pith. sign in

arxiv: 2512.17580 · v2 · submitted 2025-12-19 · ❄️ cond-mat.quant-gas

Scattering Problem in Bose-Einstein Condensates with Magnetic Domain Wall

Mei Zhao , Lijia Jiang , Tao Yang , Jun-Hui Zheng This is my paper

Pith reviewed 2026-05-16 20:51 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas
keywords Bose-Einstein condensatemagnetic domain wallscatteringBogoliubov-de Gennestwist anglequantum transportgauge transformationphonon scattering
0
0 comments X

The pith

Scattering observables in spin-1/2 BECs depend only on the magnetic domain wall's total twist angle

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

The paper establishes that a gauge transformation reduces the scattering problem for waves incident on magnetic domain walls in spin-1/2 Bose-Einstein condensates to a dependence solely on the total twist angle Θ. This independence from chirality simplifies calculations using a transfer-matrix method within the Bogoliubov-de Gennes framework. Results show an energy threshold at the Zeeman energy separating phonon and multi-channel scattering regimes, with transmission probabilities tunable by Θ. Sympathetic readers would care because this provides a way to control quantum transport through twist-engineered structures in ultracold atomic gases.

Core claim

Using a gauge transformation, the scattering observables are shown to depend solely on the total twist Θ, independent of chirality. In the Bogoliubov-de Gennes framework, a transfer-matrix method computes reflection and transmission coefficients for phonons and particles. A threshold appears at E = ħΩ0, and for large twists competition between energies leads to density modulations and Fano resonances. Transition probabilities are enhanced for odd multiples of π.

What carries the argument

The gauge transformation that eliminates chirality dependence, reducing scattering observables to dependence solely on total twist Θ

If this is right

  • Scattering coefficients are identical for walls with the same Θ regardless of chirality
  • Below the Zeeman threshold only phonon scattering occurs
  • Transition probabilities between phonon and particle channels are enhanced for odd multiples of π and suppressed for even multiples
  • For large Θ, competition between kinetic and Zeeman energies produces comb-like density modulations and Fano-like resonances

Where Pith is reading between the lines

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

  • The same gauge approach may apply directly to scattering from other spin textures in multicomponent condensates
  • Tunable transmission with Θ suggests a route to design filters or switches for atomtronic circuits
  • The threshold behavior could be tested by varying the Zeeman field strength while holding Θ fixed

Load-bearing premise

The linear Bogoliubov-de Gennes framework accurately captures the excitations around the domain wall without significant nonlinear or higher-order corrections.

What would settle it

Measuring reflection and transmission coefficients for two domain walls with identical total twist Θ but opposite chirality and finding the coefficients identical would confirm the independence claim.

Figures

Figures reproduced from arXiv: 2512.17580 by Jun-Hui Zheng, Lijia Jiang, Mei Zhao, Tao Yang.

Figure 1
Figure 1. Figure 1: FIG. 1. Spatial profiles of the domain wall’s spin rotation [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Top) Ground-state properties for a Θ = [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Ground-state profile and scattering probabilities for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Ground-state profile and scattering probabilities for [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Ground-state profile and scattering probabilities for [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

We present a comprehensive theoretical study of linear wave scattering from magnetic domain walls with varied twist angles $\Theta$ in spin-$1/2$ Bose-Einstein condensates (BECs). Using a gauge transformation, we show that scattering observables depend solely on the total twist $\Theta$, independent of chirality. Within the Bogoliubov-de Gennes (BdG) framework, we develop a transfer-matrix method to compute reflection and transmission coefficients for incident phonons and free particles. Our results reveal a scattering threshold at the Zeeman energy $E = \hbar\Omega_0$, separating a pure phonon regime from multi-channel scattering involving both collective and single-particle excitations above threshold. For large twist angles, competition between kinetic and Zeeman energies reduces the effective spin rotation, leading to comb-like density modulations and Fano-like resonances below threshold. The transition probability between phonon and particle channels is strongly tunable with $\Theta$, enhanced for odd multiples of $\pi$ but suppressed for even multiples. These findings establish twist-engineered domain walls as a versatile platform for controlling quantum transport, with implications for atomtronic devices and quantum simulation.

