Dynamical Decoupling using NMR for
Quantum Computing
A thesis submitted for the degree of Doctor rerum naturalium
at the Faculty of Physics at TU Dortmund
Mustafa Ahmed Ali Ahmed
May 14, 2013
First referee: Prof. Dr. Dieter Suter
Second referee: Prof. Dr. Götz S. Uhrig
NMR probes are very expensive so extremely care is important

Contents
1. Introduction 7
2. NMR basics 11
2.1. Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1. The nuclear Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2. Precession and Larmor condition . . . . . . . . . . . . . . . . . . . 13
2.1.3. Density matrix operator . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.4. Radio frequency pulses . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.5. Laboratory frame and rotating frame of reference . . . . . . . . . . 14
2.1.6. Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2. Introduction to Quantum computation . . . . . . . . . . . . . . . . . . . . 17
2.2.1. Quantum Bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2. Single and multiple qubit gate . . . . . . . . . . . . . . . . . . . . . 18
2.3. Liquid state NMR Quantum computing . . . . . . . . . . . . . . . . . . . . 19
2.3.1. Quantum computer in DiVincenzo view . . . . . . . . . . . . . . . 19
2.3.2. Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3. Experimental setup 21
3.1. Spectrometer configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1. Controlling the 360 MHz spectrometer . . . . . . . . . . . . . . . . 21
3.1.2. The transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.3. The RS690 word generator . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.4. The Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.5. Alternative 1H-channel demodulator/modulator . . . . . . . . . . . 28
3.1.6. The probe-head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.7. Shimming the 360 MHz spectrometer . . . . . . . . . . . . . . . . . 33
3.1.8. Noise reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.8.1. Incoherent noise . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.8.2. Coherent noise . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2. Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1. Measurement of relaxation times T1 . . . . . . . . . . . . . . . . . . 40
3.2.2. Radiation damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.3. Pure water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.4. Tune up sequences and pulse parameters optimization . . . . . . . . 44
3.2.5. Pulse visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4. Dynamical decoupling 51
4.1. Fighting against decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2. Our system and noise sources . . . . . . . . . . . . . . . . . . . . . . . . . 52
5
Contents
4.3. How to construct a robust DD sequence? . . . . . . . . . . . . . . . . . . . 53
4.4. Initial experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5. FID and Hahn echo sequence . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.6. Dynamical decoupling sequences . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6.1. Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.6.2. Robustness comparison under pulse errors . . . . . . . . . . . . . . 61
4.7. Engineering the environment by external modulation field . . . . . . . . . 65
4.7.1. Experimental modification to generate additional field perturbation 65
4.7.2. Simulation of the FID signal during the sinusoidal modulation of
the static magnetic field . . . . . . . . . . . . . . . . . . . . . . . . 67
4.7.2.1. Simulation procedure . . . . . . . . . . . . . . . . . . . . . 67
4.7.2.2. Simulation results versus experimental data . . . . . . . . 69
4.7.3. Dynamical decoupling pulse sequence performance under sinusoidal
modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7.4. Rf coil inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.8. Theoretical analysis of dynamical decoupling sequences by average Hamil-
tonian theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.8.2. Average Hamiltonian theory calculations for dynamical decoupling
sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.8.2.1. XY4(s) and XY4(a) pulse sequence . . . . . . . . . . . . . 75
4.8.2.2. CPMG and CP pulse sequence . . . . . . . . . . . . . . . 75
4.8.2.3. KDDx pulse sequence . . . . . . . . . . . . . . . . . . . . 77
5. Conclusion and outlook 79
5.1. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2. Five-spin system for dynamical decoupling . . . . . . . . . . . . . . . . . . 80
A. Signal check-up protocol in the 360 MHz Spectrometer 83
B. Interpreter pulse program: an example for multiple NMR experiments 85
C. Interpreter pulse program: an example for a single NMR experiment 87
Acknowledgments 95
6
1. Introduction
Why quantum computers?
Quantum computers can execute certain tasks more efficiently than classical computers
by processing information in a system governed by the laws of quantum mechanics. The
basic unit of quantum computation is a qubit. A qubit is typically a microscopic system
such as an atom or spin or photon [1]. A quantum mechanical two-level system (like a
spin-1/2) can be used as a computational basis by identifying their eigenstates with these
values, e.g. |0〉 for spin up and |1〉 for spin down.
One of the main advantages of quantum computers in contrast to classical computers
is "quantum parallelism". Quantum parallelism is based on the superposition principle
in quantum mechanics. The number of basis states for N qubit system is 2N . Unlike
in classical computers these states are established in superposition states. The number
of parallel function evaluations grows exponentially with the number of qubits in quan-
tum computers, whereas it increases linearly with the number of input bits in classical
computers. This powerful computation algorithm, implemented in a quantum computer,
makes it possible to solve problems which are hard or impossible for classical computers
such as the factorization of large numbers Preskill [2], Vandersypen et al. [3], DiVincenzo
[4].
Simulating quantum systems in a classical computers is awkward because the required
resources grow exponentially with the system size (e.g. superposition states do not exist in
classical computers). Inspired by Fredkin’s results about reversible computations, Feyn-
man thought in the opposite direction, that quantum simulations can be performed in a
system built of quantum mechanical elements that obey quantum laws. Thus quantum
computers can compute information more efficiently than any classical device [5, 6, 2, 7, 8].
NMR quantum computation
The principle of nuclear magnetic resonance (NMR) spectroscopy, where the nuclear
magnetic moment is directly measured experimentally and nuclear precession is described,
was first discovered by Rabi et al. in 1938. It has been fifty-nine years after this discov-
ery, then NMR was realized as a quantum computational system by two groups working
in liquid-state NMR. In that initial demonstrations the concept of bulk quantum com-
puting is introduced. Bulk quantum computing depends on addressing the ensemble of
approximately 1023 spins as one qubit [9, 10, 11].
The fascinating variety indiscipline fields of quantum mechanics in atomic physics,
quantum optics, NMR, electron spin resonance ESR, superconducting physics and quan-
tum dot research are connected to the quantum physics and quantum computation. NMR
quantum information processing (QIP) was the first approach for quantum information
systems listed by the advanced research and development activity (ARDA) based on their
underlying experimental physics sub-fields, besides other approaches (such as trapped
ions, trapped neutral atoms, cavity quantum electrodynamic, photons as optical qubits,
7
solid-state such as impurity spins and spins in quantum dots and superconducting circuits
and “unique” qubits (e.g. electrons on liquid helium, spectral hole burning etc.)) [12]. The
NMR quantum computation approach started as good testbed for the most conceptually
advanced quantum information processors developments and is expected to remain in fu-
ture applications Suter and Mahesh [8], Stolze and Suter [5], DiVincenzo [4], Vandersypen
et al. [3].
If all these spins were distinguishable the result would be a massive quantum computer.
However, partial distinguishability occurs naturally in molecules due to their electronic
environments. The ability of the fine control of these nuclear spins which is implemented
in NMR spectrometers through radio frequency pulses enables us to construct pulse se-
quences for quantum computation. This results in a great success achieved recently by
NMR quantum information processing [7, 8, 11]. Another advantage of spin 1/2 qubits is
that it exactly realizes a qubit in its two dimensional Hilbert space [8]. Since experimental
systems have similar problems in quantum control, ideas and concepts from liquid-state
NMR are adopted in a variety of other quantum computational implementations. On the
other hand quantum computing also provides a new powerful and systematic perspective
in NMR technique such as using Uhrig Dynamical Decoupling sequence (UDD) to improve
the image quality in magnetic resonance imaging (MRI) for some tissue types [2, 13]. In
hybrid systems like single nitrogen-vacancy defect (NV) center, it is used as an atomic-size
magnetic field sensor to detect NMR from nanoscale samples, this technique is based on
measuring the statistical polarization of the spin population near the NV center with a
dynamical decoupling method [14].
In quantum information processing and quantum memory applications, it is very impor-
tant to keep the information isolated from the environment: uncontrolled interactions with
the environment tend to degrade the quantum information. This environment-induced
loss of quantum information is called ‘decoherence’ [15, 16, 17]. Decoherence not only
limits the time for which quantum information can be used, but also the distance over
which quantum information can be transferred [18]. Successful quantum computations
are scaled with the decoherence time. Avoiding decoherence is simply done by choosing
systems with long decoherence times compared to gate times [4, 19]. Towards implement-
ing a fault-tolerant quantum computer in a physical system, DiVincenzo suggested five
requirements as discussed in NMR point of view in Sec. (2.3.1). In Ref. [20] the the
DiVincenzo’s criteria is generalized into
1. Scalability : the system should be able to grow in the Hilbert space exponentially
without exponential cost of resources.
2. Universal logic : this follows the DiVincenzo’s criterion where the system in the
large Hilbert space should be accessible through a finite set of control operation.
3. Correctability : in order to make operations in the initialized computer state, it
must be possible to remove the errors introduced by the influence from the outside
world of the quantum computer.
The criteria required the ability of fast switching between measurement "on" to readout
the results and "off" during computation to avoid decoherence. Therefore quantum compu-
tation looks hard because qubits should strongly interact with each other and not interact
with the environment until we measure them. In practical systems coherent control pro-
8
CHAPTER 1. INTRODUCTION
cesses suffer from imperfections in the control field of the qubits. These imperfections can
limit the computer’s performance more than decoherence itself [20, 8].
Dynamical decoupling
One of the biggest challenges for many qubit systems such as NMR is understanding
and eliminating decoherence [20]. If one is able to control the system and thereby reduces
the detrimental effect of the system-environment (SE) interaction, one can preserve the
quantum state for a longer time. This way of fighting decoherence by applying fast and
strong pulses has been termed dynamical decoupling (DD). DD techniques originate from
NMR spectroscopy [21, 22, 23, 24, 25, 26]. The main attraction of DD is that it requires
few additional resources. In contrast to quantum error correction (QEC) ,e.g., it does
not require additional qubits. QEC assumes that encoding and error correction are error
free [27, 28, 29, 30]. DD can be traced back to Hahn’s ‘spin echo’ experiment, where a
refocusing pulse induces a time reversal of the SE interaction of nuclear spins [31]. This
increases the decay time or decoherence time of the stored information in the qubit. The
technique has evolved significantly since then and its efficiency was studied, developed
and demonstrated in many different systems [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48].
It has been shown that the type as well as the spectral density of the SE interaction
play a significant role in finding the optimal DD sequences Kofman and Kurizki [49,
50], Zhang et al. [51], Gordon et al. [24], Bhaktavatsala Rao and Kurizki [52], Bylander
et al. [53], Álvarez and Suter [54], Almog et al. [43], Kotler et al. [55], Bar-Gill et al.
[56]. Furthermore, unavoidable errors in the control pulses are also an important source
of decoherence [57, 58, 36, 42, 59, 60, 61, 62, 63, 64, 65, 66, 19]. Therefore optimal DD
sequences must be able to reduce the effective SE interaction while compensating the
effects of non-ideal control fields [67, 40, 42, 66, 61, 64, 68, 65, 19]. There are several ways
to improve DD schemes, reference [66] is a recent review of this subject.
The goal of this thesis is focused on enhancement of the coherence times obtained by
DD sequences. Towards this goal we compare the performance of different DD sequences
in a system where pulse errors are the dominant source of decoherence. In our system,
pulse errors become the most significant effect because most of the effects of the SE in-
teraction have been eliminated. To this end, we prepare a system where the spectral
density of the SE coupling has two main contributions. One source of noise is almost
static and can therefore be refocused by all tested DD sequences. The other contribution
is a rapidly fluctuating noise, whose correlation time is much shorter than the time re-
quired for an inversion of the spins. This type of noise cannot be refocused by any DD
sequence. Therefore the main difference between the performance of the DD sequences is
their susceptibility to pulse errors. The optimum DD sequence should have the longest
coherence time and minimal resources.
In this work [36, 42], in particular it has been found that the Carr, Purcell, Meiboom and
Gill (CPMG) sequence [69, 70], a well known sequence in the NMR community, is strongly
initial state dependent and performed particularly well for specific initial states. In this
case the decoherence time was one order of magnitude longer than the robust sequences
that reduced decoherence symmetrically with respect to arbitrary initial states. We found
that this difference arose because the system qubit, a 13C spin, interacted with neighboring
13C spins. While the CPMG sequence was able to reduce the effect of 13C−13C couplings,
9
the robust sequences were not [54]. Since the tested DD sequences were not designed for
eliminating the effect of homonuclear couplings, the longer decoherence times for CPMG
applied to certain initial conditions cannot be taken as a measure of its performance. In
this work we therefore use a system that does not exhibit homonuclear interactions. This
allows us to show that the robust sequences can also achieve optimal decoherence time
observed under CPMG, and this performance is independent of initial condition.
The next chapters in this thesis are organized as follows. We give a brief introduc-
tion to NMR and quantum computation in the second chapter. In the third chapter the
experimental setup of the spectrometer is discussed. A new setup for the H-channel is
proposed. We have also introduced some measurement for the electronic noise in the
receiving part of the spectrometer and solutions to reduce the noise. Optimization pro-
cedures for system preparation is also described. This includes optimizing pulses using
tune-up sequences and preparing the system to avoid long experimental times and ob-
stacle for NMR experiments such as radiation damping. In chapter four we introduce
the DD sequences under our environmental consideration and compare their robustness
against pulse error effects for the prepared system. To complete the picture we give the
theoretical analysis of the underlying physics in terms of the average Hamiltonian theory
