Download Precision Spectroscopy of Atomic Lithium

OBERLIN COLLEGE
Precision Spectroscopy of Atomic
Lithium
by
Donal Sheets
in the lab of
Jason Stalnaker
Department of Physics
April 2015
Abstract
The atomic structure of lithium has aroused a significant amount theoretical and
experimental interest as a system in which precision atomic calculations and spectroscopic measurements can be combined to yield scientifically significant results.
While there have been many experimental investigations of Li spectroscopy, particularly of the isotope shift between 6 Li and 7 Li as well as the hyperfine structure
of the 2 S1/2 →2 P1/2,3/2 (D1, D2) transitions, they suffer from significant disagreements and systematic effects. By utilizing the optical-to-microwave frequency
conversion made possible by a stabilized optical frequency comb, we will be able
to measure the optical frequencies of the D1 and D2 transitions and resolve the
discrepancies.
This experiment investigates the D1 and D2 transitions by probing an atomic
beam with a single mode extended cavity diode laser at approximately 671 nm.
The difficulty in this experiment comes from precisely measuring the frequency of
this probe laser. This is accomplished by referencing our diode laser to an optical
frequency comb which acts as an optical to microwave synthesizer allowing the
precise measure of optical frequencies referenced to an SI time standard. The
optical frequency comb is generated by a Ti:Sapphire pulsed laser which which is
broadened to span an optical octave using microstructure fiber. The frequency
comb is referenced to a GPS steered Rubidium atomic clock. This system we can
measure optical frequencies with a accuracy of 1 × 10−12 .
This work reports on the progress made on the experiment. In particular, we discuss improvements to the system and the evaluation of the systematics associated
with the measurements. Systematics associated with the alignment of the beam,
Zeeman shift, and ac Stark shift have been evaluated and we have set upper limits
on their effects on the measurement. Measurements of absolute frequencies of the
D1 transition as well as isotope shift and hyperfine splitting measurements are
also presented.
Acknowledgements
First and foremost I would like to thank my advisor Jason Stalnaker. His support,
encouragement, and dedication have made this work possible. As a never ending
source of knowledge I am grateful for the opportunity to have worked with and
learned from him.
I would like to thank my lab mate Peter Elgee for his company, insight, and
support. I would also like to acknowledge the previous students that have worked
no this and related projects. This work could not have been possible with out the
contributions from but not limited to Mike Rowan, Jacob Baron, Sophia Chen,
José Almaguer, and Sean Bernfeld. The information contained in their thesis was
also a great resource throughout the project.
I would also like to thank my friends and fellow physics majors for their company
and support throughout my time at Oberlin. Lastly I would like to thank my
family for their never ending support and understanding.
ii
Contents
Abstract
i
Acknowledgements
ii
List of Figures
vi
List of Tables
viii
1 Introduction
1.1 Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Previous Measurments . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Measuring Optical Frequencies . . . . . . . . . . . . . . . . . . . . .
2 Background Theory
2.1 Notation . . . . . . . . . . . . . . . . . . . . .
2.2 Atomic Theory . . . . . . . . . . . . . . . . .
2.2.1 Hydrogen Atom . . . . . . . . . . . . .
2.2.1.1 Quantum Defect . . . . . . .
2.2.1.2 Central-Field Approximation
2.2.2 Isotope Shifts . . . . . . . . . . . . . .
2.2.3 Fine Structure . . . . . . . . . . . . . .
2.2.4 Hyperfine Structure . . . . . . . . . . .
2.2.5 Quadrupole Interaction . . . . . . . . .
2.3 Multi-Electron Atoms . . . . . . . . . . . . . .
2.4 Precision Spectroscopy: Fields and Atoms . .
2.4.1 Quantization of Electromagnetic Fields
2.4.2 Interaction of Light with Atoms . . . .
2.4.3 Emmission . . . . . . . . . . . . . . . .
2.4.4 Absorption . . . . . . . . . . . . . . .
2.4.5 Saturation . . . . . . . . . . . . . . . .
2.4.6 Zeeman Effect . . . . . . . . . . . . . .
2.5 Wigner-Eckart Theorem . . . . . . . . . . . .
2.6 Relative Transition Strengths . . . . . . . . .
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1
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Contents
2.7
iv
2.6.1 AC Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . 33
Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Frequency Comb
3.1 Overview . . . . . . . . . . . . .
3.2 Dispersion . . . . . . . . . . . .
3.3 Kerr Lens Modulation . . . . .
3.4 Oscillator Cavity . . . . . . . .
3.4.1 Pump Laser . . . . . . .
3.4.2 Oscillator Cavity . . . .
3.4.3 Lasing and Modelocking
3.4.4 Microstructure Fiber . .
3.5 Stabilization . . . . . . . . . . .
3.5.1 Repetition Rate . . . . .
3.5.2 Offset Frequency . . . .
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4 Experimental Setup
4.1 Diode laser . . . . . . . . . . . . . . . .
4.1.1 Diode Laser Theory . . . . . . . .
4.1.2 Diode Laser Experimental Setup
4.1.3 Frequency Stabilization . . . . . .
4.1.4 Fabry-Perot Interferometer . . . .
4.1.5 Optical Spectrum Analyser . . . .
4.1.6 Power Stabilization . . . . . . . .
4.1.7 Electro-Optic Modulators . . . .
4.2 Atomic Beam . . . . . . . . . . . . . . .
4.3 Vacuum System . . . . . . . . . . . . . .
4.3.1 Interaction Region . . . . . . . .
4.3.2 Ion Pump . . . . . . . . . . . . .
4.3.3 Helmholtz Coils . . . . . . . . . .
4.4 Data Aquisition . . . . . . . . . . . . . .
4.5 Collection Procedure . . . . . . . . . . .
5 Analysis and Results
5.1 Analysis . . . . . . . . . . . . . . . .
5.2 Test of Frequency Standard . . . . .
5.3 Test of locks . . . . . . . . . . . . . .
5.4 Magnetic Field Evaluation . . . . . .
5.5 Evaluation of Probe Beam Alignment
5.5.1 Correction of Misalignment .
5.6 AC Stark Measurement . . . . . . . .
5.7 Isotope Shift and Hyperfine Structure
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Splitting
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40
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6 Conclusions
99
6.1 Improvements to Apparatus . . . . . . . . . . . . . . . . . . . . . . 99
Contents
6.2
6.3
v
AC Stark Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 100
Finalize Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Bibliography
102
List of Figures
1.1
1.2
1.3
Energy Levels of Lithium . . . . . . . . . . . . . . . . . . . . . . . .
Previous Measurements of Lithium fine structure splitting and isotope shifts for the D1 and D2 transitions. . . . . . . . . . . . . . . .
Output Spectrum of Optical Frequency Comb . . . . . . . . . . . .
2.1
2.2
2.3
Fine Structure of 6 Li and 7 Li . . . . . . . . . . . . . . . . . . . . . . 12
Saturation Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Doppler Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Pulse Generation . . . . . . .
Kerr Lensing . . . . . . . . .
Oscillator Cavity . . . . . . .
Chirped Mirror . . . . . . . .
Crystal Holder . . . . . . . .
Microstructure Fiber . . . . .
Microstructure Fiber Pictures
Diagram of frep stabilization.
Diagram of f0 stabilization. .
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54
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
Experimental Setup Block Diagram . . . . . . . . . . . . . .
Extended Cavity Diode laser configurations . . . . . . . . .
Picture of ECDL Cavity . . . . . . . . . . . . . . . . . . . .
Power vs Current Plot for Diode Laser . . . . . . . . . . . .
Block Diagram for ECDL Optics . . . . . . . . . . . . . . .
Block Diagram of f0 stabilization. . . . . . . . . . . . . . .
Confocal and Planar Fabry-Perot Cavity . . . . . . . . . . .
Fabry-Perot peaks for scan of ECDL . . . . . . . . . . . . .
Response of Electro-optic modulator to applied electric field.
Diagram of Power Stabilization Optics . . . . . . . . . . . .
Diagram of Electrooptic Modulator . . . . . . . . . . . . . .
Lithium Vapor Density vs Temperature . . . . . . . . . . . .
Lithium Oven . . . . . . . . . . . . . . . . . . . . . . . . . .
Atomic Beam . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of Atomic Beams . . . . . . . . . . . . . . . . .
Interaction Region . . . . . . . . . . . . . . . . . . . . . . .
Vacuum Chamber Picture . . . . . . . . . . . . . . . . . . .
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2
3
4
List of Figures
vii
4.18 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.19 Ion Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.20 Helmholtz Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
Comparison of Data with different atomic beams
Comparisons of Frequency Standard . . . . . . . .
Magnetic Field Calibration . . . . . . . . . . . . .
Magnetic Field Correction . . . . . . . . . . . . .
Saturation dip from retro-reflected probe beam .
Saturation Data . . . . . . . . . . . . . . . . . . .
Line shape model data . . . . . . . . . . . . . . .
Data fit to interpolated line shape . . . . . . . . .
7
Li D1 F = 1 → F 0 transitions . . . . . . . . . . .
7
Li D1 F = 2 → F 0 transitions . . . . . . . . . . .
6
Li D1 transitions . . . . . . . . . . . . . . . . . .
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98
List of Tables
2.1
2.2
2.3
Quantum Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Relative transition Amplitudes . . . . . . . . . . . . . . . . . . . . . 33
Estimated AC Stark Shifts . . . . . . . . . . . . . . . . . . . . . . . 37
4.1
Conversion from applied current to magnetic field at interaction
region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Correction Current for Magnetic Fields
Magnetic Field Comparison . . . . . .
Estimated AC Stark Shifts . . . . . . .
Optical Frequencies . . . . . . . . . . .
Measurement Uncertainty Budget . . .
Isotope Shift Frequencies . . . . . . . .
Hyperfine Splitting . . . . . . . . . . .
viii
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85
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95
Chapter 1
Introduction
1.1
Lithium
Spectroscopy of atomic lithium has gained renewed interested as a system for
testing atomic theory of few electron systems as well as insight into the charge radii
of the nuclei [7] [8] [9] [10]. This has prompted a need of precision measurements of
fine structure intervals and isotope shifts. As a three electron atom with two stable
isotopes, 6 Li and 7 Li, measurements of the isotope shift, fine structure splitting,
and hyperfine structure splitting provide a window for information on charge radii
of the nucleus, magnetic dipole distribution of the nucleus, and nuclear structure.
1.2
Previous Measurments
While there have been several published high precision measurements of the D1
(22 S1/2 → 22 P1/2 ) and D2 (22 S1/2 → 22 P3/2 ) transitions (Fig. 1.1) there is still
significant discrepancy between the published results. The current theory predictions are also more precise then the experimental limit for the isotope shift. As
shown in Fig. 1.2 the discrepancy between the two most recent measurements of
the isotope shift is several times their combined uncertainties for both the isotope
shift and 7 Li fine structure measurements. The data presented by Brown et al [1]
and Sansonetti et al. [2] is the same data present with different analysis based
on the evaluation of systematics in the system. The two most recent independent
measurement are presented by Sansonetti et al. [2] and Das et al. [3].
1
Introduction
2
4d
4d
4s
3p
3p
3d
3d
3s
2p
2p
2s
2S
1/2
2PO
1/2
2P O
3/2
2D
3/2
2D
5/2
Figure 1.1: Energy levels of Lithium. This experiment is focused on measuring
the 2 S1/2 →2 P1/2 (D1) and 2 S1/2 →2 P3/2 (D2) transitions in both 6 Li and 7 Li.
The approximate transitions wavelengths are given for each transition.
The disagreement in the D2 measurements can be partially attributed to a lack
of resolution between peaks, making line shape models difficult. When multiple
resonance are separated by less then the natural line width of the transition, 6
MHz for Lithium, there is a possibility for interference in the fluorescence leading
to more complex line shapes that depend on the geometry of the experiment.
This was addressed by Sansonetti et al.[2] who investigated the effect of probe
laser polarization relative to detection angle on line shape as well as the effect
of quantum interference on atomic spectra [11]. This effect is not present in the
D1 lines and there persists significant disagreements so more investigation is still
needed in order to resolve the discrepancy. The absolute frequency measurements
of the transitions also have significant disagreements. Since these disagreements
appear as linear shifts in the optical frequencies, the isotope shift measurement
has less uncertainty as it is a relative frequency between two optical transitions.
Introduction
3
7
Fine Structure of Li
6
Fine Structure of Li
(g)
(a)
(a)
(b)
(b)
(c)
(c)
(e)
(d)
(d)
(e)
(f)
-0.8
-0.6
-0.4
-0.2
0.0
0.2
-1.2
Deviation from Weighted Mean of Measurements (MHz)
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
Deviation from Weighted Mean of Measurements (MHz)
7
6
Isotope Shift between Li and Li for the D1 Transition
(a)
(b)
(e)
(d)
0.0
0.1
0.2
0.3
0.4
0.5
Deviation from Weighted Mean of Measurements (MHz)
Figure 1.2: Previous Measurements of Lithium fine structure splitting and
isotope shifts for the D1 and D2 transitions. The red points are experimental
results and the purple are theory calculations. There is clear disagreement
between the experimental results as well as between experimental measurements
and theory predictions. Points (a) and (b) are publications of the same data
with different analysis techniques.
(a) Brown PRA 87, 032504 (2013)[1]
(b) Sansonetti PRL 107, 023001 (2011)[2]
(c) Das PRA 75, 052508 (2007)[3]
(d) Noble PRA 74, 012502 (2006)[4]
(e) Walls EPJD 22, 159 (2003)[5]
(f) Orth ZPA 273, 221 (1975)[6]
(g) Puchalski PRL 113, 073004 (2014)[7]
1.3
Measuring Optical Frequencies
The precision measurement of atomic transition frequencies relies on the ability to
determine optical frequencies of hundreds of THz and reference the frequencies to
the SI second. These measurements have long been a difficult task because the frequencies are so large. Earlier techniques including harmonic frequency chains,while
Introduction
4
Ι
νn = n frep + f0
f0
0
ν
frep
Figure 1.3: The optical frequency comb acts as a microwave to optical frequency synthesizer. The output frequency spectrum is a series of evenly spaced
comb modes, each of which can be defined by two microwave frequencies f0 and
frep as well as a mode number.
accurate, only allowed the measurement of limited frequencies and were difficult to
use. This experiment uses an optical frequency comb which acts as an optical to
microwave synthesizer, allowing the measurement of optical frequencies in terms
of two microwave frequencies (Chapter 3). A frequency comb is a light source
that has a frequency spectrum consisting of a series of discrete lines with a regular
spacing. This spacing is defined as the repetition rate of the optical frequency
comb and it is dependent on the cavity length of the oscillator. The entire mode
structure is offset by offset frequency. If the output of the comb spans an optical
octave these two frequencies can be measured and stabilized. From this we can
determine the frequency of any mode of the comb
νn = nfrep + f0 ,
(1.1)
where n is a integer mode number, frep is the repetition rate of the comb, and
f0 is the offset frequency. This ruled output is shown in Fig. 1.3. Both the
offset frequency and repetition rate are microwave frequencies which can easily be
measured to high accuracy [12]. This gives a consistent and relatively easy method
of measuring optical frequencies in terms of the SI second.
This experiment probes atomic lithium using an extended cavity diode laser locked
to the optical frequency comb. Atoms from a collimated atomic beam are excited
by tuning the frequency of the probe laser across the transition frequencies. The
fluorescence is then recorded using a photomultiplier tube. We have improved the
stability of the extended cavity diode laser as well as collimated our atomic beam
Theory
5
(Chapter 4). With this system we have established preliminary error bounds on
the systematics and present optical frequencies for the D1 transitions, isotope shift
and hyperfine splittings.
Chapter 2
Background Theory
2.1
Notation
When referring to the energy state of an atom we use the term notation
2S+1
LJ ,
(2.1)
where S is the spin quantum number, L is the the orbital angular momentum
quantum number, and J = L + S is the total electronic angular momentum.
These values are capitalized as they are the totals for the system not the state of
one electron. Since we are mainly concerned with lithium, a three electron atom,
we will use this notation. The allowed values of J are given by
|L − S| < J < L + S,
(2.2)
where all integer steps between are also allowed. The intrinsic angular momentum
or nuclear spin is represented by I. Protons and neutrons are fermions and so
have an intrinsic spin of 1/2, this means that nuclei of odd mass number will
have half integer spin. The total momentum of the atom is given by F and is the
combination of the nuclear spin I and the total angular momentum J.
When considering a one-electron atom, each state can be defined by four quantum
numbers n l j M where n is the principle quantum number, l the orbital angular
6
Theory
7
Quantum Number Desciption
n
Principal Quantum Number
Spin Angular Momentum
S
Spin Quantum Number
MS
L
Orbital Angular Momentum
I
Nuclear spin
~ +S
~
J
Total Electron Angular Momentum: J~ = L
F
Total Atomic Angular Momentum F~ = I~ + J~
Table 2.1: Table of quantum numbers.
moment, j the total momentum, and mj is the projection of the angular momentum. The n quantum number is used to refer to the shell that the electron resides
in. For each n there are n2 orbitals.
2.2
Atomic Theory
2.2.1
Hydrogen Atom
To start we will look at the simplest atom, Hydrogen which consists of a heavy
proton of charge +e and a significantly lighter electron with charge −e. The
proton is assumed to motionless. Starting with Schrödinger’s equation in spherical
coordinates
~2 1 d
1
d
δψ
1
d
δ2ψ
2 δψ
−
r
+ 2
sin θ
++ 2 2
sin θ 2 +V ψ = Eψ.
2m r2 dr
δr
r sin θ dθ
δθ
δφ
r sin θ dφ
(2.3)
Here V is the potential energy and E is the total energy and m is the mass of the
electron. We can look for solutions using separation of variables in the form
ψ(r, θ, φ) = Rl (r)Yl,m (θ, φ),
(2.4)
where Rl (r) is the radial portion and Yl,m (θ, φ) is the spherical harmonic. It can
then be shown that the radial portion of Eq. (2.3) must be equal to a constant
which is defined as l(l + 1).
1 d
2mr2
2 dR
r
− 2 [V (r) − E] = l(l + 1).
R dr
dr
~
(2.5)
Theory
8
Here the potential energy of the configuration is given by Coulomb’s law
V (r) = −
e2 1
.
4πo r
(2.6)
To simplify this further we define u(r) = rR(r)
~2 d2 u
~2 l(l + 1)
e2 1
u(r)E = −
+
u(r)
+ −
2m dr2
4πo r 2m r2
(2.7)
We now need to solve this for u(r) and determine the allowed energies. To simplify
Eq. (2.7) we divide through by E and make the following substitutions
r
ρ=
−2mr2 E
me2
and
ρ
=
0
~2
2πo ~2 ρ
(2.8)
giving
d2 u
ρ0 l(l + 1)
= 1−
+
u(r).
dρ2
ρ
ρ2
(2.9)
We can now look at solutions of the form
uρ = Ae−ρ + Beρ .
