Momentum-Expanded Heisenberg Photonic States

• Momentum-expanded Heisenberg photonic states are engineered quantum states featuring broadened momentum profiles achieved through extreme spatial confinement, gauge entanglement, and topological design.
• They exploit the Heisenberg uncertainty principle to manipulate spatial, polarization, and orbital degrees of freedom, unlocking new pathways for quantum simulation, metrology, and optoelectronic applications.
• Experimental realizations using nanoparticle decoration, HOM interferometry, and photonic quantum simulators validate these states, offering robust platforms for topologically protected light–matter interactions.

Momentum-expanded Heisenberg photonic states are photonic quantum states in which the spatial, polarization, or angular momentum degrees of freedom are engineered to create a deliberately broadened momentum-space profile of the electromagnetic field, typically as a consequence of extreme spatial confinement, gauge entanglement, topological design, or nontrivial many-photon correlations. The systematic expansion in momentum space enables manipulation of light in regimes where conventional approaches are constrained by symmetry, selection rules, or the photon's intrinsic properties, yielding novel pathways for quantum simulation, metrology, optoelectronics, and light-matter interaction. These states embody the general principle that the Heisenberg uncertainty relation, entanglement between photonic modes, and tailored gauge choices can be used to engineer the momentum distribution of photons to unlock new physical behaviors, including direct emission in otherwise forbidden systems and topologically robust manipulation of orbital angular momentum.

1. Theoretical Framework and Quantum Structure

Momentum-expanded Heisenberg photonic states are rooted in a quantum-mechanical description of photonic modes whose polarization, spatial, and orbital degrees of freedom are inseparable from their momentum content. For freely propagating photons, the momentum-space wavefunction $f(\mathbf{k}, t)$ must satisfy Maxwell's transversality condition $f(\mathbf{k}, t)\cdot\mathbf{w}=0$ (with $\mathbf{w} = \mathbf{k}/|\mathbf{k}|$), which implies that two transverse polarization components at each $\mathbf{k}$ are inherently entangled with the momentum (Li, 2017). The physical polarization wavefunction at each momentum is represented by the Jones vector, with Pauli matrices encoding the Cartesian polarization components in the local gauge-fixed basis. Gauge degrees of freedom associated with the choice of transverse axes introduce observable effects, such as rotations of the polarization about the momentum axis.

For spatially confined photonic states, the Heisenberg uncertainty principle $\Delta x\cdot\Delta k\gtrsim 1$ dictates that a sharply localized photon field (e.g., confinement within a sub-nanometer region) is described in momentum space by a broad profile, often modeled by a Gaussian momentum distribution $\psi(k)\propto k\, e^{-k^2/2\sigma_k^2}$, where $\sigma_k$ scales as the inverse of the confinement length (Noskov et al., 25 Jul 2025, Noskov et al., 16 Sep 2025). In operator language, quadrature operators $q=(a+a^\dagger)/2$ and $p=i(a-a^\dagger)/2$ satisfy $\Delta q\cdot\Delta p=(n+1/2)$ in the $n$-photon Fock state, underscoring the momentum expansion under strong confinement.

Multi-photon and lattice-based photonic states can also exhibit momentum expansion in the context of Heisenberg-spin quantum simulators and collective emission processes, where superposition and entanglement within the photonic momentum basis give rise to highly nonclassical states (Paulisch et al., 2018). These states can be mathematically formulated as

$$|\phi^{(N)}\rangle = \frac{1}{(2\pi)^N N!}\int dk_1...dk_N\,A_{k_1...k_N}\,a_{k_1}^\dagger...a_{k_N}^\dagger|0\rangle,$$

with the coefficient $A_{k_1...k_N}$ encapsulating multivariate entanglement, which is non-factorizable in general.

