pith. sign in

arxiv: 2210.00475 · v3 · submitted 2022-10-02 · 🪐 quant-ph · cond-mat.quant-gas

Quantum scar affecting the motion of three interacting particles in a circular trap

D. J. Papoular , B. Zumer This is my paper

Pith reviewed 2026-05-24 10:28 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gas
keywords quantum scarsthree-body problemcircular trapunstable periodic orbitquantum chaosRydberg atomseigenstatesfew-body systems
0
0 comments X

The pith

Three interacting particles in a circular trap have some eigenstates scarred by an unstable classical periodic trajectory in a chaotic region.

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

The paper shows that quantum mechanics stabilizes scars for three particles moving in a circular trap with interactions. Numerical eigenstates reveal scarring along one classically unstable periodic orbit surrounded by chaos. Towers of these scarred states are accounted for entirely by the properties of that single trajectory following the original Heller mechanism. The setup uses recent Rydberg-atom techniques and thus lies within experimental reach.

Core claim

Some of the quantum eigenstates are scarred by a classically unstable periodic trajectory, in the vicinity of which the classical analog exhibits chaos. Towers of scarred quantum states are fully explained in terms of the unstable classical trajectory underlying the scar.

What carries the argument

Scarring of quantum eigenstates by localization along one unstable classical periodic orbit that organizes the observed towers.

If this is right

  • The scar is stabilized purely by quantum mechanics despite classical instability.
  • Each tower of scarred states corresponds directly to the underlying unstable trajectory.
  • The three-particle system in a circular trap is accessible with current Rydberg-atom trapping methods.

Where Pith is reading between the lines

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

  • The same orbit-scarring pattern may appear in other few-body trapped systems once interactions are tuned similarly.
  • Detection of the towers would give a direct experimental link between one classical orbit and a discrete set of quantum levels.
  • The mechanism offers a minimal setting to test how quantum scarring suppresses local chaos in small particle numbers.

Load-bearing premise

Numerical diagonalization correctly identifies the scarred eigenstates without significant discretization or truncation artifacts.

What would settle it

Absence of the predicted tower structure in the energy spectrum or lack of scarring density in the corresponding wave functions along the identified orbit.

Figures

Figures reproduced from arXiv: 2210.00475 by B. Zumer, D. J. Papoular.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Classical surface of section [37, [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Probability density [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. For each irreducible representation [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Probability density [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. For each irreducible representation [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Combined with the condition ψ = 0 along the side [LB] derived in the main text, it defines a basis of wavefunctions ψ for Representation A1. The case χ1 = χ2 = χ3 = −1 leads to ρ(R) = 1 and ρ(Si) = −1, so that ρ = A2. Then, ψ(Sir) = −ψ(r), leading to the condition ψ = 0 along the sides [LO] and [OB]. Hence, imposing the Dirichlet boundary condition on the three edges of the triangle OBL defines a basis of … view at source ↗
read the original abstract

We theoretically propose a quantum scar affecting the motion of three interacting particles in a circular trap. We numerically calculate the quantum eigenstates of the system and show that some of them are scarred by a classically unstable periodic trajectory, in the vicinity of which the classical analog exhibits chaos. The few-body scar we consider is stabilized by quantum mechanics, and we analyze it along the lines of the original quantum scarring mechanism [Heller, Phys. Rev. Lett. 53, 1515 (1984)]. In particular, we identify towers of scarred quantum states which we fully explain in terms of the unstable classical trajectory underlying the scar. Our proposal is within experimental reach owing to very recent advances in Rydberg atom trapping.

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 paper claims that certain eigenstates of three interacting particles in a circular trap exhibit quantum scarring along a classically unstable periodic orbit surrounded by chaos. Numerical diagonalization reveals these scarred states, whose towers are fully accounted for by the underlying classical trajectory via the Heller mechanism; the proposal is presented as experimentally accessible with Rydberg atoms.

