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arxiv: 2502.04033 · v1 · pith:5FKR4MUDnew · submitted 2025-02-06 · 🪐 quant-ph

Hidden facts in Landau-Zener transitions revealed by the Riccati Equation

Eric P. Glasbrenner , Yannik Gerdes , S\'andor Varr\'o , Wolfgang P. Schleich This is my paper

Pith reviewed 2026-05-23 04:14 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Landau-Zener transitionRiccati equationMarkov approximationprobability amplitudesasymptotic behaviorquantum dynamicstwo-level systemnonlinear differential equation
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The pith

The Riccati equation bridges the two Landau-Zener probability amplitudes and shows why the Markov approximation gives the exact asymptotic result for one but not the other.

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

In the Landau-Zener problem the time evolution is carried by two probability amplitudes. Their dynamics can be rewritten exactly in terms of the solution of a Riccati differential equation. Dropping the nonlinear term in that equation is identical to the Markov approximation and produces the precise long-time value of one amplitude. The same Riccati equation locates the source of the Markov approximation's failure to recover the correct asymptotic value of the second amplitude. These relations are obtained from approximate analytic solutions of the Riccati equation constructed in successive time intervals.

Core claim

The dynamics of the two probability amplitudes in the elementary Landau-Zener problem can be expressed in terms of the solution of the corresponding Riccati differential equation. This solution provides the bridge between the two amplitudes. Neglecting the nonlinearity in the Riccati equation is equivalent to the Markov approximation, which yields the exact asymptotic expression for one of the probability amplitudes. The Riccati equation itself identifies the origin of the failure of the Markov approximation to provide the correct asymptotic expression for the other amplitude. The approach relies on approximate yet analytical solutions of the Riccati equation in different time regimes.

What carries the argument

The Riccati differential equation obtained from the Landau-Zener amplitudes; its solution connects the two amplitudes while its nonlinearity controls the validity of the Markov approximation.

If this is right

  • The Markov approximation supplies the exact asymptotic probability for one of the two final states.
  • The retained nonlinear term in the Riccati equation is the precise reason the Markov approximation fails for the second amplitude.
  • Separate analytic approximations to the Riccati solution in successive time regimes suffice to track the effect of nonlinearity on both final amplitudes.
  • Once one amplitude is known, the bridge property of the Riccati solution determines the other.

Where Pith is reading between the lines

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

  • The same Riccati reformulation could be applied to other two-level systems with linear time dependence to test whether the Markov-Markov distinction persists.
  • Direct numerical integration of the Riccati equation itself would provide a quantitative check on the accuracy of the piecewise analytic approximations used for the asymptotics.
  • The explicit identification of nonlinearity as the source of discrepancy suggests a systematic route for constructing improved, non-Markovian approximations in related driven two-level models.

Load-bearing premise

The approximate analytical solutions of the Riccati equation constructed separately in different time regimes correctly capture the impact of its nonlinear term on the asymptotic amplitudes.

What would settle it

A numerical solution of the original time-dependent Schrödinger equation for the Landau-Zener Hamiltonian whose long-time amplitudes deviate from the values obtained by applying the Markov approximation to the Riccati equation.

Figures

Figures reproduced from arXiv: 2502.04033 by Eric P. Glasbrenner, S\'andor Varr\'o, Wolfgang P. Schleich, Yannik Gerdes.

Figure 1
Figure 1. Figure 1: Time dependence of the probability amplitudes [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time dependence of the probability amplitudes [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical solution η of the Riccati differential equation, eq. (31), subjected to the initial condition eq. (32), visualized as a trajectory in the complex plane. The trajectory starts at τ = −τ0 from the origin marked by the grey cross and approaches rapidly without circling the origin an oscillatory regime. Indeed, the dark grey dot indicates the time τ = 0. The grey unit circle serves as a guide for the… view at source ↗
Figure 5
Figure 5. Figure 5: Emergence of the time dependence of the two probability amplitudes [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison between the exact numerical solution [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Oscillations in the early time evolution of the proba [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison between the numerical evaluation of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison between the numerical solution [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
read the original abstract