taking pulses length errors into account. Then we introduce an approach to engineer the
environment by applying additional signals to the system. For that we need to build a
new coil and provide additional devices for the setup. In the last chapter we draw some
conclusions. For future work we have suggested a molecule of five-spin system for testing
dynamical decoupling sequences. In the appendix (a) I show a protocol for testing the
NMR signal through different devices in the 360 MHz spectrometer, in which some of the
errors look trivial but are likely to occur. In (b) and (c) two interpreter pulse programs
are included with a brief description..
10
2. NMR basics
2.1. Theoretical background
Matter is composed of atoms which contains electrons and protons, these charged particles
give atoms their different physical and chemical properties. NMR spectroscopy is based
upon the fact that nuclei have angular momentum. Particularly NMR technique measure
protons angular magnetic momentum in medical applications, termed magnetic resonance
imaging (MRI), however other nuclei such as 13C are used for other applications[71, 72].
In quantum mechanics the total spin angular momentum of any physical system ~S is
quantized so the allowed value are ~
√
s(s+ 1), similar like other angular momentum,
where ~ = h/2pi, h is Planck’s constant, s is the spin quantum number which can take
the values 0, 1/2, 1, 3/2, 2, etc [73].
Substances exposed to magnetic fields possess magnetic interaction energy gives by
Eq. (2.1.1), depending on the magnetic field strength ~B0 and the magnetic moment ~µ.
E = −~µ ~B0. (2.1.1)
Since ~µ and ~B0 are vector quantities, the dot product with the negative sign indicates
that the magnetic energy will reach the minimum when the magnetic moment ~µ is parallel
to ~B0 field [74, 75].
Detection of different materials in NMR depends on their gyromagnetic ratio γ unit:
(rad/sec·T), which is the proportional constant between the magnetic moment and the
spin angular momentum ~S. As shown in Eq. (2.1.2) nuclei with higher gyromagnetic ratio
have higher magnetic moment thus higher detection sensitivity [76].
~µ = γ~S. (2.1.2)
There are two type of materials, the first type has intrinsic magnetism. It tends to
align with an external magnetic field e.g. the earth magnetic field. The second type show
induced magnetism in a magnetic field. This induced magnetism is proportional to the
magnetic field given by Eq. (2.1.3).
~µinduced =
1
µ0
V χ~B0, (2.1.3)
where: µ0 = 4pi×10−7 (Henry/m) is the magnetic susceptibility of the vacuum, V repre-
sents volume and χ is the magnetic susceptibility, a dimensionless constant that indicates
the degree of magnetization of the material when the magnetic field ~B0 is applied [74].
11
2.1. THEORETICAL BACKGROUND
2.1.1. The nuclear Zeeman effect
Application of a magnetic field breaks the degeneracy of the nuclear spin state. A nucleus
with nuclear spin s = 1/2 will split 2s+ 1 in presence of a magnetic field1. This is known
as Zeeman effect[74]. Equation (2.1.4) shows the Hamiltonian operator for a magnetic
field oriented along the z-axis in terms of the angular momentum operator along z-axis
Îz, as in Figure 2.1.1, the splitting energy is magnetic field dependent.
Hˆ = −γ~B0Iˆz. (2.1.4)
Figure 2.1.1.: Energy differences of two adjacent levels as a function of the magnetic field.
Consider a nuclear spin quantum state |ψ〉we can describe the system dynamics using
the Schrödinger Eq. (2.1.5)
i~
∂
∂t
|ψ〉 = Hˆ |ψ〉. (2.1.5)
If the energy operator Hamiltonian Hˆ is time-independent, we can solve Eq. (2.1.5)
using the evolution operator as in Eq. (2.1.7):
|ψ〉 (t) = U(t) |ψ〉 (0), (2.1.6)
where U(t) in Eq. (2.1.6), the evolution operator takes the form:
U(t) = exp(−iHˆt/~). (2.1.7)
For spin-1/2 the angular momentum operator for z components has two eigenfunctions
labeled mz = ±1/2. Using Eq. (2.1.4) we get two energy levels in a single spin-1/2 in a
magnetic field labeled Eα, Eβ:
1Nuclei which has non-spherical charge distribution, also called quadrupolar nuclei, have a spin number
s grater than 1/2 e.g. 1, 3/2, 5/2,..,etc. Spin 1/2 nuclei have two orientations while Spin 1 nuclei
have three orientations such as Deuterium (s=1).
12
CHAPTER 2. NMR BASICS
Eα = −
1
2
~γB0 , Eβ = +
1
2
~γB0. (2.1.8)
Transitions are only allowed between Eα and Eβ as in Eq. (2.1.8).
2.1.2. Precession and Larmor condition
In a typical NMR sample such as liquid water, nuclear magnetic moments are pointing
in all possible directions. If we immerse this sample in a magnetic field B, the magnetic
moment of the spin precess around the field on a cone keeping a constant angle between
the spin magnetic moment and the field.
Spin precession depends on the energy difference ∆E between the two levels in Eq. (2.1.8),
where transition are possible when the external magnetic field fulfills Eq. 2.1.9 where ω0
is called the Larmor frequency.
ω0 = γB0. (2.1.9)
To calculate the energy difference 4E we use Boltzmann’s relationship (2.1.10):
4E = e−
γ~B0
kBT ≈ 1−
γ~B0
kBT
. (2.1.10)
If we consider an NMR sample in a magnetic field 8.4555 Tesla at 300 k using Eq. (2.1.10)
for proton:
ν◦ = ω◦/2pi = 360 MHz, γ~B◦ = 360 MHz·6.62 · 10−34 J·s=2.38 · 10−25 J
The denominator gives: kBT = 1.38 · 10−23 J/K·300 K= 4.14 · 10−21 J.
The sensitivity of the NMR signal depends on this energy difference, as illustrated in
the example above, the ratio approaches ≈ 10−4 so the sensitivity to detect an NMR
signal proportional with the difference which is 1 part in 5 · 103. The Larmor frequency
for the nucleus changes slightly with different molecular environment in the NMR sample.
This small variations causes small shifts in the NMR spectrum, on the order of 1 part in
109 which known as chemical shift in NMR spectroscopy. Chemical Shift is very useful in
studying different chemical compounds and their properties 2[75].
2.1.3. Density matrix operator
Let us consider an NMR sample which contains protons such as in pure water. This bulk
sample contain approx. 1023 protons (i.e. spins) ml−1, which have different orientations
for magnetic moment and precess independently. These spins known as ensemble. We
classically i.e in the experiment, measure the average magnetization overall spins. Mag-
netization is a macroscopic quantity. The way to solve that is to use the method of
the density operator. Consider spin 1
2
, using the wave function ψ = a |α〉 + b |β〉 where
|a|2 + |b|2 = 1 and |α〉,|β〉 are the eigenstate of the operator Iˆz with eigenvalue ±12 . The
coefficients a and b are different for all spins in the ensemble. The average is taken over
the whole sample with N spins is given by Eq. (2.1.13).
The eigenvalue equation of the angular momentum operator Iˆz is written for base states
|α〉 with eigenvalue 1
2
:
2A good online book about NMR: http://www.cis.rit.edu/htbooks/nmr/inside.htm
13
2.1. THEORETICAL BACKGROUND
Iˆz |α〉 = +
1
2
|α〉 . (2.1.11)
Besides Iˆz we define the angular momentum operators Iˆy and Iˆz which do not commute
with each other according to the commutation relation Eq. (2.1.12):
[Iˆx, Iˆy] = iIˆz. (2.1.12)
We can write the density matrix ρ with the matrix elements m,n as in Eq. (2.1.13)
ρmn = 〈m |ρ|n〉 = a∗mbn =
1
N
∑
i
a∗i bi, (2.1.13)
ρ contains all information about the ensemble, so any operator Oˆ may be written as
〈
ψ
∣
∣
∣Oˆ
∣
∣
∣ψ
〉
= Tr(Oˆρˆ), (2.1.14)
where the right hand side of Eq. (2.1.14) represents the trace over the matrix product.
Further more, the variation of ρˆ with time may be written using the Schrödinger
Eq. (2.1.5) :
i
∂ρˆ(t)
∂t
=
[
Hˆ, ρˆ(t)
]
. (2.1.15)
Equation (2.1.15) known as Liouville-von Neumann equation. We can also write ρˆ(t)
(i.e. in the interaction picture) in terms of ρˆ(0) in the case of time independent-hamiltonian
Hˆ as:
ρˆ(t) = Uˆ(t)ρˆ(0)Uˆ−1(t), (2.1.16)
where Uˆ is defined in Eq. (2.1.7)[75, 71, 77].
2.1.4. Radio frequency pulses
The magnetization vector precess in a cone around the z axis due to Zeeman field as
we discuss in Section (2.1.2). This Zeeman field is usually took place by a powerful
superconducting magnet. So now we need to detect NMR signal. Applying a radio
frequency field (rf) - usually pulses - with frequency at or near the Larmor frequency
ω◦ to the NMR coil where the sample is located generates a magnetic field B1 rotating
at the Larmor frequency ω0. This magnetic field B1 is week compared to the Zeeman
field. However this rf field is able to tilt the field to the xy plane for detection when the
resonance condition is met (i.e. the rf pulses is equal to the Larmor frequency).
2.1.5. Laboratory frame and rotating frame of reference
This rf field B1 is applied to set of spins in a field. Suppose we move to a co-ordinate
system which is no more static (laboratory frame) but rotating in the same direction and
at the same rate of the field B1. This called the rotating reference frame. This treatment
removes the time dependence from the rf field B1 and let it appear stationary, in simple
14
CHAPTER 2. NMR BASICS
analogy the car driver in Figure 2.1.2 will be appear constant when the observer changes
his co-ordinate to the car co-ordinate[71].
Figure 2.1.2.: Observer reference frame and rotating frame.
Accordingly the Hamiltonian in the laboratory frame may be written as a sum of two
components from the rf field and the Zeeman field where the two components are rotating
in the same direction as spin precession where the counter rotating component is ignored
at the resonance condition ω = ω◦ due to B1  B◦ Eq. (2.1.17) :
HˆLab = −γB0Iˆz − 2γB1cos(ωt)Iˆx (2.1.17)
Since we assume we are in the rotating frame, the frequency will reduced by ω and
the apparent longitudinal field vanishes leaving the effective field in the rotating frame so
Eq. (2.1.17) becomes:
Hˆrot = −γ(B0 − ω/γ)Iˆz − γB1Iˆx (2.1.18)
2.1.6. Bloch equations
Describing spins in motion in macroscopic model, the torque is the rate of change of
angular momentum ~M/γ, so we can write the equation of motion for the magnetization
vector ~M as :
d ~M
dt
= γ ~M × ~B, (2.1.19)
where B in Eq. (2.1.19) is the magnetic field with amplitude ~B0 corresponding to the
precession of the magnetization at the Larmor frequency in static field in the z direction.
The trend of the magnetization towards its equilibrium value Mz = M0 can be describe
using the Eq. (2.1.20)
dMz
dt
= −
(Mz −M0)
T1
, (2.1.20)
15
2.1. THEORETICAL BACKGROUND
where T1is the longitudinal relaxation time which concerned the drift of the magnetiza-
tion towards their thermal equilibrium value. Applying small rf field, will add a component
to the nuclear magnetization and because the spins are not free but rather interacting
with each other and their surrounding environment, the transverse magnetization decays
also according to Eq. (2.1.21)
dMx
dt
= −
Mx
T2
,
dMy
dt
= −
My
T2
, (2.1.21)
where T2is the transverse relaxation time which concerned the decay of the coherence
between |α〉,|β〉.
For short strong rf pulses one can write the oscillatory component of the rf transverse
field
Bˆ1(t) =
−ωnut
γ
cos(φ)ˆi− (
ωnut
−γ
)sin(φ)jˆ, (2.1.22)
where iˆ, jˆ, kˆ are unit vector along x, y, z. In the rotating frame, as in the previous
Section (2.1.5), it shows that there is an effective static field given by Eq. (2.1.24) and
the nutation frequency of the rf field ωnutis calculated the flip angle and the pulse width
τp using Eq. (2.1.23)
βp = ωnutτp, (2.1.23)
~Beff = (B0 +
ω
γ
)kˆ + Bˆ1, (2.1.24)
where ω is the rotating frame frequency. Assuming that B1 has only one component in
the x direction i.e. φ = 0 we can write the effective field in Eq. (2.1.24)
~Beff =
∆ω0kˆ − ωnutiˆ
γ
, (2.1.25)
where 4ω◦ = ω−ω◦.
The total rate of change of the magnetization can be calculate by adding Eqs. (2.1.20),(2.1.21)
to the rf field Bˆ1 using the equation of motion (2.1.19) as
dM˜
dt
= γ((M˜xiˆ+ M˜y jˆ +Mzkˆ)× ~Beff )−
˜(Mxiˆ+ M˜y jˆ)
T2
−
(Mz −M0)kˆ
T1
, (2.1.26)
where M˜x, M˜y is the transversal magnetization component in the rotating frame.
From Eq. (2.1.26) we obtain the Bloch differential equations which provide a general
frame work for treating the magnetization under the previous conditions
dM˜x
dt
= −M˜x + ∆ω0M˜y, (2.1.27)
dM˜y
dt
= −∆ω0M˜x −
M˜y
T2
− ωnutMz, (2.1.28)
dMz
dt
=
−(Mz −M0)
T1
+ ωnutM˜y. (2.1.29)
16
CHAPTER 2. NMR BASICS
The transverse component Mx , My in the laboratory frame could be calculated from
the rotating component M˜x , M˜y using equations (2.1.30) , (2.1.31)
Mx = M˜xcos(ωt)− M˜ysin(ωt), (2.1.30)
My = M˜xsin(ωt) + M˜ycos(ωt). (2.1.31)
Bloch equations treats the dynamics of spins which experience rf fields and relaxation
at the same time. On the other hand transverse relaxation destroy the transverse mag-
netization, so after long time the transverse magnetization should be zero. However at
sufficient long time, the system settled down in a stationary state. As a result of this
steady state the left terms on the Bloch equations (2.1.27), (2.1.28), (2.1.29) vanishes, be-
cause the magnetization components are not changing with the time. Solving the Bloch
equations we can get the magnetization in the steady state ˜M(ss) as a function of the
offset from the resonance ∆ω0, the relaxation times and the rf strengthωnut
˜My(ss) =
−(T2ωnut)
(1 + ω2nutT1T2 + ∆ω20T
2
2 )
M◦, (2.1.32)
similarly we can write:
˜Mx(ss) =
−(T 22ωnut∆ω0)
(1 + ω2nutT1T2 + ∆ω20T
2
2 )
M◦, (2.1.33)
Equation (2.1.32) shows that the magnetization exist as long as T2 is not too short and
the nutation frequency ωnut is not too large comparing to T2, T2. To simplify equations
(2.1.32) , (2.1.33) we assume (T1=T2) and substitute (a = −∆ω0T ) and (b = −Tωnut) ,
then we can write
˜My(ss) =
b
(1 + a2 + b2)
M0, (2.1.34)
˜Mx(ss) =
ab
(1 + a2 + b2)
M0, (2.1.35)
Eq. (2.1.34) known as absorption mode where the optimum peak absorption occurs for
(b=1) and Eq. (2.1.35) known as dispersion mode where it goes through zero on the exact
resonance condition when (a=0)[75, 77, 78, 79].
2.2. Introduction to Quantum computation
Quantum computation is the study of information processing using quantum mechanical
system. Quantum computers are designed to solve problems more efficiently than classical
computer [17].
17
2.2. INTRODUCTION TO QUANTUM COMPUTATION
2.2.1. Quantum Bit
In analogy to the classical computational concept ’bit’, quantum computation uses quan-
tum bits or qubits for short. Computational basic states are |0〉and |1〉 which correspond
to the classical states 0 and 1. Qubits are realized to be actual physical systems, however,
qubits are treated as mathematical objects. So, unlike classical bit, it is also possible to
have combination of states, known as superposition:
|ψ〉 = a |0〉+ b |1〉 . (2.2.1)
In Eq. (2.1.18) a and b give us the probability of finding the the basic computational
states, so they summed to one |a|2 + |b|2 = 1.
If we add another qubit to our computational basic states, we obtain four computa-
tional basis state from two qubits denoted |00〉 ,|01〉,|10〉 and |11〉. Similar to one qubit
Eq. (2.1.18), we can write a pair of qubits as superposition of those four states
|ψ〉 = α00 |00〉+ α01 |01〉+ α10 |10〉+ α11 |11〉 , (2.2.2)
where the total probability of measuring all possible x states is unity:
∑
x
|αx|2 = 1.
2.2.2. Single and multiple qubit gate
Quantum computers are built from quantum circuits containing "wires" and quantum
gates to carry and manipulate information. As an example for single qubit gates which
acts only on one qubit the "NOT" gate, takes the state |0〉 to |1〉 and vice versa and
did not tell us what happened to the superposition of state |0〉 and |1〉 without further
knowledge about the previous state. Representing NOT gate is possible in matrix form
acting on one qubit by 2× 2matrix. The quantum gate has this form :
NOT ≡ X =
[
0 1
1 0
]
,
and the quantum state represents with the vector notation:
[
a
b
]
, the gate will act on
the state by multiplying its matrix from the left as shown in Eq. (2.2.3):
X
[
a
b
]
=
[
b
a
]
. (2.2.3)
Quantum gates have to be unitary, so the matrix describing the gate should be unitary,
satisfying the condition Eq. (2.2.4) where X∗ is the hermitian conjugate of X and I is
the identity operator .
XX∗ = I. (2.2.4)
Reverse is an important feature of unitary operations. Thus reversible logic preserved
information and consumes less power for computation in contrast to Boolean logic (which
contains only true or false values). A typical multiqubit gate is the controlled gate or
CNOT gate. UCNOT (see Eq. 2.2.5) shows the matrix representation of the CNOT gate[5,
16].
18
CHAPTER 2. NMR BASICS
UCNOT =





