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Laser Cooling and
Trapping – An
Overview
Eyal Fleminger, 38500575
December 2004
Foreword ........................................................................................................................ 3
Forces on an Atom in a Light Field ............................................................................... 3
Scattering (Radiative) Force ...................................................................................... 3
Dipole Forces ............................................................................................................. 6
Trapping of Atoms ......................................................................................................... 6
Optical Traps .............................................................................................................. 6
Magnetic Trapping ..................................................................................................... 9
Magneto-Optical Trap .............................................................................................. 12
Sub-Doppler cooling .................................................................................................... 15
Evaporative Cooling ............................................................................................ 17
Trapping on a Microchip ............................................................................................. 21
Mirror MOT ............................................................................................................. 22
Magnetic Trapping ................................................................................................... 22
Implementation of a MOT ........................................................................................... 25
Conclusion ................................................................................................................... 26
References .................................................................................................................... 27
2
Foreword
The subject of the 1997 Nobel Prize for physics, the field of laser trapping and
cooling has produced a toolbox of techniques for creating confined ensembles of
ultracold atoms. These techniques have various applications, such as the study of
ultracold matter, atom optics, Bose-Einstein Condensation (BEC), and so forth.
In this paper, I will explore the basic theory of cooling and confining atoms with
laser light. I will begin with a description of the forces induced on atoms by a laser
beam, describing how they can serve to both cool and confine atoms; following that, I
will describe several optical trapping schemes (in addition, I will describe purely
magnetic trapping schemes, as a prelude to the MOT) as well as a hybrid trap (MOT).
I will then describe the theory of sub-Doppler cooling (including evaporative
cooling). Following that will be a brief overview of a few of the variations on trapping
schemes (both hybrid and magnetic) used in creating a MOT on a microchip (part of
the subject of my group and my current work). I will conclude with a description of
one of the earlier MOT experiments.
Forces on an Atom in a Light Field
Scattering (Radiative) Force
When an atom of mass m absorbs a photon whose frequency ν matches a
resonance frequency of the atom, the photon’s energy ħν causes transition to an
excited state, while the photon’s momentum
h 
is absorbed as an addition to the
c
atom’s momentum in the direction of the photon’s movement (the converse is true for
photon emission from an atom). The momentum exchange induces a force
(1)
F
dp h p ˆ

A
dt
c
γp is the atom’s excitation (or scattering) rate (see below). In the case of
absorption, the arbitrary direction vector  is the same as the laser’s direction of
propagation; in the case of photon emission, the vector’s direction is opposite that of
the emitted photon. The change in the atom’s velocity is of the magnitude
3
v 
(2)
h 
cm
If we have a gas of atoms of mass m and temperature T in a volume (assuming
the gas is dilute or otherwise approximates an ideal gas), their velocity is governed by
the Maxwell-Boltzmann distribution
3/ 2
(3)
 m 
f (v)  4v 

 2k BT 
2
e

mv2
2k BT
The characteristic velocity for a given temperature is vrms, given by
(4)
3k BT
m
vrms 
Let us consider an atom moving in a direction opposing the laser beam (for the
moment, we will consider only the velocity component in the same axis as the beam).
If it absorbs a photon, its velocity is reduced, since the direction of Δv is directly
opposed to that of the atom’s velocity v. As the atom shifts back to its ground state, it
gives off a photon, further changing its velocity; but as the probability of the direction
of the emission is spatially symmetrical, the velocity change due to emission has a
mean value of zero over multiple instances of photon absorptions, and the total
deceleration of the atom is in the direction of the laser beam.
In order to allow absorption by the atom, ν must be equal to the atom’s resonant
frequency. There is, however, a problem; atoms moving in the same direction as the
laser beam will also absorb photons, accelerating them. To prevent this, the laser
frequency selected is slightly below the resonant frequency of the atom. From the
view point of an atom moving in the gas, the frequency is Doppler shifted by (note
that v in equation (5) is positive if its direction is the same as that of the beam’s
propagation)
(5)
v
D     
c
Thus, for atoms heading “into” the laser beam, ν is blue-shifted toward the
resonant frequency, while for atoms moving “with” the beam, ν is red-shifted and
therefore those atoms will not absorb photons.
The excitation (or scattering) rate γp depends on the laser’s detuning (designated
by δ) from resonance, defined as the difference between the atomic resonance
frequency ωa and the laser frequency ωL, and is given (for a two-level atom) by the
Lorentzian (Metcalf, van der Straten, 2003)
4
(6)
p 
s0  d
2
 2    

D


2  1s0    


d

 

