Momentum-Space Photonics

• Momentum-space photonics is a field that studies photonic modes as functions of momentum, revealing nonlocal interactions and topologically robust edge states.
• It leverages mappings between real- and momentum-space models, using techniques like the Harper–Hofstadter mapping to simulate synthetic gauge fields and complex lattice behaviors.
• The discipline enables precise quantum interference measurements, dynamic control of photonic quantum numbers, and exploration of non-Hermitian topological defects.

Momentum-space photonics is a field that investigates, engineers, and exploits the properties, dynamics, and topology of photonic modes as functions of momentum (wavevector) rather than in real space. Unlike traditional photonics, which emphasizes spatial configurations, refractive indices, or material interfaces, momentum-space photonics focuses on the structure of photonic bands, excitations, and topological features in the reciprocal (momentum) domain. This shift in perspective enables fundamentally novel phenomena, such as nonlocal interactions, topologically robust edge states, anomalous light-matter interactions, and control over photonic quantum numbers (e.g., spin, orbital angular momentum) via global band topology rather than local geometry.

1. Fundamental Principles of Momentum-Space Photonics

The foundation of momentum-space photonics is the treatment of photonic modes, band structure, and field observables as functions of momentum $\mathbf{k}$ in the Brillouin zone or continuum. Physical observables such as energy dispersion $E_n(\mathbf{k})$, Berry connection $\mathcal{A}_n(\mathbf{k})$, Berry curvature $\Omega_n(\mathbf{k})$, polarization textures, and spin angular momentum can be defined directly in momentum space. Key principles include:

• Berry phase and connection: The geometric phase $$\gamma_n = \oint_C \mathrm{d}\mathbf{k}\cdot \mathcal{A}_n(\mathbf{k})$$, where $$\mathcal{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k}) | \nabla_\mathbf{k} u_n(\mathbf{k}) \rangle$$, plays a central role in photonic band topology (Gangaraj et al., 2018).
• Chern number and momentum-space topology: The Chern number $$C_n = \frac{1}{2\pi}\int_{\mathrm{BZ}}\Omega_n(\mathbf{k})\,\mathrm{d}^2k$$ quantifies global features and predicts phenomena such as robust, unidirectional edge states (Gangaraj et al., 2018).
• Momentum-space vortices and defects: Winding of polarization or pseudospin textures can give rise to topological defects such as vortices, merons, or skyrmions, characterized by momentum-space winding numbers (Zhang et al., 2017, Rao et al., 21 May 2025, Guo et al., 2019, Yessenov et al., 15 Mar 2025, Hu et al., 18 Sep 2025).
• Nonlocality: Real-space locality transforms to momentum-space nonlocality for interactions, e.g., onsite (contact) interactions become momentum convolutions (Ozawa et al., 2014).

This conceptual framework enables reinterpretation of numerous real-space phenomena—localization, domain walls, edge states, and band gaps—in terms of their momentum-space analogs.

2. Engineering and Mapping Real- and Momentum-Space Models

Momentum-space photonics leverages the mathematical mapping between real- and momentum-space Hamiltonians to create photonic analogs of condensed matter systems, manipulate interactions, and realize synthetic gauge fields.

• Harper–Hofstadter mapping: The paradigmatic mapping of the weakly trapped Harper–Hofstadter (HH) model onto a momentum-space lattice recasts real-space periodic potentials as momentum-space “periodic potentials” $E_n(\mathbf{k})$, Berry curvature as an effective magnetic field, and the harmonic trap as momentum-space kinetic energy (Ozawa et al., 2014). The effective Hamiltonian,

$$\widetilde{\mathcal{H}} = E_n(\mathbf{k}) + \frac{\kappa}{2}\left[i \nabla_\mathbf{k} + \mathcal{A}_n(\mathbf{k})\right]^2,$$

supports tight-binding behavior, momentum-space hopping, and toroidal boundary conditions (magnetic Brillouin zone).

• Nonlocal interaction engineering: Local interactions in real space become nonlocal in momentum space. This nonlocality induces phenomena such as symmetry-breaking of condensate ground states and pattern formation (Ozawa et al., 2014).
• Momentum-to-real-space mapping: Universal mappings exist whereby the momentum-space singularities (e.g., Dirac cones, pseudospin vortices) project to real-space field profiles, enabling topological charge transfer dictated by the combined orbital and pseudospin angular momentum ($l \rightarrow l + 2s$) (Liu et al., 2019).
• Modulating momentum-space flow: Recent works establish explicit Fourier and Plancherel correspondences between spin Hamiltonians and modulated momentum-space optics. Arbitrary displacement-dependent spin interactions $J(r)$ are encoded through their Fourier transforms $V(k)$, enabling simulation of exotic magnetic phases entirely by manipulating the photonic field in $k$-space (Feng et al., 1 May 2024).

This class of mappings allows momentum-space photonics to serve as a simulator for diverse quantum many-body systems and topological phases.

3. Topological Textures, Defects, and Polarization in Momentum Space

Momentum space supports a rich variety of topological structures, directly observable in modern photonic experiments:

• Momentum-space vortices and BICs: The winding of polarization vectors in the Brillouin zone forms vortices with quantized topological charges. Such singularities underpin the existence of bound states in the continuum (BICs), enable direct design of orbital angular momentum beams, and define the local and global topological character of photonic bands (Zhang et al., 2017, Wang et al., 2019).
• Merons, skyrmions, and pseudospin textures: Photonic systems—especially when symmetry breaking, valley physics, or synthetic gauge fields are present—exhibit momentum-space meron (half-skyrmion) and skyrmion spin textures. These textures can be visualized via mapping of internal (pseudo)spin to polarization of leakage radiation, directly connecting measured Stokes parameters to underlying Berry curvature (Guo et al., 2019, Rao et al., 21 May 2025, Yessenov et al., 15 Mar 2025).
• Non-Hermitian topological defects: In systems with gain/loss, spontaneous emergence of pseudospin defects at imaginary Fermi arcs in momentum space is observed. The existence, propagation, and annihilation/protection of these defects are governed by non-Hermitian spectral degeneracies such as exceptional points, further enriching the taxonomy of momentum-space topological phenomena (Hu et al., 18 Sep 2025).
• Switchable and robust control: The sign (polarity) of the momentum-space meron can be switched by the incident polarization (e.g., RCP vs. LCP), and the robustness of the observed topological charge is maintained across broad spectral ranges (Rao et al., 21 May 2025).

