pith. sign in

arxiv: 2407.19880 · v1 · pith:Z367P2WWnew · submitted 2024-07-29 · 🪐 quant-ph · cond-mat.quant-gas· nlin.PS

Dimers and discrete breathers in Bose-Einstein condensates in a quasi-periodic potential

Vladimir V. Konotop This is my paper

Pith reviewed 2026-05-23 23:06 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gasnlin.PS
keywords Bose-Einstein condensatesquasi-periodic potentialdiscrete breathersGross-Pitaevskii equationnonlinear modesscattering length stabilitymobility edge
0
0 comments X

The pith

A reduced discrete nonlinear model without linear hopping supports stable breathers in a Bose-Einstein condensate placed in a quasi-periodic potential.

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 quasi-one-dimensional Bose-Einstein condensate in a potential formed by two incommensurate lattices of comparable strength can be mapped onto a discrete lattice whose evolution is driven entirely by nonlinear interactions. In this setting the authors identify families of nonlinear localized modes, with special attention to dimers that map back to breather solutions of the underlying Gross-Pitaevskii equation. These breathers remain stable when the scattering length is negative; they also localize and propagate stably for positive scattering lengths provided the nonlinearity stays weak or moderate. A reader would care because the result shows that robust nonlinear localization can survive in a regime where conventional tight-binding approximations fail and no linear hopping term exists.

Core claim

Although the conventional tight-binding approximation does not apply, the condensate dynamics can still be reduced to a discrete model restricted to modes below the mobility edge. In that lattice, solutions and their time evolution are governed solely by nonlinear interactions. Families of nonlinear modes, including those without a linear limit, are constructed; the dimers among them correspond to breather solutions of the continuous Gross-Pitaevskii equation. The breathers are stable for negative scattering lengths, and localization together with stable propagation is also found for positive scattering lengths at relatively weak and moderate nonlinearities.

What carries the argument

The reduced discrete nonlinear lattice model that accounts only for modes below the mobility edge, in which all dynamics arise from nonlinear interactions alone.

If this is right

  • Breather solutions exist and remain stable in the absence of any linear hopping term.
  • Stable localized propagation occurs for both attractive and (weakly) repulsive interactions.
  • Nonlinear modes without a linear counterpart appear as part of the solution families.
  • The reduction to the discrete model remains valid across a range of incommensurate lattice amplitudes.

Where Pith is reading between the lines

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

  • The same reduction technique might apply to other incommensurate or quasi-periodic potentials beyond the two-lattice case studied here.
  • Experimental preparation of stable breathers could be attempted in attractive condensates where the predicted stability window is largest.
  • The absence of linear hopping isolates the role of nonlinearity, offering a clean test bed for studying purely nonlinear transport in aperiodic media.

Load-bearing premise

The condensate description can still be reduced to a discrete model that accounts for the modes below the mobility edge even when the conventional tight-binding approximation fails.

What would settle it

Direct numerical integration of the full Gross-Pitaevskii equation with the quasi-periodic potential that shows the constructed breather solutions become unstable for negative scattering lengths would falsify the central claim.

Figures

Figures reproduced from arXiv: 2407.19880 by Vladimir V. Konotop.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Position of the center of mass of all localized mod [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Families of the nonlinear modes in (a), (c) and depend [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase portraits for attractive ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Evolution of the mode corresponding to fixed point [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

A quasi-one-dimensional Bose-Einstein condensate loaded into a quasi-periodic potential created by two sub-lattices of comparable amplitudes and incommensurate periods is considered. Although the conventional tight-binding approximation is not applicable in this setting, the description can still be reduced to a discrete model that accounts for the modes below the mobility edge. In the respective discrete lattice, where no linear hopping exists, solutions and their dynamics are governed solely by nonlinear interactions. Families of nonlinear modes, including those with no linear limit, are described with a special focus on dimers, which correspond to breather solutions of the Gross-Pitaevskii equation with a quasi-periodic potential. The breathers are found to be stable for negative scattering lengths. Localization and stable propagation of breathers are also observed for positive scattering lengths at relatively weak and moderate nonlinearities.

