Analytical Study of the Non-Hermitian Semiclassical Rabi Model

Liwei Duan; Qing-Hu Chen; Yibo Liu

REVIEW 2 major objections 5 minor 1 cited by

Analytical Study of the Non-Hermitian Semiclassical Rabi Model

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read A single similarity transformation reduces the PT-symmetric semiclassical Rabi model to an exactly solvable two-level Hamiltonian, from which the PT phase boundary, dynamics, and Bloch-Siegert shift all follow.

desk verdict A clean analytical approximation for the PT-symmetric Rabi model, honest about its limits, and worth referee time despite an uncontrolled truncation. read the letter →

arxiv 2412.02918 v2 pith:MD6IUY2C submitted 2024-12-04 quant-ph

Yibo Liu , Liwei Duan , Qing-Hu Chen This is my paper · ORCID

classification quant-ph

keywords non-HermitianRabimodelPTsymmetryexceptionalpointsrotating-waveapproximationsimilaritytransformationFloquettheoryBloch-Siegertshifttwo-levelatom

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 claims that the parity-time (PT) symmetric semiclassical Rabi model, a two-level atom driven by a classical field with purely imaginary coupling, can be solved analytically by one carefully chosen similarity transformation. The transformation turns the time-dependent Hamiltonian into an effective rotating-wave approximation with renormalized atomic frequency and coupling, and its two eigenvalues give the PT phase boundary, the dynamics of the excited-state population, and the Bloch-Siegert shift. If correct, the scheme provides a concise closed-form description of the main physics of this open atom-field system in the single-photon-resonance regime, and it locates the secondary PT-broken phases that appear at higher atomic frequencies. The central limit is that all harmonics beyond the first are discarded, so the analytical eigenvalues cannot describe the interiors of those secondary broken phases.

What carries the argument

The central object is the similarity transformation with generator $\hat S(t)=\frac{A}{2\omega}\sin(\omega t)\,\alpha\hat\sigma_x$. Expanding the transformed Hamiltonian in Bessel functions and choosing $\alpha$ through Eq. (9) cancels the counter-rotating $\sigma_y$ and $\sigma_x$ single-harmonic terms, leaving a zero-frequency $\sigma_z$ term and one rotating $\sigma_\pm$ term; a rotating frame then gives the time-independent non-Hermitian two-level model $\tilde H$. The paper also introduces a Floquet parity operator $\hat\Pi=-\hat\sigma_z(-1)^{\hat G}$ that commutes with the Floquet Hamiltonian, splitting the Hilbert space into even and odd parity subspaces and explaining why same-parity quasi-energies cannot cross, which forces the secondary PT-broken phases.

What would settle it

Take the exact Floquet matrix in Eq. (24) at $\Delta/\omega=2.5$ and locate the first exceptional point by tracking the imaginary part of the two lowest odd-parity quasi-energies; the paper's claim requires that coupling to coincide with the root of $\tilde\Delta^2=\tilde A^2/4$, so a mismatch beyond numerical resolution would invalidate the reduction. A second check is to add the smallest neglected Bessel term $J_2$ to the effective Hamiltonian and see whether the analytical dynamics at point B move toward the exact curve.

Watch

Extended reading notes

Core claim

The paper claims that the Hamiltonian $\hat H(t)=\frac{\Delta}{2}\hat\sigma_z + \frac{iA}{2}\cos\omega t\,\hat\sigma_x$ can be reduced by a single similarity transformation $\hat S(t)=\frac{A}{2\omega}\sin(\omega t)\,\alpha\hat\sigma_x$, with $\alpha$ fixed by $\Delta I_1(A\alpha/\omega)=\frac{A}{2}(1-\alpha)$, to the effective rotating-wave Hamiltonian $\tilde H=\frac{\tilde\Delta}{2}\hat\sigma_z + \frac{i\tilde A}{4}\hat\sigma_x$, where $\tilde\Delta=\Delta I_0(A\alpha/\omega)-\omega$ and $\tilde A=2A(1-\alpha)$. Its eigenvalues $\varepsilon_\pm=\pm\frac12\sqrt{\tilde\Delta^2-\tilde A^2/4}+\frac{\omega}{2}$ reproduce the numerically exact Floquet quasi-energies up to $A\simeq 6\omega$ at $\Delta=2.5\omega$, and the exceptional-point condition $\tilde\Delta^2=\tilde A^2/4$ gives the primary PT-broken phase boundary, accurate for $\Delta/\omega$ up to about 3. The same formula, through the same-parity crossing condition $\sqrt{\tilde\Delta^2-\tilde A^2/4}=2n\omega$, predicts the location of the secondary PT-broken phases that appear for $\Delta>3\omega$, and it yields a closed-form Bloch-Siegert shift whose $A^2$ term matches perturbation theory.

