Download Geometric phases in quantum systems of pure and mixed state

Geometric phases in quantum
systems of pure and mixed state
Geometriska faser in rena och blandade kvantmekaniska system
By Miran Haider
Faculty of health and science
Master’s thesis in Engineering physics, 30ECTS
Supervisor: Prof. Fuchs Jürgen
Examinator: Prof. Johansson Lars
Date: February 23, 2017
I
Abstract
This thesis focuses on the geometric phase in pure and mixed quantum states.
For the case of a pure quantum state, Berry’s adiabatic approach (4.1.10) and
Aharonov & Anandan’s non-adiabatic generalization of Berry’s approach (4.2.8)
are included in this work. Mixed quantum state involves Uhlmanns approach
(5.1.42), which is used extensively in Section 7 and Sjöqvist’s et al. approach
(5.2.22), used extensively in Section 6. Sjöqvist’s approach states that the
Uhlmann phase is an observable and provides the experimental groundwork
1
using an interferometer. The
“ geometric
` Ω ˘‰ phase for a spin- 2 system is given by
(5.2.47) γG rΓs “ ´ arctan r tan 2 , which was proven, by Du et al.[45] to
reproduce experimental data (Figure 19) on page 56.
The Uhlmann phase can be used to observe the behaviour of topological
kinks. This was tested on 3 models, the Creutz-ladder, the Majorana chain and
the SSU-model. It is found that the Uhlmann phase is split into two regimes
with the dividing parameter being the temperature. This temperature is called
1?
. If the temperathe critical temperature, Tc , and is given by Tc “ ln 2`
3q
p
ture is below the critical temperature, the Uhlmann phase yields π and if the
temperature is above the critical temperature, the Uhlmann phase yields zero.
Detta examensarbete behandlar geometriska faser i rena och blandade kvanttillstånd. I rena kvanttillstånd finner man Berrys adiabatiska behandling av den
geometriska fasen (4.1.10) och Aharonov & Anandan icke-adiabatiska generalisering av Berry fasen (4.2.8). I det blandade kvanttillstånden har Uhlmann introducerat en förlängning av den geometriska fasen som sträcker sig till det blandade kvanttillstånden (5.1.42), detta finner man i sektion 7. Senare har Sjöqvist
et al. introducerat ett alternativ till att angripa geometriska faser (5.2.22) som
beskrivs i sektion 6. Sjöqvist konstaterade att Uhlmannfasen är observerbar,
i kvantmekanisk mening, och presenterade ett experimentelt upplägg där han
visade just detta med hjälp av en interferometer. Den
fasen för
“ geometriska
` ˘‰
ett spin- 21 system ges av (5.2.47) γG rΓs “ ´ arctan r tan Ω2 , vilket senare
bevisades av Du et al.[45] där de experimentella mätvärdena stämde överens
med dem teoertiska (se figur 19 på sidan 56).
Uhlmannfasen kan även användas för att observera topologiska ”kink”-lösningar.
Detta testades för 3 olika modeller; Creutz stege formationen, Majorana kedjan
och SSU modellen. Det visade sig att Uhlmannfasen delades up i två regioner
och var starkt beroende på temperaturen. Denna temperaturen kallades för den
1?
. Om temperaturen ligger
kritiska temperaturen Tc , och ges av Tc “ ln 2`
3q
p
under eller över den kritiska temperaturen får man att Uhlmannfasen ger π eller
0 respective.
II
Acknowledgements
I would like to extend my heartfelt gratitude and appreciation to my supervisor,
Prof. Fuchs Jürgen, for constantly being there when needed and has, on multiple
occasions, guided me through several barriers. Without his persistent help and
guidance, this work would not have been possible.
I would also like to thank my family and friends for their constant love and
support.
III
Contents
Abstract
II
Acknowledgements
III
List of Figures
VII
1 Introduction
1
2 Single quantum system
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . .
2.1.1 The Qubit example and the Bloch Sphere
2.2 Density operator . . . . . . . . . . . . . . . . . .
2.3 Time-Evolution of the density operator . . . . . .
2.4 Spin operators . . . . . . . . . . . . . . . . . . .
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3 Multiple quantum systems
3.1 Entanglement and Separability for mixed states . . . . . . . . . .
3.2 Composite quantum systems . . . . . . . . . . . . . . . . . . . .
3.3 Restricted case: Pure Bipartite Quantum composite System (Pure
Bipartite qudit state) . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Purification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Tensor product spaces . . . . . . . . . . . . . . . . . . . . . . . .
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4 Geometrically induced phases for pure states
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4.1 Adiabatic Geometric phase (Berry’s phase) . . . . . . . . . . . . 13
4.1.1 Gauge invariance of the Berry phase . . . . . . . . . . . . 14
4.1.2 Informal description of the Chern number . . . . . . . . . 15
4.1.3 Example, spin 1/2 in an adiabatically rotating magnetic
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Non-adiabatic Geometric phase (Aharonov-Anandan phase) . . . 19
5 Geometrically induced phases for mixed states
5.1 A. Uhlmann’s concept of mixed geometric phase
5.1.1 Parallel amplitude . . . . . . . . . . . . .
5.1.2 Parallel transport and connection form . .
5.1.3 The Uhlmann Phase . . . . . . . . . . . .
5.2 The Sjöqvist formalism . . . . . . . . . . . . . . .
5.2.1 Parallel transport . . . . . . . . . . . . . .
5.2.2 Gauge invariance . . . . . . . . . . . . . .
5.2.3 Purification . . . . . . . . . . . . . . . . .
5.2.4 Example . . . . . . . . . . . . . . . . . . .
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6 Experimental observation of Geometric phases
using NMR interferometry
6.1 Nuclear Magnetic Resonance (NMR) . . . . . .
6.1.1 Irradiating the nucleus with RF . . . . .
6.1.2 Relaxation processes . . . . . . . . . . .
6.1.3 Spin-spin coupling (JJ-coupling) . . . .
6.1.4 General setup of CW-spectrometer . . .
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22
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29
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for mixed states
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41
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6.2
6.3
6.4
Quantum interference; Sjöqvist’s NMR interferometry model . .
6.2.1 Spatial averaging technique . . . . . . . . . . . . . . . . .
6.2.2 Gradient pulse . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 Pseudo-pure states . . . . . . . . . . . . . . . . . . . . . .
6.2.4 Non-vanishing dynamical phase . . . . . . . . . . . . . . .
Experimental observation of Geometric phases for mixed state [45]
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Uhlmann Phase as a Topological Measure
Fermion Systems
7.1 Introduction . . . . . . . . . . . . . . . . .
7.2 Topological insulators . . . . . . . . . . .
7.3 Fermionic system and Uhlmann Phase . .
7.4 Creutz-ladder model . . . . . . . . . . . .
7.5 The Critical Temperature . . . . . . . . .
7.6 Majorana-chain model . . . . . . . . . . .
7.7 Su-Schrieffer-Heeger (SSH)-model . . . . .
7.8 Conclusion . . . . . . . . . . . . . . . . .
48
48
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53
55
for One-Dimensional
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57
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8 Summary and outlook
69
8.1 One-page summary of the thesis . . . . . . . . . . . . . . . . . . 69
8.2 Contributions by the authors . . . . . . . . . . . . . . . . . . . . 70
8.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A Appendix
A.1 Space of states . . . . . . . . . . . . . .
A.1.1 Hilbert Space . . . . . . . . . . .
A.1.2 Dual vector space . . . . . . . . .
A.1.3 Operators on Hilbert spaces . . .
A.2 Operators . . . . . . . . . . . . . . . . .
A.2.1 Unitary operators . . . . . . . .
A.2.2 The closure relation . . . . . . .
A.2.3 Path/Time-ordering operator . .
A.3 Baker-Campbell-Hausdorff lemma . . . .
A.4 Matlab program used for calculating the
V
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critical temperature
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71
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76
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
[4] The Bloch sphere, a geometrical representation of the twodimensional Qudit system called the Qubit. . . . . . . . . . . . .
3
[7] The transport of a vector along the closed curve C on a 2-sphere. 12
[7] A spin- 12 particle moving in a adiabatically, rotating, θ, externally applied magnetic field B with angular velocity ω, emulating
Equation (4.1.16) . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
[7] Magnetic field, Equation (4.1.16), following the path C, expressed in parameter space . . . . . . . . . . . . . . . . . . . . . 17
[7] The path C in H being projected onto the projective state
space C 1 on PpHq. Note that the path C need not necessarily be
closed for the path C 1 to be closed. . . . . . . . . . . . . . . . . . 19
Comparison [33] between the Berry (Equation (4.1.10)) and Uhlmann
(Equation (5.1.42)) aproaches in obtaining the geometric phase. . 27
[12] A conventional Mach-Zehnder ineterferometer with two beamsplitters and two mirrors . . . . . . . . . . . . . . . . . . . . . . . 29
[?, ?] Sjöqvist’s interferometry model BS1 and BS2 are beam
splitters, M1 and M2 are Hadamard mirrors, U is a unitary operator acting on the internal states of the photons and χ is an
operator that shifts the phase. . . . . . . . . . . . . . . . . . . . . 29
Nucleus axis of rotation precessing around the magnetic field. . . 41
Energy levels splitting of an fermion due to an externally applied
magnetic field, known as the Zeeman splitting. . . . . . . . . . . 42
[55] An α pulse transforms the net magnetization, M0 , into another oriantation, Mx,y , by a degree α. A 90 degree pulse would
shift the net magnetization entierly from one axis to another. B
is the externally applied magnetic field. . . . . . . . . . . . . . . 43
[55] A spin 12 particle absorbing RF radiation with an energy
corresponding to a spin-flip from . . . . . . . . . . . . . . . . . . 44
A pictorial presentation of the JJ-coupling effect on respective
proton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
[55] General/conventional CW spectrometer. Sample solution is
placed in a rotating glass tube oriented between magnetic poles.
Radio frequency (RF) is emitted from the RF antenna, enclosing
the glass tube and a receiver to detect emitted radiation from
the sample. The receiver transmits the data to a control console
which displays the result on a screen. . . . . . . . . . . . . . . . . 47
Spin echo method . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
[57] A refocusing of spin moments, yielding a singal, echo, when
no pulse was applied. a, b, c, d, e, f . . . . . . . . . . . . . . . . . 52
[45] Quantum network describing the experiment where the top
line represents the auxiliary qubit or particle of spin-1{2 while the
lower line represents the qubit, spin 1{2 prticle, that undergoes a
cyclic evolution induced by the unitary operator U . . . . . . . . 53
The solid angle Ω is subtended by the cyclic path ABCDA. The
solid angle can be varied by changing θ, the angle of inclination
between x, y plane and the ABC plane. . . . . . . . . . . . . . . 54
VI
19
20
21
22
23
24
25
26
27
Experimental data [45] (represented by circle and squares) versus
theoretical predictions (represented by solid lines and Equation
(5.2.22)). The geometric phase ,γ, is presented versus purity, r,
for three different solid angles, Ω. . . . . . . . . . . . . . . . . . .
A sample in the presence of an externally, perpendicularly, applied magnetic field. The electrons on the sample are effected
by this magnetic field, causing electrons to orbit. Electrons close
ot the edges experience Edge states, effectively contributing to
current along the egdes, while electrons in the middle contiue to
orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Creutz ladder formation. Particles can hop both diagonally
and vertically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
[33] The Topological Uhlmann phase for the Creutz-ladder (a),
the Majorana-chain model (b) and the SSH-model(c). The topological phase is equal to π inside the green volume and zero outside. The FPB (Flat Band Points) are indicated by arrows. . . .
[34] (a): non-trivial topology and (b): trivial-topology . . . . . .
Real and imaginary part of fn (7.5.9) is plotted against the parameter n. In the figuren n appears to be continuous, however, we
are only interested in the integer numbers. Only possible integer
value, for which (7.5.9) is real, is given for n “ 0 . . . . . . . . .
Real and imaginary part of fn (7.5.9), presented in figure (24), is
plotted on top of each other. Again, only possible integer value,
for which (7.5.9) is real, is given for n “ 0 . . . . . . . . . . . . .
Top chain showing trivial phase with paired Majorana fermions
(in blue) located on the same sites on a physical lattice. Bottom chain showing non-trivial topology (topological phase) with
bound pairs located at neighboring sites resting as unpaired Majoranas at each ends, represented by red coloured balls . . . . . .
The Su-Schrieffer-Heeger model . . . . . . . . . . . . . . . . . . .
VII
56
58
61
62
62
65
65
67
68
1
Introduction
The purpose of this work is to introduce the concept of geometric phase and to
describe different variants for quantum systems which are in a pure or mixed
state. This work is divided into 5 parts. A preliminary section is intended to
prepare the reader for the mathematical concepts and reasoning implemented
in the upcoming sections, a theoretical section which introduces the concept
of geometric phase, a section discussing experiments in which the geometric
phase has been experimentally demonstrated, a more abstract section where
the relations of geometric phases are extended to topological notions and their
usage in describing behaviours of certain insulating materials is discussed, and
finally a section giving summary and mentioning further studies.
When a quantum system (see Sections 2.1 and 2.2) undergoes a cyclic unitary evolution, the state aquires a phase. This phase consists of two parts; a
dynamic phase which depends on the Hamiltonian, and a geometric phase which
depends only on the path of the evolution that the system takes in the projective
Hilbert space [9], [10], [16]. For a qudit system (see Section 2.1), the qubit in
particular, the projective Hilbert space is given by a sphere (Bloch Sphere, See
Section 2.1.1) where the geometric phase depends on the geodesical solid angle
subtended at the center of the sphere by a path by state vector. The concept
of geometric phase was first introduced in an adiabtic context [9], and later a
non-adiabatic generalized theory was proposed by Aharonov et al. [16].
The adiabatic approach requires the state vector to be parallel transported
adiabatically to ensure that the system always remains in one of the eigenstates
of the instantaneous Hamiltonian during the evolution.
The system in the non-adiabatic approach, is allowed to change abruptly.
This implies that the system reverts to its initial state through intermediate
states. The dynamical phase is eliminated from the total phase by various
methods in order to be able to experimentally measure the geometric phase. One
possibility to eliminate the dynamical phase is by using the Nuclear Magnetic
Resonance (NMR) technique called spin-echo [57] (see Section 6.2.4).
In 1986, Uhlmann approached the concept of mixed state geometric phase
[28] using the notion of amplitude (5.1.42). Here, Uhlmann has taken a large
system in a pure state and a subsystem in a mixed state. Uhlmann then obtained
the unitary evolution for the subsystem which was transported in a maximally
parallel fashion.
Recently, Sjöqvist et al. [19] provided additional insight into the nature of
geometric phases by providing a concept different to the one given by Uhlmann
(see Section 5.2), which is adapted to experimental use. In a quantum interferometer, the system undergoes multiple unitary evolutions, for which the
probability of discovering the system in one of its eigenstates behaves like a
oscillatory function. This oscilattory function, yielding a oscillation pattern of
probability, resembles an optical interference pattern. Now a shift in this interference pattern is a function of the geometric phase acquired by the quantum
system undergoing unitary evolution as well as the purity of the system (see
Equation (5.2.30)) [19]. The geometric phase can then be directly measured
from this shift in the interference pattern.
The mixed state geometric phase has been experimentally observed by by
Du et al. [45]. Du et al. obtained the geometric phase by measuring the relative
phase change of an auxiliary spin (Section 6).
1
Since the discovery of the geometric phase by Berry [10], it has played an
essential role in the study of many quantum mechanical phenomena. An example of this is the characterization of the transversal conductivity σxx in the
quantum Hall effect, by means of the integral of the Berry curvature over the
two-dimensional Brillouin zone, and its relation to the Chern number. Using
the TKNN formula [46], one can show that the Hall conductivity is related to
the Chern number (a topological invariant) by a constant e2 {h [27].
Section 7 focuses on the topological aspects of the Uhlmann phase for the
case of one-dimensional fermionic systems. The notion of topology and topological protection are explained in Section 7.2 where the latter is related to
insulators and superconductors. Mathematically, the situation is quite similar
to what happens in the work by J.M. Kosterlitz and D.J. Thouless, receivers of
the 2016 Nobel prize in physics, regarding topological phase transitions [48].
With the developments on fault-tolerant methods, it is known that, in principle, operators of a quantum computer can actively intervene to stabilize the
device from errors in a noisy environment [52]. That is, controlled encoded
quantum information can protected from errors due to its interaction with the
environment. However, in the long term, it is favourable to have an intrinsic
fault-tolerance hardware within the device. Recently, it has been proposed that
fault-tolerant (intrinsic) quantum computing can be performed with the aid of
a geometric phase [50] [51].
2
2
2.1
Single quantum system
Preliminaries
Definition 2.1.1. A Qudit system is a quantum system of d linear independent
states described by the d dimensional Hilbert space Cd
2.1.1
The Qubit example and the Bloch Sphere
Definition 2.1.2. A binary relation on a set A is said to be an equivalence
relation if and only if it is, reflexive, symmetric and transitive.
That is, for @a, b, c P A ,the relation is reflexive if a „ a, symmetric if a „ b
iff b „ a, and transitive if a „ b and b „ c then a „ c.
The Projective Hilbert space H of a complex Hilbert space H is the set of
equivalence classes (a class which contains a notion of equivalence formalized
as equivalence relation given by Definition 2.1.2) of vectors |vy P H ´ t0u for
the relation „ given by v „ w when v “ λw where λ P C ´ t0u. The equivalence classes for the relation „ is called rays which is the usual construction of
projectivication applied to a complex Hilbert space. That is, for λ P C ´ t0u,
P pλv, wq “ P pv, λwq “ P pv, wq since |vy is the same state as |wy. Thus, a ray
is a set of all vectors describing the same state
Rw “ t|vy P H|Dλ P C : |vy “ λ|wyu.
(2.1.1)
The block sphere is a geometrical representation of the pure state case of the
Qubit, the restricted Qudit system, for which one deals with the two-dimensional
complex Hilbert space C 2 . The projectivication of the two-dimensional complex
Hilbert space is the complex projective line P pC 2 q “ CP1 , which is known as
the Bloch sphere, see figure (1).
Figure 1: [4] The Bloch sphere, a geometrical representation of the twodimensional Qudit system called the Qubit.
3
By requiring that the pure state |ψy be normalized, given any orthonormal
basis (t|0y, |1yu for the case of the qubit), it can be represented as
ˆ ˙
ˆ ˙
θ
θ
iφ
|ψy “ cos
|0y ` e sin
|1y.
2
2
(2.1.2)
Then the density operator is given by
ρ “ |ψyxψ| “
¯
1´
I 2 ` ~b ¨ σ ,
2
(2.1.3)
where
σ “ pσx , σy , σz q
(2.1.4)
are the Pauli matrices, I 2 is the two-dimensional unite matrix and the Bloch
vector, ~b, is given by
~b “ psinpθq cospφq, sinpθq sinpφq, cospθqq
(2.1.5)
with θ P r0, πs and φ P r0, 2πs.
2.2
Density operator
Before discussing the density operator and density matrix some terms need to
be defined. Consider all of space to be the unbounded set S P R3 . Then a
Quantum System is a bounded subset of space, S 1 Ď S
Following the definition of a quantum system, then it is a subset of space for
which quantum mechanical studies are being acted upon. Thus, it is a system
of a size ”suitable” for experiments for which all the outcomes in S 1 reflects the
behaviour of all the outcomes in S under similar conditions.
Definition 2.2.1. The state of a quantum system is called ”pure”, or pure
Quantum State, if the system can be completely described by a single statevector |Ψy P H [1].
Pure quantum state is synonymous with the term, proposed by J. Von Neumann, Pure ensemble: collection of physical systems for which each member
can be characterized by the same state vector, say |ψy P H.
Definition 2.2.2. A mixed Quantum State is a statistical Ensemble of pure
Quantum States.
That is; a Convex Combination of ”pure” Quantum states. Suppose one
performs a measurement on a mixed ensemble of some observable A, that has a
representation on the Hilbert space H of a system. Then, the Ensemble average
of A is given by;
4
dimpHq
ÿ
xxAyy ”
ρα xψα |A|ψα y
α
dimpHq
R1
ÿ ÿ
“ lim
1
R Ñ8
dimpHq
R1
ÿ ÿ
“ lim
1
R Ñ8
ρα xψα |A|a1 yxa1 |ψα y
a1
α
ρα |xa1 |ψα y|2 a1 ,
(2.2.1)
a1
α
where the Closure relation (equation (A.2.15)) has been inserted in the first line
of equation (2.2.1) and that |a1 y is an eigenket of A. Note that the index α has
been chosen to further remind us that |ψα y need not necessarily be orthogonal. Further note the probabilistic nature of Equation (2.2.1), for which two
probability factors appear.
• ρα ; Probability factor in an ensemble of a state characterized by |ψα y.
That is, ρα is the probability factor for which the ensemble exist in the
state |ψα y
• |a1 x|ψα y|2 ; Quantum mechanical probability for the state |ψα y to be found
in an eigenstate of A, |a1 y
This motivates one to reconsider (rather re-express) the ensemble average, equation (2.2.1), using the orthonormal basis, t|b1 y, |b2 yu
xxAyy “
dimpHq
ÿ
lim
1
2
R ,R Ñ8
α
1
“
lim
2
R ÿ
R
ÿ
xψα |b1 yxb1 |A|b2 yxb2 |ψα y
b1 b2
2
R ÿ
R
ÿ
R1 ,R2 Ñ8
1
ρα
¨
dimpHq
ÿ
˝
˛
ρα xb2 |ψα yxψα |b1 y‚xb1 |A|b2 y.
(2.2.2)
α
b1 b2
Where the bracket (the α-sum) is a collection of pure ensembles. This,
further, motivates one to define the density matrix and the density operator, for
which;
ρ“
dimpHq
ÿ
pα |ψα yxψα |,
(2.2.3)
α
is defined as the density operator. One can express the density operator, ρ,
as a matrix, after a choice of basis;
xb2 |ρ|b1 y “
dimpHq
ÿ
pα xb2 |ψα yxψα |b1 y.
(2.2.4)
α
Note that equation 2.2.2 can be rewritten as xxAyy “ trpρAq, certain conditions have to be fulfilled in order for an matrix to be called a density operator.
Definition 2.2.3. An operator, Q, is a Density operator if and only if
5
• Trace; T rpQq “ 1.
• Hermitian; Q “ Q: .
• Positive definite: xφ|Q|φy ě 0,
@|φy P H.
Now, one is ready to defining Pure states using the density operator.
Definition 2.2.4. A pure ensemble is defined as; for @α P Ně0 , D!α “ α1 such
that ρα1 “ 1 and ρα “ 0, @α P Nztα1 u .
ρ “ |ψα1 yxψα1 |.
(2.2.5)
Definition 2.2.5. A mixed ensemble is an ensemble which is not pure.
After the definition of a pure density operator, one could argue that a density
operator is pure if and only if the trace, T rpρ2 q “ 1 is equal to 1. This is
immediately evident considering that the density operator of an pure ensemble
is idempotent. However, this can be proven more elegantly.
Proposition 2.1. A density operator, ρ, is pure if and only if T rpρ2 q “ 1.
Proof. Assume
ř that ρ corresponds to the density operator of a mixed ensemble.
Then ρ “ α ρα |ψα yxψα | and
2
T rpρ q “
ÿÿ
1
˜
ÿÿ
xb |
b1
b2
i
ÿÿ
“
i
i
j
ÿ
ρi
2
j
ρi ρj xψi |ψj yxψj |
˜
ÿ
¸
|b1 yxb1 | |ψi y
b1
ÿÿ
i
ρi ρj |ψi yxψi |b yxb |ψj yxψj | |b1 y
j
“
ď
¸
2
ρi ρj |xψi |ψj y|2
ÿ
ρj “ 1.
(2.2.6)
j
Since, for a mixed ensemble, 0 ď ρi ă 1 @i P N, therefore ρ2i ă ρi . This
implies that the squared density operator of a mixed ensemble cannot have a
trace equal to 1. Thus a mixed ensemble cannot be pure.
2.3
Time-Evolution of the density operator
The time-evolution of the density operator, ρptq, can be predicted using the
Schrödinger equation. Say density operator at a given time is given by;
ρptq “
dimpHq
ÿ
pα |ψα ptqyxψα ptq|.
(2.3.1)
α
Then inserting equation (2.3.1) into the Schrödinger equation, yields;
6
Bρptq
“
Bt
dimpHq
ÿ
„ˆ
ρα
α
1
“
i~
dimpHq
ÿ
˙
ˆ
˙
B
B
|ψα ptqy xψα ptq| ` |ψα ptqy
xψα ptq|
Bt
Bt
ρα rpH|ψα ptqyq xψα ptq| ´ |ψα ptqy pxψα ptq|Hqs
α
1
rHρptq ´ ρptqHs
i~
1
“ rH, ρptqs.
i~
“
(2.3.2)
The general solution (provided that the Hamiltonian is time-independent)
to the differential equation given by equation (2.3.2) is given by
ρptq “ U pt, t0 qρpt0 qU : pt, t0 q.
(2.3.3)
Where the time-shift operator, U ,is given by equation (A.2.10).
2.4
Spin operators
For a fermionic system of spin- 12 , the spin operators are given by
„
1 0
2 1
„
1 0
Iy “
2 i
„
1 1
Iz “
2 0
Ix “

