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Cited by 353Topological Materials and PhenomenaQuantum Mechanics and Non-Hermitian PhysicsAdvanced Condensed Matter Physics

The DOI matches its Crossref record; 57 of the 57 references it cites were found in a registry or an index; none of the 2 sources its claims cite is checked for support yet.

Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons

Open access
Cardano, Filippo, D’Errico, Alessio, Dauphin, Alexandre, Maffei, Maria, Piccirillo, Bruno, de Lisio, Corrado, De Filippis, Giulio, Cataudella, Vittorio, Santamato, Enrico, Marrucci, Lorenzo, Lewenstein, Maciej, Massignan, Pietro·2017·Nature Communications
DOI 10.1038/ncomms15516PMC PMC5501976
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Abstract

Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, an observable that rapidly approaches a value proportional to the Zak phase during the free evolution of the system. Then we measure the Zak phase in a photonic quantum walk of twisted photons, by observing the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent timeframe and combine the two windings to characterize the full phase diagram of this Floquet system. Finally, we prove the robustness of the measure by introducing dynamical disorder in the system. This detection method is extremely general and readily applicable to all present one-dimensional platforms simulating static or Floquet chiral systems.

Overview

AI snapshot

Written from the paper’s full text.

TL;DR

The paper introduces the mean chiral displacement as a bulk observable that converges to the Zak phase in one-dimensional chiral systems, and measures it in a photonic quantum walk of twisted photons. It then uses two inequivalent timeframes to extract the complete Floquet topological invariants and tests robustness against chiral-preserving dynamical disorder.

Key findings

  • The mean chiral displacement is independent of the initial polarization and robust against disorder, and in the long-time limit becomes proportional to the Zak phase.
  • In a seven-step photonic quantum walk using orbital angular momentum and polarization, the measured mean chiral displacement closely follows theory for two different initial polarization states, with no significant difference between them, allowing a clear Zak-phase detection after as few as seven steps.
  • Measuring a second Zak phase in a shifted inequivalent timeframe and combining the two windings yields the complete topological invariants \(C_0\) and \(C_\pi\), corresponding to 0- and \(\pi\)-energy edge states, and reproduces the full phase diagram.
  • Dynamical disorder with \(\Delta=\pi/10\) and \(\pi/5\) increases oscillations in single realizations, but ensemble averages smoothly converge to the expected theoretical bulk Zak-phase value.

Methods

  • Photonic discrete-time quantum walk: lattice sites encoded in orbital angular momentum states of a light beam, coin states in left/right circular polarization; implemented with q-plates and quarter-wave plates.
  • Protocol \(U=Q\cdot W\) realized as a seven-step walk; a second inequivalent protocol \(\tilde U\) implemented with eight q-plates total, including \(\delta/2\) q-plates at the beginning and end.
  • Detection: final polarization projections onto chiral states and OAM projections via spatial light modulator, single-mode fibre, and power meter; probabilities \(P_{i,m}\) give the chiral displacement. Analysis compares measured mean chiral displacement to theory, combines measurements in inequivalent timeframes for \(C_0\) and \(C_\pi\), and averages over disorder realizations.

Limitations

  • The paper does not present a dedicated limitations section; it states future directions including extension to chiral systems with more than two internal states, further understanding of temporal disorder, and higher spatial dimensions.
  • Evident from the text: the experimental demonstration uses only seven steps, so the measured quantity still oscillates and approaches the asymptotic Zak phase only in the long-time limit.
  • Evident from the text: disorder robustness is tested only for chiral-preserving disorder and requires disorder strength small compared to the gap to prevent interband transitions; raw data are available only upon request.

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

Integrity signals

Registry

Crossref · 10.1038/ncomms15516

Cites

57 of 57 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Verification

Claims the paper makes

0 of 2 cited sources checked · 0 supported · 0 flagged

  • Findingown data

    The mean chiral displacement is independent of the initial polarization.

    “Importantly, the result contained in equation (8) is (i) independent of the initial polarization and (ii) robust against disorder.”
    • Also stated in: § Zak phase detection in the bulk of a QW · boosted
      “Experimental points closely follow the theory curve for seven time steps (blue solid line), and no significant differences can be observed between the two different initial states, proving that this measurement is insensitive to the choice of the polarization of the photons.”
    Quote in the paperSubstantial claim
  • Findingown data

    The Zak phase was measured in an alternative, inequivalent timeframe of the same quantum walk.

