Download Quantum Theory of Atomic and Molecular Structures and Interactions

A Dissertation
entitled
Quantum Theory of Atomic and Molecular Structures and Interactions
by
Constantinos Makrides
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Doctor of Philosophy Degree in Physics
Dr. Bo Gao, Committee Chair
Dr. Svetlana Kotochigova, Committee Member
Dr. Lawrence Anderson-Huang, Committee Member
Dr. Steven Federman, Committee Member
Dr. Thomas Kvale, Committee Member
Dr. Patricia Komuniecki, Dean
College of Graduate Studies
The University of Toledo
August 2014
Copyright 2014, Constantinos Makrides
This document is copyrighted material. Under copyright law, no parts of this
document may be reproduced without the expressed permission of the author.
An Abstract of
Quantum Theory of Atomic and Molecular Structures and Interactions
by
Constantinos Makrides
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Doctor of Philosophy Degree in Physics
The University of Toledo
August 2014
This thesis consists of topics in two related areas of research that together provide
quantum mechanical descriptions of atomic and molecular interactions and reactions.
The first is the ab initio electronic structure calculation that provides the atomic and
molecular interaction potential, including the long-range potential. The second is the
quantum theory of interactions that uses such potentials to understand scattering,
long-range molecules, and reactions.
In ab initio electronic structure calculations, we present results of dynamic polarizabilities for a variety of atoms and molecules, and the long-range dispersion coefficients for a number of atom-atom and atom-molecule cases. We also present results of
a potential energy surface for the triatomic lithium-ytterbium-lithium system, aimed
at understanding the related chemical reactions.
In the quantum theory of interactions, we present a multichannel quantum-defect
theory (MQDT) for atomic interactions in a magnetic field. This subject, which is
complex especially for atoms with hyperfine structure, is essential for the understanding and the realization of control and tuning of atomic interactions by a magnetic
field: a key feature that has popularized cold atom physics in its investigations of
few-body and many-body quantum systems. Through the example of LiK, we show
how MQDT provides a systematic and an efficient understanding of atomic interaction in a magnetic field, especially magnetic Feshbach resonances in nonzero partial
iii
waves.
iv
This dissertation is dedicated to the memory of my grandfather Konstantinos Mousios.
Acknowledgments
I am forever indebted to my adviser, Dr. Bo Gao. I will always look back at
my time as a graduate student and remember his inspiration and constant support.
My interest in atomic and molecular physics is a result of his guidance and positive
attitude. Beyond the discussions on physics, the insight and perspective on many
aspects outside physics are things that I will keep close to my heart.
I also want to express my deepest gratitude to Dr. Svetlana Kotochigova. The
time I spent at the National Institute of Standards and Technology (NIST) has produced many positive experiences that I have cherished and I have learned quite a
bit from all the projects we completed together. I would also like to thank Dr. Eite
Tiesinga and Dr. Alexander Petrov for their assistance during my time at NIST.
I am grateful for the patience and guidance from the other committee members,
Dr. Lawrence Anderson-Huang, Dr. Steven Federman and Dr. Thomas Kvale.
I would also like to thank the other members of the “Toledo Rat Pack”, Ming Li,
Thomas Allen, Carl Starkey, Sam Ibdah, and Brad Hubartt. I am grateful for the
lasting friendship over the past few years and I hope for many more years to come.
Lastly, I would like to acknowledge my immediate family for their support of my
graduate studies. My parents Demetrios and Agathi Makrides, my siblings Yiannis
and Marianna Makrides and my grandmother Fereniki Mousios. It is always difficult
to leave home, especially in a family as close as our own, but I have never felt alone
or forgotten. As is evident by the mythological hero Odysseus, life often forces you
to be far away from those you love. Thank you for always being there for me.
vi
Contents
Abstract
iii
Acknowledgments
vi
Contents
vii
List of Tables
ix
List of Figures
x
List of Abbreviations
xii
1 Introduction
1.1
1
Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Ab initio Electronic Structure Calculations
2.1
2.2
2.3
2
4
Potential Energy Surface of LiY bLi . . . . . . . . . . . . . . . . . . .
6
2.1.1
Calculation of the Surface . . . . . . . . . . . . . . . . . . . .
6
2.1.2
Analytic Fit of the Data . . . . . . . . . . . . . . . . . . . . .
9
2.1.3
Technical Details and Results . . . . . . . . . . . . . . . . . .
12
2.1.4
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . .
19
Dynamical Polarizability . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.1
Brief Theoretical Considerations . . . . . . . . . . . . . . . . .
21
2.2.2
Computational Details and Results . . . . . . . . . . . . . . .
23
Dispersion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . .
33
vii
2.4
2.3.1
Theoretical Background . . . . . . . . . . . . . . . . . . . . .
34
2.3.2
Results for Select Systems . . . . . . . . . . . . . . . . . . . .
39
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3 Ultracold Atomic Interaction
46
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.2
MQDT for Atomic Interaction in a Magnetic Field
50
3.2.1
. . . . . . . . . .
Fragmentation Channels for Two Heteronuclear Group I Atoms
in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . .
50
3.2.1.1
A Single Group I Atom in a Magnetic Field . . . . .
50
3.2.1.2
Two Heteronuclear Group I Atoms in a Magnetic Field 57
3.2.2
MQDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.2.3
The K c Matrix for Heteronuclear Group I Systems . . . . . .
59
3.3
MQDT for Magnetic Feshbach Resonances . . . . . . . . . . . . . . .
61
3.4
Results for the 6 Li40 K System . . . . . . . . . . . . . . . . . . . . . .
64
3.5
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
4 Concluding Remarks
74
References
76
A Tables of Fitting Parameters for Atomic and Molecular Dynamic
Polarizabilities
85
A.1 SrLi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
A.2 LiYb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
A.3 Yb∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
A.4 NaK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
viii
List of Tables
2.1
Parameters for the three-body Potential of LiYbLi
. . . . . . . . . . . .
19
2.2
Table of Dispersion Coefficients . . . . . . . . . . . . . . . . . . . . . . .
40
3.1
Comparison of Select Feshbach Parameters for 6 Li40 K . . . . . . . . . . .
64
3.2
Table of Feshbach Parameters for LiK System . . . . . . . . . . . . . . .
65
A.1 Ground State SrLi Dynamic Polarizability Fit Parameters . . . . . . . .
86
A.2 LiYb Dynamic Polarizability Fit Parameters . . . . . . . . . . . . . . . .
87
A.3 Metastable 3 P Ytterbium Dynamic Polarizability Fit Parameters . . . .
87
A.4 NaK ground state X1 Σ Dynamic Polarizability Fit Parameters . . . . . .
88
A.5 NaK a3 Σ Dynamic Polarizability Fit Parameters . . . . . . . . . . . . . .
88
ix
List of Figures
2-1 Computational Plan of LiYbLi Surface . . . . . . . . . . . . . . . . . . .
8
2-2 Two-body Potentials of LiYb and Li2 . . . . . . . . . . . . . . . . . . . .
11
2-3 LiYbLi Potential Energy Surfaces: Isosceles Configuration . . . . . . . .
16
2-4 LiYbLi Potential Energy Surfaces . . . . . . . . . . . . . . . . . . . . . .
17
2-5 LiYbLi Three-Body Interaction Surfaces . . . . . . . . . . . . . . . . . .
18
2-6 Dynamic Polarizability of X2 Σ CaYb+ Molecule . . . . . . . . . . . . . .
26
2-7 Dynamic Polarizability of A2 Σ CaYb+ Molecule . . . . . . . . . . . . . .
27
2-8 Dynamic Polarizability of SrLi Molecule . . . . . . . . . . . . . . . . . .
30
2-9 Dynamic Polarizability of a3 Σ state of NaK . . . . . . . . . . . . . . . .
31
2-10 Dynamic Polarizability of X1 Σ state of NaK . . . . . . . . . . . . . . . .
32
2-11 Figure of Molecular Geometry . . . . . . . . . . . . . . . . . . . . . . . .
35
2-12 Vibrationally Dependent Dispersion Coefficients Between Similar
88
Sr7 Li
Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2-13 LiYb∗ Long Range Potentials . . . . . . . . . . . . . . . . . . . . . . . .
43
3-1 Picture of Two Channel Magnetic Feshbach Resonance . . . . . . . . . .
48
3-2 Energy Diagram of Group I Atom with IA = 3/2 in a magnetic field . . .
52
3-3 Energy Diagram of Group I Atom with IA = 1 in a magnetic field . . . .
53
3-4 Energy Diagram of Group I Atom with IA = 4 in a magnetic field . . . .
54
3-5 Reduced generalized scattering lengths for LiK in the MF = −5 Manifold
70
3-6 Reduced generalized scattering lengths for LiK in the MF = −4 Manifold
71
3-7 Reduced generalized scattering lengths for LiK in the MF = −3 Manifold
72
x
3-8 LiK ab channel plot of sin2 δl for p wave Resonance . . . . . . . . . . . .
xi
73
List of Abbreviations
CC . . . . . . . . . . . . . . . . . . . . . . .
CCSD . . . . . . . . . . . . . . . . . . . .
CCSD(T) . . . . . . . . . . . . . . . . .
CCSDT . . . . . . . . . . . . . . . . . . .
Coupled Cluster
Couple Cluster with Single and Double Excitations
CCSD with perturbative Triplet Excitations
Couple Cluster with Single, Double and Triple Excitations
HF . . . . . . . . . . . . . . . . . . . . . . . . Hartree-Fock
MCSCF . . . . . . . . . . . . . . . . . . . Multi Configuration Self Consistent Field
MQDT . . . . . . . . . . . . . . . . . . . Multichannel Quantum Defect Theory
PES . . . . . . . . . . . . . . . . . . . . . . Potential Energy Surface
QDT . . . . . . . . . . . . . . . . . . . . . . Quantum Defect Theory
xii
Chapter 1
Introduction
The ability to probe deeper into the quantum regime and control the interactions
of the constituents has opened the door to many investigations into ultracold gases 1 .
Since the late 1990’s many atomic species have been cooled to the ultracold regime,
which is approximately defined as below a few µK. The ability to control the interactions between atoms has been a hallmark for the field of cold atomic physics. One
of the first investigations in tuning ultracold atomic interactions was done by Inouye
et al. [2] for a
23
Na Bose-Einstein condensate. In their experiment, the sign and
strength of the interaction between the ultracold sodium atoms were shown to vary
with the applied external magnetic field. This feature is called a Feshbach resonance
and has since then been utilized and observed for a growing list of atoms, e.g. Rb [3],
Li [4], K [5], Sr [6] and Yb [7, 8]. This ability to control ultracold atoms provides a
control mechanism in the experimental studies of few and many-body systems. Essential to this control is having the best possible understanding of the atomic interactions
around a Feshbach resonance. At the present moment, most of the previous theoretical understanding of Feshbach resonances have been limited to s-wave resonances[9],
while support for non s-wave resonances has been lacking. One of the aims of this
thesis is to provide some of this understanding to higher partial waves.
1
See Chin et al. [1] and references therein.
1
A particular application of Feshbach resonances is the creation and control of ultracold polar molecules. Polar molecules are exciting to use in ultracold molecular
experiments since they posses a permanent dipole moment, which can be utilized for
various purposes such as improving trap coherence times [10] and precision measurement studies [11]. Ultracold polar molecules have been created through Feshbach
resonances; however, these “Feshbach molecules” are very weakly bound molecules.
It should be noted that direct cooling of molecules to ultracold temperatures is problematic due to the internal structure of the molecules. These “Feshbach molecules”
provide a successful alternative route in obtaining ultracold molecules. Subsequent
techniques can later bring these weakly bound molecules to the rovibrational ground
state. The successful creation of these ground state polar molecules has been realized
by Ni et al. [12] for KRb and Takekoshi et al. [13] for RbCs. In both of those cases, a
detailed knowledge of the molecular properties and structure is needed to bring these
molecules to the ground state. This information for the most part is lacking in many
of the contemporary systems and is another focus of this thesis.
1.1
Outline of the Thesis
The present thesis is organized to address the two goals previously outlined.
Chapter 2 focuses on ab initio electronic structure calculations for both atoms and
molecules. The chapter aims on addressing the lack of important atomic and molecular properties. The discussion begins by calculating the Born-Oppenheimer potentials
for the lithium-ytterbium-lithium system. This potential will be presented as part
of a work on ultracold reactive scattering. This potential, commonly called a potential energy surface, is the higher dimensional analogue of the familiar diatomic
potential energy curves. The aim of this work is to provide a surface for numerical
computations, which in turn will be used as a test of alternative descriptions of re2
active scattering (e.g. Gonzlez-Lezana [14]). In connection to the discussion on polar
molecules, LiYb has a large electric dipole moment and is being experimentally realized by [15]. Understanding and controlling the reactions between the LiYb molecules
and background Li atoms can improve the production and stabilization of LiYb polar
molecules.
The discussion on electronic structure calculations then proceeds to the calculation
of the dynamic polarizability for a number of atoms and molecules. The dynamic polarizability is important on its own merit; however in the present thesis it is necessary
in the calculation of dispersion coefficients of neutral-neutral systems.The dispersion
interactions of molecules are slightly different than their atomic (ground state) counterparts in that the anisotropy is quite significant. This anisotropy has gained much
attention in recent years especially in the case of polar molecules [16].
In chapter 3 the discussion moves to ultracold atomic interactions in the framework
of Multichannel Quantum Defect Theory(MQDT) for alkali systems. This chapter
addresses the lack of non s-wave descriptions of magnetic Feshbach resonances, by
providing precisely these descriptions. The focus is on providing this exact unified
description for all resonances of arbitrary partial wave. The discussion in this chapter
is applicable to all alkali systems; however, the emphasis will be on a test system,
6
Li40 K. We will depict these new characterization through this system.
