Classical Petermann Factor as a Measure of Quantum Squeezing in Photonic Time Crystals

Bumki Min; Changhun Oh; Kun Woo Kim; Kyungmin Lee; Young-Sik Ra; Younsung Kim

REVIEW 2 major objections 4 minor 2 cited by

Classical Petermann Factor as a Measure of Quantum Squeezing in Photonic Time Crystals

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A classical factor predicts quantum squeezing in photonic time crystals

desk verdict A mostly sound classical–quantum dictionary for PTC squeezing, but Eq. (9) is only proven at stroboscopic times and the continuous-time claims overstate it. read the letter →

arxiv 2601.17369 v2 pith:PDR74BIM submitted 2026-01-24 physics.optics

Younsung Kim , Kyungmin Lee , Changhun Oh , Young-Sik Ra , Kun Woo Kim , Bumki Min This is my paper

classification physics.optics

keywords PetermannfactorphotonictimecrystalsquantumsqueezingBogoliubov-deGennesFloquetmonodromymomentumgapparametricamplificationnon-Hermitianoptics

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 argues that the Petermann factor, a classical measure of mode nonorthogonality, directly sets the scale of quantum noise in photonic time crystals (PTCs). By showing that the classical and quantum Floquet monodromy matrices are formally equivalent, the authors derive exact relations: in stable bands the Petermann factor fixes the Bogoliubov mixing and vacuum quasiparticle population, while inside momentum gaps it multiplies the photon-number growth and enhances the initial squeezing rate. If correct, this turns classical measurements of mode nonorthogonality into quantitative predictions for squeezing and photon generation, offering a single design parameter for quantum resources in a broad class of two-mode bosonic systems. A sympathetic reader would see this as a unification of two previously separate narratives: non-Hermitian mode theory and SU(1,1) quantum squeezing.

What carries the argument

The central object is the effective Floquet Bogoliubov–de Gennes (BdG) dynamical matrix M_q^eff = σ_z H_BdG^eff, whose monodromy matrix U(T,0) generates stroboscopic evolution. The Petermann factor K_k, defined from the right and left eigenvectors of this matrix via K_k = ⟨L|L⟩⟨R|R⟩/|⟨L|R⟩|², quantifies mode nonorthogonality. In the SU(1,1) structure it maps to the two-mode squeezing parameter: in bands K_k = cosh²(2r_k), and in gaps K_k = |Δ|²/κ_k². This map converts a classical biorthogonal metric into a quantum resource parameter.

What would settle it

Measure the photon number or quadrature variance inside a photonic time crystal at a time between stroboscopic snapshots (e.g., at t = T/2) within the momentum gap. If ⟨n_k(t)⟩ does not match K_k sinh²(κ_k t) with the same prefactor at these intermediate times—or if the short-time growth rate differs from √K κ—then the Petermann factor alone does not set the continuous-time squeezing scale.

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Extended reading notes

Core claim

For a PTC, the classical and quantum monodromy matrices share the same spectrum and Petermann factor, because both are governed by the same BdG dynamical matrix. In stable bands, the Petermann factor obeys K_k = cosh^2(2r_k), where r_k is the squeezing parameter, implying the mean quasiparticle population in the vacuum is ⟨n_k,b⟩ = (√K_k - 1)/2. In momentum gaps, with Floquet growth rate κ_k, the photon number generated from vacuum is ⟨n_k(t)⟩ = K_k sinh^2(κ_k t), so the Petermann factor acts as a multiplicative gain factor; near the gap edge, the diverging K_k compensates the vanishing κ_k, yielding universal quadratic scaling |Δ|²t². The paper further shows the initial squeezing rate is en

Load-bearing premise

The central formulas are derived from the stroboscopic monodromy matrix, so they are strictly valid only at integer multiples of the modulation period; Eq. (9) is stated without this caveat in the main text, and if the physical short-time dynamics between stroboscopic snapshots deviate significantly, the claimed initial squeezing rate may not be the rate measured continuously.

