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Quantum swimming. Boris Gutkin, David Oaknin . Swimmers and pumps. Shape space=Control space. Swimming stroke. Shape and control . Velocity of ball. Force on ball. Friction coeff. Swimming and linear response. mutual forces. Control and response.

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quantum swimming

Quantum swimming

Boris Gutkin, David Oaknin

swimming and linear response

Velocity of ball

Force on ball

Friction coeff

Swimming and linear response

mutual forces

control and response
Control and response

One unknown, the position X, one equation of motion

swimming in geometric
Swimming in geometric

Swimming: when dX does not

Integrate to a function

slide8

Swimmer=scaterer

1-d quantum medium

Scattering

The swimmer controls the scattering: q, a, g

swimming at t 0
Swimming at T=0

Principle: If the swimmer can control the scattering matrix,

Its position will adjust so that the medium will think nothing has happened

quantized swimming
Quantized swimming

r-plane

Control space

Points of transparency

toolbox
Toolbox

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

energy shift
Energy shift

Dual to Wigner time delay

Ex: in 2 channel case:

sink

battery

snowplough

Avron, Elgart, Graf, Sadun

key formula

H

Key formula

r

Avron, Elgart, Graf, Sadun

slide14
BPT

Standard pumping formula

landauer formula for friction15
Landauer formula for friction

Momentum transfer to

Electron gas

conceptual issues
Conceptual issues

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

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

A function of non-commuting variables

A canonical transformation of half-line

E

p

t

x

Avron, elgart, graf, sadun

phase space
Phase space

A function of non-commuting variables

E

t

weyl calculus
Weyl calculus

symbol

operator

S(E,t) is the classical symbol of the scattering matrix

Avron, elgart, graf, sadun