1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0





. (2.2.5)
2.3. Liquid state NMR Quantum computing
NMR is concerned with measuring properties of liquids, solid and gases, determination
the structure of molecules or imaging materials or investigating biological systems and
monitoring interactions. Those many applications have led the technology of NMR to
the current status. Recently NMR spectroscopy has also been used for quantum informa-
tion processing (QIP). This new role relies on the capability of direct manipulation and
detection of nuclear spin in NMR.
2.3.1. Quantum computer in DiVincenzo view
DiVincenzo discussed five requirement for quantum computer. Here, we discuss them
from stand point of liquid state NMR :
1. Recognizing qubits is not trivial: We need for a quantum computer well charac-
terized qubits i.e. their internal Hamiltonian and other physical properties e.g.
interaction with other qubits should be well defined. In liquid NMR the bulk sam-
ple contains approx. 1023 spins can not be addressed individually. However the
resonance frequency differentiates between the spins. Nevertheless they are not
scalable.
2. State initialization: This initialization to well defined state can be done naturally
i.e. the system is cooled to reach the ground state like NMR where this is achieved
by relaxation.
3. Long decoherence times compared to gate times: Decoherence scale the computa-
tions, liquid state NMR has long decoherence times.
4. A universal set of quantum gates: A quantum algorithm is a sequence of unitary
transformations acting on qubits. This is the heart of quantum computing and this
is well done in liquid state NMR i.e these Hamiltonians are identified to physical
specifications. Physical devices are designed to be turned on and off in defined
intervals.
5. A qubit selective read out: At the end of the computational process, the result of
the qubits should be read out with high efficiency. Liquid NMR address different
qubits through their chemical shifts[5, 4].
2.3.2. Decoherence
It is the characterize the interaction of the qubit with its environment and one of the
large obstacles in front of long computation scale. We can describe the life time of non-
equilibrium classical state and quantum superposition through the relaxation times T1,T2.
19
2.3. LIQUID STATE NMR QUANTUM COMPUTING
One source of the decoherence is the phase randomization or thermalization of the spin
to their equilibrium environment which originate from the spin spin coupling, usually
represented with T1, which are all irreversible processes. However the dephasing process
results form the inhomogeneous static fields which leads the spins to presses with different
rates. This shortens the coherence time, fortunately this a reversible process. These
relaxations times formed the exponential decay in the magnetizations[4, 16].
20
3. Experimental setup
In this chapter we describe the main experimental configurations used in this work, in-
cluding the detection and noise measurements and evaluation. This is followed by the
sample preparation procedure and pulse parameters optimization.
3.1. Spectrometer configuration
It is familiar that NMR spectrometers are classified according to their field strength related
with the Larmor frequency of the 1H atom as in Eq. (2.1.9). In the 360 MHz spectrometer
the superconducting magnet has a magnetic field strength B◦ of 8.4555 Tesla, which
corresponds to the 1H Larmor frequency 360 MHz, in the z direction.
3.1.1. Controlling the 360 MHz spectrometer
In the 360 MHz spectrometer setup a personal computer PC is used to control the differ-
ent devices in the experiment. The PC (Mac-mini) controls the, direct digital synthesizers
(DDS’s), word generator, shim system and the oscilloscope using the MacExp as interface
software. MacExp is a software package designed for Mac OX system with a flexible
interface for different NMR experiment configurations, e.g. acquiring, processing and an-
alyzing NMR data, streamlined and convenience. Communication signals are transmitted
between the PC and other devices through Ethernet connections in a virtual local area
network VLAN. Whereas MacExp is also used for other experimental configurations in
our laboratories, this VLAN should be configured correctly to avoid conflicts between dif-
ferent devices which have the same IP addresses through the networks in the laboratories.
Since the DDS’s, the oscilloscope (Lecroy, Waverunner LT344), and the word generator
have GPIB ports, they are controlled via GPIB-Ethernet converter (Ethernet Proglogix).
21
3.1. SPECTROMETER CONFIGURATION
Figure 3.1.1.: Controlling the 360 MHz spectrometer.
22
CHAPTER 3. EXPERIMENTAL SETUP
Figure 3.1.2.: Main components of the 360 MHz spectrometer.
Figure 3.1.1 illustrates control windows for the most important devices in the MacExp
graphical interface. Using MacExp make it easy to run different types of experiments
automatically e.g. multidimensional NMR experiments, storing the results sequentially
and process them.
Figure 3.1.2 shows the main different components of the 360 MHz spectrometer. The
spectrometer is composed of two channels, which are labeled usually 1H-channel and X-
channel, corresponds to the Larmor frequencies for each nucleus. The frequency of the
X-channel can be varied to detect other active NMR nuclei around the 90 MHz. A block
diagram of the 360 MHz spectrometer is drawn in Figure 3.1.3, where two main parts are
separated from each other with the dashed box in the right hand side. Inside the dashed
box is the transmitter section.
23
3.1. SPECTROMETER CONFIGURATION
Figure 3.1.3.: Block diagram for the 360 MHz spectrometer.
3.1.2. The transmitter
The transmitter part generates rf pulses. Since most of the experiments in this work were
done on the 1H-channel, these rf pulses are usually close to the Larmor frequency of 1H.
In the1H-channel, the principle of mixing frequency, "intermediate frequency" is used. In
the modulator which is shown inside the dashed box (Figure 3.1.4), the DDS1(H) generates
a frequency of 60 MHz and the Hameg frequency synthesizer generates a frequency of
300 MHz, these frequencies are mixed using mixers. The mixer is a device composed
usually of diodes, it uses the nonlinearity of the diode to generate an output contains
the sum and the difference of the input signals. Mixers are called frequency translation
devices. In the modulator the two signals are mixed and the rf pulses are modulated to
be sent to the high frequency HF-amplifier, then the difference is filtered to obtain the
360 MHz[80, 81].
24
CHAPTER 3. EXPERIMENTAL SETUP
Figure 3.1.4.: Bloch diagram of the1H-channel modulator/demodulator.
25
3.1. SPECTROMETER CONFIGURATION
Consider the following equations (3.1.1), (3.1.2) to calculate the modulated frequency
from the modulator at (10), Figure 3.1.4. The DDS1(H) has two output signals with
relative 90◦ phase difference LO1, LO2. The output goes to mixer (1),mixer (2) in (see
Figure 3.1.4) and ωs from the rf Hameg frequency synthesizer (300 MHz):
LO1 = cos(ωlot), (3.1.1)
LO2 = sin(ωlot), (3.1.2)
then at the output of mixer(1) Figure 3.1.4 we get :
S1 = sin(ωst) cos(ωlot) =
1
2
[sin(ωst− ωlot) + sin(ωst+ ωlot)] , (3.1.3)
similarly for the other mixer2 we can write Eq. (3.1.4):
S2 = sin(ωst) sin(ωlot) =
1
2
[cos(ωst− ωlot)− cos(ωst+ ωlot)] . (3.1.4)
Then shifting the output at (S2) with 90
◦Eq. (3.1.4) yields:
S+902 = −
1
2
[sin(ωst− ωlot)− sin(ωst+ ωlot)] , (3.1.5)
therefore adding (S1), (S2) from Eq. (3.1.3) and (3.1.5), the output signal (S3):
S3 = sin((ωs + ωlo)t) (3.1.6)
The output pulses from the HF amplifier is filtered using a high-pass filter, then they
are sent to the probe through the duplexer. It is worth noting here that the Hameg
frequency synthesizer and the DDS1(H) should be synchronized with the master clock
to have stable rf pulses with defined phases, so the DDS1(H) should use the external
clock 10 MHz and the Hameg frequency synthesizer should be set to external trigger to
the same clock. DDS1(H) output1 and DDS1(X) output1 could be varied to change the
pulse power for both channels respectively. The duplexer keeps the rf pulses transmitted
to the probe and protect the receiver circuit against high power rf pulses. The duplexer
is composed of quarter-wave stub ‘BNC’ cable and crossed diodes, which attenuates the
power at the defined frequency, and allow the rf pulse to pass to the probe and prevent
the NMR signal from reflecting back to the modulator. These quarter waves cables with
specific length work only for specific frequency (360 MHz, 90 MHz). It behaves as a
transformer, in fact it is often referred to as a quarter-wave transformer and the input
impedance is calculated using the Eq. (3.1.7)
Zi =
Z20
Zl
, (3.1.7)
where: Zi is the input impedance, Zl is the load impedance. In a simple way when
Zl =0, in a short cable there will be standing wave when the cable length matches the
frequency and Zi =∞, and so the impedance mismatching is used effectively for this
application[74, 80, 82].
26
CHAPTER 3. EXPERIMENTAL SETUP
3.1.3. The RS690 word generator
The word generator acts as a heart of the spectrometer in time synchronization between
different devices. It controls the pulse timing (width, delays) repetition rate with respect
to internal trigger of the word generator. The minimum duration of the RS690 is 16 ns
with resolution 4 ns, however it is set by default to a minimum duration of 20 ns. The
maximum duration of the RS690 is 7500 µs, loops are used for longer durations. In
a rectangular pulse train the duty cycle is defined as the ratio between the total pulse
durations and the total period. For continuous rf irradiation the duty cycle is 100%.
The Default maximum duty cycle value is set to 1% by MacExp. This value should be
carefully adjusted taken into account not to exceed the maximum duty cycle of the probe
maximum power. Time diagram of pulse sequence is produced using pulse programs which
are read by the interpreter using the word generator. The word generator sends commands
to control the experimental pulse parameter within the modulator and the demodulator
through 16-bit TTL while other 32 ports are assigned to control the DDS’s[83].
3.1.4. The Receiver
The receiver part picks the diverted NMR signal through the duplexer to the detection
part. In the first stage the signal is amplified using the preamplifier, which should have
low noise, high sensitivity, cover a wide frequency range and recover quickly from large
overload signals. Then the signal is sent to the demodulator (see Figure 3.1.4). According
to the current electronic circuit (Figure 3.1.4) for the H-demodulator ,the DDS1 output2
(used for detection) should be set from the DDS control panel to =⇒ 1 which gives
0.5V (+7 dBm) at the Mixer inputs(LO)(4,5) where the mixer perform ideally. Similarly
according to the current electronic circuit for the X-demodulator, the DDS2 output2
should be set from the DDS control panel to =⇒ 0.74 which gives also 0.5V (+7 dBm)
at the Mixer inputs(LO). In NMR experiments, the detecting signal can be characterized
through its amplitude, phase and frequency. To detect the phase a quadrature detector
is used. The quadrature receiver combines the NMR signal which oscillates at Larmor
frequency ω◦ with a reference frequency ωrf in order to generate a new signal which
oscillates at a relative Larmor frequency, ω1 as in Eq. (3.1.8) equal to the frequency in
the rotating frame (see Sec. (2.1.2).
ω1 = ω0 − ωrf . (3.1.8)
These relative frequencies ω1 are quite low, usually in the Hz to kHz range. Cables
should have defined lengths or symmetric to adjust the phase between the two outputs
of the receiver which are 90◦ phase shifted with respect to each other, labeled as real and
imaginary components of the NMR signal. These signals are considered to be equivalent
to the projection of the transverse magnetization on to the x axis and y axis, they are
assigned Mx and My respectively. The output signal is phase sensitive detected signal,
i.e. it is DC coupled signal. However DC offset, e.g. from the amplifiers, is one of possible
artifacts associated with the rf detector[76, 82].
27
3.1. SPECTROMETER CONFIGURATION
3.1.5. Alternative 1H-channel demodulator/modulator
In the first version of DDS (DDS1), the chip is able to generate maximum frequencies
up to 120 MHz[83]. Under these conditions the DDS1 should be mixed with other high
frequency to reach the 360 MHz. Different electronic components for the alternative 1H
demodulator/modulator is shown in Figure(3.1.5).
Figure 3.1.5.: Alternative 1H-channel modulator/demodulator.
To eliminate the complexity of mixing frequencies we can use the third versions of DDS
i.e. DDS3E to generate the rf frequency. DDS3E has the advantage of controlling through
an Ethernet connection and it has a variable amplitude. Figure 3.1.6 illustrates a block
diagram circuit for an alternative demodulator/modulator can be used for the 1H channel
without need of two sources and frequencies mixing.
28
CHAPTER 3. EXPERIMENTAL SETUP
Figure 3.1.6.: Alternative1H-channel modulator/demodulator block diagram using the
DDS3E.
3.1.6. The probe-head
The NMR probe-head keeps the sample in the region of the homogenous magnetic field
which is commonly the center of the superconducting coils. The coil generates an oscillat-
ing magnetic field B1 during the rf pulse, and detect the NMR signal using the induction
principle. A poorly tuned probe reflects a lot of the power of the pulses, so that what
should be a 90◦ pulse rf is in reality only ~ 50◦ pulse. As a result probe tuning affects the
signal to noise ratio (S/N), where the signal will become worse due to mismatch between
impedances.
When radio frequency power is transmitted over transmission lines some loss can occur
29
3.1. SPECTROMETER CONFIGURATION
due to impedance mismatching between the probe and other circuits in the NMR spec-
trometer. The basic NMR probe circuit is consists of a coil which generates an oscillating
magnetic field perpendicular to the main field at the sample during the rf pulse as input
signal and then detects the precessing nuclear magnetization from the sample as output
NMR signal. In liquid-state NMR probes saddle coils are used, and two variable capac-
itors, one of them called tuning capacitor (C1), which is used to adjust the frequency
to find the resonance frequency (to enhance the current in the coil by electromagnetic
resonance), the other capacitor is used to match the impedance of the probe circuit to
50 Ω, known as matching capacitor (C2). The circuit diagram is shown in Figure 3.1.7.
The matching capacitor couples the external signals into the probe to avoid the power loss
mentioned above. To avoid distortion, different components of the probe are preferred to
be non-magnetic[84, 82, 76, 74, 73].
Figure 3.1.7.: Simple NMR probe circuit
Tuning the probe is done firstly using the network analyzer (8712ET) with changing the
matching and tuning capacitors in order to get a good resonance shape at 360 MHz and
50 Ω impedance. The reflection power should be low with comparison to the incident
power. In Figure 3.1.8 the network analyzer data shows that the reflected power was
about -25.3 dB @ 360 MHz (that means 0.29% of the power is reflected back in this case).
30
CHAPTER 3. EXPERIMENTAL SETUP
Figure 3.1.8.: Wobbling curve during probe tuning using the network analyzer for the
1H-channel
This first tuning measurement is preliminary after then we consider the probe is con-
nected with the other parts of the spectrometer, the circuit configuration will change and
the probe could be tuned further using the voltage ratio1 (reflection/incident) which is
measured with the directional coupler. A directional coupler is an asymmetric power
divider with four ports as shown in the block diagram Figure 3.1.3 for both channels of
the spectrometer. It splits the incident rf signal to a small part which goes to the scope
as incident pulse and the other part to the sender (HF) amplifier and it does the same
for the reflection signal. For a well tuned probe, the rf reflected power should not exceed
1% of the rf incident power. Figure 3.1.9 shows (a) incident pulse and (b) reflection pulse
with 2 µsec pulse width from the 1H-channel directional coupler.
1The power ratio will be the square of the voltage ratio
31
3.1. SPECTROMETER CONFIGURATION
1 2 3
-0.4
-0.2
0.0
0.2
0.4
1 2 3
-0.1
0.0
0.1
b)
A
m
p
l
i
t
u
d
e
(
m
V
)
Time( s)
a)
A
m
p
l
i
t
u
d
e
(
m
V
)
Time( s)
Figure 3.1.9.: Tuning the 1H-channel using the directional coupler : 2 µsec pulse for a)
incident pulse and b) reflected pulse
The reflected power decreases when the probe is tuned in the right direction. If the ratio
is one that means all the power is reflected and the circuits should be followed step by
step. The coupling factor (C.F.) of the directional coupler is given by the equation
C.F. = −10log
P3
P1
, (3.1.9)
where:
P1 : Input power, P3 : coupled power(from the incident part).
As an example we calculate the coupling factor (C.F.) for the directional coupler in the
X-channel (P3= 6.8 dBm, P1 =40.65 dBm)
C.F. = P3(dBm)− P1(dBm) = 33.85. (3.1.10)
32
CHAPTER 3. EXPERIMENTAL SETUP
3.1.7. Shimming the 360 MHz spectrometer
The superconducting magnet generates a magnetic field B◦, which is equivalent to Lar-
mor frequency in high frequency range. This high value makes it hard using only the
superconducting magnet to limit the variation in the magnetic field across the sample
to 10−9 . Corrections to the field homogeneity ∆B◦ are made by using the shim coils2,
shimming means adjusting the resolution of the signal by optimizing the homogeneity of
the magnetic field so that every part in the sample experiences the same field. These
inhomogeneities caused by different aspects starting from the magnet installation and
design through variations in the thickness of the sample tube, sample permeability, and
ferromagnetic materials around the magnet. Superconducting shim coils are set during
installation of the magnet. Room temperature shim coils generate small magnetic fields
for correcting gradient in all Cartesian coordinates. In the shimming unit of the 360 MHz
spectrometer there are 34 different room temperature shim coils to optimized the mag-
netic field homogeneity. Passage of current through these coils adjust the inhomogeneity
of B◦.
Before starting the shimming procedure the probe coil should be set in the middle of
the superconducting coil. Axial shims for different orders named z0, z, z2, z3, ...,z7 correct
inhomogeneities in the magnetic field along the z axis, where radial shims like x, y, ...,
etc correct inhomogeneities in the xy plane. The z0 changes the field B0. Figure 3.1.10
shows that the relative Larmor frequency variates linearly with z0 changes. However B0
is changed slightly over the time due to the small resistance of the coil, which is known
as normal magnets drift. A typical magnet drift for the 360 MHz spectrometer is 1.68 Hz
per day as shown in Figure 3.1.11. Beside that B0 exhibit variations due to magnetic
materials moving near the magnet. Installing new magnets nearby shows shift in the B0
field. Experiments are unlikely to be done during the filling process of liquid He or N,
because the superconducting coils temperature is unstable and therefore the B0 field.
-10000 -5000 0 5000 10000
-10
-5
0
5
10
frequency
  linear fit
F
r
e
q
u
e
n
c
y
 
(
K
H
z
)
z0
Frequency(Hz) = 0.73z0
Figure 3.1.10.: Relative Larmor frequency versus z0 variation.
2The name ‘shim’ comes historically earlier where small piece of nonmagnetic metal was used to improved
the homogeneity of the electromagnet by changing the distance between the halves of the magnet
33
3.1. SPECTROMETER CONFIGURATION
2 4 6 8 10 12
2.6
2.8
3.0
3.2
3.4
3.6
F
r
e
q
u
e
n
c
y
 
