High repetition rate Bose-Einstein condensate
production in a compact, transportable vacuum
systembyMatthew B. Squires
B.S., Brigham Young University, 1999
M.S., Brigham Young University, 2001

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2008This thesis entitled:High repetition rate Bose-Einstein condensate production in a compact, transportablevacuum system
written by Matthew B. Squires
has been approved for the Department of Physics

Dana Z. Anderson

Deborah S. Jin

Date

The final copy of this thesis has been examined by the signatories, and we find thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii
Squires, Matthew B. (Ph.D., Physics)
High repetition rate Bose-Einstein condensate production in a compact, transportablevacuum system
Thesis directed by Dr. Dana Z. Anderson

This thesis reports on the design and implementation of a compact (5 × 10 × 30cm) two chamber system for rapidly producing 87 Rb Bose-Einstein condensates (BEC)
on an atom chip with the future goal of a transportable BEC based sensor. We present
the first use of anodic bonding to fabricate multi-chamber vacuum systems suitable for
BEC production. Anodic bonding is a method for joining silicon and Pyrex and issuperior to epoxy based construction because anodically bonded vacuum systems canbe baked at temperatures > 300◦ C. The improved vacuum quality is a key aspectof reproducibly building compact BEC quality vacuum systems. In the two chamber
vacuum system the first chamber operates a 2D+ magneto-optical trap (MOT) at a Rbpressure of ∼ 10−7 torr and produces a flux of cold atoms that loads a 3D MOT in thesecond, low pressure (< 10−9 torr) chamber. When loaded to saturation the 3D MOT
contains ∼ 500 × 106 87 Rb atoms. The laser cooled atoms are magnetically transferred
to an atom chip where the atoms are cooled to degeneracy by forced RF evaporation ina tight magnetic trap (1.8 × 3.0 × 0.6 kHz). The system performance is demonstrated bythree modes: the rapid production of sequential BECs of 1 × 105 atoms with a periodof < 3.8 s, the number optimized production of a BEC of 4 × 105 atoms in < 10 s, andthe speed optimized production of a BEC of 5 × 104 atoms in 2.65 s. Initial experiments
demonstrating key elements of rotation sensor are also presented. Specifically splitting
atoms on a chip in a compact vacuum system and guiding atoms around a curve. Alsopresented in this thesis is a rubidium dispenser based on a gold/rubidium alloy that has
potential as a clean source of rubidium in cold atom experiments.

Dedication

To Dionne.

v

Acknowledgements

I would like to give thanks to my Heavenly Father who gave me a curious mind,amazing opportunity, inspiration, and peace when it was hard to see the road.
My wife Dionne has been an astonishing companion and friend though this process. She has been patient and supportive of my dreams, aspirations, and goals. More
importantly she has sacrificed her time, energy, and will so I could achieve my dreamof a PhD in physics. She has worked and been incredibly successful as a mother; A far
more reaching and worthwhile endeavor than all of the work I have done or will everdo. I am in constant amazement of her strength and goodness. I would write more, but
it would double the length of this thesis. Thank you Dionne!
My boys (Connor, Spencer, Trey, Drew, and Cade) have kept my grounded andhave also sacrificed their time with me. I have loved their gentle encouragement andconstant excitement about the world around them.
I would like to thank my parents, George and Jean Squires, for their constantsupport and encouragement. I would like to thank my dad for our many discussions
and pointing me toward many authors (Covey, Drucker, and Collins). I would also liketo thank my in-laws, Chuck and Joyce Clifton, for their encouragement and support. I
have also been supported by my wider family of siblings, extended family, and wonderfulfriends. There are too many to name; Their kind words and acts of kindness are too
many to list. Allen Tracy deserves special mention for finding me an internship at
Lockheed Martin. Thank you to you all!

vi
I often think about the path the brought me to this point and the teachers thattaught me. I would especially like to mention a few names: my high school physics
teacher Mike Harris for his crystal clear teaching and enthusiasm, Steve Turley at BYU
for his mentoring and example, and David Allred at BYU for his friendship, giving me
wide latitude in his lab, and teaching me to be an experimental physicist.
I would like to thank those who have had a role in this current chapter of mylife. I would like to thank Dana for letting me work on such an amazing experiment, forfinancial support, and for teaching me physics and life lessons that will long improve myprofessional and personal life. Leslie Czaia and her amazing technical talents have mademy experiments reality, and I have appreciated her friendship and advice. Alex Zozulya
on his many trips to Boulder has helped with theory. Victor Bright for teaching me
MEMS and being part of my committee. I have been lucky to work with many students:
Shane Du and I spent many days and nights together, he left a high standard to follow,
Yutaka Imai, Seth Frader-Thompson, Saravanan, Zach Walters, Ben Luey (run silent,run deep), Brian McCarthy, Ricardo Jimenez, Megan Hart, Will Holmgren, Ying-Yu
Wang, Allison Churnside, and Carl Wiedemann have all been wonderful to work with
before they moved on. Evan Salim, Steve Segal, Rick Chuang, Lynsi Aldridge, andAndrew Holmgren have helped in the final phases. Also Sterling McBride and Steve
Lipp at Sarnoff and Rob Mihailovich at Teledyne put in many hours making parts that
I would use in my experiments.
And to all of JILA, thanks for making this is one amazing place. I asked forhelp from all of my committee at some point, Eric Cornell, Debbie Jin, and Heather
Lewandowski. Also Carl Wieman for a few discussions on the way to the bus stop. The
JILA shop and their expertise, especially Todd Asnicar for all of the glass work andPaul Beckingham for the many electronics projects. The OSEP program (funded by
the NSF) for financial support and an incredible learning environment.

viiContents

Chapter
1 Introduction and background1

1.1

Brief history of BEC on a chip . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Applications of BEC and apparatus for portable BEC . . . . . . . . . .

3

1.3

BEC in a single cell at CU. . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Comparing BEC and the laser . . . . . . . . . . . . . . . . . . . . . . . .

7

1.5

My contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.6

Outline of the rest of thesis . . . . . . . . . . . . . . . . . . . . . . . . .

102 Essential theory

11

2.1

Magnetically trapping atoms . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

Basic wire magnetic trap . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2.1

Z-wire trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Dimple trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.3.1

Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.3.2

Trap frequency calculation. . . . . . . . . . . . . . . . . . . . .

21

2.3.3

Comparison of dimple and Z-wire traps . . . . . . . . . . . . . .

23

2.3.4

BEC without vias . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.3.5

Evaporation during transport . . . . . . . . . . . . . . . . . . . .

28

MOTs and BEC chips . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.3

2.4

viii
2.4.1

Basic MOT considerations . . . . . . . . . . . . . . . . . . . . . .

29

2.4.2

Chip versus six-beam MOTs . . . . . . . . . . . . . . . . . . . .

34

2.4.3

Six-beam MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.4.4

Six-beam MOTs with angled beams . . . . . . . . . . . . . . . .

37

2.4.5

2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.5

Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.6

BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.7

Evaporative cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Evaporation trajectories . . . . . . . . . . . . . . . . . . . . . . .

48

2.7.1

3 Aux Rb dispensers

51

3.1

Binary phase diagrams and vapor pressures . . . . . . . . . . . . . . . .

52

3.2

Rapid dispensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.3

Radiative heat loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.4

Optimal wire radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.5

Other geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.5.1

Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.5.2

Foil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.6

Readsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.7

Bulk versus surface effects . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.7.1

Gold pulsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

MOTs and Au/Rb dispensers . . . . . . . . . . . . . . . . . . . . . . . .

61

3.8.1

MOT N versus τ . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.8.2

Gold as a rubidium pump . . . . . . . . . . . . . . . . . . . . . .

63Gold versus other metals . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.8

3.9

4 Experimental setup and operation
4.1

Vacuum cell design and construction . . . . . . . . . . . . . . . . . . . .

66
66

ix
4.1.1

2D MOT cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.1.2

Spherical cube / pinch off . . . . . . . . . . . . . . . . . . . . . .

68

4.1.3

Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.1.4

3D MOT cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.1.5

Atom chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.1.6

Bakeout and pinch off . . . . . . . . . . . . . . . . . . . . . . . .

76

4.2

Anodic bonding for UHV construction . . . . . . . . . . . . . . . . . . .

79

4.3

Control and imaging systems . . . . . . . . . . . . . . . . . . . . . . . .

80

4.3.1

Control system . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.3.2

Imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Apparatus setup and operation . . . . . . . . . . . . . . . . . . . . . . .

87

4.4.1

2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.4.2

3D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

4.4.3

MOT debugging . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.4.4

2D → 3D MOT loading . . . . . . . . . . . . . . . . . . . . . . .

96

4.4.5

CMOT, PGC, OP . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4.6

External Z-wire magnetic trap . . . . . . . . . . . . . . . . . . . 105

4.4.7

Transfer to chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.4.8

Chip Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.4.9

Dimple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.4

4.4.10 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.11 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.5

Typical operating parameters . . . . . . . . . . . . . . . . . . . . . . . . 122

5 BEC production

124

5.1

30 shots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2

Number optimized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

x5.3

Speed optimized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.4

Vacuum lifetime, heating rates, and RF shield . . . . . . . . . . . . . . . 130

5.5

BEC production for atom interferometry . . . . . . . . . . . . . . . . . . 131

6 Towards Applications of BEC
6.1

Atom splitting and recombination

135. . . . . . . . . . . . . . . . . . . . . 136

6.1.1

Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.1.2

Atom splitting optimization . . . . . . . . . . . . . . . . . . . . . 139

6.1.3

Atom splitting and recombination data . . . . . . . . . . . . . . 142

6.2

Circular waveguiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.3

Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Bibliography

148

Appendix
A Monte Carlo for MOTs

156

xi

TablesTable
2.1

Comparison of vacuum to electrical parameters. The unit of torr will be
used instead of using particles/volume. . . . . . . . . . . . . . . . . . . .

2.2

Comparison of evaporation parameters (e.g. evaporation time) for chip
based and traditional BEC machines. . . . . . . . . . . . . . . . . . . . .

3.1

48

Specific heat and mass density of a variety of materials. Sorted by ρ csmallest to largest, or most desirable to least desirable for rapid cooling.

3.2

43

55

Calculated radius to achieve cooling in 1 s from 10% above to 20% below
the L + α phase assuming Tungsten filament for the three gold/rubidium
alloys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

56

Calculated wire width to achieve cooling in 1 s from 10% above to
20% below the L + α phase assuming Tungsten filament for the three
gold/rubidium alloys. Assume a 10:1 height to width aspect ratio. . . .

57

4.1

Summary of the essential timing for the production of the BEC. . . . . 122

5.1

Production parameters for the three BEC modes. . . . . . . . . . . . . . 124

xii

FiguresFigure2.1

Figure a.) shows a U-wire configuration for creating a quadrupole magnetic trap. Figure b.) shows a Z-wire configuration for creating a IoffePrichard magnetic trap. Figure c.) shows the wire configuration for
creating a dimple trap. By themselves these wire configurations will not
produce magnetic traps and must be augmented with external bias fields
as shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

14

Figure a.) shows the correction factor for the magnetic field and gradientof a short piece of wire as a function of distance from the wire. The
distance from the wire is scaled to the width of the wire section. The
correction is relative to the fields of an infinitely long wire. Figure b.)
shows the approximate trap depth of a 30 mm Z-wire running 5 × 130 A
as a function of distance from the center of the Z-wire. The trap depth
is the maximum magnetic field toward the edges of the Z-wire minus thefield at the center of the Z-wire. . . . . . . . . . . . . . . . . . . . . . . .

16

xiii
2.3

A calculation of the dimple trap used for BEC production. The following
parameters were used to generate this representation: Ix = 1.25 A, Iy =3.75 A, Bx = -38 G, Iy = 46 G. Figure a.) shows a 2D slice though thecenter of the magnetic trap parallel to the chip surface with the magnetic
field shown on the Z-axis and in false color. Figure b.) shows a contour
plot of the same trapping potential showing the two calculations for thetrap rotation angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

19

Dimple trap depth showing calculation of trap depth assuming depth isevaluated relative to the side of the trap. The calculation assumes Ix = 5
A, Iy = 2.5 A, and By = 80 G. Also shown are the limits assuming thedepth is evaluated at the trap center and at infinity. For values less than
20 G there is no dimple trap and the resulting potential will split the cloud. 21

2.5

Dimple trap depth as a function of distance away from the wire centerand current in the wire. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

Geometry for calculating the optimal chip width rc and wire currents tocreate BEC without vias. . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7

27

Bias fields needed to create a tight dimple magnetic trap as a function ofchip thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.9

25

Expected power dissipated in a rectangular wire as a function of chip
thickness to create trap conditions suitable for BEC production. . . . .

2.8

22

28

Numerical ODE calculation of the trajectory of an atom in a MOT starting at 0.1 m/s and the edge of the MOT. The ‘atom’ is centered in thelaser beams with the velocity pointed at the zero of the magnetic field.
This shows atoms with a low velocity will be sped up to match the opticalforces until the laser forces are balanced. . . . . . . . . . . . . . . . . . .

32

xiv
2.10 This show the reletive alignments and positions of the laser beams (dashedlines) with the cell and atom chip. Not shown are the 5th and 6th beams
that would go in and out of the page. Also shown is the position of thecoils needed to generate the MOT magnetic field. Figure a.) shows theconfiguration for a mirror MOT and Figure b.) shows the configurationof six beam MOT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.11 The angled six beam MOT configuration showing the factors that determine the distance from the atom chip to the center of the MOT. . . . .

37

2.12 Simulations of MOT number versus the angle of the 3D MOT beams.
Figure a shows a 3D calculation that includes the momentum kick fromrecoiling photons. It is interesting to note the MOT number is saturated
up to θ ≈ 20◦ . Figure b shows a 2D numerical ODE calculation that does
not take the photon recoil into consideration. . . . . . . . . . . . . . . .

39

2.13 Maxwell-Boltzmann velocity distributions and energy distribution. The
shaded area in Figure a.) shows how a significant portion of the totalenergy can be removed from an atom cloud without large number loss
by evaporating on tail of the distribution. Figure b.) shows various
distributions for various relative temperatures. . . . . . . . . . . . . . .

46

3.1

The binary phase diagram of gold and rubidium. . . . . . . . . . . . . .

52

3.2

This shows pulsed rubidium dispensing from a gold coated Nichrome foil.
This is data of a typical data run with gold evaporated on a Nichrome
foil (50 − 100 µm thick and an area of 5 × 15 mm). The foil is heated witha square pulse of current (labeled on each pulse) and as the rubidium isdepleted from the gold higher currents longer pulses are needed to achieve
the same pressure modulation. . . . . . . . . . . . . . . . . . . . . . . .

60

xv
3.3

The squares show the rubidium dispensing with the gold foil and the
diamonds show the rubidium dispensing with a SAES dispenser. The
MOT number is measured using a mirror MOT for both the SAES andgold dispensing data. The improved N τ product of the gold/rubidium
alloy indicates the improved vacuum quality using the alloy dispenser
relative to the SAES dispenser. . . . . . . . . . . . . . . . . . . . . . . .

3.4Best fit of gold pumping data showing rubidium gold interaction is well
fit by a surface interaction model. . . . . . . . . . . . . . . . . . . . . . .

4.1

63

65

Views of a two chamber cell. Figure a.) shows an isometric view of thecell construction including the placement of the ion pump and pinch off
tube. Figure b.) shows a cross section of the cell including the placementof the apertures, the NEG for the BEC region, and the configuration of
the rubidium dispenser and the 2D NEG. . . . . . . . . . . . . . . . . .

4.2

67

Schematic wire pattern of the chip used to produce BEC. Depending on
where the currents are sourced the chip can be configured as a large Z-wiretrap with many crossing wires, a variety of H-traps, a long waveguide,
etc. The wires with the filled pattern show wires used in experiment. The
experiment was also performed using wires other than the center verticalwire and minor adjustment of the other experimental parameters withthe same results. The leads that form the outermost legs of the Z-wireare connected to two vias so the maximum possible current in the Z-wireis 5 A. All other leads are connected to a single via that can support
2.5 A. On the actual chip the regions between the wires are filled with
”optical metal” that can be used as a mirror surface if necessary. The
optical metal is not shown in this figure for clarity. . . . . . . . . . . . .

75

xvi
4.3

The vacuum (a.) and air (b.) sides of the atom chip. The grey areais the size and shape of the atom chip. The light blue area just inside
the rectangles is the position where the Pyrex cell is anodically bonded
to the chip. The red boxes around the inside perimeter of the chip are
the locations of the UHV compatible vias. In Figure a.) the all of theshaded areas are electroplated copper. This image shows the location of
the optical metal between the traces. In Figure b.) the shaded traces
show gold pads that connect the vias to pads positioned over the cell
location for additional strength when connecting to the cell.

4.4. . . . . .

76

Deflection of a 2 cm square silicon chip with and without a 1 mm thick,φ = 1 cm silicon disk that is attached to the back of the chip. The disk
improves mechanical stability and reduced the deflection of the silicon
chip. Calculations performed with the COSMOSXpress tool in SolidWorks. 77

4.5

Schematic diagram of anodic bonding especially the polarity of the bondrelative to the silicon and Pyrex. . . . . . . . . . . . . . . . . . . . . . .

4.6

81

Diagram of control and imaging systems with connections to external
hardware. When the control computer starts processing a data file itsends a command over the LabVIEW Data Socket to the imaging computer to wait for a sequence of 3-4 images. After the images are taken
the control computer calculates the OD and determines if the shot waswell behaved (as determined by the atom number). If the shot was well
behaved the imaging computer triggers the control computer to run thenext experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

xvii
4.7

Example of a LabVIEW tree used to control experimental BEC production. This example shows control over major section (e.g. the entire
RFsweep1 is disabled), the use of variables (e.g. Tcollect is used several times but can be changed in the global variable section), and subelements (e.g. Xbias branch) that can be disabled without affecting othersub-elements.

4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Pictures of the experimental layout: a.) shows the relative position of
the upper 3D MOT chamber and the the lower 2D MOT chamber, b.) isa closer picture of the 3D MOT and imaging opto-mechanics, c.) shows
the optomechanics and heaters for the 2D MOT. . . . . . . . . . . . . .

4.9

88

Comparison of 2D MOT magnetic field produced with current carrying
wires and permanent magnets. Figure a.) shows the field from fourparallel wires and Figure b.) shows a similar magnetic field toward thecenter of the field diagram. . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.10 Example oscilloscope trace of optimizing the push beam power. The
MOT coils are switched on for 400 ms and then switched off for 300
ms. This leads to the pulsed trace that approximates the loading rateof the 2D MOT into the 3D MOT. At t = -15 s the push beam power
is nearly optimized, as a check the power is changed by rotating a half
waveplate in front a polarizing beam splitter cube immediately before the2D MOT cell. As the power drops the 3D MOT loading rate and number
significantly drop around -5 s. As the waveplate is rotated back the
MOT loading rate and number are optimized by adjusting the waveplate
to have the largest possible swing. . . . . . . . . . . . . . . . . . . . . .

98

xviii4.11 Images of maximally loaded MOT (a) and translated CMOT (b) in a 2 cm
cell. The translation and differences in size and shape of the two MOTs
can be seen. Additionally the atom chip can be seen sealing the top of
the cell and the spring loaded pins for making connections to the external
pads can be seen above the atom chip. Three loops of the compression
wires can be seen outside of the spring loaded pins. . . . . . . . . . . . . 102
4.12 Pictures of the external Z-wire assembly. Five flattened Z-shaped wires(originally 16 AWG) are bent around an aluminum form and then connected in series to effectively increase the total trapping by a factor offive. Spring loaded pins are press fit into a piece of Vespel, soldered toribbon cable, and then epoxied to the side of the aluminum block. A
single turn of 22-24 AWG wire forms the RF coil. The diameter of thecoil is large enough that it does not contact a 1 cm silicon disk that is
epoxied to the back of the atom chip. The compression wires are two
sets of three parallel wires just outside of the RF coil and parallel to thelegs of the external Z-wire. The top of the aluminum block is tapped foran 8-32 screw and the assembly is mounted to a 0.5” post that is thenattached to a frame around the cell. . . . . . . . . . . . . . . . . . . . . 107
4.13 Example data showing parameter search for the initial external Z-wiremagnetic trap by imaging the atoms with a resonant probe laser immediately after the external Z-wire magnetic trap is turned on. Atoms thatare positioned away from the bottom of the magnetic trap will be Zeeman
shifted out of resonance and will not contribute to the signal. The By
with the greatest signal is the starting parameter for the next optimization. In this example the external Z-wire is turned to 120 A while By isvaried in each shot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

xix
4.14 Basic timing of the initial magnetic trap, ramping atoms to the chip, adjusting the external Z-wire magnetic trap for transfer to the chip, transfer
to the chip, and compressing atoms in the dimple. This shows the calculated distance from the chip surface, the Bx and By bias fields (Bz = 0),
the currents in the external Z-wire, the compression wires, the chip Zwire, and the dimple wire. . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.15 Figure a.) Show the relative placement of the external Z-wire, the compression wires, and the chip Z-wire relative to the chip plus backing plate.
Figure b.) shows mode matching the external Z-wire field to the chip Zwire with the addition of the compression wires.

. . . . . . . . . . . . . 117

4.16 Schmatic diagram of current paths for driving interconnected wires on achip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1

Images of 30 BECs made in less than 120 s. Each BEC is produced from
MOT loading to imaging in 3.65 s. . . . . . . . . . . . . . . . . . . . . . 126

5.2

Atom number as a function of shot number in a rapid sequence (3.65 s
production time per BEC) of 30 BECs. The open circles show mild chipheating as the experiment progresses. A few drops of alcohol applied tothe back of the chip provide sufficient cooling to maintain atom number(filled circles).

5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Timing and values from decompressing a compressed magnetic trap to amagnetic trap suitable for splitting and recombining cold atom clouds.
This shows the Bx and By bias fields (Bz = 0), the currents in the external
Z-wire, the compression wires, the chip Z-wire, and the dimple wire. The
trap frequencies are calculated using the known trap parameters andequations from Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 133

xx
6.1

Setup for beamsplitting mirrors and translation stages relative to an atom
chip and cell. Not shown are magnetic field coils or the optics for the 3D
MOT or imaging beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2

Alignment of splitting beam using the reflection signature from the deformed atom chip. The dots at the end of the beams show the expected
pattern is the light is viewed with a card in the beam. Parts a.) and b.)show the signature when the splitting beam is angled relative to the flat
center section of the chip. Part c.) shows the beam with the proper angle
but too close to the chip. When the alignment is close the signatures ofa.) and c.) can both be seen. . . . . . . . . . . . . . . . . . . . . . . . . 139

6.3

Image of atoms split with a double pulse. The atoms are split and are thenpropogated in the waveguide to separate the two clouds. The atoms areimaged after a few miliseconds of TOF. While two clouds are obviously
visible the atom density does not go to zero for the optimal splitting timeand power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.4

Increased noise in the splitting signal as a function of the recombination
time. The upper graph shows the fraction of atoms in the center cloudcompared to the fraction in the outer lobes. The lower graph shows aplot of the standard deviation of the upper data. The increased standard
deviation is one sign of coherence between the atoms, however this data
is significantly affected by noise or loss of coherence. . . . . . . . . . . . 143

6.5

The chip design used for circular waveguiding. The red shaded wire isthe circular guiding wire. The green shaded wires are the endcaps of thechip Z-wire that is used for the initial magnetic trap. The blue wires arebriefly turned on when the atoms are released in the guide to increase
the velocity of the atoms. Other wires on the chip (not shaded) are forother guiding options that were not pursued. . . . . . . . . . . . . . . . 145

xxi
6.6

This shows atoms being guided 90◦ using a curved wire and a rotating
bias field. The pictures images are taken at 4 ms intervals. . . . . . . . . 146

Chapter 1

Introduction and background

1.1

Brief history of BEC on a chip
In 2001 the Hänsch group achieved Bose-Einstein condensation (BEC) in a sin-gle cell vacuum system using lithographically patterned copper wires on an aluminum
nitride chip [1]. This was significant in several ways: the critical magnetic trapping
fields were produced by lithographically patterned wires on the chip, the entire BEC
production cycle took place in a single vacuum chamber, the total production cycle took
less than 10 s with the RF evaporation as fast at 700 ms, the rubidium pressure wasmodulated with low power UV light, and they were able to transport the BEC 3 mmusing an atom motor.1
Just after the Hänsch BEC on a chip paper the Zimmerman group also publisheda paper on making BEC on a chip in a single vacuum chamber [2]. In the Zimmermann
experiment the BEC chip was suspended in a larger vacuum chamber with coils mounted
inside the vacuum chamber. The chip was positioned next to the coils for optimal opticalaccess and easy transfer of atoms to the chip trap. Instead of using UV light to modulate
the rubidium pressure a pulse of current was sent though a rubidium dispenser for a
few seconds to increase the number captured in the magneto-optical trap (MOT) [3].
Similar to the Hänsch experiment the rubidium dispensing was turned off and the MOT
1This paper was published right as I was starting my PhD and after reading it I was very interested
in BEC on a chip. Dana had just returned from a sabbatical in Germany working with the Hänschgroup and was starting his own BEC on a chip effort at CU.

2
was held to allow the vacuum to recover before moving on to evaporative cooling. The
atoms in the MOT were captured and partially cooled by RF evaporation in a magnetic
coil trap. After precooling the atoms were transfered to the chip trap for the final
evaporation to BEC.
The significance these two initial demonstration of rapidly producing BEC on
chip in a single vacuum chamber was the potential of simplifying and speeding up BEC
production. Using a lithographically patterned chip meant that preexisting technologies
could be used to develop complex chips for creating and manipulating BECs. Becauseof the compact and rapid nature of chip based BEC production there was the realpossibility of using BEC in real world applications.
BEC on a chip came 6-7 years after the 1st BEC by the groups of Carl Wieman
and Eric Cornell at JILA/University of Colorado at Boulder (JILA/CU) [4] and shortly
after by Wolfgang Ketterle at MIT [5]. The first BEC machine at CU used a single
vacuum chamber and operated at a very low rubidium partial pressure to maintaina magnetic trap lifetime long enough for efficient evaporation. Because of the low
rubidium pressure it took five minutes to load the MOT. After a sufficient number ofatoms were collected in the MOT the atoms were then transfered to a magnetic trap
and cooled to a BEC by evaporative cooling. After the initial demonstrations of BEC
in a single vacuum chamber most experiments moved to two chamber vacuum systems
where the first chamber was operated at a high alkali metal vapor pressure to quickly
gather a large MOT. The second chamber was held a low vacuum pressure to maintaina long magnetic trap lifetime for efficient evaporative cooling to BEC. Most BEC
experiments use custom vacuum systems to accommodate the various requirementsof collecting a large MOT, transferring the atoms to the BEC chamber, creating a
BEC, and then performing various experiments with the cold atoms. The lifetime of atypical BEC apparatus is typically 5–10 years and new experiments are built to takeadvantage of BEC developments. Because of this each BEC experiment tends to be a3highly customized apparatus, though there has been an effort to develop a simplified
and straight forward system for making BEC using off the shelf components [6]. The
simplifications and speed increases made possible by BEC on a chip have simplified
BEC experiments to the point where BEC could potentially be made available as a tool
to a wider spectrum of researchers and for applications of coherent atoms.

1.2

Applications of BEC and apparatus for portable BEC
Well before the experimental realization of BEC it was appreciated that building

an atom interferometer could be used for measurements with a significant increase insensitivity [7]. In particular an atom based Sagnac gyroscope is 1011 times more sensitive
to rotations that a comparable light based Sagnac gyroscope [8]. Compared to light,
atoms are very sensitive to a wide array of stimulus (gravity, static magnetic fields,etc.) and that sensitivity could be used to develop a wide variety of precision sensors
not possible using light interferometers.
In the past few funding cycles there has been a push to develop atom interferometers and to simplify the apparatus required for making BEC so atom based sensors
could be “taken on the road”. A BEC apparatus roughly consists of a vacuum chamber, a set of magnetic coils and power supplies, 3-4 lasers, an imaging system, and atiming/imaging system. When a visitor first walks into a BEC lab the jungle of lenses,
mirrors, and fiber optics normally catches the eye. Fortunately BEC optical systems canbe simplified and compacted without any significant change in paradigm. High performance, compact laser systems have been the focus of telecom companies for longer thana decade so it is reasonable to assume the laser system can be simplified and compactedusing current technologies. The same is also true of the timing and imaging systems(e.g. cell phones). Because of these advances most of the BEC on a chip effort at CU
has purposely ignored trying to modularize these systems.
The other major elements of a BEC machine is the vacuum system and the4magnetic coils. These two elements typically scale together because magnetic coils areoften mounted outside the vacuum system so the larger the vacuum system the larger
the coils and corresponding power supplies. In cold atom experiments the atoms are
typically trapped using magnetic field gradients which scale like N Ir−2 where N is thenumber of turns in the coil, I is the current, and r is the radius of the coil or the distanceaway from the wire. Typical BEC experiments use 100’s of Amps in multiturn coils togenerate the magnetic fields for transporting and/or producing BEC [9, 6]. Typical
BEC on a chip experiments use ≤ 5 A in a single wire on a chip with an area typicallyless than 2 cm2 . Because of the reduction in current and size there is the possibility
that by using a BEC chip the vacuum apparatus and the magnetic field generation canboth be compacted.
As explained above ∇B ∝ N Ir−2 , this result holds for anti-Helmholtz coil pairs
and straight wires where r is the distance from the center of the coil pair to edge of thecoil or from the center of the wire to a point outside of the wire. In a non-chip systemr ∼ 5 × 10−2 m and in a chip based system r ∼ 10−4 m. For the same magnetic-field
gradient the current required for a chip based system is ∼ 10−5 the total current ofan anti-Helmholtz coil pair. This means that the required power for chip based BECs
systems can be significantly reduced and could be powered by batteries. Though notdirectly related, using a chip can also reduce the overall size of the vacuum system andthe associated magnetic field coils.
One of the key parameters in almost any cold atom experiment is vacuum quality.
But how good does the vacuum quality need to be for making BEC and can the vacuum
apparatus be simplified? The vacuum quality required for making BEC is directly
related to how fast the atoms rethermalize in the evaporation process [1, 10], and therethermalization time τtherm is directly related to how tightly the atoms are compressedin the magnetic trap. Magnetic trap compression is expressed in terms of the classicaltrap frequencies ω of the magnetic trapping potential (think of marbles oscillating in5a bowl). It is straightforward to show that ω ∝ ∇B where B is the total magnetic
field. Increasing the magnetic-field gradient by bringing the atoms closer to the chip
wires facilitates a faster evaporation thus relaxing the vacuum requirement. Typical
trap frequencies in coil based BEC systems are on the order of 100 Hz and in chip based
systems are on the order of 1,000 Hz. Based on the relative trap frequencies there should
be a speed increase and relaxed vacuum requirement by a factor of about 10 by using
chip based BEC production versus coil based BEC production.
This speed up is directly related to the tight trapping potential made possible bytrapping atoms with wires on a chip. Additionally, the size reductions needed to make aportable BEC system are made possible by using chips to compact the vacuum system
and relax the vacuum requirement. The principle of rapid evaporation is applicable toany trap with a tight trapping potential. For example, optical dipole traps with trap
frequencies on the order of kHz have also produced BEC in a few seconds and withrelaxed vacuum requirements [11, 12].

1.3

BEC in a single cell at CU
In December of 2004 BEC on a chip was produced in a portable (30 × 30 × 15 cm)

single chamber vacuum cell at CU [13]. The details of making BEC in this system aredescribed in Reference [14] but for background the key details and history are briefly
reviewed. The vacuum cell was made by epoxying a lithographically patterned chip to
the top of a lapped quartz flourimeter cell. Because it was a single chamber cell UV
light was used to modulate the rubidium pressure in the cell for improved MOT loading
speed and number. At the beginning of the MOT phase the UV light was turned on
for a few seconds to load 6 − 10 × 106 atoms into the MOT. After the UV light was
turned off the MOT was held for 5-10 s before transferring to the magnetic trap toallow the vacuum pressure to recover from the UV induced pressure increase. Afterbackground pressure recovery the chip magnetic trap lifetime was typically 3-4 s. After6
additional optical compression and cooling 2 − 3 × 106 atoms were transfered into thechip-Z magnetic trap and after 4 s of evaporation a BEC was formed.
The vacuum cell that produced BEC was the 4th or 5th cell that had been built
and tested at CU during the experimental setup which started from a bare optics table.
Unfortunately, about a month after achieving BEC a leak formed in the cell. At thetime a new cell had already built and baked out, however UV light desorption in thereplacement cell was significantly less effective compared to previous cells. Multiple cells
were tested over a period of approximately a year with wide variability in the vacuum
quality and the rubidium modulation via UV light desorption. Also, on time scale ofa month the vacuum quality of the vacuum cells degraded to the point that the cells
could not be used to produce BEC.
Around the same time CU and Sarnoff Corporation started collaborating on new
methods of building compact ultra high vacuum (UHV) cells that would be compatible
with making BEC. Sarnoff made several cells using anodic bonding to attach a siliconchip to the end of a Pyrex cell (see Section 4.2). These anodically bonded cells showed nolong term vacuum degradation which solved a significant problem with the epoxied cells.
However UV light did not modulate the rubidium pressure in anodically bonded cells.
In the epoxied cells the UV desorption was highly variable but in the anodically bonded
cells the UV desorption was completely absent. This was attributed to the high bakeout
temperature (> 300◦ C) made possible using anodic bonding. The bakeout temperatureof the epoxied cells had been ∼ 150◦ C, so the high temperature bakeout was a key
benefit of anodic bonding because the vacuum quality was significantly improved.
Additionally anodically bonded cells showed no vacuum degradation over theperiod of several months and multiple bakeouts. While anodically bonded cells were
not amenable to rubidium pressure modulation using UV light the improvement in
the short and long term vacuum quality was significant, so there needed to be a way tocreate BEC in a compact vacuum system without using UV light desorption to modulate

7
the rubidium pressure. This thesis details the process of how BEC was achieved in acompact anodically bonded vacuum system.

1.4

Comparing BEC and the laser
It is interesting, but maybe not surprising, to consider Einstein separately con-tributed to the theory of the laser [15] and BEC [16]. Many similarities from physical[17] and technological [18] standpoints have been drawn between the laser and BEC.
From a physics standpoint both the laser and BEC depend on the statistics of Bosons
that results in a macroscopic occupation of a single quantum state. For both the photon and the atom the probability of scattering into a state depends on the occupationlevels of the system and in that sense the physics of a laser and a BEC can be directlycompared.
A BEC can also be compared to the laser from a technological development
point of view. Initially the laser started as most research efforts; It required skill andunderstanding of special techniques to make and build a laser. In our view the successof lasers as a part of everyday life can be attributed to two significant developments inthe field of lasers [19, 20]. Examining the significance of these developments and looking
at the success of lasers the general requirements and technologies that are needed tomake BEC a viable technological tool can be extrapolated.
The first is the development of the laser diode in its various forms. Replacing the
crystal lasing medium with a small piece of appropriately doped semiconductor that canbe mass produced was a significant advance in laser technology. The key improvementsof the laser diode are the reduced size, power consumption, and cost has made lasers
ubiquitous in a wide variety of products.
The second development, fiber optics, has allowed lasers to be part of everyday
life is arguably not even part of a laser. Laser light is an amazing medium but can bevery sensitive to environmental effects (e.g. smoke, clouds, moisture, etc.) especially8over long distances. Fiber optics provides a pristine and stable environment for lasers
to be sent wherever the light needs to be. The key improvement of fiber optics is theability to isolate light from the effects of its surroundings and simplify light distribution.
While the results of this thesis will not produce BEC is a package less than 1 mm3or transport BEC across the continent, they have the potential to lead to a compact
device that is portable and can produce BEC with reduced power consumption. Becauseof the two chamber system and relaxed vacuum requirement there is the possibility
that a BEC can be produced outside of the controlled laboratory environment and betransported to a variety of locations.

1.5

My contributions
My contributions to this thesis are the initial experiments with gold as rapid

rubidium dispenser in anodically bonded cells to compensate for the lack of UV lightrubidium desorption. Gold was not used as a rubidium dispenser in the final two
chamber BEC cell because the time scale of the rubidium pressure modulation is on
the order of 10’s of seconds. The gold results are included in this thesis because gold
still has excellent potential as a dispenser and pump in compact BEC systems. WhileI performed the initial experiments of gold dispensing with and without a MOT, thefinal gold pumping and dispensing experiments were performed by Ben Luey and Will
Holmgren.
Because of the difficulty of modulating the rubidium pressure in anodically bonded
cells I implemented a six-beam MOT and a magnetic trap to load and transfer at
minimum a few 106 atoms into the chip magnetic trap while maintaining a low rubidium
pressure. A six-beam MOT and magnetic transport are commonplace, but in order tofacilitate transfer to the chip the MOT is operated with angled beams to bring the
MOT closer to the chip. This angled MOT was first modeled with a basic Monte-Carlosimulation and was then implemented in the experiment. The other unique aspect of9this configuration is using a large Z-shaped wire to magnetically transport and thenadiabatically transfer atoms to the chip-Z trap.
Instead of a pulsed dispenser a two-chamber vacuum system was implemented
with a 2D MOT in a high rubidium vapor pressure chamber and a 3D MOT in a UHVchamber. Evan Salim had implemented an all glass compact two chamber system witha 2D MOT in one chamber loading a 3D MOT in a second chamber. This was not thefirst time that a 2D MOT was used in a multichamber system, but special attention
was made to not significantly increase the size compared to the single chamber design.
The experience from these experiments was applied to making a compact two chambersystem with an atom chip suitable for BEC production.
Using a 2D MOT to load a 3D MOT significantly improved the number that
could be captured in the 3D MOT. After the MOT atoms were magnetically captured
and coupled onto the atom chip BEC was produced in three significant modes. A highrepetition rate mode where 30 BECs were sequentially made with a BEC being produced
every 3.65 s. This production mode demonstrated the possibility of using BEC in a realworld application where BEC will need to be consistently and rapidly produced. The
second significant production mode was optimizing the production for the largest BEC
number. While optimizing a BEC machine for largest number is not unique, comparing
the BEC number (∼ 400 × 103 ) and production time (< 10 s) put this system within anorder of magnitude of the highest throughput rubidium BEC systems [9]. If the size ofthe vacuum system is taken into account (5×10×30 cm) this is one of the smallest BEC
systems.2

If size and BEC throughput are considered together then this apparatus

is a significant advance in BEC production. The BEC apparatus was also optimized
to produce BEC as fast as possible and is able to produce BEC from MOT loading totaking images of BEC in < 2.65 s. The final BEC mode produces BEC in a magnetic
2

Here size denotes the size of the vacuum system and magnetic coils. No efforts to simplify thelaser/control systems will be discussed in this thesis, but it is worth mentioning CU plus a few companies
are in the process of simplifying and compacting the laser system.

10
trap 300 µm from the chip surface and with trap frequencies suitable for splitting a
BEC with an optical pulse.
Additionally, there is some preliminary work on techniques that would be needed
to make a Sangac gyroscope. The first experiment is splitting a BEC or cold atoms witha standing light wave. Again splitting a cloud of atoms with a standing light wave is notnew, but splitting close to a chip that forms one wall of the vacuum chamber had notbeen previously performed. Steven Segal helped with the splitting beam setup especiallythe optomechanics around the cell. The second experiment is a curved waveguide thatwas used to guide atoms around a curve with a 1 cm radius. Guiding atoms aroundcurves has been done previously [21, 22] but this method of guiding the atoms created amoving ‘bucket’ where the atoms could be accelerated as they moved around the curve.
This was done with the help of Evan Salim and Ben Luey, plus theoretical support byAlex Zozulya.

1.6

Outline of the rest of thesis
The rest of the thesis will be organized as follows: First the basic theory ofchip magnetic traps, MOTs, vacuum considerations in small systems, BEC, and atom
splitting will be covered. These topics will be covered at a level that was useful for me indesigning and conducting cold atom experiments. Then a potential rubidium dispenser
based on the properties of the gold/rubidium alloy, however this dispenser was not
used in the final experimental configuration. While the dispenser was not used in the
final BEC configuration the technology and potential applications are still noteworthy
as a rubidium dispenser and pump. The final configuration used for BEC production
will then be detailed followed by the specifics of the various operating modes used to
produce BEC. The final chapter will be experimental results of curved waveguiding andthe splitting of a BEC/cold atoms on a simply supported chip with a standing lightfield as a groundwork for future chip based experiments.

Chapter 2

Essential theory

Theory is a strong word for this chapter. This is mostly a collection of basic
formulas and ideas needed for back of the envelope calculations related to vacuum cells,
magnetic traps made from wires and bias fields, MOTs, and BEC. Reference [10] isan excellent summary of chip based magnetic traps and BEC on a chip. Reference [14]additionally provides excellent derivations of the interaction of atoms and lasers, chiptraps, imaging, etc.

2.1

Magnetically trapping atoms
Classically atoms can be modeled as miniature bar magnets where the energy of

the atom in a magnetic field B is
→
−
→
U =−
µc · B(2.1)

where µc is the classical magnetic dipole moment. If µc is (anti-)aligned with themagnetic field then the bar magnet will be in a (high)low energy state. Assuming atoms
are bar magnets allows many magnetic trapping properties to be described, but atoms
are fundamentally quantum mechanical objects. Quantum mechanically there is not acontinuum of dipole and magnetic-field alignments but the alignments are quantized.
Equation 2.1 becomesU = µm B

(2.2)

12
where µm = µB mF gF . Here µB is the Bohr magneton, mF is the magnetic quantum
number, and gF is the Lande g-factor. The magnetic quantum number has values
F− ≤ mF ≥ F+ where F is the total atomic angular momentum (F± = J ± I) . The
Lande g-factor can be positive or negative depending on the atom and the atomic state.
Depending on the product of mF gF the atom can be put into a high-field seeking state(mF gF < 0), a clock state (mF gF = 0), or a weak-field seeking state (mF gF > 0).
Most BEC applications trap atoms in the weak-field seeking states because the Ernshaw
theorem [23] states in steady state a magnetic-field maximum cannot be created in freespace. In 87 Rb atoms are typically trapped in the |F = 1, mF = −1i (gF = −1/2) or
the |F = 2, mF = 2i (gF = 1/2) states.
The force experienced by an atom in a magnetic field is
F = −∇U = −µm ∇B.

(2.3)

This expression is useful for calculating the magnetic-field gradient needed to hold atoms
against gravity. For atoms in the |F = 2, mF = 2i(|F = 1, mF = −1i) state ∇B ≈15(30)G/cm to cancel gravity.
The other consideration when trapping atoms, especially cold atoms, is how tomaintain the alignment of the magnetic dipole moment as an atom moves in a magnetic
trapping potential. Classically a bar magnet can be given angular momentum by rotating the bar magnet. This will cause the bar magnet to precess around the directionof the magnetic field and will maintain the average alignment of the bar magnet in themagnetic field. If the direction of magnetic field is slowly varied then the orientation of
the bar magnet will follow the magnetic field. The definition of slowly is
dθ
<ω
dt

(2.4)

where θ is the instantaneous alignment of the magnet with the magnetic field and ω isthe precession frequency. For atoms the Larmor frequency is equivalent to the precession

13
frequency. The trapping requirement is nowdθµm |B|
<
= ωL
dt
h̄

(2.5)

where ωL is the Lamor frequency. This puts a minimum value on the magnetic fieldthat will prevent atoms from ‘flipping’ the value of mF . Atoms in the strong-field state
are especially sensitive to this flipping because they are aligned opposite the directionof the magnetic field and if they flip it is into an anti-trapped (high field seeking) state.
Atom loss due to mF flipping is often called spin-flip or Marjorana loses. The region
where the magnetic trapping field is too low to maintain atoms in the trapped state is
typically called a ‘hole in the bottom of the trap’ [24]. The minimum magnetic field
to avoid Marjorana losses is typically a few Gauss, but the actual value of the ‘bottom
field’ depends on the temperature (i.e. velocity) of the atoms and ∇B.

2.2

Basic wire magnetic trap
The magnitude of the magnetic field and gradient for an infinitely long wire isI
rI0
B =β 2
r
B=β

(2.6)
(2.7)

where β ≡ µ0 /(2π) = 2 × 10−7 N A−2 and all negative signs have been dropped. The
magnetic field circulates around the wire according to the right hand rule and when aperpendicular (relative to the wire) bias field is added a magnetic minimum is createdat a finite distance from the wire
Btrap = βI
− B⊥ .
r(2.8)

The gradient of the magnetic trap is the same as when there was no bias field. The
distance of the magnetic minimum from the wire and the gradient at the minimum are
I
B⊥
B2
0
Bmin
= ⊥.
βI

rmin = β(2.9)(2.10)

14
As mentioned above if the atoms pass though a zero magnetic field they will spin flip
into an untrapped magnetic state and be lost from the magnetic trap. The zero of thewire magnetic field is ‘plugged’ by adding a uniform magnetic field parallel to the wire.
The total trapping field is
Btot =

q2
Btrap+ Bk2 .

Traps are characterized by their trap frequency assuming the trap is harmonicsµm B 00ω=.
mRb(2.11)(2.12)

If Equation 2.11 is expanded around the trap minimum then
B0
ω∝p
Bk
2.2.1

(2.13)

Z-wire trap

Figure 2.1: Figure a.) shows a U-wire configuration for creating a quadrupole magnetic
trap. Figure b.) shows a Z-wire configuration for creating a Ioffe-Prichard magnetic
trap. Figure c.) shows the wire configuration for creating a dimple trap. By themselves
these wire configurations will not produce magnetic traps and must be augmented withexternal bias fields as shown.

From the magnetic field of a single wire two simple chip wire traps can be madeby bending the ends wire (see Figure 2.1). The first is the U-wire trap that make aquadrupole trap (zero minimum field), and a Z-wire trap that makes a Ioffe-Prichard
trap (non-zero minimum field). The U-wire trap can be used to make a mirror MOT [25]close to the surface of the chip and was used in first single chamber BEC experiments[18], however cold atoms need a non-zero minimum bias field to avoid Majorana losses

15
so the Z-wire trap is used for BEC production. While a U-wire was used in the first BEC
on a chip experiment [1] the current BEC production cycle exclusively uses a Z-wire
trap.
The trap frequencies of a Z-wire trap can be approximated by assuming there
is one weak axis (longitudinal or axial axis) nearly parallel with the x-axis, and twotight axes (transverse or radial axis). Assuming z  w (see Figure 2.1) then then thelongitudinal trapping field can be approximated by two half-infinite wires
B(x) = B0 + B2 x2 + . . .

(2.14)

where
IZ z0
w2
IZ (6Bx z0 + β IZ )
= 8β
IZ z0
w4 Bx + 4β 2
w

B0 =
B2

Bx + 4β(2.15)(2.16)

where z0 = βIz /By . In this expansion there is no first order term so the frequency along
the longitudinal axis is given by Equation 2.12. The transverse trap frequencies of aZ-wire trap are approximatelypω⊥ = µm /mRb

By2√

βIZ B0.

(2.17)The Z-wire trap will be also be used for magnetically transporting atoms over
several cm’s so it cannot always be assumed that z  w. The key aspects to consider
when the atoms are far away the Z-wire is maintaining the trap gradient along thetransverse axis and trap depth along the longitudinal axis. Far enough away from aZ-wire the field will largely be the field of a long straight wire except for the location of
the kink in the wire. Both the transverse and the longitudinal traps will be negligible
at z  w. The transverse magnetic field from a Z-wire can be approximated by the

16

1.1
gradient
field
1
0.9
0.8

Correction

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.5

1

1.5

2

2.5

ζ

a.)

10000

1000

Longitudinal Trap Depth (G)

100

10

1

0.1

0.01

0.001

0.0001
0

5

10

15

20
z(mm)

25

30

35

b.)

Figure 2.2: Figure a.) shows the correction factor for the magnetic field and gradientof a short piece of wire as a function of distance from the wire. The distance from thewire is scaled to the width of the wire section. The correction is relative to the fields ofan infinitely long wire. Figure b.) shows the approximate trap depth of a 30 mm Z-wirerunning 5 × 130 A as a function of distance from the center of the Z-wire. The trap
depth is the maximum magnetic field toward the edges of the Z-wire minus the field at
the center of the Z-wire.

17
magnetic field of a short section of wire
Bf inite = B∞ p0
Bf0 inite = B∞

1

1 + 4ζ 2
8ζ 2 + 1(4ζ 2 + 1)3/2(2.18)(2.19)

0 is the field of an infinitely long wire (Equation 2.7). The
where ζ = z/w and B∞

correction factors are shown in Figure 2.2a and it is interesting to note that for distancesless than w/2 the field gradient of a short wire is slightly greater than the gradient of along wire but then quickly falls off.
The trap depth of the longitudinal field is approximated by field of two long wires
separated by w and taking the difference of the maximum field versus the field at thecenter of the wires at a height zq
Bdepth = βIZ w2 +w4 + 16z02 w2
4z0 
√
− 2
.
w + 4z02
2 2wz0p(2.20)As seen in Figure 2.2b the longitudinal trap depth becomes zero for values greater than
∼ w. For atoms in the |2, 2i state 1G≈ 67 µk. For an atom cloud after the opticalmolasses stage the typical temperature is 20–50 µk and assuming the trap should be
5–7 times deeper than the temperature [10] then the minimum trap depth should be
2-5 G. As seen in Figure 2.2b that is approximately 20 mm or about 2/3 the size ofthe external Z-wire. That is for the specific parameters of the initial external Z-wiretrap of Section 4.4.6. Because the longitudinal trap depth rolls off more quickly thantransverse trapping force the width of the Z-wire is the limiting factor of the maximum
distance that atoms can be trapped by the field of a Z-wire trap.

2.3

Dimple trap
A Z-wire trap typically has two tight axes and one significantly weaker axis. The

weak axis of the trap limits the total effective trap frequency because ω̄ = (ω1 ω2 ω3 )1/3 .

18This is especially true when the chip Z-wire is wide as in this thesis (see Section 4.4.6).
To counter this a sequence of smaller chip Z-wires could be energized to compress the
weak axis of the magnetic trap [10]. This scheme is slightly more complicated because
it would require an independent current driver for each of the Z-wire legs.
A dimple trap in the center of the chip Z-wire accomplishes the same purposeby adding a relatively strong trapping potential perpendicular to the chip Z-wire field.
While the dimple trap accomplishes the same effect as compressing a sequence of Z-wires
it requires only one additional current driver.

2.3.1

Basic properties

A dimple trap is made using a wire configuration shown in Figure 2.1c plus Bx
and By bias fields to essentially overlap two magnetic waveguides. Because the dimple
trap is the vector sum of two long wire fields plus the corresponding bias fields theproperties of the dimple trap are a vector sum of the total field. This can be see in
Figure 2.3 where the trap is centered over the origin in the XY plane but the residual
traps from the Ix and Iy wire currents can be seen extending from the origin. Whilea dimple trap can achieve a higher ω̄ than a Z-wire trap the various bias fields can nolonger be considered orthogonal and independent. The behavior and applications of thedimple trap are described in the following section.

2.3.1.1

Dominant wire determination

The dimple trap is formed by taking the vector sum of two magnetic fields thateach have there own zero. One the magnetic fields zeros is dominant and determines the
position of the magnetic trap. Assume the minima are at r0 and r1 with magnetic field
gradients B00 and B10 . The dominant magnetic trap will be at the global field minimum.
Assuming the field around the minima is well described by a linear gradient so the

19

a.)
0.3

θxy
θyx

0.2

y (mm)

0.1

0

-0.1

-0.2

-0.3
-0.3

-0.2

-0.1

0
x (mm)

0.1

0.2

0.3

b.)

Figure 2.3: A calculation of the dimple trap used for BEC production. The following
parameters were used to generate this representation: Ix = 1.25 A, Iy = 3.75 A, Bx =-38 G, Iy = 46 G. Figure a.) shows a 2D slice though the center of the magnetic trap
parallel to the chip surface with the magnetic field shown on the Z-axis and in false
color. Figure b.) shows a contour plot of the same trapping potential showing the two
calculations for the trap rotation angle.

20
magnetic field at each minimum is0
B0,1 = B0,1∆r

(2.21)

where ∆r = |r0 − r1 |. So the dominant minimum is created by the field with the largest
gradient. While there is a dominant trap it is possible to have the atoms stably trappedin the secondary trap if the atoms start in the secondary trap and the energy barrier
between the dominant and secondary minima is much greater than the temperature of
the atom cloud.

2.3.1.2

Dimple trap depth

The depth of the dimple trap is defined as the difference of the magnetic field
directly above the wire crossing and at the far extent of the dominant wire. Assume
the dimple is formed at z0 so the depth of the trap is
Bdepth = B(∞, z0 ) − B(0, z0 )(2.22)

Iy z0− Bx .
x2 + z02(2.23)

where
B(x, z0 ) = β

Inserting Equation 2.23 into Equation 2.22 the trap depth is2Bx − β Izy Bx < β Izy00
Bdepth =
.


Iy
Iy

β
Bx > βz0(2.24)

z0

Thus the maximum depth of the dimple trap is determined by the wire that generates
the dimple trap and is simply the magnetic field of the dimple wire at z0 (see Figure 2.4).
The dimple trap depth is different than other wire traps where the trap depth is almost
entirely a function of the trap bias field [25]. The dimple trap depth for reasonable
parameters is significantly deep enough to trap laser cooled atoms. See Figure 2.5 fordimple trap depth versus the dimple trapping current.

2140

Dimple Trap Depth (G)

20

0

-20

-40

Trap Depth at x=5 z0< Approx.
> Approx.
0

10

20

30

40
Dimple Bias (G)

50

60

70

80

Figure 2.4: Dimple trap depth showing calculation of trap depth assuming depth isevaluated relative to the side of the trap. The calculation assumes Ix = 5 A, Iy = 2.5
A, and By = 80 G. Also shown are the limits assuming the depth is evaluated at thetrap center and at infinity. For values less than 20 G there is no dimple trap and theresulting potential will split the cloud.

2.3.2

Trap frequency calculation

As seen in Figure 2.3b the dimple trap is rotated relative to the wires that create
the magnetic field. This is due to the vector sum of two orthogonal infinite magnetic
waveguides. Calculating the trap frequencies beyond the infinite wire approximation
requires the angle of the magnetic trap to be included because the rotation angle canbe large depending on the relative current in the trapping wires [26].
The trap rotation angle is determined by minimizing the directional derivative(∇|B|·b) in the XY plane (Bz = 0) at the minimum of the trapping field. Oddly enoughthere are two similar but slightly different answers depending on the order that x andy are set to zero.
tan(θxy ) =−Ix
Iy(2.25)tan(θyx ) =1
2Bx /By + Iy /Ix(2.26)

22
220
200

z0 = 50 µm
z0 = 100 µm
z0 = 200 µm

180Dimple Depth (G)

160
140
120
100
80
60
40
20
0
0.5

1

1.5

2

2.5
I2 (A)

3

3.5

4

4.5

5

Figure 2.5: Dimple trap depth as a function of distance away from the wire center andcurrent in the wire.

where the order in which the coordinates are set to is denoted by the subscript. As seenin Figure 2.3b both approximations give a reasonable estimate of the trap rotation. For
the sake of simplicity and consistency with other derivations [27] assume θD = θxy .
By Taylor expanding around the trap minimum with the appropriate angle correction gives
1 B 02 2
t + ...
2η 2 B0
B 02 2= B0 − η(γ + η)
l + ...
B0
B0
B 02 2= B0 + By η(γ + η) z0 +z + ...
B0
2B0 0

Bt = B0 +(2.27)

Bl(2.28)

Bz(2.29)

where B0 = By |γ + η|, B 0 = By2 /(βIx ), γ = Bx /By , and η = Iy /Ix . Assuming η and γ
are less than 1 then the dimple trap will have a tight and loose axis as can been seenin Figure 2.3; t denotes the coordinate along the tight direction of the trap and l along
the loose direction of the trap.

23
The frequencies of the dimple trap (using the definition of Equation 2.12) are
ω
√0
2η
p
=
η(γ + η) ω0

ωt =(2.30)

ωl(2.31)

ωz =where ω0 =ω
√02(2.32)√µm /mRb B 0 / B0 . Because η < 1 the dimple trap compresses one of theptight axis of the dimple trap. The weak axis of the dimple trap is a function of the main
wire trap frequency. This expansion is valid for η > 0 otherwise the trap frequencies
would approach ∞. In the case of η ∼ 0 it cannot be assumed that the trap is a dimple
trap and the exact wire pattern must be considered.

2.3.3

Comparison of dimple and Z-wire traps

The dimple trap provides excellent compression in all three dimensions where a
trap made using a Z-wire configuration has excellent compression in two dimensions andit is typically weak along one axis. Compare the effective trap frequencies ω̄ to evaluate
the merits of the dimple trap versus the Z-wire trap.

2.3.3.1

Z-wire trap

The Z-wire trap is created using a wire pattern shown in Figure 2.1b. As mentioned this type of trap typically has two tightly compressed axes and one axis withweak compression. The trap frequencies of the highly compressed dimensions areω1 = ω2 =p

β IZµm /mRb 2 √ .
rz B0(2.33)The trap frequency of the weak dimension can be approximated assuming there are two
half infinite wires separated by wz .
pr
µm /mRb β Ix
ω3 ≈
wz
rz

(2.34)

24
The bottom bias field does not appear in the equation because the trap is created by
the end caps of the Z-wire trap. Equation 2.34 includes an extra factor of 1/2 to better
approximate the shape of the longitudinal axis.
The net effective trap frequency for the Z-wire trap is
!1/3
5/2
p
Ix
5/6
ω̄z = µm /mRb β.
9/2
B0 rz wz
2.3.3.2(2.35)

Dimple trap

As explained previously the dimple trap is made from two wires that cross at rightangles. For the following calculations assume that the trapping wire and the dimple wire
are at the same distance from the atoms rD and that Iy = ηIx . This form of the dimple
trap is being used instead of Equations 2.30–2.32 because it simplifies the comparison
between the two trap. Along the strong direction the trap frequencies arep
β Ixµm /mRb 2 √rD B0(2.36)β ηIxµm /mRb 2 √ .
rD B0(2.37)

ω1 = ω2 =and along the weak direction
ω3 =p

Together the effective trap frequency isω̄D =2.3.3.3p
β ID √
3µm /mRb 2 √
η.
rD B0

(2.38)

Compare dimple to Z-wire trap

Set the two effective trap frequencies equal and solve for rTs√p
Iy 3 η
rT =µm /mRb β √ .
ω̄z B0(2.39)To have the same effective trap frequency as a Z-wire trap the working distance scales
as the square root of the dimple current. Even though the dimple trap compensates forthe weak axis of the Z-wire trap the dimple current can become prohibitively large very

25
quickly as the distance from the center of the wire is increased. The values previously
used to achieve BEC [18] can be substituted into Equation 2.39 to give an approximate
experimental value for rT (η = 1/2, rz = 80 µm, wz = 2 mm, B0 = 2 G, Iz = 2.75 A)
rT ∼ 100 × 10−6

p

Iy .

(2.40)

So it may be possible to put wires on the back of a chip structure and still achieve theconditions for rapid BEC production.

2.3.4

BEC without vias
s

m

r
6
?w
6
r
?c
6

ratoms?

Figure 2.6: Geometry for calculating the optimal chip width rc and wire currents tocreate BEC without vias.

This thesis uses ultra-high vacuum compatible vias to pass current though asilicon chip to create a magnetic trap suitable for rapid BEC production. By using a
dimple trap it may be possible to create a BEC compatible magnetic trap with wires
on the air side of a thin silicon chip. The conditions used in Reference [18] for creating
BEC were IZ = 2.75 A, rz = 80 µm, with a wire size of 10 × 100 µm, and ∼ 15 mm

26
long. From these parameters the chip Z-wire dissipates about 2 W. What wire andchip configurations will have the same ω̄z but dissipate the same power? Assume thegeometry is as shown in Figure 2.6 and that the maximum current that can be easily
controlled for a few seconds without excessive circuitry is about 50 A.
First define approximate chip thickness limitations. While a 2 cm2 area and 100µm thick silicon membrane will support atmospheric pressure other issues need to beconsidered such as power dissipation and membrane deflection. For the following calculations assume that the thickness of the chip rc is at least 200 µm to assure structural
integrity.
Another issue to consider is the potential for the atoms to collide with the surfaceof the chip as they are brought close to the chip. Assuming the distance of the atoms
from the chip ratoms is 100 µm is a reasonable working distance to avoid atom loss andto facilitate direct imaging of the atom cloud.

2.3.4.1

Wire shapes

Two general wire shapes may be considered, round and rectangular. For optimized magnetic-field gradient and minimized power the rectangular wire is best. The
magnetic field from a flat wire with width w is nominally the same as a round wire aslong as the field is evaluated at r ≥ w. The maximum width of the wire that will notshow flattening effects is set by the total distance from the center of the wire to theatoms ww = ratoms + rchip + rw where rw = hw /2. The power dissipated in a rectangularwire is2
Prect = IDρL
2rw ww(2.41)

where ρ is the resistivity of the metal and L is the length of the wire. The optimal wire
shape is determined by minimizing dPrect /dID because the magnetic field is proportional

27
to ID and yields rw = (ra + rc )/2. The optimal power is
2
Prect = IDρL.
3/2(ra + rc )2(2.42)

Rearranging terms in Equation 2.39
1/6 5/6

ID = rT2

Bbot Iz
3/2

rz w1/3 β 1/6 δ 1/3.

(2.43)

Equation 2.42 becomes
1/3 5/3

27B Iz L ρPrect = 3 bot(ra + rc )2
2/3
1/3
2/3
8rz w β δ(2.44)

where the substitution rT = 3/2(ra + rc ) is made. The optimized power and current isshown in Figure 2.7 assuming the same BEC values used previously [14].
2.4

60

2.2

55

2

50

1.8

45

1.6

40

1.4

35

1.2

30

1

25

0.8

20

0.6

15

0.4

10

I(A)

P (W)

Power
Current

0.2

5
0.1

0.15

0.2

0.25
rc(mm)

0.3

0.35

0.4

Figure 2.7: Expected power dissipated in a rectangular wire as a function of chip thickness to create trap conditions suitable for BEC production.

