Transcript PPT
Statistical methods for bosons
Lecture 2.
9th January, 2012
Short version of the lecture plan: New version
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
Thomas-Fermi approximation
Dec 19
Jan 9
Infinite systems: Bogolyubov-de Gennes theory,
BEC and symmetry breaking, coherent states
Time-dependent GPE. Cloud spreading and
interference. Linearized TDGPE, Bogolyubov-de
Gennes equations, physical meaning, link to
Bogolyubov method, parallel with the RPA
2
Reminder of
Lecture I.
Offering many new details and
alternative angles of view
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
n
e ( ) 1
1
e 1
T 0
T 0
T 0
Aufbau
principle
BEC?
freezing out
1 ,1 ,
,1 , 0 ,
B
N , 0 ,0 ,
,0 ,
vac
4
Trap potential
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
5
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 6
BEC observed by TOF in the velocity distribution
Qualitative features: all Gaussians
wide vs.narrow
isotropic vs. anisotropic
7
BUT
The non-interacting model is at most
qualitative
Interactions need to be accounted for
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
9
Today:
BEC for interacting bosons
Mean-field theory
for zero temperature condensates
Interatomic interactions
Equilibrium: Gross-Pitaevskii equation
Dynamics of condensates: TDGPE
Inter-atomic interaction
Interaction between neutral atoms is
weak, basically van der Waals. Even that
would be too strong for us, but at very low
collision energies, the efective interaction
potential is much weaker.
Are the interactions important?
Weak interactions In the dilute gaseous atomic clouds in the traps, the
interactions are incomparably weaker than in liquid helium.
Perturbative treatment 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
• thermalization the atomic collisions take care of thermalization
• mean field 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
12
Fortunate properties of the interactions
The atomic interactions in the dilute gas favor formation of
long lived quasi-equilibrium clouds with condensate
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 almost elastic and spin independent:
they only weakly 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.
13
I.
Interacting atoms
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.
15
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
16
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
17
Experimental data
as
18
Experimental data
for “ordinary” gases
(K)
nm
5180
940
----73
---
3.4
4.7
6.8
8.7
8.7
10.4
1 a.u. length = 1 bohr
0.053 nm
1 a.u. energy = 1 hartree 3.16 10 05 K
19
Scattering length, pseudopotential
Beyond the potential radius, say 3 6 , the scattered wave
propagates in free space
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
20
Experimental data
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
"well behaved”; decrease increase
monotonous
seemingly erratic, very interesting physics of
Feshbach scattering resonances behind
21
II.
The many-body Hamiltonian
for interacting atoms
Many-body Hamiltonian
1 2
1
ˆ
H
pa V (ra )
2
a 2m
U (ra rb )
a b
Many-body Hamiltonian
1 2
1
ˆ
H
pa V (ra )
2
a 2m
U (ra rb )
a b
At low energies (micro-kelvin range), true interaction potential
replaced by an effective potential, Fermi pseudopotential
U (r ) g (r )
4 as
g
m
2
,
as ... the scattering length
Many-body Hamiltonian
1 2
1
ˆ
H
pa V (ra )
2
a 2m
U (ra rb )
a b
At low energies (micro-kelvin range), true interaction potential
replaced by an effective potential, Fermi pseudopotential
U (r ) g (r )
4 as
g
m
2
,
as ... the scattering length
Experimental data
for “ordinary” gases
VLT clouds
as
Many-body Hamiltonian
1 2
1
ˆ
H
pa V (ra )
2
a 2m
U (ra rb )
a b
At low energies (micro-kelvin range), true interaction potential
replaced by an effective potential, Fermi pseudopotential
U (r ) g (r )
4 as
g
m
2
,
as ... the scattering length
Experimental data
for “ordinary” gases
VLT clouds
as
Many-body Hamiltonian
1 2
1
ˆ
H
pa V (ra )
2
a 2m
U (ra rb )
a b
At low energies (micro-kelvin range), true interaction potential
replaced by an effective potential, Fermi pseudopotential
U (r ) g (r )
4 as
g
m
2
,
as ... the scattering length
Experimental data
for “ordinary” gases
VLT clouds
as
NOTES
weak attraction ok
weak repulsion ok
weak attraction
intermediate attraction
weak repulsion ok
strong resonant repulsion
Many-body Hamiltonian: summary
1 2
1
ˆ
H
pa V (ra )
2
a 2m
U (ra rb )
a b
At low energies (micro-kelvin range), true interaction potential
replaced by an effective potential, Fermi pseudopotential
U (r ) g (r )
4 as
g
m
2
,
as ... the scattering length
Experimental data
for “ordinary” gases
VLT clouds
as
NOTES
weak attraction ok
weak repulsion ok
weak attraction
intermediate attraction
weak repulsion ok
strong resonant repulsion
III.
