Novel Trapping Techniques For Shaping
Bose-Einstein CondensatesbyMicah Boyd
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
October 2006
c Massachusetts Institute of Technology 2006. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
October 24, 2006
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacAurthur Professor of Physics
Thesis Supervisor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David E. Pritchard
Cecil and Ida Green Professor of Physics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thomas J. Greytak
Professor of Physics, Associate Department Head for Education

2

Novel Trapping Techniques For Shaping Bose-Einstein
CondensatesbyMicah Boyd
Submitted to the Department of Physics
on October 24, 2006, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
A combination of radio frequency radiation and magnetic field gradients was used totrap atoms in dressed states. In a magnetic field with a quadrupole minimum, RF
fields resonant with the (|F, mf i) |1, −1i → |1, 0i transition trapped the atoms on
the surface of a sphere, and gravity caused the atoms to pool at the bottom of thesphere. BECs were transferred into this dressed Zeeman trap with 100% efficiency,with lifetimes of up to 30 s, and trapping frequencies of up to 250 Hz were measured.
A hard disk platter with a specially written magnetic pattern was used to generatemagnetic fields to confine atoms tightly. Detrimental interactions with the surface
were avoided by using an extremely thin film with a large magnetic remnant. BECs
of up to 5×104 atoms were produced in cigar shaped traps ∼40 µm above the surface,
and trap frequencies up to 5 kHz were measured. After evaporation, condensed clouds
moved closer to the surface to probe imperfections in the magnetic potential, revealing
defects at distances closer than 35 µm. Finally, BECs were dropped from a height of
350 µm in an attempt to achieve specular reflection, but a large amount of dispersion
was observed.
Finally, BECs were loaded into a three-dimensional optical lattice, and a quantum
phase transition from a superfluid to a Mott insulator was observed. Using microwave
spectroscopy, the density dependent “clock shift” was was found to depend on theoccupation number of the wells. The singly occupied lattice sites were then investigated as an atomic clock system with no density shift. Linewidths as small as 1 Hz
FWHM out of 6.8 GHz are comparable to current atomic frequency standards.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacAurthur Professor of Physics
Thesis Supervisor: David E. Pritchard
Title: Cecil and Ida Green Professor of Physics

To my wife Karin
Without whom this work would be empty.

Acknowledgments
Naivete and optimism.
Those probably best described my mental state as I walked into the empty lab on
my first day. When I joined the Ketterle group, I was assigned to the still imaginary
Rubidium lab. We had some temporary space to start work while our destination lab
was being renovated, but no optics table, no racks, no drawers, nothing. The task
before us was daunting, but eventually came success.
I started in the lab with fellow graduate student Erik Streed, and together webuilt the bulk of the machine. I’d like to thank Erik for his keen wit, awesome hat,and being a friend as well as a co-worker. We were soon joined by postdoc Yoshio
Torii, whose delicate command of the optics table impresses me to this day. Thencame Dominik Schneble. Fresh from his own Ph.D., he led the lab out of the darkages of construction into the first breaths of the golden age of experiment. I’d like tothank Dominik for his leadership, his science, but most of all his insightful advice.
Soon after Dominik came another new graduate student, Gretchen Campbell.
Quiet as a churchmouse, and humble as a nun, a few years in the Rubidium labchanged her. Now she breathes a fire of science, and everything she touches turns
to gold. Gold or a Science paper. Kind of a science Midas, except probably nicer. I
really can’t thank her enough for all her help, motivation, help, coffee breaks, help,
procrastination, and smart discussions over the years, and I’m sure that she will meet
unparalleled success in her academic career.
Jongchul Mun also deserves his thanks. Blessed with Yoshio’s skill at the optics
table, and an incredible deal of patience, I am sure that he will blaze new trails withthe lab. Patrick Medley is the youngest member in the lab, but his day of leadershipwill soon be upon him. I am certain that his enthusiasm and intelligence will bringhim much success, and I thank him for his efforts when we worked as a team. As
a postdoc, Aaron Leanhardt deserves special recognition. I learned a tremendous
amount about science, patience and experimental stamina from him, and for this hehas my thanks. I would also like to thank Luis Marcassa for all his help in the time

he has been here on his sabbatical, doing all the mundane and dirty things that a
professor thinks he will never have to do again.
The Rubidium lab is certainly not the only lab in the Ketterle group, and I have
many Sodium friends who also deserve thanks, even though they are forced to workwith a lesser element. Tom Pasquini, as of right now a new father of twins, deserves
my thanks for getting all of us a raise, giving incredibly useful feedback when reading
papers, being an awesome guy, and being a friend. I hope that he will be as patienta father as you are a scientist, and that you use your powers for good, as well as
for awesome. Dan Miller has a special place in my heart as a gracious host for some“memorable” parties, as well as keeping up the general lab morale. I am sure that theJohn Harvard statue will be relieved when Dan graduates. Jamil Abo-Shaeer left thelab quite a while ago, but his presence is still felt, and missed. He set a standard forshenanigans, tomfoolery and the fine french art of grabáss that will only be imitatedfor years to come. He was, and still is, my friend.
I would also like to thank Min Xiao and Liesl Folks at Hitachi Global Storage
Technologies. They provided invaluable guidance during my pursuit of the hard disk
platter project, and Min spent many hours preparing disks for me to put in vacuum.
Her willingness to donate her time gave the project life, and I very much appreciate
it.
I’d like to thank Jit-Kee Chin and Yingmei Liu for their patience as I repeatedly
scoured their lab for various pieces of equipment I claimed they had stolen, only to
discover that I had only hidden it from myself. Speaking of stealing, my thanks also
goes to Widagdo Setiawan for his secret stash of RF components and electronics.
Professor Martin Zwierlein, a character if there ever was one, has my thanks for hisbrilliant scientific discussions and enthusiastic renditions of Beatles songs. He also
has my thanks in advance in case I ever need a job (blackmail). My thanks goes toAndre Schirotzek for his inappropriate German comments and his chiropractic skills,
and to Christian Schunck for being both hilarious and a good travel partner. Yong-Il
Shin, Caleb Christensen, Peter Zarth, Gyu-Boong Jo, Ye-ryoung Lee, and SebastianWill, thanks for your help, discussions, lunch money, and laughs. It is people like youwho make stuff like this worth it.
Speaking of worth it, I don’t really know how to thank my wife Karin enough.
How she ever put up with my bizarre schedule, long hours, and broken experiment badmoods, I’ll never know. Hopefully, any future problems will seem small in comparison.
Thank you.
Finally, I would like to thank Dave Pritchard and Wolfgang Ketterle. From inviting me to join their group in the beginning, to fatherly career advice at the end, I
could not have asked for better advisors.

8Contents
1 Introduction

17

1.1

In the Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.2

Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . .

17

1.2.1

Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . .

18

1.2.2

Experimental Realization . . . . . . . . . . . . . . . . . . . . .

19

1.3

Lower Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . .

19

1.4

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2 Bose-Einstein Condensate Production

21

2.1

The Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.2

The Zeeman Slower . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.2.1

Slowing Theory . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.2.2

Design Considerations . . . . . . . . . . . . . . . . . . . . . .

24

2.2.3

The Slower Design Program . . . . . . . . . . . . . . . . . . .

25

2.2.4

Construction . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.2.5

Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.2.6

The Field Subtraction Slower . . . . . . . . . . . . . . . . . .

27

The Main Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.3.1

The Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . .

31

2.3.2

The Magnetic Trap . . . . . . . . . . . . . . . . . . . . . . . .

31

2.3.3

Construction . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.4.1

34

2.3

2.4

GPIB Control . . . . . . . . . . . . . . . . . . . . . . . . . . .
9

2.5

2.6

2.7

2.8

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.5.1

Time-of-Flight . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.5.2

Light Creation . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.6.1

Optical Potentials . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.6.2

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . .

37

2.6.3

Motion Algorithm . . . . . . . . . . . . . . . . . . . . . . . . .

37

The Science Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.7.1

Vacuum Preparation . . . . . . . . . . . . . . . . . . . . . . .

39

2.7.2

Technical Drawings . . . . . . . . . . . . . . . . . . . . . . . .

39

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3 RF Dressing Of a Magnetic Potential
3.1

3.2

3.3

3.4

3.5

3.6

41

The Dressed Zeeman Trap Overview . . . . . . . . . . . . . . . . . .

41

3.1.1

Adiabatic Potentials . . . . . . . . . . . . . . . . . . . . . . .

42

Dressed Potential Calculations . . . . . . . . . . . . . . . . . . . . . .

43

3.2.1

Dressed States . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.2.2

Limit Of Tight Trapping . . . . . . . . . . . . . . . . . . . . .

46

Two Dimensional Physics . . . . . . . . . . . . . . . . . . . . . . . .

47

3.3.1

The Capabilities of the Our Apparatus . . . . . . . . . . . . .

48

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.4.1

BEC Creation and Transfer . . . . . . . . . . . . . . . . . . .

49

3.4.2

RF Setup and Control . . . . . . . . . . . . . . . . . . . . . .

50

3.4.3

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.5.1

BEC Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.5.2

Trap Frequencies . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.5.3

Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Experimental Caveats . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.6.1

54

RF Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.7

3.6.2

Analog Control . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.6.3

Digital Frequency Synthesis . . . . . . . . . . . . . . . . . . .

55

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4 Atom Trapping With a Thin Magnetic film
4.1

4.2

4.3

4.4

4.5

4.6

4.7

57

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.1.2

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Surface Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.2.1

Sinusoidal Pattern . . . . . . . . . . . . . . . . . . . . . . . .

58

4.2.2

Two Dimensional Trap . . . . . . . . . . . . . . . . . . . . . .

60

4.2.3

One Dimensional Trap . . . . . . . . . . . . . . . . . . . . . .

60

Magnetic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.3.1

Hard Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.3.2

The “Out-Of-Plane” Prototype . . . . . . . . . . . . . . . . .

63

4.3.3

Vacuum Compatibility . . . . . . . . . . . . . . . . . . . . . .

63

4.3.4

Disk Properties . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.4.1

Atom Delivery to the Science Chamber . . . . . . . . . . . . .

64

4.4.2

Z-Wire Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

4.4.3

Surface Loading . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.4.4

Surface Imaging . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Surface Trap Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.5.1

Trap Frequencies . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.5.2

Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.5.3

Magnetic Potential Imperfections . . . . . . . . . . . . . . . .

71

Magnetic Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.6.1

. . . . . . . . . . . . . . . .

74

Hard Disk Platter Conclusion . . . . . . . . . . . . . . . . . . . . . .

75

4.7.1

75

Analysis of Dispersive Reflection

Special Thanks . . . . . . . . . . . . . . . . . . . . . . . . . .

11

5 The Mott Insulator Atomic Clock5.1

77

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

5.1.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Atomic Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

5.2.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.2.2

This Experiment . . . . . . . . . . . . . . . . . . . . . . . . .

78

Lattice Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

5.3.1

The Bose-Hubbard Hamiltonian . . . . . . . . . . . . . . . . .

79

5.3.2

The Mott Insulator . . . . . . . . . . . . . . . . . . . . . . . .

80

5.3.3

Density Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

82

5.4.1

BEC Creation . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

5.4.2

The Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . .

82

5.4.3

MI State Creation. . . . . . . . . . . . . . . . . . . . . . . .

83

5.4.4

Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.5.1

Pulse Bandwidth Measurement . . . . . . . . . . . . . . . . .

86

5.5.2

AC Stark Shift Cancellation . . . . . . . . . . . . . . . . . . .

87

5.5.3

Absolute Frequency Measurement . . . . . . . . . . . . . . . .

89

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.6.1

Pulse Bandwidth Limit . . . . . . . . . . . . . . . . . . . . . .

89

5.6.2

Inhomogeneous Magnetic Field Broadening . . . . . . . . . . .

91

5.6.3

Differential ac Stark Shift . . . . . . . . . . . . . . . . . . . .

92

5.6.4

Particle-Hole Pair Broadening . . . . . . . . . . . . . . . . . .

92

5.6.5

Induced Dipole-Dipole Interactions . . . . . . . . . . . . . . .

93

5.6.6

Microwave Frequency Source . . . . . . . . . . . . . . . . . . .

93

5.7

Two Photon Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5.8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5.2

5.3

5.4

5.5

5.6

6 Conclusion

95

12

A Zeeman Slower Program

97

B The Onset of Matter-Wave Amplification in a Superradiant BoseEinstein Condensate

104

C Atom trapping with a magnetic thin film

109

D A Mott insulator atomic clock

115

E Science chamber schematics

121

E.1 Hard Disk Platter Science Chamber . . . . . . . . . . . . . . . . . . . 121
E.2 Optical Lattice Science Chamber . . . . . . . . . . . . . . . . . . . . 122
E.2.1 Lattices Science Chamber Top View . . . . . . . . . . . . . . . 123
E.2.2 Lattices Science Chamber Side View . . . . . . . . . . . . . . 124
Bibliography

125

13

List of Figures
2-1 Machine overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2-2 Zeeman slower winding diagram . . . . . . . . . . . . . . . . . . . . .

26

2-3 Zeeman slower field profile . . . . . . . . . . . . . . . . . . . . . . . .

28

2-4 Atomic velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2-5 Zeeman slower error analysis . . . . . . . . . . . . . . . . . . . . . . .

30

2-6 Photo of the magnetic trap . . . . . . . . . . . . . . . . . . . . . . . .

33

2-7 Optical transport cartoon . . . . . . . . . . . . . . . . . . . . . . . .

38

3-1 Cartoon of the dressed Zeeman trap . . . . . . . . . . . . . . . . . . .

42

3-2 Zeeman energy of F = 1 state . . . . . . . . . . . . . . . . . . . . . .

44

3-3 Graph of the dressed RF potential . . . . . . . . . . . . . . . . . . . .

45

3-4 Trap frequency and lifetime vs Ω . . . . . . . . . . . . . . . . . . . .

47

3-5 Maximum ratio of energy level spacing to chemical potential versus
magnetic field gradient . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3-6 Condensate fraction vs RF power . . . . . . . . . . . . . . . . . . . .

49

3-7 RF component setup . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3-8 Large amplitude oscillations . . . . . . . . . . . . . . . . . . . . . . .

52

3-9 Lifetime of BEC in dressed trap . . . . . . . . . . . . . . . . . . . . .

53

3-10 Images of BEC in the dressed trap . . . . . . . . . . . . . . . . . . .

54

4-1 Geometry of the magnetic trap formed by a magnetic film. . . . . . .

61

4-2 Magnetic force microscopy image of a hard disk platter. . . . . . . . .

65

4-3 Photograph of the Z-wire trap . . . . . . . . . . . . . . . . . . . . . .

66

4-4 Evaporation to BEC on the disk . . . . . . . . . . . . . . . . . . . . .

68

14

4-5 Hard disk platter side imaging . . . . . . . . . . . . . . . . . . . . . .

68

4-6 Hard disk platter top imaging . . . . . . . . . . . . . . . . . . . . . .

69

4-7 Absorption image of the BEC in trap near the surface. . . . . . . . .

70

4-8 Radial trap frequency vs applied radial magnetic field. . . . . . . . .

71

4-9 Breakup of the atomic cloud as the atoms approached the surface. . .

72

4-10 Width and position of the bouncing BEC . . . . . . . . . . . . . . . .

74

5-1 Theoretical “wedding cake” density profile . . . . . . . . . . . . . . .

81

5-2 Setup of the optical lattice . . . . . . . . . . . . . . . . . . . . . . . .

84

5-3 Diagram of the lattice depth vs time . . . . . . . . . . . . . . . . . .

85

5-4 Peak location vs. bias field . . . . . . . . . . . . . . . . . . . . . . . .

86

5-5 Microwave scan of atoms in the lattice . . . . . . . . . . . . . . . . .

87

5-6 Width of the n = 1 peak vs. two photon pulse length . . . . . . . . .

88

5-7 Center frequency of the n = 1 peak vs. lattice depth . . . . . . . . . .

90

5-8 Peak width vs. bias field . . . . . . . . . . . . . . . . . . . . . . . . .

91

15

16

Chapter 1
Introduction
1.1

In the Beginning

In the beginning, there was an empty room. Construction of the 8 7Rb Bose-Einsteincondensate (BEC) machine started in the Fall of 2000, and the first BECs were
produced a little more than a year later.

1.2

Bose-Einstein Condensation

It is somewhat of an atomic physics thesis tradition to give a brief overview of BEC
in the preferred style of the writer, and I won’t break this tradition. Starting fromthe de Broglie wavelength
λT =

v
u
u 2πh̄2
t

mkB T

(1.1)

where m is the mass of one particle, kB is Boltzmann’s constant, and T is the temperature, matter-wave overlap occurs in a regime where λT becomes comparable thespacing between atoms, ∆x = n−1/3 . Thus the phase transition to BEC occurs at(or near) a phase space density of λ3T n ≈ 1. Conveniently, this simple description
is very close to the actual phase space density required (λT n−1/3 ≈ 2.612) to create
a quantum saturated vapor where all thermally accessible states are occupied. The

17

critical density required can be expressed asN
mkB T= ζ(3/2)
V
2πh̄2

!3/2

(1.2)

where ζ(3/2) ≈ 2.612 is the Riemann Zeta function. In the case of harmonically confined atoms with trap frequencies (ωx , ωy , ωz ), the excitation energies are (nx , ny , nz ) =(nx + 1/2)h̄ωx + (ny + 1/2)h̄ωy + (nz + 1/2)h̄ωz , resulting in a critical temperature
h̄ω̄
Tc =
kB

N
ζ(3)

!1/3

(1.3)

where ω̄ = (ωx ωy ωz )1/3 and ζ(3) ≈ 1.202. For a typical experimental value of N = 106
and ω̄ = 50 Hz, we must achieve Tc < 225 nK and a density of n > 4 × 1013 atoms/ cm3 . Most BEC experiments work in parameter regimes within two orders ofmagnitude of these values.

1.2.1

Chemical Potential

Once the condensate has been formed, all of the atoms are described by a single
wavefunction. Adding a term to the Schrödinger equation to account for atom-atominteractions yields the Gross-Pitaevskii equation,h̄2 2
4πh̄2 a−5 +V (~r) +|ψ(~r)|2 ψ(~r) = µψ(~r),
2m
m
!

(1.4)

where V (~r) is the magnitude of the trapping potential, a is the s-wave scattering
length between two atoms, and µ is the chemical potential. The regime where theatom-atom interaction energy is large compared to the kinetic energy is called theThomas-Fermi limit, and we may rewrite the wavefunction as
4πh̄2 a|ψ(~r)|2 ψ(~r) = µψ(~r).
V (~r) +
m
!

18

(1.5)

This equation may be solved in terms of the local density|ψ(~r)|2 = n(~r) =m[µ − V (~r)].
4πh̄2 a(1.6)

One can simply imagine a pool of atoms filling the trapping potential to a constantheight (µ). This chemical potential also sets the time evolution of the wavefunctionas Ψ(~r, t) = ψ(~r)e−iµt/h̄ ,The chemical potential µ is a measure of interaction between the atoms whichis constant throughout the cloud, and is one of the properties of the BEC most
relevant to work in this thesis. The chemical potential may be calculated quickly
from µ = 4πh̄2 an0 /m, where n0 is the peak density of the atomic cloud. The ratio
between µ and any trapping potentials or perturbations greatly affects the behaviorof the atomic system.

1.2.2

Experimental Realization

Since its first observation in dilute atomic gasses, BEC has been achieved with theentire first column of the periodic table (except for francium, in order of atomic mass
[21, 4, 12, 48, 1, 78]), metastable 4 He [53], 52 Cr [25], and 174 Yb [74]. In fact, what
was once the goal is now only a tool; BECs are used to model solid state systems,
probe interactions with surfaces, and measure properties of the atoms themselves,
sometimes with great precision.

1.3

Lower Dimensional Systems

One area of great interest in ultracold gas research is systems with reduced dimensionality. While it is easy to make a BEC1 in three dimensions, it is quite difficult tosolve many of the relevant equations of motion in three dimensions. Contrapositively,
many two and one-dimensional systems are theoretically rich, but extremely difficult
to achieve experimentally. Sometimes reducing the dimensionality of a system can1

Correction: Easy to make the second BEC.

19

make solutions to the wavefunction simpler, such as the case in two dimensions with
the Kosterlitz-Thouless transition [39] and in one dimension with the Tonks gas [50].
One characteristic of both of these phenomena is the increase of phase fluctuations
which may destroy the coherence across the atomic cloud. Studies in these regimes
thus may not deal with BEC at all in the end, but it’s a good place to start. The
three main experiments described in this thesis follow a general theme of shaping theconfining potential in such a way to remove degrees of freedom from the system.

1.4

OutlineThe outline of this thesis follows an order of reduction of atomic motion. First,
I discuss an experiment using adiabatic potentials to trap an atomic cloud in two
dimensions called the “Dressed Zeeman Trap.” Second, I describe an experimentusing magnetic thin films to trap atoms in one dimension called the “Hard Disk Platter
Trap.” Lastly, I detail work done in an optical lattice, where removing all motional
degrees of freedom eliminated detrimental effects of inter-atomic interactions, creating
the “Mott Insulator Atomic Clock”.

20Chapter 2
Bose-Einstein Condensate
Production
A detailed description of the construction of the 87 Rb BEC machine can be foundin Erik Streed’s thesis [70]. The art of creating BEC is well documented in places
too numerous to cite, so the aim of this chapter is to give a quick summary off theapparatus for reference in the rest of this thesis. I will discuss the creation of BEC
as well as delivery to the science chamber, glossing over many of the technical details
which can be found in Erik’s exhaustive work. A cartoon of the machine is shown in
Fig. 2-1.
A hot beam of rubidium atoms streams out of an effusive oven. The atoms are
first cooled in a Zeeman slower, then captured and further cooled in a magneto-opticaltrap (MOT). The MOT beams are turned off, and the cloud is then transferred toa Ioffe-Pritchard magnetic trap. Radiofrequency (RF) evaporation then cools theatomic cloud to BEC.

2.1

The Oven

A flexible bellows containing up to 5 g of rubidium metal is heated to ∼110 C undervacuum to produce the alkali vapor. Hot rubidium atoms leave the oven nozzle
at ∼300 m/s in a stream directed at the main chamber. Excess rubidium flux is21

Figure 2-1: Machine overview. A) Oven. B) Beam shutter. C) Zeeman slower.
D) Main chamber. E) Science chamber. The left / right axis is the “tweezer” or“transport” axis, and in and out of the page is referred to as the “pennywindow”
axis.

22

captured with a “cold cup” that encompasses 2π sr of the oven nozzle effusive cone.
Maintaining a cold cup temperature of -20 C prevents rubidium contamination of therest of the oven chamber, and a 5 mm hole in the center of the cup allows a collimatedbeam of approximately 1011 atoms/s to continue downstream. Each 5 g ampoule ofrubidium lasts 6-8 months (depending on the duty cycle), and adding more takes
about a day, but can take significantly longer depending on experimental skill. The
cold cup must be removed and carefully cleaned after every second rubidium change
to prevent clogging.
While the melting temperature of rubidium metal is listed as 39 C [69], in normal
laboratory conditions rubidium is a flaming liquid. Moisture in the air reacts withthe metal in an exothermic process melting it while releasing flammable hydrogen. If
rubidium metal in an enclosed space comes into contact with water, the result will bea violent explosion. In order to dispose of rubidium safely, it must never be put into a
container (such as a beaker) until completely neutralized with methanol. I write thisin the hope that my words may save some future rubidium lab member from losing
facial hair, or worse.

2.2

The Zeeman Slower

Atoms leaving the oven scatter photons from a counter-propagating laser beam, andare thus slowed as they make their way to the main chamber. This section discusses
first the theory behind Zeeman slowing, then practical design constraints for ourmachine, then the design, winding and performance of the slower.

2.2.1

Slowing Theory

Each photon delivers a momentum kick p = h̄k to the atom, so resonant scattering
at a rate of Γ/2 will impart a maximum acceleration of a = Γh̄k/2m to the atom,
where k is the wavenumber of the photon, and the excited state lifetime τ = 1/Γ,
assuming a fully saturated transition. As the velocity of the atom changes, it will
be doppler shifted out of resonance with the slowing laser beam. In fact, in only a23

few hundred photon scatterings, the atom will be completely out of resonance withthe slowing laser. While it is possible to change (chirp) the laser frequency to stay inresonance with small groups of atoms as they are slowed [18], a more efficient solution
is to use magnetic fields to tune the atomic levels back into resonance via the Zeeman
shift [57].
The magnitude of the magnetic field as a function of position then can be expressedasB(x) = B0 −

1
µB λ

s

v02 −

f Γh̄kxm(2.1)

Where B0 is a bias magnetic field, µB is the Bohr magneton, λ is the wavelength of
the slowing laser, v0 is the velocity of the atom as it exits the oven, m is the atomic
mass, and f is a safety factor to account for finite laser power and magnetic field
imperfections. If f is set too close to 1.0, small errors in design or construction of the
Zeeman slower can result in slowed atom flux reduction by an order of magnitude or
more; our design used f = 0.6.

