Observation of edge solitons in photonic graphene

Edge states emerge in diverse areas of science, offering promising opportunities for the development of future electronic or optoelectronic devices, sound and light propagation control in acoustics and photonics. Previous experiments on edge states in photonics were carried out mostly in linear regimes, but the current belief is that nonlinearity introduces more striking features into physics of edge states, leading to the formation of edge solitons, optical isolation, making possible stable lasing in such states, to name a few. Here we report the observation of edge solitons at the zigzag edge of a reconfigurable photonic graphene lattice created via the effect of electromagnetically induced transparency in an atomic vapor cell with controllable nonlinearity. To obtain edge solitons, Raman gain is introduced to compensate strong absorption experienced by the edge state during propagation. Our observations may open the way for future experimental exploration of topological photonics on this nonlinear, reconfigurable platform.


Supplementary Note 2: Experimental arrangement for edge solitons with Raman gain
The experimental arrangement for generation of the edge solitons with Raman gain is shown in Supplementary Fig. 2(a). The EIT window is induced in the subsystem |1〉 → |3〉 → |2〉, in which we demonstrate the focusing/defocusing effects (Fig. 4 in the main text) of the excited edge wave, which experiences strong absorption during propagation. The description on the three-level subsystem is given in the Methods section. = − is the detuning between the atomic resonant frequency ( , = 1,2,3) and the laser frequency of ( = 1,2,3). : probe field (horizontal polarization); : Coupling field that is formed by interfering three coupling beams (vertical polarization); : pump beam (vertical polarization); S1: adjustable rectangular slit for making a stripe probe beam; L1: lens for imaging the stripe probe beam into the zigzag boundary of the coupling lattice; PBS: polarization beam splitter; S2: adjustable rectangular slit for cutting the zigzag boundary; L2: lens for imaging the output probe beam onto the CCD camera; BS: beam splitter for introducing the reference beam.
To form edge solitons, we add a Gaussian pump field (wavelength = 780.2 nm, frequency , vertical polarization, coupling the transition |1〉 → |4〉) to drive a four-level N-type atomic configuration [Supplementary Fig. 2(b)], which can provide an amplification for the probe beam to balance the intrinsic absorption of the atomic vapor. The vertically-polarized pump beam is injected into the medium along the same direction as one of the coupling beams. Actually, the pump beam can modulate the imaginary part of the refractive index (i.e. make it negative that corresponds to gain) without affecting the real part of the refractive index under certain parametric mechanisms [1]. So the honeycomb lattice "written" by the coupling field certainly persists, when Raman gain is effectively introduced. As a consequence, the probe field with Raman gain can excite the edge state and form edge solitons by taking advantages of the double balance between dispersion and Kerr nonlinearity as well as gain and loss.

Supplementary Note 3: Atomic density versus the temperature
With the atomic density increased from to , the equivalent optical path amounts to / × , where is the effective propagation length of probe beam at . The relation between temperature and atomic density is shown in Supplementary Fig. 3.
Supplementary Figure 3. The relationship between atomic density and temperature.

