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Quantum swimming
Boris Gutkin, David Oaknin
Q-Swimming
Q
Swimmer in a quantum sea:
Photon bath
Fermi sea
C-swimming competition
Shape space=Control space
Swimming stroke=closed
path in shape space
Swimming: definition
Swimming: a map
Euclidean motions
Shape space
swimming
Untethered scatterers
Balanced scatterer
Q-Swimming
Qswimmer: a unthethered balanced periodic scatterer
qswimming
Scattering matrices
Euclidean motions
Need: a principle to fix the location
and orientation of balanced scatterer
Swimming in one dimension
Swimmer=scaterer
1-d quantum medium
r
t
On-shell scattering matrix
The swimmer controls the scattering but not its location
C-Swimming: The role of friction
Velocity of ball
Force on ball
Friction coeff
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:
snowplough
sink
Avron, Elgart, Graf, Sadun
Key formula
r
H
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