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 studies linear wave scattering from magnetic domain walls in spin-1/2 Bose-Einstein condensates for varying twist angles Θ. A gauge transformation is used to show that scattering observables depend only on the total twist Θ and are independent of chirality. Within the Bogoliubov-de Gennes framework, a transfer-matrix method is developed to obtain reflection and transmission coefficients for phonons and free particles. A scattering threshold is identified at the Zeeman energy E = ħΩ₀, separating a pure phonon regime from multi-channel scattering. For large Θ, competition between kinetic and Zeeman energies produces comb-like density modulations and Fano-like resonances below threshold, with transition probabilities between channels being tunable by Θ (enhanced for odd multiples of π).

Significance. If the gauge transformation and linear BdG results hold, the work provides a simplified description of scattering that depends only on total twist, offering a tunable platform for controlling quantum transport in atomtronic devices and quantum simulators. The identification of a Zeeman threshold and Fano resonances adds concrete predictions for experiments. The approach is parameter-free in its core mapping, which is a strength, though applicability is limited by the linear approximation.

major comments (3)
  1. [Abstract] Abstract and large-twist discussion: The central claim that scattering observables depend solely on Θ (independent of chirality) rests on a gauge transformation performed inside the linear BdG equations. For large Θ the manuscript itself notes competition between kinetic and Zeeman energies that produces comb-like modulations; this indicates the static domain-wall profile may acquire nonlinear corrections that the linearization does not capture and that the gauge map may not eliminate, potentially reintroducing chirality dependence. This is load-bearing for the applicability of the independence result.
  2. [Methods / BdG framework] BdG transfer-matrix implementation: The abstract states that reflection/transmission coefficients are computed via a transfer-matrix method, yet provides no explicit derivation, boundary-condition matching, or validation against known limits (e.g., Θ=0 or small-amplitude cases). Without these steps the numerical accuracy of the reported threshold behavior and Fano resonances cannot be assessed.
  3. [Results for large twist angles] Large-Θ regime: The claim of tunable transition probabilities (enhanced for odd multiples of π) is presented without accompanying error analysis, convergence checks, or comparison to nonlinear Gross-Pitaevskii simulations. Given the noted density modulations, the linear BdG framework may miss higher-order effects that alter the reported tunability.
minor comments (2)
  1. [Abstract] Notation for the Zeeman energy is given as E = ħΩ₀; clarify whether Ω₀ is the bare Rabi frequency or an effective value after gauge transformation.
  2. [Introduction] The manuscript would benefit from a brief comparison to prior work on domain-wall scattering in spinor BECs (e.g., references to earlier BdG studies of magnetic solitons).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below and have revised the manuscript to improve clarity, provide additional methodological details, and discuss limitations in the large-twist regime.

read point-by-point responses

  1. Referee: [Abstract] Abstract and large-twist discussion: The central claim that scattering observables depend solely on Θ (independent of chirality) rests on a gauge transformation performed inside the linear BdG equations. For large Θ the manuscript itself notes competition between kinetic and Zeeman energies that produces comb-like modulations; this indicates the static domain-wall profile may acquire nonlinear corrections that the linearization does not capture and that the gauge map may not eliminate, potentially reintroducing chirality dependence. This is load-bearing for the applicability of the independence result.

    Authors: We appreciate the referee highlighting this important subtlety regarding the regime of validity. The gauge transformation is applied directly to the linearized Bogoliubov-de Gennes equations, and within this linear framework the scattering observables depend only on the total twist Θ, independent of chirality. The comb-like density modulations for large Θ emerge from the linear solution itself due to the kinetic-Zeeman competition, while the underlying static domain-wall profile is the exact nonlinear solution (known to be chirality-independent for the spin-1/2 case). We agree, however, that strong nonlinear corrections outside the linear-wave approximation could potentially affect the mapping. In the revised manuscript we will add an expanded paragraph in the Discussion section explicitly addressing the validity range of the linear approximation, providing estimates of when nonlinear effects become significant, and noting that reintroduction of chirality dependence would require a separate nonlinear analysis. revision: partial

  2. Referee: [Methods / BdG framework] BdG transfer-matrix implementation: The abstract states that reflection/transmission coefficients are computed via a transfer-matrix method, yet provides no explicit derivation, boundary-condition matching, or validation against known limits (e.g., Θ=0 or small-amplitude cases). Without these steps the numerical accuracy of the reported threshold behavior and Fano resonances cannot be assessed.

