equals the initial OAM value of the harmonic, enabling clear identification of the OAM component. For the fourth harmonic, whose wavelength falls outside the operation range of the SLM, a series of spiral phase plates with different topological charges is used instead to provide the required wavefront modulation.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_2": "The article describes high-harmonic generation in nonlinear crystals in terms of quantum walks. Specifically, it analyzes the\nspin and orbital angular momentum (SAM and OAM) spectrum of each harmonic and shows that, as the harmonic order\nincreases, the corresponding state in the SAM–OAM space exhibits features characteristic of quantum walks when the step\nnumber is identified with the harmonic order. Experimental results are presented for output states containing up to four\nharmonic orders.\nThe system is introduced as a platform for realizing quantum walks in a non-cascaded configuration, since all step numbers\nare implemented within a single crystal rather than through a sequence of separate devices.\nThe general idea is interesting; however, in my opinion it does not meet the impact criteria required for publication in Nature\nCommunications. To justify publication, the authors should provide stronger evidence that their approach constitutes a\ngenuine alternative to standard cascaded platforms for optical information processing. Currently, this does not appear to be\nthe case: the system does not seem scalable, it cannot be applied to single-photon states, and it lacks programmability. I will\nelaborate on these points in the detailed remarks below.\nMajor remarks\n- In a quantum walk, the state at step j is obtained from the state at step j–1 through the action of a single-step unitary\noperator. In the present work, I see how the state at a given harmonic order is linked to the input state via n-photon\nabsorption and selection rules. However, is this truly equivalent to a quantum walk? Is there a well-defined analogue of the\nsingle-step operator here, linking two states at consecutive time-steps?\n- How programmable is this platform? In standard quantum-walk implementations, tuning the optical parameters of\nsuccessive layers allows one to realize arbitrary walks. How could different types of quantum walks be accessed in the\npresent scheme?\n- What is represented on the horizontal axis of Figs. 3(C–E)? If s denotes the SAM index, it should only take two values. My\nunderstanding is that the walk occurs on a one-dimensional lattice spanned by the OAM modes.\n- How scalable is this approach? What is the main factor limiting the harmonic order? How could higher orders be\naccessed?\n- The spectra shown in Fig. 4 are only theoretical. An experimental measurement of these spectra would considerably\nstrengthen the work, especially if the analysis is extended to different input states.\n- Why are both a quarter-wave plate and a q-plate required to prepare an input state different from a single Gaussian beam\nwith l=0?\n- What is the role of the crystal symmetries? Would different symmetries lead to different walks?\nFor the reasons outlined above, I cannot recommend this manuscript for publication in Nature Communications. In my view,\nits main contribution lies in the interpretation of the high-harmonic generation spectrum within the framework of a quantum\nwalk. After appropriate revisions, the work could be better published in a more specialized technical journal.", "reviewer_3": "The authors demonstrate a non-cascaded, high-dimensional random walk in the orbital angular momentum (OAM) space of\nlight using solid-state high-harmonic spectroscopy. By utilizing a highly nonlinear process in a crystal, they achieve\nsimultaneous multi-photon conversion into a series of harmonics at distinct wavelengths, with the OAM distribution governed\nby the crystal’s symmetry. This approach provides a promising route toward compact and stable integrated photonic\nplatforms for advanced information processing. As a fundamentally new strategy for realizing random walks without\ncascaded operations, this work exhibits high novelty and is likely to open new avenues for related research. Overall, the\nmanuscript presents substantial value and will attract broad interest. The paper is also well-structured and supported by\nrobust experimental data. I am pleased to recommend its publication in Nature Communications once the following major\nconcerns are adequately addressed:\n1. As is well known, the conversion efficiency typically decreases significantly with increasing harmonic order. Since the\nproposed non-cascaded random walk scheme relies on high-order nonlinear processes—where the harmonic order directly\ncorresponds to the number of steps in the random walk—it is essential to include efficiency data in the main text. Providing\neither experimental or theoretical conversion efficiencies for each harmonic order would enable readers to better evaluate\nthe practicality and scalability of this approach, and would facilitate future applications and follow-up studies by the\ncommunity.\n2. Although the authors have provided a description of the experimental methods in the latter part of the manuscript, the\nprocess remains somewhat challenging to follow—particularly given the multiple transformations involving orbital and spin\nangular momentum (OAM and SAM). To improve clarity and accessibility for a broader readership, it would be highly\nbeneficial to include a schematic figure illustrating the experimental setup, with clear visual annotations of the state evolution\nthroughout the process.