Experimental observation of bulk and edge transport in photonic Lieb lattices

We analyze the transport of light in the bulk and at the edge of photonic Lieb lattices, whose unique feature is the existence of a flat band representing stationary states in the middle of the band structure that can form localized bulk states. We find that transport in bulk Lieb lattices is significantly affected by the particular excitation site within the unit cell, due to overlap with the flat band states. Additionally, we demonstrate the existence of new edge states in anisotropic Lieb lattices. These states arise due to a virtual defect at the lattice edges and are not described by the standard tight-binding model.


Introduction
A 'flat band' is a dispersionless energy band composed of entirely degenerate states. As a result of this degneracy, any superposition of these states (even a highly localized one) is completely static, displaying no evolution dynamics whatsoever. Flat bands are manifested in a variety of condensed matter contexts, including ferromagnetic ground states of Hubbard models [1], frustrated hopping models [2], in the context of localization in the presence of magnetic fields or spin-orbit coupling [3], and the fractional quantum Hall effect [4]. With the advent of the recent experimental observations of photonic topological insulators [5,6], topological flat band models are now ripe for study in the optical domain. Indeed in optics, flat bands have direct technological relevance due to their high degeneracy, and thus high photonic density-of-states, leading to enhanced light-matter interaction [7]. Additionally, for Kagome systems single peak solitons have shown to bifurcate from the flat band at zero power threshold [8], and a non-diffractive image transmission scheme have been suggested in the linear regime [9], showing the chance for concrete applications emerging from the flat band properties.
In the Lieb lattice -a line-centered square lattice -a flat band touches two linearly dispersing intersecting bands, i.e., the flat band intersects a single Dirac point. In the low-energy regime, the dispersion is described by a quasi-relativistic equation for spin-1 fermions [10]. It is manifested in, e.g., the high-temperature tetragonal phases of the so-called cuprates: layered, ceramic materials that exhibit the phenomenon of high-temperature superconductivity [11]. In this particular system several studies were conducted only recently, experimentally probing its response to external gauge fields [12] to nematic ferromagnetism [13] and nonlinear conical diffraction [14]. The flat-band zero-energy edge modes in this lattice exhibit various peculiar features [15,16,17] and, hence, deserve particular attention.
Here, we analyze transport in Lieb lattices theoretically and experimentally, in the bulk and at the edges. We demonstrate that whether or not flat-band modes are excited strongly impacts the transport in the bulk of the lattice. Additionally, we demonstrate the presence of Tamm-like edge states that arise at a van Hove singularity in the band structure [18]. Finally, we show that our photonic realization holds promise for a thorough analysis of various features of the Lieb lattice in terms of transport at and far off the lattice edges.

