Precision Spectroscopy in
1D Optical Lattices

Diplomarbeit

zur Erlangung des Magistergrades an der
naturwissenschaftlichen Fakultätder Leopold-Franzens-Universität Innsbruckvorgelegt vonSebastian Blattdurchgeführt am JILA, Boulder
unter der Leitung von Rudolf Grimm und Jun Ye

Boulder, Juli 2005

ii

Contents

List of Figuresv

Introductionvii

Introductionvii

1 Oscillators and Feedback
1
1.1 Characterizing an oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Feedback theory and phase-locked loops . . . . . . . . . . . . . . . . . . . 4
1.2.1 Feedback theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Application to phase-locked loops . . . . . . . . . . . . . . . . . . . 7
1.2.3 Phase detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Loop filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Atomic Clocks and Precision Spectroscopy

13

3 Doppler Cooling and Magneto-Optical Traps
3.1 Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Dissipative force . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Diffusion rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Cooling limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Magneto-optical traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
15
16
16
17
17

4 1D Optical Lattices
4.1 AC Stark effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Branching ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Introducing experimental parameters . . . . . . . . . . . . . . . . .
4.2 Separating the motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Longitudinal problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Vibrational spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Parametric heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21
21
22
23
24
24
28
29

iii

Contents
4.5.1 Intensity Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Heating Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Environmental effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 AC Stark shift (probe) . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.3 AC Stark shift (blackbody) . . . . . . . . . . . . . . . . . . . . . .
4.6.4 Second order Zeeman effect . . . . . . . . . . . . . . . . . . . . . .
4.6.5 Gravitational shift . . . . . . . . . . . . . . . . . . . . . . . . . . .

29
31
32
33
33
34
36
37
39

5 Precision Spectroscopy in a 1D Optical Lattice
5.1 Atom preparation and MOT cooling . . . . . . . . . . . . . . . . . . . . .
5.2 1D optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41
43
45

6 Determining Absolute Frequency
6.1 Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Features and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Frequency Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Direct Digital Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 AD9854 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Software: Drift Compensation . . . . . . . . . . . . . . . . . . . . .

53
54
56
57
57
60
61
63
64
65
67

7 Results and Outlook

71

Bibliography

75

Index

80

Acknowledgements

83

4.6

iv

List of Figures

1.1
1.2
1.3
1.4
1.5
1.6

Oscillating voltage with varying momentaneous frequency . . . . . . . . . 2
Noise power spectrum and Allan deviation . . . . . . . . . . . . . . . . . . 4
Negative feedback loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Schematic of a Phase-Locked Loop (PLL) . . . . . . . . . . . . . . . . . . 7
Analog and digital phase detector output signals and capture range . . . . 9
Bode representation for loop filter transfer functions . . . . . . . . . . . . 11

2.1

Lowest singlet and triplet states of neutral strontium . . . . . . . . . . . . 14

3.1

MOT detunings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1

Motional sidebands in harmonic and anharmonic potentials . . . . . . . . 28

5.1 Sample preparation and pre-cooling, figure from Loftus et al. (2004a) . . .
5.2 Lowest singlet and triplet states of 87 Sr . . . . . . . . . . . . . . . . . . .
5.3 AC Stark shifts for 87 Sr . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Lattice and spectroscopy laser mount . . . . . . . . . . . . . . . . . . . . .
5.5 Sideband spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Multiple-Lorentzian sideband fit . . . . . . . . . . . . . . . . . . . . . . .
5.7 Typical spectrum of 1 S0 − 3 P0 . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Density shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Magnetic shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Lattice AC Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Lattice curvature for different ratios w0 /λ . . . . . . . . . . . . . . . . . .

42
42
44
45
46
47
48
48
49
49
50

6.1
6.2
6.3
6.4
6.5
6.6

54
58
58
59
60
62

Frequency counting setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
Frequency multiplier schematic . . . . . . . . . . . . . . . . . . . . . . . .
ADF4111 circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Loop filter circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Final Loop filter and VCO driver circuit . . . . . . . . . . . . . . . . . . .
Frequency multiplier: spectral traces . . . . . . . . . . . . . . . . . . . . .

v

LIST OF FIGURES
6.7 SPI data transfer protocol . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Piecewise linear frequency ramp . . . . . . . . . . . . . . . . . . . . . . . .
6.9 AD9854 DDS spectral traces . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Drift compensation screenshots . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Drift compensation data and Allan deviation . . . . . . . . . . . . . . . .
7.1

vi

63
64
66
68
69

Comparison of results from Takamoto et al. (2005) and Ludlow et al. (2005) 72

Introduction
easuring time has always been important to human society, but whereas it used to
be good enough to know the current date or hour, today’s increasingly technicalsocieties need better standards. This is not only true for the second, but also for allother units of measurement, represented by the système international (SI). For example,
layered semiconductors must be fabricated to specifications on the nm scale and ships
would like to know their position on the earth’s surface as accurately as possible to avoid
nautical dangers. The length scales involved in both of these applications seem very
different at first, but since the global positioning system (GPS) works by triangulating
the ship’s position from orbit, the necessary accuracy is comparable.
With the invention of clocks based on atomic phenomena in the middle of the lastcentury, time could be measured so accurately that other units of measurement were
defined with respect to the second. To state it bluntly, the atomic clock is the world’s
best caliper. . . and more. The combination of the leap in clock quality with cheap opticaloscillators (lasers) created whole branches of technology and changed the standard of
metrology to what it is today.
The question remains: Can we make it better? And if yes, should we? To answer
these questions (with “Yes” and “Of course!”), we have to quantify what makes a goodclock. A good clock must possess three attributes:

M

Accuracy It must be able to split time into pieces that match the definition of the second
as well as possible.
Stability The clock must produce ticks at constant intervals. This is often confused withthe first point, as an accurate clock must be stable, but a stable one need not be
accurate.
Reproducibility One clock in the whole world is not enough. We need at least two to beable to measure their first two attributes, as will be explained later.
We characterize a clock by its frequency and a clock’s instability is quantified by
the deviation from its desired frequency over time. This quantity can be obtained bysumming the squared deviations between two adjacent frequency measurements and has
been accepted as the international standard to compare clocks. In the case of a clockvii

Introduction
based on an atomic transition, this so-called Allan deviation 1 can be expressed as
r
T
∆ω
,
(0.1)
σy (τ ) ≈πω τ Nin the limit that the dominant noise source is quantum mechanical fluctuations (Itanoet al., 1993, Santarelli et al., 1999).
This expression tells us the basic recipe for making a good atomic clock: ∆ω is thewidth of the atomic resonance at frequency ω, so choosing a narrow atomic feature ata large frequency is desirable to minimize instability. Since we are probing a quantum
mechanical system, just taking one measurement of duration T is not enough, we mustaverage over many repetitions to obtain an accurate number. Correspondingly, τ is theaveraging time. The occurrence of τ also tells us that we can make any accurate clock
stable just by measuring long enough. Of course, this is where we gain a lot by choosing
an atomic feature with large quality factor Q := ω/∆ω, as we can trade stability forshorter averaging times. Finally, N is the number of atoms involved in producing theclock’s signal. Using more atoms is equivalent to measuring the atomic feature manytimes. This is again a trade-off we can make: more atoms means better stability, but
controlling the experimental noise involved in measuring them gets increasingly difficult.
How do the current atomic clocks work? Let us take a look at the best atomic clock
to date in the light of Eq. 0.1. The SI second is defined as an integer multiple of theoscillation period of a Cs transition. More specifically: the ground state hyperfine transition |2 S1/2 F = 3i → |2 S1/2 F = 4i in 133
Cs (I = 27 ) is probed with microwave radiation at55
9 192 631 770 Hz. The exciting microwave frequency is locked to the atomic transition
by measuring the transition probability and ensuring that it stays at its maximum. The
oscillations of the microwave oscillator (maser) can then be counted electronically and aclock signal is produced.
However, we want to know if we can improve on the Cs standard. As mentioned before:
in theory, we could always make a much better clock by choosing a narrower transition
at higher frequency, e.g. a narrow optical transition like an intercombination line. The
idea is not new, but what prevented the implementation of an optical atomic clock formany years is that counting the 105 times faster optical oscillations is impossible evenwith the best electronic device.
Technology caught up with the invention of the femtosecond-comb2 , a pulsed laser
system with an interesting frequency spectrum: Looking like the teeth of a comb it
has a component every few hundred MHz over a very wide range. The n-th tooth withfrequency ωn can be described in terms of two frequency parameters, the pulse repetition
rate νr and the offset frequency ν0 , both of which are in the microwave domain
ωn = nνr + ν0 .
By producing a comb spectrum that spans whole octave, it is possible to beat a highlying comb line against a frequency-doubled comb line and obtain a microwave beat note
1
2

The Allan deviation will be introduced more formally and in much more detail in Section 1.1.
A good summary and general reference is Ye and Cundiff (2004)viii

at ν0 that can be counted electronically.
ω2n − ωm = ν0 for 2n = m
Using a phase-locking technique (as described in Chapter 1.2) to lock the comb spectrum to the optical frequency standard ensures that the optical frequency is an exact
integer multiple of the repetition rate νr . By counting νr and determining the integer multiplication factor, the fs-comb becomes the optical equivalent of a mechanicalgear. . . with a translation ratio of 1 : 100 000. In the last five years, fs-combs havebecome the standard tool to directly count optical frequencies and many groups in theworld have started projects towards an optical frequency standard. An atomic clock
based on an optical transition with its large Q places stringent requirements on the experimental setup. Let us qualify these requirements and take a look at the difficulties
involved in the making of an all optical atomic clock:First: one needs to find an appropriate optical transition. It should be as narrow as
possible and should be accessible with a commonly available laser source, e.g. a diodelaser system.
Second: the particles must be prepared in an environment that is as immune aspossible to external perturbations. I.e. the particles must be trapped inside a controllable
potential that can be designed to the element’s specific requirements. We shall develop
a thorough understanding of this confinement in chapter 4. First, though, we will talkabout choosing (chapter 2) and preparing an apropriate sample (chapter 3).
Third: resolving a narrow spectroscopic line requires a frequency source with an evennarrower spectrum, i.e. a highly stabilized laser source.
Fourth: the optical frequency must be reduced to a microwave frequency to makeelectronic counting possible. As mentioned in the last Section, the tool to use is afemtosecond-comb.
The spectroscopy laser will be discussed throughout chapters 5, 6 and references
therein. The absolute frequency measurement by comparing against a Cs standard via
a femtosecond comb will be presented in chapter 6.
Finally, we will talk about future improvements on the measurements and new experiments in chapter 7.

ix

IntroductionxChapter

1

Oscillators and Feedback
his chapter introduces the basic vocabulary we will need when talking about oscillators and related techniques. The same principles hold true, regardless whether theoscillator is an electrical synthesizer, a pendulum clock or a laser.

T

1.1 Characterizing an oscillator
How good is an oscillator? This Section introduces a measure on the goodness of clocks,
the Allan variance. The treatment in this Section follows the NIST tutorial on oscillators
in Sullivan et al. (1990).
We are interested in oscillators producing sinusoidal output voltage with a well-definedand relatively stable frequency. We want to allow amplitude as well as phase noise, so
it is reasonable to write down the oscillator’s output as:
V (t) := V0 (1 + a(t)) sin(ω0 t + φ(t))(1.1)

with a nominal frequency ω0 , a small amplitude variation a(t) and some phase fluctuation
φ(t). The total accumulated phase at time t is Φ(t) := ω0 t + φ(t). Without loss ofgenerality, we will only consider amplitude normalized signals, i.e. V0 ≡ 1.
For non-zero phase noise φ the signal’s frequency is changing in time; can we define
an instantaneous frequency? Consider the oscillating signal in Fig. 1.1 and assume that
we measured its amplitude Vi at times ti for i = 1, 2. The instantaneous frequency isdefined as the sine wave of minimal frequency that we can fit through both of these data
points.
The phase and amplitude noise is characterized by measuring this instantaneous frequency and looking at its changes in time. When we are talking about a clock’s stability,
we are actually talking about its instability, viz. its deviation from its nominal frequency.
Actually, most measurement devices do not measure the instantaneous frequency directlyin the above sense, but are measuring phase or counting oscillations. This fact alreadyhints at an important fundamental “flaw”: to determine the frequency of an oscillator

11 Oscillators and Feedback

Figure 1.1: Oscillating voltage with varying momentaneous frequency
one always needs another oscillator. So all frequency (and time) measurements are relative to a reference which should ideally be much less noisy than the observed oscillator.
The measurement device should also introduce less noise than the sum of both theseoscillators.
It is convenient to define a fractional frequency deviation relative to the nominal
frequency
ωt − ω 0y(t) :=
ω0
because the oscillator’s time deviation can be expressed as its integral
Z tx(t) := dτ y(τ ).
0As already mentioned in its definition, measuring the instantaneous frequency inherently involves a sampling or gate time τ . Thus, the result of measuring the instantaneous
frequency can be expressed as the discretized derivate of the time deviationȳ(t) :=

x(t + τ ) − x(t).
τThere is also another time constant due to some finite data processing time after sampling
data, called the dead time τd . During this time, the measurement device is blind to theoscillator’s signal, so information is lost. This can become a problem if τ ' τd .
Direct frequency counting is often not possible, especially if the oscillator has a nominal
frequency in the optical regime, so indirect methods have to be used, several of which
will be explained in later chapters.
Mix the oscillator signal with a reference signal of comparable frequency and countthe beat signal. See chapter 6 for an elaborate example.
Phase-lock the reference to the test oscillator and observe the error signal. See
chapter 1.2 for an introduction to phase-lock techniques.
Electronically divide down the frequencies and then compare rise times with ahigh-resolution time interval counter.
These methods can be compared by what they can observe. The best method would
be one that can measure the time deviation x(t) directly, as changes in time, frequency
and frequency changes could all be derived as numerical derivates of that data. On
the other hand it is not possible to measure time deviations with a method that only
observes frequency changes since there is always a dead time involved and numericalintegration would produce erroneous data. Knowing this, the last method in the list2

1.1 Characterizing an oscillator
from above should theoretically be the best, were it not also the method fraught withthe most technical noise issues. Counting rise times involves the use of digital counters
for the frequency division which are inherently slower than analog methods based on
observing beat notes.
What is the best measure of an oscillator’s frequency instability? It needs to be astatistical estimator that is independent of the underlying noise spectrum. A lot of
research has gone into looking at these estimators (see e.g. Allan et al. (1997) and
Sullivan et al. (1990)), resulting in an internationally accepted standard for frequency
instability called the Allan variance. It is defined as the time average of the square offractional frequency deviations in the gate time interval [t, t+τ ). Its square root is called
the Allan deviation and is defined as:s1σy (τ ) :=(y(t + τ ) − y(t))2 .
(1.2)
2
Its best estimate from a sample of N fractional frequency measurements ȳ i is:
v
u
N
−1
u
X
1
sy (τ ) = t(ȳi+1 − ȳi )2 .
2(N − 1)

(1.3)

i=1

Although the Allan variance is independent of the underlying statistics, its error isnot. The yi are distributed differently for different kinds of noise processes (e.g. whitefrequency noise, 1/f phase noise, etc.) and the error depends on the noise process
responsible for the distribution of frequency data. This makes it very difficult to handle,
as one seemingly has to know the result of the measurement beforehand. Luckily, theerror can be approximated as the error of an arithmetic mean with a noise processdependent prefactor (Lesage and Audoin, 1973, Sullivan et al., 1990)
sy (τ )∆sy (τ ) = κα √N −1where κα is a scalar depending on the noise’s spectral power law Sα (f ) = κα f α . See
Tbl. 1.1 for a list.
Plotting an imaginary oscillator’s Allan deviation against the gate time τ in a loglog plot results in an intuitive estimate of the oscillator’s instability. The lower part ofNoise process
White phase noise
Flicker phase noise
White frequency noise
Flicker frequency noise
Random walk frequency noise

α
-2
-1
0
+1
+2

κα
0.75
0.77
0.87
0.99
0.99

Table 1.1: Allan deviation errors for different noise processes (Sullivan et al., 1990)

3

1 Oscillators and Feedback

Figure 1.2: Noise power spectrum due to noise processes from Tbl. 1.1 and corresponding Allan deviation for a hypothetical oscillator.
Fig. 1.2 schematically shows the Allan deviation resulting from the noise spectrum inthe upper part. Each noise source’s spectral power law translates into a unique slope in
the Allan deviation’s logarithmic plot (Sullivan et al., 1990). As long as the oscillator’serror is dominated by statistical scatter around its nominal frequency, the Allan deviation
averages down. At some point τ0 , random walk frequency noise takes over due to its τ 1/2
dependence. Thus, if one wanted to use this hypothetical oscillator in an application,
one would only trust it to remain stable on time scales shorter than τ0 .

1.2 Feedback theory and phase-locked loops
Copying of frequency and phase information from a frequency reference onto anotheroscillator is a very general and powerful tool used in almost any signal processing application. It allows synthesis of a frequency with well defined offset to the reference
and also frequency multiplication by an adjustable factor. In the field of laser physics,
synthesis of signals with highly stable frequency differences is of particular importance.
Even more so, all electro-magnetic communication relies on the ability of every device
to work in exactly the same frequency band, so every device must be referenced to a
crystal oscillator.
A phase-locked loop (PLL) is a feedback loop working on any kind of voltage-controlled
oscillator (VCO) . This VCO can be a pendulum driven by an electrical motor, an LC-

4

1.2 Feedback theory and phase-locked loops

Figure 1.3: Negative feedback loop
circuit with a voltage controlled capacitance, a quartz crystal or even a laser whose
oscillation frequency can be modified by changing a control voltage. The phase-locked
loop tries to modify the control voltage in such a way that the emitted signal’s frequency
and phase match those of a reference signal, called the local oscillator (LO).
Note that every VCO has a characteristic time constant τ0 and its output phase attime t is the integral over the control voltage from t − τ0 to t. This adds an implicit
integration step to the feedback loop.
The next two Sections will introduce the general concepts of feedback theory andmotivate why a PLL can be described in analogy to a classical feedback loop. The restof this chapter explains the building blocks of a PLL in more detail.

