Coherent momentum control of forbidden excitons

A double-edged sword in two-dimensional material science and technology is optically forbidden dark exciton. On the one hand, it is fascinating for condensed matter physics, quantum information processing, and optoelectronics due to its long lifetime. On the other hand, it is notorious for being optically inaccessible from both excitation and detection standpoints. Here, we provide an efficient and low-loss solution to the dilemma by reintroducing photonics bound states in the continuum (BICs) to manipulate dark excitons in the momentum space. In a monolayer tungsten diselenide under normal incidence, we demonstrated a giant enhancement (~1400) for dark excitons enabled by transverse magnetic BICs with intrinsic out-of-plane electric fields. By further employing widely tunable Friedrich-Wintgen BICs, we demonstrated highly directional emission from the dark excitons with a divergence angle of merely 7°. We found that the directional emission is coherent at room temperature, unambiguously shown in polarization analyses and interference measurements. Therefore, the BICs reintroduced as a momentum-space photonic environment could be an intriguing platform to reshape and redefine light-matter interactions in nearby quantum materials, such as low-dimensional materials, otherwise challenging or even impossible to achieve.


Optical microscope images of the devices
: Optical microscope images of the devices. The few-layer h-BN flake and WSe2 monolayer were transferred on the suspended photonic crystal slabs with parameters of (a) periodicity, a = 450 nm, radius of holes, r = 140 nm, and thickness of the slab, h = 265 nm, and (b) periodicity, a = 510 nm, radius of holes, r = 102 nm, and thickness of the slab, h = 233 nm, respectively. Fig. 3: Schematic of experimental setups for optical measurements. (a) the optical band-structure mode can be achieved using an image lens L3 (f = 50 mm) to project the kspace image to the spectrometer. (b) the spectra analysis mode can be achieved using a Fourier lens L4 (f = 100 mm) to Fourier transfer the k-space image to the real-space image to the spectrometer. Supplementary Fig. 4 shows the optical band structures of the device with parameters of periodicity, a = 510 nm, radius of holes, r = 102 nm, the thickness of the slab, h = 233 nm. Supplementary Fig. 4a shows the simulated result using COMSOL whereas Supplementary Fig.  4b shows the measured result using our angle-resolved reflection spectroscopy. It is clear to see the two bands avoid crossing at a wavelength of 772 nm and kx/k = 0.74 (the equivalent incident angle was 47.85°) at both simulated and measured optical band-structure mappings.

Supplementary Fig. 4:
The comparison between (a) the simulated and (b) the measured optical band-structure of the photonic crystal slab with the parameters of periodicity, a = 510 nm, radius of holes, r = 102 nm, and thickness of the slab, h = 233 nm. The index of the materials (Si3N4) is 2.23.

The on-Γ BIC for dark exciton brightening in the photonic crystal device
that supported the avoid-crossing mode BICs.
In the manuscript, we used two different devices, the first one supports a TM mode on-Γ BIC at a wavelength of 694 nm for dark exciton brightening and the second one supports a TM mode on-Γ BIC at a wavelength of 600 nm for dark exciton brightening and the avoided crossing mode BICs to couple the out-of-plane oriented dipole of the dark exciton and directionally emit the dark exciton PL signal. We firstly demonstrated that the TM mode on-Γ BIC can brighten the dark exciton with a device that just supports the on-Γ BIC and is left blank at the dark exciton energy. Next, the second device supported two types of different BICs and can brighten and directionally emit simultaneously. Supplementary Fig. 5a shows the comparison of the optical band structure between the simulation and the measurement, and they show minimal differences. It is clear to see the on-Γ BIC at a wavelength of 600 nm has a vanished line shape, indicating its infinite Q-factor. The outof-plane oriented electric field distribution mappings in the X-Y plane where the transferred WSe2 monolayer was located and, in the X-Z plane are shown in Supplementary Fig. 5b. The highly enhanced Ez has large efficiency in brightening the dark excitons at room temperature.