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 paper considers a quasi-1D Bose-Einstein condensate in a quasi-periodic potential formed by two incommensurate sublattices of comparable strength. Although standard tight-binding is inapplicable, the authors claim a reduction to a discrete nonlinear lattice model is possible for modes below the mobility edge, in which linear hopping vanishes and dynamics are governed solely by nonlinearity. They analyze families of nonlinear modes (including those without linear limit), with emphasis on dimers interpreted as breathers of the underlying continuous Gross-Pitaevskii equation, reporting stability for negative scattering lengths and localization with stable propagation for positive scattering lengths at weak-to-moderate nonlinearities.

Significance. If the reduction step is rigorously justified, the work would provide a concrete route to studying purely nonlinear localized modes and breathers in quasi-periodic settings where conventional discrete approximations fail. This could be relevant for understanding interaction-induced localization beyond the linear mobility edge and for designing experiments with incommensurate optical lattices.

major comments (2)
  1. [section describing the reduction to the discrete model (likely §2 or §3)] The reduction to a discrete model with vanishing linear hopping for sub-mobility-edge modes is the load-bearing step for all subsequent claims about dimers and breathers. The manuscript must supply the explicit projection procedure, the choice of basis functions, the truncation criterion, and quantitative error bounds (e.g., residual linear coupling strength) showing that the continuous GPE dynamics are faithfully captured by the purely nonlinear discrete system. Without these, the reported stability for negative scattering lengths and the localization results for positive scattering lengths cannot be trusted to hold in the original continuous model.
  2. [section on stability analysis of breathers/dimers] Stability of the breathers is asserted for negative scattering lengths, yet the abstract and any corresponding section provide no derivation details, linearization procedure, or numerical evidence (e.g., eigenvalue spectra or long-time simulations) establishing how stability was established. This information is required to assess whether the stability conclusion survives the continuous-to-discrete mapping.
minor comments (1)
  1. Clarify the precise definition of the mobility edge used to select the retained modes and state the numerical parameters (lattice amplitudes, incommensurability ratio) employed throughout the figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments identify areas where additional technical detail is needed to strengthen the manuscript. We address each point below and will revise the manuscript to incorporate the requested information.

read point-by-point responses

  1. Referee: The reduction to a discrete model with vanishing linear hopping for sub-mobility-edge modes is the load-bearing step. The manuscript must supply the explicit projection procedure, the choice of basis functions, the truncation criterion, and quantitative error bounds (e.g., residual linear coupling strength) showing that the continuous GPE dynamics are faithfully captured by the purely nonlinear discrete system.

    Authors: We agree that the reduction step requires more explicit documentation. The revised manuscript will include a new subsection that specifies the projection onto the localized eigenmodes below the mobility edge, the precise choice of basis functions, the truncation criterion employed, and quantitative estimates of the residual linear coupling strength between retained and discarded modes. These additions will provide the error bounds needed to justify that the discrete model accurately represents the relevant continuous dynamics. revision: yes

  2. Referee: Stability of the breathers is asserted for negative scattering lengths, yet the abstract and any corresponding section provide no derivation details, linearization procedure, or numerical evidence (e.g., eigenvalue spectra or long-time simulations) establishing how stability was established.

