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Theis: Non-Abelian Geometric Phase and Quaternionic Hopf Fibration

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Theis: Non-Abelian Geometric Phase and Quaternionic Hopf Fibration

  1. 1. 國 立 成 功 大 學 物 理 學 系 碩 士 論 文 非 Abel 幾何相與四元數 Hopf 纖維化 Non-abelian Geometric Phase and Quaternionic Hopf Fibration 研究生: 王忠斌 指導教授: 許祖斌 教授 中 華 民 國 一 零 零 年 六 月 二 十 三 日
  2. 2. 摘要 透過四元數 Hopf 纖維化𝑆4𝑙+3 /𝑆3 ≅ HP𝑙 ,我們研究了任意偶數維𝑆𝑈(2)量子幾 何相。量子態重疊函數的 Schrӧdinger 演化透過用基空間和纖維的座標的表達顯現出 𝑆𝑈(2)幾何相Te−i ∫ 𝐴𝑑𝑡 𝑇 0 和動力相Te 𝑖 ℏ ∫ H𝑑𝑡 𝑇 0 。所得的公式是複數 Hopf 纖維化 𝑆2𝑁+1 /𝑆1 ≅ CP 𝑁 的𝑈(1)幾何相在四元數 Hopf 纖維化的𝑆𝑈(2)類比。這些成果適用於 任意 Hamiltonian 和任意偶數維的量子態,也是在沒有作絕熱近似條件下對於開放或 封閉演化路徑都有效。我們用𝑆7 /𝑆3 ≅ HP1 作為一個例子在三種不同看法下得到: (1)BPST 瞬間子的幾何基礎𝑆7 /𝑆𝑈(2) ≅ 𝑆4 ,(2)交換的複數 Hopf 纖維化 𝑆7 /𝑈(1) ≅ CP3 ,以及(3)一個雙量子位元系統的糾纏態空間為𝑆7 [𝑆3 × 𝑆3 ]。我們 討論了這些看法之間的關係以及瞬間子和糾纏參數兩者的精確對應,也討論了廣義 的 Wilczek-Zee 幾何相,且示範了四元數𝑆𝑈(2)幾何聯絡及其和 Wilczek-Zee-Berry 聯絡的關係。
  3. 3. Abstract Exact non-abelian geometric phase for arbitrary even-dimensional quantum sys- tems is investigated through quaternionic Hopf fibrations S4l+3 /S3 ∼= HPl . Time evolution according to Schr¨odinger equation of the generic state is expressed in terms of quaternionic base manifold and fiber coordinates, and the non-abelian phase factor, T e−i T 0 Adt , as well as the quaternionic analog of the dynamical phase factor, T e i T 0 Hdt , are manifested in the overlap function. The formula derived is the non-abelian quater- nionic Hopf fibration S4l+3 /S3 ∼= HPl analog of the abelian geometric phase construc- tion of the complex Hopf fibration S2N+1 /S1 ∼= CPN . The result obtained holds for arbitrary even-dimensional systems which obey Schr¨odinger evolution with arbitrary Hamiltonian, and is valid without adiabatic approximation for both closed and also open paths. S7 /S3 ∼= HP1 is discussed as an explicit example viewed from three different perspectives: (1) S7 /SU(2) ∼= S4 as the geometrical basis of the Belavin- Polyakov-Schwartz-Tyupkin instanton, (2) S7 as the Hilbert space of a 4-state system for the abelian complex Hopf fibration S7 /U(1) ∼= CP3 , and (3) as the Hilbert space of a bipartite qubit-qubit system with the space of entangled states identified with S7 [S3 ×S3 ]. Explicit relations between these constructions are discussed and the pre- cise correspondence between instanton parameter and entanglement is also clarified. The derivation of non-abelian Wilczek-Zee phase is carried out in a general context and explicit illustrations of the exact non-abelian SU(2) geometric connection and its relation to the Wilczek-Zee-Berry connection is provided .
  4. 4. 誌謝 這份論文得以順利的完成,首先要感謝我的指導教授許祖斌老師,沒有老師深 入淺出的解說以及適時的指點迷津,碩士這兩年會是艱辛而迷茫的。而除了智識的 獲取之外,更重要的是在跟隨老師研究的過程中,老師能使我們漸次地體悟到學術 研究的方法與倫理,這是最為難得而讓我感到慶幸的。另外,暉程學長在研究上的 解惑和靜宜學姊的行政協助以及已畢業的昶國學長的幽默風趣也是不可或缺的支持, 我也要謝謝你/妳們。 這兩年來的生活光景是由許多人們一起拼貼構成的,其中我要感謝那些與我在 無數個餐桌或是行走的路途上,就大至人生意義小至晚餐究竟要吃什麼好而爭論的 朋友們,謝謝你/妳們的熱情與理念,著實為我展開了一幅色彩斑斕的哲學畫布。我 也要感謝台南,感謝她的盛夏和古道熱腸,讓這捲畫布的色彩基調充滿生命力。最 後,我要將這份成就獻給我的家人們,感謝你/妳們的默默支持與付出,這一切因你 /妳們才有了可能。
  5. 5. Contents 1 Introduction and overview 3 2 Geometry of Hilbert space 6 2.1 Quaternionic projective space HPl and its explicit parametrization by inhomogeneous coordinates {hα (η)} . . . . . . . . . . . . . . . . . . . . . 7 2.2 Quaternionic Hopf fibration S4l+3 /SU(2) ∼= HPl . . . . . . . . . . . . . 7 3 Schr¨odinger evolution of physical state 9 3.1 Relation between gauge potentials ACP2l+1 and AHPl of abelian and non-abelian Hopf fibrations . . . . . . . . . . . . . . . . . . . . . . . . 14 4 An explicit example: generic four-state systems 15 4.1 Explicit parametrization of S7 . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Quaternionic Hopf fibration S7 /S3 ∼= HP1 . . . . . . . . . . . . . . . . 16 4.3 Complex Hopf fibration S7 /S1 ∼= CP3 . . . . . . . . . . . . . . . . . . 19 4.4 Entangled states parametrized by S7 [S3 × S3 ] . . . . . . . . . . . . . 20 4.5 Qubit-qubit bipartite system with spin-spin interaction . . . . . . . . . 22 5 Generalization of Wilczek-Zee non-abelian geometric phase 25 5.1 Adiabatic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6 Summary 34 A Algebra of quaternions 35 B Geometry of quaternionic Hopf fibration 36 References 38 1
  6. 6. List of Figures 1 Evolution of entanglement(blue) and instanton ratio(red) of qubit-qubit bipartite system with spin-spin coupling for α = π/4. . . . . . . . . . . 24 2 Time-dependent entanglement of a four-state system considered as a bipartite system for {ω, ∆/ω} = {3, 4/3}(blue), {3, √ 3}(red). . . . . . . 33 2
  7. 7. 1 Introduction and overview The study of geometric phases in quantum mechanics is a fruitful and active endeavor (see, for instance, Ref. [1] for a review, and references therein for a selected sample of the literature and history of the subject). It reveals that rich geometrical structures are present in generic quantum systems; and the rich interplay that can exist be- tween geometrical mathematical structures (e.g. quaternionic K¨ahler manifolds) and physical solitons (monopoles and instantons, just to cite lowest dimensional examples) and quantum mechanics is both fascinating and of great pedagogical value. In 1984, Berry[2] showed that an adiabatically[3] evolved quantum state with time-dependent Hamiltonian acquires, in addition to the usual dynamical phase, a geometrical phase. Simon[4] pointed out that Berry’s phase originates from the holonomy of the Hermi- tian line bundle. Later Wilczek and Zee[5] generalized Berry’s result to systems with energy degeneracies, and non-abelian gauge structure arises in those contexts. In 1987 Aharonov and Anandan[6][7] showed that the total phase of the overlap function can be expressed as the exact sum of abelian dynamical and geometric phases even in the case of non-adiabatic systems. Non-adiabatic non-abelian geometric phase in quaternionic Hilbert space[8] which was shown by Adler and Anandan in 1996 has overlap function similar with Aharonov-Anandan’s in complex Hilbert space. Page[9] demonstrated that the Anandan-Aharonov gauge potential is related to the K¨ahler connection of CPN . In a related thesis[10] the abelian geometric phase associated with the complex Hopf fibration S2N+1 /S1 ∼= CPN was studied in detail, and explicit examples which manifested many interesting features were discussed. In this thesis, non-abelian geometric phases in quantum mechanics are investigated. The non-abelian SU(2) connection is naturally manifested in even-dimensional systems because of the existence of the series of quaternionic Hopf fibrations S4l+3 /S3 ∼= HPl . In section 2, we first show that the geometry of the Hilbert space of any even (N + 1)-dimensional system (N = 2l + 1, l ∈ N) is S4l+3 . By treating the Hilbert space S4l+3 as the total bundle space of the quaternionic Hopf fibration[11], coefficients of 3
