Geometric and Topological Phases with Applications to Quantum Computation

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(11) 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 Computation. 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 experimental 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 evolutions 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 concepts 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 approximation. 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, completely 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.

(12) To my family and grandfather.

(13) IV.

(14) 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. Sjöqvist, 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. Sjöqvist, J. Brännlund, 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. Sjöqvist, J. Brännlund, 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. Sjöqvist and M. Ericsson, Phys. Rev. A 60 (1999) 1850. V Towards a quantum Hall effect for atoms using electric fields M. Ericsson and E. Sjöqvist, 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. Sjöqvist, submitted to Phys. Lett. A. quant-ph/0209006.

(15) Other papers not included in the thesis are: i) “Density Functional Study of Chlorine-Oxygen Compounds Related to the ClO SelfReaction” T. Fängström, 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. Sjöqvist, and O. Goscinski, Proceedings of the XIV International Symposium on Electron-Phonon Dynamics and Jahn-Teller Effect (World Scientific, Singapore, 1999). . . iii) “Dlaczego kot Schrödingera la¸duje na czterech łapach?” (English title: Why Schrödinger’s cat lands on its feet) M. Ericsson and E. Sjöqvist, Delta, 11, (2001) 1. iv) “Holonomic quantum logic gates” M. Ericsson, To be published in ”Quantum Theory: Reconsideration of Foundations”, ed. by A. Khrennikov; series “Math. Modeling in Physics, Engineering and Cognitive Sciences” Växjö Univ. Press (2002). http://xxx.lanl.gov/abs/quantph/0202106. v) Reply to “Singularities of the mixed state phase” J. Anandan, E. Sjöqvist, A.K. Pati, A. Ekert, M. Ericsson, D.K.L. Oi, and V. Vedral, submitted to Phys. Rev. Lett. quant-ph/0109139..

(16) Contents 1 Introduction. 1. 2 Geometric phases 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Geometric phases in cyclic evolutions . . . . . 2.1.2 Geometric phases in non-cyclic evolutions . . . 2.1.3 Measurement of Pancharatnam’s relative phase 2.2 Mixed state phases . . . . . . . . . . . . . . . . . . . 2.3 Phases for completely positive maps . . . . . . . . . .. . . . . . .. 3 4 4 8 9 10 14. . . . . . .. 19 20 20 22 25 26 29. 4 Quantum Computation 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Geometric quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Topological quantum computation . . . . . . . . . . . . . . . . . . . . . . . .. 33 34 36 40. 5 Conclusions. 43. 3 Topological phases 3.1 Introduction . . . . . . . . . . . . . . . . . . . . 3.1.1 Aharonov-Bohm phase . . . . . . . . . . 3.1.2 Aharonov-Casher phase . . . . . . . . . 3.1.3 Fundamental electromagnetic description 3.2 Dynamics of an Aharonov-Bohm system . . . . . 3.3 Dual Landau levels . . . . . . . . . . . . . . . .. VII. . . . . . .. . . . . . .. . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . ..

(17) CONTENTS. VIII.

(18) 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 photons 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 1. Assuming that the two optical paths are of the same length.. 1.

(19) 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 photodetector, 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 reflected 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 probability 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 techniques 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.. . . . . . . . . . . . . #. $. &. . . . . . . . . .

(20) 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 aspects 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 mechanics. 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 generalisation 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 starting 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 Pancharatnam 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.