γd is an angular frequency corresponding to the excited state’s rate of decay (1/τ).
s0 is defined as the ratio between the light intensity I and the saturation intensity; the
latter is the intensity at which the rates of spontaneous emission and stimulated
emission are equal, and is given as
Is 
(7)
h 3
3c 2
Is is dependent on the atoms in question, and is typically of the order of several
mW/cm2. As intensity increases, so does the deceleration. However, at high
intensities, the rate of stimulated emission increases. Since, in the case of stimulated
emission, the photon is emitted in the same direction as the laser beam’s direction of
propagation, the momentum “kick” is in the opposite direction, nullifying the
deceleration caused by absorption. At these intensities, the atom has an equal chance
of being in the excited or ground states, and the maximum deceleration is
(8)
a max 
h d
2mc
According to equation (5),   v , the Doppler shift becomes smaller as the
atoms slow down, and eventually the shifted frequency will be too far from the
resonance frequency to allow excitation. There are various methods to compensate for
this effect (Metcalf, van der Straten, 1999). The two most common ones are changing
the laser’s frequency as cooling progresses (known as “chirping”), and spatially
varying the resonance frequency by means of an inhomogeneous magnetic field (such
as in a MOT, described later).
With a pair of counterpropagating lasers, it is possible to use this effect to slow
atoms. Atoms moving toward a laser have their momentum reduced, while atoms
moving away from it are not affected by it (due to the Doppler shift away from
resonance). This type of cooling is known as Doppler cooling, and the resulting
process is known as optical molasses.
5
Dipole Forces
Atoms in an inhomogeneous field, such as a standing wave (such as created by
two counterpropagating laser beams), experience a force derived from the spatial
gradient of their light shifts (see below).When δ>>γd, spontaneous emission may
occur less frequently than stimulated emission. The force arising from stimulated
emission is known as the dipole force and derives from the gradient of the light shift
(the Stark shift due to the electric field of the wave).
For a laser beam traveling in the  direction, the Hamiltonian is
 2  
H  *

0
2 
The Rabi frequency Ω is given by
(9)
  p
(10)
s0
2
The light shift is provided by the eigenvalues of equation (9), and is given by
(Metcalf, van der Straten, 2003)
(11)
ls 
2  2  
2
For large detuning, equation (11) is approximately
ls 
(12)
2
 2  ps0

4
8
In a standing wave, the light shift, as well as the probability of an atom
undergoing spontaneous rather than stimulated decay, varies sinusoidally from node
to anti-node. Atoms are excited by one beam and are then stimulated into emission by
the other, slowing them. The force generated is
(13)
Fx  
 d2
I  x 
8Is
where I is the total intensity distribution of the light field.
Trapping of Atoms
Optical Traps
It is possible to use the forces outlined above to spatially confine a cloud atoms.
One type of tap utilizes the dipole force.
6
Consider a Gaussian laser beam with a waist width of w0 whose intensity at the
focus is
(14)
I  r   I0e
 r 


 w0 
2
When δ<0, the ground-state light shift is negative, and has its largest value at the
beam center. Therefore, atoms in the beam experience a force toward the beam’s
center arising from the gradient of the light shift given by equation (12). For δ>>Ω
and δ>>γ this transverse force is given by equation (13), and for a Gaussian beam is
(15)
F
 I0 r
e
4 Is w 02
2
d
 r 


 w0 
2
In order to trap atoms with this force, it is necessary to overcome the radiation
pressure on the atoms in the laser beam’s direction. This can be done by selecting the
appropriate laser parameters. As per equation (6), the radiation pressure force
decreases as δ-2, while the dipole force decreases as δ-1 (equation(11)) for δ>>Ω,
choosing sufficiently large δ means that atoms spend little time in the untrapped
(repelled) excited state, since its population is proportional to δ-2. Thus, a sufficiently
large absolute magnitude of δ will produce both longitudinal and transverse trapping,
maintaining the atomic population (mostly) in the trapped ground state. The
maximum intensity, and therefore light shift and trap depth, attainable by a given laser
is proportional to the area of the beam spot πw02, requiring a large numerical aperture
for focusing (ibid).
This form of dipole trap is perhaps the simplest imaginable. One important
drawback is that negative detuning causes attraction to the regions of highest
intensity, but once their, a higher incidence of spontaneous emission is caused
(defeating the trap) unless the detuning is a large fraction of the optical frequency. In
addition, the reliance of the trap on Stark shifts (equal to the trap depth) can
compromise the use of the trap for applications such as spectroscopy. A solution to
this problem is the use of positive detuning (blue-shifted traps). In such a case, atoms
are attracted to the areas of lowest intensity. In that case, however, the simple
Gaussian beam described above could not be used, since the atoms would be drawn to
its fringes and escape. One solution (Davidson et al, 1995) is the use of two adjacent
beams, forming a trough between them where the atoms accumulate. Another possible
solution under research is the use of a hollow laser beam.
7
It is also possible to use the scattering force to confine atoms, by using a system
of six lasers, with each pair orthogonal to the other two pairs. However, the optical
Earnshaw theorem (Ashkin, Gordon, 1983) precludes such a trap from being stable,
so long as the trapping force is proportional to the light intensity. Poynting’s theorem
states
u
P  0
t
(16)
with u designating the density of the electromagnetic field and P being the
Poynting vector. For static beams (in the absence of sources or sinks of radiation), the
time derivative of u is zero, and therefore
 P  0
(17)
Since the force is proportional to the Poynting vector
Fr  
(18)
p  P  r 
c
(σp being the cross section of the particle), it follows that
 F  0
(19)
In accordance with the divergence theorem,
(20)
 F  nˆ  dS      F dV  0
S
V
In order for equation (20) to be valid, the direction of the field on the surface of a
sphere enclosing the trap must change direction (inward/outward) on at least one
point, or else be zero at all points (an alternative way of viewing this is that a
divergenceless force is represented by continuous lines, which must leave any volume
they enter). In either case, the trap is not stable; in the former case, there is a t least
one point at which the particles can escape the trap, while in the latter case, the
particles will not be kept in the trap.
In the absence of sources or sinks of radiation, the divergence of the Poynting
vector of a static laser beam is zero, and therefore so is the divergence of the force
(since it is proportional to the intensity). Therefore, there cannot be a closed surface
on which the force is inward at all points. It is possible, however, to overcome this
limitation for atoms with multiple ground states with different absorption probabilities
(Pritchard et al, 1986).
8
Magnetic Trapping
It is also possible to trap atoms with a “purely” magnetic trap. This is necessary
because the emission and absorption of photons set a lower limit (see equation (39)
below) on the achievable cooling by lasers, and cooling below that limit requires the
removal of the lasers from the system, so a trap utilizing lasers cannot be used at that
stage. This section describes the principles of magnetic traps, as well as several trap
configurations.
The magnetic moment  of an atom interacts with an inhomogeneous magnetic
field, producing a force