Mean-field treatment of interacting atoms
In the mean-field approximation, the
interacting particles are replaced by
independent particles moving in an
effective potential they create
themselves
Many-body Hamiltonian and the Hartree approximation
1 2
1
Hˆ
pa V ( ra )
2
a 2m
U (ra rb )
a b
We start from the mean
1 field approximation.
Hˆ GP
pa2 V ( ra ) VH ( ra ) const.
This is an educated way,
similar to (almost identical with) the
a 2m
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
30
Many-body Hamiltonian and the Hartree approximation
1 2
1
Hˆ
pa V ( ra )
2
a 2m
HARTREE APPROXIMATION
1 2
Hˆ GP
U (ra rb )
a b
~ many electron systems
pa V ( ra ) VH ( ra ) const.
mean field approximation
a 2m
Most of the interactions absorbed into the mean field
explicit quantum correlation corrections remain
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
31
Many-body Hamiltonian and the Hartree approximation
1 2
1
Hˆ
pa V ( ra )
2
a 2m
HARTREE APPROXIMATION
1 2
Hˆ GP
U (ra rb )
a b
~ many electron systems
pa V ( ra ) VH ( ra ) const.
mean field approximation
a 2m
Most of the interactions absorbed into the mean field
explicit quantum correlation corrections remain
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
?
32
Many-body Hamiltonian and the Hartree approximation
1 2
1
Hˆ
pa V ( ra )
2
a 2m
HARTREE APPROXIMATION
1 2
Hˆ GP
U (ra rb )
a b
~ many electron systems
pa V ( ra ) VH ( ra ) const.
mean field approximation
a 2m
Most of the interactions absorbed into the mean field
explicit quantum correlation corrections remain
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
1 2
p
V
(
r
)
V
(
r
)
H
r E r
2m
33
Many-body Hamiltonian and the Hartree approximation
1 2
1
Hˆ
pa V ( ra )
2
a 2m
HARTREE APPROXIMATION
1 2
Hˆ GP
U (ra rb )
a b
~ many electron systems
pa V ( ra ) VH ( ra ) const.
mean field approximation
a 2m
Most of the interactions absorbed into the mean field
explicit quantum correlation corrections remain
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
34
ADDITIONAL NOTES
On the way to the mean-field Hamiltonian
35
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
36
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 '
37
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
• The same can be done for a time dependent Hamiltonian
38
Many-body Hamiltonian and the Hartree approximation
HARTREE APPROXIMATION
1 2
Hˆ GP
pa V ( ra ) VH ( ra ) const.
a 2m
1 2 neglected
particle-particle
correlations
Hˆ GPpair
pa V (ra ) VH (ra )
the only relevant quantity
. . . single-particle density
a 2m
VH (ra ) drbU ( ra rb )n( rb ) g n(ra )
n(r ) n r
2
self-consistent
system
only occupied orbitals enter the cycle
1 2
p
V
(
r
)
V
(
r
)
H
r E r
2m
39
Hartree approximation for bosons 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 E00 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.
40
Hartree approximation for bosons 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 E00 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.
41
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 E00 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.
42
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 E00 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.
43
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 E00 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.
44
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 E00 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.
45
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 E00 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.
46
Gross-Pitaevskii equation – "Bohmian" form
2
n( r ) N | 0 ( r ) |2 ( r )
N [n ] N d 3 r ( r ) d 3 r n( r ) N
2
( r ) N 0 ( r ) n( r ) ei
For a static condensate, the order parameter has CONSTANT PHASE.
Then 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
47
Gross-Pitaevskii equation – variational interpretation
This equation results from a variational treatment of the
Energy Functional
E[ n ] d 3 r
2
(
n( r ) ) 2 d 3 r V ( r ) n( r ) d 3 r
2m
It is required
that E QP
E[npressure
] min
quantum
E POT
+
+
external potential
1
2
g n 2 (r )
E INT
interaction
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).
48
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).
49
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).
