cold atoms lecture 3. 17. october 2007. bec for interacting particles description of the interaction...
TRANSCRIPT
![Page 1: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/1.jpg)
Cold atoms
Lecture 3.17. October 2007
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BEC for interacting particlesDescription of the interactionMean field approximation: GP equationVariational properties of the GP equation
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3
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
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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.
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5
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.
Quasi-Equilibrium
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6
Interatomic interactions
For neutral atoms, the pairwise interaction has two parts
• van der Waals force
• strong repulsion at shorter distances due to the Pauli principle for electrons
Popular model is the 6-12 potential:
Example:
corresponds to ~12 K!!
Many bound states, too.
6
1
r
12 6
TRUE ( ) 4U rr r
-22Ar =1.6 10 J =0.34 nm
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7
Interatomic interactions
For neutral atoms, the pairwise interaction has two parts
• van der Waals force
• strong repulsion at shorter distances due to the Pauli principle for electrons
Popular model is the 6-12 potential:
Example:
corresponds to ~12 K!!
Many bound states, too.
6
1
r
12 6
TRUE ( ) 4U rr r
-22Ar =1.6 10 J =0.34 nm
minimum
vdW radius 1
62
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8
Interatomic interactions
The repulsive part of the potential – not well known
The attractive part of the potential can be measured with precision
Even this permits to define a characteristic length
6TRUE 6
( ) repulsive part - C
U rr
26
2 66
26
6
6
1/ 4
"local kinetic energy" "local potential energy
2
2
"
1
m
C
m
C
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9
Interatomic interactions
The repulsive part of the potential – not well known
The attractive part of the potential can be measured with precision
Even this permits to define a characteristic length
6TRUE 6
( ) repulsive part - C
U rr
6 2 26 6
1/ 4
For the 6 - 12 potential
4 4 2C m
26
2 66
26
6
6
1/ 4
"local kinetic energy" "local potential energy
2
2
"
1
m
C
m
C
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10
Interatomic interactions
The repulsive part of the potential – not well known
The attractive part of the potential can be measured with precision
Even this permits to define a characteristic length
6TRUE 6
( ) repulsive part - C
U rr
26
2 66
26
6
6
1/ 4
"local kinetic energy" "local potential energy
2
2
"
1
m
C
m
C
rough estimate of the last bound state energy
B collision energy of the
condensate a m
to sCk T
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11
Scattering length, pseudopotential
Beyond the potential radius, say , the scattered wave propagates in free space
For small energies, the scattering is purely isotropic , the s-wave scattering. The outside wave is
For very small energies the radial part becomes just
This may be extrapolated also into the interaction sphere
(we are not interested in the short range details)
Equivalent potential ("pseudopotential")
3
0sin( )kr
r
... the scattering leng t, hs sr a a
2
( ) ( )
4 s
U r g
ag
m
r
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12
Experimental data
as
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13
Experimental data
nm
3.44.7
6.88.78.7
10.4
nm
-1.44.1
-1.7-1.95.6
127.2
1 a.u. = 1 bohr 0.053 nm
as
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14
Experimental data
nm
3.44.7
6.88.78.7
10.4
nm
-1.44.1
-1.7-19.5
5.6127.2
1 a.u. = 1 bohr 0.053 nm
NOTES
weak attraction okweak repulsion ok
weak attractionintermediate attractionweak repulsion okstrong resonant repulsion
"well behaved; monotonous increase
seemingly erratic, very interesting physics of scattering resonances behind
as
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Mean-field treatment of interacting atoms
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16
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
Many-body Hamiltonian and the Hartree approximation
21 1ˆ ( ) ( )2 2a a a b
a a b
H p V Um
r r r
We start from the mean field approximation.
2GP
2
1ˆ ( ) ( )2
( ) d ( ) ( ) ( )
( )
a a H aa
H a b a b b a
H p V Vm
V U n g n
n n
r r
r r r r r r
r r
21( ) ( )
2 Hp V V Em
r r r r
self-consistentsystem
ADDITIONAL NOTES
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17
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
Many-body Hamiltonian and the Hartree approximation
21 1ˆ ( ) ( )2 2a a a b
a a b
H p V Um
r r r
We start from the mean field approximation.
2Hartree
2
2
1ˆ ( ) ( )2
'( ) d ( ) ( ), ( )
( )
a a H aa
H a b a b b
E
H p V Vm
eV U n U r
r
n
r r
r r r r r
r r
21( ) ( )
2 Hp V V Em
r r r r
ELECTRONS
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18
Hartree approximation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
Putting
we obtain a closed equation for the order parameter:
This is the celebrated Gross-Pitaevskii equation.
