Quantum simulation of 2D topological physics in a 1D array of optical cavities

Orbital angular momentum of light is a fundamental optical degree of freedom characterized by unlimited number of available angular momentum states. Although this unique property has proved invaluable in diverse recent studies ranging from optical communication to quantum information, it has not been considered useful or even relevant for simulating nontrivial physics problems such as topological phenomena. Contrary to this misconception, we demonstrate the incredible value of orbital angular momentum of light for quantum simulation by showing theoretically how it allows to study a variety of important 2D topological physics in a 1D array of optical cavities. This application for orbital angular momentum of light not only reduces required physical resources but also increases feasible scale of simulation, and thus makes it possible to investigate important topics such as edge-state transport and topological phase transition in a small simulator ready for immediate experimental exploration.

A s a relatively under-exploited optical degree of freedom, orbital angular momentum (OAM) of light has motivated much exciting research lately. Beams of OAM-carrying photons have an azimuthal phase dependence in the form e ilj where the OAM quantum number l can take any integer value 1 . These photon modes, which arise in the natural solutions of the paraxial wave equation in cylindrical coordinates 2 , can be manipulated and measured with high precision [3][4][5][6] . Because of the unlimited range of the angular momentum, OAM-carrying photons are recognized as a unique asset in many studies. On the application side, they are used to enable high-capacity optical communication 7,8 and versatile optical tweezers 9 . In fundamental research, they have played important roles in quantum information and quantum foundation 6,[10][11][12][13][14][15] . Although experimental study of OAM of light used to be limited to low angular momentum, there has been tremendous advance lately motivated by its great potential. This is highlighted by the remarkable recent demonstration of quantum entanglement involving angular momenta as high as hundreds 16,17 .
In spite of the many successful recent studies involving the OAM degree of freedom of light, its exploitation is still at an early stage and many novel possibilities remain unrecognized. In particular, it has not been considered useful for quantum simulation of important physics problems such as the extraordinary topological phenomena that arise in two-dimensional (2D) systems subject to external gauge fields. These include the likes of integer 18 and fractional 19 quantum Hall effect and quantum spin Hall effect 20 , which are characterized by exotic properties such as quantized conductance and edge-state transport. They are often difficult to investigate due to stringent experimental conditions required, and some theoretical predictions remain challenging to observe 20,21 . Because of this, various quantum simulation schemes based on different physical platforms such as ultracold atoms [22][23][24] and photons [25][26][27][28][29][30][31][32][33][34][35] have been suggested recently. None of them involves OAM of light whose connection to topological physics appears to be nothing but an illusion even in concept. Not surprisingly, central to most existing simulation schemes is a 2D architecture for the simulator. Many of them are still very demanding, requiring limit-pushing experimental conditions or advanced new technologies.
In this work, we show that it is not only possible, but advantageous to use the OAM of light for nontrivial quantum simulation by demonstrating theoretically how it can enable and support the study of a broad range of topological physics. In contrast to other proposals 25-33 , our system has a onedimensional (1D) structure that does not need to be large in scale, thus reducing the complexity of the simulator. Feasible scale of simulation is increased despite the simplified system, and it is so versatile that the effect of arbitrary Abelian and non-Abelian gauge fields can be studied using standard linear optics devices only, with no restriction on the form of the gauge fields 29,30,33 and no need for specially designed meta-material 31 or photonic crystal 33 . It then allows to investigate important topological problems under intense pursuit such as non-Abelian gauge field induced phase transition between a photonic normal and topological insulator. Further, we can easily probe the topological properties of our system by measuring the photon transmission coefficients which are shown to have deep connections to the essential topological invariants of the system. All this is possible because of the inherent properties of the OAM of light.

