Non-Cyclic Geometric Phase

Transcription

1 NATIONAL UNIVERSITY OF SINGAPORE Honours Project in Physics Non-Cyclic Geometric Phase Tang Zong Sheng A A Supervisor: ASSOC PROF Kuldip Singh Co-Supervisor:ASST PROF Alexander Ling Euk Jin A thesis submitted in partial fulfilment for the Bachelor of Science with Honours 2014/2015

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3 Abstract In 1983, Berry reported a phase in addition to the dynamical phase when a system undergoes a cyclic evolution under adiabatic approximation. The additional phase depends only on the geometric path taken by the system, hence it is known as the geometric phase. Soon after, Simon founds that the geometric phases could be interpreted as a the holonomy in a fibre bundle. In 1988, Samuael and Bhandari proposed a generalisation by removing the cyclic evolution condition. In this thesis, their claims as well as the gauge symmetry of non-cyclic geometric phase are analysed in the context of the Hopf Bundle. The thesis ends with some suggestions on further implications of the project.

4 Acknowledgements I would like to express my heartfelt gratitude my project supervisors Assoc Prof Kuldip Singh for offering the project and his advice during the course of the project. Under his guidance, I gained a new perspective in understanding Physics and learnt the beauty of the geometry. iv

5 Contents Abstract iii Acknowledgements iv 1 Introduction 1 2 Quantum Geometric Phase Adiabatic Approximation and the Berry Phase Gauge Symmetry of the Geometric Phase Aharonov-Anandam Phase Fibre Bundle theory Fibre Bundle Fibre Bundle and Gauge Transformations Bundle Connection Parallel Transport and the Holonomy in a Fibre Bundle Fibre Bundle and the Geometric Phase Non-Cyclic Geometric Phase Pancharatnam Connection Pancharatnam Connection and Bundle Connection Remarks on the Non-Cyclic Geometric Phase Riemannian Manifold Metric Tensor Affine Connection and Geodesics Affine Connection and Metric Tensor Hopf Bundle Hopf map Local trivialisation and inverse mapping Bundle connection and metric tensor in a local trivialisation Relation between Geodesics, Horizontal vectors and the In-Phase Condition Metric Tensor in the Horizontal Basis horizontal Lift of Geodesics v

6 vi CONTENTS 7.3 Horizontal Geodesics as a Sufficient Condition for the In-Phase Condition The In-Phase Condition as a Sufficient Condition for a Horizontal Path The In-Phase Condition as a Sufficient Condition for a Geodesic Examination of Gauge Symmetry Gauge Transformations in the Hopf Bundle Metric Tensor under Gauge Transformations Gauge Symmetry of Geodesics Gauge Symmetry of Horizontal Vector Fields Gauge Symmetry of an In-Phase Path Conclusion 54 A Pull Back of the Bundle Connection and the Metric Tensor 56 A.1 Push Forward and Pull Back action A.2 Calculation of the Metric Tensor and the Bundle Connection in Local Trivialisations B Metric Tensor in the Horizontal Basis 59 C Covariant Derivative of e θ, e φ and e χ 61 D Horizontal Lift of a Geodesic 63 D.1 General Geodesic on Base Manifold D.2 Horizontal Lift of a Geodesic Bibliography 66

7 Chapter 1 Introduction In the quantum mechanical formalism, a quantum system is described by a state vector in the Hilbert space. The dynamics of a quantum system is governed by a time dependent differential equation known as the Shrödinger equation which involves a linear operator called the Hamiltonian. In many practical situations, the Hamiltonian of a system is time dependent and relies on the configuration of the environment. This property of the Hamiltonian gives rise to a geometric based quantity in the quantum system known as the geometric phase. In 1983, Berry [2] found that when a system is transported slowly round a circuit, the system will acquire an extra phase in addition to the dynamical phase. Furthermore, the additional phase depends on the geometric path which the system takes, and is not affected by the rate at which the system moves. Due to its geometric origin, the phase factor is called the geometric phase. The geometric phase under adiabatic approximation is named as the Berry phase. Soon after the publication of the Berry phase, a number of experiments were done and recorded the observation of the geometric phase. Among these are the nuclear magnetic resonance experiment by Suter et al. [15], neutron spin experiment by Bitter and Dubbers [3] and the polarisation of light by Tomita and Chiao [16]. Experiments in more complex systems involving molecular physics have also been done. [7, 17] Shortly after Berry s discovery, Simon [14] noticed that the Berry phase can be interpreted as the holonomy in a fibre bundle, with the adiabatic condition serving as the bundle connection. This beautiful connection between mathematics and physical world motivated the application of the fibre bundle into physics, and one of the successful examples is the Yang-Mill theory.[19] 1

8 2 Chapter 1 Introduction After introducing the link between the geometric phase and the connections in an appropriate fibre bundle, several successful attempts in generalising the Berry phase were reported. In 1984, Wilczek and Zee [18] generalised Berry phase with degenerate Hamiltonian which corresponds to a non-abelian gauge field. In 1987, Aharonov and Anandan [1] showed that adiabatic condition can be removed, and the geometric phase depends only on the cyclic evolution in the projective space. In 1988, Samuel and Bhandari [13] claimed that the geometric phases can be further generalised by removing the cyclic evolution condition. They suggested that the geometric phase of an open curve can be calculated by connecting the curve from the actual evolution with a horizontal lift of a geodesic from the base space. Samuel and Bhandari adapted the idea of Phancharatnam connection in comparing two states. They claimed that when two states are connected by a geodesic which is a horizontal lift of a geodesic in the base space, they will be in phase.they suggested that the geometric phase associated with an open curve can be defined by appending a geodesic to the ends of the open curve. Here they employed the in-phase condition that arises from the Pancharatnam connection. In this thesis, we examine this condition in the context of the underlying (implicit) gauge symmetry. The thesis starts by introducing the background knowledge that underpins the geometric phase. In chapter 2, the Berry phase and the assumptions imposed in his discovery are introduced. In Chapter 3, after a brief introduction of fibre bundles, the connection between this mathematical concept and the geometric phase is shown. In Chapter 4, the generalisation of the geometric phase with non-cyclic evolution is discussed. As the ideas of metric tensor and geodesic were adapted in the non-cyclic geometric phase, some basic information about Riemannian geometry are introduced in chapter 5. In chapter 6, a particular fibre bundle called the Hopf Bundle is introduced. With the essential knowledge, various geometrical quantities that are used in the definition of the non-cyclic geometric phase are analysed. In chapter 7, we examine the relations that exist between geodesics, horizontal vector fields and the Pancharatnam connection. In chapter 8, the effect of gauge transformations on these constructs is considered. Finally, we conclude with some suggestions for the future work.

