Experimental observation of non-Hermitian phase transitions using laser-induced thermoacoustics

Domain: physics
Published: 2026-02-26
Editorial outcome: minor_revision

Abstract
Non-Hermitian physics in open systems has garnered significant attention for its exotic phenomena, particularly surrounding exceptional points that offer transformative potential for multifunctional devices. Central to this field are parity-time (PT) symmetry-defined by balanced gain and loss-and its counterpart, anti-PT symmetry. However, integrating these divergent concepts into a unified acoustic platform remains an unattainable challenge. In this study, we employ laser-induced thermoacoustics (LIT) to integrate a tunable amplifying component into a non-Hermitian system. By exciting an ultrathin carbon nanotube (CNT) film through laser irradiation, we experimentally observe the phase transitions between PT and anti-PT symmetries. Furthermore, our findings demonstrate the creation of selectable scattering states and the generation of acoustic vortex beams (VBs), facilitating both PT-symmetric scattering and the conversion of topological charges. This acoustically transparent strategy bypasses traditional, path-blocking compensation schemes, offering a versatile framework for controlled non-Hermitian phase transitions in next-generation integrated devices.

Introduction
One of the most intriguing advancements in quantum mechanics over recent decades has been the discovery of non-Hermitian Hamiltonian H that commutes with the parity-time (PT) operator, leading to the emergence of real-energy eigenvalues1. This finding has sparked significant interest in exploring its classical counterpart across various domains, with applications in photonics2–11, acoustics12–21, and beyond22–24, by utilizing balanced gain-loss regions. Anti-PT-symmetric non-Hermitian systems, serving as the conjugate counterparts, have expanded the realm of non-Hermitian symmetries and have been experimentally realized through spatially coupled flying atom beams25, electrical circuits26, and optical waveguides featuring imaginary couplings27. Nevertheless, due to the absence of an efficient experimental methodology, the establishment of an acoustic anti-PT-symmetric configuration remains exceedingly rare, hindering the phase transitions between PT and anti-PT symmetries within a single unified system-a challenge exacerbated by the incompatible forms of coupling leading to their anti-communication. Although studies on PT phase transitions, including those of anti-PT symmetry, in cavity-tube systems simulating tight-binding models have been reported28, the anti-PT phase transitions in scattering systems remain a topic worthy of further investigation.

The crucial factor in achieving the phase transition between acoustic PT and anti-PT symmetries lies in identifying dependable and controllable acoustic gain medium. While in photonics, gain can be readily induced through stimulated emission via optical or electrical excitation from an external source, or through parametric processes29–31, the realm of acoustics faces challenges. The implementation of PT symmetry in experimental configurations typically necessitates the utilization of piezoelectric transducers32, electromechanical cone loudspeakers13,14, or forced air flows16, as natural passive acoustic gain materials are absent. These mechanically-driven acoustic gain strategies present significant practical hurdles and impose constraints on achieving controlled PT symmetry due to substantial impedance mismatches with the ambient air and the inherent rigidity, leading to limited adaptability.

Nonetheless, a newer related technology, laser-induced thermoacoustics (LIT), which leverages pulsed laser light to generate acoustic waves through rapid thermal expansion may be a candidate approach for acoustic gain. This process involves three primary steps: absorption, thermal expansion, and acoustic wave generation33. In medical imaging, this principle is used in photoacoustic imaging, where specific wavelengths are chosen to target particular tissues or chromophores such as hemoglobin and melanin, enhancing contrast and providing detailed images of biological structures34. The characteristics of the generated sound waves, such as their frequency and amplitude, depend on the properties of the material and the parameters of the laser pulse, including its duration, wavelength, and intensity. This non-invasiveness is beneficial in both medical and material testing applications, where preserving the integrity of the subject is crucial. In addition, active methods for generating acoustic VB have been reported in ref. 35, and LIT may provide a viable alternative approach.

In this study, we propose a general methodology to access the complex coupling by modulating the non-Hermitian system and further facilitate the phase transitions between PT and anti-PT symmetries within a unified acoustics system using LIT. As a demonstration, we experimentally construct a compact non-Hermitian unit consisting of a pair of acoustic loss (sponge) and gain (CNT film) loaded in a waveguide. This unit provides multiple interferences in the structure when incident waves impinge on one of the components, in stark contrast to the conventional acoustic gain media like back-to-back loudspeaker pairs that absorb the incident sound field and then reradiate it14. Unlike PT-symmetric system where unidirectional reflectionless can only be demonstrated from the loss side, our method reverses the orientation at identical operating frequency. Strikingly, the proposed platform also allows for the achievement of anti-PT symmetry through the delicate interplay between gain and loss. Due to the innovation of LIT, acoustic VBs can be conveniently generated, and their PT-symmetric scattering behavior has been realized, along with the conversion of the topological charge. The combination of LIT and PT-symmetric acoustics bridges two critical aspects of non-Hermitian physics, and provides diverse routes for controlling the phase transitions of non-Hermitian Hamiltonian.

Results
Non-Hermitian phase transitions with transfer matrix
The sketch, illustrating both the optical and acoustic components, is presented in Fig. 1. The optical portion illustrates the method by which the CNT film is stimulated by a 532 nm pulsed laser. The CNT film has recently emerged as a groundbreaking platform for achieving acoustic gain in a highly controllable manner36, with significant potential for advancing acoustic devices that facilitate efficient photo-thermo-acoustic energy conversions33,37. Its ultrathin profile renders it acoustically transparent (Supplementary Note 1), allowing sound waves to propagate freely in the transverse direction with minimal impact on wave transmission, a feature distinct from conventional electromechanical loudspeakers that exhibit high levels of reflection. Consequently, the CNT film holds great promise for integration into acoustic non-Hermitian systems (Supplementary Note 2).

In the acoustic portion illustrated in Fig. 1, we consider a waveguide that supports plane wave propagation within the operation frequency range k0A < 2\pi, where A represents the side length of the waveguide cross-section, k0 = \omega/c0 is the wave number, \omega is the angular frequency, and c0 is the sound velocity in air. By introducing a pair of loss (sponge) and gain (CNT film) material, the system becomes non-Hermitian; however, the propagation for the acoustic pressure p can still be described by the one-dimensional (1D) Helmholtz equation. We decompose the pressure field outside the photoacoustic non-Hermitian system into the forward wave (denoted by “+”) and backward wave (denoted by “−”). As such, p+ and p− represent the sound pressure spreading along the forward and backward directions, respectively, originating from the loss and gain sides.

The scattering matrix Sr, which describes the relationship between incoming and outgoing acoustic waves in the 1D system, can be expressed as:

[p+L, p−G]T = Sr [p−L, p+G]T = [S11 S12; S21 S22] [p−L, p+G]T, (1)

where S11 (S22) represents the reflection coefficient from the loss (gain) side and S21 (S12) represents the transmission coefficient. Due to the transmission reciprocity, we have S12 = S21. Sr can be calculated using a transmission-matrix approach:

S11 = (u\Gamma2 + i\Lambda2 eix)/\Delta,
S22 = (u\Gamma1 + i\Lambda2 e−ix)/\Delta,
S12 = S21 = 2/\Delta, (2)

where \Delta = −u\Gamma1 + 2cosy + i\Lambda1 eix, x = k0d, and ks = \omega/\rho s = \kappa s is the wavenumber in the loss medium. \rho s and \kappa s denote the mass density and bulk modulus of the loss material, respectively. Both of them can be retrieved from the experimentally measured Sr by a parameter extraction algorithm at any operating frequency38. The variables \Gamma1, \Gamma2, \Lambda1, and \Lambda2 are all intermediate variables with respect to coupling d. Additionally, u = −YCNT/Y0 denotes the complex admittance ratio between the assembled CNT film (Supplementary Note 3) and background medium air, which can be precisely adjusted by a pulsed laser. In this model, the loss medium is a conventional bulk material (porous sponge) with a defined width D, whereas the gain medium is a piece of CNT film with negligible thickness. The geometrically asymmetric unit described in Eq. (2) exhibits reciprocity and linearity, allowing different types of EPs to emerge through the tuning of coupling d (Supplementary Note 4).

For instance, S11 = 0 (the blue curve in Fig. 2) describes a reflectionless phenomenon when waves come from the loss side, which can be observed at any coupling d by adjusting the equivalent admittance (u = −i\Lambda2 eix/\Gamma2) of the assembled CNT film. Thus, the transmission evolves to S12 = 2/\Gamma2, and its amplitude can be modulated to 1 at a coupling of d = 5.31 cm with u = 0.456 + i0.257 [see the top maps in Fig. 2a, c]. This scattering property bears resemblance to the loss-side invisibility observed at the EP in PT-symmetric systems. Notably, while the reflection from the gain side remain high in a PT-symmetric system operating at this EP14,15, the reflection from the gain side in our system is minimal [see the top map in Fig. 2b] due to the continuous interactions.

Furthermore, this setup also allows for the realization of the less commonly mentioned reversed invisibility EP in PT-symmetric systems. Similarly, the red curve in Fig. 2 represents the reflectionless phenomenon S22 = 0 from the gain side. Thus, the admittance of the CNT film is given by u = −i\Lambda2 e−ix/\Gamma1. Building upon this, the gain-side invisibility EP occurs when the amplitude of the transmission coefficient |S12| = 1/|\Gamma1| equals 1 at d = 4.34 cm with u = 0.435 + i0.291 [see the middle maps in Fig. 2b, c].

Our system, by tuning the relevant parameters d and u, is also able to achieve bidirectional scattering properties at the EP of anti-PT symmetry, where S*11 = S22 and Im(S12) = 0. When focusing on the orange curve and the six bottom maps with d = 3.88 cm and u = 0.657 + i0.24 in Fig. 2, several observations can be made: first, |S11| = |S22| in Fig. 2a, b; second, Arg(S11) = −Arg(S22) in Fig. 2d, e; and third, Arg(S12) = 0 in Fig. 2f. It should be noted that while we have implemented the EP of an anti-PT-symmetric system through the intrinsic scattering matrix, our system does not need to satisfy the condition of balanced gain and loss in the traditional sense, as the LIT-induced gain can be tuned independently. In addition, the PT symmetry and anti-PT symmetry can be generated depending on the coupling d and the laser parameters, as discussed in Notes 5 and 6.

Non-Hermitian phase transitions using LIT
In a prior study39, we effectively showcased acoustic gain through electro-thermal-acoustic effects. Expanding on this investigation, here we present the construction of an experimental system based on LIT and establish that PT-symmetric phase transitions can be achieved within a unified device through the manipulation of coupling d. In our configuration, the CNT film functions as a bidirectional acoustic transparent medium. This means that the change in sound pressure and medium velocity across the film is minimal, resulting in negligible impedance mismatch. Additionally, we deliberately set the specific sound pressure amplitude ratio and phase difference between the sound wave generated by the CNT film and the sound source. Consequently, the overall scattering matrix Sr can be derived as (see Supplementary Note 7 for details):

rL = m1 ei\phi1 ts e−ix + rs,
tL = m1 ei\phi1 ts e−ix + ts e−ix,
rG = m2 ei\phi2 rs e−i2x + 1 + rs e−i2x,
tG = m2 ei\phi2 ts e−ix + ts e−ix, (3)

Here, m1 (m2) and \phi1 (\phi2) represent the amplitude ratio and phase difference of sound pressure between the assembled CNT film and the sound source when the incident wave comes from the loss (gain) side, respectively; rs and ts represent the reflection and transmission coefficients of the assembled loss material. It is worth noting that the nonreciprocal transmission phenomenon described by the 2nd and 3rd equations in Eq. (3) originates from the introduction of the CNT film, which provides abundant possibilities to manipulate the scattering properties in our configuration. In fact, our system retains sufficient degrees of freedom to convert from transmission nonreciprocity to reciprocity, and this can be achieved effectively by adjusting the geometric and laser parameters. For illustration, all controllable parameters in this study fulfill the premise of transmission reciprocity, specifically tL = tG = t. Based on this foundation, we emphasize the importance of judiciously optimizing the contrast ratio (m1 and m2), the coupling d, and the phases (including \phi1 and \phi2) in our system.

To gain a deeper understanding of these unusual scattering responses, we analyze the scattering spectra and eigenvalues of the system’s scattering matrix, as shown in Fig. 3 as functions of coupling d. Figure 3a–c demonstrates the existence of an EP where loss-side invisibility occurs, characterized by parameters m1 = 0.262, \phi1 = −0.91, m2 = 0.335, and \phi2 = 0.942. At a coupling of d = 5.31 cm, the reflection coefficient at the loss side (rL) is zero, indicating a sudden phase change in reflection. Meanwhile, the reflection at the gain side is nonzero with rG = 0.555ei1.237, which indicates the unidirectional reflectionless characteristics in opposite directions and highlighting the unidirectional scattering properties of the photoacoustic non-Hermitian bianisotropic system.

On the other hand, the experimental results demonstrating the anti-PT-symmetric EP of bidirectional conjugated reflections and pure real transmission are shown in Fig. 4e, f. Consistent with the simulation curves, both the bidirectional reflection amplitudes and transmission amplitudes intersect at f0 = 6 kHz in Fig. 4e. Additionally, the bidirectional reflection phases are opposite to each other, while the transmission phases are close to 0 in Fig. 4f. More specifically, the measured scattering matrix at the EP is rL = 0.578ei0.401, tL = 1.256e−i0.02, rG = 0.567e−i0.405, and tG = 1.239e−i0.01, illustrating the EP behavior of the anti-PT-symmetric system.

Formation of acoustic VBs and scattering EP using LIT
The LIT method employed in this study represents a promising approach for achieving perfect acoustic VBs and enables the PT-symmetric scattering behavior illustrated in Fig. 5a. In the aforementioned configuration, the CNT film under laser irradiation functions effectively as a point sound source, since the timescale of laser-induced heat diffusion across the film surface is on the order of seconds40. When the laser spot rotates on the CNT film at a modulation frequency of fr (e.g., fr = 0.1 kHz), the diffusion time is much longer than the rotation period, ensuring that the acoustic source rotates continuously rather than discretely. As a result, the system produces perfect acoustic VBs within the waveguide, unlike the approximate VBs generated by discrete passive41 or electronical driven35 methods. This continuous, contact-free excitation is a defining advantage of LIT and fundamentally differentiates it from previous electrical approaches.

The physical mechanism and spatial distribution of the acoustic VB generated by a rotating laser spot are discussed in Supplementary Note 10. To support the propagation of first-order VBs, the waveguide was replaced with a cylindrical tube of radius R0 = 2 cm. A schematic of the LIT-generated VB is shown in Fig. 5b, in which a certain angle is set between the symmetry axis of the reflector and the rotation axis of the motor, causing the laser to trace a circular trajectory on the CNT film. Supplementary Movies 1 and 2 demonstrate the laser spot rotating on the CNT film to form a circular trajectory at rotational speeds of 150 revolutions per minute (RPM) and 6000 RPM, respectively. The corresponding simulation and experimental results, after Fourier transformation, are presented in Fig. 5c, d, confirming that the LIT system produces a stable and well-defined acoustic VB (details in Supplementary Note 11).

To understand the scattering behavior of VBs at the EP, we modeled the scattering responses for counter-clockwise (CCW) vortex excitations (see Supplementary Note 12), with the corresponding scattering matrices revealing distinct symmetry-breaking features inherent to non-Hermitian systems:

Sr = [rL tL; tL rG] = [rL1 tL1; tL1 rG1] + [rL2 tL2; tL2 rG2] (6)

where the scattering coefficients represent only the normalized ratios between the reflected or transmitted pressure amplitudes and that of the incident wave. These quantities are independent of the vortex topological charge itself; the charge merely determines the rotational phase structure of the field.