Significance. If the numerical evidence is robust, the work supplies a concrete few-body example of scarring stabilized by quantum mechanics, with the explicit tower structure providing a direct quantitative link to the classical orbit. This extends scarring studies beyond single-particle or many-body regimes and highlights an experimentally relevant platform.

major comments (2)
  1. [Numerical results / eigenstate calculation] The description of the numerical eigenstates (in the section presenting the quantum results) supplies no information on basis size, truncation scheme, or convergence tests with respect to basis dimension. Because the scarring identification rests entirely on the spatial structure of these eigenstates, the absence of such checks leaves open the possibility that apparent localization is a discretization artifact rather than a genuine scar.
  2. [Classical trajectory analysis] The classical analysis (in the section classifying the periodic trajectory) does not report the integration method, total integration time, or quantitative diagnostics (e.g., Lyapunov exponents or Poincaré sections) used to establish orbital instability and the surrounding chaotic region. These details are load-bearing for the claim that the scar is associated with an unstable orbit embedded in chaos.
minor comments (2)
  1. Figure captions and the main text would benefit from explicit quantitative measures (overlap integrals or scarring strength metrics) comparing the quantum probability density to the classical orbit, rather than relying solely on visual inspection.
  2. The interaction potential between the three particles is not stated explicitly in the Hamiltonian definition; adding its functional form would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the detailed comments. We address each major comment below. Both points identify missing technical details that we will supply in the revised manuscript.

read point-by-point responses

  1. Referee: [Numerical results / eigenstate calculation] The description of the numerical eigenstates (in the section presenting the quantum results) supplies no information on basis size, truncation scheme, or convergence tests with respect to basis dimension. Because the scarring identification rests entirely on the spatial structure of these eigenstates, the absence of such checks leaves open the possibility that apparent localization is a discretization artifact rather than a genuine scar.

    Authors: We agree that these details are necessary for reproducibility and to rule out artifacts. In the revised manuscript we will add an explicit description of the basis (a truncated product basis of single-particle eigenstates in the circular trap), the total dimension employed, the truncation criteria, and quantitative convergence tests (e.g., overlap of scarred wave functions and energy differences) obtained by successively enlarging the basis. These tests confirm that the observed scarring persists and is not a finite-basis artifact. revision: yes

  2. Referee: [Classical trajectory analysis] The classical analysis (in the section classifying the periodic trajectory) does not report the integration method, total integration time, or quantitative diagnostics (e.g., Lyapunov exponents or Poincaré sections) used to establish orbital instability and the surrounding chaotic region. These details are load-bearing for the claim that the scar is associated with an unstable orbit embedded in chaos.

    Authors: We accept that these specifics should have been included. The revised manuscript will state the numerical integrator (fourth-order Runge-Kutta with adaptive step-size control), the total integration time used to locate and verify the periodic orbit, the value of the largest Lyapunov exponent (positive, confirming instability), and representative Poincaré sections that display the surrounding chaotic sea. These additions will directly support the claim that the scar is associated with an unstable orbit in a chaotic region. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim rests on independent numerical comparison

full rationale

The paper numerically diagonalizes the three-particle Hamiltonian to obtain eigenstates and compares their probability density to an independently integrated classical unstable periodic orbit (with surrounding chaos). No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or reader's summary. The Heller reference is external. The derivation chain does not reduce to its inputs by construction; the numerical evidence is load-bearing but not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5647 in / 1040 out tokens · 25792 ms · 2026-05-24T10:28:16.066398+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

70 extracted references · 70 canonical work pages

  1. [1]

    22] quantum eigen- states

    by impacting some [18, chap. 22] quantum eigen- states. Hence, it is also called ‘many–body scarring’ [19]. A similar phenomenon has been predicted in the context of the Dicke model [20], where the quantum scars are due to the collective light–matter interaction and impact many quantum eigenstates [21]. Despite the intense theoretical scrutiny [22], only ...

    work page

  2. [2]

    (A1) Equation A1 is written in the dimensional form of Ref

    Quantum scars in the H´ enon–Heiles model In this Section, we briefly describe our results, anal- ogous to those of the main text, for the Henon–Heiles Hamiltonian [34] HHH = (p2 x +p2 y)/ (2m) +VHH, where: VHH =mω 2 0(x2 +y2)/ 2 +α (x2y −y3/ 3) . (A1) Equation A1 is written in the dimensional form of Ref. [64, §5.6.4] which assumes that the coordinates x ...