Load-bearing premise

The load-bearing premise is that all Bessel harmonic terms of order $n\ge 2$ may be discarded after the single similarity transformation, an approximation validated only by comparison with numerics, not by a controlled error estimate.

Editorial extensions

If this is right

The primary PT-broken phase boundary is available in closed form from $\tilde\Delta^2=\tilde A^2/4$, so exceptional points can be located without Floquet diagonalization for $\Delta/\omega\lesssim 3$.
The analytical Rabi frequency $\Omega_R=\frac12\sqrt{\tilde\Delta^2-\tilde A^2/4}$ gives excited-state population dynamics that match exact numerics in the PT-unbroken and PT-broken single-photon regimes and reproduce the dominant Fourier peaks in the multi-frequency oscillations.
The same-parity crossing condition $\sqrt{\tilde\Delta^2-\tilde A^2/4}=2n\omega$ predicts the positions of the second and third PT-broken phases for $\Delta/\omega>3$, giving a phase diagram accurate for $A/\omega<2$ except for the small second-phase region.
The Bloch-Siegert shift, derived from $\partial\Omega_R^2/\partial\Delta=0$, agrees with exact numerics up to $A/\omega=5$ and has the expansion $\Delta_{\mathrm{res}}=\omega+\frac{A^2}{16\omega}-\frac{5A^4}{1024\omega^3}+O(A^6)$.
The Floquet parity operator explains the emergence of PT-broken phases: quasi-energies sharing one parity cannot cross, so when the analytic levels cross a complex pair must appear.

Reading between the lines

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

The same iterative strategy could be pushed further: adding a second similarity transformation to absorb the $J_2$ term would likely close the gap at point B and could turn the qualitative prediction of the secondary broken phases into a quantitative description; that is an extension, not something the paper demonstrates.
Because the effective model is just a detuned non-Hermitian two-level system, the method should transfer to other periodically driven open two-level systems whose coupling is purely imaginary, such as dissipative spin-field models with multiple drives, as long as the first harmonic dominates.
The near-identity of the non-Hermitian Bloch-Siegert shift with the Hermitian one suggests that low-order spectral shifts in PT-symmetric versions of known atom-field models may be obtainable from the Hermitian formulas with $A\to iA$; testing this against exact Floquet data at higher order would clarify whether the resemblance persists.
A direct computational check of the truncation assumption would be to evaluate the norm of the neglected $H'_2$ term relative to the effective gap; the paper's own numerics suggest it is small at $\Delta/\omega=2.5$ but not at $\Delta/\omega=3.5$, which would explain the observed deviations.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Short version: this is a genuinely useful analytical approximation for the PT-symmetric semiclassical Rabi model, and the authors are honest about where it fails. The single similarity transformation with self-consistent α is new to me for this model; it produces an effective RWA Hamiltonian whose eigenvalues give a phase boundary that tracks the Floquet numerics well up to Δ≈3ω, and the dynamics expression (19) captures the multi-frequency oscillations in the unbroken phase. The Bloch-Siegert shift formula (22) is a nice bonus — it improves on the perturbative result and matches numerics to A/ω=5.

