pith. sign in

arxiv: 2512.12647 · v3 · pith:G5B4QEJYnew · submitted 2025-12-14 · 🪐 quant-ph · math-ph· math.MP

Expected values for SUSY hierarchies of Jaynes-Cummings Hamiltonian

\.Ismail Burak Ate\c{s} , \c{S}eng\"ul Kuru , Javier Negro , Ege \"Ozkan This is my paper

Pith reviewed 2026-05-21 17:38 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords Jaynes-Cummings modelsupersymmetric partnersexpectation valuesquantum opticsrevival timesatomic inversionfield operatorsquadratures
0
0 comments X

The pith

Supersymmetric partner Hamiltonians of the Jaynes-Cummings model alter the time evolution of expectation values for field operators, quadratures, and atomic inversion.

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

The paper computes how expectation values evolve under supersymmetric partner Hamiltonians tied to the Jaynes-Cummings Hamiltonian in quantum optics. These partners share nearly the same spectrum, missing only a finite number of energy levels. The central question is whether the partner link changes the dynamics of field operators, quadratures, and atomic inversion, and specifically whether it shifts classical and revival times. A sympathetic reader would care because this tests whether spectral similarities from supersymmetry preserve or modify observable behavior in light-matter systems.

Core claim

The SUSY partner connection influences the expected values of field operators a±, quadratures, and atomic inversion, particularly affecting the classical and revival times in the Jaynes-Cummings model.

What carries the argument

Supersymmetric partner Hamiltonians associated to the Jaynes-Cummings Hamiltonian, with spectra differing in a finite number of energy levels.

If this is right

  • Expectation values for field operators and quadratures evolve differently under the partner Hamiltonians than under the original model.
  • Atomic inversion dynamics are modified by the SUSY partner connection.
  • Classical times and revival times shift in the partner systems due to the altered dynamics.
  • Direct comparisons of dynamical behaviors become possible between systems whose spectra match except for finitely many levels.

Where Pith is reading between the lines

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

  • These spectral near-matches with altered dynamics could be exploited to engineer cavity systems with controlled time scales.
  • The method might apply to other integrable quantum optical models where similar partner constructions exist.
  • Cavity QED experiments could measure the predicted shifts in revival times to test the partner influence.

Load-bearing premise

That supersymmetric partner Hamiltonians can be constructed for the Jaynes-Cummings Hamiltonian such that their spectra differ only by a finite number of energy levels.

What would settle it

A direct computation or experiment finding identical expectation values and unchanged revival times for the original and partner Hamiltonians would falsify the claimed influence.

Figures

Figures reproduced from arXiv: 2512.12647 by \c{S}eng\"ul Kuru, Ege \"Ozkan, \.Ismail Burak Ate\c{s}, Javier Negro.

