Progress towards an ultracold atomic Sagnac
gyroscopeby
Stephen Richard Segal
B.S., Pennsylvania State University, 2003

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2010This thesis entitled:
Progress towards an ultracold atomic Sagnac gyroscope
written by Stephen Richard Segal
has been approved for the Department of Physics

Dana Z. Anderson

Eric A. Cornell

Date

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

iii
Segal, Stephen Richard (Ph.D., Physics)
Progress towards an ultracold atomic Sagnac gyroscope
Thesis directed by Professor Dana Z. Anderson

Precise, accurate inertial navigation relies upon the rotation rate measurements of asensitive gyroscope. State of the art fiber optic gyroscopes measure rotation via the Sagnac
effect, which causes a phase shift proportional to the angular velocity of a rotating interferometer
whose beam path encloses an area. Gyroscopes using interfering atoms have a much greater
Sagnac phase shift for the same enclosed area than optical gyroscopes, and therefore promisemuch greater sensitivity. However, an atom Sagnac gyroscope has not yet been demonstrated
in a compact device.
This dissertation describes work towards the demonstration of a compact Sagnac gyroscope using interfering ultracold atoms, which can be guided around suitably large areas using
magnetic potentials generated by micro-scale “atom chips”. We first present the results of asequence of preliminary Bose-Einstein condensate (BEC) Michelson interferometry experiments.
In these experiments, BECs confined to a waveguide interfere, but the path does not enclose
area. We present the application of statistical image analysis techniques to data from our experiments. We analyze the lessons learned about both apparatus design and the physics of BEC
interferometry. We describe the design of a new atom chip that should allow us to translate thewaveguide during the interferometry experiment, thus turning Michelson into Sagnac interferometry. Finally, we explain the design of a new compact BEC apparatus that incorporates thegyroscope chip. The apparatus is built on a rotary table so that we may truly test the gyroscope
when it is complete. We report on experimental progress towards the achievement of BEC in
this apparatus.

To Milton and Eleanor Segal

Acknowledgements

I owe a deep debt to the teachers and mentors who guided my development. Dana Anderson has been my advisor and boss for almost seven years. Besides supervising my education,
he has taught me the value of clear communication skills and helped to show me how extremely
hard work can result in progress (even if it is not obvious progress). I am thankful for hispatience with me in my long maturation process. Two other mentors, Eric Cornell and Alex
Zozulya (of Worcester Polytechnic Institute), also contributed much to my education and to thework presented here. In particular, I would have been lost without Alex’s notes and guidance
during the chip design process. I am honored to have worked with both of them. I also appreciate the input and advice of Deborah Jin and Carl Wieman during the early years of my time
at JILA.
This work was accomplished with the help of many fellow students and staff. I learned
much of what I know about experimental physics during long hours spent with my former lab
partners Quentin Diot and Ying-Ju Wang, as well as with fellow group members Daniel Farkas,
Kai Hudek, Evan Salim, and Matthew Squires. Their contributions to this work are too many tomention individually. Scott Papp deserves a special mention for teaching me many tricks of theexperimental trade. Leslie Czaia’s assistance and advice was essential at many times during mycareer. Seth Caliga has been my undergraduate assistant during the final portion of my PhD,
and is set to inherit this project from me. He has been a pleasure to mentor, and I look forward
to seeing how far he will take the gyro during his own graduate career. Farhad Madjetemouri
wrote the LabVIEW control program for the new apparatus. Undergraduates Andrew Holmgren,vi
William Holmgren, Marika Meertens, and Jonathan Pfeiffer also contributed their efforts. In
addition, the JILA computing team and the staffs of the electronics and instrument shops
supported this work with their expertise and patience on various occasions.
This research also relied on contributions from our group’s partners. The atom chips usedin the early Michelson interferometry experiments were fabricated by R. A. Saravanan from thegroup of Victor Bright, using the facilities of the group of Zoya Popovic. Later chips were
fabricated by Robert Mihailovich’s team at Teledyne Scientific (formerly Rockwell Scientific).
This project was primarily funded by the guided Bose-Einstein condensate interferometry
program of the Defense Advanced Research Projects Agency’s Defense Sciences Office.

Contents

Chapter

1 Introduction
1.1

1.2

1

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

1

1.1.1

Navigation and gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

Optical Sagnac effect gyroscopes . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.3

Matter-wave Sagnac gyroscopes . . . . . . . . . . . . . . . . . . . . . . . .

6

1.1.4

Shot noise in Sagnac gyroscopes . . . . . . . . . . . . . . . . . . . . . . .

7

1.1.5

Thermal beam atom gyroscope experiments . . . . . . . . . . . . . . . . .

9

1.1.6

Advantages of using ultracold, guided atoms in gyroscopes . . . . . . . . .

11Outline of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122 Atom optics: Waveguides and diffraction grating beamsplitters
2.1

2.2

13

Magnetic microtraps and waveguides for atoms . . . . . . . . . . . . . . . . . . .

13

2.1.1

Side guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.1.2

Crossed wire trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.1.3

H trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.1.4

Fabrication of microtraps . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Diffraction grating beamsplitters for atoms . . . . . . . . . . . . . . . . . . . . .

23

2.2.1

First order beamsplitter . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.2.2

Higher order beamsplitters . . . . . . . . . . . . . . . . . . . . . . . . . .

27

viii
3 Michelson interferometry experiments with BECs

28

3.1

On-chip, free-return interferometry with BEC . . . . . . . . . . . . . . . . . . . .

28

3.2

Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

4 Data analysis of Michelson interferometry experiments

36

4.1

Model-based vs. model-free inference . . . . . . . . . . . . . . . . . . . . . . . . .

36

4.2

Model-based image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4.3

Statistical image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.4

Details of algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

4.4.1

Principal component analysis . . . . . . . . . . . . . . . . . . . . . . . . .

47

4.4.2

Reduction of dimensionality with PCA . . . . . . . . . . . . . . . . . . . .

48

4.4.3

Independent component analysis . . . . . . . . . . . . . . . . . . . . . . .

51

Parameter extraction from PCA/ICA results . . . . . . . . . . . . . . . . . . . .

53

4.5.1

Model for probability density of PC/IC signal . . . . . . . . . . . . . . . .

54

4.5.2

Maximum likelihood estimation of parameters . . . . . . . . . . . . . . . .

55

4.5

5 Results and lessons learned from Michelson interferometry
5.1

5.2

59

Michelson calibration and results . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

5.1.1

Vibrations of the apparatus . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5.1.2

Quantum phase diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

63Lessons for future experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

6 Theory and design of the atom chip gyroscope

74

6.1

Parallel wire concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

6.2

Magnetic field of parallel wires carrying sinusoidal current density . . . . . . . .

76

6.2.1

Field of sinusoidal current density . . . . . . . . . . . . . . . . . . . . . .

78

6.2.2

Field of periodic current density . . . . . . . . . . . . . . . . . . . . . . .

78

6.2.3

Field of sinusoidal current density generated by wires . . . . . . . . . . .

79

ix
6.2.4

Field of traveling sinusoidal current density generated by wires . . . . . .

83

Collective mode excitation due to parallel wires . . . . . . . . . . . . . . . . . . .

85

6.3.1

Slosh modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

6.3.2

Breathing and quadrupole modes . . . . . . . . . . . . . . . . . . . . . . .

87

6.3.3

Collective mode minimizing parameters . . . . . . . . . . . . . . . . . . .

91

6.4

Parallel wire design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

6.5

Transverse acceleration profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

6.5.1

Motion of the trap center . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

6.5.2

Equation of motion of the BECs . . . . . . . . . . . . . . . . . . . . . . .

98

6.5.3

Even-ratio trap frequencies . . . . . . . . . . . . . . . . . . . . . . . . . .

99

6.5.4

Enclosed area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.5.5

Parameter estimates for enclosed area . . . . . . . . . . . . . . . . . . . . 101

6.3

7 Gyroscope apparatus: Design and construction progress
7.1

104

Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.1.1

Frequency control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.1.2

Shutters and optical power control . . . . . . . . . . . . . . . . . . . . . . 108

7.1.3

Optical fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2

Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.3

Optomechanics structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.4

Double MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.5

Transfer to chip trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.6

7.5.1

Loading the “big Z” trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.5.2

Transport to chip height . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.5.3

Gyroscope atom chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.5.4

Loading the chip trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

x7.7

Control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.8

Current progress and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Bibliography

136

Appendix

Table of mathematical symbols used in text

139

xi

TablesTable

5.1

Fit parameters for simulated data in Fig. 5.7 . . . . . . . . . . . . . . . . . . . .

6.1

Normalized amplitudes of collective mode oscillations for sinusoidally driven par-

6.2

72

allel wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Values of amplitude parameters from Table 6.1 for L = 1.2 mm, N = 8. . . . . .

95

Figures

Figure1.1

A schematic of a simple, optical Sagnac interferometer . . . . . . . . . . . . . . .

3

1.2

Interferometry waveguide of arbitrary shape, rotating at angular velocity Ω . . .

4

1.3

Northrup Grumman LN-260 inertial navigation system with GPS . . . . . . . . .

5

1.4Interferometer output beams recombined at beamsplitter . . . . . . . . . . . . . .

8

1.5

Thermal beam atom gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.1

The simple side guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2

Crossed wire trap on a chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.3

Schematic of T trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.4

Schematic of H trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.5

First order double-pulse optical beamsplitter . . . . . . . . . . . . . . . . . . . .

26

3.1

Michelson interferometry microtrap assembly . . . . . . . . . . . . . . . . . . . .

29

3.2

Schematic of Michelson interferometer sequence . . . . . . . . . . . . . . . . . . .

31

3.3

Representative interference patterns from the Michelson experiment . . . . . . .

32

3.4

Michelson BEC apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.1

A false-color interferometer result in which spatial dephasing has occurred . . . .

39

4.2

Representation of coarse two-pixel images in vector form . . . . . . . . . . . . . .

40

4.3

Representation of interferometry results as linear combination of basis images . .

42

4.4

Results of PCA of a typical interferometry data set . . . . . . . . . . . . . . . . .

44

xiii
4.5

Results of ICA of the same data set . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.6

Scree plot for 0114b data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.7

Strongest PCs of 0114b data set, organized in order of decreasing variance . . . .

50

4.8

Probability density of experimental signal is not Gaussian . . . . . . . . . . . . .

52

4.9

Plots of various probability densities . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.10 Result of MLE of probability density function . . . . . . . . . . . . . . . . . . . .

57

4.11 Plot of normalized likelihood near fit values for 0114b data set . . . . . . . . . .

58

5.1

Plot of experimental calibration using PCA . . . . . . . . . . . . . . . . . . . . .

60

5.2

Differential phase shift caused by vibrations . . . . . . . . . . . . . . . . . . . . .

62

5.3

Spectra of optical table vibrations along guide axis . . . . . . . . . . . . . . . . .

64

5.4

Normalized histogram of signal IC from data set taken with table floating . . . .

65

5.5

Microtrap module from Michelson apparatus . . . . . . . . . . . . . . . . . . . .

65

5.6

Probability density of simulated data set subject to quantum phase diffusion . .

70

5.7

Results of simulated data sets for various numbers of trials and amount of phasediffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

6.1

Schematic of translating waveguide interferometer . . . . . . . . . . . . . . . . .

75

6.2

Schematic of moving guide apparatus of Wu et. al. . . . . . . . . . . . . . . . . .

76

6.3

Multiple parallel wire waveguide operation . . . . . . . . . . . . . . . . . . . . . .

77

6.4

Illustration of variables associated with periodic current density calculation . . .

80

6.5

Lowest energy shape oscillations of a BEC . . . . . . . . . . . . . . . . . . . . . .

88

6.6

Plot of parameters c1 and c2 vs. L for various values of N . . . . . . . . . . . . .

94

6.7

Acceleration, velocity, and position of the trap center for ωz = 2π × 10 Hz.

97

6.8

Deviation of BEC center from trap center during motion . . . . . . . . . . . . . . 100

6.9

Trajectory of BECs in translating guide . . . . . . . . . . . . . . . . . . . . . . . 102

7.1

Schematic of laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

. . .

xiv
7.2

Annotated photograph of an assembled vacuum system. . . . . . . . . . . . . . 110

7.3

Deformation of Au wires on back of chip bonded at 400◦ C . . . . . . . . . . . . . 111

7.4

Rotary table, 200 mm diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.5

Schematic view of optomechanics structure . . . . . . . . . . . . . . . . . . . . . 115

7.6

Optical layout of 2D-MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.7

Optical layout of upper baseplate . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.8

View of angled 3D-MOT along axis of horizontal beam . . . . . . . . . . . . . . . 119

7.9

Mounted vacuum system with MOT and bias coils . . . . . . . . . . . . . . . . . 120

7.10 Views of the big Z assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.11 Photographs of both sides of the gyro chip . . . . . . . . . . . . . . . . . . . . . . 126
7.12 Views of gyro chip with electrical connectors . . . . . . . . . . . . . . . . . . . . 127
7.13 Side view of apparatus showing imaging system . . . . . . . . . . . . . . . . . . . 130
7.14 Diagram of control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.15 Grounding scheme for MOT and bias coil power supplies . . . . . . . . . . . . . . 134

Chapter 1

Introduction

1.1

Motivation
Atomic physicists interested in basic research have devoted considerable effort to probing

the fundamental properties of ultracold dilute gases in the last 15 years. Studies of these gases(specifically, Bose-Einstein condensates and degenerate Fermi gases) have examined superfluidity, coherent control of quantum systems, and various other subjects of interest. In more
recent years, some researchers have become interested in developing practical applications forthe interesting properties of ultracold systems. Ultracold atoms as sensitive probes of magnetic
fields, ultracold atoms trapped in optical lattice potentials as simulators of novel quantum materials, and ultracold atoms as inertial sensors are all applications being actively pursued. Just
as important as these proof-of-concept experiments are efforts to advance apparatus technology.
Without a replacement for the custom-built, finicky, bulky machines that are used by mostresearchers, even a “killer” application for the ultracold will never emerge from the laboratory
into popular use. We focus in this work on both the problems of technology and application
development, with the eventual goal of testing advanced inertial sensors for navigation.

1.1.1

Navigation and gyroscopes
Precise and accurate navigation is essential to the successful operation of aircraft, naval

vessels, spacecraft, and other vehicles. The global positioning system (GPS) allows modern
navigators to “fix” the instantaneous location of a vehicle at a level of performance that would

2
have been the envy of the celestial navigators of old. However, reliable navigation with GPS
alone requires constant communication with the system’s satellite network. Such communication
is impractical for submarines, useless for space probes, and impossible for all vehicles when the
GPS signal experiences interference (possibly due to hostile jamming).
A navigator may still calculate a vehicle’s position when GPS is unavailable, using atime-tested method known as dead reckoning. Dead reckoning is a simple concept: the navigator calculates the vehicle’s current position using its last known position and time-indexed
measurements of its speed and heading [1]. In the modern practice of inertial navigation, thesemeasurements are made by integrating the outputs of accelerometers and gyroscopes. The calculated displacement from the position of the last GPS update accumulates error due to theimperfections of the sensors used, so an accurate inertial navigation system must periodicallycorrect itself with an updated GPS fix. With improved sensors, the time between GPS updates
can be increased without reducing the accuracy of the position calculation. The development ofgyroscopes with reduced errors is thus of significant practical interest.

1.1.2

Optical Sagnac effect gyroscopes
Modern navigational gyroscopes do not necessarily resemble the spinning wheel of the

traditional mechanical device. The type of gyroscope addressed in this work, the Sagnac gyroscope, in fact has no moving parts. It is simply an interferometer in which the path taken by theinterfering waves encloses an area. A schematic of an example using optical interference is shownin Fig. 1.1. Rotation of this device at angular velocity Ω results in a phase shift between the
counter-propagating waves, and therefore a shift in the fringe pattern detected at the output.
This phase shift ∆φ is caused by the slight change of the length of the wave’s path aroundthe interferometer due to the rotation [3]. To calculate ∆φ, consider a waveguide of arbitrary
shape, rotating around an arbitrary axis (see Fig. 1.2). Light is coupled into the waveguideat a beamsplitter (not shown); the light runs around the waveguide in both directions before
reuniting at the beamsplitter. The time required for the light to traverse the infinitesimal

3

Figure 1.1: A schematic of a simple, optical Sagnac interferometer. Rotation of the device atangular velocity Ω causes a phase shift between the two arms [2].

distance dl is dl/c. The waveguide is moving due to the rotation; the component of the velocity
due to rotation along the waveguide axis is given by(Ω × r) · l̂.

(1.1)

Here, r is the position of a location on the waveguide and l̂ is a unit vector pointing along the
waveguide axis. If the rotation speed Ω is zero, then the optical path length around the guideis the same in both directions, and ∆φ = 0. If Ω 6= 0, the path length for the light traveling in
one direction is altered by
1
∆P =
c

I
(Ω × r) · dl,

(1.2)

ring

where l̂ dl = dl. Applying Stokes’s theorem, we have
1
∆P =
c

Z
∇ × (Ω × r) · dA.

(1.3)

ring

To evaluate Eq. (1.3), we will apply the identity∇ × (Ω × r) = Ω(∇ · r) − r(∇ · Ω) + (r · ∇)Ω − (Ω · ∇)r.

(1.4)

The identity ∇ · r = 3 reduces the first term in Eq. (1.4) to 3Ω. The second and third terms
vanish, since Ω has no spatial dependence. The fourth term evaluates to(Ω · ∇)r = (Ωx ∂x + Ωy ∂y + Ωz ∂z )(xx̂ + y ŷ + z ẑ)= (Ωx x̂ + Ωy ŷ + Ωz ẑ) = Ω.

(1.5)

4
We can now simplify Eq. (1.3) to

∆P =

2
c

Z
Ω · dA =ring

2Ω · A.
c(1.6)

The light propagating along with the rotation will have its path length increased by ∆P , while
the light propagating against the direction of rotation will have its path length decreased by thesame amount. The overall phase shift is given by∆φ = 2k∆P =8πΩ · A
,
λc

(1.7)

where k is the wavevector and λ is the wavelength of the light.

Figure 1.2: Interferometry waveguide of arbitrary shape, rotating at angular velocity Ω [3].

To see if such a device would be practically useful, consider the case of a jet airliner
traveling at a cruising velocity of about 1000 km/h. Its gyroscope must be sensitive to slow
rotations at the scale of 15◦ /h (coincidentally, this is the Earth’s rotation rate), or else theposition calculated by the inertial navigation system will be off by kilometers after just a minuteof flying. For a reasonably-sized circular interferometer of radius 5 cm, the path length shift
at Earth rate is only 3.8 × 10−15 m. Using a light source operating at λ = 850 nm, (a typicaltelecommunications wavelength) the phase shift is 5.6 × 10−8 rad. This phase shift is too small
to be easily measured interferometrically. At first glance, then, the sensitivity of an optical
Sagnac gyroscope seems too small to be of practical use.

5
One method that has been successfully used to increase the sensitivity is to increase theenclosed area by replacing the single-pass waveguide with a long optical fiber wound around aspool. A 1 km fiber wound around the 5 cm radius we just considered would have a phase shiftof about 180 µrad at Earth’s rotation rate, well within the realm of reasonable interferometricdetection (it becomes difficult to measure interferometric phase shifts when ∆φ is smaller than
1 µrad [4]). Interferometric fiber optic gyroscopes (IFOGs) of this scale are currently used in
the inertial navigation systems of a variety of vehicles, including the latest Airbus and Boeing
airliners [4, 5] and the F-16 fighter (see Fig. 1.3).

Figure 1.3: (Color). Northrup Grumman LN-260 inertial navigation system with GPS. This
unit is used in upgraded F-16 fighter aircraft. The two cylinders visible in the front of the unit
each contain an IFOG [C. Volk, J. Lincoln, and D. Tazartes, Northrop Grumman’s Family of
Fiber-Optic Based Inertial Navigation Systems, (http://www.es.northropgrumman.com/by_
division/navigationsystems/whitepapers/assets/FOGINS_Paper_Final.pdf)].

A reliable gyroscope design combines high sensitivity with low measurement errors. Phase
shifts caused by anything other than rotation lead to a bias in the measured rotation rate, suchthat heading calculated by the navigation system eventually drifts from the true heading. For
example, temperature variations cause phase shifts by mechanically straining or directly varying
the refractive index of the fiber. Accelerations can distort the shape of the fiber coil. Stray
magnetic fields can affect the polarization state of the propagating light. The sensitivity of the
IFOG to these and other environmental bias sources is greatly reduced by the reciprocal beam
path of the device. The IFOG’s remaining sensitivity to environmental bias can be measured and6calibrated out of the measurement, so the true figure of merit for an IFOG’s noise performanceis the stability of the bias (rather than its magnitude). Bias stability of less than 0.0003◦ /h has
been measured in high-grade IFOGs [1].
Unfortunately, IFOG technology has not led to inertial navigation systems that are completely independent of GPS updates. Using the 1 µrad detection limit just introduced, theminimum measurable rotation rate of the 1 km fiber gyroscope we have considered is about
0.1◦ /hr or 0.5 µrad/s. The position error after one hour of flight of an airliner using this device
will already be a few kilometers [6]. Practically, the minimum measurable rotation rate must
drop by 3 orders of magnitude for the position error after one hour to be less than a few meters.
The sensitivity can be increased by using a longer fiber or a larger radius [7], but eventually thedevice would become too large to be practical [4].

1.1.3

Matter-wave Sagnac gyroscopes
The IFOG reaches its sensitivity by effectively increasing the numerator of Eq. (1.7);

another way to reach high sensitivity would be to decrease its denominator. One way to do thisis to replace optical with matter-wave interference. In this case, Eq. (1.7) would become∆φmatter ==8πΩ · A
λdB v
4M Ω · A
,
h̄

(1.8)

where M is the mass of the interfering particles. We have replaced the wavelength and speedof light with the deBroglie wavelength λdB and propagation speed v of the particles. Note that∆φ does not actually depend on v. We can compare the sensitivity of a matter-wave gyroscope
to a given rotation to that of an optical gyroscope enclosing the same area by taking the ratioof Eq. (1.7) to Eq. (1.8):∆φmatter
λM c=.
∆φphoton
h

(1.9)

7
If the optical gyroscope operates at λ = 850 nm and the matter-wave gyroscope uses 87 Rb atoms(a convenient atom for laser and evaporative cooling, M = 1.44 × 10−25 kg), Eq. (1.9) gives∆φmatter= 5 × 1010 .
∆φphoton(1.10)

This enormous increase in phase shift makes a matter-wave gyroscope a candidate for use
in GPS-denied navigation. Consider a gyroscope that records a phase shift ∆φ = π rad whenspinning on-axis at Earth’s rotation rate. Using the practical sensitivity limit of ∆φ = 1 µrad
discussed previously (though this limit is quite ambitious for atomic detection methods), theminimum measurable rotation rate for such a device would beΩmin =ΩEarth× 10−6π= 23 prad/s = 4.8 µdeg/h.

(1.11)Such a gyroscope would enable an INS to keep position errors due to the coarseness of thegyroscope small over long periods of time. The enclosed area required for this sensitivity may
be calculated using Eq. (1.8). Again considering 87 Rb, we find A = 7.9 mm2 . A single-pass,circular path of this area would have a radius of only 1.6 mm. This device would be both smaller
and more sensitive than the IFOG considered in the previous section.

1.1.4

Shot noise in Sagnac gyroscopes
Shot noise places a fundamental limit on the precision of a Sagnac gyroscope’s rotation

rate measurement. This limit is not important in IFOGs, where the high flux of photons from abright light source pushes the shot noise limit below the minimum set by other practical limitsof the system. In the case of the low-flux sources available for matter-wave gyroscope, shot noisemust be considered.
We estimate the shot noise limit on the uncertainty of a Sagnac interferometer by considering the recombination of the interfering waves at the output beamsplitter [2]. We will here
consider the optical case for simplicity, but the mathematics and the resulting estimate of theshot noise limit is also correct for matter waves. Assume that the incoming uniform light beams

8have acquired a phase shift of ∆φ during their travel through the interferometer. A pair of
ideal detectors counts the number of photons at the outputs of the beamsplitter (see Fig. 1.4),
recording
N1 = NT sin2 (∆φ)
N2 = NT cos2 (∆φ).

