pith. sign in

arxiv: 2606.11822 · v1 · pith:XHPN23BTnew · submitted 2026-06-10 · 🪐 quant-ph · cond-mat.mes-hall

Large Fluctuations in Open Quantum Systems

V. Yu. Mylnikov , S. O. Potashin , A. Kamenev This is my paper

Pith reviewed 2026-06-27 09:55 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords large-deviation functionopen quantum systemsdriven dissipative systemsinstanton trajectoriesKeldysh-Lindblad actionKerr oscillatornon-analyticityrare fluctuations
0
0 comments X

The pith

In driven dissipative quantum systems the large-deviation function for rare fluctuations develops lines and surfaces of discontinuous derivatives.

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

In equilibrium the probability distribution over phase space, such as the Wigner function, is analytic. The paper shows that this analyticity is generically lost once the system is both driven and dissipative. The large-deviation function that governs the statistics of atypical steady-state outcomes then contains lines or surfaces where its derivatives jump. These discontinuities arise because the same rare fluctuation can be produced by several distinct semiclassical trajectories whose dominance switches abruptly. The effect is demonstrated for a parametrically driven Kerr oscillator coupled linearly or nonlinearly to a bath.

Core claim

We show that this property is generically lost in driven dissipative systems: their large-deviation function develops lines and surfaces across which its derivatives are discontinuous. Rare fluctuations in the amplitude and phase of the induced oscillations are governed by semiclassical instanton trajectories of the corresponding Keldysh-Lindblad action. We demonstrate that a given fluctuation can be realized through multiple distinct instanton trajectories. The competition between these trajectories leads to abrupt switching of the dominant instanton and, consequently, to non-analytic features in the large-deviation function.

What carries the argument

Competition among multiple distinct semiclassical instanton trajectories of the Keldysh-Lindblad action that realize the same rare fluctuation.

If this is right

  • The probabilities of atypical measurement outcomes in driven open systems can change abruptly across certain surfaces in phase space.
  • In the Kerr-oscillator example the statistics of amplitude and phase fluctuations exhibit these derivative discontinuities.
  • The dominant instanton trajectory switches at the locations of the non-analytic lines or surfaces.
  • The non-analyticity is presented as a generic feature of driven dissipative steady states rather than a special case.

Where Pith is reading between the lines

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

  • Similar competition between trajectories could produce non-analytic large-deviation functions in other open quantum models that admit multiple instanton solutions.
  • Numerical sampling of rare events in driven dissipative systems may need to account for the abrupt switches rather than assuming smooth interpolation.
  • The locations of the non-analytic surfaces might serve as signatures that distinguish driven-dissipative dynamics from equilibrium ones in experiments.

Load-bearing premise

Rare fluctuations are governed by semiclassical instanton trajectories of the Keldysh-Lindblad action, and a given fluctuation can be realized through multiple distinct such trajectories whose competition produces the non-analyticity.

What would settle it

A calculation or measurement of the large-deviation function for the parametrically driven Kerr oscillator that remains everywhere differentiable would show that the claimed non-analyticities do not appear.

Figures

Figures reproduced from arXiv: 2606.11822 by A. Kamenev, S. O. Potashin, V. Yu. Mylnikov.

