Entanglement-enhanced quantum sensing via optimal global control with neutral atoms in a cavity

TL;DR

This work presents a deterministic, noise-aware protocol for preparing entangled states in the symmetric Dicke subspace of $N$ spins coupled to a common cavity mode, enabling entanglement-enhanced sensing under realistic dissipation. The core approach combines a cavity-driven geometric phase gate with an analytic treatment of the noisy quantum channel and but also employs optimal control to sculpt multi-step pulse sequences that drive the system into GHZ-like or Dicke $N/2$-like probe states. Across a range of cavity cooperativities $C$ and system sizes up to $N\approx 100$, the resulting metrological precision surpasses the SQL and approaches the Heisenberg limit under favorable conditions, even in the presence of photon loss, spontaneous emission, and dephasing. The framework is validated with both analytic results and full master-equation simulations for neutral-atom cavity-QED platforms and is adaptable to other spin-boson systems, highlighting a practical route to scalable quantum-enhanced sensing.

Abstract

We present a deterministic protocol for the preparation of entangled states in the symmetric Dicke subspace of $N$ spins coupled to a common cavity mode that prepares entangled states useful for quantum sensing, achieving a precision significantly better than the standard quantum limit in the presence of photon cavity loss, spontaneous emission and dephasing. The protocol combines a new geometric phase gate which can be utilized for exact unitary synthesis on the Dicke subspace, an analytic solution of the noisy quantum channel dynamics and optimal control methods. This work opens the way to entanglement-enhanced sensing with cold trapped atoms in cavities and is extendable to other spin systems coupled to a bosonic mode.

Figures (10)

• Figure 1: (a) A register of spins with states $\{|0\rangle, |1\rangle, |e\rangle\}$ is coupled to a cavity mode with coupling strength $g$ addressing the $\ket{1}\leftrightarrow \ket{e}$ transition, with detuning $\Delta- \delta$. The cavity mode is externally driven by a laser with amplitude $|\eta(t)|$, and a global laser pulse is applied on the $\ket{0} \leftrightarrow \ket{1}$ spin transition. Panels (b1,b2): Cavity drive pulses of the optimal state preparation protocol for $N= 40$, $C= 10^4$ and $\gamma/\kappa= 0.01$, for GHZ-like and Dicke-like states, respectively. Throughout, we make a choice of the cavity drive pulse $\zeta(t)$ in the effective frame with $\mathrm{Re}(\zeta(t))= - 2 \delta \sqrt{\frac{2\phi}{3 \delta T}} \sin^{2}(\frac{\pi t}{T})$ and $\mathrm{Im(\zeta(t))}= - \partial_{t}\mathrm{Re}(\zeta(t))/\delta$ (see inverting_zeta_note and PhysRevA.110.062610). The obtained minimal measurement precision variances here are $N (\Delta \beta)^2_{\mathrm{GHZ}} = 0.03$ and $N (\Delta \beta)^2_{N/2} = 0.08$. The parameters used in optimal state preparation protocol are listed in the Supplemental Material a_suppM_note. (c1, c2): State trajectories in Husimi-Q representation of the spin states in the symmetric Dicke subspace after the application of each protocol step $j\, \forall j= 1, \dots, P$. (d) Optimal $(\Delta \beta)^2_{\mathrm{GHZ}}$ for $P=1$ and (e) $(\Delta \beta)^2_{N/2}$ for $P=3$ obtained as a function of number of qubits $N$, plotted for spin-cavity cooperativities $C= 25$ with $\gamma/\kappa= 1$, and $C= 10^2, 10^4, 10^6$ with different ratios $\gamma/\kappa = 0.01, 0.1, 1, 10, 100$, obtained for the case of $gT \rightarrow \infty$. The optimal states prepared in the presence of finite $C$ successfully surpass the SQL for single particle cooperativity values as small as $C=25$, which has already been achieved in Ref. grinkemeyerErrordetectedQuantumOperations2025.
• Figure 2: Measured $(\Delta \beta)^2$ as a function of dimensionless signal acquisition time $Jt$ by evolving the optimal probe states under a field coupled with the spins with coupling strength $J$ with local homogeneous dephasing acting on the spins with rates $\gamma_{\phi}/J= 0, 0.01, 0.1, 1.0$ for $N=40$, $C= 10^4$, $\gamma/\kappa = 1.0$. Green solid lines (darker shade for larger $\gamma_{\phi}$) correspond to GHZ-like states while red dash-dot lines correspond to the $\ket{\mathcal{D}^{N}_{N/2}}$-like states. Dotted black curves are the optimal $(\Delta \beta)^2$ obtained with analytic solution of $\mathcal{E}_{\mathrm{gpg}}$ for $\gamma_{\phi}/J = 0$.
• Figure S1: Estimates of cavity photons loss rate $\kappa$, atom- cavity coupling strength $g$ and single particle cooperativity $C$ as a function of system size $N$. Additionally we plot the ratios $\sqrt{N}g/\nu_{\rm fsr}$ (green circle markers) and $Ng^2/(\gamma \nu_{\rm fsr})$ (red diamond markers) to check the validity of the Tavis Cummings model for moderate sized systems of neutral atom spin qubits with $^{87}$Rb coupled to optical FPFC cavity mode.
• Figure S2: Minimum $gT$ required to implement a geometric gate $\hat{U} = e^{i\phi \hat{n}_1^2}$ plotted as a function of the geometric phase $\phi$ such that a $\delta$ exists in accordance with the inequality $|\zeta(t)|< g/2$, see Eq. \ref{['eq:delta_range']}.
• Figure S3: (a) Optimal $(\Delta \beta)^2_{\mathrm{GHZ}}$ for $P=1$ step obtained as a function of Cooperativity $C$ , plotted for $N= 10$ and different ratios $\gamma/\kappa$. The circle markers correspond to the results obtained with the application of unoptimised pulses referring to the pulses which prepare the ideal GHZ state with $(\Delta \beta)^2_{\mathrm{GHZ}}= 1/N^2$ for the case $\kappa= \gamma=0$. (b) $(\Delta \beta)^2_{N/2}$ for $P=1, 2, 3, 5$ steps obtained as a function of Cooperativity $C$ , plotted for $N= 10$ and different ratios $\gamma/\kappa$.