Transcript bosons fermions

Cold atoms
Lecture 1.
20. September 2006
Low temperature physics
(borrowed from an undergraduate course)
Existence absolutní nuly
• Absolutní nula teploty pro ideální plyn
definována vztahem
1
m
2
v 2  23 kBT
a podmínkou nulové kinetické energie
• Pro všechny další systémy se použije transitivnosti teploty pro tělesa v
kontaktu (nultý zákon termodynamiky)
• Absolutní nula není dostižitelná konečným procesem (3. zákon termodyn.)
S  0, Cv  0,
• Zvláštní jevy, makroskopické kvantové jevy, jako supravodivost, v blízkosti
nuly. Ovšem co je „blízkost“ ? Vysokoteplotní supravodivost, život, …
3
Teploty ve vesmíru
Stupnice
nitra hvězd
106 - 108 K
hvězdné atmosféry
103 - 104 K
komety, planety …
101 - 102 K
….
reliktní záření jako
minimum
mlhovina Bumerang
(souhvězdí Kentaura)
 2,72 K
1,15 K
4
Teploty ve vesmíru
Stupnice
nitra hvězd
106 - 108 K
hvězdné atmosféry
103 - 104 K
komety, planety …
101 - 102 K
o
….
reliktní záření jako
minimum
mlhovina Bumerang
(souhvězdí Kentaura,
objevena 1998, teplota
určena 2003)
Pozemský rekord
-89,3 C183.75 K
 2,72 K
1983 Antarktida
stanice Vostok
1,15 K
důvod: rychlá expanse
plynů z centrální hvězdy
5
Nízké teploty v laboratoři (jen výběr !!)
K
77
22
4,2
0,3
mK
K
Teplotní rekordy
1877 Pictet kapalný kyslík?
1895 von Linde kap. vzduch
1898 Dewar kapalný vodík
1905 von Linde kap. dusík
1908 Kamerlingh-Onnes
kapalné helium
odsávané helium
1933 paramagn. demagnet.
1951 H. London rozpouštěcí
refrigerátor
1956 Kurti NDR (jaderná …)
1985 Hänsch laserové
chlazení (princip)
Objevy
Teorie
1911 Kamerlingh-Onnes
supravodivost kovů
1937 Kapica supratekutost
Helia-4
1972 Osheroff supratekutost
Helia-3
1986 Müller a Bednorz
vysokoteplot. supravodivost
nK
1924 Einstein
BoseEinsteinova kondensace
1939 Landau
teorie
supratekutosti
1947 Bogoljubov teorie
supratekutosti
1956 BCS *
teorie
supravodivosti
1975 Leggett
teorie
supratekutosti Helia-3
1995 Wieman, … Ketterle
BEC v atomových parách
*Bardeen, Cooper a Schrieffer
pK
6
Naše hlavní téma
K
77
22
4,2
0,3
mK
K
Teplotní rekordy
1877 Pictet kapalný kyslík?
1895 von Linde kap. vzduch
1898 Dewar kapalný vodík
1905 von Linde kap. dusík
1908 Kamerlingh-Onnes
kapalné helium
odsávané helium
1933 paramagn. demagnet.
1951 H. London rozpouštěcí
refrigerátor
1956 Kurti NDR (jaderná …)
1985 Hänsch laserové
chlazení (princip)
Objevy
Teorie
1911 Kamerlingh-Onnes
supravodivost kovů
odsávané helium
1937 Kapica supratekutost
Helia-4
1972 Osheroff supratekutost
Helia-3
1986 Müller a Bednorz
vysokoteplot. supravodivost
nK
1924 Einstein
BoseEinsteinova kondensace
1939 Landau
teorie
supratekutosti
1947 Bogoljubov teorie
supratekutosti
1956 BCS
teorie
supravodivosti
1975 Leggett
teorie
supratekutosti Helia-3
1995 Wieman, … Ketterle
BEC v atomových parách
*Bardeen, Cooper a Schrieffer
pK
7
Nízké teploty v laboratoři (jen výběr !!)
K
77
22
4,2
0,3
mK
K
Teplotní rekordy
1877 Pictet kapalný kyslík?
1895 von Linde kap. vzduch
1898 Dewar kapalný vodík
1905 von Linde kap. dusík
1908 Kamerlingh-Onnes
kapalné helium
odsávané helium
1933 paramagn. demagnet.
1951 H. London rozpouštěcí
refrigerátor
1956 Kurti NDR (jaderná …)
1985 Hänsch laserové
chlazení (princip)
Objevy
1911 Kamerlingh-Onnes
supravodivost kovů
1937 Kapica supratekutost
Helia-4
1972 Osheroff supratekutost
Helia-3
1986 Müller a Bednorz
vysokoteplot. supravodivost
nK
pK
pokrok na
logaritmické
škále
Teorie
1924 Einstein
BoseEinsteinova kondensace
1939 Landau
teorie
supratekutosti
1947 Bogoljubov teorie
supratekutosti
1956 BCS *
teorie
supravodivosti
1975 Leggett
teorie
supratekutosti Helia-3
1995 Wieman, … Ketterle
BEC v atomových parách
*Bardeen, Cooper a Schrieffer
8
Chlazení jadernou adiabatickou demagnetisací
NDR nuclear demagnetization refrigeration
Te
elektrony
mřížkové kmity
pevná látka
TL  L
LS
jádra
jaderné spiny
Princip NDR