The load-bearing assumption is the neglect of H'_2, all Bessel components with n≥2. There is no small parameter or error bound attached to that truncation. The paper relies on the numerical comparisons, which are convincing in the primary region, but the claimed upper validity edge Δ≈3ω is exactly where the 3ω harmonic in H'_2 starts to resonate. That is not a coincidence, and the paper does not address it. The consequences show up in their own data: at point B (Δ=3.5ω, A=3ω) the analytical dynamics deviate strongly, and the secondary PT-broken phases are only predicted by the crossing condition (13), not described. The authors state these limitations clearly, so this is not an overclaiming problem; it is an incompleteness in the theory.

The derivation itself is algebraically consistent; Eq. (9) fixes α by self-consistency rather than by fitting to the target phase diagram, and the Floquet diagonalization is an independent benchmark. That keeps circularity low. The Floquet parity discussion is a helpful way to understand why the secondary phases appear, though it is more explanatory than predictive.

Who is this for? People working on driven non-Hermitian two-level systems and Floquet engineering. It won't change the world, but it gives a compact tool and a clear physical picture for the main resonance regime.

Worth a serious referee. A referee should ask for a discussion of the truncation validity — a crude estimate of the neglected term's effect near Δ≈3ω, or a second similarity transformation step — but the paper deserves review and likely publication after moderate revision. I'd cite it if I worked in this area.

Referee Report

2 major / 5 minor

Summary. The manuscript studies the PT-symmetric non-Hermitian semiclassical Rabi model H(t)=Δ/2 σz + iA/2 cos(ωt) σx. The central idea is to apply a single similarity transformation generated by S(t)=(A/2ω) sin(ωt) α σx with a self-consistently chosen α obeying Eq. (9). After neglecting the higher-harmonic part H'_2 in Eq. (6), the transformed Hamiltonian reduces to an effective rotating-wave form, and a further unitary rotation gives the time-independent non-Hermitian Hamiltonian (10). The resulting quasi-energy eigenvalues (11) are used to derive the primary PT-broken phase boundary, the lines of higher-order real-level crossings (Eq. (13)), an analytical time-evolution formula (19), and the Bloch-Siegert shift (22). All results are benchmarked against Floquet exact diagonalization.

Significance. If the approximation is reliable, the paper provides a compact analytical description of a time-dependent non-Hermitian two-level system, including the phase boundary, Rabi oscillations, and the Bloch-Siegert shift. The method is not fitted to the target phase diagram: α is fixed by the self-consistent equation (9), and the Floquet diagonalization serves as an independent numerical check. The Floquet parity operator picture gives a useful symmetry-based explanation of the emergence of PT-broken phases. The authors are candid about the limits of the scheme, noting that the secondary PT-broken phases and the dynamics at Δ=3.5ω are not quantitatively captured. The main remaining weakness is that the central rotating-wave truncation is uncontrolled, and the paper would be strengthened by a quantitative discussion of the neglected harmonics.

major comments (2)

[Section II, Eqs. (6)-(11) and Fig. 3(b)] The derivation of the effective Hamiltonian and the eigenvalues (11) rests on the complete neglect of H'_2 in Eq. (6), but the manuscript provides no quantitative estimate of the dropped Bessel components. This is a load-bearing gap because H'_2 contains the 3ω harmonic, which becomes resonant at Δ≈3ω, exactly the upper end of the claimed validity range of the primary phase boundary. The deviations visible in Fig. 2(b) and the poor agreement of the dynamics at point B in Fig. 3(b) indicate that these neglected terms are not always harmless. Please add a quantitative estimate of the leading neglected term (for example, the amplitude of the σy component at frequency 3ω, Δ J_3(iAα/ω), relative to the retained energy scale Δ̃) and discuss its effect on the exceptional-point condition in the region Δ/ω<3. Without this discussion, Eq. (11) is an uncontrolled approximation, even if the selected numerical comparisons in Figs. 1 and 2 are encouraging.
[Section IV, Eq. (13) and Fig. 1] The paper refers to the red lines obtained from Eq. (13) as "PT-broken lines" and later claims that the analytical scheme produces "a highly accurate, nearly exact phase diagram within A/ω<2 for arbitrary Δ/ω, except for the small region of the second PT-broken phase." However, Eq. (13) is the condition for two real quasi-energies to coincide; inside the actual PT-broken region the quasi-energies are complex, so this condition determines only the locus where the real parts become equal, which lies near the center of the secondary broken region and does not define its boundary. The statement that the phase diagram is nearly exact is therefore overstated. Please clarify that Eq. (13) predicts only the location of the secondary PT symmetry breaking, not the boundaries of those regions.