2.2.2

Design Considerations

Given the design parameters of our oven, with v0 = 350 m/s, and the properties of87

Rb, with λ = 780 nm, and Γ = 2π × 6 MHz for the linewidth, the resulting Zeeman

slower is ∼1 m long with a magnetic field change of ∆B = 320 Gauss. We chose alarge B0 ≈ 500 Gauss to maintain a large Zeeman splitting, preserve the polarization,and prevent unwanted hyperfine state mixing. An additional laser beam resonantwith the F = 1 → F 0 = 1 transition repumps atoms into the F = 2 state. It is also
desirable to choose a sufficiently large bias field to prevent absorption of the slowing
beam by atoms that have already exited the Zeeman slower.
The slowing laser beam is σ − polarized and resonant on the 52 S1/2 |2, −2i →52 P3/2 |3, −3i transition (|F, mF i). This polarization on the cycling transition keeps
atoms in a “stretched” state where decay into off-resonant states is unlikely. During
the process of slowing, each atom will scatter N ≈ 60,000 photons into a dipole
pattern symmetric about the axis of travel. This results in a mean random, transverse24

qvelocity of about h̄k N/3/mRb ≈ 1 m/s as atoms exit the slower. If the exit velocity
is 20 m/s, this corresponds to a 6 degree cone within which 2/3 of the atoms will fall.
The exit of the Zeeman slower must be close enough to the MOT so that most of theslowed atoms will be captured.
The design of the magnetic solenoids also presents a challenge. The high fieldsection of the slower must be near the exit, but the magnetic field must drop off rapidly
so as to not significantly effect the MOT. This demands coils with high current, but
small diameter. However, the low field section must maintain a good field shapeat very small magnetic fields, demanding low current and large diameter. Finally,cooling water must be able to enter and exit the solenoid structure at finite intervals,
perhaps necessitating multiple solenoid segments.
The ideal solution would be a series of single loops (hundreds of them), eachprovided the ideal current by its own power supply. However, this solution grossly
violates funding constraints and invites mockery. The design finally chosen consistedof three solenoid segments, each with up to 10 concentric solenoids, and one power
supply per segment.

2.2.3

The Slower Design Program

Traditionally, Zeeman slowers have been designed by hand. Each solenoid section ismodeled in a computer program, such as Mathematica, and the parameters of eachare modified by the experimentalist in an effort to converge to a viable solution. My
solution was to design a computer program which calculated the parameters governing
all possible (∼30) solenoids. The fitting method was governed by only a few handcontrolled parameters, greatly reducing the parameter space and the final design time.
A copy of this program has been provided in Appendix A. It should compile in any
ANSI C environment, and can be copied and pasted from the electronic version ofthis thesis.
The output of the program is a simple drawing which represents a cross sectionof the idea solenoid structure. This reduces guesswork and errors during the winding

25Section 1, 5.00 AmpsOOOOOOOOOOOO OOOOOOOOOOO OOOOOOO OOOOOOO OOOOOOO OOOOOOO OOOOO OOOO OOOO OOO OOOO OOO OO OOO OO OO O O O O O O
OOOOOOOOOOOO OOOOOOOOO OOOOOOO OOOOO OOOOO OOOOO OOOO OOOO OOO OOO OO OO OOO O OO O O O O O O
OOOOOOOOOOOO OOOOOOO OOOOO OOOO OOOO OOO OOO OOO OO OO OO OO O O O O O O
OOOOOOOOOO OOOOOOO OOO OOO OOO OO OO O OO O O O O O
OOOOOOOOOO OOO OOO O O OO O O
OOOOOSection 2, 10.0 AmpsOOOOOOOOOOOO OOOOOOOOOOO OOOOOOOOO OOOOOOO OOOOOOO OOOOOOO OOOOOO OOOOOO OOOOOOOOOOOOOOOOO OOOOOOOOO OOOOOOO OOOOOOO OOOOO OOOOO OOOOO OOOOO OOOO OOOO OOO OOOOOOOOOOOOOO OOOOOOO OOOOO OOOOO OOOOO OOO OOOO OOO OOOO OO OOO OO OO OO OO O O
OOOOOOOOOO OOOOOOO OOO OOO OOO OOO OOO OO OO O OO OO O O O O O O O O
OOOOOOOOOO OOO OOO O OO OO O O O O O O
OOOOOO

Section 3, 30.0 AmpsOOOOOOOOOO OOOOOO OOOOOO OOOOO OOOOO OOO OOOOO OOO OOOO OOOO OOO O
OOOOOOOOO OOOOO OOOO OOOO OOO OOO OO OOO OOO OO OO OO OO O OO O O
OOOOOOOO OOO OOO OO OO OO O OO O O O O O O O O O OOOOOOOO OO O O O O

Figure 2-2: Zeeman slower winding diagram. This figure shows the output of the Zeeman slower program, which represents cross sections of the three solenoid segments.
The circles represent cross sections of individual wires. Spaces in between wires areused to guide the winding process to lower the current density in some sections.
process, and provides a much finer control of the magnetic field shape with variable
loop spacing - effectively each solenoid has a current density which may vary with theaxial position.

2.2.4

Construction

The solenoids were wound with square, hollow core, glass insulated copper tubing,and glued in place with high temperature epoxy [70]. The wire had an overall crosssection of 3.2 mm, but including insulation when wound provided a coil spacing of
3.5 mm. The hollow core was 1.6 mm square, and high pressure water at 220 psi
provided cooling. The water flow Q through a straight tube is given byq

Q = α P/L

(2.2)

where P is the pressure, L is the length of the tube, and α = 124 m2 /kg1/2 wasexperimentally measured. Assuming that the water temperature must increase no
more than 25 C before it exits the tube, the minimum water flow through an energizedsolenoid must be
Qmin = βI 2 L

26

(2.3)

where I is the electrical current in the wire and β = 1.5 × 10−3 . Thus the maximum
lengths of any unbroken wire in the Zeeman slower is then given byα
Lmax = 
β

!2

1/3

P
I4

(2.4)

Breaks in the winding pattern were designed to not exceed this value.
Each of the three coils was wound around a 1 inch diameter brass tube, thenslid over a 0.75 inch diameter vacuum tube. This vacuum tube attaches to the main
system at each end with 1.3-inch diameter flanges. This would ordinarily prevent
sliding the slower coils over the vacuum tube, but one of the flanges was rotatable,
and the retaining ring was cut in half. The retaining ring was then installed aftersolenoid assembly.

2.2.5

Performance

The magnetic field profile of the Zeeman slower is shown in Fig. 2-3. The desired fieldcalculated from equation 2.1, the modeled solenoid field from the computer design
program, and the measured field are shown as a function of distance from the MOT.
The functionality of the slower was measured with an absorption spectrum as shownin Fig. 2-4, indicating that there was no significant loss of any velocity groups below
the calculated capture velocity (350 m/s). Limited only by the oven efficiency andlaser power, we have achieved slowed atom fluxes as high as 1011 atoms / s.

2.2.6

The Field Subtraction Slower

When designing an increasing field slower with solenoids, it is difficult to match thefield shapes due to the sign of the curvature. One possible solution is to invert thiscurvature with a “field subtraction” slower. A single, inner solenoid with a small
diameter provides a large bias field, and additional concentric solenoids of larger
radii with current flow in the opposite direction remove portions of the bias field tocreate the correct field shape. The result is a single solenoid cone that appears to be

27

300

Magnetic Field (Gauss)

250

Desired B field
Modeled B field
Measured B field

200

150

100

50

0
0

200

400
600
800
Distance From MOT (mm)

1000

1200

Figure 2-3: Zeeman slower field profile. The dotted line is the theoretically desired
field, the dashed line is the computer generated model field, and the solid line is themeasured field after construction. The dips in the field profile are a result of finite
solenoid spacing for wire insertion, and were largely corrected with additional small
solenoids. This profile does not include an additional ∼200 G bias field.

28

2.0

Absorption (%)

1.5

Thermal beam
Slowed beam
1.0

0.5

0.0
0

100

200

300
Velocity (m/s)

400

500

600

Figure 2-4: Atomic velocity profile. The absorption of the atomic beam at a 45 degree
angle is shown as a function of the velocity. The red, dotted line shoes the velocity
distribution of the atoms as they leave the oven, and the solid line shows a peak inthe velocity distribution from atoms cooled in the Zeeman slower.

29

4

Field Error (Gauss)

Conventional slower error
Subtraction slower error
2

0

-2

-4

200

400

600
Distance from MOT (mm)

800

1000

Figure 2-5: Zeeman slower error analysis. The difference between the modeled field
and the desired field is plotted as a function of distance from the MOT, showing thatdeviations from ideal are small compared to the linewidth of the slowing transition(Γ = 4.2 Gauss). The field subtraction slower design (dotted line) theoreticallyreduces field deviations by an average of ∼2.
backwards. The narrowest section is closest to the MOT, but produces the largest
field.
While not used on this machine, this design method provides several advantages.
First, at the exit off the slower, the solenoid is as small as possible, greatly reducing
stray fields which affect the MOT. Second, the use of only a single section removes
discontinuities from the need to break out wires for cooling. Finally, the minimizationof magnetic field errors ensures more robust performance. The main disadvantage isthat this design method uses much more (∼4×) power, and thus may be difficult
to cool effectively. The calculated field error for the two different design methods
is shown in Fig. 2-5, indicating a 2× reduction in the deviation from ideal. The
result is that the Zeeman slower may be designed with a safety factor f closer to 1.0,
shortening the physical length and increasing the atom flux.

30

2.3

The Main Chamber

The main trapping chamber of the rubidium machine is based on the design for a
sodium BEC machine, originally designed by Todd Gustavson. I modified Todd’s
design to include a few more features, including a never-used cold plate for catchingunslowed rubidium atoms, and both designs are discussed thoroughly in other publications [41, 71, 70]. The design uses recessed or “bucket” flanges which place themagnetic trap as close to the atoms as possible. The magnetic trap may then be
replaced without compromising the vacuum. Six optical axes intersect in the centerof the chamber, and the complexity of the vacuum flange geometry prevents ideal
optical access for some experiments.

2.3.1

The Magneto-Optical Trap

Upon arrival in the main chamber, atoms are trapped in a potential created by the
combination of radiation pressure from three laser beams and the interaction of theatoms’ magnetic dipole and a magnetic field minimum. Two electromagnets in ananti-Helmholts configuration create the magnetic field minimum, and laser beams
counter-propagating on three orthogonal axes cool the resulting atomic cloud. The
MOT operates with a field gradient of 16.5 G/cm, and a peak laser intensity of
5 mW/cm2 at a detuning of -18 MHz from the 52 S1/2 |2, 2i → 52 P3/2 |3, 3i transition.
When operating the oven at 110 C, it takes approximately 10 s to load the MOT.
Once full, the MOT is compressed by increasing the current in a coils by a factor offour in 200 ms, then the atoms are released by switching off the current. After releasefrom the MOT, the laser detuning is increased and the power decreased to further
cool the atoms for 7 ms in an optical molasses. A 3 ms pulse of “depumping” lighttransfers the atoms from the F = 2, mF = −2 state to the F = 1, mF = −1 state.

2.3.2

The Magnetic Trap

Immediately after depumping, the Ioffe-Pritchard magnetic trap is switched on. This
trap is also described in [41, 71, 70], and produces a magnetic field theoretically31described by








 x 

 0 
00 




B





 + B 0  −y  +B = B0 


 0 
2 






1

0





−xz
−yz
z 2 − 12 (x2 + y 2 )








(2.5)

In the axial (pennywindow) direction, trapping is provided by the field curvature term,
and the radial trapping is analogous to a quadrupole field zero with an additional bias
field. The B0 term prevents Majorana spin flips by removing the magnetic field zero.

2.3.3

Construction

The magnetic trap was wound with the same wire as the Zeeman slower, and castwith the same high temperature epoxy. A photograph of the magnetic trap just prior
to installation is shown in Fig. 2-6. Water is forced through the coils at 230 PSI toprovide up to 10 kW of cooling when up to 500 Amps are driven through the coils.
Flow switches (McMillan model 501) monitor the cooling water flow through every
coil, and in the rare event that they are all simultaneously functional they may belinked to the electrical interlock safety system.

2.4

Evaporation

After transfer to the magnetic trap, RF radiation is introduced which drives a transition from the trapped |1, −1i state to the untrapped |1, 0i and |1, 1i states. Sweeping
the frequency from high (f  µB B0 ) to low (f ≈ µB B0 ) evaporates the hottest atoms
from the trap in a process that cools the remaining atoms. Evaporative cooling iswell documented in the literature [29, 13, 56], and cooling the atomic cloud to BEC
in our magnetic trap takes about one minute.

32

Figure 2-6: Photo of the magnetic trap. Just prior to installation on the machine,
the magnetic trap coils are covered in high temperature epoxy and further insulatedwith Kapton film. These two protections have prevented any electrical shorts in five
years of operation.

33

2.4.1

GPIB Control

The RF signal is generated by an Agilent 33250a synthesizer, which is computer
controlled over a general purpose instruments bus (GPIB) link. The frequency canbe updated every 30 ms, or the frequency and power can be updated together every
90 ms. This method provides an advantage over voltage controlled oscillators (VCOs)
because it combines a large sweep range (80 MHz) with extremely precise frequency
control (∼ 1 mHz). However, some experimental procedures, such as rapid sweeps,
cannot be generated by this method and additional synthesizers must be used.

2.5

Imaging

All experiments in this thesis detected the atom clouds with resonant absorption
imaging. The 87 Rb D2 line has wavelength λ = 780.241 nm and a linewidth of
2π ×6.07 MHz. Atomic clouds in the F = 2 state were imaged on the F = 2 → F 0 = 3
transition, and clouds in the F = 1 state were first optically pumped into the F = 2
state along a different axis before and during imaging. Mixtures of F = 1 and F = 2
could be separately resolved by first imaging the F = 2 atoms, then adding therepumping light to image the F = 1 atoms. The duration of the imaging light pulse
was kept short to minimize atomic motion during the image, with typical durationsof 50 - 100 µs.
The imaging light was collected on a Princeton Instruments NTE/CCD-1024-EB
camera with a square pixel size of 13 µm. Three frames were taken for each image;first a probe with atoms (PWA) frame to detect the atoms, then a probe without
atoms (PWOA) frame to determine the reference laser intensity, then a dark field(DF) frame without laser illumination to determine the background noise. The final
absorption image (ABS) was then determined by
ABS =PWA − DF.
PWOA − DF(2.6)

Pixels that image no atoms have a value of around 1.0, corresponding to no absorption,34

and pixels that image an optically dense column of atoms have values that may
approach zero. The number of atoms, N , in the absorption image can be calculated
by
N =−

AX
ln ABS
σ0 x,y(2.7)

where A is the area imaged per pixel, and σ0 is the resonant scattering cross section,
but these prefactors are usually omitted when making plots of relative atom number.

2.5.1

Time-of-Flight

While held in the the magnetic trap, the BEC is typically quite small, with a typical
width and length of 10 and 100 µm. Considering that our imaging optics operate atabout f# ≈ 5, features of the BEC are barely resolved when imaged in-trap. The
solution is to suddenly release the BEC from the trap, and the chemical potentialof the cloud causes it to expand ballistically in a time-of-flight (TOF). The result isthat TOF images show the momentum distribution of the condensate after the mean
field energy is converted into kinetic energy. More detailed analysis of atomic cloud
imaging techniques can be found in [37, 47, 68].

2.5.2

Light Creation

Laser light for slowing, cooling, and imaging is produced by diode lasers on a separate optical table with diode lasers and delivered via optical fibers. Fiber coupling
eliminates the need to account for changes in the relative position of the two tables
and reduces maintenance time. Light for cooling and imaging on the F = 2 → F 0 = 3
cycling transition is produced by a grating stabilized diode laser and tapered amplifier(Toptica TA-100), repumping light on the F = 1 → F 0 = 1 transition is created byanother grating stabilized diode laser (Toptica DL-100), and both frequencies of slowing light are generated by injection locked diode lasers. Frequency locking for boththe cycling and repumping light is maintained by polarization sensitive saturated absorption spectroscopy [52, 81]. Laser beam intensities are controlled with microsecond
accuracy using acousto-optic modulators (AOMs). Further details about the optical35

setup can be found in Ref. [70].

2.6Transport

Since the main chamber is crowded with optics, magnets, and various other bulky
and sensitive equipment required to create BEC, there is little room for additional
apparatus or even optical access. In order to more closely interact with the condensate, we transport the atomic cloud into a separate “science chamber” which may bechanged entirely to more efficiently suit the needs of any particular experiment. In
order to deliver the atoms to the science chamber, an optical tweezer is used to trap
and then transport the cloud.

2.6.1

Optical Potentials

Optical forces may be used to confine atoms in situations where magnetic trapping
is unsuitable. Strong field seeking hyperfine states, as well as magnetically sensitive
mixtures of hyperfine states may be trapped in optical potentials. The optical dipole
potential is given by [2, 9]
1
I (~r) (Γ/2)2
Voptical = h̄∆ ln 1 +2Isat ∆2 + (Γ/2)2

!

(2.8)

Where I(~r) and ω are the intensity and frequency of the optical field, ∆ = ω − ω0 is
the detuning from resonance, Isat is the saturation intensity, and Γ is the width of thetransition. Thus atoms may be trapped in the local maxima of red-detuned beams,
and in the minima of blue-detuned beams. Neglecting the counter-rotating term in
the rotating wave approximation, this potential may be closely approximated by
Voptical (~r) =3πc2 Γ
I(~r),
2ω03 ∆

36(2.9)

and the spontaneous photon scattering rate isΓspont (~r) =3πc2 Γ2
I(~r).
2h̄ω03 ∆2(2.10)

Thus by choosing a large enough detuning and providing enough optical power, opticaltrapping can be used to hold the atoms with negligible heating.

2.6.2

Experimental Setup

A cartoon depicting the optics and mechanics used in the transport process is shown in
Fig. 2-7. A collimated beam with 1/e2 = 40 mm radius is focused onto the atoms witha f = 450 mm lens. An air bearing translation stage (Aerotech ABL2075) controls theaxial position of the focussing lens to sub-micron accuracy, and a computer program
controls the path of motion. Up to 800 mW of light was used to trap the atoms inorder to provide the necessary axial trapping for transport in a reasonable (3 s) time.

2.6.3

Motion Algorithm

The motion of the translation stage was controlled by specifying the transport time t,
the distance x, and the maximum acceleration (amax = δx2 /δt2 ). A computer program
then numerically determined a fixed value for the jerk (j = δx3 /δt3 ) which describedthe desired motion. Typical values were t = 2 s and x = 360 mm, stipulatingj = 1430 mm/s3 . The motion was divided into a series of reference points withδt = 4 ms for computer interpolation of the motion.

2.7

The Science Chamber

The science chamber enabled us to perform several different types of experiments
without compromising UHV in the main chamber. Venting the main chamber isconsidered a magnitude 1.0 disaster in our group, and can take many weeks to recoverin a process usually requiring removal of the magnetic trap, MOT optics, and myriad
miscellaneous apparatus, coupled with a high temperature (250 C) bakeout which
37

Transport
(j)

(k)

(l)

(i)

(m)

(c)
(h)

(g) (f)(b) (a)(n)

To laser(e) (d)To stabilization

Figure 2-7: Optical transport cartoon. Laser light with λ = 1064 nm exits the
optical fiber (a), then passes through a polarizing beam splitting cube (b) to clean thepolarization. A glass wedge (c) picks off a small portion of the light for measurement
on a photodiode (d) to stabilize the intensity. Four lenses with focal lengths 35 mm (e),150 mm (f), 35 mm (g), and 150 mm (h) form two telescopes to expand the collimatedbeam to a radius of 1/e2 = 40 mm before reaching the translation stage (i). A leadfilled extension arm and 2 inch diameter lens tube (j) hold a final f = 450 mm lensthat focuses the beam onto the atomic cloud in the main chamber (l). The translation
stage moves the lens holder to the delivery position (m), translating the focus andthe atoms 360 mm into the science chamber (n).

38may cause leaks. The removal and replacement of the science chamber has beenaccomplished several times in my time in the lab, allowing experiments to be started
in a very short time, where in other laboratories such endeavors may require thecomplete construction of a dedicated apparatus.

2.7.1

Vacuum Preparation

Before installation on the machine table, each science chamber is usually “pre-baked”
at a high temperature (again, 250 C) to reach pressures of < 10−9 Torr with a
combination of turbo and ion pumps. After a brief venting with argon or nitrogen,the science chamber can be attached to the main chamber and baked again for a much
shorter time, typically three or four days. Final bakeouts with temperatures as low at
70 C were even achieved, eventually reaching 10−11 Torr with titanium sublimation.

2.7.2

Technical Drawings

Two different science chamber designs are depicted in Appendix E, but the less interesting pump bodies are not shown. One design is for the hard disk platter experiment,
which is discussed in Chapter 4. The other design is for a future experiment with optical lattices. This chamber affords greatly increased optical access to the atoms, andshould provide for much more mechanically stable beam paths. It is likely that thenext work to be done in the rubidium lab after I leave will be done in this chamber,and I entrust it to the care of my labmates.

2.8

Conclusion

Already in its fifth year of operation, the rubidium machine has proven robust andreliable. It has survived explosive venting, magnetic trap partial meltdowns, andother, unspeakable abuses, yet it produces atom clouds with temperatures in the nK
regime with better than 5% number stability and makes measurements more accurate
than a part in 109 . I am sure that the students who follow me will push it to even39

greater achievements.

40

Chapter 3
RF Dressing Of a Magnetic
Potential
This chapter describes the use of an RF field to manipulate the effective trapping
potential in an otherwise static spherical quadrupole magnetic field. The goal was tocreate a very oblate “pancake” BEC with very low trap frequencies in two directions,and a very high trap frequency in the third for studying BEC in 2D. However, calculations showed that the maximum attainable tight trap frequency, limited by Majorana√
flops, increases smaller than B 0 , which limits its usefulness for studying 2D systems. Transfer of BECs into the dressed Zeeman trap with ∼100% efficiency was
accomplished using a specialized arbitrary function generator to provide the cleanest
possible RF signal. While this experiment did not result in a published work, a summary of the results are provided for others who may wish to investigate this in the
future.

3.1

The Dressed Zeeman Trap Overview

First proposed in [83] and demonstrated in [10] and [79], the dressed Zeeman trap, or“eggshell” trap confines atoms to the surface of a sphere centered around a magnetic
quandupole minimum. The frequency of the RF and the magnitude of the magnetic
field gradient, B 0 , determine the radius of the sphere, and the intensity of the RF
41

(a)

r

(b)
(c)

(e)
(d)

Figure 3-1: Cartoon of the dressed Zeeman trap. Magnetic coils (a) produce a gradientof B (b) that normally traps atoms at point (c). An RF field (d) is applied whichdresses the atoms, which are then trapped on a sphere of iso-B (e) of radius r. Gravity
pulls the atoms toward the bottom of the sphere.
radiation determines its thickness. For non-symmetric magnetic fields (such as ourIofffe-Pritchard trap), the RF frequency maps a surface of iso-B which defines where
the atoms are trapped. Gravity causes the atoms to live at the bottom of this potential, so the atoms are weakly but harmonically trapped in the horizontal directions
like a pendulum. Counter-intuitively, reducing the strength of the RF dressing field
increases the transverse (radial) trap frequency, but this has a severe limitation. If
the power is reduced too much, the atoms will quickly undergo Majorana flops at the
avoided crossing and will be lost from the trap.

3.1.1

Adiabatic Potentials

In this experiment, an adiabatic potential is created by the interaction of a twostate atomic system in the a magnetic gradient and a RF field. Diagonalizing the
Hamiltonian of a two level system in the presence of the coupling field results in a42

“dressed state” picture where the two states are subject to an adiabatic potential [75],on which the moving atoms will stay on as long as the coupling field is strong enoughto prevent Landau-Zener transitions. For instance, our RF evaporation creates anadiabatic crossing where the RF field is on resonance with the |1, −1i → |1, 0i and|1, 0i → |1, +1i transitions. For small magnetic fields (< 10 G), these transitions
are degenerate. We then can then map the |1, −1i, |1, 0i, and |1, +1i states onto
the dressed states φ+ , φ0 , and φ− , shown as the top, middle, annd bottom curves inFig. 3-3, respectively.

3.2

Dressed Potential Calculations

The dressed Zeeman trap is derived from first principles and discussed in great detailin [82]. In this section I will present some of the most important results from that workand discuss their relevance to the experimental process. Taking the case of an alkali
atom in the F = 1 hyperfine state, for small magnetic field values the Zeeman energyof the magnetic sublevels is linearly dependent on B. RF radiation is introduced
which resonantly couples the |1, −1i state to the |1, 0i and |1, +1i states [Fig. 3-2].
This field dresses the original atomic states into φ+ and φ− , as shown in figure 3-3,
with the form

U = mgz ± h̄

"

µB

2

q

B02 + (B 0 r)2 − ∆

#1/2
2

+Ω

(3.1)

where z is the vertical component of the distance r from the magnetic field minimum,
g is gravity, µB is the Bohr magneton, B0 is the axial bias field in the Ioffe-Pritchard
trap, ∆ is the RF frequency, and Ω is the strength (Rabi frequency) of the RF dressing
field.

3.2.1

Dressed States

If the RF field is initially turned on with ∆ > µB B0 , the atoms are trapped at thecenter of the lower potential in figure 3-3. If ∆ is then decreased, atoms with sufficient

43

30

mF = -1

Energy (MHz)20
10mF = 0

0-10

mF = +1

-20
-30
-40

-20

0
B (Gauss)

20

40

Figure 3-2: Zeeman energy of the F = 1 state vs. magnetic field. The solid lines
depict the energy dependence of the labeled states on magnetic field, and the dottedlines indicate the RF resonance for an applied field with ∆ = 30 MHz.

44

2
1.5
1

φ+

0.5
U (MHz) 0
-0.5

φ0

(a)
Ω

φ−

-1
(b)
-1.5

-0.75

-0.5

-0.25

0
r (mm)

0.25

0.5

0.75

1

Figure 3-3: Graph of the dressed RF potential. The horizontal axis of the graph isa vertical slice upwards through the center of the trap, and the top line shows thepotential for the trapped dressed state. The parameters plotted are for B 0 = 50 G/cm,
Ioffe-Pritchard bias Bbias = 1 G, ∆ = 2 MHz, Ω = 200 kHz, and including the effects
of gravity. During normal RF evaporation, atoms are confined by the lower potential
and trapped at point (b), and ejected from the trap when they reach point (a). In
the dressed trap, the atoms are confined by the upper potential and trapped at point
(b).

