Download PPT

Statistical methods for bosons
19th
Lecture 1.
December 2012
by Bedřich Velický
Short version of the lecture plan
Lecture 1
Introductory matter
BEC in extended non-interacting systems, ODLRO
Atomic clouds in the traps; Confined independent
bosons, what is BEC?
Lecture 2
Atom-atom interactions, Fermi pseudopotential;
Gross-Pitaevski equation for extended gas and a trap
Dec 19
Jan 9
Infinite systems: Bogolyubov-de Gennes theory,
BEC and symmetry breaking, coherent states
2
The classes will turn around the Bose-Einstein
condensation in cold atomic clouds
• comparatively novel area of research
•largely tractable using the mean field
approximation to describe the interactions
• only the basic early work will be covered,
the recent progress is beyond the scope
Nobelists
The Nobel Prize in Physics 2001
"for the achievement of Bose-Einstein condensation in dilute
gases of alkali atoms, and for early fundamental studies of the
properties of the condensates"
Eric A. Cornell
Wolfgang
Ketterle
Carl E. Wieman
1/3 of the prize
1/3 of the prize
1/3 of the prize
USA
Federal Republic of
Germany
USA
University of
Colorado, JILA
Boulder, CO, USA
Massachusetts
Institute of
Technology (MIT)
Cambridge, MA, USA
University of
Colorado, JILA
Boulder, CO, USA
b. 1961
b. 1957
b. 1951
4
I.
Introductory matter on bosons
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
6
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
Cannot be labelled or numbered
7
Bosons and Fermions (capsule reminder)
independent quantum postulate
Identical particles are indistinguishable
Cannot be labelled or numbered
Permuting particles does not lead to a different state
Two particles
 (x1 , x2 )  (x2 , x1 )   (x1 , x2 )
8
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 )
9
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 )
10
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 
11
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
12
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 )
Finds
justification in
the relativistic
quantum field
theory
2 1
  1
  1
fermions
bosons
antisymmetric 
symmetric 
"empirical
half-integer spin
integer spin
fact"
comes from
nowhere
13
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 )
Finds
justification in
the relativistic
quantum field
theory
2 1
  1
  1
fermions
bosons
antisymmetric 
symmetric 
"empirical
half-integer spin
integer spin
fact"
electrons
photons
comes from
nowhere
14
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 )
Finds
justification in
the relativistic
quantum field
theory
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
15
Bosons and Fermions (capsule reminder)
Independent particles (… non-interacting)
basis of single-particle states (  complete set of quantum numbers)
 
   
x     x 
   
16
Bosons and Fermions (capsule reminder)
Independent particles (… non-interacting)
basis of single-particle states (  complete set of quantum numbers)
 
   
x     x 
   
FOCK SPACE Hilbert 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
17
Bosons and Fermions (capsule reminder)
Independent particles (… non-interacting)
basis of single-particle states (  complete set of quantum numbers)
 
   
x     x 
   
FOCK SPACE Hilbert 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
18
Bosons and Fermions (capsule reminder)
Representation of occupation numbers (basically, second quantization)
…. for fermions
Pauli principle
fermions keep apart – as sea-gulls
 , , ,
3
, 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
1 ( x )1 ( x ') not allowed
1
F
2
 1 ,1 ,
,1 , 0 ,

 p ( x )
1
2
1
2
N -particle ground state
19
2
Bosons and Fermions (capsule reminder)
Representation of occupation numbers (basically, second quantization)
…. for fermions
Pauli principle
fermions keep apart – as sea-gulls
 , , ,
3
, 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
1 ( x )1 ( x ') not allowed
1
F
2
 1 ,1 ,
,1 , 0 ,

 p ( x )
1
2
1
2
2
N -particle ground state
in atoms: filling of the shells (Pauli Aufbau Prinzip)
in metals: Fermi sea
20
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 -particle ground state
 ( x1 ) ( x2 )  ( xN )
1
1
1
21
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 -particle ground state
 ( x1 ) ( x2 )  ( xN )
1
1
1
Bose-Einstein condensate
22
2
Who are bosons ?
• elementary particles
• quasiparticles
• complex massive particles, like atoms
… compound bosons
Examples of bosons
bosons
complex particles
N conserved
simple particles
N not conserved
elementary
particles
atomic nuclei
photons
atoms
quasi particles
phonons
magnons
4
He, 7 Li, 23 Na, 87 Rb
alkali metals
excited atoms
24
Examples of bosons (extension of the table)
bosons
complex particles
N conserved
simple particles
N not conserved
elementary
particles
atomic nuclei
photons
atoms
quasi particles
phonons
magnons
4
He, 7 Li, 23 Na, 87 Rb
alkali metals
excited
atoms
compound
quasi particles
ions
excitons
Cooper pairs
molecules
25
Question: How a complex particle, like an atom,
can behave as a single whole, a compound boson
ESSENTIAL CONDITIONS
1) All compound particles in the ensemble must be identical; the identity
includes
o detailed elementary particle composition
o characteristics like mass, charge or spin
2) The total spin must have an integer value
3) The identity requirement extends also on the values of observables
corresponding to internal degrees of freedom
4) which are not allowed to vary during the dynamical processes in
question
5) The system of the compound bosons must be dilute enough to make the
exchange effects between the component particles unimportant and
absorbed in an effective weak short range interaction between the
bosons as a whole
26
Example: How a complex particle, like an atom,
can behave as a single whole, a compound boson
RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC
electron configuration
1s 2 2 s 2 2 p 6 3s 2 3 p 6 3d 10 4 s 2 4 p 6 4d 10 5s1
A
87
37Rb
[Kr]5s1
2S
1
2
I  23
closed shells - spin compensation
Z
2 S 1
L0
LJ
S
1
2
J  LS
J  SL,
,SL
1
2
• single element Z = 37
• single isotope A = 87
• single electron configuration
27
Example: How a complex particle, like an atom,
can behave as a single whole, a compound boson
RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC
• single element Z = 37
• single isotope A = 87
87
37Rb
[Kr]5s1
2S
1
2
I  23
A
• single electron configuration
Z
37 electrons
total electron spin
S  12
total nuclear spin
I
37 protons
50 neutrons
3
2
• total spin of the atom decides
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
28
Atomic radius vs. interatomic distance in the cloud
http://intro.chem.okstate.edu/1314f00/lecture/chapter7/lec111300.html
COMPARE
rRb
vs.
0.244  109 m
vs.
d
n
 13
 10
1
21  3

m =0.1  106 m
in the air at 0 C and 1 atm
d
n
 13
  5.7  10
1
25  3

m =3.3  109 m
29
II.
Homogeneous gas
of non-interacting bosons
The basic system exhibiting
the Bose-Einstein Condensation (BEC)
original case studied by Einstein
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
31
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( )) / h
d
DOS Density Of States:
2 d / 2
Cd 
(d / 2  1)!
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
32
Ideal quantum gases at a finite temperature: a reminder
mean occupation
number of a oneparticle state with
energy 
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
33
Ideal quantum gases at a finite temperature
mean occupation
number of a oneparticle state with
energy 
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
34
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
mean occupation
number of a oneparticle state with
energy 
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
mean occupation
number of a oneparticle state with
energy 
high temperatures, dilute gases
fermions
bosons
chemical potential
N
FD

