Quantum boomerang effect of light

Domain: physics
Published: 2026-01-16
Editorial outcome: minor_revision

Abstract
The quantum boomerang effect is a counterintuitive phenomenon in which a wavepacket launched with finite momentum in a disordered medium returns to its origin. However, up to now, the experimental exploration of this boomerang effect remains largely unexplored. Here, we report the observation of this effect with light in an on-chip, one-dimensional (1D) disordered waveguide lattice. After benchmarking the system through Anderson localization, we launch a kinetic light beam into the system and track its center of mass (COM): it first moves away from its starting point, arrives at a maximum-valued point, reverses its direction, and returns to its original position over time, revealing the real-space observation of the photonic quantum boomerang effect. We also show two methods to accelerate and control the return: a symmetric gradient loss and time-varying coupling control to effectively increase the return velocity. Both strategies are realized experimentally and captured by our model. These results establish a controllable photonic platform for boomerang physics and open an avenue for future study in nonlinear and many-photon regimes.

Introduction
Controlling light has been at the forefront of modern optics for decades and is of fundamental interest to the photonic community. Advances in controlling light in photonics have led to the development of photonic crystals1,2 and metamaterials3,4, which enable intriguing phenomena such as the inhibition of spontaneous emission within the bandgap and the steering of light with a negative refractive index5–8. More recently, the introduction of topology into photonics, inspired by topological insulators in condensed matter physics, has opened avenues for light control9–18. The realizations leveraging artificial gauge fields in gyromagnetic photonic crystals9,10, helical waveguides13, silicon photonics14 et al., offered the conventional photonic systems one-way topological edge states that are immune to imperfections and disorder. This field has provided possibilities for robust light transport, unidirectional waveguiding, and even the realization of exotic topological phases that have no electronic counterpart, such as lasing19–22 and quantum photon sources23,24. The further interplay between the topological phases and non-Hermitian physics25–27 has led to many frontiers in photonics28–41. One paradigmatic example is the non-Hermitian skin effect42–52, where extended bulk states condense at one edge, resulting in the funneling of light53. These advances underscore the potential for photonics that leverage the unique properties of topological and non-Hermitian systems.

On the other hand, Anderson localization54 is a phenomenon where waves become trapped in a disordered medium due to interference, preventing them from diffusing through the material. This concept was originally introduced by Anderson in 1958. However, nowadays Anderson localization has been experimentally observed in various systems beyond its original electronic context, including photonic lattices2,55–65, where light waves are confined by a disordered structure, and cold atom systems, where matter waves are localized66–69. Building on this concept, the quantum boomerang effect-a counterintuitive dynamical feature beyond Anderson localization–was theoretically proposed70–76, which disproves the common wisdom that, in a highly disordered lattice, waves can only be localized. In this phenomenon, a wavepacket launched with an initial velocity in a disordered medium unexpectedly returns to its starting point over time, rather than dispersing or continuing along its trajectory. This intriguing phenomenon can be attributed to the combination of Anderson localization and parity-time symmetry of the system74,77. The former leads to a time-independent distribution of the wavepacket at long times, and the latter leads to the wavepacket’s distribution being independent of the direction of momentum [Supplementary Section I-A]. While this effect has been extensively theoretically studied, its realization was only recently reported in the momentum space with ultracold atoms77, limiting direct applications and broader transfer to other physical platforms. Altogether, these advances motivate a natural question: can the quantum boomerang effect be realized with light?

In this work, we experimentally realize the quantum boomerang effect in a 1D disordered photonic waveguide lattice, where the propagation axis of z plays the role of time and the field obeys a Schrödinger-like equation mimicking the quantum behaviors of Anderson localization. We then inject a finite-momentum wavepacket and track its COM: it moves away from its initial position, arrives at a maximum value, reverses its direction, and returns to its original position over time—consistent with the boomerang trajectory. To speed and control the return, we introduce a parity-symmetric gradient loss that acts like a restoring friction. Side-by-side tests of loss-free and loss-engineered lattices show a consistently faster COM’s comeback to the launch site. We further enhance the effect by slowly increasing the inter-waveguide coupling during the propagation, which effectively raises the return speed (see details in Supplementary Section I-D & IX). Our results not only deepen our understanding of light-matter interactions in disordered systems but also open doors to controlling light for cloaking78 and optical tweezers79.

Results
Physical model
Our model is realized in an array of evanescently coupled waveguides13,52. The waveguides that can be directly written by femtosecond lasers are equally spaced with a lattice constant of d = 15 μm. The paraxial propagation of light in this system can be well described by the Schrodinger-type equation13 (see supplementary information for details):

i∂ z Ψ = - 1/2k 0 ∇ 2 Ψ - k 0/n 0 Δn Ψ, (1)

where Ψ is the electric field amplitude, the Laplacian ∇ 2 acts on the transverse plane, k 0 = 2πn 0 /λ is the wavenumber in the background medium, and λ is the wavelength of light. Hereafter, the ambient medium for our system is the fused silica with refractive index n 0 = 1.45, and Δn(x, y, z) is the effective on-site potential, which is propagation-invariant and random in the transverse plane to construct a disordered photonic lattice. The paraxial propagation of light in this photonic system can be further governed by the tight-binding equation13:

i∂ z ψ p = c 0 ψ q + V p ψ p = Hψ p, (2)

where p ∈ [−L, L] is the lattice index, 〈pq〉 indicates the summation is taken over the neighboring sites, c 0 is the coupling strength and V p represents on-site potential, which is set to be pseudo-random values80. The potential is given by V p = δ cos [5π(p + 1) α], where δ is the disorder parameter in the realization of disorder and α is set to be 3 hereafter80. We note that the pseudo-random potential used here follows the same form as that theoretically investigated in ref. 72 in the context of the quantum boomerang effect. The detuning of the on-site potentials can be introduced by changing the writing velocity of the femtosecond laser (Supplementary Section VIII).

Anderson localization of light
To establish the platform for studying the quantum boomerang of light, we first verify Anderson localization in our disordered photonic lattice. By utilizing the beam-propagation method (BPM), the intensity profiles at the output facet with different propagation lengths are numerically simulated (See details in Supplementary Section VII), and the results are plotted in Fig. 2b with a log scale for the vertical intensity axis. In our simulations, the effective coupling strength is about c 0 = 1.55 cm−1. The strength of the pseudo-random potentials is about δ = 2.2 cm−1. The parameter of α is 3, which indicates that all the states in the system are localized80. We can see that the ensemble-averaged intensity distribution stabilizes at about z = 4 cm and remains there over propagation length. The profile of the output beam becomes exponentially localized, which is direct evidence of the Anderson localization. Note that without disorder, the output profile exhibits ballistic transport, and its width increases linearly with the propagation length (Supplementary Section II).