Supplementary Note 4: Theoretical model and numerical simulation details
Light propagation in the EIT atomic medium can be described by the Schrödinger-like paraxial wave where ∇ is the Laplacian operator, Δ is the refractive index change, = (2 )/ is the wavenumber, and the refractive index can be written as 1 + . In EIT atomic systems, the refractive index change is much smaller than 1 and can be directly written as Δ ≈ 0.5 . If we consider the first-order and third-order susceptibilities in the system, the total susceptibility can be written as tuning between the resonant transition frequency |1〉 → |3〉 (|2〉 → |3〉) and the frequency of field ( ); = | |/ℏ is the Rabi frequency for the coupling field; is the dipole momentum for transition | 〉 → | 〉; and are the spontaneous decay rates of the excited state |3〉 to the ground states |1〉 and |2〉, respectively; is the nonradiative decay rate between two ground states; and is the atomic density at the ground state |1〉.
In Supplementary Figs. 4(a,b), the profiles of the first-and third-order susceptibilities, which are honeycomb lattices with zigzag-bearded edges resulted from the three-beam interference method, are shown.
The used parameters are given in the caption. The order of ( ) is 10 and that of ( ) is By using the plane-wave expansion method and neglecting the nonlinear term in Supplementary Equation (4), one can obtain the corresponding band structure as shown in Supplementary Fig. 4(c) and edge states as shown in Supplementary Fig. 5. In Supplementary Fig. 4(c), the green curve indicates the edge state on the zigzag edge, while the red and blue curves correspond to the edge states on the bearded edge. Edge states corresponding to the red and green dots in Supplementary Fig. 4(c), where = 0.48 , are shown in Supplementary Figs. 5(a) and 5(b), respectively.
As to the nonlinear edge states, we seek for them in the region where > 0 with small . We take the edge state on the zigzag edge as our target. In Supplementary Fig. 5(c Since ( ) is dimensional, we introduce a dimensional scaling factor to ∼ . As a result, here is also dimensionless, and Supplementary Equation (4) can be rewritten as So, now we can use the Newton's method to seek for the nonlinear edge state based on Supplementary Equation (5). In Supplementary Fig. 6(a), we show the amplitude of the nonlinear edge state versus nonlinear energy . The red and black curves are obtained by setting the scaling factor = √2 × 10 mV and = √3 × 10 mV (which closely correspond to parameters (intensity levels) of the probe beams used in the experiment), respectively. One finds that the nonlinear edge states bifurcate from the linear energy ≈ 16.49 [shown by the green dot in Supplementary Fig. 4(c)]. We only find the nonlinear edge state in the band gap, so we stop iteration when the energy reaches ≈ 18.1. In Supplementary Fig. 6(b), we display the real-world amplitude of the nonlinear edge state (with dimension Vm ) versus . One finds that the real nonlinear edge state is the same under the two different scaling factors. In Fig. 3 in the main text, we choose = √3 × 10 mV .

Supplementary Note 5: Propagation dynamics of the edge wave in weakly nonlinear regime
In the absence of the third-order Kerr nonlinearity, the edge states considerably diffract along the edge during propagation [please refer to Fig. 3(g)

Supplementary Note 6: Interaction of edge solitons with lattice defects
To illustrate the robustness of the edge solitons in photonic graphene we studied their interactions with lattice defects. The defect was experimentally introduced by injecting a slim Gaussian beam (with the same frequency as the coupling beam and the diameter is ∼ 40 m) from another ECDL to cover one of the waveguides [ Supplementary Fig. 8(a)] at the edge. This operation is equivalent to removing one waveguide, and therefore a defect is created on the edge. As shown in Supplementary Fig. 8(b), the edge wave moves along one direction when there is no defect. However, when the defect is introduced, the motion direction of the edge state is clearly reversed when it meets the defect, while the shape of the state remains practically unchanged [ Supplementary Fig. 8(c)]. This is because solitons considered here are not topological, so they may bounce back when they hit the defect upon propagation. We also provide a theoretical illustration of such interaction in Supplementary Fig. 9.

Supplementary Note 7: Transition from linear diffraction pattern to soliton-like profile by changing input power
To illustrate the impact of the input power of the probe beam on propagation dynamics in the presence of losses, in Supplementary Fig. 10 we show the output edge state profiles for different input powers for the case of  1 135 MHz and for T115C. If the input power P is lower than approximately 200 µW, one can observe clear discrete diffraction with formation of the minimum (indicated by the green arrow) in the center of the output profile, so in this case the probe beam propagates in the effectively linear regime.
However, when the input power exceeds approximately 300 µW, the diffraction is clearly suppressed and one does not observe the dip in the center of the beam anymore. At this frequency detuning ( 1 135 MHz), the output intensity distributions exhibit only minor modification (except for growth of the output peak intensity) when the input power further increases from 300 µW to 600 µW. This indicates a clear transition from linear diffraction to the formation of self-sustained states, which indeed dynamically self-adjust their profiles upon propagation through the vapor cell under the unavoidable losses. Such self-adjusting states, closely resembling the states shown in Fig. 4(c) at 400 µW, clearly propagate in the nonlinear regime and even though they cannot be rigorously called solitons (in the sense adopted in conservative systems), they would propagate as solitonic objects if losses are switched off at certain distance.