(21) 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]. governed by a paramHe considered the Schrödinger evolution of a quantum state vector eter dependent Hamiltonian . Suppose that the state vector is an eigenstate of the Hamiltonian at time and that are varied along a closed path in parameter space. If the the parameters variation is slow enough for the adiabatic theorem to hold [20], the state vector remains an eigenstate of the Hamiltonian for all times . Thus, we may write . . . . . . . . . . . . . . . . . . . . . . . . . . #. . . %. . *. . . . . . . +. ,. .. 0. (2.1). 1. ). . 2. . . . . . . . is a phase factor. After a loop in parameter space during the time interval , the where phase change may be found by substituting Eq. (2.1) into the Schrödinger equation, yielding ). *. +. ,. .. 0. 1. 7. <. >. . <. M. >. . :. . D. <. >. . . . . :. ;. . O. (2.2). K. <. >. . L D. >. P. . . . . . . S. . V. X. Y. Z. S. . ]. ^. 2. . . . . a. c. 2. . . . . <. >. f. g. . :. . V. <. >. f. k. 7. #. ;. . R. is the eigenvalue of the instantaneous eigenvector of the parameter where is the dynamical phase which depends on the Hamildependent Hamiltonian. The first term tonian. The second term is the geometric phase. It depends only on the shape of the curve under the restriction that the instantaneous eigenvectors are single-valued in parameter space [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 of with respect to the choice of single valued . Furthermore, it is a real-valued quantity 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 part. Consider a non-adiabatic evolution of a state vector , which after time returns to the same state, i.e. . To find the geometric phase we introduce another state vector such that (cf. the single valued energy eigenstate vector in the above adiabatic case). This amounts to choose (modulo ) and inserting into the Schrödinger equation yields >. . . . . . . . 2. . . . . . . R. <. <. >. f. >. f. g. k. #. . q. 2. . . . :. . . . . 2. . . . . . . r. . . 2. . . . ^. 2. . 2. 2. . . . . K. t. ^. a. c. 2. . . . {. . . . . ). . . . *. :. . . .. 0. . 1. . ). . . . *. . +. . . . . . . . . 2. . . V. ^. 2. . a. c. r. . {. . . . . . ^. . . 2. . a. c. 2. :. . {. . :. . . . {. . . . . :. . D. . . . . . <. O M. O M. S K. <. . q. 2. . D. L. P. ^. . . . . S. . V. X. ^. {. . P. S. . {. S. . . <. g. V. <. k. . (2.3).

(22) 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 . . . . . . . . ". . . . . . . . . . . . . . . . . (2.4). #. !. . . . &. (. where is real-valued and run over the interval . Then the open path in Hilbert space, traced out by the cyclic state vector , is projected by to the closed path in projective Hilbert space (see Fig. (2.1)). ). *. . . . 7. ,. (. .. 0. 2. 3. . 5. 3. 9. 2π. γ. 0. H. CH Π. CP. Figure 2.1: An open curve : with the map to a closed curve 3. 5. . . . ,. 3. . 9. P. , in Hilbert space in projective Hilbert space . >. . . . . C. . . ". !. F. . . . ,. . . . . The last term in Eq. (2.3) can now be expressed as a closed line integral in . . *. 3. 9. 2. . K L. M. O. P. R. . S. R. . . is projected. according to . (2.5) (. which is gauge invariant modulo . The modulo ambiguity depends upon the choice of the , but is irrelevant when considering geometric phase factors. All curves in that cyclic state 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 and it changes sign if the path is traversed in opposite direction. real-valued for normalised .. . R. 0. .. 0. . . . . . . . . .. 0. . 7. R. .

(23) 2.1 Introduction. 6. The geometric phase can also be understood from the quantum parallel transport condition (2.6).

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(25). . . . . . . . . This condition follows from the normalisation of and from the requirement that the state vectors and should have the same phase, i.e. positive and realvalued. 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 , phase and is thus equal to the geometric phase in Eq. (2.3). For if when and , the condition (2.6) gives. . . . . . . . .

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(34). >. . :. . @. . B. (2.8). @. . . . . . . . . . >. This shows that parallel transport is a relation between the unitary operator and the initial state are parallel transported for a given . A specification of a rule that vector; not all 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 projective 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 dimensionality 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 rotated 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 geometric 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 measurements of the adiabatic geometric phase for neutron spin [28], photons [29], nuclear magnetic @. . . . . D. . .

(35) 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. γ=Ω. 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 geometric 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]..

(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 want to know the relative phase between two normalised state vectors, and , the idea is to parallel transport one of them, e.g. , along a geodesic defined by the Fubini-Study metric , where . and are now defined to be in phase, i.e. one [37], to can show that . The relative phase between and is then defined as the argument of the scalar product, i.e. . The relative Pancharatnam phase is only defined for nonorthogonal states and it is nontransitive. This latter property means that if two pairs of state vectors are in phase, and , the relative phase between and does not in general vanish. In the above two level system the relative phase between and is half the solid angle of the spherical triangle defined by the states , , and on the Bloch sphere. Thus, the relative and is equal to the geometric phase for the path defined by geodesic phase between closure of , , 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 approach to geometric phase is provided without using the Schrödinger 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 from to , according to . . . . .

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(47). ,. . . D. ,. (2.9) I. >. 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. 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 geometric phase in unitary evolution. For example, if we choose gauge so that at all times , i.e. the local phase changes of vanish, the geometric phase equals the Pancharat, nam phase. If we instead have a cyclic path and choose gauge so that . 6.

