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Cited by 83Topological Materials and PhenomenaQuantum Mechanics and Non-Hermitian PhysicsNonlinear Photonic Systems

Universal Sign Control of Coupling in Tight-Binding Lattices

Open access
Keil, Robert, Poli, Charles, Heinrich, Matthias, Arkinstall, Jake, Weihs, Gregor, Schomerus, Henning, Szameit, Alexander·2016·Physical Review Letters
DOI 10.1103/physrevlett.116.213901
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Abstract

We present a method of locally inverting the sign of the coupling term in tight-binding systems, by means of inserting a judiciously designed ancillary site and eigenmode matching of the resulting vertex triplet. Our technique can be universally applied to all lattice configurations, as long as the individual sites can be detuned. We experimentally verify this method in laser-written photonic lattices and confirm both the magnitude and the sign of the coupling by interferometric measurements. Based on these findings, we demonstrate how such universal sign-flipped coupling links can be embedded into extended lattice structures to impose a Z_{2}-gauge transformation. This opens a new avenue for investigations on topological effects arising from magnetic fields with aperiodic flux patterns or in disordered systems.

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Written from the paper’s full text.

TL;DR

The paper proposes and experimentally demonstrates a universal method to locally invert the sign of coupling in tight-binding lattices by inserting a detuned ancillary site and matching eigenmodes. Using laser-written photonic waveguide lattices, it verifies negative coupling interferometrically and embeds sign-flipped links to realize Z₂-gauge transformations and Aharonov-Bohm interference.

Key findings

  • A three-site link with detunings δ and Δ can approximate a negative coupler: setting δ = κ and Δ = κ₁²/κ − κ matches the antisymmetric mode exactly and the symmetric mode with fidelity F that grows with κ₁/κ.
  • In laser-written photonic lattices, a regular coupler had κ = 0.28 cm⁻¹; a three-site link with κ₁ ≈ 7.7κ and Δ_exp ≈ 66κ (optimal Δ_opt ≈ 58κ) yielded F_exp ≈ 0.987 and an effective coupling rate (β₊,exp − β₋,exp)/2 ≈ −0.88κ.
  • Interferometric verification: a negative-coupler section followed by a positive coupler produced revival in the input channel, with 98.0% output power in channel 2 experimentally and 99.92% in simulation; the estimated phase error was about π/70, allowing 70 π-flux cells before a gauge error.
  • In a hexagonal Aharonov-Bohm interferometer, an all-positive hexagon showed constructive interference, while replacing one link with a negative coupler caused destructive interference: 0.4% of total power at the destructive site in experiment and 0.05% in simulation, with κ = 0.27 cm⁻¹, γ = 0.1 μm⁻¹, κ₁ ≈ 4.1κ, and Δ ≈ 15κ.

Methods

  • Theoretical tight-binding Hamiltonian with magnetic phases; the sign-flip method inserts one ancillary site on the targeted link and matches the resulting three-site eigenmodes to those of an ideal negative coupler via δ = κ and Δ = κ₁²/κ − κ.
  • Experiments used laser-inscribed waveguides in fused silica: 800 nm mode-locked laser (τ = 160 fs, 100 kHz, P = 23 mW), NA 0.35, 200 μm depth, v₀ = 110 mm/min, L = 4.9 cm; 633 nm light was butt-coupled and propagation monitored by fluorescence. The hexagonal chip used L = 4.7 cm, τ = 180 fs, P = 25 mW, v₀ = 220 mm/min, v_δ = 206 mm/min, and v_Δ = 96 mm/min.
  • Measured intensity evolution was compared to numerical tight-binding solutions; coupling parameters were fitted, and non-adjacent couplings were modeled as κ(d) ∝ exp(−γd).

Limitations

  • Stated: matching of the symmetric mode is imperfect because part of it resides on the central site; fidelity tends to unity only as κ₁/κ → ∞, and the required detuning Δ grows quadratically with κ₁, making single-mode guiding increasingly challenging.
  • Stated: next-nearest-neighbour coupling across the three-site link is neglected, though the observed dynamics support this approximation; experimental deviations are attributed to fabrication inaccuracies breaking the symmetry of the structure.
  • Evident from the text: the experimental demonstration is limited to specific photonic waveguide couplers and hexagonal lattices, and sign verification relies on interferometric revival rather than direct intensity evolution alone.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevlett.116.213901

Cites

43 of 43 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 16 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 1

    The method was experimentally verified in laser-written photonic lattices.