Editorial extensions

If this is right

Classical experiments measuring decay-rate enhancement or momentum-resolved density of states can now predict the quantum squeezing and photon yield in PTCs without direct quantum characterization.
In momentum gaps, the photon number follows ⟨n_k(t)⟩ = K_k sinh²(κ_k t), so K_k acts as a tunable multiplicative gain factor alongside the Floquet growth rate.
Near the momentum-gap edge, the diverging Petermann factor compensates the vanishing growth rate, producing a universal |Δ|²t² quadratic photon-number growth that dominates the dynamics.
The unified formula K = 1/(1 − |Δ|²/g²) in bands and K = 1/(1 − g²/|Δ|²) in gaps applies to any static two-mode quadratic bosonic BdG Hamiltonian, not only PTCs.
The initial squeezing rate is enhanced by √K and remains so under symmetric Markovian loss, making the enhancement observable in realistic dissipative settings.

Reading between the lines

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

The stroboscopic restriction is a caveat the paper underplays: Eq. (9) is presented as general time dependence, but it is derived from the monodromy matrix and strictly holds at t = mT. A continuous readout within a period could reveal deviations, so testing intra-period dynamics would sharpen or limit the claim.
If the mapping holds across platforms, the Petermann factor could become a deliberate engineering knob in other two-mode BdG systems (e.g., superconducting circuits or dynamical Casimir setups), allowing classical non-Hermitian design to target quantum resource generation.
The Petermann-enhancement near gap edges suggests a metrological opportunity: squeezing is boosted exactly where the growth rate vanishes, but the divergent mode nonorthogonality may introduce additional sensitivity to perturbations—a trade-off worth exploring.
The paper uses the Euclidean inner product to define K; using the Krein (σz) inner product, natural for BdG dynamics, would yield different measures, and it remains open which metric best captures quantum noise in lossy or gainful settings.

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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

Here's my read. The punchline: the paper is a mostly sound, compact mapping between the classical Petermann factor and quantum squeezing in photonic time crystals, but the headline Eq. (9) is derived stroboscopically and the text drifts into continuous-time claims without the needed qualification. It deserves a real referee, with a request to fix that scoping.

What's new and good: the explicit formulas K = cosh^2(2r) in stable bands and K = |Delta|^2/kappa^2 in gaps, together with = (sqrt(K)-1)/2 and = K sinh^2(kappa t), are stated cleanly for PTCs. The classical–quantum monodromy equivalence in SM Sec. A is straightforward and well explained. I went through the algebra; it checks out. The paper gives a genuinely useful design parameter—one classical number, K, that sets the scale for photon pair generation in any two-mode BdG bosonic platform. That's worthwhile even if the underlying SU(1,1) mechanics isn't new.

The soft spot is exactly what the stress-test flags, though with a nuance. The main text does say 'stroboscopic evolution' immediately before introducing r_k(t), so it's not only in Fig. 1(c)'s caption. But the sentence structure and everything after—the initial squeezing rate, the short-time expansion, the loss analysis—treats Eq. (9) as a continuous-time result. That's not justified. The effective Hamiltonian is defined via the one-period monodromy, so Eq. (9) is exact only at t=mT. Including the micromotion operator P(t) gives additional phases and prefactor corrections at intra-period times, and these can be large near the MG edge where K diverges. The loss robustness argument in SM Sec. G is for a static two-mode amplifier, not the Floquet PTC; so the claim that the Petermann-enhanced rate persists under loss is only established stroboscopically, if at all. This is a real overstatement, but not a fatal one—the stroboscopic result is still correct and useful.

Minor points: novelty is moderate (the formulas follow from elementary algebra once the mapping is granted), no code or data are shipped, and the classical measurement side is inherited from prior work. Those are not flaws for a theory letter.

Bottom line: the thinking is clear and honest, the derivations are reproducible on paper, and the central defect is a scoping/packaging issue. I'd send it to peer review and ask for a revision that either extends the analysis to continuous time with micromotion or explicitly restricts the claims and the 'initial squeezing rate' to stroboscopic readouts. I'd cite it once that's done.