(2.10)
We start by looking at the boundary condition ρ → inf, we see that eρ blows up
at infinity so we set B = 0. As ρ → 0 Eq. 2.9 is dominated by the centrifugal term
l(l + 1)
d2 u
=
u(r),
dρ2
ρ2
(2.11)
which leads to solutions of the form
u(ρ) = Cρl+1 + Dρ−l .
(2.12)
As ρ → inf, ρ−l blows up and so we set D = 0 and look for solutions of the form
u(ρ) = ρl+1 e−ρ
X
cj ρ j .
(2.13)
j
We find that the eigenvalues of Eq. (2.9) are given by
ρ0 = 2n,
(2.14)
Theory
9
where n is an integer. Therefore the allowed energies are
2 2 e
1
m
E1
En = −
=
,
2~2 4πo
n2
n2
2.2.1.1
n = 1, 2, 3, 4...
(2.15)
Quantum Defect
The hydrogen system is a reasonable model for lithium as like any alkali atom
lithium has one valence electron and then completely filled inner shells. These
inner shells are extremely stable and so the excitation spectra are due to the
transitions in the outer electron. This inner shell causes a shielding effect such
that the outer electron does not see the complete charge of the nucleus and so
it is more likely to be excited into a higher energy level as the binding energy is
less. For this model the potential seen by the valence electron is no longer of the
Coulomb form used above.
When considering the potential for lithium at large distances from the nucleus the
potential is simply the Coulomb potential but near the nucleus the shielding does
not play a role
V (r) →


−e2
4π0 r
 −Ze2
4π0 r
r→∞
.
r→0
Therefore the further the electron is from the nucleus the closer to the field is to
that of hydrogen. The energy levels of any alkali atom can be described by
Enl = −
Ry
n2∗
(2.16)
where Ry is the Rydberg energy and n∗ is the effective principal quantum number.
This can be represented by
n∗ = n − ∆l ,
(2.17)
where ∆l is the quantum defect. The quantum defect is simply an angular momentum (l) dependent correction to the energy levels due to incomplete shielding
of the nuclear charges by the inner electron shell. For small values of l this has
a larger effect as the valence electron is closer to the nucleus and so sees more of
the charge. This shift removes the degeneracy found in the hydrogen atom as it is
only dependent on l not n. For lithium in the 2s, (l = 1) state the quantum defect
is found to be .41 [13]. This lowers the ionization energy for lithium in comparison
to hydrogen for low l states.
Theory
2.2.1.2
10
Central-Field Approximation
In the previous section we approximated the correction to the energy levels of
lithium by considering the valence electron orbiting a shielded core. This approach
only addressed the energy of the valence electron. Now we need to look at the
energy due to all electrons in the atom. This approximation adds the correction
due to the electrostatic interactions between the electrons. As long as the inner
shell remains closed there is no angular momentum coupling as the inner shell has
zero total angular momentum. This leads to a solution which is composed of a
product of one-electron states.
Given a system of three electrons, the Hamiltonian assuming Coulomb potential
is given by
H=
3 X
i=1
3
X
~2 2
Ze2
e2
−
∇ −
+
.
2m i 4π0 ri
4π0 rij
j>i
(2.18)
Here the first term is the kinetic energy of each electron, the second is the potential due to the interaction of each electron with the nucleus, and the third is the
repulsion between electrons. The distance between each electron is given by rij .
Next we make the assumption that the repulsion between the electrons is radially
symmetric as the interaction between the inner shell, a spherical charge distribution, and valence electron is spherically symmetric. This allows the potential
energy to be expressed such that it is only dependent on r and we assume a central
potential S(r).
VCF (r) = −
Ze2
+ S(r).
4π0 r
(2.19)
Under this approximation, the Hamiltonian for the system can then be expressed
as
HCF =
3 X
i=1
~2 2
−
∇ + VCF (ri ) .
2m i
(2.20)
From this form of the Hamiltonian the wave function can be separated into one
part for each electron ψatom = ψ1 ψ2 ψ3 leaving three equations of the form
~2 2
−
∇ + VCF ψi = Eψi .
2m
(2.21)
From this we look for solutions of the form
ψi = R(ri )Γl,m .
(2.22)
Theory
11
Again assuming that the potential is spherically symmetric we are left with
~2 d2
~2 l(l + 1)
−
R(r) = ER(r).
+ VCF (r) +
2m dr2
2mr2
(2.23)
This leads to a similar conclusion in the last section, where the energy is given by
ECF =
Zef f e
4π0 r2
(2.24)
We can place limits on Zef f although it is difficult to calculate. Once again as
r → 0 then Z= 3 and as r → ∞, Z = 1. For a derivation and more information
see Ref. [14].
2.2.2
Isotope Shifts
Changes in mass of an atom from changes in neutron number yield different isotopes of the same element with a corresponding shift in energy levels. Lithium
has two stable isotopes 6 Li and 7 Li, where 7 Li has an abundant of 92.5% and 6 Li
at 7.6%. The energy shift is caused by both a change in mass of the nucleus as
well as the distribution of charge within the nucleus know as the volume effect.
The mass shift is the dominant contributor to the isotope shift in light atoms but
decreases with mass number by A−2 . The volume shift is 10,000 times smaller in
this isotope shift but increases as Z 2 A−1/3 and so is dominant in heavier atoms
[15].
If we start by looking at a single-electron atom, hydrogen, the energy levels are
given by:
En =
Where a0 =
4π0 ~2
µe2
e2 1
4π0 a0 n2
n = 1, 2, 3, 4, ...
(2.25)
and we have assumed the mass of the proton to be much greater
then the electron and so set µ = me . If we assume that the mass of the nucleus is
not infinite then we can replace the electron mass with the reduced mass µ:
µ=
me M A
me + M A
(2.26)
where me is the mass of the electron, M is the nucleon mass, and A is the atomic
number. We can then say that since any energy level is proportional to this mass,
Theory
12
2.83 MHz
2p 2P3/2 F=0
2p 2P3/2 F=1
5.89 MHz
2p 2P3/2 F=2
9.39 MHz
2p 2P3/2 F=3
10 534.194 MHz
10 053.184 MHz
2p 2P1/2 F=2
2p 2P3/2 F=1/2
1.71 MHz
2p 2P3/2 F=3/2
92.03 MHz
2.91 MHz
2p
2P
3/2
F=5/2
10 534.039 MHz
2p 2P1/2 F=1
10 052.76 MHz
2p 2P1/2 F=3/2
26.063 MHz
2p 2P1/2 F=1/2
2s 2S1/2 F=2
2s 2S1/2 F=3/2
228.205 259 MHz
803.504 086 MHz
2s 2S1/2 F=1/2
6Li
7Li
2s 2S1/2 F=1
Figure 2.1: Fine structure for lithium D1 and D2 of 6 Li and 7 Li.
the energy levels shift due to changes in the atomic mass. For two isotopes of mass
number A and A + ∆N the frequency is shifted by
∆ω = ω0
m
m
−
+
M A + M ∆N
MA
(2.27)
where w0 is the frequency of the transition. For this experiment we are considering
transitions around 670nm and ∆N = 1 this results in a normal mass shift of ∼ 5
GHz. There is also a specific mass shift that is on the same order as the normal
Theory
13
mass shift but this is more difficult to calculate [13],[14]. When a system contains
more then two particles, the center of mass of the system is no longer the same as
the center of mass between a particular electron and the nucleus. This means the
interaction can no longer be treated as center of mass motion. This correction is
referred to as the specific mass shift.
The volume shift is due to a change in the distribution of charge in a finite volume.
Since the protons and neutrons are distributed within the nucleus we cannot assume that the electrostatic potential has the 1/r dependence of a point charge. To
estimate this effect, assume the that the change is evenly distributed in a sphere
of radius R given by
R = r0 A1/3
(2.28)
where r0 is defined to be 1.2 × 10−15 m. This model gives a potential given by
V (r) =



Ze2
(4π0 )2R

−
r2
R2
−3
r<R
(2.29)
Ze2
(4π0 )r
R < r.
Now we can use this as a perturbation to the Hamiltonian with potential given by
the Coulomb potential such that the correction to the Hamiltonian is the difference
between the Coulomb potential and the model.
H0 =



Ze2
(4π0 )2R
r2
R2
+
2R
r
−3
r<R

0
.
(2.30)
R<r
The energy shift to first order is found using the expectation value of the Hamiltonian
∆E = hψ|H 0 |ψi.
(2.31)
This is found to be [16]
∆E =
Z 5 e2 4R
∆R,
4π0 5a3µ n3
(2.32)
where aµ = a0 (m/µ) and δR is the difference in radius between the two isotopes.
For the D1 and D2 lines the total isotope shift between 6 Li and 7 Li is about 10.5
GHz as shown in Fig. 2.1.
Theory
2.2.3
14
Fine Structure
The fine structure splitting of energy levels is due to the addition of magnetic
interactions and relativistic corrections in the kinetic energy of the Hamiltonian.
~ that results in a magnetic
The electron can have an orbital angular momentum L,
dipole moment, µ
~ . The electron also has an intrinsic magnetic moment µ
~ S = g~µB S.
The coupling between these magnetic moments leads to the fine structure splitting.
In Section 2.2.1 the energy levels of hydrogen are shown to be
2 2 m
e
1
E1
En = −
=
,
2~2 4πo
n2
n2
n = 1, 2, 3, 4...
(2.33)
were the Hamiltonian was taken to be
H=−
e2 1
~2 2
∇ −
.
2m
4π0 r
(2.34)
This was derived using the classical expression for kinetic energy. To understand
the relativistic corrections we use the relativistic expression for kinetic energy
mc2
T =p
− mc2
2
1 − (v/c)
(2.35)
mv
.
p= p
1 − (v/c)2
(2.36)
and relativistic momentum
The first term in the kinetic energy expression is the total relativistic energy due
to the motion and the second term is the rest energy. We can now express the
kinetic energy in terms of the relativistic momentum to get
T =
p
p2 c2 + m2 c4 − mc2 .
(2.37)
We then Taylor expand this for small values of p/mc and get
r
p 2
1 p 2 1 p 4
2
T = mc
1 + ( ) − 1 = mc 1 + ( ) − ( ) ... − 1
mc
2 mc
8 mc
2
T =
p2
p4
−
+ ...
2m 8m3 c2
(2.38)
(2.39)
The first term is the nonrelativistic kinetic energy and the second is the relativistic
correction to lowest order. From this the relativistic correction to the En can be
Theory
15
calculated to be
Er1
4n
(En )2
−3
=−
2mc2 l + 1/2
(2.40)
where l is the orbital angular momentum quantum number [17]. Note that the
relativistic correction is 2 × 105 smaller then En and that it will be a negative shift
in energy.
Now to look at the spin-orbit coupling from the magnetic dipole moment of the
electron. When considering the electron’s rest frame the proton orbits around it
~ This field creates a torque on the electron which tries
creating a magnetic field B.
to align the magnetic moment of the electron µ with the field. The Hamiltonian
of this configuration is given by
~
H = −~µ · B.
(2.41)
From here we need to find the magnetic field created by the proton and the magnetic dipole moment of the electron. The magnetic dipole moment of electron can
be found by considering a charge spinning with angular momentum from its spin
S.
If we look at the configuration using classical electrodynamics and assume that
a charge q is distributed around a ring of radius r rotating with period τ , the
magnetic dipole moment of the configuration is
qπr2
τ
µ=
(2.42)
as the current in the ring is q/T and the area of the ring is πr2 . The angular
momentum, again found classically is given by the moment of inertia of the ring,
mr2 , times the angular velocity 2π/τ .
2πmr2
S=
τ
(2.43)
q
S.
2m
(2.44)
This classical derivation yields
µ=
However this is slightly off, in reality the magnetic moment is
µe =
e
S.
m
(2.45)
Theory
16
This factor of two is known as the g factor of the electron and was explained by
Dirac [18].
To find the magnetic field due to the relative motion of the proton we consider
the reference frame of the electron and treat the proton as a continuous charge
loop with current e/τ . The period can be found by looking at the orbital angular
momentum L = rmv =
2πmr2
.
τ
Thus the magnetic field is given by
e ~
~ = µ0 I = 1
B
L.
2r
4π0 mc2 r3
(2.46)
We can now go back to our Hamiltonian
H = −µ · B =
e2
1 ~ ~
S · L.
2
4π0 m c2 r3
(2.47)
This calculation assumes that the electron’s frame is an inertial frame, however
it is an accelerating frame and so an additional correction known as the Thomas
precession factor [17] must be added, this adds a factor of 1/2 yielding
~ =
Hso = −~µ · B
1 ~ ~
e2
S · L.
2
8π0 m c2 r3
(2.48)
~ and L,
~ this
It is important to not that the Hso depends on the orientation of S
causes a splitting in states that would be degenerate without this interaction. The
~ + S.
~ We simplify Eq. 2.48 using the
total angular momentum is defined as J~ = L
following identity
~ ·S
~
J 2 = L2 + S 2 + 2L
(2.49)
~ ·S
~ = 1 (J 2 − L2 − S 2 ).
L
2
(2.50)
The eigenvalues of L · S are given by
~2
[J(J + 1) − L(L + 1) − S(S + 1)].
2
(2.51)
l
From this we can find the correction to the energy Eso
given by
l
Eso
= hHso i =
e2
1
2
8π0 m c2 r3
~2
[J(J
2
+ 1) − L(L + 1) − S(S + 1)]
.
L(L + 1/2)(L + 1)n3 a3
(2.52)
Theory
17
By adding Eq. 2.52 and Eq. 2.40 we find the total shift to be
Efl s
En2
4n
=
3−
.
2mc2
J + 1/2
(2.53)
Here both corrections although stemming from completely different effects happen
to be on the scale of
2
En
.
2mc2
Adding this correction to the original Bohr formula,
the energy levels with the fine structure correction can be expressed as
Enj
E1
3
α
n
=− 2 1+ 2
−
.
n
n J + 1/2 4
(2.54)
Where α is the fine structure constant. For lithium, all of the states are doublets
except the 2 S states which does not split. The allowed J are therefore J = L ± 1/2
[19]. This gives the degeneracy in the transition 2 S1/2 →2 P1/2,3/2 which are the
D1 and D2 transitions we are interested in measuring.
2.2.4
Hyperfine Structure
The hyperfine structure splitting occurs because like the electron the proton and
neutron have an intrinsic magnetic dipole moment. The reason this effect is so
much smaller then the fine structure splitting is that the mass of the proton is
much greater. For the proton the magnetic moment is given by
µp =
gp e
Sp ,
2mp
(2.55)
where gp is the g-factor of the proton which is measured as 5.59 [17]. The proton
has a much more complex structure because it is made of three quarks and so the
gyromagnetic ratio is not as easily calculated. In classical electrodynamics the
magnetic field from a dipole is given by
B=
2µ0 3
µ0 3(~
µ
·
r̂)r̂
−
µ
~
+
µ
~ δ (r),
4πr3
3
(2.56)
where δ 3 is the three dimensional the delta function which is is 0 everywhere except
for (0, 0, 0) where it is one, see Ref. [20] in depth derivation of the field of a dipole.
Theory
18
The Hamiltonian for the electron in the magnetic field of the nucleus is then given
by
0
Hhf
~
3(I · r̂)(J~ · r̂) − I~ · J~
µ0 gp e2 ~ ~ 3
µ0 gp e2
+
= µe · B =
J · Iδ (r).
4πr2 mp me
r3
3mp me
(2.57)
To find the first order correction to the energy we find the expectation value of
the Hamiltonian given by
0
Ehf
=
0
hHhf
i
~
µ0 gp e2 ~ ~
µ0 gp e2
3(I · r̂)(J~ · r̂)
+
J · I |ψ(0)|2
=
2
3
4πr mp me
r
3mp me
(2.58)
For the ground state the wave function is spherically symmetric and so the expectation value of first term in the the Hamiltonian is zero. For (l=0) it is found that
the expectation value of the wave function is
|ψ100 (0)|2 =
1
πa3
(2.59)
as found in Ref. [17]. From this we can plug in to find the energy correction
0
Ehf
=
µ0 gp e 2 J
·
I
.
3mp me a3
(2.60)
This is known as the spin-spin coupling term because of the dependance on the
dot product of the spin of the the proton and electron. We can define the total
spin as F = J + I and then find J · I by squaring and simplifying to get
1
J~ · I~ = [F 2 − J~2 − I~2 ]
2
(2.61)
For Hydrogen since S for both the proton and electron are 1/2 and there are two
possible states with spins aligned or anti aligned the value for the total spin can
be 0 or 1. These are the singlet or triplet states respectively. Thus the possible
values of J · I are
J ·I =

 ~2
Triplet, F = 1
 −3~2
Singlet, F = 0
4
4
To find the shift due to the hyperfine splitting the triplet state is shifted up and
the singlet is shifted down.
Theory
2.2.5
19
Quadrupole Interaction
The other important characteristic when considering the structure of the nucleus is
the electric quadrupole moment. Since the nucleus has only positve charge it does
not have a electric dipole moment due to symmetry under time reversal however
if it is not spherically symmetric there is a quadrupole term. The Hamiltonian for
the interaction between a proton at r~p and and electron at r~e is given by
H0 = −
e2
,
4π0 |~
re − r~p |
(2.62)
where the position is taken from the center of mass of the nucleus. We can assume
that re >rp and so we will Taylor expand Eq. (2.62) in terms off rp /re . This gives
the interaction Hamiltonian to be
rp2
e2 1
rp
0
H =−
P0 (cos θ) + 2 P1 (cos θ) + 3 , P2 (cos θ)
4π0 re
re
re
(2.63)
where θ is the angle between re an rp and Pi is the Legendre polynomial. The first
term in this series is the monopole moment given by the Coulomb potential. If we
sum over all protons we see that the interaction between the nucleus and electron
is given by
0
Hmono
Ze2 1
=−
.
4π0 re
(2.64)
The second term is nuclear dipole interaction, which is zero due to the reversal invariance. The last term is the electric quadruple interaction which is a combination
of the nuclear electric quadruple moment and electric field gradient tensor.
This can be simplified to give the nuclear quadrupole moment Q as [21]
X
2
Q = hI, MI = I|
Q02 (pj )|I, MI = Ii,
e
j
(2.65)
where Qi2 is the electric quadrupole tensor and is summed over all protons. We
can then define the average field gradient at the nucleus as
δ 2 Ve
δz 2
=
X 2
JMJ = J F0 (ei )JMJ = J ,
(2.66)
i
where the sum is over the valence electrons.We can now combine the quadruple
moment and the average electric field gradient to show that the electric quadrupole
Theory
20
interaction Hamiltonian is [21]
Hquad =
where
BJ
~ 2 + 3 (I~ · J)
~ − I(I + 1)J(J + 1)],
[3(I~ · J)
2I(2I − 1)J(2J − 1)
2
δ 2 Ve
BJ = eQ
δz 2
(2.67)
(2.68)
is the electric quadrupole interaction coefficient. The energy shift due to the
quadruple interaction is then given by
0
∆E = hψ|Hquad
|ψi =
BJ K(K + 1) − I(I + 1)J(J + 1)
,
8
I(2I − 1)J(2J − 1)
(2.69)
where
K = F (F + 1) − I(I + 1) − J(J + 1).
