Topological photonic orbital angular momentum switch

TL;DR

This paper introduces a topological, robust, and efficient photonic OAM switch that enables rapid and large changes in photon orbital angular momentum states using a minimal number of optical components.

Contribution

The authors propose a novel topological process for photonic OAM switching that is faster, more efficient, and more robust than conventional methods, utilizing only a few cavities and optical elements.

Findings

Achieves large OAM change of ~10^q with q cavities and 5q cycles.

Provides a robust, fast, and efficient OAM switching mechanism.

Potential for broad applications and new topological optical device designs.

Abstract

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 novel topological design of conventional optical devices.

Equations

$${\omega }_{p,l}=n{\Omega }_{\text{F}}+(2p+|l|+1)\frac{\arccos (\frac{A+D}{2})}{2\pi }{\Omega }_{\text{F}},$$

$$H=-{\sum }_l{\sum }_{s=0}^2{J}_s{e}^{i{\varphi }_s}{a}_{l+M_s}^{†}{a}_l+h.c.,$$

$$H$$

$${\mathcal{C}}_n=\frac{1}{2\pi }{\int }_0^{T}dt{\int }_0^{2\pi }dk{\Omega }_{kt}$$

$${\Omega }_{kt}=⟨{\partial }_t{u}_n(k,t)|{\partial }_k{u}_n(k,t)⟩-⟨{\partial }_k{u}_n(k,t)|{\partial }_t{u}_n(k,t)⟩.$$

$$|\Psi (j,0)⟩=\sum _k{\psi }_k{e}^{ijk}|u_n(k,0)⟩,$$

$$|\Psi (j,t)⟩=\sum _k{\psi }_k{e}^{ijk}{e}^{-i{\int }_0^{t}d{t}^{\prime }{E}_n(k,t^{\prime })}{e}^{i{\gamma }_n(k,t)}|u_n(k,t)⟩,$$

$${\gamma }_n(k,t)=i{\int }_0^t⟨u_n(k,t^{\prime })|{\partial }_{t^{\prime }}{u}_n(k,t^{\prime })⟩d{t}^{\prime }.$$

$$\bar{j}(t)$$

$$|\tilde{\Psi }(k,t)⟩={\psi }_k{e}^{-i{\int }_0^{t}d{t}^{\prime }{E}_n(k,t^{\prime })}{e}^{i{\gamma }_n(k,t)}|u_n(k,t)⟩.$$

$$\bar{j}(t)={\sum }_{k}i{\psi }_k^{\ast }{\partial }_k{\psi }_k+{\int }_0^{t}I(t^{\prime })d{t}^{\prime }$$

$$I(t)=\sum _k|{\psi }_k{|}^2[{\partial }_{k}E(k,t)+{\Omega }_{kt}].$$

$$\Delta \bar{j}$$

$$|\Psi (j,0)⟩=\sum _k\frac{1}{\sqrt{N}}{e}^{ijk}|u_n(k,0)⟩\equiv |W_n(j,0)⟩,$$

$$\Delta \bar{j}=\frac{1}{N}\sum _k{\int }_0^T{\Omega }_{kt}dt={\int }_0^{T}dt{\int }_0^{2\pi }\frac{dk}{2\pi }{\Omega }_{kt}={\mathcal{C}}_n.$$

$$E_{\pm }$$

$${\dot{a}}_l=\frac{1}{i}[a_l,H]-\frac{\kappa }{2}{a}_l+{\delta }_{l,l_0}\sqrt{{\kappa }_{\text{e}}}{\mathcal{E}}_{\text{in}}(t),$$

$$H=-{\sum }_{l}J{e}^{i\varphi }{a}_{l+1}^{†}{a}_l+h.c.,$$

$$\dot{\rho }=\frac{1}{i}[H,\rho ]+{\kappa }_0\sum _l(a_l\rho a_l^{†}-\frac{1}{2}\rho a_l^{†}{a}_l-\frac{1}{2}{a}_l^{†}{a}_l\rho )$$

$$\rho =(1-e^{-{\kappa }_{0}t})|vac⟩⟨vac|+e^{-{\kappa }_{0}t}|\Psi (t)⟩⟨\Psi (t)|,$$

$$N(l,t)$$

$${\dot{a}}_l=\frac{1}{i}[a_l,H]-\frac{\kappa }{2}{a}_l+{\delta }_{l,l_0}\sqrt{{\kappa }_{\text{e}}}{\mathcal{E}}_{\text{in}}(t),$$