2.3.4.2

Bias field requirements

Even though the dimple wires dissipate a reasonable amount of power the bias
fields to create the dimple trap may be large. The bias fields (see Figure 2.8) arecalculated via Equation 2.6 using the current and distance from Figure 2.7. The current

28
200
By
Bx
180

160

Bias Field (G)

140

120

100

80

60

40
0.1

0.15

0.2

0.25
rc(mm)

0.3

0.35

0.4

Figure 2.8: Bias fields needed to create a tight dimple magnetic trap as a function ofchip thickness.

magnetic field generating coils can create bias fields of Bx ≈ 80 G and By ≈ 160 G(similar to rectangular coils in Reference [14]). The maximum chip thickness limited by
the current bias field configuration is rc ≈ 250 µm, ID ≈ 30 A, and P ≈ 1.4 W. These
are all reasonable experimental parameters that are currently attainable.

2.3.5

Evaporation during transport

There are currently several efforts to continuously load, evaporate, and produce
BEC in a long waveguide [28, 29]. These experiments typically employ a long (one toseveral meters) waveguide that is tightly confining in 2D. As the atoms propagate along
the guide the atomic density and collision rate is sufficient that evaporative cooling isseen in the waveguide. In one experiment the evaporation is accomplished by moving
the atoms though various evaporation zones [30].
In another experiment a chip based moving magnetic trap was implemented tomove laser cooled atoms 1.75 cm using a periodic potential [31]. Combining the twoideas and using dimple traps to maintain a high ω̄ it is plausible to consider a BEC

29“production line” where the atoms are loaded from the optical molasses and opticalpumping stage into the first dimple trap. The dimple trap array moves the atoms
though evaporation zones until the atoms are finally cooled to BEC. The rate of BEC
production would be limited by the MOT loading rate and the time in each evaporation
zone. An array of dimples similar to Figure 4.3b but with less spacing between dimple
wires could be used to make an array of traps.

2.4

MOTs and BEC chips
MOTs are the workhorse of cold atomic physics experiments and while robust to

most imperfections there are several effects contributing to a MOT so the basic physics
is not immediately intuitive. The main change to the MOT configuration in this thesis
allows a six beam MOT to be positioned about a factor of two closer to the chip surface
is explained in Section 2.4.4.

2.4.1

Basic MOT considerations

A MOT provides two different mechanisms (cooling and trapping) from the interaction of several key elements: laser polarization, laser frequencies, magnetic fieldmagnitude, magnetic field direction, and mF state. For the sake of simplicity coolingand trapping will be discussed as separate processes. First the force on an atom from asingle laser is
F=

h̄kΓ
S
2π 1 + N S + 4(δL + k · v + µm B/h̄)2 /Γ2(2.45)

where Γ is the linewidth of the transition, S is the saturation parameter I/IS , N is thetotal number of lasers shining on the atoms, and δL is the detuning of the laser awayfrom resonance. This equation can be separated into two basic parts: the momentum ofa photon (h̄k) and the number of photons scattered by the atom per unit time (the restof the equation). The force from a pair of counter-propagating beams with identical

30
parameters (S, δ, etc.) is
h̄kΓS
F=
2π



1
1
−
2
1 + N S + 4(δ+ /Γ)
1 + N S + 4(δ− /Γ)2
(2.46)

where
δ± = δL ∓ k · v ± µm B/h̄.

2.4.1.1(2.47)

Cooling

There are several regimes and limits of laser cooling that depend on a very wide
range of operating conditions, but the maximum capture velocity is a straight forward
calculation. The maximum possible force on an atom is h̄kΓ/(2π) (assume S  1 inEquation 2.45) which if divided by mRb gives an acceleration of a = 3.5 × 104 m/s2 !
From this acceleration the maximum velocity that can be stopped in a cell of width√
w is vmax = 2aw. For typical cell sizes of 1, 2, 3 cm this yields capture velocitiesof 27, 38, and 46 m/s respectively, all significantly below 240 m/s, the mean velocityof 87 Rb at room temperature. The result is only atoms in the low velocity tail of theMaxwell-Boltzmann distribution are captured in a MOT [32].
In laser cooling atoms are cooled because of an apparent velocity dependent force
that is directed against the velocity (i.e. F = −cv). This damping is greatest when
one of the lasers in nearly on resonance with the appropriate laser beam. The damping
coefficient c can be calculated by Taylor expanding Equation 2.46 around δ+ ≈ 0 and
dropping all terms other than the first orderc=dF
h̄k2 ΓS4δ−=.
dv
π (1 + N S + 4(δ− /Γ)2 )2(2.48)

Now consider the interplay between the Doppler and Zeeman shift and determine
where each effect dominates in Equation 2.47 (assume δl = 2Γ). First only consider theeffect of the Doppler shift where v = δl /k. For the assumed detuning and λ = 780 nm,
vDoppler = 9.4 m/s significantly below the maximum capture velocity of even a 1 cm
cell. Thus the Doppler effect only contributes after the atoms are cooled below 9 m/s.

31
Now only consider the effect of the Zeeman shift assuming B = B 0 z where B 0 is
the gradient of the MOT field along the strong axis. The radius where the Zeeman shift
is comparable to the typical laser detuning is
z=

2h̄Γ
.
µm B 0(2.49)For a typical MOT gradient of B 0 = 12 G/cm the distance where the magnetic field
and the laser detuning cancel at z ≈ 3 mm. This is significantly smaller than a typicalvacuum cell, but the real significance is if a low velocity atom wanders away from themagnetic field center the Zeeman shift will shift the atom onto resonance with theappropriate laser pushing the atom back to the center of the MOT. This only happens
when the atoms are near the center of the MOT.
The final consideration is when the Doppler shift and the Zeeman shift are bothmuch larger than δl but approximately the same magnitude. This is the same principle
behind the Zeeman slower [33] where atoms exiting a high temperature oven with verylarge velocities can be cooled in 1D. A significant difference between the Zeeman slower
and the MOT is the number of cooling beams, because a Zeeman slower uses a single
laser to slow the atoms in 1D. If Equation 2.45 was used instead of Equation 2.46 thenthe damping coefficient c would have the second order as the lowest order and would be
proportional to δ.
Because of this higher order dependence a Zeeman slower’s field is tailored tomatched the Doppler shift of the atoms while they are slowed to maximize the scattering
rate over the length of the Zeeman slower. In comparison cM OT only assumes δ+ is small
to achieve damping. This make the MOT easier to operate and robust over a variety ofoperating conditions, however it involves more lasers.
Because there is a balance between the two laser beams there is also an optimalcooling trajectory in a MOT. For example if an atom starts too slow it will be sped up
before it is slowed so the optimal trajectory is maintained as the atom is cooled toward32
the center of the MOT fields (see Figure 2.9). For comparison, in a Zeeman slower
atoms with a Doppler shift less than the current Zeeman shift will drift down the tube
until the Doppler and Zeeman shifts are matched and at that point the atom will be
captured by the slowing trajectory of the Zeeman slower.
16

14

12

v(m/s)

10

8

6

4

2

0

-2
0

5

10

15

20

25

t(ms)

Figure 2.9: Numerical ODE calculation of the trajectory of an atom in a MOT starting
at 0.1 m/s and the edge of the MOT. The ‘atom’ is centered in the laser beams withthe velocity pointed at the zero of the magnetic field. This shows atoms with a low
velocity will be sped up to match the optical forces until the laser forces are balanced.

In summary, there are two basic cooling mechanisms at play in a MOT dependingon the relative magnitudes of δl and k · v, and the position of the atoms in the MOT.
If k · v ≈ δl , B  δl , and z  3 mm then Doppler cooling is the dominant mechanism.
If k · v  δl and z > 3 mm then cooling takes place because of the Zeeman shiftbalances the Doppler shift. Actually MOT (Magneto-Optical Trap) does not describe
these two processes and the cooling mechanisms can be divided into two regions: theDoppler region, and the Zeeman region. The relevance of these two regions to compact
BEC systems is cooling parameters (especially the capture velocity) are related to the
MOT loading efficiency. Specifically the cooling path length directly corresponds to theminimum size of the vacuum apparatus.
The above discussion was focused on cooling atoms from a room temperature

33
thermal distribution but once the atom has been slowed to a few m/s what is the lowest
achievable temperature? When an atom is cooled by a laser it typically adsorbs aphoton from the laser that is directed opposite it’s velocity. Then the energy adsorbedby the atom is radiated away in a random direction. Averaged over many photons themomentum of the absorbed photons will have a net force in the direction of the laser.
The momentum of the emitted photons will average to zero, but at any instant of timethe emitted photons will impart a random momentum to the atom. This Brownian likeheating is canceled in a MOT by the cooling mechanisms described above. The balanceof the recoil heating rate and the low intensity Doppler cooling rate leads to the Doppler
temperature TD = h̄Γ/(2kB ) [34]. For 87 Rb TD = 143µk corresponding to a velocity
of 17 cm/s. This temperature is in agreement with the temperature of a reasonably
optimized CMOT (see Section 4.4.5).
In the early days of MOTs the Doppler limit was a well known temperature limit
but in initial laser cooling experiments the measured temperature was significantly lower
than the Doppler limit [35]. This new cooling mechanism was called sub-Doppler coolingand worked by introducing a velocity dependent force dependent on the light shift due
to the polarization of the light beams [36].

2.4.1.2

Trapping

After the atoms are cooled they need to be trapped and the MOT also performs
this function. The trapping in a MOT is assumed to have the form F = kx where k is
the effective spring constant of the trap. Assume the atoms have been cooled below a
few m/s so k · v can be ignored. The spring coefficient k can be calculated by Taylor
expanding Equation 2.46 around r ≈ 0 and dropping all terms other than the first orderk=dF(r ≈ 0)h̄k2 ΓS
16Bc δl=−2dx
π
1 + N S + 4δl2 /Γ2(2.50)

34
where Bc = µm B 0 /h̄. If the typical MOT detuning δl = Γ is used thenk=−h̄k2 S
32Bc.
π (N S + 17)2

(2.51)

Thus atoms that have been cooled will feel a force toward the center of the magnetic
field that is proportional to the magnetic field gradient. Increasing the magnetic field
gradient will compress the atoms, a key element of a compressed MOT [37]. In practicethe intensity of the optical fields will also determine the position of the MOT, but thatanaylsis would require significantly more computational effort.

2.4.2

Chip versus six-beam MOTs

a.)

b.)

Figure 2.10: This show the reletive alignments and positions of the laser beams (dashedlines) with the cell and atom chip. Not shown are the 5th and 6th beams that would goin and out of the page. Also shown is the position of the coils needed to generate the
MOT magnetic field. Figure a.) shows the configuration for a mirror MOT and Figureb.) shows the configuration of six beam MOT.

The first BEC on a chip experiment [1] and the first BEC on a chip experiment
at CU [18] both used a mirror MOT to laser cool atoms close (1-2 mm) from the chipsurface. Mirror MOTs were used for initial BEC on a chip experiments because it
simplified the transfer to the chip magnetic trap. The details of a mirror MOT can befound in References [25, 14] but the basic configuration of a chip MOT are the lasers

35
beams are reflected off the surface of a chip to form a prism of light that creates acooling force in all directions and the magnetic coils are rotated 45◦ to the plane of thechip (see Figure 2.10a). The essential characteristic of the mirror MOT is the ability toposition the MOT close to the chip. Initially the mirror MOT position is moved awayfrom the chip to capture a large number of atoms. After the atoms are gathered thecloud is compressed in a CMOT and brought close to the surface of the chip. Dependingon the laser alignment the atoms can be brought to < 1 mm from the chip surface. The
close proximity of the atoms to the chip surface makes it possible to magnetically trap
the atoms with < 5 A of current on the chip.
A “U” shaped wire on the chip plus a bias field can also create a MOT magnetic
field. This combination is called a chip mirror MOT and has the advantage of precisely
positioning the MOT relative to the chip wires and mode matching the MOT to themagnetic trap. There are many variants to the chip mirror MOT that are explained in
References [38, 39, 40].
While the benefits of the chip MOT are mode matching, proximity to the chip,
precise alignment, and low power the significant trade-off is reduced MOT number
because the stopping distance is effectively halved. The loss in MOT number is greaterthan a half because the MOT number is a strong function of the stopping distanceof the atoms [32] and the stopping distance of the mirror MOT more than halved.
Additionally the alignment of the laser fields depends on the mirror surface. Someconfigurations apply an insulated mirror directly over the trapping wires [25, 18] whileother experiments etch fine lines in a gold or silver mirror surface to create an integrated
mirrored atom chip. Both approaches have been used at CU, applying a mirror overa wire pattern is a straight forward procedure but requires careful processing. The
mirror surface is typically attached with EpoTex 353 ND vacuum compatible epoxy
that is degrades at the anodic bonding temperatures (> 400◦ C) so this process is notwell suited to anodically bonded cells but is amenable to epoxied cells. The integrated

36
mirror does not involve epoxy so it is compatible with anodic bonding and works well
for a mirror MOT away from the wire surface. However, when the atoms are moved
close to the surface the optical field is significantly affected by the gaps in the mirror
and the mirror MOT is degraded. The mirror MOT is sensitive to the cell alignment
because the lasers are reflected off the chip surface so when the vacuum cell is changed
the lasers must be carefully realigned to the new cell position. Another disadvantageof the mirror MOT is the MOT coils that are rotated by 45◦ relative to the cell. This
results in an awkward balance of optical access, coil size, coil position, and cell size thattypically results in a coil that is significantly larger than the cell (see Figure 2.10a).
These same obervation were independently published in Reference [13] around the sametime as the experiment in this thesis switched from a mirror MOT to a six-beam MOT.

2.4.3

Six-beam MOT

The more common configuration for making a MOT is the original six-beam MOT
configuration [41] rotated 45◦ to fit under the chip (see Figure 2.10b). The pros and consof using a six-beam MOT for chip based BEC experiment are almost exact opposites ofthe mirror MOT. The larger capture volume of the overlapped beams is able to capture
4-6 times the number of atoms, however the MOT is now an order of magnitude farther
away from the chip. The chip mirror MOT was able to mode match the magnetic fieldof the mirror MOT to the chip Z-wire trap, however in the six beam MOT the tightaxis of the MOT is perpendicular to the tight axis of the chip Z-wire. This can affectthe direct transfer of atom into a chip Z-wire magnetic trap [13]. While the atoms fromthe chip CMOT could be magnetically trapped with < 5W, transporting an atom cloudto the chip requires 200–1,000 W of power to generate a high magnetic-field gradient ina large volume.
Using a six-beam MOT is advantageous because the chip processing is significantly
relaxed and the alignment of the cell relative to the lasers is significantly easier because

37
the chip surface is not used as an optical surface. Not using the chip as a reflector
also relaxes the chip requirements for anodic bonding. While the MOT alignment iseasier the alignment of the transport magnetic trap requires additional optimization toefficiently couple atoms to the chip. Finally the MOT coil is an extra pair of bias coils
configured as an anti-Helmholtz pair and mounted with the bias field coils. The extra
space of the coils is minimal and improves optical access to the cell compared to the 45◦
mirror MOT coils (see Figure 2.10b).

2.4.4

Six-beam MOTs with angled beams

Figure 2.11: The angled six beam MOT configuration showing the factors that determine
the distance from the atom chip to the center of the MOT.

As mentioned above a six-beam MOT is desirable option compared to a mirror
MOT but one significant drawback is the distance of the atoms from the chip. Other cold
atom groups have worked with transfer efficiencies of ∼ 15% from their six-beam MOT
to their chip magnetic trap [42, 43], with the highest transfer efficiency of ∼ 50% [13].
In this thesis the atoms are magnetically transfered by a large Z-wire to improve mode
matching to the chip. However the external Z-wire has a limited transport distance (< 2cm). At the time it appeared that the Vuletic group at MIT used a relative beam angle
other than 90◦ (see Figure 2.11) and that it would be possible to bring the atoms close

38
to the chip without reducing the beam size (i.e. MOT number). In actuality θ = 90◦ in
the MIT experiment [44], but fortunately the idea of reducing the angle was still viable.
In an effort to improve the MOT loading efficiency Monte-Carlo (MC) simulationsof MOTs in vapor cells were used to evaluate potential mirror MOT configurations (e.g.
a 20 cm long wire plus bias field will make a long skinny mirror MOT unfortunately
the number is small). This same MC simulation shows a MOT with angled beams isnumber stable until < 20◦ (see Figure 2.12). This allows the MOT to be closer to thechip by a factor of 1.8-2 which significantly reduces the current required to transport
the atoms to the chip.
A first it is surprising that the MOT is stable to such a shallow angle but considering a few relevant MOT parameters it is not surprising. Other MOT configurations
have been successfully implemented with non-orthogonal beams [45], and as mentioned
the mechanisms of cooling the atoms and the trapping the atoms can be considered separate processes. These steps have been separated from some of the earliest laser coolingexperiments and are still in use today where the cooling is a 1D process (i.e. Zeeman
slower) followed by trapping in a MOT [9] or a shallow magnetic trap [35]. An atom
experiences a force of roughly 1,000 g during cooling so a shallow angle MOT should
hold against gravity with θ < 1◦ . Another consideration in a MOT is the depth of thetrap potential after the atoms are cooled. It is somewhat hard to define a potential
depth in an overdamped system but an atom must have a few K of energy to escapefrom a MOT [46] and the typical Doppler temperature is < 1 mK so it is possible for the
beam angle to be reduced without reducing the MOT number. The other considerationis the size of the MOT due to photon reradiation but any loss from photon reradiation
would be linked to the size of the beams which in the current case are made larger tocollect more atoms out of the background.
What are the limits and mechanisms of the shallow angle MOT? The consistencyof the MOT number over a wide range of angles can be attributed to the atom version

39

600

500

A/U

400

300

200

100

0
0

5

10

15

20
25
MOT angle (θ)

30

20
25
Beam angle (deg)

30

35

40

45

a.)

900

800

Integrated MOT number (AU)

700

600

500

400

300

200

100

0
0

5

10

15

35

40

45

b.)

Figure 2.12: Simulations of MOT number versus the angle of the 3D MOT beams. Figure a shows a 3D calculation that includes the momentum kick from recoiling photons.
It is interesting to note the MOT number is saturated up to θ ≈ 20◦ . Figure b shows a
2D numerical ODE calculation that does not take the photon recoil into consideration.

40of Lambert’s cosine law that the apparent brightness of a beam will be independentof angle if the scattering surface is perfectly random. This type of surface roughness
is know as Lambertian roughness and the light scatting will have a cos(θ) dependence
that exactly cancels the change of viewing angle. From an atomistic point of view therandomness of an atom bouncing off a wall is described by the accommodation coefficient
α. There are several sub-versions of α [47] but the simplest form isα≡Ei − Ef
Ei − Ew(2.52)

where Ei is energy of the atom before hitting the wall, Ef is the energy of the atom
after hitting the wall, and Ew is the energy of the wall. If the atom specularly reflects
off of the wall then Ef will have some dependence on Ei and α < 1, but if the atom
has no dependence on the previous state then the wall will determine the state of theatom and Ef = Ei and α = 1. Atomic specular reflections are more probable when
the atoms are light and/or non-interacting (e.g. helium or nitrogen) and α > 0.8 forspecular tending atoms incident on specially prepared surfaces. In most cases it isa good approximation to consider the walls of the vacuum chamber to be atomically
Lambertian. In this approximation the angular distribution of atoms leaving the wallsof the vacuum chamber has a cos(θ) dependence where θ is measured relative to thewall normal.1
Figure 2.12b shows a calculation of MOT number assuming probablity of atoms
coming off the wall has a cos(θ). In this calculation the capture velocity is determinedusing a 2D numerical ODE solver ignoring the random recoil of the photon leaving theatom (see Appendix A for code). This calculation has a different curve then Figure
2.12a where a 3D MC was performed using the Verlet method to integrate the atom’s
trajectory [48]. This simulation included a random recoil force that would cause loss
at lower MOT beam angles. Due to the difference in the simulations the photon recoil
1This is also the starting point for calculating the conductance of long pipes and such in the molecularflow regime.

41
appears to be the limiting factor that determines that minimum MOT beam angle thatwill support a large MOT.

2.4.5

2D MOT

A 2D MOT is more than just a high aspect ratio 3D MOT. In a 3D MOT
the magnetic field gradient is truly 3D, but in a 2D MOT there is no magnetic field
gradient along the longitudinal axis because of the symmetry of the four coil or magnet
configuration. In a 3D MOT there must be lasers with some projection onto every axis
in order to trap atoms. In a 2D MOT the lasers only propagate along two axes andleave the third axis with no optical or magnetic forces. Along the X and Y axes the 2D
MOT can be considered 1D cooling so the projection of velocity will be cooled similar
to Figure 2.9. Along the Z-axis there is no cooling and the Z-component of the velocity
acquired when the atom left the wall will carry the atoms toward one end or another ofthe 2D MOT cell. In a slightly modified version of the 2D MOT an additional laser is
propagated along the Z-axis of the 2D MOT. This laser does not provide confinement
but rather molasses cooling along the unconfined axis creating what is called a 2D+
MOT. The velocity of atoms exiting a 2D+ MOT is significantly slower than the pure
2D MOT configuration [49].
The atoms in a 2D MOT are typically extracted from the 2D MOT chamber
via an aperture. The length and width of the 2D MOT, the aperture diameter, anddistance from the 2D MOT will determine the velocity distribution of atoms exiting theaperture. Specifically there is a cutoff velocity above which atoms will not be extracted
from the 2D MOT where the maximum velocity is directly related to the diameter ofthe exit hole. This happens because atoms with greater longitudinal velocities will not
spend as much time being cooled in the transverse direction. The variance in coolingdetermines the solid angle of the beam exiting the 2D MOT. At the same time atoms
with higher angles will tend to be atoms with higher longitudinal velocities, so the exit

42
aperture of the 2D MOT cell can be used to filter high velocity atoms at the cost of total
flux. Additionally the size of the aperture determines the pressure differential between
the two systems. The velocity filtering depends on the solid angle so a small aperture
directly after the end of the 2D MOT will limit vacuum conductance and still allow ahigh atom throughput.
The optimal aspect ratio of the 2D MOT cell depends if a 2D or a 2D+ MOT isused. For a 2D MOT a longer cell will allow more atoms to be captured and be usedin the flux. If a 2D+ MOT is used a longer cell will allow the atoms more time in thecenter of the cell where the radiation imbalance will increase the atom velocity. The
2D+ MOT increases the capturable flux by a factor of 3-4 because the atoms are colder
and the radiation pressure improves the aiming of the atom flux [49]. A 2D MOT isdesirable over a 2D+ MOT when on axis light is an issue.
There are other configurations that can be used to produce a cold beam of atoms
specifically Zeeman slowers [50], a 3D MOT with a hole in an optic (LVIS)[51], and apyramid MOT with an open tip [52, 53]. Zeeman slowers tend to be long devices (> 30cm) and need a counter propagating on-axis slowing laser. The LVIS configuration isa 3D MOT with a quarter waveplate and mirror mounted inside the vacuum chamber.
The waveplate and mirror have a hole drilled in the center so the atom beam can beextracted from the MOT. From a practical device standpoint the pyramid MOT is
the best option because only one laser is needed to provide cooling in all directions,however the 2D+ MOT has the largest capturable flux relative to the physical size andcomplexity.

2.5

VacuumIn a compact BEC system a high performance vacuum system is a key element.

Compared to larger BEC systems chip based BEC experiments operate at much higher
pressures (< 10−12 versus 10−9 torr). Even though the vacuum requirement is relaxed43
VacuumOutgassing rate per area
Pressure × Pumping speed
ConductancePressure

Vacuum Units[torr][`]/[s]/[m3 ]=[atoms]/[s][torr][`]/[s]=[atoms]/[s][`]/[s][torr]= [atoms]/[m3 ]Electical
Current
Current
1/Resistance
Voltage

Table 2.1: Comparison of vacuum to electrical parameters. The unit of torr will be usedinstead of using particles/volume.

the high surface area to volume ratio of small vacuum systems and isolating a regionof high rubidium pressure (∼ 10−7 torr) less than 10 cm from a low pressure BEC
region (∼ 10−9 torr) are important design considerations. The pressure in a multichamber vacuum system can be modeled as an electrical circuit with the various vacuum
parameters directly correlating to electronic parameters (See Table 2.1).
The vacuum circuit analogy can be used to make order of magnitude estimates.
For example, the pressure in the 2D MOT chamber is ∼ 10−7 torr (largely rubidium)and the pumping rate of the upper chamber is ∼ 4 `/s assuming 2 `/s pumping eachfrom the ion pump and the rubidium sticking to the walls of the BEC region. Assuming
the pressure in the BEC chamber is 10−9 torr then the conductance between the twochambers can determined using Ohm’s law.
C =4`/s × 10−9 torr
10−7 torr= 4 × 10−2 `/s.

(2.53)(2.54)

Assuming a plate thickness of l = 1 mm what diameter hole will achieve this pumping
speed? From C = 12d3 /l [54, 55] then d ≈ 0.7 mm. This is slightly smaller than theacutal size of the pinhole used in the final two chamber design (see Section 4.1).
The electronic circuit analogy can be extended to more complicated systems.
There is one difference that is worth mentioning, while most electronic devices arethought of as fundamentally constant voltage devices while most UHV systems arefundamentally constant current devices. Specifically outgassing from the walls is the

44
major contributor in UHV systems. This outgassing rate (current) is independent of
the local vacuum pressure or conductances so the walls of a vacuum system should bemodeled as constant current sources. These calculations can be performed using themethod of Reference [56]. A key parameter in these calculations is the wall outgassing
rate. The outgassing rate of glasses and silicon are dependent on the preparation and it
is difficult to find similar conditions bakeout conditions (300◦ for weeks) in the literature[57, 58], but it may be assumed the outgassing rate is comparable to metals with similar
vacuum preparation (≤ 10−9 torr `/s/cm2 ).

2.6

BEC
The essential characteristics in the formation of a BEC can be understood using

the DeBroglie wave model (fuzzy balls in a jar). The formation of a BEC can also
be understood from the level occupation and Bosonic stimulation picture. This quantum model is used in References [59, 60] to evaluate evaporation trajectories and is areasonable explanation of the quantum formation of a BEC.
The DeBroglie wave model assumes atoms are wavelike object where the wavelength of the atom is proportional to the inverse of the atom’s momentum
λDB =

h
mv

(2.55)

where h is Plank’s constant, m is the mass of the atom, and v is the atomic velocity.
This is of course without relativistic corrections because the velocities are significantly
less than the speed of light. As the velocity of the atoms decreases via cooling theeffective size of the atom (i.e. wavefunction) increases and can become significantly
larger than the physical size of the atom. Often the cartoon picture shows atoms
initially as point particles and as the temperature drops the points become little waves
that are increasingly larger as the temperature drops [61]. But the atoms are 3D objects
and cold atoms can be thought of as fuzzy balls that don’t have well defined boundary

45
much like a wavefunction, but can still be described by an effective radius. Because
atoms are trapped in a 3D trap for most operations performed on a BEC the atoms canbe thought of being placed in a container with volume Vc filled with N atoms.
A Bose-Einstein condensate forms because Bosons are indistinguishable particles[62] and if the spacing between two Bosons is smaller than λDB of the individual Bosons
then the two particles will occupy the same space and appear to be a single particle.
Invoking the fuzzy ball picture the volume occupied by the balls is VDB ∼ N λ3DB . If
there are enough atoms at a cold enough temperature then VDB > Vc . All of the fuzzyballs will overlap inside the jar and be indistinguishable thus forming a condensate. The
key result of this analysis is that to form a BEC the atoms need to be more than justcold. The atoms also need to have a high density. The combination of high atomic
density and large wavefunction (i.e. low temperature) lead to BEC.
The concept of total wavefunction volume versus the confining volume is called
the phase space density (PSD)
P SD = nΛ3DB(2.56)

where n is the density of atoms in the trap and ΛDB is the thermal deBroglie wavelength.
A more rigorous derivation [63] shows a condensate is formed when the PSD≥ 2.61.

2.7

Evaporative cooling
As explained in the previous section a BEC is formed when there is a high atom

density and the atoms are cold. Current laser cooling techniques cannot cool atoms
at a high density, because at high densities a photon in the center of the cloud may
recoil with several atoms as exits the cloud. This outward photon pressure prevents
high densities at low temperatures. Laser cooling is key step toward BEC but must be
followed by another cooling mechanism to achieve BEC.
Evaporative or forced evaporative cooling is based on selectively removing a few

46of the hottest atoms from the Maxwell-Boltzmann (MB) distribution. As seen in Figure
2.13a the product of energy (∝ v 2 ) and the MB distribution shows the fraction of thetotal energy in the cloud is shifted toward the highest energy atoms. By removing a
few atoms from the high energy tail of the MB distribution a significant portion of theenergy in the cloud can be removed with minimal atom loss.
MB distribution
Energy Distribution

T=1.0
T=0.6
T=0.3

0.4

0.35

0.35

0.3

0.3

0.25

0.25
f(v)

f(v), E*f(v)

0.4

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05
(a.)

0

(b.)
0

0

1

2

3
|v|

4

5

6

0

1

2

3
|v|

4

5

6

Figure 2.13: Maxwell-Boltzmann velocity distributions and energy distribution. The
shaded area in Figure a.) shows how a significant portion of the total energy can beremoved from an atom cloud without large number loss by evaporating on tail of the
distribution. Figure b.) shows various distributions for various relative temperatures.