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(59) 2.1 Introduction. 9 A . . . . . . B . . . . . . . Figure 2.4: A phase in one of the interferometer beams modulates the intensity in photodetector B according to Fig. (2.5).. . I 1. 0.5. 0. 0. π/2. π. 3π /2. 2π. χ. Figure 2.5: The modulation (interference pattern) of the intensity in detector B as a function of .. 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 modified by performing a phase shift in one of the interferometer arms (see Fig. (2.4)), then the probability of reaching detector B, or the intensity in the detector B if many particles are involved, depends on the phase as. . . . . . . (2.10) .

(60) 2.2 Mixed state phases. 10 A . . . . . . B . . . . . . . Figure 2.6: operation in the upper interferometer beam and lower interferometer beam modulates the intensity according to Fig. (2.7)..

(61). . phase. . . . in the. (cf. Fig. (2.5)). Now, assume that the particle in the interferometer also carries an additional internal degree of freedom, e.g., spin, represented by . The internal state can be transformed by an operation (here and in the following subscript ” ” refers to the internal degrees of freedom) in one of the interferometer arms (see Fig. (2.6)) as . With a phase in the other arm the intensity modulation in the horizontal output channel is modified according to [43, 44] . . .

(62). . . . . . . . . . . . ". . . . . . &. (. *. . ,. . /. . (2.11). . . . . . 3. 4. 6. . 8. :. <. /. . . . . . . ". . ,. D. 3. 4. 6. . H. . (cf. Fig. (2.7)). Here is Pancharatnam’s relative phase between the pure states and and is the visibility of the phase shift. If is cyclic, i.e. takes back to the same ray, then . 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]. H. . . . . . . D. . . . D. ". . H. 2.2 Mixed state phases A quantum state can be described by a density operator that acts on the Hilbert space of the , but for mixed system. For pure states, is uniquely given by the projection operator states can be decomposed in several ways. In the spectral decomposition, is diagonal according to , where may be interpreted as the classical probability of having . A mixed state can also considered as the state of a subsystem of a larger the pure state system in an entangled state, i.e. where (Schmidt decomposition [46]). The density matrix for this pure entangled state is . If we trace over, e.g. subsystem 1, we obtain L. L. . . . /. . . (. (. (. Q. Q. L. L. L. ". P. . . . /. . . (. . (. Q. (. . . . R. . ". . P. . U. . V. . . . /. . U. . ]. U. . ". _. `. b. S. X. Z. X. L. \. ". Z. X. . \. R. . /. R. . (. Q. (. Tr U. L. . ". i. /. (. . U. . L. . . U. . ". i . . . X. /. . . X. ". L. m. X. (2.12).

(63) 2.2 Mixed state phases. 11 I 1. ν. 0.5. γ 0 0. π. π/2. 3π/2. χ. 2π. Figure 2.7: Modulation of the intensity in detector B as a function of . The Pancharatnam 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 mixed states evolving unitarily (the unitarity condition is not necessary in Uhlmann’s approach), using the language of interferometry as in the example above but with mixed internal state acting on an dimensional Hilbert space. The transformation in the upper beam gives and the intensity in detector B as a function of is now . . . . . . . . . . . . . . . . . . . . . . . . . !. #. . $. &. Tr. Tr. . . ). +. -. . . /. 1. 3. (2.13). . 4. . 6. . . 9. . . The Pancharatnam phase defined as the shift of the interference pattern, is . . . . . /. 1. Tr 3. (2.14). . . 4. 6. . /. 1. 3. =. . @. A. >. B. . and the corresponding visibility is . . Tr. (2.15). . . 9. . The total interference pattern can also be expressed as a weighted average of pure state interference profiles (cf. Eq. (2.11) (2.16) !. #. . !. . . 9. =. 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 . . . I. . . . M. . O. . . M. . . I. . . . I. B. . $. B. S. B. 9. 9. 9. . 9. (2.17).

(64) 2.2 Mixed state phases. 12. i.e. all states , that diagonalise the non-degenerated density operator , are parallel transported by the unitary operator . The condition that has to be non-degenerate is 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 realvalued, 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 . . . .

(65). . . . . .

