Control and Dynamics of Few-Body Systems at
Ultracold Temperatures
bySeth T. Rittenhouse
B.S., Western Washington University, 2002
M.S. (Mathematics), Western Washington University, 2003

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2009This thesis entitled:
Control and Dynamics of Few-Body Systems at Ultracold Temperatures
written by Seth T. Rittenhousehas been approved for the Department of Physics

Chris H. Greene

John L. Bohn

Date

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

iii
Rittenhouse, Seth T. (Ph.D., Physics)
Control and Dynamics of Few-Body Systems at Ultracold Temperatures
Thesis directed by Prof. Chris H. Greene

The role of few-body physics in ultracold gases has become increasingly important.
For instance, knowledge of two-body scattering has allowed tuning of the two-bodyinteraction strength through a Fano-Feshbach resonance in both Bose and Fermi gases.
A detailed understanding of three-body scattering provides information on the trap loss
processes which can determine the lifetime of a gas. It can also be used to explore basic
few-body behavior. The description of a gas of bosonic dimers relies on understanding
the four-body dimer-dimer scattering process. This dissertation concerns two related
concepts, the behavior of N -fermion systems with tunable two-body interactions andcontrollable few-body systems and their impact on ultracold gases. Both fall under theoverarching area of controlled few-body physics in many-body systems.
The N-fermion system is described in using the basic methods of hypersphericalcoordinates. This approach provides an intuitive picture of the qualitative behaviorof a degenerate Fermi gas by finding an effective one-dimensional potential that predicts the ground state energy and the RMS size of the gas in fair agreement with othertheoretical techniques. This study also provides an important initial connection between the techniques of few-body physics and many-body systems. By extending themethod to multi-component gases, I also predict a dynamic instability in three- andfour-component Fermi gases that is now within the range of experimental exploration.
In the second part of this dissertation, I explore three- and four-body problems
with s-wave interactions. By applying a new method based on the Lippmann-Schwinger
equation to the three-body problem, I develop scattering potentials for an arbitrary
three-body system with zero-range pseudo-potential interactions. By extending thisiv
approach to multi-channel interactions, a new class of three-body quasi-bound states
is predicted, creating true three-body Fano-Feshbach resonances. I also approximately
find the four-fermion hyperradial scattering potential. These potentials are then used tofind the zero-energy s-wave dimer-dimer scattering length to and accuracy larger thanprevious calculations. By exploring the dimer-dimer scattering potential, the full universal, energy-dependent, dimer-dimer scattering length is found, including the inelastic
processes of dimer dissociation and relaxation.

Dedication

To my family, thank you for your continuous support.

vi

Acknowledgements

First I would like to thank my advisor, Chris H. Greene. His guidance andexample over the last four years has taught me what it means to be a physicist. I would
also like to thank the people that I have gotten the pleasure of working with directly
over the last years; Jose D’Incao for many hours discussing Efimov physics; Nirav Mehta
for his helpful insight into scattering; Mike Cavagnero for guidance, encouragement andinsight into hyperspherical coordinates; Doerte Blume for help in deciphering MonteCarlo methods and their connections to the hyperspherical world; and Javier von Stecher
for Hartree-Fock calculations and the use of his four-boson potentials.
The Greene, Zach Walters, Dan Haxton, Jia Wang, Josh Dunn, Roman Curik,
Jason Pawlowski and Stefano Tonzani, have been a tremendous help in countless hours
of discussion on all topics in AMO physics. Brandon Peden took on the task of editing
this thesis and correcting the grammar, for which I and anyone else reading this must
perenially be grateful. Everyone on the theory floor in JILA have always been supportive
over the last four years. Thanks also to Evan Salim, Alan Bristow, Mike Martin andMike Thorpe for their experimental insight. Thanks to Jim Shepard for humoring myvarious crazy ideas, and introducing me to COLSYS, a new toy I plan on using for along time to come.
Thanks to Brad Johnson and Sebatian Schreiber for my first research experience.
I would especially like to thank Alexa Halford for keeping me sane, and my friends Rick
Karnesky and Brent Bailey. Last, but certainly not least, I would like to thank Pam

vii
Leland for unfailing administrative support.
This work was suppoted by funding from the National Science Foundation. Computational support was provided by the W. M. Keck foundation.

Contents

Chapter
1 Introduction

1

1.1

The many-fermion problem . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Three-body physics and the Efimov effect . . . . . . . . . . . . . . . . .

4

1.3

Dissertation structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2 Hyperspherical Coordinates

8

2.1

Defining the hyperangles: Delve’s coordinates . . . . . . . . . . . . . . .

9

2.2Hyperangular momentum and hyperspherical harmonics . . . . . . . . .

14

2.2.1

16

N bodies in a trap . . . . . . . . . . . . . . . . . . . . . . . . . .

3 N Interacting Fermions in a Trap Under the K-Harmonic Approximation

20

3.1

The K-harmonic approximation . . . . . . . . . . . . . . . . . . . . . . .

22

3.2

Bare zero-range s-wave interactions . . . . . . . . . . . . . . . . . . . . .

24

3.3

Result for bare zero-range interactions . . . . . . . . . . . . . . . . . . .

30

3.3.1

Repulsive interactions (a > 0) . . . . . . . . . . . . . . . . . . . .

31

3.3.2

Attractive interactions (a < 0) . . . . . . . . . . . . . . . . . . .

33

3.3.3

The large N limit. . . . . . . . . . . . . . . . . . . . . . . . . .

38

Density dependent interactions . . . . . . . . . . . . . . . . . . . . . . .

41

3.4.1

Operator matrix elements in the N → ∞ limit . . . . . . . . . .

42

3.4.2

Density dependent zero-range interactions . . . . . . . . . . . . .

44

3.4

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3.4.3

Finite effective interactions, kf0 a < ∞ . . . . . . . . . . . . . . .

46

3.4.4

Unitarity regime . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.4.5

Breathing mode excitations . . . . . . . . . . . . . . . . . . . . .

53

3.4.6

Multiple spin components . . . . . . . . . . . . . . . . . . . . . .

53

4 The Hypervectorial Method and Anisotropic Traps

59

4.1

The Hypervectorial Method . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.2

More density-dependent interactions . . . . . . . . . . . . . . . . . . . .

65

4.2.1

Results for the hypervectorial method . . . . . . . . . . . . . . .

67

Low energy excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.3

5 Green’s Functions and the Adiabatic Hyperspherical Method

73

5.1

The Hyperangular Green’s function . . . . . . . . . . . . . . . . . . . . .

75

5.2

The Three-Body Problem with Zero-Range Interactions . . . . . . . . .

78

5.2.1

Imposing symmetry . . . . . . . . . . . . . . . . . . . . . . . . .

82

5.2.2

Non-adiabatic couplings . . . . . . . . . . . . . . . . . . . . . . .

83

5.3

Three distinguishable interacting particles. . . . . . . . . . . . . . . . . .

85

5.4

Multi-channel interactions . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.4.1

Efimov States embedded in the continuum . . . . . . . . . . . . . 106

5.4.2

Non-adiabatic matrices for the multi-channel problem . . . . . . 111

6 Four Fermions and Dimer-Dimer Scattering
6.1

6.2

116

Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.1.1

Jacobi coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.1.2

Coalescence points and symmetry

6.1.3

Kinematic rotations . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.1.4

Hyperangular coordinates . . . . . . . . . . . . . . . . . . . . . . 124

Variational Basis Elements. . . . . . . . . . . . . . . . . 120. . . . . . . . . . . . . . . . . . . . . . . . . 134

x
6.2.1

Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.3

Four-fermion potentials and the dimer-dimer wavefunction . . . . . . . . 141

6.4

Dimer-Dimer Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.4.1

6.5

Energy dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Dimer-dimer relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7 Summary

159

Bibliography

163

Appendix
A Consructing Hyperspherical Harmonics

170

B Non-Adiabatic Corrections for the Three-Body Problem

173

C N -Body Recombination Rate

176

C.1 Scaling laws for N identical bosons . . . . . . . . . . . . . . . . . . . . . 179
C.2 Four-bosons: beyond scaling laws . . . . . . . . . . . . . . . . . . . . . . 180
D Dimer-Dimer Relaxation Rates

185

E Related Publications

189

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TablesTable
3.1

N , λ and CN /N 7/2 for the several filled shells. We can see that CN /N 7/2
pquickly converges to the Thomas-Fermi limiting value of 32 2/3/35π 3 ≈0.02408 to several digits. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

29

The possible permutation symmetries that may be imposed on the threebody system with s-wave interaction are given with the appropriate boundary conditions. B stands for a boson, F for a fermion and X for a distinguishable particle with an arbitrary mass. . . . . . . . . . . . . . . . . .

5.2

82

The possible tunable interaction regimes near the resonances of 6 Li are
given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Figures

Figure2.1

The tree that gives the standard spherical coordinates for a 3 dimensional
system is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

10

The canonical tree that gives a hyperangular parameterization for a ddimensional system is shown. . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

The tree structure used to correlate two subspaces to a single hyperradius. 12

2.4

The tree structure used to parameterize the hyperangles for N three
dimensional vectors is shown. Note that the dot at the end of eachbranch in the tree on the left stands for a spherical polar tree. . . . . . .

3.1

13

Values of CN divided by the large N limit, CN → 0.02408N 7/2 , versus
1/N are shown. The circles are the calculated value while the curve isthe fit stated in Eq. 3.23. Figure from Ref. [1]. . . . . . . . . . . . . . .

3.2

30

The dimensionless rescaled effective potential, Vef f /EN I as a function of
the rescaled hyperradius for kf0 a = 0 (solid), kf0 a = 1.19 (dashed) and
kf0 a = −0.40. (dot-dash). . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

31

The ground state energy in units of the non-interacting energy versus kf0 a
for 240 atoms calculated using the K harmonic method (curve) and using
Hartree-Fock (circles). Inset: the difference in the ground state energies
predicted be the K harmonic (EK ) and Hartree-Fock (EHF ). Clearly the
K harmonic energies are slightly higher than Hartree-Fock. . . . . . . .

32

xiii
3.4

The ground state average squared radius of the gas atoms in units of the
non-interacting rms squared radius is plotted versus kf0 a. The calculations considered 240 atoms in both the K harmonic method (curve) andHartree-Fock (squares). . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

32

The lowest breathing mode excitation (ω0 ) in units of the trap frequency
is plotted versus kf0 a for the K harmonic method (solid curve) and forthe sum rule (circles). Also shown as dashed curves are the lowest eightradial excitation frequencies predicted in the HF approximation. . . . .

3.6

33

The dimensionless rescaled potential curve Vef f /EN I is shown as a function of the rescaled hyperradius R0 for several values of kf a; kf a = kf ac(dashed), and from top to bottom kf a = −0.42, −0.66, −0.90, −1.53 (allsolid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

34

The ground state energy (in units of the non-interacting energy) versus
kf0 a for 240 atoms calculated using the K harmonic (curve) and HartreeFock (circles) methods. Inset: The difference in the ground state energies
predicted by the K harmonic (EK ) and Hartree-Fock (EHF ). Clearly the
K harmonic prediction is slightly higher. . . . . . . . . . . . . . . . . . .

3.8

35

The average squared radius of the Fermi gas ground state in units of the
non-interacting value is plotted versus kf0 a. The calculations are for 240
atoms in both the K harmonic method (curve) and Hartree-Fock (squares). 36

3.9

The frequency of the lowest energy radial transition in units of the trap
frequency versus kf0 a predicted by the K harmonic method (solid line)
and by the sum rule (circles). Also shown are the lowest eight radialtransitions predicted by Hartree-Fock. . . . . . . . . . . . . . . . . . . .

36

xiv
3.10 A portion of the energy spectrum vs kf0 a close to the critical point kf0 a =kf0 ac . Levels in the metastable region (cyan) decrease slowly while levelsin the collapse region (red) decrease very quickly. Energy levels above thebarrier in Vef f (green) live both in the collapse region and the metastableregion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.11 A plot of the location if the extrema in Eq./ 3.26 versus kf0 a is shown.
0
Examination of this plot tells us the behavior of Rminfor all allowed

values of kf0 a including the existence of the critical point kf0 ac located at
the minimum of the plot where the maximum and minimum coincide. .

39

3.12 The percentage difference between the energy found by minimizing Vef f
and the energy found by explicitly solving the hyperradial Schrödinger
equation for 240 atoms is show as a function of kf0 a. . . . . . . . . . . .

40

3.13 The breathing mode (in units of the oscillator frequency ω) in the large
N limit versus kf0 a. Note that as kf0 a → kf0 ac the frequency drops to zeroas the local minimum disappears. . . . . . . . . . . . . . . . . . . . . . .

41

3.14 The density-dependent interaction strength function ζ(kf a) is shown versus kf a. Figure from Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . .

45

3.15 The dimensionless ratio of the effective hyperradial potential to the noninteracting total energy is plotted as a function of the dimensionlessrescaled hyperradius, for several different repulsive interaction strengths.
The non-interacting limit kf0 a = 0 is shown as the sold curve; the dashedcurves show the renormalized effective potential for (bottom to top)
kf0 a = 2, kf0 a = 5 and kf0 a = 50. Also shown is the non-renormalized
effective potential with kf0 a = 5 (dotted curve). Figure from Ref. [2]. . .

47

xv
3.16 The ground state energy of the DFG in units of the non-interacting energy predicted by the K harmonic method (solid line) is plotted versusarctan kf0 a /π and compared with that predicted by the HF methodwith 2280 atoms (circles) [3]. The dashed line is the ground state energy predicted by the K harmonic method using the bare Fermi pseudopotential. Figure from Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . .

47

3.17 The average squared hyperradius of the two-component DFG groundstate in the large-N limit, divided by the non-interacting value for thisquantity, is plotted versus arctan kf0 a /π. Also shown are the values
predicted by the HF method with 2280 atoms (circles) [3]. The dashedline is the ground state energy predicted by the K harmonic method using
the bare Fermi pseudo-potential. Figure from Ref.. [2]. . . . . . . . . . .

48

3.18 The effective potential in units of the non-interacting energy is plotp
ted versus the hyperradius in units of hR2 iN I , for several interaction
strengths. The non-interacting limit kf0 a = 0 is shown as the solid curve
and the dashed curves show the renormalized effective potential for (topto bottom) kf0 a = −1 and kf0 a = −5. Also shown is the non-renormalized
effective potential with kf0 a = −1 (dotted curve). Figure from Ref. [2]. .

50

3.19 The breathing mode frequency ω0B is shown in units of the trap frequencyω versus arctan kf0 a /π . The solid curve shows the breathing modepredicted using the renormalized interaction while the dashed curve shows
the prediction based on the bare Fermi pseudo-potential. Figure from
Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1

54

The sub-hyperspherical trees that are used to parameterize the hypervectorial coordinates Rρ and Rz are shown. . . . . . . . . . . . . . . . . . .

61

xvi
4.2

The dimensionless effective hypervectorial potential is shown plotted asa function of Rρ0 and Rz0 for an interaction strength kf0 a = 0 and a trap
ratio of γ = 1/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

67

Contour plots of the dimensionless effective hypervectorial potential are
shown plotted as a function of Rρ0 and Rz0 for an interaction strength
kf0 a = −12 to 12 and a trap ratio of γ = 1/3. . . . . . . . . . . . . . . .

4.4

68

The transverse breathing mode predicted by the hypervectorial method isshown plotted versus the interaction strength kf0 a for trap ratios γ = 0.1
to 1 in steps of 0.1 from top to bottom. The kink in the seen in the γ = 1
frequency is due to a degeneracy between the longitudinal and transverse
frequencies in the non-interacting limit. . . . . . . . . . . . . . . . . . .

4.5

72

The longitudinal breathing mode predicted by the hypervectorial method
is shown plotted versus the interaction strength kf0 a for trap ratios γ = 0.1
to 1 in steps of 0.1 from top to bottom. The kink in the seen in the γ = 1
frequency is due to a degeneracy between the longitudinal and transverse
frequencies in the non-interacting limit. . . . . . . . . . . . . . . . . . .

5.1

72

All possible s-wave scattering lengths are shown for the lowest 3 Zeeman
states of Li6 from Ref. [4]. Each marked region gives a different set oflength scale discrepancies. Here a(k) is the scattering length between two
atoms in states |ii and |ji with k as the component not involved in theinteraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

xvii
5.2

(a)The the first four hyperangular eigenvalues are shown for an example
system where a(1) = a(3) and a(2) = 1000a(1) versus the hyperradius.
The solid black horizontal lines show the expected behavior for 3 identical resonantly interaction bosons. The dashed line gives the behavior of
two identical fermions interacting resonantly with a third distinguishable
particle. Dotted lines give the expected universal behavior for a single
resonant scattering length. Finally, the dot dashed line is the lowest expected free space behavior for three distinguishable free particles.(b)The
coupling strengths between the first and second (purple), the first andfourth (red), and the second and fourth (black) adiabatic potentials areshown as a function of R. . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

87

A schematic picture of the first four hyperradial potentials in region I isshown. The grey areas indicate regions where the potentials are changing
from one universal behavior to another. The blue region indicates theshort range region where the zero-range pseudo-potential not longer canbe applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

89

(a)-(e) Examples of the hyperangular eigenvalues from each region ofmagnetic field are shown as a function of the hyperradius in atomic units.
(f) The three s-wave scattering lengths are shown as a reference plottedversus the magnetic field strength. The dotted lines, from right, show themagnetic field at which the hyperangular eigenvalues from (a)-(e) were
found, B = 675, 695, 805, 820 and 845 gauss respectively. . . . . . . . . .

5.5

89

A schematic of the path for three-body recombination in region I is shown. 92

xviii
5.6

A schematic of the potentials and the path for three-body recombinationin region III is shown. Again the grey regions indicate a transition fromone universal potential behavior to another. The green line represents
the hyperradial potential for a deeply bound dimer state. The blue areais the short range region not described by zero-range interactions. . . . .

5.7

95

A schematic of the lowest hyperradial potential is shown with the pathfor three-body recombination to deeply bound states. The green line
represents the hyperradial potential for a deeply bound dimer state. Grey
areas indicate transition regions from one universal behavior to another,and the blue region indicates the short range regime. . . . . . . . . . . .

5.8

97

(a-c) The log of the three-body recombination rate from Eq. 5.58 for 6 Li
is shown for magnetic fields ranging from 834.15 to 1000 gauss is shownplotted versus a(1) , a(2) and a(3) respectively with η = 0.01 and theshort range length scale chosen to be approximately the van der Waals

5.9

length, r0 = rd ≈ 30 a.u. The beginning of an Efimov resonance can be2seen in (a) and (b) while the scaling law K3 ∝ a(3) can be seen in (c).

98

A schematic picture of the multi-channel two body interaction is shown.

100

5.10 (a) The first 300 adiabatic hyperradial potentials are shown for the multichannel three-boson problem in units of ε as a function of R/A33 . (b)The adiabatic three-body potentials near the second three-body scattering threshold are shown. The attractive Efimov diabat can be clearlyseen approaching the second scattering threshold.