The atom distribution is truncated by removing the hottest atoms but the peakvelocity of the cloud is the same. Removing more atoms from the high energy tail ofthe MB distribution will remove more energy at the cost of significantly more atoms.
However is a few high energy atoms are removed and the remaining atoms are allowed
to redistribute their energy and smooth out the velocity distribution the peak velocityof the cloud will drop (see Figure 2.13b). The dynamics of the redistribution of energy
is described by the Boltzmann transport equation [63] and knowing the energy levels ofa trap and the initial energy distribution the details of evaporative cooling trajectory
can be numerically solved [64]. While this numerical analysis can predict the number,
temperature, etc. of an atom cloud as it is evaporated Monte Carlo models showeda truncated MB distribution would essentially rethermalize if each atom on average

47
underwent three collisions [65]. From an experimental point of view the thermalization
time can be calculated from the classical collision rate
τcol = n vrel σ(2.57)

where n is the atomic density, vrel is the relative velocity of the atoms, and σ is the crosssection of the atom. This equation is directly related to the mean free path, monolayer
formation time, and other common high vacuum parameters [54].
To maintain the high energy tail of the MB distribution the rate of evaporation
τevap should not be faster than the rethermalization rate or the evaporation will beinefficient. So in generalτevap < 3 τcol(2.58)but τcol can increase or decrease during evaporation depending on several considerations.
If there is little atom loss as the atoms are cooled the atomic density will increase as theatoms settle into the bottom of the trap. The physical size of atoms in a trap (e.g. linear
or harmonic) will occupy less volume at lower temperatures than at higher temperatures
simply because the less energy the atoms have the less the atoms will push against the
spring constant of the trap. Again that assumes the relative low atom number loss.
This effect of increasing density and collision rate, even as the total number decreases,
is called runaway evaporation and is a key requirement for BEC production [65].
If there is a loss mechanism, other than evaporating only the hot atoms, it is possible that after evaporating for some time the density of the atom cloud will decrease.
In this case τcol will drop and will continue to drop because the parasitic atom loss rateis greater than the gains made by cooling the atom cloud. In most BEC experiments thedominant loss mechanisms are collisions with background thermal atoms (i.e. vacuum
pressure), resonant light scattering, and at high densities three-body loss [6]. Of thesemechanisms vacuum loss is ubiquitous in BEC experiments and the most significant incompact BEC systems. The minimum requirement is τvac < τevap for runaway evapora-

48
tion and for optimal performance τvac  τcol so the only significant source of atom loss
is forced evaporation.
Using chip based BEC systems relaxes the vacuum requirement compared to nonchip BEC experiments by trapping the atoms in a tightly compressed magnetic trap.
Tight trap compression increases τcol because compressing a gas will increase n andthe overall temperature of the cloud or vrel . If the compression is adiabatic then the
PSD will not change but the collision rate will increase allowing faster evaporation (seeEquation 2.57). The trap compression is proportional to effective trap frequency because
a high trap frequency is a result of a tighter trap. See Table 2.2 for a comparison ofchip based versus more traditional BEC machines.

τvac (s)ω̄⊥ (Hz)NM OT
TRF (s)Typical∼100∼100
109 − 1010
∼ 50

Compact∼10∼1,000107 − 108
1-4

Table 2.2: Comparison of evaporation parameters (e.g. evaporation time) for chip based
and traditional BEC machines.

2.7.1

Evaporation trajectories

Optimizing BEC number and production largely falls into three main categories:
initial number, vacuum lifetime, and evaporation trajectory. This section will cover theevaporation trajectory as the first two are covered in previous sections. As mentioned
in the previous section the rate at which atoms can be evaporated off the tail of the MB
distribution depends on how quickly the atoms thermalize or the inter-atomic collision
rate. The collision rate depends on n and vrel and as the atoms are evaporated N andvrel both drop but because the atoms are in a linear or harmonic trap the effective
volume Vef f decreases at the same rate or faster than the decrease in N and Vrel so thecollision rate is maintained.

49
Optimizing the evaporation trajectory starts by assuming the atom cloud energy
distribution is close to equilibrium and that some fraction of the atoms are evaporated
from the MB distribution tail. The fraction to evaporate is choosen to maximize thelargest energy loss while minimizing number loss. After the fraction to evaporate has
been determined the fraction is converted into the appropriate loss mechanism: change
in RF frequency for RF evaporation, decrease in laser intensity for optical evaporation[11], or distance from hard surface for surface evaporation [66]. It has been shownthat optimizing each fractional evaporation step will optimize the whole evaporation
trajectory [67]. Several optimized evaporation trajectories have been published [68, 69,
67] and a simple evaporative model that can be used for a qualitative understandingof the evaporative process [65]. These models are largely semi-classical and do nottake into account Bosonic stimulation at the end of the evaporative trajectory. Fromthese models the optimal evaporation trajectory is an exponentially decreasing energy
truncation.
Recently an evaporation trajectory was calculated taking into account the Bosonicstimulation of the atoms especially towards the end of the evaporation when the groundstate begins to have a significant population [60]. At this point in the evaporation
trajectory any two atoms that scatter will have final states that are significantly affectedby Bosonic stimulation. One of the atoms will scatter into the ground state and theother atom will receive the excess energy and will be evaporated. The cloud’s overalltemperature will drop at an accelerated rate as the condensate forms. This evaporation
has two essential trajectories: an initial exponential sweep to a point where P SD ∼ 10−1
and then a rapid linear sweep to just above the trap bottom that sweeps the atoms into
the ground state. This method improves the final number in the BEC by avoiding
vacuum related losses of a longer evaporation trajectory.
As mentioned once a BEC begins to form it dramatically changes the behaviorof the system, this similar to a laser at threshold. As the laser pump is increased above

50
threshold a single mode becomes dominant due to cavity selectivity. This dominantmode will pull gain from other potential modes and will result in a macroscopic occupation of a single mode. The behavior of a BEC is similar, once a small condensate isformed Bose statistics rapidly enhances the state with the largest occupation. In thecase of a BEC the only state that can easily have a macroscopic occupation is the groundstate of the magnetic trap where in an optical cavity the dominant mode is determinedby cavity mode with best gain/loss ratio.
In the experiment described in this thesis linear evaporation sweeps are usedinstead of exponential sweeps. Linear sweeps have been successfully used to achieve BEC
in other experiments [13]. While the exponential sweep is the most efficient trajectory
an optimized linear trajectory can also achieve BEC as long as the runaway evaporation
condition is met, especially at the end of the evaporative cooling. This can easily beachieved if the initial atom number is large so there is a reasonable number at atoms in
the final stage of the magnetic trap and the trap frequency is high so the thermalization
rate is high enough to be efficient over a large range of time scales. Typically there are
two linear evaporation stages; The first stage cools the atom cloud with higher atom
loss compared to an exponential trajectory, but still maintains a reasonable collision
rate. The final linear evaporation stage is optimized to maintain the collision rate of
the 1st sweep though the final linear sweep thus maintaining the runaway evaporation
condition to achieve BEC.

Chapter 3

Aux Rb dispensers

In 2003 CU collaborated with Sarnoff Corporation on improving cold atom technologies. One of Sarnoff efforts was an enriched 87 Rb dispenser using a gold/rubidium
alloy. As will be explained pure rubidium can be evolved from a gold/rubidium alloy
by simply heating the alloy. Sarnoff made dispensers by mixing liquid rubidium andpowdered gold and then pressing the mixture in a mold to form pellets. The alloy wasstable in air because the gold and rubidium formed a stable alloy. When the pellets
were heated in a vacuum rubidium was released because of its higher vapor pressure.
At the same time other groups also used gold used to dispense alkali metals but undersignificantly different circumstances [70].
Around the same time a CU Mechanical Engineering student was integrating
MicroElectro-Mechanical System (MEMS) into cold physics experiments [71]. In one of
the cold atom MEMS devices were little magnetic coils (< 1 mm2 ) made by patterninga gold wires on silicon. When the coils were energized in vacuum with a resonant laser
shining though the cell a brief puff of fluorescence could be seen around the coils. It
was soon realized how little power was used by the MEMS device and also how quickly
the rubidium pressure was modulated. At the time anodically bonded cells were justbeginning to be used and rapidly rubidium dispensing without UV light was an open
question. Rapid rubidium dispensing from a thin gold wire appeared to be a potential
solution. After several tests it was determined that the rubidium partial pressure did

52
not recover fast enough to be useful for rapid BEC production, however the techniqueand results are worth documenting for future dispensing needs.

3.1

Binary phase diagrams and vapor pressures
Au-Rb
1100

1064.43 °C

1000
900
L

800
730 °C

580 °C600

498 °C

500

300

AuRb

Au6Rb

400

Au2Rb

Temperature°C

700

(Au)
200

39.48 °C

100
34 °C
0
Au

10

20

30

40
50
60
Weight Percent Rubidium

(Rb)

96.8

0
70

80

90

100
Rb

Figure 3.1: The binary phase diagram of gold and rubidium [72].

Phase diagrams are a pictoral representations of the properties of a material as afunction of temperature, pressure, composition, etc. Specifically binary phase diagrams
represent the alloys, melting temperatures, and crystal phases of a two elements as afunction of the composition and the temperature (see Figure 3.1 for the phase diagramof gold and rubidium).
As seen on the left hand side of Figure 3.1 there is pure gold and as more rubidium
is added to the gold an alloy of gold and rubidium is formed. If the percentage ofrubidium by weight increases beyond ∼ 30% then an alloy is no longer formed and thetwo materials will coexist in mixture but will not be alloyed.
If pure rubidium is placed in a UHV chamber the partial pressure of rubidium

53
will typically be 10−7 torr which is significantly higher than the pressure required formaking BEC (even on a chip). In physical chemistry Raoult’s and/or Henry’s Law
describes the behavior of a solution composed of a low vapor pressure solvent (gold)
and a low concentration of a high vapor pressure solute (rubidium). In this case thevapor pressure of the solute will be suppressed by the solvent. Because of Henry’s Law
the vapor pressure of rubidium in a gold alloy is suppressed so the net vapor pressureof the gold/rubidium alloy is sufficiently low to be compatible with producing BEC.
To dispense the rubidium from the gold the alloy should be heated to the approximately to the temperature where the gold is still a solid but the rubidium has
started to liquefy. This is called the L + α phase where part of the phase is liquefied
and the other is still solid, in this case the α constituent (gold) has the higher melting
temperature. For example in Figure 3.1 at 10% rubidium the alloy will transition froma solid to the L + α phase at 580◦ C. Above this temperature the partial pressure of
rubidium will increase because of the phase change and the rubidium is be dispensed
from the alloy. No gold will be dispensensed at < 1000◦ C because the vapor pressureof gold is significantly lower than than rubidium’s, especially below the melting pointof pure gold.

3.2

Rapid dispensing
One crucial element in small BEC cell type system is quickly (< 1 s) modulating

the partial pressure of rubidium. It is important for the alloy to rapidly cool below
the dispensing point to reduce the overall vacuum pressure to a point suitable for BEC
production. Most of the cooling happens via radiative loss because the alloy and heating
element are in a vacuum with poor thermal conduction to the ambient temperature
though thin wires. Determine what wire configurations will have the most rapid wire
cooling for rapid rubidium pressure modulation.

54

3.3

Radiative heat loss
Assume at the temperatures for rubidium dispensing the significant heat loss

mechanism is radiative heat transfer. The heat loss of a system due to radiation isdQ
4= σA T 4 − TRT
dt

(3.1)

where σ is the Stefan-Boltzmann constant,  is the emissivity of the surface, and A is
the total surface area. The total heat in a system is
Q = cmT(3.2)

where c is the specific heat of the material and m is the mass of the object. Combining
Equations 3.1 and 3.2 the change in temperature of a system is

dT
σA 4
4
=
T − TRT
dt
cm

(3.3)

This can be simplified in the case of a round wire by replacing A = 2πrl and m = ρπr2 l,
where r is the radius of the wire, l is the length of the wire, and ρ is the mass density.
Equation 3.3 becomes

dT
2σ
4
=
T 4 − TRT
.
dt
cρr

(3.4)

4 so the approximate
There is no simple solution to Equation 3.4, but assume T 4  TRT

solution to Equation 3.4 is
T (t) ≈ q
3

1
−3
6σ
c ρ r t + T0

(3.5)

The only geometrical factor that comes into play is the radius of the wire. The
emissivity can be increased by making the surface as rough as possible especially whenusing a noble metal like gold. The other parameter that can be somewhat varied is thematerial dependent factor c ρ (see Table 3.1). Choosing the right material will improvethe cooling rate, but less than a factor two. Tungsten is assumed in all the calculations
because it is readily available, strong, and has a high melting temperature (3680 K).

55
Material
Ti
Al
Ag
Au
W
Mo
Pt
Pd
Cr
Cu
NiCr
Fe
SS
Ni

c(J/g/C)
0.523
0.897
0.235
0.129
0.132
0.272
0.133
0.24
0.45
0.385
0.414
0.449
0.5
0.44

ρ(g/cc)
4.5
2.7
10.5
19.3
19.3
10.2
21.4
12
7.19
8.96
8.4
7.86
7.7
8.9

ρc
2.35
2.42
2.46
2.49
2.55
2.77
2.85
2.88
3.24
3.45
3.48
3.53
3.85
3.92

Table 3.1: Specific heat and mass density of a variety of materials. Sorted by ρ c smallest
to largest, or most desirable to least desirable for rapid cooling.

3.4

Optimal wire radius
Using Equation 3.5 solve for the required wire radius given a cooling time. The

radius for a given cooling time, initial temperature, and final temperature is3 3
 σ t Tf T0
r≈6
ρ c T03 − Tf3(3.6)

Assume a 1 s cooling time and a tungsten filament. What radius is required tocool from 10% above the L + α phase to 20% below the phase? This will give a relative
estimate of what would be required to strongly modulate the Rubidium pressure in thecell. Table 3.2 shows the wire diameters to cool from the three gold/rubidium phases.
The wire diameter is small but still reasonable.

3.5

Other geometries
As seen in Equation 3.3 the heating rate is largely affected by the ratio of mass tosurface area and a round wire actually has the worst ratio of mass to surface area.

56
Tp (C)
730
580
496

d(µm)
22
14
10

Table 3.2: Calculated radius to achieve cooling in 1 s from 10% above to 20% below theL + α phase assuming Tungsten filament for the three gold/rubidium alloys.

3.5.1

Mesh

A wire mesh has a large surface area, but if the wires in the mesh are round thereis actually no gain in the surface area to mass ratio over a long wire with the sameradius. There is a small loss also because the active area is reduced where the wires
touch. Otherwise the mass to surface area is the same as a long wire of the same radius.
The potential benefit of mesh is greater structural strength, and in general thinner wires
are found in mesh or Litz type wire, so a fast cooling rate may possibly be obtained
without sacrificing structural integrity.

3.5.2

Foil

A flat wire is an easy to fabricate structure with a large surface area and smallmass. Define the aspect ratio as α = h/w where h is the height of the wire and w is thewidth of the wire. Now with A = 2w(1 + α) l and m = ρ α w2 l Equation 3.3 becomes2(1 + α) σ 4dT
4=T − TRT
.
dt
ραwc

(3.7)

The wire width as a function of cooling time assuming T  TRT isw≈6

3 3 σ(1 + α) t Tf T0.
ρ α w c T03 − Tf3(3.8)

See Table 3.3 for what wire width is required for a 1 s cooling time assuming α = 0.1
and Tungsten wire. There is an approximate factor of 10 improvement for a given wire
thickness. This improvement factor can be used to decrease the cooling time, expand
the cooling rate, or use a more robust wire.

57
Tp (C)
730
580
496

w(µm)
123
76
56

Table 3.3: Calculated wire width to achieve cooling in 1 s from 10% above to 20% below
the L+α phase assuming Tungsten filament for the three gold/rubidium alloys. Assume
a 10:1 height to width aspect ratio.

3.6

Readsorption
Another factor to consider is the readsorption of the radiated energy. Any emitted

radiation should have a direct line of sight to the walls of the chamber. The effective
area will be reduced by the any parts that face each other. The magnitude of the effectmay be determined by multiplying the interfacing areas by the emissivity. If the surface
has a low emissivity it also means the same surface has a high reflectivity, so there is
still a chance for the radiation to be adsorbed by the walls even if it hits part of thewire or foil first.

3.7

Bulk versus surface effects
The long term operation of a gold/rubidium dispenser depended if the interactionof gold and rubidium was a surface or bulk effect over the time scale of a few hours.
Based on the behavior of the bulk Sarnoff dispenser the diffusion rate would need tobe high to explain the dispenser behavior, so several experiments were performed todetermine the diffusion rate of rubidium into a gold surface on the time scale of a few
hours. The basic experimental setup is a thin piece of gold coated Nichrome in a glass
vacuum chamber with a rubidium dispenser. The process of activating the dispenser(4-5 A for 2-3 mins 3-5x) deposits an unknown amount of rubidium on the gold. Afterthe dispenser was activated the gold is heated for a few hours to remove any rubidium
deposited on the gold. The process of heating the gold does not completely remove the

58
rubidium but sufficiently cleans the gold surface to mimic typical operating conditions.
The time evolution of the gold/rubidium dispenser is determined by measuring
the absorption of a resonant laser though the glass cell. The laser beam is swept over
Doppler broadened 85 Rb peaks and the signal measured by a photodiode is sent into
a lock-in amplifier. The absorption was not calibrated to an absolute pressure for thisexperiment because the qualitative effects of surface or bulk effects could be determinedby the shape of the rubidium dispensing curve. The scanning range of the laser (New
Focus Vortex 780.24 nm) drifted slightly over the course of an hour and the drift was
manually corrected every ∼ 10 minutes.
The experiment starts by coating the surface of the gold with a layer of rubidiumby heating the SAES rubidium dispenser. The amount of rubidium deposited on the goldis qualitatively monitored via the absorption signal from the laser passing through thecell. The rubidium is allowed to diffuse into the gold by waiting for some amount of time
typically less than two hours because the dispenser reload time should be significantly
less than two hours for reasonable operating conditions. In some experiments the gold
and rubidium are heated to increase the diffusion rate of rubidium into the gold. Heating
will also increase the vapor pressure of the rubidium on the surface of the gold but
assuming the diffusion rate is higher than the increased loss then more rubidium will
be diffused into the gold.
The amount of rubidium diffused into the gold can be qualitatively assessed assuming there are two regimes of rubidium dispensing when the gold and rubidium areheated by running current though the nichrome foil. First assume a thin layer of rubidium will be sorbed onto the surface of the gold. This surface layer will produce an initial
spike of rubidium that is rapidly evaporated off the surface with an exponential decay.
After this rapid spike and decay of surface rubidium the diffused rubidium would thendominate the dispensing. This diffusion based dispenser should have a characteristic√
t dependence of a typical diffusion process. The transition between the surface and59
diffused rubidium would have a change in slope and in extreme cases a knee when allof the surface rubidium was expelled [73, 74]. Varying the diffusion time and/or the
diffusion temperature would lead to an effective diffusion rate for the given gold area
and thickness.
All of the result from the diffusion experiments appeared to have a single curveand did not display the two regimes of operation for diffusion times of several hours.
Experiments when the gold is slightly heated reduced the total amount of rubidium
dispensed from the gold including the brief burst of rubidium from the surface layer.
No rubidium is detectable if the gold was heated too much (∼50% dispensing current)
during the gold coating phase.
The qualitative room temperature results indicate significant diffusion of rubidium
into gold takes places on a time scale longer than several hours. The heated results
indicate the effective time rubidium spends on the surfaces of heated gold is significantly
less than the time required to diffuse into the bulk of the gold. While this method for
determing diffusion rates is in principle reliable the diffusion rate of rubidium in gold
is short compared to the desired experimental time scale. For short term diffusion rateother methods (i.e. sputter Auger spectrocopy [75]) could be used). After these results
it was assumed diffusion did not play a significant role in reloadable rubidium dispensers
and the surface effect would be the dominant dispensing mechanism.

3.7.1

Gold pulsing

Based on the results of Reference [71] it appeared that a rubidium dispenser
could be made with a dispensing time constant of ∼ 1 s.

The pulsed operationof a gold/rubidium dispenser is tested in using the same experimental setup as thegold/rubidium diffusion experiment. In a typical pulsed experiment the gold is coatedwith rubidium using the same process as the diffusion experiment. The gold coatedtungsten foil is then heated by a pulse of current. The rubidium pressure rise and decay

60
are monitored by the absorption of a laser passed though the cell. Immediately after
the gold is coated with rubidium a low current pulse increases the rubidium partialpressure as high as ∼ 3 × 10−8 torr (see Figure 3.2. As the dispenser is pulsed therubidium is depleted such that longer pulses or higher currents are needed to obtainthe same pressure modulation. The gold is quickly reloaded with rubidium by heatinga SAES rubidium dispenser to 4.5 A for 5 minutes and then letting the pressure recoverfor 15-20 minutes.

Figure 3.2: This shows pulsed rubidium dispensing from a gold coated Nichrome foil.
This is data of a typical data run with gold evaporated on a Nichrome foil (50 − 100 µm
thick and an area of 5 × 15 mm). The foil is heated with a square pulse of current(labeled on each pulse) and as the rubidium is depleted from the gold higher currents
longer pulses are needed to achieve the same pressure modulation.

As seen in Figure 3.2 typical pulses are a few amps at 1-5 s. The pulses decay time
constant is a few seconds and the rubidium pressure significantly dropped on the orderof 10 s. From this experiment modulating the rubidium pressure by rapidly heating agold coated foil appears to have all of the right characteristics for a BEC compatible
rubidium dispenser: a time constant of a few seconds, the pressure dropped significantlyin ∼10 s, 10-20 pulses without reloading the gold, and a fast method for reloading thedispenser with rubidium.

61

3.8

MOTs and Au/Rb dispensers
While the gold and rubidium experiment using laser absorption to measure thetransient response of the gold/rubidium dispensers the true test was putting a gold/
rubidium dispenser in a cold atom experiment. The first test used a gold coated (evaporated on both sides) stainless steel mesh about 1 × 5 cm long in a square glass cellwith a plain piece of silicon anodically bonded to the top of the cell. The rubidium
dispenser was activated before the vacuum cell was pinched off from the vacuum station so the gold was already exposed to rubidium. The MOT beams were aligned withthe rubidium dispenser at a few amps and was then turned off to allow the vacuum
to recover. The gold mesh was then pulsed for ∼ 1 s at an experimentally determinedcurrent and the MOT number immediately increased. A few seconds after pulsing themesh the MOT was emptied and the MOT loading rate was measured repeatedly over
several minutes. In UV light induced desorption experiments a few seconds was enoughtime for the rubidium pressure to drop by an order of magnitude. However, with a gold
coated mesh it took on the order of a half hour for the rubidium pressure to drop by
an order of magnitude. There were multiple hypothesis as to exactly why the recovery
time was so long but they all pointed to the large gold surface area of the wire mesh.
Several MOT based test were performed with gold coated 20 − 40 µm Nichrome
foils that were 3-5 mm in width and 10-15 mm in length. The surface area and thermal
mass of the Nichrome foils were significantly less than the gold coated mesh. The
rubidium pressure modulation using the foils was faster than the gold mesh but therubidium pressure recovery still took 10s of seconds to minutes. Additionally becauseof the a smaller surface area and faster pulsing the maximum number loaded into the
MOT significantly dropped.
In another experiment an atom chip with electroplated gold wires was used ina mirror MOT configuration. When the external mirror MOT was transfered to the62chip mirror MOT a puff of rubidium was visible around the chip because rubidium wasdispensed from the trapping wires on the chip. The possibility of using a gold coatedatom chip as a rubidium dispenser was explored but the chip must also be used tomagnetically trap the atoms and the resulting lifetime was too short ( 1 s) to produce
BEC.

3.8.1

MOT N versus τAs explained in Section 2.7 an important metric in evaluating potential feasibilityof a compact BEC system is sufficient atoms to maintain collision rate and a long
lifetime to avoid density loss. Over a reasonable range low atom number can be directlycompensated by a longer lifetime and vice-versa. This balance of number and vacuum
lifetime is called Nτ . Instead of using a mirror MOT and a SAES dispenser with alow Nτ a six-beam MOT and a gold/rubidium dispenser were used to increase both N
and τ . While the six-beam MOT loaded more atoms the long vacuum lifetime using
the gold/rubidium dispenser was critical because the magnetic trap is more sensitive to
vacuum pressure by a factor of 4-6 [14].
As seen in Figure 3.3 the Nτ product could be increased by a factor of nearly
two by using a heated gold strip to increase the rubidium pressure. The SAES rubidium dispensers were characterized using a SRS residual gas analyser. When the SAES
dispensers are heated a significant amount of hydrogen is released along with rubidium.
Using a gold/rubidium dispenser will reduce the background hydrogen by first coating
the gold foil with rubidium and then allowing the pressure to recover. When the goldis heated only rubidium is released because hydrogen will not readily sorb to the gold
surface. The improved vacuum performance seen in Figure 3.3 is attributed to reducedhydrogen background.

63
16
14

6

Atom Number (x10 )

12
10
8
6
4

Dispenser
Gold Foil

2
0
2

1

3

4 5 6 7

2

3

10
MOT Loading Time (s)4 5 6 7100

Figure 3.3: The squares show the rubidium dispensing with the gold foil and the diamonds show the rubidium dispensing with a SAES dispenser. The MOT number ismeasured using a mirror MOT for both the SAES and gold dispensing data. The improved N τ product of the gold/rubidium alloy indicates the improved vacuum quality
using the alloy dispenser relative to the SAES dispenser.

3.8.2

Gold as a rubidium pump

While I did not perform the final experiments on the gold rubidium dispensers
the results will briefly describe it here for completeness. There was evidence from
Sarnoff that a fresh layer of gold would act like a rubidium pump. In an effort tomake a gold/rubidium dispenser without bulk processing gold was sputtering onto a
Nichrome surface in the presence of rubidium vapor. The rubidium pressure was roughly
monitored by the absorption of a laser beam passed though the cell. As soon as thegold was sputtered the rubidium signal immediately dropped. The gold deposition
was stopped but the rubidium dispenser was left on. After approximately one hour therubidium pressure recovered to the original pressure. The assumed pumping mechanisms
was rubidium immediately forming an alloy with the freshly depositied gold, effectively
creating a rubidium pump.
At CU the experiment was replicated by using small gold evaporators made bymelting gold onto twisted tungsten filaments. The gold evaporator plus a Nichrome

64
strip, and a rubidium dispenser were put inside glass cell. In all of the experiments therubidium dispenser was activated before the gold was evaporated on the cell so only thesurface of the gold was exposed to rubidium. As with previous experiments the rubidium
vapor pressure was monitored by passing a laser though the cell while being swept overa rubidium transition. The rubidium dispenser was already activated so running a few
amps though the dispenser would immediately increase the rubidium pressure in the cell.
The SAES rubidium dispenser was run until the rubidium pressure in the cell reachedsteady state. With the dispenser held at the same current the gold was evaporated onto
the walls of the vacuum chamber. The experiment was built so the evaporated gold
would not coat the region where the laser passed though the cell. Immediately after thegold evaporation the rubidium signal dramatically dropped to background levels (seeFigure 3.4). After the gold was evaporated the SAES rubidium dispenser current wasincreased and gradually the rubidium pressure rose as the rubidium coverage on thegold surface reached equilibrium. This data was fit to a model that assumed there werea finite number of occupation site on the gold and in the rest of the vacuum cell, andthat the rubidium had different affinities/binding energies for each surface. Specificallyit was assumed that rubidium had a much greater affinity for gold than for the othermaterials in the system. The results of the fit are shown in Figure 3.4 [76].
From the results of the gold/rubidium diffusion experiments and the gold pumping
experiments it is reasonable to conclude that on the time scale of hours the dominantmechanism in the gold rubidium interaction is a surface effect, otherwise the gold would
act as a weak pump before the evaporation. On a final note about diffusion, the colorof the gold coated Nichrome foils has been observed to change from bright yellow toa silver color over several weeks to months. While there is no data other than visualobservation is reasonable to assume that rubidium and gold would interdiffuse on alonger timescale and that a visual change in the color due to alloy formation would beseen by eye.

65

-8

9x10

Disp 3.2 A

Rubidium Parital Pressure (Torr)

87
6
5Gold 22 A

4
3

Disp off
Disp 3.2 A

2Gold off

Disp 3.5 A

Data

Best Fit (κ=10, C=30)

1
0
00:00

00:30

01:00

01:30
02:00
Time Elapsed (hrs:min)

02:30

03:00

03:30

Figure 3.4: Best fit of gold pumping data showing rubidium gold interaction is well fit
by a surface interaction model [76].

3.9Gold versus other metals
What about the behavior of other materials for dispensing rubidium? In one

experiment two identical pieces of Nichrome foil were placed in a glass cell. One of thepieces was coated with evaporated gold and the other piece was left uncoated. The
rubidium dispenser was positioned so the dispensed rubidium hit each foil equally. The
two foils were exposed to the rubidium at a pressure of ∼ 10−7 torr for approximately
5 minutes. After the rubidium dispensing one foil is heated with a constant current
while the rubidium pressure is continuously monitored until the rubidium is emptied
from the first foil. The process is then repeated for the second foil. This procedure isrepeated several times, each time switching the order that the foils were activated toremove systematic effects related to one foil dispensing onto the other foil. On average
the Nichrome foil dispensed about 1/5th the rubidium as the gold coated foils. Thisshows rubidium will sorb onto any cold surface which can be later dispensed by heating
the substrate, but the increased rubidium dispensing from the gold shows the increased
sticking coefficient due to the gold/rubidium alloy even at short time scales.

Chapter 4

Experimental setup and operation

4.1

Vacuum cell design and construction
The design of this compact vacuum system is similar in many ways to the con-struction of chip based and larger BEC systems; however there are a few key featuresthat are noteworthy. First, the cell is assembled without the vacuum being exposed
to epoxies that were used in the original chip based BEC systems [1, 14]. The anodic
bonding process [77] makes this possible and allows the vacuum cells to be processedat much higher temperatures than was possible with earlier cell fabrication techniques[18]. Anodic bonding has been used in previous atomic physics experiments [78] but
we are not aware of anodic bonding being used in BEC systems especially to construct
a UHV system. The majority of the differential pumping between the two chambers
is accomplished by a single 0.75 mm aperture and the higher pressure region utilizes
non-evaporable getters to maintain the vacuum pressure. Also, the system features anatom chip with electrical feedthroughs or vias though the atom chip which simplifies
electrical connections to the vacuum regions of the system. The vacuum chamber (seeFigure 4.1) has four general parts: the atom chip, the BEC cell, the center six way
flange, and the 2D MOT cell. Each of these parts, their construction, and assembly will
be described. The basic process of anodic bonding will also be discussed in Section 4.2.

Figure 4.1: Views of a two chamber cell. Figure a.) shows an isometric view of the cell construction including the placement of the ionpump and pinch off tube. Figure b.) shows a cross section of the cell including the placement of the apertures, the NEG for the BEC
region, and the configuration of the rubidium dispenser and the 2D NEG.

67

68
4.1.1

2D MOT cell

The 2D MOT cell is made from a 1 cm Pyrex flourimeter cell (Starna Cells 3-Px10). The cell is lapped and polished on the open end, and a rubidium source tube is glass
blown to one of the side walls of the cell. The source tube is a 12 mm Pyrex tube thatis closed by a Pyrex disk with four pin press electrical feedthroughs [6]. Attached to thepin presses are a non-evaporable getter (SAES getters HI/7-6) and a SAES rubidium
dispenser. The getter is used to pump the hydrogen released by the rubidium dispenser
and to help maintain the vacuum in the 2D MOT cell. The 2D MOT cell is isolated fromthe ultra high vacuum regions of the system by a 1 mm thick silicon disk with a 0.75
mm diameter aperture in the center of the disk. This aperture reduces the conductance
between the 2D MOT region and the BEC chamber to < 0.1 `/s, allowing us to maintaina pressure difference of two orders of magnitude between the two chambers of the vacuum
system. One face of the silicon disk is anodically bonded to the polished end of the 2D
MOT cell and the other side of the silicon disk is bonded to a lapped and polished glass
to metal transition on a mini-conflat (1.33”) flange. The second bond is done in sucha way that it does not apply a significant electric field across the first bond to avoid
de-bonding. The details of the anodic bonding process are discussed in Section 4.2.

