Laser Spectroscopy andMagneto-Optical Trapping of
Rubidium Atoms

A Thesis Presented
byDavid E. Sproles
to
The Graduate Schoolin Partial Fulfillment of the Requirements
for the Degree ofMaster of Scientific InstrumentationinPhysics

Stony Brook University
June 2008

Stony Brook University
The Graduate School

David E. Sproles
We, the thesis committee for the above candidate for the Master of Scientific
Instrumentation degree, hereby recommend acceptance of this thesis.

James E. Lukens
Professor, Department of Physics and Astronomy

Thomas K. Hemmick
Professor, Department of Physics and Astronomy

Michael M. Rijssenbeek
Professor, Department of Physics and Astronomy

Dominik A. Schneble
Thesis Advisor and Assistant Professor, Department of Physics andAstronomy
This thesis is accepted by the Graduate School.

Graduate Schoolii

Abstract of the Thesis

Laser Spectroscopy and Magneto-Optical
Trapping of Rubidium Atoms
byDavid E. Sproles
Master of Scientific InstrumentationinPhysics
Stony Brook University
2008This thesis describes experiments on laser spectroscopy and magnetooptical trapping of 87 Rb atoms. We perform polarization spectroscopy with a novel high-power diode laser system. An ultrahighvacuum apparatus in the 10−10 Torr regime has been built which allows trapping of ultracold clouds of rubidium atoms. These atoms
can be moved in a mechanically translated magnetic trap from thetrapping region into the lower pressure region of the system withminimal losses. We demonstrate magento-optical trapping of ultracold atoms, with atom numbers up to 1010 atoms and temperatures
below 1mK, as observed by fluorescence imaging.

iii

Contents
Acknowledgements

vi

Introductionvii

1 Atoms in Light Fields
1.1 Lorentz Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Two-Level Atoms . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
5

2 Spectroscopy of Rubidium
2.1 Spectral Broadening Mechanisms . . . . . . . . . . . . . . . .
2.1.1 Spontaneous Decay . . . . . . . . . . . . . . . . . . . .
2.1.2 Collisional/Pressure Broadening . . . . . . . . . . . . .
2.1.3 Power Broadening . . . . . . . . . . . . . . . . . . . . .
2.1.4 Doppler Broadening . . . . . . . . . . . . . . . . . . .
2.2 Spectroscopy; Doppler-Free Techniques . . . . . . . . . . . . .
2.2.1 Saturation Spectroscopy . . . . . . . . . . . . . . . . .
2.2.2 Polarization Spectroscopy . . . . . . . . . . . . . . . .
2.3 Laser Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Feedback and Results . . . . . . . . . . . . . . . . . . . . . . .

8
8
9
9
11
12
14
16
16
20
21

3 Magneto-Optical Cooling and Trapping
3.1 The Scattering Force . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Doppler Cooling - Optical Molasses . . . . . . . . . . .
3.1.2 Temperature . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Sub-Doppler Cooling . . . . . . . . . . . . . . . . . . .
3.2 Magneto-Optical Trapping Theory . . . . . . . . . . . . . . .
3.2.1 One-Dimensional Model . . . . . . . . . . . . . . . . .
3.2.2 Three-Dimensional Model . . . . . . . . . . . . . . . .
3.3 MOT Loading . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Capture Velocity . . . . . . . . . . . . . . . . . . . . .
3.3.2 Capture Rate . . . . . . . . . . . . . . . . . . . . . . .

24
24
26
28
29
32
32
35
37
37
38

iv

3.3.33.3.4

Loss Mechanisms . . . . . . . . . . . . . . . . . . . . .
Loading Dynamics . . . . . . . . . . . . . . . . . . . .

39
40

4 Implementation
4.1 Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Laser Optics . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 MOT Optics . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Pressure Regimes and Assembly . . . . . . . . . . . . .
4.2.3 Pumpdown and Bakeout . . . . . . . . . . . . . . . . .
4.3 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Load Calculations . . . . . . . . . . . . . . . . . . . . .
4.3.2 Electronics and Failsafe Programming . . . . . . . . . .
4.3.3 Control Programming . . . . . . . . . . . . . . . . . .
4.4 Magnetic Coils . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Calculations . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Construction . . . . . . . . . . . . . . . . . . . . . . .

41
41
43
45
46
47
48
49
50
53
53
55
56
57
59
60

5 Cold Rubidium Clouds: MOT Performance
5.1 Imaging Techniques . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Absorption . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Fluorescence . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Temperatures: TOF . . . . . . . . . . . . . . . . . . . . . . .
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63
64
66
69
72
75

Bibliography76v

Acknowledgements
I would like to thank Prof. Schneble for all his help in the lab and with writing
this thesis. In the early days when we had just a laser, a handful of optics
and a donated table, he would both teach and discuss the relevant design concepts with me, regarding upcoming choices. Granted, I didn’t have much tocontribute to these ‘discussions’ beyond questions, but I certainly learned agreat deal. As the lab came together, he kept things organized and moving ina direction that was clear. Despite any frustrations, the lab was always goodplace to work with an exciting atmosphere. It was a wonderful experience to
see and help the system grow from a blank optical table to a working, welltuned MOT. Seeing the BEC results and more recently atom diffraction acrossa lattice is very exciting indeed!
I’d like to thank all of the students in the group for their many contributions. Especially Rebekah Schiller for her bright attitude and help with many
aspects of getting the system running, Daniel Pertot for his good nature andcertainly his patience in our collaboration on the TOP coils and last but certainly not least Hartmut Ruf for his help with collecting and analyzing dataand in general for being an entertaining and very helpful labmate. While I
have not worked closely with all the students in the lab, nearly everyone has
made a contribution to the group which in some way affected this thesis: thankyou all for your help.
I’d like to thank my parents for their interest in my work and for supportingme through my trials in all of college, both rounds. Lastly I’d like to thank myfriends for tolerating me as I worked on this thesis and I’d like to thank themfor encouraging me; in particular (and no discernable order) Mike A., Steve F.,
Evan B., Katie F., Livy K., Daniel S., Jay S., the Parsons (congratulations!),
the Lichtenbergs, Frank McW., Alice L., all of WFDL1 (may it rest in pieces)and (though only briefly...) SFWE, and most certainly all of LGBTA and its
orbiting grads. I don’t know if I could have made it this long without you.aaTriple entendre intentional.

Introduction
In the 1890s, Planck fought with equations derived from what he understoodas ‘first principles’[1]. In researching light emitted from thermal objects, if
the energy of emitted radiation at a given frequency was an integer multipleof a fundamental value, everything worked. Planck is often quoted as saying
the assumptions he made in late 1900 to arrive at his radiation law were ”anact of desperation.”b This integer multiple was his first reluctant step towards
quantization.
Louis de Broglie took the next step. In his 1924 thesis, he suggested thatmatter could behave as a wave. In 1927 Davisson and Germer[3] publishedin Nature, describing their experiment where a narrow beam of electrons was
directed at a crystal of nickel. One of two graphs in the two page paper showsthe electron intensity varying in a clear diffraction pattern.
Also in 1924, Satyendra Nath Bose made significant advances; namely
he came up with an entirely new method of deriving Planck’s formula. He
tried to publish on his own, but it required Einstein’s support (which he gaveemphatically[4]) to get it published. His paper is widely viewed as one of theseminal papers in quantum statistics. Therein the concept of degeneracy incounting photons[5] was introduced quite obliquely.
This fundamentally new concept led to Einstein writing a paper in 1924[6]on the quantum theory of monatomic ideal gases. Eventually this led to theidea of Bose-Einstein condensation, the culmination of all these works: coherent and overlapping matter wavefunctions. There were, however, serious
technical difficulties in experimentally realizing these ideas. It would takedecades to reach an entirely new way of cooling atoms to condensation.
When trying to bring atoms to Bose-Einstein condensation, there is a great
deal more involved than simply making them cold. The volume of the ‘container’ holding them becomes quite relevant. Simply compressing a balloon
changes nothing if there is no commensurate removal of heat. To quantify thisb

It is said he made this comment in reference to depending on Boltzmann’s theories atthe last moment. He was not comfortable with their tenets[2].

viiconcept, consider the de Broglie wavelength of an atom in motion,
s
r
~2
2π
L̃ =
kB T M(1)

This couples the temperature to a linear dimension. Combining this with thenumber density gives us the dimensionless phase-space density,
ρ 'nL̃3(2)

The nitrogen in the air around you has a phase-space density of roughly ∼10−10 . Liquid nitrogen (close to boiling, at atmospheric pressure) has a valueof ∼ 0.0008. To attain Bose-Einstein condensation, a phase-space density of∼ 2.6 is needed[7].c
In an experiment to create an ultracold cloud of atoms and eventually
produce a BEC, many technologies[8] are required. Extremely low pressures,
lasers for cooling[9, 10], lasers and magnetic fields for trapping (both optical[11]and magnetic) and construction of a conservative potential that can be adjusted (allowing evaporative cooling). The result is a nanokelvin cloud of
atomic gas known as a Bose-Einstein Condensate.
In this thesis I will describe a specific implementation of laser cooling of
87
Rb atoms in a novel BEC apparatus. Laser cooling[9] is performed with ahigh-power diode laser system that is locked via a polarization spectroscopy[12]technique. Atoms are caught from background vapor and trapped in a sixbeam vapor cell MOT.
This thesis is organized as follows: chapter one covers some basic theoryof the interaction of a two-level atom with light. Chapter two discusses someof the concepts of laser spectroscopy and describes our spectroscopy techniqueand its implementation. Chapter three briefly describes some of the applicable
theory of laser cooling and its application to the trapping of atoms in our setup.
Chapter four gives an overview of the experimental apparatus which was set
up in this thesis work, while chapter five briefly reviews imaging of cold atoms
and the preliminary results obtained during that time.

cInterestingly, 4 He reaches this value very close to its superfluid transition temperature.

viii

Chapter 1
Atoms in Light Fields
In this chapter I will discuss a classical model of the interaction of light withatoms. The effect of a light field on transitions will be explored and equations
will be set up for the development of spectroscopy and laser cooling in later
chapters.

1.1

Lorentz Model

A classical view of the interaction of atoms with light will suffice for manyof the effects relevant to laser cooling. A straightforward model developed by
Lorentz[13] begins with the following view of the atom: consider the electron
as bound to the nucleus by a linear restoring force characterized by a ‘spring
constant’ k. The applied electric field (from a light wave traveling in the z*

direction) is given as E(z, t). A linear restoring force and the applied driving

1force of this electric field mean we have a driven harmonic oscillatora :*me × ∂t 2 x = eE(z, t) − k x*(∂t 2 + ω0 2 )x =*q
with ω0 ≡

k
me

*

e *
E(z, t),me

as the resonance frequency. In this model the atom’s inter-actions with light fields are characterized by ω0 . Including a damping term
β in the oscillator equation will allow many more details to be revealed. The
source for this damping term will be covered later. Take the applied light field*

as E = x̂E0 e−i(ωt−kz) ,e
E0 e−i(ωt−kz)
m∂t 2 x + 2β∂t x + ω0 2 x = x̂
*

*

*

*

*

By assuming the solution has the form x = ae−i(ωt−kz) the electron’s oscillation
amplitude is given by*

a = x̂(e/m)E02(ω0 − ω 2 ) − 2iβω
*

(1.1)

*

The polarization of an atom is defined as p = ex. The polarization density**of a bulk gas (with number density N ) is P = N ex; define α(ω) such that**P = N Eα(ω). Let ω̃ 2 ≡ (ω0 2 − ω 2 ):e2 /mα(ω) = 2
ω̃ − 2iβω

a

(1.2)

The nucleus must be assumed stationary, which is reasonable given the mass ratio.

