Realization of edge states along a synthetic orbital angular momentum dimension

Yu-Wei Liao; Mu Yang; Hao-Qing Zhang; Zhi-He Hao; Jun Hu; Tian-Xiang Zhu; Zong-Quan Zhou; Xi-Wang Luo; Jin-Shi Xu; Chuan-Feng Li; Guang-Can Guo

doi:10.1364/PRJ.533602

1. INTRODUCTION

Synthetic dimensions are newly developed tools to study topological materials in recent years. The central idea of the synthetic dimension is to exploit a set of physical states based on particles’ internal degrees of freedom to simulate the motion along an extra lattice. Synthetic dimensions have been formed based on degrees of freedom such as spin [1,2], energy levels [3,4] of atoms, frequency [5 –10], orbital angular momentum (OAM) [11,12], arrival time [13 –16], and transverse spatial supermodes [17] of photons. The synthetic dimensions are independent of accurate geometric dimensions, which enables the study of higher-dimensional physics in a lower-dimensional system. Moreover, from flexible modulation and abundant detection methods, plenty of unique topological phenomena have been obtained along synthetic dimensions in atomic and photonic systems [18 –20].

OAM states, which are found functional in high-dimensional quantum information processing [21,22], represent a prominent avenue for realizing photonic synthetic dimensions. Numerous theoretical frameworks have been proposed to investigate topological physics or develop functional devices within the synthetic OAM dimension, including synthetic gauge fields [23], Weyl semimetal phases [24], all-optical devices [25], and OAM optical switches [26]. Moreover, experimental studies have extensively utilized synthetic OAM dimensions to explore diverse phenomena such as Zak phases [27], wavepacket dynamics [28], statistical moments [29], band structures [11], and non-Hermitian exceptional points [12]. Additionally, coupling the synthetic OAM system with another physical dimension enables the investigation of two-dimensional Chern topological insulators [30]. Furthermore, by incorporating atomic-mediated interactions, the synthetic OAM dimension has been used in studying interaction physics [31] and investigating Laughlin states in fractional quantum Hall effects [32].

While significant strides have been made in exploring various bulk topological features within the synthetic OAM dimension, the establishment of edges along such synthetic dimensions to explore the topologically protected edge states presents ongoing challenges. In the context of topological materials [33], the non-trivial bulk topology is typically delineated by topological invariants, such as the Chern number [34] or $Z_{2}$ index [35]. Edge states manifest at the interfaces between materials possessing distinct topological invariants, a phenomenon known as the bulk-edge correspondence [36]. These edge states are topologically protected and can be used to realize optoelectronic devices with distinct characteristics such as immunity to local defects and high transmission efficiency [37 –39]. Consequently, the creation of an edge holds paramount significance in studying topological phenomena.

In this study, we experimentally realize a Su–Schrieffer–Heeger (SSH)-like model based on synthetic OAM dimensions in a cavity. To establish an edge, we drill a circular hole in the waveplate (WP) and terminate the polarization coupling at the zero OAM mode, effectively dividing the OAM lattice into two semi-infinite chains. This methodological approach draws inspiration from theoretical frameworks advocating for boundary establishment through the use of hollow beam splitters [40]. We calculate the system’s topological invariants and delineate its phase diagram. Leveraging resonant spectrum detection, we directly observe the spectra of edge states, offering novel insights into the phase transition dynamics of their energies and illustrating the bulk-edge correspondence. Moreover, we uncover additional characteristics of topological edges that have not been directly observed in previous experimental studies. Specifically, we investigate the dynamic behaviors of edge states as they transition into the bulk by modulating the phase of the surface. Furthermore, we observe that interference near a surface results in the discretization of the detected energy spectrum.