    “Next, we measure the Zak phase in an alternative, inequivalent timeframe and combine the two windings to characterize the full phase diagram of this Floquet system.”
    • Also stated in: § Zak phase in a shifted timeframe · boosted
      “In Fig. 2d, we report the measure of the mean chiral displacement generated by the single-step operator . As in the case of protocol U, this quantity accurately follows the theory prediction, providing an unambiguous detection of the Zak phase of the infinite system after just seven steps.”
    Quote in the paperSubstantial claim
  • Findingown data

    The paper performed the first measurement of the Zak phase of a chiral quantum walk.

    “We experimentally verified our findings by performing the first measurement of the Zak phase of a chiral QW.”
    Quote in the paperSubstantial claim
  • Findingown data

    The mean chiral displacement of a particle's wavepacket in chiral 1D static and Floquet systems with spin 1/2 becomes quantized and proportional to the Zak phase in the long-time limit during free evolution, provided the initial wavefunction is localized.

    “We first introduce the mean chiral displacement, an observable that rapidly approaches a value proportional to the Zak phase during the free evolution of the system.”
    • Also stated in: § Abstract · boosted
      “Here we demonstrate that, in chiral 1D static and Floquet systems with spin 1/2 (that is, a two-state coin), the mean chiral displacement of a particle’s wavepacket becomes quantized and proportional to the Zak phase in the long-time limit.”
    • Also stated in: § Zak phase detection in the bulk of a QW · plain
      “In the limit t→∞, SΓ becomes proportional to the Zak phase, as the oscillatory correction quickly averages to zero (see Fig. 1c).”
    Quote in the paperSubstantial claim
  • Findingown data

    As few as seven steps are enough to have a clear detection of the Zak phase.

    “We note here that, although both theory and data oscillate, as few as seven steps are enough to have a clear detection of the Zak phase.”
    • Also stated in: § Discussion · boosted
      “A precise readout of the Zak phase was obtained after only seven QW steps.”
    Quote in the paperSubstantial claim
  • Probably not a substantial claim. 4
  • The quote only partly states this claim. 1
  • It may state two separate claims as one. 5

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Findingown data

    The measurements show a remarkably fast convergence towards the asymptotic limit.

    “Once again, our measurements show a remarkably fast convergence towards the asymptotic limit.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingown data

    The mean chiral displacement is robust against disorder that preserves chiral symmetry.

    “Importantly, the result contained in equation (8) is (i) independent of the initial polarization and (ii) robust against disorder.”
    • Also stated in: § Robustness to dynamical disorder · plain
      “Finally, we test the stability of the quantization of the mean chiral displacement against disorder.”
    • Also stated in: § Robustness to dynamical disorder · plain
      “an ensemble average over independent realizations smoothly converges to the expected theoretical result, which in the infinite time limit gives the bulk value of the Zak phase.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The disorder introduced in the experiment is dynamic and respects chiral symmetry because the vector vΓ defining the chiral operator does not depend on δ.

    “We note that this disorder is dynamic, in the sense that it affects independently the various q-plates crossed by the beam, but crucially it respects chiral symmetry. This can be simply understood by noting that the vector vΓ, defining the chiral operator, does not depend on δ.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The protocol U'' is topologically equivalent to U, as no gap closing happens during the rotation, so the Zak phase of U'' coincides with γ.

    “However, it is simple to show that is topologically equivalent to U, as no gap closing happens during the rotation ; therefore the Zak phase of coincides with γ.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingown data

    By combining measurements of the mean chiral displacements in two inequivalent timeframes, the invariants C0 and Cπ were computed and the complete phase diagram of the system was detected.

    “By combining our measurements of the mean chiral displacements measured in the inequivalent timeframes, we are now able to compute the invariants C0 and Cπ and detect the complete phase diagram of this system: the result is shown in Fig. 3b.”
    • Also stated in: § Discussion · plain
      “By combining the two measurements, we extracted the two invariants providing the complete bulk-edge correspondence for this driven system, that is, the one associated to the 0-energy edge state, and the one connected to the anomalous π-energy edge state.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim.

    Premisecitation

    Direct detection of topological invariants in the bulk of the system, with no need of edges, has been reported so far by very few experiments.

    “Direct detection of topological invariants in the bulk of the system (with no need of edges) has been reported so far by very few experiments272829.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites 272829

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim.

