A Circularly-Polarized Optical Dipole Trap and Other
Developments in Laser Trapping of Atoms
byKristan Lee Corwin
B.A., State University of New York at Buffalo, 1993
M.S., University of Colorado, 1997

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
1999

This thesis entitled:
A Circularly-Polarized Optical Dipole Trap and Other Developments in Laser
Trapping of Atoms
written by Kristan Lee Corwin
has been approved for the Department of Physics

Dr. Carl E. Wieman

Dr. Deborah S. Jin

Date

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

Corwin, Kristan Lee (Ph.D., Physics)
A Circularly-Polarized Optical Dipole Trap and Other Developments in Laser Trappingof Atoms
Thesis directed by Prof. Dr. Carl E. Wieman

Several innovations in laser trapping and cooling of alkali atoms are described.
These topics share a common motivation to develop techniques for efficiently manipulating cold atoms. Such advances facilitate sensitive precision measurements such asparity non-conservation and -decay asymmetry in large trapped samples, even when
only small quantities of the desired species are available.
First, a cold, bright beam of Rb atoms is extracted from a magneto-optical trap
(MOT) using a very simple technique. This beam has a flux of 5  109 atoms/s anda velocity of 14 m/s, and up to 70% of the atoms in the MOT were transferred tothe atomic beam. Next, a highly efficient MOT for radioactive atoms is described, inwhich more than 50% of 221 Fr atoms contained in a vapor cell are loaded into a MOT.
Measurements were also made of the 221 Fr 7 2 P1=2 and 7 2 P3=2 energies and hyperfine
constants. To perform these experiments, two schemes for stabilizing the frequency ofthe light from a diode laser were developed and are described in detail.
Finally, a new type of trap is described and a powerful cooling technique isdemonstrated. The circularly polarized optical dipole trap provides large samples ofhighly spin-polarized atoms, suitable for many applications. Physical processes thatgovern the transfer of large numbers of atoms into the trap are described, and spinpolarization is measured to be 98(1)%. In addition, the trap breaks the degeneracy of
the atomic spin states much like a magnetic trap does. This allows for RF and microwave cooling via both forced evaporation and a Sisyphus mechanism. Preliminary
application of these techniques to the atoms in the circularly polarized dipole trap has

iv
successfully decreased the temperature by a factor of 4 while simultaneously increasingphase space density.

Dedication

To my loving parents,
Allen George Corwin
andJulie Ann Grastorf Corwin.

Acknowledgements

Many, many people have helped make my graduate school experience a rich andrewarding one. Although it is impossible for me to thank them all, here I attempt tomention those who have had the greatest influence on my research at CU.
I have been very fortunate to learn atom trapping from a master, Carl Wieman.
He brings an honest and direct approach to both people and science, which I appreciate.
And I thank him for the mentoring me on everything from preparation for oral exams
to personal and professional advice on being a good scientist. When he adds this document to his collection, I hope he’ll say (as on the day he found the source of someparticularly annoying noise) “Superadvisor strikes again!!!”I have also had the pleasure to work with many talented postdocs. The first wasMichelle Stephens, who patiently showed me around JILA and taught me importantskills such as tuning a Ti:Sapph laser, making dry film coatings, and drilling holes.
Even after she left JILA, her continued support has been very much appreciated. After a couple months on my own, I was very happy to welcome Zheng-Tian Lu to theexperiment. Lu always encouraged my complete involvement in the experiment, andinspired me by his example to work efficiently and responsibly, always with the final
goal in mind. His love of science is contagious, and I am certain that he is now an
excellent advisor for his own students and postdocs at Argonne National Labs.
Simon Kuppens joined the experiment about 1 1/2 years ago. At that time, I
was certain that I didn’t need a postdoc, much like the eldest VanTrapp daughter invii
the Sound of Music who asserted, “I don’t need a governess!” But Simon showed meotherwise. He brought to the experiment a mathematical approach that was crucial tothe unraveling of the FORT loading puzzle. In addition, he became a close friend. Oncea visitor asked if we always run the experiment with so much laughter, and the answer
was... yes! I already miss him since his return to Holland.
The experiment is now in good hands. Kurt Miller has been doing great work on
the experiment for over two years. The majority of the evidence for cooling is largely
due to his efforts – we wouldn’t even know the FORT temperature without him. As hetakes over the experiment, he’ll have excellent help from Stephan Dürr, who only joined
the group 6 weeks ago but is already making big contributions. I thank Stephan for his
helpful comments on the cooling mechanisms and especially Landau-Zener transitions,and I’m sorry I won’t have a chance to work more with him.
Many other people have contributed directly to the work described in this thesis.
The LVIS experiment sprung from an idea of Mike Anderson, followed by an astuteobservation of Mike Renn. In addition, the DAVLL lock was developed with help from
Carter Hand and Ryan Epstein. And Tasha Fairfield contributed to the development of
the cavity lock.
The francium experiment would have been impossible without the source andhard work provided by the Berkeley team: Tim Dinneen, Harvey Gould, and Jason
Maddi. And prospects for finishing the experiment looked dim until Kurt Vogel rodein on a shining Ar+ laser (from Jan Hall) and saved the day. Thanks also to JILA for
providing us a new Ar+ laser tube, days before the final francium source arrived. Thatexperiment taught me how close colleagues can become when working on a sourcewith a 1 month decay time; I miss the sense of excitement and camaraderie, but not
the sleep deprivation. Several members of the JILA shops made major contributions
to the Fr experiment: Dave Alchenberger constructed the vacuum chamber and thehigh-precision oven manipulator, Hans Rohner built the beautiful glass cell and helped

viii
us coat it, and Terry Brown build the high-speed electronics that made the precise meter measurements possible. John Smedley, a JILA visiting fellow, also worked on
experiment construction.
Before the FORT project, I had an opportunity to bridge the gap between the oldand new teams on the “Fiber project,” under the supervision of Dana Anderson, Eric
Cornell, and Carl Wieman. Elizabeth Donley showed me the ropes, and I passed things
on to Eric Abraham, Brian DeMarco, and Dirk Müller. I enjoyed working with Brianand Dirk as new grad students, back when THEY had asked ME questions; it seems theroles are now largely reversed. I also appreciated Eric A.’s guidance at that time, evenafter I left the project in capable hands to return to the circularly polarized FORT.
The circular FORT project was originally conceived by Dong Hyung Cho, who
has continued to participate in the project during his visits to Boulder. He also graciously hosted me at his home institution, Korea University, where the collaboration
continued. Timothy Chupp also contributed to the studies of FORT loading while he
was a JILA visiting fellow in 1997-98. I appreciate Tim teaching me to take the time tostop and think on the “rare” occasions when we did not immediately understand our experimental results. I also enjoyed working and playing soccer with him. Jake Roberts
did a brief stint on the project between the Cs parity violation measurement and theBose-Einstein Condensation measurement, and his contributions then and throughout
the experiment have been very valuable. Neil Claussen has also contributed to both theearliest and most recent stages of the FORT experiment, from modeling the loading of
the FORT after its initial observation to providing us with a high-power laser systemthat he personally constructed (the MOPA).
I also want to thanks all the members of the Wieman group who contributed
greatly to the stimulating and supportive atmosphere. The old gang includes Steve
Bennett, Eric Burt, Rich Ghrist, Nate Newbury, Chris Myatt, and Chris Wood. Present
members include Nick Cizek, Simon Cornish, and John Essick. Undergraduates Dan

ix
Lowell and Travis Andrews also provided a lot of assistance in the labs.
The Wieman-Cornell-Jin “family” provides a wonderful environment in whichto do science. Eric Cornell has served as my additional advisor many times, and I appreciate his ideas and his support. Debbie Jin often has helpful insights during groupmeeting, and I thank her also for a careful reading of this manuscript with many usefulsuggestions. Other current “family” members who continue to make this a great group
include: Brian Anderson, Ian Coddington, Paul Haljan, Heather Lewendowski, Scott
Papp, Marco Prevedalli, Malcom Rickard, Peter Schwindt, and Dwight Whitaker. Previous members who have moved on to great things include Anne Curtis, Harold Gauck,
Tim Gfroerer, David Hall, Sergio Muniz, and especially Mike Matthews. Jason Ensher
deserves special mention, since he and I have been in school together 10 years, both at
UB and CU, and he has always both challenged and encouraged me.
JILA has many staff members, besides those already mentioned, who have mademy graduate career a lot easier or better. Fran Haas, Rhonda Hickey, Pam Leland, Karen
Melcher and now Linda Morris have helped keep the group and JILA running smoothly,
and they have provided me personally with a lot of support and encouragement. In the
Scientific Reports Office, Marilee Degode has improved many of the papers appearing
here, as has Laurie Kovalenko through the scientific writing course. In the Electronics
Shop, Paul Beckingham taught me his Survival class and helped me with many circuits,
always keeping me on my toes until we were “home and dry.” James Fung-A-Fat andMike Whitmore also helped us out a lot. I also thank all the instrument shop members,
including Blaine Horner, John Andru, and Hans Green. Among the computing staff,
thanks to Alan Dunwell, Joel Frahm, Chela Kunaz, and Peter Ruprecht. Also to the
supply office staff: Maryly Dole, Ed Holliness, Brian Lynch, and David Tegart. The
rest of the JILA staff has always been very helpful, and I thank you all.
My experience during the early years of graduate school was shaped predominantly by members of my class. I had an opportunity to work a problem with each andxevery one of them. I cannot list all my classmates here, but they include Brian King,
Shawn Kraut, Sarah Makoski, Jim Mooney, Keith Nordstrum, Arnold Leitner, Darren
Link, Harold Parks, Brad Paul, Orion Poplawski, Shuka Schwarz, Eric Vaandering, andOleg Vdovin. Thanks to the faculty at CU, including my committee members Katy
Garmany, Steve Pollock, and Murray Holland. Also Jinx Cooper, Alan Gallagher, andTrisha Rankin, among others.
I have been blessed with many dear friends over the last six years who reminded
me of life outside of cold atoms. Thanks to Sara Cho, Hilary Eaton, Melissa Fu, Dana
Jandzinski, Kim King, Heather Robinson, Sarah Parks, Sarah Roberts, and many others.
Special thanks to Josh Long, whose love and understanding have been very important
to me the last couple years. My extended family has also been very supportive. Finally,
I thank my mom, dad, and sister Kelly for encouraging me to come to graduate school
so far from home, for the long phone calls and many visits, and their enthusiasm formy success. I couldn’t have done it without them.

xiContents

Chapter
1 Introduction

1

1.1

Atomic Beams and the Development of Neutral Atom Traps . . . . . .

1

1.2

Trapping Radioactive Elements . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Optical Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Cooling in Dipole Potentials . . . . . . . . . . . . . . . . . . . . . . .

6

2 A Low-Velocity Intense Source of Atoms from a Magneto-Optical Trap

8

2.1

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

8

2.2

Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3

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

9

2.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5

2.4.1

Optimum Detuning . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.2

Longitudinal velocity . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.3

Collimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.4

Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Efficient Collection of 221 Fr into a Vapor Cell Magneto-optical Trap

17

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

xii
3.3

3.4

3.5

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1

Fr Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.2

The Magneto-optical Trap . . . . . . . . . . . . . . . . . . . . 19

3.3.3

Sensitive Detection . . . . . . . . . . . . . . . . . . . . . . . . 19

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.1

Trapping Efficiency . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.2

Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Spin-Polarized Atoms in a Circularly Polarized Optical Dipole Trap

25

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.4

4.3.1

The Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3.2

The Pockels Cell and Polarization Issues . . . . . . . . . . . . 30

4.3.3

Diagnostics and Loading . . . . . . . . . . . . . . . . . . . . . 31

The Elliptically and Circularly Polarized FORTs . . . . . . . . . . . . . 33
4.4.1

Loading Atoms into the Circular FORT . . . . . . . . . . . . . 33

4.4.2

Elliptically Polarized FORT Decay Rates . . . . . . . . . . . . 34

4.4.3

Model of Lifetime vs. Ellipticity . . . . . . . . . . . . . . . . . 35

4.4.4

FORT  > D1 . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5

Spin-Polarization Demonstration . . . . . . . . . . . . . . . . . . . . . 38

4.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 The Physics of Loading an Optical Dipole Trap

40

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2

The FORT potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

xiii
5.3.1
5.4

Loading and diagnostics . . . . . . . . . . . . . . . . . . . . . 44

Dynamics of the loading process . . . . . . . . . . . . . . . . . . . . . 45
5.4.1

Hyperfine repump intensity and detuning . . . . . . . . . . . . 48

5.4.2

Primary MOT light intensity . . . . . . . . . . . . . . . . . . . 50

5.4.3

The role of MOT detuning . . . . . . . . . . . . . . . . . . . . 52

5.4.4

Alignment with respect to the MOT . . . . . . . . . . . . . . . 53

5.4.5

Dependence on FORT depth . . . . . . . . . . . . . . . . . . . 54

5.4.6

The effect of elliptically polarized FORT light . . . . . . . . . . 56

5.5

Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.6

Trap Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.7

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.7.1

Analysis of density dependent losses . . . . . . . . . . . . . . . 59

5.7.2

A model for the FORT loading rate . . . . . . . . . . . . . . . 62

5.8

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.9

Enhanced and quasi continuous loading . . . . . . . . . . . . . . . . . 67

5.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6 Cooling in the Circularly Polarized Optical Dipole Trap

70

6.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2.1

Sisyphus Cooling Mechanism . . . . . . . . . . . . . . . . . . 71

6.2.2

Repumping with Spontaneous Scattering . . . . . . . . . . . . 77

6.2.3

Cooling Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2.4

Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . . 81

6.2.5

Heating Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 82

6.2.6

Ultimate Temperatures . . . . . . . . . . . . . . . . . . . . . . 82

6.2.7

Cooling with a Pulsed Drive . . . . . . . . . . . . . . . . . . . 83

xiv
6.3

6.4

6.5

Experimental Implementation . . . . . . . . . . . . . . . . . . . . . . 84
6.3.1

RF considerations . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3.2

MOPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3.3

Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4.1

Two-Photon Stimulated Raman Transitions . . . . . . . . . . . 87

6.4.2

Cooling with Stationary Frequency . . . . . . . . . . . . . . . 87

6.4.3

Cooling with Swept Microwave Frequency . . . . . . . . . . . 91

6.4.4

Heating Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Bibliography

94

Appendix
A Dichroic Atomic Vapor Laser Lock102

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.2 Diode Laser Frequency Stabilization . . . . . . . . . . . . . . . . . . . 102
A.3 The Dichroic-Atomic-Vapor Laser Lock Signal . . . . . . . . . . . . . 104
A.4 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.5 Characterization of Frequency Stability . . . . . . . . . . . . . . . . . 107
A.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B Stabilization of a Diode Laser at an Arbitrary Frequency

110

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

xv
B.3.1

Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

B.3.2

Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

B.3.3

Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

B.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
B.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xvi

TablesTable
3.1

Various rates, fluxes, and efficiencies for the transfer of 221 Fr from thesource to the trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2

The hyperfine constants and wavenumbers of 221 Fr. . . . . . . . . . . . 23

6.1

Repumping probability from the j2; +1 > state. . . . . . . . . . . . . . 78

6.2

Repumping probability from the j2; +1 > state. . . . . . . . . . . . . . 80

xvii

FiguresFigure2.1

Schematic of the LVIS system. . . . . . . . . . . . . . . . . . . . . . . 10

2.2

A typical longitudinal velocity distribution. . . . . . . . . . . . . . . . 12

2.3

The average longitudinal velocity and flux as a function of the forcing
laser intensity and detuning. . . . . . . . . . . . . . . . . . . . . . . . 13

2.4

The average longitudinal velocity and spread as a function of the forcing beam polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5

The measured divergence angle as a function of the collimation length z. 15

3.1

A diagram of the 221 Fr trap setup. . . . . . . . . . . . . . . . . . . . . 20

3.2

The atomic level diagram of 221 Fr. . . . . . . . . . . . . . . . . . . . . 21

3.3

The number of atoms loaded into a 221 Fr trap vs. time, showing a trapping efficiency of 56(10)%. . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1

Schematic level diagram for 85 Rb. . . . . . . . . . . . . . . . . . . . . 26

4.2

AC Stark shift U () in the focus of a circularly polarized Gaussian laser
beam for both hyperfine levels. . . . . . . . . . . . . . . . . . . . . . . 27

4.3

Maximum FORT Potential depth as a function of FORT wavelength. . . 28

4.4

Schematic of the experimental apparatus. . . . . . . . . . . . . . . . . 29

4.5

FORT laser beam polarization as a function of the voltage applied to
the Pockels cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

xviii
4.6

Number of atoms stored in a linearly polarized FORT as a function oftrapping time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.7

Dependence of the FORT exponential loss rateon the ellipticity  ofthe FORT laser polarization. . . . . . . . . . . . . . . . . . . . . . . . 34
4.8

Exponential decay rate

vs. FORT peak intensity for three different

FORT polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.9

FORT decay curve with FORT  = 798:4 nm. . . . . . . . . . . . . . . 37

4.10 Demonstration of spin-polarized atoms in the circular FORT. . . . . . . 39
5.1

Number of atoms N in the FORT as a function of time spent loading. . 41

5.2

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3

Number of atoms remaining in the FORT as a function of time for  =782:5 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4

Number of atoms loaded in the FORT as a function of FORT loading
stage duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.5

Number of atoms in the FORT vs time, both with and without MOT
light on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.6

FORT loading rate R0 and loss coefficient L0 as a function of hyperfine
repump light intensity. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.7

Number of atoms in FORT as a function of MOT hyperfine repump
detuning for different repump intensities. . . . . . . . . . . . . . . . . . 50

5.8

FORT loading rate R0 (a) and loss coefficient L0 (b) as a function ofprimary MOT light intensity, for a repump intensity of 5 W=cm2 . . . . 51

5.9

FORT loading rate R0 and loss coefficient L0 as a function of the primary MOT light detuning, M . . . . . . . . . . . . . . . . . . . . . . . 52

5.10 Optimum MOT detuning versus FORT depth. . . . . . . . . . . . . . . 53

xix
5.11 Number of atoms loaded into the FORT as a function of trap depth for
two different beam waists w0 . . . . . . . . . . . . . . . . . . . . . . . . 54
5.12 FORT Loading rate R0 and loss rate L0 as a function of trap depth U0 . . 55
5.13 Number of atoms in the FORT as a function of loading time for threedifferent ellipticities of the FORT light. . . . . . . . . . . . . . . . . . 56
5.14 Illustration of ballistic expansion and temperature measurement. . . . . 57
5.15 Temperature of the atoms in the FORT as a function of trap depth, for

w0 = 26 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3

5.16 Approximate intramolecular potential for Rb, with C3 = 71 Å eV. . . . 62
5.17 Radiative escape loss rate coefficient

calculated as a function of thetrap depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.18 Comparison of loading model with measured results. . . . . . . . . . . 64
5.19 Equipotential contours of the FORT. . . . . . . . . . . . . . . . . . . . 65
5.20 Demonstration of enhanced loading using a shadowed repump beam. . . 68
6.1

Two mechanisms for Sisyphus cooling. . . . . . . . . . . . . . . . . . 73

6.2

Effective Rabi frequency/(2 ) for the two-photon stimulated Raman
transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.3

Landau-Zener transition probability in the circular FORT. . . . . . . . . 76

6.4

Fraction of energy removed in one cooling cycle as a function of initialenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.5

Attenuation of rf radiation by the vacuum chamber. . . . . . . . . . . . 85

6.6

Optimum modulation of the MOPA. . . . . . . . . . . . . . . . . . . . 86

6.7

Layout of the all-optical two-photon stimulated Raman cooling scheme.

6.8

Observation of loss-inducing two-photon Raman transitions. . . . . . . 89

6.9

Cooling with a fixed microwave frequency. . . . . . . . . . . . . . . . 90

88

6.10 Cooling with a swept microwave frequency. . . . . . . . . . . . . . . . 92

xx
6.11 Heating rate of atoms stored in the circular FORT. . . . . . . . . . . . . 93
A.1 Schematic of a DAVLL system. . . . . . . . . . . . . . . . . . . . . . . 103
A.2 Comparison of the signal from a saturated absorption spectrometer anda DAVLL system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.3 Origin of the DAVLL signal shape. . . . . . . . . . . . . . . . . . . . . 106
A.4 Measured beat frequency between two DAVLL systems over a 38 h
period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B.1 Schematic of the cavity locking system. . . . . . . . . . . . . . . . . . 112
B.2 Fractional transmission (calculated) of a cavity with a finesse of 40 for
two different wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . 113
B.3 A simple computer algorithm. . . . . . . . . . . . . . . . . . . . . . . 114

Chapter 1
Introduction

The laser trapping and cooling of atoms has been a rapidly growing field since
its inception in the 1970s. A variety of optical trapping and cooling forces have been
discovered and employed successfully on atomic gases. These technologies have madepossible a host of advances, including more accurate atomic clocks, atom lithography,
atomic interferometry, and the creation of quantum degenerate gases [1]. In addition,
several precision measurements are being performed using these techniques, including
the asymmetry of nuclear -decay, the electric dipole moment of the electron, andparity non-conservation in electronic transitions of atoms.
The precision measurements listed above require large samples of rare atoms.
The most sensitive measurement of parity non-conservation [2] required many gramsof Cs per week, and 350 hrs of data collection. Such large quantities of radioactive
elements are generally unavailable; production rates of 105 /s are typical [3]. Therefore, there is a need for efficient collection and manipulation of these atoms to make
improved precision measurements possible.
The work presented here represents a series of advances that have implications
for most of the topics mentioned above. The original motivation for these advances
was to develop tools for making a precision measurement on radioactive alkali atoms.
These advances therefore include an efficient technique for transferring atoms out of an
initial magneto-optical trap (MOT) into a slow, bright atomic beam. They also include
the efficient collection of radioactive Fr into a MOT. The majority of this work is dedicated to the description of a new type of laser trap that may prove useful for precision
measurements. The properties of the trap are discussed, as well as the mechanisms developed to transfer atoms into it as efficiently as possible. Finally, cooling mechanisms
in the trap are explored. While cooling may not be necessary for precision measurements, it is very important for achieving quantum degeneracy, and may make this trap
useful for attaining Bose-Einstein Condensation using purely optical techniques.
1.1

Atomic Beams and the Development of Neutral Atom Traps

Since the beginning of modern atomic physics, effusive atomic beams have playeda central role. The study of beams has led to the development of most of the tools in
the lab used to produce this work, the first part of which (Chapter 2)is itself a usefuladvance in the technique of beam control.

2
In the 1920’s, Stern and Gerlach first discovered the quantized nature of spin
angular momentum by observing the deflection of a beam of Ag atoms [4, 5]. In the
1930s and 40s, I. I. Rabi developed the concept of nuclear magnetic resonance in beams
of Na [6], laying the groundwork for the MASER and then the LASER, the principal
tool of atomic physics as it is practiced today. In general, these beams were made byheating samples of Ag or Na in a container with a small aperture. The resulting increasein vapor pressure in the container caused the atoms to exit the aperture, forming a beam.
Beams made this way were poorly collimated and moved very fast (on the order of 1000m/s). However, they became the basis of seven generations of atomic time standards in
Boulder, Colorado and around the world.
Forty years after Rabi’s work, and with an interest in improving the time standard, Phillips and Metcalf successfully decelerated a beam of Na atoms using laser
light and a magnetic field [7]. This technique, called the “Zeeman slower”, is still used
today. The key to the success of the technique is the application of a spatially varying
magnetic field that tunes the atoms into resonance with the counter-propagating laser
as they decelerate, in order to counteract the changing Doppler shift. Atoms leaving theoven with a high velocity and therefore a large positive Doppler shift absorb photons
from a laser directed toward the beam and tuned near to the atomic resonance frequency.
This laser exerted a “spontaneous scattering force” on the atoms. Each time the atom
absorbs a photon from the laser beam, its momentum decreases by the amount of momentum carried by the photon, ~k . The photon is then reemitted in a random direction.
Therefore, on average the atoms slow down as they move toward the laser beam until
they are out of resonance with the laser. But by reducing the applied magnetic field,
the atoms are shifted back into resonance with the lasers, absorb more photons, and arefurther slowed. Unfortunately, there is no cooling in the transverse direction, and infact the spontaneous emission of photons in random directions gives rise to heating in
the transverse dimensions. Thus as the longitudinal velocities approach the transverse
velocities ( 15 m/s), the atoms diffuse randomly and densities are therefore limited.
Another implementation of the slowed beam, such as the chirp-cooled beam [8], does
not require magnetic fields. Instead, the frequency of the laser light is varied in time, inorder to stay resonant with the atoms.
An important extension of the scattering force to three dimensions occurred whena so-called optical molasses was loaded with the chirp-cooled beam [9]. Optical molasses relies on the same spontaneous scattering force described above, but in three
dimensions. Laser beams from six directions intersect at the end of the slowed beam,and their frequency is detuned several linewidths from resonance. When an atom inthe molasses region moves toward one of the laser beams, it will be Doppler shifted
into resonance with that beam and absorb photons from it until the atom slows down inthat direction. The atom must then reradiate the photon in a random direction, which
results in heating. The Doppler cooling temperature limit, or “Doppler Limit” TD isthen set by a balance between the molasses cooling and the heating that results fromthe spontaneously emitted photons. In general, this limit depends only on the natural

3
linewidthof the atomic transition [10]:

TD =

~

2kB

where ~ is Planck’s constant over 2  and kB is Boltzman’s constant. A temperature
consistent with the Doppler limit was initially observed in a cloud of Na atoms at 240
K [9].
Several years later, Lett et al. [11] cooled Na to below the Doppler limit in an
optical molasses, to 43 K. No reasonable explanation of this surprising sub-Doppler
cooling existed until two groups [12, 13] proposed a polarization gradient coolingmechanism. They explained that the counter-propagating, orthogonally-polarized molasses beams form a standing wave. In this case, the polarization of the standing wave
changes every /4. When the polarization reverses, the direction of optical pumping
and also the polarization- dependent light shifts of the atom change. An atom tends tobe optically pumped into internal states with lower potentials as it travels through thestanding wave. The time lag between traveling into the new polarization and opticalpumping causes the atoms to be continuously pumped from deep potentials into shallower ones. Thus the atoms tend to move uphill more than down, and are therefore
cooled in this so-called “Sisyphus” mechanism.
The first neutral atom trap to use the spontaneous scattering force was made in
1987 by Raab et al. [14]. To turn an optical molasses into a magneto-optical trap
(MOT), the lasers must be circularly polarized and they must intersect in the center ofa quadrupole magnetic field. This field is zero in the center of the trap, and increases
in magnitude as atoms move away from this field zero. Now the scattering force canprovide spatial confinement as well as cooling. As an example, consider what happens
to an atom in one dimension as it moves to the left in the potential. The magnetic
field in this dimension is pointed away from the center, and the laser polarization isthen arranges so that  comes from the left and + from the right. As the atom
moves to the left, the magnetic field breaks the energy degeneracy of the atomic spinorientations. Thus m = 1 transitions shift to lower frequencies, into resonance withthe red-detuned laser beam coming from the left, and further out of resonance with the
+ beam from the right. Therefore the atom is always pushed back to the trap centerand spatially confined.
Later, it was shown that the MOT could capture atoms directly from a roomtemperature atomic vapor instead of a precooled atomic beam[15]. A vapor-cell MOT
is much simpler to construct that a beam-fed MOT. However, it has the limitation ofa background pressure of atoms being trapped. Thermal background atoms limit the
lifetime of the MOT and heat the sample [16, 17]. Therefore, when long lifetimes are
required, a two-chambered MOT is constructed. In this system, atoms are pushed out of
the MOT in the high-pressure chamber with a resonant laser beam, through a magnetic
hextapole atom waveguide formed with six permanent magnets, and into a second MOT
in a chamber of lower pressure[18]. In this way, the vapor cell MOT can be used inapplications requiring very long lifetimes, such as Bose-Einstein condensation[19].
Meanwhile, the quest to create a brighter, better atomic beam continued. The