4.1.2

Spherical cube / pinch off

The 2D MOT assembly is bolted to the center vacuum flange with a solid copper
conflat gasket with a 3 mm diameter hole in the center of the gasket. In principle theflourimeter cell could be directly attached to the glass to metal with the copper gasket
providing all of the differential pumping. However, the silicon disk also acts as a mirror
surface so the cell may be used in a 2D+ MOT configuration [79].
The 2D MOT cell is joined to the BEC cell and vacuum pumps by the center vacuum flange. The flange itself is a six-way mini-conflat cube (Kimball Physics

69
MCF133-SC6). A 2 `/s ion pump mounted perpendicular to the MOT cells maintains
the pressure in the BEC cell. Additionally a second non-evaporable getter (SAES getters HI 7/6) is connected to a conflat mounted power feedthrough and is used in lieu
of a titanium sublimation pump. A copper pinch off tube is brazed to a welding lip
on a mini conflat flange. The brazing material is a UHV compatible silver solder andall brazing flux is removed to ensure low operating pressures. The conflat copper tube
assembly is bolted to one of the remaining ports on the spherical cube. The location of
the 2 `/s ion pump and copper pinch off are determined by the position of pre-existing
mirrors and beam paths.

4.1.3

Pumps

Because the vacuum system is pinched off from the pumping station any pumpused in the vacuum system must permanently capture the gas that it pumps. There arethree basic pumps in the compact vacuum system. Ions pumps for all species of gas,
getter pumps for chemically active gases like oxygen, and the walls of the system forrubidium.
Ion pumps work by ionizing gas inside the pump and then accelerating the ion
into the walls of the pump by a voltage of a few kV. In a diode pump once theions strike titanium plates in the walls they are pumped by two processes: chemical
adsorption and burial. Atoms are chemically pumped when a chemically active gas (i.e.
nitrogen, oxygen, hydrogen) is exposed to fresh titanium which is extremely reactive.
Once these nitride, carbides, etc. are formed they are assumed to be stable and will
be permanently removed from the system [80]. The surfaces of the titanium plates in
the ion pump constantly refreshed and scrubbed ion bombardment but over time thesputtered and reacted metal will cover all pump surfaces reducing the pumping speed.
This process will continue to pump until all the titanium has been chemically reactedwith active gases. Normal pump lifetime is on the order of 40,000 hours at 10−6 torr.

70
Fortunately ion pump lifetime is inversely proportional to pressure so at 10−9 torr ion
pump lifetimes are approximately 1,000 times longer (thousands of years). If there isa catastrophic failure and the ion pump is exposed to air the ion bombardment will be
stopped by the reduced mean free path at a pressure of less than 1 torr. The shutdown
will happen very quickly so a rapid venting should not significantly shorten the lifetimeof an ion pump. That being said a new ion pump will have the best performance because
all of the metal surfaces are clean and available to sorb gases.
Chemically inert or noble gases (He, Ne, Ar, etc.) are pumped by being buried in
the titanium plates. As more metal is sputtered (ejected by ion bombardment) it will
increasingly bury the gas below metal. The burial process only happens in new pumps,
if the system is exposed to a high noble gas load eventually the noble atoms in theplates will come into equilibrium with the gas pressure. For every noble gas atom thatis accelerated into the plates it will eject a previously buried atom. This equilibrium
is called saturation and to a lesser extent occur for active gases. For active gases thetitanium will continue to chemically sorb active gases, however when the plates fill witha noble gas the pumping speed is significantly reduced. This noble saturation effectcan be mitigated by using two different metals (titanium and tantalum) so there is adifferential sputtering rate that will preferentially bury the noble atoms on one sideof the pump. The other method commonly used for pumping noble gases is a triode
pump. In a triode pump the sputtered material does not fall back onto the plates but isdeposited behind the titanium plates on the walls of the pump. Because this material
is outside the sputtering zone the buried gases cannot be reemitted.
For portable vacuum systems made from glass helium diffusion is a problem worth
considering, because helium is present in the atmosphere in concentrations of ∼ 4×10−3
torr. Helium is small and non-reactive so it can diffuse into a glass envelope to asteady state pressure of a few militorr if the helium is not pumped away [57]. An
earlier experiment with increased local helium pressure around a quartz cell showed the71pressure in the cell rapidly increased and when the helium was taken away the pressurein the cell dropped [14]. The cell was made from quartz which has a higher diffusion rate
than Pyrex. It is possible that after several years the ion pump will be saturated withhelium but that can be improved by using a noble diode or triode pump. Completely
removing the ion pump is not a good option because uncompensated helium diffusion
will degrade vacuum quality.
In the current configuration a 2 `/s ion pump is attached to the side of thespherical cube and significantly increases the overall size of the vacuum system. The
position of the ion pump relative to the chip is important because the vacuum system
was retrofitted into the pre-existing opto-mechanics for a single cell vacuum system.
The total size of the vacuum cell can be reduced farther if the spherical cube and theion pump are integrated. This could be accomplished assuming a SHV connector is usedinstead of the new Varian style ion pump connector and the ion pump was integrated
into the spherical cube with a hole though the center of the ion pump interior to allow
atoms from the 2D MOT to propagate to the upper chamber. This ignores the effectof the electrical field on the atoms [81]. Another consideration in compacting the ion
pumps is shielding the ion pump’s magnetic field from the BEC experiment. Excellent
magnetic field shielding has been accomplished at CU by using a clamshell type µ-metalshield around the ion pump.
The other pump that is used in the BEC system is a non-evaporable getter (NEG).
Getter pumps work by chemically absorbing active gases into a metal with a high surface
area. The method of capturing and removing atoms from the vacuum is the samechemical pumping mechanism in ion pumps. The difference between an ion pump and a
NEG is the NEG has a high surface area achieved by sintering a finely powdered metalalloy around a heating element. In air a thin oxide layer is formed over the entire surfaceof the NEG. Once the NEG is installed in the vacuum it is activated by heating it to600 − 800◦ C. This diffuses the surface oxide into the bulk of the material and exposes

72
fresh metal to the vacuum. Any gases that are adsorbed on the fresh NEG surfaces arepermanently removed from the vacuum.
The other gas that is pumped by a NEG (and ion pumps) is hydrogen, however
its sorption is reversible because it is favorable for the hydrogen to diffuse into theNEG, but not permanent. Before, during, and after activation the NEG is heatedto a few hundred degrees celsius to drive off hydrogen. Hydrogen pumping is a key
requirement for vacuum system with rubidium dispensers because tests of dispensers
show rubidium dispensers release a significant amount of hydrogen when heated. The
rubidium dispensers release hydrogen because they are also made with a getter likematerial that traps hydrogen. Non-evaporable getters have been used in every compact
vacuum cell built at CU, so it is difficult to tell if NEGs play a significant role in thevacuum quality but in principle removing them from the vacuum system should only
degrade the vacuum quality.
The final pumping mechanism in a compact BEC system is rubidium adsorbing
onto the walls of the vacuum chamber. Rubidium has a bulk vapor pressure of ∼ 10−7
torr at room temperature, but when rubidium interacts with a clean vacuum wall therubidium is effectively pumped away. It is plausible that the rubidium interacts withthe oxides in the wall and forms RbOx . It is generally accepted that no rubidium will
be seen the first time a new vacuum chamber is exposed to rubidium because a thinlayer (presumably a few monolayers) of rubidium must coat the chamber to suppress
the pumping action of the walls. In a two chamber system the walls of the upper
chamber will act as a rubidium pump for a finite time but will eventually saturate.
This saturation effect was not seen in two different cells with both cells being operated
for several months. The presence of free rubidium in the upper cell is determined byturning the 3D MOT on with the 2D MOT off and loading for several minutes; At theend of several minutes of loading no is MOT seen. An alternative rubidium pump withincreased capacity is graphite. Atomic fountain clocks use a graphite tube to pump

73
residual cesium atoms with a large pumping capacity [82].

4.1.4

3D MOT cell

The BEC cell is made by glass blowing a custom made (New Era Enterprises)square 2 cm ID Pyrex cell to a Pyrex glass to metal mini-conflat adapter and thenanodically bonding an atom chip to the top of the glass cell. However the cells are notimmediately ready to be used when they arrive and require several processing steps.
The original joining method leaves residual stress in the edges of the cells that caused
cracking after anodic bonding. Sarnoff found the stress could be removed by fusing theedges of the cell with a torch. The entire cell is annealed after the edge fusing to remove
the stress of the glass blowing [83]. After the cells are lapped they are glass blown toa Pyrex glass-to-metal transition. The cell should be consistently aligned with the bolt
hole pattern to avoid changes later during assembly. The cell and glass to metal areannealed again to remove residual stress that would affect the lapping, polishing, or
anodic bonding.
The lapping and polishing was done by hand at Sarnoff and made a flat andsmooth surface for anodic bonding but the polished surface was not always at rightangles relative to the cell. Recently CU built an apparatus for lapping and polishingcell in a pre-aligned jig to lap cells at right angles to the conflat flange. After lapping
and polishing the cell is throughly cleaned for UHV and anodic bonding. Then a silicon
atom chip [1] with UHV electrical vias is anodically bonded to the lapped and polishedsurface.
A few notes that may be useful for future cell design. The walls of the cell are
not anti-reflection (AR) coated because of the large number of processing steps that
includes glass blowing. The few percent reflection from the walls has an impact on theimaging quality and reproducibly splitting a BEC [84]. In principle the exterior of thecell can be AR coated after the cell is anodically bonded. Coating the interior walls of

74
the cell is more difficult but could be accomplished by one of the many chemical vapor
deposition processes.
The length of the cell is chosen to be able fit a MOT with right angle beams
directly below the chip. Because of the success in a MOT with angled beams (seeSection 2.4.4) the length of the cell could be shortened. Current cells have not beenshortened because of the opto-mechanical setup based on the original size of the vacuum
cell.

4.1.5

Atom chip

The atom chip is a 400 µm thick silicon wafer with a silicon oxide layer forelectrical isolation. The wires are formed by electroplating copper into a photoresist
mold the shape of the final wire pattern. The resulting wires are typically 10 µm talland 100 µm wide. The chip is patterned with a 15 mm wide Z-wire with crossing wires
every 1 mm (see Figure 4.2). The areas between the copper wires are also electroplatedwith copper to form an “optical metal” that could be used as a mirror surface for a
mirror MOT or reflection imaging [18].
The largest chip Z-wire is connected to two vias and can support up to 5 A. The
remaining wires are each connected to a single via and can support 2.5 A. A variety
of Ioffe-Prichard (IP), quadrupole, dimple, and waveguide geometries can be configured
depending on where the currents are sourced and sunk. As mentioned previously, theelectrical connections to the vacuum side chip wires are made though the silicon chipby UHV compatible vias [85]. Each via can support 2.5 A, and the vias can be joined
together to support higher currents. The vias are made to withstand the high temperatures (> 400◦ C) required for anodic bonding and vacuum processing. The connecting
pads on the ambient side of the chip are coated in gold to prevent oxidization duringhigh temperature processing. To protect the copper on the vacuum side of the chipduring anodic bonding the bond is performed in a reducing gas atmosphere (4-6% H2 ,

75

y

x
1 mm

Figure 4.2: Schematic wire pattern of the chip used to produce BEC. Depending on
where the currents are sourced the chip can be configured as a large Z-wire trap withmany crossing wires, a variety of H-traps, a long waveguide, etc. The wires with thefilled pattern show wires used in experiment. The experiment was also performed using
wires other than the center vertical wire and minor adjustment of the other experimental
parameters with the same results. The leads that form the outermost legs of the Z-wireare connected to two vias so the maximum possible current in the Z-wire is 5 A. Allother leads are connected to a single via that can support 2.5 A. On the actual chip theregions between the wires are filled with ”optical metal” that can be used as a mirror
surface if necessary. The optical metal is not shown in this figure for clarity.

94-96% N2 ). To prevent explosions the hydrogen should be kept below 10%.
The thickness of the silicon (400-500 µm) will easily hold against ambient airpressure over 2 cm2 because chips as thin as 100 µm will hold atmospheric pressure
over a 2 cm2 area. Despite the physical strength of the silicon the atom chips flex
when heated due to current in the chip wires. This mechanical flexing interferes withimaging and the general stability of the atom trapping. The stability of the chip isenhanced by attaching a 1 mm thick disk of silicon to the back of the atom chip with ahigh temperature compatible epoxy (EpoTek 383ND). The extra support increases thethermal mass of the silicon chip and reduces the chip deflection (see Figure 4.4). The
ambient side of the chip is patterned with pads positioned directly above the cell wall
so electrical contact may be made to the pads without risk of fracturing the chip (see

76

a.)

b.)

Figure 4.3: The vacuum (a.) and air (b.) sides of the atom chip. The grey area is thesize and shape of the atom chip. The light blue area just inside the rectangles is the
position where the Pyrex cell is anodically bonded to the chip. The red boxes aroundthe inside perimeter of the chip are the locations of the UHV compatible vias. In Figurea.) the all of the shaded areas are electroplated copper. This image shows the locationof the optical metal between the traces. In Figure b.) the shaded traces show gold pads
that connect the vias to pads positioned over the cell location for additional strength
when connecting to the cell.

Figure 4.3). The pins (MILL-MAX M09922) provide good electrical contact without
applying excessive force to the chip.

4.1.6

Bakeout and pinch off

After the vacuum system has been assembled it is attached to a pumping station
via a copper pinch-off for vacuum bakeout. Bakeout is a critical part of making it
possible for a vacuum system to achieve UHV (< 10−9 torr). The main goal of abakeout is to heat the vacuum system above 100◦ C so water adsorbed on and into
the interior surfaces of the vacuum chamber will be “boiled” off and pumped out of
the system. Not only water will be boiled off but any contaminant with a high vapor
pressure. Higher temperatures are better for preparing a vacuum system for UHV
because the contaminants in the system are not only adsorbed onto the surfaces in thevacuum chamber but can also be deeply diffused into the bulk of the materials in the

77

Figure 4.4: Deflection of a 2 cm square silicon chip with and without a 1 mm thick, φ = 1
cm silicon disk that is attached to the back of the chip. The disk improves mechanicalstability and reduced the deflection of the silicon chip. Calculations performed with the
COSMOSXpress tool in SolidWorks.

chamber. Glass is known for holding a significant amount of diffused water. During
baking the cell is maintained at the highest possible temperature because the diffusion
rate increases with temperature. In the early cells the highest bakeout temperature waslimited to less than 200◦ C by the epoxy (353ND) that was used to attach the atom chip
to a quartz cell [14]. In current cells the atom chip is anodically bonded to a Pyrex cellat > 400◦ C and the cells are baked out at a lower temperature of ∼ 300◦ C to avoid
heating the cells above the bonding temperature and to reduce thermal mismatch stress.
Qualitatively the vacuum pressure is significantly improved in the vacuum cells
that are baked at higher temperatures. The rough metric for vacuum quality is the MOT
loading time and lifetime. In a single chamber configuration epoxied cells typically
had MOT loading times of 30 s and 1 − 2 × 106 atoms loaded from the backgroundrubidium after the SAES dispenser was turned off [14]. These numbers were typicalfor immediately after bakeout however the MOT loading time typically degraded as
the cells aged. Anodically bonded cells have MOT loading times 2-3 times longer thanepoxied cells and 2-4 times the atom number. The longer lifetimes and increased MOT
number are both a result of the higher bakeout temperatures made possible by anodic
bonding.
The baking process for the vacuum cells assures thermal gradients over the vac-

78
uum cell are minimized. The copper pinch-off tube on the cell is attached to a compression port (Varian FCP0075UHV) on the bakeout system. The cell is then encased in analuminum sheet metal box with an interior support structure to hold the cell within theenclosure. All electrical connections to dispenser, getter, thermocouples, etc. are madeby barrel connectors, stainless steel crimps, or wire wraps. Wire wrapping the connections is the least invasive and most reliable of all the connection methods. Solder canbe used at low temperatures but at < 300◦ C the solder melts and cannot be used inside
the bakeout box. After the electrical connections are strain relieved the aluminum box
is wrapped in heater tape and insulation. The heat from the heater tape is spread by
the aluminum and uniformly heats the whole cell. The only significant thermal gradientis the area of the copper pinch-off, but there are few problem with leaks in this area.
The temperature of the cell is gradually increased, typically at 1−2◦ C/min, up to
the bakeout temperature. Cells have been baked for as short as two days with excellent
results. For example the high duty cycle, number optimized, and speed optimized resultsof Chapter 5 were produced with a cell that was baked two to three days. Cells have also
been baked for longer periods typically two weeks with the lifetime improving by about
50%. The cell that was used for the atom splitting experiments in Section 6.1 was baked
for several weeks. During the bakeout the various dispensers, getters, gold coated pieces,
etc. are degassed and activated according to the manufactures specifications. The cellis then cooled at the same 1 − 2◦ C/min to room temperature. After the cell is cooled
the ion gauge on the bakeout station will typically have a pressure < 4 × 10−11 torr
(actually is goes off scale low at this point).
If the system does not go off scale there is typically a small leak. If the leak isin the vias it can be sealed by VacSeal (distributed by SPI) diluted with acetone andthen the system should be baked again for a few days. After the vacuum pressure is
acceptable the rubidium dispenser is turned on for 10 mins until rubidium florescence isvisible with a sweeping laser. This ensures the cell walls are coated with rubidium while

79
the cell is still attached to the pumping system. The system is then pinched off (CHA
Industries pinch off tool) leaving approximately 0.75–1 inches of clearance between thebraze joint and the pinch off to avoid cracking the braze.

4.2

Anodic bonding for UHV construction
Anodic bonding in used in the BEC cell construction because of its excellent

vacuum properties and high temperature limits. Anodic bonding has been used before
in atomic physics experiments [78] but we are not aware of anodic bonding being usedin BEC systems.
Anodic bonding is a low temperature process for bonding two dissimilar materials
using an electric field to assist in the bonding. It is considered a low temperature process
because the bond is performed below the annealing temperature of the glass. Anodic
bonding is typically performed at a temperature close to 400◦ C and requires materials
with matched coefficients of thermal expansion. Silicon and Pyrex are used in the BEC
apparatus because both materials are readily attained and UHV compatible. Ultra-highvacuum compatible Pyrex to metal transitions are available from a variety of vacuum
component distributors. Anodically bonded parts will not leak (< 10−10 torr `/s) after
repeated baking and cooling cycles. While some extra care is required to prepare thesurfaces for anodic bonding, once the parts are joined the bond is stronger than theconstituent materials. Anodic bonding can be used to build custom glass UHV vacuum
chambers without significant labor and the parts can be easily manufactured by standard
glass and silicon processing. A wide variety of UHV chambers and structures have been
constructed by anodically bonding silicon wafers to Pyrex wafers, tubes, and cells.
The process of anodic bonding starts with clean, flat, and polished surfaces. The
components are cleaned using standard optical cleaning techniques and are placed together with a slight amount of pressure (typically a mass of a few kg is used to applypressure). The parts should be flat enough to see a few interference fringes across the80
sample. Flatter is better but the electric field will pull the parts together during bonding.
The parts are then heated to 400◦ C. Heating the Pyrex is critical because it increases
the mobility of the sodium ions in the Pyrex. After heating, a voltage of approximately
1 kV (it depends on the thickness of the materials) is applied across the silicon and glass(see Figure 4.5). There are two signs of a successful bond. Because bonding involves
a migration of ions in the glass [86] there is a small current that is generated during
bonding. The current will start around a few hundred µA at the beginning of the bondprocess and will fall to the background level as the bond is completed. The other sign ofsuccessful anodic bonding can be seen by looking at the silicon/Pyrex interface. Areas
that have bonded will appear dark as the bonding draws the two parts together without
an air gap. It is possible to visually monitor the progress of the anodic bonding by
watching the progression of the dark areas. Once all of the areas to be bonded aredark and the current is steady then the parts can be cooled. The heaters can simply beturned off and the parts allowed to cool at 10 − 20◦ C/s without the bond failing or theparts cracking.
A successful anodic bond relies on a large electric field at the bond interface topromote ion transport. If the electrode is placed too far from the glass the electric field
will be weak, and the bond will be weak or fail. If the electrode is too close to the silicon
it will arc, which can be dangerous (especially in a hydrogen atmosphere) and damage
the part. When bonding cells a wire or BeCu foil is wrapped around the glass cell
2-3 mm from the bond interface. In general, anodic bonding is an irreversible process.
However, for double sided bonds it is important not to apply the field the wrong way
across a bond that has already been formed, as this can cause de-bonding.

4.3

Control and imaging systems
The control and imaging system are important from the practical point of view asthey are the interface to the operation of the experiment. A new linked control program81

Figure 4.5: Schematic diagram of anodic bonding especially the polarity of the bondrelative to the silicon and Pyrex.

and imaging software were developed for operating the BEC experiment because rapid
MOT loading and BEC production lends itself to rapid and semi-autonomous optimization [87]. The semi-autonomous operation includes a “good” shot algorithm to trigger
the next data run if the experiment was operating properly and to beep if the last data
run was poor (see Figure 4.6).

4.3.1

Control system

The production of BEC has many steps that involve controlling the multiple
aspects of the laser, magnetic, and imaging systems at various times. Along with thecomplexity there are often various modes of operation including testing and diagnostics
that require various aspects of the experiment to be turned on or off while maintaining
the behavior of the rest of the experiment.
The original control software was channel based and sometimes required multiplechanges across various channels depending on task [14]. The new control software istask based and once a task is configured in the software it is possible enable or disable
an entire task with a single click. This is achieved by using the tree structure (newin LabVIEW 7) with the main nodes used to define the main tasks (e.g. CMOT or
imaging) and the sub-nodes to define the individual parts of the main tasks (e.g. laser
detuning, bias field, etc.). As seen in Figure 4.7 the program is written so that all nodes

82

LabVIEW
Data Socket

Control Computer

Imaging Computer

LabVIEWLabVIEW

Run Next

GPIB

Producer
Loop

LabVIEW
Data Socket

Q
u
e
u
e

Arm camera

Consumer
Loop
OK

FireWire

MXI

CMOS
CameraMicro −
Controller
Imaging
shutters, etc.
PXI Chassis
trigger
PDXI−AO−32/16

Power Supplies,
Shutters,
Etc.

Agilent
33250A

Figure 4.6: Diagram of control and imaging systems with connections to external hardware. When the control computer starts processing a data file it sends a commandover the LabVIEW Data Socket to the imaging computer to wait for a sequence of 3-4
images. After the images are taken the control computer calculates the OD and determines if the shot was well behaved (as determined by the atom number). If the shot
was well behaved the imaging computer triggers the control computer to run the next
experiment.

83
can be activated or disabled so a main node like optical molasses can be disabled so
the CMOT performance can be evaluated. Disabling the optical molasses node (PGC)
is accomplished by disabling the main optical molasses node and every sub-node is
automatically disabled (see Figure 4.7). Additionally each sub-node can be individuallyenabled or disabled. For example, the dimple channel can be turned off to effectively
create a hold phase to evaluate the chip Z-wire compression.

Figure 4.7: Example of a LabVIEW tree used to control experimental BEC production.
This example shows control over major section (e.g. the entire RFsweep1 is disabled),
the use of variables (e.g. Tcollect is used several times but can be changed in the global
variable section), and sub-elements (e.g. Xbias branch) that can be disabled without
affecting other sub-elements.

As seen in Figure 4.7 variables can be used in the program to link different stagesof the experiment. Along with the tree structure there are global and local variablesections in the control program. A typical local variable would be the MOT field gradient(motMOT) because it is used only in the MOT loading section and is not needed for the84rest of the program. The global variables are typically time variables because subsequent
sections need the timing information of previous sections. In principle the timing canbe configured so when one section ends the next section begins and there is no need tocommunicate timing information between the different sections. It is practically easier
to pass timing information with global variables. The global variables are also usefulfor linking variables in optimization (e.g. fix the ratio of the chip wire current andtransverse bias field to maintain the trap height but vary the trap gradient).
The control program is able to semi-autonomously vary variables over a user
defined range. The user defines the variable and the range of values for the experiment.
The order of the values is randomized so systematic effects like heating and rubidium
pressure will be suppressed in the data. After one value is run the Imaging computer will
determine if the number of atoms falls within an acceptable range. If the experimentis well behaved then the Imaging computer will set a network variable high (via the
DataSocket connection) that triggers the next value to be run on the Control computer.
If there was a problem with the data the user has the option of repeating the experimentwith the original value or ignoring the error and continuing with the next value. Thissemi-autonomous system is very productive with a rapid BEC production system. Thissemi-autonomous system could be expanded if two technological issues were solved:
the stability of the laser locking and safety systems for the chip and external Z-wires.
Improvement in these areas could lead to a multidimensional optimization routine that
could be left to take data or find an optimal experimental condition.
The individual experiments are saved in XML like files that have an organizational
structure similar to the tree structure. The files save all of the variables and timings
exactly as they were represented in the tree. These timing files can be edited by hand
if needed and in some cases (i.e. duplicating whole stages) it is easier to edit the timing
files by hand than to use the LabVIEW tree interface.
Where it is possible a calibration is used to go between the physical units (Gauss,

85
MHz, etc.) and the control voltages. This allows the equipment in the experiment tobe switched without changing the control values. For example a new coil power supply
only needs a new calibration to create the same magnetic field as the previous power
supply. The calibration is saved in calibration file that is based on the zero and five
points of the signal and assumes the relationship between the control voltage and theoutput is linear. The basic calibration involves adjusting the control voltage until thesignal is zero (e.g. laser frequency is on resonance with a peak). The control voltage isthen adjusted until the signal is at 5 (e.g. 5 MHz detuned from the resonance peak) thenthese values are put into the zero and five columns respectively. The calibration file also
includes limits for each channel as a layer of safety to prevent accidentally overdriving
the system.

4.3.2

Imaging system

The imaging system consists of a microcontroller shutter/trigger sequencer, a
CMOS camera, and a dedicated imaging computer and software. The microcontroller
shutter/pulse controller is used to produce the same pulse train for the imaging shutter,
AOM driver, and camera trigger. A dedicated microcontroller is used because the DAC
card (UEI PDXI-AO-32/16) changes the timing resolution depending on how manychannels are being used. A PIC microcontroller was used because it easily has thetiming resolution (1 µs) and was easily programmed by the JILA electronics shop. The
imaging pulse train is triggered 10 ms before the atoms image to allow enough time forthe shutters to open before the image is taken. The shutter/pulse controller can also
be run without opening the imaging shutter allowing cleaning and dark images to betaken with the same timing.
A separate computer from the Control computer is used for taking pictures of
the BEC and processing the images. A second computer is traditionally used in BEC
experiments for imaging because the Control computer uses a large amount of computer

86
resources to run the experiment and when the camera is triggered it is important thatthe data can be downloaded as fast as possible and without glitches. The fitting routines
are typically computationally intensive so a second Imaging computer is a good practice.
The Imaging computer runs a LabVIEW program that controls the camera data
capture, the data processing, and communication with the Control computer. The imaging software is based on a Producer/Consumer model where two loops run in parallel.
The producer loop controls the camera, downloads the data, and then calculates andsaves the OD to disk. The consumer loop waits until a new OD file is available, fits the
OD to a 2D Gaussian fit, and then from the fit determines the atom number, cloud size,
peak OD, etc. The two loops run independently so in principle the producer loop canrecord images as fast as possible while the consumer/fitting loop processes the data at
a slower rate.
While it is possible for the producer loop to process data at a rate greater than the
consumer loop the two loops nominally work sequentially because part of the consumer
loop includes a rudimentary error detection calculation. After the cloud parameters are
fit the atom number is used to make sure the system is working correctly. First the atom
number must be above a minimum threshold value and once that condition is satisfied
the current atom number is compared to the distribution of previous shots. If the atom
number is within reasonable bounds then a network variable is set high and the Control
computer starts the next experimental run. If the current atom number lies more than
1.5σ outside of the distribution then a beep is sounded and the data taking stops until
there is human intervention.
The OD calculation follows most standard OD calculation methods [88, 6] butwith an extra data processing step for getting clean images close to the chip. Because
OD is a relative estimate there is a very high probability of shot noise introducing large
errors in the dark regions (i.e. chip shadows) of the images. Rogue pixels will interferewith the integrated number and if a rogue pixel has a high enough value it will interfere

87with the fitting results because each pixel is weighted equally. This spurious noise isremoved by thresholding any pixel with a count less than 10% of the maximum pixel
count (determined by the Light image) to one. Any position where the pixels are equal
will result in an OD of zero so regions of low intensity will not contribute to the fittingor the integrated number. The OD fitting will be more reliable if a reasonable starting
condition is given to the fitting algorithm. The JILA method of calculating the centerof mass of the cloud is used to get a reasonable center for the cloud and then calculating
the first moment of the cloud to get a reasonable estimate of the size of the cloud. Thismethod provides very good starting values for the fitting routine and requires no user
input.

4.4

Apparatus setup and operation

4.4.1

2D MOTThe 2D MOT beams consists of two pairs of counter propagating beams with an
aspect ratio of 5:1. The laser is split into two beams that are retroreflected to effectively
double the available cooling power. Retroreflecting the beams in a 2D MOT effectively
doubles the cooling power and works as well or better than a non-retroreflecting configuration because the 2D MOT is optically thin. A small fraction of the 2D MOT laser
power (∼3 mW) is split off and is propagated along the axis of the 2D MOT. This extra
push beam plus the mirror created by the silicon disk used at the top of the 2D MOT
chamber creates the 2D+ MOT configuration and provides molasses cooling along theaxis of the trap. This reduces the velocity of the atoms out of the jet and increases
the capturable flux out of the 2D MOT [79]. The optimal laser intensity of a 2D MOT
tends to be higher than the optimal intensity for a 3D MOT [89]. A commercial (> 300mW) and home built lasers have been used with equal success. Currently a tapered
amplifier [90] is used to deliver 60 mW to the 2D MOT.

88

a.)

b.)

c.)
Figure 4.8: Pictures of the experimental layout: a.) shows the relative position of theupper 3D MOT chamber and the the lower 2D MOT chamber, b.) is a closer pictureof the 3D MOT and imaging opto-mechanics, c.) shows the optomechanics and heaters
for the 2D MOT.

