Self-referenced, drift-tolerant dipole-resolved population inversion using degeneracy-lifted dual quasinormal modes

This paper demonstrates a self-referenced, drift-tolerant metrology scheme using degeneracy-lifted dual quasinormal modes in a hybrid microcavity to robustly quantify the relative populations of in-plane and out-of-plane excitons in monolayer WSe$_2$ without external calibration.

The Gist

The Big Problem: The "Noisy Room"

Imagine you are trying to count how many people are in a crowded room by listening to them shout. But there's a catch:

1. The Microphone Drifts: Sometimes the microphone gets louder or quieter on its own (like a battery dying).
2. The People Move: Some people shout from the back of the room where the sound is muffled, while others shout right next to the mic.
3. The Wind Changes: A draft blows the sound away.

In the world of physics, scientists try to count "excitons" (tiny energy particles in materials like a single layer of tungsten diselenide, or WSe2). Usually, they just measure how bright the light is. But just like the noisy room, if the light gets brighter, is it because there are more particles, or because the "microphone" (the detector) got better, or because the "wind" (temperature) changed?

This makes it very hard to get an accurate count, especially for "dark" particles that are very shy and don't like to shout (emit light) in the direction the scientists are looking.

The Solution: The "Twin-Engine" Strategy

The researchers built a special machine (a microcavity) that acts like a twin-engine airplane. Instead of relying on one engine (one light signal), they use two nearly identical engines (two light modes called QNM1 and QNM2) that are stuck together.

Here is how they use these two engines to solve the problem:

1. The "Common-Mode" Engine (The Weather Vane)

Both engines are connected to the same fuel tank and the same airframe.

• What happens: If the wind blows (temperature changes) or the fuel pump sputters (laser power fluctuates), both engines react exactly the same way. They both speed up or slow down together.
• The Analogy: Think of this like a weather vane. If the whole wind shifts, the vane moves. This tells the pilot, "Hey, the weather is changing," but it doesn't tell you about the specific engine trouble.
• In the paper: The scientists look at the average behavior of both light signals. This tracks the global "drift" (temperature, pump power) so they can ignore it.

2. The "Differential-Mode" Engine (The Sensitive Ruler)

While the engines are twins, they are built slightly differently inside.

• QNM1 (The Sensitive One): This engine is built with a very thin, delicate gap. If you put a tiny pebble (a local change in the material) in that gap, this engine screams and changes its pitch immediately.
• QNM2 (The Sturdy One): This engine is built with a thicker, more stable gap. That same pebble barely affects it. It stays calm.
• The Analogy: Imagine two twins standing in a wind tunnel. Twin A is holding a giant, flimsy umbrella. Twin B is holding a heavy steel rod. If a gust of wind hits, Twin A's umbrella flips wildly, but Twin B stands still. By comparing how much Twin A moves relative to Twin B, you know exactly how strong the gust was, regardless of how loud the wind tunnel fan is.

The Magic Trick: Counting the "Shy" Particles

The real goal was to count two types of excitons:

• The "In-Plane" (∥) ones: These are loud and easy to see.
• The "Out-of-Plane" (⊥) ones: These are "dark" excitons. They are shy and usually hide their light, making them impossible to count with normal tools.

The researchers used a clever trick with Temperature:

1. They cooled the material down to near absolute zero (about 50 Kelvin).
2. At this cold temperature, the "shy" dark excitons start to pile up (accumulate) because they are the lowest energy state.
3. Because QNM1 is super sensitive to the "shy" particles (it loves to amplify their light), and QNM2 is more neutral, the scientists could compare the two signals.

The Result:
By looking at the difference between the two signals (ignoring the noise that affected both), they could calculate the ratio of shy particles to loud particles.

• They found that at 50 Kelvin, there were about 200 shy particles for every 1 loud particle.
• This is a massive number! It proves that the "shy" particles were indeed piling up, something that would have been impossible to measure accurately without their "twin-engine" self-referencing system.

Why This Matters

Before this, scientists had to guess if a change in brightness was real or just a glitch in their equipment.

• Old Way: "The light got brighter! Maybe there are more particles? Or maybe my laser got stronger? I don't know!"
• New Way: "The light got brighter, but the 'sturdy' twin didn't change, and the 'sensitive' twin changed exactly as predicted. Therefore, we know for a fact there are more particles."

This method is like having a built-in ruler inside your measuring tape. It allows scientists to measure tiny, weak signals in a noisy world without needing to calibrate their equipment every five minutes. It opens the door to studying "dark" quantum states that were previously invisible to us.

Technical Summary

1. Problem Statement

Quantitative measurement of exciton populations in nanophotonic systems is currently hindered by the inability to distinguish between changes in actual particle numbers and artifacts caused by system instability.

• The Convolution Problem: Detected photoluminescence (PL) intensity ($I_{det}$) is a product of exciton population ($N$), radiative rate ($\Gamma_{rad}$), and collection efficiency ($\eta_{col}$). Variations in $\Gamma_{rad}$ (due to cavity coupling) or $\eta_{col}$ (due to alignment drift) are often misinterpreted as changes in $N$.
• Drift Sensitivity: Long-term or dynamic measurements suffer from common-mode drifts (pump power fluctuations, thermal expansion, mechanical misalignment) that corrupt intensity-based inversions.
• The Dark Exciton Challenge: Out-of-plane (OP) dipoles, such as spin-forbidden dark excitons, emit into high-in-plane-wave-vector ($k_{\parallel}$) channels that are typically outside the numerical aperture (NA) of standard detectors. While micro/nanocavities can redirect this emission, quantifying the specific population of these weakly radiative states remains difficult due to the lack of a stable internal reference.

