Microtraps and Waveguides for Bose-Einstein
Condensatesby

Aaron E. Leanhardt
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2003
c Massachusetts Institute of Technology 2003. All rights reserved.
°

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
August 29, 2003
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David E. Pritchard
Cecil and Ida Green Professor of Physics
Thesis Supervisor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacAurthur Professor of Physics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thomas J. Greytak
Professor of Physics, Associate Department Head for Education

Microtraps and Waveguides for Bose-Einstein Condensatesby
Aaron E. Leanhardt
Submitted to the Department of Physics
on August 29, 2003, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Gaseous Bose-Einstein condensates containing up to 3 × 106 23 Na atoms were loaded
into magnetic microtraps and waveguides on a microfabricated “atom chip” using optical tweezers. Single-mode propagation was observed along the waveguide. Closer tothe microfabricated surface, perturbations to the waveguide potential spatially modulated the condensate density. The condensate lifetime was ≥ 20 s and independentof the atom-surface separation, for separations ≥ 70 µm.
Condensates were coherently split by deforming an optical single-well potential
into a double-well potential. The relative phase between the two resulting condensates
was determined from the matter wave interference pattern formed upon releasing theatoms from the separated potential wells. Coherent phase evolution was observed forcondensates held separated by 13 µm for ≤ 5 ms and was controlled by applying AC
Stark shifts to either condensate. This demonstrated a trapped-atom interferometer.
Vortices and spin textures were imprinted in spinor condensates using topologicalphases. The order parameter of condensates held in a Ioffe-Pritchard magnetic trap
was manipulated by adiabatically varying the magnetic bias field along the trap axis.
Fully inverting the axial bias field imprinted vortices in F = 1 and F = 2 condensateswith 2h̄ and 4h̄ of angular momentum per particle, respectively. Reducing the axial
bias field to zero distributed the condensate population across its 2F + 1 spin states,
each with a different phase winding, and created a spin texture.
Partially condensed atomic vapors were confined by a combination of gravitational and magnetic forces. They were adiabatically decompressed, by weakening thegravito-magnetic trap to a mean frequency of 1 Hz, then evaporatively reduced in size
to 2500 atoms. This lowered the peak condensate density to 5 × 1010 atoms/cm3 andcooled the entire cloud in all three dimensions to a kinetic temperature of 450±80 pK.
Thesis Supervisor: David E. Pritchard
Title: Cecil and Ida Green Professor of Physics
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacAurthur Professor of Physics

To my family,
Dale, Sharon, and Wendi

Acknowledgments
An African proverb states that it takes a whole village to raise a child. It has been
my experience that earning a Ph.D. requires a similar level of support. While it
is inevitable that only my name will appear on this thesis and the corresponding
doctorate degree, I would like to acknowledge several important contributions to mygrowth and development as a physicist, my experiences in the laboratory, and mygeneral survival at MIT.
Dave Pritchard and Wolfgang Ketterle deserve far more than the traditional obligatory “thanks” given to most thesis supervisors. Dave always found ways to challengeme, and while the warmth with which I accepted the challenges varied wildly, I certainly benefited from facing each one. Wolfgang always found ways to encourage me,even in the midst of my many failures. In general, I would like to thank Dave andWolfgang for allowing me to explore ideas that often led the lab astray, or at least off
the beaten path, but in the end helped to develop my physical intuition.
I owe much gratitude to the people ahead of me in the lab. In particular, I thankAxel Görlitz for teaching me the ways of MIT’s original Bose-Einstein condensation(BEC) apparatus, the nuances of dye lasers, and quite generally how to be a physicist.
However, all of the work in this thesis was performed on MIT’s third generation BEC
apparatus, a.k.a. the science chamber. Todd Gustavson, Ananth Chikkatur, Axel
Görlitz, and Deep Gupta were instrumental in making the science chamber a reality,
without which much of this thesis would not exist. I am indebted to Todd for his
attention to detail, Ananth for his upbeat leadership, and Deep for his laid-backmentoring.
More recently, it has been a privilege to work with Yong-Il Shin, Tom Pasquini,
Michele Saba, and Andre Schirotzek. All of the experiments described in this thesis
were performed in collaboration with Yong. He has been both a selfless wingman anda determined leader. I thank him for supporting my ideas (as random as they were)
and teaching me his own. Tom has been a wonderful officemate with whom I haveenjoyed many spontaneous discussions, Michele’s contributions to the lab make mewish he had become our post-doc two years earlier, and Andre has brought enthusiasm
to our late night data runs. Collectively, I thank Yong, Tom, Michele, and Andre forbeing great labmates and scientific companions.
MIT’s atomic physics program certainly has strength in numbers. Over the years
I have greatly benefited from interactions with James Anglin, Micah Boyd, Gretchen
Campbell, Zoran Hadzibabic, Shin Inouye, Dan Kleppner, Roberto Onofrio, Chandra Raman, Till Rosenband, Dominik Schneble, Dan Stamper-Kurn, Johnny Vogels,
Kaiwen Xu, and Martin Zwierlein.
Lab productivity would have been greatly reduced without the aid of Mara Prentiss, Mukund Vengalattore, and MIT’s Microsystems Technology Laboratories, whohelped our research group enter the field of miniaturized atom optics, Edem Tsikata,
who provided experimental assistance and supported our lab infrastructure, and Carol
Costa, who handled our administrative needs.
While laboratory research is all that is included in this thesis, it is far from allthat is required to graduate. My early years at MIT were filled with coarse work andexams, none of which I could have survived without the help and tutelage of Jamil
Abo-Shaeer, Lia Matos, and Kendra Vant. Jamil’s “yo-man” attitude, Lia’s ability torewrite any equation in terms of a complex plane integral that I could not evaluate,
and Kendra’s meticulous organization helped to keep me sane, humble, and on-trackat a time when I needed it most.
Jamil has been a true friend and confidante ever since I arrived at MIT, and I
will continue to seek his advice even after we go our separate ways. His unwavering
support inside and outside of lab has been invaluable. Together, he and Kate DuBose
have been caring and generous roommates.
I would not be a physicist today without the guidance that I received while anundergraduate at the University of Michigan from Jean Krisch, Dave Gidley, andRich Vallery.
Finally, I thank my father Dale, my late mother Sharon, and my sister Wendi forproviding me with inspiration. Rather than trying to describe the totality of theircontributions to my life, I simply dedicate my work at MIT to them.

Contents
1 Introduction

11

1.1

BEC Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.2

Macroscopic Order Parameter . . . . . . . . . . . . . . . . . . . . . .

13

1.2.1

Density Distribution . . . . . . . . . . . . . . . . . . . . . . .

14

1.2.2

Phase Coherence . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.3

Contributions and Collaborations . . . . . . . . . . . . . . . . . . . .

16

1.4

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

16

2 Bose-Einstein Condensate Production and Transport
2.1

18

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

18

2.1.1

Condensate Production . . . . . . . . . . . . . . . . . . . . . .

18

2.1.2

Condensate Transport . . . . . . . . . . . . . . . . . . . . . .

20

2.2

Magnetic and Optical Confinement . . . . . . . . . . . . . . . . . . .

21

2.3

Optical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3 Magnetic Microtraps and Waveguides

25

3.1

Magnetic Micropotentials . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.2

Propagation in Magnetic Waveguides . . . . . . . . . . . . . . . . . .

30

3.2.1

Origins of Waveguide Perturbations . . . . . . . . . . . . . . .

32

3.2.2

Reflections from Waveguide Perturbations . . . . . . . . . . .

35

3.2.3

Excitations due to Waveguide Perturbations . . . . . . . . . .

36

Microfabricated Surface Effects . . . . . . . . . . . . . . . . . . . . .

37

3.3.1

37

3.3

Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . .
6

3.3.2

Trap Lifetimes. . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.3.3

Radio Frequency Spin-Flip Transitions . . . . . . . . . . . . .

43

3.3.4

Thermal Noise. . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.4

Waveguide Beam Splitters . . . . . . . . . . . . . . . . . . . . . . . .

45

3.5

Microfabrication Procedures . . . . . . . . . . . . . . . . . . . . . . .

47

4 Trapped-Atom Interferometry in an Optical Double-Well Potential 49
4.1

Optical Double-Well Potential . . . . . . . . . . . . . . . . . . . . . .

51

4.2

Matter Wave Interference. . . . . . . . . . . . . . . . . . . . . . . .

52

4.2.1

Relative Phase . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.2.2

Phase Coherence . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.3

Trapped-Atom Interferometry . . . . . . . . . . . . . . . . . . . . . .

57

4.4

Josephson Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

5 Vortices and Spin Textures

60

5.1

Phase Imprinting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

5.2

Vortex Formation and Detection . . . . . . . . . . . . . . . . . . . . .

63

5.2.1

Berry’s Phase . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

5.2.2

F = 1 and F = 2 23 Na Condensates . . . . . . . . . . . . . . .

66

5.2.3

Observation of Vortex Cores . . . . . . . . . . . . . . . . . . .

67

5.2.4

Angular Momentum Measurements . . . . . . . . . . . . . . .

69

5.2.5

Multiply Charged Vortices . . . . . . . . . . . . . . . . . . . .

71

Coreless Vortex Formation and Detection . . . . . . . . . . . . . . . .

72

5.3.1

Spin Textures . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5.3.2

Observation of Coreless Vortices . . . . . . . . . . . . . . . . .

76

5.4

Vortices and Spin Textures in Optical Traps . . . . . . . . . . . . . .

79

5.5

Higher Order Vortices and Spin Textures . . . . . . . . . . . . . . . .

80

5.3

6 Picokelvin Temperature Atomic Vapors

82

6.1

Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

6.2

Gravito-Magnetic Trap . . . . . . . . . . . . . . . . . . . . . . . . . .

85

7

6.2.1

Stable Confinement . . . . . . . . . . . . . . . . . . . . . . . .

87

6.2.2

Radial Symmetry . . . . . . . . . . . . . . . . . . . . . . . . .

88

6.3

Cooling Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

6.4

Adiabatic Decompression . . . . . . . . . . . . . . . . . . . . . . . . .

89

6.4.1

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

89

Picokelvin Temperatures . . . . . . . . . . . . . . . . . . . . . . . . .

90

6.5

Evaporative Atom Number Reduction

7 Conclusion

93

7.1

Superfluid Gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

7.2

Last Man Standing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

A Propagation of Bose-Einstein Condensates in a Magnetic Waveguide 97
B Bose-Einstein Condensates near a Microfabricated Surface

102

C Atom Interferometry with Bose-Einstein Condensates in a DoubleWell Potential

107

D Imprinting Vortices in a Bose-Einstein Condensate using Topological
Phases112

E Coreless Vortex Formation in a Spinor Bose-Einstein Condensate 117
F Cooling Bose-Einstein Condensates Below 500 Picokelvin

122

Bibliography

126

8

List of Figures
2-1 The science chamber apparatus. . . . . . . . . . . . . . . . . . . . . .
2-2

23

19

Na energy levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3-1 Macroscopic and microscopic magnetic traps. . . . . . . . . . . . . . .

26

3-2 Single-wire magnetic waveguide. . . . . . . . . . . . . . . . . . . . . .

29

3-3 Circular vs rectangular wire cross section. . . . . . . . . . . . . . . .

30

3-4 Z-wire magnetic trap. . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3-5 Microfabricated atom chip. . . . . . . . . . . . . . . . . . . . . . . . .

32

3-6 Scattering from waveguide perturbations. . . . . . . . . . . . . . . . .

34

3-7 Propagation in a magnetic waveguide. . . . . . . . . . . . . . . . . . .

36

3-8 Fragmentation in a magnetic microtrap. . . . . . . . . . . . . . . . .

38

3-9 Lack of fragmentation in an optical dipole trap. . . . . . . . . . . . .

39

3-10 Lifetime near a microfabricated surface. . . . . . . . . . . . . . . . . .

41

3-11 Radio frequency spin-flip transitions. . . . . . . . . . . . . . . . . . .

42

3-12 Waveguide beam splitter. . . . . . . . . . . . . . . . . . . . . . . . . .

46

3-13 Atomic wavepacket division. . . . . . . . . . . . . . . . . . . . . . . .

47

4-1 Optical double-well potential. . . . . . . . . . . . . . . . . . . . . . .

52

4-2 Matter wave interference. . . . . . . . . . . . . . . . . . . . . . . . . .

53

4-3 Phase coherence of separated condensates. . . . . . . . . . . . . . . .

55

4-4 Decoherence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4-5 Trapped-atom interferometry. . . . . . . . . . . . . . . . . . . . . . .

57

5-1 Spin rotation and Berry’s phase. . . . . . . . . . . . . . . . . . . . . .

64

9

5-2 Vortices in F = 1 and F = 2 condensates. . . . . . . . . . . . . . . .

67

5-3 Axial magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5-4 Surface wave spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . .

70

5-5 Vortex line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5-6 Planar spin textures. . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5-7 Radial dependence of the spin rotation angle. . . . . . . . . . . . . .

75

5-8 Coreless vortices in F = 1 spinor condensates. . . . . . . . . . . . . .

77

5-9 Vortices in an optical trap. . . . . . . . . . . . . . . . . . . . . . . . .

79

6-1 Gravito-magnetic trap. . . . . . . . . . . . . . . . . . . . . . . . . . .

86

6-2 Picokelvin temperature thermometry. . . . . . . . . . . . . . . . . . .

91

6-3 Partially condensed atomic vapors at picokelvin temperatures. . . . .

92

7-1 Vortex phase interferogram. . . . . . . . . . . . . . . . . . . . . . . .

95

10

Chapter 1
Introduction
The phenomenon of Bose-Einstein condensation (BEC) underlies our understandingof superfluid liquid helium and superconducting electron pairs. However, the strengthand complexity of particle-particle interactions in these systems makes controlling,
manipulating, and characterizing their condensate phases difficult. It was only after
1995, when the BEC phase transition was observed in dilute atomic vapors [40, 110],that many fundamental properties of Bose-Einstein condensates were addressed.
In the following years, the field of quantum degenerate gases has grown to include
eight species of atomic Bose-Einstein condensates, 87 Rb [6], 23 Na [46], 7 Li [21, 22],1

H [63], 85 Rb [41], 4 He∗ [167, 182], 41 K [151], 133 Cs [217], and 174 Yb [207] and two

species of quantum degenerate Fermi gases, 40 K [48] and 6 Li [213]. The field developed experimentally and theoretically in parallel, with theoretical predictions leading
to experimental discovery and experimental observations producing theoretical understanding equally often. Much of this development can be found in the multitude
of review articles, conference proceedings, and textbooks currently available, of whichRefs. [42, 94, 128, 166, 169] are a small subset.
This chapter introduces the concept of Bose-Einstein condensation as relevant
for the experiments presented in this thesis. For clarity, the presentation assumes athree-dimensional gas in thermal equilibrium.

11

1.1

BEC Phase Transition

The Bose-Einstein condensation phase transition corresponds to a macroscopic occupation of the quantum mechanical ground state of the system1 . For N particles
at a temperature T , the critical point for condensation corresponds to all thermallyaccessible being occupied, i.e. the number of quantum states with energy E ≤ kB T
equals the number of particles in the system, where kB is Boltzmann’s constant. At
the critical point, the gas is referred to as a quantum saturated thermal vapor.
An alternative, but equivalent, description of the BEC phase transition can begiven in terms of matter wave overlap. For a gas at number density2 n, matter wave
overlap occurs when the interparticle spacing, n−1/3 , equals the thermal deBroglie
wavelength, λT , of the atoms. At this point the atoms lose their individual identities
and begin to behave as a unified ensemble. This criteria can be written as nλ3T ≈ 1,

where nλ3T is the phase space density of the gas3 . Thus, the BEC phase transition
occurs at a phase space density near unity meaning that all thermally accessible states
are occupied.
Once all thermally accessible states are occupied, adding more particles to theensemble results in a macroscopic occupation of the ground state (according to Bosestatistics) and Bose-Einstein condensation. Thus, a gas cooled below the BEC phasetransition temperature consists of two components: a thermal (normal) fraction thatoccupies many quantum states and a condensed (superfluid) fraction that occupies asingle quantum state.
The number of thermally accessible states below a given temperature depends onthe excitation spectrum of the system. Free particles have a continuum of excitationenergies ²(p) = p2 /2m, where p and m are the momentum and mass of the particles,
respectively. Confinement modifies the excitation spectrum. The experiments de1

More generally, the BEC phase transition can be thought of as the macroscopic occupation ofa single quantum state, often occurring in long-lived metastable states rather than the absolute
ground state of the system.
2
All densities described in this thesis are number densities, not mass densities.
3
A phase space cell has position and momentum dimensions ∆q = n−1/3 and ∆p = h/λT ,respectively. The normalized volume of such a cell is (∆q∆p/h)3 = 1/nλ3T , where h is Planck’s
constant.

12

scribed in this thesis were performed with harmonically trapped atoms. Such particles
have excitation energies ²(nx , ny , nz ) = (nx +1/2)h̄ωx +(ny +1/2)h̄ωy +(nz +1/2)h̄ωz ,where ωx , ωy , and ωz are the trap frequencies and h̄ = h/2π.
The density, nth , or number, Nth , of thermally excited atoms is found by integrating or summing the Bose-Einstein distribution,hn(²)i =1e(²−µ)/kB T − 1

,

(1.1)

where hn(²)i gives the number of particles occupying a state of energy ² such that
N =R

d²hn(²)i, ² is the appropriate excitation spectrum, and µ is the chemical

potential of the gas. For free particles this results in√

2πmkB Th2
⇒ kB Tc =nth = ζ(3/2)
h
2πm

Ã

n
ζ(3/2)

!2/3

,

(1.2)

where Tc is the BEC phase transition temperature and ζ(3/2) ≈ 2.612 is the Riemann
Zeta function.
Likewise, for harmonically confined atoms
ÃkB T
Nth = ζ(3)
h̄ω̄

!3

Ã

N
⇒ kB Tc = h̄ω̄
ζ(3)

!1/3

,

(1.3)

where ζ(3) ≈ 1.202 and ω̄ = (ωx ωy ωz )1/3 . Typical experimental parameters fall in
the range 104 < N < 107 and 1 Hz < ω̄/2π < 100 Hz such that 1 nK < Tc < 1 µK
and 1011 atoms/cm3 < n < 1015 atoms/cm3 .

1.2

Macroscopic Order Parameter

The above discussion was for an ideal (non-interacting) Bose gas. Atom-atom interactions are essential for thermalization and efficient cooling to quantum degeneracy.
They are manifestly present under typical experimental conditions. While such interactions shift the BEC phase transition temperature by only a few percent, they
typically dominant the ground state properties of the condensate.
131.2.1

Density Distribution

The N0 = N − Nth atoms in the condensate are all described by a single orderparameter (wavefunction), Ψ(~r, t), that represents the ground state of the manybody system. The usual Schrödinger equation is modified to include a nonlinear termthat accounts for atom-atom interactions, and ψ(~r) is an eigenstate of the resulting
Gross-Pitaevskii equation,h̄2 2
4πh̄2 a−∇ + V (~r) +|ψ(~r)|2 ψ(~r) = µψ(~r),
2m
m
!

Ã

(1.4)

where V (~r) is the external potential, a is the s-wave scattering length for a binary
collision, and the time evolution of the wavefunction is given by the chemical potential,
Ψ(~r, t) = ψ(~r)e−i(µ/h̄)t .
In the Thomas-Fermi limit, the atom-atom interaction energy dominates the kinetic energy [(−h̄2 /2m)∇2 ψ(~r)] associated with the condensate wavefunction, andthe ground state condensate density isn(~r) = |ψ(~r)|2 =m[µ − V (~r)].
4πh̄2 a(1.5)

Since the chemical potential is a constant throughout the gas, the condensate wavefunction has the inverted shape of the trapping potential. For harmonically confinedatoms V (~r) = (1/2)m(ωx x2 + ωy y 2 + ωz z 2 ), and the chemical potential can be expressed as1ah̄ω̄ 15N2
aHO
1=mωi2 Ri2 ,2µ =where aHO =

µ

¶2/5

,

(1.6)
(1.7)

q

h̄/mω̄ and Ri is the Thomas-Fermi size of the condensate along thei = x, y, z axis.

More generally, taking the trap center to be the zero of energy, the chemical
potential can be expressed as µ = (4πh̄2 a/m)n0 , where n0 is the condensate density

14

at the trap center. The local mean field interaction energy is often referred to a thelocal chemical potential and is written as µ(~r) = (4πh̄2 a/m)n(~r). For 23 Na atoms in
the |F = 1, mF = −1i state, a ≈ 53a0 ≈ 2.8 nm, where a0 is the Bohr radius4 . Thus,µ ≈ kB × 66 × 10−14 nK-cm3 ≈ h × 1.4 × 10−14 kHz-cm3 .

Briefly a word about energy units. Throughout this thesis energy will be quoted
in temperature, frequency, and occasionally magnetic field units. The conversions are
kB × 1 µK ≈ h × 21 kHz ≈ (µB /2) × 30 mG, where 1 G = 10−4 T, µB is the Bohrmagneton, and atoms in the |1, −1i state have a magnetic moment of µB /2.
The treatment presented in this section is valid provided the gas is sufficiently
dilute, na3 ¿ 1, which is satisfied in all of the experiments presented in this thesis.
Two important energy scales for partially condensed atomic vapors are the thermal
kinetic energy (temperature) and chemical potential of the sample. Quite generally,µ/kB Tc ∝ (na3 )x where x = 1/3 for free particles and x = 1/6 for harmonicallyconfined atoms in the Thomas-Fermi limit. Thus, the mean field interaction energy
experienced by atoms in the condensate fraction is typically less than the kinetic
energy of atoms in the thermal fraction.

1.2.2

Phase Coherence

The condensate order parameter can be written in terms of an amplitude and phase,ψ(~r) =

qn(~r)eiS(~r) . Associating a single phase with the condensate gives rise to

its laser-like coherence [9, 25, 107, 113] and superfluid [133, 134, 210] properties.
These phase coherence properties can be destroyed in very anisotropic systems at
finite temperature [50, 83, 181] and in the presence of potential corrugations such as
optical lattices [70, 164], where localized regions of the gas have independent phases.
However, the experiments described in this thesis were performed in a regime wherea single phase accurately described the condensate.
4

For 23 Na atoms in the |F = 2, mF = +2i state, a ≈ 63a0 ≈ 3.3 nm. In general, a can be tuned
with an externally applied magnetic field at a Feshbach resonance [95]. The values quoted for a arethose far from any such resonance.

15

1.3

Contributions and Collaborations

My first two years at MIT were divided between contributing to the constructionof a new BEC apparatus5 and collaborating on experiments running on an existingone. Aside for the usual gamut of tasks given to the youngest graduate student(electronics assembly, computer programming, etc. . .), my main contribution to the
new BEC apparatus was designing and building a Ioffe-Pritchard, cloverleaf stylemagnetic trap [68, 147, 174]. In parallel, I was given the opportunity to collaborate on
two experiments performed on MIT’s original BEC apparatus. The first realized BoseEinstein condensates in lower dimensions [67], and the second transferred sodium
condensates between the lower (F = 1) and upper (F = 2) hyperfine levels [66].
While these construction and experimental efforts were invaluable in teaching me
the methods of atom cooling and condensate manipulation, they will not be describedin any detail in this thesis. Instead, I will focus on the scientific results produced in
my final two years at MIT. The title Microtraps and Waveguides for Bose-Einstein
Condensates reflects an ongoing effort in the lab to design and implement new devices
to confine and manipulate Bose-Einstein condensates, and much of the work presented
in this thesis reflects that effort. In addition, the phase coherence properties of thecondensate are emphasized with respect to atom interferometry and superfluidity.
Appendices A-F contain reprints of publications covering the main achievementsof my graduate work. This thesis aims to motivate, contextualize, summarize, andsupplement the content of these publications. General results and trends will beemphasized and discussed based on the current state of the field.

1.4

OutlineSodium condensates were produced by a novel BEC apparatus described in Ch. 2,
without which many of the following experiments would not have been possible. This
includes the trapping and guiding of condensates with magnetic potentials generated
by microfabricated current-carrying wires described in Ch. 3. A major long term
5This was MIT’s third BEC apparatus and would become known as the science chamber (Ch. 2).

16

goal of this effort is to build a trapped-atom interferometer. To this end, we identified several limitations of magnetic microtraps and worked in parallel to develop
such an interferometer using optical potentials. Chapter 4 reports on the successful demonstration of trapped-atom interferometry in an optical double-well potential.
Chapter 5 delves into the superfluid nature of Bose-Einstein condensates and describes
vortices and spin textures imprinted onto the condensate wavefunction using topological phases. Chapters 4 and 5 are complementary in the sense that Ch. 4 demonstrates
our ability to write and read uniform condensate phases, while in Ch. 5 we imprintand detect phase windings. Finally, Ch. 6 describes the cooling of partially condensed
atomic vapors to picokelvin temperatures at peak densities < 1011 atoms/cm3 . Such
low temperature and low density samples are important for spectroscopy, metrology,
and atom optics. Some concluding remarks and perspective are included in Ch. 7.
Appendices A-F include publications directly related to the chapters of this thesis.

17Chapter 2
Bose-Einstein Condensate
Production and Transport
All of the experiments described in this thesis were performed on MIT’s third generation sodium BEC apparatus, BEC-III, commonly referred to as the science chamber
for reasons that will be made obvious below. The apparatus is described briefly in
Ref. [74] and thoroughly in Ref. [33]. The aim of this chapter is to fall somewhere in
between.

2.1

The Science Chamber Apparatus

The design goal for this novel apparatus was to separate the job of producing condensates from the task of studying them. In short, we wanted to create a condensate
beamline such that every 20 − 30 s a fresh condensate would be delivered to an
experimental platform for examination. Figure 2-1 shows our realization of this goal.

2.1.1

Condensate Production

A thermal atomic beam escapes through a 4 mm aperture in an effusive oven containing liquid sodium heated to 260 C. The atomic beam is slowed by radiation pressure
in a spin-flip Zeeman slower to a mean velocity ∼ 30 m/s. The slow atom flux exiting18

Production
23
Na beam Chamber

ScienceChamber

36 cm

g

pumps

pumps

IR

slowing light
Figure 2-1: The science chamber apparatus. Gaseous Bose-Einstein condensates of
sodium atoms are created through a combination of laser and evaporative cooling in
the production chamber. Subsequently, they are captured in the focus of an infrared(IR) optical tweezers beam and transported into the auxiliary science chamber. The
acceleration due to gravity, ~g , points down the page. Drawn to scale.

the Zeeman slower is ∼ 1011 atoms/s. The circularly polarized slowing laser beam
contains ∼ 50 mW of power and is tuned 1000 MHz below the F = 2 → F 0 = 3 cyclingtransition (Fig. 2-2). 10% of the slowing laser power is transferred into repumping
sidebands near the F = 1 → F 0 = 2 transition by an electro-optic modulator (EOM)
driven near1 1743 MHz. The Zeeman slower consists of three separately powered
tapered solenoids running currents between 16 A and 105 A and produces magnetic
fields up to 700 G.
The slowed atomic beam is continuously loaded into a dark spontaneous force
optical trap (dark-SPOT [111]) type magneto-optical trap (MOT [177]) for 1 s. The
MOT consists of a spherical quadrupole magnetic field with a maximum gradientof 11 G/cm and three orthogonal pairs of counter propagating laser beams detuned
20 MHz below the F = 2 → F 0 = 3 cycling transition. The total power in all six MOT
beams is ∼ 100 mW. A ∼ 25 mW repumping beam resonant with the F = 1 → F 0 = 2
1

Empirically, this frequency can be varied over tens of megahertz without degrading the Zeeman
slower performance. This is because the magnetic field profile of the Zeeman slower is designed tokeep the F = 2 → F 0 = 3 cycling transition on resonance, but not the F = 1 → F 0 = 2 repumping
transition.

19

transition is applied to the edge of the cloud such that the atoms in the center of the
MOT remain in the (dark) F = 1 state, while the hotter atoms near the edge arecooled and transferred into the central F = 1 cloud.
Weak field seeking |F = 1, mF = −1i atoms cooled by the MOT to ∼ 100 µK
are loaded into a Ioffe-Pritchard, cloverleaf style magnetic trap [68, 147, 174]. No
intermediate sub-doppler cooling is performed although some polarization gradientcooling takes place in the MOT itself2 . One half of the magnetic trap is shown in
Ch. 3, Fig. 3-1(a). The two halves are identical and mounted outside the ultrahighvacuum chamber, symmetrically about the atoms. The trapping coils generate anaxial magnetic field curvature of 1 G/cm2 /A and a radial magnetic field gradientof 0.6 G/cm/A. The magnetic trap runs currents between 200 A and 500 A anddissipates 15 kW of power. Typical trapping parameters are an axial bias field of
1.4 G and axial and radial trap frequencies ω|| = 2π × 20 Hz and ω⊥ = 2π × 200 Hz,
respectively.
Forced radio frequency evaporation is used to cool the magnetically trapped atoms
to Bose-Einstein condensation. The radio frequency drive uses ∼ 1 W of power andits frequency3 is ramped from 30 MHz to ∼ 1 MHz in 20 s. This procedure produces
Bose-Einstein condensates containing > 107 atoms.

2.1.2

Condensate Transport

After production, the condensates are loaded into the focus of an optical tweezers
laser beam. The optical tweezers consist of a 1064 nm laser focused to a 1/e2 radiusof 26 µm, such that ∼ 50 mW of power produces axial and radial trap frequencies
ω|| = 2π×4 Hz and ω⊥ = 2π×440 Hz, respectively, and a trap depth of kB ×3 µK. The
focus is generated by passing a collimated beam through a 50 mm diameter, 500 mm
focal length lens mounted on an air bearing translation stage. Translating this lens
moves the position of the focus and hence the atoms as well. Using the opticaltweezers, condensates are transported ∼ 36 cm in 2 s into an auxiliary “science”
2
3

For 23 Na, the doppler cooling limit is TD = 240 µK.
1 MHz corresponds to a 1.4 G magnetic field.

20

chamber. Upon arrival in the science chamber the condensates contain up to 3 × 106
atoms.
The optical power for the tweezers beam is raised by a factor of 5 − 10 duringtranslation to prevent atom loss, but it is lowered again as the atoms near their finaldestination. The 1/e lifetime of condensates in the optical tweezers is 20 s. This ismost likely limited by vibrations, since the background vapor pressure (∼ 10−11 torr)
and photon scattering rate (0.01 Hz) are sufficiently low not to cause problems.
The condensate production chamber and the science chamber are separated bya gate value. This allows for the science chamber to be vented and reconfigured
while maintaining ultrahigh vacuum in the condensate production region. In addition, the science chamber is free from the many constraints of producing a condensate(laser cooling optics, Zeeman slower, magnetic trap, etc. . .) and thus has better experimental access than the production chamber. Since the first condensates arrived
in the science chamber two years ago, we have used this flexibility to demonstrateminiaturized magnetic traps [74], a continuous source of Bose-Einstein condensed
atoms [35], microfabricated magnetic traps and waveguides [122, 126], topologicaldefects such as vortices [123] and spin textures [126], a trapped-atom interferometer [199], and a weakly confining gravito-magnetic trap with picokelvin temperaturecondensates [124].

2.2
23

Magnetic and Optical Confinement

Na has a 3S1/2 ground state with F = 1 and F = 2 hyperfine levels (Fig. 2-2). At~ < 100 G), F and mF are good quantum numbers and thelow magnetic fields (|B|atomic states can be described in a basis |F, mF i. The interaction energy of thesestates with a magnetic field is~
VB = gF mF µB |B|,

21

(2.1)

F'=3

(a)

(b)
m=-3

m=-2

1/15

58.3 MHz

3P3/2

m=-1

15.9 MHzm=0

m=+1 m=+2 m=+3

1/5

2/5

3/5

2/3

1/3

8/15

8/15

1

2/3

2/5

1/5

m=-2

m=-1

m=0

m=+1 m=+2589.158 nm

F'=115.8 MHz

m=-2

m=-1

m=0

m=+1 m=+2

1/15

F=2

(c)

F'=0

1/12

F=2

1/4

664.359 MHz

3S1/2

1

1/3

F'=2
34.3 MHzF'=31771.626 MHz

1/3

1/2

1/4

m=-1

m=0

1/4

F'=2

1/2

1/4
1/12
m=+1

F=1

F=1
Figure 2-2: 23 Na energy levels. (a) The D2 line used for laser cooling and opticalimaging. Relative strength of the (b) F = 2 → F 0 = 3 and (c) F = 1 → F 0 = 2
electric dipole transitions [146].

~ has
where gF is the Landé g-factor, µB is the Bohr magneton, and the direction of B
been taken as the quantization axis. For 23 Na, gF =1 = −1/2 and gF =2 = 1/2.
Since Maxwell’s equations allow (prohibit) magnetic field minima (maxima) infree space, only the weak field seeking states |1, −1i, |2, +1i, and |2, +2i can bemagnetically trapped [148]4 . Their magnetic moments are µB /2, µB /2, and µB ,respectively.
In contrast, all spin states may be optically confined and experience the same4

Spin exchange collisions between atoms in the |2, +1i state severely limit trap lifetimes such thatmagnetically trapped sodium condensates have been realized in the |1, −1i [46] and |2, +2i [123]states only. In principle, atoms in the |2, 0i can be magnetically trapped due to the quadratic
Zeeman effect. However, they would also suffer from inelastic spin exchange collisions.