2

*

*

*

By using Maxwell’s equations and a vector identity, we have ∇2 E − 1c ∂t 2 E =*1∂ 2 P . The assumed form for the electric field reduces this to²0 c t

ω2
−ω 2
α(ω)
−k 2 + 2 = N
c
²0 c
µ ¶2
kc
α
=1+N
ω
²0
α
n2 (ω) = 1 + N
²0

(1.3)

Here n = kc
is the index of refraction of the gas; for a dilute atomic gas, α isωsmall and n is approximately one. Expanding to first order and approximatingω̃ 2 ≈ 2ω(ω0 − ω) leads to
1 N e2ω̃
1 N e22βn =1 ++i
2 ²0 m 2ω(ω0 − ω) + 4βω 2
2 ² m 2(ω0 − ω) + 4β 2 ω
{z
}
{z
}
|
| 0
dispersive=n0(1.4)

absorptive=αn = n0 + iα

The two components of this complex index of refraction are called the KramersKronig dispersion relations. The absorptive component can be rewritten interms of frequency ν ≡ ω/2π and δν0 ≡ β/2π. This introduces the Lorentzianlineshape function L(ν);δν0 /π(ν − ν0 )2 + δν0 2
N e2
a(ν) =
L(ν)
4²0 mc

L(ν) =

3

(1.5)

n'

ΑΑ0

0.5

2Β

1

2Β
Ω

Ω

Ω0

Ω0

Figure 1.1: Graphs of the absorption and dispersion curves from the
Kramers-Kronig dispersion relations. Here 2β denotes the linewidth of theresonance[14], which is ultimately limited by the natural linewidth of the transition (see Sec. 2.1.1). This width can and will be affected by many otherfactors, discussed in Chapter 2.
The plots in Fig. 1.1 are central to the upcoming sections; the lineshape ofn is an estimate of sharpness of the transition feature that can be obtained.
Several effects discussed below tend to blur this response curve and obscure
spectroscopic information.
The Lorentz model comes very close to predicting good values for opticaltransitions. The theory is patched by considering that the ‘oscillator’ has a
proscribed ‘strength’, determined experimentally for each transition.
N f i e2
e → f e : ai (ν) =L(ν)
4²0 mc22(1.6)

This patch to the theory was developed from experimental observations,but has an explanation originating in quantum mechanics. The use of oscillator
strengths provides sufficient freedom to bring the Lorentz model into good
agreement with experimental results.

4

1.2

Two-Level Atoms

The Lorentz model approximates the behavior of a gas (in the limit of low excitation). A more accurate description for stronger fields requires a semiclassicalapproach, which also captures the internal level structure of the atoms.
Let’s begin with the Schrödinger equation for a two-level system, withenergies given as E1 and E2 and interaction matrix elements with the elec*

tromagnetic field given by Ũij =< Ψi |U |Ψj >= − 12 e²̂(E0 e−iωt + c.c.) < r >ij .
Parity arguments leave the diagonal elements as zero, leaving us with










 ȧ1   E1 Ũ12   a1 
i~ 
=


Ũ21 E2ȧ2
a2
From here we set our ground state energy E1 = 0, define the transition1and define the applied light field strength
energy by its frequency ω21 ≡ E2 −E~

according to the Rabi frequency, given as³*

χij ≡ e hriij · ²̂

´E

0

(1.7)

~

Near resonance ω ' ω21 is a good approximation. Now we apply therotating wave transformation: a1 (t) = c1 (t) and a2 (t) = c2 (t)e−iωt , and obtain










0
 ċ1  i=−1/2 (χ21 + χ∗12 ei2ωt )ċ2−1/2 (χ∗21 + χ12 e−i2ωt )   c1 

ω21 − ωc2
(1.8)

The rotating wave approximation asserts that e−i2ωt will be much faster than
atomic evolution, leaving those quantities averaged out. Define the detuning5∆ as ω21 − ω and letting χ = χ21 ;ic˙1 = −1/2χ∗ c2
ic˙2 = ∆c2 − 1/2χc1

With the assumption that the Rabi frequency (and thus the electric fieldpstrength) is real, χ ∈ R, and the definition Ω =χ2 + ∆2 we can solvethese equations as
¶
µ ¶¶
Ωt
∆
Ωt
c1 (t) = cos+ i sine−i∆t/2
2
Ω
2
µ
µ ¶¶
Ωt
χ
c2 (t) = i sin
e−i∆t/2Ω2µµHere Ω is the generalized Rabi frequency; the frequency of oscillations between
the two levels.
From this point we can build the density matrix from the equations for theci (t) equations. This matrixb will allow us to include the details of populationtransfer explicitly: collisions and spontaneous emission specifically.
∗ c1 c1ρ≡

c2 c∗1c1 c∗2  ρee ρeg =c2 c∗2
ρge ρgg(1.9)

Here the index “e” (“o”) denotes the excited (ground) states of the system.
These can be solved to give more general population density equations. In
full detail, one includes loss rates from each level (out of the 2-level system) dueb

In this matrix there are only three independent quantities; Feynman[15] noticed thisand suggested a very intuitive description of the two level system.

6

to atomic collisions (Γ) and spontaneous emission rates A21 . For our timescales
and pressures, loss rates out of the system due to collisions are exceedingly
low.


ρ˙gg


∗
»1»
=»
−Γ
ρ»
11 +A21 ρee − i/2(χρge − χ ρeg ) | {z} | {z }
coll.

ρ˙ee



spont.emmis.

»

»2»
−Γ
ρ22
=»
−A21 ρee + i/2(χρge − χ∗ ρeg ) | {z} | {z}coll.

(1.10)



spont.emmis.

Through several approximations and via multiplying by atomic density, we
obtain the population rate equationsṄe = A21 Ng + σΦ(Ng − Ne )






Ṅg = −A21 Ng − σΦ(Ng − Ne ) (1.11)

where σ is the atomic absorption cross section and Φ is the photon flux.
These are the population rate equations for a two level system. Solving
yields¸N σΦ
N σΦ
e−(A21 +2σΦ)t +Ng (t) = Ng (0) −A21 + 2σΦA21 + 2σΦ·(1.12)

We can now use these equations to establish some of the principles of spectroscopy and laser cooling.

7

Chapter 2
Spectroscopy of Rubidium
In this chapter, laser stabilization and control will be discussed. Level structure
involved in spectroscopy and relevant cooling transitions will be discussed, as
will laser design allowing tight control of emitted wavelength.

2.1

Spectral Broadening Mechanisms

The stability of the lasers used will determine the final stability of the atomic
trap as well as the precision, accuracy and reliability of imaging. The lasers
produce light at hundreds of THz, must be tunable on the scale of GHz yet also
must be stable on the scale of the linewidth (several MHz) of the transitions
that will be used. To meet these requirements, high performance frequency
metrology is needed alongside electronic feedback in order to lock the lasers to
the correct transitions. The required well-characterized error signal is obtained
via spectroscopy.
Several effects will broaden the frequency response of atoms to light fields.

8While these broadening effects do not play a role in dilute, ultracold atomic
clouds, they exist in relatively high pressure spectroscopy vapor cells, andmust be thwarted to make frequency control possible. Some of these effects
are negligible and easily counteracted while some require clever techniques tomanage.

2.1.1

Spontaneous Decay

There are several ways to derive the natural linewidth of transitions; one acceptable, if pedestrian, way is to consider the uncertainty principle[14]. Excited states (energy Ee ) have decay rates and mean lifetimes (τe ). Such alifetime fits naturally into the Heisenberg relation: ∆Ee ' τ~e . Given that thisexcited state decays into the ground state, Eg , the frequency of emitted lightwill be ωdecay = (Ee − Eg )/~ with an uncertainty based on that of the excitedstate:∆ω = γ ' ∆Ee /~.

(2.1)

This gives us the natural linewidth of the transition of our two-level system.
This is a fundamental limit. For the relevant transition to cooling Rubidiuma ,the natural linewidth is 2π ∗ 6.0666(18)MHz[16].

2.1.2Collisional/Pressure Broadening

In the classical oscillator model in Ch. 1, a ‘friction’ term (β) was introduced
but little explanation provided. It eventually determined the width of the
a 2

5 S1/2 → 52 P3/2

9

resonance. As stated above, it has a minimum value, but many effects cancontribute to it. This term in the equation of motion can be justified byelectron ’dephasing’; within the framework of the model, it is reasonable tounderstand collisions with other atoms as affecting the electron’s displacement
and thus resetting the oscillator’s phase. Consider a collection of atoms all set
in motion together. Each collision will change the phase of oscillation which
will, on average across the collection, cause the electron oscillations to bedamped. This damping (i.e. ‘friction’) will broaden the frequency response,
leading to decreased accuracy in determining the specific wavelengths associated with transitions. In a spectroscopy cell, this is clearly unacceptable as
our lasers must be stabilized to the order of the natural linewidth.
The easiest solution to reduce collisional broadening is to isolate the sample of atomic gas in a low-pressure environment, as with our reference vapor
cell. Too low a pressure obviously presents its own problems with regards tomaintaining gas purity at that pressure; regardless of this difficulty, simply
decreasing the pressure in the cell is insufficient. Collisions with walls are nodifferent than collisions with other gas atoms: decreasing the pressure will
only reduce the frequency of collisions up to a point, beyond which the atoms
will be bouncing off the walls. The solution lies in noble gases, which aresymmetric and spinless. Collisions with them are significantly less likely todephase the electron’s oscillations.[17, 18] This constrains the background gas
pressure to an ideal value, maximally reducing wall collisions without serious
broadening from noble-gas collisions. This keeps the pressure in the reference
at a level requiring nothing more than the glass walls to maintain it at therequisite gas purity.
102.1.3

Power Broadening

Let us pick up 1.9 where we left off. The earlier solutions were only for ρgg andρee . If instead solved for ρge and ρeg one again pulls out β. In this context, weare only considering light effects on isolated atom(s), thus β is the relaxation
time and is dominated by 1/τ , the decay time of the excited state.
Continuing with Eq. 1.10, dropping the first term in the high-power (Φlarge) and long time limit yields:ρee =N2 (t → ∞)σΦ/A21=N
1 + 2σΦ/A21
χ2 β/2A21= 2,∆ + (β 2 + χ2 β/A21 )

(2.2)
(2.3)

with χ ∝ E0 , where E0 is the electric field strength. Here we have a Lorentzian
shape dependent on the Rabi frequency. More power results in higher Rabi
frequency; this will broaden the lineshape.
Qualitatively, power broadening is a result of transition saturation. The
light field intensity is such that many atoms are in the excited state, leaving fewer atoms available for absorption near the transition. Away from thetransition less saturation occurs, leaving more atoms available for absorption.
The easiest solution to this problem is to use weak pump and probe beams.
Detectors can be fitted with inexpensive line filters that will restrict sensitivity
to laser light only, avoiding ambient contributions.

112.1.4

Doppler Broadening

The Doppler effect refers to the frequency shift of an observed electromagnetic
wave due to atomic motion along the wave’s direction of propagation. Thiseffect is central to spectroscopy as well as cooling processes. Given light propagating in the k̂ direction and an atom’s velocity (nonrelativistic, of course)
*

as v

*

νabsorbed = νemitted (1 −v · k̂
)c(2.4)

The usual terminology is to refer to the absorbed light as blue- or red-shifted in
the reference frame of the atom; blue-shifted light increases the frequency andenergy, red-shifted vice versa. Let us briefly look at the velocity distributions
present in a gas of atoms.
Velocity Distributions
There are two ways to view the distribution of atomic velocities; the velocity ofatoms along one specific direction (orthogonal velocities count as zero along thedirection of interest; Fig. 2.1b) vs. the velocity of atoms in any given directionp(spherical, integrating all velocities as positive; Fig. 2.1a). Let ṽ ≡ kB T /M .
This is the characteristic thermal velocity of the atoms, based on temperature.
The Maxwell-Boltzmann distribution is given by
r

µ
¶
2 v2
v2
S3D (v) =exp − 2π ṽ 2
2ṽ
¶
µ
1
vx2
S1D (v) = √exp − 2
2ṽ
2πṽ

12

(2.5)
(2.6)

0.0035
0.00300.00250.00200.00150.00100.00050.00200.00150.00100.0005v @msDv @msD
100 200 300 400 500 600 700-600 -400 -200(a) S3D (v) MB distribution in 3D.

200

400

600

(b) S1D (v) MB distribution in 1D.