(2.70)
For excited states where the electron charge distribution is spherically symmetric
the electric field gradient at the nucleus vanishes and so there is not quadrupole
shift. Thus p states do exhibit the shift and s states do not.
2.3
Multi-Electron Atoms
The theory in the previous sections was all derived using hydrogen as the model
atom which only has one electron except for the quadrupole moment which is not
present in hydrogen. While this may not seem like the most useful in understanding lithium which has three electrons there are many similarity which make the
hydrogen atom a reasonable starting point. When considering lithium we know as
an alkali metal that it consists of a filled shell plus one valence electron. For this
experiment we are only looking at transitions from the ground state of lithium to
the 2 P1/2 and 2 P3/2 states which have S = 1/2 and L = 1. These states exhibit
similar characteristics to the corresponding states in the hydrogen atom.
Theory
2.4
2.4.1
21
Precision Spectroscopy: Fields and Atoms
Quantization of Electromagnetic Fields
~ r, t) in the coloumb gauge with no
Consider a field given by a vector potential A(~
free currents or charges. Thus we can set the scaler potential equal to zero. Using
~ satisfies the wave equation
Maxwells equations we find that A
~−
∇2 A
~
1 ∂ 2A
= 0.
c2 ∂t2
(2.71)
Lets now consider a single mode of light as a plane wave with polarization ˆ.
Normalizing the potential to a box with volume V in order to deal with the infinite
extent of a plane wave, the general solution to the wave equation is given by
1
ik·r
∗
∗
−ik·r
~ r, t) = √ C(t)ˆe + C (t)ˆ e
A(~
,
V
(2.72)
C(t) = C0 e−iωt .
(2.73)
where
We know that the energy of an electromagnetic wave is given by
1
E=
8π
Z
ε2 + B 2 dV
(2.74)
V
and the electric and magnetic fields can be related to the vector potential by
~
1 ∂A
iω
ik·r
∗
∗ −ik·r
ε=
= √ C(t)ˆe − C (t)ˆ e
c ∂t
c V
(2.75)
i
ik·r
∗
∗
−ik·r
~ ×A
~ = √ C(t)(~k × ˆ)e − C (t)(~k · ˆ e
B=∇
V
(2.76)
Plugging these back into our energy equation we find that
E=
2
1 ω 2 .
C(t)
2P i c2
(2.77)
Breaking E into its complex and imaginary components
E=
2 1 ω 2 + Im C(t) 2 ).
Re
C(t)
2P c2
(2.78)
Theory
22
Now lets consider the Hamiltonian for a classical simple harmonic oscillator
Hsho =
p2
mω 2 2
+
x
2m
2
(2.79)
where x is the position and p is the momentum of the particle. In order rearrange
H to make it of a similar form to Eq. 2.78 we will make the variable substitution
√
p = mωP and x = √X
. This gives
mω
Hsho =
We also know that p =
dx
m
dt
or ωP =
ω 2
(X + P 2 ).
2
dX
dt
(2.80)
and we can define P (t) and X(t) to be
X(t) = C0 cos ωt
(2.81)
P (t) = −C0 sin ωt
(2.82)
which is the same as the real and imaginary parts of C(t) as defined by Eq. 2.73.
Now we can define C(t) in terms of P and X giving
r
C(t) =
πc2
X + iP ).
ω
(2.83)
Putting this together with the Hamiltonian for the simple harmonic oscillator we
can find the Hamiltonian for a electromagnetic wave in terms of X and P.
Hem =
ω 2
X + P 2)
2
(2.84)
This is the Hamiltonian of a single mode of the electromagnetic field. As usual
the energy eigenstates of the simple harmonic oscillator have energies
1
En = ~ω(n + ).
2
(2.85)
Since each photon has energy ~ω the number of photons in a given mode of the
light field is given by En /~ω. This also shows the existence of a zero point energy
~ω/2 for a mode with no photons.
Theory
23
2.4.2
Interaction of Light with Atoms
2.4.3
Emmission
This section will derive the emission rate of an atom from an initial sate |ii to a
final state |f i.
To start we look at Fermi’s Golden Rule which gives a differential transition rate
of an atom in an external field from a initial state i to final state f .
dWf i =
2π
|hf |H 0 |ii|2 ρ(E)P (E)dE.
~
(2.86)
Here ρ(E) is the density of states or number of final states for a given energy.
P (E) is the distribution of energies which will be derived below. The density of
states refers to both the photon state, as derived in the previous section as well as
the atomic state. To find this density consider the incident light with polarization
ˆ with wave vector ~k.
For a given slice of volume the number of photon states is given by
dN =
1 d3 xd3 p
(2π)3 ~3
(2.87)
If we integrate over some volume V and substitute p~ = ~~k we get
dN =
V
k 2 dk dΩ,
(2π)3
(2.88)
where dΩ is the the differential solid angle for the emission of the photons. We
can then substitute k = ω/c = E/(~c) giving
dN =
V E 2 dE
dΩ.
(2π)3 ~3 c3
(2.89)
Now to find ρ(E) we simply want the number of photon states for a change in
energy dE so
ρ(E) =
dN
V E2
=
dΩ.
dE
(2π)3 ~3 c3
(2.90)
The next step is to find the distribution of energies which will allow the transition
to occur. Here we will assume that there is no line broadening other then the
Theory
24
natural line width given by the lifetime of the transition 1/γ, where γ is the
spontaneous emission rate.
Spontaneous emission is the process where an atom decays without being triggered
by an external field. For this process if the atom is originally in an excited state it
will naturally decay to a lower energy state and so emit a photon with energy equal
to the decay gap. This experiment detects the fluorescence due to spontaneous
emission. Since the excited state has a lifetime this results in a natural broadening
of spectral peaks given by the Heisenberg uncertainty principle.
∆E ∆T =
~
.
2
(2.91)
The average lifetime of the the D1 and D2 lines is 27 ns giving a natural line
width of 6 MHz.
In order for spontaneous emission to occur both the atomic levels and the electromagnetic field must be quantized. From this uncertainty in the energy, the decay
is then given by a Lorentzian distribution
P (ω) =
γ/(2π)
.
(ω − ω0 )2 + (γ/2)2
(2.92)
Here γ is the width of P (ω), as γ → 0 the with of the P (ω) tends to zero and the
amplitude given by (2/(πγ)) goes to infinity. This leads to P (ω) → δ(ω − ω0 ) as
γ → 0. More intuitively as the line width goes to zero the only way for excitation
to occur is through an energy exactly equal to the transition.
Plugging Eq. (2.90) and Eq. (2.92) back into Eq. (2.86) and substituting in terms
of energy
dWf i =
2π
|hf |H 0 |ii|2 ρ(E)δ(~ω0 − E)dE.
~
(2.93)
We can then integrate over the photon energies and get
Wf i =
2π
~|hf |H 0 |ii|2 ρ(~ω0 ).
~
(2.94)
To find the matrix element hf |H 0 |ii we need to find the Hamiltonian for an electron in an electric field given by H = H0 + H 0 where H0 is the the unperturbed
Hamiltonian for the electron and H 0 is the perturbation due to the electric field.
Theory
25
For a one electron atom the Hamiltonian is given by
H=
p2
Ze2
−
2m
r
(2.95)
and the perturbation due to the electric field is
H0 =
e
~
p~ · A.
mc
(2.96)
Here the binding energy between the nucleus and the electron is much stronger
then the the force due to the electric field and so we can treat the electric field as
a perturbation, for a formal proof see Ref. [22].
As stated before it is important to consider the state of both the incident light
and the atom as |ii and |f i describe the combined state. In order to find the
Hamiltonian for the entire system, we must consider the electric field state as well
as the atomic state given by 2.96.
Htotal = H0 + H 0 + Hem
(2.97)
The meaning of this Hamiltonian can be thought of as the unperturbed state
of both the atom and the electric filed plus a cross term which describes their
interaction.
We can now solve for the emission rate by plugging in the total Hamiltonian and
solving for the matrix elements. This derivation is done in detail in Ref. [22] and
gives the rate for spontaneous emission for any incident polarization is given by
γ=
4ω03 |hg, J|d|e, J 0 i|2
.
3~c3
2J 0 + 1
(2.98)
Stimulated emission occurs when an atom is already in an excited state E2 . This
process requires an incident photon with an energy equal to the energy gap between this higher energy state and some lower state E1 . The incident photon can
stimulate the upper level atom causing it to decay and emit a second photon with
the same energy.
Theory
2.4.4
26
Absorption
This section will use the notation and derivation in the previous two sections to
derive the absorption rate for a transition. Here we consider an atom in the ground
state with an incident field. When matter is irradiated with a photon there is a
probability that an atom will absorb the photon and transition to a higher energy
level. For this to occur the quantum energy of the photon must match the energy
gap between the initial energy of the atom and some final state. If there is no
such level then the matter will not absorb the light and so be transparent to the
radiation.
We will assume the incident photon is of a single mode and that we are only
considering one allowed transition. Again we start with Fermi’s Golden Rule
dWf i =
2π
|hf |H 0 |ii|2 ρ(E)P (E)dE.
~
(2.99)
To find the absorption rate for we substitute the resonance amplitude (2/~πγ)
into Eq. (2.99) to find
Weg =
4
|hf |H 0 |ii|2 .
2
γ~
(2.100)
where the initial and final states are given by
|ii = |g, F, MF i|ni
(2.101)
|f i = |e, F 0 , MF0 i|n − 1i.
(2.102)
and
Where n is the number of photons in the resonant mode of the incident light. We
now need to find the number of photons in mode n to the electric field amplitude
ε0 . The intensity of the light is given by the time averaged magnitude of the
pointing vector.
I=
cε20
8π
(2.103)
For a given flux of photons nc/V with energy ~ω the intensity can be expressed
as
I=
n
~ωc
V
(2.104)
n=
V ε20
.
8π~ω
(2.105)
and so solving for n we find
Theory
27
The matrix element can found in terms of the electric field and the rate of absorption is given by
Weg =
2.4.5
1 |he, F 0 MF0 |d~ |g, F, MF i|2 ε20
γ
~2
(2.106)
Saturation
Considering an atom in a two level system with an incident electric field, if the
incident field is strong enough the field will perturb the properties of the atoms.
This causes the system to exhibit different characteristics based on the perturbation of the the population of states. The parameter which characterizes this
perturbation is the saturation parameter κ given by [22]
κ=
excitation rate
.
relaxation rate
(2.107)
For a two level system the excitation rate is given by
Γ=
d2 ε20
,
γ0
(2.108)
where d is the dipole matrix element, ε0 is the amplitude of the incident field, and
γ0 is the spontaneous decay rate. For this system the only contribution to the
relaxation rate is the spontaneous decay rate and so the saturation parameter is
given by [22]
κ=
Γ
d 2 ε2
= 20
γ0
γ0
(2.109)
To find the fluoresence intensity we look at the number of atoms in both the excited
state Ne and ground state Ng . These can be described by the rate equations
dNg
= −ΓNg + (γ0 + Γ)Ne
dt
(2.110)
dNe
= ΓNg − (γ0 + Γ)Ne .
dt
(2.111)
and
In equilibrium it is found that the total number of atoms in the excited state is
given by
Ne =
κ
Ntot ,
1 + 2κ
(2.112)
Theory
28
Amplitude
Saturation
Parameter
— .1
—1
— 10
800
600
400
200
-4
2
-2
4
Detuning (MHz)
Figure 2.2: Line shape of atomic transition for different saturation parameters.
As the saturation parameter increases the width of the peak broadens as more
atoms are in the excited state.
where Ntot is the total number of atoms in the system. Since the intensity of the
fluorescence is proportional to the number of atoms in the upper state we see a
dependance on the line shape due to the strength of the incident electric field.
When tuning a laser across a transition from the ground to an excited state the
detuning dependent saturation parameter is given by
γ02 /4
κef f (∆) = κ 2
∆ + γ02 /4
(2.113)
where κ is the on resonance saturation parameter [22]. By substituting the effective
saturation parameter into the rate equations we can find the intensity as a function
of probe detuning from resonance given by
I(∆) ∝ γ0 Ne =
κ(∆)
γ02 /4
Ntot = κNtot 2
.
1 + 2κ(∆)
∆ + (1 + κ)γ02 /4
(2.114)
This give a Lorentzian profile with width given by
√
γ = γ0 1 + κ.
(2.115)
When tuning a laser over a transition where the value of κ is small, the line shape
Theory
29
is given by a Lorentzian with width γ0 . For large values of κ we see a broadened
line shape as shown in Fig. 2.2. This is known as the power-broadened line shape.
2.4.6
Zeeman Effect
The Zeeman effect is the splitting of spectral lines into several components in the
presence of a static magnetic field. In the presence a magnetic field the magnetic
moment of the atom shifts in energy and so there is a shift in the overall energy.
The treatment of the Zeeman effect is dependent on the strength of the external
field in comparison with the internal field associated with the spin orbit coupling.
If the external field is less then the internal then the internal field dominates and
the external field can be treated as a small perturbation to the Hamiltonian. If
the external field is much larger then the Zeeman effect dominates and the spin
orbit field becomes the perturbation.
For a single electron the perturbation to the Hamiltonian is given by a sum of the
electronic and nuclear components
~
~
~ ext ,
H = − gJ µB J + +gI µN I · B
0
(2.116)
where gJ is the Lande g factor.
For Bext << Bint the hyperfine coupling dominates with the good quantum numbers N, F, J, I. The Zeeman correction to the the energy levels is
E 0 = hN F JI|H 0 |N F JIi =
e
~
Bext · hJ~ + Ii
2m
(2.117)
To find the expectation value of J~ + I~ is not quite as simple as only F~ is conserved.
~ which we know to be constant we are left only needing
By plugging in F~ = J~ + I,
~ The time average of I~ is simply its projection
to find the expectation value of I.
along F~ .
I~ · F~ ~
I~ave =
F
F2
(2.118)
Theory
30
From this it can be found that the expectation value of J~ + I~ is given by
~ ~
~ · I~ F
·
J
F
~ =
hJ~ + Ii
+
F~
F2
F2
F (F + 1) + J(J + 1) − I(I − 1) F (F + 1) − J(J + 1) + I(I + 1) ~
+
hF i
=
2F (F + 1)
2F (F + 1)
(2.119)
as derived in Ref. [17]. Here the square bracketed term is the Lande g-factor, gj .
Finally the energy correction is
E0 =
e~
gF Bext MF ,
2m
(2.120)
where
gF = gJ
F (F + 1) + J(J + 1) − I(I − 1)
me F (F + 1) − J(J + 1) + I(I − 1)
− gI
.
2F (F + 1)
M
2F (F + 1)
(2.121)
Here we are choosing the z axis to lie along the external field and so MF can be
F, F − 1, ... − F .
2.5
Wigner-Eckart Theorem
In atomic physics many calculations require the calculation of matrix elements of
tensor operators with respect to an angular momentum basis. The Wigner-Eckart
theorem is useful because it states that the matrix elements of an irreducible
spherical tensor operator Tqκ can be expressed as [19]
F 0 −MF0
hγ 0 F 0 MF0 |Tqκ |γF MF i = (−1)
hγ 0 F 0 ||T κ ||γF i
F0
−MF0
κ
F
!
q MF
, (2.122)
where γ refers to any other quantum numbers that define the state, hγ 0 F 0 ||T κ ||γF i
is the reduced matrix element which is independent of MF , MF0 , and q. The term
on the right hand side of the equation in parentheses is the Wigner 3-j symbol
which can be related to the Clebsch-Gordan coeffients by the following [22]
J1
J2
J
M1 M2 M
!
= (−1)J1 −J2 −M √
1
hJ1 M1 J2 M2 |J − M i.
2J + 1
(2.123)
Theory
31
Here the Clebsch-Gordon coefficients are real and so the 3-j symbol is also real. For
this experiment we are interested in calculating the transition strengths which are
ˆ This is the electric dipole moment
depended on the transition dipole operator, d.
associated with the transition between states |ii and |ji. This is defined as
dˆ = dx x̂ + dy ŷ + dz ẑ
(2.124)
in Cartesian coordinates. The direction of this vector gives the polarization of the
transition and how it will interact with a external field. To use the Wigner-Eckart
Theorem we need to transform this into the spherical basis defined as
1
ê1 = − √ (x̂ + iŷ)
2
ê0 = ẑ
(2.125)
1
ê−1 = √ (x̂ − iŷ),
2
giving
1
d1 = − √ (dx + idy )
2
d0 = dz
1
d−1 = √ (dx − idy ).
2
(2.126)
We have now expressed the dipole operator as an irreducible spherical tensor
operator. In the next section we use this to find the relative strengths of different
transitions.
2.6
Relative Transition Strengths
This section derives the relative strength of transitions for lithium D1 and D2 lines.
In general the transition from a ground state |gi = |γ S L I J F MF i to and excited
state |ei = |γ 0 S 0 L0 I 0 J 0 F 0 MF0 i is determine by Fermi’s Golden Rule as given by
Eq. (2.100) which requires the calculation of he|H 0 |gi. Here the Hamiltonian is
simply the dipole matrix operator dˆ defined in the previous section. To find the
relative transition amplitude we can simplify this matrix element by applying the
Theory
32
Wigner-Eckart theorem.
hγ 0 S 0 L0 I 0 J 0 F 0 MF0 |dq |γ S L I J F MF i = (−1)F −MF hγ 0 S 0 L0 I 0 J 0 F 0 ||d||γ S L I J F i
!
F0
1 F
×
.
−MF0 q MF
(2.127)
We now want to decouple the angular momentum components. This can be done
by either applying the Wigner-Eckart theorem twice and recombining or using the
relation [19],
hn0 I 0 J 0 F 0 ||d||n I J F i = (−1)J+1+F +I hn0 J 0 |d|n Ji
(
)
0
0
0
p
J
F
I
.
× (2F + 1)(2F 0 + 1)
F J 1
(2.128)
The bracketed term is the Wigner 6j symbol. We can apply this to Eq. (2.127) to
to reduce the |J I F i to |Ji and get
hγ 0 S 0 L0 I 0 J 0 F 0 ||d||γ S L I J F i = (−1)J+1+F +I hγ 0 S 0 L0 J 0 ||d||γ S L Ji
(
)
0
0
p
J
F
I
× (2F + 1)(2F 0 + 1)
.