minor comments (5)

[Eq. (17)] The identity sin^2(Ωt/2) = -(e^{iΩt}+e^{-iΩt}+2)/4 is incorrect; the correct relation is sin^2(Ωt/2) = -(e^{iΩt}+e^{-iΩt}-2)/4. As written, the right-hand side is negative for real Ω, while the left-hand side is non-negative. Please correct the sign of the constant term.
[Section VI, Eq. (20)] The text says that ∂α/∂ω can be obtained from Eq. (9), but the displayed expression is ∂α/∂Δ. The variable should be corrected to ∂α/∂Δ.
[Throughout] The manuscript contains many typographical errors, including "by by" in the caption of Fig. 1, "with with" after Eq. (8), "Hoverer" in the Introduction, "M odel" in the title, and "th e" in the abstract. A careful proofreading is needed.
[Fig. 1 caption] The phrase "as given by by Eq. (11)" is unclear because the phase boundary is determined by the condition Δ̃² = Ã²/4, which follows from Eq. (11) but is not the equation itself. Please rephrase the caption.
[Section II, Eq. (9)] A sentence explaining the rationale behind Eq. (9) would improve readability: this condition makes the coefficients of the sin(ωt)σy and cos(ωt)σx terms equal, so that the first-harmonic part of the transformed Hamiltonian takes the rotating-wave form. Without this explanation, the self-consistency condition appears ad hoc.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analytical derivation is self-contained and benchmarked against independent Floquet numerics.

full rationale

Walking the derivation chain in Secs. II-III, Eq. (3) is an exact similarity transform and Eqs. (4)-(8) are exact Bessel decompositions. The only approximation is the explicitly stated neglect of H'_2 in Eq. (6), i.e. the truncation of all Bessel components with n>=2; this is a validity/accuracy limitation, not a hidden input. Eq. (9) is the algebraic condition that cancels the counter-rotating amplitudes in H'_1, leaving the stated RWA Hamiltonian; it is a self-consistency condition on the ansatz parameter alpha, not a fit to the target PT phase diagram or to numerical spectra. The eigenvalues (11), the primary exceptional-point condition \tildeDelta^2 = \tildeA^2/4, the secondary crossing condition (13), and the Bloch-Siegert shift (20) follow algebraically from this 2x2 effective Hamiltonian. The numerical Floquet diagonalization in Appendix A is an independent external benchmark, and the agreement in Figs. 1-5 is a genuine test of the approximation. The paper itself flags the truncation's consequences, including missed secondary PT-broken phases and poor dynamics at point B, which confirms that the analytical results are not forced by construction. Self-citations are limited to the general Rabi-model reference [4] and are not load-bearing. No circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The only hand-chosen quantity is alpha, fixed by an internal consistency condition rather than by fitting to data. The main approximation is the RWA truncation of H'_2. No new physical entities are introduced; the Floquet parity operator is a symmetry derived from the existing Hamiltonian.

free parameters (1)

alpha (similarity transformation coefficient) = solution of Eq. (9); for A to 0, alpha = omega/(Delta+omega)
Introduced ad hoc in the generator S(t) and fixed by an imposed RWA condition rather than derived from the Hamiltonian. Its value controls the renormalized detuning and coupling and hence all analytical results.

