Multipartite Entanglement and Quantum Sensing in a Spin-5/2 Heisenberg Molecular Iron(III) Triangle

TL;DR

The paper investigates a trinuclear high-spin Fe$_3$ molecular magnet as a concrete spin-$\frac{5}{2}$ Heisenberg triangle. Using exact diagonalization, it analyzes ground-state level crossings, magnetization plateaus, and entanglement quantified by bipartite and genuine tripartite negativities, revealing field-driven step-like entanglement behavior robust up to $\sim$27.5 K (bipartite) and $\sim$70 K (tripartite). It then adapts a quantum sensing protocol by initializing the system in Dicke states and applying a local field $B_x$, showing synchronized dynamics between sensor and readout spins and demonstrating quantum-enhanced precision via sequential measurements, with $F^{-1}$ shrinking approximately as $\alpha n_{\mathrm{seq}}^{-\beta}$ ($\beta>1$). Overall, Fe$_3$ emerges as a promising platform for quantum information processing and high-sensitivity quantum metrology with molecular magnets, including remote readout capabilities and potential extensions to other transition-metal clusters.

Abstract

This study provides insights into the static and dynamic quantum properties of the trinuclear high-spin iron(III) molecular complex $[\mathrm{Fe}_3\mathrm{Cl}_3(\mathrm{saltag^\mathrm{Br}})(\mathrm{py})_6]\mathrm{ClO}_4$ to be further abbreviated as Fe$_3$. Using exact diagonalization of a spin-5/2 Heisenberg triangle in a magnetic field, we model the corresponding quantum behavior of the molecular compound Fe$_3$. Our rigorous analysis employs various key metrics to explore a rich quantum behavior of this molecular compound. At sufficiently low temperatures, the bipartite negativity reveals that the pairwise entanglement between any pair of iron(III) magnetic ions of the molecular complex Fe$_3$ can be significantly enhanced by a small magnetic field. This enhancement is followed by unconventional step-like changes characterized by a sequence of plateaus and sudden downturns as the magnetic field further increases. A qualitatively similar behavior is also observed in the genuine tripartite entanglement among all three iron(III) magnetic ions in the trinuclear complex Fe$_3$. Notably, the bipartite and tripartite entanglement persist in the molecular complex Fe$_3$ up to moderate temperatures of approximately 30~K and 70~K, respectively. Additionally, we demonstrate the achievement of quantum-enhanced sensitivity by initializing the molecular complex Fe$_3$ in Dicke states. Finally, we investigated a quantum-sensing protocol by applying a local magnetic field specifically to one iron(III) magnetic ion of the molecular compound Fe$_3$ and performing readout sequentially on one of two remaining iron(III) magnetic ions.

Figures (6)

• Figure 1: Left panel displays a scheme of three exchange interactions within the trinuclear high-spin iron(III) molecular complex $[\mathrm{Fe}_3\mathrm{Cl}_3(\mathrm{saltag}^\mathrm{Br}) (\mathrm{py})_6] \mathrm{ClO}_4$. Right panel shows the molecular structure of the same complex visualized according to crystallographic data reported in Ref. Plass2023.
• Figure 2: The isothermal magnetization curves of the spin-5/2 Heisenberg triangle calculated for a few selected temperatures by assuming the coupling constant $J = 12.56\, \mathrm{cm}^{-1}$ and the gyromagnetic ratio $g = 2.0$. High-field magnetization data as reported in Ref. Plass2023 for the molecular complex Fe$_3$ at sufficiently low temperature $T = 1.8\, \mathrm{K}$ are depicted by green circles. The ground-state eigenvectors $|S_\mathrm{T}\rangle$ corresponding to intermediate magnetization plateaus are indicated by arrows.
• Figure 3: (a) Magnetic-field dependencies of the bipartite negativity ${N}_\mathrm{Bip}$ for a few different values of temperature; (b) temperature dependencies of the bipartite negativity ${N}_\mathrm{Bip}$ for a few selected values of the magnetic field. In both panels we present the theoretical prediction of the bipartite negativity ${N}_\mathrm{Bip}$ for any pair of high-spin iron(III) magnetic ions within the molecular complex Fe$_{3}$ based on the spin-5/2 Heisenberg triangle, which is given by the Hamiltonian (\ref{['H']}) with the coupling constant $J = 12.56 \, \mathrm{cm}^{-1}$ and the gyromagnetic factor $g = 2.0$ as determined from the previous experiments Plass2023.
• Figure 4: (a) Magnetic-field dependencies of the tripartite negativity ${N}_\mathrm{Trip}$ for a few different values of temperature; (b) temperature dependencies of the tripartite negativity ${N}_\mathrm{Trip}$ for a few selected values of the magnetic field. In both panels we present the theoretical prediction of the tripartite negativity ${N}_\mathrm{Trip}$ for any pair of high-spin iron(III) magnetic ions within the molecular complex Fe$_{3}$ based on the spin-5/2 Heisenberg triangle, which is given by the Hamiltonian (\ref{['H']}) with the coupling constant $J = 12.56 \, \mathrm{cm}^{-1}$ and the gyromagnetic factor $g = 2.0$ as determined from the previous experiments Plass2023.
• Figure 5: The time evolution of the local magnetization of the first and third spin of the spin-5/2 Heisenberg triangle with the coupling constant $J = 12.56 \, \mathrm{cm}^{-1}$ and the gyromagnetic factor $g = 2.0$ adjusted for a theoretical modeling of the molecular complex Fe$_3$Plass2023 by considering four different initial Dicke states: (a)--(b) $| D_0 \rangle$; (c)--(d) $| D_1 \rangle$; (e)--(f) $| D_2 \rangle$; (g)--(h) $| D_3 \rangle$. In the left panel we suppose a lower magnetic field $B_x = 1 \,\mathrm{T}$, while in the right panel we consider a higher magnetic field $B_x = 5\,\mathrm{T}$. Both spins exhibit almost identical spin dynamics.