Full text

I Introduction

Discrete degrees of freedom, such as charge, spin, valleys, etc., play a crucial role in many information encoding and device applications nielsen2002quantum ; gisin2002quantum ; culcer2012valley ; andersen2015hybrid . In this context, a fundamental degree of freedom of photons, the orbital angular momentum (OAM), possesses a unique property that an infinite number of distinctive OAM states are available allen1992orbital ; molina2007twisted ; yao2011orbital . This unique property makes photonic OAM very attractive for various applications in optical communication malik2016multi ; barreiro2008beating ; wang2012terabit , quantum simulation cardano2015quantum ; cardano2016statistical ; luo2015quantum ; zhou2017dynamically , quantum information luo2017synthetic ; fickler2012quantum ; fickler2014interface ; wang2015quantum , and quantum cryptography (e.g., key distribution) groblacher2006experimental ; mafu2013higher ; vallone2014free ; mirhosseini2015high . To fully utilize these applications, a tunable device that can rapidly and robustly switch between different OAM modes on demand is therefore highly desirable. However, many conventional OAM switching devices rely on precise parameter control and are usually limited by slow switching rates ($\sim$kHz) thalhammer2013speeding ; mirhosseini2013rapid or low purity and efficiency radwell2014high ; strain2014fast , and limited number of usable OAM modes marrucci2006optical ; slussarenko2011efficient .

The photonic OAM is a discrete degree of freedom that characterizes the topological charge (i.e., the winding of the azimuthal phase) of a photon field with cylindrical symmetry. Therefore a natural question is whether a topological process can be designed to create a robust photonic OAM switching device with high performance. Recently, the study of topological photonics has become one frontier direction in optical physics with the major focus on modulating photon propagation through topological edge states haldane2008possible ; hafezi2011robust ; fang2012realizing ; lin2014light ; lu2014topological ; yuan2016photonic ; lin2016photonic , while practical topological photonic devices for on-demand switching of photon internal degrees of freedom are still largely lacking.

In this paper, we propose a practical photonic OAM switching device through the topological adiabatic pumping of OAM states, which is robust (immune to small perturbations in system parameters), fast ($\sim$ MHz switching rate), efficient ($\sim 90\%$ efficiency and $\sim 100\%$ purity in principle), and highly tunable (on demand switch for high-OAM states). Topological pumping was initially proposed by Thouless thouless1983quantization for a periodically time-varying system, and has been recently realized in ultra-cold atom optical lattices lohse2015thouless ; nakajima2016topological and coupled optical waveguides kraus2012topological ; verbin2015topological , where particles are adiabatically pumped in real space lattices. Such “Thouless pumps” along particle’s internal degrees of freedom have not been well explored, although atomic hyperfine states have been utilized as a synthetic dimension for studying quantum Hall effects celi2014synthetic ; mancini2015observation ; stuhl2015visualizing .

Here we show how to realize Thouless pumps in the synthetic OAM space of photons where large numbers of OAM modes are available, which provides the basis for engineering a topological photonic OAM switch. Essentially, we propose a scheme to realize a tunable double well OAM lattice (two lattice sites in each unit cell) by manipulating photons in a single degenerate main cavity, where photons can be adiabatically pumped to target OAM lattice sites (i.e., the OAM states). Thanks to the topological nature of the pumping, precise switch can be realized without involving precise control of the parameters during the pumping cycle. Such robustness of the pumping against perturbations as well as its properties in the presence of photon losses are considered. Moreover, even though the pumping process is adiabatic, the pumping cycle can be very short. With realistic optical elements, it is possible to achieve a switching rate of $\sim$MHz, which is much faster than commonly used OAM switching devices thalhammer2013speeding ; mirhosseini2013rapid . Even for a large change of OAM at $\sim 10^{q}$, only $q$ degenerate main cavities and at most $5q$ pumping cycles are needed using a multistage setup, leading to exponential speedup of the OAM switch. The proposed topological photonic OAM switch only relies on a simple optical setup with a few degenerate main cavities, therefore it may become a powerful device for broad applications in quantum information and optical communications. Our proposed topological approach for device engineering may go beyond the OAM switch and inspire novel topological design of conventional optical devices.

II The system

As shown in Fig. 1(a), our system contains a main degenerate multimode cavity arnaud1969degenerate ; horak2002optical ; gopalakrishnan2009emergent ; ballantine2017meissner ; schine2016synthetic , which supports a large number of OAM modes and is coupled with three degenerate auxiliary cavities by beam splitters. The degenerate cavities used in our scheme possess fully transverse degeneracy and their design principle is based on the round-trip Gauss matrix [often called $ABCD$ ray matrix with $A,D$ ($B,C$) diagonal (off-diagonal) matrix elements] arnaud1969degenerate . We consider a ring-type cavity possessing cylindrical symmetry with respect to the optical axis, the cavity modes are Laguerre-Gaussian (LG) modes $E_{p,l}$ with radial index $p$ and azimuthal index $l$. The resonance frequencies of the LG modes are yariv2007photonics ; hodgson2005laser