N
fixes particle number N
n 
1
e  (   )  1
n 
1
e  (  )  1
BE
n 
1
e   1
36
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
mean occupation
number of a oneparticle state with
energy 
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
T 0
F
 1 ,1 ,
,1 , 0 ,
1
BE
e  (  )  1
T 0
B
 N , 0 ,0 , ,0 ,
n 
1
e   1
T 0
vac
37
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
mean occupation
number of a oneparticle state with
energy 
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
T 0
1
BE
e  (  )  1
T 0
n 
1
e   1
T 0
Aufbau
principle
F
 1 ,1 ,
,1 , 0 ,
B
 N , 0 ,0 , ,0 ,
vac
38
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
mean occupation
number of a oneparticle state with
energy 
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
T 0
1
BE
e  (  )  1
T 0
 1 ,1 ,
,1 , 0 ,
1
e   1
T 0
freezing out
Aufbau
principle
F
n 
B
 N , 0 ,0 , ,0 ,
vac
39
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
mean occupation
number of a oneparticle state with
energy 
high temperatures, dilute gases
fermions
N
FD
n 
F
bosons
N
1
n 
e  (   )  1
1
BE
e  (  )  1
n 
1
e   1
T 0
T 0
T 0
Aufbau
principle
?
freezing out
 1 ,1 ,
,1 , 0 ,
B
 N , 0 ,0 , ,0 ,
vac
40
Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
mean occupation
number of a oneparticle state with
energy 
high temperatures, dilute gases
fermions
N
FD
n 
F
bosons
N
1
n 
e  (   )  1
1
BE
e  (  )  1
n 
1
e   1
T 0
T 0
T 0
Aufbau
principle
BEC?
freezing out
 1 ,1 ,
,1 , 0 ,
B
 N , 0 ,0 , ,0 ,
vac
41
Bose-Einstein condensation:
elementary approach
Einstein's manuscript with the derivation of BEC
43
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.
44
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
45
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 ,  )
46
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?
47
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?
48
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
49
Gas particle concentration
The integral is doable:

1
N (T ,0)  V  d  
D ( )
e 1
0
use the
general formula
3
2
 2mk BT 
3
3
 V 4 
  ( 2 ) ( 2 )
2
 h

Riemann function
3
2
 2mk BT 
 V  2
  2,612
2
h


50
Gas particle 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


51
Bose-Einstein condensation:
critical temperature
Gas particle 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
53
Critical temperature
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
18 n


 1,6061  10 
  0,52725
4 uk B M
M
 2,612V 
54
Critical temperature
CRITICAL TEMPERATURE
the lowest temperature at which all atoms are still accomodated in the gas:
Tc 
2
h
4 mk B
2
3
2
3
2
3
 N 
h
n
18 n


0,52725


1,6061

10


4 uk B M
M
 2,612V 
2
A few estimates:
system
M
n
TC
He liquid
4
11028
3.54 K
Na trap
23
11020
2.86 K
Rb trap
87
11019
95 nK
55
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
56
de Broglie wave length for atoms and molekules
2

p
Thermal energies small … NR formulae valid:

2
2mEkin
At thermal equilibrium
m  Mu
... at. (mol.) mass
Ekin  32 k BT
thermal wave
length

2
3m k B

2
3u k B

1
MT
9
 2,5  10 
1
MT
Two useful equations
Ekin  32 T /11600 eV K
v
v 2  158
T
M
57
Ketterle explains BEC to the King of Sweden
58
Bose-Einstein condensation:
condensate
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
60
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
61
Condensate concentration
3
2
3
T 
N (TC ,0)
2
nG 
 BT = n   for T  TC
V
 TC 
3
3


2
2
T 
T  

n  nG  nBE  n    n 1   
  TC  
 TC 


f
r
a
c
t
i
o
n
GAS
T / TC
62
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
63
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  
64
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)
65
III.
Physical properties and discussion of BEC
Off-Diagonal Long Range Order
Thermodynamics of BEC
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
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
Beyond the thermodynamic view:
Coherence of the condensate in real space
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' for k0 = 0
 (0 )   BE (0 )
n
 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
Capsule on thermodynamics
Homogeneous one component phase:
boundary conditions (environment) and state variables
T P  dual variables, intensities
"intensive"
SV
S V N isolated, conservative
open S V 
S P N isobaric
isothermal T V N
S P  not in use
grand T V 
T P N isothermal-isobaric
not in use T P 
85
Homogeneous one component phase:
boundary conditions (environment) and state variables
T P  dual variables, intensities
"intensive"
SV
S V N isolated, conservative
The important four
isothermal T V N
grand T V 
The one we use
presently
T P N isothermal-isobaric
87
Digression: which environment to choose?
THE ENVIRONMENT IN THE THEORY SHOULD CORRESPOND
TO THE EXPERIMENTAL CONDITIONS
… a truism difficult to satisfy
 For large systems, this is not so sensitive for two reasons
•
System serves as a thermal bath or particle reservoir all by itself
•
Relative fluctuations (distinguishing mark) are negligible

Adiabatic system
SB heat exchange – the slowest
S
B
Real system
medium fast
process
• temperature lag
• interface layer
Isothermal system
the fastest
S
B
 Atoms in a trap: ideal model … isolated. In fact: unceasing energy exchange
(laser cooling). A small number of atoms may be kept (one to, say, 40). With
107, they form a bath already. Besides, they are cooled by evaporation and
they form an open (albeit non-equilibrium) system.
 Sometime, N =const. crucial (persistent currents in non-SC mesoscopic rings)
88
Thermodynamic potentials and all that
Basic thermodynamic identity (for equilibria)
dU  T d S  P dV   d N
For an isolated system,
 
T  T ( S ,V , N )  U
S
, etc.
V ,N
For an isothermic system, the independent variables are T, V, N.
The change of variables is achieved by means of the Legendre
transformation. Define Free Energy
F  U  TS , U  U (T ,V , N ), S  S (T ,V , N ),
d F  S d T  P dV   d N
 
S  F
T
, etc.
V ,N
89
Thermodynamic potentials and all that
Basic thermodynamic identity (for equilibria)
dU  T d S  P dV   d N
For an isolated system,
 
T  T ( S ,V , N )  U
S
, etc.
V ,N
For an isothermic system, the independent variables are T, V, N.
The change of variables is achieved by means of the Legendre
transformation. Define Free Energy
F  U  TS , U  U (T ,V , N ), S  S (T ,V , N ),
d F  S d T  P dV   d N
 
S  F
T
V ,N
, etc.
New variables:
perform the substitution
everywhere; this shows in
the Maxwell identities
(partial derivatives)
90
Thermodynamic potentials and all that
Basic thermodynamic identity (for equilibria)
dU  T d S  P dV   d N
For an isolated system,
 
T  T ( S ,V , N )  U
S
, etc.
V ,N
For an isothermic system, the independent variables are T, V, N.
The change of variables is achieved by means of the Legendre
transformation. Define Free Energy
F  U  TS , U  U (T ,V , N ), S  S (T ,V , N ),
d F  S d T  P dV   d N
 
S  F
, etc.
Legendre transformation:
T V , N
subtract the relevant
product of
conjugate (dual) variables
New variables:
perform the substitution
everywhere; this shows in
the Maxwell identities
(partial derivatives)
91
Thermodynamic potentials and all that
Basic thermodynamic identity (for equilibria)
dU  T d S  P dV   d N
For an isolated system,
 
T  T ( S ,V , N )  U
S
, etc.
V ,N
For an isothermic system, the independent variables are T, S, V.
The change of variables is achieved by means of the Legendre
transformation. Define Free Energy
F  U  TS , U  U (T ,V , N ), S  S (T ,V , N ),
d F  S d T  P dV   d N
 
S  F
T
, etc.
V ,N
92
A table
isolated system
U
internal energy
isothermic system
canonical ensemble
F  U  TS
free energy
isothermic-isobaric system
free enthalpy
isothermic open system
grand potential
How comes
Thus,
S ,V , N
dU T dS  P dV   dN
microcanonical ensemble
T ,V , N
d F S dT  P dV   dN
T , P, N
  U  TS  PV   N d  S dT  V d P  dN
isothermic-isobaric ensemble
T ,V , 
  U TS  N   PV d   S dT  P dV  Nd
  PV ? 
grand canonical ensemble
is additive, V is the only additive independent variable.
  
