Infrared Spectroscopy of Buffer-Gas Cooled Molecules andDynamics of Bose Gases in Spherical SymmetrybyCameron J. E. Straatsma
M.Sc., University of Alberta, 2011
B.Sc., University of Alberta, 2009

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Electrical, Computer, and Energy Engineering
2018This thesis entitled:
Infrared Spectroscopy of Buffer-Gas Cooled Molecules and Dynamics of Bose Gases in Spherical
Symmetry
written by Cameron J. E. Straatsma
has been approved for the Department of Electrical, Computer, and Energy Engineering

Heather J. Lewandowski

Juliet T. Gopinath

Date

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

iii
Straatsma, Cameron J. E. (Ph.D., Electrical Engineering)
Infrared Spectroscopy of Buffer-Gas Cooled Molecules and Dynamics of Bose Gases in Spherical
Symmetry
Thesis directed by Heather J. Lewandowski

Infrared absorption spectroscopy is a highly sensitive technique for probing molecular structure, and when combined with cold molecular beam methods it provides unparalleled spectral
resolution and absorption sensitivity to transitions in isolated gas-phase molecules. The first partof this dissertation investigates molecular beams of large polyatomic molecules produced via laser
ablation of solid targets inside a buffer-gas cooling cell. Using matrix isolation spectroscopy to
study the ablation products of graphite, we observe carbon clusters C3 to C12 produced at a rateof approximately 1011 − 1012 molecules for every pulse of the ablation laser. In a similar fashion,
we study the production of metal oxide molecules with a buffer-gas beam source by ablating puremetal in the presence of O2 gas, and find consistent production of the WO molecule. Finally, we
discuss efforts towards high resolution rotational-vibrational spectroscopy of WO in the gas phase.
The investigation of collective modes in ultracold atomic gases provides revealing information
about the nature of interactions in these systems. Of particular interest is the interplay between
a Bose-Einstein condensate and thermal excitations present at finite temperature, which effect the
dynamics of collective modes by shifting mode frequencies and introducing damping. The second
part of this dissertation investigates the dynamics of the monopole mode of a degenerate Bose gas in
an isotropic harmonic trap recently developed at JILA. Through numerical simulations within the
Zaremba-Nikuni-Griffin formalism, we identify the beating between two eigenmodes of the system,
corresponding to in-phase and out-of-phase oscillations of the condensed and noncondensed portionsof the gas. This beating leads to nonexponential collapse of the amplitude of the condensate
oscillation, followed by a partial revival. Results of the simulations are shown to agree well withexperimental data.

Acknowledgements

This dissertation represents the end to a rather interesting journey through graduate school thatwas possible due to the support and guidance of many amazing people. First and foremost, I am
indebted to Heather Lewandowski for offering her unwavering support and constant encouragement, and allowing me a great deal of freedom while pursuing research goals. Heather’s passion
for interesting physics, infinite optimism, and dedication to her students made for a wonderfulexperience conducting research in her group. I also thank my other collaborators on the monopolemode project – Victor Colussi, Dan Lobser, Matthew Davis, Murray Holland, and Eric Cornell –from whom I learned a great deal about kinetic theory and cold Bose gases through many insightful
discussions. Additionally, I thank Juliet Gopinath for acting as my titular adviser in ECEE, andfor her support and flexibility over the past year.
I would also like to acknowledge the amazing people at JILA that not only keep things
running smoothly, but readily offer their expertise to anyone in need. I thank the members of
the instrument shop, electronics shop, computing group, supply office, and administrative support
group for all their assistance over the years. In particular, I would like to thank Hans Green andTerry Brown, who have helped me with a wide variety of problems on countless occasions, andshared much of their wisdom with me. Their infinite patience, friendly demeanor, and curiosity arecontagious.
Finally, none of this would have been possible without the love and support of my amazing
family, and the wonderful friends and labmates who have helped make graduate school substantiallymore enjoyable. In particular, I am grateful for Carrie Weidner and Seth Caliga for providing

v
constant support and convincing this Canadian to finally play ice hockey, Brad Pelz for always being
eager to meet up for breakfast, climbing, snowboarding, or pretty much any crazy outdoor activity,and for Paul Ackerman and Ana Mazan for sharing many dinners and interesting conversations over
the years. Lastly, I am forever grateful for the love and constant encouragement from my partner
Courtney Peck, who is always there for me and pushes me to be the best version of myself.

Contents

Chapter
1 Introduction to Part 1
1.1

1.2

1.3

Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.1

Determination of molecular structure . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.2

Modern techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Molecular Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.2.1

Traditional beam sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.2.2

Buffer-gas beam sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 The Cryogenic Buffer-Gas Beam Source
2.1

2.2

2.3

2

14

Buffer-Gas Cooling Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1

Buffer-gas flow properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2

Collisional cooling – translational relaxation . . . . . . . . . . . . . . . . . . . 21

2.1.3

Collisional cooling – internal state relaxation . . . . . . . . . . . . . . . . . . 25

2.1.4

Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Molecular Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1

Effusive regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.2

Hydrodynamic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Comparison to a Traditional Free Jet Source . . . . . . . . . . . . . . . . . . . . . . 40

vii
3 Experimental Techniques for Infrared Absorption Spectroscopy
3.1

3.2

42

Absorption of Infrared Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1

Beer-Lambert law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.2

Absorption lineshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.3

Minimum detectable optical depth . . . . . . . . . . . . . . . . . . . . . . . . 46

Fourier Transform Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1

Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.2

Michelson interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.3

Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.4

Detection limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.5

Performance of the Nicolet iS50 FTIR spectrometer . . . . . . . . . . . . . . 55

3.3

Matrix Isolation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4

Infrared Laser Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.1

Quantum cascade lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4.2

Wavelength modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.3

Balanced subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4.4

Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.4.5

Performance of the laser absorption spectroscopy system . . . . . . . . . . . . 79

4 Production of carbon clusters with a cryogenic buffer-gas beam source

85

4.1

Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Infrared Spectroscopy of the WO Molecule
5.1

95

Expected WO Rotational-Vibrational Spectrum . . . . . . . . . . . . . . . . . . . . . 96
5.1.1

Energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.1.2

Transition wavenumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

viii
5.1.3
5.2

Population of rotational states . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Survey of WO Production with FTIR Spectroscopy . . . . . . . . . . . . . . . . . . . 101
5.2.1

Laser ablation of WO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.2

Laser ablation of W with O2 gas . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3

Expected Gas Phase Absorption Signal . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6 Conclusion to Part 1

113

6.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2

Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7 Introduction to Part 2

118

8 Background for the ZNG Method

121

8.1

Bose-Einstein Condensation in an Isotropic Harmonic Trap . . . . . . . . . . . . . . 121

8.2

Many-Body System Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.3

Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.4

8.3.1

Zero-temperature limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.3.2

Finite temperature – the ZNG method . . . . . . . . . . . . . . . . . . . . . . 128

Validity of the ZNG Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

9 Numerical Solution of the ZNG Equations in Spherical Symmetry
9.1

9.2

9.3

133

Solution of the GGPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.1.1

Derivation of the time evolution operator . . . . . . . . . . . . . . . . . . . . 136

9.1.2

Application of the time evolution operator . . . . . . . . . . . . . . . . . . . . 138

Solution of the QBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.2.1

Collisionless evolution of the test particle distribution . . . . . . . . . . . . . 140

9.2.2

Collisional evolution of the test particle distribution . . . . . . . . . . . . . . 146

Full Dynamical Evolution of the ZNG Equations . . . . . . . . . . . . . . . . . . . . 157

ix
9.3.1

Calculation of equilibrium solutions to the ZNG equations . . . . . . . . . . . 158

9.3.2

Initialization of the test particle distribution

9.3.3

Procedure for a full dynamical simulation . . . . . . . . . . . . . . . . . . . . 163

9.3.4

Results of a quench simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 164. . . . . . . . . . . . . . . . . . 160

10 Collapse and Revival of the Monopole Mode of a Degenerate Bose Gas

168

10.1 Collisionless Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.1.1 Collective modes in an isotropic harmonic trap . . . . . . . . . . . . . . . . . 169
10.1.2 Coupled-modes analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
10.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.3 Collisional Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.3.1 Simulation of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
10.3.2 Extraction of damping rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11 Conclusion to Part 2

185

11.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
11.2 Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Bibliography

187

x

TablesTable
3.1

Nicolet iS50 FTIR spectrometer specifications . . . . . . . . . . . . . . . . . . . . . . 56

3.2

Nicolet iS50 FTIR spectrometer sensitivity versus resolution . . . . . . . . . . . . . . 60

3.3

Nicolet iS50 FTIR spectrometer sensitivity versus apodization . . . . . . . . . . . . . 61

4.1

Matrix isolation results of carbon cluster spectroscopy . . . . . . . . . . . . . . . . . 89

5.1

Equilibrium constants for the Ω = 0 component of the 3 Σ− ground state of 184 WO . 98

5.2

Equilibrium constants for the Ω = 1 component of the 3 Σ− ground state of 184 WO . 99

Figures

Figure1.1

A simplified representation of vibrations and rotations in a molecule. . . . . . . . . .

1.2

Available broadband and widely tunable sources for spectroscopy in different spectral
regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

6

1.3

Example of a free jet source design that utilizes laser ablation of a solid target. . . . 11

2.1

A schematic of the buffer-gas cooling cell. . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2

Neon density inside the buffer-gas cooling cell and Knudsen number in the exit
aperture as a function of flow rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3

Reynolds number in the exit aperture of the cell as a function of flow rate. . . . . . . 20

2.4

Number of collisions between molecules and buffer-gas atoms. . . . . . . . . . . . . . 26

2.5

Adiabaticity parameter for internal state relaxation via elastic collisions. . . . . . . . 28

2.6

Cell extraction parameter as a function of neon flow rate . . . . . . . . . . . . . . . . 32

2.7

On axis temperature and density of hydrodynamic buffer-gas beam. . . . . . . . . . 39

3.1

A schematic of the Nicolet iS50 FTIR spectrometer optical system. . . . . . . . . . . 49

3.2

Typical FTIR spectrometer single beam spectrum. . . . . . . . . . . . . . . . . . . . 58

3.3

Nicolet iS50 FTIR spectrometer sensitivity versus number of scans averaged. . . . . 59

3.4

Schematic representation of the matrix isolation technique. . . . . . . . . . . . . . . 62

3.5

WMS signals for pure frequency modulation. . . . . . . . . . . . . . . . . . . . . . . 71

3.6

WMS signals including RAM effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xii
3.7

WMS signals including both RAM and etalon effects. . . . . . . . . . . . . . . . . . . 73

3.8

Auto-balanced subtractor circuit schematic. . . . . . . . . . . . . . . . . . . . . . . . 75

3.9

Dark noise of PV MCT detectors from Kolmar Technologies. . . . . . . . . . . . . . 77

3.10 Excess noise of the Daylight Solutions QCL. . . . . . . . . . . . . . . . . . . . . . . . 80
3.11 Single detector noise floor of the QCL absorption spectroscopy system. . . . . . . . . 82
3.12 Residual amplitude modulation on the Daylight Solutions QCL. . . . . . . . . . . . . 84
4.1

Infrared absorption spectrum of carbon clusters isolated in a neon matrix at 5 K . . 87

4.2

Number of each carbon cluster produced per pulse of the ablation laser as a functionof neon flow rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3

Number of each carbon cluster produced per pulse of the ablation laser as a functionof ablation pulse energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1

Calculated infrared rotational-vibrational spectrum of the Ω = 0 spin component of
the ground state of 184 WO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2

Calculated infrared rotational-vibrational spectrum of the Ω = 1 spin component of
the ground state of 184 WO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3

Infrared absorption spectrum of WO3 ablation products isolated in a neon matrix
at 5 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.4

A schematic of the buffer-gas cooling cell with secondary gas line. . . . . . . . . . . . 107

5.5

Infrared absorption spectrum of W plus O2 gas ablation products isolated in a neon
matrix at 5 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.6High resolution infrared absorption spectrum of W plus O2 gas ablation products
isolated in a neon matrix at 5 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

9.1

Example result of the method used to generate a smooth thermal cloud denisty froma swarm of test particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

xiii
9.2

Equilibrium density distributions and effective potentials for initializing a dynamic
simulation of the ZNG equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

9.3

Results from a full dynamical simulation of an equilibrium state of the ZNG equations.165

9.4

Results of a quench simulation showing a direct comparison between previously developed ZNG codes and the spherically symmetric code developed in this dissertation.167

10.1 Oscillation frequency of the in- and out-of-phase monopole modes. . . . . . . . . . . 172
10.2 Magnitude of the response of the in- and out-of-phase monopole modes to a trap
perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
10.3 Amplitude of the experimentally observed monopole mode oscillation. . . . . . . . . 176
10.4 Comparison between experimental measurements and ZNG simulations of the amplitude of the monopole mode oscillation. . . . . . . . . . . . . . . . . . . . . . . . . 180
10.5 Simulated mean-square radius of the condensate density during oscillation of themonopole mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
10.6 Damping rate of the in- and out-of-phase monopoles modes as a function of equilibrium temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Part 1: Infrared Spectroscopy of Buffer-Gas Cooled Molecules

Chapter 1

Introduction to Part 1

Molecular structure has a profound influence on the properties of materials and their applications in
the world around us. A rather striking example of this is given by the differences between diamondand graphite. Although both materials are composed entirely of elemental carbon, diamond is
one of the hardest known materials, is electrically insulating, and is visually transparent. On theother hand, graphite is soft, electrically conducting, and visually opaque. Therefore, diamond is
an excellent abrasive and an expensive, sought-after gemstone, while graphite can be used as alubricant and is so cheap that it is used to make pencils. The reason for such disparate properties
between these materials lies in differences of the underlying molecular structure.
In addition to material properties, molecular structure and bonding play a central role in
the effect of drugs on the human body, the perceived scent and taste of different compounds, aswell as in the dynamics of chemical reactions. In fact, as many as 85 − 90% of industrial chemical
processes utilize a catalyst to increase conversion efficiency and improve selectivity, examples ofwhich include oil refining, bulk chemical synthesis, and production of pharmaceuticals [1]. Catalysisof the chemical reactions that make up these processes rely on the bonding mechanism between
certain molecules and a catalyst (e.g., a metal oxide material) to lower energy barriers, allowing
particular reaction pathways to proceed. Thus, a detailed understanding of the molecular structureof catalysts and how molecules bond with them is crucial for the continued improvement of a wide
range of industrial chemical processes.
Perhaps one of the most widely used techniques for the determination of molecular structure

3
is high resolution spectroscopy of isolated molecules in the gas phase. Spectroscopic techniques
have become popular in this regard due to their extraordinary sensitivity to molecular shape,
which is governed by the bonding between the constituent atoms and their relative locations inthree-dimensional space. Through excitation of vibrational and rotational motion of a molecule atdiscrete energies, it is possible to characterize molecular bonds and the spatial arrangement of theconstituent atoms. In this dissertation, spectroscopy in the infrared region of the electromagnetic
spectrum is utilized to investigate molecular structure, and in the following section we begin witha brief historical overview of the development of spectroscopy.

1.1

Spectroscopy

One of the first accounts of an absorption spectrum dates back to 1802 when Wollaston observed
a series of dark lines interrupting the continuous spectrum of sunlight dispersed by a prism [2].
Soon after, Fraunhofer began investigating this series of dark lines from sunlight in more detail,and was able to measure the wavelength of some of the stronger lines using the first diffraction
grating [3]. In addition, Fraunhofer observed light from bright stars with a telescope and foundthat the patterns of dark lines were different than those from the sun. It was not until 1860
that Kirchhoff proposed a theory to explain the patterns of dark lines observed by Wollaston and
Fraunhofer [4]. Kirchhoff hypothesized that the dark lines in the solar spectrum were a resultof gases in the sun’s outer atmosphere absorbing continuum light at discrete wavelengths. These
ideas gave birth to the technique of spectral analysis, which was used by Kirchoff and Bunsen to
identify spectral fingerprints of a number of materials in the visible region of the electromagnetic
spectrum [5, 6].
In 1881, Abney and Festing extended the methods pioneered by Kirchhoff and Bunsen in
the visible to the infrared. They were able to obtain infrared absorption spectra of a wide rangeof compounds, and demonstrated that certain absorption bands were correlated with the presenceof certain molecular groups [7]. By the early 1900’s, the importance of infrared spectroscopy tochemical composition analysis was well-known, but commercial instruments did not become avail-4able until the 1940’s during World War II. With the advent of commercially available instruments,
the technique of infrared spectroscopy exploded in popularity due to its ability to provide a uniquespectral fingerprint for many complex molecules and information regarding molecular structure. In
tandem to the development of infrared spectroscopy, the theory of quantum mechanics was established, and the efforts of Born and Oppenheimer [8] laid the ground work for a quantum treatmentof molecular structure. This provided the necessary link between experimentally obtained infrared
spectra of a molecule and the arrangement of its constituent atoms.

1.1.1

Determination of molecular structure

The key component of the method presented by Born and Oppenheimer is to separate the molecular
Hamiltonian into its electronic and nuclear components. This separation is valid in many scenarios
due to the large mass difference between electrons and nuclei; therefore, the electronic component
can be solved under the assumption that the nuclei are stationary (i.e., neglecting the nuclear kinetic
energy) followed by solution of the nuclear motion. The former gives rise to discrete electronicenergy levels, which coincide with the energy of visible photons, whereas the latter gives rise todiscrete rotational and vibrational energy levels of the molecule, which coincide with the energy ofinfrared photons. Therefore, infrared spectroscopy effectively probes excitations of the mechanicalmotion of the constituent atoms in a given molecule. It is then possible to relate the excitationenergy of this mechanical motion to molecular properties such as bond length, orientation, andstrength.
Figure 1.1 gives a simplified representation of vibrations and rotations in a diatomic molecule,and likens these motions to the classical analogues of a simple harmonic oscillator and rigid rotor,
respectively. From the figure, it is clear that by measuring the change in vibrational or rotationalenergy upon absorption of an infrared photon it is possible to determine the natural vibrationaland rotational frequencies, ωe and Be , respectively. These parameters can then be related to
the force constant kv (i.e., bond strength) and the equilibrium distance between the constituent
atoms re (i.e., bond length), which determines the molecular structure. Now, this simplistic picture

5

Simple Harmonic Oscillator

Rigid Rotor

Bond strength

Bond length

Figure 1.1: A simplified representation of vibrations and rotations in a molecule, and their relation to the classical analogues of a simple harmonic oscillator and rigid rotor, respectively. A
measurement of the change in vibrational energy (proportional to ωe ) provides information about
the force constant kv (i.e., bond strength), while a measurement of the change in rotational energy(proportional to Be ) provides information about the equilibrium distance between the constituent
atoms re (i.e., bond length). The parameters h, c, and µ are Planck’s constant, the speed of light,
and the reduced mass of the molecule, respectively.

6

Figure 1.2: Available broadband and widely tunable sources for spectroscopy in different spectral
regions. Adapted from Ref. [11].

becomes much more complicated for larger, polyatomic molecules; however, it is still possible todetermine molecular structure through the measurement of infrared spectra when the technique
is combined with other methods such as isotopic substitution and ab initio calculations [9, 10].
Nonetheless, these methods require the precise determination of rotational constants Be for manydifferent isotopologues, hence the desire for broad spectral bandwidth, high sensitivity, and highresolution spectroscopic techniques.

1.1.2

Modern techniques

There exists a very broad range of spectroscopic techniques in use today. Therefore, the present
discussion will be limited to those used for high resolution absorption spectroscopy of isolated gasphase molecules, which are most relevant to this dissertation work. Figure 1.2 provides an overview
of currently available broadband and widely tunable sources used for high resolution spectroscopy.
Perhaps the biggest advantage of broadband spectroscopy is the ability to simultaneously acquire
high resolution spectra at many different frequencies, which greatly reduces measurement time.
At the forefront of modern broadband techniques is frequency comb spectroscopy, which combines
the benefits of broad spectral bandwidth, high spectral resolution, and high spectral intensity intoa single technology [11, 12]. Furthermore, use of a high finesse optical cavity, such as in cavity-

7
enhanced direct frequency comb spectroscopy [13], provides a substantial improvement in detection
sensitivity. However, frequency combs are still an emerging technology making them relatively
unavailable in the infrared. In this region of the electromagnetic spectrum, a typical broadband
source consists of a hot filament (e.g., globar) that acts as a blackbody; however, recent work
has demonstrated promising broadband infrared sources based on supercontinuum generation inchalcogenide fiber [14], difference frequency generation in LiGaS2 [15], and modelocked quantum
cascade lasers [16]. Notably, broadband sources have two potential drawbacks when comparedto single frequency sources used for spectroscopy – either low spectral brightness, low spectral
resolution, or both. This generally limits the achievable sensitivity in an absorption spectroscopy
experiment to roughly 10−5 − 10−4 Hz−1/2 and the spectral resolution to a few hundred MHz
or more [11]. Therefore, techniques like tunable diode laser spectroscopy offer an alternative tobroadband spectroscopy that achieves both increased sensitivity and spectral resolution.
The major benefit of tunable diode laser spectroscopy is that the full output power of thelaser is concentrated into a single spectral element with a bandwidth set by the laser linewidth(e.g., typically < 1 MHz for external cavity diode lasers). Thus, these techniques can achieve highsensitivity and high resolution, but require much longer acquisition times to cover even modest
spectral windows on the order of a few wavenumbers. This is a major disadvantage when compared
to broadband spectroscopy techniques. Nonetheless, sensitivities on the order of 10−13 Hz−1/2
have been achieved with sub-MHz resolution [17, 18], which enables investigations of weak spectralfeatures such as overtones and combination bands among other things. Furthermore, there aremany perturbations to spectra (e.g., see Refs. [19, 20]) that become prominent at the high spectral
resolution achievable with tunable diode laser spectroscopy.
To date, perhaps the most widely used experimental method for absorption spectroscopy of
isolated gas-phase molecules utilizes a free jet source combined with a tunable laser coupled intoa multipass cell or optical cavity to probe the molecules of interest. This arrangement provides
a number of advantages: the free jet source provides substantial cooling of the molecular internal
degrees of freedom, the multipass cell or optical cavity provides increased sensitivity, and the

8tunable laser provides high spectral brightness and resolution. Indeed, this method has beenused for studying a wide range of molecular systems, such as carbon clusters [21], Van der Waals
complexes [22], radicals [23], large organic molecules [24], and many others. The following section
will review some aspects of molecular beams, including the free jet source, before introducing themethod used in this dissertation: the cryogenic buffer-gas beam source.

1.2

Molecular Beams

In molecular spectroscopy, the interest and widespread use of molecular beams stems from perhaps
three very useful aspects [25]: isolation of the molecular species of interest, the ability to coolmolecular internal degrees of freedom to the order of 1 K, and the ability to control or perturb
the species of interest in a deterministic way. For spectroscopic purposes, the first two of these are
crucial for minimizing spectral line broadening and ensuring that the molecules of interest populate
only a handful of internal states – a very useful way of decongesting spectra and easing theirinterpretation. Furthermore, isolation of molecular species is critical to the study of highly reactive
free radicals. Finally, the ability to control or perturb a beam of isolated molecules is useful fortechniques like Stark and Zeeman spectroscopy, for example, which can aid in the determinationof molecular dipole moments [26].

1.2.1

Traditional beam sources

Traditional molecular beam sources largely fall into two categories: effusive sources and free jet
sources. In its simplest form, an effusive beam is formed by allowing a gas to leak out of a small
hole in a chamber that contains the molecular species of interest at a given temperature andpressure [25]. By maintaining a low pressure inside the chamber, the conditions for molecular flow
are ensured and no collisions occur between molecules as they exit the chamber to form a beam.
On the other hand, a free jet source produces a molecular beam via supersonic expansion of a highpressure gas into vacuum [25]. By maintaining a large pressure differential across a small aperture,
the conditions for continuum flow are ensured and many collisions occur between molecules as they

9form a beam. Thus, these two beam sources give rise to very different properties, and have certain
advantages and disadvantages summarized here [27]:• Effusive sources are nearly universally applicable. In addition, beam properties are welldefined and easily calculated from kinetic theory, and the molecular species has internal
state populations dictated by the equilibrium state of the source. However, many species
require high temperatures to create appreciable vapor pressure such that the molecularspecies will occupy a large number of internal states, putting an increased demand on thesensitivity of a spectroscopy system. Additionally, effusive sources have low brightness
compared to free jet sources.
• Free jet sources are also nearly universally applicable when combined with techniques likelaser ablation. In addition, beam properties are less well defined, and the molecular species
typically has non-equilibrium internal state populations. Furthermore, optimal operating
conditions tend to be empirically determined. However, the degree of translational andinternal state cooling achievable with a free jet source is unprecedented, which significantly
reduces the number of internal states populated. Additionally, free jet sources have substantially higher brightness than effusive sources.
In regards to molecular spectroscopy, perhaps the single most beneficial aspect of a free jet sourceis the degree of translational and internal state cooling that can be achieved. Not only does thedegree of cooling minimize the deleterious effects of Doppler broadening on the resolution of spectral
lines, it also reduces the demand on the sensitivity of a spectroscopy system by concentrating themolecular species of interest into a handful of internal states.
Given its widespread use in molecular spectroscopy, it is useful to consider the free jet source
in more detail. Ideally, a free jet represents an isentropic expansion from a high pressure (P ∼1 atm), high density (n ∼ 1019 cm−3 ) gas inside a chamber through a planar aperture into vacuum(P ∼ 10−8 atm). Such a process results in a rapid decrease in temperature and density, which canbe understood from the ideal gas relations Tb = T0 (Pb /P0 )2/5 and nb = n0 (Pb /P0 )3/5 , where the10subscripts indicate initial properties inside the chamber and final properties in the beam. Therefore,
we see that Tb ∼ 1 K and densities low enough to isolate molecules from collisions are readily
achievable. Typically, the molecular species of interest is introduced by seeding a carrier, or buffer
gas like argon or neon via laser ablation of a solid target placed immediately outside of the chamber.
In this way, the molecules of interest are “captured” by the supersonically expanding carrier gas,and collisions between the molecules and carrier gas reduce the molecule temperature. An exampleof this style of source is depicted in Fig. 1.3. Notably, a free jet results in a molecular beam with acarrier-gas forward velocity of several hundred m/s, and collisions between the molecular species ofinterest and the carrier gas occur over a propagation distance of a few nozzle diameters. Therefore,
the collisional cooling timescale for a traditional free jet source is on the order of 10 µs. For many
molecular species, this timescale is long enough to cool molecules to the order of 1 K. However,
it is difficult to cool large, strongly bound cluster molecules such as metal oxides and carbides
due to the additional energy they acquire during condensation in the gas phase. Such molecules
typically have binding energies of a few electron volts per bond, and this energy is deposited into
the cluster every time an atom is added [28]; therefore, the molecule heats as it forms. As a resultof this condensation heating, cluster molecules formed with a free jet source typically do not reachtemperatures on the order of 1 K. Furthermore, clusters of different sizes generally have different
temperatures as a result of larger clusters forming later in the supersonic expansion [28]. As a resultof this disadvantage, it is of general interest to investigate other methods for producing molecularbeams of cold cluster molecules, such as the cryogenic buffer-gas beam source [29].

1.2.2

Buffer-gas beam sources

Buffer-gas beam sources have a number of advantages over traditional free jet sources used inconjunction with laser ablation of a solid target, such as lower forward velocity, higher brightness,
and a longer collisional cooling timescale [29]. Notably, the free jet source readily achieves peakcarrier-gas densities 100 − 1000 times greater than a buffer-gas beam source; however, as discussed
above, molecules ablated in the vicinity of a supersonic expansion typically interact with carrier

11

Figure 1.3: Example of a free jet source design that utilizes laser ablation of a solid target to
introduce the molecular species of interest to a supersonic expansion of buffer gas. Reproduced
from Ref. [28].

12
gas of this density for times on the order of 10 µs, whereas this timescale can be on the order of
10 ms inside a buffer-gas cell [29]. Therefore, it is likely that molecules of interest in a buffer-gas
beam source experience more collisions with a high density, cold buffer gas, which could lead tolower internal state temperatures for many molecules compared to a free jet source. An additional
benefit of the larger number of collisions in a buffer-gas source is the possibility of lower internal
state temperatures for large, strongly bound clusters, such as carbon clusters or clusters of metaloxides and carbides. To date, many molecular species have been successfully cooled with a buffer-gas
beam source, including diatomic [30–35] and polyatomic [36–41] molecules, but there are relatively
few high-resolution spectroscopic studies of complex polyatomic molecules produced with a buffergas beam source (i.e., as in Refs. [41, 42]). Indeed, the ability to cool complex molecules to theirvibrational ground state will remove much of the ambiguity associated with the spectral assignment
process, and could provide validation for the large variety of theoretical techniques currently being
used in conjunction with experiments to determine molecular structures.

1.3

Dissertation Outline

This dissertation consists of two very different projects; therefore, it is separated into two distinct
parts. Part 1 of this dissertation consists of Chapters 1–6, and discusses an experimental project
regarding high resolution spectroscopy of cold molecules in the gas phase. This project involves thecharacterization of a buffer-gas beam source for the production of carbon clusters and metal oxide
molecules, and infrared spectroscopy of these molecules using different techniques. Part 2 of thisdissertation consists of Chapters 7–11, and discusses a theoretical project regarding the dynamics
of collective modes of degenerate Bose gases in spherical symmetry. This project involves theanalysis of data acquired by D. S. Lobser with the JILA TOP trap experiment, and the subsequent
theoretical analysis undertaken to explain the experimental observations.
Part I of this dissertation is organized as follows. Chapter 2 provides a discussion of the operating principles of buffer-gas beam sources, with a focus on collisional cooling and beam formationin different regimes of operation. A comparison between the buffer-gas beam source used for this13
dissertation work and a traditional free jet source is discussed at the end of Chapter 2. Chapter
3 provides an overview of some experimental techniques for infrared absorption spectroscopy. Included is a discussion of Fourier transform infrared spectroscopy, matrix isolation spectroscopy, andlaser absorption spectroscopy, as well as the techniques of wavelength modulation and balanced subtraction. Chapter 4∗ describes an experimental investigation of the production of carbon clusters
with a buffer-gas beam source, and highlights some of the non-ideal characteristics of beam formation with such a source. Chapter 5 investigates the production and infrared rotational-vibrational
spectrum of the WO molecule using a modified buffer-gas beam source and tunable diode laser
absorption spectroscopy system. Chapter 6 summarizes the first part of this dissertation, and provides a future outlook. Finally, Part 2 of this dissertation begins in Chapter 7, where a separate
introduction and outline can be found.

∗
Chapter 4 represents a publication that was a collaborative effort between the author of this dissertation, Maya
I. Fabrikant, Gary E. Douberly, and Heather J. Lewandowski [43]. The author of this dissertation was the primaryauthor of the paper, and performed the experiments and subsequent data analysis. M. I. Fabrikant constructed the
majority of the experimental apparatus, G. E. Douberly and H. J. Lewandowski provided project definition, and H.
J. Lewandowski provided guidance throughout the project.

Chapter 2

The Cryogenic Buffer-Gas Beam Source

Both supersonic expansion and buffer-gas beam sources rely on collisions with a dense, cold, andinert gas to cool hot molecules from > 1000 K into the few kelvin regime; however, their construction
is significantly different. In its simplest form, a buffer-gas beam source consists of a cell with aconstant flow of inert buffer gas (e.g., helium or neon) entering through a small aperture. The cellis held at a temperature on the order of 10 K to cool the buffer gas, into which the molecules ofinterest are introduced through laser ablation of a solid target located inside the cell, or via a warm
capillary held in the vicinity of the entrance aperture. The molecules of interest become entrainedin the flow of cold buffer gas where they rapidly cool and eventually exit the cell through a small
aperture to form a bright beam of cold molecules.
In this chapter, we discuss the principles of cryogenic buffer-gas beam (CBGB) sources witha focus on collisional cooling and extraction efficiency. As noted in the Introduction, CBGB sources
have a longer collisional cooling timescale compared to supersonic jet sources. Thus, it is expectedthat lower vibrational temperatures for complex polyatomic molecules could be achieved with a
CBGB source, which could aid in the structural determination of these large molecules with highresolution spectroscopy. Furthermore, understanding the extraction efficiency of CBGB sources
is useful in maximizing the flux of cold molecules produced with such a source. By optimizing
this process, beams of higher brightness can be produced and less demand placed on the detection
process. The extraction efficiency of a CBGB source can be estimated in a simple fashion; however,certain non-ideal behavior is encountered where these simple estimates breakdown. Hutzler et15
al. [29] have written a good introduction on the construction and performance of buffer-gas beamsources.

2.1

Buffer-Gas Cooling Cell

A schematic of the buffer-gas cooling cell used in this dissertation work is shown in Fig. 2.1. It
consists of a copper box with a cylindrical bore of length Lcell = 40 mm and diameter dcell = 20 mm.
The flow of neon buffer gas into the cell is controlled by a mass flow controller (Alicat Scientific,
MC-100SCCM-D), and first enters a small reservoir at the top of the cell to promote laminar flow
through the main compartment of the cell. A thin aperture of diameter dap = 5 mm located in thebottom face of the cell allows for beam extraction. Through a weak connection to the second stageof a two-stage cryocooler (SHI Cryogenics Group, RDK-415D), the cell is held at a temperatureof Tcell = 25 K to prevent neon gas from freezing to the cell walls. Note that the neon gas line isalso anchored to the first stage of the cryocooler to cool the neon gas from room temperature toapproximately 30 K prior to it entering the buffer-gas cell.

2.1.1

Buffer-gas flow properties

To begin, it is useful to discuss some generic properties of gas flows and the terminology used todescribe different flow regimes. For simplicity, we will assume the neon buffer gas is an equilibriummonatomic ideal gas that obeys Maxwell-Boltzmann statistics.∗

In discussing properties of gas

flows, it is useful to introduce three dimensionless quantities: the Reynolds number, the Mach
number, and the Knudsen number. The Reynolds number is typically used to identify laminar andturbulent flow regimes, and is defined as [45]
Re =

uL
,
ν

(2.1)

∗
Strictly speaking, this is not entirely justified; however, this assumption will allow us to perform simple analytic
calculations and get an idea for how the buffer-gas flow is affected by different system parameters. Rigorous treatmentof the buffer-gas flow at low temperatures requires complex fluid dynamics simulations and a more realistic equationof state. See Refs. [33, 44], for example.

16

Neon
inlet

Neon
reservoir

Target

Lcell

Ablation
laser

dcell
dap

Figure 2.1: A schematic of the buffer-gas cooling cell. It consists of a copper box with a cylindricalbore (Lcell = 40 mm and dcell = 20 mm) drilled through the center. A thin plate attached tothe bottom contains an aperture of diameter dap = 5 mm. Neon gas (blue arrows) first enters areservoir at the top of the cell before flowing into the main cell, and exits through the aperture at
the bottom. The ablation laser enters through an uncoated window attached to the outside of thecell, and is focused onto a solid target attached to the inside of the cell.

17
where u is the flow velocity, L is a characteristic length scale (i.e., aperture diameter of the buffergas cooling cell dap ), and ν is the kinematic viscosity of the gas. At low Re the flow is expected to
be non-turbulent and predominantly laminar in nature. As Re increases, the onset of turbulenceoccurs and the properties of the flow can become very complex and even chaotic. The Mach number
is typically used to identify when a flow can be treated as incompressible, and is defined as [45]Ma =where cs =

u
,
cs

(2.2)

p
γkB T /m is the speed of sound in the gas flow, γ is the ratio of specific heats for an

ideal gas (γ = 5/3 for a monatomic ideal gas like neon), kB is Boltzmann’s constant, T is the flow
temperature, and m is the mass of a gas atom. For M a . 0.3, real gas effects like compressibility
will have minimal impact on the flow and can be neglected. However, as M a approaches unity
this approximation breaks down and more sophisticated methods are required to study the flow.
Finally, the Knudsen number is typically used to determine whether or not the flow can be treatedas fluid-like, and is defined as [46]Kn =where λ =

√

2nσ

−1

λ
,
L

(2.3)

is the mean free path, n is the number density of gas atoms in the flow,and σ is the elastic collision cross section between gas atoms. When Kn is much less than unity,
the gas is in the hydrodynamic regime (i.e., many collisions occur over the length scale of thegas keeping it in local statistical equilibrium) and can be treated as a fluid. On the other hand,when Kn approaches unity the gas must be treated statistically in the so-called free molecular flow
regime because very few collisions occur over the length scale of the gas. Finally, by introducing
the relation ν = λv̄/2 that relates the dynamic viscosity to the mean free path and mean thermalpvelocity v̄ = 8kB T /πm, we derive a relationship between Kn, Re, and M a. The result is [46]rKn =πγ M a
.
2 Re

(2.4)

With the above dimensionless quantities and relations between them, we require an estimateof the equilibrium density of buffer gas inside the cooling cell to fully characterize the flow. We

18find the density inside the cell by considering that all of the gas flowing into the cell must leavethrough the exit aperture. Here, the neon flow rate into the cell is given by fi,Ne in units of sccm,
and typically lies in the range of 1 − 100 sccm. The flow of gas out of the cell is dictated by the
conductance of the aperture, which is given byC=

Aap v̄
,4(2.5)

in the molecular flow regime (i.e., Kn & 1), andAap v̄
C=
2

r

πγ
γ+1



2
γ+1

 1

γ−1

,

(2.6)

in the continuum flow regime (i.e., Kn . 0.01) [47]. In arriving at the final expression for C in thecontinuum flow regime, we have assumed choked flow (i.e., M a = 1 in the aperture), which occurs
for a large pressure difference between the buffer-gas cell and the surrounding vacuum. Thus, we
calculate the density of neon gas inside the cell using the relationnNe = 4.47796 × 1017



fi,Ne
C


,

(2.7)

where the prefactor converts fi,Ne from units of sccm to units of atoms per second.
Figure 2.2 shows the density of neon gas inside the cell as a function of flow rate, as well as
the Knudsen number in the exit aperture of the cell. To calculate the Knudsen number, we use
the density from Eq. (2.7) and the Ne-Ne collision cross section σ = 3.02 × 10−15 cm2 based on a
hard-sphere model with minimum potential radius from Ref. [48]. We find that both the density
and Knudsen number are relatively insensitive to the assumed flow, varying by roughly a factor of
two. Notably, the Knudsen number falls in the range between that for molecular and continuum
flow, implying that the buffer-gas cell operates in the transitional flow regime. Nonetheless, we will
focus our study on the limiting cases of molecular and continuum flow for simplicity. Finally, we
use Eq. (2.4) to find the Reynolds number in the exit aperture of the cell. Here, we will consider
the case of continuum flow in the aperture such that M a = 1, which will result in the highest
achievable Reynolds number in the aperture. Figure 2.3 shows the Reynolds number in the exit cellof the aperture as a function of flow rate. Comparing to flow through a pipe of constant diameter,

19

6

(a)

Molecular flow
Continuum flow

Neon density (x10 16 cm-3)

5

4

3

2

1

0
0

10

20

30

40

50

60

70

80

90

100

Flow rate (sccm)

Knudsen number

10 0

(b)

10 -1

10 -2

0

10

20

30

40

50

60

70

80

90

100

Flow rate (sccm)

Figure 2.2: (a) Density of neon gas inside the cell and (b) Knudsen number in the exit apertureof the cell assuming molecular flow (solid) and continuum flow (dashed) through the exit aperture.
The cell temperature is Tcell = 25 K and the Ne-Ne collision rate is σ = 3.02 × 10−15 cm2 . The
aperture diameter is dap = 5 mm.

20

130
120
110

Reynolds number

100
90
80
70
60
50
40
30
20
10
0
0

10

20

30

40

50

60

70

80

90

100

Flow rate (sccm)

Figure 2.3: Reynolds number of the neon gas flow through the exit aperture of the cell. The celltemperature is Tcell = 25 K, the Ne-Ne collision rate is σ = 3.02 × 10−15 cm2 , and continuum flow
has been assumed with M a = 1. The aperture diameter is dap = 5 mm.

21
the maximum Re in the buffer-gas cooling cell is well below the transition point between laminarand turbulent flow (Re ∼ 2000) [45]. Thus, it is expected that the flow through the cell will be
predominantly laminar. However, because the cell diameter changes along its length in a step-wisemanner it is possible that some large scale flow effects like eddies could form inside the cell inthe vicinity of these steps, particularly near the exit aperture of the cell (e.g., as in Ref. [44]).
Nonetheless, given the small value of Re, small scale turbulence and chaotic flow is not expected,†
but an improved design might consider smoothly tapering the cell dimensions from dcell to dap inan effort to minimize these eddies (e.g., as in Ref. [49]).

2.1.2

Collisional cooling – translational relaxation

During their residence time inside the cell, the molecules of interest will undergo many collisions
with the buffer gas, and much of their initial energy gained from the ablation process will bedissipated. We find a rough estimate of the final temperature of the translational motion of themolecules by treating the collisions classically as elastic collisions between hard spheres (i.e., billiard
balls) [29, 50]. We begin by considering a single elastic collision between a molecule of interest and abuffer gas atom with mass mm and mb , respectively. The initial and final velocities of the moleculeand buffer-gas atom are vm and vb , and um and ub , respectively. Furthermore, it is useful to define
the velocity of the center of mass (COM) V and transform to this reference frame:
v0m = vm − V,

(2.8)

v0b = vb − V,

(2.9)

u0m = um − V,

(2.10)

u0b = ub − V,(2.11)

where the prime indicates velocities in the COM reference frame. From conservation of energy andmomentum, we find that the center of mass velocity is unchanged in the course of the collision, so†

The Reynolds number indicates at which length scale dissipation of energy occurs, so small values of Re imply
that large scale motion like eddies can dissipate energy directly. On the other hand, large values of Re imply thatlarge scale motion must first cascade to smaller scales before energy dissipation can occur (i.e., turbulence).

22
we writeV=m m vm + m b v b
mm um + mb ub=.
mm + mb
mm + mb(2.12)

Furthermore, in the COM reference frame, the total momentum is equal to zero before and after
the collision, a property we use to findmm v0m = −mb v0b ,(2.13)

mm u0m = −mb u0b .

(2.14)

Finally, using this result in conjunction with conservation of energy we find
2

v0m

2

= u0m ,v0b

2

= u0b .

