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GEOMETRIC PHASE FOR OPEN SYSTEMS
LEI QIANG
B.Sc. (Hons), NUS
A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF SCIENCE
DEPARTMENT OF PHYSICS
NATIONAL UNIVERSITY OF SINGAPORE
2007
Acknowledgement
I would like to extend my gratitude to the following people, without which this project
would not have been possible.
First of all, I would like to thank my supervisors, Dr. Kuldip Singh and Prof. Oh
Choo Hiap, for offering me this masters project. I wish to express my gratitude to Dr.
Kuldip Singh for his advice and encouragement throughout my research. I am especially
grateful for his invaluable guidance and insights into the problem. His keen acumen in
mathematical details was critical in many stages of the work.
Special thanks to my ex-colleagues and supervisors in HVB who supported me on a
daily basis during my periods of study.
Last and most important, a heartfelt thanks to my wife, Chieh Mei for her constant
support and care during this very trying period, giving me moral support all along the
way.
Abstract
In this thesis, we consider the issue of a geometric phase for systems undergoing nonunitary evolutions. This pertains to systems that are interacting with an environment.
Using a relative phase concept, the nature of unitary representations of such evolutions
in the combined state space of the system and the environment are shown to be ideal for
defining a phase which we term as the geometric phase. It is well known that unitary representations of such evolutions are not unique. Here we show that this non-uniqueness in
the representations defines the gauge group and the concomitant parallel transport conditions. In particular we elucidate the nature of these conditions for a class of evolutions
that do not lead to level crossings in the eigen spectrum of the states. We also furnish a
gauge-invariant expression for the geometric phase under such maps.
Contents
Acknowledgement
i
Abstract
ii
1 Introduction
1
2 Fiber Bundle Language of Pure State Geometric Phase
5
2.1
A short Introduction to Fiber Bundle . . . . . . . . . . . . . . . . . . . .
5
2.2
Pure State Geometric Phase in Language of Fiber Bundle . . . . . . . . .
9
2.2.1
Pure State Fiber Bundle and the Connection . . . . . . . . . . . .
9
2.2.2
The Geometrical Phase . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2.3
The Dynamical Phase . . . . . . . . . . . . . . . . . . . . . . . .
15
Example of Pure State Geometric Phase . . . . . . . . . . . . . . . . . .
17
2.3
3 The Mixed States Geometric Phase
20
3.1
Review of Density Operators and Mixed States
. . . . . . . . . . . . . .
20
3.2
The Non-Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.3
The Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
CONTENTS
3.3.1
iv
Effect of Gauge Transformations . . . . . . . . . . . . . . . . . . .
4 Kinematical Approach to Geometric Phase for Open Systems
4.1
34
37
4.0.2
Brief Introduction to the Open System . . . . . . . . . . . . . . .
37
4.0.3
Amplitude Damping and Phase Damping Channels . . . . . . . .
38
Geometric Phase for Open Systems . . . . . . . . . . . . . . . . . . . . .
40
4.1.1
Gauge Freedom of the Environment . . . . . . . . . . . . . . . . .
40
4.1.2
Gauge Freedom Of the System . . . . . . . . . . . . . . . . . . . .
42
4.1.3
The Total Gauge Group . . . . . . . . . . . . . . . . . . . . . . .
44
4.1.4
Geometric Phase Expression . . . . . . . . . . . . . . . . . . . . .
45
4.1.5
Proof of Gauge Invariance . . . . . . . . . . . . . . . . . . . . . .
52
4.1.6
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
5 Applications
60
5.1
Amplitude Damping Channel . . . . . . . . . . . . . . . . . . . . . . . .
60
5.2
Phase Damping Channel . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
6 Summary
67
List of Figures
2.1
A bundle with π : E → M . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
A section s of a bundle π : E → M . . . . . . . . . . . . . . . . . . . . .
7
2.3
Pure state fiber bundle structure . . . . . . . . . . . . . . . . . . . . . . .
10
2.4
Geometrical phase by holonomy . . . . . . . . . . . . . . . . . . . . . . .
13
2.5
Pure state dynamical phase
16
2.6
A Bloch sphere representation of a state |ψ
2.7
The Bloch sphere representation of a state |ψ under rotation
. . . . . .
19
4.1
The Bloch vector in an Amplitude Damping process. . . . . . . . . . . . .
40
4.2
The block vector in a phase damping process. . . . . . . . . . . . . . . . .
41
5.1
The Amplitude Damping process coupled with a rotation about the z axis.
63
5.2
The geometrical phase for Amplitude Damping channel . . . . . . . . . .
64
5.3
The geometrical phase for Amplitude Damping channel for a rotated envi-
. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
18
ronment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
5.4
The phase damping process with a rotation Vsys = eiπtσz . . . . . . . . . .
66
5.5
The geometrical phase for evolution of Figure 5.4 . . . . . . . . . . . . .
66
Chapter 1
Introduction
The notion of a geometric phase in physical systems was first explored by Pancharatnam
in his seminal paper in 1955, where he examined phase relations between two polarizations
states of light [1]. He showed that a phase arises when the light goes through a cyclic
path in polarization state space. It is important to note that his treatment was essentially
classical. Later in 1984, M. V. Berry [2] showed that when a quantum system undergo an
evolution, it also acquires a phase that is geometric in nature. In particular, he showed
that the energy eigenfunctions acquire a phase factor which depends only on the path
of the evolution. In other words, he showed that the phase was closely related to the
geometry of the state space. Simon [3] later pointed out that the phase could be explained
in the language of fiber bundles. Essentially, he showed that the phase introduced by
Berry was nothing but the holonomy of a complex line bundle.
One short-coming of Berry’s work was his assumption of adiabatic in the evolution.
In 1987, the extension of Berry’s idea to the case of non-adiabatic evolution was carried
out by Aharonov and Anandan [4][5], where they exhibited that the trivial subtraction
of the dynamical phase from the total phase yielded a gauge-invariant quantity which
they identified as the geometric phase. This was essentially the phase that Berry had
introduced but without the audibility condition. Here they considered the holonomy of a
2
U (1) bundle over the projective Hilbert space of the system. They showed that a gauge
invariant geometrical phase could be defined for all paths which project to the same closed
curve in the projective Hilbert space of the system. The geometrical phase for non-cyclic
evolution was subsequently proposed by Samuel and Bhandari [6].
The geometric nature of Berry’s phase and its generalization has been extensively
studied [7, 8, 9] with applications even in quantum computing. Indeed, quite recently,
it was proposed that the concept of geometric phase could be used in realizing quantum
gates [10]. The importance of using geometrical phase in the construction of gates is
primarily due to its robustness towards noise. Since the geometric phase depends only on
its path in state space, it is tolerant to noise, which basically creeps into the dynamical
phase. Now, in the context of noise, it becomes important to ask whether the geometric
phase defined for pure states retain its character; especially when systems decohere.
Indeed for practical implementations, the states often considered are generally mixed
states. Not surprisingly, the notion of a geometric phase in the context of mixed states
have been the subject of intense investigation in recent years [11, 12, 13, 14, 15, 16, 17].
Uhlmann [11] has attempted to address this issue within the mathematical context of
purification, while Sj¨oqvist et.al. [12] have analyzed the problem within the context of
quantum interferometry. The authors of ref. [12] have provided some insights into how
one can understand the notion of a phase for mixed states by considering the shift in the
maximum intensity. Singh et.al. [15] have provided a geometrical interpretation to this
phase by examining the issue within the context of fiber bundles. Essentially, they have
furnished a kinematical framework that allows one to formulate the geometric phase as
a gauge-invariant entity in a manner similar to the Aharanov-Anandan framework.
Some studies on the geometrical phase of entangled states and the subsystems have also
been considered [18, 19, 20]. More recently the issue of the geometric phase for systems
undergoing non-unitary evolutions have been taken up by a few authors [21, 22, 23, 24].
Since any system interacting with an environment will essentially undergo non-unitary
3
evolution, it is important to ask what the geometric phase is for such evolutions. Indeed,
Ericsson et. al. [22] have tried to extend the interferometric scheme introduced in ref.
[12] to analyze this, while Carollo et.al. [24] have employed the quantum-jump approch
to the problem. Recently, Tong et.al [21] have proposed yet another scheme using the
notion of purification. In these works, the geometric nature of the phase is not clear as
no reference to any symmetry is made.
In this report, we are going to approach the question of mixed state geometrical phase
for non-unitary evolution from a kinematical standpoint. Here we will rely heavily on
symmetry as a guiding principle in defining, what we call, the geometric phase. Formulating the problem in the context of an appropriate fiber bundle, we will elucidate the
geometric nature of the phase by identifying the appropriate gauge group and also concomitant parallel transport conditions. It should be remarked that the approach taken
by ref. [21] is not really kinematical as they exclude the environment from the outset by
tracing it out. A full kinematical approach would require the presence of these degrees
of freedom to be manifest, with the reduction effected by the gauge symmetry. In other
words, the gauge-invariance of the geometric phase should really mean independence from
these environmental parameters even when the combined system (system + environment)
is considered. In our approach, this strictly adhered to.
In this thesis, we begin with a brief overview of some basic facts in fiber bundle theory
that will be helpful in elucidating the key concepts that underlie the geometric phase.
Central to this will be the notion of gauge invariance and the concept of a connection.
Here, we will revisit some of the essential features of the Aharanov-Anandan formulation
and show how the geometric phase is closely linked with the gauge summetry and how
parallel-transport defines the geometric phase.
In chapter 3, we review the kinematical approach developed for mixed states undergoing unitary evolution [15]. In particular, we pay special attention to systems that display
degeneracies in their density matrices. For such systems the gauge group takes on a
4
non-abelian character. The manner in which the non-abelian group is treated within the
context of the geometric phase is described at some length as it plays an important role
in the non-unitary case.
In chapter 4, we provide a generalization to the non-unitary case. We begin with a brief
introduction to open systems and review some of the usual ways that one characterizes
such systems. In particular, we show how the non-uniquness of the Krauss operators play
the role of a gauge symmetry and how many of the ideas developed in ref.[15] can be used
in defining the geometric phase. We also elucidate the nature of the parallel-transport
conditions and furnish a proof of gauge invariance for the geometric phase.
Finally, in chapter 5, we examine two simple channels; namely the Amplitude damping
and the Phase damping channels. Here, we compute the geometric phase for some paths
numerically and demonstrate that they do not depend on the environmental degrees of
freedom. We conclude with a summary of the main claims.
Chapter 2
Fiber Bundle Language of Pure
State Geometric Phase
The mathematical language of fiber bundles in differential geometry provides a powerful
tool to study and analyze the geometrical phase. It enables one to visualize this quantum
phenomenon in a geometrical way and it paves the way to other generalizations; for
instance to mixed states.
Here, we furnish a short introduction to fiber bundles that will serve both to set the
framework for other generalizations as well as introduce the notation that will be used
in this thesis. This will be followed by a review of how the pure state geometric phase
is formulated in this language, which will in turn elucidate the underlying geometric
nature of the phase. A sample calculation for a simple quantum system will serve as an
illustration.
2.1
A short Introduction to Fiber Bundle
The fiber bundle is a mathematical tool that is increasingly used in particle physics to
describe theories that have gauge degrees of freedom. Essentially, it is a basic construct
2.1 A short Introduction to Fiber Bundle
6
in modern differential geometry that incorporates various aspects of Manifold theory, Lie
algebras and group theory [26],[27],[28],[29],[30] [34].
To begin with, consider a smooth manifold M . A vector field v on M assigns to each
point p ∈ M a vector vp in the tangent space Tp M . This leads to a very simple example
of a fiber bundle. To see this, we first consider the union of all the tangent spaces of M ,
TM =
Tp M
p∈M
which we call the bundle space. Here, we denote it as E. The manifold M is called
the base space. The map that takes a vector in E to the point p ∈ M , π : E → M
is called the projection map. A bundle is a structure consisting of a base space M , a
total space E and a projection map π. A fiber over p is defined to be
Ep = {q ∈ E : π(q) = p}
It is easy to see that the total space is a union of the fibers, i.e E =
Ep .