1
1
“ σx ,
0
2

1
´i
“ σy ,
0
2

1
0
“ σz ,
´1
2
rI x , I y s “ iIz ,
(2.4.1)
(2.4.2)
(2.4.3)
(2.4.4)
where ~ has been suppressed Do not confuse the operators for the identity
operator, if both are included, a notification will be given. The eigenkets are
represented by
„ 
1
|αy “
,
(2.4.5)
0
„ 
0
|βy “
.
(2.4.6)
1
Thus, applying the spin operators onto the eigenkets yields
1
|βy,
2
i
I y |αy “ |βy,
2
1
I z |αy “ |αy,
2
1
|αy,
2
i
I y |βy “ ´ |αy,
2
1
I z |βy “ ´ |βy.
2
I x |αy “
I x |βy “
7
(2.4.7)
(2.4.8)
(2.4.9)
3
Multiple quantum systems
3.1
Entanglement and Separability for mixed states
A mentioned previously, the notion of entanglement and separability must be
defined.
Definition 3.1.1. A mixed state is called separable
if and only if the density
Ân
operator, ρYj Aj , corresponding to the system jPNą0 HAj can be expressed as
ρYj Aj “
j n
ÿ
â
ρi ρi,Aj .
(3.1.1)
j
i
Then Entanglement is defined as:
Definition 3.1.2. An Entangled state is a state that is not separable.
In other words, the state of a composite quantum system (more on this in
the next section and [14]) is called entangled (or not separable) if and only if it
cannot be represented as a tensor product of states of its subsystems.
3.2
Composite quantum systems
While entanglement is not the subject of this thesis, it will be mentioned briefly
since certain concepts are adopted
Ťn in the discussion of the bipartite quantum
system. If n P N subsystems, iPNą0 Ai , are related to each other in the sense
that they have, at some point, interacted with one another then it is generally impossible to assign a single state vector to either of the subsystems. This
principle is known as non-separability. For the case of the bipartite quantum
composite system, separability will be discussed in terms of pure and mixed
states.
The state of a quantum system is described in terms of a Hilbert space H.
Consider n subsystems tHAi uni“1 ;
Definition 3.2.1. A composite system of the subsystems tHAi uni“1 is described
as the tensor product Hilbert-space;
HŤi Ai “
â
HAi .
(3.2.1)
i
3.3
Restricted case: Pure Bipartite Quantum composite
System (Pure Bipartite qudit state)
The state vector for a pure, separable quantum system Â
consisting of ”n”,
n
d
”d “ dimpHq”-dimensional subsystems is defined as |ψy P
iPNą0 HAi . The
bipartite quantum system consists of 2 subsystems. Then the d-dimensional
d
d
Hilbert space, Hd , of the Bipartite system, A Y B, is given by HA,B “ HA
b HB
(due to the separability condition). That is if t|µyA u is an orthonormal basis
d
d
for HA
and t|γyB u is an orthonormal basis for HB
, then t|µyA b |γyB u is an
d
d
d
d
orthonormal basis for HA b HB . Thus an arbitrary pure state of HA
b HB
can
8
be expanded (this is indeed possible due to the separability of the two Hilbert
spaces for system A Y B which, in turn, implies that the the systems A Y B are
not entangled ) as;
dimpH
ÿ A q dimpH
ÿ Bq
|ψyA,B “
i
ai,j | µi yA b | γj yB
(3.3.1)
j
ř ř
Where i j |ai,j |2 “ 1, ai,j P C. Roughly speaking, in this case, a pure
state is separable if, in equation p3.3.1q, ai,j “ bi cj , such that |ψyA,B “ |ψyA b
|ψyB . Note that it
ř is possible to express equation (3.3.1) as a single sum by
denoting |γ̃yB “ j ai,j |γyB but at the expense of |γ̃yB not, in general, be
orthonormal. Say one is only interested in subsystem A, this is achieved by the
observable M A bI B , where M A is an self-adjoint operator acting on subsystem
A and I B , is the identity operator acting on subsystem B. The expectation value
of an observable M A b I B is given by;
xM A y “A,B xψ|M A b I B |ψyA,B
˜
¸
˜
¸
ÿÿ
ÿÿ
˚
“
ak,l A xµk | b B xγl | pM A b I B q
ai,j |µi yA b |γj yB
k
i
l
ÿÿÿÿ
“
k
l
i
k
i
j
b B xγl |I B |γj yB s
j
ÿÿÿ
“
a˚k,l ai,j rA xµk |M A |µi yA
a˚k,j ai,j A xµk |M A |µi yA δl,j
j
“ T r pM A ρA q ,
(3.3.2)
ř ř ř
˚
j ak,j ai,j |µk yAA xµi |,
where ρA “ T rB pρA,B q “ T rB p|ψyA,B A,B xψ|q ” k i
for which the total density operator for the entire system, A Y B is denoted as
ρA,B . That is, the pure density operator is given by
ρA,B “ |ψyA,B b |ψyA,B “ |ψyA,B A,B xψ|
(3.3.3)
Then the reduced density operator, ρA (ρB ), for subsystem A is obtained by
taking the partial trace, T rB (T rA ), over subsystem B (A). The reduced density
operator, ρA (ρB ), is self-adjoint, positive definite and its trace is equal to 1.
3.4
Purification
Generally, any result in a mixed quantum state can be viewed as the reduced
state of a pure state in a larger dimensional Hilbert space. This is the notion
of purification and has already been touched upon in the previous section. A
formal definition shall be given now [44].
n
Theorem 3.1. Let ρ be the density operator acting on the Hilbert space HA
.
n
n
n
Then there exists a Hilbert space HB and a pure state |ψyA,B P HA b HB such
n
that the partial trace of |ψyA,B A,B xψ| with respect to HB
is given by
trB p|ψyA,B A,B xψ|q “ ρA ,
where |ψy is called the purification of ρ
9
(3.4.1)
řdimpHn q
Proof. A positive semi definite density matrix, ρ, is given by ρ “ i
pi |iyxi|
n
n
for some orthonormal basis tiu. Furthermore, let HB
be another copy of HA
1
n
n
with the orthonormal basis ti u. Define |ψyA,B P HA b HB by
dimpHq
ÿ
|ψyA,B “
?
pi |iyA b |i1 yB .
(3.4.2)
i
Then, using the partial trace, one obtains
trB p|ψyA,B A,B xψ|q “
«˜
¸˜
¸ff
ÿ?
ÿ?
1
1
pI b trq
pi |iyA b |i yB
pj A xj| b B xj |
“
j
i
pI b trq
«
ÿÿ?
i
ÿÿ?
i
1
pi pj |iyAA xj| b |i yB B xj | “
j
`
˘
pi pj |iyAA xj| tr |i1 yB B xj 1 | “
j
ÿÿ?
i
ff
1
pi pj |iyAA xj|δij “ ρA .
(3.4.3)
j
This proves the claim.
The notion of purification, in the sense of the definition given by Uhlmann
and Sjöqvist, will be discussed and further elaborated in the upcoming sections.
3.5
Tensor product spaces
The wave-function, |ψy in product basis is expressed as the direct products of
wave-functions for corresponding spin.
|ψy “
N
â
(3.5.1)
|mi y.
k
That is, the direct product of two matrices A and B yields
„
AbB “
A11
A21
 „
A12
B11
b
A22
B21
 „
B12
A11 B
“
B22
A21 B

A12 B
.
A22 B
(3.5.2)
Thus the wave function |ψy in product-basis of a two-spin system is (using
the eigenkets as described in Equation (2.4.5 and 2.4.6))
10
» fi
1
„  „ 
—0ffi
1
1
ffi
|ψ1 y “ |αy b |αy “
b
“—
–0fl ,
0
0
0
» fi
0
„  „ 
—1ffi
1
0
ffi
|ψ2 y “ |αy b |βy “
b
“—
–0fl ,
0
1
0
» fi
0
„  „ 
—0ffi
0
1
ffi
“—
|ψ3 y “ |βy b |αy “
b
–1fl ,
1
0
0
» fi
0
„  „ 
—0ffi
0
0
ffi
|ψ4 y “ |βy b |βy “
b
“—
–0fl .
1
1
1
(3.5.3)
(3.5.4)
(3.5.5)
(3.5.6)
Consider the sum of the operator I 1z and I 2z (where I z is given by Equation
(2.4.3))for a two-spin system (where the superscript 1 stands for first spin and
2 for the second spin). However, the sum I 1z ` I 2z yields a 2-by-2 matrix which
is incorrect since it is a sum of one-spin operators. Formally, the operators
of a two-spin system can be calculated from the direct product of the one-spin
operators with the identity operators. That is, if one denotes one-spin operators
as 1 I, then (also for the sake of avoiding misunderstanding, the identity operator
shall be denoted E)
2 1
Iz
2 2
Iz
“ 1 I 1z b E 2 ,
2
“E b
(3.5.7)
1 2
Iz.
(3.5.8)
Thus, the sum of operators for a two-spin system is then correctly given by
(using the eigenkets described in Equation (2.4.5) and 2.4.6))
»
1
—0
2 1
2 2
1 1
1 2
I z ` I z “ I z b E2 ` E2 b I z “ —
–0
0
0
0
0
0
0
0
0
0
fi
0
0 ffi
ffi .
0 fl
´1
(3.5.9)
Thus, for a two-spin system, the operator algebra in direct product spaces
is given by:
pA b Bq p|αy b |βyq “ A|αy b B|βy.
11
(3.5.10)
4
Geometrically induced phases for pure states
The phenomenon of a geometric phase, mathematically known as holonomy
(transport of a vector along a closed curve on a manifold, in particular: transport
of a state vector within the space of states), arises due to parallel transport of
a vector on a curved area, that is, when the system undergoes a loop, see figure
2.
Figure 2: [7] The transport of a vector along the closed curve C on a 2-sphere.
Classically, it is impossible to state whether or not the system has undergone an evolution in quantum mechanics, the system retains some information
about its motion in the form of a phase factor of the state vector. In 1984
M.V. Berry addressed the issue of unitary cyclic evolution under the action of
a time-dependent Hamiltonian in a quantum system. Supposedly, the process
changes adiabatically, which implies that the time scale of the changing Hamiltonian is much larger than the time scale of the system. It was assumed that the
quantum system for a cyclic Hamiltonian would only acquire a dynamical phase
(Since it comes from the dynamics of the system), deprived of any physical
meaning, which could be eliminated by using a suitable gauge-transformation
of the from |ψy Ñ eiφ |ψy. However, M.V. Berry discovered that there is an
additional phase, beside the dynamical phase, which is purely geometrical and
depends only on the path that |ψy describes in the parameter space.
12
4.1
Adiabatic Geometric phase (Berry’s phase)
Adiabatic Process [49] involves a slow perturbation acting on the systems eigen~ of the Hamiltonian HpRq
~
state. As such, it is required that the parameter R
changes slowly for the adiabatic theorem to be in affect. This implies that the
~
system will remain in an eigenstate of HpRptq;
tq at any time t, provided that the
system initially is in an eigenstate of H. Assuming that the evolution is cyclic
~
~ qq for some period T ), then the Hamiltonian takes on its original
(Rp0q
“ RpT
and final form at T , resulting in the system returning to its initial state. The
state has been transported around a loop, according to C : t P r0, T s ÞÑ |ψy,
in parameter space with |ψy describing the instantaneous state of the system,
~
equivalent to the eigenstate |n; Rptqy
of the instantaneous Hamiltonian:
~
~
~
HpRptq;
tq|n; Rptqy
“ En ptq|n; Rptqy,
(4.1.1)
~
where En denotes the n:th energy eigenstate and |n; Rptqy
is assumed to be
normalized. The Schrödinger equation for the evolution of the system’s state
vector |ψy is given by
i~
d
~
|ψy “ HpRptq;
tq|ψy.
dt
(4.1.2)
~
~
Let the initial state |ψp0qy “ |n; Rp0qy
be an eigenstate of HpRp0q;
0q at time
t “ 0. Assuming that the solution due to adiabatic evolution, reads;
~
|ψptqy “ eiΦptq |n; Rptqy.
(4.1.3)
The phase Φptq is determined by inserting equation (4.1.3) into equation
(4.1.2):
„