2. Mechanisms for Momentum Expansion

Several physical mechanisms produce momentum-expanded Heisenberg photonic states:

• Extreme Spatial Confinement: Embedding sub–1.5 nm nanoparticles (Au, Cu, or Si) within or at the surface of silicon creates regions of intense electromagnetic confinement, broadening the photon's momentum distribution and activating otherwise forbidden optical transitions. This approach directly overcomes the indirect bandgap constraint in silicon, enabling efficient photoluminescence and electroluminescence spanning the visible to near-infrared range (Noskov et al., 25 Jul 2025, Noskov et al., 16 Sep 2025). The transition probability for a broadened momentum state $P(\sigma_k)$ aligns with the overlap between the momentum-distributed photon and the electronic band structure,

$$P(\sigma_k) = \int D(E)f(E)\exp\left[-\frac{(k(E)-\sigma_k)^2}{2\sigma^2}\right]dE,$$

where $D(E)$ is the electronic density of states and $f(E)$ the occupation function.

• Gauge-Polarization Entanglement: The gauge choice in defining the transverse polarization basis at each momentum produces observable rotations of the physical electric field about the wavevector, linking the polarization structure intrinsically to the local momentum (Li, 2017). Gauge transformation of the Jones vector is expressed as

$$\tilde{a}' = \exp[i(\vec{\sigma}\cdot\mathbf{w})\Phi(\mathbf{k})]\tilde{a},$$

with $\Phi(\mathbf{k})$ a momentum-dependent angle.

• Topological Manipulation of OAM: Photons in degenerate cavities coupled via spatial light modulators can be adiabatically pumped through synthetic lattices in orbital angular momentum space, resulting in topologically quantized momentum expansion (Luo et al., 2017). The change in the photon's center-of-mass OAM is determined by a Chern number through Thouless pumping, yielding robust multi-channel quantum states for communications and quantum information.
• Many-body Quantum Simulation and Superradiance: Photonic quantum simulators map Heisenberg spin systems onto photonic degrees of freedom via controlled interference and measurement-induced effective interactions, with expansion in momentum manifesting in the superposition and entanglement of polarizations and spatial modes (Ma et al., 2012, Paulisch et al., 2018). Dicke superradiance in one-dimensional waveguides produces highly entangled multimode states with Heisenberg phase sensitivity scaling $\Delta\phi\sim 1/N$.
• Time-varying Resonant Materials and Photonic Time Crystals: Nontrivial temporal modulation of either intrinsic or structural resonances can open giant momentum bandgaps, producing states with exponentially growing or decaying momentum components and enabling novel amplification and lasing phenomena (Wang et al., 2023).

3. Experimental Realization and Measurement

Momentum-expanded Heisenberg photonic states have been observed and engineered using several experimental techniques:

• Nanoparticle Decoration and Embedded Confiners: Silicon wafers decorated with sub-2 nm Au, Cu, or Si nanoparticles yield broadband and efficient emission attributable to momentum expansion, with the confiner size (rather than material type) being the dominant factor (Noskov et al., 25 Jul 2025, Noskov et al., 16 Sep 2025). Larger (e.g., 5 nm) particles fail to activate this effect, establishing confinement-driven momentum expansion as the critical enabling mechanism.
• Hong–Ou–Mandel (HOM) Interferometry: Manipulation of biphoton symmetry in transverse momentum using spatial light modulators enables generation and verification of states with arbitrary exchange phase, extending the class of accessible momentum-expanded states and directly observing their bunching/anti-bunching properties through HOM dips or peaks (Gao et al., 2022).
• Photonic Quantum Simulators: Tunable directional couplers and linear optics can simulate Heisenberg models by encoding spin–½ states in photon polarization, with quantum interference processes mimicking spin coupling and entanglement redistribution observed in concurrence dynamics (Ma et al., 2012).
• Matrix Product State (MPS) Simulations: Momentum-resolved MPS methods, by operating in the thermodynamic limit and targeting well-defined momentum sectors, suppress entanglement growth and enable high-precision spectral function calculations for strongly correlated systems, potentially extending to photonic lattices and cavities (Damme et al., 2022).