22

trapping potential5,6 . The interaction of a two-level atom with an optical field ofintensity I(~r) and frequency ω can be described by the optical dipole potential [10, 38](Γ/2)2I(~r)
h̄∆ωln 1 +,
Vop =2
Is ∆ω 2 + (Γ/2)2
Ã

!

(2.2)

where ∆ω = ω − ω0 is the detuning from the atomic resonance of width (FWHM) Γ
and Is is the saturation intensity of the transition. In writing Eq. 2.2, the rotating
wave approximation was made and saturation effects were neglected7 . From Eq. 2.2 itis evident that atoms are attracted to high intensity regions of red-detuned radiation(∆ω < 0). Thus, a focused red-detuned laser beam can trap atoms [38]. Conversely,
atoms are repealed from high intensity regions of blue-detuned radiation (∆ω > 0).

2.3

Optical Imaging

The 23 Na D2 line is used for laser cooling and optical imaging. It has wavelength
λ = 589.158 nm and linewidth (FWHM) Γ = 2π × 10 MHz. While non-destructive,
dispersive imaging is possible, all of the data presented in this thesis was acquired
through destructive, absorption imaging. The cross section for scattering photons is
maximal on the F = 2, mF = ±2 → F 0 = 3, m0F = ±3 cycling transitions and has

the value σ0 = 3λ2 /2π on resonance. The relative scattering cross sections for othertransitions are displayed in Figs. 2-2(b) and 2-2(c).
Absorption imaging of condensates in the F = 1 (F = 2) hyperfine level is performed on the F = 1 → F 0 = 2 (F = 2 → F 0 = 3) electric dipole transition.
Condensates in the F = 1 hyperfine level may be optically pumped into the F = 2
hyperfine level and subsequently imaged on the F = 2 → F 0 = 3 transition. The
absorption imaging process integrates the atomic density along the axis of the probe5

Spin exchange collisions between atoms in the |F = 2, mF = +1, 0, −1i states severely limit trap
lifetimes such that optically confined sodium condensates have been realized in the |F = 1, mF =+1, 0, −1i [202], |2, −2i [66], and |2, +2i [123] states only.
6
For laser detunings large compared to the fine structure splitting of the electronic excited state,
the multi-level nature of the atom and the polarization of the light field are unimportant.
7
For the optical tweezers described above, the counter-rotating term contributes an additional
30% to the trap depth in Eq. 2.2 [74, 202].

23

beam allowing only the column density of the cloud to be extracted [112]. Spatial
images are focused onto a CCD camera using pairs of lenses. The effective pixel sizeof the camera is typically ≥ 3 µm, and condensate dimensions during the imaging
process range from a few microns to a few millimeters.
Trapped condensates often contain features too small to be optically resolved.
Suddenly switching off the trapping potential allows the mean field energy, µ, to beconverted to kinetic energy, and the condensate wavefunction expands ballistically.
For prolate (cigar-shaped) condensates, the expansion is anisotropic with most of themean field energy transferred into radial kinetic energy [28]. As a result, prolate
condensates imaged during ballistic expansion have their radial features magnified
by

q

2 2
1 + ω⊥
τ , where ω⊥ is the radial frequency of the trap from which they werereleased and τ is the duration of the ballistic expansion process. This allows for small
radial features, such as vortex cores, to be optically imaged. In contrast, negligible
axial expansion occurs leading to unit magnification of axial features.
Since µ/kB Tc ¿ 1, when released from the trap the condensate fraction expands
slower than the thermal fraction and the two components can be easily distinguished.

24

Chapter 3
Magnetic Microtraps andWaveguides
The field of atom optics relies on bright sources of matter waves and techniques to coherently manipulate them. Pioneering experiments with thermal atomic beams andvapors have demonstrated precision atom interferometry [73, 219], high resolution
atom lithography [142, 209], quantum reflection from material surfaces [3, 14, 198],and strong coupling with microwave and optical cavities [15, 136]. Significant improvements to these experiments may be realized with more coherent atomic wavepackets
and advanced control over them. This invites the use of Bose-Einstein condensates,
which are well-established as fully coherent ensembles of matter waves [9, 25, 107, 113].
Likewise, microtraps and waveguides have the fine manipulation capabilities necessary
to build trapped-atom and guided-atom interferometers, bring atoms near material
surfaces for deposition purposes or interaction studies, and couple atoms into highfinesse cavities.
To emphasize the power of miniaturization, the two magnetic traps shown in
Fig. 3-1 are capable of producing harmonic confining potentials with nearly equal
frequencies1 . However, the macroscopic cloverleaf trap [Fig. 3-1(a)] requires currents∼ 500 A to duplicate the performance of the microfabricated trap [Fig. 3-1(b)] running1

The trap depth and trap volume of the macroscopic cloverleaf trap [Fig. 3-1(a)] are substantiallylarger than those of the microfabricated trap [Fig. 3-1(b)].

25

(a)

5 cm

(b)

1 cm

Figure 3-1: (Color) Macroscopic and microscopic magnetic traps. (a) One half ofthe cloverleaf trap used for condensate production (Ch. 2). (b) Microtrap used forwaveguiding (Ch. 3). To produce nearly identical harmonic confinement frequencies,
the cloverleaf trap must dissipate ∼ 10, 000 times more power than the microtrap.
∼ 1 A. The difference is that in the microtrap the atoms are only a few hundred
microns away from a 50 µm wide current-carrying wire, while in the macroscopic
cloverleaf trap they are a few centimeters away from ∼ 10 cm diameter coils.
In general terms, given a confining potential with characteristic magnitude V0
and length scale D, dimensional analysis reveals that the force on an atom in thispotential scales as V0 /D. Accordingly, reducing D increases the confinement strengthof the potential. For electromagnetic potentials, typically V0 is proportional to thefield generated at the location of the atoms and D is set by the size of the field
generating elements or the distance between them and the atoms, whichever is larger.
Thus, bringing atoms closer to smaller field generating elements inherently increases
the forces acting on them2 . To date, microtrap frequencies > 1 MHz have been
demonstrated [58].
Since the first magnetic microtrap proposal [218], several research groups havebeen active in the field of miniaturized atom optics (for an overview of the fieldsee Refs. [61, 89, 194, 208]). Thermal clouds have been trapped [60, 81, 180] and2

In practice, reducing D also increases V0 if the power driving the field generating elements is
kept constant. This further enhances the confinement strength of the potential.

26

guided [47, 154, 155] using atom chips3 , and incoherent atomic beam splitters havebeen demonstrated [27, 156].
The successful union of Bose-Einstein condensates with microfabricated devices [80,122, 165, 195] represents a significant advance for the field of atom optics. This chapter describes our contributions to this effort and expands on the work presented inRefs. [122] (App. A) and [125] (App. B).
We have demonstrated that Bose-Einstein condensates transported with opticaltweezers can be transferred into a magnetic microtrap fabricated on a substrate.
Our condensates contained over five times more atoms than other microchip-based
experiments [80, 165, 195]. Condensates were released from the microtrap into asingle-wire magnetic waveguide and propagated 12 mm before exiting the field of viewof our imaging system. The waveguide center was 450 µm below the microfabricatedsurface.
Single-mode (excitation-less) propagation was observed along homogeneous segments of the waveguide in a regime where the center-of-mass kinetic energy (velocity)of the condensate exceeded its transverse confinement energy (speed of sound). Transverse excitations were created in condensates that propagated through perturbationsin the guiding potential. These perturbations resulted from geometric deformationsof the current-carrying wires on the substrate.
The behavior of Bose-Einstein condensates near a microfabricated surface wasstudied for atom-surface separations in the range 70 − 500 µm. The condensates wereconfined at the same position relative to the surface by either a magnetic microtrap
or an optical dipole trap. Since the two traps operated on different principles andthe electromagnetic fields for each had different sources, this study provided a unique
examination of the interaction between Bose-Einstein condensates and a microfabricated surface. For example, while condensates confined near the surface in a magnetic
microtrap were found to fragment longitudinally [62, 102, 122], the clouds remained
intact under optical confinement. The measured condensate lifetime in either the3

The term “atom chip” generally refers to a microfabricated device used to confine and manipulate
ultracold atoms.

27

magnetic microtrap or the optical dipole trap was independent of the atom-surface
separation and ≥ 20 s, an order of magnitude longer than results reported elsewhere [62, 80, 195]. Radio frequency technical noise was directly observed to drive
spin-flip transitions in optically confined condensates near the atom chip. The rateof such transitions increased strongly as the atom-surface separation was reduced.

3.1

Magnetic Micropotentials

Typical magnetic waveguides are cylindrically symmetric, Ioffe-Pritchard style potentials formed by a two-dimensional quadrupole field and a uniform bias field, B|| ,along the guide axis [68, 174]. The two fields add in quadrature to produce a wavegq

uide with harmonic radial confinement at a frequency ω⊥ ∝ B 0 / B|| , where B 0 is
the radial magnetic field gradient associated with the two-dimensional quadrupole
field. The axial bias field also provides a non-zero magnetic field minimum to prevent
non-adiabatic spin-flip transitions (Majorana flops).
The simplest configuration for generating a magnetic waveguide consists of a single
microfabricated wire carrying current I and a uniform bias field, B⊥ , orthogonal to thepath of the wire but parallel to the surface of the microchip (Fig. 3-2). The magnetic
field due to the wire has magnitude Bwire = µ0 I/2πr that decays with distance, r,from its center, where µ0 is the permeability of free space. As a result, close to thewire Bwire > B⊥ but far from the wire Bwire < B⊥ . Directly beneath the wire at
r0 = µ0 I/2πB⊥ the two fields exactly cancel and a two-dimensional quadrupole field
is formed along the entire length of the wire. The radial magnetic field gradient2/µ0 I =
associated with the two-dimensional quadrupole field is B 0 = B⊥ /r0 = 2πB⊥

µ0 I/2πr02 .
The equations and scaling arguments presented thus far have assumed a circular wire cross section. However, most microfabricated wires have rectangular crosssections, characterized by a width along the microchip surface, W , and a height perpendicular to the microchip surface, H. The geometry of the wire only becomesrelevant for atom-surface separations of order W or H, whichever is larger (Fig. 3-3).
28

Bwire + B⊥ = 2D quadrupole field
Bwire > B⊥
x B||

Bwire = B⊥

Bwire < B⊥
Figure 3-2: Single-wire magnetic waveguide. The magnetic field generated by a microfabricated current-carrying wire, Bwire (thin arrows), exactly cancels an externally applied bias field, B⊥ (thick arrows), at one point in space, about which a
two-dimensional quadrupole field is formed. This creates a magnetic waveguide. The
distance between the waveguide axis and the wire scales as I/B⊥ , where I is thecurrent in the wire. An axial bias field, B|| , is applied parallel/antiparallel to the wireto prevent non-adiabatic spin-flips at the waveguide center and to make the radialconfinement harmonic.

A uniform bias field is added along the waveguide axis to create harmonic confinement and prevent non-adiabatic spin-flip transitions as discussed above. To firstorder, a uniform bias field applied perpendicular to the surface of the microchip (andhence orthogonal to both B|| and B⊥ ) shifts the waveguide axis in a plane parallel tothe surface of the microchip.
Stable three-dimensional confinement is possible by adding a small amount ofcurvature to the axial bias field. This can be accomplished by running current inmicrofabricated wire segments perpendicular to the waveguide axis (Fig. 3-4). This
style of three-dimensionally confining microtrap is referred to as a Z-wire trap due tothe shape of the three wire segments [180].
Figure 3-5 shows a schematic of the microfabricated atom chip used for the experiments described in this thesis. The microchip was mounted in a horizontal planein the science chamber such that the trapped condensates hung vertically below the29

Magnetic Field Gradient(G/cm/mA)

35
30
25
20
15
10
5
0

H
0

W
20

40
60
Distance (µm)

80

100

Figure 3-3: Circular vs rectangular wire cross section. The magnetic field gradient
per unit current generated by a circular (solid line) and rectangular (dashed line)
cross section wire is plotted as a function of distance from the wire. The rectangularwire has a width W = 50 µm and height H = 10 µm, identical to those shown in
Fig. 3-5.

device. The wires were microfabricated on a 600 µm thick silicon substrate mounted
on an aluminum block. They were 50 µm wide and electroplated with copper to aheight of 10 µm. The minimum separation distance between wires was 50 µm (100 µm
center-to-center). See Sec. 3.5 for additional technical details.
Condensates in the |F = 1, mF = −1i state containing up to 3 × 106 atoms
were delivered to the atom chip using optical tweezers (Ch. 2) and loaded into theZ-wire trap formed by currents I1 and I2 along with an orthogonal magnetic bias fieldB⊥ . An axial bias field, B|| , was applied to make the radial confinement harmonicand prevent non-adiabatic spin-flip transitions as discussed above. Transfer efficiency
from the optical tweezers to the Z-wire trap was near unity.

3.2

Propagation in Magnetic Waveguides

The Z-wire trap was operated with I1 = I2 = 1200 mA, B⊥ = 5.4 G, and 3.0 G ≤B|| ≤ 3.6 G, corresponding to a separation of 450 µm between the condensate andthe microchip surface. The measured axial trap frequency was ω|| = 2π × 6.0 Hz
and the measured radial trap frequencies were ω⊥ = 2π × (97.0, 91.1, 84.5) Hz for
30

(a)
BEC
Axial Magnetic Field

(b)

B||
B⊥
BEC
B||
Axial Position

Figure 3-4: Z-wire magnetic trap. (a) The axis of the microtrap is horizontal. Radialconfinement is provided by the horizontal wire segment and the external bias field, B⊥(Fig. 3-2). Axial confinement is provided by the magnetic field curvature created by
the vertical wire segments that adds constructively with the applied axial bias field,
B|| . The current, I, flows in the direction of the arrowheads. (b) Axial magnetic field
profile.

B|| = (3.0, 3.3, 3.6) G.
The condensate was transferred into the single-wire magnetic waveguide formedby I1 and B⊥ by ramping I2 → 0. The atoms were accelerated into the waveguide by
the remaining endcap of the Z-wire trap. Sufficiently far downstream (∼ 4 mm for
I1 = 1200 mA), the effect of this endcap was negligible and the condensate propagatedat a constant velocity of 3.0 cm/s. Further control over its propagation velocity
was possible by applying an external axial magnetic field gradient4 . With gradientsof 0 − 0.6 G/cm, the atomic velocity (kinetic energy) was varied over the range
3.0 − 6.6 cm/s [h × (25 − 121) kHz].
Single-mode (excitation-less) propagation was observed along homogeneous segments of the waveguide [Fig. 3-7(a)]. The condensates propagated with kinetic energies [h×(25−121) kHz] exceeding their transverse confinement energies (∼ h×100 Hz)
and velocities (3.0 − 6.6 cm/s) exceeding the speed of sound (c ≈ 7 mm/s) within the4

The uniform axial bias field, B|| , was applied with a pair of Helmholz coils co-axial with the
waveguide. The axial field gradient was applied by running extra current through only one of thesecoils.

31

B||
g

rf

B⊥
1

BEC

2 mm

Tweezers


1

6 mm

2

25 mm

3˚

50 µm

Si Substrate

Figure 3-5: Microfabricated atom chip. Optical tweezers loaded Bose-Einstein condensates into the Z-wire trap (Fig. 3-4) formed by currents I1 and I2 in conjunction
with the magnetic bias field B⊥ . Lowering I2 to zero released the condensates into asingle-wire magnetic waveguide. Atom flow was from left to right. The inset (dashedbox) shows the widening of the waveguide wire in the region where another wire
merges with it at a small angle. The only current flowing in the inset is I1 . Spatial
variations in the axial magnetic field were measured by driving radio frequency spinflip transitions with Irf . The condensate was trapped above the plane of the page
and the acceleration due to gravity, ~g , points out of the page. All microfabricatedfeatures are drawn to scale.

gas5 . Therefore, the atoms had enough kinetic energy to excite radial oscillations, andperturbations propagating through the condensate above a critical velocity, vc ≤ c,typically generate excitations [24, 34, 96, 163, 178]. Thus, excitation-less propagation
along the waveguide was a noteworthy achievement.

3.2.1

Origins of Waveguide Perturbations

In a single-wire magnetic waveguide, anything that causes the current path to deviate
from a straight line along the substrate generates perturbations to the waveguide
potential. Possible origins of such deviations include geometric deformations of thewire, variations in resistivity over the cross section of the wire, current leakage into
5
The condensate speed of sound, c, is given by µ = mc2 , where µ is the chemical potential andm is the atomic mass. Typical condensate densities were ∼ 1014 atoms/cm3 .

32

the substrate, and instabilities due to high current density [121].
The inset to Fig. 3-5 shows a large geometric deformation to the microfabricatedcurrent-carrying wire. To zeroth order, such a deformation causes the waveguide axis
to bend in a plane parallel to the surface of the microchip making an angle, θ, withrespect to its nominal trajectory along which B|| and B⊥ are aligned parallel andperpendicular, respectively. This bend in the waveguide axis is capable of radially
exciting propagating condensates in a plane parallel to the surface of the microchip.
With respect to the new waveguide axis, effective parallel, B||0 , and perpendicular,0B⊥, magnetic fields are found by the rotation

B||0 = B|| cos θ − B⊥ sin θ,

(3.1)

0
B⊥
= B⊥ cos θ + B|| sin θ,(3.2)

where −π/2 ≤ θ ≤ π/2. θ is taken to be a positive (negative) angle for the specificcase of atoms entering (exiting) the waveguide perturbation depicted in the inset toFig. 3-5.
Since the potential along the waveguide axis is determined by the local axial
magnetic field (due to the Zeeman interaction) and the vertical position of the guidecenter (due to the gravitational interaction), changes in the effective parallel andperpendicular magnetic fields produce variations in the potential experienced by the
propagating atoms. Variations in the effective parallel magnetic field are given by(for small angles)∆B||0 = −B⊥ sin θ.

(3.3)

Thus, atoms entering (exiting) the perturbed guiding region will encounter a magnetic
potential well (barrier). Furthermore, changes in the effective perpendicular magnetic
field cause the atom-substrate distance, r, to vary by (for small angles)∆r = −r

0
B||
∆B⊥
= −r
sin θ.
B⊥
B⊥

(3.4)

Thus, the guide center entering (exiting) the perturbed guiding region will shift to33

(a)Perturbed Guiding Region
Reflected Atoms(b)

Reflected Fraction

55

Transmitted Atoms

Kinetic Energy / h (kHz)
65
75

85

1.00
2

µ = (1/2)m(∆v)

0.75
0.50
0.25
0.00
4.4

4.8
5.2
Velocity (cm/s)

5.6

Figure 3-6: Scattering from waveguide perturbations. (a) Absorption image after 8 ms
ballistic expansion of a condensate scattered from the waveguide perturbation shownin the inset to Fig. 3-5. The imaging light propagated parallel to the microchip surface
and perpendicular to the waveguide axis. The incident velocity of the condensate was
5.0 cm/s. (b) The fraction of atoms reflected as a function of incident velocity. The
velocity spread expected due to axial mean field expansion is indicted by the solidline (Eq. 3.5). Waveguide parameters for all data were I1 = 1200 mA, B⊥ = 5.4 G,
B|| = 3.6 G, and ω⊥ = 2π × 84.5 Hz. The field of view in (a) is 0.35 mm × 4.00 mm.
wards (away from) the surface of the microchip and atoms will encounter a gravitational potential barrier (well).
For 23 Na atoms in the |1, −1i state, the Zeeman interaction energy is h×700 Hz/mG
and the gravitational interaction energy is h × 560 Hz/µm. For the waveguide perturbation depicted in the inset to Fig. 3-5, the change in gravitational potential energy
is negligible compared to the change in Zeeman interaction energy. However, theshift in atom-surface separation (Eq. 3.4) is capable of radially exciting propagatingcondensates in a plane normal to the surface of the microchip.

34

3.2.2

Reflections from Waveguide Perturbations

Equation 3.3 indicates that we should expect a large variation in the effective axial
bias field for the waveguide perturbation depicted in the inset to Fig. 3-5. The
local magnetic field in the perturbed guiding region was measured by driving radio
frequency spin-flip transitions that removed atoms from the guide. It was found thatupon entering (exiting) the perturbed region the magnetic bottom of the guiding
potential decreased (increased) by h × (50 ± 10) kHz. The signs of these shifts are
consistent with those predicted by Eq. 3.3. The magnitude of the shift in the Zeeman
energy upon exiting the guide is also consistent with the onset of transmission throughthe perturbed guiding region6 [Fig. 3-6(b)]. From Eq. 3.3 with B⊥ = 5.4 G, themaximum angular deviation, θm , of the waveguide trajectory necessary to produce ah × 50 kHz perturbation to the Zeeman energy is θm = 13 mrad. The corresponding
vertical displacement of the guide center from Eq. 3.4 with B⊥ = 5.4 G, B|| =3.6 G, and θm = 13 mrad is 4 µm. This yields a gravitational potential variation of
h × 2.2 kHz, which is small compared to the Zeeman energy shifts associated withthe perturbation.
Figure 3-6(a) shows that the reflected atoms propagated slower than the transmitted atoms, i.e. they are closer the the perturbed guiding region. The center-of-mass
velocity range over which partial reflection/transmission occurred indicates the axial
velocity spread, ∆v, of the propagating condensate. This spread can be accounted forby axial expansion due to mean field repulsion [173, 205, 214], which is approximately
1µ ≈ m(∆v)2 ,
2

(3.5)

where µ is the atomic mean field energy and m is the atomic mass. For our conditions(µ ≈ kB × 100 nK ≈ h × 2 kHz), ∆v ≈ 0.85 cm/s.

35

Excitation Amplitude (µm)

(a)
(b)
(c)

80
200

125

ω⊥ = 2π × 91.1 Hz, B|| = 3.3 G
ω⊥ = 2π × 97.0 Hz, B|| = 3.0 G

1000

150
500
100
50

100

0

0
5.4

5.8
6.2
Velocity (cm/s)

Excitation Energy / h (Hz)Kinetic Energy / h (kHz)
95
110

(d)

6.6

Figure 3-7: Propagation in a magnetic waveguide. Absorption images of condensates(a) entering and (b),(c) exiting the waveguide perturbation depicted in the inset
to Fig. 3-5. The imaging light propagated parallel to the microchip surface andperpendicular to the waveguide axis. The waveguide potential was turned off and thecondensates were allowed to expand ballistically for (a) 10 ms and (b),(c) 15 ms prior
to imaging. (a) Condensate containing 2 × 106 atoms and accelerated to a velocity
v = 6.3 cm/s. It propagated over 5 mm without excitation in a waveguide withω⊥ = 2π × 84.5 Hz and B|| = 3.6 G. Condensates transmitted through the perturbed
guiding region with v = 6.5 cm/s for (b) ω⊥ = 2π × 97.0 Hz and B|| = 3.0 G and(c) ω⊥ = 2π × 91.1 Hz and B|| = 3.3 G. (d) Peak-to-peak amplitudes of condensate
excitations after 15 ms ballistic expansion vs velocity. For all data I1 = 1200 mA andB⊥ = 5.4 G. All transmitted condensates propagated 4 mm beyond the perturbed
guiding region before being imaged. The field of view in (a)-(c) is 0.26 mm × 2.00 mm.

3.2.3

Excitations due to Waveguide Perturbations

Condensates transmitted through the waveguide perturbation depicted in the inset toFig. 3-5 were radially excited (Fig. 3-7). The imaging axis only provided sensitivity
to transverse excitations in a plane normal to the surface of the microchip. Larger
excitations are expected in a plane parallel to the surface of the microchip.
The nearly sinusoidal nature of the condensate excitations shown in Figs. 3-7(b)and 3-7(c) indicates that the dipole mode of the condensate was primarily excitedfor B|| ≤ 3.6 G. Further evidence for this comes from the fact that little variationin the excitation amplitude as a function of the propagation time after exiting the6

A 4.2 cm/s atomic velocity corresponds to a kinetic energy shift of h × 50 kHz.

36

perturbed guiding region was observed. Conversely, transmitted condensates showedstrong signs of higher order excitations when B|| was increased beyond 3.6 G corresponding to ω⊥ ≤ 2π×84.5 Hz. The absorption images deviated visibly from a smoothsinusoidal shape. In addition, the measured excitation amplitude depended strongly
on propagation time indicting the phasing and dephasing of several excitation modes.
Figure 3-7(d) shows a clear increase in the excitation amplitude for increasing
velocity at fixed B|| and ω⊥ as well as for increasing B|| (decreasing ω⊥ ) at fixedvelocity. Both trends are consistent with the expectation of increased excitations for
an increased ratio of atomic center-of-mass kinetic energy to transverse confinement
energy. The latter trend is also consistent with the expectation of increased excitations for an increased perturbation size since, from Eq. 3.4, the vertical displacementof the guide center is proportional to B|| .

3.3

Microfabricated Surface Effects

As emphasized above, bringing atoms closer to smaller current-carrying wires inherently increases the magnetic forces acting on them. This implies that finer coherent
control is possible, but it also means that atoms confined in magnetic microtraps
and waveguides will be more sensitive to perturbations and noise emanating from themicrofabricated surface.

3.3.1

Fragmentation

While single-mode waveguiding was possible 450 µm away from the microfabricatedsurface, condensates confined in the Z-wire trap were observed to fragment longitudinally when the atom-surface separation was reduced to < 200 µm (Fig. 3-8). The
axial location of the fragments was reproduced on each realization of the experiment,
and more fragments appeared as the condensates were brought closer to the microfabricated surface. This prevented smooth propagation along the waveguide close
to the microfabricated surface. While perturbations to the waveguide potential are
detrimental for atom optical applications such as interferometry, the disordered po37

(a)
(b)
(c)
(d)

Figure 3-8: Fragmentation in a magnetic microtrap. Absorption images of condensates after 5 ms ballistic expansion from the magnetic microtrap (Fig. 3-5). The
imaging light propagated parallel to the microchip surface and perpendicular to themicrotrap axis. The atom-substrate separation was (a) 190 µm, (b) 145 µm, (c)100 µm, and (d) 55 µm. For all images, the condensate started in a trap with
I1 = I2 = 540 mA, B⊥ = 5.4 G, and B|| = 0.3 G corresponding to an atom-substrate
separation of 200 µm. The condensate was translated towards the substrate by lowering the wire currents linearly over 500 ms. The atoms were held at their final position
for 100 ms prior to trap shut off. The field of view in (a)-(d) is 0.37 mm × 2.00 mm.
tential along the waveguide axis might be used to study superfluid-insulator (Boseglass) transitions [45, 59, 64, 216].
As shown above, variations in the current path along the microfabricated wires
led to perturbations in the guiding potential. Finer variations in the current path are
expected to become evident as the microtrap/waveguide center approaches the microfabricated surface. To test if irregular current flow in the microfabricated wires caused
the observed condensate fragmentation, as opposed to magnetic dopants/impurities
for example, condensates were confined in an optical dipole trap at the same location
relative to the microfabricated surface (Fig. 3-9). The magnetic and optical trapping parameters were made nominally identical (same atom-surface separation, axial
bias field, and radial trap frequencies) such that any surface contaminants (magnetic
dopants/impurities, etc. . .) would perturb the clouds identically. No fragmentation
was observed for optically confined condensates. The lack of condensate fragmentation in the optical dipole trap confirms that the longitudinal potential corrugations

38

(c)

(b)

Optical Density

(a)

1.5

microfabricated magnetic trap
optical dipole trap

1.0
0.5
0.0
0.0

0.2

0.4
0.6
Axial Position (mm)

0.8

1.0

Figure 3-9: Lack of fragmentation in an optical dipole trap. Absorption images after
10 ms ballistic expansion of condensates containing ∼ 106 atoms held at a distance of
85 µm from the microfabricated surface for 15 s in the (a) magnetic microtrap and (b)optical dipole trap. The imaging light propagated parallel to the microchip surface
and perpendicular to the microtrap/optical trap axis. (c) Radially averaged opticaldensity vs axial position for condensates shown above. Longitudinal fragmentation
occurred for condensates held in the magnetic microtrap, but not for those confined
optically at the same location with the magnetic trap off. The magnetic trap was
operated with I = 130 mA, B⊥ = 3.2 G, and B|| = 1.4 G yielding a radial trap
frequency ω⊥ = 2π × 450 Hz. The optical dipole trap had a radial trap frequencyω⊥ = 2π × 425 Hz and was operated with B|| = 1.8 G. For both condensates, thechemical potential was µ ≈ kB × 120 nK. The field of view in (a) and (b) is 0.25 mm
× 1.00 mm.
observed in the magnetic microtrap arose due to the presence of irregular current
flow in the microfabricated wires. This is in agreement with conclusions reached inRefs. [101, 121].
For the data shown in Figs. 3-8 and 3-9, the condensate chemical potential wasµ ≈ kB × 100 nK. This implies that a ∼ 3 mG “bump” in the axial magnetic field
is sufficient to cut the condensate into fragments. It should not be surprising thatmilligauss level axial magnetic field variations exist over length scales comparable tothe atom-surface separation. The magnetic field generated by the microfabricatedwires is typically a few gauss (Bwire = B⊥ = 5.4 G in Fig. 3-8 and Bwire = B⊥ =3.2 G in Fig. 3-9). Thus, 1 mrad angular deviations of the waveguide axis (Eq. 3.3)
will produce axial magnetic field variations capable of breaking the condensate intofragments.
39

Figure 3-8(d) shows that for an atom-surface separation of 55 µm, the trapping
potential developed axial variations with a characteristic length scale of 100−150 µm.
Resoundingly periodic fragmentation has been observed elsewhere with a similar period [62, 101]. Theoretical calculations have revealed that aperiodic fluctuations in
the current path along the substrate can produce quasi-periodic perturbations to the
waveguide potential with a length scale set by the atom-surface separation [216].
A microchip similar to that depicted in Fig. 3-5 was fabricated with conductors
50 µm wide and evaporated with gold to a height of 1.25 µm. Condensates confinednear its surface were observed to fragment also. To date, condensate fragmentation
has been observed near copper [62, 122], gold [125], and aluminum [102] conductors.

3.3.2

Trap Lifetimes

Confined atoms with non-zero magnetic moment are sensitive to mechanical andmagnetic field noise at multiples of their trap frequencies and magnetic field noise attheir Zeeman splitting frequency [82, 84, 85, 86, 87, 88, 102, 132]. Noise at the trap
frequency leads to heating and subsequent trap loss after the atoms acquire an energy
greater than the trap depth. Spin-flip transitions driven by radio frequency noiseat the atomic Zeeman splitting frequency distribute the atomic population acrossmagnetically confinable and unconfinable states. This causes atom loss for clouds
held in magnetic traps. However, all spin states are confined by optical dipole traps
and atoms undergoing spin-flip transitions have been directly observed (Sec. 3.3.3).
The lifetime of condensates confined in either the magnetic microtrap or optical
dipole trap at distances ≥ 70 µm from the microfabricated surface was measured tobe ≥ 20 s and independent of the atom-surface separation (Fig. 3-10).
Several experimental details altered the measured condensate lifetime. Excitations
created during the magnetic microtrap loading were found to shorten the measured
lifetime, and care had to be taken to overlap the optical and magnetic traps duringtransfer to minimize such excitations. Translating the condensate either towards oraway from the microfabricated surface by adiabatically varying I1 , I2 , and B⊥ toshift the trap center while maintaining a constant radial trap frequency was found to40

Lifetime (s)

35
30
25
20
15
10
5
0
0

100
200
300
400
Atom-Surface Separation (µm)

500

Figure 3-10: Lifetime near a microfabricated surface. The 1/e lifetime of condensatesconfined in the magnetic microtrap (solid squares) and optical dipole trap (solid circles) is shown to be independent of distance from the microfabricated surface. I1 ,I2 , and B⊥ were varied with distance to maintain the radial magnetic trap frequency
at ω⊥ = 2π × 450 Hz with B|| = 1.4 G. The vertical dashed line indicates the onset
of longitudinal condensate fragmentation in the magnetic trap. In the optical dipole
trap, the condensate was held directly below the microfabricated wire used to form
the magnetic trap with B|| = 1.8 G. No external connections were made to the microchip. The optical dipole trap had axial and radial trap frequencies ωz = 2π × 4 Hz
and ω⊥ = 2π × 425 Hz, respectively. Only atoms remaining in the |1, −1i state wereresonant with the absorption imaging light. For comparison, the distance dependenceof thermal cloud lifetimes measured in Ref. [62] is shown for atoms confined magnetically by a microstructure (open squares) and copper wire (open diamonds). Error
bars smaller than the symbol size are not included. Also, magnetically confined condensate lifetimes reported in Ref. [80] (open triangle) and Ref. [62] (open circle) are
shown for comparison.

decrease the condensate lifetime. This presumably resulted from excitations inducedby irregular current changes due to technical limitations in controlling the power
supplies connected to the microchip. As a result, microfabricated magnetic trap
lifetime data is only presented for atom-surface separations ≥ 70 µm, where theatoms were loaded into their final position directly from the optical tweezers7 .