These momentum-space topological features allow encoding, manipulating, and probing of photonic quantum numbers that are resilient to disorder and suitable for practical device applications.

4. Momentum-Space Quantum Interference, Pathways, and Measurement Techniques

Momentum-space photonics enables precise decomposition and measurement of quantum pathways and correlations via the analysis of field distributions and coherent effects in $k$-space.

• Momentum-space separation of quantum paths: Nonlinear optical processes involving multiple interfering absorption pathways (e.g., photon-photon, photon-plasmon mixing) can be distinguished by the momentum difference between constituents. The spatial Fourier transform of photoemission patterns distinctly isolates processes by their net momentum, allowing quantum path interferences to be disentangled without destroying quantum correlations (Dreher et al., 2023).
• Direct experimental access to $k$-space: Modified interferometric techniques (e.g., Fourier-transforming the two-dimensional spatial coherence function) permit practical and robust measurement of 2D momentum distributions from spatially resolved interferometry, even when samples are in cryogenic environments (Vedhanth et al., 2023).

Momentum-space observables thus form the operational basis for modern photonic tomography, coherent control, and quantum interference studies.

5. Dynamical and Non-Equilibrium Phenomena in Momentum-Space Photonics

Momentum-space photonics is not limited to static band structure but includes rich dynamical and non-equilibrium effects:

• Photonic time crystals and momentum flatbands: Temporal modulation introduces a duality where time-periodic lattices (photonic time crystals) create band gaps and flatbands not in energy but in momentum. In photonic time Moiré superlattices, isolated bands arise that are flat in momentum, not energy, supporting superluminal pulse propagation with real-valued refractive indices, enabling robust and broadband fast-light devices without gain-induced instabilities (Zou et al., 31 Oct 2024).
• Quantized Thouless pumping and interface modes: By embedding sliding or modulated periodic potentials (akin to Thouless charge pumping), bilayer photonic crystals realize quantized transport in synthetic momentum-space dimensions, robust interface (edge) states, and spectrally tunable, topologically protected lasers (Nguyen et al., 2021).
• Thermal spin photonics and persistent currents: Even at equilibrium, nonreciprocal media support persistent photon spin currents and planar heat currents—PTPS and PPHC—because of asymmetries in the statistical occupation of momentum states. Spin-momentum locking and near-field evanescent waves underpin directionality and robustness of these phenomena (Khandekar et al., 2019).

These dynamical effects exemplify the ability of momentum-space photonics to harness time-dependent, nonreciprocal, or topologically protected mechanisms for light-matter interaction and transport.

6. Photonic Momentum, Material Momentum, and the Abraham-Minkowski Controversy

Momentum-space photonics interfaces with foundational questions on the definition and partitioning of light’s momentum in material systems.

• Partitioning of momenta: Quantum theories explicitly calculate the distribution of light’s momentum between the electromagnetic field, electrons, and lattice. Notably, intrinsic DC currents and optical pulling effects—where charge carriers move antiparallel to light—emerge from band-structure-dependent momentum transfer (Pan et al., 2019).
• Canonical momentum and translation generators: The momentum operator for the electromagnetic field, defined as the generator of spatial translations, aligns with the canonical momentum (Minkowski’s form) in quantum electrodynamics. On transmission through a dielectric interface, photon momentum increases by the refractive index, clarifying key points in the Abraham-Minkowski debate (Waite et al., 4 Oct 2024).

These insights enable rigorous and quantitative design and analysis of photonic devices where the total momentum and its transfer among field and material subsystems are indispensable.

7. Applications and Future Directions

Momentum-space photonics underpins and inspires a proliferation of applications and concepts:

• Optical vortex generation and integrated spin/valley photonics: Exploitation of polarization vortices and BICs allows on-chip vortex lasers, robust orbital angular momentum channels, and topological polarization control (Wang et al., 2019, Rao et al., 21 May 2025).
• Energy-free spin-based photonics: Novel device concepts such as optical spin diodes and circulators enable the transport of spin angular momentum decoupled from energy transfer, offering a new platform for optical spintronics and energy-efficient information processing (Deriy et al., 15 May 2025).
• Quantum information processing: Momentum-space control over spin and orbital degrees of freedom opens avenues for robust and entangled photonic states suitable for quantum communication and sensing (Yessenov et al., 15 Mar 2025).
• Non-Hermitian and nonequilibrium topological photonics: Emergent pseudospin defects and topologically protected phenomena in non-Hermitian systems provide new physical regimes and experimental observables inaccessible in Hermitian (lossless) optics (Hu et al., 18 Sep 2025).
• Physical simulation and optimization: Optical simulators implementing momentum-space encodings of spin Hamiltonians solve complex many-body and combinatorial problems, enabling exploration of exotic magnetic phases and emergent dynamics (Feng et al., 1 May 2024).

Future research directions emphasize the synthesis of novel photonic quasiparticles (merons, hopfions), further exploration of synthetic momentum-space dimensions, robust device integration, and a deeper understanding of the interplay between symmetry, topology, and dynamics in engineered photonic environments.