Photonic Landau levels in an astigmatic frequency-degenerate laser | Communications Physics

Communications Physics volume 8, Article number: 111 (2025) Cite this article 499 Accesses 2 Altmetric Metrics details Landau levels refer to the quantization of electron energy levels under magnetic

Communications Physics volume 8, Article number: 111 (2025) Cite this article

499 Accesses

2 Altmetric

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Landau levels refer to the quantization of electron energy levels under magnetic fields which have recently been extended to photonic systems through the judicious design of pseudomagnetic fields. However, established approaches towards photonic Landau levels have primarily relied on complex artificial metamaterials or passive multifolded cavities. Here, utilizing an active symmetry-breaking angular momentum laser with simple displacement tuning of intracavity astigmatic elements for generating structured beams with controlled frequency-degeneracy, we demonstrate that the physical nature of such frequency-degeneracy can be modeled effectively as the quantization of photon energy levels. This work provides a compact solution to control complex quantized modes in structured light laser platform. Related potential applications of these Landau level lasers also permit advancing photonic quantum Hall effects and topological information.

Two-dimensional (2D) electron systems subjected to magnetic fields form Landau levels with quantized energy. This fundamental model in modern quantum physics has given rise to various scientific breakthroughs, such as topological insulators1,2,3,4, anyon collisions5,6, and twistronics7,8. Recent advances in synthetic quantum materials indicate that sophisticated photonic systems can emulate the behavior of electrons9, including dielectric photonic crystals10,11,12,13, continuum photon fluids14,15, topological lasers16,17,18, and merging fermionic and bosonic systems19. Now, Landau levels can be further generalized beyond the original fermionic systems to include bosonic and even anyonic systems20,21,22,23, utilizing sophisticated photonic crystal designs20,21 or exotic superconductors22. In addition to solid-state systems, the Landau levels of free-space photons have also attracted attentions with various theoretical or experimental demonstrations23, marking significant advancements in the field of fundamental physics. For instance, photonic Landau levels have been employed in the topological characterization of electromagnetic and gravitational responses24, the exploration of light-matter interactions for exotic polaritons25,26, the emulation of quantum Laughlin matter27,28, and the realization of Landau levels in acoustics29,30. However, controlling photonic Landau levels in free space still remains an open challenge. The only reported approach so far is the spatially multifolded laser resonator, which necessitates precise control of non-planar geometry to manipulate the cyclotron behavior of intracavity photons23. In contrast, the degenerate cavity with excellent capability to twist photons can offer an approach for realizing Landau levels for free-space photons31.

Here, we propose a mechanism for spatially regulating photons (the structured light modes) in Landau levels using a compact laser cavity. We control the photon distribution in astigmatically controlled frequency-degenerate laser modes as the analogy of Landau levels (omitting “frequency” hereinafter since this paper only refers to the frequency-degenerate). A crucial step in our method involves designing effective magnetic fields for the structured modes by utilizing an intracavity astigmatic mode converter to twist the structured light modes within the cavity. By manipulating the linear displacement of the cavity elements, the structured laser modes can emit photons in discrete degenerate states, fulfilling the same Hamiltonian form under specific conditions as signature in Landau energy levels. We also experimentally observe Landau quantization corresponding to a series of exotic orbital angular momentum (OAM) laser modes across different photonic Landau energy levels. The proposed scheme allows for the controllable distribution of structured modes with varying orders in photonic Landau levels by displacing cavity elements. This work reveals the potential to map various nontrivial topological effects into the structured modes, such as the Aharonov-Bohm effect, topological edge states, the Zeeman effect, and the quantum Hall effect, providing valuable insights into electron–photon analogies.