. . . . . . . . . . . . 108

5.11 The hyperangular eigenvalue corresponding to the Efimov diabat is shown
as a function of R/A33 . (inset) The hyperangular eigenvalue is shown for
A13 = 0, 0.005A33 and 0.01A33 (black, red and green respectively). As
A13 increases the width of the avoided crossings that create the Efimov
diabat can be seen to increase. . . . . . . . . . . . . . . . . . . . . . . . 111

xix
5.12 The diagonal correction, −Qnn /2µ is shown in dimensions of the twobody threshold separation ε plotted as a function of R. The red dashedline shows 1/R3 scaling while the blue dot-dashed line shows 1/R2 scaling.113
6.1

The two Jacobi coordinates configurations possible in the four body problem are shown schematically. . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2

The hyperspherical tree used to parameterize the hyperangular coordinates in the four-fermion problem is shown. See Chapter 2 or Ref. [5] fordetails. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3

Surfaces surrounding the coalescence points in the body-fixed democratic
coordinates are shown for θ1 = π/2 and θ2 =πππ(a), (b), and
(c)
4
6
12

respectively. Blue surfaces surround interaction coalescence points while
red surfaces surround Pauli exclusion nodes. . . . . . . . . . . . . . . . . 134
6.4

The hyperradial potentials are shown (solid lines) calculated for a =100 as a function of R/a. Also shown are the expectation values of thevariational basis elements used (dashed curves). . . . . . . . . . . . . . . 144

6.5

The coupling strength between the dimer-dimer potential and the lowest
dimer-atom-atom potential is shown as a function of R/a. The blue
dashed line shows the position of the coupling peak at R/a ≈ 3.5. . . . . 145

6.6

An isosurface of the dimer-dimer probability density is shown. The surfaces are found by integrating the total probability over θ1 and θ2 andplotting with respect to the remaining democratic angles (φ1 , φ2 , φ3 ).
The peak probability always occurs in planar symmetry, θ1 = π/2. The
coloring (light to dark) indicate the value of θ2 at the peak. . . . . . . . 146

6.7

The same as Fig. 6.6, but only using the 2+2 basis function. The dashedgray lines are purely for perspective. . . . . . . . . . . . . . . . . . . . . 147

xx
6.8

The isosurfaces shown here are the same as in Fig. 6.6, but in a pleasing
environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.9

The real part of the energy dependent dimer-dimer scattering length isshown in dimensions of a plotted versus the collision energy in units ofthe binding energy. The calculation if done with one, two, three, four,and five adiabatic channels (blue, black, red, green, and purple curves
respectively) from the 8 basis element calculation. The red dashed line
shows add = 0.6a, the prediction of Ref. [6]. . . . . . . . . . . . . . . . . 149

6.10 The imaginary part of the energy dependent dimer-dimer scattering lengthis shown in dimensions of a plotted versus the collision energy in unitsof the binding energy. The calculation if done with one, two, three, four,and five adiabatic channels (blue, black, red, green, and purple curves
respectively) from the 8 basis element calculation. . . . . . . . . . . . . 150
6.11 The real (red) and imaginary (green) parts of the energy dependent
dimer-dimer scattering length is shown in dimensions of a plotted versus
the collision energy in units of the binding energy. Also shown is theenergy dependent scattering length using the effective range expansion.
Figure from Ref. [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.12 The real part of the energy dependent dimer-dimer scattering length isshown plotted as a function of the two-body scattering length in atomic
units is shown calculated for several collision energies; Ecol /kb =250nK,100nK,
25nK, 10nK, 2.5nK, 1nK, 10−1 nK, and 10−2 nK. Also shown is the zeroenergy prediction (black dashed curve). Figure from Ref. [7]. . . . . . . 153
6.13 The integrand from Eq. 6.55 is shown for a = 50r0 , 64r0 , 80r0 , and 100r0
as a function of R in units of a. . . . . . . . . . . . . . . . . . . . . . . . 156
6.14 The integrand from Eq. 6.55 is shown for various values of R/a as afunction of a in units of r0 . . . . . . . . . . . . . . . . . . . . . . . . . . 156

xxi
6.15 The relaxation rate is shown as a function of scattering length (see text).
The red curve represents all possible relaxation pathways. The blue dotdashed curve represents only the contribution to relaxation for processes
occurring at R > 10r0 . The green dashed line shows the contributions
for prosses occurring at R < 5r0 . Figure from Ref. [7] . . . . . . . . . . 157
A.1 The four possible hyperangular node types are shown. . . . . . . . . . . 170
C.1 (a)-(d) Schematic of the four-boson potentials from Ref. [8] are shown,
As the scattering length becomes more negative (a-d inorder) an inner
well appears and is pulled further down until a second Efimov state isbound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
C.2 A schematic of the four-body recombination process into an Efimov stateplus a free particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
C.3 The four-body recombination rate is shown plotted as a function of |a|in units of the range of the two-body interaction. . . . . . . . . . . . . . 184
D.1 (a) A schematic of the channels involved involved inthe dimer-dimer relazaxtion process is shown. The dimer-dimer potential (red curve) goes
through an inifinite number of crossing with deeply bound states (green
dashed curves). (b) The hyperradial behavior of the outgoing wavefunction is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Chapter 1

Introduction

Since the experimental realization of the first atomic Bose-Einstein condensatesin 1995 [9, 10, 11] and later the first degenerate gas of fermionic atoms [12, 13, 14, 15,
16, 17], the field of ultracold gases has seen an explosion of experimental capabilities andtheoretical interest, becoming a veritable playground of new and interesting quantum
phenomena. Microscopic, few-body interactions in these systems plays a central role [18,19, 20, 21, 22, 23]. For instance, under the influence of a of a Fano-Feshbach resonance,
the s-wave interaction strength in an ultra-cold atomic gas can be tuned from weak tostrong and attractive to repulsive. Further, few-body correlations can play an importantrole in determining experimental observables in such systems [23, 24, 25]. With thelarge variety of knobs to turn in these systems, they provide a perfect environment for
exploring fundamental quantum mechanics as well as probing new physical phenomena.
Further understanding of the few-body interactions can lead also to an understandingof the loss processes, such as the scaling of three-body recombination with the strengthof the interaction [26, 27, 28, 29], that govern the lifetimes of ultracold gases.
As a result of this versatility, the majority of this dissertation is focused on the fewbody control and dynamics present in ultra-cold atomic gases. This is not to say that themethods introduced here are exclusively applicable to this exciting field. Hypersphericalcoordinates have a long history of use in several areas of theoretical physics ranging
from nuclear physics [30, 31, 32, 5, 33] to atomic structure [34, 35, 36, 37, 38, 39] and2fundamental few-body scattering [40, 41].
Hyperspherical methods starts with a single concept, that of the hyperradius. As
the name suggests, the hyperradius is the extension of the concept of the radius of asphere beyond three dimensions. Generally, is proportional to the square root of thesum of the squares of all of the coordinates in a d dimensional Cartesian space, i.e.
v
u d
uX
R∝t
x2i .
(1.1)
i=1

All the remaining d−1 degrees of freedom are then encapsulated in a set of hyperangles,
denoted by Ω. In the cases considered in this thesis, the hyperradius can be considered
as a coordinate describing the RMS radius of N -bodies. In this context, it is clear thatsuch a coordinate is useful. For instance, if one wanted to describe some inelastic threebody process, for example A + B + C → AB + C where A, B, and C are three particles
and AB is a bound state between particles A and B, this can be described as a systemcoming in from large hyperradius (where A, B and C are all far from each other) on one
potential curve and leaving at large hyperradius (where the distance between AB and
C is large) on another. Further, if one considers an adiabatic approach in which thehyperradius is fixed and the Hamiltonian is diagonalized in the remaining hyperangulardegrees of freedom, all the continuum behavior of the systems is encapsulated in thehyperradius [36, 42, 43]. As with most adiabatic treatments, the hyperspherical method
is a “divide and conquer” idea. By finding the eigenfunctions and eigenvalues of asystem at fixed hyperradius, a set of 1D coupled potentials can be developed, and all of
our normal Schrödinger quantum mechanics intuition can be brought to bear upon the
resulting system.

1.1

The many-fermion problem
The many fermion problem with tunable s-wave interactions has generated a great

deal of interest over the years. The two-body physics of the Fano-Feshbach resonance3
is crucial in this system for understanding the control of the s-wave scattering length,a. By using such a resonance the mean-field interaction strength can be tuned througha wide range of values ranging from weakly attractive (a < 0) to weakly repulsive(a > 0). It was suggested by Leggett in 1980 [20] and later others [44, 21, 22] that thissystem provides a unique opportunity to study the crossover behavior between BCS typesuperfluidity, caused by momentum correlated pairs of fermions, to a Bose-Einsteincondensate (BEC) of diatomic molecules. Interestingly, this connection between two
different types of superfluidity was predicted to happen smoothly, without a phasechange in between. Since these predictions, the experimental realization of this crossover
has been achieved [12, 13, 14, 15, 16, 17], and theoretical descriptions of the stronglyinteracting regime abound, from quantum Monte Carlo treatments [45, 24] to extensionsof the BCS wave function into the unitarity regime [46, 47, 48, 49]. Recently, in theoriginal work of Tan [50, 51, 52] and others [53, 54, 55], a full description of the ultracold
Fermi gas has been found in terms of a single quantity, the so called contact parameter.
Despite the intense effort directed at this system, a complete description has yet to beachieved.
Understanding the BEC side of the resonance requires knowledge of the dimerdimer scattering length. Initially, this was found in the first Born approximation [56, 21]
and through mean field theory [56, 21] to be add ≈ 2a, and later by Pieri and Strinati,through a diagrammatic approach, to be add ≈ 0.73a [57]. It wasn’t until a complete
solution to the four-fermion problem was extracted in 2004 by Petrov, Salomon and
Shlyapnikov [23, 6] that the actual dimer-dimer scattering length, in large a limit, wasfound, add = 0.60(2)a. Through correlated Gaussian methods, von Stecher, Blumeand Greene [58, 8] were able to refine this prediction and extract the effective range
correction. In a basic sense, the four-fermion problem represents the simplest systemthat could exhibit the physics of the crossover problem.

4

1.2

Three-body physics and the Efimov effect
While basic few body physics is necessary for understanding the behavior of ul-

tracold many body systems, there is still considerable interest in the behavior of purefew-body dynamics. One area that is of considerable current interest is in the famous
Efimov effect first predicted in 1970 by Vitaly Efimov [59, 32]. Efimov studies a threeboson system with short-range two-body interactions in which each two-body system isinfinitesimally away from forming a bound state; that is, the s-wave scattering length
is infinite. Under these conditions, he found that the three-body system could form
an infinite number of weakly bound trimer states with geometrically spaced binding
energies. The qualitative understanding of this effect is natural, in retrospect. If thetwo-body sub-system is barely unable to form a bound state, it is natural to think thatthe extra attraction from the third particle is enough to push the system over the top.
Quantitatively, Efimov and later others, found that this effect is described by a simple
wave function in the hyperspherical representation [60, 26, 27].
The result on resonance can easily be extended to all scattering lengths when ais much greater than the range of the two-body interaction. It was later predicted that
Efimov states should be observable through resonances in the three-body recombination rate for negative scattering lengths [26, 61]. In 2006 the first strong experimental
evidence of such a resonance was observed in an ultra-cold gas of 133 Cs by Kraemer et.
al. [62]. Unfortunately, only a single resonance has been seen to date. To confirm thisas an Efimov resonance, the characteristic geometric scaling of at least two resonances
must be seen. The resonance observed in [62] could be caused by a d-wave resonance inthe scattering cross section [63] or possibly by a three-body Fano-Feshbach resonanceattached to an excited three-body scattering threshold [29]. More recently, a resonancein the atom-dimer relaxation rate has been seen observed [64], which might be anotherindication of the presence of an Efimov state at positive scattering length. Predictions

5
focusing on the four-body loss rate [65] have recently given another handle on characterizing an Efimov resonance. The experimental realization of these predictions is stillongoing.

1.3

Dissertation structure
In Chapter 2 of this dissertation I give a brief introduction to hypersphericalcoordinates with examples of possible parameterizations of the hyperangles. The parameterization method used is common and described thoroughly in Refs. [5, 66]. I also
introduce the idea of the generalized angular momentum, or hyperangular momentum,
and give it explicitly in terms of the hyperangles. These results are then used to analyze
N non-interacting particles in an isotropic oscillator, the results of which give a usefulexample of the utility of hyperspherical coordinates.
Chapter 3 applies the results of Chapter 2 to generate a variational wavefunction, called the K-harmonic, for the two-component degenerate Fermi gas. This method
is an extension of existing studies in Bose-Einstein condensates [67, 68]. Within thisapproximation, it is shown that zero-range Fermi pseudo-potential interactions predict
an unphysical collapse behavior that is not seen in experiment or more complete quantum Monte Carlo studies of the system. I then show that this collapse can be averted
through the use of a density dependent interaction. These results are extended to allow
for multiple spin components.
In Chapter 4 the K-harmonic method is extended to allow for anisotropic trapping
potentials by breaking the hyperspherical coordinates into a hypervectorial parameterization. This method is shown to reproduce the ground state energy and RMS radiusof the gas predicted in Chapter 3. By performing a normal mode analysis the low energy excitation frequencies are extracted. These frequencies disagree both qualitatively
and quantitatively with those observed experimentally, and disagree with predictionsin the unitarity regime, a → −∞, from other theoretical methods. While this result is6disheartening, the methods presented are an initial step for further studies proposed in
the chapter.
Chapter 5 departs from the variational treatments in the previous chapters andintroduces the adiabatic hyperspherical representation. The hyperangular Green’s functions is derived here for an arbitrary number of dimensions and applied to the three-bodyproblem with regularized zero-range interactions. The resulting transcendental equation
can be used to find the adiabatic potentials for an arbitrary three-body system in which
the scattering lengths between particles is greater than the range of the interactions.
Combined with the general form of the non-adiabatic corrections, three-body scattering
at low temperatures becomes an intuitive process. These very general results are applied to the case of the lowest three hyperfine components of 6 Li. The Green’s function
method for three interacting particles is then extended to allow for multi-channel twobody interactions for identical bosons. By creating a resonance at an excited two-bodyscattering threshold, I predict a new series of Efimov states that are attached to an
excited three-body scattering threshold, and embedded in the three-body continuum.
These states can be interpreted as true three-body Fano Feshbach resonances.
Chapter 6 examines the four-fermion problem with positive s-wave scattering
lengths. By utilizing a unique variational, non-orthogonal basis set, the hyperradialpotential curves and their non-adiabatic corrections are found. These potentials are thenused to analyze elastic and inelastic dimer-dimer scattering in the large scattering length
regime. By describing dimer-dimer scattering through a complex, energy dependent
scattering length, both elastic scattering and inelastic dimer breakup are analyzed. The
results of this study are shown to compare favorably with existing theoretical predictions
and extend beyond their range of validity. I then develop a new analysis based on Fermi’s
golden rule for describing the process of dimer-dimer relaxation, the results of which
compare well with experimentally observed relaxation rates.
Chapter 7 provides a brief summary of the results of this dissertation and discusses

7
some future avenues of inquiry.

Chapter 2Hyperspherical Coordinates

Strictly speaking an overview of hyperspherical coordinates is not really needed
for this thesis, as there are many excellent existing works on the subject (see for instance[30, 31, 66, 5, 38, 67]). This section is included here for completeness, also some of theideas and derivations will be useful in later chapters. To begin, consider a d dimensional
Cartesian space whose coordinate axes are given by {xi }di=1 . For the majority of thisthesis these coordinates can be considered to be the components of a set of Jacobi
vectors or the components of a set of trap centered vectors, but for now I will proceedwith a more abstract approach. The basic concept of the hyperradius was introducedin the previous chapter. I define it explicitly here as
2

R =

d
X

x2i .

(2.1)

i=1

While this definition will be used here, often a mass scaling will be inserted. For instancein a trap centered system of equal mass atoms an extra factor of 1/N , where N is thenumber of atoms, will be used to simplify the interpretation of the hyperradius. For the
purposes of this section, though, this definition will be adequate. With Eq. 2.1, the ddimensional Laplacian can be rewritten in terms of the hyperradius [66, 5]:
∇2 =

d
X
∂2

1 ∂ d−1 ∂
Λ2
=
R
−
.
∂R R2
Rd−1 ∂R
∂x2i
i=1

(2.2)

9
In this equation Λ is called the hyperangular momentum, or grand angular momentum
operator, the square of which is given byΛ2 =X− |Λij |2 ,

(2.3)

i>j

Λij = xi

∂
∂
− xj
.
∂xj
∂xi

Already the d dimensional Laplacian has a rather pleasing form reminiscent of its 3D
counterpart. In fact if d is taken to 3, Eq. (8) reduces exactly to the three dimensional
Laplacian in spherical coordinates, and Λ becomes merely the normal spatial angularmomentum operator. To proceed from here a way of defining the remaining d−1 degrees
of freedom in terms of angles is needed.

2.1

Defining the hyperangles: Delve’s coordinates
Unfortunately, or possibly fortuitously depending on your view point, there is no

unique way to define the hyperangles in a given system. Here I will use a simple, standardized method of defining them used by many others [34, 38, 67, 5, 1, 2] in the form of
the so called Delve’s coordinates [30, 31]. I will begin by examining a well known example of hyperspherical coordinates, that of normal spherical polar coordinates. Clearlythese coordinates can be used to describe the relative motion of 2 particles in 3 dimensions or the position of a single particle in a trap-centered coordinate system, but it canalso be used in less obvious ways. For instance, spherical polar angles may be used todescribe the relative motion of 4 particles in 1 dimension.
The components of a three dimensional vector, ~r, can be written in terms of aradius and two angles as
x = r cos φ sin θ,(2.4)

y = r sin φ sin θ,(2.5)

z = r cos θ.

(2.6)

10

Figure 2.1: The tree that gives the standard spherical coordinates for a 3 dimensional
system is shown.

This parameterization can be represented in a simple tree structure shown in Fig. 2.1.
The end points of the tree represent each component of the vector ~r, and each nodein the tree is represents an angle. Also associated with each node is a sub radius.
For the lowest node the “subradius” is merely the total length of the vector, r. Forp
the upper node the subradius is merely the cylindrical radius ρ = x2 + y 2 . Using
the tree structure from Fig. 2.1, a set of rules can be developed for extracting theparameterization of Eqs. 2.4, 2.5, and 2.6. Starting at the bottom node with total
radius, r, move up through the tree to the desired coordinate. For each move throughthe tree, if you move to the left (right) from a node multiply by the sine (cosine) of theangle associated with that node. Continue until you reach the Cartesian component.
This procedure can be generalized readily from three to d dimensions. Start bybuilding a tree with d free ends and d − 1 nodes, associate an angle with each nodeand follow the above rules. Using the tree structure, starting at the bottom node withtotal hyperradius, R, move up through the tree to the desired coordinate. If you moveto the left (right) from a node, multiply by the sine (cosine) of the angle associatedwith that node. Continue until you have reach the desired Cartesian component. A
specific tree for d dimensions is shown in Fig. 2.2. Following the rules this tree gives
the hyperangular representation

11

Figure 2.2: The canonical tree that gives a hyperangular parameterization for a d dimensional system is shown.

xn = R cos αn−1d−1Ysin αj ,

(2.7)

j=n

0 ≤ αj ≤ π, j = 2, ..., d − 1
0 ≤ α1 ≤ 2π
where cos α0 ≡ 1 andd−1Q

sin αj ≡ 1. This can also be written as

j=d

qP
n
tan αn =2
j=1 xj

xn+1

,

(2.8)

n = 1, 2, 3, ..., d − 1.
This hyperspherical tree has been dubbed the canonical tree [66, 5] as it is simple toconstruct and very easy to add more dimensions to.
To avoid double counting, the range that the hyperangles take on is restricted
depending on how many free branches are attached to the node corresponding to a given
angle. If the node has two free branches, then the angle takes on the full range 0 to 2π.
If the node has one free branch attached, the angle goes from 0 to π. If the node has nofree branches attached to it, the associated angle goes from 0 to π/2. Following these12

Figure 2.3: The tree structure used to correlate two subspaces to a single hyperradius.

rules for the canonical tree gives the ranges of the angles αi ,0 ≤ α1 ≤ 2π,
0 ≤ αi ≤ π, i = 2, ..., d − 1.
Another slightly more abstract way of considering this construction is to start bybreaking the d dimensional space into two subspaces of dimension d1 and d2 , and assuming that these two subspaces are already described by two sets of sub-hypersphericalcoordinates, (R1 , Ω1 ) and (R2 , Ω2 ). With these assumptions all that remains is to correlate the sub-hyperradii. This is done by following the type of procedure describedabove using the tree structure shown in Fig. 2.3,
R1 = R sin α,(2.9)

R2 = R cos α,
where α is now the final hyperangle in the system. Using this procedure recursively,
one can define the hyperangles in the subspaces until the only remaining subspaces are
the individual Cartesian components of the total d dimensional space. The concept ofdividing the total space up into subspaces will prove very useful in later chapters.
As a final example of hyperangular parameterizations, I introduce a parameterization for N 3-dimensional vectors {~ρi }Ni=1 . One could break each vector up into its
individual components and use the canonical parameterization from Eq. 2.7, but thisremoves much of the spatial physical intuition that one could bring to bear, such as the13

Figure 2.4: The tree structure used to parameterize the hyperangles for N three dimensional vectors is shown. Note that the dot at the end of each branch in the tree on theleft stands for a spherical polar tree.

individual spatial angular momentum corresponding to each vector. Instead one canuse a variation on the canonical tree shown in Fig. 2.4. On first glance, this tree mightseem the same as the canonical tree shown in Fig. 2.2. In this case, though, the large
dot at the end of each branch represents the spherical polar sub-tree of the form shownin Fig. 2.1 for each vector ρ~i . Using this tree structure and following the rules outlined
above, 2N of the 3N − 1 hyperangles are given by the normal spherical polar angles foreach vector, (θ1 , φ1 , θ2 , φ2 , ..., θN , φN ). The remaining N − 1 hyperangles are given byqPi2
j=1 ρjtan αi =,(2.10)
ρi+1π0 ≤ αi ≤ ,2
i = 1, 2, 3, ..., N − 1.
where ρi is the length of the ith vector. It will be shown in the next section that thisparameterization is useful when spatial angular momentum plays a role in the problem ofinterest. For completeness, following the rules laid out in Appendix A, the hyperangular14
volume element that results from this parameterization is given by! N −1NYYdΩ =cos2 αj sin3j−1 αj 
dωi 
i=1

j=1

where ωi is the normal spherical polar differential volume for ρ
~i .