\n3. The manuscript does not provide a detailed explanation of the method used to measure the orbital angular momentum\n(OAM) intensity (or probability distribution) in the experimental setup. Given the central role of OAM in this study, it is\nessential to include a clear description of the measurement technique. Additionally, the potential influence of the relative\nphase among different OAM modes on the observed results should be briefly discussed. I guess the relative phase among\ndifferent OAM modes will not influence the final results, since it is irrelevant to the probability?\n4. A full characterization of polarization typically requires six measurements across the HV, ±45° (H±V), and circular (RL)\nbases. In Figure 2, however, intensity distributions are presented under only four polarization bases. Could the authors\nclarify whether a complete Stokes parameter measurement was deemed unnecessary in this context?"}, "author_response": "Response to Reviewers’ Comments:\nIn this response letter, we put the original comments by the Reviewers in italics to distinguish them\nfrom our responses in blue. Changes to the manuscript appear in red.\nReply to Reviewer #2\nThe article describes high-harmonic generation in nonlinear crystals in terms of quantum walks.\nSpecifically, it analyzes the spin and orbital angular momentum (SAM and OAM) spectrum of each\nharmonic and shows that, as the harmonic order increases, the corresponding state in the SAM–OAM\nspace exhibits features characteristic of quantum walks when the step number is identified with the\nharmonic order. Experimental results are presented for output states containing up to four harmonic\norders.\nThe system is introduced as a platform for realizing quantum walks in a non-cascaded configuration,\nsince all step numbers are implemented within a single crystal rather than through a sequence of\nseparate devices.\nThe general idea is interesting; however, in my opinion it does not meet the impact criteria required\nfor publication in Nature Communications. To justify publication, the authors should provide stronger\nevidence that their approach constitutes a genuine alternative to standard cascaded platforms for\noptical information processing. Currently, this does not appear to be the case: the system does not\nseem scalable, it cannot be applied to single-photon states, and it lacks programmability. I will\nelaborate on these points in the detailed remarks below.\nReply: We appreciate the Reviewer for recognizing the novelty of our general idea and for providing\ninsightful comments regarding scalability and programmability in our non-cascade scheme. We agree\nthat demonstrating a direct replacement for existing technologies is a high bar and would be a\nsignificant milestone. We would like to clarify, however, that the primary goal of this work is not to\npropose an immediate alternative for current devices, but to establish a new fundamental paradigm\nof non-cascade random walk via the multiphoton process of high-harmonic generation (HHG).\nThe approach proposed in our work is fundamentally distinct from standard cascaded quantum walks,\nwhich typically rely on linear optical elements arranged in cascaded interferometric networks. Such\nimplementations face inherent challenges in scalability, as system expansion demands substantial\nphysical space and highly stable environmental conditions. In contrast, our approach leverages the\nintrinsic nonlinear optical response of a solid-state medium, where HHG enables the simultaneous\ngeneration of multiple harmonic orders within a single crystal. This unique mechanism allows parallel\ninformation processing in a compact, integrated, and inherently stable configuration, without the\nneed for cascaded components.\nOur scheme circumvents several limitations of cascaded architectures and introduces a new\nparadigm for studying random-walk phenomena. This work helps bridge ultrafast science with\ninformation processing, and may inspire future developments in solid-state-based parallel\ncomputing—a point also appreciated by Reviewer 3.\nWe appreciate the Reviewer's comments regarding scalability, programmability, and applicability to\nsingle-photon states. While we will address scalability and programmability separately in subsequent\nresponses, we would like to reply to the question regarding single-photon applicability. A single-\nphoton source is not a necessary condition for constructing random walks. In optical\nimplementations of random walks, the walking unit operates identically whether considered at the\nmultiphoton level (i.e., coherent laser beams from a wave perspective) or at the single-photon level\n(i.e., single photons from a particle perspective). From the wave perspective, quantum characteristics\nin random walks are preserved through quantum interference among multiple possible pathways.