dΦ iΦptq
iΦptq d
~
~
~
~
´i~ i e
|n; Rptqy ` e
|n; Rptqy “ eiΦptq En pRptq;
tq|n; Rptqy.
dt
dt
(4.1.4)
By multiplying equation (4.1.4) with the dual of |ψy from the left and utilizing the orthogonality property of the eigenstate, one is left with
dΦ
1
B
~
~
~
“ ´ En pRptq;
tq ` ixn; Rptq|
|n; Rptqy.
dt
~
Bt
(4.1.5)
Equation (4.1.5) can be integrated with respect to time t P r0, T s, leaving
you with
Φptq “ ´
1
~
żT
żT
~
En pRptq;
tqdt ` i
0
~
xn; Rptq|
0
B
~
|n; Rptqydt.
Bt
(4.1.6)
The dynamical phase is the first integral in equation (4.1.6). It can be
written as
13
θn ptq “ ´
1
~
żT
~
~
~
xn; Rptq|Hp
Rptq;
tq|n; Rptqydt.
(4.1.7)
0
For the second integral in equation (4.1.6), the time dependence has been
~
B
dR
~
~
assumed to be implicit, that is Bt
|n; Rptqy
“ ∇R~ |n; Rptqy
dt . Inserting this
expression yields
Φptq “ ´
1
~
żT
~
En pRptq;
tqdt ` i
ż RpT q
Rp0q
0
~
~
~
xn; Rptq|∇
~ |n; RptqydR.
R
(4.1.8)
If one now considers a closed path, then Rp0q “ RpT q, and
1
Φptq “ ´
~
żT
¿
~
En pRptq;
tqdt ` i
0
~
~
~
dRxn;
Rptq|∇
~ |n; Rptqy.
R
(4.1.9)
C
That is, the Berry phase is then given by
¿
γn rCs “i
~
~
~
dRxn;
Rptq|∇
~ |n; Rptqy
R
(4.1.10)
¿C
~ ¨ dR,
~
AB pRq
“
(4.1.11)
C
~
~
where AB “ ixn; Rptq|∇
~ |n; Rptqy is called the Berry Connection. The Berry
R
phase is independent of time and only depends on the path C in parameter
space. It is also worth noting that the Berry phase, γn rCs, is real, which can
~
~
be easily shown using the normalization condition xn; Rptq|n;
Rptqy
“ 1, which
gives
∇R~ xn|ny “ xn|∇R~ ny˚ ` xn|∇R~ ny “ 0,
(4.1.12)
implying that the real part is identically zero and Equation (4.1.10) is real.
4.1.1
Gauge invariance of the Berry phase
By choosing a different phase for the eigenvectors |nptqy, one can show that
the geometric phase, γn , remains invariant, otherwise the quantity would not
be physical since the choice of phase for the eigenvectors are arbitrary at each
instant of time. That is, make the following gauge transformation
|nptqy ÞÑ |n1 ptqy “ eiαptq |nptqy.
Then, the geometric phase is expressed as
14
(4.1.13)
ż
γn1 1
“i
d
dtxn ptq| |n1 ptqy “ γn ´
dt
żτ
1
dtαptq,
(4.1.14)
0
where αpτ q ´ αp0q “ 2πn (since initial and final eigenstate can only differ by
a integer phase of 2π due to choosing single-valued eigenvalue basis). Thus
γn1 1 “ γn mod 2π.
4.1.2
Informal description of the Chern number
An informal description of the Chern number is given here. Consider an adiabatic transport of an eigenstate around a small loop in the parameter space.
When a loop is completed, the particle will be in the same eigenstates as it
started, up to a possible multiplication of a phase factor, say eiφ1 . By Stokes
theorem, φ1 should be a function describing the area inside the loop. Now, adiabatically transporting the eigenstate in the opposite direction around the loop
yields a phase factor e´iφ2 where φ2 is a function describing the area outside of
the loop. However, the phase should be the same regardless of direction. Thus
eiφ1 “ e´iφ2 or eipφ1 `φ2 q “ 1. By setting φ “ φ1 ` φ2 , one ends up with eiφ “ 1
which implies that φ must be an integer multiple of 2π. This integer is called
the Chern number [40].
15
4.1.3
Example, spin 1/2 in an adiabatically rotating magnetic field
Consider this particular example of a spin- 12 particle moving in a externally
~
applied, adiabatically rotating magnetic field Bptq
under angle θ around the
z-axis, see figure 3.
Figure 3: [7] A spin- 21 particle moving in a adiabatically, rotating, θ, externally
applied magnetic field B with angular velocity ω, emulating Equation (4.1.16)
The magnetic field is given by
fi
cospθq sinpωtq
~
~
– sinpθq sinpωtq fl
Bptq
“ |Bptq|
cospθq.
»
(4.1.15)
As the magnetic field varies, the spin´ 12 particle follows the direction of the
magnetic field in the sense that the eigenstate of Hp0q goes to the eigenstate
of Hptq at later times ,t. The interaction Hamiltonian, in rest frame, for this
system is given by
„
~
~
~
Hptq
“ µ|Bptq|~
σ “ µ|Bptq|
where the constant µ “
e
2m ~.

cospθq
e´iωt sinpθq
,
eiωt sinpθq
´ cospθq
(4.1.16)
The eigenvalue equation
Hptq|nptqy “ En |nptqy,
(4.1.17)
is solved by the normalized eigenstates of Hptq
` ˘ 
cos θ2` ˘
,
eiωt sin θ2
` ˘ 
„
´ sin `θ2 ˘
|n´ ptqy “ iωt
,
e cos θ2
„
|n` ptqy “
(4.1.18)
(4.1.19)
with the corresponding eigenvalues
~
E˘ “ ˘µ|Bptq|.
16
(4.1.20)
Furthermore, one could argue that one could indeed interpret the eigenstates
|n˘ y as spin up and spin down (for ` and ´ respectively) along the direction
of the magnetic field.
~
The allowed values for the Hamiltonian, H “ Hpθ, φptq “ ωt, rptq “ |Bptq|;
tq
2
~
is identical to the parameter-space given by S . That is, the magnetic field Bptq
traces our the curve; C : rptq “ r, θptq “ θ, φ P r0, 2πs, see figure 4.
Figure 4: [7] Magnetic field, Equation (4.1.16), following the path C, expressed
in parameter space
Thus, the gradient spanned by the magnetic field is then given by
∇|n˘ ptqy “
B
1 B
1
B
|n˘ ptqyr̂ `
|n˘ ptqyθ̂ `
|n˘ ptqyφ̂.
Br
r Bθ
r sinpθq Bφ
(4.1.21)
Inserting Equation (4.1.18 & 4.1.19) into Equation 4.1.21 yields
` ˘ 
„

1
0 ` ˘
´ 12 sin `θ2 ˘
θ̂
`
φ̂,
1 iωt
cos θ2
r sinpθq ieiωt sin θ2
2e
` ˘ 

„
1
1
0 ` ˘
´ 12 cos θ2` ˘
φ̂.
∇|n´ ptqy “
θ̂
`
r ´ 12 eiωt sin θ2
r sinpθq ieiωt cos θ2
∇|n` ptqy “
1
r
„
„
(4.1.22)
(4.1.23)
Multiplying by the corresponding bar vector from the left yields
` ˘
sin2 θ2
xn` |∇|n` y “ i
θ̂
r sinpθq
` ˘
cos2 θ2
xn´ |∇|n´ y “ i
θ̂,
r sinpθq
(4.1.24)
(4.1.25)
whereby integration along the path C yields the expression for berry phase
(4.1.10)
17
¿
γn,˘ rCs “
xn˘ |∇|n˘ yr sinpθqdθdφ “ iπp1 ¯ cospθqq,
(4.1.26)
C
with the dynamical phase (4.1.7) given by
1
θn rT s “ ´
~
żT
0
µ ~
E˘ ptqdt “ ¯ |Bptq|T.
~
(4.1.27)
Thus, the total state from initial state to final state (after on cycle) is then
given by
µ
~
|n˘ ptqy “ e´iπp1¯cospθqq e¯ ~ |Bptq|T |n˘ p0qy.
(4.1.28)
Note that the dynamical phase only depends on the period, T , of the rotation
while the geometrical phase depends on the geometry of the problem.
18
4.2
Non-adiabatic Geometric phase (Aharonov-Anandan
phase)
A generalization of the Berry phase was proposed in 1987 by Aharonov & Anandan [7]. The consideration of a closed path, unrestricted by the adiabatic theorem, allows one to bypass the notion of parameter space when describing the
evolution of the Hamiltonian. When doing so, it is important to work with the
projective state space in which the closed curve is traced by the system. The
importance of this generalization is due to the fact that in real processes, the
adiabatic condition is never exactly fulfilled.
Let us consider the projective state space, P, which is by definition, the set
of equivalence classes of all state vectors of the Hilbert space, H, with respect
to the equivalence relation (for the sake of convenience, assume that |ψy is normalized)
|ψ 1 y „ |ψy iff |ψ 1 y “ eiφ |ψy,
(4.2.1)
where φ is a real number. The projection operator, Π, from the Hilbert space
to the projective Hilbert space is defined as
Π : H Ñ PpHq,
|ψy ÞÑ |ψyxψ|.
(4.2.2)
This implies that every ray is mapped to a point in P. That is, the pure
state density operator (2.2.5) corresponds to the projective state space due to
the loss of phase information. If dimpHq “ n then the projective Hilbert space
PpHq is a manifold of dimension dimpPpHqq “ n ´ 1, see figure (5).
Figure 5: [7] The path C in H being projected onto the projective state space
C 1 on PpHq. Note that the path C need not necessarily be closed for the path
C 1 to be closed.
19
Now, consider a cyclic evolution of a state vector, |ψptqy of period T , in the
projective state space. The unitary evolution
|ψp0qy ÞÑ |ψptqy “U ptq|ψp0qy
(4.2.3)
of the state vector produces the path C : r0, T s Ñ H, which is projected on the
projective Hilbert space P pHq via the map, Π, as ΠpCq “ C 1 . However, there
are infinitely many paths in H that project to the same path in PpHq. That is,
if |ψy describes the path C and |ψ̃y “ eif ptq |ψy describes the path C̃, then for
any arbitrary real function, f ptq, they define the same path C 1 in PpHq under
the projection operator π;
eif ptq |ψy ÞÑ eif ptq |ψyxψ|e´if ptq “ |ψyxψ|.
(4.2.4)
Additionally, the evolution is cyclic if and only if the path in the projective
Hilbert space, C 1 , is closed (note that the path in the Hilbert space, C, need
not necessarily be closed for the path in C 1 to be closed), that is
|ψpT qyxψpT q| “ |ψp0qyxψp0q|.
(4.2.5)
For a closed path, the total phase Φ is given [19] as the argument of the
complex number xψp0q|ψpT qy:
ΦrCs “ argtxψp0q|ψpT qyu,
(4.2.6)
which signifies the phase shift of the system. Aharonov & Anandan [16] showed
that the geometric phase can be obtained by subtracting the dynamical phase
(reminiscent of Equation (4.1.7)):
żT
θrCs “ ´
xψptq|H|ψptqydt
0
żT
xψp0q|U : ptq
“ ´i
0
dU ptq
|ψp0qydt,
dt
(4.2.7)
From the expansion for the total phase (Equation (4.2.6)), it then follows that
the geometric phase is then given by
żT
xψp0q|U : ptq
γrC 1 s “ argtxψp0q|ψpT qyu ` i
0
dU ptq
|ψp0qydt.
dt
(4.2.8)
This is a functional of C 1 alone. The Aharonov & Anandan (AA) geometric
phase (4.2.8) can be applied on a cyclic or adiabatic Hamiltonian [18] since
the geometric phase depends only on the cyclic evolution of the system itself.
Thus, in the adiabatic limit, the AA-phase tends to the Berry phase. The
arbitrariness of f ptq allows one to completely remove the dynamical phase, that
is, equation (4.2.8) is a gauge-invariant expression for the geometric phase of a
20
şt
pure state. By choosing f ptq “ ~1 0 xψpt1 q|Hpt1 q|ψpt1 qydt, one can impose the
condition xψp0q|U : ptq dUdtptq |ψp0qy “ 0, @ψ P H, known as parallel transport.
The validity of the parallel transport condition can be easily proven, using the
gauge transformation |ψ 1 ptqy “ eif ptq |ψptqy, whereby
d 1
d
|ψ ptqy “ xψp0q|U : ptqe´if ptq U ptqeif ptq |ψp0qy
dt
dt
ˆ
˙
dU ptq if ptq
df ptq if ptq
:
´if ptq
“xψp0q|U ptqe
e
e
|ψp0q ` i
U ptq|ψp0qy
dt
dt
df ptq
dU ptq
|ψp0qy ` i
“xψp0q|U : ptq
dt
dt
˙
ˆ żt
dU
ptq
d
1
:
1
1
1
xψpt q|Hpt q|ψpt qydt
“xψp0q|U ptq
|ψp0qy ` i
dt
dt ~ 0
dU ptq
i
“xψp0q|U : ptq
|ψp0qy ` xψptq|Hptq|ψptqy
dt
~
dU ptq
dU ptq
:
|ψp0qy ´ ψp0q|U : ptq
|ψp0qy “ 0.
“xψp0q|u ptq
dt
dt
xψ 1 ptq|
(4.2.9)
(4.2.10)
(4.2.11)
(4.2.12)
(4.2.13)
(4.2.14)
Here the orthogonality condition has been used in Equation (4.2.11) and the
fundamental theorem of calculus in Equation (4.2.12). Moreover, from Equation
(4.2.11), one could easily deduce the expression of f ptq that is needed in order to
fulfil the condition of parallel transport. Thus, provided that parallel transport
is ensured, the non-adiabatic Geometric phase, formulated by Aharonov et al.
is given by Equation (4.2.6). The notion of parallel transport will be discussed
more thoroughly in Section 5.1 regarding mixed geometric phase given by A.
Uhlmann.
21
5
Geometrically induced phases for mixed states
For a pure state, Equation (4.2.6) gives the total phase (or the geometric phase
provided parallel transport is ensured) gives the total phase change during an
evolution. However, for a mixed state, this fails. The first problem lies in identifying the change of the mixed state density operator ρ. States in of identically
prepared quantum systems exhibits an uncertainty, hence the reason why they
are described as a micture of pure states.
Uhlmann, [28] [29], was among the first to develope a theory for the geometric phase for parallel transported mixed states. This was approached using
the notion of amplitudes and purification (Section (5.1.1)), which is then used,
extensively, in Section 7.
Another approach was given by Sjöqvist et al. [19] where they later showed
that, using a conventional Mach-Zehnder interferometer, it is possible to measure this phase using an interferometer by observing the intensity of the output
signal. It turns out that the Uhlmann phase is, indeed, an observable and is
later experimentally obtained in Section 6.
5.1
5.1.1
A. Uhlmann’s concept of mixed geometric phase
Parallel amplitude
The notion of parallel amplitudes is essential, for the Uhlmann case, to be able
to define a parallel transport condition. Consider the amplitudes w1 and w2 for
given states ρ1 and ρ2 respectively. The two amplitudes w1 and w2 are said
to be parallel if they minimize the Hilbert space distance in Hw , that is w1 is
parallel to w2 (w1 k w2 ) if
r1 ´ w
r2 k2 ,
k w1 ´ w2 k2 “ min k w
w
r1 ,w
r2
(5.1.1)
r1 and w
r2 which satisfies
where the minimum is taken over all w
r1 w
r1: ,
ρ1 “ w
(5.1.2)
r2 w
r2: .
w
(5.1.3)
ρ2 “
One can rewrite the minimum condition to obtain properties of the amplitudes:
“
‰
min k w
r1 ´ w
r2 k2 “ min tr pw
r1 ´ w
r2 q: pw
r1 ´ w
r2 q
w
r1 ,w
w
r 1 ,w
r2
r2
”
ı
r1: w
r1 ` w
r2 ´ w
r2 ´ w
r1
“ min tr w
r2: w
r1: w
r2: w
w
r1 ,w
r2
”
ı
r1: w
r2 ` w
r1
“ tr pρ1 q ` tr pρ2 q ´ max tr w
r2: w
w
r 1 ,w
r2
” ´
¯ı
:
r1 w
r2 ,
“2 ´ 2 max Re tr w
w
r1 ,w
r2
(5.1.4)
(5.1.5)
(5.1.6)
(5.1.7)
where in the first line the explicit expression for the Hilbert-Schmidt norm has
been used (Definition (A.1.9)). Given the condition of maximum, for Repxq ď
22
|x|, it is clear that w̃1 and w̃2 are chosen such that w̃1: w̃2 is self-adjoint and
positive definite.
A more explicit expression can be achieved by using the polar decomposition
theorem. This theorem states
a that any operator A can be decomposed into
A “ |A|U A , where |A| “ A: A and U A is a unitary operator. Now for any
unitary operator U , one has
¯
´a a
|A| |A|U U A |
Re rtr pAU qs ď | tr pAU q | “ | tr
c
” ´
¯ı
ď ptr p|A|qq tr U :A U : |AU U A | “ tr |A|,
(5.1.8)
where in the second line, the Cauchy-Schwartz inequality is used. Equlity is
obtained when U “ U :A , leaving us with
max Re rtr pAU qs “ tr |A|.
U
(5.1.9)
Now, applying Equation (5.1.9) onto the last term of Equation (5.1.7) for
?
?
the decomposition w̃1 “ ρ1 U 1 and w̃2 “ ρ2 U 2 , yields
” ´
¯ı
¯ı
” ´ ? ?
max Re tr w̃1: w̃2 “ max Re tr U :1 ρ1 ρ2 U 2
w̃1 ,w̃2
U 1 ,U 2
? ?
“ max Re rtr p ρ1 ρ2 U qs
U
b
?
?
“ tr
ρ1 ρ2 ρ1 .
(5.1.10)
Thus, inserting Equation (5.1.10) back into Equation (5.1.7), one obtains the
Hilbert space distance between two parallel amplitudes w1 and w2 which is
(equal to the Bures distance [43]) given by
kw1 ´ w2 k2 “ 2 ´ 2 tr
b
?
?
ρ1 ρ2 ρ1 .
(5.1.11)
The equality in Eqution (5.1.10) is obtained for U “ U 2 U :1 “ U :?ρ1 ,?ρ2
which is the adjoint of the unitary operator of the decomposition
? ?
? ?
ρ1 ρ2 “ | ρq ρ2 |U ?ρ1 ,?ρ2 .
(5.1.12)
This can be flipped around and be expressed as
b
U 2 U :1 “
ρ´1
2
b
b
?
?
ρ´1
ρ1 ρ2 ρ1 ,
1
(5.1.13)
Equation (5.1.13) is the relation that the unitary operators U 1 and U 2 must
satisfy to minimize the distance between associated amplitudes w1 and w2 in
order to achieve w1 k w2 .
23
5.1.2
Parallel transport and connection form
let the initial state of the trajectory, on Q (the density matrix space), be ρr0 .
?
Under polar decomposition, wr0 “ ρr0 U r0 . For simplicity, one uses the, uni?
d
tary, gauge U r0 “ I . At some point r, of the trajectory, wr “ ρr V r where
?
V r is the unitary obtained from the initial condition wr0 “ ρr0 after applying the parallel transport condition (rather the parallel amplitude condition).
Once can introduce AU prq, called the connection form of the trajectory, which
l
acts as a generator of the unitary V r . That is, if the trajectory rrptqst“1 is the
parametrization of the trajectory, V r is fully determined from the differential
#
dV rptq
dt
“ AU rrptqs V rptq ,
V rp0q “ I d .
(5.1.14)
şr
A pr 1 qdr 1
This differential equation is formally solved as V r “ Pe r0 U
. AU can
be obtained by reexpressing Equation (5.1.14) in the differential form
dV V : “ AU ,
(5.1.15)
ÿ
(5.1.16)
where
AU “
AiU dki .
i
Consider the density matrix ρ1 “ ?
ρ, and displace it by a infinitesimal trans?
lation dρ. Then w1 “ ρ and w2 “ ρ ` dρ pV ` dV q which has to fulfill the
parallel transport condition (Equation (5.1.11)) in order to obtain AU . Inserting
w1 and w2 into equation (5.1.13) yields
?
? ´1 a ´1
pV ` dV q V : “ p ρ ` d ρq
ρ
b
?
?
ρ pρ ` dρq ρ,
(5.1.17)
where one assumes that under infinitesimal translation the relation:
a
?
?
ρ ` dρ “ ρ ` d ρ
(5.1.18)
is valid. Hübner [32] introduced an auxiliary, real, parameter s, multiplied
across every differential
?
? ´1 a ´1
pV ` sdV q V : “ p ρ ` sd ρq
ρ
b
?
?
ρ pρ ` sdρq ρ,
(5.1.19)
and then expand with respect to s at s “ 0, keeping only first order terms (and
neglecting the unitary operators that arise as a result of the expansion), one
obtains
b
? ´1 a ´1 ?
? ı
d ”?
p ρ ` sd ρq
A “sdV V : “ s
ρ
ρ pρ ` sdρq ρ
ds
 s“0
„
b
? ´1 ?
?
?
d ?
´1 d
p ρ ` sd ρq
ρ`ρ
ρ pρ ` sdρq ρ
.
“s
ds
ds
s“0
24
(5.1.20)
(5.1.21)
Equation (5.1.21) has been divided into two terms. The first term can be
expressed as follows:
? ´1 ? ı
d ”?
p ρ ` sd ρq
ρ
ds
„”

