Environmentally-induced exceptional points in elastodynamics

F. M. Ellis; Ramathasan Thevamaran; T. Kottos; V. Dominguez-Rocha

REVIEW 3 major objections 4 minor 29 references

Environmentally-induced exceptional points in elastodynamics

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Two identical aluminum pillars, coupled only through their base plate, reach an exceptional point when one is subjected to purely dissipative differential loss, with the eigenfrequency splitting following the square-root law near the…

desk verdict Environment-mediated EPs in elastodynamics is a plausible and genuinely new platform, but the experimental square-root confirmation is partly model-dependent; the COMSOL simulation carries the load. read the letter →

arxiv 1908.00793 v1 pith:XKEM3QS2 submitted 2019-08-02 physics.app-ph

V. Dominguez-Rocha , Ramathasan Thevamaran , F. M. Ellis , T. Kottos This is my paper · ORCID

classification physics.app-ph

keywords exceptionalpointselastodynamicsnon-HermitianphysicsPTsymmetrycoupled-modetheorymechanicalresonatorsdifferentiallossenvironmentalmodecoupling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper demonstrates an exceptional point in a purely mechanical system: two identical aluminum pillars carved from a single block and coupled only through their common base plate. By applying a specially prepared putty layer to one pillar, the authors add damping without adding mass, creating a differential loss that tunes the two torsional resonances toward coalescence. Near the critical loss the eigenfrequency splitting follows the square-root law $\Delta f \sim \sqrt{1-\Gamma/\Gamma_\mathrm{EP}}$, the definitive signature of an exceptional point. The result matters because it transfers exceptional-point physics from photonics to elastodynamics, where the coupling itself is mediated by 'environmental' plate modes, opening the door to new mechanical sensors for surface integrity and a differential atomic force microscope.

What carries the argument

The central object is the two-mode coupled-mode theory Hamiltonian in the weak-coupling limit (Eq. 5 of the paper), which maps the unequal-loss mechanical dimer onto an effective $2\times 2$ matrix with the same square-root singularity structure as a parity-time-symmetric dimer. The pivotal linearization is $1-u^2 \approx -2\Delta$ with $u=1+\Delta$, and the Appendix derives the exceptional point position $\Gamma_\mathrm{EP}=2(\sqrt{1+2\kappa}-1)+\gamma_d$ and the splitting $\Delta\omega/\omega_0 = \tfrac{2}{3}\kappa^{3/2}$. A three-level coupled-mode model (Eq. 4), adding one explicit environmental level with detuning $\nu$ and couplings $\lambda_1,\lambda_2$, shows how a nearby plate mode qualitatively modifies the two-level exceptional-point behavior. On the experimental side, a viscoelastic boundary-layer model of the putty layer gives the key constitutive relation that damping can be added without mass loading, which is what keeps the loss contrast tunable while preserving the resonator frequencies.

What would settle it

Measure both the real and imaginary parts of the two torsional eigenfrequencies with fine steps of the loss parameter through the would-be exceptional point: the claim requires the two real parts to become exactly equal and the two imaginary parts to become exactly equal at a single value of $\Gamma$, with the splitting scaling as $|\Gamma-\Gamma_\mathrm{EP}|^{1/2}$ in the neighborhood. If instead the real parts show an avoided crossing or the imaginary parts retain a finite difference at the closest approach, the singularity is not present. This test is achievable with the paper's own transmission-fitting method, which currently loses linewidth resolution just beyond the exceptional point.

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Extended reading notes

Core claim

Using a pair of aluminum torsion pillars sharing a base plate, the paper shows that an exceptional point — a non-Hermitian degeneracy where two eigenfrequencies and their eigenvectors coalesce — can be induced by purely dissipative differential losses. The inter-pillar coupling is not a direct spring but is mediated by the elastic modes of the base plate, and the loss is introduced as a boundary layer of putty that adds damping with negligible mass loading. Experiment and COMSOL simulation both show that, as the imposed loss $\Gamma$ increases, the real parts of the two torsional eigenfrequencies merge and the splitting obeys $\Delta f \sim \sqrt{1-\Gamma/\Gamma_\mathrm{EP}}$ near the exceptional point, with the imaginary parts becoming equal in the exact phase. A coupled-mode theory in the weak-coupling limit and a three-level version including one explicit environmental mode reproduce the observed behavior, and the authors note that a strict exceptional point for unequal positive losses exists only in this weak-coupling approximation, which their device enforces with a small coupling splitting ($\kappa \approx 45$ Hz at $\omega_0 \approx 2\pi\times 16.29$ kHz).