\nThis viewpoint is supported by a pioneering study by Goyal et al. [Phys. Rev. Lett. 110, 263602 (2013)],\nwhich demonstrated that random walks can be efficiently realized using coherent laser beams,\ninterpreted via wave optics. Similar works have successfully employed coherent laser beams to\nimplement random walks and related quantum-walk phenomena, including: Phys. Rev. A 75, 052310\n(2007), Phys. Rev. Lett. 104, 050502 (2010), Phys. Rev. Lett. 106, 180403 (2011), Science 336, 55\n(2012), Phys. Rev. Lett. 110, 263602 (2013), and Phys. Rev. Lett. 121, 100502 (2018). These studies\nconfirm that the use of coherent light does not preclude the observation of essential quantum-walk\nfeatures, as the underlying interference mechanisms are captured equally in a wave-based\ndescription.\nTo make it clear to the readers, we added the following sentences in the revised manuscript.\n“Although a coherent laser source rather than single-photon state is employed in experiments, the\nquantum characteristics of the random walk are preserved and fully captured through quantum\ninterference among all possible pathways from a wave perspective (23, 29, 40-43).” (Line 17 on page\n4)\nComment #1: In a quantum walk, the state at step j is obtained from the state at step j–1 through the\naction of a single-step unitary operator. In the present work, I see how the state at a given harmonic\norder is linked to the input state via n-photon absorption and selection rules. However, is this truly\nequivalent to a quantum walk? Is there a well-defined analogue of the single-step operator here,\nlinking two states at consecutive time-steps?\nReply: We thank the Reviewer for raising these important questions. We would like to clarify two key\npoints in response.\nFirst, as emphasized earlier, the high-harmonic random walk realized in our work–arising from\nnonlinear electron dynamics in a crystal–is not equivalent to a conventional quantum walk. Rather,\nit represents a distinct class of non-cascade random walk rooted in solid-state HHG process.\nSecond, although we don’t have standard single-step operator linking different harmonic orders,\nthere exists a well-defined equivalent that connects various photon absorption processes. Specifically,\nthe HHG process can be interpreted in terms of cascaded N-photon absorption, governed by the\nfollowing relation:\n2\n𝛽(𝑁) = |∑ 𝑃 (𝑁) | .\nm1+m2+m3+m4=N m1𝜎1+m2𝜎2+m3𝜎3+m4𝜎4\nThis can be recursively expressed as\n(𝑁) (𝑁−1) (𝑁−1)\n𝑃 = 𝐶 𝑃 +𝐶 𝑃 +\nm1𝜎1+m2𝜎2+m3𝜎3+m4𝜎4 1 (m1−1)𝜎1+m2𝜎2+m3𝜎3+m4𝜎4 2 m1𝜎1+(m2−1)𝜎2+m3𝜎3+m4𝜎4\n(𝑁−1) (𝑁−1)\n𝐶 𝑃 +𝐶 𝑃 ,\n3 m1𝜎1+m2𝜎2+(m3−1)𝜎3+m4𝜎4 4 m1𝜎1+m2𝜎2+m3𝜎3+(m4−1)𝜎4\n(𝑁)\nwhere 𝑃 denotes the transition amplitude for absorbing 𝑚 photons of angular momentum 𝜎\nm 𝜎 𝑗 𝑗\n𝑗 𝑗\nin generating the N-th harmonic, and 𝐶 accounts for the number of quantum pathways leading to\n𝑗\nthe transition. This formulation captures a cascaded absorption process, analogous to a step operator\nlinking harmonic order N−1 to N. A similar interpretation has been established in the context of\nelectron-light interactions, where sequential photon absorption gives rise to a random walk in the\nelectron state space, as demonstrated in Science 373, eabj7128 (2021). In our solid-state system, the\nsituation is further enriched: beyond multiphoton absorption, the process also involves high-\nharmonic photon emission. This enables simultaneous, non-cascade emission across multiple\nharmonic orders, distinguishing our platform from conventional step-by-step quantum walks while\npreserving an underlying recursive structure.\nTo make it clear to the readers, we rewrote the following sentence in the revised manuscript, and\nadded a detailed discussion in Supplementary Materials.\n“During the multiphoton absorption process, photons carrying angular momentum superposition\nstate are absorbed indiscriminately by the electron in the crystal, which can be regarded as an analog\nto random walks in photonic systems (Detailed derivation and discussion about random-walk\nbehavior during successive photon absorption are elaborated in Supplementary Materials).” (Line 8\non page 6)\nComment #2: How programmable is this platform? In standard quantum-walk implementations,\ntuning the optical parameters of successive layers allows one to realize arbitrary walks. How could\ndifferent types of quantum walks be accessed in the present scheme?\nReply: We thank the Reviewer for this important point regarding programmability, which enables us\nto extend the approach. Indeed, our high-harmonic random walk platform can be effectively\nprogrammed using well-established optical-field-control techniques from ultrafast optics.\nA representative example is introducing a second-harmonic (SH) laser field, which actively modulates\nthe electron dynamics within the crystal and thereby enables access to different types of random\nwalks. In addition to the pathways enabled by the fundamental laser field alone, the inclusion of an\nSH field with a state such as |𝜓 ⟩ = |𝐿,+2⟩+|𝑅,−2⟩ opens new photon absorption channels\nSH\ngoverned by\nΩ = ∑ 𝑚 𝜔 +∑ 𝑛 (2𝜔) .\nn 𝑗 𝑗 𝑗 𝑘 𝑘 𝑘\nHere, 𝑚 and 𝜔 denote the number and frequency of absorbed 𝑗-th fundamental photon, 𝑛\n𝑗 𝑗 𝑘\nand (2𝜔) refer to the number and frequency of absorbed 𝑘-th SH photon.\n𝑘\nTo illustrate this programmability, we consider the generation of the third harmonic (H3) and assume\nequal contribution from two possible absorption pathways: 𝜔+𝜔+𝜔 and 𝜔+2𝜔. Under these\nconditions, the resulting H3 signal can be described as\n|𝜓 ⟩ = (5𝑖|−3⟩−|−1⟩+3𝑖|+1⟩−3|+3⟩)|𝐿⟩\nH3\n+(−3|−3⟩+3𝑖|−1⟩−|+1⟩+5𝑖|+3⟩)|𝑅⟩.\nFigure R1a illustrates the resulting OAM distributions of H3 driven by a fundamental field (single-\ncolor in blue) and a fundamental field with an SH field (two-color in red), respectively. Notably, if the\nSH field contains only a left-handed circularly polarized component, i.e., effectively implementing an\nunbalanced coin, the output distribution becomes asymmetric, as yellow bars shown in Fig. R1a.