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Real-space detection and manipulation of topological edge modes with ultracold atoms

Nature Physics volume 20, pages 1306–1312 (2024)Cite this article

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Abstract

The bulk–boundary correspondence, a fundamental principle relating the topological invariants of the bulk to the presence of edge states, is modified in periodically driven systems. Conventional bulk topological invariants are insufficient to predict the existence of topological edge modes in such systems. Although ultracold atoms provide excellent settings for clean realizations of Floquet protocols, the observation of real-space edge modes has so far remained elusive. Here we demonstrate an experimental protocol for realizing chiral edge modes in optical lattices through the periodic modulation of the tunnelling rate between neighbouring sites. In particular, we show how to efficiently prepare particles in edge modes in three distinct Floquet topological regimes in a periodically driven honeycomb lattice. Controlling the height and amplitude of the potential step, we characterize the emergence of edge modes and the dependence of their group velocity on the sharpness of the potential step. Our direct observation of topological edge modes provides a tool to study topological phases of matter in the presence of disorder and interactions, where conventional bulk observables are not applicable.

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Fig. 1: Preparation and observation of anomalous Floquet topological edge modes.
Fig. 2: Preparation and observation of edge modes in the Haldane regime.
Fig. 3: Emergence of edge modes with increasing potential height V0.
Fig. 4: Edge mode velocity for varying edge width.

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Data availability

The data that support the plots within this paper and other findings of this study are available at https://doi.org/10.17617/3.HSVMVC.

Code availability

The code that supports the plots within this paper is available from the corresponding author upon reasonable request.

References

  1. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  2. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010).

    Article  ADS  Google Scholar 

  3. Bernevig, B. A. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013).

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

    Article  ADS  Google Scholar 

  5. Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494 (1980).

    Article  ADS  Google Scholar 

  6. von Klitzing, K. The quantized Hall effect. Rev. Mod. Phys. 58, 519 (1986).

    Article  ADS  Google Scholar 

  7. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982).

    Article  ADS  Google Scholar 

  8. Niu, Q. & Thouless, D. J. Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction. J. Phys. A 17, 2453 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  9. Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185 (1982).

    Article  ADS  Google Scholar 

  10. Rammal, R., Toulouse, G., Jaekel, M. T. & Halperin, B. I. Quantized Hall conductance and edge states: two-dimensional strips with a periodic potential. Phys. Rev. B 27, 5142 (1983).

    Article  ADS  Google Scholar 

  11. MacDonald, A. H. Edge states and quantized Hall conductivity in a periodic potential. Phys. Rev. B 29, 6563 (1984).

    Article  ADS  Google Scholar 

  12. Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  13. Hatsugai, Y. Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function. Phys. Rev. B 48, 11851 (1993).

    Article  ADS  Google Scholar 

  14. Qi, X.-L., Wu, Y.-S. & Zhang, S.-C. General theorem relating the bulk topological number to edge states in two-dimensional insulators. Phys. Rev. B 74, 045125 (2006).

    Article  ADS  Google Scholar 

  15. Aidelsburger, M., Nascimbène, S. & Goldman, N. Artificial gauge fields in materials and engineered systems. C. R. Phys. 19, 394–432 (2018).

    Article  ADS  Google Scholar 

  16. Cooper, N. R., Dalibard, J. & Spielman, I. B. Topological bands for ultracold atoms. Rev. Mod. Phys. 91, 015005 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  17. Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  18. Mukherjee, S. et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nat. Commun. 8, 13918 (2017).

    Article  ADS  Google Scholar 

  19. Maczewsky, L. J., Zeuner, J. M., Nolte, S. & Szameit, A. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 (2017).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  21. Gao, F. et al. Probing topological protection using a designer surface plasmon structure. Nat. Commun. 7, 11619 (2016).

    Article  ADS  Google Scholar 

  22. Afzal, S., Zimmerling, T. J., Ren, Y., Perron, D. & Van, V. Realization of anomalous Floquet insulators in strongly coupled nanophotonic lattices. Phys. Rev. Lett. 124, 253601 (2020).

    Article  ADS  Google Scholar 

  23. Wintersperger, K. et al. Realization of an anomalous Floquet topological system with ultracold atoms. Nat. Phys. 16, 1058–1063 (2020).

