Pith. sign in

REVIEW 3 major objections 4 minor 46 references

Ballistic diffusion vs. damped oscillation of energy in a $\mathcal{PT}$-symmetric quantum kicked harmonic oscillator

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read In a PT-symmetric kicked harmonic oscillator, irrational frequency ratios produce a directed momentum current and ballistic energy growth, while rational ratios produce damped oscillations.

desk verdict The model is inconsistent as written: the harmonic potential cannot be periodic with the kick, so the discrete momentum lattice and the claimed mechanism are not justified; the non-resonant result is not new, and the resonant part is curve-fitting. read the letter →

arxiv 2501.14507 v1 pith:CJ2QXORA submitted 2025-01-24 quant-ph physics.comp-ph

Jian-Zheng Li This is my paper · ORCID
classification quant-phphysics.comp-ph MSC 81Q12 PACS 03.65.-w05.45.Mt
keywords PT-symmetrickickedharmonicoscillatornon-HermitiandrivingballisticdiffusiondirectedcurrentdampedoscillationFloquetdynamicsmomentumlattice
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

The paper asks what happens to a quantum kicked harmonic oscillator when the kicking potential is made PT-symmetric, with a real cosine plus an imaginary sine component. It claims the answer depends only on whether the ratio of the oscillator frequency to the kicking frequency is rational or irrational. For irrational ratios the system develops a linear growth of mean momentum and quadratic growth of energy, i.e. a directed current and ballistic diffusion; for rational ratios it responds like a damped oscillator, with momentum and energy oscillating as damped cosines of a common frequency. This matters because it connects non-Hermitian driving to transport and energy control in a simple, analytically guided model.

What carries the argument

The central object is the Floquet operator $U = U_\omega U_K$ acting on momentum eigenstates $|\phi_m\rangle$ with eigenvalues $m\hbar_{\text{eff}}$. The argument maps the non-resonant problem onto an effective lattice Hamiltonian (Eq. 9) in that momentum basis, where the non-Hermitian term $i\lambda\sin\theta$ becomes asymmetric nearest-neighbour hopping $a_m^\dagger a_{m+1} - a_{m+1}^\dagger a_m$; the nonzero bias of this hopping is what produces the constant force and the ballistic current. The numerical propagation uses a split-step decomposition of $U_\omega$ into alternating momentum and coordinate kicks.

What would settle it

Run the same split-step algorithm with a genuine parabolic potential on an unbounded line (no periodic boundary) and check whether $\langle p \rangle = Gt$ still holds; if the linear momentum growth disappears, the directed current is an artifact of the periodic-coordinate assumption.

Watch

Extended reading notes

Core claim

The central discovery is a dynamical phase dichotomy controlled by the frequency ratio. With an irrational ratio and nonzero imaginary kicking strength, the mean momentum grows as $\langle p \rangle = Gt$, the kinetic energy as $\langle E_k \rangle \approx \tfrac12 G^2 t^2$, the potential energy stays constant, and the total energy grows ballistically; the probability distribution remains a Gaussian moving at constant velocity, so the state behaves as a quasi-classical particle under a constant force. With a rational ratio and sufficiently strong imaginary kicking strength, $\langle p \rangle$, $\langle E_k \rangle$, $\langle E_p \rangle$, and $\langle E \rangle$ all oscillate as damped cosines with the same frequency $\omega_c$, with kinetic and potential energy oscillating in antiphase, so the system acts as an effective damped harmonic oscillator created by resonant coupling between the non-Hermitian driving and the harmonic potential.

Load-bearing premise

The paper relies on treating the harmonic potential as periodic with the same period as the kicking potential, so the coordinate is effectively taken on a circle; for a genuine unbounded parabolic trap this assumption fails, and the entire momentum-lattice picture rests on it.