  P
 V T , 
   (T ,  )  V ,  (T ,  )  
Similar consideration holds for
   (T , P)  N
93
Grand canonical thermodynamic functions
of an ideal Bose gas
 General procedure for independent particles
 Elementary treatment of thermodynamic
functions of an ideal gas
Specific heat
Equation of the state
Quantum statistics with grand canonical ensemble
Grand canonical ensemble admits both energy and particle number exchange
between the system and its environment.
The statistical operator (many body density matrix)
̂
acts in the Fock space
External variables are T ,V ,  . They are specified by the conditions
ˆ ˆ  U
Hˆ  Tr 
V  sharp
ˆ Nˆ  N
Nˆ  Tr 
ˆ ln ˆ  max
S  kB Tr 
Grand canonical statistical operator has the Gibbs' form
ˆ
ˆ
ˆ  Z 1 e  ( H   N )
ˆ
ˆ
Z (  ,  ,V )  Tr e  ( H   N )  e   (  ,  ,V ) statistical sum
 (  , ,V )  kBT ln Z (  ,  ,V )
grand canonical potential
95
Grand canonical statistical sum for ideal Bose gas
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

n
  (    )
eigenstate label


n
 e
n 
  (    )




n
 n  with
 n N

up to here trivial
1
1  e   (   )
96
Grand canonical statistical sum for ideal Bose gas
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

n
  (   )
eigenstate label


n
 e
n 
  (    )




n
 n  with
 n N

up to here trivial
1
1  e   (   )
97
Grand canonical statistical sum for ideal Bose gas
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

Z (  ,  ,V )  

  (    )
eigenstate label


n
 e
n 
1
1  e   (    )



n


  (    )

n
 n  with
 n N

up to here trivial
1
1  e   (   )
1
e

z
activity
fugacity
1  z e  
98
Grand canonical statistical sum for ideal Bose gas
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

Z (  ,  ,V )  

  (    )
eigenstate label


n
 e
n 
1
1  e   (    )



n


  (    )

n
 n  with
 n N

up to here trivial
1
1  e   (   )
1
e

z
activity
fugacity
1  z e  
99
Grand canonical statistical sum for ideal Bose gas
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

Z (  ,  ,V )  

  (    )
eigenstate label


n
 e
n 
1
1  e   (    )



n


  (    )

n
 n  with
 n N

up to here trivial
1
1  e   (   )
e

z
1
activity
fugacity
1  z e  
 (  , ,V )  kBT ln Z (  ,  ,V )


 +k T  ln 1  z e

grand canonical potential
= +kBT  ln 1  e   (   )

  
B

100
Grand canonical statistical sum for ideal Bose gas
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

Z (  ,  ,V )  

  (    )
eigenstate label


n
 e
n 
1
1  e   (    )



n


1  e   (   )

up to here trivial
e

z
1
activity
fugacity
1  z e  


 +k T  ln 1  z e

= +kBT  ln 1  e   (    )

  


n
 n N
1
 (  ,  ,V )   kBT ln Z (  ,  ,V )
B
  (    )
 n  with
grand canonical potential
valid for
- extended "ïnfinite" gas
- parabolic traps
just the same
101
Using the statistical sum
A. (Re)deriving the Bose-Einstein distribution
 n   Tr ˆ nˆ 

 
  kBT
  kBT

Z

 
Z
e
Tr e
  kBT
1
nˆ
  ( Hˆ   Nˆ )

 
 (

Tr e
   (     ) nˆ
Tr e

nˆ
   (     ) nˆ


ln Z (  ,  ,V ) 
ln 1  e  


1
 (    )
Tr e
  ( Hˆ   Nˆ )

 )

o.k.
B. Pair correlations (particle number fluctuation)
kBT

 
 n   ( n   n  )2    n  (1   n  ) bunching
For Fermions .....................   n (1   n  ) anti-bunching
For classical gas ................   n 
102
Ideal Boson systems at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
1
BE
n 
e  (  )  1
1
e   1
Equation for the chemical potential T
closes
 0the equilibrium problem:
T 0
N  N (T ,  )   n(?
j)  
e
 N , 0 ,0 , ,0 ,
j
F
 1 ,1 ,
,1 , 0 ,
B
j
1
 ( j   )
1
vac
103
Thermodynamic functions for an extended Bose gas
For Born-Karman periodic boundary conditions, the lowest level is
 (k  0 )  0
Its contribution has to be singled out, like before:

 
  (T ,  ,V )  ln(1  z )  V  d  ln(1  z e ) D ( )

N (T ,  ,V ) 
0

z
1
 V  d  1 
D ( )
1 z
z e 1
0


U (T ,  ,V )  V  d 
0

1

z e 1
activity
e z
fugacity
3D DOS

3
2
 2m 
D ( )  2  2   
h 
D ( )
* PV  23 U
104
Specific heat of an ideal Bose gas
CV 
 U / N 
T
V ,N
A weak singularity …
what decides is the coexistence of two phases
108
Specific heat: comparison liquid 4He vs. ideal Bose gas
CV 
 U / N 
T
V ,N
Dulong-Petit (classical) limit

 - singularity
linear dispersion law
Dulong-Petit (classical) limit

weak singularity
quadratic dispersion law
109
Isotherms in the P-V plane
For a fixed temperature, the specific
volume V / N can be arbitrarily small.
By contrast, the pressure is volume
independent …typical for condensation

P
vc  AT
3 / 2
T   vc / A 
2 / 3
P  BT 5 / 2
Pc  BA5 / 3  vc 5 / 3
v 
vc
N N BE
3/ 2

  g3 / 2 (1)   kBT 
V
V
P   g5 / 2 (1)   kBT 
5/ 2
110
Compare with condensation of a real gas
CO2
Basic similarity:
increasing pressure with compression
critical line
beyond is a plateau
Differences:
at high pressures
at high compressions
111
Compare with condensation of a real gas
CO2
Basic similarity:
increasing pressure with compression
critical line
beyond is a plateau
Differences:
at high pressures
at high compressions
 no critical point
Conclusion:
Fig. 151 Experimental isotherms of
carbon dioxide (CO2}.
BEC in a gas is a phase transition
of the first order
112
IV.
Non-interacting bosons in a trap
Useful digression: energy units
energy
1K
1eV
1K
k B /J
e / kB
1eV
kB / e
e/J
s -1
kB / h
e/h
s-1
h / kB
h/e
h/J
energy
1K
1eV
s-1
1K
1.38  1023
8.63  1005
2.08  1010
1eV
1.16  1004
1.60  1019
2.41  1014
s-1
4.80  1011
4.14  1015
6.63  1034
114
Trap potential (physics involved skipped)
evaporation
cooling
Typical profile
?
coordinate/ microns

This is just one direction
Presently, the traps are mostly 3D
The trap is clearly from the real world, the
atomic cloud is visible almost by a naked eye
115
Trap potential
Parabolic
approximation
in general, an
anisotropic
harmonic oscillator
usually with axial
symmetry
1 2 1
1
1
2 2
2 2
H
p  m x x  m y y  m z2 z 2
2m
2
2
2
 Hx  H y  Hz
1D
2D
3D
116
Ground state orbital and the trap potential
400 nK
level
number
200 nK
x / a0 x
 0 ( x, y, z )  0 x  x 0 y  y 0 z  z 
0 (u ) 
1
a0

e
u2
2 a 02
,

2
2
1
1
1
a0 
, E0     2  
m
2
2 ma0 2 Mum a02
u 
1
1
2 2
V (u )  m u    
2
2  a0 
2
• characteristic energy
• characteristic length
117
Ground state orbital and the trap potential
400 nK
level
number
200 nK
87 Rb
a0  1 m
 =10 nK
x / a0 x
 0 ( x, y, z )  0 x  x 0 y  y 0 z  z 
0 (u ) 
1
a0

e
u2
2 a 02
,
N ~ 106 at.