    Article  Google Scholar 

  24. Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).

    Article  ADS  Google Scholar 

  25. Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).

    Google Scholar 

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

    Article  Google Scholar 

  27. Duca, L. et al. An Aharonov–Bohm interferometer for determining Bloch band topology. Science 347, 288–292 (2015).

    Article  ADS  Google Scholar 

  28. Fläschner, N. et al. Experimental reconstruction of the Berry curvature in a Floquet Bloch band. Science 352, 1091–1094 (2016).

    Article  ADS  Google Scholar 

  29. Li, T. et al. Bloch state tomography using Wilson lines. Science 352, 1094–1097 (2016).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  31. Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).

    Article  Google Scholar 

  32. Zhou, T.-W. et al. Observation of universal Hall response in strongly interacting Fermions. Science 381, 427–430 (2023).

    Article  ADS  Google Scholar 

  33. Asteria, L. et al. Measuring quantized circular dichroism in ultracold topological matter. Nat. Phys. 15, 449–454 (2019).

    Article  Google Scholar 

  34. Tarnowski, M. et al. Measuring topology from dynamics by obtaining the Chern number from a linking number. Nat. Commun. 10, 1728 (2019).

    Article  ADS  Google Scholar 

  35. Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).

    Article  ADS  Google Scholar 

  36. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

    Article  ADS  Google Scholar 

  37. Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).

    Article  ADS  Google Scholar 

  38. 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).

    Google Scholar 

  39. Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014).

    Article  ADS  Google Scholar 

  40. Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).

    Article  Google Scholar 

  41. Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  42. 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, 1514–1518 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  43. Chalopin, T. et al. Probing chiral edge dynamics and bulk topology of a synthetic Hall system. Nat. Phys. 16, 1017–1021 (2020).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  45. Sompet, P. et al. Realizing the symmetry-protected Haldane phase in Fermi–Hubbard ladders. Nature https://doi.org/10.1038/s41586-022-04688-z (2022).

  46. Léséleuc, S. D. et al. Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms. Science 365, 775–780 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  47. Goldman, N., Beugnon, J. & Gerbier, F. Detecting chiral edge states in the Hofstadter optical lattice. Phys. Rev. Lett. 108, 255303 (2012).

    Article  ADS  Google Scholar 

  48. Goldman, N. et al. Direct imaging of topological edge states in cold-atom systems. Proc. Natl Acad. Sci. USA 110, 6736–6741 (2013).

    Article  ADS  Google Scholar 

  49. Goldman, N. et al. Creating topological interfaces and detecting chiral edge modes in a two-dimensional optical lattice. Phys. Rev. A 94, 043611 (2016).

    Article  ADS  Google Scholar 

  50. Wang, B., Aidelsburger, M., Dalibard, J., Eckardt, A. & Goldman, N. The cold-atom elevator: from edge-state injection to the preparation of fractional Chern insulators. Preprint at https://doi.org/10.48550/arXiv.2306.15610 (2023).

  51. Stanescu, T. D., Galitski, V. & Das Sarma, S. Topological states in two-dimensional optical lattices. Phys. Rev. A 82, 013608 (2010).

    Article  ADS  Google Scholar 

  52. Buchhold, M., Cocks, D. & Hofstetter, W. Effects of smooth boundaries on topological edge modes in optical lattices. Phys. Rev. A 85, 063614 (2012).

    Article  ADS  Google Scholar 

  53. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).

    Article  ADS  Google Scholar 

  54. Li, J., Chu, R.-L., Jain, J. K. & Shen, S.-Q. Topological Anderson insulator. Phys. Rev. Lett. 102, 136806 (2009).

    Article  ADS  Google Scholar 

  55. Titum, P., Lindner, N. H., Rechtsman, M. C. & Refael, G. Disorder-induced Floquet topological insulators. Phys. Rev. Lett. 114, 056801 (2015).

    Article  ADS  Google Scholar 

  56. Stützer, S. et al. Photonic topological Anderson insulators. Nature 560, 461–465 (2018).

    Article  ADS  Google Scholar 

  57. Titum, P., Berg, E., Rudner, M. S., Refael, G. & Lindner, N. H. Anomalous Floquet–Anderson insulator as a nonadiabatic quantized charge pump. Phys. Rev. X 6, 021013 (2016).

