JILA, CU Boulder/NISTETH Zurich, Department of Physics

M AST ER’S THESI S

TOWA R DS A DEGEN ER ATE FER MIGAS OF STRON TI UM-87 IN A 3D
OP TICAL L AT T ICE
by Nelson Darkwah Oppong
December 2015

Under the supervision ofProf. Dr. Jun Ye
JILA
CU Boulder/NIST, USA

Prof. Dr. Jonathan Home
Institute for Quantum Electronics
ETH Zurich, Switzerland

ABSTRACTWe have designed and constructed a new apparatus towards evaporatively coolingstrontium-87 in a far off-resonant optical dipole trap. The theory of far off-resonant
optical traps is explained and we derive equations for heating in optical dipole traps
due to intensity and pointing noise. A high power laser setup for 1064 nm lightdelivers a total power of up to 20 W for trapping and evaporatively cooling strontium87. The 1064 nm light is delivered with photonic crystal fibers, reaching up to 18 W
through a single optical fiber using a water-cooled high-power fiber connector. We
present a design of a low noise photodetector that is used in an intensity servo for the1064 nm laser system to suppress intensity noise by up to 40 dB. Preliminary resultsfrom the first attempts of evaporatively cooling strontium-87 atoms in a crossed
optical dipole trap yield a 𝑇/𝑇𝐹 = 0.5 per spin state assuming a uniform distribution
across all ten nuclear spin states.

Submitted:
Version:December 8, 2015
May 25, 2016

Please note that this is not the original version. Changes have been made to fix a few
typos. However, the content is entirely unchanged. Please send an email to the author at
nelsond@phys.ethz.ch to receive the original version.
Git commit 357476c01dd51c0f904bd29c1d99ba7a40efc306

i

ACKNOWLEDGMENTS
The last six months have been an exciting and fun experience. I have been very
fortunate to have the chance to work with Jun and his group. The people from Jun’sgroup all share exceptional technical skills and a deep understanding of physics. Jun
is not only a brilliant PI but also a truly kind person. He always takes time for his
students and I was fortunate to learn a lot of things from him. I am still fascinatedby the intuition he has for technical challenges of experiments.
I would also like to thank Jonathan Home, my adviser at ETH, who supported myapplication to Jun’s group and kindly agreed to supervise and grade my thesis.
Everyone in Jun’s group was extremely kind to me. I really appreciate the time people
took to explain physics to me. A special thanks goes to the strontium-2 gang: Sara
Campbell, Ross Hutson, Edward Marti, and Akihisa Goban. They not only shared
my appreciation for good beer but also taught me numerous things about precision
measurements, cold atoms, and science in general. Working with them has been
great fun and I will miss them. I not only enjoyed lunch and dinner with people fromstrontium-1 (Sarah Bromley, Xibo Zhang, Tobias Bothwell, Shimon Kolkowitz, andJohn Robinson) and stable lasers (Wei Zhang and Lindsay Sonderhouse) but also had
the chance to learn from them during many interesting discussions about physics
and other topics. I would like to thank Craig Benko from whom I have learned a
lot about high power optics and whose recommendations made the setup of the1064 nm laser system less painful. Thanks also to Bryce Bjork who kindly helped mewith thermal imaging and monitoring of the high power fiber connectors. The staff
from the instrument and electronics shop has helped me with numerous projects
and I am thankful for their time and hard work. I gratefully acknowledge financial
support from the Studienstiftung des deutschen Volkes (German National Academic
Foundation) for my Master’s thesis project in Jun’s group. I thank my family for theircontinuous support and Sarah Klewes for her understanding and great patience.
We thank Dylan Cotta and Prof. Stephan Kuhr for their generous help with using
photonic crystal fibers for high-power laser light.

ii

CONTENTS

1

2

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

INTRODUCTION

1

1.1

General context . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

THEORY OF OPTICAL TRAPPING3

2.1

Far off-resonant traps . . . . . . . . . . . . . . . . . . . . . . . .

3

2.2

Optical dipole trap geometries . . . . . . . . . . . . . . . . . . . .

5

2.3

Laser noise and heating . . . . . . . . . . . . . . . . . . . . . . .

8

A.

Heating due to intensity noise . . . . . . . . . . . . . . . .

9

B.

Heating due to pointing noise . . . . . . . . . . . . . . . .

11Fermions in harmonic potentials . . . . . . . . . . . . . . . . . .

13

2.4
3

EXPERIMENTAL APPARATUS

15

3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.2

The 1064 nm laser setup . . . . . . . . . . . . . . . . . . . . . . .

17A.

Seed laser and fiber amplifier . . . . . . . . . . . . . . . .

17

B.

Photonic crystal fibers . . . . . . . . . . . . . . . . . . . .

19C.

The complete 1064 nm laser setup . . . . . . . . . . . . .

24

The 813 nm laser setup . . . . . . . . . . . . . . . . . . . . . . . .

28

A.

29

3.3

Ti:sapphire laser . . . . . . . . . . . . . . . . . . . . . . .

iii

C ONT E NT S

4 EXPERIMENTAL TECHNIQUES
4.1

4.2

5

30

Intensity servo . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

A.

Photon shot noise and thermal light sources . . . . . . . .

30

B.

Photodiodes . . . . . . . . . . . . . . . . . . . . . . . . .

32

C.

Low noise transimpedance amplifier . . . . . . . . . . . .

34

D.

Analog servo loop . . . . . . . . . . . . . . . . . . . . . .

39Frequency servo . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

A.

Discrete PID controller . . . . . . . . . . . . . . . . . . .

43

B.

Computer-based frequency servo . . . . . . . . . . . . . .

46

TOWARDS QUANTUM DEGENERACY

48

5.1

Energy level structure of strontium-87 . . . . . . . . . . . . . . .

48

5.2

The blue and red MOT . . . . . . . . . . . . . . . . . . . . . . . .

49

5.3

Crossed optical dipole trap . . . . . . . . . . . . . . . . . . . . . .

51

5.4

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

51

5.5

Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . .

53

REFERENCES

57

APPENDIX

58

I

II

Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

A.

TIA schematic . . . . . . . . . . . . . . . . . . . . . . . .

58

B.

TIA PCB . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

A.

60

Digital PID controller . . . . . . . . . . . . . . . . . . . .

iv

LIST OF FIGURES
2.1

Optical dipole potentials for different detunings . . . . . . . . . .

3

2.2

Force in an optical dipole trap . . . . . . . . . . . . . . . . . . . .

4

2.3

Running and standing wave optical trap geometries . . . . . . . .

6

2.4

Optical lattice geometries . . . . . . . . . . . . . . . . . . . . . .

7

2.5

Heating processes in a harmonic potential . . . . . . . . . . . . .

8

2.6

Heating rate of atoms in 1064 nm light . . . . . . . . . . . . . . .

11

2.7

Optical dipole trap center with gravity . . . . . . . . . . . . . . .

12

2.8

Fermi energy in a harmonic potential . . . . . . . . . . . . . . . .

13

3.1

UHV chamber and laser beam paths . . . . . . . . . . . . . . . .

16

3.2

MISER Nd:YAG crystal design . . . . . . . . . . . . . . . . . . . .

17

3.3

Operating principle of a fiber amplifier . . . . . . . . . . . . . . .

18

3.4

Structure of a photonic crystal fiber . . . . . . . . . . . . . . . . .

19

3.5

Microscopic image of Holey fiber . . . . . . . . . . . . . . . . . .

20

3.6

Schematic of the high-power fiber connector . . . . . . . . . . . .

21

3.7

Photo of the high-power fiber connector . . . . . . . . . . . . . .

22

3.8

Far-field mode of the photonic crystal fiber . . . . . . . . . . . . .

22

3.9

Photonic crystal fiber coupling setup . . . . . . . . . . . . . . . .

23

3.10 Thermal images of the high-power fiber connector . . . . . . . . .

23

3.11 The 1064 nm laser setup . . . . . . . . . . . . . . . . . . . . . . .

24

3.12 Thermal lensing of a convex lens . . . . . . . . . . . . . . . . . .

25

3.13 Mezzanine optics for 1064 nm . . . . . . . . . . . . . . . . . . . .

27

3.14 The 813 nm laser setup . . . . . . . . . . . . . . . . . . . . . . . .

28

4.1

Poisson and binomial distributions . . . . . . . . . . . . . . . . .

32

4.2

Simplified transimpedance amplifier circuit . . . . . . . . . . . . .

34

4.3

Transimpedance amplifier bandwidth boost . . . . . . . . . . . .

35

4.4

Calculated transimpedance amplifier output noise . . . . . . . . .

36

4.5

Photo of the photodetector . . . . . . . . . . . . . . . . . . . . .

37

4.6

Measured transimpedance amplifier output noise . . . . . . . . .

38v

LI ST OF FIGUR E S

4.7

Closed servo loop with negative feedback . . . . . . . . . . . . . .

39

4.8

Open-loop DII and PI transfer functions . . . . . . . . . . . . . .

41

4.9

The intensity servo setup . . . . . . . . . . . . . . . . . . . . . . .

42

4.10 Intensity servo performance for 1064 nm . . . . . . . . . . . . . .

42

4.11 Discrete transfer function on z-space . . . . . . . . . . . . . . . .

44

4.12 Frequency noise supression by wavemeter lock . . . . . . . . . . .

45

4.13 Allan deviation of wavemeter lock error signal . . . . . . . . . . .

46

4.14 Wavemeter PID controller interface . . . . . . . . . . . . . . . . .

47

5.1

The energy level diagram for strontium-87 . . . . . . . . . . . . .

49

5.2

Optical dipole trap 𝑘-vectors . . . . . . . . . . . . . . . . . . . . .

50

5.3

Optical dipole trap potentials . . . . . . . . . . . . . . . . . . . .

51

5.4

Experimental sequence for evaporative cooling . . . . . . . . . . .

52

5.5

Measurement of the dimple trap frequencies . . . . . . . . . . . .

52

5.6

Time-of-flight temperature measurement . . . . . . . . . . . . . .

54

LIST OF TABLES
4.1

Photodiode performance figures . . . . . . . . . . . . . . . . . . .

33

4.2

TIA performance figures . . . . . . . . . . . . . . . . . . . . . . .

38vi

LIST OF ABBREVIATIONS

ac Alternating current

NA Numerical aperture

AOM Acousto-optical modulator

Nd:YAG Neodymium-doped yttrium
aluminium garnet

AR Anti-reflective
CNC Computer numerical control
dc Direct current
ECDL External-cavity diode lasers
FET Field-effect transistor

NIR Near-infrared
PBS Polarizing beam spliter
PCB Printed circuit board
PID Proportional-integral-derivative

FFT Fast Fourier transform

PSD Power spectral density

FIFO First in first out

QWP Quarter-wave plate

FORT Far off-resonant trap

RF Radio frequency

GBP Gain-bandwidth product

SBS Stimulated Brillouin scattering

GUI Graphical user interface

Si Silicon

HR High-reflection

SNR Signal-to-noise ratio

HWP Half-wave plate

TIA Transimpedance amplifier

InGaAs Indium gallium arsenide

UHV Ultra high vacuum

vii

1

INTRODUCTION
1.1 GENERAL CONTEXTPrevious efforts of the Strontium-2 experiment at Ye Labs/JILA were focused onhighly precise and accurate clock measurements. Fermionic strontium-87 was cooled
and loaded into a 1D magic wavelength1 optical lattice to measure the narrow 1 S0 →3
P0 transition [1]. The required stability of the clock laser at 698 nm is achieved bylocking an external-cavity diode laser (ECDL) to a 40 cm cavity with a stability of
10−16 at 1 s [2]. Future efforts are to continue the improvement of a clock with alow-entropy quantum gas, which will also help with the investigation of quantum
many-body physics. Fermions in optical lattices at low temperature allow us tosimulate solid state Hamiltonians, realize novel quantum phases, and study theirtransitions. For example, the high spectroscopic resolution given by the clock laser
might allow observations of novel magnetic phases given rise to by the (large) SU(𝑁)symmetry of the ground state of strontium-87 [3].
The complete experiment - including the UHV chamber – has been rebuilt and threemajor technical changes are introduced:(i) A new UHV chamber with more vacuum pumps is used to achieve a pressureof 10−11 Torr compared to 10−10 Torr in the old experiment.
(ii) An optical dipole trap with a wavelength of 1064 nm is used to evaporatively
cool the atoms and to achieve a low-entropy gas prior to loading into the 3D
optical lattice.
(iii) A new laser system is used for the optical lattice. The new 813 nm laser systemdelivers more power at 813 nm, which is required for the 3D optical lattice.
1

At the magic wavelength, an optical dipole trap will create identical potentials for the ground andexcited states. A superposition state, as is used in a clock, will remain coherent for an extended time ina magic wavelength trap.

1

IN TRODU CT ION

The author was mainly involved in (ii) and (iii). The work on (ii) is completed but
the laser system for the optical lattice (iii) is not completed yet. The complete outlineof this thesis is given in the next section.

1.2 THESIS OUTLINEThis thesis gives an overview of the experiment, discusses some technical changes
on the rebuild of the experiment and explains the basic theoretical frameworks. The
thesis is structured into four chapters excluding the introductory chapter. The second
chapter introduces the theoretical background for optical trapping in running andstanding wave configurations. Rate equations for heating in optical dipole traps
are derived and evaluated in this chapter as well. The third chapter explains theexperimental apparatus and discusses technical aspects of the 1064 nm laser setup,
including high-power photonic crystal fibers. The fourth chapter explains the main
experimental techniques employed for optical trapping. In this chapter, the designof the photodetectors and intensity servo are explained in detail. The fifth and lastchapter explains the experimental sequence, presents a few early experimental results
and summarizes the overall findings of this thesis. Schematics and PCB designs aswell as source code can be found in the appendices.

2

2

THEORY OF OPTICAL TRAPPING

Remarkably, trapping particles by an off-resonant optical/electric fields was first
proposed by the Soviet-Armenian physicist Askaryan in 1962 – long before laser
cooling was invented [4]. We use far-off resonant traps to capture atoms from themagneto-optical trap, evaporatively cool them towards quantum degeneracy, andload the degenerate atoms into an optical lattice. This chapter establishes the basic
theoretical background for far-off resonant traps and heating in those traps. Thisis crucial to understand the experimental techniques explained in the next chapter.
We also briefly discuss ultra-cold fermions in harmonic potentials at the end of thischapter.

2.1 FAR OFF-RESONANT TRAPS
Far off-resonant traps (FORTs) are optical traps where the frequency of the light is
far detuned from any atomic resonances. The detuning Δ = 𝜔 − 𝜔0 is defined as thedifference between the frequency of the driving electric field (𝜔) and the frequency of
the atomic resonance (𝜔0 ). The advantages of FORTs are that many different atomic
species can be trapped with the same wave length and scattering is reduced, which
increases the lifetime of atoms in the trap. We give a short introduction into thephysics of FORTs and discuss running and standing wave trap geometries in thissection.
FIGURE 2.1: Illustration of optical dipole potentials for negative (red) and positive (blue) detunings. The red-detuned
potential attracts atoms into the bright center and the bluedetuned potential pushes the atoms out of the bright into
the dark center.

3

T HE ORY OF OPT ICAL TR A PPING

−zR

9

−w0

-0.

0

FIGURE 2.2: Example plot of theforce (red arrows) in an optical dipole
trap formed by running Gaussianbeam. The normalized potential
depth is shown as a contour plot
(black). Length is defined in termsof the 1/𝑒2 beam radius 𝑤0 and theRayleigh length 𝑧R .

U dipole
U0

-0.6

r

w0

-0.3

0
z

zR

If an atom is placed into laser light, with an ac electric field 𝐸,⃗ an atomic dipole
moment 𝑝⃗ is induced. The interaction potential 𝑈dipole of the 𝐸⃗ field and the atom isgiven as [5]
1
1
⃗ =−
⃗
𝑈dipole (𝑟)⃗ = − ⟨𝑝⃗ ⋅ 𝐸(⃗ 𝑟)⟩
Re(𝛼)𝐼(⃗ 𝑟).
2
2𝜖0 𝑐

(2.1)

Here 𝛼 is the atomic polarizability defined as |𝑝|⃗ = 𝛼|𝐸|,⃗ and 𝐼 ⃗ = 𝐼 ⋅ 𝐸̂ with |𝐼|⃗ = |𝐸|2
is the intensity of the driving electric field of the laser multiplied with the unit vector⃗of the electric field 𝐸.⃗ This potential generates the dipole force 𝐹dipolewith [5]𝐹dipole (𝑟)⃗ = −∇𝑈dipole (𝑟)⃗ =1Re(𝛼)∇𝐼.⃗
2𝜖0 𝑐

(2.2)

An example optical potential and the resulting force is shown in Figure 2.2. We
directly find that for Re(𝛼) > 0 the atoms are pushed towards maximum intensityand for Re(𝛼) < 0 atoms are pushed towards minimum intensity. Furthermore, theimaginary part of the polarizability yields the scattering rate as a function of position𝑟 ⃗ [5]Γscattering (𝑟)⃗ =1⃗Im(𝛼)𝐼(𝑟).
~𝜖0 𝑐

(2.3)

Here 𝐼(𝑟)⃗ is the intensity as a function of position. The scattering rate is the rate at
which photons from the driving field are elastically scattered. For a two-level atomin the rotating wave approximation (i.e. the detuning from the resonance is small
compared to the resonance frequency) we find the dipole potential 𝑈dipole and thescattering rate Γscattering to be [5]:𝑈dipole (𝑟)⃗ =Γscattering (𝑟)⃗ =3𝜋𝑐2 Γ
𝐼(𝑟)⃗2𝜔03 Δ(2.4)

3𝜋𝑐2 Γ 2( ) 𝐼(𝑟)⃗2~𝜔03 Δ

(2.5)

4

T HE ORY OF OPT ICAL TR A PPING

Here, Γ is the decay rate from the excited state to the ground state defined in terms ofthe dipole matrix element. Thus, atoms can be trapped at a maximum of red-detuned
light (Δ < 0) and at a minimum of blue-detuned light (Δ > 0). In a multi-level atom
the expression for the atomic polarizability is more complicated and depends on alllevels accessible from the ground state. Furthermore, we find that the polarizability
depends on the spin state of the atom and polarization of the light, leading to vector
and tensor polarizabilitiy in addition to the scalar term derived above. However,even neglecting polarization and using only the scalar theory in a two level atom, wefind these important results:• The potential depth is directly proportional to the intensity 𝐼(𝑟)⃗ for a fixedatomic state and wavelength of the driving laser field. Therefore we can limit
the discussion of trap geometries in the next section to intensity distributions.
• Red-detuned light traps the atoms in maximum intensity and blue detuned
light traps the atoms in minimum intensity. Thus, atoms experience an ac Stark
shift in a red-detuned trap because the trapping potential has the minimum at
maximum intensity.
• The potential depth scales with 1/Δ but the scattering scales with 1/Δ2 . Thus,
it is favorable to choose a large detuning and power to minimize scattering
while preserving the conservative dipole force for trapping.
The optical dipole trap at 1064 nm and the optical lattice at 813 nm used in our
experiment are both relatively far red detuned from atomic resonances (compared tothe strong transitions at 461nm (for the ground state) and 679nm (for the 3 P0 excitedstate). Note that the choice of the laser wavelength for our optical dipole trap is basedon the commercial availability of low-noise high power lasers. Lasers at 1064 nm
find applications such as medicine and laser cutting and high power single-modelasers (≈ 50 W) are readily available.

