!  "  #    $   

%& '#()* *()+,

## ,)-*()#) !#.)*,) !!#.# /00/ Dissertation for the Degree of Doctor of Philosophy in Physics with Specialization in Quantum Chemistry presented at Uppsala University in 2002

Abstract Ericsson, M., 2002. Geometric and Topological Phases with Applications to Quantum Com- putation. Acta Universitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 744. 53 pp. Uppsala. ISBN 91-554-5391-0.

Quantum phenomena related to geometric and topological phases are investigated. The first results presented are theoretical extensions of these phases and related effects. Also experimen- tal proposals to measure some of the described effects are outlined. Thereafter, applications of geometric and topological phases in quantum computation are discussed. The notion of geometric phases is extended to cover mixed states undergoing unitary evo- lutions in interferometry. A comparison with a previously proposed definition of a mixed state geometric phase is made. In addition, an experimental test distinguishing these two phase con- cepts is proposed. Furthermore, an interferometry based geometric phase is introduced for systems undergoing evolutions described by completely positive maps. The dynamics of an Aharonov-Bohm system is investigated within the adiabatic approxima- tion. It is shown that the time-reversal symmetry for a semi-fluxon, a particle with an associated magnetic flux which carries half a flux unit, is unexpectedly broken due to the Aharonov-Casher modification in the adiabatic approximation. The Aharonov-Casher Hamiltonian is used to determine the energy quantisation of neutral magnetic dipoles in electric fields. It is shown that for specific electric field configurations, one may acquire energy quantisation similar to the Landau effect for a charged particle in a homogeneous magnetic field. We furthermore show how the geometric phase can be used to implement fault tolerant quantum computations. Such computations are robust to area preserving perturbations from the environment. Topological fault-tolerant quantum computations based on the Aharonov-Casher set up are also investigated. Key words: Geometric phase, topological phase, quantum computation, mixed states, com- pletely positive maps.

Marie Ericsson, Department of Quantum Chemistry, Uppsala University, Box 518, SE–751 20 Uppsala, Sweden

c Marie Ericsson 2002

ISSN 1104-232X ISBN 91-554-5391-0

Printed in Sweden by Geotryckeriet, Uppsala 2002 To my family and grandfather IV V

List of publications

This thesis is based on the following papers, which will be referred to in the text by their Roman numerals:

I Geometric phases for mixed states in interferometry E. Sj¨oqvist, A.K. Pati, A. Ekert, J.S. Anandan, M. Ericsson, D.K.L. Oi, and V. Vedral, Phys. Rev. Lett. 85 (2000) 2845.

II Mixed state geometric phases, entangled systems, and local unitary transformations M. Ericsson, A.K. Pati, E. Sj¨oqvist, J. Br¨annlund, and D.K.L. Oi, submitted to Phys. Rev. Lett. quant-ph/0206063

III Generalization of geometric phase to completely positive maps M. Ericsson, E. Sj¨oqvist, J. Br¨annlund, D. K.L. Oi, and A. K. Pati, submitted to Phys. Rev. Lett. quant-ph/0205160.

IV Mobile flux line in an Aharonov-Bohm system E. Sj¨oqvist and M. Ericsson, Phys. Rev. A 60 (1999) 1850.

V Towards a quantum Hall effect for atoms using electric fields M. Ericsson and E. Sj¨oqvist, Phys. Rev. A 65 (2002) 013607.

VI Geometric quantum computation A. Ekert, M. Ericsson, P. Hayden, H. Inamori, J.A. Jones, D.K.L. Oi, and V. Vedral, J. Mod. Opt.47 (2000) 2501.

VII Quantum computation using the Aharonov-Casher set up M. Ericsson and E. Sj¨oqvist, submitted to Phys. Lett. A. quant-ph/0209006 Other papers not included in the thesis are:

i) “Density Functional Study of Chlorine-Oxygen Compounds Related to the ClO Self- Reaction” T. F¨angstr¨om, D. Edvardsson, M. Ericsson, S. Lunell, and C. Enkvist,

Int. J. Quantum Chem. 66, (1997) 203. ¢ ii) “Degree of electron-nuclear entanglement in the E ¡ Jahn-Teller system” M. Ericsson, E. Sj¨oqvist, and O. Goscinski, Proceedings of the XIV International Symposium on Electron-Phonon Dynamics and Jahn-Teller Effect (World Scientific, Singapore, 1999).

iii) “Dlaczego kot Schr¨odingera la¸duje na czterech łapach?” (English title: Why Schr¨odinger’s cat lands on its feet) M. Ericsson and E. Sj¨oqvist, Delta, 11, (2001) 1.

iv) “Holonomic quantum logic gates” M. Ericsson, To be published in ”Quantum Theory: Reconsideration of Foundations”, ed. by A. Khren- nikov; series “Math. Modeling in Physics, Engineering and Cognitive Sciences” V¨axj¨o Univ. Press (2002). http://xxx.lanl.gov/abs/quantph/0202106.

v) Reply to “Singularities of the mixed state phase” J. Anandan, E. Sj¨oqvist, A.K. Pati, A. Ekert, M. Ericsson, D.K.L. Oi, and V. Vedral, submitted to Phys. Rev. Lett. quant-ph/0109139. Contents

1 Introduction 1

2 Geometric phases 3 2.1 Introduction ...... 4 2.1.1 Geometric phases in cyclic evolutions...... 4 2.1.2 Geometric phases in non-cyclic evolutions ...... 8 2.1.3 MeasurementofPancharatnam’srelativephase...... 9 2.2Mixedstatephases...... 10 2.3 Phases for completely positive maps...... 14

3 Topological phases 19 3.1 Introduction ...... 20 3.1.1 Aharonov-Bohmphase...... 20 3.1.2 Aharonov-Casher phase ...... 22 3.1.3 Fundamental electromagnetic description ...... 25 3.2DynamicsofanAharonov-Bohmsystem...... 26 3.3DualLandaulevels...... 29

4 Quantum Computation 33 4.1 Introduction ...... 34 4.2Geometricquantumcomputation...... 36 4.3 Topological quantum computation ...... 40

5 Conclusions 43

VII CONTENTS VIII Chapter 1

Introduction

On the scale of human perception the laws of classical physics are good approximations to phenomena around us, such as an apple falling from a tree. But on atomic scales we need to replace classical physics with quantum physics to explain results obtained in experiments. In this thesis I will focus on several fundamental quantum mechanical effects which are related to geometric and topological phases. Quantum physics is indeed very different from classical physics. To illustrate this consider an experiment in which light hits a beam-splitter which reflects half of the light and transmits the rest. If individual photons are sent towards this mirror and we put a photodetector both in the transmitted beam (B) and in the reflected beam (A), as shown in Fig. (1.1), we register a photon with equal probability in detectors A and B. We interpret this as half of the times the

A

¨ ¦

¥

¦§

©

©

¤ ¤ B

Figure 1.1: Single photon hitting a beam-splitter has equal probabilities for detection in the photodetectors A and B. photon is transmitted and half reflected. To be able to see the full quantum nature of the photon, we bring the beams together and let them intersect at a second beam-splitter to form a Mach-Zehnder interferometer, as shown in Fig. (1.2). The probability of detecting a photon behind the second beam-splitter is now one for photodetector B and zero for photodetector A 1. This is surprising since if half of the pho- tons were transmitted and half reflected by the first beam-splitter, we would expect the same to happen at the second beam splitter so that half of the photons hit photodetector A and half

1Assuming that the two optical paths are of the same length.

1 2

A

¨ ¦

¥

¦§

©

©

¤ B

¥

©

©

¤ ¤

Figure 1.2: Bringing the beams together on a second beam-splitter gives unit probability for the photon to arrive at photodetector B. hit photodetector B. This prediction does not agree with the experiment and we see that we cannot add probabilities for the different paths to have the overall probability. Instead we have to add probability amplitudes (complex-valued numbers) for every path ending in each pho-

todetector, and calculate the probability as the absolute square of this sum. If we follow these

 

rules, assuming that the photon is transmitted with probability amplitude and reflected

  with probability amplitude  , we can calculate the probabilities of detecting the photon in any of the two detectors located after the second beam-splitter. This kind of calculation gives agreement with experiments and makes the classical reasoning that the photon was either re- flected or transmitted invalid since there will be a non-trivial probability amplitude associated to each beam-splitter. This is called single-particle interference and is a fundamental quantum phenomenon. This argument shows that the fundamental object in quantum mechanics is the probability

amplitude describing interference effects for particles. We may express such a complex proba-

           

bility amplitude  according to ,where and is the associated phase factor.

 % ' (  In the above example there is a phase factor  for the reflected path and for the transmitted path. These phase factors precisely account for the constructive and destructive interference in the above interferometer example. The focus of this thesis is such phase factors. Phases can be of different origin. They can be dynamical when they depend on the speed of the evolution of the quantum system, or geometric when they depend on the geometry of state space [1]. They can also be of topological origin when they depend on the topological structure of the configuration space [2]. The focus of this thesis is on geometric and topological phases. They are described more thoroughly in Chapters 2 and 3. Why study these quantum mechanical phases? First, they help us to understand the structure of quantum physics. Secondly, they have triggered the development of many experimental tech- niques in, e.g., quantum optics, neutron optics, and atomic interferometry, since we need very accurate data to see their effects. Also, a few years ago it was proposed [3, 4] that these phases could be used to perform fault tolerant quantum computation. This latter aspect is described in Chapter 4. Chapter 2

Geometric phases

The concept of geometric phase was introduced in 1956 by Pancharatnam [5] in his studies of interference effects of polarised light waves. Independently, in molecular physics some as- pects of geometric phases were discussed by several authors [6, 7, 8, 9]. However it was Berry [10] who first realised, in 1984, that the geometric phase is a generic feature of quantum me- chanics. His approach to the Abelian geometric phase was restricted to cyclic and adiabatic evolution of non-degenerated pure quantum states, where the phase depends on the geometry of the path the Hamiltonian traces out in parameter space. Subsequently these restrictions were removed step by step. Wilczek and Zee [11] pointed out that adiabatic transport of a degenerate set of eigenstates is associated with a non-Abelian geometric phase (described in Chapter 4). Aharonov and Anandan [1] discovered the geometric phase for non-adiabatic evolutions where the phase depends on the geometry of the path in the state space. Based upon Pancharatnam’s work, Samuel and Bhandari [12] introduced the notion of non-cyclic geometric phases. The geometric phase for mixed states was studied as a mathematical concept by Uhlmann [13]. In paper I we take an operational approach when we introduce geometric phases for mixed states in the context of quantum interferometry. For good reviews on the geometric phase see, e.g., Refs. [14, 15, 16, 17, 18]. The outline of this chapter is as follows. We start with a brief introduction to geometric phases in section (2.1). The cyclic geometric phases, i.e Berry’s and the non-adiabatic generali- sation by Aharonov and Anandan, are discussed, with some illustrative examples and references to experiments, in subsection (2.1.1). In subsection (2.1.2) non-cyclic phases are described start- ing with Pancharatnam relative phase followed by the non-cyclic geometric phase by Samuel and Bhandari and its kinematic formulation by Mukunda and Simon [19]. The non-cyclic Pan- charatnam phase can be tested in interferometry, with the geometric phase as a special case, as is discussed in subsection (2.1.3). Section (2.2) is devoted to two different approaches to the geometric phase for mixed states: the one that was proposed in paper I and another one which was introduced by Uhlmann [13]. A comparison between the two approaches, including an experimental proposal of how to measure Uhlmann’s phase, is described based upon paper II. Paper I is extended to the case of completely positive maps in paper III. This extension is described in section (2.3).

3 2.1 Introduction 4

2.1 Introduction

2.1.1 Geometric phases in cyclic evolutions

In 1984 Berry discovered the geometric phase as a generic feature of quantum mechanics [10].

* + - / 1

He considered the Schr¨odinger evolution of a quantum state vector governed by a param-

+ 5 + - / /

eter dependent Hamiltonian 3 .

