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arxiv: 2412.02248 · v2 · submitted 2024-12-03 · ⚛️ physics.atom-ph

Analytical and numerical studies of periodic superradiance

Hideaki Hara , Yuki Miyamoto , Junseok Han , Riku Omoto , Yasutaka Imai , Akihiro Yoshimi , Koji Yoshimura , Motohiko Yoshimura
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Noboru Sasao
This is my paper

Pith reviewed 2026-05-23 08:29 UTC · model grok-4.3

classification ⚛️ physics.atom-ph
keywords periodic superradianceMaxwell-Bloch equationsthree-level systemEr:YSO crystalsuperradiant pulsesfield decay ratepopulation reservoir
0
0 comments X

The pith

A reduced three-level Maxwell-Bloch model produces periodic superradiance only inside a bounded parameter region that excludes the actual Er:YSO experimental values.

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

The authors reduce the system to a pair of superradiance states coupled to a population reservoir and write the Maxwell-Bloch equations for that three-level model. Linear stability analysis of the fixed point yields the condition on decay rates and coupling strength that permits periodic pulsing rather than steady or damped emission. Reduction of the dynamics to two coupled equations for coherence and population inversion supplies closed-form expressions for the pulse period, width, and total photons emitted per cycle. Numerical solutions inside the allowed parameter window reproduce the observed periodic train, yet the measured experimental numbers lie outside that window. The mismatch implies that an additional physical effect, illustrated by making the field decay rate depend on electric-field strength, is needed to bring the periodic regime into the laboratory conditions.

Core claim

Periodic superradiance is shown to arise when the eigenvalues of the linearized three-level Maxwell-Bloch system cross into the complex plane with nonzero imaginary part; the resulting two-variable equations then admit periodic solutions whose period, duration, and photon number are given by explicit algebraic expressions in the decay and pumping parameters.

What carries the argument

Two-variable equations for coherence and population difference between the superradiance states, obtained after adiabatic elimination of the reservoir population.

If this is right

  • The period, pulse duration, and number of photons per burst are given by explicit algebraic formulas once the system lies inside the periodic regime.
  • Periodic emission requires the field decay rate and coupling parameters to satisfy inequalities obtained from the eigenvalue analysis.
  • Making the field decay rate increase with electric-field amplitude extends the periodic regime to the measured experimental values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction to two dynamical variables may apply to other solid-state systems in which a reservoir state participates in the superradiant cycle.
  • An experiment that deliberately tunes the decay rate or pumping strength into the predicted periodic window would test whether the analytic formulas hold.
  • If the field-dependent decay mechanism is confirmed, cavity design could be adjusted to exploit or suppress the periodicity.

Load-bearing premise

The full atomic dynamics can be captured by a three-level system consisting of the two superradiance states and one reservoir state without significant extra decoherence or broadening channels.

What would settle it

Direct measurement of the emitted pulse period under controlled conditions inside the analytically predicted parameter region and comparison with the closed-form expression derived from the two-variable equations.

Figures

Figures reproduced from arXiv: 2412.02248 by Akihiro Yoshimi, Hideaki Hara, Junseok Han, Koji Yoshimura, Motohiko Yoshimura, Noboru Sasao, Riku Omoto, Yasutaka Imai, Yuki Miyamoto.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Energy diagram of an Er [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Reduction to an extended two-level system in our [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Streamline of Eqs. (10) and (11) in the XY [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the periodic SR regions in the (A21, P13) plane for different A32 with (A31, γ32) fixed at (102 , 107 ) Hz. Here, and below, the underline below the numerical value indicates that it corresponds to the actual experimental value shown in Table I. The blue dot￾ted, green solid, black dashed, and red dot-dashed lines are the boundaries of the periodic SR regions for A32 = 2, 4, 15, and 100 Hz, respectiv… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Example of a numerical simulation result within t [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Result of a numerical simulation with [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 4
Figure 4. Figure 4: The blue crosses and red circles represent the re [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of the analytical solution of the T2B mode [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Multilayer structure whose refractive index has [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