220 0 0 0
1( )
2p V gN E
m
r r r r
0( ) ( )N r r
221( )
2p V g
m
r r r r
This is a single self-consistent equation for a single orbital,the simplest HF like theory ever.
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19
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
Putting
we obtain a closed equation for the order parameter:
This is the celebrated Gross-Pitaevskii equation.
220 0 0 0
1( )
2p V gN E
m
r r r r
0( ) ( )N r r
221( )
2p V g
m
r r r r
single self-consistent equation
for a single orbital
![Page 20: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/20.jpg)
20
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
Putting
we obtain a closed equation for the order parameter:
This is the celebrated Gross-Pitaevskii equation.
220 0 0 0
1( )
2p V gN E
m
r r r r
0( ) ( )N r r
221( )
2p V g
m
r r r r
The lowest level coincides with the chemical potential
single self-consistent equation
for a single orbital
![Page 21: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/21.jpg)
21
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
Putting
we obtain a closed equation for the order parameter:
This is the celebrated Gross-Pitaevskii equation.
220 0 0 0
1( )
2p V gN E
m
r r r r
0( ) ( )N r r
221( )
2p V g
m
r r r r
The lowest level coincides with the chemical potential
How do we
know??
single self-consistent equation
for a single orbital
![Page 22: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/22.jpg)
22
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
Putting
we obtain a closed equation for the order parameter:
This is the celebrated Gross-Pitaevskii equation.
220 0 0 0
1( )
2p V gN E
m
r r r r
0( ) ( )N r r
221( )
2p V g
m
r r r r
The lowest level coincides with the chemical potential
How do we
know??
How much
is it??
single self-consistent equation
for a single orbital
![Page 23: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/23.jpg)
23
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
Putting
we obtain a closed equation for the order parameter:
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.
220 0 0 0
1( )
2p V gN E
m
r r r r
0( ) ( )N r r
221( )
2p V g
m
r r r r
The lowest level coincides with the chemical potential
How do we
know??
How much
is it??
single self-consistent equation
for a single orbital
![Page 24: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/24.jpg)
24
Gross-Pitaevskii equation – "Bohmian" form
For a static condensate, the order parameter has ZERO PHASE. Then
The Gross-Pitaevskii equation
becomes
0
23 3
( )
[ ] d (
( )
) d
(
(
)
)
N
n N n
n
N
N
r
r
r
r
r
r r
221( )
2p V g
m
r r r r
2 ( )( )
( )( )
2V
ng
mn
n
r
rr
r
Bohm's quantum potential
the effective mean-field potential
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25
Gross-Pitaevskii equation – "Bohmian" form
For a static condensate, the order parameter has ZERO PHASE. Then
The Gross-Pitaevskii equation
becomes
0
23 3
( )
[ ] d (
( )
) d
(
(
)
)
N
n N n
n
N
N
r
r
r
r
r
r r
221( )
2p V g
m
r r r r
2 ( )( )
( )( )
2V
ng
mn
n
r
rr
r
Bohm's quantum potential
the effective mean-field potential
![Page 26: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/26.jpg)
26
Gross-Pitaevskii equation – "Bohmian" form
For a static condensate, the order parameter has ZERO PHASE. Then
The Gross-Pitaevskii equation
becomes
0
23 3
( )
[ ] d (
( )
) d
(
(
)
)
N
n N n
n
N
N
r
r
r
r
r
r r
221( )
2p V g
m
r r r r
2 ( )( )
( )( )
2V
ng
mn
n
r
rr
r
Bohm's quantum potential
the effective mean-field potential
![Page 27: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/27.jpg)
27
Gross-Pitaevskii equation – variational interpretation
This equation results from a variational treatment of the Energy Functional
It is required that
with the auxiliary condition
that is
which is the GP equation written for the particle density (previous slide).
