Two-dimensional topological quantum walks in the momentum space of structured light

Two-dimensional topological quantum walks in the momentum space of structured light Alessio D’Errico,1,† Filippo Cardano,1,8,† Maria Maffei,1,2 Alexandre Dauphin,2,6 Raouf Barboza,1 Chiara Esposito,1 Bruno Piccirillo,1 Maciej Lewenstein,2,3 Pietro Massignan,2,4,7 AND Lorenzo Marrucci1,5 Dipartimento di Fisica “Ettore Pancini”, Università degli Studi di Napoli Federico II, ComplessoUniversitario di Monte Sant’Angelo, Via Cintia, 80126Napoli, Italy ICFO–Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860Castelldefels (Barcelona), Spain ICREA–Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys 23, 08010 Barcelona, Spain Departament de Física, Universitat Politècnica deCatalunya, CampusNord B4-B5, 08034 Barcelona, Spain CNR-ISASI, Institute of Applied Science and Intelligent Systems, Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy e-mail: alexandre.dauphin@icfo.eu e-mail: pietro.massignan@upc.edu e-mail: filippo.cardano2@unina.it


INTRODUCTION
Quantum walks (QWs) are the deterministic quantum analogues of classical random walks and describe particles (walkers) whose discrete dynamics is conditioned by the instantaneous configuration of their spin-like degree of freedom (the coin) [1]. QWs were originally introduced as versatile candidates for implementing quantum search algorithms and universal quantum computation [2,3], and have been used to model energy transport in photosynthetic processes [4]. These systems bear close analogies with electrons in periodic potentials, and it was shown that QWs can host all possible symmetry-protected topological phases displayed by noninteracting fermions in one or two spatial dimensions (1D or 2D) [5]. Practical implementations of quantum walks have been demonstrated, for instance, with ultracold atoms in optical lattices [6][7][8][9], superconducting circuits [10], and photonic systems [11][12][13][14][15][16].
In optical architectures, the lattice coordinates have been encoded in different degrees of freedom of light, such as the arrival time of a pulse at a detector [14,17,18], the optical path of the beam [11][12][13]19,20], or the orbital angular momentum [15], while the coin is typically encoded in the polarization degree of freedom or in the entrance port of beam splitters. In a remarkable series of experiments, QWs proved instrumental in studying the evolution of correlated photons [12,13], the effects of decoherence [11,17] and interactions [14], Anderson localization [21], quantum transport in the presence of disorder [22], and topological phenomena in Floquet systems [10,19,[23][24][25][26][27]. An excellent review of the state of the art on topological photonics and artificial gauge fields in quantum simulators is given in Refs. [28,29] and [30], respectively. Despite being so fruitful, experimental research on QWs has been almost entirely focused on 1D systems. Few exceptions are the studies presented in Refs. [14,18,31,32], where a 2D walk was cleverly simulated by folding a 2D lattice in a 1D chain, and in Ref. [33], where path and OAM encoding were combined. Very recently, a continuous-time walk has been realized in a 2D array of coupled waveguides [34,35].
Here, we report a novel approach to the photonic simulation of quantum dynamics on 2D discrete lattices, based on the encoding of the individual sites in the transverse wavevector (or photon momentum), which is an inherently 2D degree of freedom. In our specific case, we simulate a QW process. Unlike the common in our system the photonic evolution takes place within a single light beam that acquires a complex internal structure as it propagates. The core of our photonic QW simulator is a stack of closely spaced liquid-crystal (LC) devices, conceptually similar to standard q -plates [36,37]. We present a proof-of-principle demonstration of our platform by generating up to five steps of a 2D QW, with both localized and extended initial inputs [18,38,39].
We design the unitary evolution of our QW so that it realizes a periodically driven (Floquet) Chern insulator. To characterize this system, we first analyze the energy dispersion of one of the bands of the effective Hamiltonian by tracking the free displacement of a wavepacket. Then we probe the Berry curvature of the band by repeating the tracking under the action of a constant force that is simulated by means of simple translations of specific plates. Upon sampling uniformly across the whole band, the average transverse displacement provides us a straightforward and accurate measurement of the Chern number of that band.

A. Quantum Walks in the Transverse Wavevector of Light
A discrete-time QW on a square lattice in 2D results from the repeated action of a unitary operator U on a quantum system, the walker, and its internal spin-like degree of freedom, the coin [40]. After t discrete steps, a given initial state | 0 evolves according to | (t) = U t | 0 . The step operator U typically includes a spin rotation W, and discrete displacements of the walker along the directions x and y , generated by spin-dependent translation operators T x and T y . In the simplest case, the Hilbert space of the coin has dimension two [5,18]. In our photonic QW implementation, we encode the coin into light polarization. For definiteness, we use left and right circular polarizations (|L , |R ) as the basis states (with handedness defined from the point of view of the receiver), like in our earlier realization of a QW with twisted light [15].
The main novelty of the setup considered here lies in encoding the discrete dimensionless coordinate of the walker m = (m x , m y ) on a 2D square lattice in the transverse momentum of light. In particular, we use Gaussian modes whose mean transverse wavevector assumes the discrete values k ⊥ = k ⊥ m. The lattice constant k ⊥ is taken to be much smaller than the longitudinal wavevector component k z ≈ 2π/λ (where λ is the wavelength of the light), so that these modes propagate along a direction that is only slightly tilted with respect to the z axis. More explicitly, a generic light mode in our setup reads as follows: where A(x , y , z) is a Gaussian envelope with large beam radius w 0 in the transverse x y plane and |φ denotes the polarization state [see Sec. S1of Supplement 1 for more details]. Accordingly, arbitrary superpositions of these modes still form a single optical beam, traveling approximately along the z axis. Only in the far field, or equivalently in the focal plane of a lens, these modes become spatially separated, and their relative distribution can be easily readout [see Figs. 1(a) and 1(b)]. The parameters w 0 and k ⊥ are chosen so that these modes almost perfectly overlap spatially while propagating in the whole QW apparatus (as long as |m x | and |m y | are not too large) and have negligible crosstalk in the lens focal plane. The QW dynamics is implemented with the apparatus depicted schematically in Fig. 1(a) and described in greater detail in Sec. S2 of Supplement 1. A collimated Gaussian laser beam passes through a sequence of closely spaced LC plates, which realize both walkertranslation and coin-rotation operators. At the exit of the walk, a . Each evolution step U = T y T x W is realized with three LC devices. The walker position is encoded in the transverse momentum of photons, so that walker steps physically correspond to transverse kicks that tilt slightly the photon propagation direction. The transverse diffraction of light remains negligible across the whole setup, and the entire evolution effectively occurs in a single beam. At the exit of the walk, a lens (with focal distance equal to 50 cm) Fourier-transforms transverse momentum into position, allowing us to resolve and measure individual modes. (b) The recorded intensity pattern is a regular grid of small Gaussian spots, whose intensities are proportional to the walker's spatial probability distribution. We set the modes beam radius to w 0 = 5 mm, which corresponds to a spot size of 20 µm (radius) on the camera plane.
(c) LC optic-axis pattern for a g -plate that realizes a T x operator. The spatial period fixes the transverse momentum lattice spacing k ⊥ = 2π/ . We use = 5 mm, so that k ⊥ = 1.26 mm −1 , corresponding to a spacing between spots of 63 µm on the camera. (d) Action of a single g -plate T x on a linearly camera placed in the focal plane of a lens reads out the field intensity, providing the coordinates distribution of the walker [as in Fig. 1(b); see also Sec. S3 of Supplement 1]. If needed, the polarization components also may be straightforwardly read out (see Fig. S2 of Supplement 1). The elements yielding the QW dynamics are optical devices consisting of thin layers of LC sandwiched between glass plates. The local orientation α(x , y ) of the LC optic axis in the plane of the plate can follow arbitrary patterns, imprinted during the fabrication by a photo-alignment technique. The birefringent optical retardation δ of the LC may instead be controlled dynamically through an external electric field [37,41]. In the basis of circular polarizations |L = (1, 0) T and |R = (0, 1) T , these plates act as follows: . (2) Such plates give rise to coin rotations or walker translations, depending on the optic axis pattern. For example, a spindependent translation operator in the x direction is obtained when the local orientation α increases linearly along x : where is the spatial periodicity of the angular pattern and α 0 is a constant [see Fig. 1(c)]. This patterned birefringent structure is also known as a "polarization grating," and hence we refer here to these devices as "g -plates" (as opposed to the q -plates used in our previous works, which have azimuthally varying patterns [37]). By inserting Eq. (3) in Eq. (2), one gets the action of a g -plate: wheret x andt † x are the (spin-independent) left and right translation operators along x , acting, respectively, ast x |m x , m y , φ = |m x − 1, m y , φ andt † x |m x , m y , φ = |m x + 1, m y , φ . The spatial periodicity of the LC pattern controls the momentum lattice spacing k ⊥ = 2π/ . It is sufficient to set ∼ w 0 to avoid mode crosstalk. The action of a single g -plate T x is shown in Fig. 1(d). The T y operator is implemented analogously, imposing a gradient of the LC angle α along y . Finally, the spin rotations W are realized with uniform LC plates acting as standard quarter-wave plates, that is with constant α = 0 and δ = π/2. By using these values in Eq. (2), we get that in the basis of circular polarizations this operator acts as

B. Engineering a 2D Topological Quantum Walk
Among possible protocols obtained by combining our plates, we considered the QW generated by the unit step operator with T x and T y tuned at the same value of δ. We implemented five complete steps of this QW, which represents a generalization of the alternated protocol described in Refs. [18,38]. In particular, it realizes a periodically driven Chern insulator, exhibiting different topological phases according to the value of the parameter δ, as we discuss in detail below. We start with a localized walker state |m = (0, 0), φ , which physically corresponds to a wide input Gaussian beam with radius w 0 = 5 mm, propagating along the z direction. In Fig. 2 we show representative data for δ = π/2 and a linearly polarized input. The walker distribution remains concentrated along the diagonal m x = −m y during the whole evolution, as a consequence of the absence of coin rotation operations between every action of T x and T y . All data show an excellent agreement with numerical simulations. A quantitative comparison is provided by computing the similarity S = ( m √ P e P s ) 2 /( m P e m P s ) between simulated (P s ) and experimental (P e ) distributions. For the data shown in Fig. 2

C. Quasi-Momentum, Quasi-Energy Bands and Group Velocity
A QW can be regarded as the stroboscopic evolution generated by the (dimensionless) effective Floquet Hamiltonian H eff ≡ i ln U . The eigenvalues of H eff are therefore defined only up to integer multiples of 2π , and are termed quasi-energies. Both U and H eff admit a convenient representation in the reciprocal space q associated with the coordinate m of the walker. As discussed above, the dimensionless coordinate m = k ⊥ / k ⊥ is encoded in our setup in the transverse momentum k ⊥ of the propagating beam. As such, its conjugate variable corresponds physically to the position vector r ⊥ in the x y transverse plane. We introduce therefore the dimensionless quasi-momentum q = −2π r ⊥ / , belonging to the square Brillouin zone [−π, π ] 2 , as the conjugate variable to the walker position m. The negative sign in the definition of q provides the standard representation for plane waves m|q ∝ e im·q . In the space of quasi-momenta the effective Hamiltonian assumes the diagonal form H eff (q) = ε(q)n(q) · σ . Here n(q) is a unit vector, σ = (σ x , σ y , σ z ) represents the three Pauli matrices, and ±ε(q) yields the quasi-energies of two bands [as shown in Fig. 3(a)]. In the following, we will denote the complete eigenstates of the system by |q, φ ± (q) , where ± refers to the upper/lower band. Let us note here that, although the operator U is obtained by cascading independent displacements along the x and y axes, the overall evolution is nonseparable; that is, the effective Hamiltonian cannot be expressed as the sum of two contributions depending on a single spatial coordinate. This can be seen clearly in the expression of the quasi-energy dispersion, which is given by the following relation: The complete expression of the Hamiltonian is provided in Sec. S4 of the SM. In our experiment, we can directly explore the band structure of the system by observing the propagation of walker wavepackets | g (q 0 , ±) that are sharply peaked around a given quasi-momentum q 0 and belong to the upper/lower band. These wavepackets are physically generated as narrow Gaussian light beams (with beam radius w g ) propagating along the direction z, centered around a specific transverse position r ⊥0 = −q 0 /(2π ) at the input port of the QW, and with polarization |φ ± (q 0 ) (see Sec. S1 of Supplement 1 for further instructions for preparing these states). In the experiment, the choice of transverse position r ⊥0 is easily controlled by translating the whole QW setup (which is mounted on a single motorized mechanical holder) relative to the input laser beam. Having narrow Gaussian envelopes in the conjugate space q, these wavepackets are relatively broad Gaussians in the space of walker coordinates m. They are (approximate) eigenstates of the system and therefore preserve their shape during propagation. Their center of mass m g obeys a dynamics that semiclassically is governed by the group velocity v (±) (q 0 ) = ±∇ q ε(q)| q=q 0 [15], as shown for instance in Fig. 3(b). To measure experimentally the group velocity v (±) (q 0 ) we inject a wavepacket | g (q 0 , ±) in our QW, we detect its average displacement m as a function of time-step t, and finally we perform a linear best fit on the displacements versus time [see Fig. 3(c)].  