Topological photonic orbital angular momentum switch
TTopological photonic orbital angular momentum switch
Xi-Wang Luo,
1, 2
Chuanwei Zhang, ∗ Guang-Can Guo,
1, 3 and Zheng-Wei Zhou
1, 3, † Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China
The large number of available orbital angular momentum (OAM) states of photons provides aunique resource for many important applications in quantum information and optical communica-tions. However, conventional OAM switching devices usually rely on precise parameter control andare limited by slow switching rate and low efficiency. Here we propose a robust, fast and efficient pho-tonic OAM switch device based on a topological process, where photons are adiabatically pumpedto a target OAM state on demand. Such topological OAM pumping can be realized through ma-nipulating photons in a few degenerate main cavities and involves only a limited number of opticalelements. A large change of OAM at ∼ q can be realized with only q degenerate main cavities andat most 5 q pumping cycles. The topological photonic OAM switch may become a powerful devicefor broad applications in many different fields and motivate novel topological design of conventionaloptical devices. INTRODUCTION
Discrete degrees of freedom, such as charge, spin, val-leys, etc. , play a crucial role in many information en-coding and device applications [1–4]. In this context, afundamental degree of freedom of photons, the orbitalangular momentum (OAM), possesses a unique propertythat an infinite number of distinctive OAM states areavailable [5–7]. This unique property makes photonicOAM very attractive for various applications in opti-cal communication [8–10], quantum simulation [11–14],quantum information [15–18], and quantum cryptogra-phy (e.g., key distribution) [19–22]. To fully utilize theseapplications, a tunable device that can rapidly and ro-bustly switch between different OAM modes on demandis therefore highly desirable. However, many conven-tional OAM switching devices rely on precise parametercontrol and are usually limited by slow switching rates( ∼ kHz) [23, 24] or low purity and efficiency [25, 26], andlimited number of usable OAM modes [27, 28].The photonic OAM is a discrete degree of freedom thatcharacterizes the topological charge (i.e., the winding ofthe azimuthal phase) of a photon field with cylindricalsymmetry. Therefore a natural question is whether atopological process can be designed to create a robustphotonic OAM switching device with high performance.Recently, the study of topological photonics has becomeone frontier direction in optical physics with the major fo-cus on modulating photon propagation through topolog-ical edge states [29–35], while practical topological pho-tonic devices for on-demand switching of photon internaldegrees of freedom are still largely lacking.In this paper, we propose a practical photonic OAMswitching device through the topological adiabatic pump-ing of OAM states, which is robust (immune to small per-turbations in system parameters), fast ( ∼ MHz switch-ing rate), efficient ( ∼
90% efficiency and ∼ ∼ MHz,which is much faster than commonly used OAM switch-ing devices [23, 24]. Even for a large change of OAMat ∼ q , only q degenerate main cavities and at most5 q pumping cycles are needed using a multistage setup,leading to exponential speedup of the OAM switch. Theproposed topological photonic OAM switch only relies ona simple optical setup with a few degenerate main cavi-ties, therefore it may become a powerful device for broadapplications in quantum information and optical commu- a r X i v : . [ qu a n t - ph ] J un 𝜙 =0 ; 𝑀 =1 𝜙 ; 𝑀 =0 𝜙 ; 𝑀 =1 (b) +𝑀 𝑖 SLM −𝑀 𝑖 SLM +𝜙 𝑖 −𝜙 𝑖 BS BS Mirror Mirror BS BS |𝑉ۧ |𝐻ۧ |𝐻ۧ
EOM
PBSPBS |𝑉ۧ (a) ℰ in ℰ out TBS ℰ in ℰ out PM PM (c)
FIG. 1: (a) Experimental photonic circuit for a degeneratecavity system. The main cavity (red curved mirrors) is cou-pled with auxiliary cavities (blue curved mirrors) by beamsplitters (BSs). A tunable beam splitter (TBS) is used totune the coupling between the cavity (in H polarization) andthe input/output fields (in V polarization). (b) Optical designfor the TBS. (c) Schematic diagram of the system with threeauxiliary cavities (dashed lines), with corresponding tunnel-ing phase φ s and step M s as labeled. nications. Our proposed topological approach for deviceengineering may go beyond the OAM switch and inspirenovel topological design of conventional optical devices. THE SYSTEM