Occasionally, heating was observed for atoms in both the microfabricated magnetic
trap and optical dipole trap due to technical noise at the trap frequency, even withcare taken to eliminate ground loops and minimize cable lengths8 . Connecting a
7

The optical tweezers beam began to clip on the support structures for the microchip when placed
within 70 µm of the microfabricated surface.
8
The current supply for the microfabricated wire and the wire itself were grounded to the vacuum41mF = -1

mF = 0 mF = +1

6
4
2mF = 0 mF = +1

0.2
0.1
185 µm data

0.0

0.018

(b)

mF = -1

0.3
Transitioned
Fraction(a)Transition Rate (s-1)(c) 0.18

0

6

10

20
Time (s)

30

40

4
2

0.001
80 100200
400
Atom-Surface Separation (µm)

600

Figure 3-11: Radio frequency spin-flip transitions. Axial absorption images after
22 ms ballistic expansion of condensates held in the optical dipole trap at a distanceof 185 µm from the microfabricated surface for (a) 0 s and (b) 40 s. A magnetic field
gradient was applied during ballistic expansion to separate the different spin states.
(c) Spin-flip transition rate, Γ, vs distance, r, from the microfabricated surface on alog-log scale. A solid line Γ ∝ 1/r2 is provided as a guide for technical noise driven
spin-flip transitions. A dashed line Γ ∝ 1/r3 gives an upper limit for thermal noise
driven spin-slip transitions (Sec. 3.3.4). The inset shows the fraction of the total
atoms in the mF = 0 state with the transition rate being defined as the initial slopeof the data (before any atoms in the mF = +1 state were detected). For all data,
B|| = 1.8 G yielding a Zeeman splitting frequency of 1.26 MHz. The absorption
imaging light propagated parallel to the optical trap axis and was resonant with the
F = 2 → F 0 = 3 transition. The atoms were optically pumped into the F = 2
hyperfine level with a pulse resonant with the F = 1 → F 0 = 2 transition. This
provided equal imaging sensitivity to each magnetic sublevel. The field of view in (a)and (b) is 1.0 mm × 2.0 mm.
10 mF capacitor in parallel with the 2 Ω microfabricated wire (1/RC = 2π × 8 Hz)
eliminated such effects. Thereafter, applying radio frequency power to the microchip
at a frequency chosen to limit the trap depth for magnetically confined atoms did notconsistently alter the condensate lifetime.

42

3.3.3

Radio Frequency Spin-Flip Transitions

Weak radio-frequency noise in the microfabricated wires was observed to drive spinflip transitions. Optically confined condensates initially in the |1, −1i state [Fig. 311(a)] were found to make transitions to other magnetic sublevels [Fig. 3-11(b)]. Suchtransitions would act as a loss mechanism for magnetically confined clouds. The
transition rate was found to decrease as the square of the atom-surface separation
distance, r. Since the magnetic field of a straight wire decays as 1/r, and the power
scales as the square of the field, the 1/r2 dependence of the spin-flip transition rateis expected for atoms in the near field (r ¿ λ) of the wire, where λ ≈ 300 m is the
wavelength of ∼ 1 MHz radiation. The transition rate at large r may fall above theabove the 1/r2 decay due to low level radio frequency noise from other sources.

The transition rate vs distance data presented in Fig. 3-11(c) was taken withall connections necessary to run the microfabricated magnetic trap made to the microchip, but with no current flowing in the microfabricated wires. The atoms were
exposed to an axial bias field B|| = 1.8 G to simulate the field configuration in the
microfabricated magnetic trap. This also maximized their sensitivity to fluctuating
fields generated by wire currents since ∆mF = ±1 radio frequency transitions are
only driven by magnetic fields oscillating orthogonal to a static bias field. Spin-flip
transitions were suppressed by exposing the optically confined atoms to a bias field,
B⊥ , perpendicular (parallel) to the microfabricated wire (surface).
The radio frequency driven spin-flip transition rate depended strongly on experimental details, suggesting that antenna effects coupled radio frequency noise into thesystem. The rate was measured to be ∼ 100 times higher if care was not taken to
carefully eliminate ground loops and use minimal cable lengths. Also, with no connections to the microchip, spin-flip transitions were not detectable for condensates
held up to 60 s. The spin-flip rate presented in Fig. 3-11(c) became comparable to themeasured condensate decay rate displayed in Fig. 3-10 for atom-surface separations< 100 µm. Thus, extending long condensate lifetimes much closer to the surface willchamber at the location of the electrical feedthrough for the microchip connections. The total lengthof all (unshielded) current-carrying cables was ∼ 1 m.

43require further radio frequency shielding and/or filtering.

3.3.4

Thermal Noise

Confining and manipulating ultracold atomic vapors with electromagnetic fields allows for the microkelvin [37, 39, 172], nanokelvin [40, 110], and even picokelvin [124]temperature clouds to remain thermally isolated from their 300 K laboratory environment. Such thermal isolation is noteworthy in microtrap configurations whereultracold atoms are confined within a few microns of room temperature material
surfaces, establishing temperature gradients > 108 K/m.
For neutral atoms with a non-zero magnetic moment, limitations to this temperature gradient arise from fluctuating magnetic fields generated by thermal noise in
the material surfaces. These fluctuating magnetic fields may cause heating, spin-flip
transitions, and decoherence in condensates trapped near conducting surfaces [82, 84,
85, 86, 87, 88, 102]. Spin-flip transitions driven by thermally induced radio frequency
magnetic field noise have been observed in atomic vapors confined within ∼ 100 µm
of from macroscopic structures well approximated by conducting planes of infinite
thickness [82, 102] and within ∼ 10 µm of thinner conducting planes [132]. Typicalmicrofabricated devices (Fig. 3-5) have feature sizes smaller than the skin depth ofradio frequency radiation. This reduces the power of emitted radiation and suppresses
spin-flip transitions as compared to macroscopic conductors.
Near a conducting plane of infinite thickness, thermally induced radio frequency
noise drives the |1, −1i → |1, 0i transition at a rate [82, 87]Γ=

9 µ2B kB T 1
δ3,256π ²20 h̄2 c2 ρ r 3δ 3 + 2r3(3.6)

where the conducting material has resistivity ρ and temperature T , r is the atomsurface separation, ²0 is the permittivity of free space, c is the speed of light, andδ=s

2²0 c2 ρ
ωL

44

(3.7)

is the skin depth for radiation at the Zeeman splitting (Larmor) frequency ωL .
For r ¿ δ, Eq. 3.6 can be approximated byΓ=

3 µ2B kB T 1.
256π ²20 h̄2 c2 ρ r(3.8)

In comparison, the spin-flip transition rate near a conducting plane of finite thickness
H is [86]3 µ2B kB T 1 H
Γ=,256π ²20 h̄2 c2 ρ r r + H(3.9)

and near a circular wire of radius R is [86]Γ=

3 µ2B kB T R2.
512 ²20 h̄2 c2 ρ r3

(3.10)

Typical Zeeman splitting frequencies are ωL ≈ 2π × 1 MHz corresponding to askin depth δ = 66 µm for ρ = 1.7 × 10−6 Ω-cm (the bulk resistivity of copper9 ).

Using Eqs. 3.8-3.10 for r ≈ δ gives an upper limit for the thermally induced spin-flip
transition rate near a conducting surface. At a distance r = δ = 66 µm from room
temperature (T = 300 K) copper conductors, Eqs. 3.8-3.10 predict Γ ≈ 200 mHz,Γ ≈ 20 mHz, and Γ ≈ 2 mHz for an infinitely thick plane, a plane of thickness
H = 10 µm, and a wire of radius R = 5 µm, respectively. This demonstrates thestrong suppression of thermal noise radiated from microfabricated devices due to geometrical considerations. The thermally induced spin-flip transition rate is expected
to approach Γ = 1 Hz at distances ∼ 10 µm away from microfabricated devices with∼ 10 µm features.

3.4

Waveguide Beam Splitters

Section 3.1 described a single magnetic waveguide created by a single current-carrying
wire and bias field. A pair of waveguides can be generated by running parallel currents
in two microfabricated wires with the same bias field still applied. The separation
9

Often microfabricated conductors have resistivities up to a factor of ten higher than the bulk
resistivity of the material.

45

(a)

(b)
2d

(c)
2d

d

(d)

2d

x B
||

(e)
2d
β

B⊥

(f)
2d
β

d

2d
β
x B
||

B⊥

Figure 3-12: Waveguide beam splitter. Contours of constant magnetic field for two
wires running current I separated by a distance 2d with (a) B⊥ = 0.990 × µ0 I/2πd,(b) B⊥ = 1.000 × µ0 I/2πd,√and (c) B⊥ = 1.010 × µ0 I/2πd. Adding a central wirerunning current βI, β = ( 2 − 1)/2 ≈ 0.207 symmetrizes the potential wells split
perpendicular to the surface [(c) compared to (f)]. It also creates an extraneous
third well. (d) B⊥ = 1.226 × µ0 I/2πd, (e) B⊥ = 1.236 × µ0 I/2πd, and (f) B⊥ =1.246 × µ0 I/2πd. In (a)-(f), B|| = 0.1 × µ0 I/2πd and the magnetic field contours
are spaced in increments of 10−4 × µ0 I/2πd above B|| . In (b) and (e), the merged
potential is quartic because the radial magnetic field profile has a hexapole symmetry.

between the guide centers is a function of the applied bias field, B⊥ , wire separation,
2d, and wire currents, I (Fig. 3-12). Merging and splitting guide centers is possible
by varying these parameters and can act as a beam splitter for atomic wavepackets.
Such beam splitters form the basis of existing microchip-based atom interferometry
proposals [8, 81, 90].
Condensates split by varying I and/or B⊥ in a double waveguide geometry (Fig. 312) were heated into thermal clouds due to technical limitations (Fig. 3-13). As analternative, we have demonstrated coherent condensate beam splitting using opticalpotentials (Ch. 4).

46

(a)

x B
||

(b)

g, B⊥

2d

x B
||

g, B⊥

2d

Figure 3-13: Atomic wavepacket division. Axial absorption images of thermal
clouds split (a) horizontally and (b) vertically in a magnetic double-well potential.
I = 300 mA, B|| = 1 G, 2d = 600 µm, and (a) B⊥ < 2 G and (b) B⊥ > 2 G.
The acceleration due to gravity, ~g , points down the page. The field of view for theabsorption images is 650 µm × 650 µm.

3.5Microfabrication Procedures

The microchip in Fig. 3-5 was fabricated by MIT’s Microsystems Technology Laboratories. Their fabrication procedure is outlined here.
A 600 µm thick silicon wafer was used as a substrate to support the microfabricated wires. A ∼ 200 nm thick SiO2 layer was deposited on top of the silicon wafer toelectrically isolate the microfabricated wires from the substrate. Next a photoresist
layer was added. The microfabricated wire pattern was imaged onto the photoresist
with ultraviolet light. The photoresist was then developed such that the photoresist
was removed from the surface of the chip only where the wires were to be fabricated.
The remainder of the chip remained coated in photoresist. A ∼ 20 nm chrome adhesion layer was deposited over the wafer followed by a ∼ 200 nm copper layer. These
metal layers coated the entire wafer, but only reached the Si/SiO2 substrate where
the photoresist had been removed in the pattern of the microfabricated wires. Next,
the remaining photoresist was removed leaving a bare Si/SiO2 wafer with Cr/Cu microfabricated wires. These wires were electroplated with copper to a thickness of
10 µm.
The 50 µm × 10 µm cross section wires were ∼ 35 mm long and had a resistanceof ∼ 2 Ω. The wires could withstand currents of 1.2 A steady-state, but burnt out
47

instantly when we tried to run 1.4 A through them. The microfabricated wires wereconnected to macroscopic copper leads with a thin gold ribbon. The gold ribbon was
bonded to the microchip and macroscopic copper leads by gap welding with a shortcurrent pulse. The microchip was clamped to an aluminium block by overhanging
screw heads.
Once installed in the science chamber, the microchip was baked at 150 C for fivedays in order to achieve the necessary ultrahigh vacuum pressure (∼ 10−11 torr).
After baking, several of the microfabricated wires were no longer connected to theirleads. After removing the microchip from the chamber, many of the gold ribbons
were found to be broken.
Other fabrication techniques are currently being tested. We are testing wires up
to 2 µm thick made from evaporated gold rather than electroplated copper. Also,
we are connecting the microchip to macroscopic leads with mechanical clamps. However, the electrical isolation provided by the silicon substrate seems to degrade undermechanical stress after baking.

48

Chapter 4
Trapped-Atom Interferometry in
an Optical Double-Well Potential
Demonstrating atom interferometry [16] with particles confined by magnetic [80, 122,
165, 195] and optical [52, 53] microtraps and waveguides would realize the matterwave analog of optical interferometry using fiber-optic devices. Current proposals
for confined-atom interferometers rely on the merger and separation of two potential
wells to coherently divide atomic wavepackets [8, 81, 90]. This type of division differsfrom previously demonstrated atomic beam splitters. To date, atomic beams andvapors have been coherently diffracted into different momentum states by mechanical [26, 72, 109] and optical [72, 105] gratings, and Bose-Einstein condensates havebeen coherently delocalized over multiple sites in optical lattices [5, 30, 69, 70, 138,164]. Atom interferometers utilizing these beam splitting elements have been used tosense accelerations [5, 105, 168] and rotations [73, 129], monitor quantum decoherence [31], characterize atomic and molecular properties [56], and measure fundamental
constants [71, 168, 219].
It is difficult to compare atom interferometers and optical interferometers because
they are typically sensitive to different forces and potentials. However, one common
comparison is their rotation sensitivity. For this purpose, atom interferometers havean inherent advantage over their optical counterparts. The Sagnac phase shift of a49

~ and undergoing rotation at a rate Ω~ isgyroscope enclosing an area A∆φrot =

4π ~ ~Ω · A,
λv(4.1)

where λ and v are the wavelength and velocity of the interfering particles, respectively.
For atoms of mass m and photons of frequency ω, matter wave interferometers [73,129] can outperform optical gyroscopes [36] by a factor of mc2 /h̄ω ∼ 1010 , wherec is the speed of light. This is provided that both devices enclose the same area,
which at present is not possible. Current atom interferometers enclose far less area
than optical gyroscopes such that in practice their rotation sensitivities are nearly
equal [73]. Guided-atom interferometers have the potential to enclose large areas andtherefore to increase the ultimate rotation sensitivity of matter wave gyroscopes.
The work presented in Ch. 3 was largely motivated by the goal of creating a
miniaturized atom interferometer. However, technical limitations have currently prevented us from coherently splitting condensates with our magnetic microtraps andwaveguides (Ch. 3, Fig. 3-13). This chapter describes our parallel efforts to implement a trapped-atom interferometer using optical potentials and expands on the work
presented in Ref. [199] (App. C).
We have demonstrated a trapped-atom interferometer with gaseous Bose-Einsteincondensates confined in an optical double-well potential. Condensates were coherently
split by deforming an initially single-well potential into two wells separated by 13 µm.
The relative phase between the two condensates was determined from the spatial
phase of the matter wave interference pattern formed upon releasing the atoms fromthe separated potential wells [9, 138]. This recombination method avoids deleterious
mean field effects [201, 206] and yields an accurate measurement of applied phase shifts
on a single realization of the experiment, unlike in-trap recombination schemes [8, 81,90].
The large separation between the split potential wells allowed the phase of eachcondensate to evolve independently and either condensate to be addressed individually. An ac Stark phase shift was applied to either condensate by temporarily turning50

off the optical fields generating its potential well. The spatial phase of the resulting
matter wave interference pattern shifted linearly with the applied phase shift andwas independent of the time of its application. This verified the phase sensitivity of
the interferometer and the independent phase evolution of the separated condensates.
The measured coherence time of the separated condensates was 5 ms.
This chapter describes a trapped-atom interferometer with two interfering paths.
The two-path geometry has the flexibility to measure either highly localized potentials
or uniform potential gradients, such as those arising from atom-surface interactions or
the Earth’s gravitational field, respectively. In contrast, multiple-path interferometers
demonstrated in optical lattice systems are restricted to measurements of the latter [5,
138].

4.1

Optical Double-Well Potential

Condensates in the |F = 1, mF = −1i state containing up to 3 × 106 atoms weredelivered to the science chamber using optical tweezers (Ch. 2) and loaded into a secondary optical trap formed by a counter-propagating, orthogonally-polarized 1064 nm
laser beam shifted in frequency from the tweezers by ∼ 100 MHz to avoid interference
effects.
The secondary optical trap was formed by a collimated laser beam that passed
through an acousto-optic modulator (AOM) and was focused onto the condensate
with a lens [Fig. 4-1(a)]. The AOM was driven simultaneously by two radio frequency(rf) signals to tailor the shape the potential from single-well [Fig. 4-1(b)] to doublewell [Fig. 4-1(c)]. The separation between the potential wells was controlled by thefrequency difference between the rf drives. The 1/e2 radius of each focused beam
was 5 µm. For typical optical powers, this resulted in a single beam trap depth
U0 = h×5 kHz and a radial (axial) trap frequency ωr = 2π×615 Hz (ωz = 2π×30 Hz).

51

F

(a)

F

AOM

f1,f2

(b)

g x

d

(c)
U0

x
d

d

Figure 4-1: Optical double-well potential. (a) Schematic diagram of the optical setup
for the double-well potential. An acousto-optic modulator (AOM) was driven simultaneously by two frequencies, f1 and f2 , and diffracted a collimated beam into two
beams. The AOM was placed in the focal plane of a lens of focal length F so that thetwo beams propagated parallel to each other. The radial separation of the potential
wells, d, was controlled by the frequency difference, ∆f = |f1 − f2 |. The acceleration
due to gravity, ~g , points into the page. The absorption image shows two well-separated
condensates confined in the double-well potential diagramed in (c). The field of viewis 70 µm × 300 µm. Energy diagrams for (b) initial single-well trap with d = 6 µm
and (c) final double-well trap with d = 13 µm. In both (b) and (c), U0 = h × 5 kHz
and the peak atomic mean field energy was ∼ h × 3 kHz. The potential “dimple” in(b) was < h × 500 Hz which was much less than the peak atomic mean field energy
allowing the trap to be characterized as a single-well. The potential “barrier” in (c)was h × 4.7 kHz which was larger than the peak atomic mean field energy allowing
the resulting split condensates to be characterized as independent.

4.2

Matter Wave Interference

Condensates were initially loaded from the tweezers into a single-well trap [Fig. 41(b)]. After holding the cloud in this trap for 15 s to damp excitations, the condensate
contained ∼ 105 atoms with a peak atomic mean field energy µ ≈ h × 3 kHz. The
single-well trap was deformed into a double-well potential [Fig. 4-1(c)] by linearly
increasing the frequency difference between the rf signals driving the AOM over 5 ms.
The amplitude of the rf signals were tailored during the splitting process to guarantee
an even division of the condensate atoms and nearly equal trap depths after splitting.
Releasing the condensates from the double-well potential allowed them to ballistically expand, overlap, and interfere (Fig. 4-2). Each realization of the experiment52

(b) Optical Density

Position(a)g x
Figure 4-2: Matter wave interference. (a) Absorption image of condensates released
from the double-well potential in Fig. 4-1(c) and allowed to overlap during 30 ms
ballistic expansion. The imaging axis was parallel to the direction of gravitational
acceleration, ~g . The field of view is 600 µm × 350 µm. (b) Radial density profiles
were obtained by integrating the absorption signal between the dashed lines. Typicalimages gave > 60% contrast. The solid line is a fit to a sinusoidally-modulated
Gaussian curve from which the phase of the interference pattern was extracted (seetext). This figure presents data acquired in a single realization of the experiment, i.e.
a single shot.

produced a matter wave interference pattern with the same spatial phase. This reproducibility demonstrated that deforming the optical potential from a single-wellinto a double-well coherently split the condensate into two clouds with deterministic
relative phase, i.e. the relative phase between the two condensates was the same fromshot-to-shot.
This experiment derived its double-well potential from a single laser beam passing
through an AOM. Thus, vibrations and fluctuations of the laser beam were common
mode to both wells, and a clean and rapid trap turn off was achieved by switching off
the rf power driving the AOM. In contrast, past experiments created a double-wellpotential by splitting a magnetically trapped condensate with a blue-detuned laser
beam [9]. This previous work was unable to observe a reproducible relative phasebetween the split condensates. This was attributed to an unstable potential barrier
separating the two condensates due to pointing fluctuations in the blue-detuned laser
beam and irreproducible turn off of the high current magnetic trap that initiated53

ballistic expansion.

4.2.1

Relative Phase

The relative phase between the two separated condensates was determined by thespatial phase of their matter wave interference pattern. For a ballistic expansion timet À 1/ωr , each condensate had a quadratic phase profile [42],ψ± (~r, t) =

q

m

~

2

n± (~r, t) ei 2h̄t |~r±d/2| +φ± ,(4.2)

where ± denotes one well or the other, n± is the condensate density, m is the atomic

mass, d~ is a vector connecting the two wells, and φ± is the condensate phase. Interactions between the two condensates during ballistic expansion have been neglected.
The total density profile for the matter wave interference pattern takes the form
"

Ã

md
√
n(~r, t) = n+ + n− + 2 n+ n− cosx + φr
h̄t

!#

,

(4.3)

where φr = φ+ − φ− is the relative phase between the two condensates and d~ = dx̂.
To extract φr , an integrated cross section of the matter wave interference pattern[Fig. 4-2(b)] was fit with a sinusoidally-modulated Gaussian curve,−(x−xc )2 /σ 2

G(x) = Ae·

2π
1 + B cos(x − x0 ) + φf
λ
µ

¶¸

,

(4.4)

where φf is the phase of the interference pattern with respect to a chosen fixed x0 . Ideally, if x0 was set at the center of the two wells, then φr = φf . However, misalignmentof the imaging axis with the direction of gravitational acceleration created a constantoffset, φf = φr + δφ. With t = 30 ms the measured fringe period, λ = 41.5 µm, waswithin 4% of the point source formula prediction (Eq. 4.3), ht/md = 39.8 µm.

54

(b)
100

90

0
0

1

2
3
4
Hold Time (ms)5Standard Deviation (deg)Spatial Phase (deg)

(a)
180

80
60
40
20
0
0

1

2
3
4
Hold Time (ms)5

Figure 4-3: Phase coherence of separated condensates. (a) The spatial phase of thematter wave interference pattern is plotted vs hold time after splitting. Each data
point represents the average of eight measurements. The phase evolution was due tounequal trap depths for the two wells, which was determined from the linear fit tobe h × 70 Hz or ∼ 1% of the trap depth. (b) The standard deviation of the eightmeasurements averaged in (a) is plotted vs hold time after splitting. This represents√
the uncertainty in a single phase measurement. The error bars in (a) are 8 times
smaller. A random distribution of phases between −180 and +180 degrees would
have a standard deviation of ∼ 104 degrees (dashed line).

4.2.2

Phase Coherence

The relative phase between the separated condensates was observed to evolve linearlyin time [Fig. 4-3(a)]. This evolution was mainly caused by a difference in trap depths
between the two wells, although a slight mean field energy difference between the twocondensates contributed also.
The standard deviation of eight measurements of φr was < 90 degrees for condensates split then held separated for up to 5 ms [Fig. 4-3(b)]. Furthermore, for holdtimes up to 1 ms, the standard deviation was substantially smaller, < 40 degrees.
Since a random distribution of phases between −180 and +180 degrees would havea standard deviation of ∼ 104 degrees, the measured results quantitatively confirm
the reproducible nature of the splitting process and the coherent evolution of theseparated condensates.
Fundamental limits on the phase coherence between isolated condensates arise
due to Poissonian number fluctuations associated with the splitting process and the
55

(a)

(b)

g x

g x

Figure 4-4: Decoherence. Absorption images of condensates released from the doublewell potential and allowed to overlap during 30 ms ballistic expansion. After splitting,the condensates were held separated for (a) 0 ms [Fig. 4-2(a)] and (b) 8 ms. The
spatial phase of the matter wave interference pattern could be reliably extracted
from (a) but not (b). The imaging axis was parallel to the direction of gravitational
acceleration, ~g . The field of view in (a) and (b) is 600 µm × 350 µm.
coherent state description of the condensate [100, 131, 143]. For condensate number√
fluctuations ∆N = N , the corresponding variation in the atomic mean field energy
is
∆µ =

2 µ
√
5 N

(4.5)

where µ ∝ N 2/5 in the Thomas-Fermi limit (Ch. 1, Eq. 1.6). For our parameters(N ≈ 105 and µ ≈ h × 3 kHz), ∆µ ≈ h × 4 Hz. This sets the times scale for phasediffusion to be ∼ 250 ms.

Repulsive atom-atom interactions may lead to relative number squeezing between
the condensates in the two wells and extend the phase diffusion time [222, 223].
Also, weak Josephson coupling between the two wells may counteract phase diffusion(Sec. 4.4).
The uncertainty in determining φr at hold times > 5 ms was attributed to axial andbreathing-mode excitations created during the splitting process. These excitations led
to interference fringes that were angled and had substantial curvature, rendering adetermination of φr impossible (Fig. 4-4). Splitting the condensate more slowly inan effort to minimize excitations, but still fast compared to the phase diffusion time,56

splitting

150 µs(b) 120

expansion

τp

360
180
0
-180
-360
-540
0

20

Spatial Phase (deg)Spatial Phase (deg)(a) 54040 60
τp (µs)60
0-60

splitting
τd

-120
0

80

1 ms

200

expansion

50 µs

400 600τd (µs)800

Figure 4-5: Trapped-atom interferometry. (a) ac Stark phase shifts were applied toeither well exclusively (solid circles and open circles) or both wells simultaneously(crosses) by turning off the corresponding rf signal(s) driving the AOM for a duration
τp . The resulting spatial phase of the matter wave interference pattern scaled linearlywith τp and hence the applied phase shift. Applying the ac Stark shift to the opposite
well (solid vs open circles) resulted in an interference pattern phase shift with opposite
sign. Applying ac Stark shifts to both wells (crosses) resulted in no phase shift for
the interference pattern. This data was taken with a slightly modified experimental
setup such that the trap depth of the individual potential wells was U0 = h × 17 kHz.
(b) A 50 µs pulse induced a ∼ 70 degree shift independent of the pulse position. The
experimental setup was as described in Fig. 4-1. Solid and open circles have the same
meaning as in (a). The insets show the time sequence of the optical intensity for thewell(s) temporarily turned off.

did not improve the measured stability of φr . Since controlling axial excitations
appears critical for maintaining phase coherence, splitting condensates that are freely
propagating in a waveguide potential may be more promising (Ch. 3). In addition,
propagating the separated condensates along a waveguide prior to phase readout
would create an atom interferometer with an enclosed area, and hence with rotation
sensitivity.

4.3

Trapped-Atom Interferometry

The phase sensitivity of the trapped-atom interferometer was demonstrated by applying ac Stark phase shifts to either (or both) of the two separated condensates.
Phase shifts were applied to individual condensates by pulsing off the optical power57

generating the corresponding potential well for a duration τp ¿ 1/ωr . The spatial
phase of the matter wave interference pattern shifted linearly with the pulse duration [Fig. 4-5(a)]. Due to the inhomogeneous optical potential, U (~r), the applied ac
Stark phase shifts varied across the condensate as ∆φ(~r) = −U (~r)τp /h̄. Averaging
this phase shift over the inhomogeneous condensate density, n(~r), approximates the
expected spatial phase shift of the matter wave interference pattern as
1 Z 3
d ~r n(~r)∆φ(~r),
Nµ¶
2= U0 − µ τp /h̄,
7

(4.6)

∆φ̄ =

where N =

R

(4.7)

d3~r n(~r) is the number of condensate atoms, U0 is the potential welldepth, and µ is the atomic mean field energy. The measured phase shifts yielded
U0 = h × 5 kHz [Fig. 4-5(b)], which was consistent with calculations based on themeasured trap frequencies.

4.4

Josephson Coupling

The condensates were sufficiently separated that their phases evolved independentlyof each other to the extent that no coupling between the potential wells could be
detected. This claim is supported qualitatively by our ability to cleanly resolve theseparated, trapped condensates with absorption imaging techniques [Fig. 4-1(a)] andby the observation of high-contrast matter wave interference patterns that penetrated
the full atomic density profile with uniform spatial period and no thick central fringe[Fig. 4-2(a)] [183].
Estimating the expected Josephson coupling between the two separated wells isdifficult because tunnelling depends exponentially on the barrier height and well separation [44]. In the weak coupling limit, we expect a small population imbalance between the two wells to oscillate at the Josephson plasma frequency ωJP =q

ωJ µ/h̄,where ωJ is the single particle tunnelling rate and µ is the atomic mean field energy [201, 220]. The maximum population imbalance can be estimated as the total
58

number of particles that tunnel between the two wells in one Josephson period [220]:
N ωJ
≈N
∆N ≈
ωJP

s

ωJ
.
µ/h̄

(4.8)

The single particle tunnelling rate is given by−

ωJ = ωr e

R

dx

q

2m[U (x)−µ]
h̄2

,

(4.9)

where ωr is the radial trap frequency, U (x) is the double-well potential [Fig. 4-1(c)],and the integration is over the classically forbidden region.
For our parameters, ωJ ≈ ωr e−14 ≈ 2π × 5 × 10−4 Hz. For N = 105 atoms ineach well at an atomic mean field energy µ = h × 3 kHz, we expect ∆N ≈ 400 atoms(∆N/N = 4×10−3 ) to oscillate between the two wells at a frequency ωJP ≈ 2π×1 Hz.

Thus, on the time scale of our experiment (a few milliseconds), we do not expect toobserve the effects of Josephson coupling.
Quantitatively, the lack of coupling between the two separated condensates wasverified by measuring the evolution of their relative phase (Figs. 4-3 and 4-5). The
measured phase shifts of the matter wave interference depended only on the timeintegral of the applied ac Stark phase shifts [Fig. 4-5(b)]. This is expected for uncoupled condensates since the final relative phase, φr , should be the same on different
phase trajectories because the history of phase accumulation does not affect the total
amount of accumulated phase. In constrast, in the presence of Josephson coupling thesmall population imbalance oscillating between the two wells would cause the relative
phase between the two condensates to oscillate and produce a time dependent signal
in Fig. 4-5(b).

59Chapter 5
Vortices and Spin Textures
Shortly after the 1938 discovery of superfluidity in liquid 4 He [2, 103], the phenomenonof Bose-Einstein condensation was resurrected from its fate as an obscure theoretical subtlety and used to explain the experimental findings [133, 134, 210]. In thistreatment, the superfluid (condensate) fraction was distinguished from the normal(thermal) fraction by a macroscopic, scalar order parameter, |ψ(~r)i =where n(~r) is the superfluid density.

qn(~r)eiS(~r) ,In such a system, the superfluid velocity can be expressed as ~vs (~r) = −(h̄/m)∇S(~r),
where m is the atomic mass. Due to the single-valued nature of the condensate orderparameter, circulation in a superfluid is quantized:Ih~vs · d~` = κ , κ = 0, ±1, ±2 . . .
m

(5.1)

One ramification of Eq. 5.1 is that when a superfluid is rotated, angular momentum
enters the system through quantized vortices. In spin-less or spin-polarized condensates, line defects such as vortices have cores where the density of condensed particles
is necessarily zero to keep the order parameter single-valued [1, 78, 92, 123, 137, 224].
In Eq. 5.1, κ is known as the “winding number” or “charge” of the vortex state andrepresents the angular momentum per particle in units of h̄.
Topological defects in superfluid systems described by vector order parameters canbe more complex. In condensates with a spin degree of freedom, coreless vortices exist60as spin textures [18, 126, 141]. Such structures are referred to as skyrmions (AndersonToulouse vortices [7]) or merons (half-skyrmions, Mermin-Ho vortices [145]) depending on the boundary conditions of the system1 . Spin textures play a central role indescribing the physics of elementary particles [200], liquid 3 He-A [7, 18, 145], the quantum Hall effect [127], and gaseous Bose-Einstein condensates [91, 114, 115, 160, 225].
Superfluid phenomena become apparent when phase gradients and windings are
established. In contrast to liquid helium systems, experimental control over gaseous
Bose-Einstein condensates allows vortex and spin texture phase factors to be directly
written onto the condensate wavefunction [123, 126, 141]. Such phase windings canbe imprinted onto the condensate wavefunction either dynamically or topologically.
Dynamically, the phase of the condensate evolves according to the time integral of
its energy, which can be tailored locally with a spatially varying external potential.
Topologically, the phase of the condensate advances through adiabatic variationsin the parameters of the Hamiltonian governing the system. This phase, which issolely a function of the path traversed by the system in the parameter space of the
Hamiltonian, is known as Berry’s phase [17].
Vortices have been generated in single-component condensates by rotating thecloud with an anisotropic potential [1, 78, 92, 137], by slicing through the cloud witha perturbation above the critical velocity of the condensate [1, 96], and through the
decay of solitons [4, 55]. Engineering topological states in a Bose-Einstein condensate
has received much theoretical attention [98, 130, 140, 176, 186, 187, 221]. In spinor
condensates, vortices and spin textures have been phase-imprinted by driving spatiallyselective spin-flip transitions [141] and adiabatic magnetic field variations [123, 126].
This chapter describes vortices and spin textures imprinted in spinor condensatesusing topological phases and expands on the work presented in Refs. [123] (App. D)
and [126] (App. E).
23

Na condensates were prepared in either the lower, |1, −1i, or upper, |2, +2i,

hyperfine state and confined in a Ioffe-Pritchard magnetic trap. We created vor1

Originally, the term “skyrmion” referred to three-dimensional point-like topological defects [114,115, 200]. However, it is common to use this term when describing two-dimensional line defects aswell [91].

