Ph.D. Thesis - Physics

where V0 is the maximum signal strength and ρ is the sample density matrix. The phase of
the measurement operator can be chosen so that a given spin state produces a given spectral
line shape: for instance, for the above operator −iXk − Yk, a spin along −ˆy produces a
positive absorptive spectral line, while one along ˆy produces a negative absorptive line. A
spin along ±ˆx produces positive or negative dispersive lines. To perform this measurement
in the ˆz basis, we must first rotate all spins into the ˆx-ˆy plane, since the rf coils are only
sensitive to magnetizations in this direction. Thus a Rˆx(π/2) pulse is first applied to the
spins being measured.
The resulting free-induction decay (FID) signal oscillates at the frequency of the qubit
being measured; a Fourier transform of this signal yields a frequency spectrum, from which
one can determine the spin state along ˆz of each nucleus (using the chemical shift interaction,
Eq. 2.4). The FID decays exponentially at a rate T ∗ 2 , due to decoherence processes which
will be discussed shortly. The exponential decay in the time domain leads to a Lorentzian
lineshape in the frequency domain:
V (t)e −t/T ∗ 2 ↦→ F(ω) ∝
1
1/(2T ∗ 2 )2 1
−
+ (ω − ω0) 2 1/(2T ∗ 2 )2 , (2.21)
+ (ω − ω0) 2
where the two parts represent absorptive and dispersive lineshapes. A signal may, in general,
be of only one type or a mixture of the two. The full width at half-maximum (FWHM) of
the NMR peaks is then given by ∆f = 1/(2πT ∗ 2 ).
Although measurements in quantum information are often assumed to be strong, pro-
jective measurements, those in NMR are quite weak. The constant T ∗ 2
depends on magnetic
field inhomogeneities and other interactions between spins and with the environment; it is
noteworthy that the decay of the FID is dominated not by the interaction of the sample
with the coil, but by decoherence processes more intrinsic to the sample. This may be
contrasted with the classic “strong” measurement, in which the very act of measurement
induces sufficient decoherence to immediately effect wavefunction collapse! This weakness
of the measurement, combined with the ensemble nature of the experiment, is also the rea-
son why both ˆx and ˆy components of the magnetization can be simultaneously measured, a
feat which is forbidden (for a single quantum system) by the uncertainty principle. What
is recorded is the ensemble average of the magnetizations in each direction; each of the 10 18
or so molecules gives its own answer regarding its spin state, and these are added up.
The result of a measurement is a frequency-domain spectrum, with peaks located at all
possible locations for a given nucleus, given the various splittings due to the chemical shifts.
To summarize this section so far, we have assembled all the basic concepts and techniques
for performing quantum operations in solution-state NMR. In the next section we show how
these principles apply to the simulation of quantum systems. First, though, we address the
sources of decoherence in NMR systems.
55