Pith. sign in

REVIEW 4 minor 51 references

Coherent excitation of a two-state system by a Lorentzian field

T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For a Lorentzian field, the paper derives the power-narrowing law — linewidth ∝ 1/(Ω0 T²) at fixed T — from the complex-time DDP action, confirmed by numerical integration.

desk verdict A transparent DDP treatment of the Lorentzian two-state model that independently recovers the power-narrowing law; the pole/homotopy issue is real but does not threaten the central scaling claim. read the letter →

arxiv 2607.27466 v1 pith:7VKM7QYS submitted 2026-07-29 quant-ph

Aida Shiroyan , Genko S. Vasilev , Nikolay V. Vitanov This is my paper · ORCID
classification quant-ph
keywords two-statequantumsystemLorentzianpulseDykhne-Davis-PechukasmethodpowernarrowingtransitionprobabilitycomplexpointsconfluentHeunequationRabifrequency
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies coherent excitation of a two-state system by a Lorentzian pulse with constant detuning and develops a Dykhne-Davis-Pechukas (DDP) semiclassical description based on two interfering complex transition points. It derives explicit weak- and strong-coupling asymptotics for the DDP action, showing that near resonant odd-π pulses the transition probability takes a sech² cos² form whose scaling variable is C T√(Ω0|Δ|). The central result is that this implies a half-width scaling Δ_{1/2} ∝ 1/(Ω0 T²) at fixed duration — power narrowing — in agreement with direct numerical integration (exponent −1.0001). The paper also shows the Rosen-Zener-Robiscoe area-times-spectrum conjecture fails to reproduce this scaling, and constructs a two-point Padé approximation accurate to about 1% across the crossover. A sympathetic reader cares because this gives an independent complex-time derivation of a spectral-narrowing effect that matters for selective excitation and quantum control.

What carries the argument

The central object is the DDP action D(τ+), an analytically continued integral of the adiabatic quasienergy splitting along a complex-time contour from the origin to the upper-right transition point τ+. The two relevant transition points satisfy D(τ−)=−D*(τ+), so the generalized DDP probability takes the interference form P ∼ 4 exp(−2 ImD(τ+)) sin²(ReD(τ+)), with a unitarized sech² version. The mechanism: the strong-coupling asymptotic expansion of this elliptic integral produces the square-root dependence on Ω0Δ, which makes the near-resonant profile depend on the product α|δ| and hence yields the inverse-α linewidth law.

What would settle it

Numerically integrate the Schrödinger equation for scaled Rabi frequencies well beyond 41 (e.g., α=80, 160) and fit the half-width; if the exponent deviates from −1, the claimed inverse-α power-narrowing law fails. More directly, evaluate the DDP action along a contour that passes to the left of the pole at τ=i, or explicitly include the pole contribution, and check whether exp(−2 ImD) and the resulting linewidth law change, thereby testing whether the excluded pole is truly irrelevant.

Watch

Extended reading notes

Core claim

On the paper's own terms: the Lorentzian two-state model has two symmetry-related complex transition points τ± in the upper half-plane, plus Lorentzian poles at τ=±i. The paper claims that the DDP transition probability, with the contour chosen to pass to the right of the pole at τ=i, is governed by the action integral D(τ+)=ΔT ∫_0^{τ+} √(β²+(1+τ²)²)/(1+τ²) dτ. Expanding this action in the strong-coupling regime gives ReD ≈ (π/2)Ω0T − C T√(Ω0Δ) and ImD ≈ C T√(Ω0Δ) with C=Γ²(3/4)/√π, so for resonant odd-π pulses the central profile is P ∼ sech²(C√(αδ)) cos²(C√(αδ)). Since detuning and Rabi frequency enter only through α|δ|, any fixed probability level satisfies α|δ|=const, yielding Δ_{1/2}T ∝

Load-bearing premise

The DDP transition probability is computed from only the two transition points τ±, with the integration contour chosen to pass to the right of the Lorentzian pole at τ=i; if that pole generates additional Stokes contributions, or if the physically relevant homotopy class differs, the probabilities in Eqs. (32) and (45) — and with them the derived linewidth scaling — would change.

