High-Dimensional OAM Superposition States

Updated 18 October 2025

High-dimensional OAM superposition states are coherent combinations of orbital angular momentum modes that provide an unbounded Hilbert space compared to traditional two-dimensional systems.
They are generated via spatial light modulators, spiral phase plates, and holographic masks and characterized using interferometric and Fourier techniques for accurate mode detection.
These states enable enhanced quantum communication, metrology, and multiplexing by facilitating error-resistant high-dimensional encoding and robust state transmission.

High-dimensional orbital angular momentum (OAM) superposition states are structured photonic or electronic modes described by coherent combinations of basis states with widely differing orbital angular momentum quantum numbers. In contrast to polarization, which is inherently two-dimensional, the OAM degree of freedom supports, in principle, an unbounded Hilbert space, thereby enabling the encoding and manipulation of high-dimensional quantum information. The generation, detection, characterization, and exploitation of such states are foundational for advanced quantum communication, metrology, and information processing.

1. Generation of High-Dimensional OAM Superposition States

The preparation of high-dimensional OAM superpositions across diverse platforms leverages a variety of optical and material tools suited for precisely controlling amplitude and phase in structured spatial modes. A widely used approach is the spatial light modulator (SLM) technique, which enables the synthesis of modified Laguerre–Gaussian (LG) modes. The field at the beam waist for such modes is

$\mathrm{LG}_0(\rho, \phi) = R(\rho)\,\exp(i\ell\phi)$

with $R(\rho)$ as the radial profile and $\ell$ the OAM quantum number. To extend the accessible OAM range and encode very high quanta, the SLM-imposed phase can be tailored:

$\mathrm{LG}_N(\rho, \phi) = R(\rho)\,\exp(i N \ell \phi)$

where the "excess helicity" factor $N$ enables fine control of angular momentum per photon (Chen et al., 2013). Superpositions are then created as

$f(\rho, \phi) = \sum_n a_n \mathrm{LG}_n(\rho, \phi)$

with complex coefficients $a_n$ .

Device-based strategies include the microfabrication of spiral phase plates (SPPs) within laser cavities (e.g., in vertical-cavity surface-emitting lasers, VCSELs) to emit coherent superpositions or individual OAM modes with high efficiency directly at the source (Li et al., 2015). In the electron microscopy context, advanced holographic diffraction masks encoded by Fourier synthesis are used to compose electron vortex states of arbitrary OAM content (Eickhoff et al., 2020). In atomic systems, tightly focused non-paraxial LG beams can transfer both spin and orbital angular momentum to Bose–Einstein condensates, enabling the creation of matter-wave vortex superpositions via tailored two-photon stimulated Raman transitions (Bhowmik et al., 2016).

2. Characterization and Measurement Techniques

Characterizing high-dimensional OAM superpositions demands access to both amplitude and phase information across a broad set of OAM basis states. Interferometric schemes, such as the Mach–Zehnder interferometer with a Dove prism, offer direct visual identification: the interference of a beam with itself (or its mirror) leads to intensity patterns whose angular structure (number of stripes or "spokes") directly encodes the OAM content; a superposition of many OAM states produces multiring lattices, each with a characteristic fringe count (Chen et al., 2013).

For full spectrum quantification, broadband uniform-efficiency OAM-mode detectors use an interferometric setup with a precise image rotator, recording photon count at an EMCCD camera as a function of rotation angle $\theta$ and phase $\delta$ (Karan et al., 20 Feb 2025). This enables a Fourier cosine decomposition of the intensity modulation:

$\Delta I_\mathrm{out}(\theta) = 2|k_1||k_2|(\cos\delta_c-\cos\delta_d) \sum_\ell S_\ell\cos(2\ell\theta)$

where $S_\ell$ is the total (summed-over-radial) OAM spectrum.

Alternative approaches include principal component analysis and regression on spatial intensity images, acquiring complementary intensity measurements in different bases (e.g., after applying an OAM shift), to reconstruct the state's Bloch vector (Zia et al., 2022).

Complete projection in Hilbert space, commonly implemented using forked gratings in an SLM-based 4f system, extracts modal weights independently—analogous to vector decomposition along orthogonal axes—accompanied by a Hilbert angle metric quantifying mode purity (Shi et al., 2017). In radio-frequency domains, the spatial distribution of S21 parameters across antenna arrays provides a "characteristic maximum print" corresponding to distinct or superposed OAM content (Anufriyev, 2018).

3. Manipulation, Control, and Transformation

Manipulation of high-dimensional OAM superpositions—especially for quantum logic and communication—requires OAM mode-selective elements, controlled phase gates, and unitary transformations across the large Hilbert space. A central tool is the Dove prism (DP), which imparts an OAM-dependent phase $e^{i2\ell\alpha}$ for rotation angle $\alpha$ ; cascaded DP–half-wave plate sandwiches in Sagnac interferometers can serve as high-fidelity, mode-independent controlled-phase gates for hybrid spin-OAM states (Wang et al., 2017).

Universal unitary transformations in large OAM spaces are facilitated by interfacing path and OAM degrees of freedom: OAM-to-path interfaces demultiplex a multi-mode OAM register, enabling the implementation of arbitrary $N$ -dimensional unitary transformations using a reduced number of Mach–Zehnder interferometers, SWAP gates, and OAM-dependent beam splitters (Chen et al., 2023). This architecture enables efficient manipulation even as dimensionality grows, with precise and programmable connectivity.

Reconfigurable SLM-generated holograms also permit on-demand generation of arbitrary OAM superpositions, supporting both rapid state design and real-time adaptive quantum protocols (Chen et al., 2013).