(66). . . . . . . . . . . tot . . d . . g . (2.18) . is not valid in the mixed case. In the two level (qubit) case the mixed state geometric phase is . . . . ". #. %. (. *. ,. %. (. 0. (2.19) . 1. 3. where is the length of the Bloch vector and is the solid angle traced out. For pure states ) 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 ,. 0. ,. . 6. (2.20). 7. ?. . 9. ; #. =. >. 0. ,. >. ?. A. (. 0. ?. 1. 1. where 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 of the Bloch vector. We have considered a cyclic evolution ( ) and . Let us now consider another definition of mixed state geometric phase as proposed by and let Uhlmann [13]. Consider a density operator of a system acting on a Hilbert space be a copy of . Uhlmann then defines a ’lift’ of to be a so-called Hilbert-Schmidt operator mapping elements in to elements in and fulfilling the condition . Uhlmann’s lift is equivalent to purification by a simple isomorphism 9. 9. . 6. . . E. F. 1. 0. H. H. I. I. H. K.

(67). O. M. M. H. K. H. M. M. . M. R. I. I. K. . . I. V. . K. . M. . I. X.

(68). (2.21). M. . . R. I. K. I. K. . . I. . . K. . K. and are bases in and respectively. Clearly such a lift is not unique. From where a new lift can be constructed as where is some arbitrary unitary operator on since determines the same . Following Uhlmann, we can now define the lifts of pair of density operators and to be parallel if (2.22) . . I. . . K. H. I. M. H. K. M. M. `. `. . H. K. ^. O. M. M.

(69). ^. ^.

(70). b.

(71). ?. O. O. M. M. M. M. . b. d. . g. b. ?. ?. 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 O. M. M. . i. h. h. hermitian g. (2.23).

(72) 2.2 Mixed state phases. 13. I 1 r=1 r=0.5 r=0.1 0.5. χ. 0 0. π. π/2. 3π/2. 2π. 5π/2. 3π. Figure 2.8: The modulation of the intensity in photodetector B as a function of for unitary cyclic evolution of a qubit with (solid), (dashed), (dotted). The solid angle are given by . . . . . . . . . . .

(73). . . . . . . 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 Tr. positive and real-valued. . . . . . (2.24) . . . . by applying an extra constraint. That is, if the parallel transport condition is invariant under the , where is an arbitrary hermitian operator then Eq. (2.23) follows. transformation Uhlmann justifies this additional constraint by the minimisation of the length of the curve in the extended Hilbert space , i.e. . . . . . !. ". $. . . !. '. ). !. ,. -. .. /. Tr 1. (2.25). . . . 2. 3. . . . . 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, the geometric phase acquired during the time interval defined as 2. 8. :. . =. ?. Tr A. . . 7. 3. (2.26). . . C. D. B. F. . 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 , it can be either unitary or non-unitary. However, in order to compare Uhlmann’s approach with our approach in paper I, let us only consider unitary evolution, i.e. solutions of G. . I. H. G. . 2. K. . G. 3. . (2.27). . . . is assumed to be time-independent. In terms of Hilbert-Schmidt where the Hamiltonian operators, the solution reads (2.28) K. N. . M. C. S. O. . . . Q. G. .

(74) 2.3 Phases for completely positive maps. 14. with and , being Hermitian and assumed to be time independent but otherwise arbitrary. For the lift to be parallel it has to fulfil Eq. (2.23), yielding . . . . . . . . . . . . . . . . . . . . . . . . . $. $. which fixes . if . #. !. . . #. . . . . ( . . is of full rank. With this choice of $. . +. -. . /. 1. (2.29). $. $. !. . Tr 3. ;. 4. @. @. $. . . /. 1. 3. *. the geometric phase is . . Tr. 5. 4. . . $. $ . !. ! #. #. . . ; . (2.30). . . G. . /. 1. 3. : =. <. =. B. D. G. F. F. I. K. B. I. F. F. D. K. M. ?. 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 in Uhlmann’s approach is isomorphic to a pure state in Hilbert space of the system and an additional ancilla, . The pure state evolves under a bi-local operator written in Schmidt form as (2.31) 4. . 4. $. N. . $. F. . K. . P. R. S. P. V. S. . ;. @. . . ;. . 4. N. F. . K. . . :. F. I. K. . . S. . F. I. K. . *. . ?. where the unitary operator for the ancilla is (transpose with respect to the instantaneous eigenbasis of ). If obeys the parallel transport condition Eq. (2.29) the quantity . . . . . . . . (2.32). $. G. +. -. . /. 1. 3. B. . F. . K. *. is equivalent to Eq. (2.30). The geometric phase in paper I can also be analysed using purification, i.e. lifting to as above and then evolving only the system part with fulfilling the parallel transport condition Eq. (2.17). It can be shown that the two parallel transport conditions, Eqs. (2.17) and (2.23) coincide only in the pure state case. Moreover, underUhlmann’s geometric phase is derived from an evolution of an entangled pure state going a bi-local unitary transformation while the geometric phase from paper I is derived from 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 polarisation 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 coincident events, the interference between the short and the long path can be observed in the intensity pattern. Designing and according to the parallel transport conditions, Eqs. (2.17) and (2.23), both Uhlmann’s and paper I’s phase can be observed. . F. . . K. . P. R. S. P. V. . . . F. F. . . K. K. . . . . 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.