Bulk transport
The structure of the Lieb lattice is shown in Fig. 1(a), where it is evident that the unit cell consists of three sites A, B, and C. As in honeycomb lattices (also known as "photonic graphene") [19], the equation describing the propagation of light in this system is where z is the longitudinal propagation distance into the photonic lattice (playing the role of time in the analogous Schrödinger equation); ψ is the envelope of the electric field, defined by E(x, y, z) = ψ(x, y, z)e i(k 0 z−ωt) (E is the electric field, k 0 the wavenumber in free space, ω = ck 0 /n 0 and c is the speed of light in vacuum); ∆n(x, y) is the refractive index structure defining the photonic lattice, while n 0 is the refractive index of the material in which the lattice is embedded; ∇ 2 ⊥ = ∂ 2 x + ∂ 2 y is the transverse Laplacian operator; H cont as defined in equation (1) corresponds to the continuum Hamiltonian for wave propagation in a photonic lattice. When the refractive index structure is composed of highly confined waveguides, each with a single bound state, we employ a tight-binding approximation. Since within a given unit cell hopping takes place only between A-B and B-C nearest neighbors, the corresponding tight-binding Hamiltonian reads with a † n , b † n , c † n as the creation operators in the n-th unit cell on the A, B and C sites, respectively. The summation takes place over R n (the position of the unit cell), and δ j (the vectors connecting the neighbors via the inter-site coupling κ x,y in the x-and y-direction, respectively). In Fourier space, this becomes with k x,y ∈ [−π/2, π/2] (first Brillouin zone). From this, one readily obtains the dispersion relation, i.e., the spectrum The full spectrum (shown in Fig. 1(b) for κ x = κ y = 1) consists of three bands: a flat band (β(k) = 0) and two conically intersecting bands, with all three bands intersecting at the M point (k x = k y = π/2). Note that for each eigenstate at Bloch wavevector k in the top band with eigenvalue β, there exists an eigenstate also at k with eigenvalue −β; this is known as particle-hole symmetry [20]. The eigenfunctions of the finite Lieb lattice with periodic and vanishing boundary conditions are given in Ref. [15]. Importantly, any superposition of eigenstates belonging to the flat band is also an eigenstate of the system, meaning that it is static and does not contribute to transport. Equivalently, the group velocity in the flat band is zero at all Bloch wave vectors, therefore any wavepacket is necessarily diffraction-free. Consequently, to have transport, states in the other bands have to be excited. We also note that the flat band is not destroyed by any anisotropy κ x = κ y , being an intrinsic property of this lattice in the nearest-neighbor coupling limit [14].
In general, linear modes of any periodic structure are completely extended. However, flat band systems allow the formation of very localized eigenmodes as a destructive linear combination of extended linear wavefunctions [27]. In the case of the Lieb lattice, in any square of the lattice (formed by four B sites at the corners, two C sites at the upper and lower sides and two A sites at the right and left sides of the square) a stationary localized mode can be created by taking the four B sites strictly zero, the two C sites with an amplitude ψ C , while the last two A with an amplitude ψ A = −κ y ψ C /κ x and the rest of lattice sites are zero.
For our experiments, we fabricate Lieb lattices employing the laser direct-write technology [21]. To this end, ultrashort laser pulses (Coherent Inc. Mira/RegA, λ = 800nm, t p ≈ 200fs at 100kHz repetition rate) are tightly focused through a microscope objective (25×, NA = 0.35) into a 10cm long transparent fused silica wafer and translated by means of a high-precision positioning system (Aerotech Inc.). A microscope front view of such a lattice, which consists of 341 waveguides with nearestneighbor spacing of 20µm, is shown in Fig. 1(c). Note that due to the slightly elliptical form of our waveguides, we find that the vertical inter-site coupling is slightly stronger than the horizontal one, i.e. the lattice is effectively uniaxially strained and anisotropic with κ y ≈ 1.5κ x . In our experiments, laser light at λ = 637 nm was launched into a single waveguide at the input facet of the array using a standard microscope objective. At the output facet of the sample, intensity patterns were recorded with a CCD camera.
We begin by launching light into the bulk of the lattice, i.e., far away from the edges. As there are three elements per unit cell (see Fig. 1a), we study the propagation arising from excitations in each of the three positions A, B, and C. The corresponding diffraction patterns obtained in our experiments are shown in Figs. 2(a),(c),(e). One can clearly see that the impact of the excitation location on the diffraction pattern is significant, depending on where in the unit cell the lattice is excited. The reason lies in the unique structure of the Lieb lattice, where a single site excitation (that corresponds to a unit impulse in real space) does not excite all bands -in contrast to other lattices, like square, hexagonal or honeycomb lattices. To illustrate this effect, we plot the projections P = k, j|ψ excit of the impulse excitation |ψ excit at the A, B, or C sites on the eigenfunctions |k, j of the three bands (j is the band index) in   Fig. 2(e)], the projections on the flat band are minimal (equal to 1/(N + 1) 2 , where N is the number of complete unit cells on one side of a finite square Lieb lattice, i.e., N = 10 in our case) and, hence, transport is strong. This is clearly observed in the experiment [see 2(b)], where the light spreads across many lattice sites following the 10 cm propagation in the lattice. However, when exciting the A site the situation changes drastically [ Fig. 2(d)]. In this case, the projections on the flat band dominate and, therefore, the transport is minimal. The experiment shows exactly this: the spreading is very weak, and the light covers only a few lattice sites after propagating through the lattice [see Fig. 2(a)]. Intermediate spreading is observed when launching light into lattice site C, as shown in Fig. 2(c), as a consequence of the partial overlap with the flat band [see Fig. 2 At this point it is interesting to examine the transport through the Lieb lattice when strain is applied. Specifically, it has been shown that inhomogeneous strain applied to honeycomb lattices can give rise to Landau level splitting [22], robust Klein tunneling [23], topological transitions [24], and other unexpected phenomena. It is therefore instructive to look at the projections on the flat band as a function of the strain when exciting either the A, B, or C sites in the infinite Lieb lattice. Our calculations are summarized in Fig. 3. Interestingly, when exciting a B site, the population of the flat band always vanishes, for all values of the anisotropy. Localized flat band modes have B sites with zero amplitude, therefore launching light into this site will always result in strong transport, irrespective of the ratio κ y /κ x . However, the situation is different for an excitation of the A and C sites. Whereas for the isotropic case (κ x = κ y ), the overlap with the flat band for both sites is exactly the same (due to symmetry reasons), for the anisotropic case (κ y > κ x ) the flat band is more populated for an A-site excitation (resulting in decreasing transport) and less populated for a C-site excitation (yielding enhanced transport). (Anisotropic β = 0 modes have |ψ A | > |ψ C |, being therefore easier to excite them in sites A).

