Differential magnetometry with partially flipped Dicke states

TL;DR

The paper develops a quantum metrology framework for gradient and homogeneous-field magnetometry using two spatially separated spin ensembles. By applying local flips to Dicke states, it constructs partially flipped Dicke states that are highly sensitive to magnetic gradients, achieving Heisenberg scaling while enabling entanglement-assisted improvements over separable states. It derives sharp precision bounds for gradient estimation, analyzes both optimal and practical measurement strategies, and demonstrates robustness to partitioning noise. The results show how Dicke-state metrology can be harnessed for accurate, quantum-enhanced multiparameter estimation in a two-well setting with feasible measurement schemes.

Abstract

We study magnetometry of gradients and homogeneous background fields along all three spatial axes using two spatially separated spin ensembles. We derive trade-off relations for the achievable estimation precision of these parameters. Dicke states, optimal for homogeneous field estimation, can be locally rotated into states sensitive to magnetic gradients by rotating the spins in one subensemble. We determine bounds for the precision for gradient metrology in the three orthogonal directions as a function of the sensitivities of the homogenous field in those directions. The resulting partially flipped Dicke state saturates the bounds above, showing similar sensitivity in two directions but significantly reduced sensitivity in the third. Exploiting entanglement between the two ensembles, this state achieves roughly twice the precision attainable by the best bipartite separable state, which is a product of local Dicke states. For small ensembles, we explicitly identify measurement operators saturating the quantum Cramér-Rao bound, while for larger ensembles, we propose simpler but suboptimal schemes. In both cases, the gradient is estimated from second moments and correlations of angular momentum operators. Our results demonstrate how the metrological properties of Dicke states can be exploited for quantum-enhanced multiparameter estimation.

Figures (2)

• Figure 1: Polytopes for different quantum states showing the allowed values of ${{F}} [\varrho, J_{l,{\rm{a}}}{-}J_{l,{\rm{b}}}]$ for $l=x,y,z$, for given values of ${{F}} [\varrho, J_{l,{\rm{a}}}{+}J_{l,{\rm{b}}}]$ for $l=x,y,z$ and for $N=8$ particles, based on the set of inequalities Eq. \ref{['eq:gradient-planes']}. (red), (green) and (blue) polygons correspond to planes obtained by Eqs. \ref{['eq:x-plane']}, \ref{['eq:xy-plane']} and \ref{['eq:xyz-plane']}, respectively. (yellow ball) Actual values of ${{F}} [\varrho, J_{l,{\rm{a}}}{-}J_{l,{\rm{b}}}]$ for $l=x,y,z$ for the given quantum state. (a) Dicke state. The ball is inside the cuboid, indicating that the Dicke state does not saturate any of the bounds. (b) GHZ state, saturating the inequality \ref{['eq:xy-plane']} for $i,j,k=x,y,z$ respectively. (c) Partially flipped Dicke state, saturating the inequality \ref{['eq:xy-plane']} for $i,j,k=x,y,z$ respectively. (d) Partially flipped GHZ state, saturating the inequalities \ref{['eq:xy-plane']} for $i,j,k=y,z,x$ and $i,j,k=z,x,y$, and Eq. \ref{['eq:x-plane']} for the $z$-direction. For the partially flipped Dicke state, the ball is closer to the blue plane than in the case of the flipped GHZ state, as suggested by Eq. \ref{['eq:f-dicke-better']} (For details, see text). We compute the values obtained in each case in Appendix \ref{['app:polytope-expectation-values']}.
• Figure 2: (a-d) Reconstruction of the optimal operator $\mathcal{M}_{\rm opt}$ for $N_{{\rm{a}}}=N_{\rm{b}}=2,4,6,8,$ respectively. (transparent-boxes) Zero elements. (gray-boxes) Nonzero matrix elements. (gray-lines) Boundaries for the different $j_y$ subspaces for the $J_{y,{\rm{a}}}{+}J_{y,{\rm{b}}}$ angular momentum. The operator is a block-diagonal operator in the $J_{y,{\rm{a}}}{+}J_{y,{\rm{b}}}$ basis. All the non-zero elements are in the 3rd, 5th, 7th, 9th, etc. diagonal blocks and they are imaginary, while we have zero in the first and the last blocks, which are of size $1\times1.$