Topological photonic orbital-angular-momentum switch

The large number of available orbital-angular-momentum (OAM) states of photons provides a unique resource for many important applications in quantum information and optical communications. However, conventional OAM switching devices usually rely on precise parameter control and are limited by slow switching rate and low efficiency. Here we propose a robust, fast, and efficient photonic OAM switch device based on a topological process, where photons are adiabatically pumped to a target OAM state on demand. Such topological OAM pumping can be realized through manipulating photons in a few degenerate main cavities and involves only a limited number of optical elements. A large change of OAM at $\sim {10}^q$ can be realized with only $q$ degenerate main cavities and at most $5q$ pumping cycles. The topological photonic OAM switch may become a powerful device for broad applications in many different fields and motivate a topological design of conventional optical devices.

Figure 1

(a) Experimental photonic circuit for a degenerate cavity system. The main cavity (red curved mirrors) is coupled with auxiliary cavities (blue curved mirrors) by beam splitters (BSs). A tunable beam splitter (TBS) is used to tune the coupling between the cavity (in $H$ polarization) and the input/output fields (in $V$ polarization). (b) Optical design for the TBS. (c) Schematic diagram of the system with three auxiliary cavities (dashed lines), with corresponding tunneling phase ${\varphi }_s$ and step $M_s$ as labeled.

Figure 2

(a) Diagram of the effective lattice in the OAM dimension. Each unit cell (enclosed by the dashed square) contains two sites with detuning $\mathrm{\Delta }=-4J_{0}cos\left[{\beta }_0\left(t\right)\right]$. $J_{\pm }=J_1[1\pm e^{i{\beta }_1\left(t\right)}]$ are the inter- and intra-cell tunnelings, respectively, the dependence of which on the tunneling phase ${\beta }_1$ is shown in (b). (c) The pumping loop in the ${\beta }_0\text{−}{\beta }_1$ plane. Since the tunneling phase has a period of $2\pi$, their choice is not unique, and the two loops (red dashed and blue solid) have the same topology. (d) Illustration of the pumping process. The loop in the $\delta J\text{−}\mathrm{\Delta }$ plane can be realized by either loop shown in (c). The red solid circle represents a site occupied by a photon, which moves to the right by two sites (one unit cell) during one pumping cycle.

Figure 3

(a) The band structure in one pump cycle with Chern number $\mathcal{C}=\pm 1$ for two bands. (b) Cavity photon distribution in two pump cycles with steplike transport. $P_l\left(t\right)$ is the probability of the photon in OAM state $l$ at time $t$. (c) Center-of-mass displacements (dots) and their standard deviations (error bars), which are calculated by assuming Gaussian distributed random disorders $\delta {\beta }_{0,1}$ in the tunneling phases with standard deviations $\sigma \left(\delta {\beta }_{0,1}\right)=0.1\mathrm{rad}$. (d) The final-state purities at $\frac{m}{2}$ pump cycles ($t_m=\frac{mT}{2}$) vs different coupling ratio $J_0/J_1$. The pink stars, blue cycles, black crosses, and red squares correspond to $m=1$, 2, 3, and 4, respectively. The modulation of the pump parameters is chosen to satisfy the adiabatic condition with period $T=21/J_1$, the temporal profiles of which are shown in (c) with ${\beta }_0$ the red solid line and ${\beta }_1$ the red dashed line. (b)–(d) The initial OAM state is $l_0=0$. (a)–(c) The coupling ratio is $J_0/J_1=5$.

Figure 4

(a) Normalized cavity photon distribution and (b) input/output photon pulses during the OAM switching. The normalized input pulse is ${\mathcal{E}}_{\text{in}}\left(t\right)=e^{-i{E}_{-}t-0.2{(J_{1}t-5)}^2}$ with $l_0=0$ (i.e., only the zero-OAM Wannier orbital is excited). $E_{-}$ is the lower band energy. After the input pulse enters the cavity, we pump the state to higher OAM modes, and then release the signal by increasing the cavity loss $\kappa$ [see the inset in (a)]. Other parameters are the same as in Fig. 3.

Figure 5

Illustration of a multistage setup with $N=10$. Each half pump cycle shifts the OAM by the corresponding tunneling step.

Figure 6

Plot of frequency difference $\delta \omega$ between adjacent OAM modes normalized to the band gap $4J_1$ with respect to the width and length of the rectangular cavity. The white regions represent configurations where the cavity is unstable. Our scheme works well if $\left|l\right|\frac{{\delta }_{\omega }}{4J_1}\lesssim 1$. Other parameters are $F=10\mathrm{cm}$, $J_1=0.004{\mathrm{\Omega }}_{\text{F}}$.

Figure 7

The equivalent circuit in the OAM space, with $\varphi$ the phase imbalance between two arms of the auxiliary cavity and $a_l$ the field operator for OAM mode $l$. The red (blue) loop represents the main (auxiliary) cavity.

Figure 8

(a) Normalized cavity photon distribution and (b) input/output photon pulses in the OAM switching. The parameters are the same as in Fig. 4 in the main text, except that the photon loss is nonzero during the pumping with ${\kappa }_0=0.02J_1$.

Figure 9

(a) Center-of-mass displacements (red dots) and their standard deviations (blue error bars), which are calculated by assuming Gaussian distributed random on-site energy shift $\delta {\omega }_l=\left|l\right|\delta \omega$ due to deviations from the degeneracy. Standard deviations $\sigma \left(\delta \omega \right)=0.05J_1$ and tunneling step $M_1=1$ corresponds to the zeroth stage of the multistage switch. $\delta \omega =0.05J_1$ corresponds to $\sqrt{(X-F)(Y-F)}\simeq 10\mu \mathrm{m}$. (b) The same as in (a) except that the tunneling step $M_1=10$, corresponding to the first stage of the multistage switch.