Results
The 1D array of cavities. Shown in Fig. 1a is our simulation system. It consists of an array of N nominally identical cavities that are coupled along the x direction. The system size, N, does not need to be large; we will show that even a simulator with just a few cavities is sufficient to demonstrate topological effects. The building blocks are degenerate cavities 36,37 , which have appropriate optical design such that they can support photon modes with different OAM (Supplementary Note 1). In each cavity, we make use of clockwise-circulating OAMcarrying photons and denote their annihilation operatorâ j;l , where j (0rjrN-1) is the index of the cavity in the array and l is the OAM number of the photon mode. To manipulate the OAM state of photons, for each cavity we introduce an auxiliary cavity consisting of two beam splitters (BSs) and two spatial light modulators (SLMs). The BSs divert a portion of the light in the main cavity towards the SLMs and merge it back. When propagating between the BSs, photons can accumulate a phase. The SLMs, which can be simple spiral phase plates with very low loss 38,39 , change the OAM of photons by ±1.
As depicted in Fig. 1b, by associating the OAM number of the photon in a cavity with the site index number along the y direction of an (imaginary) lattice, we can conceptually map our 1D array of cavities to a 2D rectangular lattice system. In Fig. 1a, the BSs and SLMs of the auxiliary cavity change the OAM of a portion of the light in the main cavity by ±1, and this corresponds to hopping of a particle on the lattice site in Fig. 1b along the y direction to its neighbouring sites with a probability determined by the reflectivity of the BSs. In this hopping process, the particle can also acquire an experimentally controllable phase determined by the imbalance between the optical paths from BS j 1 to BS j 3 and backwards. As shown in the Supplementary Note 2, the Hamiltonian of the simulated system is where k is the transition rate between different OAM states, chosen to be the same with the coupling rate between neighbouring cavities, and 2pf j is the phase acquired by the photon in the jth cavity when it travels between the BSs in the auxiliary cavity. The term h.c. stands for Hermitian conjugate of the prior terms. If we set up the system such that f j is linearly dependent on the cavity index j, f j ¼ jf 0 , then H 1 describes a tight-binding model of charged particle in a 2D lattice subject to a uniform magnetic field with f 0 quanta of flux per plaquette 40 .
Therefore, by representing a spatial degree of freedom with the OAM states of photons, we can study a 2D system with a 1D simulator, greatly reducing the physical resources required for the simulation. In contrast to earlier 1D optical simulator 34 , our system performs a full and genuine 2D simulation, rather than simulate the 1D behaviour of the system at a fixed Bloch momentum in the other direction. Meanwhile, in comparison with a 2D array of coupled cavities, the size of the 2D lattice that can be simulated is markedly increased along the y direction. This is due to the fact that, unlike in an atomic system 41 where only a small number of atomic states are available for the simulation, there is no upper limit for the OAM of photons in theory. In reality, it is limited by practical factors such as the size of the optical elements and can be made very large in a proper design. In contrast, the feasible size in the y direction for a 2D cavity array would be much smaller, because nonuniformity of the cavities and local disturbances will make photons quickly lose coherence after travelling through a few cavities. This remarkable combination of reduced physical resources and increased scale of simulation makes our system very promising. Also, our system can be easily modified to support more demanding simulations by making use of additional degrees of freedom of photons. For instance, we can simulate the quantum spin Hall effect 42 in non-Abelian gauge fields 43,44 by using the horizontal and vertical polarizations of polarized photons to represent the up and down state (s ¼ ±1) of a spin. By using birefringent waveplates whose optical axes are properly aligned with respect to the horizontal and vertical polarizations, we can assign different phases to the two polarizations and cause transitions between them when they pass the waveplates (see Supplementary Note 3 for details). We can then manipulate the polarization states of the photon to mimick the spin