Jörg et al. Light: Science & Applications (2020)9:150 Official journal of the CIOMP 2047-7538 https://doi.org/10.1038/s41377-020-00385-6 www.nature.com/lsa LETTER Open Access Artificial gauge field switching using orbital angular momentum modes in optical waveguides Christina Jörg1, Gerard Queraltó 2, Mark Kremer 3, Gerard Pelegrí2,4, Julian Schulz1, Alexander Szameit 3 , Georg von Freymann1,5, Jordi Mompart2 and Verònica Ahufinger2 Abstract The discovery of artificial gauge fields controlling the dynamics of uncharged particles that otherwise elude the influence of standard electromagnetic fields has revolutionised the field of quantum simulation. Hence, developing new techniques to induce these fields is essential to boost quantum simulation of photonic structures. Here, we experimentally demonstrate the generation of an artificial gauge field in a photonic lattice by modifying the topological charge of a light beam, overcoming the need to modify the geometry along the evolution or impose external fields. In particular, we show that an effective magnetic flux naturally appears when a light beam carrying orbital angular momentum is injected into a waveguide lattice with a diamond chain configuration. To demonstrate the existence of this flux, we measure an effect that derives solely from the presence of a magnetic flux, the Aharonov- Bohm caging effect, which is a localisation phenomenon of wavepackets due to destructive interference. Therefore, we prove the possibility of switching on and off artificial gauge fields just by changing the topological charge of the input state, paving the way to accessing different topological regimes in a single structure, which represents an important step forward for optical quantum simulation. 1234567890():,; 1234567890():,; 1234567890():,; 1234567890():,; During the last decade, the growing interest in quantum edge states in crystalline topological insulators, here, we simulation has fostered the development of several tech- experimentally demonstrate that an AGF in the form of niques for implementing effective electromagnetic fields an effective magnetic flux can be induced using in systems of neutral particles1,2. In this vein, artificial Laguerre–Gauss light beams carrying OAM15. Specifi- gauge fields (AGFs) have been widely used in photonics to cally, to prove the existence of this flux, we show how control light dynamics3–5, emulating the effect of elec- Aharonov-Bohm (AB) caging naturally appears when tromagnetic fields on charged particles. Moreover, AGFs OAM modes with a specific topological charge are have also allowed the exploration of a plethora of phe- injected into cylindrical optical waveguides arranged in a nomena stemming from their close connection to topo- diamond chain configuration16,17. logical phases of matter6–9 (see Ozawa et al.10 for a recent AB caging, which was originally studied in the context review). Typically, these AGFs are introduced either by of two-dimensional electronic systems, is a single-particle geometric manipulation4,5 or by time-dependent mod- localisation effect arising from the interplay between the ulation11–13. While in Wu et al.14, wavepackets carrying lattice geometry and a magnetic flux. More specifically, a orbital angular momentum (OAM) were used to create constant magnetic flux modifies the phase relations of wavepackets, resulting in a destructive interference effect that binds the modes. Thus, it enables one to halt all Correspondence: Gerard Queraltó (gerard.queralto@uab.cat) 1 Physics Department and Research Center OPTIMAS, Technische Universität propagation by controlling the flux. This phenomenon, Kaiserslautern, 67663 Kaiserslautern, Germany 2 which can be interpreted in terms of quantum inter- Departament de Física, Universitat Auto’noma de Barcelona, E-08193 ference18,19, has been predicted to occur20–22 and Bellaterra, Spain Full list of author information is available at the end of the article experimentally verified23,24 in photonic structures These authors contributed equally: Christina Jörg, Gerard Queraltó © The Author(s) 2020 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Jörg et al. Light: Science & Applications (2020)9:150 