Referee Report

2 major / 4 minor

Summary. The Letter considers a one-dimensional photonic time crystal (PTC) with a periodically modulated permittivity, described in each (k,−k) sector by a two-mode quadratic Hamiltonian. Working in the Nambu/BdG formalism, the authors establish that the classical and quantum monodromy matrices coincide, and then introduce the Petermann factor K_k of the effective Floquet dynamical matrix. In the stable band regime they derive K_k = cosh²(2r_k), which implies a vacuum quasiparticle population ⟨n_k,b⟩ = (√K_k − 1)/2. In the momentum-gap regime they derive K_k = |Δ_k|²/κ_k² and, using the effective Floquet Hamiltonian, the photon-number growth ⟨n_k(t)⟩ = K_k sinh²(κ_k t), with an initial squeezing rate enhanced by √K_k. A Heisenberg–Langevin extension with symmetric Markovian loss gives n(t) = |Δ|²(t² − 2γt³/3) + O(t⁴). The central claim is that a classically measurable non-Hermitian geometric quantity, the Petermann factor, provides quantitative predictions for quantum squeezing and photon generation in PTCs and analogous two-mode BdG platforms.

Significance. If the central claim holds, the paper offers a compact and elegant bridge between non-Hermitian mode geometry and quantum resource generation, with potential design relevance for PTCs, parametric amplifiers, and other two-mode bosonic systems. The manuscript's strengths are its explicit SU(1,1)/BdG algebra, the absence of fitted parameters, and the closed-form, falsifiable relations connecting K_k to squeezing parameters and photon numbers. The main uncertainty concerns the temporal domain of the gap-regime prediction; this is a genuine but localizable issue that can be addressed by a careful revision rather than by invalidating the underlying algebra.

major comments (2)

[Momentum-gap regime, Eq. (9), and SM Secs. D–F] The photon-number formula ⟨n_k(t)⟩ = K_k sinh²(κ_k t) is derived from the effective Hamiltonian defined via U(T,0) = exp[−iT σ_z H_BdG^eff]. Floquet theory gives U(t,0) = P(t)e^{−itH_F}P(0)^{−1}, so the static-amplifier solution is exact only at stroboscopic times t = mT. The main text presents Eq. (9) with an unqualified continuous time variable, and the “initial squeezing rate” √K κ of SM Eq. (S56) is the slope of the stroboscopic sequence, not the physical t→0 derivative of the actual PTC evolution. Intra-period micromotion will generally produce modified prefactors and oscillations, with potentially large deviations near the MG edge where K_k diverges. Fig. 1(c) is the only place where the stroboscopic restriction is flagged. This is load-bearing because the advertised payoff is a quantitative prediction for photon generation from a classical measurement of K. Please either restrict
[SM Sec. G (loss extension)] The loss calculation inherits the same stroboscopic limitation. The statement that n(t) = |Δ|²(t² − 2γt³/3) describes the PTC under loss implicitly treats the lossy system as the static effective Hamiltonian H_eff. For the real continuous-time PTC, a Markovian bath coupled to a periodically driven system should be analyzed in the Floquet basis (or with the time-dependent Langevin equation), and the micromotion operator will enter. As written, the claim that the Petermann-enhanced initial squeezing rate “persists under symmetric Markovian damping” is supported only for the coarse-grained stroboscopic dynamics, not for arbitrary intra-period readout. The loss subsection should either be explicitly framed in the same stroboscopic sense or extended to the full Floquet–Langevin problem.

minor comments (4)