2

(2.15)(2.16)

The above relations will now help us find the change in kinetic energy for a molecule undergoing acollision. In the lab frame, the final kinetic energy of the molecule is proportional to |um |2 :|um |2 = u0m + V
= u0m

2

2

+ |V|2 + 2u0m · V,(2.17)

whereas in the COM frame, the final kinetic energy of the molecule is proportional to |u0m |2 :
u0m

2

= v0m

2

= |vm − V|2= |vm |2 + |V|2 − 2vm · V




2mb
2mm
|vm |2 + |V|2 −vm · vb .
= 1−
mm + mb
mm + mb(2.18)

Combining these expressions and expanding V in terms of the initial velocities of the molecule andbuffer-gas atom, we arrive at the following expression:



2m2b
2m2m
2mm
2
|um | =−
+ 1 |vm | +|vb |2(mm + mb )2 mm + mb
(mm + mb ]2
2mb (mm − 1)+vm · vb + 2u0m · V.
mm + mb2

(2.19)

23
Given the final velocity of the molecule in the lab frame for a single collision, we investigate theaverage change in kinetic energy for many collisions assuming the molecules and buffer-gas atoms
have Maxwell-Boltzmann velocity distributions. Under this assumption, we use the equipartition
theorem to equate the average initial and final kinetic energies to translational temperatures:
3kB Tm,f,mm
3kB Tm,i,h|vm |2 i =mm
3kB Tbh|vb |2 i =.
mbh|um |2 i =(2.20)(2.21)(2.22)

Note that we do not consider the change in kinetic energy of the buffer-gas atoms as we are working
under the assumption that on average they remain at a fixed temperature due to their constantinteraction with the buffer-gas cooling cell. Taking the statistical average of Eq. (2.19) and using
the equipartition result, we arrive at the following expression for the final molecule temperature
after a single collision:
Tm,f − Tm,i = −2mm mb(Tm,i − Tb ) ,(mm + mb )2(2.23)

where we have used the fact that the statistical average of the dot products in Eq. (2.19) are zeroRπbecause hcos θi = −π dθ cos θ = 0. We then extend this result to N collisions by rewriting as adifferential equation [29, 50]dTm
2mm mb=−(Tm − Tb ) ,dN(mm + mb )2(2.24)Tm (N ) = Tb + (Tm,i − Tb ) e−N/κ ,(2.25)

which has the solution

where κ = (mm + mb )2 /2mm mb has been introduced. Note that κ is minimum when mm = mb
implying that molecules with a similar mass to the buffer-gas atoms will be cooled most effectively.
With a simple model for the translational cooling process in hand, it is of interest to investigate its behavior for some typical parameters. The temperature of the buffer gas is wellapproximated by the temperature of the cell, which is typically about 25 K for neon buffer gas. On

24
the other hand, the initial temperature of the molecules of interest typically lie in the range of about
103 − 104 K for a laser ablation source; therefore, we make the approximation that Tm,i − Tb ≈ Tm,i .
Using this approximation, we then determine the number of collisions required for the moleculetemperature to be within X% of the buffer gas temperature:
NX% ≈ κ ln

100%
X%




Tm,i
.
Tb

(2.26)

It is necessary for this number of collisions to occur before the molecules either leave the cell or
freeze out on the cell walls. We estimate the number of collisions that occur inside the cell bycomparing the expected mean free path of molecules in the buffer gas to the average length scaleof the cell. The mean free path of a molecule moving through the buffer gas is given by
λm,b ==v̄m
nb σm,b v̄m,b

,

1
q
m Tb
nb σm,b 1 + m
mb Tm(2.27)

where σm,b is the collision cross section between a molecule and a buffer-gas atom, nb is the densityof buffer gas inside the cell, v̄i is the mean thermal velocity of the molecules (i = m) or buffer-gas
q
2 + v̄ 2 is the relative mean thermal velocity between the molecules
atoms (i = b), and v̄m,b = v̄mb
and buffer-gas atoms. Now, Tm is a function of the number of collisions (see Eq. (2.25)), so tofind the number of collisions occurring over the average length scale of the cell hLi we solve thefollowing integral equation for NhLi ,Z NhLidN λm,b (N ) = hLi,

(2.28)

0which must be done numerically.
When molecules are ablated from the target, we assume their motion through the buffer
gas is primarily unidirectional because of the large initial temperature difference. Therefore, we
assume the average distance travelled between collisions is proportional to λm,b rather than its
square root, as would be the case for a diffusive process (i.e., a random walk). Diffusion begins
once the translational temperature of the molecules is equivalent to that of the buffer-gas atoms.

25
The quantities NX% and NhLi are plotted in Fig. 2.4 for various molecule masses and assuming
neon buffer gas with mb = 20.18 amu. Thus, from the figure, we expect that molecules with a massfrom a few amu to a few hundred amu will achieve a translational temperature roughly equivalent
to that of the buffer gas by the time they travel a distance on the order of the average length ofthe cell. Notably, this translational relaxation inside the cell will occur on a timescale of roughly
100 µs. Finally, once the molecules have equilibrated with the buffer-gas atoms in the cell they will
begin a diffusive random walk towards the cell walls where they will freeze and be lost. During thisrandom walk, the molecules will continue to collide with the buffer-gas atoms at a rate given by
Rm = nb σm,b v̄m,b .

(2.29)

Thus, we calculate the number of additional collisions occurring over time τ using ND = Rm τ .
Assuming mm = 100 amu, mb = 20.18 amu, nb = 1016 cm−3 , σm,b = 10−15 cm2 , and T = 25 K
we find ND ≈ 175 − 1750 for τ = 1 − 10 ms. Here, the timescale is chosen to coincide with the
expected time it takes for molecules to be extracted from the cell, which will be discussed further
below. Therefore, by the time the molecules are extracted from the cell, we expect them to haveundergone on the order of a few hundred to a few thousand collisions with the buffer-gas atoms
depending on the buffer-gas flow rate into cell.

2.1.3

Collisional cooling – internal state relaxation

Thus far, we have been concerned with the translational temperature of the molecules, and how
it relaxes through collisional interactions with the cold buffer gas. However, molecules also haveinternal degrees of freedom that correspond to rotational and vibrational excitations, and it isexpected that many of these states will be initially populated after the laser ablation process.
For example, assuming typical rotational and vibrational transitions with ν̃ = 1 cm−1 and ν̃ =1000 cm−1 , respectively, we expect thousands of rotational states and a few vibrational states to be
populated at a temperature of 5000 K. Fortunately, this excess internal energy can be transferred tothe translational motion of the buffer-gas atoms. However, in order for this energy transfer to occur,

26

Number of collisions

10 2

N 10%
N 1%
N< L >
10 1
10 0

10 1

10 2

10 3

Molecule mass (amu)

Figure 2.4: Number of collisions required for the molecule translational temperature to be within
10% (solid) and 1% (dashed) of the buffer gas temperature, and number of collisions that occur over
the average length scale of the cell (dash-dot). Assumes Tm,i = 5000 K, Tb = 25 K, mm = 100 amu,
mb = 20.18 amu, nb = 1016 cm−3 , σm,b = 10−15 cm2 , and hLi = 30 mm.

27
the interaction must be non-adiabatic such that the interaction time between the excited moleculeand atom is short compared to the oscillation period of the molecular excited state (i.e., impulsive
interactions). We investigate the adiabaticity of typical elastic collisions between molecules andbuffer-gas atoms through the following adiabaticity parameter [51–53]:
ζ = c τc ν̃,(2.30)

where c is the speed of light in vacuum, τc is the collisional interaction timescale, and ν̃ is thetransition wavenumber corresponding to a rotational or vibrational state in the molecule. If ζ  1
the interaction is non-adiabatic and energy transfer can occur. On the other hand, if ζ  1 theinteraction is adiabatic and energy transfer is unlikely.
We estimate the collisional interaction timescale in a hard sphere collision model as
τc =

r0
,
v̄m,b

(2.31)

where r0 is the effective range of the potential between the molecule and a buffer-gas atom and v̄m,bis the relative mean thermal velocity between the molecules and the buffer-gas atoms as defined
above. The parameter ζ is plotted in Fig. 2.5 assuming a typical r0 ∼ 1 Å for interactions between
neutral atoms and molecules. From the figure, we see that higher molecule temperatures lead tosmaller values of ζ implying collisional energy transfer is more effective in the initial stages of coolinginside the buffer-gas cell. We find that ζ < 1 for ν̃ < 50 cm−1 even at a molecule temperature of
25 K; thus, excited rotational states (ν̃ ∼ 1 cm−1 for rotational transitions is typical) are likely
to relax quite efficiently even once the translational motion of the molecules has equilibrated withthe buffer-gas atoms. However, excited vibrational states (ν̃ ∼ 300 − 3000 cm−1 for vibrationaltransitions is typical) are likely to not relax very efficiently at low temperatures. Nonetheless, wefind ζ < 1 for ν̃ < 400 cm−1 at a molecule temperature of 5000 K such that some vibrational
relaxation for low frequency modes is likely to occur in the initial stages of collisional cooling inside
the buffer-gas cell.
Given this simple picture of internal state relaxation via elastic collisions with cold buffer-gas
atoms, it is of interest to compare our expectations with experimental results from the literature.

28

10 2
Tm = 25 K

Adiabaticity parameter, 1

Tm = 5000 K
10 1

10 0

10 -1

10 -2
10 0

10 1

10 2

10 3
-1

10 4

Transition wavenumber (cm )Figure 2.5: Adiabaticity parameter, ζ, for interactions between molecules and neutral atoms at
a molecule temperature of Tm = 25 K (solid) and Tm = 5000 K (dashed). It is assumed that theinteraction has an effective range of r0 = 1 Å, the molecules have a mass of mm = 100 amu, andneon buffer gas with mb = 20.18 amu at a temperature of Tb = 25 K is used.

29
Many molecular species have been investigated with buffer-gas beam sources, and it is generally
found that both the translational and rotational degrees of freedom exhibit temperatures equivalent to the buffer-gas temperature. For example, it has been observed that both translationaland rotational temperatures equilibrate with the buffer gas for diatomic metal oxides [30, 31] andfluorides [32–35], as well as for more complex polyatomic molecules [36–41]. This is the case forboth helium and neon buffer-gas sources. However, the few studies that have also investigated thevibrational degree of freedom have found it to be out of thermal equilibrium. In the case of SrF thevibrational level spacing is approximately 500 cm−1 [54]; therefore, approximately 10 vibrational
states will be initially populated for typical ablation temperatures and some initial cooling is expected based on ζ ∼ 1 at 5000 K. Indeed, Barry et al. [32] observed substantial population in only
the first few vibrational levels of a SrF buffer-gas beam implying that some vibrational relaxation
occurred inside their buffer-gas cooling cell. Similar behavior was also observed by Bu et al. [34] for
BaF, which is similar to SrF with a vibrational level spacing of approximately 450 cm−1 [55]. Basedon the results of these studies we can conclude that our expectations for internal state relaxation
based on adiabaticity arguments are reasonable, but certainly far from complete.

2.1.4

Extraction

Once the translational motion of the molecules has equilibrated with the buffer-gas atoms, themolecules will diffuse toward the cell walls and be lost unless they are extracted from the cell.
Therefore, the efficiency of the extraction process can be characterized in terms of two competingtimescales: the diffusion time and the time it takes to evacuate the cell through the exit aperture, or
pumpout time. We define a dimensionless parameter to characterize the extraction efficiency [29],γcell =

τD,τp(2.32)

where τD is the effective diffusion time for molecules to reach the cell walls and τp is the pumpout
time. When γcell . 1 the molecules diffuse to the cell walls faster than they are extracted fromthe cell, and the extraction efficiency is low. On the other hand, when γcell & 1 the molecules are30
extracted from the cell faster than they diffuse to the cell walls, and the extraction efficiency ishigh. We obtain an estimate for this parameter by looking at the diffusion and pumpout processes
in more detail.
The rate at which buffer-gas atoms leave the cell is governed by the conductance C of theaperture; therefore, the following differential equation describes the pumpout process:
Ṅb =

C Nb
,
Vcell

(2.33)

where Vcell ≈ 12.5 cm3 is the volume of the buffer-gas cooling cell. Solution of this equation is adecaying exponential with a characteristic decay time given by
τp =

Vcell
,
C

(2.34)

which defines the pumpout time of the cell. Assuming Tcell = 25 K, we find τp = 15.8 ms formolecular flow and τp = 8.7 ms for continuum flow through the exit aperture. Calculating theeffective diffusion time is slightly more complicated. For a dilute, binary gas mixture, the diffusion
coefficient Dm,b for the molecules diffusing into the buffer gas is given by the Chapman-Enskog
result [56]:3
Dm,b =
16 nb σm,bs2πkB T,µm,b(2.35)

where µm,b = mm mb /(mm + mb ) is the reduced mass. For simplicity, we will assume the buffer-gas
cooling cell is a box with an average length scale of hLi; thus, the solution to the diffusion equationin Cartesian coordinates is written in terms of modes that have characteristic decay times givenby [56]
τD,ijk =i−1
3hLi2 h
2
2
2
(2i
−
1)
+
(2j
−
1)
+
(2k
−
1)
,
π 2 Dm,b(2.36)

where i, j, k ∈ [1, ∞) are positive integers representing the mode index. The lowest order diffusionmode has molecules concentrated near the center of the cell whereas molecules spread over modesof higher order will be concentrated closer to the cell walls. Notably, the diffusion time decays
quadratically with larger mode indices; therefore, molecules occupying high order diffusion modes
will be rapidly lost because, on average, they have a shorter distance to travel before reaching the31cell walls. Now, from the simple translational cooling model, we know that the average distancetravelled by the molecules before equilibrating with the buffer gas is dependent on the mean freepath λm,b , which is inversely proportional to the buffer-gas density inside the cell. Therefore, we
expect the molecules to be distributed over the whole cell at low densities (i.e., low neon flow rate)
and the diffusion time to be dominated by the lowest order mode τD,000 = hLi2 /π 2 Dm,b . However,as the buffer-gas density increases (i.e., higher neon flow rate), we expect the molecules to be
confined closer to the ablation target at the point where they begin a diffusive random walk; thus,
the molecules will be distributed over higher order diffusion modes such that the effective diffusion
time is less than τD,000 . As this precise behavior is difficult to model, we plot γcell in Fig. 2.6
assuming the diffusion is dominated by τD,000 , which is a good approximation up to a certain point.
We see that γcell ≈ 1 at a flow rate of about 50 sccm, so we expect the extraction efficiency to notincrease further with flow rate as long as τD,000 is the dominant diffusion timescale.
Based on the above discussion, we expect the extraction efficiency to have a nonlinear behavior
with the neon flow rate. At low flow rates (i.e., low neon density), we expect the extraction efficiency
to increase approximately linearly as the diffusion time will be dominated by the lowest order mode,
which has a linear dependence on buffer-gas density. Eventually, this increase will saturate aroundγcell ≈ 1 as the diffusion and pumpout times become roughly equivalent. Finally, as the flow rateincreases further (i.e., high neon density), we expect the extraction efficiency to decrease as themolecules are confined close to the cell walls and exhibit a decreased diffusion time. Indeed, thisextraction behavior has been observed in a number of buffer-gas cooling experiments [31, 33, 44, 49,
57–59], and the higher order diffusion modes model described here was proposed and investigatedby Skoff et al. [33] and Bulleid et al. [44]. This model is also supported by the results of Harilal
et al. [60] and Farid et al. [61] who studied the expansion dynamics of laser ablation plumes atvarious ambient pressures and found the plume to be concentrated near the ablation target forhigher background pressures. Furthermore, we observed the discussed trend with neon flow rate
in our experimental study of carbon clusters with a buffer-gas cooling cell, and that data will be
discussed further in Chapter 4.

32

2
1.75
1.5

. cell

1.25
1
0.75
0.5
0.25
0
0

10

20

30

40

50

60

70

80

90

100

Flow rate (sccm)

Figure 2.6: Cell extraction parameter as a function of neon flow rate into the cell. The celltemperature is Tcell = 25 K, the molecules have a mass of mm = 100 amu, the collision crosssection is σm,b = 10−15 cm2 , and hLi = 30 mm.

33

2.2

Molecular Beam

Up to this point, we have focused primarily on the dynamics inside the buffer-gas cooling cell, butultimately we are interested in the properties of the molecular beam formed by extraction throughthe exit aperture of the cell. Given that the expected range of the Reynolds number at the exit
aperture of the cell is 1 . Re . 120 (see Fig. 2.3), it is useful to consider the formation of amolecular beam in both the effusive (Re < 1) and hydrodynamic (Re > 100) regimes, as this will
cover the full range of expected beam parameters. The effusive regime is characterized by Kn & 1,
where there are no collisions occurring in the exit aperture. Therefore, effusive beams represent asampling of the equilibrium velocity distribution inside the cell, and we expect a beam with low
forward velocity and high divergence. On the other hand, the hydrodynamic regime is characterized
by Kn . 10−3 where there are many collisions occurring in the exit aperture. Therefore, we expect
a hydrodynamic beam to have high forward velocity and low divergence as collisions near theaperture will direct the majority of atoms to leave the cell in the forward direction.

2.2.1

Effusive regime

In the effusive regime of operation, the standard kinetic theory of gases applies. We know that thefraction of buffer-gas atoms with a velocity between v and v + dv at temperature T is given by theMaxwell-Boltzmann distribution in thermal equilibrium:f (v) dv = 4πv2m2πkB T

3/2



mv 2
exp −
dv.
2kB T

(2.37)

Furthermore, given a buffer-gas density of nb inside the cell, we determine the number of atoms
per unit time leaving the cell through an aperture to be [25, 27]
J (θ, v) = nb f (v) v cos θ dv dA

dΩ
,
4π

(2.38)

where dA is the area element of the aperture, dΩ = sin θ dθ dφ is the solid angle element, and thev cos θ term gives the flux in the forward direction with respect to the aperture normal. Integrating

34
this expression over all velocities we find the average flux in the forward direction to be
J (θ) = nb v̄b cos θ dv dA

dΩ
,
4π

(2.39)

where we have used the mean thermal velocity v̄ to simplify the expression. Far from the aperture,
we make the replacement dA → A and integrate over the solid angle to find that J is equivalent to
Eq. (2.5) for the conductance of a thin aperture in the molecular flow regime as expected. Given
the flux through the aperture, we calculate the mean forward velocity of the beam from Eq. (2.38),
where we see that the velocity distribution (normalized) of the beam is given byrπmf (v) dv,
8kb T




v3
m 2
mv 2
=
exp −
dv,
2 kB T
2kB T

fJ (v) dv = v(2.40)

From this distribution, we calculate the mean forward velocity of the beam to beZ ∞ūe,f =

dv vfJ (v) dv =

0

3π
8


v̄ ≈ 1.178v̄,(2.41)

where the e subscript refers to an effusive beam. Thus, we find a forward velocity slightly larger
than the mean thermal velocity of the buffer-gas atoms inside the cell.
It is of interest to estimate the number of collisions we expect to occur in the beam as itpropagates away from the aperture. We start from the average number density in the effusive beam
exiting the cell aperture [25, 27]:ne (R, θ) =nb cos θdA,
4πR2(2.42)

where R is the distance from the cell aperture and θ is the angle with respect to the aperture
normal. To proceed, we assume an aperture shape and carrying out the integrations to find thedensity as a function of position away from the aperture. For a circular aperture with diameter
dap , we write the area element in cylindrical coordinates as dA = 2πrdr and define z = R cos θ and

35
R=

√

z 2 + r2 to find the density along the beam axis to be [25, 27]nb z
ne (z) =2

Z dap /2

r dr

3 ,
z 2 + r21z,= nb  − q2
4z 2 + d2ap
0

√(2.43)

where z > 0 is assumed and nb is the density inside the buffer-gas cooling cell. Now, assuming
the molecules of interest have equilibrated with the buffer gas inside the cell and are entrained in
the effusive beam exiting the cell, we estimate the number of collisions that occur as the beam
propagates away from the cell aperture. Here, entrained implies that the molecules and buffer-gas
atoms are moving together with a velocity given by ūe,f , and that the molecules are diffusing into
the buffer gas in this moving reference frame. We know that the number of collisions occurring in
the interval z to z + dz is given approximately by dzRm,e (z) /v̄m , where Rm,e (z) is the collision
rate in an effusive beam [62]:Rm,e (z) = ne (z) σm,b v̄m,b

√ !
7−4 2
.
4

(2.44)

Here, we use v̄m in the denominator for the number of collisions because this approximately corresponds to the forward velocity of an effusive beam. Recognizing the integrand as the mean freepath λm,b , we integrate over distance to find the total number of collisions that will occur oversome propagation distance L:


r
Z L
mm
1
z
,
NL̄,e = 0.336 1 +(nb σm,b )dz  − qmb2
0
4z 2 + d2
ap



r

p
mm nb σm,b dap = 0.336 1 +
1 + 2L̄ − 1 + 4L̄2 ,
mb
4


r
mm nb σm,b dap≈ 0.336 1 +,mb
4(2.45)

where L̄ = L/dap and the final expression is valid for L̄ > 2 − 3. We find that NL̄,e is of order unity
far from the aperture for mm = 100 amu, mb = 20.18 amu, nb = 1016 cm−3 , σm,b = 10−15 cm2 ,and dap = 5 mm. Therefore, in the effusive regime, we do not expect the molecules to continue to36exchange energy with the buffer-gas atoms once they have been extracted from the cell, which isprecisely as expected in the molecular flow regime.

2.2.2

Hydrodynamic regime

In stark contrast to the effusive regime, the kinetic theory of gases does not apply in the hydrodynamic regime; therefore, we must adopt a fluid description of the buffer gas exiting the cell. A
standard approach begins with the Bernoulli equation for isentropic flow of a compressible fluid,which is written as [52]u2+ h = h0 ,2(2.46)

where u is the flow velocity, h is the enthalpy of the fluid, and h0 is the stagnation enthalpy. For
an ideal gas, we have dh = cp dT where cp = γ/(γ − 1)(kB /m) is the specific heat at constantpressure and γ = 5/3 for a monatomic ideal gas. Assuming cp is constant with temperature, we
integrate the Bernoulli equation to find the maximum forward velocity (i.e., terminal velocity) of
the molecular beam in the hydrodynamic regime:
s


2kB
γ
ūh,f =
(T0 − T ),m
γ−1(2.47)

where T0 is the stagnation temperature. If we assume T  T0 and γ = 5/3 then we see thatūh,f ≈ 1.401v¯0 , which is roughly 20% faster than an effusive beam. To proceed, we rewrite theBernoulli equation in differential form and define stagnation parameters u0 = 0, cs,0 , P0 , and n0 , forthe flow velocity, sound velocity, pressure, and density, respectively. Then, substituting the above
ideal gas relation for dh, we find the following relations for the properties of the flow field [25, 52]:

−1/2
γ−1
2
cs = cs,0 1 +
Ma
,
2

−1


γ−1
2
Ma
,
T = T0 1 +
2



 −γ
γ−1
γ−1
2
P = P0 1 +
Ma
,
2



 −1
γ−1
γ−1
2
n = n0 1 +
Ma
,
2




(2.48)(2.49)(2.50)(2.51)

37
where the following isentropic ideal gas relations have been used to relate the pressure and density
between two points of the flow field:

P = P0n = n0Here, cs =

T
T0
T
T0

 γ

γ−1

,

(2.52).

(2.53)

 1

γ−1

p
γkB T /m is the speed of sound in the gas, P is the pressure, and the flow velocity

can be found from the relation u = M a cs . The stagnation parameters refer to the gas properties
inside the buffer-gas cooling cell such that u0 = M a0 = 0 is assumed, and n0 ≡ nb and T0 ≡ Tcell .
Thus, all relevant properties of the flow field are known as a function of M a, which depends on the
distance from the exit aperture of the cell.
Assuming a circular aperture of diameter dap , we use the results of Murphy [63] and Scoles [25]for an axisymmetric expansion of a monatomic ideal gas with γ = 5/3 to determine M a as a functionof distance z from the aperture on the beam axis:

 3
 2



1.0 + C1 dzap + C2 dzap
M a (z) =
 −1
 −2
 −3  γ−1 zzzC3 + C4 dap+ C5 dap+ C6 dzap
 dap;

0 < z < dap ,(2.54); z > dap ,where C1 = 3.337, C2 = −1.541, C3 = 3.232, C4 = −0.7563, C5 = 0.3937, and C6 = −0.0729 are
fit constants. We now use the above expressions to find the temperature and density as a functionof position away from the aperture. Unlike the effusive regime, we expect much higher densities
at the cell aperture and the temperature to decrease as the beam propagates. Notably, the above
treatment assumes continuum flow and does not consider the effect of a decreasing collision rate
as the beam propagates, which results in the continuum flow assumption becoming invalid. This
process is encapsulated by the so-called sudden freeze model [64] where the freezing, or quitting
surface is defined as the point where continuum flow transitions to molecular flow. In other words,
the quitting surface defines the distance from the exit aperture of the cell where collisions stop. An

38
approximation for this distance is given by [64]3zF = zref (1.875 Ξ) γ+2 ,!r
γ−1zref n0 σ
Ξ = 3.189
,
1/3
γ
T

(2.55)

0where n0 and T0 are the stagnation density and temperature, respectively, and zref = 0.806 dapfor γ = 5/3. For the typical neon buffer-gas beam source parameters of n0 = nb = 1016 cm−3 ,T0 = Tcell = 25 K, σ = 3.02 × 10−15 cm2 , and dap = 5 mm we find zF = 7.686 dap = 38.4 mm.
This value sets the range of validity for the continuum flow treatment; therefore, we investigate thecontinuum flow behavior up to z = zF by substituting the expression for M a (z) from Eq. (2.54)
into Eqs. (2.49) and (2.51) for the temperature and density, respectively. Figure 2.7 shows the on
axis temperature and density of the buffer-gas beam in the hydrodynamic regime. We see thatboth the temperature and density have decreased by an order of magnitude within a distance ofz = 3−5 dap from the exit aperture. Note that neither quantity is equal to unity at the exit apertureas might be expected. This is an artifact of assuming M a (0) = 1 in the aperture, and that thestagnation conditions occur far behind the aperture where the flow has zero velocity. This is notexactly correct as the buffer gas inside the cell will have a velocity component directed primarily
towards the exit aperture (e.g., see the simulation results of Bulleid et al. [44]).
In a similar fashion to the effusive regime calculation, we estimate the number of collisions
that occur in the beam before reaching the quitting surface by carrying out the following integration
over the on axis beam density:
NzF ,h =

r
Z zF
mm
dz
1+
(nb σm,b )
h


i 1 .
mb
γ−1
0
γ−1
1 + 2 M a (z)(2.56)

A numerical calculation of this integral assuming mm = 100 amu, mb = 20.18 amu, σm,b =10−15 cm2 , nb = 1016 − 1017 cm−3 , dap = 5 mm, and zF determined from Eq. (2.55) results in
NzF ,h ≈ 30 − 300. Therefore, we expect additional collisional cooling for distances up to z ≈10 − 50 dap from the cell aperture. Furthermore, we find that the total number of collisions
between the molecules of interest and the buffer-gas atoms is on the order of a few thousand when

39

1

(a)
0.9

Temperature (T/Tcell )

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

1.5

3

4.5

6

7.5

Distance from aperture (z/d ap )
1

(b)
0.9
0.8

Density (n/nb )

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

1.5

3

4.5

6

7.5

Distance from aperture (z/d ap )Figure 2.7: On axis (a) temperature and (b) density of a buffer-gas beam in the hydrodynamic
regime. Both quantities are normalized to the initial state inside the buffer-gas cooling cell, andthe distance from the exit aperture is normalized to the aperture diameter. Quantities are plotted
up to the quitting surface defined by zF in the text. Calculation assumes a monatomic ideal gas
with γ = 5/3.

40
the buffer-gas beam source is operated in the hydrodynamic regime.

2.3

Comparison to a Traditional Free Jet Source

It is of interest to compare the expected number of collisions between molecules of interest andbuffer-gas atoms in the buffer-gas beam source to that in a traditional free jet source. The closest
point of comparison is the so-called “Smalley source” [65, 66], which uses laser ablation to introduce
molecules of interest to the vicinity of a free jet expansion. Notably, the buffer-gas beam source willcertainly have an advantage in terms of the number of molecules entrained in the expansion (i.e.,
beam brightness) since the molecules are introduced inside the cooling cell rather than outside near
the aperture as in the Smalley source. However, the Smalley source has the advantage of muchhigher stagnation densities behind the exit aperture.
Typical parameters for a Smalley source are an aperture diameter of dap ∼ 0.5 − 1 mm andbacking pressure of P0 ∼ 1−5 atm [28], which translates to densities of about n0 ∼ 1019 −1020 cm−3
at a temperature of T = 295 K. Assuming that neon is used as an expansion gas, we computethe collision number using Eqs. (2.55) and (2.56) with mm = 100 amu, mb = 20.18 amu, andσm,b = 10−15 cm2 . Note that in performing the integration, we set the lower bound to be a few
millimeters rather than zero because ablated molecules are not entrained in the expanding beamat the aperture. For dap = 0.5 mm we find that NzF ,h ∼ 103 − 104 and for dap = 1 mm we findthat NzF ,h ∼ 104 − 105 . Therefore, for smaller aperture diameters it appears that the buffer-gas
beam source and Smalley source have on the order of the same number of total collisions between
the molecules of interest and the buffer-gas atoms. For larger aperture diameters, however, theSmalley source results in about an order of magnitude more collisions than the buffer-gas beamsource described here.
It is important to realize that although the source described in this dissertation is expected
to be on par with a typical Smalley source, the buffer-gas beam source provides much more design
flexibility making it possible to further optimize performance. Additionally, most of the collisions
in a buffer-gas beam source occur inside the cell at the buffer-gas temperature. This could be41very beneficial for studying metal cluster molecules, which experience substantial heating uponaggregation. The excess heat obtained during growth can be effectively dissipated by the cold
buffer-gas before the molecules of interest are extracted from the cell, resulting in a bright beam oftranslationally, rotationally, and vibrationally cold molecules.

Chapter 3

Experimental Techniques for Infrared Absorption Spectroscopy

In this chapter, we outline the experimental techniques used in much of this dissertation work.
We begin with a discussion of the absorption of infrared radiation and the Beer-Lambert law. We
then discuss the different mechanisms that effect the absorption lineshape of a given moleculartransition, and introduce the minimum detectable optical depth that can be achieved for a given
measurement setup. Next, Fourier transform infrared spectroscopy is discussed, and a detailed
overview of the spectrometer instrumentation and its expected performance is given. We thendescribe the actual performance of the spectrometer used in this dissertation work. Finally, laser
absorption spectroscopy is discussed and methods to achieve shot noise limited absorption sensitivity are presented. We finish this chapter with a discussion of the actual performance of the laser
spectroscopy system used in this dissertation work.

3.1

Absorption of Infrared Radiation

3.1.1

Beer-Lambert law

For absorption spectroscopy, the Beer-Lambert law relates the density of absorbers in a medium
to the change in intensity of light as it propagates through the medium. In the weakly absorbing
limit, and assuming scattering is negligible, the Beer-Lambert law states that the intensity of lightI (ν̃, z) propagating through an absorbing medium will decrease with distance z asdI (ν̃, z)= −α (ν̃) I (ν̃, z) ,
dz

(3.1)

43
where α (ν̃) is the absorption coefficient of the medium (assumed to be isotropic) at wavenumberν̃. Assuming the light enters the medium at z = 0 and exits the medium after a distance l we solve
the above differential equation to obtain
I (ν̃, l)= e−α(ν̃)l .
I (ν̃, 0)(3.2)

From this solution we then define the optical depth τ (ν̃) asI (ν̃, 0)= α (ν̃) l.
τ (ν̃) = ln
I (ν̃, l)(3.3)

Note that absorbance A (ν̃) is commonly encountered in place of optical depth, and the two quantities are related by A (ν̃) = log10 e τ (ν̃) ≈ 0.434 τ (ν̃).
3.1.2

Absorption lineshapes

The absorption coefficient α (ν̃) corresponding to a particular transition in an ensemble of molecules
is wavenumber dependent and has a finite extent about the transition wavenumber ν̃0 . In a typicalexperiment, various broadening mechanisms are present that give rise to different absorption lineshapes, and they are classified as either homogeneous or inhomogeneous broadening. Homogeneous
broadening mechanisms give rise to a Lorentzian lineshape such that α (ν̃) takes the formαL (ν̃) = α (ν̃0 )Γ2L
Γ2L + 4c2 (ν̃ − ν̃0 )2

,

(3.4)

where α (ν̃0 ) is the peak absorbance on resonance, ΓL is the full width at half maximum (FWHM)
of the homogeneously broadened transition in units of Hz, and c is the speed of light in vacuum.
On the other hand, inhomogeneous broadening mechanisms give rise to a Gaussian lineshape,"#
22 (ν̃ − ν̃0 )
αG (ν̃) = α (ν̃0 ) exp −4 ln 2 c
,
(3.5)
Γ2G
where ΓG is the FWHM of the inhomogeneously broadened transition in units of Hz. If homogeneousand inhomogeneous broadening mechanisms have similar magnitudes then the absorption lineshapeis a convolution of a Lorentzian and a Gaussian. This gives rise to the Voigt lineshape,Z ∞αV (ν̃; ΓL , ΓG ) =−∞dν̃ 0 αL ν̃ 0 αG ν̃ − ν̃ 0 .

(3.6)

44
The natural linewidth of an absorption line is governed by how long a particular transition
can be observed (i.e., the excited state lifetime). This can be understood via the energy-time
uncertainty principle ∆E∆t ≥ h/4π, which sets a lower bound on how well the transition energy
between a ground and excited state can be determined for a given observation time. Rearranging
slightly we write the minimum FWHM of a lifetime broadened transition as Γ0 = 1/4πτ whereτ is the excited state lifetime. As the excited state naturally decays exponentially in time due
to spontaneous emission, α (ν̃) will have a Lorentzian lineshape. The spontaneous emission rate
is proportional to ν̃ 3 [67]; therefore, excited state lifetimes are typically much shorter (Γ0 muchlarger) in the ultraviolet and visible regions of the electromagnetic spectrum compared to theinfrared and microwave regions. For example, consider the natural linewidth of the 87 Rb cyclingtransition, which is roughly 10 MHz and occurs at approximately 10, 000 cm−1 [68]. Typicalvibrational and rotational transitions occur at wavenumbers on the order of 1000 cm−1 and 1 cm−1 ,respectively, resulting in natural linewidths of roughly 10 kHz and much less than 1 Hz, respectively,
based on the scaling of spontaneous emission with ν̃ 3 . Thus, Γ0 is typically not observed ininfrared absorption spectroscopy due to finite instrument linewidths, as well as the presence ofother broadening mechanisms, which will now be discussed.
Given an ensemble of molecules with finite thermal energy, the FWHM of an absorption
line will experience Doppler broadening due to the finite translational motion of the molecules.
Assuming thermodynamic equilibrium, the ensemble of molecules will have a velocity distribution
given by the Maxwell-Boltzmann distribution

3/2


m
mv 2
fv (v) =exp −,2πkB T
2kB T

(3.7)

where m is the mass of a molecule in the ensemble, kB is Boltzmann’s constant, T is the temperatureof the ensemble, and v is the molecule velocity. Thus, for any given molecule, the transition
wavenumber will be Doppler shifted to ν̃ 0 = ν̃0 (1 + v/c) and molecules in the ensemble will acquirea distribution of transition wavenumbers given by
"
 
3/2
 0
2 #

dv
c
m
ν̃
mc2
0
fν̃ 0 ν̃ = fv (v) 0 =
−1
.
exp −
dν̃
ν̃0
2πkB T
2kB T ν̃0

(3.8)

45
Therefore, the absorption coefficient will take the form of a Gaussian function
2
mc2
ν̃ 0 − ν̃0αD (ν̃) = α (ν̃0 ) exp −2
2kB T ν̃0from which we define the FWHM (in units of Hz) of a Doppler broadened transition asr
2 ln 2kB T
ΓD = 2
ν̃0 .
m

(3.9)

(3.10)

Given the Gaussian form of the absorption lineshape, Doppler broadening is clearly an inhomogeneous broadening mechanism.
In addition to finite thermal energy, if the ensemble of molecules has a high density, or are
entrained in a buffer gas of high density then collisional interactions cannot be neglected and the
FWHM of the transition will be broadened. Unlike Doppler broadening, collisional broadening
is a homogeneous mechanism and α (ν̃) will maintain the form of a Lorentzian function. Thisis due to the fact that collisions modify the excited state lifetime of the transition and shorter
lifetimes lead to larger values of Γlt as per the energy-time uncertainty principle. The magnitude
of collisional broadening can be estimated using the classical collision rate determined from kinetic
theory Rc = nσv̄ [69]. Here, we use the collision rate defined by Eq. (2.29) because the density of
buffer-gas atoms is much larger than that of the molecules of interest. Thus, the FWHM (in units
of Hz) of a transition broadened due to collisions isΓc = nb σm,b v̄m,b ,(3.11)

where nb is the buffer-gas density, σm,b is the collision cross section between a molecule and abuffer-gas atom, and v̄m,b is the relative mean thermal velocity between the molecules and thebuffer-gas atoms.
Finally, there is one other broadening mechanism commonly encountered in absorption spectroscopy experiments that utilize a molecular beam. In such an experiment, a beam of moleculesmoving with forward velocity u is intersected by a beam of light propagating orthogonal to themolecular beam. This arrangement leads to transit time broadening, which arises as a result of
the finite time that the beam of light can probe the transition. Once again, this is due to the46
energy-time uncertainty principle and will result in broadening if the transit time of a molecule
through the beam of light is shorter than the excited state lifetime of the transition being probed.
Thus, transit time broadening is a homogeneous broadening mechanism and α (ν̃) will maintainthe form of a Lorentzian function. Assuming the light is a Gaussian beam with waist w0 , the time√
it takes a molecule to traverse the beam of light is given by t = 2 ln 2 w0 /u. Thus, the FWHM
(in units of Hz) of a transition broadened due to finite transit time is approximately
Γt = √u,2 ln 2 w0(3.12)

where u is the forward velocity of the molecular beam.

3.1.3

Minimum detectable optical depth

For absorption spectroscopy there are two main factors that contribute to the minimum opticaldepth that can be detected with a given instrument: optical power from the light source and detector
sensitivity. Shot noise present in electromagnetic radiation sets a fundamental limit on absorption
measurements because it results in fluctuations around the mean optical power incident on anabsorbing sample. The noise contribution to the optical power per unit measurement bandwidth
due to shot noise is given byσsn =p2hcν̃ P̄ ,(3.13)√
where hcν̃ is the photon energy, P̄ is the mean optical power, and σsn has units of W/ Hz.
Notably, shot noise is independent of frequency making it a form of “white” noise. In addition
to shot noise, the limited sensitivity of the detector plays an important role in determining theminimum detectable optical depth of a given instrument. Detector noise can be specified as a noise√
equivalent power (NEP) with units of W/ Hz, which defines the mean optical power that gives
a signal-to-noise ratio (SNR) of unity in a 1 Hz measurement bandwidth. Although NEP is thequantity of interest in terms of noise power, it is dependent on the detector area Ad ; therefore, thespecific detectivity D∗ is a commonly encountered metric for infrared detector performance:
√
Ad
∗
D =
,
(3.14)N EP

47√
which has units of cm Hz/W, also known as Jones. Thus, given the detector D∗ we can quantify
the noise contribution due to the detector as√
σd ≡ N EP =

Ad
.
D∗

Thus, we define the total noise power in an absorption measurement as δ P̄ =(3.15)q
2 + σ2.
σsndGiven the total noise power, we determine the minimum detectable optical depth τmin in agiven measurement bandwidth ∆f by combining Eqs. (3.13) and (3.15):
δ P̄ p∆fP̄s2hcν̃∆f
Ad ∆f
+
=
2 ,
P̄
P̄ D∗

τmin ≈(3.16)

where small optical depths have been assumed in order to expand the natural logarithm in a Taylor
series to arrive at this expression. We are fortunate that infrared detectors exist with large enoughD∗ values to reach shot noise limited performance (i.e., the detector noise contribution falls belowthat of the shot noise contribution). However, to keep noise to a minimum, particularly in the 1/f
regime, these detectors must be minimally biased. Therefore, low noise performance comes at thecost of detector linearity, which limits the incident optical power especially if detection at DC is ofinterest. This will be discussed more in Section 3.4.4

3.2

Fourier Transform Infrared Spectroscopy

Fourier transform infrared (FTIR) spectroscopy utilizes an interferometer-based spectrometer tosimultaneously measure absorption lines over a broad range of wavenumbers. The standard FTIR
spectrometer consists of a broadband source coupled to a Michelson interferometer (MI), and theoutput of the MI is passed through a sample region before being focused onto an infrared detector.
As one of the mirrors in the MI is moved, a pathlength difference is introduced and different
wavenumbers from the source experience constructive and destructive interference at the outputof the MI. By monitoring the variations in intensity at the output of the MI as a function of
the pathlength difference an interferogram can be recorded and subsequently Fourier transformed

48
to obtain the intensity of the source as a function of wavenumber. An absorbing sample placed
between the output of the MI and the detector will modify the interferogram because the source
intensity will be reduced at the transition wavenumbers corresponding to absorption by the sample.
A simplified schematic of the Thermo Fisher Scientific Nicolet iS50 FTIR spectrometer used for thisdissertation work is depicted in Fig. 3.1. This section will discuss the different aspects of the FTIR
spectrometer and provide some guidelines for optimal settings that give the best quantitative results
when investigating absorption features. An excellent resource for the theory and instrumentationof FTIR spectroscopy is the textbook of Griffiths and Haseth [70], which has been consulted formuch of the discussion presented here.

3.2.1

SourceThe source used in FTIR spectrometers is essentially a hot filament held at a temperature on theorder of 1000 K; thus, its spectral energy density is modeled as a blackbody source:
U (ν̃, T ) =2hc2 ν̃ 3

.
exp hcν̃kT − 1(3.17)

Assuming a temperature of T = 1000 K such a source has a peak spectral energy density of
approximately 570 µW/sr cm2 cm−1 at ν̃ ≈ 2000 cm−1 . One drawback to this type of source is that
it is extended with an area of approximately 1 cm2 , which limits the extent to which the light canbe collimated or focused onto a detector. However, a major advantage is that the filament source isinherently broadband, and provides light in the range of approximately 50 − 9000 cm−1 in a typical
FTIR spectrometer.

