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