3 2 22 1
2[ ] d (( ) ( ))
2( )( )n nn V g
mn
E
r r rr r
[ ] minn E
[ ]n NN
[ ] [ ] 0n n E N
ADDITIONAL NOTES
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28
Gross-Pitaevskii equation – chemical potential
This equation results from a variational treatment of the Energy Functional
It is required that
with the auxiliary condition
that is
which is the GP equation written for the particle density (previous slide). From there
3 2 22 1
2[ ] d (( ) ( ))
2( )( )n nn V g
mn
E
r r rr r
[ ] minn E
[ ]n NN
[ ] [ ] 0n n E N
[ ]
[ ]
n
n
EN
![Page 29: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/29.jpg)
29
Gross-Pitaevskii equation – chemical potential
This equation results from a variational treatment of the Energy Functional
It is required that
with the auxiliary condition
that is
which is the GP equation written for the particle density (previous slide). From there
3 2 22 1
2[ ] d (( ) ( ))
2( )( )n nn V g
mn
E
r r rr r
[ ] minn E
[ ]n NN
[ ] [ ] 0n n E N
[ ]
[ ]
n
n
EN
chemical potential by definition
![Page 30: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/30.jpg)
30
Gross-Pitaevskii equation – chemical potential
This equation results from a variational treatment of the Energy Functional
It is required that
with the auxiliary condition
that is
which is the GP equation written for the particle density (previous slide). From there
3 2 22 1
2[ ] d (( ) ( ))
2( )( )n nn V g
mn
E
r r rr r
[ ] minn E
[ ]n NN
[ ] [ ] 0n n E N
[ ]
[ ]
n
n
EN
chemical potential by definition!! !
EN
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Interacting atoms in a constant potential
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32
The simplest case of all: a homogeneous gas
2 ( )
2 ( )
n
m n
r
r
( ) ( )
V g
V
n
g n
r r
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
( ) const. ( ) const. n n V V r rNV
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33
The simplest case of all: a homogeneous gas
2 ( )
2 ( )
n
m n
r
r
( ) ( )
V g n
g n V
r r
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
( ) const. ( ) const. n n V V r rNV
![Page 34: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/34.jpg)
34
The simplest case of all: a homogeneous gas
2 ( )
2 ( )
n
m n
r
r
( ) ( )
V g n
g n V
r r
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
( ) const. ( ) const. n n V V r rNV
The repulsive interaction increases the chemical potential
If , it would be
and the gas would be thermodynamically unstable.
0g 0
n
The repulsive interaction increases the chemical potential
![Page 35: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/35.jpg)
35
The simplest case of all: a homogeneous gas
2 ( )
2 ( )
n
m n
r
r
( ) ( )
V g n
g n V
r r
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
( ) const. ( ) const. n n V V r rNV
The repulsive interaction increases the chemical potential
2 21 12 2
internal presstotal energy ure
gn gn
E
E V PV
![Page 36: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/36.jpg)
36
The simplest case of all: a homogeneous gas
2 ( )
2 ( )
n
m n
r
r
( ) ( )
V g n
g n V
r r
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
( ) const. ( ) const. n n V V r rNV
The repulsive interaction increases the chemical potential
2 21 12 2
internal presstotal energy ure
gn gn
E
E V PV
212N
EV
![Page 37: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/37.jpg)
37
The simplest case of all: a homogeneous gas
2 ( )
2 ( )
n
m n
r
r
( ) ( )
V g n
g n V
r r
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
( ) const. ( ) const. n n V V r rNV
The repulsive interaction increases the chemical potential
If , it would be
and the gas would be thermodynamically unstable.
0g 0n
2 21 12 2
internal presstotal energy ure
gn gn
E
E V PV
212N
EV
![Page 38: Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational](https://reader036.vdocuments.site/reader036/viewer/2022070412/56649ec95503460f94bd77d6/html5/thumbnails/38.jpg)
38
The simplest case of all: a homogeneous gas
2 ( )
2 ( )
n
m n
r
r
( ) ( )
V g n
g n V
r r
In an extended homogeneous system (… Born-Kármán boundary condition),
the GP equation simplifies
( ) const. ( ) const. n n V V r rNV
The repulsive interaction increases the chemical potential
If , it would be
and the gas would be thermodynamically unstable.
0g 0n
2 21 12 2
internal presstotal energy ure
gn gn
E
E V PV
212N
EV
impossible!!!
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Interacting atoms in a parabolic trap
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Reminescence: The trap potential and the ground state
level number
200 nK
400 nK
0/ xx a
2
20
0 0 0 0
2 22
0 0 0 2 20 00
( , , )
1 1 1 1( ) e , ,
2 2 2
x y z
u
a
m
x y z x y z
u a Em ma Mu aa
22 2
0
1 1( )
2 2
uV u m u
a
0
6
87
1 m
=10 nK
~ 10 at.