2. RESULTS

A. Creating Boundaries in a Synthetic OAM Lattice

We start our experiment in a degenerate optical cavity [Fig. 1(a)] [41 –43], which can support a set of modes with different OAM ( $m ℏ$ ) to resonate at the same frequency (see Appendix G for details). To form a one-dimensional (1-D) chain in OAM synthetic space, we introduce a WP made of birefringent materials and a q-plate consisting of anisotropic liquid crystal molecules into the cavity, as shown in Fig. 1(a). Compared with the platform based on a stack of q-plates and waveplates [27], the introduction of the cavity enables the reuse of optical elements, which greatly saves experimental resources. The optical modes with different spin angular momentum (SAM) [left-circular polarization $↺$ ( $s = - 1$ ) or right-circular polarization $↻$ ( $s = 1$ )] are coupled with each other by the WP with coupling strength $\sin (η / 2)$ , where $η$ is the phase delay between the ordinary and extraordinary light. On the other hand, for the spin-orbital interaction of the q-plate, modes with adjacent even OAM $m ℏ$ and $(m + 2) ℏ$ are coupled with coupling strength $\sin (δ / 2)$ and SAM is partially changed by $2 ℏ$ simultaneously, where $δ$ is an electrically controlled parameter. The coupled OAM and SAM modes form a 1-D SSH-like lattice, as shown in Fig. 1(c).

Figure 1.Degenerate cavity designed to form the OAM synthetic lattice and the topology of the bulk. (a) The degenerate cavity contains diverse OAM modes. A q-plate ( $q = 1$ ) and a waveplate are settled for coupling polarized OAM modes. WP, waveplate. (b) The WP with a centered hole in the cavity, where only optical modes with topological charge $| m | \geq 2$ pass through the WP and their polarization can be coupled. (c) The schematic of the synthetic lattice formed by OAM modes with different spins [red for left circularly polarized modes ( $↺$ ), and blue for right circularly polarized modes ( $↻$ )]; $η$ and $δ$ are coupling strengths of the waveplate and q-plate, respectively. The 1-D chain in synthetic space is cut off between the different polarized fundamental Gaussian modes ( $m = 0$ ). The unit cells labeled by index $l$ are indicated with dashed boxes. (d) The diagram of the topological phase $(ν_{0}, ν_{π})$ of our cavity system. The dashed line corresponds to $η = π / 2$ .

Download full size

View all figures

The OAM modes with $m = 0$ are fundamental Gaussian modes with the intensity peaks distributed in the center. In contrast, the OAM modes with $m \neq 0$ have doughnut-shaped intensity distributions, where the intensity peaks are located on a ring with the radius scaled as $\sqrt{m}$ (see Appendix H for the intensity profiles of the different cavity transverse OAM modes). Given the spatial distinguishability of the intensity distributions between the $m = 0$ and $m \neq 0$ modes, we can implement an $m$ -dependent operation within the cavity to truncate the OAM chain.

To establish an edge in the synthetic OAM lattice, we make a hole at the center of the WP, as shown in Fig. 1(b), to cut off the connection between two specific adjacent modes. The diameter of the hole is specially designed so that the WP will only modulate the polarization of photons with $| m | \geq 2$ . In contrast, photons with $m = 0$ can directly go through the hole and remain unchanged. With a suitable pinhole with radius $r = 0.75 ω_{0} = 125 μm$ , where $ω_{0}$ is the radius of the fundamental Gaussian mode, the undesired modal cross-coupling caused by the overlap between the cavity transverse OAM modes and the spatial aperture can be almost ignored. It is worth noting that as the sharpness of the edge is decided by the relative size of the pinhole and the beam waist, we can simply adjust the boundary from sharp to soft by tuning the position along the axis of the WP in the cavity (see Appendix H for details). As illustrated in Fig. 1(c), the polarization coupling of $m = 0$ is destroyed, and a sharp edge is created. On the other hand, the largest OAM index $m_{\max}$ (corresponding to radius $r_{\max} = ω_{0} \sqrt{m_{\max} / 2}$ ) is limited by the effective diameter $R = 6.35 mm$ of the q-plate, and the second edge appears on the sites of $m_{\max} > 5000$ . Thus, the OAM lattice can be viewed as two approximate semi-infinite lattices, where the interface between the non-trivial topological bulk and “vacuum” can support edge states.