Editorial extensions

If this is right

  • For irrational frequency ratios, the system supplies a constant momentum kick per pulse, producing a directed current whose growth rate $G$ increases with the imaginary kicking strength $\lambda$ and saturates near $2\pi$.
  • For rational frequency ratios, momentum and total energy oscillate at a common frequency $\omega_c$, with kinetic and potential energy exchanging energy in antiphase, so the system behaves as an effective damped harmonic oscillator.
  • The damped amplitude, phase shift, and saturation value of the oscillations can be tuned through the imaginary part of the kicking strength $\lambda$.
  • The non-resonant regime reproduces, in the harmonic oscillator setting, the ballistic energy growth previously seen in the non-Hermitian kicked rotor, pointing to a common mechanism.
  • The behaviour is tied to nearest-neighbour hopping on the momentum lattice: the non-Hermitian term biases the hopping and generates the current.

Reading between the lines

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

  • If the directed current is real, the same PT-symmetric kicking scheme could act as a ratchet for cold atoms in a harmonic trap, with the rational/irrational frequency ratio as a control switch between drift and ringing.
  • The mapping to nearest-neighbour hopping suggests the ballistic current should survive for other kicking potentials with a nonzero imaginary Fourier component, so the dichotomy may be generic among PT-symmetric Floquet drives.
  • The paper studies only one irrational and one rational ratio; a systematic scan of the frequency parameter would show whether the damped-oscillation frequency $\omega_c$ follows a simple function of the rational ratio, a testable prediction the paper leaves implicit.
  • Because the state is renormalized after each kick, the reported observables describe the normalized dynamics; without that renormalization the norm growth would dominate, and the claimed damped oscillation could be an artifact of the renormalization rather than intrinsic physics.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript numerically studies a PT-symmetric kicked harmonic oscillator defined by Eq. (1). For the non-resonant frequency ratio η=2π/e² it reports linear growth of mean momentum, ⟨p⟩=Gt, and ballistic energy growth, ⟨E⟩≈(1/2)G²t²+C, together with Gaussian wave packets; for the resonant ratio η=2π it reports damped cosine oscillations of momentum and energy with a common frequency ω_c, described by Eqs. (11)-(14). The paper attributes the non-resonant behavior to nearest-neighbor hopping on a discrete momentum lattice, as expressed in Eq. (9), and the resonant behavior to resonant coupling between the non-Hermitian driving and the harmonic oscillator.

Significance. If the model were well defined and the fitted laws were robust, the distinction between irrational and rational frequency ratios in a non-Hermitian kicked oscillator would be a useful addition to the Floquet literature. The paper does provide explicit fitting functions and a qualitative wave-packet characterization, which are in principle falsifiable. However, the central modeling assertion is algebraically incorrect, and the main quantitative results are fitted functions rather than derived predictions. As written, the claimed phenomena are therefore not established, and the paper's central contribution is not sound.