50
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]
51
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]
chemical potential
by definition
in thermodynamic limit
52
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
E[ n ]
!!!
N
N [ n]
53
IV.
Interacting atoms in a homogeneous gas
Gross-Pitaevskii equation – homogeneous gas
The GP equation simplifies
2
2
g r r r
2m
For periodic boundary conditions in a box with V Lx Ly Lz
1
0 ( r )
V
( r ) N 0 ( r )
N
n
V
g r r r
2
... GP equation
g r gn
2
1
E
1 3 2
1
2
2
d r
( n ) V (r )n g n g n
N
N
2
2m
2
55
The simplest case of all: a homogeneous gas
total energy
E 12 gn 2 V
internal pressure
E 1 2
P
2 gn
V
2
N
E 12 g
V
56
The simplest case of all: a homogeneous gas
total energy
E 12 gn 2 V
internal pressure
E 1 2
P
2 gn
V
2
N
E 12 g
V
57
The simplest case of all: a homogeneous gas
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.
58
The simplest case of all: a homogeneous gas
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.
59
V.
Condensate in a container with hard walls
A container with hard walls
A semi-infinite system … exact solution of the GP equation
n
2
2m
n( r )
n( r )
V ( r ) g n( r )
n( r )
61
A container with hard walls
A semi-infinite system … exact analytical solution of the GP equation
n
2
2m
n( r )
n( r )
V ( r ) g n( r )
n( r )
Particles evenly
distributed
Density high
Density gradient
zero
Internal pressure
due to interactions
prevails
62
A container with hard walls
A semi-infinite system … exact solution of the GP equation
Particles pushed
away from the wall
Density low
Density gradient
large
Quantum pressure
prevails
n
2
2m
n( r )
n( r )
V ( r ) g n( r )
n( r )
Particles evenly
distributed
Density high
Density gradient
zero
Internal pressure
due to interactions
prevails
63
A container with hard walls
A semi-infinite system … exact solution of the GP equation
Particles pushed
away from the wall
Density low
Density gradient
large
Quantum pressure
prevails
2
2m
n( r )
n( r )
V ( r ) g n( r )
healing length
4 2 as
2
n
2m
m
2
n
1
8 as n
12
n( r )
Particles evenly
distributed
Density high
Density gradient
zero
Internal pressure
due to interactions
prevails
64
A container with hard walls
A semi-infinite system … exact solution of the GP equation
Particles pushed
away from the wall
Density low
Density gradient
large
Quantum pressure
prevails
2
2m
n( r )
n( r )
V ( r ) g n( r )
healing length
4 2 as
2
n
2m
m
2
n
1
8 as n
12
n( r )
Particles evenly
distributed
Density high
Density gradient
zero
Internal pressure
due to interactions
prevails
n(r ) n tanh x / 2
65
A container with hard walls
no interactions
Interactions gradually flatten out the density in the bulk
Near the walls there is a depletion region about
wide
66
VI.
Interacting atoms in a parabolic trap
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
68
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
69
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
70
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 a03
Na02
Nas
a0
71
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" acts against
the interaction so as to form the condensate.
Still, it shrinks and becomes metastable.
Onset of three particle recombination processes
Quantitative
The decisive parameter for the "importance" of
interactions is
EINT
EKIN
gNn
N
N 2 as a03
Na02
Nas
a0
72
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" acts against
the interaction so as to form the condensate.
Still, it shrinks and becomes metastable.
Onset of three particle recombination processes
unlike in an extended
homogeneous gas !!
Quantitative
The decisive parameter for the "importance" of
interactions is
EINT
EKIN
gNn
N
N 2 as a03
Na02
Nas
a0
73
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" acts against
the interaction so as to form the condensate.
Still, it shrinks and becomes metastable.
Onset of three particle recombination processes
Quantitative
The decisive parameter for the "importance" of
interactions is
E INT
E KIN
Ngn
N
N 2as a03 Nas
2
4
Na0
a0
74
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" acts against
the interaction so as to form the condensate.
Still, it shrinks and becomes metastable.
Onset of 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 a03 Nas
2
E KIN N
4
Na0
a0
2
ma02
75
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" acts against
the interaction so as to form the condensate.
Still, it shrinks and becomes metastable.
Onset of three particle recombination processes
Quantitative
The decisive parameter for the "importance" of
interactions is
E INT
E KIN
Ngn
N
N 2as a03 Nas
2
4
Na0
a0
76
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" acts against
the interaction so as to form the condensate.