9 Chapter 2 Quantum Geometric Phase In 1984, Berry [2] found that when a system undergoes cyclic evolution under adiabatic evolution, there is an extra phase in addition to the dynamical phase. This extra phase is independent of the time taken by the system in the evolution and instead, this phase depends on the path which the system takes. Furthermore, it was found that this adiabatic geometric phase, also known as the Berry phase is a gauge invariant quantity. In other words, it cannot be eliminated by choosing a particular gauge. In 1987, Aharonov and Anandan [1] generalised the Berry phase by removing the adiabatic condition and suggested that the geometric phase is related to the geometry in the parameter space. In this chapter, the idea of the Berry phase and the generalisation to the Aharonov- Anandam phase will be introduced. 2.1 Adiabatic Approximation and the Berry Phase In quantum mechanics, the information of a physical system is encoded in a state vector in the Hilbert space. The dynamics of a state is governed by the Schödinger equation, a first order differential equation which has the form: i ψ(t) = H(t) ψ(t). (2.1) t When a system is static, meaning the Hamiltonian is time independent, an energy eigenstate state will remain unchanged during the evolution. In general, a real system cannot be isolated from the surrounding, and its Hamiltonian is affected by the changes in the environment. However, a system can be approximated as a static system under the adiabatic condition. The adiabatic approximation takes into account the dynamical effect of the Hamiltonian but in the limit of infinitely slow changes. 3

10 4 Chapter 2 Quantum Geometric Phase Consider a quantum system with a time dependent Hamiltonian H(t) that has nondegenerate energy eigenstates n(t). The action of the Hamiltonian on its eigenstate gives H(t) n(t) = E n (t) n(t). (2.2) Differentiating the equation above with respect to time, i.e., Ḣ n + H ṅ = E n n + E n ṅ (2.3) and contracting it with m, one obtains m ṅ = m Ḣ n, n m. (2.4) E n E m With this relation shown, the condition for adiabatic approximation can be introduced. The evolution generated by H(t) is said to be adiabatic if n Ḣ m E n E m T mn, (2.5) where T mn is the characteristic time of transition between state m and n. This condition requires the change in H to be slow compare to the natural time scale of the system. In the adiabatic limit, T mn 1, hence n Ḣ m 0. After the adiabatic condition introduced, we can show that a system is approximately static under this condition. Consider a quantum system with the Hamiltonian H(t) that has non-degenerate energy eigenstate n(t). Suppose the Hamiltonian is parametrised by R(t) = {R i (t)} i=1...j, where R(t) is a set of parameters which characterise the particular configuration of a changing surrounding, the state of the system ψ(t) can be expanded into the energy eigenstate basis: ψ(t) = n ( c n (t) exp i t Substituting the state into the Schrödinger equation gives 0 ) E n (τ)dτ n(r(t)). (2.6) ċ n = c n n ṅ n m c m n k exp [ i t 0 (E m (τ) E n (τ))dτ ]. (2.7)

11 Chapter 2 Quantum Geometric Phase 5 Under the adiabatic condition, the second term on the right hand side drops out and solution for c n (t) is ( c n (t) = c n (0) exp t 0 n(r(t)) d dt n(r(t)) dτ ) = c n (0)e iγn(t). (2.8) The solution of c n (t) shows that when adiabatic condition is fulfilled, the eigenstate n(t) will only acquire a phase factor under time evolution. Hence, the system is approximately static under the adiabatic condition. An important property of the phase angle γ n (t) is that it can be directly defined in term of the integral curve over a vector value function A n i (R), γ n (t) =i =i =i =i t 0 R(t) R(0)) R(t) R(0) R(t) n(r(t)) d dt n(r(t)) dt R(0) d n(r(t)) dr i n(r(t)) dri n(r(t)) n(r(t)) dr i A n i (R)dR i = R(t) R(0) A n (2.9) where A n is called the gauge potential. With the coefficient c n (t) solved, the state ψ(t) can be re-expressed as ψ(t) = n c n (0)e iγn(t) e θn(t) n(r(t)), (2.10) where θ n (t) = i t 0 E n(τ)dτ. This equation shows that under the adiabatic approximation, there is an extra phase factor γ(t) in addition to the dynamical phase factor θ(t). Notice that the extra phase factor γ is determined by the path taken during the adiabatic evolution in the parameter space, and it is independent of the rate at which the system is evolving. Hence, the phase γ is called the geometric phase due to its geometry origin. The geometric phase under the adiabatic approximation is named as the Berry phase. Another important property of the Berry phase is that it is gauge invariant under cyclic evolution. The significance of gauge symmetry in quantum mechanics and a gauge-invariant Berry phase will be shown in the next section.

12 6 Chapter 2 Quantum Geometric Phase 2.2 Gauge Symmetry of the Geometric Phase Gauge symmetry in quantum mechanics, is analogous to the gauge symmetry in particle physics. The term gauge refers to the redundant degrees of freedom in a physical system. In other words, physics is symmetric under the changing of this extra degrees of freedom. In this section, the gauge symmetry in quantum mechanics and Berry phase will be elucidated. In quantum mechanics, a physical state is represented by an element ψ in the Hilbert space. However, any two states ψ, φ H, are said to be physically equivalent ( ψ φ ) when ψ = e iα φ where α R. Hence, the proper space of describing a quantum system is the projective Hilbert space of P(H), which is defined as P(H) := H/. (2.11) Notice that, the projective Hilbert space provides a freedom in changing the phase of a state vector ψ without altering the physical system. Furthermore, the dynamics of the system is governed by the Schrödinger equation and the solution defines a trajectory in the Hilbert space ψ : t ψ(t) H. (2.12) Likewise, the projection of trajectory ψ(t) to the projective Hilbert space, P : ψ(t) P (t) P(H) (2.13) defines a solution of the von Neumann equation i d P = [H, P ], P = ψ ψ. (2.14) dt Notice that we can change H in the following way, H = H + 1α(t), (2.15) without affecting the solution of the von Neumann equation. Therefore the dynamics of the system is said to be invariant under this transformation. It means that a physical system does not uniquely correspond to a single Hamiltonian, but to an equivalence class of the Hamiltonian. In other words, there exist some freedoms in choosing the Hamiltonian and the states without altering the system. Hence, a physical quantity which