3.2.2

Michelson interferometer

The operation of a FTIR spectrometer is based on that of the standard two-beam MI, which
divides a beam of radiation in half with a beamsplitter and introduces a pathlength difference
before recombining the beams at the output of the interferometer. By varying the pathlength
difference, or optical retardation δ, a condition is created where the two beams either constructively

49

Fixed mirror
Moving mirror

Beamsplitter
SourceIris

Detector

Sample area

Figure 3.1: A schematic of the Nicolet iS50 FTIR spectrometer. The optics are designed to form
an image of the iris in the sample region with a magnification of 1.130. The image of the iris in thesample region is then imaged onto the detector with a magnification of 0.1651.

50
or destructively interfere when recombined. To understand the basic operating principle of the MI,
it is easiest to first consider a monochromatic source of wavenumber ν̃. If δ = 0 then there is nopathlength difference and the two beams will interfere constructively (i.e., all light from the source
will reach the detector). However, if the moveable mirror is displaced by a quarter of a wavelength
then δ = 1/2ν̃ and the two beams will interfere destructively (i.e., none of the light from the source
will reach the detector). Displacing the moveable mirror by another quarter of a wavelength will
result in constructive interference again; therefore, δ = m/ν̃, with m an integer, will result inconstructive interference. Thus, the intensity at the detector as a function of δ is
I (δ) =

I (ν̃)[1 + cos (2πν̃δ)] ,2(3.18)

where I (ν̃) is the source intensity. Generally, we are only interested in the cosine component suchthat the constant offset will be neglected in the following discussion.
Equation (3.18) is for an ideal MI and does not consider various instrument effects suchas beamsplitter efficiency and detector response as a function of ν̃. Therefore, it is convenient
to introduce the instrument transfer function H (ν̃) and the single beam spectral intensity of thesource seen by the detector B (ν̃) = H (ν̃) I (ν̃) /2. Thus, the interferogram intensity is given by
I (δ) = B (ν̃) cos (2πν̃δ).

(3.19)

Clearly, I (δ) and B (ν̃) form a cosine Fourier transform pair, and the above formalism can beextended to a polychromatic source to obtain
Z ∞I (δ) = 2dν̃B (ν̃) cos (2πν̃δ),(3.20)dδI (δ) cos (2πν̃δ),(3.21)0Z ∞B (ν̃) = 20

In practice, I (δ) is recorded over a finite range of δ and B (ν̃) is computed using a discrete transform.
This results in B (ν̃) having a finite resolution ∆ν̃, which is inversely proportional to the maximum
retardation δmax that the FTIR spectrometer can achieve (i.e., the maximum linear travel distanceof the moveable mirror). Furthermore, errors introduced in the sampling and digitization process

51
lead to small offsets in δ that translate into sine components being present in the interferogram.
Therefore, it is typical to record a two-sided interferogram (i.e., the retardation ranges between−δmax < δ < δmax ) and compute B (ν̃) using the fast Fourier transform algorithm. In addition,a phase correction procedure, such as that developed by Mertz [71], is applied to ensure a linear
mapping between δ and ν̃.
Given that there is a fundamental limit on the achievable resolution, it is practical to apply
an apodization function A (δ) prior to transforming the interferogram. Apodization effectively
truncates the interferogram at finite δ in a smooth way such that absorption features do notexhibit excess broadening or fictitious side lobes, and can suppress baseline fluctuations aroundweak absorption features. With the inclusion of apodization B (ν̃) becomesZ ∞B (ν̃) =dδA (δ) I (δ) cos (2πν̃δ),(3.22)−∞

which is equivalent to convolving B (ν̃) with the Fourier transform of A (δ). A variety of apodization
functions have been investigated in Refs. [72–74] and their effect on Lorentzian and Gaussian
absorption bands quantified in Refs. [75–77]. The conclusions of these studies imply that, ingeneral, no single apodization function is suitable for all applications. If quantitative informationon peak absorbance of an absorption line is important then a simple boxcar apodization function
should be used. However, it is important to realize that this will introduce baseline fluctuationsin the vicinity of weak absorption features due to the presence of large side lobes from strong
absorption features that are usually present in FTIR spectra (e.g., from atmospheric H2 O andCO2 ). Therefore, more aggressive apodization functions can be used to suppress sidelobes, and thesimple method introduced by Jalsovszky [78] can be applied to determine true peak absorbance forweak absorption features if the actual FWHM of the absorption line is known. Particularly useful
apodization functions are the Norton-Beer strong and weak functions [73], which can be used if
suppression of baseline fluctuations and high resolution are critical.
In addition to the maximum retardation and choice of apodization function, the diameter ofthe iris located between the source and the MI (see Fig. 3.1) is a critical parameter in determining

52
the achievable resolution of a FTIR spectrometer. This is a result of the fact that extended sources
cannot be well collimated∗ and the beam of light entering the MI is slightly diverging; therefore,
the extent of the source imaged onto the detector must be restricted. This can be understoodby considering the well-known circular fringe pattern observed at the output of a conventionalMI, which consists of a central bright fringe surrounded by alternating rings of dark and brightfringes [79, 80]. As the optical retardation between the two arms of the MI increases the circular
fringe pattern collapses towards the center and the central bright fringe decreases in diameter. If
the extent of the source imaged onto the detector is not restricted, the average intensity on thedetector will vary weakly as a function of δ due to the presence of many bright and dark fringes.
Therefore, the effect of beam divergence through the MI will result in an effective decrease in
the resolution of the FTIR spectrometer as a result of this self-apodization. The solution to thisproblem is to use an iris of variable diameter to restrict the image size of the source on the detector.
In this way it is ensured that the detector only sees the central bright fringe over the full range of δ
for a given resolution setting. We estimate the maximum solid angle passed by the interferometer
optical system as follows. Let the beam have a divergence angle of α through the MI such thatEq. (3.20) becomesZ ∞I (δ) = 2dν̃ B (ν̃) cos (2πν̃δ cos α).

(3.23)

0

Assuming that the circular fringe pattern has a central bright fringe then the maximum allowed
divergence angle that restricts the system to passing only this central fringe is given by the valueof α at the first dark fringe:
2πν̃δ − 2πν̃δ cos α ≈ 2πν̃δ
r
1
α=
ν̃δ

α2
2


= π,
(3.24)

where small α has been assumed. Now, assuming a resolution of ∆ν̃ = 1/δmax and a maximum∗

Optical elements have a finite field of view over which they can successfully convert diverging spherical wavefronts
into collimated planar wavefronts. If the extent of the source falls outside this field of view then portions of the lightcollected from the source will continue to diverge.

53
wavenumber of ν̃max in the source spectrum, we find that the maximum solid angle isΩmax = 2π (1 − cos αmax ) ≈

π∆ν̃
.
ν̃max

(3.25)

Therefore, we see that higher resolution (i.e., smaller ∆ν̃) and broader optical bandwidth (i.e.,larger ν̃max ) requires a smaller iris diameter to restrict Ω, which will reduce the amount of lightpassed to the detector. Given that the spectral energy density of broadband, blackbody-like sources
is already quite low, further restricting the throughput of the optical system increases the demandon detector sensitivity.

3.2.3

Detectors

The Nicolet iS50 FTIR spectrometer is equipped with two infrared detectors: a pyroelectric bolometer with a deuterated, L-alanine doped triglycine sulfate (DLaTGS) element and a quantum detector
with a mercury cadmium telluride (MCT) element. The optical bandwidth of both detectors issimilar, about 350 − 12, 500 cm−1 for the DLaTGS and about 600 − 11, 700 cm−1 for the MCT;
however, the sensitivity of the DLaTGS detector is approximately two orders of magnitude lower
than that of the MCT detector. Nonetheless, the DLaTGS detector is useful in applications thatinvolve static sample environments and high incident flux because the detector responds on thermal
timescales and remains linear over a wide range of incident flux conditions. On the other hand,
MCT detectors excel in all FTIR spectroscopy applications as long as the incident flux is kept low
because the detector response saturates and becomes nonlinear under even moderate incident flux
conditions.
An important aspect of MCT detectors used in FTIR spectrometers is that they are typicallyphotoconductive (PC) type detectors, which exhibit large 1/f noise due to the necessity of biasing
the detector element to a linear operating point. This results in low sensitivity at frequencies below
a few kHz; thus, the light incident on the detector must be modulated to achieve high sensitivity.
Fortunately, modern FTIR spectrometers make use of a rapid scan technique where the moveable
mirror is translated at constant velocity v from −δmax to δmax such that δ → δ(t) = 2vt. This

54
results in modulation of the spectral components in the source occurring at frequencies fν̃ = 2vν̃.
Considering maximum and minimum wavenumbers of 7000 cm−1 (limited by the KBr beamsplitter)and 600 cm−1 , respectively, incident on the detector, and a typical mirror velocity of about 3.2 cm/s
we find that f600 = 3.8 kHz and f7000 = 44.8 kHz. Given that the optimal frequency response of a
PC MCT detector lies in the range of approximately 1 kHz to 1 MHz we find that all wavenumbersin the source spectrum will be detected at high sensitivity. Notably, because DLaTGS detectors
have a slow response time, the mirror velocity must be much slower (typically v < 1 cm/s), which
results in substantially longer interferogram acquisition times compared to MCT detectors. Thishighlights yet another advantage of using MCT detectors for FTIR spectroscopy.

3.2.4

Detection limit

In order to determine the achievable detection limit of a given FTIR spectrometer measurement,
the optical power reaching the detector must be determined. The spectral energy density emitted
by the source is given by Eq. (3.17), but we must determine the maximum throughput, or etendue
G of the FTIR spectrometer optical system to determine the optical power at the detector. Etendue
is the product of beam area and its solid angle at any focus; thus, the MI has a maximum etendue
GM dictated by the area of the mirrors AM and the solid angle defined by Eq. (3.25):
GM =πAM ∆ν̃.
ν̃max(3.26)

Additionally, the optical system that focuses light onto the detector has a maximum etendue
Gd = Ad Ωd . Therefore, the limiting etendue of the optical system can be either GM or Gd dependingon the choice of parameters. Typically, Gd is the limiting factor for low resolution (∆ν̃ > 1 cm−1 )measurements due to the small size of MCT detector elements, whereas GM will limit the opticalthroughput at high resolutions as a result of the restricted solid angle given by Eq. (3.25). The
total signal power contributing to a measurement is thenP̄FTIR = η U (ν̃, T ) G∆ν̃,(3.27)

55
where G is chosen as the smaller of GM and Gd , and η is the efficiency of the spectrometer
optical system that incorporates losses due to beamsplitter efficiency, reflectivity of mirrors, opticalmisalignment, and detector quantum efficiency.
Given the total optical power reaching the detector, we define the detection limit of a single
beam spectrum acquired with a FTIR spectrometer using Eq. (3.16)sτmin,FTIR =2hcν̃∆f
Ad ∆f.
+
η U (ν̃, T ) G∆ν̃(η U (ν̃, T ) G∆ν̃D∗ )2(3.28)

Equation (3.28) provides a good approximation to the minimum detectable optical depth in FTIR
spectroscopy, and provides insight into the effect of different parameters on the sensitivity. For
a given source and detector, the sensitivity of a measurement is largely determined by the choiceof resolution and measurement time. Depending on whether GM or Gd limits the etendue of thesystem, the resolution can have a large effect on sensitivity. For example, if GM is the limiting
etendue, then doubling the resolution will decrease the sensitivity by a factor of 4. In order tocompensate for this loss in sensitivity, the measurement time must increase by a factor of 16.
Thus, we see that the choice of resolution and desired sensitivity strongly dictates how long a given
measurement will take. The following subsection provides details of the performance of the Nicolet
iS50 FTIR spectrometer over a range of settings and compares these experimental results with thepredictions outlined so far.

3.2.5

Performance of the Nicolet iS50 FTIR spectrometer

Based on the choice of resolution and maximum wavenumber, the Nicolet iS50 FTIR spectrometer
automatically adjusts the diameter of the iris located between the source and interferometer to setΩmax based on Eq. (3.25). This sets the value of GM , which can be calculated based on the iris
diameter and f /# of the source collimation optics. The value of Gd is fixed by the detector area andthe f /# of the detector focusing optics. In regards to the optical system, the iS50 documentation
specifies that the source collimation optics and detector focusing optics have f /# ≈ 2.9 and f /# ≈0.7, respectively, which corresponds to solid angles of approximately ΩM = 0.1 sr and Ωd = 2 sr,

56
Table 3.1: Aperture diameter and etendue of Nicolet iS50 spectrometer as a function of resolution
for a maximum wavenumber of ν̃max = 4000 cm−1 and a MCT detector with area Ad = 0.1×0.1 cm2 .





Resolution cm−1
Aperture Diameter (mm) GM cm2 sr
Gd cm2 srτmin,FTIR Hz−1/2
4
2
1
0.5
0.25
0.125

10.6
6.52
4.58
3.20
2.21
1.56

0.088
0.033
0.017
0.008
0.004
0.002

0.022
0.022
0.022
0.022
0.022
0.022

2.0 × 10−6
4.0 × 10−6
1.0 × 10−5
4.4 × 10−5
1.8 × 10−4
7.0 × 10−4

respectively. Furthermore, the MCT detector element has an area Ad = 0.1 × 0.1 cm2 resulting
in Gd ≈ 0.022 cm2 sr for the detector-limited etendue. Table 3.1 provides information on the iris
diameter as a function of resolution assuming a maximum wavenumber of ν̃max = 4000 cm−1 chosenbecause we are interested in the mid to far infrared region of the electromagnetic spectrum. We see
from Table 3.1 that the Nicolet iS50 has a detector-limited etendue for resolutions ∆ν̃ > 1 cm−1 ,and an interferometer-limited etendue for ∆ν̃ ≤ 1 cm−1 . It is instructive at this stage to calculate
the detection limit of a typical single beam FTIR spectrum at different resolutions. The following
is a list of typical parameter values used to evaluate the detection limit:• Peak spectral energy density of U (ν̃, T ) ≈ 570 µW/sr cm2 cm−1 at ν̃ = 2000 cm−1 andT = 1000 K.
• Optical system efficiency of η = 0.1 as a conservative estimate.
√• Specific detectivity of D∗ = 1010 cm Hz/W as a lower bound for a typical MCT detector.
The result of this calculation is displayed in the final column of Table 3.1. Clearly, if weak spectralfeatures are of interest, then it is advantageous to take measurements at as low a resolution as
possible. However, if high resolution is necessary then multiple interferograms can be averaged to√increase the sensitivity as N , where N is the number of interferograms averaged.
In the final portion of this section, we present measurements of the iS50 detection sensitivity
for a range of parameters. All of the measurements are performed with a maximum wavenumber

57of ν̃max = 4000 cm−1 and an interferometer mirror velocity of v = 3.1647 cm/s. Figure 3.2 shows atypical single beam spectrum acquired by averaging 100 scans at a resolution of ∆ν̃ = 1 cm−1 . There
are a number of absorption features present in this spectrum that correspond to atmospheric H2 O
between about 3500 − 3900 cm−1 and 1300 − 2100 cm−1 , and atmospheric CO2 around 2350 cm−1 .
Additionally, there is a feature between approximately 2850 − 3000 cm−1 that corresponds tohydrocarbons in an epoxy used to mount the vacuum window on the MCT detector. The spectrum
peaks around 2200 cm−1 and looks like the spectrum of a blackbody for larger wavenumbers.
However, the low wavenumber region of the spectrum is cutoff as a result of the increasing absorptionof the KBr beamsplitter and the drop off in responsivity of the MCT detector. This results in a
low wavenumber cutoff of approximately 650 cm−1 .
In order to estimate τmin from a single beam spectrum, we take δ P̄ and P̄ as the standard
deviation and mean, respectively, of the signal amplitude in a small wavenumber range near the
peak of the spectrum, but away from any absorption features. This is particularly important forboxcar apodization as the sidelobes introduced by the H2 O absorption lines will seriously degradeτmin . We then estimate the measurement bandwidth based on the scan time and calculate τmin inunits of Hz−1/2 . Figure 3.3 shows a plot of τmin as a function of the number of scans averaged aswell as the expected N −1/2 behavior. These data were acquired at a resolution of ∆ν̃ = 1 cm−1with boxcar apodization and an energy screen with 10% transmission in place to prevent the MCT
detector from saturating. We see that for an average of 10−100 scans that τmin follows the expectedbehavior, but tends towards a minimum value of approximately 5 × 10−5 Hz−1/2 after averaging
about N > 1000 scans. This indicates that other noise sources and system instabilities are present
that dominate at larger values of N . Thus, regardless of resolution the system sensitivity appears
to be limited to this level, and averaging a larger number of scans is not beneficial. In addition
to the ultimate sensitivity, it is important to point out that the sensitivity of roughly 10−3 Hz−1/2
for N = 1 is 100 times larger than the estimated value in Table 3.1. However, the estimated value
does not incorporate the energy screen, which reduces the amount of light reaching the detector
by a factor of 10, so the difference in sensitivity is really a factor of 10. This discrepancy is most

58

50

Amplitude (arb.)

40

30

20

10

0
4100

3850

3600

3350

3100

2850

2600

2350

2100

1850

1600

1350

1100

850

600

Wavenumber (cm-1)

Figure 3.2: Nicolet iS50 single beam spectrum acquired by averaging 100 scans at a resolution of∆ν̃ = 1 cm−1 . A 10% transmission energy screen was used to prevent the MCT detector from saturating, and boxcar apodization was chosen. Note the absorption features due to atmospheric H2 O
between about 3500 − 3900 cm−1 and 1300 − 2100 cm−1 , and atmospheric CO2 around 2350 cm−1 .

59=min (Hz-1/2 )

10 -3

10 -4

10 -5
10 0

10 1

10 2

10 3

Number of scans

Figure 3.3: Nicolet iS50 FTIR spectrometer sensitivity as a function of the number of scans
averaged (black points). The expected behavior is a N −1/2 trend plotted as the blue line. The
spectrometer resolution was ∆ν̃ = 1 cm−1 , boxcar apodization was used, and an energy screenwith 10% transmission was in place to prevent the MCT detector from saturating. The noise floor
appears to occur at a sensitivity of approximately 5 × 10−5 Hz−1/2 after averaging a few thousandscans.

60
Table 3.2: Single beam sensitivity of the iS50 spectrometer with boxcar apodization. For ∆ν̃ =1 cm−1 a 10% transmission energy screen is used to prevent saturation of the MCT detector. Allother resolution settings use a 30% transmission energy screen.





Resolution cm−1
τmin Hz−1/2 N = 1 τmin Hz−1/2 N = 100
1
0.5
0.25
0.125

1.26 × 10−3
3.03 × 10−3
1.23 × 10−2
3.00 × 10−2

1.12 × 10−4
3.06 × 10−4
1.21 × 10−3
2.93 × 10−3

likely due to other factors such as noise from the detector amplifier and quantization noise from thedigitization process, which are not included in the simple noise model presented here. Additionally,
the choice of boxcar apodization can degrade the noise floor due to the presence of many strong
absorption features in the single beam scan. This will be discussed more below.
It is also of interest to investigate the effect of other parameters on the sensitivity of thespectrometer. Table 3.2 gives the sensitivity as a function of the spectrometer resolution setting
for an average of N = 1 and N = 100 scans. The sensitivity of the spectrometer clearly decreases
as the resolution increases, where doubling the resolution causes the sensitivity to decrease by afactor of 2 − 4 as expected from Eq. (3.28). Additionally, averaging N = 100 scans improves
the sensitivity by about a factor of 10 for all resolutions demonstrating the expected scaling withsmall N . Another important spectrometer parameter that can affect the sensitivity is the choice of
apodization function. Table 3.3 gives the sensitivity as a function of the apodization function fora single scan (N = 1). As expected, boxcar apodization gives the worst sensitivity performance inthe presence of strong absorption features in the spectrum (e.g., those resulting from atmosphericH2 O), and the Norton-Beer strong function gives the best sensitivity performance. This is a resultof the poor sidelobe suppression of boxcar apodization compared to the other functions. However,
the Norton-Beer strong function suppresses the peak optical depth of absorption features compared
to boxcar apodization as evidenced by the data shown in the third column of Table 3.3. Other
apodization functions provide similar tradeoffs between peak optical depth suppression and sensitivity enhancement. Notably, the Happ-Genzel and Norton-Beer strong functions provide the most

61

Table 3.3: Single beam sensitivity of the iS50 spectrometer, and the peak optical depth and FWHM
linewidth of a strong H2 O absorption line at about 1617 cm−1 for different apodization functions.
The resolution is ∆ν̃ = 1 cm−1 and a 10% transmission energy screen is used to prevent saturationof the MCT detector.

Apodization Function τmin Hz−1/2
Peak OD Linewidth cm−1
BoxcarTriangular
CosineBlackman-Harris
Happ-Genzel
Norton-Beer Weak
Norton-Beer Medium
Norton-Beer Strong

1.28 × 10−3
6.89 × 10−4
9.14 × 10−4
6.84 × 10−4
7.61 × 10−4
9.11 × 10−4
8.16 × 10−4
6.38 × 10−4

0.608
0.283
0.371
0.215
0.319
0.404
0.338
0.288

0.620
0.965
0.868
1.41
1.01
0.810
0.946
1.06

62

Figure 3.4: A schematic representation of the matrix isolation technique adopted from [82]. The
inert gas host (open circles) confines molecules of interest (solid black objects) to vacancies in the
crystal lattice. Note that different types of vacancies are possible, which leads to different local
environments for the trapped molecules.

accurate representation of the spectrometer resolution setting, good sensitivity, and only moderatepeak optical depth suppression. Therefore, they are the recommended apodization functions formost studies.

3.3

Matrix Isolation Spectroscopy

The principle of matrix isolation spectroscopy is to trap molecules of interest inside a solid matrixof an inert host material, and then use a desired spectroscopic technique to probe the trappedmolecules. In this way, the trapped molecules cannot diffuse and are prevented from reacting with
one another. Perhaps the most common method of matrix isolation is to co-deposit the moleculesof interest with an inert gas (e.g., neon) onto an infrared transparent window held at a temperature
well below the melting point of the host gas. In fact, the temperature of the window should beheld at a temperature well below the point at which diffusion of trapped molecules inside the hostmatrix becomes appreciable, which is approximately 10 K for neon gas [81]. Figure 3.4 provides asimple depiction of the matrix isolation technique.
As depicted in Fig. 3.4, molecules become trapped in substitutional sites of the host matrix
where individual atoms of the host material are replaced by the molecules of interest. These substi-

63
tutional sites can differ in shape and size, and may even occur near matrix dislocation boundaries
resulting in different local environments for the trapped molecules. These environmental perturbations will lead to small frequency shifts of the molecular transitions, and can result in the observationof multiple absorption features for a single transition. In addition to these frequency shifts, themajority of molecules will be rigidly held within the host matrix such that rotational transitions
cannot occur.† Therefore, one typically observes pure vibrational transitions with matrix isolation
spectroscopy, where absorption linewidths of < 1 cm−1 are readily observed.
In conjunction with a FTIR spectrometer, matrix isolation is a useful method for obtaining
a broadband survey spectrum of a molecular source, and is a simple addition to an apparatus that
already employs a cold beam source. In this dissertation work, we use the technique of matrix
isolation spectroscopy to characterize the performance of our buffer-gas beam source as well as tooptimize the production of a given molecule of interest. This is done by situating a CaF2 or KBrwindow below the buffer-gas cooling cell at a 45◦ angle such that it intersects the molecular beam.
By connecting this window to the second stage of the cryocooler, matrix temperatures of about 5 K
are readily achieved. A major drawback of matrix isolation spectroscopy is the lack of rotational
spectra for most molecules; thus, it is of interest to study molecules in the gas phase after theirproduction has been optimized using FTIR spectroscopy.

3.4

Infrared Laser Absorption Spectroscopy

A key advantage to FTIR spectroscopy is the broadband source and multiplexed acquisition,
which makes it possible to simultaneously obtain molecular absorption spectra over thousandsof wavenumbers in a few seconds. However, major disadvantages are the low resolution of the spectrometer and low spectral energy density of the source. In the case of the Nicolet iS50 FTIR, theresolution is limited to ∆ν̃ = 0.125 cm−1 or about 3.75 GHz — this is many orders of magnitudelarger than the natural linewidth of vibrational and rotational transitions in the infrared, and can†There are certain small molecules which exhibit rotational transitions in matrix isolation spectroscopy, e.g.,
HCl [83].

64
make it difficult to fully resolve the rotation-vibration spectrum of many molecules. In addition,
the peak spectral energy density of broadband sources used in FTIR spectroscopy is so low that
only a few microwatts of power per spectral element can be used to probe molecular transitions.
This is in stark contrast to infrared laser sources, which have very high spectral energy density andnarrow linewidths on the order of 10 MHz. In addition, tunable infrared laser sources exist thatcan cover broad spectral windows on the order of 100 cm−1 ; however, acquisition is not multiplexedsuch that it can take many hours to cover this spectral window at high resolution.
As an example to compare with the Nicolet iS50 FTIR spectrometer, consider an infrared
diode laser operating at ν̃ = 2000 cm−1 with 1 mW of output power and a linewidth of 30 MHz
or about 0.001 cm−1 . Assuming the output of this laser can be focused to a diffraction limited
spot size, we find that the spectral energy density is roughly 1 MW/sr cm2 cm−1 , which is roughly
ten orders of magnitude larger than the blackbody source used in the FTIR spectrometer. Note,however, that high sensitivity, low-noise detectors used in infrared spectroscopy typically saturatewith incident powers of less than 100 µW, and this greatly limits the advantage gained from using asource with extremely high spectral energy density like a laser. Nonetheless, the narrow linewidth,wide tunability, and excellent spectral selectivity of diode laser sources offer substantial benefits forhigh-resolution spectroscopy. In addition, commercial diode lasers are practically turn-key systems
with a relatively low degree of complexity making them very straightforward to operate. Before
proceeding, it is worth mentioning here that frequency comb sources offer the best of both worlds as
they are broadband, and have high spectral energy density and narrow linewidth [11, 13]. However,frequency combs are still a developing technology and are not commercially available for the infrared
region of the spectrum that we are currently interested in.

3.4.1

Quantum cascade lasers

Standard diode lasers rely on carrier recombination across a bandgap (i.e., interband transitions)inside a semiconductor active region where the bandgap energy dictates the wavelength of laser
operation. This principle works exceptionally well in the visible to near-infrared region of the65electromagnetic spectrum, but becomes problematic when applied in the mid- to far-infrared. As
the photon energy decreases the bandgap of the active region also decreases and laser operation
becomes prohibitively sensitive to temperature fluctuations. Additionally, it becomes difficult tofabricate high quality (i.e., free of defects) semiconductor devices with small bandgaps resulting inpoor reliability [84]. These problems eventually led to the development of quantum cascade lasers(QCLs) at Bell Labs in 1994 [85].
QCLs rely on a completely different mechanism of operation that involves electrons transitioning between discrete energy levels in a quantum well (i.e., intersubband transitions) where theenergy level spacing dictates the wavelength of laser operation. This is in contrast to interband
transitions in a semiconductor material because electrons transitioning to a lower energy level in aquantum well through release of a photon remain in the conduction band allowing the electrons totunnel into subsequent quantum well regions and emit more photons. This results in a cascade effect, which makes larger quantum efficiencies possible compared to traditional semiconductor laser
diodes. Thus, by patterning many quantum wells on a single device, laser operation at larger output powers than semiconductor laser diodes can be achieved. Furthermore, by patterning quantum
wells with different energy level spacings, a single device can be created that provides gain over aspectral range of hundreds of wavenumbers. High spectral selectivity and mode-hop free tunability
can then be achieved through standard diode laser methods such as distributed feedback or an
external cavity in conjunction with temperature and injection current tuning. Good reviews on
QCL technology and their applications to spectroscopy can be found in Refs. [84, 86, 87].
The QCL used in this dissertation work is a Daylight Solutions CW-MHF system (model
21095) that provides about 100 mW of continuous wave power and continuous, mode-hop freetuning from 1020 − 1115 cm−1 . The MHF series of QCLs are external cavity lasers with grating
feedback in a modified Littrow configuration that prevents beam pointing deviation as the laser istuned. The specified linewidth is less than 30 MHz, and tuning can be achieved by changing thelaser temperature, rotating the grating via a PZT element (DC − 100 Hz bandwidth), or changing
the laser injection current (10 kHz − 2 MHz bandwidth). This QCL is used for studying cold beams

66
of molecules in the gas phase, and its narrow linewidth and broad tuning range allow for highresolution ro-vibrational spectroscopy in the mid-infrared region of the electromagnetic spectrum.
As discussed in section 3.1.3, shot noise sets a fundamental limit on the minimum detectable
optical depth in an absorption spectroscopy experiment. However, laser sources are plagued byexcess intensity noise due to spontaneous emission, thermal fluctuations, acoustic vibrations, andcurrent noise from the injection current source. Therefore, special techniques must be employedto achieve shot noise limited measurements, and, in principle, there are three common methods tominimize excess intensity noise: (1) active noise reduction through feedback stabilization of the laser
intensity, (2) use of a modulation scheme to encode the absorption signal of interest onto a frequency
band where the intensity noise is negligible, and (3) direct cancellation of the laser intensity noise
via balanced subtraction of a signal and reference beam derived from the same laser. The first ofthese methods is impractical in the mid- to far-infrared region of the electromagnetic spectrum dueto the lack, or huge expense of available devices like acousto- and electro-optic modulators. The
other two methods, however, are of interest and will be discussed further below.

3.4.2

Wavelength modulation

The simplest and perhaps most commonly used modulation technique for diode lasers is directmodulation of the injection current, which results in a frequency modulation of the laser output.
The subsequent interaction of the frequency modulated laser light with an absorbing species resultsin modulation of the laser intensity at the modulation frequency and its harmonics, from which
absorption data can be retrieved via a detector and lock-in amplifier. In regards to spectroscopy, thistechnique is broadly classified into two categories depending on whether the modulation frequency
is much greater or much less than the spectral width of the absorption feature being probed. The
former is the case of frequency modulation spectroscopy (FMS) [88], which is typically implemented
with modulation frequencies of 100 MHz to 1 GHz. The latter is the case of wavelength modulation
spectroscopy (WMS), which is typically implemented with modulation frequencies of 100 kHz or
less. Notably, QCLs can be effectively modulated via their injection current up to frequencies of only

67
about 1 MHz as a result of thermal effects being the dominant frequency modulation mechanism [89].
Therefore, the FMS technique is not applicable and only the WMS technique will be discussed here.
Application of a sinusoidal modulation to the QCL injection current results in modulation of
the laser wavenumber according toν̃ (t) = ν̃c + ν̃a cos (2πfm t),(3.29)

where ν̃c is the center laser wavenumber, ν̃a is the modulation amplitude, and fm is the modulation
frequency. In addition to this frequency modulation, the output intensity of the QCL will be
modulated as a consequence of modulating the injection current. This effect is known as residual
amplitude modulation (RAM), and results in spurious signals at the modulation frequency and its
harmonics even in the absence of an absorbing species. This is a well documented effect in WMS,
and limits the ultimate sensitivity of the technique. We write the modulated laser intensity as
follows:"
I0 (ν̃c , ν̃a , t) = I¯0 (ν̃c ) 1 +∞X#κn (ν̃) ν̃an cos (2πnfm t + φn ),(3.30)

n=1

where I¯0 (ν̃c ) is the average laser intensity at the center wavenumber, the κn (ν̃) represent thewavenumber dependent intensity modulation coefficients, and the φn represent phase shifts between
the frequency and intensity modulations. Depending on the modulation depth, terms up to n = 4
may need to be included to accurately describe RAM for QCLs [90]. Furthermore, QCLs exhibit
significantly larger linear and nonlinear RAM signals compared to diode lasers used in the nearinfrared [90, 91].
Given that both the modulated laser wavenumber and intensity are periodic signals in time,the detected signal will also be periodic in time. Thus, Fourier decomposition is a useful method
for mathematically representing the generation and detection of the harmonic components of theabsorption and background signals in an absorption spectroscopy experiment. Both Kluczynski and
Axner [92] and Schilt et al. [93] have developed rigorous theoretical descriptions of the wavelength
modulation process, so we only give the highlights here. Following the method of Kluczynski and68
Axner [92], we define the detected signal S (ν̃) in terms of background and absorption signals as
SBG = βT (ν̃) I0 (ν̃) = βI (ν̃) ,(3.31)

SA = −βα0 χ (ν̃) T (ν̃) I0 (ν̃) = −βα0 χ (ν̃) I (ν̃) ,(3.32)

where β is a detector gain factor, T (ν̃) represents the wavenumber dependent transmission of the
optical system, I0 (ν̃) is the output intensity of the laser, I (ν̃) is the intensity seen by the detector
in the absence of an absorbing species, α0 is the peak absorbance, and χ (ν̃) is a peak-normalized
absorption lineshape function. Using a lock-in amplifier, the kth harmonic of the detector signal
can be extracted, which is written in terms of even and odd Fourier components of the lineshapefunction and detected intensity aseSBG,k(ν̃c , ν̃a ) = βIke (ν̃c , ν̃a ) ,(3.33)o(ν̃c , ν̃a ) = βIko (ν̃c , ν̃a ) ,SBG,k(3.34)"e(ν̃c , ν̃a ) = −βα0SA,kk

1 + δk,0 X ee(ν̃c , ν̃a )χk−m (ν̃c , ν̃a ) Im
2
m=0

+

2 − δk,04

2 − δk,0
+
4

∞
X
m=0
∞
X

e
(ν̃c , ν̃a )χek+m (ν̃c , ν̃a ) Im#e(ν̃c , ν̃a )χem (ν̃c , ν̃a ) Ik+m,

(3.35)

m=0

"
o
SA,k
(ν̃c , ν̃a ) = −βα0

k

1X e
o
χk−m (ν̃c , ν̃a ) Im(ν̃c , ν̃a )
2
m=0

−

∞
1X

2

1
+
2

m=0
∞
X

o
χek+m (ν̃c , ν̃a ) Im(ν̃c , ν̃a )#eχem (ν̃c , ν̃a ) Ik+m(ν̃c , ν̃a ) .

(3.36)

m=0

In the above equations, the superscripts represent the even (e) and odd (o) Fourier componentsof the various quantities and the subscripts correspond to the different harmonics, and δk,0 is the
Kronecker delta function. Because the detected intensity passes through an optical system with a69
wavenumber dependent transmission function, we express the Fourier components of I (ν̃c , ν̃a ) interms of the output intensity of the laser as"k
1 + δk,0 X eee
Ik (ν̃c , ν̃a ) = −βα0
Tk−m (ν̃c , ν̃a ) I0,m(ν̃c , ν̃a )
2
m=0

+

2 − δk,04

2 − δk,0
+
4

∞
X
m=0
∞
X

e
e
Tk+m
(ν̃c , ν̃a ) I0,m(ν̃c , ν̃a )

#
e
e
Tm
(ν̃c , ν̃a ) I0,k+m(ν̃c , ν̃a ),(3.37)

m=0"
Iko (ν̃c , ν̃a ) = −βα0

k

1X e
o
Tk−m (ν̃c , ν̃a ) I0,m(ν̃c , ν̃a )
2
m=0

−

∞
1X

2

1
+
2

m=0
∞
X

o
e
(ν̃c , ν̃a ) I0,m(ν̃c , ν̃a )
Tk+m

#
e
e
(ν̃c , ν̃a )Tm(ν̃c , ν̃a ) I0,k+m.

(3.38)

m=0

Notably, only the even Fourier components of χ (ν̃c , ν̃a ) and T (ν̃c , ν̃a ) are considered because thevalues of these functions follow the modulation with no phase delay (i.e., only the cosine components
are non-zero). Thus, once we define a mathematical form for the χ (ν̃c , ν̃a ) and T (ν̃c , ν̃a ), andsubstitute the Fourier components of Eq. (3.30) for I0 (ν̃c , ν̃a ), we have a closed form descriptionof the WMS method. Assuming a Lorentzian absorption lineshape, we can the result derived by
Arndt [94] to define the Fourier components of χ (ν̃c , ν̃a ). Additionally, if we wish to investigate
the effects of unwanted etalons in the optical system (i.e., as a result of reflections from uncoatedoptical surfaces) we treat T (ν̃c , ν̃a ) as a low finesse Fabry-Perot cavity; otherwise, we can set T0e = 1
and Tne = 0 for all n > 0.
With a theoretical description of WMS in hand, it is advantageous to investigate the expected
absorption signals in the presence of RAM and unwanted etalons in the optical system. For thisanalysis, we will consider both 1f and 2f detection corresponding to the first and second harmonic,
respectively, of the detector signal. Furthermore, we will assume that the absorption lineshape is a
Lorentzian with a width limited by the laser linewidth at ΓL = 30 MHz. Finally, the modulationindex, defined as the ratio of modulation amplitude to the half-width at half-maximum absorption

70
linewidth, is chosen to be 2.0 for 1f detection and 2.2 for 2f detection, which corresponds to the
maximum achievable signal in each case [95]. Figure 3.5 shows the expected signals at the output
of a lock-in amplifier for a peak absorbance of α0 = 10−6 and the assumption of pure frequency
modulation. In the absence of RAM and etalon effects, we observe signals that resemble the firstand second derivative of the absorption lineshape for 1f and 2f detection, respectively. Notably,
the signal amplitude is reduced compared to the expected value of 10−6 for direct absorption. Thisis a result of the signal power not being completely concentrated in the sidebands of interest, butinstead spread over many sidebands. Thus, the signal amplitude will be largest for 1f detection
and decrease monotonically for schemes that detect higher harmonics of the modulation frequency.
However, higher harmonics are less sensitive to RAM and etalon effects such that 2f detectionis the most common scheme used for WMS. This is apparent in Fig. 3.6, which shows the effectof RAM on the signals of Fig. 3.5. Here, we use the results of Ref. [90] to include the effects of
non-zero κn (ν̃). Clearly, the 1f signal becomes completely obscured by the large RAM backgroundshowing the total loss of absorption sensitivity. The 2f signal also resides on a non-zero RAM
background, but due to the substantially smaller value of κ2 compared to κ1 the effect is lessdetrimental. Nonetheless, 2f detection also experiences a loss of sensitivity. Finally, we look at
the effect of unwanted etalons in the optical system. We model the etalon as a Fabry-Perot cavity
formed by two uncoated, plane-parallel KBr windows (i.e., vacuum chamber windows), which resultin a finesse of approximately 0.17. Figure 3.7 shows the expected signals at the output of a lock-in
amplifier when frequency modulation, RAM, and etalon effects are all included. With the additionof etalon effects, the 1f signal remains completely obscured while the 2f signal experiences further
degradation. Although the 2f signal is still discernible, it has lost much of its second derivative
character making it difficult to recognize.
In principle, detecting at higher harmonics of the modulation frequency will minimize thedeleterious effects of RAM and etalons. However, the signal strength also decreases for higher
harmonics such that the overall sensitivity of an absorption measurement goes down. Therefore, it
is useful to consider other methods of dealing with excess laser intensity noise when attempting to71

5#10 -7(a)4
3

Signal amplitude

2
1
0
-1
-2
-3
-4
-5
-20

-15

-10

-5

0

5

10

15

20

Detuning
5#10 -7(b)
4
3

Signal amplitude

2
1
0
-1
-2
-3
-4
-5
-20

-15

-10

-5

0

5

10

15

20

Detuning

Figure 3.5: (a) 1f and (b) 2f detection of WMS signals from a Lorentzian absorption line with
α0 = 10−6 . These signals represent the case of pure frequency modulation such that κn (ν̃) = 0 for
all n. Additionally, it is assumed that there are no unwanted etalons in the optical system suchthat T0e = 1 and Tne = 0 for all n > 0.

72

-0.0156

Signal amplitude(a)-0.0156-0.0156-0.0157
-20

-15

-10

-5

0

5

10

15

20

Detuning
1.815#10 -5(b)
1.81

Signal amplitude

1.805
1.8
1.795
1.79
1.785
1.78
1.775
1.77
-20

-15

-10

-5

0

5

10

15

20

Detuning

Figure 3.6: (a) 1f and (b) 2f detection of WMS signals from a Lorentzian absorption line with
α0 = 10−6 . These signals represent the case of frequency modulation in the presence of RAM, where
we have used average values of κ1 = 1.8 × 10−10 Hz−1 , κ2 = 1.1 × 10−20 Hz−2 , φ1 = 1.08π, and
φ2 = 1.58π based on data from Ref. [90]. Additionally, it is assumed that there are no unwanted
etalons in the optical system such that T0e = 1 and Tne = 0 for all n > 0.

73
-0.013(a)Signal amplitude-0.0135-0.014-0.0145-0.015-0.0155-0.016
-20

-15

-10

-5

0

5

10

15

20

Detuning
1.85#10 -5(b)
1.8

Signal amplitude

1.75

1.7

1.65

1.6

1.55

1.5
-20

-15

-10

-5

0

5

10

15

20

Detuning

Figure 3.7: (a) 1f and (b) 2f detection of WMS signals from a Lorentzian absorption line with
α0 = 10−6 . These signals represent the case of frequency modulation in the presence of RAM,
where we have used average values of κ1 = 1.8 × 10−10 Hz−1 , κ2 = 1.1 × 10−20 Hz−2 , φ1 = 1.08π,
and φ2 = 1.58π based on data from Ref. [90]. Additionally, a Fabry-Perot etalon with finesse ofabout 0.17 is included to model the effect of uncoated, plane-parallel KBr windows on the vacuum
chamber.

74
achieve shot-noise limited performance in an absorption spectroscopy experiment.