a
? ´ d
? ¯ı´1 ?
d
´1
“
ρ I `s ρ d ρ
ρ
ds
„´

a
? ¯´1 a ´1 ?
d
“
I d ` s ρ´1 d ρ
ρ
ρ
ds
a
a
? ı´1
?
d ” d
“
I ` s ρ´1 d ρ
“ ´ ρ´1 d ρ,
ds
(5.1.22)
(5.1.23)
(5.1.24)
(5.1.25)
where, in the last step, we have expanded wit respect to s up to the first order.
In the eigenbasis of ρ, (in the spectral basis expressed in Equation (2.2.3))
a
?
?
1
´xψi | ρ´1 d ρ|ψj y “ ´ ? xψi |d ρ|ψj y.
pi
(5.1.26)
The second term, on the other hand, requires more work. Define the followˇ
a?
?
ˇ
ing: Gpsq “
ρ pρ ` sdρq ρ. Then the last term is given by ρ´1 dGpsq
,
ds ˇ
s“0
which can be computed solving the following differential equation:
?
dGpsq ?
d
dGpsq
rGpsqGpsqs “
Gpsq ` Gpsq
“ ρdρ ρ.
ds
ds
ds
(5.1.27)
Note that Gp0q “ ρ. It follows that
ˇ
ˇ
?
?
d
dGpsq ˇˇ
dGpsq ˇˇ
rGpsqGpsqs|s“0 “
ρ
`
ρ
“ ρdρ ρ.
ds
ds ˇs“0
ds ˇs“0
(5.1.28)
Taking the matrix element of Equation (5.1.28) yields:
ppi ` pj qxψi |
ˇ
dGpsq ˇˇ
?
|ψj y “ pi pj xψi |dρ|ψj y.
ds ˇs“0
(5.1.29)
Then the matrix elements of ρ´1 dGpsq
is given by
ds
xψi |ρ´1
ˇ
ˇ
1
dGpsq ˇˇ
dGpsq ˇˇ
x|ψ
|
|ψ
y
“
|ψj y
j
i
ds ˇs“0
pi
ds ˇs“0
?
pj
“?
xψi |dρ|ψj y
pi ppi ` pj q
`
?
?
? ˘
pj
pi ` pj
?
xψi |d ρ|ψj y,
“ ?
pi ppi ` pj q
(5.1.30)
(5.1.31)
(5.1.32)
? ?
where in the last line, the following expression was used: dρ “ dp ρ ρq “
? ?
? ?
d ρ ρ ` ρd ρ. Finally, insertin Equation (5.1.26 and 5.1.32) into Equation
(5.1.21), one obtains
25
? `?
? ˘¸
pj
pi ` pj
?
1
xψi |d ρ|ψj y
xψi |A|ψj y “ ´ ? ` ?
pi
pi ppi ` pj q
ˆ?
? ˙
pi ´ pj
?
“
xψi |d ρ|ψj y,
pi ` pj
˜
(5.1.33)
(5.1.34)
which can be expressed in terms of the commutator (also inserting Equation
(5.1.16))
AU “
ÿ dimpHq
ÿ
ÿ dimpHq
µ
(where Bµ :“
5.1.3
i
B
Bkµ )
|ψik y
j
“ ? ? ‰
xψik | Bµ ρ, ρ |ψjk y k
xψj |dkµ ,
pi ` pj
(5.1.35)
and is known as the Uhlmann connection.
The Uhlmann Phase
The Uhlmann approach is roted in the concept of amplitudes. As mentioned
previously, an amplitude is a matrix, w, such that for some density matrix ρ,
one has
ρ “ ww: .
(5.1.36)
The general idea is that the amplitude form a Hilbert space Hw with the
Hilbert-Schmidt product
pw1 , w2 q :“ trpw1: w2 q.
(5.1.37)
Equation (5.1.36) suggests a U pnq-gauge freedom in the choice of amplitude:
w and wU are amplitudes of the same state for some unitary operator U . It is
worth noting the parallelism with the U p1q-gauge freedom of pure states where
one defines the projection map (similar to Equation (4.2.2) with P pHq “ Q)
such that |ψy and eiφ |ψy represent the same physical state. An amplitude,
similarly defined as π : Hw Ñ Q, is another way of describing the concept of
purification (see section 5.2.3). Using polar decomposition [31], one parametrizes
?
the possible amplitudes of the density matrix ρ as w “ ρU . The spectral
theorem (decomposition, see Equation (A.2.3) and (A.2.4)) dictates that
w“
ÿ?
pj |ψj yxψj |U,
(5.1.38)
j
with the following isomorphism between the spaces Hw and H b H
w“
ÿ?
pj |ψj yxψj |U ÐÑ |wy “
j
ÿ?
pj |ψj y b U T |ψj y,
(5.1.39)
j
where the transpose is taken with the respect of the eigenbasis of ρ. The property
given in Equation (5.1.36) can now be expressed as
26
ρ “ tr2 p|wyxw|q ,
(5.1.40)
where tr2 denotes the partial trace over the second factor of H b H. That is,
any amplitude, w, can be seen as a pure state, |wy, of an enlarged space H b H,
with the partial trace equaling ρ.
Consider the family of pure states |ψk yxψk | and a trajectory in parameter
space, tkptqu1t“0 such that initial and final states are equal. This gives rise to
the trajectory in the Hilbert space, H, |ψkptq y. Since the path on Q (set of
density matrices) is closed, the inital and final vectors are equal up to a phase
Φ, |ψkplq y “ eiΦ |ψkp0q y, provided the parallel transport condition is given by
Berry (see last term in Equation (4.2.8)), then ΦB is the Berry phase (Equation
(4.1.10). This depends only on the geometry of the path and is expressed in the
Berry Connection form (Equation (4.1.11)). Similarly, a closed trajectory of the
density matrix ρk (which may not necessarily be pure) gives a trajectory on the
Hilbert space, Hw , wkptq . The inital and final state may differ (since the path
is closed on Q) by some unitary transformation, V , such that wkplq “ wkp0q V .
Uhlmann defines the parallel transport conditon as
V “ Pe
ű
AU
U 0,
(5.1.41)
where P is the path-ordering operator (see Seciton (A.2.3)), AU is the Uhlmann
connection form and U 0 is the gauge taken at kp0q, see figure 6 for an ullustrative
difference between the approach given by Berry and Uhlmann.
Figure 6: Comparison [33] between the Berry (Equation (4.1.10)) and Uhlmann
(Equation (5.1.42)) aproaches in obtaining the geometric phase.
The Uhlmann connection form [32] is given by Equation (5.1.35), where it
is seen that trpAU q “ 0 from Equation (5.1.34). The Uhlmann geometric phase
along the closed trajectory tkptqult“0 is given by
´
¯
:
wkplq .
ΦG rks :“ argxwkp0q |wkplq y “ arg tr wkp0q
27
(5.1.42)
The polar decomposition theorem entails that the amplitudes w can be ex?
?
pressed as wkp0q “ ρkp0q U 0 and wkplq “ ρkp0q V r where U 0 is the identity
ű
operator and V r “ Pe AU such that Equation (5.1.42) can be reexpressed as
´
¯
ű
ΦG rks “ arg tr ρkp0q Pe AU .
28
(5.1.43)
5.2
The Sjöqvist formalism
A new formalism for the mixed state geometric phase in the experimental context of quantum interferometry has been provided by E. Sjöqvist et al [19], with
the aim to establish an operationally well defined notion of phase for unitarily
evolving mixed quantum states in interferometry. That is, Sjöqvist provided
an experimental approach to observing the mixed state geometric phase, thus
establishing that the geometric phase is indeed an observable quantity.
Figure 7: [12] A conventional Mach-Zehnder ineterferometer with two beamsplitters and two mirrors
Consider a conventional Mach-Zehnder interferometer, as shown in Figure
7.
Figure 8: [?, ?] Sjöqvist’s interferometry model BS1 and BS2 are beam splitters,
M1 and M2 are Hadamard mirrors, U is a unitary operator acting on the internal
states of the photons and χ is an operator that shifts the phase.
The quantum analogue to the Mach-Zehnder intereferometer is given by
figure (8). In this system the beam, separated by a beam splitter, spans the
two-dimensional Hilbert space H̃P “ spant|0̃y, |1̃yu. The state vectors |0̃y and
|1̃y are considered as wave-packets moving along two different directions specified
29
by the geometry of the interferometer. In this basis, the mirrors, beam-splitters
and relative U p1q phase shifts may be represented by the following unitary
operators:
„
0
“
1

1
Ũ M
,
0
„

1 1 i
Ũ B “ ?
,
2 i 1
„ iχ

e
0
Ũ p1q “
0 1
(5.2.1)
(5.2.2)
(5.2.3)
respectively. Consider the pure input state given by
ρ̃in “ |0̃yx0̃|.
(5.2.4)
This state transforms to the output state (reminiscent to the time-evolution
transformation of the density operator see Equation (2.3.3)) according to
ρ̃in ÞÑ Ũ ρ̃in Ũ : “ ρ̃out .
(5.2.5)
In more detail:
:
ρ̃out “ ŨB ŨM Ũ p1qŨB ρ̃in ŨB: Ũ : p1qŨM
ŨB:
„
 „ iχ

„

„
i 1 e
ieiχ 1 0
e´iχ
“
1 i
i
1
0 0 ´ie´iχ
„

1 1 ` cospχq
sinpχq
.
“
sinpχq
1 ´ cospχq
2
„
´i ´i
1
1

1
´i
(5.2.6)
For this state, the intensity along |0̃y is given by I91 ` cospχq where χ is
the relative U p1q phase which can be observed in the output signal detectors of
the interferometer. If the beam-splitter is represented by the operator
Ũ B
„
1 1
“?
2 1

1
,
´1
(5.2.7)
instead of the one present in Equation (5.2.3), which is called the Hadamardgate, the output density operator, Equation (5.2.6), changes to
ρ̃out “
„

1 1 ` cospχq
i sinpχq
.
2 ´i sinpχq 1 ´ cospχq
(5.2.8)
Furthermore, assume that the particle carries an additional internal degree
of freedom, e.g., spin. The internal spin space HiN (with index ”i” for internal)
dimpH q
is spanned by the vectors t|kyuk“1 i which are chosen such that, initially, the
density operator is diagonal:
30
ρ̃0 “
dimpH
ÿ iq
wk |kyxk|.
(5.2.9)
k
Here wk is the probability of detecting a part of the ensemble in the pure
state |ky (in other words, ρ is the density operator for a mixed state). One can
then transform the density operator inside the interferometer as
ρs0 ÞÑ U i ρs0 U :i ,
(5.2.10)
where Ui is a unitary operator acting only on the spin part. Furthermore the
internal state is unaffected by the mirrors and beam-splitters. We now introduce the operators U B and U M which act on the full Hilbert space, H̃P b Hi ,
according to
U B “ Ũ B b 1i ,
(5.2.11)
U M “ Ũ M b 1i .
(5.2.12)
The total evolution operator U corresponds to the operator U i acting on
the internal state part (the |1̃y path ) while U p1q phase χ acting on the |0̃y part
and is thus given by:
„
0
U“
0

„ iχ
0
e
b Ui `
1
0

0
b 1i .
0
(5.2.13)
The operator U can be used to generalize the notion of phase to unitarily
evolving mixed states.
Now let the incoming, separable, state
%in “ ρ̃ b ρs0 “ |0̃yx0̃| b ρsi ,
(5.2.14)
(index ”i” for internal) be coherently split by a beam splitter and recombined at
a second beam splitter upon being reflected by mirrors. Furthermore, suppose
that the operator U acts upon the beam before the mirror but after the first
beam-splitter. This implies that the incoming state would transform to the
following output state:
%out “ U B U M U U H %in U :H U : U :M U :B
`
˘
“ U B U M U U H |0̃yx0̃| b ρsi U :H U : U :M U :B .
(5.2.15)
Inserting Equations (5.2.1), (5.2.2) and (5.2.13) into Equation (5.2.15), yields
31
%out

„ iχ

˙
´
¯´
¯ ˆ„
0 0
e
0
“ Ũ B b 1i Ũ M b 1i
b Ui `
b 1i ¨
0 1
0 0
´
¯`
´
¯
˘
:
Ũ B b 1s |0̃yx0̃| b ρsi Ũ B b 1i ¨
ˆ„

„ ´iχ

˙´
¯´ :
¯
:
0 0
e
0
b U :i `
b 1i
Ũ M b 1i Ũ B b 1i
0 1
0
0
„„

„


1
1
1
1 ´1
b U i ρsi U :i `
b ρsi `
“
´1 1
4 ´1 ´1
„ „

„


1 iχ 1
1
1 ´1
e
b ρsi U :i ` e´iχ
b U :i ρsi .
´1 ´1
1 ´1
4
(5.2.16)
(5.2.17)
The intensity I along |0̃y is proportional to the partial trace of Equation (5.2.17):
´
¯
I9 trs U i ρsi U :i ` ρsi ` eiχ ρsi U :i ` e´iχ U i ρsi
9 1 ` | trpU i ρsi q| cos rχ ´ arg trpU i ρsi qs .
(5.2.18)
˚
One arrives at Equation (5.2.18) by assuming that trpρsi U :i q “ rtrpU i ρsi qs
together with Eulers formula. An important observation of Equation (5.2.18) is
evident from the oscillation of interferance produced by χ which is shifted by
φ “ arg trpUi ρsi q,
(5.2.19)
for any internal input pure or mixed state ρsi . That is, φ is regared as a relative
phase shift. Equation (5.2.18) may be understood as incoherent weighted average
of pure states interference profiles. The internal state |ky P Hi gives rise to the
interference profile [19]
Ik 91 ` vk cos rχ ´ φk s ,
(5.2.20)
where vk “ |xk|U i ptq|ky| and φk “ argxk|U i ptqρsi |ky. The total output, I, is
then the sum of all individual weighted outputs:
I“
dimpH
ÿ iq
wk Ik 91 `
dimpH
ÿ iq
k
wk vk cos rχ ´ φk s .
(5.2.21)
k
”
ı
Note that Equation (5.2.21) can be expressed in the form of 1 ` ṽ cos χ ´ φ̃ ,
By using the following expressions:
32
˛
¨
dimpH
ÿ iq
φ̃ “ arg ˝
wk vk eiφk ‚ “ arg tr rU i ptqρsi s
k
»
“ arg –
dimpH
ÿ iq
fi
wk xk|Ui ptq|kyfl “ φ
(5.2.22)
k
ˇ
ˇ
ˇ
ˇdimpH
ˇ
ˇ ÿ iq
wk vk eiφk ˇˇ “ |tr rU i ρsi s| “ v,
ṽ “ ˇˇ
ˇ
ˇ k
(5.2.23)
where Equation (5.2.22) is the total phase acquired by a mixed state under
unitary evolution. The quantity (5.2.23) is called the visibility of the inteference
pattern [19].
33
5.2.1
Parallel transport
Consider a continuous unitary transformation of a mixed state,
ρptq “ U i ptqρp0qU :i ptq.
(5.2.24)
The state ρptq is said to acquire a phase (with respect to ρsi ) if arg tr rρi ptqU i ptqs
is non-vanishing. Now if one desires to parallel transport a mixed state ρptq along
a path, Γ, then it is required that, at each instant of time, the state ρptq must
be in phase with the state ρpt ` dtq at time t ` dt. That is, the state at time t
must be connected to the state a time t ` dt in the following way;
ρpt ` dtq “ U pt ` dtqU : ptqρptqU ptqU : pt ` dtq.
(5.2.25)
Thus the phase difference between ρptq and ρpt ` dtq is given by
”
ı
arg tr ρptqU pt ` dtqU : ptq .
(5.2.26)
It ”is said that ρptq andıρpt ` dtq are in phase if and only if the number
arg tr ρptqU pt ` dtqU : ptq is real and positive. However, from Hermiticity and
”
ı
normalization it follows that the number tr ρptqU9 ptqU : ptq is purely imaginary,
similar as Equation (4.1.12) regarding the Berry phase. It follows that