61tices by adiabatically inverting the magnetic bias field along the trap axis and could
remove them by returning the bias field to its original direction. Using surface
wave spectroscopy [32, 77, 226], the axial angular momentum per particle of the|1, −1i and |2, +2i vortex states was measured to be consistent with −2mF h̄, aspredicted [98, 152, 157, 159].
The field inversion process can be decomposed into two steps. First, the axial
bias field is adiabatically reduced to zero. Then second, it is adiabatically increasedin the opposite direction. While the axial bias field was zero, spin textures with
Mermin-Ho [145] boundary conditions were produced. These spin textures were detected as coreless vortices in F = 1 spinor condensates. Reducing the axial bias
field to zero continuously transformed the initially spin-polarized condensate into a
coherent superposition of three spin states, each with a different phase winding. The
resulting angular momentum per particle varied between spin states, and the condensate evolved such that states with more angular momentum per particle circulatedaround (i.e. at a larger radius than) states with less angular momentum per particle.
Thus, the condensate had a net axial magnetization that varied with radial position.

5.1

Phase Imprinting

The spin-F condensate wavefunction can be written as |ψ(~r)i =qn(~r)|ζ(~r)i, wheren(~r) is the atomic number density and the 2F + 1 component spinor|ζ(~r)i =FX

mz =−Fζmz (~r)|F, mz i, |hζ(~r)|ζ(~r)i|2 = 1(5.2)

describes a spin texture.
A Ioffe-Pritchard magnetic trap consists of an axial bias field (with curvature)
and a two-dimensional quadrupole field in the orthogonal plane [68, 174]. Taking thez axis as the symmetry axis, the magnetic field profile of a Ioffe-Pritchard trap has

62the form~ r) = Bz ẑ + B 0 (xx̂ − y ŷ) ,
B(~
h

(5.3)
i

= Bz ẑ + B 0 r cos(2φ)r̂ − sin(2φ)φ̂ ,(5.4)

in Cartesian and cylindrical coordinates, respectively. B 0 > 0 is the radial magnetic
field gradient and quadratic terms have been neglected. The local magnetic field isnearly parallel to the z axis throughout a condensate of radial extent R confined ina Ioffe-Pritchard magnetic trap with Bz À B 0 R > 0. Accordingly, the condensate
spinor in the lab frame can be approximated by |ζ(~r)i = |F, mz = +mF i, where mz

and mF are the projection of the atomic spin, F~ , along the z axis and local magnetic

field direction, respectively.
Adiabatically varying Bz rotates the local magnetic field direction and hence theatomic spins. This establishes a spatially dependent phase across the condensate
because the spins rotate about a unit vector n̂(φ) = sin(φ)x̂ + cos(φ)ŷ that depends
on the azimuthal angle, φ, describing the atomic position [Fig. 5-1(a)]. The position
dependent phase imprinted onto the condensate wavefunction can be calculated byapplying the quantum mechanical rotation operator~|ζ(~r)i = e−i(F /h̄)·n̂(φ)β(r) |F, mz = +mF i,(5.5)~ is the spin operator such that F~ = hζ(~r)|F|ζ(~~ r)i and β(r) is the angle
where Fthrough which F~ rotated about the n̂(φ) axis.

This chapter considers two limiting cases for the variation of Bz : Bz → −Bz(Sec. 5.2) and Bz → 0 (Sec. 5.3).

5.2

Vortex Formation and Detection

Inverting Bz → −Bz rotates all atomic spins by π radians and produces spin-polarized
condensates with vortices. From Eq. 5.5 with β(r) = π, the condensate spinor in the

63

(a)

(b)

y

z'
θ'
F

φ

1

x

2
y'

1
2
b(φ)
n(φ)

x' φ'

Figure 5-1: Spin rotation and Berry’s phase. (a) The unit vectors b̂(φ) (filled arrows)
point in the direction of the two-dimensional quadrupole field providing the radialconfinement of a Ioffe-Pritchard magnetic trap. The atomic spins rotate about theunit vectors n̂(φ) (open arrows) as the axial bias field, Bz , is ramped from positive
to negative values. (b) The spin, F~ , for an atom in state |F, mF i traverses a pathon a sphere of radius |mF |h̄ as it adiabatically follows its local magnetic field. The
primed coordinate system is centered on the atomic position and has axes parallel tothose of the unprimed coordinate system in (a). For an atomic position described by
the azimuthal angle φ, F~ rotates in a half-plane defined by φ0 = −φ for mF > 0 and
φ0 = −φ + π for mF < 0 as Bz is inverted. After inverting Bz , the relative topologicalphase acquired between atoms located at positions 1 and 2 in (a) is proportional to
the solid angle subtended by the shaded surface, bounded by the contour marked
with arrowheads (see text).

lab frame after inverting Bz is~|ζ(~r)i = e−i(F /h̄)·n̂(φ)π |F, mz = +mF i,= (−1)F −mF e−i2mF φ |F, mz = −mF i.

(5.6)
(5.7)

The topological phase factor e−i2mF φ describes a vortex of winding number 2|mF | withthe sense of rotation dependent on the sign of mF . The phase singularity (angularmomentum barrier) at r = 0 requires n(0) = 0. The minimum length scale over which64√
the condensate density can vary is the healing length2,3 , ξ = 1/ 8πna, where a is thes-wave scattering length for a binary collision. Accordingly, the density depletion at
the vortex core has a radius of order ξ = 750 nm for a background condensate density
n = 1014 atoms/cm3 . The vortex core size also scales with the winding number of thevortex state [29].

5.2.1

Berry’s Phase

The result in Eq. 5.7 can be interpreted in terms of Berry’s phase [159]. Figure 5-1(b)shows the orientation of F~ in the lab frame for mF > 0. Atoms located at positionk = 1, 2 in Fig. 5-1(a) have azimuthal angle φk and angular momentum F~k . As Bz
is inverted, F~k traces path k from top to bottom on the sphere in Fig. 5-1(b). The
topological phase acquired by an atom in this process is solely a function of the pathtraversed by its angular momentum vector. Since this path depends on the azimuthal
angle describing the atomic position, a relative phase is established between spatiallyseparated atoms. The condensate spinor after inverting Bz is given by|ζ(~r)i = eiγ(φ) |F, mF i,(5.8)

where γ(φ) is the topological phase acquired by atoms with azimuthal angle φ.
For an atom in state |F, mF i, Berry’s phase, γ(C), acquired as its angular momentum vector traverses a closed contour, C, on the surface of the sphere in Fig. 5-1(b)is given by [17]γ(C) = −mF Ω(C),(5.9)

where Ω(C) is the solid angle subtended by a surface bounded by the contour C.
Calculating the relative phase, γ(φ1 ) − γ(φ2 ), with the aid of Eq. 5.9 requires closing

the contours traced by each F~k along an identical path. For clarity, we choose to
close each contour along path 2 itself and hence γ(φ1 ) − γ(φ2 ) = γ(C), where C is the2
In the liquid helium community, this length scale is referred to as the coherence length. However,in the BEC community, the coherence length is typically reserved to describe the length scale overwhich the condensate can be described by√a single phase [9, 50, 76, 83, 181, 203].
√
3
Refs. [42, 128, 166, 169] define ξ = 1/ 8πna while Refs. [67, 170, 171] define ξ = 1/ 4πna.

65contour formed by path 1 traversed from top to bottom and path 2 traversed frombottom to top, as indicated with arrowheads in Fig. 5-1(b).
A surface bounded by this contour subtends a solid angle Ω(C) = 2(φ1 − φ2 ),yielding a relative phase γ(φ1 ) − γ(φ2 ) = −2mF (φ1 − φ2 ). Thus, we make the assignment
γ(φ) = −2mF φ,(5.10)

up to an additive term independent of position. This is the same phase as in Eq. 5.7
reinterpreted in terms of Berry’s phase.

5.2.2

F = 1 and F = 2 23 Na Condensates

Condensates in the |F = 1, mF = −1i state containing up to 3 × 106 atoms weredelivered to the science chamber using optical tweezers (Ch. 2). While optically confined by the tweezers, condensates were prepared in the |2, +2i state by sweeping
through the |1, −1i ↔ |1, 0i ↔ |1, +1i radio-frequency transition with 100% efficiency, then sweeping through the |1, +1i ↔ |2, +2i microwave transition with 80%
efficiency [66]. In the science chamber, the condensate was loaded into a microfabricated Ioffe-Pritchard magnetic trap formed by a Z-shaped wire carrying current I
and an external magnetic bias field, B⊥ (Ch. 3, Figs. 3-4 and 3-5). An axial bias
field, Bz = B|| , was applied to make the radial confinement harmonic and prevent
non-adiabatic spin-flip transitions.
Typical magnetic microtrap parameters were I = 1200 mA, B⊥ = 5.4 G, andBz ≈ 1 G, resulting in a radial magnetic field gradient of B 0 = 120 G/cm. For
atoms in the |1, −1i state, the axial and radial trap frequencies were ωz = 2π × 6 Hz
and ω⊥ = 2π × 210 Hz, respectively. For atoms in the |2, +2i state, the microtrap√
frequencies (for identical magnetic field parameters) were larger by a factor of 2 dueto the larger magnetic moment. After transfer into the microtrap, condensates in the|1, −1i (|2, +2i) state contained ∼ 2 × 106 atoms (∼ 1 × 106 atoms) and had a lifetime> 10 s (> 3 s). This represented the first magnetic trapping of 23 Na condensates in theupper hyperfine level, with previous work done exclusively in optical dipole traps [66].
66

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5-2: Vortices in F = 1 and F = 2 condensates. Axial absorption images ofcondensates in the |1, −1i state after 18 ms ballistic expansion (a) prior to inverting
Bz , after inverting Bz and holding the trapped condensate for (b) 5 ms and (c)20 ms, and (d) after inverting Bz and then returning it to its original direction. Axial
absorption images of condensates in the |2, +2i state after 7 ms ballistic expansion(e) prior to inverting Bz , after inverting Bz and holding the trapped condensate for(f) 0 ms and (g) 5 ms, and (h) after inverting Bz and then returning it to its original
direction. The field of view is (a)-(d) 570 µm × 570 µm and (e)-(h) 285 µm × 285 µm.

Subsequently, 23 Na condensates have been created by direct evaporation in the |2, +2i
state [75].

5.2.3

Observation of Vortex Cores

Vortices were created in condensates held in a Ioffe-Pritchard magnetic trap by inverting Bz and could be removed by returning Bz to its original direction (Fig. 5-2).
The vortices were observed by releasing the trapped condensate and allowing theatoms to expand ballistically. They were identified by central density depletions dueto the angular momentum barrier associated with a rotating cloud. In the trapped
condensate, the vortex core size was too small (∼ ξ) to be resolved optically. However, ballistic expansion for a time τ from a trap with frequencies ωz ¿ ω⊥ radiallymagnified the condensate wavefunction, including the vortex core size, by a factorq

2 2
1 + ω⊥
τ [28, 43, 135]. This made the vortex cores large enough to be unambigu-

ously detected with standard absorption imaging techniques.
For condensates in the |1, −1i state, the best results were achieved by inverting
the axial bias field linearly from Bz = 860 mG to −630 mG in 11 ms. For atoms in
the |2, +2i state, the optimum ramp time over the same range was 4 ms. The field
67

(a)

|B(z)|(b)|B(z)|

Bz

Bz

BEC

BEC
z

z
Axial Position (arb. units)

Axial Position (arb. units)

Figure 5-3: Axial magnetic field. The Z-wire (Ch. 3, Fig. 3-4) trapping potential
along the z axis is proportional to the magnitude of the axial magnetic field shown(a) before inverting Bz and (b) after inverting Bz .

inversion process caused an atom loss of ∼ 50% due to non-adiabatic spin flips as Bz
passed through zero, in reasonable agreement with calculations in Refs. [152, 159].
The density depletions shown in Figs. 5-2(b), 5-2(c), and 5-2(g) were observed after
inverting the axial bias field and holding the trapped condensate for longer than aradial trap period. Thus, the atom loss from the center of the cloud during the field
inversion process could not be responsible for the observed density depletions. Typical|1, −1i (|2, +2i) condensates after inverting the axial bias field contained up to 1×106

atoms (0.5 × 106 atoms) with a Thomas-Fermi radius RT F = 5.4 ± 0.2 µm (RT F =4.5 ± 0.2 µm) in a trap with radial frequency ω⊥ = 2π × 250 Hz (ω⊥ = 2π × 350 Hz).
The magnetic field along the microtrap axis was~ = 0, z) = Bz + 1 B 00 z 2 ẑ,
B(r
2
µ

¶

(5.11)

where quadratic terms neglected in Eqs. 5.3 and 5.4 have been included. The axial
magnetic field curvature, B 00 , which arose from the geometry of the Z-wire, was held
constant throughout the experiment. Bz was inverted by reversing an external axial
magnetic field. Changing the sign of Bz , but not B 00 , resulted in a magnetic fieldsaddle point at the center of the cloud and axial anti-trapping of weak-field seeking
atoms (Fig. 5-3). This limited the condensate lifetime after inverting Bz to < 50 ms.
Radial excitations caused by the field inversion process may also have limited the68condensate lifetime.

5.2.4

Angular Momentum Measurements

The axial angular momentum per particle of the vortex states was measured using
surface wave spectroscopy [32, 77, 226]. A superposition of counter-rotating (m` =±2) quadrupolar (` = 2) surface waves was excited in the condensate4 , where ` andm` characterize the angular momentum and its projection along the z axis of thequadrupole modes, respectively. This created an elliptical condensate cross section
with time dependent eccentricity. In the absence of a vortex, the m` = ±2 quadrupolemodes are degenerate and the axes of the elliptical condensate cross section remain
fixed in time. This degeneracy is lifted by the presence of a vortex, causing the axes toprecess in the direction of fluid flow. The precession rate, Θ̇, is given by [32, 77, 226]
Θ̇ =

hLz i
,
2
2mhr⊥
i

(5.12)

where hLz i is the axial angular momentum per particle characterizing the vortex state,2
m is the atomic mass, and hr⊥
i = hx2 + y 2 i is the mean-squared trapped condensate

radius with vortices present.
By measuring the precession rate of the quadrupole axes and the mean-squared
radius of the condensate in the trap, the axial angular momentum per particle wasdetermined. After exciting the quadrupolar modes, the condensate evolved in thetrap for variable times in the range 0.2 − 7.4 ms. The condensate was then released
from the trap and imaged with resonant laser light after ballistic expansion [Figs. 54(a)-5-4(l)]. The resulting images were fit to an elliptical Thomas-Fermi profile todetermine the orientation of the quadrupole axes as a function of time [Fig. 5-4(m)].
To determine the mean-squared trapped condensate radius, vortices were imprinted in the condensate but quadrupolar modes were not excited. Images of ballistically expanded condensates similar to those in Figs. 5-2(b), 5-2(c), 5-2(f), and4
The surface waves were excited by vertically displacing the magnetic trap center for 200 µs.
This compressed the condensate vertically due to anharmonic terms in the microtrap potential andcreated an elliptical condensate cross section.

69

(b)

(c)

(d)

3.8 ms

5.0 ms

6.2 ms

7.4 ms

(e)

(f)

(g)

(h)

1.0 ms

2.4 ms

3.8 ms

5.2 ms

(i)

(j)

(k)

(l)

0.8 ms

1.0 ms

1.2 ms(m)
Precession Angle (rad)

(a)

π
0
-π
-2π
0

1.4 ms

π/8
π/4
3π/8
π/2
2
Time (in units of 2m〈r⊥ 〉 / h)Figure 5-4: Surface wave spectroscopy. Axial absorption images after 18 ms ballistic
expansion of |1, −1i condensates undergoing a quadrupole oscillation (a)-(d) in thepresence of a vortex and (e)-(h) in the absence of a vortex. Successive images weretaken during successive half periods of the quadrupole oscillation such that the shortand long axes of the elliptical cross section were exchanged. Images (a)-(d) showcounter-clockwise (positive) precession of the quadrupole axes, while images (e)-(h)show no precession. (i)-(l) Axial absorption images after 7 ms ballistic expansion of|2, +2i condensates undergoing a quadrupole oscillation in the presence of a vortex.
The images were taken during a single half period of the quadrupole oscillation.
Images (i)-(l) show clockwise (negative) precession of the quadrupole axes. The fieldof view is (a)-(h) 570 µm × 570 µm and (i)-(l) 285 µm × 285 µm. (m) Precession
angle vs time in the presence of a vortex for |1, −1i condensates measured after adelay of 0 ms (open circles), 5 ms (open squares), and 20 ms (open triangles) from thecompletion of the inversion of the axial bias field, in the absence of a vortex for |1, −1i(open diamonds) and |2, +2i (filled diamonds) condensates, and in the presence ofa vortex for |2, +2i condensates measured immediately upon the completion of theinversion of the axial bias field (filled circles).

5-2(g) were fit to a Thomas-Fermi profile with a circular cross section. The fitting
routine ignored the central region of the cloud where the density was depleted due tothe vortex core. The mean-squared trapped condensate radius was derived throughthe relation
2
hr⊥
i=

2
2 R⊥
,
2 2
7 1 + ω⊥τ

(5.13)

where R⊥ is the Thomas-Fermi radius of the condensate after ballistically expanding
2 2
τ accounts forfor a time τ from a trap with radial frequency ω⊥ . The factor 1 + ω⊥

the change in Thomas-Fermi radius during the expansion process [28] and the factor
70

2/7 results from averaging over the inhomogeneous condensate density distribution
assuming no vortices are present. For low angular momentum vortex states, the density depletion at the vortex core does not significantly modify the density distributionof the condensate and we expect the 2/7 factor to still be accurate [42].
For condensates in the |1, −1i state, the quadrupole oscillation was excited after
a delay of 0 ms, 5 ms, and 20 ms from the completion of the inversion of the axial
bias field. The measured axial angular momenta per particle were (+1.9 ± 0.3)h̄,(+2.1±0.3)h̄, and (+1.9±0.2)h̄, respectively. For condensates in the |2, +2i state, thequadrupole oscillation was excited immediately upon the completion of the inversionof the axial bias field. The measured axial angular momentum per particle was(−4.4 ± 0.4)h̄. For both internal states, the measurements were consistent with thepredicted axial angular momentum per particle of −2mF h̄ [98, 152, 157, 159]. The
uncertainty in the measurements arises from the linear fit to the precession angle and2
the determination of hr⊥
i.

5.2.5

Multiply Charged Vortices

Multiply charged vortices are unstable against decay into multiple, singly charged
vortices [29, 153, 158, 184]. From our experiments, we cannot determine if the condensate contained one multiply charged vortex or multiple, singly charged vortices.
If multiple vortices were present, they must be closely spaced since they were notresolved after ballistic expansion. Furthermore, if the singly charged vortices had
moved apart considerably, it would have lowered the extracted value of hLz i [32],which was not observed even with delayed probing.
It is predicted that doubly charged vortices should decay into two singly charged
vortices that are intertwined [153]. Imaging along the vortex axis and/or measuring
the angular momentum per particle of the vortex states does not efficiently differentiate between a doubly charged vortex and a pair of intertwined, singly charged
vortices. However, imaging perpendicular to the vortex axis may reveal such a decay(Fig. 5-5).
Our work represented the first observation of vortices with a single density deple71

(a)

(b)

|2,+1〉|2,+1〉|2,+2〉|2,+2〉

Figure 5-5: Vortex line. Absorption images of a condensates in the |2, +2i state
after ballistic expansion (a) before and (b) after imprinting a vortex as described inSec. 5.2. The imaging light propagated perpendicular to the condensate/vortex axis
such that the vortex core appears as a line defect running along the entire length of thecondensate. Ballistic expansion was initiated immediately after creating the vortex
state. A magnetic field gradient was applied during ballistic expansion to spatiallyseparate the remaining spin states. Spin-flip transitions to untrapped states during
the phase imprinting process led to trap loss. Spin-flip transitions to the |2, +1i state,
which is weak field seeking and remained radially confined by the magnetic trap, weredirectly observed as predicted in Ref. [152].

tion, but multiple (2 or 4) phase windings. Subsequently, long-lived vortex aggregateswith a single density depletion and 7 − 60 phase singularities have been observed [57].
Multiply charged vortices are stable in a multiply connected geometry such asa torus. This geometry can be realized by applying a repulsive potential along thevortex line with a blue-detuned laser beam [46, 98].

5.3

Coreless Vortex Formation and Detection

Section 5.2 described the formation of vortices in condensates confined in a IoffePritchard magnetic trap by inverting Bz → −Bz . During this process, the axial
bias field passed through zero. This section considers the nature of the system whenBz ≈ 0.
For condensates confined in a Ioffe-Pritchard magnetic trap, adiabatically ramping

72

y

(a)

(b)

y

x

x

Figure 5-6: Planar spin textures. Spins aligned with two-dimensional (a) quadrupoleand (b) hexapole magnetic fields produce textures with winding numbers −1 and −2,
respectively. Counter-clockwise traversal of the dashed contours in (a) and (b) leads
to clockwise (negative) spin rotation, with the winding number defined as the integer
number of revolutions made by the spin vector while circumnavigating the singularity
at the origin [144].

Bz → 0 drives the transition [98]|F, mz = +mF i →

|mF |

X

mz =−|mF |ζmz (r)e[i(mz −mF )φ] |F, mz i.

(5.14)

The condensate population is distributed across 2|mF |+1 spin states (in the lab frame)with each acquiring a different topological phase factor and angular momentum perparticle due to the variation of Berry’s phase with magnetic quantum number [17]. At
this point, the atomic spins are no longer polarized in the lab frame and a spin texture
has been created. Fully inverting Bz repolarizes the atomic spins and generates
vortices (Sec. 5.2).

5.3.1

Spin Textures

As Bz is varied, the condensate remains in the state |F, mF i with respect to the local

magnetic field provided the local Zeeman energy, ∼ µB [Bz2 + (B 0 r)2 ]1/2 , dominates

the local kinetic energy associated with the spin texture, ∼ h̄2 /mr2 , where µB is theBohr magneton and m is the atomic mass. For Bz = 0, atomic spins aligned with the73quadrupole magnetic field produce a planar spin texture [Fig. 5-6(a)]. However, theinfinite kinetic energy associated with the wavefunction singularity at r = 0 createsa nonplanar spin texture over a disc of radius ∼ (h̄2 /mµB B 0 )1/3 , with higher angularmomentum spin states residing outside those with lower angular momentum. Thus,
ramping Bz → 0 rotates the atomic spins by an amount that depends on radial
position.
For condensates initially in the |1, −1i state, adiabatically ramping Bz → 0 createsa spin texture in the lab frame described by (Eq. 5.5)|ζ(~r)i =cos2 [β(r)/2] |1, −1i +−1√sin[β(r)] e+iφ |1, 0i +
2
2
+i2φ

sin [β(r)/2] e(5.15)|1, +1i,while for condensates initially in the |1, +1i state, adiabatically ramping Bz → 0
creates a spin texture in the lab frame described by (Eq. 5.5)|ζ(~r)i =cos2 [β(r)/2] |1, +1i ++1√sin[β(r)] e−iφ |1, 0i +
2
2
−i2φ

sin [β(r)/2] e(5.16)|1, −1i.

These spin textures are classified as a skyrmions (merons) for the boundary conditions
β(0) = 0 and β(∞) = π [β(∞) = π/2].
By symmetry, we contend that the radial dependence of β(r) should be the samefor both spin textures (Eq. 5.15 and Eq. 5.16). However, since condensates initially in
the |1, +1i state are in the absolute ground state of the system5 , the radial dependenceof β(r) can be found most easily by minimizing the Gross-Pitaevskii energy functional,

E(β) =

Z





 

h̄2
c0 c2 |F~ |2  
d3~r n h∇ζ|∇ζi + V +  +
n ,
2m
2
2 h̄2

(5.17)

respect to variations in β using Eq. 5.16 as a trial spinor. Equation 5.17 uses the5

Atoms in the |1, +1i (|1, −1i) state are strong (weak) magnetic field seeking, such that in thepresence of a magnetic field the |1, +1i state is the absolute ground state of the system.

74

β(r) [rad]

π/2

π/4

0
0

1
2
2
1/3
r [in units of (h /mµBB') ]3

Figure 5-7: Radial dependence of the spin rotation angle. β(r) is plotted for Bz = 0.
For our radial magnetic field gradient, B 0 = 180 G/cm, (h̄2 /mµB B 0 )1/3 ≈ 260 nm.
Thomas-Fermi approximation, and h∇ζ|∇ζi =PF†
mz =−F ∇ζmz · ∇ζmz , V= gF µB F~ ·

~
~
B/h̄,
F~ = hζ|F|ζi,c0 = 4πh̄2 ā/m, and c2 = 4πh̄2 ∆a/m. Here ā = (2a0 + a2 )/3 and∆a = (a2 − a0 )/3 characterize two-body interactions, where a0 and a2 are scattering
lengths for binary collisions with total angular momentum F = 0 and F = 2, respectively [91]. For c2 > 0 the atomic interactions are anti-ferromagnetic (polar), as in23

Na [204], while for c2 < 0 they are ferromagnetic, as in 87 Rb [117, 193].
Using Eq. 5.16 as a trial spinor, β(r) satisfies the meron boundary conditions andvaries from β(0) = 0, due to the kinetic energy of the spin texture, to β(∞) = π/2, due
to the atomic Zeeman energy, over a characteristic length scale given by the larger of(h̄2 /mµB B 0 )1/3 and |Bz |/B 0 (Fig. 5-7). Terms proportional to (dβ/dr)2 were neglected
while varying Eq. 5.17. The trial spinor in Eq. 5.16 does not have the most generalform since it was derived by rotating a polarized spinor (Eq. 5.5) and is inherently
ferromagnetic, |F~ | = h̄. Accordingly, the spin dependent interaction term in Eq. 5.17
does not vary with β(r) and therefore does not contribute to the determination ofβ(r). This restriction is not severe since the Zeeman energy dominates the spin
dependent interaction energy in our experiment.

75

5.3.2

Observation of Coreless Vortices

The experimental setup and conditions for producing spin textures were similar to
those for producing vortices (Sec. 5.2.2). Condensates containing ∼ 2 × 106 atoms
in the |1, −1i state were confined in a Ioffe-Pritchard Z-wire trap (Ch. 3, Figs. 3-4
and 3-5) with I = 720 mA, B⊥ = 5.3 G, and Bz = B|| = 1.3 G, resulting in a radial
magnetic field gradient B 0 = 180 G/cm. The axial and radial trap frequencies wereωz = 2π × 4 Hz and ω⊥ = 2π × 250 Hz, respectively.
Coreless vortices were imprinted in F = 1 spinor condensates by adiabatically
ramping Bz → 0 (Fig. 5-8). To observe the nature of the spin texture, an axial bias
field was switched on non-adiabatically along either the negative [Figs. 5-8(a)-5-8(d)]or positive [Figs. 5-8(e)-5-8(h)] z axis. Switching the axial bias field on suddenly“froze” the atomic spins and effectively “projected” the condensate wavefunction
onto a basis quantized with respect to the local (axial) magnetic field. This allowed
the spin states to be separated by a magnetic field gradient applied during ballistic
expansion. Switching the direction of the axial projection field exchanged the roles of
the |1, −1i and |1, +1i states. With Bz ≈ 0 the condensate lifetime was ∼ 1 ms due
to non-adiabatic spin-flips (Majorana flops) at the trap center. Therefore, the axial
projection field was applied immediately after ramping Bz → 0.
Figures 5-8(a) and 5-8(e) show the coreless nature of the vortices, while Figs. 58(b) and 5-8(f) show the concentric cylinder structure resulting from the competition between the atomic Zeeman energy and the kinetic energy of the rotating spinstates. The boundary condition β(0) = 0 was maintained regardless of the sign ofBz , i.e. atomic spins along the trap axis always remained in the initial state. If
the atomic spins had simply followed the local magnetic field, β(r) would satisfy
tan[β(r)] = B 0 r/Bz . Thus, scanning Bz from slightly positive to slightly negative
would instantaneously change the boundary condition at the origin from β(0) = 0 toβ(0) = π and flip the atomic spins along the trap axis. In interpreting the absorption
images shown in Figs. 5-8(a), 5-8(b), 5-8(e), and 5-8(f), we have assumed that thetwo-dimensional (ωz ¿ ω⊥ ) ballistic expansion process radially magnifies the conden-

76

(a)

(e)

(b)

(f)

mF = -1

mF = 0mF = +1

mF = -1
1.0ñtotalñmF = -1
ñmF = 0ñmF = +1

(c)

0.8
0.6

Optical Density

Optical Density

1.0

0.4
0.2

mF = +1ñtotalñmF = -1
ñm = 0FñmF = +1

0.8
0.6
0.4
0.2
0.0
+1

(d)

M / N (µB / 2)

M / N (µB / 2)

0.0
+1

mF = 0

(g)

0
-1
0.0

0.2

0.4
0.6
0.8
Radial Position (arb. units)

1.0

(h)

0
-1
0.0

0.2

0.4
0.6
0.8
Radial Position (arb. units)

1.0

Figure 5-8: Coreless vortices in F = 1 spinor condensates. Coreless vortices were imprinted by ramping Bz → 0 and diagnosed by suddenly switching (a)-(d) Bz ¿ 0 and(e)-(h) Bz À 0. Axial absorption images display the optical density of condensates
after 20 ms ballistic expansion (a,e) without and (b,f) with a magnetic field gradientapplied to separate the different spin states. (c) and (g) Azimuthally averaged opticaldensity vs radial position for spin components shown in (b) and (f), respectively. The
radial separation of the spin states resulted from their relative phase windings and isa clear signature of the skyrmion/meron wavefunction (Eq. 5.15). (d,h) Axial magnetization per particle, M/N = (µB /2) × (ñmF =+1 − ñmF =−1 )/ñtotal , vs radial position.
The absorption imaging light was resonant with the F = 2 → F 0 = 3 transition. The
atoms were optically pumped into the F = 2 hyperfine level with a pulse resonantwith the F = 1 → F 0 = 2 transition. This provided equal imaging sensitivity to eachspin state. The field of view in (a), (b), (e), and (f) is 1.0 mm × 3.0 mm.
sate wavefunction, as it does in the expansion of a single-component condensate withvortices [28, 43, 135].
Non-zero axial magnetic field curvature, B 00 ≈ 5 G/cm2 (Eq. 5.11), along the
microtrap axis implies that the spin texture had a slight axial dependence. The axial
77bias field was ramped linearly from Bz = 1.3 G to Bz ≈ 0 in 10 ms to imprint
the coreless vortices. This compressed the condensate radially and increased thecondensate chemical potential from µ ≈ (µB /2) × 3 mG to µ ≈ (µB /2) × 27 mG.
Ramping Bz → 0 fast compared to the axial trap period (250 ms), but slow comparedto the initial radial trap period (4 ms), guaranteed that the axial magnetic field
variation remained constant throughout the experiment at ∆Bz ≈ 3 mG, while atBz ≈ 0 the radial magnetic field variation was ∆Br ≈ 27 mG. The images shown in
Fig. 5-8 integrated the atomic number density along the z axis and therefore averaged
over the minor (∆Bz ¿ ∆Br ) axial variation to the spin texture.
To project the condensate wavefunction onto a basis quantized with respect tothe local magnetic field, a ∼ 10 G axial bias field was switched on at a rate of Ḃz =2×105 G/s along either the negative or positive z axis. 100 µs later the magnetic trap
was switched off allowing the atoms to expand ballistically. For 0 ≤ Br ≤ ∆Br , theLandau-Zener non-adiabatic transition probability [185], exp(−πµB Br2 /h̄Ḃz ) ≥ 0.9,

was sufficiently close to unity that the atomic spins remained “frozen” during thesudden application of the axial bias field and the spin texture could be accurately
diagnosed. While the total condensate density monotonically decreased as a functionof radial position [Figs. 5-8(a) and 5-8(e)], the density of each spin component peaked
at a different radius signifying a variation in the angular momentum per particle
between spin states [Figs. 5-8(b) and 5-8(f)].
Applying the projection field along the positive z axis generated additional ringsof atoms in the |1, −1i and |1, 0i states [Fig. 5-8(f)]. Ramping Bz → 0 in 10 ms caused
non-adiabatic spin-flips for atoms near r = 0 resulting in an atom loss of ∼ 50% [123].
If these atoms had not left the condensate before the projection field was applied, they
may have contributed to the images displayed in Fig. 5-8(f). The additional ringsof atoms may also correspond to a low energy, radial spin-wave excitation [91, 160].
However, we could not identify any asymmetry between applying the projection fieldalong the positive z axis as compared to the negative z axis that would account forthe presence of the extra rings in Fig. 5-8(f), but not in Fig. 5-8(b).
The stability of the spin textures could not be characterized on time scales > 1 ms
78

mF = -1

mF = 0Figure 5-9: Vortices in an optical trap. Optically confined condensates were preparedin a coherent superposition of the |1, −1i and |1, 0i states by a ∼ π/2 radio frequency
pulse. An overlapping Ioffe-Pritchard magnetic trapping potential (Eqs. 5.3 and 5.4)
was turn on such that inverting the axial bias field imprinted vortices in the |1, −1istate (Eq. 5.6). No vortices were imprinted in the |1, 0i state (Eq. 5.6). The axial
absorption image was taken during ballistic expansion. The absorption imaging light
was resonant with the F = 2 → F 0 = 3 transition. The atoms were optically pumped
into the F = 2 hyperfine level with a pulse resonant with the F = 1 → F 0 = 2transition. This provided equal imaging sensitivity to each spin state. The field ofview is 1.0 mm × 2.0 mm.
due to non-adiabatic spin-flip transitions while Bz ≈ 0 and excitations created by suddenly applying the axial projection field. It is predicted that skyrmions/merons incondensates with anti-ferromagnetic (ferromagnetic) interactions are unstable (stable) [13, 97, 114, 115, 139, 149, 150, 227, 228].

