Topological characterizations of an extended Su-Schrieffer-Heeger model

Bo Yan; Bryce Gadway; Dizhou Xie; Teng Xiao; Wei Gou

arxiv: 1906.12019 · v1 · pith:LKR2P6OLnew · submitted 2019-06-28 · ❄️ cond-mat.quant-gas

Topological characterizations of an extended Su-Schrieffer-Heeger model

Dizhou Xie , Wei Gou , Teng Xiao , Bryce Gadway , Bo Yan This is my paper

Pith reviewed 2026-05-25 13:47 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas

keywords Su-Schrieffer-Heeger modeltopological phasesultracold atomsmomentum latticewinding numbermean chiral displacementtopological phase transitionedge states

0 comments

The pith

Ultracold atoms realize the four-state SSH model and measure its winding number via extended mean chiral displacement.

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

The paper establishes an experimental platform for the SSH4 model, an extension of the basic Su-Schrieffer-Heeger chain to four internal states per site. It shows that a momentum lattice of ultracold atoms can host this Hamiltonian and that the topological winding number can be extracted from the time-averaged displacement of atoms under a chiral-symmetric drive. The work also tracks the point where the winding number jumps and watches atoms initially at the edge propagate in a manner consistent with protected boundary modes. A reader would care because these steps move higher-dimensional topological invariants from theoretical proposal to direct laboratory readout.

Core claim

The SSH4 Hamiltonian is realized in a momentum-space lattice; the winding number is extracted from the mean chiral displacement measured in the four-dimensional internal space; the topological phase boundary is located by scanning lattice parameters; and quench dynamics from an edge site reveal the expected topological edge state.

What carries the argument

Momentum-lattice implementation of the SSH4 Hamiltonian, with the mean chiral displacement serving as the observable that directly yields the winding number in the higher internal dimension.

If this is right

Varying the relative strengths of the two hopping terms drives a transition between trivial and nontrivial winding numbers.
The same displacement protocol works for any even internal dimension once the appropriate chiral operator is identified.
Boundary atoms remain localized in the topological phase even after the drive is applied.
The phase diagram can be reconstructed point-by-point from displacement data alone.

Where Pith is reading between the lines

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

The same lattice technique could be used to embed other multi-band topological models whose invariants are not accessible by conventional edge spectroscopy.
If the displacement observable remains robust under moderate disorder, it offers a route to measuring topology in systems where direct band tomography is impractical.

Load-bearing premise

The physical lattice reproduces the ideal four-state SSH4 Hamiltonian closely enough that the measured displacement equals the theoretical winding number without significant distortion from calibration errors or decoherence.

What would settle it

A scan of lattice parameters in which the measured mean chiral displacement remains zero across the predicted topological transition point, or quench evolution from the boundary shows no protected propagation when the winding number is nonzero.

Figures

Figures reproduced from arXiv: 1906.12019 by Bo Yan, Bryce Gadway, Dizhou Xie, Teng Xiao, Wei Gou.

**Figure 2.** Figure 2: FIG. 2. (Color online) (a) The experimental setup for multi-frequency Raman coupling. The incoming [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (Color online) The measured total mean chiral displacement versus the tunneling ratio [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (Color online) Quench dynamics of the SSH4 model. (a) The initial state is prepared at the edge. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (Color online) (a) The experimental setup. Our BEC is created in a octagonal chamber. Two dipole [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. (Color online) Detection method. (a) shows the absorption image of atoms with 20ms free expan [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

read the original abstract

The Su-Schrieffer-Heeger (SSH) model perhaps is the easiest and the most basic model for topological excitations. Many variations and extensions of the SSH model have been proposed and explored to better understand both fundamental and novel aspects of topological physics. The SSH4 model has been proposed theoretically as an extended SSH model with higher dimension (the internal dimension changes from two to four). It has been proposed that the winding number in this system can be determined through a higher-dimensional extension of the mean chiral displacement measurement, however this has not yet been verified in experiment. Here we report the realization of this model with ultracold atoms in a momentum lattice. We verify the winding number through measurement of the mean chiral displacement in a system with higher internal dimension, we map out the topological phase transition in this system, and we confirm the topological edge state by observation of the quench dynamics when atoms are initially prepared at the system boundary.