The PL emission polarization analysis of the bright and dark excitons.
In order to analyze the PL emission polarization, we draw a schematic to illustrate the polarization mapping of the radiation pattern at the k-plane for the out-of-plane and in-plane dipoles, respectively. The out-of-plane dipole should yield a radical polarization distribution at the k-plane because of its rotational symmetry along the axis of the objective lens (Z-direction) as shown in Supplementary Fig. 6a. On the other hand, the corresponding polarization distribution for the in-plane dipole should be along the same direction (Supplementary Fig. 6b).
Supplementary Fig. 6: The polarization distribution in the k-plane of the PL signal radiated by (a) the out-of-plane oriented dipole (dark excitons) and (b) the in-plane oriented dipole (bright excitons).

Formation of the Friedrich-Wintgen BIC
While situated within the energy range of a radiative continuum, BICs possess distinctive characteristics (symmetry, polarization, and so on), forming a momentum-space or phase singularity that makes them intrinsically non-radiative. A plausible way to understand BICs is through destructive interference/coupling between different resonances, modes, or radiative channels. To illustrate the formation of the FW-BIC, we simulate and analyze the normalized outof-plane electric field (Ez) distribution of the two constituting transverse magnetic (TM) modes  Fig. 8c). In this way, we can compress more energy into the BIC mode (at 694 nm) to gain a higher enhancement factor and Ez converter efficiency. On the other hand, we used a 0.2X beam expander (GBE05-A, Thorlabs) to squeeze the laser beam in a smaller volume to compress more energy near 0° to match the on-Γ BIC mode.
Further, we calculated the average enhancement factor of the |Ez/E0| 2 at the top surface of the PhC slab (periodicity, a = 450 nm, radius of holes, r = 140 nm, thickness of the slab, h = 265 nm). Note that E0 is x polarized. We used the ratio between the integral of |Ez| 2 at the top surface of the PhC slab and |E0| 2 at the same position in the air (removing the PhC slab in simulation). The integral range in wavelengths is 692.5 nm to 695.5 nm and in K-space is -2° to 2°. We assumed the focused laser spot has a gaussian profile and used the rotational symmetry approximation (Supplementary Fig. 1). We also calculated the average |Ex/E0| 2 and average |Ey/E0| 2 and found that the ratio between |Ez/E0| 2 and |Ex/E0| 2 is ~30.5 where the ratio between |Ex/E0| 2 and |Ey/E0| 2 is ~5.2. The large |Ez/E0| 2 over |Ex/E0| 2 ratio means most of the laser energy is in the out-of-plane direction and the large |Ex/E0| 2 over |Ey/E0| 2 ratio means the photonic crystal maintains the in-plane linear polarization.
Supplementary Fig. 8: The Incident laser beam focused on the BIC. (a) schematic of the light pass with a short pass edge filter placed with a twist angle (edge at 694.8nm) and 0.2X beam expander which were used to compress the laser beam in k-space and energy space. The laser can be compressed into +/-2° in the k-space as shown in the red region in (b), the angle-resolved reflection spectra mapping. (c) The laser can be narrowed in energy space with an FWHM of 3 nm.

The calculation of the Purcell factor of the avoid-crossing BIC
The Purcell factor of the dark excitons was calculated using a commercial finite-difference time-domain (FDTD) simulation software (Lumerical). We randomly put a series of out-of-plane

Collection efficiency of the PL emission radiated by bright excitons and dark excitons
In this section, we would like to discuss the collection efficiency of the PL emission radiated by bright excitons and dark excitons. Because of their out-of-plane radiating nature, dark excitons PL emits primarily towards the in-plane direction. Even if a sufficiently large numerical aperture objective lens (as high as 0.9) was used, we still cannot collect the dark exciton emission with high efficiency. Supplementary Fig. 10: Collection efficiency of the PL emission radiated by bright excitons (red) and dark excitons (blue) using a 40X objective lens with a numerical aperture of 0.9.