As shown in Fig. 1(a), our system contains a maindegenerate multimode cavity [44–48], which supports alarge number of OAM modes and is coupled with threedegenerate auxiliary cavities by beam splitters. The de-generate cavities used in our scheme possess fully trans-verse degeneracy and their design principle is based onthe round-trip Gauss matrix [often called
ABCD ray ma-trix with
A, D ( B, C ) diagonal (off-diagonal) matrix el-ements] [44]. We consider a ring-type cavity possessingcylindrical 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 resonancefrequencies of the LG modes are [49, 50] ω p,l = n Ω F + (2 p + | l | + 1) arccos( A + D )2 π Ω F , (1)where the integer n is the longitudinal mode index thatis fixed in our scheme and Ω F is the free spectral rangeof the cavity. The off-diagonal ray matrix elements B and C only affect beam waist size. The degenerate cav-ity is obtained when the round-trip ABCD ray matrixis equal to identity [44], which can be achieved usingcurved mirrors or intracavity lenses. In this case, alltransverse LG modes E p,l have the same resonance fre-quency n Ω F , yielding a degenerate cavity supporting dif-ferent OAM modes l for photons. The tunneling be-tween different OAM states is realized by spatial lightmodulators (SLMs). The unimportant radial index p , which is related to the radial mode profile of the in-put photons, is omitted. There are many ways to con-struct a degenerate cavity, for example, we consider arectangular-shaped cavity shown in Fig. 1(a) formed byfour identical curved mirrors with focal length equal tooptical path length between adjacent mirrors (see Ap-pendix A). We use a tunable beam splitter (TBS) tocouple the cavity with the outside world, it is realizedby sandwiching an electro-optic modulator (EOM) [49]between two polarizing beam splitters (PBSs), as shownin Fig. 1(b). The EOM rotates photon’s polarizationsin a tunable way as | H (cid:105) → (cid:112) − r | H (cid:105) + r P | V (cid:105) and | V (cid:105) → (cid:112) − r | V (cid:105) − r P | H (cid:105) , with | H (cid:105) and | V (cid:105) the hor-izontal and vertical polarization states which are sepa-rated by the PBS. The tunable coefficient r P acts as thereflectivity of the TBS.The beam splitters divert a small portion of the main-cavity photons towards the s -th auxiliary cavity, andmerge them back after passing through spatial light mod-ulators (SLMs) [51, 52], which change the OAM state by ± M s . This corresponds to tunnelings between differentOAM states in the main cavity, with a tunneling rate de-termined by the reflectivities of the BSs. During the tun-neling, the photon can also acquire a phase determinedby the optical path-length difference between two armsof the auxiliary-cavity, which can be generated and tunedusing high-speed phase modulators (PMs). The systemis equivalent to a one-dimensional lattice model with thelattice sites represented by the OAM states [13–15]. TheHamiltonian is given by (see Appendix A) H = − (cid:88) l (cid:88) s =0 J s e iφ s a † l + M s a l + h.c., (2)where a l is the annihilation operator of the cavity photonin the OAM state l , and M s is the step index of the SLMsin the s -th auxiliary cavity with the tunneling amplitude J s and phase φ s = lα s + β s . The PM contains a beamrotator [53] and an EOM [49]: the beam rotator, realizedby two Dove prisms rotated by α s / l -dependent phase lα s , andthe EOM is used to tune the l -independent phase β s .The auxiliary cavities are designed as φ = 0, M = 0, M = M = 1, J = J , as shown in Fig. 1(c). Thereforethe Hamiltonian describes a generalized AAH model [54]with modulations in on-site energy, tunneling phase andamplitude, whose periods are determined by α and α .Here, we focus our discussion on α = α = 2 π · , inanalogy to the Rice-Mele model (i.e., double-well supper-lattice model) [55] with neighboring site energy detuning∆ ≡ − J cos( β ) and inter- and intra-cell tunnelings J ± ≡ J [1 ± e iβ ] [see Fig. 2(a)]. We rewrite the Hamil-tonian as H = − (cid:88) j J + b † j, b j, + J − b † j +1 , b j, + h.c. + (cid:88) j ∆2 ( b † j, b j, − b † j, b j, ) , (3)where we have introduced the unit cell index j with b j, = a j and b j, = a j +1 . After a Fourier transfor-mation b k, ∝ (cid:80) j e ijk b j, , the Hamiltonian in theBloch basis becomes H = − (cid:88) k [ b † k, , b † k, ] H k [ b k, , b k, ] T , (4)with the Bloch Hamiltonian H k ( t ) = h ( k, t ) · (cid:126)σ , where (cid:126)σ = ( σ x , σ y , σ z ) are the Pauli matrix and the real vector h ( k, t ) satisfies h x ( k, t )+ ih y ( k, t ) = − [ J + ( t )+ J ∗− ( t ) e − ik ],and h z ( k, t ) = ∆( t )2 . The band structure and the Blochwave function e ikj | u n ( k ) (cid:105) can be obtained by solving H k | u n ( k ) (cid:105) = E n | u n ( k ) (cid:105) , with band spectrum E ± = ±| h ( k, t ) | . The parameters ∆ = − J cos( β ) and J ± = J (1 ± e iβ ) depend on the time-varying phases β , ( t )[see Fig. 2(b)], so does the Hamiltonian H ( t ). TOPOLOGICAL PUMPING