13 Chapter 2 Quantum Geometric Phase 7 describe a quantum system should be invariant under these gauge transformations as well. Note that in general the Hamiltonian of the system depends on the parameters R which characterise the configuration of the surrounding. Hence, the gauge transformation depends on these set of parameters and this is an indication of the locality of the gauge symmetry. The Berry phase, under the U(1) gauge transformation, i.e. will transform in the following way: n n = e iα(r) n, γ n =i =i =i R(T ) R(0) R(T ) R(0) R(T ) R(0) n (R) n (R) dr i n(r) e iα(r) e iα(r) n(r) dr i ( n(r) n(r) + i α(r)) dr i =γ n α(r(t )) + α(r(0)). (2.16) Notice that γ is not invariant under the transformation. However, when the system return to its initial position after the time period T, i.e. R(T ) = R(0), then γ = γ α(r(t )) + α(r(0)) = γ. (2.17) Hence, Berry phase is invariant under gauge transformations under cyclic evolution. The adiabatic approximation is imposed for the Berry phase, though, adiabatic condition is not always satisfied in a quantum system. Hence a generalisation for more practical situation is required and will be shown in the next section. 2.3 Aharonov-Anandam Phase In 1987, Aharonov and Anandan [1] generalised the Berry phase by relaxing the adiabatic assumption. This generalisation showed that the geometric phase is independent of the Hamiltonian but depends only on the geometric structure of the system. The idea of the projective Hilbert space and the gauge freedom of the Hamiltonian is used to derive the generalised geometric phase, also known as the Aharonov-Anandam phase. Consider a system in a state ψ(t), and the trajectory of P (t) is closed in P(H), i.e., P (t) = P (0), this evolution is called a cyclic. Under cyclic evolution, the state ψ(t) and

14 8 Chapter 2 Quantum Geometric Phase ψ(0) projects to the same point in the projective space. Hence ψ(t) can be expressed as ψ(t) = e iϕ ψ(0) for some ϕ. Due to the freedom of choosing the Hamiltonian, one can change the Hamiltonian without affecting the system: H = H + 1α(t). (2.18) The solution of the Schrödinger equation corresponds to H will be and ( ψ (t) = exp i t 0 ) α(τ)dτ ψ(t), (2.19) ψ (t) = e iϕ ψ(0), ϕ = ϕ i t 0 α(τ)dτ. (2.20) Hence, by changing the Hamiltonian, the corresponding phase ϕ can be changed arbitrarily. According to Aharonov and Anandan [1], the total phase ϕ can be decomposed into two components ϕ = ϕ dyn + ϕ geo, (2.21) where ϕ geo is invariant under gauge transformations. By choosing a function α(t) such that, ϕ = 0, one obtains ϕ = 1 The new function ψ (t) is the solution of the Schrödinger equation t 0 α(τ)dτ. (2.22) i d dt ψ (t) = (H + 1α(t)) ψ (t). (2.23) Applying ψ (t) on the both sides, and integrating t 0 ψ (t) i d dt ψ (t) dt = i t 0 ψ (t) H ψ (t) dt + i Referring to equation (2.22), the total phase can be expressed as ϕ = t 0 ψ (t) i d dt ψ (t) dt i t The total phase can be separated into two components 0 t 0 α(t)dt. ψ (t) H ψ (t) dt. (2.24)

15 Chapter 2 Quantum Geometric Phase 9 1. The dynamical phase ϕ dyn : ϕ dyn = i t 0 ψ (t) H ψ (t) dt. (2.25) Notice that ϕ dyn depends on the Hamiltonian of the system. 2. The geometric phase ϕ geo : ϕ geo = t 0 ψ (t) i d dt ψ (t) dt (2.26) ϕ geo depends on the closed curve P (t) but not the Hamiltonian of the system. Indeed, the geometric phase depends only on the geometry of the projected curve. Let φ(t) be an arbitrary curve which differs with ψ (t) by a phase factor, i.e. φ(t) = e iξ(t) ψ (t), (2.27) such that ξ(t ) = ξ(0) The geometric phase under this transformation t 0 φ(t) i d t dt φ(t) dt =ξ(t ) ξ(0) + ψ (t) i d dt ψ (t) dt. (2.28) Hence it indicates that the geometric phase is a geometric property of the closed curve in P(H) and this generalisation of the geometric phase is called the Aharonov-Anandam phase. The example shows that the conventional dynamical phase of the system is not invariant under transformation. Hence, it suggests that geometric phase might be a more reliable quantity in measuring relative phase. 0

16 Chapter 3 Fibre Bundle theory Immediately after the discovery of the Berry s phase, Simon [14] noticed that this phase can be interpreted as a purely geometric quantity, which is the holonomy of a fibre bundle. Simon constructed a bundle system to represent the Hilbert space and he showed that the adiabatic condition resembles the connection which parallel transports a point between fibres. When a state is transported along a closed curve in the projective space, it does not return to the initial position in the Hilbert space and it acquires an extra phase factor predicted by Berry. Fibre bundle is a beautiful mathematical structure which connects mathematics and physics, it has motivated various applications in different branches in physics. In this section, fibre bundle theory will be introduced and the connection between fibre bundles and the geometric phase will be shown. 3.1 Fibre Bundle A fibre bundle [6] is a mathematical structure that consists of the following elements 1. Manifolds: a bundle space or total space E, a base space M and a fibre F. 2. A structure Lie group G which acts freely on F. i.e. given g, g G, gx = g x g = g. 3. A bundle projection, a surjective map π : E M. Such that at each point x M, the inverse image F x = π 1 (x) is homeomorphic to F, and F x is called fibre at x. 10