Figure 2.1: The Maxwell-Boltzmann distributions for 87 Rb atoms at room
temperature.
The idea of laterally moving atoms is significant to Doppler-free spectroscopy; primarily of interest to this section is the 3D distribution of velocities.
Lineshape Function
The stochastic thermal motion of atoms leads to inhomogeneous spectral
broadening via the Doppler shift; some atoms will absorb red-shifted lightand some absorb blue-shifted light. Putting the one-dimensional MaxwellBoltzmann distribution together with the Doppler shift yields the lineshapefunction for Doppler broadening. Taking T as the gas temperature in Kelvin,
M as the molecular weight of the gas and λ0 as the wavelength in Å, one
obtains[13]1S(v) =
δvDµ

4 ln 2
π

¶1/2

1
2
2
e−4 ln 2(v−v0 ) /δvDMHz
" µ ¶ #
1/2
T
1MHzδvD = 2.15 × 106
λ0 M

13

(2.7)
(2.8)

Substituting in the values for Rubidium atoms at room temperature:"δvD =2.15 × 106

17806Å

µ

298K
85.5g

¶1/2 #

δvD ≈514MHz

This calculation yields the width for an isolated transition of room-temperature
atoms as measured by one directed beam.
The linewidth of the transition to which we must stabilize is roughly 6 MHz.
While the Doppler broadening of a transition by hundreds of MHz seems severe, there are techniques to overcome this apparently large difficulty.

2.2

Spectroscopy; Doppler-Free Techniques

All methods used to overcome Doppler broadening are built on the idea ofcounter-propagating beams. Light from a laser is split into a probe and pumpbeam; these beams are sent through the gas sample in opposite directions.
Fig. 2.2 illustrates this.
In three dimensions large numbers of atoms can move orthogonally to thepump-probe axis. These atoms can interact with both beams at the rest-frame
frequency. For atoms moving along the pump-probe axis there will be Doppler
shifts in opposite directions for each beam. The result is that, for off-resonancefrequencies, each beam will interact with different classes of atoms. From thisgeneral Doppler-free setup there are two easy ways that spectroscopy data canbe obtained.

14

Absorption Ha.u.L

Absorption Profile for Counterpropagating
Beams in an Atomic Gas-∆k(a) Atoms with thermal velocity,
only (x,y) shown.

v=0

+∆k

Velocity(b) Plot of absorption vs. velocity.

Figure 2.2: Doppler shift and counter propagating beams (of wavevector ±kx̂). (a) A collection of thermal atoms with random x-y velocity (z not shown).
The atoms are shaded according to ‘observed’ frequency. Boxes are drawn
around different classes of atoms resonant with their respective beam. The
top has a right-propagating beam, the lower is left-propagating. (b) A graphof absorption dependent on velocity for a laser detuned from νres by δ. Atoms
traveling along a beam’s propagation direction will absorb according to a labframe-velocity of v = (νres − δ)/k, while atoms traveling against will absorb
at a lab-frame-velocity of (νres + δ)/k. When the Bennet holes overlap (thiscondition shown by the dashed curve), both directional beams will then interact with the same velocity-group of atoms. This drop in absorption (for eachbeam) is called a Lamb dip.

15

2.2.1

Saturation Spectroscopy

Saturation spectroscopy works by saturating the transitions via the pump
beam. The pump beam will reduce the population in the lower level of thattransition. The probe beam will then experience reduced absorption (thusmore light will pass through) near that frequency. Fig. 2.2 shows a sketchedabsorption spectrum in velocity space; the small dips in grey refer to the effectof each beam and are called Bennet holes[14]. When the laser is on resonancethese ‘holes’ in the spectrum will overlap; that drop in absorption is called a
Lamp dip[13].
By scanning the laser through the frequencies around this transition, we
can obtain the transition spectrum. This spectroscopy signal is now free of
Doppler broadening.
One of the technical difficulties with saturation spectroscopy is that thecenter of the transition is located at a point where the signal has no slope.
Scanning the laser continuously to see the resonance goes directly against theidea of a stabilized laser. To generate a signal for feedback, a dispersive shapeis required. Phase-sensitive detection (e.g. a lock-in amplifier) is needed togenerate a dispersively shaped signal. The complexity of the setup can bebypassed as discussed in the next section.

2.2.2

Polarization Spectroscopy

From one perspective polarization spectroscopy works in much the same way
as saturation spectroscopy; rather than recovering signal from the absorptive
component of Eq. 1.4, the dispersive index of refraction serves as the source.

16

PBS

PBS

λ/2

λ/2

λ/4
Balanced
Polarimeter
Figure 2.3: Balanced polarimeter. The input beam comes from the right and issent through a proportional splitter, allowing the ratio of pump/probe amplitude to be adjustable. A 50/50 non-polarizing beamsplitter is used to redirect
the pump beam backwards through the vapor cell and allow the partially attenuated probe beam through to the balanced polarimeter. The half-waveplateallows the polarimeter to be balanced according to maximum dispersive signal
strength.
From another point of view, polarization spectroscopy is entirely different inthat it makes use of the multi-level structure of atoms. This method avoids
intensive signal processing: light-induced birefringence generates a dispersive
signal that can be directly fed into the feedback loop.
The setup for polarization spectroscopy is shown in Fig. 2.3. A more intense σ + polarized pump beam pushes each lower-level electron into a ‘higher’bmagnetic sublevel. This is called optical pumping[14], leading to nearly all the
population undergoing Rabi oscillations between the highest possible magnetic
sublevels of the ground state and the possible sublevels of the excited statesbWith respect to the quantization axis defined by the pump beam.

17

2

5P

3/2

F’=3

mf
-3 -2 -1 0 1-2 -1 0

1

30
2

5

156.947(7)MHz

F’=1

3

2

266.650(9)MHz

F’=2

2

-1 0

1

3
2

5S

1/2

σ-

384.23 THz

σ+

F=2

Figure 2.4: Schematic of available transitions. Clebsch-Gordon coefficients
indicated in boxes for each transition.[9, 16](Fig. 2.4.)
Consider the appropriate sublevels of the excited states. Each transition toeither higher or lower mF will have a strength (stronger causing more absorption), determined by calculating the Clebsch-Gordon coefficients. See Fig. 2.4
for the strengths[9]. If, for one F , the mF + 1 and the mF − 1 have different
strengths, then σ + and σ − beams will experience different indices of refraction. The probe beam is linearly polarized. It can be decomposed into leftand right-circularly polarized components. The difference in the transition
coefficients determines the difference in optical ‘thickness’ for one component
versus the other. Thus, a ‘tilt’ of the linear probe beam’s polarization develops, which we will call ∆φ. Given a length of vapor L, the linear probe beam
will have its polarization rotated by∆φ = (k + − k − )L = (ωL/c)(n+ − n− )

18

(2.9)

The indices of refraction are given by the n0 component of Eq. 1.4. The
difference between transition strengths for the σ + and σ − determines both thesign and the magnitude of ∆φ.
The rotation of the pump’s polarization can be detected by blocking one
linear component and measuring the transmission of the other. If a balanced
polarimeter is used instead, the signal magnitude is increased and some background effects are diminished or removed. Some of the background is due tothe birefringence of vapor cell’s windows. This only results in a small linear
offset[19] which is easily compensated via the optics in the balanced polarimeter. This polarimeter is set up with a half-waveplate and a polarizing beamsplitter. The two photodiode signals are then subtracted. Torii and Kuga[12]give the transmissionc of this balanced polarimeter as∆It = It (45o ) − It (−45o )(2.10)= 2I0 e−αL ∆φ(2.11)

Here α is the absorptive component from Eq.1.4. The output polarization of
the linearly-polarized probe beam is rotated according to distance from thecenter of the transition. The signal for the polarimeter is dispersive in shapeand resembles the index of refraction graph in Fig. 1.1.
In practice, setting up this spectroscopy is straightforward. The angle ofthe half-waveplate in the balanced polarimeter is set experimentally according
to minimized background across the whole transition in question. If care iscPlease note that we have glossed over the effects of gas density in the vapor cell; thiswill obviously scale the effect, but does not change functional form.

19

Laser Diode & Cooling Block

G
back
Feed

al
Cryst
Piezo

rating

side
ored
Anch

ting
of graFigure 2.5: Setup for rear-coupled ECDL. Note the lever arm used by thepiezo to modulate the cavity length.
used, the diode signals can simply be wired against each other in a circuit. For
greater certainty, a simple op-amp circuit can be used to subtract the signals.
The most attractive feature of this design is the lack of equipment required.
No phase sensitive detection is needed to extract a dispersive signal; the output
can be fed directly into a laser’s control circuit, closing the feedback loop.

2.3

Laser Design

Each of the lasers in our lab is an extended cavity diode laser (ECDL) set
up roughly as in Ref. [20]. Relevant to the control of the laser system is thepiezo-mounted grating. The rapid response time and ease of incorporation
into control loops of this setup is ideal.
For the Sacher laser, one relevant modification to this ECDL setup is thatthe tuning grating is set up on the rear facet of the diode. See Fig. 2.5. The
majority of the power is output via the opposing facet. The advantage of therear-face coupled design is the lack of angular modulation of the output beam
as various frequency settings are adjusted to find a good operating point. The
Toptica laser is not rear-coupled.
This design modulates the frequency not by the variation of the angle of20

the grating, for that effect is trivial on the relevant scale. Instead, the rear
facet of the diode and the grating form a cavity, external to the laser. As thecavity length varies, so does the frequency. The cavity length must containan integer number of half-wavelengths; for small perturbations around a given
optimal longitudinal mode, the frequency will shift, rather than the numberof wavelengths in the cavity. The external cavity length (Lec ) contains many(nec ) wavelengthsd : Lec = nec λ. Thus any change in external cavity length,ec∆Lec results in a fractional wavelength change, ∆λ = ∆L
. A modest piezo
necthat would fit into the laser system shown in Fig. 2.5 can generate maximum
deflection to the scale of hundreds of nanometers, with resolution approaching(many) tenths of a nanometer: about a factor of 1000, usually more. Combined, these effects produce the requisite fractional adjustment of better than
1:1,000,000 in terms of smallest input shift to frequency modulation.

2.4

Feedback and Results

For isolated transitions, the observed spectrum will conform to the conclusions
above. If there are several closely spaced transitions, however, the observed
spectrum will require further analysis. For example, consider transitions fromground state ‘A’ to levels ‘1’ or ‘2’. Given a source laser slightly detuned suchthat the laser frequency, νl , lies between the transition frequencies of A→1 (ν1 )and A→2 (ν2 ), an atom could be moving at a velocity causing it to absorb
ν1 + νDopp and ν2 − νDopp , both of which can equal νl . The detector will observea transition where there isn’t one. This type of interaction is called a crossoverd

A 5mm external cavity would contain about 6,500 wavelengths.

21

F’=1 2 2-3
424MHzF’=3
F’=1-2 1-3
Figure 2.6: Oscilloscope trace of a balanced polarimeter signal of the 5S1/2 F =2 → 5P3/2 F 0 transition in 87 Rb. Crossover resonances are labeled. Verticalunits are photodiode voltage (thus arbitrary).

Figure 2.7: A full trace of the transitions of 87 Rb that will be used for trapping.
The width of this scope trace is ≈ 7.2 Ghz.[16] The lower traces are the separate diodes of the balanced polarimeter and clearly show Doppler broadening.

22

resonance. They were observed in our spectra and are labeled in Fig. 2.6.
Coupling the spectroscopy signal to the laser control electronics, the feedback loop is closed and the laser is locked to the correct transition. The feedback loop parameters (gain, damping, offset) are set experimentally to obtain
the best lock. See Fig. 2.7 for a full scan of all the levels discussed in Chapter
One. Figure 2.6 shows a closeup scan of the cycling transition lockpoint.

23

Chapter 3
Magneto-Optical Cooling andTrapping
Light carries momentum; light interacting with atoms allows us to engineer
forces such that cooling and confinement of atoms is possible. To accomplish
trappinga , magnetic fields will be combined with light from the carefully stabilized lasers of the previous chapter.
In this chapter we will first discuss cooling and temperature of atoms,
moving on to a simplistic model of atomic trapping. Lastly the loading ofsuch a trap is covered.

3.1

The Scattering Force

The force of laser light on atoms is a consequence of the asymmetry between
directed absorption from a laser field and random spontaneous emission. OnaAll optical traps are possible, but will not be covered in this document.