S
I.
TS
S
V rovnováze se teploty
všech podsystémů
vyrovnají.
Spin-mřížková relaxace
je pomalá!
Můžeme proto generovat
nerovnovážnou velmi
nízkou spinovou teplotu
I. KROK
izotermická magnetizace
Entropie s magnetickým polem klesá
 snižuje se orientační neuspořádanost
B0
B0
II. KROK
adiabatická demagnetizace
Teplota a vnitřní energie klesají
II.
T

9
Kryostat, kde byla dosažena rekordní teplota 100 pK
Helsinki University of Technology
YKI, Low Temperature Group
2000
1.
Předchlazení
čerpáním helia
0,7 K
2.
První stupeň: rozpouštěcí
refrigerátor
3 mK
3.
Druhý stupeň: NDR v
mědi
<0,1 mK
4.
Třetí stupeň: NDR v
samotném vzorku:
monokrystal Rh <1 nK
10
Spinová magnetická susceptibilita monokrystalu
rhodia
Curie-Weissův zákon jaderné spiny v rhodiu … antiferomagnetické uspořádání

11
12
Introductory matter on bosons
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
14
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
15
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
Permuting particles does not lead to a different state
Two particles
 (x1 , x2 )  (x2 , x1 )   (x1 , x2 )
16
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
Permuting particles does not lead to a different state
Two particles
 (x1 , x2 )  (x2 , x1 )   (x1 , x2 )   2 (x2 , x1 )
17
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
Permuting particles does not lead to a different state
Two particles
 (x1 , x2 )  (x2 , x1 )   (x1 , x2 )   2 (x2 , x1 )
2 1
  1
  1
fermions
bosons
antisymmetric 
symmetric 
half-integer spin
integer spin
18
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
Permuting particles does not lead to a different state
Two particles
 (x1 , x2 )  (x2 , x1 )   (x1 , x2 )   2 (x2 , x1 )
2 1
  1
  1
fermions
bosons
antisymmetric 
symmetric 
"empirical
half-integer spin
integer spin
fact"
comes from
nowhere
19
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
Permuting particles does not lead to a different state
Two particles
 (x1 , x2 )  (x2 , x1 )   (x1 , x2 )   2 (x2 , x1 )
2 1
  1
  1
fermions
bosons
antisymmetric 
symmetric 
"empirical
half-integer spin
integer spin
fact"
electrons
photons
comes from
nowhere
20
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
Permuting particles does not lead to a different state
Two particles
 (x1 , x2 )  (x2 , x1 )   (x1 , x2 )   2 (x2 , x1 )
2 1
  1
  1
fermions
bosons
antisymmetric 
symmetric 
"empirical
half-integer spin
integer spin
fact"
electrons
photons
everybody knows
our present concern
comes from
nowhere
21
Bosons and Fermions (capsule reminder)
Independent particles (… non-interacting)
basis of single-particle states (  complete set of quantum numbers)
 
   
x     x 
   
22
Bosons and Fermions (capsule reminder)
Independent particles (… non-interacting)
basis of single-particle states (  complete set of quantum numbers)
 
   
x     x 
   
FOCK SPACE space of many particle states
basis states … symmetrized products of single-particle states for bosons
… antisymmetrized products of single-particle states for fermions
specified by the set of occupation numbers 0, 1, 2, 3, … for bosons
0, 1
… for fermions
23
Bosons and Fermions (capsule reminder)
Independent particles (… non-interacting)
basis of single-particle states (  complete set of quantum numbers)
 