Editorial extensions

If this is right

  • For resonant odd-π Lorentzian pulses, the near-resonant half-width scales as Δ_{1/2} ∝ (Ω0 T)^{-1} at fixed duration — power narrowing — with the numerical exponent −1.0001 matching the prediction exactly.
  • In the weak-field limit, the DDP asymptotics reduce to first-order perturbation theory, P ≈ (π²/4)(Ω0T)² e^{−2|Δ|T}, reproducing the Lorentzian Fourier-tail dependence.
  • The Rosen-Zener-Robiscoe conjecture and its fixed-a Robiscoe generalization predict a constant asymptotic width of (ln2)/2, so they cannot describe Lorentzian power narrowing beyond the perturbative regime.
  • The constrained [4/2] two-point Padé approximant reproduces the exact DDP action to about 1% relative error over the crossover 0.1≤β≤100, giving a compact analytic formula for the transition probability.
  • At exact resonance, the unitarized DDP expression approaches the exact result P=sin²(πΩ0T/2) continuously, even though the transition points themselves recede to infinity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same contour-action technique could be ported to other meromorphic pulse envelopes, such as powers of Lorentzians or other algebraic-tail shapes, where the pole-versus-transition-point geometry would determine whether a generalized exponent Δ_{1/2} ∝ (Ω0T)^{-1/(λ−1)} emerges directly from the complex-time action.
  • The 26% discrepancy between the unitarized asymptotic prefactor (0.4803) and the numerical prefactor (0.6041) suggests that a uniform near-resonant DDP interpolation, rather than the sech² continuation, is needed for quantitative line-shape prediction; the exponent is robust, the prefactor is not.
  • Because the scaling variable is α|δ|, one could test the predicted collapse of line profiles on a qubit platform by sweeping peak Rabi frequency at fixed pulse duration and checking whether all odd-π central lines fall on a single universal curve when plotted against CT√(Ω0|Δ|).
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper analyzes a two-state system driven by a Lorentzian pulse with constant carrier detuning. It first notes that the Schrödinger equation can be mapped to a confluent Heun equation, but argues that no simple connection formula is available, so it turns to the Dykhne-Davis-Pechukas (DDP) complex-time method. The DDP analysis identifies two symmetry-related transition points in the upper half-plane plus poles of the coupling at τ=±i. The central technical result is the asymptotic expansion of the DDP action: in the strong-coupling/large-area regime, ReD ≈ (π/2)Ω0T − C T√(Ω0Δ) and ImD ≈ C T√(Ω0Δ), with C=Γ²(3/4)/√π (Eqs. (42)-(43)). For resonant odd-π pulses this yields a near-resonant profile P ∼ sech²(C√(αδ)) cos²(C√(αδ)), hence the linewidth scaling Δ_{1/2}T ∝ (Ω0T)^{-1}, i.e., Δ_{1/2} ∝ 1/(Ω0 T²) at fixed T. Direct numerical integration gives δ_{1/2} ≈ 0.604 α^{-1.0001}, confirming the exponent and showing a 26% prefactor discrepancy from the unitarized asymptotic prediction. The paper also contains weak-coupling asymptotics, a comparison with the Rosen-Zener-Robiscoe conjecture, and two-point Padé approximations to the action.

Significance. If the results are correct, the paper provides an independent, parameter-free DDP derivation of power narrowing for Lorentzian pulses, complementing the earlier adiabatic population-return argument. The main strengths are the matched asymptotic expansions with explicitly derived mathematical constants, the cross-check against direct numerical integration, and the unusually honest quantification of the limitations of the unitarized interpolation (0.480/α predicted vs 0.604/α numerical). The paper does not claim to discover the power-narrowing effect itself, which is already known and experimentally observed, but it gives a new complex-time perspective and useful analytic tools, including constrained Padé approximants with quantified errors. The central scaling law is robust: the near-resonant linewidth exponent is confirmed numerically, and the analytical derivation involves no fitted parameters.

