Researchers Bankim Chandra Das, Dmytro Kiselov, Lee Drori, Ariel Nakav, Alexander Poddubny and Ofer Firstenberg detail a theoretical study, published October 2, 2026, revealing a multiband dispersion governing the spatial evolution of interacting photons realized through co-propagating Rydberg polaritons. This departs from existing models that typically rely on a single-band, parabolic approximation for these interactions.

The work predicts warped vortex structures of photons resulting from interaction-induced symmetry reduction, advancing understanding of multiphoton interactions and supporting the development of future multiphoton control tools. These findings connect quantum nonlinear optics with concepts from multiband and topological quantum matter.

Rydberg Polaritons Enable Strong Multiphoton Interactions

The spatial evolution of multiple photons interacting as Rydberg polaritons is governed by a dispersion unlike those previously used to model these systems, featuring both massive and massless modes with degenerate Dirac points. This multiband dispersion departs from the standard single-band, parabolic approximation typically applied to interacting polaritons, offering a more accurate description of their behavior within atomic media. Rigorous numerical modeling confirms these analytical results, fully accounting for photon propagation within the finite atomic medium and supporting the development of tools for controlling multiple photons.

Researchers predict that these interactions will manifest as warped vortex structures of photons, a result of the internal degrees of freedom inherited from the coupled light and matter, and a reduction in symmetry during the interaction process. These predicted effects are directly measurable through photon correlation measurements, offering a pathway for experimental verification and further investigation.

Rydberg polaritons, created by coupling photons to Rydberg atoms within a dense atomic ensemble, provide a robust platform for studying these quantum interactions between a limited number of particles. Their strong interactions stem from the long-range dipolar coupling between the Rydberg constituents, enabling suppression of single-photon transmission and demonstrating large conditional phase shifts.

Previous theoretical models, including adiabatic potential approximations and quantum field theory, have offered valuable insights but fall short in fully capturing the spatial evolution of few-polariton wavefunctions in extended media. The Hamiltonian governing these interactions can be generalized to two or three polaritons, represented by multicomponent wavefunctions detailing each polariton’s state, either propagating as a photon or residing in a Rydberg state.

The symmetry, which is important to the observed effects, arises from the interplay between the multiband dispersion and the finite spatial extent of the medium, requiring both elements for its emergence in a steady state. Consequently, strongly interacting polaritons are expected to exhibit effects not predicted by traditional models relying on parabolic dispersion and contact interactions.

Multiband Dispersion Reveals Warped Vortex Structures

The internal structure of light, when strongly interacting, generates warped vortex structures absent in standard models of photon behavior, according to new theoretical work. The team’s analysis reveals a splitting of energy states, creating a fine structure sensitive to contributions from multiple states. Near a specific point, a symmetric state acquires mass, while two antisymmetric states split, forming Dirac cones, a characteristic feature also seen in materials like graphene.

Far from this point, these states mix, yielding a warped dispersion with symmetry that mirrors the trigonal warping observed in graphene, a consequence of its crystalline symmetry. Simulations show how this manifests spatially; with two photons, vortex tubes emerge reflecting two-body interactions with sixfold symmetry.

As a third photon joins, these tubes merge into a central vortex ring, clearly displaying the trigonal symmetry inherent in the underlying dispersion. The researchers found that the simplified multiband model accurately describes the dispersion of experimentally relevant bands, as demonstrated by a blue warped cone in their visualizations. The model’s ability to generalize to multiple photons, extending the Hamiltonian to describe interactions between three, supports the prediction of complex vortex formations.

Three-Photon Dynamics Exhibit Trigonal Warping

The observed reduction in rotational symmetry from six-fold to three-fold arises when distinguishing photon propagation sequences, specifically whether a single photon leads or lags a pair during interaction. This asymmetry, previously detected at a few-percent level using single atoms or effective superatoms, now manifests as a substantial warping of vortex rings formed by three-photon interactions, as detailed in recent analytical and numerical models.

Calculations reveal isosurfaces exhibiting this trigonal warping, a clear signature of the three-photon interaction captured by the employed multiband model, and are detailed in equations C4 through C7 with parameters specified in Appendix C. Simulations within a uniform medium of length, typically chosen as one for a Gaussian cloud with a width, divide space into sections based on the number of photons contained within, with the faces of the three-photon section acting as effective light-cone boundaries that warp the wavefunction.

Within this section, two distinct regions emerge: regions where one photon precedes a pair, and “pair ahead” regions where one photon trails a pair, exemplified by configurations like or for specific conditions. “The three-photon interaction ultimately produces a phase vortex ring, which the multiband model predicts to exhibit a trigonal warping,” the paper states.