major comments (3)
  1. [Sec. II, after Eq. (1)] The statement that the harmonic potential is periodic, Hp(X+b)=Hp(X) for Hp=Mω²X²/2, is false for ω≠0. This periodicity is load-bearing: it justifies the compact coordinate θ=2πX/b, the discrete momentum eigenbasis p|φ_m⟩=mħ_eff|φ_m⟩, and the nearest-neighbor hopping picture in Eq. (9). If θ is an unbounded real coordinate, the momentum spectrum is continuous and the expansion used in the paper is invalid; if θ is forced to be periodic, the term η²θ²/2 is discontinuous at θ=±π and the split-step Fourier operator in Eq. (6) does not evolve the stated Hamiltonian. The numerical results in Figs. 1-5 are therefore obtained from a model that is not well defined as written.
  2. [Sec. III B, Eqs. (11)-(14)] The damped cosine laws are not derived from the Hamiltonian; they are the functions used to fit the numerics, with adjustable saturation values, amplitudes, decay times τ, frequency ω_c, and the time-dependent phase shift tc=t0+D exp(γt). The claim that ⟨p⟩, ⟨E_k⟩, and ⟨E_p⟩ share one frequency is enforced by choosing the same ω_c=4π/15 in the fits rather than demonstrated by the dynamics, and no goodness-of-fit measures or parameter uncertainties are reported. As written, these equations are a phenomenological description and cannot support the conclusion that the system behaves as a damped harmonic oscillator with a fixed frequency.
  3. [Sec. III A, Eq. (7) and Fig. 1] The linear growth law ⟨p⟩=Gt and the ballistic law ⟨E_k⟩≈(1/2)G²t² rest on a fitted slope G whose values are not tabulated and for which no uncertainty is given. The additional claim that the growth rate G increases to 2π with increasing λ is not supported by any quantitative analysis in the text or by a dedicated figure. Without such support, the non-resonant result is a curve fit rather than a quantitative prediction.
minor comments (4)
  1. [Section heading and Introduction] The heading 'RESULTS AND DISSCUSION' contains a typo, and 'superlattices' is misspelled as 'supper-lattices' in the Introduction.
  2. [Sec. II, after Eq. (6)] The density matrix defined as ρ=(1/N)Tr(|ψ⟩⟨ψ|) is not a standard notation; if the intent is normalization after each kick, write ρ=|ψ⟩⟨ψ|/⟨ψ|ψ⟩ and define N accordingly.
  3. [Sec. II, Eq. (6)] The split-step parameter N is not specified, and no convergence study is reported; since Uω is obtained by a Trotter-type product, the error from finite N should be quantified.
  4. [Sec. III, figure captions] The figure captions give the fitted functions but omit the numerical values of many constants (e.g., t0, D, ps, Eks, Eps, A1, A2) and the fitting ranges, which makes it impossible to reproduce the fits from the text alone.

Circularity Check

4 steps flagged · score 7.0 of 10

The paper's central laws—⟨p⟩=Gt, the resonant damped cosines, and their identical frequencies—are the fitted curves with parameters chosen per data set, not consequences derived from the Hamiltonian; the nearest-neighbor-hopping mechanism is imported from the QKR model rather than derived.

  1. fitted input called prediction [Sec. III A, Eqs. (7)-(8) and Fig. 1 caption]
    "The momentum of the system grows linearly with time, ⟨p⟩ =Gt, (7) which is the characteristic feature of DC [e.g., λ = 1 in Fig. 1(a)]. Moreover, the growth rate G increases to 2π with the increase of non-Hermitian driving strength λ. Meanwhile, the kinetic energy increases with a ballistic function of time [e.g., λ = 1 in Fig. 1(b)], i.e., ⟨Ek⟩ ≈ 1/2G2t2."

    The caption to Fig. 1 states that the red lines are fitting functions. Thus G in Eq. (7) is the fitted slope of the ⟨p⟩ data, not a parameter obtained from the Hamiltonian or from the Floquet operator. The kinetic-energy law ⟨Ek⟩ ≈ 1/2 G^2 t^2 then reuses that same fitted G, and Eq. (8) repeats it for the total energy. The claimed coexistence of directed current and ballistic diffusion is therefore a restatement of the fit rather than an independently predicted consequence of the model.

  2. fitted input called prediction [Sec. III B, Eq. (11) and Fig. 3 caption]
    "When the strength of non-Hermitian driving increases sufficiently, e.g., λ = 0.5 [see Fig. 3(c)], the momentum in the system oscillates as a damped cosine function, ⟨p⟩ =ps −pam(t) · cos [ω c(t −tc)] , (11) with the saturation momentum ps, the oscillation frequency ω c, the phase shift tc =t0 +D · exp(γt), and the damped amplitude pam(t) ∝ exp(− t τ )."

    Eq. (11) is the analytic form of the red curve drawn through the numerical data, with ps, pam(t), ωc, and tc all adjusted per run; for example, the Fig. 3 caption fixes pam(t)=16.5 exp(−t/66), ωc=4π/15, and tc=t0+D exp(0.01t), and uses a different pam(t) for λ=1. No derivation of the damped-cosine form from Uω, UK, or Eqs. (2)-(6) is given. The statement that the momentum 'oscillates as a damped cosine function' is therefore equivalent to saying that a damped cosine was fitted to the simulation output.