F J 1
(2.129)
By applying this relation a second time we are left with
0
hγ 0 S 0 L0 I 0 J 0 |dq |γ S L I Ji = (−1)L+S +J+1 hγ 0 L0 |dq |γ Li
(
)
0
0
0
p
L
J
S
× (2J + 1)(2J 0 + 1)
.
J L 1
(2.130)
This leaves the only matrix element depend on the overall quantum number n and
the orbital quantum number L. Plugging the matrix elements back in we are left
with
0
hγ 0 S 0 L0 I 0 J 0 F 0 MF0 |dq |γ S L I J F MF i = (−1)2+2J+S +L+2F +I+MF hγ 0 L0 |dq |γ Li
p
p
× (2J + 1)(2J 0 + 1) (2F + 1)(2F 0 + 1)
(
)(
)
!
L0 J 0 S 0
J0 F 0 I
F0
1 F
.
J L 1
F J 1
−MF0 q MF
(2.131)
Theory
33
For both the D1 and D2 transitions all of the hyperfine levels have the same L
and n and so this term cancels when looking at relative amplitudes. We also can
say that S = S 0 and I = I 0 and so the transition rate is given by
(
×
L0
J
Wge ∝ (2J + 1)(2J 0 + 1)(2F + 1)(2F 0 + 1)
)2 (
)2
!2
J 0 S0
J0 F 0 I
F0
1 F
.
L 1
F J 1
−MF0 q MF
(2.132)
This equation can then be used to calculate the relative amplitudes of different
transitions. The atoms in this system behave according to the Boltzmann distribution and so all the group states will be evenly populated. This means that the
F states will not be equal because of the difference in degeneracies for the different states, to account for this there is an extra factor of (2F + 1) in Eq. (2.132)
which cancels the term from the denominator of Eq. (2.127). This equation is
trivial to solve using the built in 3j and 6j symbols in Mathmatica to find that
the normalized relative transition amplitudes for the D1 lines are shown in Table
2.2.
F
2
2
1
1
7
Li
6
Li 3/2
3/2
1/2
1/2
F0
2
1
2
1
Amplitude
.37
.37
.22
.05
3/2
1/2
3/2
1/2
.44
.36
.18
.02
Table 2.2: Relative amplitudes for transitions from ground state for D1 transitions.
These values align with the expectation that transitions with higher values of F
should have greater transition amplitudes. For more in-depth derivations and more
more information see Refs. [19] and [23].
2.6.1
AC Stark Effect
The Stark effect is the shift in the spectral lines of atoms in an electric field. When
a time varying electric field is applied to an atom the energy levels will be split
Theory
34
and shifted as the field polarizes the atom. This derivation will follow that of Ref.
[22] looking at the case of the electric field being near the resonance frequency of
the atoms. This will use the rotating wave approximation to find the energy level
splittings.
We will start with a two level atom with energy seperation ω0 . The atom is
subject to at time varying electric field given by ε0 sin(ωt) and the states are
coupled through the electric dipole operator
d~ = −hi|ε0 z|f i,
(2.133)
where z is the quantization axis. The Hamiltonian for this system is given by
H=
0
−dε sin(ωt)
−dε sin(ωm t)
~ω0
!
.
(2.134)
In order to find the shift due to the applied electric field we will look at the system
using the rotating wave approximation by making a unitary transformation [22] .
U=
1
0
0 e−iωm t
!
.
(2.135)
and so the Hamiltonian becomes
H 0 = U † HU.
(2.136)
The states of the atom are also transformed by
|ψ 0 i = U † |ψi.
(2.137)
In order to transform our standard time dependent Schrodinger Equation to this
basis we multiply both sides by U † and utilize the fact that U † U = 1 to transfer
bases.
∂
|ψi
∂t
∂
U † HU U † |ψi = iU † U U † |ψi
∂t
H|ψi = i
(2.138)
Theory
35
Now substitute H 0 = U † HU and |ψ 0 i = U † |ψi.
∂
U |ψ 0 i
∂t
∂ 0
0 0
† ∂U
0
H |ψ i = iU
|ψ i + U |ψ i
∂t
∂t
∂U
∂
H 0 − iU †
|ψ 0 i = i |ψ 0 i.
∂t
∂t
H 0 |ψ 0 i = iU †
(2.139)
From this we define the effective Hamiltonian of the system as
H̃ = H 0 − iU †
∂U
.
∂t
(2.140)
We can now can plug in Eq. (2.134) and (2.135) to find the effective Hamiltonian
in the rotating frame.
H̃ =
−dε
(1
2i
0
dε
(1
2i
− e2ωm t )
− e−2ωm t )
~ω0 − ~ωm
!
.
(2.141)
This Hamiltonian has two terms, an oscillatory component as well as static component. We will now make the approximation that the oscillating terms average
to zero leaving that Hamiltonian as
H̃ =
0
−dε
2i
dε
2i
~ω0 − ~ωm
!
.
(2.142)
Diagonalizing the Hamiltonian gives the shift for a two level system in an applied
electric field of ε/2 for the near resonance case. The shift for a time varying field
is given by [22]
X d2 2
ik
∆Ek =
.
Ek − Ei
i
(2.143)
For our system we have a energy difference of ~(w0 − wm ) and so our energy shift
is
∆E = ±
d 2 ε2
.
4~(ω0 − ωm )
(2.144)
Here the plus and minus correspond to the shift to the upper and lower states.
For our system we are assume the coupling is only occurring between the upper
Theory
36
states because the hyperfine splitting of the excited state is much less then the
ground state.
In order to estimate the AC stark shift we need to find the dipole matrix element
for each transition given. First we find the dipole element for each MF → MF0
given by
d~MF ,MF0 = h2 P1/2 F 0 Mf0 |d|2 S1/2 F MF i.
(2.145)
Using the Wigner-Eckart theorem outlined in the previous section, we can combine
Eq. (2.127) and Eq. (2.129) to find that
dMF ,MF0 = (−1)
f 0 −MF0
F0
1
F
!
−MF0 q MF
p
h2 P1/2 ||d||2 S1/2 i (2F + 1)(2F 0 + 1)
(
×
J0 F 0 I0
F
J
1
)
,
(2.146)
where the reduced matrix element h2 P1/2 ||d||2 S1/2 i = 3.33 ea0 and ea0 = 1.28
MHz
V/cm
[23]. Now we find the total d2 for the transition from F → F 0 by summing
overexcited states and then averaging over the ground states to find the total
dipole element. This is given by
F
X
1
d=
2F + 1 M =−F
F
"
0
F
X
f 0 −MF0
F0
1
F
!
(3.33ea0 )
−MF0 q MF
(
) #2
0
0
0
p
J
F
I
× (2F + 1)(2F 0 + 1)
F J 1
(−1)
MF0 =−F 0
(2.147)
We can now find an expected AC Stark Shift by plugging the dipole element into
Eq. (2.144) where ∆ is given as the hyperfine splitting of the upper states. The
expected stark shift is given by [24]
d 2 ε2
.
4∆
(2.148)
Theory
37
For the D1 transitions the expended Stark shift is given in Table 2.3. The shifts
for the 6 Li transitions are larger because the hyperfine splitting is ≈ 26 MHz
compared to ≈ 90 MHz for 7 Li.
Transition
6
Li
F = 1/2 → 1/2
F = 1/2 → 3/2
F = 3/2 → 1/2
F = 3/2 → 3/2
))
Expected Shift (kHz/( mW
cm2
-20.5
2.6
-10.3
10.3
7
Li
F =1→1
F =1→2
F =2→1
F =2→2
-5.5
1.1
-3.3
3.3
Table 2.3: Estimation of AC Stark shift for D1 transitions in 6 Li and 7 Li.
)) of intensity in for the incident laser
These estimates are kHz shift per ( mW
cm2
light.
2.7
Doppler Shift
As an atomic beam travels through a vacuum it diverges and we see a spread in
the velocity classes caused by different Doppler shifts. This cumulative effect is
referred to a doppler broadening. The frequency shift due to the Doppler effect is
given by
f = f0 (1 +
k̂ · ~v
)
c
(2.149)
where f is the frequency experienced by the atom, f0 is the frequency in the
rest frame, and k̂ · ~v is component of the atom’s velocity along the direction of
propagation of the light. In the limit that all of the atoms are propagating directly
perpendicular the laser this this would just be f0 . In the case of a vapor cell there
is a distribution of velocities both toward and away from the probe laser, this
results in a net effect of broadening the lines. If we take Pv (v)dv to be the fraction
of particles with a speed between v and v + dv the distribution in frequencies is
Pf (f )df = Pv (vf )
dv
df
df
(2.150)
Theory
38
(a) Probe laser on resonance with transition.
(b) Probe laser detuned from resonance.
Figure 2.3: As the probe laser is swept of the resonance of the atoms, if
the laser is slightly detuned from resonance two velocity classes will be excited
creating a broadening in line shape as more atoms are excited.
where v can be found through Eq. (2.149). For a lithium vapor cell where there
is equal chance of velocities from either direction the broadened line with is given
by the Maxwell distribution shown by
r
Pv (v)dv =
m − mv2
e 2kT dv,
2πkT
(2.151)
where m is the mass of the particle, T is the temperature, and k is Boltzmann’s
constant. Using the full width at half maximum as given by the width of a 700 K
lithium vapor would be
r
fF W HM =
8kT ln2
f0 ≈ 320 MHz.
mc2
(2.152)
For an atomic beam source the width is dependent on the geometry of the beam.
First we can find the vrms of the beam given by
r
vrms =
3kT
.
M
(2.153)
From this we can find the the transverse velocity using Eq. 2.154 which can then
be plugged into Eq. 2.149 to get the line width.
r
vT = v sin θ =
l
r
3kT
M
(2.154)
Frequency Combs
39
We tried various nozzles designed to give narrow line widths. The set up that is
currently used is a simple one hole nozzle with a collimation plate directly before
the interaction region. Without the collimation plate we observed line widths of
120 MHz. With the collimation plate the line width is 16 MHz as calculated by
finding the full width at half max of the spectral peaks, see Chapter 5 for detailed
analysis and comparison of different nozzles.
Chapter 3
Frequency Comb
This experiment uses an optical frequency comb as an optical frequency reference
to stabilize a probe laser used to excite the atomic transitions. Optical frequency
combs are light sources that consist of a series of equally spaced discrete frequencies. The frequency comb used in this experiment is generated by a mode locked
Ti:Sapphire oscillator which produces a series of pulses with a pulse duration of
tens of femtoseconds. The pulses occur at a repetition rate of ∼ 1 ns.
The output spectrum of the Ti:Sapphire oscillator is broadened using a microstructure fiber such that it spans an optical octave. This is necessary for the self
referencing and stabilization of the offset frequency as described below. The microstructure fiber is the key element to making frequency combs possible as it
allows the comb to serve as an optical reference for light in the visible spectrum
and in doing so allows for self referential stabilization. While there are some frequency combs which are octave spanning without the use of a microstructure fiber
they are more difficult to construct and maintain [25].
The frequency comb is locked to a GPS steered atomic clock that allows measurements of optical frequencies with a precision of a part in 1012 . This precision in a
relatively small system make high precision frequency measurements possible for
smaller labs. The frequency comb has become a valuable tool as a frequency standard for many different applications. Stabilized optical frequency combs were first
developed for spectroscopy in 1999 by Jones et al. [26] and have become widely
used in various fields.
40
Frequency Combs
3.1
41
Overview
The optical frequency comb is a laser source whose spectral output consists of
≈ 100, 000 evenly spaced discrete frequencies. This structure can be thought
of as as set of evenly spaced teeth. To develop this picture consider a optical
resonator with length L with pulse propagating through the cavity described by
an envelope function A(t). If we look at one pulse with carrier frequency fc
propagating through the cavity it is easy to see that A(t) = A(t + T ) where T
is found from the cavity group velocity T = 2L/vg . From this we can define our
pulse repetition time as frep = 1/T . The cavity can support light with wavelengths
of 2L/n where n is an integer. These modes correspond to frequencies that are
multiples of the repetition rate. If the waves have the proper phase relationship
this series of harmonic waves forms a pulse, as more frequencies are added the
pulse narrows this is shown in Fig. 3.1.
Ι
Ι
Ι
Ι
Pulse Formation
Spectrum of lasing light
ν
ν
ν
ν
Figure 3.1: A pulse of light in the time domain can be view in the frequency
domain as series of discrete frequencies. As more frequencies are added the
pulse becomes shorter and so ultrafast pulsed lasers contain a wide range of
frequencies.
Frequency Combs
42
Now looking at the frequency space picture of A(t) we see a spectrum of comb
modes separated by the repetition rate. This is shown in Fig. 1.3. As the pulse
propagates through the cavity the dispersion of the cavity causes a phase difference
between the phase and group velocities of the pulse. This results in a constant
phase shift between the carrier and envelope waves in successive pulses. This
constant phase shift ∆φ between pulses is equivalent to a overall offset in frequency
space. This is know as the offset frequency where f0 < frep , given by
f0 =
∆φ
frep .
2π
(3.1)
Now we are able to relate the frequency of any comb mode to frep and f0
fn = nfrep + f0 ,
(3.2)
this is know as the comb equation as described in Ref [26]. The phase slip is
difficult to experimentally detect so the offset frequency is detected directly. This
is most easily accomplished if the frequency comb spans an optical octave. If we
frequency double light given by fn = nfrep + f0 and compare it to light of twice
the frequency given by f2n = 2nfrep + f0 see that
2fn = 2nfrep + 2f0
−2fn = 2nfrep +f0
(3.3)
f0 .
This means we can directly measure the offset frequency of the frequency comb
by referencing two optical frequencies within the frequency comb separated by an
optical octave. The frequency comb in this experiment uses a Ti:Sapphire laser
and has ∼ 4 × 105 modes with a repetition rate of ≈ 920 MHz and an offset
frequency ≈ 300 MHz.
3.2
Dispersion
Dispersion is the optical phenomenon by which the index of refraction of all materials in dependent on frequency. Since the group velocity of light with a given
frequency is dependent on the index of refraction of the medium that means that
Frequency Combs
43
a packet of light with a range of frequencies will spread out in time due to a difference in optical path length due to a change in the index of refraction. Consider
the propagation constant defined as
k=
2πf n(ω)
2π
=
.
λ
c
(3.4)
Since the index of refraction is frequency dependents we will Taylor expand this
around ω0 . This gives
dk
1 d2 k
k = k(ω0 ) +
(ω0 − ω) +
(w0 − w)2 .
2
dω
2 dω
(3.5)
Here the first order term is related to the group velocity, vg ,
1
dk
= .
dω
vg
(3.6)
The second order term is known as the group velocity dispersion (GVD), and
describes how the various frequencies spread apart as a pulse propagates through
a medium,
d2 k
1 dvg
=− 2
.
2
dω
vg dw
(3.7)
In a normal medium this means that the pulses will spread out in time as each
frequency component experiences a different index of refraction. In most mediums this causes higher frequency light to travel slower then lower frequency. For
frequency combs it is necessary to have negative GVD in the cavity in order to
correct for the positive GVD caused by the gain medium. This is typically done
using chirped mirrors which have longer optical paths for lower frequencies of light
and so serve to maintain the phase relationship between different frequencies of
light.
Frequency Combs
3.3
44
Kerr Lens Modulation
The optical frequency comb relies on self lensing through the optical Kerr effect
to generate ultrafast pulses of light. The optical Kerr effect relies on the nonlinear
response of an a medium to an incident electromagnetic wave. This is given by
n = n0 + n2 I.
(3.8)
For 800 nm in a Ti:Sapphire crystal n0 = 1.76 and n2 ∼ 3 × 10−16 m2 /W [27].
Since the n2 term is negligible at anything but very high powers this term only
becomes important for pulsed operation as the peak intensity in pulsed operation
is much higher than cw, even though the average power is about the same. In cw
operation, the frequency comb has a intracavity power of ∼ 250 W, and a single
pulse in mode locked operation has a peak power of ∼ 25 MW. Thus the effect
will only be important for pulsed operation.
Figure 3.2: Hard aperture Kerr lensing. The CW and pulsed beam both enter
the medium with the same gaussian profile, since the peak intensity of the pulsed
light is much higher there nonlinear index of refraction in the medium focuses
the light through the aperture. The CW beam has lower peak intensity and so
is not effected. This effect makes the cavity favorable for pulsed operation as
more light is passed. [28]
The optical Kerr effect relies on the Gaussian profile of the beam to create a
variation in the index of refraction as a function of distance from the center of the
beam width. The index of refraction will be greatest at the center of the beam and
decrease at the tails. This serves to refocus the light when in pulsed operation. In
order to achieve pulsed operation the cavity is set up such that pulsed operation
is preferred. This can be done through either a soft or hard aperture.
Frequency Combs
45
The first designs for a mode locked laser used a hard aperture which cut off part
of the beam to make pulsed operation preferable by increasing the losses to cw
operation. This is shown in Fig. 3.2. Soft apertures use a change in the overlap
between the cavity mode and the pump beam in the gain medium which also favors
pulsed operation by increasing the overlap in pulsed operation.
3.4
3.4.1
Oscillator Cavity
Pump Laser
The pump laser used for this experiment is a Nd:YVO4 (neodymium yttrium
orthovandate) Verdi G8 manufactured by Coherent. This laser outputs 532 nm
light with a maximum power output of 8 W. Typically it is run around 5 W. The
pump beam passes through an acousto-optic modulator (AOM) and a half wave
plate before entering the oscillator cavity. The half wave plater rotates the incident
light going into the oscillator cavity from the vertically polarized light from the
pump. The AOM is used for stabilization of the offset frequency described below.
3.4.2
Oscillator Cavity
The resonator is a bowtie configuration cavity with a path length of ∼ 30 cm.
This is housed in a 10 × 20 × 30 box which is watercooled in the baseplate as well
as crystal holder to improve temperature stability. The cavity length dictates a
repetition rate of ≈ 1 GHz. The bowtie configuration is used because it allows
tighter focusing of the beam to increase the Kerr-Lensing effect. It also is compact
and so can be used for higher repetition rate cavities.
Frequency combs with repetition rates of 100-200 MHz are commercially available,
less common are frequency combs with repetition rates of ≈ 1 GHz. While higher
repetition rate combs have the advantage of higher power output per mode, the
pulse energy must decrease by 1/10, assuming the average power through the
cavity is constant. This means that the steady state pulse duration is 10 times
longer. This results in a overall pulse intensity which is 100 times less then the
100 MHz system. This makes Kerr lensing more difficult as the nonlinearity in
the crystal is depended on peak intensities. There are two ways this has been
Frequency Combs
46
M3
OC
lens
Pump
M1
M2
Figure 3.3: The oscillator cavity used is a bowtie configuration. This configuration allows for a high repetition rate because it can be made compact and
the repetition rate is based on the length of the cavity.
compensated for in this laser. The output coupler is 98% reflective as opposed
to 90% used in most 100 MHz systems. The dispersion of the cavity mirrors is
negative such that the light maintain ultra short pulses. This helps to increase
max intensity even with lower Ep .
