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Zemsky Danny
1
Outline of Talk
1) History
2) Criterion for Bose-Einstein condensation
3) How is BEC looks like?
4) Quantum description of BEC
5) GROSS- PITAEVSKII EQUATION
The Mean-Field Approximation and General Solution
6) The realization of BEC - Cooling Techniques
7) Vortices
8) Interference of two condensates.
9) Feshbach resonance.
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1. Introduction
•In the quantum level, there are profound differences
between fermions (follows Fermi-Dirac statistic) and bosons
(follows Bose-Einstein statistic).
•As a gas of bosonic atoms is cooled very close to absolute
zero temperature, their characteristic will change
dramatically.
•More accurately when its temperature below a critical
temperature Tc, a large fraction of the atoms condenses in
the lowest quantum states .
•This dramatic phenomenon is known as Bose-Einstein
Condensation (BEC).
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Bose and
Einstein
• In 1924 an Indian physicist named Bose derived the Planck law
for black-body radiation by treating the photons as a gas of
identical particles.
•Einstein generalized Bose's theory to an ideal gas of identical
atoms or molecules for which the number of particles is
conserved.
•The equations, which were derived by Einstein didn't predict
the behavior of the atoms to be any different from previous
theories, except at very low temperatures.
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•Einstein found that when the temperature is high, they
behave like ordinary gases.
•However, at very low temperatures Einstein's theory
predicted that a significant proportion of the atom in the
gas would collapse into their lowest energy level.
•This is called Bose-Einstein condensation.
•The BEC is essentially a new state of matter where it
is no longer possible to distinguish between the atoms.
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Criterion for Bose-Einstein
condensation
1. Ideal Bose gas
The Pauli principle does not apply in this case, and the lowtemperature properties of such a gas are very different
from those of a fermion gas.
The properties of BE gas follow from Bose-Einstein
distribution.
nk 
1
e  (  k  )  1
,  nk  N
k
,  1
k BT
Here T represents the temperature, kb Boltzmann constant
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and  the chemical potential.
In the Bose-Einstein distribution, the number of particles
in the energy range dE is given by n(E)dE, where
g(E)
n(E)  1 E / k BT
z e
1
z is the fugacity, defined by
ze
 / k BT
where μ is the chemical potential of the gas, and the density
of states g(E) (which gives the number of states between E
and E+dE) is given (in three dimensions) for volume V by
 2m 
3/ 2
g(E) 
4
2 3
V
E
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The critical (or transition) temperature Tc is defined as the
highest temperature at which there exists macroscopic
occupation of the ground state.
The number of particles in excited states can be calculated
by integrating n(E)d(E):

N   n(E)dE  
e
0
 2m 
4
3/ 2
2 3
V

EdE
z 1e E / k BT  1
Ne is maximal when z=1 (and thus μ=0), and for a condensate
to exist we require the number of particles in the excited
state to be smaller than the total number of particles N.
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Therefore

Ne  
0
 2m 
4
3/ 2
2 3
V

1 E / k BT
z e
 2m 

3/ 2
EdE
1
4
2 3
V
  k BT 

3/ 2

0
xdx  2mk BT  V  3   3 

    N
x
2 3
e 1
4
2 2
3/ 2
where
3 3
       2.314
2 2
 4 N 
T


2mk B  2.315V 
2
2
2/3
 Tc
Below this temperature most of the atoms will be part of the
BEC.
For example, sodium has a critical temperature of about 2μK.
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Transition temperature
The number of excited particles at temperatures below the
critical temperature can be rewritten as
T
N  N 
 Tc 
3/ 2
e
The number of particles at the ground state (and therefore in
the condensate) N0 is given by
  T 3/ 2 
N 0  N  N e  N 1    
  Tc  


In fact, the condensate fraction, i.e. how many of the
particles are in the BEC, is represented mathematically as,
T
N0
 1  
N
 TC