Page 2 of 7 implementing AGFs. Unlike the previous photonic pro- form28 posals based on geometric manipulation20–22, we show in   this work how non-zero-energy flat bands, which are     ± i‘ ϕSj ϕ0 ΨS±j ‘ rSj ; ϕSj ; z ¼ ψ ‘Sj r Sj e eiβ‘ z ð1Þ responsible for the caging effect, can be naturally and deliberately achieved by injecting light carrying OAM instead of fabricating a new sample. Therefore, our pro- where ‘ ¼ 0; 1; 2; ¼ is the topological charge, ± accounts posal enables the study of the effect of AGFs in photonic for positive  and negative circulation of the phase front, lattices just by selecting the topological charge of the ψ ‘Sj rSj is the radial mode profile given by the Bessel input beam. In this context, our proposal differs from functions15, ðrSj ; ϕSj Þ are the polar coordinates with related works where the intrinsic angular momentum, i.e., respect to the centre of each waveguide Sj in the the polarisation of the input beam, instead of the extrinsic transverse plane, z is the propagation direction, ϕ0 is an one, i.e., the OAM, was used as the AGF switching arbitrary phase origin, and β‘ is the propagation constant mechanism25. Moreover, this method also allows access to of mode ‘. Moreover, while between fundamental modes different topological regimes without the need to fabricate ð‘ ¼ 0Þ, there is only one coupling amplitude c0;0  c0 , different structures or employ high intensities, as is the between OAM modes ð‘ ≠ 0Þ with the same or opposite case for topological phase transitions realised via non- circulation directions, there are two coupling amplitudes linear optics26. c‘;‘  c1 and c‘;‘  c2 ei2‘ϕ0 29. In particular, as a proof of To experimentally visualise the AB caging effect concept, we restrict our implementation to the ‘ ¼ 0 and induced by OAM modes, we fabricate photonic lattices ‘ ¼ 1 modes by properly engineering the refractive index composed of direct laser written optical waveguides27 contrast and the width of the step-index profile presented arranged in a diamond chain configuration, as displayed in Fig. 1b. In this case, between the ‘ ¼ 1 modes with the in  Fig. 1a. The unit cell j is composed of three waveguides same or opposite circulation directions, there are two Sj ¼ Aj ; Bj ; Cj forming a triangle with a central angle θ. coupling amplitudes c1;1  c1 and c1;1  c2 ei2‘ϕ0 29. Each cylindrical waveguide sustains OAM modes of the Therefore, when dealing with OAM modes, complex a Cj Cj + 1 b Unit d 1.548 cell 0/k0 n 1.544 θ Aj Aj + 1 1/k0 1.540 Bj Bj + 1 –5 0 R 5 r (μm) z c 5 2 4 c2/c1 c (mm–1) 3 1 y 2 5 10 15 d (μm) 1 0 x 5 6 7 8 9 10 d (μm) Fig. 1 Lattice structure and optical waveguides. a Schematic representation of the structure composed of identical cylindrical waveguides arranged in a diamond chain configuration. Each unit cell j hosts three waveguides sj  Aj ; Bj ; Cj forming a triangle with central angle θ. The distances between waveguide centres are dAj Bj ¼ dAj Cj  d, dBj Cj ¼ 2d sinðθ=2Þ and dAj Ajþ1 ¼ 2d cosðθ=2Þ. The blue arrows indicate the couplings. b Refractive index profile of the waveguides, defined by ncore = 1.548, nclad = 1.540 and waveguide radius R = 1.9 μm. Field intensity of the ‘ ¼ 0 (green) and ‘ ¼ 1 (red) modes, where β‘ is the propagation constant of mode ‘, k0 = 2π/λ0 is the vacuum wavenumber and λ0 is the light wavelength in vacuum. c Numerically calculated coupling strengths for separation distances d = 5 μm, 5.5 μm, 6 μm, 6.5 μm, 7 μm and 7.5 μm using λ0 = 700 nm. In particular, c0 (crosses) accounts for the coupling between the ‘ ¼ 0 modes, and c1 (circles) and c2 (squares) account for the coupling between the ‘ ¼ 1 modes with the same or opposite circulation directions, respectively. The dashed and solid lines correspond to the exponential fittings of c0 ðdÞ  K0 expðκ 0 dÞ, c1 ðdÞ  K1 expðκ 1 dÞ and c2 ðdÞ  K2 expðκ 2 dÞ, where K0 = 387 mm−1, κ0 = 1.17 μm−1, K1 = 19.39 mm−1, κ1 = 0.52 μm−1, K2 = 56.25 mm−1 and κ2 = 0.59 μm−1. The inset in c shows c2/c1 with respect to the separation distance d Jörg et al. Light: Science & Applications (2020)9:150 Page 3 of 7 coupling amplitudes between modes with different negative circulation)16 circulation directions appear naturally. The different coupling strengths c0 ; c1 and c2 are presented in Fig. 1c E01 ðk Þ ¼ 0; E 1± ðk Þ (see Supplementary I for details on the calculations). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ffi ð3Þ Specifically, we set the phase origin ϕ0 along the Aj $ Cj ¼ ± 2 ðc21 þ c22 Þ þ ðc21  c22 Þcosðk 2dÞ direction such that c1;1 ¼ c2 is real in this direction, while c1;1 ¼ c2 ei2‘θ is complex along the Aj $ Bj The main difference between the energy bands in the direction. In particular, we fix θ = π/2, which allows the two cases is the existence of an energy gap for ‘ ¼ 1, coupling between modes propagating in next-nearest which is absent for ‘ ¼ 0, indicating the presence of an neighbour waveguides to be neglected30 (see Supplemen- AGF. By performing a basis rotation (Supplementary III), tary II for a detailed discussion). Moreover, for this the original diamond chain can be decoupled into two specific angle, a relative phase difference of π between the identical chains with three energy bands and a π flux c1;1 couplings in the Aj $ Cj and Aj $ Bj directions through the plaquettes that opens the energy gap16. appears. This phase difference introduces a π flux into the 2c,ffiffiffi in the c2 =c1 ! 1 limit, Moreover, as illustrated in Fig. p plaquettes that opens an energy gap between the the dispersive bands E 1± ! ± 2 2c1 become flat, and the dispersive bands, as discussed in detail in the following. associated supermodes are localised in the Aj ; Bj ; Bjþ1 ; Cj Assuming periodic boundary conditions, the bulk band and Cjþ1 waveguides. Therefore, if one excites Aj with a structure for the ‘ ¼ 0 modes consists of one flat and two ‘ ¼ 1 mode, then the injected intensity will oscillate dispersive bands (Fig. 2a), with energies given by20 between the central and four surrounding waveguides, as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi predicted by the AB caging effect (Supplementary III). pffiffiffi ffi To experimentally demonstrate AB caging using OAM E00 ðk Þ ¼ 0; E 0± ðk Þ ¼ ± 2c0 1 þ cosðk 2dÞ ð2Þ modes, we excite a central waveguide Aj using modes with pffiffiffi and without OAM and compare the resulting dynamics. where k is the quasi-momentum and 2d is the lattice We fabricate several samples with seven unit cells with constant. On the other hand, as presented in Fig. 2b, the different total lengths (ranging from z = 250 μm to band structure for ‘ ¼ 1 is composed of six energy bands, z = 1000 μm) and extract the output pattern intensities. A i.e., three bands with a twofold degeneracy (positive and scheme of the samples is depicted in Fig. 3. First, as a 4 E +0 E 00 E/c0 0 E –0 –4 –π –π/2 0 π/2 π k 2d b c 4 E +1 E +1 E 01 E 01 E/c1 0 E –1 E –1 –4 –π –π/2 0 π/2 π –π –π/2 0 π/2 π k 2d k 2d Fig. 2 Energy band structure. a Band structure of the considered diamond chain lattice for ‘ ¼ 0 consisting of two dispersive bands E0 ðkÞ and Eþ0 ðkÞ (dotted green lines) and one zero-energy flat band E00 ðkÞ (solid red line). Band structure of the considered diamond chain lattice for ‘ ¼ 1 when b c2/c1 = 2 and c c2/c1 = 1. In b and c, each band has a two-fold degeneracy, i.e., E1 ðkÞ  E1 ðk Þ ¼ E2 ðkÞ (dashed line), E01 ðkÞ  E3 ðkÞ ¼ E4 ðk Þ 1 (solid line) and Eþ ðkÞ  E5 ðkÞ ¼ E6 ðk Þ (dashed line) Jörg et al. Light: Science & Applications (2020)9:150 Page 4 of 7 a f C1 C7 C1 C7 A1 A7 0 µm A1 A7 B1 B7 B1 B7 5 μm 5 μm b g 250 µm 5 μm 5 μm c h 500 µm 5 μm 5 μm d i 750 µm 5 μm 5 μm e j 1000 µm 5 μm 5 μm 0 0.25 0.5 0.75 1 Intensity in arb. units Fig. 3 Aharonov-Bohm caging effect. Experimentally observed input and output intensities obtained by exciting the A4 waveguide using the OAM mode with ‘ ¼ 1 and negative circulation at a z = 0 μm, b z = 250 μm, c z = 500 μm, d z = 750 μm and e z = 1000 μm and with ‘ ¼ 0 at f z = 0 μm, g z = 250 μm, h z = 500 μm, i z = 750 μm and j z = 1000 μm. Note that the image in a is taken before entering the sample. The diamond chain lattice is composed of seven unit cells, i.e., 21 waveguides with radius R = 1.9 μm and nearest-neighbour separation d = 5.5 μm. The wavelength used is λ0 = 700 nm. The intensity distribution in each figure is normalised to the maximum intensity value of the