 9

Suppose that the state vector is an eigenstate of the Hamiltonian at time - and that

 + = ? A = A D D D / G the parameters 5 are varied along a closed path in parameter space. If the

variation is slow enough for( the adiabatic theorem to hold [20], the state vector remains an

H 9

eigenstate of the Hamiltonian for all times - . Thus, we may write

 M N O Q S

* + - / 1   T + 5 + - / / 1 A

(2.1)

 M N O Q S Y 9 A \ ]

where  is a phase factor. After a loop in parameter space during the time interval ,the

_  ^ _ + \ / e ^ _ + 9 / phase change ^ may be found by substituting Eq. (2.1) into the Schr¨odinger

equation, yielding

l m



k

^ _  e + 5 + - / / r - t w x r 5 z | T + 5 / }  T + 5 / 1  ^ _ ƒ „ + \ / t ^ _ ƒ ‡ Y G ] A

_ (2.2)

o p

T + 5 + - / / 1 + 5 + - / /

where _ is the eigenvalue of the instantaneous eigenvector of the parameter

p

_ ƒ „

dependent Hamiltonian. The first term ^ is the dynamical phase which depends on the Hamil-

_ ƒ ‡ tonian. The second term ^ is the geometric phase. It depends only on the shape of the curve

G under the restriction that the instantaneous eigenvectors are single-valued in parameter space

T + 5 + \ / / 1  T + 5 + 9 / / 1 [21], i.e. . This expression for the geometric phase can be converted to a surface integral in the parameter space using Stokes’ theorem. The important features for the phase to be called geometric are: it does not depend on the speed with which the parameters are varied (although they should vary slow enough for the

adiabatic theorem to hold), and it is gauge independent modulo an unimportant integer multiple

 T + 5 / 1

of  with respect to the choice of single valued . Furthermore, it is a real-valued quantity

T + 5 / 1 | T T 1    | }  T T 1 t | T }  T 1  9 | T }  T 1 due to the normalisation of , i.e. then is purely imaginary so that the geometric phase in Eq. (2.2) becomes real. Furthermore, the adiabatic geometric phases has been interpreted in a mathematical fashion as a holonomy of a complex fiber bundle [22]. Aharonov and Anandan [1] realised that the notion of geometric phase is independent of the adiabatic theorem. Even if the Hamiltonian is unknown and we only know the path of the state, the total phase change of a state vector after a cyclic evolution can be decomposed into a

dynamical part, expressed in terms of the expectation value of the Hamiltonian, and a geometric

* + - / 1 -  \

part. Consider a non-adiabatic evolution of a state vector , which after time returns

* + \ / 1    M * + 9 / 1

to the same state, i.e. .Tofind the geometric phase we introduce another

” + - / 1   ˜  š O Q S * + - / 1 ” + \ / 1  ” + 9 / 1

state vector such that (cf. the single valued energy

+ \ / e  + 9 /  ^

eigenstate vector in the above adiabatic case). This amounts to choose  (modulo

 ” + - / 1

 ) and inserting into the Schr¨odinger equation yields

m m

l l

r 

k

| * 3 * 1 r - t | ” ^  e ” 1 r -  ^ „ t ^ ‡ D

(2.3)

o o

r - 2.1 Introduction 5

The first term on the right hand side is the dynamical phase. The second term is of geometric nature and is called the geometric phase for the following reasons. First, in the case of adiabatic

evolution, the decomposition in Eq. (2.3) reduces to that of Eq. (2.2). Secondly, Aharonov and ‡

Anandan demonstrated a relation between ^ and the geometry of state space, i.e. the projective

+ ¤ / £ + ¤ / ¦

Hilbert space £ . is defined through the projection map as

 ¬

+ * 1 /  ¨ * © 1 « * © 1   * 1 ­ A

¦ (2.4)

Y 9 A   ] G ¯

where ® is real-valued and run over the interval . Then the open path in Hilbert space,

* + - / 1 ¦ G ± traced out by the cyclic state vector , is projected by to the closed path in projective Hilbert space (see Fig. (2.1)).

2π γ 0 C H H

Π

C P

P

 M

¯ * + 9 / 1   * + \ / 1   * + 9 / 1 ¤

Figure 2.1: An open curve G : , in Hilbert space is projected

G ± £ + ¤ / with the map ¦ to a closed curve in projective Hilbert space .

The last term in Eq. (2.3) can now be expressed as a closed line integral in ¤ according to

^ ‡ Y G ± ]  x µ | ” r ” 1 A

w (2.5)

  

which is gauge invariant modulo  . The modulo ambiguity depends upon the choice of the

” 1 ¤

cyclic state , but is irrelevant when considering geometric phase factors. All curves in that

+ ¤ /   project onto the same curve in £ have the same geometric phase (modulo ). Therefore the

geometric phase is independent of the Hamiltonian generating the evolution. Instead it depends

+ ¤ / - only on the curve in £ . This together with the fact that it is independent of the parameter

(reparametrisation invariance) guarantee that the phase is indeed a geometric quantity. Also, it is

” 1 real-valued for normalised and it changes sign if the path is traversed in opposite direction. 2.1 Introduction 6

The geometric phase can also be understood from the quantum parallel transport condition

r

| * + - / * + - / 1  9 D

(2.6)

r -

* + - / 1

This condition follows from the normalisation of and from the requirement that the state

* + - / 1 * + - t r - / 1 | * + - / * + - t r - / 1 vectors and should have the same phase, i.e. positive and real- valued. The parallel transport condition prohibits the state vector to “twist” during the evolution

and when returning to the initial state the total phase ^ has no contribution from the dynamical

‡ * + - / 1  ¶ · ¸ + ^ + - / / ” + - / 1

phase and is thus equal to the geometric phase ^ in Eq. (2.3). For if ,

” + \ / 1  ” + 9 / 1 ^ + \ / e ^ + 9 /  ^

when and , the condition (2.6) gives

l

r r r

+ - / t | ” + - / ” + - / 1  9  ^  | ” + - / ” + - / 1 r - D

^ (2.7)

r - r - r -

¾

+ - / * + 9 / 1 * ¼ + - / 1 

For unitary evolutions ¼ , so that Eq. (2.6) can be rewritten as

¾ ¿ ¾

| * + 9 / + - / + - / * + 9 / 1  9 D ¼ (2.8)

This shows that parallel transport is a relation between the unitary operator and the initial state

¾

* + 9 / 1 - / vector; not all are parallel transported for a given + . A specification of a rule that guarantees parallel transport along any path is called a connection. An illustrative example of parallel transport is given for a two level system where the pro- jective Hilbert space can be represented as the Bloch sphere. If we parallel transport a state vector along a path which projects on a curve on the Bloch sphere as shown in Fig. (2.2), then total phase of the state vector after completing the loop is equal to the geometric phase given in

Eq. (2.5), i.e. (minus) half the solid angle enclosed in £ , where the half is due to the dimen- sionality of Hilbert space. A nice mathematical analogue of this result arises when considering parallel transport of a tangent vector on a sphere (see Fig. (2.3)). During the transport the length and orientation of the vector with respect to the normal of the surface are kept constant. After completing the loop and returning to the initial point, the vector is rotated despite the fact that at no point during the parallel transportation a local rotation was performed. The vector is ro- tated by the holonomy angle, which is analogous to the geometric phase, and is dependent on the curvature of the sphere, i.e. it equals the solid angle enclosed during the parallel transport. Parallel transport of a vector on a flat surface does not result in such a rotation as the curvature vanishes there. Let us stress, however, that the analogy with parallel transport of a quantum vector is not complete: the factor one half in front of the solid angle for the geometric phase is a unique quantum feature that could not be explained by the above mathematical analogue. There are many generalisations of Berry’s geometric phase, e.g. to complex valued geo- metric phases [23] for non-Hermitian Hamiltonians and to off-diagonal geometric phases [24], the latter being introduced to uncover interference effects when the usual geometric phase is undefined. Furthermore have adiabatic geometric phases in classical physics been studied, such as the Hannay angle, [25], and the geometric phase for non-linear fields [26]. The Hannay angle has also been generalised to the non-adiabatic case [27]. Experimental verifications of the geometric phase have been performed including measure- ments of the adiabatic geometric phase for neutron spin [28], photons [29], nuclear magnetic 2.1 Introduction 7

2π γ= − Ω / 2 0

Ω

Figure 2.2: Parallel transport of a spin-half particle on a Bloch sphere with the geometric phase equal to (minus) half the solid angle enclosed.

Ω γ = Ω γ

Figure 2.3: Parallel transport of a vector on a sphere with the holonomy equal to the solid angle enclosed. resonance (NMR) [30], and nuclear quadrupole resonance (NQR) [31]. Also the adiabatic ge- ometric phase has been observed for two entangled nuclear spin systems in NMR [32]. The non-adiabatic geometric phase has been measured in NMR [33]. The adiabatic geometric phase for a classical chemical oscillator [34] has been measured, as well as for molecular systems [35]. Furthermore, the off-diagonal geometric phase has been verified in neutron interferometry [36]. 2.1 Introduction 8

2.1.2 Geometric phases in non-cyclic evolutions

Parallel transport is an important concept when defining Pancharatnam’s relative phase. If we

À 1 Á 1

want to know the relative phase between two normalised state vectors, and , the idea is

Á 1

to parallel transport one of them, e.g. , along a geodesic defined by the Fubini-Study metric

À 1 ¦ + À 1 /  ¦ + À 1 / Á 1 À 1

© ©

[37], to © ,where . and are now defined to be in phase, i.e. one

Ã Ä Å | Á À 1  9 À 1 Á 1

can show that © . The relative phase between and is then defined as the

Ã Ä Å | À Á 1  Ã Ä Å | À À 1 argument of the scalar product, i.e. © .

The relative Pancharatnam phase is only defined for nonorthogonal states and it is non-

Ä Å | Á À 1 

transitive. This latter property means that if two pairs of state vectors are in phase, Ã

9 Ä Å | Á G 1  9 À 1 G 1

and à , the relative phase between and does not in general vanish. In

À 1 G 1

the above two level system the relative phase between and is half the solid angle of

Á G

the spherical triangle defined by the states À , ,and on the Bloch sphere. Thus, the relative

À 1 G 1

phase between and is equal to the geometric phase for the path defined by geodesic

Á G closure of À , ,and . The above concept of the relative phase was used by Samuel and Bhandari [12] to introduce

a non-cyclic geometric phase. If a state vector is parallel transported along a curve whose

+ ¤ / corresponding projection in £ is open, Samuel and Bhandari realised that by closing the

curve in ¤ with a geodesic and calculating the Pancharatnam relative phase between the initial and final state vector, this phase is gauge invariant and they defined it to be the non-cyclic geometric phase. This approach is still in the spirit of Aharonov and Anandan with a geometric phase associated with a closed path in projective Hilbert space, now defined by the curve and the shortest geodesic closure. In the kinematic approach by Mukunda and Simon [19] (see also [38]) a more direct ap- proach to geometric phase is provided without using the Schr¨odinger equation as a starting point. In this treatment the geometric phase is defined as the subtraction of the accumulation

of local phase changes from the total phase (Pancharatnam relative phase), during an evolution

 9 -  \

from - to , according to

m

l

Ã Ä Å | * + - / * + - t r - / 1 ^ ‡  Ã Ä Å | * + 9 / * + \ / 1 e

o

l m

¼

 Ã Ä Å | * + 9 / * + \ / 1 t * + - / * + - / 1 r - A

| (2.9)

o

* + - / 1 where is a state vector in Hilbert space. This quantity is real-valued and geometric, i.e. it is reparametrisation invariant and depends only on the path in projective Hilbert space. On the

other hand, the total phase and the sum of local phase changes are not gauge invariant separately.

 ¬ O Q S

* + - / 1  * + - / 1   * + - / 1 ® + \ / e ® + 9 / The transformation © , adds an additional phase to both these terms. However, in the expression for the geometric phase, these additional phases cancel out. Applications of the Mukunda-Simon approach to the geometric phase are given for molecular dynamics [39], for response function in the many-body system [40, 41], and for quantal revivals [42].

The expression for the geometric phase in Eq. (2.9) contains all previous notions of geomet-

| * + - / * + - / 1  9

ric phase in unitary evolution. For example, if we choose gauge so that ¼ at all

* + - / 1

times - , i.e. the local phase changes of vanish, the geometric phase equals the Pancharat-

Ä Å | * + 9 / * + \ / 1  9 nam phase. If we instead have a cyclic path and choose gauge so that à , 2.1 Introduction 9

A

¨ ¦

¥

¦§

©

©

¤ B

¥

©

©

¤ ¤ Ì

¾

 / Figure 2.4: A + phase in one of the interferometer beams modulates the intensity in photodetector B according toÌ Fig. (2.5). I

1

0.5

0 χ 0π/2 π 3π /2 2π

Figure 2.5: The modulation (interference pattern) of the intensity in detector B as a function of

. Ì i.e. we have a closed path in Hilbert space, the geometric phase equals the sum of local phase changes for a cyclic state, as was derived by Aharonov and Anandan.