The final chapter outlines the key results from the previous chapters and will
purpose some interesting directions to the current works.
Unless otherwise noted, atomic units are used in all equations. In this system of
units, the electron charge, electron mass, ~, and the Coulomb constant ke = 1/4π0 are
set to unity.
3
Chapter 2
Ab initio Electronic Structure
Calculations
Ab initio is a Latin term that means “from the beginning”. In the present context
of atomic physics, it corresponds to the numerical procedure of solving the electronic
part of the Schrödinger Equation to make predictions without the aid of empirical
information. There have been numerous packages that provide a number of features
for one to use to conduct numerical electronic structure calculations. The field of
ab initio electronic structure calculations have been somewhat merged with quantum
computational chemistry as there is significant overlap between the two. Thus, many
of these software packages for electronic structure calculations are refereed to as quantum chemical packages. These packages have had considerable growth since the 1950s
when computer technology was becoming available for computational purposes.
An early method in the computation of the Schrödinger Equation comes from
Hartree [17] and Fock [18], which is called the Hartree-Fock Method1 (HF). Despite
being a relatively old method, the method is still used today with a few improvements
added along the way. The main limitation of the HF method is that electron correlations are completely neglected for many-electron atoms. Regardless, the HF method
1
This method is also called the Self Consistent Field (SCF) method. In the present literature
there seems to be no distinction between the two.
4
is still used as a reference wavefunction for a number of other post HF calculations.
The Møller-Plesset perturbation theory [19] and the heavily used Coupled-Cluster
method [20] are two such examples that have been heavily used in these works. It
is also note worthy to mention that the HF method results in a single determinant
approximation to the multi-electron wavefunction. The alternative is to use multiple
determinants, which in the literature is called the Multi-Configuration Self Consistent
Field [21] method2 .
The most basic results of the HF and post-HF methods are calculations of the
energy of the system. In the case of an atom, this would correspond to the energy
levels of the electrons. For diatomic molecules, the resulting calculations can produce
a potential energy surface (PES) as a function of inter-nuclear distance. As the number of atoms grows, so does the dimension of the surface. The present discussion on
ab initio single point energy calculations will focus on the calculation of the triatomic
PES of lithium-ytterbium-lithium.
The sections following the triatomic PES are devoted to the calculation of the
dynamic polarizabilities and of the dispersion coefficients. The dynamic polarizability
is another property that can be directly calculated using ab initio methods. It is a
quantity associated with the response of the object to an electric field and is important
in trapping and manipulating the internal states of atoms and molecules (e.g. [10, 22]).
This is a quantity that is very relevant in the present study and will be one of the
computations reported. The motivation for obtaining dynamic polarizabilities is that
they can be used to determine the dispersion coefficients for neutral-neutral systems,
which describe the long range behavior. These dispersion coefficients are in turn
needed for describing atomic interactions and can be used in theories such as the
Multichannel Quantum Defect Theory (MQDT) used later in chapter 3.
2
Historically this has also been called the Multi-Configuration Hartree-Fock method.
5
2.1
Potential Energy Surface of LiY bLi
As mentioned in the Introduction, the motivation for producing a PES for this
system is to approach ultracold reactions. For the present system we are interested
in the exothermic reaction LiY b + Li → Li2 + Y b. It is a simple atom exchange
reaction and has applications in controlling trap losses due to exothermic collisions.
The numerical calculations of the reaction rate require as input the PES to solve
the time independent Schrodinger equation3 . Once the rate can be produced from
this surface, other alternative calculations on the reaction rate can be tested against
this surface. The Statistical method of Gonzlez-Lezana [14] is one of the alternative
approaches that is considered for this reaction.
2.1.1
Calculation of the Surface
The potential surface was calculated at a large number of inter-particle separations
using the computational chemistry package CFOUR [24]. The grid for which the
system was calculated on is depicted in Fig. 2-1. Wernli et al. [25] have a similar
scheme implemented for their system, LiHHe+ . A key difference is that the present
system has additional geometric symmetry that reduces the computational domain.
In our scheme, the origin is placed in between the two lithium atoms and the lithium
atoms are restricted to move equally along the z axis. The addition of the ytterbium
atom to the two lithium atoms defines a plane which is labeled as the Y Z plane.
Computationally, the ytterbium atom is then placed only in the first quadrant. The
calculations with ytterbium in the other quadrants can be determined from symmetry.
The step by step procedure is then as follows.
1. Fix the distance between the two lithium atoms. Call this distance RLiLi .
3
For example of this calculation, see [23] for a calculation of H3
6
2. Place one of the lithium atoms at (0, 0, RLiLi /2) and the other at (0, 0, −RLiLi /2).
3. Place the ytterbium atom in the first quadrant in the y−z plane at (0, Y by , Y bz ),
where Y by and Y bz are both positive.
4. Calculate the ground state energy of the system.
5. Repeat moving the lithium distance and the ytterbium atom around the quadrant.
As a technical detail, placing the ytterbium atom along the z or y axis introduces
another symmetry into the geometry and in this new symmetry convergence issues
tend to arise in both the SCF and post-SCF calculations. Moving slightly off axis
would break this symmetry and return to “normal” operation. For these cases along
the axis, the symmetry of the system can once again be exploited to extrapolate to
along the axis. This is again due to the fact that the calculations on the other side
of the axis can be determined by reflections about the axis in question and in this
situation one can expect the energy calculations with ytterbium along either axis to
be either a local maximum or minimum in the immediate neighborhood. A simple
quadratic fit to the near axial calculations can determine the axial calculations with
sufficient accuracy.
7
15
10
Z axis (a.u.)
5
0
0
-5
-2
-4
-10
-6
-15
0
5
Y axis (a.u.)
10
15
-8
-10
-1
0
1
Figure 2-1: Plan of Single Point Energy Calculations of LiYbLi Potential Surface: In this figure a map is shown of the computational domain for the calculation of the PES of LiY bLi. As
indicated in the text, the two lithium atoms are placed along the
z axis and the ytterbium atom is placed in the colored first quadrant. The different colors represent regions of expected increasing computationally difficulty, with the yellow region expected
to take the longest to converge, followed by green then blue.
8
2.1.2
Analytic Fit of the Data
Following the plan depicted in Fig. 2-1, the grid was submitted to a computational cluster that calculated the energy of the system in the different geometries.
In principle, the whole computational domain could be sampled and would create a
rather large database with calculations at each grid point. The practicality of such
a situation is not ideal as the computational resource demand would be large and
the time constraints would not allow the full surface to be sampled at a reasonable
mesh. This consideration prevents a spline type interpolation to be employed in this
situation. The reason is that there would not have a small enough mesh for the spline
interpolation to accurately represent the potential surface. An alternative approach
is to fit the calculations to an analytic form that is robust enough to represent the
entire surface. The analytic form of the potential can be written as a sum of two
terms.
V (rAB , rAC , rBC ) = Vpairwise + V3Body
= V (2) (rAB ) + V (2) (rBC ) + V (2) (rAC ) + V (3) (rAB , rAC , rBC ) (2.1)
V (2) (R) is a two-body potential corresponding to a diatomic potential curve and
V (3) (rAB , rAC , rBC ) is the three-body term that describes deviations in the PES from
the pairwise potential. The pairwise potential is the sum of the two-body potential
for each pair of atoms. In the case of three atoms this corresponds to three terms.
The advantage of this separation is that the two-body potentials may be calculated
independently from the full three-body potential. This allows the usage of potentials
from experiments if desired and available. Computationally, the two-body potentials
are considerably less resource demanding. This allows one to calculate many points
in the potential curve, which has the effect of having more sophisticated analytic
9
representations. For these reasons, the focus will not be on the analytic form of
the two-body potentials. In the present case, the two-body potentials for both Li2
and LiY b were calculated to sufficient accuracy and quantity to directly use spline
interpolation for the potential curves.
To fit the three-body term, the analytic form suggested by Aguado and Paniagua
[26] was used, which has a functional form
V
(3)
(rAB , rAC , rBC ) =
M
X
dijk ρiAB ρjAC ρkBC
(2.2)
i,j,k
where for the three-body term, ρAB is defined as,
(3)
ρAB = RAB e−βAB RAB
(2.3)
The dijk serve as linear fit parameters and the three β (3) are non-linear fit parameters. These parameters are determined iteratively in the fitting routines. The input
parameter M controls the number of fitting parameters. One of the key advantages of
using this particular analytic form is the ability to dramatically increase the number
of fit parameters and still respect the short, intermediate and long range behaviors
of the potential [26].
The routine has two restrictions on M and the indices i, j, k.
1. i + j + k < M
2. i + j + k 6= i 6= j 6= k.
These conditions enforce that the three-body term goes to zero when one of the
inter-nuclear distances goes to zero [26]. In principle, increasing the value of M will
achieve a better fit to the data at the cost of increasing the number of fit parameters.
The increase in the number of linear and non-linear terms translates into longer
times to determine the parameters; however, even for fairly large values of M ∼ 8,
10
Reactants
Products
Reactants
8000
LiYb + Li
6000
Li2 + Yb
4000
-1
Potentials (cm )
2000
0
-2000
D0 LiYb
-4000
-6000
-8000
Offset = D0 Li - D0 LiYb
D0 Li
2
5
RLiYb [a.u.] 10
15
5
2
RLi [a.u.] 10
2
15
Figure 2-2: Two-body Potentials of LiYb and Li2 : The two-body potential curves for LiYb and Li2 are depicted in this figure. The
Figure gives information on the practicality of a reaction involving the initial LiYb molecule interacting with a Li atom and
exiting from the reaction as Li2 molecule with an excess ytterbium atom. The potential curves show that the reaction is an
exothermic one since the lowest bound state of the LiYb system is higher in energy than the lowest bound state of the Li2
molecule. The potential curves in this figure are real calculations
using the same basis as the LiYbLi potential surface, as opposed
to the bound states which are used for illustrative purposes.
11
the computational time is modest on contemporary personal desktop computers. The
more sensitive issue with respect to increasing M is that the number of fit parameters
increases very rapidly. Thus, there is a balance between the number of fit parameters
and the total number of calculated points available. Having too large an M can lead
to very misleading fits. For example, if the number of fit parameters is on the order
of the number of data points (but still less), one can imagine that the fitting routine
can fit each of the individual data points resulting in a delta function-like behavior
at each data point. In that case, the sum of residual squares will be close to zero
indicating a “good” fit; however, it is evident that the function in between the data
points will not be well represented. Thus, as a general principle, it is good to keep the
value M as small as possible while maintaining a low sum of the residuals squared.
2.1.3
Technical Details and Results
For the calculation of the ground PES for the LiYbLi system, a number of technical details need to be specified. The single point energy calculations were all done
at the CCSD(T) level. As mentioned earlier, the perturbative treatment of the triple
excitations is computationally cost efficient, as the full CCSDT treatment is a significant step up in performance time. For lithium we used the “aug-cc-pCVTZ” basis set
of Prascher et al. [27], which is a contracted basis that includes a number of diffuse
functions. For the calculation of the ytterbium atom we use a much larger contracted
basis set similar to the one found in Cao and Dolg [28]4 . A relativistic pseudopotential5 is also used for the inner 28 electrons, which is common practice for higher
Z atoms. The pseudopotential greatly simplifies the electron correlation part of the
calculation for these atoms as these methods can scale very poorly (∼O(n7 ))[29].
4
The actual basis set and pseudopotential used was sent via Private Communication by Dr. M.
Dolg, one of the authors of the ytterbium basis set.
5
This is the common name in physics. On the physical chemistry side, this quantity is commonly
refereed to as Effective Core Potentials.
12
To further simplify the calculation, a number of closed shell electrons are ignored
from the correlation in a process that is known as Frozen-core calculations. For the
present calculation, the lowest 15 orbitals are included in the core, which corresponds
to the 1s electrons for each of the lithium atoms and the 4s,4p,4d,5s and 5p orbitals
for the ytterbium atom. It is important to keep the 4f orbital in the correlation
as these electrons are not as deeply bound and contribute to the present case. The
exclusion of the core electrons greatly simplifies the post HF calculations that deal
with electron correlation.
To give a perspective on the computational runtime, the calculations were distributed over a number of modern Intel Xenon processors all of which were either quad
core 2.4 GHz processors or 8 core 2.1 GHz processors. The multi-processor nodes allow
multiple calculations to run concurrently on the same processor. On average 20 to 40
processors where used over a duration of two months to complete all of the purposed
calculations. To further emphasize the choice of using CCSD(T) calculations over
CCSDT, for diatomic single point energy calculations, the CCSDT calculation can
take up to 50 times longer to converge over the corresponding CCSD(T) calculation.
If one assumes this type of scaling for the three atom case, the resulting computation
would take years to complete the same set of calculations. In addition to this scaling,
the inclusion of more electrons in the correlation also scale the computation time
greatly.
The raw data points were passed through a screening procedure to eliminate data
points that may have converged to a different (possibly excited) surface. These points
should be sparse and are typically a result of the preceding SCF calculation converging
to the wrong solution. In many cases, this difference can be rather large and abrupt
and easily filtered out. The difficulty arises in certain regions where either (or both)
the SCF or CC methods take longer than usual to converge. In these cases, the
validity of the calculation can be difficult to discern by evaluating simple energy
13
differences with the neighboring data points. At worst, if other diagnostic information
in the program output fails to conclusively rule out the calculation, the calculation is
temporarily removed from the fitting process.
The rigid selection criteria results in a large amount of data points being discarded.