4. Applications in Quantum Technologies and Optoelectronics

Momentum-expanded Heisenberg photonic states underpin a series of technological advances:

• Silicon Photonics: Direct radiative recombination in silicon, typically forbidden due to its indirect bandgap, becomes practical using momentum-expanded photon states, leading not only to photoluminescence but also to electroluminescence visible to the naked eye in ambient conditions (Noskov et al., 16 Sep 2025). This compatibility with existing fabrication processes enables integration of high-efficiency silicon-based LEDs and laser sources.
• Quantum Information and Communications: Broad momentum/OAM channels provided by topological pumping and symmetry manipulation schemes enable robust, high-dimensional quantum information encoding, error suppression via topological protection, and dense optical multiplexing (Luo et al., 2017, Gao et al., 2022).
• Quantum Simulation and Metrology: Superradiant multimode states in nanophotonic waveguides and photonic lattices support Heisenberg-scaling quantum metrology, with quantum Fisher information given by $F_Q \approx 0.41N^2+N$ for large photon numbers (Paulisch et al., 2018). Measurement-induced effective Heisenberg interactions and momentum correlation maps facilitate exploration of frustrated spin systems and quantum phase transitions (Ma et al., 2012, Yannouleas et al., 2019).
• Photonic Time Crystals and Lensing: Time-modulated resonant materials and metasurfaces amplify broadband momentum components and foster photonic time crystals, enabling new forms of optical amplification and subwavelength imaging via perfect lensing of evanescent waves (Wang et al., 2023).
• Autonomous Stabilization of Topological States: Angular momentum-selective potentials and frequency-selective incoherent pumps stabilize many-photon Laughlin states and quasihole excitations, with efficiency captured analytically and error sources identified, facilitating the preparation of highly correlated, topologically nontrivial photonic states (Umucalilar et al., 2021).

5. Topology, Symmetry, and Nonlocal Photonic Effects

Topological and symmetry considerations play a central role in the momentum expansion and properties of Heisenberg photonic states:

• Polarization Singularities and Bound States in Continuum (BICs): Graph-theoretic analysis of polarization in momentum space shows that BICs can be stabilized both on and off high-symmetry lines, with their existence and robustness guaranteed by conservation of topological charge,

$$q = \frac{1}{2\pi}\oint_C d\mathbf{k}\cdot \nabla_\mathbf{k}\psi(\mathbf{k}),$$

and sum rules $\sum_i q_i = 0$ over bounded faces (Jiang et al., 2023). These features facilitate the design of nonleaky, high-Q quantum states for light localization and precision light-matter interaction.

• Protected States via Symmetric Operations: C$_2$T (twofold rotation and time-reversal) symmetry protects BICs and generalizes the landscape of momentum-expanded states, sustaining their nonradiative character and topological quantization under otherwise perturbative processes.
• Nonlocality and Quantum Fingerprinting: Momentum correlation maps, especially in strongly interacting and entangled photonic or atomic states, enable discrimination between different quantum phases and enhance control over multi-particle interference (Yannouleas et al., 2019).

6. Limitations and Future Developments

The generation and exploitation of momentum-expanded Heisenberg photonic states face several experimental and theoretical challenges:

• Confinement Control: The particle size or spatial extent of the photonic confiner is crucial; sizes below 1.5–2 nm are necessary for pronounced momentum expansion, while larger confiners revert to behavior resembling conventional quantum dots (Noskov et al., 25 Jul 2025, Noskov et al., 16 Sep 2025).
• Material and Process Integration: Techniques involving surface decoration or embedding require precise fabrication but are inherently scalable and compatible with standard semiconductor processes, owing to their material-agnostic nature.
• Photon Loss, Entanglement Scaling, and Dephasing: Robustness against photon loss, spectral diffusion, and waveguide imperfections is essential. Many schemes require maintaining coherence and high detection efficiencies, particularly in metrological or quantum simulation contexts (Paulisch et al., 2018, Damme et al., 2022).
• Topological Bandgap Engineering: In photonic time crystals, limitations due to nonlocality at large momenta, material dissipation, and achievable modulation frequencies inform practical designs for experimental realization (Wang et al., 2023).
• Measurement and Characterization: Unambiguous detection of momentum-expansion effects typically relies on precise interferometric measurements, quantum state tomography, or spectral analysis, demanding high resolution and stability.

Momentum-expanded Heisenberg photonic states thus represent a hybrid regime at the intersection of uncertainty-driven spatial confinement, gauge-controlled quantum polarization, many-body and topological physics, enabling breakthroughs in quantum simulation, metrology, optoelectronics, and photonic device engineering. Their paper continues to uncover new aspects of light–matter interaction and photonic phase structure in quantum science.