   
x     x 
   
FOCK SPACE space of many particle states
basis states … symmetrized products of single-particle states for bosons
… antisymmetrized products of single-particle states for fermions
specified by the set of occupation numbers 0, 1, 2, 3, … for bosons
0, 1
 , , ,
, p ,
 n   n1 , n2 , n3 ,
, np ,
1
2
3
… for fermions

n -particle state n  Σn p
24
Bosons and Fermions (capsule reminder)
Representation of occupation numbers (basically, second quantization)
…. for fermions
Pauli principle
fermions keep apart – as sea-gulls
 , , ,
, p ,
 n   n1 , n2 , n3 ,
, np ,
n -particle state n  Σn p
0
 0 , 0 ,0 ,
,0 ,
0-particle state vacuum
1p
 0 , 0 ,0 ,
,1 ,
1-particle
 0 , 1 ,1 ,
,0 ,
2-particle
 ( x) ( x ')   ( x ') ( x)  /
 0 , 2, 0 ,
,0 ,
2-particle
 ( x) ( x ') not allowed
1
F
2
 1 ,1 ,
3
,1 , 0 ,

 p ( x)
1
1
2
1
2
1
N -particle ground state
N
25
2
Bosons and Fermions (capsule reminder)
Representation of occupation numbers (basically, second quantization)
…. for bosons
princip identity
bosons prefer to keep close – like monkeys

 , , ,
, p ,
 n   n1 , n2 , n3 ,
, np ,
n -particle state n  Σn p
0
 0 , 0 ,0 ,
,0 ,
0-particle state vacuum
1p
 0 , 0 ,0 ,
,1 ,
1-particle
 0 , 1 ,1 ,
,0 ,
2-particle
 ( x) ( x ')   ( x ') ( x)  /
 0 , 2, 0 ,
,0 ,
2-particle
 ( x) ( x ')
1
B
2
3
 N , 0 ,0 ,
,0 ,
all on a single orbital
 p ( x)
1
2
1
2
1
1
N -částicový základní stav
 ( x1 ) ( x2 )  ( xN )
1
1
1
26
2
Examples of bosons
bosons
complex particles
N conserved
simple particles
N not conserved
elementary
particles
photons
quasi particles
phonons
magnons
atoms
4
He, 7 Li, 23 Na, 87 Rb
alkali metals
excited
atoms
27
Examples of bosons (extension of the table)
bosons
complex particles
N conserved
simple particles
N not conserved
elementary
particles
photons
quasi particles
phonons
magnons
composite
quasi particles
atoms
4
He, 7 Li, 23 Na, 87 Rb
alkali metals
excited
atoms
ions
molecules
excitons
Cooper pairs
28
Digression: How a complex particle, like an atom,
can behave as a single whole, a boson
ESSENTIAL CONDITION
the identity includes characteristics like mass of charge, but also the values
of observables corresponding to internal degrees of freedom, which are not
allowed to vary during the dynamical processes in question.
29
Digression: How a complex particle, like an atom,
can behave as a single whole, a boson
ESSENTIAL CONDITION
the identity includes characteristics like mass or charge, but also the values
of observables corresponding to internal degrees of freedom, which are not
allowed to vary during the dynamical processes in question.
30
Digression: How a complex particle, like an atom,
can behave as a single whole, a boson
ESSENTIAL CONDITION
the identity includes characteristics like mass of charge, but also the values
of observables corresponding to internal degrees of freedom, which are not
allowed to vary during the dynamical processes in question.
Rubidium
37 electrons
total electron spin
S  12
total nuclear spin
I
37 protons
50 neutrons
total spin of the atom
F SI
F  SI ,
, S  I  1, 2
Two distinguishable species coexist; can be separated by joint effect of
the hyperfine interaction and of the Zeeman splitting in a magnetic field
31
3
2
Plane waves in a cavity
Plane wave in classical terms and its quantum transcription
X  X 0e
 i t  k r 
,    (k ),   2 / k
  , p  k ,
   ( p),   h / p de Broglie wavelength
Discretization ("quantization") of wave vectors in the cavity
volume
V  Lx Ly Lz
Lz
periodic boundary conditions
ky
2
 ,
Lx
2
k ym 
 m,
Lx
2
k zn 
n
Lx
kx 
Cell size (per k vector)
Lx
Ly
 k  (2 )d / V
k x Cell size (per p vector)
 p  hd / V
In the (r, p)-phase space
d
 kV  hd
32
Density of states
ky
IDOS Integrated Density Of States:
How many states have energy less than 
kx
Invert the dispersion law
 ( p)
p( )
Find the volume of the d-sphere in the p-space
 d ( p)  Cd  p d
Divide by the volume of the cell
 ( )   d ( p( )) /  p  V   d ( p( )) / hd
DOS Density Of States:
How many states are around  per unit energy per unit volume
1 d
D ( ) 
 ( )
V d
d
d
1
d 1 d p ( )