To better characterize the lattice, we define the unit cell order $l = (m - s - 1) / 2$ . The Hamiltonian for this system in tight binding approximation ( $δ, η ≪ 1$ ) can be written as (see Appendix B for more details) $\hat{H} = \sum_{l \neq - 1} η (a_{↻, l}^{†} a_{↺, l + 1} + h.c.) + \sum_{l} δ (a_{↻, l}^{†} a_{↺, l} + h.c.),$ (1)where the $a_{↺ (↻), l}$ and $a_{↺ (↻), l}^{†}$ represent the creation and annihilation operators of the $l$ -th site with specific polarization ( $↺$ and $↻$ ).

B. Topological Phases in Synthetic OAM Lattice

The Hamiltonian of the system, owing to the spatial translational symmetry of the bulk, can be expressed as $\hat{H} = \sum_{k} a_{k}^{†} H (k) a_{k}$ , where $k$ is the quasi-momentum and $a_{k}^{†} = (a_{↺, k}^{†}, a_{↻, k}^{†})$ is the Fourier transform of the creation operator, satisfying $a_{↺ (↻), l}^{†} = \sum_{k} a_{↺ (↻), k}^{†} e^{- i k l}$ . The Hamiltonian $H (k)$ in momentum space exhibits chiral symmetry, characterized by the presence of a unitary operator $Γ = σ_{z}$ satisfying $Γ H (k) Γ = - H (k)$ , where $σ_{z}$ is the Pauli operator. Consequently, the eigenstates of the bulk crystal are constrained to rotate in a plane orthogonal to a fixed axis, hosting a topological feature in the Brillouin zone manifold. The winding number quantifies the number of rotations as $k$ traverses the entire Brillouin zone, expressed as [44] $w = \frac{1}{4 π i} \int_{- π}^{π} d k Tr [Γ H {(k)}^{- 1} \partial_{k} H (k)],$ (2)which serves as a topological invariant taking strictly integer values within the Hermitian framework.

As the parameters $δ$ and $η$ increase, the tight-binding approximation is no longer satisfied, and we have to go back to the Floquet description (see Appendix A for details). The bulk-edge correspondence in this periodically driven system relies on two invariants ( $ν_{0}$ , $ν_{π}$ ), which determine the number of 0 and $\pm π$ energy edge states [44]. These invariants are expressed as simple functions of winding numbers $w_{1}$ and $w_{2}$ in two equivalent timeframes, where the effective Hamiltonians satisfy $H_{1} (k) = - i \ln [J_{Q} (δ / 2) J_{W} (η) J_{Q} (δ / 2)]$ and $H_{2} (k) = - i \ln [J_{W} (η / 2) J_{Q} (δ) J_{W} (η / 2)]$ , respectively. $J_{Q}$ ( $J_{W}$ ) are the operators of the q-plate (WP) in the momentum space (see Appendix C for the matrix forms of the operators). As a result, the complete topological classification of such an OAM lattice can be quantified as $ν_{0} = \frac{w_{1} + w_{2}}{2}, ν_{π} = \frac{w_{1} - w_{2}}{2} .$ (3)The complete topological phases for different configurations $(η, δ)$ are illustrated in Fig. 1(d) through the calculation of the invariants of our lattice model. Moreover, we can directly detect the topological numbers in our synthetic platform, which is a unique advantage of our system (see Appendix I for experimental results).

