Simulating graphene dynamics in one-dimensional modulated ring array with synthetic dimension

A dynamically-modulated ring system with frequency as a synthetic dimension has been shown to be a powerful platform to do quantum simulation and explore novel optical phenomena. Here we propose synthetic honeycomb lattice in a one-dimensional ring array under dynamic modulations, with the extra dimension being the frequency of light. Such system is highly re-configurable with modulation. Various physical phenomena associated with graphene including Klein tunneling, valley-dependent edge states, effective magnetic field, as well as valley-dependent Lorentz force can be simulated in this lattice, which exhibits important potentials for manipulating photons in different ways. Our work unveils a new platform for constructing the honeycomb lattice in a synthetic space, which holds complex functionalities and could be important for optical signal processing as well as quantum computing.


INTRODUCTION
The honeycomb lattice, with the same geometry as the graphene [1,2], attracts great interest in condensed matter physics [3,4] and photonics [5,6]. Rich physical phenomena have been reported in photonic honeycomb lattices, taking advantages of properties of Dirac points and the valley degree of freedom [7][8][9][10][11][12][13][14] and showing excellent platform for studying topological photonics [15], which also have potential applications in the interface of nonlinear optics [16] and quantum optics [17], pointing towards topological fiber [18], topological laser [19], as well as edge and gap solitons [20]. Different platforms have been achieved to construct photonic honeycomb lattices, such as waveguide arrays [21], on-chip silicon photonics [22], semi-conductor microcavities [23], and metamaterials [24]. However, one notes that the reconfigurability and feasibility of a system are attractive for satisfying various experimental requirements and different applications, as well as relaxing the fabrication constrains, which are naturally limited in the current honeycomb systems due to their fixed configurations or structures after fabrication. Therefore, it is of significant importance to find an alternative platform, which is experimentally feasible and holds reconfigurability.
Here, we propose the construction of a highly re-configurable honeycomb lattice in a synthetic space in a modulated ring resonator system. A ring resonator under dynamic modulation has been found to be capable for creating a synthetic dimension along the frequency axis of light [25], which together with spatial dimensions, a variety of physical implementations have been suggested with two or more dimensions [25][26][27][28][29][30][31]. We show that a one-dimensional ring resonator array composed by two types of resonators under proper dynamic modulations supports a two-dimensional honeycomb lattice in a synthetic space including both spatial and frequency dimensions. Different physics associated with the photonic graphene can be simulated in this unique platform, such as Klein tunneling [13], valley-dependent edge states [12], topological edge states with the effective magnetic field [32], and valley-dependent Lorentz force [8]. The modulated ring systems can be achieved in either fiber-based system [33][34][35][36] or on-chip lithium niobate resonator [37], which brings our proposal to a flexible experimental setup with state-of-art technologies in bulk optics or integrated photonics. Our work not only broadens the current research on synthetic dimensions in photonics [38,39], but also enriches quantum simulations with topological photonics [40,41], which shows potential applications in optical signal processing [42,43] and quantum computations [44,45].

Model.