We express the dynamics of the two probability amplitudes in the elementary Landau-Zener problem in terms of the solution of the corresponding Riccati differential equation and identify three key features: (i) The solution of the Riccati equation provides the bridge between the two probability amplitudes. (ii) Neglecting the non-linearity in the Riccati equation is equivalent to the Markov approximation which yields the exact asymptotic expression for one of the probability amplitudes, and (iii) the Riccati equation identifies the origin of the failure of the Markov approximation not being able to provide us in general with the correct asymptotic expression of the other probability amplitude. Our approach relies on approximate yet analytical solutions of the Riccati equation in different time regimes, highlighting the impact of its non-linear nature on the time evolution of the system.

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 manuscript reformulates the Landau-Zener (LZ) two-level dynamics in terms of the solution to the associated Riccati differential equation. It asserts three features: (i) the Riccati solution bridges the two probability amplitudes, (ii) dropping the nonlinear term is equivalent to the Markov approximation and recovers the exact asymptotic expression for one amplitude, and (iii) the retained nonlinearity explains why the Markov approximation fails to give the correct asymptotic for the second amplitude. These conclusions are reached by constructing approximate analytical solutions of the Riccati equation separately in distinct time regimes and matching them.

Significance. If the central claims are rigorously established, the work supplies a compact mathematical link between the two LZ amplitudes and a transparent origin for the selective success of the Markov approximation. This perspective could be useful for analyzing other time-dependent two-level problems where nonlinearity in an auxiliary equation controls asymptotic accuracy. The significance is currently tempered by the absence of explicit error bounds or direct numerical/exact comparisons that would confirm the piecewise approximations transmit the nonlinear contribution without uncontrolled artifacts.

major comments (2)
  1. [Abstract] Abstract (final sentence) and the associated derivations: the claim that the regime-specific approximate solutions of the Riccati equation correctly transmit the effect of the nonlinear term onto the asymptotic amplitudes is load-bearing for all three listed features. No independent error estimate, matching-condition analysis, or direct comparison to the known exact LZ asymptotics (e.g., the standard exponential form for the transition probability) is supplied to substantiate that the piecewise construction does not introduce or mask nonlinear contributions.
  2. [Abstract] Abstract (item ii): the asserted equivalence between the linear (Markov) truncation of the Riccati equation and the exact asymptotic expression for one amplitude requires an explicit derivation showing that the neglected term vanishes identically in the relevant limit; without this step the identification of the 'origin of the failure' for the second amplitude does not follow.
minor comments (1)
  1. [Abstract] Abstract, last sentence: the phrasing 'the failure of the Markov approximation not being able to provide us in general with the correct asymptotic expression' is grammatically awkward and should be revised for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence) and the associated derivations: the claim that the regime-specific approximate solutions of the Riccati equation correctly transmit the effect of the nonlinear term onto the asymptotic amplitudes is load-bearing for all three listed features. No independent error estimate, matching-condition analysis, or direct comparison to the known exact LZ asymptotics is supplied to substantiate that the piecewise construction does not introduce or mask nonlinear contributions.

    Authors: We acknowledge that the manuscript would be strengthened by explicit validation of the piecewise construction. The matching is performed in overlapping time intervals where both approximations remain valid, which is intended to preserve continuity and carry the nonlinear contribution forward. To address the concern directly, we will add a matching-condition analysis together with direct comparisons of the resulting asymptotic amplitudes against the known exact Landau-Zener expressions and numerical integrations of the Riccati equation. revision: yes

  2. Referee: [Abstract] Abstract (item ii): the asserted equivalence between the linear (Markov) truncation of the Riccati equation and the exact asymptotic expression for one amplitude requires an explicit derivation showing that the neglected term vanishes identically in the relevant limit; without this step the identification of the 'origin of the failure' for the second amplitude does not follow.

