Universal momentum-to-real-space mapping of topological singularities

Topological properties of materials are typically presented in momentum space. Here, we demonstrate a universal mapping of topological singularities from momentum to real space. By exciting Dirac-like cones in photonic honeycomb (pseudospin-1/2) and Lieb (pseudospin-1) lattices with vortex beams of topological charge l, optimally aligned with a given pseudospin state s, we directly observe topological charge conversion that follows the rule l → l + 2s. Although the mapping is observed in photonic lattices where pseudospin-orbit interaction takes place, we generalize the theory to show it is the nontrivial Berry phase winding that accounts for the conversion which persists even in systems where angular momentum is not conserved, unveiling its topological origin. Our results have direct impact on other branches of physics and material sciences beyond the 2D photonic platform: equivalent mapping occurs for 3D topological singularities such as Dirac-Weyl synthetic monopoles, achievable in mechanical, acoustic, or ultracold atomic systems, and even with electron beams.

for three vortex beams with same topological charge l, the probe beam forms a donut-shaped triangular lattice pattern also with a net topological charge l (see Fig. 2a) before going to the crystal to probe the lattice. In momentum space, the directions of three vortex beams are matched to the three Dirac K points of the first Brillouin zone, while in real space, the triangular intensity pattern is matched to A or B sublattice to selectively excite the two pseudospin states, as illustrated in Fig. 1(a).
For the Lieb lattice, a slightly different technique is used for optical induction. A phase mask (in this case, an SLM) is used to generate two square lattice beams with different periods (9 and 18 ), and then the two induced index lattices are superimposed to form the Lieb lattice under the self-defocusing nonlinearity [2]. For probing, four vortex beams with same topological charge l are employed, which form a donut-shaped square lattice pattern also with a net topological charge l (see Fig. 4a). However, due to the non-diagonal nature of pseudospin-1 Hamiltonian, the three Lieb sublattices do not have a trivial correspondence to the three pseudospin states [illustrated in Supplementary Figure2(a)] as for the case of the HCL. As such, we elaborate here the excitation scheme for the Lieb lattice. After the Lieb lattice is written in the crystal, the four vortex beams are matched in momentum space to the four points at the corner of first Brillouin zone but with different phase winding [Supplementary Figure 2 (b)]. This phase winding shows also different phase relation between the excited lattice sites in real space [Supplementary Figure2(c)]. To excite the = 0 pseudospin state, the square lattice pattern of the probe beam is matched only to the B sublattice, with a π phase difference between the nearest excited sites. However, to excite the = 1 and = −1 pseudospin states, the square lattice pattern is matched to excite simultaneously A and C sublattices but with an opposite phase winding in a π/2 phase step. This phase requirement used in our experiment agrees perfectly with the eigenstates of the pseudospin matrix , as shown in the following theory section for the Lieb lattice.

Supplementary Note 2:
Details of theoretical derivation: The photonic honeycomb and Lieb lattices that we consider here have conical intersection points in -space (e.g., see [1][2][3][4][5] and references therein). For excitations in the vicinity of these points, the dynamics is governed by the Hamiltonian where are the components of the pseudospin angular momentum operator , which obey angular momentum commutation relations: where is the Levi-Civita symbol; the constant depends on the specific properties of the lattice. The eigenstates of the pseudospin are given by 2 , = ( + 1) , , and , = , . The Hamiltonian (Supplementary Equation 1) has 2 + 1 bands, i.e., the honeycomb lattice has 2 bands, while the Lieb lattice has 3 bands. As in the main text, we denote -components of the angular momenta with lower case letters.
Conservation of the angular momentum: The kinematical explanation of our observations involves conservation of the -component of the total angular momentum = + , where = × is the orbital angular momentum (OAM). The following commutation relations hold: It is straightforward to demonstrate [ , ] = 0: which indicates that the -component of the total angular momentum is conserved.
The initial excitations in our experiments are comprised of a single value of and . We use = 1 or = −1 for the input. The admissible values of pseudospin (z-component) are = − , − + 1, ⋯ , − 1, . Thus, we study 4 possible initial conditions for the HCL, and 6 possible initial conditions for the Lieb lattice. Maximally aligned initial condition implies maximal value of | | = | + |, and there are 2 such conditions for each lattice. The output beam has two or more values of ′ and ′, all of which obey + = ′ + ′ due to [ , ] = 0, as expected in an angular momentum conserved system.
Honeycomb lattice -dynamics via expansion over the eigenstates: Now we derive Eq. (5) in the main text, i.e., we account for the dynamics via expansion over the eigenstates. First, we consider graphene-like honeycomb lattices where = 1/2, = /2, and are the Pauli matrices. The Hamiltonian is and the Pauli matrices are: The The beam excites initially only one value of the pseudospin. Because the HCL is diagonal in the sublattice basis, this means that initially we excite only one of the two sublattices. The complex amplitude of the electric field of the initial excitation is , ( , , = 0) = 0 exp(− 2 / 0 2 ) , . We can rewrite it in momentum space as follows (since we study linear dynamics, 0 can be rescaled without affecting the results): here function ( ) depends on the transverse profile and the phase structure of the initial excitation, and the integral is taken over the whole k-space. For our input beam where defines the width of the beam in k-space. For a given , there are two possible initial excitations, , = 1 2 or , =− 1 2 , which can be written via superposition of the eigenmodes of the HCL as follows: The scalar products of the initial wavepacket with the eigenmodes are easily evaluated: The initial states are then The eigenmodes in the HCL evolve dynamically, each one with its own propagation constant: where 0 = /2. After sufficiently long time of propagation, we have far field dynamics as show below.
Let us now assume that we have excited pseudospin = 1/2. The evolving complex amplitude of the electric field is given by: . After taking the sum over the band index , we get: After substituting ( ) and • = cos( − ), we get: We now change the variable of integration from to = − to get: To clarify the mathematical structure of the output, we introduce the g-functions defined by, Supplementary Figure 3. The g-functions for the honeycomb lattice. We clearly see the total beam intensity at the output has a donut shaped structure as expected for conical diffraction. For the illustration, the parameters 0 , , , were taken as unity, and = 0.2.
Lieb lattice -dynamics via expansion over the eigenstates: In a fully equivalent manner, we can describe dynamics in the Lieb lattices. The Hamiltonian describing conical intersection of the Lieb lattice is where the , , matrices represent pseudospin 1: In this case, the propagation constants and the eigenmodes of the Hamiltonian are: Here again + = . The eigenstates of the pseudospin matrix are given by , = , : The initial excitation is described by For a given , there are three possible initial conditions: , =1 , , =0 , and , =−1 . Let us write the first of these initial conditions as a superposition of the Lieb lattice eigenstates (the calculation is fully equivalent for the other initial conditions): The scalar products of the initial wavepacket with the eigenmodes are easily found to be (the eigenmodes in Supplementary Equations 4 and 5are for this purpose used in real space as for the HCL above) For this initial state we now have The eigenmodes evolve dynamically: Next, the column matrices are expressed in terms of the pseudospin eigenstates: We collect the terms according to the pseudospin: We can use 1 − cos(2 ) = 2sin 2 ( ) and 1 + cos(2 ) = 2cos 2 ( ) to get: