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We start by introducing a general approach for the implementation of spectral photonic lattices in a nonlinear waveguide. As sketched in Fig. 1a, a spectral lattice is realized by using a shaped pump composed of several equidistant frequencies with spacing Ω. This pump can drive the interactions between the frequency components on the input signal spectrum all inside one fibre or waveguide with χ^{(3)} nonlinearity in the regime of the socalled fourwavemixing Bragg scattering^{22}. Under energy conservation and undepleted pump approximation, each pair of pump frequencies separated by nΩ drives the coupling between two signal lines with the same frequency difference nΩ, as shown in Fig. 1b. These discrete spectral lines form a lattice (Fig. 1c), where each frequency represents one lattice site. Importantly, nonlocal and complexvalued couplings can be implemented by specially tailoring the pump spectrum^{22}.
Fig. 1 Conceptual sketch of constructing multidimensional synthetic lattices in a nonlinear fibre.
a A nonlinear waveguide with χ^{(3)} nonlinearity, where a shaped pump mediates the conservative interactions between signal frequencies, giving rise to a reshaped signal spectrum at the output. b An example of a pump profile that induces crosstalk between one and two unit frequency separations. c The corresponding spectral lattice with firstorder (C_{1}) and secondorder (C_{2}) couplings driven by nonlinear interactions mediated by the shaped pump spectrum shown in b. d Synthetic twodimensional square lattice constructed using the spectral lattice in c. e Illustration of the chiraltube lattice formed by wrapping the lattice in d with the chiral periodic boundary condition pNow, we outline the key concept of exact mapping between higherdimensional lattices and a 1D spectral lattice with nonlocal couplings induced through nonlinear frequency conversion. In the example shown above in Fig. 1a–c, three pumps equally separated by Ω introduce coupling of the first and second orders (Fig. 1c). Then, the evolution of the signal spectrum along the nonlinear waveguide in the phasematching regime is governed by the Hamiltonian in terms of the creation ($\hat a_m^\dagger $) and annihilation ($\hat a_m$) operators for the discrete signal frequency components:
$$ {\bf{H}} =  \mathop {\sum }\limits_m \,\mathop {\sum }\limits_{\{ n\} } \,C_n\hat a_m^\dagger \hat a_{m + n}  {\mathrm{H}}.{\mathrm{c}}. $$ (1) where a set of positive integers {n} indicates the orders of coupling and C_{n} are the corresponding coupling constants of the nth order. 'H.c.' denotes the Hermitian conjugate, and m is an integer running through all phasematched spectral lines. The coupling constants C_{n} are given by the following expression^{22}:
$$ {\it{C}}_{\it{n}} = 2\gamma {\it{P}}\mathop {\sum}\nolimits_{\it{m}} {{\it{A}}_{\it{m}}{\it{A}}_{{\it{m}}  {\it{n}}}^ \ast } $$ (2) where γ is the effective nonlinearity and P is the average pump power. Here, A_{m} denotes the complex amplitudes of pump spectral components in the fibre, which are normalized as $\mathop {\sum}\nolimits_m {A_m^2 = 1} $. The evolution of the wavefunction governed by the Hamiltonian in Eq. (1) can be expressed as
$$ \psi (z) = {\mathrm{exp}}(iz{\bf{H}}) = {\mathrm{exp}}(izP{\bf{H}}^{\prime} ) $$ (3) where z is the propagation distance along the fibre and we denote by H′ = H/P, a normalized Hamiltonian that is independent of the total pump power. We see that the wavefunction dynamics can be observed by varying the average pump power P for a fixed fibre length z = L such that P effectively acts as the time variable.
We now consider a nontrivial and representative case of two coupling orders n = 1, 2 and show how the spectral lattice is mapped to a twodimensional square lattice. The general idea is based on the mapping of each specific order of coupling to a certain basis vector in higherdimensional space. For the example shown in Fig. 1d, which is a twodimensional space of a square lattice, there are two basis vectors, u_{x} and u_{y}. Hence, we can map the coupling order n = 1 to u_{x} and n = 2 to u_{y}. Then, we obtain a Hamiltonian in the twodimensional space that represents a square synthetic lattice:
$$ {\bf{H}}_{{\bf{sq}}} =  \mathop {\sum}\nolimits_m {\left[ {C_1\hat a_{{\bf{r}}_{\bf{m}}}^\dagger \hat a_{{\bf{r}}_{\bf{m}} + {\bf{u}}_{\bf{x}}} + C_2\hat a_{{\bf{r}}_{\bf{m}}}^\dagger \hat a_{{\bf{r}}_{\bf{m}} + {\bf{u}}_{\bf{y}}}} \right]  {\mathrm{H}}.{\mathrm{c}}.} $$ (4) where r_{m} is a vector indicating the spatial coordinate of the mth site in this twodimensional space. To provide an exact mapping, it is essential to reflect in 2D the algebraic property of the 1D lattice, where a sequence of two firstorder couplings produces the same frequency shift 2Ω as that of a secondorder coupling. This property can be satisfied by imposing a periodic boundary condition for the twodimensional synthetic space, as shown in Fig. 1d, where the orange arrow represents the wrapping vector p = 2u_{x}–u_{y}. Consequently, the twodimensional equivalent lattice is actually wrapped into a (2, −1) chiral tube connected by the dashed lines in Fig. 1d. This chiral tube is schematically illustrated in Fig. 1e.

We formulate and experimentally demonstrate an original mapping procedure for the realization of a synthetic triangular lattice. This presents a nontrivial case with nonorthogonal basis vectors, which has not been considered on any synthetic photonic lattice platform in previous studies. We show that a triangular lattice can be obtained by mapping from a spectral lattice with specially engineered simultaneously short and longrange couplings. The synthetic frequency space is sketched in Fig. 2a, where the first, third and fourth orders of the coupling are present. In the twodimensional space of the triangular lattice, as shown in Fig. 2b, the basis vectors are u_{1} = [1, 0]^{T} and ${\bf{u}}_{\bf{2}} = \left[ {1/2,\sqrt 3 /2} \right]^{\mathrm{T}}$, which are not orthogonal relative to each other. We map the firstorder coupling to the vector u_{1} and the fourthorder coupling to u_{2}. Then, we find that the thirdorder coupling is automatically mapped to u_{3} = u_{2} − u_{1}. This arrangement is used to construct the twodimensional equivalent triangular lattice sketched in Fig. 2b with the Hamiltonian
Fig. 2 Experimental observation of a quantum walk in a synthetic twodimensional triangular chiraltube lattice.
a A lattice with three coupling orders 1, 3 and 4, where only couplings to the shown sites are plotted with arrows. b The corresponding synthetic triangular lattice in twodimensional space. c 3D sketch of the lattice in b. d Experimental realization of a frequency quantum walk, where P is the power of the threepump spectral components with A_{1} = A_{2} = A_{5}. e–g Mapping of experimental data from d to a twodimensional triangular lattice with P= 0.10, 0.19, 0.28 mW, as indicated by the labels$$ {\bf{H}}_{{\bf{tr}}} =  \mathop {\sum}\nolimits_m {\left[ {C_1\hat a_{{\bf{r}}_{\bf{m}}}^\dagger \hat a_{{\bf{r}}_{\bf{m}} + {\bf{u}}_1} + C_3\hat a_{{\bf{r}}_{\bf{m}}}^\dagger \hat a_{{\bf{r}}_{\bf{m}} + {\bf{u}}_3} + C_4\hat a_{{\bf{r}}_{\bf{m}}}^\dagger \hat a_{{\bf{r}}_{\bf{m}} + {\bf{u}}_2}} \right]  {\mathrm{H}}.{\mathrm{c}}.} $$ (5) Similar to the example of the square lattice discussed above (Fig. 1d), there appears a periodic boundary condition. It is defined by the wrapping vector q = 4u_{1} − u_{2}, shown as a green arrow in Fig. 2b. Hence, the triangular lattice is effectively wrapped and connected by the dashed lines in Fig. 2b. To show this more intuitively, we sketch a threedimensional (3D) visualization of the (4, −1) chirally wrapped tube in Fig. 2c. We determine the unit cell vector of the tube lattice as l = −2u_{1} + 7u_{2} (not shown in the figure), which is the shortest vector that can connect two sites along the parallel direction of the tube (l^{T}q = 0 due to orthogonality).
We now present an experimental realization of a quantum walk in the multidimensional synthetic lattice space. Quantum walks have been observed in various types of photonic lattices using classical laser sources, where the evolution of coherent light is mathematically analogous to the quantum singleparticle dynamics^{39}. We tailor the complex amplitudes A_{m} of the pump spectral lines to induce the desired couplings in the signal frequency lattice according to Eq. (2). Specifically, we employ three pumps with equal amplitudes A_{1} = A_{2} = A_{5} to achieve the frequency lattice, as illustrated in Fig. 2a, with equal first, third and fourthorder couplings, i.e., C_{1} = C_{3} = C_{4}. We shape the pump with no phase difference between the complex amplitudes at different frequencies and, therefore, all couplings are realvalued. With a singlefrequency signal excitation, we observe a quantum walk in this frequency space, as shown in Fig. 2d. We map this experimentally realized synthetic lattice to the triangular lattice as outlined in Fig. 2b, c. The mapped quantum walk is shown in Fig. 2e–g at three representative average pump powers P = 0.10, 0.19, 0.28 mW, respectively. As mentioned above, the pump power acts as the time variable in the quantum walk. In these figures, the site of excitation is marked by a green arrow. This represents an experimental observation of quantum walks in higher synthetic dimensions. Our results agree quite well with the corresponding theoretical predictions calculated by the coupled mode equations. An animated image illustrating the dynamics of the experiment (incorporating Fig. 2e–g) and a comparison with theory is provided as Supplementary Fig. S1 in supplementary files.
To provide insight into the properties of our mapped synthetic chiraltube lattice, we also perform a theoretical analysis of the wave dispersion. We apply Bloch theorem and calculate the propagation constant as $\beta (k_1,k_2) = 2{\mathrm{Re}}[C_1{\mathrm{exp}}(ik_1) + C_4{\mathrm{exp}}(ik_2) + C_3{\mathrm{exp}}(ik_3)]$, where k_{1}, k_{2} and k_{3} are the wave numbers in the reciprocal space of the basis vectors u_{1}, u_{2}, and u_{3}, respectively, and $k_3 \equiv k_2  k_1$. In Fig. 3a, we plot a representative case for C_{1} = C_{3} = C_{4} = 1. Due to the periodic boundary condition, not all values of k_{1}, k_{2} are allowed. For the (4, −1) chiral tube discussed above, we have 4k_{1} − k_{2} = 2Nπ, where N is an integer. These allowed values are denoted as white lines in Fig. 3a. We trace out k_{1} = k_{2}/4 for the range of −π to π and present the dispersion as a 1D curve in Fig. 3b. For comparison, we also show another case with a different coupling C_{4} = i in Fig. 3c. We see that the resulting 1D dispersion shown in Fig. 3d becomes asymmetric due to the complex coupling C_{4}, which breaks the timereversal symmetry. We show in the following section that this regime is associated with the appearance of a gauge field in the mapped highdimensional lattices.
Fig. 3 Dispersion of the triangular lattice wrapped in a (4, −1) chiral tube.
a Colour density plot of the propagation constant β vs. wave numbers k_{1} and k_{2}, where the (4, −1) chiral periodic boundary condition allows only k_{1}, k_{2} values on the white lines. Here, we set C_{1} = C_{3} = C_{4} = 1. b The corresponding dispersion of a plotted in 1D along k_{1} = k_{2}/4. c, d Plots analogous to a, b but with a complex coupling C_{4} = i; the other parameters are the same 
We demonstrate that the complexvalued nature of the coupling constants in the synthetic frequency lattice can enable artificially created gauge fields. In contrast to the 1D case, in higherdimensional lattices, an important aspect of the gauge potential is associated with the induced magnetic flux that can arise in the presence of nonzero phase accumulation around a closed loop in the lattice.
To illustrate the capacity of our scheme to synthesize a nontrivial gauge field, we first revisit the mapping of a spectral lattice with three orders of coupling, as sketched in Fig. 2a. To induce these couplings, the minimum number of pumps is three, with complex amplitudes A_{1}, A_{2}, and A_{5}, as illustrated in Fig. 4a, with the corresponding couplings $C_1 \propto A_2A_1^ \ast $, $C_3 \propto A_5A_2^ \ast $ and $C_4 \propto A_5A_1^ \ast $. In Fig. 4b, we show a section of a spectral lattice implemented by the pump configuration in Fig. 4a, visualizing five sites (1–5) as an illustration. We use a oneway arrow to show each order of coupling, where the coupling to the other direction simply takes the complexconjugate value due to Hermiticity. We determine the phases of each order of the coupling along the direction of the arrows in Fig. 4b as ϕ_{1} = arg(A_{2}) − arg(A_{1}) (grey arrow), ϕ_{3} = arg(A_{5}) − arg(A_{2}) (blue arrow), and ϕ_{−4} = arg(A_{1}) − arg(A_{5}) (orange arrow). This lattice is mapped to the triangular lattice using the approach described above, which is shown in Fig. 4c for the first five sites. We find that the clockwise flux vanishes in each of the triangular cells:
Fig. 4 Implementation of the nontrivial gauge field in synthetic triangular lattices.
a Threepump configuration used to induce three orders of coupling. b The implemented synthetic frequency lattice with the first, third, and fourth orders of coupling. c The mapping of b to a twodimensional triangular lattice where the flux in each cell is zero. d–f Our specially designed scheme for implementing nonzero flux by adding an extra pump (A_{3}) and tailoring the pump phases$$ \begin{array}{*{20}{c}} {{\mathrm{\Phi }}_{1  5  2}}={\phi _{  4} + \phi _3 + \phi _1 = 0} \\ {{\mathrm{\Phi }}_{1  4  5}}={ \phi _{  4}  \phi _3  \phi _1 = 0} \end{array} $$ (6) This shows that with the minimum necessary number of three pumps, the number of free parameters is not sufficient to implement nonzero flux in any of the triangular cells of this twodimensional lattice.
We reveal that a nonzero flux can be induced by adding an extra pump, indicated with the green arrow in Fig. 4d with amplitude A_{3}. The corresponding couplings between the signal frequencies are sketched in Fig. 4e. As we are still aiming for coupling orders 1, 3, and 4, we first need to ensure that the secondorder coupling, denoted by the green dashed arrows in Fig. 4e, is cancelled out:
$$ C_2 \propto A_3A_1^ \ast + A_5A_3^ \ast = 0 $$ (7) As a sufficient condition to fulfil Eq. (7), in our experiment, we take $A_1 = A_5$, ${\mathrm{arg}}(A_3A_1^ \ast ) = \pi /4$ and ${\mathrm{arg}}(A_5A_3^ \ast ) =  3\pi /4$. Then, we calculate the phases of the other orders of coupling and find that ϕ_{3} (blue arrow) and ϕ_{−4} (orange arrow) remain the same as in the case analysed above with three pumps. This situation occurs, as each of the two orders is induced by the same pair of pumps. Importantly, the firstorder coupling acquires a different phase according to the expression
$$ C_1 \propto A_2A_1^ \ast + A_3A_2^ \ast $$ (8) This relation allows us to implement an arbitrary phase ϕ_{1} = arg(C_{1}) in the experiment, where we fix ${\mathrm{arg}}(A_3A_1^ \ast ) = \pi /4$ but freely choose the amplitudes of all three involved pumps and the phase of A_{2}. Therefore, the limitation given in Eq. (6) no longer applies and we can engineer any nonzero flux Φ_{1−5−2}. It is noteworthy that the following condition still holds:
$$ {\mathrm{\Phi }}_{1  4  5} \equiv  {\mathrm{\Phi }}_{1  5  2} $$ (9) which leads to a zero flux if one encircles a pair of neighbouring cells. This case is analogous to that of the Haldane model^{29}, where the total flux over all cells is zero, yet locally there appear locations with nonzero flux.