2.2

Hyperangular momentum and hyperspherical harmonics
The hyperangular momentum operator in terms of hyperangular coordinates canbe found by using the fact that each subset of Cartesian components is itself a Cartesian
vector space. With that in mind, consider the hyperspherical tree given by Fig. 2.3. By
writing the Laplacian for each subspace in terms of the sub-hyperradii R1 and R2 and the
sub-hyperangular momentum operators Λ1 and Λ2 the total hyperangular momentum
operator can be extracted [66]. It is−1∂2(2.11)
sin(d1 −1)/2 α cos(d2 −1)/2 αsin(d1 −1)/2 α cos(d2 −1)/2 α ∂α2Λ2 + (d1 − 1) (d1 − 3) /4 Λ22 + (d2 − 1) (d2 − 3) /4 (d − 1) (d − 3) + 1
+
−
.
+ 1
cos2 α4sin2 αΛ2 =where α is defined as in Eq. 2.9 and Λi is the sub-hyperangular momentum of thesubspace of dimension di . If one of the subspaces corresponds to a single Cartesiancomponent then the sub-hyperangular momentum for that space is zero, i.e. if di = 1
then Λ2i = 0. To find Λ21 (Λ22 ), ones needs only to apply Eq. 2.11 recursively to eachsubspace. In this way, there is a sub-hyperangular momentum operator associated witheach node in any given hyperspherical tree.
It is useful to be able to diagonalize the hyperangular momentum operator. The
eigenfunctions of Λ2 are detailed in several references (See Refs. [34, 38, 5] for example),and the method of constructing them is given in Appendix A. These functions, Yλµ (Ω),
are called hyperspherical harmonics. Their eigenvalue equation isΛ2 Yλµ (Ω) = λ (λ + d − 2) Yλµ (Ω) ,(2.12)

15
where λ = 0, 1, 2, ... is the hyperangular momentum quantum number. The index µ
enumerates the degeneracy for each λ and can be thought of as the collection of subhyperangular momentum quantum numbers that result from a given tree. Hyperspherical harmonics are also constructed as to diagonalize the sub-hyperangular momenta ofeach node in a given hyperspherical tree, e.g.
Λ21 Yλµ (Ω) = λ1 (λ1 + d1 − 2) Yλµ (Ω) ,(2.13)

where λ1 = 0, 1, 2, ... is the sub-hyperangular momentum quantum number associatedwith Λ21 . The total hyperangular momentum quantum number λ is limited by the
relation
λ = |λ1 | + |λ2 | + 2n,(2.14)

where n is a non-negative integer. The absolute values in this case are there to allow
for when either d1 or d2 are 2. In this special case the hyperangular momentum quantum number λi associated with the two dimensional subspace can be negative, as withthe magnetic quantum number, m, in spherical polar coordinates. Equation 2.14 only
applies if both d1 and d2 are greater than 1. If, for instance d2 = 1, then the restriction
takes on the form
λ = |λ1 | + n.
The behavior illustrated in Eq. 2.13 clearly demonstrates why the parameterization shown in Fig. 2.4 is useful. Each three dimensional spherical polar subtree will
have a spatial angular momentum and z-projection associated with it, e.g.
li2 Yλµ (Ω) = li (li + 1) Yλµ (Ω) ,where li2 is the square of the angular momentum operator for the ith vector. This
property allows for addition of angular momentum in the normal way, through sums
over magnetic quantum numbers and Clebsch-Gordan coefficients. Now that the hy-

16perspherical harmonics are defined, it is useful to examine a simple example applying
them.

2.2.1

N bodies in a trap

Here I examine a simple example that demonstrates the utility of hypersphericalcoordinates, that of N particles of mass m in a spherically symmetric oscillator trap.
This example can be found in Refs. [69, 1], but I show it here as the results will provevery useful. The Hamiltonian for this system is given byN X~2 2 12 2
H=−∇ + mω ri ,2m i2

(2.15)

i=1

where ~ri is the position of the ith particle with respect to the trap center, and ∇2i is thenormal 3D Laplacian with respect to ~ri . This system is trivially solved by a productof single particle oscillator states whose radial behaviors are the solutions of the single
particle Schrödinger equation,


−~2
2m



d2
l (l + 1)
−
2
dr
r2




1
2 2
+ mω r − En` rfn` (r) = 0.
2(2.16)The solution to this is well known and is given by   r l+1 l+1/2 r2,(2.17)
rfnl (r) = Anl exp −r /2`
Ln
`
`2
p
with energy En` = ~ω (2n + l + 3/2), where ` = ~/mω and n is the number of radial2

nodes in the wavefunction.
This problem can also be solved in terms of hyperspherical harmonics and theirresulting hyperangular momentum quantum number λ. The hyperradius for this 3N
dimensional system can be defined along the lines of Eq. 2.1 with an extra factor of
1/N :
N

R2 =

1 X 2
ri
N

(2.18)

i=1

giving a clear interpretation of the hyperradius as the coordinate parameterizing therms radius of the gas. For this treatment, the explicit form of the hyperangles will

17
not be needed; for completeness, though and to tie into the previous section, the full
hyperangular parameterization is given, but it is clear that they may be defined using
the parameterization described in the section 2.1.
Transformation of (2.15) into hyperspherical coordinates using Eqs. 2.18 and 2.2
yields a Schrödinger equation that separates into hyperradial and hyperangular pieces.
The hyperangular solution is a hyperspherical harmonic that diagonalizes Λ2 . Afterremoving first derivative terms in the hyperradius, the resulting hyperradial Schrödinger
equation is given by 2 2−~K (K + 1)d1
2 2−+ M ω R − E R(3N −1)/2 F (R) = 0,
2M dR2R2
2(2.19)

where K = λ + 3 (N − 1) /2 and M = N m. Comparing this with Eq. 2.16 one
can see that if the substitutions l → K, n → χ, m → M , r → R and rfnl (r) →R(3N −1)/2 FχK (R) are made, the single particle radial Schrödinger equation becomes
the N particle hyperradial Schrödinger equation. With these replacements the hyperradial solution to Eq. 2.19 is evidentlyR(3N −1)/2

 2
 K+1R
K+1/2 RLχ,FχK (R) = AχK exp −R /2L
L
L2
2



(2.20)√
where L = `/ N , and χ is the number of hyperradial nodes in the N body system.
The total energy of the N -body system is then given by




3
3N
EχK = ~ω 2χ + K += ~ω 2χ + λ +.
22(2.21)

With this result comes a different understanding of hyperspherical harmonics.
The non-interacting oscillator is separable in both the independent particle coordinates{~ri }Ni=1 and in hyperspherical coordinates {R, Ω}. This yields a relationship between
the hyperspherical quantum numbers χ and λ and the single particle quantum numbers
ni , the number of radial nodes for the ith particle, and li , the ith particle’s angularmomentum, i.e.
2χ + λ =NX
i=1(2ni + li ) .

(2.22)

18
It should be noted that this does not mean that the independent particle solution
and the hyperspherical solution are exactly same, only that the hyperspherical solution R(3N −1)/2 GχK (R) Yλµ (Ω) must be a linear combination of independent particle
solutions of the same energy, i.e.
R(3N −1)/2 GχK (R) Yλµ (Ω) =

X
k

ak

"N
Y

#
ri fnk lk (ri ) ylk mk (ωi ) ,
i i

i

i

(2.23)

i=1

where ylm (ω) is a normal spherical harmonic of the spatial solid angle ω and nki , lik , mkiare the quantum numbers associated with the ith particle in the kth state subject to
Eq. 2.22.
Equation 2.23 leads to an alternative way of constructing hyperspherical harmonics through oscillator function. Timofeyuk [70, 71] and Fabre de le Ripelle [72, 73] havedone just that by finding the linear combination of independent particle oscillator solutions that produce a hyperradially nodeless solution, χ = 0. It is also worth noting thatthe hyperradius R is invariant under any permutation of particle coordinates, meaningthat any permutation symmetry in the system (e.g. bosonic or fermionic symmetry)must be contained in the hyperspherical harmonic.
By finding a linear combination of independent particle solutions that correspondto the ground state of the system subject to a specific permutational symmetry it caneasily be shown [1] through Eqs. 2.22 and 2.23 that the N -body wavefunction must
diagonalize the hyperangular momentum operator Λ2 with eigenvalue
λGS =

N
X


2nGS
+ liGS .
i(2.24)

i=1

Here “GS” indicates that the given quantum numbers are associated with the groundstate of the system. This relationship follows from expanding a ground state independent particle N -body wavefunction in terms of noninteracting hyperspherical wavefunctions:ΨGS (~r1 , ~r2 , ..., ~rN , σ1 , σ2 , ..., σN ) =Xνcν R(3N −1)/2 Gχν Kν (R) Yλν µν (Ω, σ1 , σ2 , ..., σN ) .

19
Here I have incorporated the possibility of additional spin coordinates (σ1 , σ2 , ..., σN )to allow for the possibility of multi-component Fermi gasses. I will further assume thatthe ground state in question consists only of linear combinations of permutations of the
product of N single particle oscillator states with spin components. For instance in a
Fermi gas this would be a standard Slater determinant state. The energy of this groundstate is then given by
EGS =

N
X
i=1



3
~ω 2ni + li +.
2This combined with Eq. 2.22 gives the relation between the hyperspherical quantum
numbers and the independent particle quantum numbers,2χν + λν =NX(2ni + li )i=1

for each ν in the expansion. Now all of the ground state permutational symmetry is
contained in the hyperangular function so if R(3N −1)/2 G0Kν (R) Yλν µν (Ω, σ1 , σ2 , ..., σN )is considered, it will have the appropriate symmetry with an energy of
E=

N
X
i=1



3
~ω 2ni + li +− 2χν ,
2

which is less than than the ground state energy, a contradiction unless χν = 0. Thus
Eq. 2.24 has been shown.
The hyperspherical harmonics,
YλGS µ (Ω, σ1 , σ2 , ..., σN ) =ΨGS (~r1 , ~r2 , ..., ~rN , σ1 , σ2 , ..., σN ),R(3N −1)/2 G0KGS (R)(2.25)

corresponding to the ground state eigenvalue, λGS , are called the K harmonics, a concept that will be used heavily in Chapter 3. It is useful to note that YλGS µ (Ω, σ1 , σ2 , ..., σN )defined here is independent of the oscillator length `, a fact that will be used in later
sections. A derivation similar to the one shown here with the center of mass motion of
the system removed can be found in Ref. [1].

Chapter 3

N Interacting Fermions in a Trap Under the K-Harmonic
Approximation

The realization of the degenerate Fermi gas (DFG) in a dilute gas of fermionic
atoms has triggered widespread interest in the nature of these systems. This achievement
combined with the use of a Fano-Feshbach resonance allows for a quantum laboratoryin which many quantum phenomena can be explored over a wide range of interaction
strengths. This leads to a large array of complex behaviors including the discovery ofhighly correlated BCS-like pairing for effectively attractive interactions [13, 15, 16, 17,14]. There are many theoretical studies of these systems ranging from quantum Monte
Carlo techniques [45, 24] to extending the BCS pairing wavefunction into the stronglyinteracting regime [48, 46, 49, 74]. In this chapter I propose a simpler variational
treatment of this highly complex system. The methods presented here cannot completely
describe the full nature of the system. They do however represent an important bridge
between many-body systems and the world of hyperspherical coordinates which are often
associated with few-body techniques [34, 41, 40]. Many of the results of this chapter
can be found in Refs. [1, 2].
The starting point of the K harmonic method describes the degenerate Fermi
gas with a set of 3N − 1 hyperangular coordinates on the surface of a 3N dimensional
hypersphere of hyperradius R, where N is the number of atoms in the system. This
formulation is a variational treatment of the N -body problem in which the hyperangular21
behavior of the system is approximated by that of the non-interacting degenerate Fermi
gas presented in Section 2.2.1. At first glance this approach might seem non-intuitive,
but it is natural to assume that, in a first approximation, the behavior of the gas will be
determined by its overall spatial extent. This type of approach has been used in studying
Bose-Einstein condensates [67, 68, 75] and it has also been applied to finite nuclei[5, 71]. The theoretical approach developed here shares some mathematical kinshipwith D-dimensional perturbation theory; for instance, the N → ∞ and D → ∞ limits
both result in wavefunctions perfectly localized in the hyperradius. However, the goals
and motivations of the K-harmonic method differ for the most part from those of Ref.
[76, 77, 78].
The main goal of this chapter is to describe the motion of the gas in a single
collective coordinate R, which describes the overall extent of the gas. The benefit ofthis strategy is that the behavior of the gas is reduced to a single one-dimensional
linear Schrödinger equation with an effective hyperradial potential. The use of a realpotential then lends itself to the intuitive understanding of normal Schrödinger quantum
mechanics. This method also allows for the calculation of physical quantities such as
the energy and rms radius of the ground state; these observables agree quantitatively
with those computed using Hartree-Fock methods. The method also yields a visceralunderstanding of a low energy collective oscillation of the gas, i.e. a breathing mode.
This chapter is arranged as follows. In Section 3.1 the K-harmonic method is
introduced in general. In Section 3.2 the method is applied to a system of fermionsinteracting through a bare zero-range potential. In Section 3.3 the results from the barezero-range potential are presented and discussed. In Section 3.4, I apply the K-harmonic
method to a density dependent zero-range interaction and present the results.

22

3.1

The K-harmonic approximation
In this chapter I consider a collection of N identical fermionic particles of mass

m in a spherically symmetric oscillator trap with oscillator frequency ω, distributed
equally between two internal spin substates. The method presented here is similar tothat of Refs. [67, 68, 75] in which the K-harmonic method is applied to a Bose gas.
This study expands the method to a fermionic system. The Hamiltonian that governs
this system is given by
N

H=

N

X
X
−~2 X 2 1
∇i + mω 2
ri2 +Uint (~rij )
2m
2
i=1

i=1

(3.1)

i>j

where Uint (~r) is an arbitrary two-body interaction potential and ~rij = ~ri − ~rj . Interaction terms involving three or more bodies are ignored. In general, the Schrödinger
equation that comes from this Hamiltonian is very difficult to solve. The goal here is tosimplify the system by describing its behavior in terms of a single collective coordinate,
the hyperradius defined by Eq. 2.18. Using Eqs. 2.2 and 2.18 it is simple to write the
Schrödinger equation in terms of the hyperradius and hyperangles as∂2(3N − 3) (3N − 1) Λ2
−
− 2
∂R2
4R2
R

X
1
+ M ω 2 R2 +Uint (~rij ) − E  R(3N −1)/2 Ψ (R, Ω)
2

−~2
0=
2M




(3.2)

i>j

where Ψ (R, Ω) has been multiplied by R(3N −1)/2 to remove first derivative terms in thehyperradius. Here Λ is the grand angular momentum operator defined in Eq. 2.3. The
heart of the K-harmonic approximation lies in the variational ansatz that the wavefunction Ψ can be separated into a hyperradial function and a hyperangular function.
The hyperangular function is then fixed to the behavior of a non-interacting degenerate
Fermi gas, i.e.
Ψ (R, Ω) = F (R) Yλµ (Ω, σ1 , σ2 , ..., σN ) ,

(3.3)

23
where Yλµ is the K-harmonic described by Eq. 2.25 asP
Yλµ (Ω) =(−1)p PPNQ

fni li (ri ) yli mi (ωi ) hσi |msi i

i=1

G0K (R).

(3.4)

In this expression, fni li (ri ) is the radial solution to the independent particle harmonic
oscillator for the ith particle given byhi
2rfnl (r) = Nn` exp −r2 /2`2 (r/`)l+1 Ll+1/2
(r/`)
,
n
where Lαn (r) is an associated Laguerre polynomial with ` =p~/mω. yli mi (ωi ) is an

ordinary 3D spherical harmonic with ωi the spatial solid angle for the ith particle and|msi i is a spin ket that will allow for two spin species of atoms, |↑i and |↓i. The sum
in Eq. 3.4 runs over all possible permutations P of the N spatial and spin coordinatesin the product wavefunction. The spin coordinates in Yλµ (Ω, σ1 , σ2 , ..., σN ) have been
dropped for notational simplicity. The numerator on the right hand side of Eq. 3.4
corresponds to the ground state of the Fermi gas in which each particle successively fills
a single particle state.
To utilize the trial wavefunction, Eq. 3.3, the expectation value hYλµ |H| Yλµ i istaken, where the integration is over all hyperangles and spin coordinates at a fixed hyperradius. This approach gives a new effective linear 1D Schrödinger equation Hef f R(3N −1)/2 F (R) =ER(3N −1)/2 F (R) in terms of an effective Hamiltonian Hef f given by−~2 d2+ Vef f (R) ,2M dR2X~2 K (K + 1) 1
2 2
Vef f (R) =
+
M
ω
R
+
hYλµ |Uint (~rij )| Yλµ i .
2M R2
2
Hef f =

(3.5)
(3.6)

i>j

where Vef f (R) is an effective hyperradial potential. As in Eq. 2.19, here K = λ +3 (N − 1) /2 and λ is given by Eq. 2.24. In the following sections I develop methods
for evaluating the interaction expectation values for various two body interactions, andanalyze the resulting effective potentials in the hyperradius.

24
It will later prove convenient to rescale this effective Hamiltonian in terms of
non-interacting quantities to avoid divergences in the large N limit:
E = EN I E 0q
R = hR2 iN I R0 ,(3.7)

which introduces the dimensionless variables of energy (E 0 ) and hyperradius (R0 ). Herethe non-interacting energy EN I and average hyperradius squared R2 N I are given
explicitly by
3N
EN I = λ +
~ω
2


λ
3 2
2
R NI =+` .
N
2
Under this rescaling the effective hyperradial Schrödinger equation becomesVef f (R0 )−1 ∂ 20+− E R0(3N −1)/2 F R0 = 0
∗
02
2m ∂R
EN I

(3.8)

where m∗ = mEN I N R2 N I /~2 , and Vef f (R0 ) is the effective hyperradial potential in
Eq. 3.6 evaluated at the rescaled hyperradius.

3.2

Bare zero-range s-wave interactions
The first two-body interaction considered here will be that of the well-known

Fermi pseudo-potential [79],Uint (~r) =4π~2 a (3)δ (~r)
2µ2B(3.9)

where a is the s-wave scattering length, µ2B = m/2 is the two body reduced mass, andδ (3) (~r) is the Dirac δ-function. Normally a regularization factor of∂r appears after the∂rδ-function, but when considering a variational wave function whose derivative is regular∂∂
at r → 0 the regularization disappears, i.e. limr→0 rψ (r) = limr→0 r ψ (r) + ψ (r) =∂r∂rψ (0). The use of zero-range contact interactions to model real interactions in atomic

25
systems has a long history [79]. The interest in these interaction models arises from thesimplifications that can be made to the theory of complex systems [80, 81, 82].
Degenerate ground states cause some complications for this formulation. In thisthesis these are avoided by restricting the system to the non-degenerate ground statesthat correspond to filled energy shells of the oscillator, i.e. “magic numbers” of particles.
With moderate extensions the degeneracies can be taken into account by creating aninteraction matrix, but the magic number restriction should still give a good descriptionof the general behavior of systems with large numbers of atoms. The total numberof atoms and the hyperangular momentum quantum number λ are most conveniently
expressed in terms of the number n of single particle orbital energies filled:n (n + 1) (n + 2)
3
(n − 1) n (n + 1) (n + 2)
λ=
4
s


2mω
1
0
n+
,
kf =
~
2
N=

(3.10a)(3.10b)(3.10c)where kf0 is the peak non-interacting Fermi wave number. In the limit where N  1, λand kf0 can be expressed in terms of the total number of particles N as
(3N )4/3
,
r 4
2mω
kf0 →
(3N )1/3 .
~λ→(3.11a)(3.11b)

I next calculate the interaction matrix element by combining 3.11a with Yλµ (Ω)
from 3.4
Uef f (R)

4π~2 a X
Yλµ δ 3 (~r) Yλµ .
2µ2B
i>j

Since Yλµ (Ω) is antisymmetric under particle exchange we may do a coordinate transposition in the sum, ~ri → ~r2 and ~rj → ~r1 . Each transposition pulls out a negative sign

26
from Φλ and we are left with
Uef f (R) =4π~2 a X
Yλµ δ 3 (~r12 ) Yλµ
2µ2B
i>j

=

4π~2 a N (N − 1)2µ2B2Yλµ δ 3 (~r12 ) Yλµ .

To calculate Yλµ δ 3 (~r12 ) Yλµ it is useful to start with a more general interaction. We
assume that the interaction term in the total N body Hamiltonian is such that at a
fixed hyperradius, Uint (~rij ) is separable into a hyperradial function times a hyperangular
integral, i.e.
Uint (~rij ) ≡ VijR (R) VijΩ (Ω) .

