pith. sign in

arxiv: 2601.05783 · v2 · pith:OCAVW6VNnew · submitted 2026-01-09 · 🪐 quant-ph · cond-mat.mes-hall

Hidden time-nonlocal Floquet symmetries

Sigmund Kohler , Jes\'us Casado-Pascual This is my paper

Pith reviewed 2026-05-16 16:06 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords Floquet spectrumquasienergy crossingshidden symmetrytime-nonlocal paritydriven two-level systemperiodic drivingdetuned qubit
0
0 comments X

The pith

A hidden time-nonlocal parity symmetry produces exact quasienergy crossings in driven two-level systems when detuning is an integer multiple of the drive quantum.

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

The paper examines the Floquet quasienergy spectrum of a two-level quantum system under periodic driving with a static detuning. It shows that exact crossings between quasienergies occur precisely when the detuning equals an integer multiple of the driving field's energy quantum. These crossings arise because the system possesses a hidden parity symmetry that is nonlocal in time, allowing every Floquet mode to be labeled even or odd. Modes of the same parity repel each other while opposite-parity modes cross freely. The authors prove the symmetry exists through a scalar recurrence relation obtained from the time-periodic equations of motion and supply a numerical procedure that locates the parity labels even in more general models.

Core claim

In a periodically driven two-level system, the Floquet quasienergies display exact crossings whenever the detuning equals an integer multiple of the drive frequency. This occurs because a hidden time-nonlocal parity symmetry exists that partitions the Floquet modes into even and odd classes; crossings are allowed only between modes of opposite parity. The symmetry is established by constructing a scalar recurrence relation whose solutions directly reveal the parity of each mode, and the same relation supplies the basis for a numerical algorithm that identifies the symmetry in systems beyond the two-level case.

What carries the argument

Hidden time-nonlocal parity symmetry that classifies Floquet modes as even or odd and enforces exact crossings only between opposite-parity modes.

If this is right

  • Exact crossings appear in the quasienergy spectrum at every integer multiple of the drive quantum.
  • Floquet modes acquire definite even or odd parity labels under the hidden symmetry.
  • Same-parity quasienergies repel while opposite-parity quasienergies cross freely.
  • The recurrence relation supplies both an analytic proof and a practical numerical test for the symmetry.
  • The numerical scheme extends directly to models with more levels or additional nonlinearities.

Where Pith is reading between the lines

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

  • If similar recurrence relations can be derived for multi-level systems, the same parity classification and crossing rule may apply.
  • The symmetry could simplify the design of pulse sequences that avoid or exploit specific quasienergy degeneracies.
  • Experimental detection of the exact crossings in superconducting qubits or trapped ions would confirm the parity labels without requiring full tomography.

Load-bearing premise

The equations of motion for the driven two-level system can be reduced to a scalar recurrence relation whose solutions directly encode the even-odd parity classification.

What would settle it

Compute the Floquet quasienergies numerically for a driven two-level system with detuning exactly equal to the drive frequency; if all near-crossings remain avoided rather than becoming exact degeneracies, the hidden parity symmetry does not exist.

Figures

Figures reproduced from arXiv: 2601.05783 by Jes\'us Casado-Pascual, Sigmund Kohler.

Figure 1
Figure 1. Figure 1: Minimal quasienergy splitting as a function of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Floquet spectrum of the driven two-level sys [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Floquet spectra as a function of the driving amplitude for various integer detunings [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

We investigate the Floquet spectrum of a detuned, driven two-level system and show that it exhibits exact quasienergy crossings when the detuning is an integer multiple of the energy quantum of the driving field. This behavior can be explained by a hidden time-nonlocal parity, which allows the Floquet modes to be classified as even or odd. Then a generic feature is the emergence of exact crossings between quasienergies of different parity. A constructive proof of the existence of the symmetry is based on a scalar recurrence relation. Moreover, we present a general scheme for its numerical computation, which can be applied to models beyond the two-level system. Analytical results are illustrated with numerical data.