2
2
1
1
1
a0 
, E0     2  
m
2
2 ma0 2 Mum a02
u 
1
1
2 2
V (u )  m u    
2
2  a0 
2
• characteristic energy
• characteristic length
118
Ground state orbital and the trap potential
400 nK
level
number
200 nK
87 Rb
a0  1 m
 =10 nK
x / a0 x
 0 ( x, y, z )  0 x  x 0 y  y 0 z  z 
0 (u ) 
1
a0

e
u2
2 a 02
,
N ~ 106 at.

2
2
1
1
1
a0 
, E0     2  
m
2
2 ma0 2 Mum a02
u 
1
1
2 2
V (u )  m u    
2
2  a0 
2
• characteristic energy
• characteristic length
119
Filling the trap with particles: IDOS, DOS
1D
x
E
 ( E )  int( E /  ) E / 
D ( E )   '( E )    
1
For the finite trap, unlike in the extended gas, D ( E ) is not divided by volume !!
2D
 (E)
x
y
1
2
E 2 /(  x   y )
D ( E )   '( E )  E /(  x   y )
"thermodynamic limit"
only approximate … finite systems
better for small
E  Ex  E y
E  const.

kBTc
meaning wide trap potentials
120
Filling the trap with particles
3D
 (E)
1
6
E 3 /( x   y  z )
D ( E )   '( E )  12 E 2 /(  x   y   z )
Estimate for the transition temperature
particle number comparable with
the number of states in the thermal shell
N    kBT 
2D
3D
Tc   / kB
1
N2
Tc   / kB
1
N3
For 106 particles,
  ( x   y
1
)2
  ( x   y  z
1
)3
• characteristic energy
kBTc  102 
121
Filling the trap with particles
3D
 (E)
1
6
E 3 /( x   y  z )
D ( E )   '( E )  12 E 2 /(  x   y   z )
Estimate for the transition temperature
particle number comparable with
the number of states in the thermal shell
N    kBT 
2D
3D
Tc   / kB
1
N2
Tc   / kB
1
N3
For 106 particles,
kBTc  102 
  ( x   y
1
)2
  ( x   y  z
1
)3
• characteristic energy

important for therm. limit
122
Exact expressions for critical temperature etc.
The general expressions are the same like for the homogeneous gas.
Working with discrete levels, we have
N  N (T ,  )   n( j )  
j
j
1
e
 ( j   )
1
and this can be used for numerics without exceptions.
In the approximate thermodynamic limit, the old equation holds, only the
volume V does not enter as a factor:
N  N (T ,  ) 
In 3D,
Tc  ( (3))


1
e

 ( 0   )
1
3
1
 / kB
 V  d
1
N3
0
1
e
 (   )
1
 0.94  / kB
D ( )
  0 for T  TC
1
N3

N BE  N  1  (T / Tc )3 , T  Tc
123
How sharp is the transition
These are experimental data
fitted by the formula


N BE  N  1  (T / Tc )3 , T  Tc
The rounding is apparent,
but not really an essential feature
124
Seeing the condensate – reminder
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 

125
OPDM in the Trap
• Use the eigenstates of the 3D oscillator
• Use the BE occupation numbers
 =   n 
  ( x , y , z ),  w  0, 1, 2, 3,

1
=
 ( E   )
 ,
 = x y z
e
1
E = E x  E y  E z =  x x +  x x +  x x

• Single out the ground state
 = 000

1
e
 
1
BEC
000 


 (000)


1
e
 ( E   )
1

TERM
126
OPDM in the Trap
• Use the eigenstates of the 3D oscillator
• Use the BE occupation numbers
 =   n 
  ( x , y , z ),  w  0, 1, 2, 3,

1
=
 ( E   )
 ,
 = x y z
e
1
E = E x  E y  E z =  x x +  x x +  x x

Coherent component,
be it condensate or not.
ground
/ kB , itstate
At Tout the
contains
• Single
ALL atoms in the cloud
 = 000

1
e
 
1
BEC
Incoherent thermal component,
coexisting with the condensate.
 / kB, it freezes out
At T
and contains NO atoms
000 


 (000)


1
e
 ( E   )
1

TERM
127
OPDM in the Trap, Particle Density in Space
The spatial distribution of atoms in the trap is inhomogeneous.
Proceed by definition:
n(r )  Tr  ( rop  r )
 Tr   d r r  ( r  r ) r  Tr  r r
 r  r  r

   (r )

1
e
 ( E   )
1
 r
1
2
e
 ( E   )
1
Split into the two parts, the coherent and the incoherent phase
n(r )  r  r  r BEC r  r THERM r
128
Particle Density in Space: Boltzmann Limit
We approximate the thermal distribution by its classical limit.
Boltzmann distribution in an external field:
Two directly observable
characteristic lengths
f B (r , p )  e  (  W U ( r ))
1
3
a0  (a0 x a0 y a0 z ) 
nTHERM (r )   d 3 p  f B (r , p )
e
e
 U ( r )
R  1  m 2
T
 a0 k BT / 
 1  m(x2 x2  y2 y 2 z2 z 2 )
2
a0
  (x   y  z )
For comparison:
nBEC (r )  0 x  x  0 y  y  0 z  z 
2

m
,
1
e
3
a0 x a0 y a0 z
2
1
2
y2
x2
z2
 2  2  2
a0x a0 y a0z
e
 
1
1
e   1
anisotropy
given by analogous
definitions of the
two lengths
for each direction
134
1
3
Real space Image of an Atomic Cloud
• the cloud is macroscopic
• basically, we see the thermal distribution
• a cigar shape: prolate rotational ellipsoid
• diffuse contours: Maxwell – Boltzmann
distribution in a parabolic potential
135
Particle Velocity (Momentum) Distribution
The procedure is similar, do it quickly:
f ( p)  p  p 
=
1
p 000
 000 ( p)
p  BEC p  p THERM p
e
 
1

e    1


 (000)
known f BEC ( p)
 ( p)
1
e
 ( E   )
1
 p
1
2
e
 ( E   )
1
laborious f THERM ( p)
f BEC ( p)  0 x  px  0 y  p y  0 z  pz 
2
2
e
p
 (000)
1
2


000 p 
p 2y
px2
pz2
 2  2  2
b0x b0 y b0z
1
e
 
1
,
1
2
e    1
b0 w 
a0 w
136
Thermal Particle Velocity (Momentum) Distribution
Again, we approximate the thermal distribution by its classical limit.
Boltzmann distribution in an external field:
Two directly observable
characteristic lengths
f B (r , p )  e  (  W U ( r ))
1
3
b0  (b0 x b0 y b0 z ) 
f THERM (r )   d 3 r  f B (r , p )
e
 W
B 1 m
T
 b0 k BT / 
 1  m1 ( px2  p2y  pz2 )
e 2
a0
,
b0
Remarkable:
f BEC ( p)  0 x  px  0 y  p y  0 z  pz 
2
2
e
p 2y
px2
pz2
 2  2  2
b0x b0 y b0z
1
e
 