    Authors: We agree that an explicit demonstration of the vanishing is necessary for rigor. In the appropriate large-time limit the nonlinear term is suppressed by the asymptotic form of the solution for the amplitude in question, recovering the exact Markov result. We will insert a dedicated derivation of this limit in the revised manuscript to make the equivalence and the contrasting behavior of the second amplitude fully transparent. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from standard LZ setup; no circular reductions

full rationale

The paper starts from the elementary Landau-Zener Hamiltonian, expresses the amplitudes via the Riccati equation, and constructs approximate analytical solutions in separate time regimes to identify the listed features. No self-citations, parameter fittings, ansatzes smuggled via prior work, or renamings of known results appear as load-bearing steps. The equivalence of linear truncation to Markov approximation and the attribution of failure to nonlinearity follow directly from the equation structure and regime-specific approximations without reducing to the target asymptotics by definition. This is the common case of an independent mathematical reformulation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard two-level LZ Hamiltonian and the validity of piecewise analytical approximations to the resulting Riccati equation; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The Landau-Zener problem is governed by a pair of coupled first-order linear ODEs for the probability amplitudes.
    Standard starting point in quantum mechanics; invoked implicitly when the Riccati reformulation is introduced.
  • ad hoc to paper Approximate analytical solutions of the Riccati equation can be constructed separately in distinct time regimes and then matched.
    Explicitly stated in the final sentence of the abstract as the method used to highlight nonlinearity effects.

pith-pipeline@v0.9.0 · 5681 in / 1316 out tokens · 46254 ms · 2026-05-23T04:14:32.961606+00:00 · methodology

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Reference graph

Works this paper leans on

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

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    (33), of η into the ansatz, eq

    Probability amplitude a We start our analysis by considering the probability amplitude a and substitute this representation, eq. (33), of η into the ansatz, eq. (26), and arrive at the identity a(τ) = eiφ(τ)A(τ) (34) where the amplitude A(τ) ≡ e− Re[H(τ)] = exp  − τZ −τ0 dτ ′η(R)(τ ′)   (35) is given by the negative exponential of the integral of the ...

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    (3) as follows b = i˙a + ϵτ a, (37) and substitute eq

    Connection between a and b by η In order to find an expression for b we rearrange eq. (3) as follows b = i˙a + ϵτ a, (37) and substitute eq. (28) into this expression. We arrive at the representation b = −iηa (38) for the probability amplitude for b in terms of the complex- valued function η and the probability amplitude a. Hence, η given by the Riccati d...

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    (31), representing two real-valued functions

    Summary In summary, we have shown that the two complex- valued probability amplitudes, a and b, corresponding to four real-valued functions, are governed by the complex- valued Riccati differential equation, eq. (31), representing two real-valued functions. For the specific initial condition, eq. (32), the time-integral of the real part η(R) and the phase...

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    (38), for b, we have used the differential equation, eq

    Exponential representation In the derivation of the expression, eq. (38), for b, we have used the differential equation, eq. (3), along with the ansatz, eq. (26), for a. Equivalently, we can formally integrate the differential equation, eq. (4), for b subjected to the initial condition, eq. (6), to arrive at the formula b(τ) = −ie−iϵτ 2/2 τZ −τ0 dτ ′eiϵτ ...

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    (55) due to the presence of η in the exponent of eq

    Approximations of the non-linear integral equation It is interesting that η2 appears in eq. (55) due to the presence of η in the exponent of eq. (54). When we neglect η in this exponent, that is the non-linearity in the Riccati equation, eq. (55), we arrive at the linear differential equation ˙ηM = −2iϵτ ηM + 1 (56) with the solution ηM(τ) ≡ e−iϵτ 2 τZ −τ...