Load-bearing premise

The entire demonstration relies on the pillar-pillar coupling being weak enough that the unequal-loss system can be linearized into a strict exceptional point; if the coupling is not kept small (here about 45 Hz compared with 16.29 kHz), no true coalescence of eigenvalues occurs and the square-root splitting law is lost.

Editorial extensions

If this is right

Near the exceptional point, a small change in loss contrast $\Gamma$ produces a large, square-root response in the eigenfrequency splitting, so the structure acts as a sensitive transducer for dissipation and contact.
Because the coupling is mediated by plate modes, modifying the plate (for example, adding an indent) changes the exceptional-point characteristics without touching the pillars, giving a new control knob labeled 'environmental mode control.'
In the weak-coupling limit the unequal-loss mechanical dimer is mathematically equivalent to a PT-symmetric dimer, so established PT-symmetric sensing and transport designs can, in principle, be ported to elastodynamic structures.
The three-level model indicates that a single nearby environmental mode can qualitatively shift or distort the exceptional point, meaning the base-plate mode spectrum can be used to engineer EP behavior in mechanical devices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension the authors do not pursue: sweeping both loss terms (or loss and coupling) along a closed loop around the EP should produce a chiral mode swap, turning this pair of pillars into a mechanical analog of topological mode transfer.
An experiment the paper does not run: place a phononic band-gap lattice in the base plate so that no environmental mode lies near the pillar frequency; the model predicts the EP would then disappear or move, which would confirm the causal role of the environmental mode.
A scaling question left open: the putty boundary-layer mechanism is specific to the 16 kHz scale, and extending the same idea to MEMS resonators would require a material whose boundary-layer dynamics remains loss-dominated without mass loading at much higher frequencies.
A practical consequence the authors do not state: if the square-root law holds, the same measurement curve of $\Delta f$ versus $\Gamma$ contains both the coupling strength $\kappa$ and the intrinsic damping $\gamma_d$, so the EP calibration could be used as a non-invasive way to extract these mechanical parameters from a single transmission measurement.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Read the Domínguez-Rocha et al. paper. What is actually new: an EP in a pair of elastic pillars coupled only through the base plate, with differential loss rather than gain, and the claim that a single nearby environmental mode controls the EP. That is a legitimate extension of non-Hermitian physics to elastodynamics, and the three-level CMT model with environmental coupling is a reasonable framework.

The COMSOL study is the strongest piece. It uses a real geometry, varies the indent to steer an environmental mode, imposes balanced gain/loss in the shear modulus, and shows eigenfrequency and eigenvector coalescence. That is independent evidence for an EP. The experimental design is also thoughtful: putty as controllable loss, clamp calibration of Γ, and Eq. (6) fit with an explicit detuning check ϵ for mass loading. The putty viscoelastic modeling is a nice side contribution.

Where it gets soft: the experimental eigenfrequencies in Fig. 4(a) are not measured directly. They are extracted by fitting each transmission spectrum to Eq. (6), which is already the coupled-oscillator model whose poles carry the EP. So observing near-EP behavior in the extracted data is partly circular. The log-log plot in Fig. 4(b) bolsters the claim a bit, but the best fit gives α≈0.6 rather than the expected 0.5, with no error bars, and the paper concedes that after the EP the broader resonance linewidth cannot be resolved, leaving the broken-phase side essentially unconstrained. The appendix also admits that strict EPs with unequal positive losses exist only in the linearized weak-coupling limit; the experiment assumes that limit (κ≈45 Hz at ω0≈16.3 kHz, which is about 3×10^-3, so the assumption is plausible but not independently validated). The three-level environmental-mode model is qualitative, as the authors say.

All that said, I don't think the central physics is wrong. The COMSOL result, the weak-coupling analysis, and the rough agreement in Fig. 4(a) together make the existence of an environment-mediated EP likely. What is missing is a measurement that doesn't rely on the same model to extract the very quantity being tested, plus honest uncertainties and a fit to the full spectrum rather than a few eigenfrequency points.

This paper deserves a serious referee. The right outcome is a major-revision request: reanalyze the experimental data with a model-independent resonance extraction (e.g., phase-rotation or lineshape fitting on raw spectra), report uncertainties, and present the broken-phase linewidths or explain why they cannot be measured. I would not cite it as a definitive experimental demonstration yet, but I would file it as a useful platform paper.