    Google Scholar 

  58. Kundu, A., Rudner, M., Berg, E. & Lindner, N. H. Quantized large-bias current in the anomalous Floquet–Anderson insulator. Phys. Rev. B 101, 041403 (2020).

    Article  ADS  Google Scholar 

  59. Dutta, A., Sen, E., Zheng, J.-H., Aidelsburger, M. & Hofstetter, W. The anomalous Floquet Anderson insulator in a continuously driven optical lattice. Preprint at https://doi.org/10.48550/arXiv.2312.13589 (2023).

  60. Nathan, F., Abanin, D., Berg, E., Lindner, N. H. & Rudner, M. S. Anomalous Floquet insulators. Phys. Rev. B 99, 195133 (2019).

    Article  ADS  Google Scholar 

  61. van Houten, H. & Beenakker, C. Quantum point contacts. Phys. Today 49, 22–27 (1996).

    Article  Google Scholar 

  62. Ji, Y. et al. An electronic Mach–Zehnder interferometer. Nature 422, 415–418 (2003).

    Article  ADS  Google Scholar 

  63. Tambasco, J.-L. et al. Quantum interference of topological states of light. Sci. Adv. 4, eaat3187 (2018).

    Article  ADS  Google Scholar 

  64. Carrega, M., Chirolli, L., Heun, S. & Sorba, L. Anyons in quantum Hall interferometry. Nat. Rev. Phys. 3, 698–711 (2021).

    Article  Google Scholar 

  65. Büttiker, M. Edge-state physics without magnetic fields. Science 325, 278–279 (2009).

    Article  Google Scholar 

  66. Martínez, M. F. & Ünal, F. N. Wave-packet dynamics and edge transport in anomalous Floquet topological phases. Phys. Rev. A 108, 063314 (2023).

    Article  ADS  Google Scholar 

  67. Yao, R. et al. Observation of chiral edge transport in a rapidly-rotating quantum gas. Preprint at https://doi.org/10.48550/arXiv.2304.10468 (2023).

  68. Schneider, U. et al. Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms. Nat. Phys. 8, 213–218 (2012).

    Article  Google Scholar 

  69. Ronzheimer, J. P. et al. Expansion dynamics of interacting Bosons in homogeneous lattices in one and two dimensions. Phys. Rev. Lett. 110, 205301 (2013).

    Article  ADS  Google Scholar 

  70. Dür, W., Raussendorf, R., Kendon, V. M. & Briegel, H.-J. Quantum walks in optical lattices. Phys. Rev. A 66, 052319 (2002).

    Article  ADS  Google Scholar 

  71. Morsch, O. & Oberthaler, M. Dynamics of Bose–Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179 (2006).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank N. Ünal, M. Martínez, N. Goldman and M. Di Liberto for fruitful discussions. We further thank the University of Bonn for the lattice laser. This work was funded by the Deutsche Forschungsgemeinschaft (German Research Foundation) via Research Unit FOR 2414 under project no. 277974659. The work was further supported under Germany’s Excellence Strategy EXC-2111–390814868. J.A. was funded by the Alfried Krupp von Bohlen und Halbach Foundation. R.S.-J. has received funding from the European Union’s Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement no. 101028339.

Author information

Author notes

  1. Raphaël Saint-Jalm

    Present address: Institut de Physique de Nice, Université Côte d’Azur, CNRS, Nice, France

  2. These authors contributed equally: Christoph Braun, Raphaël Saint-Jalm.

Authors and Affiliations

  1. Fakultät für Physik, Ludwig-Maximilians-Universität München, Munich, Germany

    Christoph Braun, Raphaël Saint-Jalm, Alexander Hesse, Johannes Arceri, Immanuel Bloch & Monika Aidelsburger

  2. Munich Center for Quantum Science and Technology, Munich, Germany

    Christoph Braun, Raphaël Saint-Jalm, Alexander Hesse, Johannes Arceri, Immanuel Bloch & Monika Aidelsburger

  3. Max-Planck-Institut für Quantenoptik, Garching, Germany

    Christoph Braun, Raphaël Saint-Jalm, Alexander Hesse, Johannes Arceri, Immanuel Bloch & Monika Aidelsburger

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  1. Christoph Braun
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Contributions

C.B. and R.S.-J. carried out the experiments, and together with A.H. and J.A. maintained and improved the experimental set-up. R.S.-J. and C.B. performed the theoretical simulations. I.B. and M.A. supervised the study. All authors worked on the interpretation of the data and contributed to the writing of the manuscript.