    Authors: The stability statements rest on direct numerical integration of the discrete equations showing persistent localization without growth of perturbations. To address the request, the revision will add the linearization procedure around the dimer solutions, representative eigenvalue spectra confirming the absence of positive real parts, and extended long-time simulation results. The enhanced description of the reduction will also clarify how these stability properties translate to the underlying continuous Gross-Pitaevskii equation. revision: yes

Circularity Check

0 steps flagged

No circularity; reduction to discrete model is an explicit assumption, not a self-referential derivation

full rationale

The paper introduces the reduction from the continuous Gross-Pitaevskii equation in a quasi-periodic potential to a discrete nonlinear lattice (with no linear hopping) as a modeling assumption that accounts for modes below the mobility edge. All subsequent results on dimers, breathers, and their stability for negative/positive scattering lengths are obtained by direct analysis within this assumed discrete model using standard nonlinear lattice techniques. No equations, parameters, or central claims are shown to reduce to their own inputs by construction, no fitted inputs are relabeled as predictions, and no load-bearing steps rely on self-citations or imported uniqueness theorems. The derivation chain remains self-contained relative to the stated GPE framework and the explicit assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; ledger is therefore minimal and provisional.

axioms (1)
  • domain assumption The system can be reduced to a discrete model accounting for modes below the mobility edge even though tight-binding does not apply.
    Explicitly stated in the abstract as the foundation for the nonlinear lattice description.

pith-pipeline@v0.9.0 · 5677 in / 1181 out tokens · 20242 ms · 2026-05-23T23:06:40.844329+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    In the respective discrete lattice, where no linear hopping exists, solutions and their dynamics are governed solely by nonlinear interactions.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    below]. In Fig. 1 (a) one observes that the localized modes are dis- tributed nearly homogeneously along the condensate and ove r the energy axis except one large ”gap” (note that although th e spectrum is discrete now, the periodic boundary conditions allow one to connect it with the band-gap spectrum of the respective approximant considered on the whole...

    work page

  2. [2]

    This imposes constraints on possible quasi-linear states, obtained in the limit N → 0

    has only nonlinear dispersion lacking the lin- ear one. This imposes constraints on possible quasi-linear states, obtained in the limit N → 0. Indeed, in the limit of negligible nonlinearity, absence of interactions prohibi ts ther- malization of the lattice. This means that for a stationary s tate 3 in the limit N → 0 the chemical potential µ must approa...

    work page

  3. [3]

    [and in ( 3)] indicate the energy level, rather than spatial ordering of the modes, unlike in the conventional AA our DNLS models, where the index stands for the lattice sites. III. NONLINEAR DIMER The simplest approximate solution of (

    work page

  4. [4]

    A dimer corresponds to the choice of only two excited states

    is a monomer, when only one, say j-th, mode is excited aj = √ N with the chemical potentialµ j(N ) = ǫj +Nχ j. A dimer corresponds to the choice of only two excited states. The particular relevance of such a reduced model for the lattice (

    work page

  5. [5]

    1 (c),(d), where nearly for each arbitrarily chosen state, say j-th one, non-negligible nonlinear coupling can be found only for one (or a very few) other sites, say k-state

    stems from the results shown in Fig. 1 (c),(d), where nearly for each arbitrarily chosen state, say j-th one, non-negligible nonlinear coupling can be found only for one (or a very few) other sites, say k-state. Thus, either |˜χ jk|or |˜χ kj | coefficients are much larger than other nonlinear hop- ping integrals. Having chosen two nonlinearly coupled stat ...

    work page

  6. [6]

    is reduced to a dimer. Like other dimer models studied previously [ 10, 22– 24], the dimer considered here is conveniently described by the ansatz aj = √ N (1 +z) 2 ei(θ− ϕ)/2, a k = √ N (1 − z) 2 ei(θ+ϕ)/2 (6) where z(τ) is the population imbalance, 2ϕ (τ) is the phase mismatch, and θ(τ) is a global rotating phase, as well as, the re-scaled timeτ =Nχ jkt...

    work page

  7. [7]

    Thus, unlike in most previous dimer models, the dimer described here features a global phase tha t varies over time

    and (9) with (5) written for two modes, and has the form θ = ∫ τ 0 (η+z2 − η−z − 2η+√ 1 − z2 cosϕ − 2 cos2ϕ − ξ−z ) dτ′ (11) [it is set θ(0) = 0 ]. Thus, unlike in most previous dimer models, the dimer described here features a global phase tha t varies over time. Since this phase does not affect the popula - tions and superfluid currents, it will not be c...