4spontaneous scattering force techniques described above for three-dimensional confinement was employed to cool the two transverse dimensions. In this way, the atom
funnel was developed [20]. Similar two-dimensional compression was demonstrated
by other groups [21, 22, 23]. These systems often require a complicated apparatus orlaser configuration.
In Chapter 2, a novel method of creating a cold atomic beam is presented. While
making use of modern trapping techniques, the design is much simpler than that ofother cold beam demonstrations. Dubbed a low velocity intense source, or LVIS, it ismade by creating a small leak in a MOT through which atoms exit in a collimated beam.
Its simple design and implementation make it an excellent source for cold atoms. In
addition, it is a highly efficient source for an atomic beam, as the integrated flux in the
beam can be up to 70% of the loading rate of the MOT.
After the publication of the first LVIS results in 1996 [24], it has been used as asource of atoms for several applications. An LVIS beam has been used to load a second
MOT [25], with a transfer efficiency of 67  15%, limited only by beam divergence.
Williamson et al. [26] transferred 39 K and 40 K a distance of 35 cm from a modified
pyramidal trap [27] using an LVIS-like technique. Unfortunately, the transfer efficiency
was limited to 6%, perhaps due to poor collimation in the pyramid configuration. In
addition, atoms from the LVIS beam have been successfully guided in an optical fiber[28] and a magnetic waveguide [29].
1.2Trapping Radioactive Elements

Chapter 3 describes the efficient transfer of radioactive 221 Fr atoms from a roomtemperature vapor into a MOT. This represents a major advancement over the conventional vapor cell MOT technique described above. Typical vapor cell MOTs are
inefficient for two major reasons. First, the probability of an individual atom in thevapor to be captured by the MOT is not optimized, in part because the volume of thevacuum chamber far exceeds the capture volume of the MOT (ie. the region of overlapof all 6 laser beams). Second, the alkali atoms tend to stick to and react with metal andglass surfaces, so that the majority of the atoms of interest in the chamber are adsorbed
in a monolayer on the surface, and therefore cannot be trapped.
To circumvent the first problem, a redesigned vacuum chamber minimizes thechamber volume while maximizing the capture volume [30]. To solve the second, nonstick dryfilm coatings similar to those developed for optical pumping vapor cells areused to coat the trapping cell [31]. In addition, large, high-powered laser beams andoptimized detunings improve the capture probability [32]. By using very large laser
beams and high powers, along with a coated trapping cell almost completely illuminated with the trapping lasers, we demonstrated a 221 Fr trapping efficiency of over
50%. This represents a large improvement over previous radioactive trapping results.
In addition, this experiment offers the advantage of a laboratory source to produce Fr[33] instead of a particle accelerator.
Other groups have continued to pursue trapping of radioactive atoms since thedemonstration of the 221 Fr trap [34]. The Stony Brook collaboration has maintained an5active pursuit of spectroscopy in neutron-deficient isotopes of Fr (see for example Ref.
[35]). Large samples of radioactive 82 Rb have been trapped using a technique similar
to that described here, but with a trapping efficiency of only 0.3%. The poor trapping
efficiency was largely due to poor coating performance, most likely caused by inductive
heating of the foil to 800o C inside the cell, which damaged the dryfilm coatings. Usinga Zeeman slower, 40,000 atoms of 21 Na [36] have been trapped in preparation for a-decay asymmetry measurement.
1.3

Optical Dipole Traps

The optical dipole trap, another confinement technique developed in the 1980s,
is based on an effect quite different from the spontaneous scattering force describedabove. While the scattering force acts in the direction of laser propagation, the dipole
force acts in the direction of the gradient of the laser intensity. The optical dipole force
was first observed in 1978 (See Ref. [37]), when researchers co-propagated a tightlyfocused laser beam with an atomic beam. They observed that when the laser was tuned
close to the atomic resonance frequency, the laser beam caused deflection and focusingof the atoms. Thus the light exerted a force on the atoms perpendicular to the directionof laser propagation!
The energy shift of atoms in a light field is often referred to as the AC Stark
shift or the light shift, and the gradient of this energy shift gives the dipole force. The
dipole force can be understood simply in terms of the atom electric polarizability. In
free space, an atom in its ground state has no electric dipole moment. However, a static
electric field induces a dipole moment in the atom, reducing its potential energy. Therefore a gradient in electric field draws the atom toward the region of space with higher
field. The AC Stark shift arises in a similar fashion; the laser light is simply an electromagnetic field that reverses direction every few femtoseconds. If the laser frequency
is below the atomic resonance, the dipole induced in the atom can “keep up”, or stayin phase with the light, and atoms are drawn to regions of highest intensity. If instead
the light frequency is tuned above the atomic resonance, the dipole moment always
opposes the electric field, and atoms are expelled from regions of highest intensity.
The dipole force was used to demonstrate the first optical trap for atoms in 1986
[38]. In this experiment, about 500 Na atoms were confined at the waist of a focused
Gaussian laser beam. The laser beam contained about 220 mW, detuned about 650
GHz below the atomic resonance frequency and focused to a 10 m spot size. The
trap was loaded from an optical molasses by rapidly alternating between the opticalmolasses and the optical dipole trap. The trap laser detuning  was much larger than
the width of the transition, but not large enough to completely suppress the spontaneous
scattering force.
Soon after this experiment, the MOT was developed and could trap many more
atoms at much higher densities. Therefore dipole traps were generally abandoned until in 1993 when the far off resonance trap (FORT) was demonstrated [39]. This trap
employed more power and larger detunings, and captured a few thousand atoms. The
increased detunings allowed the much longer trap lifetimes than those of the original

6
dipole traps, in order to have strong dipole forces and simultaneously have low spontaneous scattering rates that cause negative effects. The dipole trapping force that actually
does the trapping scales as  1 , while the spontaneous photon scattering rate scales as 2 . Spontaneous scattering, although many orders of magnitude lower in the FORT
than the MOT, can still cause heating, decoherence, and scrambling of the internal
states. Thus even if atoms are spin-polarized before they are stored in the dipole trap,
the spontaneous scattering rate will tend to depolarize them [40].
Many other types of dipole traps have been developed that rely on large detunings
and high powers. Blue-detuned light traps are also used. They have the advantage that
atoms spend the majority of their time in the dark where they cannot absorb photons.
However, these typically shallow traps tend to be difficult construct and have large
volumes and unusual heating effects.
Chapter 4 and Ref. [41] presents a novel kind of dipole trap called a circular
FORT that takes advantage of an interesting aspect of the dipole force. When the dipole
trapping beam is circularly polarized, the value of the light shift depends on the internal
spin state of the atoms [42, 43]. Therefore, although we use small detunings of only a
few nm, the spontaneous scattering of the circularly polarized light always returns theatoms to the same internal spin state. Although heating and decoherence still result,the atoms are not depolarized. Therefore this trap makes an excellent source of spinpolarized atoms for precision measurements.
Demonstration of the circularly polarized FORT, or “circular FORT,” first requires the efficient loading of a large atom sample. Although many groups have described their specific techniques for loading atoms into a dipole trap, no one has offereda detailed understanding of the many physical processes that govern the loading of the
FORT. Therefore, Chapter 5 and Ref. [44] present a detailed study of the loading of
the FORT as a function of the parameters of the MOT. A simple model is developed forthe loading process and the loss processes unique to the FORT in the presence of the
MOT light. Use of this model led to techniques which increased the number of atoms
loaded into the FORT to 7106 . Samples of this size are sufficient to consider precision
measurements and novel dipole cooling schemes discussed below.
1.4

Cooling in Dipole Potentials

The spin states are non-degenerate in the circular FORT, much as they are inmagnetic traps. Magnetic traps confine atoms using potentials arising from the interaction of the atomic magnetic moment with the magnetic field. Because they do not relyon light fields, spontaneous scattering does not occur, and trap heating can be limited
to collisions with high temperature atoms in the vacuum chamber. In addition, twovery powerful cooling schemes have been demonstrated in the trap. The first is forced
evaporative cooling, in which the highest energy atoms of a thermal distribution are removed, leaving the cloud to return to thermal equilibrium at a lower temperature. Thismechanism reduces the temperature rapidly, with minimal loss of atoms.
Because a magnetic trap relies on the Zeeman effect to produce confinement,
different spin states of the atom have different dipole moments and therefore differ-7
ing strengths of confinement. Using two of these spin states, a Sisyphus cooling was
effectively demonstrated [45]. Atoms confined in a deeper potential were allowed tooscillate to their turning point, at which point they were transferred into the shallower
state. After waiting a quarter oscillation period for the atoms to reach the bottom of thepotential, they were transferred back to the deeper state. This powerful refrigeration
technique cooled the atoms to nearly the recoil limit (ie. the temperature of an atomwith the kinetic energy equivalent to 1 photon recoil). This mechanism is similar to
the Sisyphus cooling described earlier; the biggest difference is that in this case, thepotentials used for cooling are also used for confinement.
Both forced evaporation and this type of Sisyphus cooling are not feasible in traditional dipole traps. However, in the circular FORT the AC Stark shift splits the spinstates, and both these cooling mechanisms become possible. In Chapter 6 the preliminary results of cooling in the circular FORT invoking the above methods is described.
In this initial work, cooling from 280 K to 40 K is observed, due to one or both of themechansims described above. While substantial number loss accompanied this cooling,
optimization of the cooling procedure should improve these initially promising results.

Chapter 2
A Low-Velocity Intense Source of Atoms from a Magneto-Optical Trap

2.1

Introduction

Cold atomic beams are useful in a variety of applications: in ultra-high resolution
spectroscopy, as frequency standards, and in studies of cold atomic collisions [46]. An
intense beam of cold atoms is valuable for atom interferometers [47], particularly thosesensing rotational and gravitational effects. A cold atomic beam coupled into an atom
fiber guide [48] will provide much larger guided atom flux. Current experiments studying Bose-Einstein condensation [1] and trapping of radioactive atoms for fundamental
symmetry tests [49, 30] require a system of two magneto-optical traps (MOTs) with thecapturing and the measurement processes separated in space. Our experiment reveals a
simple and efficient way to transfer atoms between two MOT’s via a cold atomic beam.
Many examples of cold atomic beams have been demonstrated, such as Zeeman
slowers [50, 7], chirped-cooled beams [8], and beams slowed by broadband light [51] or
isotropic light [52]. These beams all experience serious transverse diffusion effects as<
the atoms are slowed to very low velocities (15 m/s). This causes a loss of atoms andreduced collimation. To counteract these effects, slow atoms are passed through a twodimensional MOT, or atom funnel, to compress and cool them[20, 22, 23]. To date, thebrightest slow beams employ this technique and include a beam of Na atoms traveling
2.7 m/s with a brightness of 3  1011 atoms/sr/s[20], and a beam of Ne traveling 19
m/s with a brightness of 3  1010 atoms/sr/s[22]. In contrast, we have created a lowvelocity intense source (LVIS) of atoms with a brightness of 5  1012 atoms/sr/s. Thisatomic beam, the brightest beam of atoms moving slow enough to be easily captured by<a MOT (20 m/s), offers the advantage of simplicity, for it is made merely by adding asmall modification to the simple vapor cell magneto-optical trap (VCMOT)[15].
In this Chapter we report a detailed study of the LVIS beam. We observed thelongitudinal velocity distribution by making time-of-flight studies; and we obtainedtransverse velocity distributions and absolute measures of both pulsed and continuous
brightness by adding atomic beam fluorescence measurements. Our model of the system explains most of our results quantitatively, and all of them qualitatively.

9
2.2

Description

The LVIS system is nearly identical to a standard VCMOT with six orthogonalintersecting laser beams. The only difference is that one of the six trapping laser beams
has a narrow dark column in its center. Atoms in the low-velocity tail of the thermal
vapor enter the VCMOT trapping volume and slow down. After they diffuse into thetrap center, they enter the central column (“extraction column”) and are accelerated outof the trap by the counter-propagating laser beam (“forcing beam”). The velocity of theextracted atoms is determined by the number of photons they scatter from the forcing
beam before leaving the trap. A key feature of this scheme is that these extracted atoms
are continuously apertured by laser light along the beam. Those diverging atoms thatmove out of the extraction column are recaptured and returned to the trap center. Thismechanism of recycling the diverging atoms provides a very efficient way of transferring trapped atoms into a collimated atomic beam.
The atomic beam flux is determined by the capture rate of the VCMOT. In a
conventional VCMOT, the equilibrium number of trapped atoms Neq is the steady-statesolution to the differential equation

N_ = R N=rc :Then Neq = R/rc , where R is the capture rate and rc is the collisional loss rate[15, 32].
In LVIS, most of the atoms are “lost” into the atomic beam. An additional term is thenadded to describe the LVIS beam flux F = N/rt , where rt is the rate of transferring atoms
into the beam. The steady-state solution to

N_ = R N=rc

N=rt

is given by F=R/(1+rc /rt ). Typically rc rt , e.g., 1/rc =1.0 sec and 1/rt =30 msec, soFR.
The collisional loss also affects atoms in the beam. Since it takes much lessenergy to knock an atom out of the LVIS beam than out of a VCMOT, the beam collisional loss rate is roughly 5 times that of the VCMOT. The trade-off between collection
rate of the VCMOT and collisional loss rate from the LVIS beam limits the flux anddetermines the optimum thermal vapor pressure ( 1  10 7 Torr). The low-energy
collisional cross section in Ref. [53] agrees well with the dependence of collimationand flux on Rb vapor pressure that we observe with LVIS.
2.3

Experimental Setup

A schematic of the 87 Rb LVIS apparatus is shown in Fig. 2.1. A Ti:Sapphirering laser provides about 500 mW of “trapping” light at a typical frequency 30 MHz
detuned from the 5s 2 S1=2 (F=2)!5p 2 P3=2 (F0 =3) transition. A diode laser supplies 20
mW of “repump” light, tuned to the 5s 2 S1=2 (F=1)!5p 2 P3=2 (F0 =2) transition. As in aconventional VCMOT, the trapping beam is split into three beams which intersect inside
a vacuum chamber containing Rb vapor. Each of the three beams is reflected back in10

Figure 2.1: Schematic of the LVIS system. Large shaded arrows represent the 4 cm
diameter trapping laser beams. A repump laser (not shown) illuminates the trapping
volume up to the edge of the retro-optic, which has a small hole and is placed inside
the vacuum chamber. This hole, a distance z from the trap center, creates an extraction
column through the trap center and causes atoms to accelerate out of the VCMOT. A
standing-wave light field 30 cm downstream forms the detection region. The plug is athin beam of trapping laser light; when present, it prevents atoms from leaving the trap
via the atomic beam.

the opposite direction to make six counter-propagating laser beams in a retro-reflecting
configuration. The VCMOT beams are relatively large ( 4 cm diameter) in orderto maximize R. To produce the extraction column, a millimeter sized hole is drilled
through the center of one of the retro-reflecting assemblies which consist of a quarterwave plate and mirror 1 . The atomic beam is extracted from the trapping region throughthis hole. A pair of anti-Helmholtz coils generates the quadrupole magnetic field forthe trap, with a gradient of 5 G/cm along the atomic beam direction. To position thetrap center in the extraction column, the point of zero magnetic field is moved with aset of orthogonal magnetic shim coils. In normal operation, the plug beam is blocked
by a mechanical shutter. When making measurements, the plug beam is unblocked sothat the atoms are forced out of the extraction column and returned to the center of
the trap. This capability of quickly turning the atomic beam on and off allows us tomeasure the longitudinal velocity distribution, and to run LVIS in a pulsed mode. With
a CCD camera and a photo-diode, we monitor the fluorescence emitted when the atoms
cross the detection region. This allows us to measure the flux, spatial distribution, andvelocity distribution of the atomic beam.
1

A hole was drilled in the quarter wave plate and then the back surface was coated with a reflecting
layer of gold.

11
2.4

Results

2.4.1

Optimum Detuning

Given the trap parameters described above and in Fig. 2.1, we found that thetrapping laser frequency which maximizes the LVIS beam flux is 5 detuned (whereis the natural line width) from the cycling transition, while the detuning that maximizes
N in the normal VCMOT is 3.2 . This difference presumably occurs because thetransverse light beams can heat and, if there are any imbalances, deflect the atomic
beam as it exits the trap. At lower detunings, the scattering rates and hence thesedeleterious effects increase.
2.4.2

Longitudinal velocity

The longitudinal velocity distribution in the LVIS beam, as shown in Fig. 2.2,
is measured by the time-of-flight method. Typically, we observe v15 m/s, consistent
with our simple model based on a calculation of the photon scattering rate from theforcing beam. In our model the acceleration begins when an atom enters the extraction
column. As the atom is accelerated, the scattering rate slows due to Doppler shift andZeeman shift. Scattering and acceleration cease when the atoms finally leave the regionof repump light. Figure 2.3 shows the dependence of the longitudinal velocity on theintensity and frequency detuning of the forcing beam. Both plots indicate that the final
velocity increases with the scattering rate in a manner consistent with our model. Notethat the range of useful velocities is limited because the flux decreases rapidly when theintensity and detuning are far from those which optimize the trap capturing process.
A narrow longitudinal velocity distribution is usually desired in the applicationsof cold atomic beams. The velocity spread, about 2.7 m/s FWHM, is much larger thanthe Doppler cooling limit of 0.12 m/s. The velocity spreads due to a random distribution over magnetic sublevels and statistical fluctuations in the number of scatteredphotons were both estimated to be 0.7 m/s. We believe the dominant contribution to
the longitudinal spread arises because the atoms enter the extraction column within thetrap at different positions along the beam axis, and are therefore accelerated over different distances. A calculated velocity and spread match the experimentally observed
values (v=14 m/s, FWHM=2.7 m/s) if we assume that the acceleration distance covers
the range 2.2 to 3.4 cm. This is reasonable since in this case z is 2.5 cm (see Fig. 2.1),
and the atoms trapped in the VCMOT with the plug beam unblocked form a cloud 1
cm in diameter.
The fractional FWHM of the longitudinal velocity distribution depends on theforcing beam’s polarization. Figure 2.4 shows that the fractional FWHM can be minimized by making the polarization significantly elliptical. For circularly polarized light,
the Zeeman shift causes an atom to feel an acceleration which is much larger on the
upstream side of the trap center than on the downstream side. This principle is essential
to the operation of a MOT, but in a LVIS, atoms entering the extraction column on the
upstream side experience a larger acceleration than those entering on the downstreamside. However, an elliptically polarized forcing beam makes the accelerations more

12

Figure 2.2: A typical longitudinal velocity distribution. In this case, the average velocity is 14 m/s and the FWHM is 2.7 m/s. This curve is made by first recording the shapeof the time-of-flight (TOF) signal. The derivative of the TOF signal is then taken withrespect to time, and time is converted into velocity using the known distance between
the plug and the detection region. This yields the derivative of atom flux with respect
to velocity, or the longitudinal velocity distribution.

nearly equal on the upstream and downstream sides. This results in a smaller final velocity spread. For angles <20o and >70o , the increased spatial spread in the trapped
atom cloud outweighs this decreased variation in the acceleration.
2.4.3

Collimation

Many factors contribute at some level to the transverse collimation. Initially we
expected transverse cooling and focusing within the extraction column to dominate. Instead, the measurements described below show that the transverse velocity distribution
is primarily determined by a simple geometrical collimation mechanism. Although it issimilar to the transverse velocity distribution of a conventional atomic beam collimatedwith physical apertures, the LVIS beam benefits because the apertured atoms are recycled. In LVIS, the collimation length (z) extends from the point where atoms enter the
extraction column to the mirror. The divergence angle of the atomic beam,  , is givenby  d/z, where d is the diameter of the extraction column. The spatial profile is consistent with the triangular profile expected from a geometrical collimation mechanism.
When the extraction column was produced by placing the retro-optic with a hole at 2.9
cm from the trap center, the observed divergence angle (36 mrad) agrees well with d/z(40 mrad).
To further study this collimation mechanism, the retro-optic with a hole was re-13

Figure 2.3: The average longitudinal velocity and flux as a function of (a) the forcing
laser intensity (with detuning at 32 MHz) and (b) the detuning (with I=38 mW/cm2 ). In
both cases the final velocity increases with increasing scattering rate from the forcing
beam, while the fractional spread remains nearly unchanged.

14

Figure 2.4: The average longitudinal velocity and spread as a function of the forcing
beam polarization. The quarter-wave plate at 45o gives circularly polarized light. Below
10o and above 80o the flux falls rapidly because the trap capture rate decreases, but theflux varies by less than 30% between 10o and 80o . Also, the mean velocity has a much
weaker dependence on polarization than the fractional spread.

placed by a standard retro-optic outside the vacuum chamber. We inserted a piece ofglass with opaque spots of various sizes into the laser beam in front of the retro-optic
to create the extraction column. To vary the collimation length, we varied the distance(z) over which the repump laser illuminated the atomic beam. The divergence scaledwith d/z over a wide range of conditions. Angle vs z is shown in Fig. 2.5. Note thatwhile  scales as 1/z, the measured values are consistently smaller than the opaque spot
diameter divided by z. This is presumably due to diffraction of light into the extraction
column which effectively makes d smaller than the diameter of the opaque spot.
The tightest collimation was achieved with our maximum collimation length (30cm) and a 1.6 mm diameter opaque spot. We observed a divergence angle of 5 mrad,
implying a transverse temperature of 20 K. This configuration requires careful alignment, and the atomic beam must be sent in a vertical direction to prevent gravity frompulling the atoms out of the extraction column.
To better understand the beam collimation, we replaced the conventional MOT
field gradient with a quasi-two-dimensional MOT which had a magnetic field gradientof 7 G/cm in the transverse direction and <1 G/cm in the longitudinal direction. We
kept all other conditions the same. The atomic beam width changed from 1.1 mm to0.65 mm when measured 3 cm above the trap center, but did not change when measured 30 cm above. Thus the quasi-two-dimensional configuration produced transverse
focusing but not cooling.

15

Figure 2.5: The measured divergence angle as a function of the collimation length z.
The divergence angle decreases, consistent with a geometrical collimation mechanism.
The flux density (in arbitrary units) increases until z6 cm, indicating that the atoms arebeing captured back into the trap and recycled. For these measurements, the retro-optic
was removed to the outside of the chamber, and the extraction column was generated
by a 1.6 mm opaque spot such that the atoms were accelerated vertically.

2.4.4

Flux

We measured the absolute atom flux in the LVIS beam and determined the atomtransfer efficiency by comparing this flux with the capture rate of the VCMOT. The
capture rate of the trap was determined from measurements of N and rc with the plug
beam in place. Our measurements indicated that essentially 100% of the atoms weretransferred into the atomic beam for typical values of z. However, the fraction extracted
through the hole in the retro-optic into a field-free region varied. The highest flux weachieved was 5109 /s, with an extraction efficiency of 30% ( =30 mrad, d=0.7 mm, z= 2.7 cm) which was limited by light scattering around the hole edges. By making acleaner hole through the retro-optic, ( = 36 mrad, d=0.8 mm, z=2.0 cm) we increased
the efficiency to 70%, but by that time, our deteriorating Ar+ laser tube allowed us totrap an order of magnitude fewer atoms. However, with this 70% efficiency and 500
mW of Ti:Sapphire laser power, we expect to achieve a beam flux >1010 /s. When operated at lower power the LVIS flux drops in proportion to the reduced capture rate for
the MOT. However, the beam collimation and velocity remain nearly the same so LVIS
would still produce a nice beam with low power diode lasers. Finally, we observed amuch higher peak flux when the LVIS system was operated in a pulsed mode. In thesegeometries, the VCMOT empties in 50 ms, providing nearly the same time-averaged
flux but ten times the peak flux and brightness.

16
The flux and flux density depend on the collimation angle and geometry of the
LVIS setup in the manner predicted above. Although we achieved 70% extraction efficiency when  =36 mrad, we only achieved a transfer efficiency of 20% when  =5
mrad. Figure 2.5 shows the tradeoff between flux density and collimation of the atomic
beam. Recycling causes the flux density (number s 1 cm2 ) at the detection region toincrease while the divergence decreases. Total flux decreases beyond 4.5 cm because
for a collimation this tight, the atomic beam transverse temperature becomes comparable to the temperature of atoms in the VCMOT. With tight collimations, an atom must
make many more attempts to be successfully transferred into the atomic beam. This
makes rc /rt larger, decreasing the flux as predicted.
2.5Conclusion

We have created a slow, bright beam of cold atoms. We have observed the optimum detuning, beam collimation, and flux, and find them consistent with a geometricalcollimation mechanism. A quasi two-dimensional quadrapole magnetic field did not
improve the collimation. Finally, up to 70% of the atoms loaded into the MOT were
continuously extracted into the beam, consistent with a recycling mechanism. The
simplicity, brightness, and versatility of LVIS will make it useful in a wide range ofapplications.