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 investigates the Floquet spectrum of a detuned, driven two-level system and claims that exact quasienergy crossings occur when the detuning equals an integer multiple of the driving field's energy quantum. This is attributed to a hidden time-nonlocal parity symmetry that classifies Floquet modes as even or odd. A constructive proof is given via a scalar recurrence relation for the two-level case, together with a general numerical scheme for computing the symmetry in broader models, illustrated by numerical data.

Significance. If the hidden symmetry and the exactness of the crossings are rigorously established, the result would clarify the structure of Floquet spectra in driven two-level systems and offer a symmetry-based explanation for protected degeneracies. The constructive recurrence proof and the outlined numerical scheme for extension beyond two levels are potentially useful contributions if the exactness claim can be placed on firmer analytical footing for generic models.

major comments (2)
  1. [Abstract] Abstract and the section describing the constructive proof: the scalar recurrence relation provides an explicit classification of even/odd modes only for the two-level system; the claim that exact crossings are a 'generic feature' for models beyond two levels therefore rests on an extrapolation whose validity is not demonstrated analytically.
  2. [the section presenting the general numerical scheme] The section presenting the general numerical scheme: numerical Floquet computations can detect near-degeneracies but cannot rigorously establish that splittings are exactly zero (as opposed to smaller than truncation or discretization error) without an explicit symmetry operator or recurrence that extends to multi-level or nonlinear cases.
minor comments (2)
  1. The definition and explicit construction of the time-nonlocal parity operator should be stated more clearly, ideally with its action on the Floquet modes written out.
  2. Figure captions and axis labels in the numerical illustrations could specify the truncation parameters and driving amplitude values used, to allow direct reproduction of the reported crossings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section describing the constructive proof: the scalar recurrence relation provides an explicit classification of even/odd modes only for the two-level system; the claim that exact crossings are a 'generic feature' for models beyond two levels therefore rests on an extrapolation whose validity is not demonstrated analytically.

    Authors: We agree that the scalar recurrence furnishes an explicit and rigorous classification of even/odd modes only for the two-level system. The reference to generic features for models beyond two levels is supported by the numerical scheme rather than an analytical proof. We will revise the abstract and the section describing the constructive proof to state explicitly that the analytical classification and proof of exact crossings apply to the two-level case, while the numerical scheme provides a method to identify the symmetry and the associated crossings in more general models. This revision will remove any implication of an analytical demonstration for arbitrary systems. revision: yes

  2. Referee: [the section presenting the general numerical scheme] The section presenting the general numerical scheme: numerical Floquet computations can detect near-degeneracies but cannot rigorously establish that splittings are exactly zero (as opposed to smaller than truncation or discretization error) without an explicit symmetry operator or recurrence that extends to multi-level or nonlinear cases.

    Authors: The numerical scheme constructs an approximate symmetry operator for the Floquet Hamiltonian rather than merely inspecting the spectrum. When this operator is obtained to high numerical precision and satisfies the required algebraic relations (commutation with the Floquet operator and parity classification), the symmetry argument itself guarantees that crossings between different parity sectors are exactly zero, independent of truncation errors in the quasienergy computation. We will revise the section to clarify this distinction, add quantitative diagnostics for the accuracy of the identified operator (such as the commutator norm), and explicitly note the distinction between spectrum-based near-degeneracies and symmetry-protected exact crossings. While this approach is numerical, it supplies a symmetry-based justification for exactness in the models examined; an analytical recurrence valid for general multi-level or nonlinear cases is not provided and lies outside the present scope. revision: partial

Circularity Check

0 steps flagged

No circularity: constructive recurrence proof derives symmetry directly from model equations

full rationale

The paper's central derivation starts from the driven two-level system equations, obtains a scalar recurrence relation, and uses it to constructively prove the existence of the hidden time-nonlocal parity that classifies Floquet modes as even or odd. This directly yields the exact quasienergy crossings for integer detuning multiples without any fitted parameters, self-referential definitions, or load-bearing self-citations. The general numerical scheme for multi-level models is presented as a computational extension rather than an analytical claim that reduces to the two-level case by construction. No step in the provided derivation chain exhibits the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard Floquet theory for time-periodic Hamiltonians and the assumption that the two-level detuned drive permits a scalar recurrence that encodes the parity; the hidden parity itself is introduced to explain the crossings.