(
H
z
)
Time (hours)
Magnet drift = 1.68 Hz/day 
Figure 3.1.11.: Typical magnet drift for the 360 MHz spectrometer.
Important shim coils like z, z2 and z3, where the biggest effects in the line width
occur, should be adjusted first. Passage of current adjust the inhomogeneity of B0 field.
Table (3.1.1) shows the mathematical expressions for the additional fields generated by
the shims z3, z2, z3, x, y. Figure 3.1.12 shows how these additional fields are represented
according to the equations in Table (3.1.1). Configuration of current sources is varied
over a wide range of number of channels, the 360 MHz spectrometer’s shimming unit has
a limit of ±32767 for gradient coil values, however the combined contribution from many
gradients may exceed this limit and give message that values are out of range.
Order Equation for field generated Shim
0 1 z0
1 z z
2 z2 − (x2 + y2) z2
3 z[2z2 − 3(x2 + y2)] z3
4 x x
5 y y
Table 3.1.1.: Mathematical representation of z0, z, z2, z3, x, y shims which are propor-
tional with correction field ∆B◦.
3z shim also called sometimes z1 shim but we use here the name z as in the shimming unit of the
spectrometer
34
CHAPTER 3. EXPERIMENTAL SETUP
-8 -6 -4 -2 0 2 4 6 8
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
0
20
40
60
80
-8 -6 -4 -2 0 2 4 6 8
-600
-400
-200
0
200
400
600
-8 -6 -4 -2 0 2 4 6 8
-8
-6
-4
-2
0
2
4
6
8
S
h
i
m
 
f
i
e
l
d
z
z z2
S
h
i
m
 
f
i
e
l
d
z
z3
S
h
i
m
 
f
i
e
l
d
z
y
S
h
i
m
 
f
i
e
l
d
y
Figure 3.1.12.: The curves shows magnetic field corrections due to diffident shims varia-
tions: z1, z2, z3, y.
Shimming is performed in the 360 MHz spectrometer by optimizing the best line width
of the 1H signal. Shimming is time consuming procedure and many protocols can be used
to achieve optimal results. First one may optimize z and z2 to get the narrowest line
width. Then higher order like z can be optimized. Since shim coils are interacting with
each others and lower orders contribute to higher orders, z and z2 should be optimized
a gain after performing z3. In this sequential one can optimized other shim parameters.
When all shim values are optimized, then the values are saved in text file and they can
be set again using MacExp through RS-232 interface[85, 76, 86, 71, 87].
3.1.8. Noise reduction
Battle against noise is an essential part in most NMR experiments. Noise comes from the
thermal motion of charged particle of the sample and thermal motion of electrons in the
receiver coil. The noise and the NMR signal are amplified through different amplifiers
in the circuit. In 13C NMR experiments, the detected signal is very weak due to less
natural abundance and lower gyromagnetic ratio. Comparison between the 1H which has
natural abundance 99.98% (with relative sensitivity of 1.0), and the 13C which has 1.1%
(with relative sensitivity of 1.6× 10−2), illustrates that during an NMR experiment, only
about 1.1% of the carbon atoms in the sample are able to contribute to the signal. That
causes the signal getting lost inside the noise. Many steps are used to increase the signal
to noise ratio. Noise is often a limiting factor for the performance of a device or system.
Depending on its source or its frequency bandwidth, noise can be classified into a number
of categories.
Towards reduction of noise and obtaining a well defined NMR signal we will take into
account categorizing noise, i.e. undesirable disturbance to NMR signal, into two essential
35
3.1. SPECTROMETER CONFIGURATION
parts depending on its time characteristics:
3.1.8.1. Incoherent noise
The incoherent noise is independent of the NMR pulses so it occurs randomly and its
value can not be predicted at any time. The thermal noise, also known as Johnson noise,
is an incoherent noise which is frequency independent. Its power spectral density V¯n is
proportional to the absolute temperature with Eq. (3.1.11) so it is unavoidable at non-zero
temperature[88].
V¯n = 2
√
kBT∆νR, (3.1.11)
where:
T≡ resistance absolute temperature (room temperature = 298 °K), kB ≡ Boltzmann
constant, R ≡ resistance of the noise source = 50 Ω and 4ν ≡ frequency bandwidth over
the noise which is measured in Hz.
For a given bandwidth frequency for the preamplifier 4f (500 MHz) the rms noise
given with Eq. (3.1.12), can be calculated:
V¯n = 2
√
1.38× 10−23 J/K × 298K × 500MHz × 50 Ω = 20.28µV V −81 dBm
(3.1.12)
A useful technique used to reduce thermal noise is signal averaging. Assuming that,
at the output of the oscilloscope, the signal is a compound of NMR signal SNMR(t) and
noise Snoise(t) which are given by:
S(t) = SNMR(t) + Snoise(t). (3.1.13)
Running another identical NMR experiment, SNMR(t) will add coherently to give a total
NMR signal 2SNMR(t) St. The noise Snoise which supposed to be random with standard
deviation σnoise over all sampling points. The noise amplitude in the first experiment is
given by the root mean square (rms) Eq. (3.1.17)
σ(1)noise =
〈
S2noise(1)
〉1/2
, (3.1.14)
on the assumption that the noise is stationary, its amount will be the same for the
second experiments, then the sum will be:
σnoise(1 + 2) =
〈
(Snoise(1) + Snoise(2))
2
〉1/2
, (3.1.15)
since the noise is uncorrelated between two experiments, the average term 〈Snoise(1)Snoise(2)〉
vanishes, Eq. (3.1.15) gives:
σnoise(1 + 2) ∼=
√
2σnoise(1). (3.1.16)
Extending this argument to N number of scans, we can decrease the noise by
√
N given
by Eq. (3.1.17):
St
σt
=
√
N
SNMR
σs
(3.1.17)
36
CHAPTER 3. EXPERIMENTAL SETUP
Another useful technique affects the resolution or the signal to noise ratio (S/N) is
apodization. It smooths the tail of the free induction decay (FID) signal to eliminate
the sharp edge produced by the a rectangle function, goes through with multiplying the
digitized FID by a mathematical function. Applying for example Gauss apodization
results in clipping the tail of the spectrum where the signal is weakest and leave the early
part unaffected[74, 76].
3.1.8.2. Coherent noise
This kind of noise is coherent with pulse sequences or receiving procedure. Such as
impulse noise due to the transient nature of the NMR experiment where the sender is
separated from the receiver through the duplexer which does not perform as an ideal
barrier. Transient noise consists of relatively longer duration pulses compared to impulse
noise. It could be also systematic noise such as the 50/60 Hz noise "hum" originates from
the power supplies circuits[89, 82, 90].
Removal of the artifact during quadrature detection, where two signals 90◦out of phase
are sampled (Mx, My) is achieved through phase cycling. A Familiar method in NMR
known as cyclops can be done by incrementing the rf pulse phase for four steps, every
step with 90◦. This will tilt the magnetization signals (Mx, My) in the different axises as
explained in Table (3.1.2). The receiver phase is always in the same position relative to
the magnetization. Since NMR signal change it is phase with the pulse phase, the data
(real and imaginary) added and the artifact that do not follow this phase cycling will
simply be canceled if the reviver phase is kept always the same. One should take care if
the output cables are reversed, adding the signal will give zero.
Cycle counter Phase of pulse Receiver phase Mx My
0 +X +X +Y +X
1 +Y +X -X +Y
2 -X +X -Y -X
3 -Y +X +X -Y
Table 3.1.2.: Four-steps phase cycle
If an FID is displaced from the baseline by a constant amount then the spectrum will
contain spurious resonance peaks around zero. Removing this DC offset is done by taking
the value of last 10% points and subtracting this value from each point in the FID, which
known as baseline correction[74].
Ground noise Ground noise can be characterized as coherent noise, for safety consid-
eration the chassis or enclosure of an electric equipment should be grounded. Ground
points should have well defined ground potential. In real systems the actual path taken
by the ground has different impedances. This is causes a difference in potential between
two physically separated ground points.
37
3.1. SPECTROMETER CONFIGURATION
Figure 3.1.13.: A typical ground loop circuit diagram
In Figure 3.1.13 there are two circuits sharing common wire connection to the ground.
If they share the ground at the same point, then (RG=0). In other words if V1 is isolated
from V2, then Vout=V2.
In real circuits cables with different lengths have different resistances, assuming RG 6=0,
a voltage drop from V1 occurs, then, VG = RG I1, VG applied to the second circuit and
added to the output voltage, which can be written by Eq. (3.1.18):
Vout = V2 − VG = V2 −RG(
V1
RG +R1
). (3.1.18)
Signal grounds are classified into three types as shown in Figure 3.1.14:
1. Single point grounds could be either parallel connections, also called separate ground
systems which are preferred to avoid noise or series connections which are called
common grounds and have the advantage of its wiring simplicity.
2. Multipoint grounds which is a good choice for high frequencies > 10 MHz to mini-
mize the ground impedance.
3. Hybrid grounds, this are combination between single point grounds and multipoint
grounds.
38
CHAPTER 3. EXPERIMENTAL SETUP
Figure 3.1.14.: Types of ground connections:(a_1) single point ground(common ground),
(a_2) single point ground(separated ground), b) multipoint ground, c)
hybrid ground
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
-4
-2
0
2
-750 -500 -250 0 250 500 750
0.0
4.5
9.0
13.5
18.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
-1.2
0.0
1.2
-750 -500 -250 0 250 500 750
0.00
0.65
1.30
1.95
2.60
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (sec)
 without optocoupler module
b)
Frequency (Hz)
M
a
g
n
i
t
u
d
e
 
(
a
.
u
.
)
a)
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (sec)
 with optocoupler module
Frequency (Hz)
M
a
g
n
i
t
u
d
e
 
(
a
.
u
.
)
Figure 3.1.15.: A typical ground loops signal: a) left side is a time domain signal without
using the optical coupler module and the right side shows its spectra , b)
left side is a time domain signal with using the optical coupler module and
the right side shows its spectra.
Switching all devices off, does not mean there is no source of noise active, as long as the
AC cable is connected to the power supply there could be typical ground loop noise in the
NMR circuits as shown in Figure 3.1.13 upper curves with blue color where the left part
shows the time domain noise coupled with the 50 Hz and the right side shows its spectra.
The main center peaks of the noise around 50 Hz and it is harmonic frequencies 100 Hz,
150 Hz ... etc. These ground loops are connecting through ground prongs. Removing this
third connection, i.e. prongs, would be helpful only for diagnosis where is the problem
39
3.2. SAMPLE PREPARATION
coming from, but this is not a safe solution in the long run. Connecting all sources at
the same ground point is some times hard to achieve, where the devices are very far from
each other. When the signal circuit is grounded at both end, the ground loop formed
is susceptible to noise form magnetic field or different ground voltages, then these loops
could be broken by using optical couplers.
Figure 3.1.16.: Optical coupler circuit used for reducing ground loops noise
Optical couplers behave as electrical isolators and disconnect the ground connection
from other ground loops because they transfer the signal as light as shown in the circuit
diagram in Figure 3.1.16[91]. In the 360 MHz spectrometer as we mentioned in Section
(3.1.2), the master clock is connected to the DDS and the Hameg synthesizer. The master
clock has 16 outputs generates square wave with frequency of 10 MHz for each output. All
outputs share the same ground. Ground loops are multiplied because of shared ground
with other circuits. Connecting the optical coupler module (shown in Figure 3.1.17),
which has four outputs (with separated ground) will contribute to reduce the 50 Hz noise
with a factor of two in the lower part as in Figure 3.1.13 with red color in comparison
with the upper part.
Figure 3.1.17.: Optical coupler module for the master clock
3.2. Sample preparation
3.2.1. Measurement of relaxation times T1
In pulse NMR, nuclear spins are detected through their free induction decay (FID). To
determine the maximum FID amplitude, which corresponds to a pi
2
pulse, different pulse
40
CHAPTER 3. EXPERIMENTAL SETUP
lengths have been applied and the pulse length is plotted against the FID amplitude. This
is known as a nutation experiment as shown in Figure 3.2.1. The analysis shows a pi
2
pulse
of 17.5 µs.
0 10 20 30 40 50 60 70
-4000
-2000
0
2000
4000
6000
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Pulse width ( s)
 experimental data
 sine fit of experimental data
Figure 3.2.1.: Nutation experiment
The magnetic field strength is related to the pulse power as in Eq. (3.2.1)
B1 ≈
√
(3PQ/ν0V ) ≈ 3.7
√
(PTR/V ), (3.2.1)
where:
P : transmitter power in watt, Q : quality factor, ν0 : frequency in MHz, V : coil volume
in cm3, TR : ring-down coil time in µs[92].
3.2.2. Radiation damping
For strong NMR signals (high concentrated sample with higher gyromagnetic ratio), the
strong signal induces a small rf current in the receiver coil. Because of the induction
principle, another radio frequency field Brad is created, which is always of the opposite
direction to B1, known as radiation damping field.
If the magnetization rotation is induced by the coil through applied rf field, then the
rotation is uniform given with the angular velocity:
(
dθ
dt
)
nut = γB1 = ω1, (3.2.2)
where θ is the angel between the magnetization and B1 field.
On the other hand the angular velocity of radiation damping is not uniform, but depends
on the sine function of θ given by:
(
dθ
dt
)
rad
= γBrad = −
sin(θ)
Trd
, (3.2.3)
41
3.2. SAMPLE PREPARATION
where Tr is characteristic radiation damping time which is defined by Eq. (3.2.4) and
the minus sign means the magnetization rotates in the opposite direction of the rotation
induced by B1field.
Tr = 1/(2piηγQM0), (3.2.4)
where: η is the filling factor,
Suppose we have an rf field due to rf pulse in +x-direction, the magnetization lies along
+y-direction, Brad lies along −x-direction. Brad accelerates the decay of the NMR signal
and broadens the resonance line. Thus the magnetic spin system is coupled to the coil,
which is used for detection. Radiation damping grows with the net magnetization and the
Q factor of the coil, so radiation damping is more relevant in high field magnet and rela-
tively high Q factor NMR probes, where radiation damping rate is linearly proportional
with the product QM0 (see Eq. 3.2.2). Radiation damping produces a back-action field
which tends to rotate the magnetization back to the equilibrium state[93, 94, 95, 96, 97].
The following experiments were done for different solutions to verify the radiation damp-
ing effect. In NMR experiments the bulk magnetization at thermal equilibrium is in the
z direction. Applying a pi
2
pulse, the bulk magnetization rotates from the z axis to the xy
plane. The spins need time to return to thermal equilibrium which is called the recycle
or repetition time. Usually this time is calculated to 1..5T1. Inversion recovery method
is used to measure T1 for different concentration of D2O/H2O, where the pulse sequence
is (pi, τ , pi
2
). The delay τ is varied to calculate T1. Figure 3.2.2 shows how the relaxation
rates 1/T1 are change depending on the ratios D2O/H2O. It shows a linear increment
with increasing H2O concentration. These results are matched well with the results of
Anderson et al. which are explained depending on the mutual dipole interaction among
the nuclei. [98, 76]
0 20 40 60 80 100
0.09
0.12
0.15
0.18
0.21
0.24
1
/
T
1
 
(
s
e
c
-
1
)
 
H
2
O concentration in the mixture D
2
O/H
2
O %
Figure 3.2.2.: Relaxation rate 1/T1 versus the relative volume concentration of the mix-
ture D2O/H2O.
In the case of strong radiation damping, that means Tr is too short (Tr < T ∗2 ), dif-
ficulties in defining the exact value of pi
2
pulse width arises because the nutation shape
42
CHAPTER 3. EXPERIMENTAL SETUP
deviates from the normal sinusoidal shape[99]. Figure 3.2.3 shows how the nutation curves
converges towards the normal sinusoidal function when the ratios between the mixture
D2O/H2O volumes increase. Increasing the ratio of D2O means decreasing of the proton
concentration consequently decreasing of the signal but fortunately decreasing the rate
of radiation damping. In Figure 3.2.3 at ratios 30% and 40% of D2O there is no strong
effect of radiation damping where they fit the sinusoidal curve compared to the 10% and
20% ratios.
0 20 40 60 80 100
-200
-150
-100
-50
0
50
100
150
200
 10% D
2
O in H
2
O
 20% D
2
O in H
2
O
 30% D
2
O in H
2
O
 40% D
2
O in H
2
O
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Pulse width ( s)
Figure 3.2.3.: Curves shows sine fit of nutation experiments for different mixtures of
(D2O/H2O) and symbols are experimental data.
3.2.3. Pure water
In pure water there are two types of interactions, first the intramolecular interaction which
is represented by dipolar interaction with other protons in the same molecule. Second
type of interaction is the intermolecular interaction caused by protons in neighboring
water molecules. These interactions fluctuate as water molecules diffuse in translation and
rotational motion. Paramagnetic Impurities molecules in water solution act as relaxation
centers. The dipolar interaction between the proton and the impurity ionic moment is
modulated by the relative motion of water and the ions. Adding paramagnetic ions such
as CuSO4 is useful to decrease the recycle time for H2O. In this experiment, a solution
H2O/CuSO4 with a concentration of 5 g/100 mL is added to H2O with different volume
ratios. The relaxation time T1 is inversely proportion with the H2O/CuSO4 volume ratio in
the sample respectively the CuSO4 concentration. Figure 3.2.4 shows how the relaxation
rates change with the CuSO4 concentration[75].
43
3.2. SAMPLE PREPARATION
0.01 0.1 1 10
10
100
1000
5 mg/100ml
T
1
 