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Correspondence to Monika Aidelsburger.

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Extended data

Extended Data Fig. 1 Evolution of the initial wave packet in the bulk of a static lattice.

a, Spatial distribution of the initial wave packet 10 μs after release into the lattice. The wavepacket is prepared at ω⊥/(2π) = 1.3(1) kHz. b, Real-space distribution of the atoms in the static lattice after an expansion time of 5 ms. The data shown in a and b are an average over 302 individual realizations. c, Group velocity distribution for a uniformly filled lowest band of the honeycomb lattice. d, Numerical simulation of the expansion dynamics in a tight-binding model starting from a state occupying a single site. The resulting probability distribution ∣ψ(r, t)∣2 is binned to match the number of pixels in b, the scale bar corresponds to 10a.

Extended Data Fig. 2 Comparison of topological and trivial evolution on the edge of height 17.0(3) kHz.

a, Evolution of a wave packet prepared in the unmodulated lattice close to the edge imaged 5 ms after switching off the tweezer. The left images corresponds to all parameters of the experiment equivalent to κ = + 1, the right image to κ = − 1. b, Difference image for the dataset of a showing no chiral motion. c Evolution of the wave packet 5 ms after releasing the cloud from the tweezer in the modulated lattice in the anomalous regime. The initial position is identical to the dataset shown in a. The chirality of the left image is κ = + 1, the right image κ = − 1. d Difference image for the dataset of c clearly highlighting the chiral motion in the topologically non-trivial phase. All images are the result of 67 individual realizations.

Extended Data Fig. 3 Phase gradient induced by the kick.

The velocity of the wave packet is measured as a function of the total displacement of the tweezer for two different in-plane trapping frequencies: ω⊥/(2π) = 1.3(1) kHz in light blue, and 2.0(1) kHz in dark blue. The dashed lines are the analytic predictions from the motion of a cloud in a harmonic potential, and the dotted lines are the interpolation of the measurements with an empirical function. The inset shows the averaged measurement of center of mass (CoM) of the atoms after the kick and a variable evolution time in the optical dipole trap for the lowest value of ω⊥ and displacements of 0.19 μm (circles), 0.78 μm (squares), 1.36 μm (diamonds), and 1.94 μm (pentagons). The depicted datapoints are an average of 6 individual realizations. The inset also shows the linear weighted fits from which the velocity is extracted. The associated uncertainty of the slope is used as the error bar for the main graph, and is smaller than the size of the markers. The arrow at displacement ≈ 1 μm indicates the phase gradient that was used in all measurements to populate the edge mode in the Haldane regime.

Extended Data Fig. 4 Evaluation of the fraction of atoms populating the edge mode.

Averaged absorption image after 1.5 ms of time evolution, together with the respective regions of interest used for the evaluation: close to the edge (solid black lines) and in the bulk (dashed black lines). The two background regions of half the size are indicated by the gray lines. This example corresponds to the ω⊥/(2π) = 1.3(1) kHz data point of Fig. 2 shown in the bottom right inset.

Extended Data Fig. 5 Numerical simulations.

a, Geometry of the lattice. The system consists of the sites located in the dashed red rectangle. In the y-direction the system is periodic, as indicated by the repetition of the system in pale colours. The three types of links between sites are indicated by the three shades of blue. The links that are multiplied with a complex phase factor are those that cross the dashed lines. b, Potential energy applied in the x-direction. The position of the edge is chosen in the center of the system, and its width ℓ and its height V0 are free parameters. In this example, ℓ = a. The shaded region is an example of a zone that can be used to selectively filter the spectrum. c, Full quasienergy spectrum of the system with \({J}_{1}^{{\prime} }/{J}_{1}=0.1\), ℏω = 1.5J1, ℓ = a and V0 = 2.25J1. The system is in the anomalous regime, that is, two pairs of edge modes are visible, linking the two bulk energy bands in the two gaps of the model. The gray circle indicates the eigenstate whose wavefunction is represented in e. d, Exemplary quasienergy spectrum. For this example, the same spectrum as in c is plotted with a color scale that represents the overlap with the shaded region of graph b. e, Wavefunction of the eigenstate at qy = ± π/Ly and at energy ε ≈ 0 corresponding to the introduced edge edge state. The position of the sites is represented as the black dots and the tunneling bonds are shown in gray. Around these dots, the color indicates the modulus square of the wavefunction, and the direction of the arrow indicates its phase. The position of the edge is shown as a black line.