  8. 8. quantum states can be expressed explicitly in local coordinates of the quaternionic projective space HPl which serves as the base manifold and the fiber S3 ∼= SU(2). Since each independent quaternion corresponds two independent complex numbers, the construction and identification of the Hilbert space of quantum states with complex coefficients with the total bundle space of the quaternionic Hopf fibration is exact only for even-dimensional systems. The 2-state system corresponds to only a single quaternion, so the lowest dimensional non-trivial quaternionic system (in the sense that the state can evolve from one quaternionic variable to another) is a 4-state system. The geometry of the Hilbert space, the explicit parametrization of the quaternionic projective space by inhomogeneous coordinates patched consistently by transition func- tions, the Hopf fibrations S4l+3 /S3 ∼= HPl , the explicit projection maps of the total bundle space to the base manifold, as well as a brief discussion of HPl as quaternionic K¨ahler manifolds will all be addressed in Section 2. In Section 3, time evolution according to Schr¨odinger equation of the generic even- dimensional quantum state is expressed in terms of quaternionic base manifold HPl and SU(2) fiber coordinates. The non-abelian phase factor T e−i T 0 Adt as well as the quaternionic analog of the dynamical phase factor T e i T 0 Hdt are manifested in the overlap function of the state at any time T and the initial state. Due to the non-abelian nature, time-ordering is present. The formula derived is the quaternionic analog of the abelian geometric phase result [6][10] of the Hopf fibration S2N+1 /S1 ∼= CPN for the quaternionic Hopf fibration S4l+3 /S3 ∼= HPl . The non-abelian result obtained in this thesis is valid for arbitrary even-dimensional systems obeying Schr¨odinger evolution with arbitrary Hamiltonian. It is also an exact result, without adiabatic approximation, and valid for both open and closed paths. In Section 4, the lowest dimensional non-trivial case of the quaternionic Hopf fibra- tion with l = 1 i.e. S7 /SU(2) ∼= HP1 is discussed as an example. This corresponds to generic 4-state systems and offers a fascinating case study since the total bundle space S7 can be viewed from three different contexts and perspectives: (1) as the Hilbert space 4
  9. 9. of the simplest non-trivial case of quaternionic Hopf fibration, S7 /SU(2) ∼= HP1 = S4 which forms the geometrical basis of the Belavin-Polyakov-Schwartz-Tyupkin (BPST) instanton[12], (2) as the Hilbert space of a 4-state system with abelian geometric phase which corresponds to the complex Hopf fibration S7 /U(1) ∼= CP3 , and (3) as the Hilbert space of a total 4-state system which corresponds to bipartite qubit-qubit sys- tem with the space of entangled states identified with S7 [S3 × S3 ]. The explicit relations between these constructions are discussed and the precise relation between instanton parameter and entanglement is also clarified. We consider a bipartite system with spin-spin interaction as a demonstration of our physical example. In Section 5, the Wilczek-Zee derivation is discussed in a general context before the adiabatic result and non-abelian Berry phase is derived. We also provide an ex- plicit illustration of our exact non-abelian geometric connection and its relation to the Wilczek-Zee-Berry connection and verify the relation between the entanglement parameter and the non-abelian geometric connection. A short summary of the major results is presented in Section 6. 5
  10. 10. 2 Geometry of Hilbert space Any state |Ψ(t) in a (N + 1)-dimensional Hilbert space can be expanded in time- independent orthonormal basis {|a }, a = 0, 1, ..., N, as |Ψ(t) = ca (t) |a = za (t) zb(t)¯zb(t) |a , (2.1) wherein not all za can be trivial i.e. (z0 , z1 , ..., zN ) ∈ CN+1 − {0}. In this thesis we consider systems of interest which belong to even-dimensional (≥ 4) Hilbert spaces (i.e. N = 2l + 1, l ∈ N). Two complex numbers can be paired as a quaternion by Cayley-Dickson construction (please see appendix A for details). We may set qα = zα + zα e2 = Re(zα )e0 + Im(zα )e1 + Re(zα )e2 + Im(zα )e3 , (2.2) with explicit representation eµ = σµ /i wherein σi are Pauli matrices for i = 1, 2, 3 and σ0 ≡ iI; and the indices α = 0, 1, ..., l, α ≡ α + l + 1. It follows the state can be expressed as |Ψ = l α=0 Tr(P− 1 Qα ) |α + Tr(P+ 1 (−e2 )Qα ) |α , (2.3) with Qα ≡ qα 1 2 Tr(qβq†β) = cα + cα e2 , P± 1 = 1 2 (I ± σ1 ). (2.4) Let ca = xa + iya , with xa , ya ∈ R. The normalization for the quantum state condition is 1 = N a=0 |ca |2 = N a=0 |xa + ya i|2 = N a=0 |xa |2 + N a=0 |ya |2 (2.5a) = l a=0 (|ca |2 + |cα |2 ) = l α=0 |Qα |2 . (2.5b) The first step of the equation above shows that (N + 1)-dimensional Hilbert space has geometry S4l+3 , while the second expresses the normalization condition for the 6
  11. 11. quaternions {Qα } ∈ Hl+1 . 2.1 Quaternionic projective space HPl and its explicit parametriza- tion by inhomogeneous coordinates {hα (η)} We next introduce the quaternionic projective space HPl which is defined as the collec- tion of rays in Hl+1 −{0} with the equivalence relation {q0 , q1 , ..., ql } ∼ {λq0 , λq1 , ..., λql } for all nontrivial λ ∈ H. To explicitly characterize HPl , we use inhomogeneous coor- dinates {hα (η)}, with hα (η) ≡ (qη )−1 qα in local patch U(η) wherein qη = 0. If {qα 1 } ∼ {qα 2 } are two non-vanishing quaternions in Hl+1 , from the definition of equivalence relation, we have qα 1 = λqα 2 ∀α for some nontrivial λ ∈ H; so hα 1(η) = (qη 1 )−1 qα 1 = (λqη 2 )−1 (λqα 2 ) = (qη 2 )−1 qα 2 = hα 2(η), represent the same point in HPl . Conversely, when hα 1(η) = hα 2(η) ∀α, then (qη 1 )−1 qα 1 = (qη 2 )−1 qα 2 ; so qα 1 = qη 1 (qη 2 )−1 qα 2 = λ qα 2 , thus implying {qα 1 } ∼ {qα 2 }. Hence the inhomogeneous coordinates {hα (η)}, hα (η) ≡ (qη )−1 qα ∀α, constitute a faithful and explicit description of quaternionic projective space HPl in the region wherein qη = 0. The entire HPl is the union of local patches (η) U(η) which are consis- tently patched by transition functions. In the overlap, for example U(0) ∩ U(1) , we have two representations of the same point in HPl , {hα (0)} and {hα (1)}, and they are related by hα (0) = (q0 )−1 qα = (q0 )−1 q1 hα (1), wherein (q0 )−1 q1 is the transition function for U(1) → U(0) . 2.2 Quaternionic Hopf fibration S4l+3 /SU(2) ∼= HPl At the beginning of this section, we have shown that a generic quantum state in (N+1)- dimensional Hilbert space, with N = 2l + 1, l ∈ N, can equivalently be parametrized by Qα = qα √1 2 Tr(qβq†β) , (α = 0, 1, ..., l) which belong to Hl+1 quaternionic space. The normalization condition implies the (N + 1)-dimensional Hilbert space has geometry S4l+3 . There is a precise and beautiful mathematical relation between the Hilbert space S4l+3 , the space of unit quaternions (which are elements of S3 ∼= SU(2) ), and the 7