When a uniform magnetic field is applied to an electronic system, electrons are restricted to discrete cyclotron orbits, known as Landau energy levels. This phenomenon is referred to as Landau quantization. For example, an electron moving with velocity v in a magnetic field B experiences a Lorentz force that acts towards the center of its orbit, as illustrated in Fig. 1a. The Hamiltonian for this system, according to the Fock-Darwin model32,33, is expressed as \(H={\left({{{\boldsymbol{p}}}}-e{{{\boldsymbol{A}}}}\right)}^{2}/(2m)\), where p is the canonical momentum operator, A is the electromagnetic vector potential, −e is the electron’s charge, and m is the electron’s mass. With varying cyclotron angular velocities ωc, electron orbits on discrete paths correspond to distinct energy levels given by En = (n + 1/2)ℏωc, where n = 0, 1, 2, ⋯ . The principle of Landau level splitting under a fixed magnetic field is depicted in Fig. 1b. The energy interval between any two adjacent Landau levels is a constant ℏωc, where ωc = eB/m. The highest energy level is determined by the Fermi energy \({E}_{F}={\hslash }^{2}{k}_{F}^{2}/(2m)\) of the electronic system, with kF being the Fermi wave vector.

a In a 2D electronic system subjected to a uniform magnetic field B (perpendicular to the plane and pointing outward), an electron with a charge of − e and a mass of m can rotate along discrete circular orbits (represented by the orange, green, and blue dashed lines, corresponding to energies E1, E2, and E3, respectively) at a velocity of v driven by the Lorentz force of FB. b With the emergence of a magnetic field, discrete energy levels En = (n + 1/2)ℏωc form with constant interval of ℏωc and characterized by quantum numbers (n = 0, 1, 2, 3, …). The maximum value of n is subjected to the Fermi energy EF. c, In the uncharged particle system, the reference system is a rotating system with an angular velocity ω with direction perpendicular to the paper. In this system, a particle rotates with a velocity of \({{{{\boldsymbol{v}}}}}^{{\prime} }\) along circular orbits, which is influenced by the Coriolis force Fco, analogous to the Lorentz force described in a. According to the similarity with the quantized energy levels of electron cyclotron orbits in a, the effective charge \(-{e}^{{\prime} }\) and mass \({m}^{{\prime} }\) of the uncharged particle, as well as the effective magnetic field \({{{{\boldsymbol{B}}}}}^{{\prime} }\) (which is perpendicular to the paper and points outward), can be derived. d, Besides the Coriolis force, the centrifugal force Fce is also introduced in the rotating reference system. As the centrifugal force approaches zero, the energy levels transition from independent oscillators to an energy-degenerate state. e As a photonic system, the rotating system is introduced through a designed laser cavity that contains two non-parallel cylindrical lenses (the angle between the two generatrix lines is denoted as θ). By adjusting the displacements of the elements (indicated by “⇌”), the effective centrifugal force Fce can be set to zero, allowing for the excitation of Landau levels. The wave functions of these Landau levels, represented by En = nΔE + E0, corresponds to various laser mode patterns. Under certain cavity lengths, their energy levels become degenerate, as illustrated in the right insert. The highest energy level is influenced by the gain aperture Egain, analogous to the Fermi energy EF in b.