3
2
where N0 is the number of atoms in the ground state.
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The system undergoes a phase transition and forms a
Bose-Einstein condensate, where a macroscopic number
of particles occupy the lowest-energy quantum state.
The temperature and the density of particles n at the
phase transition are related as n 3dB = 2.612.
3
 3  mkB  2 3 2
N ( T )  N 0  V  
 T
 2  2 
BEC is a phase-transition solely caused by quantum
statistics, in contrast to other phase-transitions (like
melting or crystallization) which depend on the interparticle interactions.
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The fraction of population of
atoms in different state
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2. Matter Waves and Atoms
Bose-Einstein condensation is based on the wave nature of
particles.
De Broglie proposed that all matter is composed of waves.
Their wavelengths are given by
dB = de Broglie wavelength
 = Planck’s constant
m = mass
T = temperature
dB 
2 2
mkBT
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Matter Waves and Atoms
BEC also can be explained as follows, as the atoms are cooled
to these very low temperatures their de Broglie wavelengths
get very large compared to the atomic separation.
Hence, the atoms can no longer be thought of as particles
but rather must be treated as waves.
At everyday temperatures, the de Broglie wavelength is so
small, that we do not see any wave properties of matter, and
the particle description of the atom works just fine.
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Matter Waves and Atoms
At high temperatures, a weakly interacting gas can be
treated as a system of “billiard balls”.
At high temperature, dB is small, and it is very improbable to find two
particles within this distance.
In a simplified quantum description, the atoms can be regarded as
wavepackets with an extension x, approximately given by
Heisenberg’s uncertainty relation x= h/p, where p denotes the width
of the thermal momentum distribution.
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When the gas is cooled down the de Broglie wavelength
increases.
At the BEC transition temperature, dB becomes comparable
to the distance between atoms, the wavelengths of
neighboring atoms are beginning to overlap and the Bose
condensates forms which is characterized by a
macroscopic population of the ground state of the system.
As the temperature approaches absolute zero, the
thermal cloud disappears leaving a pure Bose condensate.
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Phase Diagram
The green line is a phase
boundary. The exact
location of that green
line can move around a
little, but it will be
present for just about
any substance.
Underneath the green line there is a huge area that we
cannot get to in conditions of thermal equilibrium.
It is called the forbidden region.
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Finally, if the atomic gas is cooled enough, what results is
a kind of fuzzy blob where the atoms have the same wave
function.
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Fermions and Bosons
 Not
all particles can have BEC. This is related to the spin of
the particles.
 Single protons, neutrons and electrons have a spin of ½.
 They cannot appear in the same quantum state. BEC cannot
take place.
 Some atoms contain an even number of fermions. They have a
total spin of whole number. They are called bosons.
 Example: A 23Na atom has 11 protons, 12 neutrons and 11
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electrons.
Ground state properties of dilutegas Bose–Einstein condensates
•
•
•
•
•
•
•
•
Binary collision model
Mean-field theory
Gross-Pitaevskii equation
Thomas-Fermi approximation
Vortex states and vortex dynamics
Feshbach resonance
Atom Laser
Interference
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Binary collision model
•At very low temperature the de Broglie wavelengths of the atoms are
very large compared to the range of the interatomic potential.
This, together with the fact that the density and energy of the atoms
are so low that they rarely approach each other very closely, means
that atom–atom interactions are effectively weak and dominated by
(elastic) s-wave scattering .
•The s-wave scattering length”a” the sign of which depends sensitively
on the precise details of the interatomic potential .
•a > 0 for repulsive interactions.
•a < 0 for attractive interactions.
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Mean Field theory and the GP
equation
In the Bose-Einstein condensation, the majority of the atoms condense
into the same single particle quantum state and lose their individuality.
Since any given atom is not aware of the individual behaviour of its
neighbouring atoms in the condensate, the interaction of the cloud with
any single atom can be approximated by the cloud's mean field, and the
whole ensemble can be described by the same single particle
wavefunction.
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The Mean-Field Approximation and
General Solution
GROSS- PITAEVSKII EQUATION
In |0> , each of the N particles occupies a definite singleparticle state, so that its motion is independent of the
presence of the other particles.
Hence, a natural approach is to assume that each particle
moves in a single-particle potential that comes from its
average interaction with all the other particles.
This is the definition of the self-consistent mean-field
approximation.
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Mean-field theory
• Decompose wave function into two parts.
• One is the condensate wave function, which is the
expectation value of wave function.
• The other is the non-condensate wave function,
which describes quantum and thermal fluctuations
around this mean value but can be ignored due to
ultra-cold temperature.
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The Mean-Field Approximation
The many-body Hamiltonian describing N interacting bosons
confined by an external potential Vtrap is given, in second
quantization, by
ˆ  ( r ,t ) and ̂( r ,t ) are the boson field operators that
where 
create and annihilate particle at the position r, respectively.
V(r-r’) is the two body interatomic potential.
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The Interaction Potential
 