Figure 2.1: A bundle with π : E → M
A section of a bundle π : E → M is a function s : M → E such that for any p ∈ M .
s(p) ∈ Ep
It basically assigns to each point p ∈ M a vector in space E. We can regard the section
as a “function”, and if this is smooth, we say that the vector field defined by this section
2.1 A short Introduction to Fiber Bundle
Figure 2.2: A section s of a bundle π : E → M
is smooth.
Before we define the notion of a gauge transformation or a connection in the language
of fiber bundles, it is instructive to consider a bundle P called the Principal bundle.
On this bundle, we assume that there is group action G × P → P that fibrates the space.
Here we will assume that the group G is a Lie group. The action of each element g ∈ G
defines a diffeomorphism Rg : P → P through Rg (p) = pg such that p(g1 g2 ) = (pg1 )g2
for all g1 , g2 ∈ G and p ∈ P . We call this the right action of G on P and it is assumed
to act freely on P 1 The base space B is then the quotient space of P by the equivalence
relation induced by G; i.e. p ∼ p of p = pg for some g ∈ G. This defines the canonical
projection π : P → M such that π(pg) = π(p), ∀g ∈ G. The fiber π −1 (x) over a point
x ∈ M is a manifold that is diffeomorphic to G. The elements (P, G, M, π) defines the
principal bundle. As in the case of the tangent bundle, we can define local sections that
map open sets of M to P , i.e. si : Ui ⊂ M → P such that π(si (x)) = x, ∀x ∈ Ui . It
should be noted that, there may be no single section that covers the whole manifold M .
Rather one has to define local sections over open sets Ui ⊂ M and the collection of all
such sections will define a map from M to P .
1
An action is free if every element of P is moved away from itself by every element of G with the
exception of the unit element of G.
7
2.1 A short Introduction to Fiber Bundle
8
Now at each point p ∈ P , the fibered structure of the principal bundle distiguishes
a vertical subspace Vp ⊂ Tp P , which consists of all vectors tangent to the fiber. This
naturally leads to an assignment, also called a connection, of the horizontal space Hp .
Essentially it allows one to decompose the tangent space into a direct sum of the vertical
and horizontal subspaces, with the latter invariant under the right action of G:
(a) Tp
Vp ⊕ Hp for all p ∈ P
(b) Rg∗ (Hp ) = Hpg for all g ∈ G, p ∈ P , where Rg∗ denotes the push-forward map on
the tangent spaces induced by the right action.
Equivalently, the horizontal subspace can also be defined through the connection oneform ω. Locally, if the coordinates of P are characterized by (x, g), where x ∈ M and
g ∈ G, then the connection one-form ω has the form
ω = g −1 dg + g −1 Ag
(2.1)
where A is a one-form on M that is the pull-back of ω under s: A = s∗ ω.
A gauge transformation is an operation of G : P → P which maps a section to
another section of P ; i.e. s(p) → s (p), with s (p) = s(p)h. The group G, in this context,
is called the gauge group. It is interesting to note that under this transformation A
transforms as A → A = hAh−1 − h−1 dh, which is the familiar gauge transformations we
encounter in electrodynamics.
A connection is a smooth assignment to each point p ∈ E of a subspace Hp E and Vp M
of Tp E such that,
(a) Tp E
Vp E ⊕ Hp E for all p ∈ E
(b) Rg∗ (Hp E) = Hpg E for all g ∈ G, p ∈ E, where Rg∗ (p) := pg denotes the right action
of G on E.
2.2 Pure State Geometric Phase in Language of Fiber Bundle
2.2
9
Pure State Geometric Phase in Language of Fiber
Bundle
2.2.1
Pure State Fiber Bundle and the Connection
Here we see how Berry phase arises naturally when we consider the bundle space of
states that are undergoing unitary evolutions. To begin with we note that the space of
all normalized ket vectors {|ψ ∈ H}, i.e.
ψ(t)|ψ(t) = 1
(2.2)
is essentially S 2N −1 for an N -state system. In quantum mechanics, a physical state is
represented by a ray, which is the equivalent class of states under the equivalence relation
|ψ ∼ eiθ |ψ
(2.3)
eiθ ∈ U (1)
(2.4)
where
is a general phase. Here in fiber bundle language, the total space P is S 2N −1 with the
base space given by the quotient of P by the group action (U (1), in this case):
P (H) =
S 2N −1
U (1)
(2.5)
The fiber consists all the equivalent states and this space is isomorphic to U (1). To
define the projection map, it is not difficult to see that the map
π : |ψ(t) −→ |ψ(t) ψ(t)|
is a good map. Indeed, all “equivalent” states are mapped to the same state.
(2.6)
2.2 Pure State Geometric Phase in Language of Fiber Bundle
10
Figure 2.3: Pure state fiber bundle structure
Next we consider the horizontal subspace. Here we show that the tangent vectors take
the form of
Dµ =
∂
∂
+ Aµ
µ
∂X
∂θ
(2.7)
where X µ are the coordinates of P (H) and A = Aµ dX µ is the connection one-form. It
should be noted the vertical subspace is one-dimensional and it is characterized by the
vector
∂
.
∂θ
As we know from previous section, a connection one-form will take a form of
ω = g −1 dg + g −1 Ag.
(2.8)
For the case at hand, with g = eiθ and g −1 = e−iθ we have
ω = e−iθ d e−iθ + e−iθ Aeiθ
= i dθ + A.
(2.9)
For any point on the space of S 2N −1 , we can write the tangent vector in the form of
Xµ =
∂
∂
+ Cµ
µ
∂X
∂θ
(2.10)
where Cµ is an arbitrary component along the vertical direction. If we want this vector
X to be on the horizontal space, we should have
ω, X
= 0
2.2 Pure State Geometric Phase in Language of Fiber Bundle
∂
∂
+
C
= 0
µ
∂X µ
∂θ
∂
∂
i dθ + Aµ dX µ ,
+ Cµ
= 0
µ
∂X
∂θ
⇒ i Cµ + Aµ = 0
⇒
⇒
11
i dθ + A,
∴ Cµ = −i Aµ
(2.11)
So, basis vectors for the horizontal subspace take the form of,
Dµ =
∂
∂
− iAµ .
µ
∂X
∂θ
(2.12)
As Aµ at this point is arbitrary, we can absorb the factor i in Aµ and write instead
Dµ =
∂
∂
+ Aµ
µ
∂X
∂θ
(2.13)
With the tangent space of P regarded as a direct sum of the horizontal and vertical
subspaces, Tp P = Vp ⊕ Hp , any tangent vector at point p can be written
d
∂
=α
+ B µ Dµ
dt
∂θ
in the basis
∂
, Dµ
∂θ
(2.14)
. Hence, for any |φ(t) in the total space, we have
d
∂
|φ(t) = α
|φ(t) + B µ Dµ |φ(t)
dt
∂θ
(2.15)
In the case of Berry phase, we choose |φ(t) as the direction of vertical and take
˙
φ(t)|φ(t)
= 0 as the condition that ensures the vector remains horizontal. We call this
the parallel-transport condition. Taking the above into account, we have
φ(t)|Dµ |φ(t) = 0
⇒
φ
⇒
Aµ
∂
∂
φ =0
+
A
µ
∂X µ
∂θ
∂
∂
φ| |φ = − φ|
|φ .
∂θ
∂X µ
(2.16)
2.2 Pure State Geometric Phase in Language of Fiber Bundle
As θ is the phase term and operates on the state as eiθ , we have by writing |φ
∂
|φ
∂θ
⇒
⇒
θ
= ieiθ |φ = i|φ
12
θ
= eiθ |φ ,
θ
∂
|φ = i
∂θ
∂
Aµ = i φ|
|φ .
∂X µ
φ|
(2.17)
This shows how a choice of horizontal and vertical subspaces will fix the connections.
2.2.2
The Geometrical Phase
In this section, we will see what happens to state that is parallel transported along a
curve that starts and ends on the same fiber. In projective space, this corresponds to
an evolution that is cyclic. In the following, we show that the state may end up at a
different point on the fiber and the phase acquired is the Berry phase. In fiber bundle
jargon, we call this holonomy.
To begin with, let us choose a section |φ(t) on the fiber space with |φ(T ) = |φ(0) so
that |φ(t) is a cyclic evolution. Here |φ(t) can be regarded as the representative state
that characterizes the projective space. Then, for any open evolution |ψ(t) = eif (t) |φ(t)
we have,
˙
|ψ(t)
˙
= if˙(t)eif (t) |φ(t) + eif (t) |φ(t)
.
If we now impose the parallel-transport condition,
˙
ψ(t)|ψ(t)
˙
= if˙(t) + φ(t)|φ(t)
= 0
d
˙
f (t) = i φ(t)|φ(t)
dt
(2.18)
Integrating both sides, we obtain
T
β ≡ f (T ) − f (0) = i
0
˙
φ(t)|φ(t)
dt
(2.19)
2.2 Pure State Geometric Phase in Language of Fiber Bundle
13
where β is defined to be the geometrical phase.
In the following, we will analyze β and show that it is independent of the choice of
Figure 2.4: Geometrical phase by holonomy
section |φ(t) . In other words, it is gauge-invariant.
As mentioned earlier, the total space can be characterized by coordinates (θ, X µ ).
Hence,
d
∂
∂
|φ(t) = θ˙ |φ + X˙ µ
|φ
dt
∂θ
∂X µ
(2.20)
from which we note that
T
˙
φ(t)|φ(t)
dt
β = i
0
T
= i
0
∂
θ˙ φ| |φ dt + i
∂θ
T
= −
T
θ˙ dt + i
0
0
T
∂
X˙ µ φ|
|φ dt
∂X µ
0
∂
X˙ µ φ|
|φ dt .
∂X µ
(2.21)
Here we have used,
φ|
∂
|φ = i.
∂θ
(2.22)
2.2 Pure State Geometric Phase in Language of Fiber Bundle
14
With
T
T
θ˙ dt =
dθ = θ(T ) − θ(0) = 0 ,
0
(2.23)
0
we have
T
β = i
0
T
= i
∂
X˙ µ φ|
|φ dt
∂X µ
Aµ X˙ µ dt
0
T
= i
Aµ dX µ
0
Aµ dX µ
= i
=
A.
(2.24)
Now, let us see what happens to β under a gauge transformation. By effecting a
change of section |φ(t) → |φ (t) given by
|φ (x)
= eiθ(x) |φ(x)
where θ(x) is a real function of the coordinates X µ on base space. The connection will
transform as,
Aµ (x) = −
∂θ(x)
+ Aµ (x)
∂X µ
with the corresponding geometric phase
β
=
Aµ dX µ
=
−
= −
∂θ(x) µ
dX +
∂X µ
Aµ (x)dX µ
dθ + β
= −(θ(T ) − θ(0)) + β
(2.25)
As sections are associated with evolutions leading to closed paths, i.e. |φ(0) = |φ(T )
and |φ (0) = |φ (T ) we have, θ(0) − θ(T ) = 2πn, where n is any integer. Hence,
β
= β + 2πn
which essentially means that the geometric phase is gauge-invariant.
2.2 Pure State Geometric Phase in Language of Fiber Bundle
2.2.3
15
The Dynamical Phase
In this section, we will consider the relationship between total phase, dynamical phase
and geometrical phase. The time evolution of a state vector is given by the Sch¨odinger
equation:
i
∂
|ψ(t)
∂t
= H(t)|ψ(t)
(2.26)
Hence, for state |ψ(t) = eif (t) |φ(t) , where |φ(t) is a choice of section, we have
∂
|ψ(t)
∂t
∂ if (t)
∂
= i
e
|φ(t) + ieif (t) |φ(t)
∂t
∂t
if
(t)
if
(t)
˙
= −f˙(t)e
|φ(t) + ie
|φ(t)
.