Bright exciton PL emission decoherence at room temperature
Although the coherence of the bright exciton emission in WSe2 is well studied at both cryogenic and room temperature 1-3 . We measured the monolayer WSe2 PL pattern in k-space under p-polarized excitation. Supplementary Figs. 11a, b, and, d show the PL pattern without an analysis polarizer and with co-and cross-direction analysis polarizers. We extracted the intensity profiles from the PL patterns under co-and cross-polarized conditions, respectively ( Supplementary Fig. 11b)

Interference setup using a cylindrical lens
Inspired by the cylindrical lens enabled interference setup for demonstrating vortex beam 4 , we used the setup as shown in Supplementary Fig. 12 to demonstrate our dark excitons directional emission is coherent, i.e., emission spots with opposite phase have destructive interference pattern ( Fig. 5b in manuscript). The setup is based on our previous spectra analysis mode setup, and we just replaced the L4 Fourier Lens in Supplementary Fig. 3b with a cylindrical lens (focal length is 100 mm). In this way, we can make the directional emission spots that toward opposite directions overlap and then can observe their interference pattern.
Supplementary Fig. 12: Interference setup enabled by a cylindrical lens. Spatially distributed light waves passing through a cylindrical lens possess spatial interference patterns detectable by a spectrometer. The cylindrical lens (focal length of 100 mm) replaced the Fourier lens L4 with a focal length of 100 mm in Supplementary Fig. 3b to enable interference.

Optical band structure measurements using halogen lamp and supercontinuum laser
In order to demonstrate the 100-by-100 periods suspend photonic crystal is sufficiently large to guarantee the bound states in the continuum (BICs), we used the supercontinuum laser as the light source to observe the optical band structure. Supplementary Fig. 13 shows the comparison of measured optical band structure using a halogen lamp and supercontinuum laser. The measured optical band structures in Fig. 2b, Supplementary Fig. 4, and Supplementary Fig. 5a were obtained using a halogen lamp and we have checked the real-space iris to make sure the spectrometer received only the light reflected by the photonic crystal device (Supplementary Fig.   13a). As a comparison, we used a supercontinuum laser to carry out a similar measurement. Thanks to the high coherence of the supercontinuum laser, the laser beam can be focused into a very small volume (~10 μm, Supplementary Fig. 13b). Supplementary Fig. 13c shows the optical band

The measured refractive index of LPCVP Si3N4 film.
We used the commercially available silicon nitride /silicon wafer (Silicon Valley Microelectronics, SVM). The 300nm thick silicon nitride layer was deposited on the silicon wafer with low-pressure chemical vapor deposition (LPCVD). They claimed the silicon nitride film thickness was 3,000 Å (300 nm) and the refractive index is 2.30 +/-0.05 at 632.8 nm. In addition, the refractive index of silicon nitride with excellent stress control should be around 2.2-2.3 5 .
Supplementary Fig. 15 shows the Si3N4 refractive index as a function of the wavelength.
Supplementary Fig. 15: The measured refractive index of LPCVP Si3N4 film.

Influence of the WSe2 monolayer
Although the WSe2 (or other WSe2-like 2D materials) monolayers are semiconductors, their image part of refractive index ( ) is not negligible which may bring large damping for photonic devices. We carried out the comparison simulations for the photonic crystal with and without the WSe2 monolayer. Because of the big dimension mismatch between the thickness of WSe2 (~0.7 nm) and the periodic of photonic crystal (450 nm), the numerical simulation (COMSOL and FDTD) requires large RAM to render the mesh for accurate calculations. We estimated two different oblique angles (1° and 3°) in this simulation. Even with a high-end computer (it has two Intel Xeon Gold 6150 CPUs and 384 GB DDR4 RAM), simulation of one spectrum still requires several days. Supplementary Fig. 16b shows the reflection spectra comparisons for photonic crystals with and without the WSe2 monolayer. The introduction of the thin layer would make the Fano resonance redshift for about 1.5 nm. The refractive index of WSe2 was extracted from Ref. 6 . This redshift can be compensated by our strategy discussed in Supplementary section 17 ( Supplementary Fig. 17). The biggest problem is the large damping of the monolayer will break the Z-symmetry of the photonic crystal and result in a largely reduced Q-factor (Supplementary

