Transcript Velicky17.10.BECBrno07L3

Cold atoms
Lecture 3.
17. October 2007
BEC for interacting particles
Description of the interaction
Mean field approximation: GP equation
Variational properties of the GP equation
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
3
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.
4
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.
5
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.
6
Interatomic interactions

minimum
1
atoms,
theradius
pairwise6
vdW
2 interaction
For neutral
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.
7
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
8
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
For the 6 - 12 potential
C6  4  
6
 6    4  2m
2
2

1/ 4
9
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
kBTC  collision energy of the
condensate atoms
10
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 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 ("pseudopotential")
U (r )  g   (r )
4 as
g
m
2
11
Experimental data
as
12
Experimental data
1 a.u. = 1 bohr  0.053 nm
nm as
3.4
4.7
6.8
8.7
8.7
10.4
nm
-1.4
4.1
-1.7
-1.9
5.6
127.2
13
Experimental data
1 a.u. = 1 bohr  0.053 nm
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
14
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 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

ADDITIONAL NOTES
self-consistent
system

 1 2

p

V
(
r
)

V
(
r
)
H

  r   E   r 
 2m

16
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 absorbed into the mean field and
what remains are explicit quantum
correlation corrections
ELECTRONS
1 2
ˆ
H Hartree  
pa  V (ra )  VH ( ra )
a 2m
e '2
VH (ra )   drbU ( ra  rb ) n( rb ), U ( r ) 
r
n( r ) 

E  
  r 
2
 1 2

p

V
(
r
)

V
(
r
)
H

  r   E   r 
 2m

17
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.
18
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.
19
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.
20
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.
21
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.
22
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.
23
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
24
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
25
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
26
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

ADDITIONAL NOTES
N [ n]  N
that is
  E[ n ]   N [ n ]   0
which is the GP equation written for the particle density (previous slide).
27
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]
28
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
29
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
 E[ n ]
!!!
 
N
 N [ n]
by definition
30
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
32
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
33
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.
34
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
35
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
V
36
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
V
n
0

and the gas would be thermodynamically unstable.
37
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
V
n
0

and the gas would be thermodynamically unstable.
38
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
u 
1
1
2 2
V (u )  m u    
2
2  a0 
2
• characteristic energy
• characteristic length40
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
41
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  a0      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


42
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  a0      r  a0 

2
2
m
 2ma0

 2
2
N
  r  a0    r  3   d 3r  (r )  1
a single dimensonless
a0
parameter

2
8 as N
2
  r    r     r 
   r 
a0


43
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
44
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 compete, the condensate 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
45
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 compete, the condensate 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
gNn
N 
N 2 as a03
Na02

Nas
a0


4
46
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 compete, the condensate 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
gNn
N 
collective effect
weak or strong
depending on N
N 2 as a03
Na02

Nas
a0


4
1
weak individual
collisions
47
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 compete, the condensate 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
gNn
N 
N 2 as a03
Na02

Nas
a0


4
0
realistic
value
collective effect
weak or strong
depending on N
1
weak individual
collisions
48
The end
ADDITIONAL NOTES
On the way to the mean-field Hamiltonian
50
ADDITIONAL NOTES
On the way to the mean-field Hamiltonian
 First, the following exact transformations are performed
ˆ
ˆ
W
V
1
1
2
ˆ
H 
pa  V (ra )
2
a 2m
a

Uˆ
U (ra  rb )
a  b
Vˆ  V (ra )   d 3 r V (r )  (r  ra )   d 3r V (r )  nˆ (r )
a
1
ˆ
U
2
a

U (ra  rb ) 
a  b
1
2
3
3
d
r
d
r ' U  r  r '

particle
density operator
    r  ra    r ' rb 
a  b
TRICK!! 