assumptions (4)

ad hoc to paper The ansatz S(t) = (A/2omega) sin(omega t) alpha sigma_x with time-independent alpha is sufficient to bring the model to RWA form.
This is the key modelling choice of Sec. II; no justification is given beyond analogy with the Hermitian case.
domain assumption Higher-order harmonic term H'_2 (Bessel components with n >= 2) is negligible.
This is the rotating-wave approximation invoked after Eq. (6); it is standard but uncontrolled, and the paper shows it fails in secondary PT-broken phases.
standard math Eq. (9) has a unique solution alpha for the parameters used.
The paper states alpha can be determined by solving Eq. (9) without proving existence or uniqueness; the monotonicity of I1 and the linear right-hand side make it plausible.
domain assumption Truncation of the Floquet matrix in Eq. (24) is sufficiently large that numerical quasi-energies are converged.
The paper says a large truncated space is used but does not specify the truncation size or convergence tests, which matters for the secondary PT-broken regions near A = 0.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Analytical Study of the Non-Hermitian Semiclassical Rabi Model." pith.science (2026). https://pith.science/paper/MD6IUY2C

@misc{pith2026241202918,
  author       = {Pith},
  title        = {Pith review of: Analytical Study of the Non-Hermitian Semiclassical Rabi Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MD6IUY2C}},
  note         = {Machine review of arXiv:2412.02918}
}

abstract

The $\mathcal{PT}$ symmetric semiclassical Rabi model explores the fundamental interaction between a two-level atom and a classical field, revealing novel phenomena in open systems through the inclusion of non-Hermitian terms. We propose a single similarity transformation that yields an effective Hamiltonian in rotating-wave approximation, enabling an analytical solution. The phase boundary of the $\mathcal{PT}$-broken phase, derived from the analytical eigenvalues, closely matches the numerical exact one over a wide range of atomic frequencies, demonstrating the effectiveness of the analytical approach, especially at the main resonance. The Floquet parity operator is also introduced, providing a deeper physical understanding of the emergence of the $\mathcal{PT}$-broken phase. Furthermore, by analyzing the dynamics of excited-state population, we observe several stable oscillations in the Fourier spectrum, demonstrating the applicability of the analytical method beyond the single-photon resonance region. The Bloch-Siegert shift is also discussed and, surprisingly, resembles its Hermitian counterpart, except for the higher-order terms in the coupling strength. The present analytical treatment provides a concise and accurate description of the main physics of this non-Hermitian atom-field interaction system.

Figures

Figures reproduced from arXiv: 2412.02918 by the authors.

**Figure 2.** , and the dynamics at points A, B, and C will be presented in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3: Time evolution of the excited state population [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4: (a) and (b): The Fourier transform [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5: The resonance frequency as a function of coupling [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6: The quasi-eigenvalue spectrum, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7: Time evolution of the excited state population [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Unveiling the Self-Orthogonality at Exceptional Points in Driven $\mathcal{PT}$-Symmetric Systems
cond-mat.other 2025-07 conditional novelty 5.0 of 10

In a driven three-band PT-symmetric lattice, the Rabi frequency diverges near exceptional points, and total power oscillations can serve as an observable for self-orthogonality.

Reference graph

Works this paper leans on

53 extracted references · 28 canonical work pages · cited by 1 Pith paper

[1]

I. I. Rabi, Phys. Rev. 49, 324 (1936)

1936
[2]

(11), which directly yield the PT phase diagram directly in Fig

(11) The exceptional points are given by ˜∆ 2 = ˜A2/ 4, as de- rived from Eq. (11), which directly yield the PT phase diagram directly in Fig. 1. When ˜∆ 2 > ˜A2/ 4, the model is in a PT -unbroken phase, characterized by real eigen- values. Conversely, when ˜∆ 2 < ˜A2/ 4, the eigenvalues be- come complex and conjugate in pairs. The black dashed 4 line in ...

work page
[3]

I. I. Rabi, Phys. Rev. 51, 652 (1937)

1937
[4]

J. H. Shirley, Phys. Rev. 138, B979 (1965)

1965
[5]

Braak, Q.-H

D. Braak, Q.-H. Chen, M. T. Batchelor, and E. Solano, J. Phys. A: Math. Theor. 49, 300301 (2016)

work page 2016
[6]

C. C. Gerry and P. L. Knight, Introductory quantum op- tics (Cambridge university press, 2023)

2023
[7]

Y. N. Joglekar, R. Marathe, P. Durganandini, and R. K. Pathak, Phys. Rev. A 90, 040101 (2014)

2014
[8]