corresponding figure displayed in Fig. 3a, we inject a mode with ‘ ¼ 1 and caging is the same for any superposition of both types of negative circulation into A4 (see Supplementary IV for circulation, i.e., a lobe-shaped mode (Supplementary III). complementary results). The injected mode spreads to the In contrast, the ‘ ¼ 0 mode injected into A4 only spreads four surrounding waveguides at z = 250 μm (Fig. 3b) and transversally as it evolves along the propagation direction, recombines in the central waveguide at z = 500 μm (Fig. and no caging is observed in Fig. 3f–j. 3c). This spreading and recombination effect can be Finally, we compare the experimental observations of observed a second time at 750 μm (Fig. 3d) and 1000 μm the light dynamics with numerical calculations. Figure 4 (Fig. 3e). Even though we implement the model with shows the intensity extracted at the output port from the c2 =c1  2 due to experimental restrictions on the total A4 waveguide and its associated cage formed by size of the samples, we measure two full oscillations of the A4 ; B4 ; C4 ; B5 ; C5 . In Fig. 4a, we can observe how the AB caging effect. Since the dispersive bands are not totally experimentally measured intensity maxima in A4 asso- flat, light propagates into waveguides A3 and A5 during ciated with the caging phenomenon occur around the second oscillation, and part of the intensity escapes z = 500 μm and 1000 μm, in agreement with finite- from the cage (Fig. 3d, e). Additionally, although we try to difference method (FDM) simulations. On the other excite the donut mode with negative circulation (see the hand, in Fig. 4b, one can observe the standard decay of the input beam in Fig. 3a), the propagating mode has a lobe- intensity in A4 when the ‘ ¼ 0 mode is injected. More- shaped intensity (Fig. 3b–e) corresponding to a super- over, we also compute the light dynamics for longer dis- position of donut modes with positive and negative cir- tances using coupled-mode equations (Supplementary I culation. This lobe-shaped mode appears due to a slight and II). In Fig. 4c, one can observe how for ‘ ¼ 1, the first ellipticity of the fabricated waveguides and the influence and second intensity maxima in A4 have ~60% and 10% of of the surrounding waveguides (see Supplementary IV for the injected intensity, respectively, which can be increased a complementary discussion). Nevertheless, since the by reducing the difference between c1 and c2 (see inset of propagation of the ‘ ¼ 1 modes with positive and negative Fig. 1c). For example, for c2 =c1  1:25; i.e., d = 15 μm, the circulation results in the same flux, the observed AB first and second maxima increase up to 97% and 80%, Jörg et al. Light: Science & Applications (2020)9:150 Page 5 of 7 a 1.2 b Cage (Exp.) A4 (Exp.) Cage (Exp.) 1.0 Cage (Sim.) A4 (Sim.) Cage (Sim.) A4 (Exp.) I (arb. units) 0.8 A4 (Sim.) 0.6 0.4 0.2 0 0 0.25 0.5 0.75 1.0 1.25 0 0.25 0.5 0.75 1.0 1.25 z (mm) z (mm) c d 1.0 d = 5.3 μm d = 5.3 μm d = 15 μm d = 15 μm 0.8 I (arb. units) 0.6 0.4 0.2 0 0 1 2 3 4 0 1 2 3 4 c1z c0z Fig. 4 Light dynamics along the propagation direction. Intensity extracted from waveguide A4 (circles) and from the cage formed by A4 ; B4 ; C4 ; B5 ; C5 (squares) normalised to the intensity extracted from the entire lattice as a function of the propagation distance z when the a ‘ ¼ 1 mode and b ‘ ¼ 0 mode is injected into waveguide A4. The results shown in a are an average of the intensities extracted for ‘ ¼ 1 with positive and negative circulation. The circles and squares correspond to the experimentally extracted intensities, while the lines correspond to the best-fitting curve of the simulated results obtained using FDM numerical techniques. The error bars associated with the experimental data are estimated taking into account a refractive index error of Δn = ±0.001 in the fabrication process. Intensity propagating in waveguide A4 numerically calculated using coupled-mode equations as a function of z when the c ‘ ¼ 1 mode and d ‘ ¼ 0 mode is injected into waveguide A4. The solid lines correspond to the case with d = 5.3 μm, i.e., c2 =c1  2, while the dashed lines correspond to d = 15 μm, i.e., c2 =c1  1:25. Note that the