Fig. S7 of Supplement 1. A systematic error that can affect our setup is the possible misalignment of g -plates in both x and y directions. Our present platform permits to adjust only their position along x . As such, we can estimate the associated standard error by repeating the experiment after realigning the plates. It is not possible, however, to repeat the same procedure for the perpendicular direction. In this case, after measuring the effective displacements of the T y plates, which are determined by fabrication imperfections, we perform a Monte Carlo simulation of the propagation of our wavepacket, and we estimate the standard deviation of the final center of the mass position. The two errors are finally combined by adding their variances to obtain the error bars in Figs. 3(c), 4(b), and 4(c).

D. Measurement of the Chern Number Through the Anomalous Velocity
The energy bands of the effective Hamiltonian generally possess nonzero Berry curvature. For a 2 × 2 Hamiltonian like ours, the latter may be written as [42] (±) The integral of the Berry curvature over the whole Brillouin zone (BZ) gives the Chern number: The Chern number ν (±) of our QW depends on the optical retardation of our plates. By tuning δ, we can thus switch from a trivial to a topological Chern insulator, as shown for example in Fig. 4(a). Our QW is a Floquet evolution, and as such its complete topological classification is not based on the Chern number only but involves a more complex invariant introduced in Ref. [43]. Such classification is discussed in detail in Sec. S5 of the SM. When a constant unidirectional force is acting on the system, the Berry curvature contributes to the wave-packet displacement in a direction orthogonal to the force (as in the quantum Hall effect). Let us for definiteness consider a force F x acting along x . Within the adiabatic approximation, the semiclassical equations of motion predict that a wavepacket | g (q 0 , ±) will experience after a time t a transverse displacement along y given by [44,45] with q τ = (q 0,x + F x τ, q 0,y ). The contribution to the velocity coming from the Berry curvature is called anomalous velocity. This result is derived in the adiabatic regime, where the (dimensionless) force is much smaller than the bandgaps of the effective energy, so that interband transitions can be neglected. When we consider the overall transverse displacement of a filled band, namely when we integrate Eq. (10) over the whole Brillouin zone, the group-velocity term averages to zero, while the anomalous contributions add up to the band's Chern number [45,46] (see Sec. S4 of Supplement 1 for a detailed derivation of this result): As shown in Eq. (10), implementing a constant force in our setup requires a linear shift in time of the quasi-momentum component q x . This degree of freedom corresponds to the x coordinate in real space. Hence, we impose at each step a quasi-momentum variation by introducing a transverse spatial displacement of the light beam between each plate. Actually, rather than displacing the beam, it is equivalent (and much simpler) to displace the reference system and the setup in the opposite direction. More specifically, we shift the g -plate acting at time-step t along the x axis by an amount x t = t F x /(2π ) (see Sec. S4 in Supplement 1 for further details). Then, we sum up the measured displacements m y obtained for 11 × 11 distinct wave-packets | g (q 0 , −) , providing a homogeneous sampling of the lower band across the whole Brillouin zone and realizing a good approximation of the continuous integral yielding m y (−) . Figure 4(b) shows the mean displacement of wavepackets prepared in the lowest energy band for a QW with δ = π/2, corresponding to Chern number ν (−) = 1. The energy bandgap ≈ 1 [see Fig. 4(a)] is sufficiently larger than the applied force F x = π/20, thereby ensuring the validity of the adiabatic approximation. Experimental data (empty markers) are compared to the overall band displacement predicted by the semiclassical theory within adiabatic regime (continuous lines), namely m y (t) (−) = tν (−) F x /(2π ), m x (t) (−) = 0. While m y (−) follows the expected curve quite reasonably, the overall m x (−) is found to be not negligible. These small differences can be understood by simulating the full dynamics of the wave packet, beyond the single band approximation (see Fig. S8 in Supplement 1).
