Exotic quantum holonomy for nonadiabatic evolution

JPS 2015 Autum Meeting [Bussei]
(2015-09-16/2015-09-19)
Kansai Univ. (Senriyama campus)
16pCQ-9
Japanese title: 新奇な量子ホロノミーの非断熱拡張の例

Transcript

1. 1 / 14 Exotic quantum holonomy for nonadiabatic evolution ৽حͳྔࢠϗϩϊϛʔͷඇஅ೤֦ுͷྫ Atushi Tanaka (ాதಞ࢘), Taksu Cheon (શ୎थ)A Tokyo Metropolitan Univ. (ट౎େཧ޻), Kochi Univ. Tech. (ߴ஌޻Պେ)A 2015-09-16 JPS 2015 Autum Meeting [Bussei] (2015-09-16/2015-09-19) Kansai Univ. (Senriyama campus) 16pCQ-9
2. Introduction 2 / 14 Adiabatic passage . . . a simple and robust way to control quantum states. FIG. An adiabatic response of state vector to the dial with angle λ.
3. Introduction 3 / 14 Quantum Holonomy (adiabatic version) . . . discrepancy induced by adiabatic cycles |0(λ) |0 eiγ|0 λ Quantum holonomy (∼ geometric phase factor) |0(λ) |0 |1 λ Exotic quantum holonomy (permutation of eigenspaces)
4. Introduction 4 / 14 Example of Exotic Quantum Holonomy A kicked spin-1 2 : H(t) = 1 2 B ·σ +ϕ1−σz 2 ∑ ∞ n=−∞ δ(t−n) , where Bx and By are adiabatic parameters. We also impose (Bx,By) = B(cosϕ,sinϕ) (from AT and TC, PLA 379, 1693 (2015)). Bx By O C π a0 FIG. The adiabatic evolution of Bloch vector a = ⟨0|σ|0⟩ of a stationary state |0⟩, which is an an eigenvector of the Floquet operator.
5. Introduction 5 / 14 Outline Aim We will explain a nonadiabatic extension of exotic quantum holonomy based on the topological formulation (AT and TC, PLA 379, 1693 (2015)). Introduction Topological formulation Parameterize paths by dynamical variables Example of nonadiabatic EQH Summary
6. Topological formulation 6 / 14 Topological formulation: What is changed by exotic cycles? (△) vectors |0 eiθ1 |1 |1 eiθ0 |0 (△) projectors |0 0| |1 1| |1 1| |0 0| (⃝) ordered projectors (|0 0|, |1 1|, . . .) (|1 1|, |0 0|, . . .) (|1 1|, |0 0|, . . .) (|0 0|, |1 1|, . . .)
7. Topological formulation 7 / 14 EQH = a permutation of projectors induced by C (P0 (λ), P1 (λ), . . .) (P0 , P1 , . . .) (P1 , P0 , . . .) λ FIG. Exotic quantum holonomy in terms of ordered projectors p ≡ (P0,P1,...), where Pn is the n-th eigenprojector.
8. Parameterize paths by dynamical variables 8 / 14 Introduction Topological formulation Parameterize paths by dynamical variables Example of nonadiabatic EQH Summary
9. Parameterize paths by dynamical variables 9 / 14 For geometric phase: from Hamiltonian to a projector An adiabatic path has two equivalent interpretations (Aharonov and Anandan, PRL 58, 1593 (1987)) with a Hamiltonian or projector. λ H(λ) Parameterization of H(λ). λ P(λ) Parameterization of a projector P(λ). The latter setting is ready to extend to nonadiabatic evolution, since the projector P (≃ density operator) is a dynamical variable.
10. Parameterize paths by dynamical variables 10 / 14 Extension of AA’s P to EQH: set of eigenprojectors b We extend the single projector P to a set of eigenprojectors b ≡ {P0,P1,...}, where the order of Pn’s is disregarded. λ b(λ) = {P0 (λ), P1 (λ), . . .} In the nonadiabatic setting, Pn’s are just mutually orthogonal projectors, and b obeys the Schr¨ odinger equation. Now, for a closed trajectory of b, EQH is defined. p = ˆ P0 , ˆ P1 , . . . σ[C] (p) = ˆ Pσ[C] (0) , ˆ Pσ[C] (1) , . . . σ[C] C b(λ0 ) b(λ) = ˆ P0 (λ), ˆ P1 (λ), . . .
11. Example of nonadiabatic EQH 11 / 14 Introduction Topological formulation Parameterize paths by dynamical variables Example of nonadiabatic EQH Summary
12. Example of nonadiabatic EQH 12 / 14 Example of nonadiabatic EQH Assume that a time evolution operator U ≡ exp ← [ − i ¯ h ∫ T 0 H(t)dt ] has two eigenvalues ±z0, where U|0⟩ = z0|0⟩, and U|1⟩ = (−z0)|1⟩ hold. Two vectors |±⟩ ≡ 1 √ 2 (|0⟩±|1⟩) exhibit nonadiabatic EQH, since U|±⟩ = z0|∓⟩ holds. p = (| + + |, | − − |) σ[C] (p) = (| − − |, | + + |) σ[C] C b(λ0 ) b(λ) = {| − − |, | + + |} FIG. |±⟩ exhibit nonadiabatic EQH.
13. Example of nonadiabatic EQH 13 / 14 An implementation of the example by a kicked spin-1 2 This example can be realized in the kicked spin (we set T = 1) H(t) = π 2 σz + B 2 σy ∞ ∑ n=−∞ δ(t−n). The corresponding Floquet operator U(T) = e−π 2 σz e−i B 2 σy has eigenvalues ±i. b = {P0,P1} is represented by a director (headless vector) n ∼ {a,−a} of the Bloch vector a. n + n - n0 z x FIG. Two trajectories of director n in the real projective plane RP2.
14. Summary 14 / 14 Summary ▶ We show that the nonadiabatic extension (` a la Aharonov-Anandan) of exotic quantum holonomy (EQH) naturally emerges from its topological formulation. ▶ The central ingredients are 1. The permutation of elements in p = (P0,P1,...) is regarded as EQH. 2. The trajectory of b = {P0,P1,...} specifies a closed path both for adiabatic and nonadiabatic EQH. where Pn’s are mutually orthogonal projectors. ▶ We show an example of nonadiabatic EQH and its implementation in a quantum kicked spin-1 2 . Ref. AT and TC, PLA 379, 1693 (2015). n + n - n0 z x