Quantum Entanglement Control in Two-Spin-1/2 NMR Systems Through Magnetic Fields and Temperature

TL;DR

This work analyzes thermal entanglement in a two-spin-1/2 NMR dimer under external magnetic fields. By deriving closed-form expressions for the concurrence and a universal energy-level crossing criterion, it identifies a threshold temperature below which entanglement persists and reveals a zero-temperature quantum critical point driven by field–coupling competition. The authors connect entanglement to directly measurable NMR observables, enabling experimental quantification and control of quantum correlations in realistic spin systems. The results have practical implications for quantum-enhanced NMR, low-temperature spectroscopy, and emerging quantum technologies, providing a framework to engineer and diagnose quantum correlations via field and temperature tuning.

Abstract

We investigate quantum entanglement in two-spin-1/2 NMR systems at thermal equilibrium under external magnetic fields. We derive closed-form analytical expressions for the entanglement of the system and show how the entanglement depends on temperature and magnetic field strength, resulting in a threshold temperature beyond which entanglement vanishes. We demonstrate that at zero temperature, the system exhibits a quantum critical point, characterized by non-analytic behavior in the measure of entanglement. We further develop analytical criterion for level crossing, which serves as a condition for identifying quantum critical points in both homonuclear and heteronuclear systems, and apply it to multiple settings to analyze their quantum critical points. We establish a direct link between the quantum entanglement quantifier and experimentally accessible NMR observables, enabling entanglement to be quantified through NMR signal processing. This provides a practical framework for characterizing quantum correlations using standard NMR experiments. These findings provide insights into the thermal control of quantum features, with implications for quantum-enhanced NMR, low-temperature spectroscopy, and emerging quantum technologies.

Figures (6)

• Figure 1: (a) Zeeman energy levels of a two-spin-1/2 system, with energy values $\mathrm{E}_1$, $\mathrm{E}_2$, $\mathrm{E}_3$, and $\mathrm{E}_4$. (b) Corresponding NMR spectrum, where frequency increases from right to left. The frequency differences are represented by dashed lines. Signal intensities follow the "roofing effect", given by $1 \pm \sin 2\theta$.
• Figure 2: Concurrence as a function of temperature for different dimensionless ratio $\omega_{\Sigma}/\boldsymbol{J}$. Panels (a), (b), and (c) correspond to $\omega_{\delta}/\boldsymbol{J} = 0$ (a homonuclear system), $\omega_{\delta}/\boldsymbol{J} = 1$ (a heteronuclear system), and $\omega_{\delta}/\boldsymbol{J} = 2.5$ (a heteronuclear system), respectively. The re-scaled temperature parameter is $\tau = k_B T /\boldsymbol{J}$ in Eq. (\ref{['concurrence_General']}), and the threshold temperature is $T_t$.
• Figure 3: Concurrence as a function of the normalized frequency ratio $\omega_{\Sigma}/\boldsymbol{J}$ for varying re-scaled temperatures. Panels (a), (b), and (c) correspond to $\omega_{\delta}/\boldsymbol{J} = 0$ (a homonuclear system), $\omega_{\delta}/\boldsymbol{J} = 1$ (a heteronuclear system), and $\omega_{\delta}/\boldsymbol{J} = 2.5$ (a heteronuclear system), respectively, demonstrating how temperature influences concurrence.
• Figure 4: Quantum signatures of ground-state transitions in (left-panel) homonuclear and (right-panel) heteronuclear spin systems. The concurrence $C$ sharply drops at the critical point at low temperature ($\tau = 0.01$). Bottom panels show simulated NMR spectra for mixing angle $\theta = 45^\circ$ (homonuclear) and $\theta = 30^\circ$ (heteronuclear), with flip angle $\varphi = 5^\circ$.
• Figure 5: Field-dependent behavior of a homonuclear system (${}^{1}\text{H}-{}^{1}\text{H}$ system). The magnetic field strength increases from zero-field to low-field and finally reaches high-field. A level crossing occurs between $\mathrm{E}_3$ and $\mathrm{E}_4$.