We conduct a theoretical study to understand the periodic superradiance observed in an Er:YSO crystal. First, we construct a model based on the Maxwell-Bloch equations for a reduced level system, a pair of superradiance states and a population reservoir state. Analysis of the eigenvalues of the linearized differential equations shows that periodic superradiance can be realized only for certain parameters. We also derive two-variable equations consisting of the coherence and population difference between the two superradiance states, which contain the essential feature of the periodic superradiance. The two-variable equations clarify a mathematical structure of this periodic phenomenon and give analytical forms of the period, pulse duration, and number of emitted photons. Our model successfully reproduces the periodic behavior, but the actual experimental parameters are found to be outside the parameter region for the periodic superradiance. This result implies that some other mechanism(s) is required. As one example, assuming that the field decay rate varies with the electric field, the periodic superradiance can be reproduced even under the actual experimental condition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper constructs a reduced three-level Maxwell-Bloch model (superradiant pair plus population reservoir) for periodic superradiance in Er:YSO. Eigenvalue analysis of the linearized equations identifies the parameter region permitting periodic solutions; a further reduction to two-variable equations for coherence and population difference yields analytical expressions for the period, pulse duration, and emitted photons. The model reproduces periodic behavior inside the derived region, but the authors conclude that the cited experimental parameters lie outside it, implying that additional mechanisms are required; they illustrate one possibility by allowing the field decay rate to depend on the electric field.

Significance. If the experimental-parameter mapping is shown to be robust, the result would indicate that constant-decay-rate Maxwell-Bloch models are insufficient for this system and would motivate systematic study of field-dependent or crystal-specific effects. The algebraic derivation of the two-variable equations and the closed-form expressions for observables constitute a clear strength, furnishing a parameter-independent mathematical structure that can be tested independently of the numerics.

major comments (2)
  1. [Abstract and model-construction paragraph] Abstract and model-construction paragraph: the central claim that actual experimental parameters lie outside the periodic region (obtained from the eigenvalue analysis of the linearized system) is load-bearing for the conclusion that other mechanisms are required, yet the manuscript provides no uncertainty propagation or sensitivity analysis on the mapping of measured decay rates, inhomogeneous broadening, and coupling strengths into the three-level model; a modest systematic offset could move the operating point across the boundary.
  2. [Abstract] Abstract: the reduction to a three-level system is asserted to capture the essential dynamics, but no quantitative estimate is given for the size of neglected contributions from additional decoherence channels or crystal-specific effects; this assumption directly affects whether the derived periodic region is applicable to the experiment.
minor comments (1)
  1. Notation for the field-dependent decay rate introduced in the example should be defined explicitly when first used and its functional form stated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important aspects of robustness in our conclusions. We address each major comment below and will incorporate revisions as indicated.

read point-by-point responses

  1. Referee: [Abstract and model-construction paragraph] Abstract and model-construction paragraph: the central claim that actual experimental parameters lie outside the periodic region (obtained from the eigenvalue analysis of the linearized system) is load-bearing for the conclusion that other mechanisms are required, yet the manuscript provides no uncertainty propagation or sensitivity analysis on the mapping of measured decay rates, inhomogeneous broadening, and coupling strengths into the three-level model; a modest systematic offset could move the operating point across the boundary.

    Authors: We agree that a sensitivity analysis strengthens the central claim. In the revised manuscript we will add a dedicated paragraph (or short subsection) that propagates plausible uncertainties in the input parameters (decay rates, inhomogeneous broadening, and coupling strengths, drawn from the cited experimental literature) through the eigenvalue boundary condition. Our preliminary checks show the experimental operating point lies sufficiently far from the boundary that variations of order 20-30% do not cross into the periodic region; the added analysis will quantify this distance with explicit ranges. revision: yes

  2. Referee: [Abstract] Abstract: the reduction to a three-level system is asserted to capture the essential dynamics, but no quantitative estimate is given for the size of neglected contributions from additional decoherence channels or crystal-specific effects; this assumption directly affects whether the derived periodic region is applicable to the experiment.

    Authors: The three-level truncation is justified by the large detuning of additional Er levels relative to the superradiant transition and reservoir state. We will expand the model-construction paragraph in the revision to include an order-of-magnitude estimate of the neglected decoherence and coupling terms, showing they remain at least an order of magnitude smaller than the retained rates under the experimental conditions reported in the literature. This estimate will be based on the known level structure and linewidths of Er:YSO. revision: yes

Circularity Check

0 steps flagged

Derivation chain is algebraic from standard Maxwell-Bloch equations with no circular reductions or fitted predictions.

full rationale

The paper begins with the Maxwell-Bloch equations for a reduced three-level system (superradiant pair plus reservoir), performs linearization to obtain eigenvalue conditions, and algebraically reduces to two-variable equations whose analytic expressions for period, duration, and photon number follow directly from the differential structure. The central claim—that experimental parameters lie outside the periodic region—is obtained by substituting independently measured decay rates and couplings into those expressions, not by fitting any parameter to the observed periodicity itself. No self-citation chains, ansatzes smuggled via prior work, or self-definitional mappings appear in the derivation; the mismatch result therefore does not reduce to a quantity defined by its own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on the standard Maxwell-Bloch framework plus a domain-specific reduction to three levels; no new entities are postulated and the only free parameters are the decay and coupling rates whose values are taken from experiment or varied to locate the periodic window.

free parameters (2)
  • field decay rate
    Varied to locate the parameter region permitting periodic solutions; also allowed to become field-dependent in the final example.
  • coupling strength between superradiance states
    Determines the boundary of the periodic regime in the eigenvalue analysis.
axioms (2)
  • domain assumption The dynamics are adequately described by the Maxwell-Bloch equations for a reduced three-level system.
    Invoked at the start of model construction to justify dropping other atomic levels and crystal inhomogeneities.
  • standard math Linearization around steady states correctly identifies the onset of periodic solutions.
    Used to obtain the eigenvalue condition for periodic superradiance.