We begin with considering a pair of ring resonators (labeled as A and B) with the same circumference L undergoing dynamic modulation [see Fig. 1(a)]. The central resonant frequencies of ring A and ring B are set at ω 0 and ω 0 − Ω/2, respectively. In the absence of group velocity dispersion, the frequency of the m th resonant mode in ring A (ring B) is where Ω = 2πv g /L is the free spectral ranges (FSR) with v g being the group velocity inside both rings. We place electro-optic modulators (EOM) inside two rings, with modulation frequency Ω/2 and modulation phase φ. A synthetic frequency dimension with the effective hopping amplitude g can be constructed with spaced frequency Ω/2 in the frequency axis of light, where modes supported by rings A and B are labeled by a and b, respectively. With the building block for constructing the onedimensional synthetic frequency dimension in a pair of rings, we can then use it to construct a synthetic honeycomb lattice in a one-dimensional array of pairs of rings shown in Fig. 1(b) [see Methods]. The ring array consists groups of rings (n = 1, 2, ..., N), each of which contains two pairs of rings with different combinations, i.e., AB (labeled as α = 1) and BA (labeled as α = 2), respectively. We write the Hamiltonian of the system under the first-order approximation: n,m,1 a n−1,m,2 + b † n,m,2 b n,m,1 ) +g 2n−1 (a † n,m,1 b n,m,1 e iφ 2n−1 + a † n,m,1 b n,m+1,1 e −iφ 2n−1 ) +g 2n (b † n,m,2 a n,m−1,2 e iφ 2n + b † n,m,2 a n,m,2 e −iφ 2n )] + h.c., where a † n,m,α (a n,m,α ) and b † n,m,α (b n,m,α ) are corresponding creation (annihilation) operators, and κ is the evanescent-wave coupling strength between two rings at the same type. The Hamiltonian in Eq. (1) therefore supports a two-dimensional synthetic honeycomb lattice [see Fig. 1 The honeycomb lattice is constructed in a synthetic space including the spatial (x) and frequency (f ) dimensions. Different from conventional photonic honeycomb lattice in real space [12,13] that depends greatly on apparent geometry, the synthetic honeycomb lattice in Eq. (1) is dependent on both couplings between rings (κ) and modulations (g i ). Therefore, without loss of generality, we label the distance between two sites in the synthetic lattice with same types as d and the distance between two sites with different types as d/2 along the x-axis in the later plots of field patterns. The flexible choice of hopping amplitude g i and phase φ i provides the powerful reconfigurability towards different physical phenomena in quantum simulations. We first consider κ = g i = g and φ i = 0, and the synthetic lattice holds the unit cell with the translation symmetry including two frequency modes a and b in two rings. The band structure of this honeycomb lattice in the first Brillouin zone can be plotted in the k x -k f space, where k x and k f are wave vectors reciprocal to the spatial (x) and frequency (f ) axes, respectively. Dirac point K and K ′ at (k x , k f ) = (0, ±4π/3Ω) can be seen in the band structure in Fig. 1(d), where the zoomed-in Dirac cone shows linear dispersion near K ′ point. In the following, we will show the capability for achieving different phenomena associated to the honeycomb lattice with the current platform.
Klein tunneling. Klein tunneling, as an intriguing phenomenon in physics exploring a particle passing through a barrier higher than its energy, has experienced great interest in different platforms including graphene and photonic/phononic crystals [7,13]. To demonstrate such physics in the synthetic honeycomb lattice in Fig. 1(c), we couple the first (A 1 ) and the last (A 2N ) rings with an external waveguide such that a periodic condition along the x-axis can be naturally created. Such a design forms a carbon-nanotube-like shape [see excite the initial wave packet of the field shown in Fig. 2 The artificial square-shape barrier along the frequency axis of light is not easy to be constructed in the frequency dimension. In waveguide that composes the ring, there exists the group velocity dispersion that can introduce on-site potentials at modes with different resonant frequencies [27,46]. We now take the dispersion back into the consideration only in can be designed by the waveguide-structure engineering [47]. Note here the maximum value To verify edge states in two valleys in the synthetic honeycomb lattice, we further assume that there are boundaries at the frequency dimension, which can be achieved either by adding auxiliary rings to knock out certain modes at particular frequency [48] or by designing a sharp change in the group velocity dispersion of the waveguide that composes the ring [25].
Therefore, a lattice with the range of x∈[0, 17d] and f ∈[0, 23.5Ω] is considered in simulations.
We excite the 12 th ring on the artificial domain wall by a source field which has the Gaussian spectrum: Effective magnetic field. Photons are neutral particles, but it has been shown that, by introducing the proper distribution of hopping phases in a photonic lattice, one can create the effective magnetic field for photons [34]. In the synthetic honeycomb lattice in Fig. 1(c), we consider the modulation phase as φ 2n−1 = (2n − 1)φ and φ 2n = (2n)φ. In each unit cell, the clockwise accumulation of the hopping phase is −2φ, which naturally brings an effective magnetic field. In Fig. 4(a), we consider an infinite synthetic lattice and plot the projected band structure along φ, which gives the butterfly-like spectrum. The choice of phase can be tuned in each modulator arbitrarily. If we set φ = π/4, a projected band structure along the k f axis can be plotted by considering finite number of rings (with n∈ [1,40]). As shown in Fig. 4(b), one can see that there are 8 bulk bands, which is consistent with the fact that there are 8 sites in each unit cell once phases with φ = π/4 are considered. The middle two bulk bands have degenerate points. Meanwhile, there are 6 gaps between bulk bands, where it supports 8 pairs of topologically-protected edge states. By analyzing the distribution of the eigenstate for each edge state, we can find whether the edge state is located on the left or right edge, as shown in Fig. 4(b). Moreover, the Chern number for each bulk band counting from the highest band can be calculated as 1, 1, −3, 2, −3, 1, 1, respectively.