2.1.3 Measurement of Pancharatnam’s relative phase Relative phases can be measured and estimated from modifications of interference patterns in interferometers. As an example let us revisit the Mach-Zehnder interferometer of Chapter 1,

with an incoming particle along the horizontal direction. If the basic set up of Fig. (1.2) is

¾

 / modified by performing a + phase shift in one of the interferometer arms (see Fig. (2.4)),

then the probability of reaching detector B,Ì or the intensity in the detector B if many particles Ð

are involved, depends on the phase as Ï

Ì

t Ñ Ò Ó

 (2.10) Ì 2.2 Mixed state phases 10

A

¨ ¦

¥

¦§

¾

©

© ¤

 B

¥

©

©

¤ ¤ Ì

¾ ¾ ¾

+  / + Õ /

Figure 2.6: Ô operation in the upper interferometer beam and phase in the  lower interferometer beam modulates the intensity according to Fig. (2.7). Ì

(cf. Fig. (2.5)). Now, assume that the particle in the interferometer also carries an additional

× 1

internal degree of freedom, e.g., spin, represented by . The internal state can be transformed

¾ ¾

+ Õ /

by an Ô operation (here and in the following subscript ” ” refers to the internal degrees



¾

× 1 × 1 e  × 1 

of freedom) in one of the interferometer arms (see Fig. (2.6)) as © . With



¾

 /

a + phase in the other arm the intensity modulation in the horizontal output channel is

Ì

Ø Ð

modified accordingÏ to [43, 44]

¾ ¾

× 1 /   t Ù Ñ Ò Ó + e ^ / × 1 Ñ Ò Ó + e Ã Ä Å | × t | ×

 (2.11)

 

Ì Ì

^ × 1 × 1

(cf. Fig. (2.7)). Here is Pancharatnam’s relative phase between the pure states and © and

¾

× 1 Ù  

Ù is the visibility of the phase shift. If is cyclic, i.e. takes back to the same ray, then .

 ¾

If fulfils the parallel transport condition Eq. (2.8), ^ is the (non-)cyclic geometric phase. Instead of enforcing the parallel transport one can also arrange interferometry experiments so that dynamical phases from the arms cancel each other out, leaving only geometric phases, see, e.g., Ref. [45].

2.2 Mixed state phases

A quantum state can be described by a density operator Ü that acts on the Hilbert space of the

Ü × 1 | × Ø

system. For pure states,Ø is uniquely given by the projection operator ,butformixed

ß Ø ß Ü

states Ü can be decomposed in several ways. In the spectral decomposition, is diagonal ac-

Ü  Ý × 1 | × Ø

cording to ,where may be interpreted as the classical probabilityØ of having

ß Ø

× 1

the pure state . A mixed state can also considered as the state of a subsystem of a larger

× ? 1 à × 1 * 1  Ý | × ? á ? 1  â 㠃

system in an entangled state, i.e.  where (Schmidt

( O ( S O ( S

 * 1 | * decomposition [46]). The density matrix for this pure entangled state is Ü .Ifwetrace

over, e.g. subsystem 1, we obtain Ø

ß

Ø Ø

? + Ü /  ä | × ? Ü × ? 1  ä × 1 | ×  Ü A

Tr (2.12)

( ( ( 2.2 Mixed state phases 11

I 1

0.5 ν

γ 0 χ 0 π/2 π 3π/2 2π

Figure 2.7: Modulation of the intensity in detector B as a function of . The Pancharatnam

Ì Ù phase ^ is shown in the figure as well as the visibility . The phase shift is purely geometric if the internal state is parallel transported. which is the density matrix representation of subsystem 2, in general a mixed state. The extension of the geometric phase to the domain of mixed states was first described by Uhlmann [13], in the mathematical context of purifications. In contrast, in our paper I we

discovered another geometric phase for mixedØ states evolving unitarily (the unitarity condition

Ø ß

is not necessary in Uhlmann’s approach), using the language of interferometry as in the example

× 1 | × Ü  Õ

above but with mixed internal state Ý actingonan dimensional Hilbert space.

¿

¾ ¾ ¾

Ü Ô + Õ / Ü e  Ü 

The transformation in the upper beam gives © and the intensity in

 

 Ð

detector B as a function of Ï is now

Ì

¾ ¾

 t Ü Y Ü Ñ Ò Ó + e Ã Ä Å ] / D

Tr Tr (2.13)

 

Ì

Ø Ø

The Pancharatnam phase defined as the shift of the interferenceß pattern, is

Ø

¾

 M æ

^  Ã Ä Å Y Ü Ù  A  Ã Ä Å ä

Tr ] (2.14) 

and the corresponding visibility is

¾

Ù  Ü D

Tr (2.15) 

The total interference pattern can also be expressed as a weighted average of pure state interfer-

Ï Ð Ø Ï Ø ß

ence profiles (cf. Eq. (2.11) Ø D ä (2.16)

Let us now turn to the issue of geometric phases for a unitarily evolved mixed state in the above set up. We have seen from Eq. (2.16) that the resulting intensity is a weighted average of pure state intensities. It is therefore natural to introduce the geometric phase for mixed states by requiring that each such pure state intensity is shifted by the corresponding pure state geometric

phase. This amounts to require

¿

¾ ¾

+ - / × + 9 / 1  9 A ×   A  A D D D Õ D + - / | × + 9 /

¼ (2.17)

 

2.2 Mixed state phases 12

× + 9 / 1 Ü + 9 /

i.e. all states , that diagonalise the non-degenerated density operator , are parallel

¾

+ - / + 9 /

transported by the unitary operator . The condition that Ü has to be non-degenerate is



¾

- /

important to have a unique basis in the Hilbert space that is parallel transported. If now + 

fulfils these conditions the phase shift ^ in the interferometer above is purely geometric. This ‡ mixed state geometric phase ^ is gauge independent, reparametrisation invariant, and real- valued, just like the pure state geometric phase. Note also that, since the relative Pancharatnam phase factor for mixed states is proportional to a weighted average of pure relative Pancharatnam

phase factors, the pure state phase relation

 ^ t ^ A ^ tot d g (2.18) is not valid in the mixed case.

In the two level (qubit) case the mixed state geometric phase is

^ ‡  e Ã Ä Ñ ç à è é  ç à è ê

A (2.19)

 ë

where  is the length of the Bloch vector and is the solid angle traced out. For pure states

ê

  (  ) the geometric phase reduces to (minus) half the solid angle as expected. For mixed states, however, the relation between the geometric phase and the solid angle is non-linear. The

visibility in the context of interferometry is given by

í

(

 Ù î Ñ Ò Ó t  Ó ï è A

ê ê (

( (2.20)

 

where Ù is the pure state visibility described in the previous section. If we consider a two level system as an input in the Mach-Zehnder interferometer, the intensity modulation in the

horizontal output channel is modified according to Fig. (2.8) for different values of the length

Ù   e   of the Bloch vector. We have considered a cyclic evolution ( )and  .

Let us now consider another definition of mixed state geometricê phase as proposed by

ð ¤ ñ

Uhlmann [13]. Consider a density operator of a system acting on a Hilbert space ¤ and let

ð Ü

be a copy of ¤ . Uhlmann then defines a ’lift’ of to be a so-called Hilbert-Schmidt operator

¿

ò ò ò

 Ü ñ ¤ ð mapping elements in ¤ to elements in and fulfilling the condition . Uhlmann’s

lift is equivalent to purification by a simple isomorphism

ò ò ò ò

ð ñ × ð 1 × ñ 1 A ð ñ × ð 1 | × ñ ó 1  ä ä

 (2.21)

ð ñ ð ñ

× ð 1 × ñ 1 ¤ ð ¤ ñ

where and are bases in and respectively. Clearly such a lift is not unique. From

ò ò ô ò ô

 ¤ ñ

a new lift can be constructed as © where is some arbitrary unitary operator on

¿

ò ò

Ü ©

since © determines the same .

? Ü Following Uhlmann, we can now define the lifts of pair of density operators Ü and to be

parallel if (

¿ ¿

ò ò ò ò

 ? H 9 D

? (2.22)

( ( If the lifts are infinitesimally close to each other and parallel according to Eq. (2.22), this results

in Uhlmann’s extended parallel transport condition along a curve as

¿

ò ò 

hermitian D (2.23)

¼

Q Q 2.2 Mixed state phases 13

I

1 r=1 r=0.5 r=0.1 0.5

0 χ 0 π/2 π 3π/2 2π 5π/2 3π

Figure 2.8: The modulation of the intensity in photodetector B as a function of for unitary

Ì

    9 D ö   9 D 

cyclic evolution of a qubit with  (solid), (dashed), (dotted). The solid

 

angle are given by e . ê

This condition can also be derived from the pure state parallel transport condition Eq. (2.8)

expressed in the language of Hilbert-Schmidt operators according to

¿

ò ò

+ A

Tr / positive and real-valued (2.24)

¼

Q Q

by applying an extra constraint. That is, if the parallel transport condition is invariant under the

ò ò

 À

transformation ,where À is an arbitrary hermitian operator then Eq. (2.23) follows. Q

Uhlmann justifiesQ this additional constraint by the minimisation of the length of the curve in the

Q

÷ ø  ¤ ð à ¤ ñ

extended Hilbert space ¤ , i.e.

l

¿

ò ò

r - ù ] D

Tr Y (2.25)

¼ ¼

Q Q

Note that the parallel transport condition Eq. (2.23) involves the extended Hilbert space and is

not restricted to the Hilbert space of the system. If the path of the lift fulfils these conditions,

9 A \ ]

the geometric phase acquired during the time interval Y defined as

¿

ò ò

o

m û

^ ‡  Ã Ä Å ú Tr A (2.26) which reduces to the standard geometric phase for pure states. (For more details see also [47, 48, 49]).

So far we have not imposed any restrictions on the evolution of the mixed state Ü , it can be either unitary or non-unitary. However, in order to compare Uhlmann’s approach withQ our

approach in paper I, let us only consider unitary evolution, i.e. solutions of

k

 Y 3 A Ü ] A

Ü (2.27)

¼

Q Q

where the Hamiltonian 3 is assumed to be time-independent. In terms of Hilbert-Schmidt

operators, the solution reads

?

ò ô

o ü

' ( 

Ü (2.28)

Q Q Q

2.3 Phases for completely positive maps 14

k k

ô

ü

 ¶ · ¸ + e - 3 /  ¶ · ¸ + - 3 þ / 3

with and , þ being Hermitian and assumed to be time Q

independent but otherwise arbitrary.Q For the lift to be parallel it has to fulfil Eq. (2.23), yielding

? ?

o o

o o

' ( ' (

 Ü 3 Ü  Ü 3 t 3 Ü A þ

þ (2.29)

o

þ þ

Ü 3

which fixes 3 if is of full rank. With this choice of the geometric phase is

¿

? ?

Ø

ò ô ò

ß ß

o o ü o

' ( ' (

/ Ü Ü ^ ‡  Ã Ä Å + +  Ã Ä Å

Tr / Tr

Ø

Q Q Q

ô

ü

m m

 Ã Ä Å ä á 1 D × 1 | × | á

ã (2.30)

ƒ ã ÿ

In paper II a comparison is made between Uhlmann’s geometric phase and the geometric

phase presented in paper I. We have already mentioned that the lift ò in Uhlmann’s approach

Q

ò o

is isomorphic to a pure state in Hilbert space of the system and an additional ancilla, ó

ô

o

1 ¢ ¤ ð à ¤ ñ à ¤

. The pure state evolves under aØ bi-local operator written in Schmidt

ß

Q Q

form as Ø

ô ò

+ × 1 / à + ¤ × 1 / A 1  ä

ó (2.31)

Q Q Q Q

ÿ

ü ¨ 

where the unitary operator for the ancilla is ¤ (transpose with respect to the instantaneous

Q

Q ü

eigenbasis of Ü ). If obeys the parallel transport condition Eq. (2.29) the quantity

Q Q

o

m

‡  Ã Ä Å | 1 A ^ (2.32)

is equivalent to Eq. (2.30). The geometric phase in paper I can also be analysed using purifi-

1 ¢ ¤ ð à ¤ ñ

cation, i.e. lifting Ü to as above and then evolving only the system part with

Q Q ô fulfilling the parallel transport condition Eq. (2.17). It can be shown that the two parallel

transportQ conditions, Eqs. (2.17) and (2.23) coincide only in the pure state case. Moreover,

1 Uhlmann’s geometric phase is derived from an evolution of an entangled pure state under-

going a bi-local unitary transformation while the geometric phase from paper I is derived from

1 an evolution of an entangled state undergoing a uni-local unitary transformation, i.e. the evolution of the ancilla is trivial and thus independent of the evolution of the system. Thus, this latter geometric phase is essentially a property of the system alone; the role of the ancilla is just to make the reduced state of the system mixed. The relation between the mixed state geometric phases given by Uhlmann and the one given in paper I have also been discussed in [50]. Paper II further includes an experimental realisation of Uhlmann’s geometric phase in po- larisation entangled two-photon interferometry [51, 52, 53], see Fig. (2.9). The two outgoing paths from the source represent the system and the ancilla, respectively. By measuring co-

incident events, the interference between the short and the long path can be observed in the ô

intensity pattern. Designing and ü according to the parallel transport conditions, Eqs. (2.17) Q and (2.23), both Uhlmann’s and paperQ I’s phase can be observed.