According to the phase diagram (indicated by dashed lines), when $η = π / 2$ , the system exhibits three phases as $δ$ varies from $- π$ to $π$ . Here, we compute the theoretical eigen-energy spectrum $E$ of the effective Hamiltonians $H_{1} (k)$ and $H_{2} (k)$ for $δ \in (- π, π]$ , with $η = π / 2$ , as depicted in Fig. 2(a) (refer to Appendix A for details). The 0 and $\pm π$ energy edge states emerge within the gap at positions I and II, respectively. The energies of these edge states transition from $\pm π$ to 0 and back to $\pm π$ at the gap closing points ( $δ = \pm π / 2$ ), consistent with the predictions of the phase diagram, thus demonstrating the bulk-edge correspondence.

Figure 2.Experimental observation of the energy band structures of the edge states. (a) The theoretical energy spectrum with edge states (I and II). (b) Distributions of the edge states with 0 (I) or $\pm π$ (II) energy when $δ = 0$ or $\pm π$ . (c), (e) The schemes of the OAM lattice model when exciting different sites. Red for left circularly polarized modes ( $↺$ ), and blue for right circularly polarized modes ( $↻$ ). (d), (f) The normalized transmission intensity spectra with edge state I or II when pumping the cavity with a left circularly polarized fundamental Gaussian mode (d) or right circularly polarized $m = 2$ OAM mode (f). $ω - ω_{0}$ is the frequency detuning, and $Ω = 375 MHz$ represents the free spectral range (FSR).

Download full size

View all figures

C. Probing the Synthetic Lattice with Boundaries

Benefiting from the resonant spectrum detection of cavity output photons, such edge state energy structures can be visualized with a direct experimental perspective. The input and output relations satisfy Langevin equations [45], $\partial_{t} a_{n} (t) = - i [a_{n}, \hat{H}] - \frac{γ}{2} a_{n} (t) - \sqrt{γ} b_{in, n} (t), b_{out, n} (t) = \sqrt{γ} a_{n} (t),$ (4)where $n = s, l$ and $s \in {↺, ↻}$ . $a_{n}$ represents the light field of polarized OAM modes in the cavity and $b_{in (out), n}$ is the input (output) light field. $γ$ is the decay rate of each mode in the cavity. Transferring to the frequency domain, the input-output relation can be written as $b_{out, n} (ω) = \sum_{n^{'}} T_{n n^{'}} b_{in, n^{'}} (ω)$ , where $T_{n n^{'}} = - i ⟨ n | γ / (ω - \hat{H} + i γ / 2) | n^{'} ⟩$ represents the transmission coefficient (see Appendix D for more details). We can pre-select the input state $| n^{'} ⟩$ and reveal the edge state distribution features according to the corresponding transmission spectrum, where the edge state self-energy appears only when the corresponding edge mode is excited.

As shown in Fig. 2(b), the edge states of zero energy (I, $δ = 0$ ) are located at the edge sites, while the edge states of $\pm π$ energy (II, $δ = \pm π$ ) are located at the second nearest neighbor sites to the edge. In the experiment, we pump the cavity with a sweeping continuous-wave laser. The pre-selected input state is chosen as left circularly polarized Gaussian modes ( $m = 0$ ). The site $n = ↺,0$ on the edge along the OAM lattice is excited as shown in Fig. 2(c). The corresponding transmitted intensity spectrum $I (ω) = \sum_{n} {| T_{n n^{'}} |}^{2}$ varying with parameter $δ$ is shown in Fig. 2(d). The system energy corresponds to $E = 2 π (ω - ω_{0}) / Ω$ , where $ω - ω_{0}$ is the frequency detuning, and $Ω = 375 MHz$ represents the free spectral range (FSR) of the cavity. It is worth noting that the energy period $(- π, π]$ of the Floquet system corresponds to the frequency detuning range $(- Ω / 2, Ω / 2]$ .