The foundational model, initially formulated for electronic systems, can be readily generalized to encompass systems of uncharged particles23. For a charge-neutral particle within an introduced rotating coordinate system at cyclone velocity ω (as shown in Fig. 1c), particles exhibiting relative rotational motion to this coordinate system are influenced by the Coriolis force Fco. This force can be likened to an equivalent Lorentz force in the presence of a synthetic magnetic field \({{{{\boldsymbol{B}}}}}^{{\prime} }\). Consequently, the achievement of synthetic Landau levels becomes feasible. It is important to clarify that we use the term “synthetic” to designate parameters specific to charge-neutral particle systems, distinguishing them from those of electronic systems. The magnetic field responsible for inducing Landau levels in electronic systems is an actual, physical field, whereas the “synthetic field” that gives rise to Landau levels in neutral particle systems is a pseudomagnetic field. This synthetic field is a consequence of the system’s rotation. In addition to the Coriolis force, the centrifugal force Fce is also introduced within the rotating coordinate system. The Hamiltonian for the system can thus be expressed as \(H={\left({{{\boldsymbol{p}}}}-{e}^{{\prime} }{{{{\boldsymbol{A}}}}}^{{\prime} }\right)}^{2}/(2m)-{{{\mathbf{\Omega }}}}\cdot {{{{{\mathcal{L}}}}}}\), where \({{{{\boldsymbol{A}}}}}^{{\prime} }\) is the synthetic magnetic field potential associated with \({{{{\boldsymbol{B}}}}}^{{\prime} },{{{\mathbf{\Omega }}}}\) is the angular velocity, \({{{{{\mathcal{L}}}}}}\) is the angular momentum, and \(-{{{\mathbf{\Omega }}}}\cdot {{{{{\mathcal{L}}}}}}=\frac{1}{2}m{\omega }_{{{{\rm{trap}}}}}^{2}{r}^{2}\) represents the centrifugal force. Here, ωtrap is the trapping frequency, which is physically equivalent to the marked ω in Fig. 1c, and r is the particle’s transverse position vector. The introduction of the centrifugal force modifies the energy levels of the system, lifting their degeneracy. As the centrifugal force Fce and the trapping frequency ωtrap are gradually reduced to zero, the Hamiltonian of the system becomes mathematically identical to that of an electronic system. Concurrently, the harmonic potential, as depicted in Fig. 1d, is flattened to facilitate the achievement of Landau levels. It should be noted that ωtrap is introduced as a result of the centrifugal force, which counteracts the hypothetical centripetal force. Consequently, ωtrap could approach zero if a real force is introduced to serve as the centripetal force, effectively balancing the centrifugal force.

The photonic system, considered as a charge-neutral particle system, can achieve its synthetic Landau level within a carefully designed laser cavity, as shown in Fig. 1e. In this cavity, a pair of cylindrical lenses, positioned with an angular difference of θ between their principal axes, were introduced. These lenses functioned as an intracavity astigmatic converter, enabling the excitation of high-order structured modes34,35. The angle θ ranges from 0 to π/4, during which the amplitude of the synthetic magnetic field increases as θ grows. The excited modes are ray-wave geometric beams characterized by an intensity distribution located along several discrete rays, which can be represented by the schematic arrows in Fig. 1e. The oscillating laser rays rotate around the principal axis of the cavity, allowing for the introduction of a rotating reference system to analyze the oscillating rays within the cavity. In this rotating reference system, a hypothetical centripetal force, denoted as Fce, can be introduced. The hypothetical centripetal force can be considered analogous to the effective Lorentz force within an electronic system, with the effective magnetic field \({{{{\boldsymbol{B}}}}}^{{\prime} }\) aligned along the principal axis of the cavity. This cavity, equipped with an intracavity astigmatic converter, can be likened to a harmonic oscillator, where the eigenmodes associated with various energy levels are depicted in Fig. 1d. The hypothetical centrifugal force Fce can be adjusted to zero by manipulating the cavity mirrors and cylindrical lenses, as indicated by the black double arrows ⇌ in Fig. 1e, thereby achieving the synthetic Landau levels.

The excited modes can be decomposed into a superposition of degenerate eigenmodes, which are several eigenmodes that share the same frequency but have different indices (t, n, l), where (t, n) are two transverse indices along x- and y- directions that determines the transverse field distribution, and l is the longitudinal indices (see Supplementary Note 2 in the Supplementary Materials). The excited modes serve as an analogy for the quantum states within synthetic Landau levels. The quantum wave functions associated with various synthetic energy levels correspond to the excited modes in different degenerate states. These states can be adjusted by manipulating the off-axis displacement of the cavity elements. The energy levels of tunable excited modes are illustrated in the right inset of Fig. 1e. Here, the gain aperture restricts the highest energy level occupied by photons, denoted as Egain, which corresponds to the effective Fermi energy. For a clearer understanding, the analogies between quantized electrons and structured modes in the context of electron-photon interactions are summarized in Table 1. The motion of electrons in a magnetic field B is governed by the Lorentz force, resulting in cyclotron orbits that are quantized at discrete energy levels known as the Landau levels En. The difference in Landau levels is a constant related to the cyclotron frequency, denoted as ωc. While the structured modes in the synthetic magnetic field B (induced by the astigmatic degenerate cavity) are driven by a hypothetical centripetal force, the energy of the excited modes is quantized with a constant difference ΔE, analogous to the Landau levels. The difference ΔE in the energy levels of the structured modes is a constant related to the frequency spacing \({\omega }_{c}^{{\prime} }\), as discussed below.