Of the original 1800 calculations submitted to the computational cluster, the selection
criteria resulted in a final data set with 591 points. To a large degree, the difficulty
is primarily due to the correlation of the 4f shell of the ytterbium atom. Even at the
CCSD level of theory, the total computation time per data point can be rather large.
With regard to the fitting parameters, the best representation of the calculated
data points was found with M = 4, which corresponds to thirteen linear parameters
and two non-linear parameters. The parameters used in the final version of the
fit can be found in table 2.1. In the process of fitting the three-body potential,
numerous fits were made with M going up to seven. In principle, increasing M will
reduce the error between the fit and the data points; however, the resulting surface
tends to have distortions and oscillations that are fictitious and artifacts of the fitting
procedure. The reason for these fictitious features is difficult to discern. The root of
the issue seems to be due to the fact that there are no constraints on the parameters
in the regions where the distortions appear, thus they are free to take on any form.
Another way to look at this issue is to consider the case where the number of fit
parameters is comparable to the number of data points. In that scenario, it can be
imagined that there would be some fit parameters that do not actually have any real
constraints on them and will be able to “act” freely. As you reduce the number of
fit parameters, i.e. lower M , the fit parameters start representing more and more
data points; thus the number of free unrestricted parameters decrease until you have
just enough parameters to fit the data. In the present case, the choice of M = 4
seems to not contain any fictitious oscillations in the sampled geometries. To prevent
any unforeseen oscillations in the long range, the three-body term is forced to zero
14
via a smooth switching function past the point where the wavefunctions overlap.
This choice is again determined by the diatomic potentials and is found to be at
approximately 15 a0 for both potential curves.
In addition to the computational complexities, the potential surface for three
atoms cannot be shown in full as the surface requires a four dimensional plot. What
is often useful is to plot a three dimensional cross section by restricting the independent variables of the system. In Fig. 2-3 The atoms are arranged in an Isosceles
configuration, where the ytterbium to lithium distance is the same for both lithium
atoms. The top panel shows the complete surface and the bottom panel shows the
three-body contribution. This configuration allows us one to look at the angular profile of the molecule, which can give some insight on the nature of this interaction.
There is a global minimum in the surface that seems to be primarily due to the pairwise interaction. The minimum in the surface occurs at about 6 a.u., which is close
to the ytterbium-lithium two-body minimum as seen in Fig. 2-2.
Figures 2-3, 2-4 and 2-5 show different aspects of the PES. The various panels in
figure 2-5 show the three-body contributions to the potential at different molecular
geometries. The three-body contribution has some of the typical features that one
would expect. At sufficiently large distances, the three-body contribution tends to
zero and is reflected in all four panels in figure 2-5. There is also some variation in
a structure found in all four panels of figure 2-5. This feature is repulsive for more
acute angles, while the opposite is true for the more obtuse angles.
This structure in the three-body potential is also significant in magnitude. The
plots in figure 2-5 show the maximum of the three-body term to be on the order of
10−2 a.u. (∼ 2000 cm−1 ). In comparison, Figure 2-2 shows the minimum of the LiYb
diatomic potential to be slightly larger in magnitude. The Li2 diatomic potentials
are a bit deeper, but not large enough to justify completely ignoring the three-body
potential.
15
V(r12,r13,r23) (a.u.)
A
0.02
0
-0.02
-0.04
60
120
θLiYbLi 180
-0.05
6
-0.025
14
8 10 12
rYbLi (a.u.)
0
0.025
V(r12,r13,r23) (a.u.)
B
0.02
0
-0.02
-0.04
60
120
θLiYbLi 180
-0.05
6
-0.025
14
8 10 12
rYbLi (a.u.)
0
0.025
Figure 2-3: LiYbLi Potential Energy Surfaces: Isosceles Configuration: A cross section of the PES of LiYbLi. The top panel, the
complete potential is shown. The bottom panel has the pairwise
interaction removed leaving only the three-body interaction. The
base of each figure shows a contour plot of the surface.
16
0.02
0
-0.02
-0.04
6 9
12
rYbLi2 (a.u.) 15
-0.05
6
12 15
rYbLi1 (a.u.)
9
-0.025
0
0.025
C
V(r12,r13,r23) (a.u.)
V(r12,r13,r23) (a.u.)
B
0.02
0
-0.02
-0.04
6 9
12
rYbLi2 (a.u.) 15
-0.05
6
12 15
rYbLi1 (a.u.)
9
-0.025
0
0.025
D
0.02
0
-0.02
-0.04
6 9
12
rYbLi2 (a.u.) 15
-0.05
-0.025
6
12 15
rYbLi1 (a.u.)
9
0
V(r12,r13,r23) (a.u.)
V(r12,r13,r23) (a.u.)
A
0.025
0.02
0
-0.02
-0.04
6 9
12
rYbLi2 (a.u.) 15
-0.05
-0.025
6
12 15
rYbLi1 (a.u.)
9
0
0.025
Figure 2-4: LiYbLi Potential Energy Surfaces: Various plots of the potential energy surface of the LiYbLi system at different geometries. Each panel fixes the angle with ytterbium at the vertex.
The angles are 45◦ , 90◦ , 135◦ and 180◦ for panels A, B, C and
D, respectively.
17
0.02
0
-0.02
-0.04
6 9
12
rYbLi2 (a.u.) 15
-0.05
6
12 15
rYbLi1 (a.u.)
9
-0.025
0
0.025
C
V(r12,r13,r23) (a.u.)
V(r12,r13,r23) (a.u.)
B
0.02
0
-0.02
-0.04
6 9
12
rYbLi2 (a.u.) 15
-0.05
6
-0.025
12
9
rYbLi1 (a.u.)
0
15
0.025
D
0.02
0
-0.02
-0.04
6 9
12
rYbLi2 (a.u.) 15
-0.05
-0.025
6
12 15
rYbLi1 (a.u.)
9
0
V(r12,r13,r23) (a.u.)
V(r12,r13,r23) (a.u.)
A
0.025
0.02
0
-0.02
-0.04
6 9
12
rYbLi2 (a.u.) 15
-0.05
-0.025
6
12
9
rYbLi1 (a.u.)
0
15
0.025
Figure 2-5: LiYbLi Three-Body Interaction Surfaces: Various plots of
the the three-body interaction for the LiYbLi system. Each panel
fixes the angle with ytterbium at the vertex. The angles are 45◦ ,
90◦ , 135◦ and 180◦ for panels A, B, C and D, respectively.
18
Table 2.1: Parameters for the three-body Potential of LiYbLi: Included here are the fitted parameters used to construct the threebody potential. The notation is as in the text with the dijk being
the linear fit parameters and the β (3) being the non-linear parameters. The indices i and j correspond to powers of ρ for LiY b
distances while the index k is for powers of ρ for the LiLi distance.
i
1
1
1
2
2
0
2
1
2
2
3
3
0
j k
dijk
1 0 -0.3791234233645178 × 100
0 1 -0.1207092112030131 × 102
1 1 0.6778574385332172 × 101
1 0 0.9609698323047215 × 100
0 1 0.1808003175501403 × 102
2 1 0.4946458265430991 × 100
1 1 -0.2785537833078476 × 102
2 1 0.2029448131083818 × 101
2 0 -0.8786730695382046 × 100
0 2 0.1047435507501138 × 103
1 0 0.7103760674735782 × 10−1
0 1 0.3961820620689986 × 102
3 1 -0.1402545726485475 × 100
(3)
βLiY b = 0.7110242142956382
(3)
βLiLi = 0.2079741859771922
2.1.4
Concluding Remarks
This concludes the discussion of the PES for the LiYbLi system. Emphasized here
were the computational details of computing the PES. Some of the technical details of
fitting the data to an analytic formula were discussed as well. The fit parameters for
the three-body potential can be found in table 2.1. The pairwise potentials require
the diatomic potentials between the two lithium atoms and between lithium and
ytterbium, which can be found by ab initio means as is depicted in figure 2-2, or
experimentally. The structure of this potential does not seem to have any particular
features much different from three-body potentials of other systems found in Wernli
et al. [25]. The deciding factor on the accuracy of the PES will have to wait until an
19
experimental measurement of the reaction rate of this system can be made.
Unfortunately at the time of this thesis the question on the reaction rate cannot
be answered6 . Another question that would be of significant interest would be the
effect of the three-body term on the reaction rate. Presumably, if the three-body term
shows little to no difference on the rate, one could conceivably think about a class of
reactions that depend solely on the long range interactions [30, 14] and the conditions
to which such conditions apply. This is a question that we plan on addressing in the
near future.
2.2
Dynamical Polarizability
The dynamic polarizability is one of the most important quantities in ultracold
experiment and theory. Its theoretical foundation is quite historic and is one of the
most well known applications of perturbation theory. Since the dynamic polarizability
describes the interaction between the atom (or molecule) and a laser at different
frequencies, it is of great importance to the understanding of laser-atom interactions.
Most experimental laser frequencies are not too large due to the financial cost of those
type of lasers. A typical YAG laser has a wavelength of about λ = 1064 nm, which
translates into a laser frequency of about 9400 cm−1 . A CO2 laser has a frequency
on the order of about 1000 cm−1 . Quite often, the static polarizability is used to
approximate the polarizability at these laser frequencies; however, there are cases
where the two are quite different due to the presence of excited states nearby.
In addition, as mentioned in the Introduction to the thesis, the polarizabilities are
used in the calculation of the dispersion coefficients. This calculation is one of the
main motivations for obtaining the dynamic polarizability. Most of the polarizabilities
6
The reaction rate for this PES is currently being calculated by the group of Dr. Naduvalath at
UNLV.
20
for ground state atoms can be obtained by extracting the oscillator strengths from the
NIST ASD[31] database, as the accuracy of such states are quite high. The situation
is quite different for the molecules of interest, as the calculation of the all oscillator
strengths for a particular diatomic molecule can be quite laborious.
In the following section, the process of calculating the dynamic polarizability will
be highlighted, describing the computational details in addition to touching upon
some of the important theoretical aspects. Towards the end of the section, the fitting
process will be explained and applied to a number of systems. Appendix A will
contain the exact fitting data for the systems.
2.2.1
Brief Theoretical Considerations
In the framework of perturbation theory, the dynamic polarizability at laser frequency ω can be determined though second-order perturbation theory7 and is given
by,
X D E 2
En0 − En
0 ~
α(ω) = 2
n d · ê n (En0 − En )2 − ω 2 .
(2.4)
n0 6=n
where d~ is the dipole moment operator and ê is the polarization of the applied field.
The connection between oscillator strengths and the dynamic polarizability is embedded in Eq. 2.4. Specifically, the oscillator strength fnn‘ is defined as,
fnn‘
X D E 2
0 ~
== 2
n d · ê n (En0 − En ).
(2.5)
n0 6=n
In principle the polarizability is a tensor that relates the components of the applied
electric field to the induced dipole. For a linear molecule, the only non-zero elements
7
This qualitatively can be understood by the fact that the interaction energy of an induced dipole
in a field is proportional to the field squared and relate this to the second order correction to the
energy going also as the field squared.
21
of the polarizability tensor in the molecular frame is along the diagonal [32]. Taking
the z axis as the internuclear axis, the polarizability tensor takes the form of


α ⊥ 0 0 



α=
0
α
0
⊥




0
0 αk
(2.6)
where αk is the polarizability component along the z axis and α⊥ is the component
perpendicular to this axis. The molecule must also satisfy some atomic limits. If one
were to imagine a diatomic molecule at very large internuclear distances, it would
closely resemble two separate atoms. Thus for a diatomic molecule with two atoms,
A and B, its polarizability at large internuclear separation is,
R
→∞
αAB −−AB
−−−→ αA + αB .
(2.7)
This equation will be exploited later in the discussion on dispersion coefficients
when large separation limits are taken and examined. It is also often useful to define
for later discussions the isotropic polarizability for linear molecules,
αiso ≡
1
αk + 2α⊥
3
(2.8)
and the anisotropic polarizability for linear molecules,
∆α ≡ αk − α⊥
(2.9)
For comparative purposes, consider the polarizability of a ground state alkali
atom. The polarizability respects the symmetry of the atom and is thus completely
atom
isotropic, meaning that α̂atom = α · 1̂. As a result of this symmetry, α⊥
is zero.
In addition, the origin of the parallel and perpendicular polarizabilities is slightly
22
different. In the case of a Σ ground state, the parallel polarizability is due to the
dipole coupling to other Σ states. In the perpendicular case, it is due to dipole
coupling to Π states.
2.2.2
Computational Details and Results
Following the discussion in the previous section, the most direct way of calculating the dynamic polarizability would be to calculate the transition dipole moments
between the different states and terminate the sum in equation 2.4. This approach
is colloquially called the “sum-over-states” method. Aside from the difficulties in the
actual calculation of the transition dipole moments, there is some difficulty in determining how many terms in the sum to use as it becomes increasingly more difficult
to accurately calculate higher excited states. To get a good fit, certain features of
the dynamic polarizability need to reflected, of which the static polarizability is the
most straight forward check to make. In principle, as in the dynamic polarizability,
all the terms in the sum contribute, but in practice the static polarizability can get
reproduced by a few number of states.
Two examples of this sum-over-states method are depicted in figures 2-6 and 2-7.
They are both for the molecule CaYb+ . Fig. 2-6 is for the ground state (X2 Σ) while
Fig. 2-7 is for the first excited state A2 Σ. The CaYb+ system is different from the
other molecules in this thesis in that it is an ionic system. There is significant interest
in the charge exchange rate of this system as it has been found to be four orders of
magnitude larger than other ionic molecular systems [33]. As part of a theoretical
investigation on this rate, the dynamic polarizability of both the ground and first
excited state is required to completely describe the effects of the trapping lasers on
the ions.