2.3 Phases for completely positive maps

In paper I only unitary evolutions of mixed state in interferometry were considered. A natural extension would be to define a geometric phase for a state that evolves non-unitarily in an

2.3 Phases for completely positive maps 15

¨ ¦

§

© ©  

© ©  

¤ Ì

D ? D

Source (

 



 u v

Q Q

Figure 2.9: Polarisation entangled two-photon interferometry to measure the mixed state geo- metric phase according to Uhlmann and paper II. interferometer, i.e. undergoing a completely positive (CP) map. This represents a decohering state where information of the state is leaked out into the environment. In paper III we define

such a phase under the restriction that the unitary representation of the CP map is known.

9 1 | 9

To model a CP map we couple the internal state Ü to an environment



÷ ÷

 Ü à 9 1 | 9 A

Ü (2.33)



÷ ÷

¿

¾ ¾

Ü  Ü

and let it evolve unitarily © . The evolved density matrix of the internal part is



÷ 

obtained by tracing over the environment,÷ yielding





¿

Ü ©  Ü ©  ä   Ü 

Tr A (2.34)



÷





¾

9 1  |  ¨  1 ­

where the Kraus operators are  [54] in terms of an orthonormal basis ,



÷ ÷ ÷ ÷

 9 A D D D A  e   Õ  e Õ  ,ofthe dimensional Hilbert space of the environment, being the dimension of the internal Hilbert space. If we consider modulation of the interference pattern in a Mach-Zehnder interferometer, as

described earlier, when the incoming state is given by Eq. (2.33), we obtain

 

¾

o o

 M 

Ù    Ü  A  

Tr Ü Tr (2.35)

   

÷ ÷



o

9  1  â

where | has been used. Further interference information can be obtained if we flip

÷ ÷

  9 1

the state of the environment associated with the reference beam to an orthogonal state .

÷ ÷

o " $ "

This flip is represented by the operator ! (Fig. (2.10)) and the interference pattern becomes

 



  

¿

¾ ¾

o " $ "

 M %

Ù   Ü 9 1 | 9 ! Ü 9 1 |      Ü 

Tr Tr Tr  (2.36)

     

÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷



 M %

  A D D D A  e  ¨ Ù  ­   9 A D D D A  e  for each  .Theset , , contains maximal information

about the interference effect associated with the CP map with known ¾ ,bymeasuringonthe  system alone. ÷ 2.3 Phases for completely positive maps 16

A

¨ ¦

¥

¦§

¾

©

©

¤ 

÷ B

¥



©

©

¤ ¤ Ì

o " $ " !

Figure 2.10: Interferometer for determining complete interference information of a quantum state undergoing a CP map.

There is a geometric content of the above prescription that can be obtained from parallel

transport of the internal state. The idea is to assume a continuous (time) parameterisation of

+ - /

each  and make a polar decomposition such that

     

ô ô

ü

+ - /  ) + - /  + - /  ) + - / - / A

+ (2.37)

þ

     

ô ô ô

ü ü

+ - /  + - / + 9 /  )

  

where is Hermitian and positive, and with specified by the type

þ

ô

o

) + 9 /  â  9 /  

of decoherence, + , and we assume that . The action of each is uniquely

þ Õ

defined up to phase factors by the evolution of the system’s density operator. This ambiguity must be associated with the corresponding unitary ô as the Hermitian part is unique, and the

Õ parallel transport conditions for each interference pattern are given by





¿

ô ô

| × + - / + - / × 1  9 A ×   A D D D Õ D

¼ (2.38)

þ þ

These conditions naturally extend the unitary conditions of Eq. (2.17) to CP maps. They are

sufficient to arrive at a unique notion of geometric phase in the context of single-particle in-

 9 A D D D A  e  terferometry. The set of these geometric phases for  provides the complete geometric picture of the CP map in interferometry, given the unitary representation. Let us consider an example where a qubit is affected by a depolarisation channel. A de-

polarisation channel is a model of a decohering qubit where the probability of the qubit being

e , ,

intact is  and probability of an error to occur is . There are three possible errors with

* 1  . ? * 1 * 1  . / * 1

equal probability: the bit flip error , the phase flip error , and both of

* 1  . * 1

the previous at the same time ,where

(

 9 9 e 9 

A A . /  é A .  é ?  é

. (2.39)

(

9 e  9  9

ë ë ë 2.3 Phases for completely positive maps 17

are the standard Pauli matrices. The Kraus representation of this channel is

ô

o

   e ,

þ

ÿ

, 3

ô

 ?  . ?

î

þ

3

,

ô

  .

î

þ

( (

, 3

ô

 /  A . /

î (2.40)

þ

¾

ô ô

+  /

by appending an Ô rotation . In the case of a cyclic rotation fulfilling the parallel

þ þ

transport conditions Eq. (2.38), the geometric phases are determined by the interference patterns

o

 5 

Ù    e , + Ñ Ò Ó +  / t  Ó ï è +  / /

ê ê

ÿ

 5 8

Ù ?   9

 5 :

Ù   9

3

(

 5 =

Ù /   +  Ñ Ò Ó +  / t Ó ï è +  / /

, (2.41)

ê ê ÿ

with the solid angle enclosed by the loop of the Bloch vector. The first interference pattern

ê

 e , Ù ? Ù is precisely that obtained in paper I modified by a visibility factor  . and vanish since the corresponding errors involve bit flips. The last interference pattern obtain a non-trivial(

change in the position of  . This is due to the phase flip that introduces a relative sign between the weights of the pure state interference patterns and is a pure effect of the decoherence. 2.3 Phases for completely positive maps 18 Chapter 3

Topological phases

Electromagnetic fields do not give a complete description of electromagnetism in quantum me- chanics. Even with vanishing fields, and hence no force, the electromagnetic potentials may give rise to measurable quantum effects. In 1949 Ehrenberg and Siday [55] predicted the existence of observable quantum interference phenomena associated with stationary magnetic fluxes, but the first ones to fully describe force free electromagnetic effects were Aharonov and Bohm in 1959 [2]. They considered interference properties of a charged particle in a superposition of going on both sides of an isolated flux line. After the discovery an intensive debate about the physical significance of the effect followed. In 1975 Wu and Yang [56] made a clarifying interpretation to this end based on non-integrable (path dependent) phase factors. The gauge arbitrariness of the vector potential in the Aharonov-Bohm (AB) set up was removed by stating that the inter- action between a charged particle and an electromagnetic potential is via a certain phase factor

which depends on the closed path integral of the vector potential. This non-integrable phase  factor is gauge invariant (adding a gradient of a scalar field only adds multiples of  to the phase when we integrate over a closed path) and thus represents a physical quantity. After Aharonov and Bohm’s influential paper, other non-integrable phases have been dis- covered, such as the Aharonov-Casher (AC) effect where an electrically neutral magnetic dipole encircling a line of charge acquires a non-integrable phase. The non-integrable AB and AC phases are often referred to as topological since they only depend upon the winding number of the path around the flux/charge line. For good reviews on the AB and AC phases, see, e.g., Refs. [57, 58, 59, 60]. An introduction to the AB and AC effects is given in section (3.1). First the AB phase including its relation to the geometric phase is described in (3.1.1). Then the AC phase is discussed in (3.1.2) with a short discussion about its topological nature. In (3.1.3) the impor- tance of the vector potential is delineated. The non-integrable AB and AC phase factors are the fundamental description of the electrodynamics of moving charges and magnetic dipoles. Section (3.2) is based upon paper IV where the classical dynamics of the flux line in the AB set up is investigated in the adiabatic approximation. It is shown the adiabatic approxima- tion breaks time (motion) reversal symmetry for flux lines in the case where a full quantum mechanical treatment would restore it. In section (3.3) the AC Hamiltonian is used in order to determine the energy quantisation for electrically neutral magnetic dipoles. It is shown, based upon paper V, that for specificelectric

19 3.1 Introduction 20

fields one may acquire an equidistant quantisation, reminiscent to the Landau effect [61] for a charged particle in a homogeneous magnetic field.

3.1 Introduction

3.1.1 Aharonov-Bohm phase The first to discuss topological phases in quantum mechanics were Aharonov and Bohm in 1959

[2]. They showed that a quantum particle with charge ? circulating a magnetic flux line with

magnetic flux @ acquires a topological phase given by (SI units)

? @

k

^ A B  D (3.1)

The magnetic flux line could be generated by a very long solenoid or an array of magnetic dipoles (see Fig. (3.1)). The effect was surprising since the magnetic field vanishes outside the flux line and the charged particle is in a force free region only locally affected by a gauge de- pendent vector potential not viewed as a physical quantity. This is the reason why the AB effect is also known as a non-local effect. The effect has been observed in interferometry experiments [62, 63].

µ

µ

B µ A q µ

µ

Figure 3.1: The AB configuration with an array of magnetic dipoles.

To derive the AB phase shift let us consider the probability of finding a charged particle at À

point Á when starting at in the vicinity of a flux line, cf. Fig. (3.1), using Feynman path Á

integrals [64], i.e. adding the phase factors of all possible paths going from À to .The o

Lagrangian C of the system without the flux line acquires an additional term in the presence of

the flux line according to

o o

 C t ? E z G A C (3.2) 3.1 Introduction 21

where the last term is due to minimal coupling and E is the velocity of the charged particle and

H  } ¡ G  9

G is the vector potential consistent with the magnetic field outside the flux

o o

 Ô t J Ô

line. The additional term to the Lagrangian changes the action according to Ô , Ô

where J is given by

l

B

G z r 5 A Ô  ?

J (3.3)

A

5 À Á

r being the line element along a specific path going from to . Summing over all possible

Á J Ô paths connecting À and results in the same integral for the ones going to the left of the flux line and those going to the right, since the integral only depends on the endpoints if no

magnetic flux is enclosed. Thus, the phase difference for a particle in a superposition of going 

to the left ( á ) and to the right ( )oftheflux line, is given by

l

J Ô ã e J Ô K ? ? ?

k k k k

^ A B  H z r O  w x G z r 5  @ A

 (3.4)

N

Q G where H is the magnetic field associated with the flux and the surface enclosed by .This is a purely quantum effect since in classical physics the motion of the charged particle is given by the Lorentz force which vanishes where the particle moves in this set up. Moreover, from

this it can be seen that the vector potential cannot vanish everywhere in the field region since G the integral of G along any closed circuit that contains the flux line is non-vanishing.

The close relation to the geometric phase can be seen in the following way. As shown previ-

* 1 ously, an adiabatic cyclic transport of the state vector leads to a geometric phase according

to

‡  w x G + 5 / z r 5 A

^ (3.5)

+ 5 /  | T + 5 / }  T + 5 / 1 T + 5 / 1 where G , being the cyclic eigenstate of the Hamiltonian, is the gauge potential which determines the connection in parameter space. The AB phase, given by Eq. (3.4), may be derived from the geometric phase if we consider closed paths [10]. The common feature is that both phases are non-integrable, i.e. independent of the initial and final value of the integrand. But whereas the geometric phase is local due to dependence of local changes of the physical state, the topological phase is non-local in the sense that it cannot be defined at a point in space but only as a closed integral enclosing a magnetic flux or not, i.e. solely dependent upon a topological structure. This is the reason why there is non-cyclic geometric phases, but no non-cyclic topological phases [21]. It has been shown that one can formulate the non-cyclic geometric phase in terms of a gauge-invariant reference section [65, 66], which also shows that the geometric phase is local. A geometric illustration of the topological nature of the AB phase is given by considering parallel transport of a vector on a cone, see fig. (3.2). The cone has no intrinsic curvature, since

it can be constructed out of a flat plane by joining two straight edges at an angle ® , except at the tip where the curvature is non-vanishing. This is an analogue to the AB effect where the tip

represents the magnetic flux line defined by the angle ® and the flat space around represents the absence of magnetic field. When a vector is parallel transported on the cone without encircling the tip, no holonomy is obtained whereas when the vector encircles the tip it will be rotated when returning to the starting point, analogous to the AB phase. The holonomy (angle) is independent of the path going around the edge as illustrated in Fig. (3.2). 3.1 Introduction 22

α

α

α

Figure 3.2: Parallel transport on a cone where the cone represents the vector potential.