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.

Desk Editor's Note private letter to a colleague

First experimental realization of SSH4 in momentum lattice with mean chiral displacement check, but Hamiltonian fidelity remains the main unaddressed question.

read the letter

The main point is that this paper reports the first experimental realization of the four-band SSH4 model using ultracold atoms in a momentum lattice, along with measurements of the winding number via the higher-dimensional mean chiral displacement, a scan across the topological transition, and quench dynamics showing edge-state behavior. They set up the lattice couplings, collect the displacement data for different parameter regimes, and show that the observed values track the expected winding numbers and phase boundaries. The quench part adds a direct look at the boundary states, which is a straightforward way to cross-check the topology. That combination is new and gives the field a working platform for this extension of the SSH model. The data presentation looks clean enough on the plots described. The soft spot is the one the stress-test flags: momentum-lattice implementations rely on effective couplings from laser fields, and with four internal states the risk of residual off-resonant processes or small calibration offsets is real. The paper does not appear to include independent tomography or a quantitative error budget that would bound how closely the realized Hamiltonian matches the ideal SSH4 form, so the direct mapping from measured displacement to winding number rests on that assumption. If those controls are missing, referees will likely ask for them. This is targeted at people working on synthetic dimensions and topological quantum gases. It has enough new experimental content and testable claims to deserve peer review, even if the error analysis needs tightening.

Referee Report

1 major / 2 minor

Summary. The manuscript reports an experimental realization of the four-band SSH4 extension of the Su-Schrieffer-Heeger model using ultracold atoms in a momentum lattice. The central claims are that the winding number is verified by measuring the mean chiral displacement, that the topological phase transition is mapped out, and that the topological edge state is confirmed via quench dynamics starting from the boundary.

Significance. If the realized Hamiltonian matches the target SSH4 form to sufficient accuracy, the work supplies the first experimental test of the higher-dimensional mean-chiral-displacement protocol for extracting winding numbers and thereby extends cold-atom methods for characterizing topology in systems with internal dimension greater than two.

major comments (1)

[Experimental realization and measurement sections (implied throughout)] The manuscript provides no quantitative bounds on residual higher-order Bragg processes, rotating-wave-approximation errors, or calibration offsets in the momentum-lattice implementation. Because the equality between measured mean chiral displacement and the theoretical winding number holds only when the effective Hamiltonian exactly reproduces the ideal SSH4 form (including the precise inter-leg couplings and chiral operator), the absence of an error budget or independent Hamiltonian tomography renders the central experimental claim unverifiable from the reported data.

minor comments (2)

Clarify the precise definition of the four-dimensional chiral operator used to compute the mean chiral displacement and state whether it is identical to the theoretical operator in the SSH4 literature.
Include the raw time-of-flight images or displacement traces together with the fitting procedure that extracts the winding number from the displacement data.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the single major comment below and will revise the manuscript accordingly to strengthen the experimental claims.

read point-by-point responses

Referee: [Experimental realization and measurement sections (implied throughout)] The manuscript provides no quantitative bounds on residual higher-order Bragg processes, rotating-wave-approximation errors, or calibration offsets in the momentum-lattice implementation. Because the equality between measured mean chiral displacement and the theoretical winding number holds only when the effective Hamiltonian exactly reproduces the ideal SSH4 form (including the precise inter-leg couplings and chiral operator), the absence of an error budget or independent Hamiltonian tomography renders the central experimental claim unverifiable from the reported data.

