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Quantum swimming. Boris Gutkin, David Oaknin . Q. Q-Swimming. Swimmer in a quantum sea: Photon bath Fermi sea. C-swimming competition . Shape space=Control space. Swimming stroke=closed path in shape space. Euclidean motions. Shape space. swimming. Swimming: definition .

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quantum swimming

Quantum swimming

Boris Gutkin, David Oaknin

q swimming

Q

Q-Swimming

Swimmer in a quantum sea:

Photon bath

Fermi sea

c swimming competition
C-swimming competition

Shape space=Control space

Swimming stroke=closed

path in shape space

swimming definition

Euclidean motions

Shape space

swimming

Swimming: definition

Swimming: a map

untethered scatterers
Untethered scatterers

Balanced scatterer

q swimming6

qswimming

Euclidean motions

Q-Swimming

Qswimmer: a unthethered balanced periodic scatterer

Scattering matrices

Need: a principle to fix the location

and orientation of balanced scatterer

slide7

Swimmer=scaterer

1-d quantum medium

Swimming in one dimension

r

t

On-shell scattering matrix

The swimmer controls the scatteringbut not its location

c swimming the role of friction

Velocity of ball

Force on ball

Friction coeff

C-Swimming: The role of friction

mutual forces

control and response
Control and response

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

swimming in geometric
Swimming in geometric

Swimming: when dX does not

Integrate to a function

the invisibility principle
The invisibility principle

1-d Fermi sea, T=0

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

quantized swimming
Quantized swimming

1-d Fermi sea, T=0

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

snowplough

Avron, Elgart, Graf, Sadun

key formula

H

Key formula

r

Avron, Elgart, Graf, Sadun

quantum friction
Quantum friction

Momentum transfer to

medium

swimming equation
Swimming equation

Total force on swimmer vanishes

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