 d ( p( ) / h)  dCd h  ( p( ) / h)
d
d
33
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
34
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
fermions
N
FD
n 
1
e  (   )  1
bosons
N
n 
1
e  (  )  1
BE
n 
1
e   1
35
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
T 0
F
 1 ,1 ,
,1 , 0 ,
1
BE
n 
e  (  )  1
T 0
B
 N , 0 ,0 ,
1
e   1
T 0
,0 ,
vac
36
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
T 0
1
BE
n 
e  (  )  1
T 0
1
e   1
T 0
freezing out
F
 1 ,1 ,
,1 , 0 ,
B
 N , 0 ,0 ,
,0 ,
vac
37
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
T 0
F
 1 ,1 ,
,1 , 0 ,
B
1
BE
n 
e  (  )  1
1
e   1
T 0
T 0
?
freezing out
 N , 0 ,0 ,
,0 ,
vac
38
The Planck formula for black-body radiation
Equilibrium radiation in a cavity
Plane electromagnetic wave and its "photon" transcription
two
polarizations
E  E0 s e
 it kr 
,   ck ,   2 / k
  , p  k ,
  cp,   h / p  hc / 
CAVITY
walls at temperature T
emit and absorb radiation
inside the cavity an equilibrium distribution of
radiation … depends only on the temperature
ENERGY DENSITY PER UNIT VOLUME
 (T ,  )  2    n D ( )
polarization photon energy
population of a mode
DOS
40
Planck formula
ENERGY DENSITY PER UNIT VOLUME
 (T ,  )  2    n D ( )
polarization photon energy
population of a mode
n 
1
e   1
DOS
4
D ( )  3 3   2
ch
d
8
3
W   (T ,  )  3 3  
d
c h e 1
d
8 h  3
W   (T , )  3  h
d
c
e kBT  1
PLANCK FOMULA IN THE
STANDARD FORM:
our final result
41
Stephan-Boltzmann law
ENERGY DENSITY PER UNIT VOLUME
d
8
3
W   (T ,  )  3 3  
d
c h e 1
TOTAL ENERGY PER UNIT VOLUME



4
3
8
3
8

k

W   d  (T ,  )  3 3   d 
 T 4  3 3B   d 
c h 0 e 1
c h 0 e 1
0
W  T 4
Stefan-Boltzmann law
8 k

15c h
universal constant
5
4
B
3 3
4
15
42
Stephan-Boltzmann law
ENERGY DENSITY PER UNIT VOLUME
d
8
3
W   (T ,  )  3 3  
d
c h e 1
TOTAL ENERGY PER UNIT VOLUME



4
3
8
3
8

k

W   d  (T ,  )  3 3   d 
 T 4  3 3B   d 
c h 0 e 1
c h 0 e 1
0
W  T 4
Stefan-Boltzmann law
8 k

15c h
universal constant
5
4
B
3 3
4
15
43
Stephan-Boltzmann law
ENERGY DENSITY PER UNIT VOLUME
d
8
3
W   (T ,  )  3 3  
d
c h e 1
TOTAL ENERGY PER UNIT VOLUME



4
3
8
3
8

k

W   d  (T ,  )  3 3   d 
 T 4  3 3B   d 
c h 0 e 1
c h 0 e 1
0
W  T 4
Stefan-Boltzmann law
8 k

15c h
universal constant
5
4
B
3 3
4
15
44
Stephan-Boltzmann law
ENERGY DENSITY PER UNIT VOLUME
d
8
3
W   (T ,  )  3 3  
d
c h e 1
TOTAL ENERGY PER UNIT VOLUME



4
3
8
3
8

k

W   d  (T ,  )  3 3   d 
 T 4  3 3B   d 
c h 0 e 1
c h 0 e 1
0
W  T 4
Stefan-Boltzmann law
8 k

15c h
universal constant
5
4
B
3 3
4
15
45
Stephan-Boltzmann law
ENERGY DENSITY PER UNIT VOLUME
d
8
3
W   (T ,  )  3 3  
d
c h e 1
TOTAL ENERGY PER UNIT VOLUME



4
3
8
3
8

k

W   d  (T ,  )  3 3   d 
 T 4  3 3B   d 
c h 0 e 1
c h 0 e 1
0
W  T 4
Stefan-Boltzmann law
8 k