5.4

Vortices and Spin Textures in Optical Traps

Bose-Einstein condensates under optical confinement, as opposed to magnetic confinement, have their atomic spins decoupled from the trapping potential. For non-rotatingcondensates, spin dependent [193, 204] and magnetic field dependent [95] atom-atominteractions have been investigated. It is natural to extend these studies to rotatingcondensates where the ground state properties are predicted to depend strongly on
spin composition and atom-atom interactions [13, 23, 97, 104, 114, 115, 116, 139, 149,
150, 176, 189, 190, 192, 225, 227, 228].
By overlapping a Ioffe-Pritchard magnetic trapping potential with an optical79

dipole trap6 , vortices and spin textures were imprinted in condensates with arbitrary
initial spin (Fig. 5-9). It was technically challenging to keep both traps overlappedwhile varying the axial bias field because the current-carrying coils that produced theaxial field also produced a slight radial field. Thus, varying the axial bias field caused
a variation in the radial bias field and shifted the radial position of the Ioffe-Pritchardmagnetic trap.

5.5

Higher Order Vortices and Spin Textures

The phase imprinting technique described in this chapter can be extended to generatevortices and spin textures with higher winding numbers and angular momentum perparticle by using higher-order, axisymmetric multipole magnetic fields and/or condensates with different spin. In general, a two-dimensional axisymmetric multipole
magnetic field can be expressed ashi~ r) = Arα−1 cos(αφ)r̂ − sin(αφ)φ̂ , α > 0,B(~

(5.18)

where α = 2 describes a quadrupole field [Fig. 5-6(a)] and α = 3 describes a hexapole
field [Fig. 5-6(a)]. The topological phase factor in Eq. 5.7 is generalized to e−i2(α−1)mF φ
for inverting the axial bias field in the presence of an arbitary radial multipole magnetic field. A similar generalization is applicable for Eqs. 5.15 and 5.16.
Two-dimensional magnetic multiple fields for order α (Eq. 5.18) can be generated
by α − 1 wires and an orthogonal bias field, B⊥ . For example, a single-wire waveguide(Ch. 3, Fig. 3-2) generates a quadrupole magnetic field, while a two-wire waveguidecan generate a hexapole magnetic field for appropriate parameters [Ch. 3, Fig. 312(b)].
Additionally, vortices may be repeatedly imprinted in condensates to increase
the angular momentum per particle. Vortices were imprinted by inverting the axial
bias field in the presence of a radial multipole magnetic field, and were removed by
6

The radial confinement frequency of the optical dipole trap was approximately two times higher
than that of the magnetic trap.

80

returning the axial bias field to its original direction. If the magnetic multipole field isturned off before returning the axial bias field to its original direction, the vortices willremain in the condensate. Then, by turning back on the radial magnetic multipole
field and subsequently inverting the axial bias field additional vortices are added tothe condensate. In principle, this process can be cycled indefinitely to increase thevorticity of a condensate.

81

Chapter 6
Picokelvin Temperature Atomic
VaporsLow temperature and low density atomic vapors are used extensively for spectroscopy,
metrology, atomic clocks, and atom optics. Extending the accessible temperature anddensity ranges to even lower values can reveal phase transitions, such as Bose-Einsteincondensation [40, 110], the Kosterlitz-Thouless transition [118, 119, 120, 170, 175],and the Tonks-Girardeau transition [54, 65, 162, 171, 211] for Bosons and BCS pairingfor Fermions [12, 19, 20], transitions to reduced dimensionality [67, 69, 188, 196], andquantum reflection from material surfaces [3, 14, 198].
Temperature is a quantity that parameterizes how energy is distributed across theavailable states of a system, and effective temperatures can be defined for decoupleddegrees of freedom or subsets of particles. For example, nuclear spins isolated fromthe kinetic motion of their respective atoms have been cooled by adiabatic demagnetization to an effective temperature of 280 pK [161]. Spin ensembles have a finite
number of available states so that a spin-polarized sample, such as a magneticallytrapped condensate, would be characterized by zero effective temperature. In contrast, the motion of free particles is subject to a continuum of states, and the kinetic
temperature of an ensemble can only asymptotically approach absolute zero.
Effective temperatures in atomic vapors are defined by the widths of velocity distributions, which can be much smaller than the mean velocity of the sample. Raman82

cooling [106, 179] and velocity-selective coherent population trapping (VSCPT) [191]have generated velocity distributions with very narrow peaks, corresponding to nanokelvinand picokelvin effective temperatures. However, these temperatures were associatedwith the motion of only a subset of the atoms in the cloud and/or with atomic motion
in only one dimension.
For trapped, partially condensed atomic vapors, the condensate fraction has zero
entropy, and the kinetic temperature of the sample is determined by the velocity distribution of the thermal (noncondensed) component. When released, the condensate
fraction expands slower than the thermal component and has been characterized by
picokelvin effective temperatures for anisotropic [147] and non-interacting [217] gases.
Cooling the atomic motion of entire ensembles in all three dimensions has provendifficult. To date, kinetic temperatures of a few hundred nanokelvin have beenachieved with adiabatic and optical cooling [108, 212], and evaporative cooling techniques have produced condensates with temperatures of 3 nK [51]. This chapter describes our efforts to adiabatically and evaporatively cool partially condensed atomic
vapors to picokelvin kinetic temperatures and expands on the work presented inRef. [124] (App. F).
Spin-polarized gaseous Bose-Einstein condensates were confined by a combinationof gravitational and magnetic forces. The partially condensed atomic vapors were
decompressed, by weakening the gravito-magnetic trap to a mean frequency of 1 Hz,then evaporatively reduced in size to 2500 atoms. This lowered the peak condensate
density to 5 × 1010 atoms/cm3 and cooled the entire cloud in all three dimensions toa kinetic temperature of 450 ± 80 pK. These samples were characterized by a thermalvelocity ∼ 1 mm/s, a speed of sound ∼ 100 µm/s, and a healing length limited bythe ∼ 20 µm harmonic oscillator length of the trapping potential. The 1/e lifetimeof condensates in the gravito-magnetic trap was 80 ± 5 s.

83

6.1

Thermometry

An upper limit for the temperature of a partially condensed atomic vapor is the BoseEinstein condensation phase transition temperature, Tc , which in the thermodynamic
limit for a harmonically-trapped ideal Bose gas is (Ch. 1, Eq. 1.3)
ÃN
kB Tc = h̄ω̄
ζ(3)

!1/3

≈ 0.94h̄ω̄N 1/3 ,(6.1)

where kB is Boltzmann’s constant, ζ(n) is the Riemann Zeta function, ω̄ = (ωx ωy ωz )1/3
is the geometric mean of the harmonic trap frequencies, and N is the total number of atoms, both condensed and noncondensed. Thus, picokelvin temperatures(Tc ≤ 1 nK) are guaranteed for partially condensed atomic vapors with N ≤ 104
atoms confined at a mean frequency ω̄ ≤ 2π × 1 Hz.
Corrections to Eq. 6.1 arise due to the finite size of the system and atom-atominteractions [42]. The finite size correction always reduces the BEC transition temperature whereas repulsive (attractive) atom-atom interactions reduce (increase) it.
23

Na in the |1, −1i state has a positive s-wave scattering length, a ≈ +2.8 nm, andhence repulsive atom-atom interactions, such that for a given number of atoms andmean trap frequency Eq. 6.1 still provides an upper limit for the temperature ofa partially condensed atomic vapor. For spherically symmetric systems, the finite
size corrections scale as ∆Tc /Tc ∝ −N −1/3 and the correction due to atom-atom interactions scales as ∆Tc /Tc ∝ −(a/aHO )N 1/6 , where aHO =qh̄/mω̄ and m is theatomic mass. For atomic vapors with N = 104 atoms confined at a mean frequencyof ω̄ = 2π × 1 Hz, ∆Tc /Tc = −3 × 10−2 and ∆Tc /Tc = −3 × 10−5 due to finite size
and interaction effects, respectively. Both of these corrections are negligible for theexperiments discussed below.
Partially condensed atomic vapors contain two components, a thermal (noncondensed) component and a condensate component. The number of atoms in the thermal component, Nth , and their gaussian width in the trap, wth , as well as the numberof atoms in the condensate, N0 , are completely determined by the temperature, T ,84

and total number of atoms, N = N0 + Nth , characterizing the sample. Provided thatthe thermal energy is much larger than the trap level spacing, kB T À h̄ω̄, and thecloud is in thermal equilibrium, the following relations hold:
Nth

Ã

kB T
= ζ(3) ×
h̄ω̄
2kB T
,
mω 2
µ ¶3
T
,
= 1−
Tc

wth =
N0
N

s

!3

,

(6.2)
(6.3)
(6.4)

where ω is the trap frequency for the axis along which wth is measured. Equation 6.2
arises due to the population of all thermally accessible states, and Eq. 6.3 is a resultof the equipartition theorem. Thermometry can be performed by examining the
distribution of trapped thermal atoms (number and width) subject to the constraints
in Eqs. 6.2 and 6.3. The presence of a condensate component leads to additional
constraints on the total distribution of atoms.

6.2

Gravito-Magnetic Trap

Condensates in the |F = 1, mF = −1i state containing up to 3 × 106 atoms weredelivered to the science chamber using optical tweezers (Ch. 2) and loaded into thegravito-magnetic trap [Fig. 6-1]. A small coil carrying current IS located verticallybelow the atoms generated a vertical bias field, Bz , and supported the weak fieldseeking condensates against gravity with a vertical magnetic field gradient, Bz0 =2mg/µB ≈ 8 G/cm, where m is the atomic mass, g is the gravitational acceleration,and µB is the Bohr magneton. Additional control over Bz and Bz0 was provided by
two external coils carrying independent currents IU and IL . Weak curvature, Bz00 ,to the vertical bias field created stable vertical confinement such that a harmonic
restoring force was provided magnetically (gravitationally) for downward (upward)
vertical displacements [Fig. 6-1(b)]. A radial field gradient Br0 = ∂Br /∂r = −Bz0 /2~ = 0) was also present and added in quadrature with Bz to provide harmonic(∇ · B
85

(a)

(b)

V(z)

IU

z
IS
(c)

r

V(r)

z,g

r

IL

Figure 6-1: Gravito-magnetic trap. (a) Bose-Einstein condensates were levitatedagainst gravity ∼ 5 mm above a 1 cm diameter, 25-turn coil mounted inside theultrahigh vacuum chamber running current IS . Two 10 cm diameter, 20-turn coils
were mounted outside the vacuum chamber, and were supplied individually withcurrents IU and IL . The vertical separation between the large coils was 10 cm. The
1 cm diameter coil was mounted radially off-axis with respect to the pair of 10 cm
diameter coils by ∼ 1 cm. This broke the cylindrical symmetry of the trapping
potential. Additional bias fields ∼ 1 G were applied in the horizontal plane to cancel
the radial magnetic fields generated by IU and IL on the axis of the small coil andmaintain a stable trapping potential. The acceleration due to gravity, ~g , points down
the page. (b) Magnetic potential due to IS (short dashed line), gravitation potential(long dashed line), and joint vertical potential of the gravito-magnetic trap (solidline). (c) A radially repulsive potential magnetic potential was generated by running
IS alone (dashed line), however applying a slight anti-bias field with IU modified theradial energy profile and created a magnetic field minimum at r = 0 (solid line). In(a)-(c), the solid oval denotes the trapped condensate.

radial confinement with a restoring force proportional to (Br0 )2 /Bz [Fig. 6-1(c)].

866.2.1

Stable Confinement

The magnetic field on axis of a single coil of radius R carrying current I is1
1,
Bz (z) = µ0 IR2 22(z + R2 )3/2

(6.5)

where µ0 is the permeability of free space. Neglecting terms O(r3 ), the off axis
magnetic field takes the form (in cylindrical coordinates) [99]"2#~ z) ≈ Bz (z) − r Bz00 (z) ẑ − r Bz0 (z)r̂,
B(r,
4
2

(6.6)

where the derivatives are given by∂Bz (z)3z= − µ0 IR2 2,∂z2(z + R2 )5/2
2
2
∂ 2 Bz (z)32 4z − R
Bz00 (z) =
=
.
µ
IR
0
∂z 2
2
(z 2 + R2 )7/2
Bz0 (z) =

(6.7)
(6.8)

Atom are levitated against gravity where Bz0 = 2mg/µB . Stable three-dimenational
confinement about this point is only possible for vertical positions within a certain
range. From Eq. 6.8 Bz00 > 0 for |z| > R/2, thus stable vertical confinement is only
possible at distances greater than R/2 above the single coil. Stable radial confinement~ z)|/∂r > 0, i.e. the magnitude of the magnetic field increases
is possible only if ∂|B(r,~ z)|/∂r > 0 for (B 0 )2 − 2Bz B 00 > 0 or equivalently for |z| <off axis. ∂|B(r,zzthus stable radial confinement is only possible at distances less thanabove the single coil.

q

q

2/5R,

2/5R ≈ 0.63R

Combining these two results, we see that a stable, three-dimensional confinement
potential is achieved for weak field seeking atoms at a distance 0.50R < |z| < 0.63R
above a single current-carrying coil of radius R. For our parameters (R ≈ 5 mm),stable three-dimensional confinement is only expected over a range ∆z = 0.13R =650 µm centered 2.825 mm above the single coil.
The criterion for stable vertical confinement is not flexible and atoms may only be
trapped at distances greater than R/2 above the single coil. The criterion for stable87radial confinement, (Bz0 )2 −2Bz Bz00 > 0, is flexible since Bz in this inequality is the net
bias field along the trap axis. Bz can be lowered by applying a vertical bias field thatpartially cancels the vertical field due to the single coil, and stable radial confinement
is possible farther above the single coil than

6.2.2

q

2/5R ≈ 0.63R [Fig. 6-1(c)].

Radial Symmetry

The single coil gravito-magnetic trap has perfect radial symmetry. This is in contrast
to traditional Ioffe-Pritchard magnetic traps formed by a two-dimensional quadrupole
field and an axial bias field with curvature [68, 174]. The two-dimensional quadrupole
field can be written as~
B(x,y, z) = Br0 (xx̂ − y ŷ),(6.9)

while the axial bias field and curvature have components
1
1
1
~
B(x,
y, z) = Bz + Bz00 z 2 − Bz00 (x2 + y 2 ) ẑ − Bz00 z(xx̂ + y ŷ).
2
4
2
·

¸

(6.10)The terms −(Bz00 z/2)(xx̂ + y ŷ) in Eq. 6.10 are necessary in order to satisfy Maxwell’sequations and interfere with the two-dimensional quadrupole field (Eq. 6.9) to produce
effective magnetic field gradients Bx0 = Br0 − Bz00 z/2 and By0 = Br0 + Bz00 z/2 along thex and y axis, respectively. Thus, the radial symmetry of traditional Ioffe-Pritchardmagnetic trapping potentials is broken for planes z 6= 0.

6.3

Cooling Stages

For typical loading parameters (IS = 340 mA, IU = 5.5 A, and IL = 0) the gravitomagnetic trap was spherically symmetric with trap frequencies ωx ≈ ωy ≈ ωz ≈ 2π ×
8 Hz. Condensates loaded into the gravito-magnetic trap from the optical tweezers
were held for 5 s to allow for the damping of excitations. The resulting partiallycondensed clouds had ∼ 5×105 atoms and a BEC transition temperature Tc = 30 nK.
At all times, the atomic vapor maintained a temperature T such that 0.5 < T /Tc < 1.

88

6.4

Adiabatic Decompression

Further cooling was accomplished by adiabatically decompressing the trapping potential in two 5 s stages, with a 5 s delay in between to allow excitations to damp.
In the first stage, the vertical frequency was reduced to ωz = 2π × (1.81 ± 0.05) Hz
by simultaneously raising currents IL and IU by identical amounts and lowering thecurrent IS by a factor of 10. This kept the magnetic field gradient, Bz0 , constant,
but transferred its dominant source from the small coil to the external coils. The
external coils produced very little magnetic field curvature due to their large diameters (Eq. 6.8), thus the net magnetic field curvature was reduced by lowering IS . In
the second stage, the radial frequencies were reduced to ωx = 2π × (0.65 ± 0.05) Hzand ωy = 2π × (1.2 ± 0.1) Hz by raising IL and lowering IU by identical amounts.
The vertical magnetic field gradient and curvature remained constant, but the vertical bias field, Bz , increased. This reduced the radial confinement due to the scalingωr ∝ Br0 /Bz1/2 [68, 174]. The radial asymmetry (ωx 6= ωy ) arose because the small
coil and the large external coils were not coaxial [Fig. 6-1(a)].
For currents IS = 34 mA, IU = 14 A, and IL = 44 A, the final gravito-magnetic
trap had a mean frequency ω̄ = 2π×(1.12±0.08) Hz and an axial bias field Bz = 17 G.
The trap frequencies were measured by observing the condensate oscillate in the trap,
and the axial bias field was measured by driving the microwave transitions |1, −1i →|2, 0i, |2, −1i, |2, −2i. The residual anharmonicities of the trapping potential were
small, with ∆ω/ω = 0.1 for 500 µm displacements from the trap center. Further
radial decompression was not possible due to a finite trap depth [Fig. 6-1(c)] andsensitivity of trap stability to milligauss level radial bias fields.

6.4.1

Evaporative Atom Number Reduction

After decompression, the partially condensed atomic vapors had ∼ 2 × 105 atoms anda BEC transition temperature Tc = 3 nK. Tc was lowered further by reducing thenumber of atoms in the cloud (Eq. 6.1), while maintaining a substantial condensate
fraction at all times (0.5 < T /Tc < 1). The atom number was reduced by holding the89atoms in the gravito-magnetic trap for up to 200 s. Often, microwave radiation nearthe |1, −1i → |2, 0i transition was applied to shorten the hold time required to arrive
at lower atom number to 10 s. The atom number reduction was accompanied bycooling, during which the elastic collision rate (between thermal atoms and the condensate) dropped from 0.25 Hz to 0.01 Hz. Therefore, a few collisions were sufficient
to cause evaporation out of the finite depth trap. This cooling was not efficient in thesense of providing a gain in phase-space density, but still was capable of maintaining
thermal equilibrium and lowering the absolute temperature of the vapor (see below).

6.5

Picokelvin Temperatures

Using this technique, we cooled partially condensed vapors containing up to 30000
atoms to temperatures below 1 nK (Figs. 6-2 and 6-3). Our lowest measured threedimensional kinetic temperature was 450±80 pK for 2500 atoms at a peak condensate
density of 5 × 1010 atoms/cm3 . Under these conditions, the peak atom-atom interaction energy was µ = kB × 33 pK = h × 0.7 Hz, while the zero point energy of theharmonic trapping potential was (1/2)h̄ω̄ = kB × 27 pK. Condensates released fromthe gravito-magnetic trap would expand with energies of this order, and therefore
could be characterized by effective temperatures ∼ 30 pK.
Additional cooling would require lowering the trap frequencies further or reducing
the atom number more. However, weakly confining traps have proven technicallydifficult to control such that lowering ω̄/2π substantially below 1 Hz is challenging.
Likewise, because Tc ∝ N 1/3 , atom number reduction by an order of magnitude only
results in temperature reduction by a factor of two. Furthermore, lower temperaturesand lower densities are accompanied by collisional equilibrium times approaching
100 s.
The temperature of the atomic vapors was extracted by fitting integrated, onedimensional atomic density cross-sections by a bimodal distribution (Fig. 6-2)
Nth −x2 /w2th ,n(x) = N0 Ψ20 + √
e
πwth
90

(6.11)

(a)

30
20

0
(b)
15
10
5
0
(c)

4

Integrated Optical Density

10

2
y
0
x
0

200

400
Position (µm)

600

Figure 6-2: (Color) Picokelvin temperature thermometry. Partially condensed atomic
vapors confined in the gravito-magnetic trap with (a) 28000, (b) 16000, and (c) 2500
atoms. The one-dimensional cross sections (red solid line) were obtained by integrating the two-dimensional absorption images of the trapped clouds along the y axis.
Bimodal fits to Eq. 6.11 (blue dashed line) yielded temperatures of (a) 1.05±0.08 nK,(b) 780 ± 50 pK, and (c) 450 ± 80 pK, where the uncertainty is due to the fit of an individual image. The absorption imaging light was resonant with the F = 2 → F 0 = 3
cycling transition for the trapped atoms, and was aligned with the vertical (z) axis.
The atoms were optically pumped into the F = 2 hyperfine level with a pulse resonantwith the F = 1 → F 0 = 2 transition. The field of view for the absorption images in(a)-(c) is 460 µm × 460 µm.
where Nth , wth , and N0 are given in Eqs. 6.2, 6.3, and 6.4, respectively, and Ψ20 is abell-shaped function with width w0 that describes the condensate peak. The fitted
parameters were T , N , and w0 .
91

3

1
0.8
0.6

Atom Number

Temperature (nK)

2

0.4

105

104
0

0.2

104

Atom Number

100
200
Hold Time (s)105

Figure 6-3: Partially condensed atomic vapors at picokelvin temperatures. The temperature of over 60 partially condensed atomic vapors is plotted vs total number ofcondensed and noncondensed atoms. A solid line at the Bose-Einstein condensation
phase transition temperature (Eq. 6.1) and a dashed line at 1 nK are provided asa guides. Condensate temperatures were determined from one-dimensional fits toatomic density cross sections integrated along either the x (closed circles) or y (opencircles) axis (Fig. 6-2). Differences in the two measured temperatures for a single condensate reflect the true uncertainty of the measurement. Plotted error bars represent
the statistical uncertainty of the fit. The inset shows that the 1/e condensate lifetimein the gravito-magnetic trap was limited by one-body processes to 80 ± 5 s.
Because µ ≈ h̄ω̄, the condensates in the gravito-magnetic trap were in a density regime intermediate between the Thomas-Fermi (µ À h̄ω̄) and ideal (µ ¿h̄ω̄) gas limits. Data fit using either the Thomas-Fermi condensate wavefunction[Ψ20 = (15/16)w0−1 max(1 − x2 /w02 , 0)2 ] or the ideal gas condensate wavefunction[Ψ20 = w0−1 π −1/2 exp(−x2 /w02 )] yielded the same atom number and temperature. Also,
the fitted atom number and temperature were independent of the use of microwave
radiation to reduce the atom number.
All atomic vapors represented in Fig. 6-3 had a clear bimodal density distribution
from which a temperature was reproducibly extracted. The temperatures extracted
from one dimensional fits along both radial axis were nominally the same, empiricallyindicating that the atomic vapors remained close to thermal equilibrium at all times.
92

Chapter 7
Conclusion
As emphasized by its title, Microtraps and Waveguides for Bose-Einstein Condensates, this thesis has focused on the design and implementation of devices to advancethe field of atom optics. Connections between atom optics and photon optics are
made frequently. For example, the magnetic waveguides demonstrated in Ch. 3 are
the matter wave analog to fiber optic cables. However, matter wave theory is typically nonlinear due to atom-atom interactions (Ch. 1, Eq. 1.4). While this nonlinearity
gives rise to interesting atom optical phenomena, such as four wave mixing [49, 215],
it often plagues devices such as beam splitters [8, 81, 90, 201, 206] and produces
systematic uncertainties in precision interferometry [71].
Bose-Einstein condensates are a fully coherent source of matter waves [9, 25, 107,113], however exploiting this coherence for atom interferometric purposes has been
limited by their nonlinear interactions. The trapped-atom interferometer demonstrated in Ch. 4 successfully utilized the phase coherence associated with the condensate wavefunction, while avoiding complications arising due to wavepacket recombination in the presence of atom-atom interactions. Likewise, Ch. 6 demonstrated partiallycondensed atomic vapors at picokelvin temperatures and densities two or three ordersof magnitude lower than typical condensates. At these low densities, the atom-atominteraction energy may be low enough to pursue precision atom interferometry with
Bose-Einstein condensates [71]. An alternative approach to lowering the mean fieldenergy of the condensate is through the use of a Feshbach resonance where atom93

atom interactions can be tuned to zero with an external magnetic field [41, 95, 217].
However, performing precision interferometer with the aid of a Feshbach resonancewould require excellent magnetic field control since the atom-atom interactions vary
rapidly with the applied field.

7.1

Superfluid Gyroscope

Placing Ch. 5 (Vortices and Spin Textures) in the context of atom optics is more
subtle. Rotation sensing is a major application of atom interferometers, and theinherent rotation associated with a vortex state can be used as a superfluid gyroscope [11, 93, 197].
Consider the superposition of two spin states1|ψi = √ (|1, −1i + |2, +1i),
2

(7.1)

held in a Ioffe-Pritchard magnetic trap. The superposition |ψi was chosen because
both the |1, −1i and |2, +1i state are weak field seeking and have the same magnetic
moment. This allows for long spin coherence times even in the presence of the magnetic field gradients. The superposition |ψi has been created in 87 Rb condensateswith coherence times > 100 ms [79]. Inelastic spin-exchange collisions between atoms
in different hyperfine states are suppressed in 87 Rb as compared to 23 Na such that thecondensate lifetime in the state |ψi is a few seconds for 87 Rb [79] while it is expected
to be a few milliseconds for 23 Na [66].

Vortices created by the topological phase imprinting technique demonstrated in
Ch. 5 were shown to have a winding number that depended on the magnitude andsign of the magnetic quantum number, mF , describing the atomic state. Hence,
imprinting vortices in state |ψi produces counter rotating spin states´
1 ³|ψi → √ ei2φ |1, −1i + e−i2φ |2, +1i ,
2

(7.2)

where φ is the azimuthal angle with respect to the vortex line. We have taken
94

(a)

(b)

|〈1,-1|ψ〉|2 = sin2[2(φ - Ωτ)]|〈2,+1|ψ〉| = cos [2(φ - Ωτ)]2

2

Figure 7-1: Vortex phase interferogram. Density distribution for atoms in the (a)|1, −1i and (b) |2, +1i states for the wavefunction in Eq. 7.5. In the presence of
external rotation, these patterns rotate as τ is varied (see text).

quadrupolar radial confinement for the Ioffe-Pritchard magnetic trap [Ch. 5, Fig. 56(a)]. Higher order phase windings can be imprinted using higher order mulitpole
magnetic fields. This will increase the rotation sensitivity of the gyroscope.
To account for the presence of external rotation, the azimuthal angle φ is replacedby φ → φ − Ωt, where Ω is the rotation rate projected along the vortex line andt is time. The time dependent offset angle, Ωt, can be derived from the standard
expression for the Sagnac phase shift due to rotation (Ch. 4, Eq. 4.1)
∆φrot =

2m
ΩA,
h̄

(7.3)

where m is the atomic mass and A is the area enclosed by the rotating atoms. For thevortex states in Eq. 7.2, the superfluid velocity is |vs | = 2h̄/mr, where r is the radialdisplacement from the vortex line. The total area enclosed by the rotating atoms is
πr2 × t/(2πr/|vs |) = (h̄/m)t, where πr2 is the area enclosed while making one loop
around the condensate in a time 2πr/|vs |. Substituting this result into Eq. 7.3, we seethat ∆φrot = 2Ωt, where the factor of 2 is due to the winding number of the vortex
state.
In the presence of external rotation, the counter-rotating vortex superposition95

state evolves as´
1 ³|ψi → √ ei2(φ−Ωt) |1, −1i + e−i2(φ−Ωt) |2, +1i .
2

(7.4)

Driving a π/2 transition between the states |1, −1i ↔ |2, +1i at a time τ after creating
the vortex superposition yields the wavefunction|ψi → i sin[2(φ − Ωτ )]|1, −1i + cos[2(φ − Ωτ )]|2, +1i,

(7.5)

The angular dependence of the |1, −1i and |2, +1i state densities varies with τ according to the external rotation rate. Thus, the external rotation rate can be extracted
from the measured phase of the density distributions (vortex interferogram) as afunction of τ (Fig. 7-1).
For τ = 1 s, a rotation rate of 103 ×Ω⊕ , where Ω⊕ is the rotation rate of the Earth,would cause a phase shift of ∼ 8 degrees to the vortex interferogram. This is far fromthe rotation sensitivity achievable with other atom interferometry techniques [73, 129]
and liquid helium superfluid gyroscopes [11, 197], but demonstrates the principle ofrotation sensing with superfluids.

7.2

Last Man Standing

The science chamber apparatus (Ch. 2) has performed well over its two year life. I
am the last member of its construction team still working on the apparatus and willtake away the last memories its infancy when I depart. This includes nervous times,
like when we mistakenly vented the BEC production chamber, when the magnetic
trap sprung a water leak, when we first tried to evaporatively cool to BEC but foundthe magnetic trap bottom to be ∼ 6 G too high, and joyous times, like when we madeour first condensate and when we were finally able to translate condensates into thescience chamber.
I wish all future users the best of luck in creating their own fond memories.

96Appendix A
Propagation of Bose-Einstein
Condensates in a Magnetic
WaveguideThis appendix contains a reprint of Ref. [122]: A.E. Leanhardt, A.P. Chikkatur, D.
Kielpinski, Y. Shin, W. Ketterle, and D.E. Pritchard, Propagation of Bose-Einstein
Condensates in a Magnetic Waveguide, Phys. Rev. Lett. 89, 040401 (2002).

97

VOLUME 89, NUMBER 4

PHYSICAL REVIEW LETTERS

22 JULY 2002

Propagation of Bose-Einstein Condensates in a Magnetic WaveguideA. E. Leanhardt, A. P. Chikkatur, D. Kielpinski, Y. Shin, T. L. Gustavson, W. Ketterle, and D. E. Pritchard *Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139(Received 9 March 2002; published 2 July 2002)
Gaseous Bose-Einstein condensates of 2 3 3 106 23 Na atoms were loaded into a microfabricatedmagnetic trap using optical tweezers. Subsequently, the condensates were released into a magnetic
waveguide and propagated 12 mm. Single-mode propagation was observed along homogeneous segments of the waveguide. Inhomogeneities in the guiding potential arose from geometric deformations of
the microfabricated wires and caused strong transverse excitations. Such deformations may restrict the
waveguide physics that can be explored with propagating condensates. Finer perturbations to the guiding
potential fragmented the condensate when it was brought closer to the surface.
DOI: 10.1103/PhysRevLett.89.040401

PACS numbers: 03.75.Fi, 03.75.Be, 39.20. +q

Progress in the field of atom optics depends on developing improved sources of matter waves and advances intheir coherent manipulation. Bose-Einstein condensates ofdilute alkali vapors [1] are now used as sources of coherent atoms. Miniaturizing the current carrying structures
used to confine condensates offers prospects for finer control over the clouds [2,3]. Following the successful trapping and guiding of thermal atoms using self-supported
miniature wires [4 –8] and substrate-supported microfabricated wire arrays [9–12], recent experiments merged wire
traps on the millimeter scale [13] and microfabricated electronic devices [14,15] with Bose-Einstein condensation.
This has opened up a front on which further techniques
for coherent condensate transport and manipulation canbe explored. While condensate guiding with optical potentials may be limited fundamentally by diffraction [16],fundamental limitations to guiding condensates with microfabricated surfaces are not expected for an atom-surface
separation in excess of 1 mm [17].
In this Letter, we demonstrate that a Bose-Einstein condensate (BEC) transported with optical tweezers can betransferred into a microtrap on a substrate. Such condensates contained 5 times more atoms than those created ina similar microtrap [14]. We released the BEC from themicrofabricated magnetic trap into a single-wire magnetic
waveguide and studied its propagation. Condensates wereobserved to propagate 12 mm before exiting the field ofview of our imaging system. We observed single-mode(excitationless) BEC propagation along homogeneous segments of the waveguide in a regime where the longitudinalkinetic energy of the condensate exceeded its transverse
confinement energy. Transverse excitations were created incondensates propagating through perturbations in the guiding potential. These perturbations resulted from geometricdeformations of the current carrying wires on the substrate.
Finer imperfections were observed when trapped condensates were brought closer to the microchip as evidenced by
the longitudinal fragmentation of the cloud.
Condensates containing over 107 23 Na atoms were created in the jF, mF 典 苷 j1, 21典 state in a macroscopic Ioffe-

Pritchard magnetic trap [18], loaded into the focus ofan optical tweezers beam, and transported over 30 cm in
1250 ms into an auxiliary “science” chamber as describedin Ref. [13]. Condensates containing 2 3 3 106 atoms
arrived less than 500 mm below the microfabricated structures mounted in the science chamber. The optical tweezers consisted of 50 mW of 1064 nm laser light focused toa 1兾e2 radius of 26 mm. This resulted in axial and radial
trap frequencies of 4 and 440 Hz, respectively, and a trap
depth of 2.7 mK.
A schematic of the microchip onto which the BEC was
loaded is shown in Fig. 1. The microfabricated wires lie on
a 600 mm thick silicon substrate mounted on an aluminum
block. They are 50 mm wide and electroplated with copper to a thickness of 10 mm. The minimum separation distance between wires is 50 mm (100 mm center-to-center).
Macroscopic leads that extended to a vacuum feedthroughwere attached to the wafer using a gap welding technique. The BEC was initially loaded into a Z-wire trap
[9] formed by currents I1 and I2 along with an orthogonal
magnetic bias field B⬜ . Typical loading conditions wereI1 苷 I2 苷 1200 mA and B⬜ 苷 5.4 G, corresponding to a
separation of 450 mm between the BEC and the microchip.
The longitudinal trap frequency was vjj 苷 2p 3 6.0 Hz
and, for a longitudinal bias field Bjj 苷 3.0 G, the transverse trap frequency was v⬜ 苷 2p 3 97.0 Hz. Transfer
efficiency from the optical tweezers to the Z-wire trap wasnear unity, and BEC lifetimes over 10 s were observed withthe application of a radio frequency (rf) shield [18] which
limited the trap depth to 艐1 mK and was produced by thecurrent Irf on an auxiliary wire, as shown in Fig. 1.
The BEC was transferred into the waveguide by linearlyramping the current I2 to zero in 250 ms. The atoms wereaccelerated into the waveguide by the remaining end capof the Z-wire trap. Downstream, the effect of this end cap
was negligible and we observed BEC propagation at a constant velocity of 3.0 cm兾s after a propagation distance of
4 mm for I1 苷 1200 mA. Upon releasing the BEC fromthe Z-wire trap, its longitudinal velocity was controlled byapplying an external longitudinal magnetic field gradient

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microchip. This shift will be mirrored by a shift in the
trajectory of the guide in a plane parallel to that of the microchip [19]. Such a shift causes the guide axis to makean angle, u, with respect to its nominal trajectory along
which Bjj and B⬜ are aligned parallel and perpendicular,
respectively. As a result, the effective parallel, B0jj , andperpendicular, B0⬜ , magnetic fields are found by the rotation

FIG. 1. Microfabricated magnetic trap and waveguide. Opticaltweezers loaded a Bose-Einstein condensate into the Z-wire trap
formed by currents I1 and I2 in conjunction with the magnetic
bias field B⬜ . Lowering I2 to zero released the condensate intoa single-wire magnetic waveguide. Atom flow was from left toright. The inset shows the widening of the waveguide wire in
the region where another wire merges with it at a small angle.
The only current flowing in the inset is I1 . The condensate
was trapped above the plane of the page and the gravitational
៬ points out of the page. All microfabricatedacceleration, g,
features are drawn to scale.

created by running current through a single coil coaxialwith the waveguide axis. The field gradient was linearlyramped up and down over 6.5 ms to prevent creating excitations and was held constant for 52 ms. With gradientsof 0 0.6 G兾cm, the atomic velocity was varied over therange 3.0 6.6 cm兾s. Throughout this work, condensates
were detected via absorption imaging whereby resonant
laser light propagating parallel to B⬜ illuminated the atoms
and was imaged onto a CCD camera [18].
For all velocities, we observed single-mode condensate
propagation along unperturbed sections of the waveguide,
as shown in Fig. 3(a) below. The condensate depicted has
a smooth shape without any excitations.
Perturbations to the guiding potential arise from geometric deformations of the current carrying wires on thesubstrate. The extent to which such deformations alter the
potential experienced by the atoms depends on the atomwire separation distance, r, longitudinal extent of the perturbation, ᐉ, wire width, w, and wire thickness, t. Under
our guiding conditions (r ¿ w, t兲 the waveguide potential responds only to changes in the centroid of the current
density. In general, only wire deformations with ᐉ * r
will significantly perturb the guiding potential due to solid
angle considerations. Such a deformation (ᐉ 艐 1 mm,
r 苷 450 mm) is depicted in the inset of Fig. 1. As thewire width varies during the bifurcation process, the centroid of the current density will shift in the plane of the040401-2

B0jj 苷 Bjj cosu 2 B⬜ sinu,(1)

B0⬜ 苷 B⬜ cosu 1 Bjj sinu,(2)where 2p兾2 # u # p兾2. u is taken to be a positive(negative) angle for the specific case of atoms entering(exiting) the waveguide perturbation depicted in the inset
of Fig. 1.
Since the potential along the waveguide axis is determined by the local magnetic field (due to the Zeeman
interaction) and the vertical position of the guide center(due to the gravitational interaction), changes in the effective parallel and perpendicular magnetic fields produce
variations in the potential experienced by the propagating
atoms. Variations in the effective parallel magnetic field
are given by (for small angles)
DBjj0 苷 2B⬜ sinu .