2.2 OPTICAL DIPOLE TRAP GEOMETRIES
We generate the trapping potential at the high intensity peak using focused laser
beams, which are well described by Gaussian beams. First, we discuss the runningwave configuration and the harmonic approximation. Second, the standing wave
configuration and the periodic harmonic approximation are explained. The actual
trap configuration for the running wave optical dipole trap in the experiment can befound in Section 5.3.
The electric field of a Gaussian beam as a function of transverse distance 𝑟 and5

T HE ORY OF OPT ICAL TR A PPING

−1.0
U/U0

−0.8

Potential (U/U0 )

r

w0

−0.6

−0.4

−0.2

0.0

0
−w0
0.0

−0.5

−1.0

Harmonic
approx.

−zR

0
z

−λ

zR

0
z

λ

FIGURE 2.3: Running wave (left) and standing wave (right) normalized optical trap geometries for 𝜆 = 1064 nm and 𝑤0 = 100 µm. Length is given in units of the 1/𝑒2 beam radius 𝑤0
and the Rayleigh length 𝑧R for the running wave, and in units of 𝑤0 and the watvelength 𝜆for the standing wave. The contour plots of the 2D potentials are shown in the top row. The
white dashed lines indicate the plane for which 1D plots are shown in the bottom row.

longitudinal distance 𝑧 is given as [6, p. 77]
𝐸(𝑟, 𝑧) = 𝐸0

𝑤0
𝑟2
𝑟2
exp {𝑖𝜁(𝑧) − 2− 𝑖𝑘𝑧 − 𝑖𝑘
}.
𝑤(𝑧)
𝑤 (𝑧)
2𝑅(𝑧)

(2.6)

The 1/𝑒2 radius 𝑤0 and the wavelength 𝜆 are the only independent parameters. Allother quantities are defined as
2𝜋
𝜆
𝜋𝑤02
𝑧R =
𝜆

(Wave vector)𝑘=𝑤(𝑧) = 𝑤0 √1 + ((Rayleigh length)
𝑧 2
)
𝑧R

𝑧 2
𝑅(𝑧) = 𝑧 {1 + ( R ) }𝑧𝑧
𝜁(𝑧) = arctan ( ) .
𝑧R(1/𝑒2 radius)(Radius of curvature)(Gouy phase shift)Thus, the intensity distribution for the running wave configuration is given by6

T HE ORY OF OPT ICAL TR A PPING

1.0
λ/2

Intensity (rel.)

0.8

0.6

0
−λ/2

0.4
0.2
−w0 /2

0

w0 /2

−λ/2

0

λ/2

0.0

FIGURE 2.4: Optical lattice geometries for a single(left) and two standing waves
where the 𝑘-vectors enclosean angle of 𝜋/2 (right). The resulting potential is plotted in
the plane defined by the two
𝑘-vectors. We assume the twolaser beams are detuned.

𝐼(𝑟, 𝑧) ≡ |𝐸(𝑟, 𝑧)|2 with𝐼(𝑟, 𝑧) = 𝐼0 (

𝑤0 2
2𝑟2
) exp (− 2 ) .
𝑤(𝑧)
𝑤 (𝑧)

(2.7)

For small 𝑟 and 𝑧 we can expand 𝐼(𝑟, 𝑧) around 𝑟 = 0, 𝑧 = 0
𝐼(𝑟, 𝑧) = 𝐼0 {1 − 2(𝑧 2
𝑟 2
) − ( ) } + O(𝑟2 𝑧2 ) + O(𝑟2 ) + O(𝑧2 ).
𝑤0
𝑧𝑅

(2.8)

Thus, we can approximate the intensity distribution and the resulting potential as aharmonic potential around 𝑟 = 0, 𝑧 = 0. For atoms that are close to the trap bottom
the trapping potential looks nearly harmonic. This is the case for sufficiently cold
atoms in the lower states of the quantum harmonic oscillator. Figure 2.3 shows aplot of the exact potential and the harmonic approximation for the running wave
configuration.
The standing wave configuration is given by two overlapping Gaussian beams with 𝑘⃗vectors, which point in opposite directions. The two beams are otherwise identicaland we can simply generate this configuration by retro-reflecting a focused laser
beam. The resulting intensity distribution 𝐼sw defined by the interference pattern
between the two electric fields 𝐸− and 𝐸+ is given as
𝐼sw = |𝐸− (𝑟, 𝑧) + 𝐸+ (𝑟, 𝑧)|2
󵄨
󵄨2 󵄨
󵄨2
= 𝐸− (𝑟, 𝑧)𝐸+* (𝑟, 𝑧) + 𝐸−* (𝑟, 𝑧)𝐸+ (𝑟, 𝑧) + 󵄨󵄨󵄨𝐸− (𝑟, 𝑧)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐸+ (𝑟, 𝑧)󵄨󵄨󵄨
2
𝑤
2𝑟2
𝑟2
= 4 𝐼0 ( 0 ) exp (− 2 ) ⋅ cos2 (𝑘𝑧 − 𝑘− 2𝜁(𝑧)) .
𝑤(𝑧)𝑤 (𝑧) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2𝑅(𝑧)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
(*)

(2.9)

(**)

We can neglect the Gouy phase shift for 𝑧 ≪ 𝑧R . Close to 𝑟 = 0 the term (**) is
given as cos2 (𝑘𝑧) ≈ 1 − 𝑘2 𝑧2 around every 𝑧 = 𝜋(2𝑛 − 1)/2𝑘 with 𝑛 ∈ Z. The
overall intensity distribution is then given by the fast modulation described by theterm (**) and the envelope given in the term (*). We find a result similar to theharmonic approximation in the transverse direction of the running wave configura7

T HE ORY OF OPT ICAL TR A PPINGFIGURE 2.5: Illustration of heating processes in a harmonic potential. Atoms
that make 𝑛 → 𝑛 + 𝑗 with 𝑗 > 0 transition contribute to an increase of the totalenergy. We call this parametric heating
for 𝑗 = 2 and dipole heating for 𝑗 = 1.

n=3
2
1
0

tion. However, in the longitudinal direction there is separate harmonic potential at
every half-wavelength. This is an one-dimensional optical lattice, which is plottedin Figure 2.3. When multiple one-dimensional lattices are overlapped, two- andthree-dimensional lattices are generated. A two-dimensional plot of the resulting
potential is shown in Figure 2.4.

2.3 LASER NOISE AND HEATING
In general, potentials for optical traps are generated by laser beams. Thus, we areinterested in what influence laser noise has on the atoms in the quasi-harmonic trap.
We are specifically interested in atom heating due to presence of noise in the lightfield. Heating puts limits on the minimum achievable temperatures as well as the
lifetime in the trap and thereby on the physics we can observe in our experiment.
We derive the heating rate for intensity and pointing noise in detail and discuss the
special case of optical dipole potentials against gravity. Results from our actual laser
setup are discussed at the end of this section.
We have established a basic understanding of the potentials in running and standing
wave optical dipole traps in the last section. This allows us to estimate how noisefrom the laser contributes to heating of the trapped atoms. As long as the atoms are
sufficiently cold to occupy mainly the low motional states, the physics should be welldescribed by the quantum harmonic oscillator Hamiltonian [5],𝐻=𝑝2
1
+ 𝑀𝜔tr2 𝑥2 .
2𝑀 2

(2.10)

Here, 𝑀 is the mass of a single atom and2Re(𝛼)𝐼0 /(𝑀𝑧R2 )𝜔tr2 = {4Re(𝛼)𝐼0 /(𝑀𝑤02 )(longitudinal)(transverse)(2.11)

is the trapping frequency defined in terms of the maximum intensity 𝐼0 , the atomic
polarizability 𝛼, and the Rayleigh length 𝑧R respectively the 1/𝑒2 beam radius 𝑤0 .
The eigenstates of 𝐻 are denoted by |𝑛⟩ and their energy by 𝐸𝑛 .
8

T HE ORY OF OPT ICAL TR A PPING

Heating describes an increase in the total energy stored in the motional degree of
freedom, i.e. atoms move to higher harmonics oscillator levels over time. The two
types of laser noise we consider here are intensity noise and pointing noise. Intensity
noise modulates the effective intensity at time 𝑡 whereas pointing noise modulates theeffective trap center defined by 𝑥 = 0 at time 𝑡. Pointing noise is passively minimized
by using mechanically stable optics or transformed into intensity noise by using
optical fibers. Intensity noise is mostly determined by the type of laser and mustbe actively minimized if too high. A theoretical description of laser noise induced
heating was first given in Ref. [7]. We follow the same approach to derive the heating
rates of intensity and pointing noise but provide a comprehensive derivation.

A. HEATING DUE TO INTENSITY NOISE
First, we consider heating due to intensity noise. Let 𝐼0 be the average intensity, 𝐼(𝑡)the time-dependent intensity, and let 𝜖(𝑡) describe the relative intensity fluctuations,
𝜖(𝑡) ≡

𝐼(𝑡) − 𝐼0.
𝐼0(2.12)The time dependent Hamiltonian is then given as
𝐻(𝑡) =𝑝2
1
1+ 𝑀𝜔tr2 𝑥2 + 𝑀𝜔tr2 𝜖2 (𝑡)𝑥2 .
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2𝑀 2
2(2.13)𝐻󸀠 (𝑡)

𝐻0

Assume the atom is initially in the state |𝑛⟩. Then we are interested in the averagetransition rate 𝑊𝑚←𝑛 from state |𝑛⟩ to state |𝑚 ≠ 𝑛⟩. We assume that the populationof different harmonic oscillator states is constant within time 𝑇, which is muchlarger than the correlation time of the fluctuations 𝜖(𝑡). First-order time-dependent
perturbation theory yields then𝑊𝑚←𝑛 ==󵄨
󵄨󵄨2
1 1 󵄨󵄨󵄨 𝑇
𝑖
󵄨󵄨
󸀠
󵄨
󵄨󵄨
(𝐸
−
𝐸
)
𝑡}
𝑑𝑡
⟨𝑚|
𝐻
(𝑡)
|𝑛⟩
exp
{
∫
󵄨󵄨
𝑚
𝑛
2
󵄨󵄨
𝑇 ~ 󵄨󵄨 0~󵄨𝑇
𝑀2 𝜔tr4 1 𝑇󵄨󵄨2𝑑𝑡
𝑑𝑡󸀠 {𝜖(𝑡)* 𝜖(𝑡󸀠 )} 󵄨󵄨󵄨󵄨⟨𝑚| 𝑥2 |𝑛⟩󵄨󵄨󵄨󵄨 exp {𝑖𝜔𝑚←𝑛 (𝑡 − 𝑡󸀠 )} .
∫∫2
4~ 𝑇 00(2.14)The term 𝜖(𝑡)* 𝜖(𝑡󸀠 ) and the power spectral density (PSD) 𝑆𝜖̃ (𝜔) are related by the
Fourier transform1
𝑇𝑇1∫ 𝑑𝑡 ∫ 𝑑𝑡󸀠 {𝜖(𝑡)* 𝜖(𝑡󸀠 )} exp {𝑖𝜔(𝑡 − 𝑡󸀠 )} .
𝑇→∞ 2𝜋𝑇 00𝑆𝜖̃ (𝜔) = lim
1

This is a special case of the Wiener-Khinchin Theorem, see Ref. [8, p. 371] for a proof.

9(2.15)

T HE ORY OF OPT ICAL TR A PPING

We can use this relation to rewrite the transition probability in terms of the PSDof the noise. Note that 𝜖(𝑡)* 𝜖(𝑡󸀠 ) has only real values and the Fourier transform issymmetric to its conjugate. Thus it is practical to introduce the one-sided power
spectral density 𝑆𝜖 = 2𝑆𝜖̃ . If the noise is uncorrelated, i.e. 𝜖(𝑡)* 𝜖(𝑡 + 𝜏) = 0 for 𝜏 ≪ 𝑇,
then 𝑇 can be extended to ∞ and𝑊𝑚←𝑛 =

𝑀2 𝜔tr4 𝜋 󵄨󵄨󵄨2󵄨󵄨⟨𝑚| 𝑥2 |𝑛⟩󵄨󵄨󵄨 𝑆𝜖 (𝜔𝑚←𝑛 ) .
2
󵄨
4~ 󵄨

(2.16)

This result is especially useful since we can easily measure the PSD of the intensity
noise on the laser using a photodetector. Using the identity 𝑥 = √~/2𝑀𝜔tr (𝑎 + 𝑎† )󵄨2
󵄨for the 𝑥 operator, the term 󵄨󵄨󵄨󵄨⟨𝑚| 𝑥2 |𝑛⟩󵄨󵄨󵄨󵄨 evaluates to:~2
󵄨2󵄨󵄨󵄨󵄨⟨𝑚| 𝑥2 |𝑛⟩󵄨󵄨󵄨 ={𝛿
𝑛(𝑛 − 1) + 𝛿𝑚,𝑛+2 (𝑛 + 1)(𝑛 + 2)}󵄨󵄨
4𝑀2 𝜔2 𝑚,𝑛−2tr(2.17)

󵄨󵄨2with 𝛿𝑖,𝑗 = 1 if 𝑖 = 𝑗 and 0 otherwise. The term 󵄨󵄨󵄨󵄨⟨𝑚| 𝑥2 |𝑛⟩󵄨󵄨󵄨󵄨 vanishes unless 𝑚 = 𝑛±2.
Thus, we observe transitions for Δ𝑛 = 2, which corresponds to intensity fluctuations
at 2𝜔tr . This is a parametric process, which is fundamentally different from theresonance at 𝜔tr when driving the harmonic oscillator directly. This is similar tosquatting and standing up on a swing at twice the resonant frequency2 .
We now find the corresponding transition rates
𝜋 2
𝜔 𝑆 (𝜔
) (𝑛 + 1 ± 1)(𝑛 ± 1)
16 tr 𝜖 𝑛±2←𝑛𝜋= 𝜔tr2 𝑆𝜖 (2𝜔tr ) (𝑛 + 1 ± 1)(𝑛 ± 1).
16

𝑊𝑛±2←𝑛 =(2.18)The heating rate is now given by the average change in energy ⟨𝐸⟩̇ due to transitions𝑛 → 𝑛 ± 2. Let 𝑝(𝑛) be the occupation probability for state |𝑛⟩ at time 𝑡 and we find,⟨𝐸⟩̇ = ∑ 𝑝(𝑛) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2~𝜔tr (𝑊𝑛+2←𝑛 − 𝑊𝑛−2←𝑛 )𝑛Rate of energy change per state |𝑛⟩= ∑ 𝑝(𝑛) 2~𝜔tr𝑛𝜋 2
𝜔 𝑆 (2𝜔tr ) (4𝑛 + 2)
16 tr 𝜖1𝜋= {∑ 𝑝(𝑛) ~𝜔tr (𝑛 + )} 𝜔tr2 𝑆𝜖 (2𝜔tr )22𝑛⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(2.19)⟨𝐸⟩𝜋= ⟨𝐸⟩ 𝜔tr2 𝑆𝜖 (2𝜔tr ) = ⟨𝐸⟩ 𝜋2 𝜈tr2 𝑆𝜖 (2𝜈tr ) .
2
Thus, the energy changes exponentially with rate Γ ≡ 𝜋2 𝜈tr2 𝑆𝜖 (2𝜈tr ). It depends
quadratically on the trapping frequency 𝜈tr and intensity fluctuations at 2𝜈tr .
2

People do this on a Russian swing, see http://en.wikipedia.org/wiki/Russian_swing.

10

Heating rate Γ (Hz)T HE ORY OF OPT ICAL TR A PPING

10−1
10−3 Servo off
10−5
10−7 Servo on
10−9
mit
oise li
Shot n
10−11
10−13
100
101
102
Trapping frequency νtr (Hz)FIGURE 2.6: Heating rate of atoms inlight from the 1064 nm laser setup as
a function of trapping frequency. The
two traces show the heating rate withintensity servo turned on (thick, red)and off (thin, black). The shot noise
limit indicates the heating rate for an
intensity servo with infinite gain.