The Landau levels of structured light modes can be experimentally realized by manipulating the designed degenerate conditions of the structured laser cavity. The frequency spectrum of the excited structured modes within the cavity is expressed as ft,n,l = fz[l + (t + 1/2)fx/fz + (n + 1/2)fy/fz], where (fx, fy, fz) represent the frequency components along the x, y, z − directions. The degenerate condition refers to the phenomenon where modes with different indices exhibit the same frequency, i.e., \(\Delta {f}_{t,n,l}={f}_{t,n,l}-{f}_{{t}_{0},{n}_{0},{l}_{0}}=0\), where (t0, n0, l0) are the initial mode indices. This degenerate state can be achieved by adjusting the length of the cavity L and the astigmatism within the cavity, as the frequencies (fx, fy, fz) are influenced by both the astigmatism and the length L. Therefore, we can obtain structured light modes with the same frequency but different indices, as illustrated in Fig. 1e, where we plot how the frequency of structured light modes evolves with the cavity length L. Here, we present a simple case where fx = fy = f0, which corresponds to the absence of astigmatism (related to a plano-concave cavity36,37,38). Each line in Fig. 1e illustrates the frequency evolution of a specific structured light mode as the cavity length L is adjusted. At a particular cavity length, indicated by the black arrows in Fig. 1e, several lines converge at a single point. This convergence signifies that these structured light modes share the same frequency, indicating that the cavity has reached a degenerate state. The frequency spacing between different degenerate states is constant, denoted as ΔE. The energy of structured light modes, denoted as Et,n,l, is linearly related to the frequency by the equation Et,n,l = 2πℏft,n,l. Consequently, the frequency of structured light modes transitions from a continuous to a discrete state, indicating that Landau quantization of the energy levels has occurred due to the influence of a degenerate cavity. As illustrated in Fig. 1e, the ground state energy level of the structured light modes is \({E}_{{t}_{0},{n}_{0},{l}_{0}}=2\pi \hslash {f}_{{t}_{0},{n}_{0},{l}_{0}}\), while the energy level difference is given by ΔE = 2πℏf0 (where f0 < fz typically). This behavior is analogous to the structure of electronic Landau levels, as presented in Table 1.

As for a degenerate cavity without astigmatism (the case with astigmatism is referred to as astigmatic degeneracy), the frequencies spacings (fx, fy) in the x- and y-directions are the same: fx = fy = f0. The degenerate condition requires that f0/fz = P/Q (where P and Q are coprime integers), which can be adjusted by varying the cavity length, given by \({L}_{P/Q}=R{\sin }^{2}(\pi P/Q)\). Here, R represents the curvature of the concave mirror, and the subscript P/Q indicates the degenerate state. The relationships f0/fz = P/Q and \({L}_{P/Q}=R{\sin }^{2}(\pi P/Q)\) can be derived using the ABCD matrix (see Supplementary Note 5 in the Supplementary Materials).

These eigenmodes, corresponding to the converging lines, have the same frequency and can be coherently superposed. The superposed modes typically exhibit intensity distributions that are concentrated along several discrete rays, which are commonly referred to as ray-wave geometric beams39,40. For instance, the simulated intensity patterns of ray-wave geometric beams, superposed by HG eigenmodes, are presented in Fig. 2a. A zoomed-in view of the frequency spectrum at a specific cavity length with P/Q = 1/3 is illustrated in Fig. 2b, where the red lines are highlighted as examples, ΔL represents a tiny deviation from the degenerate cavity length L1/3, i.e., L = L1/3 + ΔL. These discrete lines correspond to a collection of eigenmodes with various indices but the same frequency. The index spacings (t − t0, n − n0, l0 − l) are also labeled in Fig. 2b. The superposed eigenmodes based on HG modes are displayed in the left area.