Chapter 3
Efficient Collection of 221 Fr into a Vapor Cell Magneto-optical Trap

3.1

Introduction

There has recently been considerable activity in the field of laser trapping of
short-lived radioactive atoms. While a wide range of isotopes are being pursued, laser
trapping of 21 Na [49], 37;38 K [54], 79 Rb [3], and 209;210;211 Fr [55] atoms has been experimentally realized. Efficient optical trapping is essential for creating large samples ofrare atoms. Such samples are very appealing for tests of the standard model including
atomic parity non-conservation (PNC), the electric dipole moment (EDM), and decay[49]. Francium, which has no long-lived isotopes, is particularly interesting for thesetests, because calculations predict PNC amplitudes and EDM enhancements to be 10
times larger in Fr than Cs [56, 57]. In this letter we demonstrate efficient trapping of
221 Fr. The general approach should work as well with any alkali isotope, and shouldmake tests of the standard model possible in rare trapped atoms.
Various techniques have been developed to collect atoms into traps [7, 8], butsince short-lived radioactive atoms are only available in limited quantities, improving the optical trap collection efficiency is a central issue. The highest efficiency yet
demonstrated used a vapor cell magneto-optical trap (VCMOT) [15] in a glass cellcoated with dryfilm. Using coated cells, Stephens et al. [30] and Guckert et al. [58]have demonstrated respectively 6% and 20% collection efficiencies of stable cesium.
Similar techniques have been applied to trapping radioactive species of K, Rb, and Fr
atoms [54, 3, 55], but with far lower trap efficiencies. We have created a highly efficient (56%) 221 Fr VCMOT in a coated cell and used it in spectroscopic measurements
on 221 Fr.
A conventional VCMOT [15] traps a very small fraction of the available atoms.
To obtain high collection efficiency, one must raise the collection rate and lower thevapor loss rate. The dependence and optimization of collection rate on various trap
parameters has been previously investigated[32]. To have a high collection rate, a trap
should have large, high-power laser beams. The trapping cell should be designed tomaximize the ratio of the trap volume (region of laser beam overlap) to the cell surface
area. To minimize the loss rate from the cell, the opening through which the radioactive
atoms enter the cell must be small enough to minimize the leak rate, and the loss of
atoms via adsorption to the glass walls must be significantly reduced. The latter isaccomplished by coating the glass surfaces with dryfilm coatings made of silicon-basedhydrocarbon polymers, which are then “cured” by exposure to alkali vapor. [31]

18
3.2

TheoryThe analysis of the capture process differs from that of a conventional VCMOT.
In a highly efficient VCMOT, the number of atoms in the vapor and the number ofatoms in the trap are so strongly coupled that the vapor density cannot be considered
constant. The time evolution of such a coupled system of Nt atoms in the trap and
Nv atoms in the vapor depends on three rates: L, the loading rate of atoms from thevapor to the trap; C, the loss rate of atoms from the trap to the vapor due to collisions
with background vapor atoms; and W, the loss rate of atoms from the vapor to the cell
walls or out of the cell. In the case where a constant flux, I, of atoms enter the cell, thedependence can be described by two coupled differential equations:d(Nt )= CNt + LNv ;dt
d(Nv )= CNt LNv W Nv + I:
dt
The time evolution of the number of trapped atoms follows a double-exponential function,

Nt (t) = a1 e k1 t + a2 e k2 t +

LI
;
WC

where k1 + k2 = L + C + W and k1 k2 = W C . Nt (t=0) and Nv (t=0) fully determine
a1 and a2 . We define trap efficiency ( ) as the probability of trapping an atom that hasentered the cell,1=
3.3

Experimental Setup

3.3.1

Fr Source

L
CNt
=
:
(L + W ) (CNt + I )The 221 Fr nuclei are produced in the decay chain
229 Th(t!

!

225 Ra(t1=2 = 7300y)
1=2 = 15d)
225 Ac(t221 Fr(t1=2 = 10d)1=2 = 4:9min):!

In order to produce a portable 221 Fr source with short-lived radioactivity, 225 Ac was firstextracted out of a long-lived 229 Th sample, and deposited onto a small piece of platinum
using one of two methods. Results reported here were obtained from a sample of 225 Ac
chemically extracted from a 229 Th solution [59] and electroplated onto a Pt ribbon.
In our preliminary measurements, we used 225 Ac implanted onto Pt via electrostaticcollection [33]. The Pt ribbon was then placed in the cavity of an orthotropic oven[33] where 221 Fr daughters were continuously produced and distilled into a collimated
1

Ref. [30] defines a recapture efficiency which is in general lower than  , but equal to  in the limitof a small collisional loss rate (CL,W)

19
atomic beam. At the beginning of the experiment, the oven was loaded with 50 Ci
 106 s 1 of 221 Fr inside the oven. The full divergence angleof the atomic beam exiting the oven was measured to be 180 mrad. By counting theparticles from the decay of the 221 Fr, we measured a 221 Fr atomic beam flux of 3:8  104
s 1 , or about 2% of the 221 Fr atom production rate inside the oven. Thirty days later, atthe end of the experiment, the 221 Fr beam flux was 5:1  103 s 1 , reflecting a decrease
matching the natural decay of 225 Ac.
225 Ac, which produced 2

3.3.2

The Magneto-optical Trap

A schematic of the apparatus is shown in Fig. 3.1. Both the francium oven andthe vapor cell were situated inside a chamber where the vacuum was maintained at
2  10 8 Torr even when the oven was heated to the operating temperature of 1050
o C. The cell was a quartz glass cube (4.4 cm inside dimension) whose top lid could
be opened and closed via a mechanical feedthrough. Following the recipe developedby Stephens et al.[31], the cell walls were coated with a short-chain dryfilm called
SC-77 (Silar Laboratories), and then cured by opening the cell lid and maintaining a
Rb vapor in the cell at about 2  10 7 Torr for ten hours. The Fr atoms from theoven entered the cell through a 2 mm diameter hole at a lower edge of the cell. Overseveral days, the coating performance deteriorated, reducing the number of trapped Fr
atoms by a factor of 3. This damage to the coatings could be attributed to heat ormaterial evaporating from the oven, and was repaired by providing a continuous, lowlevel curing with  1  10 8 Torr of rubidium in the cell.
Up to 1W of light from a Ti:Sapphire ring laser was tuned to the D2 line of
Fr at 718 nm for trapping. The laser frequency was locked to the side of a Dopplerbroadened I2 absorption peak[60] that conveniently covered the frequency of the 7S1=2 ,F=3 ! 7P3=2 , F=4 cycling transition (Fig. 3.2). In addition, an SDL 100 mW diodelaser at 817 nm was tuned to the D1 line to pump atoms out of the 7S1=2 , F=2 state. The
frequency of the diode laser was tuned to the 7S1=2 , F=2 ! 7P1=2 , F=3 transition andlocked to a Fabry-Perot cavity, which was in turn actively stabilized to the rubidium D2line. This lock is described in detail in Appendix B.
3.3.3

Sensitive Detection

We assessed the performance of the wall coatings and found the correct laser
frequencies by observing fluorescence from the room-temperature Fr vapor. We used
an optical-optical double resonance technique to separate the fluorescence from lightscattered off the cell walls. For this technique, a beam from the Ti:Sapphire laser wassent through the center of the cell, while its frequency scanned over one Doppler width(300 MHz) about the 7S1=2 , F=3 ! 7P3=2 , F=4 cycling transition. A 60 mW beamfrom the 817 nm diode laser, chopped at 100 Hz, co-propagated and overlapped theTi:Sapphire beam through the cell. The diode laser frequency was tuned close to the
7S1=2 , F=2 ! 7P1=2 , F=3 transition to pump atoms from the 7S1=2 , F=2 to the 7S1=2 ,F=3 state. The population of the 7S1=2 , F=3 state was therefore fully modulated for

20

Figure 3.1: A diagram of the 221 Fr trap setup. Both the oven and the cell were situatedinside a vacuum chamber with 10 cm diameter windows. The same oven and cell
assembly was used for the 221 Fr vapor fluorescence measurements.

the group of atoms in resonance with both laser beams. Because this was a narrowvelocity group, the resulting resonance feature was sub-Doppler (50 MHz FWHM).
The resulting modulated fluorescence at 718 nm generated near the center of the cell
was imaged onto a low noise photo-diode through a 718 nm interference filter anddemodulated with a lock-in amplifier. Thus we were capable of detecting as few as 600
atoms in the entire cell, or equivalentlyp
1 Fr atom in resonance in the viewing region,
with a signal to noise ratio (SNR) of 1/ s. By monitoring the exponential buildup of
the number after the cell lid was closed, we determined the loss rate of Fr atoms fromthe vapor (W) to be 1.59(9) s 1 , and a constant flux of 2:3  103 s 1 Fr atoms entering
the cell.
We then trapped the Fr atoms in a VCMOT using 4 cm diameter laser beams.
These beams provided a six-beam-total intensity of up to 110 mW/cm2 in the cell. In
addition, a 60 mW, 4 cm diameter beam from the diode laser at 817 nm was sent throughthe cell four times along two normal axes of the cube cell. A set of anti-Helmholtz coils
generated the MOT quadrupole field gradient of 7 Gauss/cm.
The laser light scattered from the six cell walls made detection of the 718 nm
fluorescence from the trapped Fr atoms impossible. Instead, to further avoid scattered
light and thereby increase our sensitivity, we imaged the 817 nm repump fluorescence21

Figure 3.2: The atomic level diagram of 221 Fr. In our 221 Fr trap, the trap laser (718 nm)
was tuned to  30 MHz below the 7S1=2 , F=3 ! 7P3=2 , F=4 cycling transition, and therepump laser (817 nm) was tuned to the 7S1=2 , F=2 ! 7P1=2 , F=3 transition. To detect
the trapped atoms, a 718 nm pump beam (2 mm diameter, 50 mW/cm2 ), tuned to the
7S1=2 , F=3 ! 7P3=2 , F=3 transition, was chopped to modulate the 817 nm fluorescencefrom the trap.

from the trapped atoms onto a photodetector. This light was observed through a cell
window not illuminated by any repump beams and through a 817 nm bandpass interference filter. The 817 nm fluorescence was modulated using a small, frequency-shifted
718 nm pump beam that was chopped and retro-reflected through the trap center (seeFig. 3.1). This beam increased the fractional population in the 7S1=2 , F=2 state to 70%
without affecting the trap loading rate. This detection
scheme allowed us to detect asp
few as 15 trapped Fr atoms with a SNR of 1/ s in the presence of large amounts ofscattered light. The trap contained an estimated 900 atoms, but this is a lower limit
assuming that both the 718 nm pump and the repump lasers were tuned to resonance.
This trapped atom sample could be maintained for many hours.
3.4

Results

3.4.1

Trapping Efficiency

Figure 3.3 shows a trap loading curve fitted to the double-exponential function
predicted by our model. The fit gives the trap rate constants, from which we calculate
CNt gives athe trapping efficiency to be  = (L+LW ) = 56(10)%. As a check,  = (CNt +I )
lower limit of 30% assuming the repump laser is tuned to resonance, but can be as highas 50% when plausible detunings are assumed in calculating Nt . The atomic fluxes atvarious stages of the experiment are listed in Table 3.2, showing a total efficiency of
0.4% from production to trapping. Engineering improvements in the oven and in the

22

Figure 3.3: The number of atoms (in arbitrary units) loaded into a 221 Fr trap vs. time.
In a trap with a high collection efficiency, the loading signal is expected to be a doubleexponential function. Because the signal was filtered by a low-pass network with a time
constant  = 100 ms, it was fitted with the function b1 [1-exp(-k1 t)] + b2 [1-exp(-k2 t)]  [b1 k1 + b2 k2 ][1 - exp(-t/ )]. The best fitting results were k1 = 4.8(1.0) s 1 and k2 =0.57(23) s 1 . Combining these rates with the measured result that on average a Fr atom
stayed in the vapor for 1/W = 0.63(4) s, we derived that, on average, a Fr atom stayedin the trap for 1/C = 0.58(27) s and it takes 1/L = 0.48(19) s to load a Fr atom from thevapor into the trap. These rates are consistent with those found in Rb traps, and imply
a trap efficiency of L/(L+W) = 56(10)%.

coupling into the cell would significantly improve the total efficiency. An orthotropic
source has operated with a 15% efficiency, and 50% (limited by diffusion of Fr into theoven walls) is theoretically possible[33].
3.4.2

Spectroscopy

We have made spectroscopic measurements on 221 Fr using our apparatus. In thethermal vapor, we measured the hyperfine splittings of the 7P3=2 and 7P1=2 levels. We
observed the individual hyperfine transitions by scanning the 718 nm laser, while the817 nm diode laser was set (to either the F=2!F’=3 or F=2!F’=2 transition) to selectone velocity group of atoms. We obtained the splittings by measuring the frequency differences between each hyperfine line (Table 3.1) using a high-resolution -meter[61].
By checking against known splittings and wavelengths in rubidium, we found the accuracy of the -meter to be 3 MHz or less when measuring small differences, but 50
MHz when measuring absolute frequency. This is because uncertainties in the refractive index of air and laser beam alignment are largely canceled in a frequency difference
measurement.

23
Table 3.1: Various rates, fluxes, and efficiencies on Dec. 17-18, when the source
strength was 9 Ci. Total efficiency is 0.4%. Data in row (4) are deduced from flux
entering cell and trap efficiency.
(1)
(2)
(3)
(4)

Stage
Produced in oven
Exit ovenEnter cellCollected into trap

221 Fr Flux (s 1 )3.3 105
7 103
2.3 103
1.3 103

Efficiency
2% of (1)
33% of (2)56% of (3)Using the trapped atoms as a frequency reference, we have measured the wavenumbers of the D1 and D2 transitions of 221 Fr (Table 3.1). While the wavenumber of theD1 transition measured here is in agreement with the number measured by the ISOLDE
collaboration[62, 63], our value of the D2 transition is lower by 3 ST D . As a checkon our calibration, we measured the wavenumber of the I2 line that is only 0.5 GHz
away from the D2 trapping transition and found an excellent agreement (difference =1(3)  10 3 cm 1 ) with the I2 atlas[60].
3.5Conclusions

We have demonstrated and analyzed the dynamics of highly efficient collectionof short-lived radioactive alkali atoms in an optical trap. The techniques used here, including the optimized VCMOT, the orthotropic source, and sensitive detection, can beeasily applied to other alkalis as well 2 Future applications will likely employ immediate, efficient transfer of trapped atoms out of the cell into a much longer-lived trap (see,for example, Refs. [18, 24, 64] and Chapter 2. By combining a stronger and currently
available 221 Fr source (flux  106 /s) with a double-MOT system (trap lifetime  102s), a sample of 108 trapped 221 Fr atoms could be prepared for the next generation of
Most alkalis would be easier than 221 Fr, which has an unusually small excited state splitting combined with a large ground state splitting.
2

Table 3.2: The hyperfine constants and wavenumbers of 221 Fr. The wavenumbers of
the previous work were derived from the published wavenumbers of 212 Fr and isotope
shifts [17].
ISOLDE [62, 63]
A (7P3=2 ) MHz
65.4(2.9)
B (7P3=2 ) MHz-259(16)
A (7P1=2 ) MHz
808(12)1
 (D1 ) cm
12236.6601(20) (D2 ) cm 1
13923.2118(20)Current Work66.5(0.9)-260.0(4.8)
811.0(1.3)
12236.6579(17)13923.2041(17)24high precision spectroscopy measurements to test fundamental symmetries.

Chapter 4
Spin-Polarized Atoms in a Circularly Polarized Optical Dipole Trap

4.1

Introduction

Many precision measurements require spin-polarized neutral atoms and would
be improved with trapped samples. However, no trap provides the necessary characteristics: atomic spin-polarization, tight confinement, ease of control, and a low
photon-scattering rate. An imbalanced magneto-optical trap (MOT) has some modest
polarization [65], and small spin-polarized samples have been maintained with repeated
optical pumping cycles in standard dipole traps [66]. Magnetic traps can confine spinpolarized samples, and offer the advantage of RF transitions between non-degenerate
Zeeman states that can be used to drive a variety of cooling schemes, including evaporative cooling and gravitational Sisyphus cooling [67, 45]. Unfortunately, magnetic
traps have relatively weak spring constants, and strong magnetic fields are often undesirable and difficult to rapidly control. In contrast, an optical far-off resonance trap
(FORT) [38, 39] offers superior confinement and rapid control. As noted in Ref. [43],a dipole trap made with circularly polarized laser light offers all the advantages of an
optical trap, plus energy splittings between spin states that make cooling schemes possible in an inherently spin-polarizing trap. In this chapter, we report the creation ofsuch a spin-polarizing FORT for Rb atoms using circularly polarized light.
4.2

TheoryA conventional FORT consists of a linearly polarized laser beam focused to atight waist [38, 39]. Due to a spatially varying AC Stark shift, laser light tuned below
an atom’s resonant frequency attracts the atom to regions of high intensity. For alkali
atoms in linearly-polarized light fields, the potential is the same for all internal spinstates of the atom in the ground n (S1=2 ) electronic state 1 . This degeneracy is lifted
when the light is circularly polarized; the laser field acts as a “fictitious magnetic field”[42]. When the laser is tuned between the D1 (n S 1=2 ! n P1=2 ) and D2 (n S 1=2 ! n
P3=2 ) transitions, the energy splitting between the spin states is largest (Fig. 4.1). Also,
the absorption of the circularly polarized FORT photons optically pumps the atoms into
the deepest potential of the F = 3 manifold, shown for + polarization in Fig. 4.2.
1This is true if the laser detuning is much larger than the hyperfine splitting.

F’
4

2

5 P3/2
2

26

3
2

D2

5 P½

c

b

D1

a
d

e

2

5 S½

F
3
2

Figure 4.1: Schematic level diagram for 85 Rb (I= 5=2). Laser beams are identified in
the text as (a) FORT at 784.1 nm, (b) MOT cooling, (c) MOT repump, (d) transfer, and(e) probe. In vacuum, D2 = 780.2 nm, D1 = 795.0 nm [68].

The potential energy of atomspin anpelliptically polarized
FORT may depend onp
the laser polarization vector "^  1= 2(^
x 1 +  + iy^ 1 ), with  the ellipticity. 2
The potential depth U0 (I , , F , mF , ) is then

U0 =

~ I0

24IS

[(

1
2
+ )
Æ 12 Æ 32p

gF mF 1 2 (

1
Æ 21

1
)]
Æ 32

(4.1)

where the natural line width = 2 6.1 MHz in Rb, mF is the Zeeman sublevelof the atom, gF = [F(F+1) + S(S+1) - I(I+1)]/[F(F+1)], the saturation intensity IS =2 2 ~c =(33 ), and the intensity I0 = 2P=(w02 ) in terms of power P . 3 The detuningsÆ1=2 and Æ3=2 (in units of ) represent the difference between the laser frequency andthe D1 and D2 transition frequencies, respectively. Figure 4.3 shows U0 versus the laser
wavelength for each mF level. The potentials are described in terms of the Gaussian
beam waist w0 by
U () = U0 exp( 22 =w02);(4.2)
2

 is measured by rotating an analyzing polarizer in the laser beam and recording the highest andlowest power (Ph and Pl ) transmitted through the analyzer. Then  = Ph Pl = Ph Pl .
3
Equation (4.1) is obtained by rewriting the expression for UAC in Ref. [43] in terms of experimen, Eq. (4.1) agrees with the expressions for U0 in Ref. [69].
tally accessible quantities. For (=1)( + )

27

mF= -3
F=3
Potential Energy-1.6 mK
3.03 GHz+3mF = +2ρ

F=2-2
Figure 4.2: AC Stark shift U () in the focus of a circularly polarized Gaussian laser
beam for both hyperfine levels in the 85 Rb 5 2 S1=2 ground state (g3 = 1=3) under typicalexperimental parameters, for P = 240 mW. Solid lines indicate the spin states toward
which the atoms are driven by absorption of the trapping light.

pwhere  = x2 + y 2 is the radial spatial coordinate. Radial cross-sections of thesepotentials for 85 Rb are plotted in Fig. 4.2.
4.3

Experimental Setup

4.3.1

The Lasers

The experimental apparatus is shown in Fig. 4.4. The MOT [14, 15] collects Rb
atoms from a 5  10 10 torr vapor, and contains a maximum of  3  108 atoms in steadystate with a filling time constant of 12 s. The MOT is made with two external-cavity
diode lasers stabilized to atomic lines in Rb at 780 nm with a dichroic atomic vapor
laser lock (DAVLL)(See Ref. [70] and Appendix A), which allows rapid and convenienttuning over > 100 MHz range. One laser, the “cooling” laser, provides 6 mW (divided
between three retro-reflected beams) and is tuned to 5 2 S1=2 F = 3 ! 5 2 P3=2 F 0 = 4. A
second laser, the “repump” laser, is tuned to 5 2 S1=2 F = 2 ! 5 2 P3=2 F 0 = 3. Both thecooling and repump lasers pass through acoustooptical modulators (AOMs) to control
their power.
The dipole trap with adjustable polarization is created from up to 1 W of laser

Potential Depth U0 (mK)

4

2

D1
0

D2
-2

-4
775

780

785

790

795

800

805

FORT λ (nm)28

Figure 4.3: Maximum FORT Potential depth as a function of FORT wavelength, for the F=3 manifold with the following laser parameters;
P = 600 mW, w0 = 26 m. The darkest line indicates the potentials for mF = +3, into which the atoms are optically pumped by the FORT
beam. The dashed line indicates the potential for atoms in the mF = 0 state, which are insensitive to polarization.

Probebeam

MOT laser
beams

Polarizer

o
ot

Ph

de

o
di

Mechanical
T:S Laser
Shutter~1W
Pockels
AOM
Cell
x
Rb
z
X
Intensity
cloud Aperture
X
Servo
y
gravity
Polarizer

29

Figure 4.4: Schematic of the experimental apparatus. A “transfer” laser, not shown, counter-propagates with the FORT laser. Also notshown, a two-mirror periscope after the aperture elevates the FORT laser beam and turns it 90o . (See text.)30
light from a Titanium:Sapphire laser, typically tuned to 784.1 nm. The laser beam
passes through an AOM and the output power in the first diffraction order (70%) isusually actively stabilized to prevent parametric heating in the trap[71]. This beam
passes through a Glan-Thompson polarizer and into a Pockels cell, which acts as aquarter-wave retarder when the applied voltage is about 2200 V. This voltage can beswitched in a matter of 100 ms using a Hg relay. A mechanical shutter follows that canturn the FORT off in 60s. Finally the FORT beam (!0 = 4.2 mm) is incident on anaperture and then a cemented doublet (f = 18 cm) that focuses it to w0 = 26 m.
The FORT laser beam waist w0 is measured by inserting a mirror after the doublet
lens such that the FORT laser comes to a focus outside the chamber. A razor blade isthen translated across the beam while monitoring the position of the blade x and theamount of power that is not blocked by the razor bladep P (x). The resulting data are fit
to the following function P (x) = Pmax =2[1erf( 2  (x xc )=w )] to determine thewidth w of the beam. This measurement was repeated at many longitudinal positions
until the narrowest waist w0 was found. By examining the dependence of w on z ,the effective Raleigh range could be found. This often exceeded the Raleigh range
calculated from w0 using Gaussian optics [72] by as much as a factor of 3, perhaps due
to aberrations in the lens.
4.3.2

The Pockels Cell and Polarization Issues

Establishing the proper polarization of the laser using a Pockels cell requires
precision and patience. As will be shown later, a very well-defined polarization isessential to the performance of the circular FORT. This requires careful alignment of the
Pockels cell. Our Pockels cell comes from Cleveland Crystals, model number Q1020S.
As with all Pockels cells, it relies on the Pockels effect [73], which is a birefringenceinduced in a crystal that is proportional to the applied voltage. This linear effect ispresent only in crystals that lack a center of symmetry, such as KDP and KD P.
To align the Pockels cell, first two crossed polarizers are aligned in the FORT
beam such that the first polarizer gives maximum transmission and the analyzer provides extinction. Next, the Pockels cell is inserted in the beam between the polarizers
and a large voltage (approximately the /2 voltage if safely possible) induces birefringence. Then the cell is rotated until extinction is again observed after the analyzer. Thisestablishes that the “fast axis” of the cell is aligned with the incident polarization. The
cell is then rotated such that the fast axis is at a 45o angle with respect to the incident
polarization. The applied voltage is reduced to nearly the nominal /4 value and thenadjusted until /4 voltage is obtained, and the polarization is as nearly circular as possible. Figure 4.5 shows the measured polarization ellipticity as a function of applied
voltage.
The angle that the cell’s longitudinal axis makes with the laser’s direction of
propagation is also important. One alignment trick is to place a piece of cellophane tapeacross the input port of the Pockels cell. This creates diffuse, unpolarized scattered lightat the input, some of which passes through the Pockels cell. If a piece of paper is placedalong the beam path after the analyzing polarizer, the diffuse light forms a large round31
spot. When the Pockels cell has a large applied voltage, a cross-shaped interference
pattern appears in the diffuse spot when it is viewed through the analyzing polarizer.
By tilting and translating the Pockels cell, this cross shape can be centered within the
diffuse spot. The incoming laser beam, when not passing through the transparent tape,should also strike the center of the cross. To perform this alignment, it is necessary toinsert and remove the transparent tape repeatedly.
By iteratively rotating, tilting, and translating the Pockels cell, optimum alignment can be obtained, in which the light can be switched from completely linear tocompletely circular by changing the applied voltage. However, after the Pockels cell aperiscope composed of two mirrors is required to elevate the beam so that it can enterthe vacuum chamber. This presents a problem because mirrors induce large, angle- andpolarization-dependent phase shifts in reflected beams, they are therefore not in generalpolarization-preserving. However, the polarization can be preserved by using two goldcoated mirrors each of which reflects the beam at right angles such that the final beam
is above and perpendicular to the incoming laser beam. By monitoring the polarization
after the periscope and the vacuum chamber, small changes in the periscope alignment
can be used to optimize the polarization. Residual phase shifts in circular light inducedby the periscope can also be compensated by the Pockels cell voltage, as in Fig. 4.5.
Any time the alignment of the laser through the periscope changes, one must verify the
=4 voltage. In measuring the beam polarization, one must use high-quality polarizers
such as calcite Glan-Thompson polarizers. It is also important to use the entire laser
beam when measuring the polarization, because both the periscope and the Pockels cell
can introduce non-uniformity in the polarization that can limit the lifetime of the atoms
in the circular FORT.
4.3.3

Diagnostics and Loading

The number and lifetime of atoms in the FORT are characterized using the following timing sequence. The MOT is allowed to fill for some time f before the atoms
are transferred from the MOT into the FORT. They are then stored in the FORT in theabsence of any other light sources or the quadrupole magnetic field for some time S .
To detect the atoms, the FORT is turned off and the MOT quadrupole magnetic fieldremains off while the MOT cooling and repump lasers are turned on again, tuned closer
to resonance (- =2). The number of atoms N is then inferred from the measured fluorescence [74] and plotted as a function of S (Fig. 4.6). The decay in N is governed by
the differential equation
0 N 2;
N_ = N(4.3)
where describes exponential loss processes involving a single trapped atom and 0
characterizes the density-dependent losses. The prime on 0 is used to refer to atom
number loss, instead of density loss. The data are fit to the solution [75]N = N0 et =[1 + (

0 N0

)(1 e

t )];

as shown in Fig. 4.6. From this fit, the initial number N0 , , and 0 are extracted.

(4.4)

32
0

Ellipticity (ε)

10

-1

10

-2

10

0.0

0.5

1.0

1.5

2.0

2.5

Pockels Cell Applied Voltage (kV)
Figure 4.5: Characterization of FORT laser beam polarization after Pockels cell andperiscope, as a function of the voltage applied to the Pockels cell. Ellipticity is described in Section 4.2. The black line represents a fit to  = j cos2 (=4(V=VR )sin2 (=4(V=VR )j, where VR is the desired reference voltage that provides perfectly circular light after the periscope.