Figure 5e, f demonstrated the simulated and experimentally measured scattering amplitudes and phases of the acoustic CCW VBs at the operating frequency of f0 = 6 kHz and rotation frequency of fr = 0.1 kHz, respectively. The corresponding CW characteristics are shown in Fig. 5h, i. The data reveal a remarkable behavior: when a CCW VB is incident from the loss side, the reflected VB is suppressed (rL = 0), and the transmitted VB preserves its amplitude but reverses its topological charge relative to the incident wave. In contrast, when the VB is incident from the gain side, the transmitted mode remains reversed (tG = −eiy), while the reflected field becomes a superposition of two VBs with distinct topological charges, rG = rG1 + rG2 = −ei2y/s−2 (see Supplementary Note 13 for details). Here, s is the complex reflection/transmission coefficient of the loss medium with rs = ts = s = |s|ei\alpha; |s| and \alpha are the amplitude and phase.

A more intuitive visualization is provided in Fig. 5g, j, showing the measured cross-sectional field distributions of CCW and CW VBs with a coupling d = 9.92 cm, corresponding to the vertical dashed lines in Fig. 5e, f, h, i, respectively. The fitted CNT film parameters for this EP are m11 = 1, \phi11 = \pi; m12 = 0, \phi12 = 0; m21 = 1, \phi21 = \pi; and m22 = 1/|s|, \phi22 = \pi−\alpha. The experimentally obtained scattering coefficients are rL1 = 0.068ei0.619, tL1 = 0.069ei0.57, rG1 = 1.811e−i1.803, tG1 = 0.015ei0.615, rL2 = 0.327ei0.181, tL2 = 1.016e−i3.102, rG2 = 1.982e−i3.015, and tG2 = 0.997e−i3.1 at the azimuth angle of \theta = 0. These results collectively confirm the realization of PT-symmetric scattering with acoustic VBs and the experimental observation of topological charge reversal at the EP. This finding represents the first experimental demonstration of PT symmetry in acoustic VBs, marking a substantial advance in non-Hermitian acoustics. The LIT platform not only enables the study of symmetry breaking in structured acoustic fields but also establishes a versatile route for exploring topological effects and non-Hermitian phase transitions beyond plane-wave systems.

Methods
Loss measurement
In order to verify the universality of this methodology, a common sponge sheet was selected as the acoustic loss material in our experiments. In the plane wave scenario, the measured reflection and transmission coefficients of the loss at 6 kHz are rs = 0.213ei0.585 and ts = 0.815e−i2.093. A standard extraction algorithm is used to retrieve the effective parameters of the sponge from the measured scattering matrix38. In our prototype sample, the mass density of surrounding background medium (air) is \rho0 = 1.21 kg/m3 and the bulk modulus is \kappa0 = 0.142 MPa. Thus, the retrieved mass density and bulk modulus are \rho s = 1.339ei0.144 kg/m3 and \kappa s = 0.177ei0.342 MPa, respectively. On the other hand, in the acoustic VB scenario, the reflection and transmission coefficients of the loss (thickness D = 0.4 cm) at 6.1 kHz are rs = ts = 0.559e−i1.137, and the retrieved mass density and bulk modulus are \rho s = 2.236e−i.249 kg/m3 and \kappa s = 0.12e−i0.444 MPa, respectively.

Active gain design
A CNT film irradiated by pulsed laser is used to achieve an artificial acoustic gain material since no passive acoustic gain material exists in nature. The film is placed inside the waveguide vertically and hence radiates plane waves (or VBs) facing opposite directions with identical amplitude and initial phase when pumped by laser. We control the effective transmission and reflection of the artificial CNT film to generate the predesigned effective gain medium.