To experimentally verify Anderson localization in our photonic lattice, a laser beam with a wavelength of 635 nm is initially launched into the central waveguide of the 1D optical array, fabricated via femtosecond-laser direct writing13. The optical characterization setup is shown in Fig. 2a. The measured intensity profiles at the output facet are averaged over 21 realizations at propagation lengths of z = 2, 4, 6, 8, and 10 cm, respectively. As shown in Fig. 2d, e, the light beam becomes exponentially localized at about z = 4 cm, and the measured intensity profiles closely match the simulations (Fig. 2b, c). Note that we further characterize the localization behavior by using the inverse participation ratio (IPR), and the IPR in our model finally stabilizes at a finite plateau value (IPR ~ 0.12), indicating that the wave function has relaxed into a superposition of exponentially localized eigenmodes. The result is shown in Supplementary Section VIII-D. Together, the numerical and experimental results reveal that the ensemble-averaged intensity patterns undergo a transition from ballistic transport to localization when the disorder is introduced, which confirms that our disordered photonic lattice exhibits Anderson localization of light.

Quantum boomerang effect of light
Having presented Anderson localization of light in our photonic lattice incorporating disorder, we next move forward to the interaction between the disordered lattice and the kinetic input wavepacket with nonzero momentum. The kinetic beam can be expressed in a normalized Gaussian form of ψ(x p, z = 0) / exp (- x 2 / 2σ 0 2 + ik 0 x p), where σ 0, k 0 represent the width and momentum of the initial wavepacket. Consequently, under the evolution of the photonic system, the COM of a launched wavepacket as a function of propagation distance can be calculated by x(z) = ∑ p x p |ψ(x p, z)| 2, and 〈x〉 indicates the ensemble-averaging of the COM over disorder realizations. A wavepacket propagating in the disordered lattice can be divided into two parts: a localized part and a moving part. Initially, the moving part dominates, causing the COM to move away from its original position. As the wave evolves, however, the intensity of the moving part diminishes, and the localized part begins to take over. Consequently, the COM reaches a maximum value, denoted as 〈x〉 max, before eventually returning to its initial position over time.

The next step is to select the optimal momentum value that can maximize the turning distance, thereby broadening the detectability of the phenomenon. We address this with simulations that sweep experimentally accessible parameters. In the absence of disorder, the wave packet with a fixed momentum will acquire a group velocity that is governed by the dispersion relationship of the lattice. At the output facet at z = 15 cm, the relation between the momentum of k 0 and the maximum moving distance of 〈x〉 max is shown in Supplementary Fig. S4a (Supplementary Section III). We can see that the maximum traveling distance occurs at the momentum of k 0 d = π/2 corresponding to the dispersion with the maximum group velocity (see inset in Fig. 3a). Intuitively, we envision that this momentum induces the strongest boomerang effect. This is confirmed by numerical simulations, as shown in Fig. 3a. The relationship between the 〈x〉 max and the momentum reveal that the wave packet with the initial momentum of k 0 d = π/2 travels the farthest before returning to its starting point. Therefore, we choose k 0 d = π/2 hereafter for the observability of the boomerang effect in experiments.

The disordered photonic lattices and characterization setup are identical to those in Figs. 1 and 2a. A 635 nm laser is shaped into a broad Gaussian input through an optical slit, producing a beam with width σ 0 = 3d. The momentum of the Gaussian beam can be controlled by a rotated mirror and set to be k 0 d = π/2 (Supplementary Section VIII-B). The initial input beam is then launched into the sample, centered at the optical waveguide with the index of p = 0. To reconstruct the wavepacket trajectory, we record the output-facet intensity for photonic chips with propagation lengths of 0, 0.5, 1, 1.5, 2, 3, 4, 5, 7, 10 and 14 cm through a commercial CCD camera.

The measured COM trajectory of the wave packet is shown in Fig. 3b, with the solid line (dots) representing the numerical (experimental) results. We can clearly see that the COM moves rapidly along the x direction over the first z = 1 cm, reaches a maximum value of about 1.8d, then reverses, and drifts back toward the starting position of x = 0, providing a direct real-space observation of the quantum boomerang of light. Note that in Supplementary Section I-B, we show the turning point and return time follow the mean free path and the scattering time predicted by the 1D theory, in agreement with experiment. Due to the finite sample length, the COM does not fully reach the starting position by z = 14 cm.

Enhancement approaches
A natural question arises: can we accelerate the quantum boomerang effect? Generally, loss is considered detrimental to optical experiments, particularly those involving photonic devices and quantum optics. However, surprisingly, we will show in the following that a specific design of loss configuration can facilitate the realization of the quantum boomerang effect by accelerating the return of the wave packet, meanwhile preserving the maximum excursion. The optical loss is configured to follow a symmetric gradient pattern as shown in Fig. 4b, expressed as -iγ|p|ψ p, where p, γ represent the lattice index and the strength of the gradient loss. The evolution of the COM of the launched wavepacket is traced out by numerical simulations as shown in Fig. 4a. The depth of the blue color indicates the gradient strength of 0, 1.5 × 10−3 c 0, 3.0 × 10−3 c 0. We surprisingly find that a stronger gradient causes the COM to return to its starting position of x = 0 more quickly, which facilitates the experimental realization of the quantum phenomena in our system. In Fig. 4c, we present the relationship between the strength of gradient loss and the COM at z = 10 cm, which verifies that the introduction of gradient loss can accelerate the return of the wavepacket. The experimental results agree well with the numerical ones. A general Ehrenfest analysis shows that uniform loss is inert, an antisymmetric gradient induces a drift that destroys the boomerang, while the symmetric gradient enhances the return, matching simulations and measurements. More details can be found in Supplementary Section I-C. Note that the symmetric gradient loss alone cannot lead to the quantum boomerang effect in the photonic lattice without the disorder (Supplementary Section II).