The physics involved in the transfer of atoms from the MOT to the FORT is quite
complicated and will be discussed elsewhere (See Ref. [44] and Chapter 5). Here,
we give a summary of results. For the parameters in Fig. 4.6, simple “geometricalloading,” in which the FORT laser is turned on immediately after the MOT turns off,
traps about 2 105 atoms. N0 can be increased by leaving the MOT on with different
detuning and power parameters while the FORT is loading. This increase indicates that
the MOT lasers cool atoms as they fall into the FORT. For the same FORT parameters,
the best loading is typically achieved by detuning the cooling laser to -5 and reducing
the intensity of the hyperfine pump to 10 W/cm2 for 70 ms before the MOT lasers andmagnetic fields are switched off. This procedure transfers up to 20% of the MOT atoms
into the FORT.

33

Number of FORT Atoms N

N0
10

β

6

Γ

10

5

0

1
2
3
4
FORT Storage time τS (s)5

Figure 4.6: Number of atoms stored in a linearly polarized FORT (U0 = -1.8 mK at 540mW) as a function of trapping time, after filling the MOT for f = 1 s. No data is taken
for the first 100 ms while the MOT atoms fall away. The solid line is a fit to Eq. (4.4),
giving N0 = 2.2106 atoms, 0 = 1.310 6 (atoms s) 1 , and 1/ = 3.0 s. For f = 4 s,
4  106 atoms are initially trapped in the linear FORT.

4.4

The Elliptically and Circularly Polarized FORTs

4.4.1

Loading Atoms into the Circular FORT

Optimum loading into an elliptically polarized FORT ( < 1) is different from
simply loading into the linear FORT. While geometrical loading still gives similar numbers, only about half as many atoms can be loaded using MOT cooling. The most efficient way to transfer atoms into an elliptical FORT is to first load the linear FORT,and then use the Pockels cell to quickly change the polarization from linear to the desired polarization. This procedure transfers about 70% of the atoms originally held in
the linear FORT, regardless of the time constant that governs the change in polarization (from 0.2 to 20 ms). Up to 100% of the atoms in the linear FORT are transferredto the circularly polarized FORT (“circular FORT”) by a “transfer” laser beam, circularly polarized and aligned anti-parallel to the FORT laser. This beam, tuned to the 5
2 S F = 2 ! 5 2 P F 0 transition, is applied for 20 ms after the FORT polarization
1=2
3=2
changes to circular. When + -polarized, this transfer beam assists the FORT in optically pumping the atoms into positive mF levels, and tends to keep them in the F=3

34

-1

Γ (s )

10

1

Γ (ε)
Γ ' (ε)linear0.1

circular
0.0

0.2

0.4
0.6
0.8
FORT Laser Ellipticity (ε)

1.0Figure 4.7: Dependence of the FORT exponential loss rate on the ellipticity  of the
FORT laser polarization. Crosses indicate the measured loss rate, which is small forpurely linear ( = 1) and purely circular ( = 0) polarizations. Because the circularpotentials are deeper as indicated in Fig. 4.2, ( = 0) < ( = 1). P= 280 mW, U0(mF = 0) = -0.77 mK, and U0 (mF = +3,  = 0) = -1.5 mK. The solid and dashedcurves represent models of the ground-state dipole-force fluctuation heating describedby Eq. (4.5), plus a small offset due to collisions with the background vapor.

states. Thus atoms are pumped to the deep F = 3, mF = +3 potential whence they are
not readily lost. In this fashion, 4106 atoms were loaded into a circular FORT at  =782.3 nm with U0 (mF = 0) = 2:0 mK.
4.4.2

Elliptically Polarized FORT Decay Rates

The lifetime  of the FORT depends dramatically on the polarization of the FORT
laser. A measurement of the decay rate = 1/ as a function of the ellipticity  of the
FORT laser polarization is shown in Fig. 4.7. is smallest when the polarization isperfectly linear or perfectly circular. In addition, Fig. 4.8 shows how changes with
FORT intensity for each of these polarizations. Above a trap depth of about U0 = 0:2
mK, the decay rate is independent of intensity for both linear and circular light, but
increases linearly with FORT intensity for an intermediate elliptical polarization.
The large losses at imperfect polarizations are due to ground-state dipole-force

35

2.0

-1

Γ (s )

1.5

1.0

0.5

0.0
0

50

100

150

200

250

300

350

FORT Power (mW)
Figure 4.8: Exponential decay rate vs. FORT peak intensity for three different FORT
polarizations: linear (,  = 99:995%), circular (M,  = 0:6%), and elliptical ( ,
 = 21%). FORT  = 784.2 nm, w0 = 31 m, and the maximum U0 = -0.8 mK for
linear polarization at 330 mW.

fluctuation heating that arises when atoms change internal spin state [10]. In the linear
FORT, all the ground-state potentials are degenerate, so there is no heating associatedwith changing the ground level mF state. However, when the polarization is elliptical,
these levels are no longer degenerate, and changes between the levels cause instantaneous changes in the dipole force acting on the atom. These changes result in heating,which evaporates atoms from the trap. The heating rate depends on the rate at which
the atoms change state, which in turn depends on the FORT scattering rate. Therefore
this heating mechanism increases linearly with intensity. In the circular FORT, however, only mF = +1 transitions can be excited, and the atoms are all pumped into the
F = 3, mF = +3 state in about 30 ms. After that, they no longer change their groundstate and this heating process turns off.
4.4.3

Model of Lifetime vs. Ellipticity

Agreement between the data and a simple model supports this picture. The dependence of heating on FORT polarization can be modeled with a two-level system using the two trapping potentials of the 5 2 S1=2 F = 3, mF = +3 and F = 2, mF = +2

36states, labeled ja > and jb >. These states are selected because the U0 for each isvery different, and yet they are closely coupled via off-resonant excitation by the FORT
beam. We approximate the potentials as harmonic. Atoms initially held in ja > spreadout when they are transferred to jb >, which has a smaller spring constant, and thenheated when they re-enter ja > with a larger spatial extent and therefore more potential
energy. Assuming the atoms hop between the two potentials at a constant rate h , wederive the exponential time constant as a function of FORT ellipticity:

() =

h ln

Ua () Tf=ln ;Ub () Ti(4.5)

where Ua () and Ub () are calculated for states ja > and jb > using Eq. (4.1), and Ti and
Tf are the initial and final temperatures, chosen to be 100 K and U0 =kB , respectively.
As shown by the solid curve in Fig. 4.7, Eq. (4.5) explains the rapid decrease in as
ellipticity approaches linear, but does not fit well near circular.
The decrease in near circular polarization arises from a dependence of the hopping rate on FORT ellipticity. Because the system actually contains many hyperfine
and spin states, it is difficult to model exactly. A simplified approach assumes thatabsorption of a  photon causes transfer from ja > to jb > and absorption of a +
photon transfers atoms back to ja >. This gives a new rate of hopping between potentials 0h = 2 h and modifies the expression for the exponential heating rate in thetrap to be 0 () = 2 (). Figure 4.7 shows fits to both () and 0 (), with h the
only free parameter in each case. They yield h = 13/s and h =17/s, respectively. To
calculate an optical scattering rate from these values, we simply divide by the productof the appropriate Clebsch-Gordon coefficients (0.05) for the four-photon heating process (absorption, emission to F = 2, absorption, emission to F = 3). The resulting
scattering rates (260/s and 340/s) agree well with the estimated average scattering ratein the FORT to within its 50% uncertainty.
4.4.4

FORT  > D1

The decay of atoms from the FORT behaves very differently depending on the
wavelength of the FORT laser light. All the data preceding and following this section
were taken with the FORT wavelength smaller that 795 nm. In these cases, the circular
FORT decay curves fit well to Eq. (4.4). In contrast, when the circular FORT wavelength is red of both the D1 and D2 transition, Fig. 4.9 shows that the decay of atoms
occurs over two very different exponential time scales. In Fig. 4.9, () are fit to

N = A1 e t=1 + A2 e t=2

(4.6)

yielding 1 = 65  8 ms and 2 = 3:2  0:2 s. Time constant 2 comes from the samebackground collisional loss rate observed in the decay of atoms from the CFORT tuned
between the D1 and D2 lines, and in the linear FORT. However, the new, faster decayrate 1 is due to optical pumping and evaporative loss of atoms by the circular FORT.
Optical pumping of atoms in the circular FORT only causes additional losses
when the FORT is tuned red of the D1 transition. In this case, the sign of the second

37

Number of FORT Atoms N

LinearCircular
6

10

5

10

0

1

2

3

4

5

FORT storage time τs (s)Figure 4.9: FORT decay curve with FORT  = 798:4 nm, P = 290 mW, and w0 = 31
m for both linear (, U0 = 0:7 mK) and circular [, U0 (mF = +3) = 0:3 mK]polarizations. Solid lines represent fits to Eq. (4.4) and Eq. (4.6) respectively.

term of Eq. (4.1) is such that the circular FORT tends to optically pump atoms into theshallowest potential. Measurements described in the next chapter (Section 5.5) indicatethat the FORT holds the temperature of the atoms at T = 0:4  U0 =kB . Therefore,
atoms initially held in the linear FORT have temperatures too large to be confined in
the shallower potentials of the circular FORT once they are pumped into them. Atoms
initially in the mF = 3 state evaporatively cool until they can be held by the shallow
potential. In addition, atoms in the deeper states are optically pumped into the shallowest state, and also evaporatively cool. Once all the atoms are optically pumped andhave evaporated to a reasonable temperature, this source of loss ceases to contribute to
the decay in the number of atoms.
Brief attempts to minimize the losses due to optical pumping were unsuccessful.
These included varying the time over which the polarization was switched, and using
the circularly-polarized “transfer beam” to optically pump the atoms before the potential is switched. Neither of these can eliminate the essential problem: atoms originallyheld in the linear FORT have too much kinetic energy to be retained by the shallower
F = 3, mF = 3, regardless of how they make the transition. One technique that may
work well is to reduce the linear FORT power to some fraction f of maximum at the
beginning of the cycle, where f is the ratio between U0 (mF = 3) and U0 (mF = 0);

38

f = 2 at FORT  = 800.0 nm. Then one should apply the transfer beam to opticallypump most of the atoms into the F = 3, mF = 3 state, and then simultaneously restore
the FORT power to its maximum value and turn on the Pockels cell. In this way, thedepth of the well containing the atoms would not change, and evaporative losses could
be eliminated.
4.5

Spin-Polarization Demonstration

The inherent spin-polarizing nature of the circular FORT is confirmed by measuring the fraction of atoms that populate the F = 3, mF = +3 state. Exciting theatoms in the F = 3 state with an additional laser causes loss by pumping them to theweakly trapped F = 2 state. We compare the loss when all mF states are excited to
the loss when all but mF = +3 is excited. The probe laser beam (Figs. 4.1 and 4.4)
is linearly polarized and tuned to excite 2 S1=2 F = 3 ! 2 P3=2 F 0 transitions. When
this beam is aligned as shown in Fig. 4.4, it induces either m = 0 or m = 1
transitions, depending on whether the polarization is oriented horizontally or vertically.
The probe beam is pulsed on for 20 ms at an intensity of 1 W/cm2 , 45 ms after the
MOT is switched off. We then measure the resulting decrease in the number of atoms
in the circular FORT. This signal is plotted as a function of frequency in Fig. 4.10 for
both polarizations. When the laser is tuned to the F = 3 ! F 0 = 2 transition and thepolarization is vertical (), we find that the loss saturates at 33%. However, when thepolarization is horizontal (3), unsaturated loss is only 4%. The difference in fractionalloss indicates a large population in the mF = +3 state, because horizontal polarization
does not excite atoms in this state, while vertical excites all spin states. After including
a factor of 2 for saturation, the ratio of the two losses implies that 7% of the atoms are
not in the F = 3, mF = +3 stretched state. Because these atoms are most likely to bein the mF = +2 state, the spin-polarization is 98(1)%.
4.6

Conclusion

We have demonstrated and characterized a circularly polarized FORT. We havealso shown that the trap provides a high degree of spin-polarization. The splittings of
mF levels in the circular FORT should allow RF cooling techniques in the trap similar
to methods employed in magnetic traps. Finally, we have shown that small polarization
imperfections in circular or linear FORTs can lead to heating and subsequent loss due
to ground-state dipole-force fluctuations.

39

Fraction remaining

1.0

0.9

F'=4

0.8

0.7

∆m = 0F'=2∆ m = +/-1

F'=3

0.6
-300

-200

-100

0

100

Frequency of Optical Pumping beam (MHz)Figure 4.10: Fraction of atoms remaining in the FORT as a function of the frequency ofan applied optical pumping beam for two different polarizations: (3 and ). Frequency
is measured with respect to the frequency of the unshifted 5 2 S1=2 F=3 ! 5 2 P3=2 F’ =4 transition. Signals are normalized to the number of atoms held in the trap without the
optical pumping beam (8105 atoms). Resonances are AC Stark-shifted about 30 MHz
with respect to the resonances in free atoms, and separations are consistent with the 52P0
3=2 hyperfine splitting. The difference between 3 and  for F = 2 is clear evidence
for spin-polarization. This shallow trap [P=190 mW, U0 (mF = 0) = -0.63 mK, andU0 (mF = 3) = -1.2 mK] allows resolution of the hyperfine structure. In deeper traps,
the spin-polarization should only improve.

Chapter 5
The Physics of Loading an Optical Dipole Trap

5.1

Introduction

In the last decade, many different schemes for preparing and trapping ultra coldand dense samples of atoms have been demonstrated. Of these, the optical dipole trap[38] requires no magnetic fields and relatively few optical excitations to provide a conservative and tightly confining trapping potential. These characteristics make it an appealing option for various metrology applications such as parity nonconservation and-decay asymmetry measurements. It may also be an option for reaching BEC in apurely optical trap. For these applications, large samples of atoms must be transferedinto the dipole trap. This is almost always done from a magneto-optical trap (MOT)[14]. However, the processes determining the transfer between the MOT and the optical
dipole trap are poorly understood. Here we give a detailed description and explanationof the loading process and suggest ways in which to improve the loading.
The simplest optical dipole trap consists of a focused single Gaussian laser beam.
Typically the light is detuned below the atomic resonance, from a few tenths of a nm toseveral tens of nm. The latter are called far off resonance traps (FORT) [39]. We will
use the abbreviation FORT in discussing optical dipole traps. Conceptually a FORT
works as follows: the AC Stark shift induced by the trapping light lowers the groundstate energy of the atoms proportionally to the local intensity. The spatial dependence ofthe light intensity is therefore equivalent to a spatial dependence of the atomic potential
energy. The atom has the lowest energy in the focus of the trapping beam and cantherefore be trapped there. For very large detuning, typically several nm, the photon
scattering rate becomes so low that the potential is truly conservative.
The first FORTs were running-wave Gaussian laser beams focused to a waist of
about 10 m [38, 39]. By alternating the FORT with an optical molasses that cooled
atoms into the trap [38, 39], about 500 to 1300 atoms were loaded. In later work[69, 76, 77], the FORT was loaded by overlapping it with a MOT continuously, which
improved the number of atoms that was transfered to 106 .
A key step in the loading of a FORT from a MOT is a strong reduction of the hyperfine repump in the last 10-30 ms of the overlap between the traps. It has been conjectured [76] that this reduction helps because it reduces three density limiting processes,
namely, radiative repulsion forces, photo-associative collisions, and ground state hyperfine changing collisions. However, to our knowledge, there has not yet been an ex-416

N (10 atoms)

2.0

1.5

1.0

0.5

0.0
0.0

0.1

0.2

0.3

0.4

0.5

time (s)Figure 5.1: Number of atoms N in the FORT as a function of time spent loading, for a
trap depth of -1 mK and a waist of 26 m.

tensive study of the loading process. Therefore, in this paper we present a detailed andcomprehensive investigation of the many mechanisms that govern the loading process.
We find that loading a FORT from a MOT is an interesting dynamical process richin physics. As illustrated in Fig. 5.1, the number of atoms in the FORT first increases
rapidly and nearly linearly in time until loss mechanisms set a limit to the maximum
number. The loading rate and loss processes both depend in complex ways on thelaser fields involved. One factor that determines the loading rate is the flux of atoms
into the trapping volume. This flux depends on the MOT density and temperature,i.e., average velocity of the atoms in the MOT. For the atoms to be trapped in the
FORT, cooling mechanisms must also be active in the region where the MOT and FORT
overlap Both the flux and the probability for trapping depend on the trap depth andlight-shifts inherent to the FORT.
Losses from the trap can be caused by heating mechanisms and collisional processes. Contributions to heating arise from spontaneously scattered FORT light photons, background gas collisions [16, 17], intensity fluctuations, and the pointing stability of the FORT beam [71]. However, for large numbers of atoms the losses are dominated by collisional processes [78], including photo-association, spin exchange/groundstate hyperfine changing collisions, and radiative escape. Photo-associative collisions
can be induced by the FORT light itself and lead to untrapped molecules. During aground state hyperfine changing collision, a pair of atoms gains as much as the hyperfine energy splitting in kinetic energy ( 0.14 K for 85 Rb), which is enough to eject it
out of the FORT. In radiative escape, an atom is optically excited and reemits during

42
a collision, and the attractive dipole-induced interaction between the excited and nonexcited atom leads to an increase of kinetic energy which is enough to eject an atomfrom the trap.
The loading and loss rates depend on the shape and depth of the optical potential,
as well as the intensity and detuning of the MOT light fields. By studying the loading
rate and loss rate separately as a function of these parameters, we have obtained adetailed understanding of the FORT loading process. This understanding has allowed
us to optimize parameters in order to improve our loading efficiency to high values(47% of the MOT atoms were loaded into the FORT). Although we studied the loadingof 85 Rb into a dipole trap with a detuning of a few nm, the physical processes andoptimization should be generally applicable to other alkali species and FORTs, when
the FORT trap depth exceeds the MOT atom temperature. In the opposite regime,
O’Hara et al. [79] have shown that a static equilibration model applies in CO2 laser
traps.
This Chapter is organized as follows. In Sec. 5.2 a more detailed expression forthe depth and shape of the FORT potential is given. In Sec. 5.3 our experimental setup
is discussed, including the loading of the FORT and the diagnostic tools for measuring
the number of atoms and size of the trapped sample. In Sec. 5.4 we present measurements of the loading rate and loss rate as a function of different MOT and FORTparameters. In Sec. 5.5 the temperature of the atoms in the FORT is given. In Sec.
5.6 the loss rates of the FORT in the absence of MOT light are discussed. In Sec. 5.7
we present a physical model of the loading process which explains the data presented
in Sec. 5.4. In Sec. 5.8 we discuss how our model explains the interdependencies ofthe MOT and FORT parameters. In Sec. 5.9 we demonstrate how, based on our understanding of the FORT loading process, the number of trapped atoms can be improved
using a shadowed repump beam. Finally, Sec. 5.10 contains summarizing remarks anddiscusses the general applicability of these results to other traps.
5.2

The FORT potential

The trapping potential formed by the focused laser beam has been described indetail in Section 4.2. The most general form for the spatial dependence of U is givenbyexp[ 2r2 =w(z )2 ];
(5.1)
1 + (z=zR )2with z the longitudinal coordinate and r the radial coordinate (denoted  in previous
chapter). Thenp the beam radius w(z ) is given as a function of longitudinal position,w(z ) = w0 1 + (z=zR )2 where zR is the beam Rayleigh range [72]. Equation (4.1)
gives U0 as a function of many parameters, including the laser beam polarization ellipticity .

U (r; z ) = U0

This chapter focuses primarily on the loading of a FORT made with linearlypolarized trapping light. This is because loading the linear FORT is the most efficient
process, and therefore usually a first step toward loading the circular FORT. We justify
this approach in Sec. 5.4.6, where we discuss how the loading of the FORT is affected

ph

o

di

o
ot

de

43

repump beam
MOT laser
beams

T:S Laser
~1W
AOM

Rb
cloud

apertureintensity
servo

CCD

x

z

X

X

gravityy

Figure 5.2: Experimental setup

when optical Zeeman splittings are present due to elliptical polarization of the FORT
light.
5.3

Experiment

The experimental setup is very similar to that described in the previous chapter,and shown schematically in Fig. 5.2. The starting point of all the experiments describedhere is a MOT, which collects 85 Rb atoms from a 5  10 10 Torr vapor, and contains amaximum of about 3  108 atoms in steady state with a filling time constant of  12 s.
Two extended-cavity diode lasers, both stabilized to atomic lines in 85 Rb at 780 nmwith a dichroic atomic vapor laser lock (DAVLL) are used for the MOT (see [70] andAppendix A). One laser provides 6 mW for trapping and cooling (divided between
three retro-reflected beams) and is tuned red of the 5 2 S1=2 F = 3 ! 5 2 P3=2 F 0 = 4transition by an amount denoted as M . We refer to this light as the “primary” MOT
light (previously called the “cooling” laser), and denote its total six beam intensity by
IM . The beams have a Gaussian beam radius of 9:3 mm. The other MOT laser isused for hyperfine repumping and is tuned near the 5 2 S1=2 F = 2 ! 5 2 P3=2 F 0 = 3
transition. We refer to this light as the “repump” light, and denote its detuning as R(R = 0 unless otherwise indicated). We use one single retro-reflected beam of intensity IR = 3 mW=cm2 for repumping the MOT. Both laser beams have acoustooptic
modulators (AOM) in them for temporal control of the light intensity. The magnetic
field gradient of the quadrupole field along the strong axis is 5 Gauss/cm, and can beswitched off within 3 ms.
The FORT light is generated with a home-built Titanium:Sapphire laser, witha nominal output power of 1.2 W. After passing through an AOM, the first order isintensity stabilized, collimated, and expanded. The beam is focused into the vacuum
chamber with a 20 cm focal length doublet lens to a focus of radius w0 = 26 m44(1=e2 intensity), unless stated otherwise. The detuning from the D2 line is typically
2 to 4 nm to the red. For P = 300 mW, w0 = 26m, and  = 784 nm the welldepth U0 =kB = 1:4 mK. Approximating the center region of the FORT potential asharmonic, the trap oscillation frequencies are 4:6kHz and 34Hz in the transverse andlongitudinal direction, respectively. Note that the trap as a whole is highly anharmonicat the edges. The peak photon scattering rate ( sc ) for these parameters is 1:3 kHz, ascalculated from the well known expression
sc =

1
1
I
20 1 + IS (1 + 422 )(5.2)

where the natural lifetime 0  30 ns, the saturation intensity IS  1.6 mW/cm2 , thelaser detuning is given by , and the full width half maximum of the natural linewidth 6 MHz.
5.3.1

Loading and diagnostics

The sequence for loading the FORT from the MOT and measuring the numberof atoms is as follows. The MOT fills for typically 3 s at an optimum detuning ofM = 2 , and at maximum repump intensity. This results in typically 3  107 atoms
in the MOT. We then switch on the FORT while simultaneously increasing the detuningof the primary MOT light and reducing the repump light intensity. We will refer to thisstage in the timing cycle as the “FORT loading stage,” which is typically 20 200 ms
long, depending on FORT parameters. The primary MOT light, MOT repump light andmagnetic fields are switched off, and the atoms are held in the FORT for a variable
length of time, but at least 100 ms before the detection light comes on, so that the MOT
cloud can fall out of the detection region. The atoms are then released from the FORTand detected by turning on the MOT light (primary and repump) but no magnetic field,
the primary MOT light frequency now closer to resonance M ==2. The numberof atoms N is determined from the amount of fluorescence [74] collected with a low
noise photo-diode. Under some circumstances the cloud of atoms trapped in the FORT
is so large that it extends beyond the field of view of our photo detector. In this casethe number is measured by recapturing them into a MOT. For this case the detection
must be delayed for at least 250 ms to give the MOT cloud time to fall out of the MOT
beams, thus making sure that we are detecting FORT atoms and not MOT atoms.
The number of atoms in the FORT is measured as a function of storage time, as
in the previous chapter. We call this a lifetime curve. A typical example of a lifetime
curve is shown in Fig. 5.3. Note that the number of atoms does not have a simple
exponential dependence on time. We find that the loss of atoms is well described by
Eq. (4.3) and its solution [Eq.(4.4)], as was the circular FORT. We use the analyticalsolution of Eq. (4.3) as a fit function to the data to find the number N0 initially trapped,
as well as values of and 0 (see Sec. 5.6). The prime on 0 is used to refer to atom
number loss instead of density loss.
Absorption imaging is used to measure the size and temperature of the MOT
cloud as well as the cloud of atoms in the FORT. The imaging system consists of a twoN (atoms)

45

10

6

10

5

0

1

2

3

time (s)Figure 5.3: Number of atoms remaining in the FORT as a function of time, for trap
parameters w0 = 26m, P = 290 mW ,  = 782:5 nm. The trap depth is U0 =kB =2:3 mK. The solid curve is a fit of Eq. (4.3) to the data.

lens telescope making a one to one image onto a CCD array. The lenses are 18 cm focal
length cemented doublets. The line of view is perpendicular to the FORT beam. This
allows us to observe the transverse as well as longitudinal shape of the cloud of atoms
trapped in the FORT.
5.4

Dynamics of the loading process

In Fig. 5.1 we showed that the transfer of atoms from the MOT to the FORT isa dynamical process, in which the number of atoms loaded into the FORT increases
rapidly until a competing process causes the number to reach a maximum and thendecrease at later times. Here we investigate the precise shape of the loading curve inmore detail. The number of atoms in the FORT for longer loading times is shown in
Fig. 5.4.
The shape of the loading curve is explained as follows. Initially, the number ofatoms in the FORT increases as N (t) = R0 t. But at larger times the number starts toroll over. This occurs for two reasons. First, the MOT loses atoms due to the reduced
repump intensity and different detuning of the primary MOT light. This reduces theloading rate. Second, the trap loss rates become large enough to counteract the loading.
In a similar fashion as Eq. (4.3) we find that the shape of the FORT loading curve46

6N0 ( 10 atoms )

2.0

1.5

1.0

0.5

0.0
0

1

2

3

4

time (s)Figure 5.4: Number of atoms loaded in the FORT as a function of FORT loading stage
duration, for trap parameters w0 = 26 m, P = 305mW, and  = 784:5 nm. Primary
MOT intensity IM = 8:2 mW=cm2 and MOT repump intensity IR = 4:7 W=cm2 . The
solid curve is a fit of Eq. (5.3) to the data. The loss coefficient L0 = (1:56  0:22) 
10 5 (atoms s) 1 , see text.

is well described bydN= R0 exp(dt

MOT t)

LN

0

LN

2;

(5.3)

where MOT is the rate at which the MOT loses atoms due to the change of MOT lightdetuning and repump intensity. And L and L0 characterize density-independent anddensity-dependent losses. The subscript L expresses the fact that the loss rates duringloading are generally different from those during storage in the FORT without any MOT
light present.
Four parameters determine the loading: R0 , MOT , L , and L0 . To understandthe physics of the loading process we must therefore determine these four parameters
under a variety of conditions. The initial loading rate, R0 , can be determined directlyfrom the initial slope of the loading curve. Then MOT can be determined by measuring
the rate at which the MOT fluorescence decreases during the FORT loading stage. We
confirmed this explicitly by allowing the MOT to dissipate for a variable length oftime, d , before the start of the FORT loading stage. The initial slope of the loading
curve R(d ) was observed to be R(d ) = R0 exp( MOT d ), consistent with the MOT
fluorescence measurement. The rate constants L0 and L can then be determined byfitting the numerical solution of Eq. (5.3) to the data. The solid curve in Fig. 5.4 is47such a fit, indicating how well the loading process is described by Eq. (5.3). WithR0 and MOT constrained to the directly measured values, as mentioned above, the0 2extracted values of L0 and L show that the
L N loss term clearly dominates overthe
L N term. This is most apparent in the slope of the tail of the load curve. Thus
collisional processes dominate the losses from the FORT during its loading.
The losses during the loading of the FORT can be studied independent of theloading rate. This is done as follows. The FORT is first loaded under optimum conditions and the atoms stored for 100 ms, after which the MOT lasers are switched backon, but there is no magnetic field, for the remainder of the FORT storage time. We
refer to such a measurement as a loss curve measurement. In this way we can study
the effect of the MOT light field parameters on the loss rate, i.e., we eliminated theloading term in Eq. (5.3). With the MOT lasers on, the loss rate is much larger than in
the absence of any MOT light and completely dominated by density dependent losses
0
L . The result of such a measurement is shown in Fig. 5.5 (N), which also contains
the lifetime curve () recorded in the absence of MOT light. Comparing corresponding
loading and loss curves we find the same values of L0 from the different types of data
sets to within a 30% spread. The advantage of measuring a loss curve is that we can
independently change all the MOT light field parameters without altering the numberof atoms initially loaded into the FORT.
The nice agreement between the value of L0 determined from the loading curve in
Fig. 5.4 and the loss curve in Fig. 5.5, taken for identical MOT and FORT parameters,
gives good confidence that Eq. (5.3) gives a good description of the physics involved
in loading the FORT.
The fact that collisional losses, L0 , dominate the tail of the loading curve can also
be seen in another way. If one neglects the
L N term in Eq. (5.3) and sets the loading
rate to be constant at R0 , i.e. assuming that the MOT does not lose atoms during theloading of the FORT, the steady state solution of Eq. (5.3) is

Nst =

q

R0 = L0 :(5.4)

Substituting the values of R0 and L0 obtained as discussed above, the calculated Nst
agrees to within 10% with the maxima of the loading curve.
Several consistency checks were performed on the accuracy of fitting Eq. (5.3)
to the data. One test was to set all four parameters in Eq. (5.3) free. This reproduced
the independently determined values of R0 and MOT , as well as those of L and L0 ,with L least well determined. Both MOT and L give rise to exponential decay andare therefore strongly coupled in the least squares fit. We find that most often L comesout of the fit as zero, and MOT to within 20% of the value determined directly from the
MOT fluorescence. Assuming that the value of L should at least be as big as that of
the trap in the absence of the MOT light, we set it to that value and see no significant
change in the value of L0 . Fitting the loading curves with these constraints and different
combinations of free parameters, we find a spread in the L0 values of up to 30% for agiven data set, which is more than adequate for the analysis of the data given below.
We are now in the position to investigate the loading of the FORT in terms of R0 and0
L separately, by either measuring load curves or loss curves.