We then perform simulations in a synthetic honeycomb lattice in 12 pairs of rings with f ∈[0, 11.5Ω]. To excite a specific edge state in Fig. 4(b), we choose a single-frequency source field at the frequency ω m=5,A + ω s near the 5 th mode with a small detuning ω s , i.e., there is no effective magnetic field, and choose ω s = ε 1 = 0.9g. One can see in Fig. 4(c) that the intensity distribution of the field undergoes a random-walk-like propagation in the synthetic honeycomb lattice, and bulk of the lattice is excited. Next we set φ = π/4 and introduce the effective magnetic field. In this case, we again consider the excitation ω s = ε 1 , and plot the result in Fig. 4(d). Different from Fig. 4(c), here one can see the topologicallyprotected one-way edge state propagating towards lower frequency components at the right boundary, which is consistent with the negative slope of the edge state at ε 1 in Fig. 4(b).
We further use ω s = ε 2 = 1.7g and ω s = ε 3 = 2.2g to perform simulations and plot results in Figs. 4(e) and 4(f), respectively. In both cases, the excited edge states propagate towards higher frequency components unidirectionally, corresponding to different edge states with the positive slope. Although we study phenomena only associated to the effective magnetic field with φ = π/4, the gauge field can be easily tuned in this synthetic honeycomb lattice.
Valley-dependent Lorentz force. Different from the effective magnetic field for photons introduced by modulation phases, a pseudo magnetic field can also be alternatively generated by applying non-uniform strain in the honeycomb lattice [49]. In our proposed synthetic honeycomb lattice, we can also easily simulate the effective valley-dependent Lorentz force by varying modulation strength in each ring. potential A f (x) ∝ g(x) − κ and A x = 0 in the vicinity of the Dirac point, where the relation between n and x is used and κ is assumed to be a constant. This effective gauge potential leads to a pseudo magnetic field B ∝ η and along the z direction. Therefore, one can tune η by changing modulations strengths in rings to vary B in the synthetic honeycomb lattice.
In simulations, we inject fields into multiple rings with different frequency components to excite a Gaussian-shape wave packet We first excite the vicinity of Dirac point by a wave packet s 4 with (k x , k f ) = (0, 4.4π/3Ω) which gives an initial group velocity pointing towards the negative frequency axis. Without the pseudo magnetic field (η = 0), the wave packet of the field propagates without changing the direction and its trajectory is straight, as shown in Fig. 5(a). Instead, if η = 0.004, the motion of the wave packet is bent to the clockwise side due to the pseudo magnetic field.
On the other hand, if we excite the K ′ point with (k x , k f ) = (0, −4.4π/3Ω), the motion of the wave packet is bent to the counter clockwise side under pseudo magnetic field in the synthetic lattice [see Fig. 5(b)]. In Figs. 5(c)-5(d), we also excite the vicinity of Dirac points K and K ′ with (k x , k f ) = (2.2π/3Ω, −2.2π/3Ω) and (2.2π/3Ω, 2.2π/3Ω), respectively. One see that, with a non-zero η, trajectories of the field are bent towards different directions. The direction of the pseudo magnetic field is dependent on the valley in the synthetic honeycomb lattice, which therefore results in the field bending effect by the effective valley-dependent Lorentz force.