89
The first step of aligning a 2D MOT is to center two pairs of retroreflecting beams
on the 2D MOT cell. The beams are retroreflected by placing an iris in the center of themain beam before it is split. The retroreflecting mirror is adjusted to send the reflectedbeam back though the iris. The iris reduces the size of the beam so it is easier to see
small misalignments. With an iris in the beam the power balancing of the two split
beams is balanced by adjusting the half waveplate before the splitting polarizing beam
splitter (PBS). The beams should have the same height as they pass though the cellso the force on the atoms is uniform as the atoms propagate along the axis of the 2D
MOT. The beams should not clip any part of the cell where the atoms exit the 2D
MOT chamber because the scattered light can cause the atom beam to deflect just as
it exits the 2D MOT cell.
If the 2D MOT cell is not AR coated there will be a slight beam imbalance because
the retroreflected beam has passed though four glass interfaces. Assuming there is a 4%
loss at each interface the returning power will be about 85% of the original beam. The
positive side of retroreflecting the beams is the total cooling power is nearly doubled.
Early 2D MOT experiments found the retroreflected 2D MOT configuration yieldedhigher fluxes compared to a four beam power balanced configuration using the same
initial power [91]. This indicates 2D MOTs are optimized when the total cooling poweris maximized (the flux saturates at a total 2D MOT cooling power of about 100 mW
in a 1 × 4 cm cell). The slight power inbalance due to the uncoated cell walls can becanceled by the position of the magnetic field coils or magnets.
The magnetic field of a 2D MOT is symmetric along the axis of the 2D MOT.
This allows the atoms to freely propagate along the axis of the 2D MOT. The field
can be created by long wires configured as shown in Figure 4.9a. In this experiment
the magnetic field for the 2D MOT is produced by permanent magnets configured asshown in Figure 4.9b. Sub-Doppler cooling and BEC are sensitive to stray magnetic
fields but 2D MOT permanent magnets do not affect these experimental stages because

90
the quadrupole MOT field falls off faster than (w/r)4 where w is the distance between
magnets. In the current experiment w ≈ 2 cm and r ≈ 10 cm so the effect of thecontinuous magnetic field gradient is negligible. Permanent magnets are used because
the 2D MOT field is constant during the MOT loading stage and it reduces the numberof power supplies needed to operate the experiment. From the standpoint of a practicaldevice reducing the number of power supplies in the system reduces the net weight andpower consumption. From an experimental perspective it is worthwhile to simplify thesystem but it comes at the cost of not being able to optimize the magnetic field gradientfor the greatest flux.

Figure 4.9: Comparison of 2D MOT magnetic field produced with current carrying wires
and permanent magnets. Figure a.) shows the field from four parallel wires and Figureb.) shows a similar magnetic field toward the center of the field diagram.

While the 2D MOT magnets do not significantly affect the molasses cooling or
BEC production other magnetic fields can affect the pointing of the 2D MOT flux. In
particular stray fields from the ion pump magnets have significantly altered the path of
the 2D MOT. The interference from the 2D MOT magnets is reduced by improving themagnetic shielding around ion pump magnets. There are a few key elements for shielding
the ion pump magnets. First the shielding material should have a high µ to provide alow energy path for the magnetic fields to return to the opposite magnetic pole and a

91
low residual field after being exposed to an external magnetic field. The low residual
field is important because magnetic fields will be switched many times during the courseof a BEC experiment and a varying residual field could cause fluctuations from shot toshot. The magnetic shielding should, as much as possible, completely encase the areaaround the magnets. Because the shielding material has a high µ any magnetic field
line that intersects the shielding will have a low energy path to the opposite magnet
pole. This can be called a magnetic short circuit. The field lines can escape at any
point where there is a direct line of sight.
Finally there should be space between the ion pump magnets and the magnetic
shielding. High µ-metals can provide a low energy return path for magnetic fields but
are subject to saturation. Saturation happens when all of the magnetic domains in thematerial align with the external field and cannot increase alignment when the magnetic
field is increased. Saturated magnetic shielding effectively loses its high µ properties
and will not short circuit magnetic field lines. Saturation can easily be avoided byintroducing a gap between the shielding and the ion pump magnets. The air gap allows
the magnetic field to spread out so the magnetic-field density is below saturation.
Once the stray magnetic fields have been eliminated the magnetic coils or permanent magnets should be centered horizontally and vertically around the 2D MOT cell.
A 2D MOT should be visible when the 2D MOT optics and magnetic field are both
centered on the 2D MOT cell. The 2D MOT is most easily seen by looking along thelong axis of the 2D MOT with the push beam off so there is maximum integration of
the fluoresced signal. It also help to see the 2D MOT when it’s moving so the 2D MOT
can be moved by translating the 2D MOT coils or carefully moving a magnet aroundthe area of the 2D MOT. The 2D MOT magnet assembly is aligned parallel to the axisof the 2D MOT chamber by centering the 2D MOT cell in the camera’s field of viewand then focusing the camera on one end of the long 2D MOT and then focusing on
the other end of the 2D MOT. If the MOT is parallel to the cell the two ends of the

92
MOT will be in the same place. If the MOT is angled it can be aligned by adjusting theangle of the 2D MOT magnets. The 2D MOT magnetic structure is typically mounted
on a 5-axis stage (XYZ translation + 2 rotational) that allows the the magnets to beprecisely positioned. Finally the 2D MOT should be centered relative to the pinhole inthe silicon disk.
After the 2D MOT optics and magnetic fields are aligned the push beam is turned
on. The push beam overlaps the same axis as the axial imaging camera so it is good
to finish the 2D MOT alignment before aligning the push beam. The push beam isaligned by centering the push beam on the bottom of the 2D MOT cell. An iris is theput in the push beam before the final two steering mirrors and is used to retroreflect thepush beam off of the silicon disk. The iris should be off-axis so the hole in the centerof the silicon does not affect the beam. These two mirrors are iteratively adjusted until
the push beam is both centered on the 2D MOT cell and parallel to the silicon disk.
The final 2D+ MOT alignment is best aligned with a working 3D MOT to monitor thetransfer of atoms to the upper chamber.

4.4.2

3D MOTThe 3D MOT is created with two pairs of beams that are retroreflected (angled
beams), plus two counter propagating beams (horizontal beams). While retroreflecting
in a 2D MOT improves the MOT performance the same is not true in a 3D MOT because
the optical density of a 3D MOT is much higher. A true six beam MOT number is on theorder of 10% higher than the retroreflected configuration [91], however in the 3D MOT
setup as many beams as possible are retroreflected because the performance differenceis small and optical alignment of retroreflected beams is easier. The exception is thehorizontal beams because the optical pumping beam (see Section 4.4.5) only propogatesalong one of the horizontal beam paths. The retroreflected(counter propagating) beams
are self(co)-aligned by using an iris to narrow the beam for ease of alignment similar

93
to the alignment procedure of the 2D MOT. The power in the horizontal beams isbalanced with an iris in the beam because the size of the MOT beams is ∼ 15 mm
and overfills physical size of the power meters. The center of angled beams are easily
overlapped with the center of the horizontal beams when the vacuum cell is removed.
The cross section and overlap of the beams can be seen on a card placed where thecenter of the cell would have been. The repump and optical pumping beams are double
coupled into the same polarization maintaing optical fiber and are overlapped with thehorizontal cooling beams. As mentioned the pumping beam co-propogates with oneof the horizontal cooling beams and the repump laser is overlapped with the otherhorizontal cooling beam. The total MOT cooling power is 28-32 mW, the repump is 6-8
mW, and the optical pumping beam is ≤1 mW. More cooling power is available but it
degrades the performance of PGC for the available laser detuning [92].
One difference in this MOT configuration compared to other 3D MOT arrangements is the angle of the retroreflected beams. As seen in Figure 2.11 the angled beams
are not orthogonal and in the experiment θ ∼ 20◦ . This allows the MOT to be positioned closer to the chip by a factor of 1.6-2 compared to θ = 45◦ . This reduction intransport distance is significant because the atoms are transported and transfered tothe chip with a large external Z-shaped wire. The current required for the transport depends on the transport distance as 1/r2 . The 25◦ reduction in angle reduces the current
required to magnetically trap the atoms with a large Z-wire by a factor of 1.62 −22 . Thisin turn reduces the dissipated power by 1.64 − 24 . A Monte-Carlo simulation indicatedthe beam angle could be reduced (see Figure 2.12) without a reduction in MOT number
and in practice the angled MOT has similar performance to a 3D MOT with orthogonal
beams. However there are no systematic data taken to compare the MOT performance
as a function of MOT angle.
The extent of the testing involved optimizing a six-beam MOT and noting theloading time and number in the MOT. The optical alignment was then modified so

94θ = 20◦ this included changing several mounts and realigning the retroreflecting optics.
Setting the height and angle of the retroreflecting optics is an important detail. The
height was determined by simple trigonometry and careful measurement of the initial
beam height. The angle of the retroreflecting optics were initially set with a 20◦ angle
block from the machine shop.
After the MOT was configured with angled beams it was optimized for the largest
possible MOT number by slightly adjusting the beam alignments and power balancing.
The number of atoms in the 20◦ MOT was essentially the same as a MOT with θ = 45◦ .
There is a caveat to the MOT numbers because it was measured with a CCD camera
with an auto gain function. At the time is was assumed that any reasonable MOT would
force the camera into a low gain mode making the camera be reasonably accurate for
integrated number measurements. A later comparison of MOT numbers using the CCD
camera versus a photodiode based measurement showed the CCD numbers were subject
to significant error. Given that caveat the apparent size and density of the 20◦ MOT
was the same as the 45◦ MOT. There was no effort to experimentally map out the MOT
number versus θ to compare to Figure 2.12 because of the time it took to realign the
MOT at each angle. The lowest possible temperature in sub-Doppler cooling was theother key element besides number that was in question when using angled MOT beams.
With the 20◦ beams the optical molasses stage still cools atoms to as low at 20 µK.

4.4.3

MOT debugging

If the 2D or 3D MOT is not visible there are a few common issues when searchingfor a MOT in a new setup. First verify there is rubidium in the MOT chamber by increasing the rubidium vapor pressure while illuminating the cell with cooling light. The
cooling laser should be scanned over a few hyperfine absorption peaks at approximately
10 Hz. When the rubidium pressure is high enough there should be a noticeable flashingin the MOT cell. Sometime sweeping the laser frequency will also change the intensity

95
of the laser so it is worthwhile to sweep the laser significantly off resonance to get a feelfor the intensity fluctuations of the laser. For a 3D MOT the rubidium pressure should
be decreased just below the point of seeing fluorescence. For a 2D MOT the fluorescenceshould be visible but not too dense because the glow of the background rubidium will
make it hard to see the 2D MOT. Verify the lasers are locked on the proper transitions.
The cooling laser is particularly sensitive to the locking point. The repump laser locking
point is not as sensitive and if there is at least 5-10 mW of repump power the laser canbe slowly swept over the |F = 1i → |F 0 = 2i to see a flashing MOT.
The magnetic coils or permanent magnets should create a single zero between thecoils for a 3D MOT or a line of zeros for a 2D MOT (see Figure 4.9). The zeros and themagnetic field gradient can be verified with a magnetic field probe. The 3D MOT field
gradient is optimized in the 10-14 G/cm range and the 2D MOT is optimized at slightly
higher gradients but less than 20 G/cm. The position of the magnetic field zero shouldbe aligned with the desired position of the MOT. This position of the magnetic fieldzero is most easily determined by the geometry of the coils. For a pair of 3D MOT coils
it is the center of the coil pair and for 2D MOT coils it is the center of the four wires
or permanent magnets. The width of the coil winding should be taken into accountand fixed reference point will help position the coils relative to the cell. For a mirror
MOT [26] the magnetic field zero needs to be accurate to less than 1 mm and accurate
placement is essential. For a six beam MOT or a 2D MOT in a 1 cm or larger vacuum
cell the coils can be centered by eye with reasonable results.
The laser polarization is checked with a simple circular polarization tester madewith a quarter waveplate (λ/4) and linear polarizer rotated 45◦ relative to the axis of the
waveplate. When a right-hand circular (RHC) polarization is incident on the λ/4 sideof the tester the waveplate will convert the polarization from RHC to vertically aligned
linear polarized light which will then pass though the linear polarizer. Left hand circularwill be rotated to horizontally aligned and will be blocked by the polarization tester.

96
As a double check the intensity of the beam should be constant as the tester is rotated.
If the intensity changes more than a few percent the laser beam is not pure circularpolarized but has a significant fraction that is linearly polarized. The polarizations of
the various MOT beams are determined by the direction of the MOT magnetic field.
Typically a laser polarization is picked for a direction of the magnetic field (e.g. RHC
for the magnetic field exiting the coil). Then the polarization of all of the beams thatmatch that same condition are adjusted to be the same. The opposite condition (field
entering the coil) is set to the opposite polarization (LHC). It is easy to calculate thepolarization needed for a particular magnetic field orientation, but in practice it is very
easy to make a minus sign mistake because of the various conventions. If everythingelse is aligned and there is no MOT it is easy to flip the magnetic field direction byswitching the direction of the current in the magnetic field coils. Once a MOT is visible
and optimized the magnetic field direction and strength can be measured and dependingon the configuration replaced with a set of permanent magnets.

4.4.4

2D → 3D MOT loading

When first looking for a 3D MOT being loaded from a beam out of the 2D MOT
it is important to not increase the rubidium pressure in the 2D MOT cell above thetypical operating pressure of ∼ 10−7 torr. Running the 2D MOT cell above this pressureand generally heating the whole vacuum system will force rubidium into the upper
chamber. The 2D MOT already runs at high pressure and it would defeat the effect ofisolating the two chambers if the 3D MOT chamber was overloaded with rubidium in
the initial alignment. If finding the 3D MOT loaded from the 2D MOT is difficult thenthe 3D MOT should be found and optimized using a single chamber cell at a higher
rubidium pressure. Once the optics, polarizations, frequencies, etc. are optimized thesingle chamber vacuum cell can be traded for the two chamber cell.
A poorly aligned 2D MOT should transfer some atoms into the 3D MOT. Even a

97
2D MOT with one of the beam pairs blocked has weakly loaded a 3D MOT. Once there
is some transfer between the MOTs turn on the push beam. It is not worth optimizing
the 2D MOT without the push beam because the optimization of the 2D MOT withand without the push beam is different. The 2D+ MOT is aligned by periodically
emptying the 3D MOT and then observing the 3D MOT loading (see Figure 4.10).
The MOT number is measured by focusing light from the MOT onto a photodiodeand displaying the signal on an oscilloscope for immediate feedback of the 2D+ MOTalignment. Initially the MOT was pulsed by blocking the repump laser with a card.
Quickly this transitioned into the Control computer switching the repump shutter but
a small amount of light from the repump is reflected from the cell walls which causesa large jump in the photodiode signal and MOT number. Switching the MOT current
on and off also empties the MOT and results in a clean photodiode signal. A MOT
loading time of 400 ms is good for optimizing the 2D+ MOT. It is long enough to loada noticeable MOT for good signal to noise and is fast enough for optimization. This
metric for optimizing the MOT does not optimize the saturated MOT number rather
the initial loading rate of atoms into the 3D MOT. However the saturated 3D MOT
number strongly depends on the ratio of the MOT loading rate versus the loss rate, so
any increase in the MOT loading rate should result in a larger saturated MOT number.
The ultimate MOT number not only depends on the ratio of loading and loss rates, but
photon reradiation and the maximum MOT density [32, 93].
The 2D→3D MOT optimization is an iterative process and typically starts byadjusting the position and angle of the 2D MOT magnetic field with the push beam.
The lateral position of the magnets is scanned until the maximum loading rate is found.
The angle of the magnets corresponding to the translation axis is then slightly rotated.
The lateral position is then scanned again to find the maximum. If the rotation reduced
the atom number then the magnets are rotated the opposite direction. This process isiterated until the any rotation or translation results in a reduction of the loading rate.

98
362

MOT Photo Diode Signal (mV)

360

358

356

354

352

350

348
-15

-10

-5

0
seconds

5

10

15

Figure 4.10: Example oscilloscope trace of optimizing the push beam power. The MOT
coils are switched on for 400 ms and then switched off for 300 ms. This leads to thepulsed trace that approximates the loading rate of the 2D MOT into the 3D MOT. At
t = -15 s the push beam power is nearly optimized, as a check the power is changed byrotating a half waveplate in front a polarizing beam splitter cube immediately before the2D MOT cell. As the power drops the 3D MOT loading rate and number significantly
drop around -5 s. As the waveplate is rotated back the MOT loading rate and number
are optimized by adjusting the waveplate to have the largest possible swing.

This same process is repeated for the other lateral axis of the 2D MOT magnets. It is
possible that this alignment procedure will significantly change the magnet alignment,
but this is typically not the case. If for some reason several adjustments significantly
decrease the loading rate then it is best to recenter and align the angle of the magnets
as best as possible by eye and then optimize the MOT.
Next adjust the position and angle of 2D MOT cooling beams, power balancing, and quarter waveplates. These adjustments should be incremental because of thealignment performed in Section 4.4.1, but small tweaks will typically improve the MOT
number. Any adjustment to the vertical position of the 2D MOT beams should bechecked to make sure all of the beams are at the same height and are not scattering
off of the silicon disk at the top of the 2D MOT cell. The 2D MOT push beam is
optimized by adjusting the angle and position of the push beam in small increments.

99
The optimized 2D→3D MOT loading is most sensitive to the push beam power. The
optimal push beam power is typically a few mW for a 1 cm2 beam.
After the 2D MOT beams are aligned the 3D MOT can be adjusted. The initialplacement of the MOT coils is done with the bias coil power supplies turned off. The
MOT coils should be centered and parallel to the MOT cell but can be translated to
optimize the number and shape of the MOT. After the MOT and bias coils are fixed
to the table the MOT can be moved over a wide range by applying a bias field in theappropriate direction. Before optimizing the bias field values the magnetic field should
be calibrated. This basic process is explained in Reference [14]. Once the zero point
and B vs. I slope is determined the Zero and Five points are saved in a calibration filein the Control program. See Section 4.3.1 for details of how to determine and set the
Zero and Five points. It is important to perform this calibration before optimizing theposition of the MOT because changing the bias field calibration will shift the position of
the MOT. The MOT position is adjusted for maximum number and shape by adjusting
the bias fields.
After the optimal MOT position is set by the bias fields the cooling beams, powers,
and polarizations can be slightly adjusted for MOT number and shape. Assuming the
3D MOT cooling beams have been power balanced, overlapped, retroreflected then only
small adjustments to the MOT beams should be required. The MOT number and shape
can be improved from the basic alignment but large alignment changes will affect later
laser cooling stages. In particular the power balancing of the horizontal beams should
not be adjusted because the optical molasses is sensitive to power balancing. The relative
power balancing between the horizontal and angled beams will change the aspect ratioof the MOT. The MOT should be close to spherical but can be slightly elliptical when
optimized for maximum number. All of the 3D MOT alignment is performed with the
3D MOT being loaded from the 2D MOT flux or in a single chamber cell where a highrubidium vapor pressure will not degrade the overall vacuum pressure.

100
Optimizing the overall MOT loading rate is an iterative process and all of theabove steps should be repeated a few times with finer adjustments being made after
each iteration. When the 3D MOT is loaded to maximum number sometimes the 2D
MOT beam can be seen distorting the 3D MOT. This signature is not a specific sign ofa well aligned 2D MOT flux and at the same time it is not a bad sign. The real metrics
for optimization are the MOT loading rate, the maximum MOT number, and the visualappearance of a smooth well connected MOT.
The 2D+ → 3D MOT transfer also depends on the rubidium pressure in the 2D
MOT chamber. The pressure in the 2D MOT cell should be maintain at the highest
possible rubidium pressure without significantly increasing the pressure of other gasesin the BEC cell. There is an optimal rubidium dispenser current where the ratio of
rubidium to all other gases is maximized. This operating current Iop is best found experimentally because it varies dispenser to dispenser and with thermal load. The 2D
MOT cell is also heated to increase the rubidium pressure without increasing the dispenser current above Iop . This does not significantly increase the total pressure because
the vapor pressure of rubidium is a strong function of temperature while the other gases(hydrogen, helium, etc.) effectively have negligible temperature dependence at laboratory temperatures. Thus the MOT loading rate is optimized without compromising the
MOT lifetime. This shows that the differential pumping allows the pressure in the 2D
MOT chamber to be raised to a certain point without significantly affecting the pressurein the 3D MOT cell. This depends on the pressure in the 2D MOT cell, the size of theaperature, and the pumping speed in the 3D MOT cell. Above some pressure in the
2D MOT cell the pressure differential can not be maintained by the pumps in the 3D
MOT cell and the MOT lifetime drops (see Section 2.5). In principle a longer differential pumping tube could be used to increase the pressure differential between the cells
but the current design errors on the cautious side so the atom flux is not significantly
limited by the tube or aperture.

101
4.4.5

CMOT, PGC, OP

After the 3D MOT is loaded, the atoms are compressed and cooled by a compressed MOT (CMOT) [37]. The CMOT compresses the atoms by increasing the field
gradient from 10-14 G/cm to 20-30 G/cm. At the same time the atoms are further
cooled by detuning the cooling laser frequency from 2Γ to 4 − 5Γ and reducing the repump power. During the CMOT the atoms are moved toward the chip by a Bz field to
reduce the distance the atoms must be transported to the chip. The maximum distancethat the atoms can be transported is initially determined in steady state by increasing
the magnetic-field gradient but leaving the cooling frequency at 2Γ. The maximum
displacement of the CMOT is be determined by the CMOT shape and relative number. As the CMOT is translated the X,Y, and Z bias fields are adjusted to center the
CMOT in the MOT cell. This procedure maps out the general magnetic field behavior
and the alignment of the MOT beams. The cooling and repump lasers may need to beslightly adjusted to optimize both the MOT and CMOT. It is important to optimize
the shape of the CMOT (i.e. a smooth well connected MOT) because the optical molasses stage will be optimized at this position. The atoms are displaced (5-10 mm) to
the final CMOT position in 20–40 ms. If the atoms were transfered this quickly in amagnetic trap it would introduce a significant slosh mode in the atom cloud. This does
not happen when moving the atoms in the CMOT because the CMOT is a dissipative
trap.
The bias fields needed to transport and center the CMOT can be initially determined with a steady state CMOT, but the final behavior of the CMOT is transitory andoptimization requires imaging the atoms via adsorption or fluorescence imaging. Adsorption imaging is used exclusively to evaluate the CMOT optimization. Typically theatoms are allowed to expand for a few ms to qualitatively determine the atom temperature because hotter atoms in the cloud expand more when the trapping fields are turned

102

Figure 4.11: Images of maximally loaded MOT (a) and translated CMOT (b) in a 2
cm cell. The translation and differences in size and shape of the two MOTs can be
seen. Additionally the atom chip can be seen sealing the top of the cell and the springloaded pins for making connections to the external pads can be seen above the atomchip. Three loops of the compression wires can be seen outside of the spring loaded
pins.

off. This ballistic expansion is typically known as time of flight imaging (TOF). Aftera long TOF (the linear expansion regime) the size of the cloud is directly proportional
to the temperature of the cloud. The temperature of the CMOT and optical molasses
are optimized by imaging the atoms after a long TOF and then optimizing the various
cooling parameters for the smallest cloud size while maintaining atom number. While
compressing the atoms increases the transfer into the magnetic trap it is important to
reduce the temperature of the atoms after the CMOT to ∼125 µK in preparation for
optical molasses. The temperature of the CMOT depends on the detuning of the coolinglasers, the repump power, and the total CMOT time. When the CMOT is positioned at
the center of the lasers it is relatively easy to achieve temperatures below 125 µK. However, when the CMOT is positioned in the outer extents of the lasers achieving lower

103
temperatures is not as simple. In the displaced condition the final CMOT temperature
can be reduced by increasing the cooling laser detuning to 10-11 Γ in the last 1–2 ms,with the magnetic field and repump power being held at the same values.
After the CMOT the atoms are then further cooled by optical molasses also known
as sub-Doppler cooling or polarization gradient cooling. In a typical MOT the overalllowest attainable temperature is 140 µK and is typically called the Doppler limit [94].
There is some sub-Doppler cooling near the magnetic field zero but it is only in a small
region of the MOT. The Doppler temperature is determined by the balance between
cooling via the Doppler shift and heating due to reradiated photons. In a CMOT
the temperature can be lower than the Doppler limit because the photon heating is
suppressed by detuning the laser [37]. The final temperature after optical molasses
in the current experimental setup is 40 µK. This is far above the theoretical limitof 4 µK but more than adequate for magnetic trapping on a chip. There are BEC
experiments that do not use optical molasses [6, 95]. These experiments are typicallyuse large coils to create deep magnetic traps, however in the chip and external-Z wirebased magnetic traps the trap depth is approximately an order of magnitude less thanthe macroscopic traps. Reducing the temperature of the atom cloud below 80-100 µk
significantly improves the initial loading of the magnetic trap.
Optical molasses is the most sensitive of all post-MOT loading cooling steps. It
is sensitive to zeroing the bias fields to the tenth of a Gauss level and depends on therepump power and the cooling laser power [92]. Similar to the CMOT optimization
the optical molasses is optimized by imaging the atoms after TOF (typically 5-15 ms).
The cooling laser is typically set to 11 Γ detuning, all of the bias fields are set tozero (assuming they are calibrated), the MOT magnetic field is set to zero, and theoptical molasses time is 3-5 ms. The optical molasses should be optimized with power
balanced cooling beams or the atoms will cooled in a moving reference frame. The
power balancing can be tested by increasing the optical molasses stage time to 10-15 ms

104
and looking for the atom cloud to be pushed off center. The power in the cooling beams
is adjusted until the atoms remain centered. The optical molasses is then optimized
by varying the magnetic field, laser power, laser detuning, etc. for the minimized cloudsize. The value where the size of the cloud is smallest is the new optimal value. Similar
to aligning the CMOT this is an iterative process of varying all of the parameters tofind a new set of optimal values. The same parameters are then varied again, preferably
in a different order, to further optimize the optical molasses. As stated above it should
be relatively easy to reduce the temperature to 40 µK or less.
The final step before the magnetic trap is optically pumping the atoms into thestrong field seeking state. Immediately after or a few 100 µs before the end of opticalmolasses the cooling light is turned off to avoid pumping atoms into a random mF
states. It is important to measure the time when the cooling laser switches off witha photodiode because there is an unknown delay between when the signal is sent tothe shutter to the point when the shutter occults the laser. The atoms are optically
pumped into the |2, 2i state with the pump beam slightly detuned from the F=2→F’=2transition. The laser is directed though the cloud in only one direction determined by
the Bx field. The Bx field shifts the atoms into resonance with the pump laser anddefines a quantization axis. The repump laser is left on to pump atoms out of the
F=1 state. Optical pumping is optimized after seeing at least a small magnetic trap.
Optical pumping is sensitive to Bx , the repump power, and the pumping power. WhileBx and the repump power are varied via the control software the pumping power is set
by hand in order to reduce the number of DAQ channels. Unfortunately the opticalpumping power drifts over time, especially because the optical pumping beam is double
fiber coupled with the repump laser. In the current experimental configuration there
is no passive or active monitoring of the pumping optical power. For improved BEC
production stability, on the time scale of days, the optical pumping power should bemonitored to maintain the same power.

105
4.4.6

External Z-wire magnetic trap

The initial magnetic trapping, transport, and coupling of atoms onto the chip areaccomplished with a Z-shaped wire (see Figure 4.15) positioned just above the atomchip. The main reason for using a Z-shaped wire is to improve mode matching to thechip Z-wire trap. The external Z-wire has essentially the same shape and magnetic
field as the chip Z-wire trap. In the adiabatic transfer limit the size and shape of theZ-wires would smoothly transfer from a slightly larger wire to a slightly smaller wire.
In this limit there would be no atom loss and the adiabatic compression requirements
would limit the transfer speed. In reality the change in size from the external Z-wiretrap (width∼32 mm) to the chip Z-wire trap (width∼17 mm) is not adiabatic and thetransfer efficiency is less than 100%, but has been as high as 70-80%. Other atom chip
experiments with a six-beam MOT have reported low transfer efficiency (≤ 50% often< 25%) from a quadrupole magnetic trap to the chip magnetic trap [13, 43, 42].
The external Z-wire is constructed by shaping a piece of enamel coated magnet
wire into a Z-shape with long tails and then flattening the center section with a hydraulicpress. The wires are flattened so the center of the current distribution is closer to theposition of the atoms inside the cell (see Section 2.2.1) and it is easier to stack thewires. Flattening the wires must be done with some caution, because deforming thewires beyond a certain point will crack the enamel coating on the wire. The maximum
deformation of the wires is determined empirically because it depends on the type of
coating on the wire. The total area of the wire that can be flattened depends of the
maximum pressure of the hydraulic press. In our experimental setup only the center
10 cm of the wire could be flattened. This makes a transition region where the wirechanges from rectangular to round. Despite the abrupt transition the Z-shaped wires
are robust, however any break in the wire is typically at the flat to round junction socare must be taken when bending the wires. The first wire is bent around a rectangular

106
form (see Figure 4.12a) and is held in place with tape. A second wire is then bent overthe first wire and is also taped in place. This is repeated for five wires to increase thetotal effective current for maximal magnetic field gradient and trap depth. For a given
power supply the limit to the number of wires is the resistance of the Z-wire coil and thevoltage of the power supply. After the wire are bent around the form each is checked
for shorting to the other wires and the form. The wire are epoxied (EpoTex 383ND) tothe form to maintain shape and for a mounting point. The leads of the wires are thensoldered in series with the total height of the external Z-wire and return coil is 18 cm(see Figure 4.12a). Also attached to the eternal Z-wire mounting block are the springloaded pins plus leads, the compression wires, and the RF coil (see Figure 4.12b).

4.4.6.1

Switching mode power supplies

The external Z-wire is driven by a switching mode power supply (Xantrex XHR
series 130A/7.5V). The current in the external Z-wire needs to be switched on theorder of 1 ms and to be continuously varied so the atoms can be transported from thefinal CMOT position to the chip and adiabatically transfered to the chip. The XHR
power supply has an internal current control mechanism that can be used to control thecurrent, however there is a delay of 25-75 ms between the time when the signal is sent
to the power supply to when the current output changes. In principle this delay can be
subtracted out in the control system timing file, unfortunately the power supply updates
the current every 25-75 ms leading to a chopped waveform that would be unacceptable
for transporting a cold cloud of atoms.
Instead of using the internal control circuitry the external Z-wire crrent is controlled by a current servo (similar to the circuit in Reference [96]) driving a fan cooled
MOSFET bank (IRFP3703). These MOSFETs can drive 130A for a few hundred ms but
care should be taken because MOSFETs short circuit when they fail and will continue
to conduct current until the MOSFET burns/melts. The MOSFET bank can support

107

a.)

b.)

Figure 4.12: Pictures of the external Z-wire assembly. Five flattened Z-shaped wires(originally 16 AWG) are bent around an aluminum form and then connected in series
to effectively increase the total trapping by a factor of five. Spring loaded pins are press
fit into a piece of Vespel, soldered to ribbon cable, and then epoxied to the side of thealuminum block. A single turn of 22-24 AWG wire forms the RF coil. The diameter ofthe coil is large enough that it does not contact a 1 cm silicon disk that is epoxied tothe back of the atom chip. The compression wires are two sets of three parallel wires
just outside of the RF coil and parallel to the legs of the external Z-wire. The top of
the aluminum block is tapped for an 8-32 screw and the assembly is mounted to a 0.5”
post that is then attached to a frame around the cell.

