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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

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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

Swimming: a map


Untethered scatterers

Balanced scatterer


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


Swimmer=scaterer

1-d quantum medium

Swimming in one dimension

r

t

On-shell scattering matrix

The swimmer controls the scatteringbut not its location


Velocity of ball

Force on ball

Friction coeff

C-Swimming: The role of friction

mutual forces


Control and response

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


Swimming in geometric

Swimming: when dX does not

Integrate to a function


The invisibility principle

1-d Fermi sea, T=0

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


Quantized swimming

1-d Fermi sea, T=0

r-plane

Control space

Points of transparency


Toolbox

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


Energy shift

Dual to Wigner time delay

Ex: in 2 channel case:

sink

snowplough

Avron, Elgart, Graf, Sadun


H

Key formula

r

Avron, Elgart, Graf, Sadun


Quantum friction

Momentum transfer to

medium


Swimming equation

Total force on swimmer vanishes


Conceptual issues

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


(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

A function of non-commuting variables

E

t


Adiabatic as semi-classical limit

E

t

T


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