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arxiv: 2606.19330 · v1 · pith:7APXDNG6new · submitted 2026-06-17 · 🪐 quant-ph

Floquet framework for driven polar quantum systems

Viktor Novi\v{c}enko , Piotr G{\l}adysz , Karolina S{\l}owik , Egidijus Anisimovas This is my paper

Pith reviewed 2026-06-26 20:29 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Floquet theorydriven quantum systemspolar two-level systemseffective Hamiltonianlongitudinal couplingtransverse couplingRabi modelBloch-Siegert shift
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The pith

Floquet analysis produces closed expressions for effective coupling and detuning in driven polar two-level systems.

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

The paper develops both analytical and numerical Floquet methods for two-level quantum systems subject to periodic driving with both longitudinal and transverse components. Analytically, it derives a dressed-frame effective Hamiltonian to first order in the inverse driving frequency while treating the longitudinal coupling exactly. This produces explicit formulas for the effective transverse coupling strength and detuning that are altered by the longitudinal term. The formulas reduce to the standard Rabi coupling and Bloch-Siegert shift when the longitudinal coupling is absent. A numerical flow-equation approach complements this by providing an effective time-independent Hamiltonian valid for a wider range of coupling strengths.

Core claim

We derive a dressed-frame effective Hamiltonian up to first order in the inverse driving frequency, incorporating the longitudinal coupling nonperturbatively. This yields closed expressions for the effective transverse coupling strength and the effective detuning, both of which are modified by the presence of the longitudinal interaction. In the nonpolar limit, these expressions recover the usual near-resonant Rabi coupling and the Bloch-Siegert shift. We also develop a numerical flow-equation framework that yields a time-independent effective Hamiltonian across a broad range of transverse and longitudinal coupling strengths.

What carries the argument

Dressed-frame effective Hamiltonian obtained via Floquet expansion to first order in inverse frequency with longitudinal coupling included exactly.

If this is right

  • Effective transverse coupling is renormalized by longitudinal interaction.
  • Effective detuning incorporates longitudinal effects.
  • Standard Rabi coupling and Bloch-Siegert shift recovered in nonpolar limit.
  • Numerical flow-equation method extends validity to stronger couplings.

Where Pith is reading between the lines

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

  • This approach may enable more accurate modeling of driven polar systems in superconducting circuits and optical lattices.
  • The effective Hamiltonian could be used to predict dynamics in time-dependent experiments beyond the high-frequency approximation.
  • Testing against full numerical solutions of the time-dependent system would verify the first-order results for moderate frequencies.

Load-bearing premise

The driving frequency is large compared to the coupling strengths and detuning, justifying the first-order inverse-frequency expansion.

What would settle it

Compare the oscillation frequency or effective parameters from exact time evolution of the driven system against the predictions of the derived effective Hamiltonian for nonzero longitudinal coupling.

Figures

Figures reproduced from arXiv: 2606.19330 by Egidijus Anisimovas, Karolina S{\l}owik, Piotr G{\l}adysz, Viktor Novi\v{c}enko.

Figure 1
Figure 1. Figure 1: FIG. 1. Harmonic weights obtained from the numerical inte [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: In addition to the direct Schr¨odinger evolution, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Effective Hamiltonian weights obtained numerically [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the numerically obtained micromo [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of the analytical and numerical effective Hamiltonians over the two-dimensional parameter space spanned [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We present an analytical and numerical Floquet treatment of a driven polar two-level quantum system characterized by both longitudinal and transverse coupling to a periodic field. Analytically, we derive a dressed-frame effective Hamiltonian up to first order in the inverse driving frequency, incorporating the longitudinal coupling nonperturbatively. This yields closed expressions for the effective transverse coupling strength and the effective detuning, both of which are modified by the presence of the longitudinal interaction. In the nonpolar limit, these expressions recover the usual near-resonant Rabi coupling and the Bloch-Siegert shift. As a second main result, we develop a numerical flow-equation framework that yields a time-independent effective Hamiltonian across a broad range of transverse and longitudinal coupling strengths. This dual framework is relevant for a variety of platforms, including driven polar quantum systems, optical lattices, superconducting circuits, and solids subject to surface acoustic waves.