Full-wave simulations
The simulations are performed by the full-wave finite element solver COMSOL Multiphysics, through which we validate these unusual scattering responses and obtain in-depth physical insights into the operation of this device. The pressure acoustic module and the acoustic heat source domain are used so that both loss and CNT can be fully modeled for this multi-physics process. The calculation domain is terminated by radiation boundary conditions that also include the incident field.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors report an innovative method for creating and controlling PT and anti-PT symmetries by using the technique of\nlaser-induced thermoacoustics. In particular, this approach allows to facilitate their phase transitions to be dynamically tuned\nby optical means in experiment, overcoming challenges related to incompatible coupling forms between PT and anti-PT\nsymmetries, while also ensuring that the process is fully reversible and programmable. In addition, the integration of LIT\ntechnique with ultra-thin flexible CNT film challenges the conventional view that acoustic gain requires electro-mechanical\ndevices, thus broadening potential applications. Therefore, I find this work to offer a valuable new approach in non-Hermitian\nacoustics, and contains new methodologies with potentially significant impact for future studies, as noted in the discussion\nsection of the manuscript. The physics is clearly presented with thorough derivations, and the data demonstrates strong\nalignment between theoretical predictions and experimental results. I believe these strengths make the manuscript a strong\ncandidate for publication in Nature Communications.\nI have the following comments to make to potentially improve the manuscript:\n1) In the 1st paragraph of the Introduction Section, the authors give out the incompatible forms of coupling between PT and\nanti-PT symmetries, which leads to their anti-communication. Please describe briefly how the physical model in this work\novercomes this challenge to achieve phase transition within a single unified system.\n2) The authors emphasize that “This innovative acoustically transparent strategy eliminates the need for traditional\nabsorbing-restoring compensation schemes that block the sound path.” in the Abstract. However, it seems that this point is\nnot clearly stated in the Mian Text, please make appropriate additions.\n3) In Supplementary Note VI, the photograph of the loss-gain unit should be added. The “Measuring setup” section in the\nMethods of the Main Text may be moved to Supplementary Note VI, and it will make more sense, I think.\n4) Is electro-thermoacoustic method with CNT film feasible for this concept? What’s the advantage of the LIT technology?\n5) The authors mention in the last paragraph of the Introduction Section that the three idiosyncratic EPs come from the\nmultiple interferences in the loss-gain unit, does this contradict the conclusion obtained by the transfer matrix method? After\nall, scattering is not taken into account in the transfer matrix method.\n6) In the manuscript, the scattering EP means the EPs of the scattering matrix. Could the authors provide a more intuitive\npicture of the scattering EPs? Maybe compared to EPs of a non-Hermitian Hamiltonian?\n7) In Fig. 3(a)-(b) of the Main Text and Fig. S10(a)-(b) of the Supplementary Material. Why do they show different curves,\neven though they both depict EPs for the loss-side invisibility of a PT-symmetric system? Similar problems also occur\nbetween Fig. 3(e)-(f) and Fig. S11(a)-(b) for the gain-side invisibility EPs, as well as Fig. 3(i)-(j) and Fig. S12(a)-(b) for anti-\nPT-symmetric EP.", "reviewer_2": "In this work, the authors present the design of a non-Hermitian acoustic system and the observation of non-Hermitian phase transitions. The system is modeled using the transfer-matrix method and several different phases, including PT and anti-PT phases, are identified. These non-Hermitian phases and the corresponding transitions are then verified experimentally using direct measurements of the amplitude and phase of sound. The whole structure of the manuscript is very clear and the results are self-consistent. I have the following concerns.\n1. The physics of PT symmetry, EPs and anti-PT symmetry has been well studied in various contexts. Even for acoustics, anti-PT symmetry has been studied by some of the authors (see PRL 131, 066601 (2023)). The physical phenomena in the present work are not new.\n2. The experimental implementation is also similar to what has been established by some of the authors. In their previous papers (see Nature 597, 655 (2021) and PRL 131, 066601 (2023)), the acoustic gain was also provided by a carbon nanotube film driven electronically (here the carbon nanotube film is driven by a laser).\nFor a paper to be considered for Nature Communications, I would expect, at least, that there is a certain level of significance either in physics or in experimental methodology.\nAnother technique issue: the symmetry operators were not given in the text."}, "author_response": "NCOMMS-24-62740: Response to Reviewers\nWe thank all reviewers for carefully reading our manuscript and for providing insightful comments,\nwhich have helped us to improve our manuscript. We are strongly encouraged by Reviewer #1’s\nreport stating that our work is a strong candidate for publication in Nature Communications.\nWe also sincerely appreciate the recognition of our work’s research quality and the constructive\ncomments from Reviewer #2. In response to their comments, we have conducted additional calcu-\nlations, simulations, and experiments to address the raised concerns. Below we respond in detail to\nall comments that were raised. To ensure readability, we have highlighted the manuscript changes\nin red.\nResponse to Reviewer 1\nComment P1.1 – The authors report an innovative method for creating and controlling PT and\nanti-PT symmetries by using the technique of laser-induced thermoacoustics. In particular, this\napproach allows to facilitate their phase transitions to be dynamically tuned by optical means in\nexperiment, overcoming challenges related to incompatible coupling forms between PT and anti-PT\nsymmetries, while also ensuring that the process is fully reversible and programmable. In addition,\nthe integration of LIT technique with ultra-thin flexible CNT film challenges the conventional view\nthatacousticgainrequireselectro-mechanicaldevices, thusbroadeningpotentialapplications. There-\nfore, I find this work to offer a valuable new approach in non-Hermitian acoustics, and contains\nnew methodologies with potentially significant impact for future studies, as noted in the discussion\nsection of the manuscript. The physics is clearly presented with thorough derivations, and the data\ndemonstrates strong alignment between theoretical predictions and experimental results. I believe\nthese strengths make the manuscript a strong candidate for publication in Nature Communications.\nI have the following comments to make to potentially improve the manuscript.\nResponse: Wethankthereviewerforthoroughlyreviewingourworkandhighlightingtheadvancements\nwe have made in this research field. We are delighted to address the points that were raised by the\nreviewer.\nComment P1.2 – In the 1st paragraph of the Introduction Section, the authors give out the\nincompatible forms of coupling between PT and anti-PT symmetries, which leads to their anti-\ncommunication. Please describe briefly how the physical model in this work overcomes this challenge\nto achieve phase transition within a single unified system.\nResponse: We appreciate the reviewer’s insightful comment. As noted in the first paragraph of\nthe Introduction Section, the anti-communication between PT and anti-PT symmetry arises from\ntheir fundamentally incompatible coupling forms. Specifically, for PT-symmetric elements i and j, the\ncoupling follows the relation κ = κ∗ , while for anti-PT-symmetric elements, the complex conjugate\nij ji\ncoupling is represented by κ = −κ∗ . This incompatibility requires that the gain and loss media exhibit\nij ji\nidentical geometric configurations.\ngain and loss media, which allows us to achieve phase transitions between PT and anti-PT symme-\ntries within a single unified system. The gain and loss media are intentionally designed with different\ngeometries: the loss medium consists of a bulk material (porous sponge) with a well-defined width,\nwhile the gain medium is a thin CNT film with negligible thickness activated by laser irradiation. In this\nconfiguration, the gain medium can be considered as an acoustic internal admittance. By modulating\nthe laser irradiation parameters and carefully controlling the coupling between the gain and loss media,\nwe are able to realize the distinct symmetric phases (PT, reversed-PT, and anti-PT) and observe the\ncorresponding phase transitions within the system. This approach not only resolves the issue of incom-\npatible coupling between PT and anti-PT symmetries, but also enables the investigation of various\nexceptional points (EPs) and their dynamics in a single unified system.\nFollowing the reviewer’s suggestion, we have added the sentences: “In this model, the loss medium\nis a conventional bulk material (porous sponge) with a defined width D, whereas the gain medium is a\npiece of CNT film with negligible thickness. The geometrically asymmetric unit described in Eq. 2 ex-\nhibits reciprocity and linearity, allowing different types of EPs to emerge through the tuning of coupling\nd (Supplementary Note IV).” on Page 5.\nComment P1.3 – The authors emphasize that “This innovative acoustically transparent strategy\neliminates the need for traditional absorbing-restoring compensation schemes that block the sound\npath.” in the Abstract. However, it seems that this point is not clearly stated in the Main Text,\nplease make appropriate additions.\nResponse: We appreciate the reviewer’s thoughtful comment and the opportunity to clarify this point.\nTraditional implementations of acoustic gain media typically rely on electromechanical devices such as\npiezoelectrictransducersorconeloudspeakers. Whiletheseelementscangenerateoramplifysound,their\nsolidstructuresinherentlydisruptthecontinuityoftheacousticpathway. Asaresult, mostexperimental\nrealizations of PT-symmetric acoustic systems have depended on absorbing–restoring compensation\nschemes that physically isolate the sound path, as reported in Nature Communications 6, 5905\n(2015) and Nature Communications 7, 11110 (2016). In contrast, the CNT film employed in this work\nprovides a fundamentally different approach. Owing to its ultrathin structure and high acoustic trans-\nparency, the CNT layer permits sound waves to propagate freely while simultaneously functioning as an\nactive gain element under laser excitation. This acoustically transparent configuration eliminates the\nneed for any absorbing–restoring compensation, preserving the integrity of the sound path throughout\nthe experimental domain. Such a design not only simplifies the construction of non-Hermitian acoustic\nsystems but also expands their potential applications. By maintaining an open sound channel, the sys-\ntem becomes more suitable for scenarios where both transmission and detection are crucial, including\nprecision acoustic sensing, imaging, and real-time environmental monitoring. This concept—now explic-\nitly discussed in the Main Text—represents a key innovation distinguishing our experimental strategy\nfrom previous approaches.\nAdditionally, we present further experimental evidence to more clearly illustrate the acoustically\ntransparent nature of the proposed configuration. Specifically, we measured the transmission coefficient\nof a 10 µm-thick CNT film within a waveguide, and the results, shown in Fig. R1, confirm that the\nsound wave transmission through the CNT film remains close to 100% across a wide frequency range.\n2\naa bb\n--ππ\n11\nee\ndd uu\n))dd\naa rr\ntt iinn (( ee 00\ngg ss\naa aa\nMM hh\nPP\n--ππ\n00\n33 44 55 66 77 33 44 55 66 77\nFFrreeqquueennccyy ((kkHHzz)) FFrreeqquueennccyy ((kkHHzz))\nFigure R1: Measured transmission magnitude (a) and phase (b) of a single CNT film.\nMoreover, there is no observable phase shift, indicating that the CNT film has negligible influence on\nwave propagation, thus demonstrating its transparency to sound waves. This outcome is in agreement\nwith the predictions of classical wave propagation theory. For a thin intermediate medium, such as the\nCNT film, the transmission coefficient t can be expressed by the following equation:\n2\nt = . (R1)\n(cid:18) (cid:19)\nz z\n0\n2cos(kh)+i + sin(kh)\nz z\n0\n(cid:112)\nHere, k = w/c = 2πf ρ/κ is the wave number in the CNT film, ρ ∼ 40 kg/m3 is the mass density,\nκ ∼ 36 GPa is the bulk modulus, and h = 10 µm is the film thickness. Therefore, the speed of sound\n(cid:112) √\nin the CNT film is c = κ/ρ ∼ 3×104 m/s, and the impedance z = κρ ∼ 1.2×106 kg/(m2·s).\nThe air impedance z is considered for comparison. Under the experimental conditions, we find that\n0\nthe product kh ∼ 1.2×10−5 and z /z ∼ 3.5×10−4 within the frequency range under investigation,\n0\nwhich simplifies the transmission coefficient to:\n2 2\nt ∼ ≈ . (R2)\nz\n2+i0.035\n2+i kh\nz\nates as an acoustically transparent medium, with minimal impact on wave transmission.\nFollowing the reviewer’s comment, we have now revised the manuscript to make this point clear:\n“Itsultrathinprofilerenders itacousticallytransparent(SupplementaryNote I),allowingsoundwaves to\npropagate freely in the transverse direction with minimal impact on wave transmission, a feature distinct\nfrom conventional electromechanical loudspeakers that exhibit high levels of reflection.” on Page 4.\nThe above discussions have been included in Supplementary Note I.\nComment P1.4 – In Supplementary Note VI, the photograph of the loss-gain unit should be\nadded. The “Measuring setup” section in the Methods of the Main Text may be moved to Supple-\nmentary Note VI, and it will make more sense, I think.\nof the gain-loss unit, which is now included in Fig. R2 (Fig. S19 in Supplementary Note XIV). Ad-\nditionally, we have moved the “Measuring Setup” section from the Methods in the Main Text to the\nrevised Supplementary Note XIV. This restructuring provides a clearer and more logical presentation\nof the experimental setup.\nMMiiccrroopphhoonnee\n11 22 33 44 AAnneecchhooiicc bboouunnddaarryy\nCCNNTT LLoossss\n00..005533mm dd\n00..1199mm 00..0055mm00..0055mm 00..0022mm00..0055mm00..0055mm 00..1155mm 00..66mm\n00..1155mm\nNNII PPXXII ddiiggiittiizzeerr CCoommppuutteerr mmoonniittoorr\nFigure R2: Schematic diagram of the experimental setup. Inset photograph shows a detailed view\nof the loss-gain unit outlined in the dashed box.\nComment P1.5 – Is electro-thermoacoustic method with CNT film feasible for this concept?\nWhat’s the advantage of the LIT technology?\nResponse: The reviewer has brought up a valuable point about the feasibility and advantages of\nthe LIT technology, which deserves to be carefully clarified. The electro-thermoacoustic method is\nindeed feasible for the proposed concept, and it offers distinct advantages. This method, which requires\ntwo input signals, is particularly effective when compared to traditional electromagnetic loudspeakers.\nPreliminary experiments using the electro-thermoacoustic approach [Phys. Rev. Lett. 131, 066601\n(2023); Adv. Mater. 36, 2406567 (2024)] have already demonstrated its practical viability.\nHowever, the LIT technology offers unique benefits that set it apart from conventional methods.\nOne of the key advantages of LIT is that it operates without the need for physical contact, electrical\nconnections, or magnetic fields, unlike the electro-thermoacoustic approach, which requires additional\ncontrol and power circuitry. This contactless nature makes LIT particularly advantageous in complex or\nconstrained environments, where minimizing setup complexity is crucial. Furthermore, LIT technology\nis more energy-efficient than electrical methods. In the case of LIT, almost the entire optical energy\nis converted into phonon populations due to the photoacoustic effect, which is driven by inter-cone\ntransitions in the Dirac band structure of graphene. This results in a photon-wavelength-independent\nabsorption coefficient across the visible range. The dominant electronic cooling mechanism is non-\nradiative electron–phonon scattering at these wavelengths, as demonstrated in previous work [Nat.\nMater. 10, 569 (2011)]. Additionally, studies have shown that LIT works effectively both in air and\nliquid without significant degradation of the structure [Nano Lett. 8, 4539 (2010)].