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Towards a Landau-Zener formula for an interacting Bose-Einstein condensate

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D. Witthaut, E. M. Graefe, H. J. Korsch

Quantum Physics · Submitted 20 Feb 2006 · 23 references

Pith's reading: The authors obtain an explicit Landau-Zener tunneling formula for an interacting two-mode Bose-Einstein condensate using the independent crossings approximation.

T0 · AI review

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arXiv IDquant-ph/0602163 (version 1)
Pith ID3LS6XIM5, Pith's permanent record of this paper and its review
CitationD. Witthaut, E. M. Graefe, H. J. Korsch. "Towards a Landau-Zener formula for an interacting Bose-Einstein condensate." arXiv:quant-ph/0602163 (2006). https://pith.science/paper/3LS6XIM5
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Claim paper
The paper

The paper claims

The authors obtain an explicit Landau-Zener tunneling formula for an interacting two-mode Bose-Einstein condensate using the independent crossings approximation.

The reading

Imagine a cloud of cold atoms held in a double-well trap, two pockets connected by tunneling. If you tilt the trap steadily, the energies of the two pockets move past each other. A single particle would either stay in its original pocket (diabatic) or switch pockets (adiabatic), with a probability given by the famous Landau-Zener formula. Interactions between many atoms change this: in the mean-field picture the levels can even form loops, and for strong attractive interactions the tunneling probability stops vanishing in the slow-tilting limit. Earlier approximations failed precisely in this strongly interacting, quasiadiabatic regime.

The authors go back to the full many-particle model with a fixed number N of atoms. The many-body Hamiltonian is a Bose-Hubbard chain, and the atom number is conserved because the two modes are the two wells. The initial state is all atoms in one well. As the tilt sweeps, the system passes a series of avoided crossings between the many-body levels. The paper treats these crossings as independent two-level Landau-Zener events, using the actual energy gap at each crossing as the effective coupling. Replacing sums over the N levels by integrals, they obtain closed-form expressions for the tunneling probability, Eq. (25) for the strongly interacting regime and Eq. (26) for the weakly interacting regime.

The strongly interacting formula matches numerical simulations of both the many-particle and mean-field equations, and it captures the increase of tunneling probability with interaction strength that earlier work could not reproduce. The weakly interacting formula performs worse, and the authors say so.

Extended reading notes

Core claim

The load-bearing claim is that Eq. (25), P_LZ ≈ x_c + (1-x_c) e^u u^{-(u+1)} γ(u+1,u) with u=πω^2/2α and x_c from Eq. (18), is an explicit Landau-Zener formula for the interacting two-mode BEC that 'agrees well with the numerical results' in the strongly interacting regime |g|>2v, including the previously unsolved quasiadiabatic limit α→0. If correct, this gives the first quantitative closed form for this regime.

Load-bearing premise

The independent crossings approximation (ICA), stated in Section II ('One assumes that the system undergoes a series of single, independent transitions between just two levels') and the modified ICA in the appendix, treats each avoided crossing as an isolated two-level Landau-Zener transition with the adiabatic level splitting as the effective coupling. This requires all crossings to be well separated; the paper itself notes the ICA fails for weak nonlinearity or large α, and the quality of the supercritical formula further depends on the empirically fitted constant a=1.14 in Eq. (18).

Original abstract

We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-level system, for the full many-particle system as well and in the mean-field approximation. The many-particle problem can be solved approximately within an independent crossings approximation, which yields an explicit Landau-Zener formula.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,305 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.31 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.22 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-level system, for the full many-particle system as well and in the mean-field approximation. The many-particle problem can be solved approximately within an independent crossings approximation, which yields an explicit Landau-Zener formula.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-level system, for the full many-particle system as well and in the mean-field approximation. contribution 0.65 0.76 0.07 asserted 0.70
s1 The many-particle problem can be solved approximately within an independent crossings approximation, which yields an explicit Landau-Zener formula. result 0.47 0.85 0.68 asserted 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • references and citing sentences
  • relations to other papers
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  • T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: Total energy (5) in the mean-field theory (left) and
Figure 4. FIG. 4: Landau-Zener transition probability
Figure 5. FIG. 5: Spectrum of the many-particle Hamiltonian (6) for
Figure 6. FIG. 6: Squared level splitting
Figure 7. FIG. 7: Landau-Zener tunneling probability in dependence o
Figure 8. FIG. 8: Diabatic (dash-dotted line) and adiabatic (solid li
Figure 9. FIG. 9: Diabatic (dash-dotted line) and adiabatic (solid li

Showing 7 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:quant-ph/0602163 by the authors.

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