Topological band structure via twisted photons in a degenerate cavity

Synthetic dimensions based on particles’ internal degrees of freedom, such as frequency, spatial modes and arrival time, have attracted significant attention. They offer ideal large-scale lattices to simulate nontrivial topological phenomena. Exploring more synthetic dimensions is one of the paths toward higher dimensional physics. In this work, we design and experimentally control the coupling among synthetic dimensions consisting of the intrinsic photonic orbital angular momentum and spin angular momentum degrees of freedom in a degenerate optical resonant cavity, which generates a periodically driven spin-orbital coupling system. We directly characterize the system’s properties, including the density of states, energy band structures and topological windings, through the transmission intensity measurements. Our work demonstrates a mechanism for exploring the spatial modes of twisted photons as the synthetic dimension, which paves the way to design rich topological physics in a highly compact platform.

The degenerate optical cavity has been theoretically investigated [1,2]. In experiment, the degenerate cavity consist of two plane mirrors and two lenses of focal length f = 0.1 m. The free spectral range (FSR) of the cavity is about 375 MHz, while the linewidth is about 13.6 MHz. A Q-plate with q = 1 is placed in the center of the cavity, on which the electrostatic field is controlled by an arbitrary function generator (AFG). A (η/π)-wave plate (WP) is set behind to rotate the polarization (e.g., η = π/4 for the quarter-wave plate). To scan the cavity length ∆L, a piezoelectric transducer (PZT) is pasted on the second mirror and is driven by an amplified periodic triangular wave signal generated by the AFG. The scanning frequency of the triangular wave signal is set at 40 Hz such that the system reaches a steady state at each frequency.
To reduce the cavity's dissipation (α), all the optical elements in the cavity are coated with anti-reflection films. The linewidth of the cavity can be reduced to obtain sharper transmission peaks.
The photons are out-coupled by the second mirror with the ratio of 1/99. The output photons are separated into two paths by a beam splitter (BS). The transmitted photons are detected directly by a PD, while the reflected parts are first modulated by a spatial light modulator (SLM) and post-selected by a single mode fiber (SMF). When detecting topological windings, a QWP, a half-wave plate (HWP), and a PBS in the dashed panel are set before the fiber coupler (FC) for the post-selection of polarization. All the signals detected by the PDs are recorded in an oscilloscope with a 1 GHz bandwidth, which allows readind the system's eigenenergy directly.
In our experiment, the length of the synthetic dimension is limited by the size of the minimum aperture of the Q-plate. With the increasing of topological charge number m of orbital angular momentum (OAM) modes, the transverse radius of maximum field amplitude r m will increase as r m = ω 0 m/2 [3].

II. DISPERSION RELATION OF THE CAVITY
In this section, we derive the dispersion relation for the cavity without input and output. As the photons propagate in a steady cavity, the state |φ(t) = (..., φ(t) ,m−1 , φ(t) ,m−1 , φ(t) ,m , φ(t) ,m , ...) at time t must satisfy the condition of mode self-reproduction, denoted as where β = Ω/c + iα and L is one round trip (one period) length of the cavity. Ω is the resonant frequency of the vacuum cavity and α is the attenuation coefficient. T = L/c, where c represnts the speed of light.
On the other hand, according from the method, the action of Q-plate (q = 1) is described as The action of η/π-wave plate (WP) can be described as the cavity can be described asÛ When the evolutionary period is guaranteed, the photon state in the cavity satisfies According to Eq. S1 and S5, we find Here we define an effective Hamiltonian ofĤ eff = i logÛ . In the quasi-momentum space, we define the where s = ±1 related to the SAM denotes the upper and lower energy bands. The unitary evolution in the . J Q k (δ) and J λ k (η) represent the operations of Q-plate and WP in reciprocal space, respectively, which are given by and The Eq. S7 can be rewritten as The evolution satisfiesÛ k = cos(E k )I + i sin(E k )n(k) · σ. Compared with the Eq. S10, we can obtain where E k is the energy dispersion relation of the cavity. The unit vector n(k) = [n x (k), n y (k), n z (k)] reveals the topological winding numbers of the system as discussed later. We can observe the windings of the unit vector n(k) and −n(k) for the upper and lower bands. Moreover, we can findĤ eff (k) meets ΓĤ eff (k)Γ = −Ĥ eff (k) with Γ = σ z , which means the system has chiral symmetry.
The timeframes are the time evolution with different starting points, which are unique properties in the periodically driven system. If the photons pass a round through η/π-WP firstly and then the Q-plate, denoted as the second time frame, the evolution operatorÛ k can be rewritten aŝ Similarly, the energy dispersion and unit victors of the second timeframe are given by (S13) Obviously, these two time frames have the same energy dispersion relation but the different three-dimensional unit vector n(k). The winding number of the unit vector n(k) is the topological invariant, protected by the chiral symmetry. Therefore, the two timeframes have different topological invariants and correspond to different topologies.