45

kinetic energy will be removed from the trap - this is exactly the process we use toevaporatively cool the atomic cloud when we create BEC. However, if the RF field
is turned on at ∆ < µB B0 and increased to the value greater than muB B0 , then theatoms will be confined to the upper potential of figure 3-3 and trapped at point (a).
If the field is a symmetric quadrupole, in three dimensions the atoms will be
trapped on the surface of a sphere with radius
r=

∆µB
.
B0

(3.2)

Due to the effects of gravity, the atoms will pool on the bottom of the sphere. The
trap frequency of the potential minimum, analogous to a Ioffe-Pritchard trap withbias field B0 = Ω/µB , is then given by
B 0 µB
ωz =
2π

r

gF
Ωm

(3.3)

where m is the mass and gF = 1.0 (0.5) for atoms in the mF = ±2(±1) state.
At the bottom of the sphere, atoms will be trapped horizontally by a pendulum
potential with trap frequency
r

ωx =

g
r

(3.4)

In the case of our I-P magnetic trap, the axial trapping frequency ωy is approximately
constant because the effective radius is very large, and is determined by the axial fieldcurvature. Instead of a sphere, the resulting equipotential contour is oblate.

3.2.2

Limit Of Tight Trapping

From equation 3.3, we see that decreasing the strength of the RF dressing field Ω
actually increases the trap frequency. However, Ω cannot be reduced to an arbitrarily
small value, as Landau-Zener spin flip losses through the avoided crossing will increase
with diminishing Ω. From [82] we see that the loss rate γ goes as

46

1000

8000

Lifetime (s)

10000

100

6000

10

4000

1

2000

0

200

400

600
Ω (2π ∗ Hz)

800

Trap Frequency fz (Hz)

10000

1000

Figure 3-4: Trap frequency and lifetime vs Ω. The left axis and dashed plot indicate
the theoretical lifetime vs the amplitude of the dressing field Ω. The right axis andthe solid line indicate the transverse trap frequency fz vs Ω for a fixed field gradientof 114 G/cm, 1.0 mm sphere radius, and atoms in the |2, +2i state.

−πΩ3/2γ ∼ 2 exp √
2∆

!

(3.5)

for γ  1. Mathematically, this loss is a result of off-diagonal terms in the Hamiltonian after diagonalization into the φ+ and φ− states. This particular limitation isdramatically exaggerated in experimental implementations, which will be discussed
later. Figure 3.2.2 shows the behavior of the lifetime (1/γ) and the trap frequency
wz as a function of Ω.

3.3

Two Dimensional Physics

In order to get to the 2D regime, ωz must be made large compared to both the horizontal trap frequencies and the self-energy (chemical potential, µ) of the condensate.
If µ is small compared to the harmonic oscillator level spacing h̄ωz , the vertical degree

47

30
N=10

4

25
20
h!/µ

15
N=10

5

10
N=10

5

200

400

6

600

800

1000

B’ (G/cm)

Figure 3-5: Maximum ratio of energy level spacing to chemical potential versus magnetic field gradient. The three curves are for 10,000 atoms (top), 100,000 atoms(middle), and 1,000,000 atoms (bottom), assigning a value for ∆ that produces a
sphere of r = 1 mm. For reference, our decompressed trap produces 52 G/cm andour standard compressed trap produces 114 G/cm. For this calculation, the axial
trap frequency was fixed at ωy = 2π× 3 Hz.
of freedom will be “frozen out”, and the system will begin to take on two-dimensional
characteristics. Fig. 3-5 shows the ratio h̄ω to µ for various atom numbers.

3.3.1

The Capabilities of the Our Apparatus

Assigning values relevant to our experiment, with B 0 = 114 G/cm, in our current
main chamber we could achieve h̄ω > 5µ with N=100,000. That figure is certainlyin the 2D regime, but it’s a best case scenario. Higher magnetic field gradients,
along with better imaging to more efficiently detect smaller atoms numbers could be
implemented in a specially designed science chamber. This system suffers from poor√
scaling also, since h̄ω/µ scales only approximately as B 0 .

48

0.8

Condensate Fraction

0.6

0.4

0.2

0.0
0.2

0.4

0.6

0.8
RF Power (W)

1.0

1.2

1.4

Figure 3-6: Condensate fraction vs RF power. The condensate fraction after transfer
into the dressed trap was strongly dependent on Ω. The horizontal axis is the RF
power measured just before the vacuum chamber. Too much or too little power
resulted in heating from non-adiabatic transfer. The line is a fit to the function
axe−x/b to guide the eye, with a = 4.3 ± 0.6 and b = 0.48 ± 0.06. Every change in thetrap parameters or RF signal delivery required re-optimization

3.4

Experimental Procedure

3.4.1

BEC Creation and Transfer

Each BEC was created in the standard way for our machine, direct RF evaporationof the atomic cloud in the decompressed magnetic trap in the main chamber. Afterevaporation to BEC at 1.05 MHz, an RF field was switched on with full power at
0.90 MHz and ramped to 3.49 MHz over 600 ms at a power of 1.31 W. The frequency
shape of the ramp was a half-cosine, which was important to achieve good transfer
with minimal excitations. If the ramp was too fast or the derivative δ∆/δt was too
high, large oscillations of the BEC were observed after the ramp. The gradient of
the decompressed Iofffe-Pritchard trap B 0 was measured to be 53 G/cm, and the RF
power was measured just before the vacuum chamber.

49

(i)

TTL
Controller TTL
Computer
TTL(a)VacuumChamber(h)(b)

Agilent
33250a

GPIB

(c)

TTL

RF Amplifier(d)RFMultiplexer
(e)

Tabor
1072

(g)

RF Switch(f)

RF Power
Combiner

RF
Multiplexer

Figure 3-7: RF component setup. Dashed lines indicate TTL logic signals, and solidlines indicate RF signal. The components are as follows: a) Controller computer, b)
Agilent 33250a function generator, c) Tabor 1072 DDS arbitrary function generator,d) RF multiplexer, e) RF power splitter/combiner used in the power combining direction, f) RF multiplexer (same as (d)), g) RF switch, h) RF amplifier, and finallyh) the vacuum chamber. During evaporation, the RF signal was delivered in theorder (b-d-f-g-h-i). During transfer into the dressed trap, the signal was delivered
through (c-e-f-g-h-i), and after the sweep the signal was delivered simultaneously byboth function generators ([b-d-e]+[c-e]-f-g-h-i).

3.4.2

RF Setup and Control

The setup of the RF components is depicted in figure 3-7. This system allowed
for GPIB control of the RF for evaporation, preprogrammed direct digital synthesis(DDS) of the dressing field, and the implementation of an additional RF shield in the
dressed state to prevent heating after transfer. The entire sweep and hold sequence
was programmed into the Tabor 1072 arbitrary function generator before the experimental sequence using a computer program to generate the required waveforms. The
Tabor was then triggered by a TTL signal to begin its sweep, after which the Agilent
was set to 2.110 MHz to act as an RF shield and remove thermal atoms. Nearly 100%of the atomic cloud was transferred to the dressed trap in this way, with very little
heating.

3.4.3

Imaging

After holding for some time in the dressed trap, the magnetic field and RF fields
were switched off and the atoms were allowed to expand ballistically in time of flight50

for 33 ms. After this expansion, the atomic cloud was imaged vertically on the
F = 2 → F 0 = 3 transition with repumping from F = 1 → F 0 = 2 along a different
axis. Side imaging was useful for verifying transfer into the dressed trap, and as longas ∆ was not increased so much that the atoms were lowered completely out of the
field of view, a calibration of r vs. ∆ quickly provided a precise determination of thegradient produced by the magnetic trap.

3.5

Results

3.5.1

BEC Transfer

BECs containing up to 106 atoms were transferred into the dressed RF potential withnear unity efficiency, and lifetimes of up to 11 s were observed. The efficiency of
the transfer was strongly dependent on Ω, the shape of ∆(t), and any noise sourceson the RF signal. The final value of ∆ was varied over a range from 1 - 10 MHz
without detriment to the atomic cloud, as long as the shape of ∆(t) at the point
where ∆ = µB B0 was maintained. Linear ramps of ∆ resulted in poor transfer andheating. Attempts to transfer atoms from the compressed trap (B 0 = 114 G/cm)
were similarly unsuccessful.

3.5.2

Trap Frequencies

The horizontal trap frequencies (ωx , ωy ) were measured to be 2π × (20, 3.4) Hz at∆ = 3.49 MHz. The vertical trap frequency ωz was measured with parametric heating
to be 250 Hz. In an attempt to measure the vertical trap frequency by observing
oscillations in time of flight, the atoms in the dressed trap were “kicked” by turningon the MOT coils for 50 µs. The energy delivered by this kick was large enough todrive the atoms to the top of the RF sphere, resulting in ring like structures in TOF
expansion (Fig. 3-8).

51

1 ms

3 ms

5 ms

7 ms

9 ms

11 ms

13 ms

15 ms

20 ms

23 ms

25 ms

Figure 3-8: Large amplitude oscillations in the dressed Zeeman trap. A magnetic
field gradient was applied for 50 µs, delivering kinetic energy to the atomic cloud.
The resulting pennywindow axis images are shown here as a function of time-of-flight.
Each region of interest is 4 mm × 1 mm. The densest part of the atomic cloud appears
at the top of the bubble in some images because the atoms have a relatively large
amount of kinetic energy.
52

BEC Atom Number (arb)

2000

1500

1000

500

0
0

5

10

15

20

25

Hold Time (s)Figure 3-9: Lifetime of BEC in the dressed trap. The graph shows the number ofatoms in the BEC (in arbitrary units, after subtraction of the thermal component)
as a function of the hold time in the RF trap. The lifetime was determined to be
11 ± 2 s. The parameters used were B 0 = 53 G/cm and ∆ = 2.0 MHz.

3.5.3

Lifetime

After transfer into the dressed trap, the lifetime of the BEC was measured to be
11 ± 2 s, shown in Fig. 3-9. Further improvements in the RF control increased thelifetime to 30 s, with BEC observed as long as 60 s (Fig. 3-10).

3.6

Experimental Caveats

The dressed Zeeman trap is very fragile. Almost all of the work on this project was
to determine methods to control the system without creating noise in the RF signal.
Here I will explain some of the limitations of the system, and attempts we made to
overcome them.

53

5.0 sec hold60.0 sec hold

Figure 3-10: Images of BEC in the dressed trap. Top imaging in 33 ms TOF with a
1 mm region of interest shows the condensate 5.0 s and 60.0 s after transfer into thedressed trap. The RF knife prevented the buildup of any visible thermal component.

3.6.1

RF Noise

Well designed magnetic traps benefit from common mode noise rejection, where small
current fluctuations in any one power supply relative to another do not usually disturb
the atoms. However, this is not the case with the dressed Zeeman trap. Since the
RF power Ω corresponds directly to the vertical trap frequency, and the frequency∆ controls the vertical trap position, any noise in either of these values will lead toparametric heating of the atomic cloud. Also, phase noise from signal switching (e.g.
a multiplexer) may instantly project trapped atoms into the untrapped state, leading
to loss of most of the atoms.

3.6.2

Analog Control

These losses and heating were observed in any attempt to control the RF signal
through analog devices. For example, an RF mixer (Mini-Circuits ZP-3H), which
can control the power output based on a DC voltage, was inserted into the control
system between points (c) and (e) in Fig. 3-7. The mixer was first used in a saturated
mode where output power is approximately independent of control voltage, then in
an unsaturated mode where the power is linearly proportional to the control voltage.
After adjusting the RF power in both cases to provide the same Ω, the lifetime of theatomic cloud was 7 s the saturated mode, but reduced to 900 ms in the unsaturated
mode. Attempts to ramp the control from the saturated to unsaturated regimes
54during the experiment destroyed the BEC in ∼400 ms. Any attempt to create ananalog control of the RF power, including voltage controlled oscillator (VCO) modes
with function generators such as the Agilent 33250a or the Stanford Research Systems
DS345 met with similar reductions in lifetime.

3.6.3

Digital Frequency Synthesis

After analog control of the RF properties was ruled out, digital methods were tried.
Most common (and relatively inexpensive) function generators that accept a trigger
to start a ramp will return to the start frequency as soon as the sweep is complete,
which is not useful. Mid-range synthesizers such as the Tabor 1072 and the Tektronix AFG3000 are able to precisely control either the frequency or the power, but
controlling both simultaneously over the experimental time scale required too muchmemory1 . Synthesizers able to control both parameters (digital AM and FM control
with an arbitrary waveform) cost more than $8,000, so more custom solutions were
pursued.
The last attempt to control the dressed Zeeman trap with pure digital synthesis
was to program an Analog Devices AD9952 CMOS Direct Digital Synthesizer. The
AD9952 is a single chip which synthesizes a sine wave based on real time programming.
A prototype board costing only $200 was purchased, and was to be driven by acomputer through a high speed digital I/O card. However, the decision was madeto instead focus on the development of the optical lattice project [6] and the dressed
Zeeman trap was abandoned.

3.7

Conclusion

After transfer into the dressed Zeeman trap, BEC lifetimes up to 30 s were measured,and BEC was observed in the trap for up to 60 s. Transverse trap frequencies as highas 250 Hz were measured. Future experiments implementing full digital control of the1

Sales representatives from both Tabor and Tektronix were extremely generous with the timethey let me keep their demo units.

55

RF field could investigate the limits of tight transverse trapping. While competitive
two-dimensional trapping regimes may not be attainable due to fundamental, physicallimitations, this system may still prove useful in other experiments with its precise
control of the trapping potential.

56

Chapter 4
Atom Trapping With a Thin
Magnetic film
4.1

Introduction

4.1.1

Motivation

Micrometer scale magnetic traps for Bose-Einstein condensates have been the focus
of much experimental work since their first demonstration [51, 28]. Their potential
uses in atom interferometry, precision measurements, and experiment miniaturization
have motivated many groups to develop sophisticated techniques for manufacturing
and controlling these atom chips [64, 27, 66]. However, since the earliest inceptions of
atom chips, current fluctuations and physical imperfections in the chip surface havelead to perturbations in the trapping potential that prohibit the coherent manipulation of atoms close to the surface [42, 45]. Permanent magnets offer a possible
solution to several problems inherent to current carrying wire traps. First, the magnets are almost completely decoupled from the rest of the laboratory, minimizing theeffects of environmental electrical noise. Second, they do not require current to besourced or sinked, enabling designs that would be too complicated or impossible tocreate using electromagnets. Last, extremely high field gradients are possible close
to magnetic domain boundaries, whereas traditional atom chip operation is limited57

by heat dissipation from small wires. Although a substrate of magnetizable material
may be made extremely smooth and uniform, imperfections in the process of etching [44] or writing magnetic structures could lead to some of the same problems aswith wire-based designs. It is not yet known which technique will provide the best
performance.

4.1.2

OutlineIn this chapter I will describe our investigations into atom trapping with a thin magnetic film; specifically that of a hard disk platter written with a periodic pattern.
This approach offers some advantages over previous work on neutral atom trapping
and BEC creation using permanent magnets [66, 27]. Here we use a thin metallic filmwith a large remnant magnetization from a commercial product which has alreadybeen refined to a high degree and a writing technique more accurate than anything
previously demonstrated. Cold 87 Rb atoms were first loaded into the magnetic potential formed by the disk and used RF evaporation to produce BEC. By changing
the trapping potential, the atoms could be pushed closer to the surface of the disk toprobe for imperfections in the potential. Finally, the BEC was dropped onto the disk
from a height of 2.7 mm in an attempt to produce a specular reflection of the atomic
cloud.

4.2

Surface Magnetization

4.2.1

Sinusoidal Pattern

In order to trap neutral atoms with magnetic fields, a field gradient is required. One
way to create such a gradient is to start with a surface comprising the x̂ẑ plane, thencreate on the surface a magnetization of the form
M = M0 cos (kx)ŷ.

58

(4.1)

This magnetization describes an alternating patten of up and down magnetic stripes,
where M0 is the magnetization of the material and k is the wavevector of the sinusoidal
pattern. This can be replaced with a surface current J by taking the curl
J = ∇ × M = −kM0 sin (kx) ẑ

(4.2)

Solving with physical boundary conditions [30, 33], the result is a magnetic field above
the surface given by








 Bx 
 sin (kx) 






1




 By  = µ0 M0 1 − e−kb e−ky  cos (kx)  ,




2





Bz

(4.3)

0

where b is the thickness of the magnetic material. This field is of uniform magnitudein the x̂ẑ plane and decays exponentially away from the surface as
B (y) =

1
µM
2 0 0





1 − e−kb e−ky

= B0 e−ky(4.4). In the absence of external magnetic fields, this exponential decay provides an ideal
potential for reflecting weak field seeking atoms [30], with gradientB 0 = −kB0 e−ky ,(4.5)

which for a pattern wavelength λ = 100 µm and surface field B0 = 6000 G provides
a gradient of 7000 G/cm at a height of 100 µm above the disk. Even higher field
gradients are possible with smaller patterns and closer to the disk.

59

4.2.2

Two Dimensional Trap

In the presence of gravity, if the surface is oriented horizontally, atoms may be trappedin a two-dimensional potential with vertical trap frequency
ωz =

q

gk

(4.6)

in a gravito-magnetic surface trap [30], where g is the acceleration of gravity. The
potential minimum is located above the surface a distance!

1
gF µB B0 k
h = ln
,
k
mg

(4.7)

where µB is the Bohr magneton, m is the atomic mass, and gF is 21 for the |1, −1i stateand 1 for the |2, 1i state. For example, a pattern wavelength of 5 µm would produce
a trap 10 µm above the surface with ωz = 560 Hz. A small horizontal confinement
could then be added, perhaps with an optical dipole trap, to contain the atoms in all
three dimensions.

4.2.3

One Dimensional Trap

The addition of a bias field Bx along x̂ (or By along ŷ) produces a series of quadrupole
shaped field minima above alternating tracks, depicted in Fig. 4-1. These minima may
then be used to trap weak field seeking atoms. Since each one is (to first order) aquadrupole magnetic field minimum, atoms which travel near the field zero may belost due to Majorana spin flipping. A second bias field Bz along ẑ removes this field
zero, and atoms may be stably trapped near the surface with radial trap frequencys

2πωr = kBxµB gF mF
mBz

where µB gF mF is the Zeeman energy [30, 66].

60

(4.8)

100 µm
y
x
z

N
S

S
N

50 µm

Bx

Figure 4-1: Geometry of the magnetic trap formed by a magnetic film. Magnetic
field lines resulting from the addition of an externally applied radial (x̂) bias field
are shown. The tubes represent the locations of the field minima where atoms aretrapped. The addition of an axial (ẑ) bias field of about 1.0 G prevents atom loss
from Majorana spin flips.

61

4.3

Magnetic MediaThe use of a magnetic storage medium for atom trapping is certainly not unique tothis experiment [66, 27]. Videotape and floppy disks with sinusoidal patterns had
previously been used to trap atoms in vacuum1 . However, the storage density, andthus the minimum achievable magnetic structure size, were limited to >10 µm. Also,
since they are formed of flexible materials, physical defects in the surface present
a challenge to miniaturization. It seemed that the next logical choice was to moveto a substrate that someone else had spent millions of dollars making as perfect as
possible: hard disks.

4.3.1

Hard Disks

Commercial hard drives available before 2005 all used “in-plane” magnetization,
where the magnetic field vector runs circumferentially around the disk. Information is encoded in an ac method where a change in the sign of the field means “1”and no change in the sign (where one was expected) means “0”. Unfortunately, linesof magnetic field only leave the disk in small areas where the magnetization changes
sign. This geometry is not useful for trapping atoms, as was unfortunately verified in
vacuum.
Fortunately, the commercial push towards higher and higher information densities
has very recently led to the development of hard disks using “out-of-plane” magnetization. At the time this experiment was conducted, no commercial out of planedisks were available to end users, as they were under a closely guarded process ofdevelopment at pretty much every major hard drive manufacturer. I had the good
fortune to make contact with Liesl Folks and Min Xiao at Hitachi Global Storage
Technologies in San Jose, CA, and Min agreed to provide a few out-of-plane disks
written with a specified pattern.
1

Apparently videotape and floppy disk substrates are UHV compatible, to the pleasant surpriseof the experimenters who first used them.

62

4.3.2The “Out-Of-Plane” Prototype

The magnetic media used was a prototype hard disk with a radius of 65 mm anda thickness of 0.635 mm. The magnetizable material is a dual layer system with a
20 nm thick magnetic Co-Cr-Pt based oxide layer and a 200 nm thick “magnetically
soft underlayer”, both of which are also good conductors. The exact components areapparently a trade secret, but the atoms are only affected by the field that comes out ofthe disk. The substrate is glass, the coercivity is 6.8 kOe and the magnetic remnant is
0.64 memu/cm2 . The easy axis of magnetization of this prototype “out-of-plane” disk
is aligned normal to the surface of the disk, as opposed to most modern commercial
hard disks which are magnetized in the plane of the surface. This alignment of themagnetic domains produces about 6000 G at the surface.

4.3.3

Vacuum Compatibility

One concern about the disk was that it could not be baked to the high temperatures(150 C) that we normally use to reach UHV. Digital data on similar disks is guaranteed
to 80 C, but at 100 C the data starts to degrade rapidly as the magnetic domains
randomize. The solution was to pre-bake the entire science chamber apparatus, minus
the disk, to 150 C for about a week to reach UHV conditions. The system was briefly
opened up to air, the disk was installed, and then the chamber was pumped down
again. The chamber was then heated only to 70 C for a few days, and after cooldown
the pressure was < 10−10 Torr. Apparently, hard disk platters are UHV compatible.
The signal decay rate quoted for this disk by its designers at room temperature
was 0.4% per decade, increasing to 0.8% per decade at 70 C. The decay rate is definedasδ = 100 ×

A(t) − A(t0 )1
A(t0 )log (t/t0 )(4.9)

where A(t) is the signal amplitude at time t, and A(t0 ) is the signal amplitude at t0 .
a decay rate of 0.8% per decade is considered stable for data storage, and the only
significant randomizing spin flips will occur away from the track boundaries.

63

4.3.4

Disk Properties

The disk was provided to us pre-written using a Guzik spin stand, which is essentiallya hard disk read/write head with absolute positioning capability. This method of
writing allows for the creation of truly arbitrary pattern impossible to create withlarger scale magnetic writing devices [66] or physical structures [27], and produces
smaller and cleaner structures than optical writing techniques [17]. Two patterns
were used in this experiment, written on different radial regions of the same disk.
The first region had alternating stripes of up and down magnetization with a periodof 2π/k = λ = 100.0 µm and the second had λ = 1.0 µm. The media has an almost
perfectly square hysteresis loop which precludes a purely sinusoidal magnetization.
The pattern was instead written as a square wave, and the higher Fourier harmonics
should be negligible at the height where we trap atoms [30]. Figure 4-2 shows amagnetic force microscopy image of the disk after writing2 which shows a significantly
less noisy pattern than magneto-optical thin films written with laser beams [40].
Atomic force microscopy showed a physical roughness of ∼3 nm over a region of
50 µm.

4.4

Experimental Procedure

4.4.1

Atom Delivery to the Science Chamber

Cold atoms were delivered to the surface in a two step process. First, cold 87 Rb
atoms in the F = 1, mf = −1 state were created in the main chamber Ioffe-Pritchardmagnetic trap with T > Tc where Tc is the critical temperature for Bose-Einsteincondensation (see [72, 63] for more details). Typically, if evaporation to 0.7 MHz
produced a pure BEC, we would stop evaporation at 2.0 MHz. The atoms were thentransferred to the optical dipole trap by simultaneously switching off the magnetic
trap and turning on the optical trap. The thermal cloud was then transported 36 cm
to the science chamber. The optical dipole trap was formed by a λ = 1064 nm laser
2

The image was produced by Min Xiao at HGST with a Digital Instruments NanoScope.

6450.0 µm

5.0 µm

25.0

2.5

0.0

100 nm

0.0
0.0

25.0

50.0 µm

0.0

2.5

5.0 µmFigure 4-2: Magnetic force microscopy image of a sample disk. The left image is ascan over a boundary region where the pattern wavelength changes from λ = 50 µm
to λ = 5.0 µm, showing the capability for multiple types of structures on a single disk.
The right image is a higher resolution scan over an area with λ = 5.0 µm, showing theuniformity of the track edges. The characteristic size of the roughness of the domainboundaries is about 30 nm.
focused to a 30 µm 1/e2 radius spot, and the focus (with the atoms) was translated
into the auxiliary chamber as described in [26]. By transporting atoms just above
Tc , the cloud was less sensitive to vibrations, and higher laser powers could be usedwithout causing rapid three-body losses.

4.4.2

Z-Wire Trap

After delivery to the science chamber, the final cooling was accomplished by loading
into a Z-wire trap as in [66] and RF evaporating to BEC. The Z-wire trap was formed
by a 1.2 mm diameter copper wire with a 5 mm Z center section carrying 10 Amps,
shown in Fig. 4-3. 10 G radial bias field was added to create a trap with frequencies(ωx , ωy , ωz ) = 2π × (90, 90, 20) Hz located 0.8 mm above the top of the wire. The
disk was located above the wire but below the atoms, enabling us to maneuver theatoms arbitrarily close to the surface. Initial attempts to load condensates from thewire trap into a λ = 100 µm surface trap site were unsuccessful due to a mismatch in
the trapping potentials. Gravitational sag in the wire trap caused the atoms to gain
∼300 kHz in energy as they fell into the surface potential in a “waterfall” effect as

65

Figure 4-3: Photograph of the Z-wire trap. The Z-wire trap was wound with 1.2 mm
diameter bare copper wire. The wires were used to enable trapping on two wellseparated parts of the disk, and the unused wire was used as a RF antenna for evaporation. The wire was mounted to an aluminum support that maintained the shapeand position of the wire. The support was anodized to prevent electrical shorting,and the combination of aluminum and anodization did not cause enough outgassing
to noticeably affect the vacuum. The dimensions of the support are 23 mm × 47 mm
× 5.0 mm.
the traps merged. Increasing the current in the z-wire could have lessened this effect,
but this was limited by heat dissipation rates3 . The best solution was to load theatoms directly onto the surface from the optical trap.