Following the same philosophy as previously, we now fabricate disordered photonic lattices with an engineered gradient loss and repeat the measurements. The gradient loss is implemented by inserting periodic breakpoints in the optical waveguides (Fig. 4b, inset, Supplementary Section VIII)52. The gradient strength is set to be γ = 3.0 × 10−3 c 0. The measured COM trajectory is shown in Fig. 5a. The solid curve represents the numerical result from tight-binding simulations, and the purple dots represent the measured results. By comparing this plot with that in Fig. 3b, we can see that the COM trajectory is similar, but the wave packet returns to the starting position significantly faster due to the presence of gradient loss. BPM simulations (Supplementary Section VIII-E) agree with both the tight-binding results and the measurements. In Fig. 5b–c, we compare intensity profiles at propagation lengths of z = 1 and 10 cm from BPM and experiment. At z = 1 cm, the COM of the wavepacket is displaced to the right by about 1.8d; at the propagation length of z = 10 cm, the COM approaches its original position of x = 0. Together, these data demonstrate the quantum boomerang of light and show that a designed gradient loss can substantially accelerate the return of the wavepacket.

Methods
Sample fabrication
The photonic lattices of optical waveguides are fabricated on the fused silica glass (Corning 7980) with various lengths (5, 10, 15, 20, 30, 40, 50, 70, 100, and 140 mm). This is achieved by using the femtosecond laser writing method (repetition rate 600 kHz, wavelength 515 nm, pulse duration 217 fs). The laser pulses are focused onto the sample through an objective lens (×20, NA = 0.45), and induce the refractive index contrast of up to Δn 0 = 7.5 × 10−4. The distance between the two nearest waveguides is set to be d = 15 μm corresponding to the effective coupling strength of about c = 1.55 cm−1 at the characterization wavelength of 635 nm. The incommensurate on-site potential energy and the gradient loss of the waveguide array are realized by writing with different speeds and introducing different break lengths into the waveguides, as illustrated in Supplementary Fig. S8.

Characterization setup
The experimental setup is illustrated in Supplementary Fig. S9b. The laser beam is initially shaped by the slit of S, then focused on the rotated mirror of M 2 by the objective lens of L 1 (×4, NA = 0.1). The angle of θ between the center axis of M 2 and the light path is initially set to be π/4. Next, the obtained elliptical Gaussian beam is sent to the input facet of the sample through a 4-f system (L 2 and L 3). The mirror of M 2 and the input facet of the sample are on the focal point of L 2 and L 3.