a The frequency spectrum as a function of cavity length (L) for a conventional confocal cavity reveals a series of degenerate states. The theoretical emitting orbital angular momentum (OAM) geometric modes, correspondingly marked, demonstrate the effects of tuning L. b The zoom-in of a special degenerate state with a ratio of P/Q = 1/3 reveals frequency lines that correspond to various eigenmodes, each characterized by different transverse and longitudinal mode indices. The selected Hermite-Gaussian (HG) mode patterns corresponding to the highlighted red lines are indicated. At a degenerate state, which is the intersection point of a set of eigenmodes, these eigenmodes can be superimposed to form a coherent state, as shown in the lower-left inset. These patterns can be utilized to identify the types of degenerate states associated with different P/Q ratios. c The astigmatic frequency spectrum is analyzed in relation to the length parameters d1 and d3, which are adjusted to generate various degenerate states. The frequency lines correspond to the eigenmodes characterized by different transverse mode indices and a longitudinal mode index. The cross-section in the blue frame has been specifically selected for this analysis. d The cross-section at d1 = 79.212 mm reveals various degenerate states, specifically (p1, q1) = (4, − 1), (p2, q2) = (4, 0) and (p3, q3) = (4, 1), at designated values of d1 indicated by the arrows. The LG-based superposed geometric modes corresponding to the different degenerate families are marked on the right. e The zoom-in insert of figure d at the degenerate state (p2, q2) = (4, 0), marked by the blue arrow.

The astigmatic degeneracy refers to a degenerate state characterized by astigmatism, specifically when fx ≠ fy. Astigmatism can be introduced into our designed cavity using an intracavity mode converter, where the displacement of various cavity elements offers multiple synthetic degrees of freedom (DoFs). Figure 2c illustrates the frequency spectrum in relation to two synthetic DoFs, where d1 and d3 represent the tunable lengths indicated in Fig. 1e, respectively. The frequency spacing of structured light modes is given by the equation Δft,n,l = (t−t0)fx + (n−n0)fy + (l−l0)fz where fx = f0 + qΔf and fy = f0 − pΔf. The ratios fx/fz and fy/fz can be calculated using the generalized ABCD matrix theory (see Supplementary Note 5 in the Supplementary Materials), which corresponds to the tunable parameters d1 and d3. To illustrate the relationship between the frequency spacings and the parameters (d1, d3), the astigmatic frequency spectrum is presented in Fig. 2c. In our experimental setup (see Supplementary Note 7 in the Supplementary Materials), d1 ranges from 79.19 mm to 79.23 mm, while d3 ranges from 16 mm to 16.2 mm. To clearly illustrate the frequency lines, the subspace degenerate spectrum at d1 = 79.212 mm is presented in Fig. 2d, which correspond to the blue frame in Fig. 2c. The superposed structured modes at d1 = 79.212 mm, with specific values of d1 (indicated by arrows in Fig. 2d), are decomposed into several eigenmodes due to the frequency degeneracy of (p, q) = (4, 0), (4, −1), and (4, 1). The simulated results of the superposed structured modes based on LG modes are presented in the bottom as examples. The zoomed-in figures around the frequency degeneracy with parameters (p, q) = (4, 0) (indicated by the blue arrows) are displayed in Fig. 2e.

The Landau quantization of the energy of structured light modes remains significant in an astigmatic degenerate cavity, as illustrated in Fig. 2c–e, where several frequency lines converge at a single point. It is important to note that there are two frequency spacings: a large ΔE1 and a small ΔE2, as indicated by the black arrows in Fig. 2e. Compared to the data presented in Fig. 2a, the degenerate frequency in a non-astigmatic cavity, Fig. 2e exhibits further splitting with a small frequency difference, analogous to the Zeeman effect, which is a feature of Landau levels in the presence of a magnetic field41.