24

average, spontaneous emission has no effect on an atom’s momentum, whereas
absorption from one directional laser field supplies momentum in only one
direction. These microscopic momentum exchanges will average on a macroscopic timescale to what will appear as a force.
Though it would be unrewarding to exhibit in full detail, one can derive= [H, ρ] and Eq. 1.9. From
the optical Bloch equations beginning with i~ dρdtthese the total scattering rate γp of the light field from atoms can be recovered,

γp = (1/τ )ρee

(3.1)

If one works through the optical Bloch equations, one can recover a clear
expression for ρee . The result is shown below. Let s0 = IIs be the on-resonancesaturation parameter where Is is the saturation intensity and I is the intensityof the light field, γ b the decay rate of the excited state and δ the detuning.
This gives the scattering rate as[9]γp =s0 γ/2,
1 + s0 + (2δ/γ)2(3.2)

which is proportional to the excited state population and is thus a Lorentzian
as a function of detuning.
The scattering rate indicates the frequency of events that can impart ab

We are not considering broadening or Doppler effects here and will use γ rather thanβ.

25

force. Working from Eq. 3.2,

F orce = (momentum-exchanged) · (frequency-of-change)
Fscat = ~kγp
=

(3.3)

~ks0 γ/2
1 + s0 + (2δ/γ)2(3.4)

Using γp here folds in the light field properties as they affect the atom. Again,
the excited state population is limited to 1/2, thus the scattering force has
an upper limit, Fscat−max = ~kγ/2. This limit makes sense when considering
stimulated emission. Consider an atom that has absorbed light traveling in
the x̂ direction. If the light is sufficiently intense, the light field can cause
stimulated emission into the x̂ direction: this will yield a brief change invelocity but no net momentum change.

3.1.1

Doppler Cooling - Optical Molasses

The scattering force can now be used to engineer a light field that opposes anatom’s motion in 3D. For this purpose, consider six laser beams in counterpropagating pairs. With the appropriate (negative or red) detuning of the lightfields, the Doppler effect will reduce absorption along the direction of propagation while enhancing it against the direction of propagation. Using Eq. 3.4**

and ωD = −Γδ = k · v as the Doppler shift, we can reach an approximationc**

of FOM (v) [9]:cTreating the Doppler shift as small after the fractions of F+ and F− are joined, i.e.
dropping it from the final denominator.

26

F in ÑkΓF in ÑkΓ

0.10

0.10

0.05

0.05
v in Γkv in Γk
-4

2

-2

4

-3

-2

-1

-0.05

-0.05

-0.10

-0.10

(a) s0 = 0.25, δ = −1.5

1

2

3

(b) s0 = 0.25, δ = −0.5

Figure 3.1: Plots of optical molasses force showing both beams separately andadded. Note that with large detuning (left), capture velocity can be extended
without total sacrifice of low-velocity force.

*

~kγ
s0
Fscat,± = ±2 1 + s0 + [2(δ ∓ |ωD |)/γ]2
*

*

*

*

*

⇒ FOM (v) =F+ + F−**

FOM (v) ' −s0
8~δΓk 2**2 · v = −ηf ric v
2
γ (1 + s0 + (2δ/γ) )(3.5)

Here we keep track of ωD just long enough to reorganize the fractions, thenmake the reasonable approximation that ωD /γ ' 0. This is a frictiond force;as shown in Fig. 3.1b, the linear variation with respect to small velocities leads
to the approximation that ηf ric is a constant.
A key detail here is that there is no position dependence: atoms are always
entering and leaving the trapping volume. The intersection region for thecooling beams slows atomic movement, as if it were viscous. This leads to theterm ‘optical molasses.’d

This momentum-exchange-based friction force is distinct from the dephasing frictionof Sec. 2.1.2.

27

3.1.2Temperature

The ‘temperature’ for very low-density atomic collections can be defined as theaverage kinetic energy of these atoms; 12 kB T =< Ekin >. Thermodynamics
balks at the concept of temperatures of zero for a variety of reasons. Limits
to cooling processes are clearly expected.
In optical molasses, the forces that ‘cool’ atoms are based purely on light.
Force exchange is based on scattering events; light is directionally absorbed
but randomly emitted. These random emissions may average to zero but the**

momentum exchange is conducted in units of ~k where k is the photon wave
vector; the vroot−mean−square of these exchanges is very much not zero[9]. The
consequence is random motion in the molasses region.
To analyzee the heating effect from these exchanges, the frequencies of lightabsorbed and emitted must be considered;**ωabs =ω0 + kabs · v + Er /~**©©

© · v 0 − Er /~ωem =ω0 + ©
kem| {z }
Avg’s to zero*

Here ω0 is the resonance frequency of the transition in consideration, kabs =***ωabs /c, kem = ωem /c, v and v 0 are the velocities before and after absorption. Relativistic terms in the above equations have been omitted, as thevelocities are extremely low and relativistic frequency shifts are negligible.
Er = p2 /2M = (~k)2 /2M = ~ωr is the recoil energy, the cause of the heating
effect. We find the change in energy from an absorbed to an emitted photoneThis discussion of Doppler temperature taken from [9, 10]28

easily,**∆E =~(ωem − ωabs ) = −~kabs · v − 2Er(3.6)

In order for cooling to occur, this quantity must be positive; the photons mustcarry away energy. Note that because the wave-vector and the velocity havedirection, cooling can occur. The recoil energy is unavoidable and thus leads
to a heating rate proportional to the magnitude of this recoil and the frequencyof scattering events from both directional beams:

Recoil heating =2γp × 2Er = 4γp

(~k)2
2M

(3.7)
*

*

Setting this heating rate equal to the cooling rate derived from Eq. 3.5 (Fcool · v)
and a little algebra results in
1~γ (1/2)M v 2
M v2 =
γ
2
8 2δ
+ 2δ
γ

(3.8)

The minimum energy is dependent on the detuning. Appropriately minimizing
the energy results in the Doppler temperature:

TD = ~γ/2kB

3.1.3

(3.9)

Sub-Doppler CoolingExperimentalists demonstrated optical molasses in 1985[11], but by 1988 realized a bit more had been achieved: the resultant temperature was significantly29lower than predicted by theory[21]. The source was eventually found to be a
combination of complex level structure and beam interference effects.
If we return to Eqn. 1.8, we can look at the Hamiltonian matrix of thesystem. Please note that the following describes perturbations on the groundand excited energy levels. Recall, χ = χ12 ∝ electric field (Eq. 1.7) and∆ = laser detuning.
H=~ 0 χ 2χ ∆If we diagonalize, we find energy shifts of~(−∆/2 − χ)
2~Energyground(1) = (−∆/2 + χ)
2
Energyexcited(2) =Each of these level shifts in practice is more sophisticated; Clebsch-Gordon
coefficients are required, which depend on light-field polarization as well as
magnetic quantum numbers.[9] Nonetheless, it is clear there are level-shifts
dependant on electric field intensity; that is, the Stark shift.
Let us examine the case of two linearly polarized beams with orthogonal polarization angles. The interference of these beams results in polarization gradients; light alternates on a quarter-wavelength scale between σ + , linear(+45o ),σ − and linear(−45o ). The Stark shift of the ground states causes them to vary,as shown in 3.2. As the level shifts depend on the field intensity and thus theinstantaneous polarization, the sublevels involved in cooling will interchange

30

Schematic of crossed linear polarization cooling

Eg-unshifted

Eg-shifted

0

Λ
4

Λ
2

3Λ
4

Λ

Figure 3.2: Schematic of polarization gradient cooling, with two orthogonallinear beams. The solid path is an ideal case of cooling, the atom loosing the
maximum energy every quarter-wavelength. The energy lost is accounted forin the frequency of emitted light. This effect occurs alongside Doppler cooling,
maintaining the directionality indicated by the arrows on the solid level.
repeatedly. Absorbing from the light field will increase angular momentum
whereas spontaneous emission will on average have no effect on the angularmomentum. The net result is pumping from the higher sublevel into the lower
sublevel, thus loosing energy.[9] Note that the ground state cannot be significantly Zeeman shifted, as this will reduce the amount of cooling until it cancels
out the effect completely.
This is Sisyphus cooling, named so because cooled atoms must climb the
potential induced by polarization gradients again and again. This accounts forthe experimental observation of lower-than-expected temperatures[21]. Whilethe example here was for linear polarized beams, roughly the same cooling
effects for opposed circular polarization schemes are observed. The mechanism
is understood in a somewhat different way, but the practical differences are
small and further discussion is unnecessary.

31

3.2

Magneto-Optical Trapping Theory

The forces described above have no position dependence. They can cool atoms
but there is no trapping; atoms are always diffusing in and out of the the cooling region. In fact, it is quite possible to have different thermal distributions
in different dimensions in optical molasses. By using a magnetic field it ispossible to introduce a position dependence to the forces involved in theseslowing processes and cause accumulation and interaction of atoms in such ageometry.

3.2.1

One-Dimensional Model

Let us develop a simple one-dimensional model of a magneto-optical trap
(MOT). Take an atom with spin zero in the ground state (this reduces thecomplexity of multiple magnetic ground states) and spin one in the excited*state. Take a linearly increasing magnetic field B with zero at the origin; themagnitude near the origin**

B(x) ∝|x|.

Two counter-propagating beams of equal helicity[22] (thus opposite circularpolarization) are used, as shown in Fig. 3.3. These beams are red-detuned.
When the atom encounters a significant Zeeman shift, this beam will be
brought into resonance with a magnetic sublevel and the atom will experience a force.
The energy shift due to the Zeeman effect is dependent on the magnetic

32

Energy [MHz]
m=+1
m= 0

δ

m= -1
σ-

σ+

m= 0
x [cm]
Figure 3.3: One dimensional cartoon model of a MOT. Opposing circularly
polarized beams (opposite circular polarization, identical helicity) cycle lightbetween the ground state and the lowest magnetic state. Consider the x-axis
as m=0*sublevel, mF , the field strength B, and the Lande g-factors of the relevant
levels, gF . The Bohr magneton is µB = h · 1.399MHz/G.
*∆EZeeman =µB gF mF |B|(3.10)*∆EZ =µB gF mF A|x|

Here A is the slope of the magnetic field. Only the light interacting withthe atom is relevant; a different sublevel will be used from one side to theother of the field minimum. We can account for this by using the sign ofthe displacement. The ground and excited state will both be affected by the
Zeeman shift, complicating the picture somewhat. Here the subscripts refer33

to the ground and excited states,**h · δZ (x) =µB (ge me − gg mg )A|x|*(3.11)*δZ (x) =µ̃x(3.12)

To find the total force in a MOT, we need to consider Doppler coolingas well as the contribution from the above term. To do this, we will use thedetuning observed by the atom as determined by the laser detuning, atom
velocity and position (thus magnetic field)
*

* *

*

*

δ± (v, s) =δ ∓ k · v ± µ̃x(3.13)The same scattering force equation (Eq. 3.4) will be used to start this calculation;*

F± = ±~ks0 γ/2
1 + s0 + ( γ2 δ± )2

Obviously the detuning and thus the force will be different for the different
beams. This is embodied in the detuning.
***

FM OT =F+ + F−
*

~kγ
1
1
=
s0 (
+
)
*
*
*
*
*
*
2
1 + s0 + ( 2 (δ − k · v + µ̃x)) 1 + s0 + ( 2 (δ + k · v − µ̃x))γγ

Executing the same approximation procedure used previously, e.g. rearranging
the fractions and then assuming that the detuning is small and thus that we

34

L
R

R
R

R
L

(letters indicate helicity)

Figure 3.4: Primary MOT coils shown as orange loops, field direction indicatedby same-color arrows. The coils are in an anti-helmholtz configuration. MOT
beams and their helicities indicated in red.
are close to the center of the trap, the following result is obtained[23]***

FM OT = −ηf ric v − µ̃trap x(3.14)

which is a restoring force with damping. Overdamped motion is typical insuch a trap.

3.2.2

Three-Dimensional Model

A three dimensional MOT is significantly more complicated than the above
model. Two beams for each axis must meet in the same region where a controlled magnetic field can be generated. This field is usually generated by apair of anti-Helmholtz coils (large orange/copper coils sketched in Fig. 3.4 andshown in Fig. 4.1).