\nSimilarly, the corresponding OAM distributions of the fourth harmonic as the fourth-step random\nwalk is shown in Fig. R1b.\nThis confirms that through appropriate tailoring of the driving laser fields—such as frequency,\npolarization, or intensity—our platform can be reconfigured to realize a range of distinct random-\nwalk behaviors, thereby demonstrating a meaningful degree of programmability.\nFig. R1. Probability distribution of the (a) third-steep and (b) fourth-step high-harmonic\nrandom walks driven by a single-color field (blue), a balanced two-color field (red), and an\nunbalanced two-color field (yellow), respectively.\nFollowing the suggestion from the Reviewer and making it clear to the readers, we have added the\nfollowing sentences in the revised manuscript, and added a detailed discussion in Supplementary\nMaterials.\n“Furthermore, our high-harmonic random walk exhibits a pronounced degree of programmability,\nachieved through active waveform manipulation of the driving laser field. For instance, by\nintroducing a second-harmonic field with a specific polarization state, additional photon absorption\npathways are activated, thereby modifying the output probability distribution. A theoretical\ndemonstration of this programmability—using a two-color field to reconfigure the output states—is\ndetailed in Supplementary Materials. The adoption of an unbalanced coin with only a left-handed\nsecond-harmonic field leads to asymmetric walk distributions. Such control, extendable to other laser\nparameters like intensity and phase, underscores the potential of this non-cascade system for\nimplementing diverse random walks and applications in parallel optical information processing.”\n(Line 19 on page 8)\nComment #3: What is represented on the horizontal axis of Figs. 3(C–E)? If s denotes the SAM index,\nit should only take two values. My understanding is that the walk occurs on a one-dimensional lattice\nspanned by the OAM modes.\nReply: We thank the Reviewer for the careful notice. The horizontal axis of Figs. 3(C-E) represents the\naccumulated spin angular momenta, denoted as ∑𝑆, acquired through the absorption of multiple\nphotons. To avoid any potential confusion, we have updated the axis label in the revised figures\naccordingly.\nComment #4: How scalable is this approach? What is the main factor limiting the harmonic order?\nHow could higher orders be accessed?\nReply: We thank the Reviewer for raising this question regarding scalability. The scalability of our\napproach—specifically, access to higher harmonic orders—can be effectively achieved by increasing\nthe intensity and wavelength of the driving laser field. The maximum attainable harmonic order is\nfundamentally limited by the relation 𝑁 ≈ Δ𝐸/ℏ𝜔, where ∆E represents the energy difference\n𝑚𝑎𝑥\nbetween the valence band and the relevant conduction band to which an electron can be excited,\nand ω is the driving laser frequency. For instance, harmonic generation up to the 31st order has been\ndemonstrated in MgO using such methods, as reported in Opt. Lett. 50, 1492 (2025). The use of\nintense terahertz driving lasers could further extend this cutoff. This result strongly suggests that our\nnon-cascade scheme can be directly extended to higher orders. With the ongoing advancement of\nintense mid-infrared femtosecond laser sources, accessing even higher-order harmonics is becoming\nincreasingly feasible.\nTo clearly convey this point, we have added the following sentences in the revised manuscript.\n“This architecture can be scaled to support a greater number of steps by leveraging established\ntechniques—such as increasing the driving laser intensity and wavelength (37, 39, 44, 63)” (Line 11\non page 9)\nComment #5: The spectra shown in Fig. 4 are only theoretical. An experimental measurement of these\nspectra would considerably strengthen the work, especially if the analysis is extended to different\ninput states.\nReply: We thank the Reviewer for the valuable suggestion. To further validate the proposed high-\nharmonic random walks, we have performed experimental measurements to resolve the OAM\ncomponents of individual harmonic signals.\nThe OAM states of these harmonics are analyzed using a spatial light modulator (for the second and\nthird harmonics) or custom-fabricated spiral phase plates (for the fourth harmonic) by applying a\nhelical phase factor exp(𝑖Δℓ𝜙). This phase modulation transforms the collimated harmonic beam\nfrom an initial state (𝑠,ℓ) into a modified state (𝑠,ℓ+Δℓ). The reflected or transmitted beam is\nthen Fourier-transformed by a lens and imaged onto a CMOS camera. A distinct central spot is\nobserved when the compensating topological charge Δℓ equals the initial OAM value of the\nharmonic, enabling clear identification of the OAM component.\nFig. R2. Experimental spin-orbit tomography for high-harmonic random walks. The top axis\nindicates the compensating topological charge Δℓ applied via the spatial light modulator\nor spiral phase plates. Red squares highlight the configurations with distinct central spots,\nindicating the presence of a corresponding OAM state. Quantitative analysis of each OAM\ncomponent is based on the integration of central zones in the images, marked by white\ndashed circles.