! " # $ %& '#()* *()+, ## ,)-*()#) !#.)*,) !!#.# /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.