1.2.1 Feedback theory
A feedback loop regulates an observable by measuring its current value and applying acontrol signal to the observable’s source. They are ubiquitous in any technical applicationand there is a lot of theory surrounding the implementation details of feedback loops,
such as stability and dynamic performance. We shall take a quick tour of the formalism
needed to describe a negative-feedback loop.
Feedback loops are generally described by drawing a schematic like in Fig. 1.3 where
the circuit implementing the loop is divided into different blocks. A transfer function(e.g. G(s) in the upper part of the figure) is assigned to each block. This function
describes how the block transforms an input signal X to an output signal Y . The
output Y is fed back comparing it against the input: in our case with a negative sign toproduce negative feedback (the comparator is depicted by a ⊕ in the diagram). In thatmanner, the whole feedback loop has a transfer function as well. There are some simple
rules describing how to get the total transfer function from such a schematic 1 . One justmultiplies each block’s transfer function with the input to obtain the output signal andfollows the path of the signal around the loop. In our case, this leads to the expression
Y = G(X − HY )1

A summary of Mason’s rules can be found in Franklin et al. (1994).

5

1 Oscillators and Feedback
which is an algebraic equation for the output Y as a function of the input X. We obtain
Y =

G
X.
1 + GH(1.4)

The nominator G is called the forward gain, the denominator is the loop gain. Thisequation allows the calculation of the output signal of the feedback loop for a given
input signal in the following manner:
A transfer function can be assigned to any system that is described by a linear differential equation with constant coefficients of finite order. The time-dependent transfer
function G(t) is the Green’s function of the system’s differential equation, i.e. the response of the system to a kick. Any inhomogeneous solution can be found by convolving
the Green’s function with the inhomogeneity, viz. the input signal. However, this integration is difficult, so it is convenient to transform the problem to a simpler one bychoosing an appropriate functional basis for the solution and work with the coefficients.
It turns out that the most convenient basis consists of plane waves that are exponentiallydamped and cut off at time zero. This choice ensures convergence and causality of thesolution and includes damped oscillatory behavior. The corresponding transformation
is called one-sided Laplace transform which is a generalization of the Fourier transformto complex frequencies s:
Z ∞F (s) =
dt f (t)e−st(1.5)
0
Z σ+i∞
1
f (t) =
ds F (s)est(1.6)
2πi σ−i∞with the complex integration path chosen to the right of all integrand poles.
As a generalization of the Fourier transform the Laplace transform also has the property of transforming convolutions in the time base to products in the frequency base.
Since the system’s output is a convolution between the Green’s function and the input
signal, evaluation of transfer functions in the new basis is very simple:
Z +∞y(t) = G ∗ x(t) =dτ G(t − τ )x(τ )−∞⇒ Y (s) = G(s)X(s)Once the output signal Y (s) has been calculated, it can be transformed back into
the time basis using Eq. 1.6. A table of transformation rules can be found in mostbooks dealing with ordinary differential equations, e.g. Abramowitz and Stegun (1972).
The transfer functions G and H are complex-valued. This leads to a special case whenarg(GH) ≈ π. The denominator in Eq. 1.4 suddenly becomes large and the stabilizing
effect of the negative feedback on the input signal disappears. Instead, the loop changes
to positive feedback putting more energy into the system with every feedback cycle. If nosafety guard against this effect is included, the output starts to oscillate wildly or evengrows exponentially. The cause for this effect and a remedy by choosing appropriate
forward gain will be shown in Sec. 1.2.4.

6

1.2 Feedback theory and phase-locked loops

Figure 1.4: Schematic of a Phase-Locked Loop (PLL)

1.2.2 Application to phase-locked loops
Compare the basic setup of a PLL in Fig. 1.4 to then negative feedback loop in Fig. 1.3.
The main difference is the multiplication of the analog mixer (depicted by ⊗) replacing
the addition of signals. Remember that only linear systems can be treated with themethods introduced in the last Section. As mentioned before, this loop controls frequency
by measuring phase. We will show that under certain approximations the linearity
requirement is fulfilled and that the PLL behaves exactly like a standard feedback loop
controlling the signal’s phase.
The frequency division in the feedback arm is often used to compare a frequency
divided signal to the reference, thus obtaining a phase-locked multiple of the input
frequency at the output. In the following derivation, we will omit this division ratio,
as it can easily be patched in once we have shown that the loop can be treated using a
Laplace transform.
The circuit takes an input signal xi from the LO and produces a sinusoidal signal xo
with phase θo . The mixer outputs the product of these signals
xm (t) = xi (t)xo (t)which is then fed into the loop filter. The loop filter modifies the signal so that it becomesa DC voltage able to drive the VCO:Z ∞dτ Gf (t − τ )xm (τ )y(t) =
0

Finally, the VCO converts an input voltage to a sine wave with amplitude ao and phaseθo + π/2:
xo (t) = ao sin(θo (t) + π/2) = ao cos(θo (t))
Z ∞dτ Gv (t − τ )y(t)θo (t) =0

7

1 Oscillators and Feedback
The transfer functions Gf and Gv of loop filter and mixer can be combined into a total
forward gain transfer function by convolution: G := Gf ∗ Gv . Combining the above
equations we obtain a formal expression for the output phase:Z ∞dτ G(t − τ )xm (τ )θo (t) =0

Since we are interested in locking the VCO to a reference oscillator with a well-defined
frequency, we will assume a sinusoidal input with amplitude ai and phase θixi (t) = ai sin θi (t),resulting in a mixed signal of the form
xm = ai ao sin θi cos θoai ao=(sin(θi + θo ) + sin(θi − θo ))2
Defining am := ai ao /2, the output phase reads:Z ∞θo (t) = amdτ G(t − τ ) (sin(θi + θo ) + sin(θi − θo )).

(1.7)

0

The combined transfer function G includes the loop-filter and VCO characteristics.
These electric parts include some RC combinations, maybe just due to residual capacitance and resistance of the components involved. The associated time constants will
limit the system’s memory and reduce the range of integral in Eq. 1.7 such that it will
only need to be evaluated over an interval [t − T, t] where T is the system’s memory
time.
This limitation to a finite interval allows us to approximate the sine terms under the
integral. LetdΩ+ (t) := (θi + θo )dt
be the instantaneous sum frequency and consider the sum frequency term
Z tdτ G(t − τ ) sin(θi + θo ).
t−T

If the sine function does many oscillations during the integration interval – corresponding
to T  2π/Ω+ – the integral will average to zero. Since the memory time can be set
by choosing the loop filter variables, we will assume this to be true. We will also limit
ourselves to the “locked” case where the output phase is already roughly equal to the
input phase viz. |θi − θo |  1. This allows us to linearize the difference frequency term
and further simplify Eq. 1.7 to:Z tdτ G(t − τ )(θi − θo )(1.8)
θo (t) = amt−T8

1.2 Feedback theory and phase-locked loops

Figure 1.5: Analog and digital phase detector output signals and capture range
Formally extending the integration range to infinity by including the cutoffs in G(t)allows us to simplify the expression even further: G in Eq. 1.8 is the Green’s function ofan ordinary differential equation with constant coefficients, so it can be solved using a
Laplace transform as motivated in Sec. 1.2.1. The convolution of the transfer function
with the input and output phases transforms into a product, leading to an algebraic
equation for θoθo (s) = am G(s)(θi (s) − θo (s))
am G(s)⇒ θo (s) =θi (s).
1 + am G(s)In conclusion: the analog phase-locked loop is a classical feedback loop for the signal’s
phase if the input has a well-defined frequency, sufficient loop-filter integration is included
and the phases of local oscillator and VCO are not too far apart.

1.2.3 Phase detector
As shown in the previous section, the mixer acts as a phase detector (PD) in a small range
around ωref = ωv , since the emitted signal can be approximated by a linear function inthis region. If the VCO frequency is off by too much, the error signal from the mixer
fluctuates too rapidly for the VCO to react, i.e. they happen on a faster time scale thanthe VCO’s integration time.
The VCO averages these rapid oscillations to zero, as can be seen in the upper part of
Fig. 1.5. That also means that an analog mixer is only useful in a small range around thetarget frequency: the loop does not know in which direction to regulate if the frequencies
are too far apart. This limitation can be overcome by using a digital phase-detector that9

1 Oscillators and Feedback
rails to a constant voltage when not near the resonance. The phase detector becomesa phase-frequency detector (PFD) with a very large capture range (see lower part of
Fig. 1.5), since the loop always knows in which direction to regulate the VCO frequency.
Note that there is a difference between a phase-locked loop’s capture range – thefrequency region where it is able to acquire lock – and its bandwidth, indicating its
ability to follow rapidly changing input signals.
The downside of using a digital phase-frequency detector is that the logic circuitry isinherently slower than the analog mixer. This again amounts to an effective integrationand limits the bandwidth of the feedback loop. In systems where high frequency noise
is negligible, a digital phase-frequency detector is the preferred choice. This is true formost electronic circuits. However, if the feedback loop controls an optical oscillator likea laser diode, higher bandwidth might be necessary and an analog mixer should be used.
The bandwidth of a phase-locked loop can be measured by splitting off a fraction of
the phase-detector’s output and looking at its power spectrum on an FFT (Fast Fourier
Transform) machine. By observing the difference between the noise floor of the locked
and unlocked case, one gains a quantitative measure of how well the reference signal’s
characteristics are written onto the VCO output in a spectrally resolved manner. When
the VCO is locked to the reference, the spectral power in the frequency range at which the
feedback works should decrease compared to the unlocked case. This gives an estimate
for the loop bandwidth.
As mentioned above, the bandwidth of a PLL is due to the amount of in-loop integration. Its main source is not the phase detector but the loop filter as will be explained in
the next section.

1.2.4 Loop filter
The loop filter is responsible for adapting the error signal from the phase detector tomatch the VCO’s needs. As shown in the preceding Sections it is important for theloop filter to low-pass filter the error signal. It is almost always true that the phasedetector’s signal is much too weak to drive the VCO on its own, so the loop filter needssome sort of amplification as well. Luckily, it is very easy to implement active versionsof RC combinations by using an operation amplifier. The loop filter transfer function
becomes very important at this point; to talk about it further, we need a way to image
this complex-valued function.
The complex transfer function G(s) is usually depicted in a Bode plot, where amplitude(phase) is plotted against frequency on a log-log (log-linear) scale. The Bode plot for apassive RC low-pass is shown in Fig. 1.6(a). An active low-pass filter has the same Bodeplot, but scaled by a gain factor (corresponding to adding a flat line to the Bode plot).
Combining multiple transfer functions in a Bode plot is straightforward, as the productof the amplitudes is converted to a sum in the logarithmic plot and the arguments areadded as well.
The combined transfer function of three low-pass filters with different time constants
and some proportional gain is shown in Fig. 1.6(b). Note that the combined filter’stransfer function has an amplitude that is much larger than unity at the point where101.2 Feedback theory and phase-locked loops

1.0

102
101
100

amplitude

0.5

phase

0.0

10−2
10−3
10−4

Arg(G)/π|G(ωτ0 )|

10−1

-0.5

10−5
10−6

π phase shift

10−7

-1.0

10−8
10−9 −6
10
10−5 10−4 10−3 10−2 10−1 100
ωτ0

101

102

103

104

105

-1.5
106

(a) One low-pass filter with unity gain.

1.0

107
10

5

amplitude

0.5

103
10−1

0.0

phase

10−3
-0.5

10−5
10−7

π phase shift

-1.0

10−9
10−11
103

104

Arg(G)/π|G(ωτ0 )|

101

105

106

107
ωτ0

108

109

1010

-1.5
1011

(b) Three low-pass filters in serial with large proportional gain at π-phase shift;this filter is unstable and will oscillate.

Figure 1.6: Bode representation for loop filter transfer functions

11

1 Oscillators and Feedback
its phase-shift goes through −π. At this point, the negative feedback turns into positive
feedback and the filter’s output starts to oscillate. This loop filter must be redesigned sothat it has a gain much less than unity at this point, i.e. |G(ω)|  1 for arg G(ω) = −π.
The roll-off (the steepness of the gain’s descent) and corner (the point where it starts
to decrease) can be modified by altering or adding low-pass filters. See Franklin et al.
(1994) for loop filter basics and Nash (1994) for a detailed manual on PLL loop filter
design.

12

Chapter2

Atomic Clocks and Precision
Spectroscopy
n optical atomic clock is based on a transition between two specific states of a specificelement. The element must be carefully chosen, subject to technical criteria.
The sample must produce a stable signal at a fixed frequency and should ideally only
be limited by quantum mechanical measurement noise. To obtain such a clean signal, theenvironment of the sample must be controlled to an extreme extent: Ideally, the atoms
should be as cold as possible to eliminate all temperature effects. The occurrence of thenumber of particles as N −1/2 in the formula for the atomic clock’s instability (Eq. 0.1)
suggests that using many atoms to produce a signal might be beneficial. However, there
is a trade-off between the extra noise introduced by many-body effects in the sample.
At this time, these technical reasons point to two choices: Either use a few ions in
an electro-magnetical trap or a few million neutral atoms inside an optical potential.
Technical reasons make using ions attractive.
The strong Coulomb interaction allows trapping ions in a very controlled environment provided by a Paul or Penney ion trap.
The number of particles can be decreased to one, removing all many-body effects
from the system.
On the other hand, neutral atoms have much smaller interaction strengths and allow
using millions of particles without significant many-body effects. Finally, an experimental tool to trap neutral atoms is available with the optical lattice, allowing precisioncontrol comparable to the ion trap.
In this work, an all optical neutral atom based frequency standard for 87 Sr will bepresented. This choice of sample was based on several considerations: Strontium is
an alkaline earth, i.e. a two-electron system with a ground state configuration of (Kr5s2 )1 S0 . The two most abundant isotopes are 88 Sr (82.58(1) %) and 87 Sr (7.00(1) %).
Due to the coupling between both valence electrons, the spectrum of two-electron systems is split into two parts: the singlet and the triplet states, corresponding to a total
spin S = 0 or S = 1. Electronic dipole transitions between these subsystems (intercom-

A13

2 Atomic Clocks and Precision Spectroscopy

Figure 2.1: Lowest singlet and triplet states of neutral strontium
bination linens) are very unlikely and thus are called forbidden. A low probability for a
dipole transition corresponds to a very narrow spectral line. The lowest electronic state
manifold in the triplet system (3 PJ ) can only decay to the lowest singlet state (1 S0 ). For
strontium a partial energy diagram is depicted in 2.1; the intercombination transition
between 1 S0 and 3 P1 has a natural linewidth (wavelength) of 7.6 kHz (689 nm). The
broad 1 S0 − 1 P1 transition (32 MHz, 461 nm) lends itself to standard MOT cooling into
the mK regime (as will be discussed in the next chapter). The spectrum of both isotopes
is almost identical, except that 88 Sr and 87 Sr have different nuclear spin. Whereas 88 Sr
has no nuclear spin (I = 0) and thus no hyperfine structure, 87 Sr has I = 9/2 leading tofine but interesting differences.
The hyperfine interaction makes the doubly forbidden 1 S0 − 3 P0 transition1 weakly
allowed by adding some 1 P1 character to the 3 P1 state and perturbing 3 P0 (see Sec. 4.6.4).
The transition has a theoretical linewidth of 8 mHz (Porsev and Derevianko, 2004).
Characterizing this extremely narrow transition is the first step towards an all opticallattice based frequency standard with instability on the 10−18 level.
The next two chapters will discuss the preparation of these strontium atoms, viz. weshall talk about laser cooling and trapping of neutral atoms.

1

Doubly forbidden because of ∆S = 1 and ∆J = 0.

14

Chapter3

Doppler Cooling andMagneto-Optical TrapsThe basics of laser cooling neutral atoms are introduced, leading to a discussion of
narrow-line MOTs for alkaline earths.

3.1 Doppler coolingLaser cooling of atoms relies on the mechanical effects of the photon-atom interaction.
Due to momentum conservation, absorption (emission) of a photon with momentum ~kforces the particle to increase (decrease) its own momentum by the same amount. Light
acts on matter in another distinctive way: the electric field of the photon causes anintensity dependent shift of the atom’s electronic spectrum, called the AC-Stark shift.
These mechanical light effects have been studied extensively and allow a plethora ofschemes to influence an atomic sample’s phase space distribution. I will give a brief
overview and direct the reader to Stenholm (1986), Adams and Riis (1997) or Metcalf
and van der Straten (1999) for more in-depth information.
The dynamics of laser cooling can be described in a semi-classical way by the FokkerPlanck equation 1 . It describes the evolution of an atomic ensemble’s momentum distribution W and includes two modifying terms: a dissipation term F that removes amplitude
from the distribution and a redistributing diffusion term D, viz.
∂t W (p, t) = −∇F (p, t)W (p, t) + 4D(p, t)W (p, t)(3.1)

The dissipative function F “decelerates” particles of momentum p and the diffusivefunction D “spreads” them to different momenta. In the case of laser cooling, these
momentum-changing effects are the absorption and spontaneous emission of a photon andthe atom’s corresponding momentum change. Deceleration can be achieved by providing
a preferred direction for the absorption of photons, e.g. a laser beam with wave vectork, whereas diffusion is due to the randomness in the direction of spontaneous emission.
1

See Risken (1996) and Gardiner (2004).

15

3 Doppler Cooling and Magneto-Optical Traps

3.1.1 Dissipative force
The simplest velocity-dependent force that can be used for laser cooling is provided by
the Doppler effect which moves atomic transitions in or out of resonance according to theatom’s velocity vector. The atom’s absorption probability becomes velocity dependentand so does the net force exerted through absorption processes (assuming a two-levelsystem with states |gi,|ei):
F = ~kγρeewhere γ is the Einstein A coefficient of the transition and ρee is the excited state population (the corresponding diagonal element of the system’s density matrix). Calculatinga steady-state solution for the two-level system lendsρee (t) =s0 /2
1 + s0 + (2∆/γ)2 )where the saturation parameter s0 ∈ [0, 1] is the ratio between the laser intensity I
and the saturation intensity Is that would equally distribute population between bothlevels. The parameter ∆ is the detuning between the transition frequency and the laser
frequency. The detuning is modified by the atom’s velocity v due to the Doppler effect:∆ = δ − k · v.
Putting all of the above together, we obtain an expression for the velocity-dependentdissipative force F
s0 /2
F (v) = ~kγ
2 .