35

Increasing B

Hyperfinesplitting
F=3

2

5P

267MHz

3/2

157MHzCycling

F=2
F=0,1780.24nm
25S

F=26.83GHz

Repump1/2

F=1

Figure 3.5: Shows the cycling and repump transitions for the 87 Rb MOT.
Weak-field Zeeman splitting is schematically shown. Data from [16].
Many factors serve to complicate a real-world MOT. There are Zeeman
shifts of both the ground and the excited state, requiring a calculation of
Eq. 3.10 for each. Further complicating matters, an atom’s velocity as usedin the Doppler shift of Eq. 2.4 may not be parallel to any beam. Finally,
as an atom moves around the trap, the magnetic field changes direction andintensity in a well-known but still complex way.
Beyond the subtleties of three dimensions, hyperfine structure begins toplay a role. In Fig. 3.5 we see the transition that will serve as primary means of
momentum transfer, referred to as the cycling transition. Given the complex
level structure, occasionally an atom will decay to the wrong ground state(F=1) and the repump light will move it to a state where it can decay backinto the cycling transition. This light need not be very intense, nor come fromall directions.

36

3.3

MOT Loading

When considering the loading behavior of a MOT, one reveals many interesting
parameters. The number of atoms in a MOT is governed by a simple capture
rate[24] and two significant loss mechanisms[25]: collisions of cold atoms withhot background gasf and collisions of cold atoms with other cold atoms.
While a MOT is a trap for cold atoms, atoms must find their way into
the trap. Two schemes are widely used for this purpose. One scheme cools
the atoms by means of a Zeeman slower and then deposits them into a very
good vacuum where the MOT waits. If patient and willing to trap from theminuscule low-velocity fraction of the velocity distribution, one can trap in areasonably good vacuum from background vapor. Each scheme has its advantages and disadvantages; our apparatus was designed to trap from background
vapor.

3.3.1

Capture Velocity

A precise definition of the capture velocity of a MOT is difficult, as the asymmetrical nature of the trap geometry used in nearly all systems brings heavyemphasis to path dependence. One simplified approach to estimating thisvelocity relies heavily on the one-dimensional model illustrated in Fig. 3.3.
Consider an atom on the left side of the MOT, being stopped by the beam
originating on the right. At this point, the Doppler shift of the absorbed
light works in concert with the Zeeman shift. The maximum allowed Doppler
shift will set a limit on the velocity. Beyond this velocity, the radiation forcef

It is reasonable to assume background gas is nearly all 87 Rb .

37

from the beam will become increasingly small. The shift between excited andground state of 87 Rb is roughly 1.4 MHz/G. Assume a field of 20G, whichyields a 28MHz shift. Added to this Zeeman shift (on the far-side of the beamthat is performing the slowing) is the laser detuning (assume a typical value of
20MHz); we need a Doppler shift to match 48MHz. The transition frequency
in question for rubidium is ∼384THz. Atoms must be going 37.5m/s to see ashift of 48MHz. Slower atoms will eventually find a weaker field, coming into
resonance and being slowed.
The above result is within 20% of the following method. Ref. [23] gives
vc as the velocity of an atom stopped by half the maximum scattering force
within the radius of the trap (values of ' 2cm are reasonable). The maximum
scattering acceleration (recall 3.3) is given by a ' ~γlinewidth /4m.
√
vc = 2ar(3.15)=47.5m/s

Integrating the Maxwell-Boltzmann distribution properly, one finds that only
a fraction of a percentg of the available atoms are being trapped. The resultis experimentally determined loading times of approximately 10 seconds.

3.3.2

Capture Rate

To calculate the capture rate in a transparent way one needs the surface area ofthe trapping volume (or a reasonably good estimate) and the capture velocitygIntegrating Eq. 2.5 from zero to 37.5m/s, we find only 0.28% of atoms are being captured at any given moment.

38

of the trap. If an approximately spherical trapping geometry is assumed, thecapture rate R will be found[25] by integrating the velocity distrubtion

R =πr

2

Z vc
vS3D (v)dv.

(3.16)

0

3.3.3

Loss Mechanisms

There are two significant loss mechanisms in a MOT, intra- and extra-MOT
collisions; let N (t) be the number of trapped atoms, R the capture rate, Γ theloss rate due to hot background gas and β the loss rate due to cold intra-MOT
collisions. R can be calculated[24] by considering the cross section, flux andnumber of atoms in the distribution below vc .
dN
=R − Γcoll N (t) − βN (t)dtnbg 2/3 4 q m 3
R=
V vc
2kT
2
q
BT
Γcoll = nbg σbg 8k= nbg σbg vbg
πmbg(3.17)(3.18)(3.19)

Here nbg is the density of the background gas, σbg the cross section and V thetrapping volume. A rigorous calculation of Γcoll involves careful considerationof the energy of the background gas relative to the escape energy required for
an atom in the trap. Given that the trap depth is very small compared tothe energy of the hot background gas, it is reasonable to neglect such details.
In fact, since most of the experimental parameters that vary from lab to lab
are bound up in the trap and not the room temperature background gas, acomparison of empirical values find them reasonably similar.

39

Once captured, these atoms will only escape due to collisions with background gas, which is already at a very low pressure. Eventually a balance isobtained between background pressure, detuning and total loading rate. With
the settings stated here and a moderately high background pressure (for thecapabilities of this system), loading times on the scale of a few seconds areroutine.

3.3.4

Loading Dynamics

We now have equations for the loading rates as well as the maximum numberof trapped atoms. Taking N as the population in the MOT, R as the loading
rate and G as the loss rate due to scattering from hot background gas,Ṅ =R − GNR
N = (1 − e−tG )G(3.20)High background pressure yields high loading rates. The steady state atomR
number, N = G
is independent of pressure. In this steady state condition, it

is clear that loading and loss rates are equal. Also, for other forms of trapping(i.e. magnetic trap for transport, dipole trap for experiments with no field)
high background pressure means very short lifetimes. This will play into thedesign parameters of the vacuum system in the next chapter.

40

Chapter 4
Implementation
The theory of the previous chapters is at best an exciting experimental challenge. High current for carefully constructed magnetic fields, water cooling,delicate optics, two regimes of extremely good vacuum and translation of cold
clouds between those regimes all come together in the same dense volume ofexperimental space. The end result is a trapped atomic gas with temperatures below one milikelvin. The details of these systems, choices made in theirconstruction and techniques used to build them are covered.

4.1

Laser System

We’ll begin with the full laser systems as shown in Fig. 4.2. To the left of eachlaser are the spectroscopy systems, described in detail in chapter two. From
the theory set down in chapter one, it is apparent that a variety of frequencies
will be required for magneto-optical trapping as well as the detection of any
trapped atoms. As such, two lasers are used, along with many frequency-

41

Figure 4.1: Overview of MOT-side, before construction of BEC-side. Several
components are marked in the right pane.
shifting components.
For the cycling transition light, a Sacher TEC 300a was used, with 1000mW
of available output power. As mentioned in the previous chapter, this is anextended cavity grating diode laser, with the grating in a Littrow and rear-facecoupled design.
The laser used to repump atoms back into the cycling transition is a Toptica
DL-100, with a lower output power and proven design.
Somewhat excessive mounting blocks were made for both lasers; several
inches tall and made of solid aluminum, these were chosen to maximize stability. The optical table used was donated and had some irregularities in surface
flatness. To avoid strain on the laser baseplates, each mounting block was
kinematically mounted via clamps and ball-bearings to the optical table.

a

The Tiger is one of the highest power single-stage diode lasers on the market, meaning
there is no master-slave pair of diodes. As a result, the technology for these diodes seems to
not be as mature as one would like. The decay-over-time for these diodes was unpredictable.
Several diode failures were experienced. In addition, even with the best locking parameters,
the frequency was observed to jitter by ±10MHz due to acoustic noise.

42

Abs. imaging
diagnostic~70 mW
Repumpingdiagnostic
F=2 imaging

AOM

Rb

AOM

Rb

AOM

AOM

4

4

F=1 imaging

AOM

Depump

AOM1
Toptica DL-100~110 mW~900 mW

AOM1

Sacher TEC300

780 nm~300 mW
Cycling

Figure 4.2: Overview of laser optics. A so-called ‘cat-eye’ construction wasused for the cycling AOM to avoid translation of the beam during frequency
sweeps.

4.1.1

Laser Optics

The optics systems for both lasers are motivated by power and frequency
requirements. Each laser had an output power on the scale of that requiredby the system. Losses were minimized in a variety of ways.
The very first and most crucial element in both parts of the the system
is an optical “diode”. This is a one-way optical component that preventsaccidental feedback from disrupting the mode characteristics of the laser cavity.
These diodes are Faraday isolatorsb purchased from Linos Optics, specified toreduce return light by 40dB. While the Sacher also has a built-in 38dB isolator,
another was needed. The Toptica laser did not ship with an isolator thus one
was built in to our custom mounting block.
Many half-waveplate + polarizing-beamsplitter units are used; this allowsbThis diode works via the the Faraday effect, induced by strong magnets around an
optically active crystal. Polarizing optics are part of this device; the laser output is thus
polarized.

43

easy variation of power distribution through the entire system.
Acousto-optic modulators (AOMs) drive nearly all branches of the system.
In an AOM, RF radiation induces sound waves in a crystal. Light is Braggdiffracted off of the sound waves, resulting in a frequency shift equivalent tothat of the sound waves. All our AOMs were purchased from Isomet andoperate at roughly 87MHz with > 85% efficiency. Separate branches of thesystems are labeled according to frequency in Fig. 4.2. Each AOM frequency
can vary from 60MHz to 92MHz, with reasonable efficiency in a smaller bandaround the required 80MHz. The scattering process is dependent on manyfactors including frequency and operates below ≈60MHz on the scale of 20%
efficiency.
A small amount of light from each laser is sent to a scanning Fabry-Perotcavity. This yields information about the mode structure of the lasers. The
signals from both lasers are mixed, but are clearly distinguishable on the oscilloscope readout. This allows for quick diagnosis of possible laser problems
and easy assessment of stability (e.g. multi-mode or mode hopping).
The most important branch of the Sacher system handles the cycling transition. To begin with, all of the AOMs were tested for transmission. The
highest efficiency AOM was used on the cycling branch. The AOM apertures
were significantly smaller in the vertical dimension than the unmodified beam
diameter emitted from the Sacher. Cylindrical lenses created a 1:1 telescopearound the AOM (not shown in Fig. 4.2), focusing in the vertical direction.
This allowed the beam to fit through the aperture of the AOM, while not affecting the optics of beam deflection. After the 1:1 telescope, a lens was set up
with its focus set inside the AOM. The ray exiting this lens is then parallel to44

1” beams

42” circularly
polarized beams

Figure 4.3: A schematic of the template for power distribution in the beams.
The unlabeled waveplates are λ/2 to proportionally adjust power. Line thickness is roughly proportional to power.
the first ray entering this branch, no matter the frequency offset of the AOM.
A λ/4 waveplate is included in this branch, resulting in a net 90o rotation, allowing the return beam to pass through the polarizing beamsplitter. A tilted
lens is introduced to adjust for the slight astigmatism of the beam (diodes arenotorious for horrible output modes, due to their understandably rectangularstructure). The light is sent into a single mode fiber to clean up the mode.
The beam is then partially expanded and 1” beamsplitters are used to divide
the beam (Fig. 4.3) into the necessary six orthogonal beams. The beams arethen folded around the glass chamber (Optics on the far side of Fig. 4.1) andsent into final telescopes expanding to the 2” diameter of glass chamber.

4.1.2

MOT Optics

The organization behind the trapping optics was established in collaboration
with R. Schiller and H. Ruf. A single-mode fiber connects the laser system(Fig. 4.2) to the beam distribution optics, Fig. 4.3. The beam is partiallyexpanded first, then split into the necessary components and polarizations.
The last stage is a λ/4 plate and the final 2” telescopes.

45

4.2

Vacuum System

The vacuum system was designed to be constructed in two parts, forming amagnetic-transporter type geometry[8]. The first part has a higher backgroundpressure of 87 Rb for the sake of MOT loading times. A narrow connection to
the second half of the system is maintained, keeping a high pressure differential.
This second half is maintained at such low pressures that the only available
measurement technique is based on trap lifetimes. This is where Bose-Einsteincondensates are now created.
75 l/s ion pump

Ti-subl.
pump
Rb oven
87

Rb

MOT/MT
10-12 torr(”EO3”)
Lesker
GMPS075F1B

1

TOP trap
glass cell
10mm(W) x 20mm(H)linear translation stages

l/s
25

io

n

pu

m

p

0.5m

Figure 4.4: Schematic of completed machine, to scale. Optics not shown.