Figure 3.4: Chirped Mirrors are used to compensate for the dispersion light
experiences when traveling through a medium. They are normally specially
coated dielectric mirrors which provide longer optical paths for lower frequencies
of light and so compensate for the increased group velocity. These mirrors are
used to maintain the phase relationship between multiple frequency components
of pulsed lasers.
Frequency Combs
47
In order to achieve negative GDD the cavity is tuned using negative dispersive
mirrors. These reflect different wavelengths of light from different layers of the
mirror in order to refocus the light. Longer wavelengths are reflected deeper in
the coating of the mirror. This is illustrated in Fig. 3.4.
The crystal is 2 mm × 3 mm × 1.5 mm and is placed at Breuster’s angle between
M1 and M2. Mirrors M1 and M2 are GDD oscillation compensated and have a
GDD of -50 fs2 ± 20fs2 and a radius of curvature of 3.0 cm. Mirror M3 is flat and
a GDD of -40 fs2 The output coupler is 98 percent reflective.
Ti:Sapphire
Crystal
Crystal
Holder
Cooling
Water
X-Y Translation
Stage
Figure 3.5: The crystal mount for the Ti:Sapphire oscillator is shown. The
mount is made of copper and has water channels cut through the center section
for cooling. This is mounted to an X-Y translation stage used to center the
crystal in the beam wast of the cavity. This is important as the beam is the
most focused at the beam wast and so the intensity is higher increasing the
nonlinear effects in the crystal.
The crystal is attached to a copper holder which is mounted on an x-y translation
stage to allow the crystal to be positioned at the waist of the resonator shown in
Fig. 3.5. There are channels cut in the copper block to allow cooling water to
circulate.
Frequency Combs
3.4.3
48
Lasing and Modelocking
In order to achieve lasing and mode locking of the oscillator, several aspects of the
cavity can be adjusted. The cavity is first alined to achieve cw lasing by altering
the angles of M1 and M2 as well as the crystal position. Mirror M2 is the then
adjusted using a translation stage such that the cavity is more stabile in mode
locked operation. The cavity then requires an external perturbation to interrupt
the beam in the cavity to allow the build up of pulses. This is done by taping on
M2 with the back of a screwdriver.
Once mode locked, pulses propagate in only one direction through the cavity. This
is the easiest way to tell if the laser is mode locked. The properties of the laser can
be carefully tuned by adjusting the crystal position and M2. The output power in
mode locked operation is ∼ 500 mW and ∼ 600 mW in continuos wave operation.
The output spectrum of the oscillator cavity is ∼ 50 nm centered at 790 nm.
This output does not span an optical octave which is necessarily for stabilization,
explained in section 3.5, so we use a microstructure fiber to broaden the output
of the oscillator to an optical octave.
Figure 3.6: Silicon microstructure fiber. The microstructure fiber spectrally
broadens the output of the oscillator cavity such that it spans an optical octave.
Picture taken from Ref. [29].
Frequency Combs
3.4.4
49
Microstructure Fiber
In order to stabilize the offset frequency of the frequency comb, we need it to be
self referential. This means we need to be able to compare comb light separated
by an optical octave such that light can be double and compared against itself.
The output spectrum of the comb does not span an optical octave, so to achieve
this we broaden the spectrum using a air-silica microstructure fiber.
In conventional optical fibers light is transmitted using total internal refraction.
Here the index of refraction of the core of the fiber is higher then that of the
cladding and so the light is guided through the fiber with minimal loss. Microstructure fibers use optical waveguides to transmit the light. By manipulating
the geometrical dispersion of the waveguide the material dispersion of the fiber
can be cancelled. These fibers are made up of a series of air holes in a solid core,
the arrangement of these holes can be used to tune the fiber for different purposes
an example of which is shown in Fig. 3.6. The fiber is designed to have zero GVD
as well as broadening, both of which are extremely important for our application.
Figure 3.7: The light from the Ti:Sapphire oscillator is focused into the fiber
using a microscope objective. The broadened light from the output of the fiber
is shown, on the far left there is blue light present but it is difficult to see because
of the strength of the green light.
More discussion of the development of microstructure fibers can be found in Ref.
[29] as well as detailed discussion of the particular optics setup used in this experiment in Ref. [28] and [30]. The output of the fiber spans the visible spectrum as
shown in Fig. 3.7.
Frequency Combs
3.5
50
Stabilization
The optics post cavity serve to broaden and stabilize the comb from the output of
the oscillator. The following section outlines the optics and electronics setup for
the stabilization of f0 and frep .
3.5.1
Repetition Rate
The repetition rate of the frequency comb is dependent only of the optical path
length of the cavity. Thus it can be stabilized by tuning the length of the cavity.
This is achieved using a feedback loop controlling a piezo ceramic on M3 of the
resonator cavity. Broadened laser light is picked off from the main comb beam
and sent onto a photodetector (Electro-Optics Technology ET-2030A) in order to
detect the repetition rate. The repetition rate as well as integer multiples of frep
is present in the comb light from interference of neighbor comb modes yielding a
difference frequency of frep as well as higher harmonics. This signal is then sent
through a bandpass filter to remove the higher harmonics (2frep , 3frep , ...) and then
sent to a mixer where it is mixed down with the signal from an radio frequency
generator. This generator is referenced to the atomic clock and allows the control
and tuning of the repetition rate. The output of the mixer produces an error signal
for the loop filter which can then output a correction to the high voltage amplifier
to tune the cavity length by tuning the piezo. In this application the piezo ceramic
used is a Noliac CMAR03 PZT. This then stabilizes frep at the frequency dictated
by the synthesizer. This setup is illustrated in Fig. 3.8.
This set up has yielded a consistent and stable lock which can easily stay locked
for several hours. This is the most consistent of all locks.
3.5.2
Offset Frequency
The stabilization of the offset frequency is a more complicated setup and is not as
stable as the frep lock. Unlike frep , f0 is not inherently accessible in the comb light.
In order to measure f0 we frequency double a lower frequency ν1 = n1 frep + f0 and
then beat it against a higher frequency ν2 = n2 frep + f0 where n2 = 2n1 . When
these frequencies are heterodyned together we are left with a difference frequency
Frequency Combs
51
Piezo
Ceramic
Photodetector
High Voltage
Amplifier
M3
Pump
Loop
Filter
OC
lens
M1
M2
Bandpass
Filter
Mixer
Splitter
Divide by 8
Counter
rf Signal
Generator
Figure 3.8: Diagram of frep stabilization. Light from the frequency comb is
detected using a photodetector. This signal is then amplified and filtered to
remove higher order harmonics. The signal is mixed with the output of an rf
synthesizer to get a error signal. This is sent to the loop filter used to tune the
feedback to the comb cavity. A piezo ceramic is mounted to M3 in the cavity
and can be tuned to provide small changes in cavity length and so change the
repetition rate.
of f0 . For this method to work the comb must span at least an optical octave to
have two frequencies to compare.
The comb in this experiment does not span an optical octave without the use
of a microstructure fiber. After the fiber the comb light spans a range of 530
nm (green) to 1060 nm (infrared) giving an optical octave. The 1060 nm light is
frequency doubled using a nonlinear periodically poled lithium niobate (PPLN)
crystal and then beat against the original 530 nm light. The crystal is designed
to double light at ∼ 1064 nm.
Dispersion caused by the various elements causes a relative time delay between the
two pulses. To compensate for this delay an interferometer is used to alter the path
length of one pulse. The beam is split using a mirror that reflects light between
420 nm and 630 nm and transmits frequencies between 750 nm and 1200nm. The
530 nm and 1060 nm light are seperated and the phase offset can be corrected
by adjusting the path length of the 530 nm light. The two pulses of original 530
Frequency Combs
52
nm light and doubled 1060 nm light are heterodyned onto a photodetector for the
beat note signal.
The signal from the detector first passes through a tunable bandpass filter in order
to remove some of the other frequencies present in the comb light. This is then
mixed up to 1240 MHz with the rf synthesizer and then sent through a frequency
divide by eight to give 155 MHz. This is then mixed with a synthesizer at 155
MHz in the digital phase detector to get our error signal. The process of mixing
up to 1240 MHz and then dividing by eight is used to help increase the capture
range of lock as deviation from the lock frequency will appear as a smaller fraction
of the phase of the error signal. The error signal goes through a loop filter that is
used to tune the gain response of the feedback.
The offset frequency can be tuned by changing the pump power going into the
Ti:Sapphire crystal because of the non-linear gain in the crystal. There are two
methods for tuning this input power. The first is changing the output power
of the pump beam. This is normally in the range 5 to 6 W. The second is an
acousto-optical modulator (AOM) placed in the pump beam before the cavity.
Acousto-optic modulators use radio frequency waves to vibrate a piezo ceramic
attached to a quartz crystal, producing sound waves in the glass. These act as
density waves in the glass creating regions of expansion and contraction and so
varying the index of refraction in the glass. The incoming light scatters off periodically modulated index of refraction into multiple orders of beams. The diffracted
light emerges at an angle θ dependent on the wavelength of the incident light,
λ, and the radio frequency, fm , as shown in Eq. 3.9. This diffracted light does
not couple into the resonator cavity and so by tuning the power in each order
the power to the resonator can be altered. The intensity of light in each order is
proportional to the applied rf voltage.
B(θ) =
mλ
2fm
(3.9)
The advantage of this setup is that AOM’s are much faster then any mechanical
feedback. We use a Interaction Corp. Acoustic-Optic modulator model AOM405AF1 in this experiment. The pulse rise time and depth of modulation for this
Experimental Setup
53
AOM with acoustic velocity V = 3.63 mm/µsec are given by
Tr = 177D nsec
(3.10)
and
M = e−.0936D
2f 2
m
,
(3.11)
where D is the beam size in mm and fm is the modulated frequency in MHz.
Experimental Setup
54
Pump Laser
AOM
M3
OC
lens
Pump
M1
rf Signal
Generator
155 MHz
AOM Driver
Loop
Filter
Cavity Filter
M2
Digital Phase
Detector
Divide by 8
Photodetector
Bandpass
Filter
Mixer
Splitter
Counter
rf Signal
Generator
1240MHz - fo
Figure 3.9: Diagram of f0 stabilization. The frequency doubled light and non
doubled light from the frequency comb are combined is detected with a photodetector. The beat note between these two frequencies is the offset frequency.
This signal is mixed up to 1240 MHz with the output of a rf synthesizer. This is
then frequency divided by 8 such that the signal is at 155 MHz. This is mixed
with the output of a second synthesizer at 155 MHz to get an error signal. This
is sent to a loop filter used to control the feedback to the AOM which modulates
the pump beam to the oscillator and controls the offset frequency.
Chapter 4
Experimental Setup
Figure 4.1: Overview of the experimental setup for this measurment. The
extended cavity diode laser is frequency stabilized to the optical frequency comb.
This allows the precise measurement of the frequency of the diode laser that is
used to excite the lithium atoms.
The main components of this experiment are the atomic beam, frequency comb
used as the frequency standard, and extended cavity diode laser for probing the
atoms. The lithium oven is used to create the atomic beam that propagates
through the vacuum chamber and intersects with probe laser. The frequency of the
probe laser is tuned over the transitions we are interested in measuring and then
the fluorescence from the atoms decay is recorded using a photo multiplier tube
55
Experimental Setup
56
(PMT). The probe laser is locked to the optical frequency comb which provides a
precise measurement of the optical frequency of the diode laser.
4.1
Diode laser
The extended cavity diode laser has become an extremely useful tool for atomic
and optical physics [31]. Precision atomic spectroscopy requires the ability to
excite single transitions with a tunable source. Diodes are available at many
different frequencies and so have made many transitions easily accessible. While
the frequency comb provides an absolute frequency standard, the power per comb
mode is low, making spectroscopy using the comb light directly difficult. The
diode laser provides a reasonably powerful probe beam that can then be stabilized
to the frequency comb.
Output
Mirror
Zero
Order
Diffraction
Grating
Tuning
Mirror
Zero
Order
First Order
Diode
Controller
First Order
Collimation
Lens
(a) Littrow Configuration
Diode
Controller
Diffraction
Grating
Collimation
Lens
(b) Littman-Metcaff Configuration
Figure 4.2: There widely used configurations for ECDL’s are shown. Both
configurations use a diffraction grating for feedback. The two configurations
differ in that the cavity length is tuned by displacing diffraction grating in the
Littrow configuration and a tuning mirror for the Littman-Metcaff in order to
scan the frequency with small adjustments in cavity length. This experiment
uses the Littrow configuration.
4.1.1
Diode Laser Theory
For precision spectroscopy, the diode laser must be single mode such that it outputs
a single frequency. Ideally this laser width is as narrow as possible so that the
laser line width does not broaden the spectral peak width. It is also required that
this frequency be tunable over a range sufficient to scan over the transitions.
Experimental Setup
57
Laser diodes are electrically pumped laser which uses a p-n junction of a semiconductor diode as the gain medium. Normally laser diodes has two reflective
surfaces that serve as the resonating cavity for the laser, by coating one surface of
the diode with an antireflective coating the outer surface provides no feedback and
so an external cavity is used to provide feedback. This results in a frequency tunable cavity that can be scanned over a large range of frequencies. This means that
without the external feedback the diode will not lase or have a very low output
power as the power within the cavity is much lower.
The Littrow configuration shown in Fig. 4.2a, uses a diffraction grating as the
frequency tuning element. In this configuration the first order light is reflected
back into the diode to provide optical feedback. By changing the angle of the
grating the laser can be tuned in frequency over a range of ∼ 10nm. For finer
adjustment the grating is displaced using a piezo ceramic to proved small changes
in the cavity length. The zero order is reflected off the grating as the output. One
of the limitations of this configuration is that in changing the angle of the grating
to change the frequency the alignment of the output beam is also slightly moved.
The second common configuration is the Littman-Metcaff configuration shown in
Fig. 4.2b. In this configuration the output of the diode is reflected off the grating
such that the first order light goes to tuning mirror and then light is reflected back
off the grating into the diode as optical feedback. In this configuration the grating
remains stationary and so the output alignment does not change as the beam is
scanned. The mirror position is displaced to tune the cavity length and so tune
the frequency. The limitation with this setup is the output efficiency is lower then
the Littrow configuration and it is a more complicated system.
4.1.2
Diode Laser Experimental Setup
The ECDL used in this experiment was constructed using a AR-coated diode
from Eagle Yard with a wavelength of 670nm and a specified maximum output of
18mW at 25◦ C however this is dependent on configuration. This is mounted in
a ThorLabs LDM21 TEC-cooled laser diode mount. The housing of the LDM21
has been replaced with a heavier brass shell to improve mechanical stability. The
temperature and current are controlled using a ThorLabs TED200C temperature
controller and LCD205C current controller. A lens with x-y translation stages with
z adjustment is attached to the brass housing and then an 1800 groove/mm ruled
Experimental Setup
58
Diode
Mount
Collimation Lens on
Translation Stage
Diffraction
Grating
Figure 4.3: The diode laser used in this experiment is shown. The components
are mounted to a heavy steal box which is isolated from the optics table with a
rubber pad. This is to reduce vibrations and so increase stability.
grating with 500 nm blaze is used for feedback. Here z is the axis of propagation for
the light, x is horizontal, and y is vertical. This grating was decided experimentally
as it produced higher output power than other gratings that were tried. This is
mounted to a piezo ceramic on a stainless steel disc. The disc is mounted in a
Polaris Low Drift Kinematic Mirror Mount. When a voltage is applied to the piezo
it changes the length of the cavity and so allows for tuning over a frequency range
of ∼ 8 GHz.
The diffraction grating provides frequency tunable feedback to the diode laser by
specially separating light based on wavelength. By tuning the angle at which the
light is sent back into the diode cavity the frequency can be tuned. The spacing
of the grooves determines the frequency spread as well as the amount of power
reflected into the 1st order which is used for feedback to the cavity. The grating
used in this experiment has an efficiency of 70%.
This configuration gives a maximum output power of 27 mW as shown in Fig.
Experimental Setup
59
25
Power (mW)
20
15
10
5
0
80
100
120
Current (mA)
140
Figure 4.4: The output power for the ECDL as a function of applied current
at 19◦ C is shown. This is using a 1800nm grating for external feedback. Normal
operating current is between 130 and 140mA.
4.4 however when running we typically operate with an output power of ∼ 20mW
at 671nm. A combination of temperature tuning and grading feedback is used to
bring the diode laser to the desired frequency. In order to increase the wavelength
the temperature is raised which results in lower output power. This laser normally
was run between 19◦ and 20◦ C.
The overall stability of the diode laser is very important for data taking. Both
over short times scales to be able to collect single runs of data but also over the
course of a day. The cavity has been made as small as possible in order to improve
stability. The collimation lense mount has been changed from a x-y-z stage to a
x-y stage in order to make the cavity smaller. This setup shown in Fig. 4.3.
The configuration of this diode laser has been upgraded in order to improve tuning
and stabilty. We originally used diodes that we had AR coated which output 1215mW. These diodes had more problems then the current setup with both stability
and beam profile. This led to the decision to replace the homemade AR coated
diodes with the commercially coated Eagle Yards diodes. We have also improved
the mounts in the cavity to decrease mechanical vibration and so increase stability.
This has led to an increase in mode hope free tuning range from ∼ 2 GHz to ∼8
as well as increased output power.
Experimental Setup
60
Beam Splitting
Cube
1/2 Wave Plate
EOM
ECDL
Isolators
30dB
Beam Splitting
Cube
Comb Light
Polarizer
Photodetector
Fiber
Coupler
To Fabry Perot
Figure 4.5: The controlling optics for the ECDL are shown. The output of
the ECDL is first sent through a pair of optical isolators in order to prevent
reflections from other optical elements entering the cavity which changes the
feedback. A polarization dependent beam splitter is used to divide the beam
between the beat note setup for stabilization and the interaction region. A
wave plate is used to tune the power to both arms. The EOM in the interaction
arm is used to stabilize the power to the chamber. The other arm is combined
with light from the frequency comb in order to provide a beam note for the
stabilization circuit.
4.1.3
Frequency Stabilization
The beat note lock precisely controls the frequency of the diode laser by locking it
to one mode of the comb. This is done by spatially overlapping the comb and diode
light by combining them with a polarizing beam splitting cube. The ratio of the
two beams can be adjusted using a polarizer after the cube to select a combination
of the two beams. This combined light is then measured on a photodetector.
Like the offset frequency lock, this signal is sent through a lowpass filter to remove
unwanted frequencies from the signal. The lock frequency is chosen to be 200 MHz.