1 + s0 + γ2 (δ − k · v)

Assuming that the Doppler effect is small compared to the detuning δ, i.e.
this expression can be expanded and approximated as

k·hvi
 1,
δ

F (v) ' F 0 − βv
with friction coefficient
β := −4~k 2

s0 δ/γ(1 + s0 + (2δ/γ)2 )2

.

(3.2)

Note that δ = ωeg − ωl < 0 for a red-detuned laser and that δ is the only quantity
determining the sign of the friction force. Correspondingly, a blue-detuned laser will
heat up the ensemble.

3.1.2 Diffusion rate
The diffusion function D is calculated by approximating it as its value at p = 0, which
can be obtained in a treatment analogously to Brownian motion – see Stenholm (1986).
This leads to an expression for D(p, t) ≡ D0
2D0 t = ~2 k 2 (1 + Q + ξ)Rt

16

3.2 Magneto-optical traps
which is the sum of three different contributions, where R = γρee is the scattering ratefrom above. The first summand is due to absorption and subsequent emission of aphoton, the second term contains the Mandel Q-parameterQ=h∆n2 i − hnihnithat describes the deviation of photon statistics from the Poissonian case. For typicallaser light, this is a small parameter and can be omitted. The last term contains ξ, ageometric factor for the angular distribution of spontaneous emission. We obtain
D0 = ~ 2 k 2 γ

s0 /2
1+ξ
,
2 1 + s0 + (2δ/γ)2(3.3)

which is a non-negative quantity. Thus, diffusion is always outwards and can nevershrink the momentum distribution.

3.1.3 Cooling limits
Ultimately, every laser cooling scheme relying solely on spontaneous emission for energy dissipation will be limited by the transition’s energy uncertainty γ, resulting in an
absolute lower limit of kB T = ~γ.
In the present case, the cooling is actually already limited by the counterplay of thetwo processes discussed above: dissipation cools the ensemble, diffusion heats it up. The
limiting temperature – called the Doppler temperature TD – is determined by the ratio of
both processes, represented by the friction (Eq. 3.2) and diffusion (Eq. 3.3) coefficients:
k B TD =D0
1 + ξ 1 + s0 + (2δ/γ)2
=−
~γ
β
8
δ/γ

(3.4)

3.2 Magneto-optical traps
An important improvement on the simple Doppler cooling scheme from the previous
Section is the Magneto-Optical Trap (MOT). It combines optical cooling with an inhomogeneous magnetic field designed to make use of magnetic sublevels. It not only cools
the ensemble, but also confines it spatially.
Two counter-propagating laser beams with different circular polarization σ ± cool
atoms on a J = 0 → J = 1 transition. Additionally a magnetic field B = az isapplied along the laser axis that splits the magnetic sublevels of the excited state as
depicted in Fig. 3.1. This Zeeman splitting is added to the Doppler shift in determining
the force acting on the atoms from each beam:
F ± = ±~kγ

s0 /2
1 + s0 + (2δ ± /γ)2

with the space and velocity dependent detuning δ ±δ ± = δ ∓ k · v ∓ µ0 B/~,

17

3 Doppler Cooling and Magneto-Optical Traps

Figure 3.1: MOT detunings
where µ0 = (ge me − gg mg )µB is the transition’s effective magnetic moment, assuming
σ + -polarized laser light. Adding both forces and expanding again for small Doppler shift
and Zeeman splitting cancels the common zeroth order term resulting in
F = F + + F − ' −βv − κrwith β defined as in Eq. 3.2 and a spring constant
κ=

µ0 a
β.
~k

More generally, a MOT will work for any J → J + 1 transition as long as the gfactors of both states are approximately equal. The Zeeman splitting of all magnetic
sublevels will be matched and all transitions happen at the same frequency. Note that
the Clebsch-Gordan coefficients for σ + (σ − ) transitions increase monotonically towards
the largest (smallest) magnetic quantum number, optically pumping population towards
the outer magnetic sublevels.
In the case that the g-factors and thus the magnetic splitting of both electronic states
are very disparate, the exciting laser beams cannot drive every m → m ± 1 transition
with the same efficiency. The transition frequency ωeg (x) for the σ + beam is (assumingge  gg without loss of generality)~ωeg (x) = ~ω0 + µ0 B(x)= ~ω0 + µB (me ge − mg gg )B(x)' ~ω0 + µB me ge ax.

which is completely determined by the state with the larger g-factor.
The MOT will still work if the transition is broad enough, such that the exciting laser
can still reach the magnetic sublevels with small Clebsch-Gordan coefficients without
losing trapped atoms due to photon recoil. This results in a limiting case, where theelectronic transition’s width γ is smaller than the recoil frequency εr /~, such that theabsorption of a single photon removes an atom from the trap.

18

3.2 Magneto-optical traps
By simulating the optical pumping happening in a broad-line MOT, the efficiency ofa narrow-line MOT can be significantly improved, as demonstrated by Mukaiyama et al.
(2003). This so-called Dynamic MOT (DMOT) statistically mixes magnetic sublevels
by coupling the lower electronic state to another one with comparable g-factor via a“stirring” laser. Rapid Rabi oscillations on the stirring transition shuffle population
between the magnetic sublevels. The population of the trapped state is stochastically
amplified, improving the trapping time by a factor of 6.

19

3 Doppler Cooling and Magneto-Optical Traps

20

Chapter

4

1D Optical Lattices
eutral atoms can be trapped using an optical potential called optical lattice. The
spatial intensity variation of a standing wave formed by two counter-propagating
laser beams of equal intensity and polarization produces a trap due to the AC Stark
effect.
Optical lattices have become one of the standard tool to trap neutral atoms over thelast decade due to the simplicity and controllability provided by this all-optical atomic
trap. This Chapter provides a detailed introduction to one-dimensional (1D) opticallattices, their motional structure and effect on precision spectroscopy of narrow opticaltransitions. First, we need to understand the cause of the trapping force: the AC Stark
effect.

N

4.1 AC Stark effect
The AC Stark shift of level i is the second order perturbation of the atomic spectrum
caused by an electric field E in the dipole approximation (as derived in e.g. Merzbacher,1998) and is given by:
1∆Ei = − α|E|22
where α is the polarizability of the state |ii. The dynamic dipole polarizability for an
exciting plane wave with electric field E = E ẑ and frequency ω caused by the presenceof other states |f i as defined in Bransden and Joachain (2003) is
αi (ω) =2e2 X|hf |z|ii|2,
ωf i 2~
ωf i − ω 2
f 6=i

with ~ωf i := Ef − Ei . The dipole moment zf i := hf |z|ii can be expressed in terms ofthe transition’s Einstein A coefficient (see e.g. Loudon, 1983)
Af i =

e2 ωf3 i
3π0 ~c3|zf i |2 .

21

4 1D Optical Lattices
We obtain
αi (ω) = 6π0 c3

Af i
.
2
ωf i (ωf2 i − ω 2 )
f 6=i

X

The static polarizability αis is the limit for infinite wavelength, i.e.
X Af iαis = lim αi (ω) = 6π0 c3
.
ω→0
ωf4 i
f 6=i

The Einstein A coefficients for each particular transition are usually not available in
the literature. Instead, total lifetimes are given. This is not a big problem, though,as the fraction of the lifetime due to decay to a certain level can be calculated. These
fractions are called branching ratios and will be discussed in the next section.

4.1.1 Branching ratios
Branching ratios can be obtained by using the Wigner-Eckart theorem, which allows separation of matrix elements of tensor operators (See e.g. Sakurai, 1994, Ch. 3). Applying
the theorem to the present case while ignoring hyperfine structure, we can obtain thematrix elements of the dipole operator D q for light with polarization quantum number q
in terms of geometric factors (Clebsch-Gordan coefficients) and reduced matrix elements:
hJf mf |D q |Ji mi i = hJf mf |1Ji qmi ihJf kDkJi i

p
p
J
Ji
hJf kDkJi i := δSf Si (−1)1+Sf +Ji +Lf 2Ji + 1 2Lf + 1 f
Li Lf


1
hLf kDkLi i
Siwhere the expression in curly braces is a 6j-symbol 1 , a group-theoretic factor. The
Clebsch-Gordan coefficients enforce the selection rule mf = mi + q. Since the angularmomentum is a composite of the angular momenta of two electrons, the corresponding
selection rule is Lf = Li , Li ± 1.
These expressions are very useful since the Einstein A coefficients for a transition areproportional to the square of the dipole operator’s matrix elements. Suppose we havebeen given a total inverse lifetime A(SLJ) for a certain 2S+1 LJ state and we want toknow how much of this width is due to decay to 2S+1 L0 J 0 . An expression for the total
decay rate A(SLJ) can be found by summing over the matrix elements above
A(SLi Ji ) ∝

X

Lf +S

+Jf

X

X

Lf +S

1

+1
X

Lf =|Li ±1| Jf =|Lf −S| mf =−Jf mi =−Ji q=−1JJi× (2Ji + 1)(2Lf + 1) f
Li Lf
≡

+Ji
X

X

X

Lf =|Li ±1| Jf =|Lf −S|

1
S

2

|hJf mf |1Ji qmi i|2 ×|hLf kDkLi i|2A(SLi Ji → SLf Jf )Included in Mathematica as the SixJSymbol function. See Brink and Satchler (1968) and Rotenberg
et al. (1959) for detailed information.

22

4.1 AC Stark effectUsing an orthogonality relation for Clebsch-Gordan coefficientsX
hj1 j2 m1 m2 |jmihj1 j2 m1 m2 |j 0 m0 i = δjj 0 δmm0 ,m1 m2

and noting thatP

mf 1 = 2Jf + 1, we can simplify the above toJJi
A(SLi Ji → SLf Jf ) = (2Jf + 1)(2Ji + 1)(2Lf + 1) f
Li Lf

1
S

2

|hLf kDkLi i|2 .

By dividing through the known total line width this expression lends the branching ratioA(SLi Ji → SLf Jf )J
Ji
= (2Jf + 1)(2Lf + 1) f
Li Lf
A(SLi Ji )

1
S

2

|hLf kDkLi i|2 .

The reduced matrix element involving the composite angular momenta Lf and Li depends on the decomposed angular momenta involved. Evaluating this reduced matrix
element requires knowledge of the underlying many-body wave function, as shown in
Brink and Satchler (1968, p. 83-85). We will approximate it as constant for all involved
combinations of Li and Lf , justifying this approximation by only considering small values of L: The reduced matrix element is a linear combination of radial integrals which
will vary very little except in the case of large wavefunctions (large principal quantum
number) and large values of L (many nodes), i.e. Rydberg states. In this approximation,
the branching ratio becomes a geometric factor that can be evaluated easily:A(SLi Ji → SLf Jf )J
Ji
≈ (2Jf + 1)(2Lf + 1) f
Li Lf
A(SLi Ji )

1
S

2

.

4.1.2 Introducing experimental parameters
Finally, we express the laser’s electric field in terms of experimental parameters. We
can write the field as a superposition of two counterpropagating Gaussian beams incylindrical coordinates. We assume that the lattice is formed by one beam only (withpolarization vector ε̂+ ) which is reflected back onto itself with a curved mirror withintensity loss η  1 that also rotates the polarization to ε̂− :E(r, z) = Ee
The AC Stark shift is

−

r2
w(z)2



ε̂+ e−iκz + ε̂−p
1 − η eiκz

1∆E(r, z) = − α|E|2
2
2r 2
p
2
1 2 − w(z)2ε̂+ e−iκz + ε̂− 1 − η e+iκz= − αE e
2
h
i
2r 2
η p
1 2 − w(z)22 1 − + 1 − η cos(ε̂+ , ε̂− ) cos 2κz ,= − αE e2223

4 1D Optical Lattices
resulting in

2
1
− 2r
∆E(r, z) = − αE 2 e w(z)2 4 cos2 κz
2for perfectly parallel polarization and full reflection. In the imperfect case, the trap depth
is reduced and for non-zero η, the energy origin is shifted. For the present discussion,
we will limit ourselves to the ideal case. The maximal electric field amplitude E can be
expressed in terms of the laser power P by using the maximal intensity

1
P
Imax = 0 cE 2 =.
2πw02
We find that the optical lattice potential for state |ii is∆Ei (ω) = −2
4P− 2r 2w(z) cos2 κz.
α(ω)eiπ0 cw02(4.1)

4.2 Separating the motion
The AC Stark shift depends on the electronic state of the atom. This becomes a problem in experiments involving other electronic states, e.g. when doing Rabi oscillations
between two levels. The excited state will experience a different AC Stark shift andmight not even be trapped at all. One would like to separate the motional degrees of
freedom from the electronic excitation to be able to describe the system as a tensor product between the motional and electronic state spaces. The quantity responsible for thedifferent behavior of different electronic states is the dynamic polarizability as definedin the last section. Since the polarizability is frequency-dependent, it is possible in somecases to find a magic wavelength where the AC Stark shifts of both levels are matched(Kimble et al., 1999). To decide whether this is possible for a particular combinationof states, one has to calculate the dynamic polarizabilities and plot them as function of
wavelength. For the case of 87 Sr, a scheme using a magic wavelength has been proposedby Katori (2002); the polarizabilities and numerical values will be presented in Sec. 4.6.2.
For now, let us assume that we can separate the motional degrees of freedom fromthe electronic states and try to find a quantum mechanical description of the latticepotential. However, it quickly turns out that the motional potential in Eq. 4.1 is notseparable: it is a product of the transversal (Gaussian) and longitudinal (sinusoidal)
confinement.
As a first approximation, we will ignore the transversal degrees of freedom and try tosolve the longitudinal problem. This will provide us with the lattice band structure andwave functions and allow us to estimate the effect of laser intensity noise or vibrating
optical elements.

4.3 Longitudinal problem
The one-dimensional optical lattice has traditionally been treated by approximating thepotential at each site by an harmonic oscillator. For shallow lattices, the anharmonicity

24

4.3 Longitudinal problemof the potential starts to modify the system’s dynamics drastically2 .
Setting r = 0 in Eq. 4.1, we ignore the Gaussian part for the moment and solve the 1D
Schrödinger equation for the potential (renaming the variable z to x for the remainderof this section)
V (x) = 4Vmax cos2 (κx)
Requiring translation invariance in the solution, the time-independent Schrödinger equation becomes
#
"
2

~ d
1
+ ~k + V (x) uk (x)Huk (x) =2m i dx= εuk (x)Using a modulated plane wave ansatz
uk (x) = ϕ(x)e−ikx
and the dimensionless variable ξ := κx we obtain an equation for ϕ(ξ):ϕ00 + (a − 2q cos 2ξ)ϕ = 0(4.2)

with parameters
q=

Vmax
ε
,a =
− 2q
εr
εr

and lattice recoil energy

2π 2 ~2mλ2This equation is known as Mathieu’s differential equation and also describes the motionof a particle in an ion trap or a quadrupole mass filter.
Mathieu’s equation has stable solutions only for certain combinations of the a and q
parameters. Since q is the potential height in recoil units and a + 2q is the particle’s
energy, plotting a(q) gives a list of allowed energy ranges: the band structure of the
one-dimensional optical lattice.
Finding the eigenfunctions requires a detour into the theory of ordinary differential
equations with periodic coefficients. See Abramowitz and Stegun (1972) for more information and Heuser (1995) for a more general version of the theorem.
εr =Theorem (Floquet’s theorem). The differential equation y 00 + Q(x)y = 0 with apiecewise continuous and periodic coefficient Q(x) of period π, i.e. Q(x + π) = Q(x), has
two continuously differentiable solutions y± (x) = e±iνx Pν (±x) where Pν is also periodic
with period π.
2After having developed the 1D lattice theory presented in this Chapter, it was pointed out to me
that there already are several papers based on the same ideas and sources (Jáuregui et al., 2001,
Jáuregui, 2001, Poli et al., 2002). The theory presented here still is a bit more general, due to using
the problem’s real eigenfunctions.