We firstly assembled a planar-concave cavity and adjusted the cavity length L to make a simple degenerate state laser. The frequency of resonant modes is shown in Fig. 3a, where the converged points mean degenerate states. The output power spectrum is shown in Fig. 3b, which is related to the cavity length L, where the yellow and blue curves are the experimental results and the corresponding simulation (see Supplementary Note 5 in the Supplementary Materials), respectively. When the laser cavity reaches the degenerate condition (P/Q = 1/6, 1/5, 1/4, 1/3 from left to right), the output power peaks would arise, meaning the existence of degeneracy of confined photons in the presence of Landau levels.

a The frequency spectrum as a function of cavity length (L/R) for a conventional planar-concave cavity, where R is the curvature radius of the concave mirror. Each line corresponds to a structured light mode while the converged points mean degenerate states, which locates on \(L/R={\sin }^{2}(\pi P/Q)\) where (P, Q) are coprime integers. b The output power (measured in watts) spectrum dependent on the cavity length, where the yellow and blue curves respond to the experimental and simulation results. The degenerate laser patterns (inserts) occur in the output power peaks, corresponding to P/Q = 1/6, 1/5, 1/4, 1/3, from left to right, respectively.

Then, we assembled the cavity depicted in Fig. 4a, where the two intracavity cylindrical lenses are positioned at angles of α1 and α2, respectively. The distances d1 (between the first cylindrical lens and R2) and d3 (between the second cylindrical lens and R1) are adjustable to achieve the degenerate superposed modes, as shown in Fig. 4a. The output modes from the cavity consist of degenerate superposed HG modes at position A in Fig. 4a. These modes are transformed into degenerate superposed LG modes at position B in Fig. 4b through mode conversion facilitated by the extracavity cylindrical lenses. The off-axis displacement of the first intracavity cylindrical lens adjusts the index range of the superposed HG modes in the \({x}^{{\prime} }\)-direction, while the off-axis displacement of lens R2 modifies the index range of the superposed HG modes in the \({y}^{{\prime} }\)-direction. The degenerate superposed HG (LG) modes with a tunable \({y}^{{\prime} }\)-direction index for varying Δd are presented in the right column of Fig. 4a. The top and bottom rows correspond to the simulation and experimental results, respectively. We utilize the correlation coefficient cor(A, B) to assess the relationship between simulation results (denoted as A) and experimental outcomes (denoted as B). The value of cor(A, B) ranges from 0 to 1, with larger coefficients indicating a stronger degree of correlation. The correlation coefficient cor(A, B) of our results is approximately 0.64–0.65, indicating strong agreement.

a Our experimental cavity design for synthetic Landau levels consists of two cavity mirrors (R1 and R2, with radii of curvature denoted as R1 and R2), a gain crystal (Yb:CALGO), and a pair of cylindrical lenses (with focal lengths f) positioned at rotating angles α1 and α2. The distances between R2, the two cylindrical lenses, and R1 are denoted as d1, d2, and d3, respectively. The synthetic magnetic field can be controlled by adjusting d1 and d3. The orders of HG modes can be manipulated by tuning ωtrap. Additionally, a mode converter, consisting of another pair of cylindrical lenses located outside the cavity, is employed to convert the orbital angular momentum (OAM) of the output mode from A to B, with simulated and experimental patterns displayed in the right insets. b A cross-section of the astigmatic frequency spectrum of the cavity at d3 = 16.1 mm reveals the degenerate position (DP), where distinct groups of lines (red and purple) correspond to different Landau levels. These levels can be tuned with ωtrap, and their associated modes, indicated by varying indices n0, are displayed on the right. This includes a single eigenmode represented by one line and a superposed mode represented by a set of lines. The eigenmodes correspond to one red frequency line (b1, b5) and one purple frequency line (b3, b7). The degenerate modes correspond to a group of red frequency lines (b2, b6) and a group of purple frequency lines (b4, b8). c ωtrap evolves with the tunability of d1 and d3, which represent the effects of the centrifugal force. The zero point of ωtrap corresponds to the Landau level point, which is also referred to as the degenerate point (DP). d Experimental Landau level control: the output degenerate modes of DP at Δd = 0 corresponding to patterns in the 3rd (HG-based) and 4th (superposed LG modes) columns, around which the superposed modes gradually evolve out of degeneracy by tuning d1 as the 1st, 5th, 7th columns (superposed HG modes) [the 2nd, 6th, 8th columns (superposed LG modes)], corresponding to the 1st, 3rd and 4th red (purple) vertical lines on the horizontal axis. The patterns of the lower (upper) line within the purple (red) frames represent the modes around En=0 (En=1) with the index n0 = 0 (n0 = 1) corresponding to the purple (red) vertical lines on the horizontal axis. Structured light modes with indices (t0, n0) = (4, 1) (d1-d8) and (t0, n0) = (10, 0) (d9-d16). These modes are achieved by varying the off-axis position of R1. All figures share the same intensity colorbar that in a.