The oscillator strengths and energy states were calculated at the MCSCF level of
theory using the MOLPRO suite of ab initio programs [34]. In these calculations a
23
total of 20 different electronic states were calculated and used to calculate transition
dipole moments between each other. In conjunction with the potential curves, the
transition dipole moments can be inserted directly into equation 2.4.
The MCSCF method is considered to be less accurate over the “gold-standard”
CCSD(T) methods. The difference however is that the MCSCF is a multi reference
theory and has better scaling than the CCSD(T) calculations. There are analogous post HF methods for MCSCF calculations; however these methods can be very
computationally expensive and would be a massive undertaking to compute a similar number of oscillator strengths. Regardless, the sum-over-states approach is a
straightforward way to compute the dynamic polarizability.
The other approach involves perturbative methods by adding small frequency
dependent dipolar fields to the computational Hamiltonian and to directly calculate
the polarizability at various frequencies. Once this computation is done for a sufficient
number of points, a fit can be applied and the corresponding parameters can be
extracted. This approach will be referred to as “Independent-Frequency-Fit” in this
work.
For coupled-cluster calculations of polarizabilities the details of the process can
be found in Kállay and Gauss [35]. The key aspects from the formulations are that
the end results are analytic in terms of the basis function parameters and that the
convergence times are “reasonable”. To give a small perspective on the computational runtime of some of these calculations, for the LiYb system, the calculation
at ω = 24000cm−1 took approximately three weeks to converge at the CCSD level8 .
Frequencies lower than that typically converged quicker, with the exception being if
the particular frequency happened to be close to the energy of an excited state, in
which case there would be a resonant feature in the dynamic polarizability. In the
8
It should be mentioned that the common CCSD(T) method cannot be used to calculate the
dynamic polarizability. The computational cost of CCSDT methods for the dynamic polarizability
also make it practically unusable.
24
cases near a resonant feature, the calculation may not converge or will converge very
slowly.
The independent-frequency-fit approach was used to find the dynamic polarizability for the diatomic strontium-lithium molecule. The SrLi molecule is a polar
molecule that was strongly considered experimentally for polar molecule studies. Its
dipole moment is relatively larger than that of KRb [36]; however, there was little
experimental data on its molecular properties. Individually Sr and Li have both been
cooled to the ultracold regime, thus making this system attractive to polar molecule
studies.
In figure 2-8, dynamic polarizability is plotted at various internuclear distances.
The black circles and the red squares are calculations at various frequencies as was
mentioned earlier. The solid curves are the corresponding fits to the calculations.
These fits have a form of
fg (ω) =
N
X
i
1−
Ai
2 .
(2.10)
ω
ωgi
In the above equation, it is understood that the molecule is in the state g. The
functional form of each of the sums was chosen to closely resemble the equation
of the dynamic polarizability from second order perturbation theory, eq. 2.4. The
parameters, Ai and ωi , thus have direct interpretations in this framework. Ai is
related to the oscillator strength, fgi , between the state g and the state i. The other
parameter ωgi is the frequency corresponding to the difference in energy between
states g and i. Implicitly, the state g is not included in the sum as is also the case in
the perturbative equation.
To highlight some of the more interesting features, the panel “c” in figure 2-8 shows
that the parallel and perpendicular polarizability have become indistinguishable from
one another. This is indicative that the atoms are sufficiently far apart such that
25
9000
αPara
αPerp
R=15.0 a.u.
α (a.u.)
6000
3000
0
-3000
-6000
0
5000
10000
15000
20000
-1
ω (cm )
Figure 2-6: Dynamic Polarizability of X2 Σ CaYb+ Molecule: Depicted here is the dynamic polarizability of the X2 Σ state of
CaYb+ at the MCSCF level of theory. The internuclear distance
is 15 a0 . The polarizability here is calculated by the sum-overstates. As indicated, the black curve is the parallel polarizability
and the red curve is the perpendicular polarizability. For this
calculation, a total of 20 states are considered.
26
9000
αPara
αPerp
R=15.0 a.u.
α (a.u.)
6000
3000
0
-3000
-6000
0
5000
10000
15000
20000
-1
ω (cm )
Figure 2-7: Dynamic Polarizability of A2 Σ CaYb+ Molecule: Depicted here is the dynamic polarizability of the A2 Σ state of
CaYb+ at the MCSCF level of theory. The internuclear distance
is 15 a0 . The polarizability here is calculated by the sum-overstates. As indicated, the black curve is the parallel polarizability
and the red curve is the perpendicular polarizability. For this
calculation, a total of 20 states are considered. The reason for
the perpendicular polarizability being relatively flat in the above
figure is due to the small oscillator strengths between the states
considered.
27
they can be considered as separate atoms. As mentioned in the previous section, at
this far distance the dynamic polarizability is the sum of the two polarizabilities of
the separate atoms. The other two panels give an impression of the differences in the
polarizability as the internuclear distance is changed.
The NaK system is one the polar molecules currently being looked at by experimentalists. It has a rather large permanent dipole moment of 2.7D, which is desired
for the purposed applications in quantum simulations of solids and quantum information studies. In addition, it is stable against reactions with other NaK molecules to
form homonuclear dimers of sodium and potassium. As is the case with many polar
systems, there are still efforts to get an ultracold gas of NaK molecules in the ground
state. Figures 2-9 and 2-10 show our calculated dynamic polarizability of the NaK
molecule at its equilibrium position. Also shown on those graphs is the position of
a hypothetical trapping laser at 1550 nm. If the trapping laser is nearly resonant to
molecular transition, there will be population transfers to the excited state. These
molecules in the excited state will likely not be trapped and thus lead to trap losses.
While, our polarizability calculation did not directly calculate the potentials of the
excited states, the resonant features in the polarizability indirectly give us the location
of some of these potentials. Gerdes et al. [37] have previously investigated some the
lower potentials for this system. Our resonance positions agree with the differences
in their potentials.
The static polarizability for this system was also calculated by Deiglmayr et al.
[38]. They calculated the average static polarizability by a finite field method using
a configuration interaction type computational package. For comparative purposes,
our static polarizability can be determined by evaluating our analytic functional fit
at zero frequency. Our value for the average static polarizability is 348 a.u., while
they obtained a value of 351 a.u. The difference between the two is quite small and
is likely due to the usage of different basis sets in the calculation.
28
These polarizabilities complete the description of the interaction between polar
NaK molecules and the trapping laser. One possible direction is to determine the
optimal field geometries between all the applied laser and magnetic fields as was done
for KRb [10]. The existence of “magic” geometries were found for KRb, which increased the trap coherence time by an order of magnitude . These “magic” geometries
orient the trapping laser, an external applied electric field and an external applied
magnetic field carefully such that the internal states of the molecule experience the
same trapping potential.
29
α (a.u.)
8000
A
4000
0
-4000
α (a.u.)
8000
B
4000
0
-4000
α (a.u.)
8000
C
4000
0
-4000
5000
10000
15000
-1
ω (cm )
20000
25000
Figure 2-8: Dynamic Polarizability of SrLi Molecule: The dynamic
polarizability is depicted above at three different internuclear distances. The distances for panels A, B and C are 6.7, 11.3, and
333.7 a.u. respectively. In each plot, the black (red) curve circles
(squares) represent ab initio calculations for the dynamic polarizability parallel (perpendicular) to the internuclear axis. The
subsequent lines are fits using the canonical form of the dynamic
polarizability. It is worth noting the dynamic polarizability in figure “c” has become fully symmetrical with respect to the parallel
and perpendicular directions, indicating that the atomic limits
have been reached. The calculated points were done with the
computational chemistry package CFOUR.
30
α (a.u.)
6000
4000
2000
0
-2000
-4000
-6000
0
Parallel
1550 nm
5000
10000
15000
-1
20000
25000
30000
ω(cm )
4000
Perpendicular
α (a.u.)
2000
0
1550 nm
-2000
-4000
0
5000
10000
15000
-1
20000
25000
30000
ω(cm )
Figure 2-9: Dynamic Polarizability of a3 Σ state of NaK: The polarizability of the a3 Σ state of NaK is shown here at the equilibrium distance, R= 6.59a0 . The top panel shows the parallel
polarizability while the bottom shows the perpendicular polarizability. The dashed blue line is the frequency of 1550 nm laser.
The calculated points were done with the computational chemistry package CFOUR. The QZVPP basis set was used for both
sodium and potassium.
31
α (a.u.)
2000
Parallel
1000
1550 nm
0
-1000
-2000
0
5000
10000
15000
-1
20000
25000
30000
ω(cm )
α (a.u.)
2000
Perpendicular
1000
0
1550 nm
-1000
-2000
0
5000
10000
15000
-1
20000
25000
30000
ω(cm )
Figure 2-10: Dynamic Polarizability of X1 Σ state of NaK: The polarizability of the X1 Σ state of NaK is shown here at the equilibrium distance, R= 6.59a0 . The top panel shows the parallel
polarizability while the bottom shows the perpendicular polarizability. The dashed blue line is the frequency of 1550 nm laser.
The calculated points were done with the computational chemistry package CFOUR. The QZVPP basis set was used for both
sodium and potassium.
32
2.3
Dispersion Coefficients
As eluded to in the preceding chapters, the full electronic structure sometimes
can be very computationally expensive. That being the case, practical investigations
into the dynamics of atoms and molecules in the early days of quantum mechanics
would be severely limited by the computational abilities of the day. Most of the
computational complexity resides in the correlation of many electrons in the inner
region where one would expect large wavefunction overlap. In contrast, the situation
at long range is far less complicated. The forces in the long range region are much
weaker in comparison and can be treated perturbatively. In addition, the long range
has very little wavefunction overlap, thus simplifying the situation even further. The
issue of long range molecular interaction is also rather historic beginning with London
investigating the general form of molecular interaction in the 1930s [39].
In the modern context, the long range dispersion coefficient plays an important
role in the understanding of ultracold atomic and molecular scattering [16, 40, 1].
For neutral-neutral collisions, the dominant term in the long range potential is the
dispersive van der Waals interaction, which is a second order dipole-dipole interaction.
In the case of ground rotationless molecules, this interaction is purely isotropic [16]
and has a
− 1/r6
type of behavior at long range. The importance of obtaining accurate
long range behavior appears in the subsequent chapter on ultracold scattering.
While the early investigations were mainly focused on ground state atom-atom
interactions, fairly recent experimental control of polar molecules [41, 12] and excited
atoms [42] motivates theoretical investigations for these more complex systems. The
long range interactions for these systems are no longer isotropic and have rather large
anisotropy that cannot be neglected [16, 43]. The discussion on dispersion coefficients
will be organized as follows. A theoretical derivation of the form of the potentials
will first be presented. In the derivation, the connection between the previous work
33
on dynamic polarizabilities and the present work on dispersion coefficients will be
highlighted. Following the derivation, a few results on experimentally relevant systems
will then be presented.
2.3.1
Theoretical Background
The dispersion coefficient between two neutral atoms can be determined by integrating the product of their dynamic polarizabilities over imaginary frequencies [32].
The long range dispersion interaction of two neutral molecules can be expressed
as [44]
1
1
3
2
cos θA −
=− 6
C6 + γ20 C6
+
RAB
2
2
1
3
2
cos θB −
γ02 C6
+
2
2
γ22 C6 (2 cos θA cos θB − sin θA sin θB cos φ)2
2
2
2
− cos θA − cos θB .
6
Udisp.
(RAB , θA , θB , φ)
(2.11)
The coordinates R, θA , θB and φ = φA − φB are shown in figure 2-11. The first term
in the curly brackets in equation 2.11 is called the isotropic dispersion interaction
as it does not have any angular dependence. The other terms are all anisotropic
interactions for analogous reasons.
The second and third terms in the curly brackets closely resemble the form of
the equation of two interacting dipoles [45]. Despite the lengthy appearance of the
potential, there are in fact four dispersion coefficients to determine. The isotropic
dispersion coefficient C6 and the anisotropic coefficients γ20 C6 , γ02 C6 and γ22 C6 . Thus,
the task of characterizing the long range potential reduces to determining these four
coefficients for each system of interest.
34
Figure 2-11: Figure of Molecular Geometry: Shown here are the coordinates in the molecular frame between two molecules A and B.
The distance RAB is from the center of mass of A to the center
of mass of B. The z axis is typically taken to be the internuclear axis. The angles θA and θB are the angles between the
respective molecular axis and the internuclear axis. The angle
φ = φA − φB is the relative rotational orientation about the
intermolecular axis between the two molecules.
35
The derivation of these coefficients involve applying second order perturbation
theory, treating the electrostatic interaction between the electrons of one atom or
molecule with those of the other as the perturbation. The derivation is straight forward and can be found in many sources [32, 46, 47, 48]. Each formulation reduces
the evaluation of the dispersion coefficients into integrals involving the dipole polarizability. For the isotropic coefficient, the expression is [32]
3
C6 =
π
Z
∞
dν ᾱA (iν)ᾱB (iν)
(2.12)
0
For the current discussion, the frequency dependence of the polarizability is implicitly assumed. The previous section described the details of obtaining the polarizability. In principle, experimentally obtained polarizabilities may be used if available.
There should not be too much emphasis on the physical interpretation of the polarizability at complex frequencies. Consider the integrand9 over real frequencies vs the
integrand over complex frequencies: by using complex frequencies, the denominators
all become positive in the domain ν ∈ (0, ∞), thus making the integrand a monotonically decreasing function. This has advantages if one were to determine the dispersion
coefficient by numerical quadrature. The alternative, using real frequency ω, would be
filled with singularities making numerical quadrature a much more difficult process.