minor comments (4)
  1. [Sec. IV C and Eq. (29)] The DDP contour prescription is the least rigorous premise in the paper. The pole at τ=i is closer to the real axis than either transition point, and the paper fixes a homotopy class passing to the right of the pole. The stress-test reasoning shows that this ambiguity is not load-bearing for the central scaling law: crossing the pole changes ReD by a real πα shift, which is phase-inert for resonant odd-π pulses, and ImD is homotopy-invariant; moreover, direct numerical integration supports the chosen contour. Still, a short paragraph explaining why pole-induced Stokes contributions do not alter the two-transition-point amplitude sum would remove the last foundational gap.
  2. [Sec. V C, Eqs. (47)-(51)] The 26% discrepancy between the unitarized asymptotic half-width 0.480/α and the numerical value 0.604/α is clearly disclosed, but it would be useful to state more explicitly whether this is an asymptotically constant prefactor error (because ImD is O(1) at half-maximum rather than asymptotically large) or whether it is expected to decrease slowly with α. A robustness check using only α≥11 in the fit would further strengthen the claim that the exponent is exactly −1.
  3. [Fig. 4 and Eq. (69)] The probability comparison between the exact-action DDP result and the Padé approximations is shown only for the fixed slice δ=1. Since the transition probability depends exponentially on δ times the action error, a second representative δ slice would make the validation of the [4/2] approximant more compelling.
  4. [Appendix A 3, Eq. (A24)] The strong-coupling derivation is elegant, but the branch choice for arctan(τ+) in Eq. (A24) should be stated explicitly. The principal branch of arctan gives π/2 − 1/z for large z with Re z>0, and the result depends on that choice; a one-sentence justification would avoid ambiguity for readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the power-narrowing law is derived from an un-fitted DDP action and checked against direct numerical integration.

full rationale

The paper's central derivation is self-contained. The DDP action is defined by the contour integral in Eq. (29), and the strong-coupling asymptotic expansion leading to ReD ≈ (π/2)Ω0T − CT√(Ω0Δ) and ImD ≈ CT√(Ω0Δ) is derived directly from that integral in Appendix A, with C = Γ²(3/4)/√π as a mathematical constant. No parameter is fitted to the linewidth or transition probability; the unitarized near-resonant form Eq. (45b) is explicitly identified as an interpolation rather than a fitted prediction, and the paper discloses that its prefactor differs by ~26% from numerics. The numerical fit in Eq. (51) is presented only as an independent check and is not an input to the derivation. The citations to prior power-narrowing work [25] and experimental work [26] describe previously known results and are used for comparison/context, not as load-bearing inputs; no uniqueness theorem or ansatz is imported via self-citation. The acknowledged pole/homotopy issue in Sec. IV C is an analytic-continuation premise, not a circular step, and the paper states the contour prescription explicitly. Overall the derivation chain does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation has no fitted free parameters: α, δ, β are model variables; C=Γ²(3/4)/√π and x_{1/2}=0.587... are mathematical constants; Padé coefficients are fixed by asymptotic matching. The numerical fit Eq. (51) is a check, not an input. No new physical entities are introduced. The DDP framework, the contour choice, and the unitarized interpolation are the main imported/ad hoc premises.

assumptions (5)
  • domain assumption Post-pulse probability equals |Σ_k Γ(τ_k) e^{iD(τ_k)}|² with Γ(τ±)=±1.
    Eqs. (23)-(24) and Eq. (31); this is the standard DDP/Stokes-line result imported from Refs. [37,38,41]. It is the core semiclassical framework.
  • ad hoc to paper The physical DDP action is the integral of the quasienergy along the contour C+ that passes to the right of the pole at τ=i and crosses no branch cuts, with branches fixed accordingly.
    Sec. IV C, paragraph after Eq. (27). The homotopy class is chosen, not derived; a different contour would alter ReD and ImD.
  • ad hoc to paper P_DDP^(u) = sin²(ReD)/cosh²(ImD) is a valid unitarized interpolation of the DDP asymptotics.
    Eq. (32b), justified only by reference to similar unitarizing substitutions [44]. It is used to obtain the half-width prefactor 0.480/α; the paper notes the numerical prefactor is 0.604/α.
  • standard math At exact resonance the transition probability is P=sin²(πΩ0T/2).
    Eq. (52); follows from commutation of H(t) at different times for Δ=0 (area theorem), used as boundary condition for near-resonant formulas.
  • standard math Eq. (8) maps exactly to a confluent Heun equation with local solution basis Eq. (13).
    Sec. III, Eqs. (9)-(15); used to classify the model, not to compute P.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Coherent excitation of a two-state system by a Lorentzian field." pith.science (2026). https://pith.science/paper/7VKM7QYS

@misc{pith2026260727466,
  author       = {Pith},
  title        = {Pith review of: Coherent excitation of a two-state system by a Lorentzian field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7VKM7QYS}},
  note         = {Machine review of arXiv:2607.27466}
}
abstract

We study the coherent excitation of a two-state quantum system by a field with a Lorentzian temporal envelope and constant carrier frequency. The associated differential equation admits an exact local representation in terms of confluent Heun functions. For the transition probability we develop a Dykhne-Davis-Pechukas (DDP) description based on the two relevant complex transition points and their interference. The DDP action is analyzed directly in the weak- and strong-coupling limits, yielding explicit asymptotic expressions for the oscillation phase, oscillation envelope, far-detuned line shape, and near-resonant behavior. In particular, the strong-coupling asymptotics imply a linewidth proportional to the inverse peak Rabi frequency for resonant odd-$\pi$ pulses, thereby exhibiting the power-narrowing characteristic of the Lorentzian pulse.