  12. 12. quaternionic projective space HPl : S4l+3 can be thought of as the total bundle space of the Hopf fibration (S4l+3 /S3 ∼= HPl ) with the base manifold HPl and fibre S3 ∼= SU(2). The Hopf fibration with projection S4l+3 → HPl can in fact be explicitly realized, in each local patch U(η) , by Qα → hα (η) ≡ (qη )−1 qα = (Qη )−1 Qα wherein hα (η) ∈ HPl , and consistently patched for the whole manifold. Note, for instance in U(0) patch, Qα = qα √1 2 Tr(qβq†β) = ˆq0hα (0) 1 2 Tr(hβ (0) h†β (0) ) , wherein ˆq0 = q0 |q0|≡ √ q0q0† is a unit quaternion (i.e. ˆq0 ∈ SU(2) ∼= S3 since ˆq0 can be parametrized by 2 complex numbers whose sum of absolute square is unity as ˆq0 (ˆq0 )† = I). Furthermore the fibre of the projection map is isomorphic to S3 since two points Qα and Q α on S4l+3 are projected to the same hα (0) in HPl iff (ˆq 0 )−1 Q α = (ˆq0 )−1 Qα i.e. Q α = ˆλQα with ˆλ = ˆq 0 (ˆq0 )−1 ∈ SU(2) ∼= S3 . Thus the definition of the inhomogeneous coordinates allows us to concretely realize the Hopf fibration S4l+3 /SU(2) ∼= HPl with S4l+3 as the total bundle space, HPl as the base manifold, and SU(2) as fibre. Locally S4l+3 can be thought of as the product space of the base manifold and the fibre. Thus its metric should reflect this property. By considering an infinitesimal displacement on S4l+3 , the metric is[13] ds2 S4l+3 = 1 2 Tr dQα dQ†α . After somewhat tedious calculations (see appendix B for details), we have ds2 S4l+3 = ds2 HPl + 1 2 Tr(idˆq0 ˆq†0 + ˆq0 AHPl ˆq†0 )2 . (2.6) The first term on the right hand side of the equation above is the metric of the quater- nionic K¨ahler (base) manifold HPl ds2 HPl = 1 2 Tr(dhα Gαβ dh†β ), Gαβ = δαβ I 1 2 Tr(hγh†γ) − h†α hβ [1 2 Tr(hγh†γ)]2 , G†αβ = Gβα . (2.7) The second term is the fibre metric[13] which may be expressed in terms of the SU(2)- connection 1-form AHPl ≡ hα dh†α − dhα h†α iTr(hβh†β) , (2.8) 8
  13. 13. which is the non-abelian quaternionic geometric phase factor that will be mentioned in the next section. Note that idˆq0 ˆq†0 + ˆq0 AHPl ˆq†0 is the gauge-transform of AHPl with SU(2) group element ˆq0 . 3 Schr¨odinger evolution of physical state In this section we shall derive the non-abelian geometric phase in terms of the quater- nionic Hopf fibration constructed earlier. It follows from Eq. (2.4) and the use of the explicit quaternionic Hopf fibration with qα = q0 hα that we may write (in local coordinates for U(0) patch) an arbitrary even-dimensional state as |Ψ = l α,β=0 Tr(P− 1 ˆq0 hα ) |α + Tr(P+ 1 (−e2 )ˆq0 hα ) |α 1 2 Tr(hβh†β) , (3.1) wherein ˆq0 ∈ SU(2) and {hα } ∈ HPl (when there is no confusion, we shall choose to drop the subscript (η) denoting the patch in order not to clutter the notation; the indices will be restored when it is necessary to discuss and compare results of different patches). Bearing in mind the notations and definitions of Section 2, the quantum state and its time evolution can be expressed in compact quaternionic form by using bra-ket notations through Qα = cα + cα e2 = α|Ψ + α|Ψ e2 . (3.2) Then we have Q†α = Ψ|α − e2 Ψ|α , (3.3a) ˙Qα = α| ˙Ψ + α| ˙Ψ e2 (3.3b) Contracting Eq. (3.3b) with Q†α from the right, and noting that qα = q0 hα , the 9
  14. 14. equation becomes ˙Qα Q†α = ˙qα q†α 1 2 Tr(qβq†β) − 1 2 qα q†α 1 2 Tr(qβq†β) 2 d dt 1 2 Tr(qγ q†γ ) = 2 ˙qα q†α − ( ˙qα q†α + qα ˙q†α ) Tr(qβq†β) = ˙qα q†α − qα ˙q†α Tr(qβq†β) = ˙ˆq0 ˆq†0 − iˆq0 Aˆq†0 , (3.4) wherein the connection 1-form A = Adt = hα (dh†α )/(dt) − (dhα )/(dt)h†α iTr(hβh†β) dt. (3.5) is precisely AHPl described earlier. We assume Schr¨odinger’s evolution for the quantum states i.e. i d dt |Ψ = H |Ψ . Then it can be worked out that α| ˙Ψ + α| ˙Ψ e2 Ψ|α − e2 Ψ|α = Ψ| ˙Ψ + α| ˙Ψ α|Ψ − α| ˙Ψ α|Ψ e2 = − e1 Ψ|H|Ψ + ( α|H|Ψ α|Ψ − α|H|Ψ α|Ψ ) e2 (3.6) = − e1 Ψ|H|Ψ + Ψ⊥ |H|Ψ e2 (3.7) = i    Re Ψ⊥ |H|Ψ Ψ|H|Ψ + iIm Ψ⊥ |H|Ψ Ψ|H|Ψ − iIm Ψ⊥ |H|Ψ −Re Ψ⊥ |H|Ψ    ≡ i H, (3.8) wherein we have defined Ψ⊥ ≡ −¯cα |α + ¯cα |α = l α=0 Tr(P− 1 e2 Qα ) |α + Tr(P+ 1 Qα ) |α . (3.9) The time evolution of the non-abelian phase factor ˆq0 is determined by ˙ˆq0 ˆq†0 − iˆq0 Aˆq†0 = i H. (3.10) 10
  15. 15. Thus the non-abelian phase factor ˆq0 can be solved as ˆq0 (t) = T e i t 0 Hdt ˆq0 (0)[T e−i t 0 Adt ]† , (3.11) with the symbol T denoting time-ordering which orders later times to the left of earlier times. It thus follows that in general Eq. (3.1) can be further expressed as |Ψ(t) =Tr  P− 1 T e i t 0 Hdt ˆq0 (0)[T e−i t 0 Adt ]† hα (t) 1 2 Tr(hγ(t)h†γ(t))   |α + Tr  P+ 1 (−e2 )T e i t 0 Hdt ˆq0 (0)[T e−i t 0 Adt ]† hα (t) 1 2 Tr(hγ(t)h†γ(t))   |α . (3.12) Consequently in the all-important overlap function, the relation between total phase and the dynamical and geometrical (non-abelian) phase factors can thus be expressed as Ψ(T)|Ψ(0) = ¯ca (T)ca (0) = ¯cα (T)cα (0) + ¯cα (T)cα (0) = Tr Q†α (T)P− 1 Qα (0) = Tr   h†α (T)ˆq†0 (T) 1 2 Tr(hβ(T)h†β(T)) P− 1 ˆq0 (0)hα (0) 1 2 Tr(hγ(0)h†γ(0))   = Tr   h†α (T) 1 2 Tr(hβ(T)h†β(T)) T e−i T 0 Adt ˆq†0 (0)[T e i T 0 Hdt ]† P− 1 ˆq0 (0) hα (0) 1 2 Tr(hγ(0)h†γ(0))   . (3.13) The L.H.S. is the total phase for a state which evolved according to the Schr¨odinger equation from time 0 to T, while the time-ordered factors with H and A denote the ‘dynamical’ and geometrical contributions which we shall discuss in detail later on. The above formula is the quaternionic analog of the abelian geometric phase result [6][10] with the Hopf fibration S2N+1 /S1 ∼= CPN for the quaternionic Hopf fibration S4l+3 /S3 ∼= HPl , and it is one of the major results derived in this thesis. The formula 11