RHS = i
Contracting both sides with |ψ(t) , we obtain for the LHS,
ψ(t)|i
∂
|ψ(t)
∂t
˙
= − φ(t)|e−if (t) f˙(t)eif (t) |φ(t) + i φ(t)|e−if (t) eif (t) |φ(t)
˙
= −f˙(t) + i φ(t)|φ(t)
(2.27)
while the RHS becomes, ψ(t)|H(t)|ψ(t) . Hence, we have,
˙
f˙(t) = i φ(t)|φ(t)
− ψ(t)|H(t)|ψ(t)
or after integrating
γ = f (T ) − f (0)
T
=
f˙(t)dt
0
T
= i
0
T
˙
φ(t)|φ(t)
dt −
ψ(t)|H(t)|ψ(t) dt.
(2.28)
0
γ here is the total phase acquired by the system under the evolution and the second
term on the right is the dynamical term. The latter depends on the hamiltonian. It is
˜
not difficult to see that if we have another evolution with H(t)
that produces the same
path in the projective space P as H(t), then the geometric phase is the same while both
2.2 Pure State Geometric Phase in Language of Fiber Bundle
16
the total phase and dynamical phase terms change correspondingly. Indeed, if we take
˜
|ψ(t)
= eif2 (t) |φ(t) , where new state has the same section as |ψ(t) then
γ2 = f2 (T ) − f2 (0)
T
0
The geometrical term γG = i
T
˙
φ(t)|φ(t)
dt −
= i
˜ H(t)|
˜ dt.
˜
ψ(t)|
ψ(t)
(2.29)
0
T
0
˙
φ(t)|φ(t)
dt is same for the two evolutions if they have
same path in the base space but the second term generally depends on hamiltonian.
The expression for the total phase can also be expressed in terms of the unitary operator
Figure 2.5: Pure state dynamical phase
that produces the evolution. By taking
|ψ(T ) = U (T )|ψ(0)
it is easy to see that
γT = f (T ) − f (0) = arg { ψ(0)|U (T )|ψ(0) }
(2.30)
since φ(0)|φ(T ) = 1. With the dynamical term given by
T
γD =
ψ(t)|H(t)|ψ(t) dt
0
(2.31)
2.3 Example of Pure State Geometric Phase
17
the geometrical phase is simply,
γG = γT − γD
(2.32)
We also note that if we invoke the parallel-transport condition,
˙
ψ(t)|ψ(t)
= −i ψ(t)|H(t)|ψ(t)
= 0
(2.33)
the dynamical phase
T
γD =
ψ(t)|H(t)|ψ(t) dt
0
= 0
(2.34)
which leads to
γG = γT .
(2.35)
To summarize, we would like to highlight two important results,
(1) γG is invariant under different evolution as long as they share same closed path in
the projective space.
(2) γG = γT under the parallel-transport condition.
2.3
Example of Pure State Geometric Phase
Any qubit state can be expressed as
cos 2θ
|ψ =
θ
iϕ
e sin 2
0
θ
π,
0
ϕ
2π
(2.36)
The state |ψ is a function of (θ, ϕ). Geometrically, pure states for a qubit are represented
as points on a 2-sphere called the Bloch sphere. These points also represent the projective
2.3 Example of Pure State Geometric Phase
18
Figure 2.6: A Bloch sphere representation of a state |ψ
space as it is evident from the pure state density matrix ρ = |ψ ψ| which is characterized
by points on this sphere. Consequently, the connection 1-forms can be evaluated easily,
∂
|ψ
∂θ
∂
= i ψ| |ψ .
∂ϕ
Aθ = i ψ|
Aϕ
Now
∂
|ψ
∂θ
=
∂
|ψ
∂ϕ
(2.37)
− 12
θ
2
sin
1 iϕ
e
2
cos
θ
2
0
=
iϕ
ie sin
θ
2
(2.38)
and these lead to
∂
|ψ
∂θ
θ
θ
1
θ
θ
1
= −i sin cos + i sin cos
2
2
2
2
2
2
= 0
Aθ = i ψ|
∂
|ψ
∂ϕ
θ
= − sin2
2
1
= − (1 − cos θ)
2
Aϕ = i ψ|
(2.39)
2.3 Example of Pure State Geometric Phase
19
which furnish the explicit form of the connection 1-form:
A = Aθ dθ + Aϕ dϕ
1
= − (1 − cos θ)dϕ.
2
(2.40)
Figure 2.7: The Bloch sphere representation of a state |ψ under rotation
If we consider a simple evolution consisting of a rotation of a qubit along the z axis,
(as illustrated in Figure 2.7), then the geometrical phase corresponding to this evolution
is given by,
γG =
A
1
(1 − cos θ)dϕ
2
= −π(1 − cos θ)
= −
= −
Ω
2
(2.41)
where, Ω is the solid angle corresponding to the evolution path in the Bloch Sphere
representation. This result is consistent with the expression for Berry’s phase under
adiabatic approximation [33].
Chapter 3
The Mixed States Geometric Phase
While the pure state geometric phase has been extensively studied since the 80’s, the
focus on the geometric phase for mixed state only started recently [33]. In the following
sections, we discuss how the fiber bundle approach in pure states can be extended to
mixed states. First a brief review of density operators and mixed states will be given,
and then a detail step by step approach to mixed state geometric phase will be shown.
3.1
Review of Density Operators and Mixed States
A pure state can always be written in the form |ψ which corresponds to the non-zero
vectors in the Hilbert space, H. As described in previous chapter, the physical states
form an equivalence class under the relation,
|ψ1 ∼ |ψ2
iff
|ψ2 = eiθ |ψ1
(3.1)
where θ is any real number. It is worth noting that a pure state can also be written as
density operator,
ρ = |ψ ψ|.
(3.2)
3.1 Review of Density Operators and Mixed States
21
Here the density opeartor is a projector in the sense that ρ2 = ρ. We represent the mixed
state in a similar way by taking
ρ=
|k ωk k|
(3.3)
k
where |k are eigenstates of ρ and ωk ’s are the corresponding probabilities of measuring
the state in |k . In this case the density operator is no longer a projector and ρ2 = ρ.
The density operators must satisfy three conditions, namely,
Trρ = 1,
(3.4)
ρ = ρ† ,
k|ρ|k
(3.5)
≥ 0
for all |k .
(3.6)
These properties have the following implications for the eigenvalues ωk :
(3.4) implies
ωk = 1;
(3.7)
k
(3.5) implies ωk = ωk∗ ;
(3.8)
(3.6) implies ωk ≥ 0.
(3.9)
With these restrictions on the eigenvalues ωk , it is easy to see that
< 1 for mixed states;
2
2
Trρ =
ωk
= 1 for pure states.
(3.10)
k
The expectation value of an observable A is given by
A = Tr(Aρ) =
ωk k|A|k
(3.11)
k
for any state ρ =
k
|k ωk k|. This implies that the expectation value of A is simply a
weighted sum of the expectation values of A on each eigenstate |k .
3.2 The Non-Degenerate Case
3.2
22
The Non-Degenerate Case
In this section we extend some of the formalism developed for pure state geometric
phase to mixed states. The key principle that we adhere to in the ensuing generalization
is the concept of gauge-invariance for the geometric phase. Recall that in the pure
state case, the gauge group is simply the group that keeps the projector |ψ(t) ψ(t)|
invariant. Equivalently, the unitary matrix that commutes with the projector will be the
corresponding gauge group. In the present case, we first determine what this group is.
To this end, consider a general density matrix ρ ∈ H of an N -state system,
N
ρ=
|k ωk k|,
dim(H) = N
(3.12)
k
Here we begin with a density matrix that is non-degenerate. It is easy to see that the
unitary matrix that commutes with ρ is
N
Gx = U (1) × U (1) · · · × U (1)
N
eiθk |k k|
≡
(3.13)
k=1
where {θk }k=1,2,...,N are arbitrary parameters representing phases. Note that the eigenvalues ωk are all distinct. It is instructive to note that the matrix form of this group
element in the basis {|k } is
V(t) =
eiθ1 (t)
...
eiθN (t)
Note in the diagonal basis {|k }k=1,2,...,N , it can be shown that V(t) satisfy the little
group’s property that is
[V(t), ρ(0)] = 0.
(3.14)
3.2 The Non-Degenerate Case
23
With the gauge group identified the gauge transformation for any U(t) ∈ U (N ) is
U(t) −→ U (t) = U(t)V(t)
N
eiθk |k k|
= U(t)
(3.15)
k=1
Here U(t) is the unitary operator that defines the evolution of the system. The orbit of
the density matrix ρ(0) under U(t) and the gauge transformed U(t) −→ U (t) is:
ρ(0) −→ ρ(t) = U(t)ρ(0)U † (t)
(3.16)
ρ(0) −→ ρ (t) = U (t)ρ(0)U † (t)
(3.17)
It is not difficult to see ρ(t) and ρ (t) are indeed the same.
The total phase[12] obtained from the interferometer setup for mixed under unitary evolution is
γT = arg Tr(ρ(0)U † (0)U(t))
N
ωk k|U † (0)U(t)|k
= arg
(3.18)
k=1
A simple extension of the pure state dynamical phase will give the dynamical phase for
mixed states as well:
T
dtTr(ρ(t)H(t))
γD = −
0
T
= −i
˙
dtTr ρ(0)U † (t)U(t)
0
N
T
= −i
˙
ωk k|U † (t)U(t)|k
dt
0
(3.19)
k=1
To make the dynamical phase vanish, it is clear that we need to require
˙
k|U(t)† U(t)|k
=0
∀k = 1, 2, . . . N.
(3.20)
One can interpret this equation in the following way. For the dynamical phase of a mixed
state to vanish, all the constituent pure states in the mixture must be parallel-transported
independently. There are now N parallel transport conditions.
3.2 The Non-Degenerate Case
24
One key property of geometric phase is that it should be gauge invariant under the
gauge group Gx . Let us see how the total phase and dynamic phase change under the
gauge transformation. The total phase transforms as:
γT −→ γT = arg Tr[ρ(0)U † (0)U (T )]
= arg Tr[ρ(0)V † (0)U † (0)U(T )V(T )]
n|k ωk k|U † (0)U(t)|n eiθn (T ) e−iθn (0)
= arg
k,n
ωk k|U † (0)U(t)|k eiθk (T ) e−iθk (0)
= arg
(3.21)
k
while the dyanmical phase transforms as:
T
γD −→ γD = −i
dtTr ρ(0)U † (t)U˙ (t)
0
T
= −i
˙
˙
dtTr ρ(0)V † (t)U † (t) U(t)V(t)
+ U(t)V(t)
0
T
= −i
˙
˙
dtTr ρ(0)V † (t)U † (t)U(t)V(t)
+ ρ(0)V † (t)U † (t)U(t)V(t)
0
T
= −i
˙ + ρ(0)V † (t)V(t)
˙
dtTr ρ(0)U † (t)U(t)
0
T
= −i
˙ +
dtTr ρ(0)U † (t)U(t)
0
m,n,k
T
= −i
ωm |m m|n e−iθn (t) n|k i
T
˙
dtTr ρ(0)U (t)U(t)
−i
†
0
ωk θ˙k (T ) −
= γD +
k
ωk θ˙k (t)|k k|
dtTr
0
dθk (t) iθk (t)
e
k|
dt
k
ωk θ˙k (0)
(3.22)
k
It is evident that a simple subtraction of γT −γD does not give a gauge invariant geometric
phase expression. After a careful examination of the transformation, we propose the
following functional to be the geometric phase:
ωk k|U † (0)U(T )|k e−
γG = arg
RT
0
k
It is easy to show that this is gauge invariant:
ωk k|U † (0)U (T )|k e−
γG = arg
k
RT
0
dt k|U † (t)U˙ (t)|k
dt k|U † (t)U˙ (t)|k
(3.23)
3.2 The Non-Degenerate Case
25
ωk k|U † (0)V † (0)U † (T )V † (T )|k e−
= arg
RT
0
˙ ]|k
dt k|V † (t)U † (t)[U˙ (t)V(t)+U (t)V(t)
k
ωk k|U † (0)
= arg
n
k
e−
RT
0
e−iθn (0) |n n|U † (T )
eiθm (T ) |m m|k
m
˙
dt k|V † (t)U † (t)U˙ (t)V(t)+V † (t)U † (t)U (t)V(t)|k
ωk k|U † (0)U † (T )|k eiθk (T ) e−iθk (0) e−
= arg
RT
† (t)V(t)|k
˙
˙
dt k|U † (t)U(t)+V
0
k
ωk k|U † (0)U † (T )|k eiθk (T ) e−iθk (0) e−
= arg
RT
0
RT
iθ˙k (t)dt
˙
dt k|U † (t)U(t)|k
e−
˙
dt k|U † (t)U(t)|k
e−iθk (T )+iθk (0)
0
k
ωk k|U † (0)U † (T )|k eiθk (T ) e−iθk (0) e−
= arg
RT
0
k
ωk k|U † (0)U † (T )|k e−
= arg
RT
0
dt k|U † (t)U˙ (t)|k
k
= γG
(3.24)
It is also worth noting that this will reduce to the pure state geometric phase. Indeed,
when
|k
−→ |ψ(0) ,
γG −→ arg
ωk = 1, ωj=k = 0
ψ(0)|U † (0)U † (T )|ψ(0) e−
RT
0
dt ψ(0)|U † (t)U˙ (t)|ψ(0)
Where,
U(0)|ψ(0)
= |ψ(0)
U(T )|ψ(0)
= |ψ(T )
˙
i ψ(0)|U † (t)U(t)|ψ(0)
=
ψ(t)|H(t)|ψ(t)
therefore γG = arg
ψ(0)|ψ(T ) ei
RT
0
dt ψ(t)|H(t)|ψ(t)
T
= arg { ψ(0)|ψ(T ) } + i
dt ψ(t)|H(t)|ψ(t)
(3.25)
0
which is the pure state geometric phase.