[Band regime, Eq. (6)] In the sentence preceding Eq. (6), “⟨n_k,b⟩ = sinh 2 r_k” should be “sinh² r_k” to be consistent with Eq. (6). Please check the typesetting of this superscript.
[Eq. (7) and SM Eq. (S39)] The notation “+ Iμ” is unconventional; use μI for clarity and consistency with standard matrix notation.
[Abstract and main text] The abstract’s phrase “mean bare-photon occupation of the Floquet vacuum” is not defined in the main text. The quantity computed is the expectation value of the quasiparticle number in the photon vacuum, ⟨0|b†b|0⟩ = sinh² r. Please define this terminology explicitly at the first occurrence.
[Fig. 1(c) and Eq. (9)] If the stroboscopic restriction is retained after revision, state it in the main text in the paragraph containing Eq. (9), not only in the figure caption, so that a reader cannot mistake the formula for a continuous-time law.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Petermann–squeezing relations are derived identities from a single BdG monodromy, not fitted or self-citation-forced predictions.

full rationale

The central derivation is self-contained. K_k is defined from eigenvector biorthogonality of the monodromy U(T,0) (Eq. 4 and SM Eq. S21), while the squeezing parameter r_k is defined by the SU(1,1)/Bogoliubov diagonalizer S of the same monodromy (SM Eq. S25). Equations (5) and (8), K_k=cosh^2(2r_k) and K_k=|Δ_k|^2/κ_k^2, are obtained by explicit eigenvector algebra in SM Sec. E, not assumed or fitted. The photon-number relations (6) and (9) follow from the standard mean photon number sinh^2(r) for two-mode squeezed vacuum and from the static SU(1,1) evolution formula applied to the effective BdG Hamiltonian. No parameter is fitted to a subset of data, and no 'prediction' is statistically forced. The classical-to-quantum mapping rests on the derived statement that classical and quantum Nambu dynamics share the same dynamical matrix (SM Sec. A), not on a self-citation chain. The only caveats are non-circular: Eq. (9) is presented with an unqualified continuous time variable, while the underlying effective-Hamiltonian construction U(T,0)=exp[-iT σ_z H_BdG^eff] justifies exact equality only at stroboscopic times t=mT (the Fig. 1(c) caption states 'stroboscopic evolution at integer multiples of T'); this is a scope/correctness limitation for intra-period readout, not a circularity. Also, the final claim that K_k can be inferred from classical measurements cites the authors' previous work [28,29], but that citation supports an application route, not the derivation of Eqs. (5)-(9). Score 0.

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

The central claim depends on the model Hamiltonian, the canonical quantization bridge, and the stroboscopic interpretation of the effective Floquet Hamiltonian; no free parameters are fitted and no new physical entities are introduced.

assumptions (4)

domain assumption Quantum PTC dynamics in a (k,−k) pair is governed by H_k(t)=A(t)(n_k+n_−k+1)+B(t)(a_k a_−k + h.c.) with A,B given by k[1±ε^{−1}(t)]/2 (Eq. 1).
Imported from Ref. [30]; the entire calculation rests on this model Hamiltonian, which is sketched from canonical quantization in SM Sec. A but not derived from a microscopic Lagrangian in the main text.
domain assumption Classical Maxwell dynamics and quantum BdG dynamics share the same monodromy matrix and eigenvalues after canonical quantization (SM Sec. A).
Core bridge that lets K computed classically be applied to quantum noise; the paper asserts this by promoting Poisson brackets to commutators rather than proving an error-bounded correspondence.
domain assumption The effective Floquet Hamiltonian obtained from log U(T,0) describes the physically relevant evolution at stroboscopic times; within-period dynamics is not addressed.
Needed for Eq. (9); only the monodromy determines the operator evolution at integer multiples of T, so continuous-time use of the photon-number law is an unstated extension.
domain assumption Loss, when included (SM Sec. G), is symmetric linear Markovian damping γ1=γ2=γ coupled to a vacuum bath.
Underlies the claim that loss enters only at third order; asymmetric or non-Markovian loss is not covered.