B. HEATING DUE TO POINTING NOISE
We can find an expression for pointing noise analogous to the derivation of Eqn. 2.19
in the previous section. Let 𝜒(𝑡) be the parameter that characterizes the fluctuationsof the trap center around 𝑥 = 0. Then the resulting time-dependent Hamiltonian isgiven as𝐻(𝑡) =𝑝2
1
2
2+ 𝑀𝜔tr2 𝑥2 − ⏟⏟𝑀𝜔⏟⏟⏟⏟⏟⏟⏟⏟tr⏟⏟𝑥𝜒(𝑡)⏟⏟⏟⏟⏟⏟⏟⏟⏟ +O {𝜒 (𝑡)} .
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2𝑀 2
󸀠

(2.20)𝐻 (𝑡)𝐻0Analogous to the previous derivation we use first order perturbation theory to derive
an expression for the average transition rate 𝑊𝑚←𝑛
𝑊𝑚←𝑛 =𝑇
𝑀2 𝜔tr4 1 𝑇
𝑑𝑡
𝑑𝑡󸀠 {𝜒(𝑡)* 𝜒(𝑡󸀠 )} |⟨𝑚| 𝑥 |𝑛⟩|2 exp {𝑖𝜔𝑚←𝑛 (𝑡 − 𝑡󸀠 )} .
∫∫2~2 𝑇 00(2.21)

Again, letting 𝑇 → ∞ and introducing the one-sided PSD 𝑆𝜒 (𝜔) of the pointing
noise yields
𝑀2 𝜔tr4 𝜋
𝑊𝑚←𝑛 =|⟨𝑚| 𝑥 |𝑛⟩|2 𝑆𝜒 (𝜔𝑚←𝑛 ) .
~(2.22)The term |⟨𝑚| 𝑥 |𝑛⟩|2 evaluates to|⟨𝑚| 𝑥 |𝑛⟩|2 =~{𝑛𝛿𝑚,𝑛−1 , +(𝑛 + 1)𝛿𝑚,𝑛+1 } .
2𝑀𝜔tr(2.23)

Thus, the term |⟨𝑚| 𝑥 |𝑛⟩|2 vanishes unless 𝑚 = 𝑛 ± 1. Thus, we observe transition for
Δ𝑛 = 1. This corresponds to directly driving the harmonic oscillator at the resonant
frequency 𝜔tr , which is different from parametric heating due to intensity noise.

11T HE ORY OF OPT ICAL TR A PPING

Potential (arb.)

0.0
−0.5
−1.0

g=0

−zR 0
z

g > 0, →

−zR 0
z

zR

zR

FIGURE 2.7: Optical dipole trap center without (left) and with (right) gravity for different potential depths asa function of the Rayleigh length 𝑧R .
The dashed black lines indicate the local minimums of the potentials.

Analogous to Eqn. 2.19 we find the resulting heating rate as⟨𝐸⟩̇ = ∑ 𝑝(𝑛) ~𝜔tr (𝑊𝑛+1←𝑛 − 𝑊𝑛−1←𝑛 )𝑛𝜋= {∑ 𝑝(𝑛)} 𝑀𝜔tr4 𝑆𝜒 (𝜔tr )2𝑛⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(2.24)=1𝜋= 𝑀𝜔tr4 𝑆𝜒 (𝜔tr ) = 4𝜋4 𝑀𝜈tr4 𝑆𝜒 (𝜈tr ) .
2
Thus, the energy changes linearly with the rate 𝜉 = 4𝜋4 𝑀𝜈tr4 𝑆𝜒 (𝜈tr ) and depends
on the fourth power of the trapping frequency 𝜈tr , the pointing noise at 𝜈tr , and theatomic mass 𝑀.
Last, we consider a harmonic optical dipole potential under gravity, which yields thefollowing Hamiltonian:𝐻(𝑡) =

𝑝2
1+ 𝑀𝜔tr2 𝑥2 − 𝑔𝑀𝑥
2𝑀 2(2.25)

Here 𝑔 > 0 is a constant. This potential has not a global but a local minimum at
𝑥 = 𝑔/𝜔tr2 . This minimum is a function of the trapping frequency, which depends on
the intensity (see Eqn. 2.11). Thus, intensity fluctuations can introduce trap center
fluctuations, which in turn introduce heating as described in the previous section.
The shift of the trap minimum as a function of intensity is shown in Figure 2.7.
In this section we have derived the heating rates for intensity and pointing noise inharmonic optical dipole traps. Intensity noise causes exponential growth, whereas
pointing noise causes linear growth of the motional energy. Further we have shown
how intensity noise on a combination of a harmonic and linear potential leads to trap
center fluctuations similar to those caused by pointing noise. These results motivate
the intensity servo described in Section 4.1.

12

T HE ORY OF OPT ICAL TR A PPINGFIGURE 2.8: Illustration of the Fermi
energy 𝐸F in an one dimensional harmonic potential with spin-1/2 fermions.
The lower three states are filled up except
for one fermion, which is excited in the
𝑛 = 3 level.

n=3
EF

2
1
0

2.4 FERMIONS IN HARMONIC POTENTIALSWe trap strontium-87 atoms in optical dipole traps, which essentially are threedimensional harmonic potentials. Therefore it is important how quantities like
phase-space density scale for fermions in those potentials. This section mainly
summarizes results from Ref. [9]. The interested reader can consult the reference fordetailed explanations.
First, we define a generalized three-dimensional harmonic potential, which could be
realized by one or multiple optical dipole trap potentials. Let 𝑟 ⃗ = (𝑥1 , 𝑥2 , 𝑥3 ) where𝑥𝑖 are three orthogonal directions in Cartesian space.
1
𝑉(𝑟)⃗ = 𝑀 ∑ 𝜔𝑖2 𝑥𝑖22𝑖(2.26)

Here 𝑀 is the mass of the atoms and 𝜔𝑖 is the trapping frequency along the direction
𝑥𝑖̂ . We define the (geometric) mean trap frequency 𝜔 as
1/3

𝜔 = (∏ 𝜔𝑖 ).

(2.27)

𝑖

Fermions obey the Pauli exclusion principle, i.e. two fermions can not occupy thesame quantum state in a closed system. This is described with the Fermi-Dirac
distribution, which gives the average number of fermions in the energy eigenstate 𝐸𝑖
as⟨𝑛𝑖 ⟩ = 𝑔𝑖 /{exp (𝛽 [𝐸𝑖 − 𝜇]) + 1} .

(2.28)

Here 𝑔𝑖 ∈ N is the degeneracy of the state, 𝛽−1 ≡ 𝑘B 𝑇, and 𝜇 is the chemical potential.
𝜕𝐺 󵄨󵄨󵄨
The chemical potential can be calculated from 𝜇 = 𝜕𝑁
󵄨󵄨𝑇,𝑃 where 𝐺 is the Gibbs freeenergy. The resulting phase space density distribution for an atom in the potential
described above is then given as,𝑓(𝑟,⃗ 𝑝)⃗ = 𝑔 /{exp (𝛽 [𝑝2
+ 𝑉(𝑟)⃗ − 𝜇]) + 1} .
2𝑀

13(2.29)

T HE ORY OF OPT ICAL TR A PPING

Here 𝑔 is the degeneracy of a single state in the three dimensional harmonic potential⃗ This is usually given in terms of the available spin states, e.g. 𝑔 = 10
given by 𝑉(𝑟).
for the ground state of strontium-87. Let𝑁=∫

𝑑3 𝑟 ⃗ 𝑑3 𝑝⃗𝑓(𝑟,⃗ 𝑝)⃗2𝜋~2(2.30)

be the total number of atoms in the harmonic potential. At 𝑇 = 0, the atoms are in
the lowest possible energy configuration where all 𝑁/𝑔 lowest states of the potential
𝑉(𝑟)⃗ are filled up to the energy 𝐸F . We call this energy the Fermi energy, which is
given as [9],1/3𝐸F = ~𝜔(6𝑁/𝑔)(2.31).

For the approximations in this section we assume that the Fermi energy 𝐸F is muchlarger than the energy splittings ℎ𝜔𝑖 of the three-dimensional harmonic potential.
The Fermi temperature is defined as 𝑇F = 𝐸F /𝑘B . From Eqn. 2.28 we directly find that𝜇 must be equal to 𝐸F at 𝑇 = 0 to reproduce the distribution expected from the Pauli
exclusion principle. The atom number density 𝑛 as a function of 𝑟 ⃗ is given as [9]𝑑3 𝑝⃗𝑓(𝑟,⃗ 𝑝)⃗2𝜋~2𝑔𝑝2
+ 𝑉(𝑟)⃗ − 𝜇]}) .
= 3 Li3/2 (exp {𝛽 [2𝑀
𝜆 dB𝑛(𝑟)⃗ = ∫(2.32)∞

Here, Li3/2 is a polylogarithm with Li3/2 (𝑥) = 1/Γ(3/2) ∫0 𝑑𝑞 𝑞1/2 /(𝑒𝑞 /𝑥 − 1), and𝜆 dB = √2𝜋𝛽~2 /𝑀(2.33)

is the de Broglie wavelength. The high temperature limit of 𝑛(𝑟)⃗ is the MaxwellBoltzmann distribution.
In this section, we have discussed the density for fermions in a harmonic trap.
Furthermore, important quantities like the Fermi energy have been defined. This
helps to evaluate the experimental data in Chapter 5.

14

3

EXPERIMENTAL APPARATUS

This chapter explains the experimental apparatus, including the UHV chamber andlaser systems used to trap, cool, and manipulate strontium-87 atoms. We give a shortoverview of the new UHV chamber, magnetic coils, strontium oven, and the blue
and red laser systems. The author was mainly involved in the 1064 nm and 813 nm
laser systems, which are discussed separately in Section 3.2 and Section 3.3.

3.1 OVERVIEW
The experiment uses a UHV chamber, which is shown in Figure 3.1. The chamber has
good optical access with two large viewports in the vertical and 13 viewports in thehorizontal plane. The chamber is connected to several pumps: Two ion pumps, a nonevaporative getter pump, a titanium sublimation pump, and a getter coating on theinside of the chamber (titanium-zirconium-vanadium). The pressure is 1×10−11 Torrwith the atomic beam oven turned on.
Two water-cooled magnetic coils in an anti-Helmholtz configuration generate thequadrupole field required for the MOT. The main chamber has recessed windows on
the top and bottom, which allow the magnetic coils to be placed relatively close to theatoms. The maximum field gradient at the atoms is ≈ 550 G/cm. Three Helmholtz
pairs of bias field coils are used to cancel the background magnetic field and define
the point of zero field inside the chamber. This allows us to move the MOT and alignit to the laser beams.
We use an integrated atomic beam oven and Zeeman slower from AOSense, whichfeatures a very compact design (see Figure 3.1) and does not need any water cooling
due to the use of permanent magnets in the Zeeman slower. The integrated ovenand Zeeman slower features in-vacuum optics, which allow us to get a 2D MOT bycoupling light into just one window. The atomic beam is steered towards the main15

EX PER IM EN TA L A PPAR ATUS
Gravity
150 L/sion pump

300 L/s
NEG

Lattice &clockBlue &
red MOT

Absorption
imaging probe

LatticeBlue &
red MOTZSTSPAOSenseSr oven
Gate
valve

Atoms
1 cm

2D blue
MOT

Horiz. Dimple
ODT ODT

Anti-Helmholtz coils

FIGURE 3.1: UHV chamber with main beam paths (arrows) and vacuum pumps. Combined
blue and red laser light is shown in purple. The vertical MOT, vertical lattice, vertical optical
dipole trap, and vertical imaging beam are not shown in this drawing. Those beams arelaunched through the bottom and top viewport at nearly normal angle of incidence.

chamber at an angle using a second 2D MOT. Thus, the atomic beam can be "shutoff " by blocking the light for the 2D MOT. We operate the oven at 460 ∘ C, which
gives an atom flux of ≈ 2 × 1013 atoms/s.
Most of the optics are on separate breadboards and laser light is delivered to theatoms by optical fibers. This maps pointing noise on the side of the laser setup tointensity fluctuations on the side of the atoms. We can then simply stabilize theintensity and reduce the overall noise. See Section 4.1 for a detailed description of theintensity servo scheme used. An additional advantage of the separate breadboards isthat in the event of laser failure only the optics on the side of the laser setup need
to be realigned. The beam splitting, shaping, and polarizing optics are mounted ona mezzanine, which is a ≈ 6 cm thick steel breadboard that is mounted on 25.4 cm
high aluminum posts that are filled with lead shot. This allows all the optics to beon the same height as the atoms in the chamber. Most optics are mounted on shortfixed-height steel posts to minimize drift.
The laser systems for the red and blue MOT have already been used for the oldexperiment. The red laser system uses a master ECDL locked to an ultra-stable cavity
and two slaves, which are phase locked to the master. The blue laser system uses amaster ECDL locked to a hollow cathode lamp and slave free running diode lasers,
which are injection locked to the master.

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EX PER IM EN TA L A PPAR ATUS

3.2 THE 1064 NM LASER SETUP
The 1064 nm laser setup delivers high power light for the optical dipole trap. Figure 3.11 shows the main setup and Figure 3.13 shows the optics on the mezzanine.
The first two sections discuss the seed laser/fiber amplifier and the photonic crystal
fibers used to deliver high power from the main setup to the atoms.

A. SEED LASER AND FIBER AMPLIFIER
We use a seed and amplifier configuration to generate high optical power at 1064 nm
as shown in Figure 3.11. The seed laser is a 500 mW diode-pumped Nd:YAG laser
with a monolithic design, which allows narrow linewidth and relatively high power.
Light from the seed laser is amplified using a 50 W fiber amplifier from Nufern. The
Nufern amplifies 1−15 mW of seed light to a maximum output of 50 W1 . This section
explains the basic operating principle of the seed laser and the amplifier, which helps
to understand shortcomings and specific features of the laser setup.
The Nd:YAG laser is based on the MISER (Monolithic isolated single-mode endpumped ring) design. This design uses a non-planar ring cavity, which is formed by
the Nd:YAG crystal itself. Figure 3.2 shows the beam path inside the crystal. This
design was first realized in Ref. [10] and has several advantages:• High mechanical/thermal stability because the monolithic design prevents
any differential drift of other optics with respect to the crystal.
• The beam inside the MISER cavity is a running wave. This prevents spatial
hole burning, which is the effect where spatially modulated gain in a planarcavity leads to oscillations of higher order modes.
There are some technical caveats to make the laser truly unidirectional and the output
polarization linear. When a magnetic field is applied to the Nd:YAG, the Faraday
effect leads to a 𝑘-vector dependent rotation of the polarization. This allows the laser
1

The specifications from the manufacturer quote a maximum power of 50 W output power. However,it was only operated/tested up to ≈ 40 W in our setup.

FIGURE 3.2: Monolithic isolated single-mode endpumped ring (MISER) crystal design. The pump beam
enters the crystal on the front surface, is reflected on theangled back surface on the right, the top surface, and finallyon the left angled back surface before exiting the cavity
again through the front surface. Note that the non-planar
beam path introduces a slight rotation of the polarization.
Drawing adapted from Ref. [10, Fig. 1]

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EX PER IM EN TA L A PPAR ATUS

Index profi le
n3
n2
n1

Pump diodeOutputMode stripper

Seed input

FIGURE 3.3: Basic operating principle of a fiber amplifier following the description in [6].
The index profile is shown on the left. The inner core has refractive index 𝑛1 , the outer core𝑛2 , and the outer cladding 𝑛3 . Light from the pump diode (orange) is coupled into the outercladding whereas light from the seed (red) is coupled into the quasi-single-mode core. The
expanded cross-section of the fiber is shown in the middle. Light from pump passes throughthe inner core and pumps rare-earth ions leading to amplification of the light as it travels
from left to right. The coil on the right is used to strip any higher modes from the quasi
single-mode core. The mode stripper at the output removes any remaining power in theouter core.

to operate unidirectionally. A dielectric coating on the out-coupling surface is madeless transmissive for one polarization, which introduces polarization selectivity. Theinterested reader might consult Ref. [10] for an excellent and detailed explanation.
However, the maximum power (up to 2 W lasers are commercially available) for
this design is limited by thermal lensing. As the pump power is increased, the
Nd:YAG crystal heats non uniformly along the transverse direction of the beam path.
This leads to a non-uniform modulation in the optical path length and to a mode
mismatch, i.e. the output is not single-mode anymore [11]. Thus, we use a fiber laser
amplifier to amplify the signal from the 500 mW Nd:YAG laser.
The fiber amplifier (Nufern NuAmp 50 W) uses a double cladded active fiber toamplify light from input powers on the order of a few milliwatts to output powers of≈ 50 W. The fiber has an inner core and an outer core with diameter on the order of
10 µm and 100 µm, respectively. Additionally, a second cladding - the outer cladding- surround the inner cladding. This forms two fiber interfaces:
1. The inner core together with the inner cladding forms a single-mode fiber2
with a low NA. The seed laser is coupled into this single-mode fiber.
2. The outer core together with the outer cladding forms a multi-mode fiber witha large NA. This allows the coupling of light from multiple pump diodes into
the fiber and reaching high optical powers.
The inner core of the fiber is doped with Ytterbium 3+ ions that can be pumped
2

Actually, the inner core can also be multi-mode, but the NA has to be small enough to allow
significant bend-loss for higher order transverse modes [12, p. 371]. The fiber can then be coiled andstill operate single mode while allowing for a larger mode area and therefore lower peak intensity,
which limits the overall maximum output power.

