Magnetic resonance in quantum computing and in accurate measurements of the nuclear moments of atoms and molecules

TL;DR

This work derives an exact, general rotating-wave solution for coupled nuclear and electronic spins under a rotating magnetic field, yielding closed-form wave functions that support arbitrary state transitions and quantum computing primitives. By rotating to a doubly-rotated frame, the authors obtain a time-independent bare Hamiltonian $\hat{H}_0$ and enable perturbative treatment of nuclear moments through $\hat{\delta V}_{ne}=\sum_{q=0}^{2I}K_q\big({\mathbf I}\cdot{\mathbf J}/\hbar^2\big)^q$, facilitating precise NMR/EPR measurements of nuclear moments. The paper provides explicit results for simple systems (e.g., $I=1$, $J=1/2$) and discusses numerical behavior for $\tfrac{1}{2}\le I,J\le 2$, including time-dependent and period-averaged occupation probabilities near resonance, and outlines a practical experimental framework to measure all seven moments of $^{133}$Cs. The approach promises high-precision nuclear-moment determinations and a versatile platform for quantum information processing with spin systems. The combination of exact spin dynamics, perturbative multipole handling, and near-resonant control offers a significant advance for magnetic resonance-based quantum computing and metrology.

Abstract

Modern experimental techniques can generate magnetic fields of the form H(t) = H0 z-hat + H1 [x-hat cos(ωt) + y-hat sin(ωt)], at frequencies within an order of magnitude of the nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) frequencies, ωn0 and ωe0, respectively, when acting on atoms or molecules. We derive simple closed-form expressions for the exact nuclear- and electronic-spin wave functions that enable controlled transitions between entangled states, allowing an atom or molecule to function as a quantum computer. These solutions also enable precise NMR or EPR measurements of nuclear moments in atoms and molecules. We present examples relevant to measurements of the nuclear moments of 14N, 7Li, and 133Cs. Because existing hyperfine measurements of the lowest three nuclear moments of 133Cs are mutually inconsistent, the proposed NMR/EPR experiments provide a route to measuring all seven of its nuclear moments with high precision.

Figures (10)

• Figure 1: (Color online) Plots of $P_{1/2}(\tau)$ (dotted red) and $P_{-1/2}(\tau)$ (solid blue) for $I=\frac{1}{2}$ and $0\le\tau=t/T_{n}\le1$. In (a), (c), and (e), $C_{1/2}(0)e^{-i\pi/4}=C_{-1/2}(0)=\frac{1}{\sqrt{2}}$. In (b), (d), and (f), $C_{1/2}(0)=1$ and $C_{-1/2}(0)=0$. In (a) and (b), $\omega/\omega_{n0}=0.95$, in (c) and (d), $\omega/\omega_{n0}=0.99$, and in (e) and (f), $\omega/\omega_{n0}=1$. This figure also applies for $J=\frac{1}{2}$ with $\tau=t/T_e$ and $\omega_{n0}$ and $C_m(0)$ respectively replaced by $\omega_{e0}$ and $D_{\overline{m}}(0)$.
• Figure 2: (Color online) Plots of $P_1(\tau)$ (dotted black), $P_0(\tau)$ (dotted red) and $P_{-1}(\tau)$ (solid blue) for $I=1$ and $0\le\tau=t/T_n\le1$. In (a), (c), and (e), $C_0(0)=C_{-1}(0)=e^{-i\pi/4}C_1(0)=\frac{1}{\sqrt{3}}$. In (b), (d), and (f), $C_1(0)=1$ and $C_0(0)=C_{-1}(0)=0$. In (a) and (b), $\omega/\omega_{n0}=0.95$, in (c) and (d), $\omega/\omega_{n0}=0.99$, and in (e) and (f), $\omega/\omega_{n0}=1$. This figure also applies for $J=1$ with $\tau=t/T_e$ and $\omega_{n0}$ and $C_m(0)$ respectively replaced by $\omega_{e0}$ and $D_{\overline{m}}(0)$.
• Figure 3: (Color online) Plots of $P_{3/2}(\tau)$ (dotted black), $P_{1/2}(\tau)$ (dotted red), $P_{-1/2}(\tau)$ (solid blue), and $P_{-3/2}(\tau)$ (dotted blue) for $I=\frac{3}{2}$ and $0\le\tau=t/T_{n}\le1$. In (a), (c), and (e), $C_{-\frac{3}{2}}(0)=C_{-\frac{1}{2}}(0)=C_{\frac{1}{2}}(0)=e^{-i\pi/4}C_{\frac{3}{2}}(0)=\frac{1}{2}$. In (b), (d), and (f), $C_{\frac{3}{2}}(0)=1$ and $C_m(0)=0$ for $m\ne\frac{3}{2}$. In (a) and (b), $\omega/\omega_{n0}=0.95$, in (c) and (d), $\omega/\omega_{n0}=0.99$, and in (e) and (f), $\omega/\omega_{n0}=1$. This figure also applies for $J=\frac{3}{2}$ with $\tau=t/T_e$ and $m$, $\omega_{n0}$ and $C_m(0)$ respectively replaced by $\overline{m}$, $\omega_{e0}$, and $D_{\overline{m}}(0)$.
• Figure 4: (Color online) Plots of $P_{2}(\tau)$ (solid black), $P_{1}(\tau)$ (dotted black), $P_{0}(\tau)$ (dotted red), $P_{-1}(\tau)$ (solid blue), and $P_{-2}(\tau)$ (dotted blue) for $I=2$ and $0\le\tau=t/T_{n}\le1$. In (a), (c), and (e), $C_{-2}(0)=C_{-1}(0)=C_{0}(0)=C_{1}(0)=e^{-i\pi/4}C_{2}(0)=\frac{1}{\sqrt{5}}$. In (b), (d), and (f), $C_{2}(0)=1$ and $C_m(0)=0$ for $m\ne2$. In (a) and (b), $\omega/\omega_{n0}=0.95$, in (c) and (d), $\omega/\omega_{n0}=0.99$, and in (e) and (f), $\omega/\omega_{n0}=1$. This figure also applies for $J=2$ with $\tau=t/T_e$ and $m$, $\omega_{n0}$, and $C_m(0)$ respectively replaced by $\overline{m}$, $\omega_{e0}$, and $D_{\overline{m}}(0)$.
• Figure 5: (Color online) Plots of $P_m(\tau)$ for $0\le\tau=t/T_n\le 1$. (a), (c), (e): $I=1$. (b), (d), (f): $I=2$. In both cases $C_0(0)=1$. Plots of $P_0(\tau)$ (solid blue), $P_1(\tau)$ (solid red), and $P_{-1}(\tau)$ (dot-dashed black). For $I=2$, $P_2(\tau)$ (dashed red), and $P_{-2}(\tau)$ (dotted black). In (a) and (b), $\omega/\omega_{n0}=0.95$, in (c) and (d), $\omega/\omega_{n0}=0.99$, and in (e) and (f), $\omega/\omega_{n0}=1$. This figure also applies for $J=1,2$ with $\tau=t/T_e$ and $m$, $\omega_{n0}$, and $C_m(0)$ respectively replaced by $\overline{m}$, $\omega_{e0}$, and $D_{\overline{m}}(0)$.