\nFigure R2 presents the experimental spin-orbit tomography for the high harmonic outputs. Although\nsome discrepancies are observed, attributed to inhomogeneous harmonic emission patterns and the\ncomplexity of spin-orbit components, the measured OAM distributions show a good agreement with\nthe simulated results, as summarized in Fig. R3.\nFig. R3. (a) Simulated and (b) experimental high-harmonic random walks.\nFollowing the suggestion from the Reviewer, we added the following sentences in the revised\nmanuscript, updated Figure 4, and added a detailed discussion in Supplementary Materials.\n“To determine the OAM components of the generated harmonics, we perform modal decomposition\nusing a spatial light modulator (SLM). This technique compensates for the OAM of the collimated\nharmonics by applying a helical phase factor exp(𝑖Δℓ𝜙), which transforms the initial state (𝑠,ℓ)\ninto a modified state (𝑠,ℓ+Δℓ). The reflected harmonic beam is then Fourier-transformed by a lens\nand imaged onto a CMOS camera. A distinct central spot is observed when the compensating\ntopological charge Δℓ equals the initial OAM value of the harmonic, enabling clear identification of\nthe OAM component. For the fourth harmonic, whose wavelength falls outside the operation range\nof the SLM, a series of spiral phase plates with different topological charges is used instead to provide\nthe required wavefront modulation.” (Line 26 on page 10)\n“To validate the proposed high-harmonic random walks, we have performed experimental\nmeasurements to resolve the OAM components of individual harmonic signals based on spin-orbit\ntomography (62). Figure 4b summarizes the measured OAM distributions for high harmonics, which\nshows a good agreement with the simulated results shown in Fig. 4a (see Methods and\nSupplementary Materials for more details on the spin-orbit tomography of high harmonics).” (Line 4\non page 8)\nComment #6: Why are both a quarter-wave plate and a q-plate required to prepare an input state\ndifferent from a single Gaussian beam with l=0?\nReply: We thank the Reviewer for raising the question. In our non-cascade scheme, the entire\nmultiple walk process is implemented within a single nonlinear crystal, rather than through a\nsequence of discrete optical elements. Consequently, the input state must be pre-configured to\nincorporate both the coin-flip operator in the SAM space (prepared by the quarter-wave plate) and\nthe conditional shift operator in the OAM space (prepared by the q-plate). This ensures that spin and\norbital states evolve simultaneously during the high-harmonic generation process. This approach of\ncombining a quarter-wave plate and a q-plate to prepare the initial walker state is consistent with\nestablished protocols, as demonstrated, for example, in Phys. Rev. Lett. 110, 263602 (2013).\nTo improve clarity in the revised manuscript, we have added the following sentence.\n“The absence of subsequent optical elements in non-cascade implementation necessitates pre-\nconfiguring the entire propagation operator in the input state.” (Line 18 on page 3)\nComment #7: What is the role of the crystal symmetries? Would different symmetries lead to different\nwalks?\nReply: We thank the Reviewer for raising the question. Indeed, crystal symmetry plays a fundamental\nrole in our high-harmonic random walks, as it governs the selection rules that shape the underlying\ndynamics. In solid-state high-harmonic generation, the SAM selection rule follows ∑ 𝑠 +𝑘𝑅 =\n𝑗 𝑗\n±1, where 𝑘 is an integer and 𝑅 denotes the rotational symmetry of the crystal. This rule,\ncombined with OAM conservation, determines the accessible pathways and resulting walk behavior.\nTo illustrate this, we compare three representative wide-bandgap semiconductors with distinct\nrotational symmetries: MgO (<001>) with four-fold symmetry (R=4), ZnO (<0001>) with six-fold\nsymmetry (R=6), and SiO₂ (<0001>) with three-fold symmetry (R=3). Using identical driving laser\nparameters as in the manuscript, we summarize in Table R1 the resulting OAM distributions of the\nthird harmonic signal for each crystal. These results clearly show that different symmetries lead to\ndistinct random walks—confirming that crystal symmetry serves as a key design parameter for\ntailoring random-walk behavior in our platform.\nTable R1. OAM distributions for third harmonic using different crystals.\nProbability of OAM states\nCrystal |−3⟩ |−1⟩ |+1⟩ |+3⟩\nSiO (3-fold) 1 1 1 1\nZnO (6-fold) 0 0 0 0\nFor the reasons outlined above, I cannot recommend this manuscript for publication in Nature\nCommunications. In my view, its main contribution lies in the interpretation of the high-harmonic\ngeneration spectrum within the framework of a quantum walk. After appropriate revisions, the work\ncould be better published in a more specialized technical journal.\nReply: In summary, we sincerely thank the Reviewer for recognizing the novelty of our general idea\nand for providing these constructive suggestions, which have been invaluable in strengthening the\npresentation of our work and inspiring future applications.\nFollowing the suggestions from the Reviewer, we have thoroughly revised the manuscript to:\n1) Clarify the conceptual foundation of the non-cascaded scheme;\n2) Demonstrate the scalability and programmability of the high-harmonic random walk platform;\n3) Perform OAM-resolved measurements that confirm the predicted OAM distributions.