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Anderson localization of elementary excitations in a one dimensional Bose-Einstein condensate

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Nicolas Bilas, Nicolas Pavloff

Other Condensed Matter · Submitted 27 Feb 2006 · 51 references

Pith's verdict: A mostly solid hydrodynamic result, but the all-energy transfer-matrix claim is unsupported because the Appendix B single-impurity S-matrix is internally inconsistent at ω=0.

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arXiv IDcond-mat/0602622 (version 2)
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CitationNicolas Bilas, Nicolas Pavloff. "Anderson localization of elementary excitations in a one dimensional Bose-Einstein condensate." arXiv:cond-mat/0602622 (2006). https://pith.science/paper/GDV3XHNZ
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The paper

The paper claims

The paper establishes that weak disorder in a one-dimensional Bose-Einstein condensate localizes elementary excitations with a localization length that diverges as $\omega^{-2}$ at low energy and reaches a minimum near $\hbar\omega =…

The reading

Elementary excitations of a one-dimensional Bose-Einstein condensate are sound-like at low energy and particle-like at high energy. The paper asks how weak axial disorder localizes these excitations, and answers with two complementary calculations: a low-energy hydrodynamic phase-formalism for a white-noise potential, and a transfer-matrix treatment of randomly placed point impurities valid at all energies. Both yield a localization length $L_{\mathrm{loc}}$ whose low-energy divergence behaves as $L_{\mathrm{loc}} \propto \omega^{-2}$, with a minimum near $\hbar\omega \simeq \mu$. If correct, the result gives an analytic bridge between phonon localization and single-particle Anderson localization in the same system, and concrete predictions for elongated BEC experiments.

What carries the argument

The argument runs on two equivalent tools for extracting Lyapunov exponents. The hydrodynamic phase formalism represents density oscillations as $\alpha = r\cos\theta$, $\beta = r\sin\theta$, reduces the stochastic equation for $z = \alpha/\beta$ to a Fokker-Planck equation, and gives $\gamma = (\omega/c_0)\langle z\rangle_{\mathrm{st}}$, so the low-energy expansion of the averaged phase yields $\gamma \propto \omega^2$. The transfer-matrix approach instead composes $2\times 2$ matrices for free propagation and single delta-impurity scattering, using the random-phase-averaged identity $\gamma = -n_{\mathrm{imp}}\ln|t_\lambda|$, where $t_\lambda$ is the exact transmission amplitude across one impurity computed from the Bogoliubov-de Gennes equations. The matching of the two formulas at low energy closes the argument.

Extended reading notes

Core claim

The central claim is that the localization length of elementary excitations in a weakly disordered one-dimensional BEC is explicitly computable and has a non-monotonic energy dependence. In the low-energy phonon regime, $L_{\mathrm{loc}} = \gamma^{-1}$ diverges as $(\hbar\omega/\mu)^{-2}$, with $\gamma = \xi^2 D (\hbar\omega/\mu)^2/8$ plus corrections controlled by $\omega\delta = \xi^4 D\omega/c_0$. In the one-dimensional mean-field regime with a dilute set of point impurities, the Lyapunov exponent is $\gamma = -n_{\mathrm{imp}} \ln |t_\lambda|$, where $t_\lambda$ is the single-impurity transmission amplitude obtained from the Bogoliubov-de Gennes equations; this reproduces the same low-energy $\omega^{-2}$ law and predicts a minimum of $L_{\mathrm{loc}}$ around $\hbar\omega \simeq \mu$. The same hydrodynamic treatment also covers the transverse Thomas-Fermi regime, and it yields the result that in the Tonks-Girardeau limit the envelope of low-energy density oscillations stays constant, so exponential localization is absent in that regime.

Load-bearing premise

The calculation needs the disorder correlation length to sit strictly between the healing length and the wavelength of the excitation, wide enough that the condensate order parameter stays in Thomas-Fermi form, narrow enough that the excitation sees the potential as white noise.

Editorial extensions

If this is right

  • At low energy, the localization length diverges as $(\hbar\omega/\mu)^{-2}$, so very soft phonons are almost unaffected by weak disorder; the relevant small parameter is $\omega\delta = \xi^4 D\omega/c_0$.
  • The full localization curve has a minimum near $\hbar\omega \simeq \mu$, meaning mid-energy excitations localize most strongly; in the dilute-impurity model this follows from $\gamma = -n_{\mathrm{imp}}\ln|t_\lambda|$.
  • The two approaches agree in their overlap region, with the small-$\lambda$ transfer-matrix result reducing to $\gamma = \lambda^2 n_{\mathrm{imp}} (\hbar\omega/\mu)^2/8$, matching the hydrodynamic expansion.
  • In the transverse Thomas-Fermi regime the same hydrodynamic formulas apply with rescaled $\mu$ and $\xi$, so speckle experiments can be compared directly with the closed-form estimate for $L_{\mathrm{loc}}$.
  • In the Tonks-Girardeau limit, the hydrodynamic envelope is not exponentially localized at low energy, which sets that regime apart from both the mean-field and transverse Thomas-Fermi cases.

