Geometry of Quantum Transport Yosi Avron, Martin Fraas, Gian Michele - - PowerPoint PPT Presentation

Geometry of Quantum Transport

Yosi Avron, Martin Fraas, Gian Michele Graf, Oded Kenneth November 26, 2010

• 1. Outline

◮ Motivation: QHE, adiabatic transport in open q-system ◮ Control and Response ◮ Geometry: ω Symplectic structure, g metric ◮ Main result: f −1 = γg + ω, γ = dephasing ◮ Lindbladians and dephasing ◮ Adiabatic evolutions ◮ K¨

ahler structure

◮ Examples

• 2. Motivation: Quantum Hall effect

◮ Ill characterized microscopically ◮ Quantized Hall resistivity h e2n,

n ∈ Z

◮ Accurate to 12 significant digits ◮ Resolution: Integer is a Chern

number of g.s bundle P(φ) in Hilbert space

◮ Assumption: ǫ ˙

ρ = −i[H(φ), ρ] Unitary evolution

◮ Adiabatic theory: ρ ≈ P ◮ What about open q-system?

• 3. Response and control

◮ Controls: φ = (φp, φx); ◮ Driving= control rates ˙

φ

◮ Response: ∇φH = (∂φpH, ∂φxH) ◮ Example 1: Harmonic oscillator

H(φ) = 1

2(p − φp)2 + 1 2(x − φx)2

controls=(momentum, position), response=(velocity,force)

◮ Example 2: Spin in magnetic field φ = ˆ

B, H(B) = ˆ B · σ

◮ Control=Orientation of ˆ

B, Response= Magnetic moment

x Φ p

• 4. Geometry: Metric and symplectic structure

◮ Suppose P(φ) is the ground state bundle of H(φ) ◮ Example spin 1/2:

P(φ) = 1−ˆ

B·σ 2

, φ = ˆ B

◮ Fubini-Study metric on control space

gµν(φ) = Tr P⊥

• ∂νP, ∂µP
• ◮ Symplectic structure on control space

ωµν(φ) = i Tr P⊥

• ∂νP, ∂µP
• ◮ Endows control space with geometry

◮ Geometry of q-origin

Θ Ψ Ψ

• 5. Adiabatic transport coefficients

◮ Adiabatic evolutions ǫ ˙

ρ = L(ρ), ǫ → 0

◮ Transport coefficients fµν ◮ Response & driving: Tr(ρ∂µH) = · · · + fµν ˙

φν + . . .

◮ Geometry of q-origin