[TABLE]

where the integer $n$ is the longitudinal mode index that is fixed in our scheme and $\Omega_{\text{F}}$ is the free spectral range of the cavity. The off-diagonal ray matrix elements $B$ and $C$ only affect beam waist size. The degenerate cavity is obtained when the round-trip $ABCD$ ray matrix is equal to identity arnaud1969degenerate , which can be achieved using curved mirrors or intracavity lenses. In this case, all transverse LG modes $E_{p,l}$ have the same resonance frequency $n\Omega_{\text{F}}$, yielding a degenerate cavity supporting different OAM modes $l$ for photons. The tunneling between different OAM states is realized by spatial light modulators (SLMs). The unimportant radial index $p$, which is related to the radial mode profile of the input photons, is omitted. There are many ways to construct a degenerate cavity, for example, we consider a rectangular-shaped cavity shown in Fig. 1(a) formed by four identical curved mirrors with focal length equal to optical path length between adjacent mirrors (see Appendix A). We use a tunable beam splitter (TBS) to couple the cavity with the outside world, it is realized by sandwiching an electro-optic modulator (EOM) yariv2007photonics between two polarizing beam splitters (PBSs), as shown in Fig. 1(b). The EOM rotates photon’s polarizations in a tunable way as $|H\rangle\rightarrow\sqrt{1-r_{\text{P}}^{2}}|H\rangle+r_{\text{P}}|V\rangle$ and $|V\rangle\rightarrow\sqrt{1-r_{\text{P}}^{2}}|V\rangle-r_{\text{P}}|H\rangle$, with $|H\rangle$ and $|V\rangle$ the horizontal and vertical polarization states which are separated by the PBS. The tunable coefficient $r_{\text{P}}$ acts as the reflectivity of the TBS.

The beam splitters divert a small portion of the main-cavity photons towards the $s$-th auxiliary cavity, and merge them back after passing through spatial light modulators (SLMs) oemrawsingh2005experimental ; karimi2009efficient , which change the OAM state by $\pm M_{s}$. This corresponds to tunnelings between different OAM states in the main cavity, with a tunneling rate determined by the reflectivities of the BSs. During the tunneling, the photon can also acquire a phase determined by the optical path-length difference between two arms of the auxiliary-cavity, which can be generated and tuned using high-speed phase modulators (PMs). The system is equivalent to a one-dimensional lattice model with the lattice sites represented by the OAM states luo2015quantum ; zhou2017dynamically ; luo2017synthetic . The Hamiltonian is given by (see Appendix A)

[TABLE]

where $a_{l}$ is the annihilation operator of the cavity photon in the OAM state $l$, and $M_{s}$ is the step index of the SLMs in the $s$-th auxiliary cavity with the tunneling amplitude $J_{s}$ and phase $\phi_{s}=l\alpha_{s}+\beta_{s}$. The PM contains a beam rotator leach2002measuring and an EOM yariv2007photonics : the beam rotator, realized by two Dove prisms rotated by $\alpha_{s}/2$ with respect to each other, is used to generate the $l$-dependent phase $l\alpha_{s}$, and the EOM is used to tune the $l$-independent phase $\beta_{s}$.

The auxiliary cavities are designed as $\phi_{2}=0$, $M_{0}=0$, $M_{1}=M_{2}=1$, $J_{1}=J_{2}$, as shown in Fig. 1(c). Therefore the Hamiltonian describes a generalized AAH model ganeshan2015constructing with modulations in on-site energy, tunneling phase and amplitude, whose periods are determined by $\alpha_{0}$ and $\alpha_{1}$. Here, we focus our discussion on $\alpha_{0}=\alpha_{1}=2\pi\cdot\frac{1}{2}$, in analogy to the Rice-Mele model (i.e., double-well supper-lattice model) rice1982elementary with neighboring site energy detuning $\Delta\equiv-4J_{0}\cos(\beta_{0})$ and inter- and intra-cell tunnelings $J_{\pm}\equiv J_{1}[1\pm e^{i\beta_{1}}]$ [see Fig. 2(a)]. We rewrite the Hamiltonian as

[TABLE]

where we have introduced the unit cell index $j$ with $b_{j,1}=a_{2j}$ and $b_{j,2}=a_{2j+1}$. After a Fourier transformation $b_{k,1(2)}\propto\sum_{j}e^{ijk}b_{j,1(2)}$, the Hamiltonian in the Bloch basis becomes