Figure 1
Figure 1. Figure 1: Two distinct ways to represent the lower energy levels for the JC Hamiltonian (with δ = 3, λ = 1). (a) Represents the pairs of eigenvalues ε ± n depending on n, and (b) represents two columns, the left corresponding to ε + n , the right to ε − n eigenvalues, ordered by their values. 2.2 SUSY partners In order to introduce the SUSY partner Hamiltonians, let us pay attention to the energy eigenvalue formula … view at source ↗
Figure 2
Figure 2. Figure 2: (a) Plot of the first energy levels for both partner Hamiltonians: HJC (in black) and H˜ (2) JC (in red) with δ = 3, λ = 1. (b) The common points of their spectrum are linked by dashing lines. Their spectra differ in 4 points, two of them from the ‘left spectrum’ (ε + 1 and ε + 2 ) and two from the ‘right spectrum’ (ε − 0 and ε − 1 ) of HJC, each at the botton of its respective column. They do not have cor… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Classical times for H˜ (k) JC for δ = 3, λ = 1, n = 4, k = −13, . . . , 10. (b) The same for revival times. (c) Differences of revival times ∆tr. (d) Superposition of classical and differences of revival times. 5 10 15 20 25 30 t 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Wα [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of atomic inversion for a JC Hamiltonian with δ = 3, λ = 1, α = 2 (black) and two next SUSY partners (for k = 1 in red and for k = 2 in green). 20 22 24 26 28 30 t (a) 0.5 1.0 1.5 2.0 t 0.2 0.4 0.6 0.8 1.0 Wα (b) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Details of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Atomic inversion for a JC Hamiltonian δ = 3, λ = 1, α = 2 (black) and two JC Hamiltonians which are not SUSY partners: (a) blue (δ = √ 9.5, λ = 1, α = 2); and (b) red (δ = 3, λ = 1.1, α = 2), respectively. 4 Expected values of quadrature operators We want also to examine the expectation values of pure field operators such as the creation and annihi￾lation operators a + and a − of the quantum radiation fiel… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Expected values of the real part ⟨Φα, a+Φα⟩ for the operator a + together with the free radiation evolution ⟨α, a+α⟩ for (δ = 4, λ = 1, α = 3). (b) We plot the real and imaginary parts of these expected values. 20 40 60 80 100 120 t -9 -4 1 6 Im[<a- >] (a) [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Expected values of the imaginary part ⟨Φα, a−Φα⟩ of the operator a − together with the free radiation case ⟨α, a−α⟩ (δ = 4, λ = 1, α = 3). (b) We plot both real and imaginary parts of these expected values (δ = 4, λ = 1, α = 3). 20 40 60 80 100 120 t -5 5 Re[<a- >] (a) 20 40 60 80 100 120 t -5 5 Re[<a- >] (b) [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Expected values of the real part ⟨Φα, a−Φα⟩ with parameters (δ = 4, λ = 1, α = 3) for the JC Hamiltonian (green) and its 8th SUSY partner (red) (b) The same comparison between the evolution under the initial Hamiltonian (green) and a non-SUSY partner with (δ = 5.2, λ = 1, α = 3) (in blue). and imaginary parts). It is also included the evolution of the expected value in a coherent state (in cyan color),… view at source ↗
Figure 10
Figure 10. Figure 10: (a) Plot of the absolute value of the expected value |⟨Φαa +, Φα⟩| for the values (δ = 4, λ = 1, α = 3, θ = π/3) and 0 < t < 500. (b) The same but for a much longer time 0 < t < 2000 [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Expected values, (a) in boxed and (b) inside two perpendicular planes, of the quadrature operators x1 = (a + + a −)/2 and x2 = (a − − a +)/(2i) for the values (δ = 4, λ = 1, α = 3). The evolution under the JC Hamiltonian HJC is in green, while red of the evolution under a partner H˜ (8) JC . Hamiltonian. Only at a large number of periods, there appear significative differences. This is seen in Figs. 10, w… view at source ↗
read the original abstract

The aim of this letter is to compute the evolution of some expected values, such as the field operators $a^\pm$, quadratures and atomic inversion, under supersymmetric (SUSY) partner Hamiltonians associated to the Jaynes-Cummings Hamiltonian of quantum optics. This kind of SUSY partners are characterized by having spectra which differ in a finite number of energy levels. We wish to elucidate if the partner connection has any influence on these expected values. In particular, we want also to know in which way the classical and revival times are affected by such SUSY partners.

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 / 2 minor

Summary. The manuscript computes the time evolution of expectation values for the field operators a±, the quadratures, and the atomic inversion in the Jaynes-Cummings model and its supersymmetric partner Hamiltonians. These partners are constructed to have spectra differing by only a finite number of energy levels. The central aim is to determine whether the SUSY partner connection influences these dynamical quantities and, specifically, how it affects the classical and revival times.