46

4.2.1

DesignThe main component of the MOT side of the vacuum system is the trappingcell. Many ideas came together in the design of the MOT cell. Beam distortion
from the glass surface is known to have little effect on the quality of theMOT[8]. Two angled ports (leaving optical access) into the glass cell wereplanned: one for a rubidium oven, the second for a vacuum gaugec . The large
opening on the ‘left’ side of the machine allows for good pumping of the region.
The large-diameter four-way intersection connecting the pump to the cell does
decrease pump speed slightly; however, it allows for the pump-down port,
vacuum gauge and a large flat (inexpensive) window that views the trapping
region. This window is very convenient for imaging.
Expansion into a second vacuum system with significantly lower background pressure also put constraints on the system. The pressure regimes
involved are all ballistic thus it is easy to geometrically induce a large pressure
difference between segments of the system; a long narrow tube (“differential
pumping tube”) allows three orders of magnitude pressure differential between
the two sides of the system. Rather than add the length of this differential
pumping tube to the whole system, the small-aperture tube is fed into thetrapping region. This reduces the travel required by atoms from one section of
the system to the next. The optical access of the trapping region is maximized
while travel time to lower pressure regions is minimized.

c

Never implemented, considered unnecessary.

47

4.2.2Pressure Regimes and Assembly

One of the requirements for the system is to maintain a minimum lifetime of
one second in a magnetic-trap configuration. This requires the pressure to belower than 10−8 torr in the first half of the system.
Three elements are key to maintaining low pressure and high loading rate.
The first is all-metal seals. Aside from one valve mounted opposite the pump,
all seals in the system are copper gasket, conflat seals. The second major
component is the pump itself. A 75 L/s Varian StarCell[26] ion pump is used.
The lifetime of these sealed, solid state pumps is dependent on the pressurein the system. At 10−7 torr, a lifetime of 10 years[26] is extremely reasonable.
Pressures at the pump are expected to be two orders of magnitude lower. Notethat the only pump in the MOT half of the system is an ion pump; the system
is completely closed, removing any need for roughing pumps.
During assembly, a vast quantity of gloves and single-use KimWipes wereused to ensure minimal contamination of inside surfaces. Washes of acetone
and ethanol were used to clean all surfaces before assembly. Copper gaskets
were set aside and used last if accidentally touched by anything but an unused
KimWipe. While somewhat extreme in the level of paranoia (and generating
an equally extreme mess during construction), these techniques produced asystem that worked the first time. Given the time involved in construction
and all that follows prior to a clear measure of system performance, this wasjustified.
The MOT side of the system was constructed first, with a gate valve andsmall window past this. The gate valve was brought to good vacuum with48

the MOT system. Once a MOT was established and sufficient components
were designed, manufactured, ordered and received, the gate valve was closed,
window removed and the second half of the system was added.

4.2.3

Pumpdown and Bakeout

The initial pumpdown of the system was performed via a mechanical-backed
turbopump. The turbopump used was borrowed from a neighboring lab; it
was required for at most a week for each sided . As most water- and oilsoluble contaminants were removed during assembly, the major contributor toinitial background pressure was residual water in the surfaces of the metal.
To enhance pumping of this water, the entire system was wrapped in heating
tape and aluminum foil during initial pumpdown.
Baking the water out of a vacuum system may be a ‘low-tech’ solution to
the residual water problem; however, the heating must be precisely controlled
and the pressure in the system must be monitored quite closely. The system is
primarily made of steel and borosilicate glass with almost all of the junctions
sealed via copper gaskets. The steel and glass are both rather poor conductorsof heat. If only part of the system expands, the seals would be weakened. To
avoid this the temperature was raised very slowly to ∼200◦ C over the courseof several hours and measured at dozens of points, allowing a close eye to be
kept on gradients throughout the system.
The pressure was monitored throughout the process, allowing us to estimate the level of remaining contaminants in the system. It was known thatd

The second side was constructed and pumped down many months after extensive testingand experimentation conducted in the first side.

49

Temperature During Bakeout
C
Pressure in Torr

Temp. in Celsius

250

Pressure During Bakeout
10-5

200
150
100

B
10-6
10-7

D

-8

10

-9

10

50

A
0

50

100

150

-10

10

200

Hours from beginning of bakeout

0

50

100

150

200

Hours from beginning of bakeout

Figure 4.5: Plots of temperature and pressure during bakeout. Several points
and lines are noted: (A) Line ending here indicates conclusion of controlled
cooling of 87 Rb sample. At the end, several heaters were discovered accidentally unplugged. (B) A small gas blanket was on top of the 87 Rb vial. The
turbopump was re-engaged and this was bled out, slowly. Shown here is thepeak of the pressure during this bleed-out. (C) Cooldown of system begins.
Horizontal line shown at C is 9 hours, vertical line is 170◦ C. (D) Some valvesadjusted, pressure spiked but recovered. Final values reached roughly 10−10
torr.
(roughly) the pressure would drop one order of magnitude per 50◦ C. Thisestimate gave us a feel when the system had remained at high temperaturelong enough and could be slowly cooled off.

4.3

Transport

Before the results on magneto-optical trapping are presented, I will discuss a
few components for which I did some calculations and optimizations. In theexperiment, after a MOT has been prepared, the atoms are transported into an
evaporation chamber (glass cell) using a mechanically translated pair of coils(which happen to be the same as the MOT coils). This transport is accomplished in a purely magnetic trap. While trapping monopoles is impossible,
trapping dipoles in electrostatic fields is allowable.[27] The magnetic states of50

our atoms interact with the applied magnetic field. However, these magnetic
states can be up or down with respect to the quantization according to theapplied field; there are trapped and anti-trapped states. The flipping of one
state into the other can occur at the field zero in the center of the trap, termed
Majorana losses. A larger fraction of atoms in a colder cloud will cross thezero more frequently than in a warmer one, leading to more losses. The more
brief the time in the trap, the fewer the losses. This mandates rapid transport.
The mechanical translator consists of a pair of Parker-Daedal 404XR ballscrewtranslation stages, one 60cm long and one 10cm long. See Fig. 4.6 for an
overview of the system. The stages were purchased as part of a total motion
control system which included servo motors, drivers and a controller to executeprogrammed motion.
The translation stages are mounted semi-kinematically. The stage is sensitive to flexing of its mounted surface (decays bearing lifetime significantly)
and a small increment in height was required, thus four gauge blocks werepurchased and placed in roughly even intervals under the lower stage as flat
surfaces. Eight points of pressure on the stage were established. The table onwhich the system was constructed was newly purchased upon establishmentof the lab in 2005, thus surface flaws of any significance are highly unlikely.
The entire motion control system, once satisfactorily programmed, is operated by one low-to-high edge trigger, even during startup following power
loss.

51

Figure 4.6: MOT and TOP coils shown. MOT coils attached around glass
cell in upper left. TOP coils shown with white epoxy-fiberglass composite
framework in the bottom right. Total length of the lower stage is 60cm.
52

4.3.1

Load Calculations

The lifetime of the translation stages was specified as expected travel-tillfailure, dependent on the load characteristics. Force on the four corners ofeach platform as well as any acceleration torque is calculated and comparedto tables to determine the lifetime of the system.
The translation stages have four support points to the track in which they
travel. Each of these supports have a bearing lifetime. The array of all component masses (MOT coils, mass of their cooling water, coil holder, force due
to power supply cable drag) and distances in the coordinate system of eachtranslation carriage were calculated and set up in Mathematica to allow foreasy lifetime estimates as several different coil-holder designs were considered.
After deciding on a reasonable coil-holder configuration, it was found that thestages were expected to outperform the scientific relevance of the system bymany years, even if run continuously at extremely high acceleration.

4.3.2

Electronics and Failsafe Programming

The electronics and motors driving the translation stages are quite intricate.
There are three layers to the control system; the motors, the drivers (power
the motors while managing the feedback loop) and the controller which issues
commands to the drivers.
The servo motors can resolve and hold distances on the scale of individual
counts, an internal unit of measure of which there are 4096 per revolution. Theballscrew drives are set such that one revolution is one cm, resulting in 2.4µm
per count. The controller is capable of setting many of the detailed parameters53

of the servo feedback loop, allowing extreme customization of motion. The
scripting language that runs the controller has extensive functions allowing
complex motions.
The translation path for the MOT coils is shown in Fig. 4.6; first is a long
stretch, out of the ‘high’ pressure region. Next is a jog around a corner, escaping any ballistic trajectories exiting that region. While this motion path
is somewhat elementary, it is still crucial to use controller-level interlocks to
avoid crushing delicate parts of the system. Motion programs are always very
carefully constructed, but perfection is impossible. The control programs are
written with an failsafe command that, when triggered, will stop all motion(velocity to zero) immediately and shut down the servo feedback loop. Structural aluminum guard rails (see Fig. 4.7) are secured to the underside of awell-mounted breadboard, mimicking the configuration of collision points ofthe coil holder with the machine. A steel rod is attached to the coil holder
for restraint; if a catastrophic motion is attempted, this rod will collide firstwith the structural aluminum. This rod is wound with a wire connected tothe controller which supplies a positive voltage. Upon collision of rod to thealuminum rails, the wire will be brought to ground and will trigger the failsafe command, halting all motion. If the controller failsafe command fails, thestructural aluminum will restrain the system. The motors will automaticallyshut down after a short time if the stages are physically restrained.
A handful of program adjustments have triggered the software failsafe.
In two instances known to the author, the mechanical restraint system wasnecessarye . No catastrophic translation failures have damaged the machine oreThese events were during early stages of translation stage setup.

54

Figure 4.7: In the left we see the yellow servo motor housing and the relatively
large power adaptors which are bracketing the sensor cables. The red lead isthe control wire, attached to the steel pin. The extruded aluminum bars are
secured to a breadboard segment and are held at ground with green wires
shown on the right. Where applicable, the motion path of the control pin isbracketed on two sides.
the stage in any way.

4.3.3

Control Programming

The stages were shipped with three sensors each, designed to trigger when atab on the carriage of the stage passed by. Given that power failures may
occur, calibration of our translation coordinates must not be arbitrary or dependant on volatile memory. To accomplish repeatable and safe recalibration,
an initialization routine was written.
On hardware reset, the encoders register a position of (0, 0). However, if
the program loop is terminated and restarted (when motion control parameters
are adjusted), the encoder position is not changed. The calibrated origin of thecoordinate system was chosen in the middle of the motion path, a place where
the stage would never come to rest during normal use. Thus, if the program
starts at the origin (as in the case of a power failure), the initialization routine
is activated, preventing motion until coordinates are verified.

55

Recalibration begins with a shutdown of the servo loop, allowing (andrequiring) a user to manually shift the stage to where one of the position sensors
will trigger. When the position sensor triggers, the servo loop is activated thus
freezing the stage in place. The system then waits for the user to trigger the
single control line. At this point, the drive (slowly) scans for the edges ofthe lower stage’s position sensor. On finding these edges, the lower stage’s
coordinates are set. This sensor is located where the upper stage can move infull range. The system then scans for the edges of the upper stage’s position
sensor. Once set, the stage returns to the MOT position and enters into the‘run’ loop.
The run loop is simple: on trigger it moves forwards to the BEC position
and waits, the next trigger returns it to the MOT position. A small scrap ofMathematica code was written to calculate the appropriate timing, acceleration and velocity values. The motion parameters can be quickly adjusted inthis way.