To control the momentum k x of the input Gaussian beam, we rotate the mirror of M 2 by about 0.2°. Consequently, the tilted Gaussian beam has a momentum of k x d = π/2. In the clean lattice, we observe that the COM of the beam travels farthest away from the incident position by fixing the same propagation distance. The relationship between the COM and the angle of M 2 is depicted in Supplementary Fig. S9c. The error bars represent the experimental data (averaged over 21 realizations), and the curve represents the fitting result.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors of this paper have reported on the first observation of the quantum boomerang of light.\nAlthough this is an important and beautiful experiment that could open new perspectives for the study of this phenomenon in\nmore complex situations and for possible applications in the field of photonics (perhaps), I do not think that this article meets\nthe standards to be published in Nature Communications.\nThe reasons for my statement are as follows:\n(1) As mentionned in the paper, this work is not the first observation of the quantum boomerang effect (it is the first\nobservation of the quantum boomerang effect of light).\nIndeed the quantum boomerang effect was observed for the first time three years ago (PRX 12, 011035 (2022), Ref. [77]) in\nthe context of ultracold gases in momentum space.\n(2) From a conceptual point of view, this work does not bring any new contribution. More in details:\n(2a) There is a long discussion about the fact that the boomerang effect is maximum when k0d=pi/2, but this is an evidence,\nthe group velocity being maximum at this point. Indeed it was the situation analyzed in Ref. [77] and also the one used in the\nstudy of the coherent forward scattering effect, that happens in the conjugate space with respect the quantum boomerang\neffect [arXiv:2503.01777].\n(2b) The section of the supplemental material devoted to the theoretical explanation of the quantum boomerang effect is\nidentical to that already given in Ref. [77].\n(2c) There is no discussion around the turning point and the mean-free path, neither on the time (z) needed to come back\nwith respect the theoretical model elaborated by Prat et al. (Ref[70]).\nThe only discussion concerns the role of the losses, but there is not the analysis of a physical model that explains the role of\nthe losses in the quantum boomerang effect. All the analysis are numerical.\nIn summary, this work concerns a beautiful experiment but (i) it is not the first evidence of the quantum boomerang effect in\nthe absolute, (ii) there are no new physical insights with respects the previous works on the subject, and (iii) a deep analysis\nof the experimental data with respect the quantum boomerang effect theory is missing (there is the comparison with the\nnumerical simulations, but not with the analytical theoretical curves).\nFor all these reasons even if, of course, I'm fully convinced that the results of this experiment deserve to be published, I think\nthat this draft does not meet the standards of Nature Communications.", "reviewer_2": "The work is a nice presentation of the \"quantum boomerang\" effect, here realized for light in one dimension. Originally\ndemonstrated in a cold atom setup where the kicked rotor model was implemented, the effect is here investigated using 1+1-\ndimensional setup, where one axis plays the role of time.\nOverall, I find the work very convincing, with various aspects (finite-size effects, non-linearity) explored, and well presented.\nIn particular, discussing the role of dissipation/losses in enhancing the effect is particularly interesting.\nProvided the following points are clarified, the work could fit well in the Nature Communications journal:\n1) The experimental configuration is a bit confusing, as the schematic figure (Fig.2a) comes late, and lacks some elements:\nthe z axis (equivalent to time) is not presented, I would also recommend to make explicit that the interaction [Eq.(2)] occurs\nalong the x axis. 'y' is mentioned in the refractive index, but does not play any role here. Moving this figure to Fig.1 may also\nhelp the reader to understand better the scheme from the start.\n2) Fig. 1b, which aims at showing evidence of Anderson localization, should be shown in logscale for the y-axis, to better\nconvince of the exponential decay.\n3) Why is the low IPR (Fig.1c) a signature of localization? Furthermore, isn't a (slow) spreading of the wave expected,\nalthough quite slower than for diffusion?\n4) Typically, in dielectrics, the imaginary part imag(n) encapsulates the dissipation, while in the present work it is\nproportional to |n| [see below Fig.3], can the authors comment on this point?\n5) Using 'n' for the refractive index AND as an index for the waveguides brings some confusion."}, "author_response": "Dear Reviewers,\nWe would like to sincerely thank you for the time and effort you devoted to evaluating our\nmanuscript on ’Quantum boomerang effect of light’. Your constructive comments have been in-\nvaluable and have led to substantial theoretical, experimental, and presentation improvements in\nthe revised version.\nBoth of the reviewers explicitly recognized our work as ’an important and beautiful experiment’\nand ’convincing, interesting’, confirming the validity and significance of our results. We are deeply\ngrateful for this endorsement.\nReviewer 1 acknowledged the quality of the experiment but, in our view, underestimated the\nbroader contributions of this work. In the revised version, we have included more theoretical and\nexperimental results; therefore, highlighted and expanded on the key advances:\nFirst photonic and real-space realization of the quantum boomerang effect, distinct from\nprior momentum-space cold-atom experiments, enabling direct spatial imaging and revealing a\nthreefold larger center-of-mass excursion;\nDiscovery and demonstration of two new enhancement mechanisms that accelerate the\nreturn of the wave packet [parity-symmetric gradient loss and adiabatic time-varying coupling\ncontrol (Newly studied in the 2nd round)], both experimentally implemented and theoretically\nmodeled. These enhanced methods were neither predicted nor realized in earlier works.\nDirect theoretical–experimental link, with explicit formulas for mean-free path, turning\npoint, and return time, along with a closed-form Ehrenfest analysis that explains the role of\ndifferent loss profiles.\nReviewer 2 was very supportive of our work, finding it convincing and well presented, and\nsuggested minor clarifications. We have carefully implemented all of these comments.\nFurthermore,forcomparison,wehaveincludednewtheoreticalandexperimentalresultsshowing\nthe absence of the quantum boomerang effect in the other one-dimensional disordered models.\nAltogether, these additions extend the scope of the quantum boomerang effect well beyond its\noriginalframework,showingnotonlyuniversalityacrossplatformsbutalsoopeningnewconceptual\nand practical directions—particularly in non-Hermitian photonics.\nIn light of the strong endorsement of the experiment, the substantive conceptual and analyti-\ncal advances added in revision, and the constructive feedback from both referees, we believe the\n2\nmanuscript now fully addresses all concerns and meets the high standards of Nature Communica-\ntions. We sincerely hope you will agree with us and recommend publication.\nThank you again for your insightful reviews and kind support.