15c h
universal constant
5
4
B
3 3
4
15
46
Bose-Einstein condensation:
elementary approach
Einstein's manuscript with the derivation of BEC
48
What is the nature of BEC?
With lowering the temperature, the atoms of the gas lose their energy and
drain down to the lowest energy states. There is less and less of these:
N ( E  k BT )  const  T 3/ 2
A given amount N of the atoms becomes too large starting from a critical
temperature.
Their excess precipitates to the lowest level, which becomes
macroscopically occupied, i.e., it holds a finite fraction of all atoms.
This is the BE condensate.
At the zero temperature, all atoms are in the condensate.
Einstein was the first to realize that and to make an exact calculation of the
integrals involved.
3
2
3
2
mk
T

B 
3
3
2
N G (T ) V  4 
  ( 2 ) ( 2 )  BT
2
 h

49
A gas with a fixed average number of atoms
Ideal boson gas (macroscopic system)
p2
atoms: mass m, dispersion law  ( p) 
2m
system as a whole:
volume V, particle number N, density n=N/V, temperature T.
50
A gas with a fixed average number of atoms
Ideal boson gas (macroscopic system)
p2
atoms: mass m, dispersion law  ( p) 
2m
system as a whole:
volume V, particle number N, density n=N/V, temperature T.
Equation for the chemical potential closes the equilibrium problem:
N  N (T ,  )   n( j )  
j
j
1
e
 ( j   )
1
51
A gas with a fixed average number of atoms
Ideal boson gas (macroscopic system)
p2
atoms: mass m, dispersion law  ( p) 
2m
system as a whole:
volume V, particle number N, density n=N/V, temperature T.
Equation for the chemical potential closes the equilibrium problem:
N  N (T ,  )   n( j )  
j
j
1
e
 ( j   )
1
Always  < 0. At high temperatures, in the thermodynamic limit,
the continuum approximation can be used:

N  V  d
0
1
e
 (   )
1
D ( )  N (T ,  )
52
A gas with a fixed average number of atoms
Ideal boson gas (macroscopic system)
p2
atoms: mass m, dispersion law  ( p) 
2m
system as a whole:
volume V, particle number N, density n=N/V, temperature T.
Equation for the chemical potential closes the equilibrium problem:
N  N (T ,  )   n( j )  
j
j
1
e
 ( j   )
1
Always  < 0. At high temperatures, in the thermodynamic limit,
the continuum approximation can be used:

N  V  d
0
1
e
 (   )
1
D ( )  N (T ,  )
It holds
N (T ,   0)  N (T ,0)  
For each temperature, we get a critical number of atoms the gas
can accommodate. Where will go the rest?
53
A gas with a fixed average number of atoms
Ideal boson gas (macroscopic system)
p2
atoms: mass m, dispersion law  ( p) 
2m
system as a whole:
volume V, particle number N, density n=N/V, temperature T.
Equation for the chemical potential closes the equilibrium problem:
N  N (T ,  )   n( j )  
j
j
1
e
 ( j   )
1
Always  < 0. At high temperatures, in the thermodynamic limit,
the continuum approximation can be used:

N  V  d
0
1
e
 (   )
1
D ( )  N (T ,  )
This will be
shown in a while
It holds
N (T ,   0)  N (T ,0)  
For each temperature, we get a critical number of atoms the gas
can accommodate. Where will go the rest?
54
A gas with a fixed average number of atoms
Ideal boson gas (macroscopic system)
p2
atoms: mass m, dispersion law  ( p) 
2m
system as a whole:
volume V, particle number N, density n=N/V, temperature T.
Equation for the chemical potential closes the equilibrium problem:
N  N (T ,  )   n( j )  
j
j
1
e
 ( j   )
1
Always  < 0. At high temperatures, in the thermodynamic limit,
the continuum approximation can be used:

N  V  d
0
1
e
 (   )
1
D ( )  N (T ,  )
This will be
shown in a while
It holds
N (T ,   0)  N (T ,0)  
For each temperature, we get a critical number of atoms the gas
can accommodate. Where will go the rest? To the condensate
55
Condensate concentration
The integral is doable:

1
N (T ,0)  V  d  
D ( )
e 1
0
3
2
 2mk BT 
3
3
 V 4 
  ( 2 ) ( 2 )
2
 h

use the
general formula
 /2
Riemann function
3
2
 2mk BT 
 V  2
  2,612
2
h


56
Condensate concentration
3
2
The integral is doable:

1
N (T ,0)  V  d  
D ( )
e 1
0
3
2
 2mk BT 
3
3
 V 4 
  ( 2 ) ( 2 )
2
 h

 2m 
D ( )  2  2   
h 
 /2
Riemann function
3
2
 2mk BT 
 V  2
  2,612
2
h


57
Condensate concentration
3
2
The integral is doable:

1
N (T ,0)  V  d  
D ( )
e 1
0
3
2
 2mk BT 
3
3
 V 4 
  ( 2 ) ( 2 )
2
 h

 2m 
D ( )  2  2   
h 
 /2
Riemann function
3
2
 2mk BT 
 V  2
  2,612
2
h


CRITICAL TEMPERATURE
the lowest temperature at which all atoms are still accomodated in the gas:
N (Tc ,0)  N
58
Condensate concentration
3
2
The integral is doable:

1
N (T ,0)  V  d  
D ( )
e 1
0
3
2
 2mk BT 
3
3
 V 4 
  ( 2 ) ( 2 )
2
 h

 2m 
D ( )  2  2   
h 
 /2
Riemann function
3
2
 2mk BT 
 V  2
  2,612
2
h


CRITICAL TEMPERATURE
the lowest temperature at which all atoms are still accomodated in the gas:
N (Tc ,0)  N
h2
Tc 
4 mk B
atomic mass
2
3
2
3
2
3
 N 
h2
n
19 n


 8,0306  10 
  0,52725
4 uk B M
M
 2,612V 
59
Condensate concentration
CRITICAL TEMPERATURE
the lowest temperature at which all atoms are still accomodated in the gas:
2
h
Tc 
4 mk B
2
3
2
3
2
3
 N 
h
n
19 n


0,52725


8,0306

10


4 uk B M
M
 2,612V 
2
A few estimates:
system
M
n
TC
He liquid
4
21028
1.47 K
Na trap
23
21020
1.19 K
Rb trap
87
21017
3.16 nK
60
Digression: simple interpretation of TC
Rearranging the formula for critical temperature
h2
Tc 
4 mk B
we get
V 
 
 N
 N 


2,612
V


1
3
2
3
h
mk BTc
mean interatomic
distance
thermal
de Broglie
wavelength
The quantum breakdown sets on when
the wave clouds of the atoms start overlapping
61
de Broglie wave length for atoms and molekules
2

p
Thermal energies small … NR formulae valid:

2
2mEkin
At thermal equilibrium
m  Au
... at. (mol.) mass
Ekin  32 k BT
thermal wave
length

2
1
1

 2,5 109 
3u k B
AT
AT
Two useful equations
Ekin  32 T /11600 eV K
v
v 2  158
T
A
62
Ketterle explains BEC to the King of Sweden
63
Condensate concentration
3
2
T 
N (TC ,0)
nG 
 BT = n   for T  TC
V
 TC 
3
3


2
 T 2


T
n  nG  nBE  n    n 1    
  TC  
 TC 


3
2
f
r
a
c
t
i
o
n
GAS
T / TC
64
Where are the condensate atoms?
ANSWER: On the lowest one-particle energy level
For understanding, return to the discrete levels.
N  N (T ,  )   n( j )  
j
j
1
e
 ( j   )
1
There is a sequence of energies
   0   (0)  0  1   2
For very low temperatures,  (1   0 )
1
all atoms are on the lowest level, so that
n0  N  O(e  (1  0 ) )
N 
1
 ( 0   )
e
k BT
  0 
N
1
all atoms are in the condensate
connecting equation
chemical potential is zero on the gross energy scale
65
Where are the condensate atoms? Continuation
ANSWER: On the lowest one-particle energy level
TC
For temperatures below
all condensate atoms are on the lowest level, so that
n0  N BE
N BE 
all condensate atoms remain on the lowest level
1
 ( 0  )
e
k BT
  0 
N BE
connecting equation
1
chemical potential keeps zero on the gross energy scale
question … what happens with the occupancy of the next level now?
2
Estimate:

1   0
h
2
/ m V
3
2
k BT
k BT
n0 
 O(V ), n1 
 O(V 3 ) .... much slower growth
0  
1  
66
Where are the condensate atoms? Summary
ANSWER: On the lowest one-particle energy level
The final balance equation for T
N  N (T ,  ) 
 TC is