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [1]

    trivial solution

    Linearized X2MB equations and Jacobi matrix We first rewrite the original X2MB equations, Eqs. (2) - (6), into dimensionless forms. This is done in Appendix A to obtain Eqs. (A9) - (A12). We then expand them around the equilibrium points given by Eqs. (A17) - (A20). The final form of the linearized equations is d⃗ y d˜t = J⃗ y, J =    − b00 1 0 0 ue 2 − ...

    work page

  2. [2]

    Among them, only P13 is controllable by the experi- ment

    Plot of the periodic SR region As pointed out in Sec.III C, there are five important parameters in our model; P13, A31, A32, A21, and γ32. Among them, only P13 is controllable by the experi- ment. Nevertheless, we have varied all these parame- ters to study the parameter dependence of the periodic SR region. For easier understanding, only ( A21, P 13) are ...

    work page

  3. [3]

    (A2) - (A6), solved by the Runge-Kutta- fourth-order method

    Results of the numerical simulation In this section, we present numerically integrated solu- tions of the X2MB equations, actually the dimensionless version Eqs. (A2) - (A6), solved by the Runge-Kutta- fourth-order method. We first show a typical solution both inside and outside the periodic SR regions. We then give a brief summary of our numerical simulat...

    work page

  4. [4]

    Brief summary of numerical simulation As shown above, we performed the numerical simu- lation of the X2MB model by selecting one point each from inside and outside the periodic SR region. As ex- pected from the analysis of the eigenvalues of the lin- earized X2MB equation, the periodic SR is reproduced 9 inside the periodic SR region and the SR pulses dec...

    work page

  5. [5]

    (A11) and (A12), we approximate u2 and u3 by replacing ue 2 and ue

    In Eqs. (A11) and (A12), we approximate u2 and u3 by replacing ue 2 and ue

    work page

  6. [6]

    dY dτ = − (X 2 − 1), (A26) du3 dτ = 0, (A27) The T2B model, Eqs

    Thus, the following equations are obtained. dY dτ = − (X 2 − 1), (A26) du3 dτ = 0, (A27) The T2B model, Eqs. (10) and (11), is derived by the above procedure. In addition, ρ11(= 1 − 2u3) is constant in this model. 12 Appendix B: Comparison of the analytical solution of the T2B model and numerical simulation of the X2MB model To confirm the validity of the ...

    work page

  7. [7]

    Formation of refractive index modulation The population of |3⟩ in Fig

    Principle a. Formation of refractive index modulation The population of |3⟩ in Fig. 2 is gradually increased by the pumping P13. The excited Er 3+ ions deexcite to |2⟩ with photons, which generate an electric field. The reflection of the electric field at the two crystal surfaces creates a very low finesse cavity. The surface reflection, estimated from the ref...

    work page

  8. [8]

    According to the results of Ref

    Refractive index variation ∆n We estimate the refractive index variation ∆ n for a given electric field ESR. According to the results of Ref. [21], in particular Eq.(6.3.28) [22], the electric suscepti- bility χ e is given by χ e = − α 0(0) ω 32/c δT2 − i 1 + (δT2)2 + |E|2/ |E0 S|2 , (C10) |E0 S|2 = ℏ2 4|d32|2T1T2 , (C11) α 0(0) = − ω 32 c [ N0(ρ33 − ρ22)|...

    work page

  9. [9]

    R. H. Dicke, Coherence in Spontaneous Radiation Pro- cesses, Phys. Rev. 93, 99 (1954)

    work page 1954

  10. [10]

    Skribanowitz, I

    N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, Observation of Dicke Superradiance in Opti- cally Pumped HF Gas, Phys. Rev. Lett. 30, 309 (1973)

    work page 1973

  11. [11]

    Gross, C

    M. Gross, C. Fabre, P. Pillet, and S. Haroche, Observation of Near-Infrared Dicke Superradi- ance on Cascading Transitions in Atomic Sodium, Phys. Rev. Lett. 36, 1035 (1976). 15

    work page 1976

  12. [12]