108
up to 50 A for >5 s repeatedly but have failed for times longer than 7 s. During typicaloperating conditions (< 1 s) no discernible heat is transferred from the external Z-wireto the cell structure.
Even though this high current driver can control a commercially available power
supply this setup effectively duplicates the high power control circuitry. Switch mode
power supplies (SMPS) are known for better conversion efficiency than linear power
supplies, but it comes a the cost of additional high frequency noise. Switch mode power
supplies are efficient because they chop the voltage to a transformer with a MOSFET athigh frequencies, typically 100’s of kHz, and then low pass filter the signal to remove thehigh frequency chopping. This is efficient because the switching MOSFET dissipates
no power when it is full off and dissipates very little power when the MOSFET is fullon. The output current/voltage is controlled by pulse width modulating the control
signal to the MOSFET. If the power supply is supposed to deliver full voltage then the
MOSFET will be continuously on. If the voltage is set at 50% of the max voltage thenthe MOSFET will be switched on for half of the time and off for the other half of thetime. When this chopped waveform is low pass filtered the resulting voltage is half ofthe max voltage with a small amount of ripple. From an eletrical design perspective a
SMPS is more complicated to build, but there are many companies that sell high quality
SMPS that are more cost effective than a linear power supply with the same voltage
and current.
The real issue with SMPS in BEC experiments is noise. The electrical signal is
chopped at 100’s of kHz and this noise could cause parametric heating. Fortunately
typical SMPS frequencies are in a ‘sweet’ spot for use in BEC experiments. There arebasically three regimes where noise would cause significant heating or loss. They are the
1-200 Hz range for weak magnetic traps, 1-5 kHz for tight magnetic traps, and 1-100
MHz for transitions between hyperfine levels in atoms. The SMPS frequency is at least
one order of magnitude away from all of the critical frequency bands. Because of this109
SMPS can be used for high current applications like magnetic trap transfer and may
even be suitable for BEC power supplies.
How could the current configuration be better? In the current setup the pulse
width modulation servo that drives the output of the SMPS is set to maximum voltage
and the external servo drives a MOSFET bank as a linear control so there are two servos
controlling the same power supply. The linear MOSFET operates at a low duty cycle
so there is little heating, but ideally the pulse width modulation servo would be directly
used to control the current. Actually a current to voltage converter (shunt resistor
or Hall probe) would be used to provide control feedback to the power supply. This
would most likely involve significant modification of the SMPS electronics and limits
the casual modification of a SMPS but these modification could be important for futureapplications. From the experimental point of view it is reasonable to treat the SMPS
like a high current battery and build a linear servo to control the current.

4.4.6.2

Initial external Z-wire trap

The initial external Z-wire magnetic trap condition is found by exploiting the
Zeeman shift of the atoms in the magnetic trapping field. The external Z-wire current
is turned to the maximum value and By to some value. The hold time is 1–2 ms, longenough that the fields have reached their final values but not too long such that theatoms are moving in the magnetic trap potential. An absorption image is taken withthe probe laser on resonance with the F = 2 → F 0 = 3 transition. Atoms that are in theweakest field will scatter the most light and show the greatest signal on the absorption
image. By varying By the vertical position of the magnetic trap is moved though theposition of the atoms. The By field with the largest apparent number is where theminimum of the magnetic field overlaps with the cloud center (see Figure 4.13). The
other bias fields can be optimized in the same way but they are not as important as By
in the initial alignment of the external Z-wire magnetic trap.

110
14

12

N MT

( 106 )

10

8

6

4

2

0
20

30

40

50

60

70

BY (G)

Figure 4.13: Example data showing parameter search for the initial external Z-wiremagnetic trap by imaging the atoms with a resonant probe laser immediately after
the external Z-wire magnetic trap is turned on. Atoms that are positioned away fromthe bottom of the magnetic trap will be Zeeman shifted out of resonance and will not
contribute to the signal. The By with the greatest signal is the starting parameter forthe next optimization. In this example the external Z-wire is turned to 120 A while By
is varied in each shot.

4.4.6.3

External Z-wire magnetic trap optimization

Once atoms are in a magnetic trap there are several optimization steps. The
initial magnetic trap curvature is empirically matched to the size and temperature of
the atoms. This can be accomplished two ways: the ratio of the By and the external
Z-wire can be varied together or Bx can be varied to change the trap bottom andloosen the trap frequency at the bottom of the trap. Both methods will reduce the trap
gradient and depth, but increasing Bx will relax the trap gradient only around the trap
bottom and will leave the gradient nominally the same at other locations. In the initialexternal Z-wire magnetic trap it is important to maintain a high trap gradient to holdatoms against gravity so the longitudinal bias is increased to reduce the trap frequency
because it leaves a higher gradient at regions away from the trap bottom.

111
The initial position of the atoms when they are captured in the magnetic trap isvaried by adjusting the final bias field of the CMOT. There are two critical axes: thefirst and most obvious is the Bz field that adjusts the vertical position of the CMOT.
Increasing Bz will move the atoms closer to the chip which should improve the initial external Z-wire magnetic trap. However, moving the atom up in the optical fields
changes the optical intensity experienced by the atoms and can affect atom number andtemperature after optical molasses. On the flip side if the atoms are too far away thenthe magnetic-field gradient of the external Z-wire will not be able to capture a significant fraction of the atoms. Additionally changing the position of the CMOT withoutchanging the position of the initial magnetic trap will reduce the transfer efficiency.
Given these constraints the CMOT Bz is easily optimized; A reasonable By is picked
for the initial external Z-wire magnetic trap and then the CMOT Bz is varied to map
the transfer efficiency. Change By and again sweep Bz until the parameter space has
been mapped out and the optimal final CMOT and initial magnetic trap parameters
are determined. The other parameter that affects the transfer to the chip magnetic trap
is the CMOT Bx field that determines the position of the atoms along the weak axisof the trap. Atoms that are displaced along the weak axis of the trap will be excited
into a low frequency slosh mode. While this may seem trivial because it is not along
the tight axis of the trap at low frequencies it takes several periods at the longitudinaloscillation frequency to damp out. The oscillation is most detrimental when the atoms
are transfered from the external Z-wire magnetic trap to the chip Z-wire magnetic trap
because of the width mismatch between the Z-wires. If the atoms are longitudinallysloshing they can be caught outside of the chip Z-wire width and be lost. The CMOT
Bx is optimized by measuring the longitudinal position of the cloud a various times andadjusting the CMOT Bx field. This is a necessary step before the atoms are transferedinto chip Z-wire magnetic trap. After some atoms are transfered into the chip Z-wiremagnetic trap the CMOT Bx bias is optimized by maximizing the number in the chip

112
Z-wire magnetic trap. The CMOT By value can be varied but the atom number of thevarious magnetic trapping stages is not a strong function of the transverse displacement.
The transfer efficiency into the initial magnetic trap depends on the turning on
timing of the external Z-wire and the By bias field. If the fields are ramped fasterthan the Larmor frequency it will cause significant loss because the atoms will have arandom mF projection on the new magnetic trapping field. To prevent this a small bias
field (typically 1-2 G of the Bx field from optical pumping) is maintained during theinitial magnetic trap. Because the Larmor frequency is a relatively high (1.4 MHz/G
for the |2, 2i state) a fraction of a Gauss is sufficient for atoms to adiabatically follow
the changing magnetic field. The other consideration is the different time constants toramp on the external Z-wire on versus the current in the By coil. If the currents ramp
on at different rates the position of the initial magnetic trap will be swept around thearea of the atoms. The movement of the initial magnetic trap is reduced by individually
optimizing the external Z-wire and By ramping rates for optimial number and cloud
temperature. The initial ramping time is on the order of 1 ms.

4.4.7

Transfer to chipOnce the atoms are magnetically trapped they are transfered to the atom chip
where they are evaporatively cooled. The atoms are transfered to the chip by adjusting
the ratio of IEZ and By . Initially By is increased because it moves the atoms closer tothe chip and at the same time increases the depth of the magnetic trap. Then IEZ is
reduced to bring the atoms even closer to the chip and to reduce heating in the external
Z-wire and the power MOSFETs. See Figure 4.14 for a typical currents and bias fields
during initial magnetic trap and transfer to the chip. Ramping the various currents
and bias fields is done slowly relative to the trap frequencies of the external Z-wiretrap to avoid inducing a slosh mode in the cloud. A weak transverse slosh mode will
not significantly affect the behavior of the atoms in the eternal Z-wire magnetic trap,

113
but any transverse slosh mode present during transfer to the chip Z-wire will causesignificant heating and atom loss. The Bx field is varied as the atoms move close to thechip and the external Z-wire wire to maintain a single magnetic trap. If Bx cancels thefield from the endcaps of the external Z-wire then two traps will be formed close to theendcaps of the external Z-shaped wire.
The atoms are typically transfered to approximately 300 µm from the chip inthe external Z-wire magnetic trap. While the external Z-wire can support significant
amounts of current the maximum trap compression is significantly less than compression
achievable with wires on the chip. The chip is a 400 µm thick silicon wafer with a siliconoxide layer for electrical isolation. The wires are formed by electroplating copper in aphotoresist mold that is the shape of the wire pattern. The chip is patterned with a
15 mm wide Z-wire with crossing wires every 1 mm (see Figure 4.2). The resulting
wires are nominally 10 µm tall and 100 µm wide and can continuously support 5 A for
several minutes in ambient. Previous tests had shown that approximately 10×100 µm
wires on aluminum nitride can maintain 4.5 A under vacuum before failing [14]. Basedon those results the total current in any part of the wire pattern is maintained below
4.5 A for long times (i.e. evaporation). The total current includes the wire crossing
where the current in each part of the cross may be less than 4.5 A but the total current
in the junction is greater than 4.5 A. In previous BEC experiments [1, 14] the chip
is epoxied to a quartz cell with the spaces in the wire pattern being sealed by epoxy.
In this setup the outer edge of the silicon chip is anodically bonded to the Pyrex 3D
MOT cell. Instead of the electrical connections being made though the epoxy seal theelectrical connections are made though the silicon chip by UHV compatible vias. These
vias are make by Teledyne Scientific and Imaging LLC (formerly Rockwell Scientific)
by a proprietary process. Each via is actually an array of several hundred small (a fewµm2 ) vias that are connected by an metal overlayer. Each via can support 2.5 A and thevias locations can be joined together to support higher currents. For the chip pattern

114
used in this thesis the largest chip-Z is connected to two vias and can support up to
5 A (see Figure 4.2). The remaining wires are each connected to a single via and cansupport 2.5 A. A wide variety of IP, quadrupole, dimple, and waveguide geometries canbe configured depending on where the current is sourced and sunk. The areas between
the copper wires are filled with “optical metal” (shown in Figure 4.3a) that could be
used as a mirror surface.
The UHV vias connect the vacuum side wires to the ambient air, but power
supplies need to connect with the chip in a way that is compatible with anodic bonding,the bakeout process, and does not apply stress to any unsupported region of the chip.
The air side pads are positioned directly above the cell wall so the majority of the pad is
supported by the cell wall (see Figure 4.3b). Soldering and conductive epoxy were bothconsidered but rejected because of reproducibility and high temperature stability. Silver
loaded epoxy will handle 4 A of CW current for more than a day and can be applied
after bakeout with little applied force or heat. However it is difficult to apply the epoxy
reproducibly to the pads on the back of the chip without shorting neighboring pads.
Soldering connectors to the back of the chip is a higher risk process and should be done
prior to final assembly. However typical solder melts below the bakeout temperature of300◦ C. Higher melting temperature solders were not used to avoid thermal gradients
across the chip. Additionally each of these methods permanently modifies the chippads and are not compatible with rebaking cells. Instead of permanently attachingconnectors to the chip spring loaded pins (MILL-MAX M09922), held in a machinedpiece of Vespel, are used to connect to the chip. The pins provide good contact without
excessive force and can be easily changed and reused. In the current experiment the
Vespel holder is epoxied to the external Z-wire so the entire apparatus is referenced tothe air side electrical pads (see Figure 4.3b).
Transferring from the external Z-wire to the chip Z-wire is relatively simple because the shape of the two wires is the same except for their sizes. In principle trans-

DimpleXfer to Chip Z

Xfer Prep

EZ MT Xfer

100

Initial EZ MT

115

Distance to Chip
Z (mm)

10

1

0.1
0

100

200

300

400

500

600

0

100
External Z
Compression

200

300

400

500

600

0

100
Chip Z
Chip Dimple

200

300

400

500

600

0

100

200

300
t (ms)

400

500

600

80

X-BiasY-Bias

B(G)

60
40
20
0
-20
-40

I(A)

160
140
120
100
80
60
40
20
0
5

I(A)

4
3
2
1
0

Figure 4.14: Basic timing of the initial magnetic trap, ramping atoms to the chip,adjusting the external Z-wire magnetic trap for transfer to the chip, transfer to thechip, and compressing atoms in the dimple. This shows the calculated distance fromthe chip surface, the Bx and By bias fields (Bz = 0), the currents in the external Z-wire,
the compression wires, the chip Z-wire, and the dimple wire.

116
fering from one Z-wire trap to another Z-wire trap should be optimal for a wide varietyof parameters but the transfer is surprisingly sensitive to By because for a fixed chipcurrent By will determine the final height of the chip Z-wire trap. If By is too large
the atoms will be non-adiabatically transfered into the chip magnetic trap because thetrap gradient scales as r−2 and there is a factor of four difference between the heights ofexternal Z-wire and the chip Z-wire. The rapid change in trap frequency at this pointof the experimental process is more sensitive to sloshing and breathing modes than at
any other experimental step. The causes and process for eliminating these modes is
explained in Section 4.4.6. A non-adiabatic transfer and/or sloshing/breathing modes
are manifest by heating and large number loss in the transfer to the chip magnetic trap.
The transfer between two similar Z-wires is affected by the relative change inwidth between the two Z-wires and in the current experimental setup the change inwidth is too large for optimal transfer. An additional set of wires that parallel theZ-wire endcaps aids the transfer from the external Z-wire to the chip Z-wire as seen in
Figure 4.15. The additional wires were added to the external Z-wire assembly after thechip and cell were built and baked out. The ease of adding these wires (a few hours)
demonstrates the flexibility of having a chip seal the vacuum because the atoms areclose (<1 mm) to the ambient where significant experimental modifications can be madewithout making significant changes to the vacumm system. While the compression wires
significantly improve the unoptimized transfer of atoms into the chip magnetic trap the
improvement for an optimized magnetic trap is ∼10%. The small improvement in thetransfer effeciency by using compression wires shows the robust nature of transfering
between two Z-wire magnetic traps even when the change is width is nearly a factor oftwo (32 mm to 17 mm).

117

Figure 4.15: Figure a.) Show the relative placement of the external Z-wire, the compression wires, and the chip Z-wire relative to the chip plus backing plate. Figure b.)
shows mode matching the external Z-wire field to the chip Z-wire with the addition of
the compression wires.

118
4.4.8

Chip Z

Once the atoms are transfered from the external Z-wire magnetic trap into thechip Z-wire magnetic trap they are compressed close to the final transverse trap frequencies. Even though the transfer from the external Z-wire trap happens at a modest
compression for the external Z-wire trap (ω⊥ ∼ 75 Hz, ωk ∼ 5 Hz), the initial chip magnetic trap is tight compared to macroscopic trap standards (ω⊥ ∼ 1 kHz, ωk ∼ 10 Hz).
The By bias field and the chip Z-wire current are essentially at the final trap values but
the Bx field is configured for loading the dimple trap and reduces the transverse trap
frequency at this point in the trap compression. For other experiments that use Z-wiremagnetic trap to make BEC see References [1, 10, 13, 18].

4.4.9

DimpleOnce the chip Z-wire trap has been loaded the dimple trap is turned on by increasing Bx and Id [26]. Using a dimple trap significantly increases the compression of
the longitudinal trap frequencies typically from a few Hz to nearly 1 kHz. However not
all of the atoms will be trapped in the highly compressed dimple. Atoms not trappedin the dimple will still be trapped by the chip Z-wire magnetic trap but will have a
low collision rate and will not significantly contribute to the high speed evaporation. In
some of the evaporation schemes (see Chapter 5) these atoms are quickly (< 100 ms)
evaporated away in the first evaporation sweep that starts at 50 MHz. In other evaporation schemes the upper lying atoms are ignored and the first evaporation sweep starts
in the middle of the Boltzmann distribution (typically about 33 MHz) and assumes thehigher lying atoms will eventually be ejected from the trap. A fast evaporation from 50
MHz to the mid 30 MHz improves the overall number and temperature and is worth
the extra 100 ms. A combination of the chip Z-wire and the dimple parameters define
the trap bottom and changes to either can significantly change the bottom field thus

119affecting the RF knife (see Section 2.3.1). Making large changes to the chip Z-wire or
dimple traps is avoided when performing RF evaporation to avoid changing the trap
bottom.
The optimization of chip Z-wire and dimple compression is essentially the same.
The magnetic trap parameters (chip Z-wire current, By , etc.) are chosen based on
the maximum possible trapping frequencies that maintain the chip currents below the
maximum current limits. Additionally most chip based magnetic traps perform forced
RF evaporation at a distance of 50-150 µm. Moving hot atoms closer than this to thechip typically results in a loss of atoms due to uncontrolled surface evaporation. Aftercompressing the atoms to the final trap parameters the combined magnetic trap is held
for several hundred ms to allow the atoms to thermalize and show the effects of the trap
compression. The compression times are varied to maximize the atom number and thepeak optical density (OD) of the atom cloud. At a few ms TOF the OD is proportional
to the collision rate of the atoms in the magnetic trap. Starting with and maintaininga high collision rate (i.e. OD) is a key element of efficient evaporation.
The current chip (see Figure 4.16a) has interconnected wires and the dimple
configuration needs to run current independently though the dimple wires. There are
two basic ways to drive crossing wires that require independent current. The first way
is to define a common grounding point often called a star ground. From that common
ground point determine if a lead should be driven with a positive or a negative voltage(see Figure 4.16b). The current limits in the wires and the vias are always maintained
below the damage limit. While this configuration is optimal this configuration is not
used because the current safety trips in the chip power supplies (built at JILA) are very
sensitive and trip when configured with a star ground.
The other solution works on the principle that voltage is a relative measure andas long as the references of the power supplies are independent the voltage difference
between leads on the chip can be completely arbitrary. However the power supplies are

120

Figure 4.16: Schmatic diagram of current paths for driving interconnected wires on achip.

not completely isolated as shown in Figure 4.16d. The power supplies are driven by a
DAC in an electronics chassis so the voltages that are used to control the various power
supplies are referenced to the same ground. Depending on the input configuration of
the current driver the previously isolated supplies may be referenced though the control
system. There are two basic ways to isolate the current drivers from the control ground:
an isolation amplifier or an optical isolator or optocoupler. Each has its disadvantages
in isolating analog signals because it involves chopping the input signal to capacitively
isolate the signal. Because of the noise issues isolation amplifiers are not used in thecurrent BEC system. For the data used in this thesis JILA built power supplies areused that are power isolated either by batteries or transformers. The control inputs are
not isolated with a isolation amplifier or optical coupler but are isolated to a degreeby using a differential amplifier on the control input. While the control inputs are notideally isolated the current configuration of Figure 4.16d is faithfully reproduced in theexperiment. This configuration has not been tested with other power supplies so caution
should be used when trying this with other power supplies.

121
4.4.10

Evaporation

Linear instead of exponential RF sweeps are used to make BEC because evaporation is efficient when starting with a large initial atom number and tight magnetic
trap. Additionally another group has used linear RF ramps to make BEC [13]. Because
the trap consists of a larger IP trap with a dimple trap there are two regimes of RF
evaporation and two RF sweeps. The RF sweeps are optimized at the beginning andend of each sweep for RF frequency and power and the length of each sweep is also
adjusted. The metric for optimization depends on the BEC mode, but generally any
changes made to the evaporation schedule should maintain peak OD. The process of
optimizing the RF evaporation is covered in Chapter 5. While more evaportion stages
with exponential ramp could be used two linear evapoartion stages gives a reasonableamount of room for optimization without being too complicated.

4.4.11

Imaging

The various stages of BEC production are optimized via analysis of absorption
images of the clouds in time of flight (TOF) images. The atoms are illuminated for 100µs by light resonant with the F=2→F’=3 transition. The light is sent though a long
distance microscope (Infinity Photoptical K2 CF1/B objective with doubler tube) andis then imaged on a 12 bit CMOS camera (Basler 102A).
We are not aware of other groups using CMOS cameras for imaging BEC so
we feel it warrants some discussion. CMOS cameras compared to CCD cameras are
not known for excellent noise properties because each pixel in a CMOS camera has
its own readout circuitry and in particular its own amplifier. In a single image there
can be noise from pixel to pixel because of the individual performance of each pixel
amplifier. While this noise would be significant in a single image the amplifier noise
is not a factor in absorption imaging because each pixel is directly compared to itself

122
Stage
MOT
CMOT
molasses
optical pumping
Magnetic trap/transfer
transfer to chip
dimpleevaporationTime
1.2-6s25ms
5ms
0.25ms
300ms
50ms
25ms
1.2-4sTable 4.1: Summary of the essential timing for the production of the BEC.

in absorption imaging and the amplifier gain is divided out when calculating the OD.
Another potential drawback of the current CMOS camera is a low quantum efficiency
17% at 780 nn, but other CMOS sensors can be found with higher QE but less pixel
depth. The readout of a CMOS camera is fast compared to a typical CCD camera (15f/s @ 1320×1040 vs 2 f/s @ 1024×1024 for the Andor Pixus at 1 MHz readout) but the
maximum pixel depth for CMOS cameras is 12 bits, where most CCD cameras start at
14 bits. For higher signal to noise images the CMOS camera falls behind the CCD, but
for low cost, high speed, and reasonable images the CMOS camera is a good replacement(or second diagnostic camera) for CCD cameras in atomic physics experiments. In thecurrent experimental configuration the image qualtiy is more limited by interference
fringes from uncoated optical surfaces rather than the CMOS signal to noise ratio.

4.5

Typical operating parameters
The general sequence for BEC production is as follows (see Table 4.1 for a timing

summary): the MOT is loaded for 1.2-6 s depending on the production mode (seeChapter 5 for the various BEC production modes). The 2D MOT lasers are shuttered
50 ms before the end of the 3D MOT stage to avoid light and atom beam scattering in
the other stages of the experiment. Once the atoms are captured in the 3D MOT they
are compressed and cooled by increasing the MOT field gradient to 21 G/cm, the 3D

123
MOT cooling laser is further detuned from 2Γ to 3Γ, and the repump power is reduced.
At the same time the CMOT position is shifted ∼8 mm vertically by increasing Bz to∼ 17 G. In the final 1 ms of the CMOT the cooling laser is further detuned to 10Γ.
The 3D MOT magnetic fields are then switched off for 4 ms of optical molasses. After
optical molasses the atoms are typically at a temperature of 20-40 µK. The atoms arethen optically pumped into the |2, 2i state.
The atoms are initially magnetically trapped by a current of 133 A in the EZwire, By = 46 G, and Bx = 2 G. In 325 ms the atoms are ramped to the surface bydecreasing the current in the external Z-wire to 24 A, By to 48 G, and Bx to -22 G.
The atoms are not significantly compressed during the transport and the trap depth is∼ 1 mK so there is negligible loss due to atoms spilling over the top of the trap. At
this point the atoms are about 4 mm from the chip surface. The compression wires andchip wires are ramped on as the atoms are moved to approximately 2 mm from the chipsurface. At this stage the atoms are held by a combination of the external Z-wire andthe chip Z-wire.
The atoms are transfered to the chip by fully ramping off the external Z-wire in
50 ms. After the atoms are transfered to the chip the dimple wire is ramped on in 30
ms. Forced RF evaporation is started immediately after ramping on the dimple trap.
The trap is slightly decompressed during evaporation by reducing the dimple and Bx
field commensurately. This keeps the trap bottom at nominally the same value butallows the trap frequencies to be slightly modified for optimal number or speed. The
initial and final trap frequencies for the different evaporation modes are shown in Table
5.1. Because there are several BEC production modes the specifics of the RF coolingramps and other evaporation details are detailed in the next chapter.

Chapter 5

BEC production

This chapter mainly covers the final step of BEC production (i.e. RF evaporation)and the final BEC numbers while all of the previous steps of laser cooling, magnetic
trapping, etc. that are required to get to this point are covered in the previous chapter.
Each of the BEC production modes in this chapter is slightly different because eachmode is optimized for a different goal, in particular two modes are optimized for speed
which optimization is different than optimizing for number. The first part of this chapter
covers the procedure for optimizing for speed while the later sections report the final
evaporation trajectories.
There is a direct trade off between production speed and the final BEC number.
Increasing the production speed reduces the atom number in two ways: first the MOT

MOT time (s)N MOT (106 )N MT (106 )Init. ftrap (z,ρ) (kHz)
1st RF (MHz)
1st RF (s)2nd RF (MHz)
2nd RF (s)Final ftrap (z,ρ) (kHz)N BEC (103 )High Rate
1.7∼300

N Optimized τ Optimized
6.0
1.2
∼500
∼200
30-60% of N MOT
0.84, 2.28
0.8, 2.3
0.8, 2.3
33→18
33→20
31→16
0.8
1.6
0.5
18→3.520→3.916→3.65
0.6
0.6
0.45
0.76, 2.5
0.7, 2.5
0.8, 2.4
∼90
∼400
∼50

Table 5.1: Production parameters for the three BEC modes.

125
is typically loaded for less time resulting is a reduced atom number. The second issue is evaporating faster does not allow sufficient time for thermalization so continued
evaporation will evaporate farther into the lower energy portion of the MB distribution.
This aggressive evaporation will still cool the atom cloud because an increased amountof energy is removed from the cloud but it also removes more atoms reducing the net
cooling efficiency. While the speed optimized evaporation rate is faster than the number
optimized evaporation rate it must always be significantly slower than the rethermalization rate (see Equation 2.58). The speed optimized BEC production will start theevaporation at a deeper energy cut because the evaporation will remove a greater fraction of the high energy tail of the MB distribution (compare starting RF frequencies inTable 5.1).
Rapid BEC production starts by determining the start and stop frequencies ofthe evaporative cooling sweep. The start evaporation time is initially determined byevaporating from the point where evaporation just begins to cool the atom cloud (inthis experiment ∼ 30 MHz). The stop evaporation frequency is determined by partiallycooling atoms and then determining the frequency when all of the atoms are evaporated
from the trap. This frequency is the initial trap bottom and the stop frequency is set
200-400 kHz above the trap bottom. After the initial start and stop frequencies are
determined the evaporation time is varied while maintaining the start and stop evaporation frequencies. The evaporation time that leaves ∼ 50 × 103 atoms is a reasonable
initial evaporation time.
After the RF start/stop frequencies and evaporation time are roughly determinedthe evaporation is optimized. The evaporation is optimized by splitting the evaporation
into two sweeps and optimizing each sweep for time and frequency, additionally thetrap frequencies and RF power are also varied. The optimization procedure is similar
to optimizing BEC for number, each parameter is optimized to maintain the final atom
number while maintaing peak OD. The main difference is that the final number is126choosen to maintain signal to noise while the timing and RF parameters are optimized
to cool the atoms to a BEC as fast as possible.
The optimization for each of the BEC modes in this chapter follows the above
description for the speed optimized BEC production or References [14, 96] for number
optimized BEC. Other BEC experiments have been optimized for different modes andthe modes here are an extension of those experiments with improved speed and number[1, 13].

Results
5.1

30 shots

Figure 5.1: Images of 30 BECs made in less than 120 s. Each BEC is produced from
MOT loading to imaging in 3.65 s.

In an effort to show the reproducibility of making BEC in a compact system
BEC is made 30 times sequentially. This was done without data processing between the
shots so the longer term behavior of a high repetition rate system system could be seen.
The BEC production was optimized for speed as explained above however the BEC
transition was not detected via a bimodal distribution or shape inversion. The bimodal
signature was not used because expansion of the BEC due to the mean field was on thesame order as the expansion of the thermal cloud [88] making it difficult to isolate thecondensate fraction from the thermal fraction. Shape inversion was not used to detect

127
the transition because the relatively small difference between the trap frequencies of thetight and weak axis necessitated a long TOF which was complicated by the mean field
expansion. With slightly different final trap frequencies shape inversion could be seen
but the signal to noise made it difficult to accurately determine the BEC transition.
The BEC transition was determined by a change in the effective area of the BEC cloud[97]. In the final data run the final RF frequency is well past the transition point so any
fluctuations in the atom number will not significantly affect the transition point.
In the first set 30 images the atom number decreases with each successive shot(see Figure 5.2). This number loss is attributed to the silicon chip gradually heatingfrom the Ohmic losses of the chip wires. Under normal operating conditions there issufficient time for the chip to cool between shots because the process of producing acloud of cold atoms, taking an image, and then processing takes at minimum 10-15 s.
Even with a longer time between shots the slight heating of the chip increases vacuum
losses by ∼ 10% when shots are successively taken over a period of several minutes.
The chip is not easily cooled in the current configuration because the external
Z-wire blocks access to the back of the chip (see Figure 4.11). Because the heating
rate is small the chip was first cooled by flowing air around the chip and external Zwire structure. The extra air cooling decreased the heating rate, however it increased
imaging noise by exciting mechanical vibrations. While air cooling could be a longterm solution it would most likely require significant modifications to the vacuum cellmounting and replacing the external Z-wire. Instead of air cooling the chip is cooled
over the short term with a few drops of alcohol applied to the edges of the chip. A
syringe is used to deliver the alcohol into the gap between the chip and the external
Z-wire. The additional cooling from the alcohol significantly decreases the heating rateand 30 BECs are made in rapid succession (see Figure 5.1), but a slight drop in theatom number is apparent from a linear fit of the atom number (see Figure 5.2). The
decrease is small over two minutes and may not be significant because of the variation

128
in the atom number.
The total time to make 30 BECs is less than 120 s. This time includes saving
the images, but not image processing which occurs offline. For applications of BEC thedata processing would most likely extract a single number that would use significantly
less processing resources. Another application minded issue is the shot to shot number
stability. As seen in Figure 5.2 the shot to shot variation is on the order of 20%. While
20% variation is less than desirable it is not a significant impediment to the goal ofcompacting and modularizing the BEC apparatus. Since the 30 shots and the number
and speed data were taken an etalon mode was found in the MOT cooling laser delivery
fiber that was one souce of noise. This fiber was flat polished on both ends and was
replaced with an FC to APC PM fiber which reduced the number instability to ∼ 10%.
120000100000Fit Atom Number

80000

60000

40000

20000

0
0

5

10

15
Image Number

20

25

30

Figure 5.2: Atom number as a function of shot number in a rapid sequence (3.65 s
production time per BEC) of 30 BECs. The open circles show mild chip heating as theexperiment progresses. A few drops of alcohol applied to the back of the chip provide
sufficient cooling to maintain atom number (filled circles).