    “We experimentally verify this method in laser-written photonic lattices”
    Quote in the paperSubstantial claim
  • Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 4

    In the hexagon with a negative coupling link, virtually no light (0.4% of the total power in the experiment and 0.05% in the simulation) is detected at the site where the destructive interference takes place.

    “A quite different scenario arises in the hexagon with a negative coupling link, as shown in Fig. 4 (b) : Here, virtually no light (0 . 4% of the total power in the experiment and 0 . 05% in the simulation) is detected at the site where the destructive interference takes place.”
    Quote in the paperSubstantial claim
  • Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 2

    The measured values for κ1 and ∆exp translate to a matching fidelity for the symmetric eigenmode of Fexp ≈ 0.987.

    “The measured values for κ 1 and ∆ exp translate to a matching fidelity for the symmetric eigenmode of F exp ≈ 0 . 987.”
    Quote in the paperSubstantial claim
  • Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 2

    The first structure has a rather weak detuning, such that κ1 can be unambiguously identified from the oscillation period as κ1 ≈ 7.7κ.

    “The first one has a rather weak detuning, such that κ 1 can be unambiguously identified from the oscillation period as κ 1 ≈ 7 . 7 κ .”
    Quote in the paperNot a substantial claim
  • Findingreasoning
    § Universal sign-control of coupling in tight-binding lattices · p. 2

    The antisymmetric eigenmode v- = 1/√2(1,-1,0)^T always exists with eigenvalue β- = δ, regardless of other parameters.

    “One finds that the antisymmetric eigenmode v -= 1 / √ 2(1 , -1 , 0) ᵀ always exists with an eigenvalue of β -= δ , regardless of the other parameters.”
    Quote in the paperSubstantial claim
  • May combine two claims. 18
  • Classified as probably not a substantial claim. 3

  • May combine two claims.

    Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 1

    Both the magnitude and the sign of the coupling were confirmed by interferometric measurements.

    “confirm both the magnitude and the sign of the coupling by interferometric measurements.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Universal sign-control of coupling in tight-binding lattices · p. 3

    For the parameters of Fig. 3 one obtains a phase error of about π/70, such that 70 π-flux cells could be traversed before a gauge error would occur.

    “For the parameters of Fig. 3 one obtains a phase error of about π/ 70, such that 70 π -flux cells could be traversed before a gauge error (erasing one flux cell from the system) would occur.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 4

    Attempting to reproduce the observed patterns with halved ∆ does not yield any reasonable agreement, which suggests that next-nearest-neighbour coupling is much weaker than κ in our system.

    “Attempting to reproduce the observed patterns with halved ∆ does, on the other hand, not yield any reasonable agreement, which suggests that next-nearest-neighbour coupling is much weaker than κ in our system.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 2

    Its detuning is actually slightly too large, best fitting to ∆exp ≈ 66κ, while the optimal value according to Eq. (3) would be ∆opt ≈ 58κ.

    “A close inspection reveals that its detuning is actually slightly too large, best fitting to ∆ exp ≈ 66 κ , while the optimal value according to Eq. (3) would be ∆ opt ≈ 58 κ .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § Universal sign-control of coupling in tight-binding lattices · p. 2

    In this regime, the third eigenmode v3 resides mostly on the defect node, is strongly detuned and, therefore, barely interferes with the other eigenmodes.

    “In this regime, the third eigenmode v 3 (shown in magenta in Fig. 1 (e) ) resides mostly on the defect node, is strongly detuned and, therefore, barely interferes with the other eigenmodes.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 2

    For growing ∆ the oscillation between the outer sites is slowed down, whereas the oscillation on the inner site is accelerated, but reduced in amplitude.

    “For growing ∆ the oscillation between the outer sites is slowed down, whereas the oscillation on the inner site is accelerated, but reduced in amplitude (see second group of images in (c) ).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 3

    The fidelity of this revival process is quantified by the fraction of power in channel 2 at the output of the device and is 98.0% in the experiment and 99.92% in the best-fitting numerical simulation.

    “The fidelity of this revival process is quantified by the fraction of power in channel 2 at the output of the device and is 98 . 0% in the experiment and 99 . 92% in the best-fitting numerical simulation (bottom row).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Universal sign-control of coupling in tight-binding lattices · p. 2

    This deviation from the optimal design parameter does not compromise the functionality as a negative coupler, as only the lower eigenvalue is shifted to β+,exp ≈ -0.77κ, whereas the other eigenvalue β-,exp = κ as well as the symmetries of the eigenmodes remain.