Authors: We agree that a quantitative error budget is important for verifying that the implemented Hamiltonian matches the target SSH4 model sufficiently well for the mean-chiral-displacement protocol to be reliable. In the revised manuscript we will add a dedicated subsection on experimental imperfections. This will include: (i) estimates of residual higher-order Bragg processes calculated from the measured laser intensities and detunings; (ii) a bound on rotating-wave-approximation errors given by the ratio of the two-photon Rabi frequency to the single-photon detuning; and (iii) the calibration precision of the lattice depths (typically a few percent). While a complete independent Hamiltonian tomography was not performed, the systematic agreement of the measured mean chiral displacement with the expected winding numbers across the parameter space, together with the observed phase-transition point and quench dynamics, provides supporting evidence that deviations are small. These additions will make the central claims more directly verifiable from the reported data. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental verification of SSH4 model

full rationale

The manuscript reports an experimental realization of the extended SSH4 model in a momentum lattice with ultracold atoms. It measures mean chiral displacement to verify the winding number, maps the phase transition, and observes quench dynamics for edge states. No derivation chain, fitted parameters renamed as predictions, or self-citation load-bearing steps appear in the abstract or described claims. The work is self-contained experimental confirmation against external benchmarks, with no reduction of results to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to identify specific free parameters, axioms, or invented entities; no equations or methods sections available for audit.

pith-pipeline@v0.9.0 · 5693 in / 1073 out tokens · 49829 ms · 2026-05-25T13:47:16.951949+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 1 internal anchor

[1]

& Zhang, S.-C

Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011). 8

work page 2011
[6]

Goldman, N., Budich, J. C. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639–645 (2016)

work page 2016
[16]

Jo, G.-B. et al. Ultracold atoms in a tunable optical kagome lattice. Phys. Rev. Lett. 108, 045305 (2012). 9

work page 2012
[32]

& Platero, G

Perez-Gonzalez, B., Bello, M., Gomez-Leon, A. & Platero, G. Interplay between long-range hopping and disorder in topological systems. Phys. Rev. B 99, 035146 (2019). 10

work page 2019
[39]

Cardano, F. et al. Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons. Nat. Comm. 8, 15516 (2017)

work page 2017
[42]

Atom-optics approach to studying transport phenomena

Gadway, B. Atom-optics approach to studying transport phenomena. Phys. Rev. A 92, 043606 (2015)

work page 2015
[43]

& Zhang, S.-C

Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011)

work page 2011
[44]

Wang, D. et al. Synthesis of antisymmetric spin exchange interaction and chiral spin clusters in superconducting circuits. Nat. Phys. 15, 382 (2019)

work page 2019
[45]

& Zhang, X

Wang, B., Chen, T. & Zhang, X. Experimental Observation of Topologically Protected Bound States with Vanishing Chern Numbers in a Two-Dimensional Quantum Walk. Phys. Rev. Lett. 121, 100501 (2018)

work page 2018
[46]

Chen, C. et al. Observation of Topologically Protected Edge States in a Photonic Two-Dimensional Quantum Walk. Phys. Rev. Lett. 121, 100502 (2018)

work page 2018
[47]

K., Lu, H

Stuhl, B. K., Lu, H. I., Aycock, L. M., Genkina, D. & Spielman, I. B. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1517 (2015)

work page 2015
[48]

Goldman, N., Budich, J. C. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639–645 (2016). 11

work page 2016
[49]

J., An, F

Meier, E. J., An, F. A. & Gadway, B. Observation of the topological soliton state in the Su-Schrieffer- Heeger model. Nat. Comm. 7, 13986 (2016)

work page 2016
[50]

Meier, E. J. et al. Observation of the topological Anderson insulator in disordered atomic wires. Science 362, 929 (2018)

work page 2018
[51]

Cai, H. et al. Experimental observation of momentum-space chiral edge currents in room-temperature atoms. Phys. Rev. Lett. 122, 023601 (2019)

work page 2019
[52]

Xiao, M. et al. Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11, 240–244 (2015)

work page 2015
[53]

Peng, Y .-G. et al. Experimental demonstration of anomalous Floquet topological insulator for sound. Nat. Comm. 7, 13368 (2016)

work page 2016
[54]

& Simon, J

Ningyuan, J., Owens, C., Sommer, A., Schuster, D. & Simon, J. Time- and Site-Resolved Dynamics in a Topological Circuit. Phys. Rev. X 5, 021031 (2015)

work page 2015
[55]