(
m
s
)
CuSO
4 
concentration (g/100mL) 
T
1
= 287 ms
Figure 3.2.4.: Relaxation time T1 plotted versus concentration of CuSO4 in H2O
3.2.4. Tune up sequences and pulse parameters optimization
In addition to the probe tuning procedure in Section (3.1.6) further adjustment for the
pulse parameters before applying the multiple pulse sequences is useful to minimize the
effect of pulse error and ensure that parameters are correctly set. An experimental verifi-
cation by Burum et. al. shows that the so called "tune-up" pulse sequences are useful for
minimizing phase transients[100]. Since it is hard to define the flip angles pi, pi/2 in the
nutation experiment accurately as in Section(3.2.1), identical pi, pi/2 pulse sequences are
used. The sequences are (pix − τ − pix − τ) for pi and (pi/2x − τ − pi/2x − τ) for pi/2 .
44
CHAPTER 3. EXPERIMENTAL SETUP
0 2 4 6 8 10 12
-2
0
2
4
6
8
10
12
14
16
M
a
g
n
e
t
i
z
a
t
i
o
n
 
(
a
.
u
.
)
Time (ms)
 17.0 s
 17.8 s
 18.6 s
 19.4 s
 20.3 s
 21.0 s
Figure 3.2.5.: Typical response from identical pi/2 pulse sequences (24 pulses) for different
values of pulse durations
Measuring the magnetization after each pulse for identical pi/2 pulses shows a typical
response as in Figure 3.2.5. The first pi/2 pulse rotates the magnetization from the
z direction to the xy plane (through horizontal lines above the base line for each pulse
length the maximum magnetization is measured), a second pi/2 returns the magnetization
again to the z (measure zero for the transverse magnetization i.e. the base line for each
pulse width). According to Figure 3.2.5 the optimal pi/2 value is 19.4 µs, which shows
symmetric shape around the baseline. As a result of the B1 field inhomogeneity the
magnetization values decays to minimum after 24 pulses with interpulse delay τ = 500 µs.
Similar optimization is done varying pi values and the minimum value of the magnetization
represents the optimized value of pi pulse.
The optimal responses of identical pulses pi, pi/2 sequences are shown in Figure 3.2.6,
for both pi, pi/2 sequentially, where the optimal pulse lengths are obtained.
Here are three tune-up sequences used for different purposes and their results correspond
to each sequence:
1. Flip flip flop with x prepulse sequence [pi/2x−τ − (pi/2x−τ −pi/2x−τ −pi−x−τ)] is
used to adjust the relative phase of [x] and [-x] pulses. A typical response is shown
in Figure 3.2.7
45
3.2. SAMPLE PREPARATION
0 2 4 6 8 10 12 14 16
-0.2
-0.1
0.0
0.1
0.2
0.3
M
a
g
n
e
t
i
z
a
t
i
o
n
 
(
a
.
u
.
)
Time (ms)
a)
0 2 4 6 8 10 12 14 16
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
b)
M
a
g
n
e
t
i
z
a
t
i
o
n
 
(
a
.
u
.
)
Time (ms)
Figure 3.2.6.: Typical response of identical pulses pi, pi/2 sequences
0 2 4 6 8 10 12 14 16 18 20
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
M
a
g
n
e
t
i
z
a
t
i
o
n
 
(
a
.
u
.
)
Time (ms)
Figure 3.2.7.: Typical response of Flip flip flop sequence with prepulse
2. Flip flip flop without prepulse sequence (pi/2x − τ − pi/2x − τ − pi−x − τ) is used to
eliminate asymmetric phase transition. A typical response is plotted in Figure 3.2.8
46
CHAPTER 3. EXPERIMENTAL SETUP
0 2 4 6 8 10 12 14 16 18 20
-0.5
0.0
0.5
1.0
1.5
M
a
g
n
e
t
i
z
a
t
i
o
n
 
(
a
.
u
.
)
Time (ms)
Figure 3.2.8.: Typical response of Flip flip flop sequence without prepulse
3. Flip flop sequence (pi/2x − τ − pi/2−x − τ) is used to verify the tuning of the probe
circuit. A typical response is shown in Figure 3.2.9
0 2 4 6 8 10 12 14 16 18 20
-0.5
0.0
0.5
1.0
1.5
M
a
g
n
e
t
i
z
a
t
i
o
n
 
(
a
.
u
.
)
Time (ms)
Figure 3.2.9.: Typical response of Flip flop sequence
3.2.5. Pulse visualization
A useful way to test the pulse sequences, in multiple pulse NMR experiments before
running experiments, is to see the pulses applied to the probe. This test can be done
by measuring the pulses through the demodulator, to verify if the pulses are identical
to the intended pulse sequences. Figure 3.2.10 shows pulse sequences for different DD
sequences whereas pulses are measured after the modulator using an external DDS as a
reference to the demodulator to see the phase differences4. The detected pulse sequences
are measured using phase sensitive detection method, i.e. two channels outputs which
represent different phases of the pulses in red and blue as plotted in Fig. 3.2.10. The first
4The demodulator and modulator used the same DDS for the transmitting and receiving signals. If we
want to see all phases we have to use an external frequency source, e.g. another DDS, then the change
in the phase will not depends on the external source.
47
3.2. SAMPLE PREPARATION
pi/2 pulse in each sequence does not belong to the DD sequence, it is only prepares the
initial condition for the DD sequence.
0.0 0.2 0.4 0.6
-0.4
0.0
0.4
0.0 0.2 0.4 0.6
-0.4
0.0
0.4
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-0.4
0.0
0.4
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-0.4
0.0
0.4
0.0 0.2 0.4 0.6 0.8
-0.4
0.0
0.4
0.0 0.2 0.4 0.6
-0.4
0.0
0.4
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
XY4(s)
XY4(a)
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
XY8(s)
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
XY8(a)
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
KDD
x
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
KDD
xy
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
Figure 3.2.10.: Pulse sequences time line visualization for different DD sequences through
the demodulator. The first cycle is shown for each sequence, in addition
to the pi/2 pulse, except for the KDD sequence where only half of the first
cycle is shown. The second half of the cycle is similar to the first half in
the KDD sequence . The blue pulse is the X component and Y component
is the red pulse, the phase is calculated as tan−1
(
Y
X
)
.
The phase sensitive receivers aren’t usually set to exactly 90◦ phase from each other.
To measure the phase difference of our receiver precisely, an artificial signal is used, and
the frequency difference from the reviver is detected. Figure 3.2.11 shows the measured
spectra before phase calibration with a phase error 5%. Imperfect phases lead to ghosts
"unwanted signals folded in the other region of the spectrum". The phase error effect can
be defined as the ratio between the real and ghost peak amplitudes. The phase error is
reduced to 1% by changing the cable length as shown in the inset of Fig. 3.2.11.
48
CHAPTER 3. EXPERIMENTAL SETUP
-10 -5 0 5 10
0
20
40
60
80
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Frequency offset (kHz)
 Before phase calibration
 After phase calibration
-6.0 -5.5 -5.0 -4.5 -4.0
0
2
4
Figure 3.2.11.: Experimental phase calibration for a quadrature NMR receiver, the ghost
peak has the same frequency of the real peak but with negative sign. The
inset zooms the section to compare between the ghost peak before (black)
and after calibration (red).
49

4. Dynamical decoupling
In this chapter we will describe general types of decoherence. Hence noise causes deco-
herence, we will then discuss the source of noise in our system. Next we address the DD
sequences and how they can be designed to fight decoherence. Then we investigate the
influence of pulse error on the effect of DD. Then we talk about a constructed experi-
mental setup to introduce a modulation field to engineer the environment. Under these
new environmental conditions we discuss the effect of this modulation field under a DD
sequence. To complete the picture we give a theoretical analysis by average Hamiltonian
theory for our experimental investigation under the influence of pulse errors.
4.1. Fighting against decoherence
Quantum computation on real quantum processors may be influenced by two sources: in-
ternal and external imperfections. Internal imperfections are imperfections due to "hard-
ware imperfections", such as imperfections in the setting of the initial state, gate oper-
ations and/or final measurements. Uncontrollable interactions cause the magnetization
to decay. Decoherence, can be defined as uncontrollable interactions with environmental
degree of freedom. These uncontrollable interactions lead to a deviation between exper-
iments and the ideal performance which is described by a unitary operator. There are
fundamental limits of the experimental strategy against decoherence or system isolation
that can be achieved to some degree of precision. External impact such as cosmic rays
or the gravitational force which may changes the state of a qubit, are out of our con-
trol because it is hard to avoid their effects by shielding but some external impacts like
the temperature are under control. Another experimental consideration is the amount
of the power deposited in the system which often must be kept small to avoid heating
effect on the sample1 or damage the probe [66]. We can distinguish between four kinds
of decoherence concerning the type of the coupling between the system and the environ-
ment [5, 101, 15, 30]
1. Total decoherence: this is a general case for decoherence where the coupling operator
between the system and the environment has no restrictions.
2. Independent decoherence: here the coupling operator acts on individual qubits.
3. Collective decoherence: this process is called also collective dephasing, because the
mutual phases between all qubits are destroyed, so the coupling operator acts si-
multaneous on all qubits.
1Liquid NMR samples are always subjected to evaporation, this can be avoided by well sealing the NMR
tube, e.g., flame the top of the glass tube.
51
4.2. OUR SYSTEM AND NOISE SOURCES
4. Cluster decoherence: in this case clusters of qubits decohere collectively and the
different clusters decay independently.
A simple explanation to decoherence in liquid NMR systems can be given through re-
laxation times. The spin-lattice time T1 is relevant to the energy dissipation between
the system and its surrounding environment, also called the longitudinal relaxation time
because it reduces the longitudinal magnetization. On the other hand the spin-spin re-
laxation time T2, the phase damping time, is limited in liquid NMR systems by T1. T2
is also named the transversal relaxation times because it reduces the transversal magne-
tization [30, 10].
4.2. Our system and noise sources
Our experimental system is an ensemble of non-interacting spins 1/2. They consist of
the protons of a water sample to which we added 5 mg/100 ml CuSO4 to reduce the
longitudinal relaxation time T1. Figure 4.2.1 shows a typical experimental result of the
inversion recovery pulse sequence pi(X) − τ − pi/2(X) − (tacqu) [74]. The magnetization M0
is inverted with a pi pulse, then during a time τ it allowed to relax along +z-axis. Then
the magnetization is measured by applying a pi/2 pulse. T1 is determined by fitting the
magnetization curve to
Mz(t) = M0(1− 2 exp(−τ/T1)). (4.2.1)
The horizontal axis (Figure 4.2.1) represents the increment in the time τ between the
pulses where the vertical axis represents the measured values of the magnetization by
applying pi/2 pulses each time. The experimental repetition time was 5 s, which is large
enough compared with T1.
0 400 800 1200 1600
-2
-1
0
1
2
3
A
m
p
l
i
t
u
d
e
 
(
V
)
Time (ms)
 amplitude
 exponential fit
Figure 4.2.1.: Magnetization growth of the 1H spin system after inversion recovery pulse
sequence to determine the longitudinal relaxation time T1
52
CHAPTER 4. DYNAMICAL DECOUPLING
The longitudinal relaxation time T1 in this case was 287 ms. As previously mentioned
the addition of CuSO4 results in a faster repetition times and shorter overall duration of
the experiments, to run the DD experiment. The sample was placed in a static magnetic
field along the z-direction and its Hamiltonian is
Hs = ωsSz, (4.2.2)
where ωs is the Zeeman frequency and Sz is the system spin operator along the z axis.
Firstly the inhomogeneities of the static field correspond to a static perturbation, and
molecular motion makes this perturbation time-dependent on a time scale that is slow
compared with the delays between the DD pulses used in our experiments. This condition
makes it possible to refocus this perturbation very effectively.
The second major source of noise is the fluctuating dipole-dipole interaction, whose
correlation time is the molecular reorientation time (≈ 35 fs), much faster than any
conceivable control fields for nuclear spins and therefore not amenable to DD. On the other
hand, these fluctuations are so fast that their average effect on the system is relatively
small [102].
The experiments discussed in this chapter were performed on a home-built NMR spec-
trometer on the H-channel with a 1H resonance frequency of 360 MHz (see the third
chapter for the spectrometer configurations). To cancel the normal drift of the supercon-
ducting magnet and other abnormal effects, the transmitter frequency is calibrated. All
experiments were performed on resonance unless otherwise stated. The radio frequency
field strength was 2pi · 13.3 kHz, which corresponds to a pi−pulse duration of 37.5 µs. An
initial state ∝ Ix or Iy was prepared by rotating the Iz equilibrium state with a resonant
pi/2 pulse.
4.3. How to construct a robust DD sequence?
DD sequences consist of repetitive trains of pi-pulses. Figure 4.3.1 shows a schematic
representation of a general DD pulse sequence. Errors in calibration of the pulse length
make the implementations different from expected. Our goal is to implement unitary
transformations, avoiding decoherence. The DD pulses are applied to the qubits to cancel
or minimize errors by averaging them out. The delays between the pulses and their phases
are important parameters for improving the performance of the DD sequences [69, 70, 103,
104, 21, 67, 22, 23].
53
4.3. HOW TO CONSTRUCT A ROBUST DD SEQUENCE?
 
 
 
 
 
  
φ1 𝜏𝜏0 
M 𝛑𝛑 -pulses  𝜏𝜏𝑁𝑁  
𝑡𝑡𝑐𝑐  
𝜏𝜏1 φ2 φN Initial 
state 
Final 
state 
Figure 4.3.1.: Schematic representation of a general DD pulse sequence. One cycle con-
tains N pulses and the cycle is repeated M times. The light blue squares
are pi pulses and the cycle time is tc.
In particular the relative phases, which correspond to the directions of the rotation axes,
are important for making the sequences robust against pulse imperfections and unwanted
environmental interactions [40, 36, 42].
In experimental DD sequences, the nonidealness in implementing unitary transforma-
tions is either from flip angle errors and/or offset errors. In order to construct a robust
DD sequence there are different ways: [66]
1. Improving the performance of pulses inside the DD sequence: this is can be done
by using composite pulses which are robust against rf offset errors or pulse width
errors. Another approach is to implement shaped pulses. In Ref. [105] the aim
of the work was to study the performance of different shaped pulses to implement
them in DD experiments [106]. In the previous study (i.e. Ref. [105]) the shaped
pulses were divided into two parts composite shaped pulses and continuous shaped
pulses. Because shaped pulses are often quite long, the effect of decoherence is
dramatically stronger. These results in Ref. [105] were not conclusive because in that
particular experiment radiation damping effect was not compensated. The effect of
the radiation damping precludes determining the pi pulse value when performing the
nutation experiment. Recently Grace et al. showed that it is possible to improve the
control of QIP by optimizing pulses when combining two criteria: the decoupling
pulse criteria and the optimal control theory [107].
2. Making the DD sequence robust itself: this can be done either by optimizing the
phases of the pulses by using different rotations such as the KDD sequence instead
of the identical rotations axes like the CPMG sequence, or by variation of the
delays between the pulses to be unequal. The non-equidistant pulse sequence was
firstly proposed by Uhrig, Uhrig dynamical decoupling (UDD) sequence [23]. It
worth noting that this variation between pulses can also be randomly or hybrid as
suggested by Ref. [108] [60, 66, 30, 107]. Another approach for construction the DD
sequences can be by numerical optimization based on genetic algorithm. Genetic
algorithm is an approach for optimization and solving problems based on the natural
evolution [109].
54
CHAPTER 4. DYNAMICAL DECOUPLING
4.4. Initial experiments2
We start making comparison between two dynamical decoupling sequences CPMG and
UDD. Both of them have identical rotation axes. The UDD sequence was first discovered
by Uhrig for a pure dephasing spin-boson model, describing a spin 1/2 particle (two level
system) interacting with an environment of large reservoir of bosonic field modes [110, 23].
The pulse timing in the sequence is governed by
τj = tc sin
2
(
pij
2(N + 1)
)
, (4.4.1)
where N is the UDD order which defines the number of pulses per one cycle, j takes
the numbers 1,2,...,N , tc is the cycle time.
It is obvious from Eq. (4.4.1) that UDD2 coincides with CPMG. In order to have
different conditions for the environment, we incremented z1 (shim) values from -2739 to
4390 with 325 steps, this increment changes the field inhomogeneity. The line-width is
dominated by field inhomogeneity, thus the line-width values can be a good measurement
for the field inhomogeneity. The line-width values are obtained by measuring full width at
half-height (abbreviated: FWHM) from the Fourier transform of the time domain signal.
-3000 -2000 -1000 0 1000 2000 3000 4000 5000
-20
0
20
40
60
80
100
120
140
160
F
W
H
M
 