\ntheycangenerate. LITiscapableofexcitingasubstantiallywiderrangeofacousticwaveformscompared\tothoseproducedbyelectricalmethods. Forinstance,whenCNTfilmsareelectricallydrivenatairborne\nacoustic frequencies, they behave as a purely resistive element. The current passes through the film\nquickly, resulting in uniform heating, which generates an acoustic wave that conforms to the geometric\nshape of the film. In contrast, when CNT films are excited by LIT, the heat transfer throughout the\nfilm (about 10 cm2 in this paper) takes longer than the sound wave’s period, allowing for the creation\nof more complex and continuous acoustic structures. For example, if the laser spot rotates at a specific\nfrequency (e.g., f = 0.1 kHz), the thermal diffusion time will be much longer than the rotation period,\nr\nenabling the formation of a perfect acoustic vortex beam (VB) in the waveguide. This continuous\nrotation of the acoustic vortex contrasts with the approximate vortex sound waves generated by discrete\nstructures, such as those in passive systems [Phys. Rev. Lett. 117, 034301 (2016)] or electrically driven\nsystems [Adv. Mater. 34, 202202026 (2022)].\nIn response to the reviewer’s comment, we have added a detailed discussion on the generation of\nacoustic VBs using LIT and the realization of PT-symmetric EP under the section titled “Formation\nof acoustic VBs and scattering EP using LIT” on Page 9 of the Main Text. Additionally, the specific\nmethodology for implementing this approach is now provided in Supplementary Notes X-XIII.\nComment P1.6 – The authors mention in the last paragraph of the Introduction Section that\nthe three idiosyncratic EPs come from the multiple interferences in the loss-gain unit, does this\ncontradict the conclusion obtained by the transfer matrix method? After all, scattering is not taken\ninto account in the transfer matrix method.\nResponse: We are very grateful to the reviewer for raising this important issue. The concern regarding\nthe apparent discrepancy between the description of multiple interferences in the loss–gain unit and the\nuse of the transfer matrix method (TMM) is important and merits careful clarification.\nThe three idiosyncratic EPs indeed arise from multiple interferences occurring within the loss–gain\nunit, as described in the last paragraph of the Introduction Section. These interferences capture the\nintrinsic scattering dynamics between the gain and loss media, which play a central role in shaping the\nnon-Hermitian behavior of the system. While the transfer matrix method is typically used to describe\nthe scattering properties of the entire system, it treats each component as an effective element with-\nout explicitly resolving the local scattering processes within them. Consequently, the TMM provides a\nmacroscopic description of system behavior, whereas the multiple-interference approach offers a micro-\nscopic view of the internal energy exchanges between the coupled subsystems. Despite this difference in\nperspective, the two approaches are fully consistent and complementary. Both ultimately describe the\nsame physical process—the propagation, reflection, and transmission of acoustic waves through a cou-\npled non-Hermitian structure—and lead to identical scattering characteristics. In this sense, the TMM\nand multiple-interference descriptions are equivalent in representing the global response of the system\nand its corresponding EPs. To address the reviewer’s concerns, we construct a physical model and\ndetermine the EPs of the three observed modes within the multiple-interference framework, which\nprovidesamoreintuitivepictureofhowenergyexchangebetweengainandlossdrivesthephasetransitions.\n(1) Scattering matrix formulation based on multiple interferences\n5\naa LLLooossssss GGGaaaiiinnn bb GGaaiinn LLoossss\npppp ppp\n0000 pppp2222 pppp3333 pppp4444 000 ppp222 ppp333 ppp444\npppp1111 1111 1111 1111 ppp111 111 111 111\n1111 pppp2222 pppp3333 pppp4444 111 ppp222 ppp333 ppp444\npppp1111 2222 2222 2222 ppp111 222 222 222\n2222 pppp2222 pppp3333 pppp4444 222 ppp222 ppp333 ppp444\npppp1111 3333 3333 3333 ppp111 333 333 333\n3333 333\npppp2222 pppp3333 pppp4444 ppp222 ppp333 ppp444\npppp1111 mmmm mmmm mmmm ppp111 mmm mmm mmm\nmmmm mmm\nddd dd\nFigure R3: Sketch of the multiple interferences model between gain-loss unit. (a) Loss-side inci-\ndence. (b) Gain-side incidence.\nTo illustrate this equivalence more explicitly, we derive the system’s scattering coefficients using\nthe multiple-interference approach. The incident acoustic wave undergoes repeated reflections and\ntransmissionsbetweenthegainandlosscomponents,andtheircumulativeeffectgivesrisetothedistinct\nEP behaviors observed experimentally. Let the reflection and transmission coefficients of the gain unit\n(internal admittance) be denoted by r and t , and those of the loss unit (porous sponge sheet) by r\nC C s\nand t . The two units are separated by an air layer of width d, with coupling x = k d, where k = w/c\ns 0 0 0\nis the wave number at angular frequency ω . For the case of incidence from the loss side, as shown in\n0 m\np2 , p3 , and p4 (m = 1,2,3,...), corresponding to the reflections and transmissions at the boundaries\nm m m\nof the loss and gain units. The first-order reflections and transmissions by the loss unit can be expressed\nas p1 = p r and p2 = p t . Subsequent reflections at the gain unit yield p3 = p2e−ixr = p e−ixt r\n1 0 s 1 0 s 1 1 C 0 s C\nand p4 = p2e−ixt = p e−ixt t . Higher-order reflections and transmissions can be derived recursively\n1 1 C 0 s C\nas p1 = p3e−ixt = p e−i2xt2r , p2 = p3e−ixr = p e−i2xt r r , p3 = p2e−ixr = p e−i3xt r r2,\n2 1 s 0 s C 2 1 s 0 s s C 2 2 C 0 s s C\nand p4 = p2e−ixt = p e−i3xt r t r . So we have\n2 2 C 0 s s C C\np1 = p3 e−ixt = p2 e−i2xt r = ··· = p e−i2(m−1)xt2rm−2rm−1,\nm m−1 s m−1 s C 0 s s C (R3)\np4 = p2 e−ixt = p3 e−i2xr t = ··· = p e−i(2m−1)xt rm−1t rm−1.\nm m C m−1 s C 0 s s C C\nBy summing all reflection and transmission contributions, the total reflection and transmission coeffi-\ncients of the entire loss–gain system can be analytically obtained:\nr = (cid:88) +∞ p1 /p = r +t2r e−i2x (cid:88) +∞ (r r e−i2x)m−2 = r +t2r e−i2x lim 1−(r s r C e−i2x)m−1 ,\nL m 0 s s C s C s s C m→+∞ 1−r s r C e−i2x\nm=1 m=2\nt = (cid:88) +∞ p4 /p = t t e−ix (cid:88) +∞ (r r e−i2x)m−1 = t t e−ix lim 1−(r s r C e−i2x)m .\nL m 0 s C s C s C m→+∞ 1−r s r C e−i2x\nm=1 m=1\n(R4)\nThese expressions are directly comparable to those derived from the transfer matrix method and yield\nidentical results, confirming the consistency of both theoretical descriptions.\nA similar derivation applies for incidence from the gain side, as shown in Fig. R3(b), where an\n6\nanalogous set of reflection and transmission coefficients can be determined as\nr = (cid:88) +∞ p1 /p = r +t2r e−i2x (cid:88) +∞ (r r e−i2x)m−2 = r +t2r e−i2x lim 1−(r s r C e−i2x)m−1 ,\nG m 0 C C s s C C C s m→+∞ 1−r s r C e−i2x\nm=1 m=2\nt = (cid:88) +∞ p4 /p = t t e−ix (cid:88) +∞ (r r e−i2x)m−1 = t t e−ix lim 1−(r s r C e−i2x)m .\nG m 0 s C s C s C m→+∞ 1−r s r C e−i2x\nm=1 m=1\n(R5)\nThe resulting scattering matrix comprehensively describes the acoustic response of the coupled non-\nHermitian system and the emergence of the three EPs observed experimentally:\n1−(r r e−i2x)m−1\nr = r +t2r e−i2x lim s C ,\nL s s C m→+∞ 1−r s r C e−i2x\nt = t = t t e−ix lim s C ,\nG L s C m→+∞ 1−r s r C e−i2x\n1−(r r e−i2x)m−1\nr = r +t2r e−i2x lim s C .\nG C C s m→+∞ 1−r s r C e−i2x\n(R6)\n(2) Accessing PT, reversed-PT, and anti-PT with multiple interferences\nAs detailed above, the scattering coefficients of the system are primarily governed by the magnitude\nof |r r |. The reflection coefficient of the loss unit, r = 0.213ei0.585, remains constant according to\ns C s\nexperimental measurements of the sponge sheet, while the reflection coefficient of the gain unit, r ,\nC\nvaries across different symmetry modes. In Supplementary Note IV, we have demonstrated that the\ntransfer matrix of the gain component can be expressed as [1 0;−u/z 1]. Consequently, the reflection\n0\ncoefficient is given by r = u/(2−u). This expression allows us to examine the emergence of EPs\nC\nunder PT-symmetric, reversed-PT-symmetric, and anti-PT-symmetric conditions.\nIn the case of PT symmetry, when u = 0.456+i0.257 at the EP corresponding to d = 5.31 cm,\nthe resulting reflection coefficient is r = u/(2 − u) = 0.334ei0.678, yielding |r r | = 0.071 < 1.\nC s C\nFor the reversed-PT-symmetric configuration, with u = 0.435 + i0.291 at d = 4.34 cm, we obtain\nr = 0.329ei0.773 and |r r | = 0.07 < 1. Similarly, in the anti-PT symmetry case, u = 0.657+i0.24\nC s C\nat d = 3.88 cm leads to r = 0.513ei0.527 and |r r | = 0.109 < 1. Under these conditions, the overall\nC s C\nscattering matrix of the system can be expressed as\nt2r e−i2x t t e−ix t2r e−i2x\nr = r + s C , t = t = s C , r = r + C s ,\nL s 1−r r e−i2x G L 1−r r e−i2x G C 1−r r e−i2x\ns C s C s C\n(R7)\nwhere the matrix elements are determined by the total reflections and transmissions resulting from\nmultiple interferences between the gain and loss units.\nThescatteringcharacteristicsderivedfromthismultiple-interferencemodelarepresentedinFig. R4,\ncorresponding to the three symmetry configurations of PT, reversed-PT, and anti-PT. The calcula-\ntions are performed using the respective admittance parameters u = 0.456+i0.257, 0.435+i0.291,\nand 0.657+i0.24. Remarkably, the EPs obtained through this interference-based formulation coincide\nprecisely with those predicted by the transfer matrix method in Supplementary Note IV.\nFollowing the reviewer’s constructive comment, we have now revised the manuscript on Page 6: “In\naddition, the phase transitions can be independently verified using a multiple-interference approach, as\n7\naa cc cc\n11..55\n||rrLL|| ||rrGG|| ||ttLL|| ||rrLL|| ||rrGG|| ||ttLL|| ||rrLL|| ||rrGG|| ||ttLL||\n11 11\nee dd EEPP ee dd EEPP ee dd 11\nuu uu uu\nttiinn00..55 ttiinn00..55 ttiinn\ngg aa gg aa gg aa00..55\nMM MM MM\n00 00\n00\n33 44 55 66 33 44 55 66 33 44 55 66\ndd ((ccmm)) dd ((ccmm)) dd ((ccmm))\nbb dd dd\nππ ππ ππ\nEEPP\n))dd ))dd ))dd\naa aa aa\nrr((\nee 00\nrr((\nee 00\nrr((\nee 00\nssaa ssaa ssaa\nhh\nPP\nAArrgg((rrLL)) hh\nPP\nAArrgg((rrLL)) hh\nPP\nAArrgg((rrLL))\n--ππ\nAArrgg((rrGG))\n--ππ\nAArrgg((rrGG))\n--ππ\nAArrgg((rrGG))\nAArrgg((ttLL)) AArrgg((ttLL)) AArrgg((ttLL))\n33 44 55 66 33 44 55 66 33 44 55 66\ndd ((ccmm)) dd ((ccmm)) dd ((ccmm))\nFigure R4: Scattering properties of the system with multiple interferences. (a)-(b) The magnitudes\n(a) and phases (b) of the scattering parameters with u = 0.456+i0.257, confirming the EP of PT\nsymmetry at coupling d = 5.31 cm. (c)-(d) The magnitudes (c) and phases (d) of the scattering\nparameters with u = 0.435 + i0.291, indicating the EP of reversed-PT symmetry at coupling\nd = 4.34 cm. (e)-(f) The magnitudes (e) and phases (f) of the scattering parameters with u =\n0.657+i0.24, corresponding to the EP of anti-PT symmetry at coupling d = 3.88 cm.\ndetailed in Supplementary Notes V and VI.”, and have added the above discussion into Supplementary\nNote V and VI.\nComment P1.7 – In the manuscript, the scattering EP means the EPs of the scattering matrix.\nCould the authors provide a more intuitive picture of the scattering EPs? Maybe compared to EPs\nof a non-Hermitian Hamiltonian?\nResponse: We appreciate the reviewer’s insightful remark and agree that the non-Hermitian Hamilto-\nnian offers an intuitive framework for understanding EPs, particularly in cavity-coupled PT-symmetric\nsystems where intrinsic resonance frequencies are well-defined. This approach has been widely applied\nin both optical systems [Science 373, 1261 (2021)] and acoustic systems [Phys. Rev. Lett. 131, 066601\n(2023)], providing clear insights into phase transitions. However, our study adopts a different approach\nthat does not rely on resonance modes defined by a Hamiltonian. Instead, we focus on the scatter-\ning process, where multiple interactions between the PT-symmetric bulk loss and internal admittance\nshape the reflected and transmitted waves, giving rise to scattering EPs. This method contrasts with\nthe more conventional Hamiltonian-based models but still captures the essence of non-Hermitian phase\ntransitions. In the context of scatterer-based PT-symmetric systems, significant discoveries such as\nnegative reflection [Phys. Rev. Lett. 113, 023903 (2014)] and impurity immunity [Phys. Rev. X 8,\n031035 (2018)] have demonstrated the potential of scattering EPs. The scattering matrix approach\noffers an alternative to the non-Hermitian Hamiltonian, particularly in systems where resonance modes\nare less prominent or absent.\nemphasizes the role of sound amplitudes and phases in describing the scattering properties of the sys-\ntem. While non-Hermitian Hamiltonian EPs primarily describe internal dynamics (e.g., eigenmodes),\nour scattering EPs are directly linked to the macroscopic behavior of the system, providing a clearer\npicture of the phase transitions as observed experimentally. Our theoretical models, simulations, and\nexperimental results are in strong agreement, further validating our approach. Nevertheless, we appreci-\nate the reviewer’s suggestion to explore the use of an effective Hamiltonian method for scatterer-based\nsystems and will carefully consider its potential application in future work.\nComment P1.8 – In Fig. 3(a)-(b) of the Main Text and Fig. S10(a)-(b) of the Supplementary\nMaterial. Why do they show different curves, even though they both depict EPs for the loss-side\ninvisibility of a PT-symmetric system? Similar problems also occur between Fig. 3(e)-(f) and\nFig. S11(a)-(b) for the gain-side invisibility EPs, as well as Fig. 3(i)-(j) and Fig. S12(a)-(b) for\nanti-PT-symmetric EP.\nResponse: We sincerely thank the reviewer for this valuable observation. The differences between the\ncurves shown in Fig. 3(a)–(b) of the Main Text and Fig. S10(a)–(b) of the Supplementary Material\narise from the distinct parameter settings used in each case, rather than from inconsistencies in the\nanalysis or interpretation of the EPs. In Fig. 3(a)–(b), we present the EP corresponding to the loss-side\ninvisibility of a PT-symmetric system under fixed experimental conditions. The CNT film parameters\nare set as m = 0.262, φ = −0.91, m = 0.335, and φ = 0.942. These constant values are chosen to\n1 1 2 2\nillustratetherepresentativephasetransitionbehaviorattheEP.Incontrast, Fig. S10(a)–(b)intheSup-\nplementary Material depicts a theoretical analysis in which the CNT film parameters vary as functions\nof the coupling distance d, as detailed in Fig. S10(c)-(d). This variation leads to different curve shapes\nbecause the reflection and transmission responses evolve continuously with changing coupling strength.\nDespite the differences in the curve profiles, the positions of the EPs coincide precisely, confirming the\nconsistency between the Main Text and the supplementary results. A similar rationale applies to the\ncomparisons between Fig. 3(e)–(f) and Fig. S11(a)–(b), which correspond to the gain-side invisibility\nEPs, as well as between Fig. 3(i)–(j) and Fig. S12(a)–(b), which represent the anti-PT-symmetric EPs.\nIn each case, the apparent discrepancies in curve shapes stem from whether the CNT parameters are\nfixed or coupling-dependent, while the EP locations remain identical.\nIn response to the reviewer’s comment, this clarification has been incorporated into the revised\nmanuscript to ensure consistency between the Main Text and the Supplementary Material: “The the-\noretical analysis of scattering coefficients, in which the CNT film parameters vary as functions of the\ncoupling d have been demonstrated in Supplementary Note VIII, confirming the identical positions of\nthe EPs coincide precisely.” on Page 7.\nCommentP2.1 – Inthiswork, theauthorspresentthedesignofanon-Hermitianacousticsystem\nand the observation of non-Hermitian phase transitions. The system is modeled using the transfer-\nmatrix method and several different phases, including PT and anti-PT phases, are identified. These\nnon-Hermitianphasesandthecorrespondingtransitionsarethenverifiedexperimentallyusingdirect\nmeasurements of the amplitude and phase of sound. The whole structure of the manuscript is very\nclear and the results are self-consistent. I have the following concerns.\nResponse: We would like to express our sincere gratitude to the reviewer for the careful evaluation of\nour manuscript and for the positive recognition of its clarity, structure, and self-consistent results. We\nare grateful for the reviewer’s thoughtful comments and constructive suggestions, which have helped\nus refine the manuscript and strengthen the presentation of our findings. In the following, we provide\ndetailed responses and corresponding revisions addressing each of the reviewer’s concerns.\nComment P2.2 – The physics of PT symmetry, EPs and anti-PT symmetry has been well\nstudied in various contexts. Even for acoustics, anti-PT symmetry has been studied by some of the\nauthors (see PRL 131, 066601 (2023)). The physical phenomena in the present work are not new.\nResponse: We sincerely thank the reviewer for raising this important point, which allows us to clarify\nthe novelty and fundamental distinctions of our work from previous studies, particularly Ref. 28 (Phys.