V ( r  r  ) is the two-body potential.
This full potential is commonly approximated by a simplified binary
collision pseudo-potential
treating binary collisions as hard-sphere collisions. The effective
interaction strength U0 is related to the s-wave scattering length a by
where m is the atomic mass.
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The Interaction Potential
In a dilute and cold gas, one can nevertheless obtain a proper
expression for the interaction term by observing that, in this
case, only binary collisions at low energy are relevant and
these collisions are characterized by a single parameter, the
s-wave scattering length, independently of the details of the
two-body potential.
This allows one to replace V(r’-r) with an effective interaction
 
V ( r  r  )  U 0 ( r  r  )
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The Mean-Field Approximation
The boson field operators
commutation relations:
satisfy the following
From these relations, the Heisenberg equation of motion
for
can be calculated and one obtains
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The Mean-Field Approximation
The basic idea for a mean-field description of a dilute Bose
gas was formulated by Bogoliubov (1947).
The field operators
can in general be written as a sum
over all participating single-particle wave functions
and the corresponding boson creation and annihilation
operators.
In general, this can be written as
ˆ( r ,t )   ( r ,t ) â
 i
i
i

where i ( r ,t ) are single-particle wave functions and ai are
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the annihilation operators.
The Mean-Field Approximation
The boson creation and annihilation operators obey the commutation
rules
a ,a   , a ,a   0 a

i
j
ij
i
j
,

i


,a j  0
The bosonic creation and annihilation operators a+ and a are defined
in Fock space through the relations :

âi n0 , n1 ,...., ni ,.. 
âi n0 , n1 ,...., ni ,.. 


ni  1 n0 , n1 ,...., ni  1,..
 n  n ,n ,....,n 1,..
i
0
1
i
where the ni denote the bosonic populations of the particle states.

n̂i  âi âi
Gives the number of atoms is the single-particle i state.
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Bogoliubov approximation
Since the main characteristic of BEC is that most participating
particles condense into the lowest single particle quantum state, it is
possible to separate out the condensate part
of the
generalised mean field operator.
With a total number of particles N, the population n0 of the lowest
state is macroscopic such that
n0  N0 >> 1 .
In this case (with N0  n0 ), there is no significant physical difference
between states with N0 and N0+1 so that operators
and
in the
can be replaced by.
This is well known as the Bogoliubov approximation.
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Bogoliubov approximation
Using the Bogoliubov approximation, the field
operator
is written as a sum of its expectation value
and an operator
representing the remaining
populations in thermal states, which can be considered
vanishingly small in the zero temperature BEC regime.
This decomposition leads to the following expression for
the
term in Hamiltonian.
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Gross– Pitaevskii (GP) equation
By substituting the decomposition, within the approximation, and
normalising the condensate wavefunction
to
As indicated above, all terms containing the perturbation

ˆ
operator  ( r ,t ) have been neglected
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Meaning of the Decomposition


ˆ
( r ,t )  ( r ,t )
•
is the condensate wave
function

ˆ
•  ( r ,t ) describes quantum and thermal
fluctuations around this mean value.

ˆ
• The expectation value of ( r ,t )
is zero and in the
mean- field theory its effects are assumed to be small,
amounting
to the assumption of the thermodynamic
limit (Lifshitz and Pitaevskii, 1980).