Topological pumping was first proposed by Thoulessfor fermionic systems [36]. Consider a fermionic systemwith the same single-particle Hamiltonian as H ( t ), if theHamiltonian is modulated adiabatically and periodicallywithout closing the band gap, the amount of transportedparticles (along the OAM space) for a filled band is char-acterized by the Chern number defined as the change inpolarization during one pump cycle (i.e., the center-of-mass displacement of the Wannier function), which is C n = 12 π (cid:90) T dt (cid:90) π dk Ω kt (5)withΩ kt = (cid:104) ∂ t u n ( k, t ) | ∂ k u n ( k, t ) (cid:105) − (cid:104) ∂ k u n ( k, t ) | ∂ t u n ( k, t ) (cid:105) . The Chern number also equals to the winding numberof ± h ( k, t ) surrounding the origin as t varies over oneperiod and k varies over the Brillouin zone. The twoband is gapped in the parameter space except the criticalpoint when ∆ = | J + | − | J − | = 0. It can be proven that aloop which encloses the critical point is topological non-trivial. For a clockwise loop enclosed the critical point,we find that C ± = ∓
1. The Chern number is equal tothe winding number of ± h ( k, t ) surrounding the originas t varies over one period and k varies over the Brillouinzone.Photons are non-interacting bosons, so the photonicpumping is reduced to single-particle pumping, which isdifferent from the fermionic pumping of a filled band. We consider a single photon in the n -th band | Ψ( j, (cid:105) = (cid:88) k ψ k e ijk | u n ( k, (cid:105) , (6)the modulation is adiabatic so that it will follow the ini-tial band, and also k is a good quantum number duringthe pumping. So, the final state can be written as | Ψ( j, t ) (cid:105) = (cid:88) k ψ k e ijk e − i (cid:82) t dt (cid:48) E n ( k,t (cid:48) ) e iγ n ( k,t ) | u n ( k, t ) (cid:105) , (7)with γ n ( t ) the Berry phase given by γ n ( k, t ) = i (cid:90) t (cid:104) u n ( k, t (cid:48) ) | ∂ t (cid:48) u n ( k, t (cid:48) ) (cid:105) dt (cid:48) . (8)The center of mass of the particle is¯ j ( t ) = (cid:88) j (cid:104) Ψ( j, t ) | j | Ψ( j, t ) (cid:105) = (cid:88) k (cid:104) (cid:101) Ψ( k, t ) | i∂ k | (cid:101) Ψ( k, t ) (cid:105) (9)with | (cid:101) Ψ( k, t ) (cid:105) = ψ k e − i (cid:82) t dt (cid:48) E n ( k,t (cid:48) ) e iγ n ( k,t ) | u n ( k, t ) (cid:105) . (10)So we have ¯ j ( t ) = (cid:88) k iψ ∗ k ∂ k ψ k + (cid:90) t I ( t (cid:48) ) dt (cid:48) (11)with the average current given by I ( t ) = (cid:88) k | ψ k | [ ∂ k E ( k, t ) + Ω kt ] . (12)So the displacement after one pumping cycle would be∆¯ j = ¯ j ( T ) − ¯ j (0)= (cid:90) T I ( t ) dt. (13)For the simple case with ψ k = √ N , N is the totalnumber of lattice sites, | Ψ( j, (cid:105) is reduced to the Wan-nier function | W n ( j, (cid:105) , | Ψ( j, (cid:105) = (cid:88) k √ N e ijk | u n ( k, (cid:105) ≡ | W n ( j, (cid:105) , (14)and we have∆¯ j = 1 N (cid:88) k (cid:90) T Ω kt dt = (cid:90) T dt (cid:90) π dk π Ω kt = C n . (15)We can see that the average displacement is exactly quan-tized even for a single-particle Wannier state.We consider a pumping process as shown in Fig. 2(c), OAM-mode index F r e qu e n c y s h i f t 𝐽 + 𝐽 − − ∆ 𝛽 Re 𝐽 ± / 𝐽 Im 𝐽 ± / 𝐽 𝐽 − 𝐽 + 𝜋 𝜋 𝛽 𝛽 𝛿𝐽/𝐽 Δ/𝐽 - - (a) (b) (c) (d) l l+ l+ l -1 l -2 FIG. 2: (a) Diagram of the effective lattice in the OAMdimension. Each unit cell (enclosed by the dashed square)contains two sites with detuning ∆ = − J cos[ β ( t )]. J ± = J [1 ± e iβ ( t ) ] are the inter- and intra-cell tunnelings respec-tively, whose dependence on the tunneling phase β is shownin (b). (c) The pumping loop in the β - β plane. Since thetunneling phase has a period of 2 π , their choice is not unique,and the two loops (red dashed and blue solid) have the sametopology. (d) Illustration of the pumping process. The loopin the δJ -∆ plane can be realized by either loop shown in(c). Red solid circle represents