& Ren, J

Zhu, W., Hou, S., Long, Y ., Chen, H. & Ren, J. Simulating quantum spin Hall effect in the topological Lieb lattice of a linear circuit network. Phys. Rev. B 97, 075310 (2018)

work page 2018
[56]

Jotzu, G. et al. Experimental realization of the topological haldane model with ultracold fermions. Nature 515, 237–240 (2014)

work page 2014
[57]

Becker, C. et al. Ultracold quantum gases in triangular optical lattices. New J. Phys. 12, 065025 (2010)

work page 2010
[58]

Jo, G.-B. et al. Ultracold atoms in a tunable optical kagome lattice. Phys. Rev. Lett. 108, 045305 (2012)

work page 2012
[59]

& Hemmerich, A

Wirth, G., Oelschlaeger, M. & Hemmerich, A. Evidence for orbital superﬂuidity in the P-band of a bipartite optical square lattice. Nat. Phys. 7, 147–153 (2011)

work page 2011
[60]

& Liu, W

Xu, Z.-F., You, L., Hemmerich, A. & Liu, W. V .π-ﬂux dirac bosons and topological edge excitations in a bosonic chiral p-wave superﬂuid. Phys. Rev. Lett. 117, 085301 (2016)

work page 2016
[61]

J., Jimenez-Garcia, K

Lin, Y . J., Jimenez-Garcia, K. & Spielman, I. B. Spin-orbit-coupled Bose-Einstein condensates. Na- ture 471, 83 (2011)

work page 2011
[62]

Wang, P. et al. Spin-orbit coupled degenerate fermi gases. Phys. Rev. Lett. 109, 095301 (2012)

work page 2012
[63]

Wu, Z. et al. Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates. Sci- ence 354, 83–88 (2016)

work page 2016
[64]

Dynamical detection of topological charges

Zhang, L., Zhang, L. & Liu, X. Dynamical detection of topological charges. arXiv:1807.10782 (2018). 12

work page internal anchor Pith review Pith/arXiv arXiv 2018
[65]

& Zhai, H

Wang, C., Zhang, P., Chen, X., Yu, J. & Zhai, H. Scheme to measure the topological number of a chern insulator from quench dynamics. Phys. Rev. Lett. 118, 185701 (2017)

work page 2017
[66]

Atala, M. et al. Direct measurement of the Zak phase in topological Bloch bands. Nat. Phys. 9, 795–800 (2013)

work page 2013
[67]

& Dai, X

Wang, L., Troyer, M. & Dai, X. Topological Charge Pumping in a One-Dimensional Optical Lattice. Phys. Rev. Lett. 111, 026802 (2013)

work page 2013
[68]

& Bloch, I

Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys. 12, 350 (2016)

work page 2016
[69]

Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296 (2016)

work page 2016
[70]

Leder, M. et al. Real-space imaging of a topologically protected edge state with ultracold atoms in an amplitude-chirped optical lattice. Nat. Comm. 7, 13112 (2016)

work page 2016
[71]

& Platero, G

Gómez-León, A. & Platero, G. Floquet-bloch theory and topology in periodically driven lattices. Phys. Rev. Lett. 110, 200403 (2013)

work page 2013
[72]

& Torres, L

Dal Lago, V ., Atala, M. & Torres, L. E. F. F. Floquet topological transitions in a driven one- dimensional topological insulator. Phys. Rev. A 92, 023624 (2015)

work page 2015
[73]

A., Meier, E

An, F. A., Meier, E. J. & Gadway, B. Engineering a Flux-Dependent Mobility Edge in Disordered Zigzag Chains. Phys. Rev. X 8, 031045 (2018)

work page 2018
[74]

& Platero, G

Perez-Gonzalez, B., Bello, M., Gomez-Leon, A. & Platero, G. Interplay between long-range hopping and disorder in topological systems. Phys. Rev. B 99, 035146 (2019)

work page 2019
[75]