1
e
 ( 0   )
1
V  d
0
1
e
 (   )
1
D ( )
LESSON:
be slow with making the thermodynamic limit (or any other limits)
67
Closer look at BEC
• Thermodynamically, this is a real phase transition, although unsual
• Pure quantum effect
• There are no real forces acting between the bosons, but there IS a real
correlation in their motion caused by their identity (symmetrical wave functions)
• BEC has been so difficult to observe, because other (classical G/L or G/S)
phase transitions set on much earlier
• BEC is a "condensation in the momentum space", unlike the usual liquefaction
of classical gases, which gives rise to droplets in the coordinate space.
• This is somewhat doubtful, especially now, that the best observed BEC takes
place in traps, where the atoms are significantly localized
• What is valid on the "momentum condensation": BEC gives rise to quantum
coherence between very distant places, just like the usual plane wave
• BEC is a macroscopic quantum phenomenon in two respects:
 it leads to a correlation between a macroscopic fraction of atoms
 the resulting coherence pervades the whole macroscopic sample
68
Closer look at BEC
• Thermodynamically, this is a real phase transition, although unsual
• Pure quantum effect
• There are no real forces acting between the bosons, but there IS a real
correlation in their motion caused by their identity (symmetrical wave functions)
• BEC has been so difficult to observe, because other (classical G/L or G/S)
phase transitions set on much earlier
• BEC is a "condensation in the momentum space", unlike the usual liquefaction
of classical gases, which gives rise to droplets in the coordinate space.
• This is somewhat doubtful, especially now, that the best observed BEC takes
place in traps, where the atoms are significantly localized
• What is valid on the "momentum condensation": BEC gives rise to quantum
coherence between very distant places, just like the usual plane wave
• BEC is a macroscopic quantum phenomenon in two respects:
 it leads to a correlation between a macroscopic fraction of atoms
 the resulting coherence pervades the whole macroscopic sample
69
Closer look at BEC
• Thermodynamically, this is a real phase transition, although unsual
• Pure quantum effect
• There are no real forces acting between the bosons, but there IS a real
correlation in their motion caused by their identity (symmetrical wave functions)
• BEC has been so difficult to observe, because other (classical G/L or G/S)
phase transitions set on much earlier
• BEC is a "condensation in the momentum space", unlike the usual liquefaction
of classical gases, which gives rise to droplets in the coordinate space.
• This is somewhat doubtful, especially now, that the best observed BEC takes
place in traps, where the atoms are significantly localized
• What is valid on the "momentum condensation": BEC gives rise to quantum
coherence between very distant places, just like the usual plane wave
• BEC is a macroscopic quantum phenomenon in two respects:
 it leads to a correlation between a macroscopic fraction of atoms
 the resulting coherence pervades the whole macroscopic sample
70
Closer look at BEC
• Thermodynamically, this is a real phase transition, although unsual
• Pure quantum effect
• There are no real forces acting between the bosons, but there IS a real
correlation in their motion caused by their identity (symmetrical wave functions)
• BEC has been so difficult to observe, because other (classical G/L or G/S)
phase transitions set on much earlier
• BEC is a "condensation in the momentum space", unlike the usual liquefaction
of classical gases, which gives rise to droplets in the coordinate space.
• This is somewhat doubtful, especially now, that the best observed BEC takes
place in traps, where the atoms are significantly localized
• What is valid on the "momentum condensation": BEC gives rise to quantum
coherence between very distant places, just like the usual plane wave
• BEC is a macroscopic quantum phenomenon in two respects:
 it leads to a correlation between a macroscopic fraction of atoms
 the resulting coherence pervades the whole macroscopic sample
71
Closer look at BEC
• Thermodynamically, this is a real phase transition, although unsual
• Pure quantum effect
• There are no real forces acting between the bosons, but there IS a real
correlation in their motion caused by their identity (symmetrical wave functions)
• BEC has been so difficult to observe, because other (classical G/L or G/S)
phase transitions set on much earlier
• BEC is a "condensation in the momentum space", unlike the usual liquefaction
of classical gases, which gives rise to droplets in the coordinate space.
• This is somewhat doubtful, especially now, that the best observed BEC takes
place in traps, where the atoms are significantly localized
• What is valid on the "momentum condensation": BEC gives rise to quantum
coherence between very distant places, just like the usual plane wave
• BEC is a macroscopic quantum phenomenon in two respects:
 it leads to a correlation between a macroscopic fraction of atoms
 the resulting coherence pervades the whole macroscopic sample
72
Off-Diagonal Long Range Order
Analysis on the one-particle level
Coherence in BEC: ODLRO
Off-Diagonal Long Range Order
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