48

6

N ( atoms )

10

5

10

4

10

0.0

0.2

0.4

0.6

0.8

1.0

time ( s )Figure 5.5: Number of atoms in the FORT vs time, without any MOT light () andwith MOT light on (N), primary MOT intensity IM = 8:2 mW=cm2 , and MOT repump
intensity IR = 4:7 W=cm2 . For trap parameters w0 = 26 m, P = 305mW and = 784:5 nm. The curves are fits of Eq. (4.3) to the data. The number dependent loss
rates are 0 = (1:42  0:05  10 6 (atoms s) 1 and L0 = (1:4  0:1)  10 5 (atoms s) 1 ,for the atoms stored in the FORT without any MOT and with MOT light, respectively.

5.4.1

Hyperfine repump intensity and detuning

A commonly used technique to improve the loading of a FORT is to reduce thehyperfine repump intensity. Here we show how the intensity and detuning of the repump light affect the loading rate R0 as well as the loss rate L0 . We concentrate firston the dependence of the loading rate and loss rate on the hyperfine repump intensity
at resonance (F = 2 ! F 0 = 3). The data in Fig. 5.6 show that for the maximum IM
there is a critical repump intensity below which the MOT is not sustained during the
FORT loading stage and the loading rate goes to zero. For IR above that critical value,
the loading rate decreases because the density in the MOT decreases due to radiative
repulsion. The optimum level is about IR = 5 W=cm2 . Note that the loading rate of
the FORT is as high as 5  107 atoms s 1 , which is a factor 2 higher than that of the
MOT. The loss rate L0 increases rapidly with repump intensity and starts to saturate athigh repump intensity. For lower IM , saturation sets in at lower repump intensities and0
L is smaller.
The optimum repump intensity is intimately linked to the repump detuning. This
interdependence is shown in Fig. 5.7, where the maximum number of atoms loaded
into the FORT is plotted versus R for different values of IR . Increasing IR from49

7

R0 ( 10 atoms / s)

6

4

2

(a)
0
10

20

30

40

-1

β 'L (10 (atoms s) )

0

-6

100

10

(b)
1
0

20

40

60

80

2

IR (µW/cm )Figure 5.6: FORT loading rate R0 (a) and loss coefficient L0 (b) as a function of hyperfine repump light intensity. For trap parameters w0 = 26 m, P = 300mW and = 784:5nm. The loss rate was measured for IM = 8 mW=cm2 (), reduced MOT intensity of IM = 1 mW=cm2 (2), and complete absence of primary MOT light (N). The
solid lines are derived from the model described in Sec. 5.7.1. Error bars are statisticaland do not reflect systematic uncertainties in IM .

50
1.5
F'=3

1.0
F=2

6

N0 ( 10 atoms )

F=2

F'=2

0.5

0.0
-100

-50

0

50

repump detuning ( MHz )Figure 5.7: Number of atoms in FORT as a function of MOT hyperfine repump detuningfor different repump intensities, 1:4 W=cm2 (), 4 W=cm2 (2), and 280 W=cm2(). The frequency offset is arbitrary. FORT parameters are w0 = 26 m, P =200 mW, and  = 798:1 nm. Solid curves are Lorentzians fitted to the data to guidethe eye.

1:4 W=cm2 to 4 W=cm2 , we see that the number in the trap increases, but the optimum repump detuning stays on resonance. Increasing the intensity to 280 W=cm2
causes the optimum repump detuning to shift red. Interestingly, the optimum number ofatoms is loaded when the repump scattering rate is the same for both IR = 4 W=cm2
and IR = 280 W=cm2 due to the different detuning, indicating that an optimum repump scattering rate exists for loading the FORT. However, the number of atoms drops
and the resonance broadens, which shows that the number of atoms in the FORT is notdetermined by the repump scattering rate alone. This will be explained in more detailin Sec. 5.8.
In summary, the loading rate R0 is optimal for a very low repump intensity of
about 5 W=cm2 and zero detuning. The loss rate L0 increases with repump intensity
and is larger for higher intensity of the primary MOT light.
5.4.2

Primary MOT light intensity

Both R0 and L0 depend on the primary MOT intensity, IM . The data in Fig. 5.8
show that L0 rises rapidly and then saturates with increasing IM . The MOT intensity
at which saturation sets in is higher for higher repump intensities. Combined with thedependence of L0 on repump intensity this suggests that excited state collisions leading

51

6

7

R0 ( 10 atoms / s )

(a)
4

2

0

(b)

-6

-1

β 'L ( 10 (atoms s) )

30

20

10

0
0

2

4

6

8

2

IM ( mW/cm )Figure 5.8: FORT loading rate R0 (a) and loss coefficient L0 (b) as a function of primary MOT light intensity, for a fixed repump intensity IR . Both ( ) and () represent
IR = 5 W=cm2 , while in (2) IR = 0. The solid lines are derived from the models in
Secs.5.7.2 and 5.7.1 for (a) and (b), respectively. FORT parameters are those of Fig.
5.6.

to radiative escape are responsible for the losses during loading, as is confirmed by themodel that we develop in Sec. 5.7. The solid curves in Fig. 5.6 are based on this modeland agree very well with our data.
As shown in Fig. 5.8, the loading rate R0 increases nearly linearly with IM . As
will be argued in Sec. 5.7, this implies that the loading rate strongly depends on thecooling mechanisms in the MOT.

52

(a)

-1

R0 ( 10 atoms s )

3

7

2
1
0

-6

-1

β 'L ( 10 (atoms s) )

25

(b)

20
15
10
5
0

-100

-80

-60

-40

-20

0

MOT detuning ( MHz )Figure 5.9: FORT loading rate R0 (a) and loss coefficient L0 (b) as a function of the
primary MOT light detuning, M , during loading. The repump intensity is 5W=cm2 .
For trap parameters w0 = 26 m, P = 300mW, and  = 784:5nm.

5.4.3

The role of MOT detuning

Both R0 and L0 also depend on MOT detuning, as shown in Fig. 5.9. For these
data, IR is reduced to the value that gives optimum loading into the FORT. A maximum
in R0 is observed at about M  30 MHz. At slightly larger detunings the loss
rate L0 has a minimum. The maximum number of atoms is loaded into the FORT at a
detuning that is simultaneously close to the maximum in the loading rate and close to
the minimum in the loss rate.

53

MOT detuning ( MHz )

-15

-20

-25

-30

-35
0.0

-0.4

-0.8

-1.2

trap depth ( mK )Figure 5.10: MOT detuning for maximum number of atoms loaded into the FORT
versus trap depth, w0 = 26 m.

We also observe that the optimum detuning of the primary MOT light during theloading stage depends on the depth of the FORT. This is illustrated in Fig. 5.10. For
deeper traps the optimum detuning is smaller. The FORT depth was varied by changing
the wavelength and power of the FORT light.
5.4.4

Alignment with respect to the MOT

Another factor affecting the FORT loading is the relative alignment of the FORT
with respect to the MOT. We find that the loading rate is optimum with a longitudinal
displacement between the center of the FORT and the MOT. The optimum displacement
depends on the FORT depth. Absorption imaging is used to determine the separation
between the FORT and the MOT. We observe that as trap depth increases, the displacement must increase in order to maximize the number of atoms loaded. Typicallythe displacement between the focus of the FORT and the MOT is about half a MOT
diameter.
In shallow traps the displaced loading of the atoms causes an initial sloshing of
the atoms in the longitudinal direction. The sample of atoms can be seen to make one
and a half oscillations before it thermalizes at the center of the FORT; this takes about
100 ms.

54

6

N0 ( 10 atoms )

8

6

4

2

0
0

-1

-2

-3

-4

-5

-6

trap depth ( mK )Figure 5.11: Number of atoms loaded into the FORT as a function of trap depth for
w0 = 26m () and w0 = 22m (N). For the latter data set the steady state number[see Eq. (5.4)] is calculated from R0 and L0 taken from Fig. 5.12, is plotted as Æ.

5.4.5

Dependence on FORT depth

We measured how the loading rate, loss rate, and total number of atoms trappedin the FORT depends on the trap depth. This is shown in Figs. 5.11 and 5.12. For
each data point, the MOT detuning, repump intensity, and alignment of the FORT withrespect to the MOT are optimized to give the highest number of atoms in the FORT.
The waist of the trapping beam is fixed and the trap depth is varied by changing both
the detuning and power of the FORT beam. The dependence of N0 on trap depth canbe explained as an increase in R0 and a reduction of L0 as the trap gets deeper. This is
illustrated by the data shown in Fig. 5.12.
For comparison to the data in Fig. 5.11, we calculated Nst using Eq. (5.4) byinserting the measured R0 and L0 . The calculated values Nst (Æ in Fig. 5.11) agree
well with the measured number, and depends linearly on trap depth up to 3 mK. At 6
mK, N0 lies below the extrapolated straight line because L0 cannot decrease below its
value in the absence of MOT light.
For a waist of 26 m we have observed transfer efficiencies from the MOT to
the FORT of 20%. For traps with a larger waist, w0 = 40 m, transfer efficiencies
of 40% and higher have been observed. If we compare FORTs with about the same
depth of U0  1 mK but different waists, we observe that more atoms are loaded when
the waist is bigger. This is mainly due to an increase of the loading rate (see Sec.
5.7.2). Changing the waist from 26 m to 75 m, we observed a strong reduction of

55

(a)

-1

R0 (10 atoms s )

12

7

8
4

(b)

6
4

-6

-1

β 'L ( 10 (atoms s) )

0

2
0
0

-1

-2

-3

-4

-5

-6

trap depth ( mK )Figure 5.12: FORT Loading rate R0 (a) and loss rate L0 (b) as a function of trap depth
U0 for w0 = 26 m.

56
1.6

N0 ( 10 atoms )

1.2

6

0.8

0.4

0.0
0.0

0.5

1.0

1.5

2.0

time ( s )Figure 5.13: Number of atoms in the FORT as a function of loading time for threedifferent ellipticities of the FORT light. In descending order of the curves,  =0:999; 0:915; and 0:852. The FORT parameters are w0 = 22m, P = 600 mW, and = 784:5 nm.

the lifetime of the trapped sample. We attribute this reduction to the fact that in thisregime, the scattering force becomes stronger than the trapping force and expels atoms
from the trap[38].
5.4.6

The effect of elliptically polarized FORT light

As the data in Fig. 5.13 show, the polarization of the FORT light has a profoundeffect on the number of atoms loaded into the FORT. A change of the ellipticity (See
Section 4.2) from linear polarization,  = 0:999, to slightly elliptical polarization,  =0:915, causes the number of atoms to drop by a factor of 7. At  = 0:852 the number
dropped by more than an order of magnitude. The drop in number is caused by a
combination of effects. Analysis of the shape of the load curves in Fig. 5.13 showsthat the loading rate is reduced by an order of magnitude and the exponential loss rate
L is increased by a factor of 4. The increase of L we ascribe to the ground state
dipole force fluctuations related to the optical Zeeman splittings [second term on righthand side of Eq. (4.1)] induced by the ellipticity of the light [41]. The optical Zeeman
splittings can also be expected to interfere with the functioning of the MOT and thus
reduce the loading rate. In the center of the FORT the splitting corresponds to that of amagnetic field of 13 Gauss, which is sufficient to disrupt both the cooling and trappingof the MOT.

57

σ0

t=0

time

(a)

Width σ (µm)(b) 400
300
200
100
0
0.0

0.5

1.0

1.5

2.0

FORT storage time (ms)Figure 5.14: (a) Illustration of ballistic expansion and (b) an example of a temperature
measurement.

5.5

Temperature

We measured the temperature of the atoms in the FORT for different trap depths.
The temperature T of the atoms in the FORT was determined from the rate at which thecloud of trapped atoms expands after release. The density distribution is well describedby a Gaussian of width  that increases with expansion time t, as illustrated in Fig. 5.14(a). The resulting  vs.
q expansion time is plotted as in Fig. 5.14 (b). By fitting this dataBT 2t , a temperature is readily extracted.
to the equation  = 02 + km
The temperature extracted from the transverse dimension was found to depend on
potential well depth, as shown in Fig. 5.15. Although the transverse and longitudinaltemperatures agree to within the experimental uncertainties, the transverse dimension

58

temperature ( mK )

0.8

0.6

0.4

0.2

0.0

0.0

-0.5

-1.0

-1.5

-2.0

trap depth ( mK )Figure 5.15: Temperature of the atoms in the FORT as a function of trap depth, for
w0 = 26 m. Solid line is a linear regression giving a slope T=U0 = 0:4.

is a much more sensitive probe of the temperature due to its smaller initial size. The
initial aspect ratio of the cloud is 33 to 3000 m. Note that for the deeper traps thetemperature is much higher than that of the atoms in the MOT, which is about 30 K
to 120 K, depending on the MOT detuning. Over the range of trap depths that weinvestigated the temperature is a fixed fraction 0.4 of the FORT depth.
5.6

Trap Lifetime

The number of atoms in the trap decreases after the trap is loaded, as is shown in
Fig.5.3. In the absence of any MOT light the loss of the trapped sample has a number
dependent (collisional) portion and a purely exponential part, as described by Eq. (4.3).
Typically, the exponential lifetime 1= is between 1 s and 10 s, depending on thetrap parameters. The lifetime increases with decreasing FORT scattering rate, until alimit ( 10 s) is reached that we believe is set by the background vapor pressure.
The collisional loss, characterized by 0 , depends on the wavelength and waist
size of the FORT beam. We observe clear resonances in 0 , consistent with photoassociation lines. Between these lines we see a nonzero background level that presumably
is due to ground state hyperfine changing collisions. This background level depends on
the FORT light polarization. In a circularly polarized FORT [41], the atoms are spinpolarized in one of the stretched states of the F = 3 ground state, from which hyperfinechanging collisions are suppressed. In our trap we see a factor of 2 reduction of 0
in between the photo-association lines when the polarization is changed from linear to59
circular.
5.7

Analysis

5.7.1

Analysis of density dependent losses

The main features that we have observed in our data are the strong increase(100) of the collisional loss coefficient L0 when the MOT light is present and that thisloss rate depends strongly on the MOT parameters. As we will show in this section,
density dependent losses during the loading of the FORT are mainly due to radiative
escape collisions induced by the MOT light. In the unperturbed FORT, ground-state
hyperfine changing collisions are the dominant loss process.
5.7.1.1

Conversion to density dependent loss

The typical mechanism driving the collisional loss process in the FORT is moreeasily identified when the measured rate coefficients 0 and L0 are converted to density
related, volume independent rate coefficients and L . The relation between these is= 0 V , with V the volume of the sample of atoms. The volume is found by approximating the trapped sample of atoms as a cylinder with radius and length determined by
the size of the FORT beam focus and the temperature of the atoms. The volume is thengiven by,r

V = w02 zR ln(

1

1 

)



1 

;

(5.5)

where  = kB T=U0 is shown in Sec. 5.5 to be a constant 0.4, regardless of the FORTparameters. Thus the volume only changes when w0 changes.
5.7.1.2

Density dependent loss from the FORT without the MOT present

Density dependent losses from the FORT when there is no MOT light present
are due to both photo-association and ground state hyperfine changing collisions, asmentioned in Sec. 5.6. Between photo-association lines, the values of 0 do not depend
strongly on the FORT laser power or wavelength, but change strongly with w0 . However, when 0 is multiplied by V / w04 , the resulting = (4  2)  10 12 cm3 s 1 ,independent of w0 . This value agrees well with the value of 4  10 12 cm3 s 1 reportedby Miller et al. in Ref. [80].
5.7.1.3

Density dependent loss from the FORT during loading

In the case of loading the FORT, the density dependent losses are due to lightassisted collisions. To demonstrate this, we compare with measured rates in MOTs,
where light-assisted collisions have been extensively studied. This comparison is complicated by the difference in trap depth between the MOT and the FORT (1 K vs. 1
mK), and also the difference in fractional occupation of the upper hyperfine component

60of the ground state. In the case of a MOT the fractional population of the lower hyperfine component is negligible, and the loss term is written as n2 , where n is the total
density of atoms in the MOT. Then for constant temperature, depends linearly on theintensity of the assisting light.
These light-assisted collisions, as described by Gallagher and Pritchard [81], aredue to two mechanisms: fine-structure changing collisions and radiative escape. It has
been shown that these two mechanisms contribute with the same order of magnitude
to the trap loss [82, 83] in 1 K deep MOTs. Radiative escape is predicted to scale
5=6
almost inversely with trap depth (  U0 ) [83, 84] for trap depths  1 K, but
the fractional contribution of fine structure changing collisions should decrease withdecreasing trap depth. Therefore, we can neglect the contribution from fine structurechanging collisions to in describing the losses that occur during loading of the 1 mK
deep FORT.
The fractional ground state population affects the loss rate in the following way.
For radiative escape to occur, at least one of the two colliding atoms must be in theupper hyperfine ground state (F = 3). The MOT light is too far detuned to excite an
atom from the lower ground state (F = 2) to a higher lying molecular state during acollision. Therefore, the loss term takes the form of n3 (n2 + n3 ), where n3 is thedensity of atoms in the F = 3 state, n2 is the density of atoms in the F = 2 state, andn2 + n3 = n.
The fraction of atoms in the F = 3 state depends on the relative optical pumping
rates of the primary MOT laser (F = 3 ! F 0 = 3 ! F = 2) as compared with therepump laser (F = 2 ! F 0 = 3 ! F = 3). We derive the fraction of atoms in the
F = 3 state from a simple two level rate equation model, which results in n3 =n =IR =(IR + aIM ), where a is a constant that reflects the relative optical pumping rates.
These rates depend on the AC stark shifts induced by the FORT, in addition to ClebschGordon coefficients and the frequency of each laser. The average shift in the transition
frequency for atoms in the FORT (AC ) was extracted from Fig. 5.7 and found tobe AC = 2.3 . This results in a = 0:02. The experimentally determined loss rate
coefficient is then,0

L=

K IM IR;
V IR + aIM(5.6)

where K is a constant related to the density-dependent loss rates independent of opticalpumping effects. We again use the volume V given by Eq. (5.5) to make the translationfrom the typical density-dependent loss rate to the loss coefficient that we measure in
our experiments. For w0 = 26 m we find V = 1:3  10 6 cm3 .
The behavior described by Eq. (5.6) is clearly seen in the measured dependenceof L0 on both IM and IR as shown in Figs.5.6(b) and 5.8 (b). The solid curves in theseplots are fits of Eq. (5.6) to the data, with K the only free parameter. The fits to thedifferent data sets give K = (1:4  0:7)  10 16 cm5 mW 1 s 1 , where the spread isdue to the different values we find by fitting different data sets. We see that Eq. (5.6)
gives a good description of the measured MOT and repump intensity dependencies ofthe loss rate during loading of the FORT.
For a comparison of our numbers with those normally found in MOTs we set

61n3 =n = 1, assuming all atoms are in the F = 3 ground state as they are in the MOT
studies with which we want to compare. Thus L = K IM . For IM = 10mW=cm2 wefind L = 1  10 9 cm3 /s. In contrast, a MOT with comparable powers and detunings
has a loss rate of 1 4  10 12 cm3 =s [85, 86, 87, 84], which is a factor of 250 to 1000

smaller than our measured value. As shown below, this difference is due to the smaller
depth of the FORT.
In order to explain the increased loss rate observed during loading of the FORT
as compared to a MOT we will discuss in more detail how the FORT depth affects theradiative escape loss rate. In the case of a MOT, the dependence of , due to radiative
5=6
escape, on the trap depth is predicted to go as U0
in the limit of large trap depths[83, 84]. However, some of the assumptions used to derive this dependence break down
at smaller trap depths. Therefore, to calculate L for the FORT we perform a numericalcalculation of the radiative escape process, using the semi-classical Gallagher-Pritchard(G-P) model [81]. Given the uncertainties in our measured L , a more detailed calculation is unwarranted. Therefore, we do not include angular momentum considerations
or enhanced survival probability as described by Julienne and Vigue [83] or any corrections for the hyperfine splittings as introduced by P. Lett et al. [82]. Instead, we
approximate all of the involved intramolecular potentials as a single C3 =r 3 in the GP
model as formulated by Peters et al. [88] and integrate numerically.
Figure 5.16 shows an approximate intramolecular potential for Rb. For initial
excitation at radius r0 , atoms are accelerated toward one another for some time before
spontaneous emission leaves two ground-state atoms with more kinetic energy thanthey had initially. If the kinetic energy picked up in the collision exceeds the depth of
the trap, both atoms will be lost. The radius at which the pair acquires exactly enoughenergy to be ejected from the trap is therefore called the critical radius rc , and the timerequired for the atoms to reach this is called t0 .
Once the atoms reach rc , acceleration is very fast, and the atom pair spends a
short time t1 accelerating to r = 0. If spontaneous emission does not occur during
this second time interval, the intramolecular separation will oscillate between 0 and
r0 , making multiple orbits until spontaneous emission occurs. The probability of alossy radiative escape event thus depends on the excitation probability at r0 and theprobability of decay occurring when r < rc . Following Refs.[88] and [89] we find/Z

4r02 dr0 G(r0 )PRE (r0 ) ;(5.7)

where the photon scattering rate of atoms in the F = 3 state, G(r0 ), is a function of theeffective detuning (r0 ) = C3 =(r03 ~) + M with respect to the unperturbed atomic
transition frequency. The probability of a radiative escape event resulting in trap loss is
PRE = sinh( t1 )= sinh[ (t0 +t1 )], which reflects the important contribution of multiple
orbits [88].
The result of the numerical integration of Eq. (5.7) for different trap depths is
5=6
show in Fig. 5.17. The dashed line indicates a dependence of  U0
as predicted
by Refs. [83] and [84]. This clearly tracks our numerical integration at large trap depths,
where it is expected to be valid. Examination of the change in with U0 shows that

62
0
0

energy ( µeV )

-50

t0

-1

t1

-2

-100

rc

-150

rc r0

t1
0

500

1000

r0

800

1000

t0
-200
0

200

400

600

r ( angstroms )3

Figure 5.16: Approximate intramolecular potential for Rb, with C3 = 71 Å eV. The
large graph shows the critical radius, rc , for U0 =kB = 1 K, while the inset depicts thesame parameters for U0 =kB = 1 mK.

for a change in U0 from 1 K to 1 mK, increases by a factor of 50 (for n=n3 = 1).
This ratio is at least a factor of 5 smaller than the experimentally observed ratio of 250.
However, given the large uncertainties in the experimental determination of n and n3 =n,
as well as the approximations that go into the calculations and the steep dependenceof the results, the measured and calculated numbers are not inconsistent. Therefore
we believe that the enhanced loss rates we observe during loading are consistent withlight-assisted collisions.
5.7.2

A model for the FORT loading rate

The initial loading rate is the flux of atoms into the volume of the FORT times
the probability for an atom in this volume to become trapped;

1
R0 = nMOT vAPtrap ;(5.8)
4where nMOT is the density in the center of the MOT, v = kB T=m is one dimensional
component of the root mean square thermal velocity of the atoms in the MOT, A isthe effective surface area of the FORT, and Ptrap is the trapping probability. For the
discussion below we will assume that the MOT density is constant in time, i.e. nMOT
is the density at the start of the loading stage.
The dependence of R0 on MOT detuning is plotted in Fig. 5.9. The loading
rate shows a maximum at M = 30 MHz, the same detuning at which the measured

63

-9

10

-1

β ( cm s )

-10

10

3

FORT

MOT

-11

10

-12

10

-4

10

-3

10

-2

-1

10
10
trap depth / -1 ( K )

0

10

1

10

Figure 5.17: Radiative escape loss rate coefficient calculated as a function of theabsolute value of the trap depth. When the atoms have zero initial kinetic energy, (N)correspond to values for AC + M = 2 and () to 8 . (2) depict 8 , but with an
initial relative velocity v = kB T=m, corresponding to T = 800 K. The dashed line
5=6
represents / U0
as mentioned in the text.

p

product nMOT T plotted in Fig. 5.18 has a maximum. This supports the idea that
R0 / nMOTv, assuming that APtrap is constant with MOT detuning.
The loss rate L0 has a minimum at a slightly larger detuning. The maximum
number of atoms will be loaded into the FORT at a detuning close to the maximum
in the loading rate and simultaneous minimum in the loss rate. This is illustrated bysubstituting the measured R0 and L into Eq. (5.4). Thus calculated the number ofatoms transfered into the FORT clearly shows a maximum at a detuning of 35 MHz (seeFig. 5.18). In the same figure we plotted the actual measured number of atoms. The
agreement is very good over the detuning range of interest, 0-30 MHz. The difference
between the calculated and measured numbers at large detunings is caused by the rapid
loss of atoms from the MOT, which prevents N from reaching the value predicted by
Eq. (5.4).
p
The loading rate R0 depends on A, in addition to nMOT T . For a trap witha waist of 26m the Rayleigh range is 1:5 mm. This is a few times larger than the
diameter of the cloud of atoms trapped in the MOT, DMOT . This means that the FORT
radius hardly changes over the width of the MOT. Therefore we approximate A by
the surface area of a cylinder with length given by the MOT diameter DMOT and an
effective radius re , that is a function of z and depends on the FORT waist. The radius

64

8

(a)

6

1/2

11

-3

1/2

nMOT T (10 cm µK )

10

4
2
0

6

N0 ( 10 atoms )

2.0

(b)

1.5
1.0
0.5
0.0
-100

-80

-60

-40

-20

0

MOT detuning (MHz)Figure 5.18: Product of MOT density and the square root of temperature (a) and number
loaded into FORT (b) as a function of the MOT detuning during FORT loading. The
actually measured number () and the number calculated from Eq. (5.4) (Æ) are bothshown in (b). The FORT parameters are those of Fig. 5.9.