Under the stronger parallel transport conditions, the geometric phase, it is simply the
3.3 The Degenerate Case
26
total phase:
ωk k|U † (0)U(T )|k e−
γG = arg
RT
0
dt k|U † (t)U˙ (t)|k
k
ωk k|U † (0)U(t)|k
= arg
k=1
= γT .
3.3
(3.26)
The Degenerate Case
Now, let us move to the case where the density operator has degenerate eigenvalues. We
assume that the density matrix in its diagonal form is:
ω
n
..
.
N
ωn
|k ωk k| =
ρ(0) =
ωn+1
k
..
.
(3.27)
ωN
with n degenerate eigenvalues. The rest of the N − n eigenvalues are assumed to be
distinct. The Hilbert space H can be regarded as the direct sum of two subspaces of
H = Hn ⊕ HN −n . The little group for ρ(0) is:
N −n
Gx = U (n) × U (1) × U (1) · · · × U (1)
or in the matrix form,
V(t) =
α(t)
(3.28)
iβn+1 (t)
e
..
.
eiβN (t)
(3.29)
3.3 The Degenerate Case
27
where (α(t)) ∈ U (n) is a n × n unitary matrix. It is easy to see the orbit of density
matrix ρ(0) is unchanged by V(t):
ρ (0) = V(t)ρ(0)V † (t)
=
α(t)
eiβn+1 (t)
...
×
ωn
ωn
ωn+1
...
eiβN (t)
×
α(t)
ωN
†
e−iβn+1 (t)
..
.
e−iβN (t)
ωn
=
α(t)
×
ωn+1 eiβn+1 (t)
α(t)
...
ωN eiβN (t)
†
e−iβn+1 (t)
..
...
.
e−iβN (t)
3.3 The Degenerate Case
28
=
ωn
...
ωn
ωn+1
...
ωN
= ρ(0)
(3.30)
Hence under a gauge transformation U(t) → U (t) = U(t)V(t),
ρ(t) −→ ρ (t) = U (t)ρ(0)U † (t)
= U(t)V(t)ρ(0)V † U † (t)
= U(t)ρ(0)U † (t)
= ρ(t)
(3.31)
Let us turn back to the total phase term in the non-degenerate case:
γT = arg{Tr(ρ(0)U † (0)U(t))}
(3.32)
Under the gauge transformation, the total phase become:
γT = arg{Tr(ρ(0)U † (0)U (t))}
= arg{Tr(ρ(0)V † (0)U † (0)U(t)V(t))}
= arg{Tr(ρ(0)U † (0)U(t)V(t)V † (0))}
(3.33)
We know, under the parallel transport condition, the geometric phase is equal to the
total phase. Hence, we propose a functional form of
γG = arg{Tr(ρ(0)U † (0)U(t)F(U, t))}
(3.34)
Comparing the total phase formula (3.33) with the propose geometric phase term (3.34),
we note that F(U, t) = V(t)V † (0). In addition, F(U, t) should also satisfy the following
3.3 The Degenerate Case
29
transformation:
F(U, t) −→ F (U , t) = V † (t)F(U, t)V(0)
(3.35)
so that γG is gauge invariant. Indeed, for such a transformation we have,
γG = arg{Tr(ρ(0)U † (0)U (t)F (U, t))}
= arg{Tr(ρ(0)V † (0)U † (0)U(t)V(t)V † (t)F(U, t)V(0))}
= arg{Tr(ρ(0)U † (0)U(t)F(U, t)V(0)V † (0))}
= arg{Tr(ρ(0)U † (0)U(t)F(U, t))}
= γG
(3.36)
Now let us examine V(t) under some parallel transport conditions. The little group is
given by
V(t) =
α(t)
iβn+1 (t)
e
..
.
(3.37)
eiβN (t)
where (α(t)) ∈ U (n) has n2 independent real parameters. Together with the β(t)’s, there
are a total of n2 + N − n independent real parameters.
To fix these arbitrary parameters, we impose the following parallel transport conditions:
µ|U (t)† U˙ (t)|ν = 0
µ, ν = 1, 2, · · · , n
k|U (t)† U˙ (t)|k = 0
k = n + 1, n + 2, · · · , N
(3.38)
which constitute exactly n2 +N −n constraints. Here we have denoted the the degenerate
subspace by kets indexed by greek alphabets {|µ }µ=1,...,n while the non-degenerate space
3.3 The Degenerate Case
30
is characterized by kets indexed by roman alphabets {|k }k=n+1,...,N −n . There are sufficient
conditions to fix all the parameters for V(t). We can now express these parameters in
terms of the elements of the U matrix. To see this, we first simplify the term U † (t)U˙ (t):
d
U † (t)U˙ (t) = V † (t)U † [U(t)V(t)]
dt
†
†
˙
˙
= V (t)U (t) U(t)V(t)
+ U(t)V(t)
˙
˙
= V † (t)U † (t)U(t)V(t)
+ V † (t)V(t)
Then, by invoking the parallel transport conditions, U † (t)U˙ (t) = 0, we obtain
˙
˙
V † (t)V(t)
= −V † (t)U † (t)U(t)V(t)
˙
˙
V(t)
= −U † (t)U(t)V(t)
(3.39)
If we recast above equation into a matrix form, we have:
⇒
˙
˙
V(t)
= −U † (t)U(t)V(t)
α(t)
˙
iβ˙ n+1 eiβn+1
..
.
iβ˙ N eiβN
† ˙
˙
(U U)11 · · · · · · (U U)1N
..
..
..
.
.
.
= −
...
˙ N 1 · · · · · · (U † U)
˙ NN
(U † U)
†
α(t)
eiβn+1
...
eiβN
3.3 The Degenerate Case
= −
31
n
n
˙ 1λ αλ1 · · ·
(U U)
†
λ=1
..
.
n
˙ nλ αλ1
(U † U)
λ=1
˙ 1λ αλn
(U U)
λ=1
..
.
n
† ˙
···
(U U)nλ αλn
†
λ=1
˙ n+1,n+1 eiβn+1
(U † U)
..
.
˙ N N eiβN
(U † U)
Thus, by equating the elements we get the following equations:
n
˙
U † (t)U(t)
α˙ µν (t) = −
λ=1
˙
β˙ k (t) = i U † (t)U(t)
µλ
αλµ (t)
(3.40)
(3.41)
kk
To simplify the ensuing calculations, we use the following notation:
A(t) = (α(t))
˙
Ω(t)µν = (U † (t)U(t))
µν
for µ, ν = 0, 1, · · · , n
(3.42)
˙
It should be noted that the matrix Ω(t) is a sub-matrix of (U † (t)U(t))
with indices
µ, ν = 1, 2, · · · , n. The solution for equation (3.40) is:
A(t) = Q(U, t)A(0)
(3.43)
(3.44)
where
t
i∗t
t
)+1
· · · Ω(t −
)+1
N →∞
N
N
N
t
t
(i + 1) ∗ t
)+1
· · · (Ω(0) + 1)
Ω(t −
.
N
N
N
Q(U, t) = − lim
Ω(t −
t
N
(3.45)
In these variables, the equation in question is
n
˙ µν (U, t) = −
Q
˙
U † (t)U(t)
λ=1
µλ
Qλν (U, t)
(3.46)
3.3 The Degenerate Case
32
with the initial condition,
Q(U, 0) = 1
(3.47)
The formal solution of Q(U, t) reads as
t
Q(U, t) = P exp{−
dt Ω(t )}
(3.48)
0
where the prefix P denotes path ordering. The solution for equation (3.41) is:
t
βk (t) = i
˙ )|k + βk (0).
dt k|U † (t )U(t
0
(3.49)
If we denote
t
θk (U, t) = i
˙ )|k
dt k|U † (t )U(t
0
(3.50)
then
βk (t) = θk (U, t) + βk (0).
Hence, we obtain V(t) as:
V(t) =
α(t)
eiβn+1 (t)
..
.
eiβN (t)
A(t)
ei(θn+1 (U ,t)+βn+1 (0))
=
..
.
(3.51)
ei(θN (U ,t)+βN (0))
3.3 The Degenerate Case
33
Q(U, t)A(0)
=
e
e
..
.
Q(U, t)
V(t) =
e
iθn+1 (U ,t)
..
.
eiθN (U ,t)
Q(U, t)
=
iθn+1 (U ,t) iβn+1 (0)
.
eiβN (0)
V(0)
eiθn+1 (U ,t)
..
eiθN (U ,t) eiβN (0)
A(0)
eiβn+1 (0)
..
.
(3.52)
eiθN (U ,t)
Now, recall the F(U, T ) = V(t)V † (0), therefore:
F(U, t) = V(t)V † (0)
Q(U, t)
=
e
...
eiθN (U ,t)
Q(U, t)
=
V(0)V † (0)
iθn+1 (U ,t)
eiθn+1 (U ,t)
..
.
(3.53)
eiθN (U ,t)
In the following, we will show that under a gauge transformation F(U, t) transforms in
a manner that renders the geometric phase gauge invariant.
3.3 The Degenerate Case
3.3.1
34
Effect of Gauge Transformations
In this section, we will show that F(U, t) transforms as
F(U, T ) −→ F (U , T ) = V † (T )F(U, T )V(0).
(3.54)
Recall that,
V(t) =
α(t)
..
.
eiβN (t)
Q(U, t)
F(U, t) =
eiβn+1 (t)
e
iθn+1 (U ,t)
..
.
(3.55)
eiθN (U ,t)
From the Bloch-diagonal form of F(U, t), it is instructive to break up the proof into parts:
First part:
Q(U, T ) −→ Q (U , T ) =
V † (T ) Q(U, T ) [V(0)]
(3.56)
where V † (T ) and [V(0)] denote the submatrices (with indices running from 0 to n ) of
V † (T ) and V † (0) respectively.