F   B
(21)

For a positive moment, the force drives the atom towards higher potential (such
states are referred to as high-field seekers) and for a negative moment the force drives
the atom toward lower potentials (low field seekers). A magnetic trap uses this force
to confine the atoms. There are various methods of doing so; several will be described
below.
The simplest type of trap is the quadrupole trap. Two identical coils with
identical but counterpropagating currents are used (Figure 1).
Figure 1 – Schematic of a
magnetic quadrupole trap
(Bergeman et al, 1987)
In cylindrical coordinates, the magnitude of the field is (Bergeman et al, 1987)
proportional to the coordinates as
(22)
B  2  4z 2
The field at the center of the trap is zero, and therefore will trap low field seekers.
The problem with this type of trap is that a moving atom experiences a time-varying
magnetic field, which induces state transitions. Atoms which make a transition from a
low-seeking state to a high seeking state will be ejected from the trap. This effect
becomes serious when the frequency of the time-varying magnetic field exceeds that
9
of the transitions between magnetic sublevels, which is of the order of μBB (where μB
is the Bohr magneton). Therefore, since the field magnitude is zero at the origin
(equation (22)), losses become significant as the center of the trap is approached,
imposing a limit on the time an atom can be trapped. An atom with velocity v and
mass m, passing within a minimum distance r of the center of the trap, will undergo a
nonadiabatic spin flip if the rate of change v/r is greater than the Larmor frequency
(the frequency at which the spin precesses around the magnetic field), which is
approximately (Petrich et al, 1995)
(23)
L
  r  Bg
where
(24)
Bg 
B

is the radial field gradient. Therefore, losses occur within an ellipsoid of radius
(25)
v
Bg
r0
The loss rate is given by the flux of atoms through the ellipsoid and is
(26)
1 N  r02  v

0
l3
where N is the number of atoms in the cloud and l is its radius. The viral theorem
relates mean velocity to cloud size by (ibid)
(27)
mv2   l  Bg
By isolating Bg in equation (27) and placing it in equation (25), and placing the
resultant r0 in equation (26), we get
(28)
1
N

0 m  l 2
One approach to overcome this is by use of a modified quadrupole trap known as
a time-averaged orbiting potential (TOP) trap. A rotating spatially-uniform bias field
is superimposed on the quadrupole field, changing the location of the field’s zero
point faster than the atoms can respond. For a quadrupole trap with the axis of
symmetry in the z direction, the bias field rotates in the xy planes at a frequency ωb.
The zero point of the total field orbits around the trapped cloud (Figure 2). The radius
10
of the orbit is designated R0, and equals the ratio of the magnitude of the bias field
(Bb) to the horizontal gradient of the quadrupole field
R0 
(29)
Bb
Bg
Figure 2 – The zero point (the meeting
point of the field gradient arrows) of a
TOP trap orbits around (the dotted line)
the trapped atom cloud (the grey area) at
a radius of R0 (Petrich et al, 1995)
The instantaneous potential is therefore

 

(30) U(x, y, z, t)   xBg  Bb cos b t xˆ  yBg  Bb sin b t yˆ  2zBG zˆ
The frequency ωb is chosen to be less then the Larmor frequency, thus ensuring
that the atoms will remain in the same quantum state relative to the instantaneous
field.
Averaged over one field rotation, the potential is
(31)