(3)Thus, atoms entering (exiting) the perturbed guiding region will encounter a magnetic potential well (barrier).
Furthermore, changes in the effective perpendicular magnetic field cause the atom-substrate distance to vary by (forsmall angles)
Dr 苷 2r

0
Bjj
DB⬜
苷 2r
sinu .
B⬜
B⬜

(4)

Thus, the guide center entering (exiting) the perturbed
guiding region will shift towards (away from) the surfaceof the microchip, and atoms will encounter a gravitational
potential barrier (well). For 23 Na atoms in the j1, 21典 state,
the Zeeman interaction energy is h 3 700 Hz兾mG and
the gravitational interaction energy is h 3 560 Hz兾mm,
where h is Planck’s constant.
The effects of the perturbation depicted in the inset of
Fig. 1 on BEC propagation were studied by varying the incident velocity of the atoms sent into the region. Condensates traveling below 4.5 cm兾s were totally reflected fromthe perturbation, while clouds at speeds above 5.4 cm兾swere entirely transmitted. Figure 2 shows the results forintermediate atomic velocities. The local magnetic fieldin the perturbed guiding region was measured by driving rf spin-flip transitions that removed atoms from theguide [18]. It was found that upon entering (exiting)the perturbed region the magnetic bottom of the guiding
potential decreased (increased) by h 3 共50 6 10兲 kHz.
The signs of these shifts are consistent with those predicted by Eq. (3). The magnitude of the shift in the
Zeeman energy upon exiting the guide is also consistent
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Perturbed Guiding Region
Reflected Atoms(b)

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Transmitted Atoms

Kinetic Energy / h (kHz)
65
75

85

1.00
0.75
0.50
0.25
0.00
4.4

4.8
5.2
Velocity (cm/s)

5.6

FIG. 2. Bose-Einstein condensates scattering from the waveguide perturbation shown in the inset of Fig. 1. (a) Absorption
image after 8 ms of ballistic expansion of a partially transmitted
condensate. The incident velocity was 5.0 cm兾s. (b) The fraction of atoms reflected as a function of incident velocity. Waveguide parameters for all data were I1 苷 1200 mA, B⬜ 苷 5.4 G,
Bjj 苷 3.6 G, and v⬜ 苷 2p 3 84.5 Hz. The field of view in(a) is 0.35 mm 3 4.00 mm.

with the onset of transmission through the perturbed guiding region, as shown in Fig. 2(a). From Eq. (3) withB⬜ 苷 5.4 G, the maximum angular deviation, um , of the
waveguide trajectory necessary to produce a h 3 50 kHz
perturbation to the Zeeman energy is um 苷 13 mrad. The
corresponding vertical displacement of the guide centerfrom Eq. (4) with B⬜ 苷 5.4 G, Bjj 苷 3.6 G, and um 苷
13 mrad is 4 mm. This yields a gravitational potential
variation of h 3 2.2 kHz, which is small compared to the
Zeeman energy shifts associated with the perturbation.
In the propagation regime where the longitudinalkinetic energy of the BEC is large compared to thetransverse confinement energy of the guide, perturbations
involving transverse shifts in the guide trajectory are
expected to transversely excite incident condensates. Figure 3 depicts such transverse excitations for condensatestransmitted through the waveguide perturbation depictedin the inset of Fig. 1. The incident BEC velocity was in
the range 5.4 6.6 cm兾s. Excitations were characterized
by the peak-to-peak amplitude of the transverse displacement of the BEC after ballistic expansion. The imaging
axis only provided sensitivity to transverse excitations in
the plane normal to the surface of the microchip.
The nearly sinusoidal nature of the BEC excitations
shown in Figs. 3(b) and 3(c) indicates that the dipole modeof the BEC was primarily excited. Further evidence forthis comes from the fact that little variation in the excitation amplitude as a function of the propagation time after
exiting the perturbed guiding region was observed. Conversely, transmitted condensates showed strong signs ofhigher order excitations when Bjj was increased beyond
3.6 G corresponding to v⬜ # 2p 3 84.5 Hz. The absorption images deviated visibly from a smooth sinusoidal
040401-3

FIG. 3. Bose-Einstein condensate propagation. (a) Absorption image after 10 ms of ballistic expansion of a typicalcondensate just prior to entering the waveguide perturbation
depicted in the inset of Fig. 1.
The condensate contained 2 3 106 atoms and was accelerated to a velocityy 苷 6.3 cm兾s. It had propagated over 6 mm without excitation
in a waveguide with v⬜ 苷 2p 3 84.5 Hz and Bjj 苷 3.6 G.
Absorption images after 15 ms of ballistic expansion ofcondensates transmitted through the perturbed guiding region with y 苷 6.5 cm兾s for (b) v⬜ 苷 2p 3 97.0 Hz andBjj 苷 3.0 G and (c) v⬜ 苷 2p 3 91.1 Hz and Bjj 苷 3.3 G.
(d) Peak-to-peak amplitudes of condensate excitations after
15 ms of ballistic expansion versus velocity. For all data
All transmitted condenI1 苷 1200 mA and B⬜ 苷 5.4 G.
sates propagated 4 mm beyond the perturbed guiding region before being imaged. The field of view in (a)– (c) is
0.26 mm 3 2.00 mm.

shape. In addition, the measured excitation amplitude depended strongly on propagation time, indicting the phasing
and dephasing of several excitation modes.
Figure 3(d) shows a clear increase in the excitation amplitude for increasing velocity at fixed Bjj and v⬜ as well
as for increasing Bjj (decreasing v⬜ ) at fixed velocity.
Both trends are consistent with the expectation of increased
excitations for an increased ratio of atomic longitudinal kinetic energy to transverse confinement energy. The latter
trend is also consistent with the expectation of increased
excitations for an increased perturbation size since, from
Eq. (4), the vertical displacement of the guide center is
proportional to Bjj .
Because the perturbations to the magnetic potential
above a single wire are due to its geometric deformations,
one expects such perturbations to increase as the trap
center is brought closer to the surface of the microchip.
We observed longitudinal fragmentation of the BEC as
the atoms were brought to within 150 mm of the surface(Fig. 4). At 55 mm from the surface the potential developed axial variations with a characteristic length scale
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(a)

(b)

(c)

(d)

FIG. 4. Fragmented structure of Bose-Einstein condensatesbrought close to the substrate. Absorption images after 5 msof ballistic expansion from the Z-wire trap shown in Fig. 1.
The atom-substrate separation was (a) 190, (b) 145, (c) 100,and (d) 55 mm. For all images, the condensate started in a
trap with I1 苷 I2 苷 540 mA, B⬜ 苷 5.4 G, and Bjj 苷 0.3 G
corresponding to an atom-substrate separation of 200 mm. The
condensate was translated towards the substrate by lowering
the wire currents linearly over 500 ms. The atoms were held attheir final position for 100 ms prior to trap shutoff. The field ofview in (a) – (d) is 0.37 mm 3 2.00 mm.

of 100 150 mm. These variations could be magnetic orgravitational in origin.
In conclusion, we have demonstrated the single-mode
propagation of Bose-Einstein condensates in a microfabricated magnetic waveguide and identified sources of transverse excitations. Such excitations need to be addressed torealize the full potential of integrated atom-optical components fabricated on a microchip. In this work, thecondensate density was large enough that the mean fieldenergy exceeded the transverse confinement energy. In addition, the longitudinal velocity spread was determined by
the mean field expansion of the cloud upon release fromthe Z-wire trap. The generation of more dilute clouds
with narrower longitudinal velocity distributions is possible by continually draining a constant flux of atoms fromthe Z-wire trap serving as a reservoir. Such a matter wave
source is ideal for the realization of an atom interferometeron a microchip.
We are indebted to S. Gupta for contributions to the apparatus, E. Tsikata for experimental assistance, the MIT
Microsystems Technology Laboratories for fabricating the
microchip used in this work, M. Prentiss and M. Vengalattore for valuable discussions, and A. D. Cronin and
D. Schneble for critical readings of the manuscript. This
work was funded by ONR, NSF, ARO, NASA, and the040401-4

22 JULY 2002

David and Lucile Packard Foundation. A. E. L. acknowledges additional support from NSF.
Note added.— Results similar to those in Fig. 4 also
appear in Ref. [20]*http://cua.mit.edu/ketterle_group/[1] Bose-Einstein Condensation in Atomic Gases, Proceedingsof the International School of Physics Enrico Fermi, Course
CXL edited by M. Inguscio, S. Stringari, and C. E. Wieman(IOS Press, Amsterdam, 1999).
[2] J. D. Weinstein and K. G. Libbrecht, Phys. Rev. A 52, 4004
(1995).
[3] J. H. Thywissen, M. Olshanii, G. Zabow, M. Drndić, K. S.
Johnson, R. M. Westervelt, and M. Prentiss, Eur. Phys. J. D
7, 361 (1999).
[4] J. Fortagh, A. Grossmann, C. Zimmermann, and T. W.
Hänsch, Phys. Rev. Lett. 81, 5310 (1998).
[5] J. Denschlag, D. Cassettari, and J. Schmiedmayer, Phys.
Rev. Lett. 82, 2014 (1999).
[6] M. Key, I. G. Hughes, W. Rooijakkers, B. E. Sauer, E. A.
Hinds, D. J. Richardson, and P. G. Kazansky, Phys. Rev.
Lett. 84, 1371 (2000).
[7] B. K. Teo and G. Raithel, Phys. Rev. A 63, 031402(R)(2001).
[8] P. Cren, C. F. Roos, A. Aclan, J. Dalibard, and D. GuéryOdelin, Eur. Phys. J. D (to be published).
[9] J. Reichel, W. Hänsel, and T. W. Hänsch, Phys. Rev. Lett.
83, 3398 (1999).
[10] D. Müller, D. Z. Anderson, R. J. Grow, P. D. D. Schwindt,and E. A. Cornell, Phys. Rev. Lett. 83, 5194 (1999).
[11] N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P.
Smith, M. Drndić, R. M. Westervelt, and M. Prentiss, Phys.
Rev. Lett. 84, 1124 (2000).
[12] R. Folman, P. Krüger, D. Cassettari, B. Hessmo, T. Maier,and J. Schmiedmayer, Phys. Rev. Lett. 84, 4749 (2000).
[13] T. L. Gustavson, A. P. Chikkatur, A. E. Leanhardt, A. Görlitz, S. Gupta, D. E. Pritchard, and W. Ketterle, Phys. Rev.
Lett. 88, 020401 (2002).
[14] H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, andC. Zimmermann, Phys. Rev. Lett. 87, 230401 (2001).
[15] W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel,
Nature (London) 413, 498 (2001).
[16] K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt,
W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R)(2001).
[17] C. Henkel and W. Wilkens, Europhys. Lett. 47, 414 (1999).
[18] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, in
Bose-Einstein Condensation in Atomic Gases (Ref. [1]),pp. 67–176.
[19] Analogously, the trajectory will vary in a plane normal tothe microchip for variations in the wire thickness.
[20] J. Fortágh, H. Ott, S. Kraft, and C. Zimmermann, condmat/0205310.

040401-4

Appendix B
Bose-Einstein Condensates near a
Microfabricated Surface
This appendix contains a reprint of Ref. [125]: A.E. Leanhardt, Y. Shin, A.P. Chikkatur,
D. Kielpinski, W. Ketterle, and D.E. Pritchard, Bose-Einstein Condensates near a
Microfabricated Surface, Phys. Rev. Lett. 90, 100404 (2003).

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Bose-Einstein Condensates near a Microfabricated Surface
A. E. Leanhardt, Y. Shin, A. P. Chikkatur, D. Kielpinski, W. Ketterle, and D. E. Pritchard*
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139(Received 15 November 2002; published 12 March 2003)
Magnetically and optically confined Bose-Einstein condensates were studied near a microfabricatedsurface. Condensate fragmentation observed in microfabricated magnetic traps was not observed in
optical dipole traps at the same location. The measured condensate lifetime was  20 s and independentof the atom-surface separation under both magnetic and optical confinement. Radio-frequency spin-flip
transitions driven by technical noise were directly observed for optically confined condensates andcould limit the condensate lifetime in microfabricated magnetic traps.
DOI: 10.1103/PhysRevLett.90.100404

PACS numbers: 03.75.Nt, 03.75.Be, 34.50.Dy, 39.20.+q

The manipulation of gaseous Bose-Einstein condensates with magnetic fields produced by wires microfabricated on material surfaces has opened a new frontier inthe field of atom optics [1– 4]. Magnetic confinement
using microfabricated wires is tighter and has higher
spatial resolution than is achievable in macroscopic magnetic traps [5–8]. Generally, an important feature of magnetic traps is the excellent thermal isolation between the
nanokelvin temperature clouds they confine and the 300 K
laboratory environment. However, decreasing the atomsurface separation into the micrometer range has raised
concerns that fluctuating surface currents may perturb theatoms [9,10]. While early demonstrations of trapping andguiding laser-cooled thermal atoms with microfabricateddevices reported no evidence of deleterious surface effects [11–14], recent studies using samples cooled byforced radio-frequency (rf) evaporation to & 2 K havefound corrugated potentials [3,15], large heating rates[2,15], and short trap lifetimes [2,15] for atom-surface
separations in the 100 m regime. The ultimate applicability of microfabricated devices to atom optics depends
on the characterization and elimination of such effects.
In this work, we experimentally investigate the behavior of Bose-Einstein condensates near a microfabricatedsurface. The condensates were confined at the same position relative to the surface by either a microfabricatedmagnetic trap or an optical dipole trap. Since the two traps
operate on different principles and the electromagnetic
fields for each have different sources, this study provides
a unique examination of the interaction between BoseEinstein condensates and a microfabricated surface. For
example, while condensates confined near the surface in a
microfabricated magnetic trap were found to fragment
longitudinally [3,15], the clouds remained intact under
optical confinement.
Significantly, the measured condensate lifetime in boththe microfabricated magnetic trap and the optical dipole
trap was  20 s, an order of magnitude longer than previous results [2,15], and independent of the atom-surface
separation. We have directly observed spin-flip transitions
driven by rf technical noise for condensates held in the

optical dipole trap. The transition rate increased rapidly
with decreasing atom-surface separation implying that
distance-dependent losses can occur in magnetic traps
where the products of such transitions cannot be directly
identified.
Bose-Einstein condensates containing over 107 23 Na
atoms were created in the jF  1;mF  1i state in amacroscopic Ioffe-Pritchard magnetic trap, loaded into
the focus of an optical tweezers beam, and transported
 32 cm in 2 s into an auxiliary ‘‘science’’ chamber as
described in Ref. [16]. The optical tweezers consisted of
 50 mW of 1064 nm laser light focused to a 1=e2 radiusof 26 m. This resulted in axial and radial trap frequencies !jj  2  4 Hz and !?  2  425 Hz, respectively,
and a trap depth of 2:5 K. Condensates containing
2–3  106 atoms arrived 70–500 m below the microfabricated structures mounted in the science chamber.
The atom-surface separation was varied by angling theoptical tweezers axis before translation and was limited
to distances  70 m due to the laser beam clipping on
the microchip support structures.
In the science chamber, the condensate either remained
confined by the optical tweezers or was loaded into a microfabricated Ioffe-Pritchard magnetic trap formed by a
Z-shaped wire carrying current I and an external magnetic bias field, B? , as described in Ref. [3]. An additional
longitudinal bias field, Bjj , was applied with external coils
to adjust the magnetic trap bottom and radial trap frequency. The microfabricated wires were lithographicallypatterned on a 600 m thick silicon substrate mounted on
an aluminum block. They were 50 m wide and electroplated with copper to a thickness of 10 m.
As in previous experiments [3,15], condensates confined near the surface in the microfabricated magnetic
trap were observed to fragment longitudinally [Fig. 1(a)].
The condensate density depletions appeared in the samelongitudinal position relative to the surface on each realization of the experiment, and more fragments formed as
the atoms came closer to the microchip [3]. In contrast, condensates confined optically at the same location
were not observed to fragment [Fig. 1(b)]. The same

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FIG. 1. Fragmentation of Bose-Einstein condensates. Transverse absorption images after 10 ms ballistic expansion of condensates containing  106 atoms after holding at a distance of
85 m from the microfabricated surface for 15 s in the (a) microfabricated magnetic trap and (b) optical dipole trap. (c) Radially averaged optical density vs axial position for condensates
shown above. Longitudinal fragmentation occurred for condensates held in the microfabricated magnetic trap, but not for
those confined optically at the same location with the magnetic
trap off. The magnetic trap was operated with I  130 mA,
B?  3:2 G, and Bjj  1:4 G yielding a radial trap frequency!?  2  450 Hz. The optical dipole trap had a radial trap
frequency !?  2  425 Hz and was operated with Bjj 
1:8 G. For both condensates, the chemical potential was  
kB  120 nK. The absorption imaging light was resonant with
the F  1 ! F0  2 transition. The field of view in (a) and (b)is 0:25 mm  1:00 mm.

longitudinal bias field was nominally applied to bothmagnetically and optically confined condensates so thatany surface magnetization effects would perturb the
clouds identically. The lack of condensate fragmentationin the optical dipole trap implies that the longitudinalpotential corrugations arise due to the presence of current
flow in the microfabricated wires, in agreement withconclusions reached elsewhere [17]. Deviations of thecurrent flow from a straight line would lead to suchcorrugations and could arise due to imperfect microfabrication including variations in the width and thickness of
the wire estimated to be 5% of the electroplated thickness, variations in the resistivity of the wire over its crosssection, and parasitic conductances to the substrate. Other
possible origins of irregular current flow are instabilities
at high current density [17].
It is interesting to note that in our earlier work nofragmentation was observed when condensates confinedin a macroscopic Z-shaped wire trap were brought within 10 m of the surface of the wire [16]. The wire wasmade of copper and had a circular cross section with
1:27 mm diameter. The condensates were loaded into
the wire trap 740 m from the surface of the wire andbrought closer by lowering the wire current. The experimental parameters upon closest approach were I 
920 mA and B?  2:9 G, yielding estimated axial andradial trap frequencies !jj  2  7 Hz and !?  2 
78 Hz, respectively [18]. The macroscopic wire trap contained 5  105 atoms extended longitudinally over
200 m at a chemical potential   kB  30 nK. Differ100404-2

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ences between the macroscopic and microfabricated wire
traps include vastly different fabrication techniques aswell as lower current densities in the macroscopic wire.
Confined atoms are sensitive to noise at their trap frequency and Zeeman splitting frequency [9,10]. In thiswork, typical radial trap frequencies were  500 Hz
while Zeeman splitting frequencies were  1 MHz. Noiseat the trap frequency leads to heating and subsequent trap
loss after the atoms acquire an energy greater than thetrap depth. For atoms confined in a Ioffe-Pritchard magnetic trap, radial magnetic bias field fluctuations causeradial trap-center fluctuations. The amplitude of suchtrap-center fluctuations is independent of the longitudinalbias field. However, for optically confined atoms, only
fluctuating radial magnetic field gradients cause radial
trap-center fluctuations. The effects of such gradients canbe minimized by applying a longitudinal bias field that
adds in quadrature with the fluctuating radial gradients
since it is the gradient of the magnitude of the bias field
vector that determines the force on an atom.

Lifetime (s)VOLUME 90, N UMBER 10

35
30
25
20
15
10
5
0
0

100
200
300
400
Atom-Surface Separation (µm)

500

FIG. 2. Lifetime of Bose-Einstein condensates near a microfabricated surface. The 1=e lifetime of condensates confined in
the microfabricated magnetic trap (solid squares) and optical
dipole trap (solid circles) is shown to be independent of distance from the microfabricated surface. Statistical error bars
are included. Additionally, excitations created while transferring the condensate from the optical tweezers into the microfabricated magnetic trap caused fluctuations in the measured
magnetic trap lifetime, indicated by the scatter in the data. I
and B? were varied with distance to maintain the radial magnetic trap frequency at !?  2  450 Hz with Bjj  1:4 G.
The vertical line indicates the onset of longitudinal condensate
fragmentation in the magnetic trap. In the optical dipole trap,
the condensate was held directly below the microfabricatedwire used to form the magnetic trap with Bjj  1:8 G. No external connections were made to the microchip. The optical dipole trap had axial and radial trap frequencies !z  2  4 Hz
and !?  2  425 Hz, respectively. Only atoms remaining
in the j1; 1i state were resonant with the absorption imaging
light. For comparison, the distance dependence of thermal
cloud lifetimes measured in Ref. [15] is shown for atoms confined magnetically by a microstructure (open squares) and copper wire (open diamonds). Error bars smaller than the symbol
size are not included. Also, magnetically confined condensate
lifetimes reported in Ref. [2] (open triangle) and Ref. [15](open circle) are shown for comparison.

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Spin-flip transitions driven by rf noise at the atomic
Zeeman splitting frequency distribute the atomic population across magnetically confinable and unconfinable
states. This causes atom loss for clouds held in magnetic
traps. However, all spin states are confined in an optical
dipole trap so spin-flip transitions do not lead to loss andthe products can be directly observed. Since magneticallyand optically confined condensates react differently tonoise, whether it is at their trap frequency or the atomic
Zeeman splitting frequency, a systematic study of condensate lifetimes in both magnetic and optical traps
provides better noise characterization than studies performed in either a magnetic or an optical trap exclusively.
Any atom-surface coupling, regardless of frequency,
should manifest itself as a dependence of the condensate
lifetime on the atom-surface separation. Figure 2 shows ameasurement of the magnetically and optically confinedcondensate lifetime as a function of the distance from themicrofabricated surface. No distance dependence was observed and the measured condensate lifetime was  20 s,
10 times longer than previous results [2,15]. A distanceindependent condensate lifetime indicates that atomsurface interactions are unimportant over the
70–500 m separation range. This is in contrast to resultspresented in Ref. [15], where a distance-dependent lifetime was observed for thermal atoms magnetically confined near a microfabricated surface. These data are
included in Fig. 2 for comparison.
Several experimental details altered the measured condensate lifetime. Excitations created during the microfabricated magnetic trap loading were found to shorten themeasured lifetime, and care had to be taken to overlap the
optical and magnetic traps during transfer to minimize
such excitations. Translating the condensate either towards or away from the microfabricated surface by adiabatically varying I and B? to shift the trap center while
maintaining a constant radial trap frequency was found to
decrease the condensate lifetime. This presumably resulted from excitations induced by irregular current
changes due to technical limitations in controlling thepower supplies connected to the microchip. As a result,microfabricated magnetic trap lifetime data are presented
only for atom-surface separations  70 m, where theatoms were loaded into their final position directly fromthe optical tweezers. Occasionally, heating was observed
for atoms in both the microfabricated magnetic trap andoptical dipole trap due to technical noise at the trap
frequency, even with care taken to eliminate ground loops
and minimize cable lengths [19]. Connecting a 10 mF
capacitor in parallel with the 2  microfabricated wire
(1=RC  2  8 Hz) eliminated such effects. Thereafter,
applying rf power to the microchip at a frequency chosen
to limit the trap depth for magnetically confined atoms
did not consistently alter the condensate lifetime.
The distance independent condensate lifetime presented in Fig. 2 indicates that our experiment is not
100404-3

week ending
14 MARCH 2003

currently limited by the proximity of the microfabricatedsurface. However, we have observed spin-flip transitions
driven by rf noise in the microfabricated wires. Figure 3
shows the behavior of condensates confined opticallydirectly beneath the microfabricated wire used for magnetic trapping. Condensate atoms initially in the j1; 1istate [Fig. 3(a)] were found to make transitions to othermagnetic sublevels [Fig. 3(b)]. Such transitions would act
as a loss mechanism for magnetically confined clouds.
The transition rate was found to decrease as the square of
the atom-surface separation distance, d. Since the magnetic field of a straight wire decays as 1=d, and the power
scales as the square of the field, the 1=d2 dependence ofthe spin-flip transition rate is expected for atoms in thenear field (d) of the wire, where   300 m is the
wavelength of  1 MHz radiation.
The transition rate vs distance data presented inFig. 3(c) were taken with all connections necessary to
run the microfabricated magnetic trap made to the microchip, but with no current flowing in the microfabricatedwires. The atoms were exposed to a longitudinal bias fieldBjj  1:8 G to simulate the field configuration in the
microfabricated magnetic trap. This also maximized their

FIG. 3. Spin-flip transitions driven by radio-frequency technical noise. Axial absorption images after 22 ms ballistic
expansion of condensates held in the optical dipole trap at a
distance of 185 m from the microfabricated surface for (a) 0 s
and (b) 40 s. A magnetic field gradient was applied duringballistic expansion to separate the different spin states. (c) Spinflip transition rate, , vs distance, d, from the microfabricatedsurface on a log-log scale. A line  / 1=d2 is provided as a
guide. The inset shows the fraction of the total atoms in themF  0 state with the transition rate being defined as the initialslope of the data (before any atoms in the mF  1 state weredetected). The absorption imaging light was resonant with theF  2 ! F0  3 transition. The atoms were optically pumped
into the F  2 hyperfine level with a pulse resonant with theF  1 ! F0  2 transition. This provided equal imaging sensitivity to each magnetic sublevel. The field of view in (a) and(b) is 1:0 mm  2:0 mm.

100404-3

VOLUME 90, N UMBER 10

PHYSICA L R EVIEW LET T ERS

sensitivity to fluctuating fields generated by wire currents
since rf transitions are more favorable for magnetic fields
oscillating orthogonal to a static bias field. Spin-flip transitions were suppressed by exposing the optically confined atoms to an orthogonal bias field, B? .
The rf driven spin-flip transition rate depended
strongly on experimental details, suggesting that antenna
effects coupled rf noise into the system. The rate wasmeasured to be of order 100 times higher if care was
not taken to carefully eliminate ground loops and use
minimal cable lengths [19]. Also, with no connections to
the microchip, spin-flip transitions were not detectablefor condensates held up to 60 s. The spin-flip rate presented in Fig. 3(c) became comparable to the measured
condensate decay rate displayed in Fig. 2 for atomsurface separations <100 m. Thus, extending long condensate lifetimes much closer to the surface will require
further rf shielding and/or filtering.
In conclusion, we have studied the behavior of BoseEinstein condensates near a microfabricated surface.
Condensates found to fragment while held in a microfabricated magnetic trap were observed to remain intact
while held at the same position relative to the microchip
in an optical dipole trap. A possible explanation is thatdeviations of the current path from a straight line giverise to corrugations in the longitudinal potential. The
origins of such current path deviations are under investigation. Furthermore, our work demonstrates magnetically and optically confined condensate lifetimes  20 s
at distances  70 m from a microfabricated surface.
The lifetime was measured to be independent of theatom-surface separation and 10 times longer than results
obtained elsewhere at comparable distances. Spin-flip
transitions driven by rf technical noise were directly
observed for condensates held in an optical dipole trap;
however, we found no evidence for fundamental, thermally induced noise driven processes above the level of
those attributed to technical noise. Our results demonstrate the extreme sensitivity of Bose-Einstein condensates to small static and dynamic electromagnetic fields.
This sensitivity provides a challenge for realizing microfabricated atom-optical devices, but it also emphasizes
the potential for developing new detector and instrumentation technology.
We thank T. Pasquini for experimental assistance and
M. Crescimanno for a critical reading of the manuscript.
This work was funded by ONR, NSF, ARO, NASA, andthe David and Lucile Packard Foundation. A. E. L. acknowledges additional support from NSF.
Note added. —Recently, we have observed condensate
fragmentation near conductors 50 m wide and evaporated with gold to a thickness of 1:25 m. This shows thatthe fragmentation phenomena are not limited to copper

100404-4

week ending
14 MARCH 2003

conductors, in agreement with conclusions reached elsewhere [20]. Also, Ref. [20] reports magnetic trap lifetimes
limited by thermally induced spin flips.

*Electronic address: http://cua.mit.edu/ketterle_group/[1] H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, andC. Zimmermann, Phys. Rev. Lett. 87, 230401 (2001).
[2] W. Hänsel, P. Hommelhoff, T.W. Hänsch, and J. Reichel,
Nature (London) 413, 498 (2001).
[3] A. E. Leanhardt, A. P. Chikkatur, D. Kielpinski, Y. Shin,
T. L. Gustavson, W. Ketterle, and D. E. Pritchard, Phys.
Rev. Lett. 89, 040401 (2002).
[4] S. Schneider, A. Kasper, Ch. vom Hagen, M. Bartenstein,
B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman,
L. Feenstra, and J. Schmiedmayer, Phys. Rev. A 67,
023612 (2003).
[5] J. D. Weinstein and K. G. Libbrecht, Phys. Rev. A 52,
4004 (1995).
[6] J. Schmiedmayer, Eur. Phys. J. D 4, 57 (1998).
[7] J. H. Thywissen, M. Olshanii, G. Zabow, M. Drndić,K. S. Johnson, R. M. Westervelt, and M. Prentiss, Eur.
Phys. J. D 7, 361 (1999).
[8] E. A. Hinds and I. G. Hughes, J. Phys. D 32, R119 (1999).
[9] C. Henkel and M. Wilkens, Europhys. Lett. 47, 414(1999).
[10] C. Henkel, S. Pötting, and M. Wilkens, Appl. Phys. B 69,
379 (1999).
[11] J. Reichel, W. Hänsel, and T.W. Hänsch, Phys. Rev. Lett.
83, 3398 (1999).
[12] D. Müller, D. Z. Anderson, R. J. Grow, P. D. D. Schwindt,and E. A. Cornell, Phys. Rev. Lett. 83, 5194 (1999).
[13] N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P.
Smith, M. Drndić, R. M. Westervelt, and M. Prentiss,
Phys. Rev. Lett. 84, 1124 (2000).
[14] R. Folman, P. Krüger, D. Cassettari, B. Hessmo,
T. Maier, and J. Schmiedmayer, Phys. Rev. Lett. 84,4749 (2000).
[15] J. Fortágh, H. Ott, S. Kraft, A. Günther, and C. Zimmermann, Phys. Rev. A 66, 041604(R) (2002).
[16] T. L. Gustavson, A. P. Chikkatur, A. E. Leanhardt,
A. Görlitz, S. Gupta, D. E. Pritchard, and W. Ketterle,
Phys. Rev. Lett. 88, 020401 (2002).
[17] S. Kraft, A. Günther, H. Ott, C. Zimmermann, andJ. Fortágh, J. Phys. B 35, L469 (2002).
[18] The axial and radial trap frequencies were scaled fromthe frequencies measured with I  2:0 A in Ref. [16] by
0:92 A=2:0 A 1=2 and 2:0 A=0:92 A, respectively.
[19] The current supply for the microfabricated wire and thewire itself were grounded to the vacuum chamber at thelocation of the electrical feedthrough for the microchip
connections. The total length of all (unshielded) current
carrying cables was  1 m.
[20] M. P. A. Jones, C. J. Vale, D. Sahagun, B.V. Hall, and E. A.
Hinds, quant-ph/0301018.

