Ph.D. Thesis - Physics

2.1.7 Decoherence
Decoherence, broadly speaking, is an uncontrolled evolution of a quantum state into a
classical mixture of pure states. It is caused by interactions which are outside the control of
the experimentalist, and involve noisy or random processes. Decoherence is an ever-present
limitation to the amount of time over which quantum operations may be performed, and
also to the fidelity of the operation. It is characterized by two time constants, T1 and T2,
which govern the decay of the populations (T1) and coherences (T2) of the system density
matrix. Taking ρ as the system density matrix, these decoherence channels can be described
using the following density matrix transformation model:
where
a b
b ∗ 1 − a
→
(a − a0)e −t/T1 + a0
be −t/T2
b ∗ e −t/T2 (a0 − a)e −t/T1 + 1 − a0
ρ0 =
a0
0
0 1 − a0
, (2.22)
(2.23)
is the thermal density matrix. Naturally, it is beneficial to get T1 and T2 as large as possible.
This can be done by understanding and correcting for some of the sources of decoherence,
which form two broad classes: macroscopic and microscopic.
The macroscopic sources of noise normally involve large-scale variations in the elec-
tromagnetic fields that interact with the nuclei. Inhomogeneities in B0 and B1 lead to a
type of decoherence known as inhomogeneous broadening, from the fact that the T2 value
is increased, leading to a larger spectral linewidth. The quantity that includes both inho-
mogeneous broadening and other effects is called T ∗ 2 , and is used above in discussing the
lineshape. Other macroscopic sources of noise are electromagnetic fluctuations, which may
arise from noisy amplifiers, and radiation damping, which occurs when the magnetic mo-
ment of the spins induces a voltage in the rf coils, which then create a magnetic field that
itself rotates the spins.
These sources are easier to deal with than the microscopic ones discussed below. Spin-
echo techniques can reduce the effect of inhomogeneous broadening, and in fact allow one
to measure the field inhomogeneity by comparing T2 and T ∗ 2 . Also, starting with as uniform
a B0 field as possible allows one to improve T ∗ 2
without spin-echo techniques. A technique
called shimming allows us to do this (see Sec. 3.4.4). Also, the sample is spun around the
ˆz axis at about 20 Hz, which averages out inhomogeneities in the ˆx-ˆy plane. With respect
to the other two problems, electromagnetic noise can be mitigated by “blanking” the rf
amplifiers when pulses are not being applied, and radiation damping can be reduced either
by reducing the Q value of the resonant circuit or by making the sample more dilute.
The microscopic sources have been discussed in detail in Ref. [Lev01], and we give some
important examples here. One set of decoherence sources is the various dipole-dipole inter-
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