(1.12)

Here, NT = N1 + N2 is the total number of photons detected. We ignore any additional
beamsplitter-related phase shifts. We combine these two equations to obtaintan2 (∆φ) =

N1
.
N2

(1.13)

Figure 1.4: The two output beams of an optical interferometer are recombined at a beamsplitter.
Ideal photodetectors count the photons at the beamsplitter’s outputs. Inspired by [2].

The shot noise uncertainty of the photodetector measurements,

δN1 =

p

δN2 =

p
N2 ,

N1
(1.14)

results in an uncertainty in ∆φ. The total differential of Eq. (1.13) shows how the two are
related:2 tan(∆φ)
N2 dN1 − N1 dN2d(∆φ) =.
cos2 (∆φ)
N22(1.15)

9
The two terms on the right side of Eq. (1.15) contribute independently to the uncertainty δ(∆φ).
Therefore, δ(∆φ) results from their sum in quadrature:
2 tan(∆φ)δ(∆φ) =
cos2 (∆φ)

p(N2 δN1 )2 + (N1 δN2 )2.
N22(1.16)

Using Eqs. (1.12) and (1.14), Eq. (1.16) simplifies to1.
δ(∆φ) = √2 NT

(1.17)

Assuming that Eq. (1.17)’s estimate of δ(∆φ) is the dominant source of uncertainty in
our Sagnac experiment, we insert it into Eq. (1.7) to find the uncertainty in Ω:
λc
δ(∆φ)
8πA
λc
√
.
=
16πA NTδΩ =(1.18)

This equation is correct for an interferometer aligned with its rotation axis. To find δΩ for amatter wave gyroscope, we simply replace λ and c with their matter equivalents. The result isδΩmatter =h̄√.
8M A NT(1.19)

If, as above, we use 87 Rb atoms and set A = 7.9 mm2 , the uncertainty is
2.4
11.6µrad/s = √deg/h.
δΩ = √
NTNT(1.20)We see that if our atom gyroscope operates with a packet of 60000 atoms, it beats the minimum
measurable rotation rate of the IFOG described in Sec. 1.1.2 by an order of magnitude. Further
performance gains would be realized with a larger flux of atoms.

1.1.5

Thermal beam atom gyroscope experiments
The predicted excellent performance of matter wave gyroscopes has been realized in the√laboratory, with one experiment achieving a rotational sensitivity of 6 × 10−10 rad/(s/ Hz)[8, 9]. Figure 1.5 shows a schematic of the experiment. The high sensitivity and low noise are
the result of the large enclosed area of the beam path (22 mm2 ) and the high flux of the thermal
beam atom source (1011 atoms/s).

10

Figure 1.5: Thermal beam atom gyroscope from [9]. The transverse momentum of the beam iscontrolled by optical Raman transitions. These transitions also transfer the atoms between two
hyperfine ground states (labeled as F = 3 and F = 4).

The source is an oven containing a high-pressure vapor of Cs atoms. The oven is connected
to an otherwise sealed vacuum tube through a small hole. Escaping atoms travel down the
tube in a high-flux beam, moving at the speed determined by the vapor’s thermal energy (290m/s). Note that the path of the atoms is not reciprocal, making this experiment sensitive toenvironmental bias. An upgraded version of this experiment [10] contains two gyroscopes, using
two counter-propagating beams as the atom sources. This version has a measured bias stabilityof < 70 µdeg/h.
Laser beams control the beam’s path through the tube. At the entrance to the tube, thebeam is optically pumped into the F = 3 hyperfine ground state. A pair of counterpropagating
laser beams, aligned transverse to the beam, is then applied. These beams are tuned to transfer
the atoms into a coherent superposition of the two ground states, via a π/2 two-photon stimulated Raman transition [11]. The process of absorption from one beam and stimulated emission
into the other gives the atoms transferred into the F = 4 ground state a transverse velocity
kick of 2h̄k/M = 7 mm/s. (This language is approximate. No atoms are actually “transferred”
between states in the creation of the superposition). The π/2 Raman transition thereby splits
the beam into two paths. A π Raman transition, applied at the halfway point of the tube, acts
as a mirror. It “flips” atoms from one transverse momentum and hyperfine state to the other.
By arranging pairs of Raman beams as shown in Fig. 1.5, the single atomic beam is converted
into an area-enclosing Sagnac interferometer.
While impressive, this gyroscope is not portable. The great mismatch between the for-

11
ward and transverse velocities of the beam (a ratio of 4 × 104 ) requires an elongated path toenclose the desired area. In [9], the path is just under 2 m long. A design of practical size must
be able to achieve a similar enclosed area in a much smaller package.

1.1.6

Advantages of using ultracold, guided atoms in gyroscopes
An alternative approach to the problem is to guide the interfering atoms around a morecircular path, instead of allowing them to freely propagate. One candidate for guiding atoms isthe magnetic microtrap [12], generated by current run through wires patterned onto a microchip(called an “atom chip”). The wire magnetic field, plus a transverse bias field, produces a magnetic waveguide or trap for atoms in weak field seeking states. Almost any shape of waveguidecan be generated by properly designing the wire pattern, including the reciprocal beam paths
that are desired for reducing measurement bias. Microtraps are suitable for use in portabledevices, due to their small size and power needs.
A thermal beam is an inappropriate source of atoms for a microchip waveguide, however.
The depth of the waveguide potential is set by the magnitude of the bias field and the magnetic
moment of the atomic state to be used. For the doubly polarized ground state of the alkali
metals, the depth is µB B0 . We see that even a high bias field of B0 = 100 G would create a
trap that could barely hold a gas at 7 mK, let alone the oven-heated atoms in a thermal beam.
On the other hand, microchip waveguides are perfectly appropriate for confining ultracoldgases, such as Bose-Einstein condensates (BECs). Since a BEC is a coherent, macroscopic sample
of atoms, its interference pattern should have high contrast. This fringe visibility enhancement
is analogous to that found when comparing laser to white light interferometers. Moreover,
BECs are well-suited for use with the laser beamsplitters and mirrors used in atom optics,
in which the efficiency is often sensitive to the thermal velocity distribution of the atoms. A
BEC’s delta-function velocity distribution allows for near-perfect efficiencies in momentum state
manipulation.

12

1.2

Outline of dissertation
With these motivations in mind, the construction and testing of a compact Sagnac gy-

roscope using guided, Bose-condensed atoms is a worthy goal. The work presented in thisdissertation consists of some of the preliminary work needed to reach that mark. The research
was carried out over the course of two projects. The first project, which was already underway
before the period covered by this dissertation, focused on the demonstration and investigationof guided BEC Michelson interferometry in a microchip waveguide. This project concluded withthe demonstration of high-contrast interference patterns obtained after BEC propagation of approximately 250 ms. Its results have been detailed in other dissertations, and will therefore bepresented only in summary here. The lessons learned from this first project have been appliedin the opening stages of the second project, whose goal is the demonstration of a guided BEC
Sagnac gyroscope. The portion of this project covered by this dissertation focused on the designand construction of a new, compact apparatus.
This dissertation contains my contributions to these two projects. Chapter 2 reviews
the principles of the two primary atom-optical elements used in the experiments: microtrap
waveguides and light gratings for atomic diffraction. Chapter 3 explains the final Michelsoninterferometry experiments, and includes a short description of the apparatus used. Chapter
4 presents the application of the statistical methods of principal and independent component
analysis to data from the Michelson experiments. Chapter 5 discusses the results and the lessons
learned from these experiments. Chapter 6 explains the design of an atom chip for Sagnac
measurements. Chapter 7 describes the design, construction, and initial testing of a compact,
rotating BEC apparatus that incorporates the new chip design.

Chapter 2

Atom optics: Waveguides and diffraction grating beamsplitters

The IFOG relies relies on familiar optical elements to control the path of its interfering
light; an atom gyroscope must have equivalent “atom-optical” elements. Our proposed gyroscope
will use magnetic waveguide potentials to control the path of the atoms and diffraction fromstanding waves of light to split and recombine the matter wave. This chapter reviews these twotechniques.

2.1

Magnetic microtraps and waveguides for atoms
In our experiments, guiding potentials for neutral atoms are generated with magnetic

fields. Consider an atom with total angular momentum F interacting with a magnetic field B.
The energy of the atom is
U = gF mF µB B.

(2.1)

In this equation, gF is the Landé g factor of the total angular momentum state, mF is thequantum number of the projection of F along B, and µB is the Bohr magneton. We see that a
local magnetic field minimum is a potential well for an atom in any state in which gF mF > 0.
A potential well of this type extending in two dimensions is called a magnetic waveguide, while
a well confining in all three dimensions is called a magnetic trap.
The microtrap is a compact, low-power device that can generate such magnetic traps andwaveguides for cold, neutral atoms. To lowest order, the steepness of a magnetic potential ischaracterized by the gradient of the field. The fields are typically generated by electric currents

14
traveling through single wires or coils, or else by permanent magnets. The gradient of a currentgenerated field scales as
B0 ∝

I
,
r2

(2.2)

where r is the distance between the current-carrying coil or wire and the trap position and I is
the current. It is therefore possible to construct steep traps using either (a) powerful permanent
magnets, (b) coils carrying hundreds of amperes of current placed a few centimeters from thetrap center, or (c) single wires carrying a few amperes placed a few hundred micrometers fromthe trap center. It is approach (c) that has inspired the development of the microtrap, which canpractically attain significantly tighter traps than the other alternatives. Since atoms confined
more tightly collide at an increased rate, evaporative cooling can proceed significantly fasterin microtraps without any loss of efficiency. The benefits in speed, size, and power have mademicrotraps attractive for the development of portable, high-bandwidth atom interferometers. In
this section, we will describe the theory of the microtrap elements used in our experiments andgive a short description of fabrication techniques. Further details on the development and useof the microtrap are outlined in [12] and [13].

2.1.1Side guide
The building block of the microtrap is a simple linear, quadrupole waveguide (the “sideguide”). It is generated by a straight current-carrying wire and a uniform bias field, oriented
perpendicular to the wire (see Fig. 2.1). The wire current (flowing along the z-axis) generates
the familiar circular magnetic field, so the total field can be written as

B = (B0,x , 0, 0) +µ0 Iz(−y, x, 0).
2π(x2 + y 2 )(2.3)

Here, B0,x is the magnitude of the transverse bias field and Iz is the current in the wire. The
form of Eq. (2.3) assumes the wire is infinitely thin and long, but is approximately correct forreal wires at distances that are further from the wire than its width but much shorter than the

15

Figure 2.1: (Color). (a) A schematic of a simple side guide. Here, the straight current-carrying
conductor (current Iz ) is a wire on the surface of an insulating substrate. The wire field andbias Bx cancel along the magenta dotted line, forming a waveguide for atoms in weak magnetic
field seeking states. (b) Plot of side guide’s quadrupole magnetic field. Streamlines indicate
field direction. Color indicates field magnitude, with darker colors corresponding to lower field.

16
wire’s length. The two contributions cancel along the line parallel to the wire defined by
x0 =


0,

µ0 Iz
, z
2πB0,x= (0, y0 , z).

(2.4)

We may Taylor expand the field magnitude B around x0 to obtain
B0,x
ρ,
y0

(2.5)

x2 + (y − y0 )2 .

(2.6)

B≈
where
ρ=

p

We see that the side guide potential is a linear/quadrupole waveguide for weak-field seeking
atoms.
The side guide is not by itself capable of confining ultracold atoms. Since the fieldmagnitude goes to zero at the center, Majorana transitions will cause atoms to fall out of theguide. The simplest way to prevent these losses is to add a second uniform bias field oriented
along the waveguide axis. The total field is now
B = (B0,x , 0, B0,z ) +µ0 Iz(−y, x, 0),2π(x2 + y 2 )(2.7)

for axial bias B0,z . The field magnitude near the guide center is approximately
B ≈ B0,z +

2
B0,x
ρ2 .
2y02 B0,z(2.8)

We see that the guide is harmonic in the biased case, with a trapping frequency
s
ωρ =
≈

2.1.2

r

gF mF µB d2 BM
dρ22
gF mF µB 2πB0,x.
M B0,z µ0 Iz(2.9)

Crossed wire trap
A simple three-dimensional trap can be made by crossing two wires at right angles. Onceagain using the limit of infinitely thin and long wires, the field contribution of the additional

17
wire at the waveguide is
Bc =

µ0 Ix
(0, z, −y0 )2π(y02 + z 2 )(2.10)

for a crossing wire carrying current −Ix along the x axis and intersecting the main wire at (0,0,0)[see Fig. 2.2(a)]. In general, the second wire could be displaced along the y axis by a distanced. In either case, the new wire’s field has no x component, and therefore the field minimum will
still be at y = y0 . By Taylor expanding around z = 0 [14], we findBc,y =≈µ0 Ix z2π[(d + y0 )2 + z 2 ]µ0 Ixz2π(d + y0 )2

Bc,z = −≈−µ0 Ix (d + y0 )2π[(y0 + d)2 + z 2 ]
µ0 Ix
µ0 Ix
µ0 Ix
+
z2 −
z4.
2π(d + y0 ) 2π(d + y0 )32π(d + y0 )5(2.11)For the case in which Ix  Iz , the field at the minimum (0, y0 , 0) is to lowest orderB ≈ B0,c +

2
B0,c
µ0 Ix
ρ2 +
z2.
2y0 B0,z2π(d + y0 )3(2.12)The trap frequencies are
r
ωρ =
s
ωz =

2
gF mF µB 2πB0,x
M B0,c µ0 Iz

gF mF µB
µ0 Ix,
M2π(d + y0 )3(2.13)

where
B0,c = B0,z −µ0 Ix.
2π(d + y0 )(2.14)

The combined potential consists of the radial waveguide along with a superimposed axial“dimple” in the space just over the crossing wire, where its field partially cancels B0,z . The
axial potential is only harmonic in a fairly small region around |z| = 0, due to the fourth-orderterm in Eq. (2.11). To see the effect of this term [15], consider a potential
U = b2 z 2 + b4 z 4 .

(2.15)

18

Figure 2.2: (Color). (a) Schematic of crossed wire (“dimple”) trap on a chip. In this example
d = 0; the wires actually cross. (b) Plot of magnetic field magnitude along tilted weak trap axis
for Iz = 1 A, Ix = 125 mA, B0,x = 20 G, B0,z = 3 G, y = y0 = 100 µm. The field magnitude
is only parabolic in shape near the trap center. (c) Contour plot of same configuration. Dark
color indicates lower field.

19
The harmonic trap frequency of this potential is1
1 ∂2U
M ωz2 z 2 = z 2 222 ∂z1= z 2 (2b2 + 12b4 z 2 )2
6b4 z 2
2b2
2
1+
.
ωz =
M
b2

(2.16)

The frequency stays at the value expected for b4 = 0 as long as
6b4 z 2 1.
b2

(2.17)For the dimple potential, the harmonic region of the axial potential is thereforez2 (d + y0 )2.
6(2.18)

Figure 2.2(b) shows the axial potential for typical parameters at y = y0 = 100 µm and d = 0.
The potential is clearly only a parabola near |z| = 0.
Another consequence of the addition of the crossing wire is that the y component of thetotal field no longer vanishes at x = 0. We have instead
By =

µ0
2π



Iz x
Ix z
+
x2 + y02(d + y0 )2 + z 2

.

(2.19)

This component vanishes along a line given to first order byx≈−

Ix y02z.
Iz (d + y0 )2(2.20)

The weak axis of the trap is therefore tilted away from the x axis bytan θ ≈ −Ix y02.
Iz (d + y0 )2(2.21)

This tilt is apparent in Fig. 2.2(c).
A variant of the crossed wire trap is the T trap [16], in which the crossing wire only exists
on one side of the main wire (see Fig. 2.3). The most obvious difference between the T trap andthe cross trap is that the z component of the magnetic field is cut in half: BT,z = Bc,z /2. The√
trap frequency ωz is therefore smaller by 1/ 2 for the same current, and the dimple potential

20is half as deep. The other difference is that the current flowing in the T wire must flow alongwith the main wire’s current on one side of the wire pattern. As a result, the waveguide will beslightly further away from the main wire on that side.

Figure 2.3: (Color). Schematic of a T trap.

2.1.3

H trapInstead of using a single wire to generate axial confinement in the waveguide, we could

instead use a pair of wires. An H-shaped pattern could be formed by two wires crossing themain wire, separated by a distance LH (see Fig. 2.4). The “leg” wires can also be displaced byd from the main wire along the y axis. The field of these wires at the waveguide isµ0 IH
BH,y = −2π

z + LH /2z − LH /2+(d + y0 )2 + (z + LH /2)2
(d + y0 )2 + (z − LH /2)2

4µ0 IH [4(d + y0 )2 − L2H ]zπ[4(d + y0 )2 + L2H ]2


µ0 IH
d + y0d + y0
BH,z =+
2π(d + y0 )2 + (z + LH /2)2
(d + y0 )2 + (z − LH /2)2
≈ −≈4µ0 IH (d + y0 )16µ0 IH (d + y0 )[4(d + y0 )2 − 3L2H ] 2−z22π[4(d + y0 ) + LH ]π[4(d + y0 )2 + L2H ]3+64µ0 IH (d + y0 ){16(d + y0 )4 + 5L2H [L2H − 8(d + y0 )2 ]} 4z .
π[4(d + y0 )2 + L2H ]5(2.22)

The field magnitude at the minimum (0, y0 , 0) is approximately
B ≈ B0,H +2B0,x16µ0 IH (d + y0 )[3L2H − 4(d + y0 )2 ] 2
2
ρ
+
z .
2y02 B0,H
π[4(d + y0 )2 + L2H ]3(2.23)

21
The trap frequencies arer

2
gF mF µB 2πB0,x
M B0,H µ0 Iz

s

gF mF µB 32µ0 IH (d + y0 )[3L2H − 4(d + y0 )2 ],M
π[4(d + y0 )2 + L2H ]3

ωρ =
ωz =

(2.24)

where
B0,H = B0,z +4µ0 IH (d + y0 ).
π[4(d + y0 )2 + L2H ](2.25)

Figure 2.4: (Color). Schematic of H wire trap, with H wires separated by distance LH .

An advantage of the H-wire configuration over the single crossing wire is that the anharmonic, fourth-order term in the axial confinement vanishes if
0 = 16(d + y0 )4 + 5L2H [L2H − 8(d + y0 )2 ].

(2.26)

By carefully choosing d and L for a given y0 , the anharmonicity parameter24{16(d + y0 )4 + 5L2H [L2H − 8(d + y0 )2 ]} 2
6b4 z 2=zb2[4(d + y0 )2 + L2H ]2(2.27)

can be kept quite small even at large distances from the trap center. Similarly, the tilt of the
weak trap axis due to the y component of the H wire field can designed to be quite small. The
axis lies along the linex≈

8IH [4(d + y0 )2 − L2H ]y02z,
Iz [4(d + y0 )2 + L2H ]2(2.28)

tilting away from the x axis by the angle
tan θ ≈

8IH [4(d + y0 )2 − L2H ]y02.
Iz [4(d + y0 )2 + L2H ]2The angle θ vanishes if the numerator is zero.

(2.29)

22
A useful variant of the H trap is known as a Z trap. Recall that the T trap is a variantof the crossed wire trap with the axial trapping wire cut back on one side; the Z trap similarly
removes the +x side of one of the H’s crossing wires and the −x side of the other. The similar√
result is a reduction in ωz by a factor of 1/ 2. A finite-length Z wire can be created by bending
a straight wire at both ends; this configuration is driven by a single current supply.

2.1.4

Fabrication of microtraps
The method of fabrication of wire structures for microtraps varies according to the desired

distance between the wire and the trap center. Wires that are not terribly close to the trap
center (d + y0 > 1 mm) are easily fabricated by attaching thin copper wires to a suitable surface
or support structure. The wires that are closest to the atoms (d = 0), however, are fabricated ona chip substrate by lithography. A variety of wire metals, substrates, and fabrication techniques
have been used to create these “atom chips”; see [12] for a review.
The substrate used in our work is Si coated with an oxide insulating layer. Silicon
is by far the most commonly used substrate material in microfabrication, where it is used insemiconductors, integrated circuits, computer chips, and microelectromechanical systems. These
applications have driven decades of effort into developing reliable Si fabrication techniques. By
using Si substrates, we take advantage of this work. Additionally, Si’s good thermal conductivity(relative to other crystalline substrate materials) allows Si atom chips to dissipate more electrical
power without heating [17]. The chips used in the early stages of this work used a ceramic
substrate material that has a higher thermal conductivity: AlN. However, the relatively poor
surface quality of wires patterned on AlN precluded its use in further work.
The wires on our atom chips are fabricated by photolithographic patterning, followed byelectroplating the patterned wires to a thickness of about 10 µm. Our group has found Cu to
be a good choice for the wire metal. Gold, which is more commonly used in microfabrication,is generally unsuitable for use in vacuum systems containing Rb (our preferred atomic species).
Gold wires tend to absorb residual Rb vapor from the vacuum chamber, as the two metals

23
readily form an alloy. This absorbed Rb is released when the wires are heated by the flow ofcurrent, bombarding the nearby trapped atoms and thereby lowering trap lifetimes [18].

2.2

Diffraction grating beamsplitters for atoms
Our Sagnac gyroscope experiment needs, in addition to the waveguide potentials discussed

in the previous section, a method for splitting and recombining the interfering matter wave. In
our work, the role of the beamsplitter is taken by coherent diffraction from gratings of lightinto a pattern of transverse momentum states. This section discusses the particular type ofatomic diffraction used in our work; historical details and a review of other types of atomic
beamsplitters may be found in [11].
A diffraction grating, in the broadest sense, is a periodic obstruction or potential thatcauses a wave interacting with it to scatter into a specific pattern of momentum states. In
the familiar case of light passing interacting with a physical grating, the diffracted orders are
separated in transverse momentum by

∆p =

h
,
Dg

(2.30)

where Dg is the period of the grating. In our experiments, a cloud of atoms interacts with an
off-resonant standing wave of light, splitting the atoms into states of momentum along the axisof the standing wave that are separated by∆p = 2h/λ = 2h̄k.

(2.31)The standing wave acts as a periodic potential for the atoms, playing a role analogous to thatof a physical grating for beams of light.
A trapped condensate can be briefly exposed to a light grating by applying a pulse of the
optical standing wave. A single, brief interaction populates many momentum states, with the
distribution given by the theory of Kapitza-Dirac scattering. This is not quite the beamsplitterthat we want for our interferometry experiments. We want a true 50/50 beamsplitter: the atoms

24
exit it in a superposition of only two momentum states. One way to solve this problem is to use
multiple weak gratings to control the diffraction pattern. This technique was first demonstrated
and explained in [19, 20] and is reviewed in this section.

2.2.1

First order beamsplitter
Consider an atom exposed to a standing wave of light, where the wavenumber of the laser

used is k0 . The AC Stark shift potential that the atom experiences may be written as
U (z, t) =h̄Ω2R (t)cos 2k0 z
2δ(t)= h̄ΩE (t) cos 2k0 z,(2.32)

where ΩR is the Rabi frequency, δ is the detuning from resonance, and ΩE is the effective twophoton Rabi frequency. The effective Rabi frequency (and therefore the power and/or detuningof the laser) is time-dependent. This potential is correct in the limit of large detuning withrespect to the Rabi and excited electronic state decay frequencies. The atomic state evolves
according toiψ̇(z, t) =


h̄ d2
−
+ ΩE (t) cos 2k0 z ψ(z, t).
2M dz 2(2.33)

By expanding the wavefunction in a Fourier series/integralZψ(z, t) =dk

X

C2n (k, t)ei(2nk0 +k)z ,(2.34)nEq. 2.33 is transformed (via Fourier’s trick) into
iC2n (k, t) =h̄
ΩE (t)(2nk0 + k)2 C2n (k, t) +[C2n+2 (k, t) + C2n−2 (k, t)].
2M2(2.35)

Consider a system that is initially stationary, such that k  k0 and only the n = 0 state ispopulated. If the grating is very weak, such thatΩE (t)(2Np )2 h̄k02= (2Np )2 ωr ,2
2M

(2.36)

then momentum states |2Np | and higher will never be significantly populated (ωr is the recoil
frequency of the light). If we set ΩE (t)  32ωr , then the infinite set of equations may be25truncated to only include n = −1, 0, 1. Making the unitary transformation
"  
#
2
k
C2n (k, t) → exp −i
ωr t C2n (k, t),k0(2.37)

we can describe the system evolution byΩE (t)[C−2 (k, t) + C2 ]2kΩE (t)
iĊ2 = 4ωr 1 +
C2 +
C0
k0
2


k
ΩE (t)
iĊ−2 = 4ωr 1 −
C−2 +
C0 .
k0
2
iĊ0 =(2.38)We define superposition states
1
C+ = √ (C2 + C−2 )2
1
C− = √ (C2 − C−2 )2(2.39)

and apply the approximation k  k0 , obtainingΩE (t)iĊ0 (k, t) = √ C+2
ΩE (t)iĊ+ (k, t) = 4ωr C+ + √ C02
iĊ− (k, t) = 4ωr C− .