25

4 1D Optical Lattices
This can be directly applied to our case, i.e. the Mathieu equation Eq. 4.2 has two
Floquet solutions
Fν± (ξ) = e±iνξ Pν (±ξ)
which are linearly independent for non-integer ν. The parameter ν (called the solution’s
characteristic exponent) indexes the wave function’s energy eigenvalue, that is its positionin the band structure. Note that Fν± is generally not π-periodic, but the Bloch wave
function uk (ξ) must be. It follows that there must exist a relation between the wave
number k and the characteristic exponent ν. By requiringuk (ξ + π) = uk (ξ)
we find for the n-th band, that there must exist numbers p ∈ Z and r ∈ [0, 1) such thatp + r = k/κν = n + r.
Thus the forward and backward “plane waves” of the 1D lattice corresponding to thetwo Floquet solutions read∓ipξ ±inξePn+r (±ξ).
u±k,n (ξ) = e

Here n indexes the band and r = k/κ − bk/κc determines the intra-band position corresponding to a certain quasi-momentum k (the half brackets indicate rounding down to
the next integer). Since the Bloch wave function is periodic in the lattice, a phase factore∓ipξ with integer p = bk/κc represents the same physical situation.
We are interested in localized particles in a finite lattice and want to construct a
localized Wannier basis for this problem, i.e. a basis of site- and band-orthonormal
wave functions. The Wannier functions must conform tohi, n|j, mi = δij δnm .
This can be achieved by building superpositions of Bloch waves of the same band n insuch a way that the i-th Brillouin zone of the lattice becomes their support. The mostgeneral superposition of this kind looks likewni (ξ) :=

X
p∈Z

e

−ipξ

Z 1
Z 1
X
i
i
inξ
ipξ
dr βnp(r)e−inξ Pn+r (−ξ)
dr αnp (r)e Pn+r (ξ) +
e
0

p∈Z

0

Note that the Floquet solution only exists for rational n + r, so the integral needs to beevaluated in Q. In practice, this is not a problem as numerical integration uses floating
point numbers of finite precision, i.e. rational numbers.
To simplify this expression, we will look at the lower lattice bands that lend themselves
to localization, viz. the bands are narrow and look like discrete levels (true for n √q). As the Floquet solutions are continuously dependent on the parameter ν between

26

4.3 Longitudinal problem
integral values, in this case they do not vary much over the whole range of r and we canapproximate
Pn+r (±ξ) ' Pn (±ξ),
leading towni (ξ) ' einξ Pn (ξ)

X
p∈Z

e−ipξ

Z 1
Z 1
X
i
i
dr αnp(r) + e−inξ Pn (−ξ)
eipξ dr βnp
(r).
0

p∈Z

0

We choose superposition coefficients α, β such that
X
p∈Z

e−ipξ

Z 1
Z 1
X
i
i
(r) =: cin Θi (ξ)
dr αnp(r) =eipξ dr βnp
0

0

p∈Z

with the supporting function of the i-th Brillouin zone(
1 ξ ∈ [iπ, (i + 1)π)Θi (ξ) =0 elseand normalization constants cin . The Wannier function becomeshiwni (ξ) = cin Θi (ξ) einξ Pn (ξ) + e−inξ Pn (−ξ)= 2cin Θi (ξ) cen (ξ)

where cen (ξ) is the n-th even Mathieu function as tabulated in Abramowitz and Stegun(1972). Note that these functions are again π-periodic and form a complete orthogonal
set on [0, π]. Also note that cen has no closed form and has to be evaluated numericallyas pointed out by Jáuregui et al. (2001), but that it is implemented in e.g. Mathematica
as the MathieuC function. The normalization constants are evaluated by calculating (seeAbramowitz and Stegun, 1972, p. 732)
Z
i 2
i 2
dξ | cen (ξ)|2
kwn k = |2cn |
Θi
Z π
i 2
= |2cn |dξ | cen (ξ)|2π= |2cin |202r⇒ 2cin =2π

To summarize: the localized wave functions used in this paper arer2iΘi (ξ) cen (ξ).
wn (ξ) =π

27

4 1D Optical Lattices

Figure 4.1: Motional sidebands in harmonic and anharmonic potentials

4.4 Vibrational spectroscopy
The motional structure derived in the last section can be probed using high-resolution
laser spectroscopy. Recall that we were able to separate the motional degrees of freedom
from the electronic states by matching the AC Stark shifts. In analogy to ion trapping
experiments, where ions are electromagnetically trapped in a harmonic potential, there
exists a regime where the vibration of particles around their rest position can be observed
by taking a spectral scan of the electronic transition.
This regime is known as the Lamb-Dicke regime and requires an exciting laser beamthat probes the electronic transition |gi → |ei at ωeg . If its wavelength λ is much larger
than the typical excursion x0 of the trapped particles, corresponding to a Lamb-Dicke
parameter η := 2πλ x0  1, and the electronic transition is sufficiently narrow, i.e. its
linewidth γ is less than the vibration frequency Ω, the atomic motion produces sidebandson the electronic carrier transition. The lowest order sidebands are detuned by ±Ω fromωeg .
To understand the sideband shape, one has to take a closer look at their origin. Fig. 4.1
is a cartoon of a system where the m-th sideband is composed of all inelastic transitions
from the electronic ground state and n-th motional state |g, ni to the electronically
excited state and modified motional state |e, n + mi. The left hand side depicts anharmonic potential (e.g. an ion trap), where these inelastic transitions have exactly thesame transition frequency and all the contributions add up to a Lorentzian line shape,
because the harmonic trapping spaces the motional states uniformly. This is different in
an optical lattice (as shown on the right hand side of Fig. 4.1): the potential formed by
the light shift of a standing wave is sinusoidal resulting in decreasing energy spacing forincreasing motional quantum number. By adding all contributions from n to n + m, thesideband gets smeared out towards the carrier, since the lowest transition |g, 0i → |e, mi28

4.5 Parametric heating
has the largest frequency. This results in a sideband that has a sharp flank on one side
and a smooth tail on the other. The sharp edge can be used to determine the lowest
transition’s frequency, i.e. the trap frequency obtained by approximating the lattice asa harmonic trap around the minimum.
The harmonic approximation has been made frequently to obtain simple expressions
for experimental parameters and to compare the results to the theory available for the
harmonic confinement. As motivated in the previous sections and shown by the cartoon
in Fig. 4.1, this approximation is not good enough to understand a sideband spectrum from a relatively shallow lattice. As a first order approximation omitting thetransversal degrees of freedom and quantum interference between absorption paths, aone-dimensional model of the lattice has been developed. The band structure of the 1Dlattice can be calculated from the theory in Sec. 4.3, which allows placing the individual
motional transitions on the spectrum. By assuming the population to be Boltzmanndistributed, a superposition of sub-lines with according peak heights can be fitted to the
sideband. Their centers are fixed by the band structure and their width increases dueto the broader bands for larger motional quantum number.
The individual transitions inside a sideband will be additionally broadened becauseof dissipation among the motional states. Fitting the additional broadening allows estimating heating rates due to external noise processes and provides a very accurate tool
to observe motional-state dependent effects.
Resolving these vibrational sidebands requires an electronic transition on with a width
on the kHz scale to resolve the sidebands at all, but since we are interested in <100 Hz
scale vibrational effects at frequencies of 1015 Hz, one needs an even narrower and extremely stable spectroscopy laser.
The optical lattice provides a highly stable and controllable environment against external noise sources, but cannot completely eliminate them. A study of the various error
sources is necessary and will be presented in the remainder of this chapter.

4.5 Parametric heating
A likely source of dissipation processes between motional states in the optical lattice
are parametric excitations caused by fluctuations in the lattice intensity or throughmechanical noise on optical components. The treatment in this Section follows Savard
et al. (1997) and Gardiner et al. (2000) but drops the harmonic oscillator approximation
to use the eigenfunctions derived in Sec. 4.3 to obtain classical rate equations for theparametric heating processes. The classical description is valid for an optical latticesystem with sufficient motional decoherence caused by e.g. collisions or coupling to thebackground blackbody field.

4.5.1 Intensity Noise
Assume a fractional laser intensity fluctuation
(t) =

I(t) − I0I029

4 1D Optical Lattices
that modifies the lattice potential according toṼ (x) = V0 (1 + (t)) cos2 (κx) ≡ V + V 0
and produces a small perturbation V 0 of the original potential. Transition rates between
different lattice bands are estimated in first-order perturbation theory as
1
Rm←n =T

Z T
2iωmn t0dt ehm|V |ni .
0Following the arguments in Savard et al. (1997), this integral can be approximated as
Rm←n '

1~2Z ∞2dτ eiωmn τ h(t)(t + τ )i ×−∞× hm|V0 cos2 (κx)|ni2with the noise correlation function
1h(t)(t + τ )i =T

Z T
dt (t)(t + τ ) ≡ f (τ )0Assuming f (τ ) = f (−τ ) this expression can be further simplified using the one-sided
power spectrum of the fractional noise (t)Z
2 ∞S (ω) :=dτ cos(ωτ )f (τ )π 0
to obtain a simple expression for the transition rate per particle
1
2
Rm←n = πS (ωmn ) 2 hm|V0 cos2 (κx)|ni .
~All sites are equivalent, as the evaluation of the matrix element reduces to an integral
on [0, π] for all i:Z2 π2dξ ce∗m (ξ) cos2 (ξ) cen (ξ).
hi, m|V0 cos (ξ)|j, ni = δij
π 0
The transition rate per particle reads
V2 2
Rm←n = πS (ωmn ) 02
~ π

Z π
2
dξ ce∗m (ξ) cos2 (ξ) cen (ξ) .
0Remarks(i) The Mathieu functions cen are even (odd) with respect to π/2 for even (odd) n,although they are all even with respect to 0.
(ii) cos2 ξ is even with respect to π/2, so the matrix elements vanish for |m − n| odd.
(iii) Since ce∗n = cen , Rm←n = Rn←m .

30

4.5 Parametric heating
We will use these transition rates to come up with a set of coupled linear differential
equations that determine the behavior of the system. As all particles and sites areequivalent, each rate can be scaled by the number of particles N to describe the wholeensemble. The rate equations are obtained by writing down all contributions that modify
the population of a certain state Pn :XX
Rn←m Pm (t).
Rm←n Pn (t) +Ṗn (t) = −
m6=n

m6=n

Converting this into matrix form results inṖ (t) = S · P (t)with solution
P (t) = eSt · P 0
and system matrix






S=




−

P

Rm0

m6=0

−

0

R20
..
.

0
P

R02
Rm1

m6=1

0
..
.

−

0
P

Rm2

m6=2

..
.

···




··· 


.
··· 


..
.

The diagonal elements of the S-matrix correspond to the broadening of the individual
sub-lines due to the dissipation modeled here (see Sec. 4.4).
Note that only states which are two motional quanta apart are coupled via this interaction. By experimentally providing initial conditions that put a majority of the population
in a certain state, one should be able to observe a large contrast between peak heightsof even and odd quantum number n if the dominating noise source is lattice intensity
fluctuation.

4.5.2 Phase Noise
A second source of parametric heating could be phase fluctuations of the standing wave
pattern, most likely produced by mechanical noise on a fiber tip or mirror mount. Thiskind of noise can also be described by a fluctuating function (t) with |(t)|  1. Including this noise in the lattice potential amounts toṼ (x) = V0 cos2 (κx + (t))
Expanding this expression in a Taylor expansion and neglecting terms of second or higher
order in , the potential can again be described as the original lattice potential modified
by a small perturbation:Ṽ (x) ' V0 (cos2 (κx) + (t) sin(2κx))

31

4 1D Optical Lattices
Using the same formalism as in Sec. 4.5.1, we obtain new transition rates per particle
due to lattice phase noise:Z2
V02 2 πphphRm←n = πS (ωmn ) 2dξ cem (ξ) sin(2ξ) cen (ξ)~ π 0
The only difference being in the noise spectrum and the matrix element.
Remarks(i) The matrix elements vanish for |m − n| even, since sin 2ξ is odd with respect toπ/2.
(ii) Combining the phase noise with the intensity noise transition rates from the previous section, we will obtain a system matrix without zeros, since the new rates fillin the empty spaces.
The system matrix for pure phase noise is

 P
−
Rm0
R01
0
···

 m6=0
P

Rm1
R12
··· 
R10
−




m6=1
S=
P



R
·
·
·
0
R
−
m2
21


m6=2


..
..
..
..
.
.
.
.

Intensity and phase noise show “orthogonal” behavior in coupling levels that are two
motional quanta apart, so it should be possible to distinguish between the two by looking
at sideband scans after preparing amenable initial conditions and artificially enhancing
one or the other process.

4.5.3 Heating Rates
For the harmonic oscillator approximation (see Savard et al. (1997)), a heating rate could
be defined as the coefficient Γ that appeared when evaluatingX
hĖi =~ωn Ṗn = ΓhEin

In the optical lattice potential, the equations become more difficult, and a heating rate
can only be defined in an average sense depending on the initial conditions. Finding thetime T that corresponds to
hEiT= exp(1)hEi0results in an average heating rateΓ̃ := 1/T
where the average energy needs to be evaluated by calculatingX
hEit =~ωn Pn (t).
n32

4.6 Environmental effects

4.6 Environmental effects
Doing high-precision spectroscopy requires a careful study of potential error sources.
This section estimates values for systematics such as the Doppler and Zeeman effects,
AC Stark shifts caused by the probing laser or background blackbody radiation as well
as altitude shifts. Numerical examples will be calculated for the case of 87 Sr atoms at
T = 5 K, lattice (probe) wavelength λ = 813 (698) nm and beam waist w 0 = 35 m.


4.6.1 Doppler effect
The Doppler effect is the apparent shift in transition frequency caused by an atom moving
at a certain velocity with respect to the exciting beam. In the Lamb-Dicke regime, theDoppler effect is suppressed by a factor proportional to the ratio of the particle’s meanfree path and the probing wave length (Dicke, 1953). By also probing with a beam that
copropagates with the trapping lattice laser, the first order Doppler effect is reduced tothe uncertainty limit at the focus of the Gaussian beam, corresponding to
1∆k × 2w0 ≈ .
2The uncertainty in the probing wave vector samples some of the transversal motion. We
assume the atoms to be Maxwell-Boltzmann distributed in three dimensions (i.e. we rely
on collisions to thermalize our sample). The probability density function isf(3)(v)dv = 4πv

m
2πkB T

3/2

mv 2
exp −
2kB T




dv.

Integer moments of this distribution can be computed as: (√ (n+1)!!

π 2n/2+1
n even2
2kB T n/2
n
hv i = √mπ[(n + 1)/2]! n odd
such that
hvi =

r

8kB T
πm

2
:= hv 2 i =
vrms

3kB T
.
m

This allows putting an uncertainty limit to the observable linewidth caused by theresidual first order Doppler effect as
r
3kB T
1
(1)
γD = ∆k vrms =
4w0
m
s
(1)
σ(γD )σ 2 (w0 ) σ 2 (T )σ(w0 )
.
'
=
+
2
2
(1)
4T
w0
w
γ
0
D

33

4 1D Optical Lattices
In addition to the broadening due to the residual first order Doppler effect, the transition will be shifted by the second order Doppler effect. This is easy to understand if
one remembers that the relativistic expression for the Doppler-shifted frequency due toa velocity β := v/c is:
s
ω
=
ω0

1−β
,
1+β

which can be expanded for |β|  1 asω1
= 1 − β + β 2 + O(β 3 ).
ω02Note that the second order contribution is always positive, regardless of the velocity’ssign.
We will again use the root mean square velocity vrms as the characteristic value toestimate the second order Doppler shift and its relative error:
(2)

∆ωD =
(2)

σ(∆ωD )
(2)
∆ωD

=

2
1 vrms
ω0
2 c2
s

σ(T )
σ 2 (T ) σ 2 (ω0 )
'
+
2
2
T
T
ω0

For T = 5(1) K, ω0 = 2πc/λ and λ = 698 nm, we obtain(1)γD = 2π × 43(9) Hz(2)∆ωD = 2π × 22(5) Hz.


4.6.2 AC Stark shift (probe)
The same effect that forms the lattice will also disturb the states when probing the atoms
with another laser. Since we want to scan over a spectral line, we have to introduce a
probing laser which will have a very different wavelength than the lattice laser. In ourcase, the lattice is formed by infrared light at 813 nm, whereas the investigated transitionis at 698 nm.
Recall that the maximum AC Stark shift for state |ii is∆Ei (ω) = −4Pαi (ω).
π0 cw02

We have to calculate the dynamic polarizabilities of the states involved in our spectroscopic experiment which requires knowledge of the various Einstein A coefficients. The
theoretical values for the Einstein A coefficients can be obtained from the extensive listof lifetimes in Werij et al. (1992) listed in Tab. 4.1.
Going back to the problem at hand, we are interested in branching ratios of statesthat can decay to the 1 S0 and 3 P0 states. The only state coupled to 1 S0 via dipole

34

4.6 Environmental effects

Electronic state
5s5p 1 P15s6p 1 P15s7p 1 P14d5p 1 P15s8p 1 P1
5s9p 1 P15s10p 1 P1
5s11p 1 P1
5s6s 3 S1
5s7s 3 S1
5s8s 3 S1
5s9s 3 S1
5s10s 3 S1
5s4d 3 D1
5s5d 3 D1
5s6d 3 D1
5s7d 3 D1
5s8d 3 D1
5s9d 3 D1
5p2 3 Pτ −1 (106 s−1 )
194
2.64
3.05
13.3
14.7
10.9
6.47
4.49
85
34.5
8.22
4.53
2.77
0.345
60.9
27.1
11
7.23
3.27
119

E(1015 Hz)
4.09
6.427
7.333
7.763
7.761
8.004
8.167
8.282
2.775
4.356
4.984
5.303
5.487
0.724
3.889
4.781
5.192
5.419
5.59
3.982

Table 4.1: Inverse lifetimes from Werij et al. (1992) and energies for 88 Sr from Moore(1971). The upper and lower parts of the table list the data necessary to calculate the
1 S and the 3 P AC Stark shifts, respectively.
0
0

1S
3P

0
0

∆α (a.u.)αi (698 nm) (a.u.)
340
965
635

αis (a.u.)
193
332
139

Table 4.2: Polarizabilities for 1 S0 and 3 P0 in atomic units: 1 a.u. (polarizability) =4π0 a30 .

35

4 1D Optical Lattices
interaction is 1 P1 which dipole-decays only to 1 S0 , so no branching ratio is needed. The
states coupled to 3 P0 are 5sN s 3 S1 , 5p2 3 P1 and 5sN d 3 D1 . We obtain branching ratiosof 1/9 , 1/3 and 5/9 , respectively.
Combining these values with the lifetimes from Tab. 4.1, we get the polarizabilities
listed in Tab. 4.2, resulting in an AC Stark shift due to the probing laserAC∆ωprobe= −2π × 0.6 Hz @ 100 nW.

4.6.3 AC Stark shift (blackbody)
The blackbody shift is caused by the thermal background field, resulting in different AC
Stark shifts of 1 S0 and 3 P0 . The magnitude of this effect can be evaluated by equating
the integrated blackbody radiation density
ρ(T ) =π 2 (kB T )4
15~3 c3

to the energy density of an electric field E 01
ρ = 0 |E 0 |22
lending|E 0 |2 =2 π 2 (kB T )4
' 1.38 106 V2 m−2 @ 300 K.
15 0 ~3 c3


The polarizability difference ∆α = α(3 P0 ) − α(1 S0 ) has been calculated in the previous
section to be∆α = 139 a.u. (atomic units).
The atomic unit of energy is the Hartree energy εh , defined in terms of Bohr radius
a0 = 4π0 ~2 /me e2 and electron rest mass me :1 Hartree = εh =~2
me a20We can come up with the atomic unit of electric field strength by dividing the Hartree
energy through the electron charge and the Bohr radiuse
1 a.u. (electric field strength) == 5.142×1011 V/m.
4π0 a20
The transition frequency shift and its relative uncertainty (assuming a temperature uncertainty of 1 K) can be evaluated from1~∆ωbb = − ∆α|E 0 |22
σ(ωbb )
σ(T )
=4
' 1%
ωbb
T
to be
∆ωbb = −2π × 2.39(2) Hz.