Still, it shrinks and becomes metastable.
Onset of 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 a03 Nas
2
4
Na0
a0
1
weak individual
collisions
77
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" acts against
the interaction so as to form the condensate.
Still, it shrinks and becomes metastable.
Onset of 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 a03 Nas
2
4
Na0
a0
0
realistic
value
collective effect
weak or strong
depending on N
1
weak individual
collisions
78
VIII.
Variational approach
to the condensate ground state
Idea:
Use the variational estimate
of the condensate properties
instead of solving the GP equation
General introduction to the variational method
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 E1ˆ 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 of non-interacting
particles (Hartree Ansatz, here identical with the symmetrized Hartree-Fock)
r1 , r2 ,
, rp ,
, rN 0 r1 0 r2 0 rp
0 rN
85
General introduction to the variational method
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 the total Hartree energy of the condensate
E 0 ]
2
N d r 0 r N d r V r 0 r
2
3
2m
3
2
4
1
3
N N 1 g d r 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
86
Restricted search as upper estimate to Hartree energy
Start from the Hartree energy functional
E 0 ]
2
N d r 0 r N d r V r 0 r
2
3
2m
3
2
4
1
3
N N 1 g d r 0 r
2
Evaluate for a suitably chosen orbital having the properties
normalization 0 | 0 1
variational parameters 0 (r ) f (r, 1, 2 ,
p )
Minimize the resulting energy function with respect to the parameters
E 0 ] E f ( 1, 2 ,
n
E ( 1, 2 ,
p ) 0
p )] E ( 1, 2 , p )
n 1, 2 ,
,p
This yields an optimized approximate Hartree condensate
87
Application to the condensate
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
ground state orbital
89
Rescaling the ground state for non-interacting gas
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.
90
Rescaling the ground state for non-interacting gas
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.
91
Scaling approximation for the total energy
Introduce the variational ansatz into the energy functional
1 r2
2
2b
3
0 r A e
,
A b
2
1/ 4
92
Scaling approximation for the total energy
Introduce the variational ansatz into the energy functional
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
93
Scaling approximation for the total energy
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
94
Minimization of the scaled energy
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
95
Importance of the interaction: scaling approximation
g 0
g 0
0
0
Variational estimate of the total energy of the
condensate as a function of the parameter
N 1 as
2 a0
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
in agreement with TFA
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 96
attractive atom-atom interaction
in units of a0
Now we will be interested in
dynamics of the BE condensates
described by
Time dependent Gross-Pitaevskii equation
Time dependent Gross-Pitaevskii equation
Condensate dynamics is often described to a sufficient precision by the
mean field approximation.
It leads to the Time dependent Gross-Pitaevskii equation
Intuitively,
i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) (r , t ) (r , t )
2
Today, I proceed following the conservative approach using the mean
field decoupling as above in the stationary case
Three steps:
1. Mean field approximation by decoupling
2. Assumption of the condensate macrosopic wave function
3. Transformation of the TD Hartree equation to TDGPE
101
1. Mean-field Hamiltonian by decoupling -- exact part
1 2
1
Hˆ
pa V (t , ra )
2
a 2m
a
U (ra rb )
a b
1
Hˆ Wˆ d 3r V (t , r ) nˆ ( r ) d 3 r d 3 r 'U r r ' nˆ ( r ) nˆ ( r' ) r r '
2
t n(t , r ) nˆ ( r ) t t nˆ ( r ) t
Hˆ Wˆ d 3r V (t , r ) d 3r' U r r ' n(t , r' ) nˆ ( r )
1
2
1
2
3
3
d
r
d
r 'U r r ' n(t , r )n(t , r' )
3
3
d
r
d
r 'U r r ' nˆ (t , r ) nˆ (t , r' ) nˆ ( r ) r r '
102
1. Mean-field Hamiltonian by decoupling -- approximation
Hˆ Wˆ d 3r V (t , r ) d 3r' U r r ' n(t , r' ) nˆ ( r )
1
2
1
2
3
3
d
r
d
r 'U r r ' n(t , r )n(t , r' )
3
3
d
r
d
r 'U r r ' nˆ (t , r ) nˆ (t , r' ) nˆ ( r ) r r '
Hˆ Wˆ d 3r V (t , r ) d 3r' U r r ' n(t , r' ) nˆ ( r )
1
1
2
2
3
3
VH t , r
d
r
d
r
'
U
r
r
'
n
(
t
,
r
)
n
(
t
,
r'
)
3
3
d
r
d
r 'U r r ' nˆ ( r ) nˆ ( r' ) nˆ ( r ) r r '
1 2
Hˆ GP
pa V (t , ra ) VH (t , ra ) C (t )
a 2m
103
2. Condensate assumption & 3. order parameter
Assume the Hartree condensate
N
(t ) ( ra , t )
a 1
Then
n( r, t ) N | ( r, t ) |2
and a single Hartree equation for a single orbital follows
i
r, t 1
t 0 2m
2
p V ( r, t ) gN 0 r, t 0 r, t
2
define the order parameter
r, t N ½0 r, t
The last equation becomes the desired TDGPE:
i
t
2
1 2
r, t p V ( r, t ) g r, t r, t
2m
Physical properties of the TDGPE
i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) (r , t ) (r , t )
1.