Aharonov and Bohm also described a phase effect for a charged particle due to an electric

field, known as electric AB phase (EAB). In this set up a particle in an interferometer, experi-

Y 9 A \ ]

ences a homogeneous scalar potential ” during a time interval in one of the beams giving ”

rise to a shift ? in potential energy. No force is acting on the particle when it is inside the

”  9 region where the scalar field is applied, since } . Nevertheless, there is a testable phase

difference between the beams given by

m

l

?

k

” + - / r - D R A B 

^ (3.6) o

3.1.2 Aharonov-Casher phase

In 1984, Aharonov and Casher [67] showed that the Lagrangian for a particle with charge ? and

an electrically neutral particle with magnetic dipole moment T given rise to the vector potential

G is given by

ü

 ( (

t U W t ? G + Y e 5 / z + E e [ / A 

C (3.7)

 

 E 5

where Y , ,and are the position, mass, and velocity for the charged particle and , ,and U

[ the corresponding quantities for the magnetic dipole. The first two terms in the Lagrangian

are the kinetic energy for the charged particle and magnetic dipole, respectively. From this

G + Y e 5 / z [ expression it can be noted that there is an interaction term ? for the magnetic dipole although it is electrically neutral. This term in the Lagrangian is necessary since otherwise the charged particle will experience a gauge and Galilean frame dependent force in the region with vanishing magnetic field. Note that the interaction term only depends on the relative position and velocity. Thus, the effect is independent of whether the charge is moved around the magnetic dipole or vice versa.

Therefore we should expect a dual effect to the AB effect when considering stationary charges.

5 Y

The Lagrangian for the magnetic dipole T at and a charged particle fixed at position (i.e.

 9

E ) is thus given by

(

U W

C  ? G + Y e 5 / z [ D

e (3.8)  3.1 Introduction 23

From electrodynamics we know that the vector potential at Y from a classical magnetic dipole 5

moment T at is given by [68]

 T ¡ + Y e 5 / 

G + Y e 5 /  `  e ¡ e + Y e 5 / A

T (3.9)

/

o c c

Y e 5 ?  a

( (

o

5 Y a where e is the electric Coulomb field at due to the charged particle at , is the vacuum

permittivity, and c is the speed of light. The Lagrangian now becomes

( 

U W

C  t ¡ e + Y e 5 / z [ A

T (3.10)

c

 (

which via a Legendre transformation defines the Hamiltonian

(

 

3  T ¡ e A e f

é (3.11)

c

 

ë

(

 g C g [ where f is the canonical momentum of the magnetic dipole. We can thus identify

the vector potential for the neutral particle as

h



G  ¡ e D

T (3.12)

c (

If we consider a particle with “anomalous” magnetic moment, as a spin half particle, the Hamil-

+ e ( /

tonian Eq. (3.11) is modified slightly according to (neglecting terms of i )[69]

k

(

  

e } z e A 3  T ¡ e e f

é (3.13)

c c

ë

   

( ( e

where the last term is due to the non-commutativity of f and . Also here we get the vector k potential in Eq. (3.12), but its origin is different here; it arises from a non-minimal coupling

term proportional to the field tensor ! [70, 71]. A consequence of Eq. (3.12) is the existence of an observable topological interference shift

when a magnetic dipole T encircles a line of charge (see Fig. (3.3)). This Aharonov-Casher

(AC) phase shift is given by ß

h

x

 

k k

A G z r 5  ^ A 

w (3.14)

o c

ß

a ( where is the electric charge per unit length. In order to prevent the magnetic dipole from

precession due to the electric field, i.e.



T  ¡ + E ¡ e /  9 A

T (3.15)

¼

c ( the magnetic moment has to be parallel to the line of charge and move in the plane of the electric field. If these requirements are fulfilled the dipole does not feel any force and the effect is topological in the sense that it depends only upon the winding number of the path. 3.1 Introduction 24

q

q

q

µ q

q

Figure 3.3: The AC configuration with a line of charges.

Furthermore, the expression for the AC phase Eq. (3.14) can be compared with that of the AB

phase Eq. (3.4) and we may identify a duality relationß according to



 @ ó

D (3.16)

c o

a (

This duality relation expresses the fact that the AC effect can be obtained from the AB effect by interchanging the role of the charged particle(s) and the magnetic dipole(s) in Figs. (3.1) and (3.3). Experiments to test the AC phase have been performed in neutron interferometry [72] and in atom interferometry [73, 74]. Also in the AC case a scalar effect can be found, known as the scalar AB (SAB) effect [75].

In this set up a dipole in an interferometer experiences a homogeneous magnetic field H in one

of the beams in the same way as the charged particle experiences an electric field in EAB. This

z H

gives rise to an increase in potential energy according to T (Zeeman effect) but no force

+ T z H /  9

( } ). When the beams intersect the SAB phase is given by

m

l



k

Á + - / r - A l A B 

^ (3.17) o for a dipole moment parallel to the magnetic field. There has been an intensive discussion to which extent the AC effect can be regarded as a physical analogue to the AB effect or not, i.e. topological or not. In [75] it was pointed out that the main feature of a topological effect is that it is non-dispersive, i.e. independent of the velocity of the particle. This is fulfilled for both the AB and AC phase. Another stronger definition of a topological phase was put forward in [76]. There it was claimed that a topological phase should be associated with a multiply connected region, which is true for both AB and AC, and also that the phase shift should not be possible to localise to a specificregionofthe 3.1 Introduction 25 interferometer. For the AB phase only the difference of phases between two paths encircling the flux line is gauge invariant, not the phase of an open path. In the AC effect there is an electric field where the particle moves although it does not result in a force on the particle. The presence of this field introduces other interactions such as angular momentum fluctuations and the AC phase can be explained in terms of local exchange of angular momentum between the electric field and the particle. Thus the phase can in principle be measured locally and therefore the AC effect is non-topological according to [76]. The special dependence between the AC phase and the spin has been investigated in the past [69, 77]. Using the Maxwell electromagnetic duality relations two additional topological effects can be found. The Maxwell dual to the AB set up is a hypothetical magnetic monopole encircling a line of electric dipoles proposed in [78] and the Maxwell dual to the AC set up is an electric dipole encircling a line of magnetic monopoles proposed in [79, 80]. Topological phases for higher multipole moments have also been delineated in the literature [81].

3.1.3 Fundamental electromagnetic description As shown above, in quantum mechanics we may observe electromagnetic effects on the wave function of a charged particle or an electrically neutral particle with magnetic dipole moment, also with vanishing forces. This raises the question as to what precisely constitutes a complete and intrinsic description of electromagnetism. In a paper by Wu and Yang from 1975 [56] it is claimed that “through an examination of the AB experiment an intrinsic and complete de- scription of electromagnetism in a space-time region is formulated in terms of a non-integrable

phase factor.”. So the non-integrable AB and AC phase factors,

„ 

q r s u

 M n o  p

   (3.18)

and



„ 

q r s x u

8

 M n w 

A G     T ¡ e

þ (3.19)

c ( respectively, are the fundamental description of electromagnetism for charged particles and magnetic dipoles.

Let us now consider the classical Lorentz force

z

? + E ¡ H / A

 (3.20)

E H which acts on a particle with charge ? and velocity in a magnetic field . In order to explained this force in terms of topological AB phase factors we consider the magnetic field as being build up from infinitely many flux lines and sum over all Feynman paths for the charged particle enclosing different amount of flux for different paths. The resulting interference pattern after summing over all Feynman paths is shifted due to the magnetic field. This shift of interference pattern is the same as predicted by the classical Lorentz force. This shows that the fundamental phase factor Eq. (3.18) also accounts for the classical electromagnetic forces. The same holds for the dipole case. 3.2 Dynamics of an Aharonov-Bohm system 26

3.2 Dynamics of an Aharonov-Bohm system

In the previous section we have seen that when a particle with associated magnetic flux (fluxon) encircles a charged particle it acquires a phase. Here we address the issue, what happens if the path of the fluxon encloses only a fraction of the total charge. Indeed, Aharonov and cowork- ers [82] have delineated an interesting interplay between the geometric and AB phases when the fluxon is adiabatically taken around a loop through a charged quantum cloud. This result displays a subtle aspect of the action-reaction principle in this charge-fluxon system, namely the expected AB phase is accompanied by a geometric phase due to the rearrangement of the charged quantum cloud when the fluxon moves through it. In paper IV we extend this work and let the fluxon be free to move. We examine the forces on the fluxon under the assumption that the adiabatic approximation is valid, i.e. the mass of the fluxon is assumed to be much larger than the mass of the charged particle. Let us first briefly review the ideas behind the adiabatic approximation [83]. Consider the

time-independent Schr¨odinger equation for a system which consists of light and heavy degrees 5 of freedom denoted by Y and , respectively. Such a system can, for example, be a molecule

consisting of light electrons and heavy nuclei. Due to the large difference in mass one can 5

treat the motions of Y and separately. The procedure is first to treat the problem of the light

degrees of freedom in a stationary surrounding of the heavyk degrees of freedom giving the states

5 | + Y } 5 /

of the former parametrically dependent upon , i.e. in position representation.k The

5 + 5 /

corresponding energy eigenvalues for each define the potentialk energy surfaces on

p

+ Y A 5 / which the heavy degrees of freedom move. The total state * in position representation

is in the adiabatic approximation then given by

k

k

O S

* + Y A 5 / ~ | + Y } 5 / + 5 / A

 (3.21)





k

Ì

O S

where is the solution of thek time-independent Schr¨odinger equation for the heavy degrees



Ì

5 / of freedom in the potential + . The validity of the adiabatic approximation depends upon the size of the coupling termsp between different states of the light degrees of freedom. The

accuracy of the description may be improved by adding more states according to

k

k

O S

* + Y A 5 /  | + Y } 5 / + 5 / A k

ä (3.22)



 Ì

and the exact solution is obtained when Ù is run over a complete set. The adiabatic approximation is the basis for the molecular AB (MAB) effect [7, 8, 9] where the electronic states acquire an adiabatic geometric phase when they are transported around a conical intersection (see Fig. (3.4)) in parameter space of the nuclei. This results in an AB type vector potential in the Hamiltonian for the nuclei that may give rise to a topological phase. The close similarity with the AB effect gave its name, but as pointed out in [84] the two effects are physically different as MAB is essentially local since it can be derived from the non-cyclic adi- abatic geometric phase of the corresponding electronic motion locally along the path in nuclear parameter space. Now, returning to the problem of paper IV, let us consider the time-independent Schr¨odinger equation of a light charged particle and a heavy fluxon within the adiabatic approximation. The 3.2 Dynamics of an Aharonov-Bohm system 27

Energy

X

Y

Figure 3.4: Born-Oppenheimer potential energy surfaces for the nuclei with conical intersec- ¤ tion. The  and axis represents different nuclear configuration degrees of freedom.

Hamiltonian of this system is given by (note that SI units are used here in contrast to paper IV)

k k

( ( ? ?

( (

k k

3  e G + Y e 5 / / e G + Y e 5 / / t + Y / A } € e + }  t

+ (3.23)

W

   U

where the two first terms are the kinematic energy of the charged particle and the fluxon, re-

? Y spectively, with the charged particle with mass  and charge at ,andthefluxon with mass

at 5 . The last term has been added as a confining potential for the charged particle. The

time-independentU Schr¨odinger equation for the charged particle at a fixed fluxon position reads

k

k k k

(

?