    “However, this deviation from the optimal design parameter does not compromise the functionality as a negative coupler, as only the lower eigenvalue is shifted to β + , exp ≈ -0 . 77 κ , whereas the other eigenvalue β -, exp = κ as well as the symmetries of the eigenmodes remain.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Universal sign-control of coupling in tight-binding lattices · p. 2

    To acquire a symmetric eigenmode v+ matching its counterpart for the negative coupler u-κ+, its components must satisfy v+,1 = v+,2 and its eigenvalue β+ = -κ.

    “To acquire a symmetric eigenmode v + matching its counterpart for the negative coupler u -κ + , its components must satisfy v + , 1 = v + , 2 and its eigenvalue β + = -κ .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § Universal sign-control of coupling in tight-binding lattices · p. 1

    Control over the sign can be acquired by dynamical modulation of the lattice, Raman-assisted tunneling or embedding defects in infinite arrays.

    “Control over this sign can be acquired by dynamical modulation of the lattice [16-21], Raman-assisted tunneling [22, 23] or embedding defects in infinite arrays [24].”
    Quote in the paperSubstantial claim0 of 9 sources checked
    • cites [16] D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34 , 3625 (1986). source

      Not checked in the free run.

    • cites [17] S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96 , 243901 (2006). source

      Not checked in the free run.

    • cites [18] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. 99 , 220403 (2007). source

      Not checked in the free run.

    • cites [19] P. Zhang, N. K. Efremidis, A. Miller, Y. Hu, and Z. Chen, Opt. Lett. 35 , 3252 (2010). source

      Not checked in the free run.

    • cites [20] J. Struck, C. ¨ Olschl¨ ager, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein, P. Windpassinger, and K. Sengstock, Science 333 , 996 (2011). source

      Not checked in the free run.

    • cites [21] J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. T¨ unnermann, S. Nolte, and A. Szameit, Phys. Rev. Lett. 109 , 023602 (2012). source

      Not checked in the free run.

    • cites [22] D. Jaksch and P. Zoller, New J. Phys. 5 , 56 (2003). source

      Not checked in the free run.

    • cites [23] M. Aidelsburger, M. Atala, S. Nascimb` ene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107 , 255301 (2011). source

      Not checked in the free run.

    • cites [24] J. M. Zeuner, M. C. Rechtsman, R. Keil, F. Dreisow, A. T¨ unnermann, S. Nolte, and A. Szameit, Opt. Lett. 37 , 533 (2012). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given
    § Universal sign-control of coupling in tight-binding lattices · p. 2

    Facilitating the exploration of all possible Z2-gauge fields requires an independent control over the signs of all hoppings in the lattice.

    “Facilitating the exploration of all possible Z 2 -gauge fields requires an independent control over the signs of all hoppings in the lattice.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Study design and conductown data
    § Universal sign-control of coupling in tight-binding lattices · p. 4

    Next-nearest-neighbour coupling across the three-site link is neglected, and this neglect is valid as evidenced by the observed behaviour.

    “We have so far neglected next-nearest-neighbour coupling across the three-site link. This is valid, as evidenced by the observed behaviour:”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning
    § Universal sign-control of coupling in tight-binding lattices · p. 1

    The embedding-defects method is only effective for one particular configuration, whereas dynamical modulation and Raman-assisted tunneling affect the entire lattice in a periodic manner and require dynamic control or additional pump lasers.

    “The latter method is only effective for one particular configuration, whereas the former two affect the entire lattice in a periodic manner and require dynamic control or additional pump lasers, respectively.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § Universal sign-control of coupling in tight-binding lattices · p. 4

    The optimal ∆ would be approximately half the value required by Eq. (3).

    “Hence, the optimal ∆ would be approximately half the value required by Eq. (3).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Universal sign-control of coupling in tight-binding lattices · p. 4

    This observation is fully consistent with another recent experiment, where a suppression of next-nearest-neighbour coupling in a three-site link has been measured directly.

    “This observation is fully consistent with another recent experiment, where a suppression of next-nearest-neighbour coupling in a threesite link has been measured directly [32].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [32] R. Keil, B. Pressl, R. Heilmann, M. Gr¨ afe, G. Weihs, and A. Szameit, Appl. Phys. Lett. 107 , 241104 (2015). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationcitation
    § Universal sign-control of coupling in tight-binding lattices · p. 4

    The latter structure imposes a Z2-gauge transformation corresponding to one half of a magnetic flux quantum passing through the plaquette and constitutes a static photonic Aharanov-Bohm interferometer.