The frequency spectrum, as illustrated in Fig. 4b, displays the frequency lines corresponding to the HG eigenmodes with varying transverse indices (t0, n0). We omit the longitudinal index l0 as it does not affect the intensity pattern. A set of five lines converges at certain points, indicating that the corresponding eigenmodes are degenerate when Δd1 = 0 and d1 = 79.21 mm. The converging points represent a set of discrete frequency (energy) levels, characterized by a constant difference, which corresponds to the Landau levels of structured light modes. The transverse pattern of HG modes with indices (t0, n0) = (2, 1), corresponding to a single red frequency line (marked with a red arrow), is illustrated in Fig. 4b1. In contrast, the transverse pattern of superposed eigenmodes corresponding to five red frequency lines is depicted in Fig. 4b2. In this case, the set of transverse indices is given by {(t0 + pK, n0 + qK)} with p = 1, q = 3 and K = 0, 1, 2, 3, 4. For another set of degenerate modes, the transverse pattern of HG modes with indices (t0, n0) = (2, 0), corresponding to one purple frequency line (marked with a purple arrow), is shown in Fig. 4b1. In contrast, the transverse pattern of superposed eigenmodes corresponding to five purple frequency lines is illustrated in Fig. 4b3. In this case, the set of transverse indices is {(t0 + QK, n0)} with Q = 4 and K = 0, 1, 2, 3, 4. The superposed HG modes can be converted into the superposed LG modes through astigmatism, as illustrated in Fig. 4b5 and b6.

Moreover, the Landau level can be tuned through the off-axis displacements (d1, d3) in experiments. With d1 = 79.2 mm and d3 = 16.1 mm, the trapping frequency ωtrap, which is associated with the centrifugal force, is zero (see Supplementary Note 7 in the Supplementary Materials), as illustrated in Fig. 4c. Both Fig. 4b, c illustrate that the modes become non-degenerate when the cavity parameters, including the cavity length L and the displacements d1 and d3, deviate from the degenerate position (DP). The transition from non-degenerate to degenerate states is demonstrated experimentally, as shown in Fig. 4d, where the key distinction between the patterns in the top and bottom rows is the indices n0 = 0 or n0 ≠ 0. The top row displays the patterns characterized by the indices (t0, n0) = (4, 1), which are referred to as structured light modes (Fig. 4d1–d8). These are distinct from those in the bottom row (Fig. 4d9–d16), which have indices (t0, n0) = (10, 0). When d1 is far from the DP, the degenerate modes (Fig. 4d3, d4, d11, and d12) evolve into non-degenerate modes (the other subfigures in Fig. 4d).

Landau levels represent fundamental quantum phenomena in modern electronic and solid-state physics, inspiring various research such as topological insulators, photonic crystals, and optoelectronic materials. This work extends the concept of Landau levels from electronic systems to photonic systems. We propose a high-degrees-of-design-freedom and straightforward setup for creating and manipulating photonic Landau levels of photons, which can serve as a physical mechanism for controlling on-demand symmetry breaking in OAM lasers42. Consequently, our work not only generates astigmatic structured light fields at the source with unique topological features, but also opens avenues for studying quantum Hall effects, topological phases, and quantum simulation within a platform of bosonic systems using designed degenerate laser cavities.

The proposed linear degenerate cavity platform for controlling photonic Landau levels is more robust and flexibly tunable compared to previous approaches. The extended DoFs allow us to precisely manipulate photonic Landau levels and address the trade-off between trap stability and geometric modifications. Besides, this advancement in astigmatic structured light manipulation offers the potential to access a broader range of topological phases43,44,45. We believe that combining more complex intracavity modulation, e.g., metasurfaces and spatial light modulators, in degenerate lasers could serve as an effective approach to explore advanced solid-state physics and investigate additional properties of large-scale photonic crystals within structured laser cavities, such as PT symmetry, non-Hermitian physics, topological edge states, and nonlinear dynamics46,47,48,49,50.