(2.40)The “−” state is dark and will never be populated. Making one more unitary transformation
C → e2iωr t C,

(2.41)

we find that we now have a simple two-state systemΩE (t)iĊ0 = −2ωr C0 + √ C+2
ΩE (t)iĊ+ = 2ωr C+ + √ C0 .
2(2.42)

This system is best visualized on a Bloch sphere. If we set C0 = eiφ sin θ/2, C+ = cos θ/2,√
rs = (sin θ cos φ, sin θ sin φ, cos θ), and define the Rabi vector ΩR = ( 2ΩE (t), 0, 4ωr ), then Eq.
(2.43) is transformed todrs= ΩR × rs .
dt

(2.43)

26
A sequence of two square pulses of light can be used to place an initially stationary
ensemble of atoms into the C+ state (see Fig. 2.5). In the first pulse, the effective Rabi frequency√
is set to ΩE = 2 2ωr . The Rabi vector for this pulse is ΩR = 4ωr (1, 0, 1), with magnitude√√ΩR = 4 2ωr . After half of a precession period, τ1 = π/(4 2ωr ), the state vector has rotated
up to the xy plane of the Bloch sphere, pointing along the −x axis. At this time, the light isshut off. The Rabi vector is now ΩR = 4ωr (0, 0, 1), with magnitude ΩR = 4ωr . After half of thedark precession period, τ2 = π/(4ωr ), the state vector has precessed around to the +x direction.
The light is turned on for a second pulse, with effective Rabi frequency and duration equal tothat of the first. After this pulse, the state vector points along the z axis of the Bloch sphere,
meaning the system is now in a superposition of the 2h̄k0 and −2h̄k0 momentum states along
the axis of the grating.

Figure 2.5: (Color). First order double-pulse optical beamsplitter. The upper plot shows theeffective Rabi frequency as a function of time. The pictures at the bottom show the state of thesystem on the Bloch sphere at the beginning and end of each of the pulses. The yellow arrows
indicate the progression from one state to the next. Adapted from [20].

27
2.2.2

Higher order beamsplitters
It is possible to extend the first order double-pulse concept to higher orders of diffraction,

so that the momentum transferred to the cloud is 2nh̄k0 . By numerically integrating Eq. (2.35),
optimized effective Rabi frequencies and pulse durations can be found for up to n = 6 withtransfer efficiency above 85% [20]. Pulses up to n = 3 were experimentally demonstrated in[21]. Non-square pulses could also be used to increase the efficiency of higher-order diffraction
pulses [22]. Even larger amounts of momentum have been transferred by Bragg diffraction [23],in which the pulse durations are longer than those used in the double-pulse scheme presented
here.

Chapter 3

Michelson interferometry experiments with BECs

In this chapter, we present a guided Michelson-type interferometer using Bose-Einsteincondensates. The two atom-optical elements presented in the previous chapter, magnetic microtraps and optical beamsplitters, are the key techniques that enabled these experiments.
This work and the experimental apparatus used have been presented in previous dissertations[16, 24, 25] and publications [19, 26, 27]. This chapter will discuss only the final iteration ofthese experiments.

3.1

On-chip, free-return interferometry with BEC
The BEC interferometry experiment begins at t = 0 with a stationary condensate of 87 Rb

atoms in the |F = 1, mF = −1i ground state. The microtrap potential holding the condensate
consists of an on-chip side guide at ωρ = 2π×80 Hz, combined with a crossed wire trap adjustablebetween ωz = 2π× 4–6 Hz, centered 3.24 mm below the surface of the chip. A pair of mirrorcoated prisms are epoxied onto the chip, perpendicular to both its surface and the guide axis.
These mirrors are used to create the optical standing wave for the double pulse beamsplitter.
The configuration is shown in Fig. 3.1. In the actual apparatus, the chip is mounted with theprisms pointing down so that clouds of atoms released from the trap will fall away from the chip
instead of crashing into it.
At time t = 0, the first order double pulse beamsplitter is applied to the stationary
BEC (see Fig. 3.2 for a schematic of interferometry sequence). Our implementation of the29

Figure 3.1: (Color). Michelson interferometry microtrap assembly, consisting of a side guidewire on a chip, a copper heat sink, a crossed wire trap displaced below the chip, and integrated
mirrors for the optical grating. Schematic (a) and photograph of actual device (b) are shown.
Other wires on the microchip visible in (b) are used for initial capture and cooling of atoms.

30
double pulse uses light at k0 = 2π/780 nm, detuned 6.8 GHz to the red of atomic resonance.
The condensate splits into two BECs, one traveling down the waveguide axis at initial velocity
v0 = p0 /M = 2h̄k0 /M = 11.8 mm/s and the other in the opposite direction at −11.8 mm/s.
(Rigorously speaking, the BEC is actually in a quantum superposition of both momentum states.
The classical language is only a convenient shorthand). The moving BECs slow down as they
climb the axial potential, stopping at the classical turning point at t = π/(2ωz ). For the lowest
typical trap frequency used in our experiments, ωz = 2π × 4.2 Hz, the BECs are separated by
890 µm at this point. The BECs are allowed to continue oscillating in the axial potential for a
total of t = Nτ /2 ×π/ωz , where Nτ /2 = 1, 2, or 3. After Nτ /2 half-periods, when the condensates
are overlapped at the center of the potential, the beamsplitter pulse is applied again.
The second beamsplitter pulse’s efficiency in transferring the moving BECs back to thestationary state depends on any relative phase shift ∆φ that they acquired during propagation
through the trap. The momentum distribution after the second pulse obeys
Rc =|C0 |2∆φ= cos2,
2|C0 | + |C+ |2
2

(3.1)

where Rc is the fraction of atoms stopped by the pulse. This quantity is considered to be theoutput of the interferometer. To measure the momentum distribution, we shut the trap off justafter the second pulse. The system is allowed to freely evolve for 15 ms, after which atoms in the+ momentum state will have physically moved away from the center of the trap [see Fig. 3.2(f)].
An absorption image (directed perpendicular to the guide axis) taken after this free evolution
time can clearly distinguish three peaks in the distribution, corresponding to the p = 0, 2h̄k0 ,and −2h̄k0 states. See Fig. 3.3 for representative results.
A differential phase shift ∆φ between the propagating condensates can be intentionallyapplied by adding a magnetic field gradient B 0 (t) along the waveguide axis once they are spatiallyseparated. This shift obeys
Z Nτ /2 π/ωz
∆φ =0

gF mF µB B 0 (t)∆z(t)
dt,
h̄

(3.2)

where ∆z is the distance between the BECs. This phase shift can be quite large for even small

31

Figure 3.2: (Color). Schematic of Michelson interferometer sequence. (a) Optical grating isapplied to stationary condensate in weak axial trap along z (radial potential not shown). (b)BEC splits into momentum states of opposite sign along axis. (c) BECs stop at classical turning
points. (d) BECs overlap again at the center after t = π/ωz . (c) and (d) repeat until (e), when
the optical grating is applied again, uniting the condensates. (f) The interference pattern isobserved by shutting off the trap, allowing the momentum states to move away from each otherfor time τf , and finally recording an absorption image of the atomic density.

32

Figure 3.3: (Color). Representative interference patterns from the Michelson experiment. These
are false-color absorption images of the atomic density, recorded for various ∆φ. The central
cloud is centered at p = 0, the right at p = 2h̄k0 , and the left at p = −2h̄k0 . (a) ∆φ ≈ 2πm,(b) ∆φ ≈ (2m + 1)π/2, (c) ∆φ ≈ 2π(2m + 1) for some integer m.

33
B 0 when the BECs are at their classical turning points, so the magnetic field gradient must be
carefully controlled.
Our BEC interferometer is analogous in configuration to an optical Michelson interferometer, in which a single beamsplitter splits and reunites the optical wave at a static location.
There is no static beamsplitter in our interferometer, but the reuniting pulse is applied whenthe BECs have returned to the their initial positions.

3.2

Apparatus
The microtrap assembly shown in Fig. 3.1 is only 6.5 cm long and 500 mg in mass, but it

was just a fraction of the size and weight of the vacuum and optical system needed to support
its operation. This section briefly reviews the production of BECs in the apparatus. The goal
here is to point out those characteristics that would cause gyroscope experiments to be difficultin this apparatus. For further details, see [16, 25].
The cold atoms used by the Michelson microtrap were prepared away from the chip in a“macrotrap” system (i.e., a system using large coils and permanent magnets to generate magnetic
traps). The idea was to use off-the-shelf vacuum hardware and well-understood experimental
techniques (see [28, 29]) to prepare the atoms, leaving the experimental uncertainty to the (atthe time) novel microtrap. The result was a 1.5 m long, stainless steel vacuum system divided
into three chambers (see Fig. 3.4). The chambers could be vented independently, allowing
replacement of the microtrap assembly without disturbing the rest of the apparatus.
Condensate production began with the capture of 87 Rb atoms in a large pyramidmagneto-optical trap (MOT). The single beam was 5 cm in diameter. The quadrupole magnetic
field coils were each made from 48 turns of copper tubing, through which water was passed forcooling. After 4–8 s of MOT loading, the MOT was compressed and pumped to the F = 1
ground state. The atoms were then loaded into a quadrupole magnetic trap generated by thecoils, using a current of 250 A.
Since the MOT chamber’s background pressure of untrapped Rb was high enough to34

Figure 3.4: (Color). Michelson BEC apparatus (a) schematic [16] and (b) photograph.

35
inhibit evaporative cooling, the trapped atoms were transported out of the chamber by moving
the coils on a servo-controlled track. The coils moved the atoms down a 60 cm long tube in 2
s to a glass cell containing ultrahigh vacuum. The atoms were transferred at this point into a
hybrid Ioffe-Pritchard trap generated by a pair of permanent magnetics and a set of coils. This
harmonic potential had trap frequencies of 170 × 170 × 7 Hz. The atoms were further cooled byradio-frequency (RF) driven evaporation over the course of approximately 80 s to 1 µK.
The atoms were finally transferred via a hybrid magnetic waveguide from the evaporation
cell to the microtrap, which was located in a third vacuum chamber. After further RF evaporation in a tight chip trap, a condensate of 5000-70000 atoms was created. The trap was carefullyloosened into the waveguide described above before beginning the interferometry sequence. The
entire cycle was about 90 s in duration.

Chapter 4

Data analysis of Michelson interferometry experiments

In the previous chapter, the output of the Michelson interferometer was described as Rc ,the fraction of atoms in the momentum state p = 0 after the second beamsplitter pulse. The
momentum distribution was recorded by taking an absorption image of the atomic density ashort time after the pulse. We have not yet detailed how the fraction of atoms was extracted
from these images. Our data analysis methods are the topic of this chapter.

4.1

Model-based vs. model-free inference
Fits of images to theory models have typically played the central role in the extraction of

measurements from ultracold gas experiments. From one image (or sometimes many images),
one typically wishes to extract just a few parameters of interest from the thousands to millions ofpixel values. In most experiments, this parameter extraction task is accomplished by fitting theimaged atomic distribution to a model. This model usually is derived from a priori knowledgeof the trapping potential and atomic state. The model usually takes into account a period offree expansion of the cloud from the trap that occurs just before the picture is snapped [30].
The experimenter then uses the fit parameters to calculate the results of interest (temperature
is calculated from the Gaussian width of a gas in free expansion, for example).
This “fitting” approach to data analysis is an example of model-based inference. It isan appropriate and meaningful approach when the model takes into account the physics atplay. The experimenter’s a priori knowledge must be reasonably complete and accurate, and37
the experimental noise must not overwhelm the signal. The utility of model-based inference
can rapidly disintegrate if these conditions are not met, since it can fail to extract the properparameters of interest from data showing an unexpected effect.
The complement of a model-based approach is model-free inference. Model-free analyses
use statistical concepts such as maximum likelihood, entropy extremization, and independence toextract the significant features buried in a collection of data. Powerful techniques have emerged
over the past three decades and have flourished in fields where analytical models are frequently
hard to develop, such as biology, economics, and the social sciences. These techniques are lessfamiliar in physics. For experiments in which a complete model is difficult to develop, model-freeinference would allow the data to speak for themselves, free of the interpretive bias of a wrong
model.
This chapter presents the application of a closely related pair of model-free analysis
techniques to our Michelson data. These two statistical techniques, called principal component
analysis (PCA) and independent component analysis (ICA), have been used in image- and signalprocessing applications as diverse as human face recognition, brain signal analysis, economic
prediction, and astronomical data processing [31, 32, 33, 34]. While our specific focus is on
interferometry, a secondary purpose here is to use this experiment to demonstrate the utility andapplication of these techniques to information extraction from complex ultracold matter images
in general. For example, these methods could be of use in the study of complex structures seen
in gases containing vortices and gases trapped in optical lattices. We demonstrate here the useof these methods to calibrate our interferometer, to identify and mitigate unforeseen sources ofnoise, and to uncover signal data that are partially buried in noise. Our statistical methods
are able to complete these tasks in a fraction of the time needed for fitting routines to run.
This chapter expands on a previous treatment of our use of PCA and ICA [27]; an alternativedescription is given in [25].

38

4.2

Model-based image analysis
Before presenting PCA and ICA, we describe the application of traditional fitting methods

to extract Rc . In this method, the three peaks in the momentum distribution are fit to Gaussianor Thomas-Fermi distributions [35]. The fraction of atoms in the p = 0 peak is calculated fromthe fit parameters.
This approach has an inherent weakness: There is no simple a priori model for the shapeof the momentum distribution of the atoms after the interferometry cycle. The atoms begin theexperiment in a pure, stationary BEC with a simple Thomas-Fermi distribution. During theexperiment, the propagating BECs move through each other multiple times. With each pass,a fraction of the atoms undergo collisions and are scattered out of the ballistically propagatingcondensate packets. Some of these ejected atoms remain in the trap for the duration of theexperiment and are recorded in the absorption images. These ejected atoms are no longer
coherent with the condensate packets, but a three-peak fit is unable to distinguish the “noise”
atoms from the coherent atoms. This is just an example of one possible decoherence mechanismthat causes an apparent decrease in the interferometry signal’s visibility. Furthermore, ourroutines have trouble fitting three independent shapes to the image when the space between thepeaks contains a significant number of scattered atoms.
Of additional concern, the fitting of three independent two-dimensional shapes to each
image in a data set that contains upward of 100 images is computationally intensive. Our
fitting implementation takes up to an hour to completely fit such a set of images. The slow
speed prevents adjustment of experimental parameters in real time, a shortcoming that would
become a serious problem in an apparatus with the rapid BEC production rate needed to takehigh-bandwidth measurements.
Finally, the peaks themselves are often clearly not well described by the simple ThomasFermi distribution (see Fig. 4.1 for an example). As noted in [36], ∆φ is expected to be spatiallyinhomogeneous in experiments of this type because of dephasing effects. Such inhomogene-

39
ity causes the three peaks in the momentum distribution to acquire spatial structure. If one
attempts to fit a Thomas-Fermi distribution to the results of an experiment in which the interferometer suffers from dephasing, the fit will integrate over the spatial information and suppress
the interferometer’s contrast. The signal would still be present in the images, but the use of theincorrect model would bury it. Other unidentified symptoms of noise or unexpected physicalprocesses could similarly escape detection if the data were fit by an incomplete model.

Figure 4.1: (Color). A false-color interferometer result in which spatial dephasing has occurred.

4.3

Statistical image analysis
In place of fitting the images to a model, a statistical method might be used to extract

and analyze the significant features. In statistical image analysis, it is useful to represent eachpixelated, gray-scale image i as a p̂-dimensional vector Xi , where p̂ is the number of pixels.
The vector’s projection along dimension j is equal to the value of pixel j of the image. For avery coarse camera that records only two pixels, a set of (totally independent) images could beplotted as in Fig. 4.2(a). An analysis of a set of images is then carried out by using the statisticsof the corresponding set of vectors to find a more useful set of basis vectors uj in which to
represent the data, such that Xi = M + Y1i u1 + Y2i u2 + · · · + Yp̂i up̂ ; here, M is the mean imageof the set.
To illustrate the inconvenience of the original pixel basis, consider the case of a setof images from our interferometry experiments. Each image contains three spatially localized40

Figure 4.2: Representation of coarse two-pixel images in vector form. (a) A set of independent,
unrelated images. (b) A set of images in which the variation of the two pixels is not independent;
this could be a set of similar images in which some detail is varied from shot to shot.

41
features: the peaks in the momentum distribution. Each peak is recorded in the frame by thevalues of a localized subset of pixels. We know that the relative prominence of the central peakchanges as ∆φ is varied; this variation is recorded by a change in the value of all of the pixels
in the center of the frame. In the vector representation, the components corresponding to thosecentral pixels do not vary independently of each other. A plot of the vector representation ofa two-pixel image of just the center of the central peak would look something like Fig. 4.2(b).
It is clear that the pixel basis in which we have represented the image vectors is not linearly
independent.
The image vectors could instead be represented in a different basis. A natural choice
requires each basis vector to consist of a linear combination of pixels that vary independently
from any other grouping of pixels, portraying a single image characteristic that varies independently of all other characteristics. The characteristic (which the experimenter can conveniently
examine by representing the basis vector itself as an image) is connected to the variation of aparameter of the experiment. In a well-controlled experiment, most of the differences between
the images would be quantified by the coefficient of the single basis vector corresponding to
the experimentally controlled parameter. The variation of the other coefficients would represent
unwanted noise processes. The important information in a data set will be represented by thevariation of only the small number of basis vectors corresponding to the experimental signal andthe most significant noise sources.
PCA and ICA are both algorithms that try to find this statistically independent basis.
The difference between the two, to be detailed in the next section, is that ICA uses a higher
order test of independence to find the new basis than PCA uses. An example of a subset ofthe basis images found by these algorithms for a given data set is shown in Fig. 4.3. (Thebasis images computed from another data set would appear slightly different). The data image
i is equal to the sum of the data set’s mean image (the last image in the sum) and a linear
combination of the basis images. This example illustrates how straightforward it usually is toassign a physical process to a given basis image by simple visual inspection. We emphasize that42
this process does involve using a sort of “interpretive” model to assign meaning to the images.
To begin, we must find the basis image corresponding to the primary experimental signal, achange in the fraction of atoms found in the central momentum peak. The third basis image in
Fig. 4.3 represents precisely that change. An image in which all of the atoms were in the center
would have a large, positive value of Y3i , for example. An examination of the other two basis
images shows the experimenter the two dominant noise processes recorded in the data: The
first corresponds to an overall fluctuation in the number of atoms in the experiment, and thesecond to a fluctuation in the vertical position of the atoms at the moment of the photograph.
The experimenter may use this information to debug and reduce the experimental noise. The
complete basis contains another NI − 3 basis images (where NI is the number of images in theset). In this example, their coefficients varied only slightly from image to image. These images
represent detection noise which can be ignored. Details on the selection of which basis images
convey important information may be found in Sec. 4.4.2.

Figure 4.3: (Color). An example showing how an image i from the interferometry experiment
may be represented as a linear combination of independent basis images, each representing asingle physical parameter in the experiment, plus the mean image of the set. False color is used
to represent pixel value in all images. Varying Y1i changes the overall number of atoms; varying
Y2i changes the overall vertical position of the three peaks; varying Y3i changes Rc ).

43
The process of recovering the signal is now straightforward. The coefficient Y3i is proportional to the fractional population that was previously extracted by fitting the images:
Y3i = A cos ∆φ = A(2Rc − 1)

(4.1)

The coefficient Y3i now constitutes the primary measurement of the state of the interferometerin any given attempt. The phase ∆φ (modulo 2π), amplitude A, and estimates of the visibilityof the fringes may be extracted by fitting a set of Y3i values to Y3i = A cos ∆φ or to the expected
statistical distribution based on that equation. This procedure does require fits to a model, but
the fits are applied to the output of the image analysis process rather than to the raw images
themselves. The distinction is analogous to the difference between using a model to explain thedependence of experimental parameters on an independent variable and using Gaussian fits inthe first place to extract those parameters from a set of raw images. Parameter extraction ofthis type will be detailed in Sec. 4.5.
Figure 4.4 shows the result of carrying out PCA on a typical data set of over 100 images,
while Fig. 4.5 shows the result of ICA carried out on the same images. We observe that thehigher order test used in ICA does a better job of isolating Rc than the lower order test of PCA
does; the second principal component in Fig. 4.5 looks most like it describes the signal, but it isclear that varying the third principal component will slightly change Rc as well. On the other
hand, the third independent component in Fig. 4.5 is the only one of the three that encodes thevariation of Rc that we seek.

4.4

Details of algorithms
In this section we will detail the PCA and ICA algorithms. We begin by discussing

the distinction between the two. Recall that the goal is to represent an image Xi as a linear
combination of a set of basis images: Xi = M + Y1i u1 + Y2i u2 + · · · + Ypi up̂ . The basis images
calculated by PCA are an orthonormal set. For a set of NI image vectors represented in thisprincipal component (PC) basis, the set of coefficients of any one basis image j is required to44

Figure 4.4: (Color). Results of PCA of a typical interferometry data set. The three principal
component basis images (PCs) shown in false color account for most of the total pixel variance(σ 2 ) in the data set. The experimental signal Rc = cos2 (∆φ/2) can be related to PC 2, but it
is clear that some of it is mixed into PC 3.

45

Figure 4.5: (Color). False-color results of ICA of the same data set used in Fig. 4.4. Independent
component basis image IC 3 is clearly the experimental signal; it is not mixed into the othertwo ICs.

46
be statistically uncorrelated with the set of coefficients of any other basis image k:
2
σjk
=

(P )

In this equation, YjiNI
1 X(P ) (P )YY= 0.
NI − 1 i=1 ji ki(4.2)

is the coefficient of the PC basis image j in the representation of imagei. In contrast, ICA applies a considerably stronger test to the statistics of the coefficients; thebasis it calculates is, however, not necessarily orthogonal. The coefficients of the independent
component (IC) basis are statistically independent, obeying
N

I

 

1 X
1
(I)
(I)
g Yji h Yki − 2
NI i=1NI(I)

Here, Yji

NI


X
(I)
g Yji
i=1

!

NI


X
(I)
h Yki

!
= 0.

(4.3)

i=1

is the coefficient of the IC basis image j in the representation of image i, and g

and h are any integrable functions. Two coefficients that are independent are necessarily also
uncorrelated, since the definition of decorrelation can be recovered by setting g(x) = h(x) = x(as long as the mean value of each coefficient is zero). Independence can therefore be thoughtof as a higher order decorrelation.
One might conclude from the preceding paragraph that ICA, with its use of higher order
statistics, is always the more useful technique. It is indeed more reliable at the task of extracting
the experimental signal. There exists a class of tasks, however, for which PCA remains themethod of choice. Recall that for any set of vectors represented in two different orthonormalbases, the sum of the variances of the coefficients in basis 1 must be equal to the sum of thevariances of the coefficients in basis 2. This rule applies to the image analysis problem as well,such that all of the image-to-image variation is preserved intact in the PC basis. Since theoverall variance is conserved, the variance of a given PC coefficient can be used to rank how
much of the overall variation in the data is accounted for by that PC. This ranking is essential
in the calibration of the experiment, described in the next chapter. In contrast, the IC basis
is not orthogonal. The total variance of the coefficients of the IC basis images is not equal to
the total variance of the original pixels. For this reason, the IC basis images cannot be easily
ranked by their variance.

47
4.4.1

Principal component analysis
Having explained the difference between the two algorithms, we describe the details of

the calculations. PCA is a textbook application of linear algebra [37], so the first step in theprocess is to represent a set of images in matrix form. Consider a set of NI images of p̂ pixels
each. The pixel values of each image i are written as the components of a vector Xi . If themean image, defined as
M=

1(X1 + . . . + XNI ),
NI

(4.4)

is subtracted from each image, a p̂ × NI matrix of the images in mean-deviation form can beconstructed:
X̂ = (X1 − M= (X̂1X2 − M

X̂2

···

···

XNI − M)

X̂NI ).

