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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 l.jpg

Quantum swimming

Boris Gutkin, David Oaknin


Q swimming l.jpg

Q

Q-Swimming

Swimmer in a quantum sea:

Photon bath

Fermi sea


C swimming competition l.jpg
C-swimming competition

Shape space=Control space

Swimming stroke=closed

path in shape space


Swimming definition l.jpg

Euclidean motions

Shape space

swimming

Swimming: definition

Swimming: a map


Untethered scatterers l.jpg
Untethered scatterers

Balanced scatterer


Q swimming6 l.jpg

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 l.jpg

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 l.jpg

Velocity of ball

Force on ball

Friction coeff

C-Swimming: The role of friction

mutual forces


Control and response l.jpg
Control and response

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


Swimming in geometric l.jpg
Swimming in geometric

Swimming: when dX does not

Integrate to a function


The invisibility principle l.jpg
The invisibility principle

1-d Fermi sea, T=0

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


Quantized swimming l.jpg
Quantized swimming

1-d Fermi sea, T=0

r-plane

Control space

Points of transparency


Toolbox l.jpg
Toolbox

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


Energy shift l.jpg
Energy shift

Dual to Wigner time delay

Ex: in 2 channel case:

sink

snowplough

Avron, Elgart, Graf, Sadun


Key formula l.jpg

H

Key formula

r

Avron, Elgart, Graf, Sadun


Quantum friction l.jpg
Quantum friction

Momentum transfer to

medium


Swimming equation l.jpg
Swimming equation

Total force on swimmer vanishes


Conceptual issues l.jpg
Conceptual issues

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


E t as a canonical pair l.jpg
(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 l.jpg
Phase space

A function of non-commuting variables

E

t



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