4.4.3

Surface Loading

The atoms were loaded onto the surface in the λ = 100 µm region from the optical trapby translating the focus of the optical trap to a position parallel to and 50 µm above
the surface. The Z-wire trap was ramped up in 200 ms to provide axial confinement,
and a small Bx created radial trapping on the surface. The optical trap was ramped
off linearly over 2 seconds, transferring the atoms with almost unity efficiency. The
3

Running 15 Amps CW outside of the vacuum made the wire way too hot to touch, and would
have demagnetized parts of the platter due to heating above the Curie temperature66

disk was positioned along k̂ (or x̂) to maximize the transfer of atoms into a single
trap, The narrow width of the optical trap enabled loading of >90% of the atoms
into a single surface trap site. If misaligned, or if an especially large thermal cloud
was transported, up to three surface traps could be loaded. RF Evaporation over 20
seconds from 1.200 MHz to 0.890 MHz produced a BEC with approximately 50,000
atoms in a trap with (ωx , ωy , ωz ) = (390, 390, 9) × 2π Hz and a lifetime of ∼ 30 s
(Fig. 4-4). Observation of the bimodal density distribution of the cloud in trap was
the most reliable method to optimize evaporation, and condensate fractions of >80%
were observed.

4.4.4

Surface Imaging

The atomic cloud was detected with on-resonance absorption imaging (Fig. 4-7) on
the F = 2 → F 0 = 3 transition. The atoms were optically pumped along anotheraxis from the F = 1 state to the F = 2 state prior to and during the imaging pulse.
The platter is a good reflector (>95%) for 780 nm light, so grazing incidence imaging
was used to measure the distance from the atoms to the surface (Fig 4-5). In thisconfiguration, probe light reflects off the disk at an angle of only a few degrees abovehorizontal. Two images of the atoms are formed on the camera: One image fromthe beam portion which passes through the atoms and then reflects off the disk, andanother image from the beam portion which first bounces of the disk and then passes
through the atoms. In the limit of a small angle of incidence, the distance between
the atoms and the surface is easily calculated as half the distance between the cloud
images.
Normal incidence imaging was also used, in which case the imaging light passed
through the atoms twice, bouncing off the disk (Fig. 4-6). Since it was difficult toperfectly align the beam to normal incidence, images of atoms held far above thedisk (>200 µm) sometimes showed a double image where the two clouds were notseparable. Those images were difficult to analyze. The lack of physical structures on
the surface resulted in good image quality, even compared to normal, direct imaging.
The top and side imaging beam paths were used interchangeably as pump and probe.
671.400 MHz

1.300

1.200

1.100

1.000

0.970

0.950

0.930

0.920

0.915

0.910

0.905

0.900

Figure 4-4: Evaporation to BEC on the disk. This series of normal incidence images
shows the transition from a thermal cloud to a BEC as the RF knife is lowered.
Since releasing the atoms for ballistic expansion in time-of-flight was impossible, thebimodal appearance of the trapped cloud was used as a key signature for BEC. The
width of the image is 1.2 mm.

Probe beam

Image

Atoms

Reflective surface

Figure 4-5: Hard disk platter side imaging. Probe light reflects off the surface of thedisk at a grazing incidence and creates an absorption image at the camera.
68

PBS

!/4
Mirror(in vacuum)

Probe beam

Atoms
Image
Reflective surface

Figure 4-6: Hard disk platter top imaging. Circularly polarized imaging light reflects
off the surface of the disk at normal incidence. The returning light is separated fromthe probe light with a polarization beam splitting cube. The mirror above the disk
is fixed in vacuum.
Attempts to release the atoms from the trap and observe ballistic expansion in timeof flight were hindered by the geometry of the system and the static nature of themagnetic surface, so all of the imaging was done in-trap.

4.5

Surface Trap Results

4.5.1

Trap Frequencies

The axial trap frequency in the surface trap was measured by imaging oscillations of
the atomic cloud, and radial trap frequencies were measured by parametric heating[Fig. 4-8]. Parametric heating of the atomic cloud was driven by small current oscillations in the secondary Z-wire, and observed by measuring atom loss. The Z-wire trap
was left on at all times to provide axial trapping. Ramping up the current in external
electromagnets increased Bx , and ωr as high as 2π × 5 kHz was measured. Using adisk with λ = 10.0 µm, we have measured ωr as high as 2π × 16 kHz. High transverse
trap frequencies are desired for studies of 1D systems [38], but atom heating and loss
69

a)

b)

Figure 4-7: Absorption image of the BEC in trap near the surface. a) Top imaging.
The field of view is 0.58 mm × 0.58 mm. b) Side imaging. The field of view is 0.56 mm
× 0.56 mm. The double image comes from the reflection of the imaging beam in theplatter surface. The trap frequencies were 2π×(400, 400, 10) Hz. The configurationof the experiment prevented ballistic expansion, but the bimodal distribution in trap
is still clear.
at higher trap frequencies prevented such studies here.

4.5.2

Lifetime

While trapped 40 µm above the surface, the BEC had a lifetime of ∼30 s. Previous experiments with atomic clouds magnetically trapped near conducting surfaces
showed atom loss resulting from spin flips driven by Johnson noise [35, 45, 61]. This
noise is a result of thermal eddy currents in the metallic surface which drive magnetictransitions into the untrapped state. In the regime where the thickness of the metallic
film is much less than the skin depth of the resonant radiation, the decay rate isΓF,m =

2
CF,mC0
d (1 + d/t) (1 + 2d/w)(4.10)

where t and w are the thickness and width of the surface, and d is the distance
between the atom and the surface. C0 = 88 s−1 µm × (T /300 K)(ρCu /ρ) where
T is the temperature of the metal, ρ is the resistivity of the surface, and ρCu =2
1.7 × 10−8 Ω m is the resistivity of copper. CF,m
= |hF, m − 1|S− |F, mi|2 is thestrength of the spin changing transition where S− is the spin lowering (or raising)operator [45]. In the case for atoms in the |1, −1i state over a surface with w  t
70

Trap Frequency (kHz)

5
4
3
2
1
0
0.0

1.0
2.0
Applied Bias Field (Gauss)

3.0

Figure 4-8: Radial trap frequency vs applied radial magnetic field. Atoms were loaded
into a single trap site on the surface and evaporated to BEC. The trap frequencies
were measured with parametric heating, but above 5 kHz other heating effects madeit difficult to resolve reliably. The deviation from linearity at low applied fields ismost likely due to a small, off-axis, residual bias field.
and w  d, this reduces to
2
C0
C1,−1
Γ1,−1 =d (1 + d/t)

(4.11)

Assuming ρ = ρCu , and taking the rest of our experimental parameters t = 200 nm,
2= 1/8, we calculate a spin flip rate of only 0.1 s−1 . Combiningd = 40 µm, and C1,−1

this loss process with losses from collisions with background gasses (Γbg ) results in a
net lifetime of

τ=

1
1
+
Γbg Γ1,−1(4.12)

Any spin flip loss was below the level of our 1-body losses at a vacuum pressureof ∼ 5 × 10−11 Torr.

4.5.3

Magnetic Potential Imperfections

The BEC was also used to probe imperfections in the trapping potential. After evaporation, the axial trapping frequency was reduced from 10 Hz to ∼1 Hz to allow the
71

a)
b)
c)
d)
e)
Figure 4-9: Breakup of the atomic cloud as the atoms approached the surface. The
thermal cloud was evaporated to T ≈ Tc , then the axial trapping frequency was
reduced to from 10 Hz to 1 Hz to allow the atoms to expand in one dimension.
The axial bias field was then increased to push the atoms closer to the surface, andthe atoms were imaged in trap. The grazing incidence imaging used here produces adouble image of the atoms from the primary and secondary reflections of the imaging
beam off the disk. The field of view is 0.29 mm × 2.3 mm, and each frame is anaverage of five distinct images to reduce noise for clarity. At distances smaller than
40 µm, significant breakup was observed. Height for the images shown: a) 35 µm b)
29 µm c) 25 µm d) 22 µm e) 19 µm.
cloud to expand slightly in one dimension to increase the sensitivity and the measurement area, and the radial bias field (Br ) was increased to push the atoms closer to thesurface. At distances closer than 40 µm, breakup of the atomic cloud was observed(Fig. 4-9), and the strength and spatial frequency of the perturbations increased as
the atoms neared the surface, similar to [42, 7]. The magnitude and size scale ofthese imperfections can be attributed to the sputtering process used to create thefilm. Magnetic grain nucleation occurs randomly, resulting in small angular misalignments of the axes of anisotropy of individual magneto-crystalline domains (typicallya few degrees), the individual magnetic grains can vary in size by about 25% of their∼7 nm diameter, and the magnetic moments have a distribution of magnitudes. Allof these phenomena create imperfections in atomic trapping potentials.

72

While the presence of these perturbations does not necessarily preclude the creation of neutral atom waveguides using this type of substrate, it does limit its usefulness. High trapping frequencies and single transverse mode confinement require close
proximity to the surface and will suffer from the observed imperfections. However,for use as a high reliability, low noise, and low cost waveguide at distances >50 µm,
commercial metallic, magnetic thin films are ideal.

4.6

Magnetic Mirror

The magnetized surface was also examined for its usefulness as an atom mirror, asin [58, 65, 44]. In order to minimize residual and time dependent magnetic fields(from magnetic trap turnoff), all optical trapping was first used to create a BEC. The
optical dipole trap provided only weak axial confinement, so a second, 200 mW laser
beam with λ = 1064 nm and 200 µm waist was added in a crossed configuration, andallowed more efficient evaporation. To evaporate to BEC the power was reduced over
2.0 s. An axial bias field Bz = 1.2 G was maintained throughout the experiment, andBx and By were minimized to <20 mG.
The region of the disk used as a magnetic mirror was written with λ = 1.0 µm.
This value was chosen to be large compared to the magnetic writing precision, but
small compared to the extent of the atomic cloud, and hopefully small compared tothe deBroglie wavelength of the atoms. The BEC was released from the crossed ODT
2.7 mm above the disk. The atoms fell under gravity for 23.5 ms and then reflected off
the magnetic potential. Up to three bounces off the surface were observed (Fig. 4-10).
The atomic cloud was imaged (in separate experiments) from both the top and side
after various times of flight. Side imaging showed that the reflection from the disk did
not significantly affect the axial or vertical velocity distributions of the atomic cloud.
Top imaging, however, showed significant spreading along the vector of the magnetic
pattern after reflection, analogous to bouncing off a rough mirror [55]. The magnitudeof this dispersion was minimized by fine tuning Bx and By , but was impossible toeliminate in our apparatus. Before bouncing off the disk, the x̂ width of the cloud
73

350

Width or Height (mm)

300
250
200
150
100
50
0
0

10

20
Time (ms)30

40

Figure 4-10: Width and position of the bouncing BEC. After release from the crossed
ODT trap above the region of the disk with λ = 1.0µm, the BEC was imaged after
various times. Side imaging showed the height of the BEC above the disk as it
bounced (squares), and top imaging showed the width of the atomic cloud (fit to a
Gaussian) as it expanded (circles). At times <8 ms, the expansion is purely due tothe chemical potential of the atoms. Between 8 and 24 ms, the expansion rate of thecloud is increased due to dispersive effects. After the second bounce at ∼24 ms, theexpansion rate increased dramatically. The line is a fit to y0 + A |cos (2πx/τ + φ)|with y0 = 21 ± 3 µm, A = 318 ± 4 µm, τ = 32.44 ± .08 ms, and φ = −0.03 ± .01 rad.
expanded by 1 mm/s, and after bouncing the x̂ width increased by 34 mm/s. The
reflection was not performed on the region with λ = 100.0 µm. Experiments using a
different disk with λ = 10.0 µm showed similar effects.

4.6.1

Analysis of Dispersive Reflection

The magnetic properties of the hard disk surface are one possible source of the observed expansion after reflection. While the theory presented earlier (Eq. 4.1) applies to a sinusoidally magnetized surface, the square hysteresis loop and directional
anisotropy of digital recording media prevent recording of a pure sine wave, instead
forming a square wave approximation of that sine, as discussed in [30]. While the74higher harmonics resulting from this approximation do not necessarily adversely effect
BEC trapping, they create a corrugation of the planar equipotential that prevented
the specular reflection of a macroscopic atomic cloud. Another contribution to theroughness of the reflection potential is the existence of small, stray magnetic fields.
Any nonzero component of B in the x̂ŷ plane creates a regular corrugation of thepotential in the plane, further inhibiting specular reflection.

4.7

Hard Disk Platter Conclusion

In conclusion, we have demonstrated BEC production on the surface of a hard disk
platter. Small scale imperfections in the magnetization caused condensate fragmentation close to the surface, prohibiting its use as a neutral atom waveguide. The disk
was also used as an atom mirror, and specular reflection was observed on two axes.
Microtraps based on permanent magnets, and in particular magnetic metallic thinfilms, may become an alternative to atom chips using current-carrying wires if thefabrication can be further improved, e.g. by using molecular beam epitaxy, and thewriting process improved by using write heads optimized for recording DC structures
on media with B field perpendicular to the surface.

4.7.1

Special Thanks

I would like to especially thank Min Xiao at HGST for writing several disks for us
and providing the MFM images, this work would not have been undertaken without
her assistance. She provided three disks which we put into vacuum, and each one
took many hours for her to prepare.

75

76

Chapter 5
The Mott Insulator Atomic Clock5.1

Introduction

5.1.1

Abstract

We implemented a proof-of-principle atomic clock using a 87 Rb Bose-Einstein condensate confined in an 3D optical lattice. Several typical sources of systematic error were
eliminated by preparing the atoms in the Mott Insulator (MI) phase before measurement. The suppression of tunneling in the MI phase permitted longer measurements
than possible with other optical confinement techniques, as well as elimination of thedensity shift due to the discretized density resulting from integer filling factors. The
hyperfine transition of 87 Rb was measured with a FWHM linewidth of 1.2 Hz and an
uncertainty of 0.2 Hz out of 6.8 GHz.

5.2

Atomic Clocks

Precision frequency and time measurements provide a fundamental reference for thebasis of nearly every physical measurement. Modern measurements have improved theaccuracy of atomic clocks to the 10−15 level [49, 16, 54, 14], and recent experimentswith optical transitions in trapped, neutral atoms promise precision at the 10−18
level [73]. In fact, as the most accurate measurement of any kind, atomic clocks are77

being used to study the time dependence of fundamental constants such as α. Great
care must be taken to protect the atomic clock system from perturbations from theenvironment. Since forces used to trap the atoms may perturb the system, fountainclocks which measure resonance frequencies with the atoms in free-fall have been thestandard. However, the interrogation time, and thus the bandwidth, is limited bygravity and the size of the system. Recent experiments have made progress withmeasuring trapped neutral atoms and ions, as the tools necessary to trap without
perturbation have been developed. As these trapped atom clocks reach new records
of precision, a new problem emerges; interactions within the atomic cloud begin toaffect the measurement.

5.2.1

Motivation

Unwanted perturbations to an atomic clock system can come from external sources,
such as external trapping for cold atoms or stray electromagnetic fields in the laboratory, or from internal sources, such as the atomic cold-collision shift [19, 54, 67]
or dipole-dipole interactions [19, 60]. Although systems can be shielded from external perturbations through clever engineering, it is more difficult to control the effectof interactions, as they are “built in” to the atoms. Simultaneously interrogating a√large number of atoms N can increase the signal to noise ratio by N , but increasing
the atom number typically comes at a cost of increased density, leading to greater
uncertainty due to interactions. For example, the microwave hyperfine transition ofcold 87 Rb at a typical density of 1013 cm−3 will be perturbed by the “clock” shift at
a level of 2 Hz. One way to minimize such collisions is to hold each atom apart fromthe others in an optical lattice.

5.2.2

This Experiment

In this work we implement an atomic clock by measuring a microwave hyperfine
transition in a 87 Rb Bose-Einstein condensate (BEC) confined in a 3D optical lattice.
Atomic clocks using neutral atoms in an optical lattice have been demonstrated previ-78

ously [73, 3, 46], but here we eliminate a key source of systematic error by increasing
the depth of the lattice potential to confine the atoms in the Mott insulator (MI)regime. In the MI phase, tunneling is suppressed due to the inter-site interactionenergy barriers, reducing the lattice depth required to minimize interactions. This
reduced lattice depth avoids complications such as higher order light shifts [43]. As
recently shown [6], the microwave spectrum of the MI shows discrete peaks corresponding to the integer filling factor of individual lattice sites. This discretizationof the density shift eliminates density dependent errors in the measurement of thetransition frequency. Systematic shifts due to trapping laser intensity may also be
extrapolated to zero intensity by varying the laser power over a wide range within
the MI regime.

5.3

Lattice Theory

5.3.1

The Bose-Hubbard Hamiltonian

Bosons held in a 3D optical lattice with repulsive interactions are described by the
Bose-Hubbard Hamiltonian [20, 34]
Ĥ = −JX
1 X
â†i âj + U
n̂i (n̂i − 1) + (i − µ)n̂i ,
2
i
i
hi,ji
X

(5.1)

where J is the tunneling term between nearest neighbors, n̂i = â†i âi is the numberof atoms in the ith lattice site, U is the repulsive onsite interaction, i = Vext (ri ) is
the energy due to the external confinement, and µ is the chemical potential. The
onsite interaction is expressed as U = (4πh̄2 a/m) |w(x)|4 d3 x, where a is the sR

wave scattering length, m is the atomic mass, and w(x) is the single particle Wannierfunction. The first term in the Hamiltonian tends to de-localize atoms over the lattice
due to tunneling, while the second term tends to localize atoms to lattice sites due
to repulsive atom-atom interactions. The ratio J/U is thus the primary descriptor of
the behavior of the atoms. For small lattice depths, the ratio J/U is large and the79system behaves as a superfluid.

5.3.2

The Mott Insulator

As the lattice depth is increased, J/U increases, the localizing effects of the interaction
begin to dominate, and the system undergoes quantum phase transition to a MI phase.
The MI phase is characterized by Fock states, where atoms are localized to individual
lattice sites with integer occupation number. With the spherically symmetric external
confinement
1
2
ri2 ,
i = V (ri ) = mωtrap
2

(5.2)

where ri is the distance to the lattice site i from the center of the trap, and thechemical potential of the condensate in the Thomas-Fermi approximation
2/5



3
3
U
15 (λ/2) m3/2 N ωtrap

√
µ=
16
π 2

,

(5.3)

the peak density in a given lattice site is then11
2
µ − mωtrap
ri2 . [24]
ρ(ri ) =
U
2




(5.4)

In the MI phase, this describes a series of concentric shells, each with uniform occupationn=bµ − ic
U(5.5)

where bxc is the largest integer less than or equal to x. It is this discrete filling which
we utilize to eliminate the density shift of our atomic clock.

5.3.3

Density Shifts

The density dependent shift in 87 Rb is determined by the difference in the scattering
lengths a11 and a21 , where a21 (a11 ) is the scattering length between an atom in the
F = 1, mF = −1 state and one in the F = 2, mF = +1 (F = 1, mF = −1) state. The

80

Figure 5-1: Theoretical “wedding cake” density profile. This figure shows a single
slice through the center of the MI for our experimental parameters with N = 4 × 105
and a lattice depth of 35 Erec . The horizontal axes are in µm and the vertical axis is
the filling factor. Each grid square represents approximately one lattice site.

81

difference in the scattering lengths results in a mean field energy shift∆E =h2(a21 − a11 )
πmρ

(5.6)

will then resolve to discrete frequency shifts for integer lattice filling factors, where ρis the atomic density. The resulting microwave spectrum for a BEC in the MI regime
is a series of peaks, and the frequency difference between neighboring MI shells is
then
δν =

U (a21 − a11 )
.
h
a11

5.4

Experimental Procedure

5.4.1

BEC Creation(5.7)

For our experiments, the BEC was created using forced radio-frequency evaporationof a 87 Rb atomic cloud in the |F, mF i = |1, −1i state, as described previously in [71].
The trapping potential was formed by the combination of a Ioffe-Pritchard magnetic
trap and a retro-reflected, cross polarized optical dipole trap. The optical trap wasoriented perpendicular to the weak axis of the magnetic trap to provide a more spherical geometry for better mode matching to the optical lattice [6]. BECs containing∼ 1.5 × 106 atoms with no discernible thermal component were held for 5 seconds
with a constant RF knife, reducing shot-to-shot atom number fluctuations to below
5%. For most measurements, a fast RF sweep through the RF resonance was used toreduce the atom number by ∼75%.

5.4.2

The Optical Lattice

A 3D optical lattice was then adiabatically ramped on using an exponential increase
with a 40 ms time constant. Two lattice axes were formed by increasing the intensity in fully retro-reflected beams, while the third axis was formed by rotating thepolarization of the retro-reflected optical dipole trap using a liquid crystal waveplate82

(Fig. 5.4.2). The lattice beams were all derived from the same laser with λ = 1064 nm,
and were detuned from each other by acousto-optic modulators. The beams also all
had roughly equivalent 1/e2 beams waist radii of 150 µm, focused on the atoms. The
lattice depth was increased up to 35 Erec where Erec = h̄2 k 2 /(2m) and k = 2π/λ isthe wavevector of the lattice1 . After ramping up the lattice, the atoms were held for
50 ms before measurement to compensate for transient nematic effects in the liquid
crystal polarization rotator. The resulting MI structure was calculated to containapproximately 7 × 104 atoms in the n = 1 shell.

5.4.3

MI State Creation

As the lattice is ramped up, the ac Stark shift and the density shift begin to broaden
and move the microwave resonance to lower frequencies. As the lattice depth isincreased, the tunneling rate decreases until it is completely suppressed at depths
greater than approximately 15 Erec due to the transition to the MI phase. Discrete
peaks begin to appear, corresponding to cold collision density shifts resulting from
integer filling factors within lattice sites. The observation of discrete peaks is direct
evidence for the suppression of tunneling by interactions when the MI transition isreached. Tunneling of atoms between sites of different occupation number during the
microwave pulse would smear out the discrete peaks and eventually lead to peaks
reflecting the clock shift at an averaged density. We estimate the tunneling time in
the absence of interactions using a double-well model to be h/4Jnz, where z is thenumber of next neighbors. For n = 1, the tunneling time is ∼5 ms at 20 Erec , 13 ms
at 25 Erec and 60 ms at 35 Erec . A very deep lattice would localize atoms even for athermal or non-interacting gas. However, for our lattice depths no discrete spectrum
was observed when the cloud was heated during the ramp up by misalignment of
the lattice beams. Our spectra were taken under conditions where the condensate
fraction after ramp down of the lattice was greater than 80%.
1This corresponds to a peak intensity in each beam of ∼240 mW/cm2 for 87 Rb with λ = 1064 nm.

83

To Telescope& FiberTo Telescope& Fiber

a)
b)
c)

d)

e)

f)
g)

h)
Pennywindowi)

Tweezer

Figure 5-2: Setup of the optical lattice. The light is delivered to the machine table
via optical fibers for each axis, then the beams are expended with telescopes before
the optical setup depicted in the cartoon. a) 300 mm lens. b) Boundary of vacuum
chamber. c) Retro mirror. d) Liquid crystal waveplate. e) 500 mm lens. f) 400 mm
lens. g) Atomic cloud. h) 200 mm lens. i) Motorized flipper mirror for MOT axis
utilization. The vertical axis, which coincides with the imaging axis, is not pictured.
The waveplate is used to adjust the polarization of the reflected tweezer axis beam tozero contrast during evaporation and maximum contrast during the lattice rampup.

84

Lattice Depth (Erec)

30

20

(a)

(b)

10

(c)

τ

0
0

100

200
Time (ms)Figure 5-3: Diagram of the lattice depth vs time. (a) The lattice is ramped up toU = 35Erec in 193 ms, then held for 50 ms. The shape of the ramp is an exponential
with a time constant of τramp = 40 ms. (b) The two photon pulse is applied for time
τ , then (c) the atoms are released from the trap and imaged in 3 ms time of flight.

5.4.4

Detection

A single two-photon pulse resonant with the |1, −1i → |2, 1i transition was applied
to the atoms in the lattice for some time τ , as depicted in Fig. 5.4.2. The pulse
consisted of a microwave photon at 6.33 GHz and a RF photon at ≈1.6781 MHz.
The microwave synthesizer was an Agilent E8257D running off its internal timebase.
The RF synthesizer was an Agilent 33250a, and was synchronized to the 10 MHz
timebase of the microwave synthesizer. Magnetic field dependent line shifting andbroadening was avoided by working at a bias field of ∼3.23 Gauss, where the twostates experience the same first order Zeeman shift. The effect of the magnetic fieldon the resonance frequency is shown in Fig. 5.4.4. The power of the pulse was adjusted
to produce a π pulse for each hold time τ . Immediately after the pulse, the confining
potentials were switched off and the atomic cloud was allowed to expand for 3 ms to
reduce the column density. Atoms in the |2, 1i state were detected with absorption
imaging using light resonant with the F = 2 → F 0 = 3 transition.
85n=1 Peak Center (f - 1678000 Hz)

104

103

102

101

100

99

98
3.10

3.15

3.20
3.25
Magnetic Field (Gauss)

3.30

3.35

Figure 5-4: Resonance of the n = 1 peak vs magnetic field. The line is a fit to
the parabolic field minimum. Determination of the resonance minimum to a level of
10 mG provided an uncertainty of 0.1 Hz in the frequency.