[TABLE]

with the Bloch Hamiltonian $H_{k}(t)=\mathbf{h}(k,t)\cdot\vec{\sigma}$, where $\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli matrix and the real vector $\mathbf{h}(k,t)$ satisfies $h_{x}(k,t)+ih_{y}(k,t)=-[J_{+}(t)+J_{-}^{\ast}(t)e^{-ik}]$, and $h_{z}(k,t)=\frac{\Delta(t)}{2}$. The band structure and the Bloch wave function $e^{ikj}|u_{n}(k)\rangle$ can be obtained by solving $H_{k}|u_{n}(k)\rangle=E_{n}|u_{n}(k)\rangle$, with band spectrum $E_{\pm}=\pm|\mathbf{h}(k,t)|$. The parameters $\Delta=-4J_{0}\cos(\beta_{0})$ and $J_{\pm}=J_{1}(1\pm e^{i\beta_{1}})$ depend on the time-varying phases $\beta_{0,1}(t)$ [see Fig. 2(b)], so does the Hamiltonian $H(t)$.

III Topological pumping

Topological pumping was first proposed by Thouless for fermionic systems thouless1983quantization . Consider a fermionic system with the same single-particle Hamiltonian as $H(t)$, if the Hamiltonian is modulated adiabatically and periodically without closing the band gap, the amount of transported particles (along the OAM space) for a filled band is characterized by the Chern number defined as the change in polarization during one pump cycle (i.e., the center-of-mass displacement of the Wannier function), which is

[TABLE]

with

[TABLE]

The Chern number also equals to the winding number of $\pm\mathbf{h}(k,t)$ surrounding the origin as $t$ varies over one period and $k$ varies over the Brillouin zone. The two band is gapped in the parameter space except the critical point when $\Delta=|J_{+}|-|J_{-}|=0$. It can be proven that a loop which encloses the critical point is topological non-trivial. For a clockwise loop enclosed the critical point, we find that $\mathcal{C_{\pm}}=\mp 1$. The Chern number is equal to the winding number of $\pm\mathbf{h}(k,t)$ surrounding the origin as $t$ varies over one period and $k$ varies over the Brillouin zone.

Photons are non-interacting bosons, so the photonic pumping is reduced to single-particle pumping, which is different from the fermionic pumping of a filled band. We consider a single photon in the $n$-th band

[TABLE]

the modulation is adiabatic so that it will follow the initial band, and also $k$ is a good quantum number during the pumping. So, the final state can be written as

[TABLE]

with $\gamma_{n}(t)$ the Berry phase given by

[TABLE]

The center of mass of the particle is

[TABLE]

with

[TABLE]

So we have

[TABLE]

with the average current given by

[TABLE]

So the displacement after one pumping cycle would be

[TABLE]

For the simple case with $\psi_{k}=\frac{1}{\sqrt{N}}$, $N$ is the total number of lattice sites, $|\Psi(j,0)\rangle$ is reduced to the Wannier function $|W_{n}(j,0)\rangle$,

[TABLE]

and we have

[TABLE]

We can see that the average displacement is exactly quantized even for a single-particle Wannier state.

We consider a pumping process as shown in Fig. 2(c), where the pumping cycle corresponds to a loop in the 2D parameter space spanned by $\beta_{0}$ and $\beta_{1}$. The two bands of the system are gapped in the parameter space except at the critical point $\beta_{0}=\beta_{1}=\pi/2$ with $\Delta=\delta J=0$ and $\delta J\equiv|J_{+}|-|J_{-}|$. A loop which encloses this critical point is topologically non-trivial, and the topology of the pump is invariant under deformation of the loop without cutting through the critical point. Therefore it is more convenient to consider the pump in the $\Delta$-$\delta J$ plane, with corresponding pump loop and photon movement illustrated in Fig. 2(d).

The band structures ($E_{\pm}=\pm|\mathbf{h}(k,t)|$) along the pumping loop are shown in Fig. 3(a), with a smallest bandgap $4J_{1}$. The two gapped bands have different transport properties due to their different topologies, characterized by Chern number $\mathcal{C}=\mp 1$ respectively. We start the pumping at $\delta J=2J_{1}$ and $\Delta=-4J_{0}$ with $J_{0}\gg J_{1}$, where the Wannier functions are well localized at a single OAM state (since the inter-cell tunneling $J_{-}$ is vanished and intra-cell tunneling $J_{+}$ is very weak compared to detuning $\Delta$). We consider a lower-band state with a single photon initialized at OAM state $l_{0}$ (this can be realized by resonantly feeding a single-photon pulse carrying the corresponding OAM into the cavity), which will be pumped to $l_{0}+2$ after one cycle (each unit cell contains 2 sites). The photon distribution and displacement of its center-of-mass are shown in Figs. 3(b) and (c). We can see a clear step-like displacement as expected. The transport is topologically protected and robust against perturbations in the parameter modulation loop. Fig. 3(c) shows the center-of-mass displacement and its variation caused by the random shifts in the phase $\beta_{0}$ and $\beta_{1}$, demonstrating that the quantized transport is almost immune to such small errors.