Figure 1
Figure 1. Figure 1: (a,b) Stationary Wigner function W0(α, α∗ ) and effective potential − ln W0(α, α∗ ) (c,d) vs. phase-space coordinates Re(α) and Im(α). The first row presents the case of zero dephasing (κϕ = 0), where the exact (a,c) and WKB (d) solutions present (see Eqs. (15)-(17) and Appendix A). The second row shows the nonzero dephasing case (κϕ = 0.2), which does not admit the holomorphic decomposition (15) and is ca… view at source ↗
Figure 2
Figure 2. Figure 2: The real-time instanton trajectories, Eq. 37, spanning the fluctuation Riemann surface, [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Stokes lines (red) on the complex α plane, as de￾fined by Eq. (47). Within the inner Stokes region (blue), only the instanton with S− action contributes to the Wigner func￾tion. Within the outer Stokes region (green), both instantons with S± contribute. The two actions equalize along the anti￾Stokes lines (black), given by Eq. (23). The large deviation function is non-analytic along these lines. One possib… view at source ↗
Figure 4
Figure 4. Figure 4: (a,b) The large-deviation function from the in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The α-plane flow and the phase-slip instanton path (orange line) in the bistable regime. The phase-slip path starts at the classical fixed point α = α0 and ends at the saddle point α = 0 (red dots). From the saddle point to the opposite stationary point, the system evolves classically (i.e., in χ = 0 subspace), according to Eq. (9), with no action ac￾cumulated along this part of the phase-slip path. The sy… view at source ↗
Figure 6
Figure 6. Figure 6: (a-b) The large deviation function S(α, α¯) = min{S−(α, α¯), S+(α, α¯)} vs. phase-space coordinates Re(α) and Im(α). It is calculated numerically for zero dephasing rate κϕ = 0 (a) and finite dephasing κϕ = 0.03 (b) using Eq. (57). Red and blue curves correspond to nonanalytic lines, where two instanton actions are equal S−(α, α¯) = S+(α, α¯). The system parameters are set to G = 5, ∆ = 2, κ = 0.4, η = 1, … view at source ↗
Figure 7
Figure 7. Figure 7: Stokes regions near the turning point z = 1. Within the inner Stokes region (blue), there is only one WKB solution Ψ−. Within the outer Stokes region (green), the solution is the sum of two WKB terms Ψ− + Ψ+. The Stokes lines are shown in red. A possible branch cut is shown in blue dashed line. in [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

We study statistics of atypical measurement outcomes in the steady states of driven open quantum systems. In equilibrium, the probability distribution over the phase space, as encoded in, e.g., the Wigner function, is analytic in the phase-space coordinates. We show that this property is generically lost in driven dissipative systems: their {\it large-deviation function} develops lines and surfaces across which its derivatives are discontinuous. As an illustrative example, we consider a parametrically driven Kerr oscillator coupled linearly and/or nonlinearly to a dissipative bath. Rare fluctuations in the amplitude and phase of the induced oscillations are governed by semiclassical instanton trajectories of the corresponding Keldysh-Lindblad action. We demonstrate that a given fluctuation can be realized through multiple distinct instanton trajectories. The competition between these trajectories leads to abrupt switching of the dominant instanton and, consequently, to non-analytic features in the large-deviation function.

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 claims that in driven dissipative quantum systems the large-deviation function for atypical steady-state measurement outcomes develops lines and surfaces across which its derivatives are discontinuous, in contrast to the analytic phase-space distributions (e.g., Wigner function) found in equilibrium. This non-analyticity is attributed to competition among multiple distinct semiclassical instanton trajectories of the Keldysh-Lindblad action; the claim is illustrated with a parametrically driven Kerr oscillator coupled linearly or nonlinearly to a bath, where abrupt switching between dominant instantons produces the kinks or cusps.

Significance. If substantiated, the result would establish a qualitative distinction between equilibrium and driven open quantum systems in the structure of large-deviation functions, with potential consequences for the statistics of rare events in quantum optics and related platforms. The mechanistic identification of instanton competition as the origin of the non-analyticity supplies a concrete, falsifiable picture within the standard Keldysh-Lindblad formalism.

major comments (2)
  1. [Kerr-oscillator example (semiclassical treatment)] The central claim that non-analyticities survive generically rests on the semiclassical saddle-point approximation; the Kerr-oscillator example supplies no explicit bound on the magnitude of subleading fluctuation determinants or interference terms that could round the discontinuities across the switching surface.
  2. [Abstract and illustrative example] Generality is asserted from a single model without a parameter-free argument showing that the same multiple-trajectory competition mechanism persists in other driven-dissipative systems; the abstract states the property is 'generically lost' but the supporting analysis is model-specific.
minor comments (1)
  1. Notation for the large-deviation function I(observable) and the precise definition of the observable (amplitude/phase) should be introduced earlier for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, clarifying the scope of the semiclassical analysis and the generality of the proposed mechanism.

read point-by-point responses

  1. Referee: [Kerr-oscillator example (semiclassical treatment)] The central claim that non-analyticities survive generically rests on the semiclassical saddle-point approximation; the Kerr-oscillator example supplies no explicit bound on the magnitude of subleading fluctuation determinants or interference terms that could round the discontinuities across the switching surface.