& Lim, L.-K

Sun, N. & Lim, L.-K. Quantum charge pumps with topological phases in a Creutz ladder. Phys. Rev. B 96, 035139 (2017)

work page 2017
[76]

Jünemann, J. et al. Exploring interacting topological insulators with ultracold atoms: The synthetic creutz-hubbard model. Phys. Rev. X 7, 031057 (2017)

work page 2017
[77]

& Massignan, P

Maffei, M., Dauphin, A., Cardano, F., Lewenstein, M. & Massignan, P. Topological characterization of chiral models through their long time dynamics. New J. Phys. 20, 013023 (2018)

work page 2018
[78]

Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356 (2019)

work page 2019
[79]

Celi, A. et al. Synthetic Gauge Fields in Synthetic Dimensions. Phys. Rev. Lett. 112, 043001 (2014)

work page 2014
[80]

Xiao, L. et al. Higher winding number in a nonunitary photonic quantum walk. Phys. Rev. A 98, 063847 (2018)

work page 2018
[81]

Cardano, F. et al. Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons. Nat. Comm. 8, 15516 (2017). 13 Energy 01 2-1-2N= (a) (b) (c) Ψ(n) / |Ψ(n)|2 ta tb tc td A1 A2B1 B2 FIG. 1. (Color online) (a) The diagrammatic sketch of the SSH4 model. tunneling terms are periodic with a unit cell of 4 sites. (b) The eigenen...

work page 2017
[82]

J., An, F

Meier, E. J., An, F. A. & Gadway, B. Atom-optics simulator of lattice transport phenomena. Phys. Rev. A 93, 051602 (2016)

work page 2016
[83]

& Yan, B

Xie, D., Wang, D., Gou, W., Bu, W. & Yan, B. Fast production of rubidium Bose-Einstein condensate in a dimple trap. J. Opt. Soc. Am. B-Opt. Phys. 35, 500–503 (2018)

work page 2018
[84]

Atom-optics approach to studying transport phenomena

Gadway, B. Atom-optics approach to studying transport phenomena. Phys. Rev. A 92, 043606 (2015). 14  nff0 0f BEC      n n f    2 (a) (b) 1 2 3 4-3 -2 -1 0 -4  kp/2 E FIG. 2. (Color online) (a) The experimental setup for multi-frequency Raman coupling. The incoming Raman beam passes the BEC, then passes two AOMs, and ﬁnally propagates back t...

work page 2015

[1] [1]

& Zhang, S.-C

Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011). 8

work page 2011

[2] [6]

Goldman, N., Budich, J. C. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639–645 (2016)

work page 2016

[3] [16]

Jo, G.-B. et al. Ultracold atoms in a tunable optical kagome lattice. Phys. Rev. Lett. 108, 045305 (2012). 9

work page 2012

[4] [32]

& Platero, G

Perez-Gonzalez, B., Bello, M., Gomez-Leon, A. & Platero, G. Interplay between long-range hopping and disorder in topological systems. Phys. Rev. B 99, 035146 (2019). 10

work page 2019

[5] [39]

Cardano, F. et al. Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons. Nat. Comm. 8, 15516 (2017)

work page 2017

[6] [42]

Atom-optics approach to studying transport phenomena

Gadway, B. Atom-optics approach to studying transport phenomena. Phys. Rev. A 92, 043606 (2015)

work page 2015

[7] [43]

& Zhang, S.-C

Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011)

work page 2011

[8] [44]

Wang, D. et al. Synthesis of antisymmetric spin exchange interaction and chiral spin clusters in superconducting circuits. Nat. Phys. 15, 382 (2019)

work page 2019

[9] [45]

& Zhang, X

Wang, B., Chen, T. & Zhang, X. Experimental Observation of Topologically Protected Bound States with Vanishing Chern Numbers in a Two-Dimensional Quantum Walk. Phys. Rev. Lett. 121, 100501 (2018)

work page 2018

[10] [46]

Chen, C. et al. Observation of Topologically Protected Edge States in a Photonic Two-Dimensional Quantum Walk. Phys. Rev. Lett. 121, 100502 (2018)

work page 2018

[11] [47]

