Quantum swimmingQuantum swimming

Boris Gutkin, David Oaknin Boris Gutkin, David Oaknin

Q-SwimmingQ-Swimming

Q

Swimmer in a quantum sea :

Photon bath

Fermi sea

C-swimming competition C-swimming competition

Shape space=Control space

Swimming stroke=closedpath in shape space

Swimming: definition Swimming: definition

Swimming: a map

Shape space Euclidean motions

swimming

Untethered scatterersUntethered scatterers

Balanced scatterer

Q-SwimmingQ-Swimming

Qswimmer: a unthethered balanced periodic scatterer

Scattering matrices Euclidean motions

qswimming

Need: a principle to fix the location and orientation of balanced scatterer

Swimming in one dimensionSwimming in one dimension

Swimmer=scaterer

1-d quantum medium

The swimmer controls the scatteringbut not its location

tr

On-shell scattering matrix

C-Swimming: The role of frictionC-Swimming: The role of friction

mutual forcesFriction coeff

Force on ball

Velocity of ball

Control and responseControl and response

One unknown, the position X. Equation of motion: equilibrium

Swimming in geometricSwimming in geometric

Swimming: when dX does notIntegrate to a function

The invisibility principle The invisibility principle

The position adjusts so that the medium will think nothing has happened

1-d Fermi sea, T=0

Quantized swimming Quantized swimming

r-plane

Control space

Points of transparency

1-d Fermi sea, T=0

ToolboxToolbox

Avron, Buttiker, Elgart, Gutkin, Graf, Oaknin, Pretre, Sadun, Thomas

Energy shiftEnergy shift

Avron, Elgart, Graf, Sadun

Dual to Wigner time delay

Ex: in 2 channel case:

snowplough sink

Key formulaKey formula

Avron, Elgart, Graf, Sadun

H

Quantum friction Quantum friction

Momentum transfer to medium

Swimming equation Swimming equation

Total force on swimmer vanishes

Conceptual issuesConceptual issues

Avron, Buttiker, Elgart, Gutkin, Graf, Oaknin, Pretre, Sadun, Thomas

(E,t) as a canonical pair (E,t) as a canonical pair

Avron, elgart, graf, sadun

A function of non-commuting variables

A canonical transformation of half-line

x

p E

t

Phase spacePhase space

A function of non-commuting variables

E

t

Adiabatic as semi-classical Adiabatic as semi-classical limit limit

t

E

T