36

4.6 Environmental effects

4.6.4 Second order Zeeman effect
The second order Zeeman effect is caused by the second order perturbation of the linear
Zeeman effect and the diamagnetic term of the 87 Sr atoms. The Zeeman effect caused
by a magnetic field B up to second order is modeled by the HamiltonianµB B
e2 2 2 2(Lz + 2Sz ) +B r sin θ~8me

H = H0 +where µB is the Bohr magneton and Lz (Sz ) is the z-component of the total angularmomentum (electron spin) operator. Both contributions to the second order Zeeman
effect are small and can be treated in perturbation theory. The diamagnetic term willbe treated in first order, whereas the linear term contributes in second order.
Linear term in second order
The linear term
H 0 ≡ µB B(Lz + 2Sz )/~ = µB B(Jz + Sz )/~produces a second order energy correction∆Ei = −

X |hf |H 0 |ii|2
f 6=i

~ωf i

.

Ignoring hyperfine structure for the moment and working with the fine structure eigenfunctions |LSJmJ i ≡ |JmJ i, the matrix element reduces to hf |Sz |ii, as the diagonalterm is excluded from the sum. By expanding in terms of uncoupled angular momentum
functions |LSmL mS i
|JmJ i =X

mL ,mS|LSmL mS ihLSmL mS |JmJ i,

the matrix element can be evaluated as a sum over Clebsch-Gordan coefficients hLSm L mS |JmJ i:
hJmJ |Sz |J 0 mJ 0 i = ~δLL0 δSS 0X

mL ,mSmS hJmJ |LSmL mS ihLSmL mS |J 0 mJ 0 i

to obtain2Xµ2 B 2 X −1
∆Ei = − B
ωf i δ L f L i δ S f S i
mSi hJf mJf |Li Si mLi mSi ihLi Si mLi mSi |Ji mJi i .
~
m ,m
f 6=i

Li

Si

Note that each term of the first sum contributes with the inverse energy difference
between the two states |f i and |ii. Contributions will come only from close lying stateswith the same L and S quantum numbers. For the 1 S0 state the shift vanishes, whereas37

4 1D Optical Lattices3
for 3 P0 , only thep P1 state produces a non-zero contribution at mJ = 0. The sum
evaluates to − 2/3 and
2 µB B 2 3
3
ω( P1 − 3 P0 )−1∆ω( P0 ) =3~' 16 mHz @ 100 mGauss.

Note that this calculation omits hyperfine structure. This can be remedied by applying
the hyperfine mixing theory in the last part of this Section.
Diamagnetic term in first order
The diamagnetic term causes a first order energy shift∆Ei =e2 B 2
hi|r2 sin2 θ|ii.
8me

Its magnitude can be estimated easily when knowing that for the low 87 Sr states,p
hr2 i ≈ 5 − 10 × a0 . We obtaine2 B 2100a208me' 150 Hz @ 100 mGauss

∆Ei ≈



Including hyperfine mixing
The hyperfine interaction mixes the singlet and triplet P states, making the clock transition possible. This means that the pure states are not the right candidates to use in
the calculation of the Zeeman shift. Instead, we have to find the mixing coefficients andwork with superpositions. The treatment in this Section follows Peik et al. (1994) wherea clock transition in In+ was examined. The idea is to find the mixing coefficients forthe modified 1 P1 and 3 P1 states caused by hyperfine interaction. Then calculate thesuperposition corresponding to the 3 P0 state in terms of these coefficients in first order000
perturbation theory. The final states will be 1 S0 , 1 P1 , 3 P0 and 3 P1 , where the prime
denotes a modified state.
The theory relating hyperfine constants and mixing coefficients was developed by Breit
and Wills (1933) and a good mathematical reference is Lurio et al. (1962). The hyperfine
interaction mixes the 1 P1 and 3 P1 states as follows:0|3 P1 i = α|3 P1 i + β|1 P1 i
0

|1 P1 i = −β|3 P1 i + α|1 P1 i√
where normalization enforces β = 1 − α2 . These coefficients can be obtained from thelifetimes and transition wavelengths asτ (1 P1 )λ3 (3 P1 − 1 S0 )
β2
=
.
α2
τ (3 P1 )λ3 (1 P1 − 1 S0 )38

4.6 Environmental effects
The 3 P0 state is perturbed through the dipole part of the hyperfine interaction operator
K (1) as defined in Lurio et al. (1962)
00

0|3 P 0 i = | 3 P 0 i +h1 P1 |K (1) |3 P0 i 1 0
h3 P1 |K (1) |3 P0 i 3 0|Pi+| P1 i1
E(3 P0 ) − E(3 P1 0 )E(3 P0 ) − E(1 P1 0 )00≡ | 3 P 0 i + α 0 |3 P 1 i + β 0 |1 P 1 i= |3 P0 i + (α0 α − β0 β)|3 P1 i + (α0 β + β0 α)|1 P1 i.
Expanding the inner products, calculation of the matrix elements reduces to computing
h3 P1 |K (1) |3 P0 i and h1 P1 |K (1) |3 P0 i for which algebraic expressions can be obtained fromthe same paper. They reduce to
h c
i
p
√
2,1
hI 3,1 P 1 F m|K (1) |I 3 P0 F mi = I(I + 1) ±(as − a1/2 ) + c1,2 2a000 .
2
The constants c1 and c2 are related to α and β by√  
  r 
1 1
2
α
c1
√
=
.
β
c2
2 −1
3

The a numbers describe single-electron hyperfine coupling and can be calculated fromthe hyperfine interval spacings A(2S+1 LJ ) for J 6= 0. The equations are underdeterminedand some relativistic correction parameters have to be calculated from theory to obtain
numerical values for the a numbers as listed in Heider and Brink (1977). Thus, the final
states are known and the differential second order Zeeman shift between 1 S0 and 3 P0
can be calculated in perturbation theory as demonstated in the previous two Sections.
Note that Lurio et al. (1962) also list algebraic expressions for all first order Zeeman
matrix elements.

4.6.5 Gravitational shift
Absolute frequency measurements are comparisons against some sort of frequency standard. If the frequency standard is located at a different altitude than the experiment,
the difference in gravitational field amplitude amounts to a time dilation and thus a
global shift in frequency. In the present experiment, absolute frequencies are measured
by comparing against an optical-fiber-transferred signal locked to a Cs-standard at NIST
Boulder. The Cs clock is located at a larger altitude than the experiment, i.e. under
the influence of a gravitational field of lower magnitude. The relative change of the rest
wavelength λ0 in a gravitational field produced by a mass M at distance r is
r
2GM
λ
= 1−
λ0
rc2
with the gravitational constant G. This translates into a relative change of frequency
between our lab frequency ωl and the Cs clock frequency ωc (situated at rc − h and rc

39

4 1D Optical Lattices
respectively)vuu 1 − 2GM
ωl
rc c2
=t
2GM
ωc
1 − r c2 (1−h/r )ccwhich can be expanded in the small parameter ε := h/rc as
ωl
=1+
ωc



r c c2−2GM

−1

ε + O(ε2 ).

Using rc = 6371.01 km + 1640 m, earth’s mass M = 5.9742 × 1024 kg and an estimate of
the height difference h = 14(3) m, we obtain a relative correction
ωl − ω c= 1.5(3) × 10−15 .
ωcTo get a feeling for its magnitude,∆ωG ' 2π × 0.64(13) Hz @ 698 nm.

40

Chapter

5

Precision Spectroscopy in a
1D Optical Lattices an application of the theory developed in the previous chapter, precision spectroscopy of the 87 Sr 1 S0 − 3 P0 clock transition in a 1D optical lattice will be presented.
Every spectroscopic experiment can be separated into three conceptual parts: probe,
sample and detector corresponding to the 1 S0 − 3 P0 spectroscopy laser (698 nm), the87 Sr atoms confined in a magic-wavelength 1D optical lattice and fluorescence detection
on 1 S0 − 1 P1 (461 nm) with a photo-multiplier tube (PMT). In addition, various trappingand utility lasers help to prepare the sample in a perturbation-free environment.
The spectroscopic data is obtained by scanning the 698 nm laser over the 1 S0 − 3 P0
resonance. This data must then be referenced to a frequency standard: this experimentuses the NIST F1 Cs standard available via an optical fiber link to a Cs-referencedhydrogen-maser at NIST Boulder. The spectroscopic data is compared to the reference
signal by an octave-spanning femtosecond comb. The details of the absolute frequency
metrology will be presented in the next chapter.
This chapter will focus on the issues associated with a magic wavelength 1D opticallattice for 87 Sr; the sample preparation and pre-cooling steps are described in more detailin several papers from the Ye group (Xu et al., 2002, 2003a,b, Loftus et al., 2004a,b, Ido
et al., 2005).

A

5.1 Atom preparation and MOT coolingPreparation and initial trapping is described in Loftus et al. (2004a) and we shall only
summarize the most important steps; see Fig. 5.1 for a detailed picture. Refer to Fig. 5.2
for a detailed energy diagram of 87 Sr and the relevant laser wavelengths.
The strong 1 S0 − 1 P1 transition at 461 nm is used for all of the following preparation of the initial atomic sample. This blue light is produced by frequency-doubling a
Ti:Sapphire laser at 922 nm. Acousto-Optical Modulators (AOMs) detune the light with41

5 Precision Spectroscopy in a 1D Optical Lattice

FIG. 1. (Color online) (a) Pa
level diagram. Numbers in parenstein A coefficients (in s−1). Wa
vacuum. (b) Top (x-y plane) vie
ing and trapping apparatus. B
represent 461 nm (689 nm, 679
trapping (trapping and repump
mirror; l / 4, dual wavelength qul / 2, half-wave plate; DBS, dichr
PBS, polarization beamsplitter; B
SMPM, single-mode polarizati
ECDL, external-cavity diode las
Alpert vacuum gauge; NI, nude
TSP, titanium sublimation pumption coil.

to motivate
the experimentallyobserved 3D l
Alternatively,
in regime
where D . GEfigure, the atomFigure 5.1: tions.
Samplepreparation
and(I)pre-cooling,from Loftus
et al.
(2004a)light interaction is dominated by single-beam photon scattering and trap thermodynamics become intimately linked toboth the velocity and the spatial dependence of the force.
Here, gravity plays an essential role as the ratio R for the1
S0-3 P1 transition is only ,16. Consequently, the atoms sag
to vertical positions where the Zeeman shift balances d, leading to d-independent equilibrium temperatures.
Narrow line cooling also displays a unique set of d . 0
thermal and mechanical dynamics. For these experiments,
the atomic gas is first cooled to mK temperatures and then dis suddenly switched from d , 0 to d . 0. Subsequently, thesample evolves from a thermal distribution to a discrete setof momentum-space packets whose alignment matches lattice points on a three-dimensional (3D) face-centered-cubic
crystal [16]. Up to 26 independent packets are created withd- and s-dependent lattice point filling factors. Note this surprising behavior occurs in the setting of incoherent excitationof a nondegenerate thermal cloud. To obtain qualitative insight into the basic physics, we begin with an analytic solution to the one-dimensional (1D) semiclassical radiative
force equation. Here, we show that d . 0 excitation enables“positive feedback” acceleration that efficiently bunches theatoms into discrete sets of d- and s-dependent velocity spacegroups. A simple generalization of the 1D model is then used

This intuitive picture is then confirmed withculations of the final atomic velocity and stions. Using the numerical calculations, we
d . 0 momentum-space crystals are a unive
standard Doppler cooling and that observat
possible, although increasingly impractical, w
molasses.
Finally, we demonstrate that R directly ithermodynamics, enabling cooling arouns-dependent velocity v0 where gravity balancforce. Observed values for v0 agree well withdictions while cooling is evident in distinccloud spatial distributions that appear in both
culations of the cooling process and the expe
momentum crystal formation, gravitationallycooling is universal to Doppler limited system
typical case where R , 105, however, equilib
tures sv0d are on the order of hundreds os100 m / sd rather than the more usefuls,10 cm/ sd values achieved with narrow lin
The remainder of this paper is organized ation II gives an overview of the 461 nm 1S0to precool 88Sr to ,2.5 mK. 3 P2 metastab
magnetic traps that are continuously loaded

063413-2

Figure 5.2: Lowest singlet and triplet states of 87 Sr

42

5.2 1D optical latticerespect to the 1 S0 − 1 P1 transition in an offset lock using saturated absorption feedback.
The 87 Sr atoms are heated in an effusion oven (see upper left of Fig. 5.1) and form
a thermal atomic beam that is filtered by an aperture. After passing the aperture, thebeam is transversally cooled in a 2D optical molasses. A gate valve and an electrical
shutter separate the production region with its hot thermal background from the trap
region. The beam is then further cooled in a Zeeman-slower, i.e. a coil producing
a magnetic field gradient along the beam direction and a head-on σ − -polarized laser
beam. Finally, a standard six-beam MOT captures the atoms at the trap center, where
three pairs of Helmholtz coils allow minimizing residual magnetic fields.
Approximately 3 × 107 atoms are captured and cooled by the blue MOT to Tblue '2.5 mK. Fluorescence light from the blue MOT is captured on a CCD camera, allowing tomonitor the MOT alignment by observing its shape. The metastable 3 P2 state (τ ∼ 500 s)
acts as a sink for the MOT population as it removes atoms from the cooling cycle via
the spontaneous decay channel 1 P1 − 1 D2 − 3 P2 for a long time (see Fig. 5.2 and Xu
et al. (2003b)). To counteract the losses into the 3 P2 state, population can be pumped
to 3 S1 from where it subsequently decays to 3 P1 (Loftus et al., 2004a).
To further cool the sample, the atoms are then transferred to a narrow-line MOT on
the 1 S0 − 3 P1 transition at 689 nm. This narrow line allows MOT-cooling into the K
regime (TD = 180 nK, Tr = 450 nK). The six beams are introduced into the same beam
path as the blue MOT with dichroic mirrors and fluorescence at 689 nm is monitoredwith a PMT. We manage to transfer roughly 25% of the atoms into the narrow-line
MOT, producing a sample at temperatures as low as 250 nK.
As discussed in chapter 3, trapping fermionic isotopes of alkaline earths is very different
from the textbook description of an alkali metal MOT. The g-factors of the electroniclevels are very different: the upper states 1 P1 and 3 P1 have much larger g-factors than
the ground state 1 S0 due to hyperfine interaction (see Sec. 4.6.4). But since the bluetransition at 461 nm has a natural linewidth of 32 MHz, the 1 S0 − 1 P1 MOT still works
very well. On the other hand, the red MOT uses the 1 S0 − 3 P1 transition with a linewidthof 7.1 kHz comparable to the recoil frequency of 4.8 kHz. This makes it necessary to
implement the DMOT approach discussed in Sec. 3.2. Two External Cavity stabilized
Diode Lasers (ECDLs) in a Littman setup were built, one being injection-locked toanother diode laser to form a strong trapping beam and one being used as stirring laser(see section 3.2).
The red MOT (689 nm) trapping laser is detuned to the red of the |1 S0 , F = 9/2i →3| P1 , F = 11/2i transition, as shown in the inset on the right hand side of Fig. 5.2,
depicting the hyperfine manifold of 3 P1 . The stirring laser is tuned below the higher
lying |3 P1 , F = 9/2i state with its smaller g-factor (Mukaiyama et al., 2003).


5.2 1D optical lattice
After having cooled the atoms into the K regime, they are slow enough to be loaded
into a far-off-resonance trap (FORT) or optical lattice. As shown in chapter 4, it isconvenient to use a trapping wavelength that matches the AC Stark shifts of 1 S0 and

435 Precision Spectroscopy in a 1D Optical Lattice

Figure 5.3: AC Stark shifts of the relevant 87 Sr levels, figure courtesy of J. Ye
3 P . In Fig. 5.3, the magic wavelength is shown to lie in the vicinity of 800 nm, with an0

accuracy limited by the knowledge of the atom’s spectrum and oscillator strengths. We
experimentally determined the magic wavelength to be 813.418(10) nm by observing thevariation of the clock transition linewidth while varying the lattice wavelength.
The figure also shows the AC Stark shift of the 3 P1 state. Since it is also shifted
negatively, the presence of the strong lattice laser light does not hamper the red MOT
cooling process, even in the region where both wavelengths are incident on the atoms.
This greatly enhances the lattice loading efficiency since the atoms trapped in the red
MOT are forced into the much smaller lattice while still being cooled (transfer efficiency
10 %, lattice lifetime 1 s).
The lattice and spectroscopy lasers are introduced into the trapping chamber via thecontraption depicted in Fig. 5.4 and form Gaussian laser beams that are tilted at a small
angle with respect to gravity. This vertical mounting ensures that hot sample atoms
drop away under the influence of gravity, but also makes it necessary to use a large
enough trap depth for the optical lattice. The infrared lattice laser beam leaves fibertip
A and gets combined with the spectroscopy beam from fiber B on dichroic mirror D.
The copropagation of both beams is of paramount importance to cancel the first order
Doppler effect: if the probing beam is not parallel to the trapping beam, it will sample
transversal motion and thus broaden the line. The maximal beam divergence is 5 mrad,resulting in an uncertainty of the optical frequency measurement of ∼ 1.9 Hz. After
entering the trapping chamber through focusing lens L, both beams get reflected back by
the highly reflective mirror M . The reflections from this mirror are coupled back into therespective fibers and are used to implement a fiber-noise cancellation protocol. By using
the reflection from the mirror situated behind the sample, the fiber-noise cancellation
not only reduces noise due to thermal and acoustic vibrations of the fiber, but also of all
optical elements up to mirror M . To prevent fiber tip reflections from influencing thenoise cancellation, the fiber tips were polished at an angle.
The polarizations of trapping and probing beams need to be matched as well, to avoid44

5.3 Spectroscopy

Figure 5.4: Lattice and spectroscopy laser mount (see text).
additional line broadening due to the tensorial nature of the sample’s polarizability. The
quality of the optical elements, especially the focusing lens, is the main factor introducing
mismatch in beam polarizations.