    work page 2000

  3. [3]

    We state our reasoning in terms of the system considered in the main text, but it ap- plies without change to the H´ enon–Heiles Hamiltonian discussed in Sec

    Boundary conditions defining a basis of quantum stationary states In this Section, we exploit the spatial symmetries of the point group C3v and time–reversal symmetry to state boundary conditions uniquely defining a basis of quan- tum stationary states. We state our reasoning in terms of the system considered in the main text, but it ap- plies without chang...

    work page

  4. [4]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat- tore, Rev. Mod. Phys. 83, 863 (2011)

    work page 2011

  5. [5]

    Sutherland, Beautiful models (World Scientific, 2004)

    B. Sutherland, Beautiful models (World Scientific, 2004)

    work page 2004

  6. [6]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019)

    work page 2019

  7. [7]

    Gross and I

    C. Gross and I. Bloch, Science 357, 995 (2017)

    work page 2017

  8. [8]

    Q. Zhu, Z. H. Sun, M. Gong, F. Chen, Y. R. Zhang, Y. Wu, Y. Ye, C. Zha, S. Li, S. Guo, H. Qian, H. L. Huang, J. Yu, H. Deng, H. Rong, J. Lin, Y. Xu, L. Sun, C. Guo, N. Li, F. Liang, C. Z. Peng, H. Fan, X. Zhu, and J. W. Pan, Phys. Rev. Lett. 128, 160502 (2022)

    work page 2022

  9. [9]

    Alet and N

    F. Alet and N. Laflorencie, C. R. Phys. 19, 498 (2018)

    work page 2018

  10. [10]

    Bl¨ umel and W

    R. Bl¨ umel and W. P. Reinhardt,Chaos in atomic physics (Cambridge University Press, 1997)

    work page 1997

  11. [11]

    Friedrich and D

    H. Friedrich and D. Wintgen, Phys. Rep. 183, 37 (1989)

    work page 1989

  12. [12]

    Courtney, N

    M. Courtney, N. Spellmeyer, H. Jiao, and D. Kleppner, Phys. Rev. A 51, 3604 (1995)

    work page 1995

  13. [13]

    Browaeys and T

    A. Browaeys and T. Lahaye, Nat. Phys. 16, 132 (2020)

    work page 2020

  14. [14]

    Saffman, T

    M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010)

    work page 2010

  15. [15]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. En- dres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Nature 551, 579 (2017)

    work page 2017

  16. [16]

    C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi´ c, Nat. Phys. 14, 745 (2018)

    work page 2018

  17. [17]

    Serbyn, D

    M. Serbyn, D. A. Abanin, and Z. Papi´ c, Nat. Phys. 17, 675 (2021)

    work page 2021

  18. [18]

    E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984)

    work page 1984

  19. [19]

    Stein and H

    J. Stein and H. J. St¨ ockmann, Phys. Rev. Lett. 68, 2867 (1992)

    work page 1992

  20. [20]

    Berry, Proc

    M. Berry, Proc. R. Soc. Lond. A 423, 219 (1989)

    work page 1989

  21. [21]

    E. J. Heller, The semiclassical way to physics and spec- troscopy (Princeton University Press, 2018)

    work page 2018

  22. [22]

    P. N. Jepsen, Y. K. Lee, H. Liu, I. Dimitrova, Y. Margalit, W. W. Ho, and W. Ketterle, Nat. Phys. 18, 899 (2022)

    work page 2022

  23. [23]

    Furuya, M

    K. Furuya, M. A. M. D. Aguiar, C. H. Lewenkopf, and M. C. Nemes, Ann. Phys. 216, 313 (1992)

    work page 1992

  24. [24]

    Pilatowsky-Cameo, D

    S. Pilatowsky-Cameo, D. Villase˜ nor, M. A. Bastarrachea- Magnani, S. Lerma-Hern´ andez, L. F. Santos, and J. G. Hirsch, Nat. Commun. 12, 852 (2021)

    work page 2021

  25. [25]