To get rid of this spurious contribution, which arises mainly from residual group-velocity effects, we consider also the "inverse protocol" generated by the step operator U −1 = W −1 T −1 x T −1 y . The bands of U −1 have the same dispersion as the bands of U but feature opposite Chern numbers. In this way, if filling the same band, we expect to observe identical contributions from the group velocity dispersion, while the anomalous displacement should be inverted. The step operator U −1 can be easily implemented by swapping the T y and W operators and changing suitably their retardation (see Sec. S4 and Fig. S2 in Supplement 1). In Fig. 4(b) we show with filled markers the difference (divided by two) of the data obtained with the protocols U and U −1 . This procedure reduces significantly the overall displacement along x , while in the y direction we observe a very nice agreement between our data and the semiclassical predictions. The measured value of the Chern number is ν (−) = 1.19 ± 0.13, consistent with the theoretical value of 1 (errors are given at one standard deviation). A similar behavior is also observed for larger values of the force, as shown in Fig. S9 in Supplement 1. In Fig. 4(c), we replicate the same experiment for a QW with δ = 7π/8, when the Chern numbers are zero, even though the presence of edge states witnesses nontrivial topology [43] (see also Sec. S5 of Supplement 1). In agreement with the prediction of vanishing anomalous displacement, the average wavepacket motion in both directions is observed to be negligible, yielding a Chern number ν (−) = 0.10 ± 0.15.

CONCLUSION
In this work we have experimentally demonstrated a conceptually new scheme for the realization of a 2D discrete-time QW, that relies on encoding the walker and the coin systems into the transverse momentum of photons and in their polarization, respectively. The coin rotation and shift operators are implemented by suitably engineered LC plates, whose number scales linearly with the number of time-steps. They are arranged in a compact setup, in which multiple degrees of freedom can be controlled dynamically, such as the plates' optical retardation δ or their transverse position, allowing one to study several QW architectures. If needed, different LC patterns could be written onto the plates, yielding different types of quantum dynamics. The platform accurately simulates the dynamics dictated by the QW protocols that we tested, as witnessed by the good agreement between measured distributions and numerical results. We investigated 2D walks of both localized and extended inputs, with and without an external force. The 2D protocol we presented here simulates a Floquet Chern insulator. We probed the associated topological features by preparing wave-packets that well approximate the eigenstates of the QW Floquet Hamiltonian and detecting their average anomalous displacement arising when a constant force is applied to the system. The setup has been designed to minimize the decoherence effects caused by light diffraction and walkoff phase delays occurring when the walker follows different paths (see Sec. S7 of Supplement 1). There is no fundamental limitation to scaling up our setup to a much larger number of steps. Reflection losses at each LC plate ( 15%), representing the main current limitation to the setup efficiency, could be largely reduced to the level of 1%-2% by applying a standard antireflection coating. As such, while our experiment is carried out in a classical-wave regime, the proposed setup is perfectly suitable for single-photon quantum experiments, similarly to what was already demonstrated in Ref. [15]. These represent one of the most appealing future prospects for our system, particularly in view of the very large number of input and output modes that can be easily addressed.
The demonstration of a new platform for 2D QWs opens new avenues for the experimental study of the rich quantum dynamics in two dimensions. In prospect, diverse directions could be investigated with our platform, such as the realization of 2D lattices with more complex topologies (for example, hexagonal), or experiments in the multiphoton regime, for instance in the context of QW applications to Boson sampling. Direct access to both walker position and quasi-momentum could be exploited to study complex dynamics in the regime of spatial disorder. By combining topology and our dynamical control of the system parameters, we could investigate dynamical quantum phase transitions in QWs [47][48][49]. Finally, by introducing losses for specific polarization states, this platform could be used to investigate topological features of 2D non-Hermitian systems [50][51][52][53].