„
dU ptq :
U ptq “ 0.
tr ρptq
dt
(5.2.27)
This is the parallel transport condition for a mixed state under unitary transformation.
Now we are in a position to define the Geometric phase. Let the state trace out
the open path
Γ:
t P r0, τ s ÞÑ ρptq “ U i ptqρsi U :i ptq,
(5.2.28)
in the space of density operators, where the end-points are given by ρp0q “ ρsi
and ρpτ q. The evolution need not necessarily be cyclic, i.e. we allow for ρp0q ‰
ρpτ q (as mentioned in Section 4.2). A geometric phase γG rΓs can naturally be
assigned to this path once it can be shown that the dynamical phase vanishes,
which we are going to prove now. The dynamical phase can be expressed as
(the first term in Equation (4.1.7)
γd “ ´
1
~
żτ
dt tr rρptqHptqs “
0
1
~
żτ
0
„

dU ptq
dt tr ρsi U : ptq
.
dt
(5.2.29)
However, the term in the bracket contains the expression for parallel transport
(Equation (5.2.27)) and must be equal to zero. Thus the total phase (given by
34
Equation (5.2.22)) is the mixed state geometric phase under parallel transport
condition (Equation (5.2.27)), and we denoted the geometric phase by
»
γG rΓs “ arg tr rU i ptqρsi s “ arg –
dimpH
ÿ iq
k
35
fi
wk xk|Ui ptq|kyfl .
(5.2.30)
5.2.2
Gauge invariance
If a unitary evolution operator, U ptq, does not fulfil the parallel transport condition (5.2.27), then the dynamical phase remains present in arg tr rU i ρsi s as
an accumulative term and the total phase will no longer be purely geometric.
The dynamical phase is gauge dependent, therefore one can get rid of it by
constructing a gauge invariant quantity.
For an N -dimensional Hilbert space, the density matrix ρ is an N ˆN -matrix
and the gauge group is the U pnq. If all eigenvalues of ρ are non-degenerate, the
a gauge transofrmation with a matrix of the form ([17] , Eq.(19))
GN ptq “
N
ÿ
eiφn ptq |nyxn| P
N
ą
Un p1q,
(5.2.31)
n
n“1
where tφn uN
n“1 are phases and the vectors t|nyu leaves ρi invariant (Equation
(2.2.3)). It follows that there is an infinite number of orbits in G that correspond
to the same path ρ, that is the gauge transformation is given by
U ptq P U pN q ÞÑ U 1 ptq “ U ptq
N
ÿ
eiφn ptq |nyxn|,
(5.2.32)
n“1
then
ρp0q Ñ ρ1 ptq “ U 1: ptqρp0qU 1 ptq
N
ÿ
“
e´iφn ptq |nyxn|U : ptqρp0qU ptq
n
N
ÿ
eiφm ptq |myxm|
m
N ÿ
N
ÿ
“
eipφm ptq´φn ptqq U : ptq|nyxn|ρp0q|myxm|U ptq
n m
“ U : ptqρp0qU ptq.
(5.2.33)
That is, there is an infinite number of orbits in G that correspond to the
same path in ρptq. Unlike the pure state case, the removal of the dynamical
phase does not leave the functional (Equation 4.2.8) gauge invariant. Under
gauge transformation (5.2.32) The total phase (5.2.22) transforms as follows:
“
‰
γT ÞÑ γT1 “ arg tr ρsi U 1i ptq
“ lim arg
R
ÿ
RÑ8
c
“ lim arg
“ arg –
N
ÿ
eiθn ptq |nyxn|cy
n
dimpH
ÿ iq
R
ÿ
RÑ8
»
xc|ρsi U i ptq
xc|
c
dimpH
ÿ iq
wk |kyxk|U i ptq|cyeiθc ptq
k
fi
wk xk|U i ptq|kyeiφk ptq fl ,
k
36
(5.2.34)
while the dynamical phase transforms as follows:
żτ
γd ÞÑ
γd1
„

dU 1i ptq
s 1:
“ ´i
dt tr ρi U ptq
dt
0
¸
żτ « ˜
N
ÿ
tr ρsi U :i ptq e´iθn ptq |nyxn| ¨
“ ´i
0
˜
¨
dU i ptq
dt
n
N
ÿ
e
iθn ptq
|nyxn| ` iU i ptq
n
N
ÿ
dθn ptq
n
dt
¸ff
e
iθn ptq
|nyxn|
 dimpH
„
żτ
ÿ iq
dθk ptq
dU i ptq
`
wk
dt
dt tr ρsi U :i ptq
dt
dt
0
0
n
„
 dimpH
żτ
ÿ iq
dU i ptq
s :
wk θk pτ q.
(5.2.35)
`
dt tr ρi U i ptq
“ ´i
dt
0
k
żτ
“ ´i
It is evident from Equation (5.2.34) and (5.2.35) that the θ-dependence cannot be removed by simply subtracting the dynamical phase term. This problem
is solved by finding an explicit expression for θ. That is, similar to Equation
(4.2.11), a gauge invariant functional can be expressed in the form
γG rγs “ arg tr rρsi U i ptqs
»
fi
ˆ żτ
˙
dimpH
ÿ iq
dU
ptq
i
“ arg –
wk |xk|U i ptq|ky| exp ´
dtxk|U :i ptq
|ky fl ,
dt
0
k
(5.2.36)
where
żτ
dtU i ptqU9 i ptq.
θ“i
(5.2.37)
0
The expression (5.2.36) is manifestly gauge invariant, which assures us that
it only depends upon the path in the state space.
37
5.2.3
Purification
Any mixed state of a quantum system can be obtained by tracing out suitable
degrees of freedom of a larger composite system thas has been prepared in a
pure state. Consider the pure state
dimpHq
ÿ
|ψp0qy “
?
wi |µi yA b |γi yS P HA b HS ,
(5.2.38)
i
in a product state space HA b HS , where for simplicity it is assumed that
dimpHA q “ dimpHS q and |µi yA |γi yS are orthonormal basis of HA and HS ,
respectively. Applying the operator U “ I dA b U S ptq (where I is the identity
operator) onto (5.2.38), yields
´
¯ dimpH
ÿ Aq ?
d
wi |µi yA b |γi yS
|ψptqy “ U|ψp0qy “ I A b U S ptq
i
dimpH
ÿ Aq
“
?
wi |µi yA b U S ptq|γi yS .
(5.2.39)
i
The inner product of the initial state, |ψp0qy, and final state, |ψptqy, is given
by
dimpH
ÿ Aq
xψp0q|ψptqy “
?
wkA xµk | b S xγk |
dimpH
ÿ A q dimpH
ÿ Aq
“
dimpHq
ÿ
“
?
wi |µi yA b U S ptq|γi yS
i
k
k
dimpH
ÿ Aq
?
wi wk xµk |µi yA b xγk |U S ptq|γi yS
i
wk rxγk |U S ptq|γk yS s
k
“
‰
“ tr U S ptqρSi S ,
(5.2.40)
whereby the partial trace with respect to subsystem ”S” yields a purification
in the sense a mixed state on a finite Hilbert space can be viewed as a reduced
state of a pure state. Note that, the index i in ρSi stands for ”internal” (first
introduced in Equation (5.2.14)).
38
5.2.4
Example
Consider the qubit case of a system with spin-1/2. The density operator for
such a system can be expressed as in Equation (2.1.3), that is
ρ“
1
pI d ` ~r ¨ σq ,
2
where ~r is the three-dimensional Bloch vector and σ are the Pauli matrices.
For the sake of calculations (and convenience), consider a Bloch vector of the
particular form ~r “ pr sinpθq, 0, r cospθqq, such that Equation (2.1.3) can be
expressed as
„

1 1 ` r cospθq
r sinpθq
ρ“
.
r sinpθq
1 ´ r cospθq
2
(5.2.41)
Note that r P r0, 1s; the system is pure if r “ 1 while the system is in a mixed
state when r ă 1. The system is then subjected to a unitary transformation
(evolution) given by Equation (2.3.3), that is
ρp0q ÞÑ ρptq “ U ptqρp0qU : ptq,
it
U ptq “ e´ 2 σ3 .
(5.2.42)
This means that the Bloch vector is precessed about the z-axis at constant
polar angle, θ. Consider a loop on the Bloch sphere with parameter t P r0, 2πs.
The state of the system can be described by the eigenvalue equation:
ρ ¨ |ky “ λ|ky.
(5.2.43)
The eigenvalues, λ, and eigenvectors, |ky, are obtained by solving det rρ ´ λI d s “
0 and pρ ´ λI d q |ky “ 0, respectively, which yields the eigenvalues
„
1 1`r
ρλ “
0
2

0
.
1´r
(5.2.44)
(5.2.45)
and eigenvectors
` ˘
` ˘ 
cos ` θ2 ˘
sin `θ2 ˘
r|ky` , |ky´ s “
,
sin θ2
´ cos θ2
„
(5.2.46)
Inserting the eigenvalues and eigenvectors into the gauge invariant mixed state
geometric phase (Equation (5.2.36)), one obtains
39
„

1 ` r iπ cospθq 1 ´ r ´iπ cospθq
γG rΓs “ arg
e
`
e
2
2
„
1`r
pcospπ cospθqq ` i sinpπ cospθqqq `
“ arg
2