simulations were performed considering N = 7 unit cells and λ0 = 700 nm with a correction of Δd = –0.2 μm with respect to the expected experimental distance d = 5.5 μm. This difference may originate from slight variations in the position during the writing process (±0.05 μm) and small changes in the refractive index contrast respectively, achieving 100% in the flat-band limit. However, direct laser written cylindrical waveguides in a diamond larger separations between waveguides require longer chain configuration. Using this structure, we showed how samples, which were not feasible in our experiments. an energy gap is opened between the dispersive bands of Alternatively, for ‘ ¼ 0, the intensity in A4 exponentially the system when light carrying OAM is injected, analo- decays independent of the waveguide separation, confirm- gous to the effect produced by an artificial gauge field20. ing the different origins of the oscillations. Finally, note that Moreover, we proved how non-zero-energy flat bands, the agreement between the experimental results obtained which yield the AB caging effect, can be achieved by with different samples and waveguides and injecting modes properly tuning the geometry of the unit cells and the with both types of circulation shown in Supplementary IV separation between waveguides. The agreement between confirms the robustness of the AB caging effect since each the dynamics revealed by the coupled-mode equations, measurement includes slight parameter variations (see the FDM simulations and the experiments confirms the Supplementary V for more details). validity of the presented model, which constitutes a step In summary, we demonstrated that an artificial gauge towards accessing different topological regimes in an field in the form of an effective magnetic flux could be active way by controlling the input states. Moreover, the induced in a photonic lattice by exploiting the orbital inherently infinite dimensionality of OAM modes15 can angular momentum carried by light beams. Specifically, be potentially exploited to increase the transmission we demonstrated the appearance of this synthetic flux by capacity by using mode multiplexing31, paving the way experimentally measuring the photonic analogue of the towards combining integrated spatial multiplexing32 with Aharonov-Bohm caging effect for an arrangement of topological protection10. Jörg et al. Light: Science & Applications (2020)9:150 Page 6 of 7 Materials and methods negative circulation or a Gaussian-shaped intensity pro- Sample fabrication file with ‘ ¼ 0 and constant phase (depending on the The waveguide samples were fabricated via direct laser hologram that we loaded). writing (DLW)33 using a commercial Nanoscribe system The reflection of the input mode was imaged via a and the photoresist IP-Dip. To create waveguides in a beamsplitter onto camera 1. The use of white light from a single writing step, the inside of waveguides was written common torch lamp allowed additional imaging of the with higher laser power (60%) than the surrounding sample input facet onto camera 1 at the same time to material (35%), which resulted in a refractive index con- overlay the input mode with the waveguide position. The trast Δn of ~0.008. The scan speed used was 20 mm/s. output intensity at the sample output facet was imaged Multiple samples were fabricated (each on its own onto camera 2. The intensity distributions for the differ- substrate) with different total lengths corresponding to ent outputs were normalised to the maximum value to z = 250 μm, 500 μm, 750 μm and 1000 μm. We used a increase the visibility. Moreover, the recorded images waveguide radius of R = 1.9 μm and a centre-to-centre were post-processed to reduce noise. This was achieved distance of d = 5.5 μm. In contrast to common methods, by overlaying the pictures with a mask of the waveguide where the sample is placed in isopropanol after writing to structure at the position determined by a convolution. In remove the non-polymerised resist, here, the sample was this way, the intensities within the waveguides and in the not developed. Excess resist on the sample output facet surroundings were separated. The noise