To obtain a final state with a high purity (defined as the probability of finding the photon in the desired OAM state), a large $J_{0}/J_{1}$ is required to make the Wannier function well localized at a single OAM state. This is because, at the beginning (end) of the pumping, we have two flat bands with $E_{\pm}=\pm\sqrt{4J_{0}^{2}+4J_{1}^{2}}$, and the eigenvectors $|u_{-}(k,0)\rangle=[\cos(\theta),-\sin(\theta)]$ $|u_{+}(k,0)\rangle=[\sin(\theta),\cos(\theta)]$, with $\tan(2\theta)=J_{1}/J_{0}$ independent of $k$. The Wannier functions of the lower band is $|W_{-}(0)\rangle=\cos(\theta)|l=0\rangle-\sin(\theta)|l=1\rangle$, which is well localized on $l=0$ for $J_{1}/J_{0}\ll 1$. As a result, the pumping of state $|l=0\rangle$ is characterized by the lower band which gives a quantized transport of $\mathcal{C}_{-}=1$. In addition, different from the pumping of a whole filled band, the final state is not a simple displacement of the initial state for single-particle pumping, this is because there exists diffusion during the pumping due to the dynamical phase $e^{-i\int_{0}^{T}dtE_{n}(k,t)}$, which decreases the purity of the final state. The diffusion leads to a wider profile of its density distribution, and induces minor populations of unwanted OAM states. To reduce such effect, we need to make the band as flat (in momentum space) as possible during the pump, then the dynamical phase becomes a constant phase independent of $k$. During the pump, the bands are given by

[TABLE]

which is always flat for $\beta_{1}=0,\pi$. As we modulate $\beta_{1}$ from [math] ($\pi$) to $\pi$ (0), such that $J_{+}$ ($J_{-}$) changes from $2J_{1}$ to [math] while $J_{-}$ ($J_{+}$) changes from [math] to $2J_{1}$, the bands are approximately flat in the limit $J_{1}/J_{0}\ll 1$, and the diffusion effect is negligible. The band gap is also very large during this modulation, and the diffusion effect can be reduced further by increasing the modulating speed properly. These effects are verified by our numerical calculation of the purity for different values of $J_{0}/J_{1}$ [see Fig. 3(d)]. The purity can be close to 100% by using a larger value of $J_{0}/J_{1}$.

Each pumping cycle shifts photon’s OAM by 2, therefore even switching numbers can be realized by integer pumping cycles. However, we notice that photon’s OAM state is also well localized at half-integer pumping cycles with OAM shifted by an odd number [see Figs. 3(b)-(d)], therefore odd switching numbers can be realized by half-integer pumping cycles. Alternatively, we can design the synthetic double-well lattice such that two sites in each unit cell are represented by different polarization states with the same OAM, and each pumping cycle changes photon’s OAM by 1. Similarly, we can put the photon in the upper band, then the above pumping will change its OAM by $-1$ or $-2$, alternatively, this can also be achieved by considering a counterclockwise pumping loop and a photon in the lower band. The system is linear (with no photon-photon interaction), as a result, our scheme can simultaneously switch multiple photons in different OAM modes, each pumping cycle shifts the OAM by $+2$ ($-2$) for photons in the lower (upper) band.

IV Topological OAM switch

Such quantized transport offers a robust way to switch the OAM states of photonic signals in three steps: (i) Input—the input photon pulse enters the cavity and the $l_{0}$ cavity-mode is excited; (ii) Topological pumping—the photon is pumped to the desired OAM state; (iii) Output—photon pulse is released out of the cavity. The input and output are realized by a tunable beam splitter (TBS) which couples the cavity with the outside world. The tunability of the input/output beam splitter is crucial for improving the efficiency of the OAM switch because the coupling between the cavity and the outside world need be turned on during input/output so that the signals can get in/out, and turned off during the pumping to avoid unwanted photon losses. The dynamics are characterized by walls2008quantum

[TABLE]

where $\mathcal{E}_{\text{in}}(t)$ is the input photonic field in $l_{0}$-OAM state, $\kappa$ is the total photon loss, and the tunable coupling strength between the cavity and the input/output fields is $\sqrt{\kappa_{\text{e}}}\simeq|r_{\text{P}}|\sqrt{\frac{\Omega_{\text{F}}}{2\pi}}$ hernandez1986fabry with $r_{\text{P}}^{2}\ll 1$ and $\Omega_{\text{F}}$ the free spectral range (FSR) of the main cavity.