65
80

radius reff ( µm )

60

40

20
(a)

(b)

(c)

(d)

0
0

5

10

position z ( mm )Figure 5.19: Equipotential contours of the FORT for E=kB =1; 2; 3; and 4 mK depth , (a)-(d) and w0 = 26 m.

100 K and U0 =kB =is set by the coordinates at which the FORT potential becomes the same size as thetemperature of the atoms in the MOT, U (re ; z ) = Uc  kB T . The effective radiusre (z ) of the FORT, depending on z and Uc is then



 12

U
1
re (z ) = w0 (1 + (z=zR )2 ) ln( 0)Uc 2(1 + (z=zR )2

:

(5.9)

In Fig. 5.19 this equipotential contour is plotted for different values of U0 and U (re ; z ) =100K. The bow tie shape of these contours explains why the best loading is achieved
when the FORT and MOT are displaced. The radius has a maximum away from the focus. Moreover, for deeper traps the z at which the maximum radius occurs shifts tobigger z values, just as in the data.
A typical initial MOT density during the FORT loading stage is 2  1011 cm 3 ,and the temperature ranges from 30 K to 100 K depending on detuning of the primary MOT light. A reasonable value for the average speed in one dimension is thusv = 5 cm=s. The MOT diameter is 500 m (FWHM). For U0 = 1 mK and Uc =100 K the area A = 1  10 3 cm2 . Using these numbers the loading rate is
2:5  108 s 1 if the trapping probability Ptrap = 1. The loading rate measured forsuch a trap is 3  107 s 1 , implying Ptrap  0:1.
The data in Fig. 5.8 show that the loading rate increases nearly linearly with
MOT intensity. However, it is known that the MOT temperature and density in the SubDoppler regime scale as intensity and the inverse square root of intensity, respectively[74, 90].

66
The product nMOT v is thus expected to show no strong dependence on MOT
intensity. We have experimentally confirmed this by measuring the density and temperature of the cloud of MOT atoms as a function of MOT intensity. Over the rangeof MOT intensities as in Fig. 5.8 we find that nMOT vDMOT is nearly constant. Therefore, we conclude that the observed increase of the loading rate with increasing MOT
intensity arises from an intensity dependence of the trapping probability Ptrap . From
the data we find Ptrap  0:1
We estimate Ptrap by using the following simple model. The atoms come into
the trapping volume with a narrow distribution of velocities around an average velocityv. To be trapped, an atom’s energy has to be reduced to below the edge of the trapping
potential. The time scale for this to happen must be about half an oscillation period =2of the FORT in the strong direction, which is 150 s for the above mentioned trap. The
scattering of MOT laser photons affects the atomic velocity in two ways. First, the atom
scatters photons that leads to heating, that is, a broadening
q of its velocity distribution.
The spread in velocity due to photon scattering is  = 13 ( ~mk )2 scat =2. Second, thefrictional component of the scattering leads to a reduction of the atom’s average kinetic
energy. The rate at which the atom loses
is E_ =v2 . This damping leads topenergya change in average velocity of v = v m in the characteristic time =2.
The probability of being captured is the integral of the shifted and broadened
distribution
Z 1

Ptrap =

v v

p1

2

exp[

v2
]dv :
2 2

(5.10)The integral leads to1v
Ptrap = 1 erf[ p (1
2
2

r


)] :
m

(5.11)For M = 30 MHz and IM = 6 mW=cm2 the MOT scattering rate scat = 9:5  105 s 1 ,which makes  = 6 cm=s.
We obtain a value of from the formula given in Ref.[84]. The damping rate
calculated from this formula increases linearly with MOT intensity for the velocities
we are interested in. The values of and  also depend on MOT detuning. Thereis a strong spatial dependence of the effective MOT detuning due to the light-shift
induced by the FORT. We make the crude approximation that the MOT is shifted out of
resonance by half the FORT light-shift. This results in v= = 2 and =m = 200 s 1 .
The result of this model is plotted as a solid curve in Fig.5.8 and shows good agreement.
This is a very approximate treatment but it clearly supports our general interpretationof the loading process.
5.8

Discussion

From our understanding of the loss process and the loading rate we are now ina position to explain the dependencies presented in Sec. 5.4. We saw that more atoms

67
are loaded into deeper FORTs, because the loading rate increases and the loss rate decreases. The loading rate increases because the effective FORT radius increases withtrap depth, which enhances the flux of atoms into the trap. The loss rate decreases because the probability of light-assisted collisional loss decreases when the trap is deeper.
In addition, the light-shift of the trap increases the effective detuning of the MOT lightand repump light for the atoms in the trap, which reduces their excitation rate, andtherefore reduces radiative escape.
The data in Fig. 5.10 shows that the MOT detuning at which loading is optimized is smaller for deeper FORTs. A deeper trap means larger light shifts of theatoms. The MOT detuning and the repump detuning are both shifted blue, so that the
detuning of both lasers from the atoms becomes more negative (red) in both cases. Dueto the increased effective detunings the MOT cooling rate decreases and thus the trapping probability decreases. By choosing a smaller MOT detuning the loading rate isincreased.
The dependence on MOT detuning of the L0 that we measured for a given trap
depth (see Fig. 5.9) is also well described by Eq. (5.6). The detuning dependence entersthrough the parameter a. At small MOT detunings, the MOT excitation rate is highest
and therefore L0 is large. By increasing the MOT detuning further a becomes smaller.
Therefore aIM becomes negligible compared to IR and L0 becomes independent of
detuning.
In Sec. 5.4.1 it was shown that loading is optimized for repump light of very
low intensity that is tuned on resonance. We saw that the number of atoms in the
FORT dropped when the repump intensity was increased but the repump scattering rate
was maintained by changing the repump detuning. Increasing the repump intensity
increases the loss rate. Although the detuning is increased for the atoms in the MOT
such that the scattering rate is the same for the MOT atoms, the increased intensity stillcauses an increased scattering rate for the atoms trapped in the FORT.
5.9

Enhanced and quasi continuous loading

The previous data and analysis shows that L0 is very small when the repump
power is zero, and reducing the primary MOT intensity also helps to reduce the loss rate.
However, reducing these light intensities also reduces the loading rate. It is possible toimprove the ratio of loss to loading rate by changing the beam geometry. The loss rate
only has to be eliminated in the volume where the FORT is, which can be done byinserting obstructions to block portions of the MOT beams and the repump beam.
To demonstrate this concept, we make a shadow in the repump beam. For thispurpose an additional repump beam is used that co-propagates with the FORT beam.
A 0:6 mm opaque disk in this beam is imaged into the MOT and FORT region, suchthat the shadow covers the length of the FORT. The peak intensity in this repump beam
is increased from 5 W=cm2 to 460 W=cm2 , which is sufficient to prevent the MOTfrom being lost while loading the FORT. The MOT is still loaded as before using a
repump beam without a shadow. During the loading of the FORT, that repump beam isswitched off, and the shadowed beam switched on. In Fig. 5.20 we show two measured68

6N0 ( 10 atoms )

8

6

4

2

0
0

1

2

3

4

time ( s )Figure 5.20: Number of atoms loaded into the FORT as a function of loading time,using normal loading (Æ) and enhanced loading with a shadowed repump beam ().
The FORT parameters are w0 = 58 m, P = 580mW and  = 783:2nm. The solid
curves are fits of Eq. (5.3) to the data.

loading curves. One shows loading using the conventional method without the shadow
and with reduced repump power, and the second curve shows the new method withthe shadowed beam and increased repump power. The maximum number of loaded
atoms doubles from 3:9  106 to 7:7  106 . The transfer efficiency from MOT to FORT
increased from 21% to 42 %.
The increase in the number is caused by a decrease of the loss rate from L0 =(3:5  0:5)  10 6 s 1 to L0 = (1:7  0:3)  10 6 s 1 and a simultaneous increase of
the loading rate, from R0 = 5:8  107 s 1 to R0 = 10:2  107 s 1 , respectively. The
loading rate is increased due to the fact that the MOT density increases by reducing therepump intensity in the center of the MOT.
The loss rate L0 is still larger during loading with the shadow than in the absenceof all MOT light, where the loss rate 0 = 2:3  10 7 s 1 . This is due to the fact that
the MOT light is not shadowed and the FORT acts as a very weak repump laser. Thus
putting additional shadows in the main MOT light may improve the loading even more,
but this requires a more complicated optical setup. By reducing L0 to the limit of no
MOT light, while maintaining the same loading rate, the steady state number of atoms
would be 2  107 , which would be nearly all of the MOT atoms.

69
5.10

Conclusions

Our study can be summarized as follows. Deeper traps load more atoms and thetemperature is a fixed fraction of the trap depth. Optimum loading is achieved for very
low repump scattering rate (IRP = 5 W=cm2 ), and a MOT detuning that depends
on the FORT depth. Controlling the geometry of the overlap of the MOT and FORT
beams gives substantial improvement. To maximize the number of atoms trapped in the
FORT, these are the parameters to adjust.
However, underneath this recipe lies a lot of interesting physics. Loading the
FORT from a MOT is a dynamical process, governed by a loading rate R0 and density
dependent losses characterized by L0 . During the loading the MOT light fields increase
the FORT loss rate considerably. The main loss mechanism during loading of the FORTis radiative escape collisions induced by the MOT light fields. The loss rate is higher
than in a MOT because the FORT is only on the order of a mK deep, whereas a typical
MOT is a K deep. In addition, the light-shift due to the FORT changes the balance of
optical excitation of the MOT trapping and cooling light, which is responsible for theradiative escape, and the hyperfine repump rate. Taking these effects into account wewere able to model how the loss rate depends on the intensities and detunings of the
primary MOT and repump lasers, as well as the FORT depth. This model explains theexperimentally observed dependencies of the loss rate on these parameters very well.
The loading rate can be described in two parts: a flux of atoms into the volume of the
FORT times a probability of being trapped. Both parts depend on the MOT and FORTparameters, including the size of the FORT.
Here we studied the loading of a FORT with a radius  60 m, which is muchsmaller than the radius of the MOT. Even with this mismatch of size between the MOT
and the FORT we were able to transfer more than 40% of the atoms initially trappedin the MOT to the FORT. For FORTs with a radius comparable to that of the MOT thedescription of the loading rate changes, and also the loss rate may behave differently.
However, we believe that radiative escape processes may still be enhanced as compared
to a MOT. Working with a small waist enables one to make deeper traps and achieve
tighter confinement. Such conditions may make it possible for instance to reach Bose
Einstein condensation at high temperatures of tens of micro-Kelvin.

Chapter 6
Cooling in the Circularly Polarized Optical Dipole Trap

6.1

Motivation

Cooling of atoms in a trap using completely optical techniques is an important
pursuit of atomic physicists. Optical traps offer rapid control and tight confinement.
Optical cooling techniques allow the cooling process to be decoupled from the thermal
collisional process, allowing very low temperatures to be achieved while minimizing
potentially lossy collisions. A circularly-polarized dipole trap offers a variety of nondegenerate states in which to work, much like a magnetic trap does. Thus many coolingmechanisms previously available only in magnetic traps are available in the circular
FORT, and can be implemented optically.
Many schemes exist to cool atoms in red-detuned optical dipole potentials. Evaporative cooling via relaxation of the potential has been used to reach temperatures of 4
K [76], but so-called run-away evaporation was not possible in this scheme because
the density of atoms decreases as the potential increases. Raman cooling has also been
demonstrated in a dipole trap formed with YAG lasers [91].
Many techniques are employed in optical lattices as well. They include Raman
cooling, evaporative cooling, adiabatic expansion, sideband and degenerate sideband
cooling [92, 93]. However, optical lattices can be difficult to construct and leave atoms
in many different potential wells instead of the same potential well.
Sisyphus cooling refers to any cooling process in which an atom dissipates kinetic energy by physically moving into a region of space in which it loses kinetic energy
to potential energy, then changes its internal state to one with smaller potential energy
before it leaves the elevated potential. In this way, the atom can be described as continually rolling up hill, much like the mythical Greek Sisyphus was forced to push aboulder eternally uphill. Many different types of Sisyphus cooling exist. It is an essential part of polarization-gradient cooling, which manifests itself as the sub-Doppler
cooling mechanism in MOTs [12, 13]. In addition, gravitational Sisyphus cooling [45]employed the Sisyphus cooling concept to two potentials with nearly the same curvature, but displaced from one another. Evanescent wave cooling [94, 95] employs themechanism as atoms bounce off an evanescent wave atom mirror. Sisyphus coolingis in general much more efficient than Doppler cooling, because a huge fraction of an
atom’s total energy can be carried away for the cost of 1 recoil photon. The ratio of thepotential energies of the two internal states involved sets the amount of energy that can

71
be extracted in an event. A Sisyphus cooling mechanism clearly presents itself in thecircular FORT, where internal energy levels are non-degenerate and separated by much
more than the energy of one photon recoil.
Blue Sisyphus cooling (BSC) has been successfully employed to load a dipole
trap [69]. Atoms were successfully loaded into linearly polarized YAG traps with temperatures of 2 – 10 K. In this way, phase space densities of over 10 3 were achieved
with 104 atoms in an individual potential. The BSC scheme is more similar to the Sisyphus cooling of MOTs, in that the potentials used for cooling are applied by additionallasers, and not supplied by the confining light.
Evaporative cooling is also a very powerful cooling mechanism. By removing
the highest energy atoms and allowing the remainder to rethermalize, temperatures ofalkali atoms have been reduced many orders of magnitude. In fact, to date it is the only
cooling mechanism that remains powerful enough over many orders of magnitude intemperature to achieve Bose Einstein condensation [1]. Evaporative cooling has also
been demonstrated in optical dipole potentials, but not without at the same time reducing the spring constant of the trap [76]. Unfortunately, this reduction simultaneously
reduces the confinement of the potential, which therefore also reduces the density of
the atoms and the collision rate of the atoms. Forced evaporation in magnetic traps,
in contrast, can maintain the shape of the potential while removing the fastest atoms
from the trap. Such a scheme is possible in the circular FORT just as it is in magnetic
traps, and allows for close control of the rate of evaporation, which is necessary for
optimization. Forced evaporative cooling has not yet been demonstrated in an optical
dipole trap, to our knowledge.
In this chapter, we demonstrate significant cooling in the circular FORT coupledwith an increase in phase-space density. The mechanism involved is most likely acombination of forced evaporative cooling and Sisyphus cooling. Preliminary results
indicate that these cooling mechanisms are viable. Although these promising results arefar from optimized, they point to a powerful new cooling technique for optical dipole
traps.
6.2

Theory6.2.1

Sisyphus Cooling Mechanism

Any Sisyphus cooling mechanism requires that an atom roll up a steep conservative potential gradient, undergo a transition to a different internal state, roll back downa shallower conservative potential, and then be “recycled” to the initial state. Thiscan be accomplished in the circular FORT as shown in Fig. 6.1. Atoms start in the
jF; mF >= j3; +3 > state, labeled ja > and described with a potential depth Ua . An
atom rolls up the side of the potential, converting kinetic energy to potential. Near theturning point of its oscillation, the atom is driven into an internal state with a smaller
AC Stark shift labeled jb > and described with a potential depth Ub . An example withjb > = j3; +2 > is shown in Fig. 6.1 (a). The atom then rolls down the shallower
potential, in which it gains less kinetic energy than it originally started with. By re-72
pumping the atom back to the j3; +3 > state at the bottom of the potential, the atom
returns to its initial position and internal state with less kinetic energy, closing the cooling cycle. It is then ready to be cooled again. The energy has been carried away by theelectro-magnetic fields involved in the transitions.
Many types of transitions can be used to drive this cooling mechanism. Here they
are discussed in two categories: the “drive” transitions that excite the atoms to shallower
potentials near their turning points, and the “repumping” transitions used to return theatom to the deeper potentials at the bottom of the potential. Figure 6.1 (a) depicts one
embodiment of the cooling scheme using a “rf drive”, in which a rf photon drives theatoms into the j3; +2 > state, and then the atoms absorb a photon from the circular
FORT laser beam to repump them. Figure 6.1 (b) depicts a similar mechanism, where
the drive is now an optical two-photon stimulated Raman transition. This transition
is induced by an additional “Raman” laser with two frequency components separated
by the ground state hyperfine splitting of 3035 MHz plus the difference in their AC
Stark shifts at the turning point of the atom. This “microwave drive” can be usedin conjunction with the FORT repump described above. Or, an additional repumping
laser can be added to provide better state selectivity and more preferentially repump theatoms back to the j3; +3 > state to complete the cycle.
The rf and microwave drives each have their advantages and disadvantages. The
rf drive is conceptually simpler than the microwave drive as a means to initiate theSisyphus cooling, since only one photon is required to cause a transition between neighboring mF states. No additional spontaneous photon scattering is added using the RF
drive, whereas the lasers used to create the two-photon Raman transitions also cause additional light scattering. This scattering can cause both simple recoil heating and alsoheating via the ground state dipole force fluctuations described in Chapter 4. However,
one rf cycle generally extracts smaller amounts of energy because the potential energy
differences between the levels involved in the rf cycle are generally smaller than thoseof the microwave cycle.
To implement cooling using the microwave drive, one must maximize the frequency of cooling cycles while minimizing the additional heating rates. Figure 6.2
shows both the spontaneous scattering rate and the resonant Rabi frequency induced by
the Raman laser as a function of both the Raman laser frequency and intensity. The resonant Rabi frequency is an important factor in determining the probability that an atom
undergoes the two-photon Raman transition. If the atoms were continually resonantwith the microwave modulation, they would undergo Rabi flopping between the twostates with this frequency. Because the microwave transition actually involves a twophoton stimulated Raman process, eff
mf mi depends on the intensity of each frequency
component as [96]:eff = 7:4
mf mi



1013

pI I

1 2



fA (mf mi );(6.1)

where fA is a numerical factor depending on the Clebsch-Gordon coefficients between
the initial and intermediate states and between the final and intermediate states. For
m = +1 transitions, fA  0:2. I1 and I2 are the intensities of the two frequency

73
2

a)

P3/2

2

P1/2

Potential Energy

F=3λ FORTmF= -3RFphoton

+3
2

S1/2

F=2

+2
-2
ρ
2

b)

P3/2

2

P1/2

RepumpPotential Energy

F=3

mF= -3

MOPA

+3
2

S1/2

F=2

3.03 GHz+2-2
ρFigure 6.1: Schematic of Sisyphus cooling mechanism utilizing (a) Radio-Frequency
transitions to drive the cooling and the FORT spontaneous scattering to repump, and(b) “microwave” two-photon stimulated Raman transitions to drive and an additionallaser at 795 nm to repump. The microwave drive shown in (b) will also yield coolingin conjunction with the FORT spontaneous scattering repump shown in (a), as is later
demonstrated.

74
components, in units of mW/cm2 . Both eff
mf mi and the detuning of the laser fromresonance with the intermediate state  are in units of rad/s.
The effective Rabi frequency changes with laser detuning and intensity, as shownin Fig. 6.2 with a solid line. A dashed line demonstrates the photon scattering rate ofatoms in the FORT under the same conditions. In order to use the two-photon Rabi
frequencies for cooling, one must find a regime in which the Rabi frequencies are highenough while the heating due to photon scattering is low. To find the optimum detuning
and intensity, we must evaluate the drive transition rate.
The rate of atoms making a transition between j3; +3 > and j2; +1 > is strongly
influenced by the presence of the AC Stark shifts. As the atom oscillates in the potential,
it sweeps into and out of resonance with the drive field at a fixed microwave frequency.
This is exactly analogous to a Landau-Zener transition in which the atom is stationary
and the drive frequency is swept from 1 to 1. The more adiabatically (slowly) the
frequency is swept, the larger the transfer of population from one state to another in theabsence of dephasing (see for example Ref. [97]). This is expressed mathematically by
writing the probability PLZ of transferring atoms during one sweep through resonanceas [98]
PLZ = 1 exp( 2 LZ )(6.2)
where the Landau-Zener parameter LZ = 2eff (4d!rf =dt) 1 . Then d!rf =dt is the rateof change of the atom’s resonant frequency due to the change in position of the atomwith time:d!rf 1 dd=[U (3; +3; ) U (2; +1; )] :
(6.3)

dt

~ d

dt

U (F; mF ; ) is given in Eq. (4.1) and Eq. (4.2). The probability of an atom changing
state while oscillating through resonance and back again is 2PLZ (1 PLZ ).
The velocity of atoms and slope of the potential are continually changing. Therefore, the probability of an atom changing state as it oscillates depends on the frequencyof the microwave modulation or analogously on the location in the trap at which thedifference in AC Stark shifts between the two states exactly equals the microwave drive
frequency. Figure 6.3 shows the probability of an atom changing state, assuming thatthe microwave frequency is different at each point so as to be resonant with the difference in AC Stark shifts at each point. The maximum probability of a transitionis achieved near the turning points of the oscillation, but unfortunately in that case
2PLZ (1 PLZ ) approaches zero, and no atoms return to the bottom of the well in the
jb > state. Within a few eff of the turning point, it is unclear if decoherence will
permit transitions to the jb > state to occur at the turning point. If not, the Raman
lasers may have to be pulsed on and off so that the transition is no longer adiabatic. The
majority of calculations in this Chapter are made assuming a large population transfer
at the turning points. Then in Section 6.2.7, modifications brought about by a pulsed
scheme will be discussed.
To select the optimum detuning and intensity, it is helpful to examine some general scaling issues. Heating rates are minimized when the spontaneous scattering rate2
sc is unsaturated,  must be large and sc / It = [see Eq. (5.2)]. However, two

75

6

10

5

10

4

10

3

10

2

10

1

rate (Hz)

a) 10

0.0

0.5

1.0

1.5

2.0

detuning (nm)

6

10

5

10

4

10

3

10

2

10

1

10

0

rate (Hz)

b) 10

10

3

10

4

10

5

10

6

intensity (mW/cm^2)
Figure 6.2: Effective Rabi frequency/(2 ) (solid curve) for the two-photon stimulated
Raman transition (a) versus the laser detuning for total intensity It = 1:18  105
mW/cm2 and (b) versus the laser intensity at a fixed detuning of 0.4 nm. Broken curves
represent the spontaneous photon scattering rate of atoms in the FORT due to the Raman laser beam. The intensities describe typical experimental conditions. Therefore,
It = I1 + 2I2 , and the intensity of second frequency
component I2 = 12 I1 , where It isppused to calculate the spontaneous rate and I1 I2 = It = 8 for calculation of the Rabi
frequency.

Landau-Zener transition probability

76

10

10

0

-1

0

5

10
15
20
Position (microns)

25

30

Figure 6.3: Landau-Zener transition probability as a function of position in the circular
FORT for 1 atom oscillating in the FORT potential, with a maximum amplitude maxof 30 m. The microwave frequency is different at each position so as to be resonantwith the difference in AC Stark shifts at each point. It = 7:5  104 mW/cm2 (solid),
3:8  104 mW/cm2 (dashed), and 1:9  104 mW/cm2 (dotted).

different regimes must be considered for the Landau-Zener probability. Often, two photon Raman transitions can be employed with high Rabi frequencies and long interaction
times. In this case, the Landau-Zener model no longer applies and atoms can undergo
as much as a  -pulse. The time t required for a  -pulse scales as 1= eff  =It ,and therefore the ratio of two-photon transition rate to spontaneous scattering rate is
proportional to . This scaling also applies to the Landau-Zener transitions when LZ
approaches 1. However, when atoms are resonant with the drive frequency for a small
fraction of the Rabi oscillation period, LZ is small. Then PLZ / 2eff  It2 =2 .
Therefore the ratio of Landau-Zener transition rate to spontaneous scattering rate goeswith It . To demonstrate the Raman transitions, we operate in the small LZ regime.
Thus high intensities are critical for observing Raman transitions due to the Raman
laser over the background due to spontaneous scattering of the laser.
However, to drive both forced evaporative cooling and Sisyphus cooling, it is important that the microwave frequency stay resonant with atoms near the turning points,
where LZ is not small and therefore the first scaling characteristics apply. Thus asatoms are removed from the largest  values, the frequency must be decreased in order
to interact with atoms at slightly lower energies. The fastest the frequency can be swept
and still interact with all atoms is the width of the oscillation every quarter-period of
oscillation osc =4. The width of the resonance is approximately e . To cool all the77atoms, the drive frequency must vary at a rate not to exceed ( e =osc ), and it must
interact with atoms at each AC Stark shift. Therefore the duration of the fastest sweepthat cools each atom is then the potential depth in MHz divided by the sweep rate in
MHz/s. The number of photons spontaneously scattered during this time then scaleswith ratio of the scattering rate to the Rabi frequency, / 1=. Thus the number of photons scattered during an optimized sweep is independent of the Raman laser intensity,
assuming there is enough intensity to prevent the Landau-Zener approximation frombreaking down.
6.2.2

Repumping with Spontaneous Scattering

Repumping is also an important part of the cooling cycle. For the cycle to beirreversible, it must involve a spontaneous transition. For the cycle to remove anyenergy, the repumping must occur at smaller differential Stark shifts than the drive
transition does. For the circular FORT potentials, the amount of energy extracted per
cycle depends on the initial energy Ei , where Ei is the total energy of the atom withrespect to the energy of the bottom of the trapping potential U0 .