DISCUSSION
The highly tunable parameters of modulated ring resonators are of apparent significance in our design for achieving the synthetic honeycomb lattice, which can be realized in potential experiments based on established platforms with fiber loops [33][34][35][36], and lithium niobite technologies [37,53]. For the fiber-based ring resonator, the modulation frequency is ∼10 MHz for a fiber length of ∼10 m. 2×2 fiber couplers with a high-contract splitting ratio can be used to couple two rings. As for the on-chip lithium niobate device, fields in nearby resonators are coupled through evanescent wave, where the modulation frequency can reach to ∼10 GHz when the ring radius is ∼2-3 mm [37]. As an important note, the proposed method that we use to build the synthetic honeycomb lattice through staggered resonances also provides a new perspective for further constructing other complicated lattice structures with C 3 symmetry, such as triangular lattice and kagome lattice, both of which hold rich physics in photonics [54][55][56][57].
In summary, we use an array of ring resonators composed by two types of rings undergoing dynamic modulations to form a two-dimensional honeycomb lattice in a synthetic space including one spatial dimension and one frequency dimension. We demonstrate a highly reconfigurable synthetic honeycomb lattice which can be used to simulate various phenomena including Klein tunneling, valley-dependent edge states, topological edge states with effective magnetic field, and field bending with the valley-dependent Lorentz force. Our work shows not only the capability for simulating quantum phenomena, valley-dependent physics, and topological states in a modulated ring system, but also points out an alternative way to control the frequency information of light with synthetic dimensions, which potentially enriches quantum simulations of graphene physics with photonic technologies.

METHODS
The construction of the synthetic honeycomb lattice. In Fig. 1(a), ring A (B) supports a set of resonant modes with frequency ω m,A = ω 0 + mΩ (ω m,B = ω 0 − Ω/2 + mΩ), which is plotted in blue (cyan) color. The effective hopping amplitude between the nearby resonant modes a and b in two rings is formed through a two-step process: the resonant mode a in ring A couples to a corresponding non-resonant mode in ring B via evanescent-wave coupling, and then couples to the resonant mode b in ring B via the dynamic modulation, vice versa. Hence the effective coupling strength g is composed by both the evanescent-wave coupling strength and the modulation strength in EOM [58], and hence such connections construct a synthetic frequency dimension in a pairs of modulated rings. The construction of effective couplings between resonant modes in a pair of rings with the AB type can also be generalized to the pair of rings with the BA type by mirror symmetry [see the pair of rings labelled by n = 1 and α = 2 as an example in Fig. 1(c)]. Therefore, in an array of pairs of rings arranged with alternate combinations AB and BA as shown in Fig. 1(b), the spatially nearby resonant modes at the same frequency can be coupled through the evanescent wave at the coupling strength κ. Following this procedure, a synthetic honeycomb lattice can be constructed in a space shown in Fig. 1(c) with the longitudinal frequency dimension and the horizontal spatial dimension.
Simulation method. We expand the field inside each ring as |ψ(t) = n,m (C n,m,α a † n,m,α + D n,m,α b † n,m,α )|0 , where C n,m,α and D n,m,α are the field amplitude at the m th mode in the corresponding ring. Schrödinger equation i|ψ(t) = H|ψ(t) is then used in simulations [25]. For different simulations, we take different source s to excite particular rings at specific modes, which we show details in the main text. The excitation process is done by coupling each ring with external waveguides where the light can be injected. The field amplitudes in rings can then also be readout through waveguides. The coupling equation between the external waveguide and the ring is   C n,m,α (D n,m,α )(t + ) where γ is the coupling strength between the external waveguide and the ring through evanescent wave. E C(D),m,α (t − ) (E C(D),m,α (t + )) is the corresponding field component injected (collected) through the external waveguide, and t ± = t + 0 ± .
An effective on-site potential induced by frequency shift. For a ring, the resonant frequency for the m th resonant mode is ω m = ω 0 + mΩ = ω 0 + m · 2πv g /L, where ω 0 is a reference optical frequency and is much larger than Ω (which is usually in the regime of GHz where I n,m,α = |C n,m,α | 2 or |D n,m,α | 2 , x n,m,α and f n,m,α are the corresponding position along the x-axis and frequency dimension in the synthetic lattice, respectively.