(75) 2.3 Phases for completely positive maps. 15. . . . . . . . . . . . . . D. D. Source. u. . . . v. . . . . . . . Figure 2.9: Polarisation entangled two-photon interferometry to measure the mixed state geometric 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. To model a CP map we couple the internal state to an environment. . . .

(76). . . . . .

(77). . . . . . (2.33) . . and let it evolve unitarily . The evolved density matrix of the internal part is obtained by tracing over the environment, yielding . . . . .

(78). .

(79). . . . . Tr . (2.34). . . . ". . . .

(80) .

(81). . [54] in terms of an orthonormal basis , where the Kraus operators are , of the 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 . . . .

(82). . . . . +. +. +. .

(83). .. 0. . 1.

(84). . $.

(85). . . '. . .. 1. Tr. 4. 3. . . 7. . 5. . Tr. . .

(86).

(87) . . . 4. . .

(88) . (2.35). . .

(89). . where has been used. Further interference information can be obtained if we flip the state of the environment associated with the reference beam to an orthogonal state . This flip is represented by the operator (Fig. (2.10)) and the interference pattern becomes 4. . . . . . ;. . . . . . . 4. . . . . . . .

(90). 3. 7. Tr. . 5. . Tr. . . . . 4.

(91).

(92) . . . . . . . . . . . . . . .

(93). .

(94) . . . . . . . . Tr . .

(95) . . (2.36).

(96). . for each . The set , , contains maximal information about the interference effect associated with the CP map with known , by measuring on the system alone.. . 0. . +. +. +. .

(97). .. 0. $. 3. 5.

(98). 7. . '. . . . +. +. +. .

(99). .. 0. .

(100). .

(101) 2.3 Phases for completely positive maps. 16 A . . . . . . B .

(102). . . . . . . . . . . . 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 . . . . . . . . . . . (2.37). . . . . . . . . . . . . . . . . . . . . . . . . . . . where is Hermitian and positive, and with specified by the type , and we assume that . The action of each is uniquely of decoherence, defined up to phase factors by the evolution of the system’s density operator. This ambiguity must be associated with the corresponding unitary as the Hermitian part is unique, and the parallel transport conditions for each interference pattern are given by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . !. . . . . #. . . $. . . $. . (2.38). . *. . %. '. . ) . . . . . . ). '. /. . . . '. . !. . 6. 6. 6. $. 6. +. . 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 interferometry. 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 depolarisation channel is a model of a decohering qubit where the probability of the qubit being intact is and probability of an error to occur is . There are three possible errors with equal probability: the bit flip error , the phase flip error , and both of the previous at the same time , where !. =. . . . . 6. 6. 6. .

(103). =. !. . ). ). ?. /. A. ?. /. A. . ). . ?. C. ). ?. /. ). ?. /. A. . . ). ?. /. /. F. . . C. . !. H. . . =. ! L. H. . . . . H. F. !. L . I. . I. =. !. I. (2.39).

(104) 2.3 Phases for completely positive maps. 17. are the standard Pauli matrices. The Kraus representation of this channel is . . . . . . . . . . . . . .

(105). . . . . .

(106). . . . (2.40). . . . . . . .

(107). . rotation . In the case of a cyclic rotation fulfilling the parallel by appending an transport conditions Eq. (2.38), the geometric phases are determined by the interference patterns . . . . . . . . . . . .

(108). . . . . . . . . . #. . . %. '. ). +. ,. . ". #. . . . ". . . . . . . . . 0. . 0. . . . . . . . . (2.41). . . #. . ). . . . #. ". . . %. '. +. ,. . #. . . . ". . with the solid angle enclosed by the loop of the Bloch vector. The first interference pattern 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. ". . . . . . 7. ). .