E (Ei ; )
= (1
Ei

Ub
 2
)(1 (
) );
Ua
max

The turning point of the atom’s oscillation max can be found by solving Eq. 4.1 for :

r

Ei
)= 2:
U0

max (Ei ) = w0 ln(1(6.4)

The probability that an atom is repumped at a particular  depends on the time intervaldt() that it spends at a particular , and dt() is given by energy conservation222
dt() = d[ Ub (e 2max =w0m

e 2 =w0 )] 2 ;221

where m is the mass of the Rb atom [99].
Because the detuning and intensity can vary with position, the photon scattering
rate sc described in Eq. (5.2) also acquires a  dependence. When the FORT light isused as the repump, the detuning  is so large that it is essentially uniform across thetrap and the rate just depends on the local intensity. In that case,int = Ge 22 =w02 ;sc

where G is an arbitrary constant.
An additional circularly polarized laser tuned close to resonance with the P1=2 orP3=2 can be added to perform the repumping. Even when it is not focused, selectivity is
still provided by the spatial dependence of the detuning provided by the AC Stark shift.
det () =sc

2
2 + 4[Æ + U0 (1he 22 =w02 )]2

78

Table 6.1: Repumping from the j2; +1 > state. Tabulated are the percent probabilities
that an atom decays from the excited state listed across the top to the ground state listedon the left side. These excited states are accessible with a m = +1 transition from thej2; +1 > state. Based on calculations by Christopher J. Myatt.
5 2 P3=2

j3; +3 >j3; +2 >j3; +1 >j2; +2 >j2; +1 >5 2 P1=2

j30; +2 > j20; +2 > j30; +2 > j20; +2 >
14
19
23
15
30

16
5
1
52
26

11
15
19
19
37

56
19
4
15
7

where w0 is the FORT laser beam waist, the P state linewidth = 6 MHz, and Æ =detuning of the laser from the AC Stark-shifted atoms at the bottom of the trap. The
optimum Æ = =2 for maximum selectivity.
The average fractional energy change per cooling cycle can then be computed
as a function of the atom’s initial energy Ei :Rmax
E (Ei ;)E (Ei )=0R dt() sc ()
Ei
=<
>=
;
max
Ei
=0 dt() sc ()(6.5)

where max is a function of Ei as in Eq. (6.4). The result of this integral as a functionof initial energy for both rf and microwave drives, as well as both types of repump
transitions, is plotted in Fig. 6.4. From this plot, it is clear that the microwave drive
connecting j3; +3 > and j2; +1 > can remove much more energy per cycle than therf scheme connecting to j3; +2 >. In addition, the spatial selectivity of the repumping
is not essential for removing energy, as shown by curves 2 and 6. Finally, an optical
repumping technique can provide spatial selectivity using detuning. This technique
works best when Ub is large compared with the linewidth of the repumping transition,
as in curve 3 of Fig. 6.4.
Another large influence on the effectiveness of the repump is the probability thatan atom spontaneously decays to the j3; +3 > state after being repumped. This depends
on quantum numbers of the initial state jb >, the polarization of the repumping photon(always + ), and the excited state. Examination of the appropriate Clebsch-Gordon
coefficients tabulated in Table 6.1 and Table 6.2 reveals some important distinctions
between repumping schemes.
The RF cooling scheme excites atoms to the j3; +2 > from which they need tobe repumped. The circularly polarized FORT laser can do this quite effectively. When
it is tuned close to the D2 transition, 84% of the absorbed photons excite atoms to thej40; 3 > state. Of these, Table 6.1 shows that 1/4 decay to the j3; +3 > state, while
3/4 decay back to the j3; +2 > state in which they started. They then have anotherchance to make it back to the j3; +3 > state while being penalized only 1 photon
recoil. Averaging over all transitions, this cycle is 92% closed.

79

Fractional Energy Removed

a)

0.12
0.11

1

0.10
0.09
0.08
0.07
0.0

2

0.2

0.4

0.6

0.8

1.0

b)

Fractional Energy Removed

initial energy / U0 (F=3, mF=3)

0.55

3

0.50
0.45

4
0.40
0.35

5

0.30

6

0.0

0.2

0.4

0.6

0.8

1.0

initial energy / U0 (F=3, mF=3)
Figure 6.4: Evaluation of Eq. (6.5) as a function of initial energy, for various repumping
schemes. (a) For ja >= j3; +3 > and jb >= j3; +2 > as in the rf cooling scheme
depicted in Fig. 6.1 (a). (b) shows the cooling efficiency for ja >= j3; +3 > and jb >=j2; +1 > the scheme depicted in Fig. 6.1 (b). Curves 1 and 5 represent repumping
by the FORT as described by int
sc (). Curves 2 and 6 show that energy can still be
extracted when no spatial dependence of the repump scattering rate is employed, i.e.
when sc = 1. When repumping is done with an additional repump laser, detsc () isused in the integral. Curve 4 is shown for U0 (a) = -1 mK, and curve 3 for U0 (a) = -10
mK.

80

Table 6.2: Repumping from the j3; +2 > state. Tabulated are the percent probabilities
that an atom decays from the excited state listed across the top to the ground state listedon the left side. These excited states are accessible with a m = +1 transition from thej3; +2 > state. Based on calculations by Christopher J. Myatt.
5 2 P3=2

j3; +3 >j3; +2 >j2; +2 >5 2 P1=2

j40; +3 > j30; +3 > j30; +3 >
25
75
0

42
14
44

33
11
55

The circular FORT is not as effective when repumping atoms from the j2; +1 >state, where they are driven in the microwave cooling scheme. If the FORT is closest
to the D2 transition, an average over the appropriate Clebsch-Gordon probabilities indicates that only 15% of the repumping transitions return the atoms to the j3; +3 > state
to complete a closed cycle. 28% return the atoms to the j2; 1 > state, and again no net
cooling results. Unfortunately, 34% return to the shallowest j2; +2 > state from which
they are most likely lost before they can be pumped to the j3; +3 > state.
In contrast, an additional circularly polarized repump laser tuned from the 5 2 S1=2
j2; +1 > to the 5 2 P1=2 j20; +2 > state should be much more efficient. As shown in Table
6.2, 56% of the repump transitions result in a closed cycle. For this additional laser todominate the repumping process, the scattering rate of atoms in the j2; +1 > statemust be several times higher than the FORT. However, this will not cause significant
additional heating because the vast majority of atoms are in the j3; +3 > state where
they are detuned several GHz from resonance with the additional repump laser.
6.2.3

Cooling Rates

Once the efficiency of the cooling process is understood, the maximum coolingrate is readily calculated. Because the Landau-Zener probability is very high at theturning points of the oscillation, after half an oscillation period no more atoms will beresonant with the drive frequency, because they will either be cooled or lost. Therefore,
to continue cooling, the microwave or rf drive frequency must increase so as to interact with atoms of lower kinetic energy. As described in the previous subsection, theoptimum rate at which to sweep the frequency  is

d
4 eff
=
dt osc 2

(6.6)

where aosc is the period of oscillation of atoms in the deeper of the two circular FORT
potentials involved in the cooling cycle.
The maximum cooling rate is the ratio of the amount of energy carried away
in one cooling event to the time between cooling events. As plotted in Fig. 6.4, thefraction ( ) of energy removed in one cooling cycle depends on the initial energy Ei ;

81

E =

Ei . The time between events t, for a constant frequency sweep, also

depends on the initial energy. The more energy an atom gives up in one cooling event,the longer it must wait before the drive frequency is resonant with its turning point
again and it can experience another cooling cycle.

t =E d 1( ) :h dt

The resulting cooling rate is surprisingly constant in time, independent of the atom’s
energy:dE 4 eff ~
= osc
dt
a

(6.7)

dE
=
dt

(6.8)

This calculation must break down when the time for an atom to be repumped (tR )exceeds t above. In essence, that means that the atoms can no longer keep up withthe sweep rate of the drive frequency. If tR is limited by the time required for the atom
to oscillate back to the center of the trap, then tR = 1=4bosc , where bosc is the periodof oscillation in the shallow potential. Cooling then crosses into a different regime, inwhich the sweep rate is limited so that the atom is repumped before the drive comes
into resonance with the frequency of the new turning point of the atom. This gives an
exponentially decreasing cooling rate:

4
E osc
b

and a resulting time-dependent energy ofE (t) = Ei e 4 t=b :
osc

(6.9)

For typical experimental parameters of eff = 2 50 kHz and aosc = 1 ms, dE=dt= 10 mK/s. The energy (Et ) at which the expression for dE=dt changes from a constantto an energy-dependent expression is then Ei = eff ~bosc = aosc . For the values above,bosc = 2 ms, and = 0.3, EB =kB = 17 K.
6.2.4

Evaporative Cooling

Forced evaporative cooling is also a powerful technique that can be highly efficient in the limit of rapid collisions and long lifetimes. The speed with which thepotential depth can be reduced and hence the cooling can be driven depends on thecollision rate of the atoms. Because the circular FORT has much tighter confinement
and higher collision rates, evaporation can proceed at much faster rates in this trap thanin magnetic traps. For the best evaporative cooling configuration, one should operate
the trap at a wavelength in which the jb > state is unbound. This is the case for j2; 1 >state between 786 nm and 795 nm.

82
6.2.5

Heating Mechanisms

A variety of heating mechanisms can limit the performance of both Sisyphus andevaporative cooling. The heating due to collisions with background gasses is predicted
by Beijerinck to be a modest 12 K/s [17] in a 1 mK trap with a collisionally limited
lifetime of 10 s.
Larger contributions come from simple photon scattering in the circular FORT.
Even with perfect polarization, each photon absorbed on the cycling transition (j3; +3 >!
j40; +4 >) gives a recoil kick. The heating rate due to this process is a simple random
walk in momentum space. The resulting heating rate isdT
v2 m= sc Rdt
3 kB
where vR is the recoil velocity (0.5 cm/s in Rb), m is the atomic mass, and kB is Boltzman’s constant. Conveniently in Rb, for T in K and v in cm/s, m=kB  1. Typicalexperimental parameters give 500 photons/s absorbed and spontaneously scattered fromthe FORT laser beam, and therefore a heating rate of 40 K/s.
The FORT atoms will also absorb photons from the Raman laser beams, andbecause these are not circularly polarized, they tend to randomize the mF states. The
resulting dipole force fluctuations can give rise to large heating rates, as shown in Chapt.
4, Fig. 4.7. In that chapter, an expression was derived for the exponential lifetime as afunction of initial and final temperatures and the depths of the potentials involved. Using the same arguments, a heating rate can be directly derived. The resulting expression
for energy as a function of time becomes:
Ua

E (t) = Ei e Ub h t(6.10)

where h is defined previously as the hopping rate between potentials. The resulting
1/e time constant is then [ UUab h ] 1
For the Raman lasers detuned 0.4 nm, typical experimental parameters give 200
photon/s scattering rates and therefore h  10/s. The derivative of Eq. 6.10 evaluatedfor an initial energy of 0.4 mK gives a large initial heating rate of 4 mK/s, nearly equal
to the cooling rate.
Imperfect FORT polarization can still contribute to heating in the same way described in Chapt. 4. Therefore, the FORT laser polarization must be carefully controlled
to keep the dipole force fluctuation heating from limiting the cooling.
Often, radiation trapping is cited as a limit to effective optical cooling schemes.
This should not limit the performance of the schemes proposed here, because the FORT
photons are very far from resonance, and therefore unlikely to be reabsorbed as they
exit the atom cloud. Even the addition of a repump at 795 nm will not cause additional
heating, because it is several GHz detuned from the majority of atoms in the trap.
6.2.6

Ultimate Temperatures

The limits of the optimized Sisyphus cooling process are found by examining
the balance between the heating and cooling rates. The largest heating rate calculated

83
in the preceding section is caused by dipole force fluctuations induced by spontaneous
scattering of photons from the Raman laser. However, this heating rate decreases withtemperature, whereas the Sisyphus cooling rate is independent of temperature above
about 20 K. Therefore, if the temperature is low enough to start the cooling process,this heating rate can only affect the Sisyphus cooling below the crossover temperature,
where Sisyphus cooling starts to turn off. In that case, the differential equation that
describes the competition between heating and cooling is simply

dE
=(
dt

U
+ h a )E:
osc
b
Ub
4

For the parameters given in the previous section, the cooling term outweighs the heatingby a factor of 15, and therefore this heating mechanism does not greatly impact theSisyphus cooling rate.
Recoil heating due to FORT scattering of 40 K/s does not limit the final temperature. Because this heating rate is independent of the atom’s energy, and less thanthe constant cooling rate of 10 mK/s, this cannot limit the final temperature until thecooling turns off in the exponential regime. By setting the magnitudes of the coolingand heating rates equal to one another as above, the limiting temperature is 70 nK.
Therefore, this heating rate will not limit the final temperature of the atoms, as this is
far below the recoil limit of the cooling process. The constant heating rate due to background collisions of 12 K/s should also not limit the final temperature, as that heating
rate balances the cooling rate at an even lower temperature: 20 nK.
Finally, the cooling process itself causes some recoil heating, due to the spontaneous emission of photons in all directions. If a few repumping events are on average
required to close the cooling cycle, the final temperature is about 250 nK. Thus therecoil limit seems to be the ultimate cooling limit of the Sisyphus process.
The above limits were established under the assumption of ideal repumping. If
the cooling cycle is only 92% closed, then about 10% of the atoms are lost during eachcooling event, and the number of atoms left after j cooling cycles will be 0.9 j . Thus
to cool atoms a factor of 10 with = 0.3, cooling a factor of 10 results in a loss of
50% of the atoms. Even this loss should turn off as the atoms get colder, because atoms
will no longer be lost from the very shallow j2; +2 > state. Of course, as temperatures
decrease, densities increase and therefore collisional losses will increase. Increased
losses are not calculated in this document, but will undoubtedly be important.
6.2.7

Cooling with a Pulsed Drive

As mentioned on page 74, it may not be possible to drive transitions between
ja > and jb > continuously due to the adiabatic interaction of the atoms with the drive
frequency. In this case, a pulsed implementation similar to that used in gravitational
Sisyphus cooling may be necessary. To do this in the circular FORT, the MOPA driveis turned on for about 1/4 aosc in order to induce Rabi flopping between ja > and jb >that results in 50% of the population in each state. Then 1/4 aosc later, if the repumping
is induced by an additional laser, the repump laser should be turned on for an additional

84
1/4 aosc . This cycle should then be repeated every 3/4 aosc so that the drive can interact
with atoms in every phase of oscillation. Therefore the optimum sweep rate is a factorof 3 slower than that given by Eq. (6.6) and therefore so is the optimum cooling rate
[Eq. (6.7)] from 10 mK/s to 3 mK/s.
One of the heating rates scale with the time that the Raman laser beams illuminate
the trapped sample, specifically the dipole force fluctuation heating induced by the
spontaneous scattering of Raman laser light as described on pg. 82. The reduced
duty cycle of the Raman lasers changes the heating from 4 mK/s to 1.3 mK/s. The
other heating rates are not reduced with the Raman laser duty cycle. Therefore thetemperatures at which the cooling and heating balance will increase by up to a factor of
3. Fortunately the final temperature appears to be limited by the recoil heating due tospontaneous emission during the cooling cycle, which remains the same as previously
estimated.
6.3

Experimental Implementation

6.3.1

RF considerations

To implement the Sisyphus cooling scheme, we first explored the rf drive. We
attempted to couple rf into the vacuum chamber by placing a drive coil about 2 in. dia.
on a 6 in. flange (4 in. clear aperture window), and drove it at frequencies between 1
and 10 MHz. Unfortunately, the stainless steel vacuum chamber is an excellent shieldof rf. To quantify that statement, we measured the voltage across a pick-up coil as afunction of the distance between the pick-up coil and the source coil, in free space andin the presence of a 4 in. dia. pipe, mimicking our vacuum chamber. The results, shownin Fig. 6.5, indicate that the rf that reaches the atoms at about 16 cm from the coil issuppressed by 3 orders of magnitude in amplitude, and therefore 6 orders of magnitudein power. Consequently, we concluded that the only way to couple rf into our vacuum
chamber is to break vacuum and insert a coil very close to the atoms.
6.3.2

MOPA

Rather than open the vacuum chamber, we chose to pursue the two-photon stimulated Raman transition option. This was made easier by the existence of a microwavemodulated master oscillator power amplifier MOPA system already constructed by Neil
Claussen. A typical MOPA system [100] consists of a grating-tuned diode laser injected
into a tapered amplifier. Multiple diode laser beams at different frequencies can also
be injected into the amplifiers [101]. Our MOPA relies on microwave modulation [96]to create 2 frequencies in the master oscillator (MO) diode laser. The length of thecavity is chosen to be resonant with the modulation frequency, so that large modulation
depths are achieved with small injected powers. Typically two frequency sidebands are
created with powers up to about 50% of the power remaining in the carrier, and 25% of
the initial power in each sideband.
The output from the MO is then tightly focused and mode-matched into the power
amplifier (PA) that is driven at between 1.5 and 1.8 Amps. The output from the PA

voltage across inductive coil (V pk-pk)

85

1000

100

10

0

5

10

15

20

25

distance between source and coil (cm)Figure 6.5: Demonstration of the difficulty of coupling rf (frequency = 3.7 MHz) radiation into a vacuum chamber from outside. In free space (), the field detected by an
inductive coil decreases as 1/r3 . However, when the rf radiation must propagate along
the axis of a metal cylinder (dia. = 4 in) ( ), the field decays with a e 1 length of 1.3
cm.

contains all the frequency components from the MO. Unfortunately, as discussed indetail in Ref. [100], the output from the PA also contains a lot of broadband radiation
spread over 20 - 30 nm. The power density of this radiation decreases by more thanan order of magnitude when the MO is injected into the PA, making it about 1.5% of
the power at the amplified frequency per 0.1 nm [100]. If the MOPA is focused to a
total intensity of 1:5  105 mW/cm2 , the amount of intensity within a linewidth of the
Rb resonance is about 3:4  10 4 mW/cm2 , resulting in a scattering rate of 2.5 103
photons per s, which is larger than the scattering due to the amplified power at thedesired wavelength by a factor of 10 when the MO is tuned 0.5 nm from resonance.
The effect of the scattering induced by the broad-band power emitted from the PA wasclearly observable. The lifetime of FORT atoms in the presence of the MOPA lightdecreased from 7 s to 200 ms at MOPA total intensities as small as 104 mW/cm2 .
This broad-band radiation is difficult to filter out using conventional filters, because we want to block broadband radiation on resonance with the atoms, but completely pass the amplified power about 0.5 nm away. Typical interference filters do
not have narrow enough bandpass widths or sharp enough cut-offs for this application.
Fabry-Perot cavities might be made to work, but would require a lot of effort. Therefore, the most logical filter is an absorption cell of the appropriate species, in this case
Rb. Therefore, a Rb cell was placed in the MOPA beam path and heated such that thecoldest point in the cell is about 550 C , giving a vapor pressure of 6 10 6 Torr, 30
times that of room temperature. The effect of broadband MOPA light on the FORT

Effective Rabi frequency (a.u.)

86

0.4

0.3

0.2

0.1

0.0
0.0

0.1

0.2

0.3

0.4

0.5

Fraction of power in first sideband
Figure 6.6: Calculated relative Rabi Frequency vs. the efficiency of MOPA modulation.
The MOPA modulation is defined as the ratio of the power in one sideband over the
power at the carrier frequency, assuming the power is equal between the +1 and -1
order modulation peaks.

lifetime became much less significant. Without any injected light, the lifetime of theatoms changes only about 10% when illuminated by the PA broadband radiation underconditions in which the maximum MOPA intensity is a total of 105 mW/cm2 .
The optimum modulation amplitude of the MOPA light in order to drive twophoton Raman transitions is readily calculated. Because both sidebands are equally
populated, but the transition is driven by an interference between one sideband and thecarrier, over-modulation can deplete the carrier and reduce the transition rate. Thus the
optimum requires that half the power remain in the carrier and 25% in each sideband.
However, it is not a steep function, so the requirements on modulation are not stringent,as shown in Fig.6.6.
6.3.3

Procedure

A schematic of the experiment is shown in Fig. 6.7. The procedure is very
similar to that described in Chapter 4. Primary differences include the absence of arepump beam to help with loading. Also, we have the capability to image the atoms

87
as in Chapter 5. Thus we are able to measure the temperature of the atoms in ballistic
expansion as the atoms are cooled with the MOPA laser beam. Finally, we include the
MOPA laser, focused to 1.5 105 mW/cm2 , counterpropagating the FORT laser. This
choice of geometry allows the highest intensities for driving the two-photon transitions,but only allows m = -2 transitions. Therefore state jb > in the cooling scheme mustbe j2; +1 > because j2; +2 > is inaccessible, although it would provide slightly higher
values of . Thus with an infinite intensity, directing the MOPA beams perpendicular
to the FORT may be preferable. The MO is modulated at frequencies near 3035 MHz,
as described above. We have the capability to trigger a ramp of that frequency, linear
to within about 10%, with control over the ramp rate, duration, beginning and ending
frequencies.
6.4

Preliminary results

6.4.1

Two-Photon Stimulated Raman Transitions

To verify that the MOPA laser could actually excite two-photon stimulated Raman transitions, we monitored the number of atoms stored in the FORT for a fixed
FORT storage time while the MOPA illuminated the atoms for the majority of thattime. The frequency of the modulation was changed from shot to shot, and the number
remaining in the trap was monitored. The resulting loss is shown in Fig. 6.8. This plot
presents many interesting features that give information about the internal states andkinetic energies of atoms in the FORT and the depth of the FORT. No transitions occur
blue of the 3035 MHz Rb resonance, implying that no atoms are being driven to deeper
potentials, only shallower ones. This is a weak indication of the spin-polarization wemeasured previously in Chapter 6. The depth of the potential can be measured by theshift of the steep edge. This occurs 35 MHz red of the Rb unshifted resonance, implying
a 50 MHz AC stark shift at the bottom of the j3; 3 > potential, or 2.5 mK. Calculated
values for this trap are 1 mK, assuming w0 = 60 m This value decreased the next day
by 10 MHz, and these are the conditions under which the remaining data were taken.
6.4.2

Cooling with Stationary Frequency

Figure 6.9 shows observed temperature as a function of time, for the frequency
fixed at a detuning of 22.5 MHz below the Rb resonance. During the cooling, thenumber of atoms in the FORT decreased between a factor of 20 and 50. The uncertainty
arises because the peak optical depth after expansion is monitored, but not the total
number. The conversion of the peak optical depth relies on knowledge of the scaling of
the length of the FORT. In (b),  are calculated assuming the longitudinal dimension
does not contract and is therefore not in thermal equilibrium with the other dimensions,whileshow N assuming thermal equilibrium in all 3 dimensions. Finally, phasespace density is computed assuming the sample is in thermal equilibrium, and is seen
to increase by a factor of about 15.
The mechanism here is most likely primarily evaporative. Because the frequency
is not changing with time, most atoms are not excited at the turning point of theirPolarizer

MOT laser
beams

MOPA andCircular Repump
beams

Ph

oto

Rb
cloud
dio

de

Pockels
Cell

x

z

X

X

gravityy

T:S Laser~1W
AOM

Intensity
Servo

88

Figure 6.7: Layout of the all-optical two-photon stimulated Raman cooling scheme. Primary differences between this and Fig. 4.4 include
the addition of a MOPA laser beam counter-propagating the FORT. The additional repump shown to counter-propagate the FORT laser
beam is only proposed, and has not yet been implemented.

89

Atoms Remaining (a.u.)

2

1

0

0
10
20
30
Microwave Frequency (MHz) with 3GHz offset

40

Figure 6.8: Loss from the circular FORT due do ground-state changing transitions as afunction of the microwave frequency driving the transition. The resonance frequency
for atoms in free space is 3035 MHz. () represent a FORT storage time of 250 ms anda cooling duration of 150 ms, while ( ) show the same data for a longer FORT storage
time (500 ms) and a longer cooling duration of 480 ms. Because at longer times thesignal drops all the way to zero, the shape of the curve is distorted and the slope of thecurve near the bottom of the trap goes to zero.

90
300

temperature (uK)

a)

100

30
0

100

200

300

400

0

100

200

300

400

0

100

200

300

400

b)
number of atoms x 10

6

1

0.1

phase space density x 10

-9

c)

10

1
duration of mcrowave drive (ms)Figure 6.9: While the microwave drive frequency was held fixed at 3012.5 MHz, thetime during which it was applied was varied simultaneously with the FORT storage
time. (a) Shows the change in temperature, (b) number, and (c) the deduced phase spacedensity, assuming the cloud remains in thermal equilibrium. In b), ()’s represent the
number left assuming the longitudinal dimension of the trapped sample does not changeand ( )’s represent the number assuming all dimensions are in thermal equilibrium.

91
oscillation. Instead they are excited to state jb > with large kinetic energies, fromwhich they are lost. However, at very low temperatures, this mechanism must turn off
because all states in the circular FORT are trapped (ie. U0 < 0) . Perhaps this is thecause of the turnoff in cooling, along with substantial number loss.
6.4.3

Cooling with Swept Microwave Frequency

Once two-photon Raman transitions were demonstrated and cooling was demonstrated at a fixed frequency, cooling with a swept drive frequency was demonstrated.
The frequency ramp began at the unshifted resonance of the ground-state splitting, andwas swept at a constant rate for a variable time. Therefore the duration of the sweep also
represents a “deeper cut” into the cloud, or allows interactions with increasingly colder
atoms. The results are shown in Fig. 6.10. Phase space density increases by a factor of
5. In addition, the number loss is about a factor of 7.5, which could be ascribed to inefficient evaporation. At the lowest temperatures, however, evaporation seems unlikely
because all states in the circular FORT are trapped at this wavelength. Thus additional
losses are again likely to be due to collisional losses caused by increased density.
A comparison of the cooling results is at first glance perplexing. The fixed frequency data gives lower temperatures (30 K vs. 80 K), and higher phase space density increase (factor of 10 vs. factor of 5). Even in reaching 80 K, the fixed frequency
cooling gave comparable number loss and slightly higher (factor of 1.5) phase spacedensity increase. Perhaps the fixed frequency cooling is initiating an evaporative cooling mechanism, while the swept frequency is poorly optimized for Sisyphus cooling.
In both cases, the Sisyphus cooling may be turning off as the atoms get colder and theLandau-Zener transitions become increasingly adiabatic. Further study is warranted.
6.4.4

Heating Rates

Once cooling is shown to work, heating rates are easily measured by applying
a cooling pulse and then watching the trapped sample heat up with time. As shownin Fig. 6.11, rates of 150 K/s are measured in the circular FORT in the absence of
any additional laser light. This rate is consistent with a 5-10 s lifetime in a 1 mK trap.
The expected heating rate in this trap, calculated earlier, is only 46 K/s. However,the FORT polarization is not likely to be perfectly circular, and additional dipole force
fluctuation heating, in conjunction with the heating already described, can easily cause
the higher measured rate.
6.5

Conclusion

Sisyphus cooling and forced evaporative cooling schemes are theoretically very
powerful cooling mechanisms in the circular FORT. Sisyphus cooling rates of up to
10 mK/s are possible, and limiting temperatures approaching the recoil limit may bepossible. Initial results show that the temperature can be reduced by a factor of 4 to80 K while the number decreases by only a factor of between 2 and 6. Significant

92a)temperature (µK)

300

100

30
150

200

250

300

350

400

450

150

200

250

300

350

400

450

200

250

300

350

400

450

number of atoms * 10

6

b)

1

0.3

phase space density *10

-9

c)

100

30
150

duration of µ -wave sweep (ms)Figure 6.10: Atoms in the FORT are continually illuminated by the Raman beams while
the microwave frequency was swept at a constant rate, starting from the unshifted resonance of 3035 MHz. As time increases, the end point of the sweep changes, effectively
cutting deeper into the cloud. 150ms corresponds to a sweep of 9.8 MHz. Plotted are(a) the temperature of the atoms after cooling, (b) the number of atoms remaining after
cooling, and (c) the resulting phase space density, assuming the sample is in thermal
equilibrium. In b), ()’s represent the number left assuming the longitudinal dimension of the trapped sample does not change, ( )’s represent the number assuming all
dimensions are in thermal equilibrium, and finally the ()’s show the time-dependenceof atoms in the FORT during that time without any cooling.

