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Quantum Shock Waves
Hydrodynamic Singularities in Degenerate 1D quantum systems
P. Wiegmann
Together with Alexander Abanov
How does a wave packet propagate in degenerate Fermi gas?
degenerate Bose gas?
Free fermions in 1D
A smooth bump in density or momenta:
all gradients << Fermi scale
A single particle:
Wave packet consisting of a single particle diffuses
Does quantum coherence
(or Fermi sea)
make an impact?
Can this question be answered by elementary means?
Free Fermions
Hydrodynamics of quantum coherent systems
(traditionally called bosonization):
String theory
Methods:
Integrable hierarchies /matrix models
Hydrodynamics: to express particles (fermions or bosons)
through hydrodynamics (bosonic) modes:
ivtxu x ),(
Bose FieldBose Field
0||)()()( dxxcxcxV
eV
)()( xVx
)(|),(| xVetxV iHt
)(|)()(|)(),(),( tVxcxctVtxtxu x
Initial (coherent) Initial (coherent) statestate
Evolving stateEvolving state
Matrix Matrix elementelement
Integrability - Free FermionsIntegrability - Free Fermions
0||0|0,...),()()()(
21
dxxcxcxVeVtt
k
kk xtxV
0
)(
Tau-function of KP-Tau-function of KP-hierarchyhierarchyMiwa, Jimbo, Miwa, Jimbo, DateDate
kkk JtdxxcxcxV )()()(
nn
nkk ccJ Virasoro currentsVirasoro currents
;;
nn
nkk ccJ Virasoro currentsVirasoro currents
n
nnk
k ccnH Currents of -Currents of -algebraalgebra
1W
KP-hierarchyKP-hierarchy
Benjamin-Ono hierarchyBenjamin-Ono hierarchy
bosonization - linear hydrodynamics:
Linearisation of the spectrum:
Shape does not change!?
Dispersion -
asymmetry between particles and holes
Benjamin-Ono-equationFree Fermions =Free Fermions =
Quantum degenerate (or coherent) systems
obey dispersive non-dissipative
hydrodynamics
dispersive non-dissipative hydrodynamics
0 xxxxt uuuu
KdV - equationKdV - equation
Non-periodic, non-soliton solutionNon-periodic, non-soliton solution
Hopf equation
Wave equation- a linearized version
Shock-wave solution
Burgers
Semiclassics:
single particle: quantum mechanics
Burgers
Hopf -Riemann
Benjamin-Ono
Fermi-sea: quantum field theory
Witham ModulationWitham Modulation
Witham modulation
Periodic solution
Modulation
Shock wave
True, non-linearized hydrodynamics Hamiltonian
Jevicki, Sakita, Polchinsky, .........
Free fermions:
==
Non-linear bosonsNon-linear bosons
Distribution of solitons is sensitive to initial data
Arena for observation:
cooled alkali atomic gases
Chain of rolling cloudsMorning glory
South Australia Believed to be Benjamin-Ono eq
Morning Glory