5.3 Spectroscopy
Doing spectroscopy on the ultranarrow 1 S0 − 3 P0 transition makes it necessary to use alaser with a spectral width comparable to the natural linewidth. Building laser oscillatorsthat are stable on the sub-Hz scale requires a highly stable reference cavity. These highly
stable cavities are mounted on massive tables which are suspended from the ceilingwith rubber bands to minimize sensitivity to accelerations. Acoustic noise is usuallyattenuated by encasing these mounts with acoustical isolation. All these noise sources
perturb the cavity’s resonance frequency by varying the cavity’s optical length. There isa new scheme on how to get around these large and massive platforms: by mounting thecavity vertically and attaching it to the mount at specific points, the differential change
in optical length below and above the mounting point can be compensated to first order.
The spectroscopy laser is an ECDL locked to the reference cavity in a Pound-DreverHall locking scheme. In a beat experiment against another sub-Hz laser system, thespectroscopy laser has shown a linewidth <3 Hz (Notcutt et al., 2005). Since the cavity
spacer is made out of Zerodur, it has very good short-term stability, but shows a residual
mechanical drift rate on the order of 7 Hz/s. This would make the laser unusable forspectroscopy of such a narrow line, so the drift has to be compensated for. A driftcompensation scheme that also allows scanning the laser frequency over the 1 S0 − 3 P0
resonance will be presented in detail in chapter 6. For the remainder of this chapter, we
will assume that we have a perfectly stable and tunable probe that is referenced to afrequency standard and focus on the issues arising from the confinement geometry andthe specifics of 87 Sr.
The spectroscopy signal is obtained by observing the fluorescence on the strong 1 S0 −1 P transition as a measure of the 1 S population. Doing a Rabi flop on the narrow
10
1 S − 3 P transition is inefficient since the Rabi frequency varies over the radial extent
00of the probing beam. Thus, the metastable 3 P2 state is used to store population in the45

5 Precision Spectroscopy in a 1D Optical Lattice

Lattice Band StructureBlue SB

160

3000
2000

Red SB
Carrier

1000

E/Erecoil @ 813 nm

Photon counts

4000Ω/2π ~
80 kHzBackgroundcounts0(a) -120-80 -40140

Etrap~20 µK

120
100
80
60
40
20

0

40

(b) 0
80 120698nm probe frequency (kHz)

3
2
1
0

(n)

Signal amplitude (a.u.)

0.5
0
5

Decomposed fit
0.6

0.7

0.8

0.9

1

1.1

1.2

Signal Amplitude (Arb. Units)

4600 sidebands,
Figure 5.5: Typical spectrum Dataof the 1 S0 − 3 P0 resonance with vibrational
1
2
Multiple Lorentzian fit
figure from Ludlow et al. (2005).
4400

4200
following way: A π-pulse at the maximal Rabi frequency is shone onto thesample. Then,3
33
population is excited to P2 by driving the P0 − S1 transition which decays to 3 P1 and4000
3 P (γ = 2π × 1.4 MHz). By repeating this probing and shelving cycle 20
times for each2
point in the spectral scan, the fluorescence signal can be nulled almost completely
in the3800
101resonant case. This works because the atoms in the S0 state have a finite temperatureand rethermalize on a much shorter timescale than the 3 P2 lifetime (500s). 177 Hz
3600
When first starting to measure narrow spectra of the transition in question,
we foundLinewidth
vibrational sidebands as discussed in section 4.4. These sidebands are3400
very convenientfor quickly checking important0.6
systemThe distanceof both the red and0.7 parameters.
0.8 0.9 1.0
1.1
3200
the blue sideband’s outer flankfromthecarrieristhetransition
frequency-1.0from-0.5 the0.0 0.5 1.0
nn + 1 transition frequency
motional ground state to the firstexcitedstate.
Sincetheharmonic
approximation
fits
698 nm probe frequency [kHz]1 )(normalized against 0
best for the lowest-lying motional states, fitting a half-Lorentzian to the outer flank andfitting a Lorentzian to the carrier returns the longitudinal trap frequency ω z . Next, therelative height of the blue and red sideband returns the temperature in the harmonic
approximation, assuming a Boltzmann distribution. Finally, the ratio of carrier andsideband peak heights is related to the Lamb-Dicke parameter η (Wineland and Itano,1979).
Doing this preliminary analysis on the data shown in Fig. 5.5 gives ωz = 2π × 80 kHz,T = 5 K and η = 0.23.
Applying the theory developed in chapter 4, we can do a bit better, though: By fitting
a superposition of Lorentzians to the sideband we can get the lattice parameters moreaccurately, due to being able to calculate the motional spectrum beforehand. As was
discussed in section 4.4, we can use the trap frequency determined by the sharp flanks ofthe sidebands to calculate the q-parameter which completely determines the spectrum.
This has been done for ωz = 2π × 80 kHz in Tbl. 5.1. Thus we can fixate the individual
sub-lineshapes relative to the sharp flank and fit a superposition of Lorentzians – withdecreasing peak heights corresponding to a Boltzmann distribution – to the sideband.
1

5

2

5

3

5

4

5

5

(c)



46

(d)

5.3 Spectroscopy
140
Lattice Band Structure

160|6iE/Erecoil @ 813 nm

120
Energy (Erec )E(Erec )4000Γ(Erec ) E(kHz) Γ(Hz)−4 SB
11.23 10−8
38.97 10Blue−6
33.16300010
115.06 0.009−453.97 10187.24 0.360
Red SB
73.5320000.002
255.13 8.384
91.68 0.036Carrier 318.09Ω/2π124.406~
kHz
108.0110000.359
374.7680 1243.010
Backgroundcounts
121.42 2.185
421.28 7580.502
Photon counts

n
0
1
2
3
4
5
6

100

80
60

140|5i120|4i100|3i

Etrap~20 µK

80|2i

3
2
20
40
|0i
1
20
0
0
(b) 0
0
(a) -120
0 lattice4080(n)kHzTable 5.1: Band structure of
the-80
1D -40
opticalfor a120trap frequency of 80
40

60|1i

698nm probe frequency (kHz)

0.6

0.7

0.8

0.9

1
1

1.1

5
1
5
2
5

0

3

1


5
4
5
5

0.9 1.0 1.1(c) n 0.6 n0.7+ 1 0.8transition
frequency(normalized against 0



1

4600

2


)

1.2

Signal Amplitude (Arb. Units)

Signal amplitude (a.u.)

0.5
0

Data
Multiple Lorentzian fit
Decomposed fit

4400
4200
4000
3800
3600 177 Hz
3400
3200

Linewidth-1.0 -0.5 0.0(d)0.5 1.0698 nm probe frequency [kHz]Figure 5.6: Multiple-Lorentzian fit to the blue sideband in Fig. 5.5, figure from Ludlow
et al. (2005)
The fit lends a more accurate measure of the longitudinal temperature; an example forthe blue sideband is shown in Fig. 5.6. The bumpy structures especially apparent in the
red sideband of Fig. 5.5 are very well matched with the centers of the sub-lineshapes.
The temperature obtained from the Boltzmann distribution of the peak heights also
matches the 5 K obtained by taking expansion pictures. The trap depth calculated fromthe observed trap frequency is ∼20 K matching the design parameters very well. We
see significant (∼ kHz) broadening of the sub-lineshapes in addition to the broadening
given by the band structure, indicating strong motional dissipation. These dissipative
processes make our model of incoherently adding Lorentzians even more valid, as thesystem’s coherences decay rapidly.
In conclusion: motional sideband spectroscopy is an accurate tool to obtain trap
characteristica. It is doubly beneficial because it allows monitoring of experimental
parameters without modifying the setup. By doing high-resolution scans of the motional
sidebands it will be possible to quantify state-specific effects, such as dissipation processes
broadening each motional state (see Sec. 4.5).
Finding the sideband structures was serendipical and not the main goal of the experi47

E/Erecoil

CarrierΩ/2π ~
80 kHz

80

Phot

3
2
1000 Background40
1
counts200(b) 00(a) -120-80 -40 Spectroscopy
0
40
80 in
120a 1D Optical(n)5 Precision
Lattice60

698nm probe frequency (kHz)Signal amplitude (a.u.)

0.5

0.6

0

0.7

0.8

0.9

1
1

1.1

5
1
5
2
5

0

3

1

5
4
5
5

(c) n

0.6

0.7 0.8 0.9 1.0 1.1
n + 1 transition frequency(normalized against 0

1

4600

2

)

1.2

Signal Amplitude (Arb. Units)

Data
Multiple Lorentzian fit
Decomposed fit

4400
4200
4000
3800
3600 177 Hz
3400
3200

Linewidth-1.0 -0.5 0.0(d)0.5 1.0698 nm probe frequency [kHz]Figure 5.7: Typical spectrum of the 1 S0 − 3 P0 transition, figure from Ludlow et al.
(2005)
ment. We proceeded to zoom in on the carrier and tried to minimize its width. A typicalscan with a linewidth of less than 200 Hz is shown in Fig. 5.7. The measured linewidth ismuch larger than the theoretically predicted 8 mHz and the uncertainty limit of ∼ 50 Hz,
due to several systematic effects.
Many of the broadening and shifting effects have already been discussed in Sec. 4.6, butsome have to be determined experimentally, because the theory lacks accurate knowledgeof important atomic parameters such as the scattering length.
An example for such an experimentally determined systematic is the density shift:at K-temperatures, scattering becomes a purely quantum-mechanical process. Theinteratomic molecular potentials vary depending on the average particle spacing, causingtransition frequencies to shift. This is usually a very small effect (∼ mHz) and only
noticable due to the extremely high precision necessary for an optical clock experiment(Gibble and Chu, 1993, Legere and Gibble, 1998). As shown in Fig. 5.8, we varied theatomic density inside the lattice by a factor of 50 and fitted a linear function to the center
frequency data to obtain an upper limit on the density shift: 1.0(6.4) Hz/(10 12 cm−3 ).
The second experimentally determined systematic is the Zeeman shift due to residual
magnetic fields or an anisotropic population distribution among the lower state’s magnetic substates. By using the three pairs of compensation coils, we determined a firstorder Zeeman shift of -32(7) Hz/Gauss (see Fig. 5.9). The magnetic field uncertainty isless than 400 mGauss, corresponding to an uncertainty of 12 Hz in the frequency measurement.
Finally, the AC Stark shifts due to the incident lattice and probe lasers were determined experimentally. By inserting an EOM into the probe laser path, we were able tovary the probing intensity while retaining a strong off-resonant carrier frequency to use
for the fiber-noise compensation discussed in the last section. We obtained -17(8.3) and
7.0(0.9) Hz for the lattice and probe AC Stark shifts, as shown in Fig. 5.10 for the shift
caused by the lattice laser.


48

@ λmagic: 0(3) Hz/I0

10

100

0

-50

30

Lattice Stark shift0@ λmagic: 0(3) Hz/I0-100 -10

(a)
0.6

0.8

1.0

Lattice Intensity [I/I0]

1.2

(b)
10

4x10
-20

10

11

10

12

3

Atom ic D ensity [atom s/cm ]ν - 429228004229900 [Hz]

50

20

40

Density shift12-3
1.0(6.4) H z / (10 cm )150

Lattice Stark shift0ν - 429228004229900 [Hz]sr
νsr - 429228004229900 [Hz]
νsr - 429228004229900 [Hz] νsr - 429228004229900 [Hz]5.3 Spectroscopy

4x10

Density shift12-3
1.0(6.4) H z / (10 cm )

150
100
50
12

0

-50

νsr - 429228004229900 [Hz]

sr

80-30
Ref. from[9] Ludlow et al.-100(b)Figure 5.8: Density
shift, figure(2005)
(a)
60
-40
300
Magnetic Shift
10
11
12
12
4x10
10
10
4x10
0.6
0.8
1.0
1.2
40
3
-32(7) Hz / G
Lattice Intensity [I/I0]
Atom ic D ensity [atom s/cm ]
200
20400
80
100
Ref. [9]060300Magnetic
Shift
0
-20
40
-32(7) Hz / G
200
100
-40
20
This work(c)
100(d)-60
0200-4 -3 -2 -1 0 1 2 3 4
5/30 5/17 5/31 6/14 6/28 7/12 7/26
-20
axial magnetic field [G]Date (2005)-100-40
This work
(d)
(c)
-60
-200
-4 -3 -2 -1 0 1 2 3 4
5/3 5/17 5/31 6/14 6/28 7/12 7/26
axial magnetic field [G]Date (2005)@ λmagic: 0(3) Hz/I0

-10
-20
-30

(a)
-40

0.6

0.8

1.0

Lattice Intensity [I/I0]

1.2

Density shift12-3
1.0(6.4) H z / (10 cm )

150
100
50
0
-50
-100

sr

Lattice Stark shift0ν - 429228004229900 [Hz]

νsr - 429228004229900 [Hz]Figure 5.9: Magnetic shift, figure from Ludlow et al. (2005)

(b)

4x10

10

10

11

10

12

3

4x10

12

Atom ic D ensity [atom s/cm ]νsr - 429228004229900 [Hz]80400
Ref. [9]Figure 5.10: Lattice300
AC StarkMagnetic
shift, figureal. (2005)
Shiftfrom Ludlow et 60
40-32(7) Hz / G
200
20
100
0
0
49
-20
-100
-40
This work
(d)
(c)
-60
-200
-4 -3 -2 -1 0 1 2 3 4
5/3 5/17 5/31 6/14 6/28 7/12 7/26
axial magnetic field [G]Date (2005)
νsr - 429228004229900 [Hz]400

5 Precision Spectroscopy in a 1D Optical Lattice

3

2

2

1

1

r/w0

w0/λ = 1

3r/w0

w0/λ = 0.5

0

0
1

−1

−1

−2

−2

0.9
0.8
−3

−3
−2

−1

0

1

2

3

−6 −5 −4 −3 −2 −1 0

z/ λ2

1

2

3

4

5

z/ λ2

6

0.7
0.6
0.5
0.4

3

0.3

2

2

0.2

1

1

r/w0

w0/λ = 40

3r/w0

w0/λ = 50

V (r, z)/V0

−3

0.1
0

0

−1

−1

−2

−2
−3

−3
−30

−20

−10

0

z/ λ2

10

20

30

−200

−100

0

100

200

z/ λ2

Figure 5.11: Lattice curvature for ratios w0 /λ = {0.5, 1, 5, 40}
Another concern is that the geometry of the 1D optical lattice might adversely affectthe width of our spectra. To answer this question, we will try to get a better feeling forthe lattice shape by deriving some additional system parameters. Expanding the latticepotential Eq. 4.1 to second order around a minimum lends the ratio between trapping
frequenciesλωr=√' 5 × 10−3 ,
(5.1)
ωz
2πw0
obtaining a transversal trap frequency ωr = 2π × 0.4 kHz at a lattice waist w0 = 35 m.
The lattice has ∼400 sites with typically 250 atoms per site, i.e. spatial densities of
2 × 1012 cm−3 . Each populated lattice site looks the same. This is true because theratio of w0 /λ ' 43 produces a cylindrically symmetric standing wave in the longitudinal
dimension for the populated region around the focal point of the lattice beam. This
amounts to approximating w(z) ' w0 in Eq. 4.1. In Fig. 5.11, the lattice potential
is plotted for different ratios w0 /λ. Note that the z-axis is scaled differently for each
subplot and that only the curvature of the envelope changes.
Still, a one-dimensional optical lattice is not really a one-dimensional system. This
becomes very apparent when comparing the relevant energy scales to the transversal levelspacing: due to the non-separability of the Gaussian beam potential into longitudinal andtransversal degrees of freedom, the transversal trapping frequency will always be several
50

5.3 Spectroscopyorders of magnitude less than its longitudinal counterpart. Being in the longitudinalground state cannot guarantee being in the transversal ground state. Even worse, by
focusing the lattice beam more tightly, this inequality deteriorates even more as shown
by Eq. 5.1. This means that atoms in the longitudinal ground state are highly excited in
the transversal degrees of freedom. In the sense of the experiments on Tonks-Girardeau
gases1 , this is not a one dimensional system. But since the transversal level splitting isso small, we can treat the particle as being quasi-free in the transversal direction, i.e.
small changes in the particle’s transversal energy do not affect its longitudinal state.
The separation into two different harmonic oscillators is also not correct: the potential
is not separable after all, even though all sites might look the same. This means thatthe longitudinal trap frequency changes when moving away from the lattice axis. These
geometric effects make an accurate determination of the motional sideband spectrum
difficult, but do not affect the carrier line width to first order for the following reasons:
The transversal motion is not sampled to first order, i.e. the first order Doppler
effect is cancelled by using a copropagating probe beam.
The sample temperature is ∼ 5 K, resulting in an average longitudinal quantum
number hnz i ' 1. Therefore, a large percentage of the population is in the longitudinal ground state, which shows much less dissipative broadening than the excited
motional states.
Over the course of several months, we were consistently able to produce spectra of
the 1 S0 − 3 P0 transition which were referenced to the NIST F1 Cs-fountain clock using
the scheme described in the next chapter. The low instability of the reference allowed
relatively rapid averaging of the transition frequency. A summary of our results and an
outlook towards improvements and future experiments will be presented in chapter 7.
1

A quasi one-dimensional quantum gas of bosons, produced by transferring a BEC into a 3D opticallattice with high aspect ratio. These systems achieve ωr /ωz ' 10 such that ~ωr is the largest energy
scale in the system, making it highly unlikely to be radially excited by any longitudinal process(Paredes et al., 2004).

51

5 Precision Spectroscopy in a 1D Optical Lattice52

Chapter

6

Determining Absolute
Frequency
hat do we mean by measuring frequency? Recall the discussion in Sec. 1.1: every
frequency measurement is a comparison of one oscillator against another oscillator
which is defined as the frequency standard. The frequency standard used in determining
the optical frequencies in the 87 Sr spectroscopy experiment as discussed in the previous
chapter is a Cs-fountain clock at NIST Boulder. Since the standard is not situatedin the same lab, its signal needs to be transferred from NIST, which is several km
away. Frequency transfer via a frequency-modulated hydrogen maser transmitted overan optical fiber link has proven to be sufficiently stable (Ye et al., 2003).