T. E. Lee and Y. N. Joglekar, Phys. Rev. A 92, 042103 (2015)

2015

Show all 53 references

[9]

Gong and Q.-h

J. Gong and Q.-h. Wang, 11 Phys. Rev. A 91, 042135 (2015)

2015
[10]

Q. Xie, S. Rong, and X. Liu, Phys. Rev. A 98, 052122 (2018)

2018
[11]

Quijandr ´ ıa, U

F. Quijandr ´ ıa, U. Naether, S. K. ¨Ozdemir, F. Nori, and D. Zueco, Phys. Rev. A 97, 053846 (2018)

2018
[12]

H. Wang, X. Zhang, J. Hua, D. Lei, M. Lu, and Y. Chen, Journal of Optics 23, 123001 (2021)

2021
[13]

X. Lu, H. Li, J.-K. Shi, L.-B. Fan, V. Mangazeev, Z.-M. Li, and M. T. Batchelor, Phys. Rev. A 108, 053712 (2023)

2023
[14]

Wei and L

B.-B. Wei and L. Jin, Sci. Rep. 7, 7165 (2017)

2017
[15]

Longhi, Phys

S. Longhi, Phys. Rev. Lett. 122, 237601 (2019)

2019
[16]

F. E. ¨Ozt¨ urk, T. Lappe, G. Hellmann, J. Schmitt, J. Klaers, F. Vewinger, J. Kroha, and M. Weitz, Science 372, 88 (2021)

2021
[17]

Yoshida, K

T. Yoshida, K. Kudo, and Y. Hatsugai, Sci. Rep. 9, 16895 (2019)

2019
[18]

Ochkan, R

K. Ochkan, R. Chaturvedi, V. K¨ onye, L. Veyrat, R. Gi- raud, D. Mailly, A. Cavanna, U. Gennser, E. M. Han- kiewicz, B. B¨ uchner,et al. , Nat. Phys. 20, 395 (2024)

2024
[19]

Reisenbauer, H

M. Reisenbauer, H. Rudolph, L. Egyed, K. Hornberger, A. V. Zasedatelev, M. Abuzarli, B. A. Stickler, and U. Deli´ c, Nat. Phys.20, 1629 (2024)

2024
[20]

Brighi and A

P. Brighi and A. Nunnenkamp, Phys. Rev. A 110, L020201 (2024)

2024
[21]

Castagnino and S

M. Castagnino and S. Fortin, J. Phys. A: Math. Theor. 45, 444009 (2012)

2012
[22]

S. Dey, A. Raj, and S. K. Goyal, Phys. Lett. A 383, 125931 (2019)

2019
[23]

Z. G. Yuto Ashida and M. Ueda, Adv. Phys. 69, 249 (2020)

2020
[24]

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987)

1987
[25]

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)

1998
[26]

V. V. Konotop, J. Yang, and D. A. Zezyulin, Rev. Mod. Phys. 88, 035002 (2016)

2016
[27]

El-Ganainy, K

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

2018
[28]

C. M. Bender, in TIME AND SCIENCE: Volume 3: Physical Sciences and Cosmology (World Scientiﬁc, 2023) pp. 285–310

2023
[29]

Moiseyev, Non-Hermitian quantum mechanics (Cam- bridge University Press, 2011)

N. Moiseyev, Non-Hermitian quantum mechanics (Cam- bridge University Press, 2011)

2011
[30]

C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, American Journal of Physics 81, 173 (2013)

2013
[31]

Z.-P. Liu, J. Zhang, i. m. c. K. ¨Ozdemir, B. Peng, H. Jing, X.-Y. L¨ u, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, Phys. Rev. Lett. 117, 110802 (2016)

2016
[32]

J. Li, R. Yu, C. Ding, and Y. Wu, Phys. Rev. A 93, 023814 (2016)

2016
[33]

Farhat, M

M. Farhat, M. Yang, Z. Ye, and P.-Y. Chen, ACS Photonics 7, 2080 (2020)

2020
[34]