    “The latter structure imposes a Z 2 -gauge transformation corresponding to one half of a magnetic flux quantum passing through the plaquette [2] and constitutes a static photonic Aharanov-Bohm interferometer.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [2] D. R. Hofstadter, Phys. Rev. B 14 , 2239 (1976). http://dx.doi.org/10.1103/PhysRevB.14.2239 source

      Not checked in the free run.

  • May combine two claims.

    Interpretationnone given
    § Universal sign-control of coupling in tight-binding lattices · p. 4

    The method of eigenmode matching does not require any assumptions on the surrounding and the outer sites remain in their positions.

    “Two key advantages of the method of eigenmode matching are that it does not require any assumptions on the surrounding and that the outer sites remain in their positions.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § Universal sign-control of coupling in tight-binding lattices · p. 1

    The concept is based solely on static adjustments and can be universally applied to any tight-binding system, given that additional sites can be inserted and the on-site potential can be controlled.

    “The concept is based solely on static adjustments and can be universally applied to any tight-binding system, given that additional sites can be inserted and the on-site potential can be controlled.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation
    § Universal sign-control of coupling in tight-binding lattices · p. 3

    Consequently, the light evolution in the second section will exactly reverse the one in the first and a full revival must occur in the initially excited channel.

    “Consequently, the light evolution in the second section will exactly reverse the one in the first and a full revival must occur in the initially excited channel [31].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [31] S. Longhi, Opt. Lett. 33 , 473 (2008). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § Universal sign-control of coupling in tight-binding lattices · p. 4

    If one of the links is replaced by an effective negative coupler, the phase shift across that link will be -π/2, leading to a phase difference of ∆Φ = π between the two paths and causing destructive interference.

    “If, however, one of the links is replaced by an effective negative coupler (Fig. 4 (b) ), the phase shift across that link will be -π/ 2, leading to a phase difference of ∆Φ = π between the two paths and causing destructive interference.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § Universal sign-control of coupling in tight-binding lattices · p. 3

    If the couplings in both sections had the same sign, the light would continue tunneling from site 2 to site 1 and no revival would occur at z = L.

    “Note that if the couplings in both sections had the same sign, the light would continue tunneling from site 2 to site 1 and no revival would occur at z = L .”
    Quote in the paperSubstantial claim
21 of 21

  • The paper relies on this without support.

    A direct implementation of large magnetic fluxes in solids exceeds current experimental capabilities by orders of magnitude.

    “However, a direct implementation of such large magnetic fluxes in solids exceeds current experimental capabilities by orders of magnitude.”
1 of 1

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Due to recent progress in implementing waveguide elements that invert the sign of hybridization, the complexity of this model could recently be captured by a photonic cavity lattice structure.

    “Due to recent progress in implementing waveguide elements that invert the sign of hybridization 21 , the complexity of this model could recently be captured by a photonic cavity lattice structure 22 .”

    From Topolectrical-circuit realization of topological corner modes (2018) · cites this paper as [21]

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Related

Authors
Keil, R., Poli, C., Heinrich, M., Arkinstall, J., Weihs, G., Schomerus, H., & Szameit, A.
Affiliations
  • Universität Innsbruck (AT)
  • Lancaster University (GB)
  • Friedrich Schiller University Jena (DE)
ORCID
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
2016-05-27
Volume
116
Issue
21
Pages
213901-213901
ISSN
  • 0031-9007 (ISSN-L)
  • 1079-7114
DOI
10.1103/physrevlett.116.213901
OpenAlex
W2187390651
Cited by
83
References
43
Citations by year
2026: 3 · 2025: 9 · 2024: 5 · 2023: 9 · 2022: 8 · 2021: 7 · 2020: 12 · 2019: 10 · 2018: 9 · 2017: 6 · 2016: 5
FWCI
4.70
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Lancaster EPrints (Lancaster University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • European Research Council (257531)
  • Canadian Institute for Advanced Research
  • Bundesministerium für Bildung und Forschung (03Z1HN31)
  • Deutsche Forschungsgemeinschaft (SZ276/7-1)
  • Engineering and Physical Sciences Research Council (EP/J019585/1, EP/L01548X/1)
  • Austrian Science Fund (M 1849)
  • German National Academy of Sciences (LPDR 2014-03)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Nonlinear Photonic Systems
Keywords
tight-binding lattice, topological effects, disordered systems, interferometric measurement

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