    Authors: The large-deviation function is defined as the leading exponential rate I = -lim (1/N) log P, obtained from the saddle-point evaluation of the Keldysh-Lindblad path integral. This rate is exactly the minimum action among competing instantons; at a switching surface the rate function is the lower envelope of two smooth branches and is therefore non-analytic whenever the gradients differ. Sub-exponential corrections arising from fluctuation determinants or interference enter only the prefactor and cannot remove the non-analyticity of the leading rate function itself. In open dissipative systems the environment further suppresses coherent interference between macroscopically distinct trajectories. We will add a short clarifying paragraph on this point in the revised manuscript. revision: partial

  2. Referee: [Abstract and illustrative example] Generality is asserted from a single model without a parameter-free argument showing that the same multiple-trajectory competition mechanism persists in other driven-dissipative systems; the abstract states the property is 'generically lost' but the supporting analysis is model-specific.

    Authors: The mechanism follows from the general structure of the Keldysh-Lindblad action for driven systems: the drive term renders the effective potential non-Hermitian, permitting multiple distinct instanton solutions that can cross in action. In equilibrium the corresponding action reduces to a form whose minimizing trajectory is unique for each observable, restoring analyticity. This distinction is independent of the specific Kerr parameters and holds for any driven dissipative system whose steady-state manifold supports multiple attractors. The Kerr oscillator serves only as an explicit illustration; we will revise the abstract and introduction to separate the general structural argument from the concrete example. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard Keldysh-Lindblad instanton analysis to derive non-analyticity

full rationale

The paper's central result follows from minimizing the Keldysh-Lindblad action over semiclassical trajectories for the driven Kerr oscillator model; multiple distinct instantons compete to produce switching and kinks in the large-deviation function as a direct consequence of the variational problem. No step reduces by definition to the target non-analyticity, no parameters are fitted to the observable being predicted, and no load-bearing self-citation or imported uniqueness theorem is required. The derivation remains self-contained within the established formalism.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the semiclassical instanton approximation within the Keldysh-Lindblad framework for large deviations; no free parameters or new entities are introduced in the abstract description.

axioms (1)
  • domain assumption Semiclassical instanton trajectories of the Keldysh-Lindblad action govern rare fluctuations in driven open quantum systems.
    Invoked directly in the abstract as the method for describing atypical measurement outcomes.

pith-pipeline@v0.9.1-grok · 5686 in / 1228 out tokens · 22642 ms · 2026-06-27T09:55:58.840345+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

71 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Leghtas, G

    Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi, Phys. Rev. Lett.111, 120501 (2013)

    2013

  2. [2]

    Mirrahimi, Z

    M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, New J. Phys. 16, 045014 (2014)

    2014

  3. [3]

    Grimm, N

    A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, Nature584, 205 (2020)

    2020

  4. [4]

    S. Puri, S. Boutin, and A. Blais, npj Quantum Inf.3, 18 (2017)

    2017

  5. [5]

    Goto, Scientific Reports6, 21686 (2016), publisher: Nature Publishing Group

    H. Goto, Scientific Reports6, 21686 (2016), publisher: Nature Publishing Group

    2016

  6. [6]

    Zhang and M

    Y. Zhang and M. I. Dykman, Physical Review A95, 053841 (2017), publisher: American Physical Society

    2017

  7. [7]

    Ch´ avez-Carlos, M

    J. Ch´ avez-Carlos, M. A. Prado Reynoso, R. G. Corti˜ nas, I. Garc´ ıa-Mata, V. S. Batista, F. P´ erez-Bernal, D. A. Wisniacki, and L. F. Santos, Quantum Science and Tech- nology10, 015039 (2024), publisher: IOP Publishing