1´r
`
pcospπ cospθqq ´ i sinpπ cospθqqq .
2
“ arg rcospπ cospθqq ` r sinpπ cospθqqs
„
ˆ ˙
Ω
“ arctan rr tan π cos pθqs “ ´ arctan r tan
,
2
(5.2.47)
where Ω is the solid angle, Ω “ 2πp1 ´ cospθqq. The visibility (as defined in
Equation (5.2.23) is given by
d
v“
|trpU i qρsi |
“
cos2
ˆ ˙
ˆ ˙
θ
θ
` r2 sin2
.
2
2
(5.2.48)
Note that, if the system is maximally pure (r “ 1), then the visibility is
equal to 1.
40
6
Experimental observation of Geometric phases
for mixed states using NMR interferometry
6.1
Nuclear Magnetic Resonance (NMR)
NMR [8], which stands for Nuclear Magnetic Resonance, is a research technique
that utilizes the magnetic properties of certain atomic nuclei. That is, NMR
determines the structure of a molecule based on the response of the nuclei with
a non-zero spin. That response is influenced by the environment that the nuclei
experience within the molecule.
Subatomic particles, such as electrons, protons and neutrons, can be imagined
as spinning about an axis [22].The nuclei of atoms such as carbon,12 C, have
no overall spin since the spins of their nucleons are paired against each other.
However, some atoms such as 1 H and 13 C, have a nucleus which does indeed
possess a overall spin. The processes that determines these outcomes are as
follows. For an element N `P A, with N neutrons and P protons, the nucleus
possesses (for n P N)
• no overall spin if and only if N and P are both even.
• half integer spin, 1{2, 3{2,..., p2n ` 1q{2 if the sum of N ` P is odd.
• integer spin, 1,2,3, ...n, if and only if N and P are both odd.
A spinning charge generates a magnetic field. The spin magnetic moment µ,
is induced by the spin and is proportional to it. The nucleus is spinning around
its axis, and in the presence of an externally applied magnetic field, this axis of
rotation will precess around the magnetic field with a frequency called Larmor
frequency, see Figure 9.
Figure 9: Nucleus axis of rotation precessing around the magnetic field.
41
The Larmor frequency is given by
ω “ γB,
(6.1.1)
where B is the magnitude of the externally applied magnetic field and, for charge
eg
is the gyromagnetic ratio with g the ”g-factor” and m is the mass
´e, γ “ ´ 2m
of the precessing system. When an external magnetic field (B0 ) is applied on
the nucleus, two spin-states emerge with spin-component ˘ 21 ~ (for simplicity,
we suppress the factor ~). The magnetic moment of the lower energy, state ` 21 ,
is aligned with the external field (sometimes called α-state), see Figure (10),
while the magnetic moment of the higher energy, state ´ 21 , is aligned against
the field (sometimes called the β-state). The difference between the two energy
state is highly dependent on the strength of the externally applied field.
Figure 10: Energy levels splitting of an fermion due to an externally applied
magnetic field, known as the Zeeman splitting.
This phenomenon is known as the Zeeman effect (also known as weak-field
effect as opposed to the Paschen-Back effect known as the strong-field effect),
where the energy levels of an atom, in the presence of an externally applied
magnetic field, split into several states. For the case of a two-spin state system,
the energy difference (taking the first order correction of perturbation theory) is
given by [21]
∆E “ ml
e~
B.
2m
(6.1.2)
As Equation (6.1.2) entails, the energy difference is proportional to the
strength of the magnetic field. This separation can be further explained using
the notion of Boltzmann distribution. Now, the magnetic moment of the spinning nucleus precesses with a characteristic frequency called Larmor frequency,
as mentioned above. A particle can absorb a photon of the electromagnetic
radiation if its in a uniform periodic motion.
Now if the particle moves uniformaly and periodically, i.e. it precesses, the
the energy absorbed would be E “ hvprecess . For a spin-1{2 prticle, the energy
gap between the two states, caused by the externally applied magnetic field,
42
is given by Equation (6.1.2). In order for absorbtion to occure, the radiation
frequency of the photon must match the precession frequency,
vprecess “ vphoton ,
(6.1.3)
which is the origin of the term Resonance used in NMR.
6.1.1
Irradiating the nucleus with RF
Before RF irradiation of the nuclei in some sample, the nuclei in both spin states
are randomly oriented (i.e. both are completely out of phase) and precessing
around the same axis, with characteristic frequency. In figure (9), the net magnetization is in the z-direction due to a higher population being in the spin up
direction, whereas the magnetization in x and y direction are zero.
As irradiation occurs, all of the individual nuclear magnetic moments become
phase coherent (note that in density matrices, coherence is represented in the
off-diagonal matrix elements). This forces the net magnetization to precess
around the z-axis, effectively introducing magnetizational components in x and
y directions. This can be expressed as
Mx,y “ M sinpαq,
(6.1.4)
where α is the tip angle determined by the power and duration of the electromagnetic radiation, see figure 11.
Figure 11: [55] An α pulse transforms the net magnetization, M0 , into another
oriantation, Mx,y , by a degree α. A 90 degree pulse would shift the net magnetization entierly from one axis to another. B is the externally applied magnetic
field.
Naturally, this will flip back to its normal position with the net magnetization
aligned with the externally applied magnetic field. However, if enough energy
is introduced, i.e. Equation (6.1.3) is fulfilled, the spin jumps (from m “ 1{2)
to a higher energy level m “ ´1{2, see figure (10), which corresponds to a
180o -pulse, see figure 12.
43
Figure 12: [55] A spin 12 particle absorbing RF radiation with an energy corresponding to a spin-flip from
However, molecules themselves have circulating electrons which will induce
an internal magnetic field (stimulated by the externally applied magnetic field)
opposing the external one. This phenomenon is known as shielding where the
electrons ”shield” the molecule from an external magnetic field with an internal
one. Thus, the total magnetic field that the molecule ”feels” is given by
Beffective “ Bexternal ´ Binternal .
(6.1.5)
This causes the NMR signal frequency to shift depending on the chemical
nature of the molecule. The absolute RF cannot be determined to an absolute
accuracy of ˘1Hz [25], however this does not apply to the relative positions
of two signals in the NMR spectrum. Thus one introduces a reference signal
and the difference between these two signals is known as chemical shift which
is unique to each element. Thus by detecting this shift, using the NMR, it is
possible to quantify what element is being sampled.
After irradiation ceases (irradiation comes in pulses), the population of the
states reverts to a Boltszmann distribution and the individual nuclear magnetic
moments lose their phase coherence and return to a randomized arangement
around the z-axis. This process is recorded in the NMR spectroscopy and is
called the relaxation process, for which there are two types.
6.1.2
Relaxation processes
The relaxation process of the nuclei in higer energy requires a more complicated
answer than simply ”emission of radiation” since at RF (low-energy), re-emission
is negligible. The relaxation process is better explained thermodynamixally.
There are two types of relaxation processes
• Spin-Lattice (longitudinal) relaxation
• Spin-Spin (transverse) relaxation
44
For the Spin-Lattice case, the nuclei in the NMR experiment are kept in a
sample called the lattice. The nuclei in the lattice are in motion of vibration and
rotation which creates its own magnetic field. The magnetic field caused by the
motion of the nuclei in the lattice is called the lattice field. Some of the components of this lattice field equals in frequency and phase to the larmo frequency
of the nuclei. These components can act with the nuclei and exhange energy,
causing the nuclei to lose its energy. The nucleis loss of energy is transfered
to the lattice as vibration and rotation, effectively increasing the temperatur of
the sample. The relaxation time T1 is the average lifetime of the nuceli in the
higher energy state which is also dependet on the mobility of the lattice.
The spin-spin case describes the interaction between neighbouring nuclei
with different magnetic quantum states but identical precessional frequencies,
that is the nuclei hasa possibility to exhange states with another. A nuclei in
the lower energy state will be stimulated by exciting to a higer energy state,
while the other excited nuclei relaxes to a lower energy state. Note that there
is no net change in the populations of energy states, but the average lifetimes
of a nucleus in the excited state will decrease.
6.1.3
Spin-spin coupling (JJ-coupling)
The spin Hamiltonian for a scalar coupling between spins for sites 1 and 2 is
given by (assuming a two-spin system) [26]
H 1,2 “ 2πJ1,2 I 1z b I 2z
(6.1.6)
This term gives rise to a splitting to the chemical shift, this is further evaluated in section 6.2.1, (observed frequency shift in the NMR relative to a reference element of known resonance freqyency) depending on the orientation of the
spin-angular momentum. The energy of the spin system is increased if the spin
angular momenta are parallel and decreased if opposite. This can be explained
pictoraliy, see figure 13.
Part (a) of the picture shows two imginary protons A and B in the presence of
an externally applied magnetic field (in the figure, the external magnetic field
is defined as B for simplicity) with, what one expects to observe in a NMR
spectra, a singlet peak at their respective chemical shifts. Let us observe how
proton B affects proton A. The magnetic moment of Proton B can be either
aligned with or against the externally applied magnetic field. In (b), proton B
has a magnetic moment which is aligned with the externally applied magnetic
field, resulting in a higher effective magnetic field, B 1 , which acts on proton A
(in addition to the applied field already acting on proton A). That is proton A
”feels” B ` B 1 . A higher applied magnetic field implies a stronger Zeeman splitting which further implies a bigger energy difference between α- and β-state
resulting in stronger chemical shift from its initial expected singlet position.
Similarly, in (c), if proton B has a magnetic moment aligned against the applied
field, a weaker effective magnetic field is created B ` B 2 , B 2 ă B 1 , which acts
on proton A, resulting in a lowering of the chemical shift from its original singlet
position. Proton A affects proton B in a similar manner, leading to (d), which is
45
what one actually observes. The splitting distance (from its singlet position to
J
a doublet peak) can be calculated as d « 1,2
2 , where J1,2 is called the coupling
constant. Thus the effective resonance frequency ω 1 can be seen as as shift from
its regular resonance frequency, ω (unhindered by the spin-spin coupling), by
the scalar coupling constant J.
ω 1 “ ω ˘ πJ1,2 ,
(6.1.7)
where ω 1 is called the chemical shift where the energy is shifted ˘πJ1,2 by the
coupling constant (can be calculated for hydrogen [56] to correspond to about
42MHz ˘ 6.1Hz). This is explained in figure 13.
Figure 13: A pictorial presentation of the JJ-coupling effect on respective proton
6.1.4
General setup of CW-spectrometer
The NMR spectrometer is specifically tuned to a nucleus, such as the proton
that constitutes nucleus of a hydrogen atom. The simplest procedure is called
the Continuous Wave (CW) method. A general CW-spectrometer is shown
schematically in figure (14). A solution of the sample is plced in a glass tube
oriented between the poles of a magnet and is spun to average any tube imperfections and any variations in the magnetic field. The tube is irradiated, from
the antenna coil, with Radio Frequency (RF) of appropriate energy. Naturally,
a proton irradiated with energy will be excited. As de-excitation occurs, a receiver coil surrounding the sample tube will detect the emission of RF from the
sample and be displayed using electronic devices and computer. A NMR spectra is obtained by sweeping the magnetic field over a range while observing the
recieved RF signal from the sample. Another technique is to vary the frequency
whilst keeping the magnetic field constant.
46
Figure 14: [55] General/conventional CW spectrometer. Sample solution is
placed in a rotating glass tube oriented between magnetic poles. Radio frequency (RF) is emitted from the RF antenna, enclosing the glass tube and a
receiver to detect emitted radiation from the sample. The receiver transmits
the data to a control console which displays the result on a screen.
47
6.2
Quantum interference; Sjöqvist’s NMR interferometry model
Basically [13] an interferometer device utilizes the principle of superposition to
extract some meaningful property or a pattern (fringes in the case of photons).
The incoming, coherent, wave is split into two parts such that one part acquires
a different phase than the other. These two waves are then recombined and
either interferes constructively or destructively.
Now that the NMR and interferometer has been mentioned, Sjöqvist’s interferometry model is, rougly speaking, a combination of NMR spectroscopy and
general interferometry, yielding the name of NMR interferometry. The principle is illustrated in figure 8. The photons, entering the interferometer along a
horizontal path, are split into two parts using a beam splitter (BS1 ). Photons
along the horizontal path are globally phase shifted by the phase shifter, χ,
whereas along the vertical path gthe internal states of the photons undergo a
unitary evolution U . The photons then recombine at the beam splitter, BS2 ,
after they have been reflected at the two mirrors M1 and M2 . The detector only
detects the horizontal beam and the intensity shows an interference pattern as
a function of phase shift (This is explained, in detail, in Section (5.2)).
In order to simmulates these processes, spatial averaging techniques are being
utilized.
6.2.1
Spatial averaging technique
The key factor to making the experiment possible is heavily rooted in the Nuclear Magnetic Resonance spectroscopy (NMR or NMRs) being capable of emulating a many of the capabilities of a quantum computer without having the
total collapse of the wave-function. This is possible due to two facts.
• In a solution, the spins of a molecule are isolated by surface-to-volume
considerations and from spins from neighbouring molecules, effectively
making each nuclei an independent quantum computer.
• The discovery of Pseudo-spin-states, a manifold of statistical spins states,
which has similar transformation properties of a true pure state.
Consider, the simplest case of a liquid consisting of identical molecules each
containing two coupled spin-1{2 of same nuclei H 1 . Due to the rotational motion
of the molecules in the liquid, the dipolar-coupling between the spins is averaged
to zero (hence is called the scalar coupling). Calculations will be simplified if
one assumes weak-coupling in the sense that the coupling constant J 12 is small
compared to the difference |ω1 ´ ω2 | in the resonance frequency of the two
spins. With the magnetic field along the z´direction, the two-spin Hamiltonian
approximately expressed as
H “ ω1 2 I 1z ` ω2 2 I 2z ` 2πJ 12 2 I 1z b 2 I 2z ,
(6.2.1)
where for a I bz : a numbers the spin system, and b denotes the numbered spin in
that spin system. Considering that the subject at hand will be interily in a twospin system, the superscript entailing the spin-system shall be hidden to simplify
further calculations (it will otherwise be mentioned if this constraint is lifted).
48
Since the energy difference are small compared to KB T at room temperature
´ H
(RT), the Boltzmann approximation can be applied to the exponential e kB T «
1 ´ kH
. Thus, the density matrix at equilibrium takes the form of Equation
BT
(3.5.9). For example, a soft radio pulse (RP) whose frequency range spans only
the resonance frequency of only the first spin with sufficient energy to rotate it
by an angle of γ about the y-axis is given by
rγs1y
I 1z ÞÑ cos pγq I 1z ` sin pγq I 1x .
(6.2.2)
The density matrix evolves according to Equation (2.3.3). Because all the
terms in the Hamiltonian commutes. The propagator (Equation (A.2.10)) factors into a chemical shift and a scalar coupling. The chemical shift for the first
spin is given by
ˆ
e
itω1 I 1z
“ cos
ω1 t
2
˙
ˆ
2
E ` 2i sin
ω1 t
2
˙
I 1z ,
(6.2.3)
where the identity matrix is expressed as E. The transformation using Pauli
matrix algebra (or BCH-lemma Equation (A.3.3) together with angular momentum commutation relations given by Equations (2.4.1- 2.4.4)), yields
itω1 I 1
1
1
I 1x ÞÑ z e´itω1 I z I 1x eitω1 I z “ cos pω1 tq I 1x ` sin pω1 tq I 1y .
(6.2.4)
Repeating this for the y and z components, one end up with
itω1 I 1
I 1x ÞÑ z cos pω1 tq I 1x ` sin pω1 tq I 1y ,
(6.2.5)
itω1 I 1
I 1y ÞÑ z
(6.2.6)
I 1z
cos pω1 tq I 1y ´ sin pω1 tq I 1x ,
itω1 I 1z
ÞÑ I 1z .
(6.2.7)
This propagator only affects terms involving the first spin and remains invariant under terms involving the second spin. The scalar coupling can be
expanded as
ˆ
e
2πJ 12 I 1z I 1z
“ cos
πJ 12 t
2
˙
ˆ
2
E ` 4i sin
πJ 12 t
2
˙
I 1z I 2z .
(6.2.8)
Similarly, transofrming along x,y and z components yield
I 1x
I 1y
I 1z
2πJ 12 tI 1z I 2z
ÞÑ
2πJ 12 tI 1z I 2z
ÞÑ
2πJ 12 tI 1z I 2z
ÞÑ
cos pπJ 12 tq I 1x ` 2 sin pπJ 12 tq I 1y I 2z
(6.2.9)
cos pπJ 12 tq I 1y ´ 2 sin pπJ 12 tq I 1x I 2z
(6.2.10)
I 1z
(6.2.11)
With the expressions being analogous for terms invovling
“ 1the
‰ second spin
(Ik2 , k “ 1, 2, 3). Note that, a pulse of any operators, such as 2J
, is considered
a time interval also known as a delay.
49
6.2.2
Gradient pulse
On the other hand, a Gradient pulse produces a transient field inhomogenity
along the z-axis. The Hamiltonian is given by
H grad “ γ~r ¨ ∇Bz ,
(6.2.12)
where ~r is the position vector in the sample. A RF-pulse changes the choerence
of a spin from one coherence to another (see section 6.1.1). This effectively
destroyes the transverse magnetization across the sample, leading to mixing of
spins with different initial positions. That is, it makes it possible to render
the magnetization unobservable due to the selected spins [23]. An example is
provided here
I 1z ` I 2z
(6.2.13)
r s 1
ÞÑ I z ` I 2x
(6.2.14)
rGsz 1
ÞÑ I z
(6.2.15)
π 2
2 y
A more extensive look into coherence and gradient pulses is provided by [24].
6.2.3
Pseudo-pure states
Assume one generates all three of the operators I 1z , I 2z and 2I 1z I 2z simultaneously
in the same sample. The density matrix for such a state is then given by
»3
2
—0
ρ “ I 1z ` I 2z ` 2I 1z I 2z “ —
–0
0
0
´ 21
0
0
0
0
´ 12
0
fi
0
0 ffi
ffi ,
0 fl
´ 12
(6.2.16)
which is not a pure state. However, by shifting the matrix by 12 I 4 (the 4
dimensional identity operator I 4 ), one could obtaine a pseudo-pure matrix
»
2
—0
—
–0
0
0
0
0
0
0
0
0
0
fi
0
0ffi
ffi
0fl
0 .
This can be further expressed as an exterior product
» fi
1
—0ffi “
ffi
2 p|0y b |0yq px0| b x0|q “ 2 —
–0fl 1 0 0
0
(6.2.17)
‰
0 .
(6.2.18)
The expression I 1z ` I 2z ` 2I 1z I 2z is then called a Pseudo-pure state (expressed
in product operator notation) due to its pure-state like behaviour upon transformation.
50
The transformation of a two-spin state, using RF and gradient pulses sequences, into a Pseudo-pure state is given by the following algorithm using the
expression for spin transformation (Equation (A.3.1)) BCH lemma (Equation
(A.3.3)) and the commutation relation for the spin operators (Equation (2.4.4))
I 1z ` I 2z
(6.2.19)
?
π 2
3 x
r s 1 1 2
3 2
ÞÑ I z ` I z ´
I
2
2 y
rGsz 1
1
ÞÑ I z ` I 2z
2
1
r π4 sx 1 1 1 2
1
ÞÑ ? I z ` I z ´ ? I 1y
2
2
2
”
1
2J 12
(6.2.20)
(6.2.21)
(6.2.22)
ı
?
1
1
ÞÑ ? I 1z ` I 2z ` 2I 1x I 2z
(6.2.23)
2
2
1
r´ π4 sy 1 1 1 2 1 1
(6.2.24)
ÞÑ I z ` I z ´ I x ` I 1x I 2z ` I 1z I 2z
2
2
2
rGsz 1 1
1
(6.2.25)
ÞÑ I z ` I 2z ` I 1z I 2z ,
2
2
which simulates the reduced Hamiltonian of a two-spin system (up to an approximation). An extensive descirpiton of Pseudo-pure states, its resemblens to
quantum computing and its limitations can be found in [23].
6.2.4
Non-vanishing dynamical phase
For the case where the unitary operator, U (in Equation (5.2.22)), cannot be
parallel transported, a method, known as spin-echo [47], can be used to eliminate the dynamical part. For convenience, observe the Total phase obtained
by Berry’s approach, which is explicitely calculated at the example, Equation
(4.1.28). We shall express it as
µ
~
`
`
| Ò, n` ptqy “ e´iπp1´cospθqq e´ ~ |Bptq|T | Ò, n` p0qy “ eiγn pφq eiθ | Ò, n` p0qy,
(6.2.26)
µ
~
´
´
| Ó, n´ ptqy “ e´iπp1`cospθqq e ~ |Bptq|T | Ó, n´ p0qy “ eiγn pφq eiθ | Ó, n` p0qy.
(6.2.27)
Upon rotation of of the magnetic field with direction ´~npπ ´ φq, one obtains
´
´
`
´
| Ò, n` ptqy “ eiγn pπ´φq eiθ | Ò, n` p0qy “ eiγn pφq eiθ | Ò, n` p0qy,
| Ó, n´ ptqy “ e
`
iγn
pπ´φq iθ `
e
| Ó, n` p0qy “ e
´
iγn
pφq iθ `
e
| Ó, n` p0qy.
(6.2.28)
(6.2.29)
The net effect of the two periods then yields (adding Equation (6.2.26) with
(6.2.28) and (6.2.27) with (6.2.29))
`
| Ò, n` ptqy “ e2iγn pφq | Ò, n` p0qy,
| Ó, n` ptqy “ e
´
2iγn
pφq
51
| Ó, n` p0qy,
(6.2.30)
(6.2.31)
absent of any dynamical phase. What is happening in the NMR when spin-echo
method is applied?
The phenomena of spin-echo was, first, discovered by Erwin Hahn and then
further developed by Carr and Purcell [57], see figure 15 which is in the same
reference frame as figure 11
Figure 15: Spin echo method
Figure 16: [57] A refocusing of spin moments, yielding a singal, echo, when no
pulse was applied. a, b, c, d, e, f
(a): The vertical arrow represents the average magnetic moment of a group of
spins.
(b): A 90o degree pulse is applied, flipping the arrow to the x ´ y axis.
(c): Due to the magnetic field being inhomogenous, as the net moment precesses, some spins are either slowed down or sped up due to different local
field strength.
(d): A 180o degree puls is applied, reversing the speeding process of spins such
that slower spins are faster and the faster spins are now slower.
(e): At this point, the faster spins align themself with the lower spins at the
main moment.
(f): A complete refocusing has occured.
Thus leading to an output signal, without any input. This phenomena is
similar to a sound echo, which is where the name ”spin-echo” originates from.
52
6.3
Experimental observation of Geometric phases for mixed
state [45]
In the experiment, one measures the geometric phase, given by Equation (5.2.30)
using an auxiliary spin-1{2 particle as a phase-reference, see Equation (5.2.14).
The Sjöqvist intereferometry model has ben explained in section 6.2 and is
pictoraly expressed in figure 8. The phase shifter, χ (or U ´operation after the
operators described in Equation (5.2.1, 5.2.2 and 5.2.3)), is the central part of
the experiment. A simplified figure 17 shows the operation of the experiment
in regard to the auxiliary qubit 1 and qubit 2.
Figure 17: [45] Quantum network describing the experiment where the top
line represents the auxiliary qubit or particle of spin-1{2 while the lower line
represents the qubit, spin 1{2 prticle, that undergoes a cyclic evolution induced
by the unitary operator U .
The reference basis for qubit 1 and 2 has been chosen to be given by the
states | Òy and | Óy which describes the spin states aligned either with or gainst
the externally applied, static, magnetic field in the z-direction. In this basis,
1
|˘y “ ? p| Òy ˘ | Óyq ,
2
(6.3.1)
with the initial state of the auxiliary qubit being |`y. ρ2 traces out the loop
(closed path) C : t P r0, τ s Ñ ρptq “ U ptqρp0qU : ptq on the Bloch sphere with
a solid angle Ω if and only if the auxiliary qubit is in a state of | Òy1 (spin
`1{2) and ρ2 remains unaffected if the auxiliary qubit is in the state | Óy. This
evaluation is possible with the NMR due to the spin-spin coupling (mentioned
in section 6.1.3). This introduces a relative phase shift between the initially
prepared superposition with already known phase of the states | Òy and | Óy
of the auxiliary qubit (this is equivalent to the initially singlet peaks with the
shifted doublet peaks) which can be read out directly using the NMR. The
unitary operation completes the loop along the two geodesics ABC and CDA,
see figure 18, which satisfies the parallel transport condition (Equation (5.2.27)),
effectivley eliminating the dynamical phase (Equation (5.2.29)).
53
Figure 18: The solid angle Ω is subtended by the cyclic path ABCDA. The solid
angle can be varied by changing θ, the angle of inclination between x, y plane
and the ABC plane.
In the experimental article [45], a NMR spectrometer was used with. A
0.5ml, 200mol sample of 13 C labeled chloroform (Cambridge isotope) in d6 acetone was used as the auxiliary qubit while the 1 H nucleus was used as the qubit
which undergoes cyclic evolution. The reduced Hamiltonin for such a system
is given by Equation (6.2.1) where the first two terms describs the free spin
precession of spin 1 (13 C) and 2 (1 H) around the magnetic field B0 with larmor
ωb
frequencies ω2πa « 100MHz and 2π
« 400. The I 1,2
are the z-components of
z
angular momentum operators for the qubits 1 and 2 respectively. The last term
describes the scalar spin-spin coupling with J “ 214.5Hz.
The first stage to conducting the experiment is to prepare the initial state.
Initially, the two qubits are in thermal equilibrium with the environment. Spatial averaging techniques (Section 6.2.1) is used to create the Pseudo-pure state
(the effective pure state) | Òy1 b | Òy2 which in density operator form is given by
1
1
1
2
2 pI ` σ z q b 2 pI ` σ z q. The sequence of transformation leading to this state is
quite similar to the transformations happening in Equation (6.2.19) to Equation
(6.2.25), with the sequence being
R2x
´π¯
´ π¯
´ π¯
1
´ Gz ´ R2y ´
´
´ R2y ´
´ Gz ,
3
4
2J1,2
4
(6.3.2)
σx
where R2x pαq “ e´iα 2 denotes a puls that rotates spin 2 around the x-axis by
an angle α. Furthermore, the subsequent puls
R2x
´ nπ ¯
12
´ Gz ´ R1y
generates the initial state
54
´π¯
2
´ R2y
´π¯
2
,
(6.3.3)
ρ1 p0q b ρ2 p0q “
˘
1`
1
I2 ` rσ 2z ,
pI2 ` σ 1z q b
2
2
(6.3.4)
` ˘
where r “ cos nπ
n “ 0, 1, 2, ..., 11. This makes it possible to measure the
24 ,
geometric phase, γG rΓs, for 12 values of r. This is set by rotating the angle, nπ
12 ,
` ˘
of the selective pulse R2x nπ
.
12
Regarding the controlled U-operator, The Hamiltonian for gubit 2 is set in
a rotating frame with angular frequency ω21 “ ω2 ´ πJ1,2 , yielding
`
˘
RH 2 R´1 9 ω2 ´ ω21 ˘ πJ1,2 I 2z ,
(6.3.5)
where the sign ˘ is determined by the state of qubit 1. If qubit 1 is in a state of
| Òy1 , H 2 p0q “ 0, else H 2 p0q “ 2πJ1,2 I 2z . Moreover, one implements the cyclic
evolution using the following sequence
R2x p´θq ´
1
1
´ R2x p2θ ´ πq ´
,
2J1,2
2J1,2
(6.3.6)
where θ “ Ω4 in figure (18) is the solid angle. The two eigenstates of ρ2 p0q
(|˘y2 ) traces out a path which subtends the solid angle Ω with the acquired
Ω
geometric phase |˘y2 ÞÑ empi 2 |˘y2 . Since the parallel transport condition is
fulfilled, the dynamical phase vanishes and the geometric phase is then, upon
averaging, given by Equation (5.2.30) (or Equation (5.2.47) in particular).
The sequences (6.3.2) allow one to control the rate of purity (rather the first
term in that sequence does so) in the system and the sequence (6.3.6) allows
on to control the cyclic evolution of the system. Thus, using the NMR, one
can measure the geometric phase, γG rΓs, versus the purity, r, of the system for
different solid angles, Ω. This result is presented in figure 19 in Section 6.4.
6.4
Result
All the experiments were conducted at room temperature and room pressure
on a Bruker AV-400 spectrometer. In the experiment, all the pulses are square
and of several microseconds of duration. The spin-spin relaxation time is 0.3s
for carbon and 0.4s for proton. The time used for cyclic parallel transport
is « 4.7ms which is wihtin decoherence time, the results are shown in figure
19. The solid lines represents actual theoretical predictions (Equation (5.2.30))
while the ”circle” and respectively ”squares” are experimental data. The small
errors can be attributed to imperfect pulses which further leads to deviation
of required purity. Furthermore, pulse errors can lead to the evolution of the
mixed state not being entirely cyclic. Other errors reside in the inhomogeneity
of the static and the RF magnetic fields.
55
Figure 19: Experimental data [45] (represented by circle and squares) versus
theoretical predictions (represented by solid lines and Equation (5.2.22)). The
geometric phase ,γ, is presented versus purity, r, for three different solid angles,
Ω.
56
7
7.1
Uhlmann Phase as a Topological Measure for
One-Dimensional Fermion Systems
Introduction
A fundamental problem with geometric phases lies in its extension from pure
quantum states (Berry phase) to a mixture of quantum states described by density matrices. This was first adressed by Uhlmann [28] and later given a solution
[29]. A renewed interest for studying geometric phases has emerged in the area
of quantum information [19] which has led to the first experimental measurement of Geometric phases along a mixture of states using NMR techniques (See
Section 6).
In the present section we discuss the Uhlmann geometric phase and its endowment with a topological structure when it is applied to a one-dimensional
fermion system. That is, (i): How Uhlmann phase allows one to characterize topological insulaters at both zero and finite temperature. (ii): To find a
finite critial temperature, Tc , for which the Uhlmann phase is constant and nonvanishing. (iii): study one-dimenisonal such as the Creutz ladder (CL) [30], The
Majorana chain (MC) [41] and the Polycetylene (SSH).
We shall now focuses on the the Uhlmann phase in one-dimensional fermion
model [33]. For such a system, k is the one-dimensional momentum existing in
the S 1 -circle BZ. However, first we shall mention Topological insulators and its
importance in this subject.
7.2
Topological insulators
The notion of topoplogical insulators does not refer to its general shape of a
material, but rather how current travels within the material. A topological
insulator is a material that is an insulater in the bulk area but can carry electrons
along its surface. The current exhibits behaviour best explained with quantum
mechanics. How is this achieved?
By applying an externally, perpendicular, magnetic field onto a material that
supports the Quantum Hall-Effect, causes electrons far from the ”edges” (i.e.
the surfaces) of the material (say a semiconductor) to move in a circular orbit.
However, electrons close to the edges will have their orbit interupted by the
surface causing them to bounce back into the bulk whereby the magnetic field
will cause it to orbit circularly, resulting in another collision with the surface.
This causes the electrons close to the edges to effectively trace a path along the
surface of the material and to be in a state called edge state, see figure 20.
57
Figure 20: A sample in the presence of an externally, perpendicularly, applied
magnetic field. The electrons on the sample are effected by this magnetic field,
causing electrons to orbit. Electrons close ot the edges experience Edge states,
effectively contributing to current along the egdes, while electrons in the middle
contiue to orbit.
The Quantum Hall state has properties similar to the properties of a Topological insulator. the presence of a magnetic field makes the entire process
symmetric under time-reversal. That is, without the magnetic field, electrons
at opposite ends (i.e. surfaces) of the material would move in opposite direction, which is a violation (it is said that the Hall-conductivity is odd under
time-reversal). The magnetic field restores this symmetry because it reverses
its direction under time-reversal.
Now, a topological insulator is a material that exhibits the quantum Hall
state behaviour but without the presence of an externally applied magnetic
field. This is possible due to the electrons having an intrinsic angular momentum (spin) which, effectively, solves the time-reversal symmetry problem.
That is, when time is reversed, the electron spin reverses (flips). However this
phenomenon is material dependent. more on this can be found in [37] and [38].
Interestingly, surface contamination, slight edge (surface) deformation or
surface disorder causes the edge-states to deform in path to accomodate the
change, however the conduction persists in the edges. That is, the edge-states
are ”topologically protected” or ”Symmetry protected topological phases”. This
can be explained, analogously, by the ”hole in the donut” case for which, the
shape of the donut may change but the hole in the center survives. This is a
basic aspect of topology, more on this can be found on [39].
It follows in particular that it is possible to place two different surfaces
close to each other with opposite magnetic orientation. This can be achieved
by having edge-states in one of the surfaces going counterclockwise, inducing
a magnetic field say upwards and the other surface having edge-states going
clockwise producing a magnetic field going downwards. That is, the Hall-current
is uninterupted since the edge-states are topologically protected.
58
7.3
Fermionic system and Uhlmann Phase
In condensed matter, the crystalline momentum ~k, defines the BZ due to translational invariance, and characterizes the eigenstate of the system. The phase
~ where G
~
of the system, due to parallel transport condition, from ~k0 to ~k0 ` G,
is the reciprocal lattice vector, is gauge invariant and a geometric phase [36].
Consider a two-band Hamiltonian with the spinor representation Ψk “
´
¯T
âk , b̂k
where âk and b̂k are fermionic operators. For superconductors, the
spinors Ψk out of Nambu transformation [35] of paired fermions with oposite
crystalline
momentum. The Hamiltonian for such a system is given by H “
ř
Ψ
H
Ψ
k k where the 2x2 matrix H k is given by
k k
H k “ f pkqI2 `
∆k
~nk ¨ σ,
2
(7.3.1)
where f pkq is some function depending on k, σ “ p~σx , ~σy , ~σz q are the pauli spin
matrices,
~nk “ psinpθpkqq cospφpkqq, sinpθpkqq sinpφpkqq, cospθpkqqq is called the winding
vector and is given in terms of spherical coordinates depending on k, and ∆k
corresponds to the gap in H k . Equation (7.3.1) can be written in the form of
an eigen-equation.
H k uk “ λk uk ,
(7.3.2)
where H k in matrix form is given by
„
f pkq ` ∆2k cospθpkqq
H k “ ∆k
iφpkq
2 sinpθpkqqe
∆k
2