level of the sur- was observed to distort the images during measurements. roundings was then subtracted from the original recorded Therefore, this resist was removed by carefully dabbing picture. All resulting negative values were set to zero. To the sample facet with a tissue wetted by isopropanol. extract the intensities shown in Fig. 4, we subsequently During the writing process, the laser intensity towards integrated over a circle that covered almost the whole the edges of the sample decreased due to vignetting of the mode at the position of each waveguide. The circles were writing objective lens. At the same time, the proximity as large as possible such that they touched at the effect34 had less influence at the edges of the sample than diagonals. in the centre. Both phenomena led to a non-uniform Acknowledgements refractive index profile of the sample, with a higher index G.Q., J.M., and V.A. acknowledge financial support from the Spanish Ministry of in the centre and a lower index at the edges. Preliminary Science and Innovation MICIU (contract no. FIS2017-86530-P) and Generalitat results27 led us to assume that the index does not increase de Catalunya (contract no. SGR2017-1646). A.S. thanks the Deutsche Forschungsgemeinschaft for funding this research (grants BL 574/13-1, SZ 276/ linearly with the used writing power but saturates for high 15-1 and SZ 276/20-1). G.v.F. acknowledges support by the Deutsche powers below the threshold for resist destruction. As a Forschungsgemeinschaft through CRC/ Transregio 185 OSCAR (project result, the waveguides written with a high laser power number 277625399). were less prone to refractive index changes by vignetting Author details and the proximity effect than the material surrounding 1 Physics Department and Research Center OPTIMAS, Technische Universität the waveguides (written with a low laser power). The Kaiserslautern, 67663 Kaiserslautern, Germany. 2Departament de Física, refractive index contrast between the waveguides and Universitat Auto’noma de Barcelona, E-08193 Bellaterra, Spain. 3Institut für Physik, Universität Rostock, Albert-Einstein-Straße 23, 18059 Rostock, Germany. surrounding material is therefore supposed to increase 4 Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, towards the edges of the sample. Therefore, the mea- UK. 5Fraunhofer Institute for Industrial Mathematics ITWM, 67663 surements were performed on the central waveguides (A3 Kaiserslautern, Germany and A4). Author contributions G.Q., G.P., C.J., and M.K. developed the theory. C.J. and J.S. fabricated the Measurement samples and performed the measurements. A.S., G.v.F., J.M. and V.A. supervised the project. All authors discussed the results and co-wrote the paper. The full setup can be seen in Supplementary VI. Laser light from a white light laser (NKT photonics) Data availability was sent through a VARIA filter box to select a wave- All experimental data and any related experimental background information length of 700 nm. The beam was linearly polarised, not mentioned in the text are available from the authors upon reasonable request. expanded and sent to a spatial light modulator (SLM). We loaded a hologram onto the SLM that consisted of a Conflict of interest phase-only vortex, with an added blazed grating to shift The authors declare that they have no conflict of interest. the pattern to the first diffraction order. Other orders Supplementary information is available for this paper at https://doi.org/ were blocked by a pinhole. The beam was circularly 10.1038/s41377-020-00385-6. polarised and imaged onto an objective lens, which Fourier transformed the phase hologram to create a Received: 14 April 2020 Revised: 26 July 2020 Accepted: 11 August 2020 donut-shaped intensity profile with ‘ ¼ 1 and positive/ Jörg et al. Light: Science & Applications (2020)9:150 Page 7 of 7 References 18. Vidal, J., Mosseri, R. & Douçot, B. Aharonov-Bohm cages in two-dimensional 1. Dalibard, J. et al. Colloquium: artificial gauge potentials for neutral atoms. Rev. structures. Phys. Rev. Lett. 81, 5888–5891 (1998). Mod. Phys. 83, 1523–1543 (2011). 19. Vidal, J. et al. 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