2 more flagged steps
  1. self definitional [Sec. III B, text after Eq. (12) and after Eq. (14)]
    "Moreover, the oscillation frequency ω c and the phase shift tc are identical to those of the corresponding momentum ⟨p⟩. ... Here the oscillation frequency ω c and the phase shift tc are identical to those of the ⟨Ek⟩ and ⟨p⟩."

    The identical-frequency and identical-phase claims are asserted, but the figure captions for ⟨p⟩, ⟨Ek⟩, and ⟨Ep⟩ all insert the same fitted values ωc=4π/15 and tc=t0+D exp(0.01t) into Eqs. (11)-(13), and Eq. (14) then declares the total energy to have the same ωc and tc. Because the same fitted constants are used in every fit by construction, the equality of frequencies and phase shifts is not an emergent dynamical discovery; it is enforced by the fitting ansatz.

  2. renaming known result [Sec. III A, text before Eq. (9)]
    "We find obviously that, except the constant ⟨Ep⟩, the evolutions of ⟨p⟩, ⟨Ek⟩, and M in this system are similar to those of the QKR system [34, 35] respectively in the non-Hermitian phase. According to the mapping method in the references [40, 41], therefore, the effective Hamiltonian of the PTQKHO system under the non-resonant conditions is written as ..."

    Eq. (9) is not derived from the PTQKHO Hamiltonian (2); it is the QKR nearest-neighbor-hopping Hamiltonian transplanted by declaring the dynamics 'similar to those of the QKR system' and invoking a mapping method, with the paper's own prior work [35] cited for the similarity. The later conclusion that directed current and ballistic diffusion 'originate from the nearest-neighbor hopping between the momentum eigenstates' then re-reads the hopping terms that were inserted into Eq. (9). The mechanism is an imported ansatz presented as an explanation, not a consequence obtained from the stated Hamiltonian.

full rationale

The numerical simulations are self-contained and may be reproducible, but the paper's central analytical claims are not derived from those simulations; they are the fitting functions. Under non-resonance, ⟨p⟩=Gt is a fit to the red line, and ⟨Ek⟩≈1/2 G^2 t^2 reuses that fitted G. Under resonance, Eqs. (11)-(14) are the damped-cosine fitting forms with amplitudes, phase shifts, and ωc chosen per data set, and the statement that the frequencies are identical is enforced by inserting ωc=4π/15 and tc=t0+D exp(0.01t) into every fit. These are fitted inputs presented as dynamical laws, so the central dichotomy (ballistic diffusion versus damped oscillation) reduces by construction to curve fitting. The nearest-neighbor-hopping explanation (Eq. 9) is imported from the QKR model by analogy and self-citation rather than derived, so the mechanism claim is not independent either. Separately, the model as written is internally inconsistent: Hp=Mω^2 X^2/2 cannot be periodic with the kick potential (Hp(X+b)=Hp(X) is false for ω≠0), which undermines the discrete momentum basis used in Sec. II and in Eq. (9). I regard that as a correctness issue rather than a circularity, and it does not by itself raise the circularity score. Overall score 7: partial circularity because the headline behaviors are the fitted functions themselves.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the compact momentum basis, the state renormalization convention, and a set of fitted functional forms. The most serious item is the asserted periodicity of the harmonic potential, which is algebraically false as written. No new particles or fields are introduced; the 'constant force' f=G in Eq. (10) is just the time derivative of the fitted momentum and is not a new physical entity.