While it would have been sufficient to simply integrate the polarizabilities since
they are fitted to a function described by equation 2.10, the specific functional form
of the polarizability allows an analytic representation of the integral to be used.
Substituting each atom’s polarizability with the functional form in equation 2.10 into
equation 2.12 yields
9
In this case it would be simply the product of the two polarizabilities.
36
3
C6 =
π
Z
∞
dν
0
Ni
X
i=1
1+
Nj Z ∞
Ni X
3X
dν
=
π i=1 j=1 0
Nj
X
Āi
2 ·
ν
νAi
1+
Āi
2 ·
Z ∞
Nj
Ni X
3X
dν
Āi B̄j
=
π i=1 j=1
0
1+
j=1
ν
νAi
1+
1+
1
ν
νAi
B̄j
2
(2.13)
ν
νBj
B̄j
2
2 ·
(2.14)
ν
νBj
1+
1
ν
2 .
(2.15)
νBj
In the above equations the coefficients Ai and Bj do not depend on the frequency ν;
thus they may be pulled out of the integral. Once in this form, the integration can be
done analytically. The details will not be included here. The outline of the solution
involves splitting the integrand into two terms that have functional form 1/1 + ν 2 and
then recognizing that each of those integrals can be done via contour integration or
by relating it to a special function10 . The solution of the integral becomes
Z
∞
dν
I=
−∞
1+
1
ν
νAi
2 ·
1+
1
ν
2 = π
νAi νBj
π
=1
.
νAi + νBj
/νBj + 1/νAi
(2.16)
νBj
Since the integrand is symmetric over ν with respect to positive and negative
values, we can take half since the original integral is from 0 to ∞, and substitute it
back into the equation for C6 to get our final result.
N
Nj
i X
3X
Ai Bj
C6 =
1
2 i=1 j=1 /νAi + 1/νBj
(2.17)
This result is analogous to the Slater-Kirkwood formula [49, 50]. To reiterate,
10
The special function is called the Student’s t-distribution with the characteristic parameter being
ν = 1. The form of the integral is obtained by integrating the distribution from −∞ to +∞ and by
definition it is equal to unity.
37
the main motivation here was to determine the analytic representation of the C6
coefficient in terms of the parameters used to fit the individual polarizabilities. The
advantage is that there is no loss of precision due to numerical issues as the numerical
quadrature is not used. In addition, the current formulation makes use of the fact
that the integrand is a monotonically decreasing function of ν to minimize the error
in C6 due to the cutoff in the sums for the polarizabilities. The cutoff reflects the
computational difficulty in obtaining very high frequency calculations. The integrand
in those regions has already decayed appreciably in comparison to low frequencies.
Thus the contributions to the value of C6 from high frequencies is much less.
The other anisotropic dispersion coefficients γ20 , γ02 and γ22 are derived in analogous manners to the isotropic case. The only difference is that the average polarizabilities, ᾱ, are replaced by the “difference” polarizability ∆α [32].
Z
1 ∞
dν ∆αA (iν)ᾱB (iν),
γ20 C6 =
π 0
Z
1 ∞
γ02 C6 =
dν ᾱA (iν)∆αB (iν),
π 0
Z
1 ∞
γ22 C6 =
dν ∆αA (iν)∆αB (iν).
π 0
(2.18)
(2.19)
(2.20)
If the set of parameters for ᾱ can be denoted as {Āi , ν̄Ai } and the set of parameters
for ∆α as {Ãi , ν̃Ai }, the resulting expressions for the anisotropic parameters become
38
N
Nj
i X
1X
γ20 C6 =
2 i=1 j=1
N
Nj
i X
1X
γ02 C6 =
2 i=1 j=1
N
Nj
i X
1X
γ22 C6 =
2 i=1 j=1
Ãi B̄j
1 ,
+
ν̃Ai
ν̄Bj
(2.21)
Āi B̃j
1 ,
+ ν̃Bj
ν̄Ai
(2.22)
Ãi B̃j
1 .
+ ν̃Bj
ν̃Ai
(2.23)
1
1
1
In the case of ground state alkali and alkaline earth atoms, the polarizability tensor
in the atomic frame is proportional to the identity matrix. As a result, the difference
polarizability for these atoms is equal to zero.
This concludes the derivation of the dispersion coefficients in terms of the fit
parameters of the polarizability. The following section uses the polarizabilities and
equations 2.17 and 2.21 to obtain the dispersion coefficient for a number of molecular
systems.
2.3.2
Results for Select Systems
The main results can be found in table 2.2. The parameters for the polarizability
of the molecules and excited atoms can be found in appendix A. The systems in
table 2.2 all involve polar molecules with SrLi and LiYb being mentioned in the
previous subsections. The system LiYb∗ is a unique system as it involves a nonground state atom, Yb∗ . In addition, the Yb atom is about 30 times more massive
than the Li atom making this a highly mass-mismatched mixture. The creation of
these types of mass imbalanced molecules are important in the future investigations
on Effimov trimers [51].
A few remarks can be made on the specific systems. In every case, the isotropic
dispersion coefficient has the largest value, followed by the γ20 C6 or γ20 C6 . In both
cases, SrLi and LiY b, the γ22 C6 coefficient is the weakest; however, it is large enough
39
and cannot be ignored.
Table 2.2: Table of Dispersion Coefficients:
Presented here are the
values for the isotropic and anisotropic dispersion coefficient. For
each system, the molecule is placed at the equilibrium position
and its dynamic polarizability is calculated via ab initio methods.
The ground state atoms use parameters found in the NIST Atomic
Spectra Database [31] and conform to the functional form in the
text. The Yb∗ polarizability is also calculated using ab initio
programs. The value of the dispersion coefficient is in a.u. (Eh a60 )
System
SrLi+Li
SrLi+Li
SrLi+Sr
SrLi+Sr
SrLi+SrLi
SrLi+SrLi
SrLi+SrLi
LiYb+Li
LiYb+Li
LiYb+Yb
LiYb+Yb
LiYb+LiYb
LiYb+LiYb
LiYb+LiYb
Li+Yb∗
Li+Yb∗
Quantity
C6
γ20 C6
C6
γ20 C6
C6
γ20 C6
γ22 C6
C6
γ20 C6
C6
γ20 C6
C6
γ20 C6
γ22 C6
C6
γ02 C6
Value
3130.80
623.42
4683.19
854.61
7308.33
1419.91
989.18
3086.36
775.76
3469.65
779.79
7120.64
1731.36
1432.56
2219.73
240.34
In the case of SrLi the dispersion coefficient is determined for a number of internuclear radii. This permitted investigations into the vibrational dependencies of
the dispersion coefficient. This process can be determined by evaluating matrix elements over the vibrational wavefunction. Figure 2-12 depicts this dependence of
the isotropic and anisotropic dispersion coefficient for a pair of
88
Sr7 Li molecules in
the same vibrational state. The calculation was done using the program VIBROT,
which is a part of the MOLCAS quantum chemistry package [52]. The plot shows
a few features that are consistent with some basic asymptotic limits. For example,
40
8000
C6 Iso
γ20C6
γ22C6
Limit
C6 (a.u.)
6000
4000
2000
0
0
2
4
6
Vibrational Quantum Number
Figure 2-12: Vibrationally Dependent Dispersion Coefficients Between Similar 88 Sr7 Li Molecules: The leading coefficient of
the dispersion interaction between two similar 88 Sr7 Li molecules
is plotted as a function of vibrational quantum number ν. The
solid black line is the isotropic dispersion coefficient, while the
red and blue solid curves represent anisotropic coefficients as
described in the text. The dashed line is a flat line at 8436
a.u. that represents the limit as the constituent atoms are
all very far apart from one another. The value is determined
from a combinatory relationship between all four of the atoms,
C6limit = C6LiLi + 2C6SrLi + C6SrSr .
41
equation 2.7 explains that as the intermolecular distance increases, the dynamic polarizability reduces to the sum of the atomic values. Substituting this into the equation
for the dispersion coefficient, Eq. 2.12 will obtain the large internuclear separation
limit of the dispersion coefficient. For two SrLi molecules, the resulting limit is
lim
RSrLi →∞
C6,SrLi+SrLi
Z
3 ∞
= lim
dν ᾱSrLi ᾱSrLi
RSrLi →∞ π 0
Z ∞
3
dν (ᾱSr + ᾱLi ) (ᾱSr + ᾱLi )
=
π 0
Z
3 ∞
=
dν (ᾱSr ᾱSr + 2ᾱSr ᾱLi + ᾱLi ᾱLi )
π 0
(2.24)
= C6,Sr2 + 2C6,SrLi + C6,Li2 .
The result is a simple pairwise combinatory expression. In an analogous manner, one
would expect the anisotropic values to also tend to zero as the ground state atomic
anisotropic coefficients are equally zero. Figure 2-12 for the most part reflects both of
these limits, although technical issues prevented results for highly vibrational states
to explicitly show this. Specifically, there are certain hard coded parameters in the
VIBROT program that prevented the calculation of highly vibrational states11 .
To give an impression of the potentials at long range, figure 2-13 is an example
of a number of long range potentials constructed using the calculated dispersion
coefficients for the 6 Li174 Yb∗ system. The Zeeman and hyperfine interactions12 are
also included in this figure to give a accurate impression of the potential landscape.
It should be noted that this specific isotope of ytterbium does not have any hyperfine
structure. The result of the Zeeman and hyperfine interactions greatly complicate
the asymptotic channel structure, as there are many asymptotic channels in a small
11
These parameters were changed in the source code and the routine was recompiled; however,
other instabilities occurred likely connected to these hard coded values.
12
See section 3.2.1.1 for more information on the Zeeman and Hyperfine interactions.
42
Figure 2-13: LiYb∗ Long Range Potentials: Depicted here are some of
the long range potentials for the LiYb∗ system at a magnetic
field of 100G. These potentials all have even ` ≤ 6 and are
constructed using the long range coefficients found in table 2.2.
Included in these potentials are the Zeeman and hyperfine interactions, which increase the number of asymptotic channels.
There are 85 different potentials included in this figure.
43
energy range. The key feature of figure 2-13 is the energy scales associated at regions.
The figure shows the long range on the order of 10−7 a.u., while the short range would
be on the order of 10−2 a.u. This is not to imply that the long range is not important,
but emphasizes that there is a dramatic difference in energy scales between the two
regions. It is this physical picture that our MQDT formulation takes advantage of to
simplify the theoretical description and will be discussed in chapter 3.
2.4
Chapter Summary
Presented in the preceding chapter were some results in ab initio electronic structure calculations. The discussion is predominantly application driven with a variety
of systems presented. The chapter was divided into three topics. The first went
through the process of constructing a PES for the LiYbLi system. The motivation in
that work was to produce a surface that would be used to explore reactive scattering.
This surface would be used as a standard to test the validity of a number of theories
that attempt to explain the reaction dynamics without relying on computationally
expensive numerical techniques.
Also presented were the dynamic polarizabilities for a variety of systems. Two
separate methods of obtaining the polarizability were highlighted. The first, sumover-states way was shown for the CaYb+ system, which calculated the dynamic
polarizability over a large number of states. The other method added dipolar fields
to the computational Hamiltonian and calculated the polarizability at a number of
frequencies. A fitting procedure is then employed to extract the parameters needed
to characterize the polarizability.
Lastly, the polarizabilities were employed to calculate the long range interaction
between atoms and molecules. It was emphasized that while numerical quadrature
techniques could be used, an analytical solution to the integral was used to obtain
44
dispersion coefficients. These computations permitted the construction of potentials
that contained the correct long range behavior, which can then be used as input in
numerical studies.
45
Chapter 3
Ultracold Atomic Interaction
In this chapter a multichannel quantum defect theory for magnetic Feshbach resonances in the interaction of two heteronuclear alkali atoms is presented. Feshbach
resonances are one of the principle mechanisms in controlling cold atomic gases. A
Feshbach resonance is illustrated in Figure 3-1. It occurs when a bound state of a
closed channel energetically crosses the scattering state of an open channel. The scattering state in the ultracold regime is energetically right above the threshold of the
open channel, with zero energy being right at threshold. The open channel is sometimes called the entrance channel in some of the literature. In the present discussion,
an external magnetic field is applied to the system to provide a control mechanism on
the collisional process. The magnetic field controls the energy difference between the
bound state and the collisional energy, as is indicated by the red arrow in figure 3-1.
This ability to manipulate the energy difference allows one to tune to a magnetic field
around a Feshbach resonance. Essential to this magnetic control would be to have the
best possible description of the interaction around these resonances to probe different
aspects.
Historically, both numerical and other theoretical studies have been able to partially characterize s-wave Feshbach resonances with support for higher partial waves
lacking in comparison. The present work provides a systematic and unified descrip-
46
tion for resonances in any partial waves. The following chapter will go through the
MQDT treatment of the complexities caused by the inclusion of an external magnetic
field and present the new description through a sample system, 6 Li40 K. The focus
here is on the application of theory to zero energy magnetic Feshbach resonances.
3.1
Introduction
Magnetic Feshbach resonance [53] plays an important role in studies and applications of cold atoms (see [1] and references therein). As mentioned, in the context of
few-body and many-body physics, it is known for being the key mechanism for controlling and tuning of atomic interactions. In the context of more traditional atomic
physics, it represents one of the most precise experimental measurements that can be
carried out in the sub-thermal temperature regime and can be used to calibrate the
understanding of atom-atom interactions (see, e.g., Refs. [54, 55, 56]).