In an ideal case, all optical elements are perfectly designed, and the only photon loss channel is the TBS, so we have $\kappa=\kappa_{\text{e}}$. With proper modulation of $\kappa_{\text{e}}$ (i.e., $r_{\text{P}}$), the input signal pulse can enter the cavity with an efficiency as high as $90\%$. Then it is pumped to the desired OAM state and finally released from the cavity. The evolution of the photon field inside the cavity, as well as the temporal profiles of input/output fields and photon loss $\kappa$ are shown in Figs. 4(a) and (b). We see that photons are almost perfectly transported to the desired OAM state, but the intensity is slightly reduced due to photon losses during input and diffusion during pumping.

V Experimental consideration

Typically the FSR of the main cavity is $\Omega_{\text{F}}\sim 2\pi\times 1$GHz, and the tunneling strength, given by $J_{s}=\Omega_{\text{F}}\frac{|r_{s}|^{2}}{2\pi(1+|t_{s}|^{2})}$ with $r_{s}$ ($t_{s}$) the reflectivity (transmissivity) of the corresponding beam splitters hafezi2011robust ; luo2015quantum ; zhou2017dynamically ; luo2017synthetic , can be up to tens of MHz (e.g., $J_{0}\sim 2\pi\times 20$MHz and $J_{0}\gg J_{1}\sim 2\pi\times 4$MHz). The pump period $T\sim 20/J_{1}$ leads to a switching time of the order of $\mu$s. Notice that the bandwidth of the input signal pulse should be smaller than the initial band gap $4J_{0}$, leading to a bandwidth of the order of $10$MHz for the input photon, and such photon source can be realized using cavity-enhanced parametric downconversion PhysRevLett.101.190501 . The upper limit ($l_{\text{max}}$) that the OAM state can be switched to, is determined by the aperture of the optical elements because the beam size increases with the OAM number $l$. For typical mirror size, $l_{\text{max}}$ can be very large (hundreds), leading to a switchable OAM range $l\in[-l_{\text{max}},l_{\text{max}}]$. Imperfections such as extra photon losses may be introduced by cavity mirrors, TBS, phase modulators and SLMs. Such losses only reduce the intensity of the output field without affecting the quantized switching (see Appendix B). To reduce such extra losses, we can make use of high-efficiency intra-cavity elements (e.g., EOM, SLM, etc.) with high transmission (through anti-reflection coating), it is possible to make these extra losses as small as a few tens of kHz (much smaller than the switching rate of MHz) nagorny2003collective ; oemrawsingh2005experimental ; marrucci2011spin ; raut2011anti . Also, the defects in the SLMs would induce imperfect switch, fortunately, only two high-efficiency SLMs (which can be fabricated with high precision oemrawsingh2005experimental ) are used and photons may pass the SLMs many times to realize high switch distance. Deviations from cavity degeneracy lead to random on-site energy shifts $\delta\omega_{l}\sim|l|\delta\omega$ (see Appendix C). Due to the topological protection, our scheme works well as long as the shift for the maximal OAM state (hundreds) is smaller than the smallest band gap $l_{\text{max}}\delta\omega\lesssim 4J_{1}$ (see Appendix C). This requires that the accuracy of the mirror (lens) distance should be of the order of $\mu$m which can be realized using high-precision translation stage. Thanks to the degenerate-cavity setup with all OAM modes sharing the same optical paths, disorders in the tunneling coefficients are negligible since all tunnelings are realized by the same sets of auxiliary cavities and can be controlled simultaneously. Other imperfections, such as a global shift in the resonance frequency of the degenerate cavity, will not affect the pumping. Therefore, our scheme should be feasible even in the presence of realistic imperfections.

VI Multistage generalization

To obtain high OAM states, large numbers of pump cycles may be required, which slow down the switching rate. To accelerate the switching rate, we consider a multistage setup with several cascaded degenerate-cavity systems as shown in Fig. 5. In the $n$-th stage, we choose the tunneling step of the SLMs as $M_{1}=M_{2}=N^{n}$ with $N$ an integer, and $\alpha_{0}=\alpha_{1}$ satisfying $\text{mod}(\alpha_{0}N^{n},2\pi)=\pi$. For an arbitrary switching distance $\Delta l=\sum_{n}c_{n}N^{n}$ (with $c_{n}$ the expansion coefficients), the corresponding pump cycle of the $n$-th stage is $c_{n}/2$, which gives the total switching time $T\sum_{n}c_{n}/2$. For example, consider $\Delta l=512$ and $N=10$, the total switching time is only $4T$. Both the maximum number of stages $\leq\log_{N}l_{\text{max}}$ and the maximum switching time $\leq\frac{TN}{2}\log_{N}l_{\text{max}}$ are logarithmic, yielding exponential speedup for the OAM switch.