(

Y e k G + Y e 5 / / t + Y / ] | + Y } 5 /  + 5 / | + Y } 5 / D } € t

+ (3.24)

p

W

 

The fluxon problem is thereafter given by

k k

O S O S

3 + 5 /  5 / A

eff + (3.25)

p



 

Ì Ì

with the effective Hamiltonian

k

k k k k k

(

(

3  | | + 5 / 3 | + 5 / 1  e + }  e G + 5 / / t + 5 / t + 5 / A þ

eff þ (3.26)

p p



U

k k

z 1 Y

where | stands for integration over only. The total state of the system can, in the adiabatic

k

O S O S

* + Y } 5 / „ | + Y } 5 / + 5 / 3 5 /

regime, now be written as , + being the fluxon state. In eff,



  Ì

the adiabatic vector potential is Ì

k k k

?

k

G / | + 5 / 1 G + 5 /  | | + 5 / + }  t þ (3.27)

3.2 Dynamics of an Aharonov-Bohm system 28

k

5 /

and there is a correction to the standard potential energy + of the form

p

k

k k k k k

? ? (

k G / | + 5 / z +  e | + 5 / 1 | | + 5 / / + }  t k G / | + 5 / 1 D + 5 /  | + }  t

þ (3.28)

p



k

U G

It is further shown in paper IV that the adiabatic vector potential þ is invariant under a local,

Y e 5 / i.e. + dependent, gauge transformation.

To illustrate the above situation, let us consider a harmonic confining potential centered at 5

e , i.e.

 

( ( ( ( ( (

+ Y /  ‡ + Y t 5 /   ‡ +  t = / t  ‡  = Ñ Ò Ó + ‰ e Š / A

 (3.29)

W

 

= A Š / +  A ‰ / 5 Y ‡ where + and are the polar coordinates of and , respectively, and is the angular frequency of the oscillator. The fluxon is situated at the origin in this choice of coordinate

system. In Coulomb gauge we may write the vector potential as

‹

k

‰

+ Y /  ® A

G (3.30)

 

where ® is proportional to the flux.

?

 ® G

Let us focus on the semi-fluxon case characterised by ® . For this -value, and

( G

e only differ by a single-valued gauge. In other words, the system obeys time (motion)

 e G / (  +  t G / (  +  e G t } / ( }

reversal symmetry since + ,where is a single- Ì

valued gauge that does not change the physics. The charged particle’sÌ wave function is obtained

‡ (  = Ñ Ò Ó + ‰ e Š / by diagonalising the perturbation term  in Eq. (3.29), in the unperturbed

oscillator basis characterised by the angular momentum quantum number á and the principal

quantum number T . It is found that the potential energy surfaces of the fluxon are degenerate at

 9 á  9 á   5 5 for and , and furthermore have the same form as in Fig. (3.4) when moves away from the origin. Inserting the energy eigenfunctions of the charged particle into Eq. (3.27)

we obtain the adiabatic vector potential

‹

Š  ‡ 

?

O ' ( S

k ”

 G _

þ é

_ ƒ ‘

î (3.31)

 = ’ 

ë ” where _ is determined by properties of the oscillator eigenfunctions (for further details, see

Eq. (19) in paper IV) and the index – indicates the diagonal basis states. Integrating the adia-

batic vector potential around a closed curve G in the fluxon parameter space the phase is

 ‡ 

?

O ' ( S

k ”

^   – _ ˜ A

_ ƒ ‘

î (3.32)

 

where ˜ is the length of the loop. The first term on the right-hand side in this expression is the adiabatic geometric phase. The appearance of the second term shows that there have to be a force on the fluxon and that time-reversal symmetry does not hold. This effect originates from the AC modification. The broken time-reversal symmetry in this charge-fluxon system can be compared with the breaking of the full rotational symmetry in the adiabatic treatment of a molecular system. In the adiabatic approximation this continuous symmetry is lowered to a point group description of the molecule. The main point in paper IV is that an adiabatic approach is generally inconsistent with time-reversal symmetry and thus may also break symmetries of discrete anti-linear/anti- unitary nature. 3.3 Dual Landau levels 29

3.3 Dual Landau levels

The standard Landau levels [61] are the energy solutions for a particle with charge ? moving in

e ¤ H the  plane perpendicular to a uniform magnetic field . This system is described by the

Hamiltonian

k

(



(

3  š + e } e ? G / D

 (3.33)

    G where  is the mass of the particle and is the vector potential. To find the energies of the

system we identify the only non-vanishing commutation relation as

k

¦ A ¦ œ ]   ‡ A

Y (3.34)

ø

 ? Á   e ¤ where ‡ . It follows that the motion in the plane can be transformed into a one

dimensional harmonic oscillator accompanying free motion in the  direction. Thus, the energy

e ¤

eigenvalues for the  motionaregivenby

k

k



‡ A Ù  9 A  A D D D ž Ù t

 (3.35)

p 

It can be noted that the energies are independent of the orbit centre and this results in an energy

degeneracy associatedk with the quantisation of the corresponding orbit centre operator. The

* ¡ ƒ ­ ¡ £ degenerate set ¨ , which runs over the orbit centre eigenvalues ,defines a Landau level. These Landau levels play an important role in condensed matter physics as, e.g., in the quantum Hall effect. In ref. [85], the existence of a dual quantum Hall effect for neutral atoms was pointed out when considering anyonic excitations in rotating Bose-Einstein condensates. In paper V we take another approach towards an atomic quantum Hall effect by the developing the theory of

dual Landau levels, i.e. Landau levels for electrically neutral particles with magnetic moment

k

T   ¤ e e } e

in an electric field , where the Hamiltonian is given by Eq. (3.13). Here 

š

c

 ˜ ( + ¤ ¡ e /

is the kinematic momentum. If we consider the commutation relation between ¦

ø œ and ¦ we find the similar relation as Eq. (3.34) for certain electric field configurations with

respect to the dipole direction ¤ . The conditions to be fulfilled are

/

+ no dipole precession

/

+ electrostatics

/ H  } ¡ G þ

+ homogeneous eff .

+  A ¤ / A œ +  A ¤ / A 9 /  +

The first two requirements are fulfilled for an electric field e ,where

p p

ø

g  9 ¤  + 9 A 9 A  /  e ¤ e g œ

œ , and a dipole pointing in direction andmovinginthe plane.

p p

ø ø

o

¤ + / Ü With this e and , condition leads to a non-vanishing uniform volume charge density

where the dipole moves. Under these conditions we obtain the commutation relation

x

k

¦ A ¦ œ ]   ‡ A A

Y (3.36) ø

where the cyclotron frequency is

o o

x

 Ü

.  Ü

‡ A  D

 (3.37)

c c o o

  a a

( (

3.3 Dual Landau levels 30

 –  Here, we have introduced . which determines the revolution direction of a classical

dipole in this e . Proceeding as in the standard case we obtain the energy eigenvalues as

k

x

k



ž

+  t . /  Ù t ‡ A A Ù  9 A  A D D D

(3.38)

p 

It is surprising to note that the energy eigenvalues depend on the revolution direction . ,in

contrast to the standard case described above. Thus the set of Landau levels separates into two

k

O ¦ S

¡ ƒ

­ ¡ ¨ * classes, one for each value of . and denoted by . The energy separation between the dual Landau levels can be found from the duality relation between the AB and AC effect Eq. (3.16) as follows. The separation of the energies for the

standard Landau levels,

k

 ? Á  A

J (3.39)

p

k

 ? @ Ô  Á  @ Ô

can be expressed as J , where the magnetic field is expressed as , p

Ô being the area where the flux is measured. Now, using Eq. (3.16) we may write the energy ß

separation for the dual case as

k

ß

o c

(

a A   Ô 

J (3.40)

p

o

Ü  Ô where gives the dual energy separation we already derived. It is interesting to note that for dual Landau levels we can have two different experimen- tal configurations which are related by a gauge transformation of the AC vector potential and therefore give the same physical effect. This is due to the physical significance of the AC vector

potential, i.e. its dependence upon physical quantities as the electric field and the direction of ¤ the dipole moment. So two pairs of e and may yield the same effective magnetic field. For

example, this gauge freedom can be realised by considering a uniform volume charge density

o

 e ¤  Ü in a region with different shaped cross sections and a magnetic dipole in direction. If we have an infinitely long cylindrically shaped region (Fig. (3.6)), the electric field inside the

cylinder is given by

o

Ü

+  A ¤ A 9 / D 

e (3.41)

o

 a

Z

Y

X o

Figure 3.5: Cylindrically shaped uniform volume charge density Ü . 3.3 Dual Landau levels 31

The corresponding vector potential is

o

x

Ü

G A  e ¤ A  A 9 / A

+ (3.42)

c o

 a (

which is the AC analog to symmetric gauge. For a uniform volume charge density inside a plate

¤ 

with finite width in  direction but infinite in and directions (Fig. (3.7)), the electric field is

o

Ü

+  A 9 A 9 / D 

e (3.43)

o a

Z

Y

X o

Figure 3.6: A plate with uniform volume charge density Ü .

The corresponding vector potential is

o

x

Ü

G A  9 A  A 9 / A

+ (3.44)

o c

a (

which is the AC analog to Landau gauge. The two vector potentials differ by a gauge function



 ¤ ª

 .

: ¬



( « Ì The dependence of the Landau levels on the revolution direction in the AC case can be

understood from supersymmetry (SUSY). To do this let us introduce extended Fock states

?

+  t . / £ A T B A T ­ 1 T B  Ù T ­  with the number of bosons and the number of fermions . The bosonic degree of freedom is associated with the oscillator and the fermionic one( with the

spin, thus increasing Ù increases the number of bosons and flipping the spin creates or annihi-

£ A T B A T ­ 1 ­ ¡ lates a fermion. A Landau level can now be defined as a set ¨ of extended Fock

3.3 Dual Landau levels 32

£ A 9 A 9 1 ­ ¡ states. The lowest Landau level corresponds to ¨ , in which the number of fermions and bosons both vanish. Higher levels are divided into two classes containing one or zero fermion and one can go between the first to the second of these two classes by simultaneously adding a boson without changing the energy. This expresses the property of unbroken SUSY. Chapter 4

Quantum Computation

Quantum computation uses quantum phenomena, such as interference and entanglement, for information processing. Feynman [86] posed the first question in this direction in 1982 when he asked if the behavior of every quantum mechanical system can be efficiently simulated by a computer or a classical simulator. In 1985 Deutsch [87] proposed the first quantum algorithm and also described a universal quantum computer. After Deutsch several quantum algorithms have been proposed, culminating with the important discovery of Shor’s algorithm [88] in 1994. With this quantum algorithm it was shown that quantum computers, at least in principle, can factor large numbers in such a way that the execution time of the algorithm grows polynomially with the number of digits of the number to be factored. In comparison for any known classical algorithm the execution time grows exponentially. Another quantum algorithm, discovered by Grover in 1996 [89], makes it possible to search in an unsorted data base for an element and is faster when compared to the classical algorithms by the square root of the number of elements in the data base. The theory for a quantum computer is in principle set, the main obstacle is practical, i.e. to maintain the coherence in the system when we increase the number of computational units. An important method in fighting the impurities in quantum systems is the error correction method proposed by Deutsch [90]. He suggested a symmetrisation procedure (for details see [91]). This idea has been step by step improved, e.g. by Palma et. al. who introduced decoherence free subspaces [92], and in particular by Shor [93] who proposed a quantum analog of classical error correcting codes. In this method it is possible to use redundant encoding of qubits to detect and correct quantum gate errors due to interaction with the environment. Another method to reach coherent quantum computing is to use geometric or topological quantum computation. For good reviews see, e.g., [94, 95, 96, 97]. In section (4.1) an introduction to quantum computation is given. Also experimental real- isations are mentioned. Section (4.2) deals with geometric quantum computation, based upon paper VI. Section (4.3) describes an idea how to use the Aharonov-Casher effect from Chapter 3 to build a conditional phase gate. The section is based upon paper VII.

33 4.1 Introduction 34

4.1 Introduction

The basic building block in quantum computation is the qubit, i.e. any quantum mechanical

9 1  1 two level system. If we denote the orthonormal basis for the system as and (called the

computational basis) we can express an arbitrary normalised state of the qubit as

‰ ‰

 M

* 1  Ñ Ò Ó 9 1 t Ó ï è  1 D

 (4.1)

 

Since a qubit can be in a superposition it can store two values at the same time, e.g. the Boolean

9 1  1 9  states and , whereas a classical bit is restricted to only one value, or . A qubit can be realised with an atom, nuclear spin, or a polarised photon. If we have several qubits, the

amount of information they can store in a given moment of time increases exponentially with

* 1    + 9 9 1 t 9  1 t  9 1 t   1 / the number of qubits. If we have 2 qubits in state ,

then at a given moment of time we have stored 4 numerical values (with equal probability

 ®

amplitude). In general Õ qubits can store numerical values (in a quantum superposition) at

®  any moment of time whereas classical Õ bits can store only one out of numerical values. Another important quantum phenomena is entanglement. Two (or many) qubits are entangled,

non-separable, if the total state cannot be written as a tensor product of the two individual qubit

* ? ƒ 1  * ? 1 à * 1

states, i.e. .

