Symmetric Dicke States as Optimal Probes for Wave-Like Dark Matter

TL;DR

The paper tackles the challenge of detecting wave-like dark matter with networks of quantum sensors and shows that symmetric Dicke states provide the optimal probe in an ensemble-averaged metrological framework, delivering a robust $N_d^2$ enhancement for short-baseline arrays. It develops a detailed model incorporating stochastic DM phases, derives the ensemble-averaged QFI, and demonstrates that Dicke probes maintain their advantage under realistic amplitude damping, unlike GHZ states. For two detectors at coherence length, inter-detector correlations with a phase factor further boost precision beyond Dicke-level performance. The framework is broadly applicable to stochastic bosonic fields, including gravitational waves, and is compatible with current quantum platforms such as superconducting qubits, atomic ensembles, and NV centers, signaling a practical path toward noise-robust quantum-enhanced DM searches.

Abstract

We identify symmetric Dicke states as the optimal quantum probes for distributed sensing of wave-like dark-matter fields. Within an ensemble-averaged quantum-metrological framework that incorporates the field's random phases and finite coherence, they maximize the Fisher information for short-baseline arrays with $N_d$ sensors and realize a robust $N_d^2$ enhancement. They also retain this collective advantage under amplitude-damping noise, whereas GHZ-type probes are highly fragile and rapidly lose their sensitivity once such noise is included. For two sensors at separations comparable to the dark-matter coherence length, the optimal entangled state acquires an additional spatial-correlation phase and outperforms both Dicke and independent probes. Our framework applies broadly to stochastic bosonic fields, including gravitational waves, and can be implemented with superconducting qubits, atomic ensembles, and NV centers.

Figures (2)

• Figure 1: (a) Dicke state $|j=4,j_z=0\rangle$ for $N_d=8$ detectors depicted as a thin blue ring along the equator of the collective Bloch sphere. (b) The GHZ state appears as a series of blue rings that coherently connect the north and south poles of the Bloch sphere. It can be expanded in the Dicke basis as $\Sigma_{j_z} c_{j_z} |j,j_z\rangle$, where the vertical coordinate represents the magnetic quantum number $j_z$ and the ring radius encodes the amplitude $|c_{j_z}|$. (c) $N_d=8$ independent detectors at their ground states $|g_1g_2\dots g_8\rangle$, shown as a small blue ring near the south pole of the Bloch sphere. The DM field drives the states to the configuration shown in green. The red arrows show the effects of amplitude damping and phase damping on the detector states.
• Figure 2: Normalized QFI for three initial states: the optimal state $\ket{\psi}^{(\mathrm{max})}_{\mathrm{2-det}}$ (dashed), the symmetric Dicke state $\ket{\psi}^{(\mathrm{Dicke})}_{\mathrm{2-det}}$ (dotted), and two independent detectors $\ket{\psi}^{(\mathrm{ind})}_{\mathrm{2-det}}$ (solid). The QFI is normalized by that of a single detector, and the baseline $d$ is expressed in units of the DM coherence length $\lambda_c$.