W

Once the maser signal arrives at the lab, its modulation frequency at 950 MHz is detected. With the help of a femtosecond comb, this radio frequency is compared against
the spectroscopy laser at 698 nm. The beat between the maser signal and the downscaled optical frequency is counted on a frequency counter (∼ 10 kHz). This beat notecan be scaled up again to the optical regime, thereby putting a frequency scale on the
spectroscopic data.
This chapter will introduce the elements of the setup, but will concentrate on theelectronics and software used for counting the optical frequency. Additionally, a feedback
scheme for dynamical compensation of residual cavity frequency drift (∼ 7 Hz/s) will bepresented. The other elements such as the femtosecond comb, the spectroscopy laser
and the reference cavity are explained in much greater detail elsewhere (see references
throughout the chapter).
In Fig. 6.1, the setup for doing the frequency comparison is depicted schematically.
The next section will be devoted to going through the whole figure and explaining indetail the different parts involved in counting optical frequencies.

53

6 Determining Absolute Frequency

Figure 6.1: Schematic of the frequency metrology setup, figure courtesy of S. Foreman

6.1 Schematic
We shall begin in the upper left corner of Fig. 6.1: the spectroscopy laser system consistsof a laser diode in a Littman setup emitting light at 698 nm. Two feedback loops control
the wavelength by modifying diode current and backcoupling mirror position (by changing the voltage on a Piezo crystal). After roughly matching the laser’s wavelength to apre-stabilizing cavity by observing the transmitted light on a CCD camera, these feedback loops stabilize the diode wavelength using the Pound-Drever-Hall locking technique(see Hall et al., 2001, for more details on such a setup).
The pre-stabilized laser beam passes an AOM, a small part gets split off and is introduced to the reference cavity. Again, by monitoring the cavity transmission and steering
the pre-stabilization cavity length with a Piezo crystal, the first cavity and thus thediode laser wavelength is matched to the transmission frequency of the reference cavity.
Two feedback loops on the AOM and the pre-stabilization cavity length lock the laser
frequency to the reference at νc .
The reference cavity is made out of a Zerodur spacer and uses a novel vertical mounting
scheme that minimizes cavity length sensitivity to vertical accelerations such as gravity.

546.1 SchematicThis scheme is so effective that the cavity can be placed on a simple spring-compensation
platform and makes large-scale large-mass setups unneccessary (Notcutt et al., 2005).
Due to using a Zerodur spacer, the cavity has a residual mechanical frequency driftof 7 Hz/s that makes it very difficult to use the laser to do spectroscopy on the 10 Hz
level. To counter this frequency drift, a digital feedback loop measures the drift andcompensates for it by controlling the AOM placed before the 87 Sr sample on the righthand side of the figure.
First, the optical frequency has to be measured, though: the spectroscopy laser signal issplit and coupled into two optical fibers which transport the signal to two different optical
tables. Each of these fiber links introduces noise on the signal mostly due to acoustic
noise and thermal effects. This noise is compensated for by a fiber noise cancellation
protocol using an AOM to dynamically counter any noise detected on the reflection fromthe tip on the far end of the fiber. The AOM used to transmit to the fs-comb (atomic
sample) shifts the signal by −fne1 (+fne2 ). These frequencies are derived from an RF
oscillator running at frf = 155.46464 MHz and obey frf = fne1 + fne2 .
On the comb table1 , the signal at νs := νc − fne1 is heterodyned against a tooth of the
comb at νN := N frep + f0 , producing a beat note νs − νN ≈ 25 MHz which is detected ona photo detector. The beat note is electronically mixed with the signal fDDS ≈ 75 MHz
from the cavity drift compensating oscillator. The mixed signal is again mixed downwith fSRS = 100 MHz from another RF oscillator. The final signal then controls a Piezo
crystal in the femtosecond comb with a feedback loop, locking the comb’s repetition rate
frep , such that(νs + fDDS ) − (N frep + f0 ) = fSRS .
(6.1)
The comb’s other free parameter f0 is also locked with a feedback loop to 70 MHz. By
fixing both of the comb’s free parameters, it works as a translation gear between an
optical frequency νc and a microwave frequency frep .
The repetition rate of a femtosecond comb can easily be detected if its spectrum spans
a whole octave: by frequency-doubling a low-lying comb tooth and beating it against ahigh-lying one, one obtains a beat note at frep and its harmonics. In the present case,the tenth harmonic of the repetition rate is detected with a photo detector. This signal
is then beat against the maser frequency at 950 MHz, resulting in a signal at f c ≈ 10 kHz
which is then counted with an electronic frequency counter. It can be converted backinto an optical frequency by inserting Eq. 6.1:
fc = 10frep − fm =10[(νs + fDDS ) − f0 − fSRS ] − fm .
N

(6.2)

The frequency counter is controlled with a Labview program that takes a chunk offrequency data every 30 s and fits a linear function to it to determine a frequency drift
rate due to the drifting reference cavity. The program then programs a programmableoscillator (with output frequency fDDS , see right hand side of the figure) to emit acorresponding linear frequency ramp starting at roughly 75 MHz. This frequency ramp
1

A description of the octave-spanning femtosecond comb system can be found in Fortier et al. (2003).

55

6 Determining Absolute Frequency
is then mixed with a signal from a VCO which is phase-locked to another programmableoscillator (fAg ) with a frequency multiplication factor of 20.
The mixed signal (1.12 GHz) controls the previously mentioned drift compensating
AOM, such that the final frequency νSr introduced to the atomic sample isνSr = (νs + fDDS ) + 20fAg + fne1 + fne2 .

(6.3)

6.2 Features and Remarks
Since so many oscillators, counters and lasers are involved in comparing the NIST standard against the 87 Sr transition frequency, one has to worry about the accuracy of very
many parts. Thus it is very important that every oscillator and electronic counter isreferenced to the same highly stable electronic reference. We use a commercial Cs clock
situated in the lab, whose 10 MHz signal is split to all the electronic parts with a distributing amplifier.
The most critical electronic part is the frequency counter counting the beat between
the maser signal and the comb repetition rate. It is an HP53132a counter by Agilent 2 .
These counters take data at a their reference clock frequency of 10 MHz and thus obtaina whole array of frequency data during the gate time of 1 s. A sine function is theninternally fitted to that data allowing a measure of the counted frequency on the Hzlevel within 1 s. The counter’s accuracy has been tested by independently counting thesame signal with two of the counters and comparing their result.
The counter must be so accurate because the data it produces will be converted into
an optical frequency by inverting Eq. 6.2 and inserting into Eq. 6.3 to obtain


νSr =

N
(fc + fm ) + f0 + fSRS + 20fAg + frf ,
10

(6.4)

thus multiplying the counted frequency with the huge comb index N ≈ 4524000. If thecounter is off by 1 Hz, the resulting optical frequency will be off by half a MHz. The
same level of accuracy is also required of the maser frequency fm for the same reason.
In contrast to that, it does not matter too much if the other RF frequencies involved
in determining νSr are off by 1 Hz, since their inaccuracy directly translates into theoptical regime. Also note that frep and fDDS do not show up at all in Eq. 6.4, i.e. they
are in-loop variables and do not need to be accurately known at all.
Also note that the comb index N might change each time the whole system is re-locked.
Actually, N is determined by guessing νs + fDDS = ν 0 from its value on the previous
day of measurement and a typical cavity drift of 7-8 Hz/s. To find the spectroscopytransition again, the system is locked up and some preliminary data at fc is counted.
By using Eq. 6.2, a value N 0 is estimated as
N0 =2

See http://www.agilent.com.

56

10(ν 0 − f0 − fSRS )fc + f m

6.3 Frequency Multiplier
which never differed by more than 10−3 from an integer. So N is calculated by rounding
N 0 and a corresponding value of fAg is calculated from prior knowledge of νSr by using
Eq. 6.4. Then the programmable oscillator producing fAg is tuned until a spectroscopy
signal is found. This is convenient because fAg is also used to step the spectroscopy laser
over the transition to obtain a spectrum.
The remainder of this chapter will focus on the implementation of the drift-compensationelectronics and software, i.e. the programmable oscillator producing f DDS , the programmable frequency multiplier scaling fAg and the software controlling them.

6.3 Frequency MultiplierAs another technical application of the phase-locked loop theory developed in Sec. 1.2,
a USB programmable frequency multiplier with a target range of 500 MHz to 1 GHz
is presented. The small box can be plugged into any personal computer running theWindows operating system and provides convenient access to the GHz regime.
The box uses the ADF4111 PLL chip by Analog Devices3 . Although it was originallydesigned for cell phone applications and thus is a tiny surface-mount device, it has several
interesting features that make its use worthwhile:
digital phase-frequency detector as discussed in Sec. 1.2.3
error signal provided by a charge-pump with programmable amplitude
two dividers with large programmable division ratios on the reference and VCO
inputsaccepts 3.3 V or 5 V power supply voltage
A schematic of the box can be found in Fig. 6.2. The ADF4111 chip is programmed
via a USB digital I/O card from DeVaSys4 . The error signal provided by the ADF4111
chip is filtered and amplified with a custom loop filter to match the requirements of the
Minicircuits MCL-POS1000W VCO. The VCO output is split into two parts: one is theoutput signal and one is fed back into the ADF4111 to be divided down and comparedagainst a fraction of the reference clock. The division ratios as well as several locking
schemes can be programmed via the USB board.

6.3.1 Circuit
The experiment required a programmable frequency source around 0.9-1.0 GHz, so theparts were chosen accordingly. The ADF4111 is specified for output frequencies between
80 MHz and 1.2 GHz, the Minicircuits MCL-POS1000W is a wide band linearly tunable
VCO working between 0.5 and 1.0 GHz. The splitter is a Minicircuits MCL ZFSC-2-4
for frequencies between 0.2 and 1000 MHz.
The circuit diagram for the ADF4111 board is shown in Fig. 6.3. The RF inputs J 1(reference) and J2 (VCO) are AC-coupled and can be set to 50 Ω input impedance byadding 51 Ω SMD resistors at R1 and R2 . There is an additional RF input RFINB (pin 5)
3
4

Analog Devices ADF4111, see http://www.analog.com
DeVaSys USB I2C/IO card, see http://www.devasys.com/usbi2cio.htm

57

6 Determining Absolute Frequency

Figure 6.2: Frequency multiplier schematic

Figure 6.3: ADF4111 circuit

58

6.3 Frequency Multiplier

Figure 6.4: Loop filter circuit
which gets subtracted from the VCO input RFINA (pin 6); it is unused and AC-coupled
to the ground plane. The supply voltage for the PLL chip on J6 was chosen to be 3.3 V
as the USB I/O card programming the chip uses 3.3 V CMOS logic. All power supply
pins use 1 nF decoupling capacitors. The chip is programmed via the four lines of J 4
and offers a programmable monitor at J5 . As an indicator that the signal is locked, J5
is connected to an LED. The error signal is provided at J3 and its output range is set bychoosing R3 — 4.7 kΩ are used corresponding to a maximum current from 0.63 mA to
5.00 mA as set by the software. No loop filter is included on this board, but as the VCO
accepts positive voltages up to 16 V it becomes necessary to amplify the error signal.
The schematic for the loop filter is given in Fig. 6.4. The ADF4111 error signal output
at J3 is a charge-pump, i.e. a current source. This current has to be converted into acontrol voltage using the parallel combination of capacitances C1 and C2 . By adding
an additional resistor R1 , the resulting voltage is already low-pass filtered. This can beseen easily, as the impedance of the conversion stage is5
Z12 =

1
1
sτ1 + 1
k (R1 +)=
sC2
sC1
s(C1 + C2 )(sτ12 + 1)

with time constants τ1 := R1 C1 and τ12 := R1 (C1 k C2 ). The voltage after the input
stage is v = Z12 icp with icp being the current from the charge-pump to ground throughthe input stage. The input signal is the charge-pump current which is proportional tothe phase error detected by the PLL chip. The output signal is the voltage v after thestage, so the impedance Z12 is the transfer function determining the frequency responseof this sub-circuit.
The second part of the loop-filter is low-pass filtering v with a time-constant τ 3 :=R3 C3 resulting in a control voltage which is amplified with a non-inverting amplifier
configuration of gain g = 1 + R4 /R5 . The total transfer function is read off from
vctrl = g
51sτ1 + 1v=g
icp .
1 + sτ3
s(C1 + C2 )(sτ12 + 1)(sτ3 + 1)

The operation k is defined as x k y := xy/(x + y).

59

6 Determining Absolute Frequency

Figure 6.5: Final Loop filter and VCO driver circuit
The effect of the loop filter is: an input current is converted into a voltage which is
low-pass filtered two times and then amplified with a constant gain factor.
An amplification factor of ∼ 5 proved sufficient to cover the desired frequency range,
so the circuit uses R4 = 10 kΩ and R5 = 3.3 kΩ k 8.7 kΩ = 2.4 kΩ. The combination
producing the least amount of ringing and fastest locking time was found to be C 2 =100 pF, C1 = 15 nF, R1 = 1 kΩ, resulting in a corner frequency f12 = (2πτ12 )−1 =1.6 MHz. The low-pass filter (R3 , C3 ) was finally omitted, since the loop characteristics
did not seem to improve.
An existing VCO driver board for the MCL-POS series was modified to accommodate
the loop filter and the power supply connections. The final circuit is depicted in Fig. 6.5.
The ±15 V power supply is decoupled from VCO and operation amplifier with slowstarter circuits. When switching on the supply voltage, the large 68 F capacitors slowly
open the transistors on a time-scale τ = RC = 147 ms to minimize VCO noise fromsuddenly turning it on. A 3 dB attenuator was included to match the VCO output
power to the reference power.


6.3.2 Output
Typical spectral traces of the box’ output are shown in Fig. 6.6. These scans weretaken while using the Cs-referenced DDS oscillator described in Sec. 6.4 at 50 MHz asa reference to produce a signal at 1 GHz (multiplication factor 20). Each scan uses asmaller scan to zoom in on the center frequency, until the minimal bandwidth of 100 Hz
is reached in Fig. 6.6(c) which was taken with the max-hold setting over more than 60 s.
Note that the spectrum analyzer was not calibrated, the signal is at 1 GHz with the60

6.3 Frequency Multiplieraccuracy of the 10 MHz signal produced by the Cs standard. The broad undergroundespecially apparent in Fig. 6.6(c) is due to using the relatively inexpensive Minicircuits
VCO. For the current experiment, the quality of the output sufficed, but higher quality
wide-band VCOs6 will be used in the next version of the circuit, promising to improve
the spectrum drastically.

6.3.3 Software
Programming the ADF4111 requires a communication protocol called SPI which is afour-wire serial protocol. The library implements this protocol on the I/O pins of theUSB board. The protocol timing is shown in Fig. 6.7: A zero on the CE (chip enable)
pin shuts down the chip’s output and resets all registers. Then, the program produces aclocking signal on the SCLK (serial clock) pin, causing the ADF4111 chip to read a bitfrom DATA on each rising edge. Finally, the transfer is completed by strobing LE (latch
enable) telling the receiver to latch the accumulated data from its input buffer into thecore. A rising edge on CE enables the chip again and completes the SPI transfer. 7
A program is provided that uses the above protocol to expose all ADF4111 registers to
the script language Lua8 to simplify modification of parameters. A script that completely
configures the device is shown in Lst. 6.1 (everything after “--” up to the end of the line
is considered a comment).
-- pll . lua-- USB
serial
sclk
data
le
ce

I2C I / O serial number and bitmasks for pins
= "05070303"= 0 x00000010= 0 x00000020= 0 x00000040= 0 x00000008-- Dataf_latchr_latchn_latchto write= {0 x12 ,0 x00 ,0 x02 }= {0 x00 ,0 x00 ,0 x00 }= {0 x00 ,0 x00 ,0 x01 }-- Do something to the data-- ...
-- Power down the device , write data , power up
lib = ADF4111 . new ( sclk , data , le , ce , serial )lib : ce (0)lib : write ( f_latch )lib : write ( r_latch )lib : write ( n_latch )lib : ce (1)6E.g. the VCO-107TC available from http://www.sirenza.com
This is not a standard SPI transfer: usually, the CE pin is replaced with a CS (chip select) pin using
negative logic. The device will not be turned off by a falling edge on CS but will not accept data
otherwise. This allows omitting multiple versions of the SCLK, DATA and LE wires.
8
Lua is a fast embeddable script language with low memory footprint, see http://www.lua.org.
761

6 Determining Absolute Frequency

Power (dBm)

0
−20
−40
−60
−80
998

999

1000
1001
Frequency (MHz)1002(a) bandwidth 1 kHz

Power (dBm)

−10

−30

−50

−70
999.9

999.951000
1000.05
Frequency (MHz)(b) bandwidth 100 Hz

Power (dBm)

−10

−30

−50

−70
999.994

999.996999.998
1000
Frequency (MHz)1000.002(c) max-hold >60 s, bandwidth 100 Hz

Figure 6.6: Spectral traces from the output of the frequency multiplier box producing
a signal at 1 GHz. The traces zoom in on the peak; note that the spectrum analyzer
was not calibrated and that the signal is at the specified 1 GHz with the accuracy of
the reference used.

62

6.4 Direct Digital Synthesis

Figure 6.7: SPI data transfer protocol; used to transfer data from the controllingcomputer to the ADF4111 PLL chip-- pll . lua ends here

Listing 6.1: Example Lua script for ADF4111.
A script like the above is executed by calling the embedded LUA interpreter with thescript’s path as argument: interp.exe pll.lua.

6.4 Direct Digital Synthesis
Direct digital synthesis (DDS) is a method to produce phase-continuous wave forms
while tuning the synthesized frequency. In analog synthesizers, this is not possible as
they use different phase-locked loops with specific frequency bands internally, causing thephase and even amplitude to jump when switching bands. When using the synthesized
frequency as e.g. a moving reference for a phase-locked loop, this behavior will cause
the lock to break. These analog devices are also very large and expensive. DDS offers
a compelling alternative, as the synthesizers become much smaller, more effective andcheaper. They are also easier to program, since the whole board has to use CMOS logic
anyway.
A DDS synthesizer has an internal array of memory that is filled with phase information. The chip then iterates over this array and produces a sine wave with corresponding
phase at the output. Modifying the output frequency is as easy as modifying the iteration speed. The iteration speed is a fraction of the DDS chip’s reference clock, making
the synthesized frequency as accurate and stable as the reference clock is.
Commercial DDS systems that can be programmed via the GPIB bus are available
from Agilent9 . They are still quite expensive, and not very flexible in their programmability. We were looking for a DDS system able to fulfill the following list of requirements:
Target frequency range: 50-100 MHz
Must be referenced to 10 MHz Cs standard
Able to produce linear frequency ramps with small slopes (10 Hz/s ) over long timescales (a whole day)9

E.g. the 80 MHz Agilent 33250A, see http://www.home.agilent.com

63

6 Determining Absolute Frequency

Figure 6.8: Piecewise linear frequency ramp
Controllable with a Labview script to integrate well into the existing data acquisition system
Support USB, since GPIB is too complicated and slow and RS232 or the parallel
port are not flexible enoughMore than one DDS system can be connected to the same computer
The whole system should cost much less than a commercial one, which retail forabout USD 4500 at the moment
The most promising candidate was the Analog Devices AD9854 DDS chip 10 , supporting reference frequencies up to 300 MHz. Evaluation boards with convenient SMC
connectors are available for USD 300, so designing a high frequency printed circuitboard was not necessary. The same USB digital I/O card (costs USD 80) as for the PLL
frequency multiplier from Sec. 6.3 was used; the implementation of the chip’s specificrequirements was done by modifying the existing library. Labview bindings were writtenby wrapping calls to library functions in separate sub-VIs.