1
,
2
1 B  b0 k BT / 
  T
e 1
R  a0 k BT / 
T
b0 w 
b0
a0
Thermal and condensate
a0 wlengths in the same ratio
for positions and momenta
137
Three crossed laser beams: 3D laser cooling
20 000 photons
needed to stop from
the room temperature
braking force proportional to velocity:
viscous medium,
"molasses"
For an intense laser a
matter of milliseconds
temperature measurement: turn off the
lasers. Atoms slowly
sink in the field of
gravity
simultaneously, they
spread in a ballistic
fashion
the probe laser beam
excites fluorescence.
the velocity
distribution inferred
from the cloud size
and shape
138
Three crossed laser beams: 3D laser cooling
20 000 photons
needed to stop from
the room temperature
braking force proportional to velocity:
viscous medium,
"molasses"
For an intense laser a
matter of milliseconds
Temperature
measurement: turn off
the lasers. Atoms
slowly sink in the field
of gravity
simultaneously, they
spread in a ballistic
fashion
the probe laser beam
excites fluorescence.
the velocity
distribution inferred
from the cloud size
and shape
139
Three crossed laser beams: 3D laser cooling
20 000 photons
needed to stop from
the room temperature
braking force proportional to velocity:
viscous medium,
"molasses"
For an intense laser a
matter of milliseconds
Temperature
measurement: turn off
the lasers. Atoms
slowly sink in the field
of gravity
the probe laser beam
excites fluorescence.
the velocity
distribution inferred
from the cloud size
and shape
140
Three crossed laser beams: 3D laser cooling
20 000 photons
needed to stop from
the room temperature
braking force proportional to velocity:
viscous medium,
"molasses"
For an intense laser a
matter of milliseconds
Temperature
measurement: turn off
the lasers. Atoms
slowly sink in the field
of gravity
simultaneously, they
spread in a ballistic
fashion
the probe laser beam
excites fluorescence.
the velocity
distribution inferred
from the cloud size
and shape
141
Three crossed laser beams: 3D laser cooling
20 000 photons
needed to stop from
the room temperature
braking force proportional to velocity:
viscous medium,
"molasses"
For an intense laser a
matter of milliseconds
Temperature
measurement: turn off
the lasers. Atoms
slowly sink in the field
of gravity
simultaneously, they
spread in a ballistic
fashion
the probe laser beam
excites fluorescence.
the velocity
distribution inferred
from the cloud size
and shape
142
BEC observed by TOF in the velocity distribution
143
BEC observed by TOF in the velocity distribution
Qualitative features:  all Gaussians
wide vs.narrow
 isotropic vs. anisotropic
144
Importance of the interaction – synopsis
Without interaction, the
condensate would occupy the
ground state of the oscillator
(dashed - - - - -)
In fact, there is a significant
broadening of the condensate
of 80 000 sodium atoms in the
experiment by Hau et al. (1998),
The reason … the interactions
experiment perfectly reproduced
by the solution of the Gross –
Pitaevski equation
146
Next time !!!
The end
149
V.
Interacting atoms
Are the interactions important?
In the dilute gaseous atomic clouds in the traps, the interactions are
incomparably weaker than in liquid helium.
That permits to develop a perturbative treatment and to study in a
controlled manner many particle phenomena difficult to attack in HeII.
Several roles of the interactions
• the atomic collisions take care of thermalization
• the mean field component of the interactions determines most of the
deviations from the non-interacting case
• beyond the mean field, the interactions change the quasi-particles and
result into superfluidity even in these dilute systems
151
Fortunate properties of the interactions
1.
Strange thing: the cloud lives for seconds, or even minutes at
temperatures, at which the atoms should form a crystalline
cluster. Why?
For binding of two atoms, a third one is necessary to carry
away the released binding energy and momentum. Such
ternary collisions are very unlikely in the rare cloud, however.
2.
The interactions are elastic and spin independent: they do not
spoil the separation of the hyperfine atomic species and
preserve thus the identity of the atoms.
3.
At the very low energies in question, the effective interaction
is typically weak and repulsive … which enhances the
formation and stabilization of the condensate.
152
Fortunate properties of the interactions
1.
Strange thing: the cloud lives for seconds, or even minutes at
temperatures, at which the atoms should form a crystalline
cluster. Why?
For binding of two atoms, a third one is necessary to carry
away the released binding energy and momentum. Such
ternary collisions are very unlikely in the rare cloud, however.
2.
The interactions are elastic and spin independent: they do not
spoil the separation of the hyperfine atomic species and
preserve thus the identity of the atoms.
3.
At the very low energies in question, the effective interaction
is typically weak and repulsive … which enhances the
formation and stabilization of the condensate.
153
Interatomic interactions
For neutral atoms, the pairwise interaction
has two parts
• van der Waals force
1
 6
r
• strong repulsion at shorter distances due
to the Pauli principle for electrons
Popular model is the 6-12 potential:
   12   6 
U TRUE (r )  4       
 r 
 r  

Example:
Ar  =1.6 10-22 J  =0.34 nm

corresponds to ~12 K!!
Many bound states, too.
154
Interatomic interactions
minimum

vdW radius
1
6
2 
For neutral atoms, the pairwise interaction
has two parts
• van der Waals force
1
 6
r
• strong repulsion at shorter distances due
to the Pauli principle for electrons
Popular model is the 6-12 potential:
   12   6 
U TRUE (r )  4       
 r 
 r  

Example:
Ar  =1.6 10-22 J  =0.34 nm

corresponds to ~12 K!!
Many bound states, too.
155
Interatomic interactions
The repulsive part of the potential – not well known
The attractive part of the potential can be measured with precision
U TRUE (r )  repulsive part -
C6
r6
Even this permits to define a characteristic length
"local kinetic energy"  "local potential energy"
2
1
2m 
2
6

C6
 66
 6   2mC6
2

1/ 4
156
Interatomic interactions
The repulsive part of the potential – not well known
The attractive part of the potential can be measured with precision
U TRUE (r )  repulsive part -
C6
r6
Even this permits to define a characteristic length
"local kinetic energy"  "local potential energy"
2
1
2m 

2
6

C6
 66
 6   2mC6
2

1/ 4
rough estimate of the last bound state energy
compare
with
kBTC  collision energy of the
condensate atoms
157
Scattering length, pseudopotential
Beyond the potential radius, say
propagates in free space
3 , the scattered wave
For small energies, the scattering is purely isotropic , the s-wave
scattering. The outside wave is
sin(kr   0 )

r
For very small energies,
k  0 , the radial part becomes just
r  as , as ... the scattering length
This may be extrapolated also into the interaction sphere
(we are not interested in the short range details)
Equivalent potential ("Fermi pseudopotential")
U (r )  g   (r )
4 as
g
m
2
158
Experimental data
as
159
Useful digression: energy units
energy
1K
1eV
1K
k B /J
e / kB
1eV
kB / e
e/J
s-1
kB / h
e/h
a.u.
k B / Ha
e / Ha
s-1
a.u.
h / kB
Ha / k B
h/e
Ha / e
h/J
Ha / h
h / Ha
Ha / J
energy
1K
1eV
s-1
a.u.
1K
1.38  1023
8.63  10 05
2.08  1010
3.17  10 06
1eV
1.16  1004
1.60  1019
2.41  1014
3.67  10 02
s-1
4.80  1011
4.14  1015
6.63  1034
1.52  1016
a.u.
3.16  10 05
2.72  10 01
6.56  10 15
5.88  1021
160
Experimental data
1 a.u. = 1 bohr  0.053 nm
for “ordinary” gases
VLT clouds
nm as
3.4
4.7
6.8
8.7
8.7
10.4
nm
-1.4
4.1
-1.7
-19.5
5.6
127.2
NOTES
weak attraction ok
weak repulsion ok
weak attraction
intermediate attraction
weak repulsion ok
strong resonant repulsion
"well behaved; monotonous increase
seemingly erratic, very interesting physics of scattering
resonances behind
161
Experimental data
1 a.u. = 1 bohr  0.053 nm
for “ordinary” gases
 (K)
5180
940
----73
---
VLT clouds
nm as
3.4
4.7
6.8
8.7
8.7
10.4
nm
-1.4
4.1
-1.7
-19.5
5.6
127.2
NOTES
weak attraction ok
weak repulsion ok
weak attraction
intermediate attraction
weak repulsion ok
strong resonant repulsion
"well behaved; monotonous increase
seemingly erratic, very interesting physics of scattering
resonances behind
162
Experimental data
1 a.u. = 1 bohr  0.053 nm
for “ordinary” gases
VLT clouds
nm as
3.4
4.7
6.8
8.7
8.7
10.4
nm
-1.4
4.1
-1.7
-19.5
5.6
127.2
NOTES
weak attraction ok
weak repulsion ok
weak attraction
intermediate attraction
weak repulsion ok
strong resonant repulsion
"well behaved; monotonous increase
seemingly erratic, very interesting physics of scattering
resonances behind
163
VI.
Mean-field treatment of interacting atoms
Many-body Hamiltonian and the Hartree approximation
1 2
1
ˆ
H 
pa  V (ra ) 
2
a 2m
 U (ra  rb )
a  b
We start from the mean field approximation.
This is an educated way, similar to (almost identical with) the
HARTREE APPROXIMATION we know for many electron systems.
Most of the interactions is indeed absorbed into the mean field and
what remains are explicit quantum correlation corrections
1 2
ˆ
H GP  
pa  V (ra )  VH (ra )
a 2m
VH (ra )   drbU (ra  rb ) n(rb )  g  n(ra )
n(r )   n   r 
2
self-consistent
system