2
In the MFA, the present coincides with the order parameter
defined from the one particle density matrix and (at zero temperature)
all particles are in the condensate
( r, r; t ) ( r, t ) * ( r , t )
2.
TDGPE is a non-linear equation, hence there is no simple
superposition principle
3.
It has the form of a self-consistent Schrödinger equation: the same
type of initial conditions, stationary states, isometric evolution
4.
The order parameter now typically depends on both position and time
and is complex. It is convenient to define the phase by
(r , t ) n0 (r , t ) ei ( r ,t )
5.
The MFA order parameter (macroscopic wave function) satisfies the
continuity equation
105
Physical properties of the TDGPE
time-independent
Stationary state of the condensate for V r , t
Make the ansatz
(r , t ) ( r ) e i t /
Then satisfies the usual stationary GPE
2
2m
V r g r r r
2
This is a real-valued differential equation and has a
real solution
n0 r
All other solutions are trivially obtained as
(r ) n0 r ei
106
Physical properties of the TDGPE
Continuity equation
The TDGPE and its conjugate:
(r , t ) 2m V (r , t)
i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) (r , t ) ( r , t )
2
i t *
2
(r , t ) g (r , t ) *(r , t ) *(r , t )
*
Just like for the usual Schrödinger equation, we get
t n0 j0 0
2
n0 (r, t ) (r, t )
j0 (r, t ) 2mi * *
( r, t )
Only complex fields are able to carry a finite current (quite general)
i ( r ,t )
Employ the form (r , t ) n0 (r , t ) e
The current density becomes
j0 (r, t ) n0 (r, t ) m (r, t ) n0 (r, t ) v0 (r, t )
107
Physical properties of the TDGPE
Continuity equation
The TDGPE and its conjugate:
(r , t ) 2m V (r , t)
i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) (r , t ) ( r , t )
2
i t *
2
(r , t ) g (r , t ) *(r , t ) *(r , t )
*
Just like for the usual Schrödinger equation, we get
t n0 j0 0
2
n0 (r, t ) (r, t )
j0 (r, t ) 2mi * *
( r, t )
Only complex fields are able to carry a finite current (quite general)
i ( r ,t )
Employ the form (r , t ) n0 (r , t ) e
phase
r, t dependent
The current density becomes
j0 (r, t ) n0 (r, t ) m (r, t ) n0 (r, t ) v s (r, t )
108
Physical properties of the TDGPE
Continuity equation
The TDGPE and its conjugate:
(r , t ) 2m V (r , t)
i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) (r , t ) ( r , t )
2
i t *
2
(r , t ) g (r , t ) *(r , t ) *(r , t )
*
Just like for the usual Schrödinger equation, we get
t n0 j0 0
2
n0 (r, t ) (r, t )
j0 (r, t ) 2mi * *
( r, t )
Only complex fields are able to carry a finite current (quite general)
i ( r ,t )
Employ the form (r , t ) n0 (r , t ) e
superfluid
phase
velocity field
gradient
The current density becomes
j0 (r, t ) n0 (r, t ) m (r, t ) n0 (r, t ) v s (r, t )
109
Physical properties of the TDGPE
Continuity equation
The TDGPE and its conjugate:
(r , t ) 2m V (r , t)
i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) (r , t ) ( r , t )
2
i t *
2
(r , t ) g (r , t ) *(r , t ) *(r , t )
*
Just like for the usual Schrödinger equation, we get
t n0 j0 0
2
n0 (r, t ) (r, t )
j0 (r, t ) 2mi * *
( r, t )
Only complex fields are able to carry a finite current (quite general)
i ( r ,t )
Employ the form (r , t ) n0 (r , t ) e
superfluid
phase
velocity field
gradient
The current density becomes
j0 (r, t ) n0 (r, t ) m (r, t ) n0 (r, t ) v s (r, t )
The superfluid flow is irrotational (potential):
rot vs 0
But only in simply connected regions – vortices& quantized circulation!!!
