Multi-Photon Ring Structures in Black Holes & Quantum Optics

• Multi-photon ring structures are defined as sequences of geometrically ordered light rings formed by unstable photon orbits in strong gravitational fields and by engineered quantum optical systems.
• They are characterized by measurable metrics such as the Lyapunov exponent and ring scaling relations, which quantify exponential ring narrowing and time delays in photon orbits.
• These structures serve as diagnostic tools to probe black hole properties, test general relativity, and enable controlled quantum state engineering in photonic circuits.

A multi-photon ring structure refers, in modern usage, to the sequence of highly lensed, geometrically ordered light rings (subrings) appearing in the image plane around ultra-compact objects (including black holes and exotic horizonless solutions), and more broadly, to analogous structures in collective quantum optical systems with ring geometries. The terminology arises predominantly in the context of strong gravitational lensing, where each ring is tied to photons that have completed a specific number of half-orbits around the compact object before escaping to a distant observer. In quantum optical systems (as in engineered Rydberg atomic rings or quantum oligomers), similar ring symmetries give rise to multi-photon states with distinct lifetime, angular momentum, and entanglement properties.

1. Geometric and Dynamical Origin of Multi-Photon Ring Structures

In gravitational lensing scenarios, the multi-photon ring structure is generated by the set of unstable photon orbits—often called the photon sphere (Schwarzschild) or photon shell (Kerr and more general geometries)—that encircle the compact object (Johnson et al., 2019, Hadar et al., 2022). Each bound null geodesic in this region acts as an attractor for photons with critical impact parameters, so that the lensed emission creates a sequence of exponentially narrowing rings, labeled by the integer $n$ denoting the number of (half-)orbits completed prior to escape (Hadar et al., 2020, 2206.12066).

The characteristic properties of these rings—such as their width, flux, and spacing—are determined by the instability (Lyapunov exponent) of the underlying photon orbit, which dictates the exponential demagnification and radial contraction of higher-order rings:

$$\delta r_n \sim e^{\gamma n}\, \delta r_0, \qquad \text{and} \qquad \frac{F^{(n+1)}}{F^{(n)}} \approx e^{-\gamma}$$

with $\gamma$ the Lyapunov exponent specific to the null geodesic structure (Johnson et al., 2019, Hadar et al., 2022, Salehi et al., 22 Nov 2024).

In ring-like quantum optical systems, the origin lies in the collective emission or coupling of quantum emitters/photons arranged on a ring lattice or oligomer, where symmetry and collective effects produce tailored angular momenta or long-lived nonradiant states (Olmos et al., 2010, Ustimenko et al., 2023). The circulant interaction matrices and orbital momentum eigenmodes give rise to spatial and spectral multi-ring emission patterns or subradiant multiphoton states.

2. Mathematical Characterization and Critical Parameters

The photon ring and its subring structure are precisely characterized by the critical curve on the observer's sky (the “shadow edge”), onto which the infinite sequence of $n$-photon images converge (Johnson et al., 2019, Achour et al., 11 Jun 2025). For Kerr and related axisymmetric metrics, this is obtained by solving for the conserved quantities (impact parameter $\xi$, Carter constant $\eta$) associated with spherical photon orbits:

$R(r) = 0, \quad \frac{\partial R}{\partial r} = 0$

where $R(r)$ is the radial geodesic potential, with the observed photon ring mapped parametrically via

$$(\alpha_c, \beta_c) = \left( -\frac{\xi(r)}{\sin \theta_o}, \pm \sqrt{\eta(r) + a^2 \cos^2 \theta_o - \xi^2(r) \cot^2 \theta_o} \right)$$

(Gußmann, 2021, Achour et al., 11 Jun 2025).