100404-4

Appendix C
Atom Interferometry with
Bose-Einstein Condensates in a
Double-Well Potential
This appendix contains a preprint of Ref. [199]: Y. Shin, M. Saba, T.A. Pasquini,
W. Ketterle, D.E. Pritchard, and A.E. Leanhardt, Atom Interferometry with BoseEinstein Condensates in a Double-Well Potential, arXiv:cond-mat/0306305 (2003).

107

Atom interferometry with Bose-Einstein condensates in a double-well potential
Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt∗
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139(Dated: July 17, 2003)

arXiv:cond-mat/0306305 v2 17 Jul 2003

A trapped-atom interferometer was demonstrated using gaseous Bose-Einstein condensates coherently split by deforming an optical single-well potential into a double-well potential. The relative
phase between the two condensates was determined from the spatial phase of the matter wave interference pattern formed upon releasing the condensates from the separated potential wells. Coherent
phase evolution was observed for condensates held separated by 13 µm for up to 5 ms and wascontrolled by applying ac Stark shift potentials to either of the two separated condensates.
PACS numbers: 03.75.Dg, 39.20.+q, 03.75.-b, 03.75.Lm

Demonstrating atom interferometry with particles confined by magnetic [1, 2, 3, 4] and optical [5] microtraps and waveguides would realize the matter wave analog of optical interferometry using fiber-optic devices.
Current proposals for confined-atom interferometers relyon the merger and separation of two potential wells tocoherently divide atomic wavepackets [6, 7, 8]. This
type of division differs from previously demonstrated
atomic beam splitters. To date, atomic beams and vapors have been coherently diffracted into different momentum states by mechanical [9, 10] and optical [11]gratings, and Bose-Einstein condensates have been coherently delocalized over multiple sites in optical lattices [12, 13, 14, 15, 16, 17]. Atom interferometers utilizing these beam splitting elements have been used tosense accelerations [12, 18] and rotations [19, 20], monitor quantum decoherence [21], characterize atomic andmolecular properties [22], and measure fundamental constants [18, 23].
In this Letter, we demonstrate a trapped-atom interferometer with gaseous Bose-Einstein condensates confinedin an optical double-well potential. Condensates were
coherently split by deforming an initially single-well potential into two wells separated by 13 µm. The relative
phase between the two condensates was determined fromthe spatial phase of the matter wave interference patternformed upon releasing the atoms from the separated potential wells [17, 24]. This recombination method avoids
deleterious mean field effects [25, 26] and detects applied
phase shifts on a single realization of the experiment, unlike in-trap recombination schemes [6, 7, 8].
The large separation between the split potential wells
allowed the phase of each condensate to evolve independently and either condensate to be addressed individually. An ac Stark phase shift was applied to either condensate by temporarily turning off the optical fields generating its potential well. The spatial phase of the resulting matter wave interference pattern shifted linearly withthe applied phase shift and was independent of the timeof its application. This verified the phase sensitivity of
the interferometer and the independent phase evolutionof the separated condensates. The measured coherencetime of the separated condensates was 5 ms.
The present work demonstrates a trapped-atom interferometer with two interfering paths. This geometry has
the flexibility to measure either highly localized potentials or uniform potential gradients, such as those arising from atom-surface interactions or the earth’s gravitational field, respectively. In contrast, multiple-pathinterferometers demonstrated in optical lattice systems
are restricted to measurements of the latter [12, 17].
Bose-Einstein condensates containing over 107 23 Na
atoms were created in the |F = 1, mF = −1i state ina magnetic trap, captured in the focus of a 1064 nm optical tweezers laser beam, and transferred into an auxiliary “science” chamber as described in Ref. [27]. In
the science chamber, the condensate was loaded from theoptical tweezers into the interferometer’s single-well optical trap formed by a counter-propagating, orthogonallypolarized 1064 nm laser beam shifted in frequency fromthe tweezers by ∼ 100 MHz to avoid interference effects.
A schematic diagram of the setup for the interferometer’s optical trap is shown in Fig. 1(a). The optical potentials were derived from a collimated laser beam thatpassed through an acousto-optic modulator (AOM) and
was focused onto the condensate with a lens. The AOM
was driven by two radio frequency (rf) signals to create
the double-well potential. The separation between thepotential wells was controlled by the frequency difference
between the rf drives. The 1/e2 radius of each focused
beam was 5 µm. For typical optical powers, this resulted
in a single beam trap depth U0 = h × 5 kHz, where his Planck’s constant, and a radial (axial) trap frequency
fr = 615 Hz (fz = 30 Hz).
The condensate was initially loaded into the singlewell trap shown in Fig. 1(b). After holding the cloud inthis trap for 15 s to damp excitations, the peak atomic
mean field energy was µ̃0 ≈ h × 3 kHz. The single-welltrap was deformed into the double-well potential shownin Fig. 1(c) by linearly increasing the frequency difference
between the rf signals driving the AOM in 5 ms. The amplitude of the rf signals were tailored during the splitting

2
F

(a)

(a)

F

(b) Integrated Optical Density

AOM
d
g

Positionx

f1, f2
(b)

(c)
U0

x
d

d

FIG. 1: (a) Schematic diagram of the optical setup for the
double-well potential. An acousto-optic modulator (AOM)
was driven by two frequencies, f1 and f2 , and diffracted a collimated beam into two beams. The AOM was placed in the
focal plane of a lens of focal length F so that the two beams
propagated parallel to each other. The radial separation of
the potential wells, d, was controlled by the frequency difference, ∆f = |f1 − f2 |. ~g denotes the direction of gravitational
acceleration. The absorption image shows two well-separated
condensates confined in the double-well potential diagramedin (c). The field of view is 70 µm × 300 µm. Energy diagrams for (b) initial single-well trap with d = 6 µm and (c)final double-well trap with d = 13 µm. In both (b) and (c),U0 = h × 5 kHz and the peak atomic mean field energy was∼ h × 3 kHz. The potential “dimple” in (b) was < h × 500 Hz
which was much less than the peak atomic mean field energy
allowing the trap to be characterized as a single-well. The
potential “barrier” in (c) was h × 4.7 kHz which was larger
than the peak atomic mean field energy allowing the resulting
split condensates to be characterized as independent.

process to guarantee an even division of the condensate
atoms and nearly equal trap depths after splitting.
The key achievement of this work was the reproducibility of the spatial phase of the matter wave interference
pattern on each realization of the experiment. Figure 2
shows a typical matter wave interference pattern formedby the condensates released from the double-well potential. The reproducibility directly confirmed that deforming the optical potential from a single-well into a doublewell coherently split the condensate into two clouds withdeterministic relative phase. While past work sufferedfrom an unstable potential barrier separating the twocondensates and irreproducible turn off a high current
magnetic trap to initiate ballistic expansion [24], the current experiment derived its double-well potential from asingle laser beam. Thus, vibrations and fluctuations of
the laser beam were common-mode to both wells and aclean and rapid trap turn off was achieved.
The condensates were sufficiently separated that theirphases evolved independent of each other to the extent
that no coupling between the potential wells could be
detected. This claim is supported qualitatively by theabsorption image in Fig. 1(a) and the observation of high-

g

x

x

FIG. 2: Matter wave interference. (a) Absorption imageof condensates released from the double-well potential in
Fig. 1(c) and allowed to overlap during 30 ms of ballistic expansion. The imaging axis was parallel to the direction ofgravitational acceleration, ~g . The field of view is 600 µm ×
350 µm. (b) Radial density profiles were obtained by integrating the absorption signal between the dashed lines, andtypical images gave > 60% contrast. The solid line is a fit
to a sinusoidally-modulated Gaussian curve from which thephase of the interference pattern was extracted (see text).

contrast matter wave interference patterns that penetrated the full atomic density profile with uniform spatial
period and no thick central fringe [28], and quantitatively
by measurements of the phase evolution (Figs. 3 and 4).
The relative phase between the two separated condensates was determined by the spatial phase of their matterwave interference pattern. For a ballistic expansion timet  1/fr , each condensate
had a quadratic phase propm
~ 2 + φ± ),file [29], ψ± (~r, t) = n± (~r, t) exp(i 2h̄t|~r ± d/2|
where ± denotes one well or the other, n± is the condensate density, h̄ = h/2π, m is the atomic mass, and φ±is the condensate phase. This resulted in a total density
profile for the matter wave interference patternmd√
x + φr )),n(~r, t) = (n+ + n− + 2 n+ n− cos(
h̄t

(1)

where φr = φ+ − φ− is the relative phase between
the two condensates and d~ = dx̂. To extract φr ,the integrated cross section shown in Fig. 2(b) was fit
with a sinusoidally-modulated Gaussian curve, G(x) =A exp(−(x − xc )2 /σ 2 )(1 + B cos( 2πλ (x − x0 ) + φf )), where
φf is the phase of the interference pattern with respect
to a chosen fixed x0 . Ideally, if x0 was set at the centerof the two wells, then φr = φf . However, misalignmentof the imaging axis with the direction of gravitational acceleration created a constant offset, φf = φr + δφ. Witht = 30 ms the measured fringe period, λ = 41.5 µm,
was within 4% of the point source formula prediction[Eq. (1)], ht/md = 39.8 µm.

Spatial Phase (deg)

3

180
90
0
0

1

2
3
Hold Time (ms)45FIG. 3: Phase coherence of the separated condensates. The
spatial phase of the interference pattern is plotted versus holdtime after splitting. Each point represents the average of eightmeasurements, and the error bars are ± one standard deviation. The phase evolution was due to unequal trap depths forthe two wells, which was determined from the linear fit to be
h × 70 Hz or ∼ 1% of the trap depth.

The coherent phase evolution of the split condensates
is displayed quantitatively in Figs. 3 and 4. The relative phase, φr , between the separated condensates wasobserved to evolve linearly in time and the standard deviation of eight measurements of φf was < 90 degrees up
to 5 ms after splitting (Fig. 3). Furthermore, for holdtimes between 0 and 1 ms, the standard deviation was
substantially smaller, < 40 degrees. Since φr distributed
randomly between −180 and +180 degrees would have astandard deviation of ∼ 104 degrees, the results in Figs. 3
and 4 clearly demonstrate that the separated condensates
had a reproducible relative phase after splitting. The linear time evolution of φr was due to a chemical potential
difference between separated condensates and could becontrolled by varying the trap depths of the individual
potential wells after splitting.
Fundamental limits on the phase coherence between
isolated condensates arise due to Poissonian number fluctuations associated with the coherent state description of
the condensate [30, 31, 32]. For our experimental parameters, the time scale for phase diffusion was ∼ 200 ms.
The uncertainty in determining φf at longer hold times> 5 ms is attributed to axial and breathing-mode excitations created during the splitting process. These excitations lead to interference fringes that were angled andhad substantial curvature, rendering a determination of
φf impossible. Splitting the condensate more slowly in aneffort to minimize excitations, but still fast compared tothe phase diffusion time, did not improve the measured
stability of φf . Since controlling axial excitations appears critical for maintaining phase coherence, splitting
condensates that are freely propagating in a waveguide
potential may be more promising [3].
The phase sensitivity of the trapped-atom interferometer was demonstrated by applying ac Stark phase shifts
to either (or both) of the two separated condensates.
Phase shifts were applied to individual condensates bypulsing off the optical power generating the corresponding potential well for a duration τp  1/fr . Figure4(a) shows that the spatial phase of the matter wave
interference pattern shifted linearly with the pulse duration, as expected. Due to the inhomogeneous optical potential, U (r), the applied ac Stark phase shifts
varied across the condensate as ∆φ(r) = −U (r)τp /h̄.
Averaging this phase shift over the inhomogeneous condensate density, n(~r), approximates the expected spatial phase shift
R of the matter wave interference patternas ∆φ̄ = N1 d3~r n(~r)∆φ(~r) = (U0 − 27 µ̃0 )∆t/h̄, where
N is the number of atoms, and U0 and µ̃0 are the trap
depth and mean field energy at the center of each potential well, respectively. The measured phase shifts yielded
U0 = h × 5 kHz [Fig. 4(b)], which was consistent withcalculations based on other measured trap parameters.
The measured phase shifts of the matter wave interference depended only on the time-integral of the applied ac
Stark phase shifts [Fig. 4(b)], as expected for uncoupledcondensates. The final relative phase, φr , should be thesame on different phase trajectories because the historyof phase accumulation does not affect the total amount ofaccumulated phase. For coupled condensates, Josephson
oscillations between the wells would cause the relative
phase to vary nonlinearly with time [25, 29] and produce
a time dependent signal in Fig. 4(b). Due to the large
well separation and mean field energy h × 1.7 kHz below
the barrier height, the single-particle tunnelling rate in−3
our system was extremely low (<∼ 10 Hz) [29], and thecondensates were effectively uncoupled.
In conclusion, we have performed atom interferometry with Bose-Einstein condensates confined in an optical double-well potential. A coherent condensate beam
splitter was demonstrated by deforming a single-well potential into a double-well potential. The large spatial
separation between the potential wells guaranteed thateach condensate evolved independently and allowed for
addressing each condensate individually. Recombination
was performed by releasing the atoms from the doublewell potential and allowing them to overlap while expanding ballistically. Implementing a similar readout schemewith magnetic potentials generated by microfabricatedcurrent carrying wires should be possible and would eliminate deleterious mean field effects inherent in proposals using in-trap wavepacket recombination. Propagating the separated condensates along a waveguide prior tophase readout would create an atom interferometer withan enclosed area, and hence with rotation sensitivity.
We thank W. Jhe, C. V. Nielsen, and A. Schirotzek
for experimental assistance and S. Gupta, Z. Hadzibabic, and M. W. Zwierlein for critical comments on
the manuscript. This work was funded by ARO, NSF,
ONR, and NASA. M.S. acknowledges additional support
from the Swiss National Science Foundation.

4

Spatial Phase (deg)(a)

splitting

540
360

150 µsfree expansion

τp

180
0
-180
-360
-540
0

(b)

20

40
τp (µs)60

80

Spatial Phase (deg)

120
60
0
splitting

1 ms

free expansion

-60
τd

50 µs

-120
0

200

400
τd (µs)

600

800

FIG. 4: Trapped-atom interferometry. (a) ac Stark phaseshifts were applied to either well exclusively (solid circles andopen circles) or both wells simultaneously (crosses) by turning off the corresponding rf signal(s) driving the AOM for aduration τp . The resulting spatial phase of the matter wave
interference pattern scaled linearly with τp and hence the applied phase shift. Applying the ac Stark shift to the opposite
well (solid versus open circles) resulted in an interference pattern phase shift with opposite sign. Applying ac Stark shifts
to both wells (crosses) resulted in no phase shift for the interference pattern. This data was taken with a slightly modified
experimental setup such that the trap depth of the individual potential wells was U0 = h × 17 kHz. (b) A 50 µs pulse
induced a 70 degree shift independent of the pulse position.
The experimental setup was as described in Fig. 1. Solid andopen circles have the same meaning as in (a). The insets
show the time sequence of the optical intensity for the well(s)temporarily turned off.

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[23] S. Gupta, K. Dieckmann, Z. Hadzibabic, and D. E.
Pritchard, Phys. Rev. Lett. 89, 140401 (2002).
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Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637(1997).
[25] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy,
Phys. Rev. Lett. 79, 4950 (1997).
[26] J. A. Stickney and A. A. Zozulya, Phys. Rev. A 66,
053601 (2002).
[27] T. L. Gustavson, A. P. Chikkatur, A. E. Leanhardt,
A. Görlitz, S. Gupta, D. E. Pritchard, and W. Ketterle,
Phys. Rev. Lett. 88, 020401 (2002).
[28] A. Röhrl, M. Naraschewski, A. Schenzle, and H. Wallis,
Phys. Rev. Lett. 78, 4143 (1997).
[29] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. 71, 463 (1999).
[30] M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489
(1996).
[31] J. Javanainen and M. Wilkens, Phys. Rev. Lett. 78, 4675
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[32] C. Menotti, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys.
Rev. A 63, 023601 (2001).

Appendix D
Imprinting Vortices in a
Bose-Einstein Condensate using
Topological Phases
This appendix contains a reprint of Ref. [123]: A.E. Leanhardt, A. Görlitz, A.P.
Chikkatur, D. Kielpinski, Y. Shin, D.E. Pritchard, and W. Ketterle, Imprinting Vortices in a Bose-Einstein Condensate using Topological Phases, Phys. Rev. Lett. 89,
190403 (2002). Axel Görlitz and I worked on this experiment during April and May,
2001 with no success. At that time we attempted to “thread” the magnetic fieldzero of a three-dimensional (spherical) quadrupole field through an optically confinedcondensate. The quadrupole field had a 10 − 20 G/cm gradient and was normallyused for the MOT. The optical trap had a disc shape and had just been used to create
two-dimensional condensates [67]. We were very concerned about spin-flip transitions
as the quadrupole zero passed through the condensate so we pierced the center of thecloud with a blue-detuned laser beam to create a toroidal shaped condensate. Ultimately the experiment sat dormant for nearly a year before it was revitalized late onenight in March, 2002. The results of that revitalization are contained in this Letter.

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Imprinting Vortices in a Bose-Einstein Condensate using Topological Phases
A. E. Leanhardt, A. Görlitz,* A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard, and W. Ketterle†
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139(Received 16 June 2002; published 22 October 2002)
Vortices were imprinted in a Bose-Einstein condensate using topological phases. Sodium condensates held in a Ioffe-Pritchard magnetic trap were transformed from a nonrotating state to one withquantized circulation by adiabatically inverting the magnetic bias field along the trap axis. Using
surface wave spectroscopy, the axial angular momentum per particle of the vortex states was found to
be consistent with 2h or 4h,
 depending on the hyperfine state of the condensate.
DOI: 10.1103/PhysRevLett.89.190403

PACS numbers: 03.75.Fi, 03.65.Vf, 67.40.Db, 67.40.Vs

As superfluids, Bose-Einstein condensates support rotational flow only through quantized vortices. The atomic
velocity field is proportional to the gradient of the phaseassociated with the macroscopic wave function. This
phase winds through an integer multiple of 2 radiansaround a vortex line. Such a phase winding can be imprinted onto the condensate wave function either dynamically or topologically. Dynamically, the phase of thecondensate evolves according to the time integral of its
energy, which can be tailored locally with a spatiallyvarying external potential. Topologically, the phase of
the condensate advances through adiabatic variations in
the parameters of the Hamiltonian governing the system.
This phase, which is solely a function of the path traversed by the system in the parameter space of the
Hamiltonian, is known as Berry’s phase [1].
In this Letter, we implement the proposal of Refs. [2 –5]
and demonstrate the use of topological phases to imprint
vortices in a gaseous Bose-Einstein condensate. Previously, vortices have been generated in two-component
condensates using a dynamical phase-imprinting technique [6] and in single-component condensates by rotating the cloud with an anisotropic potential [7–10], byslicing through the cloud with a perturbation above thecritical velocity of the condensate [8,11], and through the
decay of solitons [12,13]. In this work, 23 Na condensates
were prepared in either the lower, jF; mF i  j1; 1i, or
upper, j2; 2i, hyperfine state and confined in a IoffePritchard magnetic trap. Vortices were created by adiabatically inverting the magnetic bias field along the trap
axis and could be removed by returning the bias field to
its original direction. Using surface wave spectroscopy[14 –16], we measured the axial angular momentum perparticle of the j1; 1i and j2; 2i vortex states to beconsistent with 2mF h as predicted [2 –5].
A Ioffe-Pritchard magnetic trap consists of an axial
bias field (with curvature) and a two-dimensional quadrupole field in the orthogonal plane [17,18]:

dratic terms have been neglected. For a condensate ofradial extent R, inverting Bz from Bz
B0 R > 0 to Bz
0B R < 0 rotates the atomic angular momentum, F~ , by 
radians. While all atomic angular momenta rotate
through the same angle, a relative phase is establishedacross the condensate because the angular momenta rotate about a unit vector n^   sinx^  cosy^ that depends on the azimuthal angle, , describing the atomic
position [Fig. 1(a)].
As Bz is inverted, F~ adiabatically follows B~ x; y; z,and the condensate remains in the state jF; mF i withrespect to the local magnetic field. However, in a basis

(a)

(b)

y

z'
θ'
F

φ

1

x

2
y'

1
2
b(φ)
n(φ)

x' φ'

where B0 is the radial magnetic field gradient and qua-

FIG. 1. Geometry of the rotating magnetic field for imprinting topological phases. (a) The unit vectors b^  point in the
direction of the two-dimensional quadrupole field providing
the radial confinement of a Ioffe-Pritchard magnetic trap. The
atomic angular momenta rotate about the unit vectors n^  as
the axial bias field, Bz , is ramped from positive to negativevalues. (b) For an atom in state jF; mF i, its atomic angularmomentum, F~ , traverses a path on a sphere of radius jmF jh as it
adiabatically follows its local magnetic field. The primed
coordinate system is centered on the atomic position and hasaxes parallel to those of the unprimed coordinate system in (a).
For an atomic position described by the azimuthal angle , F~
rotates in a half-plane defined by 0   for mF > 0 and0     for mF < 0 as Bz is inverted. After inverting Bz ,the relative topological phase acquired between atoms locatedat positions 1 and 2 in (a) is proportional to the solid angle
subtended by the shaded surface, bounded by the contour
marked with arrowheads (see text).

190403-1

 2002 The American Physical Society

B~ x; y; z  Bz z^  B0 xx^  yy^ ;(1)0031-9007=02=89(19)=190403(4)$20.00

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PHYSICA L R EVIEW LET T ERS

fixed in the lab frame, the condensate makes the transition jF; mz  mF i ! jF; mz  mF i, where mF and mz
are the projection of F~ along the local magnetic field
direction and z axis, respectively. Applying the quantum
mechanical rotation operator gives the condensate wave
function in the lab frame after inverting Bz asq
~  n^ j i  eiF =h(2)
~r jF; mz  mF i;q
 1FmF ~r ei2mF  jF; mz  mF i; (3)where F~ is the angular momentum operator such thatF~  hF~ i and ~r  is the number density of condensed
atoms. The topological phase factor ei2mF  describes a
vortex of winding number 2jmF j with the sense of rotation dependent on the sign of mF .
This result can be interpreted in terms of Berry’s phase[4]. Figure 1(b) shows the orientation of F~ in the lab frame
for mF > 0. Atoms located at position k  1; 2 in Fig. 1(a)have azimuthal angle k and angular momentum F~ k . As
Bz is inverted, F~ k traces path k from top to bottom on thesphere in Fig. 1(b). The topological phase acquired by anatom in this process is solely a function of the pathtraversed by its angular momentum vector. Since thispath depends on the azimuthal angle describing theatomic position, a relative phase is established between
spatially separated atoms. The condensate wave function
after inverting Bz is given byq
j i  ~r ei jF; mF i;(4)where  is the topological phase acquired by atoms
with azimuthal angle .
For an atom in state jF; mF i, Berry’s phase, C,acquired as its angular momentum vector traverses a
closed contour, C, on the surface of the sphere in
Fig. 1(b) is given by [1]C  mF C;(5)

where C is the solid angle subtended by a surface
bounded by the contour C. Calculating the relative phase,1   2 , with the aid of Eq. (5) requires closing
the contours traced by each F~ k along an identical path.
For clarity, we choose to close each contour along path 2
itself, and hence 1   2   C, where C is thecontour formed by path 1 traversed from top to bottom
and path 2 traversed from bottom to top, as indicated witharrowheads in Fig. 1(b).
A surface bounded by this contour subtends a solid
angle C  21  2 , yielding a relative phase1   2   2mF 1  2 . Thus, we set  2mF ;(6)

up to an additive term independent of position. This yields
a reinterpretation of Eq. (3) in terms of Berry’s phase.
In this work, Bose-Einstein condensates containing
over 107 23 Na atoms were created in the j1; 1i state in
190403-2

4 NOVEMBER 2002

a magnetic trap, captured in the focus of an opticaltweezers laser beam, and transferred into an auxiliary‘‘science’’ chamber as described in Ref. [19]. While optically confined by the tweezers, j2; 2i condensates weregenerated by sweeping through the j1; 1i $ j1; 0i $j1; 1i radio-frequency transition with 100% efficiency,
then sweeping through the j1; 1i $ j2; 2i microwave
transition with 80% efficiency [20]. In the science chamber, the condensate was loaded into a microfabricated
Ioffe-Pritchard magnetic trap formed by a Z-shaped
wire carrying current I and an external magnetic bias
field, B? , as detailed in Ref. [21]. Condensates weredetected via axial absorption imaging whereby resonant
laser light propagating along the z axis illuminated theatoms and was imaged onto a CCD camera.
Typical wire-trap parameters were I  1200 mA,
B?  5:4 G, and Bz  1 G, resulting in a radial magnetic
field gradient of B0  120 G=cm. For j1; 1i (j2; 2i)
condensates, the axial and radial trap frequencies were!z  2  6:0 Hz (!z  2  8:5 Hz) and !?  2 
210 Hz (!?  2  300 Hz), respectively. After transfer into the wire trap, condensates in the j1; 1i (j2; 2i)
state contained over 2  106 atoms (1  106 atoms) andhad a lifetime in excess of 10 s (3 s) with an applied radiofrequency shield. This represents the first magnetic trapping of 23 Na condensates in the upper hyperfine level,with previous work done exclusively in optical dipole
traps [20].
Along the wire-trap axis, the magnetic field is
B~ x  0; y  0; z  Bz  12B00 z2 ^z ;(7)where quadratic terms neglected in Eq. (1) have been
included. The axial magnetic field curvature, B00 , whicharises from the geometry of the Z wire, was held constantthroughout the experiment. By reversing an external axial
magnetic field, we inverted Bz . Changing the sign of Bz ,but not B00 , resulted in a magnetic field saddle point at thecenter of the cloud and axial antitrapping of weak-fieldseeking atoms. This limited the condensate lifetime after
inverting Bz to & 50 ms.
Vortices created by inverting Bz were observed after
ballistic expansion and were identified by central density
depletions due to the angular momentum barrier associated with a rotating cloud (Fig. 2). These vortices could be
removed by returning Bz to its original direction.
For j1; 1i condensates, the best results were achievedby inverting the axial bias field linearly from Bz 
860 mG to 630 mG in 11 ms. For j2; 2i condensates,
the optimum ramp time over the same range was 4 ms.
The field inversion process caused an atom loss of  50%
due to nonadiabatic spin flips as Bz passed through zero[4,5]. The density depletions shown in Figs. 2(b), 2(c),and 2(g) were observed after inverting the axial bias field
and holding the trapped condensate for longer than aradial trap period. Thus, the atom loss from the centerof the cloud during the field inversion process could not be
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PHYSICA L R EVIEW LET T ERS

FIG. 2. Observation of vortices formed by imprinting topological phases. Axial absorption images of condensates in thej1; 1i state after 18 ms of ballistic expansion (a) prior to
inverting Bz , after inverting Bz , and holding the trapped condensate for (b) 5 ms and (c) 20 ms, and (d) after inverting Bz
and then returning it to its original direction. Axial absorption
images of condensates in the j2; 2i state after 7 ms of ballistic
expansion (e) prior to inverting Bz , after inverting Bz , andholding the trapped condensate for (f) 0 ms and (g) 5 ms, and(h) after inverting Bz and then returning it to its original
direction. The field of view is (a) –(d) 570 m  570 m and(e) –(h) 285 m  285 m.

responsible for the observed density depletions. Typical j1; 1i (j2; 2i) condensates after inverting theaxial bias field contained up to 1  106 atoms (0:5 
106 atoms) with a Thomas-Fermi radius RTF 
5:4  0:2 m (RTF  4:5  0:2 m) in a trap with radialfrequency !?  2  250 Hz (!?  2  350 Hz).
The axial angular momentum per particle of the vortex
states was measured using surface wave spectroscopy[14 –16]. A superposition of counterrotating (m‘  2)
quadrupolar (‘  2) surface waves was excited in thecondensate by radially displacing the magnetic trap center for 200 s. Here ‘ and m‘ characterize the angularmomentum and its projection along the z axis of thequadrupole modes, respectively. This created an ellipticalcondensate cross section with time-dependent eccentricity. In the absence of a vortex, the m‘  2 quadrupolemodes are degenerate and the axes of the elliptical condensate cross section remain fixed in time. This degeneracy is lifted by the presence of a vortex, causing the axes
to precess in the direction of the fluid flow. The precessionrate, _ , is given by [14 –16]
hLz i
_ 
;
(8)
2Mhr2? iwhere hLz i is the axial angular momentum per particle
characterizing the vortex state, M is the atomic mass, andhr2? i  hx2  y2 i is the mean-squared trapped condensate
radius with vortices present.
By measuring the precession rate of the quadrupoleaxes and the mean-squared radius of the condensate in
the trap, the axial angular momentum per particle wasdetermined. After exciting the quadrupolar modes, thecondensate evolved in the trap for variable times in therange 0:2–7:4 ms. The condensate was then released fromthe trap and imaged with resonant light after ballistic
190403-3

4 NOVEMBER 2002

expansion as shown in Figs. 3(a) –3(l). The resulting
images were fit to an elliptical Thomas-Fermi profile todetermine the orientation of the quadrupole axes. The
orientation angle is plotted as a function of time in
Fig. 3(m). To determine the mean-squared trapped condensate radius, vortices were imprinted in the condensate
but quadrupolar modes were not excited. Images of ballistically expanded condensates similar to those in
Figs. 2(b), 2(c), 2(f), and 2(g) were fit to a ThomasFermi profile with a circular cross section. The fitting
routine ignored the central region of the cloud where
the density was depleted due to the vortex core. The
mean-squared trapped condensate radius was derived
through the relation
hr2? i 

R2?
2;
7 1  !2? !2(9)

where R? is the Thomas-Fermi radius of the condensate
after ballistically expanding for a time ! from a trap withradial frequency !? . The factor 1  !2? !2 accounts forthe change in Thomas-Fermi radius during the expansion
process [22], and the factor 2=7 results from averaging
over the inhomogeneous condensate density distribution
assuming no vortices are present. For low angular momentum vortex states, the density depletion at the vortex
core does not significantly modify the density distribution of the condensate, and we expect the 2=7 factor tostill be accurate [23].
For j1; 1i condensates, the quadrupole oscillation was
excited after a delay of 0, 5, and 20 ms from the completion of the inversion of the axial bias field. The
measured axial angular momenta per particle were1:9  0:3h, 2:1  0:3h, and 1:9  0:2h, respectively. The uncertainty in the measurement arises
from the linear fit to the precession angle and the determination of hr2? i. For j2; 2i condensates, the quadrupole
oscillation was excited immediately upon the completionof the inversion of the axial bias field. The measured axial
angular momentum per particle was 4:4  0:4h.
 Forboth internal states, the measurements are consistent withthe predicted axial angular momentum per particle of2mF h [2 –5].
Multiply charged vortices are unstable against decay
into singly charged vortices [24]. From our experiments,
we cannot determine if the condensate contained one
multiply charged vortex or multiple, singly charged vortices. If multiple vortices were present, they must be
closely spaced since they were not resolved after ballistic
expansion. Furthermore, if the singly charged vortices
had moved apart considerably, it would have lowered theextracted value of hLz i [15], which was not observed evenwith delayed probing.
In conclusion, we have used topological phases toimprint vortices in a Bose-Einstein condensate. Higher
angular momentum states can be generated by using
higher-order, axisymmetric multipole magnetic fields.
In general, this phase-imprinting technique opens the190403-3

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PHYSICA L R EVIEW LET T ERS

FIG. 3. Surface wave spectroscopy. Axial absorption images
after 18 ms of ballistic expansion of j1; 1i condensates undergoing a quadrupole oscillation (a) –(d) in the presence of avortex and (e) –(h) in the absence of a vortex. Successive images
were taken during successive half periods of the quadrupole
oscillation such that the short and long axes of the ellipticalcross section were exchanged. Images (a) –(d) show counterclockwise (positive) precession of the quadrupole axes, while
images (e) –(h) show no precession. (i) –(l) Axial absorption
images after 7 ms of ballistic expansion of j2; 2i condensates
undergoing a quadrupole oscillation in the presence of a vortex.
The images were taken during a single half period of thequadrupole oscillation. Images (i) –(l) show clockwise (negative) precession of the quadrupole axes. The field of view is(a) –(h) 570 m  570 m and (i) –(l) 285 m  285 m.
(m) Precession angle vs time in the presence of a vortex forj1; 1i condensates measured after a delay of 0 ms (opencircles), 5 ms (open squares), and 20 ms (open triangles)
from the completion of the inversion of the axial bias field,in the absence of a vortex for j1; 1i (open diamonds) andj2; 2i (filled diamonds) condensates, and in the presence of avortex for j2; 2i condensates measured immediately upon thecompletion of the inversion of the axial bias field (filled circles).

potential for studying the stability of multiply charged
vortices and the dynamics of vortex-vortex interactionsat short separations.
We thank J. R. Abo-Shaeer for a critical reading of themanuscript. This work was funded by ONR, NSF, ARO,
NASA, and the David and Lucile Packard Foundation.
A. E. L. acknowledges additional support from NSF.