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

0 major / 2 minor

Summary. The manuscript presents an analytical and numerical Floquet treatment of a driven polar two-level quantum system with both longitudinal and transverse couplings to a periodic drive. Analytically, a dressed-frame effective Hamiltonian is derived to first order in the inverse driving frequency, with the longitudinal coupling incorporated exactly (nonperturbatively), producing closed expressions for the renormalized transverse coupling strength and effective detuning. These expressions recover the standard near-resonant Rabi coupling and Bloch-Siegert shift in the nonpolar limit. As a second result, a numerical flow-equation method is developed that produces a time-independent effective Hamiltonian over a broad range of coupling strengths. The framework is positioned as relevant to driven polar systems in optical lattices, superconducting circuits, and solids under surface acoustic waves.

Significance. If the derivations hold, the work supplies a practical analytical tool for effective Hamiltonians in driven polar qubits that extends beyond the rotating-wave approximation while treating longitudinal drive exactly. The dual analytical-numerical approach, the explicit recovery of known limits, and the absence of free parameters in the closed-form expressions are strengths. The numerical flow-equation validation across parameter regimes adds robustness for applications in superconducting circuits and related platforms.

minor comments (2)
  1. The abstract states the expansion is taken to first order while treating longitudinal coupling exactly; the manuscript should explicitly state the validity condition (driving frequency large compared with couplings and detuning) in the main text near the derivation, with a brief remark on the expected error scaling.
  2. Notation for the dressed-frame transformation and the flow-equation generator should be introduced with a short table or equation list to aid readers unfamiliar with the specific conventions used for polar systems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation to accept. The report accurately summarizes our analytical and numerical contributions.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives an effective Hamiltonian via a standard dressed-frame Floquet transformation that removes the longitudinal drive exactly, followed by a first-order high-frequency expansion on the transverse terms. This starts from the driven Hamiltonian and recovers the known Rabi and Bloch-Siegert limits in the nonpolar case. An independent numerical flow-equation method is presented separately. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated procedure. The derivation chain is self-contained and externally verifiable against standard limits.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The framework relies on standard Floquet theory and the two-level approximation implicit in the model.

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Reference graph

Works this paper leans on

27 extracted references · 7 linked inside Pith

  1. [1]

    Fujii and T

    K. Fujii and T. Suzuki, Rotating wave approximation of the Law’s effective Hamiltonian on the dynamical Casimir effect, International Journal of Geometric Meth- ods in Modern Physics11, 1450003 (2014)

    2014

  2. [2]

    Fujii, Introduction to the rotating wave approxima- tion (RWA): Two coherent oscillations, Journal of Mod- ern Physics8, 2042 (2017)

    K. Fujii, Introduction to the rotating wave approxima- tion (RWA): Two coherent oscillations, Journal of Mod- ern Physics8, 2042 (2017)

    2042

  3. [3]

    O. V. Kibis, G. Y. Slepyan, S. A. Maksimenko, and A. Hoffmann, Matter coupling to strong electromagnetic fields in two-level quantum systems with broken inversion symmetry, Phys. Rev. Lett.102, 023601 (2009)

    2009

  4. [4]

    Macovei, M

    M. Macovei, M. Mishra, and C. H. Keitel, Popula- tion inversion in two-level systems possessing permanent dipoles, Phys. Rev. A92, 013846 (2015)

    2015

  5. [5]

    G. Y. Kryuchkyan, V. Shahnazaryan, O. V. Kibis, and I. A. Shelykh, Resonance fluorescence from an asymmet- ric quantum dot dressed by a bichromatic electromag- netic field, Phys. Rev. A95, 013834 (2017)

    2017

  6. [6]

    M. A. Ant´ on, S. Maede-Razavi, F. Carre˜ no, I. Thanopu- los, and E. Paspalakis, Optical and microwave control of resonance fluorescence and squeezing spectra in a polar molecule, Phys. Rev. A96, 063812 (2017)

    2017

  7. [7]

    N. N. Bogolubov and A. V. Soldatov, Fluorescence by incoherently pumped polar quantum system driven by a strong off-resonant external field, Physics of Particles and Nuclei57, 1 (2026)

    2026

  8. [8]

    Schweizer, F

    C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero, E. Demler, N. Goldman, I. Bloch, and M. Aidelsburger, Floquet approach to Z2 lattice gauge theories with ultra- cold atoms in optical lattices, Nature Physics15, 1168 (2019)

    2019

  9. [9]

    Niemczyk, F

    T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. H¨ ummer, E. Solano, A. Marx, and R. Gross, Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nature Physics6, 772 (2010)

    2010

  10. [10]

    Yoshihara, T

    F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime, Nature Physics13, 44 (2017)

    2017

  11. [11]