Figures

Figures reproduced from arXiv: 2607.27466 by the authors.

Figure 1
Figure 1. FIG. 1. Representative Stokes-line structure for the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Scaled positive half-width at half maximum ∆ [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Numerically integrated transition probability versus [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Transition probability versus the scaled Rabi fre [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Normalized errors [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 4 linked inside Pith

  1. [1]

    (A5) along the contourC + gives D(τ +) = ∆T√1−iβ h −(i+β)E(φ +, m) +βF(φ+, m) −iβ 2Π(n;φ +, m) i

    Evaluation at the transition points For the upper-right transition pointτ + = √−1 +iβ, define φ+ =iarcsinh s −1 +iβ 1 +iβ ! .(A6) Continuation of Eq. (A5) along the contourC + gives D(τ +) = ∆T√1−iβ h −(i+β)E(φ +, m) +βF(φ+, m) −iβ 2Π(n;φ +, m) i . (A7) For the upper-left transition pointτ − =− √−1−iβ, the same mapping givesφ − =−π/2 on the branch connect...

  2. [2]

    W eak-coupling limit A direct matched expansion of the contour integral dis- plays the origin of both the logarithm and its constant term. For the + branch, introduce w= 1+τ 2, R +(β)≡ D(τ +) ∆T = 1 2 Z iβ 1 p w2 +β 2 w√w−1 dw, (A10) where the contour and square roots are inherited from the physical prescription and √w−1 =iatw= 0. Choose an intermediate s...

  3. [3]

    Put ϵ=β −1, τ= p β z, z + = τ + √β ,(A20) so thatz + →e iπ/4

    Strong-coupling limit The leading strong-coupling coefficients are obtained most transparently from the contour integral itself. Put ϵ=β −1, τ= p β z, z + = τ + √β ,(A20) so thatz + →e iπ/4. Equation (29) becomes R+(β) = p β Z z+ 0 p 1 + (z2 +ϵ) 2 z2 +ϵ dz.(A21) Separate the singular inner contribution according to p 1 + (z2 +ϵ) 2 z2 +ϵ = 1 z2 +ϵ +G(z, ϵ)...

  4. [4]

    Majorana, Nuovo Cimento9, 43 (1932)

    E. Majorana, Nuovo Cimento9, 43 (1932)

    1932

  5. [5]

    (A24)–(A28) gives R+(β) = π 2 β+ (−1 +i)C p β+O(β −1/2),(A30) because−e −iπ/4 +e −iπ/4(1− √ 2C) = (−1+i)C

    Combining Eqs. (A24)–(A28) gives R+(β) = π 2 β+ (−1 +i)C p β+O(β −1/2),(A30) because−e −iπ/4 +e −iπ/4(1− √ 2C) = (−1+i)C. Finally, R−(β) =− π 2 β+ (1 +i)C p β+O(β −1/2),(A31) follows from the exact branch symmetry. Equa- tions (A30) and (A31) provide an independent deriva- tion of the strong-coupling coefficients used in Sec. V B, including the constantC....

  6. [6]

    B. W. Shore,The Theory of Coherent Atomic Excitation (Wiley, New York, 1990)

    1990

  7. [7]

    I. I. Rabi, Space quantization in a gyrating magnetic field, Physical Review51, 652 (1937)

    1937

  8. [8]

    L. D. Landau, Zur theorie der energieubertragung. ii, Physikalische Zeitschrift der Sowjetunion2, 46 (1932)

    1932

Show all 51 references
  1. [9]

    F. T. Hioe and J. H. Eberly,n-level coherence vector and higher conservation laws in quantum optics and quantum mechanics, Physical Review Letters47, 838 (1981)

    1981

  2. [10]