4. Propagation, Robustness, and Practical Transmission

A historical obstacle for high-dimensional OAM states has been modal dispersion and decoherence during propagation, due to asynchronous diffraction and mode-dependent Gouy phases. Encoding exclusively in LG modes with fixed overall order $N=2p+|\ell|$ creates a $d=N+1$ dimensional subspace in which all superposed components share identical propagation properties (Gouy phase and curvature), ensuring that the state is propagation-invariant and immune to asynchronous modal distortion (Mao et al., 2021). This secures orthogonality and high-fidelity detection even after transmission over free-space links.

For fiber-based quantum communication, specially designed air-core fibers support robust propagation of multiple OAM states and their superpositions over kilometer-scale distances with extinction ratios >18 dB and state fidelities >90%. Compensation for differential group delays and advanced mode sorting at detection ensures minimal cross-talk (Cozzolino et al., 2018).

In turbulent free-space channels, adaptive optics (AO) employing real-time wavefront sensing and correction by deformable mirrors can suppress turbulence-induced crosstalk and stabilize high-dimensional OAM entanglement, as evidenced by maintained Bell inequality violations and improved signal trace under simulated strong scintillation (Sorelli et al., 2018).

5. Applications in Quantum and Classical Information

High-dimensional OAM superpositions underpin a variety of advanced protocols:

Quantum Key Distribution (QKD): Encoding in multilevel OAM "qudits" allows larger Hilbert spaces per photon, increasing secret key rates, noise robustness, and security thresholds (Cozzolino et al., 2018). Multiple mutually unbiased bases (MUBs) extend cryptographic functionality.
Entanglement Generation and Bell-State Measurement: Controlled, postselection-free generation and tailoring of entanglement spectra in up to 150-dimensional bases have been demonstrated, with generation accuracy exceeding 98% for smooth spectra (Karan et al., 2022). Complete OAM Bell-state measurement using ancillary polarization enables superdense coding with success probability ~82% and channel capacity of 1.1(4) bits per detection event (Kong et al., 2017).
Classical and Quantum Multiplexing: Interferometric protocols separate even/odd OAM superpositions, enabling near-lossless and reversible channel multiplexing with fidelities >0.98, supporting classical and quantum data streams in parallel (Rodenburg et al., 2015).
On-Chip/Free-Space Interfaces and Quantum Networks: Mode sorters and quantum interfaces allow conversion between path-encoded and OAM-encoded entangled states, bridging chip-scale quantum processors and free-space or fiber links in hybrid networks (Fickler et al., 2014).

Specialized applications include optical trapping and tweezing, metrology, high-capacity short-reach data interconnects, and robust error-resistant quantum state transfer.

6. Synthesis, Mode Selection, and Advanced Structures

Structured OAM states with custom symmetries and composite OAM spectra are synthesized using Fourier synthesis with holographic masks or bichromatic multiphoton ionization pathways. In electron beams, such masks generate spatial patterns with defined rotational symmetries and fractional effective topological charge (Eickhoff et al., 2020). In photonic systems, rotationally symmetric superpositions of chiral states, realized through pinhole sieve masks or spiral zone plates, provide mode-combing and filtering, selectively enhancing OAM components (only topological charges fulfilling $\ell = m \times M$ for $m$ -fold symmetry), and enabling the generation of high-dimensional compound vortex beams for advanced information encoding (Yang et al., 2017).

7. Challenges and Future Perspectives

Key challenges include:

Uniform Detection and Fidelity: Overcoming mode-dependent detection efficiencies and coupling losses; recent developments in interferometric, polarization-corrected detectors achieve detection fidelities >98% for up to 100 modes in a few minutes (Karan et al., 20 Feb 2025).
Mode Purity and Scalability: OAM mode purity is sensitive to fabrication, alignment, and environmental effects; advances in nanoimprinting and active cavity control (e.g., in OAM-VCSELs or geometric-phase lasers) directly address these constraints (Li et al., 2015, 1505.02256).
Automated Synthesis and Tomography: Efficient computational and experimental schemes for state regression (e.g., PCA-based image analysis) enable rapid characterization and tomography, even in the presence of inherent LG mode symmetries (Zia et al., 2022).
Propagation and Environmental Noise: Adaptive optics, propagation-invariant encoding, and dynamic subspace selection are critical for joint robustness and high dimensionality, with open directions in active AO system design and dynamic channel adaptation (Sorelli et al., 2018, Mao et al., 2021).

Future directions include integrating universal quantum gates directly on photonic chips for high-dimensional OAM, extending postselection-free entanglement to non-Schmidt bases, developing lossless sorting and detection over all relevant degrees of freedom, and deploying OAM-based protocols in field scenarios for quantum networking, sensing, and metrology.

Key Experimental Approaches and Capabilities

Approach	Dimensionality (d)	Notable Metric(s) / Feature
SLM-based LG superpositions + Mach–Zehnder ID	d $\gg$ 10	Direct OAM readout via spoke-count in interferogram
Broadband uniform-efficiency OAM detector	d ≈ 100	Fidelity >98%, time per measurement ≈ minutes
Controlled-phase OAM module (PMM, Sagnac, DP)	d: arbitrary	Controlled-phase gate, sorting fidelity ≈ 96–98%
Fiber transmission via air-core fiber	d = 4	Fidelity ≈ 94–97%, extinction >18 dB, μs delay/CFO
Postselection-free high-d OAM entanglement	d up to 150	Generation accuracy >98%, arbitrary Schmidt spectra
Rotationally symmetric electron/optical masks	d: designable	User-defined OAM selection, multi-ring structure

High-dimensional OAM superposition states provide an enabling platform for quantum and classical applications requiring large state spaces, complex logic, and robust, information-rich encoding—supported by a growing ecosystem of generation, manipulation, detection, and characterization methods attuned to photonic and electronic quantum systems.