5.5

Results

5.5.1

Pulse Bandwidth Measurement

By using long measurement pulses, we narrowed the bandwidth of the scan andincreased the resolution the n = 1 and n = 2 peaks from that reported in ourprevious work [6] (Fig. 5.5.1). Using a double Gaussian fit to the number of F = 2
2

atoms detected versus the RF frequency of the form y = yo + A1 e−((x−x1 )/σ1 ) +2

A2 e−((x−x2 )/σ2 ) , the n = 1 peak was more than 10σ2 removed from the n = 2 peak.
This indicated that any frequency shift from higher densities to the singly occupied
sites was removed at a level better than 10−5 . Shifts from superfluid layers are
assumed to be negligible for our experimental parameters [15], leaving only the n > 2
MI states to be considered as perturbations to the n = 1 sites. Thus by using the MIphase to separate density shifts into carrier and sideband peaks, we were able avoid
density dependent shifts that affect other atomic clock experiments. By increasing
the duration of the microwave pulse, and thus narrowing the bandwidth, we were

86

1200
1100

n=1

Number of F=2 Atoms (Arb)

1000
900
800
700
600

n=2

500
400
300
200
100
0
60

65

70

75
80
85
RF Frequency (f - 1678000 Hz)

90

95

100

Figure 5-5: Microwave scan of atoms in the lattice. The n = 2 peak is centered at
1,678,067.2 ± 0.1 Hz with a FWHM of 4.2 ± 0.8 Hz, and the n = 1 peak is centeredat 1,678,095.03 ± 0.03 Hz with a FWHM of 2.85 ± 0.08 Hz. The fit is a double
Gaussian, and shows that the n = 2 peak is move than five linewidths removed fromthe n = 1 peak, eliminating the density shift. The lattice depth was a 35 Erec , andthe measurement time was 300 ms.
able to measure successively smaller linewidths of the singly occupied sites down to1.0(2) Hz FWHM (Fig. 5.5.1). Measurement times longer than 1200 ms did notfurther decrease the observed width.

5.5.2

AC Stark Shift Cancellation

Since the F = 1 and F = 2 states have different complex polarizabilities, the interaction with the lattice light field will produce a frequency shift in the microwave
resonance that is dependent on the lattice depth. In order to correct for this frequency
shift induced by the ac Stark effect, the location of the n = 1 peak was measured fora variety of lattice depths in the MI regime. Since the ac Stark shift scales linearlywith laser power, measuring the resonance as a function of lattice depth allowed us
to quantify this effect. After a linear fit to the data (Fig. 5.5.3), extrapolation to adepth of 0 Erec revealed the resonance frequency without any linear laser intensity
87

10
9
8

)zH( htdiW kaeP 1=n

7
6
5
4

3

2

1
9
8
5

6

7

8

9

2

3

4

100

5

6

7

8

9

2

1000
Pulse Length (ms)Figure 5-6: Width of the n = 1 peak vs. two photon pulse length. After carefullattice alignment and bias field calibration, the width of the single occupation density
peak was measured as a function of the interrogation pulse length. The two photon
intensity was adjusted to provide a π pulse at each hold time. The line represents
the theoretical bandwidth of the pulse. Pulses longer than 1200 ms resulted in larger
peak widths due to technical noise.

88dependent shifts. This extrapolation increased the uncertainty of the measurement
by about a factor of two over that of a typical single frequency measurement. The
minimum lattice depth required to clearly resolve the n = 1 peak was 20 Erec , and amaximum depth of 40 Erec was used.

5.5.3

Absolute Frequency Measurement

After extrapolation, the |1, −1i → |2, +1i hyperfine transition frequency was measured to be 6,834,678,015.20(18) Hz (statistical). After applying a 4497.3 Hz correction predicted from the Breit-Rabi formula for the finite magnetic field [69], this is98.4 Hz from the accepted value of 6,834,682,610.90434(3) Hz [59], but the focus of
this work was only to control systematic errors resulting from inter-atomic interactions, not to make a high accuracy measurement. This discrepancy is explained in
the analysis section as a result of drift of our microwave frequency source.
The relatively long experimental cycle time is another limiting factor in the ultimate precision. It takes approximately 10 data points to make an accurate determination of the line center, and each data point (BEC) takes up to 90 seconds
to produce in our machine. 15 minutes is a long time in a laboratory environment
with elevators, other experiments nearby, and even the subway possibly affecting eachmeasurement [11]. Experiments designed to produce cold atoms clouds much more
frequently would be better suited to make precision clock measurements.

5.6

Analysis

5.6.1

Pulse Bandwidth Limit

The primary determinant of the peak width was the bandwidth of the two-photon
pulse. In the case of a bandwidth limited pulse, the lineshape should be a sincsquared with a FWHM of 0.8859/τ . This is shown as the solid line in Fig. 5.5.1,
which agrees well with the data for τ < 1 s. For times longer than 1 s, several sourcesof inhomogeneous broadening begin to affect the lineshape.

89

103
102

Resonance (Hz - 1678 kHz)

101
100
99
98
97
96
95
94
93
20

22

24

26

28
30
32
Lattice Depth (Er)

34

36

38

40

Figure 5-7: Center frequency of the n = 1 peak vs. lattice depth. The centerof the single occupation peak was measured as a function of lattice depth in order to extrapolate to zero laser intensity. The line is a linear fit which gives azero intercept of 110.04 ± 0.18 Hz, indicating an ac Stark shift free frequency of
6,834,678,110.04 ± 0.18 Hz. The error bars have a typical value of ±0.10 Hz.

908n=1 Peak Width (Hz)

6

4

2

0
3.15

3.20

3.25
Bias Field (Gauss)

3.30

3.35

Figure 5-8: FWHM of the n = 1 peak vs magnetic field. Control of the magnetic
field at a level of <20 mG was necessary to minimize inhomogeneous magnetic fieldbroadening, and τ = 800 ms was used for this data. The line is a parabolic fit toguide the eye.

5.6.2

Inhomogeneous Magnetic Field Broadening

The largest source of inhomogeneous broadening in our experiment is due to a magnetic field gradient across the atomic cloud. Our magnetic trap is operating at the
low limit of holding the atoms against gravity, and thus we assume that the entire
cloud lives in a constant vertical magnetic field gradient of 15 G/cm. Taking a n = 1
shell radius of 19 µm, we calculate a field variation of 58 mG vertically across theentire cloud, implying a broadening of the hyperfine transition of ∼0.36 Hz due tothe inhomogeneous magnetic field. The effect of the magnetic bias field on the peakwidth is shown in Fig.5.6.2. Since the magnetic field broadening depends on the sizeof the atomic cloud, a large atom number could lead to larger linewidths. Within thestatistical uncertainty of the fits, the width of the n = 1 peak was not found to besignificantly sensitive to atom number in the range 105 < N < 106 .

91

5.6.3Differential ac Stark Shift

Another source of inhomogeneous broadening is the differential ac Stark shift. If
the clock measurement samples atoms which are trapped in regions of different laser
intensity, the resonance will be broadened. However, the shell structure of the MIphase will tend to minimize effects from the differential ac Stark shift as all of theatoms in the measured shell live in approximately the same laser intensity. If theatomic cloud is small compared to the diameter of the lattice beams, this effect will
also be small. Assuming that the measured atoms consist of a shell with radius equal
to the Thomas-Fermi radius of the BEC, we calculate that the differential ac Stark
shift across the shell results in a broadening of <0.2 Hz.
This broadening scales linearly with the inverse of the beam diameters, so narrow
beams used for all optical trapping would adversely affect the measurement. For
example, 20 µm waist beams would lead to a differential ac Stark shift broadened
peak width of ∼2 Hz, dominating other sources of broadening currently observed.
This broadening, as well as the total ac Stark shift may be reduced by the use ofa “magic” wavelength lattice. By carefully choosing λ so that the two interrogatedlevels are shifted equally by the trapping laser, the ac Stark shift may be reduced
by several orders of magnitude [73]. In the absence of a magic wavelength, a bluedetuned lattice laser could be used, trapping the atoms in regions of minimum laser
intensity [5].

5.6.4

Particle-Hole Pair Broadening

One possible source of homogeneous broadening is the existence of particle-hole pairs
within the n = 1 number state[22]. If we rewrite the Hamiltonian in the MI phasewith the tunneling term as a perturbation to the interaction term, to first order wehave [23]|Ψ(1) i ≈ |ΨiMI +J X †â âj |ΨiMI .
U hi,ji i(5.8)

The tunneling term of the Hamiltonian acts as a perturbation on the otherwise uniform array of Fock states in the MI phase, adding a small superposition of n = 092and n = 2 noise to the n = 1 signal. This manifests itself in our measurement asa small shift of the n = 1 peak towards n = 2 due to the slightly increased averagedensity. At a lattice depth of 20 Erec , a 3.5% mixture could generate a shift as large as
1.0 Hz. However, this effect scales as J/U , and for a lattice depth of 35 Erec , the 0.2%
mixture generates a shift of 0.05 Hz and would be further reduced for larger lattice
depths. The measurement of center frequency versus lattice depth in Fig. 5.5.3 does
not indicate any nonlinear effects at lower lattice depths, suggesting that particle-hole
pair broadening plays a smaller role than the theory describes.

5.6.5

Induced Dipole-Dipole Interactions

Another systematic shift which depends on interactions is the induced dipole-dipole
shift [8, 36], where the measurement electromagnetic field drives dipole oscillationsin the atoms that affect neighboring lattice sites. Since the measurement wavelength(λµw ≈50 mm) is very long compared to the lattice spacing (d = 523 nm), there will
be a dipole-dipole interaction in the near field regime. This interaction is given by [5]
Vdd =

sΓ
3
kmeas
d3

(5.9)

where s is the saturation parameter, Γ is the linewidth of the transition, kmeas is
the wavenumber of the measurement field, and d = λlattice /2 is the spacing between
atoms. However, the very narrow natural linewidth of the |1, −1i → |2, 0i transition(∼ 10−8 Hz), as well as the relatively large single photon detuning (420 Hz) imply
an interaction energy shift of < 0.02 Hz. Experiments using measurements in the
optical regime will benefit from a reduced interaction since the energy shift scales as(λmeas /d)3 [36], and careful selection of the lattice geometry can even further suppress
this effect [8].

5.6.6

Microwave Frequency Source

The main source of systematic error in this experiment was the microwave frequency
source. Our synthesizer was not linked to any external reference clock, such as GPS,93

and the specified drift rate of 4.5 ppb per day corresponded to 31 Hz per day at
6.8 GHz. Indeed, frequency jumps as large as 20 Hz over a two day period were observed when temperature stability in the lab was not maintained. Future experiments
would certainly implement better master oscillator stabilization.

5.7

Two Photon Pulses

Using a two-pulse Ramsey technique was investigated, but phase jitter due to synthesizer frequency noise made accurate fitting of the resulting signal difficult. The useof a double pulse method is limited only by the fact that each of the two pulses musthave a bandwidth narrow enough to not address any atoms in sites with occupation
greater than one, and could improve the accuracy of future experiments. However,as the bandwidth of the pulse is increased, sites with higher than unity occupation
will begin to affect the signal, and precision will be lost.
In one experiment, very short (5 ms) pulses were used. The bandwidth of thesepulses was large enough (>150 Hz) to address all the atoms in the lattice. If the time
between the pulses was varied (instead of the frequency), the resulting fringe patternconsisted of a beat note between the atoms in the n = 1, 2, and 3 sites.

5.8

Conclusion

In conclusion, we have implemented an atomic clock with an atomic cloud trapped in
the MI phase of a 3D optical lattice. While our demonstration is not competitive withcurrent, state-of-the-art atomic clocks, the density discretization technique shown
here could be used to increase the precision of optical frequency Ca, Yb, or Sr clocks
currently under development [49, 76, 77, 80, 31, 32].

94

Chapter 6
Conclusion
Looking back on my work here I feel satisfied that I tackled three extremely different
projects and had at least some success with each. The hard disk platter project
took well over a year, with many trials and tribulations. It was very much my baby
from start to end, and while I take pride in what I was able to accomplish with it,
I can only wish that the disk itself had been more cooperative. While it is indeed
a significant improvement over many other atom chip implementations, it certainly
wasn’t the next-generation leap I had been hoping for.
The dressed Zeeman trap was almost a fling in comparison. Trying to accomplish
the proof-of-principle project for almost zero dollars was a wonderful exercise in begging, borrowing, and stealing. I was in the middle of writing a computer program tocontrol the synthesizer chip for the next phase of the experiment when we had somediscussions as a group and decided that the possible scientific payoff wasn’t great
enough to delay the optical lattice project any longer. I still feel that it was the rightdecision, but it was definitely a fun project while it lasted, and achieving mid-term
results almost an order of magnitude better than reported anywhere else didn’t hurt.
The optical lattice clock was a great note on which to end the show. The machine
was never really stable enough until after about 2 A.M., so I had plenty of time alone
to meditate with the machine and contemplate my existence in the dark. It wasalso a full-bodied chance to make a precision measurement - well, at least precisioncompared to my other, even more proof of principle experiments.
95

An entirely new crew of students now runs the machine: Jongchul Mun, Patrick
Medley, David Hucul, and soon post-doc David Weld. Hopefully not too many components will turn into mysterious black boxes, and I hope my contributions have been
lasting ones.

96

Appendix A
Zeeman Slower ProgramThis appendix contains the source code for the program I used to design the Zeeman
slower. The constants in the header must be modified to suit any particular new
applications, and can be compiled in any ANSI C environment. It outputs several files,
two of which are of particular importance. The file “field7.dat” describes the desired
and theoretically created magnetic field profiles, and can be imported into Excel or
any other spreadsheet program using a tab delimited format. The file “windings.dat”
is text file which contains a drawing of the proposed slower cross section. This drawing
is for use during the winding of the solenoid, and was used to produce the Zeeman
slower used on our machine.
#i n c l u d e <math . h>#i n c l u d e <s t d i o . h>#i n c l u d e <s t d l i b . h>/∗∗∗∗∗∗∗∗∗∗ MATH CONSTANTS ∗∗∗∗∗∗∗∗∗∗∗/#d e f i n e PI 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9#d e f i n e MU0 1 2 . 5 6 6 3 7 0 6 1 4 4/∗ = 4 xPI
g a u s s ∗ mm / A ∗/#d e f i n e CPI 2 . 5 4/∗ cm p e r i n c h ∗//∗ a l l l e n g t h s i n mm ∗//∗∗∗∗∗∗∗∗ COIL PARAMETERS ∗∗∗∗∗∗∗∗∗∗∗∗/#d e f i n e INSIDE RADIUS 1 9 . 7/∗ The r a d i u s o f t h e i n n e r m o s t c o i l ∗/#d e f i n e COIL SPACING 3 . 5/∗ The t h i c k n e s s o f one l o o p ∗/#d e f i n e MAX SPACING 2 1 . 0/∗ 5 t u r n s i n between w i n d i n g s maximum ∗/#d e f i n e SPACING INC 0 . 5/∗ i n u n i t s o f w i r e d i a m e t e r s ∗/#d e f i n e MAX TURNS 345/∗ Maxmimum c u r r e n t l o o p s t o u s e i n s i m u l a t i o n ∗/#d e f i n e MAX SOLENOIDS 45/∗ Maxmimum s o l e n o i d s t o u s e i n s i m u l a t i o n ∗/#d e f i n e CURRENT 1 5 . 0/∗ C u r r e n t ( Amps ) i n s o l e n o i d 1 ∗/#d e f i n e CURRENT 2 1 0 . 0#d e f i n e CURRENT 3 3 0 . 0#d e f i n e RATIO 1 0 . 9 8/∗ F i e l d t a p e r p e r c e n t a g e ( e m p i r i c a l ) ∗/#d e f i n e RATIO 2 0 . 9 8#d e f i n e RATIO 3 0 . 9 8/∗ L a s t l o o p o f low c u r r e n t s e c t i o n ( i n memory ) ∗ /#d e f i n e LOW C 15
#d e f i n e MED C 30/∗ L a s t l o o p o f medium c u r r e n t s e c t i o n ( i n memory ) ∗ /#d e f i n e HIGH C 45/∗ L a s t l o o p o f h i g h c u r r e n t s e c t i o n ( i n memory ) ∗ /#d e f i n e LOW P 0 . 4 5/∗ P e r c e n t a g e l e n g t h o f low c u r r e n t s e c t i o n ( e m p i r i c a l ) ∗/#d e f i n e MED P 0 . 3/∗ P e r c e n t a g e l e n g t h o f medium c u r r e n t s e c t i o n ∗/#d e f i n e SLOWER START 150/∗ S t a r t i n g p o s i t i o n o f t h e s l o w e r i n mm from t h e MOT c e n t e r ∗/#d e f i n e SLOWER LENGTH 920/∗ Length i n mm o f t h e t o t a l s l o w e r ∗/#d e f i n e MAX LENGTH 1200/∗ Length i n mm o f p h y s i c a l s i m u l a t i o n ∗/#d e f i n e FIT BUFFER 25/∗ Sample r a n g e where t h e sim w a t c h e s t h e B f i e l d from 1 c o i l ∗/97/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗∗∗∗∗∗∗∗ FIELD PARAMETERS ∗∗∗∗∗∗∗∗∗∗∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗ S t a r t i n g p o i n t o f t h e B f i e l d t h a t m a t t e r s i n#d e f i n e Z ZERO 200#d e f i n e Z ONE 1070/∗ Ending p o i n t o f t h e B f i e l d t h a t m a t t e r s ∗/#d e f i n e B BIAS 0 . 0/∗ Amount o f b i a s f i e l d t o add ( n o t u s e d ) ∗/t h e sim ∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗∗∗∗∗∗∗∗ EXPERIMENTAL PARAMETERS ∗∗∗∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗ a c c e l e r a t i o n s a f e t y f a c t o r ∗/#d e f i n e A MAX 0 . 6#d e f i n e FINAL VELOCITY 2 0 . 0/∗ m/ s ∗/#d e f i n e ACCELERATION 1 1 2 1 2 2 .
/∗ m e t e r s / s e c ˆ2 from photon r e c o i l ∗//∗ h e r t z p e r g a u s s ∗/#d e f i n e BOHR MAGNETON 1 4 0 0 0 0 0 . 0#d e f i n e WAVELENGTH 0 . 0 0 0 0 0 0 7 8 0/∗ m e t e r s ∗/#d e f i n e GAUSSMS 1 . 0 / (BOHR MAGNETON ∗ WAVELENGTH) / ∗ g a u s s p e r m e t e r p e r s e c o n d ∗/#d e f i n e EPSILON 0 . 0 0 0 0 0 1/∗ some s m a l l v a l u e ∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ FUNCTIONS ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
void i n i t i a l i z e ( void ) ;
void
void
void
void
int

make field ( float ∗ f i e l d ) ;make array field ( float ∗ f i e l d );copy field ( f l o a t ∗ target , f l o a t ∗ source ) ;f i t s o l e n o i d s ( void ) ;fit solenoid ( int a , float ratio );

void
void

add dz section ( int radius ,m a k e t r u e f i e l d ( void ) ;

float
float
float
float
float
float
float

eval accel ( float ∗ field );accel ( int z , float ∗ f i e l d );eval b ( float ∗ f i e l d ) ;getmax ( i n t s t a r t , i n t end , f l o a t ∗ f i e l d ) ;g e t m i n ( i n t s t a r t , i n t end , f l o a t ∗ f i e l d ) ;m i n d e l t a b ( i n t s t a r t , i n t end , f l o a t ∗ h i g h , f l o a t ∗ low ) ;b r a t i o ( i n t s t a r t , i n t end , f l o a t ∗ h i g h , f l o a t ∗ low , f l o a t

void
void
void
void

s a v e s o l e n o i d s ( char ∗ filename ) ;s a v e f i e l d ( char ∗ filename , f l o a t ∗ f i e l d ) ;s a v e s o l u t i o n ( char ∗ filename ) ;draw solenoid ( char ∗ filename ) ;i n t pos ,float

spacing ,float

current ,float ∗ field );bias ) ;f l o a t rndnum ( v o i d ) ;typedef struct {f l o a t current ;f l o a t s p a c i n g [MAX LENGTH ] ;int layer ;int start z ;i n t max length ;int length ;} solenoid type ;/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ GLOBAL VARIABLES ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
s o l e n o i d t y p e s o l e n o i d [ MAX SOLENOIDS ] ;
float
float
float
float
float

f i e l d [MAX LENGTH ] ;/∗ The w o r k i n g copy o f t h e B f i e l d ∗/t r u e f i e l d [MAX LENGTH ] ;/∗ The B f i e l d g e n e r a t e d from s c r a t c h ∗/d e s i r e d [MAX LENGTH ] ;/∗ The t h e o r e t i c a l , d e s i r e d f i e l d ∗/bzero ;
w i n d i n g [MAX TURNS ] [ MAX SOLENOIDS+ 1 ] ;i n t main ( ) {initialize ();/∗ make d e s i r e d f i e l d and d e f i n e s o l e n o i d c u r r e n t s ∗/fit solenoids ();s a v e s o l u t i o n (” f i e l d 7 . dat ” ) ;s a v e s o l e n o i d s (” s o l e n o i d s . dat ” ) ;s a v e f i e l d (” f i e l d . dat ” , f i e l d ) ;/∗ S a v e s d e s i r e d and a c t u a l f i e l d s , and e r r o r . ∗/d r a w s o l e n o i d (” windings . dat ” ) ;/∗ Drawn s o l e n o i d w i n d i n g p a t t e r n . ∗/}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗FIT ALL SOLENOIDS TO THE FIELD
void f i t s o l e n o i d s ( void ) {∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/

int a ;
int b;
i n t pause ;i n t check ;int lastlayer ;check = 1 ;/∗∗∗ FIT LOW CURRENT SOLENOIDS ∗∗∗/
f o r ( a =0; a<LOW C && c h e c k == 1 ; a++) {s o l e n o i d [ a ] . s t a r t z = SLOWER LENGTH ∗ ( 1 − LOW P) + SLOWER START ;98s o l e n o i d [ a ] . m a x l e n g t h = SLOWER LENGTH ∗ LOW P ;solenoid [ a ] . layer = a ;s o l e n o i d [ a ] . c u r r e n t = CURRENT 1 ;solenoid [ a ] . length = 0;c h e c k = f i t s o l e n o i d ( a , RATIO 1 ) ;i f ( c h e c k ) p r i n t f ( ” S o l e n o i d %d : Length : %d\n ” , a +1 , s o l e n o i d [ a ] . l e n g t h ) ;i f ( c h e c k == 2 ) p r i n t f ( ” \ t H i g h End Trimmed . \ n ” ) ;}check = 1 ;/∗∗∗ FIT MEDIUM CURRENT SOLENOIDS ∗∗∗/
f o r ( a=LOW C; a<MED C && c h e c k == 1 ; a++) {s o l e n o i d [ a ] . s t a r t z = SLOWER LENGTH ∗ (1 −(LOW P+MED P) ) + SLOWER START ;s o l e n o i d [ a ] . m a x l e n g t h = SLOWER LENGTH ∗ MED P ;s o l e n o i d [ a ] . l a y e r = a−LOW C;
s o l e n o i d [ a ] . c u r r e n t = CURRENT 2 ;solenoid [ a ] . length = 0;c h e c k = f i t s o l e n o i d ( a , RATIO 2 ) ;i f ( c h e c k ) p r i n t f ( ” S o l e n o i d %d : Length : %d\n ” , a +1 , s o l e n o i d [ a ] . l e n g t h ) ;i f ( c h e c k == 2 ) p r i n t f ( ” \ t H i g h End Trimmed . \ n ” ) ;}check = 1 ;/∗∗∗ FIT HIGH CURRENT SOLENOIDS ∗∗∗/
f o r ( a=MED C ; a<MAX SOLENOIDS && c h e c k == 1 ; a++) {s o l e n o i d [ a ] . s t a r t z = SLOWER START ;s o l e n o i d [ a ] . m a x l e n g t h = SLOWER LENGTH ∗ (1 −(MED P+LOW P ) ) ;s o l e n o i d [ a ] . l a y e r = a−MED C ;s o l e n o i d [ a ] . c u r r e n t = CURRENT 3 ;solenoid [ a ] . length = 0;c h e c k = f i t s o l e n o i d ( a , RATIO 3 ) ;i f ( c h e c k ) p r i n t f ( ” S o l e n o i d %d : Length : %d\n ” , a +1 , s o l e n o i d [ a ] . l e n g t h ) ;i f ( c h e c k == 2 ) p r i n t f ( ” \ t H i g h End Trimmed . \ n ” ) ;}}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗FIT AN INDIVIDUAL SOLENOID TO THE DESIRED FIELD WITHIN GIVEN LIMITS ∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
int fit solenoid ( int a , float ratio ) {int layer ;int z ;
int modified ;int buffer ;int under buffer ;i n t pos ;int beginning ;f l o a t temp [MAX LENGTH ] ;f l o a t spacing ;f l o a t current ;float bratio ;float limit ;float crit ratio ;float radius ;float bias ;f l o a t d e l t a [MAX LENGTH ] ;f l o a t t e m p f i e l d [MAX LENGTH ] ;modified = 0;s p a c i n g = COIL SPACING ;current = solenoid [ a ] . current ;crit ratio = 0.0;
r a d i u s = s o l e n o i d [ a ] . l a y e r ∗ COIL SPACING + INSIDE RADIUS ;f o r ( z =0; z<MAX LENGTH; z++) {temp field [ z ] = 0.0;
delta [ z ] = desired [ z ] − f i e l d [ z ] ;}f o r ( z = 0 ; ( s p a c i n g < MAX SPACING) && ( z < s o l e n o i d [ a ] . m a x l e n g t h ) ; z++) {p o s = s o l e n o i d [ a ] . s t a r t z +z ;i f ( z < FIT BUFFER ){i f ( a >= MED C) {b u f f e r = FIT BUFFER∗2 − z ;u n d e r b u f f e r = FIT BUFFER ;}elseb u f f e r = FIT BUFFER ;u n d e r b u f f e r = FIT BUFFER ;beginning = 1;}
else {
b u f f e r = FIT BUFFER ;u n d e r b u f f e r = FIT BUFFER ;beginning = 0;}/∗ t r yi f ( z + b u f f e r > s o l e n o i d [ a ] . max length )