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Pith. "Pith review of Classical Petermann Factor as a Measure of Quantum Squeezing in Photonic Time Crystals." pith.science (2026). https://pith.science/paper/PDR74BIM

@misc{pith2026260117369,
  author       = {Pith},
  title        = {Pith review of: Classical Petermann Factor as a Measure of Quantum Squeezing in Photonic Time Crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PDR74BIM}},
  note         = {Machine review of arXiv:2601.17369}
}

read the original abstract

Photonic time crystals realize a continuum of momentum-resolved SU(1,1) parametric amplifiers. We show that a classical quantity, the Petermann factor of the effective Floquet Bogoliubov de Gennes (BdG) dynamical matrix, sets the scale of their quantum noise. In stable bands it fixes the Bogoliubov mixing and hence the mean bare-photon occupation of the Floquet vacuum, while in momentum gaps it sets the photon-number prefactor and enhances the squeezing dynamics, with the Floquet growth rate setting the time scale. This converts classical measurements of mode nonorthogonality into quantitative predictions for squeezing and photon generation, and offers a compact design parameter for engineering quantum resources in two-mode BdG platforms.

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Forward citations

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Works this paper leans on

41 extracted references · 1 linked inside Pith · cited by 2 Pith papers

[1]

A. L. Schawlow and C. H. Townes, Infrared and optical masers, Phys. Rev. 112, 1940 (1958)

1940
[2]

A. E. Siegman, Excess spontaneous emission in non- hermitian optical systems. i. laser ampliﬁers, Phys. Rev. A 39, 1253 (1989)

1989
[3]

A. E. Siegman, Excess spontaneous emission in non- hermitian optical systems. ii. laser oscillators, Phys. Rev. A 39, 1264 (1989)

1989
[4]

Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain- induced waveguiding, IEEE Journal of Quantum Elec- tronics 15, 566 (1979)

K. Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain- induced waveguiding, IEEE Journal of Quantum Elec- tronics 15, 566 (1979)

1979
[5]

M. V. Berry, Mode degeneracies and the petermann excess-noise factor for unstable lasers, Journal of Mod- ern Optics 50, 63 (2003)

2003
[6]

Wiersig, Petermann factors and phase rigidities near exceptional points, Phys

J. Wiersig, Petermann factors and phase rigidities near exceptional points, Phys. Rev. Res. 5, 033042 (2023)

2023
[7]

Schomerus, Eigenvalue sensitivity from eigenstate geometry near and beyond arbitrary-order exceptional points, Phys

H. Schomerus, Eigenvalue sensitivity from eigenstate geometry near and beyond arbitrary-order exceptional points, Phys. Rev. Res. 6, 013044 (2024)

2024
[8]

Wang, Y.-H

H. Wang, Y.-H. Lai, Z. Yuan, M.-G. Suh, and K. Va- hala, Petermann-factor sensitivity limit near an excep- tional point in a brillouin ring laser gyroscope, Nature Communications 11, 1610 (2020)

2020

Show all 41 references

[9]

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljaˇ ci´ c, and S. G. Johnson, General theory of spontaneous emission near exceptional points, Opt. Express 25, 12325 (2017)

2017
[10]

H. P. Yuen, Two-photon coherent states of the radiation ﬁeld, Phys. Rev. A 13, 2226 (1976)

1976
[11]

Wodkiewicz and J

K. Wodkiewicz and J. H. Eberly, Coherent states, squeezed ﬂuctuations, and the su(2) and su(1,1) groups in quantum-optics applications, J. Opt. Soc. Am. B 2, 458 (1985)

1985
[12]

C. M. Caves, Quantum-mechanical noise in an interfer- ometer, Phys. Rev. D 23, 1693 (1981)

1981
[13]

G. Gwak, C. Roh, Y.-D. Yoon, M. S. Kim, and Y.-S. Ra, Completely characterizing multimode second-order nonlinear optical quantum processes, Nature Photonics 10.1038/s41566-025-01787-x (2025), published online 11 Nov 2025; volume/pages may be assigned later

2025 doi
[14]

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, 30 years of squeezed light generation, Physica Scripta 91, 053001 (2016)

2016
[15]

Aasi et al

J. Aasi et al. , Enhanced sensitivity of the ligo gravita- tional wave detector by using squeezed states of light, Nature Photonics 7, 613 (2013)