sinpθpkqqe´iφpkq
,
f pkq ´ ∆2k cospθpkqq
(7.3.3)
∆k
with the eigenvalues
given
by λ˘
k “ f pkq ˘ 2 (assuming that the auxiliary pa´
¯
rameter t “ cos θpkq
when obtaining eigenvector). The eigenvalue containing
2
f pkq is convenient since f pkq vanishes when obtianing the eigenvectors which is
given by
¨
´
¯ ˛
θpkq
´iφpkq
e
sin
˝
´
¯2
‚,
|u`
ky“
θpkq
cos 2
¯ ˛
´
θpkq
´iφpkq
´e
cos
´
˝
¯2
‚.
|u´
ky“
θpkq
sin 2
¨
(7.3.4)
If the thermalization process preservs the particle number with theś
fermi
energy set in the middle of the gap, the equilibrium state is given by ρβ “ k ρkβ
where
ρkβ “
Hk
„
ˆ
˙

e´ T
1
∆k
´ H ¯“
I
´
tanh
~
n
¨
σ
,
2
k
k
2
2T
tr e´ T
(7.3.5)
where T “ 1{β denotes the temperature. Inserting the eigenvectors (Equation
(7.3.4)) into The Uhlmann connection (Equation (5.1.35)) for ρβk , the ”new”
form of Uhlmann connection can be expressed as [33]
59
`
´
`
AkU “ mk1,2 xu´
k |Bk uk y|uk yxuk |dk ` h.c.
(7.3.6)
The symmetries of the models, considered in this article, imposes a constriction on ~nk to some plane as a function of k such that only two components
nik and njk (with I ‰ j) are different from zero. That is, the expression can
be simplified by fixing a gauge in the eigenvectors such that the off-diagonal
`
overlap xu´
k |Bk uk y and the winding vector components are connected such as
`
xu´
k |Bk uk y “
Bk nik
.
2njk
Thus one obtains a non-trivial mapping from S 1 Ñ S 1
characterized by a winding number ω1
1
ω1 :“
2π
¿
1
dα “
2π
¿ ˜
Bk nik1
njk1
¸
dk 1 ,
(7.3.7)
´
¯
(which is an integer) where α :“ arctan nik {njk is the angle covered by nk as it
winds around the unit circle S 1 . Using Equations ((7.3.4), (7.3.7) with (7.3.6))
and inserting these into the Uhlmann Phase (Equation (5.1.42)), one obtains
[33]
«
˜ ¿ ˜
¸
¸ff
ˆ
˙
1
∆k 1
Bk nik1
ΦU “ arg cos pπω1 q cos
sech
dk 1
,
2
2T
njk1
(7.3.8)
In particular, lim ΦTU “ arg cos pπω1 q, and for ω1 “ 0, one has lim ΦTU “ 0.
T Ñ0
T Ñ0
However, for the non-trivial case of ω1 “ ˘1
lim ΦTU “ π.
T Ñ0
(7.3.9)
As suspected, due to the nature of the argument, arg, the phase can only
take the values 0 or π.
60
7.4
Creutz-ladder model
Figure 21: The Creutz ladder formation. Particles can hop both diagonally and
vertically.
The Creutz-ladder is a model for topological insulaters (see section 7.2 for a
brief explanation of Topological Insulators) and describes the dynamics of two
spinless electrons moving in a ladder-formation as illustrated in figure (21). The
Hamiltonian [30] is given by
HCL “ ´
L ” ´
¯
ÿ
R e´iΘ a:n`1 an ` eiΘ b:n`1 bn `
n“1
¯
ı
´
R b:n`1 an ` a:n`1 bn ` M a:n bn ` h.c. ,
(7.4.1)
where an and bn are fermionic annihilation operators associated to the n:th site
of the upper and lower chain and a:n and b:n are the corresponding creation operators. Movement along diagonal or horizontal links (see figure 21) is governed
by R“ ą 0 and
over vertical links by M ą 0. Furthermore, a magnetic flux
‰
Θ P ´ π2 , π2 is induced by a perpendicular magnetic field. For, Θ ‰ 0, nonzero
M
magnetic flux and small vertical hopping m “ 2R
ă 1, the system has localized edge states at the two ends of the ladder. The Creutz-ladder Hamiltonian
(7.4.1) can be expressed in the form of Equation (7.3.1) by denoting (in units
of 2R “ 1):
2
pm ` cospkq, 0, sinpΘq sinpkqq ,
~nk “
∆
a k
∆k “ 2 pm ` cospkqq2 ` psinpΘq sinpkqq2 ,
(7.4.2)
(7.4.3)
where φ “ 0, π in order for the spinor decomposition to have the form expressed
in Equation p7.3.1q. The Results of the modified Uhlmann phase (Equation
(7.3.8)) was computed as a function of the parameters Θ, m and T , by O.
Viyuela et al [33] and can be seen in figure 22a.
61
Figure 22: [33] The Topological Uhlmann phase for the Creutz-ladder (a), the
Majorana-chain model (b) and the SSH-model(c). The topological phase is
equal to π inside the green volume and zero outside. The FPB (Flat Band
Points) are indicated by arrows.
At T Ñ 0, the topological region coincides
‰ with the topological phase (Equa“
tion (7.3.9)) for m P r0, 1s and θ P ´ π2 , π2 . However, there exists a ”critical
temperature”, Tc , for any values of the parameters, at which the topological
phase goes abruptly to zero, ΦU “ 0. The physical meaning of this critical
temperature is rooted in the existence of some critical momentum kc splitting
the phase into two inequivalent parts ΦU pk ă kc q “ 0 and ΦU pk ą kc q “ π.
Let us interpret the existence of the critical temperature Tc in a pictorial way.
For the sake of illustration, indicate the amplitude (purification, see Equation
(5.1.40)) as arrows with fixed length. There are 2 different ways for which the
amplitudes are paralell transported, in the Uhlmann sense (Equation (5.1.11)),
along a closed loop that covers the whole Brillouin zone, k P r0, 2πs. The angle
between the two arrows will be given by the relative phase between the two
amplitudes (Equation (5.1.42)). The behaviour of Uhlmann phase, when paralell transported from k0 “ 0 to kf “ 2π, presents two situations. Non-trivial
topology highlited by a topological kink at kc and trivial topology in which there
is no a topological kink, see figure 23.
Figure 23: [34] (a): non-trivial topology and (b): trivial-topology
For the trivial-topological case (figure 23 (b)), the arrow is transported pointing the same direction along the loop, thus the relative phase between the amplitudes remains zero. That is, the initial and final arrow are parallel yielding
ΦU “ 0 for all T . For the case of non-trivial topology (figure 23(a)), there exists
62
2 different situtations (a third situation exists which reverts you back to trivialtopology), all highily dependent on the temperature T . For T “ 0, the Uhlmann
phase remains zero (arrows are parallel) up to a critical point, kc , where the arrows reverse are reversed yielding the Uhlmann phase ΦU “ π which constitutes
a topological kink. For 0 ă T ă Tc , the critical momentum becomes dependent
on the temperature T , kc “ kc pT q where kc shifts towards kf as T increases with
ΦU pk ă kc pT ă Tc qq “ 0 and ΦU pk ą kc pT ă Tc qq “ π. The reason why T ą Tc
is not included in non-trivial topology is because one recovers the behaviour
seen in figure 23b with no phase change (ΦU “ 0) since the singular momentum
point kc has reached the end of the loop and the topological kink cannot occur
along the path k P r0, 2πs. Thus the effect of temperature in Uhlmann paralell
transport can be seen as a displacement of the topological kink along k-space,
and the critical temperature, Tc can be viewed as corresponding to a maxium
amount of disorder allowed for displaying a topological king along the Uhlmann
holonomy. Further note that ΦU pk ă kc q “ π and ΦU pk ą kc q “ 0 cannot
be smoothly connected, since the momentum k has to pass a singular point kc
(hence the topological kink). Thus, the notion of topological protection is only
present in the non-trivial topological case.
For m “ 0 and Θ “ ˘ π2 (giving, in Equation (7.4.2) and (7.4.3)), ∆k “ 2
and ~nk “ pcos k, 0, ˘ sin kq), the edge-state becomes independent of the system
dynamics, resulting in Flat band (FB) behaviour; these points are indicated by
the arrows in figure 22. At these flat band points (FBP), the critical temperature
can be computeted analytically.
7.5
The Critical Temperature
We now describe how to calculate the critical temperature, Tc . At FBP, ΦU “ 0
in Equation (7.3.8) for the non-trivial topological case (winding number ω1 “
˘1), the Uhlmann phase yields either 0 or π depending on the temperature.
What we are interesetd in is, what this critical temperature is that divides
the two regions of the phase. According to figure 23, the critical temperature
dependence is given by
„
ˆ
ˆ ˙ ¿ ˙
1
1
dk
ΦU “ arg ´ cos ˘ sech
2
Tc
„
ˆ
ˆ ˙˙
1
“ arg ´ cos π sech
“ 0,
Tc
(7.5.1)
(7.5.2)
where the first cosine term inside the argument is equal to ´1. Note that, for
x P R,
argpxq “
π
p1 ´ signpxqq .
2
(7.5.3)
This implies that, in order to obtain zero Uhlmann phase (7.3.8), we need
„
ˆ
ˆ
sign ´ cos π sech
63
1
Tc
˙˙
“ 1.
(7.5.4)
This further implies that
ˆ
ˆ
˙˙
1
cos ˘π sech
ď 0.
Tc
ˆ ˙
1
1
3
2n ` ď sech
ă 2n ` ,
2
Tc
2
(7.5.5)
(7.5.6)
where n P Z. Equation 7.5.6 requires additional reasonings to obtain the critical
temperatur, Tc and a value for the prameter n. We start by applying the inverse
hyperbolic secant;
„
arcsechpxq “ ln
1`
?
1 ´ x2
x

(7.5.7)
on Equation (7.5.6), giving us
„
ln
2`
?
?