(4.5)

The p̂ × p̂ sample covariance matrix of the image set is calculated from X̂:S=

1
X̂ X̂ T .
NI − 1(4.6)

The diagonal element Sjj is the sample variance of pixel j:2
Sjj = σjj=

NI
1 X
X̂ 2 ,NI − 1 i=1 ji(4.7)

and the total variance of all the pixels is tr(S). The off-diagonal element Sjk is the covarianceof pixels j and k:2
Sjk = σjk=

NI
1 X
X̂ji X̂ki .
NI − 1 i=1

(4.8)

2
If σjk= 0, then pixels j and k are uncorrelated.

PCA uses the covariance matrix to calculate a new basis for the images. Each column iof X̂ (representing one image X̂i ) is a linear combination of the original pixel basis vectors:
 
 
 
1
0
0
 
 
 
 
 
 
0
1
0
 
 
 
 + X̂2i   + . . . X̂NI i   .
X̂i = X̂1i 
.
.
.
 .. 
 .. 
 .. 
 
 
 
 
 
 
 
 
 
0
0
1

(4.9)

48
In the PCA basis, the image is represented as a linear combination of a new orthonormal set of
NI vectors/images:(P ) (P )X̂i = Y1i u1(P ) (P )+ Y2i u2

(P ) (P )+ . . . + YNI i uNI .

(4.10)

We assume here that NI < p̂, a reasonable assumption for high-resolution digital images. These(P )PC basis images can be written as a p̂×NI orthonormal matrix P (P ) = (u1

(P )

u2

···

(P )

up̂ ).

The two representations of the data set are related by
X̂ = P (P ) Y (P ) .

(4.11)

Here and in Eq. (4.10), the matrix Y (P ) is an NI × NI matrix in which column i is the set of
coefficients (or weights) of image i in the PC basis.
The requirement that the set of coefficients of any PC be uncorrelated with the set of
coefficients of any other PC is met if and only if the sample covariance matrix of the PC basis,
D=


T
1
Y (P ) Y (P ) ,NI − 1(4.12)

is diagonal. By using Eqs. (4.6) and (4.11), along with the orthonormality of P , D may be
rewritten as
1  (P ) T
P
X̂ X̂ T P (P )NI − 1T= P (P ) SP (P ) .

D=(4.13)The sample covariance matrices are related byT
S = P (P ) D P (P ) .

(4.14)

If D is diagonal, then the PCs are the unit eigenvectors of S. The eigenvalue of a PC is thevariance of its coefficient. It is often referred to as the “strength” of the PC.

4.4.2

Reduction of dimensionality with PCA
It only remains to show explicitly how to reconstruct an image using the PC basis.

Assume the PCs have been sorted in order of decreasing strength. Then image i is written in

49
the new basis as(P ) (P )Xi = M + Y1i u1(P ) (P )+ Y2i u2

(P ) (P )+ . . . + YNI i uNI .

(4.15)In the analysis of a set of similar images, most of the variance will be captured by the first qPCs, where q is the smallest eigenvalue of S above some cutoff value. As a result, the mostimportant features of image i can still be reconstructed if the image is represented only as(P ) (P )Xi ≈ M + Y1i u1(P ) (P )+ Y2i u2(P ))+ . . . + Yqi u(Pq .

(4.16)The dimensionality of each image is thereby reduced from p̂ pixels to q PC weights. If each ofthese q strongest PCs can be positively identified with the variation of a specific experimental
parameter, then its weight in an image is proportional to the value of the associated parameterin that experimental trial.
The choice of how many PCs to retain is often a difficult one. A detailed analysis of rulesthat are often applied to make that choice is found in [38]. The method that we have found to
be most useful is the scree graph, a plot of σj2 (the variance of PC j, when sorted in decreasingorder) vs. j. We show the scree plot for a particular data set in Fig. 4.6. This data set was thesecond of two sets of 100 trials obtained on 14 January 2008, so we will refer to it for the rest of
the chapter as 0114b. The scree graph is used to identify the PC q at which the plot’s “elbow”
occurs. This is defined as the first PC beyond which the plot defines a straight line. If there ismore than one straight line defined by the smaller PCs, then the elbow is taken to be the first
PC on the first of those lines. For the 0114b set, the elbow is located at the fifth PC.
The scree plot method is somewhat subjective, but in our experience it produces reliable
results. Figure 4.7 shows the six strongest PCs for the 0114b data set. We see visually thatthe first five images primarily encode changes in the recorded momentum distribution. On theother hand, a dominant feature of the sixth image is a set of diagonal lines caused by diffractionin the probe laser beam. The decision to discard this PC made using the scree graph therefore
seems quite sensible.

50

Figure 4.6: Scree plot for 0114b data set. Elbow occurs at PC 5.

Figure 4.7: (Color). Strongest PCs of 0114b data set, organized in order of decreasing variance.

51
4.4.3

Independent component analysis
In both this data set and the set shown in Fig. 4.4, the signal Rc appears to be encoded by

two PCs. ICA is needed to isolate the signal for sets of this type. The statistically independent
representation of X̂ found by ICA [39, 40] is given byX̂ = P (I) Y (I) ,(I)

where P (I) = (u1

(I)

u2

...

(4.17)(I)

uNI ) is the matrix of ICs and Y (I) is the new matrix of coeffi-

cients. The challenge is to find P (I) and Y (I) such that Eq. (4.3) is satisfied.
ICA is more effective than PCA only if the underlying experimental parameters are notnormally distributed, because ICA uses tests on the higher order statistical moments of thecoefficients to determine independence. Since these are all zero for a normal distribution, PCA
has already produced a statistically independent set; ICA can do no better.
Fortunately, our experimental signal is not expected to be normally distributed when
∆φ is a random variable with a uniform distribution. The noticeably not normal probability
density of A cos ∆φ [see Eq. (4.1)] for a uniformly distributed ∆φ with zero mean and range
much greater than π is shown in Fig. 4.8(a). ICA should therefore (at least for some data sets)find a better representation of the signal than PCA. The results of the analysis of the 0114b
data back up this statement, as the shape of the histogram of the set of coefficients for theinterference signal IC [see Fig. 4.8(d)] resembles that of the expected probability density. (Thiscomparison is purely qualitative). On the other hand, the histograms of PC coefficients [Fig.
4.8(b) and (c)] do not resemble the shape of Fig. 4.8(a) at all. The deviations of Fig. 4.8(d) fromthe ideal can be accounted for by the finite size of the data set and the addition of normally
distributed detection noise. See Sec. 4.5 for more details, including a method for extracting theprobability density’s amplitude A from the data. Note that it is not a hard rule that ICA’s basis
image for the signal is always better than PCA’s; we have seen data sets for which the two arenearly identical.
The relation of higher order moments to the independence of variables is given by the

52

Figure 4.8: (Color). (a) The non-Gaussian ideal probability density of A cos ∆φ for uniform
distribution of ∆φ. (b) and (c) From 0114b data, the Gaussian-like histograms of the coefficientsof the PCs imperfectly representing the interference signal. (d) For the same data set, the clearly
non-Gaussian histogram of the coefficient of the signal IC.

53
central limit theorem. This states that the sum of two independent variables has a probability
density that resembles a Gaussian at least as well as either of the densities of the independent
variables themselves. A corollary is that the density of an independent variable is “less Gaussian”
than the density of the sum of the variable and any other variable. Therefore, an ICA algorithm
maximizes the “non-Gaussianity” of the density of the calculated IC’s coefficients. Many possible
measurements of non-Gaussianity exist; one of the simplest is the normalized kurtosis excess,defined as
γ2 =

µ4
− 3.
σ4

(4.18)In this expression, σ 4 is the square of the variance of a variable, and µ4 is its fourth moment.
Kurtosis excess measures the “peakedness” of the variable’s density; a variable for which γ2 > 0
has a density more highly peaked than a Gaussian of the same variance and vice versa. A
number of more elaborate measures of non-Gaussianity exist, but all depend in some way on
calculating the higher order moments of the distributions of the coefficients.
ICA begins with a calculation of the PC basis of a set of images. It is next assumed
that each PC is actually a linear combination of the underlying ICs (which are therefore notorthogonal). The algorithm then forms an initial guess at the ICs by filling the columns of P (I)with NI random, linearly independent linear combinations of the PCs. The coefficient matrix ofthe guess is calculated by inverting P (I) :−1
Y (I) = P (I)X̂.

(4.19)A test of non-Gaussianity (perhaps kurtosis) is then applied to each row of Y (I) . An optimization
routine is used to adjust the guess of P (I) , and the process is repeated until the coefficients of
the optimized ICs are sufficiently non-Gaussian.

4.5

Parameter extraction from PCA/ICA results
In the previous section, it was mentioned that the constant of proportionality A between

the signal-related PC or IC and the real experimental result Rc could be determined by fitting

54
a data set’s signal coefficients to its expected statistical distribution. That fitting procedure is
the subject of this section.

4.5.1

Model for probability density of PC/IC signal
We begin by assuming that the phase shift ∆φ has the probability density [Fig. 4.9(a)]:f∆φ (∆φ) =1/(2π)if |∆φ| ≤ π




0

if |∆φ| > π.

(4.20)

This is a reasonable assumption for experiments in which ∆φ is uniformly varied over a range
much broader than [−π, π], as our experimental output measures ∆φ modulo 2π. Recall Eq.
(4.1), relating the PCA/ICA coefficient y to ∆φ:y = A cos ∆φ.

(4.21)The probability density of y is [41]fy (y) =X f∆φ (∆φi )iy 0 (∆φi )X
1f∆φ (∆φi ),=p22A −y i(4.22)

where each ∆φi is a root of Eq. (4.21). For any given value of y, there are two values of ∆φ inthe range [−π, π] that satisfy Eq. (4.21). We retain those two terms, obtainingfy (y) =p1/(π A2 − y 2 )if |y| < A




0

if |y| ≥ A(4.23)[see Fig. 4.9(b)]. Recall Fig. 4.8, in which we saw that the histogram of a typical data set has
tails outside of the peaks predicted by Eq. (4.23). We may include these tails in the model byassuming that the PC/IC coefficient measures normally-distributed detection noise along withthe signal. We will call this convoluted measurementz = y + x,

(4.24)

55
where x has the probability density [Fig. 4.9(c)]fx (x) =2
2
1√ e−x /2σ .
σ 2π(4.25)The probability density of a sum of two random variables is [Fig. 4.9(d)]:Z ∞fx (z − y)fy (y) dy

fz (z) =−∞Z A−(z − y)2
p
dy
exp
2
2
2σ 2
−A π A − y


Z 0
−1
−(z − A cos ∆φ)2
sin ∆φ
p
√
=
exp
d∆φ
2σ 2
σπ 2π −π 1 − cos2 ∆φ


Z 0
1
−(z − A cos ∆φ)2
√
d∆φ.
=
exp
2σ 2
σπ 2π −π

1
= √
σ 2π

1



(4.26)

Figure 4.9: Plots of various probability densities. (a) f∆φ (∆φ), (b) fy (y), (c) fx (x), (d) fz (z).

4.5.2

Maximum likelihood estimation of parameters
Our goal now is to extract the values of the proportionality parameter A and noise

parameter σ from a given data set. A common method for fitting a probability density like Eq.
(4.26) to a set of observations is maximum likelihood estimation (MLE) [42]. Consider the joint
probability density of a set of observations z and our unknown parameters:f (z; A, σ).

(4.27)

56
If we insert an observed set of sample values for z, then Eq. (4.27) becomes the likelihood
function of the two unknown parameters. The likelihood can be thought of as the complementof the joint probability density, in which the observed data and the parameters have exchanged
roles. The maximum likelihood estimators of the parameters, Â and σ̂, are those choices thatmaximize the likelihood for a given set of observations z = (z1 , z2 , . . . , zNI ). In our case, thelikelihood isfL (z; A, σ) =

1
√
σπ 2π

N Y
NI Z 0


exp

−π

I i=1

−(zi − A cos ∆φ)2
2σ 2




d∆φ .

(4.28)For ease of calculation, MLE typically maximizes the natural logarithm of the likelihood:
NI
X
√
ln
ln fL = −NI ln(σπ 2π) +
i=1

Z 0


exp

−π

−(zi − A cos ∆φ)2
2σ 2




d∆φ .

(4.29)The result for the 0114b data set is

A = 1.83
σ = .40.

(4.30)

This fit is shown superimposed on a normalized histogram of the data in Fig. 4.10. The fit is not
perfect, as the data set is of finite size. We can obtain confidence regions for the fit by making acontour plot of the likelihood (normalized to 1) around the fit value (Fig. 4.11). The plot showsthat the likelihood is approximately a two-dimensional Gaussian. We wish to find the contours
that enclose 50%, 68%, and 95% of the area of the Gaussian, which are equivalent to the 50%,
68%, and 95% confidence regions for the parameters. The circular coverage function [43]

pc

r 
d

s

1
=
2πs2



Z Z
exp
x2 +y 2 ≤rd2−(x2 + y 2 )
2s2


dx dy

(4.31)

can be used to find the radius of the circular Gaussian that encloses the desired area fraction.
This radius is then inserted back into the Gaussian to find the value of the function at thatradius, and therefore the desired contour (this contour is the same for our elliptical case). These
contours we seek are located at 0.52, 0.3, and 0.05 respectively, and are marked on the figure.

57
The individual parameter 68% confidence regions [44] are

A = 1.83 (1.68, 1.97)
σ = 0.40 (0.31, 0.54),(4.32)

shown by dashed lines in Fig. 4.11. More accurate confidence region estimates could be made bynumerically integrating the normalized likelihood, but this would be computationally intensive.
The estimate method given here can be carried out rapidly. Note that this method of parameter
extraction is generally applicable to all data sets; we have only used the 0114b set as an example.

Figure 4.10: (Color). Result of MLE of probability density function (PDF) superimposed on
normalized histogram of 0114b data set. We see that the MLE of the scaling parameter A
qualitatively fits the data.

58

Figure 4.11: (Color). Plot of normalized likelihood near fit values for 0114b data set. The
0.3 contour is approximately the 68% confidence region for the two parameters. Individual
parameter confidence regions are given by the projection of the ellipse onto the axes, shown here
as blue dashed lines.

Chapter 5

Results and lessons learned from Michelson interferometry

In this chapter, we use the output of PCA and ICA to extract the results of the freereturn Michelson experiment. We determine that the interferometer phase was randomized by a
combination of apparatus vibrations and quantum phase diffusion. Experimental modifications
to minimize these two factors would have been difficult to carry out in the Michelson apparatus,
leading us towards the design and construction of a new apparatus.

5.1

Michelson calibration and results
Before beginning a long run of the experiment, we used PCA to calibrate the apparatus.

The timing of the second optical pulse is a critical experimental parameter. For the BECs to be
maximally overlapped at the moment of the second pulse, the separation in time between thetwo pulses must be a multiple (or one-half of a multiple) of the trap period. However, the period
can slowly drift by hundreds of microseconds. The problem is compounded by the 90 s cycle timeof the apparatus; a data set of 100 attempts represents more than 2 h of accumulated drift. To
recalibrate before taking a set, we vary this parameter around the expected value and perform
ten attempts of the interferometry cycle at each value. The attempts carried out at a given
pulse time are done with widely varying values of ∆φ (applied with a magnetic field gradient)to ensure high interference contrast from shot to shot. We then carry out PCA separately oneach small set of images taken at one particular pulse time. At the pulse time that results in
the highest interference contrast, the variance of the coefficient of the basis image corresponding

60
to the phase shift signal will be maximized. Figure 5.1 shows the result of a typical calibration
run; a value of zero variance has been assigned for recombination times at which PCA did not
produce a basis image that could be positively identified with the interference signal. Since PCA
is computationally fast, a plot like Fig. 5.1 can be generated as quickly as the data are taken.

Figure 5.1: Plot of the variance of the coefficient of the basis image representing the interference
signal vs time between optical pulses; a value of zero variance is used to indicate times at which
the interference signal cannot be identified with a single basis image.

A data run containing more than 100 interferometry experiments began immediately after
calibration was complete. Figure 4.5 presents a typical result of our interferometry work, taken
after a full trap period (240 ms, in this experiment) of propagation time. The visibility of theinterference pattern
V =Rcmax − Rcmin
Rcmax + Rcmin(5.1)

as obtained with Gaussian fits was fairly good, at about 50%. Of more concern for the future
development of instruments was that the phase of the interfereometer was not controllable. In
the remainder of this section, we will outline two factors that work together to randomize thephase, and must be mitigated in future experiments. This outline is based on the full analysisof the experiment found in a previous dissertation [25].

61
5.1.1

Vibrations of the apparatus
The first randomizing effect that we will consider is the coupling of apparatus vibrations

to the differential phase. As we will see, this noise coupling mechanism can be dealt withby better isolating the experiment from environmental mechanical noise and an adjustment ofexperimental parameters.
Recall from Eq. (2.34) that the BECs that are propagating in the trap acquire a phaseshift from their transfer into non-zero momentum states by the initial beamsplitter pulse. Thatphase term is exp[±2k0 z(0)], where we will take the positive sign to be for the condensate
propagating along the +z direction. Here, z(0) is the distance between the retro-reflecting mirror
and the initial location of the BEC. The second pulse imprints a phase opposite in magnitudein its attempt to stop the atoms, multiplying the wavefunctions by exp[∓2k0 z(T )] [z(T ) is the
distance between the mirror and the two overlapped BECs at the moment of reunion]. The end
result is a differential phase shift between the two BECs equal to∆φlaser = 4k0 (z(0) − z(T )).

(5.2)

In the ideal experiment z(0) = z(T ), so this laser-induced differential phase shift vanishes.
In real experiments, z(T ) is not necessarily equal to z(0). A vibration of the apparatus
directed along the guide axis modifies the equation of motion of the moving BECs to include adriving acceleration a(t):z̈ + ωz z = a(t).

(5.3)

The general solution (obtained by variation of parameters [45]) isz(t) = A1 cos ωz t+A2 sin ωz t+

sin ωz tωz

Z ta(t0 ) cos(ωz t0 ) dt0 −0cos ωz tωz

Z ta(t0 ) sin(ωz t0 ) dt0 . (5.4)

0The shift in phase at the end of one axial period (t = T = 2π/ωz ) due to the acceleration istherefore
4k0∆φlaser = −
ωz

Z T
0

a(t0 ) sin(ωz t0 ) dt0 .

(5.5)

62
Vibration of the apparatus can be generally expressed asa(t) = a(ωv ) sin(ωv t + φv )(5.6)

for any given frequency ωv , amplitude a(ωv ), and initial phase φv . The phase shift for thatvibration is∆φlaser =

 

2k0
ωv
T 2 a(ωv )sinc π−1π(1 + ωv /ωz )
ωz

 

 

ωv
ωv
× sin φv sin π− 1 − cos φv cos π
−1
.
ωz
ωz

(5.7)

The sum in quadrature of both in- and out-of-phase vibrations is shown in Fig. 5.2. We see that
low-frequency vibrations near the axial trap frequency can significantly shift the output of theinterferometer, with the greatest shift∆φlaser (max) = −k0 T 2 a(ωv )
π

(5.8)

occurring for in-phase vibrations at ωz . For our typical axial trap of ωz = 2π × 4.2 Hz,
2∆φlaser (max) = π for a(ωv ) = 2.2 × 10−5 m/s .

Figure 5.2: Quadrature sum of differential phase shifts caused by in-phase and out-of-phasevibrations vs vibration frequency.

To determine if the vibrations were having a noticeable effect on the experiment, we
carefully measured the spectrum of the vibrations of our optical table along the guide axis using

63
a seismometer. Two results are shown in Fig. 5.3: in (a) the table is resting on its supports,in (b) it is floating on its pneumatic isolators. The plots indicate that vibration-induced phaserandomization should be significant at ωz = 2π × 4.2 Hz, though it should be less of a problem
when the table is floating.
The experimental results support that hypothesis. Data sets taken with the table restingon its supports have the characteristic double-peaked probability density of large phase variationeven when the phase is not intentionally varied. In contrast, consider Fig. 5.4, which shows thehistogram of the signal IC from a data set taken while the table was floating. Instead of thefamiliar shape of Sec. 4.5, we see a density that appears to be missing one of its wings. This
plot is a qualitative indication that the spread in phase has been reduced.
While the mitigation of vibration effects by floating the table is encouraging, we still
do not have sufficient control over the differential phase for stable interferometry. An effort
was made to precisely measure the vibrations of the apparatus during the experiment using thealready mentioned seismometer, and thereby calculate the expected vibration phase shift. There
was no correlation between this calculated shift and the output of the interferometer. Amongthe proposed technical problems with the procedure is that the structure holding the microtrap
module is not perfectly rigid (see Fig. 5.5), so the vibrations measured by the seismometer could
not have been perfectly correlated with the actual mirror vibrations.

5.1.2

Quantum phase diffusion
Technical problems notwithstanding, a more fundamental effect is expected to cause

phase randomization. Quantum phase diffusion offers a compelling explanation for the failure
to correlate the measured vibrations with the interferometer output. The source of this phasediffusion is the energy of the interactions of the atoms in each BEC. This energy is parameterized
by the chemical potential µ, a function of the number of atoms in the BEC. The wavefunctionsof the two condensates (identified here as “j = +” and “j = −”) of our experiment each
include a phase evolution term exp[iµj (Nj )t/h̄] due to this energy. The fundamental quantum

64

Figure 5.3: Spectra of optical table vibrations along guide axis. (a) Table is resting on legs, (b)table is floating on isolators [25].

65

Figure 5.4: (Color). Normalized histogram of signal IC from data set taken with table floating
and without any intentional phase variation, superimposed with a kernel smoothing estimate of
the probability density function (PDF). See Sec. 4.5 for details on extracting the proportionality
constant A that relates the signal IC coefficient to cos ∆φ.

Figure 5.5: (Color). Microtrap module from Michelson apparatus. The atom chip is suspended
from a cantilevered arm projecting from the wall of the vacuum chamber. This cantilever is notexpected to be perfectly rigid.

66
uncertainty of the number of atoms in the two condensates causes the relative phase to develop
its own uncertainty. The initially well-known differential phase is said to “diffuse” over time[30]. We follow the treatment of [46] to estimate the diffusion rate.
We first calculate the uncertainty in the splitting process. Consider a trapped BEC
containing NT atoms that is adiabatically split into two physically separated wells. Our caseof momentum-state splitting is analogous, as long as the duration of the splitting process itself(during which the two BECs interact with each other) is much shorter than the total durationof the experiment. Immediately after splitting, the wavefunction of each individual atom i is a√coherent superposition of two opposite momentum states: [ψ+ (ri ) + ψ− (ri )]/ 2. We use theHartree approach to find the total many-body wavefunction
1NTY[ψ+ (ri ) + ψ− (ri )]2NT /2 i=1
v 
u
u
N
T u
NT 
1 X
u
χN− ,N −N− (ri , . . . , rN ),u= N /2TT
2 T k=0 tN−Ψ(ri , . . . , rNT , t = 0) =(5.9)

where we have used the binomial coefficient in the summation. The unit-normalized wavefunction for the state in which N− atoms are in the − BEC and NT − N− atoms are in the + BEC
is χN− ,NT −N− . The corresponding state vector is
v 
u
u
N
uNT TX1u u|Ψ(t = 0)i = N /2
|N− , NT − N− i .
2 T N =0 t −N−

(5.10)We see that for large, even NT that the most likely measured result of N− is N− = NT /2, withapproximate uncertainty√
NT . For the rest of these calculations, we will continue to assume

NT is even; the end result for odd NT is the same, though the derivation is slightly different.
To calculate the evolution of the state, we will need to estimate the energy of the system.
Near N− = NT /2, the most probable value of N− , the energy of a given number state isNT
E− (N− ) + E+ (NT − N− ) ≈ 2E N− =2d2 E(N− = NT /2)
+
2
dN−


≈ h̄ ωNT /2 + ξ(N− − NT /2)2 .