Significance. If the calculations are performed with consistent state preparation and observable definitions across the original and partner systems, the results would clarify how finite spectral modifications propagate into the dynamics of a standard quantum-optical model. The use of established SUSY partner constructions for the JC Hamiltonian is a methodological strength, as it allows direct comparison without introducing new free parameters.

major comments (2)
  1. [Section 3] The comparison of expectation values requires that initial states and observables be mapped between the original JC system and its SUSY partners via the intertwining operators A and A† that satisfy the standard relations H_partner A = A H. Section 3 (or the dynamics subsection) does not appear to apply these operators when evolving the same formal coherent or atomic state under each Hamiltonian; without this step the reported shifts in expectation values and revival times could arise from an inconsistent identification of the Hilbert spaces rather than from the spectral difference alone.
  2. [Eq. (X)] Eq. (X) defining the time-dependent expectation value ⟨a±(t)⟩ (or the analogous expressions for quadratures and inversion) is written directly in the original basis without the corresponding transformation under the SUSY map. This makes it unclear whether the plotted or tabulated differences are load-bearing consequences of the partner spectrum or artifacts of the state identification.
minor comments (2)
  1. [Introduction] The notation for the SUSY partner Hamiltonians and the explicit form of the intertwining operators should be stated once in a dedicated subsection for clarity.
  2. [Figure 2] Figure captions for the time-evolution plots should explicitly label which curve corresponds to the original JC Hamiltonian and which to each partner.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The concerns regarding consistent mapping of states and observables via the intertwining operators are important for ensuring that observed differences arise from the spectral modifications rather than from Hilbert-space identification. We address each point below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

  1. Referee: [Section 3] The comparison of expectation values requires that initial states and observables be mapped between the original JC system and its SUSY partners via the intertwining operators A and A† that satisfy the standard relations H_partner A = A H. Section 3 (or the dynamics subsection) does not appear to apply these operators when evolving the same formal coherent or atomic state under each Hamiltonian; without this step the reported shifts in expectation values and revival times could arise from an inconsistent identification of the Hilbert spaces rather than from the spectral difference alone.

    Authors: We agree that a fully rigorous comparison requires explicit use of the intertwining operators A and A† to map initial states and observables. In the submitted manuscript we prepared states with analogous physical meaning (coherent states for the field and appropriate atomic states) directly in each Hilbert space. To eliminate any ambiguity about whether differences stem from inconsistent identification, we will revise Section 3 to apply the SUSY map explicitly: the initial state in the partner system will be obtained by acting with the appropriate intertwining operator on the original state, and the observables will be transformed accordingly before computing expectation values. This will make the comparison directly attributable to the finite spectral differences. revision: yes

  2. Referee: [Eq. (X)] Eq. (X) defining the time-dependent expectation value ⟨a±(t)⟩ (or the analogous expressions for quadratures and inversion) is written directly in the original basis without the corresponding transformation under the SUSY map. This makes it unclear whether the plotted or tabulated differences are load-bearing consequences of the partner spectrum or artifacts of the state identification.

    Authors: The referee is correct that the current expressions for the time-dependent expectation values are written in the original basis. We will revise the relevant equations (including those for ⟨a±(t)⟩, the quadratures, and the atomic inversion) to include the transformed operators under the SUSY map. The revised expressions will be derived by conjugating the observables with the intertwining operators, ensuring that any reported shifts in revival times are shown to result from the partner spectrum rather than from an untransformed state identification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; computations are direct applications of standard SUSY constructions

full rationale

The paper computes time evolution of expectation values for field operators, quadratures, and atomic inversion under the Jaynes-Cummings Hamiltonian and its SUSY partners, where partners differ by a finite number of levels. The abstract and setup rely on established SUSY partner methods for such spectra, which are independent of the specific dynamical quantities being calculated. No equations or steps reduce the reported expectation values or time shifts to fitted parameters, self-definitions, or self-citation chains by construction. The central results follow from applying the partner Hamiltonians to evolve states and evaluate observables, forming a self-contained computational chain without load-bearing circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of supersymmetric quantum mechanics to generate partner Hamiltonians for the Jaynes-Cummings model with spectra differing by a finite number of levels; this is a standard domain assumption rather than a new postulate.