99

{

to

f i t max f i e l d∗/b u f f e r = s o l e n o i d [ a ] . max length − z ;}b r a t i o = b r a t i o ( pos−u n d e r b u f f e r ,p o s+b u f f e r ,delta ,

temp field ,0.0);

i f ( ( b e g i n n i n g ) && ( b r a t i o >= 0 . 3 ) && ( a < MED C) ) {modified = 2;
break ;}i f ( b r a t i o >= 1 . 0 )break ;i f ( bratio > c r i t r a t i o ) {s p a c i n g = COIL SPACING / pow (1.0 − b r a t i o ,c r i t r a t i o = bratio ;}i f ( s p a c i n g > MAX SPACING)
add dz section ( radius ,add dz section ( radius ,0.5);

s p a c i n g = MAX SPACING ;solenoid [ a ] . start z + z ,solenoid [ a ] . start z + z ,spacing ,spacing ,current ,current ,field );
temp field );solenoid [ a ] . spacing [ z ] = spacing ;modified = 1;}
i f ( ( m o d i f i e d ) && ( z >= 1 0 ) ) {solenoid [ a ] . length = z ;f o r ( ; z<MAX LENGTH; z++)solenoid [ a ] . spacing [ z ] = 0 . 0 ;return ( modified ) ;}p r i n t f ( ” ( Some S o l e n o i d s Unused ! ) \ n ” ) ;solenoid [ a ] . length = 0;
solenoid [ a ] . current = 0 .0;
return ( 0 ) ;}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗ ADD THE FIELD FROM A TINY SOLENOID SECTION ∗//∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
v o i d a d d d z s e c t i o n ( i n t r a d i u s , i n t pos , f l o a t s p a c i n g , f l o a t c u r r e n t , f l o a t ∗ f i e l d ) {int z ;
f l o a t junk ;f l o a t rsq ;f l o a t twospacing ;rsq = radius ∗ radius ;j u n k = MU0 ∗ c u r r e n t ∗ r s q ;twospacing = 2 ∗ spacing ;f o r ( z =0; z<MAX LENGTH; z++)f i e l d [ z ] += j u n k / ( t w o s p a c i n g ∗ pow ( r s q + pow ( z−pos , 2 ) , 1 . 5 ) ) ;}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗ MAKE THE WHOLE FIELD FROM SCRATCH
void m a k e t r u e f i e l d ( void ) {∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/

int a ;
int z ;
i n t pos ;float

radius ;p r i n t f ( ” Remaking

field . . . \ n”);

f o r ( a =0; a<MAX LENGTH; a++)truefield [a] = 0.0;
f o r ( a =0; a<MAX SOLENOIDS ; a++)i f ( solenoid [ a ] . current > 0.0) {p r i n t f ( ” \ tAdding s o l e n o i d %d . . . \ n ” , a + 1 ) ;r a d i u s = s o l e n o i d [ a ] . l a y e r ∗ COIL SPACING + INSIDE RADIUS ;f o r ( z =0; z<s o l e n o i d [ a ] . l e n g t h ; z++)i f ( ( s o l e n o i d [ a ] . s p a c i n g > 0 ) && ( s o l e n o i d [ a ] . s p a c i n g [ z ] <= MAX SPACING ) )a d d d z s e c t i o n ( r a d i u s , s o l e n o i d [ a ] . s t a r t z +z ,solenoid [ a ] . spacing [ z ] , solenoid [ a ] . current , t r u e f i e l d ) ;}p r i n t f ( ” . . . done . \ n ” ) ;}/∗∗∗∗∗∗∗∗∗∗FIND MAXIMUM RATIO BETWEEN DESIRED AND ACTUAL FIELDS∗∗∗∗∗∗∗∗∗∗∗∗/
f l o a t b r a t i o ( i n t s t a r t , i n t end , f l o a t ∗ h i g h , f l o a t ∗ low , f l o a t b i a s ) {int z ;
f l o a t temp ;f l o a t max ;i f ( s t a r t < 0)start = 0;100

i f ( end > MAX LENGTH) end = MAX LENGTH;
max = 0 . 0 ;f o r ( z=s t a r t ; z<end ; z++) {i f ( ( h i g h [ z ] − b i a s ) == 0 ) c o n t i n u e ;temp = ( low [ z ]− b i a s ) / ( h i g h [ z ]− b i a s ) ;i f ( temp > max ) max = temp ;}r e t u r n ( max ) ;}/∗∗∗∗∗∗∗∗FIND MAXIMUM DIFFERENCE BETWEEN DESIRED AND ACTUAL FIELDS
f l o a t m i n d e l t a b ( i n t s t a r t , i n t end , f l o a t ∗ h i g h , f l o a t ∗ low ) {∗∗∗∗∗∗∗∗∗∗∗/

int z ;
f l o a t temp ;f l o a t min ;i f ( s t a r t < 0) s t a r t = 0 ;i f ( end > MAX LENGTH) end = MAX LENGTH;
min = 0 . 0 ;f o r ( z=s t a r t ; z<end ; z++) {temp = h i g h [ z ] − low [ z ] ;i f ( temp < min ) min = temp ;}i f ( min > 0 ) min = 0 ;r e t u r n ( min ) ;}/∗∗∗∗∗∗∗∗ GET MAXIMUM FIELD IN RANGE ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
f l o a t getmax ( i n t s t a r t , i n t end , f l o a t ∗ f i e l d ) {int z ;
f l o a t temp ;i f ( s t a r t < 0) s t a r t = 0 ;i f ( end > MAX LENGTH) end = MAX LENGTH;
temp = f i e l d [ s t a r t ] ;f o r ( z=s t a r t ; z<end ; z++)i f ( f i e l d [ z ] > temp ) temp = f i e l d [ z ] ;r e t u r n ( temp ) ;}/∗∗∗∗∗∗∗∗ GET MINIMUM FIELD IN RANGE ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
f l o a t g e t m i n ( i n t s t a r t , i n t end , f l o a t ∗ f i e l d ) {int z ;
f l o a t temp ;i f ( s t a r t < 0) s t a r t = 0 ;i f ( end > MAX LENGTH) end = MAX LENGTH;
temp = f i e l d [ s t a r t ] ;f o r ( z=s t a r t ; z<end ; z++)i f ( f i e l d [ z ] < temp ) temp = f i e l d [ z ] ;r e t u r n ( temp ) ;}/∗∗∗∗∗∗∗∗∗∗∗ EVALUATE THE CURRENT FIELD∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗//∗ T h i s f u n c t i o n r e t u r n s t h e Chi ˆ2 e r r o r sum betwe en t h e d e s i r e d and a c t u a lfloat eval b ( float ∗ f i e l d ){f i e l d s . ∗/int z ;
float errorsq = 0.0;
f o r ( z=SLOWER START ; z<Z ONE ; z++)e r r o r s q += pow ( f i e l d [ z ]− d e s i r e d [ z ] , 2 ) / 1 0 0 ;return ( errorsq ) ;}/∗∗∗∗∗∗∗∗∗∗∗ EVALUATE THE ACCELERATION FIT
float eval accel ( float ∗ field ) {∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/

float accl ;int z ;
f l o a t temp ;accl = 0;
f o r ( z=Z ZERO ; z<Z ONE ; z++) {temp = a c c e l ( z , f i e l d ) ;i f ( temp > ACCELERATION ∗ A MAX)a c c l += pow ( temp − ACCELERATION ∗ A MAX, 2 ) / 1 0 0 0 0 ;}return ( accl ) ;}/∗∗∗∗∗∗∗ CALCULATE THE ACCELERATION AT ONE POINT∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
float accel ( int z , float ∗ f i e l d ) {r e t u r n ( 1 0 0 0 . 0 ∗ WAVELENGTH ∗ BOHR MAGNETON ∗ ( f i e l d [ z ]− f i e l d [ z + 1 ] ) ∗ ( b z e r o − f i e l d [ z ] ) ) ;}101/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ COPY THE FIELD∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
v o i d c o p y f i e l d ( f l o a t ∗a , f l o a t ∗b )
{
int

i;

f o r ( i =0; i <MAX LENGTH; i ++)a[ i ] = b[ i ];}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ MAKE A RANDOM NUMBER 0 − 1
f l o a t rndnum ( v o i d ) {f l o a t r = RAND MAX;r e t u r n ( rand ( ) / r ) ;}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗INITIALIZE VARIABLES
void i n i t i a l i z e ( void ) {∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/

int a ;
int b;/∗∗∗∗∗∗∗∗∗∗∗ MAKE DESIRED FIELD ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
b z e r o = B BIAS + GAUSSMS ∗ s q r t ( pow ( FINAL VELOCITY , 2 ) +2 ∗ ACCELERATION ∗ A MAX∗ ( Z ONE−Z ZERO ) / 1 0 0 0 ) ;f o r ( a =0; a<Z ZERO ; a++)d e s i r e d [ a ] = b z e r o − GAUSSMS ∗ a b s (FINAL VELOCITY ) ;f o r ( ; a<Z ONE ; a++)d e s i r e d [ a ] = b z e r o − GAUSSMS ∗ s q r t ( pow ( FINAL VELOCITY , 2 ) +2 ∗ ACCELERATION ∗ A MAX ∗ ( a−Z ZERO ) / 1 0 0 0 ) ;f o r ( ; a<MAX LENGTH; a++)d e s i r e d [ a ] = B BIAS ;/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
f o r ( a =0; a<MAX LENGTH; a++) {field [a] = 0.0;
truefield [a] = 0.0;}/∗

initialize

initialf o r ( a =0; a<MAX SOLENOIDS ; a++) {/∗ f i l l s o l e n o i dsolenoid [ a ] . length = 0;
solenoid [ a ] . current = 0. 0;
f o r ( b =0;b<MAX LENGTH; b++)s o l e n o i d [ a ] . s p a c i n g [ b ] = COIL SPACING ;
}

field

data with∗/d e f a u l t s ∗/}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗ SAVE SOLENOID CONFIGURATION
void s a v e s o l e n o i d s ( char ∗ filename ) {∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/

int a ;
int b;
int inc ;FILE ∗ o u t f i l e ;o u t f i l e = f o p e n ( f i l e n a m e , ”w ” ) ;f p r i n t f ( o u t f i l e ,”%d\ t%d\n ” ,MAX SOLENOIDS,MAX LENGTH ) ;f o r ( a =0; a<MAX SOLENOIDS ; a++) {inc = 9;f p r i n t f ( o u t f i l e ,”%d\ t %3.0 f \ t ” , s o l e n o i d [ a ] . l e n g t h , s o l e n o i d [ a ] . c u r r e n t ) ;f o r ( b =0;b<s o l e n o i d [ a ] . l e n g t h ; b++) {i n c ++;i f ( i n c >= 1 0 ) {f p r i n t f ( o u t f i l e ,”%4.1 f \ t ” , s o l e n o i d [ a ] . spacing [ b ] ) ;i n c −= 1 0 ;}}f p r i n t f ( outfile ,”\n ”);}
fclose ( outfile );}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗ SAVE ANY FIELD∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
void s a v e f i e l d ( char ∗ filename , f l o a t ∗ t e m p f i e l d ) {int a ;FILE ∗ o u t f i l e ;o u t f i l e = f o p e n ( f i l e n a m e , ”w ” ) ;f p r i n t f ( o u t f i l e ,”%d\ t %8.2 f \n ” ,MAX LENGTH, b z e r o ) ;f o r ( a =0; a<MAX LENGTH; a++)f p r i n t f ( o u t f i l e ,”% f \n ” , t e m p f i e l d [ a ] ) ;fclose ( outfile );}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ SAVE THE GENERAL SOLUTION
void s a v e s o l u t i o n ( char ∗ filename ) {∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/

int a ;FILE ∗ o u t f i l e ;102

o u t f i l e = f o p e n ( f i l e n a m e , ”w ” ) ;f p r i n t f ( o u t f i l e , ” z \ t f i e l d \ t d e s i r e d \ t d e l t a b \ t t r u e f i e l d \ t a c c e l e r a t i o n \ t e v a l a c c e l \n ” ) ;f o r ( a =0; a<MAX LENGTH; a++)f p r i n t f ( o u t f i l e ,”%d\ t%f \ t%f \ t%f \ t%f \ t%f \ t%f \n ” ,a , f i e l d [ a ] , d e s i r e d [ a ] , t r u e f i e l d [ a ] , f i e l d [ a]− d e s i r e d [ a ] ,a c c e l ( a , f i e l d ) ,ACCELERATION ) ;fclose ( outfile );s a v e f i e l d (” d e s i r e d f i e l d . dat ” , d e s i r e d ) ;}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DRAW THE SOLENOID STRUCTURE ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
void draw solenoid ( char ∗ filename ) {int a ;
int z ;
f l o a t curr ;f l o a t pos ;f l o a t inc ;FILE ∗ o u t f i l e ;o u t f i l e = f o p e n ( f i l e n a m e , ”w ” ) ;c u r r = CURRENT 1 ;f p r i n t f ( o u t f i l e , ” % 5 . 2 f \n ” , s o l e n o i d [ 0 ] . c u r r e n t ) ;f o r ( a =0; a<MAX SOLENOIDS ; a++) {i f ( solenoid [ a ] . current > 0.0){
pos = 1 . 0 / s o l e n o i d [ a ] . s p a c i n g [ 0 ] ;inc = 0;i f ( c u r r != s o l e n o i d [ a ] . c u r r e n t ) {f p r i n t f ( o u t f i l e , ”C%5.2 f \n ” , s o l e n o i d [ a ] . c u r r e n t ) ;curr = solenoid [ a ] . current ;}f p r i n t f ( o u t f i l e , ”O” ) ;f o r ( z =0; z<s o l e n o i d [ a ] . l e n g t h ; z++) {p o s += 1 . 0 / s o l e n o i d [ a ] . s p a c i n g [ z ] ;i f ( i n c >= COIL SPACING ) {i n c −= COIL SPACING ;i f ( p o s >= 1 . 0 ) {f p r i n t f ( o u t f i l e , ”O” ) ;pos −−;}
else{
fprintf ( outfile ,” ”);}}
i n c ++;}
f p r i n t f ( outfile ,”\n ”);}}
fclose ( outfile );}103

Appendix B
The Onset of Matter-Wave
Amplification in a SuperradiantBose-Einstein Condensate
This appendix contains a reprint of Ref. [62]: Dominik Schneble, Yoshio Torii, Micah
Boyd, Erik W. Streed, David E. Pritchard, and Wolfgang Ketterle, The Onset of
Matter-Wave Amplification in a Superradiant Bose-Einstein Condensate, Science 300,
475 (2003). While not discussed in detail in this thesis, this was one of the first experiments performed on the the rubidium BEC machine.

104

REPORTS

The Onset of Matter-Wave
Amplification in a SuperradiantBose-Einstein Condensate
Dominik Schneble,* Yoshio Torii,† Micah Boyd, Erik W. Streed,David E. Pritchard, Wolfgang Ketterle
The interaction of short and strong laser pulses with an atomic Bose-Einsteincondensate is found to generate patterns of recoiling atoms that are different
from those seen in previous light-scattering experiments. This phenomenon can
only be explained by optical stimulation, showing that the previous descriptionof superradiance as atomic stimulation is incomplete and that matter-wave
amplification in Bose-Einstein condensates is suppressed at short times. Our
experiments clarify the nature of bosonic stimulation in the four-wave mixingof light and atoms.
Superradiance is the collective emission oflight from an ensemble of excited atoms andinvolves the spontaneous buildup of coherence in a macroscopic ensemble of atoms.
Dicke, in his seminal work (1), showed thatthe spontaneous emission of photons from anensemble of N localized atoms is enhanced
by a factor of N over the single-atom emission rate when the atoms are in a cooperative
state with all the transition moments in phase.
If, however, the atomic ensemble extends
over more than the wavelength of the radiation, the situation becomes more complex.
Superradiant emission is then only possible
along certain directions where a phasematching condition is fulfilled. Furthermore,
there are now two physically distinct picturesof the collective emission process in which
the distinction between spontaneous emission
and stimulated emission is blurred (2). In the
viewpoint introduced by Dicke, the entire
extended ensemble interacts with empty
modes of the electromagnetic field, and all
photons are emitted spontaneously, albeit in acooperative way. In the other picture, some
atoms emit spontaneously, and this radiation
stimulates other atoms to emit in phase. The
equivalence of these two pictures emphasizes
that, if a system emits many photons cooperatively, a strict distinction between spontaneous and stimulated emission is no longer
possible. The phenomenon of superradiancethus deepens and clarifies our understandingof the interaction of matter and light.
Because of their coherent nature, BoseEinstein condensates readily exhibit collective superradiant behavior. Controversy reMassachusetts Institute of Technology–Harvard Center for Ultracold Atoms, Research Laboratory of Electronics and Department of Physics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA.
*To whom correspondence should be addressed. Email: schneble@mit.edu†Present address: Department of Physics, Universityof Tokyo, 3-8-1 Komaba, Meguro-Ku, Tokyo, Japan.

garding the role of quantum statistics in superradiance was resolved in (3, 4) in which it
was shown that processes attributed tobosonic stimulation can also be expected insuperradiant Fermi systems. An important
application of superradiance has been the coherent amplification of matter waves (5, 6).
We study superradiance in the short-pulse
limit and show that amplification is attenuated at short times, leading to a more profoundunderstanding of matter-wave amplification.
When a Bose-Einstein condensate is illuminated by a single laser beam ( pump beam),light can be scattered into an empty mode.
The condensate at rest, “dressed” by thepump beam, corresponds to the excited electronic state in the usual superradiance. Spontaneous emission is now replaced by a spontaneous Rayleigh scattering process that leads
to a recoiling atom. This atom interferes withthe atoms at rest and creates a periodic density modulation that, in turn, leads to anincrease of the scattering rate, thus amplifying itself. Because the square of the density
modulation is proportional to the number ofrecoiling atoms, one can describe this process
as matter-wave amplification due to bosonicstimulation by atoms (5–7). Here, superradiance corresponds to the diffraction of thepump beam from a matter-wave grating in thecondensate volume. In an elongated condensate, the gain is strongest when the photons
leave the condensate along its long axis in theso-called endfire mode (1), whereas the atoms recoil at an angle of 45° with respect tothe axis and emerge as a distinct atomic cloud(forward peak) with a momentum component
parallel to the pump beam. The theoretical
literature on superradiant scattering in condensates (5, 7–12) corroborates this description, but here we show that it is incomplete.
Our new experimental results suggest analternative picture for superradiance in which
the endfire beam and the pump beam form an
optical standing wave and diffract atoms in a

stimulated scattering process between momentum states. The ambiguity between spontaneous and stimulated emission now reappears as the ambiguity of whether light isdiffracted from a matter-wave grating orwhether atoms are diffracted from a lightgrating. In the latter picture, the forward
peaks in superradiance correspond to Bragg
diffraction of atoms (13, 14). In the present
work, we observed the diffraction of atoms
into additional states with momentum components antiparallel to the pump beam (backward peaks). This is possible in the regime ofshort pulses in which several recoil states are
quasi-degenerate [Kapitza-Dirac diffraction(15, 16)]. The backward peaks result fromthe reabsorption of spontaneously scatteredphotons and stimulated emission back into
the pump beam. This off-resonant process,which cannot be described by the Dicke picture of collective spontaneous emission, both
limits the growth of the endfire mode andsuppresses matter-wave amplification by destructive interference.
The experiments discussed here were performed with a 87Rb condensate in the F ! 1,
mF ! "1# hyperfine ground state, where F
and mF are the quantum numbers for the total
spin and its z component, respectively. We
collected 1010 atoms in a magneto-opticaltrap from a slow atomic beam with a flux of1011 s"1. The atoms were cooled further indark optical molasses and evaporatively
cooled in a cloverleaf-type Ioffe-Pritchardmagnetic trap. The trapped, almost-pure condensates contained about two million atoms
and had Thomas-Fermi radii of 100 $m and
7.5 $m along the axial and radial directions,
respectively. To induce superradiant scattering, we illuminated the condensate with an
off-resonant pump laser beam (center intensity of 63 m W cm"2 and e"2 radius of 1.6mm) for a variable pulse duration %. The
linearly polarized beam was incident horizontally along a radial direction with its electricfield vector perpendicular to the axial direction (compare Figs. 1 and 2A). Immediately
after applying the laser pulse, the atomic
momentum distribution was analyzed byswitching off the magnetic trap and imaging
the atomic ensemble after 30 ms of ballistic
expansion. Absorption images were obtainedwith the use of a vertical probe beam after theatoms had been optically pumped into theF ! 2 ground state (Fig. 1). In case A for
short pulses, the Rayleigh scattering rate &sc
was chosen to be comparable to the singlephoton recoil frequency 'r ! 2.4 ( 104 s"1
so that a large fraction of condensate atoms
were scattered on a time scale that was shortcompared to the recoil time 'r"1 ! 42 $s. In
this case, we observed a nearly symmetric,
X-shaped pattern. For comparison, when the
detuning was chosen to be one order of magnitude larger (case B), the pattern evolved on

www.sciencemag.org SCIENCE VOL 300 18 APRIL 2003

475

REPORTS
a time scale that was one order of magnitude
above the recoil time, and the familiar superradiant scattering fan appeared, as in (8).
The scattering processes that give rise to theobserved symmetric pattern are illustrated inFig. 2A. In lowest order, forward peaks [label(1)] are generated when a pump photon with
momentum !k is scattered off a condensate
atom (zero momentum), thereby producing ascattered photon with momentum !(k " q) anda recoiling atom with momentum !q, q # $2
k. In contrast, backward peaks [label (2)] areproduced when a previously scattered photon isscattered back into the pump mode, thereby
transferring a condensate atom into a recoilingatom with momentum "!q. Whereas in thefirst case energy is conserved, the energy mismatch in the latter case is twice the recoil
energy, %E # 4!&r. Therefore this process canonly take place as long as the energy uncertainty !/' ( %E, that is, for pulse lengths ' up to)10 *s. At an early stage, when the endfire
modes do not yet dominate completely, one canclearly recognize the dipole pattern of spontaneous photon emission (Fig. 2B) as a halo ofrecoiling atoms with radius !k on the right-handside of the condensate. In addition, one also
observes another halo on the left-hand side.
This halo results from stimulated scattering of
the spontaneously emitted photons back into
the pump laser beam.

It is important to notice the slight left-rightasymmetry of the X-shaped scattering patternof Fig. 1A. There exist two endfire modes, q+
and q", propagating in opposite directions dueto the axial symmetry of the condensate, andtherefore both the backward and forward peaks
are grouped in pairs. However, the separation of
the backward peaks in the axial direction islarger than that of the forward peaks. Thissuggests that the scattering processes for eachof the two endfire modes occur near the edgesof the condensate (Fig. 2C) (17).
This observation allows one to distinguishthe two simple pictures in which light is diffracted from atoms or atoms are diffracted fromlight. The fact that the scattering process isstrongest near the edges of the condensate suggests that atoms are diffracted from an opticalstanding wave formed by the pump and endfire
modes: The intensity of the endfire modes andthus the intensity of the standing wave arehighest near the edges of the condensate. Conversely, if the scattering processes could simply
be described by diffraction of light from amatter-wave grating, one would expect the recoiling atoms to originate in the center where
the atomic density and the diffraction efficiencyof the grating are highest. This is indeed thecase in the long-pulse limit where the ThomasFermi profile of the condensate shows depletion in the center (inset of Fig. 1B).

Fig. 1. Superradiant scattering of a laser beam (arrow) from a Bose-Einsteincondensate in the shortpulse (A) and long-pulse(B) limit. Absorption images of the atomic density distribution were taken
after 30 ms of ballistic expansion. In case A, % was"420 MHz and ' was 6*s. In case B, % was"4400 MHz and ' was
800 *s. The inset in (B) isa magnification of the region around the condensate showing depletion of its center; for better visibility the contrast has
been enhanced by a factor of 1.5. Also shown is a profile of the average optical density in the centerregion. The field of view of both images is 2.0 mm by 2.0 mm, and that of the inset is 120 *m by
270 *m.

Fig. 2. Scattering processes (A) and scattering pattern (B) for a short (' # 3 *s) light pulse. The
circles and lines in (C) illustrate the origin of the dipole emission halos and the Kapitza-Diracdiffraction orders.

476

A more detailed analysis of the observed
recoil patterns confirms the picture of optical
diffraction. For short times &r' ,, 1, one canneglect the frequency difference between thepump light (+y direction) and the endfire
modes (-z direction), and their interference
leads to a modulated light intensity
I mod # 2E k E k!q" cosk. y " z/
" 2E k E k!q! cosk. y ! z/
" E k!q" E k!q! cos2kz(1)where Ek"q+ and Ek"q" are the field amplitudes of the two endfire modes withwavevectors k " q+ and k – q", respectively,
and Ek is the amplitude of the pump beam.
The first and second terms give rise to diffraction of atoms into momenta !qn # n$2!k along axes tilted at -45° with respect tothe long axis, with populations (15)
P n # J n 2(02R'), n # 0, -1, -2, . . . (2)Here the Jn are Bessel functions of the firstkind; 02R # &R,k &R,k"qi/2% is the twophoton Rabi frequency, given by the Rabi
frequencies &R,k and &R,k"qi (i # +, –) of
the laser beam and the two endfire modes;
and % is the detuning of the beams. Scattering
processes that involve more than one endfire
mode (such as those induced by the third termin Eq. 1) can be neglected as long as theendfire mode intensities are smaller than theintensity of the pump mode.
With the use of this model, we can determine the intensity of the individual endfire
modes. Above 5 *s, the population in theminus-second-order peak exceeds 50% ofthat in the minus-first-order peak. At 6 *s,
the ratio is 63%, which implies a two-photon
Rabi frequency 02R # 3.9 1 105 s"1 corresponding to an endfire-mode intensity of
0.8 m W cm"2. This intensity is more than
two orders of magnitude larger than the intensity of spontaneous Rayleigh scattering
into this mode (18).
A comparison of an extracted pattern to a
calculated Kapitza-Dirac pattern is given in Fig.
3B. There is good general agreement; however,a major discrepancy exists for the zeroth order,that is, for the condensate itself. The depletion of
the condensate is far less than predicted by themodel. This is easily explained by the fact thatthe scattering processes take place mainly near
the edges, leaving many condensate atoms in thecentral region. The model assumes an external
endfire laser beam, whereas the endfire mode is
produced in the condensate volume itself. Another discrepancy appears at longer times when' approaches the two-photon recoil time: The
resonant first-order peak p1 # $2 !k becomesdominant. The emergence of this asymmetry isa clear signature for the transition from the
Kapitza-Dirac to the Bragg regime, that is, forthe onset of energy conservation.