Rba
N
• characteristic energy
• characteristic length
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41
Parabolic trap with interactions
GP equation for a spherical trap ( … the simplest possible case)
Where is the particle number N? ( … a little reminder)
2
22 21
2 2m r g
m
r r r
23 3d ( ) d ( )n N r r r r
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42
Parabolic trap with interactions
GP equation for a spherical trap ( … the simplest possible case)
Where is the particle number N? ( … a little reminder)
2
22 21
2 2m r g
m
r r r
23 3d ( ) d ( )n N r r r r
Dimensionless GP equation for the trap 0 energy = energya r r
2222 2 2
0 0 0 020
230 3
0
22
0
41
22
d ( 1)
8
s
s
am a r a a a
mma
Na
a
a Nr
a
r r r
r r r r
r r r
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43
Parabolic trap with interactions
GP equation for a spherical trap ( … the simplest possible case)
Where is the particle number N? ( … a little reminder)
2
22 21
2 2m r g
m
r r r
23 3d ( ) d ( )n N r r r r
Dimensionless GP equation for the trap 0 energy = energya r r
2222 2 2
0 0 0 020
230 3
0
22
0
41
22
d ( 1)
8
s
s
am a r a a a
mma
Na
a
a Nr
a
r r r
r r r r
r r r
a single dimensonless
para e
2
met r
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44
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
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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
2 3INT 0
2KIN 0 0
s sE N a a NagNn
E N Na a
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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
Quantitative
The decisive parameter for the "importance" of interactions is
2 3INT 0
2KIN 0 0 4
s sN a a NagNn
N Na a
EE
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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
Quantitative
The decisive parameter for the "importance" of interactions is
weak individual collisions
1collective effect weak or strong depending on N
can vary in a wide range
2 3INT 0
2KIN 0 0 4
s sN a a NagNn
N Na a
EE
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48
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
weak individual collisions
collective effect weak or strong depending on N
can vary in a wide range
realistic value
02 3INT 0
2KIN 0 0 4
s sN a a NagNn
N Na a
EE
1
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The end
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50
On the way to the mean-field Hamiltonian
ADDITIONAL NOTES
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51
On the way to the mean-field HamiltonianADDITIONAL NOTES
First, the following exact transformations are performed
TRICK!!
2
3 3
3 3
3 3
1 1ˆ ( ) ( )2 2
( ) d ( ) ( ) d ( )
1 1( ) d d ' ' '
2 2
1d
ˆ
d ' '
ˆ ˆ
'
ˆ
ˆ
ˆ
)
2
(
'
a a a ba a a b
a aa a
a b a ba b a b
a ba b
H p V Um
V V V
U U
U
U
U
V
W V
n
r r
r r r
r r r r r r r
r r r r r r r r r r
r r r r r r r r
r
ˆ( )n r' eliminates SI (self-interaction)
particle density operator
3 3 3ˆ ˆ ˆ(1ˆ d ( ) d d ' ' '2
) ) )ˆ ( (n nU nH VW r r rr 'r r rr r r r
ˆ( )n r
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52
On the way to the mean-field HamiltonianADDITIONAL NOTES
Second, a specific many-body state is chosen, which defines the mean field:
Then, the operator of the (quantum) density fluctuation is defined:
The Hamiltonian, still exactly, becomes
3 3
3 3
3 3
ˆ d ( ) d ' ( )
1d d ' ' ( ) ( )
21
d d ' ' '
ˆ( )
ˆ2
ˆ ˆ( ) ( ) ( )
ˆH V U n
U n
W n
n nU n
n
r r r' r r r'
r r r r
r
r r' r
r r'
r r r r r r
ˆ ˆ( ) ( ) ( )n n n r r r
ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
n n n
n n n n n n n n n n
r r r
r r' r r' r r' r r' r r'
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53
On the way to the mean-field HamiltonianADDITIONAL NOTES
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:
3 3
3 3
3 3
ˆ d ( ) d ' ( )
1d d ' ' ( ) ( )
2
1d d ' ' '
ˆ( )
ˆ2
ˆ ˆ( ) ( ) ( )
ˆH V U n
U n
W n
n nU n
n
r r r' r r r'
r r r r
r
r r' r
r r'
r r r r r r
REMARKS
• Second line … an additive constant compensation for double-counting of the Hartree interaction energy
• In the original (variational) Hartree approximation, the self-interaction is not left out, leading to non-orthogonal Hartree orbitals
substitute back
and integrate
(ˆ( )) aa
n r rr HV r
BACK
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54
ADDITIONAL NOTES
Variational approach to the condensate ground state
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Variational estimate of the condensate properties
ADDITIONAL NOTES
VARIATIONAL PRINCIPLE OF QUANTUM MECHANICS
The ground state and energy are uniquely defined by
In words, is a normalized symmetrical wave function of N particles. The minimum condition in the variational form is
HARTREE VARIATIONAL ANSATZ FOR THE CONDENSATE WAVE F.