(
H
z
)
z1 (units)
y=a+bx
Figure 4.4.1.: The line-width (static field inhomogeneity) changes versus gradient strength
values (z1). The black curves are experimental data while the blue lines are
the calculated data.
Figure 4.4.1 shows the change of the z1 (negative and positive values) versus the FWHM
values. We measured the slopes of the tangent of the curves (black) to define the change
in the inhomogeneity. For instance the minimum value (shown with the green arrow in
Fig. 4.4.1 @ z1=1390) represent the highest homogeneity of the magnetic field along z
2These initial experiments were done before the described opti-
mization procedures (i.e. the tune-up sequence, shorten T1by
adding CuSo4, the Pulse visualization)
55
4.5. FID AND HAHN ECHO SEQUENCE
axis. Table 4.4.1 gives the calculated slopes values and their intercepted values fitted to
Eq. 4.4.2.
y = a+ bx, (4.4.2)
where: y≡ FWHM, x≡ z1 values, a≡ intercept from the y axis, b≡ slope.
Intercept Slope
Symbol a b
Line (left side) 6.8 -0.034
Line (right side) -28.3 0.020
Table 4.4.1.: Slope calculations according to z1 changes versus line-width values
50 100 150 200 250 300 350 400
0.5
1.0
1.5
2.0
D
e
c
a
y
 
t
i
m
e
 
(
s
e
c
)
Delay between pulses (ms)
 CPMG (Z1=1390)
 CPMG (Z1=3000)
 CPMG (Z1=4000)
 UDD3 (Z1=1390)
 UDD3 (Z1=3000)
 UDD3 (Z1=4000)
Figure 4.4.2.: Decay times comparison between CPMG and UDD3 under different condi-
tions of field inhomogeneity (z1)
Figure 4.4.2 shows a comparison between the UDD pulse sequence to the third order
(UDD3) and CPMG pulse sequence. Changes of z1 values cause a drift in the decay
curves for different delays between pulses. However the CPMG sequence still outperforms
the UDD3 sequence. The UDD sequence performance is strongly environmental spectral
density dependent. As an example, experimental studies of the UDD sequence in a model
atomic system shows that UDD sequence can dramatically outperform CPMG sequence
in the presence of noise with an ohmic system and sharp cut-off [111, 34]. This is in good
agreement that the UDD sequence cannot outperform the CPMG sequence in a systems
with a soft cut-off frequency [36].
4.5. FID and Hahn echo sequence
Spin echo was discovered first by Erwin Hahn when two rf pulses are applied at a maximum
rf field B1 (pi/2 pulses) separated by time intervals τ [112]. The physics behind the
56
CHAPTER 4. DYNAMICAL DECOUPLING
experiment was explained in Hahn’s famous paper in 1950 [31]. The pattern of the echo is
always symmetric regardless of the spacing between the pulses. Refocusing is an important
principle related with spin echo. The refocusing causes the phase differences of all spin
vectors (with respect to the initial orientation established by the first pulse) to canceled
effectively at the time where the echo is maximum. . The echo can never repeat again
without applying another pulse. Figure 4.5.1 shows the Hahn echo pulse sequence applied,
the experimental data shows the magnetization amplitude (black on y-axis and red on
x-axis) after the pi/2 pulse (blue rectangle above) and the pi-pulse (green rectangle above).
The second small echo, marked with the blue arrow in Fig. 4.5.1 at the end of the time
line, is possibly related to the radiation damping effect. This can be explained as follows.
Radiation damping acts as a new "pulse" after the pi pulse and causes this second small
echo.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-6
-4
-2
0
2
4
6
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Evolution time (ms)
Figure 4.5.1.: Hahn echo experiment and the pulse sequence. The second echo at the end
of the time evolution is induced by the radiation damping effect.
The free evolution decay of the transversal magnetization of our system has a decay
time of 2.9 ms (FID). A simple Hahn-echo sequence [31] increases this time to 106 ms,
as shown in Fig. 4.5.2.
57
4.6. DYNAMICAL DECOUPLING SEQUENCES
 Hahn echo decay
 Exponential fit (HE)
 Free evolution
0.1 1 10 100 1000
0.00
0.25
0.50
0.75
1.00
T
HE
2
=106 ms
N
o
r
m
a
l
i
z
e
d
 
m
a
g
n
e
t
i
z
a
t
i
o
n
Evolution time (ms)
T
FID
2
=2.9 ms
Figure 4.5.2.: Decay of the magnetization of the 1H spin system under free evolution and
in a Hahn spin-echo (HE) sequence.
4.6. Dynamical decoupling sequences
In earlier works using these sequences, the conditions were chosen such that the dominant
perturbation was the environmental noise [36, 42, 62, 64, 65]. In this work, we focus on a
system that allows us to make a comparison between these sequences in a regime where all
sequences perform equally well at eliminating the environmental noise and any differences
in their performance can be attributed directly to their robustness, i.e. to their efficiency
in suppressing the effect of pulse imperfections.
58
CHAPTER 4. DYNAMICAL DECOUPLING
 
2 
2   
0࢓
 
30
࢓
 
   
30
࢓
 
0࢓
 
90
࢓  
 
       
    
    
12
0࢓ 
    
    
90
࢓ 
    
    
18
0࢓
    
    
90
࢓ 
12
0࢓30࢓0࢓90࢓0࢓30࢓
  
x yy
x
 
y
 
x
 
y
 
y
 
x
 
y
 
x
 Ini
tia
l   s
tat
e  
 pr
ep
ara
tio
n 
x
 
CPMG 
XY4(s) 
XY8(s) 
KDDx 
XY8(a) 
KDDxy 
XY4(a) 
ૌ૛ ૌ૛ 
߬ ߬ ߬ 
߬ 
߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬
 ߬ ߬ ߬ ߬ ߬ ߬ ૌ૛ ૌ૛ ૌ૛ ૌ૛ 
߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ߬ ૌ૛ ૌ૛ ૌ૛ 
    
    
    
    
12
0 ૌ૛ 
y
x yy
x
 
yx
 
y
x
y x y x
 
y
Figure 4.6.1.: Dynamical decoupling pulse sequences.
Figure. 4.6.1 gives an overview of the sequences that we examined to see the effect of
pulse error on the performance of DD sequences. It shows a single cycle for each sequence,
which is repeated as often as required. τ is the delay between the pulses. The Carr -
Purcell (CP) sequence [69] and the version by Meiboom and Gill [70], well known in
NMR community (CPMG), use the same sequence of refocusing pulses; they differ only
with respect to the state to which they are applied. In the case of the CP sequence,
the initial state is perpendicular to the rotation axis of the inversion pulses, while in the
CPMG version, it is parallel. Errors in the flip angles leave the longitudinal component
unscathed, but they destroy the perpendicular component [70, 36].
The sequence XY4 was introduced by Maudsley [103] and it reduces the effect of pulse
imperfections for arbitrary initial states [103, 104, 36]. It consists of four pulses with
59
4.6. DYNAMICAL DECOUPLING SEQUENCES
phases x − y − x − y (Fig. 4.6.1). An asymmetric version of the XY-4 sequence was
introduced by Viola et al. [21], which is designated XY4(a). The XY8-sequences are sym-
metrized versions of the XY4 sequences [104, 64]. Two DD sequences that are particularly
robust against flip-angle and resonance offset errors are the KDDx and KDDxy sequences.
KDD sequences are inspired from robustness of composite pulses [42, 65, 66]. KDDxy is
designed by combining the rotation pattern of the XY4 sequence with that of a robust
composite pulse [113].
4.6.1. Data analysis
The extracted echoes (blue squares in Fig. 4.6.2) were fitted with an exponential function
to obtain the decay time of the magnetization. Experiments were repeated for different
delays between pulses, different DD sequences and initial conditions. Then the decay
times plotted as a function of the delay between pulses. Figure 4.6.2 shows the echo train
of a CPMG sequence. From these data, we extracted the signal (Ix magnetization in this
example) at the end of each DD cycle (marked with blue squares in Fig. 4.6.2).A Matlab
(Version 7.6) code is written to extract these points. Actually there were two different
protocols to extract the values:
1. The echo maxima value during each cycle: this is easy but has the disadvantage
that the transition or noise maybe greater than the echo maximum values.
2. Middle point between the pulses: this the most precise method and need precise
calculations of the cycle time and the position of the echoes.
The decay of the echoes was mostly exponential fit to Eq. (4.6.1), with some exceptions
discussed below.
M = M◦ exp(
−t
T2
), (4.6.1)
where:
M0 : the maximum magnetization and T2 : the transversal relaxation time decay
constant.
60
CHAPTER 4. DYNAMICAL DECOUPLING
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
N
o
r
m
a
l
i
z
e
d
 
m
a
g
n
e
t
i
z
a
t
i
o
n
Evolution time (sec)
t
c
Figure 4.6.2.: Time evolution of the spin-system magnetization under the application of
a CPMG sequence. The black solid line shows the evolution of the mag-
netization and the blue squares mark the echo amplitude at the end of a
CPMG cycle. The echo maxima are used in this case for measuring the
CPMG decay time. The cycle time was tc = 32 ms.
4.6.2. Robustness comparison under pulse errors
To compare the sensitivity of the sequences to pulse imperfections, we prepared two
orthogonal initial states Ix and Iy and then measured their decay as a function of time
under the application of the different DD sequences described in Sec. (4.6).
0 2 0 4 0 6 0 8 0 1 0 0 1 2 00 . 0 00 . 0 50 . 1 0
0 . 1 50 . 2 00 . 2 50 . 3 0 F r e e  e v o l u t i o n  H a h n  e c h o  Decay time (s) D e l a y  b e t w e e n  p u l s e s  ( m s )
 C P M G C P X Y 4 ( s ) X Y 4 ( a ) X Y 8 ( s ) X Y 8 ( a ) K D D x K D D x y
Figure 4.6.3.: Average decay times as a function of the delay τ between pulses for different
DD sequences .
Figure 4.6.3 compares the decay times of different DD sequences as a function of the
delay between pulses. For the CPMG sequence, we present the decay of the Ix and Iy
magnetization separately, marked as CP and CPMG, respectively. Thus CPMG, CP
sequences are very sensitive to the initial condition. For the other sequences, their perfor-
61
4.6. DYNAMICAL DECOUPLING SEQUENCES
mance is quite symmetric with respect to the initial condition. For long delays between
the pulses, the observed decay times reach a limiting value of ≈ 276 ms, irrespective of
the sequence and the initial condition, and very close to the measured value of T1. This
is a verification of the assumption that all sequences can effectively decouple the slowly
fluctuating environment.
For shorter pulse delays (i.e. more pulses in a given time interval), the signal decays
more rapidly. This is most prominent for the CP sequence. In this situation, pulse
imperfections add coherently and generate a rapid loss of magnetization [36].
As the pulse delays become shorter than 0.5 ms, which corresponds to 864 pulses during
the 0.5 s measurement time, the other sequences also start to generate shorter decay times,
and their decays become nonexponential. Figure 4.6.4 shows a representative example of
such a signal. It can be fitted with a double exponential,
M(t) = a e−t/T
f
2 + b e−t/T
s
2 (4.6.2)
with two decay times T f2 and T
s
2 .
0.0 0.1 0.2 0.3 0.4 0.5
0.1
1
be
-t/T
s
2
Slow decay
N
o
r
m
a
l
i
z
e
d
 
m
a
g
n
e
t
i
z
a
t
i
o
n
Evolution time (sec)
Fast decay
ae
-t/T
f
2
Figure 4.6.4.: Normalized spin magnetization as a function of the evolution time for short
delays (τ = 100µs) between the pulses for an XY8(s) sequence. Pulse errors
dominate here, inducing a multi-exponential decay. The red solid line is a
fit to eq. (4.6.2).
62
CHAPTER 4. DYNAMICAL DECOUPLING
0.1 0.2 0.3 0.4 0.5
1
10
100
D
e
c
a
y
 
t
i
m
e
 
(
m
s
)
Delay between pulses (ms)
 CPMG
 CP
         T
s
2
 XY4(s)
 XY4(a)
 XY8(a)
 XY8(s)
 KDD
x
         T
f
2
 XY4(s)
 XY4(a)
 XY8(s)
 XY8(a)
 KDD
x
Figure 4.6.5.: Fitted decay times for the double-exponential decay
Figure 4.6.5 shows the decay times fitted with eq. (4.6.2) for different average delays
between pulses. For the fast decay times (T f2 ), which are represented by empty symbols in
Fig. 4.6.5, the performance of all DD sequences is quite similar. For the slow component
(T s2 , represented by filled symbols), XY8, CPMG, and KDDx perform better than XY4.
As we stated before, the decay time is dominated by the effect of pulse errors for
short delays. The resulting average Hamiltonian projects the magnetization onto its
eigenbase; this results in the fast decay component. After this projection, the remaining
magnetization, which is not significantly affected by the pulse imperfections, decays on a
slower timescale, which is dominated by the environment.
For a quantitative comparison of the different pulse sequences, we calculate the average
magnetization decay resulting per pulse of the sequence. For this evaluation, we consider
only the short-time component described by T f2 .
Since the pulse error is the dominant source of decay under these conditions, we quantify
its effect by measuring the fractional decay of the magnetization per pulse. The pulses
are the same for all the DD sequences, but their effect, averaged over full cycles, shows
how well the sequence is able to cancel the imperfections of the individual pulses.
Figure 4.6.6 shows the average decay per pulse for the different sequences, plotted
against the number of pulses. For these data, the interpulse delay was τ = 100 µs. The
most conspicuous feature is that CP performs very badly and CPMG very well. The
compensated sequences lie between these two extremes, and we find that the higher order
sequences (XY8, KDD) perform better than the lower order sequences (XY4). For un-
known initial conditions, KDD shows the best performance. Under the present conditions,
sequences that differ only with respect to time reversal symmetry perform quite similarly,
in contrast to other cases discussed earlier [64].
63
4.6. DYNAMICAL DECOUPLING SEQUENCES
 XY4(a)
 XY4(s)
 XY8(a)
 XY8(s)
 KDD
x
 KDD
xy
0 10 20 30 40 50 60 70 80
10
-4
10
-3
10
-2
10
-1
CPMG
A
v
e
r
a
g
e
 
e
r
r
o
r
 
p
e
r
 
p
u
l
s
e
 Number of pulses
CP
Figure 4.6.6.: Average error per pulse for different DD sequences with delay τ =100µs.
30 32 34 36 38 40 42
0.0
0.1
0.2
0.3
0.4
D
e
c
a
y
 
 
t
i
m
e
 
(
s
e
c
)
Pulse length ( s)
 CP
 CPMG
 XY4(a)
 XY4(s)
 XY8(a)
 XY8(s)
 KDD
x
Figure 4.6.7.: Decay times of different DD sequences as a function of the pulse length of
the DD pulses. The delay between pulses is 16 ms.
The DD sequence performance is sensitive to flip angle errors. We performed an ex-
periment where we changed the pulse length and measure the decay times for each DD
sequence. Figure 4.6.7 shows a comparison between decay times for the DD sequences
versus small deviation in pulse length from the optimal pi-pulse value where the average
delay between pulses is 16 ms. The light gray rectangle (Fig. 4.6.7) marks the optimal
values for the pulse length which correspond to the longest decoherence times.
64
CHAPTER 4. DYNAMICAL DECOUPLING
4.7. Engineering the environment by external modulation
field
The z shim generates a constant field along z axis. It would be interesting to have
time-dependent field along z axis, for this purpose we change the experimental setup by
building a new coil to generate an external field as explained below.
4.7.1. Experimental modification to generate additional field
perturbation
In order to build an additional coil to generate an external time-dependent field along
the z axis, the position of the rf coil inside the probe is determined. In liquid NMR
spectrometer’s 5 mm probes (i.e. NMR sample tube diameter) the sample length should
be above and below the active region of the probe’s coil, i.e. longer than 2 cm (length
of the rf coil), in this case the sample-to-sample variations on the field homogeneity
will be negligible. Otherwise the air has a different susceptibility which can affect the
magnetic field homogeneity in the active region of the rf field. Another solution is to use
susceptibility plugs matched with the NMR sample.
10 15 20 25 30
50
100
150
200
250
300
S
i
g
n
a
l
 
a
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Sample hight inside the NMR probe-head (mm)
NMR     tube
2 mm
5 mm
Plugs
Figure 4.7.1.: NMR signal amplitude at different positions in the NMR probe-head along
the z axis. In the right side is the NMR tube where plugs (yellow) are used
to restrict an NMR sample of 2 mm length.
First the position of the rf coil is determined by moving the sample vertically along the
NMR probe and measuring the NMR signal at different positions. Magnetic susceptibility
plugs of polyetherimide is used to restrict the water sample to which we added 5 mg/100 ml
CuSO4. The length of the sample was 2 mm. Figure 4.7.1 shows the experimental data
for an NMR signal at different positions in the NMR probe-head along the z axis, the
highest signal located when the sample passes through to the coil position inside the NMR
probe-head (in the yellow cylinder Fig. 4.7.2). Figure 4.7.2 shows the construction method
of building the new coil to generate an additional time-dependent field along the z axis.
65
4.7. ENGINEERING THE ENVIRONMENT BY EXTERNAL MODULATION FIELD
The new coil is a Helmholtz coil: a pair of circular coils on a common axis with equal
currents flowing in the same sense is built at the center of the probe where the sample is
located vertically along the z direction. An arbitrary wave generator (AWG) is used to
generate a time-dependent signal, which is amplified and sent to the new coil to generate
the desired field.
New coil
rfsaddlecoil
B1Ampl
ifier
-DC
P26
0/30
P
(Ser
vow
att)
AW
G 
(Tab
orW
W5
064
) Z 
Figure 4.7.2.: Block diagram of the rf coil inside the NMR probe-head and the new coil is
winded around the probe head and the devices for generating an additional
time-dependent field (arbitrary wave generator [AWG] and an amplifier).
Figure 4.7.3 shows experimental data for calibration of the new setup to generate an
additional filed using the new coil. Increment the DC voltage at the input of the amplifier
(i.e. output of the AWG) results in linear increment of the offset frequency and current
through the new coil.
66
CHAPTER 4. DYNAMICAL DECOUPLING
0 10 20 30 40 50 60
0
2
4
6
8
10
  Offset frequency shift
 Amplifier output current
Input voltage (mV)
O
f
f
s
e
t
 