  d 3 r d 3 r 'U  r  r '     r  ra     r ' rb     r  r ' 
2
a
b
SI
eliminates
nˆ (r' )
nˆ (r ) 
1
(self-interaction)
1
Hˆ  Wˆ   d 3 r V (r )  nˆ (r )   d 3 r d 3 r 'U  r  r '  nˆ (r ) nˆ (r' )    r  r ' 
2
51
ADDITIONAL NOTES
On the way to the mean-field Hamiltonian
 Second, a specific many-body state is chosen, which defines
the mean field:
  n(r )  nˆ (r )   nˆ (r ) 
Then, the operator of the (quantum) density fluctuation is defined:
nˆ  r   n  r    nˆ  r 
nˆ  r  nˆ  r'   nˆ  r  n  r'   n  r  nˆ  r'    nˆ  r   nˆ  r'   n  r  n  r' 
The Hamiltonian, still exactly, becomes


Hˆ  Wˆ   d3 r V (r )   d 3r' U  r  r '  n(r' )  nˆ (r )


1
2
1
2
3
3
d
r
d
r 'U  r  r '  n(r ) n( r' )

3
3
d
r
d
r 'U  r  r '  nˆ (r ) nˆ (r' )  nˆ (r )  r  r ' 

52
ADDITIONAL NOTES
On the way to the mean-field Hamiltonian
 In the last step, the third line containing exchange, correlation
and the self-interaction correction is neglected. The mean-field
Hamiltonian of the main lecture results:


Hˆ  Wˆ   d 3 r V (r )   d 3 r' U  r  r '  n(r' )  nˆ (r )


1
d r d r 'U  r  r '  n( r ) n( r' )

2
1
2
3
3
VH  r 
substitute back
nˆ (r )    (r  ra )
a
and integrate
3
3
d
r
d
r 'U  r  r '  nˆ (r ) nˆ (r' )  nˆ (r )  r  r ' 

REMARKS
• Second line … an additive constant compensation for doublecounting of the Hartree interaction energy
• In the original (variational) Hartree approximation, the self-interaction
is not left out, leading to non-orthogonal Hartree orbitals
BACK
53
ADDITIONAL NOTES
Variational approach
to the condensate ground state
54
ADDITIONAL NOTES
Variational estimate of the condensate properties
 VARIATIONAL PRINCIPLE OF QUANTUM MECHANICS
The ground state and energy are uniquely defined by
E   Hˆ    ' Hˆ  ' for all  '  H NS ,  '  '  1
In words,  ' is a normalized symmetrical wave function of N particles. The
minimum condition in the variational form is
  Hˆ   0 equivalent with the SR Hˆ   E 
 HARTREE VARIATIONAL ANSATZ FOR THE CONDENSATE WAVE F.
For our many-particle Hamiltonian,
1 2
1
ˆ
H 
pa  V (ra ) 
2
a 2m
 U (ra  rb ),
U (r )  g   (r )
a  b
the true ground state is approximated by the condensate for non-interacting
particles (Hartree Ansatz, here identical with the symmetrized Hartree-Fock)
  r1 , r2 ,
, rp ,

 
, rN  0  r1 0  r2  0 rp
0  rN 
55
ADDITIONAL NOTES
Variational estimate of the condensate properties
Here,  0 is a normalized real spinless orbital. It is a functional variable to be
found from the variational condition
 E 0 ]   0 ] Hˆ 0 ]  0 with 0 ] 0 ]  1  0 0  1
Explicit calculation yields
E 0 ]
2
2m
N  d 3 r 0  r   N  d 3 r V  r  0  r   
2
2
4
1
N  N  1 g  d 3r 0  r  
2
Variation of energy with the use of a Lagrange multiplier:

 N 1E 0 ]   0 0

0  0  r  , 0  0  r 
BY PARTS
2 2 3
4
3
3
3

d
r





2
d
r


V
r




N

1
g
d
r











0
0
0
0
0
0


2m 
2
This results into the GP equation derived here in the variational way:
2
 1 2
p  V (r )   N -1 g 0  r  0  r   0  r 