    Forn-D´ ıaz, L

    P. Forn-D´ ıaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Reviews of Modern Physics91, 025005 (2019)

    2019

  12. [12]

    A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nature Reviews Physics1, 19 (2019)

    2019

  13. [13]

    M. V. Gustafsson, T. Aref, A. F. Kockum, M. K. Ekstr¨ om, G. Johansson, and P. Delsing, Propagating phonons coupled to an artificial atom, Science346, 207 (2014)

    2014

  14. [14]

    Burgess, M

    A. Burgess, M. Florescu, and D. M. Rouse, Optical po- laron formation in quantum systems with permanent dipoles (2023), arXiv:2303.03996 [quant-ph]

    arXiv 2023

  15. [15]

    G ladysz and K

    P. G ladysz and K. S lowik, Light interactions with polar quantum systems, Phys. Rev. A111, 053704 (2025)

    2025

  16. [16]

    G ladysz, P

    P. G ladysz, P. Wcis lo, and K. S lowik, Propagation of optically tunable coherent radiation in a gas of polar molecules, Scientific Reports10, 17615 (2020)

    2020

  17. [17]

    Floquet, Sur les ´ equations diff´ erentielles lin´ eaires ` a coefficients p´ eriodiques, Annales scientifiques de l’Ecole Normale Superieure 2,12, 47 (1883)

    G. Floquet, Sur les ´ equations diff´ erentielles lin´ eaires ` a coefficients p´ eriodiques, Annales scientifiques de l’Ecole Normale Superieure 2,12, 47 (1883)

  18. [18]

    Sambe, Steady states and quasienergies of a quantum- mechanical system in an oscillating field, Phys

    H. Sambe, Steady states and quasienergies of a quantum- mechanical system in an oscillating field, Phys. Rev. A 7, 2203 (1973)

    1973

  19. [19]

    Bukov, L

    M. Bukov, L. D’Alessio, and A. Polkovnikov, Univer- sal high-frequency behavior of periodically driven sys- tems: from dynamical stabilization to Floquet engineer- ing, Adv. Phys.64, 139 (2015), arXiv:1407.4803 [cond- mat.quantum-gas]

    Pith/arXiv arXiv 2015

  20. [20]

    Eckardt and E

    A. Eckardt and E. Anisimovas, High-frequency approx- imation for periodically driven quantum systems from a Floquet-space perspective, New J. Phys.17, 093039 (2015), arXiv:1502.06477 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2015

  21. [21]

    Mikami, S

    T. Mikami, S. Kitamura, K. Yasuda, N. Tsuji, T. Oka, and H. Aoki, Brillouin-Wigner theory for high-frequency 9 expansion in periodically driven systems: Application to Floquet topological insulators, Phys. Rev. B93, 144307 (2016), arXiv:1511.00755 [cond-mat]

    Pith/arXiv arXiv 2016

  22. [22]

    Rahav, I

    S. Rahav, I. Gilary, and S. Fishman, Effective Hamilto- nians for periodically driven systems, Phys. Rev. A68, 013820 (2003), arXiv:nlin/0301033

    Pith/arXiv arXiv 2003

  23. [23]

    Guerin and H

    S. Guerin and H. R. Jauslin, Control of quantum dynam- ics by laser pulses: adiabatic Floquet theory, Adv. Chem. Phys.125, 147 (2003)

    2003

  24. [24]

    Goldman and J

    N. Goldman and J. Dalibard, Periodically driven quantum systems: Effective Hamiltonians and engi- neered gauge fields, Phys. Rev. X4, 031027 (2014), arXiv:1404.4373 [cond-mat]

    Pith/arXiv arXiv 2014

  25. [25]

    Noviˇ cenko, E

    V. Noviˇ cenko, E. Anisimovas, and G. Juzeli¯ unas, Flo- quet analysis of a quantum system with modulated periodic driving, Phys. Rev. A95, 023615 (2017), arXiv:1608.08420 [cond-mat]

    Pith/arXiv arXiv 2017

  26. [26]

    Noviˇ cenko, G

    V. Noviˇ cenko, G. ˇZlabys, and E. Anisimovas, Flow- equation approach to quantum systems driven by an amplitude-modulated time-periodic force, Phys. Rev. A 105, 012203 (2022)

    2022

  27. [27]

    Verdeny, A

    A. Verdeny, A. Mielke, and F. Mintert, Accurate effective Hamiltonians via unitary flow in Floquet space, Phys. Rev. Lett.111, 175301 (2013), arXiv:1304.3584 [quant- ph]

    Pith/arXiv arXiv 2013