  16. 16. obtained is valid for arbitrary even-dimensional systems obeying Schr¨odinger evolution and is independent of the particular form of the Hamiltonian. It is also an exact result, without adiabatic approximation and valid for both open and closed paths. Note that when the system traces a ‘closed path’ in HPl i.e. hα (T) = hα (0) ∀α, the overlap or total phase Ψ(T)|Ψ(0) acquires a non-abelian geometric phase factor T exp{−i T 0 Adt} = T exp{−i {hα(T)} {hα(0)} dA} = T exp{−i dA}, (3.14) which, as discussed earlier, depends only on geometrical quaternionic K¨ahler prop- erties of the HPl manifold (as opposed to the ‘dynamical’ Hamiltonian contribution T e i T 0 Hdt to the total phase). The path traced out by the system during evolution is however dependent on the Hamiltonian. T e i T 0 Hdt is the analog in the quaternionic Hopf fibration of the dynamical phase[6][10] of the abelian Hopf fibration which is to be subtracted from the total phase to obtain the Anandan-Aharonov geometrical phase. Since ˆq(0)T e−i T 0 Adt ˆq†0 (0) = T e−i T 0 A dt where A is gauge equivalent to A, for closed paths the total phase Ψ(T)|Ψ(0) is effectively the trace of the product of the geo- metrical T e−i T 0 A dt and dynamical [T e i T 0 Hdt ]† factors with the projection operator P− 1 (which is due to our particular choice of associating two complex numbers with a quaternion in Section 2. Other choices would presumably lead to different projection operators but isomorphic results). In the overlap which is just a complex number, only the σ1 projection is selected, but the non-abelian nature of A may be manifested by different choices of ˆq(0) since A is ‘gauge-rotated’ by the different choices of ˆq(0). In the overlap of different patches describing the same state, say patches 0 and 1 with hα 1 = (q1 )−1 qα = (q1 )−1 q0 hα 0 and transition function (q1 )−1 q0 = (|q0 |/|q1 |)v, with v = (ˆq1 )−1 ˆq0 ∈ SU(2) , the connection 1-forms for the respective patches are gauge-related by the induced gauge transformation such that A1 = hα 1 dh†α 1 − dhα 1 h†α 1 iTr(hβ 1 h†β 1 ) = idvv† + v hα 0 dh†α 0 − dhα 0 h†α 0 iTr(hβ 0 h†β 0 ) v† = idvv† + vA0v† , (3.15) 12
  17. 17. and the gauge field strengths by F1 = vF0v† . (3.16) Hence the geometric phase factor transforms as T exp{−i T 0 A1dt} = v(T)T exp{−i T 0 A0dt}v† (0). (3.17) Furthermore, the relation between total, dynamical, and geometrical non-abelian phase factors is invariant with respect to patch transition in the following sense: by using hα 0 = (|q1 |/|q0 |)v† hα 1 and noting that i H = −e1 Ψ|H|Ψ + Ψ⊥ |H|Ψ e2 is already patch-invariant, we have Ψ(T)|Ψ(0) = Tr   h†α 0 (T) 1 2 Tr(hβ 0 (T)h†β 0 (T)) T e−i T 0 A0dt ˆq†0 (0)[T e i T 0 Hdt ]† P− 1 ˆq0 (0) hα 0 (0) 1 2 Tr(hγ 0(0)h†γ 0 (0))   (3.18) = Tr h†α 1 (T)v(T) 1 2 Tr(hβ 1 (T)h†β 1 (T)) × v† (T)T e−i T 0 A1dt v(0)× ˆq†0 (0)[T e i T 0 Hdt ]† P− 1 ˆq0 (0) × v† (0)hα 1 (0) 1 2 Tr(hγ 1(0)h†γ 1 (0)) (3.19) = Tr   h†α 1 (T) 1 2 Tr(hβ 1 (T)h†β 1 (T) T e−i T 0 A1dt ˆq†1 (0)[T e i T 0 Hdt ]† P− 1 ˆq1 (0) hα 1 (0) 1 2 Tr(hγ 1(0)h†γ 1 (0))   . (3.20) In other words the total phase or overlap is invariant, and so are its respective relations to the local quantities (which ‘justifies’ a posteriori our suppression of the patch index). 13
  18. 18. 3.1 Relation between gauge potentials ACP2l+1 and AHPl of abelian and non-abelian Hopf fibrations Exact abelian geometric phase can be derived naturally (analogous to our non-abelian construction) using the complex Hopf fibration S2(2l+1)+1 /U(1) ∼= CP2l+1 , wherein CP2l+1 denotes complex projective space[9][10]. From this perspective[10], the quan- tum Hilbert space of interest has an alternative expression as the total bundle space whose coordinates can be expressed in terms of local inhomogeneous complex projective space coordinates ζa ≡ za /z0 ∈ CP2l+1 and coordinates of the fibre S1 parametrized by 0 ≤ φ0 < 2π thus giving rise to |Ψ(t) = 2l+1 a=0 ca (t) |a = 2l+1 a,b=0 za (t) zb(t)¯zb(t) |a = 2l+1 a,b=0 eiφz0 (t) ζa (t) ζb(t)¯ζb(t) |a , (3.21) wherein φz0 (t) = φz0 (0) − ζ(t) ζ(0) ACP2l+1 − 1 t 0 Ψ|H|Ψ dt. Details of this construction and its investigations can be found in [10]. We would like to point out a precise relation between the abelian and non-abelian connections ACP2l+1 = dζa ¯ζa − ζa d¯ζa 2iζb ¯ζb , AHPl = hα dh†α − dhα h†α iTr(hβh†β) , (3.22) for the conventions we adopted. The relation, assuming c0 = 0, the expression in the U(0) patch, with q0 = 0 and using za = z0 ζa , qα = q0 hα , and cβ = (Qβ −e1 Qβ e1 )/2, cβ = (Qβ + e1 Qβ e1 )(−e2 )/2, is −i¯ca dca = ¯za dza − d¯za za 2i¯zbzb = dφz0 + ACP2l+1 = ACP2l+1 (3.23) = −iTr(P− 1 ¯ca dca ) = i 2 Tr(σ1 dQα Q†α ) = i 2 Tr σ1 dqα q†α − qα dq†α Tr(qβq†β) = 1 2 Tr σ1 (idˆq0 ˆq†0 + ˆq0 AHPl ˆq†0 ) = 1 2 Tr σ1 AHPl . (3.24) 14
  19. 19. Thus the gauge equivalent class of the abelian connection is just the σ1 component of the gauge equivalent class of the non-abelian connection. 4 An explicit example: generic four-state systems The Hilbert space of a four-state system is S7 which has several interesting properties from different geometrical perspectives: (1) it corresponds to the Hilbert space of the simplest non-trivial quaternionic Hopf fibration, S7 /[SU(2) ∼= S3 ] ∼= HP1 ∼= S4 which is the geometrical basis of the BPST instanton[12] (2) the Hilbert space for abelian geometric phase construction which corresponds to the complex Hopf fibration S7 /[U(1) ∼= S1 ] ∼= CP3 . (3) the Hilbert space of a bipartite qubit-qubit system (by choice of an appropriate division of a total 4-state system), with the space of entangled states identified as S7 [SU(2) × SU(2)] with SU(2) ∼= S3 . The key to the precise relations between these highly interesting objects is the com- mon total bundle space S7 which can be parametrized explicitly for the computations of the relevant objects. To wit, we describe the total bundle space as follows: 4.1 Explicit parametrization of S7 S7 can be defined by points (Q0 , Q1 ) ∈ H2 obeying |Q0 |2 + |Q1 |2 = I. Letting Q0 = u cos θ 2 , Q1 = uv sin θ 2 , wherein u and v are unit quaternions belonging to SU(2) and are explicitly parameterized as u = ee1(γ1+β1)/2 cos α1 2 + ee1(γ1−β1)/2 sin α1 2 e2 , (4.1) v = ee1(γ2+β2)/2 cos α2 2 + ee1(γ2−β2)/2 sin α2 2 e2 , (4.2) with ranges 0 ≤ θ ≤ π, 0 ≤ α s ≤ π, 0 ≤ β s ≤ 2π, 0 ≤ γ s ≤ 4π. Identifying the two quaternions Q0 and Q1 above with what was described in Section 2, the corresponding 15