18 APRIL 2003 VOL 300 SCIENCE www.sciencemag.org

REPORTS
Before discussing the interplay between
optical and atomic stimulation in our shortpulse experiment, we want to emphasize thequalitative equivalence of these two picturesin the long-pulse limit. The superradiant gain
has been successfully interpreted as matterwave stimulation (7, 8), that is, as the diffraction of photons from a self-amplifying matter-wave grating. However, by extrapolating
the observed short-time behavior to longer
times, a purely optical interpretation of superradiance emerges: The endfire modes and thepump beam form a standing light wave fromwhich the atoms can diffract. For !r" ## 1,
the stimulated backscattering of endfire mode
photons into the pump beam is energeticallysuppressed, the endfire mode grows exponentially in time, and so does the diffraction
efficiency of the standing light wave. In thisinterpretation, superradiance in the longpulse limit is nothing but (self-stimulated)
Bragg scattering.
The equivalence of the two descriptions in
the long-pulse limit can be obtained in a morequantitative way by considering the rate for
Bragg scattering from the standing light wave
formed by the pump beam and an endfire
mode (neglecting propagation effects) (7):2/% 2
W B ! N$ 2R(3)Here N is the number of atoms in the condensate, and %2 is the width of the Bragg
resonance (19). The Rabi frequency of thepump beam can be expressed through its
Rayleigh scattering rate %sc as !R,k & 2'(%sc/%)1/2, where % is the natural linewidth. The
number of photons scattered in the coherencetime 1/%2 is identical to the number of recoiling atoms Nq inside the condensate volume.
This gives the Rabi frequency of the endfire
mode as !R,k(q & %[Nq%2)!/(2IsA)]1/2,
where Is & *)!%/3+2 is the saturation intensity and A is the cross-sectional area of thecondensate (+ is the wavelength, and ! is thefrequency of the transition). The result
W B ! % SC

3 +2
NN q ! GN q
2* A

(4)

is, up to a numerical factor, identical to theresult for the superradiant gain coefficient G
obtained in (8), in which the interference term
NNq was interpreted as the modulation depthof the matter-wave grating from which thepump photons diffract.
This equivalence between optical andatomic stimulation was not properly recognized in earlier treatments (7, 8) that werebased on the argument that the number ofphotons in the condensate volume was smaller than one. However, as already pointed outin (7), the relevant photons (and recoilingatoms) are all those produced within the coherence time 1/%2. Although for the atoms
the relevant mode volume is given by thecondensate itself, it is much bigger for the

scattered photons because the coherence
length of the endfire mode greatly exceeds
the length of the condensate. The fact thatphotons and recoiling atoms are produced inpairs implies that their effective numbers areequal and that atomic stimulation in one picture can be replaced by optical stimulation inanother picture.
We now apply these concepts to the shortpulse limit. At early times ", the scattering
pattern is symmetric, and therefore the creation of endfire-mode photons ( positive orders or forward peaks) is as probable as theirabsorption (negative orders or backward
peaks), resulting in a fully suppressed superradiant gain. Spontaneous Rayleigh scattering is only amplified at longer times when thepattern develops an asymmetry. We describe
this by attributing a Lorentzian lineshapewith a linewidth % , 1/" to the first andminus-first order (neglecting coherences).
The former is resonant, and the latter is detuned by the two-photon recoil shift - &'E/). This model predicts a suppression of
the gain by a factor of (2-/%)(2 , (-")(2.
With the use of a coherent model based on
driven two-level systems, we find a suppression by a factor of 12(-")(2, which has thesame quadratic time dependence. The suppression of the gain at short times can also be
understood in a purely atomic picture. An
optical standing wave creates a phase gratingin the atomic cloud, which transforms into adensity grating only after the inverse recoil
frequency, that is, the time it takes the atoms
to move half a wavelength. For a given number of recoiling atoms, the atomic gain is
initially zero and grows as (-")2, in agreement with the picture of optical amplification. It is the matter wave grating that gets
amplified, rather than the number of recoilingatoms itself: The density grating created by
the population Nq destructively interfereswith the population N(q in the backward
peak, and the gain is reduced.
Although all three models are qualitative,
they consistently predict a suppression of thegain by around two orders of magnitude forpulse lengths around 1 .s. For the typicalcondensate geometry in (8) and the present
experiment, the ratio of G as defined in Eq. 4
to %sc is about 4 / 103. At " 0 1 .s, we
expect an (attenuated) gain coefficient G 1
106 s(1 for %sc 0 !r. The observed magnitude for the endfire-mode intensity is consistent with such a low gain. The attenuation is
also evident when the scattering rate for the
first order, Nq 1 1010 s(1, is compared to theglobal Rayleigh scattering rate of the pump
beam. In the short-pulse limit, it is of thesame order, whereas in the long-pulse limit
an enhancement of one order of magnitude
was observed (8).
This new physical picture of superradianceat short times can be applied to the atom am-

plifier (5, 6) where a condensate illuminated bya pump beam is seeded with Nq(0) recoilingatoms, resulting in a density grating. This grating diffracts the pump light into the endfire
mode. At early times, the density modulation is
not amplified because the number of atoms
added to mode q is accompanied by an equal
number in mode (q, and the two additional
matter-wave gratings interfere destructively.
Therefore, the gain equation is Ṅq (t) & GNq(0),which implies linear growth. Exponential
growth only starts after the inverse recoil frequency, when the atoms in mode (q havedisappeared because of energy conservation.

Fig. 3. Comparison of superradiant scattering inthe short-pulse regime with Kapitza-Dirac diffraction. (A) The atom numbers in the image of Fig.
1A were integrated over a thin strip that contains
the condensate and the diffraction orders. The
strip is oriented along the direction from thebottom right to the top left of the image in Fig.
1A. The solid curve is a Gaussian representing thebroad background of incoherently scattered andresidual thermal atoms. a.u., arbitrary units. (B)Diffraction patterns for pulse durations " between
4 and 8 .s. These patterns were obtained bysubtracting the background from images similar
to those in Fig. 1A. At 6 .s, the corresponding
Kapitza-Dirac pattern for a standing light wave
formed by two crossed laser beams with a $2R of
3.9 / 105 s(1 is shown as a solid line. The widthsof the peaks (approximated by Gaussians) werechosen to fit the minus-first-order peak.

www.sciencemag.org SCIENCE VOL 300 18 APRIL 2003

477

REPORTS
In conclusion, by exploring the short-pulse
limit of superradiant Rayleigh scattering, wehave observed a previously unknown phenomenon in the scattering of a single laser beam byatoms: self-stimulated Kapitza-Dirac diffraction. The symmetry of the diffraction pattern
implies a reduced gain for both atom and light
amplification at early times. This realizes anovel regime of superradiance in which theatomic system has three almost equally spaced
levels with all of the initial population in the
middle one. This system has allowed us toillustrate important concepts of stimulation andsuperradiance. In the Dicke picture, all photons
are emitted spontaneously and simply leave thesystem. However, we directly observed stimulation by and absorption of these photons in theform of Kapitza-Dirac scattering, and therefore
the Dicke picture is incomplete for the shortpulse regime. None of the theoretical descriptions of superradiance in Bose-Einstein condensate (5, 7–12) included transitions to momentum states that appear as negative orders in
Kapitza-Dirac scattering.
Extrapolating our observations of
Kapitza-Dirac scattering to longer pulses
establishes the equivalence of self-stimulated Bragg diffraction and matter-wave amplification. There is a widespread opinion
that bosonic stimulation is a purely quantum-statistical effect based on symmetry,
whereas (3, 4 ) showed that bosonically enhanced scattering can always be traced
back to enhanced fluctuations. Here wehave directly shown how photons mediate
an effective interaction between the atoms
that, after elimination of the photon field,
results in bosonic stimulation of atoms.
However, at early times, the “stimulating
effect” of the population in mode q interferes destructively with the effect of atoms
in mode !q and the basic concept ofbosonic stimulation by atoms fails, whereasa simple picture of optical stimulation provides the correct answer (20).
References and Notes
1. R. H. Dicke, Phys. Rev. 93, 99 (1954).
2. M. Gross, S. Haroche, Phys. Rep. 93, 301 (1982).
3. W. Ketterle, S. Inouye, Phys. Rev. Lett. 86, 4203
(2001).
4. M. G. Moore, P. Meystre, Phys. Rev. Lett. 86, 4199
(2001).
5. S. Inouye et al., Nature 402, 641 (1999).
6. M. Kozuma et al., Science 286, 2309 (1999).
7. W. Ketterle, S. Inouye, in Bose-Einstein Condensates
and Atom Lasers, A. Aspect, J. Dalibard, Eds., vol. IV of
Comptes Rendus de l’Académie des Sciences Paris(Elsevier, Paris, 2001), pp. 339 –380.
8. S. Inouye et al., Science 285, 571 (1999).
9. M. G. Moore, P. Meystre, Phys. Rev. Lett. 83, 5202
(1999).
10. Ö. E. Müstecaplioglu, L. You, Phys. Rev. A 62, 063615(2000).
11. N. Piovella, R. Bonifacio, B. W. J. McNeil, G. R. M.
Robb, Opt. Commun. 187, 165 (2001).
12. E. D. Trifonov, J. Exp. Theor. Phys. 93, 969 (2001)[translation from E. D. Trifonov, Zh. Eksp. Teor. Fiz.
120, 1117 (2001)].

478

13. P. J. Martin, B. G. Oldaker, A. H. Miklich, D. E. Pritchard, Phys. Rev. Lett. 60, 515 (1988).
14. M. Kozuma et al., Phys. Rev. Lett. 82, 871 (1999).
15. P. L. Gould, G. A. Ruff, D. E. Pritchard, Phys. Rev. Lett.
56, 827 (1986).
16. Y. B. Ovchinnikov et al., Phys. Rev. Lett. 83, 284(1999).
17. For the 30-ms time of flight, the condensate expands by only 10% in the axial direction, as compared to more than a factor of 15 in the radial
direction.
18. The fractional solid angle subtended by the condensate is " # (d2/L2 )/4$, where d is its diameter and L
its length. The contribution of spontaneous Rayleighscattering to the endfire mode intensity thus is on
the order of " % (N&sc'() % 1/d2 ) 2.5 % 10!3 m
W cm!2.
19. J. Stenger et al., Phys. Rev. Lett. 82, 4569 (1999).
20. We would like to thank G. Campbell, A. Leanhardt,
and J. Steinhauer for a critical reading of the manuscript and P. Meystre and S. Inouye for discussions.
This work was supported by NSF.
7 February 2003; accepted 18 March 2003
Published online 27 March 2003;
10.1126/science.1083171
Include this information when citing this paper.

Mantle Fault Zone Beneath
Kilauea Volcano, Hawaii
Cecily J. Wolfe,1* Paul G. Okubo,2 Peter M. Shearer3
Relocations and focal mechanism analyses of deep earthquakes (!13 kilometers) at Kilauea volcano demonstrate that seismicity is focused on an active
fault zone at 30-kilometer depth, with seaward slip on a low-angle plane, andother smaller, distinct fault zones. The earthquakes we have analyzed predominantly reflect tectonic faulting in the brittle lithosphere rather than magma
movement associated with volcanic activity. The tectonic earthquakes may
be induced on preexisting faults by stresses of magmatic origin, althoughbackground stresses from volcano loading and lithospheric flexure may also
contribute.
Earthquake characteristics at the active Kilauea
volcano have long been used to construct models of this shield volcano and to constrain the
structure of its magmatic system (1–4). Basedon hypocenters in the catalog of the U.S. Geological Survey Hawaiian Volcano Observatory(HVO) seismic network, it has been suggested
that the earthquakes at Kilauea outline a pipelike magma conduit that extends through thelithosphere to depths as great as 60 km (1–4),
thereby delineating the magma pathway fromits source to the surface. However, the accuracy of routinely determined hypocenters islimited by factors including the network geometry, the available arrival time picks, picking errors, and the presence of three-dimensional velocity heterogeneity. Thus, relocations of earthquakes, using relative location
methods, which reduce the effects of velocity
heterogeneity, combined with precise crosscorrelation measurements, have produced
more accurate locations (5–11) and delineated fault structures. Here, we present relocations of deep earthquakes, focusing on distinguishing magmatic versus tectonic events tounderstand how the growing volcano affects
the crust and upper mantle.
1
Hawaii Institute of Geophysics and Planetology, University of Hawaii at Manoa, Honolulu, HI 96822, USA.
2
Hawaiian Volcano Observatory, U.S. Geological Survey, Hawaii National Park, HI 96718, USA. 3Scripps
Institution of Oceanography, University of California
at San Diego, La Jolla, CA 92093– 0225, USA.

*To whom correspondence should be addressed. Email: wolfe@hawaii.edu

Our data consist of waveforms from 1988 to
1998 in the catalog of routinely located deep
earthquakes recorded by the HVO seismic network, amounting to 14,604 events spread
across and around Hawaii. The voluminous
Pu*u *O*o-Kupaianaha eruption of Kilauea,
which began in 1983 and still continues, wasoccurring throughout the study period. The
seismic events at Kilauea can be classified into
three types, (i) high-frequency earthquakes thatindicate shear faulting, and (ii) long-period (LP)earthquakes and (iii) tremor (12, 13) that likely
reflect resonance in fluid-filled conduits. The
data set contains 2376 LP events, with 90%
above 20-km depth, and 135 tremor events,
with 90% below 30-km depth (14).
Precise arrival time differences for earthquake pairs were obtained using waveform crosscorrelation with parameters appropriate for highfrequency earthquakes (15), and these data wereable to relocate 7034 hypocenters, or 48% of theanalyzed events. This calculation is a minimum
estimate of the number of similar events, because pairs will not correlate highly if the signalto-noise ratio is low: small magnitude events thatare displaced far from stations, such as small,
deeper earthquakes, are more difficult to constrain. Families of highly correlated earthquakes
occur when there are closely spaced clusters withsimilar focal mechanisms, as at fault zones.
Hence, our relocated clusters identify fault zones
in the lithosphere beneath Hawaii.
The subset of 2522 earthquakes relocated in
the vicinity of Kilauea is shown in Figs. 1 and
2, and fig. S1 displays the total set of 7034

18 APRIL 2003 VOL 300 SCIENCE www.sciencemag.org

Appendix C
Atom trapping with a magnetic
thin film
This appendix contains a reprint of the hard disk platter paper, based on work described in Chapter 4. It was submitted to Physical Review A, and is currently underreferee review.

109

Atom trapping with a thin magnetic film
Micah Boyd, Erik W. Streed, Patrick Medley, Gretchen K. Campbell, Jongchul Mun, Wolfgang Ketterle, David E. Pritchard∗
MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics and Department of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA(Dated: October 16, 2006)

We have created a 87 Rb Bose-Einstein condensate in a magnetic trapping potential produced by a hard disk
platter written with a periodic pattern. Cold atoms were loaded from an optical dipole trap and then cooled to
BEC on the surface with radiofrequency evaporation. Fragmentation of the atomic cloud due to imperfectionsin the magnetic structure was observed at distances closer than 40 µm from the surface. Attempts to use thedisk as an atom mirror showed dispersive effects after reflection.
PACS numbers: 03.75.Be, 03.75.Lm, 75.50.Ss, 75.70.-i

Micrometer scale magnetic traps for Bose-Einstein condensates have been the focus of much experimental work since
their first demonstration [1, 2]. Their potential uses in atom interferometry, precision measurements, and experiment miniaturization have motivated many groups to develop sophisticated techniques for manufacturing and controlling these atom
chips [3–5]. However, since the earliest inceptions of atom
chips, physical imperfections in the chip surface have lead to
perturbations in the trapping potential that prohibit the coherent manipulation of atoms close to the surface[6, 7]. Permanent magnets offer a possible solution to several problems inherent to current carrying wire traps. First, the magnets arealmost completely decoupled from the rest of the laboratory,
minimizing the effects of environmental electrical noise. Second, they do not require current to be sourced or sinked, enabling designs that would be too complicated or impossible
to create using electromagnets. Last, extremely high field
gradients are possible close to magnetic domain boundaries,
whereas traditional atom chip operation is limited by heat dissipation from small wires. Although a substrate of magnetizable material may be made extremely smooth and uniform,
imperfections in the process of etching [8, 9] or writing magnetic structures could lead to some of the same problems aswith wire-based designs. It is not yet known which techniquewill provide the best performance.
In this paper we investigate atom trapping with a thin magnetic film; specifically that of a hard disk platter written witha periodic pattern. This approach offers some advantages overprevious work on neutral atom trapping and BEC creation using permanent magnets [4, 5, 10]. Here we use a thin metallic film with a large remnant magnetization from a commercial product which has already been refined to a high degree
and a writing technique more accurate than anything previously demonstrated. Cold 87 Rb atoms were first loaded into
the magnetic potential formed by the disk and used RF evaporation to produce BEC. By changing the trapping potential,
the atoms could be pushed closer to the surface of the disk toprobe for imperfections in the potential. Finally, the BEC was
dropped onto the disk from a height of 2.7 mm in an attempt
to produce a specular reflection of the atomic cloud.

∗ URL: http://cua.mit.edu/ketterle group/

The form of the written magnetization on the disk is of critical importance. A surface magnetization of the formM(x) = M0 cos (kx)ŷ,(1)

where M0 is the magnetization of the material and k is the
wavevector of the sinusoidal pattern produces a magnetic field
above the surface(Bx , By , Bz ) = B0 e−ky (− cos (kx), sin(kx), 0) ,(2)

which is of uniform coplanar magnitude and decays exponentially away from the surface [11]. This is an ideal potential forreflecting weak field seeking atoms.
The addition of a bias field Bx along x̂ (or By along ŷ)
produces a series of quadrupole shaped field minima above
alternating tracks, depicted in Fig. 1. A second bias fieldBz along ẑ removes the magnetic field zero and weak fieldseeking atoms may be trapped near the surface with radial trap
frequency!
µB gF mF(3)
2ωr = kBx
mBz
where m is the atomic mass, and µB gF mF is the Zeeman
energy [5, 11].
The magnetic media used in this work was a prototype harddisk with a radius of 65 mm and a thickness of 0.635 mm.
The substrate is glass, and the magnetizable material is a dual
layer system with a 23 nm thick magnetic Co-Cr-Pt based oxide layer and a 200 nm thick “magnetically soft underlayer”,
both of which are also good conductors and provide a totalmagnetization of 470 emu/cm3 and a coercivity of ∼6000 Oe
according to the manufacturer. The easy axis of magnetization of this prototype “out-of-plane” disk is aligned normal
to the surface of the disk, as opposed to most modern commercial hard disks which are magnetized in the plane of thesurface. The disk is also covered in a ∼1 nm layer of perfluoropolyether lubricant (Z-Tetraol). When written with patternof λ = 100 µm (1 µm), the magnetic field at the surface isapproximately 75 G (4200 G).
The disk was provided to us pre-written using a Guzik spinstand, which is essentially a hard disk read/write head withabsolute positioning capability. This method of writing allows for the creation of truly arbitrary pattern impossible to2$"!%µ&#"$

FIG. 1: Geometry of the magnetic trap formed by a magnetic film.
Magnetic field lines resulting from the addition of a radial (x̂) bias
field to the magnetized surface. The tubes represent the locations of
the field minima where atoms are trapped. The addition of an axial
bias field of about 1.0 G along ẑprevents atoms loss from Majorana
spin flips.

create with larger scale magnetic writing devices [5] or physical structures [4], and produces smaller and cleaner structures
than optical writing techniques [12]. Two patterns were usedin this experiment, written on different radial regions of thesame disk. The first region had alternating stripes of up anddown magnetization with a period of 2π/k = λ = 100.0 µm
and the second had λ = 1.0 µm. The media has a very squarehysteresis loop which precludes a purely sinusoidal magnetization. The pattern is instead written as a square wave,
and the higher Fourier harmonics should be negligible at theheight where we trap atoms [11]. Figure 2 shows a magnetic
force microscopy image of the disk after writing which showsa significantly less noisy pattern than magneto-optical thinfilms written with laser beams [13]. Atomic force microscopy
showed a physical roughness of ∼3 nm over a region of 50µm.
One concern about the disk was that it could not be baked
to the high temperatures (200 C) that we normally use to reach
UHV. Digital data on similar disks is guaranteed to 80 C, but
at 100 C the data starts to degrade rapidly as the superparamagnetic effect causes the magnetic domains to randomly reorient. The solution was to pre-bake the entire science chamberapparatus, without the disk, to 150 C for about a week to reach
UHV conditions. The system was briefly opened up to air, thedisk was installed, and then the chamber was pumped down
again. The chamber was then heated only to 70 C for a few
days, and after cooldown the pressure was < 10−10 Torr. The
recording signal decay rate quoted for this disk by its manufacturer was 0.4% per decade at room temperature, increasing
to 0.8% per decade at 70 C. The effect of domain randomization would be most pronounced in the center of the written
stripes, but minimal at the stripe edges where the configuration is more stable.
In our experiment, cold atoms were delivered to the surface in a two step process. First, cold 87 Rb atoms in the
F = 1, mf = −1 state were created in a Ioffe-Pritchardmagnetic trap with T >∼ Tc where Tc is the critical temper-

!"!
!"!

#"$

$"!%µ&

FIG. 2: Magnetic force microscopy image of the disk used in thisexperiment. This scan is over the region with λ = 1.0 µm. The trackedges are of the same uniformity in the λ = 100.0 µm region. The
characteristic size of the roughness of the domain boundaries is about
30 nm.

ature for Bose-Einstein condensation. The atoms were thentransferred to an optical dipole trap and transported 36 cm toa separate auxiliary chamber (see [14] for more details). The
optical dipole trap was formed by a λ = 1064 nm laser focused to a 30 µm 1/e2 radius spot, and the focus (with theatoms) was translated into the auxiliary chamber as describedin [15]. By transporting atoms just above Tc , the cloud wasless sensitive to vibrations, and higher laser powers could be
used without causing rapid three-body losses.
The atoms were loaded onto the surface from the opticaltrap by translating the focus of the optical trap to a position
parallel to and 50 µm above the region with λ = 100 µm. A
Z-shaped wire below the disk provided axial confinement, anda small Bx created radial trapping on the surface. The optical trap was ramped off over 2 seconds, transferring the atoms
with almost unity efficiency. The optical trap enabled loading of >90% of the atoms into a single surface trap site. RF
Evaporation over 20 seconds from 1.200 MHz to 0.890 MHz
produced a BEC with approximately 50,000 atoms in a trapwith (ωx , ωy , ωz ) = (390, 390, 9) × 2π Hz. The spatial distribution was clearly bimbodal, and condensate fractions of>80% were observed.
The atomic cloud was detected with on-resonance absorption imaging (Fig. 3). The platter is a good reflector (>95%)
for 780 nm light, so grazing incidence imaging was used tomeasure the distance from the atoms to the surface. Normal
incidence imaging was also used, in which case the imaging
light passed through the atoms twice, bouncing off the disk.
The lack of physical structures on the surface resulted in good
image quality.
The axial trap frequency in the surface trap was measured
by imaging oscillations of the atomic cloud, and radial trap
frequencies were measured by parametric heating [Fig. 4].

3
!"

#"

!"
#"
$"
%"

FIG. 3: Absorption image of the BEC in trap near the surface. a) Top
imaging. The field of view is 0.58 mm × 0.58 mm. b) Side imaging. The field of view is 0.56 mm × 0.56 mm. The double image
comes from the reflection of the imaging beam in the platter surface.
The trap frequencies were 2π×(400, 400, 10) Hz. The configuration of the experiment prevented ballistic expansion, but the bimodal
distribution in trap is still clear.

Trap Frequency (kHz)

5
4
3

&"
FIG. 5: Breakup of the atomic cloud as the atoms approached thesurface. The thermal cloud was evaporated to T ≈ Tc , then the axialtrapping frequency was reduced to from 10 Hz to 1 Hz to allow theatoms to expand in one dimension. The axial bias field was then increased to push the atoms closer to the surface, and the atoms wereimaged in trap. The grazing incidence imaging used here produces
a double image of the atoms from the primary and secondary reflections of the imaging beam off the disk. The field of view is 0.29 mm
× 2.3 mm, and each frame is an average of five distinct images toreduce noise for clarity. At distances smaller than 40 µm, significant
breakup was observed. Height for the images shown: a) 35 µm b) 29µm c) 25 µm d) 22 µm e) 19 µm.

2
1
0
0.0

1.0
2.0
Applied Bias Field (Gauss)

3.0

FIG. 4: Radial trap frequency vs applied radial magnetic field.
Atoms were loaded into a single trap site on the surface and evaporated to BEC. The trap frequencies were measured with parametricheating, but above 5 kHz other heating effects made it difficult to resolve reliably. The deviation from linearity at low applied fields ismost likely due to a small, off-axis, residual bias field.