17 Chapter 3 Fibre Bundle theory 11 With the bundle projection defined, a map from the base manifold M back to the bundle space E is required. The inverse map s is called a cross-section and it is defined as s : U π 1 (U) (x) (x, f), x U M, f F. (3.1) When the whole manifold can be covered by one open set, the cross section is called a global section; otherwise it is called a local section. Furthermore, we can define a local section s as s : (x) (x, e), x M, e F, (3.2) where e is the identity element in the structure group. This section can serve as a reference local section which generates the other local sections with an appropriate group action. s α (x) = s(x)g α (x). (3.3) A fibre bundle is said to be trivial if there exist a map φ such that, φ : M F E. (3.4) this means that the bundle space can be expressed as a simple Cartesian product of the base manifold and the fibre. In general, a fibre bundle has a complex global structure such as twisting, hence local trivialisations are required for the purpose of explicit calculations. A local trivialisation is a set {(U α, ϕ α )}, where {U α } is an open covering (a chart) of M, and ϕ α is a map such that ϕ α : π 1 (U α ) U α F. (3.5) When moving between local patches, e.g. from U i to U j, for which U i U j, there exists an induced map t ji, such that: t ji : F F. (3.6) The map t ji is the transition function between U i, U j. It turns out to be the composition of ϕ j with ϕ 1 i, t ji = ϕ j ϕ 1 i (3.7)

18 12 Chapter 3 Fibre Bundle theory and it is an element of the structure group G. Essentially it provides a mean of changing from one fibre point f that belongs to (U i, ϕ i ) to another fibre point f in (U j, ϕ j ). f (x) = f(x)t ji (x). (3.8) The transition function encodes the information about the gluing of U i F. In this thesis, a particular type of bundle called a principle bundle is used. A bundle P is called principle G-bundle when the structure group G acts freely on E. The freedom of G-actions implies that each orbit is homeomorphic to G, hence the fibre is homeomorphic to G. 3.2 Fibre Bundle and Gauge Transformations With the principle bundle defined, we are able to define gauge transformations in the context of fibre bundles. In physics, a gauge is a redundant degree of freedom which does not affect the physical system. In mathematics, a gauge is a choice of a local section in the fibre bundle. Gauge transformations correspond to the transformation between local trivialisations. As shown in equation (3.3), a local section can be transformed to the another by an appropriate group action. However, before discuss about the mapping between two local trivialisations, we need to introduce the concpet of the principle bundle map. A principle bundle map is a map (F, f) between a pair of bundle (G, P (G, P π M ) which preserves the group action in the bundle, i.e. π M) and F(pg) = F(p)g, p P, g G. (3.9) Here, F : P P and f : M M. Under gauge transformation, the principle bundle map (F, id M ) maps local sections in the following way: U G F U G π U id M U π

19 Chapter 3 Fibre Bundle theory 13 where F is the bundle automorphism F : M G M G. Furthermore, the bundle automorphism F has an induced map γ : M G such that F(x, g) = (x, γ(x)g) (3.10) Indeed, the induced map γ satisfies equation (3.9). F(x, g)g = (x, γ(x)g)g = (x, γ(x)gg ) = F(x, gg ). (3.11) After introducing gauge transformations in the language of fibre bundles, we will examine their relevance for the geometric phases that arise in the context of fibre bundles. 3.3 Bundle Connection According to Simon [14], the geometric phase can be interpreted as the phase that arises when we consider parallel transport between fibres. The basic idea of parallel transport is to compare the points in the neighbouring fibre. This idea suggests to look at the vector field that is orthogonal to the fibre. The obvious natural field on P is the vector field induced by the right action of the structure group on the fibre. However, these fields are generated by the actions which move along the fibre. Hence, they are pointing in the direction of the fibre rather than away from it. More precisely, this vector field belongs to the vertical subspace V p (P ) of T p (P ), which is defined as: V p P := {τ T p P π τ = 0}. (3.12) In order to construct a vector pointing to another fibre, the concept of the bundle connection is required. A bundle connection assigns H p P to each point p P such that T p P =H p P p V p P R g (H p P ) =H pg P, g G, p P (3.13) where R g (p) := pg denotes the right action of G on P. The bundle connection can be associated by a g(lie algebra)-valued one-form ω which projects the tangent space into the vertical subspace, i.e., ω : T p (P ) V p (P ). (3.14)

20 14 Chapter 3 Fibre Bundle theory Since the bundle connection projects a vector field into the vertical subspace, a horizontal vector field can be defined as the vector field that goes to zero after the projection, i.e., H p P := {v T p P ω p (v) = 0}. (3.15) Furthermore, the g-valued one-form satisfies 1. ω p (X A ) = A, p P, and X A is the vector field which is generated by A g. 2. R gω = Ad g 1ω, where Ad g 1 is an adjoint map which is defined as Ad g (g ) = gg g 1 (3.16) An important property of g-valued one-form is that locally it can be determined by a g-valued one-form on the base manifold. This is defined by the pull-back of ω by the local section: s (ω) = A s (3.17) Using equation (3.16), and equation (3.3), the local connection one-form satisfy A s (x) = g 1 (x)a s g(x) + g 1 dg(x) (3.18) 3.4 Parallel Transport and the Holonomy in a Fibre Bundle With the bundle connection defined, the horizontal vector field which points away from the fibre can be generated. Since there is an isomorphism between the horizontal vector space and vector space on the base manifold M. hence, to each vector field X on M, there exists a unique vector field X in P called the horizontal lift of X such that π X = X, the bundle projection maps the lifted vector field to the original vector field. ver( X) = 0, the lifted vector field does not have a vertical component. Similarly, a curve on the base manifold can be horizontally lifted in the same manner. Given a smooth curve C on M, there exist a horizontal lifted curve C, such that π C = C.