 U  b
2
2  / b

U(t)dt  Bb 
0
Bg2
4Bb

2

 8z 2  ....
Therefore, the time-averaged field never vanishes, and there is no “hole” in the
trap.
Another way to remove the “hole” is to use a magnetic trap with a non-zero
minimum. Consider a trap similar to the quadrupole trap, but here the current in both
coils flows in the same direction. At the center of the system, the potential is given by
(Pethick, Smith, 2002)
z
U   A l r l Pl  

(32)
where Pl are Legendre polynomials and Al are coefficients. In the vicinity of the
origin, the magnetic field is
(33)


2  
B(, , z)   3A 3z, 0, A1  3A 3  z 2   
2 


If A1 and A3 have the same sign, the magnetic field increases with the distance
from the origin in the z direction. This configuration is known as a magnetic bottle,
11
and is used in plasma physics to confine charged particles. However, the field does
not have a local minimum at the origin, and neutral atoms moving perpendicular to
the field are not constrained. One way of creating a local minimum is known as the
Ioffe trap. Four conducting bars (with identical current flowing in each of them) are
placed parallel to the axis of symmetry (Figure 3).
Figure 3 - Ioffe trap, side
(left) and frontal (right)
view (Bergeman et al, 1987)
The potential near the origin must be of the form
(34)
U
2
Ce2i  C*e2i
2


The constant C is determined by the current in and geometry of the bar. If the
azimuthal angle φ is zero between two adjacent bars, C is real. Symmetry requires the
potential function be proportional to x2-y2. The field generated by the bars is therefore
(35)
BI  x, y, z    C,C,0 
And the magnitude of the total field (expanded to the second order) is
(36)

2  C 2 2
B  A1  3A3  z 2   
2  2A1

Therefore, the magnitude of the field will have a minimum for
C 2  3A1A 3
It is possible to generate fields with different degrees of curvature in the axial and
radial directions by varying the current in the bars.
Magneto-Optical Trap
The most common method of trapping atoms is the magneto-optical trap (MOT).
As its name suggests, this traps utilizes a combination of lasers and a magnetic field to
trap and cool the atoms.
12
Figure 4 – Schematic of a MOT
The field used for a MOT is a linearly inhomogeneous field, such as that
generated by a quadrupole trap (equation (22)). For each axis, two counterpropagating
lasers with opposite circular polarizations are used. Each laser is detuned by δ below
the atomic resonance for zero magnetic field (Figure 5).
Figure 5 – (a) Schematic
view of the field and beams
in a MOT. (b) The
transitions induced by each
beam. (c) Influence of the
magnetic field on the atomic
sublevels (Pethick, Smith,
2002)
At z=0, both laser beams have an equal probability of being absorbed by an
atom. However, at z>0 the frequency of transition to the m-1 substate is reduced, and
is closer to the laser frequency, while the frequency of the transition to the m=1
substate is tuned farther away from the lasers’ frequency. Therefore, atoms at z>0
have a higher probability of absorbing photons from the σ- beam than photons from
the σ+ beam, and therefore the net force is toward z=0. At z<0, the situation reverses
itself. Therefore, atoms throughout the trap tend to move toward its center.
13
Figure 6 – Force operating on
an atom in a MOT. The dotted
lines represent the force from a
single laser, while the dark line
is the net force.
As can be seen in Figure 6, the net force is almost linear, and takes the form of
the restoring force
(37)
F   kz
This description is simplified. In practice, MOTs tend to be more complicated
than described above. One problem is that many atoms have multiple hyperfine states.
Since the laser does not match the resonances between all the levels, the population of
excited atoms at certain levels may grow to the point the MOT ceases to function. For
example, the 3S1/2 ground state of sodium has hyperfine levels F=1,2, while the
excited state 3P3/2 has hyperfine levels F’=0,1,2,3. If the laser frequency is resonant
with the F=2→F’=3 transition, some atoms will be excited to F’=2, and will then
decay either to F=2 or F=1. Since the laser does not match the resonance of the
F=1→F’=2 transition, there will be a buildup in the population of atoms in the F=1
substate compared to the F=2 substate. This process is known as optical pumping.
Eventually, the number of atoms in the F=2 level (referred to as the bright state) will
be sufficiently low that the MOT will cease functioning. To avoid this, radiation
resonant with the F=1→F’=2 transition is applied, thus depopulating the F=1 level in
a process known as repumping (Ketterle et al, 1993).
In evaporative cooling (discussed later) it is necessary to achieve a high density
of atoms. However, the emission of photons from excited atoms creates a force which
drives the atoms apart at high densities. In addition, at high densities, the atom cloud
becomes opaque to the laser. One way to reduce the impact of these effects is to
reduce the level of repumping radiation so that only a small proportion of the atoms
are in a level where the laser beams can induce a transition. Though this reduces the
effective force of the trap, the attainable density increases. One method (ibid) of doing
so is repumping light preferentially to the outer portions of the cloud, causing atoms
14
on the fringes to be driven inward, while atoms in the center will experience a high
degree of optical pumping, reducing the radiation forces in the interior. This
configuration is known as a dark-spot MOT, and can achieve densities two orders of
magnitude greater than a conventional MOT (Pethick, Smith, 2002).
Sub-Doppler cooling
Doppler cooling can only reduce the temperature of an atom gas so far. As the
velocity of the atoms decreases, so does the differential cooling rate, and at some
point it becomes small enough to be counteracted by the random momentum kicks
imparted by the atoms’ photon emissions. The temperature at which this occurs
(known as the cooling limit or the Doppler temperature) is given by
(38)
TD 