Being intrinsically a multichannel phenomenon that often involves a large number
of channels, a thorough understanding of magnetic Feshbach resonances, or more
generally the understanding of atomic interactions in a magnetic field, remains a
complex task. Around an s wave resonance, the parametrization of the scattering
length [57, 58, 1]
al=0 (B) = abgl=0
∆Bl=0
1−
B − B0l=0
,
(3.1)
in terms of three parameters (the resonance position B0l=0 , the background scattering
length abgl=0 , and the width parameter ∆Bl=0 ) has greatly simplified the description of
ultracold atomic interaction in a magnetic field and greatly facilitated its applications
in cold-atom physics [1]. For resonances in nonzero partial waves (e.g. Refs. [59, 3,
60, 61, 62, 63, 64, 65, 56]), however, similar parametrization has not existed until very
recently [66], due to the fact that they cannot be described using the standard effect
range theory [67, 68, 69]. Furthermore, even for an s wave resonance, the description
47
Figure 3-1: Picture of Two Channel Magnetic Feshbach Resonance:
A simple two channel depiction of a Feshbach resonance. In this
figure, the black channel is the open channel and the red channel is the closed channel. The closed channel supports a bound
state right depicted by the red horizontal line. The Feshbach
Resonance occurs when a bound state of the closed potential
crosses the collisional energy. The collision energy for the ultracold regime is near the threshold of the open channel. The
magnetic field comes into play by moving the position of the
bound state relative the the threshold. This “tuning” of the
closed bound state allows one to realize the Feshbach resonance.
48
in Eq. (3.1) is generally insufficient to fully characterize ultracold atomic interactions
if the resonance is not of the “broad” type [70, 71, 58, 1, 66].
In Ref. [66], a uniform parametrization of magnetic Feshbach resonances in arbitrary partial waves is derived with analytical descriptions of ultracold atomic interactions around them. The theory is based on the MQDT of Ref. [72] and on the QDT
expansion of Ref. [73]. It has been shown that a magnetic Feshbach resonance in an
arbitrary partial wave l, and the ultracold atomic interactions around the resonance,
can be fully characterized using a set of five parameters. They can either be the set
c0
of B0l , Kbgl
, gres , dBl , and sE (or C6 ), or the set of B0l , e
abgl , ∆Bl , δµl and sE (or C6 ),
with the second set being the more direct generalization of the s wave parametrization of Eq. (3.1) . Here e
abgl is a generalized background scattering length, and is
well defined for all l, δµl is a differential magnetic moment, and sE is the energy
scale associated with the long-range van der Waals interaction, −C6 /R6 , which was
discussed in the previous chapter. Used with the QDT expansion, or the generalized
effective-range expansion derived from it [73, 66], they give accurate analytic descriptions of atomic interactions around a magnetic Feshbach resonance, not only of the
scattering properties, but also of the binding energies of a Feshbach molecule and of
scattering at negative energies [74]. The parametrization works the same for both
broad and narrow resonances, and resonances of intermediate characteristics.
The present work gives a more detailed presentation of MQDT for heteronuclear
atomic interactions in a magnetic field [9, 66], with a focus on its application in
the description of Feshbach resonances, especially in the determination of resonance
parameters. Using the example of 6 Li40 K [65, 54, 9], it will be shown that MQDT
provides an efficient and uniform description of a large number of resonances, in
different partial waves and over a wide range of fields, from the same few parameters
that characterize atomic interactions in the absence of the field [72]. All resonances
49
in aa, ab, and ba channels1 , in partial waves s (l = 0) through h (l = 5), and in fields
of 0 G through 1000 G, are fully characterized using the parameters introduced in
Ref. [66].
3.2
MQDT for Atomic Interaction in a Magnetic
Field
In the subsequent sections, the MQDT formulation will be presented. the asymptotic channel structure will be defined and formalized incorporating the effects of
an external magnetic field. The remainder of this section formulates the MQDT
for atomic interaction accounting for the Zeeman splitting, hyperfine interaction and
dispersion interaction.
3.2.1
Fragmentation Channels for Two Heteronuclear Group
I Atoms in a Magnetic Field
The following section will define and construct the channel structure used throughout. The channel structure will gradually be built up by first considering the case of
a single atom in a magnetic field and then applying those results for the two atom
channel description.
3.2.1.1
A Single Group I Atom in a Magnetic Field
The Zeeman effect of a group I atom in a magnetic field B is well known [75]. The
results will be summarized here with the intention of defining the notation.
A group I atom, with an electronic angular momentum of JA = 1/2 and a nuclear
1
The channel labeling will be explained in the next section.
50
spin2 of IA in a magnetic field has 2(2IA + 1) states that are fully split. The states are
labeled by |FA , mf i in a weak field, where FA = JA + IA , and by |JA , mJA ; IA , mIA i
in a strong field. For intermediate fields, both labeling schemes can in principle still
be used to uniquely identify a state, but are no longer ideal since quantum numbers
other than mf = mJA + mIA are generally no longer good quantum numbers.
The states will presently be identified using the notation of |(mf )α i, to emphasize
that if there is interest in a wide range of magnetic fields, only mf is a good quantum
number, whereas different FA states are generally coupled. Specifically, in the singleatom JA IA basis, with basis states |JA , mJA ; IA , mIA i to be denoted by a simpler
notation of |mJA ; mIA i, the states of a group I atom in a magnetic field B are given,
for |mf | = IA + 1/2, by
|(IA + 1/2)0 i = |1/2; IA i ,
(3.2)
|(−IA − 1/2)0 i = |−1/2; −IA i .
(3.3)
For mf = −IA + 1/2, · · · , IA − 1/2, there are two states corresponding to each mf .
They are distinguished by α = 1, 2, and are given by
|(mf )1 i = sin(θB /2) |1/2; mf − 1/2i
− cos(θB /2) |−1/2; mf + 1/2i ,
(3.4)
|(mf )2 i = cos(θB /2) |1/2; mf − 1/2i
+ sin(θB /2) |−1/2; mf + 1/2i .
2
(3.5)
The nuclear spins have been well studied for many of the experimentally viable systems. All of
the nuclear spins have been found in tables such as in [76].
51
IA = 3 /2
1 .5
1 .0
g
(+ 2 )0
(+ 1 )2
(0 )2
(-1 )2
h
f
0 .5
0 .0
d
E
A
/ | ∆E
h f
A
|
e
-0 .5
(-2 )
c
b
-1 .0
(-1 )1
(0 )1
(+ 1 )1
a
-1 .5
0 .0
0 .5
0
1 .0
1 .5
2 .0
2 .5
|x |
Figure 3-2: Energy Diagram of Group I Atom with IA = 3/2 in a
magnetic field: Illustration of the energy diagram for a group
I atom with IA = 3/2 in a magnetic field. It applies to 7 Li,
23
Na, 39 K, and 87 Rb, upon ignoring the small correction due to
gI . Here x = (gJ −gI )µB B/∆EAhf is a scaled magnetic field. Both
the alphabetic labeling and the more detailed labeling of states
are illustrated.
52
IA = 1
1 .5
(+ 3 /2 )
0
(+ 1 /2 )
1 .0
(-1 /2 )
2
2
0 .0
E
A
/ | ∆E
h f
A
|
0 .5
-0 .5
(-3 /2 )
0
-1 .0
(-1 /2 )
(+ 1 /2 )
-1 .5
0 .0
0 .5
1 .0
1 .5
2 .0
1
1
2 .5
|x |
Figure 3-3: Energy Diagram of Group I Atom with IA = 1 in a magnetic field: Illustration of the energy diagram for a group I
atom with IA = 1 in a magnetic field, applicable to 6 Li.
53
IA = 4
1 .5
(-7 /2 )1
(-5 /2 )1
1 .0
(+ 5 /2 )
1
(+ 7 /2 )
(+ 9 /2 )
0
0 .0
E
A
/ | ∆E
h f
A
|
0 .5
1
-0 .5
(+ 7 /2 )
(+ 5 /2 )
2
2
-1 .0
(-7 /2 )2
(-9 /2 )0
-1 .5
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
|x |
Figure 3-4: Energy Diagram of Group I Atom with IA = 4 in a magnetic field: Illustration of the energy diagram for a group I
atom with IA = 4 in a magnetic field, applicable to 40 K.
54
In these states, θB (mf ) takes values between 0 and π and is defined by
r
1−
tan θB (mf ) =
2mf
2IA +1
2mf
2IA +1
2
,
+x
(3.6)
the quantity x is defined by x = (gJ − gI )µB B/∆EAhf , in which µB is the Bohr magneton and gJ and gI are the gyromagnetic ratios for the electron and the nucleus, respectively
[77].
The
hyperfine
splitting
∆EAhf
is
defined
as
∆EAhf
≡
EA (FA = IA + 1/2) − E(FA = IA − 1/2) in the absence of a magnetic field. It is
positive for most alkali-metal atoms, but negative for
40
K. In defining these states, a
phase convention is chosen such that
B→0
|(mf )1 i ∼ |FA = IA − 1/2, mf i
B→0
|(mf )2 i ∼ |FA = IA + 1/2, mf i .
The energies of the states |(mf )α i are given, for |mf | = IA + 1/2, by
EA ((IA + 1/2)0 ) =
∆EAhf
EA ((−IA − 1/2)0 ) =
∆EAhf
1 gJ + 2IA gI
IA
+
x ,
2IA + 1 2 gJ − gI
1 gJ + 2IA gI
IA
−
x ,
2IA + 1 2 gJ − gI
(3.7)
(3.8)
and for other states by [75]
1
1
2mf gI
hf
EA ((mf )1 ) =
∆EA −
+
x
2
2IA + 1 gJ − gI
1/2 #
4mf
− 1+
x + x2
,
2IA + 1
55
(3.9)
1
1
2mf gI
hf
EA ((mf )2 ) =
∆EA −
+
x
2
2IA + 1 gJ − gI
1/2 #
4mf
.
x + x2
+ 1+
2IA + 1
(3.10)
For ∆EAhf > 0, the state (mf )2 has higher energy than (mf )1 . The reverse is true for
∆EAhf < 0
The other common labeling of states of a group I atom in a magnetic field is
alphabetic ordering, a, b, c, . . . , in the order of increasing energy [1]. This labeling is
very concise and can be convenient, but is, at times, not sufficiently informative. Both
sets of notation are utilized for convenience and for connection with earlier work, but
I will defer to the |(mf )α i notation when necessary.
Figure 3-2 illustrates the energy diagram for alkali-metal atoms 7 Li,
23
Na,
39
K,
and 87 Rb, all with IA = 3/2. It shows that for ∆EAhf > 0, the lowest atomic state, the
a state, is always |(IA − 1/2)1 i, and the b state is always |(IA − 3/2)1 i. Figure 3-3
depicts the energy diagram for a 6 Li atom, which has an IA = 1. Figure 3-4 depicts
the energy diagram for a
40
K atom, which has an IA = 4. It illustrates that for
∆EAhf < 0, the lowest atomic state, the a state, is always |(−IA − 1/2)0 i, and the b
state is always |(−IA + 1/2)2 i.
Equations (3.2)-(3.5) define a unitary transformation UAB , with elements
hmJA ; mIA |(mf )α i, that relates the single atom JA IA basis to the single-atom states
in a magnetic field, which diagonalizes both the hyperfine and the magnetic interactions. It is a one dimensional unit matrix for |mf | = IA + 1/2, and is given by


B
B
 sin(θ /2) − cos(θ /2) 
UAB = 
 ,
B
B
cos(θ /2)
sin(θ /2)
for mf = −IA + 1/2, · · · , IA − 1/2.
56
(3.11)
3.2.1.2
Two Heteronuclear Group I Atoms in a Magnetic Field
The fragmentation channels for atomic interactions in a magnetic field are determined by two-atom states in a magnetic field. For two heteronuclear alkali-metal
atoms, labeled 1 and 2, in a magnetic field, there are 2(2I1 + 1) × 2(2I2 + 1) number of
channels for each partial wave l. They can each be identified by |(mf 1 )α1 , (mf 2 )α2 , li,
with |(mf x )αx i characterizing the state of atom x in a magnetic field in a way as
specified in Sec. 3.2.1.1. The channel threshold energies are given by E1 ((mf 1 )α1 ) +
E2 ((mf 2 )α2 ).
Ignoring the weak magnetic dipole-dipole [78, 57] and second-order spin-orbit
interactions [79, 80, 81], MF = mf 1 + mf 2 is conserved. The interaction and the
scattering matrices are block-diagonal with each block labeled by MF . The number
of (coupled) channels in each block is independent of l, and is determined by |MF |.
For instance, MF = ±(I1 + I2 + 1) blocks both have only a single channel, MF =
±(I1 + I2 ) blocks both have 4 channels, and blocks with |I1 − I2 | < |MF | < I1 + I2
have 4(I1 + I2 + 1 − |MF |) channels, etc. The total number of channels for each l adds
up to 2(2I1 + 1) × 2(2I2 + 1).
3.2.2
MQDT
MQDT for atomic interactions in a magnetic field is formally the same as MQDT
for atomic interactions in the absence of external fields. The theory takes full advantage of the physics that both the energy dependence [82] and the partial wave
dependence [83] of the atomic interaction around a threshold are dominated by effects of the long-range potential, which are described by a set of universal QDT
functions [74]. The short-range contribution is isolated to a short-range K c matrix
that is insensitive to both the energy and the partial wave. The details of the short
range K c matrix will be explained in the next subsection.