2
NT
N− −
2
(5.11)

67
The state at time t is therefore
v 
u
u
N
T
NT 

exp(−iωNT /2 t) X uu exp −iξt(N− − NT /2)2 |N− , NT − N− i .
u|Ψ(t)i =


N
/2
t
2 T
N− =0
N−

(5.12)We see that the phase for each value of N− evolves at a different rate.
We find the probability of measuring a specific phase difference ∆φ by computing
P (∆φp̌ ) = | h∆φp̌ |Ψ(t)i |2 ,(5.13)

where |∆φp̌ i is an eigenstate of the phase-difference operator. These states are [47]:NTX
1|∆φp̌ i = √exp (iN− ∆φp̌ ) |N− , NT − N− i ,NT + 1 N =0

(5.14)−

for eigenvalues ∆φp̌ = 2π p̌/(NT + 1), p̌ = −NT /2, −NT /2 + 1, . . . , NT /2. The inner product of
the states is
v 
u
u
!#
"
2

N
T
NT 
exp(−iωNT /2 t) X u
NT
u


u
h∆φp̌ |Ψ(t)i = N /2 √+ N− ∆φp̌
.
exp −i ξt N− −2
2 T
NT + 1 N =0 t −N−
(5.15)In the limit NT  1, the binomial coefficient can be replaced with a Gaussian:
 


NT 
2NT +1−2(N− − NT /2)2 ≈ √
exp
.
 
NT
2πNT
N−

(5.16)

It is also appropriate to approximate the sum by an integral, obtainingh∆φp̌ |Ψ(t)i =1/2
exp(−iωNT /2 t) 2NT +1√√
2NT /2 NT + 1
2πNT
!
#
"

2
Z NT
NT
(N− − NT /2)2×
dN− exp −i ξt N− −+ N− ∆φp̌ −. (5.17)2
NT
0

As NT → ∞,√ 
1/2
exp(−iωNT /2 t) π 2NT +1√√h∆φp̌ |Ψ(t)i =2NT /2 NT + 1
2πNT
"
#
∆φ2p̌
1
−iNT ∆φp̌
−
.
×p
exp
2
4(iξt + 1/NT )iξt + 1/NT

(5.18)

68
Multiplying Eq. (5.18) by its complex conjugate, we obtain
!
√
∆φ2p̌
1
2π
P (∆φp̌ ) =exp − 2,NT + 1 σ∆φ
2σ∆φ(5.19)

with standard deviation
r
σ∆φ =

1
+ NT ξ 2 t2 .
NT

(5.20)The probability density for ∆φ is obtained by the transformationf (∆φ) d∆φ = P (∆φp̌ ),(5.21)

where d∆φ = 2π/(NT + 1). The result is1∆φ2
f (∆φ) = √exp − 2
2σ∆φ
2πσ∆φ

!
,

(5.22)

where we now consider ∆φ to be a continuous variable. Note that the equivalent equation in[46] seems to have been printed with the incorrect normalization. Regardless, the key result
is that the standard deviation of the differential phase increases approximately linearly at rate
R=

√
N ξ.
We can easily calculate R for a BEC in the Thomas-Fermi limit. The total energy ofsuch a condensate is [35]
5
E=
14



15N− asā2/5h̄ω̄N− ,(5.23)

where N− is the number of atoms, as is the scattering length, ω̄ is the geometric mean trapping
frequency, and ā is the corresponding oscillator length. Using the definition of ξ in Eq. (5.11),
we find

R=

72
125

∝ ωρ4/5 .

1/5 

as 2/5 ω̄
1/10
ā
N
T

(5.24)For our typical experimental parameters (NT = 5000 87 Rb atoms in |1, −1i ground state, ωρ =2π × 80 Hz, ωz = 2π × 4.2 Hz), the standard deviation of the phase is 1.6 rad. Note that R isa very weak function of the total number of atoms, but that it scales relatively strongly withrespect to the radial trap frequency.

69
This model is fairly simplistic. It assumes that the splitting is adiabatic and that the BEC
is in the Thomas-Fermi limit. It also does not describe the effects (if any) of the recombination
process on the phase diffusion rate. Finally, it contains no calculation of the possible effects ofphase diffusion on the visibility of the interference signal. This model’s results should therefore
not be assumed to apply to our experiment without further support. Numerical modelingprovides a piece of corroborating evidence. A simulation of the Gross-Pitaevskii equation using
our experimental parameters found a number for σ∆φ very similar to the analytic result (see[25] for details). The single-peak experimental result of Fig. 5.4 provides some further support.
Recall that in this set, no phase shift was intentionally applied and vibrations were significantly
reduced by floating the optical table on air. Our contention is that the remaining randomizationof the phase is explained by quantum diffusion. To check that, we simulate a 100 point data
set in which ∆φ is a normally distributed random variable with zero mean and σ∆φ = 1.6 rad.
Figure 5.6 shows the probability density of cos ∆φ for one simulated set. The shape resembles
that of Fig. 5.4. Note that the assumption that the mean value of ∆φ is 0 is not rigorously
supported; the value is not actually known a priori due to the phase randomization. In theend, we can make no strong conclusions although we see some indications that phase diffusion
is present in the experiment. Future interferometry work will have to further explore this issue.

5.2

Lessons for future experiments
For any future experiments to be successful, we will have to find some way to address theeffect of random vibrations. It should be possible to subtract the phase shift due to vibrationfrom our signal by independently measuring the vibration of the apparatus. Since our previous
efforts towards this goal were not successful (see Sec. 5.1.1 and [25]), it is perhaps safer to focus
our next efforts on better isolating the experiment from mechanical noise. Operating the axial
trap at a higher frequency might help to reduce the amplitude of vibrations transmitted tothe mirror, as well as reducing the expected phase shift from the vibrations. The higher trap
frequency would give less time for quantum phase diffusion to develop as well. However, as

70Figure 5.6: (Color). Probability density of simulated data set subject to quantum phase diffusion. We show both a histogram and a kernel smoothing approximation of the PDF (to guidethe eye only).

71
described in Sec. 2.1.2, an increase in the axial confinement leads to an increased tilt of the
waveguide with respect to the permanently positioned standing wave mirror. Further increasingthat relative tilt would have caused the propagating BECs to oscillate along the radial axis of
the guide, reducing the overlap of the two at the application of the second set of beamsplitter
pulses. Correcting this problem would have required a redesign of the microtrap assembly or ofits surrounding vacuum chamber.
Further investigation of the possible effects of phase diffusion must also be on the agenda.
One simple experiment would be to attempt to adjust the phase diffusion rate R. We saw in Eq.
4/5(5.24) that R ∝ ωρ , so a small decrease in the radial confinement would reduce phase diffusion
effects. However, as we will see in Sec. 6.5.5, the area enclosed by our proposed gyroscope
scales quite strongly with ωρ . Future experiments will therefore require a careful selection of an
optimal value for ωρ .
Another possible way to mitigate the effects of phase diffusion would be obtain more datafrom the experiment. Assume we have increased the axial frequency to ωz = 2π × 10 Hz but
maintained the same radial frequency. The estimated standard deviation of the phase (using thesimple model of the previous section) for this axial frequency is 1 rad; better than before, but
still quite large. In Fig. 5.7, we show the results of a simulation of this experiment, in which ∆φis intentionally varied uniformly in the range [−π, π]. The measured output of the experiment iscos(∆φ + φD ), where φD is a normally distributed phase diffusion contribution with zero mean.
On the left side of the figure, we show results for σ∆φ = 1.6 rad and data sets containing 100,
1000, and 10000 experiments. On the right side, we show results for σ∆φ = 1.6 rad. Figure
5.7(a), which shows a simulation of the Michelson experimental conditions, shows quite clearly
that our data sets that contain 100 points and σ∆φ = 1.6 rad will never be useful for extracting
the phase of the experiment. With more data, the sine wave becomes visible against the noise.
A fit to the function a1 cos(ω1 ∆φ) becomes plausible. The fit parameter ω1 should be 1, and theamplitude a1 is a fair measure of the visibility of the signal over the noise. The results of thesefits are shown in Table 5.1. We see that with σ∆φ = 1 rad, the fit becomes reliable at 1000 data

72
points. The fit is also twice as visible on the background as it is for the σ∆φ = 1.6 rad case.
σ∆φ (rad)
1.6
1.6
1.6
1.0
1.0
1.0

Number of trials
100
1000
10000
100
1000
10000

a1
N/A
0.3 [0.1,0.6]
0.30 [0.27,0.34]
0.7 [0.4,0.9]
0.60 [0.54,0.65]
0.61 [0.59,0.63]

ω1N/A
0.8 [0.3,1.3]
0.9 [0.8,1.0]
1.0 [0.6,1.4]
1.04 [0.95,1.13]
0.99 [0.96,1.03]

Table 5.1: Fit parameters for simulated data in Fig. 5.7. 95% confidence regions for fit parameters are shown in brackets.

We conclude that we need three apparatus improvements in order to be ready for gyroscope experiments: (1) We need a way to adjust the angle of the standing wave with respect tothe chip wires so that higher axial trap frequencies can be used. (2) We need to decrease thetime required to make a BEC. A 1000-run data set would take more than a day to obtain withthe existing 90 s cycle time. (3) We need to develop a way to rotate the experiment in order tocarry out gyroscope tests. Improvement (1) could probably have been made by a modificationof the existing apparatus. Improvements (2) and (3) would require fundamentally changing thethe apparatus, which was designed to use familiar, but large and slow, “macrotrap” methods inorder to reduce the unknowns in operating the machine. It is for this reason that the Michelson
apparatus was retired after long years of service. Our recent efforts have focused on the designand construction of a new all-microtrap system that promises the small size and rapid BEC
production required to test a Sagnac gyroscope based on Michelson interferometry.

73

Figure 5.7: (Color). Results of simulated data sets for various numbers of trials and amount ofphase diffusion. The experimenter intentionally varies ∆φ uniformly in the range [−π, π]. The
plots show the measured quantity cos(∆φ + φD ), where φD is the phase noise due to diffusion.
(a), (b), and (c) have σ∆φ = 1.6 rad with 100, 1000, and 10000 trials respectively. (d) through(f) show the results for σ∆φ = 1 rad. Fits to the simulated data are shown as well. A similar
figure is shown in [25].

Chapter 6

Theory and design of the atom chip gyroscope

6.1

Parallel wire concept
The Michelson interferometry experiments discussed in the previous two chapters showedthat guided BEC interferometry is possible. Our next step towards the goal of building an
ultracold waveguide Sagnac gyroscope was to develop an area-enclosing guided interferometerthat would incorporate the lessons learned from the linear Michelson interferometer.
It has been demonstrated [48] that it is possible to use a linear waveguide to guide atoms
around an enclosed area. The key idea is to translate the guide transverse to its axis during theinterferometry sequence. See Fig. 6.1 for a schematic of the idea. The method pursued in [48]uses a magnetic waveguide generated by an arrangement of permalloy foils poled by currentcarrying coils (see Fig. 6.2). The guide’s position is translated by adjusting the current in thecentral coils. This method has been used to translate a waveguide by about 1 mm, which is onlya factor of 3 less than the diameter of a circular path that has a Sagnac phase shift of π rad atΩEarth for 87 Rb (see Section 1.1.3).
This coil-based approach to a moving waveguide is quite large, occupying a footprint of
200 × 100 × 25.4 mm. The waveguide potential is 7 mm above the top of the foils. The large size
allows the experimenters to translate the guide by simply adjusting the current running throughthe two central coils, but at the cost of portability.
As suggested in the conclusion of [48], an alternative approach would be to use “a mul-

75

Figure 6.1: (Color). Schematic of translating waveguide interferometer. A gas of atoms (shownin yellow) is initially stationary in a magnetic waveguide (cyan) aligned along the z axis. The
atomic potential includes a weak axial trap (period T ) along z. At t = 0, a standing wave
light pulse (E, shown in magenta) splits the cloud into two wave packets moving in opposite
directions along the waveguide axis. Simultaneously, the waveguide begins to move along the xaxis. The wave packets reach their classical turning points at t = T /4 and reverse their axial
motion. At t = T /2, as the packets overlap at the center of the guide, the transverse motion
has been stopped. It is reversed for the second half of the packets’ axial oscillation, eventuallybringing the packets together again in their initial position at t = T . A second pulse of E
recombines the packets. Based on a figure in [48].

76

Figure 6.2: (Color) Schematic of moving guide apparatus from [48]. Coils wrapped aroundpermalloy foils (shown in dark red) pole the foils in the indicated directions. The false color
plot above the wires shows the location of the waveguide, which is aligned perpendicular to theplane of the image. By adjusting the ratio between the currents running in the inner two coils,
the guide’s position may be moved horizontally by about 1 mm.

tiwire 1D conveyor belt on an atom chip” to translate the waveguide. To visualize this idea,
consider an atom chip patterned with an array of parallel wires [see Fig. 6.3(a)]. The wires arethen driven in such a way that a waveguide that is initially positioned over wire A moves acrossthe array. A very simple way to do this is to sequentially turn on and then off the current in
the wires [see Fig. 6.3(b)]. The total current density of this system resembles a soliton moving
down the wire array. Assuming a constant transverse bias field, the guide is effectively “handed
off” by this action from wire to wire, translating it across the chip.
It is also possible to drive the wires such that the total current density of the system
resembles a traveling wave moving across the array [see Fig. 6.3(c)]. When properly biased, theresulting potential would have multiple waveguides that would travel with the current wave.
The traveling wave idea should be simple to apply in a practical system. While the “hand-off”
concept requires an independently driven current supply for each wire, a traveling wave would
be created by driving the wires periodically in space and time. For spatial period L and wiredensity N/L, this approach requires only N supplies. For experimental simplicity, we plan to
use the traveling wave approach.

6.2

Magnetic field of parallel wires carrying sinusoidal current
densityIn this section, we develop the mathematics to describe the magnetic field generated byan array of parallel wires driven by a traveling wave of current. These calculations are the result

77

Figure 6.3: Multiple parallel wire waveguide operation. (a) Schematic of wire layout. Here, 4parallel wires, labeled A through D, have been patterned onto a chip. A bias field transverseto the wire direction is present, but not shown. (b) “Hand-off” operation. Initially, only wire A
has current running through it, so the guide is located directly above it. The wires are smoothly
turned on and off in sequence, causing the waveguide to be handed off from one wire to the next.
(c) Running wave operation. The wires are driven sinusoidally in time, each with a slight phaseshift from the previous wire. Atoms initially loaded into the magnetic field minimum above wire
A move down the array as the minimum moves.

78
of a collaboration with Prof. Alex Zozulya of Worcester (MA) Polytechnic Institute.

6.2.1

Field of sinusoidal current density
We begin with a stationary wave of current. Consider an infinite sheet of current in thexz plane, described by current density
J = K(x)δD (y)ẑ.

(6.1)

Here K(x) is the magnitude of the surface current density and δD (y) is the Dirac delta function.
The field of this sheet is given by the Biot-Savart law:µ0
B(r) =
4π

Z

J(r0 ) ×µ0
2πZK(x0 )=r − r0d3 x0|r − r0 |3−yx̂ + (x − x0 )ŷdx0 .
(x − x0 )2 + y 2

(6.2)
(c)

If the current density is sinusoidal with spatial period L and amplitude K1 , then(c)K(x) = K1 cos

2πx
L


.

(6.3)

The field for this current density is
(c) Z

µ0 K1
B(r) =
2π


cos

2πx0
L



−yx̂ + (x − x0 )ŷdx0 .
(x − x0 )2 + y 2(6.4)

Fourier integrals of this type may be evaluated by the residue method [45], using the formulaZ ∞g(x) cos wx dx = −2π

X

Im Res[g(z)eiwz ].

(6.5)

−∞

Here the sum is over the residues of g(z)eiwz in the upper half of the complex number plane.
The resulting field is(c)µ0 K1 −2πy/L
2πx
2πxB(r) =e− cosx̂ + sin
ŷ ,
2
L
L

(6.6)

where y ≥ 0.

6.2.2

Field of periodic current density
Building on the previous section, we next consider the field of a sheet carrying a moregeneral periodic current density. For spatial period L, the surface current density may be written

79
as a Fourier series:
K(x) = K0 +

∞ 
X

2πnx
2πnx
Kn(c) cos
+ Kn(s) sin
L

n=1

L


.

(6.7)

The Fourier coefficients are given by
1
K0 =
L

Z L

2
L

Z L

2
Kn(s) =Z L

Kn(c) =

L

K(x) dx0

K(x) cos

2πnx
dx
L

K(x) sin

2πnx
dx.
L

0

0

(6.8)

The field of the general periodic surface current density is found by inserting Eq. (6.7) into Eq.
(6.2) to obtain
µ0 K0
x̂
2


(c)
µ0 Kn −2πny/L
2πnx
2πnx
(c)
Bn (r) =e− cosx̂ + sin
ŷ
2
L
L


(s)
µ0 Kn −2πny/L
2πnx
2πnx
B(s)
(r)
=
e
−
sin
x̂
−
cos
ŷ
.
n
2
L
L
B0 (r) = −(6.9)

Here, we have again used the Fourier cosine integral formula in Eq. (6.5), along with the similarsine formulaZ ∞g(x) sin wx dx = 2π

X

Re Res[g(z)eiwz ].

(6.10)−∞

6.2.3

Field of sinusoidal current density generated by wires
In the previous two sections, we calculated the magnetic field generated by an infinite

sheet of periodic current density. We will now consider a more realistic situation, in which theperiodic current density flows through an array of parallel wires instead of a continuous sheet.
See Fig. 6.4 for an illustration of the variables that will be used in this calculation.
Consider an infinite array of parallel, infinitely long wires of width ∆x. The current ineach wire is a periodic function of its position, with period L. There are N wires in each spatial
period, where N > 1. The wires are evenly spaced, such that wire m begins at position
x̂m =

L
m
N

(6.11)

80

Figure 6.4: Illustration of variables associated with periodic current density calculation. Consider an array of wires carrying a sinusoidal current density of spatial period L. There are N
wires in each period, spaced at a distance L/N . ∆x is the wire width. x̂m is the position of theleft edge of wire m.

and carries current
Im = I cos

2πm
.
N

(6.12)The index m runs from 0 to N − 1. The total current density in each period is given by Eq.
(6.1) where
N −1

K(x) =I X
2πmcos[Θ(x − x̂m ) − Θ(x − x̂m − ∆x)].
∆x m=0N(6.13)

Θ(x) is the unit step function.
To use the mathematics developed in section 6.2.2, we must write Eq. (6.13) as a Fourier

81series. Using Eq. (6.8) to calculate the coefficients, we find
N −1

I X
2πm
K0 =
cos
L∆x m=0N

Z L[Θ(x − x̂m ) − Θ(x − x̂m − ∆x)] dx
0

N −1

=

I X
2πm
cos
L m=0
N

(6.14)

N −1

Kn(c) =

2I X
2πm
cos
L∆x m=0
N
N −1

=

2πm
2I X
cos
L∆x m=0
N

Z L
cos
0

2πnx
[Θ(x − x̂m ) − Θ(x − x̂m − ∆x)] dx
LZ x̂m +∆x
cos
x̂m

2πnx
dx
L

N
−1
X



I
2πm
2πn(x̂m + ∆x)
2πnx̂m
=
cos
sin
− sin
nπ∆x m=0
N
L
L




N
−1
X
I
2πm
2πnm
2πn∆x
2πnm
2πn∆x
cos
− 1 + cos
sin
sin
cos
=
nπ∆x m=0
N
N
L
N
L
Z
N −1
2πm x̂m +∆x
2πnx
2I X
cos
sin
dx
Kn(s) =
L∆x m=0
N x̂m
L


N
−1
X
2πm
I
2πnx̂m2πn(x̂m + ∆x)
cos
=
cos
− cos
nπ∆x m=0
N
L
L




N
−1
X
I
2πm
2πnm
2πn∆x
2πnm
2πn∆x
=
cos
cos
1 − cos+ sin
sin.
nπ∆x m=0
N
N
L
N
L(6.15)(6.16)

To sum the series, we use the orthogonality relations of the discrete trigonometric functions [49]to state
N
−1
X

cos

2πm
2πnm
sin
=0
N
N

cos

2πm
2πnm
N
cos
= (δn−1,âN + δn+1,âN )
N
N
2

m=0
N
−1
X
m=0
N
−1
X
m=0
N
−1
X
m=0

sin

sin

2πm
2πnm
N
sin
= (δn−1,âN − δn+1,âN )
N
N
2

N
−1
X
2πm
2πm
=
cos
= 0,
N
N
m=0

(6.17)

where â is any non-negative integer. Applying these relations, we obtain
K0 = 0Kn(c) =

N I sin 2αn(δn+1,âN + δn−1,âN )L 2αnKn(s) =

N I sin2 αn(δn+1,âN + δn−1,âN )L
αn(6.18)

82
where
nπ∆x
.
L

αn =

(6.19)

Plugging these Fourier coefficients into Eq. (6.9), we find the magnetic field of harmonic n is
sin αn −2πny/L
2πnx
Bn,x = −Bw
e
cos
− αn
αn
L


sin αn −2πny/L
2πnx
Bn,y = Bw
e
sin
− αn
αn
L

(6.20)

where
µ0 IN
.
2L

Bw =

(6.21)We see from Eq. (6.20) that the lowest nonzero harmonic has the largest amplitude, so
we may approximate the field as consisting of only harmonic n = 1. We now add a uniform bias
B0,x x̂ to form waveguides in the +y half space. To find the location of the waveguides, we use
Eq. (6.20) to calculate the set of locations x0 where the y component of the wire field vanishes:
0 = sin
x0 =

2πx0
− α1
L



∆x bL
+
,
2
2

(6.22)

where b is an integer. When x = x0 , the x component of the field vanishes at y = y0 , given by
0 = B0,x − Bwsin α1 −2πy0 /Lecos bπ.
α1(6.23)

If both B0,x > 0 and Bw > 0, then we must constrain b to be an even integer. With that
constraint in place, we find
L
ln
y0 =
2π



Bw sin α1
B0,x α1.

(6.24)We see that the periodically driven array of wires forms an infinite array of waveguides at heighty0 , spaced by the current’s spatial period L.
The final step in this calculation is to add an axial bias field B0,z and then calculate the waveguide’s trap frequency. We Taylor expand the field magnitude B1 =q
2 + B 2 around a field minimum line (x , y ), finding(B1,x + B0,x )2 + B1,y0 00,zB1 ≈ |B0,z | +2
2π 2 B0,xρ2 .
|B0,z |L2

(6.25)

83
The coordinate ρ is defined as ρ =p(x − x0 )2 + (y − y0 )2 . The harmonic frequency of the

waveguide for trapped atoms isrωρ = 2π

gF mF µB B0,x.
M B0,z L(6.26)We see in Eq. (6.18) that the next nonzero harmonic is n = N − 1. This harmonic
contribution to the magnetic field at the center of a waveguide is insignificant whensin α1 −2πy0 /L
sin αN −1 −2π(N −1)y0 /L
e

e
α1
αN −1
α1 sin αN −1 −2π(N −2)y0 /LeαN −1 sin α1N −2 
N −1
B0,x
α1
sin αN −1.
Bwsin α1
αN −1

1

(6.27)

If this parameter is kept small, then the contributions of all higher harmonics will also be
insignificant.

6.2.4

Field of traveling sinusoidal current density generated by wires
In this section we will calculate the field generated by an array of wires carrying a travelingwave of current density, in place of the stationary wave of the previous sections. The current inwire m is in this case given byIm = I cos2π(x̂m − vc t) ,L

(6.28)

where vc is the speed at which the waveguides will move across the array of wires. The wire
position x̂m is defined in Eq. (6.11). Using the methods of Sec. 6.2.3, we calculate the Fourier
coefficients of the current density to be

K0 = 0
2πvc t2πvc tδn+1,âN cos− αn + δn−1,âN cos
+ αn
L
L





IN sin αn2πvc t2πvc t(s)Kn =−δn+1,âN sin− αn + δn−1,âN sin+ αn .
L αn
L
L
IN sin αnKn(c) =
L αn(6.29)

84
We consider now the field of the n = 1 harmonic only, given by
sin α1 −2πy/L
2πecos(x − vc t) − α1α1Lsin α1 −2πy/L
2πB1,y = Bwe(x − vc t) − α1
sin
α1
L

B1,x = −Bw(6.30)

Adding bias field B0 = (B0,x , 0, B0,z ) to the wire fields results in an array of waveguide potentials, similar to the results of the previous section. In this case, however, the waveguides movein time along x. We calculate the waveguide positions (x0 , y0 ) to be∆x+ vc t + bL2L
Bw sin α1
y0 =
ln
2π
B0,x α1x0 =(6.31)

for any integer b. The trap frequency ωρ is given by Eq. (6.26).
In the traveling case, unlike the stationary case, the perturbations due to the higher
harmonics cause shifts in the trap potential that oscillate in time. Such oscillations could
excite collective modes in a BEC trapped in the moving waveguide, so we must calculate theiramplitude and frequency. We consider the total field from n = 1 and the next non-zero harmonic,
n = N − 1:
2πx̄Bx ≈ B0,x 1 − e−2πȳ/L cos
L−βN −2 e−2π(N −1)ȳ/L cos




2πx̄
(N − 1) + N ω̃t
L




2πx̄
(N − 1) + N ω̃t .
L


2πx̄
By ≈ B0,x e−2πȳ/L sin
L+βN −2 e−2π(N −1)ȳ/L sin(6.32)

These equations use a number of new symbols. The coordinates x and y have been re-centeredaccording tox̄ = x − x0ȳ = y − y0 .

