Multi-ensemble Superradiance for Distributed Quantum Sensing

TL;DR

This work develops a driven multi-ensemble superradiant framework that intrinsically generates inter-ensemble entanglement through collective dissipation, producing dark-state metastructures (MEDS) whose properties depend on initialization. It provides analytical covariance matrices for MEDS in the large-N limit via Holstein–Primakoff mapping and introduces the multiparameter squeezing coefficient as a variational tool to optimize distributed, multiparameter metrology. A spin-based multimode interferometry protocol is formulated, showing how optimal estimation of linear combinations of parameters is achieved by aligning the measurement direction with the minimum-eigenvalue eigenvector of the spin-squeezing matrix, with explicit results under uniform coupling. The study also analyzes finite-size effects and scaling, highlighting practical relevance beyond the thermodynamic limit and offering a concrete route toward multimode quantum interferometry in driven-dissipative quantum systems.

Abstract

Multi-ensemble superradiance extends Dicke superradiance to multiple ensembles and supports dark states whose properties depend on the initial state. In the large-\(N\) limit, we derive analytical covariance matrices for these dark states, revealing inter-ensemble entanglement that enhances quantum metrology. The minimum eigenvalue, determined by the curvature of the superradiance potential, corresponds to the optimal multiparameter spin-squeezing coefficient, which is given by the \emph{Rayleigh quotient} of the spin-squeezing matrix, linking metrological sensitivity to the geometric structure of the underlying dynamics. The multiparameter squeezing coefficient provides a variational framework for optimizing metrological performance. These results enable optimal estimation of arbitrary linear combinations of multiple parameters, offering a concrete protocol for distributed quantum sensing and a promising route toward multimode quantum interferometry.

Figures (4)

• Figure 1: (a) Schematic of a driven multi-ensemble superradiant system. A total of $N$ identical particles are distributed into $M$ spatially separated ensembles, which undergo collectively driven-dissipative dynamics. (b) Spin-based multimode quantum interferometry. The unknown parameters $\theta_1, \ldots, \theta_M$ are encoded onto the probe state MEDS via the unitary operator $\hat{U}(\boldsymbol{\theta}) = \exp(-i \boldsymbol{\theta}^\top \hat{\mathbf{S}}^X)$. The parameters are probed by measuring the observables $\hat{S}_1^Y, \ldots, \hat{S}_M^Y$ ,and subsequently classically post-processed to infer a global parameter of interest. (c) $M$ Bloch spheres with mode entanglement. For the $j$th Bloch sphere ($j=1,2,\ldots,M$), the transverse spin components $S_j^X$ and $S_j^Y$ lie in the plane perpendicular to the Bloch vector $\mathbf{S}_j$, with $S_j^X$ chosen to be parallel to the global $x$ axis. The Bloch vector $\mathbf{S}_j$ forms an angle $c_j\theta$ with respect to the $-z$ axis, where $\theta$ is a common parameter shared by all Bloch spheres. The squeezed modes within each ensemble, together with the metrologically useful inter-ensemble entanglement, enhance the measurement precision.
• Figure 2: Superradiance potential $V(\theta)$ and curvature $\mathcal{C}(\theta)$ for the two-ensemble superradiant system, with $\eta_1=\eta_2 = 1/2$, $c_1 = 3/5$, and $c_2 = 4/5$. In the undriven multi-ensemble superradiant system, stability is achieved only at discrete values of $\theta$. However, with the introduction of a driving field, a steady state can be reached via the drive–dissipation balance within the stable region.
• Figure 3: (a) The minimum eigenvalue $\lambda_{\mathrm{min}}$ of the spin-squeezing matrix as a function of curvature $\mathcal{C}$ for different particle numbers. (b) The logarithm of $\lambda_{\mathrm{min},N}$ as a function of the logarithm of particle numbers $N$. Here, $\lambda_{\mathrm{min},N}$ denotes the smallest value of $\lambda_{\mathrm{min}}$ for all curvatures at a given $N$. The exponent $\alpha$ is defined via $\lambda_{\mathrm{min},N} \propto N^{\alpha}$. $\alpha = 0$ and $\alpha = -1$ correspond to the standard quantum limit (SQL) and the Heisenberg limit (HL), respectively.
• Figure 4: Graphical representation of the minimum eigenvalue of $\bm{\Gamma}_Q^a(\infty)$.