57
For an N -channel problem and at energies where all channels are open, MQDT
gives the physical K matrix, in the present case the K MF l , as [72]
c
c
K MF l = −(Zfc c − Zgc
K c )(Zfc s − Zgs
K c )−1 ,
(3.12)
c
c
c
(si , l) being the Zxy
s are N × N diagonal matrices with elements Zxy
where Zxy
functions [74] evaluated at scaled energies si = (E − Ei )/sE relative to the respective
channel threshold Ei . Here sE = (~2 /2µ)(1/β6 )2 and β6 = (2µC6 /~2 )1/4 are the
characteristic energy and the length scales, respectively, associated with the −C6 /2R6
van der Waals potential, with µ being the reduced mass. At energies where No
channels are open, and Nc = N − No channels are closed, MQDT gives [72]
c
c
c
c −1
K MF l = −(Zfc c − Zgc
Keff
)(Zfc s − Zgs
Keff
) ,
(3.13)
c
c
c
c −1 c
Keff
= Koo
+ Koc
(χc − Kcc
) Kco ,
(3.14)
where
c
c
in which χc is a Nc × Nc diagonal matrix with elements χcl (si , l) [74], and Koo
, Koc
,
c
c
, are submatrices of K c corresponding to open-open, open-closed, closed, and Kcc
Kco
open, and closed-closed channels, respectively. Equation (3.13) is formally the same
c
as Eq. (3.12), except that the K c matrix is replaced by Keff
that accounts for the
effects of closed channels. From the physical K matrix, the physical S matrix is
obtained from [40]
S MF l = (1 + iK MF l )(1 − iK MF l )−1 ,
(3.15)
from which all the scattering properties can be determined. At energies where all
channels are closed, the bound spectrum can be determined either from
det[χc (E) − K c ] = 0 ,
58
(3.16)
where K c is the full N × N K c matrix, or from equivalent effective single-channel or
effective multichannel problems, as discussed in Ref. [66].
The presence of a magnetic field changes the channel structure for atomic interactions, including both the channel wave functions and the channel threshold energies.
It is through these changes that atomic interactions depend on the magnetic field.
The change of channel threshold energies is discussed in Sec. 3.2.1. The change of
channel wave functions leads to a field-dependent K c matrix. This aspect is discussed
in the following subsection.
3.2.3
The K c Matrix for Heteronuclear Group I Systems
The short-range K c matrix for two heteronuclear group I atoms in a magnetic
field can be obtained through a frame transformation from the two single-channel
K c : KSc (, l) for spin singlet X 1 Σ state, and KTc (, l) for spin triplet A3 Σ state, which
are the same parameters describing atomic interactions in the absence of the magnetic field [72]. Specifically, the K c matrix, being a short-range K matrix, has much
simpler representation in condensation channels that diagonalize the short-range interactions [40, 72]. For interactions of group I atoms, a convenient choice of condensation channels is the JI coupled basis, in which a state is labeled by |(JMJ , IMI )li
where J = J1 + J2 and I = I1 + I2 . The short-range K c matrix is to an excellent
approximation diagonal in this basis with elements given by
h(J 0 MJ0 , I 0 MI0 )l|K c(JI) |(JMJ , IMI )li = KJc (, l)δJ 0 J δMJ0 MJ δI 0 I δMI0 MI ,
(3.17)
c
c
where KJ=0
corresponds to KSc , and KJ=1
corresponds to KTc [72].
From K c(JI) , the K c matrix in fragmentation channels in a magnetic field, as
59
defined in Sec. 3.2.1, is given by
K c (B) = U B† K c(JI) U B ,
(3.18)
U B = U JI (U1B ⊗ U2B ) .
(3.19)
where
Here U1B and U2B are unitary transformations, discussed earlier, describing the recoupling of atom 1 states and atom 2 states by the magnetic field, respectively. U JI
is a unitary transformation with elements, hJMJ |J1 mJ1 , J2 mJ2 ihIMI |I1 mI1 , I2 mI2 i,
given by the Clebsch-Gordon coefficients describing the coupling of J = J1 + J2 and
I = I1 + I2 .
The applicability of frame transformation depends critically on K c being a shortrange matrix. It is not applicable to the physical K matrix except at energies much
greater than the hyperfine and the magnetic splittings. The K c matrix, when regarded
as an operator associated with short-range atomic interactions, is dominated by electrostatic and exchange interactions, is to an excellent approximation not changed by
external fields and other weak interactions such as hyperfine interaction. External
fields and weak interactions only change the basis in which K c needs to be represented,
not the operator itself.
From the unitarity of U B , one can show rigorously that all off-diagonal elements
of K c (B) are proportional to ∆K c ≡ KSc − KTc . It is a general feature applicable
both in and in the absence of the magnetic field [72]. It implies that the coupling
of different fragmentation channels, which is responsible both for the existence of
Feshbach resonances and for inelastic scattering, is due to the difference between the
singlet and triplet scattering, as reflected, e.g., in the difference in s wave scattering
lengths ∆al=0 ≡ aSl=0 − aTl=0 , where aSl=0 and aTl=0 are the s wave scattering lengths for
the singlet and the triplet state, respectively.
60
3.3
MQDT for Magnetic Feshbach Resonances
The MQDT formulation of the previous section provides a systematic understanding of atomic interactions in a magnetic field, including both elastic and inelastic
processes, and over a wide range of energies and fields. It works the same for all
heteronuclear group I atoms, and uses the same parameters, KSc (, l) and KTc (, l),
that one would use for interactions in the absence of the field [72].
The 6 Li40 K system will be used to illustrate the theory. To set the benchmark
for future investigations and to keep the focus on concepts, the results will be limited to baseline MQDT results that ignore the energy and the partial-wave dependencies of the short-range parameters, namely in the approximation of KSc (, l) ≈
KSc ( = 0, l = 0) and KTc (, l) ≈ KTc ( = 0, l = 0). In this baseline description, all
aspects of cold atomic interactions, including parameters for all magnetic Feshbach
resonances in all partial waves, are determined from three parameters [72, 9]: the C6
coefficient, the singlet s wave scattering length aSl=0 , and the triplet s wave scattering
length aTl=0 , in addition to well known atomic parameters such as the atomic mass
and hyperfine splitting.
6
Li40 K is an interesting quantum system of two fermionic atoms. The magnetic
Feshbach resonances have been investigated experimentally by Wille et al. [65], and
theoretically by Hanna et al. [9]. 6 Li has I1 = 1 with a hyperfine splitting of ∆E1hf /h =
228.205 MHz and nuclear g-factor3 of gI1 = −4.477 × 10−4 [77].
40
K has I2 = 4 with
a hyperfine splitting of ∆E2hf /h = −1285.79 MHz and gI2 = 1.765 × 10−4 [77]. The
three parameters used in the present baseline MQDT description are C6 = 2322
a.u. [84, 54], aSl=0 = 52.50 a.u., and aTl=0 = 63.70 a.u. The singlet and the triplet
s wave scattering lengths are slightly adjusted from the values of Hanna et al. [9]
for better agreement with experimental s wave Feshbach resonance positions at low
3
The nuclear g-factors are on the order of melectron /mproton times smaller than the electron
g-factor.
61
fields. In terms of quantum defects, the adopted scattering lengths correspond to
µcS ( = 0, l = 0) = 0.3938 for the singlet state and µcT ( = 0, l = 0) = 3202 for the
triplet state.
The s wave scattering lengths give KSc (, l) ≈ KSc ( = 0, l = 0) = −16.91 and
KTc (, l) ≈ KTc ( = 0, l = 0) = 5.748 [72, 74], from which the K c matrix in a magnetic
field is constructed as discussed in Sec. 3.2.3. From the K c matrix, the MQDT of
Sec. 3.2.2 provides a complete description of cold atomic interactions in a B field.
The C6 coefficient gives a length scale of β6 = 81.56 a.u. and a corresponding energy
scale of sE /kB = 2.490 mK for 6 Li-40 K van der Waals interactions.
In understanding magnetic Feshbach resonances, the focus is on atomic interactions near the lowest threshold for each MF . For all energies below the second
lowest threshold, both scattering and bound states can be described by an effective
single-channel problem with [66]
c
c
c
c −1 c
Keff
= K11
+ K1c
(χc − Kcc
) Kc1 ,
(3.20)
or with its corresponding Klc0 matrix [66]
Klc0 (, B) =
c
Keff
(, l) − tan(πν0 /2)
.
c
1 + tan(πν0 /2)Keff
(, l)
(3.21)
In Eq. (3.20), “1” refers to the lowest-energy channel for the MF manifold, and “c”
refers to all other (closed) channels. In Eq. (3.21), ν0 = (2l + 1)/4, and = E − E1
is the energy relative to the lowest threshold.
c
In terms of Keff
, scattering below the second threshold, where only elastic scat-
tering is possible, is described by the physical K matrix
c
c
c
c −1
) ,
tan δl = (Zgc
Keff
− Zfc c )(Zfc s − Zgs
Keff
62
(3.22)
from which the elastic scattering cross section can be calculated in the standard
manner.
In terms of the value of Klc0 (, B) at zero energy (namely at the lowest threshold),
defined by
Klc0 (B) ≡ Klc0 ( = 0, B) ,
(3.23)
the resonance locations, B0l , of zero-energy magnetic Feshbach resonances in any
partial wave l, can be determined by the roots of Klc0 (B), namely as the solutions of
[66]
Klc0 (B) = 0 .
(3.24)
The generalized scattering length as a function of magnetic field, for an arbitrary
partial wave l, is given by [66]
e
al (B) = āl
1
(−1) + c0
Kl (B)
l
,
(3.25)
where āl = āsl β62l+1 is the mean scattering length (with scale included) with
āsl =
π2
,
24l+1 [Γ(l/2 + 1/4)Γ(l + 3/2)]2
being the scaled mean scattering length for partial wave l [73].
63
(3.26)
Table 3.1: Comparison of Select Feshbach Parameters for 6 Li40 K: Selective magnetic Feshbach resonances for the 6 Li-40 K system that
have been found experimentally [65], and investigated in an earlier
MQDT theory [9]. Before the work of Ref. [66], the width parameter ∆Bl was not rigorously defined for nonzero partial waves, and
was therefore not provided in the previous theory [9] for p wave
resonances.
Identification
Channel Mf
ba
-5
aa
-4
aa
-4
aa
-4
ab
-3
ab
-3
ab
-3
3.4
l
0
0
0
1
0
0
1
Present MQDT
B0l (G) ∆Bl (G)
215.0
0.245
157.6
0.141
167.6
0.107
247.1
0.447
149.7
0.242
158.8
0.425
259.1
0.692
Previous MQDT [9]
B0l (G) ∆Bl (G)
213.6
0.28
159.3
0.22
170.1
0.07
258.0
153.1
0.42
159.6
0.16
260.8
-
Experiment [65]
B0l (G)
215.6
157.6
168.2
249.0
149.2
159.5
263.0
Results for the 6Li40K System
This MQDT-based formalism provides a unified and a uniform understanding
of magnetic Feshbach resonances in all partial waves, as illustrated in Figs. 3-5-3-7
for 6 Li-40 K. Figure 3-5 depicts the reduced generalized scattering lengths, e
al (B)/āl ,
versus magnetic field for 6 Li40 K in the ba channel, for all partial waves s (l = 0)
through h (l = 5). Here ba refers to 6 Li in state b, namely the |(−1/2)1 i state (c.f.
Fig. 3-3), and 40 K in state a, namely the |(−9/2)0 i state (c.f. Fig. 3-4). It is the lowest
energy channel of the MF = −5 manifold with 4 coupled channels. Note that by
plotting the dimensionless reduced generalized scattering lengths, the understanding
of different partial waves are put on the same footing and can be depicted in the same
figure. The plots are grouped into an even parity group and an odd parity group, as
they are not coupled and can in principle be experimentally probed independently.
Figures 3-6 and 3-7 show similar results for the aa channel (MF = −4 manifold with 8
coupled channels) and the ab channel (MF = −3 manifold with 11 coupled channels),
64
Table 3.2: Table of Feshbach Parameters for LiK System: A complete
list of magnetic Feshbach resonances and their parameters for 6 Li40
K system in ba, aa, and ab channels, in partial wave s (l = 0)
through h (l = 5), and in magnetic fields from 0 through 1010 G.
A large number of resonances are predicted in the h wave (l = 5).
The columns B0l , ∆B and dBl are all in Gauss while the other
columns are unitless.
Mf
-5
-5
-4
-4
-4
-4
-4
-4
-4
-4
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
l
0
5
0
0
1
5
5
5
5
5
0
0
0
1
1
5
5
5
5
5
5
B0l
215.0
206.7
157.6
167.6
247.1
18.22
184.3
201.1
981.7
999.8
149.7
158.8
165.2
7.721
259.1
1.265
22.43
211.7
228.1
237.1
1009
∆B
e
abgl /āsl
0.245
1.631
2.189
3.509
0.141
1.648
0.107
1.608
0.447
3.415
0.506
3.477
0.053
3.705
1.100
3.468
0.028
3.578
0.596
3.458
0.242
1.697
0.425
1.579
0.004
1.490
0.092
1.557
0.692
3.284
0.0004 3.551
0.670
3.466
0.088
3.942
1.155
4.100
1.593
3.051
1.103
4.384
δµl /µB
1.776
1.769
1.660
1.800
0.152
2.854
1.678
1.830
1.890
1.956
1.588
1.758
1.815
1.005
0.163
2.308
2.372
1.510
1.672
1.760
1.901
65
c0
Kbgl
1.586
0.222
1.543
1.644
0.227
0.223
0.213
0.224
0.218
0.224
1.435
1.726
2.043
0.391
0.233
0.220
0.224
0.202
0.196
0.247
0.186
gres
-0.03036
-0.08129
-0.01605
-0.01379
-0.00142
-0.03027
-0.00190
-0.04213
-0.00111
-0.02440
-0.02528
-0.05489
-0.00059
-0.00152
-0.00234
-0.00002
-0.03329
-0.00285
-0.04189
-0.05696
-0.04604
ζres
dBl
0.006383 -0.6336
-0.003133 -1.703
0.003467 -0.3583
0.002795 -0.2839
-0.001255 -0.3460
-0.001158 -0.3932
-0.000077 -0.0420
-0.001609 -0.8535
-0.000043 -0.0217
-0.000930 -0.4624
0.005874 -0.5901
0.01060
-1.157
0.000096 -0.0121
-0.000776 -0.05597
-0.002008 -0.5308
-0.000001 -0.00034
-0.001271 -0.5203
-0.000121 -0.07011
-0.001826 -0.9286
-0.001972 -1.199
-0.002118 -0.8978
respectively.