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    (54), into the form η = i b a (60) following from eq

    A different integral equation for η In order to cast η given by the non-linear integral equation, eq. (54), into the form η = i b a (60) following from eq. (38), we take the term a−1(τ) out of the integral which yields the expression η(τ) = e−iϵτ 2 exp   τZ −τ0 dτ ′η(τ ′)   × τZ −τ0 dτ ′eiϵτ ′2 exp  − τ ′ Z −τ0 dτ ′′η(τ ′′)   . (61) Next, we decom...

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    An integral identity Indeed, we now use the non-linear integral equation, eq. (62), to establish the identity |I(τ)| = p 1 − A2(τ) (64) 9 where we have introduced the integral I(τ) ≡ τZ −τ0 dτ ′eiλ(τ ′)A(τ ′). (65) From the derivative d dτ |I|2 = d dτ (II ∗) = ˙II ∗ + I ˙I ∗ (66) of the absolute value squared of I, we find with the help of eq. (65) the id...

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    Summary We are now in the position to make the connection between the non-linear integral equation, eq. (62), for η and the representation eq. (47). Indeed, the identity, eq. (64), leads to the representation e−iλ(τ) τZ −τ0 dτ ′eiλ(τ ′)A(τ ′) = p 1 − A(τ)2eiϕη(τ) (73) where the phase ϕη(τ) is determined by the argument of the left-hand side of this equati...

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    Schr¨ odinger picture When we insert the formal solution, eq. (49), for b into the differential equation, eq. (3), we obtain the differential equation ˙a(τ) = iϵτ a(τ) − e−iϵτ 2/2 τZ −τ0 dτ ′eiϵτ ′2/2a(τ ′) (76) of Lippmann-Schwinger type [28] for the probability am- plitude a. Up to this point, our derivation remains exact, and no approximation has been ...

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    Interaction picture Next, we compare and contrast this expression with the one derived from the Markov approximation in the interaction picture, which was previously obtained in Ref. [24]. From eq. (13), we find with the initial condition, eq. (15), the expression for ˜b which when substituted into eq. (12) yields the integral differential equation ˙˜a(τ)...

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    Emergence of oscillations We start our discussion with the case of a vanishing initial condition, eq. (32), at a finite but large negative time τ = −τ0. Given that η is initially zero, we expect it to remain small during the early-time dynamics. This assumption allows us to linearize the Riccati equation by Figure 7. Oscillations in the functions η and ηM...

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    No oscillations Deeper insight into the oscillations emerges when we take the first derivative in time of of ηM given by eq. (98). The first term, proportional to 1 /τ turns into 1/τ 2 which we neglect. However, due to the quadratic chirp the oscillatory term creates an additional term linear in τ, and as a result, the amplitude is now of the order τ /τ0,...

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    Connection formula From the definition, eq. (79), of ηM we find the expres- sion ηM(−τ) = e−iϵτ 2 −τZ −τ0 dτ ′eiϵτ ′2 (113) which with the new integration variable ¯τ ≡ − τ ′ takes the form ηM(−τ) = e−iϵτ 2 τ0Z τ d¯τ eiϵ¯τ 2 . (114) Next, we subtract and add an appropriate integral to 16 obtain the relation ηM(−τ) = e−iϵτ 2   − τZ −τ0 d¯τ eiϵ¯τ 2 + τ0Z...

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    Expression for probability amplitude a We start our discussion by recalling the linear second order differential equation d2 dz2 a(z) − z2 4 + i 2ϵ + 1 2 a(z) = 0 (A1) for the probability amplitudes a with the complex scaling z ≡ √ 2ϵτei π 4 . (A2) We emphasize, that z differs from the corresponding quan- tity in Ref. [35] by the factor √ϵ. According to R...

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    Expression for probability amplitude b In order to derive an expression for b we recall eq. (3) which in the complex scaling, eq. (A2), reads i d dz a(z) = i z 2 a(z) − e−i π 4 √ 2ϵ b(z), (A26) and cast it into the form b(z) = √ 2ϵe−i π 4 d dz a(z) − z 2 a(z) . (A27) Together with the expressions, eq. (A3) and eq. (A8), for a and its derivative, we find b...

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