    E. C. G. St¨ uckelberg, Helvetica Physica Acta5, 369 (1932)

    1932

  3. [11]

    Zener, Non-adiabatic crossing of energy levels, Pro- ceedings of the Royal Society of London

    C. Zener, Non-adiabatic crossing of energy levels, Pro- ceedings of the Royal Society of London. Series A137, 696 (1932)

    1932

  4. [12]

    Rosen and C

    N. Rosen and C. Zener, Double stern–gerlach experiment and related collision phenomena, Physical Review40, 502 (1932)

    1932

  5. [13]

    Allen and J

    L. Allen and J. H. Eberly,Optical Resonance and Two- Level Atoms(Wiley, New York, 1975)

    1975

  6. [14]

    F. T. Hioe and C. E. Carroll, Two-state problems in- volving arbitrary amplitude and frequency modulations, Physical Review A32, 1541 (1985)

    1985

  7. [15]

    Bambini and P

    A. Bambini and P. R. Berman, Physical Review A23, 2496 (1981)

    1981

  8. [16]

    Y. N. Demkov and M. Kunike, Vestnik Leningrad Uni- versity, Physics and Chemistry16, 39 (1969), see also F. T. Hioe and C. E. Carroll, Phys. Rev. A 32, 1541 (1985); J. Zakrzewski, Phys. Rev. A 32, 3748 (1985); K.- A. Suominen and B. M. Garraway, Phys. Rev. A 45, 374 (1992)

    1969

  9. [17]

    Y. N. Demkov, Soviet Physics JETP18, 138 (1964)

    1964

  10. [18]

    E. E. Nikitin, Optics and Spectroscopy13, 431 (1962), see also Discuss. Faraday Soc. 33, 14 (1962); Adv. Quan- 14 tum Chem. 5, 135 (1970)

    1962

  11. [19]

    C. W. S. Conover, Effects of pulse shape on strongly driven two-level systems, Physical Review A84, 063416 (2011)

    2011

  12. [20]

    N. V. Vitanov and B. M. Garraway, Landau–zener model: Effects of finite coupling duration, Physical Review A53, 4288 (1996), see also Erratum, Phys. Rev. A 54, 5458(E) (1996)

    1996

  13. [21]

    provide useful comparisons with the meromorphic Lorentzian model considered here. Finite-duration ef- fects for shaped pulses have likewise been analyzed sys- tematically [22, 23], while approximate analytic treat- ments of sinusoidal near-resonant dynamics provide an- other c...

    2026 arXiv

  14. [22]

    A. M. Ishkhanyan and A. E. Grigoryan, Fifteen classes of solutions of the quantum two-state problem in terms of the confluent heun function, Journal of Physics A: Mathematical and Theoretical47, 465205 (2014), arXiv:1402.1326

    2014 arXiv

  15. [23]

    Xie and X.-L

    Q.-T. Xie and X.-L. Liu, Exact analytical results for a two-level quantum system under a lorentzian-shaped pulse field, Chinese Physics B29, 060305 (2020)

    2020

  16. [24]

    P. R. Berman, L. Yan, K.-H. Chiam, and R. Sung, Nonadiabatic transitions in a two-level quantum system: Pulse-shape dependence of the transition probability for a two-level atom driven by a pulsed radiation field, Phys- ical Review A57, 79 (1998)

    1998

  17. [25]

    G. S. Vasilev and N. V. Vitanov, Coherent excitation of a two-state system by a gaussian field, Physical Review A70, 053407 (2004)

    2004

  18. [26]

    G. S. Vasilev and N. V. Vitanov, Coherent excitation of a two-state system by a linearly chirped gaussian pulse, The Journal of Chemical Physics123, 174106 (2005)

    2005

  19. [27]

    I. I. Boradjiev and N. V. Vitanov, Control of qubits by shaped pulses of finite duration, Physical Review A88, 013402 (2013)

    2013

  20. [28]

    I. S. Mihov and N. V. Vitanov, Qubit dynamics driven by smooth pulses of finite duration, Physical Review A 110, 052609 (2024), arXiv:2404.12236 [quant-ph]

    2024

  21. [29]

    L. P. Yatsenko, S. Gu´ erin, and H. R. Jauslin, Pulse-driven near-resonant quantum adiabatic dynamics: Lifting of quasidegeneracy, Physical Review A70, 043402 (2004)

    2004

  22. [30]