4.4

Magnetic Coils

Several sets of magnetic coils are used for both the MOT and the BEC apparatus. The primary set used for the MOT is attached to a translation stage
and is also used in the BEC portion of the system.
The primary MOT coils are set up in an anti-Helmholtz configuration (SeeFig. 3.4) and serve three purposes. First, as a source for the low-level fieldrequired for the MOT. Secondly, as a pure magnetic trap used to hold theatoms for transport to the BEC side of the machine. Finally, as part of the56

moving-coil time-orbiting-potential trap (McTOP trap). An approximate formof the MOT coil field (valid only near the origin) is given below with Bq0 as
the slope of the field.
*

B(r) ∝ − 2Bq0 z ẑ + Bq0 xx̂ + Bq0 ŷ

4.4.1

*

(4.1)

Calculations

Several key constraints went into the first generation of coils; the amount ofheat that could be dissipated via available water cooling, the cost of materialsf
and the available power supplies.
When it was decided that some variant of a TOP trap would be used,
there was a need for more detailed simulations. The author wrote a set of
Mathematica routines to calculate these fields. Several models for this code
had been attempted but were too slow in calculation times. The calculation
scheme used is built around an effective implementation of the Biot-Savart
law;*µ0 dl × r̂
dB =
4π r2
*

(4.2)

and is outlined in Fig. 4.8. There are two key features: first it calculates thefield only for a small grid of specified points, saving calculation time; second it
will break a wire segment into smaller pieces according to a scale parameter.
The coils are supplied to the code in the form of lists of points. Indeed,fSome groups have used square cross-section coils in order to gain performance; thistypically requires a special distributor or a 500lb order directly from the supplier.

57wi = wire path defined by points

Coil Simulation
Overview

Li = distance from wi to wi+1
distance from mj to wi = rijmjcalculation mesh(find field here)

S defines a scale factor for radius towire point separation.
If rij/Li < S, wi to wi+1 will be split.

Figure 4.8: An outline for the structure of the simulation code. Two lists ofpoints are required, for the wire generating the field and for the points where
the field information is required. The code scales according to the factor S,
attempting to increase efficiency without sacrificing accuracy; it is assumed
that the wire points sufficiently define wire path. A simple Biot-Savart law isused.
‘building’ the coils in this way took a non-trivial amount of time. Code wasused to build them from pieces of functions, so the number of points could
be specified (e.g. the precision could be adjusted). Point density could be
increased where the wire bent more or was close to calculation regions, whereas
it could decrease for more straight runs or for regions far away.
Each pair of points is treated by the code as portions of straight wire
(hereafter referred to as coil-lines). These are further broken down into subsegments such that the nearest sub-segment has a given ratio to the distancefrom the calculation points. Coil-lines very far from the calculation points havefewer sub-segments, where coil-lines nearer to calculation points have many
more sub-segments. The ratio of sub-segment length to calculation distancecan be adjusted to increase or decrease accuracy: poor accuracy leads to short
calculation times, while high accuracy leads to somewhat increased calculation

58

Figure 4.9: Calculation framework for magnetic simulation code. The largest
box is the volume MOT coils cannot impinge upon. The smallest box is the
final glass cell. The large arrays of dots define the MOT coils. The striateddark loops are the final magnetic trap. On the scale of this image, simulations
were carried for a volume of space about the size of a period.
times.
These simulations allowed us to verify the details of all the sets of coils, aswell as their required precision in manufacture and assembly.
The nature of the TOP trap as a simple bias field applied to the quadrupole
field lent itself to an easy method of simulation. The fields of each separate coilwere calculated and superimposed, with the TOP coils having modulations ofsine and cosine functions as would be implemented with electronics.

4.4.2

DesignThe parameters that dictated the very first primary MOT coil design are heat,current flow and cost. The initial power supplies were bought at huge discountand were intended to give us a start rather than serve for the life of the system.

59

Also, the purpose of the primary MOT coils was expected to (and did) change
during the system’s development. A small infrastructure was progressively
developed for the generation of MOT coils. At least four sets of MOT coils
were created, with major participation by the author for all.
A variety of shapes were considered for the TOP coils; the simplicity ofrectangular coils eventually won. The key feature of Helmholtz coils is thevanishing curvature in a large region between them. D. Pertot worked out theproper separation of two rectangular coils (dependent on the ratio of widthand height) necessary to zero field curvature at the center of the coils. Twointerlocking pairs of rectangles were chosen in the final version and tested
extensively with the simulation code.

4.4.3

Construction

While the applied magnetic field intended to generate the MOT is significantly
stronger than the Earth’s field outside of the trapping region, the MOT is not
the only trapping phase. In the optical molasses phase, the field will be turned
off. If there is a residual magnetic field due to the Earth, the beam intensity
will need to be adjusted to offset the Zeeman effect such that there is no
average velocity in the cloud of atoms once released (or transferred to anothertrap). Compensation coils designed for low optical occlusion are oriented on
three orthogonal axesg . When the MOT field is reduced, the uncompensatedzero of the superposition of Earth and MOT fields will move, allowing for aneasy diagnostic to calibrate the compensation field.
gHelmholtz pairs are typically used. However, the region that requires compensation isvery small. In our system only three coils are needed.

60

MOT Coils
Using specialized copper tube for the MOT coils was determined to be prohibitively expensive. Tube easily obtainableh from any hardware store wasused along with basic epoxy and Kapton insulating tape, wrapped very carefully around the tube. A brass form was created around which the coils were
wound, using a massive and time-proven lathe. The initial designs (before
custom power supplies were ordered) were alternating layers of four and fivewindings, four layers thick. The final version was five and six windings, sixlayers thick. The water cooling allowed extremely high current, roughly 450A,
to flow without difficulty. This configuration produced gradients of roughlyG.
0.8 cm·A

TOP Coils
The TOP coils required significantly more effort to construct. Given the highresponse characteristics required by magnetic fields varying on the kHz scaleand smooth predictable potential surfaces in the trap, any metal in the coilholding construct was completely out of the question. The coil forms were
cut by the author out of fiberglass-epoxy composite material. Small copper
wire was wound into a special heavy-duty epoxy chosen specifically for thispurpose.
Two generations of forms were created, both wound. The first generation
was very basic and only one pair of coils was built. These eventually failed
due to metal fatigue of the wire after being used for several tests on the basichRefrigerator tube, 1/4” OD

61

concept. The second generation (shown in Fig. 4.6)were designed around thepairs of coils being directly interlocked and glued together, then mounted toan external gantry of the same material.
Tolerances on the forms were kept as tight as possible. The material andthe dust created by cutting is highly abrasive; each cut would reduce the sizeof the cutter on a measurable level. Measurements were required every few
cuts to maintain compliance.

62

Chapter 5
Cold Rubidium Clouds: MOT
Performance
The performance of a MOT is determined by the time to load the MOT, thenumber of atoms available for experiment, their temperature and the reproducibility of these numbers. In this chapter a brief summary of the quantitative results obtained during the author’s time in the lab are presented. Dataacquisition techniques are discussed and some images are included.
Many parameters affect these numbers; optimization is nontrivial. Shortloading times indicate high background pressures, while high background pressures can reduce atom number if the cloud is not transported out of the highpressure region fast enough. Extremely rapid movement can fatigue the transport system or even heat the cloud. During the author’s time in the lab,
optimization of trapped atom number was performed, but more in a spirit ofdiagnostics (e.g. can it perform) than an attempt to push or even establish
the limits of the system. Diode failures in the midst of data sets, rearrange-

63

MOT Loading Curve
Photodiode Signal
0.05

fluorescence @a.u.D

0.04
0.03
0.02
0.01
0.00
0

1

2

3

4

time @sD

Figure 5.1: Loading curve as recorded by photodiode in a simple fluorescenceconfiguration. Intensity units are arbitrary; data is clearly offset by ambient
light. The 87 Rb background pressure was at 3 ∗ 10−9 Torr. An exponential fit
was performed on data from 0.5s onwards. ‘Error’ (data minus fit) is plottedon the same graph. It was concluded that the power supply in use at the time
could not regulate quickly at low currents. A new power supply eventually replaced it, after the author’s time in the lab. Loading constants were extracted
from this data; the e−1 time was 0.47 s.
ment/restructuring of the cooling sequence and commissioning of the second
lowest-pressure portion of the system necessitated further optimization, notcovered here.

5.1

Imaging Techniques

Simple loading curves (Fig. 5.1) were obtained by placing a photodiode near
the cold cloud. A short telescope using a pinhole spatial filter was set up
between the diode and the MOT in order to increase signal-to-noise ratio. As
this curve was intended only for purposes of extracting the loading time, thedistances and solid-angles involved are unnecessary.
Beyond this simple diagnostic photodiode measurement, two primary meth-64

Figure 5.2: Two methods of imaging used. The telescopes involved were wellcharacterized, giving relevant solid angles.
ods of imaging are used: absorption and fluorescence (See Fig. 5.2). In bothcases, a high-speed large-well-depth PIXIS camera manufactured by Princeton
instruments was used. The quantum efficiency of this camera was measured
by B. Dixon[28].
To measure the quantum efficiency of the camera, the conversion fromphotons to counts was required. A known intensity of light was sent into thecamera; this was measured with a power meter and then reduced in a known
way (to avoid camera saturation) via neutral density filters. Several camera
configurations were explored during the construction and testing of the firsthalf of the system.
To manage the recording of images, the author wrote several simple visual basic scripts. One example would prompt for a region of interest, thencontinuously take images every time the camera received a TTL trigger. Allof the scripts would process the image (and only the region of interest) as
follows. Three separate images were recorded: with atoms (‘data’), without
atoms (‘background’) and without light (‘dark’). Rather than import all of65

these into our analysis pipeline and then perform normalization, this was takencare of within the imaging program, post-crop.a

5.1.1

Absorption

For absorption imaging, a collimated beam of light is projected onto thecloud, then reduced and projected onto the camera. The data is then a twodimensional map of number-of-atoms in a column. To extract the information,
we process the data image as follows.

Iresult =Idata − Idark
Ibg − Idark

This is done on a pixel-by-pixel basis, normalizing the result image to relative
intensity values. The absolute beam intensity is unnecessary information (aslong as the atoms are not saturated by the imaging light), as we will see.
The absorption of light by dilute media (each column of atoms along thez-direction) was first discussed by Rayleigh long ago; particles in the media
are assigned a scattering cross section σcs and each collision results in loss oflight, as the directionality of the light is very likely changed:***dI(x) = − α(x)I(x)dz
*

(5.1)

*

α(x) = σcs N (x)*where I is the intensity of light, N (x) is the number density of atoms and α isthe media-dependant absorption coefficient. The scattering cross section canaThis image pipeline saved many hours of time during calibration of the system parameters.

66

(a) Absorption image of MOT. Summation ofcolumns shown below. Note ripple-like distortions
due to multiple pieces of glass between source andcamera.

(b) Plot of uncalibrated atom number. Columns above summed
into a single row.

Figure 5.3: Uncalibrated absorption image taken in Sept. 2006. Images taken
during this period were an attempt to get consistent, predictable data from
MOT to MOT. In order to calibrate this image, a great deal more information
regarding the instantaneous detuning of the laser as well as the geometryof the temporary setup employed would be required. This imaging method
was neglected during the author’s time in the lab due to various technicaldifficulties.
67

be derived straightaway from the scattering rate:
γp =s0 γ/2
1 + s0 + (2δ/γ)2[3.2]

The cross section (for one atom) combined with intensity will give the scatteredenergy; the scattering rate (for one atom) and the energy of the scattered lightwill also give the scattered energy, linking the two.

σcs I =~ωγpNote that the scattering rate depends on the ratio of intensity to saturation
intensity. Imaging light will be as weak as is reasonable, avoiding saturation
by a wide margin. Let us derive the atom density from the two-dimensional*dN
intensity map recovered by the CCD. With N (x) = dAdz
, we have*dI(x) = − σcs

dN
*
½
I(x)½
dz
½
dA½
dz

We will be using pixels of the camera as the units of area in this measurement.
Naturally, via telescopes shown in Fig. 5.2, the image will be expanded to fill a
sufficient fraction of the CCD that variation across the pixels will be minimal.
*−A dI(x)dN =*σcs I(x)I−Aln( )N (x, y) =
σcs
I0

(5.2)

As the telescopes and the specifications of the camera are theoretically known,
so A is theoretically known. The earlier mentioned image normalization leads
to direct knowledge of I and I0 . We now know the number of atoms in each68

pixel-column. In practice, during our first-attempt setups, fine control overa beam traversing the entire machine was somewhat more difficult than expected.
The image is a projection of the 3-d density cloud into a two-dimensional
image. Integrating over the cloud density yields the total number of atoms.
One problem with this method is fast control of light. The absorption
method requires trapping beams to be inactive, thus the imaging light must be
turned on rather quickly. The duration of the imaging pulse must not saturate
the camera or allow the atoms to significantly fall during image capture. Much
light-switching is accomplished via the AOMsb , but there are several shuttersin the system. As the scattering rate is dependant on the detuning of the light,
the mechanical shock from these shutters was a significant effect in disrupting
the ‘instantaneous’ lock of the laser frequency; that is, the detuning of the laser
during image capture. This problem was somewhat solved by ‘soft-mounting’the shutters on absorptive material.c While many of these bugs were being
worked out in the first months of imaging, we relied on fluorescence for our
data.