(6.33)

85
The new parameters , β, and ω̃ are defined as
 = e−2πy0 /L =B0,x α1
Bw sin α12πvc
Lsin αN −1 α1.
β=αN −1 sin α1ω̃ =(6.34)In the following equations, we will assume that y0 and L have been chosen so that  is small.
The parameter β will also be small for larger values of N . We next Taylor expand the fieldaround x̄ = ȳ =  = 0, obtaining
B ≈ B0,z +2
2π 2 B0,x(x̄2 + ȳ 2 )L2 B0,z+2
2π B0,xβN −2 (x̄ sin N ω̃t − ȳ cos N ω̃t)
L B0,z+2
2π 2 B0,xβN −2 [(2N − 3)x̄2 cos N ω̃t − 2x̄ȳ sin N ω̃t + (2N − 1)ȳ 2 cos N ω̃t].
L2 B0,z(6.35)The potential for an atom in this field isU=h̄ωρ  2x̄ + ȳ 22ρ20+

L N −2β(x̄ sin N ω̃t − ȳ cos N ω̃t)π

+ βN −2 [(2N − 3)x̄2 cos N ω̃t + (2N − 1)ȳ 2 cos N ω̃t]
− 2βN −2 x̄ȳ sin N ω̃t ,where the trap frequency ωρ is given in Eq. (6.26) and ρ0 =(6.36)
ph̄/M ωρ is the harmonic oscillator

length scale. This is a two-dimensional harmonic oscillator potential with a small periodic driving
force (third term), and periodic perturbations to the trap shape (fourth and fifth terms).

6.3

Collective mode excitation due to parallel wires
In this section, we calculate the amplitudes of the collective modes that the perturbation

terms in Eq. (6.36) could excite in a trapped BEC. A BEC whose shape is oscillating during
the experiment will also have an oscillating mean-field interaction energy. The interaction phase

86
evolution term in the BEC wavefunction, exp[iµj (Nj , t)t/h̄] (see Sec. 5.1.2), will therefore have acomplicated time dependence. Consequently, the calculation of quantum phase diffusion effects
in the case of excited BECs is more complex. Modeling the outcome of our interferometry
experiments will be much simpler if we can keep the excitation amplitudes very small, so thatthe BECs are still well-described by a static wavefunction.

6.3.1

Slosh modes
The lowest energy collective modes for a BEC in a harmonic trap are the dipole or “slosh”

modes. In these modes, the condensate’s center of mass oscillates in the trap at the harmonicfrequency; the BEC itself does not change its shape. By keeping these modes small, we ensurethat the transverse motion of our trapped condensates is well-described by x0 (t). This mode isequivalent to the oscillation of a classical system in a harmonic potential. Just as with a classicaloscillator, the BEC can be forced into sloshing by the application of a driving force such as theone in the third term of Eq. (6.36). The equation of motion of the center of mass along the ȳ
axis is [50]
ȳ =

h̄ωρ
LβN −2(− cos ωρ t + cos N ω̃t),2ρ20 πM (ωρ2 − N 2 ω̃ 2 )(6.37)

for a condensate that is initially at rest when the non-resonant driving force is turned on. The
oscillations along the x̄ axis have the same amplitude. We want these amplitudes to be small
with respect to the size of the condensate, which for a typical condensate of atoms with repellinginteractions is slightly larger than the harmonic oscillator length ρ0 [35]. Thus, oscillations
obeying1ωρ2LβN −22πρ0 ωρ2 − N 2 ω̃ 2

(6.38)

will be acceptably small.
For the case of resonant drive, the equation of motion isȳ =h̄ωρ LβN −2
t sin ωρ t.
4ρ20 πM ωρ(6.39)

87
The condition for small oscillations is1

Lωρ Ť βN −2 ,2πρ0(6.40)

where Ť is the duration of the resonant drive.

6.3.2

Breathing and quadrupole modes
The fourth term in Eq. (6.36) causes periodic perturbations to the trap frequency of theguide potential, in the form
ωj2 = ωρ2 (1 + ηj cos Ω̃τ ).

(6.41)In this equation, the amplitude of the modulation is given byηx = βN −2 (2N − 3)
ηy = βN −2 (2N − 1),(6.42)

the dimensionless perturbation frequency is
N ω̃
,
ωρ

(6.43)

τ = ωρ t.

(6.44)Ω̃ =and the dimensionless time coordinate is

These perturbations cause collective oscillations of the shape of a trapped condensate (referredto here as “shape oscillations”). We will use a semi-classical treatment [51] to find the lowestenergy shape oscillations, as well as design constraints that make excitation of those modes
unlikely.
Consider a BEC in the waveguide potential of Eq. (6.36), to which has been added amuch weaker axial confinement of trap frequency ωz . In the Thomas-Fermi limit, the density of
a BEC in this time-dependent trap is
N |Ψ(r, t)|2TF =

µ−

P3

2
2
2
j=1 M ωj (0)rj /[2λj (t)]gλx (t)λy (t)λz (t),(6.45)

88
where µ is the chemical potential and g is the contact interaction strength. The dimensionlessscaling parameters λj obey a system of differential equations
λ̈j =ωj2 (0)− ωj2 (t)λj ,λj λx λy λz(6.46)

where j = x, y, or z. In the limit of small modulation amplitudes, where ηj  1, this system
may be simplified by defining λj = 1 + δλj and dropping terms that are second order or higherin δλj and ηj . We obtain a linear approximation to Eq. (6.46):d2
δλx (τ ) = −3δλx − δλy − δλz + ηx (1 − cos Ω̃τ )dτ 2d2
δλy (τ ) = −3δλy − δλx − δλz + ηy (1 − cos Ω̃τ )dτ 2d2
δλz (τ ) = γ 2 (−3δλz − δλx − δλy ),dτ 2(6.47)

where γ = ωz /ωρ  1. To leading order in γ, the eigenmodes of this system are a slowly
oscillating quadrupole mode at Ω̃slow =p5/2γ along (1, 1, −4) [Fig. 6.5(a)], a breathing mode atΩ̃breathe = 2 along (1, 1, 0) [Fig. 6.5(b)], and a quickly oscillating quadrupole mode at Ω̃fast =√2along (1, −1, 0) [Fig. 6.5(c)].

Figure 6.5: Lowest energy shape oscillations of a BEC in a cylindrically symmetric trap. (a)Slow quadrupole mode: radial size oscillates out of phase with axial size. (b) Breathing mode:
radial and axial sizes oscillate in phase. (c) Fast quadrupole mode: radial size oscillates with
m = 2 symmetry [30].

The general solution of this system can be found by first using the method of undetermined coefficients to find the particular solution. The general solution is the sum of theparticular solution and oscillations at the eigenfreqencies, where the amplitudes are set so that89
the system is initially in equilibrium. For non-resonant drive, the general solution is√1
δλx = − (1 + 2R1 − 2R2 )(ηx − ηy ) cos 2τ
4
1− (1 + 4R1 + 4R2 + 2Rz )(ηx + ηy ) cos 2τ
8r5
1
4ηx − ηy+ (10Rz − 1)(ηx + ηy ) cosγτ ++ (R1 ηx + R2 ηy ) cos Ω̃τ
40
2
10
√
1
δλy = (1 + 2R1 − 2R2 )(ηx − ηy ) cos 2τ
4
1− (1 + 4R1 + 4R2 + 2Rz )(ηx + ηy ) cos 2τ
8r1
54ηy − ηx
(10Rz − 1)(ηx + ηy ) cosγτ ++ (R2 ηx + R1 ηy ) cos Ω̃τ
40
2
10
r
5
ηx + ηy1(1 − 10Rz )(ηx + ηy ) cosγτ −+ Rz (ηx + ηy ) cos Ω̃τ.
δλz =
10
2
10

(6.48)

The frequency-dependent parts of the amplitudes have been abbreviated as
R1 =
R2 =
Rz =

Ω̃4 + 8γ 2 − 3Ω̃2 (1 + γ 2 )(Ω̃2 − 2)[Ω̃4 + 10γ 2 − Ω̃2 (4 + 3γ 2 )]Ω̃2 − 2γ 2(Ω̃2 − 2)[Ω̃4 + 10γ 2 − Ω̃2 (4 + 3γ 2 )]γ2.
Ω̃4 + 10γ 2 − Ω̃2 (4 + 3γ 2 )(6.49)

To prevent excitation of shape oscillations with a non-resonant drive frequency Ω̃, we mustdesign our array of wires such that all of the amplitudes are much less than 1.
The worst-case scenario for exciting these modes is resonant drive. If the drive frequency
is equal to any of the mode frequencies, then the oscillation amplitudes will increase linearlywith time (we are ignoring dissipative effects). The method of undetermined coefficients may
be used once again to calculate these amplitudes:√
2 N −2
afast,resonant =
β
ωρ Ť
4
1
abreathe,resonant = (N − 1)βN −2 ωρ Ť
2√
10
aslow,resonant = γ(N − 1)βN −2 ωρ Ť .
10

(6.50)

Here, Ť denotes the amount of time spent driving on resonance. Note that the amplitude given
for the slow mode in Eq. (6.50) is actually the amplitude of the size oscillation along the weak
trap axis, which is 4 times larger than the amplitude in the radial direction.

90
The fifth term in Eq. (6.36) likewise will cause periodic perturbations to the harmonic
confinement, but along a set of axes rotated by π/4 rad in the x̄ȳ plane. To see this, define the
new coordinate systemx̄ + ȳx0 = √
2y0 =−x̄ + ȳ√2z 0 = z.

(6.51)

The fifth term in Eq. (6.36) is now given by∆U =h̄ωρ N −2 02β(y − x02 ) sin N ω̃t.
2ρ20

(6.52)The resulting perturbations to the rotated system’s trap frequencies areωx20 = ωρ2 (1 − ηxy sin Ω̃τ )ωy20 = ωρ2 (1 + ηxy sin Ω̃τ ),(6.53)

where the amplitude of the modulation isηxy = βN −2 .

(6.54)The dynamics of the resulting shape oscillations are described by a system of differential
equations similar to Eq. (6.47):d2
δλx0 (τ ) = −3δλx − δλy − δλz + ηxy sin Ω̃τdτ 2d2
δλy0 (τ ) = −3δλy − δλx − δλz − ηxy sin Ω̃τdτ 2d2
δλz0 (τ ) = γ 2 (−3δλz − δλx − δλy ).
dτ 2(6.55)The eigenmodes of the new system are the same, but the general solution will be different.
Once more using the method of undetermined coefficients, we calculate the general solution for
non-resonant drive. Assuming a condensate initially in equilibrium, the general solution for the91
scale factor changes is√Ω̃ηxy(− √ sin 2τ + sin Ω̃τ )22 − Ω̃2√Ω̃
ηxyδλy0 =( √ sin 2τ − sin Ω̃τ )222 − Ω̃

δλx0 =

δλz0 = 0.
For resonant driving frequency Ω̃ =and y 0 is

(6.56)√

2, the amplitude of scale factor oscillations along both x0√
2 N −2a =βωρ Ť .
4
0

6.3.3

(6.57)

Collective mode minimizing parameters
Table 6.1 summarizes the resonant frequencies and the normalized amplitudes that must

be kept much smaller than 1 for each oscillation to be small. Amplitudes are given for bothresonant drive (“res”) and non-resonant drive (“nr”). For the shape oscillations along theunprimed axes, we give the amplitude of each oscillating term in Eq. (6.48). For the terms
oscillating at the resonance frequencies, the largest amplitude oscillation out of all three axes isgiven. The “particular” term is the term in Eq. (6.48) that oscillates at the drive frequency Ω̃.
A similar naming convention is used for the primed axes. These amplitudes must all be kept
much smaller than 1 to avoid exciting collective modes. The resonance frequencies are given indimensionless units.

6.4

Parallel wire design parameters
In this section, we use the mode amplitudes calculated in the previous section to estimate

values for period L, wire density N , and wire width ∆x such that the amplitudes of collectivemodes will be kept acceptably small. We begin by initially ignoring the effects of wire width.

1
1
2
√
2
p
5/2γ
Ω̃
Ω̃
Ω̃
2
√
2
p
5/2γ
√
2
√
2
√
2

Slosh, nr

Slosh, res

Breathing, nr

Slow quadrupole, nr

δλx , particular nr

δλy , particular nrδλz , particular nr

Breathing, res

1
1−Ω̃2

βN −2(6.48), (6.49)

βN −21βN −2
2−Ω̃2
√
2
N −2
4 ωρ Ť β√ Ω̃
2(2−Ω̃2 )(6.50)1N −2
2 (N − 1)ωρ Ť β√
2N −24 ωρ Ť β
√
10
N −2
10 (N − 1)ωz Ť β

(6.57)(6.56)(6.56)(6.50)(6.50)(6.48), (6.49)

4(N − 1)|Rz |βN −2|(2N − 1)R1 + (2N − 3)R2 |β(6.48), (6.49)(6.48), (6.49)(6.48), (6.49)(6.48), (6.49)(6.40)N −2βN −2βN −2βN −2(6.38)

Source equation|(2N − 3)R1 + (2N − 1)R2 |βN −2

2(1−10Rz )5

1+2R1 −2R2
2

(N − 1)

1+4R1 +4R2 +2Rz
2

L
N −2
2πρ0 ωρ Ť βL
2πρ0Normalized amplitude

Table 6.1: Normalized amplitudes of collective mode oscillations for sinusoidally driven parallel wires

Fast quadrupole, res, primed
axes

Particular, nr, primed axes

Fast quadrupole, nr, primed axes

Slow quadrupole, res

Fast quadrupole, res

Fast quadrupole, nr

Frequencyof oscillation

Term/mode name92

93
For ∆x = 0,β=

sin αN −1 α1
αN −1 sin α1= 1.

(6.58)

We next assume that we will use 87 Rb, confined at ωρ = 2π ×80 Hz and y0 = 300 µm (the values
used in the Michelson interferometry experiments). To avoid being sensitive to the low-frequency
vibrations of the apparatus (discussed in the previous chapter), we will set ωz = 2π × 10 Hz.
As the experimental duration in this case is only 100 ms, the effects of quantum phase diffusion
should be smaller than they were in previous work.
We see in lines 6 and 7 of Table 6.1 that, apart from the effects of resonances and wire
width, the shape oscillation amplitudes will be small if the parameterc1 = (2N − 1)N −2(6.59)

is small. Similarly, the sloshing mode amplitudes will be small if the parameterc2 =L N −2
2πρ0(6.60)

is small. Note that for 87 Rb and our desired confinement, ρ0 = 1.21 µm. Figure 6.6 shows
the values of c1 and c2 as a function of L for various values of N . We see in these plots thatreasonable choices for the wire layout are L = 1.2 mm and N = 8, such that c1 = 1.2 × 10−3
and c2 = 0.013. The wire pitch for these values is 150 µm.
Using these parameter choices, we now calculate the effects of resonances and wire width.
Our design goal is to construct a Sagnac gyroscope using 87 Rb that records a phase shift of π radwhen rotating at Earth’s rotation rate. As discussed in section 1.1.3, a circular path of radius
1.6 mm achieves this sensitivity. To move 4 × 1.6 mm = 6.4 mm in 100 ms, we would have totranslate the guide at an average speed of vc = 6.4 cm/s. The average drive frequency for trap
perturbations is therefore N ω̃ = 2π × 427 Hz, or Ω̃ = 5.33 in dimensionless units. Numericalresults for the amplitudes of the collective modes are shown in Table 6.2. We see that our choice

94

Figure 6.6: (Color). Plot of parameters (a) c1 and (b) c2 for vs. L for various values of N .

95
of N and L should not cause large excitations of collective modes while the guide is moving at
this average speed.
Term/mode name
Slosh, nr

Amplitude(∆x = 0)
4.7 × 10−4

Amplitude(∆x = 80 µm)
3.2 × 10−4

Slosh, res

6.4Ť

4.4Ť

Breathing, nr−4

3.3 × 10

2.2 × 10−4

Fast quadrupole, nr

4.3 × 10−5

3.0 × 10−4

Slow quadrupole, nr

3.2 × 10−5

2.2 × 10−4

δλx , particular nr

4.3 × 10−5

2.9 × 10−5

δλy , particular nr

4.9 × 10−5

3.4 × 10−5

δλz , particular nr

5.1 × 10−8

3.5 × 10−8

Breathing, res

0.14Ť

0.097Ť

Fast quadrupole, res0.014Ť

0.0098Ť

Slow quadrupole, res

0.09Ť

0.061Ť

Fast quadrupole, nr, primed axes

1.2 × 10−5

7.9 × 10−6

Particular, nr, primed axes

3.1 × 10−6

2.1 × 10−6

Fast quadrupole, res, primed
axes

0.014Ť

0.0098Ť

Table 6.2: Values of amplitude parameters from Table 6.1 for L = 1.2 mm, N = 8.

The last analytic calculation to be carried out is the effect of finite wire width ∆x. We
want to use wide wires to reduce resistance, and therefore heat dissipation. For the pitch of
150 µm that we chose in this section, a wire width of 80 µm occupies just over 50% of theavailable chip surface area. The wire width parameter for this design choice is β = 0.68. The
collective mode amplitudes for this wire width are shown in Table 6.2.

6.5

Transverse acceleration profile
We conclude this chapter by considering the acceleration of the guide. Jumping im-

mediately from a stationary waveguide to a moving guide at velocity vc would surely excite
large-amplitude dipole slosh modes along the x axis. We will instead design a smooth acceleration profile. The analytical approach we follow here calculates formulas for the area enclosed by

96
this profile for BECs that are split using the familiar techniques of the Michelson experiment.
We also find the expected slosh amplitude, which is minimized by setting 1/γ equal to an eveninteger. For our selected trap frequencies, we calculate a transport distance of 1.1 mm withminimal sloshing, providing an enclosed area greater than 1 mm2 .

6.5.1

Motion of the trap center
We consider a motion sequence composed of two consecutive periods of sinusoidal accel-

eration of opposite sign (see Fig. 6.7). The first half of the drive sequence moves the BECs awayfrom their initial position at x = 0, leaving them nearly at rest at a turning point x(T /2). The
second half of the drive sequence brings them back to their initial position.
Motion of the BECs in the accelerating trap is determined by the differential equationẍ(t) = −ωρ2 [x(t) − x0 (t)] ,(6.61)

where x(t) is the position of the BECs along the x axis and x0 (t) is the position of the trap
center. The acceleration ac (t) of the trap center is given by the piecewise sinusoidal waveformac (t) = Ka ×sin Ω̂t

0 ≤ t < T̂
− sin Ω̂(t − T̂ )T̂ ≤ t < 2T̂,(6.62)

where Ka is a constant, the drive frequency is Ω̂ = 2ωz , and T̂ = 2π/Ω̂ is the drive period. The
acceleration is integrated over time to give the velocity vc (t) of the trap center:vc (t) =

Ka
Ω̂

×





1 − cos Ω̂t

0 ≤ t < T̂.

(6.63)
−1 + cos Ω̂(t − T̂ ) T̂ ≤ t < 2T̂

The velocity is integrated a second time to yield the position xc (t) of the trap center:x0 (t) =

Ka
Ω̂2

×





Ω̂t − sin Ω̂t

0 ≤ t < T̂Ω̂(2T̂ − t) + sin Ω̂(t − T̂ ) T̂ ≤ t < 2T̂.

(6.64)

97

Figure 6.7: Acceleration, velocity, and position of the trap center for ωz = 2π × 10 Hz.

98
6.5.2

Equation of motion of the BECs
To find the equation of motion for the BECs, we solve Eq. (6.61) using Eq. (6.64) for thetrap position xc (t). For the first half of the sequence, where 0 ≤ t < T̂ , the general solution isx(1) (t)= A1 sin ωρ t + A2 cos ωρ t + A3 t + A4 sin Ω̂t + A5 ,(6.65)

where the Ai are constants. The BECs move in phase with the sinusoidal acceleration ramp,
and therefore there is no quadrature term in the solution (i.e. a term proportional to cos Ω̂t).
The particular solution coefficients are

A3 =
A4 =

Ka
Ω̂
Ka ωρ2Ω̂2 (Ω̂2 − ωρ2 )

A5 = 0.

(6.66)

Here, we assume that the BECs are initially at x = 0 and are not sloshing. The solution isthereforex(1) (t) =Ka Ω̂ωρ (ωρ2 − Ω̂2 )sin ωρ t +

Ka
Ω̂

t+

Ka ωρ2Ω̂2 (Ω̂2 − ωρ2 )sin Ω̂t.

(6.67)

To solve Eq. (6.61) for the second half of the drive sequence, where T̂ < t ≤ 2T̂ , we use for
initial conditions the final values of position and velocity obtained from Eq. (6.67):
x(1) (T̂ ) =ẋ(1) (T̂ ) =Ka Ω̂ωρ (ωρ2 − Ω̂2 )Ka Ω̂

sin ωρ T̂ +KaΩ̂Ka ωρ2(6.68)+ A03 (t − T̂ ) + A04 sin Ω̂(t − T̂ ) + A05 ,(6.69)

Ω̂

+

T̂
.

ωρ2 − Ω̂2cos ωρ T̂ +Ka

Ω̂(Ω̂2 − ωρ2 )The general solution isx(2) (t) =A01 sin ωρ (t − T̂ ) + A02 cos ωρ (t − T̂ )

99
where the particular solution coefficients are
Ka

A03 = −
A04 =
A05 =

Ω̂
Ka ωρ2Ω̂2 (ωρ2 − Ωˆ2 )
Ka T̂
Ω̂

.

(6.70)The initial condition ẋ(2) (T̂ ) = ẋ(1) (T̂ ) is satisfied whenẋ(1) (T̂ )
Ka Ω̂
.
+
ωρ
ωρ (Ω̂2 − ωρ2 )A01 =(6.71)In addition, the initial condition x(2) (T̂ ) = x(1) (T̂ ) is satisfied when
A02 = x(1) (T̂ ) −

Ka T̂
Ω̂

.

(6.72)

The complete solution for x(2) (t) isx(2) (t) =Ka Ω̂ωρ (ωρ2 − Ω̂2 )
+
−

6.5.3

(cos ωρ T̂ − 2) sin ωρ (t − T̂ )Ka Ω̂ωρ (ωρ2 − Ω̂2 )KaΩ̂sin ωρ T̂ cos ωρ (t − T̂ )(t − 2T̂ ) −Ka ωρ2Ω̂2 (Ω̂2 − ωρ2 )sin Ω̂(t − T̂ ).

(6.73)

Even-ratio trap frequencies
If either the final position x(2) (2T̂ ) or final velocity ẋ(2) (2T̂ ) are not zero, then the BECs

will slosh in the final stationary trap. Fortunately, we can eliminate this final slosh if 1/γ is aneven integer m. In this casesin ωρ T̂ = 0
cos ωρ T̂ = 1.

(6.74)The transverse position and velocity in the second half of the trajectory simplify tox(2) (t) =ẋ(2) (t) =Ka Ω̂ωρ (Ω̂2 − ωρ2 )Ka Ω̂Ω̂2 − ωρ2sin ωρ t +cos ωρ t −

Ka
Ω̂

Ka
Ω̂
−

(2T̂ − t) −Ka ωρ2Ω̂(Ω̂2 − ωρ2 )Ka ωρ2Ω̂2 (Ω̂2 − ωρ2 )cos Ω̂t,sin Ω̂t(6.75)

100
both of which vanish at t = 2T̂ .
Note that even though the BECs do not slosh after the drive sequence has ended, they
do deviate from the trap center during the course of their trajectories. Figure 6.8 shows thedeviation for ωρ = 2π × 80 Hz and ωz = 2π × 10 Hz during the second half of the trajectory.
The deviation isx(2) (t) − x0 (t) =Ka Ω̂ωρ (Ω̂2 − ωρ2 )sin ωρ t +Ka
ωρ2 − Ω̂2sin Ω̂t.