The energy spectrum of bulk states agrees well with theoretical predictions in Fig. 2(a). Moreover, only the zero energy edge when $δ \in (- π / 2, π / 2)$ can be obtained with site $n = ↺,0$ excited, which demonstrates the zero energy edge state mainly distributed on the edge, as predicted in Figs. 2(a) and 2(b) (I). Then, we change the input modes to the right circularly polarized OAM modes with $m = 2$ , and the corresponding site $n = ↻,0$ is excited [Fig. 2(e)]. Only the $\pm π$ energy edge modes when $δ \in (- π, - π / 2)$ or $δ \in (π / 2, π)$ appear [Fig. 2(f)], which demonstrates energy edges with $\pm π$ energy mainly distributed on the second-nearest edge, as predicted in Figs. 2(a) and 2(b) (II). Also, we can reveal the distribution of edge states equivalently by post-selecting the output state $| n ⟩$ on the OAM basis, denoted as projective measurements.

D. Edge State Behaviors with Edge Perturbations

The edge state is the topological feature in the SSH model and is protected by chiral symmetry (see Appendix E for details). Therefore, the edge state will be significantly affected when the symmetry changes. To investigate the robustness of the edge state energy, we probe the band structure with perturbations on the edge.

One advantage of photonic synthetic dimensions is that we can dynamically change the lattice structures by flexibly adjusting the optical parameters. By tilting the angle of the WP with a central hole, the thickness of the birefringent crystal changes continuously. The light of high-order OAM modes ( $m \neq 0$ ) passing through the crystal accumulates a roundtrip phase $e^{- i ϕ}$ compared with the fundamental Gaussian mode passing through the hole, which introduces an effective phase shift for the edge sites, as illustrated in Fig. 3(a).

Figure 3.Experimentally measured edge state behaviors with edge perturbations and the spectrum discretization. (a) The schemes of the OAM lattice model when modulating the phase $ϕ$ of the states on the edge. Red for left circularly polarized modes ( $↺$ ), and blue for right circularly polarized modes ( $↻$ ). (b) The detected transmission spectra with different $ϕ$ when $δ = - π / 3$ . The edge state (indicated by the arrow) moves from $ω = ω_{0}$ into bulk states as $ϕ$ varies from 0 to $0.2 π$ . $ω - ω_{0}$ is the frequency detuning, and $Ω = 375 MHz$ represents the FSR.

Download full size

View all figures

The experimentally transmitted intensity spectra $I (ω)$ with $δ = - π / 3$ of different $ϕ$ are shown in Fig. 3(b). As $ϕ = 0$ , a small transmitted peak corresponding to the zero energy edge state (marked by the black arrow) is located in the center of the topological band gap. As the parameter $ϕ$ changes from 0 to $0.2 π$ , we observe the zero energy edge state move to the bulk states (green regions) and the topological protection is effectively destroyed. Here, the semi-infinite chain has an open boundary. The roundtrip phase applied by WP causes a shift of the quasi-energies for the edge site compared with the bulk modes, which breaks the chiral symmetry of the model.

E. Spectrum Discretization for the Interference

As an analog of our single boundary model, we consider an emitter close to a surface. The density of states of emitted light will be modified because of the interference between the forward propagating and the reflected wave [46]. According to Bloch’s theorem, the mode propagating on a periodic lattice is a modified plane wave $u_{k} (x) e^{i k x}$ , where $k$ is the momentum, $u_{k} (x)$ is the periodic amplitude, and $x$ is the axis. Here we consider a light source that will release two Bloch waves traveling in opposite directions $e^{\pm i k (x - L)}$ , where $L$ represents the distance to the surface [Fig. 4(a), left panel]. Waves traveling to the left will be reflected and result in the state $Ψ (x) = e^{- i k (x - L)} + e^{i k (x - L)} + r e^{i k L} e^{i k x},$ (5)where $r \approx - 1$ is the reflection coefficient. The interference between the leftward and the reflected waves causes an oscillation of the intensity distribution $| Ψ (x {) |}^{2}$ . For example, the wave intensity at the position of emitter $| Ψ (L {) |}^{2} = 5 - 4 \cos 2 k L$ along with momentum $k$ is shown in Fig. 4(a) (right panel), where the oscillating period is determined by $π / L$ .