3.4.3

Balanced subtraction

The balanced subtraction technique is the principle used in differential absorption spectroscopy,
where the output of the laser is split into a signal and reference beam. The signal beam interacts
with the absorbing species of interest before being passed to a detector while the reference beam ispassed directly to a separate detector. By subtracting the resulting detector signals it is possible tocompletely eliminate any correlated excess noise present in the measurement. However, to achieve
shot-noise limited sensitivity, the two signals must be balanced to a high degree of precision prior
to subtraction. For example, to achieve 40 dB of noise subtraction over the full measurement
bandwidth the two signals must be balanced to a part in 102 , or 1% over that same bandwidth.
This is challenging to do optically, but straightforward to do electronically.
In the early 1990’s, Philip Hobbs at the IBM Thomas J. Watson Research Center developed anovel electronic noise cancellation circuit for differential absorption spectroscopy measurements [96,97]. The basic principle of the circuit is to employ negative feedback to automatically adjust thereference detector signal to achieve near perfect subtraction of the signal and reference detector
signals at DC. As long as the intensity noise in the signal and reference beams remains correlated,
subtraction at DC effectively cancels the excess intensity noise at all frequencies because this noise
is directly proportional to the average, or DC intensity. A monograph by Hobbs [98] provides adetailed discussion of this circuit along with its limitations and some variations that are useful fordifferent measurement setups. Notably, use of this circuit can readily achieve shot-noise limited
absorption sensitivity.
In this dissertation work, we employ an auto-balanced subtractor circuit based on the principle developed by Hobbs. This circuit was developed by Terry Brown from the JILA electronics
shop, and was designed for use with infrared detectors. Figure 3.8 provides the schematic for thiscircuit. The input stages allow for adjustable gain and offset trimming of both the signal and reference channels, and the feedback bandwidth can be adjusted based on the integrator time constant.

A

B

C

D

-VREF

+ VREF

0.1uF
C42

0805

VSRTN

0805

VRRTN

0.1uF
C43

100R
R67

1%
0805

0805
0.1%

Figure 3.8: Schematic of the high CMRR auto-balanced subtractor circuit.

8

0805
0.1%

R59
24.9k

Vre f > 0

0805
0.1%

R60
100k

C38
0.1uF

0805
0.1%

R58
249k

0805
0.1%

R56
4.99k

0805
0.1%

R55
4.99k

0805
0.1%

R57
24.3k

0805
0.1%

C34
R52
24.9k 0.1uF

R53
249k

0805
0.1%

R9
24.3k

0805
0.1%

0805
0.1%

C40
33uF

C39
33uF

C6
0.1uF

0805
0.1%

0805
0.1%

R12
249k

R5
249k

C9

+

3

0805
0.1%

R64
24.3k

C5
0.1uF

U2

8

5
6
AD829JN

7

0.1uF
C13

U8
1

-V1

4

+

-

0805
0.1%

R65
24.3k

C19
0.1uF

C32
2.7-10pF

C41
2.7-10pF

3

2

7

+ V1

6

0805
1%

R51
51.1

0805
0.1%

R50
1.00k

0805
0.1%

R10
1.00k

0805
0.1%

R11
1.00k

6

0805
0.1%

R7
1.00k

R19
50K

4

+

-

7

1

U7
8

5
6

-V1

OFFSET

CW

R49
20K

AD829JN

C30
2.7-10pF

0805
0.1%

R63
1.00k

1

-VREF

+ VREF

R47
51

6

R32
50K

-V1

0805
1%

R23
750R

R15
51

0805
0.1%

MF
1%

1

0805

-3Vd

-Vd

+ V1
C27
0.1uF

BAV19W-7
SOD-123

D4

BAV19W-7
SOD-123

D1

Q3-B

DMMT3904W

L2
600ohm

-2Vd

C16
10nF

DMMT3904W

Q4-B

U3

-V1

4

-

+

7

8

5

0805
0805 0.1%

R33
49.9k

4

1

8

5
6
AD829JN

0.1uF
C26

U5

-V1

4

+

-

7

5

Q6-A

DMMT3904W

R31

0805
0.1%

R29
1.00k

D6
SOD-123

0805

M1
4-40

M4
4-40

0805
0.1%

R25
49.9k

0805
0.1%

R30
499k

3

R39
51

2

5

4

+

3

2

11
CW

10

9

8

7

6

5

4

3

2

1
1nF

1uF

0.47uF

0.22uF

0.1uF

47nF

22nF

10nF

3.3nF

3

@Sig Le ve l = 1.7V DC, TC is ~ 40% fa s t e r

POS_1: 122us
POS_2: 244us
POS_3: 520us
POS_4: 1.4ms
POS_5: 3ms
POS_6: 5.8ms
POS_7: 13ms
POS_8: 26ms
POS_9: 60ms
POS_10: 122ms

Me a s ure d TC (t a u)
@Sig Le ve l = 1.25V DC

TO_J6-1

0805

R34
51

ELECTRONICS

6.95V
LM399H

VR2

6.95V
LM399H

VR1

SIGNAL
RTN

2

-VREF

+ VREF

-VREF

-V1

2

0.1uF
C37

4

SW1-B
1 MAX4510EUT

C4
0.1uF

TTL_RTN2+ V1

TTL_IN HIGH= HOLD

1

DATE: 04/ 10/ 2017

DN: YJ0179A1

V3

V3

T. BROWN

YE / B. SPAUN1SHEET 1 OF 1

DRAWN BY:BUILT FOR:CIRCUITVERSION:

T. BROWN

2

C31
33uF

C22
33uF

J7
INTEGRATOR HOLD++

+ VREF

1

HIGH CMRR AUTO ADJUST SUBTRACTOR

0805
1%

R42
10.0k

0805
1%

R43
51.1R

OMIT R34 WHEN USED1J4OPTIONAL LOCK-IN INPUTUPDATE: 2/ 22/ 2018

DESIGN BY:

1.00k
R48

1%
MF

-

+

-

+

MF
1%

R38
1.00k

J3
DIFFERENCE OUTPUT

-V1

+ V1

2

100K
R46

CU-Boulder & NIST

0805
0.1%

3

CW

5

6

R22
51

JILA

R44
1.00k
0805
0.1%

0805
0.1%

R45
1.00M

SW1-A
MAX4510EUT

R40
100k

C28
1.0uF

TO_J6-2

FEEDBACK Time Cons t a nt Front Pa ne l Rot a ry Swit ch

-V1

1

-

7

C33
0.1uF

+ V1

U6

0.1uF
C29

LF356N

6

C24
1nF

0805
0.1%

1

0805

6

366uV/ rt (Hz)AD829JRC36
0.1uF

Int e gra t or Ze ro Adjus t me nt :While s ys t e m is locke d, i.e . Int e t ra t or is
ON a nd e qua l pos it ive DC volt a ge s int o
Signa l a nd Re fe re nce input s . Adjus t
R46 s uch t ha t t he volt a ge a t t he out putof U6 = 0V.

J5
Ra t iome t ric Out put

4

POLE
COMMON

1

8

5

0805
0.1%

R26
2.49k

0805

C35
0.1uF

U4

-V1

4

+

-

7

+ V1

R41
100

3

2

0805

C15
10pF

J6
INTEGRATOR ROT SWITCH

0.1uF
C18

U5 Ze ro Adjus t :Wit h out put of U8 = 0V, a nd t he s hort e s twire s hort ing D3's a node t o Q5-A's e mit t e r.
Adjus t R32 unt il volt a ge diffe re nce
be t we e n pins 2 a nd 6 of U5 is ze ro.

0805
0.1%

-4Vd 1.00k
DMMT3904W

Q5-A

D5
SOD-123

0805
1%

-V1
Q5-B

0805

C44
10nF

R28
DNI

CMRR a djus t .
CW

0805
0.1%

2W
50K
R36

366pA/ rt (Hz)

0805
0.1%

0805
0.1%

R21
1.00k

R20
1.00k

-VREF

+ VREF

BAV19W-7 BAV19W-7

R68
750R

0805
0.1%

R24
25.5R
0805
0.1%

R27
1.00k

DMMT3904W DMMT3904W

Q6-B

Q3-A

DMMT3904W

L1
600ohm

Q4-A

DMMT3904W

Q1-A
DMMT3906W

Q2-A

DMMT3906W

DMMT3906W

DMMT3906W

Q1-B

Q2-B

AD829JN

6

D2, D3 compe ns a t e s for
Vbe of Q3-B

3

2

C23
2.7-10pF

D2
D7
BAV19W-7 BAV19W-7 -3Vd

1.0uF
C21

R37
619

D3
BAV19W-7

C8
0.1uF

2

3

C10
2.7-10pF

J2
SIGNAL MONITORJ8REFERENCE MON

R35
499k

-V1

OFFSET

CW

MF
1%

R16
1.24k

0805

AD829JN

6

R2
20K

8

5

CW

U1

-V1

4

+

3
C7
0.1uF

-

7
2

C2
2.7-10pF

0805
0.1%

R62
1.00k

0805
0.1%

0805
0.1%

C12
1.0uF

R17
49.9k

R18
499k

+ V1
C17
0.1uF

CW

+ V1

C11
0.1uF

wire s hort ing Q2-B colle ct or t o Q1-B e mit t e r.
Adjus t R19 unt il volt a ge diffe re nce
be t we e n pins 2 a nd 6 of U3 is ze ro.

U3 Ze ro Adjus t :Wit h out put of U2 = 0V, a nd t he s hort e s t

U7 ze ro Adjus t :Wit h out put of U8 = 0V, a djus t R49
unt il out put of U7 is ze ro.

C20
0.1uF

3

2

C25
0.1uF

+ V1

5

The Input DC le ve ls we re t oo s ma ll a nd t he curre nt mirror t ra ns is t ors we re ope ra t ing in a highly non-line a r ra nge
I incre a s e d t he ga in of t he input a mplifie r t o 5x, or cha nge d R64, R9, R65, a nd R57 from 4.99 kohm t o 24.3 kohm.
Addit iona lly I a dde d ga in a t t he V t o I curre nt mirror circuit cha nging R16 t o 1.24 kohm, a nd R37 t o 619 ohms . The ne t ga in cha nge ~ 10x.

U1 ze ro Adjus t :Wit h out put of U2 = 0V, a djus t R2
unt il out put of U1 is ze ro.

AD829JN

C14
0.1uF

8

5

-VREF

+ VREFU2 Ze ro:Short J1 input ;Adjus t R1 for 0V a t Pin 6 U1

C3
2.7-10pF

1

-V1

4

-

2

7

+ V1

7

U8 Ze ro:Short J9 input ;Adjus t R61 for 0V a t Pin 6 U12

2.7-10pF

C1
0.1uF

0805
0.1%

R3
24.9k

R4
100k

R54
100k

-V1

+

+

0805
0.1%

R14
24.9k

J9
REFERENCE INPUT

4-40
M3

CW

3

R61
10k

2

+ -RTN-15V IN

+ V1

0805
0.1%

1

0.1%
0805

0805
0.1%

100R
R66

R13
100k

+ 15V IN

CW

R6
4.99k

R1
10k

1% 4.99k
0805 R8

SIGNAL INPUT
J1

Vs ig > 0

-VREF

+ VREF

J10
Powe r Input

M2
4-40

8

Lock Ra nge (Vs ig/ Vre f) = 0.1 - 1.1

A

B

C

D

75

76
Additionally, either the subtracted output or the ratiometric output can be used. The former is ahigh-pass filtered version of the balanced difference between the signal and reference channels while
the latter is a low-pass filtered version of the balanced difference divided by the signal channel. The
corner frequency of these filters are set by the integrator. Bench tests of the circuit demonstrate
greater than 60 dB of common-mode rejection at frequencies in excess of 100 kHz. The circuit issetup in such a way that it can be used for either direct absorption spectroscopy measurements at
DC or for WMS measurements using a lock-in detection scheme. For the direct absorption spectroscopy case, it is crucial to eliminate DC offsets on the signal and reference detectors as well as
to prevent any unwanted intensity modulation on either beam, which can result from poor focusingof the beams onto the detector elements or beam clipping elsewhere in the optical setup. For WMS
measurements these factors are less crucial as long as the unwanted intensity modulation is not at
the modulation frequency or its harmonics.

3.4.4

Detectors

In order for the balanced subtraction technique to work for direct absorption spectroscopy, thedetectors used must have low noise down to DC. This precludes the use of PC detectors because
they exhibit large 1/f noise characteristics. Instead, photovoltaic (PV) detectors are preferred,and PV MCT detectors from Kolmar Technologies are some of the best on the market with acharacteristic D∗ on the order of 1011 Jones at 50 kHz and extremely low noise transimpedance
amplifiers. We use two such detectors from Kolmar (model KLD-0.5-J1/DC/10) with a detector
area of Ad = 0.05 × 0.05 cm2 , bandwidth of DC-1 MHz, responsivity of greater than 5 × 105 V/W,‡
a cutoff wavlength above 10 µm or 1000 cm−1 , and field of view of 15◦ . Figure 3.9 shows thedark noise spectrum for both of these detectors, where the vertical axis has been converted intocurrent noise density using the specified transimpedance of the detector amplifier. Given the white
noise floor of approximately 3 pA/Hz−1/2 above 10 kHz, these detectors exhibit a NEP of about‡
The responsivity of the MCT element is in the range of 4 − 6 A/W, and the transimpedance amplifier has a gainof 105 V/A.

77

10 3

Current noise density (pA/Hz-1/2 )Serial number 4346-3
Serial number 4346-4

10 2

10 1

10 0
10 1

10 2

10 3

10 4

10 5

Frequency (Hz)Figure 3.9: Current noise density of the PV MCT detectors from Kolmar Technologies measured
with an SR780 FFT spectrum analyzer. The white noise floor above 10 kHz is equivalent to a NEPof about 0.6 pW/Hz−1/2 or D∗ ∼ 1011 Jones assuming a responsivity of 5 A/W.

78
0.6 pW/Hz−1/2 or D∗ ∼ 1011 Jones assuming a responsivity of 5 A/W. Plugging the detector
parameters into Eq. (3.16), we find a detection limit of approximately 10−7 Hz−1/2 for an incident
power of about 10 µW. Notably, this noise floor occurs at frequencies above 10 kHz such that a
combination of WMS and balanced subtraction will be required to reach the ultimate detection
sensitivity. Unfortunately, the 1/f noise appears to be less than optimal on these detectors, which
could be a result of the high transimpedance gain or detector degradation.§
The MCT detectors used here likely have far too large of a transimpedance gain (105 V/A),
which greatly limits the amount of optical power that can be used – at 5 × 105 V/W the transimpedance amplifier will saturate at about 20 µW (about 10 V amplifier output for ±12 V rails).
Furthermore, considering the typical series resistance of the MCT element at liquid nitrogen temperature to be 5 − 10 Ω, and a typical bias voltage to be 100 mV, we expect linear operation
for incident powers greater than 1 mW. Therefore, a more reasonable transimpedance would be< 104 V/A, which would greatly improve our shot-noise limited sensitivity. However, it may bebetter to work with a lower bias voltage to minimize 1/f noise if DC balanced detection is themethod of choice.
An important note about the MCT detectors is their sensitivity to room temperature blackbody radiation. From Wien’s displacement law (λp ≈ 3000 µm K), we find that the room temperature (T ∼ 300 K) blackbody spectrum peaks at a wavelength of approximately 10 µm, which
is precisely where the responsivity of the MCT detectors peak. Therefore, small changes in thetemperature of the area surrounding the detectors can cause large DC offsets, which are detrimental to the functionality of the auto-balanced subtraction circuit. It is necessary to shield thespectroscopy setup from background blackbody radiation (e.g., warm bodies moving around theroom) and frequently check the DC offsets on each detector to ensure that the local environment
has not changed.
§

We have borrowed these detectors from the Nesbitt group in JILA, and they were manufactured in 2011.

79
3.4.5

Performance of the laser absorption spectroscopy system

Here, we discuss the performance of the laser absorption spectroscopy system, and investigate
both the balanced subtraction and wavelength modulation techniques to maximize the absorption
sensitivity of the setup. Due to the low optical powers required to avoid saturating the detectors,
we use crossed wire grid polarizers with an extinction ratio of > 104 at 1060 cm−1 or 9.4 µm.
Furthermore, we focus the laser onto the detectors using anti-reflection coated, ZnSe plano-convex
lenses with a focal length of 20 mm. Figure 3.10 shows the excess noise present on the QCL usedin this dissertation work. As can be seen in the figure, the noise is dominated by the detector up
to about 100 Hz, at which point the QCL noise begins to dominate. The broad feature around200 − 250 Hz is due to a mechanical resonance of the QCL laser cavity, which is most likely related
to mounting of the feedback grating. The majority of the other noise spikes up to a few kHz appear
to be feedthrough of power line harmonics (e.g., harmonics of 60 Hz), and the large spike near
30 kHz is likely from a switching power supply used in the QCL laser controller. At the time ofthis measurement, a function generator and PZT driver were connected to the modulation inputs
on the QCL laser head; therefore, it is possible that the power line noise is a result of ground loops.
Besides the noise spikes, the noise floor from about 0.5 − 5 kHz is about 2 − 3 times larger thanthe detector noise floor. This is far too low for the auto-balanced subtraction circuit to be useful,because, at best, it can achieve a noise floor that is the quadrature sum of the detector noise floors.
Looking at Fig. 3.9, we see that this quadrature sum will be larger than the low frequency noiseof the laser itself. Indeed, attempts to use the auto-balanced subtractor resulted in a higher noise
floor than without it, and the circuit appeared to do very little for the many spikes present in thenoise spectrum. Therefore, we conclude that auto-balanced subtraction is not beneficial for directabsorption spectroscopy measurements with the current setup.
It is of interest, then, to investigate the sensitivity we can achieve with a single detector. For
the measurements presented here we use approximately 15 µW of laser power, which corresponds
to a DC output voltage of 8 V from the detector. Furthermore, we connect a 10 kHz low pass

80

Current noise density (pA/Hz-1/2 )

10 3

10 2

10 1

Shot noise for V out = 5.52 V

10 0
10 1

10 2

10 3

10 4

10 5

Frequency (Hz)Figure 3.10: Excess noise of the Daylight Solutions QCL for approximately 10 µW power incident
on detector 4346-4, which corresponds to a DC output voltage of 5.52 V. The red, dashed line is
the shot noise photocurrent density of 4.2 pA/Hz−1/2 corresponding to 5.52 V output voltage. Also
shown is the dark noise floor of detector 4346-4.

81
filter between the detector output and oscilloscope input, and AC couple the signal with a 10 Hz
high pass filter. To simulate an actual experiment, we trigger the oscilloscope at the ablation laser
repetition rate of 10 Hz, and acquire 40 ms of data at a rate of 1 MS/s. Note that we expect themolecule signal to consist of a pulse with a rise time of a few hundred microseconds and a fall timeof a few milliseconds, which corresponds to the time it takes after a pulse of the ablation laser formolecules to be entrained in the buffer-gas flow and extracted from the cell. Figure 3.11 shows theresults of this measurement for a single acquisition and an average of 16 acquisitions. From thefigure, we find that the standard deviation of the laser noise is about 2 × 10−4 V and 4 × 10−5 V fora single acquisition and an average of 16 acquisitions, respectively. In a 10 kHz bandwidth thesevalues correspond to approximately 20 pA/Hz−1/2 and 4 pA/Hz−1/2 equivalent photocurrent noise
densities, respectively, which are roughly 4 times larger than the shot noise photocurrent density
at a DC output voltage of 8 V from the detector. In terms of minimum detectable optical depth,
the laser noise corresponds to about 2.5 × 10−7 /Hz−1/2 and 5 × 10−8 /Hz−1/2 for a single acquisition
and an average of 16 acquisitions, respectively. Notably, the detector noise is about 2 − 3 times
larger than shot noise. Therefore, it appears that we are very close to the detector limited noise
floor, which is also very close to the shot noise limited noise floor. Notably, this likely represents
the best we can achieve with this setup at DC, and the performance will likely get worse once thelaser is aligned through the vacuum chamber and all of the vacuum pumps are running.
Although we have not utilized any specialized techniques to achieve the above noise floor,
it is important to mention that it was crucial to vibration isolate the optical system as much as
possible. Thus, the optical breadboard is mounted on sorbothane and the QCL is mounted to asecond optical breadboard that is isolated from the main breadboard through more sorbothane.
Additionally, care must be taken with filling the MCT detector liquid nitrogen dewar. Once thedetector has cooled down a ball driver is inserted into the dewar to “quiet” the liquid nitrogen inside,otherwise bubbling can vibrate the detector. This shows up as noise around 120 Hz, which can easily
be mistaken for power line noise. Furthermore, all electronics are connected through power line
filters in an effort to isolate the system from the main building power as much as possible. Finally,

82

1

(a)
0.8
0.6

Voltage (mV)

0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

5

10

15

20

25

30

35

40

Time (ms)
1

(b)
0.8
0.6

Voltage (mV)

0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

5

10

15

20

25

30

35

40

Time (ms)Figure 3.11: Single detector noise floor of the QCL absorption spectroscopy system for (a) asingle acquisition and (b) an average of 16 acquisitions. The black curve is the dark noise with thedetector (4346-4) blocked, and the blue curve is with the laser incident on the detector. A 10 Hz
high pass filter and 10 kHz low pass filter are used to set the measurement bandwidth, and data is
acquired at a rate of 1 MS/s. The DC output voltage from the detector is 8 V, which corresponds
to approximately 15 µW of laser power.

83
the current limitation is likely that the optical setup is not on a proper floating optical table, butinstead a normal table with a wood top. Therefore, low frequency vibrations around 30 Hz andbelow get transmitted to the optical system because the sorbothane does not provide isolation fromthese frequencies. Fortunately, the period of these vibrations are long compared to our expectedsignal duration, so we do not expect this to have a substantial impact on measurements.
Although the single detector performance appears to be quite good at DC, it is worth investigating wavelength modulation as a method to improve measurement sensitivity and experimentrobustness (i.e., insensitivity to vibration noise). For this measurement, we current modulate thelaser with a 0.1 Vpp sinusoid at 42.345 kHz, and investigate the RAM with a spectrum analyzer.
Figure 3.12 shows the RAM at the fundamental of the modulation frequency. From the figure, we
see that the RAM peak corresponds to an equivalent current noise density of about 40 nA/Hz−1/2 ,which is well above the detector and shot noise floors. The two sets of sidebands correspond tomodulation at about 30 Hz and 220 Hz, which correspond to low frequency vibrations transmitted to the optical system and a mechanical resonance of the QCL cavity, respectively. Notably,
the noise floor away from the RAM peak and the modulation sidebands is equivalent to about
5 pA/Hz−1/2 , which is very close to the shot noise floor of about 4 pA/Hz−1/2 for a DC output
voltage of 5.52 V from the detector. Therefore, if we use a high pass filter to eliminate frequencies
below about 30 Hz and do not introduce any additional RAM in the optical system, we will be able
to make shot noise limited measurements with wavelength modulation spectroscopy. However, adetailed characterization of the QCL is necessary to find the optimal modulation depth for a given
absorption lineshape, which adds complexity to a given measurement.

84

Current noise density (pA/Hz-1/2 )

10 5

10 4

10 3

10 2

10 1

10 0
42

42.1

42.2

42.3

42.4

42.5

42.6

42.7

Frequency (kHz)

Figure 3.12: Residual amplitude modulation on the Daylight Solutions QCL for current modulation
with a 0.1 Vpp sinusoid at 42.345 kHz, and a DC output voltage of 5.52 V from the detector.

Chapter 4
Production of carbon clusters with a cryogenic buffer-gas beam source∗

In this work, we investigate the production of carbon clusters with a neon buffer-gas beam source,which is in contrast to previous studies of small carbon clusters [99] that primarily utilized supersonic expansion sources. Using matrix isolation spectroscopy, we observe clusters ranging in size
from C3 to C12 , and estimate the number of each cluster produced. Clusters in this size range havebeen previously produced using a supersonic expansion source with a small growth channel [100–102]. This chapter is organized as follows. Section 4.1 provides an overview of the apparatus andthe experimental procedure used to characterize the buffer-gas beam source. Section 4.2 investigates the production of carbon clusters as a function of various system parameters, and describes
a method for quantifying the number of each cluster produced. Section 4.3 concludes and provides
a future outlook.

4.1

Experimental Methods

Carbon clusters are produced inside the buffer-gas cell (see Fig. 2.1) by laser ablation of a graphitetarget located at the edge of the cylindrical bore halfway along the length of the cell. To ablate
graphite, the output of a Nd:YAG laser (Spectra Physics, Quanta-Ray Lab-150, 532 nm) is focused
onto the target to a spot roughly 30 µm in diameter using a biconvex lens with a 400 mm focal
length. To prevent the ablation laser from drilling a hole into the target [28], the beam is constantly
scanned in two dimensions using a motorized mirror mount. Without scanning, we observe rapid∗

A version of this chapter has been published in Ref. [43].

86
degradation in the number of molecules produced per laser pulse.
Following ablation, the carbon clusters undergo collisions with the cold neon buffer gas fora few milliseconds before becoming entrained in the molecular beam extracted from the cell. The
properties of this buffer-gas beam are determined by the cell geometry, the aperture diameter, andthe flow rate of neon gas into the cell [29]. The first two variables are fixed in this experiment,
whereas the neon flow rate can be varied in the range 0 − 100 sccm using the mass flow controller.
Notably, both cluster formation and collisional cooling of the ablation products occur during theinteraction time inside the cell.
To measure the number and composition of carbon clusters produced inside the neon buffergas cell, we use a matrix isolation spectroscopy setup [82]. The buffer-gas beam exiting the cell isfrozen onto a CaF2 window (T ∼ 5 K) located approximately 2 cm from the cell aperture. Infraredabsorption spectra of matrix isolated carbon clusters are acquired in the range 4000 − 1000 cm−1
at 1 cm−1 resolution using a Fourier-transform infrared spectrometer (Thermo Fisher Scientific,
Nicolet iS50). We average 36 scans for each acquired spectrum resulting in a dynamic range ofapproximately 70 dB. This leads to a signal-to-noise ratio of about 925 : 1 for the ν6 mode of C9
as depicted in the spectrum of Fig. 4.1(a), which is acquired after 30 minutes of deposition.
For a given set of system parameters, which include the neon flow rate QNe , laser pulse
energy Ep , and laser repetition rate fr , we ablate continuously for two minutes then acquire aspectrum. During the acquisition of each spectrum, the neon flow is turned off and the ablation
laser is physically blocked by a mechanical shutter. This process is repeated 15 times for a total
deposition time of 30 minutes to determine the rate of increase in optical depth (i.e., absorbance)
for each observed peak. Given this change in optical depth, we quantify the performance of ourbuffer-gas beam source by estimating the number of each carbon cluster produced per pulse of theablation laser. The following section discusses the results of this experiment as a function of QNe ,Ep , and fr , and describes our method for estimating molecule production per pulse from the rateof increase in optical depth.

87

0.3

Optical depth

0.25

C9

(a)

C3

C5
C7

0.2
0.15
0.1

C8

0.05

C 12

C 11

C9

0
2200

2150

2100

C7

C6

C 10

C 10
2050

2000

1950

1900

C 11
1850

1800

Wavenumber (cm-1)
0.1

(b)
C4

Optical depth

0.08

0.06

0.04

0.02

0
1750

C8

1700

C9

1650

1600

C5

1550

1500

1450

C6

1400

1350

1300

1250

1200

1150

Wavenumber (cm-1)

Figure 4.1: Infrared absorption spectrum of carbon clusters isolated in a neon matrix at 5 K. The
system parameters were QNe = 40 sccm, Ep = 2 mJ, and fr = 7.5 Hz. Note the difference in thevertical scale for the spectral range (a) 2200 − 1800 cm−1 and (b) 1750 − 1150 cm−1 . Reproduced
from Ref. [43].

88

4.2

Results and Discussion

A typical infrared spectrum of carbon clusters isolated in a neon matrix, acquired using the apparatus described above, is shown in Fig. 4.1. We observe vibrational modes of linear carbon chains
ranging in size from C3 up to C12 , where peak assignments are based on previous matrix isolation spectroscopy results [103–114]. Notably, we have not observed signatures of C60 or carbon
cluster ions isolated in the matrix based on the absence of known vibrational transitions in thesemolecules [115, 116]. As a consistency check, we have calculated correlation coefficients for all
clusters with multiple absorption lines (C5 to C11 ) using data over a wide range of system parameters, and find r > 0.995 in all cases. This indicates that our peak assignment based on previous
work is valid. A summary of the data depicted in Fig. 4.1 is given in the first three columns ofTable 4.1. Note that the spectrum in Fig. 4.1 is acquired after 30 minutes of deposition onto thematrix window held at a temperature of about 5 K. Because 5 K is below the neon matrix annealing temperature of about 10 K, we conclude that the larger clusters observed spectroscopically areformed within the buffer-gas cell. This is in contrast to other matrix isolation studies of carbon
clusters [103–105, 107, 110], where an additional annealing step was necessary to promote diffusive
aggregation of smaller clusters into larger clusters [118, 119].
To quantify the number of each cluster present in a given spectrum, we start from Beer’s law:

I0 (ν)
τ (ν) = ln
,
I(ν)

(4.1)

where I0 (ν) and I(ν) are the incident and transmitted intensity, respectively, and τ (ν) is thefrequency-dependent optical depth of the sample (absorbance is related to optical depth by A ≈0.434τ ). From Beer’s law, we define the integrated molar absorption coefficient, or infrared intensity
S as [67]
1
S=
nl

Z
dν τ (ν),(4.2)

where n is the molar concentration of the absorbing species, l is the absorption length, and theintegration is over the entire absorption line. Assuming that we probe an area of uniform density,

89

Table 4.1: Representative data for QNe = 40 sccm, Ep = 2 mJ, and fr = 7.5 Hz. The center
frequency of each absorption line, ν0 , is given along with theoretical infrared intensities, S, peak
optical depth per ablation pulse, ∆τpp , and the number of each carbon cluster produced per ablation
pulse, Npp . Quoted uncertainties are statistical.
Molecule Mode ν0 cm−1 ∗ S (km/mol)† ∆τpp 10−6
Npp 1011
C3

ν32036.59(1)612.0

21.6(4)10.8(2)C4

ν31547.07(1)321.0

6.44(8)6.12(8)C5

ν3
ν4

2166.36(3)1444.24(3)1648.897.3

20.4(3)1.38(6)

3.78(6)4.3(2)

C6

ν4
ν5

1958.65(1)1199.29(2)

800.8
60.3

8.7(1)
0.91(7)3.33(4)4.6(4)

C7

ν4
ν5

2135.08(2)1897.31(1)

2695
677.4

16.4(2)9.16(6)

1.86(2)4.13(3)C8

ν5
ν6

2067.69(1)1707.47(5)1366.3984.8

4.20(4)1.61(8)0.937(9)0.50(3)

C9

ν5
ν6
ν7

2081.13(3)2010.02(1)1602.74(1)

2324
3030
376.5

1.99(4)22.1(2)
1.68(5)0.261(5)2.22(2)
1.36(4)C10––2075.01(3)1915.85(5)––1.97(4)1.24(9)

–
–

C11

ν7
ν8

1938.72(3)1853.86(5)––2.88(6)1.75(5)

–
–

C12 ‡

ν8

2003.47(8)–4.08(6)–∗ Line centers are determined from a Gaussian fit to each peak, and uncertainties represent 95%

confidence bounds on the fit.
† Infrared intensities from Ref. [117].
‡ Tentative assignment based on Ref. [112].

90
we estimate the number of molecules in the matrix as

N≈

NA Ap
S


∆ν τp ,

(4.3)

where NA is Avogadro’s constant, Ap is the approximate area of the buffer-gas beam at the matrix
window, and the integral has been approximated as the product between the width of the absorptionline, ∆ν, and the peak optical depth, τp . For the calculations here, we use Ap ∼ 5 cm2 , and assume∆ν ∼ 1 cm−1 as a typical value for matrix isolated species [120].
Given Eq. (4.3), we calculate the number of molecules produced per ablation pulse in thefollowing way. We first extract τp for each observed peak in the spectrum after every two-minutedeposition cycle, then use linear regression to calculate the time rate of change of τp for each peak.
Finally, we convert this time rate of change to a change in peak optical depth per ablation pulse,∆τpp , using the repetition rate of the laser. Thus, we arrive at the following expression for thenumber of molecules produced per ablation pulse:
19

Npp ≈ 3 × 10



∆τpp
S


,

(4.4)

where S is in units of km/mol, and we have substituted Ap and ∆ν for the values given above.
Table 4.1 shows the results of this analysis for a representative data set with QNe = 40 sccm,
Ep = 2 mJ, and fr = 7.5 Hz. Note that this calculation depends on knowledge of the infraredintensity for a given molecular transition. As this parameter is difficult to measure directly, we rely
on results from density functional theory calculations for linear C3 to C9 [117]. We neglect C10 toC12 in our analysis as we have been unable to find consistent theoretical values of S for the observed
absorption lines. For C5 to C8 , we find reasonable agreement (i.e., same order of magnitude) for
the value of Npp calculated from different absorption lines of the same molecule, which implies good
relative accuracy of the theoretical infrared intensities. However, for C9 , the value of Npp calculated
for the ν5 mode is off by an order of magnitude from the corresponding values for the ν6 and ν7
modes. This inconsistency likely stems from a neglect of electrical anharmonicity in the theoreticalmethod used to determine S.

91
We now apply the analysis described above to investigate carbon cluster production in thebuffer-gas cell as a function of the system parameters. For clusters that have multiple absorption
lines, we use the line with the largest infrared intensity (corresponding to largest peak opticaldepth) to calculate Npp . In Fig. 4.2, the number of each carbon cluster produced per pulse isplotted versus the flow rate of neon buffer gas. For all detected clusters, we observe a linear
increase in production with flow rate up to about QNe = 40 sccm. This increase then saturates
and begins to decrease for higher flow rates, an effect that has been observed in a number of otherbuffer gas experiments [31, 33, 44, 57, 58].
The effect of neon flow rate on the number of each cluster detected in the matrix can beunderstood by considering the two relevant timescales that characterize the buffer-gas cell dynamics:
the molecule diffusion time to the cell walls, td , and the cell pumpout time, tp . If td  tp thenmolecules produced by ablation diffuse to the cell walls and are lost before they are extracted fromthe cell. On the other hand, if td  tp , we expect that the molecules are extracted from thecell before being lost to the cell walls resulting in a larger number of molecules detected inside thematrix. The cell pumpout time is governed by the conductance of the cell aperture, and is therefore
purely geometrical. The diffusion time, however, depends on the density of neon buffer gas inside thecell, which is linearly proportional to the flow rate. Thus, we characterize the extraction behavior
using the dimensionless quantity [29] γcell ≡ td /tp . It is typically assumed that td is dominated by
the lowest order diffusion mode in the cell (a good approximation at low buffer-gas density [33]), andincreases linearly with the neon density in the cell [56]. For our system parameters, we find γcell ≈ 1
for QNe ≈ 25 sccm, which corresponds to td = tp ≈ 8 ms. This correlates well with the observed
linear increase in molecules per pulse up to about QNe = 30 − 40 sccm shown in Fig. 4.2. As γcell
becomes much larger than unity, we would expect the number of detected clusters to saturate as
all molecules should be extracted from the cell before diffusing to the cell walls. However, at highbuffer-gas densities (i.e., high flow rate) the diffusion process is no longer dominated by the lowest
order mode [33]. Since higher-order diffusion modes have smaller time constants [56], the result isan effective decrease in γcell , and therefore in the extraction efficiency. Additionally, it is possible

92

12

(a)

C3
C4

Molecules per pulse (10 11 )

10

C5
C6

8

6

4

2

0
0

10

20

30

40

50

60

70

80

90

Neon flow rate (sccm)
2.5

(b)

C7

Molecules per pulse (10 11 )

C8

2

C9

1.5

1

0.5

0
0

10

20

30

40

50

60

70

80

90

Neon flow rate (sccm)

Figure 4.2: Number of each carbon cluster produced per pulse for (a) C3 to C6 and (b) C7 to C9
as a function of neon flow rate for Ep = 2 mJ and fr = 7.5 Hz. Error bars represent statisticaluncertainty propagated from the determination of ∆τpp . Lines are guides for the eye. Reproduced
from Ref. [43].

93that the neon flow becomes turbulent [44], and negatively impacts the extraction efficiency in asimilar way. We observe these effects as a reduction in the number of each carbon cluster detectedin the matrix at neon flow rates in excess of about 40 − 50 sccm.
In addition to flow rate, carbon cluster production as a function of the ablation laser pulse
energy and repetition rate is investigated, the former of which is plotted in Fig. 4.3. We observea linear increase in the production of all carbon clusters with increasing ablation pulse energy up
to Ep = 4 mJ, at which point a substantial amount of carbon dust was present in the vacuum
chamber. Therefore, we did not explore higher pulse energies. We observed no variation in cluster
production rate for laser repetition rates of fr = 3, 5, and 7.5 Hz, implying local heating of thesubstrate does not effect cluster production at these low repetition rates.

4.3

Summary

We have produced carbon clusters C3 to C12 using a cryogenic buffer-gas beam source. For eachof the C3 to C9 molecules, we estimate a production rate on the order of 1011 − 1012 molecules per
pulse of the ablation laser based on infrared intensities determined from density functional theory
calculations. Similar to other buffer-gas beam experiments, we observe an initial linear increase inmolecule production with neon flow rate followed by a decrease at high flow rates (i.e., high neon
density in the cell). We attribute this decrease at high flow rates to a reduction in the diffusion
time for molecules to be lost to the cell walls.
Based on the direct observation of large numbers of carbon clusters containing more than a
few atoms, the buffer-gas beam source may be a viable method for the production of more exotic
cluster systems such as metal oxides and carbides. Furthermore, given the increased number ofcollisions in comparison to a supersonic expansion source, we anticipate the buffer-gas beam to be auseful source for the production of vibrationally cold cluster systems and other complex polyatomic
molecules. Used as the source in a high-resolution infrared spectroscopy system, it could then be
possible to determine molecular structures to a high degree of precision.

94

15

(a)

C3

Molecules per pulse (10 11 )

C4

12

C5
C6

9

6

3

0
1.8

2.2

2.6

3

3.4

3.8

4.2

Pulse energy (mJ)
3

(b)

C7
C8

Molecules per pulse (10 11 )

2.5

C9

2

1.5

1

0.5

0
1.8

2.2

2.6

3

3.4

3.8

4.2

Pulse energy (mJ)

Figure 4.3: Number of each carbon cluster produced per pulse for (a) C3 to C6 and (b) C7 to C9
as a function of ablation pulse energy for QNe = 40 sccm and fr = 7.5 Hz. Error bars represent
statistical uncertainty propagated from the determination of ∆τpp . Lines are guides for the eye.
Reproduced from Ref. [43].

Chapter 5

Infrared Spectroscopy of the WO Molecule

A detailed understanding of the bonding and electronic structure in transition metal containing
molecules is of interest to a variety of fields including catalysis [121], semiconductor science [122],
and astrophysics [123, 124]. Typically, such an understanding comes from laboratory-based spectroscopic studies, where analysis of high-resolution spectra is used to deduce the geometrical arrangement of the individual atoms in a molecule, as well as the strength of bonds between thoseatoms. Notably, transition metal containing molecules typically possess rich spectra due to theircomplicated electronic structure and multitude of low-lying, interacting states, which serve to perturb and further complicate the interpretation of spectra. In fact, WO is an excellent example ofthis complexity. Until the work of Ram et al. [125] in 2001, the electronic structure of WO was
not well characterized and even its ground state was not known with certainty. Starting from theinitial observation of WO bands by Gatterer and Krishnamurty [126], through the experimental
work of Weltner and McLeod [127], Green and Ervin [128], Samoilova et al. [129], and Lorenz andBondybey [130], as well as the theoretical work by Nelin and Bauschlicher [131], there was muchdebate about whether WO possessed a 3 Σ− or 5 Π ground state. Through extensive electronic
spectroscopy and complex ab initio calculations, Ram et al. provided a detailed analysis of the
184 WO molecule [125, 132].

We make use of the results of Ram et al. in this chapter to predict

the infrared rotational-vibrational spectrum of the WO molecule for all four naturally occurringisotopes of tungsten. Following this, we develop a method for reliably producing WO molecules in
the gas phase and optimize WO production using FTIR matrix isolation spectroscopy.

96

5.1

Expected WO Rotational-Vibrational Spectrum

5.1.1

Energy levels

Ram et al. [125, 132] carried out extensive characterization of the electronic structure of 184 WO,
and computed rotational and vibrational constants with anharmonic and centrifugal distortion
corrections for a variety of electronic states. We are interested in the ground electronic state of
WO, which is a 3 Σ− state with the lowest energy belonging to the Ω = 0 spin component. Here,Ω = Λ+Σ and represents the projection of the electronic orbital and spin angular momentum along
the internuclear axis [67]. Thus, we have a minimum total angular momentum of J = 0, and therotational energy levels follow the standard expression for a diatomic rigid rotor with centrifugal
distortion [67, 133]:
Fv (J) = Bv J (J + 1) − Dv J 2 (J + 1)2 ,(5.1)

where the v subscript indicates the vibrational level, Bv is the rotational constant, and Dv isthe centrifugal distortion constant. In general, the above polynomial expansion can be continued
to higher-order corrections; however, these are not necessary to describe rotational-vibrationaltransitions in the electronic ground state of WO. Note that the rotational and centrifugal distortion
constants are dependent on the vibrational energy level because as v increases so does the averageinternuclear separation of the atoms in the molecule (i.e., the bond length increases with v). The
lowest vibrational level corresponds to v = 0, and the vibrational energy levels follow the standard
expression for an anharmonic oscillator [67, 133]:






1
1 2
1 3
G (v) = ωe v +− ωe xe v ++ ωe ye v +
,
2
2
2

(5.2)

where ωe is harmonic oscillation frequency, and ωe xe and ωe ye are constants that represent quadratic
and cubic corrections to the harmonic oscillator result. In general, and similar to the rotationalenergy levels, the above polynomial expansion for vibrational energy levels can be continued tohigher-order corrections, but it is not necessary here. We find the total energy of a given molecular

97
state to be
T (v, J) = Fv (J) + G (v) .
Note that ωe is proportional to the force constant kv through the following relation:
s
1
kv
ωe =
,
2πc µ

(5.3)

(5.4)

where µ is the reduced mass of the WO molecule; therefore, measurement of vibrational spectra
provides information on the strength of the molecular bond. Now, given an expression for thevibrational energy levels, we write the rotational and centrifugal distortion constants as [67, 133]
1,
Bv = Be − αe v +21
Dv = De + βe v +
,
2

(5.5)
(5.6)

where Be is the equilibrium rotational constant, and αe represents a correction due to vibrational
excitation. Here, Be is proportional to the moment of inertia Ir through the relation
Be =

~
,
4πcIr

(5.7)

where Ir = µre2 for a diatomic molecule, and re is the equilibrium separation between the atoms.
Therefore, measurement of rotational spectra provides information on the length of the molecularbond. In addition to ωe and its corrections, the equilibrium rotational constant and its corrections
are quantities found through measurement of infrared rotational-vibrational spectra. On the other
hand, the equilibrium centrifugal distortion constant De and its correction βe are derived quantities
based on the following expressions [133]
4Be3
,
ωe2


8ωe xe 5αeαe2 ωeβe = De
−
−
.
ωe
Be
24Be3
De =

(5.8)
(5.9)

Table 5.1 provides equilibrium constants for the Ω = 0 component of the 3 Σ− ground state of 184 WO
as reported by Ram et al. [132]. From these values, we expect the fundamental vibrational transition
from v = 0 to v = 1 for the Ω = 0 component to occur at approximately ν̃0 = 1057.57 cm−1 .