  20. 20. coefficients of the four-state system then worked out to be c0 = 1 2 Tr(P− 1 Q0 ) = cos θ 2 cos α1 2 ei(γ1+β1)/2 , (4.3) c2 = 1 2 Tr(P+ 1 (iσ2 )Q0 ) = cos θ 2 sin α1 2 ei(γ1−β1)/2 , (4.4) c1 = 1 2 Tr(P− 1 Q1 ) = sin θ 2 ei(γ1+γ2+β1+β2)/2 cos α1 2 cos α2 2 − ei(γ1−γ2−β1+β2)/2 sin α1 2 sin α2 2 , (4.5) c3 = 1 2 Tr(P+ 1 (iσ2 )Q1 ) = sin θ 2 ei(γ1+γ2+β1−β2)/2 cos α1 2 sin α2 2 + ei(γ1−γ2−β1−β2)/2 sin α1 2 cos α2 2 . (4.6) 4.2 Quaternionic Hopf fibration S7 /S3 ∼= HP1 In this subsection, we consider S7 as a non-trivial fibre bundle over quaternionic projec- tive space HP1 with SU(2) ∼= S3 gauge group. Following the earlier parametrization of S7 , in U(0) patch, coordinates of HP1 are h0 = (Q0 )−1 Q0 = I, h1 = (Q0 )−1 Q1 = v tan θ 2 , respectively. Hence the SU(2)-connection of HP1 is AHP1 = 1 α,β=0 hα dh†α − dhα h†α iTr(hβh†β) = 1 β=0 h1 dh†1 − dh1 h†1 iTr(hβh†β) = i sin2 θ 2 dvv† . (4.7) We digress to verify and clarify the bundle relation S7 /S3 ∼= HP1 mentioned in Section 2. After somewhat tedious calculation, we have ds2 S7 = 1 2 Tr 1 α=0 dQα dQ†α = 1 4 (dθ)2 + (dγ1)2 + (dβ1)2 + 2 cos α1dγ1dβ1 + (dα1)2 + sin2 θ 2 [(dα2)2 + (dγ2)2 + (dβ2)2 + 2 cos α2dβ2dγ2 + 2(dγ2 + cos α2dβ2)(cos α1dγ1 + dβ1) + 2 cos(β1 + γ2)(dα1dα2 − sin α1 sin α2dγ1dβ2) + 2 sin(β1 + γ2)(sin α1dα2dγ1 + sin α2dα1dβ2)] , (4.8) 16
  21. 21. ds2 HP1 = 1 2 Tr( 1 α,β=0 dhα Gαβ dh†β ) = 1 2 Tr(dh1 G11 dh†1 ) = 1 4 sin2 θ 2 cos2 θ 2 (dγ2)2 + (dβ2)2 + 2 cos α2dγ2dβ2 + (dα2)2 + 1 4 (dθ)2 , (4.9) wherein G11 = δ11 I 1 2 Tr( 1 β=0 hβh†β) − h†1 h1 [1 2 Tr( 1 β=0 hβh†β)]2 = cos4 θ 2 I, (4.10) and 1 2 Tr(id ˆQ0 ˆQ†0 + ˆQ0 AHP1 ˆQ†0 )2 = 1 4 sin4 θ 2 (dγ2)2 + (dβ2)2 + 2 cos α2dγ2dβ2 + (dα2)2 + (dγ1)2 + (dβ1)2 + 2 cos α1dγ1dβ1 + (dα1)2 + sin2 θ 2 [2(dγ2 + cos α2dβ2)(cos α1dγ1 + dβ1) + 2 cos(β1 + γ2)(dα1dα2 − sin α1 sin α2dγ1dβ2) + 2 sin(β1 + γ2)(sin α1dα2dγ1 + sin α2dα1dβ2)] . (4.11) They indeed satisfy the relation ds2 S7 = ds2 HP1 + 1 2 Tr(id ˆQ0 ˆQ†0 + ˆQ0 AHP1 ˆQ†0 )2 . (4.12) Note that AHP1 is the gauge potential of an instanton with charge one, i.e. BPST instanton, which can be verified by calculating the second Chern number C2. To show this, we first calculate the gauge field and then follow the computations detailed below: F = DA = dA + iA ∧ A = i sin θ 2 (dθ ∧ dvv† − sin θ 2 dv ∧ dv† ) (4.13) ⇒ F ∧ F = sin3 θ 8 (dθ ∧ dvv† ∧ dv ∧ dv† + dv ∧ dv† ∧ dθ ∧ dvv† ) = − sin3 θ 4 dθ ∧ (dvv† )3 . (4.14) 17
  22. 22. By dvv† = Ωi ei , i = 1, 2, 3, we have Ω1 = 1 2 Tr(−e1 dvv† ) = 1 2 (dγ2 + cos α2dβ2), (4.15) Ω2 = 1 2 Tr(−e2 dvv† ) = 1 2 (cos γ2dα2 + sin γ2 sin α2dβ2), (4.16) Ω3 = 1 2 Tr(−e3 dvv† ) = 1 2 (sin γ2dα2 − cos γ2 sin α2dβ2). (4.17) · · · Tr(dvv† )3 = Ωi ∧ Ωj ∧ Ωk Tr( σi i σj i σk i ) = iΩi ∧ Ωj ∧ Ωk Tr((i ijm σm + δij )σk ) = −Ωi ∧ Ωj ∧ Ωk Tr( ijm (i mkl σl + δmk )) = −Ωi ∧ Ωj ∧ Ωk Tr( ijk ) = −2 ijk Ωi ∧ Ωj ∧ Ωk = −2 · 3!Ω1 ∧ Ω2 ∧ Ω3 = 3 2 sin α2dα2 ∧ dβ2 ∧ dγ2. (4.18) · · · C2 = − 1 8π2 S4 Tr(F ∧ F) = 1 8π2 S4 sin3 θ 4 dθ ∧ Tr(dvv† )3 = 1 8π2 S4 sin3 θ 4 dθ ∧ ( 3 2 sin α2dα2 ∧ dβ2 ∧ dγ2) = 3 64π2 π 0 sin3 θdθ π 0 sin α2dα2 2π 0 dβ2 4π 0 dγ2 = 1. (4.19) We need at least two local charts U(0) and U(1) to cover the whole HP1 . In patch U(1) , hα (1) = (Q1 )−1 Qα fails at θ = 0 since Q1 = uv sin θ 2 = 0. And in patch U(0) , hα (0) = (Q0 )−1 Qα fails at θ = π since Q0 = u cos θ 2 = 0. In their overlap U(0) U(1) , coordinates transform according to hα (0) = (Q0 )−1 Q1 hα (1), where (Q0 )−1 Q1 = v tan θ 2 is the transition function which is a large gauge transformation with winding number 1 with the gauge potential transforming as A(0) = idvv† +vA(1)v† . This can be seen from the alternative way to calculate the second Chern number C2: by using F(0) = vF(1)v† and Tr(F ∧F) = dTr(F ∧A− i 3 A3 ) expressed as the exterior derivative of the Chern- 18
  23. 23. Simons 3-form, we have C2 = − 1 8π2 S4 Tr(F ∧ F) = − 1 8π2 U(0) Tr(F(0) ∧ F(0)) − 1 8π2 U(1) Tr(F(1) ∧ F(1)) = − 1 8π2 S3 Tr(F(0) ∧ A(0) − i 3 A3 (0)) + 1 8π2 S3 Tr(F(1) ∧ A(1) − i 3 A3 (1)) = 1 24π2 S3 Tr(dvv† )3 = 1. (4.20) To conclude this subsection, the generic four-state system for S7 /SU(2) ∼= HP1 = S4 can be expressed as |Ψ(t) = 1 α,β=0 Tr P− 1 ˆQ0 (t)hα (t) |α + Tr P+ 1 (iσ2 ) ˆQ0 (t)hα (t) |α 1 2 Tr(hβ(t)h†β(t)) , (4.21) wherein ˆQ0 (t) = T e i t 0 Hdt ˆQ0 (0)[T e−i h(t) h(0) AHP1 ]† , and the geometric connection is pre- cisely the BPST instanton gauge potential! 4.3 Complex Hopf fibration S7 /S1 ∼= CP3 In this subsection the abelian geometric phase is calculated from the perspective of abelian complex Hopf fibration. Using the coefficients of the generic four-state system stated earlier, the coordinates of complex projective space CP3 are listed below as: ζ0 = c0 /c0 = 1, (4.22) ζ1 = c1 /c0 = tan θ 2 cos α2 2 ei(γ2+β2)/2 1 − e−i(γ2+β1) tan α1 2 tan α2 2 , (4.23) ζ2 = c2 /c0 = tan α1 2 e−iβ1 , (4.24) ζ3 = c3 /c0 = tan θ 2 sin α2 2 ei(γ2−β2)/2 1 + e−i(γ2+β1) tan α1 2 cot α2 2 . (4.25) 19
  24. 24. The U(1)-connection is then worked out to be ACP3 = 3 a,b=0 ¯ζa dζa − ζa d¯ζa 2iζb ¯ζb = − sin2 α1 2 dβ1 + 1 2 sin2 θ 2 {sin(β1 + γ2) sin α1dα2 + [cos α1 cos α2 − sin α1 sin α2 cos(β1 + γ2)]dβ2 + cos α1dγ2}. (4.26) The SAME generic four-state system for this construction is |Ψ(t) = 3 a,b=0 eiφz0 (t) ζa (t) ¯ζb(t)ζb(t) |a , (4.27) wherein φz0 (t) = φz0 (0) − ζ(t) ζ(0) ACP3 − 1 t 0 Ψ|H|Ψ dt. The universal coordinates of S7 allow us to explicitly and precisely relate the (ζa , φ0) and (hα , ˆq0 ) coordinates. 4.4 Entangled states parametrized by S7 [S3 × S3 ] The quantum state of any qubit-qubit bipartite system can be written as |Ψ = cij |i ⊗ |j , (4.28) wherein i and j assume values ±. The measure of entanglement can be taken to be | det c|2 which is monotonic to all other reasonable measures, as has been shown in[14]. By singular value decomposition, cij = [UDV ]ij, wherein U and V are unitary 2×2 matrices, and the diagonal real positive matrix D = diag.[λ, √ 1 − λ2] follows from the normalization condition Tr(cc† )=1. Furthermore, the overall phase factor of U and V , | det U| and | det V |, can be absorbed into the definition of the basis. Consequently, without loss in generality, cij = [µDν]ij with µ, ν ∈ SU(2). Hence the entanglement parameter is 0 ≤ | det c|2 = | det D|2 ≤ λ2 (1 − λ2 ) ≤ 1 4 , with the lower bound corresponding to λ = 0, 1 and upper bound to λ = 1√ 2 . To explicitly show when the state is separable (i.e. | det | = 0 ⇔ λ = 0, 1) iff it 20