ˆ
 ( r ,t ) is negligibly small in the equation
• The effects of
because of zero temperature (i.e., pure condensate).
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The Mean-Field Approximation


( r ,t )  ˆ ( r ,t )
Its modulus fixes the condensate density through

 2
n0 ( r ,t )  ( r ,t )
The function (r,t) is a classical field having the meaning of an
order parameter and is often called the ‘‘wave function of the
condensate.’’
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The time-independent GP equation
In certain cases, i.e. for eigenstates of a harmonic trap, the
wavefunction ( r ,t ) can be separated into parts of spatial
and time dependence

 i t
( r ,t )  ( r )e
with eigenvalue  representing the chemical potential of the system at
zero temperature.
Substituting into the time-dependent GP equation leads to the time
independent GP equation
36
Numerical results
37
Thomas-Fermi Approximation
The time independent GP equation, with nonlinearity C , and for a
harmonic trapping potential
r2
Vtrap 
4
can be simplified in the so-called ``Thomas-Fermi Approximation"
In BEC, the kinetic energy term
becomes small compared to the
high self-energy and can be neglected
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s-Wave Scattering
M1
M2
New coordinate system -> scattering of a
particle of mass  in a potential U(r)
M 1M 2

M1  M 2

, U( r )  U( r )
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s-Wave Scattering

exp( ikr )
( r )  exp( ikz )  f (  , )
r
incident wave
simply a plane wave
outgoing wave
The differential cross section;
d (  , )
2
 f k (  , )
d
k – the momentum of the incident wave.
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s-Wave Scattering
In the case of a central potential we expand the wave
function as:

1
 k , l , m ( r )  uk , l ( r )Yl m (  , )
r
where uk,l are the solutions of the radial Schrödinger
equation:
  2 d 2 l (l  1) 2

 2k 2

 U ( r )uk , l ( r ) 
uk , l ( r )

2
2
2 r
2
 2 dr

with
for large r
uk , l ( r  0 )  0
 d2
2
 2  k uk , l ( r )  0
 dr

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s-Wave Scattering (cont.)
and the solution is
uk , l ( r )
r 
l


 sin  kr 
 l 
2


After expansion of the plane wave exp(ikz) in terms os
spherical harmonics, we get

fk ( )  
l 0
1 
f k , l (  )   4 ( 2l  1 ) ei l sin l Yl 0 (  )
k l 0
and the total cross section is
d
4
2
   d   f k (  ) d  2
d
k

2
(
2
l

1
)
sin
(l )

l 0
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S-scattering (cont.)
From now on we discuss on the case of particles so slow that
kr0  1
r0 is the range of the potential U (r).
For cold enough collisions, only the l=0, or s-wave, partial
wave will contribute to the scattering cross section.
4
  2 sin 2 (  0 )
k
In the low-energy regime, one has approximately:
 l  ( kr0 )2l 1  1 ( kr0  1 )
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S-scattering (cont.)