Second part:
θk (U, T ) −→ θk (U , T ) = −βk (T ) + θk (U, T ) + βk (0)
(3.57)
For the first part, we know that
n
˙ µν (U, t) = −
Q
˙
U † (t)U(t)
i=1
µi
Qiν (U, t)
(3.58)
3.3 The Degenerate Case
35
and for the gauge-transformed counterpart,
n
˙ (U , t) = −
Q
µν
U † (t)U˙ (t)
i=1
µi
Qiν (U , t)
(3.59)
By recasting the above equation into a matrix form, (and noting that we are only working
in the subspace), we have
dQ (U
dt
dQ (U
dt
dQ (U
dt
dQ (U
dt
, t)
, t)
, t)
, t)
= − U † (t)U˙ (t) Q (U , t)
= − V † (t)U †
d
(U(t)V(t)) Q (U , t)
dt
= − V † (t)U † (t)
˙
˙
U(t)V(t)
+ U(t)V(t)
Q (U , t)
˙
˙
= − V † (t)U † (t)U(t)V(t)
+ V † (t)V(t)
Q (U , t)
dQ (U , t)
˙
˙
+ V † (t)V(t)
Q (U , t) = − V † (t) U † (t)U(t)
[V(t)] Q (U , t)
dt
dQ (U , t) d [V(t)]
˙
[V(t)]
+
Q (U , t) = − U † (t)U(t)
[V(t)] Q (U , t)
dt
dt
d ([V(t)] Q (U , t))
˙
= − U † (t)U(t)
[V(t)] Q (U , t)
dt
(3.60)
Comparing equation (3.58) with (3.60), we note that both Q(U, t) and [V(t)] Q (U , t)
satisfy the same differential equation, hence,
[V(t)] Q (U , t) = Q(U, t)C
(3.61)
where C is a time-independent non-singular n×n matrix. We can fix C by considering the
initial condition. Recall from (3.47) that Q(U, 0) = 1. If we also require that Q (U , 0) = 1
at time zero then
C = [V(0)]
(3.62)
Hence,
[V(t)] Q (U , t) = Q(U, t) [V(0)]
V † (t) Q(U, t) [V(0)]
Q (U , t) =
For the second term, we have
t
θk (U, t) = i
0
˙ )|k
dt k|U † (t )U(t
(3.63)
3.3 The Degenerate Case
36
and
t
θk (U , t) = i
dt k|U † (t )U˙ (t )|k
0
t
= i
dt k|V † (t )U †
0
t
= i
d
[U(t )V(t )] |k
dt
˙ )V(t ) + U(t )V(t
˙ ) |k
dt k|V † (t )U † (t ) U(t
0
t
= i
˙ )V(t ) + V † (t )V(t
˙ )|k
dt k|V † (t )U † (t )U(t
0
t
= i
dt k|U † (t )U˙ (t )|k
0
N −n
t
= i
dt
k|
0
N −n
|l e
−iβl (t )
N −n
|l e−iβl (t ) l|
+ k|
0
t
=
|m i
m=1
l=1
t
m|eiβm (t ) m|k
m=1
l=1
N −n
= i
l|U U˙
†
dβm (t ) iβm (t )
e
m|k
dt
t
˙ )|k + i
dt k|U (t )U(t
†
dt i
0
dβk (t )
dt
˙ )|k − βk (t) + βk (0)
dt k|U † (t )U(t
0
= θk (U, t) − βk (t) + βk (0)
(3.64)
Thus,
Q(U, T ) −→ Q (U , T ) =
V † (T ) Q(U, T ) [V(0)]
θk (U, T ) −→ θk (U , T ) = −βk (T ) + θk (U, T ) + βk (0).
(3.65)
F(U, T ) −→ F (U , T ) = V † (T )F(U, T )V(0)
(3.66)
Therefore,
which furnishes the transformation property of F(U, T ).
Chapter 4
Kinematical Approach to Geometric
Phase for Open Systems
In this section, we extend the kinematic approach to the geometric phase, developed in
the previous chapters, to systems that are open. Here the system that will be of interest
will be coupled to an environment, with the combined system undergoing some unitary
evolution. In the first part, some salient features of open systems will be discussed.
This will also serve as an introduction to the notation that will be used in the rest of this
thesis. Qubit systems undergoing non-unitary evolutions, namely through the Amplitude
damping channel and the phase damping channel will be discussed as examples.
4.0.2
Brief Introduction to the Open System
Here we will denote the state space of the system as Hs and that of the environment as
He . The state space of the total space can be taken as Hse = Hs ⊗ He and we denote the
unitary evolution in Hse as U : Hse → Hse . Denoting the initial state of the combined
system as
ρ¯se (0) = ρs (0) ⊗ |0e 0e |
(4.1)
38
where ρs (0) is the initial state of the system and |0e 0e | is the initial state of environment,
the total state will evolve as:
ρ¯se (0) −→ ρse (t) = U (t)¯
ρse (0)U † (t).
(4.2)
For the subsystem we have,
ρs (t) = Tre U (t)(ρs (0) ⊗ |0e 0e |)U † (t)
L
µe | U (t)(ρs (0) ⊗ |0e 0e |)U † (t) |µe
=
µ=0
L
µe |U (t)|0e ρs (0) 0e |U † (t)|µe
=
µ=0
L
Mµ (t)ρs (0)Mµ† (t)
=
(4.3)
µ=0
where {Mµ }, µ = 1, 2, . . . L are the Krauss operators,
Mµ (t) =
µe |U (t)|0e .
(4.4)
Here we have assumed that dim(Hs ) = N and dim(Hs )= L.
The Krauss operators satisfy the completness relation:
Mµ† (t)Mµ (t) = 1.
(4.5)
µ
It is worth noting that in a basis {|is ⊗ |µe } of Hs ⊗ He , Mµ (t) can be represented by
(Mµ (t))ij =
4.0.3
is , µe |U |js , 0e
(4.6)
Amplitude Damping and Phase Damping Channels
Amplitude Damping Channel
The Amplitude Damping is often used in describing systems that undergo spontaneously
emissions. Let us consider a qubit system dim(Hs )=2 that is coupled to an environment
39
also of dimension 2 (i.e. dim(He )=2). We can denote the ground state of the system as
|0s and its excited state as |1s . Now if the qubit is in |0s state, then nothing happens.
On the other hand, if the qubit is in state |1s , then we assume that there is a probability
p that it will decay to state |0s by emitting a photon. This puts the environment in
state |1e from its original |0e state. Hence we have
U |0s , 0e
= |0s , 0e
U |1s , 0e
=
1 − p|1s , 0e +
√
p|0s , 1e
It is clear that there are two Kraus operators if we assume the environment to be two
dimensional. From equation (4.6) we have,
M0 =
1
0
0
√
1−p
√
0
p
=
0 0
M1
If we put p = 1 − e−t , then as t → ∞, p → 1. In this case, for a system decribed by a
Bloch vector (rx , ry , rz ), where
1
ρs = (1 + r · σ)
2
an initial state with Bloch vector (rx , ry , rz ) = (0, 1, 0) will tansform to (rx , ry , rz ) →
(0, 0, 1) as t → ∞ (see Figure (4.1)).
Phase Damping Channel
Another interesting channel often used in describing decohering systems is the phase
damping channel. This channel models the noise process for which there is loss of quantum
information without loss of energy. The Kraus operators for this channel are given by,
1
0
M0 =
√
0
1−p
4.1 Geometric Phase for Open Systems
40
Figure 4.1: The Bloch vector in an Amplitude Damping process.
M1 =
0
0
.
√
0
p
With p = 1 − e−t , a state with Bloch vector (rx , ry , rz ) = (0, 1, 0), transforms to a state
with Bloch vector (rx , ry , rz ) → (0, 0, 0) as t → ∞ ( see Figure (4.2)).
4.1
4.1.1
Geometric Phase for Open Systems
Gauge Freedom of the Environment
To see what the symmetry group is for a system undergoing non-unitary evolution, we
first note that there is an equivalent set of Krauss operators, {Mµ },
Mµ −→ Mµ =
Wµν Mν
ν
(4.7)
4.1 Geometric Phase for Open Systems
41
Figure 4.2: The block vector in a phase damping process.
where {Wµν } are elements of an L × L unitary matrix W , that leave the system invariant:
L
Mµ (t)ρs (0)Mµ† (t)
ρs (t) −→ ρs (t) =
µ=0
∗
Wµi (t)Mi (t)ρs (0)Mj† (t)Wµj
(t)
=
µ,i,j
Mi (t)ρs (0)Mj† (t)
=
∗
Wµj
(t)Wµi (t)
µ
i,j
Mi (t)ρs (0)Mi† (t)
=
i
= ρs (t).
(4.8)
It is interesting to see what the corresponding transformation is for the unitary operator
on the combined space Hse = Hs ⊗ He . By noting that (see eqn. (4.6))
U |is ⊗ |0e
=
Mµ |is ⊗ |µe
µ
we have
U |is ⊗ |0e
=
Mµ |is ⊗ |µe
µ
(4.9)
4.1 Geometric Phase for Open Systems
=
42
Wµν Mν |is ⊗ |µe
µν
=
Mν |is ⊗
ν
=
Wµν |µe
µ
Mν |is ⊗ W |νe
ν
= (1 ⊗ W )
Mν |is ⊗ |νe
ν
= (1 ⊗ W )U |is ⊗ |0e
(4.10)
from which we surmise that
U (t) −→ U (t) = (1 ⊗ W (t))U (t).
(4.11)
Equation (4.11) reflects the symmetry of the environment in the combined space.
4.1.2
Gauge Freedom Of the System
In the previous section we examined only the symmetry due to the environment. Here we
consider the full symmetry of the combined system; i.e we include the symmetry as well.
Now in a unitary evolution case, the eigenvalues of the density matrix will not change.
For example if the degeneracy of a density matrix is initially two, then it will remain two
through out the evolution. This has implications for the symmetry group of the system.
Indeed, the gauge group depends on the little group that preserves the density matrix
at some time t. In the case of non-unitary evolutions, this is no longer true. A system
starting with no degeneracy may end up with degenerate eigenvalues, with a gauge group
that has essentially changed. For instance for a qutrit (3-state) system, a non-degenerate
density matrix will have U (1) × U (1) × U (1) as its little group. Over time, this may
evolve into a state with two of its eigenvalues degenerate, in which case, the little group
becomes U (2) × U (1). We call this level crossings. However, here, we are going to restrict
our attention to cases where this does not occur. In particular for the two channels that
we consider, this problem does not occur.
4.1 Geometric Phase for Open Systems
43
We begin with a density operator of the system given by
ρs (t) = R(t)ρsD (t)R† (t)
where ρsD (t) denotes the diagonal matrix in a time-independent basis {|ks
(4.12)
k=0,1,...,N −1 }
of
Hs and R(t) ∈ U (N ) is some unitary matrix that diagonalizes the density operator. For
a non-degenerate state ρ(t), this decomposition is unique with:
ρsD (t) =
ωk (t)|ks ks |
(4.13)
and
ωk (t)|ks ks | R† (t)
ρs (t) = R(t)
k
ωk (t)|ks (t) ks (t)|
ρs (t) =
(4.14)
k
where {ωk (t)}k=0,1,...N −1 are the eigenvalues (which we assume are non-degenerate) and
{|ks (t) = R(t)|ks }k=0,...N −1 are the corresponding time-dependent ket vectors that are
unitarily related to the time-independent ones.
Suppose the little group element associated with ρs (0) is Ξ(t),
Ξ(t)ρs (0)Ξ† (t) = ρs (0)
(4.15)
then it is not difficult to see that
S(t) = R(t)Ξ(t)R† (t)
(4.16)
is the little group element for ρs (t). Indeed, we have
ρs (t) −→ ρs (t) = S(t)ρs (t)S † (t)
= R(t)Ξ(t)R† (t)R(t)ρsd (t)R† (t)R(t)Ξ(t)R† (t)
= R(t)Ξ(t)ρsd (t)Ξ(t)R† (t)
= R(t)ρsD (t)R† (t)
= ρs (t).
(4.17)
4.1 Geometric Phase for Open Systems
44
Following the unitary evolution case in the previous chapter, the little group Ξ(t) can be
written as
N −1
eiθk (t) |ks ks |
Ξ(t) =
(4.18)
k=0
with the correspponding element S(t) given by
S(t) = R(t)Ξ(t)R† (t)
N −1
eiθk (t) |ks ks | R† (t)
= R(t)
k=0
N −1
eiθk (t) |ks (t) ks (t)|
=
(4.19)
k=0
It is easy to see that in the full space Hs ⊗ He space, this symmetry can be represented
as,
U (t) −→ U (t) = (S(t) ⊗ 1)U (t).
4.1.3
(4.20)
The Total Gauge Group
We are now ready to consider the full gauge group. From the above discussions, it is
clear that the system and the environment remain invariant under the following transformations:
For the environment
U (t) −→ U (t) = (1 ⊗ W (t))U (t)
For the system
U (t) −→ U (t) = (S(t) ⊗ 1)U (t).