 2m

eliminates self-interaction
BACK
56
ADDITIONAL NOTES
Variational estimate of the condensate properties
 ANNEX Interpretation of the Lagrange multiplier 
The idea is to identify it with the chemical potential. First, we modify the notation
to express the particle number dependence
1 2
1


E N  ] N  
p    V    N  1 g  d3r  4 
2
 2m

2
 1 2
EN  E N 0 N ], 
p  V ( r )   N - 1 g 0 N  r   0  r    N 0 N  r 
 2m

The first result is that  is not the average energy per particle:
1 2
1
p 0 N  0 N V 0 N   N  1 g  d3 r 0 N 4
2m
2
1 2
 N  0 N
p 0 N  0 N V 0 N   N  1 g  d3r 0 N 4
2m
EN / N  E N 0 N ]/ N  0 N
from the GPE
57
ADDITIONAL NOTES
Variational estimate of the condensate properties
Compare now systems with N and N -1 particles:
EN  E N 0 N ]  E N 1 0 N ]   N  E N 1 0, N 1 ]   N  EN 1   N
N … energy to remove a particle
use of the variational
principle for GPE
without relaxation of the condensate
In the "thermodynamic"asymptotics of large N, the inequality tends to equality.
This only makes sense, and can be proved, for g > 0.
BACK
Reminescent of the Koopmans’ theorem in the HF theory of atoms.
Derivation:
E N  ]
N  V   2 N  N  1 g  d3 r  4
N
1
2m
p2  
E N 1  ]  N  1 
1
2m
p 2   N  1  V   2  N  1 N  2  g  d 3 r  4
E N  E N 1 
1
2m
1
1
p 2    V   2  N  N  1   N  1 N  2   g  d 3 r  4
1
 N for 
0 N
58
ADDITIONAL NOTES
Variational estimate of the condensate properties
 SCALING ANSATZ FOR A SPHERICAL PARABOLIC TRAP
The potential energy has the form

V  r   12 m02  r 2  12 m02 x 2  y 2  z 2

Without interactions, the GPE reduces to the SE for isotropic oscillator
 1 2 1
2
2
3   r
p

m


r

r



0
0 0 

 0
2
2
 2m

The solution (for the ground state orbital) is
1 r2
  2
2 a0
3
00  r   A0 e
, a0 
m0
,
0 
2
A0 
ma02
 
1/ 4
2
a0 
We (have used and) will need two integrals:
I1   




du e
u2

2
   , I 2   




du e
u2
2
u 2  12  3 
59
ADDITIONAL NOTES
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.
Next, the total energy is calculated for this orbital
E 0 ]
2
2m
N  d 3 r 0  r   N  d 3r V  r  0  r   
The solution (for)
2 is
1
6  a0
 0 NA  4  d3 r e
2
b

2
r2
 2
b
r2 
4
1
N  N  1 g  d 3r 0  r  
2
2r 2 
2
 2
ma

r 2   N  1 A6 20 g  d 3 r e b 


2
1
3
d
re
2 
a0
r2
 2
b
60
ADDITIONAL NOTES
Variational estimate of the condensate properties
For an explicit evaluation, we (have used and) will use the identities:
2
m
E 0 ]

0 a02 ,
m02
0
4 2 as
1
1
 2 , A 

, g
I1  b  b 
m
a0
2
The integrals, by the Fubini theorem, are a product of three:

2
I
b
 3I 2  b   1   
 0 N 
2
2
b

I
b

 1  

 a02
 4
1
 2    N  1
a0 
2b 
b
ma02
1
3 3/ 2
2
4
2
m
as
 
I1 b / 2

 I1  b  
3
3





 3  a02 b 2   N  1 as a03 
E 0 ]  0 N   2  2  
 3   0 N  E  
a0 
2 a0 b 
 4  b
3 1
1
b dimension-less

dimension-less
E     2   2     3


orbital size
energy per particle
4 

a0

Finally,
This expression is plotted in the figures in the main lecture.
BACK
61
The end