f
r
e
q
u
e
n
c
y
 
(
k
H
z
)
0.0
0.1
0.2
0.3
0.4
0.5
O
u
t
p
u
t
 
c
u
r
r
e
n
t
 
(
A
)
Figure 4.7.3.: Experimental data of the offset frequency for a proton (H2O) NMR spec-
trum (blue squares) and current through the new coil (red squares) upon
variation of the input voltage to the amplifier (used to generate current
through the new coil).
4.7.2. Simulation of the FID signal during the sinusoidal modulation
of the static magnetic field
The above discussion shows how the setup is optimized, e.g., to avoid the non-linearity in
the output field generated by the new coil. In this subsection we will describe briefly the
simulation procedure and compare simulation results with experimental data.
4.7.2.1. Simulation procedure
A numerical simulation program using Matlab Version 7.6 for calculating the NMR time-
domain signal or frequency-domain spectrum involves the following steps:
If we consider spin 1/2 system, the spin operators are :
Sx = 0.5
[
0 1
1 0
]
; Sy = 0.5
[
0 −i
i 0
]
; Sz = 0.5
[
1 0
0 −1
]
. (4.7.1)
The transverse spin magnetization Sm , is the sum of the magnetization on x axis (Sx)
and the magnetization on y axis (Sy) by Eq. (4.7.2):
Sm = Sx + iSy. (4.7.2)
The quantum state of the entire ensemble of the isolated spins 1/2 can be described us-
ing the spin density operator matrix ρ without referring to the individual spin states. Us-
ing the differential Liouville-von Neuman equation mentioned in chapter two, Eq. (2.1.5)
we can rewrite
67
4.7. ENGINEERING THE ENVIRONMENT BY EXTERNAL MODULATION FIELD
d
dt
ρ(t) = −i[H(t), ρ(t)], (4.7.3)
which can be solved similarly as Eq. (2.1.16) to give
ρ(t) = U(t)ρ(0)U−1.(t) (4.7.4)
However U is defined by Eq. (4.7.5):
U(t) = T exp
{
−i
ˆ t
0
H(t′)dt′
}
, (4.7.5)
where
U−1 is the complex conjugate transpose of the operator U .
T is the Dyson time ordering operator, which is defined to solve the exponential function
at different times when they do not commute with each other.
If we introduce an additional perturbation which can be defined by a sinusoidal function
oscillating along the static magnetic field then we can write the Hamiltonian as
H = A(sin(Ωt′ + φ))Sz + Ω0Sz, (4.7.6)
where:
Ω, φ, A = are consecutively the frequency, the initial phase, and the amplitude of the
sinusoidal perturbation.
Ω0 = ωl−ωoff is the offset difference between the Larmor frequency and the transmitter
rf frequency.
Assuming that Ω◦ = 0, we can write:
ˆ t
0
H(t′)dt′ =
ASz
Ω
[− cos(Ωt′ + φ)]t
0
, (4.7.7)
where: the integration is limited by the time interval of the sine wave t, this gives:
ˆ t
0
H(t′)dt′ =
ASz
Ω
(− cos(Ωt+ φ) + cosφ) . (4.7.8)
We can write the evolution operator from Eq. (4.7.5) using Eq. (4.7.8):
U(T ′) = e
−iASz
Ω (− cos(Ωt+φ)+cosφ). (4.7.9)
If the spin density operator at thermal equilibrium state which is along the z axis (Sz)
we can write :
ρ(0) = Sz. (4.7.10)
Applying pi/2 pulse:
Sz → Sx. (4.7.11)
.
We can calculate the FID by matrix multiplication, we use the evolution operator from
Eq. (4.7.9). Using Eq. (4.7.11) and Eq. (4.7.4), we can calculate the density matrix ρ(t).
Then the expectation values of the magnetization are calculated by tracing the product of
68
CHAPTER 4. DYNAMICAL DECOUPLING
the observable Sm as in Eq. (4.7.2) and the calculated density matrix ρ(t). Then the re-
sult is the calculated value for the magnetization without including relaxation. To include
relaxation we multiply the magnetization with an exponential function with a decay con-
stant T ∗2 . T
∗
2 is a decay time combined of two decay times: molecular interaction (T2) and
variations in B0(i.e. magnetic field inhomogeneities = T2inh.) given by Eq. (4.7.12) [76]:
1/T ∗2 = 1/T2 + 1/T2inh.. (4.7.12)
T ∗2 can be measured directly from the FID decay.
4.7.2.2. Simulation results versus experimental data
The FID signal is measured under the sinusoidal modulation field with a frequency of 60 Hz
as shown in Fig. 4.7.4. The sinusoidal modulation (i.e. modulated field) changes the phase
of the FID. To show this change clearly, the data are converted from Cartesian coordinate
to polar coordinate. The contour plots in Fig. 4.7.5 show the polar representation (i.e.
amplitude/phase) for the simulation and the experimental data of the spin magnetization
under the sinusoidal modulation. The spin magnetization is measured versus evolution
time after a pi/2 pulse synchronized with the pulse. During the sinusoidal modulation the
spin magnetizations are measured experimentally for different frequency values (i.e. for
the sine wave modulation): [Ω = 5 Hz, 60 Hz, 1 kHz] with an amplitude: A=10 mV(pp)
= 3.54 mV (rms). In this comparison we are able to have consistence between simulation
and experiments. For simulations we used the same parameters in the experiments (the
amplitude for the sine wave: 3.54 mV (rms), which is equivalent to 528 Hz offset when
applying DC with 3.54 mV, and the exponential decay time T ∗2 = 0.020 sec, we have an
offset of 70 Hz).
0 10 20 30 40 50
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
 x-magnetization
 y-magnetization
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
 
Evolution time (ms)
Figure 4.7.4.: Effect of sinusoidal modulation field on the FID where the modulated sinu-
soidal field has an effective amplitude of 3.54 mV and a frequency of 60 Hz.
The magnetization is measured in both x axis (blue line) and y axis (red
line).
69
4.7. ENGINEERING THE ENVIRONMENT BY EXTERNAL MODULATION FIELD
0 10 20 30 40
0.0
0.5
1.0
0 10 20 30 40
0.0
0.5
1.0
0 10 20 30 40
0.0
0.5
1.0
0 10 20 30 40
0.0
0.5
1.0
0 10 20 30 40
0.0
0.5
1.0
0 10 20 30 40
0.0
0.5
1.0
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
5 Hz
5 Hz
60 Hz
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
60 Hz
Simulation
1 kHz
1 kHz
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (ms)
Experiemnt
(a)
0 5 10 15 20 25 30
0
20
40
60
0 5 10 15 20 25 30
0
20
40
60
0 5 10 15 20 25 30
0
20
40
60
z
 Exp
 Sim
P
h
a
s
e
 
(
r
a
d
)
Evolution time (ms)
z
 Exp
 Sim
P
h
a
s
e
 
(
r
a
d
)
Evolution time (ms)
k z
P
h
a
s
e
 
(
r
a
d
)
Evolution time (ms)
 Exp
 Sim
(b)
Figure 4.7.5.: Time evolution of magnetizations for experimental data and simulations
during the sinusoidal modulation field with frequencies of 5 Hz, 60 Hz,
1 kHz. . The magnetizations are shown in the polar coordinate with the
amplitude (Fig. 4.7.5a) and phase (Fig. 4.7.5b).
4.7.3. Dynamical decoupling pulse sequence performance under
sinusoidal modulation
Noise spectroscopy in quantum computation systems uses the advantage that qubit can
be used as a noise probe. An unknown spectral density can be determined by using DD
sequences. The decay rates of the DD sequences are given by the spectral density at
discrete frequency [54, 53].
70
CHAPTER 4. DYNAMICAL DECOUPLING
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035
-6
-4
-2
0
2
4
6
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Time (sec)
(a) Sinusoidal signal used for the
field modulation to engineer
the environment.
0 20 40 60 80 100
0.00
0.05
0.10
0.15
0.20
0.25
20.1 Hz
D
e
c
a
y
 
t
i
m
e
 
(
s
e
c
)
Delay between pulses(ms)
 CPMG
 CPMG ( 60Hz)
58.8 Hz
11.8 Hz
Figure 4.7.6.: Decay times as a function of the delay τ between pulses for the CPMG
sequences (black) compared with the CPMG sequence under the sinusoidal
modulated field (red). The sinusoidal field has a frequency of 60 Hz with a
magnetic field strength of 0.0124 mT for longitudinal initial condition.
In our case we apply, instead of a noise function, a direct modulation sinusoidal signal
with a define frequency to engineer the environmental spectral density. The CPMG
sequence is applied to our system under a sinusoidal perturbation on the z axis with
a frequency of 60 Hz and a magnetic field strength of 0.0124 mT. Figure 4.7.6 shows
the decay times of the CPMG sequence (black) due to delay between pulses without the
sinusoidal modulation signal (shown in Fig. 4.7.6a). The CPMG sequence performance
deviates from the usual performance when applying the modulation field. The green arrow
shows the first dip in Fig. 4.7.6 at 17 ms which corresponds to a modulation frequency
of 58.8 Hz, which is near to our sinusoidal modulation frequency ≈60 Hz. This small
difference between the modulation frequency and the measured one by the CPMG could
be result from the uncontrolled noise coupled with the modulation sinusoidal field.
The main peak in Fig. 4.7.6 at the modulation frequency (60 Hz) has some satellite
71
4.7. ENGINEERING THE ENVIRONMENT BY EXTERNAL MODULATION FIELD
peaks at lower frequencies. The decay of the main peak and its satellite peaks for the
CPMG sequence under the modulation (sinusoidal field with a frequency Ω) can be given
by
R(t) = Ω/n, (4.7.13)
where n=1,3,5.
These experimental results obviously show that we can see the second and the third dip
near the third and matched exactly with the fifth fractions of the modulation frequency.
4.7.4. Rf coil inhomogeneity
The rf is power is strongly dependent on the geometrical construction of the rf coils [30].
For a given system, the field inhomogeneity can be measured from the nutation experiment
where the pulse width is increased with the decay of the magnetization till it decays to
zero. The field inhomogeneity can be determined from the spectral line shapes. The
nutation experiment results in Figure 4.7.7 shows that the magnetic field inhomogeneity
is ≈12%, where a) is the time domain data and b) its spectrum.
72
CHAPTER 4. DYNAMICAL DECOUPLING
0.00 0.25 0.50 0.75 1.00 1.25
-9
-6
-3
0
3
6
9
0 10 20 30 40 50
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Evolution time (ms)
b)
f kHz
A
m
p
l
i
t
u
d
e
 