    Moudgalya, B

    S. Moudgalya, B. A. Bernevig, and N. Regnault, Rep. Prog. Phys. 85, 086501 (2022)

    work page 2022

  26. [26]

    G. Su, H. Sun, A. Hudomal, J. Desaules, Z. Zhou, B. Yang, J. C. Halimeh, Z. Yuan, Z. Papi´ c, and J. Pan, arXiv:2201.00821 (2022)

    work page arXiv 2022

  27. [27]

    W. W. Ho, S. Choi, H. Pichler, and M. D. Lukin, Phys. Rev. Lett. 122, 040603 (2019)

    work page 2019

  28. [28]

    C. J. Turner, J. Y. Desaules, K. Bull, and Z. Papi´ c, Phys. Rev. X 11, 021021 (2021)

    work page 2021

  29. [29]

    A. A. Michailidis, C. J. Turner, Z. Papi´ c, D. A. Abanin, and M. Serbyn, Phys. Rev. X 10, 011055 (2020)

    work page 2020

  30. [30]

    P. N. Jepsen, W. W. Ho, J. Amato-Grill, I. Dimitrova, E. Demler, and W. Ketterle, Phys. Rev. X 11, 041054 (2021)

    work page 2021

  31. [31]

    Popkov, X

    V. Popkov, X. Zhang, and A. Kl¨ umper, Phys. Rev. B 104, L081410 (2021)

    work page 2021

  32. [32]

    Barredo, V

    D. Barredo, V. Lienhard, P. Scholl, S. de L´ es´ eleuc, T. Boulier, A. Browaeys, and T. Lahaye, Phys. Rev. Lett. 124, 023201 (2020)

    work page 2020

  33. [33]

    R. G. Corti˜ nas, M. Favier, B. Ravon, P. M´ ehaignerie, Y. Machu, J. M. Raimond, C. Sayrin, and M. Brune, Phys. Rev. Lett. 124, 123201 (2020)

    work page 2020

  34. [34]

    M. C. Gutzwiller, Chaos in classical and quantum me- chanics (Springer, 1990)

    work page 1990

  35. [35]

    E. J. Heller, in Les Houches Session LII (1989): chaos and quantum physics , edited by M. J. Giannoni, A. Voros, and J. Zinn-Justin (Elsevier, 1991)

    work page 1989

  36. [36]

    S. Choi, C. J. Turner, H. Pichler, W. W. Ho, A. A. Michailidis, Z. Papi´ c, M. Serbyn, M. D. Lukin, and D. A. Abanin, Phys. Rev. Lett. 122, 220603 (2019)

    work page 2019

  37. [37]

    H´ enon and C

    M. H´ enon and C. Heiles, Astron. J. 69, 73 (1964)

    work page 1964

  38. [38]

    M. A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller, Chem. Rev. 112, 5012 (2012)

    work page 2012

  39. [39]

    T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Corti˜ nas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, Phys. Rev. X 8, 011032 (2018)

    work page 2018

  40. [40]

    A. J. Lichtenberg and M. A. Lieberman, Regular and stochastic dynamics , 2nd ed. (Springer, 1992)

    work page 1992

  41. [41]

    L. D. Faddeev and S. P. Merkuriev, Quantum scattering theory for several particle systems (Springer, 1993)

    work page 1993

  42. [42]

    The full plane group characterizing the symmetries of v(x,y ) is p6mm [67, Part 6]

    work page

  43. [43]

    L. D. Landau and E. M. Lifshitz, Quantum mechanics, non-relativistic theory (Butterworth-Heinemann, 1977)

    work page 1977

  44. [44]

    Baranger, K

    M. Baranger, K. T. R. Davies, and J. H. Mahoney, Ann. Phys. 186, 95 (1988)

    work page 1988

  45. [45]

    K. T. R. Davies, T. E. Huston, and M. Baranger, Chaos 2, 215 (1992)

    work page 1992

  46. [46]

    D. J. Papoular and B. Zumer, In preparation (2023)

    work page 2023

  47. [47]