Referee Report

3 major / 4 minor

Summary. The manuscript reports a non-Hermitian elastodynamic platform in which two identical torsion pillars on an aluminum plate are coupled only through plate (environmental) modes. The authors use a coupled-mode model with one environment mode, COMSOL eigenfrequency simulations with complex shear moduli, and transmission measurements with variable putty-induced differential loss to argue that the system exhibits an exceptional point. The central experimental claim is that the measured frequency splitting scales as Δf ∼ sqrt(1 − Γ/Γ_EP), confirming an EP in the weak-coupling regime. The paper also proposes applications in atomic force microscopy and surface-integrity sensing.

Significance. If the central claim is correct, this is a useful extension of EP physics to mechanical/elastodynamic systems and supports a distinct route to EP control via environmental modes. The strengths are the physically credible geometry, the independent COMSOL calculations, explicit weak-coupling analytic results in the Appendix, and the careful calibration of putty-induced damping. The experimental verification is weakened by the model-dependent extraction of eigenfrequencies and by an exponent mismatch, so the current manuscript does not yet provide a quantitatively decisive confirmation of the square-root singularity.

major comments (3)

[Section IV, Fig. 4(a), Eq. (6)] The experimental eigenfrequencies in Fig. 4(a) are not directly measured quantities; they are the poles of Eq. (6), the transmission function of the same weak-coupling coupled-oscillator model that contains the exceptional point. Fitting each spectrum to this function therefore builds the EP structure into the extracted eigenvalue trajectories, and the good agreement with the COMSOL simulation does not remove the circularity for the experimental confirmation. I recommend an independent check: report the raw transmission data at selected Γ values together with the COMSOL prediction using the measured Γ and no free resonance parameters, or extract resonance positions and linewidths directly from the spectra and compare them with the CMT/COMSOL curves.
[Section IV, Fig. 4(b)] The claimed square-root law is not actually demonstrated by the fit shown. The text states Δf ∼ sqrt(1 − Γ/Γ_EP), but the best linear fit to the log-log plot is reported as (1 − Γ/Γ_EP)^α with α ≈ 0.6, and no error bars, residuals, or confidence intervals are given for α, Γ, or Δf. Given that α = 0.6 differs from 0.5 by 20%, the data are at best consistent with a near-square-root singularity, not a confirmation of the square-root law. The authors should (i) report the uncertainty in α, (ii) fit the data with a fixed exponent 0.5 and show the residuals, and (iii) mark the broken-phase points that are not constrained because the linewidths are unresolved, as stated in Section IV.
[Appendix, after Eq. (8)] The Appendix explicitly states that a strict exceptional point exists only when γ0 = 0, and that in the experimental situation with unequal positive losses the EP is restored only in the weak-coupling limit where the frequency dependence of the loss terms is linearized. The experiment's validity therefore depends on the smallness of the neglected terms in Eq. (8), but the paper does not quantify this. I ask for a quantitative check: using the fitted values of κ, γd, and Γ, evaluate the exact eigenvalues of Eq. (8) and compare their coalescence behavior with the CMT prediction of Eq. (5); or estimate the relative magnitude of the neglected γΔ and κ^2 terms. Without this check, one cannot distinguish a true branch-point singularity from an avoided crossing rounded by the neglected terms.

minor comments (4)

[Throughout] There are typographical errors including 'EXEPTIONAL' in the Section IV heading, 'epansion' near Eq. (2), and 'diﬀerential' in the abstract; these should be corrected.
[Section IV] The value of Poisson's ratio used in the COMSOL simulations is said to have been fixed earlier but is never reported; please state it explicitly in Section IV or in the Methods.
[Fig. 1] Fig. 1(c)–(e) are difficult to read because the panels share axes and the labels overlap; the caption also does not clearly identify which plotted quantities correspond to the real and imaginary eigenfrequency axes in each panel. Please redraw or annotate.
[Eq. (6)] The fitting parameters in Eq. (6) (t0, A, φ, ε, γd, κ, Γ) are numerous; at minimum the uncertainties in κ and Γ, which enter the EP position and the exponent α, should be reported.