It is then amplified and then mixed up with a 760 MHz signal from a synthesizer
to 960 MHz. This signal is then sent through a tunable bandpass filter to further
Experimental Setup
61
Output
Mirror
Current
Controller
Zero
Order
Diffraction
Grating
Diode
Controller
First Order
Proportional
Gain
Collimation
Lens
rf Signal
Generator
60 MHz
High Voltage
Amplifier
Digital Phase
Detector
Loop
Filter
Divide by 16
Photodetector
Counter
Splitter
Lowpass
Filter
Mixer
Bandpass
Filter
rf Signal
Generator
760 MHz
Figure 4.6: The beat not of the heterodyned comb and diode light is detected
using an amplified photodiode. This is then filtered and amplified before being
mixed up to 960 MHz with the output of the synthesizer. This is sent through
a divide by 16 before being sent to the feedback control. This is to increase the
capture range as the effect of noise frequency noise on the diode will decreased
through the mix up and division. This is sent through the controllable gain
feedback which tunes the diode with both proportional gain current feedback
and tunable PID gain from the loop filter which controls the grating.
filter the signal and then the frequency is divided by sixteen. This again serves
to increase the capture range of the lock. Here we use a divide by 16 instead of
a divide by 8 as the frequency noise in the beat note is larger and so by dividing
by 16 we increase the capture range which leads to a better lock. The signal
is then mixed down with another synthesizer at 60 MHz to get our error signal.
The error signal is then split and sent to the loop filter which controls the low
frequency feedback as well as a current feedback for higher frequencies. The loop
Experimental Setup
62
filter outputs to a high voltage amplifier which controls the piezo attached to the
grating.
This system stabilizes the frequency of the diode to one comb mode. To scan
the frequency of the diode the repetition rate of the comb is scanned by changing
the frequency of the synthesizer used in the repetition rate lock, this changes the
frequency of the comb mode and so the diode.
4.1.4
Fabry-Perot Interferometer
Fabry-Perot interferometers have many uses in both spectroscopy and other fields.
A Fabry-Perot cavity in its simplest form is made up of a pair of parallel mirrors
separated by a distance L. Coherent light will be transmitted through the cavity
if its frequency is such that standing waves can be maintained inside the cavity.
When the frequency of the light is not an integer multiple of the free spectral range
of the cavity the light will destructively interfere and so not be transmitted.
The power transmission of a Fabry-Perot cavity can be calculated using the phase
difference acquired during the light traveling one cycle through the cavity. This
calculation is performed in Ref. [32] and yields the Airy function
T =
1+
[4r2 /(1
1
,
− r2 )2 ] sin2 (δ/2)
(4.1)
where δ = 2kd is the phase shift acquired through one round trip through the
cavity. The square bracketed term in Eq. 4.1 is called the coefficient of fitness, F
which is dependent only on the reflectivity of the mirrors r.
F =
4r2
,
(1 − r2 )2
(4.2)
As r varies from 0 to 1, F varies from 0 to infinity. Thus the transmission through
the cavity given in Eq. 4.1 has a maximum when sin(δ/2) = 0 and min when
sin(δ/2) = ±1. Thus the cavity will have a maximum when δ = m2π. For a
planar cavity the wavelengths at which the cavity has maximum transmission are
given by
2L = mλ.
(4.3)
Experimental Setup
63
In frequency space the separation of the transmission peaks is known as the free
spectral range given by
νF SR =
c
.
2L
(4.4)
Note that this means that the peak spacing is only a function of cavity length
and so this allows Fabre-Perot cavities to act as an indicator or laser frequencies.
Equation 4.4 assumes a planar cavity, this experiment uses a confocal cavity that
has two curved mirror where the radius of curvature of the mirrors is equal to the
cavity length.The free spectral range given by
νF SR =
c
.
4L
(4.5)
The additional factor of 1/2 is because the round trip path of the light in a confocal
cavity is twice as long as that of a planar cavity, shown in Fig. 4.7. The advantage
of a confocal cavity is that for a given path length the confocal cavity is smaller
and so more stabile. Similarly for a given cavity length a confocal cavity has twice
the resolution of a planar cavity.
Path length = 4L
Path length = 2L
L
L
Figure 4.7: Confocal and planar Fabry-Perot Cavities. The round trip path
length for the confocal cavity is twice that of the planar cavity.
The cavity in this experiment is constructed with a tube of length of 30 cm. The
mirrors have radii of curvature equal to the length of the cavity. This gives a free
spectral range of 250 MHz. Figure 4.8 shows the transmission of the Fabry-Perot
cavity as a function of relative frequency. The spikes in transmission occur when
the laser is a integer multiple of the free spectral range of the cavity.
Fabry-Perot cavities are useful tools for characterization of the diode and its ability
to scan over a frequency range. We cannot use it to give absolute frequency
measurements of the diode laser but it is a good measure on overall stability
Fabry-Perot Transmission (V)
Experimental Setup
64
1.2
1.0
0.8
0.6
0.4
0.2
0
2
4
Frequency (GHz)
6
8
Figure 4.8: The transmission through the Fabry-Perot cavity is shown vs
relative laser frequency. Each peak is separated by 250 MHz, the free spectral
range of the cavity. This shows that the ECDL is single mode over an ∼ 8 GHz
tuning range.
and scanning range. If the peaks are clean with no noise it is an indicator that
the laser is single mode. If the peak spacing changes that implies that there is
nonlinearity in the piezo response. When characterizing the diode laser’s stability
and tuning characteristics, the piezo attached to the grating on the diode laser
is scanned with a 100 Hz triangle wave which modulates the frequency of the
diode laser. Transmission through the Fabry-Perot cavity is then recorded with
a photodetector. This has shown that we can get 8 GHz mode hope free tuning
with the diode laser as shown in Fig. 4.8. This is a useful tool for insuring that
the alignment of the ECDL cavity is robust over a large range. This is important
for data collection stabilized to the comb, if the diode laser is not stabile then it
will not be able to scan the transitions.
4.1.5
Optical Spectrum Analyser
In order to set the ECDL to the appropriate wavelength we use an optical spectrum
analyzer to calibrate the laser on a coarse scale. Diode laser light is split from the
main beam and coupled into the OSA before entering the vacuum chamber. This
analyzer is only accurate to ≈ .1 nm and exhibits a systematic offset. We normally
observe transitions at 671.13 nm when the true wavelength is 670.94 nm. While
Experimental Setup
65
this does not give a precise method of measuring the frequency it is a useful tool
for rough calibrations that would be difficult otherwise.
4.1.6
Power Stabilization
When scanning the frequency of the diode over the transitions there are power
variations as the feedback and alignment changes. For any precision spectroscopy
it is important to have consistent power incident on the atoms as the number of
atoms stimulated is depended on the power. In order to solve this problem we use
an electro-optic modulator to stabilize the beam power prior to the interaction
region. Using a EOM to control the power is a better option then direct feedback
to the laser as the EOM has no direct impact on the frequency stabilization of the
laser while modulating the power with current would.
Power Transmitted (mW)
2.5
2.0
1.5
1.0
-1.5
-1.0
-0.5
0.0
Monitor Voltage (V)
0.5
1.0
Figure 4.9: Response of Electro-optic modulator to applied electric field. The
transmitted power is shown which follows a sin curve with applied voltage. The
power can be cut to ∼ 30% of the maximum power.
When locking the power signal we lock somewhere in the middle of EOM range
shown in Fig. 4.9 using the offset on the high voltage amplifier, this way if the
power pulls we have a fairly large stabilization range. The power signal is picked
off from the probe laser directly before entering the chamber as shown in Fig. 4.10.
The signal from the photodetector is sent to a loop filter where is is combined with
a bias voltage. The bias is set to the negative of the DC voltage level of Fig. 4.9,
Experimental Setup
66
this gives an error signal of around zero volts. The output of the loop filter is sent
to a high voltage amplifier which goes to the EOM.
To Interaction
Region
Fiber
Coupler
Fiber
Coupler to
OSA
Polarizing
Beam Splitting
Cube
70:30
Splitter
Photodetector
Figure 4.10: The optics pictured are the post fiber coupled optics for the
probe beam. These are used to generate a signal for the power stabilization
which is directly before the interaction region.
4.1.7
Electro-Optic Modulators
Electo-optic modulators consist of an electrooptic crystal with a refractive index
which is sensitive to an applied electric field. Electro-optic modulators are used
to tune the amplitude of an optical beam by introducing a phase shift between
different components of the light. The change in index of refraction for a electrooptic crystal in an applied electric field is given by [33]
∆n = n30 r
E
,
2
(4.6)
where E is is the applied electric field, n0 is the index of refraction without the
presence of an electric field, and r is the component of the electro-optic tensor.
The electro-optic tensor describes the crystals interaction with the electric field,
typically r is very small. This effect introduces a phase shift in the light traveling
Experimental Setup
67
Electooptical
Crystal
L
x
y
Input
Polarizer
z
DC Voltage
Output
Polarizer
Figure 4.11: Diagram of an Electrooptic Modulator. Electrooptic Modulators
use a crystal with indices of refraction that are dependent on the applied field.
By changing the applied voltage the EOM induces a phase difference between the
two paths which causes destructive interference and so amplitude modulation.
The input and output polarizers are used to split and recombine the light from
going through the crystal.
through the crystal given by
∆φ =
πn30 rV l
,
λ d
(4.7)
where V is the voltage applied across distance d in the crystal and l is the length
of the crystal.
A standard EOM is shown in Fig. 4.11, the main components are a pair of polarizers set 90◦ apart and an electro-optic crystal. When an electric field is applied to
the electro-optic crystal acts as a variable wave plate. When a voltage is applied
the electro-optic effect changes the index of refraction for the two crystal axis to
a different degree, changing the optical path length of the two components of the
light.
This experiment uses a Conoptics 360-80 EOM, this is not designed for 671 nm
light and so there is more loss in overall amplitude then a crystal designed for 671
nm light, however it works well enough for our application.
Experimental Setup
4.2
68
Atomic Beam
The most common atomic spectroscopy is done using a vapor cell. Here a sealed
glass tube of the substance being investigated is heated and then probed with
the excitation laser. These vapor cells are easy to use and do not require constant
pumping to maintain vacuum. This makes them an easier solution then the atomic
beam source for most experiments. The limitations with vapor cells is they have a
large Doppler width meaning the transitions we are looking at will not be resolved.
An additional limitation specific to lithium is that when heated it reacts and
corrodes glass.
Lithium Vapor Density
16
10
15
10
14
10
13
-3
Atomic Density [cm ]
10
12
10
11
10
10
10
9
10
8
10
7
10
6
10
450
500
550
600
650
700
750
800
850
900
950
Temperature [K]
Figure 4.12: Atomic Denisty of Lithium as a function of Temperature. For
this experiment the temperature of the oven is 400◦ C.
This experiment uses an atomic beam source from a lithium oven. In order to
create the beam we heat small pellets of lithium to ≈ 400◦ C. Heating the chamber
this high gives a vapor pressure of ∼ 1013 atoms/cm3 as seen from the vapor
pressure curve in Fig. 4.12.
The oven was constructed using a 2 1/8” conflat nipple that was welded to a blank
flange on one side and and a flange with a .5” hole on the other. This was welded
Experimental Setup
69
instead of using standard copper gaskets as previous ovens had problems with
leaking lithium after it corroded through the copper gasket. Threaded rods were
then welded to both flanges in order to connect the oven to the vacuum chamber.
The nozzle was not welded so that the lithium could be refilled.
Threaded Rods to Attach
to Vacuum Chamber
Lithium Chamber
Nozzle
Heater Coils
Figure 4.13: The lithium oven is constructed from 2 1/8” conflat nipple welded
to a blank flange on one side and a flange with a half inch hole on the other.
Threaded rods were then welded to the end to attach it to the vacuum chamber.
A nozzle is attached to the hole on the other side. The oven is heated with
tantalum heater wire wrapped in coils.
The oven is heated using coiled tantalum heater wire. The wire is coiled and then
placed in ceramic tubes to prevents shorting between the heaters and vacuum
chamber. Originally the oven was set up to allow heating of both the chamber and
the nozzle as a precaution against clogs. We have stopped using the nozzle heater
and have not had any issues with clogging. The voltage to the heater is controlled
by a Variac, normally in the 18-22 volt range. A thermocouple is connected to
the outside of the oven which gives a ballpark reading for the temperature in the
oven.
One of the requirements for the atomic beam is that its width be as close to the
natural line width of 6 MHz as possible. This is important as the spacing of
the transitions we are looking at is such that we are limited in precision by the
resolution of the peaks. If the line width of the atomic beam is large the peaks
will not be resolved and so line shape modeling will be much more difficult. We
tried many different nozzle designs attempting to collimate the beam before this
setup. The previous nozzles all had large doppler widths and so were not sufficient
for the level of precision requited in the measurement.
The second method used to limit the effect of the doppler width and line asymmetries on the precision of our measurements is to reflect the probe beam back
through the chamber parallel to the initial beam. This helps to symmetrize the
Experimental Setup
70
peaks from any misalignment between the probe and atomic beams. This is done
using a ThorLabs retroreflector prism that reflects the incoming beam 180◦ with
part in 105 uncertainty.
This is also useful for doppler-free saturated fluorescence spectroscopy. When
the probe beam is tuned slightly off resonance it will excite two velocity classes at
equal angles away from perpendicular. When the probe laser is the same frequency
as the transition it will only excite the velocity class of atoms which are directly
perpendicular to the probe beam. This means that there will be a dip at resonance
as less atoms will be excited. This dip can be used to accurately find the centers
in the case where the doppler width is large as so hides features.
Collimation
Plate
Lithium Oven
Probe Beam
Figure 4.14: The atomic beam is collimated using a single channel nozzle on
the lithium oven and then a collimation plate directly before the interaction
region. This gives line widths of ∼ 16 MHz.
Currently we are using a single channel nozzle attached to the oven and then
a collimation plate bolted to the inside of the vacuum cube which houses the
interaction region. This solution in has given line widths of ∼ 16 MHz. This is
substantial improvement over the 120 MHz line widths that we had previously
recorded, this is shown in Fig. 4.15
4.3
Vacuum System
In order to precisely measure the transitions in lithium it is important that it is
done at high vacuum. Like all alkali atoms, lithium is highly reactive and will
Experimental Setup
Amplitude (Scaled Voltage)
30x10
-3
71
Collimated Atomic Beam
Original Atomic Beam
25
20
15
10
5
0
1450
1500
1550
1600
1650
Relative Frequency (MHz)
1700
1750
Figure 4.15: Data showing the narrowed width of the current atomic beam
setup. The blue data was taken with the original setup and has line width of
120 MHz. The red data is taken with the improved setup which has a line width
of 16 MHz.
react in air. Also you cannot have an atomic beam in air as the atoms will collide
with gas and disperse. The atomic beam and interaction region are housed in a 8”
expanded spherical cube from Kimball Physics. Attached to this are are various
nipples and bellows to connect the other components as shown in Fig. 4.16 and
Fig. 4.18. Attached to the South side is a 4.50” nipple containing the oven and
associated heaters and thermocouples. Opposite that is the ion pump used to
pump down the system. Perpendicular to this are two adjustable bellows with
AR coated windows on either side. The probe beam passes through from the east
side and then is retro-reflected back through the interaction region using a corner
cube.
The chamber is kept at 10−9 Torr when the ovens are not on and gets to 10−7
Torr when running. All elements that are placed in the chamber are cleaned and
baked prior to installation to minimize contamination of the chamber. Even with
these precautions the chamber takes about 10 days to pump down after switching
nozzles or other components.
Experimental Setup
72
PMT with
Filter
Figure 4.16: The atomic beam and probe laser intersect in the center of the
vacuum cube this houses a set of lenses used to magnify the fluorescence. The
PMT is placed on the top of the cube and has a pass band filter at 670nm to
filter the scattered light.
4.3.1
Interaction Region
The fluorescence of the atoms is detected using a ET Enterprises 51mm photomultiplier tube (PMT). A 10 nm bandpass interference filter at 670 nm is placed
in front the PMT to reduce ambient light entering the chamber. This is mounted
at the top of the vacuum cube as to be perpendicular to both the atomic beam
and probe laser. Inside the vacuum chamber a pair of 60 mm plano convex lenses
are suspended above the interaction region as shown in Fig. 4.16. These serve
Experimental Setup
73
Figure 4.17: Picture of the vacuum chamber and associated components. The
over is visible on the right of the picture. The probe beam enters from the far
side of the chamber and exits through the window in the foreground.
to increase the detection efficiency by increasing the area of detection. The input
windows are also attached to bellows such that any reflection from passing through
the windows can be adjust away from the interaction region. This helps to reduce
the scattered light level in the chamber.
4.3.2
Ion Pump
This experiment uses an ion pump to maintain a pressure of
10−7 Torr while
taking data with the ovens hot. In general ion pumps are able to achieve pressures
of 10−11 Torr by ionizing the gas molecules and then catching them in the pump
walls. Ion pumps consist of a set of charged anode rings and a grounded cathode
plates and a set of strong magnets (Fig. 4.19). A strong magnetic field is applied
through the center of the coils. As free electrons enter the anode rings they are
trapped in the string magnetic field and begin to precess in the rings and as more
electrons are trapped this creates an electron cloud. When stray gas molecules
enter the pump there is now a high probability the will collide with a free electron
and become a charged ion. This is then repelled by the anode to the neutral
cathode plate where it trapped.
Experimental Setup
74
Lithium
Oven
Atomic Beam
Collimation
Chopper
Wheel
Retroreflected,
chopped
Laser Light
Corner
Cube
Ion
Pump
Figure 4.18: The vacuum setup with oven and ion pump is shown. The
collimation plate is connect to the inside of the vacuum cube. The probe laser
passes through the chamber and then is retroreflected back using a corner cube.
When taking data with the lockin amplifier the chopper wheel is used to chop
the retro beam.
Ion pumps must be have a start pressure of 10−4 Torr or lower. We rough the
system using a turbo-molecular pump backed by a mechanical pump. The ion
pump used in this experiment is a Kurt J. Lesker LION ion pump.
4.3.3
Helmholtz Coils
The presence of background fields in our interaction region must also be accounted
for and trimmed out. This is done using a set of three Helmholtz coils in the northsouth, east-west, and up-down axis. The coils are much large then the interaction
region and are outside the vaccum chamber. The advantage of this setup is the
Experimental Setup
75
Vacuum Chamber
Magnetic Field
Magnet
Magnet
Anode
Rings
Cathodes
Figure 4.19: The ion pump pumps by ionizing the gas and then accelerating
the gas into the cathode plate where it reacts and is stuck.
coils create a region of nearly uniform magnetic field with little gradient. This
set up does not use exactly a helmhotz configuration which specifies that the coils
should be seperated by the radius of the coils but it is close. This is restricted by
our experimental setup in that the coils need to fit around the other components
of the setup. This effect should be small, less the 1% change in total compensation
field.
We can find the field from these coils simply in terms of geometric descriptors and
applied current. Starting from the on-axis field from a sing loop of wire.