a site occupied by a photon,which moves to the right by two sites (one unit cell) duringone pumping cycle. where the pumping cycle corresponds to a loop in the 2Dparameter space spanned by β and β . The two bandsof the system are gapped in the parameter space exceptat the critical point β = β = π/ δJ = 0and δJ ≡ | J + | − | J − | . A loop which encloses this crit-ical point is topologically non-trivial, and the topologyof the pump is invariant under deformation of the loopwithout cutting through the critical point. Therefore it ismore convenient to consider the pump in the ∆- δJ plane,with corresponding pump loop and photon movement il-lustrated in Fig. 2(d).The band structures ( E ± = ±| h ( k, t ) | ) along thepumping loop are shown in Fig. 3(a), with a smallestbandgap 4 J . The two gapped bands have different trans-port properties due to their different topologies, charac-terized by Chern number C = ∓ δJ = 2 J and ∆ = − J with J (cid:29) J ,where the Wannier functions are well localized at a sin-gle OAM state (since the inter-cell tunneling J − is van-ished and intra-cell tunneling J + is very weak comparedto detuning ∆). We consider a lower-band state witha single photon initialized at OAM state l (this canbe realized by resonantly feeding a single-photon pulsecarrying the corresponding OAM into the cavity), whichwill be pumped to l + 2 after one cycle (each unit cellcontains 2 sites). The photon distribution and displace-ment of its center-of-mass are shown in Figs. 3(b) and(c). We can see a clear step-like displacement as ex-pected. The transport is topologically protected and ro-bust against perturbations in the parameter modulation (a) (c) (d) C =-1 C =+1 𝐸 / 𝐽 𝑙 𝛽 , / 𝜋 𝑃 𝑚 ( 𝑡 𝑚 ) 𝑃 𝑙 𝐽 𝑡 𝐽 𝑡 (b) 𝑙 𝐽 /𝐽 𝐽 𝑡 FIG. 3: (a) The band structure in one pump cycle with Chernnumber C = ± P l ( t ) is the prob-ability of the photon in OAM state l at time t . (c) Centerof mass displacements (dots) and their standard deviations(error bars), which are calculated by assuming Gaussian dis-tributed random disorders δβ , in the tunneling phases withstandard deviations σ ( δβ , ) = 0 . m pump cycles ( t m = mT ) versus different couplingratio J /J . The pink stars, blue cycles, black crosses and redsquares correspond to m = 1, 2, 3 and 4 respectively. Themodulation of the pump parameters is chosen to satisfy theadiabatic condition with period T = 21 /J , whose temporalprofiles are shown in (c) with β the red solid line and β thered dashed line. In (b-d) the initial OAM state is l = 0 andin (a-c) the coupling ratio is J /J = 5. loop. Fig. 3(c) shows the center-of-mass displacementand its variation caused by the random shifts in the phase β and β , demonstrating that the quantized transportis almost immune to such small errors.To obtain a final state with a high purity (definedas the probability of finding the photon in the desiredOAM state), a large J /J is required to make the Wan-nier function well localized at a single OAM state. Thisis because, at the beginning (end) of the pumping, wehave two flat bands with E ± = ± (cid:112) J + 4 J , and theeigenvectors | u − ( k, (cid:105) = [cos( θ ) , − sin( θ )] | u + ( k, (cid:105) =[sin( θ ) , cos( θ )], with tan(2 θ ) = J /J independent of k .The Wannier functions of the lower band is | W − (0) (cid:105) =cos( θ ) | l = 0 (cid:105) − sin( θ ) | l = 1 (cid:105) , which is well localized on l = 0 for J /J (cid:28)
1. As a result, the pumping of state | l = 0 (cid:105) is characterized by the lower band which givesa quantized transport of C − = 1. In addition, differ-ent from the pumping of a whole filled band, the finalstate is not a simple displacement of the initial state forsingle-particle pumping, this is because there exists dif-fusion during the pumping due to the dynamical phase e − i (cid:82) T dtE n ( k,t ) , which decreases the purity of the finalstate. The diffusion leads to a wider profile of its densitydistribution, and induces minor populations of unwantedOAM states. To reduce such effect, we need to makethe band as flat (in momentum space) as possible duringthe pump, then the dynamical phase becomes a constantphase independent of k . During the pump, the bands are 𝑙 (a) (b) 𝑡 / 𝑡 𝜅 / 𝐽 𝑡 / 𝑡 𝑙 |ℰ| 𝑃 𝑙 𝑡 / 𝑡 FIG. 4: (a) Normalized cavity photon distribution and (b) Input/output photon pulses during the OAM switching. Thenormalized input pulse is E in ( t ) = e − iE − t − . J t − with l = 0 (i.e., only the zero-OAM Wannier orbital is excited). E − isthe lower band energy. After the input pulse enters the cavity, we pump the state to higher OAM modes, and then release thesignal by increasing the cavity loss κ [see the inset in (a)]. Other parameters are the same as in Fig. 3(b). given by E ± = ± (cid:113) J + | J ∗ + + J − e ik | (16) (cid:39) ± (cid:26) J + J J + J J [cos( k ) − cos( k + 2 β )] (cid:27) , which is always flat for β = 0 , π . As we modulate β from 0 ( π ) to π (0), such that J + ( J − ) changes from 2 J to 0 while J − ( J + ) changes from 0 to 2 J , the bandsare approximately flat in the limit J /J (cid:28)