\nFundamentally, this work extends—for the first time—the concept of a random walk into the high-\nharmonic generation regime, realizing a frequency-resolved, non-cascade walk within a single\nnonlinear crystal. It enables a robust form of walking that operates at the femtosecond timescales\nintrinsic to HHG. This approach establishes a new paradigm for ultrafast walking simulations in\nsolids, beyond conventional cascaded architectures.\nFurthermore, our results represent a novel demonstration of algorithmic simulation based on light–\nmatter interaction, effectively bridging ultrafast physics with quantum information science. By\nleveraging the scalability and chip compatibility of solid-state systems—including bulk crystals, 2D\nmaterials, and metasurfaces—this platform opens promising pathways toward future on-chip\nquantum computation and information processing in the frequency domain.\nWe hope that the detailed responses and substantially improved manuscript adequately address\nthe Reviewer’s concerns, and we believe the work in its current form offers a timely and distinctive\ncontribution suitable for publication in Nature Communications.\nReply to Reviewer #3\nThe authors demonstrate a non-cascaded, high-dimensional random walk in the orbital angular\nmomentum (OAM) space of light using solid-state high-harmonic spectroscopy. By utilizing a highly\nnonlinear process in a crystal, they achieve simultaneous multi-photon conversion into a series of\nharmonics at distinct wavelengths, with the OAM distribution governed by the crystal’s symmetry.\nThis approach provides a promising route toward compact and stable integrated photonic platforms\nfor advanced information processing. As a fundamentally new strategy for realizing random walks\nwithout cascaded operations, this work exhibits high novelty and is likely to open new avenues for\nrelated research. Overall, the manuscript presents substantial value and will attract broad interest.\nThe paper is also well-structured and supported by robust experimental data. I am pleased to\nrecommend its publication in Nature Communications once the following major concerns are\nadequately addressed:\nReply: We sincerely appreciate the Reviewer for the positive evaluation of our manuscript and for\nrecommending its publication in Nature Communications. We are greatly encouraged by the\nReviewer's assessment that our approach “provides a promising route toward compact and stable\nintegrated photonic platforms for advanced information processing” and represents “a\nfundamentally new strategy for realizing random walks without cascaded operations.”\nIn response to the Reviewer’s valuable comments, we have provided detailed point-by-point\nresponses below and have accordingly revised the manuscript to fully address all the suggestions\nraised.\nComment #1: As is well known, the conversion efficiency typically decreases significantly with\nincreasing harmonic order. Since the proposed non-cascaded random walk scheme relies on high-\norder nonlinear processes—where the harmonic order directly corresponds to the number of steps in\nthe random walk—it is essential to include efficiency data in the main text. Providing either\nexperimental or theoretical conversion efficiencies for each harmonic order would enable readers to\nbetter evaluate the practicality and scalability of this approach, and would facilitate future\napplications and follow-up studies by the community.\nReply: We thank the Reviewer for this helpful suggestion. Following the suggestion, we have included\nthe experimental conversion efficiencies for each harmonic order (prior to normalization) in the\nharmonic spectrum presented in Figure 2 and rewrote the following sentence in the revised\nmanuscript.\n“The bottom inset is the normalized high-harmonic spectrum measured in experiments, and the\nnormalization coefficients as the signature of conversion efficiencies are indicated on the right.” (Line\n12 on page 20)\nComment #2: Although the authors have provided a description of the experimental methods in the\nlatter part of the manuscript, the process remains somewhat challenging to follow—particularly given\nthe multiple transformations involving orbital and spin angular momentum (OAM and SAM). To\nimprove clarity and accessibility for a broader readership, it would be highly beneficial to include a\nschematic figure illustrating the experimental setup, with clear visual annotations of the state\nevolution throughout the process.\nReply: We thank the Reviewer for this helpful suggestion. Following the suggestion, we have\nadded a schematic figure (shown here as Fig. R4) illustrating the experimental setup with clear\nvisual annotations of state evolution in Supplementary Materials.\nFig. R4. Schematic of the experimental setup. An incident linearly polarized Gaussian beam\nis converted into a radially polarized beam by a q-plate (QP). The beam is then passed\nthrough a quarter-wave plate (QWP), resulting in a complex vector beam whose polarization\nellipticity varies continuously with the azimuthal angle, cycling from linear to circular and\nback to linear. This vector beam is focused by a lens onto a bulk α-quartz crystal to generate\nhigh harmonics. Each harmonic beam can be selected using a band-pass filter. The\npolarization state of the selected harmonic is analyzed using a combination of a QWP and a\npolarizer. Meanwhile, its OAM components are characterized by diffraction from a spatial\nlight modulator (SLM). The resulting beam profile is finally focused onto a CMOS camera for\ndetection.