(
a
.
u
.
)
Frequency (kHz)
 FFT (nutation)
f kHz
a)
Figure 4.7.7.: Long nutation experiment to determine the field inhomogeneity: a) The
nutation experiment b) Its spectrum (Fourier transform of (a))
4.8. Theoretical analysis of dynamical decoupling
sequences by average Hamiltonian theory
4.8.1. Introduction
Manipulation of Hamiltonian requires an external time-dependent or an independent per-
turbation acts on the system. Time-independent perturbation changes can be, e.g., pres-
sure or temperature. On the other hand time-dependent perturbations can be applied or
removed quickly such as sample spinning or radio frequency pulses.
The average Hamiltonian theory (AHT) was first introduced by Haeberlen and Waugh
73
4.8. THEORETICAL ANALYSIS OF DYNAMICAL DECOUPLING SEQUENCES
BY AVERAGE HAMILTONIAN THEORY
to interpret cyclic pulse sequences in the aim of seeking experimental methods for accom-
plishing high resolution solid NMR [114]. AHT describes the effect of the time-dependent
perturbation Hamiltonian of multiple pulse sequences applied to the system during an
interval t1 < t′ < t2. The primary aim from AHT was to eliminate static dipolar line
broadening in solids. Generally AHT can help to control the system efficiently without
full knowledge of the system dynamics [30]. There is a minimum limit for the pulse spac-
ing τ , for which it is possible to detect the FID signal between the pulses. This minimum
is determined by a variety of technical limitations, such as rise and fall times, widths of
square pulses and recovery time of the sensing system (0 τ ∞).
If the evolution of the system is governed by a time-dependent Hamiltonian H(t), the
effective evolution can be described by an average Hamiltonian H˜(t1, t2). The Hamiltonian
H(t) is periodic with cycle time tc. The ratio tc/T2 is made usually small so that the system
develops over long times Mtc, where M is the number of cycles.
4.8.2. Average Hamiltonian theory calculations for dynamical
decoupling sequences
Now we present the calculations of the AHT terms for the DD pulse sequences: XY4, XY8,
CPMG and KDDx. We consider a DD sequence with only flip angle errors. To simplify
the calculation procedure, the Hamiltonians are transformed to the toggling frame (or the
interaction picture), [115].
The remaining environmental noise fluctuates so rapidly that its effects cannot be re-
duced by DD as described in Sec. (4.2). It is a good approximation to describe these
rapid fluctuations as a classical field affecting the precession frequency of the spins [77].
Thus Eq. (4.2.2) can be written
HSE = ∆ωz(t)Sz, (4.8.1)
where the average of the random precession frequency is 〈∆ωz(t)〉 = 0 for every t. The
effects of the environment will be an exponential attenuation e−t/T2 independent of the
delay between pulses.
We can write the pulse propagator as a composition of the product of the ideal pulse
propagator Rφ = e−ipiSφ with a phase φ and two additional evolutions for flip angle errors
as
Rφ = e
−i(1+)piSφ = e−iHφ(tp/2)e−ipiSφe−iHφ(tp/2), (4.8.2)
where Hφ = pitp Sφ and tp is the pulse length,
Sφ = cos(φ)Sx + sin(φ)Sy. (4.8.3)
The zero- and first-order average Hamiltonians terms are given by Eqs. 4.8.4, 4.8.5
under these two conditions:
1) H(t) is periodic: H(t) = H(t+ tc).
2) The observation is stroboscopic and synchronized with the period of the Hamiltonian
tc, the evolution operator is exp
{
−iH˜(0, tc)Mtc
}
where H˜is the effective Hamiltonian [114,
116].
74
CHAPTER 4. DYNAMICAL DECOUPLING
H˜0 =
1
tc
∑
n
Hnτn, (4.8.4)
where n = 1, 2, 3, ... and τn is the delay between pulses.
H˜1 =
−i
2tc
{H2τ2,H1τ1] + [H3τ3,H1τ1] + [H3τ3,H2τ2] + ...}. (4.8.5)
In the next calculations we will assume that the environment Hamiltonian HE=0.
4.8.2.1. XY4(s) and XY4(a) pulse sequence
For the sequences XY4(s) and XY4(a) considering the effect of the flip angle error the
zeroth-order average Hamiltonian vanishes and the first-order term for both sequences
is [64]
H˜1
XY 4(s)
= H˜1
XY 4(a)
=
52pi2
16τ
Sz, (4.8.6)
where the second term shown in Ref. ([64]) vanishes, because the environment Hamil-
tonian (as in Eq. (4.6.2)) commutes with the system Hamiltonian.
This shows that there is no difference between symmetric and asymmetric sequences of
the sequence XY4 up to first order terms of the average Hamiltonian, which is in good
agreement with the experimental results of Fig. 4.6.6.
We consider now the XY8(s) and XY8(a) sequences. The zero-order and the first-order
average Hamiltonian terms are null considering only the flip angle error effects [64]. Then
the second-order term, is equal for both version of the XY8 sequence, is given by [64]
H˜2
XY 8(s)
= H˜2
XY 8(a)
=
133pi3
1536τ
(Sx + Sy). (4.8.7)
This is also in good agreement with Fig. 4.6.6 where the symmetric and asymmetric
versions of the sequence XY8 behave similarly, moreover they are more robust than the
XY4 sequences.
4.8.2.2. CPMG and CP pulse sequence
Figure 4.8.1 shows a schematic representation of the CPMG pulse sequence and the ro-
tation of the initial frame to the toggling frame due to the pulses.
75
4.8. THEORETICAL ANALYSIS OF DYNAMICAL DECOUPLING SEQUENCES
BY AVERAGE HAMILTONIAN THEORY
Figure 4.8.1.: Schematic representation of the CPMG dynamical decoupling sequence and
the terms of the average Hamiltonian in the toggling frame. In each cycle
of total length tc= 2τ , the two pulses spaced by τ and lead to a rotated
coordinate system, known as toggling frame. The average Zeeman Hamilto-
nian is obtained by averaging the transformed Hamiltonian in the toggling
frame.
Using Eqs. (4.6.2), (4.8.2), (4.8.3) the effective Hamiltonian terms can be written as
τ1H1 = τ1(HSE) +
tp
2
Hy =
τ
2
(ωzSz) +
pi
2
Sy, (4.8.8)
τ2H2 = τ2(−HSE) + tpHy = τ(−ωzSz) + piSy, (4.8.9)
τ3H3 = τ3(HSE) +
tp
2
Hy =
τ
2
(ωzSz) +
pi
2
Sy. (4.8.10)
Explicit calculations for the above terms using Eqs. (4.8.4, 4.8.5) give the zero-order
term:
H˜CPMG0 =
2piSy
tc
, (4.8.11)
and the first-order
H˜CPMG1 = 0. (4.8.12)
In the CPMG experiment, the initial condition is ∝ Sy, which commutes with the
average Hamiltonian zero-order term (Eq. 4.8.11) and is therefore not affected by pulse
errors. In the CP experiment, the initial condition is ∝ Sx, which is dephased by the
pulse errors, in agreement with the data in Fig. 4.6.6 [36].
76
CHAPTER 4. DYNAMICAL DECOUPLING
4.8.2.3. KDDx pulse sequence
The KDDx cycle is composed of ten pulses as in Fig. 4.6.1, thus we can write the effective
Hamiltonian terms as
τ1H1 = τ1(HSE) +
tp
2
(H30), (4.8.13)
τ2H2 = τ2(−HSE) +
tp
2
(H30 +H60), (4.8.14)
τ3H3 = τ3(HSE) +
tp
2
(H60 +H150), (4.8.15)
τ4H4 = τ4(−HSE) +
tp
2
(H150 +H−120), (4.8.16)
τ5H5 = τ5(HSE) +
tp
2
(H−120 +H−y), (4.8.17)
τ6H6 = τ6(HSE) +
tp
2
(H−y), (4.8.18)
τ7H7 = τ7(−HSE) +
tp
2
(H−y), (4.8.19)
τ8H8 = τ8(HSE) +
tp
2
(H−120 +H−y), (4.8.20)
τ9H9 = τ9(−HSE) +
tp
2
(H150 +H−120), (4.8.21)
τ10H10 = τ10(HSE) +
tp
2
(H60 +H150), (4.8.22)
τ11H11 = τ11(−HSE) +
tp
2
(H30 +H60), (4.8.23)
τ12H12 = τ12(+HSE) +
tp
2
(H30), (4.8.24)
where:
Hx = Sx
pi
tp
, Hy = Sy
pi
tp
, (4.8.25)
H30 =
(√
3
2
Sx +
1
2
Sy
)
pi
tp
, (4.8.26)
H60 =
(
1
2
Sx +
√
3
2
Sy
)
pi
tp
, (4.8.27)
H150 =
(
−
√
3
2
Sx +
1
2
Sy
)
pi
tp
, (4.8.28)
77
4.8. THEORETICAL ANALYSIS OF DYNAMICAL DECOUPLING SEQUENCES
BY AVERAGE HAMILTONIAN THEORY
H−120 =
(
−
1
2
Sx −
√
3
2
Sy
)
pi
tp
, (4.8.29)
H−90 = −Sy
pi
tp
. (4.8.30)
Explicit calculations of the previous twelfth Hamiltonian terms and using Eqs. (4.8.4, 4.8.5)
give the zero-order
H˜0
KDDx
= 0, (4.8.31)
and the first-order as
H˜1
KDDx
= 0. (4.8.32)
The higher order terms were kept small by the design of the sequence. As shown in
Ref. [117], this makes it robust against several systematic errors because it is a geometric
quantum gate. In Ref. [42], a numerical simulation and experimental data showed that
the KDDxy sequence is more robust against flip angle errors than the other DD sequences
tested [113, 118]. Overall, the experimental comparison between the different sequences is
in good agreement with the numerical simulations and analytical results based on average
Hamiltonian theory.
78
5. Conclusion and outlook
5.1. Conclusion
In this project the experimental setup of the liquid NMR experiment (i.e. spectrometer
configuration) was discussed. In addition, we were able to analyze the main sources
of the electronic noise emitted from different devices. Also, we could reduce a kind of
coherent noise (i.e. 50Hz noise) by separating common ground connections. Moreover,
we constructed and tested a new setup configuration for the H-channel to avoid the
complexity of using different frequency sources for generating rf pulses.
Optimization procedures such as tune-up sequences were applied towards performing
high fidelity NMR quantum computing experiments. Furthermore, suitable repeated ex-
perimental times were optimized by adding CuSO4 to the NMR sample to avoid obstacles
(e.g. radiation damping). For quantum memory purposes, we have tested the robustness
of different DD sequences by comparing them in an environment that interacts with the
spins in such a way that the decoherence time under the application of DD sequences
with ideal pulses is independent of the delay times between the pulses. This allowed us
to study the robustness of the different DD sequences and thereby isolate the effects of
the pulse errors. We found that the decoherence times of the most robust sequences,
the KDD family, is the longest for arbitrary initial states. This is consistent with the
measured error per pulse averaged over many cycles of a DD sequence, where the KDD
sequences have the lowest effective error. In the regime studied, the time symmetrization
of the cycles does not play a significant role in decoherence reduction, whereas only the
phases of the pulses are important for reducing the effect of pulse errors.
In order to study the environmental decoherence sources and find the appropriate DD
sequence for quantum computing, an additional coil was built in the experimental setup.
Through this coil, using an AWG and a pulse amplifier, we were able to generate a time-
dependent field which is synchronized with the pulse timing. We were able to achieve
agreement with the simulated FID and experimental data under the effects of this sinu-
soidal modulation. The sine function modulation is applied to the system synchronized
with the CPMG pulse sequence. The sinusoidal modulation frequency value is matched
with the value measured by the experimental CPMG sequence decay time. Finally we
calculated the average Hamiltonian terms for CPMG and KDDx sequences taking into
account flip angle errors. Comparing the expression of both versions (symmetric and
asymmetric) of the sequences XY4 and XY8 showed no differences. Our experimental re-
sults for pulse errors are consistent with numerical simulations and predictions of average
Hamiltonian theory.
Our results can be useful for better understanding of experimental implementations
of DD sequences, and the influence of errors on quantum control of spin systems. Our
findings strongly support that liquid NMR works as a good testbed for DD sequences
and studying environmental decoherence effects. Our system can be a helpful tool for
79
5.2. FIVE-SPIN SYSTEM FOR DYNAMICAL DECOUPLING
simulating other unknown systems by inserting different kind of noise, which can be
useful to better understanding for the decoherence effects.
5.2. Five-spin system for dynamical decoupling
Now we summarize some preparation experiments on the proposed molecule, 1,2-difluoro-
4-iodobenzene for quantum computation experiments. The experiments were done on a
Bruker Avance II 500 MHz spectrometer. The above mentioned molecule (1,2-difluoro-4-
iodobenzene) composed of two fluorine atoms (F1, F2) can be regarded as a two qubit
system and three protons (H1, H2, H3) can be regarded as environment. The molecule
(structure is shown in Fig. 5.2.1) was dissolved in the liquid-crystal solvent ZLI-1132 to re-
tain the dipolar couplings between the spins. To have a homogenous mixture the following
procedure was performed: the mixture was heated above its clearing point≈343 K, then
properly stirred and cooled down. This procedure was repeated several times thoroughly.
To have a stable and homogenous magnetic field, an external deuterium lock signal was
implemented. The lock channel of the 500 MHz spectrometer requires a significant signal
to maintain the rf stability. Therefor we used an insert tube of 5 cm length. According to
the data from Ref. [119] we recalculated the Hamiltonian under controlled temperature of
295 K. Table 5.2.1 shows the parameters of the spin qubits in the molecule. Figure 5.2.2a
shows the fitted spectrum of protons compared with experimental data after a pi/2 hard
pulse while Fig. 5.2.2b shows the fluorine spectra.
H2
F1
F2
H1
I
H3
Figure 5.2.1.: Chemical structure of the molecule 1,2-difluoro-4-iodobenzene
80
CHAPTER 5. CONCLUSION AND OUTLOOK
F1 F2 H1 H2 H3
F1 2603.7 -228.6 -292.1 -616.4 -321.8
F2 -1621.9 -2198.0 45.0 -55.3
H1 122.1 -35.8 52.0
H2 -37.4 -2605.5
H3 36.0
T∗2 0.13 0.12 0.18 0.21 0.13
Table 5.2.1.: Parameters of the spin qubits in the molecule 1,2-difluoro-4-iodobenzene.
The chemical shifts with respect to the transmitter frequencies of the proton
and fluorine spins are shown as the diagonal terms and the dipolar couplings
between spins are shown as the off-diagonal terms in units of Hz. The effec-
tive relaxation times T2 are determined by fitting the peaks in the spectra.
We calculated some pulses, to use them in DD sequences. These pulses are strongly
modulated pulses (SMP) for their robustness against flip angle errors and frequency offset
errors. Table 5.2.2 shows some strong modulated pulses which are calculated for fluorine
spins with different lengths and their theoretical fidelities1.
no pulse pulse length (ms) Fidelity %
1 pi 1 97
2 pi 1 97
3 pi 1 97
4 pi 1 99
5 pi/2 2 97
6 pi/2 2 97
7 pi/2 2 97
8 pi/2 2 97
9 pi/2 0.496 97
Table 5.2.2.: Strongly modulated pulses calculated for fluorine spins with different lengths
and their theoretical fidelity
1The pulses are located in: afp://chaos._afpovertcp._tcp.local/Pulseprograms/
Pulseprograms500/mustafa_SMP_pulses
81
5.2. FIVE-SPIN SYSTEM FOR DYNAMICAL DECOUPLING
−4000 −3000 −2000 −1000 0 1000 2000 3000 4000
0
10
20
30
40
50
60
Frequency (Hz)
Am
plit
ude
 (a.u
.)
 
 
fit
experiment
 Impurities peaks from the lock−signal (Deuterium)  
(a)
−4000 −3000 −2000 −1000 0 1000 2000 3000 4000
0
10
20
30
40
50
Frequency (Hz)
Am
plit
ude
 (a.u
.)
 
 fit
experiment
(b)
Figure 5.2.2.: a) Fitting results of the proton spins from the thermal states by a hard
pulse. b) Fitting results of the fluorine spins from the thermal states by a
hard pulse.
82
A. Signal check-up protocol in the 360
MHz Spectrometer
 
No NMR signal ?
Channel connected to 
oscope (H, X)?
See reflected and 
incident pulse?
DDS output 
(correct 
frequency)?  
Other source 
frequency mixing 
(Hameg) ?
Word 
generator/ 
pulse 
programm? 
Reflected 
pulse 
equal to 
incident? 
Probe 
tuning?
Filters: 
frequency 
band-width?
Enough 
sample?
83

B. Interpreter pulse program: an
example for multiple NMR
experiments
1 ########################################################
2 ### In t e r p r e t e r pu l s e program : mul t ip l e pu l s e programs
3 ### Auther : Mustafa Ahmed Al i Ahmed
4 ### Sample : H2O+CuSO4
5 ### 360MHz spectrometer
6 ########################################################
7 2D_Interpreter
8 #c a l l the i n t e r p r e t e r to run a mul t ip l e 2D exper iments
9 pulse_program_name_1
10 #c a l l the pu l s e program , t h i s should be in the same f o l d e r
11 DATA_filename_1 . 1 d
12 #save the data as DATA_filename_1 . 1 d 2D_Interpreter
13 pulse_program_name_2 DATA_filename_2 . 2 d
85

C. Interpreter pulse program: an
example for a single NMR
experiment
1 ####################################################
2 ### In t e r p r e t e r pu l s e program : CPMG pul se sequence
3 ### Auther : Mustafa Ahmed Al i Ahmed
4 ### Sample : H2O+CuSO4
5 ### 360MHz spectrometer
6 #####################################################
7 ! s e t : number_of_steps=1
8 #de f i n e the number o f d i f f e r e n t exper iments by changing a
parameter ( data0 )
9 ! s e t : number_of_scans=1
10 #de f i n e the number o f scans f o r every experiment
11 ! cyc l ops
12 #de f i n e the phase cy c l e by increment ing the r f pu l s e phase f o r
f our steps ,
13 #every step with 90 degree
14 ! d i g i t a lPhaseCor r : 289 , 379 , 469 , 559 ;
15 #de f i n e the d i g i t a l phase c o r r e c t i o n
16 ! const : pw=35.6
17 #de f i n e a constant p i pu l s e
18 ! const : pw_2=17.5
19 #de f i n e a constant p i /2 pu l s e
20 ! set_2D_exp_parameters
21 #de f i n e 2D experiment parameters
22 ! s e t : MaxPulseDurationH=2000000
23 #de f i n e the maximum pul se durat ion
24 ! s e t : MaxDutyCycleH=0.09
25 #de f i n e the maximum duty cy c l e
26 delay ; 1
27 #s t a r t the pu l s e program a f t e r 1 microsecond delay
28 r e c y c l e d e l a y ; 3
29 #3 second r e p e t i t i o n time
30 +ddsH : phase1 =cyc l e0 ; 0 . 2 6
31 #the p lus s i gn means t h i s s tep w i l l be programmed p a r a l l e l with
the prev ious command , s e t the DDS(H) phase to zero with de lay
0 .26 micro sec
87
32 pulseH ; [ pw_2 ]
33 #pi /2 pu l s e on H channel
34 +ddsH : phase1 = 0/4 ; 0 . 26
35 #se t the DDS(H) to zero phase with de lay 0 .26 micro sec
36 t r i g g e r ; 1
37 #send t r i g g e r s i g n a l to the oscope to s t a r t d e t e c t i on
38 (2000
39 #s t a r t a loop repeated 2000 t imes
40 acquireH ;50
41 #open the r e c e i v e r f o r 50 micro sec to de t e c t a s i g n a l
42 +ddsH : phase1 =1/4+cyc l e 0 ; 0 . 2 6
43 #change the DDS phase four t imes by 90 degree every time
44 pulseH ; [ pw ]
45 +ddsH : phase1 = 0/4 ; 0 . 26
46 acquireH ;100
47 +ddsH : phase1 =1/4+cyc l e 0 ; 0 . 2 6
48 pulseH ; [ pw ]
49 +ddsH : phase1 = 0/4 ; 0 . 26 acquireH ;50
50 )
51 #end o f the loop
52 delay ; 1
53 #end o f the pu l s e program
88
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Acknowledgments
Thanks God, perhaps it is good to write this verses from the Nobel Quran: ". . . And if
you would count the blessings of Allah you would not be able to count them. . . " [The
Nobel Quran 14:34].
I would like to express my earnest gratitude to my supervisor Prof. Dr. Dieter Suter
for giving me an opportunity to broaden my horizon. I would not have completed this
thesis without his guidance, encouragement, advices and patience.
I want to sincerely thank Dr. Gonzalo A. Álvarez who introduced me to the field of
quantum computation and played an essential role in completion of this work. I truly
appreciate the help and valuable advices of Dr. Jingfu Zhang and Prof. Dr. Alexander
Martins de Souza.
I would like to thank Prof. Dr. Götz S. Uhrig for reading this thesis and giving valuable
comments. Thanks are also extended to the examination committee.
It is a pleasure for me to thank Dr. Reiner Küchler who helped me to carry out my first
NMR experiment, six years ago, and further supported me in experimental techniques. I’m
grateful to the technical and administrative support of Angelika Sommer, Anja Sapsford,
Walerian Goralcyk and Horst Rombeck as well as answering personal questions.
I would like to acknowledge the friendly atmosphere and scientific cooperation from
all current and previous members of E3 group. I appreciate the help of Andranik Bagh-
dasaryan, Rahuman Sheriff and Wulayimu Maimaiti in grammar and misspelling correc-
tions while writing this thesis.
The workshops at the Physics Department allowed technical implementation of many
projects. For the effective cooperation, I would like to thank all employees of Development
of electronics, Electronics Workshop, Mechanical Workshop, Preparation laboratory and
low temperature unit.
I would also like to acknowledge the financial support from the International University
of Africa in Sudan.
I’m grateful to my mother Maryam Mergani who always keeps praying for me. Thanks
for the love and patience of my wife, Sarah Saeed and my children, Abdelrahman and
Maryam (who were born during the period of this work). Lastly I would like to thank
everyone who directly or indirectly helped me to finish this thesis.
95