Our synthetic Landau level cavity incorporates two mirrors and two cylindrical lenses. The cylindrical lenses are strategically introduced to disrupt symmetry and facilitate the generation of high-order modes. To establish a rotating frame for the photons within the cavity, the generatrices of the pair of intracavity cylindrical lenses are intentionally set to be non-parallel. Specifically, the cavity is meticulously engineered with a pair of mirrors, R1 (with a radius of curvature R1) and R2 (with a radius of curvature R2), as well as a gain crystal Yb: CALGO, and a set of cylindrical lenses, both possessing the same focal length f. These cylindrical lenses are deliberately positioned at angles α1 and α2 to disrupt the symmetry within the cavity, thereby enabling the emission of 2D modes. This design leverages the principles of geometrical optics and the properties of cylindrical lenses to achieve the desired mode distribution, which is a critical aspect of creating a synthetic Landau level cavity. The distances between the components of the cavity are carefully defined: the distance from mirror R1 to the first cylindrical lens is d1, from the first cylindrical lens to the second cylindrical lens is d2, and from the second cylindrical lens to mirror R2 is d3. The synthetic magnetic field can be adjusted by varying the distances d1 and d3. This is achieved by moving the first cylindrical lens and mirror R1 along the optical axis, respectively. Additionally, the high-order HG eigenmodes can be generated by aligning the first cylindrical lens and mirror R2 off-axis within the cavity. This precise control over the distances and alignment is essential for fine-tuning the synthetic magnetic field and achieving the desired eigenmode patterns.

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

The codes that support the findings of the study area vailable from the corresponding author upon reasonable request.

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This work is supported by National Natural Science Foundation of China (62275137), Beijing Natural Science Foundation (JQ23021); Y. Shen acknowledges the support from Nanyang Technological University Start Up Grant, Singapore Ministry of Education (MOE) AcRF Tier 1 grant (RG157/23), MoE AcRF Tier 1 Thematic grant (RT11/23), and Imperial-Nanyang Technological University Collaboration Fund (INCF-2024-007).

Yuan Meng

Present address: Mechanical Engineering & Materials Science, Washington University in St. Louis, Saint Louis, USA

These authors contributed equally: Jing Pan, Zhaoyang Wang.

Department of Precision Instrument, Tsinghua University, Beijing, China

Jing Pan, Zhaoyang Wang, Yuan Meng, Xing Fu & Qiang Liu

State Key Laboratory of Precision Space-Time Information Sensing Technology, Beijing, China

Jing Pan, Zhaoyang Wang, Xing Fu & Qiang Liu

Key Laboratory of Photonic Control Technology (Tsinghua University), Ministry of Education, Beijing, China

Jing Pan, Zhaoyang Wang, Xing Fu & Qiang Liu

Centre for Disruptive Photonic Technologies, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

Yijie Shen

School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore

Yijie Shen

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Y.S. conceived the idea and drafted the initial manuscript. J.P. performed the initial experiment and theoretical modeling. Z.W. contributed to the complementary experiment. J.P., Z.W., and Y.M. revised the manuscript. X.F., Y.S., and Q.L. supervised the project. All authors took part in discussions, interpretations of the results, and revisions of the manuscript.

Correspondence to Xing Fu, Yijie Shen or Qiang Liu.

The authors declare no competing interests.

Communications Physics thanks Bo Wang, Haoran Xue and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

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Pan, J., Wang, Z., Meng, Y. et al. Photonic Landau levels in an astigmatic frequency-degenerate laser. Commun Phys 8, 111 (2025). https://doi.org/10.1038/s42005-025-02013-4

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Received: 06 July 2024

Accepted: 21 February 2025

Published: 22 March 2025

DOI: https://doi.org/10.1038/s42005-025-02013-4

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