flips and spin-dependent phase delays caused by non-Abelian gauge fields, as illustrated in Fig. 2. The simulated Hamiltonian is (Supplementary Note 3) whereâ y j;l ¼ ða y j;l;$ ; a y j;l;l Þ is a two-component (the horizontal and vertical polarization) photon creation operator, and l j is an effective on-site energy. The tunnelling phases that correspond Dove prism e il e il ( +ϑ) ϑ / 2 Dove prism BR 2 Figure 1 | A 1D array of degenerate cavities for simulating a 2D rectangular lattice in a magnetic field. (a) The optical design for simulating H 1 . Each main cavity has an auxiliary cavity consisting of two BSs (BS j 1 and BS j 3 ) and two SLMs (SLM j 1 and SLM j 2 ). There is also a coupling cavity (made of BS j 2 and BS j þ 1 4 ) between adjacent main cavities (It can be replaced with a simple BS to reduce the number of optical elements in experiments). The length of both the auxiliary and coupling cavity is chosen for destructive interference, and most light remains in the main cavity. The cavities at the two ends of the array can be coupled to realize periodic boundary condition, or uncoupled for open boundary condition. (b) Mapping of the 1D simulator array in (a) to a 2D rectangular lattice in a magnetic field. (c) The coupling cavity (left) for simulating H 5 and the optical design (right) for the beam rotators BR 1 and BR 2 with opposite rotation angles ± W ¼ ± 2pf 0 . The main cavity and auxiliary cavity require no modification, except that the phase difference between the arms containing the SLMs is set to 0. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8704 ARTICLE to the potentials of the associated gauge fields 22 , are given bŷ where f j is the spin-independent part of the phase, and a, b j ,ŝ 1 andŝ 2 are determined by the Jones matrices 2 of the waveplates as shown in Fig. 2. By selecting appropriate waveplates and manipulating the polarization of the photon accordingly, we can engineer non-commuting tunnelling phasesŷ x andŷ y , and thus simulate the effect of an arbitrary non-Abelian gauge field.
Probing scheme. Since we represent a spatial degrees of freedom with OAM states of photons, the measurement of our system involves manipulation and detection of the OAM states. Specifically, we pump the j i th cavity using a probing light with a definitive OAM l i and measure in the steady state how much ends up in the OAM mode l o in the j o th cavity by leaking a small amount of light out of each cavity, as shown in Fig. 1a. It is determined by the transmission coefficient 45 where o is the detuning of the probing light from the cavity frequency, g is the photon loss of the system, and H SYS is the simulated Hamiltonian. When non-Abelian gauge fields are concerned, the polarization indexes s i and s o should also be included for the input and output modes. Generation and detection of OAM-carrying photons can be accomplished very reliably 3,6 . By a coherent measurement, we can determine both the amplitude and phase of T jo;lo j i ;l i ðoÞ. Thanks to the 1D structure of our system and the use of OAM states, we can perform this measurement between any pair of (j i , l i ) and (j o , l o ), equivalent to measuring the transmission coefficient between any pair of sites in the simulated 2D lattice. Such powerful probing capability is key to the demonstration of various topological effects in our system.
Feasible measurement and clear demonstration of topological properties is the topic of many recent studies 21,31,32,46-48 since generally speaking it is a very challenging task. Remarkably, in our system it is straightforward and requires no more than measuring the photon transmission coefficient in equation (4). As we will show, there is a deep connection between the photon transmission coefficient and the essential topological invariants, which can be exploited to demonstrate topological behaviour in optical systems.
System spectrum and density of states. As can be seen in equation (4) ð Þ is sensitive to the energy mismatch between the frequency of the probing light and the energy of the system. Because of this, we can study the system's spectrum by measuring the transmission coefficient as a function of the frequency of the probing light, where T jo;lo For a system in an Abelian gauge field described by H 1 , we calculate and plot in Fig. 3a the system spectrum which is the well-known Hofstadter butterfly 40 . We see that the main characteristics of the system spectrum are clearly identifiable even in a small simulator with just a few cavities.