Attempts to release the atoms from the trap and observe ballistic expansion in time of flight were hindered by the geometry of the system and the static nature of the magnetic surface,
so all of the imaging was done in-trap. The Z-wire trap wasleft on at all times to provide axial trapping, but its effect on
the radial trapping was negligible. Ramping up the current
in external electromagnets increased Bx , and ωr as high as
2π × 5 kHz was measured. Using a disk with λ = 10.0 µm, we
have measured ωr as high as 2π × 16 kHz. High transverse
trap frequencies are desired for studies of 1D systems [16],but atom heating and loss at higher trap frequencies prevented
such studies here.
While trapped 40 µm above the surface, the BEC had alifetime of ∼30 s. Previous experiments with atomic clouds
magnetically trapped near conducting surfaces showed atom
loss resulting from spin flips driven by Johnson noise [7, 17,18]. For this experiment however, the conducting layer was
extremely thin (200 nm), and the atoms were relatively far
from the surface. These two parameters reduced any spin flip

loss below the level of our 1-body losses at a vacuum pressureof ∼ 5 × 10−11 Torr.
The BEC was also used to probe imperfections in the trapping potential. After evaporation, the axial trapping frequency
was reduced from 10 Hz to ∼1 Hz to allow the cloud to expand slightly in one dimension to increase the sensitivity andthe measurement area, and the radial bias field (Br ) was increased to push the atoms closer to the surface. At distances
closer than 40 µm, breakup of the atomic cloud was observed[Fig. 5], and the strength and spatial frequency of the perturbations increased as the atoms neared the surface, similar to[6, 19]. The magnitude and size scale of these imperfections
can be attributed to the sputtering process used to create thefilm. Magnetic grain nucleation occurs randomly, resulting insmall angular misalignments of the axes of anisotropy of individual magneto-crystalline domains (typically a few degrees),
the individual magnetic grains can vary in size by about 25%of their ∼7 nm diameter, and the magnetic moments have a
distribution of magnitudes. Finally, the recording head used
to write the magnetic pattern can introduce unwanted magnetizations from edge effects. All of these phenomena create
imperfections in atomic trapping potentials.
While the presence of these perturbations does not necessarily preclude the creation of neutral atom waveguides using
this type of substrate, it does limit its usefulness. High trapping frequencies and single transverse mode confinement require close proximity to the surface and will suffer from theobserved imperfections. However, for use as a high reliability, low noise, and low cost waveguide at distances >50 µm,
commercial metallic, magnetic thin films are ideal.
The magnetized surface was also examined for its usefulness as an atom mirror as first proposed in [20] and demonstrated in [8, 21–23]. In order to minimize residual and time

4
dependent magnetic fields (from magnetic trap turnoff), all
optical trapping was first used to create a BEC. The optical
dipole trap provided only weak axial confinement, so a second, 200 mW laser beam with λ = 1064 nm and 200 µm waist
was added in a crossed configuration, and allowed more efficient evaporation. To evaporate to BEC the power was reduced over 2.0 s. An axial bias field Bz = 1.2 G was maintained throughout the experiment, and Bx and By were minimized to <20 mG.
The region of the disk used as a magnetic mirror was written with λ = 1.0 µm. This value was chosen to be large compared to the magnetic writing precision, but small comparedto the extent of the atomic cloud. The BEC was released fromthe crossed ODT 2.7 mm above the disk. The atoms fell under gravity for 23.5 ms and then reflected off the magnetic
potential. The atomic cloud was imaged (in separate runs)
from both the top and side after various times of flight. Side
imaging showed that the reflection from the disk did not significantly effect the axial or vertical velocity distributions of
the atomic cloud. Top imaging, however, showed significant
spreading along the vector of the magnetic pattern after reflection, analogous to bouncing off a rough mirror [24]. The
magnitude of this dispersion was minimized by fine tuning Bx
and By , but was impossible to eliminate in our apparatus. Before bouncing off the disk, the x̂ width of the cloud expandedby 1 mm/s, and after bouncing the x̂ width increased by 34
mm/s. The reflection was not performed on the region with λ= 100.0 µm, but experiments using a different disk with λ =10.0 µm showed similar effects.
The magnetic properties of the hard disk surface are one
possible source of the observed expansion after reflection.
While the theory presented earlier (Eq. 1) applies to a sinusoidally magnetized surface, the square hysteresis loop and directional anisotropy of digital recording media prevent recording of a pure sine wave, instead forming a square wave approximation of that sine, as discussed in [11]. While thehigher harmonics resulting from this approximation do notnecessarily adversely effect BEC trapping, they create a cor-rugation of the planar equipotential that may prevent the specular reflection of a macroscopic atomic cloud. However, field
perturbations from harmonics or order n are of magnitude
B0 (1 − e−nkb )e−nky , and the effect of this corrugation at thereflection height (∼ 4 µm) should be negligible (< 1 mrad).
Another contribution to the roughness of the reflection potential is the existence of small, stray magnetic fields. Any
nonzero component of B in the x̂ŷ plane creates a regularcorrugation in the plane, further inhibiting specular reflection.
While the bias field in the Bx and By planes was minimized
at the atoms, magnetic field gradients from external sources,
such as a nearby ion pump, were not compensated. However,an overestimate of the residual magnetic field gradient still underestimates the amount of dispersion observed after reflection. The primary cause of the dispersion is thus unknown.
Diffraction peaks that might result from reflection off such a
corrugated surface were not observed.
In conclusion, we have demonstrated BEC production on
the surface of a hard disk platter. Small scale imperfections in
the magnetization caused condensate fragmentation close to
the surface, prohibiting its use as a neutral atom waveguide.
The disk was also used as an atom mirror, and specular reflection was observed on two axes. These results are a substantial
improvement over that for early wire based experiments, andthe possibility for more complex structures enables trap geometries impossible for electromagnets. Microtraps based on
permanent magnets, and in particular magnetic metallic thinfilms, may become an alternative to atom chips using currentcarrying wires if the fabrication can be further improved, e.g.
by using molecular beam epitaxy, and the writing process
improved by using write heads optimized for recording DC
structures on perpendicular media.
The authors would like to thank Aaron Leanhardt for experimental assistance, Tom Pasquini for a critical reading of themanuscript, and Min Xiao at Hitachi Global Storage Technologies for writing several disks for us and providing theMFM images, this work would not have been undertaken
without her assistance.

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Appendix D
A Mott insulator atomic clock
This appendix contains the most recent draft of the Mott insulator atomic clock
paper, based on work described in Chapter 5. It is still being revised, and will be
submitted to Physical Review A upon completion. It was my last experimental workat MIT.

115

A Mott Insulator Atomic ClockRb Lab∗MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics and Department of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA(Dated: September 13, 2006)
We have implemented a proof-of-principle atomic clock using a 87 Rb Bose-Einstein condensate confined in
an 3D optical lattice. Several typical sources of systematic error were eliminated by preparing the atoms in
the Mott insulator (MI) phase before measurement. The suppression of tunneling in the MI phase permitted
longer measurements than possible with other optical confinement techniques for the same optical power, as
well as elimination of the density shift due to the discretized filling factor. The hyperfine transition of 87 Rb wasmeasured with a FWHM linewidth of 1.2 Hz and an uncertainty of 0.2 Hz out of 6.8 GHz.
PACS numbers:

Precision frequency and time measurements provide a fundamental reference for the basis of nearly every physical measurement. Modern measurements have improved the accuracy of atomic clocks to the 10−15 level [1–5], and recent experiments with optical transitions in trapped, neutral atoms
promise precision at the 10−18 level [6]. As the properties ofclock atoms are measured more precisely, care must be taken
to prevent the environment, and in some cases the atoms themselves, from affecting the measurement.
Unwanted perturbations to an atomic clock system cancome from external sources, such as external trapping forcold atoms or stray electromagnetic fields in the laboratory,or from internal sources, such as the atomic cold-collision
shift [1, 7, 8] or dipole-dipole interactions [7, 9]. Althoughsystems can be shielded from external perturbations, it is moredifficult to control the effect of interactions. Simultaneouslyinterrogating a large numberof atoms N can increase the√signal to noise ratio by N , but increasing the atom number typically comes at a cost of increased density, leading togreater uncertainty due to interactions. For example, the microwave hyperfine transition of cold 87 Rb at a typical densityof 1013 cm−3 will be perturbed by the “clock” shift at a levelof 2 Hz. One way to minimize such collisions is to hold theatoms in an optical lattice.
In this work we implement an atomic clock by measuring a
microwave hyperfine transition in a 87 Rb Bose-Einstein condensate (BEC) confined in a 3D optical lattice. Atomic clocks
using neutral atoms in an optical lattice have been demonstrated previously [6, 10, 11], but here we eliminate a key
source of systematic error by increasing the depth of the lattice potential to confine the atoms in the Mott insulator (MI)regime. In the MI phase, tunneling is suppressed due to theonsite interaction energy, reducing the lattice depth required
to minimize interactions. This reduced lattice depth avoids
complications such as higher order light shifts [12]. As recently shown [13], the microwave spectrum of the MI with87
Rb shows discrete peaks corresponding to the integer filling
factor of individual lattice sites. This discretization of the density shift eliminates density dependent errors in the measure-ment of the transition frequency, and should also not be limited by phase separation or collisions [14]. Systematic shifts
due to trapping laser intensity may also be extrapolated to zerointensity by varying the laser power over a wide range within
the MI regime.
Bosons held in a 3D optical lattice with repulsive interactions are described by the Bose-Hubbard Hamiltonian [15, 16]
Ĥ = −J

!

#i,j$

!
1 !
n̂i (n̂i −1)+(!i −µ)n̂i , (1)â†i âj + U
2iiwhere J is the tunneling term between nearest neighbors,n̂i = â†i âi is the number of atoms in the ith lattice site, U
is the repulsive onsite interaction, !i = Vext (ri ) is the energy due to the external confinement, and µ is the chemical potential." The onsite interaction is expressed as U =(4πh̄2 a/m) |w(x)|4 d3 x, where a is the s-wave scattering
length, m is the atomic mass, and w(x) is the single particle Wannier function. The first term in the Hamiltonian tends
to delocalize atoms over the lattice due to tunneling, while
the second term tends to localize atoms to lattice sites due
to atom-atom interactions. For low lattice depths, the ratio
J/U is large and the system behaves as a superfluid. As thelattice depth is increased, the localizing effects of the interaction begin to dominate, and the system undergoes quantum
phase transition to a MI phase. The MI phase is characterized by Fock states, where atoms are localized to individual
lattice sites with integer filling factor. Due to the external confinement and the initial density distribution of the condensate,
this filling factor is a maximum in the center of the trap, anddecreases to the edge of the atomic cloud. The result is a series of concentric shells, each with uniform filling. It is thisdiscrete filling which we utilize to eliminate the density shiftof our atomic clock.
The density dependent shift in 87 Rb is determined by thedifference in the scattering lengths a11 and a21 , where a21 is
the scattering length between an atom in the F = 2, mF =+1 state and one in the F = 1, mF = −1 state. The resulting
mean field energy shift∆E =∗ URL: http://cua.mit.edu/ketterle group/h2(a21 − a11 )
πmρ

(2)

will then resolve to discrete frequency shifts in the microwave

|1, −1" → |2, 1" hyperfine transition for integer lattice filling factors, where ρ is the atomic density. The resulting microwave spectrum for a MI is then a series of peaks, andthe frequency difference between neighboring MI shells isthen [17]U (a21 − a11 )
.
δν =
h
a11

(3)

For our experiments, the BEC was created using forced radiofrequency (RF) evaporation of a 87 Rb atomic cloud in the|F, mF " = |1, −1" state, as described previously in [18]. The
trapping potential was formed by the combination of a IoffePritchard magnetic trap and a retro-reflected, cross polarized
optical dipole trap. The optical trap was oriented perpendicular to the weak axis of the magnetic trap to provide a more
spherical geometry for better mode matching to the opticallattice [13]. BECs containing ∼ 1.5 × 106 atoms with no discernible thermal component were held for 5 seconds with aconstant RF shield to reduce shot-to-shot atom number fluctuations to below 5%. For most measurements, a fast RF
sweep through the RF resonance was used to reduce the atom
number by ∼75%. A 3D optical lattice was then adiabatically ramped on using an exponential increase with a 40 ms
time constant. Two lattice axes were formed by increasing
the intensity in retro-reflected beams, while the third axis wasformed by rotating the polarization of the retro-reflected optical dipole trap using a liquid crystal waveplate. The lattice
beams were all derived from the same laser with λ = 1064 nm,
and were detuned from each other by acousto-optic modulators. The beams also all had roughly equivalent 1/e2 beams
waist radii of 150 µm, focused on the atoms. The lattice depth
was increased up to 35 Erec where Erec = h̄2 k 2 /(2m) andk = 2π/λ is the wavevector of the lattice. After ramping up
the lattice, the atoms were held for 50 ms before measurement
to compensate for transient nematic effects in the liquid crystal polarization rotator. The resulting MI structure was calculated to contain approximately 7 × 104 atoms in the n = 1
shell.
A single two-photon pulse resonant with the |1, −1" →|2, 1" transition was applied to the atoms in the lattice forsome time τ , as depicted in Fig. 1. The pulse consistedof a microwave photon at 6.83 GHz and a RF photon at≈1.6781 MHz. The microwave synthesizer was an Agilent
E8257D running off its internal timebase. The RF synthesizer
was an Agilent 33250a, and was synchronized to the 10 MHz
timebase of the microwave synthesizer. Magnetic field dependent line shifting and broadening was avoided by working at
a bias field of ∼3.23 Gauss, where the two states experience
the same first order Zeeman shift [17]. The power of the pulse
was adjusted to produce a π pulse for each hold time τ . Immediately after the pulse, the confining potentials were switched
off and the atomic cloud was allowed to expand for 3 ms to
reduce the column density. Atoms in the |2, 1" state were detected with absorption imaging using light resonant with the
52 S1/2 |2, 1" → 52 P3/2 |3, 1" transition.
As the lattice is ramped up, the ac Stark shift and the density shift begin to broaden and move the resonance of the|1, −1" → |2, 1" microwave transition to lower frequencies.

%&''()*+,*-'.+/01*)2

2

$!
#!
/&2

/62

"!

/)2

τ

!
!

"!!

#!!
3(4*+/452

FIG. 1: Diagram of the experimental procedure. (a) The lattice isramped up to U = 35Erec in 193 ms, then held for 50 ms. (b) The
two photon pulse is applied for time τ , then (c) the atoms are released
from the trap and imaged in 3 ms time of flight.

As the lattice depth is increased, the tunneling rate decreases
until it is completely suppressed at depths greater than approximately 15 Erec due to the transition to the MI phase. Discrete
peaks begin to appear in the microwave spectrum, corresponding to cold collision density shifts resulting from integer filling factors within lattice sites. The observation of discrete
peaks is direct evidence for the suppression of tunneling byinteractions when the MI transition is reached. Tunneling ofatoms between sites of different occupation number during the
microwave pulse would broaden the discrete peaks and eventually lead to peaks reflecting the clock shift at an averaged
density. We estimate the tunneling time in the absence of interactions using a double-well model to be h/4Jnz, where z
is the number of next neighbors, and n is the filling factor.
For n = 1, the tunneling time is ∼5 ms at 20 Erec , 13 ms at
25 Erec and 60 ms at 35 Erec . A very deep lattice would localize atoms even for a thermal or non-interacting gas. However,for our lattice depths no discrete spectrum was observed when
the cloud was heated during the ramp up by misalignment of
the lattice beams. Our spectra were taken under conditions
where if the lattice were immediately ramped back down, condensate fraction remaining was greater than 80%.
By using long two-photon pulses, we narrowed the bandwidth of the scan and increased the resolution the n = 1 andn = 2 peaks from that reported in our previous work [13] (Fig.
2). Using a double Gaussian fit of the form y = yo +2222
A1 e−(x−x1 ) /σ1 + A2 e−(x−x2 ) /σ2 , the n = 1 peak was morethan 10σ2 removed from the n = 2 peak, indicating thatany higher density contribution to the singly occupied sites
was removed at a level better than 10−5 . Thus by using theMI phase to separate density shifts into carrier and sideband
peaks, we were able avoid density dependent shifts that affect other atomic clock experiments. By increasing the length,
and thus narrowing the bandwidth of the microwave pulse,
we were able to measure successively smaller linewidths ofthe singly occupied sites down to 1.0(2) Hz FWHM (Fig. 3).
Pulse lengths longer than 1200 ms did not further decrease thewidth of the transition.

Resonance (Hz - 1678 kHz)

Number of F=2 Atoms (Arb)

3
1200
1000
800
600
400
200
0
60

70
80
90
100
RF Frequency (f - 1678000 Hz)

FIG. 2: Microwave scan of atoms in the lattice. The n = 2 peak iscentered at 1,678,067.2 ± 0.1 Hz with a FWHM of 4.2 ± 0.8 Hz,
and the n = 1 peak is centered at 1,678,095.03 ± 0.03 Hz with a
FWHM of 2.85 ± 0.08 Hz. The fit is a double Gaussian, and showsthat the n = 2 peak is move than five linewidths removed from then = 1 peak, eliminating the density shift. The lattice depth was a 35
Erec , and the measurement time was 300 ms.

FIG. 3: Width of the n = 1 peak vs. two photon pulse length. Afterloading into the lattice, the width of the single occupation density
peak was measured as a function of the interrogation pulse length.
The two photon intensity was adjusted to provide a π pulse at eachhold time. The line represents the theoretical bandwidth of the pulse.
Pulses longer than 1200 ms resulted in larger peak widths due totechnical noise, primarily from our microwave synthesizer.

In order to correct for the frequency shift induced by theac Stark effect, the location of the n = 1 peak was measured for a variety of lattice depths in the MI regime. Since
the ac Stark shift scales linearly with laser power, measuring the resonance as a function of lattice depth allowed us toquantify this effect. After a linear fit to the data (Fig. 4), extrapolation to a depth of 0 Erec revealed the resonance frequency without any linear laser intensity dependent shifts.
This extrapolation increased the uncertainty of the measurement by about a factor of two over that of a typical single
frequency measurement. The minimum lattice depth required
to clearly resolve the n = 1 peak was 20 Erec , and a maximum depth of 40 Erec was used. After extrapolation, the|1, −1" → |2, +1" hyperfine transition frequency was measured to be 6,834,678,015.20(18) Hz (statistical). After ap-

102
100
98
96
94
20

24

28
32
Lattice Depth (Er)36

40

FIG. 4: Center frequency of the n = 1 peak vs. lattice depth. The
center of the single occupation peak was measured as a function oflattice depth in order to extrapolate to zero laser intensity. The line isa linear fit which gives a zero intercept of 110.04 ± 0.18 Hz, indicating an ac Stark shift free frequency of 6,834,678,110.04 ± 0.18 Hz.
The error bars are not fully resolved on this graph, with a typicalvalue of ±0.10 Hz.

plying a 4497.3 Hz correction predicted from the Breit-Rabi
formula for the finite magnetic field [19], this is 98.4 Hz fromthe accepted value of 6,834,682,610.90434(3) Hz [20], but
the focus of this work was only to control systematic errors
resulting from inter-atomic interactions, not to make a highaccuracy measurement.
The primary determinant of the peak width was the bandwidth of the two-photon pulse. In the case of a bandwidth
limited pulse, the lineshape should be a sinc squared with a
FWHM of 0.8859/τ . This is shown as the solid line in Fig. 3,
which agrees well with the data for τ < 1 s. For times longer
than 1 s, several sources of inhomogeneous broadening begin
to affect the lineshape. The largest source of inhomogeneous
broadening in our experiment is due to a magnetic field gradient across the atomic cloud. Our magnetic trap is operating
at the low limit of holding the atoms against gravity, and thus
we assume that the entire cloud is trapped in a constant vertical magnetic field gradient of 15 G/cm. Taking a n = 1 shellradius of 19 µm, we calculate a field gradient of 58 mG vertically across the entire cloud, implying a broadening of thehyperfine transition of ∼0.36 Hz due to the inhomogeneous
magnetic field. The effect of the magnetic bias field on thepeak width is shown in Fig. 5. Since the magnetic field broadening depends on the size of the atomic cloud, a large atom
number could lead to larger linewidths. However, within thestatistical uncertainty of the fits, the width of the n = 1 peak
was not found to be significantly sensitive to atom number inthe range 105 < N < 106 .
Another source of inhomogeneous broadening is the differential ac Stark shift. If the clock measurement samples atoms
which are trapped in regions of different laser intensity, theresonance will be broadened. However, the shell structureof the MI phase will tend to minimize effects from the differential ac Stark shift as all of the atoms in the measured
shell are trapped in approximately the same laser intensity.
If the atomic cloud is small compared to the diameter of the4
our measurement as a small shift of the n = 1 peak towardsn = 2 due to the slightly increased average density. However,this effect scales as J/U , and for a lattice depth of 35 Erec ,the 0.2% mixture generates a shift of 0.05 Hz and would befurther reduced for larger lattice depths.

n=1 Peak Width (Hz)

8
6
4
2
0
3.15

3.20
3.25
3.30
Bias Field (Gauss)

3.35

FIG. 5: FWHM of the n = 1 peak vs magnetic field. Control ofthe magnetic field at a level of <20 mG was necessary to minimize
inhomogeneous magnetic field broadening. The line is a parabolic fit
to guide the eye.

lattice beams, this effect will also be small. Assuming thatthe measured atoms consist of a shell with radius equal to
the Thomas-Fermi radius of the BEC, we calculate that thedifferential ac Stark shift across the shell results in a broadening of <0.2 Hz. This broadening scales linearly with theinverse of the beam diameters, so narrow beams used for all
optical trapping would adversely effect the measurement. For
example, 20 µm waist beams would lead to a differential ac
Stark shift broadened peak width of ∼2 Hz, dominating othersources of broadening currently observed. This broadening,as well as the total ac Stark shift may be reduced by the useof a “magic” wavelength lattice. By carefully choosing λ sothat the two interrogated levels are shifted equally by the trapping laser, the ac Stark shift may be reduced by several ordersof magnitude [6]. In the absence of a magic wavelength, ablue-detuned lattice laser could be used, trapping the atoms inregions of minimum laser intensity [21].
One possible source of homogeneous broadening is theexistence of particle-hole pairs within the n = 1 number
state[22, 23]. The tunneling term of the Hamiltonian acts asa perturbation on the otherwise uniform array of Fock statesin the MI phase, adding a small superposition of n = 0 andn = 2 noise to the n = 1 signal. This manifests itself in[1] F. Pereira Dos Santos, H. Marion, S. Bize, Y. Sortais, A. Clairon, and C. Salomon, Phys. Rev. Lett. 89, 233004 (2002).
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Nature 435, 321 (2005).

Another systematic shift which depends on interactions is
the induced dipole-dipole shift [24, 25], where the measurement field drives dipole oscillations in the atoms which effectneighboring lattice sites. Since the measurement wavelength(λµw ≈50 mm) is very long compared to the lattice spacing(d = 523 nm), there will be a dipole-dipole interaction in thenear field regime. However, the very narrow linewidth of the|1, −1$ → |2, 0$ transition (∼10−8 s−1 ), as well as the relatively large single photon detuning (420 Hz) imply an interaction energy shift of < 0.02 Hz [21]. Experiments using measurements in the optical regime will benefit from a reduced
interaction since the energy shift scales as (λ/d)3 [25], andcareful selection of the lattice geometry can even further suppress this effect [24].
The main source of systematic error in this experiment was
the microwave frequency source. Our synthesizer was not
linked to any external reference clock, such as GPS, and thespecified drift rate of 4.5 ppb per day corresponded to 31 Hz
per day at 6.8 GHz. Indeed, frequency jumps as large as 20 Hz
over a two day period were observed when temperature stability in the lab was not maintained. Future experiments would
certainly implement better master oscillator stabilization. Using a two-pulse Raman technique was investigated, but phasejitter due to synthesizer frequency noise made accurate fittingof the resulting signal difficult. The use of a double pulse
method is only limited by the fact that each of the two pulses
must have a bandwidth narrow enough to not address any
atoms in sites with occupation greater than one, and would
improve the accuracy of future experiments.
In conclusion, we have implemented an atomic clock with a
BEC trapped in the MI phase of a 3D optical lattice. While ourdemonstration is not competitive with current atomic clocks,
the density discretization technique shown here could be used
to increase the precision of optical frequency Ca, Yb, and Srclocks currently under development [2, 26–30].

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Appendix E
Science chamber schematics
This appendix contains drawings of the science chambers used on the rubidium machine.

E.1

Hard Disk Platter Science Chamber

The hard disk platter chamber consisted of a six inch radius sphere faced with viewports. A vacuum bellows manipulator sat on top of the chamber to hold and position the hard disk platter. Position control in the pennywindow horizontal axis
was achieved with micro-positioner translation stages, and tilt and vertical control was maintained by three jackscrews. The diagram below shows the chamberwith the manipulator section rotated 90 degrees about the vertical axis for clarity.

121

3/8-40 threaded rod
5.97
2.75
2.25

Feedthrough flange
4.0
3.0

0.25

6 Conflat, clearance holes, rotatablewelded bellows

2.56

1/4-20 tapped

T-Stage

T-Stage
6 Conflat, clearance holes, fixededges milled flat
5.5

0.375

6 CFF
Tapped holes

About 3.75

4.5 CFF
Tapped holes

1.94
6.125 OD
6.00 ID
SphereTapped flange
2.75 Hole pattern

6 CFF
Tapped holes

E.2

Optical Lattice Science Chamber

While not used in this work, this science chamber will be installed on the rubidium machine shortly after the completion of this thesis. It was designed to afford a
maximum of optical access, while maintaining UHV capability.

122E.2.1

Lattices Science Chamber Top View

This science chamber design accommodates six axes; three for lattices and one for
imaging, and one for optical transport from the main chamber.

1.25” OD

1.25” OD
4.0” OD pipe

3.86”

2.5”
2.0” radius
3.75”
3.86”
3.75”
1.25” OD
1.5” OD

123E.2.2

Lattices Science Chamber Side View

Imaging with aperture close for f = 1 is achieved through the top viewport.

1.75”
2.5”
3.75”
1.75”

4.0”

124

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