1, and thediffusion effect is negligible. The band gap is also verylarge during this modulation, and the diffusion effect canbe reduced further by increasing the modulating speedproperly. These effects are verified by our numerical cal-culation of the purity for different values of J /J [seeFig. 3(d)]. The purity can be close to 100% by using alarger value of J /J .Each pumping cycle shifts photon’s OAM by 2, there-fore even switching numbers can be realized by integerpumping cycles. However, we notice that photon’s OAMstate is also well localized at half-integer pumping cycleswith OAM shifted by an odd number [see Figs. 3(b)-(d)], therefore odd switching numbers can be realized byhalf-integer pumping cycles. Alternatively, we can de-sign the synthetic double-well lattice such that two sitesin each unit cell are represented by different polariza-tion states with the same OAM, and each pumping cyclechanges photon’s OAM by 1. Similarly, we can put thephoton in the upper band, then the above pumping willchange its OAM by − −
2, alternatively, this can alsobe achieved by considering a counterclockwise pumpingloop and a photon in the lower band. The system islinear (with no photon-photon interaction), as a result,our scheme can simultaneously switch multiple photonsin different OAM modes, each pumping cycle shifts theOAM by +2 ( −
2) for photons in the lower (upper) band.
TOPOLOGICAL OAM SWITCH
Such quantized transport offers a robust way to switchthe OAM states of photonic signals in three steps: (i)Input—the input photon pulse enters the cavity and the l 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. Theinput 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 cru-cial for improving the efficiency of the OAM switch be-cause the coupling between the cavity and the outsideworld need be turned on during input/output so that thesignals can get in/out, and turned off during the pump-ing to avoid unwanted photon losses. The dynamics arecharacterized by [56]˙ a l = 1 i [ a l , H ] − κ a l + δ l,l √ κ e E in ( t ) , where E in ( t ) is the input photonic field in l -OAM state, κ is the total photon loss, and the tunable couplingstrength between the cavity and the input/output fieldsis √ κ e (cid:39) | r P | (cid:113) Ω F π [57] with r (cid:28) F the freespectral range (FSR) of the main cavity.In an ideal case, all optical elements are perfectly de-signed, and the only photon loss channel is the TBS, sowe have κ = κ e . With proper modulation of κ e (i.e., r P ),the input signal pulse can enter the cavity with an effi-ciency as high as 90%. Then it is pumped to the desiredOAM state and finally released from the cavity. The evo-lution of the photon field inside the cavity, as well as thetemporal profiles of input/output fields and photon loss κ are shown in Figs. 4(a) and (b). We see that photons arealmost perfectly transported to the desired OAM state,but the intensity is slightly reduced due to photon lossesduring input and diffusion during pumping. Pump cycles: 𝑐 Tunneling step: Pump cycles: 𝑐 Tunneling step: Pump cycles: 𝑐 𝑛 Tunneling step: 𝑛 𝑛 -th Stage ℰ in ℰ out … … Switch distance: 𝑐 + 𝑐 × 10 + … + 𝑐 𝑛 × 10 𝑛 + … FIG. 5: Illustration of a multistage setup with N = 10. Each half pump cycle shifts the OAM by the corresponding tunnelingstep. EXPERIMENTAL CONSIDERATION