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EX PER IM EN TA L A PPAR ATUSa

Propagation

FIGURE 3.4: Structure of a photonic crystal fiber fiber with the hole diameter 𝑎 andhole spacing Λ. The unit cells are hexagonsof which three are shown on the left. Light
propagates in the defect at the center of thefiber along the longitudinal direction (redbeam at the center).

between 800 nm and 1064 nm and can emit at 1064 nm (in silica) [13]. This allows theinner core to act as the gain medium. While the light from the seed laser determines
the phase and wavelength of the output light, the high intensity light from the pumplaser diodes allows a large gain (up to 46 dB for our amplifier). A mode stripper at theoutput end of the fiber strips the higher-order modes in the outer core and turns theexcess optical power into heat. Figure 3.3 shows the basic operating principle of thefiber laser. The single mode with high intensity at the output can then be collimatedusing a lens. The mode quality of the out-coupled beam is very good compared tothe mode of a laser diode. The fiber amplifier is thermally stabilized using a chiller(setpoint at 23 ∘ C) to prevent excess heat induced damages.

B. PHOTONIC CRYSTAL FIBERS
We use photonic crystal fibers to deliver power from the 1064 nm laser system to theatoms. This section describes limitations of standard step-index fibers for high laser
power and explains how photonic crystal fibers can overcome these limitations.
For high optical powers, dielectric materials show electrostriction – an effect thatcauses the local density to be modulated3 by the electric field strength [14]. This is
the origin of stimulated Brillouin scattering (SBS) in optical fibers. Acoustic noise- present in all fibers - can make forward-traveling photons scatter into a phonon
and a backwards-traveling photon (of slightly different frequency)4 . This forms
an interference pattern in the fiber that modulates the density and this acoustic
wave further enhances back-scattering, so this is a stimulated process. SBS is a nonlinear process and there exists a threshold where the back-scattered power becomescomparable to the input power. This threshold depends on the fiber length, dielectricmaterial, core diameter, and the wavelength of the incident light [14]. For a 5 m
step-index, single-mode (fused silica) fiber with a 5 µm core, and 1064 nm light thethreshold is given by ≈ 2.75 W using the approximation given in Ref. [15]. These
parameters are typical of those for other fibers in the experiment.
When using step-index fibers for high power delivery, it is necessary to increase
3
4

The modulation strength is proportional to the intensity of the electric field.
Forward scatter is of course also possible but less likely.

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EX PER IM EN TA L A PPAR ATUS

FIGURE 3.5: Microscopic image of the Holey fiber(NKT Photonics LMA-PM-15) used in the experiment. The scale is based on the measurements givenin the datasheet (OD of cladding given as 230 µm ±5 µm [16]). Stress rods are vaguely visible on the top
left and bottom right. This allows this fiber to be polarization maintaining. The contrast of the image isslightly increased to provide better visibility of thedifferent features.
Image kindly provided by NKT Photonics/USA.

10 um

the SBS threshold value. The straight-forward approach would be to to increase
the core diameter, i.e. reduce the peak power density inside the fiber. But, we wantto maintain single-mode operation. The constraint for single-mode operation of astep-index fiber is given as [6, p. 337],
𝑉=

1/2
𝜋𝑎 2
2
≤ 2.405.
(𝑛core − 𝑛cladding )⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝜆0

(3.1)

NA

Here, 𝑎 is the fiber core diameter and 𝜆 0 the wavelength of interest. The 𝑉-parameteris defined in terms of the number of modes 𝑁m [6, p. 338]
𝑁m ≈

4 2
𝑉 .
𝜋2

(3.2)

Thus, for larger 𝑎, the NA has to be reduced to maintain single-mode operation.
However, decreasing the NA far below 0.07 leads to significant bend loss rendering
the fiber useless. Therefore standard step-index fiber cannot be used to deliver highpower single-mode laser light.
Photonic crystal fibers consist of a solid single-material (usually fused silica) fiberthat has a two-dimensional periodic structure of air gaps on the plane transverse tolight propagation. The fiber we use has a triangular pattern of circular hole with adefect at the center. This particular type of fiber is called the Holey fiber. The structureis shown in Figure 3.4 and a microscopic image of the cross-section is shown in
Figure 3.5. The Holey fiber has an effective 𝑉-parameter 𝑉eff (for which Eqn. 3.1
applies) given as [17]𝑉eff =

2𝜋 1/2 2
2 1/2
Λ𝐹 (𝑛0 − 𝑛air
) .
𝜆0

(3.3)

Here, 𝑛0 is the refractive index of the fiber material, Λ is the spacing between neighboring holes (see Figure 3.4), and 𝐹 is the air filling fraction of a (hexagonal) unit

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EX PER IM EN TA L A PPAR ATUS

Water-cooled
copper feruleCollapsed
air holes

Holey
fiber

~1 mm

Fused silica
end cap

Fused silica lens(f = 10 mm)

FIGURE 3.6: High-power fiber connector with different features and a focused laser beam
entering the fiber from the right. The glowing fiber in the copper ferrule shows the escape of
miscoupled light from the fiber. Note that none of this schematic is to scale and part of theHoley fiber as well as the end cap are magnified to make the details visible. An additional
glass ferrule for protection of the delicate fiber is not shown in the drawing. Compare withthe photo of the connector in Figure 3.7 for the relative sizes of the all features.

cell, given as
√3
𝜋
𝑎
𝑎 2
( ).
𝐹 = 𝜋 ( ) /( Λ2 ) =
√32 Λ
2
2

(3.4)

Here 𝑎 is the diameter of a circular hole in the unit cell (see Figure 3.4). The lighttravels in the defect at the center of the fiber and the mode area is approximately
proportional to Λ/𝑎 [17]. This allows us to increase the mode area by choosing
large spacing Λ and a relatively small hole diameter 𝑎 while maintaining singlemode operation in terms of Eqn. 3.1 [18]. Thus, the SBS threshold is higher and theHoley fiber can deliver high-power single-mode laser light. This solves the problem
described in the previous paragraph. The periodic holes in the fiber "cladding" of theHoley fiber present a band gap for the fundamental mode so it cannot exist outside
the center defect region. However, higher order modes can exit the core. They can
simply escape the defect at the center of the fiber and propagate to the outside of
the fiber [17]. Thus, only the fundamental mode is guided, whereas significant loss
for higher order modes is introduced. The fundamental mode of the Holey fiber
has a similar intensity profile as a Gaussian. Figure 3.8 shows the far field intensity
distribution of the Holey fiber used in the experiment. Holey fibers are made frompure fused silica compared to doped fused silica in standard step-index fibers. Thisis another advantage as defects and undesired scattering are reduced.
We use the Holey fiber LMA-PM-15 from NKT Photonics. It has a 1/𝑒2 mode
diameter of 12.6 µm ± 1.5 µm at 1064 nm and is made from pure fused silica [16].
We use a special high power connector (HP2.5 from Tratech Fiberoptics) to prevent
damage of the fiber surface and the fiber in general. The connector design, shown in
Figure 3.6, has several features to prevent damage of the fiber and allow transmissionof high optical powers:
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EX PER IM EN TA L A PPAR ATUS

Silica ferrule

FPC connector
Copper ferrule

Fiber
Silica end cap

FIGURE 3.7: Photo of the high-power
fiber connector. The inset on the bottom
left shows a magnification of the fiber tip
and silica end cap. The copper ferrule aswell as the cut-outs are visible on the right.
1 mm• The Holey fiber is fused to a solid silica end cap of 1 mm ± 100 µm length. The
laser beam continues to focus down onto the fiber tip inside the end cap (seeFigure 3.6). This allows a larger mode diameter and correspondingly a lower
power density on the air to end cap interface than it would be possible on
an air to fiber tip interface. This makes the fiber connector less likely to bedamaged due to surface defects or trapped dust particles.
• The fiber end cap and Holey fiber are in good thermal contact with a copper
ferrule. This allows us to cool heat from miscoupled light.
• Cutouts in the copper ferrule allow miscoupled light to exit the ferrule withoutcausing more heating. This is actually visible on an IR viewer (we are using
the fibers with 1064 nm light).
We use a custom-made copper block that holds the copper ferrule of the fiber connector for water-cooling. Thermocouples connected to the copper ferrule close to thefiber tip allow continuous monitoring of the fiber temperature and will be connectedto the interlock, which can redirect the high-power laser beam to a beam dump
using an AOM. With no light the fibers are at a temperature of ≈ 17.5 ∘ C, which isrelatively close to the mean water temperature of ≈ 16 ∘ C. When operating the fiber
at high power (i.e. ≥ 12 W) the temperature can rise to up to 22 ∘ C depending on

w0

−w0 0

w0

0

−w0

0.0
0.5
1.0
Intensity (rel.)

FIGURE 3.8: Photo of the far-field mode of the photonic crystal fiber after collimation. The white contour lines indicate thesame intensity. A hexagonal pattern is visible in the intensity
distribution. Here, 𝑤0 refers to the 1/𝑒2 beam radius of a twodimensional Gaussian fit. Note that defects on the camera filter
slightly distort the image and the contour lines.

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EX PER IM EN TA L A PPAR ATUS

Fiber
Lens tube(protects fi ber tip)

Lens mountFiber connector
Beam path

Cooling block

Water lines

Th ermocouple head3D transl. stage

1 cm

FIGURE 3.9: Photo of the high power photonic crystal fiber coupling setup with watercooling, thermocouple probe, and 3D translation stage. The thermocouple probe is connected
to a read-out device. Note that the setup is nearly identical for all high power beam paths in
the 1064 nm setup.

the fiber coupling efficiency5 . Figure 3.9 shows the complete in-coupling setup for asingle Holey fiber. When initially testing the maximum power handling capability of
the fiber, the temperature was monitored continuously with an IR camera. Thermal
images from the camera with no light and high power light present are shown in
Figure 3.10. The maximum output power achieved is ≈ 18 W at 86 % fiber coupling
efficiency without any AOMs in the beam path. Usually, the fiber coupling is adjusted
continuously while ramping up the power to account for change in the AOM modeand thermal lensing of optics in the beam path. The lens used for fiber coupling
5
The fiber coupling efficiency is mainly determined by the mode overlap. AOMs introduce ellipticity
to the beam, which reduce the mode overlap.

0W

23
22
21
20
19
18
17
16

21.3 W (86 %)18.9○ C○C

17.7○ C23

FIGURE 3.10: Thermal image ofthe input high-power fiber connector with (left) and without light(right). Three strips of Kepton tape
are used to record the black body
spectrum of the copper, which isotherwise highly reflective. The
strips are located on the cooling
block and the back as well as frontof the copper ferrule (as shown Figure 3.9).

EX PER IM EN TA L A PPAR ATUS

HP fi berf = 10 mm

Beam dump

6 - 12 W

Mirror-30.2 MHz

+30 MHz

Mephistoseed laser

Mirrorf = 15 cm

AOM

PBS

2:1

HWP

HWP
PBS

PBS

35 dB OI

HWP

450 mW

HWP

To wavemeter
35 dB
OI

+25.6 MHzf = 10 mm

HWP

0.5 - 1 W

HWP

Razer bladestack

1-50 W

6 - 12 W

3D transl. stage
Step-index fi ber

HWP
PBS

10 mW

Nufernfi ber amplifi er

Iris

FIGURE 3.11: The complete 1064 nm laser setup. The beam path is shown in red with arrows
indicating the propagation direction. Light red lines refer to the 0th orders of AOMs. Laser
power is denoted in red font. Coiled colored and black lines refer to optical fibers (green isunspecified, orange is multi-mode, blue is step-index, and black is Holey).

is mounted on a three-dimensional differential translation stage, which allows fineadjustment in all three directions.

C. THE COMPLETE 1064 NM LASER SETUP
The 1064 nm main setup is on a steel breadboard that sits on rubber spacers, which
are on a floating table. This allows decoupling of vibrational noise from the table.
Light from the main setup is delivered to the mezzanine with three fibers. The main
setup consists of two parts: the seed laser optics and the fiber amplifier optics. We
discuss those separately by following the beam path shown in Figure 3.11. Section 3.2
discusses the optics on the mezzanine.
The seed laser delivers 500 mW of 1064 nm light. A 35 dB (specified from manufacturer) fixed-wavelength isolator (Thorlabs IO-5-1064-VHP) is used to reducethe intensity of back-reflections. Approximately 90 % (measured) transmission canbe achieved with maximized rejection. We use a half-waveplate and an AR-coatedwindow to pick off 1 − 5 % of the light, which is coupled into a multi-mode fiber
going to the wavemeter. This allows to monitor the wavelength and detect potential
multi-mode emission of the laser. A half-wave plate and a PBS cube allows us to24

EX PER IM EN TA L A PPAR ATUS

f’
f

FIGURE 3.12: Thermal lensing of a convex lens: The transverse intensity distribution of laser beams (in general) is given
by a Gaussian. Absorption of laser light by the optics leads toheating and expansion of the glass, but this expansion is nonuniform due to the Gaussian intensity distribution. Thus, thefocal length (𝑓 → 𝑓󸀠 ) of the lens is decreased as a functionof incident laser power. This introduces instability, especiallyfor fiber coupling, which strongly depends on the beam 1/𝑒2
radiuschoose the amount of power, which is coupled into the fiber and the amount that
goes into to fiber amplifier.
All optics after the fiber amplifier are exposed to high optical powers (30 − 40 W) ata relatively small 1/𝑒2 beam radius (1.1 − 0.5 mm). Thus, thermal lensing is an issue –see Figure 3.12 for an illustration of the effect. Undesired reflections from surfaces
are also more problematic for safety and stability at high power. We use high-power
optics to minimize these effects:• All transmissive optics are made from fused silica. Fused silica has bettertransmittance of 1064 nm light (i.e. less absorption) than BK7 and a lower
coefficient of thermal expansion (5.5×10−7 / ∘ C compared to 7.1×10−6 / ∘ C) [19].
This reduces thermal lensing.
• All optics are V-coated with a narrow-band coating that is HR (mirrors) or AR(lenses) for 1064 nm. This minimizes undesired back-reflection from lenses
and transmission of light through HR coated mirrors.
• Thin-film polarizing beam splitters (CVI TFPN-1064-PW-1025-UV) are usedinstead of regular polarizing beam splitter cubes. They use a coating to reflects and transmit p-polarization. Regular PBS cubes are epoxied at the beam
splitter interface. This can easily be damaged by high power laser beams.
The output fiber of the fiber amplifier is connected to a fixed collimator from Nufern
yielding a collimated beam with a 1/𝑒2 radius of approximately 1.1 mm.6 . We use a
35 dB (specified from manufacturer) fixed-wavelength isolator (Thorlabs IO-5-1064VHP) to protect the fiber amplifier from back-reflections. The isolator transmits
89 % (measured) of the light when rejection is maximized. The isolator uses two
Brewster windows (polarization analyzers) at the input and output. This leads to
non-normal output angles of the rejected light which escapes through windows. We
6

The output fiber of the Nufern fiber amplifier has an unusual connector (LD-80 from Leonie
Fiberoptics). We tried to use a custom collimator made from a solid piece of aluminum with a lens
yielding a smaller collimated beam radius. However, it was not possible to get the fiber tip perfectly
centered with respect to the lens which yielded an elliptical beam. This does not to seem to be a problem
with the non-adjustable collimator made by Nufern.

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EX PER IM EN TA L A PPAR ATUS

use two custom-made beam dumps for safety and avoiding heating the breadboard.
The beam dumps are made from aluminum that is sand-blasted and anodized inblack to minimize potential reflection. Additionally, a v-groove in the beam dumps
makes sure rejected light is trapped and cannot escape. Due to the relatively large
thermal load of the beam dumps, their temperature increases only a few ∘ C underload.
A HWP and a PBS directly after the optical isolator allow us to dump most of thepower into the beam dump during initial alignment or for realignment. The modeof the fiber amplifier changes with the output power so that it is useful to always run
it at full power and just drop most of the power directly after the optical isolator.
We use a 2:1 (𝑓1 = 200 mm and 𝑓2 = −100 mm) telescope to reduce the 1/𝑒2 beam
radius since the 1.1 mm output beam from the Nufern collimator is too large forthe active aperture of the AOMs7 . After the telescope, the beam is split into three
beams using three pairs of HWPs and PBSs. The first beam is for the horizontal
optical dipole trap, the second beam is for the vertical optical dipole trap, and thethird beam is for the dimple trap with relative low power (1 − 2 W) compared to thetwo other beams.
The optics for each beam path are identical, but the beam path for the dimple beamuses regular BK7 instead of fused silica optics. AOMs with a center frequency of
30 MHz (IntraAction AOM-30AF3) are used to control the intensity of the beams bycontrolling the amplitude of the RF signal driving the transducer in the AOM. We
use the ±1st order and detune the AOM frequencies relative to each other to prevent
undesired interference patterns at the atoms. The diffraction efficiency of the AOMs
into the ±1st order is approximately 85 %. If we use the ±1st orders, then the 0th and∓1st orders still have significant power (15 % combined). For safety reasons we pick
off the unused orders with a slitted mirror. A 3.5 mm slit is ground into the mirror
allowing only one order to propagate straight while all other orders are reflected intoa beam dump (see Figure 3.11). A HWP before the fiber coupling lens is used to alignthe polarization axis to the axis of the polarization-maintaining fiber. This reduces
polarization fluctuations from acoustic and thermal noise. We use a V-coated fusedsilica lens (𝑓 = 10 mm, Thorlabs LA4280-YAG) to focus the beam down onto thefiber tip. From Gaussian beam propagation (ABCD-matrices) we know that the 1/𝑒2
radius 𝑤𝑓 of the focused beam is given as 𝑤𝑓 = 2𝑓𝜆 0 /(𝜋𝑤0 ). So for our configurationof 𝑤0 = 0.5 mm 𝑤𝑓 = 6.8 µm, which is as close as we can get to the 1/𝑒2 mode radius(6.3(0.75) µm [16]) of the photonic crystal fiber with commercially available lenses.
The lens is glued into an adapter that is mounted inside a custom-made lens holder.
The photonic crystal fiber and the coupling process are described in the previous
section. The beam path for the dimple beam is identical (see Figure 3.11), but a regularpolarization-maintaining step-index fiber (Thorlabs P1-1064PM-FC-5) is used.
The optics on the mezzanine are limited to collimation and shaping of the beam from
7

The AOM still diffracts the beam, but the beam quality is degraded and the diffraction efficiency
is reduced.