129

5.2

Number optimized
After the demonstration of 30 consecutive BECs the apparatus was optimized to

produce the largest possible condensate. The first and most obvious change to optimize
the number is loading the MOT to saturation including a few extra seconds past the
MOT loading exponential roll off to assure the largest possible number is captured in
the MOT. Secondly, the RF evaporation sweeps in the optimal configuration were about
two times longer than the high repetition rate optimization. However, increasing theevaporation time beyond a factor of two and/or reducing the trap frequencies to avoid
three body loss did not improve the BEC number. The limited time optimization was
traced to the chip magnetic trap lifetime of ∼2 s. Compared to the 7 s lifetime in theexternal Z-wire magnetic trap the chip Z-wire lifetime is considerably shorter.
This short lifetime in the chip Z-wire magnetic trap prevents a slower, more
efficient evaporation that would lead to a larger BEC number. The vacuum cell usedin this experiment was baked at 300◦ C for three days however a cell baked for 10–14
days had a slightly longer external Z-wire magnetic trap lifetime (∼ 10 s) but effectively
the same chip magnetic trap lifetime. Initially it would seem that chip heating would
directly lead to the short chip magnetic trap lifetime but further experiments did notconclusively identify chip heating as the only source of atom loss. Other possible chip
related loss mechanisms are collisions with the surface, Johnson current noise, andparametric heating in tight magnetic traps. None of these mechanisms individually was
the most significant contributor to the number loss, so it appears that the loss may be
a sum of plausible loss mechanisms.
As mentioned previously there is an approximate 10% atom loss due to residual
heating from taking sequential shots every 15-30 s. For the maximum number there wasa 10 minute cooling period between mapping out the BEC transition and taking thenumber optimized shots. The loss associated with residual heating was insignificant in130
the second vacuum cell and the improved performace is attributted to the longer baking
time.

5.3

Speed optimized
In optimizing the system to produce BEC as fast as possible the rubidium dis-penser current was increased from 3.75 A to 3.85 A and a halogen bulb was placed close
to the 2D MOT cell without interfering with the optical fields to generally heat the whole
2D MOT cell. These two changes increase the flux from the 2D MOT by 10-20%, but
shorten the vacuum lifetime. The shorter vacuum lifetime did not affect the ability to
produce BEC because the evaporation time was commensurately reduced maintaining
the relative time scales (see Section 2.7). The evaporation trajectory was optimized for
rapid evaporation following the procedure explained in the first section of this chapter.
See Table 5.1 for final trap and RF evaporation parameters and BEC numbers. The
transition to BEC was determined by the appearance of a bimodal cloud distribution.
The bimodal shape is more easily seen in the speed optimized BEC production because
the reduced atom number leads to a reduced mean field energy.

5.4

Vacuum lifetime, heating rates, and RF shield
As mentioned previously the external Z-wire magnetic trap lifetime is 7 s withthe dispenser at 3.75 A and the 2D MOT cell heated to 80 − 90◦ C. The 7 s lifetime
is assumed to be primarily limited by the background vacuum pressure1 . Below arubidium dispenser current of 3.75 A the 2D MOT flux is a function of the dispenser
current but the magnetic trap lifetime is constant. This is a result of the differential
pumping between the high pressure and low pressure chambers being able to maintaina pressure differential of ∼ 103 . Above 3.75 A the 2D MOT flux continues to increase
1

Stray resonant light was a source of loss after the initial 2D MOT laser setup. An IR viewer wasused to search for light leaks that were not obvious. These light leaks were easily remedied by putting
curtains around both the lasers and the experimental setup.

131
but the vacuum lifetime drops because the flux of rubidium atoms reaching the upper
chamber is greater than the net rubidium pumping speed.
As mentioned previously the vacuum lifetime depends on the how recently thechip had been used to trap atoms. This vacuum dependence is due to the overall heatingof the chip from Ohmic heating. Additionally the chip trap lifetime is 2-3 s shorter thanthe vacuum limited lifetime. This may be due to chip heating causing a temporary
vacuum pressure increase while the chip is energized but efforts to isolate the short chipmagnetic trap lifetime to heating were inconclusive.
The heating rate of atoms held in the magnetic trap just above the BEC transition temperature is 2.5(4) µK/s for the compressed(decompressed) trap. It is odd
that the decompressed trap would have a higher heating rate but the decompressed
trap frequencies may overlap with a mechanical/acoustic/electrical resonance in the experiment. Any heating is eliminated by applying an RF shield to maintain the cloud
temperature while maintaining 95% of the atoms compared to no RF shield. When theatoms are evaporated to BEC the 1/e lifetime of the condensate is ∼300 ms with an
optimized RF shield. The RF shield frequency is determined by looking at a long TOF(10 ms) and reducing the frequency while monitoring size and number. The optimal
frequency will maintain the cloud size with the maximum possible number.

5.5

BEC production for atom interferometry
The purpose in compacting and simplifying the experimental aspects of a BEC

machine is to enable future applications of BECs, in particular atom based interferometric sensors. Using light pulses to split and recombine atoms has been one of the mostreliable implementations of an atom interferometer to date [98, 99, 100, 101, 102]. As apreliminary experiment a light pulse interferometer is implemented in the compact BEC
system. A magnetic trap with low trap frequencies (< 100 Hz) is optimal for splitting
and recombining atoms with light pulses and some groups have taken considerable effort

132
to produce the weakest magnetic traps that can still hold against gravity [103]. The
magnetic trap frequencies for manipulating atoms with light pulses is diametrically opposed to the conditions for producing BEC. Additionally atoms that have been cooled
on a chip must be moved away from the chip (typically < 100 µm) to approximately
300 µm. Translating cold atoms by a factor of three seems trivial but the trap frequencies scale proportional to 1/r2 so the trap frequencies can easily change by an order ofmagnitude during the transport. The atoms are first evaporated nearly to a BEC close
to the chip and are then moved to the final position where the final evaporation step
creates a BEC.
The magnetic trap is moved by adjusting the relative chip currents and bias
fields, however moving the atoms and changing the trap frequency must be adiabatic(dω/dt < ω 2 ) or sloshing and breathing modes will be excited in the atom cloud. The
magnetic trap is transported with the trap frequencies as high as possible to avoid
exciting other modes in the atom cloud. Once the atoms are translated to the newposition a final evaporation sweep cools the atoms to a condensate. The final evaporation
sweep is performed with a moderately tight trap to maintain a reasonable collision rate
at the final magnetic trap position. The evaporation parameters are optimized similar
to any other evaporation stage only the evaporation rate is not as fast.
The final step of preparing a BEC to be split with optical pulses is to decompress
the atoms to the final magnetic trap frequencies while maintaining the position of themagnetic trap. After the trap has been decompressed to the final values the atoms are
split with an optical standing wave. The trap frequencies of the final magnetic trap
were (10, 90) Hz as measured by exciting a slosh mode and then fitting the data to asine wave.
The timing and values of the currents and bias fields for transporting and decompressing the magnetic trap are shown in Figure 5.3. Additionally shown in the top graphof Figure 5.3 are the calculated trap frequencies. The ramp times for each operation are

133

Move

Evaporate

Decompress

Trap Freq. (Hz)

10000

ωt
ωl

1000

100

10
0

50

100

150

200

250

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200
t (ms)250

z (µm)

300

300
350
Distance from Chip

400

250
200
150

B(G)

100
50
40
30
20
10
0
-10
-20
-30
3
2.5

350
Y-Bias
X-Bias400

350
Chip Z
Chip Dimple

400

300

400

I(A)

2
1.5
1
0.5
0
350

Figure 5.3: Timing and values from decompressing a compressed magnetic trap to amagnetic trap suitable for splitting and recombining cold atom clouds. This shows the
Bx and By bias fields (Bz = 0), the currents in the external Z-wire, the compression
wires, the chip Z-wire, and the dimple wire. The trap frequencies are calculated using
the known trap parameters and equations from Chapter 2.

134
optimized by ramping the currents and bias fields to the new values and then holding
the atoms at that stage for 0.3-1 s. This allows any breathing or sloshing modes to bedamped out and converted to heat. The ramp times are then optimized to minimize
heating and preserve number.

Chapter 6

Towards Applications of BEC

As explained in the introduction the purpose of compacting and simplifying a
BEC was to enable applications of BEC. Applications involving coherent atom interferometry has attracted a significant amount of interest, specifically for building a rotation
sensor based on the Sagnac effect. A Sagnac interferometer implemented with atoms
should have an approximated 1011 increase in sensitivity compared to a comparable
light interferometer [8].
Assuming the atoms already exhibit phase coherence, implementing a Sagnac
interferometer requires three to four key components. The first component is a methodto split the atoms in a way to preserves the phase for later measurement. The second
component is a method to move the split clouds on a path that encloses area. The
third component is a method to recombine the atoms in such a way that the output
is dependent on the relative phase between the two packets. The final component is amethod to detect the relative number of atoms in the output ports.
Each of these steps has been demonstrated previously but generally the steps ofsplitting, recombination, and detection have been performed without enclosing area.
The one notable exception is an rotationally sensitive atom interferometer made byMark Kasevich using a Bragg diffraction on a cloud or beam of laser cooled atoms [98].
There have been a few demonstrations of guiding atoms on a curved path [21, 22], but
guiding atoms though curves has not recently received much attention.

136
There have been several demonstrations of atom interferometers that do not enclose area. The splitting and recombination mechanisms fall into a few broad categories:
physical gratings [104], magnetic field potentials [105], optical double well potentials[106], dressed RF dressed potentials [107], and optical standing waves [99]. Several ofthese methods have drawbacks that would make it difficult to use these splitting methods
in a broadly applicable device. Physical gratings require precise micro/nano machining,coherently splitting with magnetic fields requires precise control of the magnetic fields,
and splitting with optical potentials does not exhibit phase reproducibility. Splittingwith RF dressed potentials is straight forward and has demonstrated coherent phasereproducibility. Splitting with an optical standing wave is a good splitting method interms of simplicity and flexibility. The laser frequency requirements are not stringent
and a few milliwatts of power is more that enough power to split the atoms. Anotherbenefit of splitting with optical fields is the momentum of the split atoms is >10 mm/s.
This is significant because the act of splitting the atoms will separate the atom clouds
and if properly configured could enclose area.
While these are the requirements of creating an atom interferometer building aninterferometer is not the subject of this dissertation however there are a few preliminary
splitting/recombining and interferometry experiments. Additionally a separate curved
waveguiding experiment that will be discussed. These demonstrations are preliminary
and are meant to be basic demonstration of the technologies required for an atominterferometer with an enclosed area and to debug any potential difficulties due to thecompact nature of the apparatus.

6.1

Atom splitting and recombination
As mentioned previously not all methods of splitting atoms split with a repro-

ducible phase. A pulsed light grating has been very successful at coherently splitting
and recombining atoms with a reproducible phase up to a few hundred miliseconds when

137
decoherence or noise become significant [99, 102, 101]. This method is also very straightforward in its setup and operation. The theory of the splitting and recombination canbe found in Reference [66]. While splitting with light pulses can be compared to alight transmission grating that diffracts atoms into various orders the behavior of the
splitting requires considering the quantum mechanical nature of the atoms.
The setup and the basic operation of the splitting and recombination follow Reference [66]. One significant difference between this experiment and other atom chip
based splitting experiments is the use of the atom chip to seal the vacuum system. The
chip is slightly bowed in due to the external air pressure (see Figure 4.4). This bowing
is unique to the compact vacuum system and is a consideration in the alignment of the
splitting beam.

6.1.1

Optical setup

The optics setup for Bragg splitting atoms a few hundred microns from the chip
starts with a beam with a waist smaller than half the distance from the chip to prevent
light scattering from the chip. A lens with F = 500 mm and an input beam size of
1.5 mm yields a beam with a theoretical beam waist of 280 µm and a Rayleigh rangeof 40 mm. The atoms are held at 300 µm from the chip to match the experimental
parameters of [66], the beam waist of 280 µm should easily pass below the chip withnegligible scattering off the chip. The Rayleigh range of 40 mm is chosen to maintaina small beam and flat wavefront over two times the distance between the center of thechip and the retroreflecting mirror (see Figure 6.1). Directly after the focusing optic apair of waveplates sets the polarization of the beam parallel to the chip.
There are no easy reference marks on the chip and because the BEC is split in a
weak dimple trap the trap axis is rotated relative to the chip wires. The position andangle of the splitting beam are adjusted using the optical setup shown in Figure 6.1.
The lateral position of the splitting beam is adjusted by translating mirror M1. The

138

Figure 6.1: Setup for beamsplitting mirrors and translation stages relative to an atom
chip and cell. Not shown are magnetic field coils or the optics for the 3D MOT or
imaging beams.

vertical position and angle of the beam are adjusted by translating and rotating mirror
M2. As mentioned one difference between this alignment and previous splitting beamalignments [99, 102] is the atom chip is bowed due to external air pressure. Fortunately
the bowing of the atom chip creates a unique reflection signature that can be used toalign the splitting beam parallel to the center region of the chip (see Figure 6.2).
Once the splitting beam is aligned parallel to the chip in the vertical direction thehorizontal angle of the splitting beam is purposely set 2 − 3◦ relative to the cell wall for
two reasons. The first is to better match the angle of the dimple trap with the splitting
beam. The other reason is to avoid interference effects from the partial reflections frommultiple passes though the cell walls. The angle is achieved by appropriately adjusting

139

Figure 6.2: Alignment of splitting beam using the reflection signature from the deformed
atom chip. The dots at the end of the beams show the expected pattern is the lightis viewed with a card in the beam. Parts a.) and b.) show the signature when the
splitting beam is angled relative to the flat center section of the chip. Part c.) shows
the beam with the proper angle but too close to the chip. When the alignment is close
the signatures of a.) and c.) can both be seen.

M2 remembering that rotating a mirror by θ results in a beam rotation of 2θ. Afterthe splitting beam is aligned relative to the chip and cell wall the retroreflecting mirror(M4) is added and aligned to retroreflect the splitting beam. The final alignment isperformed by holding a BEC in the splitting position and turning the splitting beam on
for 10 ms. The splitting beam is aligned with the magnetic trap by adjusting M1 andM2 for maximal splitting loss. The power in the splitting beam is adjusted so there isalways a small number of atoms at the minimum to better identify the optimal position.
This method is sensitive to 20-40 µm which is about 10% of the beam diameter. Otheralignment methods [108] improve on the loss based alignment by adjusting the beam
position to minimize the splitting power needed to split the atoms without leaving atoms
in the zero momentum order. For all of the data in the splitting section the splitting
beam position was only optimized for loss with a continous 10-20 ms pulse.

6.1.2

Atom splitting optimization

The atoms are split by using a double pulse scheme [66] that allows the atoms tobe split with high fringe contrast [84] into ±h̄k orders. For diagnostics and testing both
the double pulse format and a single pulse format were used to split the atoms. The
splitting signature of the single pulse format qualitatively matches the splitting signalof Reference [66]. The agreement is only qualitative because the splitting power is notcharacterized for all AOM voltages. The maximum splitting power is ∼1.5 mW.

140
The splitting of the double pulse versus power also qualitatively agrees with dataof Reference [66]. The power of the splitting beam varied over the time scale of a day
so before each data set the splitting power of the double pulse sequence is adjusted
to minimize the number of atoms in the central peak. Additionally during splitting
and recombination data runs the splitting optimization is checked by turning off therecombination pulse and then taking several shots to verify the splitting condition is
not drifting. The splitting condition typically did not drift over the period of a few hours
so it is assumed the splitting and recombination data does not suffer from splitting power
variations.
The objective of using a double pulse format instead of a single pulse format
is to optimize splitting into the ± orders with no atoms in the zeroth order. In thisexperiment after the atoms are split the atom clouds sit on top of a background thatmakes it hard for the splitting power and time to be optimized (see Figure 6.3). Despite
the background there it is possible to optimize the splitting such that there is no peak left
in the middle of the distribution. This background can be attributed to three sources:
collisions due to high atom density, a residual thermal background, and splitting beam
inhomogeneities. Scattering due to high atomic density is a probable cause and isquickly solved by reducing the MOT loading time to reduce the final atom number.
While reducing the MOT time is a quick fix it assumes the evaporation trajectory is
still optimized for lower starting number. In this experiment reducing the MOT loading
time reduced the atom number without sacrificing BEC production.
The other method for reducing the density of the atoms is to relax the trap
frequencies to reduce the density of the cloud while maintaining atom number. This isthe better way to deal with density related scattering because it maintains the atom
number for signal to noise issues. However decompressing to a different trap frequency isslightly more involved because decompressing too fast can introduce breathing modes.
The first trap used for atom splitting had trap frequencies of ωt,w = (18, 180) Hz or141

Figure 6.3: Image of atoms split with a double pulse. The atoms are split and are thenpropogated in the waveguide to separate the two clouds. The atoms are imaged after a
few miliseconds of TOF. While two clouds are obviously visible the atom density does
not go to zero for the optimal splitting time and power.

ω̄ ≈ 80 Hz with 20 − 30 × 103 atoms. Compared to previous splitting experiments of
10 × 103 atoms at ω̄ ≈ 37 Hz [66] and 8 × 103 atoms at ω̄ ≈ 81 Hz [27] the densityof the initial trap is too high. The trap was then relaxed to ωt,w = (12, 130) Hz withapproximately the same atom number for ω̄ ≈ 60 Hz. Reducing the trap frequency madethe splitting slightly better but did not notably change background atoms. Additionallythe atom background is cleaner after the the atoms has been through one oscillation
cycle. If the background was due to density collision then the atom clouds would
have collided twice and should have doubled the scattered atom signal The improved
background after two cycles seems to indicate atom density is not the cause of the atombackground signal.
A background of thermal atoms is another possible cause of the background signal.
While this is a plausible cause thermal atoms (beam and cloud) have been split andrecombined using an optical standing wave [98, 100]. To test the splitting of a condensate
versus a non-condensate the RF knife is held off the bottom by up to a few MHz.
Increasing the RF knife (i.e. the atoms are hotter) by a few hundred kHz does not
significantly change the appearance of the atom background. Increasing the RF knife
further increases the temperature of the atom cloud significantly above the transition
temperature. When the splitting pulse is applied to this hotter cloud the atoms still

142
split but the splitting modulation is on the order of 10-20% of the peak OD. It is obvious
there is splitting and it is also obvious the background is significantly larger.
This extra background only happens when the cloud is significantly hotter. The
temperature of the hot atoms was not measured, but small changes in the atom temperature around the BEC transition do not significantly change the background. Additionally there is no ‘phase transition’ from clean splitting to poor splitting when the cloudswitches from a BEC to a thermal cloud. Thus a thermal cloud does not significantly
contribute to the splitting background when the atoms a near condensate temperatures.
Another possible cause of the poor splitting is poor beam quality due to passing
though the uncoated cell walls up to three times during the splitting. This was determined to be the cause of system instability in other BEC interferometry experiments[84]. Spatial intensity variations from interference can cause different parts of the BEC
to experience different intensities and have different optimal splitting times. Variationsin the splitting beam due to interference effects should be small because the beam is
purposely aligned a few degrees off normal from the cell walls. Intensity variations could
come from imperfections from the glass blowing, hand polishing, and anodic bonding
procedures. There are no deep scratches in the area where the splitting beam passes
though the cell but there are some areas of the cell that are obviously scratched. There
are several obvious remedies: following References [99, 84] add optics to the chip toavoid passing light outside the cell, AR coat the cell walls, and/or improve the qualityof the glass finishing near the chip.

6.1.3

Atom splitting and recombination data

The end goal of splitting and recombining the atom cloud is to make an interferometric measurement of some measurable quantity (e.g. rotation or gravity). A key
requirement of any type of interferometer is the coherence length of the light, atoms,
neutrons, etc. must be longer than the total length of the interferometer arms. The

143
0.35
Center Fraction

0.3
0.25
0.2
0.15
0.1
0.05

1st day
2nd day
44

45

46

47

48

49

44

45
46
47
48
Recombination Time (ms)

49

0.06

Std. Dev.

0.05
0.04
0.03
0.02

Figure 6.4: Increased noise in the splitting signal as a function of the recombination
time. The upper graph shows the fraction of atoms in the center cloud compared to thefraction in the outer lobes. The lower graph shows a plot of the standard deviation of
the upper data. The increased standard deviation is one sign of coherence between theatoms, however this data is significantly affected by noise or loss of coherence.

longest coherent time for the current state of the art is approximately one second, however phase can be recovered only out to ∼150 ms [109]. This can be compared to a lightinterferometer where for short arm separations there is a well defined fringe pattern at
the output ports. When the arms are lengthened there is shaking fringe pattern indicating there is still coherence but the fringe variability is a result of significant noise in
the system. If the light was incoherent when it reached the end of the interferometer
there would be no fringes rather a constant intensity at the output ports. Thus anincrease in noise at the correct recombination time indicates the split BEC packets are144still coherent but overwhelmed by technical noise.
As mentioned above one signature of coherence is increased noise when the atoms
are recombined at the proper time. In an effort to see coherence effects in a BEC cloudthe experiment of Reference [102] is recreated with a single laser frequency. In the caseof a single splitting laser frequency the atoms are split near the bottom of the magnetic
trap and then propagate out to the classical turning points where the atoms turn around.
The atoms propagate back to the center of the weak dimple trap where a second double
pulse is applied to recombine the atoms. The recombination time is initially determinedby measuring the size of the cloud as a function of time in the waveguide without arecombination pulse. The apparent size of the cloud is minimized when the clouds areoverlapping. The recombination time is varied around the overlapping recombination
time. At each time multiple shots are taken at each time and the standard deviation of
the point is plotted to better quantify the increased noise (see Figure 6.4).

6.2

Circular waveguiding
The other key element of implementing a rotation sensor is moving the atoms ona path that encloses area. One method for guiding atoms in a circle was implemented
using a chip pattern shown in Figure 6.5. In this experiment the atoms are magneticallytrapped and transported after being laser cooled. In principle a BEC could be transported with a similar moving trap assuming the acceleration is slow enough to avoid
exciting low energy modes that are not visible with laser cooled atoms.
The trap confinement is the dot product of the external magnetic field with thespatially varying magnetic field created by the curved wire. This creates a curved wiremagnetic trap
B(θ) = βIcos(θ) − B⊥
z0

(6.1)

where θ determines a position along the curve of the wire. The trap can also be param-

145

B
Figure 6.5: The chip design used for circular waveguiding. The red shaded wire is thecircular guiding wire. The green shaded wires are the endcaps of the chip Z-wire that isused for the initial magnetic trap. The blue wires are briefly turned on when the atoms
are released in the guide to increase the velocity of the atoms. Other wires on the chip(not shaded) are for other guiding options that were not pursued.

eterized along the arc of the curveI
B(a) = β
z0


1−

a
rw

2 !
− B⊥

(6.2)

where a is the coordinate along the arc and rw is the radius of the wire. The trap
curvature is
B 00 (a) = β

2I.
r w z0(6.3)

The compression of the atoms in the curved trap is proportional to the field at thetrapping height and the radius of the trapping wire. The trap is moved by rotating theangle of B⊥ . As long at the wire has the same radius the trapping potential will always
have the form of Equation 6.2 because of cylindrical symmetry.
The atoms are transported in the curved wire trap by a varying the angle ofB⊥ relative to the arc of the guiding wire. The angular rotation rate of the trap is
simply the angular rotation rate of the magnetic field. Accelerating the atoms in a
curved waveguide could be a very important part of an atom gyroscope because BEC

146
coherence times may be limited to a few hundred miliseconds. Atoms that are split with
optical standing waves into ±h̄k have velocities of ∼ 12 mm/s. Using an interferometerwith a 1 cm2 enclosed area it would take 3-4 s to make one revolution; Far longer thanthe coherence time of the atoms. Actively accelerating the atoms after splitting couldin principle enclose the same area on a much faster time scale.
In the curved waveguiding experiment the atoms are first captured in the chip
Z-wire magnetic trap in the center of the wire pattern (see Figure 6.5). The atoms arelaunched into the curved waveguide by turning one leg of the chip Z-wire off and at
the same time turning the curved wire current on. Turning off one side of the Z-wireallows creates a potential the pushes the atoms toward the curved wire. When theatoms reach the edge of the curve B⊥ is rotated by changing the ratio of the Bx and By
fields to match the velocity of the atoms in the waveguide. The atoms rotate the full
extent of the curve (slightly more than a quarter circle) and at the end of the guide arerecompressed because the guide abruptly ends. Figure 6.6 shows absorption images of
the atom movement taken by reflecting the imaging beam off the chip at a near incident
angle and then steering the light into a camera with mirrors.

Figure 6.6: This shows atoms being guided 90◦ using a curved wire and a rotating bias
field. The pictures images are taken at 4 ms intervals.

6.3

Future directions
The natural extension of this work is a vacuum system with more than two cham-bers so a MOT can be loaded while atoms are being evaporated in a light isolated regionof the vacuum system. As seen in Table 4.1 the MOT loading and RF evaporation are

147both accomplished in ∼ 1 s. With a few improvements the BEC duty cycle could befaster than 1 Hz. There are several design problems related to achieving this. One issue
to solve is a method to transport the atoms between light isolated chambers. This could
involve a moving shutter similar to Reference [110], but preferably a chamber could bedesigned that involved no moving parts but still maintain good light baffling. Anodic
bonding could be used to construct a multichamber system and maintain a compact
vacuum system.
The other extension of this work would be to implement an atom interferometerthat encloses area and that can be used to detect rotation. There are several issues thatneed to be addressed. The first is a method to move atoms so they enclose a significant
area (∼ 1 cm2 ). The other fundamental issue to be addressed is the coherence timeof cold atoms in a magnetic trap. Reproducible phase has been demonstrated out
to several hundred ms [109, 101], but at long times the interference fringes have low
contrast or are overwhelmed by noise [102, 111]. This could be a technical noise issue
and would require isolating the atoms from external influences or the dephasing could
be a fundamental issue of atom interferometry [111].

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Appendix A

Monte Carlo for MOTs

Here is the front end for setting up the MOT MC

clear all;%%

The front end to vary the starting position and velocitiesof the experiment% Assume the atoms are in a box of width w and laser beams have a%diameter d_laser (defined in the called function);% How long the solver should run and size of the cellt=linspace(0,.5,1000);
w = 6e-2;% Define all of the relevant constants once so they can be called%from a subroutine. The variable must be globals to be%accessed from inside the called function
global g = 2*pi*6e6;
global hbar = 1.05e-34;
global lambda = 780e-9;
global k=2*pi/lambda;
global I0=4.1;
global I=6;
global S = I/I0;
global N=2;
global dl = 2*g;
global mohb = 8.8e10;
global Bp = 0.15;
global m = 87*1.66e-27;
global r=6e-3;
global o=2.5e-3;

157% some combined values
global prefactor=k*hbar*g*I/2/I0/m;
global OpNS = 1+N*S;
global mohbTBp = mohb*Bp;
global Fdgs = 4/g/g;
global thetaBeam;
global wCell = w;
% The range of values that will be calculated
y_start = 0;
y_end = w/2;
theta_start = -80;
theta_end = 80;t1=time();xnot = -w/2;% Loop over the angle of the MOT beams
for thetaB=0:5:90%thetaBeam = 45;
thetaBeam = thetaB*pi/180;% Dummy variable to keep track of the output datam = 1;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Only look along the upper left hand verticle wall %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Position of atoms along wallfor ynot=y_start:w/20:y_end% Angle of atoms leaving the wall
for thetaDeg=theta_start:10:theta_end
theta = thetaDeg*pi/180;
upper = 150;
lower = 0;
for n=1:8

158% Perform 8 steps of the bisection method to find the%capture velocity. Use bisection because the capture%
velocity transition is binaryv = ( upper + lower )/2;%x=[xnot,v*cos(theta),ynot,v*sin(theta),thetaBeam,w];x=[xnot,v*cos(theta),ynot,v*sin(theta)];
xf = lsode("MOT2DAngledTopHatBeams",x,t);
final = length (xf);final_v=sqrt( xf(final,2)^2 + xf(final,4)^2 );final_r=sqrt( xf(final,1)^2 + xf(final,3)^2 );% Check to see if the atoms have been cooled below a small value%
and are near the center of the laser beams
if ( (final_v < .05) && (final_r < 1e-3) )lower = v;
else
upper = v;
endif
endforv = ( upper + lower )/2;
% take into account the last iteration for the final value%of the capture velocity and then save the data
output(m,1) = xnot;
output(m,2) = ynot;
output(m,3) = theta;
output(m,4) = v;
m=m+1;
endfor
endforname1 = "VcaptureFullAngle_wCell_6cm__dbeam_10cm__theta_beams_";
name2 = "deg.dat";
name = strcat(name1, dec2base(thetaB,10), name2);save("-text", name, "output");
endfortotal_time=(time() - t1)/60

159
Here is the function called by the ODE

function xdot = MOT2DAngledTopHatBeams (x, t)
%
%
%
%
%

SPEED NOTES
Dot products in Matlab/Octave are incredibly slow comparedto multiplying and adding by hand. Because the vectors
are in 2D it is just as easy to write out the whole form
and get a speed bump%%%

Referencing and building arrays is slow so if a variable
is called several times it is worth it to declare avariable so the array is only reference once%%%

To speed up the computation all of these variables should
be declared in the main calling program but need to be
defined here so they can be referenced

global k;
global dl;
global prefactor;
global OpNS;
global mohbTBp;
global Fdgs;
global thetaBeam;
global wCell;%theta = x(5);xX = x(1);xY = x(3);%wCell = x(6);tan_theta = tan(thetaBeam);cos_theta = cos(thetaBeam);
sin_theta = sin(thetaBeam);
wB=10e-2;% First determine if the particle is in the plus beam by looking at%
the sign of the position relative to the extents of the beam
upper = xX*tan_theta+wB/2/cos_theta;
lower = xX*tan_theta-wB/2/cos_theta;% if the particle is inside the plus beam this value will be positive
inplus = (xY-upper)*(lower-xY);% Now the same for the minus beam

160
upper = -xX*tan_theta+wB/2/cos_theta;
lower = -xX*tan_theta-wB/2/cos_theta;
inminus = (xY-upper)*(lower-xY);% Now make sure the particle is inside the cell width
in_x = ( -wCell/2-xX )*( xX-wCell/2 );in_y = ( -wCell/2-xY )*( xY-wCell/2 );inside = in_x > 0 && in_y > 0;r= sqrt(xX^2+xY^2); % Assume MOT is centered at orgin% Calculate force from plus beam if conditions are rightif ( inside*inplus > 0 )% determine the alignment of the magnetic field (quantization axis)%
with the direction of the laser field and calculate oncealignP= ( cos_theta*xX + sin_theta*xY )/r;
dotkpv = k*( cos_theta*x(2) + sin_theta*x(4) );Fp=prefactor*(1/(OpNS+Fdgs*(dl+dotkpv+alignP*mohbTBp*r)^2) \-1/(OpNS+Fdgs*(dl-dotkpv-alignP*mohbTBp*r)^2));
else
Fp=0;
endif
% Repeat for the minus beamif ( inside*inminus > 0 )alignM= ( cos_theta*xX - sin_theta*xY )/r;
dotkmv = k*( cos_theta*x(2) - sin_theta*x(4) );Fm=prefactor*(1/(OpNS+Fdgs*(dl+dotkmv+alignM*mohbTBp*r)^2) \-1/(OpNS+Fdgs*(dl-dotkmv-alignM*mohbTBp*r)^2));
else
Fm=0;
endif
% Now update the xdot vector for the next iteration of the ODE
xdot(1) = x(2);xdot(2) = Fp*cos_theta+Fm*cos_theta;xdot(3) = x(4);xdot(4) = Fp*sin_theta-Fm*sin_theta-9.8; % Include gravity%xdot(5) = 0;
%xdot(6) = 0;
endfunction