21 Chapter 3 Fibre Bundle theory 15 The tangent vectors w t T P to C has no vertical components and project to tangent vectors v t T M of C. With the definition of horizontal lift introduced, the horizontal lift of a curve can be calculated. Consider the following set up. Let a curve C in base manifold, C : [0, 1] M. The horizontal lift of C, C : [0, 1] P and a bundle P with a complete set of local trivialisation {(U i, ϕ i )}. For simplicity, consider the curve lies in a single chart which reduces the complexity of transiting between charts. Using equation (3.15), and expressing the bundle connection in local coordinates (equation(3.18)), one obtains g s (t) 1 A s (X)g s (t) + g s (t) 1 dg s(t) dt =0. (3.19) where X T P and X T M, and applying g s (t) on the both side gives dg s (t) dt = A s (X)g s (t). (3.20) The solution of the g s (t) is [ g s (t) = exp C(t) C(0) A s (X) ]. (3.21) Writing the local connection one-form and the vector field in the component form: A s = ia j (x)dx j X = dc(t) dt = dxk dt k, (3.22) equation (3.21) can be expressed as: [ g s (t) = exp i [ = exp = exp [ i i t 0 C(t) C(0) C(t) C(0) ] dt dt A j (C(t )) dxj (t ) A j (x)dx j ] A(x) ], x C. (3.23) The solution of g s (t) provides a group element to generates the horizontal lift of the curve C. Hence a horizontal lift can be written as C = s(c)g s (t). (3.24)

22 16 Chapter 3 Fibre Bundle theory A simplification can be made when C is a closed curve, i.e. C(T ) = C(0). The lifted curve is C(T ) = C(0)g s (T ). (3.25) Which shows that there is an extra phase between the two end points. The integrals in equation (3.23) becomes [ ] g s (T ) = exp i A. (3.26) C Different g s (T ) can be obtained by choosing different closed curves in the base manifold. The collection of all group element forms a subgroup in G called the holonomy group of the bundle connection ω. The Berry phase can be interpreted as an element of the holonomy group with the adiabatic condition serving as the bundle connection. 3.5 Fibre Bundle and the Geometric Phase The correspondence between the fibre bundles and the geometric phase was first noticed by Simon[14]. He constructed a fibre bundle which resembles the Hilbert space in quantum mechanics. Here the Hilbert space represents the principle bundle P, with the parameter space assuming the base manifold. The states with a relative U(1) phase project to the same point on the base manifold. Hence they can be represented by points on fibre with the U(1) structure group. The time evolution of an eigenstate defines a curve in the bundle space and this is governed by the Shrödinger s equation. The curve in the bundle space is a horizontal lift of the curve on the base manifold and the adiabatic condition provides a connection called adiabatic connection A. Locally, adiabatic connection can be represented as A = i ψ d ψ (3.27) Notice that when transforming the states to another point of the fibre, i.e. ψ ψ = e iξ ψ (3.28)

23 Chapter 3 Fibre Bundle theory 17 The adiabatic connection after the transformation read as A =i ψ d ψ =i ψ e iξ de iξ ψ =i ψ e iξ e iξ d ψ dξ ψ e iξ e iξ ψ =A dξ (3.29) the holonomy element h of the can be obtained by integrating the connection along a closed curve C. h = A = γ(c) (3.30) C The example shows that the Berry phase is a natural interpretation of the holonomy in the fibre bundle. Note that the geometric phase here arises under cyclic evolution and in the next section, the generalisation to the non-cyclic case will be considered.

24 Chapter 4 Non-Cyclic Geometric Phase In 1988, Samuel and Bhandari [13] claimed that geometric phase can be further generalised by removing the requirement of cyclic evolution. In their paper, the concept of the Pancharatnam connection was adapted and they proposed a condition where the Pancharatnam connection reduces to a bundle connection. Consequently, geometric phase which arises from bundle connection can be calculated in terms of the Pancharatnam connection and geometric phase of open curve can be defined. A brief introduction of the Pancharatnam connection and the concept of non-cyclic geometric phase will be discussed in the following sections. 4.1 Pancharatnam Connection In quantum mechanics, all states of a physical system are represented by element of the Hilbert spaceh. Two states ψ and φ represent the same physical state when ψ = e iα φ, where α R. When two states are equivalent, the phase factor α has no physical meaning. However, when ψ and φ represent two different states, the relative phase factor between two states become significant. In 1956, Pancharatnam [10] provided the physical interpretation and the definition of the relative phase between two non-orthogonal polarization states of light. He showed that the relative phase has a quantum counterpart which was related to the geometry of the projective Hilbert space. Consider two non-orthogonal states ψ 1, ψ 2. The relative phase between two states is defined as the phase factor of the inner product between two states. ψ 1 ψ 2 = re iα. (4.1) 18

25 Chapter 4 Non-Cyclic Geometric Phase 19 Two states are said to be in-phase when α = 0, or ψ 1 ψ 2 R. It is interesting to note that in-phase condition is non-transitive, i.e., if ψ 1 is in phase with ψ 2 and ψ 2 is in-phase with ψ 3, then ψ 3 is not always in-phase with ψ 1. A simple example of the non-transitivity property of in-phase condition is illustrated by the following three normalised states: ( ) ψ 1 = 1 1, ψ 2 = 2 1 The relative phase between three states are ( ) 1 0 ( ), ψ 3 = 1 1. (4.2) 2 i ψ 1 ψ 2 = 1 2, ψ 2 ψ 3 = 1 2, ψ 1 ψ 3 = i 2. (4.3) Clearly ψ 1 and ψ 3 are not in-phase, even when ψ 1 and ψ 3 exhibit in-phase relationship with ψ 2. The rule of defining relative phase is called the Pancharatnam connection. 4.2 Pancharatnam Connection and Bundle Connection Two connections in the fibre bundle have been introduced, which are Bundle connection: Defines horizontal vector fields which point from one fibre towards a neighbouring fibre. Pancharatnam connection: Defines the relative phase between two points in bundle space of which projects to different point of the base manifold. From the descriptions above, these two connections operate differently in fibre bundles. However, according to Samuel and Bhandari [13], the Pancharatnam connection between two states will reduce to the bundle connection when the following condition is fulfilled. If two states ψ 0 and ψ 1 lie on a geodesic in the bundle space. phase β = arg [ ψ 1 ψ 0 ] is given by then the relative β = 1 0 A s ds (4.4) [ where A s = Im ψ 1 d ] ds ψ 0 is the bundle connection and ψ 0 = ψ ( 0), ψ 1 = ψ ( 1). Moreover when the geodesic is also horizontal (relative to the bundle connection) then