    Premisecitation

    One-dimensional systems with chiral symmetry are characterized by the Zak phase, the Berry phase accumulated by an eigenstate during its parallel transport through the whole Brillouin zone.

    “One-dimensional (1D) systems with chiral symmetry are characterized by the Zak phase, that is, the Berry phase accumulated by an eigenstate during its parallel transport through the whole Brillouin zone31.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [31] Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989). source

  • Probably not a substantial claim. It may state two separate claims as one.

    Premisereasoning

    The winding number γ/π assumes strictly integer values and counts the number of times the unit vector n rotates around the unit vector vΓ as k traverses the whole Brillouin zone.

    “The winding number γ/π assumes strictly integer values and counts the number of times the unit vector n rotates around the unit vector vΓ as k traverses the whole Brillouin zone.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    The method achieves detection of topological properties by observing the free evolution of a single particle initially localized on a single site in the bulk, unlike former methods that require uniform filling, external forces, loss mechanisms or fine-tuning.

    “While formerly known methods for detection of topological properties require a uniform filling of the band of interest, external forces, loss mechanisms or fine-tuning so that only edge states are populated, the method proposed here quite remarkably achieves this goal by observing the free evolution of a single particle, initially localized on a single site in the bulk.”
    Quote in the paperSubstantial claim
9 of 9

  • The paper relies on this without support.

    The quasi-energy spectrum of the Floquet system is independent of the choice of timeframe.

    “While the dispersion E(k) is equal in all timeframes, the effective Hamiltonian, its eigenstates and symmetries and the resulting dynamics crucially depend on the timeframe40.”
    “As remarked above, the spectrum is independent of the timeframe.”
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?

  • Robust localized states associated with nontrivial winding numbers were first demonstrated in a photonic setup realizing a 1D quantum walk.

    “In this broader context, robust localized states, associated with nontrivial winding numbers (Kitagawa, Rudner et al., 2010), were first demonstrated in a photonic setup realizing a 1D quantum walk (Kitagawa et al., 2012); a similar setup was recently explored in view of directly extracting winding numbers through Zak-phase measurements performed in the bulk (Cardano et al., 2017).”

    From Topological photonics (2019) · cites this paper as Cardano et al., 2017

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Metadata

Authors
Cardano, F., D’Errico, A., Dauphin, A., Maffei, M., Piccirillo, B., de Lisio, C., De Filippis, G., Cataudella, V., Santamato, E., Marrucci, L., Lewenstein, M., & Massignan, P.
Affiliations
  • University of Naples Federico II (IT)
  • Institute of Photonic Sciences (ES)
  • Institute of Applied Science and Intelligent Systems (IT)
  • Institució Catalana de Recerca i Estudis Avançats (ES)
Corresponding author
  • Cardano, Filippo
  • Dauphin, Alexandre
ORCID
Venue
Nature Communications
Publisher
Springer Science and Business Media LLC
Published
2017-06-01
Volume
8
Issue
1
Pages
15516-15516
ISSN
2041-1723 (ISSN-L)
DOI
10.1038/ncomms15516
PMC
PMC5501976
OpenAlex
W2620775208
Cited by
351
References
57
Citations by year
2026: 20 · 2025: 37 · 2024: 41 · 2023: 37 · 2022: 34 · 2021: 47 · 2020: 39 · 2019: 51 · 2018: 28 · 2017: 17
FWCI
17.38
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Communications (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (acceptedVersion)
  • RECERCAT (Consorci de Serveis Universitaris de Catalunya) (acceptedVersion, other-oa)
  • UPCommons institutional repository (Universitat Politècnica de Catalunya) (submittedVersion, cc-by-nc-nd)
  • arXiv (Cornell University) (submittedVersion)
  • HAL (submittedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) (submittedVersion, cc-by-nc-nd)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • CINECA IRIS Institutional Research Information System (University of Bari Aldo Moro) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Fundación Cellex (SEV2015-0522)
  • Centres de Recerca de Catalunya
  • Institut de Ciències Fotòniques (SEV-2015-0522)
  • Generalitat de Catalunya (SEV-2015-0522, SGR874)
  • Horizon 2020 Framework Programme (641122, H2020-FETPROACT-2014)
Topics
  • Physical Sciences › Physics and Astronomy › Condensed Matter Physics › 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 › Condensed Matter Physics › Advanced Condensed Matter Physics
Keywords
Zak phase, topological invariants, twisted photons, topological insulators, periodically driven systems, dynamic disorder, photonic quantum walks

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