98Table 5.1: Equilibrium constants (in units of cm−1 ) for the Ω = 0 component of the 3 Σ− groundstate of 184 WO. Data reproduced from Ref. [132].
ωe1065.5951(71)ωe xe
4.0103(40)ωe y e-0.00268(60)Be
0.4165548(60)αe0.0020237(23)An interesting feature of the 3 Σ− ground state is the additional spin component Ω = 1,
which represents a doublet separated from the Ω = 0 spin component by an amount of 2λ ≈414 − 1898 cm−1 [132], where λ is the spin-spin coupling constant. The lower bound on λ is fromexperimental observations, while the upper bound is from ab initio calculations; however, Ram etal. note that their estimate from experimental observations makes the assumption of an isolated
3 Σ− ground state, which is likely incorrect.

Furthermore, ab initio calculations on a system suchas WO are challenging, so the precise value of λ is practically unknown. Thus, it is of interest
to investigate the expected rotational-vibrational spectrum of the Ω = 1 spin component of the
3 Σ− ground state, as it is possible that we will have some population of WO molecules in thisstate. Since Ω = 1 represents a doublet, excitation of rotational transitions in the molecule will lift
the degeneracy through spin-orbit interactions. Therefore, we modify the rotational energy level
expression to include a J-dependent splitting [132]:Fv (J) = Bv J (J + 1) − Dv J 2 (J + 1)2 ± qv J (J + 1) ,(5.10)

where qv is a lambda-doubling constant that in general depends on the vibrational energy level.
Note that this expression is valid only for Ω = 1 component, and the vibrational energy levels follow
Eq. (5.2). Table 5.2 provides equilibrium constants for the Ω = 1 component of the 3 Σ− groundstate of 184 WO as reported by Ram et al. [132]. From these values, we expect the fundamental
vibrational transition from v = 0 to v = 1 for the Ω = 1 component to occur at approximatelyν̃0 = 1059.93 cm−1 , which is about 2 cm−1 higher than that of the Ω = 0 component.

99
Table 5.2: Equilibrium constants (in units of cm−1 ) for the Ω = 1 component of the 3 Σ− groundstate of 184 WO. Values of qv (in units of cm−1 ) for the v = 0 and v = 1 vibrational states are also
included. Data reproduced from Ref. [132].


ωe
ωe xe
ωe y eBeαeq0 ×104
q1 ×104
1067.9939(34) 4.0314(6)0.4172526(31) 0.002022(15) 1.6254(56) 1.6228(36)5.1.2Transition wavenumbers

With expressions for the rotational and vibrational energy levels in hand, we now investigate theobservable transitions between these energy levels. Selection rules for diatomic molecules dictatethat allowed transitions have ∆v = ±1 and ∆J = ±1 [67]; therefore, we are primarily interested intransitions from v = 0 to v = 1 with ∆J = ±1. Since ∆J = 0 is forbidden, we do not expect tosee the Q-branch in the spectrum. However, both the R- and P -branches will be observable, whichcorrespond to ∆J = +1 and ∆J = −1, respectively. To find the wavenumber corresponding to atransition between J in the v = 0 state and J 0 in the v = 1 state, we use Eq. (5.3):ν̃J−J 0 = T 1, J 0 − T (0, J) ,





2

3αe
3βe
0
0
= Be −J J + 1 − De +J 02 J 0 + 1 ± q1 J 0 J 0 + 1
2
2




αe
βe
− Be −J (J + 1) − De +J 2 (J + 1)2 ∓ q0 J (J + 1)22
13
+ ωe − 2ωe xe + ωe ye .
4

(5.11)

100
Now, for the R-branch we have J 0 = J + 1 and for the P -branch we have J 0 = J − 1, which leads
to the following expressions for the transition wavenumbers:ν̃R (J) = 2Be (J + 1) − αe (J + 1) (J + 3)
− 4De (J + 1)3 + βe J 4 + 8J 3 + 19J 2 + 18J + 6

± q1 (J + 1) (J + 2) ∓ q0 J (J + 1)
+ ωe − 2ωe xe +13
ωe y e ,4(5.12)

ν̃P (J) = −2Be J − αe J (J − 2)
+ 4De J 3 + βe J 2 J 2 − 4J + 1

± q1 J (J − 1) ∓ q0 J (J + 1)
+ ωe − 2ωe xe +13
ωe y e ,4(5.13)

where the J value corresponds to the v = 0 state, and the minimum value is J = 0 and J = 1
for the R- and P -branch, respectively. For the Ω = 0 component, we set q0 = q1 = 0 in theabove expression. As previously mentioned, the W atom has four naturally occurring isotopes —182 W (26.50%), 183 W (14.31%), 184 W (30.64%), and 186 W (28.43%), where the values in parentheses

indicate the natural abundance. Therefore, each rotational-vibrational transition observed willconsist of a quartet for the Ω = 0 component and an octet for the Ω = 1 component. Using ωeand Be for the 184 WO molecule from Tables 5.1 and 5.2, and Eqs. (5.4) and (5.7), we estimate
isotopic splittings on the order of roughly 0.2 − 0.5 cm−1 or 6 − 15 GHz, which is resolvable with
our apparatus.

5.1.3

Population of rotational states

Assuming all of the molecules are initially in the ground electronic and vibrational states, then therelative intensity of the transitions in the R- and P -branches depends on the ground state populationof the rotational states. We calculate the fractional population of WO molecules in each J state

101
using classical Boltzmann statistics and the fact that each state has a 2J + 1 degeneracy:
n 
h


io

βe
αe
hc
2 (J + 1)2
(2J
+
1)
exp
−
B
−
J
(J
+
1)
−
D
+
J
e
e
k
T
2
2
NJ
B r
= ∞


io ,
n 
h

P
N
2
βe
αe
hc
2
(2J + 1) exp − kB Tr
Be − 2 J (J + 1) − De + 2 J (J + 1)

(5.14)

J=0

where the denominator is the rotational partition function. We now have all of the necessary
expressions to calculate the expected infrared rotational-vibrational spectrum of WO. Figure 5.1
shows a representative spectrum for the Ω = 0 component of 184 WO molecules with a rotationaltemperature of 10 K. From the figure, we see that the strongest transitions (i.e., highest fractional population) occur for J = 2 − 3 in the R- and P -branches at this temperature. At higher
temperatures, the peak will shift to larger values of J and the relative intensity of each transition
will decrease as the molecules are spread over more states. The opposite will occur at lower temperatures. Finally, Fig. 5.2 shows a representative spectrum for the Ω = 1 component of 184 WO
molecules with a rotational temperature of 10 K. Other than the slight shift to higher wavenumbers, the spectrum appears identical to that of the Ω = 0 component. However, for Ω = 1 eachline is in fact a doublet (see inset of Fig. 5.2), hence the reduced intensity of each transition by afactor of two. The splitting of the doublet depends on the value of the doubling constant qv , andis approximately 0.002 cm−1 or 60 MHz for 184 WO, which is resolvable with our apparatus.

5.2

Survey of WO Production with FTIR Spectroscopy

In this section, we outline two different techniques that were investigated to produce WO in thegas phase. The first method involves pressing WO3 powder∗ into a small pellet, and ablating ata few mJ of pulse energy. The second technique involves ablating pure W metal† in the presenceof O2 gas. In each case, characteristic matrix isolation spectra are presented to highlight the key
differences between the two methods. Note that for this work, we replaced the CaF2 matrix isolation
window and vacuum chamber windows with KBr to increase the spectral range from approximately
1000 cm−1 to 650 cm−1 , which is limited by the Nicolet iS50 FTIR spectrometer. This is necessary∗†

Sigma-Aldrich, part number 95410-100G, 99.9% purity
A piece of a thermal evaporation boat from R. D. Mathis, part number S1-.010W, 99.95% purity

102

Relative intensity

0.24

0.18
R-branchP-branch

0.12

0.06

0
1068

1066

1064

1062

1060

1058

1056

1054

1052

1050

1048

Wavenumber (cm-1)

Relative intensity

0.12

0.09
R-branchP-branch

Relative intensity

Figure 5.1: Calculated infrared rotational-vibrational spectrum of the Ω = 0 spin component of
the ground state of 184 WO assuming a rotational temperature of Tr = 10 K.

0.12

01058.263
1058.261
Wavenumber (cm -1 )

0.06

0.03

0
1068

1066

1064

1062

1060

1058

1056
-1

1054

1052

1050

1048

Wavenumber (cm )Figure 5.2: Calculated infrared rotational-vibrational spectrum of the Ω = 1 spin component of
the ground state of 184 WO assuming a rotational temperature of Tr = 10 K. The inset shows the
doublet corresponding to the J = 2 transition of the P -branch.

103
for investigating metal oxide compounds because many of the characteristic vibrational modes lie
below 1000 cm−1 . In addition, we reduced the temperature of the buffer-gas cooling cell to Tcell =16 K, which is approximately the lowest temperature that neon gas maintains an appreciable vapor
pressure. This was motivated by our desire to achieve the lowest possible molecule temperature forrotational-vibrational spectroscopy, and did not affect the absolute production of WO as determinedby matrix isolation spectroscopy.

5.2.1

Laser ablation of WO3

One approach to producing WO molecules in the gas phase involves laser ablation of hydrostaticallypressed WO3 powder. Due to the bright yellow color of WO3 , we mix approximately one part ofpure W metal shavings with three parts of the WO3 powder (i.e., 1:3 mixing ratio) prior to pressing.
This leads to slightly more consistent results, and higher yield of WO, which we believe is a resultof the increased absorptivity of the material (the W metal shavings darken the appearance of thetarget). Once the two components are well mixed, we place a small amount of the powder mixture
in a homemade die setup and apply a force of approximately 10, 000 psi for about 5 minutes. The
die consists of a stainless steel cylinder with a central bore 0.375 inches in diameter, which wasmade to be a snug fit for two hardened steel dowel pins. The powder to be pressed is then placed
between the dowel pins inside the stainless steel cylinder, and the whole assembly is placed in a
hydraulic press. The resulting targets are typically quite brittle, but hold together well enough tobe epoxied to the target holder‡ and installed inside the buffer-gas cooling cell.
To study the production of WO from a pressed target, we ablate the target at 532 nm withpulse energies between Ep = 2 − 8 mJ and a repetition rate of fr = 10 Hz. As in the case ofgraphite ablation, we focus the beam to a spot size of roughly 30 µm using a biconvex lens witha focal length of 400 mm. Furthermore, we constantly scan the ablation beam in two dimensions
across the target to prevent localized target degradation. Figure 5.3 shows a typical FTIR spectrum
acquired as described in Section 4.1 with 5 mJ ablation pulse energy. In addition to unidentified‡

We use Armstrong A-12 epoxy, which works well under vacuum and at cryogenic temperatures.

104

0.06
WO 2

W 3O 9Optical depth

0.05
0.04
WO 3
0.03
0.02

WO

0.01
0
1080

1050

1020

990

960

930

900

870
-1

840

810

780

750

720

Wavenumber (cm )Figure 5.3: Infrared absorption spectrum of WO3 ablation products isolated in a neon matrix at
5 K. The system parameters are QNe = 40 sccm, Ep = 5 mJ, and fr = 10 Hz.

105
peaks, we observe vibrational modes of WO, WO2 , WO3 , and W3 O9 , where peak assignments are
based on previous matrix isolation spectroscopy results [127, 128, 130, 134, 135]. Note that the
spectrum in Fig. 5.3 is acquired after 7.5 minutes of deposition onto the matrix window held at atemperature of about 5 K. An interesting feature of the spectrum of WO3 ablation products are thetwo broad transitions centered at about 850 cm−1 and 1020 cm−1 . It is thought that these spectralfeatures represent a practically solid film of a tungsten oxide compound that results from the dusty
nature of the ablation target. Indeed, upon visual inspection of the neon matrix after deposition,
there is a clear signature of a localized dark spot corresponding to the expected location that thebuffer-gas beam hits the matrix window. Additionally, this dark spot sometimes sticks to the matrix
window as a deposited film that can only be removed by gently brushing it off. Furthermore, large
amounts of dust are found inside the buffer-gas cooling cell after a few runs of ablating a pressedtarget. Therefore, we believe these broad spectral features are a result of the dust produced during
the ablation process. It would be interesting to investigate the ablation products entrained in thebuffer-gas beam using a mass spectrometer as it is likely that the dust being produced is made up
of metal oxide cluster molecules of various sizes. Notably, we observe very similar characteristics
and spectral features from laser ablation of a V2 O5 sputtering target; thus, these seemingly messy
targets could be an excellent source of more complex metal oxide cluster molecules.
Notably, our goal with the current study is not to identify signatures of new molecules, but
to find system parameters that maximize the production of the WO molecule. To this end, we
repeated the above study for a range of flow rates and pulse energies, and found that a neon flowrate of about 40 − 60 sccm and a pulse energy of Ep = 8 mJ gave the best results. Increasing thepulse energy further leads to rapid target degradation, which is thought to be due to the brittle
nature of the pressed targets. With optimum parameters, we found that we could produce a peakchange in optical depth per ablation pulse of approximately ∆τpp ∼ 3 × 10−6 for WO. However, wefound the WO production from ablation of pressed powder targets to be very inconsistent between
experimental runs, and we found it difficult to produce “good” targets in a consistent fashion — infact, some targets resulted in no WO being observed in the matrix. This motivated us to develop

106
a more consistent method for producing WO.

5.2.2

Laser ablation of W with O2 gas

In order to improve the consistency of WO production, we switched from ablating pressed powder
targets to ablating pure W metal in the presence of O2 gas. This is not trivial at cryogenic
temperatures because O2 freezes at a temperature of about 54 K, which is well above the typicaltemperature of the buffer-gas cooling cell. Therefore, we cannot simply mix O2 gas with the neon
buffer gas. Instead, we installed a secondary gas line that injects O2 into the vicinity of the ablation
target, and thermally insulated the gas line from the buffer-gas cell using a piece of Vespel.§ With
such a spacer material, the additional gas line is held at a temperature of approximately 100 K.
Notably, this configuration is not restricted to O2 , and can be used for a wide variety of reactive
gases. The addition of nichrome wire wrapped around the gas line allows heating as necessary.
Figure 5.4 depicts the upgraded buffer-gas cooling cell. The cell is identical to that depicted inFig. 2.1 except for the addition of the Vespel plug and stainless steel gas line with a 0.125 inch outer
diameter. The plug is machined to sit nearly flush with the inner wall of the buffer-gas cooling cell,and the gas line is fixed with a teflon set screw such that it also sits nearly flush with inner wall of
the cell. The flow of O2 gas into the cell is controlled by a mass flow controller (Alicat Scientific,
MC-0.5SCCM-D), and is arranged such that it intersects the ablation plume.
To study the production of WO in this configuration, we ablate W metal at 532 nm with pulse
energies up to Ep = 20 mJ and a repetition rate of fr = 10 Hz while flowing QO2 = 0.1 − 0.2 sccm
of O2 gas. As in other experiments, we focus the beam to a spot size of roughly 30 µm using abiconvex lens with a focal length of 400 mm. Furthermore, we constantly scan the ablation beamin two dimensions across the target to prevent localized target degradation. Figure 5.5 shows atypical FTIR spectrum acquired as described in Section 4.1 with 20 mJ ablation pulse energy. Here,
we observe vibrational modes of WO, WO2 , and WO3 in addition to O3 , where peak assignments§
Vespel is a polyimide material that has one of the lowest thermal conductivities at cryogenic temperatures among
readily available materials.

107

Oxygenline

Vespelinsulator

Figure 5.4: A schematic of the buffer-gas cooling cell with a secondary gas line for in-cell reactions.
The cell is identical to that depicted in Fig. 2.1 except for the addition of a thermally insulated gasline for flowing oxygen gas into the vicinity of the ablation plume. A piece of Vespel acts to isolate
the gas line from the cold cell.

18#10 -3O3

16

Optical depth

14
12

WO 3

WO
10
8
6
WO 2

4
2
0
1080

1060

1040

1020

1000

980

960
-1

940

920

900

880

Wavenumber (cm )Figure 5.5: Infrared absorption spectrum of W plus O2 gas ablation products isolated in a neon
matrix at 5 K. The system parameters are QNe = 40 sccm, QO2 = 0.15 sccm, Ep = 20 mJ, andfr = 10 Hz.

108
are based on previous matrix isolation spectroscopy results [127, 128, 130, 134–136]. Note that the
spectrum in Fig. 5.5 is acquired after 5 minutes of deposition onto the matrix window held at atemperature of about 5 K. Clearly, the broad spectral features present in the ablation of pressed
WO3 powder targets are completely absent. In addition, upon visual inspection of the neon matrix
after deposition, there is no signature of a dark spot, and no dust is found inside the buffer-gas
cooling cell after a few runs of ablating pure W metal. Thus, we find that ablating W metal in thepresence of O2 gas is a much cleaner method of producing WO.
We repeated the above study for a range of flow rates and pulse energies, and found thata neon flow rate of about 40 − 60 sccm and a pulse energy of Ep = 20 mJ gave the best results.
Increasing the pulse energy further leads to burning of either the vacuum chamber window or the
buffer-gas cooling cell window, or both. To circumvent this issue, a substantially longer tube isrequired to move the buffer-gas cooling cell window far from the cell and very close to the vacuum
chamber window. This modification would likely allow pulse energies up to 50 mJ or more to
be used, which could be very advantageous for ablating a high melting point material like W.
Regardless, with optimum parameters, we found that we could produce a peak change in opticaldepth per ablation pulse of approximately ∆τpp ∼ 3 − 4 × 10−6 for WO, which is similar yield to thepressed WO3 target. However, we found the WO production to be substantially more consistent
between experimental runs, and that a piece of W metal about 1 − 2 mm thick lasts many months.
Before proceeding to a discussion of the expected gas phase spectrum of WO, it is of interest
to investigate the FTIR matrix isolation spectra of W plus O2 gas ablation products at higher
resolution to see if we can identify the isotopic splitting due to the four naturally ocurring isotopesof W. Here, we increase the FTIR spectrometer resolution to its maximum value of 0.125 cm−1and acquire a single beam spectrum of the matrix isolated ablation products. Figure 5.6 shows ahigh resolution FTIR spectrum in a narrow band around the WO and O3 transitions. We clearlyobserve the isotopic splitting of the WO vibrational transition, as well as the matrix site shift of
the antisymmetric stretch vibrational mode of the O3 molecule [136]. Note that the spectrum in
Fig. 5.6 is acquired after 20 minutes of deposition onto the matrix window held at a temperature of

109

0.07
0.06

Optical depth

0.05
0.04
WO
0.03
O3
0.02
0.01
0
1060

1058

1056

1054

1052

1050

1048

1046
-1

1044

1042

1040

1038

1036

Wavenumber (cm )Figure 5.6: High resolution infrared absorption spectrum of W plus O2 gas ablation products
isolated in a neon matrix at 5 K. The system parameters are QNe = 40 sccm, QO2 = 0.1 sccm,
Ep = 10 mJ, and fr = 10 Hz.

110
about 5 K. Based on the center frequency of the vibrational transition for each WO isotopologue,
which is determined from a Gaussian fit of each peak, we use Eq. (5.4) to obtain kv = 9.6 ± 0.5
mdyne/Å from the slope of a linear fit. This value is in good agreement with previous work by
Lorenz and Bondybey [130]; thus, we conclude that our identification of matrix isolated WO isvalid and consistent with previous work.

5.3

Expected Gas Phase Absorption Signal

Given the matrix isolation spectroscopy results, we now aim to estimate the change in peak opticaldepth per pulse of the ablation laser for WO molecules in the gas phase. The first step is to estimate
the FWHM absorption linewidth of a WO transition by calculating the contributions of the various
broadening mechanisms discussed in Section 3.1.2. Here, we will assume that the molecular beam
is probed very close to the exit aperture of the buffer-gas cooling cell, and that the molecules andbuffer-gas atoms have equal translational temperatures. The following properties will be assumed
to estimate the gas phase absorption linewidth:• Molecule mass of mm = 200 amu and neon buffer gas with mb = 20.18 amu.
• Transition wavenumber of ν̃0 = 1057.57 cm−1 .
• Buffer-gas density of nb = 1016 cm−3 and collision cross section of σm,b = 10−15 cm2 .
• Transverse temperature of T = 25 K and forward velocity of u = 200 m/s for the molecularbeam.
• Probe laser beam waist of w0 = 100 µm.
Using these values, we find that Doppler broadening, collisional broadening, and transit time broadening result in linewidths of ΓD ∼ 8 MHz, Γc ∼ 150 kHz, and Γt ∼ 2 MHz, respectively. Therefore,
we expect that the finite QCL linewidth on the order of 10 − 30 MHz over our estimated integration
time of about 5 ms will be the dominant broadening mechanism for laser absorption spectroscopyof gas phase WO.

111
The second step is to convert the matrix isolation spectroscopy result of ∆τpp ∼ 3 − 4 × 10−6
into an equivalent value for gas phase spectroscopy. We can estimate this value in the gas phase byconsidering Eq. (4.2) for the infrared intensity, which is a conserved quantity. If we approximateRthe integral as dν̃ τ (ν̃) ≈ ∆ν̃ τp , and assume that the number of molecules probed in the matrix
is equivalent to that in the gas phase then we obtain the following expression:τpp,GP =

AMI
AGP



∆ν̃MI
∆ν̃GP


τpp,MI ,(5.15)

where AMI and AGP are the areas probed in matrix isolation and gas phase spectroscopy, respectively, and ∆ν̃MI and ∆ν̃GP are the absorption linewidths in matrix isolation and gas phase spectroscopy, respectively. For matrix isolation, we assume that the molecules are uniformly distributed
over the cold window of area AMI ∼ 5 cm2 and that the absorption linewidth is that of the FTIR
spectrometer ∆ν̃MI = 1 cm−1 or 30 GHz. For gas phase, we assume a beam velocity of 200 m/s,
a beam diameter of 5 mm, and a molecule pulse length of 5 ms, which corresponds to an effective
area of AGP = 50 cm2 . Furthermore, we assume a QCL linewidth of ∆ν̃GP = 30 MHz, which leads
to a peak change in optical depth per pulse of the ablation laser of ∆τpp,GP ∼ 3 − 4 × 10−4 for gas
phase spectroscopy.
Finally, the distribution of WO molecules among rotational and vibrational states must be
considered. This distribution is strongly dependent on the temperature of the molecules entrainedin the buffer-gas beam. Assuming the molecules are in the ground electronic state with Ω = 0, we
write the rotational and vibrational partition functions in a simplified form as [27]

Zr ≈

kB Tr
hcBe


exp




hcBe
,
3kB Tr

(5.16)

1,(5.17)Zv =2 sinhhcωe
2kB Tv

respectively, where the centrifugal distortion and anharmonic correction terms have been neglected
for simplicity. Here, we have defined separate rotational and vibrational temperatures, denotedby Tr and Tv , respectively, because the rotational degree of freedom is typically in equilibriumwith the buffer gas while the vibrational degree of freedom is not (see discussion in Section 2.1.3).

112
Therefore, it is difficult to precisely estimate the number of populated vibrational states. Using
the spectroscopic constants for 184 WO from Ref. [132] we find Zr ∼ 40 for Tr = 25 K, and Zv ∼ 5
for the worst case scenario of Tv = 5000 K (i.e., no vibrational relaxation occurs after ablation).
Furthermore, the molecule population will be spread over the four naturally occurring isotopes ofW; thus, in the worst case situation, we could have molecules spread over approximately 800 states.
However, in a more likely scenario we will have Zv ∼ 1 and Zr ∼ 10 due to additional collisionalcooling in the molecular beam, and the molecule population will be spread over approximately
40 states. Thus, we expect a change in peak optical depth per pulse of the ablation laser in therange of ∆τpp,GP = 10−6 − 10−5 . Notably, this is on the order of 2 − 20 times larger than theminimum detectable optical depth of τmin ≈ 5 × 10−7 for 20 µW incident on the detectors and a
5 ms integration time, as determined by Eq. (3.16).

5.4

Summary

We have investigated the production of WO using a cryogenic buffer-gas beam source, and foundthat ablating pure W metal in the presence of O2 gas is a reliable and consistent source of WO
molecules. Using matrix isolation spectroscopy, we identified a number of different molecules present
in the buffer-gas beam including WO2 , WO3 , and larger polyatomic tungsten oxides, as well as anappreciable amount of O3 . To date, we have been unsuccessful in observing rotational-vibrational
spectra of WO in the gas phase using the tunable diode laser absorption spectroscopy systemdescribed in Section 3.4.5. We attribute this to a lack of sensitivity; thus, it is necessary to bothincrease the production of WO by increasing the ablation laser pulse energy, and decrease thenoise floor of the spectroscopy system through incorporation detectors with lower 1/f noise or a
modulation scheme.

Chapter 6

Conclusion to Part 1

6.1

SummaryThis first part of this dissertation has investigated the application of a buffer-gas beam source tothe production of large polyatomic molecules, including carbon clusters and metal oxide molecules.
The original motivation for this work was based on the longer collisional cooling timescale available
with buffer-gas beam sources compared to traditional free jet sources. Indeed, the possibility ofachieving lower internal state temperatures for large, strongly bound clusters is of general interest,
and would aid in determining the structure of these complex molecules. Used in conjunction witha high resolution infrared spectroscopy system, such a source would help to remove much of theambiguity associated with the spectral assignment process, and provide validation for the large
variety of theoretical techniques currently being used in conjunction with experiments to determine
molecular structures.
In Chapter 2 and 3, we outlined the major experimental techniques used in this dissertation
work. Chapter 2 provided a detailed background on collisional cooling and beam formation incryogenic buffer-gas beam sources, and compared the source used in this dissertation to a traditional free jet source. It was found that the number of collisions between buffer gas atoms and themolecules of interest is likely lower in our buffer-gas beam source compared to a traditional free jet
source; however, the buffer-gas beam source has much more design flexibility such that improvements can be made. Chapter 3 provided a detailed background on infrared absorption spectroscopy,
including the techniques of Fourier transform infrared spectroscopy, matrix isolation spectroscopy,

114
wavelength modulation spectroscopy, and direct absorption spectroscopy with a tunable diode laser
system. The performance of various aspects of the experimental apparatus was also presented in
Chapter 3.
In Chapter 4, we discussed production of carbon clusters C3 to C12 using a cryogenic buffergas beam source. For each of the C3 to C9 molecules, we estimated a production rate on the orderof 1011 −1012 molecules per pulse of the ablation laser based on infrared intensities determined fromdensity functional theory calculations. Similar to other buffer-gas beam experiments, we observed
an initial linear increase in molecule production with neon flow rate followed by a decrease at highflow rates (i.e., high neon density in the cell). We attribute this decrease at high flow rates toa reduction in the diffusion time for molecules to be lost to the cell walls. Notably, the resultspresented in Chapter 4 highlight an interesting aspect of the buffer-gas beam source – it performs
very similarly for a wide range of molecule masses from 36 amu to 144 amu.
In Chapter 5, we investigated the production of WO using a cryogenic buffer-gas beam source,
and found that ablating pure W metal in the presence of O2 gas is a reliable and consistent source of
WO molecules. Using matrix isolation spectroscopy, we identified a number of different molecules
present in the buffer-gas beam, including WO2 , WO3 , and larger polyatomic tungsten oxides, aswell as an appreciable amount of O3 . To date, we have been unsuccessful in observing rotationalvibrational spectra of WO in the gas phase using the tunable diode laser absorption spectroscopy
system described in Section 3.4.5. We attribute this to a lack of sensitivity; thus, it is crucial toboth increase the production of WO by increasing the ablation laser pulse energy, and decrease
the noise floor of the spectroscopy system either through incorporation of detectors with lower 1/f
noise or developing a modulation scheme free from residual amplitude modulation effects.

6.2

Future Outlook

In moving forward on this project, there are a number of apparatus upgrades that could be madeto improve system performance. Firstly, the buffer-gas cooling cell geometry can be optimized
to increase the number of collisions as well as extraction from the cell and beam formation. For

115
example, increasing the inner cell dimensions would allow more collisions to occur inside the cell
before molecules are extracted. Additionally, tapering the cell at the exit aperture would promote
laminar flow and prevent eddies from forming, which can trap molecules and force them towards
the cell walls. Secondly, the signal-to-noise ratio of the tunable diode laser absorption spectroscopy
system could be improved dramatically by locking the laser to a tunable cavity. This could largely
eliminate much of the low frequency noise that plagues the Daylight Solutions QCL, and ultimately
limits the detection sensitivity. Furthermore, this could reduce the laser linewidth to the sub-MHz
regime, which is generally advantageous for spectroscopy. Additionally, MCT detectors with alower transimpedance gain would be beneficial as more laser power could be used to probe moleculartransitions, which effectively reduces the shot noise limited detection sensitivity. The limiting factor
in this regard is the linear operating regime of the MCT detector – balanced subtraction does notwork if the signal and reference beams are not linearly proportional. Finally, incorporation of abetter multipass cell would increase the signal by a factor of 10 − 100.
In regards to science, there is a plethora of interesting molecular systems that can be studied
with the apparatus described in this dissertation. For example, it would be very interesting to
optimize the buffer-gas beam source to produce larger metal oxide molecules such as V4 O10 , which
has a vibrational transition around 1030 cm−1 ; thus, in principle rotational-vibrational transitions
could be probed in the gas phase with the QCL system. Notably, such a study would provide
unprecedented resolution for the infrared spectra of large metal oxide cluster molecules, the investigation of which has been largely confined to action spectroscopy techniques [137]. In addition, othermolecules, like metal carbides and hydrides, for example, could be produced by flowing reactive
gases into the buffer-gas cell via the secondary gas line installed to produce WO.
Another interesting research avenue would be to investigate the production of free radicals
with the buffer-gas beam source. Coupled with the matrix isolation spectroscopy technique, chemical reactions involving free radicals could be investigated, and undetermined reaction intermediates
could be isolated and probed with the Fourier transform spectrometer. For example, CH could be
produced via ablation of graphite in the presence of hydrogen gas, and trapped in a neon matrix

116with excess oxygen to study intermediates of the reaction CH + O2 , which has a number of different
product channels [138]. Such reactions have widespread applicability, particularly to the chemistryof the interstellar medium where many reactions occur at low temperature.

Part 2: Dynamics of Bose Gases in Spherical Symmetry

Chapter 7
Introduction to Part 2∗

In strongly interacting systems, the study of collective modes of a quantum many-body problem
provides revealing information about the nature of the underlying Hamiltonian. In ultracold atomic
gases, due to the fact that binary interactions between particles at low energy are well understood,
the experimental measurement of collective modes provides a means of evaluating many-body theoretical methods used to describe these systems. Experiments probing the collective modes ofultracold gases were carried out shortly after the demonstration of Bose-Einstein condensation ina dilute atomic vapor [140, 141]. In early experiments at JILA [142] and MIT [143], the low-lying
quadrupole modes of a nearly pure Bose-Einstein condensate (BEC) were excited, and the observed
oscillation frequencies showed good agreement with the Bogoliubov spectrum [144, 145]. Experiments were then conducted over a range of temperatures below the critical point, and temperaturedependent shifts in the oscillation frequencies and damping rates were observed [146, 147]. At thetime, existing theoretical models were unable to reproduce the experimental observations.
The observation of these unexplained temperature-dependent shifts motivated further exploration of collective-mode behavior at finite temperature, where experiments probed the interaction
between the condensate and thermal component of the gas (i.e., noncondensate). Experiments on
the scissors modes [148] provided an alternative means of measurement of temperature-dependent
shifts of the mode frequencies and damping rates through observation of the angle oscillations of thecondensate and noncondensate, and reasonable agreement with existing theories was found. Fur∗

A version of this chapter has been published in Ref. [139].

119
thermore, a study of the transverse breathing mode in an elongated harmonic trap [149] found uncharacteristically small damping rates and observed that the mode frequency was quasi-independentof temperature.
In order to address the unexpected behavior of the experiments, models were initially developed to explain the anomalous temperature dependence of the quadrupole mode found in Ref. [146].
Early efforts assuming a static noncondensate were unable to reproduce the experimental results;
however, inclusion of the dynamics of the noncondensate lead to a consistent framework thatmatched the experiment. Using a semi-classical coupled-modes model, Stoof, Bijlsma, and Al
Khawaja [150, 151] described the coupled dynamics of the condensate and noncondensate in termsof in-phase and out-of-phase eigenmodes, which are collisionless analogs of first and second soundhydrodynamic modes [152–154]. They concluded that the anomalous behavior found in Ref. [146]
was the result of simultaneous excitation of both eigenmodes of the system. Numerical simulationof the Zaremba-Nikuni-Griffin (ZNG) equations by Jackson and Zaremba [155–157] confirmed thispicture, and Morgan, Rusch, Hutchinson, and Burnett provided additional analysis in an extensionof their previous work [158–162]. These efforts highlighted the important role of the noncondensate
dynamics in the behavior of collective modes at finite temperature.
Experiments to date have operated with anisotropic trapping geometries, which lead to anincreased degree of complexity in the collective-mode spectrum. An isotropic harmonic trap simplifies the mode spectrum due to its spherical symmetry, and allows for a detailed comparison
between experimental measurements and existing theoretical models. Furthermore, a spherical
trapping geometry eases the computational burden of sophisticated numerical studies such as thesimulation of the ZNG equations. However, experiments in this regime face the technological hurdleof minimizing asphericities in the trapping potential, which, previous to the work described here,
has prevented the study of collective modes in such a simplified geometry. Thus, fundamental
comparisons between theoretical predictions for the collective-mode spectrum of condensates, aswell as their frequency shifts and damping rates at finite temperature, had not been made.
Prior to the work presented here, Lobser et al. [163] investigated the monopole mode of a120classical gas of 87 Rb atoms confined in an isotropic harmonic trap in an effort to test a prediction
from Boltzmann that the monopole mode does not damp in time. Lobser et al. found thatthe monopole mode was practically undamped, with a small residual damping rate due to minor
imperfections in the trap geometry. A detailed theoretical analysis of the experiment found thatthe residual damping rate of the monopole mode was caused primarily by anharmonicity along one
axis of the trap, and identified a range of collision rates that minimized the residual damping [164].
Given the success of the experiment and theoretical analysis for a classical gas, we were motivated
to investigate the behavior of the monopole mode in an isotropic harmonic trap below the BoseEinstein condensation transition temperature. In this regime, it is expected that the interplay
between the condensed and non-condensed portions of the gas will give rise to dynamics not observed
above the transition temperature. Indeed, the results of D. S. Lobser’s experiment below thetransition temperature showed interesting behavior — the oscillation amplitude of the monopolemode exhibited damping and a characteristic modulation in time [165]. Thus, the work presented
here describes our methods, analysis, and interpretation of the experimental results.
The second part of this dissertation begins by outlining the theoretical basis of the ZNG
method. Following this background material, a detailed description of a code developed to numerically solve the ZNG equations in an isotropic harmonic trap is given. Given the spherical symmetryof such a problem, the dimensionality of the ZNG equations can be reduced, resulting in a fast andaccurate code for investigating the unique problem of the monopole mode of a finite temperature
Bose gas confined in an isotropic harmonic trap. Thus, the final chapter in the second part ofthis dissertation provides a detailed analysis of this problem including both experimental data andnumerical results from the developed spherically symmetric ZNG code.††

The final chapter of this dissertation represents a publication that was a collaborative effort between the authorof this dissertation, Victor E. Colussi, Matthew J. Davis, Dan S. Lobser, Murray J. Holland, Dana Z. Anderson,
Heather J. Lewandowski, and Eric A. Cornell [139]. The author of this dissertation was the primary author of
the paper, and performed numerical simulations using the ZNG method and analyzed the experimental data. V.
E. Colussi performed a theoretical analysis of the monopole mode problem in the collisionless regime, M. J. Davis
performed numerical simulations using classical field methods, and D. S. Lobser conducted the experiments. M. J.
Holland, H. J. Lewandowski, and E. A. Cornell provided project definition and guidance throughout. All authors
contributed to interpretation of the results.

Chapter 8

Background for the ZNG Method

This chapter provides an outline of the theoretical basis for the ZNG equations. It begins with abrief discussion of the phenomenon of Bose-Einstein condensation in an isotropic harmonic trap,
given in the context of an ideal gas of non-interacting particles. To move beyond this simple
treatment, the system Hamiltonian for a gas of weakly interacting particles is introduced and an
equation of motion for the condensate at zero temperature is derived. In order to move beyond thezero temperature limit, fluctuations about the condensed state are introduced perturbatively andan equation of motion that governs their evolution is given. Thus, the end result is a closed systemof self-consistent equations that describe the dynamical evolution of a degenerate Bose gas at finite
temperature.

8.1

Bose-Einstein Condensation in an Isotropic Harmonic Trap

At room temperature, a gas of identical particles behave classically, and are well described bya Boltzmann distribution function. In this limit, the thermal de Broglie wavelength Λ is small
compared to the interparticle spacing (∝ n−1/3 , where n is the density) and quantum effects can beignored. As the temperature decreases, however, quantum effects become relevant and the natureof the individual particles becomes important. For bosons, there exists a critical temperature Tc
at which point Λ becomes comparable to the interparticle spacing and a phase transition known as
Bose-Einstein condensation can occur. In contrast to the room temperature case, where individualenergy states of the system have a mean occupation much less than unity, in the presence of Bose-

122
Einstein condensation there exists a macroscopic occupation of the lowest energy state of the system(i.e., the condensate).
The temperature at which Bose-Einstein condensation occurs can be calculated for an isotropic
harmonic trap via the equilibrium Bose-Einstein distribution function
f0 (i ) =1eβ(i −µ) − 1

,

(8.1)

where β = 1/kB T with kB Boltzmann’s constant and T the temperature of the gas, i is the energyof state i in the system, and µ is the equilibrium chemical potential. At room temperature, thechemical potential is small and negative such that µ  0 , where 0 is the lowest energy state of
the system. In this regime, the Bose-Einstein distribution is approximately equal to the classicalBoltzmann distribution. As the temperature is lowered, the chemical potential increases to thepoint where µ = 0 and the mean occupation of the lowest energy state diverges. This is precisely
the point at which Bose-Einstein condensation occurs; thus, we can define Tc as the temperature
at which µ = 0 and all particles in the gas occupy excited states such that
Z ∞
N=
0

dg (),eβ − 1

(8.2)

where N is the number of particles in excited states, and g () = 2 /2 (~ω)3 is the density of states
for an isotropic harmonic trap with frequency ω [166]. Assuming N is large, the zero point energyof the harmonic trap is neglected and µ = 0 ≈ 0 is used in the above equation. Evaluating the
integral and equating Tc ≡ T gives the critical temperature for a Bose gas confined in an isotropic
harmonic trap:kB Tc = ~ω

N
ζ (3)

1/3
,

(8.3)

where ζ (s) is the Riemann zeta function. We can then use this result to define the number ofatoms in the condensate, Nc , below the critical temperature to be" 3 #
T
Nc = N 1 −
.
Tc

(8.4)

Clearly, unless T = 0 some of the atoms remain in higher energy states of the system (i.e., thenoncondensate). Thus, to obtain a full description of the system at finite temperature it is necessary

123
to include the dynamics of both the condensate and noncondensate, as well as the interactions
between them. To accomplish this task, we must go beyond the above analysis, which assumes anideal, non-interacting gas of identical particles. The following sections outline a theory for finite
temperature Bose gases in the weakly interacting limit.

8.2

Many-Body System HamiltonianIn the dilute, weakly interacting limit of Bose gases only binary collisions (i.e., two-body interactions) are relevant, and the Hamiltonian for a system of N interacting particles is written in second
quantization as [167]
Z
Ĥ =
+

d3 r Ψ̂† (r, t) Ĥ0 Ψ̂ (r, t)
Z Z



1
d3 r d3 r0 Ψ̂† (r, t) Ψ̂† r0 , t V r − r0 Ψ̂ r0 , t Ψ̂ (r, t) ,
2

(8.5)

where Ĥ0 = ~2 ∇2 /(2m) + Vext (r, t) is the Hamiltonian for a single particle of mass m confined in
an external potential Vext (r, t) and V (r − r0 ) represents the two-body interaction potential. Here,Ψ̂† (r, t) and Ψ̂ (r, t) are Bose field operators that represent the addition (creation) or removal(annihilation), respectively, of a particle at position r and time t. Such field operators can beconstructed from a linear combination of suitable basis statesΨ̂ (r, t) =Xâi (t) φi (r, t) ,

(8.6)

X †
âi (t) φ∗i (r, t) ,

(8.7)

i

Ψ̂† (r, t) =i

where the summation is carried over a complete set of single-particle basis states φi (r, t), andâ†i (t) and âi (t) are creation and annihilation operators, respectively, for state i. For bosons, these
operators satisfy the usual commutation relations:hiai , a†j = δij ,hiΨ̂ (r, t) , Ψ̂† r0 , t = δ r − r0 ,hia†i , a†j = [ai , aj ] = 0,
h
i h
i
Ψ̂† (r, t) , Ψ̂† r0 , t = Ψ̂ (r, t) , Ψ̂ r0 , t = 0.

(8.8)
(8.9)
(8.10)(8.11)

124
In this dissertation, we are concerned with dilute gases at low temperature such that binary
collisions can be treated approximately as localized, perfectly elastic collisions like those between
billiard balls. In this so-called s-wave scattering regime, the two-body interaction potential isapproximated by [166]V r − r0 ≈ g δ r − r0 ,(8.12)

where g = 4π~2 as /m characterizes the interaction strength, and as is the s-wave scattering length
for a given atomic species. Making this substitution in the many-body Hamiltonian of Eq. (8.5)
results in:
Z
Ĥ =
+

d3 r Ψ̂† (r, t) Ĥ0 Ψ̂ (r, t)Zgd3 r Ψ̂† (r, t) Ψ̂† (r, t) Ψ̂ (r, t) Ψ̂ (r, t) .
2

(8.13)

This Hamiltonian represents the starting point for the subsequent discussion of the dynamical
evolution of the Bose field operator Ψ̂ (r, t).