( ( If we have a register, i.e. a collection, of qubits in some initial state, we can perform quan- tum operations on these according to some prescribed procedure called a quantum algorithm.

The computation is then performed in a parallel manner on the different basis states in the super-

 ®

position. If we have Õ qubits, i.e. parallel basis states, we can run one parallel computation ® that would require  computations on a classical computer. After the computation the final state of the qubits carries the output information. Because measurements destroy superpositions it is not possible to restore all information from the calculation. But with quantum interference we may increase the probability amplitude of the desired output so that the measurement gives the correct result in most of the cases. A fixed unitary quantum operation performed on a selection of qubits is called a quantum logic gate. Several logic gates, synchronised in time, form a quantum network where the size of the network is the number of gates it contains. The one-qubit Hadamard gate and the two-qubit

conditional phase gate together forms a universal set of gates, i.e. any Õ -qubit unitary operation can be simulated exactly using networks made out of the two gates. This is not a unique set, almost any two qubit gate that can entangle the qubits can be used as a universal gate [98, 99]. Here follows a description of the universal set with the Hadamard gate and the conditional phase shift gate. A single phase gate is also considered since it is not efficient to use a condi-

tional phase gate for single qubit phase shift operations. ¯

The Hadamard gate is the single qubit gate ° performing the unitary transformation

? ?

± ±

9 1   1  9 1 t  1 / + 9 1 e  1

known as the Hadamard transform defined by + , ), (

i.e. creating superpositions. We can also write the Hadamard( gate as



 

? ?



é

±

S ² ´ ´ ˜ O ˜

ø µ · ø µ ¹

D  1

H H (4.2)

 e 

ë

  (

4.1 Introduction 35

9 1 A  1 ­

The matrix is written in the basis ¨ and the diagram on the right provides a

 1   9 A  schematic representation of the gate 3 acting on a qubit in state , with .A physical realisation of a Hadamard gate could be a beam-splitter in an interferometric set

up. ¯

Using the same notation we define the phase shift gate º as a single qubit gate such

 M

9 1 » 9 1  1 »   1 ^ that and , for some prescribed . The matrix notation and the

structure diagram are given by

^

¼

 9



é

 1 ½   M  1 D

ø (4.3)

9   M ë

A physical realisation of a Phase shift gate can be a phase-shifter in one of the beams of an interferometer. With the above single qubit gates, the Hadamard and the phase shift gate, any single qubit transformation can be realised (up to a global phase) according to a simple network shown

below, in diagrammatic representation,

%

t ^

‰

¾ ¾

(

?

Ñ Ò Ó ¿  1  1 t   M O ˜ ( S Ó ï è ¿  e  1

H H ø

( (

9 9 D D D 9 1 * ? * D D D * 1

With this network we can transform the Õ -qubit state to ,where

( ®

9 1  1

each * is an arbitrary superposition of and . These are so called product states 

or separable states, which is a restricted set of Õ -qubit states. To transform to the non-

separable states, i.e. the entangled states, we need two-qubit gates.

¯

9 9 1 » 9 9 1 9  1 » 9  1  9 1 »

The two-qubit conditional phase gate is defined as: , ,

 M

 9 1   1 »    1 ^

, , for some prescribed © . The matrix notation and structure diagram

9 9 1 A 9  1 A  9 1 A   1 ­

are given by (the matrix written in the computational basis ¨ )

ÃÄ Æ Ç

É Ê

ÊË

Ä Ç

 9 9 9

½

 1

9  9 9

œ

Ê

Á + ^ / 

 M

ø

^   1 ¤ 1 D

© (4.4)

Ê

Å È

9 9  9

Ì

9 9 9   M

½

¤ 1

This gate makes it possible to entangle two (or more) qubits.

The universal set of gates cannot be constructed from phase shifts (regular or conditional) alone. There is a need for operations that mix the computational basis, such as the Hadamard gate. In order to implement such gates in a geometric way we need to be able to construct non-Abelian geometric and topological gates. Examples of experimental realisations of gates have been performed using, e.g., nuclear magnetic resonance, ion traps, cavity QED, and photonic systems. For an overview of experi- mental techniques in quantum information science, see [95]. 4.2 Geometric quantum computation 36

4.2 Geometric quantum computation

To achieve fault tolerant quantum computation we can use the geometric phase, described in Chapter 2, to implement quantum gates, so called geometric quantum computation. Such phases are, as mentioned before, fault tolerant to state space area preserving operations. I will now go through a universal set of gates based on the adiabatic geometric phase. The description of the single qubit phase gate and a two qubit conditional phase gate is based on paper VI whereas the Hadamard gate description is based on Refs. [100, 101]. Since the Hadamard gate is based on non-Abelian adiabatic geometric phase description we begin with a brief overview of non-Abelian geometric phases. The adiabatic geometric phase was considered in Chapter 2. If an energy eigenstate of a

quantum system depends on a set of external parameters 5 then an adiabatic cyclic variation of these parameters returns the system to its original state. The final state vector turns out to

be related to the initial state vector via the product of a dynamical phase factor and a geometric

· ¸ + ^ ‡ /

phase factor [10] ¶ , which depends only on the shape of the path in parameter space. If

* 1 + - /

the state belongs to a degenerate subspace it remains in the degenerate subspace during

* + \ / 1

an adiabatic evolution. However, the system returns in general to a final state related to

¾

‡ * + 9 / 1

the initial state by some unitary operator

m

l

¾

k

* + \ / 1  ‡ * + 9 / 1 ¶ · ¸ Í e - / r - Ï A

+ (4.5)

o p

¾ ¾ ‡

where ‡ depends only on the shape of the path in parameter space. The matrix is a gen-

· ¸ + ^ ‡ / eralisation of the geometric phase (multiplication by the ¶ factor) into non-Abelian

cases [11].

¾ ‡

In order to evaluate let us choose a set of Õ -fold energy degenerate reference states

T + 5 / 1 ­

¨ , being local in parameter space. Then at any point on the adiabatic path in the pa-

+ - /

rameter space 5 we can write

¾

ã _ + - / T + 5 + - / / 1 D * ã + - / 1  ä

(4.6)

_

¾

* ã + \ / 1  Ý ‡ Ð ã _ + \ / T 1

In particular, for a closed loop the final state _ is related to the initial

¾

á 1 _ + \ / state, chosen to be ,via ã . These matrix elements can be evaluated as the path ordered

line integral

l

Q

¾

r 5

r - ¶ · ¸ À + 5 /  Ï

P Í (4.7)

o

r -

with the gauge potential defined as

g

”

À ñ Ñ + 5 /  | Ò 1 D

(4.8)

g 5 In the Abelian case, i.e. the geometric phase, the path ordering is not necessary and the integral reduces to a regular line integral. An experimental test of the non-Abelian adiabatic geometric phase has been performed in [31]. The non-Abelian geometric phase has also been generalised to non-Adiabatic [102] and non-cyclic evolutions [103].

4.2 Geometric quantum computation 37

T + ‰ A | / 1 The simplest geometric gate is a single qubit phase gate. The reference state can

be written as

‰ ‰

 Ó

T + ‰ A | / 1  Ñ Ò Ó 9 1 t  Ó ï è  1 A

(4.9)

 

 ¨ ‰ A | ­ where 5 are the spherical polar angles of the Bloch vector. The Abelian gauge

potential reads

g

À  | T T 1  9 A

(4.10)

¿

g ‰

g 

À  | T T 1  e +  e Ñ Ò Ó ‰ / D

(4.11)

Ó

g |  Suppose that a qubit, for example a spin half nucleus in a slowly varying magnetic field, under-

goes a cyclic conical evolution with cone angle ‰ . Then the line integral of the gauge potential

?

^ ‡  – – –  +  e Ñ Ò Ó ‰ /

gives the Abelian cyclic adiabatic geometric phase  ,wherethe ê

signs depend on whether the system is in the eigenstate aligned( with or against the field, and

ê

9 1  1

is the solid angle subtended by the conical circuit. Thus the two qubit states and may ^ end up with geometric phases of opposite sign, which gives a phase gate with shift  between the two states.

The most common experimental realisation of the geometric phase shift gate is a qubit in o a static bias magnetic field coupled to an oscillating electromagnetic field [30]. If ‡ is the

transition frequency of the qubit in the bias field, ‡ is the frequency of the oscillating field, and

? ‡ ‡ ? ‡ is the amplitude of the oscillating field, then by controlling and one can effectively

implement the conical circuit equivalent to that of slowly varying magnetic field with ‰ given

by

o

‡ e ‡

Ñ Ò Ó ‰ 

D (4.12)

o

+ ‡ e ‡ / t ‡

(

( ?

Note that any deformation of the path ofÿ the spin which preserves this solid angle leaves the phase unchanged. Thus the phase is not affected by the speed with which the path is traversed; nor is it very sensitive to random fluctuations about the path. For an experimental realisation of this scheme see, for example, [30].

For a conditional geometric phase gate we may consider a system of two non-interacting

ñ Ô Ñ spin-half particles Ô and as described in paper VI. In a reference frame aligned with the

static bias field, the Hamiltonian reads

k k

o

3 ‡ ñ Ô ñ Õ à Ö Ñ t ‡ Ñ Ö ñ à Ô Ñ Õ A

 (4.13)

ñ   ‡ Ñ  

where the frequencies ‡ and are the transition frequencies of the two spins and we

 .  ‡ ñ ‡ Ñ

have used the scaled Pauli operators Ô . From now on we assume that and are

 

ñ H ‡ Ñ very different with ‡ . If the two particles are sufficiently close to each other, they will interact, creating additional splittings between the energy levels. In the case of two spin-half

particles, the magnetic field of one spin may directly or indirectly affect the energy levels of the

k ×



other spin; the energy of the system is increased by  if the spins are parallel and decreased

k ×

 by  if the spins are anti-parallel. The Hamiltonian of the system taking into account this

interaction reads

k ×

o

 3 t   Ô ñ Õ à Ô Ñ Õ D 3 (4.14)

4.2 Geometric quantum computation 38

¥

Ù Ú Ù Û % Ý

· ·

Ù Ú Ù Û

·

Þ Þ 1

ñ Ñ

(

Ü

( Ü

‡

·

Ù Ú ˜ Ù Û

Þ Ø 1

¥

ñ Ñ

Ù Ú ˜ Ù Û ˜ % Ý

( ß

ß

(

‡

˜

˜ Ù Ú Ù Û

·

Ø Þ 1

ñ Ñ

˜ Ù Ú Ù Û ˜ % Ý

·

( ß

ß

(

˜ Ù Ú ˜ Ù Û % Ý

·

˜ Ù Ú ˜ Ù Û

Ø Ø 1

ñ Ñ

(

Ü

( Ü

Figure 4.1: The energy diagram of two interacting spin-half nuclei. The transition frequency of

the first spin depends on the state of the second spin.

Ô œ

Due to the bias field, Ô and are small for both particles. Fig. (4.1) illustrates the energy

ø

Ñ Þ 1 Ô ñ

levels of the system. When spin Ô is in state , the transition frequency of the spin is

×

 ‡ ñ t  A

‡ (4.15)

·

Ñ Ø 1 Ô ñ

whereas when spin Ô is in state , the transition frequency of the spin is

×

 ‡ ñ e  D

‡ (4.16) ˜ Now suppose that in addition to the static field, we apply a rotating field that is slowly varied as mentioned in the previous section. We have seen that the Berry phase acquired by a spin

depends on its transition resonance frequency as given by Eq. (4.12). Therefore, at the end of a ñ

cyclic evolution, the Berry phase acquired by the spin Ô will be different for the two possible

Ñ Ô Ñ Þ 1 Ô ñ

states of spin Ô . Indeed, when spin is in state , the Berry phase acquired by the spin

  +  e Ñ Ò Ó ‰ / Ô ñ

is ^ , with the sign negative or positive depending on whether spin is up

· ·

or down,’ respectively, and

‡ e ‡

·

Ñ Ò Ó ‰ 

D (4.17)

·

+ ‡ e ‡ / t ‡

(

(

?