 1 2

p

V
(
r
)

V
(
r
)
H

  r   E   r 
 2m

165
Hartree approximation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
2
 1 2
p

V
(
r
)

gN

r
0    0  r   E00  r 

 2m

Putting
This
is a single self-consistent equation for a single orbital,
theory
(r )  ever.
N  0 (r )
the simplest HF like
we obtain a closed equation for the order parameter:
2
 1 2
p  V (r )  g   r    r     r 

 2m

This is the celebrated Gross-Pitaevskii equation.
166
Hartree approximation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
2
 1 2
p

V
(
r
)

gN

r
0    0  r   E00  r 

 2m

Putting
This
is a single self-consistent equation for a single orbital,
theory
(r )  ever.
N  0 (r )
the simplest HF like
we obtain a closed equation for the order parameter:
2
 1 2
p  V (r )  g   r    r     r 

 2m

This is the celebrated Gross-Pitaevskii equation.
167
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
2
 1 2
p

V
(
r
)

gN

r
0    0  r   E00  r 

 2m

Putting
 (r )  N  0 (r )
we obtain a closed equation for the order parameter:
2
 1 2
p  V (r )  g   r    r     r 

 2m

This is the celebrated Gross-Pitaevskii equation.
168
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
2
 1 2
p

V
(
r
)

gN

r
0    0  r   E00  r 

 2m

Putting
 (r )  N  0 (r )
we obtain a closed equation for the order parameter:
2
 1 2
p  V (r )  g   r    r     r 

 2m

This is the celebrated Gross-Pitaevskii equation.
• has the form of a simple non-linear Schrödinger equation
• concerns a macroscopic quantity 
• suitable for numerical solution.
169
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
2
 1 2
p

V
(
r
)

gN

r
0    0  r   E00  r 

 2m

Putting
 (r )  N  0 (r )
we obtain a closed equation for the order parameter:
The lowest level
coincides with the
chemical potential
2
 1 2
p  V (r )  g   r    r     r 

 2m

This is the celebrated Gross-Pitaevskii equation.
• has the form of a simple non-linear Schrödinger equation
• concerns a macroscopic quantity 
• suitable for numerical solution.
170
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
2
 1 2
p

V
(
r
)

gN

r
0    0  r   E00  r 

 2m

Putting
 (r )  N  0 (r )
we obtain a closed equation for the order parameter:
The lowest level
coincides with the
chemical potential
2
 1 2
p  V (r )  g   r    r     r 

 2m

This is the celebrated Gross-Pitaevskii equation.
• has the form of a simple non-linear Schrödinger equation
• concerns a macroscopic quantity 
• suitable for numerical solution.
171
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
2
 1 2
p

V
(
r
)

gN

r
0    0  r   E00  r 

 2m

Putting
 (r )  N  0 (r )
we obtain a closed equation for the order parameter:
The lowest level
coincides with the
chemical potential
2
 1 2
p  V (r )  g   r    r     r 

 2m

This is the celebrated Gross-Pitaevskii equation.
• has the form of a simple non-linear Schrödinger equation
• concerns a macroscopic quantity 
• suitable for numerical solution.
172
Gross-Pitaevskii equation – "Bohmian" form
For a static condensate, the order parameter has ZERO PHASE.
Then
 ( r )  N  0 ( r )  n( r )
N [ n]  N   d 3 r  ( r )   d 3 r  n( r )  N
2
The Gross-Pitaevskii equation
2
 1 2
p  V (r )  g   r    r     r 

 2m

becomes

2
2m
 n( r )
n( r )
Bohm's
quantum
potential
 V ( r )  g n( r )  
the effective
mean-field
potential
173
Gross-Pitaevskii equation – "Bohmian" form
For a static condensate, the order parameter has ZERO PHASE.
Then
 ( r )  N  0 ( r )  n( r )
N [ n]  N   d 3 r  ( r )   d 3 r  n( r )  N
2
The Gross-Pitaevskii equation
2
 1 2
p  V (r )  g   r    r     r 

 2m

becomes

2
2m
 n( r )
n( r )
Bohm's
quantum
potential
 V ( r )  g n( r )  
the effective
mean-field
potential
174
Gross-Pitaevskii equation – "Bohmian" form
For a static condensate, the order parameter has ZERO PHASE.
Then
 ( r )  N  0 ( r )  n( r )
N [ n]  N   d 3 r  ( r )   d 3 r  n( r )  N
2
The Gross-Pitaevskii equation
2
 1 2
p  V (r )  g   r    r     r 

 2m

becomes

2
2m
 n( r )
n( r )
Bohm's
quantum
potential
 V ( r )  g n( r )  
the effective
mean-field
potential
175
Gross-Pitaevskii equation – variational interpretation
This equation results from a variational treatment of the
Energy Functional
2


1
3
2
2
E[n]   d r  ( n(r ) )  V (r )n(r )  g n (r ) 
2
 2m

It is required that
E[n]  min
with the auxiliary condition
N [ n]  N
that is
  E[ n ]   N [ n ]   0
which is the GP equation written for the particle density (previous slide).
176
Gross-Pitaevskii equation – variational interpretation
This equation results from a variational treatment of the
Energy Functional
2


1
3
2
2
E[n]   d r  ( n(r ) )  V (r )n(r )  g n (r ) 
2
 2m

It is required that
E[n]  min
with the auxiliary condition
N [ n]  N
that is
  E[ n ]   N [ n ]   0
LAGRANGE MULTIPLIER
which is the GP equation written for the particle density (previous slide).
177
Gross-Pitaevskii equation – chemical potential
This equation results from a variational treatment of the
Energy Functional
2


1
3
2
2
E[n]   d r  ( n(r ) )  V (r )n(r )  g n (r ) 
2
 2m

It is required that
E[n]  min
with the auxiliary condition
N [ n]  N
that is
  E[ n ]   N [ n ]   0
which is the GP equation written for the particle density (previous slide).
From there
 E[ n ]

 N [ n]
178
Gross-Pitaevskii equation – chemical potential
This equation results from a variational treatment of the
Energy Functional
2


1
3
2
2
E[n]   d r  ( n(r ) )  V (r )n(r )  g n (r ) 
2
 2m

It is required that
E[n]  min
with the auxiliary condition
N [ n]  N
that is
  E[ n ]   N [ n ]   0
which is the GP equation written for the particle density (previous slide).
From there
chemical potential
 E[ n ]