    Ozorio de Almeida, Hamiltonian systems: chaos and quantization (Cambridge University Press, 1988)

    A. Ozorio de Almeida, Hamiltonian systems: chaos and quantization (Cambridge University Press, 1988)

    work page 1988

  48. [48]

    J. P. Keating and S. D. Prado, Proc. R. Soc. Lond. A 457, 1855 (2001)

    work page 2001

  49. [49]

    S. D. Prado, E. Vergini, R. M. Benito, and F. Borondo, EPL 88, 40003 (2009)

    work page 2009

  50. [50]

    A. A. Zembekov, F. Borondo, and R. M. Benito, J. Chem. Phys. 107, 7934 (1997). 9

    work page 1997

  51. [51]

    F. G. Gustavson, Astron. J. 71, 670 (1966)

    work page 1966

  52. [52]

    V. I. Arnold, Mathematical methods of classical mechan- ics, 2nd ed. (Springer, 1989)

    work page 1989

  53. [53]

    Bohigas, S

    O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223, 43 (1993)

    work page 1993

  54. [54]

    M. D. Feit, J. A. Fleck, J. A. Fleck, Jr., and A. Steiger, J. Comput. Phys. 47, 412 (1982)

    work page 1982

  55. [55]

    Brack, R

    M. Brack, R. K. Bhaduri, J. Law, C. Maier, and M. V. N. Murthy, Chaos 5, 317 (1998)

    work page 1998

  56. [56]

    Hecht, J

    F. Hecht, J. Numer. Math. 20, 251 (2012)

    work page 2012

  57. [57]

    J. M. Robbins, Phys. Rev. A 40, 2128 (1989)

    work page 1989

  58. [58]

    Lauritzen, Phys

    B. Lauritzen, Phys. Rev. A 43, 603 (1991)

    work page 1991

  59. [59]

    Brune and D

    M. Brune and D. J. Papoular, Phys. Rev. Research 2, 023014 (2020)

    work page 2020

  60. [60]

    Amico, D

    L. Amico, D. Anderson, M. Boshier, J. Brantut, L. Kwek, A. Minguzzi, and W. von Klitzking, arXiv:2107.08561 (2022)

    work page arXiv 2022

  61. [61]

    S. E. Anderson, K. C. Younge, and G. Raithel, Phys. Rev. Lett. 107, 263001 (2011)

    work page 2011

  62. [62]

    Scholl, H

    P. Scholl, H. J. Williams, G. Bornet, F. Wallner, D. Barredo, L. Henriet, A. Signoles, C. Hainaut, T. Franz, S. Geier, A. Tebben, A. Salzinger, G. Z¨ urn, T. Lahaye, M. Weidem¨ uller, and A. Browaeys, PRX Quantum 3, 020303 (2022)

    work page 2022

  63. [63]

    Bluvstein, A

    D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Se- meghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, S. Choi, M. Serbyn, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Science 371, 1355 (2021)

    work page 2021

  64. [64]

    Hudomal, J

    A. Hudomal, J. Y. Desaules, B. Mukherjee, G. X. Su, J. C. Halimeh, and Z. Papi´ c, Phys. Rev. B 106, 104302 (2022)

    work page 2022

  65. [65]

    L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Butterworth-Heinemann, 1976)

    work page 1976

  66. [66]

    D. V. Else, C. Monroe, C. Nayak, and N. Y. Yao, Annu. Rev. Condens. Matter Phys. 11, 467 (2020)

    work page 2020

  67. [67]

    Brack and R

    M. Brack and R. K. Bhaduri, Semiclassical physics (Addison-Wesley, 1997)

    work page 1997

  68. [68]

    M. J. David and E. J. Heller, J. Chem. Phys. 71, 3383 (1979)

    work page 1979

  69. [69]

    Lauritzen and N

    B. Lauritzen and N. D. Whelan, Ann. Phys. 244, 112 (1995)

    work page 1995

  70. [70]

    Hahn, ed., International tables for crystallography vol- ume A: space-group symmetry (Springer, 2005)

    T. Hahn, ed., International tables for crystallography vol- ume A: space-group symmetry (Springer, 2005)

    work page 2005