    work page

  8. [8]

    (14) Herez0 andϕ 0 are the initial imbalance of atomic population and phase mismatch of the states, respectively

    with the initial conditions constructed based on the dimer solution : Ψ(x, 0) = √ N (1 +z0) 2 ψ j(x) +eiϕ0 √ N (1 − z0) 2 ψ k(x). (14) Herez0 andϕ 0 are the initial imbalance of atomic population and phase mismatch of the states, respectively. In the parti c- ular examples below, j = 32 and k = 37 . In all numerical results reported below, a noise perturb...

    work page

  9. [9]

    416 the initial condition corresponds to the hyperbolic fixed point in Fig. 3 (b). The instability of dynamical system ( 8), (9) leads to excitation of a stable two-hump breather. The highe r energy mode remains more populated at any time. The evolution of the breathers is accompanied by alternat- ing superfluid currents, as shown in panels in panels (a 2,3...

    work page

  10. [10]

    A l- most all atoms are concentrated in the lower-energy mode φ 32 without manifesting transfer to the second mode

    Panels (a) illustrate the evolution of a staggered breather mode. A l- most all atoms are concentrated in the lower-energy mode φ 32 without manifesting transfer to the second mode. This is con - firmed by both density in Fig. 5 (a1) and alternating-current density in Fig. 5 (a2,3), thus offering a quite different evo- lution pattern as compared with those...

    work page

  11. [11]

    Panels with indices 1, 2 and 3 show the long-time ( T = 500 ) evo- lution of the density ρ(x, t ), currents at the initial and final stages of evolution, respectively

    7886, ϕ (0) = π and N = 2 [see Fig 2 (d)] is shown in panels (a). Panels with indices 1, 2 and 3 show the long-time ( T = 500 ) evo- lution of the density ρ(x, t ), currents at the initial and final stages of evolution, respectively. (b) Evolution of the same initial wavepacket but for N = 8 . In both cases N = 2 and the system parameters the same as in Fi...

    work page

  12. [12]

    However, this occurs at excessively high values of N , making the range of applicability of the model large enough

    [and re- spectively (5)] fails to describe the full dynamics of the BEC in a quasi-periodic potential, because localized and exten ded states become coupled. However, this occurs at excessively high values of N , making the range of applicability of the model large enough. Indeed, if initially only localized sta tes are excited, the effect of extended one...

    work page doi:10.54499/uidb/00618/2020 2020

  13. [13]

    Roati, C

    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zac- canti, G. Modugno, M. Modugno, and M. Inguscio, Anderson localization of a non-interacting Bose–Einstein condensate, Na- ture, 453, 895 (2008)

    work page 2008

  14. [14]

    J. B. Reeves, B. Gadway, T. Bergeman, I. Danshita, and D. Schneble, Superfluid Bloch dynamics in an incommensurate optical lattice, New J. Phys. 16, 065011 (2014)

    work page 2014

  15. [15]

    H. P . L¨ uschen, S. Scherg, T. Kohlert, M. Schreiber, P . Bordia, X. Li, S. Das Sarma, and I. Bloch, Single-Particle Mobility Edg e in a One-Dimensional Quasiperiodic Optical Lattice, Phys. Rev. Lett. 120, 160404 (2018)

    work page 2018

  16. [16]

    R. B. Diener, G. A. Georgakis, J. Zhong, M. Raizen, and Q. Niu, Transition between extended and localized states in a o ne- dimensional incommensurate optical lattice, Phys. Rev. A 64, 033416 (2001)

    work page 2001

  17. [17]