(3.12)

From properties of the δ-function and Eq. 2.10 it is easy to see that the Uint (~r) =4π~2 a 3
δ (~r) fits this criterion. While VijΩ (Ω) might have some very complex form, it
2µ2B
will be seen shortly that only the form of VijR (R) and Uint (~rij ) will matter. This meansthat the hyperradial behavior of Uef f (R) will be entirely determined by V12R (R) , i.e.
Uef f (R) =N (N − 1)
N (N − 1)hYλµ |Uint (~r12 )| Yλµ i = ζ
V12R (R) ,2
2(3.13)

where ζ = hYλµ |VΩ (Ω)| Yλµ i. To find ζ, the definition Yλµ (Ω) from Eq. 3.4 is used,then R3N −1 G0K (R)2 is multiplied on both sides. Integrating over R and using Eq. 3.12
to replace V12R (R) V12Ω (Ω) gives a simple equation that may be solved for ζ.
N (N − 1)ζα =β,(3.14a)2Z
α = R3N −1 G0K (R)2 V12R (R) dR,(3.14b)ZNNYN ( N − 1)  Y 3  Xβ=d rj(−1)p1 P1
fni li (ri ) yli mi (ωi ) hσi |msi i ,
2
j=1

i=1

P1

(3.14c)× Uint (~r12 )

X
P2

p2

(−1) P2NY

fni li (ri ) yli mi (ωi ) hσi |msi i ,i=1

where I have used the fact that R3N −1 dRdΩ ∝NQ

d3 rj [66]. The integral in β is aj=1

diagonal, determinantal matrix element and may be drastically simplified by using the27orthogonality of the single particle basis functions (for details see [83], §6-1):
N

1 X
β=
2

Z

d3 r1 d3 r2(3.15)i,j=1

× ψi∗ (~r1 ) ψj∗ (~r2 ) Uint (~r12 ) ψi (~r1 ) ψj (~r2 )− δmsi msj ψi∗ (~r2 ) ψj∗ (~r1 ) Uint (~r12 ) ψi (~r1 ) ψj (~r2 )i,where the sum runs over every independent particle state in the ground state wavefunction ψi (~r). At this point it is convenient to specify Uint (~r) as the pseudo-potential, Eq.
3.9. The pseudo-potential gives VR12 (R) = 1/R3 , and using the definition of G0K (R)in Eq. 2.20 this yields
3 (N − 1)
Γ λ+
2

 .
α=
3N
L3 Γ λ +2
Substituting the pseudo-potential into Eq. 3.15 simplifies the sum considerably asβ=4π~2 a
2µ2B

Z

d3 r

N1
X
ν=1

|ψν (~r)|2N2X|ψµ (~r)|2 ,(3.16)µ=1

where the sum over ν (µ) is over all N1 (N2 ) of the spatial states occupied by particles
in spin state 1 (2). In the case considered here, N1 = N2 = N/2, but it is clear how thiscan be generalized to polarized gases. Putting everything together with Eq. 3.5 yields
an effective Schrödinger equation in the hyperradius:ER−~2 d2+ Vef f (R) R(3N −1)/2 F (R) ,F (R) =
2M dR24π~2 a CN~2 K (K + 1) 1
2 2
+
M
ω
R
+
,
Vef f (R) =2M R2
2
2µ2B N 3/2 R3(3N −1)/2

where
`3 Γ
CN =





PN/2
3N R 3 PN/2
λ+d r ν=1 |ψν (~r)|2 µ=1 |ψµ (~r)|22.
3 (N − 1)
Γ λ+2(3.17)(3.18)

There are some interesting things to note about the effective potential Vef f . For
N = 1 , CN = 0, K = l and the effective Schrödinger equation become the Schrödinger

28
equation of a single particle in a trap. Note that the form of Vef f is very similar to
the effective potential found for bosons by the authors of Ref. [67]. What may besurprising is the extra term of 3 (N − 1) /2 contained in K. The kinetic energy term in
Vef f is controlled by the hyperangular momentum, which in turn reflects the total nodal
structure of the N -fermion wavefunction. This added piece of effective hyperangularmomentum summarizes the energy cost of confining N fermions in the trap.
The large N behavior of CN can be found by realizing that ψv in Eq. 3.16 is a
non-interacting independent particle oscillator state, and the sum of |ψν |2 over ν is thedensity of a single component,ρ(1)(~r) =N/2X|ψν (~r)|2 .

(3.19)

ν=1

In the limit N → ∞, the Thomas-Fermi approximation [84], which should become exact
for non-interacting oscillator states in the large N limit, gives

 
3/2
2mµχ 3/2
1mω 2 r2(χ)ρ (~r) = 2
1−
,
6π
~2
2µχ
Z
Nχs = d3 rρ (~r) ,(3.20)(3.21)

where ρ(χ) (~r) is the density for the χth component of the gas, and µχ is the noninteracting chemical potential set by the condition 3.21.Inserting this into Eq. 3.18
gives


3N
Γ λ+
2 256
2
3/2 
.
N
(3.22)CN →
3 (N − 1)
3 315π 3Γ λ+2Inserting Eqs. 3.11a and 3.11b the leading order behavior in N can be extracted:r
2 32N 7/2
0.049 0.277
CN →1 + 2/3 − 4/3 + ... ,(3.23)
3 35π 3NNrwhere the higher order terms in 1/N have been found by fitting the numerically calculated values of CN . Table 3.1 gives CN /N 7/2 calculated numerically for the several
filled shells while Fig. 3.1 shows both the calculated values of CN /N 7/2 for the first 100
filled shells and the values from the fit in Eq. 3.23 versus 1/N.

n

N

λ

1

2

0

2
3
4
5
...
15
30
100

8
20
40
70
...
1360
9920
343400

6
30
90
210
...
14280
21576025497450

CN /N 7/2
1
≈ 0.0127
8π 20.06370.02510.02440.02435...
0.02410.024090.0240829

7/2
Table 3.1: N , λ and CN /N 7/2 for the several filled shells. We
p can see3that CN /Nquickly converges to the Thomas-Fermi limiting value of 32 2/3/35π ≈ 0.02408 to
several digits.

In both Table 3.1 and Fig. 3.1 the convergence to the large N behavior, CN →√√

232N 7/2 / 335π 3 = 0.02408N 7/2 , can clearly be seen.
In the large N limit the effective potential in Eq. 3.8 can be written using Eqs.
3.11a, 3.11b and 3.17
0

σkf a
Vef f (R0 )1
1→+ R02 + 03
02
EN I
2R
2
R

(3.24)

where σ = 4096/2835π 2 . In the large N limit the effective mass m∗ in the hyperradialderivative term in Eq. 3.8 becomesm∗ →1(3N )8/3 .
16(3.25)For large numbers of particles, the second derivative term becomes negligible, a fact thatwill be useful in later sections. It can be noted that the only parameter that remains in
the effective potential is the dimensionless quantity kf0 a.
The behavior of Vef f (R0 ) versus R0 is illustrated in Fig. 3.2 for various valuesof kf0 a. For kf0 a = 0 (solid curve), the non-interacting limit, the curve is exact, andthe ground state solution is given by Eq. 2.20. For non-zero values of a, Vef f acquires
an attractive (kf0 a < 0) or repulsive (kf0 a > 0) 1/R03 contribution as indicated by the
dot-dashed and dashed lines respectively. For kf0 a < 0 the DFG is metastable in a region

30

1.006

7/2

CN/(0.0241N )

1.005
1.004
1.003
1.002
1.001
1
0.999
0

0.0005 0.001 0.0015 0.002 0.0025
1/N

Figure 3.1: Values of CN divided by the large N limit, CN → 0.02408N 7/2 , versus 1/N
are shown. The circles are the calculated value while the curve is the fit stated in Eq.
3.23. Figure from Ref. [1].

which has a repulsive barrier which it may tunnel through and emerges in the region ofsmall R0 where the interaction term is dominant. It should be noted, though, that small
R0 means the overall size of the gas is small. Thus the region of collapse corresponds
to a very high density in the gas. In this region several of the assumptions made inthis section can fall apart, most notably the assumption dealing with the validity of thetwo-body, zero-range potential [85]. For kf0 a > 0 the positive 1/R03 serves to strengthen
the repulsive barrier and pushes the gas further out.

3.3

Result for bare zero-range interactions
Examination of Eq. 3.24 shows that effectively attractive interactions (a < 0)and effectively repulsive interactions (a > 0) have qualitatively different behavior. With
this in mind I will examine these two cases separately. With the effective hyperradialpotential in hand, the hyperradial behavior of Eq. 3.3 can be variationally optimized
by solving the effective Schrödinger Eq. 3.17. The results reported here are found bysolving Eq. 3.17 for a fixed number of particles using the exact effective potential, butmuch of the intuition may be extracted from the large N limit potential in Eq. 3.24.

31
65
Veff HRLENI

4
3
2
1
0
0

0.5

1
1.5
2
!!!!!!!!!!!!!!!!!!
!
2
R < R >NI

2.5

3

Figure 3.2: The dimensionless rescaled effective potential, Vef f /EN I as a function of
the rescaled hyperradius for kf0 a = 0 (solid), kf0 a = 1.19 (dashed) and kf0 a = −0.40.
(dot-dash).

3.3.1

Repulsive interactions (a > 0)

For effective repulsive interaction the predicted energy from the K harmonic
method is expected to deviate from experimental values, since the trial wavefunction does not allow any fermions to combine into molecular pairs as has been seen
in experiments.[13, 15, 16, 17] This method only can describe the normal degenerate
Fermi gas. The strong repulsive barrier for repulsive interactions shown in Fig. 3.2
arises as the gas pushes against itself which increases the energy and rms radius of theground state. Figures 3.3 and 3.4 compare the ground state energy and average radiussquared respectively of 240 trapped atoms, plotted as a function of kf0 a with a HartreeFock (HF) calculation. The inset in Fig. 3.3 shows that the K harmonic energies areslightly above the HF energies; since both methods are variational upper bounds, we
can conclude that the HF solution is a slightly better representation of the true solution
to the full Schrödinger equation with δ-function interactions.
An added benefit of the K harmonic method is in providing an intuitively simple
way to understand the energy of the lowest radial excitation of the gas, i.e. the breathingmode frequency. Figure 3.2 shows that as kf0 a increases the repulsion increases the

32

-4
-4

3×10

-4

1.15

2×10

-4

E/ENI

1×10
0

1.1 0

0.2 0.4 0.6 0.8
kfa

1(EK-EHF)/ENI

4×10

1.2

1.05
1
0

0.2

0.4

0.6
kfa

0.8

1

1.2

Figure 3.3: The ground state energy in units of the non-interacting energy versus kf0 a
for 240 atoms calculated using the K harmonic method (curve) and using Hartree-Fock(circles). Inset: the difference in the ground state energies predicted be the K harmonic(EK ) and Hartree-Fock (EHF ). Clearly the K harmonic energies are slightly higher
than Hartree-Fock.
1.25

1.15

2

<R >/<R >NI

1.2

2

1.1

1.05

1

0

0.2

0.4

0.6

0.8

1

1.2

kfa

Figure 3.4: The ground state average squared radius of the gas atoms in units of the
non-interacting rms squared radius is plotted versus kf0 a. The calculations considered
240 atoms in both the K harmonic method (curve) and Hartree-Fock (squares).

curvature at the local minimum, whereby stronger repulsion causes the breathing mode
frequency to increase. Figure 3.5 compares the breathing mode frequency calculatedusing the K harmonic method to the sum rule prediction [86] based on HF orbitals,

33

ω0/ω

2

1.9

1.8
0

2

4

6
kfγ

8

10

Figure 3.5: The lowest breathing mode excitation (ω0 ) in units of the trap frequency
is plotted versus kf0 a for the K harmonic method (solid curve) and for the sum rule(circles). Also shown as dashed curves are the lowest eight radial excitation frequencies
predicted in the HF approximation.

and also the lowest eight radial excitation frequencies predicted by Hartree-Fock. As
anticipated, the K harmonic method and the sum rule method agree that the breathingmode frequency will increase with added repulsion. Interestingly both the K harmonicand sum rule methods disagree qualitatively with all eight of the lowest HF excitations.
This difference is attributed to the fact that Hartree-Fock on its own can only describe
single particle excitations while both the sum rule and the K harmonic methods
describe collective excitations in which the entire gas oscillates coherently.

3.3.2

Attractive interactions (a < 0)In this section the behavior of the gas under the influence of attractive s-wave interactions (kf0 a < 0) is examined. For attractive interactions the gas lives in a metastableregion and can tunnel through the barrier shown in Fig. 3.2. Figure 3.6 shows the behavior of Vef f for several values of kf0 a. The location of the local minimum gets pulled
down with stronger attraction as the gas pulls in on itself and deeper into the center of
the trap. Further, as the strength of the interaction increases, the height of the barrier

34
decreases. In fact beyond a critical interaction strength ac the interaction becomes sostrong that it always dominates over the repulsive kinetic term. At this critical point
the local extrema disappear entirely and the gas is free to fall into the inner “collapse”
region. The value of ac can be calculated approximately by finding the point where
Vef f loses its local minimum and becomes entirely attractive. This critical interaction
strength is given by
kf0 ac = −189π 2 1≈ −1.22.
1024 51/4

This is not exact as the gas will have some small zero point energy that will allow it totunnel through or spill over the barrier before the minimum entirely disappears.
5
Veff HRLENI

4
3
2
1
0
0.5

1
1.5
!!!!!!!!!!!!!!!!!!!
R < R2 >NI

2Figure 3.6: The dimensionless rescaled potential curve Vef f /EN I is shown as a functionof the rescaled hyperradius R0 for several values of kf a; kf a = kf ac (dashed), and from
top to bottom kf a = −0.42, −0.66, −0.90, −1.53 (all solid)0Just before the minimum disappears, its location is given by Rmin
= 5−1/4 , with√0
an energy of Vef f (Rmin) = 5EN I /3 ≈ 0.75EN I . This means that if the gas, in

actuality, is mechanically stable for all values of the two body scattering length, i.e.
a → −∞, this approximation must be altered to include a renormalization cutoff in thestrength of the δ-function such that kf0 a > −1.22 for all a. With this in mind, we begin
to examine the behavior of the DFG for the allowed values of kf0 a.
Figures 3.7 and 3.8 show a comparison of the ground state energy and rms radiusof the gas versus kf0 a down to kf0 ac as calculated in the K harmonic and Hartree-Fock35methods. Again HF does just slightly better in energy, which we interpret as HartreeFock giving a slightly better representation of the actual ground state wavefunction.
The energy difference becomes largest as the interaction strength approaches the criticalvalue. This increase is due to the fact that Hartree-Fock predicts that collapse occurs
slightly earlier with kf0 ac ≈ −1.12. As the interaction strength increases, the energy
and rms radius of the gas decrease, and as kf0 a approaches kf0 ac the overall size of thegas decreases sharply.

-3

2×10

-3

1×10

E/ENI

0.95
0.9

-4

-1.2 -1 -0.8 -0.6 -0.4 -0.2
kfa

0

5×10
0

(EK-EHF)/ENI

-3

2×10

1

0.85
0.8
-1.2

-1

-0.8

-0.6
kfa

-0.4

-0.2

0

Figure 3.7: The ground state energy (in units of the non-interacting energy) versus
kf0 a for 240 atoms calculated using the K harmonic (curve) and Hartree-Fock (circles)
methods. Inset: The difference in the ground state energies predicted by the K harmonic(EK ) and Hartree-Fock (EHF ). Clearly the K harmonic prediction is slightly higher.

While the local minimum present in Vef f only supports metastable states, andanything outside of the kf0 a  1 regime is suspect, it is still informative to examine
the behavior of the energy spectrum versus kf0 a, beginning with the breathing mode
frequency. As the interaction strength becomes more negative Fig. 3.6 shows thatthe curvature about the local minimum in Vef f decreases. This “softening” of thehyperradial potential leads to a decrease in the breathing mode frequency in the outerwell. Figure 3.9 shows the breathing mode vs. kf0 a predicted by the K harmonic (curve)
method and also using the sum rule with Hartree-Fock orbitals (circles). Also shown in

36

1

2

0.8

2

<R >/<R >NI

0.9

0.7
0.6
0.5
-1.2

-1

-0.8

-0.6
kfa

-0.4

-0.2

0

Figure 3.8: The average squared radius of the Fermi gas ground state in units of the
non-interacting value is plotted versus kf0 a. The calculations are for 240 atoms in boththe K harmonic method (curve) and Hartree-Fock (squares).

Fig. 3.9 are the lowest eight Hartree-Fock excitation frequencies for a filled shell of 240
atoms. Again, the K harmonic method agrees quite well with the sum rule, while bothdiffer qualitatively from the HF prediction. The sharp decrease in the breathing mode
frequency that occurs as kf0 a → kf0 ac is a result of the excited mode “falling” over thebarrier into the collapse region as the barrier is pulled down by the interaction.

ω0/ω

2.5

2

1.5

-1.2

-1

-0.8

-0.6
kfa

-0.4

-0.2

0

Figure 3.9: The frequency of the lowest energy radial transition in units of the trap
frequency versus kf0 a predicted by the K harmonic method (solid line) and by the sum
rule (circles). Also shown are the lowest eight radial transitions predicted by HartreeFock.

37
Figure 3.10 displays some energy levels in the metastable region as functions of
kf0 a near kf0 ac . Because of the singular nature of the 1/R3 behavior in the inner region,an inner repulsive 1/R12 barrier has been added to truncate the infinitely many nodesof the wavefunction in the inner region. The behavior of the wavefunction is not correct
within this region anyway because recombination becomes dominant, and in any casethe zero-range interaction is suspect beyond kf0 a ∼ 1, and it must be renormalized.
0.95

EENI

0.90
0.85
0.80
0.75
0.70
-1.3

-1.2

-1.1
kfa

-1.0

-0.9

Figure 3.10: A portion of the energy spectrum vs kf0 a close to the critical point kf0 a =kf0 ac . Levels in the metastable region (cyan) decrease slowly while levels in the collapse
region (red) decrease very quickly. Energy levels above the barrier in Vef f (green) live
both in the collapse region and the metastable region.

Figure 3.10 shows three distinct types of energy level. Levels that are containedin the local minimum (shown in cyan) are decreasing, but not as quickly as the others;
levels that are in the collapse region (shown in red) have a very steep slope as they are
drawn further in toward R = 0; and energy levels that are above the barrier in Vef f(shown in green) have wavefunctions in both the collapse region and the local minimum.
As kf0 a decreases, the higher energy levels fall over the barrier into the collapse region
earlier, until finally just before kf0 ac is reached the first excited metastable level falls
below the “ground state”. This corresponds to the breathing mode behavior seen in
Fig. 3.9. Of course all of this applies only if there is no further hyperradial dependenceof a. If the interaction becomes density dependent, a concept that will be explored in

38section 3.4 and has been used in several other treatments [87, 3, 88, 2], a change in thehyperradius will change the density and thus change the effective scattering length, i.e.
a → a (R).

3.3.3

The large N limit

This section deals with the degenerate Fermi gas in the large N limit. To do this,the fact that the ∂ 2 /∂R2 term in the effective Hamiltonian, the “hyperradial kinetic
energy”, becomes negligible can be exploited. In this limit the total energy of thesystem is merely given by E = Vef f (Rmin ), as is the case in dimensional perturbation
theory [89]. To find the ground state energy one must merely find the minimum (local
minimum for a < 0) value of Vef f . Accordingly, we find the roots ofdVef f= 0. Using
dR0

Eq. 3.24 we simplify this to
kf0 a =


1 0
04
Rmin Rmin−1 ,3σ(3.26)

0is the hyperradial value that minimizes Vef f and σ is defined in the same
where Rminway as in Eq. 3.24. The solutions to Eq. 3.26 are illustrated graphically in Fig. 3.11; for0that gives that value. The exact solution
any given kf0 a only look for the value of Rminof Eq. 3.26 cannot be determined analytically for all values of kf0 a, but Fig. 3.11 shows0that satisfies Eq. 3.26. Thisthat for kf0 a > 0 there is always only one positive, real Rmincorresponds to the global minimum discussed for repulsive interactions. For kf0 a < 0things are a bit more complicated. Figure 3.11 shows that for kf0 ac < kf0 a < 0 there are
two solutions to 3.26. The inner solution is a local maximum and corresponds to thepeak of the barrier seen in Fig. 3.6; the outer solution corresponds to the local minimum
where the DFG lives. The local minimum is the state of concern here as this will givethe energy and hyperradial expectation values of the metastable Fermi gas. The valueof kf0 a where these two branches merge is the place where the local maximum merges
with the local minimum, namely the critical value, kf0 ac . There is no solution to 3.26

39
for kf0 a < kf0 ac , and thus there is no region of stability. For kf0 a = 0, the non-interacting
000
limit, there are two solutions, Rmin
= 0 and Rmin
= 1. The solution Rmin
= 0 must be

discounted as there is a singularity in Vef f at R = 0. Thus in the non-interacting limit0Rmin→ 1 and Vef f (Rmin ) → EN I , as should be expected.