B(x) =
µo IR2
(4.8)
3
2(R2 + x2 ) 2
Where x is the distance from the coil to the point and R is the radius of the coil.
From this we can multiply by N for the number of turns in each coil and then
substitute the distance between the coil and the interaciton region to find our field
from the coils at the interaction region.
B(x) =
µo IR2 N
3
2(R2 + (− 2s )2 ) 2
+
µo IR2 N
2(R2 +
s 2 32
)
2
,
(4.9)
Experimental Setup
76
Up Down
East West
North South
Figure 4.20: Diagram of Helmholtz coils used to cancel the magnetic field in
the center of the interaction region. Large diameter coils are used in order to
have minimal gradients in the interaction region.
Where s is the separation distance between the coils. Table 4.1 shows the conversion for all coils.
Field
North South
East West
Up Down
Conversion
6.8 G/A
4.1 G/A
3.3 G/A
Table 4.1: Conversion from applied current
to magnetic field at interaction region.
4.4
Data Aquisition
Data is recorded using a National Instruments PCI/MIO-16XE, 16 bit analog
to digital converter. The data are recorded using LabView. The computer is
connected to the synthesizer which the repetition rate is locked. To scan the
Experimental Setup
77
frequency of the diode laser the computer steps the rep rate synthesizer which
steps the frequency of frep and so the shifts frequency of the comb mode which
the diode is locked to. LabView synchronizes the incoming data with the the
scanned repetition rate. When scanning, the computer steps the synthesizer in 1
Hz increments and then averages for a set timescale. This is normally 500-1000
samples per channel when taking data. The scans increase over the range and
then decrease back to the start value giving two scans for each run. The PMT
signal, diode power, Fabry-Perot transmission, and beat note error signals for the
offset frequency and diode beat note are recorded.
4.5
Collection Procedure
The process of data collection requires the entire system to be stable for a period
of a few hours. This has proved to be difficult because of the overall instability
in the system. The diode laser and frequency comb are both highly sensitive to
mechanical noise and well as changes in the conditions in the lab. The frequency
comb oscillator must be kept extremely clean in order to maintain mode locked
operation. Any condensation or dust on the crystal or mirrors results in a loss of
power and so it will not mode lock. Both lasers are also temperature sensitive as
changes in the ambient temperature change the cavity. This would not be an issue
if the lab was reasonably stabile however we have recorded fluctuations of 15◦ F
over the course of 24 hours.
The first step in data collection is to adjust the wavelength of the diode laser to
the frequency of the atomic transition. To find atoms the grating is modulated
using a ±2 V triangle wave. The output of the PMT is viewed on an oscilloscope
while changing the various parameters of the diode to tune the frequency. The
grating is adjusted to get the approximate wavelength as measured by the OSA.
The horizontal adjustment is used to tune the wavelength on the nm scale. The
vertical adjustment is then used to adjust the feedback in order to increase the
power as well as improve scanning response. The wavelength can also be adjusted
using the temperature controller for the diode head. This is normally set around
19.2◦ C for the current Eagle Yard diode however this can operate in the range
18 − 22◦ C. By increasing the temperature the wavelength is increased but the
power output decreases. Once we find the transitions we are interested in, we use
the Fabry-Perot interferometer to make sure that the diode is scanning without
Data and Analysis
78
mode hops and is stable for changes in the piezo. This is important as the locks
become much less stabile if there are mode hops near the resonances.
Once the offset frequency and repetition rate of the comb have been stabilized to
the frequency standard, the beat note stabilizing the diode laser to the frequency
comb must be found and stabilized. This is done by adjusting the specially overlap
of the diode laser and comb light as well as adjusting the power of the comb such
that it produces more light in the 670 nm region. Getting the f0 beat note and fb
beat note to lockable levels is one of the continual challenges of the experiment as
the comb tends produce more light in one frequency band but not the other. We
have made improvements by increasing the sensitivity of the detection for both
locks but this is still an area that would benefit from more work. If the beat notes
are larger the system is more stable and the locks are less likely to jump, meaning
data collection is more consistent.
With diode laser locked to the frequency comb, data is collected by tuning the
repetition rate of the frequency comb using the synthesizer. The repetition rate
is scanned from an initial to final frequency and then back down to the initial
frequency giving two data points for every frequency step. This scans the diode
laser over the peaks and the PMT signal, repetition rate, power stabilization photodector, and Fabre-Perot signal are recorded. The increasing and decreasing scan
provides a check for systematics such as lags in detection which might cause an
offset.
Chapter 5
Analysis and Results
5.1
Analysis
The data recorded contain the fluorescence signal as a function of repetitional
rate. To convert to absolute optical frequency we determine which transition we
are observing by comparing the peak separation and relative amplitude to the
known splittings. For each scan the f0 beat note frequency is recorded and from
that and the repetition rate the mode number can be found. This then allows
conversion to absolute frequency. The sign of f0 and fb are unknown and so this
gives four possibilities for the frequency
ν1 = nfrep + f0 + fb
ν2 = nfrep + f0 − fb
(5.1)
ν3 = nfrep − f0 + fb
ν4 = nfrep − f0 − fb .
Here the sign of the offset frequency dependent on whether we see a sum or difference frequency of the doubled and normal light used to detect the offset frequency.
The sign in the beat note for the diode laser stabilization is dependent on whether
the diode laser is locked below or above one tooth of the frequency comb. We can
compare this with the approximate frequency of the transition for the identified
peaks to uniquely determine the mode number.
With the scans converted to absolute frequency we then fit the transitions to a
line shape model. For non saturated peaks the line shape can be modeled by a
79
Data and Analysis
80
combination of a Lorentzian and a Gaussian for each peak. This line shape is
used because the atomic beam shape is given by a Gaussian distribution and the
natural line width of the peak is a Lorentzian. This is not a perfect model but
empirically it provides reasonable fits with unstructured residuals. The line shape
model for a single transition is given by
f (ν) = A fL
γL2 /4
γL2 /4 + (ν − ν0 )2
+e
−
(ν−ν0 )2
2γ 2
G
+ B.
(5.2)
Here A is the overall amplitude of the peak, γL is the fractional lorentzian, γL is the
width of the Lorentzian peak, γG is the width of the Gaussian peak, ν0 is the peak
center, and B is an over offset used to fit the baseline. For spectra with multiple
peaks we simply add more of the bracketed term and allow the amplitude and
center frequency to vary for each peak. We keep the peak widths and fractional
Lorentzian constant for multiple peaks as the width should not be dependent on
transition only on the atomic beam an natural line width.
Li7 D1
F = 2 -› 1
Li7 D1
F = 2 -› 2
50
Data 2012
Data 2013
Saturated Fluorescence
Data 2015
Li7 D1
F = 1 -› 2
40
Li7 D1
F = 1 -› 1
x10
-3
30
20
10
0
800
1000
1200
1400
1600
1800
Figure 5.1: This plot shows the improvement in line shape due to the collimated beam. The red curve is the data from this work. The blue curve is data
taken with the original atomic beam. The green is data taken using saturated
fluorescence. The original plan for the measurement of these transitions was to
use saturated fluorescence spectroscopy and so the broad line shape would not
be a limiting factor. The full width at half max of the previous beam was 120
MHz and the current setup has a width of 16 MHz.
Data and Analysis
81
This line shape model provides reasonable fits for peaks which are symmetric with
no saturation. When there are asymmetries in the data we see clear structure in
the residuals. This model also relies on the peaks being resolved as the line shape
becomes more complicated when the peaks overlap. The atomic beam has bean
narrowed yielding peaks with a full width at half maximum of ≈ 16 MHz meaning
that all of the D1 lines are mostly resolved. The 6 Li hyperfine splitting in the
excited state is 26 MHz meaning the tails of the peaks overlap however the peaks
are resolved enough to easily get line shapes which accurately model the spectra.
Figure 5.1 shows the line profile for data taken with a previous atomic beam. The
blue curve is the initial atomic beam setup which gave line widths ∼ 120MHz. The
broadness of the atomic beam resulted in many features of the spectra being hidden. This setup was initially designed to use saturated fluorescence spectroscopy
which is not Doppler width dependent.
The green curve shows the saturated fluorescence data that was taken in order
to resolve the peaks. This was motivated by the idea that saturated fluorescence
should result in Doppler width free features and resolve the transitions. This
method was unsuccessful because the lines shape was not modeled with sufficient
accuracy and crossover resonance complicated the line shape, see Ref. [34] for
detailed analysis of this method.
Figure 5.1 shows the data taken for this thesis with the new atomic beam in red.
Here it is clear that the peaks are mostly resolved and the line width is ∼ 16 MHz.
This is a large improvement over the previous data. Features such as the 6 Li D2
peak at the bottom of the 7 Li D1, F = 2 → 1 peak are now visible and can be
accurately fit with the line shape given by Eq. (5.2).
5.2
Test of Frequency Standard
This experiment uses a GPS steered SRS rubidium atomic clock as the frequency
standard to reference the optical frequency comb. The atomic clock has very high
stability in short term but will drift at longer time scales. To correct for this drift,
the clock is steered to a GPS clock which on long time scales is very accurate.
To measure the stability a frequency standard the most common technique is
using an Allan deviation. Frequency standards tend to drift over long time scales,
the Allen deviation compares successive sets of points in order to determine an
Data and Analysis
82
3
GPS A 10 MHz Output vs SRS Rb Clock steered to GPS B
Lucent Rb Clock vs SRS Rb Clock steered to GPS B
Fractional Instability
2
-11
10
9
8
7
6
5
4
3
2
-12
10
0
1
10
2
10
3
10
10
Averaging Time (s)
4
10
Figure 5.2: Allan deviation comparing multiple frequency standards in the
lab. The SRS atomic clock which is used as the frequency standard for the
frequency comb has a accuracy when compared to the Lucent Clock of a part
in 101 1 for the time scales that we collect data on resonance. This translates to
a < 1 kHz uncertainty in the frequency standard.
accuracy over long time periods. For a given frequency standard let ν(t) be the
output of the frequency standard and νn be the ideal frequency. We can then
define the fractional frequency as
y(t) =
ν(t) − νn
.
νn
(5.3)
The average fraction deviation for a given time interval from 0 to τ is then defined
as the sum of y(t) over an interval. This can also defined as a continuously given
by
1
x̄(t, ) =
τ
Z
τ
y(t)dt.
(5.4)
0
Since y(t) is the derivative of x(t) we can write the nth average over some small
time interval τ as
ȳn =
xn+1 − xn
.
τ
(5.5)
The Allen variance is then defined as [35]
1
1
σy2 (τ ) = h(ȳn+1 − ȳn )2 i = 2 h(xn+2 − 2xn+1 + xn )2 i.
2
2τ
(5.6)
Data and Analysis
83
This is result is normally presented as the Allen deviation given as the square root
of the variance
r
q
1
σy (τ ) = σy2 (τ ) =
h(xn+2 − 2xn+1 + xn )2 i.
2τ 2
(5.7)
The Allen deviation is different from a standard deviation as it characterizes the
stability by comparing the variance in a set of points as opposed to some fixed
value.
The SRS rubidium atomic clock in this experiment is compared to a second Lucent
atomic clock as well as a GPS clock in Fig. 5.2. The stability in the SRS atomic
clock steered by GPS gives an uncertainty of < 1 kHz when averaging for around
200 seconds.
5.3
Test of locks
There are multiple checks for the quality of the locks. This is important as if the
locks are not tight and consistent systematic shifts will be present in the data. The
repetition rate lock is very consistent when counted and has very little frequency
noise. The offset frequency and diode stabilization locks are less stabile as there is
more frequency noise then on the repetition rate beat note. The lock error signals
are recorded to give a indicator if they are have large frequency noise or if there
are other issues such as consistent offsets.
The first check of the locks is viewing the beat notes using an rf spectrum analyzer.
When looking at the offset frequency the lock may appear to have frequency jitter
if the beat note is not large enough. This will appear in the error signal as a
large deviation from zero as the lock electronics correct. The other check is to
look at the width of the beat note in frequency space. The narrower the beat note
the tighter the lock, if the gain on the feedback is set to high the beat note will
broaden and so have more frequency instability.
A second check of the locks while collecting data is counting the beat notes using a precision frequency counter referenced to the atomic clock. The counter is
accurate for rf frequencies on the hz scale. The beat notes for both the offset
frequency and probe laser are counted, if the locks are good there is less then
Data and Analysis
84
hz variation in counted frequency. These checks insure low uncertainty in the
frequency stabilization of the probe laser.
5.4
Magnetic Field Evaluation
The presence of Zeeman splitting from a non zero external magnetic field is an
important systematic in this experiment. In order to achieve high accuracies in our
measurement want to have as near to zero magnetic field as possible. The external
field is mostly due to the Earth’s magnetic field. In order to compensate for this
we use three Helmhotz coils which we use to zero the magnetic field in all three
axis. We have made these coils about .5 meters in diameter to have as small a
Residuals (mG)
gradient as possible. We have used two methods to set zero field for the coils. The
0.4
0.0
-0.4
10
Magnetic Field (mG)
0
-10
-20
Fit Coefficients
Background Field = -44.0 mG
Slope = 1.4 G/A
-30
-40
0
10
20
Current (mA)
30
40
Figure 5.3: Magnetic field at interaction region as a function of applied current
through up down coils. Data is fit to a line to find zero field as well as calibration
between applied current to coils and field at interaction region.
first was using a flux gate magnetometer and zeroing the field in the interaction
region before the chamber was in vacuum. This was done twice, a year apart with
increased accuracy on the second iteration giving a residual magnetic field of 10
mG. The main source of uncertainty for these measurements is in the placement
of the probe. In our first measurement we moved the probe one centimeter either
direction in the axis of the field and saw a 30 mG shift in magnetic field. This gave
us a very large uncertainty for the first measurement. We improved this in the
second measurement to give an error of about 10 mG. These two measurements
Data and Analysis
85
0.10
0.00
-0.10
-0.20
Coefficient values ± one standard deviation
a
=2.7313 ± 0.198
x0
=0.12379 ± 0.0187
b
=16.681 ± 0.0568
19.0
Peak Width (MHz)
18.5
18.0
17.5
17.0
16.5
-0.6
-0.4
-0.2
0.0
Current (A)
0.2
0.4
0.6
Figure 5.4: Magnetic field at interaction region as a function of applied current
through up down coils. Data is fit to a line to find zero field as well as calibration
between applied current to coils and field at interaction region.
agree with each other as shown in 5.1 which shows a stability in the system as
these measurements were taken about a year apart.
Axis
Magnetic Field
North-South
672.2 mG
35.4 mG
East-West
Up-Down
107 mG
First Zero +- 30 mG
154 mA
-8 mA
167 mA
Second Zero +- 10 mG
127 mA
-12 mA
157 mA
Table 5.1: Magnetic Field Measured with flux gate magnetometer. The two
measurements were taken a year apart. The uncertainty is from the probe
position in the chamber.
The second method used to evaluate the magnetic fields of our system was varying
the magnetic field and then looking at the affect on line width of our transition.
In this case the affect of non-zero magnetic field should appear as a broadening of
our peak width.
Our second method shows a clear increase in peak width with increased magnetic
field in all three axis. Figure 5.4 shows an example of the correction for one axis of
Data and Analysis
86
Field
Minimum Using Meter (mA) Minimum from width data (mA)
Up-down
157(10)
159 (8)
127(10)
124 (8)
North-South
-12(10)
-7 (8)
East-West
Table 5.2: Zero magnetic field compensation current values for calibration
using flux gate magnetometer and minimum based on broadening of spectral
peaks.
the field. This is fit to a parabola to find a minimum for each field. From this we
are able to compare our two means of evaluating the magnetic field. This method
is more precise the the probe method because when using the probe it is not certain
that the probe is exactly where the atoms are interacting with the probe light.
With the Helmholtz coils this effect should be small because the gradients in the
center of the coils are near assumed to be near zero. The two methods agree for
the value of the compensation current in order to have zero external field in all
axes.
5.5
Evaluation of Probe Beam Alignment
The alignment of the atomic beam with respect to the probe beam is extremely
crucial for having symmetric line shapes such that the line shape models can
accurately represent the data. This has been the most difficult systematic to
address for several reasons.
One of the issues in the alignment between the probe and atomic beam is that the
sensitivity in the adjustment in the angle of the probe beam has less resolution than
our ability to detect the misalignment. The angle of the probe beam is adjusted
using the final mirror prior to the vacuum chamber. Originally we used a standard
ThorLabs kinematic mirror mount with 1/4 − 80 threaded micrometers with a
resolution of .1◦ . This mount had both stability issues and limited resolution. We
changed to a Gimbal high stability mount which improved overall stability and
gave slightly better resolution. This mount has a resolution of .02◦ . This spacing
is still to large to accurately set the angle of the mirror to the level of detection.
The first problem was finding a method to evaluate the symmetry of the peak in
order to test the alignment. To do this we use a lock in amplifier with chop wheel
to look at the contribution of the retro beam to the PMT signal. The probe beam
87
10
5
0
-5
-10
-15
x10
-3
Data and Analysis
1.6
Coefficient values ± 1 SD
A1 =0.38491 ± 0.000832
C1 =666.01 ± 0.00061
WG =8.826 ± 0.00173
WL1 =27.068 ± 0.0501
FL =2.137 ± 0.0141
A2 =0.75853 ± 0.00304
C2 =666.31 ± 0.000652
WL2 =14.069 ± 0.0156
B
=0.43338 ± 3.5e-005
M
=0.0020008 ± 3.32e-006
Lockin PMT Data
Fit
Fit Residuals
Lineshape Function
Fluorescence Dip
1.4
1.2
1.0
0.8
0.6
620
640
660
680
700
720
Figure 5.5: This data is an example of the data used to align the probe laser
with the atomic beam. The retro-reflected probe beam is chopped before the
interaction region. This means we can use a locking amplifier to look at just the
contribution of the retroreflected beam. This data is shown in red. If the probe
beam is perpendicular to the atomic beam the saturation dip should be exactly
symmetric in the line shape and the peak center for the line shape as well as
the saturation dip should be the same. The data is fit in blue to a combination
of the normally line shape composed of a Gaussian and Lorentzian (shown in
black) minus the line shape for the saturation dip in green.
is chopped after the vacuum chamber, before being reflected back into chamber
shown in Fig. 4.18. The chopper wheel introduces some mechanical noise to the
system and so was suspended from the unistrut such that it would not vibrate the
optics table.