(109) 2.3 Phases for completely positive maps. 18.

(110) Chapter 3 Topological phases Electromagnetic fields do not give a complete description of electromagnetism in quantum mechanics. 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 interaction 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 discovered, 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 importance 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 approximation 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 specific electric . 19. .

(111) 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) . . . (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 dependent 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 point when starting at in the vicinity of a flux line, cf. Fig. (3.1), using Feynman path integrals [64], i.e. adding the phase factors of all possible paths going from to . The of the system without the flux line acquires an additional term in the presence of Lagrangian the flux line according to (3.2) . . . . . . . . . .

(112).

(113). . .

(114) 3.1 Introduction. 21. where the last term is due to minimal coupling and is the velocity of the charged particle and is the vector potential consistent with the magnetic field outside the flux line. The additional term to the Lagrangian changes the action according to , where is given by . . . . . . . . . .

(115). . . . . . . . . . (3.3) . being the line element along a specific path going from to . Summing over all possible for the ones going to the left of the paths connecting and results in the same integral 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 ( ) and to the right ( ) of the flux line, is given by . . . . . . . . . . . . . . !. . . " . " . #. $. . . . . " . (3.4). . . . . . . " . . . where 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 the integral of 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 previleads to a geometric phase according ously, an adiabatic cyclic transport of the state vector to (3.5) . . . (. (. ). . -. . 0. #. $. . 3. . 6. . . *. . ,. . where , 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. . 6. . 0. <. >. 3. . 6. ). . A. >. 3. . 6. ,. ). >. 3. . 6. ,. E. E.

(116) 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, experiences a homogeneous scalar potential during a time interval in one of the beams giving in potential energy. No force is acting on the particle when it is inside the rise to a shift region where the scalar field is applied, since . Nevertheless, there is a testable phase difference between the beams given by . . . . . . . . . . (3.6). . . . . . . . . . . . . . . 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 given rise to the vector potential is given by . . (3.7). !. " ". $. . #. $. . . . '. (. . +. . . '. . . . #. where , , and are the position, mass, and velocity for the charged particle and , , and 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 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. The Lagrangian for the magnetic dipole at and a charged particle fixed at position (i.e. ) is thus given by . . (. . . . . . . (. '. (. . +. . . . ". #. '. . . . '. (. . +. . . (3.8).

(117) 3.1 Introduction. 23. From electrodynamics we know that the vector potential at moment at is given by [68] . from a classical magnetic dipole . . . . . . . . . . . . . . . (3.9). . . . . . . . . . .

(118). . . . . .

(119). . where is the electric Coulomb field at due to the charged particle at , permittivity, and is the speed of light. The Lagrangian now becomes . . . . is the vacuum. . . . (3.10). . . . . . . . . . . . . . . which via a Legendre transformation defines the Hamiltonian (3.11). . . !. . . . . . . ". . is the canonical momentum of the magnetic dipole. We can thus identify where the vector potential for the neutral particle as. . . . $. . . . (3.12). . . . . '. . If we consider a particle with “anomalous” magnetic moment, as a spin half particle, the Hamiltonian Eq. (3.11) is modified slightly according to (neglecting terms of ) [69] . . . . +. . . !. !. . . . . , . . . . . (3.13). " . . . . where the last term is due to the non-commutativity of and . Also here we get the vector 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 encircles a line of charge (see Fig. (3.3)). This Aharonov-Casher (AC) phase shift is given by . . $. %. #. . 4. . .. !. + -. '. +. /. . 2. . . (3.14). 4. . . where is the electric charge per unit length. In order to prevent the magnetic dipole from precession due to the electric field, i.e.. . . . . ). . . . :. . (3.15). 5. . 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..

(120) 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 relation according to . (3.16). . . . . . . . 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 in one of the beams in the same way as the charged particle experiences an electric field in EAB. This gives rise to an increase in potential energy according to (Zeeman effect) but no force ( ). When the beams intersect the SAB phase is given by . . . . . . . . . . . . . . . . . . . . !. (3.17). . 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 specific region of the.

(121) 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 description 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, . . . . . . . . . (3.18). . . and . . . . . . . . . . . .

(122) . . . . . . (3.19). . . respectively, are the fundamental description of electromagnetism for charged particles and magnetic dipoles. Let us now consider the classical Lorentz force . . . . . . .

(123). (3.20). 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. . .