6.4.1 AD9854
The AD9854 has an on-chip phase-locked loop which can multiply the reference clock
frequency by a programmable integer from 4 to 20. Using the 10 MHz Cs reference,
this provides us with a virtual reference clock of 200 MHz. The DDS output frequency
is limited by the Nyquist criterion, i.e. fout ≤ fref /2. We operate the DDS system atroughly 75 MHz, so this multiplication factor proved sufficient.
The DDS features different frequency modulation and ramping schemes; in our case
we are using it in a piecewise linear fashion by setting the starting frequency once to
75 MHz and then updating the ramp slope, which is about 10 Hz/s every 30 s.
This will produce a signal as in Fig. 6.8 where the frequency increment per fref cycle
and the update interval in fref cycles are set in software.
Typical spectra of the output are presented in Fig. 6.9 for various output frequencies
with and without on-chip reference clock multiplier. The output frequency was tuned toshow why there is a huge sideband to the right of the intended output frequency (figures(a)) and to determine whether the background depends on the DDS board’s settings
10

See http://www.analog.com

64

6.4 Direct Digital Synthesis(figures (b)). For all scans presented, the board was referenced to a 10 MHz Cs clock
and the spectrum analyzer’s bandwidth is noted in the figure caption. The (b) spectra
were taken with the spectrum analyzer’s max-hold setting turned on for more than aminute. Note that the (b) figures are still bandwidth limited, even at the lowest settingof 100 Hz and that the background only deteriorates a bit when turning on the reference
clock multiplier.
In Fig. 6.9(a), the reference clock was multiplied up by a factor of 20 using the onchip
PLL multiplier to produce a virtual reference of 200 MHz. In the subsequent (a) scans,this multiplier was turned off and the output frequency was tuned from 4 MHz to 1 MHz,
covering a large range of the output between DC and the Nyquist frequency f Ny =fref /2 = 5 MHz. The large sideband to the right of the main peak is the beat between
the output frequency and the reference clock.

6.4.2 Caveats
There are several caveats that should be mentioned. Improvements and possible solutions
are presented, but most have not been implemented as the system works sufficiently well.
The chip produces sidebands on the output signal as a beat between the various
system frequencies: 10 MHz reference, 200 MHz virtual reference and 75 MHz output(see Figs. 6.9(a), 6.9(c), 6.9(e) and 6.9(g)). These sidebands have to be filtered out so
the output signal is amplified and band-pass filtered before being used in the experiment.
The driver for the USB board is supplied by the manufacturer11 and writes a single
bit mask to the digital I/O pins per command. Since the DDS board uses a serial SPI
protocol as well, transferring one bit takes 2 USB commands plus 4 control commands
per write. To configure the device completely from scratch, at least 20 bytes have to betransferred. The time it takes to perform one write operation is limited by the operating
system’s time resolution: ∼4 ms for a typical 32 bit Windows system. That means thatwriting to the DDS system is very costly and several optimizations were included to limit
the number of bytes written on each update to the bare minimum. Still, it takes almost
a second to update the slope data. During this time the DDS continues ramping using
its old settings as the data is buffered and then latched into the chip. The problem isthat updating more often than every few seconds will not work with the current setup.
In our experiment this does not matter currently, since the update cycle is limited by
how much data can be obtained during one cycle. At the moment, a 30 s update cycle
is used.
This problem could be eliminated by writing a custom USB driver for the chip usedon the USB I/O board — a Cypress EZUSB microcontroller12 . Another more powerful solution would be to include some logic chips in the DDS system, like a sufficiently
advanced microprocessor running a real-time operating system. This would allow arbitrary programs to be written which could be executed with a much faster update rate.
11
12

Driver available for download at http://www.devasys.com.
See http://www.cypress.com

65

6 Determining Absolute Frequency−10Power (dBm)

Power (dBm)

−20
−30
−50

−40

−60

−70
0

25

50

75
100 125 150
Frequency (MHz)

175

200

74.99674.9987575.002Frequency (MHz)75.004(b) 75 MHz, multiplier ×20, max-hold>60 s, bandwidth 100 Hz(a) 75 MHz, multiplier ×20, bandwidth 10 kHz
0Power (dBm)

Power (dBm)

−20
−20
−40
−60

−40

−60

−80
−100
0

2

4
6
Frequency (MHz)

8

3.996

10

3.998
4
4.002
Frequency (MHz)4.004(d) 4 MHz, multiplier off, max-hold>60 s, bandwidth 100 Hz(c) 4 MHz, multiplier off, bandwidth
10 kHz−10

Power (dBm)

Power (dBm)

−20
−30
−50
−70

−40

−60

−90
0

2

4
6
Frequency (MHz)

8

10

1.996

1.998
2
2.002
Frequency (MHz)2.004(f) 2 MHz, multiplier off, max-hold>60 s, bandwidth 100 Hz(e) 2 MHz, multiplier off, bandwidth
10 kHz−10

Power (dBm)

Power (dBm)

−20
−30
−50
−70

−40

−60

−90
0

2

4
6
Frequency (MHz)8(g) 1 MHz, multiplier off, bandwidth
10 kHz

10

0.996

0.998
1
1.002
Frequency (MHz)1.004(h) 1 MHz, multiplier off, max-hold>60 s, bandwidth 100 Hz

Figure 6.9: AD9854 DDS spectral traces for different output frequencies

66

6.4 Direct Digital Synthesis
An interesting candidate would be an ARM microprocessor13 running Clinux14 . This
would also require that relevant data could be read directly by the microprocessor; thus
adding some analog or digital inputs to the DDS system would be necessary.
Another bonus of this more advanced and work-intensive approach would be that thesystem is fast and stable enough so that arbitrary non-linear frequency ramps and pulse
sequences could be synthesized — with a precision limited only by the reference clock.


6.4.3 Software: Drift Compensation
To compensate the residual mechanical drift rate, the DDS system has to be programmed
to emit a frequency ramp. The system is based on Labview and uses a HP53132a counterto count the beat between NIST’s hydrogen maser and the femtosecond comb which islocked to the spectroscopy laser. By measuring this beat frequency over a time interval
τ , the program obtains an array of frequency data, to which a linear function is fitted.
The slope in Hz/s corresponds directly to the mechanical drift rate of the cavity. The box
is then programmed to emit a frequency slope to compensate for the drift. The program
retains knowledge of all prior slope settings and can use them along with the newest
fitted slope to form a new slope value. By calculating a weighted mean of the latest nslopes, the loop filter simulates the memory time of an analog circuit. Safeguards areincluded to protect against outliers and sudden jumps in laser frequency due to breaking
locks. Screenshots of the controls are shown in Fig. 6.10: All parameters determining
the behavior of the digital loop filter can be adjusted with the first panel (Fig. 6.10(b)),
real-time display and analysis is shown below (Fig. ??).
One of the first drift compensation measurements is depicted in Fig. 6.11(a). Initially,
the optical frequency is drifting at a large rate; after 2 min, the device manages to keep
the optical frequency within several hundred Hz for half an hour. The Allan deviationof this frequency measurement is plotted below, averaging down to a relative instabilityof 10−13 at 20 s.
The maximum time of a continuous drift compensation run is limited by the laser
systems becoming unlocked. The Allan deviation is mostly determined by the instabilityof the hydrogen maser. The digital loop filter was improved by adding an integration
step in analogy to the memory time of an analog filter. This allowed averaging down
the instability faster as can be seen in the considerably longer compensation run inFig. 6.11(b), reaching 7 × 10−14 at 70 s.
A typical spectrum of the 1 S0 − 3 P0 transition takes about 100 s, so the optical frequency maximally drifts by ∼ 30 Hz, which proved to be sufficient for the current experiment. The sign of the cavity’s drift stayed the same for several months and theaverage rate was always between 4 and 9 Hz. The spectral scans were taken alternating
the direction of the scan each time; viz. alternatively with increasing and decreasing
frequency. Since the linewidth did not show any dependency on the direction of the
13
14

E.g. Atmel AT91 ARM series, see http://www.atmel.com
A real-time Linux operating system with ports for a variety of microprocessors and -controllers. See
http://www.uclinux.org

67

6 Determining Absolute Frequency(a) Digital loop filter setup(b) Compensation run

Figure 6.10: Drift compensation screenshots686.4 Direct Digital Synthesis
scan, we are confident that the drift compensation does not have any effect on our dataat the current resolution.
In future work, the drift compensation will work as an aside to help maintain thecorrect frequency, but the spectra will be referenced to the atomic transition itself.

69νopt − hνopt i (Hz)6 Determining Absolute Frequency

500
250
0
−250
−500
−750
−1000
0

5

10

15

20
t (min)

25

30

35

σy (τ )

10−12

10−13
100

101

102

103

τ (s)

(a) Early drift compensation run

νopt − hνopt i (Hz)

800
600
400
200
0
−200
−400
−600
0

10 20 30 40 50 60 70 80 90 100 110 120 130 140
t (min)σy (τ )

10−12

10−13
100

101

102

103

τ (s)

(b) Improved digital loop filter

Figure 6.11: Drift compensation data and Allan deviation
70

Chapter

7

Results and Outlook
The first spectroscopic measurement of the 1 S0 − 3 P0 clock transition in 87 Sr was reported by the Tokyo group (Takamoto et al., 2005). We presented an second independentmeasurement, experimentally determining important systematics (instead of giving theoretical estimates) and achieving a better statistical accuracy. We report a 1 S0 − 3 P0transition frequency of (429 228 004 229 867 ± 20 sys ± 2.8 stat) Hz, disagreeing with theTokyo group’s result by 85 Hz. The results of the Tokyo group and our measurement arecompared in Fig. 7.1, where the blue (red) bars indicate systematic (statistical) errors.
Their GPS-referenced measurement (instability ∼ 10−10 @ 1 s) would have required averaging times of more than 106 s to achieve the reported instability of 3 × 10−14 (thiswas not done). In contrast, we used a hydrogen-maser referenced itself to NIST’s F1
Cs-fountain clock: A reference that has an instability of 2 × 10−13 @ 1 s and typicallyshows instabilities of 7 × 10−14 at our scan time of 100 s (see Fig. 6.11(b)). A budget ofthe error sources discussed in sections 4.6 and 5.3 is summarized in Tbl. 7.1.
The drift compensation scheme presented in this work allowed compensating for an
unstable reference cavity while maintaining various laser locks. This made it possible
to consistently measure spectra below the 200 Hz level, almost achieving the quantumContributor
AC Stark shift (lattice)
AC Stark shift (probe)
AC Stark shift (blackbody)
Zeeman shift
Recoil / Doppler shift
Density shift
Gravitational shift
Cs-maser calibration
Systematic total

Correction (Hz)
-17
7.0
2.4
0
0
-2.0
-0.64
46.8
36

Uncertainty (Hz)
8.3
0.9
0.02
12
1.9
13
0.13
0.5
20

Section in this work
5.3 on page 45
4.6.2 on page 34
4.6.3 on page 36
5.3 on page 45
5.2 on page 43
5.3 on page 45
4.6.5 on page 39(Data from NIST)

Table 7.1: Error budget for 1 S0 − 3 P0 spectroscopy, modified from Ludlow et al. (2005)71ν - 4292280

νsr - 42922800

-20

0.6

-100

Lattice Intensity [I/I0]

400

Magnetic Shift-32(7) Hz / G

300
200
100
0

-100

-200

(b)

10
and
0.8 7 Results
1.0
1.2Outlook 4x10(c)
-4 -3 -2 -1 0

1

2

3

axial magnetic field [G]4νsr - 429228004229900 [Hz]

νsr - 429228004229900 [Hz]

(a)
-40

-50

sr

-30

0

10

11

10

12

3

4x10

12

Atom ic D ensity [atom s/cm ]80
Ref. [9]
60
40
20
0
-20
-40
This work(d)-60
5/3 5/17 5/31 6/14 6/28 7/12 7/26
Date (2005)

Figure 7.1: Comparison of results from Takamoto et al. (2005) (upper part) and Ludlow
et al. (2005) (lower part), figure from the latter paper
mechanical uncertainty limit of 50 Hz for the current 1D optical lattice setup. We also
demonstrated an unprecedented level of control on large timescales of an all opticaltrap for neutral atoms, allowing measurements previously only doable for ions such as
vibrational spectroscopy. In contrast to the ion trap experiments, the anharmonicity ofthe optical potential and its matched shape for both spectroscopic levels allow stateresolved studies of vibrational dissipation. A semi-classical model describing the effectof external noise sources has been given in this work.
In future experiments, the spectroscopic scans will be referenced directly to the 1 S0 −3 P transition. The 1D lattice will be replaced by a three-dimensional version, promising0
to eliminate several systematics such as the aforementioned uncertainty limit as well as
the tensorial polarizability of the electronic states.
A scheme to use 88 Sr for an optical lattice clock by simulating the ultranarrow 1 S0 −3 P0transition in an Electromagnetically-Induced Transparency (EIT) scheme has been proposed by Santra et al. (2005). Using the bosonic 88 Sr would remove all the hyperfine
structure effects and the EIT scheme would provide a controllable transition width.
Alternatively, measuring the 87 Sr hyperfine constants with the current precision could
improve the understanding of hyperfine mixing by providing the theory with more parameters.
The current setup also allows photo-associative spectroscopy (PAS) experiments which
will lead to better understanding of the 87 Sr two-body interactions, such as getting ameasure of the scattering length. PAS has never been done on such a narrow line as
the 87 Sr 1 S0 − 3 P0 transition, although several groups have demonstrated PAS on the
singlet transition 1 S0 − 1 P1 in 88 Sr (Nagel et al., 2005, Yasuda et al., 2005). Work on
this narrow-line PAS is in progress.
All these experiments and improvements will help further understanding atomic strontium and alkaline earths in general. The all optical atomic clock with an instability of
10−18 is an important application, but by far not the only one. The ultrahigh precision72

and low instability required for these experiments also opens up completely new fields,such as the possibility to test the evolution of fundamental constants such as the finestructure constant α.

73

7 Results and Outlook

74

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79

Index6j-symbol, 22
AC-Stark shift, 15
Acousto-Optical Modulators, 41
Allan deviation, viii, 3Allan variance, 1, 3
AOMs, see Acousto-Optical Modulators
blackbody shift, 36
Bode plot, 10
Bohr magneton, 37
Bohr radius, 36
characteristic exponent, 26
Clebsch-Gordan coefficients, 37
convolution, 9
DDS, see Direct digital synthesis
dead time, 2
detuning, 16
Direct digital synthesis, 63
DMOT, see Dynamic MOT
Doppler effect, 16
second order, 34
Doppler temperature, 17
Dynamic MOT, 19

gate time, 2
global positioning system, vii
GPS, see global positioning system
Green’s function, 6
Hartree energy, 36
Lamb-Dicke parameter, 28
Lamb-Dicke regime, 28
LO, see local oscillator
local oscillator, 5magic wavelength, 24
Magneto-Optical Trap, 17
Mandel Q-parameter, 17
Maxwell-Boltzmann, 33
probability density function, 33
MOT, see Magneto-Optical Trap
nominal frequency, 1
one-sided Laplace transform, 6
optical atomic clock, viii
optical lattice, 21

PAS, see photo-associative spectroscopy
PD, see phase detector
ECDLs, see External Cavity stabilized Diode Lasers PFD, see phase-frequency detector
phase detector, 9effective magnetic moment, 18
phase-frequency detector, 10
Einstein A coefficient, 16
EIT, see Electromagnetically-Induced Transparency phase-locked loop, 4
analog mixer, 7Electromagnetically-Induced Transparency, 72
bandwidth, 10External Cavity stabilized Diode Lasers, 43
capture range, 10
feedback loop, 5loop filter, 7
forward gain, 6
photo-associative spectroscopy, 72
loop gain, 6
photo-multiplier tube, 41
Floquet solutions, 26
PLL, see phase-locked loop
Fokker-Planck equation, 15
PMT, see photo-multiplier tube

80

INDEX
reduced matrix elements, 22
saturation intensity, 16
saturation parameter, 16SI, see système international
système international, vii
time deviation, 2Tonks-Girardeau gas, 50
transfer function, 5
VCO, see voltage-controlled oscillator
voltage-controlled oscillator, 4Wannier basis, 26
Wigner-Eckart theorem, 22
Zeeman effectdiamagnetic term, 37
second order, 37
Zeeman splitting, 17

81

INDEX82

Acknowledgements

Many people have directly and indirectly contributed to this text: I thank Rudi Grimm
for accepting me as a diploma student and Jun Ye for making my work at JILA possible,
as well as supporting and encouraging me along the way. I learned a great deal about
what can be achieved with the right idea.
My special thanks go to all the members of the Sr Gang (in alphabetical order: Marty
Boyd, Seth Foreman, Tetsuya Ido, Andrew Ludlow and Tanya Zelevinsky) for letting
me play with the most sensitive parts of their increasingly complex setup. You provided
a great work atmosphere and were always ready to help.
I also want to mention the amazing members of the JILA electronics shop, especially
Terry Brown and James Fung-a-Fat who gave great advice and laid out all the printed
circuit boards I used.
Without my parents’ help and constant support all of this would never have been possible.

83