The probe beam is operated in the saturation region such that the retro beam
creates a dip at the center frequency because only one velocity class is excited
when the laser is tuned on resonance (see Chapter 2). The dip is symmetric in
the main peak when the probe and atomic beam are orthogonal. The center of
the saturation peak and the main peaks should be the same if the two beams are
orthogonal. Figure 5.5 shows one example of data taken with the lockin amplifier
where the alignment is as close as possible given the resolution of the angle between
Data and Analysis
88
0
x10
10-3
2
-3
4
-2
0.35
PMT Voltage (V)
0.30
0.25
0.20
Coefficient values ± 1 SD
A1 =0.019708
\ ± 6.04e-005
C1 =573.88 ± 0.00255
WL =14.246 ± 0.0388
WG =7.7752 ± 0.00292
FL =7.7085 ± 0.204
A1S =0.094549 ± 0.00351
DS =0.12527 ± 0.00338
WLS
=11.545 ± 0.0382
A2 =0.10669 ± 0.000146
C2 =665.83 ± 0.00194
A2S =0.66031 ± 0.0209
B =0.076665 ± 5.06e-006
M =0.0020027 ± 4.09e-005
0.15
0.10
560
580
600
620
640
660
Frequency (Hz - 446800000)
680
700
Figure 5.6: Example of the PMT combined PMT signal when collecting data
for the alignment between the probe and atomic beam. The saturation dip
is clear in the second peak but not present in the first peak because of the
difference in saturation intensities for the two transitions.
the beams. Here the dip is fairly centered in the PMT signal however there is still
a visible offset between the saturation dip center and the line shape center. Ideally
the saturation peak and line shape peak would have the center value, this would
indicate that the probe beam and atomic beam are perpendicular to each other
and there is no asymmetry in the line shape of the atomic beam.
This data is fit to a Lorentzian plus a Gaussian for the main line shape, shown in
black in Fig. 5.5 minus a Lorentzian for the saturation dip, shown in green. This
is given by
γL2 /4
f (ν) = A1 F L 2
γL /4 + (ν − ν1 )2
−
+e
(ν−ν1 )2
2γ 2
G
− A2
2
γL2
/4
+ B,
2
(γL2
/4 + (ν − ν2 )2 )
(5.8)
where the first term is the line shape for the peak without the saturation dip and
Data and Analysis
89
the second term is the lorentzian used to model the saturation dip. The center of
the saturation dip is ν2 and the amplitude is given by A2 . A lorentzian is used
for the saturation peak because the line shape does not depend on the velocity
distribution of the atomic beam.
Figure 5.6 shows the PMT signal for the combined beams. This is an example of
how the saturation peak center changes for small changers in the beam alignment.
This is as close to aligned as possible with the resolution in the mirror before the
chamber.
5.5.1
Correction of Misalignment
The misalignment in the probe beam with the atomic beam has been reduced as
much as possible with the current setup and the alignment process outlined in the
previous section. Unfortunately this still results in a shift in the data due to the
misalignment. There are two ways to address this shift. The first is to correct
aspects of the system in order to minimize the effect which will be addressed in
the following section. The second is to correct for the shift when analyzing the
data.
When data are taken with both the probe beam and retro beam the recorded
line shape is a combination of the contribution from both beams. If the power
was equal in both beams then this would completely symmetrize the peak and
remove any shift from the misalignment. Unfortunately because of the non-coated
windows used for this data, the power in the retro beam is ≈ 60% of the incoming
beam because of the losses to the windows. This makes fitting the line shape
more difficult as it is the contribution of two non-symmetric peaks with different
amplitudes.
We can account for this asymmetry by using the line shape from data taken without the retro-reflected beam. This line shape is asymmetric because of either
a asymmetric in the distribution of the atomic beam or a misalignment in the
probe beam relative to the atomic beam. We can then fit this line shape using
an interpolating function. This is used because the the peak cannot be fit it to
a combination of a Lorentzian and a gaussian as used before as the asymmetries
cause structure in the residuals. The contribution from the retro-reflected beam
has this asymmetry mirrored because the propagation direction is reversed. In
Data and Analysis
90
PMT Voltage (V)
0.05
— No RetroReflection Data
— Retro-Reflection
Data
0.04
0.03
0.02
0.01
620
640
660
680
700
720
Frequency (MHz - 446800000)
Figure 5.7: Example of data used to fit the shift in from the misalignment
between the probe beam and atomic beam. The Blue data is used as the base
function for the interpolation function which is then used to fit the RetroReflection data. This is used to find a correction to the center frequency for the
data taken with including the retro-reflected probe beam.
order to fit the peak with contributions from both beams we start with line shape
found from the peak from just one beam. This serves as a model for the line shape
of the peak from both beams. Figure 5.7 shows the line shape model data in blue
used for the interpolation.
To fit the peak with with contributions from both beams the model line shape is
mirrored over a center point and added to the line shape for single beam. This fit
is given by
L(ν) = A[Lm ((ν − δ) − ν0 ) + fR Lm (−(ν + δ) − ν0 )] + B
(5.9)
where Lm is the interpolated line shape from the non-retroreflection data, A is
the overall applitude, fR is the fraction of the retroreflected beam, ν0 is the center
frequency, δ is the mirror point, and B is a baseline offset. With this line shape
model we can then fit the line shape of the data with both beams. With this
method we were able to find fits which appear to have no structure to the residuals
Data and Analysis
91
0.010
Residuals
0.0010
0.005
0.0005
50
100
150
-0.005
PMT Voltage (V)
-0.0005
0.05
— Data
— Interpolated
Fit
0.04
0.03
0.02
0.01
620
640
660
680
700
Frequency (MHz - 446800000)
Figure 5.8: Data shown is taken with contributions from both the probe beam
and retro reflected probe beam. This is fit to the interpolated line shape for one
beam that has been mirrored over a center value and multiplied by a relative
amplitude to account for the contribution from both beams. This fit appears
to have no structure to the residuals.
as shown in Fig. 5.8. From this we can find the true frequency of the transition
which is free of the shift present from the beam misalignment. When using this
method the fractional retro contribution remains constant at .57 ± .02 for all fits.
This agrees with the contribution expected as the loss in power due to the windows
on the vacuum chamber is ≈ 40%. The peak center frequency is also independent
of the frequency used as the mirror point. We also checked the fit by using different
base fines to generate the line shape model. These fits were consistent between
different base files in both retro beam contribution and center frequency.
Applying this line shape model to our data we see that the scatter in center
frequency is less then 100 kHz. We re fit the data using multiple different line shape
model scans and then take the standard deviation of the center frequencies found
with different models plus the statistical uncertainty in weighted mean of this data
Data and Analysis
92
for our error. Using this method we find a center frequency of 446800666.439±.009
MHz for the 7 Li F = 1 → 2 transition. This gives our error bound for beam
misalignment can be reduced to 9 kHz. The overall shift from the standard line
shape model using a combination of a Lorentzian and a Gaussian is found to be
-200 kHz. This correction is added to the optical frequencies presented.
5.6
AC Stark Measurement
The expected AC Stark shift for the D1 is estimated in Section 2.6.1. For these
data the intensity of the beam is 1.5 mW/cm2 , using this intensity we find that
the upper bound in the uncertainty due to the AC Stark shift for each transition
is given in Table 5.3. These uncertainties can be further limited by experimentally
measuring the AC Stark shift and correcting for the shift to the optical frequencies.
Transition
6
Li
F = 1/2 → 1/2
F = 1/2 → 3/2
F = 3/2 → 1/2
F = 3/2 → 3/2
Expected Shift (kHz ± 10%
-30
4
-15
15
7
Li
F =1→1
F =1→2
F =2→1
F =2→2
-2
8
-5
5
Table 5.3: Estimated AC Stark shifts for beam intensity of 1.5 mW/cm2 .
These values are a reasonable upper bound on the uncertainty due to the AC
Stark shift. The effect is larger in 6 Li as the hyperfine splitting of the upper state
is smaller and so the effect is stronger.
5.7
Isotope Shift and Hyperfine Structure Splitting
The isotope shift is calculate by finding all of the D1 transition frequencies for 6 Li
and 7 Li. Examples of the data used to find these optical frequencies are shown in
Data and Analysis
93
Transition
6
Li D1
6
Li D1 cog
7
Li D1
6
Li D1 cog
Fg
3/2
3/2
1/2
1/2
Fe
1/2
3/2
1/2
3/2
Frequency (MHz)
446 789 502.585 (27)
446 789 528.723(24)
446 789 731.834 (44)
446 789 756.973 (32)
446 789 596.093(42)
2
2
1
1
1
2
1
2
446 799 771.089 (12)
446 799 862.950(12)
446 800 574.507(17)
446 800 666.439(11)
446 800 129.800(26)
Table 5.4: Optical Frequencies for the D1 transition infor 6 Li and 7 Li.
Fig. 5.9, 5.10 and 5.11. These were fit using the line shape of a combination of a
Lorentzian and Gaussian given by Eq. (5.8). Each scan is fit using a weighted fit
where the weighting is given by the error in each point. By taking multiple scans
of each transition and then averaging the data using a weighted mean we are able
to further reduce our statistical uncertainty. We then correct for the misalignment
between the atomic beam and probe laser as outlined above. This appears as a
linear shift to all the peaks. The shift from the misalignment is subtracted from
the optical frequencies of the peak centers. This systematic shift will not change
the isotope shift measurement or fine structure measurement as it appears as a
linear shift in all the data. The corrected optical frequencies given in Table 5.4.
Systematic
Statistical
Beam Alignment
Zeeman Effect
Stark Shift
Reference
7
Li D1 F = 1 → 2 6 Li F = 3/2 → 1/2
4
20
9
9
<1
<1
2
15
<1
<1
Total Uncertainty
11
27
Table 5.5: An example of the uncertainty budget for each center frequency.
This is the error budget for 7 Li D1 F = 1 → 2. The errors are added in
quadrature in order to get the total error in value.
The data for the 7 Li transitions have a better signal to noise ratio then 6 Li because
the peak amplitude is much smaller because the difference in natural abundance.
The statistically biggest issue in these data is that the F = 1/2 → 1/2 peak is not
resolved and has a much smaller amplitude and so the line shape model has higher
Conclusions
94
uncertainty. For determining the isotope shift we will set the hyperfine splitting
of the 6 Li F = 1/2 → F 0 equal to that determined from the 6 Li F = 3/2 → F 0
transition. This gives a hyperfine splitting of 26.138(35) and an corrected optical
frequency for 6 Li F = 1/2 → 1/2 of 446 789 730.834(44). This value is used for
the calculation of the isotope shift.
Transition
D1
Shift (MHz)
Reference
10533.707(50) This Work
10533.763(9) Sansonetti [2]
10534.26(13) Walls [5]
10534.039(70) Noble [4]
10534.215(39) Das [3]
Table 5.6: Isotope Shift Frequency for this work and previously published
measurements.
To calculate the isotope shift, the center of gravity for each transition must be
calculated to find the overall D1 line for both isotopes. This is calculated by
weighting the frequency of each transition by the degeneracy of the state. The
optical frequencies used for this calculation are given in Table 5.4. An example of
the error budget for two optical transitions is outline in Table 5.5. The uncertainty
in the 6 Li transitions is significantly greater then the 7 Li. This is due to both
the increased statistical uncertainty and larger AC Stark shift in 6 Li due to the
smaller hyperfine splitting of the excited state. In calculating the isotope shift the
uncertainty in the 6 Li center of gravity dominates the uncertainty in the isotope
shift. We determined that the isotope shift for the D1 transition is 10533.707(50)
MHz. This is within the combined uncertainty of the most recent measurement
published by Sansonetti et al. [2] but does not agree with the other published
values shown in Table 5.6. In order to resolve this discrepancy we need to further
limit our systematic uncertainty.
The hyperfine splitting measurements of both the ground state and excited state
of the D1 transition are shown in Table 5.7. The ground state measurements
agree with Sansonetti et al. [2] within the uncertainty in our measurement. It
also agrees with the rf spectroscopy measurement by Beckmann [36] for 7 Li but
not for 6 Li. The excite state measurements agree with the previous measurements
within the combined uncertainty for each measurement except from Das et al [3].
These results would indicate that there is more work left in understanding the
uncertainties present in the measurement, especially for 6 Li.
Conclusions
95
Interval
7
Li 2s 2 S hfs
Splitting (MHz) Reference
803.489(16)
This Work
803.493(14)
Sansonetti [2]
803.504 086 6(10) Beckmann [36]
6
Li 2s2 S hfs
228.250(35)
228.215(17)
228.205 261(3)
This Work
Sansonetti [2]
Beckmann [36]
7
Li 2p2 P1/2 hfs
91.896(16)
91.887(9)
92.020(50)
92.047(6)
This Work
Sansonetti [2]
Walls [5]
Das [3]
6
Li 2p2 P hfs
26.138(35)
26.111(17)
26.079(46)
26.091(6)
This Work
Sansonetti [2]
Walls [5]
Das [3]
Table 5.7: Hyperfine splitting for the D1 transition frequencies for 6 Li and
7 Li.
Conclusions
96
x10
-6
Coefficient values ± one standard deviation
A1 =0.0016551 ± 3.6e-05
A2 =0.0081582 ± 0.000183
C1 =1574.8 ± 0.0418
C2 =1666.6 ± 0.0129
WG =7.6771 ± 0.0805
WL =14.498 ± 0.121
FL =2.371 ± 0.0792
B
=-0.00084804 ± 1.16e-05
600
400
200
0
-200
-400
-3
25x10
Li7 D1
F = 1 -› 2
Data
Fit
Residuals
PMT Voltage (V)
20
15
10
5
Li7 D1
F = 1 -› 1
0
1550
1600
1650
Frequenecy (MHz - 446799000)
1700
Figure 5.9: Data for 7 Li D1 F = 1 → F 0 transitions. Peaks were fit to a
combination of a Lorentzian and a Gaussian.
Conclusions
97
x10
-3
Coefficient values ± one standard deviation
A1
=0.053591 ± 0.000594
C1
=771.26 ± 0.01
A2
=0.0082337 ± 9.82e-05
C2
=803 ± 0.0392
A3
=0.052368 ± 0.000582
C3
=863.09 ± 0.0098
WG =7.949 ± 0.0374
WL
=17.878 ± 0.105
FL
=1.5505 ± 0.03
B
=0.018799 ± 5.36e-05
2
1
0
-1
-2
0.14
Li7 D1
F = 2 -› 1
Data
Fit
Residuals
Li7 D1
F = 2 -› 2
PMT Voltage (V)
0.12
0.10
0.08
Li6 D2
F = 1/2 -› 1/2, 3/2
0.06
0.04
750
800
Frequency (MHz - 446799000)
850
Figure 5.10: Data for 7 Li D1 F = 2 → F 0 transitions. Peaks were fit to
a combination of a Lorentzian and a Gaussian. We see some structure to the
residuals but they are mostly small variations.
Coefficient values ± one standard deviation
a1 =0.0095482 ± 0.000762
c1
=502.85 ± 0.038
a2 =0.011436 ± 0.000911
c2
=529.09 ± 0.0315
a3 =0.0012391 ± 0.000107
c3
=732.05 ± 0.27
a4 =0.0084791 ± 0.000679
c4
=757.11 ± 0.0402
wg =6.611 ± 0.0503
wl
=26.11 ± 1.57
fl
=0.88997 ± 0.152
b
=0.040679 ± 1.87e-05
Conclusions
98
x10
-3
1.0
0.0
-1.0
Li6 D1
F = 3/2 -› 1/2
60x10
Li6 D1
F = 3/2 -› 3/2
-3
PMT Voltage (V)
Li6 D1
F = 1/2 -› 3/2
55
Data
Fit
Residuals
50
Li6 D1
F = 1/2 -› 1/2
45
40
500
600
700
Frequency (MHz - 446800000)
800
Figure 5.11: Data taken on 6 Li D1 transitions. This transition has a much
smaller amplitude then the 7 Li D1 transitions and so the signal to noise ratio
is not a good as the 7 Li measurements. Each peak is fit to a combination of a
Lorentzian and gaussian.
Chapter 6
Conclusions
The measurements of the D1 transition in the previous section show reasonable
agreement with the previously published measurements. There is more work to
be done to fully understand the uncertainties associated with the frequency measurements. With the measurement of the D1 complete we will move on to the D2
transitions. Most of the systematics associated with the D2 transitions are the
same as those for the D1 transitions however there are other effects that were not
present in the D1 lines. These include unresolved hyperfine structure as well as
polarization effects outlined by Brown et al [1].
The hyperfine splitting of the D2 exited state is less the natural line width of
lithium. When probing these transitions the peaks are not resolvable and so
the line shape is complex. There is also interference effects dependent on the
polarization of the probe laser relative to the detection.
6.1
Improvements to Apparatus
In order to improve our error limit on the alignment between the probe beam and
atomic beam we have replaced the input windows to the vacuum chamber which
the probe beam travels through. If the retro reflected beam and probe beam have
the same power then the contribution from both beams will be the same. Since
the beams propagate from opposite directions this will symmetrize the spectral
peaks. This will eliminate the asymmetry in the peak and put a lower error bound
on the uncertainty. The windows that were used for the data in this work were
99
Conclusions
100
glass windows that reflects ≈ 20% of the laser light. Since the retro reflected
probe beam travelled through the windows of the vacuum chamber twice before
interacting with the atoms, the power in the retro-reflected beam was ≈ 60% of
the input beam the spectral peaks. While we can correct for this using line shape
models a truly symmetric peak would mean that our error bound would be lower.
The amplitude of the beat notes for the the offset frequency of the comb and the
diode laser stabilization are a consistent challenge in this experiment. While the
frequency comb spans a frequency an optical octave, the power output per mode
is not uniform over this range. Both the output of the Ti:Sapphire laser as well as
the alignment of the micro-structure fiber effect the distribution of power through
the output spectrum. This causes an constant compromise between tuning the
alignment in order to get more power for the offset frequency beat note and the
diode stabilization. To improve this we are working to improve the detection of
both beat notes in order to make the power requirements lower. We are working
on a new detection setup for the offset frequency using a Menlo Systems avalanche
photodetector which is more sensitive the previous amplified photodetector. This
should improve the detection of the beat note making it easier to stabilize.
6.2
AC Stark Measurement
The only systematic that has not been fully evaluated for the D1 transition is the
AC Stark shift. To measure this effect we will scan each transition with multiple
powers as derived in Chapter 2 the shift will be greater for the 6 Li transitions. The
shift should be linear with respect to power so can be extrapolated to zero. Measuring this effect will limit the uncertainty further then the estimation provided
in the previous section.
6.3
Finalize Uncertainties
In this work we have set an upper bound on all the uncertainties related to the
measurement of the optical frequencies for the D1 transition. Several of these
bounds can be improved with future work. The statistical error, particularly in
the 6 Li data can be improved by taking more data to limit the uncertainty. Taking
Conclusions
101
more data over the course of several weeks will also insure that the system is self
consistent over long time periods.
The new AR coated probe beam windows on the vacuum chamber should also
provide a lower error on the uncertainty in the alignment between the probe laser
and atomic beam. With these improvements we should be able to reduce the total
uncertainty in the isotope shift measurement to less the 20 kHz from the 50 kHz
uncertainty found in this data.
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