2013
[16]

J. R. Zurita-S´ anchez, P. Halevi, and J. C. Cervantes- Gonz´ alez, Reﬂection and transmission of a wave inci- dent on a slab with a time-periodic dielectric function (t), Phys. Rev. A 79, 053821 (2009)

2009
[17]

Wang, Z.-Q

N. Wang, Z.-Q. Zhang, and C. T. Chan, Photonic ﬂoquet media with a complex time-periodic permittivity, Phys. Rev. B 98, 085142 (2018)

2018
[18]

Galiﬃ, R

E. Galiﬃ, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huido- bro, M. G. Silveirinha, R. Sapienza, A. Al` u, and J. B. Pendry, Photonics of time-varying media, Advanced Pho- tonics 4, 10.1117/1.ap.4.1.014002 (2022)

2022 doi
[19]

J. G. Gaxiola-Luna and P. Halevi, Temporal photonic (time) crystal with a square proﬁle of both permittivity ε(t) and permeability µ(t), Phys. Rev. B 103, 144306 (2021)

2021
[20]

M. A. Salem and C. Caloz, Temporal photonic crystals: Causality versus periodicity, in 2015 International Con- ference on Electromagnetics in Advanced Applications (ICEAA) (2015) pp. 490–493

2015
[21]

Park and B

J. Park and B. Min, Spatiotemporal plane wave expan- sion method for arbitrary space–time periodic photonic media, Opt. Lett. 46, 484 (2021). 5

2021
[22]

T. F. Allard, J. E. Sustaeta-Osuna, F. J. Garc ´ ıa-Vidal, and P. A. Huidobro, Broadband dipole absorption in dis- persive photonic time crystals (2025), arXiv:2508.04619 [physics.optics]

2025
[23]

M. M. Asgari, P. Garg, X. Wang, M. S. Mirmoosa, C. Rockstuhl, and V. Asadchy, Theory and applications of photonic time crystals: a tutorial, Adv. Opt. Photon. 16, 958 (2024)

2024
[24]

G. T. Moore, Quantum theory of the electromagnetic ﬁeld in a variable-length one-dimensional cavity, Journal of Mathematical Physics 11, 2679 (1970)

1970
[25]

Dodonov, Fifty years of the dynamical casimir eﬀect, Physics 2, 67 (2020)

V. Dodonov, Fifty years of the dynamical casimir eﬀect, Physics 2, 67 (2020)

2020
[26]

J. E. Sustaeta-Osuna, F. J. Garc ´ ıa-Vidal, and P. A. Huidobro, Quantum theory of photon pair creation in photonic time crystals, ACS Photonics 12, 1873 (2025)

2025
[27]

J. Bae, K. Lee, B. Min, and K. W. Kim, Quantum elec- trodynamics of photonic time crystals, Nature Commu- nications 17, 858 (2026), published online 19 Dec 2025; version of record 22 Jan 2026

2026
[28]

J. Park, K. Lee, R.-Y. Zhang, H.-C. Park, J.-W. Ryu, G. Y. Cho, M. Y. Lee, Z. Zhang, N. Park, W. Jeon, J. Shin, C. T. Chan, and B. Min, Spontaneous emis- sion decay and excitation in photonic time crystals, Phys. Rev. Lett. 135, 133801 (2025)

2025
[29]

K. Lee, M. Kyung, Y. Kim, J. Park, H. Lee, J. Choi, C. T. Chan, J. Shin, K. W. Kim, and B. Min, Spon- taneous emission and lasing in photonic time crystals (2025), arXiv:2507.19916 [physics.optics]

2025
[30]

Lyubarov, Y

M. Lyubarov, Y. Lumer, A. Dikopoltsev, E. Lustig, Y. Sharabi, and M. Segev, Ampliﬁed emission and las- ing in photonic time crystals, Science 377, 425 (2022)

2022
[31]