For our many-particle Hamiltonian,
the true ground state is approximated by the condensate for non-interacting particles (Hartree Ansatz, here identical with the symmetrized Hartree-Fock)
Sfor allˆ ˆ' ' ' , ' ' 1NE H H H'
equivalent with the SR ˆ ˆ0 H H E
21 1ˆ ( ) ( ), ( ) ( )2 2a a a b
a a b
H p V U U gm
r r r r r
1 2 0 1 0 2 0 0, , , , ,p N p N r r r r r r r r
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56
10 0 0
23 3 3 3
0 0 0 0
0 0 0 0 0 0
]
2 4d 2 d 1 d
2
,
2
N
V N gm
BY PARTS
r r
r r r r
E
Variational estimate of the condensate properties
ADDITIONAL NOTES
Here, is a normalized real spinless orbital. It is a functional variable to be found from the variational condition
Explicit calculation yields
Variation of energy with the use of a Lagrange multiplier:
This results into the GP equation derived here in the variational way:
0 0 0 0 0 0 0with ˆ] ] ] 0 ] ] 1 1 H E
0
2
2 2 43 3 30 0 0 0
1] d d 1 d
2 2N N V N N g
m r r r r r r r
E
220 0 0
1( )
2p V N g
m
r r r r-1
eliminates self-interaction
BACK
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57
Variational estimate of the condensate properties
ADDITIONAL NOTES
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
The first result is that is not the average energy per particle:
2 3 40 0 0 0 0 0
2 3 40 0 0 0 0from
1 1/ ]/ 1 d
2 21
the GPE 1 d2
N N N N N N N N
N N N N N N
E N N p V N gm
p V N gm
r
r
E
2 3 4
220 0 0 0
1 1] 1 d
2 2
1], ( )
2
N
N N N N N N
N p V N gm
E p V N gm
-1
r
r r r r
E
E
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58
Variational estimate of the condensate properties
ADDITIONAL NOTES
Compare now systems with N and N -1 particles:
0 1 0 1 0 1, 1] ] ]N N N N N N N NNN NE E E E E
N … energy to remove a particle without relaxation of the condensate
use of the variational principle for GPE
In the "thermodynamic"asymptotics of large N, the inequality tends to equality. This only makes sense, and can be proved, for g > 0.
Reminescent of the Koopmans’ theorem in the HF theory of atoms. Derivation:
1 12 3 42 2
1 12 3 41 2 2
1 12 3 41 2 2
0
] 1 d
] 1 1 1 2 d
1 1 2 d
o f r
N m
N m
N N m
N N
N p N V N N g
N p N V N N g
p V N N N N g
r
r
r
E
E
E E
BACK
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2 2 2 2 2 20 0
1 12 2V m r m x y z r
Variational estimate of the condensate properties
ADDITIONAL NOTES
SCALING ANSATZ FOR A SPHERICAL PARABOLIC TRAP
The potential energy has the form
Without interactions, the GPE reduces to the SE for isotropic oscillator
The solution (for the ground state orbital) is
We (have used and) will need two integrals:
2 2 20 0 0 0
312 2
1
2p m r
m
r r
2
20
2 1/ 43 200 0 0 0 0 02
0 0
12
e , ,
r
aA a A am ma
r
2 2
2 2 2 31 2
12d e , d e
u u
I u I u u
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60
The condensate orbital will be taken in the form
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
The solution (for) is
2
2 1/ 43 20
12e ,
r
bA A b
r
Variational estimate of the condensate properties
ADDITIONAL NOTES
SCALING ANSATZ
2 2 2
2 2 2
22 2 43 3 3
0 0 0 0
2 26 3 2 3 2 6 30 0
0 4 2 20
2
1] d d 1 d
2 2
1 1d e d e 1 d e
2
r r r
b b b
N N V N N gm
a maNA r r N A g
b a
r r r r r r r
r r r
E
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Variational estimate of the condensate properties
ADDITIONAL NOTES
22
2 2 200 0 0 2
10
41 1, , , saa m A g
m I b ma b
0
32 2 2 2 112 0 0
0 2 4 2 3 3 / 2 2 301 1
]
/ 23 41 11
22sI bI bI b a ma a
N Nmb a bb I b I b
E
2 320 0
0 0 02 2 30 0
22 3
0
13]
4 2
3 1 1
4
sNa a abN N E
b a a b
bE
a
E
dimension-less energy per particle
dimension-less orbital size
For an explicit evaluation, we (have used and) will use the identities:
The integrals, by the Fubini theorem, are a product of three:
Finally,
This expression is plotted in the figures in the main lecture.
BACK
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The end