Figure 4.Detected spectrum discretization. (a) Left panel: an emitter near the surface. Right panel: oscillation of energy distribution $| Ψ (L {) |}^{2}$ caused by the interference. (b) Schematic of the lattice dynamics when exciting the left circularly polarized OAM mode with topological charge $m = 6$ . Photons travel along the purple arrow, and the black cross indicates that the edge causes the reflection of the left-traveling photons. Red for left circularly polarized modes ( $↺$ ), and blue for right circularly polarized modes ( $↻$ ). (c), (e) The experimental (c) and numerically simulated (e) transmission intensity spectra. (d), (f) The experimental (d) and numerically simulated (f) transmission intensity spectra when $δ = 0.35 π$ , corresponding to the dotted lines in (c) and (e). $ω - ω_{0}$ is the frequency detuning, and $Ω = 375 MHz$ represents the FSR.

Download full size

View all figures

This effect will cause spectrum discretization in our OAM lattice system. We pump the cavity with a high-order OAM mode ( $m = 6$ ) and excite the right semi-infinite lattice site $n = ↻,2$ as shown Fig. 4(b). The experimental and simulated transmitted intensity spectra $I (ω)$ with different $δ$ are shown in Figs. 4(c) and 4(e), respectively. Since the left-traveling photons and reflected photons interfere in three unit cells (corresponding to $L = 3$ ) between the edge ( $n = ↺,0$ ) and the exciting sites ( $n = ↻,2$ ), six discrete bands of the bulk are obtained in the transmitted spectra (see Appendix F for more simulation results). To clarify the distinct peaks for the transmitted spectra, we illustrate the contour of the transmitted intensity $I (ω)$ with $δ = 0.35 π$ as shown in Fig. 4(d) (experiment) and Fig. 4(f) (simulation), which show six discrete peaks in the bulk clearly. Notably, the experimental transmitted peak has a wider peak width than the simulation results, which is more indistinguishable due to the loss of the cavity.

3. CONCLUSION AND DISCUSSION

In conclusion, we have experimentally demonstrated the edge along a synthetic OAM lattice in a single optical cavity and explored the bulk-edge correspondence of our system by measuring the topological invariants. We further investigate the behavior of the edge state under perturbations, and we find that the spectrum discretization observed in our OAM lattice system aligns well with the theory of emitters near the surface.

Compared with schemes of topological edge states in other synthetic dimensions based on atoms [1,3,4], photonic frequency [8,47,48], and spatial supermodes [17], we form a unique approximate semi-infinite lattice and explore corresponding spectrum properties, which provides a new perspective to attain topological edge effects in the SSH model. Notably, the properties of finite lattice and semi-finite lattice exhibit significant differences in non-Hermitian systems [49 –52]. It is convenient to explore the non-Hermitian topology of edge states in our cavity system by inserting a partially polarized beam splitter. Moreover, the degenerate-cavity-based scheme makes it easier to support the couplings of multiple synthetic dimensions [53] and cavities [23], which provides opportunities to construct high-dimensional topological insulators for exploring nontrivial topological edge phenomena, such as the chiral edge states along the OAM lattice. The long-range couplings along the OAM lattice can also be introduced via a J-plate [54], which can perform arbitrary spin-to-orbital angular momentum conversion. Drilling a hole in optical elements to distinguish different OAM modes can also build all-optical devices, such as all-optical memory [25]. Furthermore, the degenerate cavity offers the possibility of coupling atoms [32], allowing for the exploration of effective many-particle systems by introducing photon-photon interactions via atomic mediums [55,56]. Our work provides a new topological photonic structure that may benefit topological devices such as topological lasers.

Category: Quantum Optics

Received: Jul. 2, 2024

Accepted: Sep. 17, 2024

Published Online: Dec. 20, 2024

The Author Email: Jin-Shi Xu (jsxu@ustc.edu.cn)

DOI:10.1364/PRJ.533602

CSTR:32188.14.PRJ.533602