Direct observation of photonic Landau levels and helical edge states in strained honeycomb lattices

We report the realization of a synthetic magnetic field for photons and polaritons in a honeycomb lattice of coupled semiconductor micropillars. A strong synthetic field is induced in both the s and p orbital bands by engineering a uniaxial hopping gradient in the lattice, giving rise to the formation of Landau levels at the Dirac points. We provide direct evidence of the sublattice symmetry breaking of the lowest-order Landau level wavefunction, a distinctive feature of synthetic magnetic fields. Our realization implements helical edge states in the gap between n = 0 and n = ±1 Landau levels, experimentally demonstrating a novel way of engineering propagating edge states in photonic lattices. In light of recent advances in the enhancement of polariton–polariton nonlinearities, the Landau levels reported here are promising for the study of the interplay between pseudomagnetism and interactions in a photonic system.


BANDS
shows a coloured version of the tight-binding band-structure used in Fig. 2(b) of the main text for the fit of the experimental data where the colour of each state is defined by the mean position of its wavefunction. Close to E s 0 and the Dirac points K and K , the wavefunction of the Landau level n = 0 is localised in the bulk of the lattice (black region in Fig. S2). On the contrary, the bands extending away from the Dirac points correspond to states localised at the left and right zigzag edges.   Figure S3 shows a coloured version of the tight-binding band-structure used in Fig. 3(b) and in Fig. 3(c) of the main text for the fit of the p bands where the colour of each state is defined by the mean position of its wavefunction.  In s bands (Fig. S4(c)), despite the strain being twice as large as in Fig. 2

V. ENERGY SPECTRUM OF LANDAU LEVELS IN P BANDS
To calculate the energy spectrum of Landau levels in the p bands, we assume that the hopping between p orbitals oriented perpendicularly to the link between adjacent micropillars to be zero 1 . Considering the hopping between orbitals oriented parallel to the links to be t 1 for horizontal links, and t 2 = t 3 ≡ t for angled links, and a lattice spacing a = 1, the momentum-space Hamiltonian is: The bare site energy is set to zero and the reduced Planck constant to unity ( = 1). When The eigenvalues of this Hamiltonian at the Dirac point are 0, 0, −3t/2, and 3t/2. The corresponding normalized eigenvectors, in this order, are : In order to write the Hamiltonian in terms of the basis of these eigenvectors, we introduce a unitary matrix : And then write the Hamiltonian in this basis as : To understand what happens around the Dirac point, we just need to focus on H 0 . Now letting t 1 be different from t, and expanding H 0 around the Dirac point as (k x , k y ) = (0, −4π/3 √ 3) + (q x , q y ), one obtains, up to linear order in (q x , q y ) : Writing this in a Dirac form v F [(q x + eA x )σ y + (q y + eA y )σ x )], we can read off that v F = 3t/4 and : If we assume that t 1 depends linearly on x coordinate, as in the experiment : one can find the expression of the pseudovector potential eA y = −2xτ /9 and the resulting pseudomagnetic field eB = −2τ /9.
The off diagonal elements of (6) can be expressed as the following operator : Using the canonical equation [x,q x ] = i, we have By definingâ = −V /(t √ τ /2), one can find the harmonic oscillator commutation relations Setting This eigenvalue equation can be solved in a way similar to the s band Hamiltonian in Ref. 2 .
The result is that the eigenvalues are relativistic Landau levels: where n is any natural number.

VI. SIMULATION OF PROPAGATION THROUGH HELICAL EDGE STATES
To confirm the presence of the p bands helical edge states in our system, we performed driven-dissipative simulations of polariton lattices with the parameters of two strained lattices studied in our experiments.
We consider the strained lattices coherently driven by a monochromatic resonant pump with strength F p and frequency ω p . The dissipation is considered uniform for all lattice sites with a rate of γ. We search the steady states of the equation: where H p is the real space tight-binding Hamiltonian of the p bands and ψ is the vector containing the polariton amplitude on each site. To probe the propagating edge state of the zeroth Landau level, the pump is placed on one pillar near the zigzag edges and ω p is set in between the n = 0 and n = 1 Landau level.   is the same as the one in lattice shown in Fig. 3(f,h) of the main text. The pump frequency is located in between the n = 0 and n = 1 Landau levels of the p-bands (ω p = ( p 0+ + p 1 )/3) and the dissipation rate is equal to γ = 0.2t (t = 0.17 meV). c,d Same calculations for τ = 0. For each panel, the pumped site is indicated by an arrow. The strength of the pump is equal on p x and p y sub-states.