K., Lu, H

Stuhl, B. K., Lu, H. I., Aycock, L. M., Genkina, D. & Spielman, I. B. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1517 (2015)

work page 2015

[12] [48]

Goldman, N., Budich, J. C. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639–645 (2016). 11

work page 2016

[13] [49]

J., An, F

Meier, E. J., An, F. A. & Gadway, B. Observation of the topological soliton state in the Su-Schrieffer- Heeger model. Nat. Comm. 7, 13986 (2016)

work page 2016

[14] [50]

Meier, E. J. et al. Observation of the topological Anderson insulator in disordered atomic wires. Science 362, 929 (2018)

work page 2018

[15] [51]

Cai, H. et al. Experimental observation of momentum-space chiral edge currents in room-temperature atoms. Phys. Rev. Lett. 122, 023601 (2019)

work page 2019

[16] [52]

Xiao, M. et al. Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11, 240–244 (2015)

work page 2015

[17] [53]

Peng, Y .-G. et al. Experimental demonstration of anomalous Floquet topological insulator for sound. Nat. Comm. 7, 13368 (2016)

work page 2016

[18] [54]

& Simon, J

Ningyuan, J., Owens, C., Sommer, A., Schuster, D. & Simon, J. Time- and Site-Resolved Dynamics in a Topological Circuit. Phys. Rev. X 5, 021031 (2015)

work page 2015

[19] [55]

& Ren, J

Zhu, W., Hou, S., Long, Y ., Chen, H. & Ren, J. Simulating quantum spin Hall effect in the topological Lieb lattice of a linear circuit network. Phys. Rev. B 97, 075310 (2018)

work page 2018

[20] [56]

Jotzu, G. et al. Experimental realization of the topological haldane model with ultracold fermions. Nature 515, 237–240 (2014)

work page 2014

[21] [57]

Becker, C. et al. Ultracold quantum gases in triangular optical lattices. New J. Phys. 12, 065025 (2010)

work page 2010

[22] [58]

Jo, G.-B. et al. Ultracold atoms in a tunable optical kagome lattice. Phys. Rev. Lett. 108, 045305 (2012)

work page 2012

[23] [59]

& Hemmerich, A

Wirth, G., Oelschlaeger, M. & Hemmerich, A. Evidence for orbital superﬂuidity in the P-band of a bipartite optical square lattice. Nat. Phys. 7, 147–153 (2011)

work page 2011

[24] [60]

& Liu, W

Xu, Z.-F., You, L., Hemmerich, A. & Liu, W. V .π-ﬂux dirac bosons and topological edge excitations in a bosonic chiral p-wave superﬂuid. Phys. Rev. Lett. 117, 085301 (2016)

work page 2016

[25] [61]

J., Jimenez-Garcia, K

Lin, Y . J., Jimenez-Garcia, K. & Spielman, I. B. Spin-orbit-coupled Bose-Einstein condensates. Na- ture 471, 83 (2011)

work page 2011

[26] [62]

Wang, P. et al. Spin-orbit coupled degenerate fermi gases. Phys. Rev. Lett. 109, 095301 (2012)

work page 2012

[27] [63]

Wu, Z. et al. Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates. Sci- ence 354, 83–88 (2016)

work page 2016

[28] [64]

Dynamical detection of topological charges

Zhang, L., Zhang, L. & Liu, X. Dynamical detection of topological charges. arXiv:1807.10782 (2018). 12

work page internal anchor Pith review Pith/arXiv arXiv 2018

[29] [65]

& Zhai, H

Wang, C., Zhang, P., Chen, X., Yu, J. & Zhai, H. Scheme to measure the topological number of a chern insulator from quench dynamics. Phys. Rev. Lett. 118, 185701 (2017)

work page 2017

[30] [66]

Atala, M. et al. Direct measurement of the Zak phase in topological Bloch bands. Nat. Phys. 9, 795–800 (2013)

work page 2013

[31] [67]

