Experimental observation of Aharonov-Bohm caging using orbital angular momentum modes in optical waveguides

The discovery of artificial gauge fields, controlling the dynamics of uncharged particles that otherwise elude the influence of standard electric or magnetic fields, has revolutionized the field of quantum simulation. Hence, developing new techniques to induce those fields is essential to boost quantum simulation in photonic structures. Here, we experimentally demonstrate in a photonic lattice the generation of an artificial gauge field by modifying the input state, overcoming the need to modify the geometry along the evolution or imposing the presence of external fields. In particular, we show that an effective magnetic flux naturally appears when light beams carrying orbital angular momentum are injected into waveguide lattices with certain configurations. To demonstrate the existence of that flux, we measure the resulting Aharonov-Bohm caging effect. Therefore, we prove the possibility of switching on and off artificial gauge fields by changing the topological charge of the input state, paving the way to access different topological regimes in one single structure, which represents an important step forward for optical quantum simulation.


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During the last decade, the growing interest in quantum simulation has fostered the development of several techniques to implement effective electromagnetic fields in systems of neutral particles 1,2 . In this vein, artificial gauge fields (AGF) have been widely used in photonics to control light dynamics [3][4][5] emulating the effect of electromagnetic fields on charged particles. Moreover, AGF have also allowed to explore a plethora of phenomena stemming from its close connection to topological phases of matter [6][7][8][9] (see Ozawa et al. 10 for a recent review). Typically, these AGF are introduced either by geometry manipulation 4,5 or by time-dependent modulation [11][12][13] . In contrast, we experimentally demonstrate that an AGF in the form of an effective magnetic flux can be induced using orbital angular momentum (OAM) states 14,15 . Specifically, we show how Aharonov-Bohm (AB) caging appears naturally when light beams carrying OAM 16 are injected into cylindrical optical waveguides arranged in a diamond chain configuration.
AB caging, which was originally studied in the context of two-dimensional electronic systems, is a single-particle localization effect arising as a consequence of the interplay between the lattice geometry and the magnetic flux. This phenomenon, which can be interpreted in terms of quantum interference 17 , has been predicted to occur 18-20 and experimentally verified 21,22 in photonic structures implementing AGF. Unlike the previous photonic proposals based on geometry manipulations 18-22 , we show in this work how non-zero energy flat bands, which are responsible for the caging effect, can be achieved naturally by injecting light carrying OAM. This enables to study the effects of AGF just by selecting the topological charge of the input light beam. Moreover, this method also allows to access different topological regimes without the need of fabricating different structures or employing high intensities, as it is the case for topological phase transitions realized via nonlinear optics 23 .
To experimentally visualize the AB caging effect induced by OAM modes, we fabricate photonic lattices composed of direct laser written optical waveguides 24 arranged in a 3 diamond chain configuration, as displayed in Fig. 1a. The unit cell is composed of three waveguides ( = , , ) forming a triangle with a central angle . Each cylindrical waveguide sustains OAM modes of the form 25 where ℓ = 0,1,2, … is the topological charge, ± accounts for positive and negative circulations of the phase front, ℓ ( ) is the radial mode profile given by the Bessel functions 16 , ( , ) are the polar coordinates with respect to the center of each waveguide in the transverse plane, is the propagation direction, 0 is an arbitrary phase origin, and ℓ is the propagation constant of mode ℓ. Besides, we restrict our implementation to ℓ = 0 and ℓ = 1 modes by properly engineering the refractive index contrast and the width of the step-index profile represented in Fig. 1b. While between ℓ = 0 modes there is only one coupling amplitude 0,0 ≡ 0 , between ℓ = 1 modes with equal or opposite circulations there are two coupling amplitudes 1,1 ≡ 1 and 1,−1 ≡ 2 2 0 , respectively 26 . Therefore, when dealing with OAM modes, complex coupling amplitudes between modes with different circulations appear naturally. The different coupling strengths 0 , 1 and 2 are represented in Fig. 1c (see Supplementary I for details on the calculations). Specifically, we set the phase origin 0 along the ⟷ direction such that 1,−1 = 2 is real in that direction, while 1,−1 = 2 2 0 = 2 − 2θ is complex along the ⟷ direction. In particular, we fix = /2, which allows to neglect the coupling between modes propagating in and since − = √2 and the coupling decays exponentially as shown in Fig. 1c. Moreover, for this specific angle, a relative phase difference of between the 1,−1 couplings in the ⟷ and ⟷ directions appears. This phase difference introduces a flux in the plaquettes that opens an energy gap between the dispersive bands, as discussed in detail as follows.
Assuming periodic boundary conditions, the bulk band structure for ℓ = 0 modes consists of one flat and two dispersive bands (see Fig. 2a), with energies given by 18 4 where is the quasi-momentum and √2 is the lattice constant. On the other hand, as represented in Fig. 2b, the band structure for ℓ = 1 is composed of six energy bands, i.e., three bands with a two-fold degeneracy (positive and negative circulations) 14 The main difference between the energy bands in both cases is the existence of an energy gap for ℓ = 1 which is absent for ℓ = 0, indicating the presence of an AGF. By performing a basis rotation (see Supplementary II), the original diamond chain can be decoupled into two identical chains with three energy bands and a flux through the plaquettes that opens the energy gap 14 . Moreover, as it is illustrated in Fig. 2c, in the 2 / 1 → 1 limit the dispersive bands ± 1 → ±2√2 1 become flat and its associated supermodes are localized in , , +1 , and +1 waveguides. Therefore, if one excites with a ℓ = 1 mode, the injected intensity will oscillate between the central and the four surrounding waveguides, as predicted by the AB caging effect (see Supplementary II for details).
To experimentally demonstrate AB caging using OAM modes, we excite a central waveguide using modes with and without OAM and compare the resulting dynamics.
We fabricate several samples with a total number of 7 unit cells with different total length (ranging from = 250 μm to = 1000 μm) and extract the output pattern intensities. A scheme of the sample is depicted in Fig. 3. First, as it is displayed in Fig In summary, we demonstrated that an artificial gauge field of the form of an effective magnetic flux can be induced in a photonic lattice by exploiting the orbital angular