93

temperature (uK)

250

200

150

100
0

200

400

600

800

1000

storage time after cooling (ms)Figure 6.11: Heating rate of atoms stored in the circular FORT. Temperature is plotted
as a function of time the atoms are stored in the FORT after a 50 ms cooling pulse isapplied.

improvements in the Sisyphus contribution to this cooling scheme may be realized byutilizing an additional circularly polarized repump on the D1 line, tailoring the frequency sweep to coincide with the optimum cooling rate, and pulsing the the MOPA on
and off rapidly. Evaporation has not been optimized, and most likely the evaporative
cooling effects could be improved significantly by tailoring the evaporation sweep rate
as is done in magnetic traps. [67]

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Appendix A
Dichroic Atomic Vapor Laser Lock

A.1

Introduction

Lasers with stable frequencies are essential in many fields of research. In addition, they are used commercially in precision machining tools, gravimeters, and laser
vibrometers. He-Ne lasers have been the industry standard for many years,[102] but
they are bulky, energy inefficient, and have limited tube lifetime. Diode lasers offer an
improvement in all these areas and moreover can be stabilized to atomic transitions.
Typical methods of stabilization [103, 104] although practical in some laboratory settings, are not reliable enough for use in commercial equipment. Using a techniqueoriginally demonstrated with a LNA (La(1 x) Ndx MgAl11 019 ) laser in helium [105] wedeveloped a robust diode laser stabilization scheme that will be useful in both commercial instruments and research laboratories.
A.2

Diode Laser Frequency Stabilization

The frequency of a diode laser with grating feedback depends on the current,
temperature, and external diffraction grating position. With the laser cavity in a Littrow
configuration (see Fig. A.1), the output beam reflects off the grating, while the firstorder beam diffracts back into the laser diode. The optical feedback from the gratingis spectrally narrowed and peaked at a frequency that can differ from the bare diodecentral frequency. Thus this feedback narrows the laser linewidth to <1 MHz andforces the central frequency to nearly that of the feedback signal. To tune the laser
central frequency, the grating is tilted by applying a voltage to a piezoelectric transducer(PZT). Over time, the central frequency will drift because of temperature, current, andmechanical fluctuations. This drift can be reduced by stabilizing the laser to an externalreference. In addition, small, rapid fluctuations in laser frequency, which contribute to
the laser linewidth, can be reduced by rapidly controlling the diode laser current.
In one popular method of stabilizing the diode laser frequency, some of the output light is sent into a saturated absorption spectrometer. The diode laser frequency isthen locked to either the side or the peak of the narrow saturated absorption features,[103, 104, 106] shown in Fig. A.2. These narrow lines offer the advantage of a steep
slope, where the slope is the change in the fractional absorption signal with laser frequency. Side-locking to this slope is accomplished by electrically controlling the PZTFigure A.1: Schematic of a DAVLL system. Here we show the entire beam passing through the lock, but in actuality, only a small amountof power is picked off from the main beam and enters the locking apparatus.

103

104

Figure A.2: Oscilloscope trace of (a) the signal from a saturated absorption spectrometer and (b) the DAVLL signal, as the diode laser is scanned across Rb resonanceswith the PZT. A laser can be locked to either of the two circled zero crossings of the
DAVLL signal. These features are due to the 87 Rb F = 2 ! F 0 = 1; 2; 3 and the 8 5Rb
F = 3 ! F 0 = 2; 3; 4 transitions. The frequency of the lock point can be tuned optically by rotating the quarter-wave plate, or electronically by adding an offset voltage tothe signal.

voltage so that the saturated absorption signal is maintained at a particular level. However, a disadvantage of side-locking is that fluctuations in beam alignment and intensity
will alter the lock point and cause drift in the laser frequency. Peak-locking is lesssensitive to these fluctuations, but has its own disadvantages related to phase-sensitive
detection: Either the output of the laser is modulated directly or expensive electro-optic
components are used to modulate only the light entering the spectrometer. A further
disadvantage of both peak and side locks is their small capture range, which prevents
them from recovering from perturbations that shift the laser frequency by more than
-30 MHz.
A.3

The Dichroic-Atomic-Vapor Laser Lock Signal

To overcome the aforementioned disadvantages with the conventional locks, wedeveloped a dichroic-atomic-vapor laser lock (DAVLL). This technique employs a weak

105
magnetic field to split the Zeeman components of an atomic Doppler-broadened absorption signal and then generates an error signal that depends on the difference in absorption rates of the two components. The subtraction technique minimizes the frequency
drifts that are due to changes in line shape and absorption that typically limit the utility of Doppler-broadened absorption features for frequency stabilization. The DAVLL
lock offers the advantages over saturated absorption: large recapture range, simplicity,
low cost, and no need for frequency modulation.
As shown in Fig. A.3(a), a Doppler-broadened absorption feature is detected
when a laser beam (with wave vector = k z^) passes through a Rb vapor and the laser’s
frequency is scanned across a transition. In the absence of a magnetic field, we obtainthe same signal regardless of the laser polarization (c). However, if a uniform magnetic
field ( = B z^) is present and the laser is circularly polarized ( = + ), the central
frequency of the absorption feature increases [Fig. A.3(b)]. If the laser has the opposite
polarization ( =  ) [Fig. A.3(c)], the central frequency decreases. By subtracting
the two absorption profiles [Fig. A.3(d)], we obtain an antisymmetric signal that passes
through zero and is suitable for locking.
A DAVLL signal with a steep slope causes the lock to be less sensitive to noise
sources that mimic laser frequency changes, such as laser intensity noise. A roughcomparison with a typical saturated absorp- tion setup in our laboratory shows that the
DAVLL slope is comparable with that of the saturated absorption lines. This may seem
surprising at first, because the linewidths of the saturated absorption lines (FWHM-20 MHz) are much smaller than those of the DAVLL lines (-500 MHz peak-peak)[Fig. A.2(b)]. However, the heights of the saturated absorption features range from
1/3 to 1/30 of the on-resonant Doppler-broadened absorption fraction, whereas the
DAVLL signal height is twice that absorption fraction. By approximating the slope
as the linewidth divided by the signal height, we estimate that the slope of the largest
saturated absorption peak is only four times bigger than the DAVLL slope.
The slope of the DAVLL signal is also affected by the magnetic field. The separation of the two Zeeman-shifted absorption peaks must be large enough to give asizable capture range, but small enough to give a large slope through the unshifted resonance. In addition, the Zeeman-shifted absorp- tion peaks broaden with increasing
field, because the various transitions contained within one Doppler- broadened feature
shift different amounts. We found that 100 G maximizes the slope, and therefore represents the best compromise between increased separation and increased broadening.
However, the dependence of slope on the magnetic field is not strong, so if B is varied
by a factor of 2 it should not significantly alter the lock performance.

k

B

^

^

A.4

Apparatus

A schematic of the diode laser and optics used to generate the DAVLL signal isshown in Fig. A.1. The SDL 780 nm diode laser is tuned by use of a diffraction grating,as described above. The output beam from this laser passes through a beam splitter, anda small amount of power is split off to be used for locking. After passing through a small
aperture, the resulting beam passes through a linear polarizer. Pure linear polarization

106

Figure A.3: Origin of the DAVLL signal shape. (a) A Doppler-broadened transition in
Rb in the presence of no magnetic field. (b) The same transition, Zeeman shifted in a
100 G magnetic field, when circularly polarized light is incident on the vapor. (c) The
same as (b), but with the opposite circular polarization. (d) The difference between (c)and (b) giving the DAVLL signal. In this idealized case, the arrow indicates that theoff-resonant signal is zero.

107
is equivalent to a linear combination of equal amounts of two circular polarizations.
This beam (2.5-mm diameter, 0.5 mW) next passes through a cell-magnet combination, consisting of a glass cell filled with Rb vapor and a 100 G magnetic field. The
magnet is made of rings of rubber-embedded permanently magnetic material, spaced
appropriately and glued together concentrically around the glass cell.1 To generate
the DAVLL signal, the absorption profiles of the ^+ light must be subtracted from thatof the ^ . To accomplish this, after exiting the cell, the two circular polarizations areconverted into two orthogonal linear polarizations by passing through a quarter-wave
plate. Then the two linear polarizations are separated by a polarizing beam splitter, andthe resulting two beams are incident on two photodetectors whose photocurrents are
subtracted. As the frequency of the laser is scanned across an atomic transition, an antisymmetric curve is generated, as shown in Figs. A.2 and A.3. The diode laser is thenlocked by feeding back a voltage to the PZT so that the DAVLL signal is maintained at
the central zero crossing.
We align the optics by orienting the fast axis of the quarter-wave plate at 45oto the axis of the output polarizing beam splitter, so that equal intensities are incident
on the two photodetectors when the laser is far detuned (>1 GHz) from the Rb resonances [see Fig. A.3(d)]. The DAVLL system is least susceptible to drifts when theoff-resonant signal gives no net photocurrent, and the lock is therefore very near thecenter of the unshifted resonance, as shown in Figs. A.2 and A.3. We tuned the locked
laser frequency either by adding an electronic offset or by rotating the quarter-wave
plate. The latter optical method changes the frequency by weighting one circular polarization more heavily than the other. This type of offset is more stable than the electronic
offset because the lock point is always at a zero in net photocurrent, which occurs when
the powers incident on the two photodetectors are equal. Thus with optical offsets, thelock point maintains its insensitivity to laser intensity fluctuations.
A.5

Characterization of Frequency Stability

To monitor the frequency stability of the laser lock, we stabilized two separate lasers each to their own DAVLL system. We locked them to the same Dopplerbroadened feature (85 Rb F = 3 ! F 0 ) with different optical offsets, typically approximately 25 MHz apart. A portion of the light from each laser was combined ata beam splitter and copropagated onto a fast photodetector (125 MHz). The resulting
beat note, corresponding to the difference between the two laser frequencies, was fed
into a high-speed counter. By reading the counter every 5 s, a computer monitored thelaser stability over periods ranging from 12 to 38 h.
In this way, the difference between the two laser frequencies was monitored overmany days, under different conditions. The beat frequency was stable to 2.0 MHz peakpeak while the temperature of the laboratory, and therefore of the optical components,
1
We used the material with part number PSM1-250-3X36C from the Magnet Source, 607-T S.
Gilbert St., Castle Rock, Colo. 80104, 1-800-525-3536. Although uniformity is not critical to stability, we minimized variations to 5 along the field axis of symmetry by spacing the inside rings closer
together than the outer ones.

%

108

Figure A.4: Measured beat frequency between two DAVLL systems over a 38 h period.
Variations in the beat frequency indicate the limits of the laser stability to be approximately 500 kHz peak-peak. These data show a stability of 27 kHz rms during an 11 h
period at night when environmental factors such as room temperature and air currents
are more stable. The discontinuities at the end of the run are due to incomplete shielding of the detection photodiodes from room lights. The run was stopped when a lasermode hopped, but after we adjusted the current to return the laser to the proper mode,
it returned to the same frequency.

varied a couple of degrees throughout the day. When the cells (with attached magnets) were enclosed in a copper pipe and crudely temperature stabilized, the stability
improved to 500 kHz peak-peak over 38 h, as shown in Fig. A.4. The cell-magnet
combinations have measured dependences of 1.0 MHz/o C and 1.7 MHz/o C. We attribute this drift to a temperature-dependent birefringence of the cell windows, because
the lock point is more sensitive to birefringence than to any other parameters. This is
expected and observed, as discussed below. To confirm that optical offsets are morestable than electronic, we used an optical offset to tune one laser 120 MHz away andfound that the drift rate was still comparable. When similar frequency offsets wereapplied electronically, the drift increased to 3 MHz peak-peak.
If the two lasers drift in a correlated manner, then the difference frequency remains constant so the above measurement is insensitive to it. To confirm that this was
not occurring, we measured the stability of one DAVLL system by beating it against
a second diode laser that was locked to a peak of a saturated absorption feature. Because the physics of the two locks is quite different, we expect drifts in the two systems
to have different dependencies. In this case, we observe a stability of 200 kHz peakpeak over 12 h, which is consistent with the result previously described. From this weconclude that the two DAVLL systems were not drifting in a correlated manner, and109
the stability of the beat frequency can be interpreted as the stability of the absolute
frequency.
The frequency stability of the lock can also be predicted without comparing two
separate systems. We can convert the stability of the off-resonant signal level (Fig. A.3)
to an equivalent frequency stability by multiplying the fluctuations in photocurrent by
the slope of the central resonant DAVLL signal. This calculation reliably predicts thefrequency stability of the locked system and is therefore a simple, useful diagnostic.
The agreement between the predicted and measured stability also indicates that the
primary source of drift is changing birefringence of the optical components, becausebirefringence equally affects the signal levels both on and off resonance. As a final
testament to the lock’s stability, we used these lasers to maintain a Rb magneto-optic
trap for many days without adjusting the lasers that were locked to DAVLL systems.
The above results were obtained by use of zero-order glass/polymer retarders,
calcite Glan-Thompson input polarizers, and calcite Wollaston prism beam splitters.
Comparable stability was also found when we used less expensive optics, including
a plastic film polarizer, a plastic film retarder (/4 at 540 nm), and a single calcite
crystal (used as a polarizing beam splitter). In contrast, we found that some dielectric
polarizing beam-splitting cubes give a large temperature dependence.
The DAVLL lock was found to be robust because of the very broad locking signal.
In fact, we applied mechanical perturbations to the optical table as high as the table’s
damage threshold (including banging on the table with a hammer) and were unable toknock the lasers out of lock. The lasers jumped once every couple days, apparently
because one of the lasers jumped to a different mode of the laser chip. These jumps
were usually attributable to temperature drifts in the laser chip, but could occasionallybe caused by a fast electromagnetic pulse such as that produced by our turning on alarge nearby argon-ion laser. These types of mode hops are not observed in diodes
with good antireflection coatings because the chip resonances are greatly suppressed.
Therefore a DAVLL system constructed with such diodes would likely never lose lock.
A.6

Conclusion

We have shown that the DAVLL lock provides an effective method for stabilizing
a diode laser to a very broad, stable atomic reference. In comparison with saturated
absorption locks, this system stays locked for much longer periods of time and requires
fewer optics, less electronics, and less laser power. It can also be quite compact andinexpensive. This simple, robust stabilization scheme should work for a number of
atomic and molecular species at a variety of wavelengths and is an appealing option
whenever a continuous stable laser frequency is desired.

Appendix B
Stabilization of a Diode Laser at an Arbitrary Frequency

B.1

Introduction

We describe a general technique for stabilizing a diode laser to an arbitrary wavelength. This involves the use of a stable diode laser locked to an atomic transition at adifferent wavelength, and a low-finesse scanning Fabry-Perot cavity. The resulting lockis very robust and tunable. Therefore this lock can be employed in situations where no
atomic or molecular absorption lines exist.
This development was motivated by the Fr MOT described in Chapt.3. For that
trap, two lasers needed to be stabilized to two different transitions, one at the D2 transition and the other at the D1 . This is more challenging than locking to the correspondinglines in Rb because no stable isotopes of Fr exist, and therefore a Fr absorption cell
is not an option. While molecular iodine lines exist near the 221 Fr D2 line, none areavailable near the D1 .
The cavity lock was therefore developed to meet some unusual demands. We
knew the frequency of the 221 Fr D1 line only to within about 100 MHz, so our lockreference had to allow for a large scanning range. Because we had access to a -meterwith precision of a few MHz, long-term (day-to-day) stability was not particularly necessary. However, a robust lock was essential, because we needed to find both laser
frequencies simultaneously by fixing the D1 diode laser and scanning and the D2 laser.
If the laser frequently lost lock, we would never find the Fr laser frequencies.
Thus we devised a scheme that offered all of the above benefits with relatively
simple components. In short, a commercial scanning Fabry-Perot cavity was used totransfer the stability of a diode laser locked to a Rb line (at 780 nm) to a diode laser
near the Fr D2 line at 817 nm.
B.2

Description

A schematic of the lock is shown in Fig. B.1. The idea is relatively straightforward. First a diode laser at wavelength 1 is stabilized to a readily accessible atomic
transition. A beam from this laser is combined at a beam splitter (B.S.) with a beamfrom another laser at 2 , the wavelength of interest. Following the beam splitter, a polarizer (Pol.) and quarter-wave retarder (/4) provide some optical isolation from thelight that reflects off the cavity. The combined beam is then focused into a Fabry-Perot

111
cavity with mirrors that reflect at both wavelengths. The power transmitted throughthe cavity is partially collimated with another lens before it strikes a diffraction grating(D. G.) and diffracts into two separate beams, each of which is collected with a lensand focused onto a separate photodetector. This allows the transmitted power at eachwavelength to be monitored separately. To lock the length of the cavity to 1 , the powertransmitted through the cavity at 1 is monitored, and this signal is fed back to the cavity PZT to control the length. This effectively makes the cavity a length standard, equalin length to a half-integral number of wavelengths (1 ).
The frequency of the second diode laser is in turn locked to the cavity at 2 . To
do this, the transmitted power at 2 is monitored by an additional photodetector. Thenthe signal is fed back to the second diode laser to stabilize its frequency to the side of
the transmission peak of 2 s. In this way the stability of the first laser can be transferredto the second laser via the cavity. In addition, 2 can be tuned in a controllable way bymaking small changes to the frequency of the first laser. The cavity length will change,
and therefore 2 will change by nearly the same amount as 1 of the second laser’s lockpoint will also change by nearly the amount that the first laser was adjusted.
This scheme will only work, however, when the cavity transmits both wavelengths. This condition is of course not met at every cavity length, and in fact becomesless likely as the cavity finesse increases. As an example, consider the case for 1 = 780
nm and 2 = 817 nm. The transmitted power as monitored by the two photodetectors
is shown in Fig. B.2 for these wavelengths. One can see in Fig. B.2 (a) that there are
only a limited set of cavity mirror spacings for which power at both wavelengths willbe transmitted. Fig. B.2 (b) and (c) show enlarged portions of (a). The distance one
must translate the rear cavity mirror in order to observe coincident fringes depends on
the two wavelengths and the finesse of the cavity. If the two wavelengths share a large
least common denominator, then only a few fringes separate coincident fringes. For
example, when 1 = 500 nm and 2 = 600 nm, then one must only scan 6 fringes at
1 or 5 at 2 in order to observe exact coincidence. In general, however, coincidence
is only approximate, and depends on the width of the cavity resonance, or fringe. This
fringe width is given by the cavity free spectral range (FSR) divided by the finesse F .
The number of fringes that separate coincident fringes can be easily calculatedusing a simple computer program, as shown in Fig. B.3. This simple algorithm, written
in IDL,1 will calculate the number of fringes that separate coincident fringes. The only
input parameters consist of the two wavelengths and the finesse of the cavity. We havenot been able to write down a simple analytical expression for the results of this routine,
but one may certainly exist. For 1 = 780 nm and 2 = 817 nm, n1 = 22 (22 < F < 260),
n1 = 265 (260 < F < 850) and n1 = 780 for larger F .
1written by Kurt Miller

112

Figure B.1: Schematic of the cavity locking system used for transferring the stabilityof one laser to another.

113

(a) 1

0
0

2

4

6

8

10

(b) 1

0
4.0
(c) 1

4.5

5.0

5.5

0
8.5

9.0

9.5

change in cavity length (microns)
Figure B.2: Fractional transmission (calculated) of a cavity with a finesse of 40 for two
different wavelengths. The dotted line represents the transmission of 780 nm, while thesolid line represents 817 nm. Sections of plot (a) are enlarged below to show regions inwhich fringes are not coincident (b) and where fringes are coincident (c).

PRO cavity;initialize
n1
lambda1
lambda2

F

flag

114
= 0L
= 780D= 817 D
= 130D=0

while (flag eq 0) do begin
n1 = n1 + 1
n2 = n1  (lambda1=lambda2)
if ( abs(n2-round(n2)) lt (1/F ) ) then flag = 1
endwhile
print, ’n1=’,n1, ’ n2=’,n2END
Figure B.3: A simple computer algorithm (written in IDL) to calculate the distanceone must change the length of the cavity (in units of cavity fringes) between coincident
maxima in cavity transmission. Input parameters include the 2 wavelengths (lambda1
and lambda2) and the finesse of the cavity (F). The output parameters, n1 and n2,
represent the number of fringes (for each wavelength) through which the cavity mustscan before two fringes coincide at the same length, to within the width of the fringe.

B.3

Implementation

B.3.1

Components

The diode laser at 780 nm was stabilized to the Rb D2 transition using the DAVLL
scheme described in Appendix A and Ref. [70]. In fact, the requirements of insensitivity to disturbances motivated the development of the DAVLL lock. The output of thisstabilized laser then entered the scanning Fabry-Perot cavity. The voltage applied to
the PZT of the cavity was electronically controlled by monitoring the transmission of
the 780 nm light through the cavity, and side-locking to one of the cavity fringes.
The SDL laser at 817 nm was employed without an external cavity, in order toextract the most power possible for the Fr MOT. As a result, up to 80 mW were sent
to the trap from the 100 mW nominal output diode. The frequency of the diode wascontrolled by applying a DC voltage to the servo input of the current supply. Thus
the voltage was used to tune the current of the diode laser. A small amount of powerfrom this laser was sent into a -meter, as described in Chapt. 3. It had a precision ofa few MHz, and therefore allowed the frequency of the 817 nm diode to be precisely
determined. Without active frequency stabilization, the frequency drifted at 32 MHz/hr

115
or more.
The cavity we used was a commercially available Tropel scanning Fabry-Perotcavity. The mirror set we used was not particularly good at 817 or 780 nm, and as aresult, the cavity finesse was small, typically about 60. We used a high frequency, highvoltage amplifier built in the JILA electronics shop to drive the PZT of the cavity. The
amplifier’s high frequency response allowed us to use it in a servo loop to control thecavity length without inducing more phase shift than that inherent in the PZT-mirror
spring-mass system alone. The position of the rear mirror could also be coarsely controlled by rotating the mirror mount so as to screw the mirror in and out.
B.3.2

ProcedureOnce both lasers are optimally aligned into the cavity, the following procedure
is employed to stabilize the frequency of the 817 nm diode laser. First, the 780 nm
diode laser is locked to the zero-crossing of Fig. A.2, where it is least susceptible todrift. Then the 817 nm laser is tuned to the desired frequency by adjusting both its
temperature and current while monitoring the -meter. A saw-tooth ramp applied tothe high-voltage amplifier sweeps the cavity across several fringes, while the output
of each photodetector is monitored on an oscilloscope as simulated in Fig. B.2. By
simultaneously monitoring the fringes, the cavity can be adjusted such that the cavity
is nearly confocal and the fringes are therefore as symmetric as possible. Even so, thefringes are often not completely symmetric. Then small turns of the screw are applied
to find a pattern of coincident and nearly coincident fringes, as shown in Fig. B.2 (c).
Once the correct coincident fringes are found, the cavity is locked to the side of
the transmission fringe observed by photodetector 1. At this point, one must recheck thefrequency of the 817 nm laser, to ensure that it has not drifted too far, and if necessary,
relock the cavity to a different fringe of the 780 nm diode. One must also keep inmind the desired tuning range of the 817 nm laser when selecting the cavity fringe. At
last, the 817 nm diode frequency is side-locked to the transmitted power detected byphotodetector 2. This frequency can then be scanned by changing the set point voltageof the DAVLL-locked diode laser.
B.3.3

Polarization

As a note, we also attempted to separate the two wavelengths using polarization
instead of wavelength. In this scheme, the two wavelengths enter the cavity with opposite polarization, and are then separated at the output with a polarizing beam splitter.
This method might prove useful in cases where 1 and 2 are too close together tomake the diffraction grating useful. Unfortunately, this technique is very sensitive toslight mirror birefringence, especially for high finesse. This made it more difficult toeliminate cross-talk between the two polarizations, as discussed below. Therefore, theremainder of this Appendix will deal with the wavelength separation.

116
B.4

Results

The cavity lock performed very well throughout the Fr experiment. The robust
behavior of the DAVLL lock has already been discussed. During the experiment, itessentially never lost lock. The 817 nm diode laser lost lock when it mode hopped, andthis occurred every couple days, due to the unfortunate relative inaccessibility of ourtransition to our diode. The lock would also jump when the table was bumped too hard,because the cavity was sensitive to mechanical perturbations. Day-to-day, however,the cavity was also quite stable. Once the length was set, the coincident fringe would
consistently be found at the same voltage, 200 V.
One possible limitation to the stability of the lock would be cross-talk between
the two lasers. This would occur if some light detected by photodetector 2 was at 1
and vice versa. This leakage was less than 0.3%, implying that at least 99.7% of thelight striking photodetector 2 was at 2 .
Finally, all evidence indicates that the laser was stable to within a few MHz forhours. We were able to maintain a Fr MOT for hours at a time, and the MOT should besensitive to changes of 3 MHz or less, especially operating with limited repump power
as we were. The -meter readings were also stable to within its resolution.
B.5

Conclusion

In conclusion, we successfully transferred the stability of the DAVLL lock to an
additional diode laser via a cavity. This technique has minimal technological requirements, including a low-quality diffraction grating and a low-finesse cavity. It allows forcontrolled tuning over a wide range of laser frequencies, up to several hundred MHz.
Therefore this locking scheme is generally applicable to a large number of wavelengths
for a variety of applications.