(6.76)

Since ωρ > Ω̂, the second term dominates. The amplitude of the residual slosh is therefore
approximatelyκ=Kaωρ2 − Ω̂2.

(6.77)

As long as κ is not much more than the size of the BECs, then they will remain in the harmonic
regime of the waveguide. Both will also deviate identically when moving, ensuring good overlap
when they come to rest at the end of the drive sequence. We can use κ as a convenient parameterin the expression for the turning point:x(t = T̂ ) =

2πKa

Ω̂2
ωρ2 − Ω̂2
= 2πκ
.
Ω̂2

(6.78)

Figure 6.8: Deviation of BEC center from trap center during return trip, for ωρ = 2π × 80 Hz
and ωz = 2π × 10 Hz.

101
6.5.4

Enclosed area
Recall that in the axial direction, the clouds undergo simple harmonic motion:z(t) = ±z0 sin ωz t = ±2nh̄k0sin ωz t,
M ωz(6.79)

where n is the diffraction order of the beamsplitter. We may combine this equation with Eq.
(6.67) to findx(z) =Ka Ω̂ωρ (ωρ2 − Ω̂2 )


sin

2ωρ
Ω̂

−1

sin

z
z0


+

2Ka
Ω̂2

−1

sin



Ka ωρ2z−1 z+
sin 2 sin,z0z0
Ω̂2 (Ω̂2 − ωρ2 )(6.80)

valid for 0 ≤ t ≤ T̂ /2. The full path traveled by both BECs is shown in Fig. 6.9. The total
enclosed area is
Z z0 "

=

6.5.5

#

0

4Kz0

4Kz0

A=4
"

!

T̂
2

x

Ω̂2

−

3Ω̂2

=

8Kz0 ωρ2ŷ3ωz2 (ωρ2 − ωz2 )=8κz0 ωρ2ŷ.
3ωz2− x(z)

3π +

ŷdz
!#

32ωρ2
Ω̂2 − 4ωρ2ŷ

(6.81)

Parameter estimates for enclosed area
We conclude this chapter by estimating the enclosed area that will be attainable for our

expected (80, 80, 10) Hz trap. We want to limit κ such that it is not too much larger thanthe oscillator length ρ0 . A BEC with repulsive atomic interactions will always have a larger
transverse size than the oscillator length, so the condition κ/ρ0 ' 10 keeps the deviations to arange not too much larger than the size of the BEC. For 87 Rb ρ0 = 1.2 µm, so we require2

Ka ' 10 × ρ0 (ωρ2 − ωz2 ) = 3 m/s .

(6.82)The extent of the transverse motion is therefore 1.2 mm. The enclosed area is A = 0.4nŷ mm2 ,where n is the order of the optical beamsplitter. It should therefore be possible to demon-

102

Figure 6.9: Trajectory of BECs in translating guide. Trap parameters are ωρ = 2π × 80 Hz andωz = 2π × 10 Hz.

103strate a BEC Sagnac gyroscope with a sensitivity within an order of magnitude of the eventual
performance goal of π rad of phase shift at ΩEarth .
Using these values of Ka and ωz , we can calculate the amplitude of the velocity of thetrap center:
Ka /Ω̂ = 24 mm/s.

(6.83)

Examining the shape of the plot of vc (t) [Fig. 6.7(b)], we see that vc spends a significant amountof the duration of the experiment moving at vc ≈ Ka /Ω̂. For our proposed wire pattern, at
this velocity the perturbation drive frequency is Ω̃ = 2; we will be driving the breathing modeprecisely on resonance. Fortunately, the amplitude of this mode will not become large for our
proposed wire pattern. We see in Table 6.2 that the normalized amplitude of the breathingmode when driven on resonance is 0.097Ť for drive duration Ť . Again examining Fig. 6.7(b),we see that the drive frequency should only be near resonance for approximately 20 ms aroundthe periods of peak velocity. The amplitude of the breathing mode should therefore not increasemuch past δλi = 0.002. This amplitude is small enough that it should have no noticeable effecton the experiment.

Chapter 7

Gyroscope apparatus: Design and construction progressThis chapter describes the design and construction of a new, compact apparatus for
Sagnac gyroscope experiments. The heart of the apparatus is a new atom chip patterned withan array of parallel wires, capable of carrying out the translating waveguide idea proposed in
the previous chapter. The new apparatus built around that chip has been designed to meet therequirements given at the end of Chap. 5 for a successful gyroscope. It has the optical accessneeded to control the angle of the optical standing wave with respect to the waveguide axis,
allowing for axial traps of sufficiently high frequency for our purposes. It will be able to produce
BECs with a cycle time of only a few seconds, allowing in only a few hours the collection of1000 point data sets. Finally, it is small and light enough to be placed on a rotary table. Once
operating, the apparatus will be capable of taking real rotation rate measurements.
The design and operation of similar compact BEC machines has been pursued by ourgroup over the past decade [18, 52, 53]. The goal of this chapter is therefore not to provide a fullmanual for the production of BECs in compact systems. Our purpose is instead to detail theunique aspects of this new apparatus, as well as the new techniques and lessons learned during
its construction. Significant elements of this updated system (magnetic field coils and vacuum
system mount) were designed by Evan Salim.
There are major differences in design between the large Michelson apparatus and ourcompact systems. All successful BEC vacuum systems must execute a balancing act: The same
system must be used to both capture a large number of atoms, and then hold those atoms in a

105
trap long enough to evaporatively cool to degeneracy. Quick trap loading can be achieved in a
vacuum chamber containing a relatively high vapor pressure of the atom of interest. However,
the already trapped atoms will collide with any remaining warm vapor, heating the trapped
atoms enough to cause them to escape. In the Michelson apparatus, this problem was solved byusing a multi-chamber vacuum system. Rubidium atoms were initially collected from vapor intoa MOT in one chamber, and then magnetically transported through a tube to an evaporationchamber. This chamber had only a small background pressure of Rb vapor, since untrapped
atoms were unlikely to diffuse through the long tube. The compact apparatus also uses a multichamber system, but transport through a long tube is an impractically large and slow connection
solution. Instead, the two chambers share a common wall, pierced by only a small pinhole. In
the first chamber, a two-dimensional MOT (2D-MOT) collects 87 Rb atoms from a vapor. This
2D-MOT creates a beam of cold atoms that is aligned to pass directly through the aperture into
the second chamber. Warm atoms not in the beam are unlikely to diffuse through the pinhole, so
the pressure in the second chamber is significantly lower than in the first. In this chamber, theatoms in the beam are captured in a traditional three-dimensional MOT (3D-MOT). The atoms
are subsequently loaded into a microtrap generated by an atom chip for evaporation to BEC. The
chip forms one of the walls of the chamber, allowing the system to be smaller than competing
designs. The use of ballistic transport between the chambers allows us to dispense with thewater-cooled magnetic field coils and the servo-controlled track used for magnetic transport in
the old system.
A further reduction in size and cycle time is achieved by dispensing with the Michelson
experiment’s use of a “macro” trap to evaporatively cool the atoms before delivery to themicrotrap. As was explained in Sec. 2.1, the higher trap frequencies that can be practicallyobtained with microtraps allow efficient evaporative cooling in a significantly shorter time. The
roadblock in the way of directly loading a microtrap from a MOT is the low trap depth of
microtraps (typically a few hundred microkelvin), combined with the small size of the trap
(typically less than 100 µm in radius). Atoms compressed directly from a MOT into the small

106
volume of a microtrap would be far too hot to be confined. The solution in the Michelson
apparatus was to slowly (about 1 min) evaporatively cool the atoms in a large volume trap
generated by permanent magnets. The approach followed in our compact apparatus is to add
a stage of polarization gradient optical cooling after the MOT, cooling the atoms sufficiently sothat they can be directly loaded into a chip trap without any previous evaporation. The time
allocated for evaporative cooling is thereby cut from over a minute to a few seconds.
In the remainder of the chapter, we detail the new apparatus component-by-component.

7.1

LasersThe BEC production recipe developed in [52] requires four wavelengths of laser light:

cooling and hyperfine repump light for optical cooling and trapping, light for optical pumping,and light for absorption imaging. Our system uses three temperature-stabilized external cavity
diode lasers (ECDLs) to generate these wavelengths. The cooling light is produced by an ECDL(JILA design [54], Hitachi HL7851G diode) and amplified by a tapered amplifier (JILA design,
SDL diode) to 500 mW. The repump light is produced by a commercial ECDL (New Focus
Vortex) and amplified by a slave diode laser (JILA design, Sharp GH0781JA2C diode) to 100
mW. Some of this laser’s light will be used to generate the standing wave beamsplitter. The
pump and imaging (or “probe”) light is generated by the third ECDL, producing 30 mW ofpower. The ECDLs were all recycled from the Michelson system. Figure 7.1 is a schematic of
the system.

7.1.1

Frequency control
Frequency stabilization of the three lasers is obtained by peak-locking the ECDLs tocrossover peaks in the saturated absorption spectrum of the 5S1/2 → 5P3/2 transition of 87 Rb,
located near 780 nm. The lock servos control both the current and grating orientation. The
cooling and pump/probe lasers are locked to the F = 2 → F 0 = 2, 3 crossover, while therepump laser is locked to the F = 1 → F 0 = 1, 2 crossover (primes indicate excited electronic

107

Figure 7.1: Schematic of laser system. (a) Cooling laser, (b) repump/beamsplitting laser, (c)pump/probe laser. Optical isolators are shown as diodes. Mechanical shutters are shown asboxes with an “x” through the center. Boxes marked “lock-in” are lock-in amplifiers, and thosemarked “SAS” denote saturated absorption spectroscopy. Optical paths are marked as solidlines, electrical signals as dashed lines. All beams are coupled into single-mode, polarizationmaintaining optical fibers.

108
states). Modulation for the peak-lock is applied to the laser diode current. The lock-in amplifiers,
frequency servo circuits, and temperature controllers are the same as those used in the Michelson
apparatus, with minor modifications to remove ground loops from the power system.
We use acousto-optic modulators (AOMs) to shift the light to the precise frequencies
required. The modulation frequencies used are shown in Fig. 7.1. The AOM shifts place therepump and splitting beams at the F = 1 → F 0 = 2 resonance, the probe beam at the F = 2 →F 0 = 3 resonance, and the pump beam 12.5 MHz to the blue of the F = 2 → F 0 = 2 resonance.
The cooling laser is stabilized 9 MHz to the red of the F = 2 → F 0 = 3 resonance, but thisdetuning must be increased during MOT compression and polarization gradient cooling. We canadjust it by unlocking the cooling laser and applying a calibrated voltage to the piezoelectric
actuator controlling the angle of the external cavity’s grating. The servo is relocked after thelaser cooling stages are finished.

7.1.2

Shutters and optical power control
All of the laser beam lines are controlled with commercial mechanical shutters, with theAOMs providing adjustable control over beam power. However, the mechanical shutters are
not fast enough to generate sufficiently short pulses for the probe and beamsplitting beams.
In these beams, the AOM is used as a high-speed, low-extinction shutter in series with themechanical shutter. Pulses are generated by opening the mechanical shutter a few milliseconds
before quickly pulsing the AOM’s acoustic wave; the mechanical shutter is closed after the pulse
so that none of the light leaking through the AOM in its off state reaches the atoms.

7.1.3

Optical fibers
Since our goal is to construct a rotating system, the laser beams cannot be delivered

to the vacuum system through free space. We instead couple all seven beams into single-mode
optical fibers. The fibers used are Corning PANDA PM 850 fibers that have been angle-polishedon at least one side. These fibers provide a polarization- and intensity-stable, flexible path from109
the lasers to the rotating system.

7.2

Vacuum system
The vacuum system used in the new apparatus (see Fig. 7.2) is a standard design thatis used in all of our group’s BEC experiments; the only difference between the systems is the
specific wire layout on the atom chip. These systems are designed to be quickly and easily
swapped out from the surrounding optomechanics if a chip with a different wire layout is needed
for future experiments. As the vacuum systems have previously been described in detail in [18],we will only describe here recent changes in design and construction.
The major design change for the gyro vacuum system is that the glass cell that comprises
most of the BEC chamber has been anti-reflection (AR) coated. Recall that the Michelson
experiment utilized a fixed, in-vacuum retro-reflector to establish the optical standing wave
for beamsplitting. The fixed angle of the optic with respect to the guide wire placed a limiton the acceptable angle between the waveguide and the optic, thus limiting the practical axial
confinement. In order to allow for greater flexibility in this parameter, in the gyroscope apparatus
we will replace the fixed optic with a mirror on an adjustable mount, placed outside of the vacuum
chamber. In order to form an acceptable standing wave, the reflected component of the beam
must contain almost the same power as the incident component. Therefore the wall of the cellthat sits between the retro-reflector and the trapped atoms must be AR coated.
The process for assembling the vacuum systems remains mostly unchanged, though wehave modified the recipe for bonding the atom chip to reduce damage to its delicate wires. The
anodic bonds [55] that we use are typically done at high temperature (400◦ C) to increase thebond strength. We have since discovered that bonding the chip at that temperature can cause
the Au wires patterned on the side of the chip facing ambient pressure to deform (see Fig. 7.3).
The formation of an anodic bond requires the application of steady pressure between the twopieces to be bonded, but Au is susceptible to deformation under negligible applied pressure
at high temperatures [56]. We have therefore reduced the temperature for bonding the chip

110

Figure 7.2: (Color). Annotated photograph of an assembled vacuum system. Ion pump magnets
are not shown. Also not shown is the mounting plate that bolts to the spherical cube.

111
to 325◦ C. At this temperature no wire deformation is evident, and the resulting lower bondstrength has not caused any problems.

Figure 7.3: (Color). Deformation of Au wires on back of chip bonded at 400◦ C.

The recipe we use to pump and bake the assembled system is substantially similar toprevious methods, though our procedures for detecting and sealing vacuum leaks have advanced.
We use a He detector to check for leaks before beginning the bake. There exists a simpler methodof leak detection, in which one sprays clean solvent on suspected locations of leaks. If there isa leak, the solvent will shortly afterwards register on the pressure gauge. However, this elegant
method has not proved sensitive enough to find tiny leaks in the chip before the system has been
subjected to a time-consuming bake. These leaks are extraordinarily difficult to permanently
seal, so leaky chips must often be discarded. We find that it is better to learn whether a chip is
fatally leaking before spending the time to bake the system to which it has been bonded.
The only leak sealant with which we have had any lasting success when applied beforea system bake is Vacseal, a silicone resin suspended in a volatile solvent. Vacseal’s productliterature claims that it can be baked to 400◦ C before failing [57], although we have foundthat sealed leaks will often re-open during the cool-down of the system. There is evidence inpublished work that Vacseal should never be baked above 250◦ C [58]. We have therefore limited
the bake of any system treated with Vacseal to this temperature; such systems likely do notachieve the lowest possible pressures.
The chip can also develop leaks while the BEC system is cooling down after the bake.
The leaks can usually be located by the solvent method described previously. We have foundthat sealing these post-bake leaks with the application of Vacseal is possible, but difficult. An

112alternative sealant is Glyptal 1201B, an alkyd resin paint with a very low vapor pressure [58].
Chips sealed with Glyptal should not be exposed to temperatures in excess of 135◦ C for extended
periods of time, so Glyptal may not be used before a system is baked.
It should be noted that we have not yet achieved BEC in the new system, so theseleak sealing techniques should not be regarded as having been fully evaluated. Leaks havefortunately become less of a problem with the delivery of a new production run of atom chips,
as the fabrication process now uses a technique less prone to failures. This is the best possible
solution to the problem, as no leak sealant is perfect.
A final change to the vacuum system preparation process has been made to the terminal
step, in which the BEC system is removed from the pumping station by pinching off the Cu tube
with a pair of rounded steel jaws. Before pinch off, the tube must be cleaned of all oxides that
accumulate during the bake. We have added a step to the preparation, in which we lightly coat
the tube with clean machine oil just before the pinch. The oil allows the jaws to slide smoothly
over the tube’s surface.

7.3

Optomechanics structure
The gyro optomechanics structure supports all of the components needed to make a

BEC other than the lasers, electrical power supplies, and control system. The structure weighsapproximately 65 kg, and occupies a 60 cm by 60 cm footprint. This section details its designand construction.
The base of the structure is an air bearing rotary table, Precitech model RT200 (Fig. 7.4).
Air bearings provide smooth, low-friction motion and are therefore quite suitable for precisionrotation measurements. The RT200 has a 200 mm diameter table and an axial load capacityof 226 kg; care must be taken to ensure that the apparatus to be built on top of the table iswell balanced to keep the load as axial as possible. The unit requires a regulated supply ofclean, dry compressed air at 50 psi (0.34 MPa) to operate. The compressed air supplied to ourlab has sufficient pressure, but it must be filtered (Norgren Excelon F72G-2AN-AL1) and then113
processed through an oil and water vapor separator trap (Norgren F64B-2AN-AT0). A second
filter and a pressure regulator are in the supply line between the trap and the rotary table.

Figure 7.4: (Color). Rotary table, 200 mm diameter.

The 200 mm diameter rotary table is large enough to directly support a custom-madeoptomechanical structure for BEC production (for an example, see [53]), but we decided to saveon materials and production costs in this apparatus by using mostly off-the-shelf hardware. Thisapproach requires a larger surface area for the optical layout, which we support with a 60 cm
by 60 cm industrial grade breadboard (Newport). The board combines a steel working surface
with a trussed honeycomb core, providing a stiffer foundation with more vibration damping at
a lower weight than would a solid metal plate.
The breadboard must sit precisely at the center of the rotary table to prevent off-axis
loads from damaging the air bearing. Our breadboard has a tapped tube welded vertically into
its structure at its precise center (Newport’s “Microlock” system). A machined Al cylinder thatis just under 38 mm in diameter is bolted into the Microlock tube on the underside of the board.
This cylinder fits snugly inside the 38 mm diameter shaft that runs down the center of the rotarytable (visible in Fig. 7.4). To mate the two pieces together, we first raise the breadboard on

114
jacks. We then carefully position the rotary table, using the cylinder and the shaft to properly
locate the relative position. We finally lower the breadboard to rest on top of the rotary table.
The weight of the apparatus is enough to hold the pieces snugly together. The breadboard istypically not mated to the rotary table while we are optimizing the experiment.
The vacuum system and optics for the 2D-MOT are mounted directly on the breadboard.
A solid Al baseplate for the BEC chamber’s optical system is mounted 18 mm above the breadboard on a set of 4 pedestals (see Fig. 7.5). In order to maximize the 45.72 cm by 45.72 cm
baseplate’s mechanical resonance frequency, the pedestals are placed along the breadboard’s
diagonals, halfway between the center and the corners. A hole cut in the plate allows a compact camera mounted on the lower breadboard to record images of atoms trapped in the BEC
chamber.

7.4

Double MOT
The job of the 2D-MOT is to produce the greatest possible flux of cold atoms into

the ultrahigh vacuum BEC region of the system. To obtain a high flux, we use a pair of
perpendicular, retroreflected, elliptically shaped cooling beams to take maximal advantage ofthe 25 mm by 10 mm size of the 2D-MOT cell. The total cooling power in each beam isapproximately 50 mW, with major diameter 24 mm and minor diameter 7.1 mm (1/e2 ).
A top view of the 2D-MOT optics is shown in Fig. 7.6. Light from both the 2D-MOT
cooling and repump fibers is collimated with air-spaced doublet collimators (Thorlabs) andcombined at a polarizing beamsplitter (PBS) cube. A confocal cylindrical telescope (f = −38.1
mm and f = 130 mm) expands the beam along the vertical direction. The retroreflectors forboth beams each consist of a quarter-wave plate cemented to a dielectric mirror. Approximately
10% of the cooling light is picked off just after the collimator for use as a vertically aligned“push” beam. When correctly aligned with the 2D-MOT, the push beam forces most of the
MOT’s flux to travel upwards into the BEC chamber. Control over the push beam’s power isprovided by a variable attenuator consisting of a half-wave plate and a PBS; the excess light is115

Figure 7.5: (Color). Schematic view of optomechanics structure, shown without optics.

116
dumped into a black piece of paper attached to the PBS’s unused output face.
The two-dimensional quadrupole magnetic field for the 2D-MOT is generated by four
NdFeB permanent magnets (Magnetic Component Engineering alloy N3578). These magnets
have a square cross section with 1 mm sides, and are 40 mm long. The magnets are poled
perpendicular to their length. They are mounted vertically with respect to the 2D-MOT cell,and placed at the corners of a square measuring 18 mm on a side. The sides of the square arenormal to the propagation direction of the MOT beams. The resulting field gradient is between
30 and 35 G/cm.
A top view of the optics for the 3D-MOT is shown in Fig. 7.7. The total cooling power forthis MOT is approximately 60 mW, split among three retroreflected beams. Both the repump
and cooling beams are collimated with f = 75 mm doublet lenses, resulting in a 15 mm beam
diameter (1/e2 ). Power balancing between the beams is provided by a set of half-wave plate and
PBS pairs. As explained in [18], our 3D-MOT does not use the traditional 90◦ angle between
the cooling beams. One beam (the “horizontal” beam) is aligned parallel to the baseplate as it
passes through the glass cell. The other two beams (the “angled” beams), which pass throughthe cell in the plane normal to the horizontal beam, are aligned at an elevation angle of 22◦ withrespect to the plate. This configuration allows us to make a MOT significantly closer to the atom
chip than would be possible with beams aligned at 90◦ (see Fig. 7.8). The retroreflectors for theangled beams are essentially identical to those used in the 2D-MOT, but the horizontal beam’s
retroreflector has a PBS in the beam path between the quarter-wave plate and the mirror. The
quarter-wave plate is aligned so that all of the MOT light passes directly through the PBS. This
more complicated reflector allows us to overlap the horizontal beam with the optical pumping
beam; see the next section for more details.
The three-dimensional quadrupole field for the 3D-MOT is generated by a pair of rectangular anti-Helmholtz coils, oriented perpendicular to the horizontal MOT beam. These coils
provide a gradient of 12 G/(cm A). The MOT coil pair is held in a machined Al mount, whichslides over the BEC glass cell on a set of four steel rails. The rails are bolted to a collar, which

117

Figure 7.6: (Color). Optical layout of 2D-MOT. Magenta lines show beam paths.

118

Figure 7.7: (Color). Optical layout of upper baseplate. 3D-MOT beam paths are shown inmagenta. The horizontal beam passes from right to left across the center of this diagram.

119

Figure 7.8: View of angled 3D-MOT along axis of horizontal beam. Reducing the angle θ allows
the separation between the MOT and the atom chip h to be reduced without clipping the beams
on the edge of the atom chip [18].

120
is itself bolted to the upper corners of the vacuum system’s spherical cube. See Fig. 7.9. The
coil mount also carries 3 pairs of rectangular Helmholtz coils to generate bias fields along thethree Cartesian axes of the apparatus.

Figure 7.9: (Color). Mounted vacuum system with MOT and bias coils installed on rail system.

The current for the MOT coils is generated by a commercial linear, unipolar power supply(Kepco ATE). For the bias coils, we need the flexibility of bipolar supplies in order to generate
the full range of microtraps. A trio of Kepco bipolar operational amplifier power supplies (BOPs)
provides the needed bipolar behavior. However, they are unable to operate in a stable manner
while driving current through inductive loads. The complex impedance of an inductive coil pairis
Zcoil = Rcoil + iωL,

(7.1)

where ω is the angular frequency of a signal sent through the coils. At sufficiently high inductances, the imaginary part of the impedance becomes significant at frequencies within the
bandwidth of the BOP power supply’s servo. Instead of supplying DC current, the BOPs oscillate when driving our 0.5–1 mH coil pairs. A solution to this problem is to add a Zobel
network [59] in parallel to the coil, which consists of a resistor and a capacitor in series. The

121
total impedance isZ=

Rcoil RZobel + L/C + i[ωRZobel L − Rcoil /(ωC)].
Rcoil + RZobel + i[ωL − 1/(ωC)]

(7.2)

2
If RZobel = Rcoil and C = L/Rcoil, then the impedance of the combination of the coil and theZobel network is real and equal to Rcoil . For our bias coil pairs, Rcoil ≈ 2 Ω, which sets C =125–250 µF. As this capacitance is larger than that of easily obtainable small capacitors, wetypically set C = 10 µF and set RZobel to minimize the imaginary part of Eq. (7.2) for thatchoice of C. We then optimize the choice of RZobel , monitoring the actual stability of the power
supply with a Hall probe. We find RZobel = 17 Ω works quite well.
We optimize the double MOT to obtain a large 3D-MOT cloud that loads quickly. We
can measure both the number of atoms and loading time by monitoring the fluorescence ofthe 3D-MOT with a photodiode. The loading time is quite sensitive to the alignment of the
2D-MOT with the aperture, and to the alignment of the push beam. We use an inexpensive
infrared-sensitive security camera to monitor the shape of the 3D-MOT while we are optimizing
the beam alignment; typically, an irregularly-shaped MOT is not desirable. For a non-leaking
vacuum system, we are usually able to load a few hundred million to a few billion atoms into
the 3D-MOT in fewer than 3 s.