(124) 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 coworkers [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 Schrödinger equation for a system which consists of light and heavy degrees of freedom denoted by 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 treat the motions of and separately. The procedure is first to treat the problem of the light degrees of freedom in a stationary surrounding of the heavy degrees of freedom giving the states of the former parametrically dependent upon , i.e. in position representation. The define the potential energy surfaces on corresponding energy eigenvalues for each which the heavy degrees of freedom move. The total state in position representation is in the adiabatic approximation then given by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.21). . . . . .

(125).

(126). . . is the solution of the time-independent Schrödinger equation for the heavy degrees where of freedom in the potential . The validity of the adiabatic approximation depends upon the size of the coupling terms between different states of the light degrees of freedom. The accuracy of the description may be improved by adding more states according to . . .

(127). . . . . . . . . . . . . . . . . . . . . . . . . (3.22).

(128).

(129). . and the exact solution is obtained when 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 adiabatic 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 Schrödinger equation of a light charged particle and a heavy fluxon within the adiabatic approximation. The .

(130) 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 intersection. 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) . . . . . . . . (3.23). . . . . . . . . . . . . . . . . . . . . . . where the two first terms are the kinematic energy of the charged particle and the fluxon, respectively, with the charged particle with mass and charge at , and the fluxon with mass at . The last term has been added as a confining potential for the charged particle. The time-independent Schrödinger equation for the charged particle at a fixed fluxon position reads . . . . . . . . . . !. . . . . . . . . . . . '. . . . . . . . . . . . .. (3.24). +. . . The fluxon problem is thereafter given by . eff . . 0. 1. 0. . . (3.25). 1. . . . +. 2. 2. 2. /. /. with the effective Hamiltonian . . . . . . eff . . . 6. . . . 8. . 8. . . . :. . . . . . . >. . . . . +. . . >. . . (3.26). +. . where stands for integration over regime, now be written as the adiabatic vector potential is 6. @. :. B. . . . only. The total state of the system can, in the adiabatic , being the fluxon state. In eff , . . . 0. . . . . 1. . 0. . 1. . . . . 2. 2 2. /. . /. . . . >. . . . . 6. . . . 8. . . . . . . . . :. (3.27).

(131) 3.2 Dynamics of an Aharonov-Bohm system. 28 . and there is a correction to the standard potential energy . . of the form . . . . . . . .

(132). . . . . . . (3.28). . . . . . . . . . . . . . . . . . #. . . . . . . . . . . . . . . . . . . . #. +. . . is invariant under a local, It is further shown in paper IV that the adiabatic vector potential i.e. dependent, gauge transformation. To illustrate the above situation, let us consider a harmonic confining potential centered at , i.e. . . . . . . . . . (3.29). .

(133). . . .

(134). . . .

(135). . .

(136). . .

(137). 1. .

(138). 2. . . . . 1. 2. 5. 6. 8. . . . . <. . and are the polar coordinates of and , respectively, and is the angular where 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 . 2. <. . . . 1. <. . . . . . (3.30). . . . . C. <. @. 1. where is proportional to the flux. Let us focus on the semi-fluxon case characterised by . For this -value, and only differ by a single-valued gauge. In other words, the system obeys time (motion) reversal symmetry since , where is a singlevalued gauge that does not change the physics. The charged particle’s wave function is obtained by diagonalising the perturbation term in Eq. (3.29), in the unperturbed oscillator basis characterised by the angular momentum quantum number and the principal quantum number . It is found that the potential energy surfaces of the fluxon are degenerate at 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 C. E. C. C. .

(139). . . . . .

(140). G. . . . . .

(141). . . . . . .

(142). . J. . .

(143). 1. 2. 5. 6. 8. . . . J. . M. N. . P. M. P. M. . . . . . (3.31). . E. T. U.

(144). X. . . Y. [. % Y. \. !. ]. 2. #. ^ _. is determined by properties of the oscillator eigenfunctions (for further details, see where Eq. (19) in paper IV) and the index indicates the diagonal basis states. Integrating the adiabatic vector potential around a closed curve in the fluxon parameter space the phase is %. Y. &. `. . . E. T. U.

(145). X. a. Y. [. ^. &. % Y. !. ]. ). <. (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/antiunitary nature. ).

(146) 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 the plane perpendicular to a uniform magnetic field . This system is described by the Hamiltonian . . . . . (3.33). . . . . . . . .

(147). . . .

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