free parameters (7)
  • G (non-resonant momentum growth rate) = varies with λ; approaches 2π for large λ
    Fitted slope of ⟨p⟩ vs t, appears in Eq. (7) and Eq. (10); no analytical expression from the Hamiltonian is given.
  • ω_c (resonant oscillation frequency) = 4π/15
    Chosen to match oscillations in ⟨p⟩, ⟨E_k⟩, ⟨E_p⟩; used identically for all three observables, so the 'same frequency' claim is partly imposed by the fit.
  • τ (damping time) = 66 (λ=0.5), 600 (λ=1)
    Fitted from data; controls decay of momentum and energy oscillations in Eqs. (11)-(14).
  • Damped amplitude prefactors and power-law factors = pam=16.5 exp(-t/66) for λ=0.5; pam=30 exp(-t/600)(2/πt)^(1/4) for λ=1
    Ad hoc amplitude functions chosen to match the numerical figures.
  • Phase shift parameters t0, D, γ = γ=0.01; t0 and D left unspecified
    The time-dependent phase shift tc=t0+D exp(γt) has no physical explanation and is fitted to delay the oscillations.
  • Energy saturation and decay constants (Eks, A1, A2, μ1, μ2, Eps, B1, B2, μ, amplitude prefactors) = e.g., A1/A2, μ1=323, μ2=2730, Ekm=330/550, Epm=50/80, etc.
    Numerous constants in Eqs. (12)-(14) and figure captions are all fitted to make the damped-cosine curves pass through the numerical data.
  • Wave-packet width exponent α and prefactor β (M=βt^α) = α≈0.8 for λ=0; α unspecified for λ≠0
    Fitted power-law exponents for the non-resonant wave-packet width in Fig. 1(d).
assumptions (5)
  • standard math Momentum eigenstates |φ_m⟩ with integer m form a complete basis and [θ,p]=iℏ_eff.
    Used throughout; standard for quantum kicked rotor on a circle, but requires θ to be a compact angular coordinate.
  • domain assumption The harmonic potential is periodic with the same period as the kicking potential, Hp(X+b)=Hp(X).
    Stated in Sec. II after Eq. (1); false for a real harmonic potential, but necessary for the discrete momentum basis and nearest-neighbor hopping model.
  • domain assumption Normalizing the state to unit norm after each kick gives the physically relevant observables.
    Sec. II density matrix definition; a standard but not uniquely justified choice in non-Hermitian dynamics.
  • ad hoc to paper The damped cosine ansatz with a common frequency ω_c and time-dependent phase shift tc=t0+D exp(γt) describes the resonant dynamics.
    Eqs. (11)-(14); the functional forms are assumed, not derived from the Hamiltonian.
  • domain assumption The mapping method of refs [40,41] applies, so the effective Hamiltonian under non-resonant conditions can be written as the tight-binding form Eq. (9).
    Sec. III.A; no derivation of the mapping conditions for this specific system is provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Ballistic diffusion vs. damped oscillation of energy in a $\mathcal{PT}$-symmetric quantum kicked harmonic oscillator." pith.science (2026). https://pith.science/paper/CJ2QXORA

@misc{pith2026250114507,
  author       = {Pith},
  title        = {Pith review of: Ballistic diffusion vs. damped oscillation of energy in a $\mathcalPT$-symmetric quantum kicked harmonic oscillator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CJ2QXORA}},
  note         = {Machine review of arXiv:2501.14507}
}
abstract

We numerically study the quantum dynamics of a $\mathcal{PT}$-symmetric kicked harmonic oscillator. We observe that directed current of momentum and ballistic diffusion of energy coexist under the non-resonant conditions, whereas both the momentum and energy oscillate as damped cosine functions with identical frequencies under the resonant conditions. The research shows that the directed current of momentum and ballistic diffusion of energy arise from nearest-neighbor hopping between momentum eigenstates with the non-Hermitian driving, while the damped oscillations of momentum and energy originate from resonant coupling between the non-Hermitian driving and the harmonic oscillator. Our findings indicate that the non-Hermiticity and the frequency characteristic of this system collectively result in these distinctive dynamical behaviors.

Figures

Figures reproduced from arXiv: 2501.14507 by the authors.