\nComment #3: The manuscript does not provide a detailed explanation of the method used to\nmeasure the orbital angular momentum (OAM) intensity (or probability distribution) in the\nexperimental setup. Given the central role of OAM in this study, it is essential to include a clear\ndescription of the measurement technique.\nReply: We thank the Reviewer for this helpful suggestion. To prove the reliability of our results, we\nhave added the description of OAM measurement technique in the revised manuscript.\n“To determine the OAM components of the generated harmonics, we perform modal decomposition\nusing a spatial light modulator (SLM). This technique compensates for the OAM of the collimated\nharmonics by applying a helical phase factor exp(𝑖Δℓ𝜙), which transforms the initial state (𝑠,ℓ)\ninto a modified state (𝑠,ℓ+Δℓ). The reflected harmonic beam is then Fourier-transformed by a lens\nand imaged onto a CMOS camera. A distinct central spot is observed when the compensating\ntopological charge Δℓ equals the initial OAM value of the harmonic, enabling clear identification of\nthe OAM component. For the fourth harmonic, whose wavelength falls outside the operation range\nof the SLM, a series of spiral phase plates with different topological charges are used instead to\nprovide the required wavefront modulation.” (Line 26 on page 10)\nComment #4: Additionally, the potential influence of the relative phase among different OAM modes\non the observed results should be briefly discussed. I guess the relative phase among different OAM\nmodes will not influence the final results, since it is irrelevant to the probability?\nReply: We thank the Reviewer for raising the question. In the multiphoton absorption process of HHG,\nthe phase coherence of the fundamental driving laser is inherently preserved and directly transferred\nto the emitted harmonics. Taking the fundamental beam as an example, when the rotation angle of\nQP is denoted as 𝜃 and the rotation angle of QWP is denoted as 𝛽, the constructed fundamental\nbeam can be expressed as\n|𝜓 ⟩ = 𝑖 𝑒𝑖𝜃|𝐿,+1⟩+ 1 𝑒−𝑖(𝜃+2𝛽)|𝐿,−1⟩+ 1 𝑒𝑖(𝜃−2𝛽)|𝑅,+1⟩+ 𝑖 𝑒−𝑖𝜃|𝑅,−1⟩.\nH1\n2 2 2 2\nThe resulting second harmonic beam is given by\n|𝜓 ⟩ = (−𝑒−2𝑖𝜃|−2⟩+2𝑖𝑒−2𝑖𝛽|0⟩+𝑒𝑖(2𝜃−4𝛽)|+2⟩)|𝐿⟩+\nH2\n(𝑒−𝑖(2𝜃−4𝛽)|−2⟩+2𝑖𝑒2𝑖𝛽|0⟩−𝑒2𝑖𝜃|+2⟩)|𝑅⟩.\nAs correctly noted by the Reviewer, although the relative phase among OAM modes influences the\nspatial structure of the harmonic field—resulting in a global rotation of the far-field intensity profile,\nas illustrated in Fig. R5—it does not alter the probability distribution across OAM modes. This is\nbecause the observable probability in the OAM basis depends only on the squared amplitude of each\nmode and remains invariant under global rotation or relative phase shifts. Therefore, while the\nrelative phase can modulate the spatial arrangement of the output, it does not affect the core\nstatistical outcomes of the random walk, which are the focus of this study.\nFollowing the suggestions from the Reviewer, we have added the following sentence in the revised\nmanuscript and added a detailed discussion in Supplementary Materials.\n“Here, the relative phase among different angular momentum components is determined by the\nrotation angles of the QP and QWP. Although it introduces a global rotation in the far-field intensity\nprofile (see Supplementary Materials for more details on the effect of relative phase), the probability\ndistribution across OAM modes remains invariant, as it depends only on the square amplitude of\neach mode.” (Line 10 on page 4)\nFig. R5. Influence of relative phase among OAM modes on the resulting second-harmonic\nintensity profiles. White dashed lines indicate the corresponding global rotation of the far-\nfield patterns.\nComment #5: A full characterization of polarization typically requires six measurements across the\nHV, ±45° (H±V), and circular (RL) bases. In Figure 2, however, intensity distributions are presented\nunder only four polarization bases. Could the authors clarify whether a complete Stokes parameter\nmeasurement was deemed unnecessary in this context?\nReply: We thank the Reviewer for this valuable suggestion. We fully agree that a complete\npolarization characterization requires Stokes parameter measurements across six distinct bases.\nFollowing the suggestion from the Reviewer, we have performed additional measurements at ±45°\n(H±V) bases to complement the original HV and RL basis data and have updated accordingly in the\nrevised manuscript (see Fig. R6 below for reference). This complete measurement provides a rigorous\ncharacterization of the polarization state for each harmonic order.\nFig. R6. Experimental observations and theoretical predictions of non-cascade high-\nharmonic random walks across six polarization bases.\nResponse to Reviewers’ Comments:\nWe thank the referees for reviewing our manuscript and for their thoughtful comments and\nsuggestions. We have revised the manuscript point by point according to their recommendations. In\nthis response letter, we put the original comments by the Reviewers in italics to distinguish them from\nour responses in blue. Changes to the manuscript appear in red.