For beyond-Kerr metrics (e.g., Kerr Off Shell), analytic parametric forms link the critical curve to symmetry-preserving deformations, with the sequence of subrings converging on this critical curve (Achour et al., 11 Jun 2025). The scaling relations for the radii and widths of photon subrings are

$$\frac{\rho_{n+1} - \rho_p}{\rho_n - \rho_p} \approx e^{-\gamma_p}, \qquad t_{n+1} - t_n \approx \tau_p, \qquad \psi_{n+1} - \psi_n \approx \delta_p$$

with $\gamma_p$ the Lyapunov exponent, $\tau_p$ the time delay per half-orbit, and $\delta_p$ the net azimuthal rotation, all functions of the underlying spacetime and observer's inclination (Salehi et al., 22 Nov 2024).

In quantum ring systems, the symmetry yields a set of well-defined eigenmodes $\mathcal{M}_{\gamma k}$ for the collective emission, which, via mapping coefficients $g_{\alpha q \lambda}$, determine the angular momentum structure and interference pattern in the observed photon emission ring (Olmos et al., 2010).

3. Physical Mechanisms and Observational Manifestations

In strong gravitational lensing, the physical origin of multi-photon ring structures lies in the exponential focusing and time delays accumulated by photons as they execute multiple orbits in the strong-field region. For black holes, the structure is self-similar and universal, with the Lyapunov exponent and azimuthal shifts encoding the instability of the photon shell and the degree of frame dragging, respectively (Johnson et al., 2019, Hadar et al., 2022, Salehi et al., 22 Nov 2024).

Reflection-asymmetric traversable thin-shell wormholes in Palatini $f(R)$ gravity generate a richer multi-photon ring structure because both sides of the throat host photon spheres at distinct locations (typically with different effective potentials). The result is a more intricate set of photon rings, including rings formed by light crossing the throat, whose positions and brightness diverge from canonical black hole predictions. These can be especially pronounced in images with accretion disks situated on both sides of the wormhole, leading to multiple narrow bright rings and a strong reduction in the central brightness depression (i.e., “shadow”) region (Macedo et al., 22 Oct 2025). This stands in contrast to the exponentially suppressed, self-similar photon subrings characteristic of black holes.

Tables summarizing the key differences as observed in (Macedo et al., 22 Oct 2025):

Multi-Ring Property	Thin-Shell Wormhole	Kerr Black Hole
Number of photon spheres	Two (one each side of throat)	One (photon shell)
Subring scaling	Not universally exponential	Exponential, $e^{-\gamma n}$
Central brightness depression	Greatly reduced (especially with disks on both sides)	Pronounced (shadow)
Cross-throat ring luminosity	Can be bright, especially two-disk	Negligible

In quantum photonic systems, the ring structure manifests as single-photon states with well-defined orbital angular momentum (OAM) and multi-photon entangled states where collective symmetry “locks” OAM or protects double excitations from decay (Olmos et al., 2010, Ustimenko et al., 2023).

4. Impact of Observer Inclination, Disk Geometry, and Horizonless Surfaces

The completeness and detectability of the photon ring structure depend sensitively on the compact object's surface properties, the observer’s inclination, and the accretion/emission geometry. When the surface radius $r_s$ lies within the photon region, the photon ring is closed and continuous. As $r_s$ increases and begins to occlude the photon region, the ring becomes broken, fragmentary, or disappears entirely from the image. These effects are more pronounced at high observer inclinations, where certain geodesics are blocked or Doppler-shifted beyond detection (Wang et al., 8 May 2024). The critical behavior is governed by the respective functions in impact parameter space: $\tilde{\eta}(\tilde{\xi})$ for the photon region, $\eta_o(\xi_o)$ for the observer, and $\eta_s(\xi_s)$ for the surface.

Photon ring features are thus strong diagnostics for distinguishing black holes from horizonless ultra-compact objects (UCOs), as ring disruptions or the absence of expected subrings point to the presence of a non-trivial surface rather than an event horizon (Wang et al., 8 May 2024).

5. Polarization, Quantum Signatures, and Nontrivial Topologies

The photon ring structure encodes polarization and quantum signatures from both spacetime and field-theoretic effects. In black holes pierced by cosmic axion strings, Aharonov–Bohm-like topological effects cause additional, quantized polarization rotations in the subrings, with the phase shift per orbit given by $\Delta \Phi_{2n} = \pm C n \delta(r)$, where $C$ is set by the anomaly coefficient (Gußmann, 2021). Measurement of relative polarization angles in different subrings thus allows direct probe of UV physics tied to axion-photon couplings.