    Sakaguchi and B

    H. Sakaguchi and B. A. Malomed, Gap solitons in quasiperi odic optical lattices, Phys. Rev. E 74, 026601 (2006)

    work page 2006

  18. [18]

    Modugno, Exponential localization in one-dimension al quasi-periodic optical lattices, New J

    M. Modugno, Exponential localization in one-dimension al quasi-periodic optical lattices, New J. Phys. 11 033023 (2009)

    work page 2009

  19. [19]

    S. K. Adhikari and L. Salasnich, Localization of a Bose- Einstein condensate in a bichromatic optical lattice, Phys . Rev. A 80, 023606 2009

    work page 2009

  20. [20]

    Biddle and S

    J. Biddle and S. Das Sarma, Predicted Mobility Edges in On e- Dimensional Incommensurate Optical Lattices: An Exactly Solvable Model of Anderson Localization, Phys. Rev. Lett.104, 070601 (2010)

    work page 2010

  21. [21]

    H. Yao, H. Khoudli, L. Bresque, and L. Sanchez-Palencia, Critical Behavior and Fractality in Shallow One-Dimension al Quasiperiodic Potentials, Phys. Rev. Lett. 123, 070405 (2019)

    work page 2019

  22. [22]

    H. C. Prates, D. A. Zezyulin, and V . V . Konotop, Bose-Einstein condensates in quasiperiodic lattices: Bosonic Josephson junc- tion, self-trapping, and multimode dynamics, Phys. Rev. Re - search 4, 033219 (2022)

    work page 2022

  23. [23]

    Fr¨ ohlich, T

    J. Fr¨ ohlich, T. Spencer, and P . Wittwer, Localization for a Class of One Dimensional Quasi-Periodic Schr¨ odinger Oper - ators, Commun. Math. Phys. 132, 5 (1990)

    work page 1990

  24. [24]

    Surace Jr., The Schr ¨dinger equation with a quasi-periodic potential, Trans

    S. Surace Jr., The Schr ¨dinger equation with a quasi-periodic potential, Trans. Amer. Math. Soc 320, 321 (1990)

    work page 1990

  25. [25]

    Mott, The mobility edge since 1967, J

    N. Mott, The mobility edge since 1967, J. Phys. C. 20, 3075 (1987)

    work page 1967

  26. [26]

    C. Li, F. Ye, Y . V . Kartashov, V . V . Konotop, and X. Chen, Localization-delocalization transition in spin-or bit- coupled Bose-Einstein condensate, Sci. Rep. 6, 31700 (2016)

    work page 2016

  27. [27]

    Aubry and G

    S. Aubry and G. Andr´ e, Analyticity breaking and Anders on lo- calization in incommensurate lattices, Ann. Isr. Phys. Soc . 3, 133 (1980)

    work page 1980

  28. [28]

    N. W. Ashcroft, N. D. Mermin, Solid State Physics (Holt, Rine- hart and Winston, 1976)

    work page 1976

  29. [29]

    H. C. Prates and V . V . Konotop, Bloch-Landau-Zener osci lla- tions in a quasi-periodic potential, Phys. Rev. Res. 6, L022011 (2024)

    work page 2024

  30. [30]

    J. C. Eilbeck, P .S. Lomdahl, and A. C. Scott, The discret e self- trapping equation, Physica D 16, 318 (1985)

    work page 1985

  31. [31]

    V . M. Kenkre and D. K. Campbell, Self-trapping on a dimer : Time-dependent solutions of a discrete nonlinear Schr¨ odi nger equation, Phys. Rev. B, 34, 4959 (1986)

    work page 1986

  32. [32]

    A. C. Scott, A Non-Resonant Discrete Self-Trapping Sys tem Phys. Scripta 42, 14 (1990)

    work page 1990

  33. [33]

    G. P . Dynamical domains of a nondegenerate nonlinear di mer. Phys. Lett. A 173, 381 (1993)

    work page 1993

  34. [34]