Thus, the total gauge freedom can be written as:
U (t) −→ U (t) = (S(t) ⊗ W (t))U (t)
where an element of the gauge group can be expressed as
V (t) = S(t) ⊗ W (t)
(4.21)
4.1 Geometric Phase for Open Systems
=
45
R(t)Ξ(t)R† (t) ⊗ W (t)
= [R(t) ⊗ 1] [Ξ(t) ⊗ W (t)] [R(t) ⊗ 1]†
= [R(t) ⊗ 1] Ω(t) [R ⊗ 1]† (t).
(4.22)
Here we have introduced an element
N −1
Ω(t) = Ξ(t) ⊗ W (t) =
M −1
|ks e
iθk (t)
ks | ⊗
|µe Wµν νe |
µ,ν=0
k=0
N −1 M −1
|ks , µe eiθk (t) Wµν ks , νe |
=
k=0 µ,ν=0
which essentially parametrizes an element of the gauge group of the combined system.
Here we have assumed that dim(He )=M . Writing,
N −1 M −1
|ks , µe Ωkµν ks , νe |
Ω(t) =
(4.23)
k=0 µ,ν=0
with
Ωkµν = eiθk (t) Wµν
(4.24)
the matrix form for the element the gauge group is
iθ0 (t)
[W ]
e
eiθ1 (t) [W ]
Ω(t) =
...
(4.25)
eiθN −1 (t) [W ]
which is a Bloch diagonal form. Here we can regard the matrix W as an element of
SU (M ) instead of U (M ) since the phase factor can be absorbed into the θ’s which are
themselves arbitrary.
4.1.4
Geometric Phase Expression
The approach for evaluating the geometric phase here, will be very similar to the unitary
evolution case. Since for the combined system (system + environment), the evolution is
4.1 Geometric Phase for Open Systems
46
unitary, the total phase term is simply,
γT = arg{Tr(¯
ρse (0)U † (0)U (t))}
(4.26)
where the trace is taken over both the system and the environment. Now, under a gauge
transformation U → U = V U , the total phase will transform as
ρse (0)U † (0)U (t))}
γT −→ γT = arg{Tr(¯
= arg{Tr(¯
ρse (0)U † (0)V † (0)V (t)U (t))}
(4.27)
which essentially is not gauge invariant. Following the unitary case, we propose the
following form for the geometric phase:
γG = arg{Tr(¯
ρse (0)U † (0)F(U, t)U (t))}
(4.28)
with the functional F(U, t) satisfying:
F(U, t) −→ F (U , t) = V † (0)F(U, t)V (t)
(4.29)
so that γG is rendered gauge invariant. It is interesting to note that, similar to the
unitary case, the geometric phase and the total phase can be identified when the some
U (t) = V (t)U (t) is chosen such that it satisfies the parallel transport conditions. These
conditions basically fix the free parameters in V (t). The latter then has a functional
dependence on U (t) and the term V † (0)V (t) in eqn.(4.27) can be identified with the
functional F(U, t) in (4.28). It will be shown later that this transforms as (4.29).
In the present case, there are altogether N free real number parameters in Ξ(t) and
M 2 −1 free real number parameters in W (t). In total, there are M 2 +N −1 free parameters
to be fixed. First, let us define U = V U . Then we have
U˙
⇒
⇒
= V˙ U + V U˙
U˙ U † = (V˙ U + V U˙ )U † V † = V˙ V † + V U˙ U † V †
iU˙ U † = iV˙ V † + V (iU˙ U † )V †
(4.30)
4.1 Geometric Phase for Open Systems
47
With
V (t) = S(t) ⊗ W (t)
we obtain,
V˙
⇒
˙
= S˙ ⊗ W + S ⊗ W
˙ †⊗1+1⊗W
˙ W†
V˙ V † = SS
˙ † ⊗ 1 + 1 ⊗ iW
˙ W†
iV˙ V † = iSS
⇒
(4.31)
Next, we perform a subtrace over the environmental degrees of freedom in equation (4.30):
µ|iU˙ U † |µ
˙ † ⊗ (tre 1) + 1 · tre [iW
˙ W †] +
= iSS
µ
µ|V (iU˙ U † )V † |µ . (4.32)
µ
˙ W † is a generator of SU (M ), the trace term on the right handside in
By noting that iW
eqn. (4.32) drops out. If we further impose
µ|iU˙ U † |µ
= 0
µ
then eqn. (4.32) reduces to
˙ † = −
iM SS
µ|V (iU˙ U † )V † |µ .
(4.33)
µ
To simplify the expression, we introduce A = iU˙ U † and write
˙ † = −
iM SS
µ|V AV † |µ
µ
Since
eiθk (t) |k k|R(t)†
S(t) = R(t)
k
we have, with T (t) =
k
eiθk (t) |k k|,
S(t) = R(t)T (t)R(t)†
⇒
˙ R† + RT˙ R† + RT R˙ †
S˙ = RT
(4.34)
4.1 Geometric Phase for Open Systems
⇒
48
˙ † = (RT
˙ R† + RT˙ R† + RT R˙ † )RT † R†
SS
˙ † + RT˙ T † R† − (RT )R† R(RT
˙
= RR
)†
˙ † + RT˙ T † R† − (RT R† )RR
˙ † (RT R† )†
= RR
⇒
˙ † = iRR
˙ † + R(iT˙ T † )R† − (RT R† )iRR
˙ † (RT R† )†
iSS
(4.35)
Next, we contract both sides eqn. (4.35) by the ket |k(t) :
˙ † |k(t)
k(t)|iSS
˙ † |k(t) + k(t)|R(iT˙ T † )R† |k(t)
k(t)|iRR
=
˙ † (RT R† )† |k(t) .
− k(t)|(RT R† )iRR
(4.36)
Let us analyze each term independently. The 1st term on the RHS gives,
˙ † |k(t)
k(t)|iRR
˙ † R|k
= i k|R† RR
˙
= i k|R† R|k
where we have used |k(t) = R(t)|k . The 2nd term on the RHS gives,
k(t)|R(iT˙ T † )R† |k(t)
= i k|T˙ T † |k
= −θ˙k .
(4.37)
Finally, the 3rd term on the RHS gives,
˙ † (RT R† )† |k(t)
− k(t)|(RT R† )iRR
˙ † RT † R† R|k
= −i k|R† RT R† RR
˙ † |k
= −i k|T R† RT
eiθm |m m| R† R˙
= −i k|
m
= −i
e
i(θm −θn )
e−iθn |n n| |k
n
˙
k|m m|R R|n
n|k
†
m,n
˙
= −i k|R† R|k
(4.38)
Hence, we obtain
˙ † |k(t)
k(t)|iSS
˙
˙
= i k|R† R|k
− θ˙k − i k|R† R|k
= −θ˙k .
(4.39)
4.1 Geometric Phase for Open Systems
49
Contracting both sides of eqn.(4.33) with |k(t) and noting the above relation, we obtain
M θ˙k =
k(t), µ|V AV † |k(t), µ
(4.40)
µ
where A = iU˙ U † . Let us expand the RHS of the equation,
k(t), µ|V AV † |k(t), µ
µ
k(t), µ|(S ⊗ W )A(S † ⊗ W † )|k(t), µ
=
µ
k, µ|(R† RT R† ⊗ W )A(RT † R† R ⊗ W † )|k, µ
=
µ
=
eiθm |m m|R†
k, µ|
µ
⊗
m
e−iθn |n n|
R
Wα∗2 β2 |β2 α2 |
⊗
n
=
Wα1 β1 |α1 β1 |
A
α1 ,β1
|k, µ
α2 ,β2
e
(θm −θn )
Wα1 ,β1 Wα∗2 ,β2 k, µ|m, α1 m(t), β1 |A|n(t), β2 n, α2 |k, µ
µ,m,n,α1 ,β1 ,α2 ,β2
∗
Wµ,β1 Wµ,β
k(t), β1 |A|k(t), β2
2
=
µ,β1 ,β2
Recall that W ∈ SU (M ), that is
µ
∗
Wµ,β1 Wµ,β
= δβ1 β2 . This reduces eqn. (4.40) to
2
M θ˙k =
k(t), µ|V AV † |k(t), µ
µ
=
k(t), β1 |A|k(t), β1
β1
=
k(t)|tre A|k(t) .
(4.41)
Integrating both sides with respect to t, we obtain
1
θk (τ ) − θk (0) =
M
τ
dt k(t)|tre A|k(t)
(4.42)
0
Next, let us evaluate W . From equation (4.30) and (4.31), we have,
˙ † ⊗ 1 + 1 ⊗ iW
˙ W † + V AV † .
iU˙ U † = iSS
(4.43)
If we impose
k(t), α|iU˙ U † |k(t), β
= 0
(4.44)
4.1 Geometric Phase for Open Systems
50
for a particular k and any α, β, these additional parallel transport conditions imply that
˙ † ⊗ 1|k(t), β + α|iW
˙ W † |β + k(t), α|V AV † |k(t), β
k(t), α|iSS
= 0
The 1st term on the LHS is,
˙ † ⊗ 1|k(t), β
k(t), α|iSS
˙ † |k(t) δαβ
k(t)|iSS
=
= −θ˙k δαβ .
(4.45)
The 2nd term on the LHS is,
˙ W † |β
α|iW
˙ αγ W ∗
iW
βγ
=
γ
˙ W†
= i W
(4.46)
αβ
The 3rd term on the LHS is,
k(t), α|V AV † |k(t), β
=
k(t), α|(S ⊗ W )A(S † ⊗ W † )|k(t), β
=
k, α|(R† RT R† ⊗ W )A(RT † R† R ⊗ W † )|k, β
eiθm |m m|R†
k, α|
=
⊗
m
µ
e−iθn |n n|
R
n
A
Wα∗2 β2 |β2 α2 |
⊗
|k, β
α2 ,β2
(θm −θn )
=
Wα1 β1 |α1 β1 |
α1 ,β1
e
Wα1 ,β1 Wα∗2 ,β2 k, α|m, α1 m(t), β1 |A|n(t), β2 n, α2 |k, β
m,n,α1 ,β1 ,α2 ,β2
∗
Wα,β1 Wβ,β
k(t), β1 |A|k(t), β2 .
2
=
(4.47)
β1 ,β2
Defining
Ak =
k(t)|A|k(t)
and writing
k(t), β1 |A|k(t), β2
= Akβ1 β2
(4.48)
4.1 Geometric Phase for Open Systems
51
the third term (4.47) reduces to
k(t), α|V AV † |k(t), β
∗
Wα,β1 Wβ,β
Akβ1 β2
2
=
β1 ,β2
=
W Ak W †
αβ
.
(4.49)
Putting all the terms together, we obtain,
˙ W†
−θ˙k δαβ + i W
αβ
+ W Ak W †
˙ W†
⇒ −i W
αβ
αβ
= 0
=
W (Ak − θ˙k 1)W †
αβ
˙ = iW (A − θ˙k 1).
W
k
⇒
To simplify this somewhat, it is instructive to define B k = Ak − θ˙k 1, with
˙ = iW B k .