Circularity Check

1 steps flagged · score 6.0 of 10

The square-root EP confirmation is partly inherited from the fitting model: the eigenfrequencies in Fig. 4(b) are extracted from Eq. (6), a coupled-oscillator response whose un-driven poles already contain the EP.

fitted input called prediction [Section IV, Eq. (6) and Fig. 4]
"The experimental transmission spectra from the weakly damped side to the damped side was measured, and fit to the function … (6) … The eigenfrequencies, extracted from the equivalent un-driven modes using the same parameters are reported with blue dots in Fig. 4(a). … By considering the logarithmic behavior of this curve, we find that the frequency splitting Δf near the exceptional point scales as Δf∼√(1−Γ/Γ_EP), thus confirming the existence of an EP singularity at a critical value Γ_EP in the case of weak coupling."

Equation (6) is the response function of the same two-damped-oscillator CMT model whose un-driven poles undergo an EP with square-root splitting in the weak-coupling limit. The paper fits each measured transmission spectrum to Eq. (6) and then reads off the eigenfrequencies from the equivalent un-driven modes using the same fitted parameters. Those blue-dot eigenfrequencies are therefore not directly measured resonances; they are the poles of the assumed model, and they carry the model's EP branch structure by construction. Plotting those extracted points as Δf versus Γ and announcing Δf∼√(1−Γ/Γ_EP) returns the fitting function's own pole law as an empirical finding.

full rationale

The paper's broad claim of an environment-mediated EP in elastodynamics is not wholly circular: the COMSOL solution of the continuum elastodynamic structure is an independent numerical check, and the raw transmission spectra show resonance broadening and merging near the expected parameter range. However, the specific quantitative confirmation of the square-root law in Fig. 4(b) is obtained from eigenfrequencies that are outputs of a fit to Eq. (6), the coupled-oscillator response function whose un-driven poles already contain the EP. Thus the Δf∼√(1−Γ/Γ_EP) statement is partly an output of the fitting model rather than an independent measurement. The Appendix's own limitation—that a strict EP exists only in the weak-coupling, linearized limit when γ0=0—further constrains but is not itself a circular step. No load-bearing self-citation chain or imported uniqueness theorem is present. Score 6 reflects the model-inherited square-root confirmation while crediting the independent COMSOL support and direct spectral evidence.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central results depend on a small number of fitted CMT parameters (kappa, lambda, nu) that are chosen to reproduce the COMSOL spectra, and on per-spectrum fit parameters used to extract eigenfrequencies. The material constants for putty and the Poisson ratio are also fitted to calibration data. No new physical entities are introduced; the environmental mode is a real plate resonance of the structure.

free parameters (6)

kappa (CMT inter-oscillator coupling) = 6.14e-3 (COMSOL, Fig. 1e); ~2.8e-3 (experiment, 45 Hz / 16.29 kHz)
Used in Eq. (4) to reproduce the COMSOL EP and in Eq. (6) to fit transmission spectra. Its smallness defines the weak-coupling regime required for the loss-only EP.
lambda1, lambda2 (torsion-environment couplings) = ±0.9e-3
Chosen so that |lambda2| = |lambda1| to reproduce the COMSOL three-level EP in Fig. 1(e).
nu (environmental mode detuning) = -2e-3
Chosen to match the COMSOL level spacing at r_hole ~ 3.8 cm.
Transmission fit parameters t0, A, phi, epsilon, gamma_d, kappa, Gamma = Not tabulated; fitted per spectrum
Eq. (6) fits each experimental transmission spectrum to extract eigenfrequencies; epsilon checks mass loading.
Putty viscoelastic constants G, eta0, eta1 = G=15.8 MPa, eta0=72.6 MPoise, eta1=54.9 MPoise
Fitted to the frequency shift and linewidth data of Fig. 3 to validate the boundary-layer model.
Poisson ratio of aluminum = Not stated numerically, previously fixed to best match
Tuned in COMSOL to match the damped pillar and plate modes, affecting the simulation frequencies.

assumptions (5)

domain assumption Weak-coupling linearization u = 1 + Delta, keeping only first order in Delta, kappa, and gamma (Appendix, Eq. 7).
Required to arrive at the Liouvillian CMT form and to restore an exceptional point in the presence of unequal positive losses (Eq. 8 discussion).
ad hoc to paper The environment is represented by a single nearby mode plus a spectrally distant background sea parameterized by kappa (Eq. 4).
This modeling choice is specific to the paper and is shown to reproduce COMSOL only qualitatively; the authors admit it fails near the EP.
domain assumption Pillar torsion modes are described by a 1D wave equation with solid-body rotation of each cross section (Eq. 14).
Used in the putty boundary-layer model; ignores 3D warping and other pillar modes.
standard math Aluminum is an isotropic elastodynamic medium with shear modulus G0 = 25 GPa (Section III).
Standard material assumption used in COMSOL simulations.
domain assumption Putty behaves as a combined Maxwell-Voigt viscoelastic medium (Appendix).
Needed to explain the near-zero mass loading and loss proportional to contact area; relies on fitting constants.