8.3

Equations of Motion

In the Heisenberg picture, we can derive an equation of motion for Ψ̂ (r, t) as follows:i~∂ Ψ̂ (r, t)
∂t

=

h

Ψ̂ (r, t) , Ĥi

= Ĥ0 Ψ̂ (r, t) + g Ψ̂† (r, t) Ψ̂ (r, t) Ψ̂ (r, t) .

(8.14)

To proceed in the presence of Bose-Einstein condensation, it is convenient to separate the condensate
contribution from the remainder of Ψ̂ (r, t). Assuming a single state of the system is macroscopicallyoccupied with N0  1 atoms (i.e., the condensate) the Bose field operator is split into two parts,Ψ̂ (r, t) = φ̂ (r, t) + δ̂ (r, t) ,(8.15)

where φ̂ (r, t) is a field operator for the atoms in the state occupied by the condensate and δ̂ (r, t)
represents atoms occupying all other states of the system (i.e., the noncondensate). Once again,

125
these new field operators can be represented in a suitable basis asφ̂ (r, t) = â0 (t) φ0 (r, t) ,(8.16)

X(8.17)δ̂ (r, t) =âi (t) φi (r, t) ,i6=0

where it is assumed that the condensate occupies the lowest state (i = 0) of the system andthe noncondensate occupies all other states (i 6= 0). Since we are assuming that state i = 0
is macroscopically occupied and N0  1, we can make a simplification based on the following
argument. Let state i = 0 be represented by |N0 i in the occupation number representation suchthatâ†0 |N0 i =pp
N0 + 1 |N0 + 1i ≈ N0 |N0 i ,â0 |N0 i =pp
N0 |N0 − 1i ≈ N0 |N0 i ,where N0 ± 1 ≈ N0 holds true as long as N0  1. Under this assumption â†0 and â0 approximately√
commute, and we can make the replacement â†0 ≈ â0 ≈ N0 , which is commonly referred to as the
Bogoliubov approximation [168]. Thus, the field operator Ψ̂ (r, t) can be decomposed asΨ̂ (r, t) ≈ φ (r, t) + δ̂ (r, t) ,where the condensate wavefunction is given by the complex number φ (r, t) =(8.18)√

N0 φ0 (r, t) andall operator dependence is contained in δ̂ (r, t). Thus, δ̂ (r, t) represents fluctuations of the system
about the condensed state. Note that the Bogoliubov approximation does not conserve particle
number since we have explicitly assumed that adding or removing an atom from the condensate
does not change the state of the system.
In what follows, equations of motion for both the condensed and non-condensed portionsof the Bose gas will be derived. To aid with readability, the following variables and notation
will be used to describe properties of the system: n (r, t) and N represent the total density andnumber of atoms, respectively, in the system; nc (r, t) and Nc represent the density and number ofatoms, respectively, in the condensate; ñ (r, t) and Ñ represent the density and number of atoms,

126
respectively, in the noncondensate (therefore, n (r, t) = nc (r, t) + ñ (r, t) and N = Nc + Ñ ); thesubscript “c” will be used to denote condensate quantities; a tilde (∼) over symbols will be used
to represent non-condensate quantities; and the subscript “0” will be used to denote equilibrium
quantities. Given this notation, and working within the Bogoliubov approximation, the densities
and number of particles in the condensed and non-condensed portions of the system are given bync (r, t) = |φ (r, t)|2 ,(8.19)ñ (r, t) = hδ̂ † (r, t) δ̂ (r, t)i,

(8.20)

where the total number of particles in the system is N =

8.3.1

R

d3 r [nc (r, t) + ñ (r, t)] = Nc + Ñ .

Zero-temperature limit

In the limit that T = 0, all atoms in the Bose gas occupy the condensate such that δ̂ (r, t) =δ̂ † (r, t) = 0. Taking an ensemble average of Eq. (8.14) in this limit, and using the fact thathΨ̂ (r, t)i = φ (r, t), results in the Gross-Pitaevskii equation (GPE) [166],i~∂φ (r, t)~2 2= −∇ + Vext (r, t) + g |φ (r, t)|2 φ (r, t) ,∂t
2m

where φ (r, t) is normalized to the total number of atoms in the condensate –(8.21)
R

d3 r |φ (r, t)|2 =Nc . The GPE is a nonlinear Schrödinger equation that describes the dynamical evolution of thecondensate wavefunction. The potential acting on the atoms in the condensate is the sum of
the external potential and a nonlinear term that represents the mean-field potential produced by
two-body interactions between all atoms in the condensate.
Stationary states of the GPE can be found by minimizing the energy of the system:hφ∗ (r)| Ĥ |φ (r)i
 2

Z
g
~
2
2
4
3
|∇φ (r)| + Vext (r) |φ (r)| + |φ (r)| .
=
d r
2m
2

Ec (φ) =(8.22)

However, care must be taken as a result of the Bogoliubov approximation, which violates particle
number conservation. The energy minimization must occur subject to the constraint that the127number of particles in the condensate remains constant. Therefore, the energy of the system
is minimized in the grand-canonical ensemble, Ec − µc,0 Nc , where µc,0 is the equilibrium chemical
potential. The minimal energy state is found by setting the functional derivative δEc −µc,0 δNc = 0.
This procedure results in the time-independent GPE:


~2 2
2
µc,0 φ0 (r) = −∇ + Vext (r) + g |φ0 (r)| φ0 (r) ,2m(8.23)

where the chemical potential µc,0 represents the eigenvalue of the condensate rather than the energy
per particle as in the case of the standard Schrödinger equation; these two quantities are equivalent
only in the limit of non-interacting particles.
Before proceeding to a discussion of the finite temperature ZNG theory, it is advantageous to
introduce an alternative system of equations that describe the dynamical evolution of the condensate
in terms of its density and the gradient of its phase. This leads to two equations of motion akin to
the hydrodynamic equations for a classical fluid, and provides a description of local quantities of thecondensate that arise in the ZNG theory. To proceed, we can write the condensate wavefunctionin terms of its density and phase,φ (r, t) =p
nc (r, t) eiθ(r,t) ,(8.24)

where nc (r, t) = |φ (r, t)|2 as defined above and θ (r, t) = arg [φ (r, t)] = arctan [φ∗ (r, t) /φ (r, t)].
Making this substitution in Eq. (8.21) and equating real and imaginary parts results in the following
two coupled equations [166]:∂nc (r, t)+ ∇ · [nc (r, t) vc (r, t)] = 0,∂t∂pc (r, t)− ∇c (r, t) = 0,∂t(8.25)(8.26)

where the following local properties of the condensate have been introduced:µc (r, t) = −p~2
p∇2 nc (r, t) + Vext (r, t) + gnc (r, t) ,2m nc (r, t)

pc (r, t) = ~∇θ (r, t) ,c (r, t) = µc (r, t) +(8.27)(8.28)1|p (r, t)|2 .
2m c(8.29)

128
Here, µc (r, t) is the local chemical potential (i.e., potential energy) of the condensate, pc (r, t) is
the local superfluid momentum (i.e., kinetic energy) of the condensate, and c (r, t) represents thetotal energy of an atom in the condensate as the sum of the local potential and kinetic energies.
These localized quantities are a necessary component of the ZNG theory, which is described in thenext section.

8.3.2

Finite temperature – the ZNG method

When T > 0 we can no longer neglect the fluctuations of the system described by the operatorδ̂ (r, t). Taking an ensemble average of Eq. (8.14) and using the fact that hΨ̂ (r, t)i = φ (r, t) once
again results in an equation of motion for the condensate wavefunction:∂φ (r, t)
~2 2
i~
= −
∇ + Vext (r, t) φ (r, t) + ghΨ̂† (r, t) Ψ̂ (r, t) Ψ̂ (r, t)i,∂t2m(8.30)

where the operator triplet represents the effects of two-body interactions with contributions fromboth the condensate and non-condensate. The operator triplet on the right-hand side of theabove equation can be expanded and simplified by first utilizing the Bogoliubov approximationof Eq. (8.18),
Ψ̂† Ψ̂Ψ̂ = |φ|2 φ + φ∗ δ̂ δ̂ + 2 |φ|2 δ̂ + φ2 δ̂ † + 2φδ̂ † δ̂ + δ̂ † δ̂ δ̂,(8.31)and then taking an ensemble average of the resulting expression:hΨ̂† Ψ̂Ψ̂i = nc φ + m̃φ∗ + 2ñφ + hδ̂ † δ̂ δ̂i,

(8.32)

where the space and time dependence has been omitted for clarity, nc and ñ are given by Eqs. (8.19)
and (8.20), respectively, and the property hδ̂i = hδ̂ † i = 0 has been used to arrive at the final
expression. Equation (8.32) includes contributions from the anomalous density m̃ = hδ̂ δ̂i and athree-field correlation function of the non-condensate hδ̂ † δ̂ δ̂i. The standard ZNG approach makes
the Popov approximation (m̃ (r, t) = 0) to avoid complexity associated with the anomalous density
term. This approximation is typically valid in the case of repulsive two-body interactions andisotropic trapping potentials since m̃ (r, t) is negligible in this regime [168]. However, the threefield correlation function is kept to first-order in the interaction strength g, and it will be shown129
that this leads to a term describing particle exchange between the condensate and noncondensate.
Thus, we have the following equation of motion for the condensate wavefunction – the generalized
GPE (GGPE):∂φ (r, t)
i~
∂t



~2 2
= −
∇ + Vext (r, t) + gnc (r, t) + 2gñ (r, t) φ (r, t)
2m+ ghδ̂ † (r, t) δ̂ (r, t) δ̂ (r, t)i.

(8.33)

Notably, compared to the zero temperature GPE, the GGPE contains an additional mean-field
contribution due to the density of atoms in the noncondensate.
In order to evaluate the three-field correlation function of the noncondensate, it is necessary
to find an equation of motion that describes the evolution of the noncondensate field operatorδ̂ (r, t). In the case of the ZNG method, this is a lengthy and involved procedure that has been
covered in detail previously (e.g., see Refs. [169–171]). Therefore, only the main points will begiven here. The noncondensate can be represented by the Wigner operatorfˆ (r, p, t) =

Z


 

r0
r0
δ r + ,t δ r − ,t ,220 ip·r0 †dr e(8.34)

where r and r0 are the center-of-mass and relative coordinates, respectively, and p is the momentum. Taking the expectation value of the Wigner operator yields the Wigner distribution function,f (r, p, t), the evolution of which is governed by a QBE [169–171]∂f (r, p, t)p+· ∇r f (r, p, t) − ∇r Ũ (r, t) · ∇p f (r, p, t)∂tm= C12 [f (r, p, t) , φ (r, t)] + C22 [f (r, p, t)] ,(8.35)

where Ũ (r, t) = Vext (r, t) + 2gnc (r, t) + 2gñ (r, t) is the effective potential for the noncondensate
in the Hartree-Fock approximation. The left-hand side of Eq. (8.35) describes the free evolution of
the distribution function in phase space as a result of Liouville’s theorem while the right-hand sidedescribes the modification of the distribution function as a result of interatomic collisions. Note

130that in this formalism the density and number of atoms in the noncondensate are defined asZñ (r, t) =dp
f (r, p, t) ,(2π~)3(8.36)

dr ñ (r, t) ,(8.37)

ZÑ =respectively. The two collision processes in Eq. (8.35) represent a collision between a condensed
atom and a non-condensed atom (C12 ) and two non-condensed atoms (C22 ). The former process isrelated to the three-field correlation function by [171]hδ̂ † (r, t) δ̂ (r, t) δ̂ (r, t)i ≈ −i~∗2gφ (r, t)

Z

dp
C12 [f (r, p, t) , φ (r, t)] ,(2π~)3(8.38)

where only the first-order in g has been kept. Notably, this quantity is purely imaginary andwill contribute a non-Hermitian source term to the GGPE of Eq. (8.33). Therefore, the C12collision process leads to the addition or removal of particles from the condensate, and causes thenormalization of φ (r, t) to change in time. In terms of the noncondensate distribution function andthe local condensate properties of Eqs. (8.27)–(8.29), the C12 collision process is described by [171]σ |φ|2C12 [f, φ] =πm2

Zdp2 dp3 dp4 δ (pc + p2 − p3 − p4 ) δ (c + ˜2 − ˜3 − ˜4 )× [δ (p − p2 ) − δ (p − p3 ) − δ (p − p4 )] [(1 + f2 ) f3 f4 − f2 (1 + f3 ) (1 + f4 )] ,(8.39)

where the space and time dependence has been omitted for clarity, σ = 8πa2s is the Bose-enhanced
scattering cross-section, and ˜ ≡ |p|2 /2m + Ũ (r, t) is the energy of a non-condensed particle within
the Hartree-Fock approximation. In a similar fashion, the C22 collision process is described by [171]σ
C22 [f ] =πh3 m2Z
dp2 dp3 dp4 δ (p + p2 − p3 − p4 ) δ (˜ + ˜2 − ˜3 − ˜4 )× [(1 + f ) (1 + f2 ) f3 f4 − f f2 (1 + f3 ) (1 + f4 )] .

(8.40)In Eqs. (8.39) and (8.40) the delta functions ensure conservation of energy and momentum in acollision, and fi represents the value of f (r, p, t) at the phase-space coordinates (ri , pi ). The (1+fi )terms represent Bose enhancement of the scattering process (i.e. particles have a higher probabilityof scattering into occupied states).

131
8.3.2.1

Thermodynamic equilibrium

In thermodynamic equilibrium, both the condensate and noncondensate are stationary in time, andno net particle exchange occurs between the two components of the gas. Therefore, the C12 andC22 collision integrals vanish and the distribution function for the non-condensate takes the formof a stationary Bose-Einstein distribution:
f0 (r, p) =1eβ0 (˜0 −µ̃0 ) − 1,(8.41)

where β0 = 1/kB T0 is the equilibrium inverse temperature and µ̃0 is the equilibrium chemical
potential of the noncondensate. Noting that ˜0 = |p|2 /2m + Ũ0 (r), we can use Eq. (8.36) to findthe equilibrium noncondensate density:ñ0 (r) =where Λ =1
g [z0 (r)] ,Λ3 3/2(8.42)p2π~2 /kB T0 m is the thermal de Broglie wavelength, z0 (r) = e−β0 [Ũ0 (r)−µ̃0 ] is the

local fugacity, and g3/2 [z] is the polylogarithm function of order s = 3/2 defined by the series (seeRef. [69], for example)
gs [z] =

∞
X
zn
n=1

ns

,

(8.43)

which is convergent if and only if |z| < 1. For z = 1, gs [1] = ζ (s), where ζ (s) is the Riemann zetafunction. Note that the equilibrium state of the noncondensate depends on that of the condensate
through the mean-field interaction term contained in the definition of the noncondensate potential
energy Ũ0 (r) = Vext (r)+2gnc,0 (r)+2gñ0 (r). Thus, the two components of the gas remain coupledin the absence of the C12 collision process.
In equilibrium, the GGPE takes on a time-independent form similar to that of Eq. (8.23),
but modified by the mean-field of the non-condensate:~2 2∇ + Vext (r) + gnc,0 (r) + 2gñ0 (r) φ0 (r) ,µc,0 φ0 (r) = −2m(8.44)

where the term related to the three-field correlation function vanishes in thermal equilibrium. For
the two components of the gas to remain in diffusive equilibrium (i.e., no net particle exchange) they

132
must have the same chemical potential; therefore, we have the condition µ̃0 = µc,0 in equilibrium.
Thus, equilibrium solutions to the ZNG equations can be found by solving Eqs. (8.42) and (8.44)
self-consistently subject to the conditions that µ̃0 = µc,0 and the temperature T0 and total numberof atoms in the gas N remain fixed.

8.4

Validity of the ZNG Method

In the ZNG approach, the fluctuations about the condensed state are represented semiclassicallywith a phase-space distribution function. Physically, this implies that atoms in the noncondensate behave like billiard balls that interact via elastic collisions, and move around in an effective
Hartree-Fock potential that consists of the external trapping potential and contributions from boththe condensate and noncondensate mean-field potentials. Such a treatment completely neglects
any quantum fluctuations of the condensed state such as Bogoliubov quasi-particle excitations;
therefore, the ZNG method is valid only in certain regimes. The use of a semiclassical approximation for the noncondensate is valid as long as the condensate appears locally homogeneous on thescale of variation in the noncondensate distribution function. This is typically the case at higher
temperatures close to Tc and as long as the external trapping potential is not highly anisotropic.
Furthermore, use of the Hartree-Fock excitation spectrum to described the fluctuations about thecondensed state as opposed to the Bogoliubov spectrum is generally valid for trapped Bose gases as
the dominant fluctuations are well described in the Hartree-Fock limit down to very low temperatures [172]. Therefore, the ZNG approach should provide an accurate representation of the problem
studied in this dissertation, which involves a finite temperature Bose gas close to Tc confined in ahighly isotropic harmonic trap.

Chapter 9

Numerical Solution of the ZNG Equations in Spherical Symmetry

The ZNG equations consist of a nonlinear Schrödinger equation (the GGPE of Eq. (8.33)) coupled
to a collisional Boltzmann equation (the QBE of Eq. (8.35)). These styles of equations have been
widely investigated; thus, there exists a variety of numerical methods for treating the dynamical
evolution of such equations independently. However, coupling them in a self-consistent fashion ischallenging as the numerical methods used to solve each equation differ significantly. A numericalsolution to the full ZNG equations is presented by Jackson and Zaremba in Ref. [173], and thischapter focuses on the adaptation of their technique to problems with spherical symmetry.
The goal of this chapter is to outline the specific numerical methods used to propagate an
initial state of the system forward in time through subsequent time steps denoted by ∆t. The
numerical methods presented are second-order accurate in time (i.e., the leading error term isO ∆t3 ), which provides a good balance between accuracy and algorithm complexity. Furthermore, the focus will be on spherically symmetric problems such that the number of degrees of
freedom needed to accurately describe the system is greatly reduced. The problem investigated inthis dissertation involves an isotropic harmonic trap; thus, the problem can be reduced from six
dimensions to three – a radial displacement r ≡ |r|2 , a momentum magnitude p ≡ |p|2 , and anangular variable cos θ that describes the orientation of p with respect to r. This choice of variables
reduces the GGPE to an equation in one variable and greatly simplifies the implementation of thenumerical method for solving the QBE, as only three variables need be considered in the Monte
Carlo sampling routines rather than the usual six. This can lead to significant speedup of the134numerical solution to the ZNG equations. Of course, a caveat to this is that only problems withspherical symmetry can be investigated.
This chapter begins by outlining the numerical method used to propagate an initial state of
the GGPE forward in time. A very popular class of numerical methods for nonlinear Schrödinger
equations are split operator pseudospectral methods, where the potential and kinetic energy portions of the Hamiltonian operator are separated and applied to a given wavefunction in succession.
This allows the potential energy operator to be applied as a simple point-wise multiplication whereas
the kinetic energy operator can be applied in momentum space using the fast Fourier transform(FFT) [174], or in position space using the method of finite differences (e.g., the Crank-Nicolson
method [175]).
Following a discussion of the GGPE, a numerical method for solving the QBE with sphericalsymmetry is given. Compared to a nonlinear Schrödinger equation, the collisional Boltzmann
equation is significantly more complex and nearly impossible to solve directly due to its highdimensionality. A method applicable to a wide range of problems that has found widespread use is
the direct simulation Monte Carlo (DSMC) method [176]. The key aspect of the DSMC method is
to represent the phase space distribution function f (r, p, t) as a swarm of point-like test particles
that evolve in time based on Liouville’s theorem and sample the available phase space. Monte
Carlo techniques are then utilized to incorporate the effect of collisions on the test particles, andthe approximate evolution of the distribution function is obtained by averaging over the motion of
all test particles. The main advantage to this method is that it avoids having to store and update asix-dimensional quantity on a discrete grid – a nearly impossible task even for modern computers.
The final section of this chapter will discuss the full dynamical simulation of the ZNG equations. The first step of this process is to generate an equilibrium state of the system that is dictatedby the temperature of the gas and the total number of atoms it contains. This is done in a selfconsistent fashion such that equilibrium densities for both the condensate and noncondensate are
obtained. As the noncondensate is represented by a distribution of test particles in a dynamical
simulation, an initial position and momentum must be calculated for each test particle based on the

135equilibrium properties of the gas. The simulation then proceeds based on the update schemes thatwill be outlined below. Finally, to test the numerical method developed for spherical symmetry, aquench simulation is performed and directly compared to previous results in the field.
Throughout this chapter, some examples of the numerical methods developed will be given.
An example problem with an equilibrium temperature of T0 = 200 nK, total atom number of
N = 50, 000 87 Rb atoms, and a harmonic trap frequency of ω = 2π × 187 rad/s will be used forthese numerical examples. The mass and scattering length used are m = 1.44 × 10−25 kg andas = 5.29 nm, respectively, which correspond to 87 Rb atoms. Regarding the numerical simulations, it is advantageous to identify characteristic properties of a given problem that can be used
to convert a system of equations to an equivalent, dimensionless form. This avoids limitations ofcomputer arithmetic that give rise to overflow and underflow errors, which arise through the addition, subtraction, multiplication, and/or division of very small and/or very large numbers (e.g.,
Planck’s constant h = 6.626 × 10−34 m2 kg/s, or it’s inverse). For a degenerate Bose gas confinedby a harmonic trap with oscillator frequency ω, the following characteristic properties are used:
r

~
,
mω

(9.1)

~mω,

(9.2)

~ω
,
2

(9.3)

aho =
√
pho =
ho =

which correspond to the harmonic oscillator length, momentum, and ground state energy, respectively. The results of numerical examples will be given in these dimensionless units.

9.1

Solution of the GGPE

This section will outline the numerical method for propagating the solution of the GGPE forward
in time. The GGPE takes the following form in spherical symmetry:  2
~1 ∂∂φ (r, t)2 ∂= −r+ Vext (r, t) + gnc (r, t) + 2gñ (r, t) − iR (r, t) φ (r, t) ,i~
∂t2m r2 ∂r
∂r
(9.4)

136
where the function R (r, t) has been introduced to represent the effect of the three-field correlation
function on the condensate wavefunction:~
R (r, t) =2nc (r, t)

Z

dp
C12 [f, φ] .
(2π~)3(9.5)

In an isotropic harmonic trap of frequency ω, the external potential is given by Vext (r, t) ≡ 21 mω 2 r2
and the GGPE can be cast in an alternative form with the substitution φ (r, t) → ϕ (r, t) /r,
  2 2
∂1∂ϕ (r, t)~
2 2+ mω r + gnc (r, t) + 2gñ (r, t) − iR (r, t) ϕ (r, t) .
i~
= −
∂t
2m ∂r2 2(9.6)

The advantages of this substitution are the simplification of the Laplacian term to a single second
derivative and a simplification of the boundary conditions since ϕ (r, t) → 0 as r → 0, ∞. This
comes at the expense of a slightly more complicated nonlinear term as nc (r, t) ≡ |ϕ (r, t)|2 /r2 ;however, the advantages outweigh this minor inconvenience. This is the form of the GGPE that isassumed for application of the numerical methods outlined below.

9.1.1

Derivation of the time evolution operator

To proceed with a derivation of the numerical method, it is convenient to write the GGPE in thefollowing form:i~∂ϕ (r, t)= Ĥ (t) ϕ (r, t) ,
∂t

(9.7)

where Ĥ (t) ≡ T̂ + V̂ (t) is the time-dependent Hamiltonian operator with the kinetic and potential
energy operators defined asT̂ ≡ −

~2
2m



∂2
,
∂r2

1
V̂ (t) ≡ mω 2 r2 + gnc (r, t) + 2gñ (r, t) − iR (r, t) ,
2

(9.8)
(9.9)

respectively. A formal solution to this equation is given by application of the time evolution operatorÛ (t; t0 ) to some initial state ϕ (r, t0 )ϕ (r, t) = Û (t; t0 ) ϕ (r, t0 ) ,(9.10)

137
where t ≡ t0 + ∆t and application of Û (t; t0 ) evolves the wavefunction over the interval ∆t. In general, the time evolution operator takes the following form for a time-dependent Hamiltonian [177]:
 Z
Z tn−1
Z t2
∞ 
X
1 n t
dtn Ĥ (t1 ) Ĥ (t2 ) · · · Ĥ (tn ) .
(9.11)
dt2 · · ·Û (t; t0 ) = 1 +
dt1
i~
t0
t0
t0
n=1

As this expression represents an infinite sum of terms, the present task is to find a closed form
expression that has a leading error term of O ∆t3 . With this in mind, we proceed by expanding
the Ĥ (tn ) terms of Eq. (9.11) in a Taylor series:∂ Ĥ (t)1 ∂ 2 Ĥ (t)(tn − t) +(tn − t)2 + . . .
∂t
2 ∂t2
α2
≡ α0 + α1 (tn − t) +(tn − t)2 + . . . ,2Ĥ (tn ) = Ĥ (t) +(9.12)

where the shorthand form αn = ∂ n Ĥ (t) /∂tn has been introduced. Expanding the first two termsof the sum in Eq. (9.11) and substituting the Taylor series expansion for Ĥ (tn ) results in
Z
h
i
α2
1 t
dt1 α0 + α1 (t1 − t) +Û (t; t0 ) = 1 +(t1 − t)2 + . . .
i~ t02
 2 Z t
Z t1
h
i
α2
1
dt1
dt2 α0 + α1 (t1 − t) +(t1 − t)2 + . . .
+
i~
2
t0
t0
h
i
α2
× α0 + α1 (t2 − t) +(t2 − t)2 + . . . + . . . .
2(9.13)

Carrying out the integration, making the substitution ∆t ≡ t − t0 , and keeping only terms up to∆t2 then gives 2 21 
α1 2 
1
α0 2
Û (t; t0 ) = 1 +α0 ∆t −∆t +∆t + . . .
i~
2
i~
2



α1 
i
≈ exp −
α0 +∆t ∆t + O ∆t3 ,~2(9.14)

Finally, using the Taylor series expansion for Ĥ (tn ) an expression for α1 can be found by a backward
finite-difference approximation:
α1 ≡Ĥ (t) − Ĥ (t − ∆t)∂ Ĥ (t)≈+ O (∆t) .
∂t∆t(9.15)

Therefore, the exponent in the time evolution operator can be written asα1∆t23Ĥ (t) − Ĥ (t − ∆t)=,2H̃ (t) ≡ α0 +(9.16)

138
and the second-order accurate time evolution operator isÛ (t; t0 ) ≈ e−iH̃(t)∆t/~ + O ∆t3 .

(9.17)

Note that the backward finite-difference approximation for α1 has a leading error term proportional
to ∆t, which would appear to spoil the second-order accuracy of this method. However, α1 ismultiplied by ∆t2 in the exponent of the time evolution operator; therefore, the approximation forÛ (t; t0 ) maintains second-order accuracy.

9.1.2

Application of the time evolution operator

Given the form of Û (t; t0 ), a numerical method for applying the exponential operator of Eq. (9.17)
to a given wavefunction is required. A standard approach that maintains second-order accuracy
makes use of the symmetric operator splitting first introduced by Strang [178]:Û (t; t0 ) ≈ e−iH̃(t)∆t/~ + O ∆t3 ≈ e−iT̂ ∆t/2~ e−iṼ (t)∆t/~ e−iT̂ ∆t/2~ + O ∆t3 ,(9.18)

hiwhere Ṽ (t) = 3V̂ (t) − V̂ (t − ∆t) /2 is the potential energy operator evaluated at the midpointof a full time step, t = ∆t/2. To apply the time evolution operator to a given wavefunction
each exponential term is applied in succession, which is mathematically equivalent to solving threepartial differential equations consecutively:∆tt0 → t0 +,2∂ϕ (r, t)i~= T̂ ϕ (r, t) ,∂t↓i~

∂ϕ (r, t)= Ṽ (t) ϕ (r, t) ,∂t

(t0 → t0 + ∆t) ,↓∂ϕ (r, t)i~= T̂ ϕ (r, t) ,
∂t



∆t
t0 +
→ t0 + ∆t ,2

where t0 is the initial time point, t0 + ∆t is the final time point for both operators, and it isunderstood that the solution of the first equation is used in the second equation and the solutionof the second equation is used in the third equation.

139
To implement this numerical method on the computer, all functions of space are represented
on a discrete grid of points spaced uniformly by ∆r and extending from r = 0 to r = rmax , wherermax is chosen large enough that the boundary condition ϕ (∞, t) = 0 is accurately represented.
Therefore, the equation involving the potential energy operator is straightforward to solve and theexponential operator can be applied directly to the wavefunction as a point-wise multiplication.
However, the equations involving the kinetic energy operator include a derivative such that amore complex approach is needed. Due to the spherical symmetry of the problem standard FFT
pseudospectral methods are not applicable, so the Crank-Nicolson finite-difference method is chosen
instead.
The Crank-Nicolson method is an implicit finite-difference numerical scheme that is unconditionally stable and second-order accurate in both space and time. In order to use the Crank-Nicolson
method to apply an exponential operator to a given wavefunction, Cayley’s form [175] is used toobtain a finite-difference representation of the kinetic energy portion of the GGPE:




i∆t
i∆t
n+1/2
1+
= 1−
T̂ ϕj
T̂ ϕnj ,4~
4~(9.19)

where the space and time dependence of ϕ is represented by j, a discrete spatial point, and n, a
discrete point in time, respectively. Expanding T̂ and using a second-order accurate finite-difference
approximation for the second derivative results in




i~∆t
n+1/2
n+1/2
n+1/2
n+1/2
=
ϕj
−
ϕj+1 − 2ϕj+ ϕj−1
8m∆r2





i~∆t
n
n
n
n
ϕj+1 − 2ϕj + ϕj−1 ,
ϕj +
8m∆r2

(9.20)

which is a system of linear equations that represents a tridiagonal problem. A solution for ϕn+1
atj
all j can be obtained efficiently using the well known Thomas algorithm [175] in conjunction withthe boundary conditions ϕn0 = ϕnjmax = 0 for all n.

140

9.2

Solution of the QBE

This section will outline the numerical method for propagating the solution of the QBE forward
in time. The premise of the DSMC method is to sample the distribution function f (r, p, t) overa large number of points in phase space, and then evolve each of these points forward in time.
These phase space points are referred to here as test particles, and their relation to the distributionfunction is given byf (r, p, t) ≈ γ (2π~)3Xδ [r − ri (t)] δ [p − pi (t)] ,(9.21)iwhere ri (t) and pi (t) are the position and momentum, respectively, of test particle i, and thedelta functions imply that the distribution function is represented only at the discrete phase spacepoints of each test particle. The weighting factor γ = Ñ /Ntp describes the relationship between
the number of physical particles in the non-condensate Ñ and the number of test particles Ntp
used to represent them. Thus, a solution to the QBE is found by evolving the distribution oftest particles forward in time and using their properties to obtain a local estimate of f (r, p, t)
and the effect of collisions. To begin, it is convenient to separate the collisionless evolution of
the test particle distribution from the effects of collisions. This separation is accurate as long as
the position and momentum of the test particles change over a time that is less than the averagecollisional timescales.

9.2.1

Collisionless evolution of the test particle distribution

Similar to the approach used to derive a numerical method for the GGPE, it is convenient to write
the collisionless QBE in the following form:∂z= −iL (t) z,∂t(9.22)

where z ≡ (r, p) is a phase space coordinate and L is the Liouville operator [179] defined as
iL (t) ≡ { · , H (t)} =∂H (t)
∂p



∂
−
∂r



∂H (t)
∂r



∂
,
∂p

(9.23)

141
where { · } denotes a Poisson bracket, and H (t) is the classical Hamiltonian. Note that only asingle pair of canonically conjugate coordinates, (r, p), are considered here given the assumption of
spherical symmetry. Thus, the usual summation over all coordinates is neglected in the definitionof L (t). In a completely analogous fashion to the approach used for the GGPE, a second-orderaccurate update scheme consists of finding a closed form expression for the time evolution operator,
which takes the familiar formU (t; t0 ) ≈ e−iL̃(t)∆t + O ∆t3 ,(9.24)

where L̃ (t) = [3L (t) − L (t − ∆t)] /2. To apply this operator to a given initial state the Strang
splitting approach is again used to obtain
U (t; t0 ) ≈ e−iL̃(t)∆t + O ∆t3 ≈ e−iLT ∆t/2 e−iL̃V (t)∆t e−iLT ∆t/2 + O ∆t3 ,(9.25)

where the classical kinetic and potential energy operators are defined as
p ∂
,
m ∂r
∂ Ṽ (t) ∂L̃V (t) ≡ −
,
∂r ∂p
LT ≡

(9.26)(9.27)

respectively, and Ṽ (t) = [3V (t) − V (t − ∆t)] /2 is the potential energy operator defined at the
midpoint of a full time step, t = ∆t/2. To apply the time evolution operator to a given state of
the system, each exponential term is applied in succession, which is mathematically equivalent tosolving three partial differential equations consecutively:p (t)∂r (t)=,∂tm∆t
t0 → t0 +2,↓∂p (t)∂ Ṽ (t)
=−
,
∂t
∂r

(t0 → t0 + ∆t) ,↓∂r (t)
p (t)
=
,
∂t
m




∆t
t0 +
→ t0 + ∆t ,2

142
where t0 is the initial time point, t0 + ∆t is the final time point for both operators, and it isunderstood that the solution of the first equation is used in the second equation and the solutionof the second equation is used in the third equation. These equations are found by expanding thephase space coordinate z and writing equations of motion for r and p separately. Notably, LT
operates only on the position of the test particles while LV operates only on the momentum of thetest particles.
To implement this numerical method on the computer, each test particle is represented by adiscrete phase space coordinate zi ≡ (ri , pi,x , pi,y , pi,z ) and finite difference approximations are used
to estimate the derivatives in the above equations. To update the position of each test particle
in spherical symmetry, the method outlined by Bird [176] is used, and only the radial position ofeach test particle is stored. However, the complete motion of each particle in 3D space must be
tracked such that three momentum components are stored for each particle. At the beginning ofeach time step the spherical symmetry of the problem is utilized to arbitrarily align the position
vector of each particle with the x-axis. The action of momentum components directed along the yand z-axes is to then push the particle away from the x-axis. It is straightforward to calculate the
new radial position of the particle; however, the off-axis motion causes a rotation of the particle
trajectory and the momentum components must be rotated accordingly. The new particle positionon the x-axis after a half time step of ∆t/2 isx = rin +

pni,x
∆t,
2m

(9.28)

where n represents a discrete time point, rin is the initial radial position of particle i, and pni,x it’s
momentum along the x-axis. The action of the off-axis momentum components moves the particle
by a distance
d=

s
 n
pi,y
2m

2
∆t

+

 n
pi,z
2m

2
∆t ,

(9.29)

such that the new radial position of particle i is
n+1/2

ri

=

p
x2 + d2 .

(9.30)

143
The rotation of the particle trajectory is defined byn+1/2

sin ϑ = d/ri,

(9.31)

n+1/2(9.32)cos ϑ = x/ri,

and an azimuthal angle chosen at random such that ϕ ∈ [0, 2π]. Thus, the rotated momentum
components are given by
n+1/2
pi,x
= pni,x cos ϑ +

r

pi,y

n+1/2

= pi,cn+1/2= pi,c

pi,z

pni,y

2

 2
+ pni,z sin ϑ,(9.33)

n+1/2

sin ϕ,(9.34)

n+1/2

cos ϕ,(9.35)

n+1/2
where the circumferential momentum component is pi,c
= −pni,x sin ϑ +

r

pni,y

2

 2
+ pni,z cos ϑ.

This completes a single update of the particle position. Note that although three momentum
components are stored for each particle in addition to the radial position, the algorithm is effectively
three-dimensional since the azimuthal angle ϕ is randomized at each time step. Thus, the only
unique momentum coordinates are pi,x and the angle ϑ.
Following a single update of the position for each test particle, the momentum along the xaxis is updated based on the effect of the potential energy operator. The x component is updated
since this is the axis that is chosen to coincide with the position of the test particle at the beginningof each time step. The update scheme for each test particle is then
"
n+1/2
pn+1
− ∆t
i,x = pi,x

∂ Ṽ (t)∂r#n+1/2,(9.36)

r=ri

where the derivative term is calculated at the recently updated position of test particle i. In practice,Ṽjn is represented on the same discrete grid as the condensate wavefunction, ϕnj . Therefore, thespatial derivative of Ṽjn is approximated using a fourth-order finite difference approximation,n − 8Ṽ n + 8Ṽ n − Ṽ nṼj−2
∂ Ṽ (t)
j−1
j+1
j+2
≈
,
∂r
12∆r

(9.37)and linear interpolation is used to obtain an estimate at the position of the test particle. Finally,

144
a second update of each test particle’s position over a half time step ∆t/2 is performed, and hiscompletes the update scheme for the test particle ensemble at each time step.
Before proceeding to a discussion of collision effects it is important to mention a complicationthat arises with the above numerical update scheme when estimating the spatial derivative of Ṽ (t).
As shown in Chapter 8, the noncondensate – and therefore the test particle distribution – evolves
in an effective Hartree-Fock potential given by1
V (t) ≡ mω 2 r2 + 2gnc (r, t) + 2gñ (r, t) .
2

(9.38)

This potential includes a term proportional to the noncondensate density, which must be estimatedfrom the test particle distribution at each time step. This is challenging because a smooth, differentiable function must be constructed from a set of discrete points in phase space to employ
Eq. (9.37), and this function must be represented on the same discrete grid as the condensate
wavefunction. The following section outlines a method for accomplishing this task in a sphericallysymmetric geometry.

9.2.1.1

Estimation of ñ (r, t) from the test particle distribution

Given the positions of each test particle in the distribution, a method for numerically generating asmooth and differentiable form of ñ (r, t) at each time step is required. This is typically accomplishedwith a cloud-in-cell scheme [180], which is commonly used in particle-in-cell simulations of plasmas.
In a 1D simulation, application of the cloud-in-cell scheme consists of assigning each particle to its
two nearest grid points with a linear weight factor given by the distance of the particle to each grid
point. However, such a linear weighting scheme introduces errors in a spherical coordinate system,
particularly near the boundaries of the discrete grid [181]. To avoid these errors a volume weighting
scheme is employed where the test particle weight factors are given by the volume of space between
the particle and a given grid point [182, 183] rather than the linear distance.
The first step in estimating ñ (r, t) is to break up the spatial domain into a series of bins,
where each bin extends over a small portion of the computational domain. The edges of these bins

145then make up a discrete grid that ñ (r, t) can be represented on. Not only is this necessary for
obtaining an estimate of ñ (r, t), it is also required to estimate the effects of collisions, which will be
discussed further in the next section. Due to the spherical symmetry of the problem and the choiceof a volume weighting scheme, the test particles are binned into radial shells of constant volume.
Given the size of the simulation domain, rmax , and the total number of desired bins, Nb , the radial
position of the start of each bin is given by
rb,k = rmax

k
Nb

1/3
,

(9.39)

where k ∈ [0, Nb ] is an integer representing the bin index. For a given time step, the two nearest
bin edges, denoted by l and l + 1, are computed for each test particle by inverting Eq. (9.39) andsolving for k. Each test particle is then weighted to the grid positions rl and rl+1 using the following
volume weighting schemeñl = ñl +ñl+1 = ñl+1 +1Vb

3 − r3rl+1i
3 − r3
rl+1
l



1
Vb



!

ri3 − rl3
3 − r3rl+1
l,(9.40)!
,(9.41)

where ñl is the noncondensate density at the bin edges, Vb is the volume of each bin, and ri is
the radial position of test particle i. This procedure is applied to all test particles and an initial
estimate of ñ (r, t) is obtained. In order to find ñ (r, t) on the same discrete grid used to represent
the condensate wavefunction, a standard 1D linear interpolation routine is used to compute thediscrete function ñj .
The initial estimate of ñj based on the cloud-in-cell scheme with volume weighting is generally
not smooth and differentiable. Therefore, it is necessary to smooth ñj using a Gaussian low-pass22

filter, which is accomplished by convolving ñj with a finite extent Gaussian function Sj = e−rj /η .
This procedure is accomplished by Fourier transforming ñj using a FFT routine, multiplying theresult by the Fourier transform of Sj , and inverse Fourier transforming to obtain the smoothedversion of ñj . The simulation results are not strongly dependent on the choice of smoothing parameter, η, which is chosen empirically to provide just enough smoothing to make ñj differentiable.

146
Furthermore, for a given grid spacing ∆r the smoothing parameter remains fixed throughout asimulation.
Figure 9.1 provides an example of the procedure outlined above for generating ñj from thetest particle distribution at each time step. The initial cloud-in-cell estimate accurately captures
the overall shape of ñj , but the discrete nature of the particle distribution contributes substantial
fluctuations. Application of the Gaussian low-pass filter effectively eliminates these fluctuations
and results in a ñj that closely resembles the theoretical density distribution. The next section
describes the numerical implementation of the C12 and C22 collision processes.

9.2.2

Collisional evolution of the test particle distribution

Collisionless evolution of the distribution function is a conservative process in that the number ofparticles contained in the infinitesimal phase space volume element drdp/(2π~)3 does not change
in time. The collision integrals of Eqs. (8.39) and (8.40) represent the number of particles enteringor leaving drdp/(2π~)3 per unit time, and therefore result in a change of f (r, p, t). If the typicalcollision timescale is much larger than the time step for collisionless evolution, ∆t, then the effect ofcollisions can be treated separately at the end of a given time step. In this limit, the probability of atest particle undergoing a collision in time ∆t is much less than unity, and an accurate representationof the collision integrals can be obtained via Monte Carlo sampling. The present task is then to
calculate the rate of each collision process and define a collision probability for each test particle inthe distribution. We begin by considering the C22 collision integral (Eq. (8.40)), which corresponds
to the scattering of two atoms in the noncondensate.