Reading between the lines

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

  • Because the same $\omega^{-2}$ law emerges from two different disorder models, the universal low-energy prediction is likely robust for any weak short-correlated disorder; measuring only the exponent separates universal physics from model details.
  • The paper's Tonks-Girardeau result, absence of exponential localization in the hydrodynamic limit, suggests that strong effective interactions protect long-wavelength excitations; a one-dimensional gas tuned across a confinement resonance would be a direct place to test this.
  • The transfer-matrix formula connects naturally to the single-particle regime at high energy, so the crossover at $\hbar\omega \approx \mu$ could serve as an interaction-tunable switch between phonon localization and standard Anderson localization.
Original abstract

We study the elementary excitations of a transversely confined Bose-Einstein condensate in presence of a weak axial random potential. We determine the localization length (i) in the hydrodynamical low energy regime, for a domain of linear densities ranging from the Tonks-Girardeau to the transverse Thomas-Fermi regime, in the case of a white noise potential and (ii) for all the range of energies, in the ``one-dimensional mean field regime'', in the case where the randomness is induced by a series of randomly placed point-like impurities. We discuss our results in view of recent experiments in elongated BEC systems.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,031 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.02 · states quantitative result 0.11 · compares with prior results 0.18 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We study the elementary excitations of a transversely confined Bose-Einstein condensate in presence of a weak axial random potential. We determine the localization length (i) in the hydrodynamical low energy regime, for a domain of linear densities ranging from the Tonks-Girardeau to the transverse Thomas-Fermi regime, in the case of a white noise potential and (ii) for all the range of energies, in the ``one-dimensional mean field regime'', in the case where the randomness is induced by a series of randomly placed point-like impurities. We discuss our results in view of recent experiments in elongated BEC systems.

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 study the elementary excitations of a transversely confined Bose-Einstein condensate in presence of a weak axial random potential. contribution 0.92 0.63 0.04 asserted 0.73
s1 We determine the localization length (i) in the hydrodynamical low energy regime, for a domain of linear densities ranging from the Tonks-Girardeau to the transverse Thomas-Fermi regime, in the case of a white noise potential and (ii) for all the range of energies, in the ``one-dimensional mean field regime'', in the case where the randomness is induced by a series of randomly placed point-like impurities. contribution 0.73 0.96 0.44 asserted 1.00
s2 We discuss our results in view of recent experiments in elongated BEC systems. interpretation 0.99 0.38 0.04 unclear 0.74

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.95.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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. Lloc as a function of the energy ~ω of an ele￾mentary excitation in logarithmic scale (µ is the chemical potential of the system, and Lµ is the value of Lloc when ~ω = µ). The curve has been drawn within the model used in Section 4, employing formulas (40) and (61). This yields λ 2 nimpLloc = 4[(~ω/µ) 2 + 1]/[ p (~ω/µ) 2 + 1 − 1] (the meaning of the parameters λ and nimp in this for￾mula is explained in Section 4)
Figure 2. γ as a function of ω in rescaled units. The solid line is the numerical evaluation of γ using formulas (16) and (22). The dashed lines are the small and large ωδ approximations [Eqs. (23) and (24)]. The quantity Jω is also of interest for itself, because it gives informations on the density of states of the excita￾tions. It is show in Appendix A that, if N(ω) denotes the integrated density of state per unit length, one has N(ω) = ω c0 Jω . (25)
Figure 3. Scattering channels used for writing the trans￾fer matrices Tλ and T0(ℓ) of (33). In the case of unper￾turbed motion over a length ℓ one has xout−xin = ℓ. In the case of scattering by a delta peak located at (xin+xout)/2, one should take ξ ≪ xout − xin ≪ n −1 imp. The coefficients uω and vω in
Figure 4. γ as a function of ω in rescaled units. The plot is drawn for λ = 1 and ξ nimp = 0.02. The dots are the results of the numerical simulation and the solid line is the analytical result from Eq. (40). The inset displays a blowup of the figure at low energy. As shown in Ref. [38], in the limit nimp ≪ q, one can obtain an analytical estimate of γ. From the relation TN+1 = Tλ × T0(ℓN ) × TN one gets hln |tN+1|i = ln |tλ| + hln |tN |i −
Figure 5. γ as a function of ω in rescaled units. The plot is drawn for λ = 0.2 and ξ nimp = 0.03. The dots are the results of the numerical simulation and the solid line is the analytical result from Eq. (40). The dashed line is the approximate result (41). Formula (41) is interesting because it is identical to the first term of expansion (23) which has been obtained in Section 3 in a completely different framework, and this permits to bridge the gap between the hydrodynamical approach and the present transfer matrix method. As just mentioned, formula (41) is restricted to small values of λ,…
Figure 6. Transmission probability Tλ across a poten￾tial U(x) = λ µ ξ δ(x) as a function of ω (in rescaled units). The solid line is the exact result (57) in the cases λ = 1 and λ = 2. The dashed line are the corresponding small λ approximations (61). The exact formula for Tλ [from (57)] is compared on

Figures reproduced from arXiv:cond-mat/0602622 by the authors.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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