The resonance positions, determined from Eq. (3.24), are part of the parameters
tabulated in Tables 3.1 and 3.2. Table 3.1 provides a subset of the present results
for magnetic Feshbach resonances that have been measured experimentally by Wille
et al. [65] and investigated theoretically by Hanna et al. [9]. The resonance positions,
which are the only parameters measured, are found to be consistent with the previous
theory and in good agreement with experiment.
Around each resonance, the dependence of the generalized scattering length on
the magnetic field can be characterized by [66]
e
al (B) = e
abgl 1 −
∆Bl
B − B0l
,
(3.27)
where e
abgl is the generalized background scattering length, and ∆Bl is one measure
of the width of the resonance. They can be computed from
e
al (B = B0l + ∆Bl ) = 0 ,
(3.28)
1
e
abgl = −e
al (B = B0l + ∆Bl ) ,
2
(3.29)
and
for an isolated resonance, and can be further refined, when necessary, by fitting e
al (B)
to Eq. (3.27) at B fields sufficiently close to B0l .
Except for the case of a “broad” resonance, the set of parameters B0l , e
abgl , and
∆Bl are generally still insufficient to characterize atomic interactions away from zero
energy [70, 71, 58, 1, 66]. For a complete characterization of ultracold atomic interactions around a magnetic Feshbach resonance, another parameter is generally needed,
the differential magnetic moment, δµl , which is the difference of the magnetic mo-
66
ments in interacting and non-interacting states [1, 66]. It is given by
dl (B) δµl =
,
dB B=B0l
(3.30)
and can be found from solutions of Klc0 (, B) = 0 at energies slightly different from
zero.
The parameters, B0l , e
abgl , ∆Bl , δµl , along with the C6 coefficient, form one set
of parameters that provide a complete description of atomic interaction around a
magnetic Feshbach resonance. It is the most direct generalization of the s wave
characterization [1] to other partial waves, and is convenient at zero energy (the
threshold). The parameters for 6 Li40 K in ba, aa, and ab channels are provided in
Table 3.2 for all Feshbach resonances that have been found in partial wave s through
h and in the field range of 0 G through 1009 G4 .
For descriptions of atomic interactions at energies away from the threshold, it is
c0
more convenient to use a different set of parameters, B0l , Kbgl
, gres , dBl , and sE (or
C6 ). They differ from the first set in three parameters that are related by [66]
c0
Kbgl
=
gres
1
,
e
abgl /āl − (−1)l
e
abgl /āl
=−
e
abgl /āl − (−1)l
dBl = −
δµl ∆Bl
sE
e
abgl /āl
∆Bl .
e
abgl /āl − (−1)l
(3.31)
,
(3.32)
(3.33)
With this set of parameters, which are also provided in Table 3.2, the Klc0 parameter
4
The reason for the unusual magnetic field range is due to a resonance for the Mf = 4 channel
at 999.8 G with a width of 0.596 G. A larger range was needed to examine this particular Feshbach
resonance.
67
is given by [66]
Klc0 (s , Bs )
=
c0
Kbgl
gres
1+
s − gres (Bs + 1)
,
(3.34)
around a Feshbach resonance, where Bs = (B −B0l )/dBl . This expression of Klc0 , as a
function of both energy and magnetic field, goes into the QDT expansion of Ref. [66]
to give an analytical description of atomic interactions in an arbitrary partial wave l
and in a B field around B0l , not only at the threshold but also in a range of energies
away from the threshold. Figure 3-8 depicts an example of the p wave phase shift,
more precisely sin2 δl=1 , around a p wave resonance in the ab channel located around
B0l ≈ 259 G. It shows that the results obtained using parameters of Table 3.2 and the
analytical QDT expansion of Ref. [66] are in good agreement with those computed
numerically from Eq. (3.22).
Also tabulated in Table 3.2 is a derived parameter
ζres ≡
gres
,
c0
(2l + 3)(2l − 1)Kbgl
(3.35)
which distinguishes a “broad” resonance (ζres 1) that follows the single-channel
universal behavior from a “narrow” resonance (ζres 1) that deviates strongly from
such behavior [66]. All resonances for 6 Li40 K are found to be narrow, a result that is
closely related to the fact that ∆al=0 ≡ aSl=0 − aTl=0 , and thus ∆K c is small for 6 Li40 K.
3.5
Chapter Summary
In conclusion, a multichannel quantum defect theory is presented for the interaction of heteronuclear group I atoms in a magnetic field, and applied to develop a theory
for magnetic Feshbach resonances in such systems. The theory provides a systematic
and a uniform description of resonances in all partial waves, and enables the descrip68
tion of a large number of resonances in terms of very few parameters. The simplified
description and the parameters provided by the theory will facilitate further theoretical investigations of two-body, few-body, and many-body systems around magnetic
Feshbach resonances, especially those in nonzero partial waves. These parametrizations were explicitly used in the presented 6 Li40 K system. In the future, it would
be interesting to explore other heteronuclear alkali-metal systems to look for broad
resonances in nonzero partial waves.
69
Reduced Generalized Scattering Length
4
Even Parity
MF = -5
l=0
l=2
l=4
3
2
1
0
Odd Parity
6
l=1
l=3
l=5
4
2
0
-2
0
100
200
300
400
Magnetic Field (G)
Figure 3-5: Reduced generalized scattering lengths for LiK in the
MF = −3 Manifold: Reduced generalized scattering lengths,
e
al (B)/āl , versus magnetic field for 6 Li40 K in the ba channel (corresponding to the manifold of MF = −5), for partial waves s
through h.
70
Reduced Generalized Scattering Length
4
Even Parity
MF = -4
l=0
l=2
l=4
3
2
1
0
Odd Parity
6
l=1
l=3
l=5
4
2
0
-2
0
100
200
300
400
Magnetic Field (G)
Figure 3-6: Reduced generalized scattering lengths for LiK in the
MF = −4 Manifold: Reduced generalized scattering lengths,
e
al (B)/āl , versus magnetic field for 6 Li40 K in the aa channel (corresponding to the manifold of MF = −4), for partial waves s
through h.
71
Reduced Generalized Scattering Length
4
Even Parity
MF = -3
l=0
l=2
l=4
3
2
1
0
Odd Parity
6
l=1
l=3
l=5
4
2
0
-2
0
100
200
300
400
Magnetic Field (G)
Figure 3-7: Reduced generalized scattering lengths for LiK in the
MF = −3 Manifold: Reduced generalized scattering lengths,
e
al (B)/āl , versus magnetic field for 6 Li40 K in the ab channel (corresponding to the manifold of MF = −3), for partial waves s
through h.
72
1
ab Channel
B = 260.0 G
p wave
0.6
2
sin δl=1
0.8
0.4
0.2
0
-3
4.020×10
-3
4.022×10
-3
4.024×10
-3
4.026×10
-3
4.028×10
εs
Figure 3-8: LiK ab channel plot of sin2 δl for p wave Resonance: Plot
of sin2 δl for p wave interaction in the ab channel as a function
of scaled energy s at a magnetic field of 260.0 G. The solid
line is the result of a numerical calculation from Eq. (3.22). The
symbols are results from the analytical QDT expansion [66] using
the parameters specified in Table 3.2 for the p wave resonance
located at B0l ≈ 259 G.
73
Chapter 4
Concluding Remarks
In the present thesis, two aspects of atomic and molecular investigations were conducted. Aspects of electronic structure calculations were discussed in chapter 2 and
in the following chapter, MQDT of magnetic Feshbach Resonances were presented.
In the chapter on ab initio electronic structure calculations a few topics were presented. A full PES for the ground state of lithium-ytterbium-lithium was constructed
in an effort to explore ultracold reactive scattering. A calculation of the reaction
rate is currently being carried out for the exchange reaction LiYb + Li → Li2 + Yb.
Pending the results of this calculation, we hope to be able to address the applicability
of less computationally intensive theories in the description of the reaction dynamics.
In addition, control over this reaction would be of experimental importance. In the
creation of ultracold molecules, there are often background atoms not used to form
molecules. It would be of great interest to minimize the effects of these background
atoms, which in most cases results in the loss of molecules from the trap.
Dynamic polarizabilities were also calculated for a variety of atoms and molecules.
The results of these fitting procedures are contained in appendix A. The last section
used the new information on polarizabilities in addition to well known polarizabilities
of ground state atoms to derive a number of dispersion coefficients for a variety of
atom-atom, atom-molecule and molecule-molecule systems. These calculations, which
74
include anisotropic dispersion coefficients, provide a landscape for both numerical and
analytic theories.
A multichannel quantum defect theory was also presented for atoms in an external
magnetic field. This enhanced understanding is manifested in the new description of
Feshbach resonances for arbitrary partial waves. While the present work focused on
the 6 Li40 K system, the presented material is systematic and aside from differences in
a few number of parameters would apply to all alkali systems.
In the future, our multichannel quantum defect theory can be used to better
constrain some of the input parameters such as the dispersion coefficient and the
two s-wave scattering lengths. In principle the resonance locations, and possibly
the widths, are dependent on these parameters. Direct experimental measurement
of a few resonance locations can be used with a fitting algorithm to fit the input
parameters. This is one potential direction to apply our multichannel quantum defect
theory.
Another interesting direction would be to examine systems with energy and/or
partial wave dependent s-wave scattering lengths. In our current formulation, the
s-wave scattering lengths are assumed be energy and partial wave independent. In
our test system, this assumption works well as all of the Feshbach resonances can be
reproduced using our theory. This may not be the case for other systems, especially
for some of these mass imbalanced systems. Addressing the energy and partial wave
dependence can potentially open the door to higher temperature investigations.
75
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Appendix A
Tables of Fitting Parameters for
Atomic and Molecular Dynamic
Polarizabilities
Presented in the following sections are the parameters used in 2.10 for a number
of diatomic molecules related to the present studies. Also included are the basis sets
used for each atom and the level of theory the calculations were conducted at. As
a reminder, the commonly used CCSD(T) level of theory does not permit dynamic
polarizability calculations. In each case, the parallel and perpendicular polarizabilities
are presented.
A.1
SrLi
See table A.1.
A.2
LiYb
See table A.2.
85
Table A.1: Ground State SrLi Dynamic Polarizability Fit Parameters: Def2-QZVPP basis set was used for both lithium and
strontium. This basis set can be found in the standard CFOUR
basis library. The parameters were fit to data points calculated
at the CCSD level of theory. The associated pseudopotential for
the inner 28 electrons was used for the strontium atom.
i
1
2
3
4
1
2
3
4
A.3
Ai (a30 )
αk
144.016
130.659
271.714
21.6014
α⊥
45.6356
8.73391
173.903
33.9479
Yb∗
See table A.3.
A.4
NaK
See tables A.4 and A.5
86
νi (cm−1 )
9690.39
12663.6
14719.6
18887.1
11994.2
14383.8
22799.8
25031.7
Table A.2: LiYb Dynamic Polarizability Fit Parameters: A specialized basis set was used for ytterbium, sent via private communication from the authors of [28]. The Def2-QZVPP basis found
in the CFOUR basis library was used for lithium. The parameters were fit to data points calculated at the CCSD level of theory.
The associated pseudopotential for the inner 28 electrons was also
used for the ytterbium atom. For this calculation, the molecule
is at the equilibrium separation
i
1
2
3
4
1
2
3
4
Ai (a30 )
αk
293.414
191.438
60.5147
42.3083
α⊥
17.3537
40.3263
12.5494
165.183
νi (cm−1 )
10596.9
15356.8
20016.0
23561.9
8420.48
15446.2
18391.7
26377.0
Table A.3: Metastable 3 P Ytterbium Dynamic Polarizability Fit Parameters: A specialized basis set was used for the ytterbium
atom sent via private communication from the authors of [28].
The parameters were fit to data points calculated at the CCSD
level of theory. The associated pseudopotential for the inner 28
electrons was also used.
i
1
2
3
1
2
Ai (a30 )
αk
218.874
113.29
24.8098
α⊥
162.31
80.7938
87
νi (cm−1 )
10354.5
15076.7
27851.4
10442.3
27620.9
Table A.4: NaK ground state X1 Σ Dynamic Polarizability Fit Parameters: The QZVPP basis was used for both sodium and
potassium. The parameters were fit to data points calculated at
the CCSD level of theory. The molecule is at the equilibrium
position for this calculation.
i
1
2
1
2
Ai (a30 )
αk
495.192
21.3802
α⊥
228.684
34.6618
νi (cm−1 )
13322.2
19813.6
17683.6
21595.1
Table A.5: NaK ground a3 Σ Dynamic Polarizability Fit Parameters:
The QZVPP basis was used for both sodium and potassium. The
parameters were fit to data points calculated at the CCSD level
of theory. The molecule is at the equilibrium position for this
calculation.
i
1
2
3
4
1
2
3
Ai (a30 )
αk
833.262
30.7737
6.15024
4.00357
α⊥
25.0094
226.393
114.856
88
νi (cm−1 )
9996.36
15694.0
23221.1
19027.4
4350.54
14330.7
20789.5