9.2.2.1

Probability of the C22 collision process

Since the quantityR

drdp/(2π~)3 C22 [f ] represents the total number of particles entering or leaving

the volume drdp/(2π~)3 per unit time, we can define the mean collision rate at point r by integrating

147

14
Cloud-in-cell
Smoothed
Actual12~ (r) (a!3
n
ho )

10

8

6

4

2

0
0

2

4

6

8

10

12

14

16

r (aho )Figure 9.1: Example of generating ñj using the procedure described in the main text. The cloudin-cell profile (blue, solid) is constructed from Ntp = 100, 000 test particles. The smoothed profile(red, solid) is an accurate representation of the actual equilibrium density profile obtained withthe method outlined in Section 9.3.1 (black, dashed), and does not exhibit the fluctuations present
in the cloud-in-cell profile. The actual noncondensate density is the equilibrium solution ñ0 (r) for
temperature T0 = 200 nK, total atom number N = 50, 000, trap frequency ω = 2π × 187 rad/s,
and 87 Rb atoms with a scattering length as = 5.29 nm.

148
C22 [f ] over momentum:
Γout
22 (r, t) =σ
πh6 m2Z
dpdp2 dp3 dp4 δ (p1 + p2 − p3 − p4 ) δ (˜ + ˜2 − ˜3 − ˜4 )× f1 f2 (1 + f3 ) (1 + f4 ) ,(9.42)

where the 1 and 2 subscripts represent the incoming (i.e., pre-collision) momenta of the two atoms
and the 3 and 4 subscripts represent the outgoing (i.e., post-collision) momenta of the two atoms.
In the above equation, only the half of the collision integral that corresponds to particles leaving thevolume drdp/(2π~)3 is considered, hence the usage of the “out” superscript. This is valid because
the “in” process is symmetric and will lead to the same value for the mean collision rate at point r,
which we only need in order to define a probability that a given test particle undergoes a collisionat each time step. Furthermore, the C22 collision process conserves the total number of atoms in
the noncondensate.
To simplify the expression for Γout
22 (r, t) it is convenient to transform the momentum coordinates to the center of mass frame by defining
p1 + p2√,2
p −pprel ≡ 1√ 2 ,2+ppp0cm ≡ 3√ 4 ,2
p −pp0rel ≡ 3√ 4 .
2
pcm ≡(9.43)(9.44)(9.45)(9.46)The expression can then be simplified using conservation of energy and momentum in an elastic
collision (i.e., p0cm = pcm and |p0rel | = |prel |) and properties of the delta function. The end result isΓout
22 (r, t) =

σ
m

Z

dp1
f1
(2π~)3Zdp2f2(2π~)3ZdΩ|p − p2 | (1 + f3 ) (1 + f4 ) .
4π 1(9.47)

Thus, calculation of this rate involves integrals over all possible incoming momenta and all possible
scattering angles defined by the solid angle Ω. In the center of mass frame, the outgoing momenta

149
are given in terms of the incoming momenta by
pcm + |prel | û (Ω)√,
2
p − |p | û (Ω)
p4 = cm √rel
,
2
p3 =

(9.48)(9.49)

where û (Ω) is a unit vector in the direction determined by the solid angle Ω.
To determine the probability that a given test particle in the distribution undergoes a C22
type collision, the Monte Carlo sampling technique [175] can be utilized to estimate the integrals
in Eq. (9.47). Notably, the random phase space points necessary for Monte Carlo sampling areconveniently provided by the test particle distribution. As discussed in section 9.2.1.1, at each time
step the test particle distribution is binned in position space using radial shells of equal volume.
The particles within each bin are then paired at random, and a probability for a pair of particles
to undergo a collision for the time step ∆t is [173]σñijPij22 =
mZ
pi − pjdΩ(1 + f3 ) (1 + f4 ) ∆t,
4π(9.50)

where i and j represent a pair of particles and ñij represents the local noncondensate density. The
integration over Ω is estimated by randomly generating the unit vector û (Ω) such that cos ϑ ∈[−1, 1] and ϕ ∈ [0, 2π]. This results in a random scattering angle ΩR and the following collision
probability:
Pij22 =




σñij
pi − pj 1 + f3ΩR 1 + f4ΩR ∆t,m(9.51)

which depends on the value of the distribution function evaluated at the outgoing momenta f3ΩR and
f4ΩR . As the collision probability depends on a local estimate of the distribution function at specificmomenta, the test particle distribution within each spatial bin must also be binned in momentum
space. This procedure will be discussed in detail below after the probabilities associated with theC12 collision process are introduced.

150
9.2.2.2

Probability of the C12 collision process

In an analogous fashion to the C22 process, mean collision rates at point r can be found by integrating C12 [f, φ] over momentum to obtain the corresponding “out” and “in” rates:
Z
σnc
Γout
(r,
t)
=
dp2 dp3 dp4 δ (pc + p2 − p3 − p4 ) δ (c + ˜2 − ˜3 − ˜4 )12
πh3 m2× f2 (1 + f3 + f4 ) ,ZσncinΓ12 (r, t) =dp2 dp3 dp4 δ (pc + p2 − p3 − p4 ) δ (c + ˜2 − ˜3 − ˜4 )πh3 m2× f2 f4 ,(9.52)(9.53)

where the cubic terms f2 f3 f4 formally cancel and have therefore been dropped from the expressions
for each rate. The “out” process represents an atom from the noncondensate scattering off of thecondensate to produce two non-condensed atoms (i.e., an atom is removed from the condensate
and added to the noncondensate). On the other hand, the “in” process represents the scattering of
two atoms in the noncondensate to produce one atom in the noncondensate and one atom in thecondensate (i.e, a particle is removed from the noncondensate and added to the condensate). These
two processes must be considered independently because neither conserves the number of atoms in
the condensate or noncondensate alone.
To simplify the expression for the “out” rate, we once again transform to the center of massframe, and utilize conservation of energy and momentum in an elastic collision and properties ofthe delta function to obtainσnc
Γout
12 (r, t) =
m

where

pout
rel

Z

dp2
f2
(2π~)3ZdΩ outp(1 + f3 + f4 ) ,4π rel(9.54)q= |pc − p2 |2 − 4gmnc is the relative momentum of the incoming atoms corrected

for the additional local mean-field energy of the noncondensate. Similar to the C22 process, theoutgoing momenta are given in terms of the incoming momenta byoutpout
cm + prel û (Ω)√,
2pout − poutû (Ω)√relp4 = cm,2

p3 =(9.55)(9.56)

151√
where pout
cm = (pc + p2 ) / 2. In Eq. (9.54), it is understood that the integration over p2 is restricted
2
by the condition that poutrel is a real, positive number. This leads to the constraint |pc − p2 | >4gmnc , which implies that only noncondensate atoms satisfying this constraint can undergo an“out” collision. The physical meaning behind this constraint can be understood from the differencein potential energy between a condensed and non-condensed atom, which differ by a factor of gnc .
Therefore, to conserve energy, the atom in the noncondensate with momentum p2 must have excess
kinetic energy to compensate for the potential energy gained by an atom leaving the condensate
and entering the noncondensate.
The probability that a test particle located in a given spatial bin undergoes an “out” collision
for the time step ∆t is [173]
Piout =


σnc,i out prel 1 + f3ΩR + f4ΩRm

(9.57)

where nc,i is the condensate density at the position of particle i, and ΩR represents a random
scattering angle once again determined by randomly generating the unit vector û (Ω) such thatcos ϑ ∈ [−1, 1] and ϕ ∈ [0, 2π]. Similar to the C22 process, Piout depends on the value of the
distribution function evaluated at the outgoing momenta.
Unlike the collision processes already discussed, the momenta of the incoming non-condensate
atoms are restricted for “in” collisions. Conservation of energy and momentum requires that(pc − p2 ) · (pc − p4 ) = mgnc , where 2 and 4 label the incoming atoms. To simplify the expression
for the “in” rate, we once again transform to the center of mass frame, and utilize conservation ofenergy and momentum in an elastic collision and properties of the delta function to obtain
Γin
12 (r, t) =

σnc
πm

Z

dp2
f2
(2π~)3

Z
dp⊥

f4
,
pin
rel

(9.58)

where pinrel = |p2 − pc | and the second integral is a 2D integral over a momentum vector p⊥ thatlies in a plane normal to pin
rel . The constraint mentioned above leads to the requirement that thesecond incoming noncondensate atom, labelled 4, has a momentum
p4 = p⊥ + pc +gnc inp̂rel ,
pinrel

(9.59)

152
inwhere p̂inrel is a unit vector in the direction of prel . The momentum of the outgoing non-condensate

atom is thenp3 = p⊥ + p2 +gnc in
p̂rel
pin
rel

= p4 + p2 − pc ,(9.60)

as required by momentum conservation. Finally, the probability that a test particle in a given
spatial bin undergoes an “in” collision for the time step ∆t is [173]Piin =

σnc,i AppR
f ⊥ ∆t,in 4
πm |p|rel(9.61)

where pR
⊥ is a randomly generated vector that lies within a momentum plane of area Ap . Note that
although the choice of Ap is arbitrary, its exact value has little effect on the total mean collision
rate as long as Ap is large enough to completely sample the region of occupied phase space. In
practice, Ap = πp2max , where pmax is the maximum momentum magnitude found in the test particle
distribution being sampled. This will be discussed in more detail in section 9.2.2.3.
As discussed in section 9.1, the dissipative term in the GGPE is governed by the C12 collision
integral (see Eq. (9.5)) and can be rewritten in terms of the mean collision rates:
R (r, t) =

 out

~
Γ12 (r, t) − Γin
12 (r, t) .
2nc (r, t)(9.62)In a simulation, R (r, t) is represented on the same discrete grid as the condensate such that an
estimate of Rj is required at each time step. The discrete function Rj is constructed from the test
particle distribution in a similar fashion as the function ñj , and is calculated based on the value of
Ri for each test particle:
Ri =


γ~
Piout − Piin ,2nc,i Vb ∆t(9.63)

where γ gives the scaling between test particles and real noncondensate atoms, nc,i is evaluatedat the location of particle i, and Vb is the volume of the spatial bin which contains particle i.
Generation of this quantity at each time step will be discussed in more detail in section 9.2.2.3.

153
9.2.2.3

Implementation of the collision processes

The collision processes are implemented at the end of a time step after the test particle positions andmomenta are updated. In order to sample the collision processes, the local test particle distribution
in each spatial bin is considered independently in succession. The following procedure outlines thesteps for each spatial bin:
(1) A 2D momentum grid is defined with the momentum magnitude p ∈ [0, pmax ] along one
dimension and cos θ ∈ [−1, 1] along the other dimension. Here, pmax is the maximum
momentum magnitude found in the local test particle distribution. With the momentum
grid set, a local estimate of the distribution function at the current time step is found using
the cloud-in-cell scheme to weight each test particle to the momentum grid. Similar tothe scheme used to compute ñ, a 2D volume weighting scheme is used to construct the
distribution function on the (p, cos θ) grid points. Here, pi ≡ |pi | and cos θi ≡ pi,x / |pi | forparticle i. This results in a discrete representation of f (r, p, cos θ) in the current spatialbin.
(2) With a local estimate of f (r, p, cos θ) we proceed by calculating the probabilities for theC12 collision process for the test particles in the local distribution. Therefore, the following
sequence is repeated for each test particle in the current spatial bin:(a) First, the local condensate momentum and density are calculated at the positionof the test particle using linear interpolation. Next, the outgoing momenta for the“out” process are generated by randomly choosing two numbers∗ such that X1 ∈[−1, 1] ≡ cos ϑ and X2 ∈ [0, 2π] ≡ ϕ. The unit vector describing the random scatteringppangle is then given by û (ΩR ) = −X1 , 1 − X12 sin X2 , 1 − X12 cos X2 and theoutgoing momenta p3 and p4 are calculated using Eqs. (9.55) and (9.56), respectively.
Note that the unit vector is defined in such a way as to make the x component∗

Random numbers are denoted by the subscripted variable X, and they are generated using the MT19937
routine. Each time a unique random number is introduced the subscript is incremented. In practice, a generated
random number is never used more than once.

154align with the radial position of the particle. Given the outgoing momenta, values off3ΩR and f4ΩR are found at the momentum coordinates (|p3 | , cos θ3 = p3,x / |p3 |) and(|p4 | , cos θ4 = p4,x / |p4 |) by interpolating from the function f (r, p, cos θ) using bilinear
interpolation. Finally, the probability Piout is calculated for the current test particle
using Eq. (9.57).
(b) The next step is to generate the incoming momentum p4 for the “in” process, which
depends on the value of a randomly generated momentum vector pR⊥ that lies in a planeorthogonal to p̂inrel . This vector is constructed by randomly choosing two numbers suchthat X3 ∈ 0, p2max and X4 ∈ [0, 2π] ≡ ϕ. Note that the range of X3 is chosen to
ensure that p⊥ samples the full area Ap defined in Eq. (9.61); otherwise, the “in”
collision rate will not be properly reproduced. Thus, the random number X3 defines√
the vector magnitude as pR
X3 , and the components of pR⊥ =⊥ are given byppR⊥,x = − X3 cos X4 sin θ2 ,(9.64)pR⊥,y =p
X3 (cos X4 cos θ2 sin φ + sin X4 cos φ) ,(9.65)pR⊥,z =p
X3 (cos X4 cos θ2 cos φ − sin X4 sin φ) ,(9.66)

where cos θ2 = p̂in
rel,x and sin θ2 =p
1 − cos θ2 2 , andcos φ =

sin φ =

p̂rel,z sin
θ2 , if sin θ2 > 0



1,
otherwisep̂rel,y sin
θ2 , if sin θ2 > 0



0,

(9.67)(9.68)

otherwise

Note that cos θ2 is chosen to lie along the x component of the relative momentum unit
vector to coincide with the routine for updating the test particle positions, where thex component of the particle momentum is chosen to align with the radial position of
the particle. This is correct since the condensate momentum only has a component in

155
the radial direction due to spherical symmetry, and therefore the relative momentum
unit vector must point in the radial direction. Given pR⊥ the incoming momentum
pR

p4 is calculated using Eq. (9.59) and the value of f4 ⊥ is found at the momentum
coordinate (|p4 | , cos θ4 = p4,x / |p4 |) by interpolating from the function f (r, p, cos θ)using bilinear interpolation. Finally, the probability Piin is calculated for the current
test particle using Eq. (9.61).
(c) Given Piout and Piin for the current test particle, the dissipative term of the GGPE
is calculated using Eq. (9.63). This is done in the same way as described in section 9.2.1.1 where the cloud-in-cell scheme with volume weighting is used followed bylinear interpolation and smoothing with a Gaussian low-pass filter. This procedure
results in an estimate of the discrete function Rj for the current time step.
(d) Finally, the type of collision that the current test particle undergoes, if any, is determined by comparing the calculated probabilities to a random number X5 ∈ [0, 1]. Oneof three outcomes is possible:(i) If X5 < Piout then an “out” collision occurs. The current test particle’s momentum is updated to p3 (Eq. (9.55)) and a new test particle is created at the same
position as the current test particle, but with momentum p4 (Eq. (9.56)). The
current test particle is flagged as having undergone a collision, and the new test
particle data is temporarily stored such that it can be added to the distribution
after all current test particles have been considered for collisions. The routine
then advances to the next test particle.
(ii) If Piout < X5 < Piout +Piin then an “in” collision occurs. A test particle in the current spatial bin with momentum p4 (Eq. (9.59)) that has not already undergone
a collision must be located and eliminated from the test particle distribution. If
no test particles matching this criteria exist in the momentum bin that contains
p4 then no collision occurs and the routine advances to the next test particle.

156
Otherwise, the current test particle’s momentum is updated to p3 (Eq. (9.60))
based on the momentum of the actual test particle found (not the ideal value ofp4 calculated with Eq. (9.59), which would not conserve momentum). Then, thecurrent test particle is flagged as having undergone a collision, and the test particle to be eliminated is temporarily stored such that it can be eliminated fromthe distribution after all current test particles have been considered for collisions.
(iii) If Piout + Piin < X5 then no collision occurs and the routine advances to the next
test particle.
(3) Once each test particle in the local distribution has been considered for a C12 collision,a list of randomly paired test particles is generated to calculate the probability for theC22 collision process. Therefore, the following sequence is repeated for each pair of test
particles in the current spatial bin:(a) First, the local noncondensate density is estimated as ñij ≈ γNtp,k /Vb , where γ gives
the scaling between test particles and real noncondensate atoms, Ntp,k is the total
number of test particles in the current spatial bin, and Vb is the bin volume. Next,
the outgoing momenta are generated by randomly choosing two numbers such that
X6 ∈ [−1, 1] ≡ cos ϑ and X7 ∈ [0, 2π] ≡ ϕ. The unit vector describing the randomppscattering angle is then given by û (ΩR ) = −X6 , 1 − X62 sin X7 , 1 − X62 cos X7
and the outgoing momenta p3 and p4 are calculated using Eqs. (9.48) and (9.49),
respectively. Given the outgoing momenta, values of f3ΩR and f4ΩR are found at the
momentum coordinates (|p3 | , cos θ3 = p3,x / |p3 |) and (|p4 | , cos θ4 = p4,x / |p4 |) by interpolating from the function f (r, p, cos θ) using bilinear interpolation. Finally, theprobability Pij22 is calculated for the current test particle pair using Eq. (9.51).
(b) It is determined if the current test particle pair undergoes a collision by comparing
the calculated probability to a random number X8 ∈ [0, 1]. One of two outcomes ispossible:157(i) If X8 < Pij22 then a collision occurs. The current test particle pair’s momenta
are updated to p3 (Eq. (9.48)) and p4 (Eq. (9.49)). The routine then advances
to the next test particle pair.
(ii) Pij22 < X8 then no collision occurs and the routine advances to the next test
particle pair.
After each spatial bin has been considered the test particle distribution must be updated basedon the temporarily stored data of particles to be added and eliminated based on the C12 collision
process. This procedure changes the total number of test particles in the simulation, and therefore
the total number of real noncondensate atoms because γ remains fixed for a given simulation.
However, the total number of atoms in the simulation is conserved because the normalization of
the condensate wavefunction changes due to R (r, t), which is also a result of the C12 collision
process. In practice, the total atom number is conserved only to within statistical fluctuations√∝ N because the noncondensate atom number changes in discrete jumps while the condensate
atom number changes continuously.
We have now outlined the numerical procedures used to propagate both the GGPE and QBE
forward in time. The next section provides a detailed description of how to initialize and perform
a full dynamical simulation that evolves a computed equilibrium state forward in time. Simulation
results are then given that directly compare the presented numerical implementation of the ZNG
equations with previous work in the field.

9.3

Full Dynamical Evolution of the ZNG Equations

A full dynamical simulation begins by computing an equilibrium solution to the ZNG equations
given a temperature and total atom number. Given an equilibrium state, the test particle distribution is generated by randomly choosing positions and momenta for each test particle that
accurately reproduce the equilibrium phase-space distribution function. This equilibrium state isthen propagated forward in time using the routines developed in sections 9.1 and 9.2. As an exam-

158
ple, results of a quench simulation are presented that demonstrate growth of the condensate, decayof the noncondensate, and conservation of total atom number. These results are directly compared
to previous work in the field and provide a validation for the numerical implementation of the ZNG
equations described in this dissertation.

9.3.1

Calculation of equilibrium solutions to the ZNG equations

As discussed in section 8.3.2.1, an equilibrium state of the system can be found by solving Eqs. (8.42)
and (8.44) self-consistently subject to the conditions that µ̃0 = µc,0 and the temperature T0 andtotal number of atom in the gas N remain fixed. Therefore, the starting point for any simulation
is to set T0 and N to the desired values, set the isotropic harmonic trap frequency ω to the desired
value, set ñ0 (r) = 0, and then make an educated guess of the functional form of the condensate
wavefunction. A standard approach is to neglect the kinetic energy term in Eq. (8.44) and use theso-called Thomas-Fermi solution for the condensate wavefunction [166]. However, this approach
can lead to numerical instabilities due to the discontinuity present at the edge of the condensate.
Therefore, it is preferable to use the non-interacting ground state wavefunction with Gaussian
widths broadened by the effect of interatomic interactions.†

The condensate wavefunction ϕ0 (r)is then normalized to the number of atoms in the condensate Nc,0 , which is initially estimated using
the critical temperature and ideal gas relations of Eqs. (8.3) and (8.4), respectively. Finally, an
initial estimate of the chemical potential is obtained by integrating Eq. (8.44) over all space:
4πµc,0 ≈
Nc,0

Z ∞
drr
0

2




1
2 2
mω r + gnc,0 (r) |ϕ0 (r)|22(9.69)

Note that the kinetic energy portion of the Hamiltonian has been neglected to coincide with themethod used to obtain a variational estimate of the condensate wavefunction.
With an initial estimate for the ground state of the time-independent GGPE, the numericalroutine for calculating an equilibrium state of the system proceeds as follows:(1) Calculate the effective potential for the noncondensate using Ũ0 (r) = 21 mω 2 r2 +2gnc,0 (r)+†See Ref. [166], section 6.2.1 for a variational estimate of this wavefunction159
2gñ0 (r).
(2) Calculate the noncondensate density ñ0 (r) using Eq. (8.42), and calculate the number ofatoms in the noncondensate Ñ0 by integrating over all space.
(3) Calculate the number of atoms in the condensate using Nc,0 = N − Ñ0 , and renormalize
the condensate wavefunction to this value.
(4) Calculate the effective potential for the condensate using Uc,0 (r) = 12 mω 2 r2 + gnc,0 (r) +2gñ0 (r)(5) Propagate the GGPE in imaginary time using the Crank-Nicolson method (see section 9.1.2),
renormalize ϕ0 (r) to Nc,0 , and calculate a new value for the chemical potential.
(6) Compare the updated chemical potential and noncondensate atom number to the previous
values and proceed as follows:(a) If the change in µc,0 and Ñ0 is less than some predefined tolerance then stop thecalculation because the equilibrium state has been found.
(b) Otherwise, return to step (1).
There are a few subtleties with the above method that must be addressed. First, if the chemical
potential exceeds the minimum value of Ũ0 (r) then it must be artificially set equal to this minimum
value. Otherwise, the fugacity z0 (r) > 1 and the polylogarithm function becomes divergent. Thisanomaly is typically encountered when temperatures close to Tc are studied, and currently limits theapplication of the above procedure. Second, the numerical evaluation of the polylogarithm function
with double precision is not straightforward. To address this issue, a Chebyshev expansion method
introduced by MacLeod [184] is employed. Finally, it is useful to explain the reason for propagating
the GGPE in imaginary time. Let the condensate wavefunction be expanded in an orthonormal
basis of eigenstates χn,0 (r):ϕ0 (r) =∞X
n=0

cn e−iEn t/h χn,0 (r) ,(9.70)

160
where it is assumed that E0 corresponds to the ground state and En < En+1 for all n. If thesubstitution t → −iτ is made then clearly the above expansion becomes a summation over decaying
exponential terms. Therefore, if an initial estimate for ϕ0 (r) is propagated over many imaginary
time steps, and renormalized at every step, eventually only the ground state contribution to theabove summation will remain. This is a result of the fact that the decay time for each state
decreases as n increases.
An example equilibrium solution for the characteristic problem described at the beginningof this chapter is shown in Fig. 9.2. The computed state contains Nc,0 = 26, 324 atoms in thecondensate and Ñ0 = 23, 676 atoms in the noncondensate, which is significantly different than theinitial estimate from the ideal gas relation. This difference arises due to the mean-field repulsion
terms present in the expressions for the effective potentials felt by the condensate and noncondensate, the effect of which is particularly evident in the density distributions of each component
depicted in Fig. 9.2(a). The noncondensate experiences a repulsive force at r = 0 due to the large
density of the condensate at this location, and ñ0 (r) obtains a maximum value near the edge of thecondensate where nc,0 (r) approaches zero. The condensate, on the other hand, experiences only aweak repulsive force near its edge where the two components have a similar density, resulting in aslight narrowing of the condensate density distribution. These effects are clear when examining theeffective potentials plotted in Fig. 9.2(b), where Ũ0 (r) is peaked at r = 0 while Uc,0 (r) is effectively
flat in this region, and Ũ0 (r) is minimal at the edge of the condensate.

9.3.2

Initialization of the test particle distribution

With an equilibrium state computed it is necessary to find initial positions and momenta for the test
particle distribution such that the test particles accurately represent the equilibrium distributionfunction of the noncondensate. Furthermore, the positions and momenta of the test particles
must be chosen at random such that they sample all of the available phase space. A simple,
generally applicable algorithm that generates random deviates which follow a given distribution isthe acceptance-rejection method [175], and it is used here to initialize the test particle distribution.

161

160

(a)

nc;0 (r)n~ 0 (r)nc;0 (r) + n~ 0 (r)140n(r) (a!3
ho )

120
100
80
60
40
20
0
0

2

4

6

8

10

12

r (aho )150(b)Uc;0 (r)~0 (r)
U
125

U (r) (Eho )

100

75

50

7c;0
25

0
0

2

4

6

8

10

12

r (aho )Figure 9.2: (a) Equilibrium density distributions for the condensate (blue), non-condensate (red),and total system (black). (b) Equilibrium effective potentials felt by the condensate (blue) andnon-condensate (red). The black dashed line indicates the equilibrium chemical potential, which lies
below the minimum value of the noncondensate effective potential. The temperature is T0 = 200 nK,
the total atom number is N = 50, 000, the trap frequency is ω = 2π × 187 rad/s, and 87 Rb atoms
are assumed with a scattering length as = 5.29 nm.

162
The algorithm employed here first finds a suitable position for a test particle based on theequilibrium noncondensate density ñ0 (r). Once a position has been determined the equilibrium
fugacity is calculated and a suitable momentum magnitude for the test particle is found based on
the equilibrium Bose-Einstein distribution. This process is repeated until the desired number of test
particles have been initialized. The following procedure outlines the steps for each test particle:(1) Generate two random numbers such that X9 ∈ [0, rmax ] and X10 ∈ 0, r2 ñ0 (r) max , where
the “max” subscript indicates the maximum value of the function r2 ñ0 (r).
(2) Compare X10 to X92 ñ0 (X9 ):(a) If X10 < X92 ñ0 (X9 ) then assign ri = X9 as the position of the test particle andproceed to find it’s momentum(b) Otherwise, return to step (1).
(3) Calculate the equilibrium fugacity at the position of the test particle z0 (ri ).
(4) Generate two random numbers such that X11 ∈ [0, pmax ] and X12 ∈ 0, p2 f0 (ri , p) max ,where the “max” subscript indicates the maximum value of the function
p2 f0 (ri , p) =p2 z0 (ri ).
eβ0 p2 /2m − z0 (ri )(9.71)

2 f (r , X ):(5) Compare X12 to X110 i
112 f (r , X ) then assign |p | = X(a) If X12 < X110 i
11i
11 as the momentum magnitude of thetest particle and proceed to find the momentum components.
(b) Otherwise, return to step (4).
(6) Generate two random numbers such that X13 ∈ [−1, 1] ≡ cos ϑ and X14 ∈ [0, 2π] ≡ ϕ and163
calculate the momentum components as
pi,x = −X11 X13 .
q
2 sin X ,pi,y = X11 1 − X1314q
2 cos X .
pi,z = X11 1 − X13
14(9.72)(9.73)(9.74)

Note that in the above procedure, values of functions (e.g., ñ0 (r)) at arbitrary r and p are foundusing standard 1D linear interpolation. Once each test particle has been assigned a position andmomentum it is necessary to subtract the average momentum of the entire test particle distribution
along each axis from each individual test particle’s momentum components. This ensures that
there is no net momentum of the distribution function at equilibrium, and is necessary because ofnumerical errors introduced by the acceptance-rejection method.

9.3.3

Procedure for a full dynamical simulation

Finally, we have all of the necessary components to perform a full dynamical simulation of the ZNG
equations in spherical symmetry. The following procedure outlines the necessary steps to evolve
the ZNG equations over a single time step ∆t:(1) Set T0 , N , and ω, and calculate the equilibrium state (see section 9.3.1).
(2) Initialize the test particle distribution (see section 9.3.2).
(3) Update the test particle positions by ∆t/2 (see section 9.2.1).
(4) Construct ñnj (see section 9.2.1.1) and Ṽ = 3Ũjn − Ũjn−1 /2 where1Ũjn = mω 2 rj2 + 2gnnc,j + 2gñnj .
2(5) Update the test particle momenta by ∆t (see section 9.2.1).
(6) Update the test particle positions by ∆t/2 (see section 9.2.1).
(7) Check for C12 collisions and construct Rjn (see section 9.2.2.3).

(9.75)

164(8) Check for C22 collisions (see section 9.2.2.3).
(9) Update the number of test particles in the simulation (see section 9.2.2.3).
n − U n−1 /2 where(10) Construct Ṽc = 3Uc,j
c,j
1
n
Uc,j
= mω 2 rj2 + gnnc,j + 2gñnj − iRjn .
2(9.76)(11) Update the condensate wavefunction by ∆t (see section 9.1.2).
(12) Return to step (3).
An example of this procedure applied to the characteristic problem described at the beginning of
this chapter is shown in Fig. 9.3. This simulation represents the propagation of an equilibrium state
forward in time, which should remain unchanged. We observe no readjustment of the number ofatoms in the condensate and noncondensate, which indicates the equilibrium state is accurate andthe update procedure outlined above is functioning correctly. Note that due to the difference in how
the number of condensate and noncondensate atoms change in a dynamical simulation there aresome noticeable fluctuations in these quantities. Nonetheless, the total atom number is conserved√
to within N as depicted by the shaded area in Fig. 9.3. The next section will investigate thenon-equilibrium performance of the spherically symmetric ZNG code by comparing results of aquench simulation to previous work.

9.3.4

Results of a quench simulation

In order to better characterize the spherically symmetric ZNG code, it is important to performa non-equilibrium simulation to ensure the mean field interaction between the condensate andnoncondensate, in addition to the collision processes, drive the system towards equilibrium. A
simple non-equilibrium test case is realized by taking an equilibrium state and suddenly reducing
the temperature by a factor of two. This method has been used previously to test the original ZNG
code developed by Jackson and Zaremba [156], as well as a parallelized ZNG code developed by

165

2.65
Nc
~
N
N=2

Atom number (x10 4 )

2.6

2.55

2.5

2.45

2.4

2.35
0

2

4

6

8

10

Time (ms)Figure 9.3: Condensate (blue), noncondensate (red), and total (black) atom numbers as a functionof time for a full dynamical simulation of an equilibrium state. The gray√ shaded area represents
the expected region of total atom number conservation given by N/2 ± N /2. The temperature
is T0 = 200 nK, the total atom number is N = 50, 000, the trap frequency is ω = 2π × 187 rad/s,
and 87 Rb atoms are assumed with a scattering length as = 5.29 nm. The simulation uses 200, 000
test particles, 8000 bins in position space, 20 bins for the momentum magnitude, 10 bins for the
angular variable, and a time step of 10−3 . Parameters were found empirically.

166
Märkle [185]. By directly comparing the results of this quench simulation to previous work, we canensure the spherically symmetric ZNG code is consistent with codes developed by others.
The quench simulation begins with the equilibrium state of the characteristic problem described at the beginning of this chapter (see section 9.3.1). Following the procedure outlined in
section 9.3.2, the quench is performed by initializing the test particle distribution to a distributionfunction with the substitution β0 → 2β0 :p2 f0 (ri , p) =e

p2 z0 (ri ).
− z0 (ri )2β0 p2 /2m(9.77)

This results in a test particle distribution with half the temperature of the equilibrium state; therefore, the test particles momenta are too low to remain in equilibrium and substantial particle
transfer from the noncondensate to the condensate should occur. Furthermore, the noncondensate
will collapse towards the center of the trap before being repelled by the large mean field potentialof the condensate. This will lead to small oscillations of both the condensate and noncondensate,
the effect of which is evident in the evolution of the number of atoms in each component of thegas. This behavior is depicted in Fig. 9.4, which shows the growth of the condensate, decay of
the noncondensate, and conservation of total atom number as the system evolves forward in time.
Notably, the results of the spherically symmetric ZNG code exhibit slightly more atoms transferredfrom the noncondensate to the condensate; however, the total particle number experiences less fluctuations and appears to be slightly more stable. The small, periodic oscillations in the condensate
and noncondensate atom numbers show good agreement between all three codes demonstratingthat the mean field potentials are accurately reproduced. Given the consistency of the results for aquench simulation, we conclude that the developed spherically symmetric ZNG code is consistent
with previous work.

167
3.2

3

Atom number (x10 4 )

(a)

Nc
~
N
N=2

2.8

2.6

2.4

2.2

2

1.8
0

5

10

15

20

25

30

Time (ms)
2.54

(b)

JacksonMB
aerkleThis work

Atom number (x10 4 )

2.52

2.5

2.48

2.46
0

5

10

15

20

25

30

Time (ms)Figure 9.4: (a) Time evolution of the condensate (blue), noncondensate (red), and total atom number (black) after a sudden quench of the system temperature. (b) Comparison of total atom number
conservation for ZNG codes developed by Jackson and Zaremba [156] (dotted) and Märkle [185](dashed), and the spherically symmetric code developed here (solid). The gray√ shaded area represents the expected region of total atom number conservation given by N/2 ± N /2. The temperature of the initial state (before the quench) is T0 = 200 nK, the total atom number is N = 50, 000,
the trap frequency is ω = 2π × 187 rad/s, and 87 Rb atoms are assumed with a scattering lengthas = 5.29 nm. The simulation uses 200, 000 test particles, 8000 bins in position space, 20 bins forthe momentum magnitude, 10 bins for the angular variable, and a time step of 10−3 . Parameters
were found empirically.

Chapter 10

Collapse and revival of the monopole mode of a degenerate Bose gas in an
isotropic harmonic trap∗

In this chapter, we present experimental measurements and analysis of the monopole mode of a finite
temperature BEC confined in an isotropic harmonic trap. We observe a collapse and partial revivalof the condensate oscillation, and compare these results to the predictions of finite temperature
BEC models. We set the scene in Sec. 10.1 by providing an overview of the limiting cases forthe collective modes of a BEC and ideal gas in an isotropic harmonic trap. This is followed by atheoretical analysis of the spectrum of coupled modes of the condensate and noncondensate at finite
temperatures in Sec. 10.1.2. This provides a framework for understanding the collapse and revival
time scales observed in the experiment. In Sec. 10.2 we provide a description of the experimental
procedure and the main results of this chapter. In Sec. 10.3 we investigate the damping observed
experimentally through numerical simulations within the ZNG formalism. After discussing theresults of the numerical simulations, we summarize in Sec. 10.4.

10.1

Collisionless Dynamics

Here, we provide an overview of the collisionless dynamics of a trapped Bose gas, beginning witha discussion of the limiting cases for collective modes in an isotropic trap. We then discuss themonopole mode of a finite temperature BEC through application of the semiclassical collisionless model from Ref. [150, 151] to a spherically symmetric trapping geometry, and show how the∗A version of this chapter has been published in Ref. [139].

169
monopole mode response can be cast in terms of two eigenmodes of the system. This analysis provides a framework for understanding the collapse and revival behavior of the condensate oscillation
observed in the experimental results.

10.1.1

Collective modes in an isotropic harmonic trap

In an isotropic harmonic trap, the collective modes of a Bose gas are well understood in two limits.
In the Thomas-Fermi (TF) limit at zero temperature, the ratio of the kinetic to interaction energy
is small when the number of atoms in the BEC is large; thus, the kinetic energy can be neglected.
The collective mode frequencies of a BEC in a three-dimensional isotropic harmonic trap in thislimit can be estimated using a hydrodynamic approach [145]. The mode frequencies depend on theprinciple quantum number n and angular quantum number l according toω 2 = ω02 l + 3n + 2nl + 2n2 ,(10.1)

where ω0 is the harmonic trap frequency and ω is the frequency of the collective mode. For the
spherically symmetric monopole, or breathing mode (n = 1, l = 0), the mean-square radius of√
the condensate oscillates at ω = 5ω0 , and the motion is undamped. Above the BEC criticaltemperature, Tc , mean-field effects can be neglected and the gas can be described by a classicalBoltzmann equation. In this case, the mode oscillates at ω = 2ω0 in both the collisionless andhydrodynamic regimes [186], and the motion is undamped. In the collisionless regime individual
atoms may undergo many oscillations before experiencing a collision while the hydrodynamic regime
implies the gas is in local statistical equilibrium.

10.1.2

Coupled-modes analysis†

To obtain insight into the behavior of the monopole mode at finite temperature in an isotropic trap,
we apply a model previously developed by Bijlsma and Stoof [150]. This methodology introduces
a dynamical scaling ansatz for the condensate and noncondensate that successfully reproduces the†
The analysis presented in this section was done by V. E. Colussi under the supervision of M. J. Holland. The
author of this dissertation made use of the results as part of the work published in Ref. [139].

170
limiting cases of the monopole mode behavior described in the previous section. The condensate
and noncondensate are described by a Gross-Pitaevskii equation (GPE) and a collisionless quantum
Boltzmann equation (QBE), respectively, which are coupled by their mean-field interaction. The
analysis here assumes a small amplitude perturbation of the system, and a linear response suchthat the effects of damping are absent.
In the following calculation, a scaling ansatz is made for the time evolution of the condensate
density,nc (r, t) =1 0r
n,
λ3 c λ(10.2)and the Wigner distribution function of the noncondensate,1f (r, p, t) = 6 f 0
ᾱ





r α
mα̇
,
p−
,
αᾱ ᾱ
α(10.3)

which are written in terms of a Gaussian density profile n0c for the condensate and a saturated
Bose-Einstein distribution f 0 for the noncondensate: mω 1/2 mω0 2
0e− ~ r ,n0c (r) = Nc
π~


~ω0
0
f (r, p) = ÑkB T ζ(3)

"  2
#−1
×

e

p
+ 12 mω02 r2
2m

/kB T

−1

.

(10.4)The number of atoms in the condensate and noncondensate are denoted by Nc and Ñ , respectively,and ζ(s) is the Riemann zeta function. The scaling parameters, λ(t) and α(t), capture the oscillationof the widths of the two components, and the bar denotes the equilibrium value. Inserting the scaling
ansatz into the GPE and QBE results in a set of coupled equations of motion for the condensate
and noncondensate characteristic widths:ü + ω02 u = v(u),where the vector u contains the scaling parameters


 λ 
u=
,
α

(10.5)(10.6)

171
and v(u) is a nonlinear vector function describing the spreading of the cloud due to kinetic energy
and the effects of nonlinear interactions (see Ref. [150] for details).
In the limit of a small amplitude oscillation, the total density of the system can be writtenasn(r, t) = n̄(r) + δn(r)eiωt ,(10.7)

where the perturbation is generated by modulating the trap frequency with amplitude :ω0 (t) = 1 + eiωt ω0 .

(10.8)In this limit Eq. (10.5) can be linearized:− ω 2 δu + ω02 δu = [∇u v] ū · δu − 2ω02 ū,(10.9)and the eigenfrequencies ωn and eigenmodes u(n) of the homogeneous part of Eq. (10.9) can beextracted with a solution of the form
δu = 2ω02

X u(n) · ūn

ω 2 − ωn2u(n) .

(10.10)From this solution, we find two eigenmodes that we refer to as the in-phase and out-of-phase modesof the system. The in-phase mode corresponds to the condensate and noncondensate monopolemodes oscillating together with a phase difference of φ = 0, and the out-of-phase mode corresponds
to a phase difference of φ = π. In Fig. 10.1, the frequencies of the in-phase and out-of-phase modes
as a function of temperature are shown.
Given the eigenmodes of the system, the time-averaged work done by a perturbation of thetrap frequency can be used to characterize the response of the system:
W =

X
n

bn
,
2
ω − ωn2(10.11)

where the bn are a measure of the magnitude that each eigenmode responds with when the system
is perturbed. Figure 10.2 shows the bn as a function of temperature for the two modes discussed
above. For temperatures T > 0.2 Tc both modes of the system will be excited by a perturbation of

172
2.3
Thomas-Fermi

!=!0

2.2

?=0

2.1

?=:

Ideal gas
2

1.9
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T =Tc

Figure 10.1: Oscillation frequency of the in-phase (φ = 0) and out-of-phase (φ = π) modes
as a function of temperature from the coupled-modes√ analysis (solid lines) and ZNG simulations(crosses). The dashed lines represent the TF limit ( 5ω0 ) and ideal gas limit (2ω0 ) for the monopolemode frequency of the condensate and noncondensate, respectively. Assumptions made regarding
the form of excitations become invalid for T . 0.2 Tc . Reproduced from Ref. [139].
30
?=0
?=:

Mode Response (arb.)

25

20

15

10

5

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T =Tc

Figure 10.2: Magnitude of the response of the in-phase (φ = 0) and out-of-phase (φ = π) modes toa trap frequency perturbation as a function of temperature. The amplitude of the trap frequency
modulation is  = 0.01. Reproduced from Ref. [139].

173
the trap frequency.
The coupled-modes analysis suggests that the oscillation of a single component of the gas(e.g., condensate) is described by a superposition of two eigenmodes oscillating at slightly different
frequencies. Therefore, we expect measurements of the condensate width as a function of time fortemperatures T > 0.2 Tc to beat at a frequency corresponding to the frequency difference between
the two eigenmodes, ∆ω/ω0 ∼ 0.2–0.25 (see Fig. 10.1). Given this result, we present experimental
observations of the monopole mode in an isotropic trap in the next section.

10.2

Experiment‡

The experimental system is a Bose gas of 87 Rb atoms cooled to quantum degeneracy via forced
radio-frequency evaporation in a time-averaged, orbiting potential (TOP) trap [187]. A standard√
TOP trap configuration results in an oblate harmonic trap with an aspect ratio of ωz /ωr = 8,
where ωz (ωr ) is the axial (radial) trapping frequency. Here, the overall harmonic confinement of
the trap is reduced and the trap minimum is allowed to sag under the force of gravity. This causes
the curvature of the magnetic field along the z axis to decrease, which effectively decreases theratio ωz /ωr . The end result is an isotropic harmonic trap with ω0 ≡ ωr = ωz = 2π × (9.03(2) Hz)
with a residual asphericity of less than 0.2%. This system was used in previous work to study themonopole mode of a Bose gas above the BEC critical temperature [163], and a detailed descriptionof the apparatus can be found in Ref. [165].
We excite the monopole mode of the system below the BEC critical temperature in the range
of approximately 0.75–0.9 Tc . The experimental procedure parallels that of Ref. [163]—beginningfrom a system at equilibrium, the trap frequency is sinusoidally modulated at a driving frequencyωD ≈ 2π × 18–19 Hz for four periods with an amplitude  ≈ 0.1:
ω(t) = [1 +  sin (ωD t)] ω0 .