Figure 3.11: A plot of the location if the extrema in Eq./ 3.26 versus kf0 a is shown.
0
Examination of this plot tells us the behavior of Rminfor all allowed values of kf0 a
including the existence of the critical point kf0 ac located at the minimum of the plot
where the maximum and minimum coincide.

Substitution of 3.26 into 3.24 gives the energy of the ground state in the large N
limit, as a function of the size of the gas:04
Vef f (Rmin )1 + 5Rmin
=
.
02
EN I
6Rmin

(3.27)The solutions to Eq. 3.26 immediately give the ground state energy of the gas versus
kf0 a. Fig. 3.12 shows the percentage difference of the ground state energy found by thisminimization procedure and that of 240 particles found by solving Eq. 3.17.
Another result from 3.8 is the fact that in the large N limit the commutator[Hef f , R] → 0. Thus for any operator that is solely a function of the hyperradius Ô (R)Energy difference (%)

40
0

-0.5

-1

-1.5
-1.2

-0.8

-0.4

0
kfa

0.4

0.8

1.2

Figure 3.12: The percentage difference between the energy found by minimizing Vef f
and the energy found by explicitly solving the hyperradial Schrödinger equation for 240
atoms is show as a function of kf0 a.

the ground state expectation value in the large N limit is given by the operator evaluatedDE
00
at Rmin, i.e. Ô (R0 ) = Ô (Rmin). This tells us that the large N limit wavefunction is
2
given by R(3N −1)/2 G (R) = δ (R − Rmin ). This can be perturbed slightly by assuming a ground state hyperradial wavefunction approximated by a very narrow Gaussian
centered at Rmin . The width of this Gaussian can be found by approximating Vef fabout Rmin as a harmonic oscillator with mass m∗ and frequency ω0 . By comparing theoscillator potential with the second order Taylor series about Rmin in Vef f , ω0 can befound:vuu 1 1
ω0 = t ∗m EN I∂ 2 Vef f∂R02 R0 =R0!
.

(3.28)

min

The breathing mode frequency is now simply ω0 . The frequency in Eq. 3.28 is inunits of the non-interacting energy; to get back to conventional units, ω0 must be
multiplied by EN I /~. From Eq. 3.25, m∗ = mEN I N R2 N I /~2 , and noting thatN R2 N I = `2 EN I /~ω yieldsvuu 1
ω0B = tEN I

∂ 2 Vef f∂R02 R0 =R0!
,(3.29)minfor ω0B , the breathing mode frequency in units of the trap frequency. Using Eq. 3.24

41
and substituting in Eq. 3.26 to evaluate at the minimum gives thats1
ω0B = 5 − 04 .
Rmin

(3.30)

Note that this is now dependent only on the value of kf0 a, i.e. for a fixed kf0 a thepredicted breathing mode frequency is independent of the number of atoms in thesystem in the large N limit.
Figure 3.13 shows the breathing mode frequency predicted by Eq. 3.30 versus
kf0 a.

Bω 0 (oscillator units)

2
1.5
1
0.5
0

-1.2

-0.8

-0.4

0
kfγ

0.4

0.8

1.2

Figure 3.13: The breathing mode (in units of the oscillator frequency ω) in the large
N limit versus kf0 a. Note that as kf0 a → kf0 ac the frequency drops to zero as the local
minimum disappears.

The same behavior can be seen in this plot as was seen in Fig. 3.9 where thebreathing mode frequency dives to zero as kf0 a → kf0 ac .

3.4

Density dependent interactions
Unfortunately, in strongly-interacting or high-density systems, the overly singularnature of the δ-function often poses a problem [85]. Another problem is that dynamic
instability arises when the non-regularized zero-range interaction is used variationallyin a two component degenerate Fermi gas, as seen in the previous section, which is not42seen in experiment [1, 13, 15, 16, 17, 14]. One method of avoiding these problems is touse a density-dependent interaction. Refs. [3, 87, 2] do just that, introducing a densitydependent interaction strength for a zero-range interaction. In this section I apply thehyperspherical K-harmonic method to this type of interaction potential. The goal will
be to evaluate the interaction potential matrix element in Eq. 3.6 to find the effective
hyperradial potential Vef f (R0 ).

3.4.1

Operator matrix elements in the N → ∞ limit

Before specifying the exact interaction used here, a method for evaluating themore general interaction matrix elements must be found. In this section I develop amethod for calculating hyperangular matrix elements of an operator in the large N
limit, e.g.,Õ R0 =Z∗(Ω) Ô R0 , Ω Yλµ (Ω) dΩ.
Yλµ(3.31)

Here Ô (R0 , Ω) is a general operator that is a function of the rescaled hyperradius R0
from Eq. 3.7 and the hyperangles Ω. Here Yλµ (Ω) = Yλµ (Ω, σ1 , σ2 , ..., σN ) is the K–
harmonic, defined by Eq. 3.4, for N fermions distributed equally between two internal
spin states. Again the spin coordinates (σ1 , σ2 , ..., σN ) have been suppressed for notational simplicity. To allow us to integrate over all of the 3N dimensions of the space,
both sides of Eq. 3.31 are multiplied by a δ-function in the hyperradius and integrated
over R0 :
Õ

R00



Z
=

 ∗

δ R0 − R00 Yλµ(Ω) Ô R0 , Ω Yλµ (Ω) dΩdR0 .

(3.32)

A clever choice for the δ-function representation will aid in this calculation. To
create the δ-function, consider the function
R0(3N −1)/2 GN R


0

= AN

!λ+3N/2−1/2!
p
R02 N R2 N I
N hR2 iN I R0
exp −
,
`R00
2`2 R002(3.33)

43
where AN is a normalization constant, and ` =p~/mω is the oscillator length definedin Eq. 2.17. This should look very familiar, as it is the hyperradial solution for N
non-interacting fermions in a symmetric oscillator from Eq. 2.20 with oscillator length`R00 . It is easy to show that, in the limit where N → ∞, Eq. 3.33 becomes
lim

N →∞

h

R0(3N −1)/2 GN R0

i2

= δ R0 − R00 .

Inserting this into Eq. 3.32 and playing slightly fast and loose with exchange of theorder of the limit and the integral yields
Õ

R00



= lim

N →∞

Z h

R0(3N −1) GN R0

i 2


∗
(Ω) O R0 , Ω Yλµ (Ω) dΩdR0 .
YλµReferring to Eq. 3.4 and remembering that the K-harmonic Yλµ (Ω) is independentof the oscillator length scale, it follows that the wave function GN (R0 ) Φλ (Ω) is merelya Slater determinant of non-interacting single particle oscillator states with oscillator
length,`ef f = R00 `0 .

(3.34)

Further, Ref. [66] gives that R0(3N −1) dR0 dΩ is the full volume element for the 3N
dimensional space. All of this implies that in the large N limit, the hyperangular
operator expectation value hYλµ |O (R, Ω)| Yλµ i is approximated by the full expectation
value of the operator for a trial wavefunction consisting of a Slater determinant of
non-interacting oscillator states, i.e.
Õ R00 = D`ef f (~r1 , ~r2 , ..., ~rN ) O R0 , Ω D`ef f (~r1 , ~r2 , ..., ~rN ) 3N

(3.35)

where D`ef f (~r1 , ~r2 , ..., ~rN ) is a Slater determinant of oscillator states with oscillator
length `ef f and the subscript 3N is to indicate that the matrix element is taken overall 3N spatial and N spin degrees of freedom.

44
3.4.2

Density dependent zero-range interactions

The results of the previous section can now be applied to the density-dependent
renormalized zero-range interactions presented in Ref. [3], in which a zero-range interaction is used whose strength is dependent on the density of the gas,
Uint (~rij ) =4π~2 ζ (kf (~ri ) a)
δ (~rij ) ,2µ2B kf (~ri )(3.36)

where a is the two-body s-wave scattering length and the Fermi wave number, kf =
1/3
kf (~r) = 6π 2 ρ(1) (~r), is defined 1 in terms of the single spin component density,
ρ(1) (~r). The effect of the density dependent interaction is to impose the appropriate
short-range behavior on clusters of particles in a manner similar to that of Ref. [90].
In other words, if a cluster of particles is very close together with rms radius much lessthan the scattering length, the effective interaction energy they experience is the sameas if the scattering length was infinite. This short-range behavior is extracted in Ref.
[3] by considering a two-particle system, but the general method may be extended to alarger number of particles. The dimensionless renormalized function ζ (kf a) from Ref.
[3] is approximated byζ (kf a) = A + B arctan (Ckf a − D) ,(3.37)

A = 0.3949,
B = 1.1375,
1 + tan2
C=



A
B



= 0.9942,
 B
A
D = tan= 0.3618.
B
1This interaction is appropriate when the Fermi surfaces of two components coincide. When this is
not the case, for instance when there is a population imbalance, the different components have different
masses or trap frequencies, or in the case of multi-component gases, the interaction must be averaged
over the different components.

45
Two of the fitting parameters A and B are found by fitting the asymptotic behavior ofζ (kf a) as kf a → ±∞, which are given in ref. [3] bylim ζ (kf a) = 2.1817,

kf a→∞

lim

kf a→−∞

ζ (kf a) = −1.3919.

The constants C and D in Eq. 3.37 are determined by matching the Fermi pseudopotential in the |kf a|  1 limit [79, 82], i.e.
4π~2 ζ (kf a)
4π~2 a→.
2µ2B kfm(3.38)

Fig. 3.14 shows the behavior of this interaction as a function of kf a.
2

ζ(kfa)

1

0

-1
-10

-5

0

10

5

kfa

Figure 3.14: The density-dependent interaction strength function ζ(kf a) is shown versus
kf a. Figure from Ref. [2].

The density of a single spin component in a Slater determinant state is given by
Eq. 3.20, and in the large N limit, the Thomas-Fermi approximation can be used:1(1)ρ`ef f (~r) = 2 3 (2µ)3/2
6π `ef f

1−

r2
2`2ef f µ

!3/2,

(3.39)

where `ef f is given by Eq. 3.34, and µ = (3N )1/3 is the chemical potential at zerotemperature of N non-interacting fermions divided equally between two different spin

46
substates. Combining this with Eq. 3.35 and some integration substitutions gives aneffective interaction potential,*
Uef f R


0

=

+

Yλµ

X

Uint (rij ) Yλµ

i>j

f

kf0 a
R0

!

Z 1
≡
0

p
y6 1 − y2ζ

kf0 a
R0

kf0 a

64N 4/3= 2/3 2 02 f
3 π R

R0

!
,

(3.40)!
y dy.

Here kf0 is the peak Fermi wave number for N non-interacting atoms given by Eq. 3.11b.
Observe that, again, the only parameter in this expression is kf0 a which is dimensionless.
Inserting Eq. 3.40 into Eq. 3.6 now gives the final effective hyperradial potential in theN  1 limit,
Vef f (R0 )
1
1
256
→
+ R02 + 2 02 f
02
EN I
2R
2
9π R

kf0 a

!

R0

.

(3.41)For kf0 a  R0 the integral may be evaluated exactly giving
4096kf0 a
Vef f (R0 )
1
1 02
→
+
R
+
EN I
2R02 22835π 2 R03
which is exactly the effective potential found in Eq. 3.24 for the bare zero-range pseudopotential. It should also be noted that Eq. 3.40 holds for any density-dependent zerorange interaction in the form of Eq. 3.36, i.e. for any function ζ.

3.4.3

Finite effective interactions, kf0 a < ∞Here I explore the behavior of the DFG under a repulsive effective potential where
the two-body scattering length, a, is positive. The renormalized description of theinteractions used here and in Ref. [3] is only accurate if the real two-body interactions
are purely repulsive, meaning that the gas is somehow prevented from coalescing intomolecular dimer states. In other words, with the present initial formulation, only agas of atoms can be described, not of molecules. Figure 3.15, which shows Vef f forseveral positive two-body scattering lengths, also depicts an example of the bare nonrenormalized effective potential. As with the bare pseudo-potential in Sec. 3.2, the47repulsive interactions cause the gas to push out against itself and against the trap walls,
which increases the overall energy and size of the gas.
3.5

Veff/ENI

3
2.5
2
1.5
1
0.5

1

2

1.5
2

R/(<R >NI)

2.5

1/2

Figure 3.15: The dimensionless ratio of the effective hyperradial potential to the noninteracting total energy is plotted as a function of the dimensionless rescaled hyperradius, for several different repulsive interaction strengths. The non-interacting limit
kf0 a = 0 is shown as the sold curve; the dashed curves show the renormalized effective
potential for (bottom to top) kf0 a = 2, kf0 a = 5 and kf0 a = 50. Also shown is the
non-renormalized effective potential with kf0 a = 5 (dotted curve). Figure from Ref. [2].

1.4

E/ENI

1.2

1

0.8
0.1
-0.5 -0.4 -0.3 -0.2 -0.1 0
arctan(kfa)/π

0.2

0.3

0.4

0.5

Figure 3.16: The ground state energy of the DFG in units of the non-interacting
 energy0
predicted by the K harmonic method (solid line) is plotted versus arctan kf a /π andcompared with that predicted by the HF method with 2280 atoms (circles) [3]. The
dashed line is the ground state energy predicted by the K harmonic method using thebare Fermi pseudo-potential. Figure from Ref. [2].

48
1.6

1.2

2

2

<R >/<R >NI

1.4

1
0.8
0.6
0.4
0.1
-0.5 -0.4 -0.3 -0.2 -0.1 0
arctan(kfa)/π

0.2

0.3

0.4

0.5

Figure 3.17: The average squared hyperradius of the two-component DFG ground statein the large-Nlimit,
 divided by the non-interacting value for this quantity, is plotted0
versus arctan kf a /π. Also shown are the values predicted by the HF method with
2280 atoms (circles) [3]. The dashed line is the ground state energy predicted by the K
harmonic method using the bare Fermi pseudo-potential. Figure from Ref.. [2].

Using the large N limit, the second derivative term in Eq. 3.8 becomes negligible, and as before, the ground state energy and hyperradius can be found by simply
minimizing the effective potential, Vef f (R0 ). Figures 3.16 and 3.17 show the energy andaverage squared hyperradius of the minimum of Vef f as functions of kf0 a, compared tothose same values calculated using the bare non-renormalized effective potential fromSec. 3.2. Also shown are the ground state energy and average squared hyperradiusthat are predicted by the HF method using the same density-dependent interaction. As
the interaction gets stronger the renormalized energies and hyperradii flatten out andapproach a constant in the unitarity limit. For kf0 a  1 the Fermi pseudo-potential
approximation is in good quantitative agreement with the renormalized interactions,but diverges dramatically as kf0 a → ∞. This dramatizes the breakdown of the nonrenormalized zero-range approximation, which overestimates the interaction strength
as the unitarity regime is approached.
Figure 3.18 shows the behavior of the effective potential for some attractive values of the two-body scattering length, along with an example of the non-renormalized

49
effective potential. The decisive qualitative importance of the renormalization is now
apparent. The renormalized effective potential is repulsive as R0 → 0, and the groundstate of the gas is in a global minimum. In contrast to the bare pseudo-potential, thisnow has no collapse phenomenon.
Figures 3.16 and 3.17 show the ground state energy and average hyperradiussquared of the system compared to the non-renormalized values. The effects of densitydependence for attractive interaction are even more striking than in the repulsive interaction case. With renormalization, as kf0 a → −∞ the energy and average hyperradius squared go towards a fixed value. Figures 3.16 and 3.17 also show the groundstate energy and average squared hyperradius predictions from the HF method withdensity-dependent interactions. They show that for kf0 a  1 the non-renormalized
and renormalized values are in good agreement, but as kf0 a → kf0 ac = −1.22 thebare Fermi-pseudo-potential results diverge away from the renormalized interaction
results. Just before the point of collapse the ground state energy from Eq. 3.27 is√
E = 5EN I /3 = 0.75EN I . Not only does the renormalized effective potential eliminate
the collapse, it also allows the gas to reach a lower energy than would be possible without the density dependence. In other words, if the interaction coefficient in Uint (~rij )were not density-dependent, but merely involved a cut off as kf0 a → −∞, the gas would
not be able to reach the unitarity energy before collapsing.

3.4.4

Unitarity regime

In this section, the behavior of Vef f for density-dependent interactions in thestrong interaction regime, i.e. a → ±∞, is explored. In contrast to the bare pseudopotential, which had energies that diverged in the kf0 a → ∞ limit and produced acollapse when kf0 a ≤ −1.22, the effective hyperradial potential produced using density
dependent interactions produces finite energies in the unitarity regimes. Examining Eq.

50
3Veff/ENI

2.5
2
1.5
1
0.5
0.5

1

2

1.5
2

R/(<R >NI)

2.5

1/2

Figure 3.18: The effective potential
p in units of the non-interacting energy is plottedversus the hyperradius in units of hR2 iN I , for several interaction strengths. The noninteracting limit kf0 a = 0 is shown as the solid curve and the dashed curves show therenormalized effective potential for (top to bottom) kf0 a = −1 and kf0 a = −5. Also
shown is the non-renormalized effective potential with kf0 a = −1 (dotted curve). Figure
from Ref. [2].

3.40 shows that the unitarity limit appears in Vef f when kf0 a  R0 , i.e.
hYλµ |Vint (R00 , Ω)| Yλµ i256ζ±→ 2 02
EN I
9π R
5ζ±
=
9πR02

Z 1

y6

p
1 − y 2 dy0where ζ± is the maximum (+) or minimum (−) value acquired by the interaction function ζ (kf a). This gives a total effective potential ofVef f (R0 )1
1/2 + 5ζ± /9π→ R02 +
EN I
2
R02

1 02 0.886



R +
for kf0 a  R0


R02
 2
=
.



 1 02 0.254
 R +for −kf0 a  R0
2R02

(3.42)The hyperradius is a collective coordinate so that, as R → 0, all of the atoms in thesystem are forced to the center of the trap, which increases the density of the system.
Thus, for small hyperradii, Vef f is expected to act like Eq. 3.42. In fact, the dashed51
curves in Figs. 3.15 and 3.18 show that as R0 → 0 the renormalized effective potential
curves all start to behave the same, independently of a. Alternatively if the two-bodyscattering length approaches −∞, e.g. near a resonance, then Vef f can be expected toapproach Eq. 3.42 for all hyperradii.
In the case where kf0 a → ±∞ the effective potential takes on the form of Eq.
3.42. Minimization of Vef f as a function of R0 gives a ground state energy:pE= 1 + 10ζ± /9π
EN I 1.331 f or a → ∞=.
 0.712 f or a → −∞(3.43)

The average hyperradius of the gas is described by the value of the hyperradius at thisminimum which is given by0
Rmin= (1 + 10ζ± /9π)1/4 1.154 f or a → ∞= 0.844 f or a → −∞(3.44)

At first glance this may seem strange, one might expect the behavior to be smoothacross a resonance, and the energy to connect smoothly from the a → −∞ limit to thea → ∞ limit [45]. But the density-dependent renormalization used here only applies
to a gas of atoms and does not allow for the incorporation of higher order correlations,i.e. the formation of diatomic molecules. Presumably there is another branch in therenormalization that will match continuously with the a → −∞ limit (for a more
complete discussion see section II of Ref. [3]).
Another quantity of interest is the chemical potential of the interacting gas at
unitarity [3, 25, 91], given byµu =~2 kf2 (0)2m(1 + β) ,where β is a universal parameter. From the single spin component density given in Eq.

52
3.39 the interacting peak Fermi wavenumber is
kf (0) =

kf0
02
Rmin

.

(3.45)

Further, from Eqs. 3.43 and 3.44, the ratio of the chemical potential of the interacting
unitarity-limit gas to that of the non-interacting gas can be written in terms of therescaled hyperradius as:µu
E02= Rmin.
=µEN I(3.46)

Solution of Eqs. 3.45 and 3.46 in the a → −∞ limit yields
β = 10ζ− /9π = −0.49.

(3.47)

This value of β coincides with the value predicted by the renormalized HF calculationof Ref. [3]. This agreement can be explained by realizing that the interaction potential
in the HF approximation with the density dependent interaction in the unitarity regime
takes on the form Vint ∝ ρ2/3 [3]. By considering the density from Eq. 3.39 thisgives an interaction potential that can be approximated by an oscillator-like potential,
2i.e. Vint ∝ 1 − r2 /2µlef
f . Even though neither Ref. [3] nor the present treatment
explicitly incorporates Cooper-type fermion pairing, this unitarity limit β is in fairagreement with quantum Monte Carlo estimates that have obtained β = −0.58 (Ref.
[45]) and −0.56 (Ref. [24]). Also, observe that this gives a significant improvement
over the value of β = −0.41 expected on the basis of BCS theory [48]. Further thevalue of β found here is lower than the β = −0.44 [92] found by quantum Monte-Carloon a normal state Fermi gas. Even though the trial wavefunction used here would be
more appropriate for describing a normal state Fermi gas, the lower value of β foundusing the density-dependent interaction indicates that a simplistic wavefunction cangive surprisingly good results.