\nZhaoju Yang\nOn behalf of all authors\nReviewer 1 (Remarks to the Author):\nReviewer Comments:\nThe authors of this paper have reported on the first observation of the quantum boomerang of light.\nAlthough this is an important and beautiful experiment that could open new perspectives for the\nstudy of this phenomenon in more complex situations and possible applications in the field of pho-\ntonics (perhaps), I do not think that this article meets the standards to be published in Nature\nCommunications.\nAuthors Response:\nWe sincerely thank the reviewer for recognizing our experiment as ’an important and beautiful\nexperiment’ and for acknowledging its potential to open new research avenues in photonics.\nIn the following, we will address the reviewer’s comments point by point.\nReviewer Comments: The reasons for my statement are as follows:\n(1) As mentioned in the paper, this work is not the first observation of the quantum boomerang\neffect (it is the first observation of the quantum boomerang effect of light). Indeed, the quantum\nboomerang effect was observed for the first time three years ago (PRX 12, 011035 (2022), Ref.\n[77]) in the context of ultracold gases in momentum space.\nAuthors Response:\nWe fully acknowledge the pioneering work of Ref. [77] as the first experimental observation of\nthe quantum boomerang effect in momentum-space cold atoms.\nOur contribution is fundamentally different in both platform and physical settings:\n1. We report the first photonic realization of the effect in static real-space disorder, enabling\ndirect spatial imaging and revealing a three-times larger center-of-mass (COM) displacement\n(1.8d vs. 0.6d in ultra-cold atoms, where d is the lattice constant).\n2. This distinct realization not only broadens the scope of the phenomenon but also uncovers\ntwo previously unreported enhancement mechanisms—parity-symmetric gradient loss [Fig.\n4 & 5, see more details in Supplementary Materials (SM), Section I-C] and adiabatic time-\nvarying coupling control (Fig. S15, see details in SM, Section I-D & IX)—which were\n4\nneither predicted nor explored in earlier works.\nFurthermore,thecross-platformrealizationoffundamentalphysicalconceptsisapowerfuldriver\nof discovery: by transplanting ideas such as topological phases into photonic or acoustic systems,\nor implementing Anderson localization in optics, one not only verifies their universality but also\nexploits the unique capabilities of the new platform - often revealing phenomena and applications\ninaccessibleintheoriginalcontext. Forexample, beyonditsnoveltyinphotonics, ourexperimental\nrealization of the quantum boomerang effect of light opens new perspectives to exploring non-\nHermiticity-induced enhancement that is difficult to reach in ultracold atoms or condensed matter\nphysics,andshowspotentialinnovelphotonicdevicessuchaslasers,cloaking,andopticaltweezers.\nWe believe these advances significantly extend the reach and controllability of the quantum\nboomerang effect beyond the original cold-atom experiment and will be of interest to the wide and\ninterdisciplinary audience of Nature Communications.\nReviewer Comments:\n(2) From a conceptual point of view, this work does not bring any new contribution. More in\ndetails:\nAuthors Response:\nWe respectfully disagree with the reviewer on this point. On the other hand, we have to say it\nwas our fault that we failed to convey the main message of our work.\nWhile fully acknowledging Ref. [77] as the seminal momentum-space cold-atom realization,\nwe respectfully note that our study advances the field in two original and substantial ways:\n1. First real-space and photonic realization – We demonstrate the quantum boomerang\neffect of light in a real-space disordered photonic lattice, a fundamentally different physical\nsetting that enables direct spatial imaging and reveals a ∼ 3× larger COM displacement (1.8d\nvs 0.6d), thereby broadening both the scope and detectability of the phenomenon.\n2. Two novel enhancement mechanisms – We introduce and experimentally verify two\nmethods to enhance the quantum boomerang effect: parity-symmetric gradient loss - non-\nHermitianengineeringthatacceleratesCOMreturnviarestoringfriction(Fig. 4&5, seemore\ndetails in SM, Section I-C) and adiabatic time-varying coupling - Hermitian protocol that\neffectively increases the return velocity v during propagation (Fig. S15, see details in SM,\nre\nin earlier experiments. These methods provide unprecedented control over the boomerang\ndynamics by accelerating the COM’s return, and open routes for studying non-Hermitian and\nspatial-temporal effects in disordered systems. The results are also presented in Fig. R1 and\nFig. R2.\nIn addition, we have conducted more simulations and experiments, which focus on the addi-\ntional models, such as the Aubry-Andre (AA), the generalized AA model, and mosaic models for\ncomparison. The results are shown in SM, Section X & XI, Fig. S16-S18.\nWe sincerely hope these advances — spanning novelty of platform, conceptual innovation, and\nquantitative-theoretical connection — address the reviewer’s concern on significance and place our\nwork within the standards of Nature Communications.\nIn the revised version, we have rewritten large parts of the manuscript, added substantial\ntheoretical/experimental results, and improved presentation details. We show all changes in blue.\narray with breaks (the displayed length in the z axis is 1 mm). The bluescale represents the strength of\nthe random potential. b, Profile of the loss with γ = 3.0×10−3c .\n0\nc, The relationship between the COM\nand strength of gradient loss at a fixed propagation length of z =10 cm (purple dashed line in panel d).\nd,\nCOM trajectories with different strengths of gradient loss. The comeback is accelerated, and the maximum\nexcursion is preserved. e, COM trajectory in the disordered lattice with γ =3.0×10−3c .\nThe blue curve\n0\n(purple dots) represents the numerical (experimental) result, that averaged over 103 (50) realizations. The\nexperimental data agree with the simulations.\ntime-varying photonic lattice. The inter-waveguide coupling increases monotonically along the propagation\ndirection (implemented by tapering the spacing, d > d ). The bluescale indicates the disorder strength.\n1 2\nb, COM trajectories for three coupling protocols; the inset shows the corresponding coupling strength over\ntime. c, With an increasing coupling c(z): 1.55→3.97 cm−1, the COM reverses and returns to the launch\nsite sooner than in the static case. The blue curve (purple dots) represents the numerical (experimental)\nresult, that averaged over 103 (50) realizations. The experimental data agree well with the simulations.\n(2a) There is a long discussion about the fact that the boomerang effect is maximum when k d =\n0\nπ/2, but this is an evidence, the group velocity being maximum at this point. Indeed it was the situa-\ntionanalyzedinRef. [77]andalsotheoneusedinthestudyofthecoherentforwardscatteringeffect,\nthat happens in the conjugate space with respect the quantum boomerang effect [arXiv:2503.01777].\nAuthors Response:\nWe fully agree with the reviewer that, in a clean tight-binding lattice, k d = π/2 corresponds\n0\nto the maximum group velocity, and that this condition was indeed employed in Ref. [PRA 99,\n023629 (2019), PRX 12, 011035 (2022)] and in the coherent forward scattering (CFS) studies\n[arXiv:2503.01777].