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arxiv: 2604.23279 · v2 · submitted 2026-04-25 · ❄️ cond-mat.mes-hall

Analytical Treatment of Noise-Suppressed Klein Tunneling in Graphene with Possible Implications for Quantum-Dot Qubits

Kamal Azaidaoui , Ahmed Jellal , Hocine Bahlouli , A. Al Luhaibi , Michael Vogl This is my paper

Pith reviewed 2026-05-08 07:36 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords grapheneKlein tunnelingwhite noiseLindblad equationtransmission probabilityquantum dotsqubitsdissipation
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The pith

Noise in a fluctuating graphene barrier induces a complex wavevector that suppresses Klein tunneling even at normal incidence.

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

The paper treats a time-varying potential barrier in graphene whose height is driven by Gaussian white noise. It converts the stochastic dynamics into an equivalent time-independent Lindblad equation for the density matrix, which yields closed-form solutions. Inside the barrier the longitudinal wavevector becomes complex, with an imaginary part set by the noise intensity. This produces exponential damping of the transmission probability instead of the perfect transmission that occurs in the noiseless case. The result suggests that controlled noise could serve as a tunable knob for electron transport in graphene devices and quantum-dot qubits.

Core claim

By modeling the barrier height as Gaussian white noise and mapping the problem onto a Lindblad master equation, the transmission through the barrier acquires an exponentially decaying factor whose rate grows with noise strength. Consequently the perfect normal-incidence transmission characteristic of Klein tunneling is replaced by strong suppression, independent of barrier height.

What carries the argument

The exact mapping of the stochastic time-dependent Schrödinger equation onto a time-independent Lindblad equation for the density matrix, which replaces the real wavevector inside the barrier by a complex value whose imaginary part produces damping.

If this is right

  • Transmission probability decays exponentially with barrier width instead of oscillating with Fabry-Pérot resonances.
  • Klein tunneling is eliminated at normal incidence for any barrier height once noise is present.
  • Noisy barriers function as tunable dissipative elements that can be engineered into graphene circuits.
  • The same formalism supplies an analytical route to design graphene quantum dots with reduced unwanted tunneling for spin-qubit applications.

Where Pith is reading between the lines

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

  • The Lindblad mapping may extend directly to other Dirac materials whose low-energy Hamiltonian is linear.
  • Controlled noise on electrostatic gates could provide a practical handle for suppressing leakage currents in graphene-based quantum information devices.
  • Time-resolved transport measurements in noisy junctions would test whether the predicted complex wavevector appears in the differential conductance.

Load-bearing premise

The fluctuating barrier can be replaced by an exactly solvable Markovian Lindblad equation under the white-noise and weak-coupling approximations.

What would settle it

Fabricate a graphene junction with a gate whose voltage is deliberately fluctuated at white-noise frequencies and measure the angle-resolved transmission; if the transmission at normal incidence remains near unity and independent of noise amplitude, the predicted suppression is ruled out.

Figures

Figures reproduced from arXiv: 2604.23279 by A. Al Luhaibi, Ahmed Jellal, Hocine Bahlouli, Kamal Azaidaoui, Michael Vogl.

Figure 1
Figure 1. Figure 1: FIG. 1. Quantum tunneling across a barrier of static height view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Sample realization of Gaussian white noise fluctua view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic of the nine regions in the ( view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Transmission and absorption for a non-relativistic view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Transmission and absorption probabilities for a non view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Transmission (a) and absorption (b) probabilities view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Angular dependence of transmission and ab view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Illustration of Quantum confinement in two cases:(a) view at source ↗
read the original abstract

We study quantum tunneling through a potential barrier whose height fluctuates in time and is modeled by Gaussian white noise. We map the stochastic dynamics onto an equivalent time-independent Lindblad equation for the density matrix, allowing fully analytical solutions. For Schr\"odinger particles, noise introduces dissipation that suppresses Fabry-P\'erot oscillations and yields an exponentially decaying transmission. Applying the same formalism to graphene, we demonstrate that noise induces a complex longitudinal wavevector within the barrier, leading to a strong suppression of transmission and Klein tunneling, even at normal incidence. Our approach promises improved control over Klein tunneling. These results demonstrate that noisy barriers can act as tunable dissipative elements, offering a pathway to enhanced control of electron transport in graphene-based devices. We also briefly discuss how our results could guide the design of graphene quantum dots for potential use in spin qubit devices.

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

3 major / 2 minor

Summary. The paper develops an analytical framework for quantum tunneling through a time-fluctuating potential barrier modeled as Gaussian white noise. It maps the stochastic Dirac (or Schrödinger) equation to a time-independent Lindblad master equation for the density matrix under white-noise and Markovian assumptions, yielding closed-form transmission probabilities. For graphene, the noise generates a complex longitudinal wavevector inside the barrier that exponentially suppresses transmission, including Klein tunneling at normal incidence; the authors suggest this offers a route to tunable control of transport and improved graphene quantum-dot qubit designs.

