Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation

We show that a single ring resonator undergoing dynamic modulation can be used to create a synthetic space with an arbitrary dimension. In such a system, the phases of the modulation can be used to create a photonic gauge potential in high dimensions. As an illustration of the implication of this concept, we show that the Haldane model, which exhibits nontrivial topology in two dimensions, can be implemented in the synthetic space using three rings. Our results point to a route toward exploring higher-dimensional topological physics in low-dimensional physical structures. The dynamics of photons in such synthetic spaces also provides a mechanism to control the spectrum of light.

Figure 1

(a) A single ring resonator undergoes dynamic modulation. (b) The structure in (a) can be described by a one-dimensional tight-binding model with nearest-neighbor coupling along the frequency dimension.

Figure 2

(a) A single ring resonator undergoes dynamic modulation with a format as described by Eq. (6) with $N=4$. (b) Such a design supports a one-dimensional model along the frequency dimension with the long-range connectivity. (c) Such a one-dimensional model is equivalent to a two-dimensional $4\times 4$ square lattice structure with additional connections labeled by dashed-dotted curves. (d) The lattice in (c) can alternatively be represented by a two-dimensional lattice on a cylindrical surface with a twisted boundary condition. In (c) and (d), the blue and red lines correspond to the coupling induced by the first and second terms in Eq. (6), respectively.

Figure 3

(a) A single-unit cell of the Haldane model consists of a honeycomb lattice with nearest-neighbor coupling, and the coupling between the next-nearest neighbors along the $y$ directions, which carries a direction-dependent coupling phase. (b) The projected band structure for the Haldane model of (a) in a stripe geometry. The stripe is infinite along the $y$ direction and has 12 unit cells along the $x$ direction. (c) Two-ring resonators $A$ and $B$ with resonant frequencies ${\omega }_{A,m}$ and ${\omega }_{B,m}$ respectively. Each ring has a phase modulator (PM) inside it. In between, there is an auxiliary ring $C$ with resonant frequencies ${\omega }_{C,m}$ with two phase modulators inside it. (d) A one-dimensional synthetic lattice along the frequency axis is created by the dynamic modulators placed in rings as shown in (c). Each unit cell of the lattice consists of a pair of the $m\mathrm{th}$ resonant mode $\left({\omega }_{A,m},{\omega }_{B,m}\right)$ in the $A$ and $B$ rings. (e) The one-dimensional model in (d) is equivalent to a two-dimensional honeycomb lattice structure with additional connections labeled by dashed-dotted curves. (f) The two-dimensional lattice in (e) can be represented by a honeycomb lattice on a cylindrical surface with a twisted boundary condition.

Figure 4

(a) Same as Fig. 3, except with external waveguides coupling to rings $A$ and $B$. (b) The time sequence for the input source (in black curve) with the temporal width $T_S$, the modulation (in red curve) with $T_M$, and the signal collection (in green curve) with $T_C$ in the proposed experiment. All of these characteristic time windows have a turn-on/-off time $T_O$.

Figure 5

(a) Output spectra in external waveguide $A$ for the system as shown in Fig. 4, with $T_M=T,2T,3T,4T$, respectively. The spectra are normalized to the range $[0,1]$. Only 22 resonant modes are plotted. The amplitudes for other modes are close to zero. (b) Output spectra are replotted in the synthetic two-dimensional space for $T_M=T, 2T, 3T$, and $4T$, respectively. Red arrows show the frequency of the input excitation. Blue arrows denote the unidirectional transport in the synthetic space.