\nReply to Reviewer #2\nThe authors have provided thorough and constructive responses to all comments raised by the\nreferees. The manuscript has been significantly improved and now presents its main results in a\nclearer and more coherent manner. While I remain only partially convinced by the claims concerning\nthe broader impact of the results, I find the core contribution solid and well supported.\nI appreciate the novelty of the approach and the insight that the dynamics of photonic degrees of\nfreedom in high harmonic generation can be naturally framed in terms of quantum walks. The\nexperimental procedures appear sound and technically robust. On this basis, I recommend publication.\nThat said, I encourage the authors to revise the contextualization of their results. Beginning with the\nabstract, the comparison with the state of the art is presented from a somewhat misleading\nperspective. Although it is correct that cascaded systems face significant limitations for large scale\nintegrated photonic circuits, I believe the present work should not be framed as a scalable alternative\nto such platforms. I am skeptical that step numbers exceeding those achievable with cascaded\napproaches can be realistically realized in this setting.\nA clearer emphasis on the conceptual and physical innovation of the work, namely the emergence of\na mechanism that can be interpreted as a quantum walk, together with the distinctive\nprogrammability enabled by pulse shaping techniques, would strengthen the manuscript and place\nits contributions in a more accurate context.\nReply: We thank the Reviewer for the insightful summary of our modifications and recommending\nthe publication of our manuscript in Nature Communications. Following the kind suggestions from\nthe Reviewer, we have rewritten the abstract in the revised manuscript as below to emphasize the\nnatural framework of high-harmonic generation instead of the alternative to conventional cascaded\nplatforms.\n“This approach reveals the dynamics of photonic degrees of freedom in high-harmonic generation\ncan be naturally framed as an ultrafast, high-dimensional random walk, paving the way for compact,\nhighly stable photonic platforms tailored for solid-state information processing.” (Abstract on page\n1)\nReply to Reviewer #3\nAll my concerns have been fully addressed, and the manuscript now meets the high standards\nrequired for publication in Nature Communications.\nIn terms of novelty, the authors leverage the intrinsic nonlinear optical response of a solid-state\nmedium, using high-harmonic generation to simultaneously produce multiple harmonic orders within\na single crystal. This unique approach enables parallel information processing in a compact,\nintegrated, and inherently stable configuration, eliminating the need for cascaded components. The\nlevel of innovation demonstrated here is substantial enough to warrant publication.\nTherefore, I strongly recommend that the manuscript be accepted for publication in Nature\nCommunications.\nReply: We thank the Reviewer for kindly recommending the publication of our work in Nature\nCommunications."}, {"round": 2, "reviewer_comments": {"reviewer_2": "The authors have provided thorough and constructive responses to all comments raised by the referees. The manuscript has\nbeen significantly improved and now presents its main results in a clearer and more coherent manner. While I remain only\npartially convinced by the claims concerning the broader impact of the results, I find the core contribution solid and well\nsupported.\nI appreciate the novelty of the approach and the insight that the dynamics of photonic degrees of freedom in high harmonic\ngeneration can be naturally framed in terms of quantum walks. The experimental procedures appear sound and technically\nrobust. On this basis, I recommend publication.\nThat said, I encourage the authors to revise the contextualization of their results. Beginning with the abstract, the comparison\nwith the state of the art is presented from a somewhat misleading perspective. Although it is correct that cascaded systems\nface significant limitations for large scale integrated photonic circuits, I believe the present work should not be framed as a\nscalable alternative to such platforms. I am skeptical that step numbers exceeding those achievable with cascaded\napproaches can be realistically realized in this setting.\nA clearer emphasis on the conceptual and physical innovation of the work, namely the emergence of a mechanism that can\nbe interpreted as a quantum walk, together with the distinctive programmability enabled by pulse shaping techniques, would\nstrengthen the manuscript and place its contributions in a more accurate context.", "reviewer_3": "All my concerns have been fully addressed, and the manuscript now meets the high standards required for publication in\nNature Communications.\nIn terms of novelty, the authors leverage the intrinsic nonlinear optical response of a solid-state medium, using high-\nharmonic generation to simultaneously produce multiple harmonic orders within a single crystal. This unique approach\nenables parallel information processing in a compact, integrated, and inherently stable configuration, eliminating the need\nfor cascaded components. The level of innovation demonstrated here is substantial enough to warrant publication.\nTherefore, I strongly recommend that the manuscript be accepted for publication in Nature Communications."}, "author_response": ""}]

Decision letter