 N [ n]
by definition
179
Interacting atoms in a constant potential
The simplest case of all: a homogeneous gas
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
N
n( r )  n 
 const.
V

2
2m
 n( r )
n( r )
V (r )  V  const.
 V ( r )  g n( r )  
g n   V
184
The simplest case of all: a homogeneous gas
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
N
n( r )  n 
 const.
V

2
2m
 n( r )
n( r )
V (r )  V  const.
 V ( r )  g n( r )  
g n   V
185
The simplest case of all: a homogeneous gas
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
N
n( r )  n 
 const.
V

2
2m
 n( r )
n( r )
V (r )  V  const.
 V ( r )  g n( r )  
g n   V
The repulsive interaction increases the chemical potential
The repulsive interaction increases the chemical potential
If
g  0, it would be
n
0

and the gas would be thermodynamically unstable.
186
The simplest case of all: a homogeneous gas
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
N
n( r )  n 
 const.
V

2
 n( r )
2m
n( r )
V (r )  V  const.
 V ( r )  g n( r )  
g n   V
The repulsive interaction increases the chemical potential
total energy
E  12 gn 2  V
internal pressure
E 1 2
P 
 2 gn
V
187
The simplest case of all: a homogeneous gas
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
N
n( r )  n 
 const.
V

2
 n( r )
2m
n( r )
V (r )  V  const.
 V ( r )  g n( r )  
g n   V
The repulsive interaction increases the chemical potential
total energy
E  12 gn 2  V
internal pressure
E 1 2
P 
 2 gn
V
2
N
E  12 g
V
188
The simplest case of all: a homogeneous gas
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
N
n( r )  n 
 const.
V

2
 n( r )
2m
n( r )
V (r )  V  const.
 V ( r )  g n( r )  
g n   V
The repulsive interaction increases the chemical potential
total energy
E  12 gn 2  V
If g
 0, it would be
internal pressure
E 1 2
P 
 2 gn
V
2
N
E  12 g
V
n
0

and the gas would be thermodynamically unstable.
189
The simplest case of all: a homogeneous gas
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
N
n( r )  n 
 const.
V

2
 n( r )
2m
n( r )
V (r )  V  const.
 V ( r )  g n( r )  
g n   V
The repulsive interaction increases the chemical potential
total energy
E  12 gn 2  V
If g
 0, it would be
internal pressure
E 1 2
P 
 2 gn
V
2
N
E  12 g
V
n
0

and the gas would be thermodynamically unstable.
190
VII.
Interacting atoms in a parabolic trap
Reminescence: The trap potential and the ground state
400 nK
level
number
200 nK
87 Rb
a0  1 m
 =10 nK
x / a0 x
 0 ( x, y, z )  0 x  x 0 y  y 0 z  z 
0 (u ) 
1
a0

e
u2
2 a 02
,

N ~ 106 at.
2
2
1
1
1
a0 
, E0     2  
m
2
2 ma0 2 Mum a02
193
Reminescence: The trap potential and the ground state
400 nK
level
number
200 nK
87 Rb
a0  1 m
 =10 nK
x / a0 x
 0 ( x, y, z )  0 x  x 0 y  y 0 z  z 
0 (u ) 
1
a0

e
u2
2 a 02
,
N ~ 106 at.

2
2
1
1
1
a0 
, E0     2  
m
2
2 ma0 2 Mum a02
u 
1
1
2 2
V (u )  m u    
2
2  a0 
2
• characteristic energy
• characteristic length
194
Parabolic trap with interactions
GP equation for a spherical trap ( … the simplest possible case)
2

2
1
2 2
  m r  g   r    r     r 

2
 2m

195
Parabolic trap with interactions
GP equation for a spherical trap ( … the simplest possible case)
2

2
1
2 2
  m r  g   r    r     r 

2
 2m

Where is the particle number N? ( … a little reminder)
3
3
d
r

(
r
)

d

 r  n(r )  N
2
196
Parabolic trap with interactions
GP equation for a spherical trap ( … the simplest possible case)
2

2
1
2 2
  m r  g   r    r     r 

2
 2m

Where is the particle number N? ( … a little reminder)
3
3
d
r

(
r
)

d

 r  n(r )  N
2
Dimensionless GP equation for the trap
r  r  a0
energy = energy  
2
2

2
4

as
1
2 2 2
  m a0 r 
  r  a0    r      r  a0 

2
2
m
 2ma0

  r  a0    r 
N
a03

 d r  (r )
3
2
1

2
8 as N
2
  r    r     r 
   r 
a0


197
Parabolic trap with interactions
GP equation for a spherical trap ( … the simplest possible case)
2

2
1
2 2
  m r  g   r    r     r 

2
 2m

Where is the particle number N? ( … a little reminder)
3
3
d
r

(
r
)

d

 r  n(r )  N
2
Dimensionless GP equation for the trap
r  r  a0
energy = energy  
2
2

2
4

as
1
2 2 2
  m a0 r 
  r  a0    r      r  a0 

2
2
m
 2ma0

 2
2
N
  r  a0    r  3   d 3r  (r )  1 a single dimensionless
a0
parameter

2
8 as N
2
  r    r     r 
   r 
a0


198
Importance of the interaction – synopsis
Without interaction, the
condensate would occupy the
ground state of the oscillator
(dashed - - - - -)
In fact, there is a significant
broadening of the condensate of
80 000 sodium atoms in the
experiment by Hau et al. (1998),
perfectly reproduced by the
solution of the GP equation
199
Importance of the interaction – synopsis
Without interaction, the
condensate would occupy the
ground state of the oscillator
(dashed - - - - -)
In fact, there is a significant
broadening of the condensate of
80 000 sodium atoms in the
experiment by Hau et al. (1998),
perfectly reproduced by the
solution of the GP equation
guess:
internal pressure due to repulsion
competes with the trap potential
200
Importance of the interaction
Qualitative
for g>0, repulsion, both inner "quantum pressure" and
the interaction broaden the condensate.
for g<0, attraction, "quantum pressure" and the
interaction together allow the condensate to form. It
shrinks and becomes metastable. Onset of instability
with respect to three particle recombination processes
Quantitative
The decisive parameter for the "importance" of
interactions is
EINT
EKIN
gNn
N 
N 2 as a03
Na02

Nas
a0
201
Importance of the interaction
Qualitative
for g>0, repulsion, both inner "quantum pressure" and
the interaction broaden the condensate.
for g<0, attraction, "quantum pressure" and the
interaction together allow the condensate to form. It
shrinks and becomes metastable. Onset of instability
with respect to three particle recombination processes
Quantitative
The decisive parameter for the "importance" of
interactions is
EINT
EKIN
gNn
N 
N 2 as a03
Na02

Nas
a0
202
Importance of the interaction
Qualitative
for g>0, repulsion, both inner "quantum pressure" and
the interaction broaden the condensate.
for g<0, attraction, "quantum pressure" and the
interaction together allow the condensate to form. It
shrinks and becomes metastable. Onset of instability
with respect to three particle recombination processes
Quantitative
The decisive parameter for the "importance" of
interactions is
EINT
EKIN
gNn
N 
N 2 as a03
Na02

Nas
a0
203
Importance of the interaction
Qualitative
for g>0, repulsion, both inner "quantum pressure" and
the interaction broaden the condensate.
for g<0, attraction, "quantum pressure" and the
unlike in an extended
interaction together allow the condensate to form. It homogeneous gas !!
shrinks and becomes metastable. Onset of instability
with respect to three particle recombination processes
Quantitative
The decisive parameter for the "importance" of
interactions is
EINT
EKIN
gNn
N 
N 2 as a03
Na02