W
This can be integrated formally:
τ
W (t) = W (0)P exp i
B k (t)dt
(4.50)
0
where the integral is a t-ordered integral. With the θ(t) and W (t) obtained, we can write,
V (τ ) = S(τ ) ⊗ W (τ )
eiθn (τ ) |n(t) n(τ )| ⊗ W (τ )
=
n
i
e(iθn (0)+ M
=
Rτ
0
dt n(t)|tre A|n(t)
τ
) |n(τ ) n(τ )| ⊗ W (0)P exp i
B k (t)dt
0
n
V (0)
eiθm (0) |m(τ ) m(τ )| ⊗ W (0)
=
m
i
eM
Rτ
0
dt n(t)|tre A|n(t)
τ
|n(τ ) n(τ )| ⊗ P exp i
B k (t)dt
0
n
= V (0)
e
n
i
M
Rτ
0
dt n(t)|tre A|n(t)
τ
|n(τ ) n(τ )| ⊗ P exp i
0
B k (t)dt
(4.51)
4.1 Geometric Phase for Open Systems
52
Hence, we have,
Fk (U, τ ) = V (0)† V (τ )
i
=
eM
Rτ
0
dt n(t)|tre A|n(t)
τ
|n(τ ) n(τ )| ⊗ P exp i
(4.52)
0
n
4.1.5
B k (t)dt
Proof of Gauge Invariance
Here we prove that the the geometric phase is indeed gauge invariant. To begin, we first
examine the manner with which the functional,
Fk (U, τ ) =
e
i
M
Rτ
0
dt m(t)|tre A|m(t)
τ
|m(τ ) m(τ )| ⊗ P exp i
B k (t)dt
0
m
where
A = iU˙ U †
Bk =
k(t)|A|k(t) −
1
k(t)|tre A|k(t) 1
M
i
transforms under a gauge transformation. First, let’s see how e M
transforms. Effecting
U (t) −→ Ω(t)U (t) ≡ U (t)
where
eiψn (t) |n(t) n(t)| ⊗ β
Ω(t) =
n
β ∈ SU (M )
we note that A = iU˙ U † transforms as,
A
= iU˙ U
†
d
[ΩU ](ΩU )†
dt
˙ + ΩU˙ )U † Ω†
= i(ΩU
= i
˙ † + iΩU˙ U † Ω†
= iΩΩ
˙ †.
= ΩAΩ† + iΩΩ
Rτ
0
dt m(t)|tre A|m(t)
|m(τ ) m(τ )|
4.1 Geometric Phase for Open Systems
53
It follows then that
m(t)|tre A |m(t)
˙ † )|m(t)
m(t)|tre (ΩAΩ† + iΩΩ
=
˙ † |m(t), µ
m(t), µ|ΩAΩ† + iΩΩ
=
µ
˙ † |m(t), µ .
i m(t), µ|ΩΩ
m(t), µ|ΩAΩ† |m(t), µ +
=
µ
µ
Now, the first term can be simplified as,
m(t), µ|ΩAΩ† |m(t), µ
µ
m(t), µ|{eiψn (t) |n1 (t) n1 (t)| ⊗ βλ1 λ2 |λ1 λ2 |Aλn32λn43 |n2 (t) n3 (t)| ⊗ |λ3 λ4 |
=
µ
e
−iψn4 (t)
|n4 (t) n4 (t)| ⊗ βλ∗5 λ6 |λ6 λ5 |}|m(t), µ
ei(ψn1 −ψn4 ) βλ1 λ2 Anλ32λn43 βλ∗5 λ6 m(t)|n1 (t) n1 (t)|n2 (t) n3 (t)|n4 (t) n4 (t)|m(t)
=
µ|λ1 λ2 |λ3 λ4 |λ6 λ5 |µ
∗
= βµλ2 Amm
λ2 λ4 βµλ4
∗
= βµλ2 βµλ
Amm
λ2 λ4
4
= Amm
λ2 λ4 ≡
m(t), µ|A|m(t), µ
µ
=
m(t)|tre A|m(t)
For the second term we first evaluate,
Ω = RT R† ⊗ β
˙ R† ⊗ β + RT˙ R† ⊗ β + RT R˙ † ⊗ β + RT R† ⊗ β˙
Ω˙ = RT
˙
˙ † = [RT
˙ R† ⊗ β + RT˙ R† ⊗ β + RT R˙ † ⊗ β + RT R† ⊗ β][RT
ΩΩ
R† ⊗ β]
˙ † ⊗ 1 + RT˙ T † R† ⊗ 1 + RT R˙ † RT † R† ⊗ 1
= RR
which, upon substituting in the second term yields,
˙ † |m(t), µ
m(t), µ|ΩΩ
i
µ
(4.53)
4.1 Geometric Phase for Open Systems
54
(2a)
(2b)
˙ † ⊗ 1|m(t), µ + i
m(t), µ|RR
= i
µ
m(t), µ|RT˙ T † R† ⊗ 1|m(t), µ +
µ
(2c)
(2d)
˙ † |m(t), µ .
m(t), µ|1 ⊗ ββ
m(t), µ|RT R˙ † RT † R† ⊗ 1|m(t), µ + i
i
µ
µ
Let us consider the four terms (2a)-(2d) separately:
2a-term:
˙ † ⊗ 1|m(t), µ
m(t), µ|RR
i
µ
˙
= i m|R† R|m
M
2b-term:
m(t), µ|RT˙ T † R† ⊗ 1|m(t), µ
i
µ
= i m|T˙ T † |m M
2c-term:
m(t), µ|RT R˙ † RT † R† ⊗ 1|m(t), µ
i
µ
= i m|T R˙ † RT † |m M
m|eiψn1 |n1 n1 |R˙ † Re−iψn2 |n2 n2 |m M
= i
n1 ,n2
i(ψn1 −ψn2 )
= e
m|n1 n1 |R˙ † R|n2 n2 |m M
= i m|R˙ † R|m M
=
˙
−i m|R† R|m
M
(cancels with the 2a-term)
2d-term:
˙ † |m(t), µ
m(t), µ|1 ⊗ ββ
i
µ
˙ † |µ
µ|iββ
=
µ
= 0
4.1 Geometric Phase for Open Systems
55
since generators of SU (M ) are traceless. Thus, the second term reduces to
˙ † |m(t), µ
m(t), µ|ΩΩ
i
µ
= i m|T˙ T † |m M
iψ˙ n1 eiψn1 m|n1 n1 |e−iψn2 |n2 n2 |m M
= i
n1 ,n2
= −ψ˙ m M.
Hence
m(t)|tre A |m(t)
˙ † |m(t), µ
m(t), µ|ΩAΩ† |m(t), µ + i m(t), µ|ΩΩ
=
µ
=
m(t)|tre A|m(t) − ψ˙m M.
Substituting into
i
Rτ
i
Rτ
i
Rτ
eM
= eM
= eM
0
0
0
m(t)|tre A |m(t) dt
i
m(t)|tre A|m(t) dt − M
e
Rτ
0
ψ˙ m (t)M dt
m(t)|tre A|m(t) dt −iψm (τ ) iψm (0)
e
i
= eiψm (0) [e− M
Rτ
0
e
m(t)|tre A|m(t)
]e−iψm (τ )
we have
i
Rτ
i
Rτ
eM
m
=
eM
0
0
m(t)|tre A |m(t)
m(t)|tre A|m(t)
|m(τ ) m(τ )|
eiψm (0) e−iψm (τ ) |m(τ ) m(τ )|
m
i
eiψn1 (0) |n1 (τ ) n1 (τ )|
=
eM
n1
Rτ
0
n2 (t)|tre A|n2 (t)
n2
e−iψn3 (τ ) |n3 (τ ) n3 (τ )| .
n3
Next we consider P exp i
Bk =
τ
0
Bk (t)dt. Now,
k(t)|A |k(t) −
1
k(t)|tre A |k(t) 1
M
|n2 (τ ) n2 (τ )|
4.1 Geometric Phase for Open Systems
=
=
56
1
{ k(t)|tre A|k(t) − ψ˙ k M }1
M
1
k(t)|A |k(t) −
k(t)|tre A|k(t) 1 − ψ˙ k 1
M
k(t)A |k(t) | −
First-term:
˙ † , we have,
From (4.53) A = ΩAΩ† + iΩΩ
k(t)|A |k(t)
=
˙ † |k(t)
k(t)|ΩAΩ† + iΩΩ
(∗∗)
(∗)
=
˙ † |k(t)
k(t)|ΩAΩ† |k(t) + i k(t)|ΩΩ
Consider (*):
k(t)|eiψn1 (t) |n1 (t) n1 (t)| ⊗ βλ1 λ2 |λ1 λ2 |Aλn32λn43 |n2 (t) n3 (t)| ⊗
|λ3 λ4 |e−iψn4 |n4 (t) n4 (t)| ⊗ βλ∗5 λ6 |λ6 λ5 |k(t)
= ei(ψn1 −ψn4 ) k(t)|n1 (t) n1 (t)|n2 (t) n3 (t)|n4 (t) n4 (t)|kt
βλ1 λ2 Anλ32λn43 βλ∗5 λ6 |λ1 λ2 |λ3 λ4 |λ6 λ5 |
∗
= βλ1 λ2 Akk
λ2 λ4 βλ5 λ4 |λ1 λ5 |
= (β k(t)|A|k(t) β † )λ1 λ5 |λ1 λ5 |
= β k(t)|A|k(t) β †
(**):
˙ † |k(t) ), we note that
To simplify the term (i k(t)|ΩΩ
˙ †.
˙ † = RR
˙ † ⊗ 1 + RT˙ T † R† ⊗ 1 + RT R˙ † RT † R† ⊗ 1 + 1 ⊗ ββ
ΩΩ
Then,
˙ † |k(t)
i k(t)|ΩΩ
(3a)
˙ † |k(t) 1
= i k(t)|RR
(3b)
+ i k(t)|RT˙ T † R† |k(t) 1
4.1 Geometric Phase for Open Systems
57
(3c)
+ i k(t)|R T R˙ † RT † R† |k(t) 1
(3d)
˙ †
+ i k(t)|k(t) ββ
where the terms (3a) - (3d) are given by
˙ 1
(3a) = i k|R† R|k
(3b) = i k|T˙ T † |k 1 = −ψ˙k 1
˙ 1 = −(3a)
(3c) = i k|T R˙ † RT |k = −i k|R† R|k
˙ †.
(3d) = iββ
Hence
˙ † − 1 k(t)|tre A|k(t) 1 + ψ˙ k 1
Bk = β k(t)|A|k(t) β † − ψ˙ k 1 + iββ
M
†
†
˙
= βBk β + iββ
which shows that Bk transforms as a gauge-field!
It is easy to show that (see Appendix)
τ
P exp i
τ
Bk dt −→ P exp i
0
Bk dt
0
=
τ
β(0) exp i
Bk dt β † (τ ).
0
Thus, combining the results we have,
Fk (U, τ ) −→ Fk (ΩU, τ )
i
=
eM
Rτ
0
m(t)|tre A |m(t) dt
τ
|m(t) m(t)| ⊗ P exp i
0
m
i
e−iψm (τ ) eiψm (0) e M
=
Rτ
0
m(t)|tre A|m(t) dt
Bk dt
τ
|m(t) m(t)| ⊗ β(0) P exp i
0
m
Ω† (τ )
Ω(0)
=
eiψn (0) |n(τ ) n(τ )| ⊗ β(0)] Fk (U, τ ) [
[
n
=
Bk dt β † (τ )
eiψm (τ ) |m(τ ) m(τ )| ⊗ β(τ )]†
m
†
Ω(0)Fk (U, τ )Ω (τ )
4.1 Geometric Phase for Open Systems
58
Thus under U (τ ) −→ Ω(τ )U (τ )
φG
=
arg{Tr[ρse (0)U † (0)Fk (U, τ )U (τ )]}
†
−→ arg{Tr[ρse (0)U (0)Fk (U , τ )U (τ )]}
=
arg{Tr[ρse (0)U † (0)Ω† (0)Ω(0)Fk (U, τ )Ω† (τ )Ω(τ )U (τ )]}
=
arg{Tr[ρse U † (0)Fk (U, τ )U (τ )]}
=
φG
which shows that φG is gauge invariant.
4.1.6
Appendix
In this appendix we consider the transformation rule for
τ
Bk (t)dt.
ωk (τ ) = P exp i
0
˙ † then
In particular we want to show that if Bk −→ Bk = βBk β † + iββ
τ
P exp i
τ
Bk (t)dt −→ β(0) P exp i
0
Bk (t)dt β † (τ ).
0
Now, in a differential form ωk (t) satisfies
ω˙ k (t) = iωk (t)Bk (t)
(1A)
If Bk −→ Bk , then we assume that ωk −→ ωk , with
ω˙ k = iωk Bk
˙ †)
= iωk (βBk β † + iββ
⇒ ω˙ k β = iωk βBk − ωk β˙
⇒ ω˙ k β + ωk β˙ = i(ωk β)Bk
⇒
d
[ω β] = i(ωk β)Bk
dt k
(2A)
4.1 Geometric Phase for Open Systems
Comparing (1A)and (2A), we can write
ωk β = β(0)ωk
then
d
[ω β] = β(0)ω˙ k
dt k
= iβ(0)ωk Bk
= iωk βBk
which is (2A).