110
Physical properties of the TDGPE
Isometric evolution
Three formulations
The norm of the solution of TDGPE is conserved:
t d r (r, t ) 0
2
t n0 j0 0
2
S
t d r (r, t ) d S j0 (rS , t )
0
This follows from the continuity eq.
The condensate particle number is conserved (condensate persistent)
N0 d r n0 (r, t ) const.
Finally, isometric evolution
2
t
d r (r, t ) const.
2
I avoid saying "unitary", because the evolution is non-linear
111
A glimpse at applications
1.
Spreading of a cloud released from the trap
2.
Interference of two interpenetrating clouds
1. 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
113
1. 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
114
1. 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
115
1. 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
116
1. Example of calculation
Even without interactions and at zero
temperature, the cloud would spread
quantum-mechanically
The result would then be identical with a
ballistic expansion, the width increasing
linearly in time
The repulsive interaction acts strongly at
the beginning and the atoms get "ahead
of time". Later, the expansion is linear
again
From the measurements combined with
the simulations based on the
hydrodynamic version of the TDGPE, it
was possible to infer the scattering
length with a reasonable outcome
Castin&Dum,
PRL 77, 5315 (1996)
To an outline of a simplified treatment
117
2. Interference of BE condensate in atomic clouds
Sodium atoms form a macroscopic wave function
Experimental proof:
Two parts of an atomic cloud separated and then interpenetratin again
interfere
Wavelength on the order of hundreth of millimetre
experiment in the Ketterle group.
Waves on water
118
Basic idea of the experiment
1. Create a cloud in the trap
2.Cut it into two pieces by laser light
3. Turn off the trap and the laser
4. The two clouds spread,
interpenetrate and interfere
119
2. Experimental example of interference of atoms
Two coherent condensates are interpenetrating and interfering.
Vertical stripe width …. 15 m
Horizontal extension of the cloud …. 1,5mm
120
2. Calculation for this famous example
It is an easy exercise to calculate
the interference of two spherical
non-interacting clouds
Surprisingly perhaps, interference
fringes are formed even taking into
account the non-linearities – weak,
to be sure
To account for the interference,
either the full Euler eq. including
the Bohm quantum potential – OR
TD GPE, which is non-linear: only
a numerical solution is possible (no
superposition principle)
Theoretical contrast of almost 100
percent seems to be corroborated
by the experiments, which appear
as blurred mostly by the external
optical transfer function
A.Röhrl & al.
PRL 78, 4143 (1997)
121
2. Calculation for this famous example
It is an easy exercise to calculate
the interference of two spherical
non-interacting clouds
Idealized
theory
Surprisingly perhaps, interference
fringes are formed even taking into
account the non-linearities – weak,
to be sure
Experim.
profile
Realistic
theory
To account for the interference,
either the full Euler eq. including
the Bohm quantum potential – OR
TD GPE, which is non-linear: only
a numerical solution is possible (no
superposition principle)
Theoretical contrast of almost 100
percent seems to be corroborated
by the experiments, which appear
as blurred mostly by the external
optical transfer function
A.Röhrl & al.