Nas
a0
204
Importance of the interaction
Qualitative
for g>0, repulsion, both inner "quantum pressure" and
the interaction broaden the condensate.
for g<0, attraction, "quantum pressure" and the
interaction together allow the condensate to form. It
shrinks and becomes metastable. Onset of instability
with respect to three particle recombination processes
Quantitative
The decisive parameter for the "importance" of
interactions is
E INT
E KIN
Ngn
N 
N 2as a03 Nas



2
4
Na0
a0
205
Importance of the interaction
Qualitative
for g>0, repulsion, both inner "quantum pressure" and
the interaction broaden the condensate.
for g<0, attraction, "quantum pressure" and the
interaction together allow the condensate to form. It
shrinks and becomes metastable. Onset of instability
with respect to three particle recombination processes
Quantitative
2
4

a
s
The decisive
g  parameter for the "importance" of
interactions is m
E INT
Ngn N 2as a03 Nas



2
E KIN N 
4
Na0
a0

2
ma02
206
Importance of the interaction
Qualitative
for g>0, repulsion, both inner "quantum pressure" and
the interaction broaden the condensate.
for g<0, attraction, "quantum pressure" and the
interaction together allow the condensate to form. It
shrinks and becomes metastable. Onset of instability
with respect to three particle recombination processes
Quantitative
The decisive parameter for the "importance" of
interactions is
E INT
E KIN
Ngn
N 
N 2as a03 Nas



2
4
Na0
a0
207
Importance of the interaction
Qualitative
for g>0, repulsion, both inner "quantum pressure" and
the interaction broaden the condensate.
for g<0, attraction, "quantum pressure" and the
interaction together allow the condensate to form. It
shrinks and becomes metastable. Onset of instability
with respect to three particle recombination processes
can vary
in a wide range
Quantitative
The decisive parameter for the "importance" of
interactions is
E INT
E KIN
Ngn
N 
collective effect
weak or strong
depending on N
N 2as a03 Nas



2
4
Na0
a0
1
weak individual
collisions
208
Importance of the interaction
Qualitative
for g>0, repulsion, both inner "quantum pressure" and
the interaction broaden the condensate.
for g<0, attraction, "quantum pressure" and the
interaction together allow the condensate to form. It
shrinks and becomes metastable. Onset of instability
with respect to three particle recombination processes
can vary
in a wide range
Quantitative
The decisive parameter for the "importance" of
interactions is
E INT
E KIN
Ngn
N 
N 2as a03 Nas



2
4
Na0
a0
0
realistic
value
collective effect
weak or strong
depending on N
1
weak individual
collisions
209
Variational method of solving the GPE
for atoms in a parabolic trap
Reminescence: The trap potential and the ground state
400 nK
level
number
200 nK
87 Rb
a0  1 m
 =10 nK
x / a0 x
 0 ( x, y, z )  0 x  x 0 y  y 0 z  z 
0 (u ) 
1
a0

e
u2
2 a 02
,

N ~ 106 at.
2
2
1
1
1
a0 
, E0     2  
m
2
2 ma0 2 Mum a02
212
Reminescence: The trap potential and the ground state
400 nK
level
number
200 nK
87 Rb
a0  1 m
 =10 nK
x / a0 x
 0 ( x, y, z )  0 x  x 0 y  y 0 z  z 
0 (u ) 
1
a0

e
u2
2 a 02
,

N ~ 106 at.
2
2
1
1
1
a0 
, E0     2  
m
2
2 ma0 2 Mum a02
ground state orbital
213
Reminescence: The trap potential and the ground state
400 nK
level
number
200 nK
87 Rb
a0  1 m
 =10 nK
x / a0 x

N ~ 106 at.
0 ( x, y, z )  0 x  x 0 y  y 0 z  z 
u2
 2
1
0 (u ) 
e 2a 0 ,
a0
2
2
1
1
1
a0 
, E0     2  
m
2
2 ma0 2 Mum a02
ground state orbital
214
Variational estimate of the condensate properties
SCALING ANSATZ
The condensate orbital will be taken in the form
1 r2
  2
2b
3
0  r   A e
,
 
A b 
2
1/ 4
It is just like the ground state orbital for the isotropic oscillator, but with a
rescaled size. This is reminescent of the well-known scaling for the ground
state of the helium atom.
215
Variational estimate of the condensate properties
SCALING ANSATZ
The condensate orbital will be taken in the form
variational
parameter
b
1 r2
  2
2b
3
0  r   A e
,
 
A b 
2
1/ 4
It is just like the ground state orbital for the isotropic oscillator, but with a
rescaled size. This is reminescent of the well-known scaling for the ground
state of the helium atom.
216
Importance of the interaction: scaling approximation
Variational ansatz: the GP orbital is a scaled ground state for g = 0
1 r2
  2
2b
3
0  r   A e
,
 
A b 
2
1/ 4
217
Importance of the interaction: scaling approximation
Variational ansatz: the GP orbital is a scaled ground state for g = 0
1 r2
  2
2b
3
dimension-less
energy per particle
E  0N E
0  r   A e
,
 
A b 
2
3 1
1

E     2   2     3
4 


Variational estimate of the total energy of the
condensate as a function of the parameter 
Variational parameter is the orbital width


1/ 4
dimension-less
orbital size

 N  1 as
2  a0 
b
a0
self-interaction
ADDITIONAL NOTES
in units of a0
218
Importance of the interaction: scaling approximation
Variational ansatz: the GP orbital is a scaled ground state for g = 0
1 r2
  2
2b
3
dimension-less
energy per particle
E  0N E
0  r   A e
 
A b 
2
,
3 1
1

E     2   2     3
4 


Variational estimate of the total energy of the
condensate as a function of the parameter 
Variational parameter is the orbital width

 N  1 as
2  a0



1/ 4
dimension-less
orbital size

 N  1 as
2  a0 
b
a0
self-interaction
ADDITIONAL NOTES
in units of a0
1
2(2 )
3/ 2

219
Importance of the interaction: scaling approximation
Variational ansatz: the GP orbital is a scaled ground state for g = 0
1 r2
  2
2b
3
dimension-less
energy per particle
E  0N E
0  r   A e
,
 
A b 
2
3 1
1

E     2   2     3
4 


Variational estimate of the total energy of the
condensate as a function of the parameter 
Variational parameter is the orbital width


1/ 4
dimension-less
orbital size

 N  1 as
2  a0 
b
a0
self-interaction
ADDITIONAL NOTES
in units of a0
minimize the energy
E
0

 5    2  0
220
Importance of the interaction: scaling approximation
g 0
g 0
Variational estimate of the total energy of the
condensate as a function of the parameter 
Variational parameter is the orbital width


 N  1 as
2  a0
in units of a0
221
Importance of the interaction: scaling approximation
g 0
Variational estimate ofminimize
the total energy
the
the eneofrgy
condensate as a function of the parameter 
g 0
 N  1 as

E
2  a0
 0  5    2  0
 is the orbital width  in units of a0
Variational parameter
222
Importance of the interaction: scaling approximation
g 0
g 0
Variational estimate of the total energy of the
condensate as a function of the parameter 
Variational parameter is the orbital width
The minimum of the E ( ) curve
gives the condensate size for a
given  . With increasing  , the
condensate stretches with an
1/ 5
asymptotic power law  min  


 N  1 as
2  a0
in units of a0
223
Importance of the interaction: scaling approximation
g 0
g 0
Variational estimate of the total energy of the
condensate as a function of the parameter 
Variational parameter is the orbital width
The minimum of the E ( ) curve
gives the condensate size for a
given  . With increasing  , the
condensate stretches with an
1/ 5
asymptotic power law  min  


 N  1 as
2  a0
in units of a0
For 0.27    0, the condensate
is metastable, below c  0.27, it
becomes unstable and shrinks to a
'zero' volume. Quantum pressure no
more manages to overcome the 224
attractive atom-atom interaction
The end