2k B
where γ is the natural width associated with the resonance frequency. For
example, for sodium, TD is approximately 240 μK (Philips et al, 1988)
In 1988, it was discovered that the temperature of Doppler-cooled atoms was
well below the Doppler limit (ibid). This is caused by the inhomogeneity of the light
field as a result of the opposing lasers’ polarization, and the fact that while the
discussion above referred to a simple two-level atomic model, in reality atoms are
more complex, containing sublevels as well (such as those cause by Zeeman shifts).
Figure 7 – Polarization of the
superposed field of two
counterpropagating and
perpendicularly polarized lasers.
The field has a minimum at 0, a
maximum at λ/8, and both
maxima and minima have a
sinusoidal periodicity of λ/8
(Metcalf, van der Straten, 2003)
If the lasers are linearly polarized perpendicular to one another, the total electric
field potential’s magnitude varies sinusoidally along the lasers’ axis of propagation.
Due to the light shifts, each of the ground state substates (e.g. m=±1/2) has a
maximum at the other substate’s minimum, and vice versa. The electric field’s
polarization also varies, from linear at the minima to circular at the maxima, with
alternating directions (Figure 7).
15
This mechanism is dependent on the fact that optical pumping between two
sublevels takes a finite and nonzero amount of time. If an atom begins in a potential
“valley” (say at the substate m=1/2), it can move a certain distance, climbing the
potential “hill”. When it reaches the maximum, the light is now polarized in the σdirection, optically pumping the atom to the m=-1/2 substate. The potential difference
between the levels is emitted as a photon in the transition. The atom is now at the
minimum for the m=-1/2 substate, and climbs the potential to the next maximum,
where the polarization is now σ+, inducing a transition to m=1/2 (Figure 8). In such a
fashion, the atom continues to climb potential “hills” without ever descending them,
translating its kinetic energy into potential energy (Dalibard, Cohen-Tannoudji, 1989).
The process repeats until the kinetic energy is too small to climb the next “hill”. This
cooling mechanism is known as Sisyphus cooling, after the mythological Greek figure
who was condemned to eternally roll a boulder up a hill. Through this mechanism,
very low temperatures can be reached (e.g. 35 μK for sodium).
In the case that the lasers are circularly polarized, the resulting electric field is
linearly polarized everywhere and of constant magnitude, but the polarization’s
orientation rotates through an angle of 2π over one wavelength. In this case, an effect
similar to the force in a MOT (described above) occurs. The m=1 sublevel (with m
being the magnetic quantum number) will have a higher population for atoms moving
in the positive direction (Metcalf, van der Straten, 1999), while in atoms moving in
the negative direction the m=-1 sublevel will have a greater population. Because of
the different Clebsch-Gordan coefficients involved in the various transitions (ibid,
Appendix D), the m=1 sublevel scatters σ+ at an efficiency six times greater than σphotons. Therefore, atoms moving against the σ+ beam scatter more of its photons and
experience a greater momentum shift in the negative direction, while atoms moving in
16
the negative direction are preferentially pumped to the m=-1 sublevel and recoil in the
positive direction. Though difficult to quantify, the final cooling derived from this
mechanism is on the same order as for Sisyphus cooling (Dalibard, Cohen-Tannoudji,
1989).
In all the methods involved, there is constant absorption and emission of photons
by the atoms. That sets a lower temperature limit due to the fact that each time an
atom emits a photon, it receives a “kick” in some direction. At low temperatures, the
velocity changed caused by emission is of the same order as the atom’s total velocity,
and thus the atom cannot be cooled further. This limit, known as the recoil limit, is
given by
(39)
Tr
 h 

2
m
While optical methods have been developed to cool atoms beneath this limit,
description of those methods is beyond the scope of this paper. Further material can
be found in the advanced background notes for the 1997 Nobel Prize in physics.
Evaporative Cooling
Laser cooling has a limit to temperatures of several μK. At higher densities,
photons emitted by a given atom are absorbed by other atoms, causing a repulsion
effect. In addition, as density increases, so does the incidence of interatomic
collisions. As these are inelastic in nature, increasing the density increases the heat.
Evaporative cooling is a method based on the preferential removal (from an
ensemble of trapped atoms) those atoms possessing a higher energy. The principle
used is the same as that which exists when a hot liquid (e.g. a cup of coffee) cools.
Particles whose energy is higher than average are removed from the ensemble, thus
lowering its average energy (and hence temperature). Eventually, the particles tend to
occupy the lowest energy states of the trap at high densities.
I will describe here a simple model of evaporative cooling (Metcalf, van der
Straten, 1999). It makes the following assumptions:
1. The gas possesses sufficient ergodicity – the phase space distribution of atoms is
dependent on their atoms and the trap alone.
2. The gas is governed by classical statistics
3. Evaporation preserves the thermal nature of the distribution.
17
4. Atoms escaping from the trap do not exchange energy with the remaining atoms.
Consider an ensemble of N atoms with a temperature of T held in an infinitely
deep trap. The model (Davis et al, 1995) assumes the evaporation is composed of
discrete cycles. The trap depth is lowered to a depth ηkBT (η is known as the
truncation parameter), allowing the escape of atoms with a higher temperature; the
atoms still inside the trap then achieve a new mean temperature as a result of mutual
collisions.
We define N’ as the number of atoms in the trap at the end of this cycle, and T’
as the temperature; for convenience, we define υ=N’/N. The parameter β is a measure
of the temperature decrease, and is defined