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EX PER IM EN TA L A PPAR ATUS

~ 0.2 W

1D transl. stage

dimplebeam

step-index
Achromat f = 33 mm f = 250 mm
fi berf = 1000 mm
HP fi berHP fi ber

99/1

~ 6.5 W

99/1

hrzt.
beam

f = 100 mm 1:6 cylindrical~ 2.5 W

vert.
beam

f = 33mm QWP f = 500 mm

Atoms

PD
Beam
dump

ND
fi lter
PD

Beam
dump

ND
fi lterPD
ND
fi lter

FIGURE 3.13: 1064 nm mezzanine optics for beam shaping of the three optical dipole trap
beams and intensity stabilization. Two mirrors per beam path allow to fully control direction
and angle of incidence at the atoms. Note that the beam paths are not in the same plane
and the three beams hit the atoms at different angles (dotted lines). See Section 5.3 for a
description of the different beam directions and angles.

the fiber to achieve the desired trap geometry. The optics are shown in Figure 3.13.
The optics for each beam are nearly identical as follows• Dimple dipole trap: We use a 𝑓 = 33 mm achromat to collimate the beam
and a 𝑓 = 250 mm to focus the beam down to a 1/𝑒2 radius of approximately
17 µm at the atoms. The Rayleigh length is relatively short (853 µm) for thisbeam and we use a 1D translation stage to adjust the focus of the beam.
• Horizontal dipole trap: We use a 𝑓 = 100 mm lens to collimate the beam,a 𝑓 = 300 mm and a 𝑓 = 50 mm cylindrical lens to make the beam shapeelliptical with a 6:1 horizontal to vertical ratio. A 𝑓 = 1000 mm is used to focus
the beam down to a 1/𝑒2 radius of approximately 240 µm × 40 µm (horizontal
× vertical) at the atoms.
• Vertical dipole trap: We use a 𝑓 = 33 mm lens to collimate the beam and a𝑓 = 500 mm to focus the beam down to a 1/𝑒2 radius of of approximately
106 µm at the atoms. A QWP before the atoms is used to circularly polarize
the beam. This allows more efficient cooling into the optical dipole using the
red MOT beams [20].
Optics for intensity stabilization are placed after the atoms on the mezzanine andshown in Figure 3.13. We use thin-plate 99:1 beam splitters to dump most of the highpower light into beam dumps. The beam dumps are water cooled with a custommade copper block to prevent heating of the mezzanine and drift of optics. The
remaining power is focused onto photodetectors connected to the intensity servos.
See Section 4.1 for a discussion of the intensity servo and its components.
27

EX PER IM EN TA L A PPAR ATUS
Piezzo-mounted
mirror
EtalonOutputcoupler

Birefringent
fi lter

Mirror6.2 W

Verdi V18
pump laser

Optical
diode

18 W

Input coupler Ti:Sapphire crystal M SQUARED
SolsTiS

Periscope

12 cm to 6.5 cm

1:2

Iris
AOM
~1W

step-index
air-gapped fi ber

f = 50 cm+230 MHz~2W

5:3 cylindrical
HWP

To wavemeter

PBS
PBS HWP

Beam dump

FIGURE 3.14: The 813 nm laser setup for one lattice beam; The beam path is shown in red(green: 532 nm pump) with arrows indicating the propagation direction. Light red lines referto the 0th order of an AOM. Laser power is denoted in red font. Coiled blue lines denote
step-index fibers (orange: multi-mode fibers). The internal beam path and optical elementsof the Ti:sapphire laser are shown on the top (from Ref. [21]). The elements surrounded by adashed line will be duplicated for all three beams (indicated by the red dotted line). Notethat the size of the SolsTiS laser is actually much smaller than the pump laser head.

In summary, the 1064 nm laser system is able to deliver a maximum total power ofapproximately 20 W at the atoms using Holey fibers. This makes our setup robust
against pointing noise from the fiber amplifier and allows us to swap the fiber laser
in an event of failure quickly without the need to realign the optics relative to theatoms.

3.3 THE 813 NM LASER SETUPThe 813 nm laser setup is not completed as of writing this thesis. However, a basic
overview of the setup is given in this section since some work described in the
Section 4.2 refers to the 813 nm laser setup. We limit the discussion to Figure 3.14,
which shows the setup for a single beam path and the Ti:sapphire laser itself.

28

EX PER IM EN TA L A PPAR ATUS

A. TI:SAPPHIRE LASER
We use a Ti:sapphire laser (M SQUARED SolsTiS) pumped by 18 W of 532 nm light(Coherent Verdi V18). The SolsTis laser features a sealed and alignment-free designthat is supposed to deliver frequency stable light. However, first measurements
showed drift of the output power, pointing noise, and frequency drift after a cold
start. We therefore usually turn the laser on two hours before we intend to use it. The
Ti:sapphire laser can be tuned by controlling the angle of a birefringent plate that
introduces wavelength-dependent loss and by adjusting the length of an etalon in the
beam path inside the bow-tie cavity (see Figure 3.14). The etalon can be locked using
a dither lock, which stabilizes the (short-term) frequency of the laser. The maximum
output of the Ti:sapphire laser at 813 nm is 6.2 W (with a single mirror in the beampath). The laser controller is connected via Ethernet and all settings of the laser canbe directly controlled from a web browser on any computer in the lab. Additionally,
the controller has a JSON (JavaScript Object Notation)8 based interface that we use
to lock the frequency.

8

Data-interchange format and readily implemented in most modern programming languages.

29

4

EXPERIMENTAL TECHNIQUESThis section contains information on techniques employed in the experiment. We
explain the intensity servo used for the 1064 nm (and 813 nm) laser system as wellas the wavemeter based frequency servo for the 813 nm laser.

4.1 INTENSITY SERVO
The discussion in Section 2.3 showed that laser intensity noise leads to heatingof atoms in optical dipole traps. We use a servo loop to suppress intensity noise.
The details of the servo loop are discussed in this section. First, we are interested
in fundamental limits of the minimum laser noise floor and explain the photon
shot noise limit in Subsection A. This allows us to understand the design of thephotodetector.

A. PHOTON SHOT NOISE AND THERMAL LIGHT SOURCESFirst we establish a basic understanding of photon number fluctuation in an ideal
light source. Assume we have a light source with constant intensity, i.e. constant
power 𝑃. This light has a mean photon flux 𝑛 of
𝑛=

𝑃
.
~𝜔

(4.1)

Here 𝜔 is the angular frequency of the photons. The standard deviation is given as√𝑛, which we call photon shot noise [6, p. 778]. Furthermore, the signal-to-noise30

E XPE R IME NTAL T ECHNIQU E S

ratio SNR is given as
SNR =

𝑛
𝑃
= √𝑛 = √ .
√𝑛
~𝜔

(4.2)

The SNR is a function of the power 𝑃, so in general high signal-to-noise ratio can beachieved using high power. Quantum mechanics sets a fundamental limit on how
little noise light can have. Most light sources, such as lasers, are coherent states, witha photon number noise set by √𝑛𝜏 where 𝜏 is the averaging time. Photons from a
coherent state follow Poissonian statistics described in the equations above. The SNR
in Eqn. 4.2 is the theoretical limit on the performance of the intensity servo.
More interestingly, we can also show that photons from a thermal noise source canfollow Poissonian statistics. This result allows us to easily measure photon shot noisewith a photodetector. We know that the probability 𝑝(𝑛) for 𝑛 photons at temperature
𝑇 is given by the Bose-Einstein distribution [6, p. 466]
1
𝑛 𝑛
(
)
𝑛+1 𝑛+1(4.3)

1.
exp(~𝜔/𝑘B 𝑇) − 1(4.4)

𝑝(𝑛) =
with𝑛=

The Bose-Einstein distribution has standard deviation √𝑛 + 𝑛2 and thus is superPoissonian. However, we have only considered one mode out of the quasi-continuumof thermal modes in the visible spectrum. Photodetectors can be broadband anddetect a multitude of modes. Before we proceed, consider these two remarks frombasic probability theory [22, p. 372]:
1. The Bose-Einstein distribution is a geometric distribution of the form 𝑝(𝑛) =(1 − 𝑝)𝑛 𝑝 for (1 − 𝑝) ≡ 𝑛/(𝑛 + 1).
2. The distribution of a sum of 𝑁 independent geometric distributions is a negative binomial distribution given as
𝑁+𝑛−1 𝑛Γ(𝑁 + 𝑛)𝑝𝑁 (𝑛) = () 𝑝 (1 − 𝑝)𝑁 =𝑝𝑛 (1 − 𝑝)𝑁 .
Γ(𝑛 + 1)Γ(𝑁)
𝑛

(4.5)

Here Γ is the Gamma-function with Γ(𝑛) ≡ (𝑛−1)!. The mean of the geometric
distribution is given by 𝑝𝑁/(1 − 𝑝). So for constant mean 𝐶 (independent of𝑁) 𝑝 = 𝐶/(𝑁 + 𝐶).
Back to physics – let 𝑁 = 𝑁m where 𝑁m is the number of thermal modes and assume
𝑁m → ∞, but while keeping the intensity and therefore the mean of 𝑝𝑁m (𝑛) is31

E XPE R IME NTAL T ECHNIQU E S

0.15

Neg. Bin., N = 1
Neg. Bin., N = 10
Neg. Bin., N = 100
Poisson

Probability (x = n)

n

0.10

0.05

0.00

0

5

15

10

20

25

FIGURE 4.1: Poisson andnegative binomial distributions for different 𝑁(see Eqn. 4.5) and constantmean 𝑛. Note that for
𝑁 = 1 the negative binomial distribution is equal
to the Bose-Einstein distribution.

nconstant. We denote this average by 𝑛. Then 𝑝 = 𝑛/(𝑛 + 𝑁m ) and we find𝑁mΓ(𝑁m + 𝑛)𝑛𝑛
1
𝑝∞ (𝑛) = lim)(
𝑁m →∞ 𝑛! ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Γ(𝑁m )(𝑁𝑚 + 𝑛)𝑛 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
1 + 𝑛/𝑁m𝑁m →∞󳨀󳨀󳨀󳨀󳨀󳨀→ 1=

𝑛𝑛 exp(−𝑛)
.
𝑛!

(4.6)

𝑁m →∞

󳨀󳨀󳨀󳨀󳨀󳨀→ exp(−𝑛)This is the Poisson distribution. See Figure 4.1 for a plot of the negative binomial
distribution for different 𝑁m compared to the Poisson distribution. If we consider
many thermal modes, the light is Poissonian in the limit of infinitely many modes.
This condition is usually fulfilled in the visible spectrum. Thus, a thermal light sourceis shot noise limited in terms of Eqn. 4.2 and is suitable for measuring photon shotnoise for a given dc intensity.

B. PHOTODIODES
For detection of photons and measuring laser intensity a suitable detector is required.
Semiconductor photodiodes work well for visible to near-infrared light. They havehigh quantum efficiency (∼ 0.9) compared to photo multiplier tubes and high bandwidth (> 10 GHz) compared to thermal detectors. We are interested in two aspects ofphotodiodes: Signal-to-noise ratio and bandwidth. This section explains limitationsof both.
Photodiodes consist of a p-n junction, which is an interface of a p-type (excess ofholes) and a n-type (excess of electrons) semiconductor. At the interface electrons
and holes can diffuse and the depletion region is formed. Photons that hit a p-njunction can create an electron-hole pair if their energy ~𝜔 is larger than the band
gap of the semiconductor used. Those electron-hole pairs contribute to a current thatcan be measured on the terminals of the p-n junction. This current is called photocurrent. The probability that a photon creates an electron-hole pair as a function of
the photon wavelength 𝜆 is the quantum efficiency 0 ≤ 𝜂(𝜆) ≤ 1 of the photodiode [6,p. 753]. The responsitivity R = 𝜂𝜆/1.24 is quantity that is defined in terms of 𝜂 and 𝜆.
32

E XPE R IME NTAL T ECHNIQU E STABLE 4.1: Performance figures of photodiodes from Hamamatsu used for intensity stabilization. All values are quoted from the manufacturer data sheet [25, 26].
Operating conditions
Model

Semiconductor

DiameterWavelength

Responsitivity

Bias volt.

Capacitance

Dark curr.

S3072
G12180SiliconInGaAs 3 mm
 2 mm813 nm1064 nm≈ 0.55 A/W≈ 0.59 A/W

−12 V
−5 V

≈ 8 pF≈ 200 pF

0.12 nA
2 nA

It is usually quoted in units A/W and allows direct calculation of the photo current
from the incident light power. Silicon photodiodes are inefficient at wavelength& 1000 nm (𝐸(𝛾, 1000 nm) = 1.23 eV) in terms of 𝜂 and speed. The reason for this isthat silicon (band gap ≈ 1.1 eV at 300 K [23]) is almost transparent for NIR photons.
Thus, fewer photons are absorbed (smaller 𝜂) and the photons are absorbed outside
the depletion region. Once the electron-hole pair is created outside the depletion
region, it can take a substantial amount of time before they drift to the junction
and contribute to the photocurrent. InGaAs (band gap ≈ 0.75 eV at 300 K [24])photodiodes are a better choice for 1064 nm – our optical dipole trap wavelength(𝐸(𝛾, 1064 nm) = 1.17 eV). Using Eqn. 4.2 and not considering internal sources1 ofnoise, the SNR of the photodiode for a shot noise limited light source is just given as
SNRPD = √𝜂

𝑃
.
~𝜔

(4.7)

So the SNR is reduced by √𝜂 since only 𝜂 photons out of a single photon contributeto the photo current. For a given wavelength we want to maximize 𝜂(𝜆) and 𝑃 to
maximize the SNR. However, we can not measure arbitrarily large powers 𝑃 due
to the damage threshold of the semiconductor and the non-linearity of the p-njunction. Photodiodes generate a photocurrent proportional to the incident light
power. We need to convert this current to a voltage that can be measured. The
simplest current-to-voltage converter can be realized using a resistor 𝑅. If the resistor
is connected to the two terminals of the photodiode we can measure the voltage
𝑈ph = 𝑅𝐼ph = 𝑅𝜂𝑃/~𝜔. The resistor sets the gain, so for high gain a large resistance 𝑅
is required. The bandwidth, which sets the maximum speed the voltage will respondto a change in light intensity, is then given by the 𝑅𝐶-constant 𝑓−3 dB ≈ 1/2𝜋𝑅𝐶j .
Here 𝐶j is the capacitance of the p-n junction, which basically acts as a parallel plate
capacitor separated by the thickness of the depletion region. Usually 𝐶j is lower
for Silicon than InGaAs photodiodes and 𝐶j increases with the active area for thesame semiconductor. From electrostatics, the diode junction capacitance 𝐶j is given
1

Internal sources of noise are Johnson (thermal) noise from the shunt resistance of the diode andcurrent shot noise from the dark current.

33

E XPE R IME NTAL T ECHNIQU E S

𝐶f

𝑅f

−

𝑈out
𝐶s

𝐼D

+

−𝑉B
FIGURE 4.2: Simplified transimpedance amplifier circuit. The photodiode is shown with asimplified equivalend circuit: Current source 𝐼D and capacitance 𝐶s . Note that we can include
differential and common-mode capacitance of the terminals of the operational amplifier in𝐶s since the positive terminal of the operational amplifier is connected to ground.

by [27]:𝐶j2 ∝1.
(𝜙𝑖 − 𝑈)

(4.8)

Thus, when we reverse bias photodiodes, i.e. 𝑈 < 0 V, the capacitance is reduced,
which is favorable for high speed applications. The decrease of the capacitance canbe explained by considering that the depletion region thickness is increased underreverse bias voltage. Additionally, a reverse bias voltage also helps to operate thediode in the linear regime. However, the dark current increases as a function ofreverse bias, which limits the minimum detectable light level. Table 4.1 lists theperformance figures of the photodiodes used for intensity stabilization. We use
 3 mm silicon photodiodes for intensity stabilization of the 813 nm laser for the
optical lattice and  2 mm InGaAs photodiodes for intensity stabilization of the1064 nm laser for the optical dipole trap. The large active area of the photodiodes
allows us to be less sensitive to pointing noise.

C. LOW NOISE TRANSIMPEDANCE AMPLIFIER
Operational amplifier based transimpedance amplifiers (TIA) can be used to increase
the bandwidth of photodiodes for a given gain and junction capacitance. This allows
higher bandwidth than discussed in the last section. We discuss the principle ofoperation for these transimpedance amplifiers in this section.
34

E XPE R IME NTAL T ECHNIQU E S

2
S3 0 7843/
OPA

2
S3 0 7
RC/

80

57
105

80

58

G 12 1
843/
OPA

59

12 1
RC/G

Gain (dB)

60

106
107
Frequency (Hz)108

FIGURE 4.3: Bandwidth boost byusing a transimpedance amplifier
for the photodiodes used in the experiment. The frequency response
for simple resistor (RC) and transimpedance (OPA843) current-tovoltage conversion is shown forthe operational amplifier OPA843.
Model numbers refer to the photodiodes in Table 4.1.

A simplified transimpedance circuit with a reverse-biased photodiode is shown in
Figure 4.2. We know that the output voltage 𝑈out of a non-ideal operational amplifier(gain < ∞) as a function of the inverting 𝑈− and non-inverting 𝑈+ input is given
as [28, p. 233]
𝑈out = 𝐴(𝑈+ − 𝑈− ).