See Supplemental Material for details in Secs. A–G, including the classical–quantum mapping of the mon- odromy matrix (A), su(1, 1) algebra and stability clas- siﬁcation (B–C), derivation of the stroboscopic eﬀec- tive Hamiltonian (D), Petermann-factor formulas and band/gap sq...
[32]

K. S. Eswaran, A. E. Kopaei, and K. Sacha, Aspects of quantum geometry in photonic time crystals (2025), arXiv:2511.17317 [physics.optics]

2025
[33]

Wakeﬁeld, A

R. Wakeﬁeld, A. Laing, and Y. N. Joglekar, Non- hermiticity in quantum nonlinear optics through sym- plectic transformations, Applied Physics Letters 124, 201103 (2024)

2024
[34]

Figueiredo Roque, A

T. Figueiredo Roque, A. A. Clerk, and H. Ribeiro, Engi- neering fast high-ﬁdelity quantum operations with con- strained interactions, npj Quantum Information 7, 28 (2021)

2021
[35]

J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori, Dynamical casimir eﬀect in a superconducting coplanar waveguide, Phys. Rev. Lett. 103, 147003 (2009)

2009
[36]

C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing, Observation of the dynamical casimir eﬀect in a superconducting circuit, Nature 479, 376 (2011)

2011
[37]

K. W. Murch, S. J. Weber, K. M. Beck, E. Ginossar, and I. Siddiqi, Reduction of the radiative decay of atomic coherence in squeezed vacuum, Nature 499, 62 (2013)

2013
[38]

Hudelist, J

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, Quantum metrology with parametric ampliﬁer-based photon correlation interferometers, Na- ture Communications 5, 3049 (2014). Supplemental Material: Classical Petermann Factor as a Measure of Quantum Squeezing in Ph...

2014
[39]

Wünsche, Journal of Optics B: Quantum and Semiclassical Optics 2, 73 (2000)

A. Wünsche, Journal of Optics B: Quantum and Semiclassical Optics 2, 73 (2000)

2000
[40]

J. E. Sustaeta-Osuna, F. J. García-Vidal, and P. A. Huidobro, ACS Photonics 12, 1873 (2025)

2025
[41]

Inglima and B

S. Inglima and B. J. Schroers, Lett. Math. Phys. 109, 1433 (2019) , arXiv:1804.05782 [hep-th]

2019 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

A. L. Schawlow and C. H. Townes, Infrared and optical masers, Phys. Rev. 112, 1940 (1958)

1940

[2] [2]

A. E. Siegman, Excess spontaneous emission in non- hermitian optical systems. i. laser ampliﬁers, Phys. Rev. A 39, 1253 (1989)

1989

[3] [3]

A. E. Siegman, Excess spontaneous emission in non- hermitian optical systems. ii. laser oscillators, Phys. Rev. A 39, 1264 (1989)

1989

[4] [4]

Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain- induced waveguiding, IEEE Journal of Quantum Elec- tronics 15, 566 (1979)

K. Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain- induced waveguiding, IEEE Journal of Quantum Elec- tronics 15, 566 (1979)

1979

[5] [5]

M. V. Berry, Mode degeneracies and the petermann excess-noise factor for unstable lasers, Journal of Mod- ern Optics 50, 63 (2003)

2003

[6] [6]

Wiersig, Petermann factors and phase rigidities near exceptional points, Phys

J. Wiersig, Petermann factors and phase rigidities near exceptional points, Phys. Rev. Res. 5, 033042 (2023)

2023

[7] [7]

Schomerus, Eigenvalue sensitivity from eigenstate geometry near and beyond arbitrary-order exceptional points, Phys

H. Schomerus, Eigenvalue sensitivity from eigenstate geometry near and beyond arbitrary-order exceptional points, Phys. Rev. Res. 6, 013044 (2024)

2024

[8] [8]

Wang, Y.-H

H. Wang, Y.-H. Lai, Z. Yuan, M.-G. Suh, and K. Va- hala, Petermann-factor sensitivity limit near an excep- tional point in a brillouin ring laser gyroscope, Nature Communications 11, 1610 (2020)

2020