    Carlson, D

    N. Carlson, D. Jackson, A. Schawlow, M. Gross, and S. Haroche, Superradiance triggering spectroscopy, Optics Communications 32, 350 (1980)

    work page 1980

  13. [13]

    Florian, L

    R. Florian, L. Schwan, and D. Schmid, Superradiance and high-gain mirrorless laser activity of O − 2-centers in KCl, Solid State Communications 42, 55 (1982)

    work page 1982

  14. [14]

    Rain` o, M

    G. Rain` o, M. A. Becker, M. I. Bodnarchuk, R. F. Mahrt, M. V. Kovalenko, and T. St¨ oferle, Superfluorescence from lead halide perovskite quantum dot superlattices, Nature 563, 671 (2018)

    work page 2018

  15. [15]

    Angerer, K

    A. Angerer, K. Streltsov, T. Astner, S. Putz, H. Sumiya, S. Onoda, J. Isoya, W. J. Munro, K. Nemoto, J. Schmied- mayer, and J. Majer, Superradiant emission from colour centres in diamond, Nature Physics 14, 1168 (2018)

    work page 2018

  16. [16]

    Braggio, F

    C. Braggio, F. Chiossi, G. Carugno, A. Or- tolan, and G. Ruoso, Spontaneous formation of a macroscopically extended coherent state, Phys. Rev. Research 2, 033059 (2020)

    work page 2020

  17. [17]

    K. Cong, Q. Zhang, Y. Wang, G. T. Noe, A. Belyanin, and J. Kono, Dicke superradiance in solids, J. Opt. Soc. Am. B 33, C80 (2016)

    work page 2016

  18. [18]

    H. Hara, J. Han, Y. Imai, N. Sasao, A. Yoshimi, K. Yoshimura, M. Yoshimura, and Y. Miyamoto, Periodic superradiance in an Er:YSO crystal, Physical Review Research 6, 013005 (2024)

    work page 2024

  19. [19]

    Benedict, A

    M. Benedict, A. Ermolaev, V. Malyshev, I. Sokolov, and E. Trifonov, Super-radiance: Multiatomic Coherent Emission (CRC Press, 1996)

    work page 1996

  20. [20]

    Sanchez, P

    F. Sanchez, P. L. Boudec, P.-L. Fran¸ cois, and G. Stephan, Effects of ion pairs on the dynamics of erbium-doped fiber lasers, Phys. Rev. A 48, 2220 (1993)

    work page 1993

  21. [21]

    S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engi- neering (Studies in Nonlinearity) (2001)

    work page 2001

  22. [22]

    Bonifacio and L

    R. Bonifacio and L. A. Lugiato, Cooperative radia- tion processes in two-level systems: Superfluorescence, Phys. Rev. A 11, 1507 (1975)

    work page 1975

  23. [23]

    R. M. Corless, G. H. Gonnet, D. E. Hare, D. J. Jeffrey, and D. E. Knuth, On the lambert W function, Advances in Computational mathematics 5, 329 (1996)

    work page 1996

  24. [24]

    Petersen, High-resolution spectroscopy of praseodymium ions in a solid matrix: towards single-ion detection sensitivity , Ph.D

    L. Petersen, High-resolution spectroscopy of praseodymium ions in a solid matrix: towards single-ion detection sensitivity , Ph.D. thesis, ETH Zurich (2011)

    work page 2011

  25. [25]

    Carvalho, J.-M

    N. Carvalho, J.-M. Le Floch, J. Krupka, and M. E. To- bar, Multi-mode technique for the determination of the biaxial Y 2SiO5 permittivity tensor from 300 to 6 K, Ap- plied Physics Letters 106, 192904 (2015)

    work page 2015

  26. [26]

    B¨ ottger, Y

    T. B¨ ottger, Y. Sun, C. W. Thiel, and R. L. Cone, Spectroscopy and dynamics of Er 3+:Y2SiO5 at 1.5 µm, Phys. Rev. B 74, 075107 (2006)

    work page 2006

  27. [27]

    Shang, T

    J. Shang, T. Feng, S. Zhao, J. Zhao, Y. Zhao, Y. Song, and T. Li, An investigation into self-pulsing behavior in an Er-doped ring laser, Applied Physics Express 13, 112006 (2020)

    work page 2020

  28. [28]

    G. R. Fowles, Introduction to Modern Optics (Courier Corporation, 1989)

    work page 1989

  29. [29]

    R. W. Boyd, Nonlinear Optics, Third Edition , 3rd ed. (Academic Press, Inc., USA, 2008)

    work page 2008

  30. [30]

    It assumes a near-resonant, steady-state two-level sy s- tem

    work page