(40)
ln  T '/ T 
ln  N '/ N 

ln  T '/ T 
ln 
Therefore, the decrease in temperature is the power-law function
T '  T
(41)
However, for atoms in a trap of potential U,
 2m 
g(E) 
3/ 2
(42)
4

2 3
E  Ud 3r
The fraction of atoms remaining in the trap (for chemical potential μ) equals
1

N
(43)
k BT


g(E)e
(E  )
k BT
dE
0
Let us assume a potential which can be described as a power law function
(Bagnato et al, 1987) so that
(44)
x
U  x, y, z   1
a1
s1
y
 2
a2
s2
z
 3
a3
s3
We further define
(45)

1 1 1
 
s1 s 2 s3
For this potential, the volume is proportional to the temperature so that
(46)
V  T
18
Property
Number of
Sy
mbol
q
N
1
Temperature
T
β
Volume
V
βξ
Density
N
1- βξ
atoms
Phase space
density
Collision rate
ρ
K

2


2

1    3
1    1
Table 1 – scaling of thermodynamic properties, where for property X X’=Xυ q
Table 1 shows the exponential scaling for various thermodynamic properties.
At infinite η, υ=1, from which the chemical potential is determined (Bagnato, et
al, 1987). Equation (43) becomes

(47)
       e d
0
where the reduced energy is defined as

(48)
E
k BT
and the reduced density of states is
(49)
  


 1
2
  3
2

Using the incomplete gamma function, equation (47) can be written as
(50)


inc   3 , 
2

3
 
2


Therefore, the fraction of remaining atoms is dependent solely on the truncation
parameter and on the trap parameter ξ.
19
The average reduced energy before truncation is (Metcalf, van der Straten, 1999)


(51)
     e
0

   e


d
 
d
3
2
0
while the average reduced energy after truncation is

     e
(52)



d
inc   5 , 
2
'

   3 ,

2
     e d inc
0



0
Due to equation (48),
 ' T'

 

T
(53)
The energy carried away by each evaporated atom is
(54)
l 


'
1  1
  3
2 1 
1 
For large values of η, the denominator in equation (54) is small and therefore

(55)
l
1
 3
2
In such a case, β represents the energy above the mean energy, which is carried
away.
As ξ grows (stronger potential), the decrease in volume (and therefore the
increase in density) as the temperature decreases becomes larger. Elastic collision
rates also increase for a larger potential, speeding up the rethermalization. At weak
potentials, the collision rate decreases and the cooling eventually stops.
The speed of evaporation can be estimated using the principle that elastic
collisions produce atoms whose energy is greater than ηkBT at a rate determined by
the number of atoms with energy above this divided by their collision rate (Ketterle,
van Druten, 1996). The velocity of atoms with energy ηkBT is
(56)
v
2k BT
3

v
m
2
For large values of η, the fraction of atoms with energy greater than ηkBT is
(57)
f      e
20
3
2
The elastic collision rate is defined as
(58)
k el  nv
where the collisions’ cross section for scattering length a is
(59)
  8a 2
Therefore, the rate of evaporation is
(60)
dN
N
  Nf     k el  nve  N  
dt
ev
Since the average value of kel depends on the relative rather than the mean
velocity of the atom,
(61)
k el 
4nv
3
Thus, the ratio of the evaporation time τev and the collision time τel increases
exponentially with η
(62)
ev
2e