Figure 2
Figure 2. FIG. 2: (a) Probability density distribution [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1: Time evolution of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 4
Figure 4. show the dynamical behaviors of kinetic energy in the system. In the Hermitian system (i.e., λ = 0) [see [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5: Time evolution of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 38 canonical work pages

  1. [1]

    Huang, A

    Z. Huang, A. Clerk, and I. Martin, Phys. Rev. Lett. 126, 100601 (2021). DOI: https://doi.org/10.1103/PhysRevLett.126.100601

    work page doi:10.1103/physrevlett.126.100601 2021

  2. [2]

    Fleckenstein, and M

    C. Fleckenstein, and M. Bukov, Phys. Rev. B 103, L140302 (2021). DOI: https://doi.org/10.1103/PhysRevB.103.L140302

    doi:10.1103/physrevb.103.l140302 2021

  3. [3]

    C. Ma, Y. S. Wang, and J. H. An, Phys. Rev. A 97, 023808 (2018). DOI: https://doi.org/10.1103/PhysRevA.97.023808

    work page doi:10.1103/physreva.97.023808 2018

  4. [4]

    Zhang, and G

    Y. Zhang, and G. Chen, Phys. Rev. A 107, 023520 (2023). DOI: https://doi.org/10.1103/PhysRevA.107.023520

    work page doi:10.1103/physreva.107.023520 2023

  5. [5]

    D ´ ıaz-Fern´ andez, E

    A. D ´ ıaz-Fern´ andez, E. D ´ ıaz, A. G´ omez-Le´ on, G. Platero, and F. Dom ´ ınguez-Adame,Phys. Rev. B 100, 075412 (2019). DOI: https://doi.org/10.1103/PhysRevB.100.075412

    doi:10.1103/physrevb.100.075412 2019

  6. [6]

    A. F. Adiyatullin, L. K. Upreti, C. Lechevalier, C. Evain , F. Copie, P. Suret, S. Randoux, P. Delplace, and A. Amo, Phys. Rev. Lett. 130, 056901 (2023). DOI: https://doi.org/10.1103/PhysRevLett.130.056901

    work page doi:10.1103/physrevlett.130.056901 2023

  7. [7]

    Reitter, J

    M. Reitter, J. N¨ ager, K. Wintersperger, C. Str¨ ater, I. Bloch, A. Eckardt, and U. Schneider, Phys. Rev. Lett. 119, 200402 (2017). DOI: https://doi.org/10.1103/PhysRevLett.119.200402

    work page doi:10.1103/physrevlett.119.200402 2017

  8. [8]

    J. H. S. Toh, K. C. McCormick, X. Tang, Y. Su, X. W. Luo, C. Zhang, and S. Gupta, Nat. Phys. 18, 1297 (2022). DOI: https://doi.org/10.1038/s41567-022-01721-w

    doi:10.1038/s41567-022-01721-w 2022

Show all 46 references
  1. [9]

    S. S. Maurya, J. B. Kannan, K. Patel, P. Dutta, K. Biswas, J. Mangaonkar, M. S. Santhanam, and U. D. Rapol, Phys. Rev. E 106, 034207 (2022). DOI: https://doi.org/10.1103/PhysRevE.106.034207

    2022 doi

  2. [10]

    Petiziol, and A

    F. Petiziol, and A. Eckardt, Phys. Rev. Lett. 129, 233601 (2022). DOI: https://doi.org/10.1103/PhysRevLett.129.233601

    2022 doi

  3. [11]

    Su, and M

    K. Su, and M. J. Lawler, Phys. Rev. B 106, 214310 (2022). DOI: https://doi.org/10.1103/PhysRevB.106.214310

    2022 doi

  4. [12]

    G. A. Kells, J. Twamley, and D. M. Heffernan, Phys. Rev. E 70, 015203(R) (2004). DOI: https://doi.org/10.1103/PhysRevE.70.015203

    2004 doi

  5. [13]