Typically the FSR of the main cavity is Ω F ∼ π × J s =Ω F | r s | π (1+ | t s | ) with r s ( t s ) the reflectivity (transmissiv-ity) of the corresponding beam splitters [13–15, 30], canbe up to tens of MHz (e.g., J ∼ π × J (cid:29) J ∼ π × T ∼ /J leads to a switching time of the order of µ s. Notice thatthe bandwidth of the input signal pulse should be smallerthan the initial band gap 4 J , leading to a bandwidth ofthe order of 10MHz for the input photon, and such pho-ton source can be realized using cavity-enhanced para-metric downconversion [58]. The upper limit ( l max ) thatthe OAM state can be switched to, is determined bythe aperture of the optical elements because the beamsize increases with the OAM number l . For typical mir-ror size, l max can be very large (hundreds), leading to aswitchable OAM range l ∈ [ − l max , l max ]. Imperfectionssuch as extra photon losses may be introduced by cavitymirrors, TBS, phase modulators and SLMs. Such lossesonly reduce the intensity of the output field without af-fecting the quantized switching (see Appendix B). To re-duce such extra losses, we can make use of high-efficiencyintra-cavity elements (e.g., EOM, SLM, etc.) with hightransmission (through anti-reflection coating), it is possi-ble to make these extra losses as small as a few tens of kHz(much smaller than the switching rate of MHz) [51, 59–61]. Also, the defects in the SLMs would induce imper-fect switch, fortunately, only two high-efficiency SLMs(which can be fabricated with high precision [51]) areused and photons may pass the SLMs many times torealize high switch distance. Deviations from cavity de-generacy lead to random on-site energy shifts δω l ∼ | l | δω (see Appendix C). Due to the topological protection, ourscheme works well as long as the shift for the maximalOAM state (hundreds) is smaller than the smallest bandgap l max δω (cid:46) J (see Appendix C). This requires thatthe accuracy of the mirror (lens) distance should be of theorder of µ m which can be realized using high-precisiontranslation stage. Thanks to the degenerate-cavity setupwith all OAM modes sharing the same optical paths, dis-orders in the tunneling coefficients are negligible sinceall tunnelings are realized by the same sets of auxiliarycavities and can be controlled simultaneously. Other im- perfections, such as a global shift in the resonance fre-quency of the degenerate cavity, will not affect the pump-ing. Therefore, our scheme should be feasible even in thepresence of realistic imperfections. MULTISTAGE GENERALIZATION
To obtain high OAM states, large numbers of pump cy-cles may be required, which slow down the switching rate.To accelerate the switching rate, we consider a multistagesetup with several cascaded degenerate-cavity systems asshown in Fig. 5. In the n -th stage, we choose the tun-neling step of the SLMs as M = M = N n with N aninteger, and α = α satisfying mod( α N n , π ) = π . Foran arbitrary switching distance ∆ l = (cid:80) n c n N n (with c n the expansion coefficients), the corresponding pump cycleof the n -th stage is c n /
2, which gives the total switchingtime T (cid:80) n c n /
2. For example, consider ∆ l = 512 and N = 10, the total switching time is only 4 T . Both themaximum number of stages ≤ log N l max and the max-imum switching time ≤ T N log N l max are logarithmic,yielding exponential speedup for the OAM switch. DISCUSSION AND CONCLUSION
Conventional spatial light modulator [23] and digitalmicro-mirror device [24] are limited by the switchingrates of ∼ kHz. Higher switching rates can be achievedby combing acousto-optic (electro-optic) modulator withSLMs (q-plate) [25, 28], or using on-chip resonators [26].However, the acousto-optic modulator would induce un-wanted change in wavelength, and the on-chip switchinghas a very low efficiency. Moreover, all these approachesrequire precise control of experimental parameters andthe number of usable OAM modes is usually very limited.In contrast, our scheme is robust against perturbationsdue to its topological feature, and also able to rapidlyswitch to high OAM modes with high efficiency. Ourresults of single-photon pumping can be generalized tomulti-photon states or even classic coherent states. Sincethe system is linear with no interaction, every photon ispumped independently.In summary, we proposed a topological photonic OAMswitch which is fast, robust, efficient and accessible to ex-ponentially large OAM states. The proposed topologicalpumping in the OAM-based synthetic dimension offers apowerful platform to study 1D topological physics of thegeneralized AAH model. The simple optical setup for thetopological photonic OAM switch opens a wide range ofexperimental opportunities and may find important ap-plications in quantum information processing and opticalcommunications. Acknowledgments : This work is funded byNNSFC (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
The cavity stability and effective Hamiltonian.