5.1.2

Fluorescence

Fluorescence imaging works by collecting the scattered light from atoms andfocusing an image on the camera. The image processing is slightly different;

Iresult = Idata − Ibgbcsee Sec. 4.1.1
A variety of sorbothane pieces were used.

69

This washes out dark current without requiring a specific image for that purpose. We found fluorescence imaging to be more robust, in contrast to absorption.
To calculated the atom density from this resultant image (pixels containingc(x, y) counts), we must know the appropriate solid angle (Ω), losses due tooptics (0.96r w/ r = number of non-AR surfaces), the scattering rate (Eq. 3.2),
exposure time (texp ), the quantum efficiency of the camera (q) and the conversion from photon energy to number of electrons excited p.

N (x, y) =4π c(x, y)q.
Ω pγp 0.96r

As the cloud is very small compared to the Rayleigh range of the telescope
system, the entire cloud should be sufficiently in focus and the image is a
map of column counts. The total atom number can be calculated easily bysumming all the pixels.
One problem with this method is reflections from the apparatus itself; lightscattered away from the camera and thus not expected by this calculation
may be reflected by components back into the imaging system, registering
false counts. For the most part, this was limited to small areas that could be
ignored by analysis software.

dThis discussion follows [28].

70(a) Fluorescence image of MOT. Summation of columns shown below. Noteblocked out reflections.

Calibrated Atom #Bimodal fit to atom number
800000600000400000200000
0
0

100
200
300
pixel HarbitraryL400(b) Plot of atom number. Columns above summed into a single row.

Figure 5.4: This data is from the first set of MOT coils in an early imaging
configuration, April 2006. Most of the population is in the outer region. Atom
numbers for this data were estimated at 3 × 109 in the central feature, with
11 × 109 total atoms. Typical parameters were better tuned, placing nearly
all atoms in the central feature. This particular MOT image was chosen to
exemplify the two-component regime. Sub-Doppler cooling is based on theshifts of sublevels due to the surrounding light. When the shifts from magnetic
fields in a MOT are comparable to the light shifts, sub-Doppler cooling occurs
is no longer effective. Thus a MOT can have two cooling regions, a sub-Doppler
volume and an outer region which is less-dense with higher temperature.[7]

71

5.2

Temperatures: TOF

The two invaluable numbers that can be extracted from these pictures are
atom number and temperature. To obtain the former, all that is required issimple calibrated addition of pixel values in one image; the latter requires asequence of images in time. Free expansion of a trapped cloud was used todetermine the temperature: by finding the rate of expansion of the cloud, we
can discern the average velocity and thus the temperature of the cloud. Thisis one version of time-of-flight (TOF) temperature determination.
It is a reasonable assumption that the three dimensional density distribution of trapped atoms is very close to a gaussian in form,*nt*n(s, t = 0) = √( 2πσs )3

*

s·s
−
2
e 2σs .

Here nt is the total number of atoms and σs is the width of the distribution
in a given dimension. This can, of course, be decomposed into three separate
gaussian distributions which can be integrated separately. This would allow
for different sizes in different dimensions, as is the case with our MOT: thefield is twice as strong in the vertical dimension. This is no matter, however,as each dimension is independent of the other.
It is straightforward to find the distribution of atoms after a given time t.
Using a Dirac-delta function and the relevant Maxwell-Boltzmann distribution,72

we can find the atoms able to make it to any given location:Z∞*n(s, t) =
−∞

Z∞
*

dv

−∞

N
n(s, t) = √( 2πσsi )3
*

*

*

*

*

*

*

dsi n(si )f (v)δ(si − s + vt)
r³
Z∞
Z∞
**
**
s ·s
m ´3
− i i2 − mv·v
*
*
*
*
*
2σs i
e 2kT δ(si − s + vt)
dv
dsi e
2πkT
−∞

−∞

*

where si is some dimension. If one wishes to find the full expression, eachdimension will have its own integral. It is possible to use only one dimension,*

as done with the analysis used for Fig. 5.5b. Here f (v) is an amalgam ofthree one-dimensional Maxwell-Boltzmann distributions, all given the same
temperature. After a fair bit of algebra, one recovers*n(s, t) = √

N
3

sx 2
e 2σs (t)−2π σs (t)3q
σs (t) = kT
t2 + σsi 2
m

(5.3)
(5.4)

This is another Gaussian, but with variable width σs (t) which will allow us tomeasure the temperature via a well-timed sequence of images of the MOT’s
expansion.
The obvious problem to this method is the reproducibility of a cold cloud atgiven experimental parameters; the available imaging methods are destructive
to such a free-expansion MOT. With care it is possible to consistently produce
similar clouds. As such, sequences were taken with varied delay between MOT
release and image capture.

73

(a) Sequence of multiple fluorescence images of MOT. Different times from different
MOTs. Sigma was found for each frame.

(b) Plot of sigma versus time. Fit resulted in an estimated temperature of 1.3µK.

Figure 5.5: Time-of-flight image taken in October, 2006. The images in 5.5a
represent a thin slice through the center of the cloud. At t=15ms, light reflected from the differential pumping tube behind the cloud is clearly visible.
This ring-shaped region was excluded from the fit. This fit estimated the atom
number at 3.4 ∗ 107 . Many runs were made with a variety of detuning values(for the cycling light) and for repump power values. Trends were established,allowing for rudimentary optimization. This is one of the earliest optimizations
based on these parameters and was repeated later.

74

5.3

Conclusions

Working with Prof. Schneble and fellow students H. Ruf, R. Schiller and S.
Albert, the first complete MOT system was characterized. Traps capable ofholding ∼ 108 atoms were observed in early configurations, with temperaturesin the micro-kelvin range without a great deal of fine-tuning required. Many
improvements from these first estimates were made during the author’s timein the lab, and surely have been made since.
Some advantages and strong points of the system were fortuitous. The
physically separate tables for lasers and the trapping apparatus lent the laser
lock easy sustainability to within fractions of the natural linewidth, even during
use of the mechanical translator. Some were hard fought, such as the exact
configuration of the MOT coils and coil holder, currently on versions four andtwo respectively. The imaging system is flexible and has served in multiple
configurations, seamlessly integrated with data collection. Some struggles havebeen ongoing, most significantly the quality of the laser diodes supplied by thevendors.
With a MOT ready and in the secondary cooling system in roughly one
minute, the system performs quite well. By the conclusion of the author’s timein the lab, the BEC portion of the system was pumped down below measurable
levels and tests were being performed on the quality of transport scheme andthe trap lifetimes possible there.

75Bibliography[1] Helge Kragh. Max Planck: The Reluctant Revolutionary. Physics World,
December 2000.
[2] Summary of proceedings of Mini-symposium: The Birth of Quantum
Theory, December 2000. URL http://physics.ucsc.edu/~michael/planck.html.
[3] C. Davisson and LH Germer. The scattering of electrons by a single
crystal of nickel. Nature, 119:558–560, 1927.
[4] O. Theimer and Budh Ram. The beginning of quantum statistcs.
American Journal of Physics, 44(11), November 1976.
[5] M. Delbruck. Was Bose-Einstein statistics arrived at by serendipity?
Journal of Chemical Education, 57:467–470, 1980.
[6] A. Einstein. Quantentheorie des einatomigen idealen Gases. Akad. d.
Wiss. Berling, Sitzungsber Phys. Math. Kl., 22, 1924.
[7] C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot,A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard. Phase-spacedensity in the magneto-optical trap. Phys. Rev. A, 52(2):1423–1440, Aug
1995.
[8] H. Lewandowski, D. Harber, D. Whitaker, and E. Cornell. Simplified System for Creating a Bose–Einstein Condensate. Journal of Low
Temperature Physics, 132(5):309–367, 2003.
[9] Harold J. Metcalf and Peter van der Straten. Laser Cooling and Trapping.
Springer-Verlag, 1999.
[10] D. J. Wineland and Wayne M. Itano. Laser cooling of atoms. Phys. Rev.
A, 20(4):1521–1540, Oct 1979.

76

[11] S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable. Experimental observation of optically trapped atoms. Phys. Rev. Lett., 57(3):314–317, Jul
1986.
[12] Utaka Yoshikawa, Takeshi Umeki, Takuro Mukae, Yoshio Torii, andTakahiro Kuga. Frequency stabilization of a laser diode with use of lightinduced birefringence in an atomic vapor. Applied Optics, 42(33):6645–9,
2003.
[13] Joseph H. Eberly and Peter W. Milonni. Lasers. John Wiley & Sons,
1988.
[14] Wolfgang Demtröder. Laser Spectroscopy. Springer-Verlag, 3rd edition,
2003.
[15] R.P. Feynman, F.L. Vernon Jr, and R.W. Hellwarth. Geometrical Representation of the Schrödinger Equation for Solving Maser Problems.
Journal of Applied Physics, 28:49, 1957.
[16] Daniel A. Steck. Rubidium 87 D Line Data. URL http://steck.us/alkalidata/. oct 2003.
[17] Irina Novikova, Andrey B. Matsko, and George R. Welch. Influence ofa buffer gas on nonlinear magneto-optical polarization rotation. J. Opt.
Soc. Am. B, 22(1):44–56, 2005.
[18] S. Brandt, A. Nagel, R. Wynands, and D. Meschede. Buffer-gas-induced
linewidth reduction of coherent dark resonances to below 50Hz. Phys.
Rev. A, 56(2):R1063–R1066, Aug 1997.
[19] C P Pearman, C S Adams, S G Cox, P F Griffin, D A Smith, andI G Hughes. Polarization spectroscopy of a closed atomic transition:
applications to laser frequency locking. Journal of Physics B: Atomic,
Molecular and Optical Physics, 35(24):5141–5151, 2002. URL http://stacks.iop.org/0953-4075/35/5141.
[20] L. Ricci, M. Weidemüller, T. Esslinger, A. Emmerich, C. Zimmermann,
V. Vuletic, W. König, and T. Hänsch. A compact grating-stabilized diodelaser system for atomic physics. Optics Communications, 117:541–9, 1995.
[21] Paul D. Lett, Richard N. Watts, Christoph I. Westbrook, William D.
Phillips, Phillip L. Gould, and Harold J. Metcalf. Observation of atoms
laser cooled below the doppler limit. Phys. Rev. Lett., 61(2):169–172, Jul
1988.
77

[22] Luis A. Orozco. Laser cooling and trapping of neutral atoms. In S. Hacyan, R. Jauregui, and R. L. Pea, editors, The XXXI Latin-American
School of Physics New Perspectives on Quantum Mechanics, pages 67–
90. AIP Conference Proceedings 464, 1999.
[23] A. M. Steane, M. Chowdhury, and C. J. Foot. Radiation force in themagneto-optical trap. J. Opt. Soc. Am. B, 9(12):2142–58, dec 1992.
[24] C. Monroe, W. Swann, H. Robinson, and C. Wieman. Very cold trapped
atoms in a vapor cell. Phys. Rev. Lett., 65(13):1571–1574, Sep 1990.
[25] Arjen Amelink. Photoassociation of Ultracold Sodium Atoms. PhD thesis,
Utrecht University, feb 2000.
[26] VacIon Plus 150 pumps. Varian Inc., April 2006.
[27] Alan L. Migdall, John V. Prodan, William D. Phillips, Thomas H. Bergeman, and Harold J. Metcalf. First observation of magnetically trapped
neutral atoms. Phys. Rev. Lett., 54(24):2596–2599, Jun 1985.
[28] B. Dixon. MOT Characterization with a CCD Camera. Optics Rotation
Presentation at SUNY Stony Brook, March 2006.

78