53
3.4.5

Breathing mode excitations

By using the effective hyperradial potential from Eq. 3.41, the lowest radial
excitation frequency can be obtained from Eq. 3.29. Figure 3.19 shows the breathingmode frequency in units of the oscillator frequency compared to that calculated using
non-renormalized interactions from Sec. 3.2. Of course the breathing mode frequency inthe non-interacting limit is ω0B = 2ω, and for weak interactions, kf0 a < 1, the breathingmode frequency follows the same behavior for both bare the interaction and densitydependent interactions. Somewhat surprisingly, though, the frequency predicted fromthe density-dependent interactions turns over and returns to the non-interacting value
as kf0 a → ±∞. Inserting the second derivative of Eq. 3.42, the effective potential as
kf0 a → ±∞, givesv
uu 1
ω0 = ω t
EN I

1 + 10ζ± /9π
1+ 3
R04
R0 =R0!
.

(3.48)

min

When the minimum hyperradius from Eq. 3.44 is inserted, the resulting equation implies
that the unitarity limits for the breathing mode frequency are both ω0 = 2ω. This is a
general property of any hyperradial potential of the form V (R0 ) = A/R02 + BR02 in thelarge N limit where A, B > 0. This unitarity behavior has also been predicted in Ref.
[90, 93].

3.4.6

Multiple spin components

The realization of degenerate multi-component atomic Fermi gases is now possibleusing present-day technology [94, 95]. Neglecting for the moment possible losses, theoccupation of more than two different hyperfine states of the same species requires only
moderate changes to the previous analysis. A degenerate gas consisting of the lowest
three hyperfine states of 6 Li has already been created [94, 95], and the coexistence of
three hyperfine states has already been demonstrated for 40 K [96]. Alternatively, a number of groups are presently pursuing the simultaneous trapping of three different atomic

54
2.06
2.04

ω0/ω

2.02
2
1.98
1.96
1.94
-0.4

-0.2

0

0.2

0.4

0

arctan(kf a)/π

Figure 3.19: The breathingmode frequency ω0B is shown in units of the trap frequencyω versus arctan kf0 a /π . The solid curve shows the breathing mode predicted using
the renormalized interaction while the dashed curve shows the prediction based on thebare Fermi pseudo-potential. Figure from Ref. [2].

species [97]. In the case of the same atom in different hyperfine states, the atomic
masses of all components are equal, whereas in the case of different atomic species,
the atomic masses of the different components differ. In either of these realizations ofmulti-component Fermi gases, all or some of the interspecies scattering lengths may
be tunable thanks to the possible existence of magnetic or optical Fano-Feshbach resonances. With these new developments, it is of considerable interest to investigate thesesystems. This section provides an initial look at such a multiple component systemfrom the K-harmonic approximation with density dependent interactions, specificallythe dynamic stability of such systems.
Next consider what happens when the atoms in the gas are equally distributed
among an arbitrary number χ of spin substates. In order to limit the large parameter
space, that the s-wave scattering length between two atoms in any two different spinstates is assumed to have the same value, a. Also, for calculational simplicity, thepossibility of inelastic collisions, e.g. of the type:
3
m1 =
2



3
+ m2 = −
2


→

1
m01 =

2


+

1
m02 = −255
is neglected. To proceed, the question of what density should go into the renormalizedinteractions must be addressed. A particle in spin state i cannot interact with anyother particle in the same spin state by the zero range approximation, but the densitythat determines kf (r) in the renormalization function ζ might be chosen in various
alternative ways, and a unique criterion to specify the appropriate renormalization inthis context has not yet been developed. As an initial exploration, I make the assumptionthat the appropriate interaction is merely the average of the two density dependent
scattering lengths,ihi h (j)(i)
ζ kf (~ri ) a
ζ kf (~rj ) a δ (~ri − ~rj )Uint (rij ) =
+
(j)
(i)
2m
kf (~ri )kf (~rj )
4π~2


1/3
(j)
where kf (~r) = 6π 2 ρ(j) (~r)
is the Fermi wave number of the spin component thatparticle j belongs to.
The derivation following this assumptions is the same as that for the two-component
gas, up to Eq. 3.40. The only additional pieces of information needed are the common
density of each component in the effective trap with oscillator length, `ef f , the chemical
potential, the non-interacting ground state energy and the average hyperradius squared
for the system with χ spin substates in the large N limit:
1(2µ)3/22
6π `3ef f
6N 1/3µ=,χ(i)ρ`ef f (~r) =2`2ef f µ!3/2,

(3.49)(3.50)(6N )4/3,
χ1/3 8(3.51)~ (6N )4/3.
mω χ1/3 8N(3.52)EN I = ~ωR2 N I =1−

r2

Evaluating the matrix element in Eq. 3.15 involves basic well known methods. With
some algebra and a change of integration variables, the effective hyperradial potential

56
becomes
Vef f (R0 )
1
1
256
=
+ R02 + (χ − 1) 2 02 f
02
EN I
2R
2
9π R
! Z
!
1
p
kf0 a
kf0 a
6
2ζ
f
≡
y
y dy.
1
−
y
R0
R0
0

kf0 a
R0

!
,

(3.53)

Comparison with the effective potential for the two component gas, Eq. 3.41, demonstrates that the extra spin components increase the strength of the interaction by afactor of χ − 1. To analyze the stability of the gas, it is sufficient to examine the
kf0 a → −∞ limit, i.e.
Vef f (R0 )1
1 + 10 (χ − 1) ζ− /9π→ R02 +
.
EN I
2
2R02

(3.54)Taking χ = 3 yields
Vef f (R0 )1
1 + 20ζ− /9π→ R02 +
,
EN I
2
2R02
1
0.00772= R02 +.
2
R02

(3.55)In this limit the barrier preventing the gas from falling in to the center of the trap, i.e.
R0 → 0, is very weak. The unitarity energy and average hyperradius are given bypE= 1 + 20ζ− /9π = 0.124,
EN I
R
p min = (1 + 20ζ− /9π)1/4 = 0.352.
hR2 iN I
Since the K harmonic method is intrinsically a variational calculation, it is entirely
possible that a better trial function, for example one based on the HF method, mightshow that the 3 component gas becomes mechanically unstable in the unitarity limit[25]. In other words the three component gas might collapse in a manner similar tothat of the bosenova [67, 98, 99, 10, 100]. Further, if the asymptotic behavior of thedensity dependence in the two-body interaction is modified to agree with the β for thetwo-component gas predicted by Ref. [45] the barely repulsive unitarity potential in Eq.
3.55 becomes attractive, predicting a collapse of the three-component gas [88]. If the

57
asymptotic behavior of the density-dependent interaction of Eq. 3.37 are modified sothat the value for β in Eq. 3.47 matches the quantum Monte-Carlo results of Ref. [45],β = −0.58, at unitarity the effective hyperradial potential for a three component gas
becomes
Vef f (R0 )1
0.08→ R02 − 02
EN I
2R
which provides no stable region, implying that the three-component Fermi gas is unstable in this regime. The critical scattering length that determines the collapse point canbe calculated numerically:
kf0 ac = −1.413.

(3.56)

Recently, degenerate gases of 6 Li have been created in the three lowest hyperfine stateswith peak densities of approximately 6 × 1011 atoms/cm3 [94, 95]. If it is assumed
that all three components interact with the same scattering length, a = 3000 a.u., areasonable assumption for magnetic fields near 1000 G [4], this predicts a bosenova-likecollapse at a central density of approximately
ρ (0) =

1
0 3
≈ 1.4 × 1013 atoms/cm3 .
k
f
2
6π

It is likely that the predicted instability can be achieved at a lower density by using a
different magnetic field where the three scattering lengths are larger, but not approximately equal.
Setting χ = 4 in Eq. 3.54, the effective potential as kf0 a → −∞ becomes entirely
attractive, i.e.
Vef f (R0 )1
1 − 30ζ− /9π→ R02 +
EN I
2
2R02
1
0.238
= R02 −,
2
R02

(3.57)

meaning that the gas is predicted to collapse down toward R0 → 0. Presumably somevery rich and complex dynamics (cluster formation, inelastic collisions, etc.) occur
during this process, but the K-harmonic trial wave function is too simple to describe

58
these phenomena. When the local minimum in Vef f (R0 ) becomes a saddle point thegas is no longer mechanically stable and is free to collapse. This occurs at a criticalinteraction strength of kf0 ac = −0.657.

Chapter 4

The Hypervectorial Method and Anisotropic Traps

Most theoretical studies of these DFGs look either at a homogeneous gas directly,
which can be related to a harmonically trapped gas through the use of the local density
approximation, or at a gas in a spherically symmetric oscillator trap as in Chapter 3.
While these studies can lead to interesting predictions of phenomena in such a system,
experimentally the gas is often held in an anisotropic “cigar-shaped” trap [13, 15, 16,
17, 14].
The starting point for this study is akin to the hyperspherical K-harmonic method
presented in the previous chapter in which the DFG was described by a set of 3N − 1
angular coordinates on the surface of a 3N dimensional hypersphere of radius R where
N is the total number of particles in the system. To incorporate an anisotropic trapin that formulation would require a very high order in hyperspherical harmonics, andwould result in a complex system of coupled 1D differential equations. To avoid thesecomplications here I implement a hypervectorial formulation, which is based on thedivision of the total 3N dimensional space into physically meaningful subspaces in amanner similar to the hyperspherical tree depicted in Fig. 2.3. The division can bethought of as describing the gas by 3N − 2 angular coordinates on the surface of a
3N dimensional “hyper-cylinder” with height Rz and cylindrical radius Rρ . These
coordinates have been used previously to describe a Bose-Einstein condensate in cigarshaped and completely anisotropic traps [68]. Their method is extended in this chapter

60
to describe the degenerate Fermi gas.
This chapter is arranged as follows; In Sec 4.1 I introduce the hypervectorial
method. In Section 4.2 I apply this method to the case of the two-component degenerate
Fermi gas with density-dependent, zero-range interactions and analyze the resulting
potential surfaces. Finally, Section 4.3 uses the potential surface to extract the frequencyof low-energy excitations.

4.1

The Hypervectorial Method
The use of hyperspherical coordinates is well suited to an isotropic oscillator trap

because the trapping potential is easily written in terms of the hyperradius, for instancein Eq. 3.2. This is not the case in an anisotropic trap, where different Cartesian coordinates in the trap are associated with different oscillator frequencies. This section
will deal with a cylindrically symmetric “cigar” shaped trap, but with minor modifications, the methods presented here carry over to a completely anisotropic trap. The
Hamiltonian for this system is given by
H=−

N

N

i=1

i=1


 X
~2 X 2 1 X 2 2
Uint (rij ) ,∇i + mω⊥ xi + yi2 + ωz2 zi2 +
2m
2

(4.1)

i>j

where ωz and ω⊥ are oscillator frequencies in the longitudinal and transverse directions,
respectively. Here xi , yi , and zi are the Cartesian coordinates of the ith atom and rij
is the inter-particle distance between particles i and j. Now consider two collective
coordinates, Rz and Rρ , given by the rms longitudinal and transverse size of the gas,
respectively, i.e.
N

Rz2 =
Rρ2 =

1 X 2
zi ,
N

(4.2)

1
N

(4.3)

i=1
N
X


x2i + yi2 .

i=1This choice of collective coordinates is equivalent to the hyperspherical tree shown pictorially in Fig. 2.3 with R1 = Rρ , and R2 = Rz . Here, though, the final step of correlating

61

Figure 4.1: The sub-hyperspherical trees that are used to parameterize the hypervectorial coordinates Rρ and Rz are shown.

these two collective coordinates is omitted. The remaining 3N − 2 spatial degrees of
freedom in the gas are described by angles. For completeness, the Delves parameterizations shown in Fig. 4.1 can be used, i.e. N angles are merely the cylindrical polar
angles for each atom {φi }Ni=1 , while the remaining 2N − 2 angle are given byqPi2
j=1 zjtan βi =,zi+1r
Pi  2
2
x
+
y
j=1
j
j
tan αi =
,
ρi+1

(4.4)

(4.5)

where 0 ≤ βi , αi ≤ π/2 and 1 ≤ i ≤ N − 1. Collectively these 3N − 2 hyperangles will bereferred to as Ω = (Ω1 , Ω2 ) where Ω1 corresponds to the sub-hyperangles describing thetransverse coordinates and Ω2 the longitudinal sub-hyperangles. In these coordinates
the sum of the Laplacians in Eq. 4.1 can be found using Eq. 2.2:
N
X
i

∇2i =

1
N




Λ2⊥
∂
1
∂
Λ2z
2N −1 ∂
N −1 ∂
Rρ
− 2 + N −1
R
− 2 ,
∂Rρ
Rρ
∂Rz z ∂Rz
Rz
Rρ2N −1 ∂Rρ
Rz
1

(4.6)

where Λ2⊥ and Λ2z have been defined by the Eq. 2.3 in each of the two subspaces.
Combining Eqs. 4.4, 4.3 and 4.6 the Hamiltonian in Eq. 4.1 can be rewritten in terms

62of the hypervectorial coordinates:

Λ2⊥
∂
1
2N −1 ∂2 2Rρ
− 2 + M ω⊥Rρ +2N −1 ∂R
∂R
R
2
Rρ
ρ
ρ
ρ


X
∂
1
Λ2z
1
~2
N −1 ∂
R−+ M ωz2 Rz2 +−Uint (rij ) ,
z
N
−1
2
2M Rz
∂Rz
∂Rz
Rz
2
i>j

~2
H =−
2M



1

(4.7)

where M = N m.
The key to the hypervectorial method presented here is similar to the K-harmonic
approximation, that of a variational ansatz wavefunction,
Ψ (Rρ , Rz , Ω1 , Ω2 , σ1 , σ2 , ..., σN ) = F (Rρ , Rz ) Yλ⊥ λz µ (Ω1 , Ω2 , σ1 , σ2 , ..., σN )(4.8)

where Yλ⊥ λz µ (Ω1 , Ω2 , σ1 , σ2 , ..., σN ) is the lowest allowed harmonic for the N fermion
system. Here (σ1 , σ2 , ..., σN ) are the spin coordinates. It would be convenient if Yλ⊥ λz µwere merely a product of two K-harmonics, one from each subspace in the system,
but this is not the case.

The permutational symmetry of the system mixes sub-hyperspherical harmonics together. Fortunately, examining Eq. 4.7 with Uint (rij ) = 0,
it is clear that the anisotropic oscillator is separable in the hypervectorial coordinates.
A similar derivation to the one in Section 2.2.1 shows that the hyperangularbehavior of the noninteracting system is given by
Yλ⊥ λz µ (Ω1 , Ω2 , σ1 , σ2 , ..., σN ) =D (~r1 , ~r2 , ..., ~rN , σ1 , σ2 , ..., σN ),(2N −1)/2 ⊥(N −1)/2 zRρ
G0K⊥ (Rρ ) Rz
G0Kz (Rz )(4.9)

where D (~r1 , ~r2 , ..., ~rN , σ1 , σ2 , ..., σN ) is a ground state Slater-determinant of independent
particle states, i.e.:D (~r1 , ~r2 , ..., ~rN , σ1 , σ2 , ..., σN ) =

X
P

√

p

(−1)

N
Y

ψi (~ri ) hσi |msi i(4.10)i=1
ρi ψi (~r) =An⊥i nzi mi exp −ρ2 /2`⊥ − z 2 /2`2z exp (imφ)
 2  ρ |m|+1/2
ρz|m|+1/2
×
Ln⊥
Hnzi
2
i
`⊥
`z
`⊥

where the sum runs over all permutations, P , of the N spatial and spin coordinates,pLγn (x) is a Laguerre polynomial, Hn (x) is a Hermite polynomial, `⊥ = ~/mω⊥ , `z =63p~/mωz , and An⊥i nzi mi is a normalization constant. In Eq. 4.9, K⊥ = λ⊥ + N − 3/2,
Kz = λz + N/2 − 3/2, and the sub-hyperradial solutions to the noninteracting oscillatorzG⊥χK⊥ (Rρ ) and GχKz (Rz ) are given by!
Rρ K⊥ +1 K⊥ +1/2 Rρ2
Lχ
,
L⊥
L2⊥
!
 
 Rz Kz +1 K +1/2 Rρ2(N −1)/2 zz
2.
Rz
GχKz (Rz ) = AχKz exp −Rz /2Lz
Lχ ⊥
Lz
L2⊥

⊥
Rρ(2N −1)/2 G⊥
χK⊥ (Rρ ) = AχK⊥ exp−Rρ2 /2L⊥





√
√
z
Here L⊥ = `⊥ / N , Lz = `z / N and A⊥
χK⊥ and AχKz are normalization constants.
Even though Yλ⊥ λz µ (Ω1 , Ω2 , σ1 , σ2 , ..., σN ) is not a product of two sub-hypersphericalharmonics, it still satisfies the relevant eigenvalue equation:
Λ2⊥ Yλ⊥ λz µ (Ω1 , Ω2 ) = λ⊥ (λ⊥ + 2N − 2) Yλ⊥ λz µ (Ω1 , Ω2 ) ,Λ2z Yλ⊥ λz µ (Ω1 , Ω2 ) = λz (λz + N − 2) Yλ⊥ λz µ (Ω1 , Ω2 ) ,where the spin coordinates have been omitted for notational simplicity.

The sub-

hyperangular momentum eigenvalues, λ⊥ and λz , are determined by the number ofoscillator quanta in D (~r1 , ~r2 , ..., ~rN , σ1 , σ2 , ..., σN ) in the longitudinal and transverse
directions respectively:
λ⊥ =

λz =

N
X
i=1
N
X

(2n⊥i + |mi |) ,n zi .

i=1

For this treatment I will again only consider nondegenerate, filled energy shells.
The nondegenerate ground state is found by filling every state in the noninteracting
system up to a Fermi energy, F . In the large N limit, which will be the focus of thischapter, the Fermi energy is given in terms of the number of atoms by2
f → ~ 3N ω⊥
ωz

1/3

,

(4.11)

64
where it has been assumed that there are enough atoms to occupy many longitudinal
and transverse modes, i.e. the system is still three dimensional. The energy of the noninteracting system is given in terms of the hyperangular momentum quantum numbersas
EN I = ~ω⊥ (λ⊥ + 1) + ~ωz (λz + 1/2) .
In the large N limit, this becomes
4f

2ω
~ 3N ω⊥z=
EN I → 3 22ω
4~ ω⊥ ωz
4ω⊥
z

4/3
.

(4.12)The hyperangular momentum quantum numbers, λ⊥ and λz , in the large N limit are
given by2 ω 4/3
3N ω⊥
4F
z
λz →
,
2 =
2 ω2
12~4 ωz2 ω⊥12ω⊥z2 ω 4/3
3N ω⊥
4F
z
.
λ⊥ → 43 =3ω
6~ ωz ω⊥6ω⊥z

(4.13)(4.14)

To employ the variational principle the hyperangular expectation value of the
Hamiltonian given in Eq. 4.7 must be taken, leaving an effective Schrödinger equationin the collective coordinates, Rρ and Rz :ERz(N −1)/2 Rρ(2N −1)/2 F (Rρ , Rz ) =


 2

~2
∂
K⊥ (K⊥ + 1)−−
(4.15)
2M ∂Rρ2
2M Rρ2 2
~2∂
Kz (Kz + 1)−−
2M ∂Rz2
2M Rz21
1
2 2
2 2+ M ω⊥
Rρ + M ω⊥
Rz
2
2

X
+
hYλ λz µ |Uint (rij )| Yλ λz µ i Rz(N −1)/2 Rρ(2N −1)/2 F (Rρ , Rz ) .
⊥

⊥

i>j
(N −1)/2

Here F (Rρ , Rz ) has been multiplied by a factor of Rz(2N −1)/2Rρ

to remove first

derivative terms. To avoid divergences in the large N limit, it is convenient to rescale

65
the effective Schrödinger equation by noninteracting values:
E = EN I E 0 ,q
Rρ =Rρ2 N I Rρ0 ,q
Rz =Rρ2 N I Rz0 ,where EN I is the noninteracting energy, and Rρ2 N I is the expectation value of thetransverse collective coordinate squared given in the large N limit by~Rρ2 N I =
Mω
→



1
λ⊥ + N −2⊥(4.16)(3N γ)4/3 2
`⊥ ,
6N γ

with γ = ωz /ω⊥ . In the large N limit, with this rescaling, Eq. 4.15 becomes
"
#

 2


Vef f Rρ0 , Rz01∂∂2E0(N −1)/2 0(2N −1)/2000= − ∗
+
+
−
R
R
F
R
,
R
.
z
ρ
ρ
z
2m
∂Rρ02 ∂Rz02
EN I
EN I
(4.17)

0Here m∗ = M EN I Rρ2 N I /~2 , and the effective potential, Vef f Rρ0 , Rz , is given by
Vef f Rρ0 , Rz0
1
1
1
1= 02 + Rρ02 ++ γ 2 Rz02
2
02
EN I
3Rρ
3
12γ Rz3EDPYλ⊥ λz µU(r)
Yλ⊥ λz µi>j int ij
+
.
EN I

(4.18)

with γ = ωz /ω⊥ . All that remains is to calculate the hyperangular expectation value ofthe interaction.