There are many ways to construct a degenerate cav-ity, and its stability may depend on specific configura-tion. for example, we consider a rectangular-shaped cav-ity shown in Figure 1(a) in the main text with width X and length Y . Four curved mirrors are identical withthe focal length F and the fully transverse degeneracycan be obtained for X = Y = F . The cavity is stable for( X − F )( Y − F ) > = 0 and unstable for ( X − F )( Y − F ) < w = λF π (cid:113) X − FY − F with λ the wavelength, and w is ofthe order of 1mm. These properties are similar to thosefor confocal Fabry-Perot cavity [49, 50]. Because the cav-ity possesses cylindrical symmetry with respect to the op-tical axis, ellipsoidal (rather than spherical) mirrors areused to correct astigmatism caused by non-normal reflec-tions. Alternatively, this can be done by replacing curvedmirror with an intracavity lens and a plane mirror, or bychanging the geometry of the cavity.Inserting beam splitters (BSs) may slightly adjust theoptical path length of the cavity and thereby the ABCD ray matrix, which can be restored easily to identity bymodifying X or Y . The beam splitters divert a small por-tion of photons in the main cavity to the auxiliary cavitycontaining two SLMs, which induce tunnelings betweendifferent OAM modes. To show this, we first consider asingle auxiliary cavity with two ± H = − (cid:88) l Je iφ a † l +1 a l + h.c., (A1) (X − F)/μ m ( Y − F ) / μ m 𝛿𝜔/4𝐽 FIG. 6: Plot of frequency difference δω between adjacentOAM modes normalized to the band gap 4 J with respectto the width and length of the rectangular cavity. The whiteregions represent configurations where the cavity is unstable.Our scheme works well if | l | δ ω J (cid:46)
1. Other parameters are F = 10cm, J = 0 . F . 𝑙 𝑙 + 1 BS 𝜙 −𝜙 𝑎 𝑙 𝑎 𝑙+1 …… FIG. 7: The equivalent circuit in the OAM space, with φ the phase imbalance between two arms of the auxiliary cavityand a l the field operator for OAM mode l . Red (blue) looprepresents the main (auxiliary) cavity. where a l is the annihilation operator of the cavity photonin the OAM state l , J and φ are corresponding tunnelingamplitude and phase, which are determined by reflectiv-ities of the BSs and the optical path length of two armsof the auxiliary cavity, respectively. It is straightforwardto generalize the results to multi auxiliary cavities. Us-ing three properly designed auxiliary cavities, we obtainthe Hamiltonian Eq. (1) in the main text. Without tun-nelings, the system has a single trivial flat band sinceall OAM modes are degenerate. The tunneling and in-terference between different OAM modes lead to non-trivial topological band structures, based on which wecan switch OAM states through topological pumping. Appendix B
Effects of photon losses . Typically, the optical pathlength of the cavity is about tens of centimeters whichleads to a FSR Ω F = 2 πc/L ∼ π × L is theoptical path length of the cavity and c the speed of light.The reflectivity r P of the TBS can be tuned by the EOM.If the tuning rate is much smaller than the FSR Ω F , itseffect is well characterized by a time-dependent coupling (a) (b) J 𝑡 𝜅 / 𝐽 J 𝑡 𝑙 |ℰ| 𝑃 𝑙 ( t ) 𝑙 J 𝑡 FIG. 8: (a) Normalized cavity photon distribution and (b) Input/output photon pulses in the OAM switching. The parametersare the same as in Fig. 4 in the main text, except that the photon loss is nonzero during the pumping with κ = 0 . J . (cid:112) κ e ( t ) (cid:39) | r P ( t ) | (cid:113) Ω F π , where κ e can be tuned from 0 toa few MHz within a few µ s and vice versa. In realisticexperiments, there are other photon losses due to factorssuch as the finite Q-factor of the cavity, the intrinsic lossof the SLMs and phase modulators. Such photon losscan be made as low as tens of kHz (much smaller thanthe switching rate) using high performance optical ele-ments. For extremely high OAM states, the beam sizebecomes comparable with the aperture of the optical el-ements, which gives a upper limit that the OAM can beswitched to. We consider the pumping between OAMstates smaller than the upper limit, the dynamics of thelossy system is characterized by the master equation˙ ρ = 1 i [ H, ρ ] + κ (cid:88) l ( a l ρa † l − ρa † l a l − a † l a l ρ ) (A2)with photon loss rate κ . For the pumping of a singlephoton state, the solution is simply given by ρ = (1 − e − κ t ) | vac (cid:105)(cid:104) vac | + e − κ t | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | , (A3)with | Ψ( t ) (cid:105) the solution of non-dissipative case. The pho-ton density distribution N ( l, t ) = Tr[ ρ ( t ) a † l a l ] (cid:80) l Tr[ ρ ( t ) a † l a l ]= (cid:104) Ψ( t ) | a † l a l | Ψ( t ) (cid:105) (A4)is the same as the non-dissipative case, except that theprobability to find the photon inside the cavity is reducedto e − κ t p with t p the total pumping time. Our systemis linear with no interactions, thus the results of singlephoton pumping can also be generalized to multi-photonstates and even classic coherent states.When the cavity is coupled with the input/output sig-nals, the dynamics are characterized by the Langevinequation ˙ a l = 1 i [ a l , H ] − κ a l + δ l,l √ κ e E in ( t ) , (A5)with κ = κ + κ e and E in ( t ) the input optical field in l -OAM state which can be either a single-photonpulse or a classic coherent pulse. The dynamics of bothsingle-photon and coherent input signal are described byEq. (A5), with a l being the coherent (single-photon) am-plitude of OAM state l . Typically, the intrinsic pho-ton loss κ is of the order of tens of kHz (which can bemade even smaller by improving the performance of theoptical elements), and it only reduces the switching ef-ficiency without affecting the quantized transport (evenfor a strong loss), as shown in Fig. 8 with a large κ ( ∼ π × Appendix C
Effects of deviations from degeneracy point.
Inthe ideal case, the cavities always stay at the degen-eracy point when
ABCD matrix is equal to identity.Deviations from the degeneracy point lead to an fre-quency shift (2 p + | l | + 1)Ω F arccos( A + D ) to the LGcavity modes. This would hardly affect tunnelings be-cause the BS reflectivity is not affected and the opticalpath length of the auxiliary cavity is chosen for destruc-tive interference (photons spend most of their time inthe main cavity). For the OAM states considered inour system, the radial index p is of the same order ofmagnitude as | l | (though each OAM mode may containmultiple p components). Overall, deviations from de-generacy lead to a random energy shift δω l a † l a l to eachOAM mode, with δω l (cid:39) | l | δω and δω (cid:39) arccos( A + D )2 π Ω F (unimportant global shift is ignored). Thanking to thetopological protection, we find that our scheme worksvery well as long as the energy shift is smaller than thesmallest band gap | l | δω J (cid:46)
1, as confirmed by our numer-ical simulation shown in Fig. 9. Without loss of gener-ality, we consider above rectangular-shaped cavity with F = 10cm. δω J as a function of X and Y is shown inFig. 6, and | l | δω J (cid:46) | l | (cid:112) ( X − F )( Y − F ) (cid:46) µ m. If we encode the quantum information in thespace l ∈ [ − l max , l max ] (i.e., the maximal OAM to beswitched to is | l | = l max ), our scheme works well when 𝑙 / 𝑀 𝐽 𝑡 𝐽 𝑡 𝑙 / 𝑀 FIG. 9: (a) Center of mass displacements (red dots) and their standard deviations (blue error bars), which are calculatedby assuming Gaussian distributed random on-site energy shift δω l = | l | δω due to deviations from the degeneracy. Standarddeviations σ ( δω ) = 0 . J and tunneling step M = 1 corresponds to the 0-th stage of the multi-stage switch. δω = 0 . J corresponds to (cid:112) ( X − F )( Y − F ) (cid:39) µ m. (b) The same as in (a) except that the tunneling step M = 10, corresponding tothe 1-th stage of the multi-stage switch. l max (cid:46) µ m √ ( X − F )( Y − F ) . Using standard micrometric lin-ear translation stages, one can easily achieve an accuracywith ( X − F ) ∼ ( Y − F ) ∼ µ m, thus corresponding l max can be up to hundreds. Nanometer-precision linear trans-lation stage (e.g. M-714.2HD, PI, Germany) can reachan accuracy of 0.05 µ m, which yields a maximal usableOAM of several thousands in principle. ∗ Email: [email protected] † Email: [email protected][1] M. A. Nielsen and I. L. Chuang,
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