fk ( )  
l 0
with
Y00 (  ) 
1 
f k , l (  )   4 ( 2l  1 ) ei l sin l Yl 0 (  )
k l 0
1
4
we will get,
f k ,0 
0
  4 a 2
k
where a is the scattering length
tan  0 ( k )
k 0
k
a   f   lim
44
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The Strange State of BEC
 When
all the atoms stay in the condensate, all the atoms are absolutely
identical. There is no possible measurement that can tell them apart.
 Before
condensation, the atoms look like fuzzy balls.
 After
condensation, the atoms lie exactly on top of each other
(a superatom).
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How Is BEC Made?
Laser beam
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Experimental realizations of BEC
1.First 108 - 1011 atoms are collected and precooled to 10100K at densities around 1010 -1011 cm-3 using laser cooling
techniques.
2.This point is typically reached with 104 -107 atoms at 100 nK1K and densities around 1014 cm-3
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3.The whole experimental cycle typically takes between 10 and
Cooling Techniques
• Laser cooling in a magneto-optical trap.
• Evaporative cooling
process.
• Subsequent rethermalisation.
49
Laser cooling in a magnetooptical trap
• The gas sample first optically trapped and cooled
using laser light .
50
Magneto-Optical Trap (MOT)
• A typical magneto-optical
trap configuration.
•Three pairs of counterpropagating laser beams
with opposite circular
polarizations are
superimposed on a
magnetic quadrupole field
produced by a pair of
anti-Helmholtz coils.
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MOT
• The Zeeman sublevels
of an atom are shifted
by the local magnetic
field in such a way that
(due to selection rules)
the atom tunes into
resonance with the laser
field propagating in the
opposite direction to
the atom’s displacement
from the origin
•Temperatures ~10K, densities ~1011 .
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Evaporative cooling
•Briefly, this cooling technique is based on the
preferential removal of atoms with an energy higher than
the average energy.
g F  B B   RF
Vtrap  mF g F  B B( r )  B( 0 )
E   mF  RF   0 
53
Evaporative cooling process
54
55
Cloverleaf configuration of
trapping coils
by Mewes et al. (1996a)
E   mF  RF  0 
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Hot atoms escape
57
What Does a Bose-Einstein Condensate
Look Like?
 There
is a drop of condensate at the center.
 The condensate is surrounded by uncondensed gas atoms.
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60
Atom LASER
•An atom laser is a device which generates an intense coherent
beam of atoms through a stimulated process.
•It does for atoms what an optical laser does for light.
•The atom laser emits coherent matter waves whereas the
optical laser emits coherent electromagnetic waves.
•Coherence means, for instance, that atom laser beams can
interfere with each other.
•An atom laser beam is created by stimulated amplification of
matter waves.
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The rf output coupler for an atom la
(a) A Bose condensate
trapped in a magnetic trap.
All the atoms have their
(electron) spin up, i.e.
parallel to the magnetic
field.
(b) A short pulse of rf
radiation tilts the spins of
the atoms.
(c) Quantum-mechanically, a tilted spin is a superposition of
spin up and down. Since the spin-down component experiences
a repulsive magnetic force, the cloud is split into a trapped
cloud and an out-coupled cloud.
(d) Several output pulses can be extracted, which spread out
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and are accelerated by gravity.
Atom Laser
Laser of light: all the photons are exactly the same in color, direction and
phase (positions of peaks and valleys).
 Laser of atoms: all the atoms in the condensate are exactly the same.

63
Interference Pattern

When two Bose-Einstein condensates spread out, the interference pattern
reveals their wave nature.
64
Interference between two
condensates
65
Resonance
Experiment in which a “particle” is scattered from a “target”.
In an elastic scattering experiment the energy of the “particle” is
conserved.
In a non-elastic scattering experiment there is an energy
exchange between the “particle” and intrinsic degrees of freedom
of the “target”.
66
67
Feshbach Resonance




a  abg 1 
 BB 
peak 

68
69
70
Bpeak
71




a  abg 1 
 BB 
peak 

A changes a sign at
B=Bpeak
72
73
74
Molecular BEC from Fermions
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Molecular BEC from Fermions
This is evidence for
condensation of pairs of 6Li
atoms on the BCS side of
the Feshbach resonance.
The condensate fractions
were extracted from
images like these, using a
Gaussian fit function for
the ‘‘thermal’’ part and a
76
Vortices
When the condensate is rotated, vortices appear. The
angular momentum of each of them has a discrete value.
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What Is Bose-Einstein Condensation
Good For?
This is a completely new area. Applications are too early to predict.
The atom laser can be used in:
 atom optics (studying the optical properties of atoms)
 atom lithography (fabricating extremely small circuits)
 precision atomic clocks
 other measurements of fundamental standards
 hologram
 communications and computation.
 Fundamental understanding of quantum mechanics.
 Model of black holes.


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References
Homepage of the Nobel e-Museum (http://www.nobel.se/).
 BEC Homepage at the University of Colorado
(http://www.colorado.edu/physics/2000/bec/).
 Ketterle Group Homepage (http://www.cua.mit/ketterle_group/).
 The Coolest Gas in the Universe (Scientific American, December 2000, 92-99).
 Atom Lasers (Physics World, August 1999, 31-35).
 http://cua.mit.edu/ketterle_group/Animation_folder/TOFsplit.htm
 http://www.colorado.edu/physics/2000/bec/what_it_looks_like.html.
 http://www.colorado.edu/physics/2000/bec/lascool4.html.
 http://www.colorado.edu/physics/2000/bec/mag_trap.html
 Pierre Meystre Atom Optics.

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