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 = γ g + ω, γ = dephasing ◮ Lindbladians and dephasing ◮ Adiabatic evolutions ◮ K¨ ahler structure ◮ Examples

2. Motivation: Quantum Hall effect ◮ Ill characterized microscopically h ◮ Quantized Hall resistivity e 2 n , 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 p ◮ Controls: φ = ( φ p , φ x ); Φ ◮ Driving= control rates ˙ φ x ◮ Response: ∇ φ H = ( ∂ φ p H , ∂ φ x H ) ◮ Example 1: Harmonic oscillator 2 ( p − φ p ) 2 + 1 H ( φ ) = 1 2 ( x − φ x ) 2 controls=(momentum, position), response=(velocity,force) ◮ Example 2: Spin in magnetic field φ = ˆ H ( B ) = ˆ B , B · σ ◮ Control=Orientation of ˆ B , Response= Magnetic moment

4. Geometry: Metric and symplectic structure ◮ Suppose P ( φ ) is the ground state bundle of H ( φ ) P ( φ ) = 1 − ˆ φ = ˆ B · σ ◮ Example spin 1/2: , B 2 Ψ � ◮ Fubini-Study metric on control space Ψ � � g µν ( φ ) = Tr P ⊥ ∂ ν P , ∂ µ P Θ ◮ Symplectic structure on control space � � ω µν ( φ ) = i Tr P ⊥ ∂ ν P , ∂ µ P ◮ Endows control space with geometry ◮ Geometry of q-origin

5. Adiabatic transport coefficients ◮ Adiabatic evolutions ǫ ˙ ρ = L ( ρ ) , ǫ → 0 ◮ Transport coefficients f µν ◮ Response & driving: Tr ( ρ∂ µ H ) = · · · + f µν ˙ φ ν + . . . ◮ Geometry of q-origin