„
3 ´ 16n2 ´ 4n
1
1 ` i 16n2 ` 12n ` 5
ď
.
ă ln
4n ` 1
Tc
4n ` 3
(7.5.8)
Notice that the upper limit and lower limit is complex valued depending on
the integer n with the exception that the upper limit is allways complex valued.
It is possible to apply Principal value on the upper limit in order to obtain a
more refined expression.
However, only the lower limit is interesting as it provides the minimum value
for the critical temperature. We are only interested in real valued temperaturs
and as such n P Z must be chosen such that the lower limit must be real valued.
The limit is real valued if and only if the expression inside the natural logarithm
is real valued. That is, let the lower limit be defined as lnpfn q where
fn “
2`
?
3 ´ 16n2 ´ 4n
.
4n ` 1
This can be analyzed more thoroughly, see figure 24.
64
(7.5.9)
Figure 24: Real and imaginary part of fn (7.5.9) is plotted against the parameter
n. In the figuren n appears to be continuous, however, we are only interested
in the integer numbers. Only possible integer value, for which (7.5.9) is real, is
given for n “ 0
Figure 25: Real and imaginary part of fn (7.5.9), presented in figure (24), is
plotted on top of each other. Again, only possible integer value, for which (7.5.9)
is real, is given for n “ 0
As evident from figure 24 and (25), the only allowed value for n, for which
there are no complex contributions from equation (7.5.9), is n “ 0. Furthermore
The only physically acceptable temperatures are real ones. As such, Inequalities
involving imaginary parts are unacceptable leaving us with n “ 0. Thus, the
65
temperature is bounded below by a real entity and the temperature is given by
T ě
1
ˇ
? ˇ,
ln ˇ2 ` 3ˇ
(7.5.10)
where equality is yielded for the critical temperature T “ Tc . That is, Equation
(7.5.10) is the temperature for which the Uhlmann phase (7.3.8) is equal to zero.
Evidently, all 3 models share the same critical temperature at the Flat Band
Points.
Thus, a material has topological behaviour (a topological insulater that has
properties similar to the Quantum hall state effect) when the temperature is
within T P r0, Tc q
As a side note, the existence of complex temperature is not common, but
also no unheard of. Complex temperatures, in this case, presents itself for n ‰ 0,
as can be seen in figure (24).
66
7.6
Majorana-chain model
Consider a model of spinless fermions with p-wave (i.e. angular momentum
l “ 1) superconducting pairing (cooper pair with l “ 1, m “ 0, ˘1 yielding
the natural spherical harmonics at Yl,m “ Y1,˘1 ) hopping on a L-site (perfectly
distributed atoms) one-dimensional chain. The Hamiltonian for such a system
is given by [41]
HM C “
L ´
¯
ÿ
µ
´Ja:j aj`1 ` M aj aj`1 ´ a:j aj ` h.c. ,
2
j“1
(7.6.1)
where µ ą 0 is the chemical potential, J ą 0 is the hopping amplitude, the superconducting gap given by M “ |M |eiΘ , aj (a:j ) are the annihilation operators
(creation) fermionic operators. For convenience, define the following parameters
m :“
µ
2|M |
c :“
J
,
|M |
(7.6.2)
and chose Θ “ 0. It can be shown [41] that the system has two non-local
Majorana modes at the two ends of the chain (as shown in the bottom chain in
figure (26)) if m ă c which corresponds to non-vanishing winding number ω1 .
The Majorana chain Hamiltonian (7.6.1) can be written in the form of Equation
´
¯T
(7.3.1) using the Nambu spinors Ψk “ ak , a:´k , yielding (7.3.1) with
2
p0, ´ sinpkq, c cospkq ´ mq ,
∆k
b
∆k “ 2 pc cospkq ´ mq2 ` sin2 pkq,
~nk “
(7.6.3)
(7.6.4)
in units of |M | “ 1. This implies that φ “ ˘ π2 in Equation (7.3.1). Analogously to Creutz ladder , we calculated the Uhlmann phase as a function of the
parameters m, c and temperature T , see figure 22b.
At T Ñ 0, the topological region is similar the topological phase (Equation
(7.3.9)). Furthermore, as in the case of Creutz Ladder, a critical temperature
exists and is given by Equation (7.5.10).
Figure 26: Top chain showing trivial phase with paired Majorana fermions
(in blue) located on the same sites on a physical lattice. Bottom chain showing
non-trivial topology (topological phase) with bound pairs located at neighboring
sites resting as unpaired Majoranas at each ends, represented by red coloured
balls
67
7.7
Su-Schrieffer-Heeger (SSH)-model
The Hamiltonin for system of a Su-Schrieffer–Heeger model (SSH model, see
figure 27) is given by [42]
HSSH “ ´
ÿ`
ÿ`
˘
˘
a:n an ´ b:n bn .
J1 a:n bn ` J2 a:n bn´1 h.c. ` M
(7.7.1)
n
n
The fermionic operators act on the adjacent sites of a dimerized chain. In
momentum space, HSSH can be written in the form of Equation (7.3.1) by
defining:
2
p´J1 ´ J2 cospkq, J2 sinpkq, 0q ,
∆k
b
∆k “ 2 J12 ` J22 ` 2J1 J2 cospkq,
~nk “
(7.7.2)
(7.7.3)
which implies fixing θ “ ˘ π2 @k. The Uhlmann phase is then plotted as a
function of the hopping prameters J1 and J2 and the temperature T , see figure
22c.
Similarly, at T Ñ 0, topologcial phase (Equation (7.3.9)) is similar to the
topological region (for J1 ă J2 ). Additionally, there exists a critical temperature, Tc , which, at the Flat Band Points (corresponds to J1 “ 0 and J2 “ 1
leading to ∆k “ 2) is given by Equation (7.5.10).
Figure 27: The Su-Schrieffer-Heeger model
7.8
Conclusion
In conclusion, it has been shown that the Uhlmann phase provides the notion of
symmetry-protected topological order in fermions system outside the realm of
pure states. This is in effect when studying dissipative effects like thermal baths.
When applied to the three models (CL, MJ and SSH) of topological insulators,
a finite discontinuity presents itself in the the form of a critical temperature, Tc
which limits the region of topological behaviour. Note that there has not been
any mention in regards to which model is superior over the other. This section
only provides the reader with the additional knowledge that a geometric phase
can be used to determine certain topological aspects that once were believed to
be unknown.
68
8
8.1
Summary and outlook
One-page summary of the thesis
The principal goal of this thesis was to understand the nature of geometric
phases in pure and mixed states, how they can be obtained experimentally and
how they can be utilized in other areas.
A geometric phase arises as a consequence of parallel transport along a path
on a curved manifold. For a quantum system in a pure state, if the geometric
phase is obtained by an adiabatic process, it is called the Berry phase (Equation
(4.1.10)) and is given by
¿
~
~
~
dRxn;
Rptq|∇
~ |n; Rptqy
R
γn rCs “i
C
A non-adiabatic generalization was given by Aharonov and Anandan (Equation (4.2.8)):
żT
xψp0q|U : ptq
γrC 1 s “ argtxψp0q|ψpT qyu ` i
0
dU ptq
|ψp0qydt.
dt
For a Quantum system in a mixed state, the geometric phase was given
by Uhlmann, who approached this using the notion of amplitudes (Equation
(5.1.42)):
´
¯
:
ΦG rks :“ argxwkp0q |wkplq y “ arg tr wkp0q
wkplq .
The mixed state geometric phase was later approached, by Sjöqvist (et al.),
in a different manner using an interferometer. The theory stated that the
Uhlmann phase is indeed an observable and can be observed using the interferometer. Sjöqvist et al. geometric phase is given by (Equation (5.2.30))
»
γG rΓs “ arg tr rU i ptqρsi s “ arg –
dimpH
ÿ iq
fi
wk xk|Ui ptq|kyfl .
k
it has later been experimentally obtained by Du et al. for which, the results
coincides with the theory. These results can be seen in figure 19.
The notion of geometric phases can also be utilized in various areas. As
an illustration we have covered the Uhlmann phase as a topological measure
for 3 different models. Here the Uhlmann phase has been used to observe the
behaviour of topological kinks. The Uhlmann phase, adapted to these models,
is given by (Equation (7.3.8))
«
˜ ¿ ˜
¸
¸ff
ˆ
˙
1
Bk nik1
∆k 1
ΦU “ arg cos pπω1 q cos
sech
dk 1
.
2
2T
njk1
69
Results for the phase are given in figure 22. The critical temperature, i.e.
the temperature for which the Uhlmann phase yields zero, is calculated at the
flat band points (FBP) and is given by (Equation (7.5.10))
Tc “
1
ˇ
? ˇ.
ln ˇ2 ` 3ˇ
Evidently, the critical temperature is universal in the sense that all 3 models
share this expression at the flat band points.
8.2
Contributions by the authors
Various calculations, that appear in this thesis, were not presented in the articles [33] [45] which is not surprising considering that these articles are written
by experts and for experts in their respective fields. As such, many of these calculations were performed by the author of this thesis. As an example, here are
but a few of the equations/expressions obtained by the author: (5.2.6), (5.2.17),
(5.2.40), (5.2.47) (6.2.16), (7.3.4), (7.5.10).
Moreover, the author feels like this work, compared to original material, may
be more accessible for readers with limited knowledge in geometric phases since
the author has included many details needed to understand them.
As of now, the only way to observe and manipulate the geometric phase is
through the use of NMR which is its greatest strength. The ability to manipulate magnetic properties of the nuclei makes it possible to not only detect the
geometric phase, but also manipulate it. However there are still aspects that can
be improved. One such example lies in the RF pulses, in addition to inhomogeneous static and RF magnetic field, which are difficult to control and can lead
to a deviation of the result from theoretical predictions (Figure 19). Suggested
areas of research could implement a more refined and controllable pulse as well
as improving the homogeneity of the stating and RF magnetic field.
8.3
Outlook
Geometric phases and their usage is an interesting topic which will most likely
inspire a lot of research within areas such as the fault-tolerant quantum computing or as topological measures.
There are several articles which utilize the properties of the geometric phase.
One article, written by J.D. Budich and S. Diehl [53] uses the Uhlmann phase
to unravel the topological properties of various gauge structures over space of
density matrices. Another article, written by O. Viyuela, A. Rivas and M.A.
Martin-Delgado [54], continued their work from [33] (Section 7) by applying it
to two-dimensional fermionic systems.
Another interesting area is the nature of the critical temperature. Certain
functions can be analyzed more thoroughly by making it analytical. Extending the critical temperature to the complex plane may reveal more interesting
properties regarding the nature of the topological kink.
Undoubtedly further research is required to fully understand the importance
and usage of the geometric phases.
70
A
Appendix
A.1
Space of states
Before discussing quantum states, ensembles and operators, one has to specify
the mathematical framework in which these concepts are formulated.
A set, V “ tvi unPN
i“1 , is a collection of distinct elements, vi , in which ordering
and multiplicity is of no significance. Now, one is ready to define a vector space
over the complex numbers.
Definition A.1.1. A vector space, pV , `, ¨q over the complex numbers C, is a
set V endowed with two binary operators ` and ¨ that must satisfy the following
axioms:
Closure: @~u, ~v P V , @c P R then ~u ` ~v P V and c~u P V .
(1)
Commutative law: @~u, ~v P V ,
(2)
(3)
Associative law: @~u, ~v , w
~ P V , ~u ` p~v ` wq
~ “ p~u ` ~v q ` w.
~
~
~
Additive identity: D0 P V such that ~v ` 0 “ ~v ., @v P V
(4)
Additive inverse: @~v P V , D!~v 1 P V such that ~v ` ~v 1 “ ~0.
(5)
Distributive law: @~u, ~v P V , @c P C, cp~u ` ~v q “ c~u ` c~v .
(6)
Distributive law: @~u P V , @c, d P C, pc ` dq~u “ c~u ` d~u.
(7)
~u ` ~v “ ~v ` ~u.
Associative law: @~u P V , @c, d P C, cpd~uq “ pcdq~u.
(8)
Unitary law: 1~u “ ~u,
1PC
@~u P V .
Here we use the BraKet notation proposed by P.A.M. Dirac without defining
them. We shall define them later when the notion of dual-vector space has been
discussed.
Definition A.1.2. An inner product space, pV , x¨ | ¨yq, is a vector space over
C with the additional structure of the inner product to C, x¨ | ¨y : V ˆ V Ñ C,
satisfying the following axioms;
(1)
(2)
(3)
Conjugate: @~u, ~v P V , xu | vy “ xv | uy˚ .
Linearity: @~u, ~v P V , @c P C, xu | pc | vyq “ cxu | vy.
Positive definite: @~u P V , xu | uy ě 0.
The inner product is anti-linear in the first argument as a result of linearity
in the second argument. The norm as a result of the inner product is given by
||~u|| :“
A.1.1
a
xu | uy.
(A.1.1)
Hilbert Space
Definition A.1.3. A sequence tui u with ui P C is a Cauchy sequence if
@ P Rą0 DN P N such that @n, m P NąN ||un ´ um || ă .
Definition A.1.4. A vector space with a norm, || ¨ ||, is complete if every
Cauchy sequence converges with respect to this norm to a limit in the space.
71
Now, one is ready to define the Hilbert space.
Definition A.1.5. A Hilbert space H is a complete, complex, inner product
space with a norm induced by the inner product.
Definition A.1.6. A Hilbert space is separable if and only if H has a countable
orthonormal basis U Ă H.
A.1.2
Dual vector space
Definition A.1.7. The algebraic dual space V ˚ of a vector space V over the
field F is the set of all linear maps (linear functionals). f : V Ñ F.
Lemma A.1.1. The double algebraic dual pV ˚ q˚ is isomorphic to V if and
only if V is finite-dimensional.
A.1.3
Operators on Hilbert spaces
Definition A.1.8. Let H̃ be the vector space of operators acting on the Hilbert
space H. Then H̃ forms a Hilbert space in itself.
Definition A.1.9. A bounded linear operator B on a separable Hilbert space
with
basis tei u is called a Hilbert-Schmidt operator if and only if
ř orthonormal
2
k
Be
k
ă
8
with
the Hilbert-Schmidt norm given by
i
i
´
¯
k B k2 “ tr B : B
(A.1.2)
Definition A.1.10. For any bounded linear operator A : H1 Ñ H2 , @u P H1
@v P H2 ;
xu | A: vy :“ xAu | vy,
(A.1.3)
defines another bounded linear operator A: : H2 Ñ H1 called the adjoint of A
and satisfies pA: q: “ A.
Definition A.1.11. The adjoint A: of a densly defined linear operator A on
H is the linear operator on H whose domain DpA: q consist of all α P H such
that the linear map β ÞÑ xα|Aβy is a continuous linear functional on H.
Definition A.1.12. If α P DpA: q, then D!γ P H such that xα|Aβy “ xγ|βy
@β P DpA: q
Definition A.1.13. A (bounded or unbounded) operator A on a hilbert space
H is called symmetric if and only if it satisfies, @u, v P DpAq;
xu | Avy “ xAu | vy.
(A.1.4)
Definition A.1.14. A linear operator, A, on a Hilbert space is called selfadjoint if and only if A: “ A.
that is equation (A.1.4) is fulfilled and DpA: q “ DpAq. In physics literature,
a self-adjoint operator is known as an hermitian operator.
72
A.2
Operators
An observable is an quantity that can be measured in an experiment of a physical
system. In quantum mechanics, this is a self-adjoint operator on the state space
H. An operator, A, ia a linear map taking domain of H to H;
A : |ψy ÞÑ A|ψy such that
A pa|ψy ` b|ψyq “ aA|ψy ` bA|ψy,
@a, b P C.
(A.2.1)
The adjoint, A: , of A is defined by
xuA|vy “ xu|Avy
@u P DpA: q, v P DpAq.
(A.2.2)
A self-adjoint operator, A, in a Hilbert space H has the spectral decomposition
ÿ
an P n ,
(A.2.3)
A“
n
where its eigenstate from a complete orthonormal basis on H, where an is an
eigenvalue of A and;
P n “ |nyxn|,
(A.2.4)
(if an is non-degenerate) is the projection operator into the space of eigenvectors with eigenvalue an and has the following properties;
P n P m “ δm,n P n
(A.2.5)
P :n
(A.2.6)
“ Pn .
73
A.2.1
Unitary operators
An operator U is said to be unitary if
U U : “ U : U “ I,
(A.2.7)
I is the identity operator. In quantum mechanics, unitary operators are
useful since they leave the length of a vector unchanged and thus can be used
to describe symmetries. That is, any operator can be expressed in terms of two
self-adjoint operators A and B as follows;
U “ A ` iB
:
U “ A ´ iB
(A.2.8a)
(A.2.8b)
Applying the unitary condition given by equation (A.2.7) to equations (A.2.8a
& A.2.8b) yields:
U U : “ A2 ` B 2 ´ i rA, Bs “ I
:
2
2
U U “ A ` B ` i rA, Bs “ I
(A.2.9a)
(A.2.9b)
From equation (A.2.9a & A.2.9b) one can immediately see that A commutes
with B and A2 ` B 2 “ I. This further motivates one to assign A “ cospCq and
B “ sinpCq, for which the operator U can be re expressed as U “ exp riCs. As
an example, the Time-evolution operator, is given by
i
U pt, t0 q “ e´ ~ Hpt´t0 q
74
(A.2.10)
A.2.2
The closure relation
Assume that t|ϕα u is a set of orthogonal vectors
xϕβ |ϕα y “ δα,β ,
(A.2.11)
then, |ψy can be expanded in this basis as
|ψy “ lim
R
ÿ
RÑ8
aα |ϕα y,
(A.2.12)
α“1
thus
xϕβ |ψy “ lim
RÑ8
R
ÿ
aα xϕβ |ϕα y “ lim
RÑ8
α“1
R
ÿ
δα,β aα “ aβ ,
(A.2.13)
α“1
in which it follows that
R
ÿ
|ψy “ lim
RÑ8
|ϕα yxϕα |ψy,
(A.2.14)
α“1
for which, the identity operator, also known as the closer relation, is given by:
I “ lim
RÑ8
A.2.3
R
ÿ
|ϕα yxϕα |
(A.2.15)
α“1
Path/Time-ordering operator
The path-ordering operator (or time-ordering operator), P (T ) is an proceduring
operator, for which orders a product of operators according to their value of
parameter. That is,
P pO1 pσ1 q, O2 pσ2 q, ..., ON pσN qq ” Op1 pσp1 qOp2 pσp2 q ¨ ¨ ¨ OpN pσpN q.
The process is similar for the time-ordering operator, T .
75
(A.2.16)
A.3
Baker-Campbell-Hausdorff lemma
Spin operators involves unitary transformations of the form
Bpλq “ e´iλA BeiλA ,
(A.3.1)
where the operators A and B are hermitian operators and λ is a real parameter.
Equation A.3.1 can be expanded into a series using the The Baker-CampbellHausdorff (BCH) lemma, which goes as follows.
Bpλq “B ` iλ rA, Bs `
piλq2
rA, rA, Bss ` ...
2!
R
ÿ
piλqk k
A tBu,
RÑ8
k!
k“0
“ lim
(A.3.2)
(A.3.3)
where the commutation superoperator Ak t u “ rA, s, and for k “ 0, the superoperator is equal to the identity operator, A0 t u “ Id .
A.4
Matlab program used for calculating the critical temperature
The allowed value for the integer parameter n in the determination of the critical temperature has been calculated with the aid of the program Matlab. The
following Matlab syntax was used.
1
2
3
%% Master t h e s i s Geometric p h a s e s
% Miran Haider
%
4
5
6
7
%% P r e p a r a t i o n
clear
clc
8
9
10
11
12
n = l i n s p a c e (´pi , pi , 1 e3 ) ;
%y = 1 . / ( 1 ´ c o s ( x ) + i ∗ 4 ) ;
y = (2+ s q r t (3 ´16∗n.ˆ2 ´4∗n ) ) . / ( 4 ∗ n+1) ;
y2=(2+ s q r t (´16∗n.ˆ2 ´12∗n´5) ) . / ( 4 ∗ n+3) ;
13
14
15
16
17
18
%% P l o t a b s o l u t e v a l u e and phase
%f i g u r e ;
%s u b p l o t ( 2 , 1 , 1 ) ; p l o t ( x , abs ( y ) ) ;
%s u b p l o t ( 2 , 1 , 2 ) ; p l o t ( x , a n g l e ( y ) ) ;
19
20
21
22
%% P l o t r e a l and i m a g i n a r y p a r t s o f l o w e r l i m i t
figure ;
subplot (2 ,1 ,1) ; plot (n , r e a l (y) ) ;
76
23
24
25
26
27
28
29
30
31
t i t l e ( ’ Real Part o f f n ’ )
xlabel ( ’n ’ ) ,
y l a b e l ( ’ Re f n ’ )
a x i s ([ ´ p i p i ´40 4 0 ] )
s u b p l o t ( 2 , 1 , 2 ) ; p l o t ( n , imag ( y ) ) ;
t i t l e ( ’ Imaginary Part f n ’ )
xlabel ( ’n ’ ) ,
y l a b e l ( ’ Im f n ’ )
a x i s ([ ´ p i p i ´1.5 1 . 5 ] )
32
33
34
35
36
37
38
39
40
41
42
43
44
%% P l o t Real and Imaginary p a r t s o f upper l i m i t
%f i g u r e ;
%s u b p l o t ( 2 , 1 , 1 ) ; p l o t ( x , r e a l ( y2 ) ) ;
%t i t l e ( ’ Real Part o f upper Ln ( . . . ) ’ )
%x l a b e l ( ’ n ’ ) ,
%y l a b e l ( ’ Ln ( n ; . . . ) ’ )
%a x i s ([ ´ p i p i ´40 4 0 ] )
%s u b p l o t ( 2 , 1 , 2 ) ; p l o t ( x , imag ( y2 ) ) ;
%t i t l e ( ’ Imaginary Part o f upper Ln ( . . . ) ’ )
%x l a b e l ( ’ n ’ ) ,
%y l a b e l ( ’ Ln ( n ; . . . ) ’ )
%a x i s ([ ´ p i p i ´40 4 0 ] )
45
46
%% EXP
47
48
49
50
51
52
53
54
55
56
figure ;
p l o t ( n , r e a l ( y ) , ’ r ’ , n , imag ( y ) , ’ b ’ )
%h o l d on
%p l o t ( n , imag ( y ) , ’ b ’ )
%h o l d o f f
xlabel ( ’n ’ )
t i t l e ( ’ r e a l and i m a g i n a r y p a r t o f f n ’ )
a x i s ([ ´ p i p i ´5 5 ] )
l e g e n d ( ’ Re f n ’ , ’ Im f n ’ )
77
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