26 20 Chapter 4 Non-Cyclic Geometric Phase the two states are said to be in-phase. Essentially they define the non-cyclic geometric phase as follows. They consider a state vector ψ(0) and evolve it along a horizontal curve C 1 (in the bundle space) to the state ψ(τ). If ψ(0) is not orthogonal to the ψ(τ), then they pose the question, what is the phase different between them? They use the following scheme to address this: 1. The curve C 1 is first projected to the base manifold C 1 = π( C 1 ). 2. The end point of C 1 are then joined by a geodesic C 2 (in the base manifold) that results in a closed curve C 2 C The curve C 2 C 1 is then horizontally lifted (relative to the bundle connection) such that the horizontal lift of C 1 segment coincides with C 1. The resultant curve is in general an open curve in the bundle space with the end points, ψ(0) and ψ(t ) are in the same fibre. 4. If this open curve C 2 C 1 is closed by a curve C 3 along the fibre by joining ψ(t ) to ψ(0) then ϕ g = A s ds = A s ds + C 1 A s ds + C 2 A s ds C 3 (4.5) represents the phase difference between ψ(t ) and ψ(0). 5. Since C 2 C 1 is horizontal, C1 A s ds = C2 A s ds = 0, leaving ϕ g = A s ds = C 3 χ(τ) χ(0) dχ = χ(τ) χ(0). (4.6) Here, χ denotes the fibre coordinate). The phase ϕ g is then regraded as the geometric phase associated with the curve C 1 or C 1 in the base manifold. A schematic diagram shows the setup of the non-cyclic geometric phase is shown in figure (4.1).

27 Chapter 4 Non-Cyclic Geometric Phase 21 Figure 4.1: A schematic diagram of the non-cyclic geometric phase. C 1 is the projection of C 1 and C 2 is a geodesic on the base manifold. C1 and C 2 are the horizontal lifts of C 1 and C 2 respectively. C2 C 1 is closed by C 3 along the fibre by joining ψ(t ) to ψ(0). The non-geometric phase can be represented as the phase difference between ψ(t ) and ψ(0). 4.3 Remarks on the Non-Cyclic Geometric Phase Some remarks are in order here. The motivation (or justification) for regrading ϕ g as the geometric phase of a non-cyclic curve C 1 stems from the in-phase condition adsorbed to the curve C 2. The rationale being that with ψ(τ) assuming an in-phase relation with ψ(t ), a measure of the phase difference between ψ(t ) and ψ(0) provides a measure of the phase difference between ψ(0) and ψ(τ). Moreover, the claim that is gauge invariant is attributed to ϕ g = A s ds being gauge-invariant; as cyclic evolution are always gauge invariant. An important consideration that appears to be lacking in this scheme in whether the in-phase criterion is a gauge-invariant concept. That is whether two states that are in-phase remain in-phase after a gauge transformation. Underlying this is also whether geodesics in the bundle space are gauge invariant. In the following we will analyse these issue in greater detail. In this thesis, a particular fibre, namely Hopf Bundle is used as a tool to analyse the problems suggested. Basic concepts of Riemannian geometry, geodesic and Hopf bundle will be introduced in the following chapters as well.

28 Chapter 5 Riemannian Manifold According to Samuel and Bhandari [13], the Pancharatnam connection reduces to a bundle connection when two states are connected by a geodesic. In order to define a geodesic, the concept of Riemannian manifold with a metric tensor is required. A Riemannian manifold is a smooth manifold M with a tensor field g of type T (2,0) M, named after the German mathematician Bernhard Riemann [12]. In a Riemannian manifold, the geometrical properties are encoded in a metric tensor: i.e. knowing the metric tensor allows one to calculate angles between tangent vectors, length of a differetiable curve, curvature of the surface and other much complicated geometric notions [8]. In this section, some definitions and operations of the metric tensor are introduced. 5.1 Metric Tensor A metric tensor g is a tensor field of type T (2,0) M is a map g p : T p (M) T p (M) R, p M (5.1) with the following properties: 1. g is a bilinear, i.e. U, V, W T p (M), a, b R g p (au + bv, W ) = ag p (U, W ) + bg p (V, W ) g p (W, au + bv ) = ag p (W, U) + bg p (W, V ). and 22

29 Chapter 5 Riemannian Manifold g is symmetric, i.e. U, V T p (M), g p (U, V ) = g p (V, U). 3. g is non-degenerate i.e. g p (U, V ) = 0 U T p (M), V = 0. For simplicity, g p (U, V ) will sometime be written as U, Y. If {ϕ i } 1=1,2,... are basis one-forms, a metric tensor can be written locally as g = g ij ϕ i ϕ j, (5.2) where g ij = g ji, is symmetric in its indices and depends smoothly on p M. The coefficient matrix is defined by g ij = e i, e j, where {e i } is the dual basis of ϕ i, i.e. ϕ i e j = δ i j. 5.2 Affine Connection and Geodesics In differential geometry, a geodesic can be regarded as the generalisation of a straight in Euclidean space. An alternative definition of a geodesic is that of a curve that minimises arclength between nearby points. In this thesis, the first definition is adapted and different properties of straight are generalised in Riemannian manifold. A curve in Euclidean space is a straight when its acceleration is zero, and this property is chosen as a property to define a geodesic on Riemann manifold. To define the notion of acceleration in a Riemannian manifold, the concepts of an affine connection and covariant derivative are required. An affine connection is a map : X(M) X(M) X(M) (X, Y ) X Y X(M) (5.3) which satisfy the following conditions: 1. X (Y + Z) = X Y + X Z,

30 24 Chapter 5 Riemannian Manifold 2. X+Y Z = X Z + Y Z, 3. fx = f X Y, 4. X (fy ) = X(f)Y + f X Y, where f C (M), and X, Y, Z X(M). Here X(M) represents the set of all vector fields in M. In a chart φ : U R of M, the affine connection coefficients Γ λ µν are defined as eµ e ν = Γ λ µνe λ (5.4) { } where {e λ } = x λ, and x λ are the coordinates relative to chart φ. For simplicity, x λ is written as λ henceforth. In general, for vector fields V = V µ e µ, W = W ν e ν, V W = V µ e µ W ν e ν =V µ eµ W ν e ν =V µ [ e ν W ν e ν + W ν eµ e ν ] =V µ [ µ W ν e ν + W ν Γ λ µνe λ ] =V µ [ µ W λ + W ν Γ λ µν ] e λ (5.5) V W is known as the covariant derivative of a vector fields W along the direction of V. The covariant derivative specifies the derivative along a tangent vector in a manifold and it can be treated as a generalised directional derivative in Riemannian space. Given a curve in manifold γ : t γ(t), a covariant derivative of a vector fields Y along the curve γ can be expressed as X Y, where X is a vector field such that one of its integral curve coincide with γ. The vector field Y is said to be parallel transported along the curve when X Y = 0. (5.6) When a vector field X is parallel transport along itself, i.e., X X = 0, (5.7)

