Entanglement Witnesses of Condensation for Enhanced Quantum Sensing

TL;DR

The work addresses the challenge of surpassing the standard quantum limit in spin-based sensing by introducing a condensation-inspired entangled state of particle-hole pairs among $N$ spin qubits with strong dipole interactions. It formalizes an entanglement witness based on the largest eigenvalue $\lambda$ of the modified particle-hole RDM and links $\lambda^{1/2}$ to the spin-transition amplitude $A$, predicting an $\mathcal{O}(\sqrt{N})$ enhancement when a collective mode forms. The results show that $A$ and $\lambda$ increase with system size in the presence of dipole interactions, with the strongest effects when qubits are aligned along the microwave-propagation axis, and that 2D extensions and realistic noise modulate but do not wholly erase the enhancement. This work provides a design principle for robust, entanglement-assisted quantum sensing in spin-based platforms, including potential applications to molecular arrays and fluorescent proteins for improved ODMR contrast in noisy environments.

Abstract

Quantum phenomena such as entanglement provide powerful resources for enhancing classical sensing. Here, we theoretically show that collective entanglement of spin qubits, arising from a condensation of particle-hole pairs, can strongly amplify transitions between ground and excited spin states, potentially improving signal contrast in optically detected magnetic resonance. This collective state exhibits an $\mathcal{O}(\sqrt{N})$ enhancement of the transition amplitude with respect to an applied microwave field, where $N$ is the number of entangled spin qubits. We computationally realize this amplification using an ensemble of $N$ triplet spins with magnetic dipole interactions, where the largest transition amplitudes occur at geometries for which the condensation of particle-hole pairs is strongest. This effect, robust to noise, originates from the concentration of entanglement into a single collective mode, reflected in a large eigenvalue of the particle-hole reduced density matrix -- an entanglement witness of condensation analogous to off-diagonal long-range order, though realized here in a finite system. These results offer a design principle for quantum sensors that exploit condensation-inspired entanglement to boost sensitivity in spin-based platforms.

Figures (6)

• Figure 1: Schematic of model system. Three qubits, shown in gray, are spaced equidistant along the Y axis with spins aligned along the Z axis. The microwave, illustrated in blue, is applied along the Y axis (wavelength not to scale). Inter-qubit spacing can be scaled, and qubits can be rotated about the polar ($\theta$) and azimuthal ($\phi$) axes. While the zero field splitting parameter used in the model Hamiltonian corresponds to the NV center parameter in this work, similar models could easily be constructed to represent fluorescent protein or molecular lattice-based spin qubits.
• Figure 2: (a) Largest transition amplitudes (green) and $\lambda$ value for the corresponding excited spin state (blue) for three centers spaced at increasing distances from each other along the axis of microwave propagation. (b) Transition amplitude plotted against $\lambda$ for three centers with increasing interaction strengths spaced at decreasing distances from each other along the axis of microwave propagation. The best-fit curve is obtained via nonlinear least-squares fit, and shows a square-root dependence of transition amplitude on $\lambda$, as predicated by equation (\ref{['eq:lambdarad']}) .
• Figure 3: The (top) largest transition amplitude and (bottom) $\lambda$ value for the corresponding excited spin state for systems with increasing numbers of spins, with and without dipole interactions. Spins are spaced along the Y axis with 5.125 Å separation. For the non-interacting case, the maximum transition amplitudes and $\lambda$ values are completely unchanging with increasing numbers of qubits. In the presence of dipole interactions, both the transition amplitudes and the values of $\lambda$ display an approximately radical dependence on the number of qubits.
• Figure 4: Schematic of model system for probing 2D interactions. Spin qubits are spaced equidistant along the Y axis, with additional row of qubits added offset 5.125 Å in Z (or X) direction. The microwave, illustrated in blue, is applied along the Y axis (wavelength not to scale).
• Figure 5: (top) The largest transition amplitude for systems with increasing numbers of spins for the ZY and XY arrangements. (bottom) The $\lambda$ value corresponding to the largest transition $A$ and $\lambda$ values for a 1D chain of spins spaced along the Y axis with 5.125 Å separation are included for comparison.