Above-Unity Coherent Cooperativity of Tin-Vacancy Centers in Diamond Photonic Crystal Cavities

TL;DR

The study demonstrates above-unity coherent cooperativity for tin-vacancy (SnV) centers coupled to diamond photonic crystal cavities (PCCs) using a quasi-isotropic undercut fabrication, achieving $C_{\mathrm{coh}} = 8.3 \pm 1.2$ and $C = 20.6 \pm 1.1$. Gas-tuning and cryogenic CQED measurements show strong light-matter interaction with cavity-transmission extinction up to $\Delta T \approx 0.988$ and strong Purcell-enhanced dynamics, validating SnV-PCC interfaces for quantum networks. The work confirms robust room-temperature fabrication yields ($Q \approx 1.0 \pm 0.3 \times 10^{4}$ across ~327 cavities) and detailed single-emitter coupling parameters ($g$, $\kappa$, $\gamma$) that underpin protocols for entanglement generation and photon-mediated processing. It also analyzes the external beta factor $\\beta_e$ and highlights design paths to approach near-unity coupling, including increased outcoupling and reduced intrinsic losses, which could enable high-rate, scalable quantum networking with solid-state spins.

Abstract

The tin-vacancy center in diamond (SnV) has emerged as a compelling building block for realizing next-generation quantum networks thanks to its excellent optical and spin properties. Coupling to photonic crystal cavities (PCCs) promises to further enhance the SnV light-matter interface and unlock a diverse range of entanglement generation protocols. Recent pioneering experiments showing Purcell enhancement of SnV centers in PCCs underscore this potential. However, optical coupling that is coherent - the key ingredient for use in quantum protocols - has so far remained elusive. Here, we demonstrate above-unity coherent cooperativity of SnV centers embedded in photonic crystal cavities. We fabricate free-standing PCCs using a quasi-isotropic undercut. Across two samples, we conduct room-temperature characterizations, measuring resonances for 327 cavities, with an average quality factor exceeding $Q = 1.0(3) \times 10^4$. Two cavity-coupled emitters are examined in detail, exhibiting quality factors up to $Q = 25.4(4) \times 10^3$ and Purcell-reduced lifetimes corresponding to cooperativities up to $C = 20.6(11)$. Furthermore, the single SnVs are observed to strongly modulate the cavity transmission with an extinction contrast up to $98.8(4) \%$ on resonance. Finally, SnV linewidth measurements reveal above-unity coherent cooperativities in both devices, with the highest value being $C_\mathrm{coh} = 8.3(12)$. These results open the door to using cavity-coupled SnV centers as efficient, coherent light-matter interfaces for future quantum networks.

Figures (17)