31 Chapter 5 Riemannian Manifold 25 then the curve is a geodesic. In this context, covariant derivative is treated as the acceleration of a vector fields along a curve. This is intuitively in agreement with the idea that if acceleration is zero along the curve, the vector field will remain in same direction along the curve for all time. The integral curve of this vector field will be a straight line with respect to the manifold. Consider the above expression in local coordinates, a vector field X can be written as X = X i e i, X i = dxi dt. (5.8) By substitute this expression into equation (5.5) with the geodesic condition (equation(5.7)), one can obtain, [ ] X X =X µ µ X λ + X ν Γ λ µν e λ [ dx µ dx λ = dt x µ dt + dxµ dx ν dt [ d 2 x λ = dt 2 + dxµ dx ν dt dt Γλ µν d2 x λ dt 2 + dxµ dt dt Γλ µν ] e λ = 0 ] e λ dx ν dt Γλ µν = 0. (5.9) The set of equations corresponding to different indices λ is known as the geodesic equations in the local coordinate frame. The curve that satisfy the geodesic equations is called a geodesic. 5.3 Affine Connection and Metric Tensor In the previous section, the definition of geodesic was introduced through affine connection. However, notice that the definition in equation (5.7) is independent of the metric tensor. In the following, we show how the metric tensor comes into the formulation. In order to define the affine connection from a metric tensor, the affine connection is required to be compatible with g, i.e. X Y, Z = X Y, Z + Y, X Z, X, Y, Z X(M). (5.10) It turns out that the compatibility condition is not enough to determine a unique connection as given two covariant derivative and, these exists a tensor for transforming

32 26 Chapter 5 Riemannian Manifold from one to the other a x b = ax b C c ab x c (5.11) This arbitrariness can be removed by setting the torsion tensor τ defined as τ(x, Y ) = X Y Y X [X, Y ] (5.12) to be zero. Indeed, forx, Y, Z X(M), equation (5.10) under cyclic permutations of X, Y, Z reads X Y, Z = X Y, Z + Y, X Z, Y Z, X = Y Z, X + Z, Y X, (5.13) Z X, Y = Z X, Y + X, Z Y. Substituting equation (5.12), into equation(5.13), one has X Y, Z = X Y, Z + Y, Z X + Y [X, Z], Y Z, X = Y Z, X + Z, X Y + Z [Y, X], (5.14) Z X, Y = Z X, Y + X, Y Z + X [Z, Y ]. Adding the first two these equations and abstracting the third gives 2 X Y, Z =X Y, Z + Y Z, X Z X, Y X[Y, Z] + Y [Z, X] + Z[X, Y ]. (5.15) This equation is known as the Koszul formula. Suppose there are two connections and which satisfy equation (5.15), since the right hand side is independent of the connections, it follows that X Y XY, Z = 0, X, Y, Z. Hence, =, in other words, the affine connection is uniquely defined for a given metric tensor.

33 Chapter 6 Hopf Bundle In this thesis, the statements suggested by Samuel and Bhandari [13] are studied systematically in the context of fibre bundles. The particular bundle used for the analysis is the Hopf Bundle, with S 3 as the bundle space, S 2 as a base space and a fibre isomorphic to the group U(1). Hopf Bundle was discovered by Heinz Hopf ([5])in 1931 and it is one of the earliest and simplest non-trivial fibre bundles.interestingly, Hopf Bundle has some counterparts in physics; one of the examples being the Bloch sphere which represents all the states in a two-level system. In this chapter, the definition of Hopf Bundle will be introduced and the calculations for the necessary geometric structures will be shown. 6.1 Hopf map Hopf map is a map π from S 3 to S 2. which is defined as π : C 2 R 3 (α, β) (αβ + αβ, i ( αβ αβ ), α 2 β 2 ). (6.1) By embedding unit three-sphere into C 2, any points on sphere can be expressed as S 3 = {(α, β) C 2 α 2 + β 2 = 1}. (6.2) All the points on the unit three-sphere are mapped via π to a unit two-sphere in R 3 and they exhibit U(1) invariance, i.e. π(e iλ α, e iλ β) = π(α, β). (6.3) 27

34 28 Chapter 6 Hopf Bundle Hence, this important property allows the construction of a principle U(1) bundle which has the form (U(1), S 3 π S 2 ). (6.4) This bundle is known as Hopf bundle. Introducing real coordinates one has α = ξ 1 + iξ 2 β = ξ 3 + iξ 4 (6.5) with α 2 + β 2 = ξ ξ ξ ξ 2 4 = 1 (6.6) which is a unit three-sphere in R 4. The Hopf map π : (α, β) (x 1, x 2, x 3 ) gives x 1 = 2(ξ 1 ξ 3 + ξ 2 ξ 4 ), x 2 = 2(ξ 2 ξ 3 ξ 1 ξ 4 ), (6.7) x 2 = ξ ξ 2 2 ξ 2 3 ξ 2 4, which characterises S 2 in R 3. In term of charts, S 2 can be parametrised by several coordinates systems and in this thesis, spherical coordinates and stereographic coordinates are used. Spherical coordinates (θ, φ) are defined as x 1 = sin θ cos φ x 2 = sin θ sin φ (6.8) x 2 = cos θ On the other hand, stereographic coordinates are defined by a map that projects a sphere (minus a point) onto a plane. The projection needs at least two charts to cover whole sphere. Here north pole and south pole are chosen as points of projection and the plane of projection is the plane containing equator of the sphere. The two patches are shown below. The stereographic coordinates of the north patch (u, v), which project the sphere from south pole is u + iv = x 1 ix x 3 = ξ 3 + iξ 4 ξ 1 + iξ 2 = β α. (6.9) Similarly, the stereographic coordinates of the south patch (u, v ) which project the sphere with respect to north pole is u + iv = x 1 + ix 2 1 x 3 = ξ 1 + iξ 2 ξ 3 + iξ 4 = α β. (6.10)