4.2

More density-dependent interactions
It was shown in Chapter 3 that the bare Fermi pseudo-potential in Eq. 3.9

predicted an unphysical collapse of the two component degenerate Fermi gas. With thisin mind I will jump directly to using the density-dependent interaction of Eq. 3.36 in thelarge N limit. In the K harmonic approximation, the hyperangular expectation value
in the large N limit is equivalent to a simple expectation value of a Slater-determinant,

66
ground state, independent particle wavefunction in an effective oscillator trap taken overall 3N spatial degrees and N spin degrees of freedom. A similar derivation here yields
the same result, i.e.:*0Vint Rρ0 , Rz =+X

Y λ⊥ λz µ

Uint (rij ) Yλ⊥ λz µ(4.19)

i>j

+

*
=

X

D`ef f⊥ `ef fz,

Uint (rij ) D`ef f⊥ `ef fz

i>j

3N

where D`ef f⊥ `ef fz (~r1 , ~r2 , ..., ~rN , σ1 , σ2 , ..., σN ) is a ground state Slater-determinant wavefunction of N independent noninteracting fermions in an anisotropic cigar-shaped trapwith effective oscillator lengths,`ef f⊥ = Rρ0 `⊥ ,q
`ef fz = Rz0 p(4.20)
Rρ2 N IhRz2 iN I`z .

Here Rz2 N I is the expectation value of the square of the longitudinal collective coordinate:~
Rz2 N I =
Mω
→


z

N
1
λz +
−
2
2


(4.21)(3N γ)4/3 2` .
12N γ 2 z

(1)

In the large N limit the density, ρ`ef f `ef f (ρ, z), of a single spin component in the⊥z

ground state of the effective oscillator is given by [84](1)ρ`ef f `ef f (~r) =
z
⊥

1

3/2

6π 2 `ef fz `2ef f⊥

1−

(2µ)

2/3 z 2 /`4ef fz + ρ2 /`4ef f⊥`ef fz `2ef f⊥
2µ

!3/2
,
(4.22)

where µ = (3N )1/3 is set byR(1)ρ`ef f `ef f (~r) d3 r = N/2.
⊥

z

Evaluating the expectation value in Eq. 4.19 with Uint given by Eq. 3.36 and67

Figure 4.2: The dimensionless effective hypervectorial potential is shown plotted as afunction of Rρ0 and Rz0 for an interaction strength kf0 a = 0 and a trap ratio of γ = 1/3.


1/3
(1)
kf (r) = 6π 2 ρ`ef f `ef f (~r)
gives the effective hypervectorial interaction potential
⊥

z




0a
k
Vint Rρ0 , Rz0
256
f
=
√
2/3 f  √1/3  ,EN I
9π 2
2γRz0 Rρ20
2γRz0 Rρ02
Z 1 p
f (x) =
y 6 1 − y 2 ζ (xy) dy.

(4.23)

0

One should note that the function f (x) is the same as in the isotropic trap hyperradial effective potential, Eq. 3.40. Putting everything together yields a total effective
hypervectorial potential,
Vef f Rρ0 , Rz0
1
1 02
1
1 2 02 Vint Rρ0 , Rz0=+ R ++ γ Rz +.
EN I
3Rρ02 3 ρ
12γ 2 Rz02 3
EN I(4.24)An example of the effective potential is shown in Fig. 4.2 for a trap ratio γ = 1/3 andan interaction strength of kf0 a = −1.
4.2.1

Results for the hypervectorial method

Here I analyze the behavior of the effective hypervectorial potential for various
values of kf0 a. Figure 4.3 shows contour plots of Vef f for evenly spaced values of kf0 a from68
−12 to 12 for a system with trap ratio γ = 1/3. For attractive interactions (a < 0),
the minimum is seen to be pulled into the center as the gas pulls in on itself. For
repulsive interactions (a > 0) the minimum is pushed out away from the center. It isalso interesting to see the low lying contours behavior as it gets twisted towards theorigin for attractive interactions and away for repulsive. This behavior will be studiedin more detail later.
k0 a = −12

k0 a = −9fk0 a = −6

f

f

2

2

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0

0

1

2

3

4

0

0

1

0

2

3

4

0

2

2

1.5

1.5

1

1

1

0.5

0.5

0.5

1

2

3

4

0

0

1

0

Rρ’

2

1

2
Rz’

4

0

3

4

2

2
1.5

1

1

0.5

0.5
0

1

2

3

4

0

1

2

3

4

3

4

k0f a = 12

1.5

0

1

3

k0f a = 9

kf a = 6

0
0

2

2
0

kf a = 3201

kf a = 0

1.5

0

0

0

kf a = −3

3

4

0

0

1

2

Figure 4.3: Contour plots of the dimensionless effective hypervectorial potential are
shown plotted as a function of Rρ0 and Rz0 for an interaction strength kf0 a = −12 to 12
and a trap ratio of γ = 1/3.

In this section the ground state energy and average squared collective coordinates
for the DFG in an anisotropic trap are found, i.e. E/EN I , Rz02 andRρ02 . In thesame way as the hyperspherical K harmonic method, these quantities can be found by
minimizing Vef f , i.e. by solving∂Vef f∂Vef f== 0. Finding the minimum of Eq.
0
∂Rρ
∂Rz0

4.24 cannot be done analytically, but because the interaction is function of Rz0 Rρ02 only,
the relationship between the minimum longitudinal coordinates, Rz0 min , and transverse

69
coordinate, Rρ0 min , can be found:Rρ0 minRz0 min = √2γ(4.25)

This indicates that the DFG with density dependent interactions will always maintainthe same aspect ratio. This behavior is likely due to the fact that the hyperangularbehavior was frozen to the noninteracting behavior so that oscillator quanta cannot be
exchanged between the longitudinal and transverse directions, i.e. λz and λ⊥ are fixed.
If a more complex formulation were to be used, by allowing λz and λ⊥ to be functionof the hypervectorial coordinates, a task beyond the scope of this study, it is likely thatthe aspect ratio of the gas would change with repulsive and attractive interactions. The√
minimum in Vef f must lie along the line, Rz0 min = Rρ0 min / 2γ, and evauates to



Rρ0
0
0
Vint Rρ , Rz = √
1
1
1
1
256
2γ
=
+ Rρ02 ++ γ 2 Rz02 + 2 02 f
02
02
EN I
2Rρ
2
6Rz
6
9π Rρ

kf0 a
Rρ0

!
.

(4.26)

This is the exact same functional form as the effective potential found for the hyperspherical treatment in an isotropic trap, Eq. 3.41, meaning that large N expectation
for E/EN I and Rρ2 / Rρ2 N I in the ground state will be exactly the same as E/EN I
and R2 / R2 N I shown in Figs. 3.16 and 3.17.

4.3

Low energy excitations
One of the benefits from the hyperspherical K-harmonic was the simple extrac-

tion of the low lying radial excitations. Similarly, in the hypervectorial picture, low
lying excitations can be extracted as well. The difference is that now there are two distinct types of excitation corresponding to transverse and longitudinal breathing modes.
In the noninteracting limit these two modes decouple, but as interactions are turned
on, excitations in the two trap axes become coupled. This behavior can be visually
understood by examining the ellipses made by the low lying contours shown in Fig. 4.3.

70
A more quantitative view of the low lying excitations can be extracted by normalmode type of analysis which begins by approximating the effective potential about theminimum as a harmonic oscillator potential,∂ 2 Vef f  ∂Rρ02
EGS 1  0≈
+ Rρ − Rρ0 min , Rz0 − Rz0 min  ∂ 2 Vef f
EN I 2∂Rρ0 ∂Rz00

Vef f Rρ0 , Rz
EN I



∂ 2 Vef f00Rρ − Rρ min ∂Rρ0 ∂Rz0 2∂ Vef f 
0 − R0
R
z
z min
∂Rz02
(4.27)

where EGS is the ground state energy found by minimizing the effective potential. The
oscillator frequencies about this minimum can be extracted by finding the eigenvaluesof the Hessian matrix in Eq. 4.27:
r

1
ν1 ,
m∗
r
1
ω2 =
ν2 ,
m∗
ω1 =

where ν1 and ν2 are the eigenvalues. These breathing modes are in units of the noninteracting energy; to get back to conventional units, the frequencies must be multiplied
by EN I /~. From Eq. 4.17, m∗ = mEN I N Rρ2 N I /~2 , and noting that N Rρ2 N I =2`2⊥ EN I /3~ω⊥ givesr

3 √ω⊥ ν1 ,
2
r
3 √
ω2B =
ω⊥ ν2 ,
2
ω1B =(4.28)The eigenvectors corresponding to the eigenvalues in this equation have a direct meaning, as the directions in which the gas oscillates. Roughly, one of these frequencies
corresponds to the transverse breathing mode, while the other to longitudinal. I willtake ω1 (ω2 ) as the transverse (longitudinal) mode, i.e. in the noninteracting limit
ω1 = 2ω⊥ .
Figures 4.4 and 4.5 show the two breathing mode frequencies as a function of kf0 a
for trap ratios γ = 0.1 to γ = 1. For γ = 1, a cusp can be seen at kf a = 0 in both frequencies. This is due to the degeneracy of the two breathing modes in the isotropic trap.

71
Again, it should be mentioned that both the variational trial wavefunction and the density dependent interaction cannot describe a gas of bosonic dimers for positive scattering
lengths. There are several worrying things that can be seen in the predicted behaviorof the breathing modes. First, in the unitarity regime, kf0 a → −∞, both the transverseand longitudinal breathing modes are greater than the noninteracting frequencies forγ < 1, and the frequencies in this regime depend on γ. This is in disagreement withp
the unitarity predication from superfluid hydrodynamic models of ω1B = 10/3ω⊥ andp
ω2B = 12/5ωz [101]. Second, and most importantly, the frequencies predicted here differ, both quantitatively and qualitatively, from those found in experiment [102]. Thesedisagreements are likely due to the overly simplistic variational trial wavefunction, Eq.
4.8, used in the hypervectorial picture. By fixing the hyperangular behavior to thatof the noninteracting Fermi gas, the wavefunction cannot take into account the higher
order correlations, such as BEC-like pairing, that are present in the system.
While this study does not “hit the nail on the head”, this is seen as a stepping
off point for future studies that hopefully will begin to incorporate the more complex
nature of this system. This chapter may also be seen as a simple example of the hypervectorial method. While the idea was applied here to a degenerate Fermi gas in an
anisotropic trap, this is by no means the only possible application of the technique.
For instance, in a Fermi gas of distinguishable particles, e.g. a Fermi gas where thecomponents have unequal masses, unequal numbers, or different trapping frequencies,
the hypervectorial approach can be applied with each part of the hypervector corresponds to a sub-hyperradius for each component in the gas. This might allow for higher
order fluctuations, phenomena like phase separation, or a “beating” mode where twocomponents oscillate out of phases.

72

2.4

2.2

Β

ω2 /ωperp

2.3

2.1

2

-0.4

-0.2

0
0
kf a

0.2

0.4

Figure 4.4: The transverse breathing mode predicted by the hypervectorial method isshown plotted versus the interaction strength kf0 a for trap ratios γ = 0.1 to 1 in stepsof 0.1 from top to bottom. The kink in the seen in the γ = 1 frequency is due to adegeneracy between the longitudinal and transverse frequencies in the non-interacting
limit.

2.2

B

ω1 /ωz

2.1

2

1.9

1.8

1.7

-0.4

-0.2

0
arctan(kfa)/π

0.2

0.4

Figure 4.5: The longitudinal breathing mode predicted by the hypervectorial method isshown plotted versus the interaction strength kf0 a for trap ratios γ = 0.1 to 1 in stepsof 0.1 from top to bottom. The kink in the seen in the γ = 1 frequency is due to adegeneracy between the longitudinal and transverse frequencies in the non-interacting
limit.

Chapter 5

Green’s Functions and the Adiabatic Hyperspherical Method

In the previous chapters, hyperspherical coordinates were used to create simple
variational trial wavefunctions that then could be used to create an effective low dimensional Schrödinger equation. In this way, complex, highly correlated systems could beanalyzed using the intuition of normal Schrödinger quantum mechanics. In contrast,
this chapter will deal with exact hyperspherical solutions to small systems. Rather thanusing a simple variational approach to approximate the hyperangular behavior of a system, I will employ a Lippmann-Schwinger (LS) equation-based method. Some of thework presented here is developed from the work of Mehta et. al. [29].
The adiabatic hyperspherical method has proven useful for analyzing many fewbody systems [35, 37, 66, 103, 104, 105]. The heart of this method lies in treating
the hyperradius adiabatically, and diagonalizing the remaining hyperangular behavior.
The result takes a d dimensional partial differential equation to a set of coupled onedimensional differential equations:
 2 2

~ d
~2 (d − 3) (d − 1)(d−1)/2ER
Fn (R) = −++ Un (R) R(d−1)/2 Fn (R)2µ dR2 2µ
4R2


~2 X
d
2Pnm
− Qnm R(d−1)/2 Fm (R) ,
−
2µ m
dR

(5.1)

where d is the total number of dimensions, and Un (R) is found by solving the fixed R
adiabatic Schrödinger equation,

 2 2
~ Λ
+ V (R, Ω) Φn (R; Ω) = Un (R) Φn (R; Ω) .
2µ R2

(5.2)

74
The non-adiabatic matrices P and Q are found by inserting the expansion,Ψ (R, Ω) =X

R(d−1)/2 Fn (R) Φn (R; Ω) ,ninto the Schrödinger equation and are given by∂
Pmn = Φm (R; Ω) |Φn (R; Ω) ,
∂R


∂2
Qmn = Φm (R; Ω) | 2 Φn (R; Ω) .
∂R

(5.3)
(5.4)

The integrals in Eqs. 5.3 and 5.4 are taken over the hyperangles Ω.
Approximate solutions can be found by solving the uncoupled system of equations 2 2~2 (d − 3) (d − 1)
~2
~ d
+
+
Q
+
U
(R)
R(d−1)/2 Fn (R)ER(d−1)/2 Fn (R) = −nnn2µ dR2 2µ
4R2
2µ
(5.5)
Ground state solutions to this give a variational minimum to the full solution to Eq.
5.1. Another method is that of the Born-Oppenheimer approximation in which the
diagonal correction to the potential~2
Qnn is ignored. These two approximations will2µ

be the main focus of this chapter, while the non-adiabatic couplings Pnm will be used
to describe transitions between the different hyperradial channels Un (R).
The sticking point of the adiabatic hyperspherical method lies in solving the adiabatic Schrödinger equation, Eq. 5.2. Often solving this equation is as hard as solving
the total d dimensional Schrödinger equation in the first place. Having a variety ofmethods available is helpful for this reason. The benefit of using the adiabatic hyperspherical method comes from the simple final interpretation that can often be applied
to the resulting coupled set of one-dimensional equations [106]. For instance, in thethree-body problem, which I will discuss later in this chapter, if two particles can forma bound state, then one of the resulting scattering channels consists of an atom anda dimer colliding. In the adiabatic hyperspherical method this type of fragmentation
channel arises naturally with a simple interpretation.

75
In Section 5.1 I derive a general form of the hyperangular Green’s function fora d-dimensional system. In Section 5.2 I apply the Green’s function to the problem ofthree particles with regularized, zero-range, s-wave interactions. Section 5.3 applies thisresult to the three lowest hyperfine states of 6 Li. Finally, Section 5.4 extends the Green’s
function methods to the three-body problem with multi-channel two-body interactions,and analyzes the resulting system of hyperradial potentials.

5.1

The Hyperangular Green’s function
In this section I derive the free space hyperangular Green’s function for an ar-bitrary d dimensional space, for example, d = 3 (N − 1) for an N body system withthe center of mass coordinate removed. This Green’s function can then be used in a
LS equation to solve Eq. 5.2. The d dimensional Laplacian written in hypersphericalcoordinates is given in Eq. 2.2 as∇2 =where R =∂ 2 (d−1)/2 (d − 1) (d − 3) Λ2
R
−
− 2
4R2
R
R(d−1)/2 ∂R2
1

(5.6)

qP
d

2
i=1 xi . The hyperangular Green’s function is given as the solution to

 2



Λ − ν (ν + d − 2) G Ω, Ω0 = δ d Ω − Ω0 .

(5.7)

Here Ω stands for the d − 1 hyperangular coordinates needed to describe the surface of ad dimensional hypersphere and δ d (Ω − Ω0 ) is the Dirac δ-function in the hyperangularR
coordinates, i.e. δ (Ω − Ω0 ) = 0 if Ω 6= Ω0 and δ d (Ω − Ω0 ) dΩ = 1. The Green’s function can be found in several forms, including the full hyperspherical harmonic expansion[107], and in closed form given by Szmytkowski in Ref. [108].
The simplest derivation of the Green’s function relies on the completeness of
hyperspherical harmonics:
X
λµ

 ∗

∗
Yλµ
Ω0 Yλµ(Ω) = δ d Ω − Ω0 .

(5.8)

76
This relationship can be used in conjunction with Eq. 2.12 to find G [107]:0G Ω, Ω =X
λµ∗ (Ω0 ) Y ∗ (Ω)
Yλµλµλ (λ + d − 2) − ν (ν + d − 2)

.

(5.9)

Unfortunately, eigenfunction expansions of Green’s function often have slow convergencewith respect to the number of eigenfunctions, often making them unsuitable for numericcalculations. The closed form of the Green’s function from Ref. [108] is given asG Ω, Ω0 =−πCν(d−2)/2 −R̂ · R̂0 ,(d − 2) Sd sin πν

(5.10)

where Cνα is a Gegenbaeur function, Sd is the surface area of the d dimensional unitRhypersphere: Sd = dΩ = 2π d/2 /Γ (d/2), and R̂ · R̂0 is the cosine of the angle between
the two normalized hypervectors R̂ and R̂0 . Here ν is defined by equation 5.7. While
this does have a pleasing, compact form it is often divergent at critical points. For
instance if ν is non-integer valued then G (Ω, Ω0 ) diverges as R̂ · R̂0 → 1.
For these reasons, it is convenient to find a third form of the Green’s function.
The first step in this derivation relies on the division of the total d dimensional space
into two subspaces in the same hyperspherical tree structure as is shown in Fig. 2.3.
For the purposes of this thesis I will assume that the dimension of the two subspaces
are both greater than 2: i.e. d1 , d2 ≥ 2. The hyperangular momentum can be written
in terms of the sub-hyperangular momenta using Eq. 2.11 asΛ2 =∂2(sin α)(d1 −1)/2 (cos α)(d2 −1)/2(d1 −1)/2(d2 −1)/2 ∂α2(sin α)(cos α)−1+(5.11)

Λ21 + (d1 − 1) (d1 − 3) /4 Λ22 + (d2 − 1) (d2 − 3) /4 (d − 1) (d − 3) + 1+−.
cos2 α4sin2 αThe Green’s function can be expanded using the completeness of the sub-hypersphericalharmonics, i.e.
 X X



G Ω, Ω0 =g α, α0 Yλ∗1 µ1 Ω01 Yλ1 µ1 (Ω1 ) Yλ∗2 µ2 Ω02 Yλ2 µ2 (Ω2 ) ,λ1 µ1 λ2 µ2(5.12)

77
where
Λ21 Yλ1 µ1 (Ω1 ) = λ1 (λ1 + d1 − 2) Yλ1 µ1 (Ω1 ) ,(5.13)

Λ22 Yλ2 µ2 (Ω2 ) = λ2 (λ2 + d2 − 2) Yλ2 µ2 (Ω2 ) .
Inserting this expansion, Eq. 5.7 is satisfied if and only if"∂2δ (α − α0 )−1=(sin α)(d1 −1)/2 (cos α)(d2 −1)/2
d1 −1
d2 −1
(d1 −1)/2(d2 −1)/2 ∂α2(sin α)(cos α)(sin α)(cos α)λ1 (λ1 + d1 − 2) + (d1 − 1) (d1 − 3) /4
sin2 αλ2 (λ2 + d2 − 2) + (d2 − 1) (d2 − 3) /4+
cos2 α(d − 1) (d − 3) + 1
2−− ν (ν + d − 2) gλd11,dν; α, α0 ,
,λ
2
4

+

where δ (α − α0 ) is a Dirac δ-function and the denominator on the left hand side arises
from the hyperangular volume element associated with the angle α (See Refs. [66, 108]for details). The right hand side can be manipulated into the standard Sturm-Liouvilleform,"δ (α − α0 )d1 −1(sin α)(cos α)

d2 −1

=

−1

∂
(5.14)∂α(sin α)(cos α)
λ1 (λ1 + d1 − 2)λ2 (λ2 + d2 − 2)
d1 ,d2
0
+
+
−
ν
(ν
+
d
−
2)
g
ν;
α,
α
.
λ1 ,λ2cos2 αsin2 αd1 −1∂

d2 −1 ∂α(sin α)d1 −1 (cos α)d2 −1

The general solution for any differential equation of this form is given by
02
gλd11,d,λ2 ν; α, α =f + (α< ) f − (α> )(sin α)d1 −1 (cos α)d2 −1 W [f + , f − ],(5.15)

where W [f1 , f2 ] = f1 f20 − f2 f10 is the Wronskian (see chapter 3 of your favorite edition ofJackson [109] for details) and α<(>) = min (α, α0 ) (max (α, α0 )). The functions f + (α)
and f − (α) are regular at α = 0 and α = π/2 respectively and satisfy the homogeneous
equation,"0=(sin α)

−1
d1 −1

∂
d2 −1 ∂α(cos α)(sin α)d1 −1 (cos α)d2 −1
λ2 (λ2 + d2 − 2)+− ν (ν + d − 2) f (±) (α) .
cos2 α∂λ1 (λ1 + d1 − 2)+∂αsin2 α(5.16)

78
The solutions to this are given in Ref. [110] asf(±)(α) = (sin α)

λ1(cos α)

λ2


2 F1

λ1 + λ2 − ν ν + λ1 + λ2 + d − 2
d± 1 ∓ cos 2α,; λ± +;
2
2
2
2(5.17)



 

d1
d2
−2Γ λ1 +Γ λ2 +
 + −
22 ,W f ,f =ν + λ1 + λ2 + d − 2
λ1 + λ2 − ν(sin α)d1 −1 (cos α)d2 −1 ΓΓ2
2
(5.18)
where 2 F1 (a, b; c, x) is a hypergeometric function, λ+ = λ1 , d+ = d1 , λ− = λ1 , andd− = d2 .