    Authors: We agree that the transfer-matrix implementation requires fuller documentation for reproducibility and assessment of accuracy. In the revised manuscript we will expand the Methods section with a dedicated subsection deriving the transfer-matrix method in detail, including the explicit form of the asymptotic solutions, the boundary-condition matching procedure across the domain wall, and the numerical implementation. We will also add validation against the known analytic limits: for Θ=0 the reflection coefficient vanishes and transmission is unity for all energies, and for small-amplitude scattering we recover the expected perturbative results. These additions will allow direct verification of the reported threshold at E=ℏΩ₀ and the Fano resonances. revision: yes

  3. Referee: [Results for large twist angles] Large-Θ regime: The claim of tunable transition probabilities (enhanced for odd multiples of π) is presented without accompanying error analysis, convergence checks, or comparison to nonlinear Gross-Pitaevskii simulations. Given the noted density modulations, the linear BdG framework may miss higher-order effects that alter the reported tunability.

    Authors: We thank the referee for emphasizing the need for quantitative controls. In the revision we will include error analysis and convergence checks for the transition probabilities, with supplementary figures demonstrating stability against variations in discretization step size, integration cutoff, and basis truncation. Regarding comparison to nonlinear Gross-Pitaevskii simulations, our work is confined to the linear scattering regime where the BdG framework is the appropriate tool; full nonlinear simulations lie outside the present scope. We will add an explicit statement in the Discussion acknowledging this limitation and noting that higher-order nonlinear effects could modify the tunability for very large Θ, suggesting such comparisons as a natural direction for future work. revision: partial

Circularity Check

0 steps flagged

No circularity: standard gauge transformation and BdG transfer matrix are independent of inputs

full rationale

The derivation begins from the spin-1/2 BEC Gross-Pitaevskii equations, applies a standard gauge transformation to absorb the domain-wall twist into the phase, and linearizes to the BdG equations. The transfer-matrix method then solves the resulting linear scattering problem for phonons and particles. None of these steps reduce to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation whose content is unverified. The claim that observables depend only on total twist Θ follows directly from the gauge map applied to the linearized equations; it is not imposed by construction. The paper remains self-contained against external benchmarks (standard BdG scattering techniques) and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on established theoretical tools for BECs without introducing new free parameters or postulated entities in the abstract.

axioms (1)
  • domain assumption The Bogoliubov-de Gennes framework accurately describes linear excitations in the spin-1/2 BEC with domain wall.
    Invoked to develop the transfer-matrix method for computing reflection and transmission coefficients.

pith-pipeline@v0.9.0 · 5501 in / 1241 out tokens · 60938 ms · 2026-05-16T20:51:07.431460+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [1]

    Next we consider a left-incident particle (PT)

    for phonon and particle channels. Next we consider a left-incident particle (PT). The boundary conditions become A1 = 0, A 5 = 1, A ′ 2 = 0, A ′ 6 = 0,(37) A3 = 0, A ′ 4 = 0, A 7 = 0, A ′ 8 = 0.(38) The sameM ′ applies, but with D=D PT = (M15, M25, M35, M45,· · ·, M 85)T .(39) The resultingCyields the reflection coefficientsA 2, A6 and transmission coeffi...

    work page

  2. [2]

    With these coefficients, the scattering matrix also can be constructed. D. Numerical Calculation We now turn to the discrete model required for numerical work. Space is discretised on a uniform grid xi =i∆x,fori= 0,1, . . . , N−1,(40) 5 and the wave-function is denoted byχ i ≡χ(x i) = U(x i) V(x i) for the BdG equations (10). Starting from the continuous ...