(4.9)

Here 𝐴 is the open-loop gain of the amplifier. It approximately follows the frequency
and phase response of a first-order low-pass filter [28, p. 191]
𝐴(𝜔) =𝐴 DC.
1 + 𝑖𝜔/𝜔𝑐(4.10)

Here 𝐴 DC is the open-loop gain at DC and 𝜔𝑐 is the frequency at which the gain is
3 dB lower than at DC. If we use Kirchhoff ’s current law to solve for 𝑈out in Figure 4.2
we easily find𝑖𝜔𝐶s −1
1 + 𝑖𝜔𝐶f 𝑅f
1
𝑈out (𝜔) = R𝑃 {
(1 +
)−
} .
𝑅f
𝐴(𝜔)
𝐴(𝜔)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟(4.11)

Gain as a function of the frequency 𝜔Here, R is the responsitivity of the photodiode, 𝑃 is the power of the light incident
on the photodiode, and all other quantities are defined in Figure 4.2. The large openloop gain 𝐴(𝜔) of the photodiode enhances the bandwidth of the photodetector as
shown in Figure 4.3. The bandwidth, which is defined as the frequency of the −3 dB
point of the gain, can be estimated as [29, p. 627]:
𝑓−3 dB ≈√𝑓RC 𝑓GBP
𝑓= √ GBP .
2
8𝜋𝑅f 𝐶s(4.12)

Here, 𝑓RC is the cutoff frequency defined by the feedback resistor 𝑅f and the diodecapacitance 𝐶j . The gain-bandwidth product of the operational amplifier is given by
𝑓GBP .
The feedback capacitor 𝐶f is required to make the circuit stable. The optimal value
35

E XPE R IME NTAL T ECHNIQU E S√
Output noise (nV/ Hz)

1000

Shot noise
P = 10 mW
P = 1 mW

100
10

P = 0.1 mW

Total TIA noisef = 100 Hz
f = 1 MHz

1
0.1 2
10

Voltage noise

oise
ent n
Curr

103
Feedback resistance R f (Ω)

104

FIGURE 4.4: Output voltage noise from the operational amplifier OPA843 in the transimpedance circuit as a function of the feedback resistance 𝑅𝑓 for two frequencies (100 Hz
and 1 MHz). The photon shot noise is calculated for quantum efficiency 𝜂 = 0.6. The region
where the detector is shot noise limited can be estimated from this plot. All values arecalculated from the specifications given in the manufacturer data sheet [30].

is given by 1/2𝜋𝑅f √𝑓GBP 𝑓RC [29, p. 628]. However, stray capacitances present on
printed circuit boards usually require an experimental selection of the exact value.
This can be done by increasing the size of the capacitor until the output of the TIA
stops ringing and then using a slightly larger capacitance to compensate for possible
variations in component specifications.
We found the three key factors that determine the maximum bandwidth for a fixedtransimpedance gain 𝑅f :• Source capacitance (𝐶s )• Feedback capacitance (𝐶f )• Gain-bandwidth product of the operational amplifier (𝑓GBP ).
Table 4.2 lists performance figures for the operational amplifiers used for the TIAs.
Increasing the bandwidth of photodetectors using an operational amplifier is great but there is no free lunch. Like most other electronic devices, operational amplifiers
introduce noise. Using Kirchhoff ’s circuit laws, assuming that all noise is uncorrelated and that the shunt resistance of the photodiode 𝑅shunt is much larger2 than
the feedback resistance 𝑅f , the total mean-squared noise of the TIA 𝑢N in units of2This is a valid assumption for moderate gain.

36

E XPE R IME NTAL T ECHNIQU E S

PCBAluminum case

FIGURE 4.5: Photo of
the photodetector with thedifferent features. The lidof the aluminum case isshown as half transparent
to reveal the PCB inside
the case. The case measures 6.9 cm × 4.9 cm ×2.54 cm.

Power connector

PhotodiodeLED
OutputLens tube (0.5 in)nV/√Hz is given as [31]
√
√
√
(2𝑒𝑖
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
dark +

𝑢N =

(i)

√

2
4𝑘B 𝑇
4𝑘B 𝑇
2
2 + 𝑒2 (1 + 𝑅f ) .
++ ⏟⏟⏟𝑖⏟⏟
)
𝑅
𝑁⏟⏟
𝑁⏟⏟
𝑓 ⏟⏟⏟⏟⏟
𝑅shunt𝑅f
𝑅shunt⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
(ii)

(iii)

(iv)

(4.13)(v)

The different noise terms are the following:(i) Current shot noise of the dark current 𝑖dark(ii) Johnson noise of the shunt resistance 𝑅shunt(iii) Current noise of the operational amplifier 𝑖N(iv) Johnson noise of the feedback resistor 𝑅f at temperatue 𝑇 (= 300K)(v) Voltage noise of the operational amplifier 𝑒N .
Note that in general the noise is white, i.e. flat across all frequencies. However,most operational amplifiers have 1/𝑓 voltage and current noise, which introduce afrequency dependence of Eq. 4.13. Field-effect transistor (FET) based operational
amplifiers usually have higher voltage noise but lower current noise whereas the
opposite applies to bipolar operational amplifiers. The output voltage noise depends
on the voltage noise of the operational amplifier plus the current noise multiplied by
the gain 𝑅f . Thus, a rule of thumb is that for low gain bipolar and for high gain FET
operational amplifiers yield the lowest overall noise. Since the available power for theintensity servo is large, we do not require high gain and choose a bipolar operational
amplifier. Figure 4.4 shows total noise and the different contributions for the TIA
used for the photodetectors. To fully characterize and control the amplitude noiseof the light, the circuit’s electronic noise floor must be below the photon shot noise.
Additionally, we are ultimately limited by the photon shot noise of a constant intensity37

E XPE R IME NTAL T ECHNIQU E S√
Noise (nV/ Hz)

10000
1000
100
10
10−1

100

101 102 103
Frequency (Hz)

104

105

FIGURE 4.6: PSD of the TIA noise(without photocurrent) using theoperational
amplifier
OPA843.
At ≈ 200 Hz the noise level reaches
the noise floor of 10 nV/√Hz. Thisagrees well with the calculated values.
The noise peaks at 60 Hz and 180 Hz
are due to noise on the power supply.
The origin of the noise peaks at≥ 40 kHz are unknown but seem tobe electronic pickup from sources in
the lab since they are not present in
the spectrum for measurements at
another location.

light source. If the noise floor of our photodetector is below the amplified shot noisefor a specific light power 𝑃0 we call it shot noise limited for 𝑃0 . When measuring
intensity noise we have to make sure that we are actually measuring laser noise andnot detector noise. For verification we could measure the noise of the photodetector
while blocking light to see the dark response. However, if the frequency response ofour detector has unusual resonances, we would not be able to see them in the dark
response. We can use a result from the previous section and measure the intensity
noise of a thermal light source. The detector is not mode selective and therefore
the thermal light is Poissonian. We choose the power of the thermal light sourcesuch that the detector gives the same dc value as for the actual laser intensity noise
measurement. The shot noise of the light is amplified by the transimpedance gain𝑅f . Thus, the expected PSD of the shot noise has a value of √2𝑒𝑈/𝑅f (in units ofV/√Hz) where 𝑒 is the charge of the electron and 𝑈 the measured dc voltage. We
can then compare the thermal intensity noise level to the noise floor of the dark
response and evaluate whether the detector is shot noise limited or not. We perform
this measurements for all photodetectors used in the experiment. If the detector isnot shot noise limited for a given power 𝑃0 we increase the power until the detector
is shot noise limited.
We are using the operational amplifier OPA843 from TI as the TIA in the photodetecTABLE 4.2: Performance figures for the operational amplifiers (OPA843) and TIA used
for intensity stabilization. The photodiodes are those described in Table 4.1. All values arequoted from the manufacturer datasheet [30].
Calc. TIA bandwidth for 1 kΩ
Model

GBP

Voltage noise (freq.)

Current noise (freq.)

S3072 - Si

G12180 - InGaAs

TI OPA843

800 MHz

9 nV/√Hz (100 Hz)

7 pA/√Hz (100 Hz)

55.0 MHz12.2 MHz38

E XPE R IME NTAL T ECHNIQU E S

𝑁(𝑠)
𝑅(𝑠)
−

C(s)

G(s)

+

𝑌(𝑠)

H(s)

FIGURE 4.7: Closed servo loop with negative feedback. The quantities are given as: input
𝑅(𝑆), system 𝐺(𝑆), noise 𝑁(𝑆), loop filter/controller 𝐶(𝑠), feedback fraction 𝐻(𝑠), and output
𝑌(𝑠).

tors. It offers a pretty large gain-bandwidth product of 800MHz while offering gooddc performance in terms of noise (9 nV/√Hz at 100 Hz). This is useful, since we use
large area photodiodes (i.e. large junction capacitance) to reduce sensibility to pointing noise and make the setup more robust, i.e. not requiring frequent realignmentof optics. The performance figures for both operational amplifiers and the resulting
TIA can be found in Table 4.2. The TIA circuit used for detection of intensity noise
is similiar to the basic circuit depicted in Figure 4.2. The full circuit for the TIA used
for detection of intensity noise can be found in the Appendix. Additionally, variable
regulators allow selection of the reverse biasing voltage by just two resistors. We use
the OPA843 with an unbalanced power supply to increase the maximum positive
voltage swing of the ±5𝑉 supply operational amplifier. The PCB is designed witha 50 Ω transmission line (coplanar) on the output of the TIA to allow interfacingwith 50 Ω systems, e.g. RF mixers. A solid aluminium case is used to shield theelectronics from RF and other noise sources present in the laboratory. The case is acustom design to fullfill the strict space requirements on the optical table. It features
a LED power indicator and a Hirose power connector that matches the standard
power cable used in the lab. All connectors are on the top of the case to prevent
cables from blocking optical beam paths. The aperture for the photodiode is tapped(0.535 inch) to allow attachment of Thorlabs 0.5 inch lens tubes. This allows to easily
block background radiation and adding or swapping of neutral density filters. The
case is made from two pieces of solid block aluminium machined on a CNC mill.
Figure 4.5 shows a photo of the full assembly.

D. ANALOG SERVO LOOP
We measure the PSD of the intensity noise to calculate the resulting heating rate
as a function of trapping frequency. The laser light is focused onto a low-noise

39

E XPE R IME NTAL T ECHNIQU E S

photodetector and the spectrum is recorded with a FFT machine3 , which fouriertransforms the time domain signal. Figure 4.9 shows the basic measurement setup
and Figure 4.10 the results for the 1064 nm laser system.
We use an analog servo loop to reduce laser intensity noise. This is necessary tominimize heating as described in Section 2.3. We use the JILA intensity servo, which
has been developed by the electronics shop at JILA. First, we discuss how this loop
filter works. Secondly, we explain the intensity stabilization setup and report theperformance.
We use the Laplace transform to understand the behavior of a basic closed loop
time-invariant system. This allows us to evaluate the JILA AOM intensity servo loop
filter. The Laplace transform of a basic closed loop system is shown in Figure 4.7.
Simple algebra yields the following expression for the output 𝑌(𝑠)𝑌(𝑠) =

𝐶(𝑠)𝐺(𝑠)𝑅(𝑠)𝑁(𝑠)+.
1 + 𝐺(𝑠)𝐶(𝑠)𝐻(𝑠) 1 + 𝐺(𝑠)𝐶(𝑠)𝐻(𝑠)(4.14)

Here all quantities are defined in Figure 4.7. Note that 𝑠 = 𝑖𝜔 where 𝜔 is an angularfrequency. We find two conditions:• For large 𝐺(𝑠)𝐶(𝑠)𝐻(𝑠) the noise 𝑁(𝑠) can be suppressed significantly on the
output 𝑌(𝑠), i.e. in general we want 𝐺(𝑠)𝐶(𝑠)𝐻(𝑠) ≫ 1 – especially at frequencies to which the atoms are sensitive, i.e. at twice the trapping frequency4 .
• For 𝐺(𝑠)𝐶(𝑠)𝐻(𝑠) = −1 the output is unbound, i.e. the system becomes unstable. Thus, |𝐺(𝑠)𝐶(𝑠)𝐻(𝑠)| ≪ 1 or |𝐺(𝑠)𝐶(𝑠)𝐻(𝑠)| ≫ 1 if arg {𝐺(𝑠)𝐶(𝑠)𝐻(𝑠)} =𝜋.
Controlling the laser intensity requires introducing a nonlinear acousto-optical orelectro-optical component. We use an acousto-optical modulator (AOM) to control
the laser light intensity. The AOM is a delay limited system and we discuss theinfluence on stability of the servo loop here. A piezo-electric transducer drives
a sound wave in the AOM crystal. This generates a modulation of the index ofrefraction in the crystal as a function of the sound wave intensity. The laser beam
enters the crystal perpendicular to the sound wave and is diffracted into multipleorders. This effect is similar to diffraction on a grating5 . The intensity of the soundwave determines the intensity of the different orders relative to the undiffracted (0th )3

FFT machines are real-time spectrum analyzers that have no lag due to a frequency sweep. This
allows to observe dynamics in the frequency space. However, we are only using the FFT machine here
because it is available in the lab and has low noise close to dc (1 Hz) and hight resolution (800 FFT
points).
4Note that the noise cannot be suppressed below the noise floor of our detector. We might see thaton an in-loop detector but an out-of-loop detector will show the absolute/real noise suppression.
5Note that the light is frequency shifted since the diffraction occurs on a moving and not a stationary
modulation in the refractive index [6].

40Phase (deg)Gain (dB)

E XPE R IME NTAL T ECHNIQU E S

80
40
0
−40

0
−45
−90
−135
−180
10−2

DII: 1/s2 + 1/s

PI: 1/s + 1

10−1
100
101
Frequency (Hz)102

FIGURE 4.8: Bode plots of openloop transfer functions for the double
integrator plus single integrator (DII,
thick line) and proportional plus integrator (PI, thin dashed line). The
DII has a gain boost close to DC
and compared to the PI. The −180∘phase shift of the DII close to DC does
not cause instability since the gain ismuch larger than one. Note that thefrequency and gain scale is arbitrary
and does not reflect the values in our
setup.

order. Thus, we can modulate the intensity by changing the RF drive power. The
AOM has two speed limitations, which are given as function of the speed of soundin the crystal 𝑣s :• The sound wave has to travel from the transducer to the position of the beam,
the delay as a function of the distance 𝑑 is given as 𝜏 = 𝑑/𝑣s .
• The sound wave has to propagate through the non vanishing beam waist(assuming a Gaussian laser beam), the rise time as function of the 1/𝑒2 beam
radius 𝑤 is given as 𝜏 = 𝛼𝑤/𝑣s . Here 𝛼 is a constant that is defined based on
the definition of the rise time (i.e. from 10 % to 90 %).
If the RF drive power is changed, there will be a finite time delay until we see a changeof the intensity distribution across the diffracted orders.6 This time delay limits theservo loop, since it takes a duration of 𝜏 to write a correction onto the laser light.
The time delay can be described by a phase shift of −𝜔𝜏 where 𝜔 is the frequency ofinterest. If this phase shift equals 𝜋 the feedback becomes positive and the system
unstable. We cannot compensate for this phase shift since the AOM introduces atime delay, which would require us to predict the intensity noise at a point in the
future in order to write the correction onto the driving RF signal ahead of time. Thisis fundamentally different from a pure (constant) phase shift that is introduced bye.g. an integrator. For most systems the gain is below 1 before the time delay phaseshift is 𝜋. However, AOMs have a relatively flat gain across a wide bandwidth, which
is mainly determined by the bandwidth of resonance circuit driving the transducer.
Thus, the gain does not roll off before the time delay phase shift is 𝜋. We need an
integrator (𝐾/𝑠) in the loop filter 𝐶(𝑠) to make sure the gain is below 1 at this point.
This system can then be controlled by a PI-controller, which results in a controller
6Note that the speed of light is much larger than the speed of sound, which is ≈ 3000 m/s for the
AOMs used in the experiment. Thus, the delay is solely determined by the AOM

41

E XPE R IME NTAL T ECHNIQU E S

AOM

LO

RF

Beam dump±1st HP fiber Atoms
1 + 1/(sCR)PD

out-of-loop

99/1

50/50

-1/(sCR)PD

in-loopIF
Output

Setpoint

FIGURE 4.9: The intensity servo setup; the JILAAOM intensity servo (loopfilter) is shown in the graybox (simplified), where thetransfer functions of thetwo operational amplifier
circuits are shown above.

JILA AOM intensity servo

with a double integrator and a single integrator:
𝐶(𝑠) =𝐾𝑖𝑖 𝐾𝑖+𝑠2
𝑠(4.15)

Here 𝐾𝑖𝑖 , 𝐾𝑖 are constants. This is how the JILA AOM intensity servo loop filter
works. It allows higher gain close DC compared to a pure PI-controller as you cansee in the Bode plots in Figure 4.8).
Figure 4.9 shows the setup we use to cancel intensity noise and also measure theperformance of our servo loop. The overall intensity is controlled with an AOM and
the ±1st order is coupled into an optical fiber that delivers the light to the atoms. We
use a 99/1 beam splitter to dump most of the power into a beam dump. The remaining
power is then split and focused onto two photodetectors. One of them is connectedto the intensity servo (in-loop) and one is used to measure the performance of theservo (out-of-loop). The signal of the in-loop photodetector can be teed off and also
be measured on the FFT machine. The output of the intensity servo is fed into the
IF port of an mixer, which controls the LO/RF port isolation and thereby allows
to control the RF power level that drives the AOM. The intensity servo setpoint is
controlled by a voltage from the DAC board that runs the experimental sequence.
This allows to ramp the intensity up or down in an arbitrary way.
Figure 4.10 shows the performance of the intensity servo for the 1064 nm. The overall
−20
−40
−60
−80
−100
−120
−140
−160

RIN (dBc/Hz)

Servo unlocked (OL)
Detector (shot) noise
Servo locked (OL)Servo locked (IL)

100

101
102
Frequency (Hz)

103

42

FIGURE 4.10: Intensity servo
performance for 1064 nm laser
system. The detector shot noise
is measured with a thermal
light source yielding the samedc light level. (OL) = out-ofloop and (IL) = in-loop.