    E. G. Arrais, J. S. Sales, and N. G. D. Almeida, J. Phys. B: At. Mol. Opt. Phys. 49, 165501 (2016). DOI: 10.1088/0953-4075/49/16/165501

    2016 doi

  6. [14]

    Wimberger, R

    S. Wimberger, R. Mannella, O. Morsch, and E. Ari- mondo, Phys. Rev. Lett. 94, 130404 (2005). DOI: https://doi.org/10.1103/PhysRevLett.94.130404

    2005 doi

  7. [15]

    M. S. Cust´ odio, W. T. Strunz, M. W. Beims, Physica A 486, 337 (2017). DOI: https://doi.org/10.1016/j.physa.2017.05.094

    2017 doi

  8. [16]

    Dana, Phys

    I. Dana, Phys. Lett. A 197, 413 (1995). DOI: https://doi.org/10.1016/0375-9601(94)01007-H

    1995 doi

  9. [17]

    Gupta, and M

    S. Gupta, and M. Bandyopadhyay, Phys. Rev. E 84, 041133 (2011). DOI: https://doi.org/10.1103/PhysRevE.84.041133

    2011 doi

  10. [18]

    T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bu- jkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001). DOI: https://doi.org/10.1103/PhysRevLett.87.046803

    2001 doi

  11. [19]

    N. D. Varikuti, A. Sahu, A. Lakshminarayan, and V. Madhok, Phys. Rev. E 109, 014209 (2024). DOI: https://doi.org/10.1103/PhysRevE.109.014209

    2024 doi

  12. [20]

    A. A. Chernikov, R. Z. Sagdeev, D. A. Usikov, M. Y. Zakharov, and G. M. Zaslavsky, Nature 326, 559 (1987). DOI: https://doi.org/10.1038/326559a0

    1987 doi

  13. [21]

    R. E. Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Na- ture Phys. 14, 11 (2018). DOI: https://doi.org/10.1038/nphys4323

    2018 doi

  14. [22]

    Ashida, Z

    Y. Ashida, Z. P. Gong, and M. Ueda, Adv. Phys. 69, 249 (2020). DOI:https://doi.org/10.1080/00018732.2021.1876991

    2020

  15. [23]

    Franca, V

    S. Franca, V. K¨ onye, F. Hassler, J. V. D. Brink, and C. Fulga, Phys. Rev. Lett. 129, 086601 (2022). DOI: https://doi.org/10.1103/PhysRevLett.129.086601

    2022 doi

  16. [24]

    K. Ding, Z. Q. Zhang, and C. T. Chan, Phys. Rev. B 92, 235310 (2015). DOI: https://doi.org/10.1103/PhysRevB.92.235310

    2015 doi

  17. [25]

    M. L. N. Chen, L. J. Jiang, S. Zhang, R. Zhao, Z. Lan, and W. E. I. Sha, Phys. Rev. A 104, 033501 (2021). DOI: https://doi.org/10.1103/PhysRevA.104.033501

    2021 doi

  18. [26]

    Zhu, and J

    W. Zhu, and J. Gong, Phys. Rev. B 108, 035406 (2023). DOI: https://doi.org/10.1103/PhysRevB.108.035406

    2023 doi

  19. [27]

    Kreibich, J

    M. Kreibich, J. Main, H. Cartarius, and G. Wunner, Phys. Rev. A 90, 033630 (2014). DOI: https://doi.org/10.1103/PhysRevA.90.033630

    2014 doi

  20. [28]

    P. He, H. T. Ding, and S. L. Zhu, Phys. Rev. A 103, 043329 (2021). DOI: https://doi.org/10.1103/PhysRevA.103.043329

    2021 doi

  21. [29]

    Zhang, C

    X. Zhang, C. Liu, Z. Zhu, and Y. Zhang, Phys. Rev. B 109, 184426 (2024). DOI: https://doi.org/10.1103/PhysRevB.109.184426