el

So far, it has been assumed all collisions are elastic. This is not necessarily true,
however, which may cause problems, since the release of internal energy can heat the
atoms, and the state of the atoms may be changed to a new state which is no longer
trapped. Thus, the final limit cooling is dependent on the ration of elastic to inelastic
collisions. The magnitude of inelastic collisions varies with atomic species; for alkali
atoms, the final temperature is on the order of picokelvins (ibid).
Trapping on a Microchip
There are advantages to small trap sizes. Large field gradients and curvatures can
be achieved with small currents. This means high gradients can be achieved for less
energy dissipation, which leads (Table 1) to faster rethermalization and therefore
faster evaporative cooling rates. In addition, by their nature small traps allow the
entire device to be small, which is important for various practical applications. There
is considerable research being carried out in the field of creating traps on a microchip;
this section will discuss a few MOT and magnetic trap configurations specific to such
chips.
21
Mirror MOT
The small volume of a chip trap presents certain drawback. Its small size presents
difficulties in loading the atoms into the trap; in addition, the proximity of the trap to
the substrate on which it resides presents difficulties when implementing a six-laser
MOT, since the laser must be positioned between the substrate and the trapping area
(otherwise, the substrate blocks the beam). One solution (Reichel et al, 1999) for this
problem is a mirror MOT (Figure 9).
Figure 10 – Schematic view of a mirror
MOT (Reichel et al, 1999)
This type of MOT uses four lasers instead of six. By reflecting two of the lasers
from a mirror, their polarizations are reversed, and for a particle above the mirror,
there are effectively four lasers
Figure 11 – Path of the lasers in a
mirror MOT (there are two
additional lasers, not indicated
here, perpendicular to the page)
Magnetic Trapping
As discussed above, once the atom cloud reaches the limit for laser cooling, it
must be confined by a purely magnetic trap for evaporative cooling. For obvious
reasons of size, normal traps such as the quadrupole or Ioffe traps described above
cannot be used on a chip (or at least, fabrication is extremely difficult). It is possible,
however, to achieve similar potentials by means of the use of wires on the chip, in
22
conjunction with an external homogenous bias field. Here, I will describe two
configurations to do so.
According to the law of Biot-Savart, the field created by a wire1 of length 2a
placed on the y axis (in such a way that it is bisected by the origin), through which a
current I is flowing, is given by
B  x, y, z  
(63)
0 I
4
ay
x 2   a  y 2  z 2

ay
x 2   a  y 2  z 2
x 2  z2
 z, 0, x 
One method of trapping is to bend the wire into a U shape. An example of a field
generated by such a wire configuration can be seen in Figure 112 (using arbitrary
values for current I and crosspiece length 2a).
Y
Z
a
Z
b
2
4
4
1
3
3
0
2
2
-1
1
1
-2
X
-2
-1
0
1
2
0
-2
Y
-1
0
1
2
0
-2
c
X
-1
0
1
2
Figure 12 – U trap field (without bias field from (a) above (b) in front (c) the side
The field generated by this configuration is
(64)
 ay ay



x 
x
x
x


  
1 
z
  a  y  1     a  y  1  
 1  

 
 

0 I   




 yˆ 
B  x, y, z  
xˆ  z 


2
2
2
  a  y 2  z 2
4 
x 2  z2
  a  y   z2 a  y   z2 
a  y   z2









ay ay 
 x       
 
  zˆ 
 
x 2  z2
 
 
where
(65)
  x 2   a  y   z 2
2
By inserting the (x,y,z) coordinates for the desired trap location into equation
(64), an appropriate magnitude for the bias field can be calculated, such that the total
1
Because of the scale, unless stated otherwise, I will assume here the wires are one-dimensional
The plots in this section were generated with Mathematica. Magnitude ascends from red through
orange hues, then greens, blues, purples, and red again for maximal intensity. All the traps lie in the xy
plane, with orientations as shown in Figure 11(a), Figure 12(a) Figure 13(a), and Figure 14(a). For
other orientations, the values x,y,z in equations (64) and (66) are transposed as needed.
2
23
field will be zero at some point. Doing so results in a quadrupole trap, as seen in
Figure 12
Y
Z
a
Z
b
2
4
4
1
3
3
0
2
2
-1
1
1
X
-2
-2
-1
0
1
2
0
-2
Y
-1
0
1
2
0
-2
c
X
-1
0
1
2
Figure 13 – Quadrupole trap generated by a U-wire trap in a bias field, from (a)
above, (b) the front, (c) the side
A second configuration for a wire trap is the Z-trap, so named because the wires
are twisted into a Z shape. The field caused by that configuration is shown in Figure
13.
Y
Z
a
Z
b
2
4
4
1
3
3
0
2
2
-1
1
1
-2
X
-2
-1
0
1
2
Y
0
-2
-1
0
1
2
0
-2
c
X
-1
0
1
2
Figure 14 – (a) top (b) front and (c) side views of the field magnitude of a Z-trap
The field for this trap is
(66)
 ay ay



x 
x
x
x


  
1 
z
  a  y  1     a  y  1  
 1  

 


