Dynamically Manipulating Topological Physics and Edge Modes in a Single Degenerate Optical Cavity

We propose a scheme to simulate topological physics within a single degenerate cavity, whose modes are mapped to lattice sites. A crucial ingredient of the scheme is to construct a sharp boundary so that the open boundary condition can be implemented for this effective lattice system. In doing so, the topological properties of the system can manifest themselves on the edge states, which can be probed from the spectrum of an output cavity field. We demonstrate this with two examples: a static Su-Schrieffer-Heeger chain and a periodically driven Floquet topological insulator. Our work opens up new avenues to explore exotic photonic topological phases inside a single optical cavity.

Figure 1

(a) Illustration of the experiment setup about the degenerate cavity. (b) The effective photonic circuit of (a). $\varphi$ is the imbalanced phase between the two arms of the auxiliary cavity.

Figure 2

(a) Proposed experimental setup of a single degenerate cavity to simulate a 1D finite lattice using composite modes induced by two SLMs with $\delta l=\pm L_m$. Two hollow beam splitters are employed so that only modes $l=0$ can transmit through the hole. (c) is the effective photonic circuit of (a). (b) Normalized intensity profiles $I_l(r)=|E_l(r){|}^2$ for different $l$ modes calculated using the parameters discussed in Ref. [34]. The vertical lines indicate the possible center pinhole sizes in each BS for different hopping steps $n=1$, 3, and 5, with the corresponding fraction of $l=0$ photon intensity inside the pinhole as 78%, 96%, and 99%, respectively.

Figure 3

Schematic diagram of simulating a 1D SSH chain. (a) shows the skeleton inside of the main cavity with two-auxiliary-cavity circuits depicted in (b) and (c), where optical circuits related to the hopping $J_0\cos \varphi$ and $J_1$ are shown. (d) is the diagrammatic representation of the Hamiltonian $H$. BS, beam splitter. HBS, BS with a pinhole in the center. SLM, spatial light modulator. HWP, half-wave plate. PBS, polarization beam splitter which transmits vertical polarized photons and reflects horizontal polarized photons.

Figure 4

(a) and (b) show the spectrum $\tau (\omega )$ of model (1) for $J_1=1$ and $L_m=49$ with the decay rates ${\gamma }_j=0.05(1+e^{-j/\sqrt{25}}+e^{-|j-L_m|/\sqrt{25}})$ for $n=2$ and $n=4$, respectively, where the influence of $H^{\prime }$ in the remaining lattices is also taken into account. (c) shows the output spectrum of a prolonged chain using the same decay as in (a), which corresponds to the soft boundary condition discussed in the context. (d) shows the dynamics of $N_0$ with hopping step $n=2$ for an input pulse $a(t{)}_{\mathrm{in},0}=\exp [-(t-3{)}^2/8]/\sqrt{2\sqrt{\pi }}$ at site $j=0$ with the decay given in (a) for fixed $J_0^{\prime }=0.9$ and 1.1, respectively. (e) plots the amplitude $N_0$ for different $n$ at $t=15$ along with $J_0^{\prime }$, which drops to zero across the phase transition point.

Figure 5

(a) The spectrum $T(\omega )$ within the Floquet zone $(-\mathrm{\Omega }/2,\mathrm{\Omega }/2)$ for different $\mathrm{\Omega }$ for hopping step $n=4$ with the same decay used in Fig. 4. Other parameters are $[J_0,J_1,\lambda ]=[2,1,0,1.6]$ and $L_m=49$. (b) and (c) show the spectra $T(0)$ and $T(\mathrm{\Omega }/2)$ along with $\mathrm{\Omega }$, where a topological transition is manifested by jumps around the phase transition points. $v_0$ and $v_{+}$ are their corresponding numbers of edge modes defined in Ref. [34].