& Dai, X

Wang, L., Troyer, M. & Dai, X. Topological Charge Pumping in a One-Dimensional Optical Lattice. Phys. Rev. Lett. 111, 026802 (2013)

work page 2013

[32] [68]

& Bloch, I

Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys. 12, 350 (2016)

work page 2016

[33] [69]

Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296 (2016)

work page 2016

[34] [70]

Leder, M. et al. Real-space imaging of a topologically protected edge state with ultracold atoms in an amplitude-chirped optical lattice. Nat. Comm. 7, 13112 (2016)

work page 2016

[35] [71]

& Platero, G

Gómez-León, A. & Platero, G. Floquet-bloch theory and topology in periodically driven lattices. Phys. Rev. Lett. 110, 200403 (2013)

work page 2013

[36] [72]

& Torres, L

Dal Lago, V ., Atala, M. & Torres, L. E. F. F. Floquet topological transitions in a driven one- dimensional topological insulator. Phys. Rev. A 92, 023624 (2015)

work page 2015

[37] [73]

A., Meier, E

An, F. A., Meier, E. J. & Gadway, B. Engineering a Flux-Dependent Mobility Edge in Disordered Zigzag Chains. Phys. Rev. X 8, 031045 (2018)

work page 2018

[38] [74]

& Platero, G

Perez-Gonzalez, B., Bello, M., Gomez-Leon, A. & Platero, G. Interplay between long-range hopping and disorder in topological systems. Phys. Rev. B 99, 035146 (2019)

work page 2019

[39] [75]

& Lim, L.-K

Sun, N. & Lim, L.-K. Quantum charge pumps with topological phases in a Creutz ladder. Phys. Rev. B 96, 035139 (2017)

work page 2017

[40] [76]

Jünemann, J. et al. Exploring interacting topological insulators with ultracold atoms: The synthetic creutz-hubbard model. Phys. Rev. X 7, 031057 (2017)

work page 2017

[41] [77]

& Massignan, P

Maffei, M., Dauphin, A., Cardano, F., Lewenstein, M. & Massignan, P. Topological characterization of chiral models through their long time dynamics. New J. Phys. 20, 013023 (2018)

work page 2018

[42] [78]

Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356 (2019)

work page 2019

[43] [79]

Celi, A. et al. Synthetic Gauge Fields in Synthetic Dimensions. Phys. Rev. Lett. 112, 043001 (2014)

work page 2014

[44] [80]

Xiao, L. et al. Higher winding number in a nonunitary photonic quantum walk. Phys. Rev. A 98, 063847 (2018)

work page 2018

[45] [81]

Cardano, F. et al. Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons. Nat. Comm. 8, 15516 (2017). 13 Energy 01 2-1-2N= (a) (b) (c) Ψ(n) / |Ψ(n)|2 ta tb tc td A1 A2B1 B2 FIG. 1. (Color online) (a) The diagrammatic sketch of the SSH4 model. tunneling terms are periodic with a unit cell of 4 sites. (b) The eigenen...

work page 2017

[46] [82]

J., An, F

Meier, E. J., An, F. A. & Gadway, B. Atom-optics simulator of lattice transport phenomena. Phys. Rev. A 93, 051602 (2016)

work page 2016

[47] [83]

& Yan, B

Xie, D., Wang, D., Gou, W., Bu, W. & Yan, B. Fast production of rubidium Bose-Einstein condensate in a dimple trap. J. Opt. Soc. Am. B-Opt. Phys. 35, 500–503 (2018)

work page 2018

[48] [84]

Atom-optics approach to studying transport phenomena

Gadway, B. Atom-optics approach to studying transport phenomena. Phys. Rev. A 92, 043606 (2015). 14  nff0 0f BEC      n n f    2 (a) (b) 1 2 3 4-3 -2 -1 0 -4  kp/2 E FIG. 2. (Color online) (a) The experimental setup for multi-frequency Raman coupling. The incoming Raman beam passes the BEC, then passes two AOMs, and ﬁnally propagates back t...

work page 2015