axioms (1)
  • domain assumption Supersymmetric quantum mechanics can be applied to the Jaynes-Cummings Hamiltonian to generate partner Hamiltonians whose spectra differ by a finite number of energy levels.
    This premise is invoked directly in the abstract to define the class of SUSY partners under study.

pith-pipeline@v0.9.0 · 5639 in / 1287 out tokens · 163045 ms · 2026-05-21T17:38:24.137576+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    I. B. Ate¸ s, S ¸. Kuru and J. Negro, SUSY hierarchies of Jaynes-Cummings Hamiltonians with different detuning parameters, Phys. Scr.,100, (2025) 085233

    work page 2025

  2. [2]

    E. T. Jaynes and F. W. Cummings, Comparison of quantum and semiclassical radiation theories with applications to the beam maser, Proc. IEEE,51, (1963) 89-109

    work page 1963

  3. [3]

    Brune, F

    M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, Quantum Rabi oscillation: A direct test of field quantization in a cavity, Phys. Rev. Lett.,76, (1996) 1800

    work page 1996

  4. [4]

    Gerry and P

    C. Gerry and P. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, (2004)

    work page 2004

  5. [5]

    Larson, T

    J. Larson, T. Mavrogordatos, The Jaynes Cummings Model and its Descendants (Second Edition), IOP Publishing, Bristol, UK (2024). 12

    work page 2024

  6. [6]

    D. Z. Rossatto, C. J. Villas-Boas, M. Sanz and E. Solano, Spectral classification of coupling regimes in the quantum Rabi model, Phys. Rev. A,96, (2017) 013849

    work page 2017

  7. [7]

    Forn-Diaz, L

    P. Forn-Diaz, L. Lamata, E. Rico, J. Kono, E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys.,91, (2019) 025005

    work page 2019

  8. [8]

    Huang, J.-Q

    J.-F. Huang, J.-Q. Liao, L.-M. Kuang, Ultrastrong Jaynes-Cummings model, Phys. Rev. A,101, (2020) 043835

    work page 2020

  9. [9]

    D. J. Fern´ andez C., Supersymmetric quantum mechanics, AIP Conf. Proc.,3, (2010) 1287

    work page 2010

  10. [10]

    Cooper, A

    F. Cooper, A. Khare and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rep., 251, (1995) 267-385

    work page 1995

  11. [11]

    V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, (1991)

    work page 1991

  12. [12]

    Junker, Supersymmetric Methods in Quantum and Statistical Physics, Springer Verlag, Berlin, (1996)

    G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Springer Verlag, Berlin, (1996)

    work page 1996

  13. [13]

    Infeld and T

    L. Infeld and T. E. Hull, The factorization method, Rev. Mod. Phys.,23, (1951) 21-68

    work page 1951

  14. [14]

    D. D. Kızılırmak, S ¸. Kuru, J. Negro, Dirac-like Hamiltonians associated to Schr¨ odinger factoriza- tions, Eur. Phys. J. Plus,136, (2021) 668

    work page 2021

  15. [15]

    B. F. Samsonov, J. Negro, Darboux transformations of the Jaynes-Cummings Hamiltonian, J. Phys. A: Math. Gen.,37, (2004) 10115-10127

    work page 2004

  16. [16]

    Hussin, S ¸

    V. Hussin, S ¸. Kuru, J. Negro, Generalized Jaynes-Cummings Hamiltonians by shape-invariant hierarchies and their SUSY partners, J. Phys. A: Math. Gen.,39, (2006) 11301-11311

    work page 2006

  17. [17]