• Figure 1: Tin-vacancy centers in diamond photonic crystal cavities (PCC). (a) Schematic of the device. The PCC design is illustrated from left to right: strong mirror, defect region, weak mirror and taper end. We collect the light with a tapered optical fiber in a lensed configuration. (b) Simulated electric field of the fundamental cavity mode normalized to the maximum $u(\mathbf r)$, see Appendix \ref{['sec:cavity-simulations']}. (c) Simplified level structure of the combined emitter-cavity system. The emitter is described by a transition frequency $\omega_a$ between the ground state $|g\rangle$ and the excited state $|e\rangle$. The spontaneous emission rate is denoted $\gamma_0$ and $\gamma_\mathrm{dep}$ is the dephasing rate. The state $|c\rangle$ denotes a photon in the cavity mode and the emitter in the ground state. The cavity mode has a tunable frequency $\omega_c=\omega_a+\Delta$ and its total decay rate is $\kappa =\kappa_i+\kappa_e$. The coupling strength between the cavity and the emitter is $g$. (d) SEM image of a free-standing PCC device, taken at a 0° angle. The device regions (false colored) correspond to the schematic in (a). (e) SEM image of the diamond sample taken at a 65° angle. It displays multiple arrays of cavities, where each array contains six PCC devices.
• Figure 2: Room-temperature PCC characterization. (a) The cavity resonance is measured by taking a reflection spectrum with a white supercontinuum laser. The spectrum is fitted with a Lorentzian plus linear background, from which the resonance wavelength $\lambda_c$ and full width at half maximum (FWHM) $\Delta\lambda$ are extracted to determine the quality factor $Q = \lambda_c / \Delta\lambda$. (b) Cavity resonance vs. device scaling factor for Sample 1. We vary the geometrical parameters of the PCCs by a common scaling factor, changing the hole diameter, lattice constant and device width. Devices within the indicated tuning range can be brought into resonance with the SnVs zero-phonon line (ZPL) at 619nm by gas deposition. (Appendix \ref{['sec:gas-tuning']}). We use two different e-beam exposure doses, 265µCcm and 275µCcm, (dark and light blue, respectively). (c) Distribution of quality factors for Sample 1 (top panel) and Sample 2 (bottom panel). The average quality factor is $Q = 1.0 \pm 0.3\times 10^4$. Due to the limited spectrometer resolution of 0.021nm the quality factors reported are a lower bound.
• Figure 3: Cavity-emitter coupling. (a) Detected counts on a spectrometer as the cavity is tuned. This experiment is used to identify cavity-coupled SnVs. Initially, the cavity resonance is shifted to $\sim \! 619.5nm$ using gas deposition. Then, the cavity is tuned backwards by evaporating the deposited nitrogen with a green laser. When the cavity is on-resonance with the SnV (Emitter 2 indicated by the gray marks), it increases in brightness. The inset shows the spectrometer counts at this frequency ($618.96nm$) as a function of cavity detuning $\Delta$. (b) The SnV is excited off-resonantly with a pulsed green laser ($70ps$ pulse length). The collected ZPL counts are recorded on a timetagger and the result is fitted with a single exponential decay to obtain the lifetime $\tau$. (c) Lifetime as a function of detuning fitted with $\tau = \tau_0 / (C \, f(\Delta) + 1)$, leading to $C = 20.6 \pm1.1$.
• Figure 4: Coherent SnV-cavity interactions. (a) The collected light is split into ZPL and PSB using a dichroic mirror. (b) The cavity and SnV are tuned on resonance where $\Delta \equiv \omega_a - \omega_c = 0$. The red laser is scanned across the cavity resonance, PSB and ZPL are recorded as a function of the laser-SnV detuning $\Delta_a \equiv \omega - \omega_a$. The top panel shows the collected PSB counts: A Purcell broadened lineshape that is fitted by a Lorentzian with amplitude $\alpha$ and linewidth $\gamma^\prime$. The bottom panel shows the emitter-modulated cavity transmission reiserer_cavitybased_2015. For this detuning, Equation \ref{['eq:transmission']} predicts a symmetric dip. Fitting the data reveals a contrast of $\Delta T =0.988 \pm 0.004$. (c) When the SnV is in a dark state, we measure the Lorentzian transmission of a bare cavity, where $Q = 25.4 \pm 0.4\times 10^3$. (d) When the SnV is in the bright state, the cavity transmission is modulated by the interaction with the emitter. The transmission is shown for four different detunings $\Delta \in [0, 5, 12, 20] \, GHz$. (e) The top panel shows the emitter linewidth $\gamma^\prime$ as a function of detuning. Due to the Purcell effect, the linewidth broadens as the emitter and cavity are tuned on-resonance. The data is fitted with $\gamma^\prime = \gamma (f(\Delta) \, C_\mathrm{coh} + 1)$, and yields $C_\mathrm{coh} = 8.3 \pm 1.2$. Conversely, the bottom panel shows that the amplitude $\alpha$ reduces. This can be explained by a lower excited state population on resonance (Appendix \ref{['sec:cqed_modelling']}).
• Figure 5: Influence of the external coupling. (a) The external beta factor $\beta_e = \frac{\kappa_e}{\kappa}\frac{C}{C+1}$ as a function of the emitter location for two different external couplings $\kappa_e/\kappa \in \{0.21, 0.31\}$, corresponding to Cavity 1 and Cavity 2. Within each curve, we mark the point corresponding to the investigated emitters. (b) The external beta factor as a function of the loss rate into the waveguide $\kappa_e$ and other losses $\kappa_i$. The dashed lines mark a Purcell reduced lifetime of 0.5, 2 and 4 ns to a natural lifetime of $\tau_0 = 6.5ns$. The solid lines an external beta factor of $0.2, 0.5$ and $0.8$. A similar plot for the silicon vacancy center (SiV) can be found in knall_efficient_2022. (c) Calculated cavity reflection as a function of laser-cavity detuning $\Delta_c$ for four different emitter frequencies. The broad dip corresponds to the response of the cavity and the peak-dip pairs correspond to the emitters' response, similar to those found in Fig. \ref{['fig:4']}, but in reflection. The dashed black lines mark the envelope of the peaks and the dips. We use the C-QED parameters of Emitter 2.