·

ÿ

Ñ Ø 1 Ô ñ

Similarly, when spin Ô is in state , the Berry phase acquired by the spin is

  +  e Ñ Ò Ó ‰ /

^ ,where

˜ ˜

’

‡ e ‡

˜

Ñ Ò Ó ‰ 

D (4.18)

˜

+ ‡ e ‡ / t ‡

(

(

?

˜

ÿ

e ^

In paper VI it was pointed out that the geometric phase difference ^ depends on the

˜ k

· ?

amplitude of the oscillating magnetic field ‡ in such a way that it has a maximum for a non-

? ‡ ? vanishing value of ‡ . Thus, if is chosen to be close to this value, fluctuation errors are of second order and the implementation of the conditional phase gate is intrinsically fault tolerant,

4.2 Geometric quantum computation 39

× ×

^ + ‡ ñ e ‡ /  ‡ ?  Figure 4.2: Differential phase shift J as a function of and . see Fig. (4.2). This mechanism effectively implements the conditional phase gate such as the one demonstrated experimentally using the nuclear magnetic resonance technique [32]. The Hadamard gate requires non-Abelian holonomies. They have been analyzed from a theoretical point of view in [104, 3, 105, 106]. However, probably the simplest, experimen-

tally viable construction has been presented in [100, 101]. In this construction two degenerate

”

9 1  1 1 Ò 1 qubit states and and two ancilla states and are used together with an interaction

Hamiltonian

k

ß ß

o c

”

 Y Ò 1 + ‡ | 9 t ‡ ? |  t ‡ ñ | / t ) D D ]

3 (4.19)

k

ƒ  9 / ƒ á  –

with the degenerate eigenvalues ? and the non-degenerate eigenvalues ,

(

ê

?

o

o

‡  Á Ó ï è ‰ Ñ Ò Ó | ‡ ?  Á Ó ï è ‰ Ó ï è |  + ‡ ( t ‡ ( t ‡ ( / ' ( ñ

where ? . If we parameterise , ,and

ê

ñ  Á Ñ Ò Ó ‰

‡ the eigenvectors for the degenerate eigenvalue read

?  Ó ï è | 9 1 e Ñ Ò Ó |  1

Ì

”

Ñ Ò Ó ‰ Ñ Ò Ó | 9 1 t Ñ Ò Ó ‰ Ó ï è |  1 e Ó ï è ‰ 1 D

 (4.20)

(

Ì

 9 A cyclic adiabatic evolution in the parameter space, starting and ending at | and with fixed

‰ , generates a non-Abelian holonomy of the form

l

Q

¾

Ñ Ò Ó ^ e Ó ï è ^

A | ¼ Ñ Ò Ó ‰ r \ Ï  é ¶ · ¸ Í . œ

 (4.21)

o

Ó ï è ^ Ñ Ò Ó ^

ë

Q

o

^  å | Ñ Ò Ó ‰ r \ ¨ ‰ A | ­

where ¼ equals the swept solid angle in the space of parameters .For

`

  ^ the non-Abelian phase matrix will be the Hadamard gate. In all experimental realisations, in addition to the geometric phases there will also be dy- namical phases, which depend on experimental details. In principle these could be calculated and corrected for using, for example, a conventional spin echo technique. We have implicitly as- sumed adiabatic schemes for implementations of the geometric phase gates. This does not have to be the case. For a non-adiabatic scheme of the conditional phase gate see, e.g., Ref. [107].The phase shift gates, both the single qubit and the conditional phase shift gates have been imple- mented in nuclear magnetic resonance [32] and in Josephson junctions [108]. The Hadamard gate, which requires non-Abelian holonomies is more difficult to implement. However, it has been suggested theoretically for ion-traps [101] and for neutral atoms in Cavity QED [109]. 4.3 Topological quantum computation 40

4.3 Topological quantum computation

Another way of achieving fault tolerant quantum gates would be to use topological phases for the gate implementation since they are robust against deformations of the physical path. There are suggestions of fault-tolerant quantum computation based on topological phases in anyonic systems [4, 110] and non-Abelian Aharonov-Bohm effects [111]. In paper VII we use the topological phase in the two-dimensional Aharonov-Casher (AC) set up [67] as the basic building block for one- and two-qubit phase shift gates. This implemen- tation would be fault tolerant as the AC effect is only dependent upon the winding number of the physical path. To have a universal set of gates we have to combine these phase gates with a non-topological one-qubit partial swap gate. This set up demonstrates topological quantum computation using electromagnetic interactions between elementary systems.

In Chapter 3 the AC set up was discussed in a three dimensional system. In this approach ?

we use the two dimensional set up where a magnetic moment  and a point charge are free to

e ¤ g À œ e g œ À

move in the  plane, say. In this two-dimensional set up, the curl of the vector

ø ø

 + À A À œ / T

potential G vanishes except at the origin, but when encircling the two particles ø

times around each other along the path G there is a phase of the form

Ð

?

k x

 w G + è / z r è T  ? A

^ (4.22)

 Y e 5

where è . This phase is topological in the sense that it only depends upon the winding G number T , which makes it insensitive to small deformations of the path . Now we use this AC phase to implement fault tolerant one- and two-qubit phase shift gates. th

Consider the é qubit with the computational basis given by the different spatial Aharonov-

” ”

9 1 ê   + / ? + Ò / 1 ê Casher configurations, i.e. with the magnetic moment localised at site

and the charge localised at site Ò , respectively, and the reverse for the orthogonal state, i.e.

”

 1 ê  ? + /  + Ò / 1 ê

, see Fig. (4.3). We assume that the states are sufficiently localised so that

9  1 ê  9 é | for each . A quantum register is built up from such qubits arranged along a line in

the two-dimensional plane. Let us assume that all the qubits contain the same magnetic moment ?  and the same charge .

ab a b q µµq

|0 j |1 j

9 1 ê  1 ê Figure 4.3: The computational basis and , stored on different spatial Aharonov-Casher

configurations.

”

+ ^ / é A controlled phase shift gate Á can be achieved by when the particle at site of qubit

4.3 Topological quantum computation 41

Ò é G

encircles the particle at site of qubit © , along the path . This results in

 M

Á + ^ / « 9 9 1 ê ê ì e   9 9 1 ê ê ì

Á + ^ / « 9  1 ê ê ì e  9  1 ê ê ì

Á + ^ / «  9 1 ê ê ì e   9 1 ê ê ì

 M

+ ^ / «   1 ê ê ì e     1 ê ê ì A Á (4.23) as illustrated in Fig. (4.4). This gate is topological as it is insensitive to any deformations of the

path G under the assumption that charge-charge and dipole-dipole interactions can be neglected

between all pairs of AC-qubits.

+ ^ / Universal quantum computation can be achieved by combining Á with the one-qubit

logic gates



¾ ^

ê



A + ^ /  ¶ · ¸ e . A

Õ (phase shift gate)





¾

‰ ê

ê



l í A î + ‰ ê /  ¶ · ¸ e A A .

œ (swap gate) (4.24)



ê ê

. .  9 1 ê | 9 ê e  1 ê |  ê  e 9 1 ê |  ê t  1 ê | 9 ê œ

where Õ and . The one-qubit phase shift gate

¾

^ /

+ is achieved by encircling the two sites within a single AC-qubit around each other. The

¾

í A î l gates could be realised in principle by exposing beam-splitters to each of the AC set

ups.

¾

^ /

While the + gate is topological and thereby fault tolerant to path deformations, the

¾

í A î l gate would not be expected to be fault tolerant as it relies on the detailed interaction between the AC-qubit and the beam-splitter. 4.3 Topological quantum computation 42

q µ q µ

|00 e i γ |00 jj’ jj’

q µ µ q

|01 |01 jj’ jj’

µ q q µ

|10 |10 jj’ jj’

µ q µ q

i γ |11 e |11 jj’ jj’

Figure 4.4: Controlled phase gate based on the Aharonov-Casher set up. The particle at site ”

é Ò é of qubit encircles the particle at site of qubit © . There is a resulting AC phase if the moving particle have different character than the encircled one. Chapter 5

Conclusions

Geometric and topological phases in quantum mechanics have been successfully investigated and tested over the last decades. In this thesis we have focused on theoretical aspects of these phases and related effects. We have extended the geometric phase concept to unitarily evolved mixed states and to states undergoing completely positive maps. In addition, we have proposed experiments to test these phases. One possible technique to test the mixed state geometric phase would be to use a source of polarised entangled photons as demonstrated in Ref. [52]. For modeling states undergoing a completely positive map we need to control both the system and the environment, which may be difficult in practice. We have moreover proposed a test of Uhlmann’s mixed state geometric phase using two entangled photons in a Franson interferometer. In this set up, one of the photons is representing the system degree of freedom and the other the ancilla degree of freedom. We have also investigated the dynamics of the Aharonov-Bohm system using the adiabatic approximation when considering a heavy fluxon and a light, charged particle. It has been shown that the time-reversal symmetry for a semi-fluxon, a particle with an associated magnetic flux which carries half a flux unit, is unexpectedly broken due to the Aharonov-Casher modification in the adiabatic approximation. This is explicitly demonstrated for a harmonic confining po- tential of the charged particle. Further studies in this direction would be to investigate how to restore the time-reversal symmetry in the approximation to have an accurate treatment of the charge-fluxon system, and also to take into account other confining potentials. Another new result in this thesis is the discovery of an Aharonov-Casher effect dual to the standard Landau levels. We here consider electrically neutral particles with magnetic dipole moment in a certain electric field configuration. The quantisation of energy levels for the dipoles resembles that of the standard set up of charged particles in homogeneous magnetic fields. The effect is essentially relativistic and hence very small. An extension would be to study the Maxwell dual of this proposed effect, i.e. investigate reminiscent Landau quantisations for electric dipoles in magnetic field configurations. In the area of quantum computation we have discussed an adiabatic geometric conditional phase gate. This gate been implemented experimentally [32]. In order to have a universal set of gates the next step would be to implement a Hadamard gate by geometric means. This have been suggested theoretical for ion-traps [101] and for neutral atoms in Cavity QED [109]. We

43 44 have also proposed a conditional phase gate with use of the topological Aharonov-Casher set up. An experimental realisation of such Aharonov-Casher based quantum computation is difficult to accomplish since the corresponding qubit is represented by magnetic dipoles and charged particles that may cause unwanted non-topological Coulomb and dipole-dipole interactions. Acknowledgments

I would like to express my deepest gratitude to my supervisor Erik Sj¨oqvist for never ending support and excellent guidance. His broad and deep knowledge of physics together with great pedagogical skills and cool attitude have inspired me enormously. It has been a privilege to be your student! I am also greatly indebted to my co-supervisor Osvaldo Goscinski for support, numerous inter- esting discussions and also for creating a wonderful atmosphere at the department. During my time as a graduate student I spent some months at the Centre for Quantum Com- putation at Oxford University. I would like to thank Artur Ekert for the warm hospitality, for introducing me to the field of quantum computation, and for scientifichelp. I would also like to acknowledge various collaborators during these years: Jeeva S. Anadan, Johan Br¨annlund, Patrick Hayden, Hitoshi Inamori, Jonathan A. Jones, Daniel K.L. Oi, Arun K. Pati, and Vladko Vedral. My warmest thanks go to past and present members of the Department of Quantum Chemistry for both science related issues and nice coffee breaks. Especially, I would like to thank Mauritz Andersson and Bj¨orn Hessmo for a lot of scientific help and for being such good friends. I would also like to thank Johan Br¨annlund for cool cool-aborations, Bo Durbeej for splendid linguistic support, Stefan Eriksson for course collaborations, Asa˚ Granstr¨om for following the same trajectory for such a long time, Magnus Jansson for explaining everyday physics, Per- Erik Larsson for trying to make me intellectually updated, Maria Lundqvist for lightening up the department, Johan Aberg˚ for mathematical help, and the quantum chemistry football team for keeping me fit. I would also like to thank Christina Rasmundson for administrative help and Martin Agback for solving computer issues. I would also like to thank my best friends Karin Mel´en and Malin Abrahamsson for all your warm support. Also thanks to Helena Fr¨ostadius, Asa˚ Granstr¨om, Sara Johansson, Sara Nilsson, Sara Nordgren, Eva Piscator, and Witek Sadowski for many memorable occasions. Mamma, pappa, and Jenny, thank you for your unconditional support! Finally a big hug to Yoda for everything.

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