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Cite this review

Pith. "Pith review of Environmentally-induced exceptional points in elastodynamics." pith.science (2026). https://pith.science/paper/XKEM3QS2

@misc{pith2026190800793,
  author       = {Pith},
  title        = {Pith review of: Environmentally-induced exceptional points in elastodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XKEM3QS2}},
  note         = {Machine review of arXiv:1908.00793}
}

read the original abstract

We study the nature of an environment-induced exceptional point in a non-Hermitian pair of coupled mechanical oscillators. The mechanical oscillators are a pair of pillars carved out of a single isotropic elastodynamic medium made of aluminum and consist of carefully controlled differential losses. The inter-oscillator coupling originates exclusively from background modes associated with the "environment", that portion of the structure which, if perfectly rigid, would support the oscillators without coupling. We describe the effective interaction in terms of a coupled mode framework where only one nearby environmental mode can qualitatively reproduce changes to the exceptional point characteristics. Our experimental and numerical demonstrations illustrates new directions utilizing environmental mode control for the implementation of exceptional point degeneracies. Potential applications include a new type of non-invasive, dfferential atomic force microscopy and hypersensitive sensors for the structural integrity of surfaces.

Figures

Figures reproduced from arXiv: 1908.00793 by the authors.

**Figure 2.** FIG. 2: (a) Photograph of the experimental set-up. The pillar dimer with its plate environment is machined from a single [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: (a) Experimental measurement of the frequency [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: (a)Numerical calculations (black dots) and experi [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

L. Feng, R. El-Ganainy, L. Ge, Non-Hermitian photonics based on paritytime symmetry, Nat. Photonics 11, 752762 (2017)

work page 2017
[2]

R, El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, D. N. Christodoulides, Non- Hermitian Physics and PT symmetry , Nat. Phys. 14, 11 (2018)

work page 2018
[3]

Alu, Exceptional points in optics and pho- tonics, Science 363, 42 (2019)

M Ali-Miri, A. Alu, Exceptional points in optics and pho- tonics, Science 363, 42 (2019)

work page 2019
[4]

Hodaei, A

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points , Nature 548, 187 (2017)

work page 2017
[5]

J. L. Miller, Exceptional points make for exceptional sen- sors, Phys. Today 70, 30 (2017)

work page 2017
[6]

K ¨Ozdemir, G Zhao, J Wiersig, L Yang, Exceptional points enhance sensing in an optical micro- 8 cavity, Nature 548, 192 (2017)

W Chen, S. K ¨Ozdemir, G Zhao, J Wiersig, L Yang, Exceptional points enhance sensing in an optical micro- 8 cavity, Nature 548, 192 (2017)

work page 2017
[7]

J. Wiersig, Enhancing the sensitivity of frequency and en- ergy splitting detection by using exceptional points: appli- cation to microcavity sensors for single particle detection, Phys. Rev. Lett. 112, 203901 (2014)

work page 2014
[8]

Hodaei, M.-A

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, M. Khajavikhan, Parity-time- symmetric microring lasers , Science 346, 975 (2014)

work page 2014

Show all 29 references

[9]

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, X. Zhang, Single-mode laser by parity-time symmetry breaking , Sci- ence 346, 972 (2014)

2014
[10]

Hodaei, MA Miri, A

H. Hodaei, MA Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, M. Khajavikhan, Sin- gle mode lasing in transversely multimoded PTsymmet- ric microring resonators Laser Photonics Rev. 10, 494 (2016)

2016
[11]

Lin, et al

Z. Lin, et al. , Unidirectional invisibility induced by PT- symmetric periodic structures , Phys. Rev. Lett. 106, 213901 (2011)

2011
[12]

Peng, et al., Paritytime-symmetric whispering-gallery microcavities, Nat

B. Peng, et al., Paritytime-symmetric whispering-gallery microcavities, Nat. Phys. 10, 394 (2014)

2014
[13]

J Schindler, Z Lin, JM Lee, H Ramezani, FM Ellis, T Kottos, PT -symmetric electronics , J. Phys. A: Math. Theor. 45, 444029 (2012)

2012
[14]

S Assawaworrarit, X Yu, S Fan, Robust wireless power transfer using a nonlinear paritytime-symmetric circuit , Nature 546, 387 (2017)