\nWerespectfullynotethatourworkwasnotmeanttore-establishthiselementaryfact[boomerang\neffect is maximum when k d = π/2], but rather to demonstrate — both theoretically and experi-\n0\nmentally, that this condition remains the optimum for observing the quantum boomerang effect\nin a real-space disordered photonic lattice, where finite disorder strength, loss, and system size\ncould in principle shift or suppress this optimum value.\nFrom the theoretical side, rigorous analysis of the 1D Anderson model shows that the peak\nCOM displacement scales with the scattering mean free path l(k ), which is maximized at the\n0\nband center (k d = π/2) due to the combined effect of maximal v and longest localization length.\n0 g\nOur numerical momentum scans [see Fig. 3a in the main text] confirm this prediction: the COM\nexcursion follows a sin(k d)-like dependence, dropping to zero at k = 0,π. Experimentally,\n0 0\nwe directly mapped the boomerang signal at k d = π/2 and found a clear maximum of the drift\nspace observation of the momentum dependence of quantum boomerang effect in any platform,\ncomplementing the momentum-space cold-atom realization of Ref. [77].\nIn summary, while the group-velocity argument explains why k d = π/2 is expected to be\n0\noptimal in the ideal case, our contribution is to verify and quantify this optimum under realistic\nexperimental conditions in a different physical setting.\nReviewer Comments:\n9\n(2b) The section of the supplemental material devoted to the theoretical explanation of the quantum\nboomerang effect is identical to that already given in Ref. [77].\nAuthors Response:\nWe thank the reviewer for highlighting this issue.\nWeacknowledgethatthesupplementalderivationinsectionI-Abuildsonthetheoreticalframe-\nwork of Ref. [77] (ultracold atom, PRX 2022) and Ref. [70] (Prat model PRA 2019), but our goal\nwas not to claim novelty in the basic formalism, but to provide a self-contained and accessible\nexplanation for the photonics community, so that readers can follow the logic without having to\nconsult prior work in a different platform.\nIn the revised manuscript, we have rewritten this section substantially and added more results\nto ensure originality and depth:\n1. We have rewritten SM, Section I-A, while providing explicit citations to Ref. [77] and to the\nanalytical derivations in Refs. [70–72].\n2. We have added three more subsections I-B, I-C and I-D. In subsection I-B, the discussions\nabout the turning point, the mean-free path, and the returning time are included. We adapt\nthe framework to the case of real-space photonic lattices with static disorder, and present\nthe momentum-dependence analysis [showing the sin(k d)-like scaling and the suppression\n0\naway from k d = π/2]. In subsections I-C and I-D, we provide a detailed analysis of the\n0\nenhanced quantum boomerang effect by introducing non-Hermitian gradient loss and time-\nvarying couplings.\nOverall, we believe that these aspects, together with our experimental realizations, go beyond\nRef. [77] as well as the earlier works, and strengthen the link between theory and real-space\nobservation.\nReviewer Comments:\n(2c) There is no discussion around the turning point and the mean-free path, neither on the time\n(z) needed to come back with respect the theoretical model elaborated by Prat et al. (Ref. [70]). The\nonly discussion concerns the role of the losses, but there is not the analysis of a physical model that\nexplains the role of the losses in the quantum boomerang effect. All the analysis are numerical.\nWe thank the reviewer for this insightful suggestion.\nIn the revised manuscript, we have substantially expanded the theoretical discussion to address\nexactly the turning point, mean-free path, and return time, which builds on the framework of Ref.\nPrat et al. Specifically, we now provide analytical expressions linking the COM dynamics to the\n4c2\nscatteringmeanfreepathl = 0dandmeanscatteringtimeτ = l/v , andshowthatourmeasured\nδ2 0\nturning point ⟨x⟩ ≈ 1.8d and return time z ∼ τ agree quantitatively with theory.\nmax TP\nImportantly, we developed a closed-form Ehrenfest analysis for non-Hermitian Hamiltonians\nthat explains the distinct roles of different loss profiles: (i) uniform loss is inert, (ii) antisymmetric\nloss destroys the boomerang, and (iii) symmetric gradient loss provides a restoring “friction” that\naccelerates return. These analytic results go beyond purely numerical observations and are fully\nconsistent with our experiments.\nWe believe these additions now establish a clear and rigorous theoretical link between the\nturning point, mean free path, loss mechanisms, and the observed quantum boomerang dynamics,\nthereby addressing the reviewer’s concern.\nWe have added the details on this issue accordingly in Supplementary Materials, section I-B,\nC, D:\nB. Turning point, mean-free path, and return time\nThe evolution of light in our waveguide array is governed by the tight-binding equation:\n(cid:88)\ni∂ ψ = c ψ +V ψ = Hψ\nz p 0 q p p p (R1)\n⟨pq⟩\nwhere p ∈ [−L,L] is the lattice index along the transverse dimension x, ⟨pq⟩ indicates the\nsummationistakenoveralltheneighboringsites, c isthecouplingstrengthandV represents\n0 p\non-site pseudo-random potential. We set v as the mean wave-packet velocity, which is given\n0\nby:\nv = 2c d·sin(k d) (R2)\n0 0 0\nTo describe the temporal evolution of the center of mass (COM), we introduce the mean\nscattering time:\nl\nτ = (R3)\nv\n0\n11\nwhere l is the mean free path, which in the 1D model is equal to one-fourth of the localization\nlength l . For weak disorder, the localization length can be computed analytically, which can\nloc\nbe written as (Refs. [72]):\n1 1 ⟨V p 2⟩ (cid:88) ∞ ⟨V p V p+l ⟩\n= [1+ cos(2lkd)] (R4)\nl d8c2sin2(kd) ⟨V2⟩\nloc 0 l=1 p\nIn our system, ⟨V V ⟩ = 0, ⟨V2⟩ = δ2 , and the momentum distribution of the wave-packet\np p+l p 2\n∼\nis sufficiently narrow (only momenta close to k = k contribute to the dynamics). Therefore,\n0\nthe mean free path is given by:\n4c2\nl = 0d (R5)\nδ2\nUnder the experimental parameters, we obtain the scattering mean-free path l = 1.99d,\nand the scattering mean-free time τ = 0.64 cm. In our Hermitian device (no gradient loss),\nthe COM attains its maximum at ⟨x⟩ ≈ 1.8d when z ∼ 1.3 cm, in agreement with the\nmax TP\ntheoretical prediction ⟨x⟩ ≃ l and z ≃ τ (see Fig. 3(b) of the main text).\nmax TP\nThe accessible propagation length in our device is limited, which means that a full return\nof the center of mass (COM) to its original position cannot be observed in a purely Hermitian\nlattice. To address this limitation and to test the robustness of the quantum-boomerang\neffect, we implemented two acceleration strategies: 1. Introducing symmetric gradient loss; 2.\nIntroducing time-varying coupling during propagation, as detailed below.\nC. Analytic model for the role of loss.\nIn our revised manuscript, we add a comprehensive analysis of the trajectory of QBE\nunder loss, and explain why a symmetric gradient loss can enhance the return of COM.\nThe\nfull Hamiltonian can be written as:\n(cid:88)\nH = H −iV, V = γ f(p)a†a\n0 p p (R6)\np∈[−L,L]\nwhere H is the Hermitian part of the disordered system, while f(p) encodes the loss profile.\nd\n⟨O⟩ = i⟨[H ,O]⟩−⟨{V,O}⟩ (R7)\n0\ndt\nwhere O is any time-independent operator. We define the normalizing factor as N(t) =\n⟨ψ(t)|ψ(t)⟩. The evolution of N(t) can be obtained with the general Ehrenfest formulas as:\ndN\n= −2N⟨V⟩ (R8)\ndt\nd\n⟨X⟩ = i⟨[H ,X]⟩−2⟨XV⟩ (R9)\n0\ndt\nThe COM of the wave-packet is defined as ⟨x⟩ = ⟨X⟩/N(t). Combinate with eq.