  25. 25. belongs to [S3 × S3 ], we note that generic SU(2) matrices can be expressed as µ =    a b −¯b ¯a    , ν =    c d − ¯d ¯c    , (4.29) wherein |a|2 + |b|2 = |c|2 + |d|2 = 1 (thus SU(2) ∼= S3 ). Then the state |Ψ = (b |+ + ¯a |− )⊗(− ¯d |+ + ¯c |− ) for λ = 0 and |Ψ = (a |+ −¯b |− )⊗(c |+ +d |− ) for λ = 1. In both instances, the state indeed belongs to generic elements of the product separable qubit-qubit Hilbert spaces S3 × S3 . From the Schmidt decomposition displayed above, the total bipartite Hilbert space S7 is entangled iff λ = 0, 1 ⇐⇒ | det | = 0 i.e. iff the state belongs to S7 [S3 × S3 ]. Hence entanglement is determined completely by the parameter | det c|. To relate the entanglement to our construction, we consider the generic four-state system as a total qubit-qubit bipartite system which is parameterized by two quater- nions Q0 = C0 + C2 e2 = u cos θ 2 , Q1 = C1 + C3 e2 = uv sin θ 2 , u, v ∈ SU(2), 0 ≤ θ ≤ π. (4.30) The relation between the total four-state system and the qubit-qubit bipartite system can be constructed through the mapping between the respective bases as |Ψ = cij |i ⊗ |j = Ca |a ⇒ cij = ij|a Ca = Ua ijCa , (4.31) with Tr(Ua U†b ) = Ua ij(U†b )ji = Ua ij(Ub ij)∗ = ij|a b|ij = b|ij ij|a = δab . Then det c = 1 2 ij kl cikcjl = 1 2 (iσ2)ij (iσ2)kl cikcjl = 1 2 Tr(σ2 cT σ2 c) = 1 2 Tr(σ2 (Ua Ca )T σ2 (Ub Cb )) = 1 2 Tr(σ2 (Ua )T σ2 Ub )|Ca ||Cb |, (4.32) wherein Ua ≡ eiφa Ua , Tr(Ua U†b ) = ei(φa−φb) Tr(Ua U†b ) = ei(φa−φb) δab = δab . Then 21
  26. 26. we choose Ua = ua σa √ 2 , with u0 = i, u1 = 1, u2 = i, u3 = 1 adopted to satisfy the previous orthonormalization condition for Ua . Using σ2 (σa )T σ2 = −(σa )† , yields the entanglement parameter as det c = − 1 4 Tr((σa )† σb )ua ub |Ca ||Cb | = − 1 2 ((u0 )2 |C0 |2 + (u1 )2 |C1 |2 + (u2 )2 |C2 |2 + (u3 )2 |C3 |2 ) = 1 2 (|C0 |2 + |C2 |2 − |C1 |2 − |C3 |2 ) = 1 2 (cos2 θ 2 − sin2 θ 2 ) = 1 2 cos θ. (4.33) Therefore, for this construction we have identified the entanglement parameter | det c|2 = 1 4 cos2 θ which is related by a simple elegant formula to the SAME θ parameter of the BPST instanton discussed earlier (see also Ref.[15]). Note that the instanton connec- tion is AHP1 = i sin2 θ 2 dvv† = i r2 r2+s2 dv(x)v† (x), wherein x is the coordinate of the base manifold S4 , r ≡ |x| is the distance to the instanton center, and s is the size of the in- stanton. It also follows that r/s = tan θ 2 . So maximum entanglement occurs at θ = 0, π corresponding to r/s = 0, ∞ respectively. On the other hand, minimum entanglement is reached when θ = π/2, which corresponds to r/s = 1. We shall return to discuss of these interesting relations in explicit examples in later sections. 4.5 Qubit-qubit bipartite system with spin-spin interaction We now consider a qubit-qubit bipartite system with spin-spin coupling as our physical example wherein non-trivial evolution of entanglement and its relation to the instanton picture can be illustrated. The Hamiltonian of spin-spin coupling is assumed to be H = JS1 · S2 = J 2 (S2 − S2 1 − S2 2), (4.34) with S = S1 + S2. For the bipartite system, in total-spin basis, {|↑↑ , |↑↓ +|↓↑ √ 2 , |↓↓ , |↑↓ −|↓↑ √ 2 }, the Hamil- tonian is diagonal, [H] = diag.[J 4 2 , J 4 2 , J 4 2 , −3J 4 2 ]. We may choose the orthonormal 22
  27. 27. qubit-qubit bipartite basis as          |++ |+− |−+ |−−          =          ei J 4 t |↑↑ ie−i J 4 t |↑↓ e−i J 4 t |↓↑ ei J 4 t |↓↓          , (4.35) which is related to the total-spin basis by the matrix transformation [T] =          ei J 4 t 0 0 0 0 ie−i J 4 t / √ 2 0 ie−i J 4 t / √ 2 0 e−i J 4 t / √ 2 0 −e−i J 4 t / √ 2 0 0 eiJ 4 t 0          . (4.36) Thus the evolution operator in qubit-qubit bipartite basis is [U]bipartite = exp (− i THT† − T d dt T† )t =          1 0 0 0 0 cos J 2 t sin J 2 t 0 0 − sin J 2 t cos J 2 t 0 0 0 0 1          . (4.37) Obviously there is non-trivial evolution when |+− or |−+ is involved in the initial state. As an example let us consider the case that initial state is |Ψ(0) = cos α |+− + sin α |−+ . (4.38) Then the evolved quantum state is |Ψ(t) = cos α cos J 2 t + sin α sin J 2 t |+− − cos α sin J 2 t − sin α cos J 2 t |−+ . (4.39) Following the discussion of entanglement of bipartite system in Subsection 4.4, the 23
  28. 28. entanglement parameter is computed to be 4 |det c|2 = sin2 (2α − J t), (4.40) and the corresponding instanton ratio is r/s = 1 + sin(2α − J t) 1 − sin(2α − J t) . (4.41) 0 Π 2 Π 3 Π 2 2 Π 5 Π 2 3 Π J t Sin2 2 Α 1 2 3 Figure 1: Evolution of entanglement(blue) and instanton ratio(red) of qubit-qubit bipartite system with spin-spin coupling for α = π/4. 24
  29. 29. 5 Generalization of Wilczek-Zee non-abelian geo- metric phase In this section we discuss the non-abelian Berry phase of Wilczek and Zee[5] in a general context. Unlike our non-abelian geometric phase formula which originates from the exact Hopf fibration and which is valid for arbitrary even-dimensional systems, the Wilczek-Zee non-abelian Berry phase derivation and formula is predicated upon the particular nature of the degeneracy of the Hamiltonian (which determines the gauge group of the non-abelian Berry connection) and is valid only for adiabatic systems. This means that there can be no generic correspondence between our Hopf fibration construction and the Wilczek-Zee approach, although specific examples with particular Hamiltonians can exhibit similarities between the two formalisms. We next discuss the derivation of the Wilczek-Zee non-abelian geometric phase in a context which is slightly more general than the original derivation[5]. The state |Ψ(t) can be expanded in time-dependent energy-eigenstates {|Eαi (t) } of Hamiltonian of the system, |Ψ(t) = α,i aαi (t)e− i t 0 Eα(t )dt |Eαi (t) , (5.1) wherein the subscript α denotes the energy eigenvalue and i is the degeneracy index i.e. states |Eαi (t) have energy Eα for all i. Substituting into the Schr¨odinger equation yields daβj (t) dt = i α,i aαi (t)(M)βj,αi , (5.2) wherein (M)βj,αi = e i t 0 (Eβ(t )−Eα(t ))dt (A)βj,αi , (A)βj,αi = i Eβj (t) d dt |Eαi (t) . (5.3) 25
  30. 30. Thus the state can be expressed as |Ψ(t) = α,β,i,j aβj (0) T ei t 0 Mdt αi,βj e− i t 0 Eα(t )dt |Eαi (t) . (5.4) The Hermitian Hamiltonian can be written as H(t) = V (t)E(t)V † (t) such that Eβj (0)|E(t)|Eγk (0) = Eγ(t)δβj,γk (5.5) which is diagonal. Then the eigenvalue equation H(t) |Eαi (t) = Eα(t) |Eαi (t) becomes Eβj (0)|H(t)|Eγk (0) Eγk (0)|Eαi (t) = Vβj,σl Eσl,ηm V † ηm,γk Eγk (0)|Eαi (t) = Eα(t) Eβj (0)|Eαi (t) (5.6) ⇒Eσl,ηm Wηm,αi = Eσ(t)Wσl,αi = Eα(t)Wσl,αi , Wσl,αi = V † σl,βj Eβj (0)|Eαi (t) (5.7) ⇒(Eσ(t) − Eα(t))Wσl,αi = 0 (5.8) ⇒Wσl,αi (t) = 0 for (σ = α), i.e. block − diagonal. (5.9) Thus, from Eq. (5.7), we have Eβj (0)|Eαi (t) = Vβj,σl Wσl,αi ⇒ |Eαi (t) = V (t)W(t) |Eαi (0) . (5.10) So starting the evolution with energy-eigenstate |Eαi (0) , the overlap between the evolved quantum state |Ψαi (t) = U(t) |Eαi (0) and energy-eigenstate at time t (from Eqs. (5.4) and (5.10)) is Eβj (t)|Ψαi (t) = T ei t 0 Mdt βj,αi e− i t 0 Eβ(t )dt = Eβj (0)|W† V † U|Eαi (0) . (5.11) 26