    2024

  8. [8]

    Venkatraman, R

    J. Venkatraman, R. G. Corti˜ nas, N. E. Frattini, X. Xiao, and M. H. Devoret, Proceedings of the National Academy of Sciences121, e2311241121 (2024), publisher: Proceed- ings of the National Academy of Sciences

    2024

  9. [9]

    R´ eglade, A

    U. R´ eglade, A. Bocquet, R. Gautier, J. Cohen, A. Marquet, E. Albertinale, N. Pankratova, M. Hall´ en, F. Rautschke, L.-A. Sellem, P. Rouchon, A. Sarlette, M. Mirrahimi, P. Campagne-Ibarcq, R. Lescanne, S. Je- zouin, and Z. Leghtas, Nature629, 778 (2024), pub- lisher: Nature Publishing Group

    2024

  10. [10]

    Marquet, S

    A. Marquet, S. Dupouy, U. R´ eglade, A. Essig, J. Co- hen, E. Albertinale, A. Bienfait, T. Peronnin, S. Jezouin, R.Lescanne, andB.Huard,PhysicalReviewApplied22, 034053 (2024), publisher: American Physical Society

    2024

  11. [11]

    Rugar and P

    D. Rugar and P. Gr¨ utter, Physical Review Letters67, 699 (1991), publisher: American Physical Society

    1991

  12. [12]

    R. B. Karabalin, R. Lifshitz, M. C. Cross, M. H. Ma- theny, S. C. Masmanidis, and M. L. Roukes, Physical Review Letters106, 094102 (2011), publisher: American Physical Society

    2011

  13. [13]

    Eichler, T

    A. Eichler, T. L. Heugel, A. Leuch, C. L. Degen, R. Chi- tra, and O. Zilberberg, Applied Physics Letters112, 233105 (2018)

    2018

  14. [14]

    Eichler and O

    A. Eichler and O. Zilberberg,Classical and Quantum Parametric Phenomena(Oxford University Press, Ox- ford, 2023)

    2023

  15. [15]

    J. Guo, Q. He, and M. Fadel, Physical Review A109, 052604 (2024), publisher: American Physical Society

    2024

  16. [16]

    Lescanne, M

    R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Delbecq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas, Nat. Phys.16, 509–513 (2020)

    2020

  17. [17]

    Touchette, Physics Reports478, 1 (2009)

    H. Touchette, Physics Reports478, 1 (2009)

    2009

  18. [18]

    Assaf and B

    M. Assaf and B. Meerson, Journal of Physics A: Mathe- matical and Theoretical50, 263001 (2017)

    2017

  19. [19]

    Graham and A

    R. Graham and A. Schenzle, Zeitschrift f¨ ur Physik B Condensed Matter52, 61 (1983)

    1983

  20. [20]

    M. I. Dykman, M. M. Millonas, and V. N. Smelyanskiy, Physics Letters A195, 53 (1994)

    1994

  21. [21]

    M. I. Freidlin and A. D. Wentzell, inRandom perturba- tions of dynamical systems(Springer, 1998) pp. 15–43

    1998

  22. [22]

    Graham and T

    R. Graham and T. T´ el, Phys. Rev. A31, 1109 (1985)

    1985

  23. [23]

    Graham and T

    R. Graham and T. T´ el, Physical review letters52, 9 (1984)

    1984

  24. [24]

    Graham and T

    R. Graham and T. T´ el, Journal of statistical physics35, 729 (1984)

    1984

  25. [25]

    Dembo and O

    A. Dembo and O. Zeitouni,Large deviations techniques and applications, Vol. 38 (Springer Science & Business Media, 2009)

    2009

  26. [26]

    P. C. Martin, E. D. Siggia, and H. A. Rose, Physical Review A8, 423 (1973)

    1973

  27. [27]

    Wigner, Physical review40, 749 (1932)

    E. Wigner, Physical review40, 749 (1932)

    1932

  28. [28]