The transmission spectroscopy is also very valuable for studying physics associated with a non-Abelian gauge field. As an example, in equation (3) which is a topic of intense research because of its importance for understanding the properties of graphene and other exotic systems 23,24,50,51 . Characteristic of H 3 are four conical singularities at the Dirac points 51 in the spectrum, which give rise to massless relativistic particles. As the energy deviates from the Dirac points, the change of the dispersion relation from relativistic to non-relativistic is revealed by the Van Hove singularities in the density of states. When the decay rate g is small, the density of states can be inferred from the photon transmission spectrum which is shown in Fig. 3b. The Dirac point at o ¼ 0 and two Van Hove singularities near o ¼ ± 2k are observed, confirming Dirac physics related behaviour in the system.    Edge states and topological protection. One of the most remarkable phenomena in topological physics is the existence of topologically protected chiral edge states in the band gaps of a finite lattice. In our system, we can study the edge states by pumping the cavity at the end of the 1D simulator array using a probing light beam with a definitive OAM. It is equivalent to driving a site on the edge of a 2D lattice. When the frequency of the probing light falls in a band gap, excitation of gapless edge states dictates that the light can only propagate along the edge of the simulated system. This is clearly demonstrated in Fig. 4a,b, where chiral edge-state transport is observed in a small simulator.
To study the robustness of the edge states against disorder, we introduce the average OAM 'displacement' for the transport process defined as where T ji;0 j 2 , and P ji2edge refers to summation over the sites close to one edge of the lattice where the amplitude of the edge states is significant. As proved in the Supplementary Note 5, when the frequency of the probing light o falls in a large band gap, l e has the interesting property that it is equal to the total Chern number for the bands below the gap. Also, the value of l e is mainly determined by states roughly in resonance with o. Consequently, how l e is disturbed by disorder is a measure for the robustness of these states. Shown in Fig. 4c are l e and its variation caused by a random shift in the cavity resonance frequency. It can be concluded that the edge states are almost immune to the disorder when the band gap is large compared with the photon loss and random cavity frequency shift, whereas the in-band states are strongly affected.
In addition to its fundamental interest, edge-state transport is also very useful for probing the topological behaviour of a system. One such example is the observation of the relativistic quantum Hall effect which arises in the Dirac Hamiltonian H 3 with small but nonzero magnetic flux f 0 . As shown in Fig. 4d, l e experiences a double-step leap from 2 to À 2 around the Dirac point at o ¼ 0 caused by a sudden change in the topological property of the system. Such exotic behaviour 43,44 was predicted and observed in graphene 52,53 .
Topological quantum phase transition. By measuring the system spectrum and edge-state transport, we can study nontrivial physics such as topological quantum phase transitions driven by non-Abelian gauge fields, which are important for understanding novel quantum states of matter such as topological insulators and superconductors 21,23,24,43,44,54,55 . In our system with non-Abelian gauge field, if we chooseŝ 1 (3), the Hamiltonian in equation (2) becomes which describes an effective spin in a non-Abelian gauge field characterized by spin-dependent magnetic field and strong spinorbit coupling. Also present is a periodically modulated on-site potential l j . In the simulation system, the horizontal and vertical polarizations with degenerate on-site energies flip to their counterpart when the photon tunnels between cavities and acquire opposite phases when the photon goes around a plaquette in the simulated lattice in the same direction. This is the same behaviour with that of the spin up and down in an electronic system, which has time-reversal symmetry, and polarized photonedge states analogous to spin edge states can emerge in our system. The two polarized edge states are associated with opposite Chern numbers, and thus their total Chern number C is 0 whereas the difference 2n can be nonzero. The properties of such a photonic topological insulator are in contrast with those of a normal insulator in which both C and n are 0 and photon transport of both polarizations is strongly suppressed. A topological quantum phase transition can be induced in the system by adjusting the value of the non-Abelian gauge field. In Fig. 5a, it is shown how the band structure of the system