Two-dimensional Thouless pumping of light in photonic moiré lattices

Continuous and quantized transports are profoundly different. The latter is determined by the global rather than local properties of a system, it exhibits unique topological features, and its ubiquitous nature causes its occurrence in many areas of science. Here we report the first observation of fully-two-dimensional Thouless pumping of light by bulk modes in a purpose-designed tilted moiré lattices imprinted in a photorefractive crystal. Pumping in such unconfined system occurs due to the longitudinal adiabatic and periodic modulation of the refractive index. The topological nature of this phenomenon manifests itself in the magnitude and direction of shift of the beam center-of-mass averaged over one pumping cycle. Our experimental results are supported by systematic numerical simulations in the frames of the continuous Schrödinger equation governing propagation of probe light beam in optically-induced photorefractive moiré lattice. Our system affords a powerful platform for the exploration of topological pumping in tunable commensurate and incommensurate geometries.


Supplementary Text
Configuration of laser beams and tilted moiré lattice. The tilted moiré pattern used in the experiment was created by two interfering lattice-forming TE beams, each one formed by four plane waves: where the wavevectors k m are given by k 1,2 = (∓k x cos θ, ±k x sin θ, k z ), k 3,4 = (∓k y sin θ, ∓k y cos θ, k z ) and k ′ m = R x (α)R z (θ)k m , where R ν (φ), ν = x, z, stand for the standard operators of 3D rotations through an angle φ with respect to ν-axis and m = 1, ..., 4. The amplitude of the waves, E, in our experiment had the value of E ∼ 460 V/m. Thus, the lattice intensity in main text Eq. (1), determined by the mechanism of the photorefractive response, is approximated by the optical potential Eq. (2) of main text (where we neglected terms of the order of α 2 y and, α 3 z and all higher orders) with the reference lattice V (r) = E[cos(k x x) + cos(k y y)] The z−independent primitive translation vectors of the so created photonic lattice at α > 0 and m = 2 and n = 1, used in the experiment, are e 1 = 2i − j and e 2 = i + 2j.
Tight-binding approximation. We assume that a complete basis of well localized 2D Wannier functions (or quasi-Wannier functions) w ν (r − e, z) exists [1,2] (as a matter of fact, this requirement can weakened, since we are dealing only with the upper occupied band, and, say, eventual crossings of other empty bands would not affect the observed effects). Substituting the respective expansion in Eq. (1) of the main text, and using the definition of the Wannier functions is the Fourier coefficient of the expansion of the propagation constant that is periodic in the reciprocal space, and In the linear case, when only the upper band with ν = 1 is excited, i.e., when c νe (z = 0) = 0 for all ν > 1, all these coefficients remain zero if the evolution is adiabatic and no band crossing occurs (these are condition of our experiments). Furthermore, from (7) one obtains that B νν (0, z) ≡ 0, while B νν (e, z) with nonzero e is negligible due to the tight localization of the Wannier functions, and has additional smallness due to adiabatic variation of the Wannier functions.
On the other hand, flatness of the bands implies smallness of the Fourier coefficientŝ β ν (e, z) at all z. This justifies weak diffraction: recall that the diffraction shown in the main text while being visible, corresponds to extremely long (in dimensionless units) propagation distances. Therefore the quantized pumping of the center of mass is determined by the adiabatic evolution of the Bloch states determining the Chern numbers.
Beam dynamics. Using the Wannier-function expansion (3) it is straightforward to compute the representation r c = χ(z)R(z) + ρ t (z) + ρ ib (z), used in the main text with the explicit expression for r and asterisk stands for the complex conjugation. The first term in this expression describes the transition between the bands while the second term describes the diffraction of the beam due to the energy redistribution in the first upper band (for a highly symmetric input beam, like the one used in experiments this term almost vanishes). The evolution of the vector R(z) for the band ν = 1, can be expressed in terms of the Bloch functions through the Berry connection where Ω L is the primitive cell of the tilted moiré lattice. One computes where BZ stands for the reduced Brillouin zone of the tilted moiré lattice. The Bloch functions are normalized to one: ⟨u νk |u νk ⟩ = 1.
Assuming that R(0) = 0, we obtain the shift after one period Z as R(Z). For the Pythagorean lattice used in the experiment (m = 2 and n = 1) we have |e 1 | = |e 2 | = √ 5a and in the system of coordinates defined by the translation vectors e 1,2 we compute R(Z) = C 1 e 1 + C 2 e 2 with C 1 = −1 and C 2 = 1 (computed numerically).
This ensures that the first finite gap in the spectrum of this lattice remains open at all distances z. This lattice also has the smallest primitive cell as compared to other square Pythagorean moiré lattices. However, tilted Pythagorean moiré lattices corresponding to other triples may exhibit gap closing at specific propagation distances z. This unwanted effect may preclude observation of topological transport due to coupling between different bands. In Supplementary Fig. 2 we illustrate this transformation of bandgap structure for a lattice associated with the triple (5,12,13) (m = 3 and n = 2).
Light propagation in three different types of lattices. Supplementary Fig. 3 shown below compares the evolution of light beam in three different types of lattices, which are: (a) moiré lattice used in our experiment and created by two mutually twisted and tilted sublattices, (b) usual periodic lattice created by two non-twisted and non-tilted sublattices, i.e. lattice obtained at θ = α = 0, (c) the lattice created by two tilted, but non-twisted sublattices at θ = 0 and α = 0.015. In all three cases the same input Gaussian beam ψ(r) = exp(−r 2 /r 2 0 ) of width r 0 = 0.7, covering approximately one cell of the lattice, was applied (the beam is illustrated by the red spot superimposed on the lattice profile in the first column). One can readily make two observations. First, light diffraction is significantly suppressed in the moiré lattice ( Supplementary Fig. 3a). In this lattice even the pattern after one or two full pumping cycles is narrower than diffraction patterns in two other lattices at a quarter and a half cycle (their quantitative comparison is shown in Fig.2h of the main text). Second, no directed displacement of the light pattern is seen in the lattice produced by tilted, but non-twisted sublattices ( Supplementary Fig. 3c). Propagation up to larger distances z in this type of lattice confirms this observation.