  31. 31. On the other hand, for |Ψ(t) = U(t) |Ψ(0) , Schr¨odinger equation can be written as i d dt U = HU ⇒ ( d dt V )(V † U) + V d dt (V † U) = − i V (t)E(t)V † (t)U(t) (5.12) ⇒ d dt (V † U) = (− i E(t) − V † d dt V )(V † U) (5.13) ⇒(V † U)(t) = T exp t 0 (− i E(t ) − V † d dt V )dt V † (0). (5.14) And (A)βj,αi = i Eβj (t) d dt |Eαi (t) = i Eβj (0) (V W)† d dt (V W) |Eαi (0) (5.15) = i(W† d dt W)βj,αi + W† βj,γk (A )γk,σl Wσl,αi , (A )γk,σl = i(V † d dt V )γk,σl . (5.16) In general, bearing in mind Eqs. (5.14) and (5.16), Eq. (5.11) becomes Eβj (t)|Ψαi (t) = T ei t 0 Mdt βj,αi e− i t 0 Eβ(t )dt = Eβj (0) W† (t)T exp i t 0 (− 1 E(t ) + A )dt V † (0) |Eαi (0) (5.17) = Eβj (0) W† (t)T exp i t 0 (− 1 E(t ) + A )dt W(0) |Eαi (0) (5.18) = Eβj (0) T exp i t 0 (− 1 E(t ) + A)dt |Eαi (0) , (5.19) wherein (M)βj,αi = e i t 0 (Eβ(t )−Eα(t ))dt (A)βj,αi with A = iW† d dt W + W† A W, and the properties W(t) is block-diagonal and E(t) is proportional to the identity in each block have been used. Different choices of W lead to gauge-equivalent A. Note that up to this present point, the formula above is exact and we have made no approximations. This formula is more general than that presented in the Wilczek-Zee derivation and a non-abelian gauge potential A is already present. We shall show that the result reduces to the Wilczek-Zee formula in the adiabatic limit. Following the construction which lead to Eq. (3.13), we have the above relation in 27
  32. 32. quaternionic form as Eαi (t)|Ψ(t) = Tr  P− 1 T e i t 0 Hdt ˆq0 (0)[T e−i t 0 Adt ]† hβ (t) 1 2 Tr(hγ(t)h†γ(t)) (V W)β,αi (t) − (V W)β,αi (t)e2   , (5.20) where β, γ = 0, 1, ..., l. 5.1 Adiabatic limit When adiabatic limit is taken i.e. when the condition | Eβj (t) d dt |Eαi (t) | = Eβj (t) dH(t) dt |Eαi (t) Eβ(t) − Eα(t) ≈ 0 for (α = β), (5.21) is satisfied, the relevant quantities obey A → Aadia, (Aadia)βj,αi = i (V W)† d dt (V W) adia βj,αi ≈ 0 for (α = β) (5.22) ⇒ (Madia)βj,αi ≈ (Aadia)βj,αi and [ 1 E(t), Aadia] ≈ 0, (5.23) and the exact result of Eq. (5.19) in this limit becomes Eβj (t)|Ψαi (t) adia ≈ Eβj (0) e− i t 0 E(t )dt T ei t 0 (Aadia)dt |Eαi (0) , (5.24) wherein T ei t 0 (Aadia)dt is precisely the Wilczek-Zee’s non-abelian geometric phase[5]. 28
  33. 33. 5.2 Examples As a simple example, consider the case that V (t)W(t) = e−iKt , wherein K is time- independent and Hermitian. This results in (A)βj,αi = i (V W)† d dt (V W) βj,αi = (K)βj,αi . (5.25) Then the adiabatic limit simplifies to the requirement that K is block-diagonal with respect to the degeneracy of H(t) i.e. | Eβj (t) d dt |Eαi (t) | = |Kβj,αi | ≈ 0 for (β = α), K → Kadia. (5.26) Hence [1 E(t), Kadia] ≈ 0, and T exp t 0 (− i E(t ) + iKadia)dt ≈ exp − i t 0 E(t )dt exp {iKadiat} , (5.27) wherein, as proven earlier, exp{iKadiat} is the non-abelian geometric Wilczek-Zee phase factor. To make the example even more explicit, consider a Hamiltonian with (3,1) degen- eracy, with H(t) = V (t)E(0)V † (t), and in energy-eigenstate basis at t = 0, E(0) =          E0 0 0 0 0 E0 0 0 0 0 E0 0 0 0 0 E1          , V (t)W(t) = e−iKt , K = i          0 α1 −α2 −ω1 −α1 0 α3 −ω2 α2 −α3 0 −ω3 ω1 ω2 ω3 0          . (5.28) Substituting into Eq. (5.19), we have Eβj (t)|Ψαi (t) = T ei t 0 Mdt βj,αi e− i t 0 Eβ(t )dt (5.29) = Eβj (0) exp − i E(0)t + iKt |Eαi (0) , (5.30) 29
  34. 34. wherein (M)βj,αi = e i (Eβ−Eα)t (A)βj,αi , (A)βj,αi = (K)βj,αi . The adiabatic condition, |Kαi,βj | ≈ 0 for (α = β) is in this case realized by taking wi ≈ 0, i = 1, 2, 3 thus implying [1 E(0), Kadia] ≈ 0. Then the above equation reduces to Eβj (t)|Ψαi (t) adia ≈ Eβj (0) e− i E(0)t eiKadiat |Eαi (0) = e− i Eβt (eiKadiat )βj,αi . (5.31) To specialize to a particular case as an illustration of Eq. (5.20) and the Wilczek- Zee’s non-abelian geometric phase, we adopt the following K (hence V (t)W(t)) and consistently the corresponding Hamiltonian of the form K = i          0 α 0 0 −α 0 0 0 0 0 0 −ω 0 0 ω 0          , H =          ∆ + Σ 0 0 0 0 ∆ + Σ 0 0 0 0 Σ + ∆ cos 2ωt ∆ sin 2ωt 0 0 ∆ sin 2ωt Σ − ∆ cos 2ωt          , (5.32) with Σ = (E0 + E1)/ and ∆ = (E0 − E1)/ . Then the energy-eigenstates at time t (in energy-eigenstate basis at t = 0) are {|Eµ(t) } =             1 0 0 0          ,          0 1 0 0          ,          0 0 cos ωt sin ωt          ,          0 0 − sin ωt cos ωt             = {|0 , |1 , cos ωt |2 + sin ωt |3 , − sin ωt |2 + cos ωt |3 }, (5.33) which correspond respectively to energy eigenvalues {E0, E0, E0, E1}. For this type of 30
  35. 35. system, the time evolution operator can be solved exactly as U(t) =                e−it(∆+Σ) 0 0 0 0 e−it(∆+Σ) 0 0 0 0 e−iΣt cos ωt cos Ωt −(i∆ cos ωt−ω sin ωt) Ω sin Ωt e−iΣt − sin ωt cos Ωt +(ω cos ωt−i∆ sin ωt) Ω sin Ωt 0 0 e−iΣt sin ωt cos Ωt −(ω cos ωt+i∆ sin ωt) Ω sin Ωt e−iΣt cos ωt cos Ωt +(i∆ cos ωt+ω sin ωt) Ω sin Ωt                , (5.34) with Ω = √ ω2 + ∆2. There is obviously non-trivial evolution when |2 or |3 is involved in the initial state. As a specific non-trivial example, we choose |2 to be our initial state (which is the third eigenstate at t = 0 of the block of 3 degenerate eigenstates). Then we have the exact evolved state as |Ψ2(t) = e−iΣt cos ωt cos Ωt − (i∆ cos ωt − ω sin ωt) Ω sin Ωt |2 + sin ωt cos Ωt − (ω cos ωt + i∆ sin ωt) Ω sin Ωt |3 . (5.35) The corresponding quaternions Q0 and Q1 are Q0 = e−e1Σt cos ωt cos Ωt − (e1 ∆ cos ωt − ω sin ωt) Ω sin Ωt e2 , (5.36) Q1 = e−e1Σt sin ωt cos Ωt − (ω cos ωt + e1 ∆ sin ωt) Ω sin Ωt e2 , (5.37) and the coordinates of HP1 , h0 and h1 , are h0 =I, (5.38) h1 = ∆2 sin 2ωt + ω2 sin 2ωt cos 2Ωt − ωΩ cos 2ωt sin 2Ωt + (2ω∆ sin2 Ωt)e1 ω2 + 2∆2 cos2 ωt + ω2 cos 2ωt cos 2Ωt + ωΩ sin 2ωt sin 2Ωt e2 (5.39) =v tan θ 2 , (5.40) 31
  36. 36. where tan θ 2 = |Q1 |/|Q0 |, sin2 θ 2 = |Q1 |2 , and v = [(∆2 + ω2 cos2 2Ωt) sin 2ωt − ωΩ cos 2ωt sin 2Ωt]I + (2ω∆ sin2 2Ωt)e1 ((∆2 + ω2 cos2 2Ωt) sin 2ωt − ωΩ cos 2ωt sin 2Ωt)2 + (2ω∆ sin2 2Ωt)2 . (5.41) Then the exact geometric connection 1-form and the quaternionic Hamiltonian quantity in the dynamical phase, H, are explicitly A = −ω∆ 4ω (∆2 + ω2 cos2 Ωt) cos 2ωt sin2 Ωt − Ω (∆2 + ω2 cos 2Ωt) sin 2ωt sin 2Ωt Ω2(ω2 + 2∆2 cos2 ωt + ω2 cos 2ωt cos 2Ωt + ωΩ sin 2ωt sin 2Ωt) σ1 (5.42) =i sin2 θ 2 ( d dt v)v† , (5.43) = (t sin 2∆t − 2 sin2 ∆t ∆ )ω2 + O(ω3 ) σ1 , (5.44) H = Σ + ∆3 + ω2 ∆ cos 2Ωt Ω2 σ1 . (5.45) Hence the R.H.S. of Eq. (5.20) is computed to be E2(t)|Ψ2(t) = e−i(Σ+∆)t 1 − 1 + 2i∆t − e2i∆t 4∆2 ω2 + O(ω3 ) . (5.46) Note that this is the exact result containing ω and demonstrates the correction to the lowest-order ω-independent adiabatic contribution. The adiabatic result is obtained by taking ω → 0. Following the methods discussed earlier, the full matrix calculated according to R.H.S. of Eq. (5.20) is Eβj (t)|Ψαi (t) =          e−i E0t cos αt −e−i E0t sin αt 0 0 e−i E0t sin αt e−i E0t cos αt 0 0 0 0 e−i E0t [1 + O(ω2 )] e−i E1t e−i∆t sin ∆t ∆ ω + O(ω3 ) 0 0 e−i E0t −ei∆t sin ∆t ∆ ω + O(ω3 ) e−i E1t [1 + O(ω2 )]          . (5.47) 32