    work page

  3. [3]

    Using the Hermiticity ofV, the left-hand side becomes ˜Ψ†T ˜Ψ−(T ˜Ψ)† ˜Ψ = 0, leading to a total derivative equation∂ x[ ˜Ψ†Dx ˜Ψ−(D x ˜Ψ)† ˜Ψ] = ˜Ψ†D2 x ˜Ψ−(D 2 x ˜Ψ)† ˜Ψ =

    work page

  4. [4]

    This requires the currents at left and right sides should be equal,J L =J R

    The conserved current is therefore J(x) = ℏ 2mi h ˜Ψ†Dx ˜Ψ−(D x ˜Ψ)† ˜Ψ i (50) = ℏ 2mi U †DxU−(D xU) † U+V †DxV−(D xV) † V . This requires the currents at left and right sides should be equal,J L =J R. For a left-incident phonon withE >ℏΩ 0 the currents on the left and right are JL = (U 2 k1 +V 2 k1)(1− |A 2|2) ℏk1 m − |A6|2 ℏk3 m ,(51) JR =|A ′ 1|2(U 2 k...

    work page 2023

  5. [5]

    N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys.51, 591 (1979)

    work page 1979

  6. [6]

    S. S. P. Parkin, M. Hayashi, and L. Thomas, Magnetic domain-wall racetrack memory, Science320, 190 (2008)

    work page 2008

  7. [7]

    Nagaosa and Y

    N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nat. Nanotechnol.8, 899 (2013)

    work page 2013

  8. [8]

    Kawaguchi, M

    Y. Kawaguchi, M. Kobayashi, M. Nitta, and M. Ueda, Topological excitations in spinor Bose-Einstein condensates, Prog. Theor. Phys. Suppl.186, 455 (2010)

    work page 2010

  9. [9]

    M. O. Borgh and J. Ruostekoski, Topological interface engineering and defect crossing in ultracold atomic gases, Phys. Rev. Lett.109, 015302 (2012)

    work page 2012

  10. [10]

    M. O. Borgh, J. Lovegrove, and J. Ruostekoski, Imprinting a topological interface using Zeeman shifts in an atomic spinor Bose–Einstein condensate, New J. Phys.16, 053046 (2014)

    work page 2014

  11. [11]

    J.-y. Choi, W. J. Kwon, and Y.-i. Shin, Observation of topologically stable 2D Skyrmions in an antiferromag- netic spinor Bose-Einstein condensate, Phys. Rev. Lett. 108, 035301 (2012)

    work page 2012

  12. [12]

    M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Vortices in a Bose- Einstein condensate, Phys. Rev. Lett.83, 2498 (1999)

    work page 1999

  13. [13]

    Hallacy, N

    L. Hallacy, N. J. Martin, M. J. Mehrabad, D. Hallett, X. Chen, R. Dost, A. Foster, L. Brunswick, A. Fenzl, E. Clarke, et al., Nonlinear quantum optics at a topological interface enabled by defect engineering, npj Nanophotonics2, 9 (2025)

    work page 2025

  14. [14]

    Kawaguchi and M

    Y. Kawaguchi and M. Ueda, Spinor Bose–Einstein condensates, Phys. Rep.520, 253 (2012)

    work page 2012

  15. [15]

    A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D. Lett, Spinor dynamics in an antiferromagnetic spin-1 condensate, Phys. Rev. Lett.99, 070403 (2007)

    work page 2007

  16. [16]

    Bighin, P

    G. Bighin, P. A. Murthy, N. Defenu, and T. Enss, Resonantly enhanced superconductivity mediated by spinor condensates, Phys. Rev. Res.7, L022070 (2025)

    work page 2025

  17. [17]

    J. M. McGuirk, Spin transport in coupled spinor Bose gases, Phys. Rev. A82, 011612 (2010)

    work page 2010

  18. [18]

    X. Cao, C. Jia, H. Lyu, Y. Hu, and Z. Liang, Transport of vector solitons in spin-dependent nonlinear Thouless pumps, Phys. Rev. A111, 023329 (2025)

    work page 2025

  19. [19]

