Quantum Simulation of NMR Spin Echo

Project Title: Monte Carlo Simulation of NMR Signal Recovery

Problem Statement: When running an NMR spectroscopy experiment on a chemical sample, since each nuclei is subject to a slightly different alignment magnetic field, the spins of the nuclei become unaligned and the measured magnetic moment decays to zero, a phenomenon called the free induction decay. In this project, we demonstrate how a 180° RF pulse can recover the original signal.

Method: We generated a Monte Carlo simulation of 1000 nuclei, each receiving a constant alignment magnetic field of strength drawn from a Gaussian distribution of mean 1 T and standard deviation of 0.2 T to simulate realistic conditions where each nuclei in a sample receives a slightly different alignment field strength due to geometrical and shielding variances. An RF pulse of strength 10 T was applied perpendicular to that of the constant magnetic field for

seconds. The simulation was then run for an additional 30 seconds during which the aggregate magnetic field generated by the nuclear spins was measured along the axis of the applied RF pulse. This time period allowed for the observed magnetic field to decay 0 due to the slight variances in the alignment field causing each nuclear spin to destructively interfere with each other.

After these 30 seconds, a second RF pulse of the same strength (10 T) was applied for seconds and the simulation was run for another 60 seconds during which the aggregate magnetic field was again measured.

Results: The following plot shows the aggregate observed magnetic field strength along the axes of the applied RF pulse.

Note that for the first 30 seconds we observe a decaying, oscillating signal that eventually dies out due to the destructive interference between the nuclear spins. However, applying a second 180 degree pulse at 30 seconds reverses this phenomena and temporarily recovers the original signal. A more intuitive understanding of this phenomena can be found through observing the average angular momentum vector from each nucleus. While the angular momentum wavefunction is not localized to a single point, by taking the average of the angular momentum probability distribution, we can visualize the effects of the two pulses in a classical sense. The following movie shows the angular momentum vectors of five randomly chosen nuclei along with the recorded NMR signal and applied RF field strength.

The variances in the causes the individual angular momentum vectors to become dis-aligned due to their varying procession frequencies. The second 180 degree pulse causes the angular momentum to "reverse" their procession causing the prior dis-alignment to be similarly reversed, allowing for the magnetic field signal to be read again.