    M. V. Berry, Philosophical Transactions of the Royal So- ciety of London. Series A, Mathematical and Physical Sciences287, 237 (1977)

    1977

  29. [29]

    Dykman, M

    M. Dykman, M. Marthaler, and V. Peano, Physical Review A—Atomic, Molecular, and Optical Physics83, 052115 (2011)

    2011

  30. [30]

    M. I. Dykman and V. N. Smelyanskii, Sov. Phys. JETP 67, 61 (1988)

    1988

  31. [31]

    Roberts, A

    D. Roberts, A. Lingenfelter, and A. Clerk, PRX Quan- tum2, 020336 (2021)

    2021

  32. [32]

    Roberts and A

    D. Roberts and A. A. Clerk, Phys. Rev. X10, 021022 (2020)

    2020

  33. [34]

    Kamenev,Field theory of non-equilibrium systems (Cambridge University Press, 2023)

    A. Kamenev,Field theory of non-equilibrium systems (Cambridge University Press, 2023)

    2023

  34. [35]

    Bartolo, F

    N. Bartolo, F. Minganti, W. Casteels, and C. Ciuti, Physical Review A94, 033841 (2016)

    2016

  35. [36]

    C. H. Meaney, H. Nha, T. Duty, and G. J. Milburn, EPJ quantum technology1, 7 (2014)

    2014

  36. [37]

    V. Y. Mylnikov, S. O. Potashin, G. S. Sokolovskii, and N. S. Averkiev, Nanomaterials12, 2543 (2022)

    2022

  37. [38]

    Downing and A

    C. Downing and A. Vidiella-Barranco, Scientific reports 13, 11004 (2023)

    2023

  38. [39]

    V. Y. Mylnikov, S. Potashin, G. Sokolovskii, and N. Averkiev, Physical Review Research7, 013061 (2025)

    2025

  39. [40]

    Walls and G

    D. Walls and G. Milburn,Quantum Optics(Springer Cham, 2025)

    2025

  40. [41]

    M. O. Scully and M. S. Zubairy,Quantum optics(Cam- bridge university press, 1997)

    1997

  41. [42]

    K. E. Cahill and R. J. Glauber, Phys. Rev.177, 1882 (1969)

    1969

  42. [43]

    Husimi, Proceedings of the Physico-Mathematical So- ciety of Japan

    K. Husimi, Proceedings of the Physico-Mathematical So- ciety of Japan. 3rd Series22, 264 (1940)

    1940

  43. [44]

    R. J. Glauber, Physical Review131, 2766 (1963)

    1963

  44. [45]

    P. D. Drummond and C. W. Gardiner, Journal of Physics A: Mathematical and General13, 2353 (1980)

    1980

  45. [46]

    Kinsler and P

    P. Kinsler and P. D. Drummond, Physical Review A43, 6194 (1991)

    1991

  46. [47]

    Marthaler and M

    M. Marthaler and M. Dykman, Physical Review A—Atomic, Molecular, and Optical Physics76, 010102 (2007)

    2007

  47. [48]

    Dykman, C

    M. Dykman, C. Maloney, V. Smelyanskiy, and M. Sil- verstein, Physical Review E57, 5202 (1998). 17

    1998

  48. [49]

    Dykman, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics75, 011101 (2007)

    M. Dykman, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics75, 011101 (2007)

    2007

  49. [50]

    Dykman, Fluctuating Nonlinear Oscillators , 165 (2012)

    M. Dykman, Fluctuating Nonlinear Oscillators , 165 (2012)

    2012

  50. [51]

    Q. Su, R. G. Corti˜ nas, J. Venkatraman, and S. Puri, Phys. Rev. A112, 042202 (2025)

    2025

  51. [52]

    N. E. Frattini, R. G. Corti˜ nas, J. Venkatraman, X. Xiao, Q. Su, C. U. Lei, B. J. Chapman, V. R. Joshi, S. Girvin, R. J. Schoelkopf,et al., Physical Review X14, 031040 (2024)