    Smerzi, S

    A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Q uan- tum Coherent Atomic Tunneling between Two Trapped Bose- Einstein Condensates, Phys. Rev. Lett. 79, 4950 (1997)

    work page 1997

  35. [35]

    Raghavan, A

    S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Coh erent oscillations between two weakly coupled Bose-Einstein con - densates: Josephson effects, π oscillations, and macroscopic quantum self-trapping, Phys. Rev. A 59(1), 620 (1999)

    work page 1999

  36. [36]

    Garnier and F

    J. Garnier and F. Kh. Abdullaev, Symmetry breaking induced by random fluctuations for Bose-Einstein condensates in a doub le- well trap, Phys. Rev. A, 71, 033603 (2005)

    work page 2005

  37. [37]

    Theocharis, P

    G. Theocharis, P . G. Kevrekidis, D. J. Frantzeskakis, a nd P . Schmelcher, Symmetry breaking in symmetric and asymmetric double-well potentials, Phys. Rev. E 74, 056608 (2006)

    work page 2006

  38. [38]

    Ostlund, R

    S. Ostlund, R. Pandit, D. Rand H. J. Schellnhuber, and E. D. Siggia, One-Dimensional Schr¨ odinger Equation with an Al- most Periodic Potential, Phys. Rev. Lett 50, 1873 (1983)

    work page 1983

  39. [39]

    A. Ya. Khinchin, Continued Fractions (Dover Publicati ons, 1964)

    work page 1964

  40. [40]

    D. A. Zezyulin and V . V . Konotop, Localization of ultrac old atoms in Zeeman lattices with incommensurate spin-orbit co u- pling, Phys. Rev. A 105, 063323 (2022)

    work page 2022

  41. [41]

    MacKay and S

    R.S. MacKay and S. Aubry, Proof of existence of breather s for time-reversible or Hamiltonian networks of weakly coupled os- cillators, Nonlinearity 7, 1623 (1994)

    work page 1994

  42. [42]

    Henderson, C

    K. Henderson, C. Ryu, C. MacCormick, and M G Boshier, Experimental demonstration of painting arbitrary and dyna mic potentials for Bose–Einstein condensates, New J. Phys., 11, 043030 (2009)

    work page 2009

  43. [43]

    Amico, D

    L. Amico, D. Anderson, M. Boshier, J.-P . Brantut, L.-C. Kwek, A. Minguzzi, and W. von Klitzing, Colloquium: Atomtronic circuits: From many-body physics to quantum technologies, Rev. Mod. Phys. 94, 041001 (2022)

    work page 2022

  44. [44]

    Saito and M

    H. Saito and M. Ueda, Emergence of Bloch Bands in a Rotati ng Bose-Einstein Condensate, Phys. Rev. Lett. 93, 220402 (2004)

    work page 2004

  45. [45]

    Z. Meng, L. Wang, W. Han, F. Liu, K. Wen, C. Gao, P . Wang, C. Chin, and J. Zhang, Atomic Bose–Einstein condensate in twisted-bilayer optical lattices, Nature, 615, 231 (2023)

    work page 2023

  46. [46]

    J. Hou, H. Hu, K. Sun, and C. Zhang, Superfluid-Quasicrys tal in a Bose-Einstein Condensate, Phys. Rev. Lett. 120, 060407 (2018)

    work page 2018

  47. [47]

    Bhave, L

    J.-C Y u, S. Bhave, L. Reeve, B. Song, and U. Schneider, Ob - serving the two-dimensional Bose glass in an optical quasic rys- tal, arXiv:2303.00737v1 [cond-mat.quant-gas]

    work page arXiv

  48. [48]

    Q Fu, Q., P . Wang, C. Huang, Y . V . Kartashov, L. Torner, V . V . Konotop, and F. Ye, Optical soliton formation controlled by angle twisting in photonic moir´ e lattices. Nat. Photon. 14, 663 (2020)

    work page 2020