Asymmetric Coupling Splits Localized Modes in Photonic Crystal Emitters

Two V-type three-level emitters embedded in a one-dimensional photonic crystal can retain long-lived entanglement through a combination of coherent excitation exchange and non-Markovian feedback from the structured photonic reservoir. A new theoretical study shows that making the two optical exchange channels unequal does more than shift the dynamics: it pushes the two dominant localized spectral poles in opposite directions, eventually removing one of them into the radiation continuum. The result separates three commonly conflated behaviors—persistent entanglement, high-entanglement lifetime and late-time revivals—and demonstrates that each responds differently to the coupling asymmetry and the relative orientation of the transition dipoles.

The authors model two identical V-type emitters, each with two optical transitions sharing a ground state, coupled to an isotropic one-dimensional photonic crystal with a quadratic band-edge dispersion. This spectral structure produces a square-root singularity in the density of states, allowing the photonic field to retain information about the emitters and return it to them. The simulations begin from a bright, single-excitation state that loads the two channels symmetrically. They then vary an asymmetry parameter while holding the mean coherent resonant dipole-dipole exchange fixed, allowing the imbalance to be studied independently of the overall interaction strength.

Rather than treating the reservoir as a simple irreversible sink or retaining only a selected set of spectral poles, the study reconstructs the dynamics through exact Bromwich inversion of the full Laplace-domain amplitude equations. This procedure keeps the localized poles, leaky contributions and band-edge branch cut, with no Markov approximation or pole-truncation procedure. The distinction is crucial for long-time behavior because the existence, position and residue of each pole determine not only whether a persistent oscillatory signal survives, but also how much entanglement remains in the reduced state of the two emitters.

At the symmetric coupling point, the bright sector contains two dynamically relevant localized poles. Their frequencies correspond to two persistent components of the emitted field, and cross terms between those components produce late-time entanglement beating. When the two channels become asymmetric, one branch moves farther into the photonic band gap while the other travels toward the highest band edge. The first-order pole-displacement expression derived from the exact bright-sector denominator reproduces the local motion in the weak-asymmetry regime. At a geometry-dependent merger point, the continuum-seeking pole reaches the square-root branch point and ceases to be an isolated pole. This is a spectral merger, not a thermodynamic phase transition: the system consists of only two finite-dimensional emitters, and the boundary marks a change in the number of isolated singularities in the continued resolvent.

Losing the second isolated pole has a direct dynamical consequence. With two poles, the amplitudes contain two nondecaying frequencies whose interference creates persistent late-time revivals. After the merger, the surviving pole can still retain a stationary component of atomic excitation, allowing a nonzero entanglement plateau to persist. Its common phase, however, cancels from the reduced density matrix, so the two-frequency beating is lost. The result means that a visible plateau does not necessarily imply robust oscillatory feedback. Conversely, strong revivals do not guarantee that the entanglement will remain above a specified threshold for a long time. The study therefore evaluates the tail-averaged logarithmic negativity, the time required to fall below a chosen high-entanglement level and the amplitude of late-time revival as separate diagnostics.

Several dipole arrangements produce sharply different landscapes. Near-antiparallel and antiparallel configurations preserve the largest late-time plateaus and longest high-entanglement lifetimes, while near-parallel configurations lose protection rapidly. The asymmetry parameter alone cannot organize these outcomes because the relative dipole angle controls how the two channels interfere in the structured reservoir. The cross-memory kernel changes the residues and phases of the localized amplitudes, meaning that the same degree of channel imbalance can be processed quite differently by different geometries. Numerical checks against finer Bromwich calculations, pole-tracking refinements and alternative operational thresholds support the robustness of these qualitative patterns, although the exact crossover locations depend on the observation window and chosen threshold.

The orthogonal configuration provides an exact test of the proposed mechanism. At a relative dipole angle of 90 degrees, the cross-memory kernel vanishes and the two V-shaped channels decouple exactly. Entanglement can nevertheless decline as the coupling asymmetry grows, ruling out interchannel routing as the explanation. Instead, the two independent bright sectors are dressed differently by the photonic reservoir. Their channel populations become unequal, and their combined retained atomic weight falls. In this decoupled case, the logarithmic negativity depends exactly on the total localized atomic weight, while the population imbalance serves as a fingerprint of the unequal dressing. This benchmark demonstrates that asymmetry can suppress protected entanglement even when no excitation is transferred between the two optical channels.

The study also links the return of quantum distinguishability to the observed dynamics using a symmetry-adapted lower bound on the Breuer–Laine–Piilo non-Markovianity functional. The researchers maximize the accumulated positive trace-distance variation over a fixed collection of bright, dark, channel-resolved, localized-transfer and ground-state contrasts. Because the search is restricted to this finite probe set, the resulting value is explicitly a lower bound rather than the globally optimized functional. Across the parameter grid, it shows a strong monotonic association with the late-time entanglement-revival amplitude. It has essentially no global association with the high-entanglement lifetime, however. Antiparallel configurations can preserve long-lived high entanglement at backflow values at which other geometries have already lost that protection.

The physical explanation is that information backflow measures the renewed distinguishability of selected emitter states, but it does not determine how the returning coherence is divided between the two atoms or the two optical channels. Entanglement depends on those bipartite coherences, whose conversion is controlled by the dipole-dependent cross kernel. In practical terms, reservoir memory and geometry act as separate controls: memory determines whether revival-supporting dynamics are available, while orientation determines how effectively that feedback becomes persistent emitter-to-emitter entanglement. The conclusions apply to the study’s zero-temperature, single-excitation, phase-matched model and benchmark coupling regime. They also come with clear limits: nonguided emission, nonradiative relaxation and pure dephasing are omitted from the baseline model. Weak independent loss turns the ideal permanent plateau into a metastable feature, while dephasing suppresses the inter-emitter coherence needed for entanglement. The work therefore offers a controlled theoretical account of how channel asymmetry reorganizes localized modes, rather than a prediction for a specific device without measurements of its reservoir coupling, loss and dephasing properties.

Subject of Research: Non-Markovian entanglement dynamics of two asymmetrically coupled V-type emitters in a one-dimensional photonic crystal

Article Title: Entanglement, localized poles, and non-Markovian crossovers of asymmetrically coupled V-type emitters in a photonic crystal

Article References: Ismail, G. M., Altuijri, R., Kadry, H., Alharthi, N. S., Ghorashi, S. A., Shaaban, S. M., Voon, B. W. N., Hafez, M., & Abdel-Aty, A.-H. (2026). Entanglement, localized poles, and non-Markovian crossovers of asymmetrically coupled V-type emitters in a photonic crystal. Frontiers in Physics, 14, Article 1914618. https://doi.org/10.3389/fphy.2026.1914618

Image Credits: AI Generated

DOI: 10.3389/fphy.2026.1914618

Keywords: quantum entanglement, photonic crystal, non-Markovian dynamics, V-type emitters, dipole coupling, localized spectral poles, information backflow, photonic reservoir, Entanglement, localized, poles, non-Markovian