(R9), the\nevolution of ⟨x⟩:\nd i⟨[H ,X]⟩−2⟨XV⟩+2⟨X⟩⟨V⟩\n⟨x⟩ = 0 (R10)\ndt N\nFirstly, we set f(p) = 1 (uniform loss). Under this condition, ⟨XV⟩ = γ⟨X⟩, ⟨V⟩ = γ. Using\neq.(R10), the COM of the wave-packet can be written as:\nd i⟨[H ,X]⟩−2γ⟨X⟩+2γ⟨X⟩\n0\n⟨x⟩ =\ndt N\n(R11)\ni⟨[H ,X]⟩\n0\n=\nN\nTherefore,theuniformlosscancelsoutafternormalization–itneitheracceleratesnorhinders\nthe boomerang; it only multiplies the wave-function by an overall factor e−γt.\nThen,wesetf(p) = p(non-symmetricgradientloss). Underthiscondition,⟨XV⟩ = γ⟨X2⟩,\n⟨V⟩ = γ⟨X⟩. Using eq.(R10), the COM of the wave-packet can be written as:\nd i⟨[H ,X]⟩−2γ⟨X2⟩+2γ⟨X⟩2\n0\n⟨x⟩ =\ndt N\n(R12)\ni⟨[H ,X]⟩ 2γ\n0\n= − Var(X)\nN N\nSince Var(X) > 0, the extra term is a constant drift (of sign −γ) once localization freezes\nthe variance. The COM keeps drifting, and the boomerang return is destroyed.\nFinally, we set f(p) = |p| (symmetric gradient loss). Under this condition, ⟨XV⟩ =\nγ⟨X|X|⟩, ⟨V⟩ = γ⟨|X|⟩. Using eq.(R10), the COM of the wave-packet can be written as:\nd i⟨[H ,X]⟩−2γ(⟨X|X|⟩−⟨X⟩⟨|X|⟩)\n⟨x⟩ = 0 (R13)\ndt N\nIf the packet is perfectly symmetric, the second term vanishes; hence, the packet is stable\nwhen⟨x⟩ = 0. For⟨x⟩ ≠ 0, thesecondtermopposes⟨x⟩anddampensitonthetime-scaleγ/2.\nThis’restoringfriction’acceleratestheboomerangreturn,meanwhilepreservingthemaximum\nexcursion. The numerical simulations under different loss profiles are depicted in Fig. R3).\nD. Acceleration via time-dependent coupling.\ntrajectory indistinguishable from the lossless case (grey). A symmetric gradient loss (blue) introduces a\nrestoring friction that accelerates the return to the origin, whereas an antisymmetric gradient loss (green)\ninduces a net drift that breaks the boomerang effect.\nBeyond the symmetric gradient loss, we identify a second, purely Hermitian route to speed\nup the return of the wave packet by adiabatically increasing the coupling c(z) during propa-\ngation. In the long-time regime, the center of mass follows (Ref. [70]):\nln(z/4τ)τ2\n⟨x(z)⟩ ≈ 64l (R14)\nz2\nwhere l,τ are the mean-free path and time, respectively.\nThis form identifies a characteristic return distance z ≈ 4τ, which leads to a characteristic\nre\nreturn rate v ∼ l/τ. Combined with the eq.(R3) and eq.(R5), we obtain v ∝ c . Therefore,\nre re 0\nas the coupling strength, denoted as c(z), is increased adiabatically, the average returning speed\nv increases, enhancing the returning of the wave-packet. Note that the coupling ramp should be\nre\nadiabatic to avoid spurious forward/backward components.\nReviewer Comments:\nIn summary, this work concerns a beautiful experiment but (i) it is not the first evidence of the\nquantum boomerang effect in the absolute, (ii) there are no new physical insights with respects the\nprevious works on the subject, and (iii) a deep analysis of the experimental data with respect the\nquantumboomerangeffecttheoryismissing(thereisthecomparisonwiththenumericalsimulations,\nbut not with the analytical theoretical curves).\nForallthesereasonevenif, ofcourse, I’mfullyconvincedthattheresultsofthisexperimentdeserve\n14\nto be published, I think that this draft does not meet the standards of Nature Communications.\nAuthors Response:\nWe sincerely thank the reviewer for recognizing the quality of our experiment.\nWe fully acknowledge that Ref. [77] reported the first-ever observation of the quantum\nboomerang effect in momentum-space cold atoms. Our contribution is fundamentally different: we\npresent the first photonic and real-space realization, which enables direct spatial imaging and\nreveals a threefold larger COM excursion compared to ultracold atoms, thereby broadening the\nscope and detectability of the phenomenon.\nConceptually, our study introduces two previously unreported control mechanisms —\nparity-symmetricgradientlossandtime-varyingcouplingcontrol—whicharebothexperimentally\nrealized and analytically modeled. These strategies accelerate and enhance the boomerang effect,\nexpanding its scope to non-Hermitian and time-dependent regimes not considered in earlier works.\nFinally,wehavenowaddedadirectanalytical–experimentallink: explicitformulasforthemean-\nfree path, turning point, and return time, benchmark our data against theory, while a closed-form\nEhrenfest analysis explains the role of different loss profiles. These additions provide precisely the\n’deep analysis’ requested and firmly establish the theoretical basis of our observations.\nIn light of the platform, conceptual, and analytical advances, we believe the revised manuscript\ndo meet the standards of Nature Communications. We thank the reviewer again for the oppor-\ntunity to clarify the significance of our work and sincerely hope the reviewer will now be happy to\nrecommend the publication.\nReviewer Comments:\nThe work is a nice presentation of the ”quantum boomerang” effect, here realized for light in\none dimension. Originally demonstrated in a cold atom setup where the kicked rotor model was\nimplemented, the effect is here investigated using 1+1-dimensional setup, where one axis plays the\nrole of time.\nOverall, I find the work very convincing, with various aspects (finite-size effects, non-linearity)\nexplored, and well presented. In particular, discussing the role of dissipation/losses in enhancing\nthe effect is particularly interesting.\nProvided the following points are clarified, the work could fit well in the Nature Communications\njournal:\nAuthors Response:\nWesincerelythankthereviewerforthepositiveandencouragingevaluationofourwork, andfor\nrecognizing the novelty of realizing the quantum boomerang effect with light. We also appreciate\nthe reviewer’s recognition of our exploration of finite-size effects, nonlinearity, and in particular,\nthe interesting role of dissipation and losses.\nWe are grateful for the reviewer’s constructive suggestions, which have substantially improved\nboth the clarity and the rigor of the manuscript, and we believe the revised version now fully meets\nthe standards of Nature Communications.\nIn the following, we will address the reviewer’s comments point by point.\nReviewer Comments:\n1) The experimental configuration is a bit confusing, as the schematic figure (Fig.2a) comes late,\nandlackssomeelements: thezaxis(equivalenttotime)isnotpresented, Iwouldalsorecommendto\nmake explicit that the interaction [Eq.(2)] occurs along the x axis. ’y’ is mentioned in the refractive\nindex, but does not play any role here. Moving this figure to Fig.1 may also help the reader to\nunderstand better the scheme from the start.\nAuthors Response:\nWe thank the reviewer for this constructive suggestion regarding the schematic.\nadded explicit axes (z as the propagation direction equivalent to time, and x as the transverse\ninteractiondirectionofEq. (2)),andclarifiedtheirrolesinthecaption. Notethatourexperimental\nsystem is built on optical waveguides (a real photo of the sample in the x−y plane is shown in Fig.\nS9a). Thus, the refractive index can be expressed as n(x,y,z). In the schematic, for simplicity, we\nhave removed the axis y, because our model is a 1D system and the y direction does not play any\nrole.\nWe believe these revisions (see Fig. R4) resolve the confusion and make the experiment trans-\nparent from the start.\nFIG. R4. Schematic for the quantum boomerang effect of light. The disordered optical lattice consists of\nequidistant waveguides with a lattice constant of d. The varying bluescale represents the strength of the\nrandom potential. The axis of z is the propagation direction equivalent to time, and the axis of x indicates\nthe transverse interaction axis of Eq. (2). A wave packet with momentum k is launched into the lattice.