axioms (2)
  • standard math Floquet theory applies to periodically driven quantum systems and yields quasienergies
    Invoked implicitly as the framework for the spectrum analysis.
  • domain assumption The driven two-level Hamiltonian with detuning admits a scalar recurrence relation for its solutions
    Required for the constructive proof of the parity symmetry.
invented entities (1)
  • hidden time-nonlocal parity no independent evidence
    purpose: Classifies Floquet modes as even or odd to explain exact quasienergy crossings
    Postulated to account for the observed crossings; no independent falsifiable prediction outside the recurrence is given.

pith-pipeline@v0.9.0 · 5403 in / 1488 out tokens · 36808 ms · 2026-05-16T16:06:03.578436+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

22 extracted references · 22 canonical work pages

  1. [1]

    Solution of the Schrödinger equation with a Hamiltonian periodic in time

    Jon H. Shirley. “Solution of the Schrödinger equation with a Hamiltonian periodic in time”. Phys. Rev.138, B979 (1965)

    work page 1965

  2. [2]

    Steady states and quasiener- gies of a quantum-mechanical system in an oscillating field

    Hideo Sambe. “Steady states and quasiener- gies of a quantum-mechanical system in an oscillating field”. Phys. Rev. A7, 2203 (1973)

    work page 1973

  3. [3]

    Driven quantum systems

    Peter Hänggi. “Driven quantum systems”. In Quantum Transport and Dissipation. Chap- ter 5, pages 249–286. Wiley-VCH, Wein- heim (1998)

    work page 1998

  4. [4]

    Coherent destruc- tion of tunneling

    Frank Grossmann, Thomas Dittrich, Peter Jung, and Peter Hänggi. “Coherent destruc- tion of tunneling”. Phys. Rev. Lett.67, 516 (1991)

    work page 1991

  5. [5]

    Local- ization in a driven two-level dynamics

    Frank Großmann and Peter Hänggi. “Local- ization in a driven two-level dynamics”. Eu- rophys. Lett.18, 571 (1992)

    work page 1992

  6. [6]

    Landau-Zener-Stückelberg interfer- ometry of a single electron charge qubit

    J. Stehlik, Y. Dovzhenko, J. R. Petta, J. R. Johansson, F. Nori, H. Lu, and A. C. Gos- sard. “Landau-Zener-Stückelberg interfer- ometry of a single electron charge qubit”. Phys. Rev. B86, 121303(R) (2012)

    work page 2012

  7. [7]

    Characterization of qubit dephasing by Landau-Zener-Stückelberg- Majorana interferometry

    F.Forster, G.Petersen, S.Manus, P.Hänggi, D. Schuh, W. Wegscheider, S. Kohler, and S. Ludwig. “Characterization of qubit dephasing by Landau-Zener-Stückelberg- Majorana interferometry”. Phys. Rev. Lett. 112, 116803 (2014)

    work page 2014

  8. [8]

    Continuous-time monitoring of Landau- Zener interference in a Cooper-pair box

    Mika Sillanpää, Teijo Lehtinen, Antti Paila, Yuriy Makhlin, and Pertti Hakonen. “Continuous-time monitoring of Landau- Zener interference in a Cooper-pair box”. Phys. Rev. Lett.96, 187002 (2006)

    work page 2006

  9. [9]

    Amplitude spectroscopy of a solid- state artificial atom

    David M. Berns, Mark S. Rudner, Sergio O. Valenzuela, Karl K. Berggren, William D. Oliver, Leonid S. Levitov, and Terry P. Or- lando. “Amplitude spectroscopy of a solid- state artificial atom”. Nature (London)455, 51 (2008)

    work page 2008

  10. [10]