Figs. 16c and d).
In this work, the on-Γ BIC converts the normal incident laser with in-plane polarization into near-field energy with out-of-plane polarization and has a giant enhancement. We further checked the conversion efficiency of the degenerated on-Γ BIC to make sure the lower Qfactor resonances still can serve as the efficient convertor. Supplementary Fig. 16e shows the absolute value of out-of-plane electric field (Ez) distributions when the incident angle is 1°. It is worth noting that the two Fano resonances shown in Supplementary Fig. 16e have different wavelength scales and we just want to label the positions for each E-field distribution. Although the Fano resonance in the case without WSe2 has higher E-field intensity (the highest intensity at the 3 rd position, we used its maximum intensity to do the normalization for each E-field distribution), its smaller peak width has a lower chance to enhance the incident laser (5 nm width in wavelength). We use the equation shown below to compare the conversion efficiencies of two cases.
where the 0 demotes the center wavelength of the pump, Δ is the wavelength width of the pump, ave( ) denotes the average of the out-of-plane electric field intensity in the X-Y plane at the top surface of the photonic crystal, and 1 or 2 are the relative dielectric constant for each case (with or without WSe2). The ratio R (1°, with over without WSe2) is 2.53, and the ratio R (3°) is 0.76 when the wavelength width of the pump is 5 nm, indicating the lower Q-factor has higher overall covert efficiency, counterintuitively.
In summary, the WSe2 monolayer has a relatively large image part of the refractive index, which will largely reduce the Q-factor of the on-Γ BIC. It is true that a higher Q-factor of the Fano resonance has a larger electric field intensity, the sharp resonance peak, however, will reduce the coupling efficiency between resonance and pump laser, which would decrease the overall efficiency.
Supplementary Fig. 16: The integration of the WSe2 monolayer influences the photonic crystal. (a) schematic of the simulation. (b) reflection spectra of a photonic crystal with and without WSe2 monolayer cases, respectively. A 1.5 nm red shift will be introduced for the integration of WSe2. (c, d) Q-factors of photonic crystal devices for with and without WSe2 monolayer cases, respectively. (e) Electric field distributions when the incident angle is 1°.

Influence of h-BN flake and compensation strategy
Supplementary Fig. 17 shows the comparison between the designed structure and the h-BN flake on top with compensation. Thanks to the refractive index of h-BN being close to the silicon nitride, this strategy works very well and only a 0.15 nm difference in the resonance wavelength was observed. We found that the Z-symmetry was slightly broken by the h-BN layer but the Fano resonances with high-Q-factor were still supported (Supplementary Fig. 17c). To note that the

Modal overlap between on-Γ BIC and FW-BIC
We estimated the modal overlap (also known as the mode-matching factor) between on-Γ BIC and FW-BIC used in this work. We used the eigenfrequency model in COMSOL to extract the modes of the two BICs and used the equation below for the modal overlap estimation.
= | ∫ 1 * * 2 | 2 ∫ | 1 | 2 * ∫ | 2 | 2 We would like to note that E1 and E2 are absolute values instead of the original value 7 . In the normal case for modal overlap, the energy transfer would happen to two different modes directly. However, in this case, the WSe2 monolayer serves as the medium to connect the two modes. Specifically, excitons can be excited because of the energy in the first mode and then recombined resulting in a photoluminescence (PL) signal. The PL signal is coupled with the second mode and can be directionally emitted. In the whole process, only the absolute value of the E-field contributes to the mode coupling whereas the phase of the E-field does not. Thus, we used the absolute value of the E-field for the modal overlap calculation. The distribution of the absolute value of the E-field for two modes is shown in Supplementary Fig. 18. The modal overlap is 45.73%. Supplementary Fig. 18: Electric field distributions in the X-Y plane at the top surface of the photonic crystal device of on-Γ BIC and FW-BIC.

Comparison of different dark excitons brightening methods.
Compared with current state-of-the-art dark excitons brightening methods, the method in this work has both high brightening efficiency and high collection efficiency, thus the high overall efficiency. The method in this work also demonstrated room temperature and CMOS compatibility.