E XPE R IME NTAL T ECHNIQU E S

noise beyond 100 Hz is reduced to −140 dB, which is close to the shot noise limitof the laser light. Shortcomings of our current intensity servo are that the responseof the mixer is not linear, i.e. over a wide range of setpoint voltages it would be
advantageous to linearize the mixer behaviour. This could be done using a digital
instead of an analog loop filter. Further improvements include adding feed-forward
during intensity ramps to decrease the error signal and improve the settling time.

4.2 FREQUENCY SERVO
The 813 nm laser we are using for the optical lattice has frequency drift on longtimescales (see Figure 4.12) even when locked using the built-in etalon dither lock7 .
This introduces two potential problems:• The magic wavelength condition is not fulfilled anymore and a differential ac
Stark shift is introduced. This is especially a problem for high precision, i.e.
clock measurements.
• All three lattice beams are retro-reflected and the mirrors define a fixed nodeof the electro-magnetic wave. Thus, when the wavelength changes, not only
the spacing of the atoms is modulated but also the trap position. This canintroduce heating as discussed in Section 2.3.
This is the motivation for building a simple frequency servo for the 813 nm laser.
Luckily, the 813 nm laser offers an all digital interface, which allows changing thepiezo voltage on one of the mirrors in the ring cavity. We can use a wavemeter thatis regularly calibrated to the clock laser at 698 nm (stability of 10−16 at 1 s [2]) as afrequency standard. This should in principle allow to transfer the stability of theclock laser to the 813 nm laser. This section establishes a basic understanding of
discrete PID controllers, explain the implementation in our setup, and the resulting
performance of the servo.

A. DISCRETE PID CONTROLLER
Continuous linear time invariant systems are usually analyzed using the Laplacetransform of the transfer function. This allows to evaluate transfer function of
combined systems and stability criteria easily. However, digital controllers operate
7

We do not now the exact way the dither lock is implemented in the M SQUARED SolsTiS laser.
However, we can describe the general operating principle of a dither lock: the laser frequency ismodulated with frequency Ω when the dither lock is engaged. The etalon transmission is measured
and a lock-in amplifier with frequency Ω produces an error signal that is proportional to the offset
from the etalon resonance. This error signal can be used to feed back onto the laser frequency using amirror mounted on a piezo.

43

E XPE R IME NTAL T ECHNIQU E S

on discrete and not continuous space. This section discusses a method to derive a
discrete transfer function from a continuous transfer function. Specifically, we aretransforming the continuous transfer function of the PID controller.
The transfer function 𝐶PID (𝑠) of the continuous PID controller in s-space is given
as [32]
𝐶PID (𝑆) = 𝐾p +𝐾i
+ 𝐾d 𝑠.
𝑠(4.16)

Here, 𝐾p , 𝐾i , 𝐾d is the proportional, integral, and derivative gain. We use the ztransform to discretize this controller, which is given as [32, p. 194]
𝑠←

2 𝑧−1
.
𝑇𝑧+1

(4.17)

Here, 𝑇 is the time step in the discrete system. Thus, we can obtain a transfer functionin z space by replacing 𝑠 with the expression in Eqn. 4.17. A full derivation of thez-transform is not given here but a very instructional derivation can be found in [33].
The z-transform takes a function from complex s-space to the (discrete) z-space,which is given by the unit circle in C [33, p. 561]. We can then express the transfer
function in terms of a recursive function on discrete time space that can be easily
implemented in a computer program or hardware logic, i.e. FPGAs. It is useful tobring the transfer function 𝐻[𝑧] in z-space to a general form[33, p. 563], whichallows straight forward definition of the recursive function 𝑦[𝑛] on discrete time
space. If the form of 𝐻[𝑧] is given as
𝐻[𝑧] =−𝑖𝑎0 + ∑𝑁
𝑖=1 𝑎𝑖 𝑧−𝑖
1 − ∑𝑁𝑖=1 𝑏𝑖 𝑧.

(4.18)

Then 𝑦[𝑛] is simply given as
𝑁𝑁

𝑦[𝑛] = (∑ 𝑎𝑖 𝑥[𝑛 − 𝑖]) + (∑ 𝑏𝑖 𝑦[𝑛 − 𝑖]) .
𝑖=0(4.19)

𝑖=0

Here, 𝑦[𝑛] is the output of the system and 𝑥[𝑛] is the input of the system as shown in
Figure 4.11. Thus, for the continuous PID controller in Eq. 4.16 we find (after sometedious calculations)
𝐻[𝑧] =

𝑥[𝑛]𝑎 +∑𝑁 𝑎 𝑧−𝑖
𝐻[𝑧] = 0 𝑁𝑖=1 𝑖 −𝑖1−∑𝑖=1 𝑏𝑖 𝑧

𝑎0 + 𝑎1 𝑧−1 + 𝑎2 𝑧−2.
1 − 𝑏2 𝑧−2

𝑦[𝑛]

44

(4.20)FIGURE 4.11: Discrete transfer function
𝐻[𝑧] on z-space. The input and output are
given by 𝑥[𝑛] and 𝑦[𝑛], both are discrete
time series.

E XPE R IME NTAL T ECHNIQU E S

Error (MHz)

35

0

1

Time (hr)
2

4

3

0
Servo on-35

Servo off

-350

Frq. Noise (dB)

0

−15

−30

−45

−60

10−4

10−3
10−2
Frequency (Hz)10−1

FIGURE 4.12: Wavemeter lock error signal in time space (top) and frequency
space (bottom). The thick black traceshows the in-loop error signal with PID
feedback every 𝑇 = 5 s enabled. The red
trace shows the error signal of the unlocked laser. Both data traces were taken
one hour after a cold start of the laser.
Note that the y-axis in the plot on the
top has piece-wise linear scale from 35
to −35 MHz and from −35 to −350 MHz.
The error signal in frequency space (bottom) is normalized to the maximum amplitude in the spectrum of the unlocked
signal.

And thus the recursive function 𝑦[𝑛] is given as
𝑦[𝑛] = 𝑎0 𝑥[𝑛] + 𝑎1 𝑥[𝑛 − 1] + 𝑎2 𝑥[𝑛 − 2] + 𝑏2 𝑦[𝑛 − 2].

(4.21)

The coefficients in terms of the proportional, integral, and derivative gain as well as
the discrete time step 𝑇 are given as
𝑇
2
+ 𝐾𝑑
2
𝑇
4
𝑎1 = 𝐾𝑖 𝑇 − 𝐾𝑑𝑇𝑇2𝑎2 = 𝐾𝑖 − 𝐾𝑝 + 𝐾𝑑
2𝑇
𝑏2 = 1.

𝑎0 = 𝐾𝑝 + 𝐾𝑖

(4.22)So all we need to implement is simple multiplication with constants given by Eqn. 4.22,
and two FIFO buffers, which each can store two values (either floating or fixed point
numbers of the desired precision).
It is important to note that although the number of poles and zeros of the transfer
function are not changed with the z-transform, their position in frequency spacechanges. This effect is called frequency warp and can be compensated for if an exacttransform from s to z-space is required (e.g. for digital filter design). However, weare tuning the PID controller directly in the discrete form and therefore do not needto compensate for frequency warp.

45

Allan deviation

E XPE R IME NTAL T ECHNIQU E S

10−8

10−9

10−10

10−11
1s

1m
t

1h

FIGURE 4.13: Allan deviation for the
813 nm laser with the digital PID lock engaged. The gray area denotes the statisticalerror. Note that this is the in-loop error
signal and does not include drift of thewavemeter.

B. COMPUTER-BASED FREQUENCY SERVO
The laser for the 813 nm optical lattice features a network interface. This allows us todesign an all digital feedback controller running on the computer that is connectedto the wavemeter. For a description of the 813 nm laser system refer to Section 3.3.
We implement the digital PID controller as a class in the programming language
Python (see Appendix for source code) and use a simple GUI to tune and monitor
the PID controller. Additionally an interface to the 813 nm laser and the wavemeter
control software8 is implemented in a simple wrapper class. Thus we have a detector(wavemeter), actuator (piezo/laser network interface), and a loop filter (digital PID
controller). This is a full laser frequency servo when combined. The GUI allows
to select different channels of the multiplexed wavemeter as well as entering the
network configuration for the 813 nm laser. The other Strontium experiment at Yelabs can potentially use the same program to lock their 813 nm laser simultaneously.
The maximum rate of feedback is mainly limited by the network interface of the laser,which - even with an established connection - takes between 300 ms and 500 ms tocomplete a request. The sample respectively feedback rate can be controlled fromthe GUI.
The 813 nm laser has relatively large drift (compare with Figure 4.12) after turning on.
The servo is able to suppress these fluctuations by as much as 40 dB. The servo allows
the laser to reach a long term stability of 10−10 at one hour in terms of the Allan
deviation (see Figure 4.13). Note that this stability is only relative to the detector anddoes not consider drift of the wavemeter.

8

The wavemeter manufacturer offers a proprietary DLL that can be addressed using python ctypes.
The library uses shared memory to exchange information with third-party tools. This is extremely
useful as it allows real-time access to the current value with higher precision than the wavemeter
software.

46

E XPE R IME NTAL T ECHNIQU E SFIGURE 4.14: Wavemeter PID controller interface running on Windows 7. The numbers
correspond to (1) current value written to the piezo voltage of the 813 nm laser, (2) wavemeterchannel selection, setpoint, and enable/disable button, (3) PID parameters and samplinginterval, (4) plot of the error signal as a function of time steps, and (5) logging feature.

47

5

TOWARDS QUANTUM DEGENERACYThis chapter discusses the elements of a typical experimental sequence, and summarizes the preliminary results from the first attempts of evaporatively cooling ofstrontium-87 in out crossed optical dipole trap. First, we explain the energy levelstructure of strontium-87 and discuss the blue and red MOT, which set the initialconditions for the evaporative cooling. Second, we explain the optical dipole trap
geometry and show results from evaporative cooling in this trap. Last, results fromthis thesis are reviewed and an outlook is given.

5.1 ENERGY LEVEL STRUCTURE OF STRONTIUM-87
We give a short overview on the properties of strontium-87 that are necessary tounderstand the blue and red MOT described in the next section. Strontium is analkaline-earth element, so there are two electrons in the outer electron shell. Thus,
the atomic energy structure is split into a singlet (√1/2 {|↑↓⟩ − |↓↑⟩}) and triplet
manifold (|↑↑⟩, |↓↓⟩, √1/2 {|↑↓⟩ + |↓↑⟩}). Transitions within the singlet or triplet
manifolds are dipole-allowed and strong but transitions between manifolds are dipoleforbidden. Weak transitions with narrow linewidths are still allowed due to mixingof the singlet and triplet states. In particular, we use the weak transitions between the1
S0 ground state and 3 P𝐽 states for improved cooling and precision spectroscopy. The
clock transition is the doubly-forbidden transition between the 1 S0 ground state andthe 3 P0 excited state (698 nm) and has a linewidth of 1 mHz [35]. All relevant energy
levels and transitions are shown in Figure 5.1. The nuclear spin of strontium-87 is 9/2.
The spin of the ground state is purely nuclear and the resulting nuclear spin states are
given as 𝑚𝐹 = −9/2, −7/2, . . . , 7/2, 9/2. The ten nuclear spin states can for example
enhance s-wave scattering, which can be useful for efficient evaporative cooling inan optical dipole trap.
48

TOWA RD S QUA NTUM DE GE N ER ACY

Singlet1

TripletP1

1

679 nmD2

688 nm

3

S1

3

P𝐽

707 nm𝐽

Leakage2
1
0461 nmΓ=2𝜋×32 MHz

689 nm698 nmΓ=2𝜋×7.5 kHz

Clock

1

S0

FIGURE 5.1: Energy level diagram for strontium-87 showing relevant singlet and triplet manifolds in the spectroscopic notation (2𝑆+1 𝐿 𝐽 ). Level spacing values and transition linewidths
are from Ref. [34]. Note that all level spacings are to scale except the 3 P𝐽 spacings.

5.2 THE BLUE AND RED MOT
We use a three-dimensional MOT to capture, cool, and spatially confine the atoms
in three dimensions.
Before the MOTs are described in detail, consider the theoretical limits for theminimum temperature achievable in a MOT [36, 37]:
𝑇Doppler =𝑇recoil =

~Γ
2𝑘B
ℎ2
.
2𝑀𝑘B 𝜆2(Doppler limit)(Recoil limit)

Here Γ is the linewidth and 𝜆 is the wavelength of the cooling transition. 𝑀 is
the mass of a single strontium-87 atom. The Doppler limit is proportional to thelinewidth of the transition. Thus, we expect lower temperatures for transition thatare narrow. However, as the linewidth decreases, the capture velocity also decreases.
Thus, the ability to capture relatively hot atoms from the atomic beam is decreased
as the cooling laser photons are Doppler-shifted out of resonance for sufficiently hot
atoms. Combining the best of both worlds, we employ a scheme where we first use
49

TOWA RD S QUA NTUM DE GE N ER ACY

a blue MOT (1 S0 → 1 P1 [32 MHz]) and then a red MOT (1 S0 → 3 P1 [7.5 kHz]) to
reduce the temperature of the atoms in the cloud further [38].
The transition 1 S0 → 3 P1 is spin-forbidden and therefore the linewidth is muchnarrower than for the 1 S0 → 1 P1 transition (see Figure 5.1). This results in a much
lower Doppler limit for the red MOT (180 nK) than the blue MOT (768 µK). Thus,
the Doppler limit and the recoil limit (230 nK) are very similiar for the red MOT.
First, the atoms are captured from the atomic beam using the blue MOT on the1
S0 → 1 P1 (461 nm) transition. Some atoms may decay into the long-lived 3 P2 state.
A laser at 707 nm repumps those atoms to the ground state through the intermediate
state 3 P2 → 3 S1 → 3 P1 → 1 S0 . In the process, atoms may also be pumped to thelong-lived 3 P0 clock state. Those atoms are pumped back to the ground state by asecond repump laser at 679 nm.
After approximately 1 s the atoms are transferred to the red MOT on the 1 S0 → 3 P1(689 nm) transition. The initial stage of the red MOT, light is frequency modulatedwith an amplitude of 1.5 MHz. This allows for the capture of atoms that are detuned
from resonance due to their thermal Doppler shift. The amplitude of the frequency
modulation is reduced as a function of time to efficiently cool atoms in the red
MOT. Relatively high transfer efficiency of approximately 10 % is achieved using
this technique. The transfer efficiency is mainly limited by the total available coolinglaser power.
The magnetic moment of the ground state of strontium-87 is purely nuclear. Thus,
the magnetic moment for the ground and 3 P1 states differ significantly [39]. This,
combined with the narrow linewidth of the 1 S0 → 3 P1 transition, results in some𝑚𝐹 states that are not trappable after being excited to the 3 P1 state. Thus, a single red
MOT laser on the 1 S0 (𝑚F = 9/2) → 3 P1 (𝑚F = 11/2) transition is only able to coola certain class of 𝑚F states, so a second laser, red detuned from the 1 S0 (𝑚F = 9/2) →3
P1 (𝑚F = 9/2) transition, is needed to remix or "stir" the 𝑚F states so that every
atom spends some time in a state that can be cooled [34].
The atoms are transferred from the red MOT to a crossed optical dipole trap, which
is described in the next section.
gz

Vertical~10°xImaging
Horizontal

y

~20°

FIGURE 5.2: Optical dipole trap 𝑘-vectors (red) andimaging probe beam direction. Note that the angles are
not to scale and only approximate values.

Dimple50

TOWA RD S QUA NTUM DE GE N ER ACY−48 µK200

0 −9 µK −6 −3−24

y (µm)

100
0
−100

−200

−100

0
100
x (µm)−100

0
100
x (µm)0

FIGURE 5.3: Optical dipole trap potential (𝑈/𝑘B ) in the x-y plane at 𝑧 =0 before (left) and after (right) evaporation. Features from all three laser
beams are clearly visible in the left
plot. See Figure 5.2 for definition of
coordinates. All values are calculatedusing the beam 1/𝑒2 radii and intensities.

5.3 CROSSED OPTICAL DIPOLE TRAP
We use three laser beams for our optical dipole trap. We start with a crossed dipole
trap to capture atoms from the red MOT, and use a third, dimple, laser beam to
improve evaporation.
Laser light from the 1064 nm laser setup, described in Section 3.2, is used to generate
the optical dipole potential. All optical dipole laser beams are intensity stabilizedusing the methods described in Chapter 4.
The confinement of a single Gaussian beam dipole trap is different in the transverseand longitudinal direction (compare Section 2.2). Therefore, we use two beams(horizontal and vertical) to provide comparable confinement along both directions.
The horizontal beam (1/𝑒2 radius of 240 × 40 µm) provides good spatial overlap
with the red MOT and helps to maximize the transfer efficiency. The vertical beam
(1/𝑒2 radius of 106 µm) provides confinement in the longitudinal direction of thehorizontal beam. An additional beam, the dimple beam (1/𝑒2 radius of 17 µm), isused to provide very tight confinement for the coldest atoms. This helps to increase
the efficiency of evaporation through faster thermalization. The 𝑘-vectors of thedifferent beams are shown in Figure 5.2 and Figure 5.3 shows a cross-section of the
optical dipole potential of all three beams.
The next section presents first results from evaporation in the optical dipole trap.