Extended Data Fig. 6 Determination of the edge mode velocity.

a, In situ images of atoms in the optical tweezer. The position of the tweezer is varied from top to bottom by steps of 0.48 μm in the x direction (perpendicular to the orientation of the edge), as emphasized by the vertical dashed line. The pictures are an average of five individual experimental realizations, performed without the edge potential in order to better see the displacement of the tweezer. The scale bar corresponds to 10a. b-c, Evolution of the cloud for the two chiralities. The first line shows the average of the absorption images after an increasing evolution time, displayed with the same colorscale. The second line shows the result of the Gaussian fit that is performed on the averaged images. On these fits, the center of the Gaussian is indicated as a black dot. The error bar, which stems from a bootstrap analysis, is smaller than the marker. d, The absolute distance Δr between the center-of-mass positions of the time-evolved clouds with the two different chiralities is plotted as a function of the time. The five different markers correspond to the five initial positions, and the corresponding solid lines are linear fits. e, The slope of the fit is divided by a factor of two to obtain the average velocity of the edge mode. The measured velocity is plotted as a function of the initial position, and the final value that is selected is the maximum value of these points. The markers are the same as d. For the leftmost tweezer position the error bar is very large: the atoms are released on top of the potential step and no reliable velocity can be extracted.

Extended Data Fig. 7 Calibration of the height of the repulsive potential.

a, Measurement of the summed optical density in the regions corresponding to the zeroth (circles) and the first (diamonds) diffraction orders. The error bars correspond to the standard deviation of the population in the respective order for images taken with the same parameters. The solid lines show the sinusoidal fits from which the frequency is extracted. The inset shows an exemplary image, where the different regions of interest for evaluating the optical density are indicated (light blue box: first diffraction order; dark blue box: zeroth order). b, Height of the repulsive potential evaluated using sinusodial fits as shown in a as a function of the voltage used to control the laser intensity for three values of dsq: 1.0 μm (circles), 1.3 μm (diamonds) and 1.7 μm (pentagons). The error bars correspond to the uncertainty of the fit, and the dashed black line is a weighted linear fit of all the measured points, which has a slope of 11.12(3) kHz/V.

Extended Data Fig. 8 Optical setup to vary the edge width.

The DMD is imaged on the atomic plane (p2). An iris of variable diameter d is added in the Fourier plane to vary the width of the potential step. The DMD light at 638 nm is overlapped with the tweezer beam (tw.) at 1064 nm with a dichroic plate and focused with a microscope objective. The imaging beam (img.) at 767 nm is propagating in the opposite direction and passing through the dichroic plate to reach the camera (not shown).

Extended Data Fig. 9 Optical potential created by the DMD and edge width evaluation.

a, Optical potential in the intermediate plane (p1 in Extended Data Fig. 8) generated with the DMD and recorded with a camera. The diameter of the iris, and thus the Fourier plane, is slightly larger than the corresponding size of the objective. The gray rectangle indicates the area integrated for the curve shown in c. b, Same as a, but the diameter of the iris is closed as much as possible to a minimum diameter of 0.91(1) mm. The brown rectangle indicates the area integrated for the curve shown in c. The scale bar in a and b is 10a in the atomic plane p2. c, Resulting edge width. The edge width is extracted for a position close to the atomic cloud. We show the data of the integrated signal highlighted by the rectangles in a and b together with the corresponding fit. The iris diameter corresponds to a diameter slightly larger than the diffraction limit (a) and the smallest iris diameter from Fig. 4b (b).

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Braun, C., Saint-Jalm, R., Hesse, A. et al. Real-space detection and manipulation of topological edge modes with ultracold atoms. Nat. Phys. 20, 1306–1312 (2024). https://doi.org/10.1038/s41567-024-02506-z

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