\nRev. Lett. 131, 066601 (2023)). Although our research shares conceptual ground in exploring PT\nsymmetry, EPs, and anti-PT symmetry, the underlying physical mechanisms, theoretical frameworks,\nand experimental implementations are entirely different.\nUnlike Ref. 28, which investigates a cavity-tube system modeled as a topological SSH chain\nusing a tight-binding Hamiltonian, our study is based on a scattering system described through the\ntransfer matrix formalism derived from transmission-line theory. The cavity-tube model focuses on\nthe intrinsic modal properties of coupled resonators, where PT-symmetry breaking is expressed through\neigenvalue coalescence in the Hamiltonian spectrum. In contrast, our work examines the extrinsic\nscattering behavior of acoustic waves, where PT and anti-PT symmetries manifest through reflection\nand transmission characteristics governed by the spatial distribution of gain and loss.\nThis distinction is further exemplified in the anti-PT case. In Ref. 28, anti-PT symmetry arises\nfrom the coupling of eigenmodes within the Hamiltonian framework, reflecting internal energy dynam-\nics. In our system, however, anti-PT symmetry is observed directly in the scattering response, where\nconjugate reflection and purely real transmission coexist under wave excitation—an experimentally ver-\nifiable signature distinct from the internal energy distribution reported previously. To the best of our\nknowledge, this represents the first experimental realization of such conjugate reflection and real-valued\ntransmission in an acoustic anti-PT configuration.\nBeyond the conventional PT and anti-PT regimes, our system also supports a reversed-PT-\nsymmetric phase, a phenomenon rarely reported in acoustics. This state occurs when the system\nis excited from the gain side, yet it still exhibits unidirectional transparency—a reversal of the usual\nPT scattering behavior, where excitation from the loss side leads to transparency. The realization of\nreversed-PT symmetry underscores the versatility of our laser-induced thermoacoustic (LIT) platform,\n10\nwhichallowsprecisecontrolofgain–lossinteractionsandfielddistribution,capabilitiesdifficulttoachieve\nwith traditional loudspeaker-based or cavity-coupled systems.\nA key strength of our approach lies in its integration of PT, reversed-PT, and anti-PT sym-\nmetries within a single unified experimental system. To our knowledge, this is the first time these\ndistinct non-Hermitian phase transitions and their mutual conversions have been experimentally realized\ntogether. While our current focus is on plane-wave excitations, the same principle can be extended\nto arbitrary acoustic wavefronts, including PT-symmetric acoustic vortices, which are discussed in the\nsubsequentsection. Thisunifiedframeworknotonlydemonstratesthecoexistenceofmultiplesymmetry\nregimes but also provides a flexible foundation for designing advanced acoustic devices with reconfig-\nurable non-Hermitian functionalities.\nIn light of this clarification, we have revised the relevant description in the manuscript on Page 2\nas: “Nevertheless, due to the absence of an efficient experimental methodology, the establishment of\nan acoustic anti-PT-symmetric configuration remains exceedingly rare, hindering the phase transitions\nbetween PT and anti-PT symmetries within a single unified system—a challenge exacerbated by the\nincompatible forms of coupling leading to their anti-communication. Although studies on PT phase\ntransitions, including those of anti-PT symmetry, in cavity-tube systems simulating tight-binding mod-\nels have been reported [28], the anti-PT phase transitions in scattering systems remain a topic worthy\nof further investigation.”\nComment P2.3 – The experimental implementation is also similar to what has been established\nby some of the authors. In their previous papers (see Nature 597, 655 (2021) and PRL 131, 066601\n(2023)), the acoustic gain was also provided by a carbon nanotube film driven electronically (here\nthe carbon nanotube film is driven by a laser).\nResponse: We thank the reviewer for this thoughtful observation. It is true that both our present\nlaser-driven approach and the previously reported electrically driven method [Nature 597, 655 (2021);\nPhys. Rev. Lett. 131, 066601 (2023)] utilize CNT films as non-Hermitian acoustic gain elements. From\na physical standpoint, both mechanisms share the same fundamental origin: the rapid electron–phonon\nrelaxation process following carrier excitation increases the lattice temperature of the CNT film, gener-\nating thermal waves that are subsequently transferred to the surrounding air. The resulting expansion\nof the adjacent air layers acts as an acoustic piston, producing sound radiation.\nAlthough the underlying thermoacoustic principle is common, the driving mechanisms and re-\nsulting acoustic behaviors differ profoundly. In the electrically driven configuration, the CNT film\nis excited by alternating current, leading to periodic Joule heating. The generated acoustic signal is\ntypically at twice the driving frequency, with the waveform determined by the uniform resistive heating\nprofile of the film. This approach is well suited for producing geometrically defined sound fields—for\ninstance, cylindrical wavefronts when the film is wrapped around a tubular substrate, as demonstrated\nin Nature 597, 655 (2021).\nIn contrast, the laser-pumped configuration adopted in this study is contactless, electrically and\nmagnetically passive, and free from external wiring or circuitry, which greatly simplifies system in-\ntegration, especially in confined or high-interference environments. Beyond practicality, laser excitation\noffers significant physical advantages. Nearly all absorbed optical energy contributes to phonon genera-\ntion due to the wavelength-independent optical absorption of graphene-based CNTs and the dominance\nofnon-radiativeelectron–phononscattering[Nat. Mater. 10,569(2011)]. Thisresultsinhigherconver-\n11\nsion efficiency compared to electrical pumping. Furthermore, CNT films maintain excellent mechanical\nand thermal stability in both air and liquid environments without structural degradation [Nano Lett. 8,\n4539 (2010)].\nA particularly important distinction lies in the diversity of acoustic waveforms accessible under\nlaser excitation. Because the characteristic thermal diffusion time of the CNT film (on the order of 10\ncm2 area in this work) is much longer than the acoustic period, spatiotemporal modulation of the laser\nbeamcandynamicallyshapetheemittedsoundfield. Forinstance, whenthelaserspotisrotatedaround\nthe central axis of the waveguide and at a frequency such as f = 0.1 kHz on the film, the continuous\nr\nheating trajectory gives rise to a stable, coherent acoustic vortex beam (VB) within the waveguide.\nThis continuous rotation produces a genuine vortex field, as opposed to the discrete vortex-like struc-\ntures previously realized using passive [Phys. Rev. Lett. 117, 034301 (2016)] or electrically driven\n[Adv. Mater. 34, 202202026 (2022)] configurations. To verify this effect, as a proof-of-concept, we\ndeveloped theoretical models and performed experiments demonstrating the generation of a first-order\nacoustic vortex beam under rotating laser excitation. This result highlights the unique capabilities of\nlaser-induced thermoacoustics for achieving controllable, broadband, and geometrically versatile sound\nfields—capabilities unattainable with the earlier electrically driven CNT systems.\n(1) Theoretical models for acoustic vortex beams realized by laser-induced thermoacoustics\nThe method employed in this work—laser-induced thermoacoustics (LIT)—provides a simpler and\nmore efficient alternative to the electrically driven CNT films used in previous studies. While previous\nresearch (Nature 597, 655, 2021; Phys. Rev. Lett. 131, 066601, 2023) has demonstrated acoustic gain\nthrough CNT films driven electrically, our approach harnesses the unique properties of laser excitation,\neliminating the need for complex electronic circuit designs. For example, the generation of acoustic VBs\ncan be accomplished by simply introducing a rotating laser beam, which induces a rotational Doppler\neffect. This stands in contrast to the more intricate methods required for electrically pumped CNT film\nsystems.\nHere, we present an experimental setup that significantly improves upon earlier implementations.\nInstead of the stationary laser excitation typically used, we introduce a rotating laser beam that traces\na circular path on the CNT film. This configuration aligns the center of the circular path along the\naxis of the cylindrical waveguide, as depicted in Fig. R5(a). Additionally, the square waveguide in our\nprior experiments has been replaced with a cylindrical one, with a radius of R = 2 cm, providing a\n0\nmore suitable environment for vortex beam generation. The LIT effect serves as the basis for both the\ngeneration of the sound source and the amplification of the acoustic wave. We consider a laser beam\nwhose center on the CNT film rotates along a circular trajectory,\nr (t) = (0,r cos(ω t),r sin(ω t)), (R8)\ns s r s r\nwherer istheradiusofthelaserpathandω = 2πf istheangularfrequencyofrotation. Therotational\ns r r\nmotionofthelaserspotinducesatime-varyingheatsource,whichinturngeneratesamodulatedpressure\nfield through the thermoacoustic effect. In the global coordinate system, the distribution of the rotating\nGaussian acoustic source is given by\nS(r,t) = Qe\n−|r−r\nr\ns\nc 2\n(t)|2\ne−iω0t, (R9)\nwhere Q represents the intensity at the center of the laser beam, r characterizes the Gaussian decay\nc\nconstant, andω = 2πf correspondstothelaserrepetitionfrequency. TheGaussiantermdescribesthe\n0 0\n12\nspatial confinement of the laser spot on the CNT surface, which acts as a localized source of periodic\nthermal expansion. The exponential term can be further expanded as\ne\n−r2+rs 2−2rr\nr\ns\nc 2\ncos(θ−ωrt)\n= e\n−r2\nr\n+\nc 2\nrs 2\ne\n2\nr\nr\nc 2\nrs cos(θ−ωrt)\n, (R10)\nwhere θ is the azimuth angle. By applying the well-known Bessel function identity\n∞\n(cid:88)\neαcosφ = I (α)eimφ, (R11)\nm\nm=−∞\nwith I (α) being the modified Bessel function of the first kind, the angular dependence of the source\nm\ncan be expanded as\n∞ (cid:18) (cid:19)\ne 2 r r c 2 rs cos(θ−ωrt) = (cid:88) I m 2 r r 2 r s eim(θ−ωrt). (R12)\nm=−∞ c\nSubstitutingthisintotheoriginalsourceexpressionyieldsthetime-dependentsoundsourceincylindrical\ncoordinates (r, θ, X):\nS(r,θ,X,t) = Qδ(X)e −r2 r + c 2 rs 2 (cid:88) ∞ I m (cid:18) 2 r r 2 r s (cid:19) eimθe−i(ω0+mωr)t, (R13)\nm=−∞ c\nwhere X is the axial distance, Q is the intensity of the sound source, and δ(X) denotes the Delta\n0\nfunction representing the thin CNT layer along the propagation axis. This expression reveals that the\nrotating laser excitation decomposes the sound field into multiple angular momentum modes, each as-\nsociated with a distinct frequency component ω = ω +mω . The mode index m corresponds to the\n0 r\ntopological charge of the resulting vortex beam. Through this mechanism, the LIT approach not only\nprovides tunable and broadband acoustic gain but also enables precise control over the spatial phase\nstructure of the generated sound field. Such flexibility highlights the fundamental distinction between\nlaser-induced thermoacoustics and the earlier electrically driven CNT systems, establishing LIT as a\nversatile platform for exploring non-Hermitian acoustics and related phenomena.\nNext, we present the acoustic field and mode analysis in the cylindrical waveguide. The acoustic\npressure field inside the cylindrical waveguide follows the Helmholtz equation under rigid-wall bound-\nary conditions. Owing to the cylindrical symmetry, the pressure distribution can be expressed as a\nsuperposition of orthogonal eigenmodes,\n∞ ∞\np(r,θ,X,ω) = (cid:88) (cid:88) A mn (ω)J m (k r mnr)eimθe−ik X mn(ω)X, (R14)\nm=−∞n=1\nwhere J is the m-th order Bessel function and kmn denotes the radial wave number, which satisfies\nm r\n(cid:112)\nthe boundary condition of J(cid:48) (kmnR ) = 0. The axial wavenumber is defined as kmn = k2−(kmn)2,\nm r 0 X r\nwith k = ω/c representing the total wavenumber and c the speed of sound in air. The integers m\n0 0\nand n correspond to the azimuthal and radial mode numbers, respectively, defining the field topology\ninside the waveguide. The modal coefficients A (ω) are obtained by applying the boundary conditions\nmn\nand exploiting the orthogonality of the Bessel functions. The acoustic field satisfies the inhomogeneous\nHelmholtz equation,\n(∇2+k2)p = −iωρ S, (R15)\n0\n13\nwhere ρ is the mass density of air. The orthogonality of eigenmodes gives\n0\n(cid:90) 2π(cid:90) R0\nJ (kmnr)e−imθ ·J (km(cid:48)n(cid:48) r)eim(cid:48)θrdrdθ = N δ δ ,\nm r m(cid:48) r mn mm(cid:48) nn(cid:48)\n0 0\nwith the normalization coefficient\n(cid:34) (cid:35)\n(cid:90) R0 (cid:18) m (cid:19)2\nN = 2π [J (kmnr)]2rdr = πR2 1− [J (kmnR )]2.\nmn m r 0 kmnR m r 0\n0 r 0\n(R17)\nThis orthogonality allows the modal coefficient for each frequency component to be expressed as\n−iωρ (cid:90) 2π(cid:90) R0\nA (ω) = 0 S(r,θ,ω)J (kmnr)e−imθrdrdθ,\nmn N m r\nmn 0 0\n(R18)\nwhere S(r,θ,ω) is the Fourier transform of the time-domain sound source S(r,t):\n1 (cid:90) ∞\nS(r,θ,ω) = S(r,t)eiωtdt.\n2π\n−∞\n(R19)\nSubstituting the frequency-domain expression for the rotating laser-induced source yields\nA mn (ω) = −2π N iωρ 0 Q e −r r s c 2 2 (cid:90) R0 e −r rc 2 2I m (cid:18) 2 r r 2 r s (cid:19) J m (k r mnr)rdr.\nmn 0 c\n(R20)\nThe propagation characteristics are further corrected using the frequency-domain Green’s function\nG(ω):\nA mn (ω) = −2π N iωρ 0 Q e −r r s c 2 2 (cid:90) R0 e −r rc 2 2I m (cid:18) 2 r r 2 r s (cid:19) J m (k r mnr)rdr·G(ω).\nmn 0 c\n(R21)\nThe mode structure is constrained by the cutoff frequencies ωmn = c kmn. Under the experimental\nc 0 r\ncondition of f = 6 kHz, only the first-order vortex modes with azimuthal indices m = ±1 and\n0\nradial index n = 1 remain propagating. These two degenerate modes possess opposite orbital angular\nmomenta, resulting in helical acoustic wavefronts of opposite rotation directions. For these propagating\nmodes,\nA ±1,1 = −2πi(ω N 0 ±ω r )ρ 0 Q e − r r c s 2 2 (cid:90) R0 e −r rc 2 2I 1 (cid:18) 2 r r 2 r s (cid:19) J 1 (k r 11r)rdr·G(ω 0 ±ω r ),\n11 0 c\n(R22)\nwhere the Green’s function is given by\n0 r (cid:112)\n(ω ±ω )2−(ω11)2\n0 r c\nHere,ω11 = c k11representsthecutofffrequencyofthefundamentalpropagatingmode. Whenobserved\nc 0 r\nalong the propagation direction (the positive X-axis), the phase helicity of the two vortex beams is\nreversed, such that\n∞ ∞\np(r,θ,X,ω) = (cid:88) (cid:88) A mn (ω)J m (k r mnr)e−imθe−ik X mn(ω)X.\n(R24)\n14\n((aa))\n((bb)) ff 00 -- ff rr ff 00 ff 00 ++ ff rr ((cc)) ff 00 -- ff rr ff 00 ff 00 ++ ff rr\n11 11\nAA AA\nmm mm\n22 ccmm pp pp\nttiill ttiill\nuu uu\ndd dd\nee ee\n..mm 00 ..pp 00\niiSS ππ xx EE ππ\nPP PP\nhh hh\naa aa\nss ss\nee ee\n--ππ --ππ\nFigureR5: ExperimentaldemonstrationofacousticVBs. (a)SchematicoftheLIT-basedgeneration\nof acoustic VBs. A certain angle is set between the symmetry axis of the reflector and the rotation\naxis of the motor (left inset), causing the laser to trace a circular trajectory on the CNT film with\na spot diameter of 1.5±0.1 cm (right inset). Simulated (b) and experimental (c) demonstration of\nthe acoustic VBs by CCW-rotating laser spot.\nConsequently, for the counter-rotating modes, the modal coefficients take the form\nA ±1,1 = −2πi(ω N 0 ∓ω r )ρ 0 Q e − r r c s 2 2 (cid:90) R0 e −r rc 2 2I 1 (cid:18) 2 r r 2 r s (cid:19) J 1 (k r 11r)rdr·G(ω 0 ∓ω r ).\n11 0 c\n(R25)\n.\n(2) Experimental demonstration of acoustic vortex beams\nWe experimentally demonstrate the generation of first-order acoustic VBs using a rotating laser-\ninducedheatsource. Asestablishedinourtheoreticalanalysis,arotatingheatsourcecansimultaneously\nproduce two acoustic vortex components carrying opposite topological charges m = ±1. When the heat\nsource oscillates at an intrinsic frequency of f and rotates counterclockwise (CCW) with a modulation\n0\nfrequency of f , three distinct acoustic components emerge: a VB at f −f with a clockwise (CW)\nr 0 r\nrotating wavefront and m = 1; a plane wave at f with m = 0; and another VB at f +f with a CCW\n0 0 r\nrotating wavefront and m = −1. The total pressure field results from the linear superposition of these\nthree components. Their spatial distributions can be effectively separated and analyzed through Fourier\ntransformation, as detailed in [Phys. Rev. Lett. 134, 137001 (2025)].