Subradiant collective states for precision sensing via transmission spectra

TL;DR

The paper addresses precision sensing with quantum emitters by leveraging subradiant collective states to produce sharp features in transmission spectra. It develops a unified waveguide-QED and free-space array framework, showing that global and local frequency perturbations imprint narrow transmission features whose width scales as $\\Gamma_{\\text{sub}}$, enabling enhanced sensitivity and potential atomic clock applications. It analyzes two coupling schemes—near-Dicke-limit 1D waveguides and detuned dark-bright mode mixing in both 1D and 2D configurations—demonstrating site-resolved metrology under symmetry-breaking detunings and outlining robustness to imperfect coupling, motion, and missing atoms. The work provides quantitative precision estimates and shows that subwavelength arrays could approach optical clock performance with ultranarrow transitions, offering practical routes for high-precision sensing in current experimental platforms.

Abstract

When an ensemble of quantum emitters interacts with a common radiation field, their emission becomes collective, giving rise to superradiant and subradiant states, characterized by broadened and narrowed linewidths. In this work, we propose to harness subradiant states for quantum metrology; such states naturally arise in subwavelength-spaced atomic arrays in free space and in small ensembles of emitters coupled to one-dimensional waveguides. We demonstrate that their collective optical response yields sharp, narrow features in the transmittance spectrum, which can be used to enhance sensitivity to external perturbations. This improved sensitivity can be applied to atomic clock operation, spatially resolved imaging of emitter positions, and enables precise detection of both global and spatially varying detunings (such as those induced by electromagnetic fields or gravitational gradients).

Figures (8)

• Figure 1: Experimental setups. Schematic of metrological protocols based on collective light scattering. A weak probe laser illuminates (a) a one-dimensional array of emitters coupled to a waveguide, or (b) a two-dimensional atomic array in free space with a spatially-varying atomic frequency shift $\delta_0(x,y)$. The measured transmittance depends on the collective scattering properties of the emitters. Enhanced sensitivity arises from the narrowed linewidth of subradiant states. The system allows detection of frequency shifts induced by environmental perturbations and can operate as an atomic clock, where changes in the laser frequency manifest as a shift in the transmittance spectrum. In the latter case, the detuning $\delta$ inferred from the transmittance data is used in a feedback loop to adjust and stabilize the laser frequency.
• Figure 2: Two atoms coupled to a waveguide near the Dicke limit. We consider the system driven by a weak field detuned $\Delta_L$ from the atom frequency $\omega_0$. (a) is a level diagram depicting the collective ground state $\ket{G}$, and the one-excitation eigenstates ---the antisymmetric $\ket{A}$ and symmetric $\ket{S}$ states--- with their decay rates and Rabi frequencies. (b) Transmittance as a function of the laser detuning. The green line is the transmittance of two atoms separated $0.04\lambda_0$, which is very similar to the transmittance only due to the bright mode (dashed line) except for a narrow feature caused by the subradiant mode. The yellow line represents the transmittance when an additional detuning $\delta=0.2\Gamma_{1D}$ is applied to both atoms. The narrow peak provides high sensitivity to global detunings. (c) Maximum of the sensitivity with respect to the laser detuning as a function of spacing $a$ between atoms. For comparison, the dashed line shows the maximum sensitivity attainable by a single atom.
• Figure 3: Probing dark states via spatially dependent detunings. We consider systems that have a perfectly dark state, such as two atoms coupled to a waveguide in the Dicke limit and an infinite 2D array in free space. In both cases, as depicted in the diagram (a), the dark mode $\ket{D}$ will be coupled to a bright mode $\ket{B}$ by applying a control space-dependent detuning $\delta_0$. The bright mode has a decay rate $\Gamma_B$ and Rabi frequency $\Omega_B$. $\ket{G}$ is the collective ground state. (b) Transmittance as a function of the laser detuning for two atoms coupled to a waveguide in the Dicke limit with detunings $\pm\delta_0$ (in green), and for a 2D array in free-space with spacing $a = 0.55\lambda_0$ and periodic detuning $\delta_0(\boldsymbol{r}) = \delta_0 \cos(\pi/a \cdot (x+y))$ (in yellow). We choose $\delta_0 = 0.1\Gamma_0$. The dashed line is the transmittance of the two atoms in the Dicke limit without detuning. (c) Maximum sensitivity of an infinite 2D array, optimized over all laser detunings $\Delta_L$, as a function of the lattice spacing, shown for different values of the detuning amplitude.
• Figure 4: Sensitivity to spatially dependent perturbations. We demonstrate that applying an asymmetric control detuning enables site-resolved sensing in atomic arrays coupled to a waveguide. Panels (a) and (b) present the sensitivity of a 4-atom chain to (a) spatially varying detunings and (b) atomic displacements, as a function of the lattice spacing $a$, for different choices of control detuning: no detuning, $\delta_0(r) = 0$ (blue); periodic detuning, $\delta_0(r) = 0.1 \sin(\pi r / N a)$ (yellow); and linear detuning, $\delta_0(r) = 0.1 / [(N{-}1)a] r$ (red). The case with no detuning is included for comparison, though it is not experimentally feasible because the transmittance spectrum is not injective and thus cannot uniquely resolve local perturbations.
• Figure 5: Sensitivity of two atoms coupled to a waveguide including experimental errors. Maximum sensitivity as a function of interatomic distance $a$. (a) Effect of imperfect atom-waveguide coupling for different values of the decay into non-guided modes, $\Gamma'$. (b) Effect of atomic motion for different values of zero-point motion, $\sigma$. The dashed lines correspond to the case of a single atom coupled to the waveguide.