    Watabe, Y

    S. Watabe, Y. Kato, and Y. Ohashi, Excitation transport through a domain wall in a Bose-Einstein condensate, Phys. Rev. A86, 023622 (2012)

    work page 2012

  20. [20]

    Jim´ enez-Garc´ ıa, A

    K. Jim´ enez-Garc´ ıa, A. Invernizzi, B. Evrard, C. Frapolli, J. Dalibard, and F. Gerbier, Spontaneous formation and relaxation of spin domains in antiferromagnetic spin-1 condensates, Nat. Commun.10, 1422 (2019)

    work page 2019

  21. [21]

    Kudo and Y

    K. Kudo and Y. Kawaguchi, Magnetic domain growth in a ferromagnetic Bose-Einstein condensate: Effects of current, Phys. Rev. A88, 013630 (2013). 10

    work page 2013

  22. [22]

    R. Dum, J. I. Cirac, M. Lewenstein, and P. Zoller, Creation of dark solitons and vortices in Bose-Einstein condensates, Phys. Rev. Lett.80, 2972 (1998)

    work page 1998

  23. [23]

    V. R. Kumar, R. Radha, and M. Wadati, Collision of bright vector solitons in two-component Bose–Einstein condensates, Phys. Lett. A374, 3685 (2010)

    work page 2010

  24. [24]

    C. Qu, L. P. Pitaevskii, and S. Stringari, Magnetic solitons in a binary Bose-Einstein condensate, Phys. Rev. Lett.116, 160402 (2016)

    work page 2016

  25. [25]

    A. L. Fetter, Rotating trapped Bose-Einstein conden- sates, Rev. Mod. Phys.81, 647 (2009)

    work page 2009

  26. [26]

    J. Jin, S. Zhang, and W. Han, Spin domain wall in rotating two-component Bose–Einstein condensates, J. Phys. B: At. Mol. Opt. Phys.44, 165302 (2011)

    work page 2011

  27. [27]

    Coen and M

    S. Coen and M. Haelterman, Domain wall solitons in binary mixtures of Bose-Einstein condensates, Phys. Rev. Lett.87, 140401 (2001)

    work page 2001

  28. [28]

    K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett.84, 806 (2000)

    work page 2000

  29. [29]

    L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, Spontaneous symmetry break- ing in a quenched ferromagnetic spinor Bose–Einstein condensate, Nature443, 312 (2006)

    work page 2006

  30. [30]

    Saito, Y

    H. Saito, Y. Kawaguchi, and M. Ueda, Topological defect formation in a quenched ferromagnetic Bose-Einstein condensate, Phys. Rev. A75, 013621 (2007)

    work page 2007

  31. [31]

    Atala, M

    M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Observation of chiral currents with ultracold atoms in bosonic ladders, Nat. Phys.10, 588 (2014)

    work page 2014

  32. [32]

    R. G. Scott, A. M. Martin, T. M. Fromhold, S. Bujkiewicz, F. W. Sheard, and M. Leadbeater, Creation of solitons and vortices by Bragg reflection of Bose- Einstein condensates in an optical lattice, Phys. Rev. Lett.90, 110404 (2003)

    work page 2003

  33. [33]

    A. P. Malozemoff and J. C. Slonczewski, Magnetic domain walls in bubble materials: advances in materials and device research, Vol. 1 (Academic Press, 2013)

    work page 2013

  34. [34]

    Emori, U

    S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach, Current-driven dynamics of chiral ferromagnetic domain walls, Nat. Mater.12, 611 (2013)

    work page 2013

  35. [35]

    Kuratsuji, Domain wall dynamics in the spinor Bose- Einstein condensates, Europhys

    H. Kuratsuji, Domain wall dynamics in the spinor Bose- Einstein condensates, Europhys. Lett.133, 47002 (2021)

    work page 2021

  36. [36]