Hence
ωk = β(0)ωk β † (t)
which completes our proof.
59
Chapter 5
Applications
In this chapter, we will explore two well known examples of systems undergoing nonunitary evolutions. The evolutions, sometimes referred to as quantum channels, that
we will consider are the Amplitude damping and the phase damping channels. Here
we examine some of the details pertaining to the kinematical approach and furnish the
geometrical phase for the two channels.
5.1
Amplitude Damping Channel
As mentioned in section (4.0.3), the amplitude damping channel models a spontaneous
emission or decay process. In either of the two cases the system, an unstable state always
tries to reach the ground state through some emission process. For example an atom
excited will tend to go down to its ground state by emitting a photon. Similarly in a
nuclear reaction, a spontaneous decay such as as alpha emission can also be modeled by
this channel.
In this model, the system undergoing decay has two degrees of freedom - the ground
state |0s and the excited state |1s . The environmental degrees of freedom refer to the
number of states of the photon being released. Here we will assume that the system
5.1 Amplitude Damping Channel
61
either emits no photon (|0e ) or one photon (|1e ). Thus we have a two dimensional
environment.
The Krauss operators that describe such a
1
M0 =
0
0
M1 =
0
process are given by
0
√
1−p
√
p
0
To facilitate the calculations, we first need to find the unitary evolution of the combined
system (system + environment) such that the corresponding Krauss operators for the
system are as above. This is sometimes referred to as a “purification”. This is of course
not unique, as evident from the theory developed earlier. To begin, we only need one
explicit form, as the formalism guarantees that all gauge equivalent unitary evolutions
will lead to the same geometric phase. It is not difficult to find, for a qubit environment,
the following U :
0
0
1
0 √1 − p −√p
U =
√
√
0
p
1−p
0
0
0
0
0
0
1
(5.1)
For the evolution that we are interested in, it is convenient to set p = 1 − e−t , so that at
t = 0, p = 0 and as t → ∞, p tends to 1. Hence, we have,
0
0
1
√
√
0
e−t
− 1 − e−t
U (t) =
√
0 √1 − e−t
e−t
0
0
0
0
0
0
1
(5.2)
For a qubit system, we can represent the initial state by a Bloch vector r ≡ (r1 , r2 , r3 ):
ρs (0) =
1
(1 + r · σ)
2
5.1 Amplitude Damping Channel
62
=
1 + r3
1
2 r + ir
1
2
r1 − ir2
(5.3)
1 − r3
As the environment we are interested in is a qubit system, we can start with an initial
state of the combined system as
ρse (0) = ρs (0) ⊗ |0e 0e |
(ρs )11 (ρs )12 0
1
(ρs )21 (ρs )22 0
=
2 0
0
0
0
0
0
1 + r3 r1 − ir2
1
r1 + ir2 1 − r3
=
2
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
(5.4)
With the above, we could see the total phase is,
γT = arg{Tr(ρse (0)U † (0)U (t))}
(5.5)
However, it is not difficult to see that the total phase γT is zero, as the term Tr(ρse (0)U † (0)U (t))
is a real number. This evolution yields zero phase. Next we combine a time dependent
rotation about the z-axis with the amplitude damping channel. To this end, we consider
iπt
Vsys = exp ( σz )
80
iπt
0
e 80
=
− iπt
0 e 80
(5.6)
(5.7)
where σz is one of the three Pauli matrices. The operation for the combined system will
be,
Vse = Vsys ⊗ 1
(5.8)
5.1 Amplitude Damping Channel
63
iπt
80
e
0
=
0
0
0
0
0
− iπt
80
0
0
0
e 80
iπt
0
0
0
e− 80
e
iπt
(5.9)
Then the combined evolution U (t) is given by,
Figure 5.1: The Amplitude Damping process coupled with a rotation about the z axis.
iπt
80
e
0
U (t) =
0
0
0
0
√
iπt √
1
e− 80 iπt 1 − p −e 80 p
1
iπt √
√
e− 80 iπt p
e 80 1 − p
0
0
0
0
0
1
(5.10)
e− 80 iπt
It should be noted that this unitary evolution is not gauge-equivalent to the evolution
given by (5.2). With both U (t) and ρse (0), we can implement a numerical scheme to
evaluate the geometric phase for the system. The results is given in Figure 5.2.
It is instructive to test the gauge invariance of the geometric phase under a rotation
of the environment. Indeed, by introducing
cos θ sin θ
Re =
− sin θ cos θ
5.1 Amplitude Damping Channel
64
Figure 5.2: The geometrical phase for Amplitude Damping channel
Rse
0
0
cos θ sin θ
− sin θ cos θ
0
0
= 1 ⊗ Re =
0
0
cos θ sin θ
0
0
− sin θ cos θ
(5.11)
the geometric phase remains unchanged (see Figure 5.3).
Figure 5.3: The geometrical phase for Amplitude Damping channel for a rotated environment
5.2 Phase Damping Channel
5.2
65
Phase Damping Channel
The Krauss operators for Phase Damping channel are given by,
1
0
M0 =
√
0
1−p
0 0
.
M1 =
√
0
p
Here again the environment has two degrees of freedom. For a qubit system, a unitary
evolution that realizes this channel is given by
0
1
0 √1 − p
U =
0
0
√
0
p
0
0
0
√
− p
1
0
0
√
.
(5.12)
1−p
Introducing time dependence through p = 1 − e−t , we have
0
0
0
1
√
√
0
e−t
0 − 1 − e−t
U (t) =
0
0
1
0
√
√
1 − e−t 0
e−t
0
.
(5.13)
Following the previous example, we assume that intial state for the system (ρs (0)) and
the combined system (ρse (0)) are given by (5.3) (5.4) respectively. Here, as in the pure
amplitude damping channel, the total phase γT = arg{Tr(ρse (0)U † (0)U (t))} is again
zero. Hence, we apply a small time dependent rotation about the z axis. The resulting
evolution is shown in Figure (5.4). The geometric phase for the system undergoing such
a non-unitrary evolution is shown in Figure (5.5). To check whether the geometrical
phase of this channel is gauge invariant, we test it in the same manner as we did for the
amplitude damping case; by effecting a rotation on the environment. Again the geometric
phase remains unchanged under this rotation demonstrating the gauge-invariance of the
phase.
5.2 Phase Damping Channel
Figure 5.4: The phase damping process with a rotation Vsys = eiπtσz .
Figure 5.5: The geometrical phase for evolution of Figure 5.4
66
Chapter 6
Summary
In summary, we had proposed a kinematic approach to the geometrical phase of a mixed
state undergoing non-unitary evolutions. In the approach taken here, we include the
environmental degrees to facilitate a direct application of the well established results for
unitary evolutions. Following closely the unitary case we have identified the appropriate
bundle structure and the underlying symmetry to define the geometric phase. The gauge
symmetry essentially renders the geometric phase to be dependent only on the paths
taken in system’s state space; thus justifying its geometric character. The parallelity
conditions are shown to be easily derivable in this framework. Finally, we furnish explicit
examples of the geometric phases for the Amplitude damping and the Phase damping
channels.
(LQ)
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iii
[...]... evolution path in the Bloch Sphere representation This result is consistent with the expression for Berry’s phase under adiabatic approximation [33] Chapter 3 The Mixed States Geometric Phase While the pure state geometric phase has been extensively studied since the 80’s, the focus on the geometric phase for mixed state only started recently [33] In the following sections, we discuss how the fiber... parallel transport condition, the geometric phase is equal to the total phase Hence, we propose a functional form of γG = arg{Tr(ρ(0)U † (0)U(t)F(U, t))} (3.34) Comparing the total phase formula (3.33) with the propose geometric phase term (3.34), we note that F(U, t) = V(t)V † (0) In addition, F(U, t) should also satisfy the following 3.3 The Degenerate Case 29 transformation: F(U, t) −→ F (U , t) =... the geometric phase is gauge-invariant 2.2 Pure State Geometric Phase in Language of Fiber Bundle 2.2.3 15 The Dynamical Phase In this section, we will consider the relationship between total phase, dynamical phase and geometrical phase The time evolution of a state vector is given by the Sch¨odinger equation: i ∂ |ψ(t) ∂t = H(t)|ψ(t) (2.26) Hence, for state |ψ(t) = eif (t) |φ(t) , where |φ(t) is a choice... same path in the projective space P as H(t), then the geometric phase is the same while both 2.2 Pure State Geometric Phase in Language of Fiber Bundle 16 the total phase and dynamical phase terms change correspondingly Indeed, if we take ˜ |ψ(t) = eif2 (t) |φ(t) , where new state has the same section as |ψ(t) then γ2 = f2 (T ) − f2 (0) T 0 The geometrical term γG = i T ˙ φ(t)|φ(t) dt − = i ˜ H(t)|... and Vp M of Tp E such that, (a) Tp E Vp E ⊕ Hp E for all p ∈ E (b) Rg∗ (Hp E) = Hpg E for all g ∈ G, p ∈ E, where Rg∗ (p) := pg denotes the right action of G on E 2.2 Pure State Geometric Phase in Language of Fiber Bundle 2.2 9 Pure State Geometric Phase in Language of Fiber Bundle 2.2.1 Pure State Fiber Bundle and the Connection Here we see how Berry phase arises naturally when we consider the bundle... way For the dynamical phase of a mixed state to vanish, all the constituent pure states in the mixture must be parallel-transported independently There are now N parallel transport conditions 3.2 The Non-Degenerate Case 24 One key property of geometric phase is that it should be gauge invariant under the gauge group Gx Let us see how the total phase and dynamic phase change under the gauge transformation... State Geometric Phase The mathematical language of fiber bundles in differential geometry provides a powerful tool to study and analyze the geometrical phase It enables one to visualize this quantum phenomenon in a geometrical way and it paves the way to other generalizations; for instance to mixed states Here, we furnish a short introduction to fiber bundles that will serve both to set the framework for. .. parallel-transport condition 2.3 Example of Pure State Geometric Phase Any qubit state can be expressed as cos 2θ |ψ = θ iϕ e sin 2 0 θ π, 0 ϕ 2π (2.36) The state |ψ is a function of (θ, ϕ) Geometrically, pure states for a qubit are represented as points on a 2-sphere called the Bloch sphere These points also represent the projective 2.3 Example of Pure State Geometric Phase 18 Figure 2.6: A Bloch sphere representation... (2.18) Integrating both sides, we obtain T β ≡ f (T ) − f (0) = i 0 ˙ φ(t)|φ(t) dt (2.19) 2.2 Pure State Geometric Phase in Language of Fiber Bundle 13 where β is defined to be the geometrical phase In the following, we will analyze β and show that it is independent of the choice of Figure 2.4: Geometrical phase by holonomy section |φ(t) In other words, it is gauge-invariant As mentioned earlier, the total... total phase[ 12] obtained from the interferometer setup for mixed under unitary evolution is γT = arg Tr(ρ(0)U † (0)U(t)) N ωk k|U † (0)U(t)|k = arg (3.18) k=1 A simple extension of the pure state dynamical phase will give the dynamical phase for mixed states as well: T dtTr(ρ(t)H(t)) γD = − 0 T = −i ˙ dtTr ρ(0)U † (t)U(t) 0 N T = −i ˙ ωk k|U † (t)U(t)|k dt 0 (3.19) k=1 To make the dynamical phase vanish, ... Transformations Kinematical Approach to Geometric Phase for Open Systems 4.1 34 37 4.0.2 Brief Introduction to the Open System 37 4.0.3 Amplitude Damping and Phase. .. Therefore, which furnishes the transformation property of F(U, T ) Chapter Kinematical Approach to Geometric Phase for Open Systems In this section, we extend the kinematic approach to the geometric. .. the formalism developed for pure state geometric phase to mixed states The key principle that we adhere to in the ensuing generalization is the concept of gauge-invariance for the geometric phase