III. DIRECT MEASUREMENT OF THE DENSITY OF STATES
In this section, we turn to an open system and demonstrate the method to directly measure the density of states (DOS) from the cavity output. The coupling of the cavity mirror can be described by [4,5]  where κ = i|κ| and r = |r|. m represents the OAM topological charge. φ in (φ out ) represents the input Combining Eq. S14 and S15, we find and φ out, ( ),m = 1 r * (κe −iβL a ( ),m + φ in, ( ),m ). (S17) Representing the states as the state vectors, Eq. S16 and Eq. S17 become and where |φ in = (..., φ in, ,m−1 , φ in, ,m−1 , φ in, ,m , φ in, ,m , ...), so are |a , |b and |φ out .
By taking |b =Û |a into Eq. S18, we can get where n represents the loop number of the photons running in the cavity. Note that |a n−1 = |a n if n → ∞. Initially, there is no photon in the cavity, which means |a 0 = 0. After n loop number, we get Combining the Eq. S19 and S21, we can get the output state as The first term on the right-hand side represents the direct reflection of |φ in . The second term represents the transmission of the field, and we redefine the |φ out as The eigenstates |φ s k = |ψ s k |k of the HamiltonianĤ eff (k) form a set of complete basis for expanding |φ out . We set the input field to be |φ in = |φ s in |m 0 with |m 0 representing a special momentum state. The transmission field |φ out can be written as where sE k represents the eigenenergy of the HamiltonianĤ eff . Taking ∆L= L − 2nπ/β (n ∈ N + and β∆L < 2π), we define the transmission coefficient T s k as By choosing an appropriate input state |φ s in , s | k|m 0 ψ s k |φ s in | 2 could be independent on k. For instance, the input state in our experiment with the Gaussian mode |m 0 = 0 is prepared to the maximally mixed polarization state of 1/2(| | + | |), the total intensity I o becomes On the other hand, the density of states related to volume V is defined as where E k represents the energy band along momentum k. In our experiment, only sE k = β∆L mainly contribute to the transmission intensity I o (β∆L) in Eq. S27. When |r| → 1, 1 1+|r| 2 −2|r|x will very close to the δ(x − 1) function (x = cos(sE k − β∆L)). Thus, I o can be approximated as where Γ is the normalised coefficient. Regarding Γ as the volume V, I o is denoted as density of the states

IV. THE TRANSMISSION MODES OF THE CAVITY
Here we illustrate more details of the output modes of the cavity with η = π/4, which is shown in Fig.   S2. The input photons are on the Gaussian mode (m = 0) with horizontal polarization. When δ = 0, the transverse mode of light is always kept in the Gaussian mode (m = 0), but the polarization changes periodically. As a result, there are the splittings of the transmission peaks, which are twice that in the vacuum cavity. When δ > 0, the high order angular momentum modes begin mixed, and the spectra of the system getting more and more complicated, which satisfies the dispersion relation described in Eq. S11. Especially when δ = π/2, the transverse modes are restricted to the angular momentum modes

V. DIRECT MEASUREMENT OF THE ENERGY BAND SPECTRUM
With the post-selected state |k , the transmission intensity becomes which illustrates the distribution of E k . By scanning the state |k , the energy band spectrum can be directly demonstrated.

VI. PHOTON DISTRIBUTIONS AFTER THE MODULATION VIA SLM
The photon distributions after modulated by SLM of different (k exp , N ) are shown in Fig. S3. The number of the "petals" of the interfered patterns is 2N , while the patterns rotate with k exp .

VII. DIRECT MEASUREMENT OF THE TOPOLOGICAL WINDING
Under the measurement basis |k k| ⊗ (σ x , σ y , σ z ), the output result gives, where and The topological windings can be revealed by the variations of transmitted peaks.

VIII. EDGE EFFECT AND DISORDER EFFECT
Edge states are topologically protected, an outstanding feature of topological physics. Though the edge effect is weak in our current experiment, the edge states can be investigated in our platform by engineering the operation on different optical modes. For instance, a QWP with a ping hole on its center, as shown in can support edge states. The numerical DOS without disorder is shown in Fig.S4c, in which the edge states can be clearly seen at E k = 0 or ±π.
The disorder is further introduced from the imperfect degeneracy of the cavity, which is given by a random phase e i∆θm (|∆θ m | < 0.1π) on each optical mode. Such kind of disorder corresponds to a distribution of energies around the main energy and makes the edge energy move to bulk bands. The simulated result is shown in Fig. S4d. The edge state will merge into the bulk state with increasing the disorder strength. † jsxu@ustc.edu.cn c.The numerical DOS without disorder. d.The numerical DOS with disorder. ‡ smhan@ustc.edu.cn § cfli@ustc.edu.cn