    2024 doi

  22. [30]

    K. Shi, L. Qiao, Z. Zheng, and W. Zhang, Phys. Rev. A 110, 022222 (2024). DOI: https://doi.org/10.1103/PhysRevA.110.022222

    2024 doi

  23. [31]

    K. Q. Huang, W. L. Zhao, and Z. Li, Phys. Rev. A 104, 052405 (2021). DOI: https://doi.org/10.1103/PhysRevA.104.052405

    2021 doi

  24. [32]

    K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008). DOI: https://doi.org/10.1103/PhysRevLett.100.103904

    2008 doi

  25. [33]

    C. E. R¨ uter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature Phys. 6, 192 (2010). DOI: https://doi.org/10.1038/nphys1515

    2010 doi

  26. [34]

    W. L. Zhao, J. Wang, X. Wang, and P. Tong, Phys. Rev. E 99, 042201 (2019). DOI: https://doi.org/10.1103/PhysRevE.99.042201

    2019 doi

  27. [35]

    J. Z. Li, W. L. Zhao, and J. Liu, Phys. Rev. A 107, 032208 (2023). DOI: https://doi.org/10.1103/PhysRevA.107.032208

    2023 doi

  28. [36]

    Lemari´ e, J

    G. Lemari´ e, J. Chab´ e, P. Szriftgiser, J. C. Garreau, B . Gr´ emaud, and D. Delande, Phys. Rev. A 80, 043626 (2009). DOI: https://doi.org/10.1103/PhysRevA.80.043626

    2009 doi

  29. [37]

    W. L. Zhao, L. B. Fu, and J. Liu, Phys. Rev. E 90, 022907 (2014). DOI: https://doi.org/10.1103/PhysRevE.90.022907

    2014 doi

  30. [38]

    Cherroret, B

    N. Cherroret, B. Vermersch, J. C. Garreau, and D. De- lande, Phys. Rev. Lett. 112, 170603 (2014). DOI: https://doi.org/10.1103/PhysRevLett.112.170603

    2014 doi

  31. [39]

    Ermann, and D

    L. Ermann, and D. L. Shepelyansky, J. Phys. A-Math. Theor. 47, 335101 (2014). 7 DOI: 10.1088/1751-8113/47/33/335101

    2014 doi

  32. [40]

    Russomanno, M

    A. Russomanno, M. Fava, and R. Fazio, Phys. Rev. B 103, 224301 (2021). DOI: https://doi.org/10.1103/PhysRevB.103.224301

    2021 doi

  33. [41]

    F. M. Surace, A. Russomanno, M. Dalmonte, A. Silva, R. Fazio, and F. Iemini, Phys. Rev. B 99, 104303 (2019). DOI: https://doi.org/10.1103/PhysRevB.99.104303

    2019 doi

  34. [42]

    F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, Eur. J. Phys. 21, 27 (2000). DOI:https://doi.org/10.1088/0143-0807/21/1/305

    2000 doi

  35. [43]

    Gadway, J

    B. Gadway, J. Reeves, L. Krinner, and D. Schneble, Phys. Rev. Lett. 110, 190401 (2013). DOI: https://doi.org/10.1103/PhysRevLett.110.190401

    2013 doi

  36. [44]

    D. Xie, T. S. Deng, T. Xiao, W. Gou, T. Chen, W. Yi, and B. Yan, Phys. Rev. Lett. 124, 050502 (2020). DOI: https://doi.org/10.1103/PhysRevLett.124.050502

    2020 doi

  37. [45]

    Jahanpanah, Commun

    J. Jahanpanah, Commun. Nonlinear SCI 128, 107614 (2024). DOI: https://doi.org/10.1016/j.cnsns.2023.107614

    2024

  38. [46]

    G. Ji, K. Bose, A. C. Balram, and B. Yang, Phys. Rev. B 110, 075113 (2024). DOI: https://doi.org/10.1103/PhysRevB.110.075113

    2024 doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.