Next, we show a representative set of experimental results that demonstrate these nontrivial gauge potentials. We intentionally make $C_4$ slightly larger than $C_1 = C_3$ to more clearly observe the features associated with the artificial gauge potential. Specifically, we choose the pump profiles with four frequencies to obtain the coupling constants $C_1:C_3:C_4 = 3:  3:5{\mathrm{exp}}(i\alpha )$. This arrangement effectively corresponds to a phase of π − α along the u_{2} basis vector (positive direction) if we use a gauge transformation to make all couplings other than those along u_{2} realvalued. In the two sets of experiments presented in Fig. 5, we realize quantum walks with a singlesite excitation in the synthetic triangular lattice with α = −π/2 and α = π/2. In Fig. 5a–c, we show the case with α = −π/2, where the yellow arrow indicates the direction along which there is a positive π/2 phase in the coupling. For the gradually increasing pump powers, as indicated in Fig. 5a–c, we find that the evolution of the singlesite excitation exhibits an asymmetric behaviour along the direction of the effective gauge field. This situation can be clearly seen by a comparison with the case of α = π/2, which is shown in Fig. 5d–f for the same pump powers as in Fig. 5a–c, respectively. In particular, the patterns formed in the quantum walk, as shown in Fig. 5c, f, look like two arrows pointing in opposite directions. Comparisons of the experimental results with theory for Fig. 5a–c, d–f are shown using two animated images in the supplementary files; see Supplementary Figs. S2 and S3, respectively.
Fig. 5 Experimental observation of quantum walks in synthetic dimensions with artificial gauge fields.
a–c Experimental results for an effective complex coupling phase π/2 along the direction of the yellow arrow for pump average powers P = 0.10, 0.19, 0.28 mW, respectively. d–f The corresponding cases of a–c with the opposite direction of the π/2 phase 
We now discuss how 3D cubic lattices can be constructed through mapping and how the periodic boundary conditions wrap the cubic lattice into a 4D analogue of the 3D chiral tubes considered above. We keep using the lattice configuration in Fig. 2a as an example, which involves coupling orders 1, 3 and 4; however, we perform a different mapping procedure. We map the firstorder coupling to the basis vector u_{x}, the thirdorder coupling to u_{z} and the fourthorder coupling to −u_{y}; see Fig. 6a. By doing so, we effectively realize the Hamiltonian
Fig. 6 Construction of a fourdimensional analogue of chiraltube lattices.
a Sites of a cubic lattice constructed from the coupling orders 1, 3 and 4, which is wrapped in a fourth dimension, forming a chiral lattice. b Zoomedin view of the dashed panel in a, with the sites in one unit cell numbered from 1 to 26$$ {\bf{H}}_{{\bf{cub}}} =  \mathop {\sum}\nolimits_m {\left[ {C_1\hat a_{{\bf{r}}_{\bf{m}}}^\dagger \hat a_{{\bf{r}}_{\bf{m}} + {\bf{u}}_{\bf{x}}} + C_4\hat a_{{\bf{r}}_{\bf{m}}}^\dagger \hat a_{{\bf{r}}_{\bf{m}}  {\bf{u}}_{\bf{y}}} + C_3\hat a_{{\bf{r}}_{\bf{m}}}^\dagger \hat a_{{\bf{r}}_{\bf{m}} + {\bf{u}}_{\bf{z}}}} \right]  {\mathrm{H}}.{\mathrm{c}}.} $$ (10) The mapped cubic lattice is subject to a nontrivial periodic boundary condition. In contrast to the 2D case, where the periodic boundary is described by a wrapping vector, as discussed above, the condition can be expressed here as a wrapping plane s, given by the equation x − 4y + 3z = const. Within each wrapping plane, we keep lattice sites with no repetitions, which gives rise to the lattice structure shown in Fig. 6a. We determine the unit cell vector as l_{par} = u_{x} − 4u_{y} + 3u_{z}, which is shown as a red arrow in Fig. 6a. The colour map in Fig. 6a shows the coordinate of each lattice site in a unit cell along l_{par}. We further zoom in the dashed panel in Fig. 6a and show it as Fig. 6b, where we number all 26 sites in one unit cell. It is noteworthy that although this lattice structure is visualized in 3D, there are couplings (connections) enabled by a fourth dimension analogous to the wrapping of a tube. For example, in this 3D layout, site 15 has only two neighbouring sites, i.e., 14 and 19 (see Fig. 6b); however, it actually also interacts with sites 11, 12, 16, 18 and 19 (connections not shown) via a fourth dimension. This is the first example of using a synthetic lattice to realize a 4D generalization of tube lattices. We note that the geometry of this structure is equivalent to an open 3torus^{40}, which is one of the important models used to study the topology of the universe^{41}.