Edge transport
Notwithstanding the bulk transport, we pay particular attention to the diffraction of light at the edge of the lattice. It is important to note that, within tight-binding model (Eq. 2) there are no edge states present, in the isotropic as well as in the anisotropic case. However, we surprisingly find that edge states arise in anisotropic Lieb lattices in the continuous model (Eq. 1). In Fig. 4(a) we present a plot of the band structure at the edge of the Lieb lattice by diagonalizing H cont as given by Eq. 1, showing clearly the existence of states residing in the band gap that are localized at the lattice edge. The underlying mechanism for the formation of these edge states is similar to the one recently reported for "photonic graphene" [18]. Note that these calculations are performed by full diagonalizing H cont , as given in Eq. 1, as opposed to simply examining the spectrum of the tight-binding model.
To explain these edge states in the Lieb lattice, one has to take into account the fact that the effective on-site propagation constant of each mode is modified by its neighbouring sites. This modification differentiates between the B-sites in the bulk, where they have four neighbours, and the edge, where they have only three. Although this difference is small and is usually neglected in tight-binding theories, it comes into play where the modes are highly degenerate at the van Hove singularity. At the degeneracy point, any slight edge perturbation takes the edge mode out of the band and creates a Tamm-like edge state as shown in Fig. 4(b). We emphasize again that this state does not result from any real surface perturbation but rather from the specific surface structure along the edge. We classify this state as a "Tamm state" [25] as opposed to a "Shockley state" [26], because it does not arise due to a band crossing, the criterion for the emergence of Shockley states. It is important to note that these states do not have a topological origin, and only arise from the 'effective defect' at the edge. Although Tamm states are conventionally associated with surface perturbations or defects (that are inherent in the system, or that arise from modulation [28]), in the present case no defects whatsoever are present. Instead, what happens here is that the edge itself acts as a sufficiently strong defect to localize light on the edge. This effect can not be accounted for in the standard tight-binding model and appears only in the continuum model. The experimental observation of strong edge diffraction in the anisotropic Lieb lattice is presented in Fig. 4(c), the simulations using Eq. 1 are shown in Fig. 4(d).

Conclusions
In conclusion, we presented a theoretical and experimental study of transport of light in photonic Lieb lattices. We find that the transport in the bulk is significantly affected by which site in the unit cell is excited, due to different overlap with the flat band (since any superposition of flat band states does not diffract and can form localized bulk states). Importantly, an excitation of the B-site in the unit cell leaves the flat band essentially empty, resulting in maximal transport. We also show that the population of the different bands is a function of the anisotropy of the lattice. Moreover, we demonstrate the existence of new edge states in anisotropic Lieb lattices that arise due to a virtual defect at the lattice edges and are not manifested in the standard tight-binding model. Our findings experimentally prove that the Lieb lattice exhibits various peculiar and unique features concerning wave transport, which might find their way into new concepts for imaging, routing and switching.