    Li, Q.-Y

    Z.-D. Li, Q.-Y. Li, P.-B. He, J.-Q. Liang, W. M. Liu, and G. Fu, Domain-wall solutions of spinor Bose-Einstein condensates in an optical lattice, Phys. Rev. A81, 015602 (2010)

    work page 2010

  37. [37]

    D. T. Son and M. A. Stephanov, Domain walls of relative phase in two-component Bose-Einstein condensates, Phys. Rev. A65, 063621 (2002)

    work page 2002

  38. [38]

    Zhang, D

    W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman, and L. You, Dynamical instability and domain formation in a spin-1 Bose-Einstein condensate, Phys. Rev. Lett. 95, 180403 (2005)

    work page 2005

  39. [39]

    Y. X. Chen, Vector peregrine composites on the periodic background in spin-orbit coupled spin-1 Bose-Einstein condensates, Chaos, Solitons Fractals169, 113251 (2023)

    work page 2023

  40. [40]

    Kittel and P

    C. Kittel and P. McEuen, Introduction to solid state physics (John Wiley & Sons, 2018)

    work page 2018

  41. [41]

    Zhu, S.-Y

    H. Zhu, S.-Y. L¨ u, X.-Y. Shi, and W.-M. Liu, Magnetic domain walls in ferromagnetic Spin-2 Bose–Einstein condensates, Ann. Phys.170241(2025)

    work page 2025

  42. [42]

    Yu and P

    X. Yu and P. B. Blakie, Dark-soliton-like magnetic domain walls in a two-dimensional ferromagnetic super- fluid, Phys. Rev. Res.3, 023043 (2021)

    work page 2021

  43. [43]

    H. M. Hurst and I. B. Spielman, Measurement-induced dynamics and stabilization of spinor-condensate domain walls, Phys. Rev. A99, 053612 (2019)

    work page 2019

  44. [44]

    Ohashi and Y

    T. Ohashi and Y. Kawaguchi, Perfect transmission and perfect reflection of Bogoliubov quasiparticles in a dynamically unstable Bose–Einstein condensate, J. Phys. Soc. Jpn.90, 034501 (2021)

    work page 2021

  45. [45]

    M. O. D. Alotaibi and L. D. Carr, Scattering of a dark– bright soliton by an impurity, J. Phys. B: At. Mol. Opt. Phys.52, 165301 (2019)

    work page 2019

  46. [46]

    Gaul and C

    C. Gaul and C. A. M¨ uller, Anisotropic scattering of Bogoliubov excitations, Europhys. Lett.83, 10006 (2008)

    work page 2008

  47. [47]

    Dalfovo, S

    F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys.71, 463 (1999)

    work page 1999

  48. [48]

    A. J. Leggett, Bose-Einstein condensation in the alkali gases: some fundamental concepts, Rev. Mod. Phys.73, 307 (2001)

    work page 2001

  49. [49]

    Dalibard, F

    J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. ¨Ohberg, Colloquium: Artificial gauge potentials for neutral atoms, Rev. Mod. Phys.83, 1523 (2011)

    work page 2011

  50. [50]

    A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Fano resonances in nanoscale structures, Rev. Mod. Phys.82, 2257 (2010)

    work page 2010

  51. [51]

    R. A. Vicencio, J. Brand, and S. Flach, Fano blockade by a Bose-Einstein condensate in an optical lattice, Phys. Rev. Lett.98, 184102 (2007)

    work page 2007

  52. [52]

    K. A. Yasir and W.-M. Liu, Controlled electromag- netically induced transparency and Fano resonances in hybrid BEC-optomechanics, Sci. Rep.6, 22651 (2016)

    work page 2016

  53. [53]

    Zheng and M

    J.-H. Zheng and M. A. Cazalilla, Nontrivial interplay of strong disorder and interactions in quantum spin-Hall insulators doped with dilute magnetic impurities, Phys. Rev. B97, 235402 (2018)

    work page 2018

  54. [54]

    Zheng, A

    J.-H. Zheng, A. Brataas, M. Kl¨ aui, and A. Qaiumzadeh, Theory of domain-wall magnetoresistance in metallic antiferromagnets, Phys. Rev. B102, 184413 (2020)

    work page 2020