5.4 EVAPORATIVE COOLING
We use the optical dipole trap to evaporatively cool strontium atoms. This section
first reports the results from evaporation and then discusses the methods. Note thatthe results presented in this section are preliminary.
We start evaporation with 5 × 105 atoms at 1 µK and a trap depth of approximately
30 µK. The temperature is reduced to 70 nK and the atom number to 2×104 as the trap
51

TOWA RD S QUA NTUM DE GE N ER ACY

2000

10 0 0 ms

6 .4 W2 × 10 0Horz.

0
2 .5 WVert.

0
114 mW
180exp(-t/5 0 0 ms)

DimpleFIGURE 5.4: Experimental sequence for evaporative cooling. The change ofthe power in the three optical dipole trap beams isshown. All waveforms arelinear/step ramps except for
the evaporation, which follows an exponential interpolated with ten equispaced
linear ramps. Note thatthe experimental sequence
for the MOTs (before) andthe imaging (after) is notshown.

depth is lowered to approximately 1.5 µK during evaporation. We reach a minimumof 𝑇/𝑇F of ≈ 0.5 per spin state assuming the population is a uniform mixture often spin states (the actual spin distribution is currently under investigation). If thenuclear spins are populated across only a few selected states, then 𝑇/𝑇F per spin state
will be lower.
Evaporation in our experiment is done in two steps: We start with all three optical
dipole beams at constant intensity. First, the horizontal beam is ramped down within
one second. Second, we exponentially (exp(−𝑡/500 ms)) ramp down the dimple trap
over two seconds while holding the vertical beam at constant intensity. See Figure 5.4
for the exact experimental sequence and beam powers. Note that the experimental
sequence is not fully optimized yet.

Amplitude (µm)

Absorption imaging is used to estimate the number of atoms [40]. The probe beamis resonant with the blue transition (1 S0 → 1 P1 ). For the results in this thesis, theside imaging system with a resolution on the order of 10 µm was used. It allows us
to track the atomic cloud in time of flight images. All images are triggered at the

120
60
0

−60
−120

Fit: 187(5) Hz

Fit: 537(11) Hz

0 1 2 3 4 5Time (ms)0

5 10 15 20Time (ms)52

FIGURE 5.5: Measurement oftrap frequencies in the dimple trap
along z by observing the centerof mass motion in time of flightimages. The trap frequencies areshown before (left) and after evaporation (right). The data is fitted to an exponentially decaying
sine. The signal-to-noise ratio forthe data before the evaporation isbetter due to a higher number ofatoms.

TOWA RD S QUA NTUM DE GE N ER ACY

end of the evaporation sequence and analyzed by the Matlab program edcam1 . This
allows us to extract atom number, position, and the size of the atomic cloud. We
fit the atomic cloud to a Gaussian, which is a valid approximation in the regime ofoperation (see Section 2.4) and calculate temperature by fitting to the atomic size as
a function of time of flight. Figure 5.6 shows a time of flight image sequence and thetemperature fit to a Gaussian distribution with𝑘 𝑇
𝜎(𝑡) = √𝜎02 + 𝑡2 B .
𝑀

(5.1)

Here, 𝑡 is the time of light, 𝜎(𝑡) is the size of the atomic cloud at time 𝑡, 𝜎0 the size ofthe atomic cloud (convolved with the point spread function of the imaging system)
at 𝑡 = 0, 𝑇 the temperature of the atoms, and 𝑀 the mass of a strontium-87 atom.
The trap frequencies are measured by increasing the dimple trap intensity momentarily (10 ms) and recording the position of the atomic cloud after turning the dimple
dipole trap back on. The atoms are displaced from the center of mass due to gravity
as the trap depth is increased. As soon as the trap is turned back on, the centerof mass oscillates around the minimum of the harmonic potential at the trapping
frequency. This allows us to see the oscillations of the center of mass. We can only
measure the trapping frequency against gravity with this method. However, we
assume a symmetric dimple beam. Thus, this method gives a lower bound on thetransverse trapping frequency as the vertical optical dipole trap enhances the trapping frequency along one direction. We estimate the longitudinal trapping frequency
to be 60 Hz, which is set by the properties of the two crossed beams. We measure a
trapping frequency along 𝑧 of 537 Hz at the beginning and 187 Hz at the end of theevaporation. This allows us to calculate the Fermi energy and Fermi temperature as
described in Section 2.4.
For the results presented in this section, magnetic spin distributions were not measured. Thus the presented value for 𝑇/𝑇F is an upper bound for an equal distribution
across all ten nuclear spin states of the ground state.

5.5 CONCLUSION AND OUTLOOKThe physics of optical trapping were explained and we derived rate equations forheating in optical dipole traps. We presented a high power laser system that is able
to deliver up to approximately 20 W of 1064 nm light to the atoms via photonic
crystal fibers and is stable as well as robust. This allows us to trap and evaporatively
cool atoms in optical dipole traps. We built an intensity servo and photodetectors,
which are used to suppress the intensity noise up to 40 dB. Finally, we presented first1

See http://bitbucket.org/amoberkeley/edcam for source code and details.

53

TOWA RD S QUA NTUM DE GE N ER ACY

σz (µm)50
25

τ = 0 ms

5

10

15

Fit: T = 68(10) nK0z (mm)

8
-0.5

6
4

-1.0

2
τ = 0 ms

5 ms

10 ms

15 ms

Number of atoms per µm2

10

0.0

0

FIGURE 5.6: Temperature measurement after evaporation using time of flight images(bottom). Note that the images are averaged over four shots and the image at 𝜏 = 0 ms is
distorted/saturated due to high density of the atomic cloud. The plot (top) shows a fit toa Gaussian density distribution, which yields the value for the temperature. The statisticalerror bars are smaller than the datapoints and not shown in the plot.

experimental results from evaporative cooling in the optical dipole trap where wereach 𝑇/𝑇F of 0.5 per spin state, assuming equal distributions across all ten nuclear
spin states.
As a next step, evaporative cooling has to be optimized to reach lower temperaturesand larger number of atoms. Ref. [20] has demonstrated cooling of strontium-87
to 𝑇/𝑇F = 0.1 per spin state, which should be achievable in our setup as well. After
finishing 813 nm laser setup, we should be able to load atoms into the lattice andoptimize the transfer efficiency and lifetime in the lattice. The first completly new
application of this experiment could be a clock measurement in a three-dimensional
optical lattice and characterization of ac Stark shifts therein.

54

REFERENCES[1]

T. L. Nicholson et al. “Systematic evaluation of an atomic clock at 2 × 10−18 total
uncertainty”. In: Nature Communications 6 (Apr. 2015).

[2]

M. J. Martin. “Quantum Metrology and Many-Body Physics: Pushing the Frontier of
the Optical Lattice Clock”. PhD thesis. JILA/CU Boulder, 2013.

[3]

A. V. Gorshkov et al. “Two-orbital SU(N) magnetism with ultracold alkaline-earth
atoms”. In: Nature Physics 6.4 (Apr. 2010), pp. 289–295.

[4]

G. A. Askar’yan. “Effects of the Gradient of a Strong Electromagnetic Beam on Electrons and Atoms”. In: J. Exptl. Theoret. Phys. (U.S.S.R) 42.1567–1570 (Dec. 1962).

[5]

R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov. “Optical Dipole Traps for Neutral
Atoms”. In: Advances In Atomic, Molecular, and Optical Physics. Ed. by B. Bederson
and H. Walther. Vol. 42. Academic Press, 2000, pp. 95–170.

[6]

B. E. A. Saleh and M. C. Teich. Fundamentals of Photonics. 2nd ed. Wiley, 2007.

[7]

T. A. Savard, K. M. O’Hara, and J. E. Thomas. “Laser-noise-induced heating in far-off
resonance optical traps”. In: Physical Review A 56.2 (1997), pp. 1095–1098.

[8]

Miller and Childers. Probability and random processes. 1st ed. Academic Press, 2004.

[9]

W. Ketterle and M. W. Zwierlein. “Making, probing and understanding ultracold
Fermi gases”. In: ArXiv e-prints (Jan. 2008). arXiv: 0801.2500 [cond-mat.other].

[10]

T. J. Kane and R. L. Byer. “Monolithic, unidirectional single-mode Nd:YAG ring laser”.
In: Opt. Lett. 10.2 (Feb. 1985), pp. 65–67.

[11]

M. Arrigoni, O. Haupt, and V. Leonhardt. Ultra-Low Noise and Narrow Linewidth. url:
https://www.coherent.com/downloads/WP1_MephistoNPRO_Final.pdf(visited on 11/17/2015).

[12]

V. V. Ter-Mikirtychev. Fundamentals of Fiber Lasers and Fiber Amplifiers. Springer
International, 2014.

[13]

H. Pask et al. “Ytterbium-doped silica fiber lasers: versatile sources for the 1-1.2 mu;mregion”. In: IEEE Journal of Selected Topics in Quantum Electronics 1.1 (Apr. 1995),
pp. 2–13.

[14]

A. Kobyakov, M. Sauer, and D. Chowdhury. “Stimulated Brillouin scattering in opticalfibers”. In: Adv. Opt. Photon. 2.1 (Mar. 2010), pp. 1–59.

[15]

R. G. Smith. “Optical Power Handling Capacity of Low Loss Optical Fibers as Determined by Stimulated Raman and Brillouin Scattering”. In: Appl. Opt. 11.11 (Nov. 1972),
pp. 2489–2494.

55

R EFE R ENCE S[16]

NKT. LMA-PM-15 Datasheet. url: http : / / www . nktphotonics . com / wp content/uploads/2015/01/LMA-PM-15.pdf (visited on 11/20/2015).

[17]

P. Russell. “Photonic Crystal Fibers”. In: Science 299.5605 (2003), pp. 358–362.

[18]

J. Knight et al. “Large mode area photonic crystal fibre”. In: Electronics Letters 34(June 1998), pp. 1347–1348.

[19]

CVI/idex. Material Properties. url: https://marketplace.idexop.com/store/
SupportDocuments/MaterialProperties.pdf (visited on 11/19/2015).

[20]

S. Stellmer, R. Grimm, and F. Schreck. “Production of quantum-degenerate strontium
gases”. In: Phys. Rev. A 87 (1 Jan. 2013).

[21]

M-SQUARED. SolsTis Datasheet. url: http : / / www . m2lasers . com / media /74139/solstis_datasheet_1014.pdf (visited on 11/20/2015).

[22]

J. Pitman. Probability. Springer New York, 1993.

[23]

P. J. Collings. “Simple measurement of the band gap in silicon and germanium”. In:
Am. J. Phys 48.3 (1980), pp. 197–199.

[24]

T. Pearsall. “Ga0.47In0.53As: A ternary semiconductor for photodetector applications”. In: IEEE Journal of Quantum Electronics 16.7 (July 1980), pp. 709–720.

[25]

Hamamatsu. G12180 Datasheet. July 2015. url: http://www.hamamatsu.com/
resources/pdf/ssd/g12180_series_kird1121e.pdf (visited on 11/12/2015).

[26]

Hamamatsu. S3072 Datasheet. Nov. 2014. url: http : / / www . hamamatsu . com /resources/pdf/ssd/s3071_etc_kpin1044e.pdf (visited on 11/12/2015).

[27]

B. V. Zeghbroek. Principles of Semiconductor Devices - The p-n junction capacitance.
2011. url: http://ecee.colorado.edu/~bart/book/pncap.htm (visited on
11/11/2015).

[28]

P. Horowitz and W. Hill. The Art of Electronics. 2nd ed. Cambridge University Press,
1989.

[29]

P. C. D. Hobbs. Building Electro-Optical Systems: Making it all Work. 1st ed. Wiley,
2000.

[30]

TI. OPA843 Datasheet. Dec. 2008. url: http://www.ti.com/lit/ds/symlink/
opa843.pdf (visited on 11/13/2015).

[31]

R. Scott, C. Langrock, and B. Kolner. “High-dynamic-range laser amplitude and phasenoise measurement techniques”. In: Selected Topics in Quantum Electronics, IEEE
Journal of 7.4 (July 2001), pp. 641–655.

[32]

G. Franklin, J. D. Powell, and A. Emami-Naeini. Feedback Control of Dynamic Systems.
5th ed. Prentice Hall, 2005.

[33]

S. W. Smith. The Scientist & Engineer’s Guide to Digital Signal Processing. 1st ed. California Technical Pub, 1997.

[34]

M. M. Boyd. “High precision spectroscopy of strontium in an optical lattice: Towards
a new standard for frequency and time”. PhD thesis. JILA/CU Boulder, 2007.

[35]

M. M. Boyd et al. “Optical Atomic Coherence at the 1-Second Time Scale”. In: Science
314.5804 (2006), pp. 1430–1433.

[36]

D. J. Wineland and W. M. Itano. “Laser cooling of atoms”. In: Physical Review A 20.4
(1979), pp. 1521–1540.

56

R EFE R ENCE S[37]

Y. Castin, H. Wallis, and J. Dalibard. “Limit of Doppler cooling”. In: J. Opt. Soc. Am.
B 6.11 (1989), pp. 2046–2057.

[38]

H. Katori et al. “Magneto-Optical Trapping and Cooling of Strontium Atoms down to
the Photon Recoil Temperature”. In: Physical Review Letters 82.6 (1999), pp. 1116–1119.

[39]

S. Stellmer. “Degenerate quantum gases of strontium”. PhD thesis. University of
Innsbruck, 2013.

[40]

T. Pyragius. “Developing and building an absorption imaging system for Ultracold
Atoms”. In: ArXiv e-prints (Sept. 2012). arXiv: 1209.3408 [physics.ins-det].

57

APPENDIXI ELECTRONICS
A. TIA SCHEMATIC
Schematic of the TIA circuit. Note that C1, R1, and R2 need to be chosen accordingly
to the capacitance of the photodiode D1 and the desired reverse bias.
−Vcc

IN

ADJ

OUT

1

1

Vcc

+6V
OUT

2
2

ADJ

LM337

OUT

U4

3

D1

10pF

R6
4.3k

1uF

omit C6!

2

C9

C6

R5
1k

10nF

3

0R

C7

OPA843

6

U1
4

TP2

TP4 TP5 TP6

+15V

−15V

−4V

100nF

R4
50R

M3

M3

TITLE
−4V

+6V

OPA843 TIA

FILE:
PAGE 1REVISION:
OF 1

58

OUT
2

CON1
1 +15V
2 GND

C11
1uF

CON2
1

C12

−4V

TP1

−15V

100nF

7

−Vcc

1

120R
R12

1

R10
120R

1
IN

R7

2
1

ADJ

430R
R9

2
−15V

C1

TP3

+6V

1

1

C10

C4 100nF

C3

PD1

2

+15V

IN

R11
270R

10uF

1uF

2

LM317
3

10uF

C5

R2
120R

R8 1k

100R

U3

D3

C8

10uF

1

2

R1
168R

C2

+15V

R3

−Vbias

3

1

2

1uF

U2

2

2

−15V

−Vbias

LM337

Vcc

D2

1.1A

DRAWN BY: N. Darkwah Oppong

3 −15VAPPEN DIX

B. TIA PCB
Top and bottom layer of the PCB for the TIA. The silkscreen layer is shown in gray.
Size of the PCB: 2000mil × 1500 mil (not shown to scale).

59

APPEN DIX

II SOURCE CODE
A. DIGITAL PID CONTROLLER
Python class for the digital PID controller described in Section 4.2.
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from collections import dequeclass PID:
def __init__(self, current_value=0., setpoint=0., sampling_interval=1.,
overall_gain=1., prop_gain=1., int_gain=1., diff_gain=0., min_max=(float(’inf’),float(’inf’)), offset=0):
self.rbuffer = deque([0., 0.], maxlen=2)
self.current_value = current_value
self.setpoint = setpoint
self.overall_gain = overall_gain
self.min_max = min_max
self.offset = offset
self.sampling_interval = 0.
self.prop_gain = 0.
self.int_gain = 0.
self.diff_gain = 0.
self.error = 0.
self.set_params(sampling_interval=sampling_interval,
prop_gain=prop_gain,
int_gain=int_gain,
diff_gain=diff_gain)
def set_params(self, sampling_interval=False, overall_gain=False, prop_gain=False, int_gain=False, diff_gain=False):
self.sampling_interval = sampling_interval or self.sampling_interval
self.prop_gain = prop_gain or self.prop_gain
self.int_gain = int_gain or self.int_gain
self.diff_gain = diff_gain or self.diff_gain
self.overall_gain = overall_gain or self.overall_gain
self.update_filter_coefficients()
def update_filter_coefficients(self):
G = self.overall_gain
T = self.sampling_intervalp = G*self.prop_gain
i = G*self.int_gain
d = G*self.diff_gain
self.filter_coefficients = {’b_0’: p + i*T/2. + d*2./T,’b_1’: i*T - d*4./T,’b_2’: i*T/2. - p + d*2./T,’a_1’: 0.,’a_2’: -1.
}
def tick(self, current_value=None):
if current_value != None:60

APPEN DIX
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self.current_value = current_value
self.error = self.current_value - self.setpoint
a_1 = self.filter_coefficients[’a_1’]
a_2 = self.filter_coefficients[’a_2’]
b_0 = self.filter_coefficients[’b_0’]
b_1 = self.filter_coefficients[’b_1’]
b_2 = self.filter_coefficients[’b_2’]
w = self.rbuffer
x = self.error# bilinear transform of continous PID transfer functionv = x - a_1*w[-1] - a_2*w[-2]y = b_0*v + b_1*w[-1] + b_2*w[-2]w.append(v)# offsety += self.offset# clamp
if y < self.min_max[0]: y = self.min_max[0]if y > self.min_max[1]: y = self.min_max[1]self.output = yreturn y
def reset(self):
for i, e in enumerate(self.rbuffer): self.rbuffer[i] = 0.
self.current_value = 0.
self.output = 0.

The complete source code for the GUI and the laser network interface can be found athttps://github.com/yesrgang/m2-frequency-servo.

61