    Dynamical control of matter-wave tunnel- ing in periodic potentials

    H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo. “Dynamical control of matter-wave tunnel- ing in periodic potentials”. Phys. Rev. Lett. 99, 220403 (2007)

    work page 2007

  11. [11]

    Discontinuities in driven spin- boson systems due to coherent destruction of tunneling: Breakdown of the Floquet- 8 Gibbs distribution

    Georg Engelhardt, Gloria Platero, and Jian- shu Cao. “Discontinuities in driven spin- boson systems due to coherent destruction of tunneling: Breakdown of the Floquet- 8 Gibbs distribution”. Phys. Rev. Lett.123, 120602 (2019)

    work page 2019

  12. [12]

    Quantum dissipation at conical intersections of quasienergies

    Sigmund Kohler. “Quantum dissipation at conical intersections of quasienergies”. Phys. Rev. A110, 052218 (2024)

    work page 2024

  13. [13]

    Quantum signatures of chaos

    Fritz Haake, Sven Gnutzmann, and Marek Kuś. “Quantum signatures of chaos”. Springer Series in Synergetics. Springer. Cham (2018). 4th edition

    work page 2018

  14. [14]

    Dynamical quasidegeneracies and quantum tunneling

    Asher Peres. “Dynamical quasidegeneracies and quantum tunneling”. Phys. Rev. Lett. 67, 158 (1991)

    work page 1991

  15. [15]

    Nonadiabatic electron pumping: Maximal current with minimal noise

    Michael Strass, Peter Hänggi, and Sigmund Kohler. “Nonadiabatic electron pumping: Maximal current with minimal noise”. Phys. Rev. Lett.95, 130601 (2005)

    work page 2005

  16. [16]

    Two-level systems driven by large-amplitude fields

    S. Ashhab, J. R. Johansson, A. M. Zagoskin, and Franco Nori. “Two-level systems driven by large-amplitude fields”. Phys. Rev. A75, 063414 (2007)

    work page 2007

  17. [17]

    Nonadiabatic Landau- Zener-Stückelberg-Majorana transitions, dy- namics, and interference

    Oleh V. Ivakhnenko, Sergey N. Shevchenko, and Franco Nori. “Nonadiabatic Landau- Zener-Stückelberg-Majorana transitions, dy- namics, and interference”. Phys. Rep.995, 1 (2023)

    work page 2023

  18. [18]

    Attempt to find the hidden symmetry in the asymmetric quantum Rabi model

    S. Ashhab. “Attempt to find the hidden symmetry in the asymmetric quantum Rabi model”. Phys. Rev. A101, 023808 (2020)

    work page 2020

  19. [19]

    The hid- den symmetry of the asymmetric quantum Rabi model

    Vladimir V. Mangazeev, Murray T. Batch- elor, and Vladimir V. Bazhanov. “The hid- den symmetry of the asymmetric quantum Rabi model”. J. Phys. A: Math. Theor.54, 12LT01 (2021)

    work page 2021

  20. [20]

    Remarks on the hid- den symmetry of the asymmetric quantum Rabi model

    Cid Reyes-Bustos, Daniel Braak, and Masato Wakayama. “Remarks on the hid- den symmetry of the asymmetric quantum Rabi model”. J. Phys. A: Math. Theor.54, 285202 (2021)

    work page 2021

  21. [21]

    Dynamical symmetries and symmetry- protected selection rules in periodically driven quantum systems

    Georg Engelhardt and Jianshu Cao. “Dynamical symmetries and symmetry- protected selection rules in periodically driven quantum systems”. Phys. Rev. Lett. 126, 090601 (2021)

    work page 2021

  22. [22]

    Nonstandard symmetry classes in meso- scopic normal-superconducting hybrid struc- tures

    Alexander Altland and Martin R. Zirnbauer. “Nonstandard symmetry classes in meso- scopic normal-superconducting hybrid struc- tures”. Phys. Rev. B55, 1142 (1997). 9

    work page 1997