\nReviewer Comments:\n2) Fig. 1b, which aims at showing evidence of Anderson localization, should be shown in logscale\nfor the y-axis, to better convince of the exponential decay.\nAuthors Response:\nWe thank the reviewer for this valuable suggestion. In the revised manuscript, we have re-\nplotted Fig. 1b on a logarithmic y-axis, which clearly reveals the expected exponential decay of\nintensity and thereby provides unambiguous evidence of Anderson localization. We believe this\n17\nlog-scalepresentationsignificantlyimprovestheclarityandrigorofthelocalizationdemonstration.\nReviewer Comments:\n3) Why is the low IPR (Fig.1c) a signature of localization? Furthermore, isn’t a (slow) spreading\nof the wave expected, although quite slower than for diffusion?\nAuthors Response:\nWe thank the reviewer for raising this important point. The Inverse Participation Ratio (IPR)\nis widely used to distinguish between extended and localized states. In our case, the IPR starts at\nunity(single-siteexcitation),decreasesduringtheinitialtransientasthewaveexploresneighboring\nsites, and then stabilizes at a finite, nonzero plateau (IPR∼0.12). This saturation reflects that the\nwavefunction has relaxed into a superposition of exponentially localized eigenmodes. In contrast,\nin a diffusive or ballistic system, the IPR would continue to decay toward zero (∼ 1/L), reflecting\nunbounded spreading.\nRegarding the possibility of ’slow spreading’: indeed, at early propagation, the packet tran-\nsiently broadens, but Anderson localization implies that this broadening halts at long distances.\nTheexponentiallylocalizedeigenstatesalwaysenforceafiniteparticipationratio,andhencenosus-\ntained diffusion (even sub-diffusion) is present in the long-time limit. The observed finite plateau\nof the IPR is thus a direct hallmark of localization, rather than continued spreading.\nTo make this explicit and avoid any possible ambiguity, we have moved the IPR results to the\nSupplementary Materials VIII-D.\nReviewer Comments:\n4) Typically, in dielectrics, the imaginary part imag(n) encapsulates the dissipation, while in the\npresent work it is proportional to |n| [see below Fig.3], can the authors comment on this point?\nAuthors Response:\nInconventionaldielectrics,dissipationindeedoriginatesfromthematerial’simaginaryrefractive\nindex Im(n), leading to uniform intrinsic absorption. In contrast, in our work, the non-Hermiticity\nof loss is not a uniform material property but an engineered, site-dependent term −iγ|p|ψ†ψ ,\np p\nwhere p is the site index and γ represents the strength of gradient loss.\ning breaks in the waveguides during femtosecond-laser writing (please see Fig. R5, Fig. S8, and\nSM, Section VIII-A for more details), such that the effective attenuation grows with |p|.\nThis\nartificial non-Hermitian profile creates a restoring friction that enhances the boomerang return —\nfundamentally different from homogeneous absorption in standard dielectrics. We have clarified\nthis distinction in the revised manuscript and revised the notation (using p, q for site indices) to\navoid confusion with the refractive index n.\nFIG. R5. Schematic of the waveguide array with breaks (the displayed length in z direction is 1 mm).\nThe disordered optical lattice consists of equidistant waveguides with a lattice constant of d. The bluescale\nrepresents the strength of the random potential. The breaking length in the waveguide is proportional to\nthe strength of loss.\nReviewer Comments:\n5) Using ’n’ for the refractive index AND as an index for the waveguides brings some confusion.\nAuthors Response:\nWe sincerely thank the reviewer for pointing out this confused notation. In the revised\nmanuscript, we now use p, q to denote waveguide site indices, while n is reserved for the re-\nfractive index."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have taken into account all criticisms and suggestions, increasing considerably the quality and the impact of the\nmanuscript, by improving the physical discussion of their results and also by adding new results.\nI congratulate them for this revision work.\nI support the publication of this new version of the manuscript in Nature Communications.\nI have only a last remark: the authors should mention that the potential they have used (expression in the paragraph below\nEq. 2) is the same as the one studied (theoretically) in Ref. [72] always in the context of the quantum boomerang.", "reviewer_2": "The authors have satisfactorily answered my comments to clarify their manuscript.\nThe first referee points out that this is only the second observation of quantum boomerang effect - the first being reported a\nfew years ago in cold atoms. The present one has been achieved with light, and as pointed out by the authors,\ndemonstrating it on another platform, and a different carrier (light instead of atoms) contribute to showing the universality of\nthe process. In addition, the present setup with light exhibits features not present with atoms, and the discussion on non-\nHermitianity brings interesting aspects to the description of the phenomenon.\nThus in my view the results and the manuscript bring in sufficient novel results to deserve the publicity of a pulication in\nNature Communications."}, "author_response": "Response Letter\nReviewer 1 (Remarks to the Author):\nReviewer Comments:\nThe authors have taken into account all criticisms and suggestions, increasing considerably the\nquality and the impact of the manuscript, by improving the physical discussion of their results and\nalso by adding new results. I congratulate them for this revision work.\nI support the publication of this new version of the manuscript in Nature Communications.\nI have only a last remark: the authors should mention that the potential they have used (expression\nin the paragraph below Eq. 2) is the same as the one studied (theoretically) in Ref. [72] always in\nthe context of the quantum boomerang.\nAuthors Response:\nWe thank the reviewer for the positive evaluation and for recommending publication of our\nrevised manuscript.\nAs suggested, we have added the following statement in the paragraph below Eq. 2: ’We note\nthat the pseudo-random potential used here follows the same form as that theoreti-\ncally investigated in Ref. [72] in the context of the quantum boomerang effect.’\nThe authors have satisfactorily answered my comments to clarify their manuscript.\nThefirstrefereepointsoutthatthisisonlythesecondobservationofquantumboomerangeffect-the\nfirst being reported a few years ago in cold atoms. The present one has been achieved with light, and\nas pointed out by the authors, demonstrating it on another platform, and a different carrier (light\ninstead of atoms) contribute to showing the universality of the process. In addition, the present\nsetup with light exhibits features not present with atoms, and the discussion on non-Hermitianity\nbrings interesting aspects to the description of the phenomenon.\nThus in my view the results and the manuscript bring in sufficient novel results to deserve the\npublicity of a pulication in Nature Communications.\nAuthors Response:\nWe sincerely thank the reviewer for the positive assessment and for evaluating our revised\nmanuscript. We appreciate the reviewer’s supportive recommendation for publication in Nature\nCommunications."}]

Decision letter