2017
[15]

H. Li, M. Chitsazi, R. Thomas, F. M. Ellis, and T. Kot- tos,PT -Symmetry and Non- Hermitian Wave Transport in Microwaves and RF Circuits , in Parity-Time Sym- metry and Its Applications , Springer Tracts in Modern Physics 280, Edt. D. Christodoulides and J. Yang (2018)

2018
[17]

Fleury, D

R. Fleury, D. L. Sounas, A. Al´ u, Parity-Time Symme- try in Acoustics: Theory, Devices, and Potential Appli- cations, IEEE Journal of Selected Topics in Quantum Electronics 22, 5000809 (2016)

2016
[18]

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, X. Zhang, Accessing the exceptional points of parity-time symmetric acoustics , Nature Communica- tions 7, 11110 (2016)

2016
[19]

Aur´ egan and V

Y. Aur´ egan and V. Pagneux, PT-Symmetric Scattering in Flow Duct Acoustics , Phys. Rev. Lett. 118, 174301 (2017)

2017
[20]

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang and Y. Xiao, Anti-paritytime symmetry with ﬂying atoms , Nature Phys. 12, 1139 (2016)

2016
[21]

C. Hang, G. Huang, Parity-time symmetry with coherent atomic gases, Advances in Physics 2, 737 (2017)

2017
[22]

Kato, Perturbation theory for linear operators (Springer-Verlag, Berlin, 1966), p.p

T. Kato, Perturbation theory for linear operators (Springer-Verlag, Berlin, 1966), p.p. 63-64

1966
[23]

W. D. Heiss, The physics of exceptional points , J. Phys. A: Math. Theor. 45, 444016 (2012)

2012
[24]

C. M. Bender, S. Boettcher, Real spectra in non- Hermitian Hamiltonians having PT symmetry , Phys. Rev. Lett. 80, 5243 (1998)

1998
[25]

Bender, Making sense of non-Hermitian Hamiltoni- ans, Rep

M. Bender, Making sense of non-Hermitian Hamiltoni- ans, Rep. Prog. Phys. 70, 947 (2007)

2007
[26]

Moiseyev, Non-Hermitian Quantum Mechanics , (Cambridge Univ

N. Moiseyev, Non-Hermitian Quantum Mechanics , (Cambridge Univ. Press, 2011)

2011
[27]

K. F. Graﬀ, Wave motion in elastic solid (Dover Publi- cations, Inc., New York, 1991)

1991
[28]

St¨ ockmann, Quantum Chaos: An Introduction , Cambridge University Press (March 5, 2007)

H.-J. St¨ ockmann, Quantum Chaos: An Introduction , Cambridge University Press (March 5, 2007)

2007
[29]

R. N. Thurston, in Physical Acoustics (Academic Press, New York, 1964), V. I, Part A., p. 89

1964

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

L. Feng, R. El-Ganainy, L. Ge, Non-Hermitian photonics based on paritytime symmetry, Nat. Photonics 11, 752762 (2017)

work page 2017

[2] [2]

R, El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, D. N. Christodoulides, Non- Hermitian Physics and PT symmetry , Nat. Phys. 14, 11 (2018)

work page 2018

[3] [3]

Alu, Exceptional points in optics and pho- tonics, Science 363, 42 (2019)

M Ali-Miri, A. Alu, Exceptional points in optics and pho- tonics, Science 363, 42 (2019)

work page 2019

[4] [4]

Hodaei, A

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points , Nature 548, 187 (2017)

work page 2017

[5] [5]

J. L. Miller, Exceptional points make for exceptional sen- sors, Phys. Today 70, 30 (2017)

work page 2017

[6] [6]

K ¨Ozdemir, G Zhao, J Wiersig, L Yang, Exceptional points enhance sensing in an optical micro- 8 cavity, Nature 548, 192 (2017)

W Chen, S. K ¨Ozdemir, G Zhao, J Wiersig, L Yang, Exceptional points enhance sensing in an optical micro- 8 cavity, Nature 548, 192 (2017)

work page 2017

[7] [7]

J. Wiersig, Enhancing the sensitivity of frequency and en- ergy splitting detection by using exceptional points: appli- cation to microcavity sensors for single particle detection, Phys. Rev. Lett. 112, 203901 (2014)

work page 2014

[8] [8]

Hodaei, M.-A

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, M. Khajavikhan, Parity-time- symmetric microring lasers , Science 346, 975 (2014)

work page 2014