Sample fabrication
The waveguide samples were fabricated via direct laser writing (DLW) 29 , using a commercial Nanoscribe system and the photo-resist IP-Dip. To create waveguides in a single writing step, the inside of waveguides was written with more laser power (60%) than the surrounding material (35%), which results in a refractive index contrast Δ of approximately 0.008. The used scan speed was 20 mm/s. Multiple samples were fabricated (each on its own substrate) with different total length corresponding to = 250 μm, 500 μm , 750 μm 1000 μm. We used a waveguide radius of = 1.9 μm and a center-to-center distance of = 5.5 μm. In contrast to common methods, where the sample is put in isopropanol after the writing to remove the non-polymerized resist, here, the sample was not developed. Excess resist on the sample output facet was seen to distort the images during measurements. Therefore, this resist was removed by carefully dabbing onto the sample facet with a tissue wetted in isopropanol.
During the writing process, the laser intensity towards the edges of the sample decreased due to vignetting of the writing objective lens. At the same time, the proximity effect 30 had less influence at the edges of the sample than in the center. Both processes led to a non-uniform refractive index profile of the sample, with higher index in the center and lower index at the edges. Preliminary results 24 led us to assume that the index does not increase linearly with the used writing power, but saturates for high powers below the destruction of the resist. As a result, the waveguides that were written with high laser power, were less prone to refractive index changes by vignetting and proximity effect than the material surrounding the waveguides (written with low laser power). The refractive index contrast between waveguides and surrounding material is therefore supposed to increase towards the edges of the sample. Due to this, the measurements were performed on the central waveguides ( 3 and 4 ).     We consider a unit cell of the diamond chain configuration formed by three identical cylindrical waveguides ≡ , , forming a triangle with central angle represented in Figure S1. Here, we focus on the subset of OAM modes with ℓ = 1 topological charge. In this subset, the coupling amplitude between modes with equal circulations is given by 1,1 ≡ 1 (black solid arrows in Figure S1), while the coupling amplitude between modes with opposite circulations is given by 1,−1 ≡ 2 2 0 = 2 − 2θ , with 0 being the phase origin 1 . Specifically, we set 0 along the ⟷ direction such that the coupling amplitude is real in the ⟷ direction and complex in the ⟷ direction, as represented by the blue dashed and red dotted arrows in Figure S1, respectively. Moreover, the coupling between and can be neglected for > /3 since the coupling amplitudes decay exponentially with the distance. Note that, for the evanescently coupled waveguides with a small index contrast here employed, the spin-orbit interaction can be neglected and it does not affect the light dynamics 2 . Therefore, light dynamics can be , ( 2) where ± is the propagation constant of mode ℓ = 1 with positive and negative circulations in waveguide . Note that + = − , and, since we consider identical waveguides, the diagonal elements can be factorized introducing a global phase into the dynamics. Imposing = /2 in (S2) and considering unit cells, coupled-mode equations (S1) read d ± d = 1 ± + 1 ( ± + +1 ± + ± + +1 ± ) + 2 ( +1 Finally, let us momentarily consider a unit cell with the in-line configuration ( = ), which allows to calculate the coupling strengths, 1 and 2 , between waveguides in a very convenient way, as described in the following. In this configuration, the symmetric | ⟩ and antisymmetric | ⟩ supermodes in the central waveguide read and it is straightforward to check that they are only coupled to |K j + ⟩ = 1 2 (| + ⟩ + | − ⟩ + | + ⟩ + | − ⟩) and |K j − ⟩ = 1 2 (| + ⟩ − | − ⟩ + | + ⟩ − | − ⟩), ( 5) with coupling strengths + = √2( 1 + 2 ) and − = √2| 1 − 2 |, respectively. Therefore, by injecting the symmetric (antisymmetric) supermode in waveguide , one can measure the beating length + = 2 + ⁄ ( − = 2 − ) ⁄ . From + and − the dependence of the 1 and 2 with respect to the distance between waveguides can be characterized, see Table S1.