VII Discussion and Conclusion

Conventional spatial light modulator thalhammer2013speeding and digital micro-mirror device mirhosseini2013rapid are limited by the switching rates of $\sim$kHz. Higher switching rates can be achieved by combing acousto-optic (electro-optic) modulator with SLMs (q-plate) radwell2014high ; slussarenko2011efficient , or using on-chip resonators strain2014fast . However, the acousto-optic modulator would induce unwanted change in wavelength, and the on-chip switching has a very low efficiency. Moreover, all these approaches require precise control of experimental parameters and the number of usable OAM modes is usually very limited. In contrast, our scheme is robust against perturbations due to its topological feature, and also able to rapidly switch to high OAM modes with high efficiency. Our results of single-photon pumping can be generalized to multi-photon states or even classic coherent states. Since the system is linear with no interaction, every photon is pumped independently.

In summary, we proposed a topological photonic OAM switch which is fast, robust, efficient and accessible to exponentially large OAM states. The proposed topological pumping in the OAM-based synthetic dimension offers a powerful platform to study 1D topological physics of the generalized AAH model. The simple optical setup for the topological photonic OAM switch opens a wide range of experimental opportunities and may find important applications in quantum information processing and optical communications.

Acknowledgements.

Acknowledgments: This work is funded by NNSFC (Grant Nos. 11574294, 61490711), NKRDP (Grant Nos.2016YFA0301700 and 2016YFA0302700), the ”Strategic Priority Research Program(B)” of the CAS (Grant No.XDB01030200), ARO (W911NF-17-1-0128), AFOSR (FA9550-16-1-0387), and NSF (PHY-1505496).

Appendix A Appendix A

The cavity stability and effective Hamiltonian. There are many ways to construct a degenerate cavity, and its stability may depend on specific configuration. for example, we consider a rectangular-shaped cavity shown in Figure 1(a) in the main text with width $X$ and length $Y$. Four curved mirrors are identical with the focal length $F$ and the fully transverse degeneracy can be obtained for $X=Y=F$. The cavity is stable for $(X-F)(Y-F)>=0$ and unstable for $(X-F)(Y-F)<0$ (see Fig. 6). In the stable region, the beam waist is given by $w_{0}^{2}=\frac{\lambda F}{2\pi}\sqrt{\frac{X-F}{Y-F}}$ with $\lambda$ the wavelength, and $w_{0}$ is of the order of $1$mm. These properties are similar to those for confocal Fabry-Perot cavity yariv2007photonics ; hodgson2005laser . Because the cavity possesses cylindrical symmetry with respect to the optical axis, ellipsoidal (rather than spherical) mirrors are used to correct astigmatism caused by non-normal reflections. Alternatively, this can be done by replacing curved mirror with an intracavity lens and a plane mirror, or by changing the geometry of the cavity.

Inserting beam splitters (BSs) may slightly adjust the optical path length of the cavity and thereby the $ABCD$ ray matrix, which can be restored easily to identity by modifying $X$ or $Y$. The beam splitters divert a small portion of photons in the main cavity to the auxiliary cavity containing two SLMs, which induce tunnelings between different OAM modes. To show this, we first consider a single auxiliary cavity with two $\pm 1$-SLMs. Our system is equivalent to a 1D lattice of coupled cavity array with lattice sites represented by different OAM modes luo2015quantum ; zhou2017dynamically ; luo2017synthetic , as shown in Fig. 7. The optical path length of the auxiliary cavity is chosen for destructive interference so that photons spend most of the time in the main cavity (small imperfections of the auxiliary cavity can be ignored because of this). The system is characterized by the following tight-binding Hamiltonian hafezi2011robust

[TABLE]

where $a_{l}$ is the annihilation operator of the cavity photon in the OAM state $l$, $J$ and $\phi$ are corresponding tunneling amplitude and phase, which are determined by reflectivities of the BSs and the optical path length of two arms of the auxiliary cavity, respectively. It is straightforward to generalize the results to multi auxiliary cavities. Using three properly designed auxiliary cavities, we obtain the Hamiltonian Eq. (1) in the main text. Without tunnelings, the system has a single trivial flat band since all OAM modes are degenerate. The tunneling and interference between different OAM modes lead to non-trivial topological band structures, based on which we can switch OAM states through topological pumping.

