Bose-Einstein
Condensation and
Superfluidity
Gordon Baym
University of Illinois, Urbana
東京大学
January 2004
Fermions (Fermi-Dirac, 1926):
Particles that obey the exclusion principle
(Pauli, 1925). Can’t put two in same state at
the same time.
Bosons (Bose-Einstein, 1924-5): Particles
that don’t obey the exclusion principle.
Can put many in the same state at the
same time
S. Bose
A. Einstein
S.N. Bose 1924: concept of light quanta as particles with
2 polarization states. New statistics => Planck distribution:
A. Einstein 1924: Extension to monoatomic ideal gases:
Condensation:
Condensate
I maintain that in this case a
number of molecules steadily
growing with increasing density
goes over in the first quantum state
... a separation is affected; one part
‘condenses,’ the rest remains a
saturated ideal gas.
A. Einstein, 1925
Bose-Einstein Condensation
Hot atoms (bosons) in a box
Cool below Bose-Einstein
transition temperature
Gravity
Bose-Einstein condensate
At absolute zero temperature
motion “ceases”
Free Bose gas
Box
Potential well (trap)
In condensed system have macroscopic occupation of
single (generally lowest) mode
: ground state
: flow state (vortex)
MANY-PARTICLE WAVE FUNCTION
= condensate wave function
FINITE TEMPERATURE
Thermal wavelength
No. condensed particles
Which “statistics” apply to nature?
i.e., is ordinary matter made of
fermions or bosons?
The [Fermi-Dirac] solution ... is probably the
correct one for gas molecules, since it is
known to be the correct one for electrons in
an atom, and one would expect molecules to
resemble electrons more closely than light
quanta.
P.A.M. Dirac, 1926
With a heavy heart I have
become converted to the idea
that Fermi … , not EinsteinBose, is the correct statistics
[for electrons].
W. Pauli to E. Schrödinger, Nov. 1926
Superfluid 4He:
The first Bose-Einstein condensate
W.H. Keesom and Miss A.P. Keesom (1935):
specific heat of liquid helium
F.London (1938): Spectroscopic data => 4He obeys
Bose-Einstein statistics:
“The strange change of state of liquid helium at 2.19 o
abs., even though it occurs in the liquid and not in the
gaseous state, is due to the condensation mechanism
of the Bose-Einstein gas.”
“It seems difficult not to imagine a connexion with
the condensation phenomenon of the BoseEinstein statistics.” (London, 1938)
Superfluid Liquid Helium
Temperatures below “Lambda point”
2.17o above absolute zero
Flows through tiny capillaries without
friction
Flows around a closed pipe forever
1938
(Tony Leggett)
Spin bucket of
superfluid
helium slowly. Helium
liquid remains at rest!
Spin fast enough.
Form vortex in center
of liquid!
L. Landau (1941): rejects suggestion “that
helium-II should be considered as a degenerate
ideal Bose gas.” Importance of interactions!
ROLE OF STATISTICS:
Sydoriak, Grilley, and Hammel (1948) liquified 3He.
Osborne, Winstock, and Abraham (1948): no superflow
down to 1.05 K.
Bose character critical to superfluidity
Order parameter of Bose-condensed system
-- 0 in normal system
-- constant in BEC
= complex order parameter
Free particle
state, |N>
If |N> and |N-1> differ only by number of particles in
condensate then
In weakly interacting Bose gas:
Time dependent order parameter
condensate wave function
condensate density
superfluid velocity
chemical potential
superfluid acceleration eqn.
Equilibrium:
Flow and superfluidity
Complex order parameter:
=> flow
Superfluid velocity
Momentum density of superfluid flow = rs vs
Superfluid mass density =
Normal mass density =
Condensate density differs from superfluid mass density:
At T=0 in 4He, rs = r, n0 = 0.09 n
BOSE CONDENSED SYSTEMS
Low temperature systems of bosons:
liquid 4He
trapped bosonic atoms
excitons in semiconductors (?)
Nuclear matter
pion condensation
kaon condensation
Vacuum as Bose condensed state
Chiral symmetry breaking
Gluon condensation
Higgs condensation
Graviton condensation, gmn
PION CONDENSED MATTER
Softening of collective spin-isospin oscillation of nuclear matter
Above critical density have transition to new state with
nucleons rotated in isospin space:
with formation of macroscopic pion field
Important, if it exists, for enhanced cooling of neutron
stars by neutrino emission
Transition density very sensitive to effective particle-hole
interactions (Landau g’) and D-hole interactions
Analogous neutral pion condensate
can coexist with
STRANGENESS (KAON) CONDENSATES
Analogous to p condensate
Chiral SU(3) X SU(3) symmetry of strong interactions
=> effective low energy interaction
Kaplan and Nelson (1986),
Brown et al. (1994)
“Effective mass” term lowers K energies in matter
=> condensation
Rotate u and s quark states:
Form
Admix
condensate
in n;
in p
Results very sensitive to Kinteractions in matter
(Pandharipande, Pethick and Vesteinn, 1995)
-
* Would soften equation of state and lower maximum
neutron star mass to ~ 1.5 solar masses
* Would enhance neutrino luminosity and cooling of neutron
stars
Can also form
condensate => macroscopic η field
Condensates in vacuum
EXPERIMENTAL BOSE-EINSTEIN
DECONDENSATION
Ultrarelativistic heavy ion collisions:
2000: RHIC 100 GeV/A + 100 GeV/A colliding beams
2007?: LHC 2600 GeV/A + 2600 GeV/A
Relativistic Heavy Ion Collider (RHIC)
(Brookhaven, NY)
Break chiral symmetry in
different state? (Disordered
chiral condensate?)
Np ~104, V ~ 103 fm3 :
p- BEC unlikely; entropy too high
Applications in Biology
A strong proponent of the idea that
Bose-Einstein condensation may provide
the “unitary sense of self” that seems to be
characteristic of consciousness, in relation
to Fröhlich’s ideas is Ian Marshall (1989) …
R. Penrose, Shadows of the Mind (1994)
Application to the Movies
Information, Adaptive Contracting, Distributional
Dynamics, Bayesian Choice, Bose-Einstein Statistics
and the Movies