PRL 78, 4143 (1997)
122
Linearized TDGPE
Small linear oscillations around equilibrium
i t (r , t ) 2m V (r) (r , t ) g * (r , t ) (r , t ) (r , t )
2
I. Stationary state of the condensate
0 ( r , t ) ( r ) e i t /
Stationary GPE
2
2m
V (r ) g r r r
2
All solutions are obtained from the basic one as
(r ) n0 r ei
I
124
Small linear oscillations around equilibrium
i t (r , t ) 2m V (r) (r , t ) g * (r , t ) (r , t ) (r , t )
2
I. Stationary state of the condensate
0 ( r , t ) ( r ) e i t /
Stationary GPE
2
2m
V (r ) g r r r
2
All solutions are obtained from the basic one as
(r ) n0 r ei
II. Small oscillation about the stationary condensate
(r , t ) 0 (r , t ) (r , t )
Substitute into the TDGPE and linearize
(autonomous oscillations, not a linear response)
125
Small linear oscillations around equilibrium
III. Linearized TD GPE for the small oscillation
i t (r , t ) 2 m V (r) (r , t )
2
2
2 g 0 (r , t ) (r , t ) g 0 ( r , t ) 2 * ( r , t )
where
0 (r , t ) ei t / n0 r ei
DROP
THIS
126
Small linear oscillations around equilibrium
III. Linearized TD GPE for the small oscillation
i t (r , t ) 2 m V (r) (r , t )
2
2
2 g 0 (r , t ) (r , t ) g 0 ( r , t ) 2 * ( r , t )
where
0 (r , t ) ei t / n0 r ei
DROP
THIS
127
Small linear oscillations around equilibrium
III. Linearized TD GPE for the small oscillation
i t (r , t ) 2 m V (r) (r , t )
2
2
2 g 0 (r , t ) (r , t ) g 0 ( r , t ) 2 * ( r , t )
where
0 (r , t ) ei t / n0 r ei
DROP
THIS
IV. Ansatz for the solution
(r , t ) e
i t /
u (r )e
i t
v (r )e
i t
suggestive of Bogolyubov, excit. energy with respect to
Substitute into the linearized TDGPE and its conjugate
to obtain coupled diff. eqs. for u (r ), v (r )
(which are just Bogolyubov – de Gennes equations)
128
Small linear oscillations around equilibrium
V. Bogolyubov – de Gennes equations
2
2m
v r g r u r v r
V (r ) 2 g r u r g r v r u r
2
V (r ) 2 g r
2m
2
(r , t ) e
2
i t /
2 *
*
u(r )e
i t
2
v (r )e
*
i t
precisely identical with the eqs. for the Bogolyubov transformation
generalized to the inhomogeneous case (a large yet finite system with
bound states).
VI. Bogolyubov excitations of a homogeneous condensate
For a large box, go over to the k- representation to recover the famous
Bogolyubov solution for the homogeneous gas
k ( r, t ) e
i t /
uk e i kr e
i t
v k e i kr e
i t
129
Excitation spectrum
DISPERSION LAW
( k ) ( k )
2
asymptotically
merge
2
(k )
2m
k
2
2
k gn
GP
2
(k ) c k
k
c
k
2
high energy region
2
2m
2
2
2
2
k
gn
gn
k
k
2 gn
2m
2m
2m
2
gn
m
quasi-particles are
nearly just GP particles
sound region
quasi-particles are
collective excitations
cross-over
k
4 mgn
2
2
defines scale for k in terms of healing length
2
2 2 m2
1
2
gn
More about the sound part of the dispersion law
Entirely dependent on the interactions, both the magnitude of the velocity
and the linear frequency range determined by g
Can be shown to really be a sound:
V VV E
c "
"
,
mn
BULK MODULUS
1: V1
(k ) c k
c
gN 2
E
2V
p
V
,
gn
m
p VE
Even a weakly interacting gas exhibits superfluidity; the ideal gas does not.
The phonons are actually Goldstone modes corresponding to a broken
symmetry
The dispersion law has no roton region, contrary to the reality in 4He
The dispersion law bends upwards quasi-particles are unstable, can
decay
Small linear oscillations around equilibrium
VII. Interpretation of the coincidence with Bogolyubov theory
Except for Leggett in review of 2001, nobody seems to be surprised.
Leggett means that it is a kind of analogy which sometimes happens.
A PARALLEL
Electrons (Fermions)
Atoms (Bosons)
Jellium
Homogeneous gas
Hartree
Stationary GPE
Linearized TD Hartree
Linearized TD GPE
equivalent with RPA
equivalent with Bogol.
plasmons
Bogolons (sound)
ne nJ , VH 0
n0 , VH gn0
132
The end
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 1023
8.63 10 05
2.08 1010
3.17 10 06
1eV
1.16 1004
1.60 1019
2.41 1014
3.67 10 02
s-1
4.80 1011
4.14 1015
6.63 1034
1.52 1016
a.u.
3.16 10 05
2.72 10 01
6.56 10 15
5.88 1021
134