74
Coherence in BEC: ODLRO
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

75
Coherence in BEC: ODLRO
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

76
Coherence in BEC: ODLRO
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

77
Coherence in BEC: ODLRO
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

78
OPDM for homogeneous systems
In coordinate representation
 (r , r ')   r k nk k r'
k

1
ei k ( r  r ') nk

V k
• depends only on the relative position (transl. invariance)
• Fourier transform of the occupation numbers
• isotropic … provided thermodynamic limit is allowed
• in systems without condensate, the momentum distribution is smooth and
the density matrix has a finite range.
CONDENSATE
lowest orbital with
k0
79
OPDM for homogeneous systems: ODLRO

CONDENSATE
lowest orbital with
1
3
k0  O(V )  0
1 i k0 ( r  r ')
1
 (r  r ')  e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
  BE (r  r ')
FT of a smooth function
has a finite range
 G (r  r ')
80
OPDM for homogeneous systems: ODLRO

CONDENSATE
lowest orbital with
1
3
k0  O(V )  0
1 i k0 ( r  r ')
1
 (r  r ')  e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
  BE (r  r ')
FT of a smooth function
has a finite range
 G (r  r ')
DIAGONAL ELEMENT r = r'
 (0 )   BE (0 )
 nBE
 G (0 )
+ nG
81
OPDM for homogeneous systems: ODLRO

CONDENSATE
lowest orbital with
1
3
k0  O(V )  0
1 i k0 ( r  r ')
1
 (r  r ')  e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
  BE (r  r ')
FT of a smooth function
has a finite range
 G (r  r ')
DIAGONAL ELEMENT r = r'
 (0 )   BE (0 )
 nBE
 G (0 )
+ nG
DISTANT OFF-DIAGONAL ELEMENT | r - r' |
|r  r '|
 BE (r  r ') 
 nBE
|r  r '|
G (r  r ') 
 0
|r  r '|

 nBE
 (r  r ')
Off-Diagonal Long Range Order
ODLRO
82
From OPDM towards the macroscopic wave function

CONDENSATE
k0  O(V )  0
lowest orbital with
 (r  r ') 
1
3
1 i k0 ( r  r ')
1
e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
FT of a smooth function
has a finite range
 (r )  (r' )

dyadic
1
i k ( r  r ')
e
nk

V k  k0
MACROSCOPIC WAVE FUNCTION
 (r )  nBE  ei( k r + ) , 
0
an arbitrary phase
• expresses ODLRO in the density matrix
• measures the condensate density
• appears like a pure state in the density matrix, but macroscopic
• expresses the notion that the condensate atoms are in the same state
• is the order parameter for the BEC transition
83
From OPDM towards the macroscopic wave function

CONDENSATE
k0  O(V )  0
lowest orbital with
 (r  r ') 
1
3
1 i k0 ( r  r ')
1
e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
FT of a smooth function
has a finite range
 (r )  (r' )

dyadic
1
i k ( r  r ')
e
nk

V k  k0
MACROSCOPIC WAVE FUNCTION
 (r )  nBE  ei( k r + ) , 
0
an arbitrary phase
• expresses ODLRO in the density matrix
• measures the condensate density


• appears like a pure state in the density matrix, but macroscopic

• expresses the notion that the condensate atoms are in the same state
• is the order parameter for the BEC transition
84
From OPDM towards the macroscopic wave function

CONDENSATE
k0  O(V )  0
lowest orbital with
 (r  r ') 
1
3
1 i k0 ( r  r ')
1
e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
FT of a smooth function
has a finite range
 (r )  (r' )

dyadic
1
i k ( r  r ')
e
nk

V k  k0
MACROSCOPIC WAVE FUNCTION
 (r )  nBE  ei( k r + ) , 
0
an arbitrary phase
• expresses ODLRO in the density matrix
• measures the condensate density

? why bother?

• appears like a pure state in the density matrix, but macroscopic

• expresses the notion that the condensate atoms are in the same state
• is the order parameter for the BEC transition
? what is it?
? how?
85
F.Laloë: Do we really understand Quantum mechanics,
Am.J.Phys.69, 655 (2001)
86
The end
Problems
Some problems are expanding on the presented subject matter and are voluntary… (*)
The other ones are directly related to the theme of the class and are to be worked out within a
week. The solutions will be presented on the next seminar and posted on the web.
(1.1*) Problems with metastable states and quasi-equilibria in defining the temperature and
applying the 3rd law of thermodynamics
(1.2*) Relict radiation and the Boomerang Nebula
(1.3) Work out in detail the integral defining Tc
(1.4) Extend the resulting series expansion to the full balance equation (BE integral)
(1.5) Modify for a 2D gas and show that the BE condensation takes never place
(1.6) Obtain an explicit procedure for calculating the one-particle density matrix for an ideal
boson gas [difficult]

89