2. Methodology

The authors propose a dual-channel self-referenced scheme utilizing a hybrid microcavity composed of a silica microsphere (MS) placed on a gold (Au) substrate, with a monolayer of Tungsten Diselenide (WSe$_2$) inserted in the nanogap.

• Degeneracy-Lifted Quasinormal Modes (QNMs): The system supports two nearly degenerate, TM-like quasinormal modes (QNM1 and QNM2).
  • QNM1 (Sensitive Channel): Its electric field ($|E|^2$) is strongly concentrated in the lateral non-contact regions of the nanogap. It is highly sensitive to local dielectric perturbations (e.g., bending of the WSe$_2$ layer) and exhibits strong selectivity for out-of-plane (OP) dipoles.
  • QNM2 (Reference Channel): Its field is concentrated at the contact center (pinned region). It is weakly perturbed by local gap changes but shares the same optical path as QNM1, making it sensitive to global drifts (temperature, pump power) but insensitive to local geometric deformations.
• Self-Referencing Strategy:
  • Common-Mode Observables: The average frequency and total intensity track global system drifts (pump fluctuations, thermo-optic shifts).
  • Differential-Mode Observables: The spectral splitting ($\Delta \omega$) and the intensity ratio ($R = I_1/I_2$) encode local perturbations and dipole-specific radiative weights, effectively canceling out common-mode noise.
• Experimental Control: The researchers used optical heating to induce controlled bending of the WSe$_2$ monolayer within the gap, creating a differential perturbation that lifts the mode degeneracy. They further utilized temperature tuning (7 K to 297 K) to manipulate the relative populations of bright (in-plane) and dark (out-of-plane) excitons.

3. Key Contributions

1. Novel Metrology Architecture: The first demonstration of a self-referenced, drift-tolerant population inversion scheme in a nanogap photonic system using a single hybrid cavity with two distinct QNMs.
2. Decoupling of Drift and Physics: The method successfully separates instrumental drifts (pump power, alignment) from intrinsic physical changes (exciton population dynamics) by utilizing the ratio of two simultaneously measured modes.
3. Dipole-Resolved Quantification: The system exploits the distinct polarization selectivity of the two modes (QNM1 favors OP dipoles; QNM2 enhances both OP and IP dipoles comparably) to mathematically invert and quantify the relative populations of out-of-plane ($N_\perp$) and in-plane ($N_\parallel$) excitons without external absolute calibration.
4. Validation of Non-Thermal States: The technique revealed a significant discrepancy between the measured population ratio and the expected Boltzmann distribution, suggesting non-thermal steady states in dark exciton manifolds.

4. Key Results

• Mode Characterization: Simulations and experiments confirmed that QNM1 is highly sensitive to local gap dielectric changes (participation factor $\approx$ 2x higher at the substrate surface), while QNM2 acts as a stable reference.
• Degeneracy Lifting: Optical heating induced WSe$_2$ bending, causing a spectral splitting of $\sim$1.3 meV. QNM1 (low-energy branch) showed a cumulative redshift, while QNM2 (high-energy branch) remained nearly stationary, validating the differential sensitivity.
• Polarization Selectivity:
  • QNM1: Nearly unpolarized, narrow angular distribution ($\pm 20^\circ$), strong enhancement for OP dipoles ($F_{r,\perp} : F_{r,\parallel} \approx 7.2$).
  • QNM2: Stable TM-like polarization, narrow radiation band ($\pm 30^\circ$), comparable enhancement for both orientations ($F_{r,\perp} : F_{r,\parallel} \approx 1.98$).
• Population Inversion:
  • By cooling the sample from 297 K to 7 K, the ratio of out-of-plane to in-plane exciton populations ($N_\perp/N_\parallel$) was extracted.
  • At $\sim$50 K, the ratio reached $N_\perp/N_\parallel \approx 200$.
  • This result coincides with the accumulation of excitons in the out-of-plane dark manifold.
• Non-Equilibrium Insight: The extracted energy scale for this ratio ($\sim$22 meV) was significantly lower than the theoretical spin-splitting energy ($\sim$40 meV). This indicates that the system does not reach a thermal equilibrium between the bright and dark manifolds, likely due to suppressed inter-manifold scattering at low temperatures.

5. Significance

This work provides a robust solution to the "drift problem" in nanophotonics, enabling quantitative, absolute-free metrology of weakly radiative states.

• Practical Application: It allows for long-duration, dynamic monitoring of local dielectric environments and exciton populations without the need for complex external calibration or active stabilization.
• Broad Impact: The self-referenced dual-mode approach is generalizable to other nanogap platforms (plasmonic cavities, metasurfaces, hybrid resonators). It opens new avenues for:
  • Drift-tolerant population thermometry.
  • In-situ monitoring of strain and temperature variations in 2D materials.
  • Calibrated readout of other weak emitters, such as defect centers in diamond or molecular transitions, in complex environments.

By transforming a single-mode intensity measurement (prone to error) into a differential-mode ratio measurement (inherently stable), this study establishes a new standard for precision optical sensing in quantum and nanophotonic systems.