Polarimetric interferometry can directly separate and detect the photon ring (n = 1) from the direct emission (n = 0) by exploiting the sharp phase reversal in the observable $\hat{\beta}_2$, a property exploited for robust VLBI detection even in the presence of calibration errors or scattering (Palumbo et al., 2023).

In the quantum optical context, ring configurations enable the preparation of robust, long-lived subradiant multi-photon states (in ring oligomers) through engineering of symmetry-induced interference, and quantum phase transitions such as Bose-Einstein condensation into ring eigenstates with fixed phase relations (Redmann et al., 2023).

6. Theoretical and Observational Implications

High-resolution imaging of multi-photon ring structures, especially with next-generation VLBI and space-based platforms (e.g. BHEX), enables precise measurement of parameters such as the Lyapunov exponent (from ring spacing), the time delay (from light-curve autocorrelations), and the azimuthal shift (from ring twisting), allowing direct constraints on black hole spin, the ergosphere, and even deviations from general relativity through model-independent metrics (e.g., Johannsen–Psaltis, Kerr off shell) (Salehi et al., 22 Nov 2024, Achour et al., 11 Jun 2025, Galison et al., 17 Jun 2024). For traversable thin-shell wormholes, the detection of a rich, anomalous multi-photon ring structure, including the reduction of the shadow region and the emergence of cross-throat rings, serves as a prospective “smoking gun” for non-black-hole compact objects (Macedo et al., 22 Oct 2025).

In quantum optical systems, controlled multi-photon ring states have parallel implications for quantum information, enabling on-chip non-Gaussian state generation, protected quantum excitations, and high-dimensional OAM-entangled photonic states (Olmos et al., 2010, Ustimenko et al., 2023, Banic et al., 2021).

7. Summary Table: Core Properties Across Contexts

Context	Origin of Rings	Control Mechanism	Detectable Features	Diagnostic Value
Black hole	Near-horizon null geodesics (photon shell)	Spacetime metric, inclination, spin	Exponential ring narrowing, visibility oscillations, time delays, polarimetric phase	Tests GR, spin measurement, horizon test
Wormhole	Two photon spheres on opposite sides of throat	Mass/charge asymmetry, disk geometry	Numerous/luminous cross-throat rings, reduced shadow, irregular ring scaling	Discriminates from black holes
Quantum optics	Symmetric emission/eigenmodes of ring lattice	Atomic spacing, ring geometry, excitation structure	Well-defined OAM, long-lived subradiant states, entanglement	Quantum memory, photonic circuits

• (Olmos et al., 2010) Collective photon emission from symmetric states created with Rydberg atoms on a ring lattice
• (Johnson et al., 2019) Universal Interferometric Signatures of a Black Hole's Photon Ring
• (Gußmann, 2021) Polarimetric signatures of the photon ring of a black hole that is pierced by a cosmic axion string
• (Hadar et al., 2022) Holography of the Photon Ring
• (2206.12066) Images and photon ring signatures of thick disks around black holes
• (Ustimenko et al., 2023) Nonradiant multiphoton states in quantum ring oligomers
• (Redmann et al., 2023) Bose-Einstein Condensation of Photons in a Four-Site Quantum Ring
• (Wang et al., 8 May 2024) Is a photon ring invariably a closed structure?
• (Salehi et al., 22 Nov 2024) Influence of Observer Inclination and Spacetime Structure on Photon Ring Observables
• (Achour et al., 11 Jun 2025) Black hole photon ring beyond General Relativity: an integrable parametrization
• (Macedo et al., 22 Oct 2025) Multi-photon ring structure of reflection-asymmetric traversable thin-shell wormholes

This body of research establishes the multi-photon ring structure as a key probe of ultra-compact object properties, quantum optical phenomena, and strong gravity, with utility spanning from quantum state engineering to empirical tests of the Kerr hypothesis and the nature of compact astrophysical objects.