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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