Appendix B Appendix B

Effects of photon losses. Typically, the optical path length of the cavity is about tens of centimeters which leads to a FSR $\Omega_{\text{F}}=2\pi c/L\sim 2\pi\times 1$GHz, where $L$ is the optical path length of the cavity and $c$ the speed of light. The reflectivity $r_{\text{P}}$ of the TBS can be tuned by the EOM. If the tuning rate is much smaller than the FSR $\Omega_{\text{F}}$, its effect is well characterized by a time-dependent coupling $\sqrt{\kappa_{\text{e}}(t)}\simeq|r_{\text{P}}(t)|\sqrt{\frac{\Omega_{\text{F}}}{2\pi}}$, where $\kappa_{\text{e}}$ can be tuned from 0 to a few MHz within a few $\mu$s and vice versa. In realistic experiments, there are other photon losses due to factors such as the finite Q-factor of the cavity, the intrinsic loss of the SLMs and phase modulators. Such photon loss can be made as low as tens of kHz (much smaller than the switching rate) using high performance optical elements. For extremely high OAM states, the beam size becomes comparable with the aperture of the optical elements, which gives a upper limit that the OAM can be switched to. We consider the pumping between OAM states smaller than the upper limit, the dynamics of the lossy system is characterized by the master equation

[TABLE]

with photon loss rate $\kappa_{0}$. For the pumping of a single photon state, the solution is simply given by

[TABLE]

with $|\Psi(t)\rangle$ the solution of non-dissipative case. The photon density distribution

[TABLE]

is the same as the non-dissipative case, except that the probability to find the photon inside the cavity is reduced to $e^{-\kappa_{0}t_{\text{p}}}$ with $t_{\text{p}}$ the total pumping time. Our system is linear with no interactions, thus the results of single photon pumping can also be generalized to multi-photon states and even classic coherent states.

When the cavity is coupled with the input/output signals, the dynamics are characterized by the Langevin equation

[TABLE]

with $\kappa=\kappa_{0}+\kappa_{\text{e}}$ and $\mathcal{E}_{\text{in}}(t)$ the input optical field in $l_{0}$-OAM state which can be either a single-photon pulse or a classic coherent pulse. The dynamics of both single-photon and coherent input signal are described by Eq. (A5), with $a_{l}$ being the coherent (single-photon) amplitude of OAM state $l$. Typically, the intrinsic photon loss $\kappa_{0}$ is of the order of tens of kHz (which can be made even smaller by improving the performance of the optical elements), and it only reduces the switching efficiency without affecting the quantized transport (even for a strong loss), as shown in Fig. 8 with a large $\kappa_{0}$ ($\sim 2\pi\times 100$kHz).

Appendix C Appendix C

Effects of deviations from degeneracy point. In the ideal case, the cavities always stay at the degeneracy point when $ABCD$ matrix is equal to identity. Deviations from the degeneracy point lead to an frequency shift $(2p+|l|+1)\Omega_{\text{F}}\arccos(\frac{A+D}{2})$ to the LG cavity modes. This would hardly affect tunnelings because the BS reflectivity is not affected and the optical path length of the auxiliary cavity is chosen for destructive interference (photons spend most of their time in the main cavity). For the OAM states considered in our system, the radial index $p$ is of the same order of magnitude as $|l|$ (though each OAM mode may contain multiple $p$ components). Overall, deviations from degeneracy lead to a random energy shift $\delta\omega_{l}a_{l}^{\dagger}a_{l}$ to each OAM mode, with $\delta\omega_{l}\simeq|l|\delta\omega$ and $\delta\omega\simeq 3\frac{\arccos(\frac{A+D}{2})}{2\pi}\Omega_{\text{F}}$ (unimportant global shift is ignored). Thanking to the topological protection, we find that our scheme works very well as long as the energy shift is smaller than the smallest band gap $\frac{|l|\delta\omega}{4J_{1}}\lesssim 1$, as confirmed by our numerical simulation shown in Fig. 9. Without loss of generality, we consider above rectangular-shaped cavity with $F=10$cm. $\frac{\delta\omega}{4J_{1}}$ as a function of $X$ and $Y$ is shown in Fig. 6, and $\frac{|l|\delta\omega}{4J_{1}}\lesssim 1$ leads to $|l|\sqrt{(X-F)(Y-F)}\lesssim 1000\mu$m. If we encode the quantum information in the space $l\in[-l_{\text{max}},l_{\text{max}}]$ (i.e., the maximal OAM to be switched to is $|l|=l_{\text{max}}$), our scheme works well when $l_{\text{max}}\lesssim\frac{1000\mu\text{m}}{\sqrt{(X-F)(Y-F)}}$. Using standard micrometric linear translation stages, one can easily achieve an accuracy with $(X-F)\sim(Y-F)\sim\mu$m, thus corresponding $l_{\text{max}}$ can be up to hundreds. Nanometer-precision linear translation stage (e.g. M-714.2HD, PI, Germany) can reach an accuracy of 0.05$\mu$m, which yields a maximal usable OAM of several thousands in principle.