\nFigureR5(a)presentstheschematicoftheLIT-basedgenerationofacousticVBs, inwhich, acertain\nangle is set between the symmetry axis of the reflector and the rotation axis of the motor (left inset),\ncausing the laser to trace a circular trajectory on the CNT film with a spot diameter of 1.5±0.1 cm\n15\n(right inset). Supplementary Video I and Video II demonstrate the laser spot rotating on the CNT\nfilm to form a circular trajectory at rotational speeds of 150 RPM and 6000 RPM, respectively. Both\nsimulated and experimental acoustic VBs obtained with a CCW-rotating laser spot have been demon-\nstrated in Figs. R5(b) and (c). The upper and lower panels depict the amplitude and phase profiles\ncorresponding to the different frequency components. The experiments were conducted at an operating\nfrequency f = 6 kHz and a modulation frequency f = 0.1 kHz, with the sound field measured at a\n0 r\ndistanceofl = 6.7λ = 6.7c /f fromthesource. Theobservedvortexstructuresareinexcellentagree-\n0 0 0\nment with theoretical predictions, confirming that the rotating laser excitation generates well-defined\nacoustic VBs. Compared with acoustic VBs generated via well-designed structures [Phys. Rev. Lett.\n117, 034301 (2016)], our LIT approach offers a simpler and more flexible method to achieve precisely\ntailored vortex fields. This tunability enables the realization of scattering exceptional points based on\ndynamically controllable sound-field topologies.\nTo conclude, while both the present and earlier systems employ CNT films as thermoacoustic trans-\nducers, the laser-pumped configuration represents a fundamentally different mechanism from\nthe previously reported electrically driven designs [Nature 597, 655 (2021); Phys. Rev. Lett. 131,\n066601 (2023)]. In the earlier electrically driven systems, the CNT film acted as a resistive heater,\nwhere the thermoacoustic response was constrained by electrode geometry and thermal diffusion lim-\nits. In contrast, the laser-pumped system operates entirely contact-free, using optical energy to directly\nmodulatethelocaltemperaturefield. Thisapproachallowsspatially programmable excitation, higher\nmodulation bandwidth, and a much greater degree of control over the acoustic field topology. These\nadvantages make the laser-pumped thermoacoustic platform not merely a variation, but a significant\nextension of the previous technology—opening new possibilities for broadband, reconfigurable, and non-\ncontact sound exciation.\nTo clarify this distinction, we have revised the manuscript and added the following statement: “The\nphysical mechanism and spatial distribution of the acoustic VB generated by a rotating laser spot are\ndiscussed in Supplementary Note X. To support the propagation of first-order VBs, the waveguide\nwas replaced with a cylindrical tube of radius R = 2 cm. A schematic of the LIT-generated VB\n0\nis shown in Fig. 5(b), in which, a certain angle is set between the symmetry axis of the reflector\nand the rotation axis of the motor, causing the laser to trace a circular trajectory on the CNT film.\nSupplementary Video I and Video II demonstrate the laser spot rotating on the CNT film to form a\ncirculartrajectoryatrotationalspeedsof150revolutionsperminute(RPM)and6000RPM,respectively.\nThe corresponding simulation and experimental results, after Fourier transformation, are presented in\nFigs. 5(c)-(d),confirmingthattheLITsystemproducesastableandwell-definedacousticVB(detailsin\nSupplementary Note XI).”. This addition, now included on Page 9, explicitly highlights how the laser-\ndriven thermoacoustic excitation differs from earlier electronic approaches and experimentally verifies\nthe generation of a robust, geometrically structured sound field unique to the LIT platform.\nComment P2.4 – For a paper to be considered for Nature Communications, I would expect,\nat\nleast, that there is a certain level of significance either in physics or in experimental methodology.\nResponse: We sincerely thank the reviewer for their insightful feedback and for highlighting the\nimportance of both physical significance and methodological innovation for consideration in Nature\nmeets these criteria in both aspects.\nFrom a physics perspective, this study reports the first experimental realization of non-Hermitian\nphase transitions in an open scattering system. Unlike our previous work [Phys. Rev. Lett. 131, 066601\n(2023)], which described non-Hermitian behavior within a closed cavity-tube network modeled by an\neffective Hamiltonian, the current system is formulated directly through scattering matrices belong-\ning to the PT-symmetric family. This distinction is fundamental: the present approach captures the\nfull wave dynamics of an open system, revealing three distinct symmetry regimes—PT, reversed-PT,\nand anti-PT—that have not been observed together in a single experimental platform. While each\nsymmetry has been discussed separately in earlier theoretical reports, our experiments demonstrate their\ncoexistenceandthetransitionsbetweenthem,includingthemaskingandunmaskingprocessesnearEPs.\nThis comprehensive observation of symmetry transitions establishes a new experimental framework for\nexploring non-Hermitian physics in scattering systems.\nIn terms of experimental methodology, our approach represents a major step beyond the electri-\ncally driven CNT film systems used in earlier studies [Nature 597, 655 (2021)]. Electrically driven CNT\nfilms were limited by electrode configurations and thermal diffusion, constraining the spatial flexibility\nof sound generation. In contrast, the laser-induced thermoacoustic (LIT) platform developed here en-\nables non-contact, programmable, and broadband acoustic excitation. For instance, by rotating a laser\nbeam across the CNT film, we can dynamically sculpt the spatial phase of the generated acoustic fields,\nproducing vortex beams with well-defined topological charges. This capability not only enhances exper-\nimental controllability but also makes it possible to realize non-Hermitian phase transitions for arbitrary\nwavefronts, extending beyond the plane-wave configurations that dominated previous PT studies. The\nsuccessful observation of an EP associated with an acoustic vortex beam marks, to our knowledge, the\nfirst such demonstration in acoustics.\nTo address the reviewer’s concern and provide a clear proof-of-concept, we further modeled the\nquasi-PT-symmetric scattering behavior of vortex beams at the exceptional point, simulating and ex-\nperimentally confirming the phase transitions induced by CCW vortex incidence (m = 1). The corre-\nsponding scattering matrices for both CCW and CW vortex excitations are presented, revealing distinct\nsymmetry-breaking behaviors. Furthermore, we derived the parameter conditions for achieving unidi-\nrectional transparency and experimentally observed the reversal of the transmitted beam’s topological\ncharge—an effect that directly evidences non-Hermitian topology in vortex-field space.\ni) Theoretical model for quasi-PT-symmetric scattering of vortex beams\nTo provide theoretical clarity, the full derivation of the scattering matrix for vortex beams interact-\ning with the CNT film is now presented in Supplementary Note XII. In this model, the introduction of\nangular momentum (m = ±1) splits the acoustic field within the waveguide into two coupled vortex\nmodes. Unlike the rotating-source analysis in Supplementary Note X—where m = ±1 corresponds to\nfrequency shifts at ω = ω ±ω —the present derivation focuses on a single frequency ω = ω +ω ,\n0 r 0 r\nwhere coexisting CW and CCW modes interact. When the laser beam traces a circular tangential path\non the CNT film, both CW and CCW vortex modes are generated simultaneously. Under dual-beam\nexcitation, twocounter-rotatinglasersgiverisetofourdistinctvortexmodespropagatingwithinthesys-\ntem. These field components are denoted as p± , where the subscripts indicate the propagation region\ni,m\n(i = 1,2,3) and the topological charge (m = ±1). As shown in Fig. R6, the forward and backward\n17\nacoustic VBs in region 1 (incident region) are denoted as p+ , p− and p− ; the forward and backward\n1,1 1,1 1,−1\nVBs in region 2 (interact region) are denoted as p+ , p+ , p− , and p− ; the forward VBs in region\n2,1 2,−1 2,1 2,−1\n3 (emergent region) are denoted as p+ , and p+ , respectively, at the corresponding positions. The\n3,1 3,−1\nresulting configuration allows investigation of both loss-side and gain-side incidence, thereby capturing\nthe complete symmetry evolution across the non-Hermitian parameter space.\nLoss-side incidence:\nFigure R6 illustrates the configuration of the prototype under loss-side incidence, where the in-\ncoming wave is an acoustic VB featuring a CCW rotating wavefront propagating along the positive\nX-direction, with a pressure amplitude a . When a CW rotating laser beam interacts with the gain\n1\nmedium, it excites a CW-propagating acoustic VB directed leftward and a CCW-propagating VB di-\nrected rightward, each possessing an identical sound pressure amplitude b . Conversely, when the gain\n11\nmedium is driven by a CCW-rotating laser beam, a CCW-propagating VB emerges toward the left and\na CW-propagating VB toward the right, both characterized by a pressure amplitude b . In this con-\n12\nfiguration, a , b , and b are each proportional to A J (k11r) at r = R . Therefore, The acoustic\n1 11 12 11 1 r 0\nfields introduced by the primary sound source (p ) and by the bias excitations (p , p ) within the\n0 11 12\nwaveguide—at a specific azimuthal position θ and evaluated in the time domain—can be expressed as\np (0) = a , p (l+D+d) = b eiφ11, p (l+D+d) = b eiφ12, (R26)\n0 1 11 11 12 12\nwhere φ (φ ) represents the relative phase differences between the CW- (CCW-) rotating laser beams\n11 12\nirradiating on the internal and source CNT films. The axial component of the wave vector at frequency\n(cid:112)\nω is given by k11 = (ω/c )2−(k11)2, with y = k11d. By enforcing the intrinsic continuity of the\nX 0 r X\nacoustic pressure fields at X = l, X = l+D, and X = l+D+d, we obtain two independent sets of\nboundary conditions corresponding to the CCW (m = 1) and CW (m = −1) vortex components:\np+ 1,1 (l) = p 0 (0)e−ik X 11l = a 1 ,\np− (l+D) = p (l+D+d)e−iy = b eiφ11e−iy,\n2,1 11 11\np− (l) = p+ (l)r +p− (l+D)t = a r +b eiφ11t e−iy,\n1,1 1,1 s 2,1 s 1 s 11 s\np+ (l+D) = p− (l+D)r +p+ (l)t = b eiφ11r e−iy +a t ,\n2,1 2,1 s 1,1 s 11 s 1 s\np+ (l+D+d) = p+ (l+D)e−iy +p (l+D+d) = b eiφ11r e−i2y +a t e−iy +b eiφ12,\n3,1 2,1 12 11 s 1 s 12\nrreeggiioonn 11 LLoossss rreeggiioonn 22 GGaaiinn rreeggiioonn 33\nppp--- ppp+++\n111,,,---111 222,,,111 pppp++++\nppp+++ 3333,,,,1111\nSSoouurrccee ppp+++ 222,,,---111\n111,,,111 ppp---\n222,,,111 pppp++++\nppp--- ppp--- 3333,,,,----1111\n111,,,111 222,,,---111\nXX == 00 ll DD dd\nFigureR6: Sketchoftheprototypeimplementedinourexperimentunderloss-sideincidence(VBs).\n18\nand\np− (l+D) = p (l+D+d)e−iy = b eiφ12e−iy,\n2,−1 12 12\np− (l) = p− (l+D)t = b eiφ12t e−iy,\n1,−1 2,−1 s 12 s\n(R28)\np+ (l+D) = p− (l+D)r = b eiφ12r e−iy,\n2,−1 2,−1 s 12 s\np+ (l+D+d) = p+ (l+D)e−iy +p (l+D+d) = b eiφ12r e−i2y +b eiφ11.\n3,−1 2,1 11 12 s 11\nIn this incidence mode, assuming the sound pressure amplitude ratio m = b /a and m =\n11 11 1 12\nb /a , and taking into account the topological charges, the reflection (r ) and transmission (t )\n12 1 L L\ncoefficients of the loss-gain coupling are derived as\np− (l) p− (l) (cid:16) (cid:17) (cid:16) (cid:17)\nr = r +r = 1,1 + 1,−1 = m eiφ11t e−iy +r + m eiφ12t e−iy ,\nL L1 L2 p+ (l) p+ (l) 11 s s CCW 12 s CW\n1,1 1,1\np+ (l+D+d) p+ (l+D+d)\n3,1 3,−1 (R29)\nt = t +t = +\nL L1 L2 p+ (l) p+ (l)\n1,1 1,1\n(cid:16) (cid:17) (cid:16) (cid:17)\n= m eiφ11r e−i2y +t e−iy +m eiφ12 + m eiφ12r e−i2y +m eiφ11 .\n11 s s 12 12 s 11\nCCW CW\nIn Eq. R29, the subscripted terms distinguish between the CCW and CW contributions to the scattering\nparameters. This formulation enables a complete description of the asymmetric scattering behavior\narising under loss-side excitation. By explicitly considering the coupled CW and CCW vortex modes\ngenerated by the rotating laser field, the analysis reveals how the interplay between topological charge\nand non-Hermitian coupling governs the overall transmission and reflection properties of the system.\nGain-side incidence:\nrreeggiioonn 11 GGaaiinn rreeggiioonn 22 LLoossss rreeggiioonn 33\nppp--- ppp+++\n111,,,---111 222,,,111 pppp++++\nppp+++ 3333,,,,1111\nSSoouurrccee ppp+++ 222,,,---111\n111,,,111 ppp---\n222,,,111 pppp++++\nppp--- ppp--- 3333,,,,----1111\n111,,,111 222,,,---111\nXX == 00 ll dd DD\nFigureR7: Sketchoftheprototypeimplementedinourexperimentundergain-sideincidence(VBs).\nFigure R7 shows the schematic of the prototype under gain-side incidence, where the incident\nacoustic wave is a VB with a CCW rotating wavefront. When a CW-rotating laser beam interacts\nwith the gain medium, it simultaneously generates a CCW-propagating VB traveling to the right and\na CW-propagating VB traveling to the left, both with a sound pressure amplitude of b . Conversely,\n21\na CCW-rotating laser beam generates a CCW-propagating VB directed leftward and a CW-propagating\nVB directed rightward, each characterized by an amplitude of b . In both cases, the wave generation\n22\nresults from the thermoacoustic modulation of the CNT film, where the rotating laser field dynamically\nsculpts the spatial phase of the emitted sound.\n(p ) and bias (p , p ). Their complex amplitudes depend on the relative phase differences, φ and\n0 21 22 21\nφ , between the CW- and CCW-rotating laser beams applied to the internal and source CNT films,\n22\nrespectively:\np (0) = a , p (l) = b eiφ21, p (l) = b eiφ22,\n0 2 21 21 22 22\n(R30)\nwhere the spatial and temporal dependencies are determined by the same harmonic structure as in the\nloss-side configuration. At the interfaces located at X = l, X = l+d and X = l+d+D, the continuity\nof the acoustic pressure fields must be satisfied for both topological components. For the CCW mode\n(m = 1), this yields\np+ 1,1 (l) = p 0 (0)e−ik X 11l = a 2 ,\np+ (l) = p+ (l)+p (l) = a +b eiφ21,\n2,1 1,1 21 2 21\np− (l) = p+ (l)r e−i2y = a r e−i2y +b eiφ21r e−i2y,\n2,1 2,1 s 2 s 21 s\np− (l) = p− (l)+p (l) = a r e−i2y +b eiφ21r e−i2y +b eiφ22,\n1,1 2,1 22 2 s 21 s 22\np+ (l+D+d) = p+ (l)t e−iy = a t e−iy +b eiφ21t e−iy,\n3,1 2,1 s 2 s 21 s\nand for the CW mode (m = −1), this yields\np+ (l) = p (l) = b eiφ22,\n2,−1 22 22\np− (l) = p+ (l)r e−i2y = b eiφ22r e−i2y,\n2,−1 2,−1 s 22 s\n(R32)\np− (l) = p− (l)+p (l) = b eiφ22r e−i2y +b eiφ21,\n1,−1 2,−1 21 22 s 21\np+ (l+D+d) = p+ (l)t e−iy = b eiφ22t e−iy.\n3,−1 2,−1 s 22 s\nTo characterize the coupling between the gain and loss regions, we define the amplitude ratios m =\n21\nb /a and m = b /a . Incorporating the distinct topological charges of the interacting vortex\n21 2 22 22 2\nmodes, the corresponding reflection (r ) and transmission (t ) coefficients for gain-side excitation are\nG G\nderived as\np− (l) p− (l)\n1,1 1,−1\nr = r +r = +,\nG G1 G2 p+ (l) p+ (l)\n1,1 1,1\n(cid:16) (cid:17) (cid:16) (cid:17)\n= m eiφ21r e−i2y +m eiφ22 +r e−i2y + m eiφ22r e−i2y +m eiφ21 ,\n21 s 22 s 22 s 21\nCCW CW (R33)\np+ (l+D+d) p+ (l+D+d)\n3,1 3,−1\nt = t +t = +,\nG G1 G2 p+ (l) p+ (l)\n1,1 1,1\n(cid:16) (cid:17) (cid:16) (cid:17)\n= m eiφ21t e−iy +t e−iy + m eiφ22t e−iy .\n21 s s 22 s\nCCW CW\nThese coefficients quantify the balance between amplified and attenuated components within the non-\nHermitian system. At the operating frequency f, the overall scattering behavior of the system can be\ncompactly expressed through the scattering matrix formalism:\n(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)\nr t r +r t +t r t r t\nS = L G = L1 L2 G1 G2 = L1 G1 + L2 G2 ,\n(R34)\nr t r t +t r +r t r t r\nL G L1 L2 G1 G2 L1 G1 CCW L2 G2 CW\n20\nwhere the matrix elements are defined by\nr = m eiφ11t e−iy +r ,\nL1 11 s s\nt = m eiφ21t e−iy +t e−iy,\nG1 21 s s\n(R35)\nt = m eiφ11r e−i2y +m eiφ12 +t e−iy,\nL1 11 s 12 s\nr = m eiφ21r e−i2y +m eiφ22 +r e−i2y,\nG1 21 s 22 s\nand\nr = m eiφ12t e−iy,\nL2 12 s\nt = m eiφ22t e−iy,\nG2 22 s\n(R36)\nt = m eiφ11 +m eiφ12r e−i2y,\nL2 11 12 s\nr = m eiφ21 +m eiφ22r e−i2y.\nG2 21 22 s\nThe subscripts “CCW” and “CW” denote the propagation modes of the corresponding vortex compo-\nnents.\nIt is worth emphasizing that the scattering coefficients represent only the normalized ratios between\nthe reflected or transmitted pressure amplitudes and that of the incident wave. These quantities are\nindependent of the vortex topological charge itself; the charge merely determines the rotational phase\nstructure of the field. This distinction underscores the versatility of the laser-induced thermoacoustic\nplatform, where both CW and CCW vortex excitations can be treated within a unified scattering frame-\nwork. The ability to generate and control such dual-topology modes under optical excitation is central\nto realizing the experimentally observed non-Hermitian phase transitions reported in this study.\n(2) Accessing PT symmetry in acoustic vortex beams\nAcoustic vortex beams provide a rich platform for exploring non-Hermitian physics because their\nintrinsic angular momentum introduces additional degrees of freedom beyond conventional plane waves.\nIn this work, we present the first theoretical and experimental realization of quasi-PT-symmetric\nscattering behavior in a system excited by acoustic vortex beams. This achievement marks a key\nstep forward in extending PT symmetry from plane-wave systems to structured wavefronts carrying\norbital angular momentum.