    2024

  52. [53]

    D. K. J. Boneß, W. Belzig, and M. I. Dykman, Phys. Rev. Res.7, 023188 (2025)

    2025

  53. [54]

    Putterman, K

    H. Putterman, K. Noh, R. N. Patel, G. A. Peairs, G. S. MacCabe, M. Lee, S. Aghaeimeibodi, C. T. Hann, I. Jar- rige, G. Marcaud, Y. He, H. Moradinejad, J. C. Owens, T. Scaffidi, P. Arrangoiz-Arriola, J. Iverson, H. Levine, F. G. S. L. Brand˜ ao, M. H. Matheny, and O. Painter, Phys. Rev. X15, 011070 (2025)

    2025

  54. [55]

    K. S. Dubovitskii, D. M. Basko, J. S. Meyer, and M. Houzet, Phys. Rev. B110, 024505 (2024)

    2024

  55. [56]

    K.S.Dubovitskii,PhysicalReviewA111,012617(2025)

    2025

  56. [57]

    Chamberland, K

    C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Campbell, C. T. Hann, J. Iverson, H. Putterman, T. C. Bohdanowicz, S. T. Flammia, A. Keller, G. Refael, J. Preskill, L. Jiang, A. H. Safavi-Naeini, O. Painter, and F. G. Brand˜ ao, PRX Quantum3, 010329 (2022)

    2022

  57. [58]

    F.-X. Sun, Q. He, Q. Gong, R. Y. Teh, M. D. Reid, and P. D. Drummond, New Journal of Physics21, 093035 (2019)

    2019

  58. [59]

    C.-W. Lee, P. Brookes, K.-S. Park, M. H. Szyma´ nska, and E. Ginossar, Phys. Rev. A112, 012216 (2025)

    2025

  59. [60]

    Thompson and A

    F. Thompson and A. Kamenev, Physical Review Re- search4, 023020 (2022)

    2022

  60. [61]

    S´ epulcre, Phys

    T. S´ epulcre, Phys. Rev. Res.8, L012058 (2026)

    2026

  61. [62]

    Carde, R

    L. Carde, R. Gautier, N. Didier, A. Petrescu, J. Cohen, and A. McDonald, Phys. Rev. Lett.136, 100402 (2026)

    2026

  62. [63]

    Thompson, D

    F. Thompson, D. K. J. Boneß, M. Dykman, and A. Kamenev, Phys. Rev. A113, 043712 (2026)

    2026

  63. [64]

    L. M. Sieberer, M. Buchhold, and S. Diehl, Reports on Progress in Physics79, 096001 (2016)

    2016

  64. [65]

    Analytic Continuation Of Chern-Simons Theory

    E. Witten, arXiv preprint arXiv:1001.2933 (2010), https://doi.org/10.48550/arXiv.1001.2933

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1001.2933 2010

  65. [66]

    C. M. Bender and S. A. Orszag,Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Meth- ods and Perturbation Theory, Vol. 1 (Springer, 1999)

    1999

  66. [67]

    Xiang, W

    Y.-X. Xiang, W. Li, Z. Bai, and Y.-Q. Ma, National Science Review , nwag146 (2026)

    2026

  67. [68]

    Drummond and D

    P. Drummond and D. Walls, Journal of Physics A: Math- ematical and General13, 725 (1980)

    1980

  68. [69]

    Marthaler and M

    M. Marthaler and M. I. Dykman, Phys. Rev. A73, 042108 (2006)

    2006

  69. [70]

    Vicentini, F

    F. Vicentini, F. Minganti, R. Rota, G. Orso, and C. Ciuti, Physical Review A97, 013853 (2018)

    2018

  70. [71]

    L. M. Sieberer, M. Buchhold, J. Marino, and S. Diehl, Reviews of Modern Physics97, 025004 (2025)

    2025

  71. [72]

    Olver,Asymptotics and special functions(Academic Press, New York, 1997)

    F. Olver,Asymptotics and special functions(Academic Press, New York, 1997)

    1997