Characterization of the beam displacement.
To characterize the displacement of the light beam we used the well-defined coordinate of the center of mass (COM), which accounts for the total integral intensity of the light beam. The COM trajectory, shown by the green curve in Supplementary Fig. 4b, predicts somewhat smaller displacement after one pumping cycle in comparison with ideal pumping (which is obtained also in Fig.2d and Fig.3b of the main text). Such deviation from the ideal pumping is mainly due to the diffraction of the pattern. Indeed, if the small-amplitude diffracted field is discarded upon calculation of the COM, then nearly perfect shift is obtained. This is illustrated by the red curve in Supplementary Fig. 4b, where upon calculation of COM we discarded the contribution of the field below 9% of the peak intensity. Furthermore, if the displacement of the pattern is characterized by the position of the global intensity maximum (blue curve of Supplementary Fig. 4b), we again observe oscillations of displacement around the idealized curve, but at the same time a perfect match with the idealized pumping is obtained after a full pumping cycle. In this case, the position of the main lobe of the diffraction pattern after one pumping cycle matches exactly the position predicted by the idealized theory, as shown in Supplementary Fig. 4a.
2D Thouless pumping for negative twisting angle. In the main text, we present the pumping of the light when the twisting angle θ between two sublattices is equal to θ = arctan(2mn/(m 2 − n 2 )) = arctan(4/3) with integer m = 2 and n = 1. In this case the two Chern numbers determining pumping direction of light are given by (C 1 , C 2 ) = (− 1,1) and in accordance with the formula R(Z) = C 1 e 1 + C 2 e 2 , the light exhibits pump into the second quadrant ( Supplementary Fig. 5a). If, on the other hand, the twisting angle θ changes its sign, θ = − arctan(4/3), that corresponds to (m, n) = (2, −1), the above Chern numbers become (C 1 , C 2 ) = (1,1), and the light is found to pump into the first quadrant ( Supplementary Fig. 5b). Direct comparison between θ and −θ cases shows that the magnitude of the COM displacement is exactly the same in both cases, but the direction of displacement changes from +18 o to −18 o with respect to the pumping direction of the lattice. Intensity distributions after one and two pumping cycles for these two twisting angles are depicted in Supplementary Fig. 5c and 5d.
Approaching the adiabatic regime of the Thouless pumping. A perfect pumping is only achieved when the pumping cycle (z-period of the lattice Z = 2a/α) tends to infinity.
In reality, the sample length is finite (2 cm long in our case) that imposes a practical limitation with respect to how close one can approach the adiabaticity. In our experiment we used an angle α = 0.015, yielding lattice period of 1.2 cm, which is of course not very close to the mathematical adiabatic limit, but still, we observed an excellent agreement between the direction of the beam pumping and prediction based on the topology of the system. This confirms that we are working in the quasi-adiabatic regime. To illustrate how pumping dynamics depends on the angle α (z-period of the lattice) and to give an idea on transition to adiabatic regime, in supplementary Figure 6 we show the results of simulations of the displacement of the beam center of mass upon pumping in moiré lattice for several progressively decreasing angles α (results for two such angles are also included into main text in Fig. 2) at fixed dc field E 0 . These results clearly show that pumping regime becomes nearly adiabatic (i.e. COM displacement for light beam practically exactly coincides with prediction based on the Chern numbers) in samples with lengths ∼ 5-10 cm that in principle can be manufactured. On the other hand, when cyclic modulation becomes too fast, it would result in significant inter-band transitions. In this regime the direction/magnitude of displacement of the beam depends on the details of modulation (rather than determined by