\nTo establish the PT-symmetric condition, we begin by considering a loss medium characterized\nby a reflection/transmission coefficient r = t = s = |s|eiα, where |s| and α denote the amplitude\ns s\nand phase, respectively. The scattering characteristics associated with PT symmetry imply that the\nreflection coefficient r of the system must be zero, yielding r = r = 0. Under this constraint, the\nL L\ncoupling relations produce\nr = 0 → m ei(φ11−y)+1 = 0 → m = 1 & φ = y+π+2j π,\nL1 11 11 11 1\n(R37)\nr = 0 → m = 0,\nL2 12\nin which j is an integer. The corresponding transmission coefficients for CCW and CW modes are\n1\nobtained as\n(cid:104) (cid:105)\nt = m ei(φ11−y)+1 se−iy +m eiφ12 = 0,\nL1 11 12\n(R38)\nt = m eiφ11 = −eiy.\nL2 11\nThese relations show that, for an incident VB with topological charge m = 1, no reflected wave is\ngenerated within the system. The transmitted wave emerges as a vortex beam of opposite handedness\n21\nm = −1 but identical amplitude, and its phase shift varies proportionally with the coupling distance d.\nThis mode conversion reflects the interplay between non-Hermitian coupling and topological phase, a\nphenomenon unattainable in ordinary acoustic media.\nThe reciprocity principle allows us to determine the parameters of the CNT film when a CCW vortex\nbeam is incident from the gain side, leading to\nt = t = 0 → m eiφ21 +1 = 0 → m = 1 & φ = π+2j π,\nG1 L1 21 21 21 2\n(R39)\nt = t = −eiy → m eiφ22 = −ei2y/s → m = 1/|s| & φ = 2y−α+π+2j π,\nG2 L2 22 22 22 3\nwhere j and j are integers. Under these conditions, the reflection coefficients can be expressed as\n2 3\n1\nr = (m eiφ21 +1)se−i2y +m eiφ22 = − ei(2y−α),\nG1 21 22\n|s| (R40)\nr = m eiφ21 +m eiφ22r e−i2y = −1−1 = −2,\nG2 21 22 s\nand the overall scattering matrix of the system takes the form\n   \n(cid:20)\nr t\n(cid:21) 0 −eiy 0 0 (cid:20)\n0\n−eiy(cid:21)\nS r = t L r G =  −eiy − 1 ei(2y−α)−2  =  0 − 1 ei(2y−α) + −eiy −2 .\nL G |s| |s| CW\nCCW\n(R41)\nThis scattering matrix naturally separates into two components. The first corresponds to the vortex\nmode sharing the same topological charge as the incident beam (m = 1), which behaves as a unidirec-\ntional total-reflection system. The second corresponds to the opposite-handed mode (m = −1), whose\nscattering matrix follows the characteristics of a typical PT-symmetric configuration. Under optimized\nparameters, the overall system exhibits PT-symmetric scattering at an EP—featuring zero reflection\nfrom one side and finite reflection from the other, while the transmission amplitudes remain balanced\nin both directions. Importantly, the transmitted beam undergoes a reversal of topological charge,\nlinking non-Hermitian phase transitions to angular-momentum conversion and opening possibilities for\nadvanced acoustic information processing.\nThe experimental verification of this phenomenon is shown in Fig. R8. To facilitate the propagation\nof first-order VBs, the waveguide was replaced with a cylindrical tube of radius R = 2 cm.\nR8(a)–(b) present the simulated and measured scattering amplitudes and phases for CCW VBs at an\noperating frequency f = 6 kHz and rotational modulation f = 0.1 kHz; Figs. R8(c)–(d) show the\n0 r\ncorresponding CW results. The data reveal that, when a CCW VB is incident from the lossy side, no\nreflected component is observed (r = 0), while the transmitted VB can be expressed as t = −eiy.\nL L\nThe transmitted beam retains the same amplitude but exhibits an opposite topological charge rela-\ntive to the input. When the beam enters from the gain side, the transmitted mode remains reversed\n(t = −eiy), whereas the reflected field becomes a superposition of two vortex modes with distinct\nG\ntopological charges, r = r +r = −ei2y/s−2 (see Supplementary Note XIII for details). Here,\nG G1 G2\ns = |s|eiα = r = t is the reflection/transmission coefficient of the loss medium. The field patterns\ns s\nin Figs. R8(e) and (f) visualize this asymmetric scattering at the coupling distance d = 9.92 cm,\ncorresponding to the vertical dotted lines in Figs. R8(a)-(b) and (c)-(d), respectively. The optimized\nfilm parameters at the EP are m = 1, φ = π; m = 0, φ = 0; m = 1, φ = π; and\n11 11 12 12 21 21\nm = 1/|s|, φ = π−α. The experimentally extracted scattering parameters are r = 0.068ei0.619,\n22 22 L1\nt = 0.069ei0.57, r = 1.811e−i1.803, t = 0.015ei0.615, r = 0.327ei0.181, t = 1.016e−i3.102,\nL1 G1 G1 L2 L2\n22\n((aa)) CCCCWW ((cc)) CCWW ((ee)) rrLL11 ttLL11 rrGG11 ttGG11\n22\nee\ndd uu ttiillpp\nmm\n0\n1\n0\n1\n.\n.\n.\n.\n4\n8\n4\n8\nsssiiimmm...\nrrrLLL111 tttLLL111 rrrGGG111 tttGGG111\nee\ndd uu ttiillpp\nmm 11\neee\nsssiii\nxxx\nmmm\nppp\n...\n...\nrrrLLL222 tttLLL222 rrrGGG222 tttGGG222\nWW\n..mm\nCC\niiSS\n00 ee dd\nuuttiill pp mm\nAA eeexxxppp... AA CC ππ\n00..22 PP\nhh\n00\n..pp\nxx EE ee ss\naa\n00 --ππ\n((bb)) ((dd))\nsssiiimmm...\nrrrLLL222 tttLLL222 rrrGGG222 tttGGG222 ((ff)) rrLL22 ttLL22 rrGG22 ttGG22\nππ ππ eeexxxppp... 22\nAA\nmm\n..mm pp\nee\nss aa hh PP 00\nee\nss aa hh PP 00 WW\niiSS\n00 ee dd\nuuttiill\nCC ππ\n--ππ\nsssiiimmm...\nrrrLLL111 tttLLL111 rrrGGG111 tttGGG111 --ππ ..pp\nxx aa\nhh PP\neeexxxppp... EE ee ss\n44 77 1100 1133 44 77 1100 1133 --ππ\ndd ((ccmm)) dd ((ccmm))\nFigure R8: Realization of the scattering EP using LIT. (a)-(b) Simulated (curves) and\nexperimentally measured (symbols) scattering amplitudes and phases of counterclockwise (CCW)\nvortex beams at a rotation frequency of fr = 0.1 kHz. (c)-(d) Same as (a)-(b), but for clockwise\n(CW) vortex beams, obtained under identical conditions. (e) Comparison of simulated (top) and\nmeasured (bottom) field amplitudes and phases of CCW vortex beams at a coupling distance of\nd = 9.92 cm, corresponding to the data in panels (a)-(b). (f) Same as (e), but for CW vortex\nbeams, corresponding to panels (c)-(d).\nr = 1.982e−i3.015, and t = 0.997ei3.183 at the azimuth angle of θ = 0. These results collectively\nG2 G2\nconfirm that the system realizes PT-symmetric scattering with acoustic vortex beams and achieves\nexperimentally observable topological charge conversion at the exceptional point. The observation\nconstitutes the first experimental demonstration of PT symmetry in acoustic vortex beams, establishing\na versatile platform for studying non-Hermitian phase transitions in structured acoustic fields.\nIn summary, we would like to note that the present study advances the field of non-Hermitian\nacoustics in two dimensions.\nFrom a physics perspective, it reports the first experimental observation of the PT, reversed-PT,\nandanti-PT phasetransitionswithinasinglescatteringsystem. Thisunifiedrealizationestablishesanew\nexperimental framework for investigating non-Hermitian phenomena that extend beyond conventional\nHamiltonian models, revealing the full spectrum of symmetry transitions accessible in open acoustic\nsystems.\nFrom a methodological standpoint, the work introduces a laser-driven thermoacoustic platform\nthat enables precise control over broadband and geometrically complex sound fields. Through rotating\nlaser excitation, we demonstrate the generation and manipulation of first-order acoustic vortex beams\nat exceptional points—an achievement that cannot be realized with earlier electrically driven CNT\nconfigurations. This platform therefore offers an unprecedented level of flexibility for engineering non-\nTogether, these advances substantially deepen the physical insight and broaden the experimental\ncapabilities of non-Hermitian acoustics, directly addressing the reviewer’s concern regarding the level of\nsignificance required for publication.\nIn light of this feedback, we have also added a new section titled “Formation of acoustic VBs and\nscatteringEPusingLIT”totheMainText(Page9),whichprovidesfurthertheoreticalandexperimental\nvalidation of these findings. The Fig. R9 here is Fig. 5 in the revised manuscript.\nFormation of acoustic VBs and scattering EP using LIT. The LIT method employed in this study\nrepresents a promising approach for achieving perfect acoustic VBs and enables the PT-symmetric\nscattering behavior illustrated in Fig. 5(a). In the aforementioned configuration, the CNT film under\nlaser irradiation functions effectively as a point sound source, since the timescale of laser-induced heat\ndiffusion across the film surface is on the order of seconds [40]. When the laser spot rotates on the\nCNT film at a modulation frequency of f (e.g., f = 0.1 kHz), the diffusion time is much longer than\nr r\nthe rotation period, ensuring that the acoustic source rotates continuously rather than discretely. As\na result, the system produces perfect acoustic VBs within the waveguide, unlike the approximate VBs\ngenerated by discrete passive [41] or electronical driven [35] methods. This continuous, contact-free\nexcitation is a defining advantage of LIT and fundamentally differentiates it from previous electrical\napproaches.\nThephysicalmechanismandspatialdistributionoftheacousticVBgeneratedbyarotatinglaserspot\nare discussed in Supplementary Note X. To support the propagation of first-order VBs, the waveguide\nwas replaced with a cylindrical tube of radius R = 2 cm. A schematic of the LIT-generated VB\n0\nis shown in Fig. 5(b), in which, a certain angle is set between the symmetry axis of the reflector\nand the rotation axis of the motor, causing the laser to trace a circular trajectory on the CNT film.\nSupplementary Video I and Video II demonstrate the laser spot rotating on the CNT film to form a\ncirculartrajectoryatrotationalspeedsof150revolutionsperminute(RPM)and6000RPM,respectively.\nThe corresponding simulation and experimental results, after Fourier transformation, are presented in\nFigs. 5(c)-(d), confirming that the LIT system produces a stable and well-defined acoustic VB (details\nin Supplementary Note XI).\nBuilding on this foundation, we modeled the quasi-PT-symmetric scattering behavior of VBs at the\nEPandverifiedtheresultsexperimentally. Thescatteringresponsesforcounter-clockwise(CCW)vortex\nexcitations were formulated (see Supplementary Note XII), with the corresponding scattering matrices\nrevealing distinct symmetry-breaking features inherent to non-Hermitian systems:\n(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)\nr t r t r t\nS = L G = L1 G1 + L2 G2 ,\n(R42)\nr t r t r t r\nL G L1 G1 CCW L2 G2 CW\nin which\nr = m eiφ11t e−iy +r ,\nL1 11 s s\nt = m eiφ21t e−iy +t e−iy,\nG1 21 s s\n(R43)\nt = m eiφ11r e−i2y +m eiφ12 +t e−iy,\nL1 11 s 12 s\nr = m eiφ21r e−i2y +m eiφ22 +r e−i2y,\nG1 21 s 22 s\n24\n((aa)) CCCCWW CCWW\nLLoossss GGaaiinn LLoossss GGaaiinn ((ee)) ((hh))\n22\n((bb\nii\n))\nnncciiddeenntt iinncciiddeenntt eedduuttiillppmm 0\n1\n0\n1\n.\n.\n.\n.\n4\n8\n4\n8\neee sssiii xxx mmm ppp ... ... rrrLLL111tttLLL111rrrGGG111tttGGG111 eedduuttiillppmm 11 eee sssiii xxx mmm ppp ... ...\nrrrLLL222tttLLL222rrrGGG222tttGGG222\nAA00..22 AA\n00 00\n((ff)) ((ii))\nππ ππ\neessaahhPP\n00\neessaahhPP\n00\n--ππ --ππ\n((cc)) ((dd)) 44 77 1100 1133 44 77 1100 1133\ndd ((ccmm)) dd ((ccmm))\nSSiimmuullaattiioonnss EExxppeerriimmeennttss\nff00 -- ffrr ff00 ff00 ++ ffrr ff00 -- ffrr ff00 ff00 ++ ffrr ((gg)) rrLL11 ttLL11 rrGG11 ttGG11 ((jj)) rrLL22 ttLL22 rrGG22 ttGG22\nAAAbbbsss(((ppp)))::: 000 111 AAbbss((pp)):: 00 11\nssttnneemm\niirreeppxxEE AAmmpplliittuuddee::00 22\nssttnneemm\niirreeppxxEE AAAmmmpppllliiitttuuudddeee:::000 222\nAArrgg((pp)):: --ππ ππ AArrgg((pp)):: --ππ ππ PPhhaassee:: --ππ ππ PPhhaassee:: --ππ ππ\nFigure R9: Formation of acoustic VBs and realization of the scattering EP using LIT.\n(a) Illustration of the PT-symmetric system for acoustic VBs. (b) Schematic of the LIT-based\ngeneration of acoustic VBs. (c)-(d) Simulated and experimental demonstration of the acoustic VBs\nby CCW-rotating lasr spot. (e)-(f) Simulated (curves) and experimentally measured (symbols)\nscattering amplitudes and phases of CCW VBs at a rotation frequency of f = 0.1 kHz.\nr\n(g)\nMeasured scattering field amplitudes and phases of CCW VBs with a coupling of d = 9.92 cm,\ncorresponding to the data in panels (e)-(f). (h)-(i) Same as (e)-(f), but for CW VBs, obtained\nunder identical conditions. (j) Same as (g), but for CW VBs, corresponding to panels (h)-(i).\nand\nr = m eiφ12t e−iy,\nL2 12 s\nt = m eiφ22t e−iy,\nG2 22 s\n(R44)\nt = m eiφ11 +m eiφ12r e−i2y,\nL2 11 12 s\nr = m eiφ21 +m eiφ22r e−i2y.\nG2 21 22 s\nThe parameters r , t , t , r describe the CCW VB relative to the incident wave at an azimuthal\nL1 G1 L1 G1\nangle of θ = 0, meanwhile r , t , t , r characterize the clockwise (CW) VB response. Here, the\nL2 G2 L2 G2\nfrequencies of the VBs are all given by f = f +f . m and φ represent the amplitude ratios and\n0 r ij ij\nphase differences between the CNT film components excited by CW (j = 1) and CCW (j = 2) rotating\nlaser spots and the incident CCW VB coming from the loss (i = 1) and gain (i = 2) sides, respectively.\n(cid:112)\nThe propagation constant is expressed as k11 = (ω/c )2−(k11)2 with y = k11d, in which k11 is the\nX 0 r X r\nfirst order radial wave number of the waveguide. It is worth emphasizing that the scattering coefficients\nrepresent only the normalized ratios between the reflected or transmitted pressure amplitudes and that\noftheincidentwave. Thesequantitiesareindependentofthevortextopologicalchargeitself; thecharge\nmerely determines the rotational phase structure of the field. This distinction underscores the versatility\nof the laser-induced thermoacoustic platform, where both CW and CCW vortex excitations can be treated\nwithin a unified scattering framework. The ability to generate and control such dual-topology modes under\noptical excitation is central to realizing the experimentally observed non-Hermitian phase transitions reported in this study.\nFigures5(e)-(f)demonstratedthesimulatedandexperimentallymeasuredscatteringamplitudesand\nphases of the acoustic CCW VBs at the operating frequency of f = 6 kHz and rotation frequency of\n0\nf = 0.1 kHz, respectively. The corresponding CW characteristics are shown in Figs. 5(h)-(i). The\nr\ndata reveal a remarkable behavior: when a CCW VB is incident from the lossy side, no reflection occurs\n25\n(r = 0), and the transmitted beam takes the form t = −eiy. The transmitted VB preserves its\nL L\namplitude but reverses its topological charge relative to the incident wave. In contrast, when the VB is\nincidentfromthegainside, thetransmittedmoderemainsreversed(t = −eiy), whilethereflectedfield\nG\nbecomes a superposition of two VBs with distinct topological charges, r = r +r = −ei2y/s−2\nG G1 G2\n(see Supplementary Note XIII for details). Here, s is the complex reflection/transmission coefficient of\nthe loss medium with r = t = s = |s|eiα; |s| and α are the amplitude and phase of s.\ns s\nA more intuitive visualization is provided in Figs. 5(g) and (j), showing the measured cross-sectional\nfield distributions of CCW and CW VBs with a coupling d = 9.92 cm, corresponding to the vertical\ndashed lines in Figs. 5(e)-(f) and (h)-(i), respectively. The fitted CNT film parameters for this EP are\nm = 1, φ = π; m = 0, φ = 0; m = 1, φ = π; and m = 1/|s|, φ = π−α. The experi-\n11 11 12 12 21 21 22 22\nmentally obtained scattering coefficients are r = 0.068ei0.619, t = 0.069ei0.57, r = 1.811e−i1.803,\nL1 L1 G1\nt = 0.015ei0.615, r = 0.327ei0.181, t = 1.016e−i3.102, r = 1.982e−i3.015, and t = 0.997ei3.183\nG1 L2 L2 G2 G2\nat the azimuth angle of θ = 0. These results collectively confirm that the system realizes PT-symmetric\nscattering with acoustic VBs and achieves experimentally observable topological charge conversion at the exceptional point.\nThe observation constitutes the first experimental demonstration of PT symmetry in acoustic vortex beams,\nestablishing a versatile platform for studying non-Hermitian phase transitions in structured acoustic fields.\nIn summary, we would like to note that the present study advances the field of non-Hermitian\nacoustics in two dimensions.\nFrom a physics perspective, it reports the first experimental observation of the PT, reversed-PT,\nandanti-PT phasetransitionswithinasinglescatteringsystem. Thisunifiedrealizationestablishesanew\nexperimental framework for investigating non-Hermitian phenomena that extend beyond conventional\nHamiltonian models, revealing the full spectrum of symmetry transitions accessible in open acoustic\nsystems.\nFrom a methodological standpoint, the work introduces a laser-driven thermoacoustic platform\nthat enables precise control over broadband and geometrically complex sound fields. Through rotating\nlaser excitation, we demonstrate the generation and manipulation of first-order acoustic vortex beams\nat exceptional points—an achievement that cannot be realized with earlier electrically driven CNT\nconfigurations. This platform therefore offers an unprecedented level of flexibility for engineering non-\nTogether, these advances substantially deepen the physical insight and broaden the experimental\ncapabilities of non-Hermitian acoustics, directly addressing the reviewer’s concern regarding the level of\nsignificance required for publication.\nIn light of this feedback, we have also added a new section titled “Formation of acoustic VBs and\nscatteringEPusingLIT”totheMainText(Page9),whichprovidesfurthertheoreticalandexperimental\nvalidation of these findings. The Fig. R9 here is Fig. 5 in the revised manuscript.\nFormation of acoustic VBs and scattering EP using LIT. The LIT method employed in this study\nrepresents a promising approach for achieving perfect acoustic VBs and enables the PT-symmetric\nscattering behavior illustrated in Fig. 5(a). In the aforementioned configuration, the CNT film under\nlaser irradiation functions effectively as a point sound source, since the timescale of laser-induced heat\ndiffusion across the film surface is on the order of seconds [40]. When the laser spot rotates on the\nCNT film at a modulation frequency of f (e.g., f = 0.1 kHz), the diffusion time is much longer than\nr r\nthe rotation period, ensuring that the acoustic source rotates continuously rather than discretely. As\na result, the system produces perfect acoustic VBs within the waveguide, unlike the approximate VBs\ngenerated by discrete passive [41] or electronical driven [35] methods. This continuous, contact-free\nexcitation is a defining advantage of LIT and fundamentally differentiates it from previous electrical\napproaches.\nThephysicalmechanismandspatialdistributionoftheacousticVBgeneratedbyarotatinglaserspot\nare discussed in Supplementary Note X. To support the propagation of first-order VBs, the waveguide\nwas replaced with a cylindrical tube of radius R = 2 cm. A schematic of the LIT-generated VB\n0\nis shown in Fig. 5(b), in which, a certain angle is set between the symmetry axis of the reflector\nand the rotation axis of the motor, causing the laser to trace a circular trajectory on the CNT film.\nSupplementary Video I and Video II demonstrate the laser spot rotating on the CNT film to form a\ncirculartrajectoryatrotationalspeedsof150revolutionsperminute(RPM)and6000RPM,respectively.\nThe corresponding simulation and experimental results, after Fourier transformation, are presented in\nFigs. 5(c)-(d), confirming that the LIT system produces a stable and well-defined acoustic VB (details\nin Supplementary Note XI).\nBuilding on this foundation, we modeled the quasi-PT-symmetric scattering behavior of VBs at the\nEPandverifiedtheresultsexperimentally. Thescatteringresponsesforcounter-clockwise(CCW)vortex\nexcitations were formulated (see Supplementary Note XII), with the corresponding scattering matrices\nrevealing distinct symmetry-breaking features inherent to non-Hermitian systems:\n(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)\nr t r t r t\nS = L G = L1 G1 + L2 G2 ,\n(R42)\nr t r t r t r\nL G L1 G1 CCW L2 G2 CW\nin which\nr = m eiφ11t e−iy +r ,\nL1 11 s s\nt = m eiφ21t e−iy +t e−iy,\nG1 21 s s\n(R43)\nt = m eiφ11r e−i2y +m eiφ12 +t e−iy,\nL1 11 s 12 s\nr = m eiφ21r e−i2y +m eiφ22 +r e−i2y,\nG1 21 s 22 s\n24\n((aa)) CCCCWW CCWW\nLLoossss GGaaiinn LLoossss GGaaiinn ((ee)) ((hh))\n22\n((bb\nii\n))\nnncciiddeenntt iinncciiddeenntt eedduuttiillppmm 0\n1\n0\n1\n.\n.\n.\n.\n4\n8\n4\n8\neee sssiii xxx mmm ppp ... ... rrrLLL111tttLLL111rrrGGG111tttGGG111 eedduuttiillppmm 11 eee sssiii xxx mmm ppp ... ...\nrrrLLL222tttLLL222rrrGGG222tttGGG222\nAA00..22 AA\n00 00\n((ff)) ((ii))\nππ ππ\neessaahhPP\n00\neessaahhPP\n00\n--ππ --ππ\n((cc)) ((dd)) 44 77 1100 1133 44 77 1100 1133\ndd ((ccmm)) dd ((ccmm))\nSSiimmuullaattiioonnss EExxppeerriimmeennttss\nff00 -- ffrr ff00 ff00 ++ ffrr ff00 -- ffrr ff00 ff00 ++ ffrr ((gg)) rrLL11 ttLL11 rrGG11 ttGG11 ((jj)) rrLL22 ttLL22 rrGG22 ttGG22\nAAAbbbsss(((ppp)))::: 000 111 AAbbss((pp)):: 00 11\nssttnneemm\niirreeppxxEE AAmmpplliittuuddee::00 22\nssttnneemm\niirreeppxxEE AAAmmmpppllliiitttuuudddeee:::000 222\nAArrgg((pp)):: --ππ ππ AArrgg((pp)):: --ππ ππ PPhhaassee:: --ππ ππ PPhhaassee:: --ππ ππ\nFigure R9: Formation of acoustic VBs and realization of the scattering EP using LIT.\n(a) Illustration of the PT-symmetric system for acoustic VBs. (b) Schematic of the LIT-based\ngeneration of acoustic VBs. (c)-(d) Simulated and experimental demonstration of the acoustic VBs\nby CCW-rotating lasr spot. (e)-(f) Simulated (curves) and experimentally measured (symbols)\nscattering amplitudes and phases of CCW VBs at a rotation frequency of f = 0.1 kHz.\nr\n(g)\nMeasured scattering field amplitudes and phases of CCW VBs with a coupling of d = 9.92 cm,\ncorresponding to the data in panels (e)-(f). (h)-(i) Same as (e)-(f), but for CW VBs, obtained\nunder identical conditions. (j) Same as (g), but for CW VBs, corresponding to panels (h)-(i).\nand\nr = m eiφ12t e−iy,\nL2 12 s\nt = m eiφ22t e−iy,\nG2 22 s\n(R44)\nt = m eiφ11 +m eiφ12r e−i2y,\nL2 11 12 s\nr = m eiφ21 +m eiφ22r e−i2y.\nG2 21 22 s\nThe parameters r , t , t , r describe the CCW VB relative to the incident wave at an azimuthal\nL1 G1 L1 G1\nangle of θ = 0, meanwhile r , t , t , r characterize the clockwise (CW) VB response. Here, the\nL2 G2 L2 G2\nfrequencies of the VBs are all given by f = f +f . m and φ represent the amplitude ratios and\n0 r ij ij\nphase differences between the CNT film components excited by CW (j = 1) and CCW (j = 2) rotating\nlaser spots and the incident CCW VB coming from the loss (i = 1) and gain (i = 2) sides, respectively.\n(cid:112)\nThe propagation constant is expressed as k11 = (ω/c )2−(k11)2 with y = k11d, in which k11 is the\nX 0 r X r\nfirst order radial wave number of the waveguide. It is worth emphasizing that the scattering coefficients\nrepresent only the normalized ratios between the reflected or transmitted pressure amplitudes and that\noftheincidentwave. Thesequantitiesareindependentofthevortextopologicalchargeitself; thecharge\nmerely determines the rotational phase structure of the field. This distinction underscores the versatility\nof the laser-induced thermoacoustic platform, where both CW and CCW vortex excitations can be treated\nwithin a unified scattering framework. The ability to generate and control such dual-topology modes under\noptical excitation is central to realizing the experimentally observed non-Hermitian phase transitions reported in this study.\nFigures5(e)-(f)demonstratedthesimulatedandexperimentallymeasuredscatteringamplitudesand\nphases of the acoustic CCW VBs at the operating frequency of f = 6 kHz and rotation frequency of\n0\nf = 0.1 kHz, respectively. The corresponding CW characteristics are shown in Figs. 5(h)-(i). The\nr\ndata reveal a remarkable behavior: when a CCW VB is incident from the lossy side, no reflection occurs\n25\n(r = 0), and the transmitted beam takes the form t = −eiy. The transmitted VB preserves its\nL L\namplitude but reverses its topological charge relative to the incident wave. In contrast, when the VB is\nincidentfromthegainside, thetransmittedmoderemainsreversed(t = −eiy), whilethereflectedfield\nG\nbecomes a superposition of two VBs with distinct topological charges, r = r +r = −ei2y/s−2\nG G1 G2\n(see Supplementary Note XIII for details). Here, s is the complex reflection/transmission coefficient of\nthe loss medium with r = t = s = |s|eiα; |s| and α are the amplitude and phase of s.\ns s\nA more intuitive visualization is provided in Figs. 5(g) and (j), showing the measured cross-sectional\nfield distributions of CCW and CW VBs with a coupling d = 9.92 cm, corresponding to the vertical\ndashed lines in Figs. 5(e)-(f) and (h)-(i), respectively. The fitted CNT film parameters for this EP are\nm = 1, φ = π; m = 0, φ = 0; m = 1, φ = π; and m = 1/|s|, φ = π−α. The experi-\n11 11 12 12 21 21 22 22\nmentally obtained scattering coefficients are r = 0.068ei0.619, t = 0.069ei0.57, r = 1.811e−i1.803,\nL1 L1 G1\nt = 0.015ei0.615, r = 0.327ei0.181, t = 1.016e−i3.102, r = 1.982e−i3.015, and t = 0.997ei3.183\nG1 L2 L2 G2 G2\nat the azimuth angle of θ = 0. These results collectively confirm that the system realizes PT-symmetric\nscattering with acoustic VBs and achieves experimentally observable topological charge conversion at the exceptional point.\nThe observation constitutes the first experimental demonstration of PT symmetry in acoustic vortex beams,\nestablishing a versatile platform for studying non-Hermitian phase transitions in structured acoustic fields.\nIn summary, we would like to note that the present study advances the field of non-Hermitian\nacoustics in two dimensions.\nFrom a physics perspective, it reports the first experimental observation of the PT, reversed-PT,\nandanti-PT phasetransitionswithinasinglescatteringsystem. Thisunifiedrealizationestablishesanew\nexperimental framework for investigating non-Hermitian phenomena that extend beyond conventional\nHamiltonian models, revealing the full spectrum of symmetry transitions accessible in open acoustic\nsystems.\nFrom a methodological standpoint, the work introduces a laser-driven thermoacoustic platform\nthat enables precise control over broadband and geometrically complex sound fields. Through rotating\nlaser excitation, we demonstrate the generation and manipulation of first-order acoustic vortex beams\nat exceptional points—an achievement that cannot be realized with earlier electrically driven CNT\nconfigurations. This platform therefore offers an unprecedented level of flexibility for engineering non-\nTogether, these advances substantially deepen the physical insight and broaden the experimental\ncapabilities of non-Hermitian acoustics, directly addressing the reviewer’s concern regarding the level of\nsignificance required for publication.\nIn light of this feedback, we have also added a new section titled “Formation of acoustic VBs and\nscatteringEPusingLIT”totheMainText(Page9),whichprovidesfurthertheoreticalandexperimental\nvalidation of these findings. The Fig. R9 here is Fig. 5 in the revised manuscript.\nTo maintain the logical coherence of the manuscript, we have included relevant sentences ”Fur-\nthermore, we verified that LIT can be effectively utilized to generate acoustic vortex beams (VBs),\nachieve their PT-symmetric scattering behavior, and enable the conversion of the topological charge\nin such VBs.” on Page 2, ”In addition, active methods for generating acoustic VB have been reported\nin Ref. [35], and LIT may provide a viable alternative approach.” on Page 3, ”Due to the innovation\nof LIT, acoustic VBs can be conveniently generated, and their PT-symmetric scattering behavior has\nbeen realized, along with the conversion of the topological charge.” on Page 4, and ”The realization of\nPT-symmetricscattering behaviorinacousticVBsmarks akeystepforwardinextendingPT symmetry\nfrom plane-wave systems to structured wavefronts carrying orbital angular momentum.” on Page 11.\nComment P2.5 – Another technique issue: the symmetry operators were not given in the text.\nResponse: We sincerely thank the reviewer for this valuable comment. The clarification regarding\nsymmetry operators is indeed important, as it helps to distinguish between the different theoretical\nframeworks used to describe non-Hermitian acoustic systems. In the field of PT-symmetric acoustics,\nexisting research can generally be divided into two main categories: the cavity-tube model and the\nscattering model. The cavity-tube model, as discussed for example in [Phys. Rev. Lett. 131, 066601\n(2023)], focuses on the intrinsic resonant modes of coupled cavities. In this framework, the system is\ndescribed by a non-Hermitian Hamiltonian whose eigenvalues correspond to the resonance frequencies,\nandthePT symmetryisdefinedthroughexplicitsymmetryoperatorsactingonthesystem’seigenstates.\nThis model is particularly effective for studying mode coalescence and internal energy exchange between\ncoupledresonators. Bycontrast, thescatteringmodel, asdevelopedin[Nat. Commun. 6, 5905(2015)],\nemphasizes the input–output behavior of waves interacting with a PT-symmetric medium. Instead of\nanalyzing intrinsic modes, this approach characterizes how incident waves are reflected and transmitted\nthrough the system. The analysis is based on the system’s transfer matrix and the resulting scattering\nmatrix, which encapsulate the effects of balanced gain and loss within the medium. In this framework,\nthe notion of PT symmetry is expressed through the spatial distribution of gain and loss, rather than\nthroughsymmetryoperatorsactingoneigenmodes. Consequently,theuseofexplicitsymmetryoperators\nis not required for describing the underlying physics.\nIn summary, the cavity model focuses on the internal modal properties of a system and relies on\nHamiltonian symmetry operators to reveal PT behavior, whereas the scattering model concentrates on\nthe external wave dynamics governed by the transfer and scattering matrices. Our work follows the\nlatter framework, where symmetry arises from the balanced gain–loss configuration rather than from\noperator-based transformations. For this reason, the symmetry operators are not explicitly given in the\nmanuscript.\nWe sincerely thank the reviewer for his/her insightful and constructive feedback, which has greatly\ncontributed to the improvement of our manuscript. With great respect for the reviewer’s evident\nexpertise and deep familiarity with the developments in the PT-symmetric acoustics field, we hope the\nrevisionswehavemadealignwiththereviewer’sexpectationsforpublicationinNatureCommunications,\nand we kindly request a reassessment of his/her verdict.\n27"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I carefully read the authors' detailed responses. The authors have done an excellent job in addressing the previous comments thoroughly and clearly. I am also pleased to see that their revisions went beyond my initial expectations, especially the new experimental results on LIT-based acoustic vortex PT control, which are quite impressive.\nThe enhanced capabilities of the LIT platform suggest that it could be developed into a novel technical platform for modulating more complex acoustic fields. I have also noted that this approach holds potential for generating tailored structured acoustic fields, much like an SLM in optics. Although this exciting direction falls outside the scope of the current study, it could open up valuable opportunities for future work.\nOverall, the authors have strengthened the manuscript significantly through their careful revisions and additional insights. I think the current version of the paper is now well-suited for publication in Nature Communications.", "reviewer_2": "I thank the authors for the extensive revisions. Especially, the demonstration of the VBs connects non-Hermitian physics to structured waves, which is novel and makes the paper more attractive to a broader readership. Although I still think the novelty in the experimental implementation is not high, the current version, overall, can meet the standard of Nature Communications."}, "author_response": "NCOMMS-24-62740A: Response to Reviewers\nWe express our gratitude to all reviewers for their acceptance of our responses to their comments\nand for their invaluable contributions to the peer review process of this paper.\nResponse to Reviewer 1\nComment P1.1 – I carefully read the authors’ detailed responses. The authors have done an\nexcellent job in addressing the previous comments thoroughly and clearly. I am also pleased to see\nthat their revisions went beyond my initial expectations, especially the new experimental results on\nLIT-based acoustic vortex PT control, which are quite impressive.\nThe enhanced capabilities of the LIT platform suggest that it could be developed into a novel\ntechnical platform for modulating more complex acoustic fields. I have also noted that this approach\nholds potential for generating tailored structured acoustic fields, much like an SLM in optics. Al-\nthough this exciting direction falls outside the scope of the current study, it could open up valuable\nopportunities for future work.\nOverall, the authors have strengthened the manuscript significantly through their careful re-\nvisions and additional insights. I think the current version of the paper is now well-suited for\npublication in Nature Communications.\nResponse: We thank the reviewer for thorough review of our responses and the endorsement of our\npaper for publication in this esteemed journal of high quality.\nResponse to Reviewer 2\nComment P2.1 – I thank the authors for the extensive revisions. Especially, the demonstra-\ntion of the VBs connects non-Hermitian physics to structured waves, which is novel and makes\nthe paper more attractive to a broader readership. Although I still think the novelty in the experi-\nmental implementation is not high, the current version, overall, can meet the standard of Nature\nCommunications.\nResponse: We thank the reviewer for the constructive reassessment. It is particularly encouraging\nto see the novelty of linking non-Hermitian physics with structured acoustic waves highlighted, as this\nconceptual bridge is central to our work’s broader appeal.\nWe are grateful for the reviewer’s supportive conclusion and are pleased that the manuscript is now\ndeemed suitable for publication in Nature Communications.\n1"}]

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