This contrasts with the two-photon wavefunction, which displays a symmetric vortex-antivortex pair, and highlights the influence of the multiband dispersion on the spatial evolution of the interacting photons. The work demonstrates how the difference between the single-band Schrödinger and multiband Dirac-like dispersions manifests in real-space wavefunctions, offering a new perspective on polariton dynamics and potentially enabling more precise control of multiphoton systems.

Analytical Model Captures Few-Polariton Symmetry

The analytical model accurately describes the symmetry observed in few-polariton states, a departure from established single-component approximations. Numerical and analytical methods confirm these results, revealing phenomena absent in simpler models. The team’s approach centers on a multiband Dirac-like dispersion, which governs the relationship between the wavevector and the center-of-mass motion of the polaritons. This dispersion features a single massive mode alongside gapped antisymmetric modes, a configuration that fundamentally differs from the parabolic dispersion typically used to describe interacting polaritons.

The model accounts for the propagation of polaritons within a three-level ladder-type atomic system, where the wavefunction consists of amplitudes representing states with zero, one, or two atoms in excited states. This splitting is sensitive to contributions from these states and appears in higher-order interactions. Analysis reveals that the dual-band model naturally enforces light-cone limits on the phase pattern, a constraint not present in single-band approximations.

The model predicts a forward shift in the vortices, attributable to these imposed limits, and this effect is amplified when considering three photons. “The symmetry of the wavefunction becomes lower than that of the interaction potential,” the researchers report, highlighting a key distinction from previous work. This analytical approach provides a foundation for understanding the complex interplay between the polariton’s internal structure and its spatial propagation.

The team’s full numerical model for three photons further validates the predictions, illustrating the evolution within a non-uniform, finite medium. The model predicts the formation of symmetry-reduced warped vortex rings in the phase of the interacting photons, a phenomenon not captured by the single-band model.

By combining analytical models and numerical simulations, the study offers insights into the behavior of strongly interacting photons and lays the groundwork for controlling their quantum correlations. The team’s approach provides a robust method for analyzing multi-polariton dispersion and spatial propagation, offering a new perspective on the dynamics of light-matter interactions.

Single-Band Approximations Fail to Predict Key Effects

The established method of modeling interactions between photons relies on approximations that fail to capture newly observed effects, according to recent work on Rydberg polaritons. Specifically, calculations reveal that single-band approximations, commonly used to simplify complex interactions, cannot accurately predict the symmetry reduction and warped vortex structures arising from the interplay of light and matter. These structures emerge due to the internal degrees of freedom inherent in the system, a factor overlooked by simpler models.

The team’s analysis of two- and three-photon interactions demonstrates a significant departure from traditional single-band predictions, particularly in the dispersion of polaritons. While the single-band Schrödinger approximation provides a basic understanding, it fails to capture the Dirac-like dispersion exhibited by the full multiband model, and is unable to account for the trigonal warping observed in three-photon systems.

This warping, visualized through isosurface calculations, manifests as a vortex ring, a hallmark of the interaction accurately predicted by the more comprehensive model. The calculations involved 1069 vectors and were performed using a system of linear equations derived from the model. The limitations of the single-band approach extend beyond the shape of the resulting structures, influencing the dynamics of photon correlation. The team’s approach involved transforming the initial equations into a system of linear equations, then diagonalizing them to find eigenvectors and eigenvalues.

This allowed for a detailed examination of the spatial dispersion outside the Rydberg blockade region, the area where interactions are suppressed. By adjusting boundary conditions to model a uniform medium, researchers were able to observe the accumulation of phase as photons propagate, revealing the formation of vortex-antivortex pairs. The single-band model, however, retains the original symmetry, failing to replicate this detailed behavior. The findings emphasise the importance of incorporating multiband effects when modeling strongly interacting photons.

Photon Correlations Reflect Reduced Symmetry

This multiband dispersion directly impacts photon correlations, reducing the rotational symmetry of the wavefunction from fourfold to threefold by distinguishing between photons propagating ahead of or behind others. These regions evolve differently, with single-ahead photons experiencing interaction at an earlier point in propagation than their pair-ahead counterparts.

This timing difference amplifies correlations within single-ahead configurations, further diminishing the overall symmetry of the wavefunction. This difference in origin and timing is key to understanding the observed warping. In contrast, a single-band model fails to replicate this detailed behavior, retaining the original symmetry.

👉 More information
🗞 Multiband Dispersion and Warped Vortices of Strongly Interacting Photons
✍️ Bankim Chandra Das, Dmytro Kiselov, Lee Drori, Ariel Nakav, Alexander Poddubny and Ofer Firstenberg
🧠 DOI: http://link.aps.org/doi/10.1103/mndb-nzqd

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