Synthesizing arbitrary lattice models using a single degenerate cavity

We propose a general method to simulate arbitrary lattice models using a single degenerate cavity. Such a cavity supports a large number of degenerate optical modes with different angular momenta. Couplings between different optical modes can be readily controlled. These features allow us to simulate lattice models that are not convenient to realize using other systems, particularly models in high dimensions and with complicated hopping amplitudes. As a concrete example, we demonstrate how to construct two topological lattice models: the two-dimensional Haldane model and a four-dimensional time-reversal invariant model. For the latter case, we show how topological properties can be detected from the outputs of the cavity, where the second Chern number can be extracted. In the presence of open boundaries, the chirality of the Weyl edge modes can also be detected using the input-output formalism of the cavity modes.

Figure 1

Basic components to realize the arbitrary lattice using the artificial degrees of freedom with a concrete example in Fig. 2. The basic components contain three steps including one-dimensionalization, hopping term mapping, and hopping amplitude controlling. One-dimensionalization is just marking each DOF in the lattice system with an integer corresponding to that in the artificial system. (a) Components of hopping term mapping. Hopping term mapping is to create hopping terms between the artificial DOFs, which correspond to the lattice system DOF, using spatial light modulators. The SLM induces hopping terms between the light with orbital angular momenta $L$ and $L+\delta L$. (b) Components of hopping amplitude controlling. The hopping amplitude controlling process is to realize arbitrary hopping amplitudes, where beam rotators (BRs) induce OAM-dependent hopping amplitudes. The ${\mathrm{BR}}_1$ and ${\mathrm{BR}}_2$ induce $B_{1}exp\left(iL{\theta }_1\right)$ and $B_{2}exp\left(iL{\theta }_2\right)$ hopping amplitudes, respectively, for the OAM mode $L$. The hopping amplitudes for the OAM modes $L_1$ and $L_2$ are $A_{L_1}$ and $A_{L_2}$, respectively, which can be controlled by modifying $B_1$ and $B_2$, arbitrarily.

Figure 2

(a) Brick lattice structure with the nearest hopping terms and next-nearest hopping terms (not shown), realized by the OAM synthetic system. The brick lattice contains two types of internal states, including $A$ (closed circle) and $B$ (open circle), for each unit site, which is marked by the red dashed oval. For this case, we choose the number of unit sites in the first dimension $N_1=5$, so the lattice contains ten states in the first line. The circles $a–e$ are the long-distance hopping terms on the edge induced by the hopping terms in the OAM system. (b) All hopping terms of the Haldane model (brick lattice). The system contains nearest hopping terms, marked by the black solid lines $t_1$, and next-nearest hopping terms $t_2=texp\left(i\varphi \right)$ and $t_3=texp(-i\varphi )$, marked by red dashed lines and green dash-dotted lines, respectively. The blue dash-dotted oval shows the hopping terms with hopping step $\delta L=9$.

Figure 3

Scheme for the $\delta L=9$ hopping step for Haldane's model in the OAM degeneracy cavity system. (a) Structure of cavities including a main cavity and two auxiliary cavities. The main cavity contains four beam splitters (BSs), which connect the main cavity and the two auxiliary cavities. Each auxiliary cavity contains a pair of SLMs to modify the OAM of photons. After crossing ${\text{SLM}}_1$ and ${\text{SLM}}_3$ (${\text{SLM}}_2$ and ${\text{SLM}}_4$), the photons increase (decrease) nine OAMs. One auxiliary cavity contains additionally a pair of beam rotators. The BRs rotate the light through the propagating direction, where the photons with OAM $L$ gain a phase $exp\left(iL\theta \right)$, where $\theta$ is the angle of rotation. The angle $\theta$ of ${\text{BR}}_3$ (${\text{BR}}_4$) equals $\pi$ ($-\pi$). (b) Effective cavity structure of $\delta L=9$ hopping steps of Haldane's model including a main cavity chain and auxiliary cavities. The main cavity chain is realized by the main cavity in (a) through the OAM DOF. The top row auxiliary cavities have the same hopping amplitude $B_{9,0}$, but the hopping amplitudes of the bottom row cavities are modified periodically by $B_{9,1}$ and $-B_{9,1}$, as the gain of phases through BRs is periodic for the OAM $L$. (c) Lattice structure of $\delta L=9$ hopping steps, where each unit cell contains two internal DOFs. The hopping amplitudes are $A_{(1,-1,1),0}=t_1$ and $A_{(-1,0,1),1}=0$, which are determined by Eq. (7).

Figure 4

Approximate second Chern number varying with the mass parameter $m$, the number of sites in each dimension $N$, and the cavity loss rate $\gamma$ in discrete momentum space shown in Eq. (E8) with $L_{\mu }=N$. (a) Second Chern number ${\text{Ch}}_2$ varying with the mass parameter $m$ with $N=15$ and $\gamma =1\times {10}^{-5}$. The mass $m$ varies from $-5$ to 5. (b) Chern number varying with the number of sites in each dimension $N$. We set the mass $m=-1$ and $\gamma =1\times {10}^{-5}$. (c) Chern number varying with the loss rate $\gamma$ with $m=-1$ and $N=15$.

Figure 5

Isosurface of the detection chirality $\tilde{C}\left(\vec{k}\right)={\langle \vec{x}\rangle }_1·{\langle \vec{x}\rangle }_2\times {\langle \vec{x}\rangle }_3$, which is the approximation of Eq. (28) using ${\langle \vec{x}\rangle }_a$ to replace ${\vec{v}}_a$, of Weyl chiral edge states with (a) the mass $m=-3$ and Chern number ${\text{Ch}}_2=1$ and (b) the mass $m=-1$ and Chern number ${\text{Ch}}_2=-3$, where ${\langle \vec{x}\rangle }_a$ is defined by Eq. (29), $|{\tilde{C}}_m|$ is the maximum value of $|\tilde{C}|$ in momentum space $\vec{k}$, the frequency of input light is $\omega =0.5$, the loss rate is $\gamma =0.1$, the number of unit cells in each direction is $N=11$, and the width of the wave packet is $\mathrm{\Omega }=1$.

Figure 6

Beam rotator. The beam rotator includes two Dove prisms intersecting with each other at an angle $\theta$. It induces a extra phase $exp\left(iL\theta \right)$, when a photon with OAM $L$ crosses it.

Figure 7

(a) Scheme of the experimental setup used to simulate the 1D lattice with an open boundary condition on the left edge. The main cavity contains two beam splitters (${\text{BS}}_1$ and ${\text{BS}}_2$) with one hole to block the hopping on the light with OAM $l=0$. (b) Effective cavity line of (a). The holes in BSs block the hopping on the light with OAM $l=0$. (c) Strength of light of the Laguerre Gaussian with $p=0$ and OAM $l$. (d) Effective hopping factor ${\eta }_l$ varies with the OAM $l$ by choosing $r_h$ to guarantee ${\eta }_{4N^3}=1-{\eta }_{4N^3+1}$, where the number of unit cells in one dimension $N=11$.