Significance. If the mapping and resulting expressions are valid, the work supplies a rare closed-form treatment of dissipative effects in relativistic tunneling, treating noise as a controllable parameter rather than a perturbation. This could be relevant for mitigating Klein tunneling in graphene mesoscopic devices and for qubit architectures where barrier fluctuations are inevitable. The approach also recovers known limits for Schrödinger particles (damped Fabry-Pérot resonances and exponential decay), lending internal consistency when the approximations hold.

major comments (3)
  1. [§2] §2 (Model and Lindblad mapping): The derivation of the time-independent Lindblad equation from the stochastic Dirac equation invokes both delta-correlated (white) noise and the Markovian limit without comparing the noise correlation time to the barrier traversal time ħ/|E−V|. In tunneling regimes this traversal time can be comparable to or longer than realistic fluctuation timescales in graphene devices, so non-Markovian corrections could alter the effective complex k_x and the claimed suppression of normal-incidence transmission.
  2. [§3] §3 (Graphene transmission): The central claim that noise induces a complex longitudinal wavevector leading to strong suppression even at normal incidence rests on the Lindblad-derived dispersion inside the barrier. No explicit check is shown that the transmission probability recovers the known perfect Klein tunneling (T=1 at θ=0) in the zero-noise limit; without this limit test the result risks being an artifact of the mapping.
  3. [§4] §4 (Numerical validation): The manuscript presents only the analytical Lindblad solution; no direct comparison with stochastic time-dependent simulations of the original fluctuating barrier is provided to quantify the error introduced by the white-noise/Markovian approximations.
minor comments (2)
  1. [Abstract] The abstract and introduction use “fully analytical solutions” without specifying which quantities are obtained in closed form versus those requiring numerical root-finding of the Lindblad eigenvalues.
  2. [§2–§3] Notation for the noise strength and the resulting imaginary part of k_x should be unified between the Schrödinger and Dirac sections to avoid reader confusion.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and indicate the revisions made.

read point-by-point responses

  1. Referee: [§2] §2 (Model and Lindblad mapping): The derivation of the time-independent Lindblad equation from the stochastic Dirac equation invokes both delta-correlated (white) noise and the Markovian limit without comparing the noise correlation time to the barrier traversal time ħ/|E−V|. In tunneling regimes this traversal time can be comparable to or longer than realistic fluctuation timescales in graphene devices, so non-Markovian corrections could alter the effective complex k_x and the claimed suppression of normal-incidence transmission.

    Authors: We agree that the Markovian approximation requires the noise correlation time to be much shorter than the barrier traversal time ħ/|E−V|. Our derivation is performed strictly in the white-noise limit (zero correlation time), under which the Lindblad mapping is exact. We have revised §2 to include an explicit discussion of the validity regime, stating that for finite correlation times comparable to the traversal time, non-Markovian corrections could arise and would necessitate a different formalism. This addition clarifies the applicability to graphene devices without altering the core analytical results. revision: yes

  2. Referee: [§3] §3 (Graphene transmission): The central claim that noise induces a complex longitudinal wavevector leading to strong suppression even at normal incidence rests on the Lindblad-derived dispersion inside the barrier. No explicit check is shown that the transmission probability recovers the known perfect Klein tunneling (T=1 at θ=0) in the zero-noise limit; without this limit test the result risks being an artifact of the mapping.

    Authors: We thank the referee for this important observation. When the noise strength vanishes, the Lindblad dissipator term is identically zero and the master equation reduces to the unitary von Neumann evolution under the Dirac Hamiltonian. We have added an explicit calculation in the revised §3 demonstrating that the longitudinal wavevector becomes purely real as the noise amplitude approaches zero, with the transmission probability recovering T(θ=0)=1. This limit test confirms that the suppression is due to noise and not an artifact of the mapping. revision: yes

  3. Referee: [§4] §4 (Numerical validation): The manuscript presents only the analytical Lindblad solution; no direct comparison with stochastic time-dependent simulations of the original fluctuating barrier is provided to quantify the error introduced by the white-noise/Markovian approximations.

    Authors: We acknowledge that direct stochastic simulations would provide useful error quantification. However, numerical integration of the stochastic Dirac equation with delta-correlated white noise is technically challenging and requires specialized techniques such as stochastic unraveling to avoid instabilities. Our work is analytical and the mapping is exact within the stated white-noise and Markovian assumptions. We have added a paragraph in the revised §4 discussing the expected agreement in the white-noise limit and suggesting avenues for future numerical validation, but we do not include new simulations in this revision. revision: partial

Circularity Check

0 steps flagged

No circularity; mapping and solution are independent of target result

full rationale

The derivation begins from the stochastic Dirac equation with Gaussian white-noise barrier fluctuations, applies the standard Markovian averaging procedure to obtain a time-independent Lindblad equation for the density matrix, and then solves the resulting ODEs analytically to extract transmission probabilities. The complex longitudinal wavevector inside the barrier emerges directly from the noise-induced dissipative terms in that Lindblad equation; it is not presupposed or fitted. No self-citations are invoked as load-bearing uniqueness theorems, no parameters are tuned on a data subset and then relabeled as predictions, and the Klein-tunneling suppression is a calculational output rather than a definitional input. The white-noise and Markovian approximations are stated explicitly as modeling choices whose validity can be checked externally, leaving the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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    3, becauseU(x) andV(x) are piecewise constant

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