changes with b 0 . As b 0 increases, the first band gap near o ¼ À 1.6k closes and opens again. Initially, when b 0 is small, the topological index n of the system is n ¼ 1, and the system is in a topological insulator state. Correspondingly, there are a pair of photon-edge states with opposite polarizations propagating in opposite directions as shown in Fig. 5b,c. These polarized-edge states are protected as long as the local noise does not disturb the symmetry between the two polarizations so that their on-site energies stay degenerate and their phases around a plaquette remain opposite to each other. When the energy gap opens again with a large b 0 , n changes to 0, and the system becomes a normal insulator. This is confirmed by the disappearance of the photon-edge states in Fig. 5d,e. ARTICLE Measurement of the chern number. The Chern number is the ultimate quantum invariant to classify topological states and characterize their behaviour 21 . As shown in Fig. 4c, in a finite lattice the Chern number can be measured via the average OAM displacement l e for edge-state transport. In an infinite system, the Chern number is equivalent to the TKNN index 56 . For its measurement, we insert a pair of beam rotators (BRs) with opposite rotation angles ±W ¼ ±2pf 0 in the coupling cavities, as shown in Fig. 1c. A BR with a rotation angle W is made of two Dove prisms rotated by W/2 with respect to each other and can   (10), defined by dividing the Brillouin zone into B1, whose boundary is marked by the red line and where u 1 0 is always nonzero, and B2, which contains all zero points of u 1 0 but where u 1 3 does not vanish. In one phase convention, u 1 0 is real in B1. In the other convention, u 1 3 is real in B2. The Chern number is determined by the integration of the gradient of w(k x , k y ) on the boundary of B1, qB1 (Supplementary Note 6). In a and b, the size of the simulator N ¼ 10. The OAM included in the calculation is lA[ À 48, 48). Periodic boundary conditions are used in both the x and y directions. The photon loss rate g ¼ 0.1k. change the azimuthal dependence of the OAM mode from e ilf to e il(f þ W) . We also balance the two paths of the auxiliary cavities containing the SLMs. The simulated Hamiltonian becomes which is related to H 1 by a gauge transformation and helps keep the size of the simulator small (Supplementary Note 2). In Fig. 6a, the amplitude of the photon transmission coefficients jT j;l 0;0 j 2 is shown for a system with a rational magnetic flux f 0 ¼ 1/6. Since the first energy band of this system is very narrow (see Supplementary Note 6), in a lossy cavity the probing light will be in resonance with the entire first energy band 57 . This allows us to determine the in-band Bloch eigenstates e ik x j e ik y l u 1 l k x ; k y À Á ð10Þ from the Fourier transforms of T j;l 0;0 , where k x A[ À p,p], k y A[0,2p/6] define the Brillouin zone and u m l ðk x ; k y Þ ¼ u m l þ 6 k x ; k y À Á for the mth band is a periodic function. There is a Chern-number-conserving gauge freedom in the phase choices of u 1 l ðk x ; k y Þ, as shown in Fig. 6b. w(k x ,k y ), the phase mismatch of u 1 3 resulting from the two different phase conventions in Fig. 6b, can be used to calculate the Chern number (Supplementary Note 6). Our numerical calculation using w(k x ,k y ) yields the Chern number 1 for the related band.

Discussion
By mapping the OAM states of photons to spatial coordinates in a lattice, we have found a promising scheme for studying nontrivial 2D topological physics in a 1D physical simulator. Our method relies on only linear optics and manipulation of OAM states, and thus it can be realized with any physical systems that provide these elements or their equivalent, though longer wavelengths may have an advantage in coupling a large number of cavities.
Our system is ready for immediate experimental exploration, because the key elements in our scheme, such as reliable manipulation of photon modes with high angular momenta 4,16 , precise measurement of the OAM states 5,6 , design and operation of degenerate cavities 36,37 and locking of multiple optical cavities 58 , have all been realized. Our idea may also be used to simulate 1D problems with OAM modes in a single cavity [59][60][61] , and it can lead to novel photonic effects with practical applications 25 . Above all, by demonstrating the counterintuitive application of photonic OAM in quantum simulation, our work deepens our understanding of the OAM degree of freedom and advances our view of photonic quantum simulation. Building on the presented ideas, we can then leverage the extreme flexibility and reliability in the design and operation of optical circuits for quantum simulation of various topological problems. All these issues and possibilities provide exciting opportunities for further investigation.