    I. I. Boradjiev and N. V. Vitanov, Power narrowing in coherent atomic excitation by smoothly shaped pulsed fields, Optics Communications288, 91 (2013)

    2013

  23. [31]

    I. S. Mihov and N. V. Vitanov, Defying conventional wisdom in spectroscopy: Power narrowing on IBM quantum, Physical Review Letters132, 020802 (2024), arXiv:2308.14187

    2024

  24. [32]

    I. S. Mihov and N. V. Vitanov, Observation of power su- perbroadening of spectral line profiles on an IBM quan- tum processor, Physical Review A112, 062617 (2025), arXiv:2506.08748 [quant-ph]

    2025

  25. [33]

    R. T. Robiscoe, Perturbative solution to the time- dependent two-level problem and the validity of the rosen–zener conjecture, Physical Review A27, 1365 (1983)

    1983

  26. [34]

    G. F. Thomas, Validity of the rosen–zener conjecture for gaussian-modulated pulses, Physical Review A27, 2744 (1983)

    1983

  27. [35]

    I. S. Mihov and N. V. Vitanov, Pulse-shape effects in qubit dynamics demonstrated on an IBM quan- tum computer, Physical Review A108, 042604 (2023), arXiv:2301.10004 [quant-ph]

    2023

  28. [36]

    N. V. Vitanov, B. W. Shore, L. P. Yatsenko, K. B¨ ohmer, T. Halfmann, T. Rickes, and K. Bergmann, Power broad- ening revisited: Theory and experiment, Optics Commu- nications199, 117 (2001)

    2001

  29. [37]

    Halfmann, T

    T. Halfmann, T. Rickes, N. V. Vitanov, and K. Bergmann, Lineshapes in coherent two-photon exci- tation, Optics Communications220, 353 (2003)

    2003

  30. [38]

    N. V. Vitanov and B. W. Shore, Quantum transitions driven by missing frequencies, Physical Review A72, 052507 (2005)

    2005

  31. [39]

    F. T. Hioe and C. E. Carroll, Journal of the Optical So- ciety of America B2, 497 (1985)

    1985

  32. [40]

    I. S. Mihov and N. V. Vitanov, Isoprobability models of qubit dynamics: Demonstration via time-dependent phase control on IBM quantum (2025), arXiv:2506.19572 [quant-ph]

    2025 arXiv

  33. [41]

    E. W. Leaver, Journal of Mathematical Physics27, 1238 (1986)

    1986

  34. [42]

    A. M. Dykhne, Soviet Physics JETP11, 411 (1960), see also Sov. Phys. JETP 14, 941 (1962)

    1960

  35. [43]

    J. P. Davis and P. Pechukas, The Journal of Chemical Physics64, 3129 (1976)

    1976

  36. [44]

    M. V. Berry and R. Lim, Universal transition prefac- tors derived by superadiabatic renormalization, Journal of Physics A: Mathematical and General26, 4737 (1993)

    1993

  37. [45]

    Suominen, B

    K.-A. Suominen, B. M. Garraway, and S. Stenholm, The adiabatic limit of level crossing models, Optics Commu- nications82, 260 (1991)

    1991

  38. [46]

    A. Joye, G. Mileti, and C.-E. Pfister, Physical Review A 44, 4280 (1991)

    1991

  39. [47]

    Fukushima and T

    K. Fukushima and T. Shimazaki, Lefschetz-thimble in- spired analysis of the dykhne–davis–pechukas method and an application for the schwinger mechanism, Annals of Physics415, 168111 (2020)

    2020

  40. [48]

    J. V. Armitage and W. F. Eberlein,Elliptic Func- tions(Cambridge University Press, 2006) see also M. Abramowitz and I. A. Stegun (eds.), Handbook of Math- ematical Functions, 10th printing, National Bureau of Standards (1972)

    2006

  41. [49]

    D. S. F. Crothers, Journal of Physics A5, 1680 (1972), see also J. Phys. B 6, 1418 (1973)

    1972

  42. [50]

    Banks and T

    T. Banks and T. J. Torres, Two point pad´ e approximants and duality (2013), arXiv:1307.3689 [hep-th]

    2013 arXiv

  43. [51]

    G. A. Baker, Jr. and P. R. Graves-Morris,Pad´ e Approx- imants, 2nd ed., Encyclopedia of Mathematics and its Applications, Vol. 59 (Cambridge University Press, Cam- bridge, 1996)

    1996

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.