0 I   




 yˆ  
B  x, y, z  
xˆ  z 


2
2
2
  a  y 2  z 2
4 
x 2  z2
  a  y   z2 a  y   z2 
a  y   z2








Introducing a suitable bias field, we get an Ioffe trap (Figure 14)
24

ay ay 
 x       
 
  zˆ 
 
x 2  z2
 
 
Y
Z
a
Z
b
2
4
4
1
3
3
0
2
2
-1
1
1
-2
X
-2
-1
0
1
2
Y
0
-2
-1
0
1
2
0
-2
c
X
-1
0
1
2
Figure 15 – (a) Top (b) front and (c) side views of the resulting trapping potential of the
Z-trap + bias field
Implementation of a MOT
This section will describe the construction of a “basic” MOT. The experiment in
question is described by Monroe et al (1990).
Figure 16 – The
vapor cell used in
the experiment
(Monroe et al, 1990)
The experiment’s configuration is shown in Figure 14. The vapor cell (containing
a vapor of cesium atoms) is a vertical cylinder with two crosswise cylinders placed at
the top, with a window at each of the six openings (note that henceforth, the “center”
of the cell will refer to the intersection of this cross, rather than the cell’s physical
center halfway down the vertical cylinder). Two coils are placed above and below the
center (the larger coil shown in Figure 15 is used to create a purely magnetic trap later
in the experiment, and will not be discussed here). Two smaller tubes are attached to
the vertical tube. One contains a reservoir of cesium, whose temperature (which can
25
be reduced to -23ºC) controls the vapor pressure in the cell. The second tube leads to
an ion pump which removes any residual gas which may diffuse through the walls of
the cell.
A diode laser, tuned to the 6S1/2,F=4→6P3/2,F=5 transition is split into three
beams, which are circularly polarized and directed to intersect perpendicularly in the
center of the cell. After leaving the cell, each beam passes through a quarter-wave
plate, and is then reflected back into the cell, thus reversing its polarization (since it
passes the cell twice). A second laser, tuned to the 6S1/2,F=3→P3/2,F=4 transition
illuminates the intersection to prevent atoms from accumulating in the F=3 ground
state. The coils produce a magnetic field gradient of about 10 Gauss/cm.
When the apparatus was activated, with the laser red-detuned by 1-10 MHz
below resonance (optimal results were attained for about 6 MHz below resonance), a
cloud of atoms with a volume of less than 1 cubic millimeter appeared in the center of
the cell. By measuring the resulting fluorescence, it was determined that the cloud
contained about 1.8x107 atoms, at a temperature of about 30 μK. Varying the pressure
from about 10-7 Torr to about 1.5x10-9 Torr reduced the number of trapped atoms
(though by less than 30%), presumably because of the loss due to collisions with noncesium atoms. At high pressures, the number of atoms in the trap also decreases, as
the vapor attenuated the laser beams.
Conclusion
This overview is, of course, by no means exhaustive; just the variations on a few
of the traps shown here would each merit a substantial article of their own.. Further
variations and techniques are created constantly. Likewise, various aspects of the
theory can be expanded at great length. However, this paper goes over the principles
involved, giving an overview of the basics of this field. Further information can be
found in the references given below, as well as the web sites and publications of
various groups and researchers involved in this field.
26
References
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Journal of the Optical Society of America B, volume 20, 887
2) Ashkin, A., Gordon, J.P. (1983) Stability of Radiation-Pressure Traps: An
Optical Earnshaw Theorem, Optics Letters, volume 8, 511
3) Pritchard, D.E., Raab, E.L., Bagnato, V. (1986) Light Traps Using Spontaneous
Forces, Physical Review Letters, volume 57, 310
4) Davidson, N., Lee, H.J., Adams, C.S., Kasevich, M., Chu, S. (1995) Long
Atomic Coherence Times in an Optical Dipole Trap, Physical Review Letters,
volume 74, 1311
5) Smith, H., Pethick, C.J. (2002) Bose-Einstein Condensation in Dilute Gases,
Cambridge University Press
6) Metcalf, H.J., van der Straten, P. (1999) Laser Cooling and Trapping, Springer
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(1988) Observation of Atoms Laser Cooled Below the Doppler Limit, Physical
Review Letters, volume 61, 169
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9) Nobel Prize in physics website (http://www.nobel.se/physics/index.html)
10) Bergeman, T., Erez, G., Metcalf H. (1987) Magnetostatic Trapping Fields for
Neutral Atoms, Physical Review A, volume 35, 1535
11) Petrich, W., Anderson, M.H., Ensher, J.R., Cornell, E.A (1995) Stable, Tightly
Confining Magnetic Trap for Evaporative Cooling of Neutral Atoms, Physical
Review Letters, volume 74, 3352
12) Ketterle, W., Davis, K.B., Joffe, M.A., Martin, A., Pritchard, D.E. (1993) High
Densities of Cold Atoms in a Dark Spontaneous-Force Optical Trap, Physical
Review Letters, volume 70, 2253
13) Bagnato, V., Pritchard, E.E., Kleppner D. (1987) Bose-Einstein Condensation in
an External Potential, Physical Review A, volume 35, 4354
14) Davis, K.B, Mewes, M.O., Ketterle, W. (1995) An Analytical Model for
Evaporative Cooling of Atoms, Applied Physics B, volume 60, 155
27
15) Ketterle, E., van Druten, N.J. (1996) Evaporative Cooling of Trapped Atoms,
Advances in Atomic, Molecular and Optical Physics, volume 37, 181
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Atoms in a Vapor Cells, Physical Review Letters, volume 65, 1571
28