35 Chapter 6 Hopf Bundle 29 As the Hopf bundle is a non-trivial bundle, the bundle space cannot be expressed as a global Cartesian product of the base manifold and fibre space. Thus necessitate the introduction of local trivialisation to facilitate the explicit calculation. 6.2 Local trivialisation and inverse mapping The Hopf bundle is a non-trivial bundle, and posses a complex global structure. However, locally it has simple product structure, which is a local trivialisation. To explicate these trivialisations, the maps for north patch and south patch are φ N : π 1 (U N ) U N U(1) ( β (α, β) α, α ) α eiχ, (6.11) φ S : π 1 (U S ) U S U(1) ( α (α, β) β, β ) β eiψ, (6.12) In order to calculate the metric tensor and bundle connection in a trivialisation, one requires a map φ 1 that inversely maps a local Cartesian product space to the bundle space: φ 1 N : U N U(1) π 1 (U N ) (u, v, χ) (ξ 1, ξ 2, ξ 3, ξ 4 ). (6.13) The inverse map can be calculated by solving e iχ = ξ 1+iξ 2 ξ 2 1 +ξ2 2 and β α = u + iv, α = ξ 1 + iξ 2 = e iχ ξ ξ2 2 β = α(u + iv) = e iχ ξ ξ2 2 (u + iv) = ξ 3 + iξ 4. (6.14) Comparing real and imaginary parts, one obtains ξ 1 = ξ1 2 + ξ2 2 cos χ ξ 2 = ξ1 2 + ξ2 2 sin χ ξ 3 = ξ1 2 + ξ2 2 (u cos χ + v sin χ) (6.15) ξ 4 = ξ1 2 + ξ2 2 (u sin χ v cos χ)

36 30 Chapter 6 Hopf Bundle Since ξ ξ2 2 + ξ2 3 + ξ2 4 = 1, summing the square of the terms above yields ξ ξ 2 2 = 1 u 2 + v (6.16) ξ 1 = f cos χ ξ 2 = f sin χ ξ 3 = f(u cos χ + v sin χ) (6.17) ξ 4 = f(u sin χ v cos χ) where f = u 2 + v Bundle connection and metric tensor in a local trivialisation With the maps defined above, the metric tensor and the bundle connection can be calculated. The metric tensor in S 3 is given by g = dξ 1 dξ 1 + dξ 2 dξ 2 + dξ 3 dξ 3 + dξ 4 dξ 4, (6.18) Subjected to the constraint ξ 2 1 +ξ2 2 +ξ2 3 +ξ2 4 = 1. According to [4], the bundle connection in the S 3 is given in the form ω =αdα + βdβ (6.19a) or equivalently ω =ξ 2 dξ 1 ξ 1 dξ 2 + ξ 4 dξ 3 ξ 3 dξ 4 (6.19b) The metric tensor ḡ and bundle connection ω in U N U(1) can be calculated by applying pull back operation φ 1 N on metric tensor g and bundle connection ω in S3. ḡ = φ 1 N g (6.20) ω = φ 1 N ω (6.21)

37 Chapter 6 Hopf Bundle 31 The calculations of the metric tensor, the bundle connection and a brief introduction about push forward and pull back action are given in the appendix A. Here we summarise them in table (6.1). With the metric tensor and bundle connection in a trivialisation obtained, the validity of the claim by Samuel and Bhandari can be studied. S 3 U N S 1 Metric tensor g = dξ 1 dξ 1 + dξ 2 dξ 2 + dξ 3 dξ 3 + dξ 4 dξ 4 ḡ = f 2 ( v ) du du + f 2 ( u ) dv dv f 2 uv(du dv + dv du) + f 2 v(du dχ + dχ du) + f 2 u(dχ dv + dv dχ) + dχ dχ Bundle connection ω = αdα + βdβ ω = f( vdu + udv) + dχ Table 6.1: The table summaries the metric tensor and the bundle connection in the S 3 and S 1

38 Chapter 7 Relation between Geodesics, Horizontal vectors and the In-Phase Condition In previous chapter, the conditions for a non-cyclic geodesic phase have been shown. In this chapter, the relation between geodesics, horizontal vector fields and the inphase condition is examined. In order to facilitate calculations, the metric tensor is first expressed in the horizontal basis to ensure the curve always satisfies the condition of being a horizontal vector field. 7.1 Metric Tensor in the Horizontal Basis From the previous section, the metric tensor in M E is shown in equation (6.20) and it can be written in the matrix form: f 2 ( v ) f 2 uv f 2 v ḡ = f 2 uv f 2 ( u ) f 2 u. (7.1) f 2 v f 2 u 1 This matrix has several off-diagonal terms which complicate the calculation and simplification can be obtained when it is expressed in terms of the horizontal basis. A vector field is horizontal when it satisfy the following relation: ω( X) = 0. (7.2a) 32

39 Chapter 7 Relation between Geodesics, Horizontal vectors and the In-Phase Condition 33 Let X be an arbitrary horizontal vector X = A u + B v + C χ, (7.2b) then equation (7.2a) gives ω( X) =A( fv) + Bfu C =0. (7.3) Solving C in term of A and B, we obtain C = A( fv) + B(fu). Upon substituting C in equation (7.2b) gives X = A( u fv χ ) + B( v + fu ). (7.4) χ The two horizontal basis can be identified as e u = u fv χ and e v = v + fu χ. In order to have a complete set of basis in M E, e χ = is added as a vertical basis. χ With the horizontal basis defined, the corresponding one-form basis can be calculated by taking the dual of the vector field basis. The result is summarised in table (7.1) vector field basis e u = u fv χ e v = v + fu χ e χ = χ one-form basis E u = du E v = dv E χ = dχ + fvdu fudv Table 7.1: Horizontal vector field basis and its corresponding one-form basis in the stereographic coordinates. In the horizontal basis, the metric tensor has a simpler form: g = f 2 E u E u + f 2 E v E v + E χ E χ (7.5) Besides the stereographic coordinates, the basis can be written in the spherical coordinates for further simplification. The result of horizontal basis in the spherical coordinates is shown in table (7.2)