190403-4

4 NOVEMBER 2002

*5th Physikalisches Institut, University of Stuttgart,
70550 Stuttgart, Germany.
†
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Phys. Rev. Lett. 88, 020401 (2002).
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Vol. II.

190403-4

Appendix E
Coreless Vortex Formation in a
Spinor Bose-Einstein Condensate
This appendix contains a reprint of Ref. [126]: A.E. Leanhardt, Y. Shin, D. Kielpinski, D.E. Pritchard, and W. Ketterle, Coreless Vortex Formation in a Spinor BoseEinstein Condensate, Phys. Rev. Lett. 90, 140403 (2003). I am grateful to K. Machidafor bringing the spin textured examined in this Letter to my attention and for his
hospitality during my subsequent visit to Okayama.

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week ending
11 APRIL 2003

Coreless Vortex Formation in a Spinor Bose-Einstein Condensate
A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle*
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139(Received 20 December 2002; published 9 April 2003)
Coreless vortices were phase imprinted in a spinor Bose-Einstein condensate. The three-component
order parameter of F  1 sodium condensates held in a Ioffe-Pritchard magnetic trap was manipulated
by adiabatically reducing the magnetic bias field along the trap axis to zero. This distributed thecondensate population across its three spin states and created a spin texture. Each spin state acquired adifferent phase winding which caused the spin components to separate radially.
DOI: 10.1103/PhysRevLett.90.140403

Spin textures play a central role in describing thephysics of elementary particles [1], liquid 3 He-A [2– 4],
the quantum Hall effect [5], and gaseous Bose-Einsteincondensates [6 –10]. Topological defects vary between
superfluid systems described by scalar and vector orderparameters. In spinless or spin-polarized condensates,
line defects such as vortices have cores where the densityof condensed particles is necessarily zero to keep theorder parameter single valued [11–13]. However, in condensates with an internal, spin degree of freedom, coreless vortices exist as spin textures [4,14]. Such structures
are referred to as skyrmions (Anderson-Toulouse vortices[3]) or merons (half-skyrmions, Mermin-Ho vortices [2])depending on the boundary conditions of the system.
In this Letter, we study spin textures in a Bose-Einsteincondensate. Coreless vortices were created in F  1spinor condensates held in a Ioffe-Pritchard magnetic
trap by adiabatically reducing the magnetic bias fieldalong the trap axis to zero. This continuously transformed
the initially spin-polarized condensate into a coherent
superposition of three spin states, each with a different
phase winding. The resulting angular momentum perparticle varied between spin states and the condensate
evolved such that states with more angular momentum
per particle circulated around states with less angularmomentum per particle. Thus, the condensate had a net
axial magnetization that varied with radial position.
Previous work on vortices in a two-component systemused laser, microwave, and radio frequency fields to spatially and temporally control the interconversion between
components [14]. However, without these applied fields
the two components evolved independently as distinguishable fluids. In our work, the spin states can freely
interconvert at all points in space and time such that the
spin texture would continually heal itself even in thepresence of state-dependent losses.
In cylindrical coordinates, the spin-F condensate
wave
functioncan be written as jr; ; zi p
nr; ; zjr; ; zi, where n is the atomic number density
PF and the 2F  1 2 component spinor ji mz F mz jF; mz i; jhjij  1 describes a spin texture.
140403-1

0031-9007=03=90(14)=140403(4)$20.00

PACS numbers: 03.75.Lm, 03.65.Vf, 03.75.Mn, 67.57.Fg

A Ioffe-Pritchard magnetic trap consists of an axial
bias field (with curvature) and a two-dimensional quadrupole field in the orthogonal plane [15,16]:^ ;
B~ r; ; z  Bz z^  B0 r cos2^r  sin2

(1)

where B0 is the radial magnetic field gradient and quadratic terms have been neglected. For a condensate ofradial extent R confined in a Ioffe-Pritchard magnetic
trap with Bz
B0 R > 0, ji  jF; mz  mF i, where mz
and mF are the projection of the atomic spin along the zaxis and local magnetic field direction, respectively.
Adiabatically ramping Bz from Bz
B0 R > 0 to zerorotates the atomic spin about the position-dependent
axis n^   sincosy^ , and drives the transitionPx^ FjjF; mz  mF i ! jm exp imz  mF  jF; mz i
mz jmF j mz
[17,18]. Thus, the condensate population is distributed
across 2jmF j  1 spin states with each acquiring a different topological phase factor and angular momentum perparticle due to the variation of Berry’s phase with magnetic quantum number [19].
The condensate remains in the state jF; mF i with respect to the local magnetic field provided the local
Zeeman energy, gF B B2z  B0 r2 1=2 , dominates the
local kinetic energy associated with the spin texture,h 2 =mr2 , where gF is the Landé g factor, B is theBohr magneton, and m is the atomic mass. For Bz  0,
atomic spins aligned with the quadrupole magnetic field
produce the planar spin texture in Fig. 1(a). However, theinfinite kinetic energy associated with the wave function
singularity at r  0 creates a nonplanar spin texture overa disk of radius h 2 =mgF B B0 1=3 , with higher angularmomentum spin states residing outside those with lower
angular momentum.
Bose-Einstein condensates containing over 107 23 Na
atoms were created in the j1; 1i state in a magnetic trap, captured in the focus of an optical tweezers
laser beam, and transferred into an auxiliary ‘‘science’’chamber as described in Ref. [21]. In the science chamber, the condensate was loaded into a microfabricated
Ioffe-Pritchard magnetic trap formed by a Z-shaped 2003 The American Physical Society

140403-1

(a)

y

(b)

y

x

x

FIG. 1. Planar spin textures. Spins aligned with twodimensional (a) quadrupole and (b) hexapole magnetic fields
produce textures with winding numbers 1 and 2, respectively. Counterclockwise traversal of the dashed contours in (a)and (b) leads to clockwise (negative) spin rotation, with thewinding number defined as the integer number of revolutions
made by the spin vector while circumnavigating the singularity
at the origin [20].

50 m  10 m electroplated copper wire carrying current I and an external magnetic bias field, B? , as described in Ref. [22]. Typical wiretrap parameters wereI  720 mA, B?  5:3 G, and Bz  1:3 G, resulting in aradial magnetic field gradient B0  180 G=cm. This produced axial and radial trap frequencies !z  2  4 Hz
and !?  2  250 Hz, respectively. Condensates in thewiretrap had  2  106 atoms, a Thomas-Fermi radiusof 5 m, and a lifetime  25 s.
Coreless vortices imprinted onto the condensate wave
function by adiabatically ramping Bz ! 0 are shown in
Fig. 2. To observe the nature of the spin texture, an axial
bias field was switched on nonadiabatically along either
the negative [Figs. 2(b) –2(d)] or positive [Figs. 2(e) –2(h)] z axis. Switching the axial bias field on suddenly‘‘froze’’ the atomic spins and effectively ‘‘projected’’ thecondensate wave function onto a basis quantized withrespect to the local (axial) magnetic field. This allowed
the spin states to be separated by a magnetic field gradientapplied during ballistic expansion. Switching the direction of the axial projection field exchanged the roles of thej1; 1i and j1; 1i states. Figures 2(a) and 2(e) show thecoreless nature of the vortices, while Figs. 2(b) and 2(f)show the concentric cylinder structure resulting from thecompetition between the atomic Zeeman energy and the
kinetic energy of the rotating spin states. We assume thatthe two-dimensional (!z  !? ) ballistic expansion
process simply magnifies the condensate wave function,
as it does in the expansion of a single-component condensate with vortices [23,24].
Along the wiretrap axis, the magnetic field was
B~ r  0; ; z  Bz  12B00 z2 ^z ;(2)where quadratic terms neglected in Eq. (1) are included.
Nonzero axial magnetic field curvature, B00  5 G=cm2 ,implies that the spin texture had a slight axial depen140403-2

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PHYSICA L R EVIEW LET T ERSVOLUME 90, N UMBER 14

dence. The axial bias field was ramped linearly fromBz  1:3 G to Bz  0 in 10 ms to imprint the corelessvortices. This compressed the condensate radially andincreased the condensate chemical potential from  B =2  3 mG to   B =2  27 mG. Ramping
Bz ! 0 fast compared to the axial trap period (250 ms),
but slow compared to the initial radial trap period (4 ms),
guaranteed that the axial magnetic field variation remained constant throughout the experiment at Bz 
3 mG, while at Bz  0 the radial magnetic field variationwas Br  27 mG. The images shown in Fig. 2 integrated the atomic number density along the z axis andtherefore averaged over the minor (Bz  Br ) axial
variation to the spin texture.
To project the condensate wave function onto a basis
quantized with respect to the local magnetic field, an10 G axial bias field was switched on at a rate of B_ z 2  105 G=s along either the negative or the positive z
axis. 100 s later the magnetic trap was switched off
allowing the atoms to expand ballistically. For 0  Br Br , the Landau-Zener nonadiabatic transition probability, expB B2r =h B_ z   0:9, was sufficiently close
to unity that the atomic spins remained ‘‘frozen’’ during
the sudden application of the axial bias field and the spin
texture could be accurately diagnosed. While the total
condensate density monotonically decreased as a function of radial position [Figs. 2(a) and 2(e) ], the densityof each spin component peaked at a different radiussignifying a variation in the angular momentum perparticle between spin states [Figs. 2(b) and 2(f)].
Applying the projection field along the positive z axis
generated additional rings of atoms in the j1; 1i andj1; 0i states [Fig. 2(f)]. Ramping Bz ! 0 in 10 ms caused
nonadiabatic spin flips for atoms near r  0 resulting in
an atom loss of  50% [18]. If these atoms had not left thecondensate before the projection field was applied, they
may have contributed to the images displayed in Fig. 2(f).
The additional rings of atoms may also correspond to a
low energy, radial spin-wave excitation [6,7]. However,
we could not identify any asymmetry between applying
the projection field along the positive versus negative zaxis that would account for the presence of the extra rings
in Fig. 2(f), but not in Fig. 2(b).
Engineering topological states in a Bose-Einsteincondensate has received much theoretical attention[17,25–30]. The evolution of a condensate confined in a
Ioffe-Pritchard magnetic trap while ramping Bz ! 0 is
described by a position-dependent spin rotation about then^  axis through an angle r [18]~

 n^ rjr; ; zi  eiF =hj0 i;(3)

where F~ is the spin operator and j0 i  jF; mz  mF i is apolarized spinor. For j0 i  j1; 1i, Eq. (3) gives thecondensate spinor in the laboratory frame as
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FIG. 2. Coreless vortex formation in a spinor Bose-Einstein condensate. Coreless vortices were imprinted by ramping Bz ! 0 anddiagnosed by suddenly switching (a) –(d) Bz  0 and (e) –(h) Bz
0. Axial absorption images display the optical density ofcondensates after 20 ms of ballistic expansion (a),(e) without and (b),(f) with a magnetic field gradient applied to separate thedifferent spin states. (c),(g) Azimuthally averaged optical density vs radial position for spin components shown in (b) and (f),respectively. The radial separation of the spin states resulted from their relative phase windings and is a clear signature of the~ mF 1 =~n mF 1  nn total , vsskyrmion/meron wave function [Eq. (4)]. (d),(h) Axial magnetization per particle, M=N  B =2  ~radial position. The absorption imaging light was resonant with the F  2 ! F0  3 transition. The atoms were optically pumped
into the F  2 hyperfine level with a pulse resonant with the F  1 ! F0  2 transition. This provided equal imaging sensitivity toeach spin state. The field of view in (a), (b), (e), and (f) is 1:0 mm  3:0 mm.

jr; ; zi  cos2 r=2 j1; 1i
1 p sin r ei j1; 0i
2
 sin2 r=2 ei2 j1; 1i:(4)0  0 and skyrmions (merons) are described by theboundary condition 1   [1  =2].
The radial dependence of r is determined by requiring the Gross-Pitaevskii energy functional,
E

Z


 
h 2
c0 c2 jF~ j2
d rn
hrjri  V 

n ;
2m
2
2 h 2(5)
3~

be stationary with respect to variations in . Equation (5)uses the Thomas-Fermi approximation, and hrjri PFyV  gF B F~  B~ =h,F~ mz F rmz  rmz ,hjF~ ji, c0  4h 2 a =m, and c2  4h 2 a=m. Herea  2a0  a2 =3 and a  a2  a0 =3 characterize
two-body interactions, where a0 and a2 are scattering lengths for collisions with total angular momentum
F  0 and F  2, respectively [6]. For c2 > 0, the atomic
interactions are antiferromagnetic (polar), as in 23 Na [31],while for c2 < 0 they are ferromagnetic, as in 87 Rb [32].
140403-3

Using Eq. (4) as a trial spinor, we find that r satisfies
the meron boundary conditions and varies from 0  0,
due to the kinetic energy of the spin texture, to 1 
=2, due to the atomic Zeeman energy, over a characteristic length scale given by the larger of h 2 =mgF B B0 1=3
and jBz j=B0 . We observed that the boundary condition
0  0 was maintained regardless of the sign of Bz ,i.e., atomic spins along the trap axis always remained in
the initial state. If the atomic spins had simply followed
the local magnetic field, r would satisfy tanr 
B0 r=Bz . Thus, scanning Bz from slightly positive toslightly negative would instantaneously change theboundary condition at the origin from 0  0 to0   and flip the atomic spins along the trap axis.
The trial spinor in Eq. (4) does not have the mostgeneral form since it was derived by rotating a polarized
spinor [Eq. (3)] and is inherently ferromagnetic, jF~ j  h.

Accordingly, the spin-dependent interaction term in
Eq. (5) does not vary with  and therefore does notcontribute to the determination of r. This restriction
is not severe since the Zeeman energy dominates the spindependent interaction energy in our experiment.
In future experiments, it should be possible to overlap the condensate with an optical dipole trap so that,after ramping Bz ! 0, relaxing the radial magnetic field
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PHYSICA L R EVIEW LET T ERS

gradient would allow for the spin-dependent interactions
to determine the evolution of the condensate. It is predicted that skyrmions/merons in condensates with antiferromagnetic (ferromagnetic) interactions are unstable(stable) [9,10,33–39]. Presently, excitations created during the field ramping process have prevented a study ofthe stability of coreless vortices in an antiferromagnetic
23
Na condensate. However, we were able to imprint the
spin texture in the presence of an optical dipole trap, as
well as produce vortices with a 4 phase winding in an
optical dipole trap by fully inverting the axial bias field[18]. At zero magnetic field, it may be possible to observe
multiply charged vortices in a spinor condensate ‘‘unwind’’ themselves as predicted in Ref. [6]. The unwinding
process is precisely the position-dependent spin rotation
demonstrated in previous work [18], with the spin texture
investigated here as an intermediate state [37].
In conclusion, we have demonstrated a robust technique for creating coreless vortices in a Bose-Einsteincondensate. Our technique can be extended to generate
spin textures with arbitrary winding number and variable
angular momentum per particle by using higher-order,
axisymmetric multipole magnetic fields and condensateswith different spin. This work opens up the opportunity to
study the stability of topological defects in spinor BoseEinstein condensates.
We are grateful to K. Machida for bringing the spin
texture studied here to our attention. We thank T. Pasquini for experimental assistance and J. R. Anglin and
M. Crescimanno for valuable discussions. This work was
funded by ONR, NSF, ARO, NASA, and the David andLucile Packard Foundation.

*Electronic address: http://cua.mit.edu/ketterle_group/[1] T. H. R. Skyrme, Proc. R. Soc. London A 260, 127 (1961).
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Rev. Lett. 43, 214 (1979).
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Kielpinski, Y. Shin, D. E. Pritchard, and W. Ketterle,
Phys. Rev. Lett. 89, 190403 (2002).
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Phys. Rev. Lett. 88, 020401 (2002).
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T. L. Gustavson, W. Ketterle, and D. E. Pritchard, Phys.
Rev. Lett. 89, 040401 (2002).
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Rev. A 62, 013602 (2000).
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63, 063603 (2001).
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Appendix F
Cooling Bose-Einstein Condensates
Below 500 PicokelvinThis appendix contains the galley proofs for Ref. [124]: A.E. Leanhardt, T.A. Pasquini,
M. Saba, A. Schirotzek, Y. Shin, D. Kielpinski, D.E. Pritchard, and W. Ketterle,
Cooling Bose-Einstein Condensates Below 500 Picokelvin, Science in press, (2003).
I was allowed to append these proofs with the following disclaimer: Confidential /Embargoed: Not for Release Until 2 pm ET Thursday, Sept. 11, 2003 / Not for
Re-distribution.

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NO. RE1088827/CJH/PHYSICS
rich4/se-sci/se-sci/se-orig/se1850d03r byrds S⫽8 8/25/03 22:56 Art: Fig. 1 - live B&W

REPORTS
Cooling Bose-Einstein
Condensates Below 500 Picokelvin
A. E. Leanhardt,* T. A. Pasquini, M. Saba, A. Schirotzek, Y. Shin,
D. Kielpinski, D. E. Pritchard, W. Ketterle
Spin-polarized gaseous Bose-Einstein condensates were confined by a combination of gravitational and magnetic forces. The partially condensed atomic
vapors were adiabatically decompressed by weakening the gravito-magnetic
trap to a mean frequency of 1 Hz, then evaporatively reduced in size to 2500
atoms. This lowered the peak condensate density to 5 ⫻ 1010 atoms/cm3 andcooled the entire cloud in all three dimensions to a kinetic temperature of 450 ⫾80 pK. Such spin-polarized, dilute, and ultracold gases are important for spectroscopy, metrology, and atom optics.
The pursuit of lower temperatures is motivated by the quest to observe phenomena that
occur on very low energy scales, in particular, phase transitions to new forms of matter.
The achievement of temperatures near 1 K in
solids and in liquids led to the discoveries ofsuperconductivity (1) and superfluidity (2),respectively. The advent of laser cooling resulted in microkelvin temperature atomic vapors (3–5), subsequently cooled to nanokelvin temperatures by evaporative cooling toform dilute Bose-Einstein condensates (6, 7)
and quantum degenerate Fermi gases (8).
Collectively, these low-temperature systems
have a host of applications, including superconducting quantum interference devices (9),superfluid gyroscopes (10, 11), and atomic
clocks (12).
Temperature is a quantity that parameterizes how energy is distributed across theavailable states of a system, and effective
temperatures can be defined for decoupleddegrees of freedom or subsets of particles.
For example, nuclear spins isolated from thekinetic motion of their respective atoms havebeen cooled by adiabatic demagnetization toan effective temperature of 280 pK (13). Spinensembles have a finite number of available
states, such that a spin-polarized sample, as in
our work, would be characterized by zero
effective temperature. In contrast, the motionof free particles is subject to a continuum ofstates, and the kinetic temperature of an ensemble can only asymptotically approach absolute zero.
Effective temperatures in atomic vapors
are defined by the widths of velocity distriDepartment of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge,
MA 02139, USA.
*To whom correspondence should be addressed. Email: ael@mit.edu

butions, which can be much smaller than themean velocity of the sample. Raman cooling(14, 15) and velocity-selective coherent population trapping (16) have generated velocity
distributions with very narrow peaks, corresponding to nanokelvin and picokelvin effective temperatures. However, these temperatures were associated with the motion of only
a subset of the atoms in the cloud and/or withatomic motion in only one dimension.
For trapped, partially condensed atomic
vapors, the condensate fraction has zero entropy and the kinetic temperature of the sample is determined by the velocity distributionof the thermal (noncondensed) component.
When released, the condensate fraction expands more slowly than the thermal component and has been characterized by picokelvin effective temperatures for anisotropic (17) and noninteracting (18) gases.
Cooling the atomic motion of entire ensembles in all three dimensions has proven difficult.
To date, kinetic temperatures of a few hundred
nanokelvin have been achieved with adiabaticand optical cooling (19, 20), and evaporative
cooling techniques have produced condensateswith temperatures of 3 nK (21). By adiabatic
expansion and subsequent evaporation, we
have cooled partially condensed atomic vapors to picokelvin kinetic temperatures.
Our thermometry is calibrated by the
Bose-Einstein condensation (BEC) phasetransition temperature, Tc, which in the thermodynamic limit for a harmonically trapped
ideal Bose gas is (22)k B T e ⫽ ប␻¯

冉 冊

N 1/3
⬇ 0.94ប␻¯ N 1/3
␨ (3)

(1)

where kB is Boltzmann’s constant, ប is Planck’s
constant h divided by 2␲, ␨(n) is the Riemann
Zeta function, ␻ ⫽ (␻x␻y␻z)1/3 is the geometricmean of the harmonic trap frequencies, and N is
the total number of atoms, both condensed andnoncondensed. Thus, the atom number and thetrap frequencies set an upper limit for the temperature of a confined Bose-Einstein condensate. In our work, adiabatically weakening thetrapping potential to a mean frequency of␻ ⫽ 2␲ ⫻ (1.12 ⫾ 0.08) Hz guaranteedthat partially condensed atomic vapors withN ⱕ 8000 atoms had picokelvin temperatures (Tc ⱕ 1 nK).
Bose-Einstein condensates containing
more than 107 23Na atoms were created in the
weak field seeking ⱍF ⫽ 1, mF ⫽ ⫺1典 state ina magnetic trap, captured in the focus of an
optical tweezers laser beam, and transferred
into an auxiliary “science” chamber as described in (23). In the science chamber, condensates containing 2 ⫻ 106 to 3 ⫻ 106 atoms
were transferred from the optical tweezers
into a gravito-magnetic trap (Fig. 1A). A
small coil carrying current IS generated avertical bias field Bz and supported the condensates against gravity with a vertical magnetic field gradient, B⬘z ⫽ 2 mg/␮B ⬇ 8
G/cm, where m is the atomic mass, g is theFig. 1. Gravito-magnetic trap. (A) Bose-Einsteincondensates were levitated against gravity ⬃5
mm above a 1-cm diameter, 25-turn coil
mounted inside the ultrahigh vacuum chamber
running current IS. Two 10-cm diameter, 20turn coils were mounted outside the vacuum
chamber and were supplied individually withcurrents IU and IL. The vertical separation between the large coils was 10 cm. The 1-cm
diameter coil was mounted radially off-axis
with respect to the pair of 10-cm diameter coilsby ⬃1 cm. This broke the cylindrical symmetryof the trapping potential. Additional bias fieldsof ⬃1 G were applied in the horizontal plane tocancel the radial magnetic fields generated by
IU and IL on the axis of the small coil and tomaintain a stable trapping potential. 3
g denotes
the direction of gravitational acceleration. (B)Magnetic potential due to IS (short dashed line),
gravitation potential (long dashed line), andjoint vertical potential of the gravito-magnetic
trap (solid line). (C) A radially repulsive potential magnetic potential was generated by running IS alone (dashed line); however, applying aslight antibias field with IU modified the radial
energy profile and created a magnetic field
minimum at r ⫽ 0 (solid line). In (A) to (C), thesolid oval denotes the trapped condensate.

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REPORTS
gravitational acceleration, and ␮B is the Bohrmagneton. Additional control over Bz and B⬘z
was provided by two external coils carrying
independent currents IU and IL. Weak curvature, B⬘⬘z, to the vertical bias field createdstable vertical confinement such that a harmonic restoring force was provided magnetically (gravitationally) for downward (upward) vertical displacements (Fig. 1B). A
radial field gradient, B⬘r ⫽ ⳵Br / ⳵r ⫽ ⫺B⬘z/2,
was also present and added in quadrature
with Bz to provide harmonic radial confinement with a restoring force proportional to(B⬘r)2/Bz (Fig. 1C). The trapping potential
does not have fundamental radial asymmetries as do previously demonstrated IoffePritchard magnetic traps (17, 24, 25). In principle, stable three-dimensional confinement
is possible above a single coil in the presenceof gravity without the aid of external bias
fields or gradients.
For typical loading parameters—IS ⫽ 340
mA, IU ⫽ 5.5 A, and IL ⫽ 0 —the gravitomagnetic trap was spherically symmetric,
with trap frequencies ␻x ⬇ ␻y ⬇ ␻z ⬇ 2␲ ⫻8 Hz. Condensates loaded into the gravitomagnetic trap from the optical tweezers were
held for 5 s to allow for the damping of
excitations. The resulting partially condensed
clouds had ⬃5 ⫻ 105 atoms and a BEC
transition temperature Tc ⫽ 30 nK. Through-Fig. 2. Picokelvin temperature thermometry.
Partially condensed atomic vapors confined in
the gravito-magnetic trap with (A) 28,000,(B) 16,000, and (C) 2,500 atoms. The onedimensional cross sections (red) were obtainedby integrating the two-dimensional absorption
images of the trapped clouds along the y axis.
Bimodal fits to Eq. 2 (blue) yielded temperatures of (A) 1.05 ⫾ 0.08 nK, (B) 780 ⫾ 50 pK,
and (C) 450 ⫾ 80 pK, where the uncertainty is
due to the fit of an individual image. The absorption imaging light was resonant with theF ⫽ 2 3 F⬘ ⫽ 3 cycling transition for thetrapped atoms and was aligned with the vertical (z) axis. The atoms were optically pumped
into the F ⫽ 2 hyperfine level with a pulse
resonant with the F ⫽ 1 3 F⬘ ⫽ 2 transition.
The field of view for the absorption images in(A) to (C) is 460 ␮m ⫻ 460 ␮m.

2

out our work, the atomic vapor maintained atemperature T such that 0.5 ⬍ T/Tc ⬍ 1.
Further cooling was accomplished by adiabatically decompressing the trapping potential in two 5 s stages, with a 5 s delay inbetween to allow excitations to damp. In thefirst stage, the vertical frequency was reduced
to ␻z ⫽ 2␲ ⫻ (1.81 ⫾ 0.05) Hz by simultaneously raising currents IL and IU by identicalamounts and lowering the current IS by afactor of 10. This transferred the dominantsource of magnetic field gradient from thesmall coil to the external coils, with the remaining vertical curvature still due to the
reduced IS. In the second stage, the radialfrequencies were reduced to ␻x ⫽ 2␲ ⫻(0.65 ⫾ 0.05) Hz and ␻y ⫽ 2␲ ⫻ (1.2 ⫾ 0.1)
Hz by raising IL and lowering IU by identicalamounts. The vertical magnetic field gradientand curvature remained constant, but the vertical bias field, Bz, increased. This reduced
the radial confinement due to the scaling ␻r ⬀B⬘r / B1/2z (24, 25).
For currents IS ⫽ 34 mA, IU ⫽ 14 A, andIL ⫽ 44 A, the final gravito-magnetic trap hada measured mean frequency ␻ ⫽ 2␲ ⫻(1.12 ⫾ 0.08) Hz and axial bias field Bz ⫽ 17
G. The residual anharmonicities of the trapping potential were small, with ⌬␻/␻ ⫽ 0.1
for 500 ␮m displacements from the trap center. Further radial decompression was notpossible because of a finite trap depth (Fig.
1C) and sensitivity of trap stability to milligauss level radial bias fields.
After decompression, the partially con-Fig. 3. Bose-Einstein condensates at picokelvintemperatures. The temperature of more than
60 partially condensed atomic vapors is plottedversus total number of condensed and noncondensed atoms. A solid line at the Bose-Einsteincondensation phase transition temperature (Eq.
1) and a dashed line at 1 nK are provided as
guides. Condensate temperatures were determined from one-dimensional fits to atomic
density cross sections integrated along either
the x (closed circles) or y (open circles) axis (Fig.
2). Differences in the two measured temperatures for a single condensate reflect the true
uncertainty of the measurement. Plotted error
bars represent the statistical uncertainty of thefit. The inset shows that the 1/e condensate
lifetime in the gravito-magnetic trap was limited by one-body processes to 80 ⫾ 5 s.

densed atomic vapors had ⬃2 ⫻ 105 atoms anda BEC transition temperature Tc ⫽ 3 nK. Tc
was lowered further by reducing the number ofatoms in the cloud (Eq. 1), while maintaining asubstantial condensate fraction at all times(0.5 ⬍ T/Tc ⬍ 1). The atom number was reduced by holding the atoms in the gravitomagnetic trap for up to 200 s. Often, microwave
radiation near the ⱍ1, ⫺1典3 ⱍ2, 0典 transition
was applied to shorten the hold time required toarrive at lower atom number to 10 s. The atom
number reduction was accompanied by cooling,
during which the elastic collision rate (between
thermal atoms and the condensate) dropped
from 0.25 Hz to 0.01 Hz. Therefore, a few
collisions were sufficient to cause evaporationout of the finite depth trap. This cooling was notefficient in the sense of providing a gain inphase-space density but still was capable ofmaintaining thermal equilibrium and lowering
the absolute temperature of the vapor.
Using this technique, we cooled partial
condensed vapors containing up to 30,000
atoms to temperatures below 1 nK (Figs. 2
and 3). Our lowest measured threedimensional kinetic temperature was 450 ⫾80 pK for 2500 atoms at a peak condensate
density of 5 ⫻ 1010 atoms/cm3. Under these
conditions, the peak atom-atom interactionenergy was ␮ ⫽ kB ⫻ 33 pK, while the zero
point energy of the harmonic trapping potential was (1/2)ប ␻ kB ⫻ 24 pK. Condensates
released from the gravito-magnetic trap
would expand with energies of this order andtherefore could be characterized by effective
temperatures ⬃30 pK.
Additional cooling would require lowering the trap frequencies further or reducing
the atom number more. However, weakly
confining traps have proven technically difficult to control such that lowering ␻/2␲
substantially below 1 Hz is challenging.
Likewise, because Tc ⬀ N1/3, atom number
reduction by an order of magnitude only results in temperature reduction by a factor of
two. Furthermore, lower temperatures andlower densities are accompanied by collisional equilibrium times approaching 100 s.
Because ␮ ⬇ ប␻, the condensates in thegravito-magnetic trap were in a density regime intermediate between the ThomasFermi (␮ ⬎⬎ ប ␻) and ideal (␮ ⬍⬍ ប␻) gas
limits. No simple approximation describes
the condensate wave function, but the number of thermal atoms, Nth ⫽ ␨(3) ⫻ (kB Tប␻)3, and the width of their distribution,
wth ⫽ (2kB T / m␻2 )1/2, in any spatial direction can be related to the temperature, provided that the thermal energy is much larger
than the trap level spacing, kB T ⬎⬎ ប ␻,
where ␻ is the trap frequency for the axis
along which wth is measured (22). The ratioof atoms in the condensate, N0, to the total
number of atoms, N ⫽ N0 ⫹ Nth, is also
related to the temperature through N0/N ⫽● MONTH 2003 VOL 0 SCIENCE www.sciencemag.org

REPORTS
3

1 ⫺ (T/Tc) . N0, Nth, and wth are therefore
completely determined by T and N.
The temperature of the atomic vapors was
extracted by fitting integrated, one-dimensional atomic density cross sections to a bimodal distribution (Fig. 2)
n共 x兲 ⫽ N 0⌿20 ⫹

2
N th
2/w
e ⫺ x th , (2)冑␲w thwhere ⌿20 is a bell-shaped function withwidth w0 that describes the condensate peak[⌿20 ⫽ (15 / 16) w0⫺1 max(1 ⫺ x2 / w20,0)2 fora Thomas-Fermi gas and ⌿20 ⫽ w0⫺1␲⫺1/2
exp(⫺ x2 / w20 ) for an ideal gas]. The fitted
parameters were T, N, and w0. We checkedthat the fitted temperature did not depend on
the exact choice of the condensate wave function (inverted parabola or gaussian) or theapplication of microwave radiation to reduce
atom number.
All atomic vapors represented in Fig. 3
had a clear bimodal density distribution fromwhich a temperature was reproducibly extracted. The temperatures extracted from onedimensional fits along both radial axes were
nominally the same, empirically indicatingthat the atomic vapors remained close to thermal equilibrium at all times.

In conclusion, we have created long-lived(80 ⫾ 5 s), low-temperature (450 ⫾ 80 pK),
and low-density (5 ⫻ 1010 atoms/cm3) partially condensed atomic vapors using a weakly confining [␻ ⫽ 2␲ ⫻ (1.12 ⫾ 0.08) Hz]
gravito-magnetic trap. These samples arecharacterized by a thermal velocity ⬃1 mm/s,
a speed of sound ⬃100 ␮m/s, and a healing
length limited by the ⬃20-␮m harmonic oscillator length of the trapping potential. Lowtemperature and low-density ensembles areimportant for spectroscopy, metrology, andatom optics. In addition, they are predicted toexperience quantum reflection from material
surfaces (26–28).
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7 July 2003; accepted 12 August 2003

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