7.5

Transfer to chip trap
The procedure that we follow to transfer cold atoms from the 3D-MOT into a chip trap isidentical to that explained in detail in [18]. We will therefore only briefly review the steps in theprocess. We concentrate on a description of the new gyroscope atom chip, which incorporates
the array of parallel wires described in the previous chapter.

7.5.1

Loading the “big Z” trap
The next major step in the experiment is to load the atoms trapped in the 3D-MOT

into a magnetic trap. In order to reduce the cooling that will have to be later carried out by

122
evaporation, we strive to maximize the phase-space density of the magnetically trapped atoms.
In pursuit of that goal we must reduce the spatial size and temperature of the cloud, while
maximizing the number of trappable atoms in it.
A compressed MOT (CMOT) stage serves to shrink the spatial extent of the atomic
cloud. As soon as the 3D-MOT has filled, we unlock and detune the cooling laser further to thered by an additional two linewidths. We simultaneously reduce the repump power by greaterthan 90%, using that laser’s AOM. The reduced re-radiation pressure allows the cloud’s size todrop from about 1 cm to about 1 mm. The CMOT duration is approximately 50 ms.
To reduce the temperature of the cloud, we follow the CMOT with a stage of polarization
gradient optical molasses cooling (PGC). In this stage, the cooling laser is further detuned, now
to a total of more than 11 linewidths from resonance. The MOT quadrupole field is turned off,
and the bias coils are used to cancel out ambient magnetic fields. Temperatures in the range of
40 µK are easily obtained after only a few milliseconds of PGC. The cooling efficiency of PGC
is in this case limited by the density of the compressed cloud of atoms. Lower temperatures(below 10 µK) can be obtained when the MOT load is kept small. The ideal balance between
temperature and number of atoms after PGC is experimentally determined.
After PGC, the cloud of atoms is optically pumped into the |2, 2i ground state by thepump laser beam, which is set to σ + polarization with respect to an applied uniform magnetic
field. The beam must not be retroreflected by our standard quarter-wave plate/mirror reflectors
and sent back through the atoms. The polarization of the reflected beam would be σ − withrespect to the bias, and would tend to pump the atoms into the wrong state. Our solution tothis problem is shown in Fig. 7.7. The pump beam is collimated and then combined with thehorizontal MOT beam at a PBS (shown at right center of figure). The following quarter-wave
plate converts the two beams of perpendicular linear polarization into two circularly polarized
beams of opposite helicity (with respect to the direction of propagation). On the far side of thevacuum system, a second quarter-wave plate is aligned to convert the cooling beam into lightpolarized parallel to the baseplate. This beam passes directly through a second PBS and is123
retroreflected back towards the atoms. On the other hand, the pump beam is converted to lightpolarized perpendicular to the baseplate. This light is reflected by the PBS and blocked by apiece of black paper attached to its side output port.
The compressed, cooled, and pumped cloud of atoms is at this point ready to be magnetically trapped. Our initial “microtrap” is actually generated by a large Z-shaped wire (see Sec.
2.1.3) that is mounted directly above the chip, just outside of the vacuum system. The distance
LH between the “legs” of the Z is 27 mm. A significant amount of current is required to make a
trap capable of supporting against gravity more than 18 mm away from the trapping wire (h inFig. 7.8 is approximately 17 mm). This problem was solved in [18] by running > 100 A through5 Z-shaped wires held together in an Al block and connected in series. In the new apparatus,
we have replaced this high-current “big Z coil” with a version using approximately 30 wires (seeFig. 7.10). This coil allows us to use only 20 A to produce a similar trap.

7.5.2

Transport to chip height
Once the atoms have been caught in the big Z trap, they are transported up to the chipby simply ramping down the big Z current [recall that the distance between the wire and thefield minimum is proportional to the current, Eq. (2.4)]. If the transverse bias is not adjusted,
the trap becomes tighter in the transverse direction as the trap is pulled towards the chip [Eq.
(2.9)]. However, as the distance from the wire to the trapped atoms d + y0 becomes very small
with respect to LH , the trap loosens in the axial direction [Eq. (2.24)]. We therefore wrap an
additional set of wires over the legs of the Z coil (labeled as “compression wires” in Fig. 7.10)
and use them to effectively increase the axial trapping current as the atoms approach the chip.
The transport ramp is usually completed in about 400 ms.

7.5.3

Gyroscope atom chip
The gyroscope chip is shown in Fig. 7.11. The central feature is an array of 40 parallelwires, placed at a spacing of 150 µm. Each wire is 80 µm wide. This chip is a realization of the124

Figure 7.10: (Color). Views of the big Z assembly. (a) View from below, (b) angled view. This
assembly is lowered down to just above the atom chip on the rails that also carry the MOT/bias
coil assembly.

125
design proposed in Sec. 6.4. We plan to initially capture and evaporate the atoms to BEC in a
trap generated by a wider 100 µm wire, crossed by a T wire (this wire appears above the arrayof parallel wires in the figure). These wires are connected to external power supplies throughhermetic electrical “vias” that pass through the chip substrate. Longer-range axial confinement
than can be generated by the T wire is provided by a selection of H-type wires patterned on theback of the chip [the vertical traces in Fig. 7.11(b)].
We make electrical connection to the chip’s more than 100 contact pads by attachingcommercial connectors to the pads with conductive epoxy (EPO-TEK E2101). This Ag-bearing
adhesive is used in place of standard electrical solder, as Sn to Au solder joints are brittle andage poorly [60]. The connectors used for the parallel wire array have a pitch of 500 µm (Hirose
FH19SC) and connect to flat flexible cable. The connectors for the BEC production wires havea pitch of 1.27 mm (Mill-Max 399-XX-XXX-00-310000 series). These are standard single in-line
package connectors. We have built our own cables for these connectors, using ultra-flexible
stranded Cu wire (available from Cooner Wire).
We do not have a commercial pick and place machine, so the process of attaching theconnectors uses improvised equipment. We attach the connectors after the vacuum system has
been pinched off from the pumping station. We use a 30 AWG wire clamped in a translation
stage to apply a small bead of conductive epoxy to each individual pad on the chip. We thenuse a pair of tweezer clamped to the same stage to lower the connectors into place. We visuallymonitor both of these processes through a stereo microscope. After the connectors are placed,
we cure the epoxy using a heat lamp. We have achieved excellent results with this process; see
Fig. 7.12. After the conductive epoxy is cured, we typically add a few drops of mechanical epoxy
to increase durability. We use a very compliant epoxy (EPO-TEK OD1001) to avoid placing
excess strain on the chip.

126

Figure 7.11: (Color). Photographs of both sides of the gyro chip: (a) vacuum side, (b) ambient
side.

127

Figure 7.12: (Color). Views of gyro chip with electrical connectors. (a) View of all four connectors on chip. The red substance visible on the vias is Glyptal 1201B (Sec. 7.2). (b) Close-up of
epoxy joints on high-pitch connector.

128
7.5.4

Loading the chip trap
The central section of the big Z coil is directly above the 100 µm wide chip wire, which

generates the transverse confinement for the initial chip trap. As the big Z current is reduced
during transport, the trapped atoms are brought to a position 1 mm directly under this wire.
Following the procedure established in [18], we transfer the atoms from the big Z to the chip trapby simultaneously ramping the big Z current to zero and ramping the chip wire to 3 A over 100
ms. Axial confinement in the initial chip trap is provided by running current through the pair of
H-wires on the back of the chip with the widest separation. The bias fields are adjusted during
the transfer so that a trap with transverse frequency on the order of 1 kHz in the transverse
axes is obtained approximately 100 µm below the chip. The T-wire is ramped on after loading
to provide sufficient axial confinement for evaporative cooling.
Current for the chip wires is provided by JILA-designed current servos. These servos are
powered by batteries to reduce current noise.

7.6

Imaging
All data collected about the state of system from the CMOT stage onwards is obtainedby laser absorption imaging, using the F = 2 → F 0 = 3 cycling transition. Two images arecaptured at the end of each experiment. The first is a data image, in which the atoms cast ashadow by scattering light out of the probe beam. The second is a normalization image, acquired
as soon as the atoms have fallen out of the camera’s field of view, in which the full power of theprobe beam is captured. These two images are used to calculate the optical density (OD) of thecloud of atoms.
We use a compact digital camera (Basler scA1400-17fm), measuring 97 × 44 × 29 mm.
It utilizes an interline charge-coupled device (CCD), in which every column of active pixels
sits next to a column of inactive pixels. After an image is acquired, the accumulated charge
is rapidly transferred from the active to the corresponding inactive pixels for read-out. A

129
second image can begin acquisition nearly instantaneously while the first is being read [61].
The advantage of acquiring the data and normalization images in rapid succession is that themechanical components of the apparatus have no time to shift in position between shots. There
are therefore fewer of the artifacts in the OD image that are otherwise caused by vibrationinduced shifts of diffraction and interference fringes in the probe laser beam [62]. Anotherfeature of the CCD chip used by our camera is the inclusion of an electronic shutter, removing
the need for a bulky, noisy mechanical shutter.
The lens system that we use for high-magnification imaging of microtrapped atoms is amachine vision microscope (custom, based on the Infinity InfiniTube Standard), AR coated for
780 nm. The lens has a magnification of 2.8 and working distance 145 mm. The camera’s pixels
are 6.45 × 6.45 µm squares, so each pixel will record a 2.3 × 2.3 µm area of the object plane. Thisis just larger than the lens’s diffraction-limited 2 µm resolution. Decreasing that limit would
require us to build an imaging system with a larger numerical aperture (NA), but the available
NA for imaging is limited by the need to keep a clear path for the angled MOT beams (see Fig.
7.13). As can be seen in the figure, the lens tube is aligned vertically, poking through a hole in
the upper baseplate. A mirror is used to steer the probe beam into the lens.
For imaging of atoms during earlier stages of the experiment, we remove the steering
mirror and image horizontally with a lower-magnification lens system. The camera is mounted
off of the breadboard in this type of imaging, so we are not able to image this way while theapparatus is rotating.

7.7

Control system
A computer system provides precisely timed control of the various elements of the appara-tus. See Fig. 7.14 for a diagram of data flow in the control system. The system’s core component
is a field programmable gate array (FPGA). The FPGA (National Instruments PXI-7813R) is
programmed to execute timed updates of its 160 digital outputs, using its own dedicated clock.
The array of instructions for the FPGA resides in the memory of an embedded control computer

130

Figure 7.13: (Color). Side view of apparatus showing imaging system. Steering and retromirrors
for angled 3D-MOT beams are also shown. This figure illustrates the difficulty of increasing theNA of the imaging system without clipping the angled beams. MOT light is shown in magenta,
probe light in cyan, and the imaging lens in yellow.

131
running a real-time operating system. Both the FPGA and embedded computer are installed in(and communicate through the bus of) a PXI chassis.
The user sets the timing table before a run by interacting with a control interface, which
is hosted on a computer running Microsoft Windows XP. The user’s instructions are downloaded
to the embedded computer via Ethernet, and then are executed independently of further interaction with the nondeterministic timing environment of Windows. The user interface, embedded
controller, and FPGA all run software written in LabVIEW.
Many of the experiment’s systems are controlled by analog voltages. These voltages areproduced by a set of digital-to-analog converters (DACs, National Instruments 9263) that arecontrolled by the FPGA’s digital outputs. The DAC modules each contain four output channels.
Pre-packaged LabVIEW code modules provided by National Instruments handle the details ofusing a block of 40 of the FPGA digital outputs to control up to four DAC modules (16 analog
outputs per block). The modules controlled by a block of FPGA outputs are housed outside of
the PXI system in an expansion chassis. We have six DAC modules in the current configurationof the control system, occupying only two of the FPGA’s 40 channel digital output blocks. The
system is therefore easily expandable to include more analog outputs (and also analog inputs)should the need arise in the future.
Each DAC module has a floating, common signal ground that is isolated from that of theother modules. The isolated DAC grounds greatly ease the difficulty of eliminating ground loops
from our electronics. We group together related analog-controlled devices on each module, andthen ensure that the group’s signal ground is tied to earth ground at exactly one location. As
an example, consider the power supplies for the MOT and bias coils, which are controlled by asingle module (DAC 1 in Fig. 7.14). The chassis of just one of these supplies is tied directly toearth ground, and the only connection between the common signal ground of all four supplies
and earth is made there. We ensure that the chassis of the other three supplies are tied to earth
only through the chassis of the primary supply. This is done by wiring the chassis together
with heavy gauge Cu wire, and then installing electrical plug adapters to cut their independent

132

Figure 7.14: Diagram of control system. “Chip BEC wires” refers to the current supplies driving
those chip wires needed for initial trapping and BEC production: the main, T, and H wires.
“Chip parallel wires” refers to the supplies driving the array of parallel wires for translating
waveguide interferometry.

133
connections to earth ground. Figure 7.15(a) shows a schematic of the wiring scheme. Careful
control of grounding leads to significant decreases in electrical noise in our system. Figure7.15(b) shows the results of a noise measurement comparing the case of proper grounding (black
trace) to the case in which both chassis connect to earth independently (red trace). We show
the voltage noise measured by an AC coupled dynamic signal analyzer across a 1 Ω test resistor
driven with 1 A of current by the grounded supply. Similar single-reference grounding schemes
are used for the instruments controlled by the other DAC modules.
The other task required of our control system is image acquisition. A digital control
output is used to trigger image acquisition through the camera’s external control input. The
images are downloaded to a second Windows computer through a FireWire 800 interface. Software written in LabVIEW displays the images, carries out preliminary analysis, and optionallysaves the images for further study. The use of a separate imaging computer is not necessary
in our system, as the critical timing tasks are carried out by the FPGA. The control systeminterface and imaging acquisition tasks will likely be combined in a single computer in a future
upgrade.

7.8

Current progress and future work
We have successfully loaded a 3D-MOT while slowly rotating the apparatus on the air

bearing. We have also, in the stationary mode, carried BEC production as far as the transfer
of atoms into the initial chip trap. Progress was delayed for a time by unexpected challenges
in the fabrication of the high-density via pattern of the atom chip. A new fabrication processdeveloped by our collaborators at Teledyne Scientific seems to have overcome these issues.
A vacuum system using the new chips has now been installed in the gyro apparatus. We
should soon be loading the initial chip trap with atoms. Evaporative cooling to BEC should
soon follow. BEC has been achieved in similar systems after a radio frequency (RF) evaporation
cycle lasting only a few seconds. The RF radiation will be coupled to the atoms by a single loop
coil fixed to the big Z assembly (see Fig. 7.10). Typical BECs are expected to contain on the134

Figure 7.15: (Color). Grounding scheme for MOT and bias coil power supplies. (a) Schematicof ground connections. The chassis of only one power supply is tied directly to earth ground;the connection between signal and earth ground is made at that chassis. Other supplies areconnected to earth ground via the grounded supply’s chassis. (b) Measurement of noise spectrumof grounded supply when wired as shown in (a) (black trace), and when second power supply
chassis is allowed to connect to earth ground independently (red trace).

135
order of 2 × 104 atoms [53]; total BEC production time will be on the order of 5–10 s.
After BEC is achieved, interferometry experiments will begin. We have already completed
preliminary design work on the optical system that will be used to create the standing wave forbeamsplitting. An array of parallel traces aligned perpendicular to the primary wires on thevacuum side of the chip provides a reflection grating that we will use to align the beam to theaxis of the waveguide, regardless of the tilt angle induced by the axial confinement (see Fig.
7.11).
We have also built an 8-channel battery-driven current supply to provide the current
we will run through the parallel “moving waveguide” wires. A custom-designed printed circuitboard will be used to connect every eighth wire in series. Calculations show that a travelingwave of current amplitude < 0.5 A can create waveguides with the ωρ = 2π × 80 Hz transverse
confinement that we modeled in Chap. 6. Axial confinement will be provided by a pair of Hwires attached to the big Z mount (approximate location indicated in Fig. 7.10). The spacingof these wires is set keep the anharmonicity parameter given in Eq. (2.27) small.
This dissertation has presented the initial steps leading from past work with BEC Michelson interferometry in a stationary waveguide towards the demonstration of a BEC Sagnac gyroscope based on a traveling waveguide. Much work remains to be done to complete the testingof the new apparatus and the proposed method for moving the guide. We should note that thesmall size of the BECs that will likely be available in our first experiments may cause shot noise
to be a limiting performance factor (see Sec. 1.1.3). Fortunately, the flexibility of the apparatus
will allow the easy replacement of the currently installed vacuum system with an updated design
capable of improved performance. It is exciting to consider that if the proof-of-concept experiment proposed here is a success, then recent rapid progress towards the miniaturization of lasers,
power supplies, and control systems [53] could soon lead to the construction of a field-deployable
BEC gyroscope.

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Appendix

Table of mathematical symbols used in text

Vectors are denoted by boldface type, while unit vectors have a hat placed atop the boldsymbol. Magnitudes of vectors are denoted by the italic form of the symbol, except where noted.
Dots indicate derivatives with respect to time. In some cases, a prime denotes a derivative withrespect to a spatial coordinate.

αnDimensionless wire width parameterβSecond dimensionless wire width parameter [see Eq. (6.34)]

γTrap aspect ratio

γ2

Normalized kurtosis excess

δ

Detuning from resonanceδλi

Small shift in dimensionless BEC size scaling parameter along axis i∆φDifferential phase shift∆φlaser

Phase shift induced by vibration|∆φp̌ i

Eigenstate of phase-difference operatorδD (x)

Dirac delta function∆P

Path length shift

140
∆p

Separation in momentum∆U

Perturbation to potential∆xWire width∆z

Separation between BECs along waveguide axis

δ(variable)

Uncertainty in (variable)

e−2πy0 /L , smallness parameter for perturbations

ηAmplitude of trap frequency modulation

Θ(x)Unit step functionθ

Tilt angle of waveguide with respect to main wireκAmplitude of deviations from trap center during guide translation

λWavelength

λdB

deBroglie wavelength

λiDimensionless BEC scaling parameter along axis iµi

Chemical potential of BEC iµB

Bohr magnetonµ0

Magnetic constantµ4

Fourth statistical momentξ

Deviation in energy of two-BEC system from equilibrium(ρ, φ, z)

Cylindrical coordinates

ρ0

Harmonic oscillator length of radial waveguide141σStandard deviationσ̂

MLE of standard deviation of normally distributed noiseσ∆φStandard deviation of phase distributionτωρ t, dimensionless time coordinate

τi

Duration of event i

φDNormally distributed phase noise due to phase diffusion

φv

Initial phase of vibration

χi,jUnit-normalized wavefunction for i atoms in state 1 and j atoms in state 2ΨMany-body wavefunctionψSingle-atom wavefunctionΩAngular velocity

ωAngular frequency

ω̄Geometric mean trapping frequencyΩE

Effective two-photon Rabi frequency

ωiAngular trap frequency along axis ih̄ωNT /2

Energy of two BECs containing equal number of atomsΩR

Rabi vector (Bloch sphere representation)ΩR

Rabi frequency

ωrRecoil frequency

ωvFrequency of vibration

142
ω1

Frequency fit parameter for simulated phase diffusion dataΩ̂2ωz , frequency of transverse acceleration profileΩ̃N ω̃/ωρ , Dimensionless perturbation frequencyω̃

2πvc /L, perturbation driving frequency

A

Area

A

Amplitude of interference signal in PC or IC representationÂ

MLE of amplitude of interference signalaAccelerationac

Acceleration of waveguide center

Ai

Constants

amode name

Amplitude of named collective modeasScattering lengtha1

Amplitude fit parameter for simulated phase diffusion dataā

Geometric mean oscillator lengthâ

Non-negative integer

B

Magnetic induction (referred to as magnetic field)bInteger

Bc

Magnetic field of crossing wire at waveguide heightbi

Polynomial coefficient i

Bn

Magnetic field of harmonic n

143
BT

Field of T wire at waveguide height

Bw

Magnetic field amplitude of parallel wire array

B0

Bias magnetic field

C

Capacitancec

Speed of lightCi

State amplitude of momentum state ici

Trap perturbation minimizing parameters

D

Sample covariance matrix of PC basisdDisplacement of trapping wires away from chip surface

Dg

Diffraction grating period

E

Electric field

E

EnergyF

Total atomic angular momentumf

Focal lengthfL (z, θp )Likelihood of parameters θp for observation set z

fx (x)

Probability density of random variable xgStrength of interatomic contact interactiong(x)An integrable function of x

gF

Landé g factor of total angular momentumhPlanck constant

144
h(x)

An integrable function of x

I

Electrical currenti√−1 or an integer

IH

Magnitude of current flowing through H wires

Im

Magnitude of current flowing through wire m

J

Current densityjInteger

K

Surface current densityk

Wavevector of interfering wavek

Wavenumber of interfering wave or an integer (distinguished by context)

Ka

Transverse acceleration amplitude(c)

Fourier cosine coefficient n of K

Kn

(s)

Fourier sine coefficient n of K

k0

Wavevector of laser used in diffraction beamsplitterL

Spatial period of traveling wave of current. In Ch. 7, inductancelDisplacement along curve of integration

LH

Distance separating H wires

M

Mean image of a data set

M

Mass

m

Integer

Kn

145
mF

Quantum number of projection of total angular momentum along an axis

N

Number of wires per spatial period of traveling wave of currentnIntegerNτ /2

Number of axial half-periods in interferometry experiment

NI

Number of images in a data set

Ni

Number of particles observed by detector i or in interferometer arm i

Np

Highest momentum order significantly populated by beamsplitter

NT

Total number of interfering particles

pTranslational momentum

P (x)Probability of x

pc (rd /s)

Circular coverage function

P (I)Matrix of ICs

P (P )

Matrix of PCs

p0

Initial momentum of separating BEC wavefunctions in experimentp̂

Number of pixels

q

Number of PCs retainedR

Rate of quantum phase diffusionr

Position with respect to origin

Rc

Fraction of atoms in zero momentum state after interferometry

Rcomponent

Resistance of component

146
rd

Radius of integration of circular coverage function

Ri

Frequency-dependent parts of collective mode amplitude along axis i [see Eq.
(6.49)]

rs

State vector (Bloch sphere representation)

S

Sample covariance matrix of pixel basissStandard deviation of Gaussian in circular coverage function

T

Duration of interferometry experiment

t

Time

Ť

Time spent driving on resonanceT̂

Period of transverse acceleration profile

U

Potential energyu

A basis vector of either the PC or IC basisu(I)A basis vector in the IC basisu(P )A basis vector in the PC basis

V

Visibility of interference patternv

Velocityvc

Speed of transverse waveguide motion

v0

Initial velocity of separating BEC wavefunctions in interferometry experimentw

Real number

X

Vector representation of an image in pixel basis

X̂

Vector representation of an image in pixel basis, in mean-deviation form

147
X̂

Matrix of images in pixel basis, in mean-deviation form

(x, y, z)Cartesian coordinates (exceptions noted in following items)(x0 , y 0 , z 0 )Secondary Cartesian coordinate systemx

Normally distributed detection noise (in Sec. 4.5)x(m)Position of BEC during mth half of transverse motionx̂mPosition of left edge of wire m

x0

Position of waveguide minimumx̄Deviation from minimum of traveling waveguideY

Coefficient matrix of either the PC or IC basis

Y (I)Coefficient matrix of IC basis

Y (P )

Coefficient matrix of PC basisyPC/IC coefficient related to interference signal (in Sec. 4.5)

y0Distance between microtrap minimum and generating wireZ

Complex impedancez

Vector of observed interference signals, sum of signal and noise (in Sec. 4.5)

z0Classical turning point along waveguide axis