  37. 37. Eq. (5.20) indeed converges to the result given by Eq. (5.31) in the adiabatic limit of ω → 0. We may furthermore compute, following the method discussed in Section 4.4, the entanglement parameter of the evolving state of Eq. (5.35) to be 4 |det c(t)|2 = ∆ Ω 2 cos 2ωt + ω Ω sin 2ωt sin 2Ωt + ω Ω 2 cos 2ωt cos 2Ωt 2 (5.48) = cos2 θ. (5.49) It is indeed related to the connection 1-form A in Eq. (5.42) by the formulae of Eq. (5.43) and Eq. (5.49). Π 3 Π 2 2 Π 3 Π 4 Π 3 3 Π 2 t 0.2 0.4 0.6 0.8 1.0 Figure 2: Time-dependent entanglement of a four-state system considered as a bipartite system for {ω, ∆/ω} = {3, 4/3}(blue), {3, √ 3}(red). 33
  38. 38. 6 Summary We summarize the major results of this thesis: We discussed in detail, and explicitly realized, the quaternionic Hopf fibration series S4l+3 /S3 ∼= HPl with explicit projection map and inhomogeneous coordinates for the base manifold; and derived the generic overlap function wherein the non-abelian geo- metric phase factor of the quaternionic K¨ahler connection is manifested. In addition we also derived the quaternionic analog of the dynamical phase factor. Our construction is exact and valid for arbitrary (finite) even-dimensional systems with arbitrary Hamil- tonian obeying Schr¨odinger evolution and holds for both closed and open paths. This set of results is the precise quaternionic analog of the Anandan-Aharonov geometric phase formalism (in which the total phase is the sum of geometrical and dynamical phases) which corresponds to the abelian Hopf fibration S2N+1 /S1 ∼= CPN . The generic 4-state quantum system is investigated and correlated from three dif- ferent perspectives: (1) as the S7 /S3 ∼= S4 description of the BPST instanton in which the SU(2) instanton gauge potential is precisely the geometric K¨ahler connection of the geometric phase (2) as the abelian geometric phase manifestation of the complex Hopf fibration S7 /S1 ∼= CP3 , and (3) as a qubit-qubit bipartite system with the space of entangled states identified with S7 [SU(2)×SU(2)], and the entanglement measure precisely related to the instanton parameter. Exact non-adiabatic non-abelian geometric phase is derived in a general sense by considering degeneracies in the Hamiltonian, and the Wilczek-Zee’s result is obtained in the adiabatic limit. Explicit examples which served to illustrate, clarify, and verify our constructions are provided and worked out in detail. The formalism we presented is valid for very generic systems and Hamiltonians and can be applied to entanglement generation and the control of entanglement in quantum systems. Our work also reveals and illustrates the rich interplay that can exist between geometrical mathematical structures, physical solitons, and generic quantum mechanical systems. 34
  39. 39. A Algebra of quaternions Quaternions were discovered and introduced by the Irish mathematician-physicist Sir W. B. Hamilton in 1844[16][17]. The algebra is associative but non-commutative. A quaternion is characterized by four real numbers {x0, x1, x2, x3}, q = x0 + x1i + x2j + x3k, x s ∈ R, (A.1) wherein {1, i, j, k} is the basis for quaternions, and they obey Hamilton’s famous for- mula, i2 = j2 = k2 = ijk = −1. (A.2) In this thesis, we realize {1, i, j, k} with explicit matrices {e0 , e1 , e2 , e3 }, wherein eµ = σµ /i, σi are Pauli matrices for i = 1, 2, 3 and σ0 ≡ iI, and it can be checked explicitly that they indeed obey Hamilton’s formula (A.2). Quaternions q and p = y0 + y1i + y2j + y3k, obey the following rules: Addition: q + p = (x0 + x1i + x2j + x3k) + (y0 + y1i + y2j + y3k) = (x0 + y0) + (x1 + y1)i + (x2 + y2)j + (x3 + y3)k. (A.3) Multiplication: qp =(x0 + x1i + x2j + x3k)(y0 + y1i + y2j + y3k) =(x0y0 − x1y1 − x2y2 − x3y3) + (x0y1 + x1y0 + x2y3 − x3y2)i + (x0y2 + x2y0 + x3y1 − x1y3)j + (x0y3 + x3y0 + x1y2 − x2y1)k. (A.4) Conjugation: ¯q = x0 − x1i − x2j − x3k, qp = ¯p¯q. (A.5) 35
  40. 40. Norm: |q| = x2 0 + x2 1 + x2 2 + x2 3, |q|2 = q¯q, |qp| = |q| |p| . (A.6) Note: |q| = 1 for unit quaternion, denoted by ˆq. In fact, a unit quaternion can be expressed as ˆq = q |q| . Inverse: q−1 = ¯q |q|2 . (A.7) A quaternion can be built from two complex numbers by the Cayley-Dickson con- struction as shown below: q = u + vj = x0 + x1i + x2j + x3k, where u = x0 + x1i, v = x2 + x3i, x s ∈ R (A.8) ⇒ ¯q = ¯u − j¯v = ¯u − vj = x0 − x1i − x2j − x3k, where ¯u = x0 − x1i. (A.9) B Geometry of quaternionic Hopf fibration Defining the inhomogeneous coordinates {hα } of HPl by hα ≡ (q0 )−1 qα = (Q0 )−1 Qα , the metric[13] of an infinitesimal displacement on S4l+3 is ds2 S4l+3 = dca d¯ca = 1 2 Tr dQα dQ†α . (B.1) Using Qα = qα √1 2 Tr(qβq†β) and qα = q0 hα , we have dQα dQ†α = d   ˆq0 hα 1 2 Tr(hβh†β)   d   h†α ˆq†0 1 2 Tr(hγh†γ)   (B.2) =   dˆq0 hα 1 2 Tr(hβh†β) + ˆq0 dhα 1 2 Tr(hβh†β) − 1 2 ˆq0 hα [1 2 Tr(hβh†β)] 3 2 d 1 2 Tr(hγ h†γ )   (c.c.) (B.3) 36
  41. 41. =dˆq0 dˆq†0 + dhα dh†α 1 2 Tr(hβh†β) + 1 4 d(1 2 Tr(hγ h†γ )) 2 [1 2 Tr(hβh†β)]2 I − 1 2 ˆq0 dˆq†0 d 1 2 Tr(hγ h†γ ) 1 2 Tr(hβh†β) − 1 2 ˆq0 hα dh†α ˆq†0 d 1 2 Tr(hγ h†γ ) [1 2 Tr(hβh†β)]2 + dˆq0 hα dh†α ˆq†0 1 2 Tr(hβh†β) − 1 2 dˆq0 ˆq†0 d 1 2 Tr(hγ h†γ ) 1 2 Tr(hβh†β) + ˆq0 dhα h†α dˆq†0 1 2 Tr(hβh†β) − 1 2 ˆq0 dhα h†α ˆq†0 d 1 2 Tr(hγ h†γ ) [1 2 Tr(hβh†β)]2 (B.4) =dˆq0 dˆq†0 + dˆq0 (hα dh†α − dhα h†α )ˆq†0 Tr(hβh†β) + ˆq0 (dhα h†α − hα dh†α )dˆq†0 Tr(hβh†β) + dhα dh†α 1 2 Tr(hβh†β) − 1 4 d(1 2 Tr(hγ h†γ )) 2 [1 2 Tr(hβh†β)]2 I (B.5) =ˆq0 ˆq†0 dˆq0 dˆq†0 ˆq0 + ˆq†0 dˆq0 (hα dh†α − dhα h†α ) Tr(hβh†β) + (dhα h†α − hα dh†α )dˆq†0 ˆq0 Tr(hβh†β) ˆq†0 + dhα dh†α 1 2 Tr(hβh†β) − 1 4 d(1 2 Tr(hγ h†γ )) 2 [1 2 Tr(hβh†β)]2 I (B.6) =ˆq0 ˆq†0 dˆq0 + dhα h†α − hα dh†α Tr(hβh†β) dˆq†0 ˆq0 − dhα h†α − hα dh†α Tr(hβh†β) ˆq†0 + ˆq0 hα dh†α − dhα h†α Tr(hβh†β) 2 ˆq†0 + dhα dh†α 1 2 Tr(hβh†β) − 1 4 d(1 2 Tr(hγ h†γ )) 2 [1 2 Tr(hβh†β)]2 I (B.7) = idˆq0 ˆq†0 − ˆq0 dhα h†α − hα dh†α iTr(hβh†β) ˆq†0 2 + dhα dh†α 1 2 Tr(hβh†β) − dhα h†α hγ dh†γ [1 2 Tr(hβh†β)]2 . (B.8) Hence we obtain ds2 S4l+3 = ds2 HPl + 1 2 Tr(idˆq0 ˆq†0 + ˆq0 AHPl ˆq†0 )2 , (B.9) where AHPl = hα dh†α − dhα h†α iTr(hβh†β) , (B.10) ds2 HPl = 1 2 Tr(dhα Gαβ dh†β ), Gαβ = δαβ I 1 2 Tr(hγh†γ) − h†α hβ [1 2 Tr(hγh†γ)]2 , G†αβ = Gβα . (B.11) 37
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