NMR Basics

Many atomic nuclei have a magnetic moment, which means that, like small bar magnets, they respond to and can be detected by their magnetic fields. Although single nuclei are impossible to detect directly by these means with currently available technology, if sufficiently many are available so that their contributions to the magnetic field add, they can be observed as an ensemble. In liquid-state NMR, the nuclei belong to atoms forming a molecule, a very large number of which are dissolved in a liquid. An example is ${}^{13}C$-labeled trichloroethylene (TCE) (Fig. 1). The hydrogen nucleus (that is the proton) of each TCE molecule has a relatively strong magnetic moment. When the sample is placed in a powerful external magnetic field, each proton's spin prefers to align itself with the field. It is possible to induce the spin direction to ``tip'' off-axis by means of RF pulses, at which point the effect of the static field is to induce a rapid precession of the proton spins. In this introduction, precession refers to a rotation of a spin direction around the main axis, here the $z$-axis as determined by the external magnetic field. The precession frequency $\omega$ is often called the Larmor frequency and is linearly related to the strength $B$ of the external field: $\omega = \mu B$, where $\mu$ is the magnetic moment. For the proton, the magnetic moment is $42.7{\mathchoice{\mbox{Mhz}}{\mbox{Mhz}}{\mbox{\small Mhz}}{\mbox{\tiny Mhz}}}\,/{\mathchoice{\mbox{T}}{\mbox{T}}{\mbox{\small T}}{\mbox{\tiny T}}}\,$. ( ${\mathchoice{\mbox{Mhz}}{\mbox{Mhz}}{\mbox{\small Mhz}}{\mbox{\tiny Mhz}}}\,$ stands for ``megahertz'', which is a frequency unit equal to $10^6$ rotations per second. ${\mathchoice{\mbox{T}}{\mbox{T}}{\mbox{\small T}}{\mbox{\tiny T}}}\,$ stands for ``Tesla'', a magnetic field unit.) At a typical field of $B=11.7{\mathchoice{\mbox{T}}{\mbox{T}}{\mbox{\small T}}{\mbox{\tiny T}}}\,$, the proton's precession frequency is $500{\mathchoice{\mbox{Mhz}}{\mbox{Mhz}}{\mbox{\small Mhz}}{\mbox{\tiny Mhz}}}\,$. The magnetic field produced by the precessing protons induces oscillating currents in a coil judiciously placed around the sample and ``tuned'' to the precession frequency, allowing observation of the entire ensemble of protons by ``magnetic induction''. This is the fundamental idea of NMR. The device that applies the static magnetic field and RF control pulses and that detects the magnetic induction is called an NMR spectrometer (Fig. 2).

FIG. 1: Schematic of trichloroethylene, a typical molecule used for QIP. There are three useful nuclei for realizing qubits. They are the proton (H), and the two carbons (${}^{13}$C). The molecule is ``labeled'', which means that the nuclei are carefully chosen isotopes. In this case, the normally predominant isotope of carbon, ${}^{12}$C (a spin-zero nucleus), is replaced by ${}^{13}$C, which has spin ${1\over 2}$.

FIG. 2: Schematic of a typical NMR spectrometer (not to scale). The main components of a spectrometer are the magnet, which is superconducting, and the console , which has the electronics needed to control the spectrometer. The sample containing a liquid solution of the molecule used for QIP is inserted into the central core of the magnet, where it is surrounded by the ``probe''. The probe (shown enlarged in the insert to the right) contains coils for applying the radio frequency (RF) pulses and magnetic field gradients.

Magnetic induction by nuclear spins was observed for the first time in 1946 by the groups of E. Purcell [3] and F. Bloch [4]. This achievement opened a new field of research, leading to many important applications, such as molecular structure determination, dynamics studies both in the liquid and solid state [5], and magnetic resonance imaging [6]. The application of NMR to QIP is related to methods for molecular structure determination by NMR. Many of the same techniques are used in QIP, but instead of using uncharacterized molecules, specific ones with well-defined nuclear spins are synthesized. In this setting, one can manipulate the nuclear spins as quantum information so that it becomes possible to experimentally demonstrate the fundamental ideas of QIP.

Perhaps the clearest example of early connections of NMR to information theory is the spin echo phenomenon [7]. When the static magnetic field is not ``homogeneous'' (that is, it is not constant across the sample), the spins precess at different frequencies depending on their location in the sample. As a result, the magnetic induction signal rapidly vanishes because the magnetic fields produced by the spins are no longer aligned and therefore do not add. The spin echo is used to ``refocus'' this effect by inverting the spins, an action that effectively reverses their precession until they are all aligned again. Based on spin echoes, the idea of using nuclear spins for (classical) information storage was suggested and patented by A. Anderson and E. Hahn as early as 1955 [8,9].

NMR spectroscopy would not be possible if it were not for relatively long ``relaxation'' times. Relaxation is the process that tends to re-align the nuclear spins with the field and randomize their phases, an effect that leads to complete loss of the information represented in such a spin. In liquid state, relaxation times of the order of seconds are common and attributed to the weakness of nuclear interactions and a fast averaging effect associated with the rapid, tumbling motions of molecules in the liquid state.

Currently, ``off-the-shelf'' NMR spectrometers are robust and straightforward to use. The requisite control is to a large extent computerized, so most NMR experiments involve few custom adjustments after the sample has been obtained. Given that the underlying nature of the nuclear spins is intrinsically quantum mechanical, it is not surprising that, soon after P. Shor's discovery of the quantum factoring algorithm, NMR was studied as a potentially useful device for QIP.