(10.12)After driving, we find that the peak TF radius of the condensate is 10–15% larger than the equi‡

The experiment was conducted by D. S. Lobser under the supervision of H. J. Lewandowski and E. A. Cornell.
The author of this dissertation made use of the experimental data as part of the work published in Ref. [139].

174
librium value for all of the experimental data sets. The system is then allowed to freely evolve inthe static isotropic trap for a time t before six nondestructive phase-contrast images record theintegrated column density of the cloud at intervals of 10 ms or 17 ms, sampling between 1 and 1.5
oscillation periods of the monopole mode. This experimental procedure is repeated between 2–4
times for each t, and for times up to t ≈ 1.5 s.
Each phase-contrast image is analyzed using a 2D bimodal fit to the atomic column density.
The fitting function is the sum of a Gaussian and integrated TF function [188]:
" 

 #

x − xc 2z − zc 2
ncol (x, z) = AG exp −
−
σG,x
σG,z
"


 #3/2z − zc 2x − xc 2−+ AT F 1 −σT F,xσT F,z+ Ccol ,(10.13)where AG and AT F are the amplitudes of the Gaussian and TF functions, respectively, xc and zc
are the center points of the cloud, σG,i are the Gaussian widths, σT F,i are the TF widths, and Ccol
is a constant offset. Note that the TF function is defined to be zero if the argument in brackets isnegative.
The dynamics of the condensate monopole mode are captured by the spherically symmetricquantity2
σM= σT2 F,x + σT2 F,y + σT2 F,z /3.

(10.14)

During the data runs for this experiment, images were consistently taken in the xz plane. In
earlier measurements described in Ref. [163], data were also taken along the xy plane, but technicaldifficulties were encountered with the imaging system along this axis during the course of theexperiments discussed here. However, the limited data available from the xy plane suggests that thecloud was highly symmetric [165]. Therefore, we set σT F,y = σT F,z in Eq. (10.14) when calculating
the amplitude of the condensate monopole mode. Although we observe excitation of other collectivemodes (dipole and quadrupole), we find that the key features of the experimental results for themonopole mode are independent of whether this assumption is made or σT F,y is simply excluded175
from Eq. (10.14).
We determine the instantaneous amplitude of the condensate monopole mode by fitting afixed frequency sine wave to each set of six consecutive time points. The fitting function is of theformgσ (t) = Aσ cos (2πνt) + Bσ sin (2πνt) + Cσ ,(10.15)where ν = 19 Hz, and Aσ , Bσ , and Cσ are fit parameters. This functional form is chosen because weare concerned with the amplitude of the mode, not the frequency, which enables a straightforward
linear regression analysis for computing Aσ , Bσ , and Cσ . Finally, we present the data in the formof a fractional amplitude given by
AM =A2σ + Bσ2,
Cσ2

(10.16)

where Aσ , Bσ , and Cσ correspond to the fit parameters of Eq. (10.15). The results of this analysis
are shown in Fig. 10.3, where time t = 0 is defined as the point at which the modulation of thetrap frequency ceases.
A central feature of the data is that the amplitude of the condensate monopole mode does
not decay exponentially. Across the five data sets, there is a consistent collapse in the amplitude of
the monopole mode between t = 0.3–0.5 s, and a partial revival around t = 0.7–0.9 s. Motivated by
the results of the coupled-modes analysis, we fit the data to an envelope function that represents
the superposition of two sinusoidal modes, and we include an overall exponential damping factor
to represent the loss of amplitude with time:ge (t) = Ae cos∆ω 2 −Γe tt e,2(10.17)

where Ae is the initial amplitude, ∆ω is the beat frequency, and Γe is the damping rate of theenvelope. The results of this fit are overlaid with the data in Fig. 10.3. From the fit, we find∆ω/ω0 ∼ 0.13 on average, and damping rates in the range Γe ∼ 1.5–3.5 s−1 . The observed beatfrequency is less than ∆ω/ω0 ∼ 0.2 as expected from the coupled-modes analysis. We attribute
this disagreement to the naive form of the fitting function, which assumes that the two sinusoidal
modes damp at the same rate, respond equally to the trap frequency perturbation, and have no176

Figure 10.3: Amplitude of the monopole mode oscillation (squares) for atom numbers of (a)N = 8.9 × 105 , (b) N = 9.7 × 105 , (c) N = 6.7 × 105 , (d) N = 5.4 × 105 , and (e) N = 7.9 × 105 .
Each frame is labeled with the condensate fraction (Nc /N ) and temperature (T /Tc ). Error bars
represent the statistical uncertainty of multiple realizations of the experiment at each time point.
The dashed lines are a fit of the data to Eq. (10.17), which represents the envelope function for thesuperposition of two sinusoids. From the fit it is found that the beat frequency is ∆ω/ω0 ∼ 0.13
on average and the damping rate lies in the range Γe ∼ 1.5–3.5 s−1 . Reproduced from Ref. [139].

177
phase difference between them. These assumptions are investigated further in the next section
where the two sinusoidal modes are identified with the in-phase and out-of-phase modes predictedby the coupled-modes analysis.
Before moving on, it is important to note that in Ref. [164] it was shown that anharmoniccorrections to the trap geometry were likely responsible for the anomalous exponential dampingof the monopole mode observed above the critical temperature [163]. As shown in Ref. [163], thisdamping is < 0.2 s−1 for clouds with a full width at half maximum (FWHM) of < 125 µm. In
this work, the FWHM of the cloud below the critical temperature satisfies this criterion; thus, we
neglect anharmonic corrections to the trap geometry as the observed damping rate is approximately
an order of magnitude larger.

10.3

Collisional Dynamics

We now investigate the damping observed in the experimental data through numerical simulations
within the ZNG formalism. The coupled-modes analysis ignores collisions and exchange of particles
between the two components, as well as nonlinear mean-field effects. However, below the criticaltemperature, these interactions between the condensate and noncondensate can shift the frequenciesof collective modes and cause damping. Collisions that exchange energy and particles between thetwo components of the gas cause collisional damping [171, 173], whereas mean-field effects lead to
Landau damping (see Refs. [189–196] for further discussion) and Beliaev damping [197]. Landau
damping describes a process where a collective mode decays due to its interaction with a thermal
distribution of excitations, and it is expected to dominate at higher temperatures approaching thecritical temperature. On the other hand, Beliaev damping is a process where a collective mode
decays into two lower energy excitations, which is suppressed for the lowest energy collective modesof a trapped gas due to the discretization of energy levels. Thus, the Beliaev process is absent forthe monopole mode, and is therefore excluded from our analysis in this paper.

178
10.3.1

Simulation of the experiment

To model the experiment, we simulate a gas of N = 8 × 105 87 Rb atoms in a spherically symmetricharmonic trap with ω0 = 2π × 9 Hz. Using the algorithm outlined in Ref. [171], we generate
equilibrium initial states of the condensate and noncondensate for temperatures ranging from 0.1–
0.9 Tc .§We then directly simulate the excitation of the monopole mode as in the experiment bysinusoidally modulating the frequency of the trapping potential at ωD = 2ω0 for four periods. We√√
find that our results are essentially unchanged for drive frequencies of (1 + 5/2)ω0 or 5ω0 . We
use trap frequency modulation amplitudes of  = 0.02, 0.03, or 0.04, and then allow the system to
evolve freely for t = 2 s. We find that this range of  excites the monopole mode of the condensate
with an amplitude comparable to that observed in the experiment (i.e. 10–15% peak increase in the
TF radius of the condensate from equilibrium). We note that these values are somewhat less than
the quoted experimental value of  ≈ 0.1, and speculate that this discrepancy is a result of multiple
collective modes being excited in the experiment due to the difficulty of driving the trap perfectly
spherically. This is in contrast to the simulations where only the monopole mode is excited, andtherefore less energy is required to be added to the system to achieve the same level of excitationof the condensate.
We record the mean-square radius of the condensate as a function of time, along with snapshots of the individual density profiles. Although the experimental data sets have total atom
numbers that range between about 6 × 105 and 1 × 106 , we find simulations for 8 × 105 atoms
represent the features of interest, namely the collapse and revival behavior and damping rate. In
order to compare directly with the experimental data, we generate 2D column densities from thesimulation results, and determine the TF radii using the same bimodal fitting routine described inSec. 10.2. Equation (10.14) is used to calculate the amplitude of the condensate monopole mode,and Eq. (10.15) is fit to single periods of the oscillation corresponding to a window of approximately
53 ms. The results of this analysis are overlaid with the experimental data in Fig. 10.4 for the three§
As T → Tc the numerical method for calculating the equilibrium state of the gas becomes unstable, and 0.9 Tc
is an empirical upper bound.

179
different values of . The timescale of the first collapse and revival observed in the simulation
results show good agreement with the experiment.

10.3.2Extraction of damping rates

The prediction of the coupled-modes analysis and results of the ZNG simulations show good agreement with the collapse and revival behavior observed in the experimental data (see Figs. 10.3
and 10.4). In addition, the damping observed in the results of the ZNG simulations agrees wellwith experimental observations. Therefore, due to the limited and noisy experimental data available, we use the results of the ZNG simulations instead of experimental data to get an estimate ofthe damping rates for the in-phase and out-of-phase eigenmodes predicted by the coupled-modes
analysis.
We fit the simulated evolution of the condensate mean-square radius, Rc2 =R

dr r2 nc (r),by the sum of two sine waves with decaying amplitudes
gc (t) = A1 sin (2πν1 t + φ1 )e−Γ1 t+A2 sin (2πν2 t + φ2 )e−Γ2 t + Cc ,(10.18)where Ai , νi , φi , and Γi are the amplitudes, frequencies, phases, and damping rates, respectively,of the two eigenmodes, and Cc is a constant offset. Figure 10.5 shows typical results of this fitting
procedure for simulation results with a trap frequency modulation amplitude of  = 0.03. Timet = 0 is defined as the point at which the modulation of the trap frequency ceases. We choose to fit
to the mean-square radius of the condensate as its time evolution is most sensitive to the presenceof both eigenmodes across the temperature range investigated. The mean-square radius of thetotal density becomes dominated by the noncondensate at higher temperatures, and any signatureof a second eigenmode is lost. Similar behavior is observed in the evolution of the noncondensate
mean-square radius.
The mode frequencies extracted from this fitting procedure show excellent agreement with theresults of the coupled-modes analysis across the temperature range simulated. Simulation results

180

Figure 10.4: (Color online) Amplitude of the monopole mode oscillation for experimental atom
numbers of (a) N = 8.9 × 105 , (b) N = 9.7 × 105 , (c) N = 6.7 × 105 , (d) N = 5.4 × 105 , and(e) N = 7.9 × 105 . Each frame is labeled with the condensate fraction (Nc /N ) and temperature(T /Tc ), and the legend denotes the different modulation amplitudes used in the ZNG simulations.
Error bars on the experimental data represent the statistical uncertainty of multiple realizationsof the experiment at each time point. (f) Condensed fraction vs temperature for the ideal Bosegas, Nc /N = 1 − (T /Tc )3 (solid line), the equilibrium state of the ZNG simulations (blue points),
and the experimental data (red crosses). All simulations are performed with N = 8 × 105 atoms.
Reproduced from Ref. [139].

(μm2)

(μm2)

(μm2)

181

Figure 10.5: (Color online) Simulated mean-square radius of the condensate density for a trap
frequency modulation amplitude of  = 0.03 at (a) T = 0.4 Tc , (b) 0.6 Tc , and (c) 0.8 Tc (black
circles) and resulting fit of Eq. (10.18) (blue line). The density of simulated points has been reduced
for clarity. Reproduced from Ref. [139].

182
at temperatures of T = 0.1 Tc and 0.2 Tc are fit with a single decaying sinusoid due to the absenceof a second mode. This also agrees with the prediction of the coupled-modes analysis, where onlya single mode responds to a trap frequency perturbation for T ≤ 0.2 Tc (see Fig. 10.2). A notablefeature in Fig. 10.5 is a downward shift of the fast oscillations of Rc2 (i.e., carrier frequency)with increasing temperature, an effect also observed in Ref. [173] for the monopole mode. As thetemperature of the system increases the out-of-phase mode begins to get excited in conjunction
with the in-phase mode, and the fast oscillations shift to a lower frequency because they represent
a weighted average of the two independent mode frequencies.
The damping rate of each mode determined from the fitting procedure is shown in Fig. 10.6
as a function of temperature. In the temperature range of the experiment, there is a mismatch of
the damping rates between the two eigenmodes. This mismatch, along with the beating between
the two modes, captures the behavior seen experimentally of strong collapse and subsequent revivalof the condensate oscillation. At lower temperatures, the in-phase mode dominates and the out-ofphase mode is strongly damped, and the inverse is true at higher temperatures. Thus, the in-phasemode appears to be dominated by the condensate while the out-of-phase mode consists primarilyof the noncondensate.
Based on the results of the coupled-modes analysis, one may suspect that the particulardrive frequency used to excite the system has a large effect on the nature of the response due to thepresence of two resonant excitation frequencies. However, results from ZNG simulations showed
little dependence on the drive frequency, which can be attributed to the presence of damping.
Damping in the system effectively broadens the resonances such that both modes are appreciably√
excited when the system is driven in the range 2ω0 to 5ω0 . Thus, the main characteristics of thecondensate collapse-revival behavior are relatively insensitive to the particular drive frequency.

10.4

Summary

In conclusion, we have experimentally observed non-exponential collapse and subsequent revivalof the monopole mode of a finite temperature BEC in an isotropic magnetic trap. A coupled-

183

5
?=0
?=:

Damping Rate (s -1)

4

3

2

1

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T =Tc

Figure 10.6: Damping rate of the in-phase (squares) and out-of-phase (triangles) mode as a resultof fitting Eq. (10.18) to the simulated evolution of Rc2 at each temperature for a trap frequency
modulation amplitude of  = 0.03. The solid lines are guides to the eye. Representative fits areshown in Fig. 10.5. Reproduced from Ref. [139].

184
modes analysis was used to study the linear response of the system to external perturbation, theresults of which are the identification of two eigenmodes of the system corresponding to in-phaseand out-of-phase oscillations of the condensate and non-condensate. These modes appear to becollisionless analogues to the first and second sound modes as previously discussed in Ref. [150].
Simultaneous excitation of these two modes results in the observed collapse and partial revivalof the condensate monopole mode, which has a timescale compatible with the mismatch in the
eigenfrequencies. Damping of the oscillations was also observed experimentally, and simulations
within the ZNG formalism resulted in good agreement with the data.

Chapter 11

Conclusion to Part 2

11.1

Summary

The second part of this dissertation has investigated the dynamics of the monopole mode of adegenerate Bose gas confined in an isotropic harmonic trap. Previous to the work described here,
experiments investigating the collective mode spectrum of degenerate Bose gases were confined toanisotropic trapping geometries, and predictions about the dynamics of collective modes in spherical symmetry could not be proven. Thus, fundamental comparisons between different theoreticalpredictions for mode frequencies and damping rates as a function of temperature had not beenmade.
In Chapter 8, the theoretical background for the ZNG method was outlined. It was emphasized that the ZNG method treats excitations about the condensed state semiclassically in the
Hartree-Fock limit, and completely neglects quantum fluctuations like Bogoliubov quasi-particle excitations. Therefore, the ZNG method is not applicable to the study of many physical phenomena;
however, it is expected to be a valid approach to the problem investigated in this dissertation. As
shown in Chapter 10, the ZNG method was able to accurately reproduce experimental observations
in a regime close to the BEC transition temperature.
In Chapter 9, a detailed discussion of a numerical solution to the ZNG equations in sphericalsymmetry was given. We outlined the solution of the GGPE using the Crank-Nicolson method,
as well as the solution of the QBE using a modified DSMC method that included both mean-fieldinteractions and collisional interactions between the condensate and non-condensate. In addition,

186
explicit details were given on the Monte Carlo sampling method used to incorporate the effects ofcollisions, as well as on how to generate equilibrium states and perform a full dynamical simulationof the ZNG equations. Finally, the developed spherically symmetric ZNG code was compareddirectly to previously codes in the literature, and excellent agreement was found.
In Chapter 10, we demonstrated the experimental observation of non-exponential collapse
and subsequent revival of the monopole mode of a finite temperature BEC in an isotropic harmonic
trap. A coupled-modes analysis was used to study the linear response of the system to external
perturbation, the results of which were the identification of two eigenmodes of the system corresponding to in-phase and out-of-phase oscillations of the condensate and non-condensate. These
modes appear to be collisionless analogues of the first and second sound modes as previously discussed in Ref. [150]. Simultaneous excitation of these two modes resulted in the observed collapse
and partial revival of the condensate monopole mode, which has a timescale compatible with themismatch in the eigenfrequencies. Damping of the oscillations was also observed experimentally,and simulations within the ZNG formalism resulted in good agreement with the data.

11.2

Future Outlook

In moving forward from the work described in Part 2 of this dissertation, it would be interesting toobtain more detailed experimental results that cover a larger range of temperatures below the BEC
transition temperature. This would allow direct analysis of the damping rates and mode frequencies
as a function of temperature, which could then be directly compared to existing theories such as
the ZNG method or the classical field method. Besides additional experimental results, it would beinteresting to investigate modification of the ZNG or classical field methods to include some of themissing physics. In regards to the ZNG method, this would involve going beyond the Hartree-Fockand local density approximations for the treatment of the excitations about the condensed state
and their interactions. In regards to the classical field method, modifications would include a way ofdescribing excitations beyond the Bogoliubov quasi-particle limit; therefore, it would be necessary
to couple this method with a QBE in a similar fashion to the ZNG method.

Bibliography[1] J. C. Védrine, Catalysts 7, 341 (2017).
[2] W. H. Wollaston, Philosophical Transactions of the Royal Society of London 92, 365 (1802).
[3] J. Fraunhofer, Annalen der Physik 56, 264 (1817).
[4] K. G., Annalen der Physik 185, 148 (1860).
[5] K. G. and B. R., Annalen der Physik 186, 161 (1860).
[6] K. G. and B. R., Annalen der Physik 189, 337 (1861).
[7] C. Abney and L.-C. Festing, Philosophical Transactions of the Royal Society of London 172,887 (1881).
[8] B. M. and O. R., Annalen der Physik 389, 457 (1927).
[9] M. D. Harmony, Accounts of Chemical Research 25, 321 (1992).
[10] D. W. Pratt, Annual Review of Physical Chemistry 49, 481 (1998).
[11] K. C. Cossel, E. M. Waxman, I. A. Finneran, G. A. Blake, J. Ye, and N. R. Newbury, Journalof the Optical Society of America B 34, 104 (2017).
[12] S. T. Cundiff and J. Ye, Reviews of Modern Physics 75, 325 (2003).
[13] M. Thorpe and J. Ye, Applied Physics B 91, 397 (2008).
[14] C. R. Petersen, U. Møller, I. Kubat, B. Zhou, S. Dupont, J. Ramsay, T. Benson, S. Sujecki,
N. Abdel-Moneim, Z. Tang, D. Furniss, A. Seddon, and O. Bang, Nature Photonics 8, 830(2014).
[15] I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, T. Paasch-Colberg,
I. Znakovskaya, M. Pescher, W. Schweinberger, V. Pervak, E. Fill, O. Pronin, Z. Wei,
F. Krausz, A. Apolonski, and J. Biegert, Nature Photonics 9, 721 (2015).
[16] J. Faist, G. Villares, G. Scalari, M. Rösch, C. Bonzon, A. Hugi, and M. Beck, Nanophotonics
5, 272 (2016).
[17] J. Ye, L.-S. Ma, and J. L. Hall, Journal of the Optical Society of America B 15, 6 (1998).

188[18] L.-S. Ma, J. Ye, P. Dubé, and J. L. Hall, Journal of the Optical Society of America B 16,2255 (1999).
[19] J. M. Hollas, High Resolution Spectroscopy, 2nd ed. (John Wiley and Sons, New York, 1998).
[20] J. Brown and A. Carrington, Rotational Spectroscopy of Diatomic Molecules (Cambridge
University Press, Cambridge, 2003).
[21] J. R. Heath, A. L. Cooksy, M. H. Gruebele, C. A. Schmuttenmaer, and R. J. Saykally, Science
244, 564 LP (1989).
[22] H. Piest, G. von Helden, and G. Meijer, The Journal of Chemical Physics 110, 2010 (1999).
[23] F. Dong, S. Davis, and D. J. Nesbitt, The Journal of Physical Chemistry A 110, 3059 (2006).
[24] B. E. Brumfield, J. T. Stewart, and B. J. McCall, The Journal of Physical Chemistry Letters
3, 1985 (2012).
[25] G. Scoles, Atomic and Molecular Beam Methods I (Oxford University Press, New York, 1988).
[26] J. T. Brice, T. Liang, P. L. Raston, A. B. McCoy, and G. E. Douberly, The Journal ofChemical Physics 145, 124310 (2016).
[27] H. Pauly, Atom, Molecule, and Cluster Beams I (Springer-Verlag, New York, 2000).
[28] M. A. Duncan, Review of Scientific Instruments 83, 041101 (2012).
[29] N. R. Hutzler, H.-I. Lu, and J. M. Doyle, Chemical Reviews 112, 4803 (2012).
[30] S. E. Maxwell, N. Brahms, R. DeCarvalho, D. R. Glenn, J. S. Helton, S. V. Nguyen, D. Patterson, J. Petricka, D. DeMille, and J. M. Doyle, Physical Review Letters 95, 173201 (2005).
[31] N. R. Hutzler, M. F. Parsons, Y. V. Gurevich, P. W. Hess, E. Petrik, B. Spaun, A. C. Vutha,
D. DeMille, G. Gabrielse, J. M. Doyle, and D. DeMille, Physical Chemistry Chemical Physics
13, 18976 (2011).
[32] J. F. Barry, E. S. Shuman, D. DeMille, P. W. Hess, E. Petrik, B. Spaun, A. C. Vutha,
D. DeMille, G. Gabrielse, J. M. Doyle, and D. DeMille, Physical Chemistry Chemical Physics
13, 18936 (2011).
[33] S. M. Skoff, R. J. Hendricks, C. D. J. Sinclair, J. J. Hudson, D. M. Segal, B. E. Sauer, E. A.
Hinds, and M. R. Tarbutt, Physical Review A 83, 023418 (2011).
[34] W. Bu, T. Chen, G. Lv, and B. Yan, Physical Review A 95, 032701 (2017).
[35] G. Z. Iwata, R. L. McNally, and T. Zelevinsky, Physical Review A 96, 022509 (2017).
[36] L. D. van Buuren, C. Sommer, M. Motsch, S. Pohle, M. Schenk, J. Bayerl, P. W. H. Pinkse,and G. Rempe, Physical Review Letters 102, 033001 (2009).
[37] C. Sommer, L. D. van Buuren, M. Motsch, S. Pohle, J. Bayerl, P. W. H. Pinkse, G. Rempe,
J. Petricka, D. DeMille, and J. M. Doyle, Faraday Discussions 142, 203 (2009).

189[38] L. Santamaria, V. Di Sarno, I. Ricciardi, M. De Rosa, S. Mosca, G. Santambrogio, P. Maddaloni, and P. De Natale, The Astrophysical Journal 801, 50 (2015).
[39] X. Li, L. Xu, Y. Yin, S. Xu, Y. Xia, and J. Yin, Physical Review A 93, 063407 (2016).
[40] B. Spaun, P. B. Changala, D. Patterson, B. J. Bjork, O. H. Heckl, J. M. Doyle, and J. Ye,
Nature 533, 517 (2016).
[41] S. K. Tokunaga, R. J. Hendricks, M. R. Tarbutt, and B. Darquié, New Journal of Physics
19, 053006 (2017).
[42] P. Asselin, Y. Berger, T. R. Huet, L. Margulès, R. Motiyenko, R. J. Hendricks, M. R. Tarbutt,
S. K. Tokunaga, and B. Darquié, Physical Chemistry Chemical Physics 19, 4576 (2017).
[43] C. J. E. Straatsma, M. I. Fabrikant, G. E. Douberly, and H. J. Lewandowski, The Journalof Chemical Physics 147, 124201 (2017).
[44] N. E. Bulleid, S. M. Skoff, R. J. Hendricks, B. E. Sauer, E. A. Hinds, M. R. Tarbutt, J. E.
Stalnaker, M. Zolotorev, G. Gabrielse, J. M. Doyle, and D. DeMille, Physical Chemistry
Chemical Physics 15, 12299 (2013).
[45] P. J. Pritchard, Fox and McDonald’s Introduction to Fluid Mechanics, 8th ed. (John Wileyand Sons, Hoboken, 2011).
[46] Y. Sone, Molecular Gas Dynamics - Theory, Techniques, and Applications (Birkhäuser,
Boston, 2007).
[47] J. F. O’Hanlon, A User’s Guide to Vacuum Technology, 3rd ed. (John Wiley and Sons,
Hoboken, 2003) pp. 25–33.
[48] J. Farrar, Y. Lee, V. Goldman, and M. Klein, Chemical Physics Letters 19, 359 (1973).
[49] V. Singh, A. K. Samanta, N. Roth, D. Gusa, T. Ossenbrüggen, I. Rubinsky, D. A. Horke,
and J. Küpper, Physical Review A 97, 032704 (2018).
[50] R. DeCarvalho, J. Doyle, B. Friedrich, T. Guillet, J. Kim, D. Patterson, and J. Weinstein,
The European Physical Journal D 7, 289 (1999).
[51] L. Landau and E. Teller, Phys. Z. Sow. 10, 34 (1936).
[52] G. Sanna and G. Tomassetti, Introduction to Molecular Beams Gas Dynamics (Imperial
College Press, London, 2005).
[53] E. E. Nikitin and J. Troe, Phys. Chem. Chem. Phys. 10, 1483 (2008).
[54] P. Colarusso, B. Guo, K.-Q. Zhang, and P. Bernath, Journal of Molecular Spectroscopy 175,
158 (1996).
[55] B. Guo, K. Zhang, and P. Bernath, Journal of Molecular Spectroscopy 170, 59 (1995).
[56] J. B. Hasted, Physics of Atomic Collisions, 2nd ed. (Elsevier, New York, 1972).
[57] A. O. Sushkov and D. Budker, Physical Review A 77, 042707 (2008).

190[58] M.-J. Lu, K. S. Hardman, J. D. Weinstein, and B. Zygelman, Physical Review A 77, 060701(2008).
[59] M.-J. Lu and J. D. Weinstein, New Journal of Physics 11, 055015 (2009).
[60] S. S. Harilal, C. V. Bindhu, M. S. Tillack, F. Najmabadi, and A. C. Gaeris, Journal ofApplied Physics 93, 2380 (2003).
[61] N. Farid, S. S. Harilal, H. Ding, and A. Hassanein, Journal of Applied Physics 115, 033107(2014).
[62] D. M. Lubman, C. T. Rettner, and R. N. Zare, The Journal of Physical Chemistry 86, 1129
(1982).
[63] H. Murphy, The Effects of Source Geometry on Free Jet Expansions, Ph.D. thesis, Universityof California, San Diego (1984).
[64] H. C. W. Beijerinck and N. F. Verster, Physica B+C 111, 327 (1981).
[65] T. G. Dietz, M. A. Duncan, D. E. Powers, and R. E. Smalley, The Journal of Chemical
Physics 74, 6511 (1981).
[66] V. E. Bondybey and J. H. English, The Journal of Chemical Physics 74, 6978 (1981).
[67] P. F. Bernath, Spectra of Atoms and Molecules, 2nd ed. (Oxford University Press, New York,
2005).
[68] D. A. Steck, “Rubidium 87 D Line Data,” (2003).
[69] K. Huang, Statistical Mechanics, 2nd ed. (John Wiley & Sons, New York, 1987).
[70] P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (John Wileyand Sons, New York, 1986).
[71] L. Mertz, Infrared Physics 7, 17 (1967).
[72] A. S. Filler, Journal of the Optical Society of America 54, 762 (1964).
[73] R. H. Norton and R. Beer, Journal of the Optical Society of America 66, 259 (1976).
[74] J. K. Kauppinen, D. J. Moffatt, D. G. Cameron, and H. H. Mantsch, Applied Optics 20,
1866 (1981).
[75] R. J. Anderson and P. R. Griffiths, Analytical Chemistry 47, 2339 (1975).
[76] R. J. Anderson and P. R. Griffiths, Analytical Chemistry 50, 1804 (1978).
[77] S. F. Parker and P. B. Tooke, Spectrochimica Acta Part A: Molecular and Biomolecular
Spectroscopy 53, 2245 (1997).
[78] G. Jalsovszky, Journal of Molecular Structure 114, 127 (1984).
[79] E. V. Loewenstein, in Aspen International Conference on Fourier Spectroscopy, edited by
G. A. Vanasse, A. T. Stair, and D. J. Baker (1970) pp. 3–18.

191[80] R. J. Nordstrom, in Fourier, Hadamard, and Hilbert Transforms in Chemistry (Plenum Publishing, New York, 1982) pp. 421–451.
[81] H. E. Hallam and G. F. Scrimshaw, in Vibrational Spectroscopy of Trapped Species (John
Wiley and Sons, New York, 1973) pp. 11–66.
[82] I. R. Dunkin, Matrix-isolation techniques: a practical approach (Oxford University Press,
Oxford, 1998).
[83] L. J. Schoen, D. E. Man, C. M. Knobler, and D. White, The Journal of Chemical Physics
37, 1146 (1962).
[84] R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky,
G. Wysocki, and F. K. Tittel, Chemical Physics Letters 487, 1 (2010).
[85] J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264,
553 LP (1994).
[86] F. Capasso, C. Gmachl, R. Paiella, A. Tredicucci, A. Hutchinson, D. Sivco, J. Baillargeon,
A. Cho, and H. Liu, IEEE Journal of Selected Topics in Quantum Electronics 6, 931 (2000).
[87] C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, Reports on Progress in Physics 64,
1533 (2001).
[88] G. C. Bjorklund, Optics Letters 5, 15 (1980).
[89] L. Tombez, F. Cappelli, S. Schilt, G. Di Domenico, S. Bartalini, and D. Hofstetter, Applied
Physics Letters 103, 031111 (2013).
[90] X. Chao, J. B. Jeffries, and R. K. Hanson, Applied Physics B 106, 987 (2012).
[91] H. Li, G. B. Rieker, X. Liu, J. B. Jeffries, and R. K. Hanson, Applied Optics 45, 1052 (2006).
[92] P. Kluczynski and O. Axner, Applied Optics 38, 5803 (1999).
[93] S. Schilt, L. Thévenaz, and P. Robert, Applied Optics 42, 6728 (2003).
[94] R. Arndt, Journal of Applied Physics 36, 2522 (1965).
[95] P. Kluczynski, J. Gustafsson, Å. M. Lindberg, and O. Axner, Spectrochimica Acta Part B:
Atomic Spectroscopy 56, 1277 (2001).
[96] P. C. D. Hobbs, Proc. SPIE 1376 , 216 (1991).
[97] K. L. Haller and P. C. D. Hobbs, Proc. SPIE 1435 , 298 (1991).
[98] P. C. D. Hobbs, Applied Optics 36, 903 (1997).
[99] A. Van Orden and R. J. Saykally, Chemical Reviews 98, 2313 (1998).
[100] J. R. Heath and R. J. Saykally, The Journal of Chemical Physics 94, 1724 (1991).
[101] P. Neubauer-Guenther, T. Giesen, U. Berndt, G. Fuchs, and G. Winnewisser, Spectrochimica
Acta Part A: Molecular and Biomolecular Spectroscopy 59, 431 (2003).

192[102] L. Belau, S. E. Wheeler, B. W. Ticknor, M. Ahmed, S. R. Leone, W. D. Allen, H. F. Schaefer
III, and M. A. Duncan, Journal of the American Chemical Society 129, 10229 (2007).
[103] W. Weltner, P. N. Walsh, and C. L. Angell, The Journal of Chemical Physics 40, 1299
(1964).
[104] W. Weltner and D. McLeod, The Journal of Chemical Physics 40, 1305 (1964).
[105] K. R. Thompson, R. L. DeKock, and W. J. Weltner, Journal of the American Chemical
Society 93, 4688 (1971).
[106] L. N. Shen and W. R. M. Graham, The Journal of Chemical Physics 91, 5115 (1989).
[107] J. Szczepanski and M. Vala, The Journal of Physical Chemistry 95, 2792 (1991).
[108] A. M. Smith, J. Agreiter, M. Härtle, C. Engel, and V. E. Bondybey, Chemical Physics 189,
315 (1994).
[109] R. H. Kranze, P. A. Withey, C. M. L. Rittby, and W. R. M. Graham, The Journal of Chemical
Physics 103, 6841 (1995).
[110] J. Szczepanski, S. Ekern, C. Chapo, and M. Vala, Chemical Physics 211, 359 (1996).
[111] R. H. Kranze, C. M. L. Rittby, and W. R. M. Graham, The Journal of Chemical Physics
105, 5313 (1996).
[112] P. Freivogel, M. Grutter, D. Forney, and J. P. Maier, Chemical Physics 216, 401 (1997).
[113] L. Lapinski and M. Vala, Chemical Physics Letters 300, 195 (1999).
[114] X. D. Ding, S. L. Wang, C. M. L. Rittby, and W. R. M. Graham, The Journal of Physical
Chemistry 112, 5113 (2000).
[115] B. Kern, D. Strelnikov, P. Weis, A. Böttcher, and M. M. Kappes, The Journal of Physical
Chemistry A 117, 8251 (2013).
[116] J. Szczepanski, S. Ekern, and M. Vala, The Journal of Physical Chemistry A 101, 1841
(1997).
[117] J. Hutter, H. P. Luethi, and F. Diederich, Journal of the American Chemical Society 116,
750 (1994).
[118] J. Szczepanski, R. Pellow, and M. Vala, Z. Naturforsch. 47a, 595 (1992).
[119] M. Grutter, P. Freivogel, D. Forney, and J. P. Maier, The Journal of Chemical Physics 107,5356 (1997).
[120] M. E. Jacox, Chemical Society Reviews 31, 108 (2002).
[121] C. L. Kingsbury, S. J. Mehrman, and J. M. Takacs, Current Organic Chemistry 3, 497(1999).
[122] C. N. R. Rao, Annual Review of Physical Chemistry 40, 291 (1989).

193[123] W. Weltner, Science 155, 155 (1967).
[124] A. J. Merer, Annual Review of Physical Chemistry 40, 407 (1989).
[125] R. S. Ram, J. Liévin, G. Li, T. Hirao, and P. F. Bernath, Chemical Physics Letters 343, 437(2001).
[126] A. Gatterer and S. G. Krishnamurty, Nature 169, 543 (1952).
[127] W. Weltner and D. McLeod, Journal of Molecular Spectroscopy 17, 276 (1965).
[128] D. W. Green and K. M. Ervin, Journal of Molecular Spectroscopy 89, 145 (1981).
[129] A. Samoilova, Y. Efremov, and L. Gurvich, Journal of Molecular Spectroscopy 86, 1 (1981).
[130] M. Lorenz and V. E. Bondybey, Chemical Physics 241, 127 (1999).
[131] C. J. Nelin and C. W. Bauschlicher, Chemical Physics Letters 118, 221 (1985).
[132] R. S. Ram, J. Liévin, and P. F. Bernath, Journal of Molecular Spectroscopy 256, 216 (2009).
[133] G. Herzberg, Molecular Spectra and Molecular Structure - I. Spectra of Diatomic Molecules,
2nd ed. (D. Van Nostrand Company, Inc., Princeton, 1950).
[134] W. D. Bare, P. F. Souter, and L. Andrews, The Journal of Physical Chemistry A 102, 8279
(1998).
[135] M. Zhou and L. Andrews, The Journal of Chemical Physics 111, 4230 (1999).
[136] P. Brosset, R. Dahoo, B. Gauthier-Roy, L. Abouaf-Marguin,Physics 172, 315 (1993).

and A. Lakhlifi, Chemical[137] K. R. Asmis, Physical Chemistry Chemical Physics 14, 9270 (2012).
[138] P. Bocherel, L. B. Herbert, B. R. Rowe, I. R. Sims, I. W. M. Smith, and D. Travers, The
Journal of Physical Chemistry 100, 3063 (1996).
[139] C. J. E. Straatsma, V. E. Colussi, M. J. Davis, D. S. Lobser, M. J. Holland, D. Z. Anderson,
H. J. Lewandowski, and E. A. Cornell, Physical Review A 94, 043640 (2016).
[140] M. H. Anderson, J. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science
269, 198 (1995).
[141] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. StamperKurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).
[142] D. S. Jin, J. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.
77, 420 (1996).
[143] M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Stamper-Kurn, D. S. Durfee, C. G.
Townsend, and W. Ketterle, Phys. Rev. Lett. 77 (1996).
[144] M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, Phys. Rev. Lett.
77, 1671 (1996).

194[145] S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).
[146] D. S. Jin, M. R. Matthews, J. Ensher, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.
78, 764 (1997).
[147] D. M. Stamper-Kurn, H.-J. Miesner, S. Inouye, M. R. Andrews, and W. Ketterle, Phys. Rev.
Lett. 81, 500 (1998).
[148] O. Maragò, G. Hechenblaikner, E. Hodby, and C. Foot, Phys. Rev. Lett. 86, 3938 (2001).
[149] F. Chevy, V. Bretin, P. Rosenbusch, K. W. Madison, and J. Dalibard, Phys. Rev. Lett. 88,
250402 (2002).
[150] M. J. Bijlsma and H. T. C. Stoof, Phys. Rev. A 60, 3973 (1999).
[151] U. Al Khawaja and H. T. C. Stoof, Phys. Rev. A 62, 053602 (2000).
[152] E. Zaremba, A. Griffin, and T. Nikuni, Phys. Rev. A 57, 4695 (1998).
[153] V. Shenoy and T.-L. Ho, Phys. Rev. Lett. 80, 3895 (1998).
[154] G. M. Kavoulakis, C. J. Pethick, and H. Smith, Phys. Rev. A 57, 2938 (1998).
[155] B. Jackson and E. Zaremba, Phys. Rev. Lett. 88, 180402 (2002).
[156] B. Jackson and E. Zaremba, Laser Phys. 12, 93 (2002).
[157] B. Jackson and E. Zaremba, Laser Phys. 13, 1004 (2003).
[158] S. A. Morgan, J. Phys. B.: At. Mol. Opt. Phys. 33, 3847 (2000).
[159] M. Rusch, S. A. Morgan, D. A. W. Hutchinson, and K. Burnett, Phys. Rev. Lett. 85, 4844
(2000).
[160] S. A. Morgan, M. Rusch, D. A. W. Hutchinson, and K. Burnett, Phys. Rev. Lett. 91, 250403(2003).
[161] S. A. Morgan, Phys. Rev. A 69, 023609 (2004).
[162] S. A. Morgan, Phys. Rev. A 72, 043609 (2005).
[163] D. S. Lobser, A. E. S. Barentine, E. A. Cornell, and H. J. Lewandowski, Nat. Phys. 11, 1009
(2015).
[164] V. E. Colussi, C. J. E. Straatsma, D. Z. Anderson, and M. J. Holland, New Journal of
Physics 17, 103029 (2015).
[165] D. S. Lobser, Observation of a Persistent Non-Equilibrium State in an Extremely Isotropic
Harmonic Potential, Ph.D. thesis, University of Colorado (2015).
[166] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, 2nd ed. (Cambridge
University Press, Cambridge, 2008).
[167] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (Dover Publications, Mineola, 1971).

195[168] N. P. Proukakis and B. Jackson, J. Phys. B.: At. Mol. Opt. Phys. 41, 203002 (2008).
[169] T. R. Kirkpatrick and J. R. Dorfmann, J. Low Temp. Phys. 58, 301 (1985).
[170] E. Zaremba, T. Nikuni, and A. Griffin, J. Low Temp. Phys. 116, 227 (1999).
[171] A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperature (Cambridge University Press, New York, 2009).
[172] S. Giorgini, L. P. Pitaevskii, and S. Stringari, J. Low Temp. Phys. 109, 309 (1997).
[173] B. Jackson and E. Zaremba, Phys. Rev. A 66, 033606 (2002).
[174] M. D. Feit, J. J. A. Fleck, and A. Steiger, J. Comput. Phys. 47, 412 (1982).
[175] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in
Fortran 77 – The Art of Scientific Computing, 2nd ed. (Cambridge University Press, New
York, 1992).
[176] G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, 2nd ed. (Clarendon, Oxford, 1994).
[177] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd ed. (Addison-Wesley,
Redwood City, 2011).
[178] G. Strang, SIAM J. Numer. Anal. 5, 506 (1968).
[179] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison-Wesley, Redwood
City, 2000).
[180] C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation (Taylor andFrancis, Boca Raton, 2004).
[181] D. J. Larson, D. W. Hewett, and A. B. Langdon, Comput. Phys. Commun. 90, 260 (1995).
[182] J. P. Verboncoeur, J. Comput. Phys. 174, 421 (2001).
[183] C. Cornet and D. T. K. Kwok, J. Comput. Phys. 225, 808 (2007).
[184] A. J. MacLeod, Computers in Physics 11, 385 (1997).
[185] J. Märkle, Hybrid systems at finite temperatures, Ph.D. thesis, Universität Tübingen (2014).
[186] D. Guéry-Odelin, F. Zambelli, J. Dalibard, and S. Stringari, Phys. Rev. A 60, 4851 (1999).
[187] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74, 3352
(1995).
[188] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, in Bose-Einstein Condensation in
Atomic Gases, Proceedings of the International School of Physics “Enrico Fermi”, edited by
M. Inguscio, S. Stringari, and C. E. Wieman (1999) pp. 67–176.
[189] P. C. Hohenberg and P. C. Martin, Ann. Phys. 34, 291 (1965).
[190] P. Szépfalusy and I. Kondor, Ann. Phys. 82, 1 (1974).

196[191] W. V. Liu, Phys. Rev. Lett. 79, 4056 (1997).
[192] L. P. Pitaevskii and S. Stringari, Phys. Lett. 235, 398 (1997).
[193] P. O. Fedichev, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 80, 2269 (1998).
[194] M. Guilleumas and L. P. Pitaevskii, Phys. Rev. A 61, 013602 (1999).
[195] K. Das and T. Bergeman, Phys. Rev. A 64, 013613 (2001).
[196] B. Jackson and E. Zaremba, New J. Phys. 5, 88 (2003).
[197] S. T. Beliaev, Zh. Eksp. Teor. Fiz. 34, 433 (1958)