Y. C. Hu and T. L. Hughes, Phys. Rev. B 84, 153101 (2011)

2011
[35]

X. Ni, D. Smirnova, A. Poddubny, D. Leykam, Y. Chong, and A. B. Khanikaev, Phys. Rev. B 98, 165129 (2018)

2018
[36]

Ezawa, Phys

M. Ezawa, Phys. Rev. Res. 3, 043006 (2021)

2021
[37]

Fritzsche, T

A. Fritzsche, T. Biesenthal, L. J. Maczewsky, K. Becker , M. Ehrhardt, M. Heinrich, R. Thomale, Y. N. Joglekar, and A. Szameit, Nat. Mater. 23, 377 (2024)

2024
[38]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009)

2009
[39]

L. Feng, R. El-Ganainy, and L. Ge, Nat. Photonics 11, 752 (2017)

2017
[40]

Kawabata, K

K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys. Rev. X 9, 041015 (2019)

2019
[41]

Fring and T

A. Fring and T. Frith, Phys. Rev. A 95, 010102 (2017)

2017
[42]

X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, Phys. Rev. Lett. 110, 243902 (2013)

2013
[43]

Doppler, A

J. Doppler, A. A. Mailybaev, J. B¨ ohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moi- seyev, and S. Rotter, Nature 537, 76 (2016)

2016
[44]

A. U. Hassan, B. Zhen, M. Soljaˇ ci´ c, M. Khajavikhan, and D. N. Christodoulides, Phys. Rev. Lett. 118, 093002 (2017)

2017
[45]

T. T. Koutserimpas, A. Al` u, and R. Fleury, Phys. Rev. A 97, 013839 (2018)

2018
[46]

Zhang, Q.-h

D.-J. Zhang, Q.-h. Wang, and J. Gong, Phys. Rev. A 100, 062121 (2019)

2019
[47]

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, Nat. Commun. 10, 855 (2019)

2019
[48]

Hausinger, Phys

J. Hausinger, Phys. Rev. A 81 (2010)

2010
[49]

L¨ u and H

Z. L¨ u and H. Zheng, Phys. Rev. A 86, 023831 (2012)

2012
[50]

Y. Yan, Z. L¨ u, and H. Zheng, Phys. Rev. A 91, 053834 (2015)

2015
[51]

Y. Han, S. Zhang, M. Zhang, Q. Guan, W. Zhang, and W. Li, Phys. Rev. A 109, 053704 (2024)

2024
[52]

E. K. Twyeﬀort Irish and A. D. Armour, Phys. Rev. Lett. 129, 183603 (2022)

2022
[53]

Bloch and A

F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940)

1940

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

I. I. Rabi, Phys. Rev. 49, 324 (1936)

1936

[2] [2]

(11), which directly yield the PT phase diagram directly in Fig

(11) The exceptional points are given by ˜∆ 2 = ˜A2/ 4, as de- rived from Eq. (11), which directly yield the PT phase diagram directly in Fig. 1. When ˜∆ 2 > ˜A2/ 4, the model is in a PT -unbroken phase, characterized by real eigen- values. Conversely, when ˜∆ 2 < ˜A2/ 4, the eigenvalues be- come complex and conjugate in pairs. The black dashed 4 line in ...

work page

[3] [3]

I. I. Rabi, Phys. Rev. 51, 652 (1937)

1937

[4] [4]

J. H. Shirley, Phys. Rev. 138, B979 (1965)

1965

[5] [5]

Braak, Q.-H

D. Braak, Q.-H. Chen, M. T. Batchelor, and E. Solano, J. Phys. A: Math. Theor. 49, 300301 (2016)

work page 2016

[6] [6]

C. C. Gerry and P. L. Knight, Introductory quantum op- tics (Cambridge university press, 2023)

2023

[7] [7]

Y. N. Joglekar, R. Marathe, P. Durganandini, and R. K. Pathak, Phys. Rev. A 90, 040101 (2014)

2014

[8] [8]

T. E. Lee and Y. N. Joglekar, Phys. Rev. A 92, 042103 (2015)

2015