Mathematics

The hamiltonian $\hat{H}$ we use for NMR systems is of the form

$$\begin{align} \hat{H} &= -\sum_\ell \gamma_\ell B^z \left( 1 + \delta_\ell\right) \hat{I}^z_\ell + 2 \pi\sum_{k < l} J_{kl} \mathbf{\hat I}_k \cdot \mathbf{\hat I}_l \end{align}$$

with gyromagnetic factors $\gamma_\ell$, chemical shifts $\delta_\ell$ of nuclear spin $\ell$, coupling between spins $k$ and $l$ denoted as $J_{kl}$, and $\hat{I}^\alpha = \hat{S}^\alpha / \hbar$, with $\hat{S}^\alpha$ being the usual $su(2)$ spin operators.

Within NMR, we have a strong magnetic field $B_z$ in the $z$-direction, and electromagnetic pulses / oscillating fields are applied to flip the spins into the $x/y$ plane. Since typically $B_z$ is of the order of 500Mhz, the pulses of 10kHz bandwidth, and the required resolution is sub 1Hz, we refrain from modeling the explicit time dependence of the pulses. Instead, we model the pulses directly by calculating the spectral function, i.e., time-dependent correlations between the corresponding $\hat{M}^\pm$ operators.

The spectrum measured in an NMR experiment corresponds to the spectral function, which is the Fourier transform of the correlation function of the operators $\hat{M}^\pm$, calculated by the quantum program created by the HQS Qorrelator App.

Calculation of the spectral function

The spectral function of the NMR problem is given by the Fourier transform

$$\begin{align} \\ \cal{G}(\omega) &= \mathcal{FT} G(t) \\ G(t) &= -i \langle \hat{M}^-(t) \hat{M}^+(0) \rangle \\ \hat{I}^\pm &= \hat{I}^x \pm i \hat{I}^y \\ \hat{M}^\alpha &= \sum_\ell \gamma_\ell \hat{I}^\alpha_\ell \end{align}$$

where the $\hat{M}^\alpha$ operators, $\alpha \in {x, y, z, +, -}$, contain the gyromagnetic factors for convenience, and $t$ is the real time dependence.

The contribution of an individual nuclear spin $\ell$ to the full NMR spectrum is obtained via

$$\begin{align} G_\ell(t) &= -i \langle \gamma_\ell \hat{I}_\ell^-(t) \hat{M}^+(0) \rangle, \end{align}$$

while the full NMR signal is the sum of individual contributions.