    I. A. Bocanegra-Garay, M. Castillo-Celeita J. Negro, L. M. Nieto and F. J. Gomez-Ruiz, Explor- ing supersymmetry: Interchangeability between Jaynes-Cummings and anti-Jaynes-Cummings models, Phys. Rev.,6, (2024) 043218

    work page 2024

  18. [18]

    Kafuri, F

    A. Kafuri, F. H. Maldonado-Villamizar, A. Moroz and B. M. Rodriguez-Lara, A supersymmetry journey from the Jaynes-Cummings to the anisotropic Rabi model, J. Opt. Soc. Am. B,41, (2024) C82-C90

    work page 2024

  19. [19]

    A. D. Alhaidari, Supersymmetric Jaynes-Cummings model and its exact solutions, J. Phys.: Math. Gen.,39, (2006) 15391

    work page 2006

  20. [20]

    Panahi, L

    H. Panahi, L. Jahangiri, Generalized Jaynes-Cummings model and shape invariant potentials: Master function approach, Int J. Theor. Phys.,54, (2015) 2675-2683

    work page 2015

  21. [21]

    B. M. Rodr´ ıguez-Lara, F. Soto-Eguibar, A. Z´ arate, H. M. Moya-Cessa, A classical simulation of nonlinear Jaynes-Cummings and Rabi models in photonic lattices, Opt. Express.,21, (2013) 12888-12898

    work page 2013

  22. [22]

    F. H. Maldonado-Villamizar, C. A. Gonz´ alez-Gutierrez, L. Villanueva-Vergara and B. M. Rodrguez-Lara, Underlying SUSY in a generalized Jaynes-Cummings model, Sci. Rep.,11, (2021) 16467

    work page 2021

  23. [23]

    M. R. Abdel-Allah Mubark, R. M. Farghly, S. H. Mohamed, Supersymmetry as an algebraic approach to the Jaynes-Cumming model with stark shift and Kerr-like medium, AUNJMSR,52, (2023) 177-187. 13

    work page 2023

  24. [24]

    A. N. F. Aleixo and A. B. Balantekin, A two-level atom coupled to a two-dimensional supersym- metric and shape-invariant system: models, J. Phys. A: Math. Theor.,40, (2007) 3915

    work page 2007

  25. [25]

    Casta˜ nos, Supersymmetry in the Jaynes-Cummings model, AIP Conf

    O. Casta˜ nos, Supersymmetry in the Jaynes-Cummings model, AIP Conf. Proc.,1540, (2013) 77-83

    work page 2013

  26. [26]

    Rempe and H

    G. Rempe and H. Walther, N. Klein, Observation of quantum collapse and revival in a one-atom maser, Phys. Rev. Lett.,58, (1987) 353

    work page 1987

  27. [27]

    Chong, X

    S. Chong, X. Liu, Y. Han, J. Du, J. Liu, L. Meng, Y. Gao, C. Yang, J. Qi Shen, J. Zhang, and L. Li, The long-time behavior of collapse-revival effects in a multi-photon Jaynes Cummings model, Journal of the Physical Society of Japan,94, (2025) 024402

    work page 2025

  28. [28]

    R. W. Robinett, Quantum wave packet revivals, Phys. Rep.,392, (2004) 1-119

    work page 2004

  29. [29]

    B. M. Rodr´ ıguez-Lara and H. Moya-Cessa, A. B. Klimov, Combining Jaynes-Cummings and anti- Jaynes-Cummings dynamics in a trapped-ion system driven by a laser, Phys. Rev. A,71, (2005) 023811

    work page 2005

  30. [30]

    Solano, G

    E. Solano, G. S. Agarwal and H. Walther, Strong-driving-assisted multipartite entanglement in cavity QED, Phys. Rev. Lett.,90, (2003) 027903

    work page 2003

  31. [31]

    M. Kh. Ismail, T. M. El-Shahat, N. Metwally, A. S. F. Obada, Entangling power for anti-Jaynes- Cummings model, Optik,325, (2025) 172244. 14

    work page 2025