5.2

The Three-Body Problem with Zero-Range Interactions
In this section I show the utility of the Green’s function developed in the previous

section by applying it to the three body problem with regularized, zero-range, s-wave,
pseudo-potential interactions. This problem is well studied by a variety of sources[111, 26, 112, 113]. The full Hamiltonian for the untrapped system is given by
Htot =

3X
i=1−~2 2 X∇ +V (rij ) ,2mi i(5.19)

i>jwhere ~ri is the position of the ith particle, and ∇2i is the Laplacian for ~ri . Theinteraction is given by
V (rij ) =4π~2 aij (3)∂δ (~rij )
rij ,
2µij
∂rij

(5.20)

where aij is the s-wave scattering length between particles i and j and µij is the twobody reduced mass, µij = mi mj / (mi + mj ). The pseudo-potential defined in this way
applies the Bethe-Peierls boundary condition to the two-body wave function as r → 0,i.e. ψ (r) → (1 − aij /r) C [114]. The center of mass can be removed from this system by
converting to a system of Jacobi vectors. Jacobi vectors are created for this system by


,

79
considering the separation vector between two of the three particles and then a second
vector from the center of mass of that two body system to the third. The final vector
is then just the center of mass coordinate. The choice of Jacobi vectors is not unique.
In this study I consider three different Jacobi coordinates each of which is convenientfor describing one of the three possible two-body interactions V (rij ) . In the “odd manout” notation these are given by(k)ρ
~1 = (~ri − ~rj ) /dk ,mi~ri + mj ~rj(k)ρ~2 = dk− ~rk ,
mi + mj(5.21)(m1~r1 + m2~r2 + m3~r3 ),m1 + m2 + m3
(mk /µ) (mi + mj )d2k =.
m1 + m2 + m3

~rCM =

Here µ is the three-body reduced mass given byr
m1 m2 m3µ=.
m1 + m2 + m3
In these Jacobi coordinates the total Hamiltonian can be rewritten in terms of the Jacobi
coordinates and the center of mass as
Htot = H + HCM ,(5.22)

−~2 2
∇ ,
2M CM
2
X
~2 X 2H=−∇ρi +V (rij ) .
2µ

HCM =

i=1

i>j

Transforming the Jacobi coordinate piece of the Hamiltonian in Eq. 5.22 into hyperspherical coordinates using Eqs. 2.2 and 2.1 yields
H=−



15~2
~2 Λ2 X
~2 1 ∂ 2 5/2
(k)
R
+
+
+
V
d
ρ
.
ij
k
1
2µ R5/2 ∂R2
8µR2 2µR2(5.23)

i<jTo apply the adiabatic hyperspherical formulation, the hyperangular adiabatic Schrödinger
equation must be solved:




2
X
(k)
Λ2 + 2µR
Vij dk ρ1− ν (ν + 4) Φ (R; Ω) = 0.
2
~
i<j

80
This can now be accomplished with the use of the hyperangular Green’s function in the
Lippmann-Schwinger equation,
2µR2
Φ (R; Ω) = − 2~(k)0where ρ
~1

Z





X


(k)0
dΩ0 G Ω, Ω0 
Vij dk ρ1  Φ R; Ω0 ,(5.24)

i<jis the kth Jacobi vector parameterized by (R, Ω0 ). The hyperradial Hamil-

tonian from Eq. 5.1 in the absence of the non-adiabatic couplings P and Q is given interms of the hyperangular eigenvalue ν as∂2
(ν + 2)2 − 1/4
−
∂R2
R2

−~2
HR =
2µ

!
.

(5.25)

To evaluate the integrals in the LS equation, the Green’s function from Eq. 5.12
is rotated into the appropriate Jacobi coordinate set for each interaction term in thesum with the hyperangles defined as
o
n
(k)
(k)
Ω(k) = ω1 , ω2 , α(k) ,(k)where ωi(5.26)(k)is the spherical polar angular coordinates for ρ~i . The remaining hyperangle

α(k) is defined as in Eq. 2.9, i.e.
(k)

ρ1 = R sin α(k) ,(5.27)(k)ρ2 = R cos α(k) .
With this choice of hyperangles, it is clear that d1 = d2 = 3 and the hyperspherical(i)harmonics Yλi µi (Ωi ) in Eq. 5.12 reduce to normal spherical harmonics yLi Mi (ωi ).
The δ-function implies that the two-body boundary condition for each two-bodyinteraction can be applied and the third particle can be considered to be far away, i.e.
!


a(k)
(k)
(k)
lim Φ (R; Ω) = 1 − (k) yLM ω2 CLM(5.28)
rr(k) →0(k)where yLM is a spherical harmonic describing the free space behavior in ω2

and carries

the total angular momentum of the system, and the superscript k is indicates the odd

81
man out notation. Inserting Eq. 5.28 into Eq. 5.24 gives the hyperangular eigenfunction,f − α(k)2µ X a(k)(k)(k)Φ (R; Ω) =yLM ω1 CLM.
R
2µij d3k(sin α)2 (cos α)2 W [f + , f − ]k(5.29)
√
3− πΓ L +


2
 

(sin α)d1 −1 (cos α)d2 −1 W f + , f − = 
L−ν
ν+L+4
Γ
Γ
2
2
(k)0

where the orthonormality of spherical harmonics has been used to evaluate the ω1(k)0

and ω2integrals. The δ-function in Eq. 5.20 implies that the integral in α(k)0 can be(k)accomplished by evaluating at α< = α(k)0 = 0.
The analytic equation for the hyperangular eigenfunction in Eq. 5.29 is not very
useful without knowing the eigenvalue ν. To obtain an equation for ν the boundary
condition given in Eq. 5.28 must be applied again. Applying it to both sides of Eq.
5.29 yields

 0

∂
(k )
(k0 )
(k0 )
(k0 )
=
y
ω2
CLM
α
Φ
R;
Ω
LM
0)
(k
0
α(k ) →0 ∂α




∂
2µ X a(k)(k)(k)(k0 ) −
(k)
N
y
ω
=
C
lim
α
f
α
,
Lν LM
2
LM k0
(k0 )
R
2µij d3k
α( ) →0 ∂αlimk(5.30)
Γ
NLν = −

 

L−ν
L+ν+4
Γ
2
2


.
√
3
πΓ L +
2

To evaluate the limit on the right hand side of this the k Jacobi coordinates when
(k0 )

ρ1

→ 0 must be understood. Eqs. 5.21 and 5.27 give, for k 6= k 0 ,lim α
0α(k ) →0(k)
(m1 + m2 + m3 ) µ0= βkk0 = arctan; lim ρ(k) ∝ −ρ(k ) .
0
mk mk0α(k ) →0Note that if f + is regular at βkk0 , then


∂
(k0 )
(k)
α
f
α
→ f2 (βkk0 ) .
2
(k0 )
0
α(k ) →0 ∂αlim(5.31)

82(k)Using this and evaluating limits in Eq. 5.30 yields a matrix equation for CLM :
(k0 )

CLM =

X

(k)

MkLν
0 k CLM ,(5.32)

k



 

L
−
ν
ν
+
L
+
4



2Γ
Γ
0


a(k )
2µ
2
2



 


L−ν−1
L + ν + 3 2µij d3k
RLν
Mk 0 k =
Γ
Γ

2
2




2µ
a(k) −L
f (βkk0 )+ (−1)NLνR
2µij d3k

for k = k0.
for k 6= k 0

The hyperangular eigenvalue, ν, is found by solving the closed form transcendental
equation, det (M − 1) = 0, for any given total angular momentum L and any set ofs-wave scattering lengths, a(k) .

5.2.1

Imposing symmetry

The hyperangular eigenvalues for the general three-body problem with arbitraryexchange symmetry can be found by solving the transcendental equation implied by Eq.
5.32, but the system can be simplified by considering different permutation symmetries(k)and imposing those symmetries on the boundary conditions CLM . For example, if
the particles in question are identical bosons, permutation cannot have any affect on
the wave function. Thus, if a particle is exchanged in the two-body subsystem, the
(1)

(2)

(3)

boundary condition must remain the same, i.e. CLM = CLM = CLM = CLM anda(1) = a(2) = a(3) = a. A complete list of the possible exchange symmetries is given in
Table 5.1. To illustrate this post-symmetrization, I apply the identical boson symmetry
X1 X2 X3
BBB
BBX
FFX

C (1)
C
C1
C

C (2)
C
C1
−C

C (3)
C
C2
0

Table 5.1: The possible permutation symmetries that may be imposed on the three bodysystem with s-wave interaction are given with the appropriate boundary conditions. B
stands for a boson, F for a fermion and X for a distinguishable particle with an arbitrary
mass.

83
with L = 0 to Eq. 5.32 resulting in the well known transcendental equation for ν[115, 111, 26, 112, 113],

R
=
a

−31/4



8
√ sin
3






π (ν + 2)
π (ν + 2)− (ν + 2) cos
6
2


.
√
π (ν + 2)
2 sin
2(5.33)In the limit where R  |a| the first solution to this transcendental equation gives asupercritical attractive 1/R2 effective potential,
U (R) =

~2 −s20 − 1/4,
2µR2(5.34)

s0 = 1.00624.
This attractive potential is the source of the famous Efimov effect, where an effective
dipole potential supports an infinite set of three-body bound states that accumulate at
the non-interacting three-body threshold, E = 0.

5.2.2

Non-adiabatic couplings

As in any adiabatic treatment, the effective hyperradial potentials are coupled
by non-adiabatic terms that arise from the hyperradial dependence of the hyperangulareigenfunctions. The full system described by Eq. 5.1 can be written as a coupled systemof ordinary differential equations of the form,



~2 X
∂2
∂
0=−
δmn
+ 2Pmn+ Qmn Fn (R)
2µ n
∂R2
∂R
"
#
~2 (νm (R) + 2)2 − 1/4+− E Fm (R)
2µ
R2
where



∂
Φn (R; Ω) ,
∂R


∂2
Qmn = Φm (R; Ω)Φn (R; Ω) .
∂R2
Pmn =

Φm (R; Ω)(5.35)

84
To find the non-adiabatic coupling matrices, the methods of Ref. [116], in which
derivatives of the adiabatic Schrödingier equation are considered, can be employed byconsidering two matrix elements,Φ0nΛ2 − εmΦm = 0,−ε0n hΦm |Φn i + Φm

Λ 2 − εnΦ0n = 0

(5.36)

where εn = νn (νn + 4) is the hyperangular eigenvalue of the nth adiabatic eigenfunction,and the prime indicates a hyperradial derivative has been taken. Taking the differenceof these leads to an equation for the non-adiabatic coupling matrix element Pmn form 6= n:Φ0n Λ2 Φm − Φm Λ2 Φ0n − (εm − εn ) Pmn + δmn ε0n = 0.

(5.37)The difference Φ0n Λ2 Φm − Φm Λ2 Φ0n is given by the boundary conditions of the
wave functions Φm and Φn at the coalescence points:
Φ0n

2

2

Λ Φm − Φm Λ

Φ0n

X a(k)

=

∂ (k)
(k) ∂
Cm
C − Cm
dk R
∂R n
∂R

k

X

=

(k) (k)CmCnka(k) (k)C
dk R n!
(5.38)

a(k)
.
dk R2
(k)

Here the LM subscript in the boundary values CLM have been suppressed. Inserting
this into Eq. 5.37 yields two equations,
P

(k)

(k) a

(k)

k Cm Cn

dk R2
for n 6= m
(εm − εn )
2 a(k)X
−ε0n =
Cn(k)
.
dk R2

Pmn =

(5.39)k

Because the hyperangular eigenfunctions are orthonormal, the diagonal part of the P
matrix is zero, i.e. Pnn =1 ∂hΦn |Φn i = 0. The second equation gives the normaliza2 ∂Rtion condition for Φn , with an overall phase that is free. This overall phase is chosen

85
here so that(k)k Cn is positive. A similar derivation provides the matrix elements Qmn :

P

 0

εn + Rε00n + An
ε0n Pmn + Bmn
ε000nQmn = δmn
+
2
(1
−
δ
)
+
,
mn
R2 ε0n6ε0n(εm − εn )
0  2
X a(k) 
(k)
An =
Cn
,
dk
k
!
#
"

0
(k)
(k)
X
a
a
(k) (k)(k)Cn(k) − Cm
Cn
.
Bmn =
Cm
dk R2
dk R3

(5.40)k

When the symmetries given in Table I are used, there can be a considerable
simplification of the expressions for Pmn and Qmn . For a system of identical bosons
(1)

(2)

(3)

where a(1) = a(2) = a(3) = a, d1 = d2 = d3 = d and Cn = Cn = Cn = Cn , Pmn and
Qmn are given byp
ε0m ε0n(5.41)Pmn =(εm − εn )
s
"
#
"
#
p
 
2ε0n ε0n ε0m
1 ε00n 2 1 ε000
ε00n
ε0m
n
Qmn = δmn −++ (1 − δmn )
−
,
4 ε0n
6 ε0n
(εm − εn ) ε0n(εm − εn )2which are in agreement with previously calculated nonadiabatic corrections for the threeidentical boson system [111].

5.3

Three distinguishable interacting particles.
In this section the results of the previous section are applied to the case of three

distinguishable equal-mass particles. This system could be realized, for instance, in an
ultracold three component Fermi gas of 6 Li atoms [94, 95]. The scattering lengths near
the resonance positions used here, as a function of magnetic field, are given in Ref.

86
Region
I
II
III
IV
V

r0  a(3) . a(1)  a(2)r0  a(3) ∼ a(1)  a(2)r0  a(2)  a(1) , a(3)r0  a(2)  a(1) , a(3)r0  a(2)  a(1)  a(3)a(1) , a(2) , a(3) > 0
a(2) < 0; a(1) , a(3) > 0
a(2) < 0; a(1) , a(3) > 0
a(2) , a(1) < 0; a(3) > 0
a(1) , a(2) , a(3) < 0

Table 5.2: The possible tunable interaction regimes near the resonances of 6 Li are given.

[4, 94, 95] by
(k)

a


= ab


∆
[1 + α (B − B0 )] ,1−
B − B0(5.42)

for k = 1 :ab = −1450a0 , B0 = 834.15 G,∆ = 300 G and α = 4 × 10−4 G−1 ;for k = 2 :ab = −1727a0 , B0 = 690.4 G∆ = 122.2 G and α = 2 × 10−4 G−1 ;for k = 3 :ab = −1490a0 , B0 = 811.22 G,∆ = 222.3 G and α = 3.95 × 10−4 G−1 ;where a0 is the Bohr radius. The Fano-Feshbach resonances in this system allow for alarge variety of tunable interactions.
In this section I consider five different regions of magnetic field, shown in Fig.
5.1, near the three resonance positions which all have distinct behavior. In all five
regions the scattering length is much larger than the effective range, allowing for theuse of the zero-range interaction assumptions. Table 5.2 shows the various length scale
discrepancies in these regions.
Figure 5.2(a) shows an example of the lowest four hyperangular eigenvalues (ν + 2)2

87

Figure 5.1: All possible s-wave scattering lengths are shown for the lowest 3 Zeeman
states of Li6 from Ref. [4]. Each marked region gives a different set of length scale
discrepancies. Here a(k) is the scattering length between two atoms in states |ii and |jiwith k as the component not involved in the interaction.

Figure 5.2: (a)The the first four hyperangular eigenvalues are shown for an example
system where a(1) = a(3) and a(2) = 1000a(1) versus the hyperradius. The solid black
horizontal lines show the expected behavior for 3 identical resonantly interaction bosons.
The dashed line gives the behavior of two identical fermions interacting resonantly witha third distinguishable particle. Dotted lines give the expected universal behavior fora single resonant scattering length. Finally, the dot dashed line is the lowest expected
free space behavior for three distinguishable free particles.(b)The coupling strengths
between the first and second (purple), the first and fourth (red), and the second andfourth (black) adiabatic potentials are shown as a function of R.

obtained from solving Eq. 5.32 for a(1) = a(3) and a(2) = 1000a(1) . This is provided

88
as an example that is qualitatively similar to the behavior of the system in region I.
When the hyperradius is in a region where all other length scales are much different,
the hyperangular eigenvalue (ν + 2)2 becomes constant, or, in the case of 2-body boundstates, becomes proportional to R2 . This behavior can be interpreted as giving a universal set of potential curves from Eq. 5.25. For example in region I where r0 
a(3) . a(1)  a(2) there are three hyperradial regions: r0  R  a(3) . a(1)  a(2) ;r0  a(3) . a(1)  R  a(2) ; and r0  a(3) . a(1)  a(2)  R. In each region
the hyperangular eigenvalues take on the universal value that is expected for resonantinteractions [60, 111, 112].
Figure 5.3 schematically shows the behavior of the first few hyperradial potentials
Un (R) =~2 (νn + 2)2 − 1/4.
2µR2(5.43)The grey areas are the regions where potentials are transitioning from one universal
behavior to the next. The zero-range pseudo-potential cannot describe the short range
details of the interaction, meaning that the potentials found here are only valid forR  r0 where r0 is a short range parameter shown schematically in the blue region ofFig. 5.3.
2 /2µ (E − E ), between the differFigure 5.2(b) shows the coupling strength, Pνµνµent potentials. The places where this coupling peaks are the points where a transition
between curves is the most probable. Figures 5.4(a-e) are examples of the hyperangulareigenvalues found in each region. The magnetic field at which each set of eigenvalues
are found is shown as dotted lines in Fig. 5.4(f) from left to right for Fig. 5.4(a-e)
respectively. In each figure the hyperangular eigenvalue can be seen flattening out toa universal constant in each region of length scale discrepancy. As the magnetic field
is scanned through each resonance, one two body bound state becomes a virtual state.
This behavior can be seen in the hyperangular eigenvalues that diverge toward −∞. Aseach resonance is crossed, one of the hyperangular eigenvalue curves goes from diverging

89

Figure 5.3: A schematic picture of the first four hyperradial potentials in region I isshown. The grey areas indicate regions where the potentials are changing from one
universal behavior to another. The blue region indicates the short range region where
the zero-range pseudo-potential not longer can be applied.

30

20

20
2

30
10
0
-10

0
-10

3

(a)

10

4

10
R (a. u.)

5

10

-20
2
10

6

10

30

20

20
2

30
10
0
-10
-20
2
10

3

(c)

10

4

10
R (a. u.)5

10

(10 a. u.)

5

10

6

10

3

(d)

10

4

10
R (a. u.)

5

10

6

10

8
4

3

10
0

(k)

2

4

10
R (a. u.)

0
-20
2
10

6

10

20
(ν+2)

10

-10

30

a

-10
-20
2
10

3

(b)

10

(ν+2)

(ν+2)

2

-20
2
10

10

(ν+2)

(ν+2)

2

to −∞ to converging to (ν + 2)2 → 4.

3

(e)

10

4

10
R (a. u.)

5

10

6

10

0
-4
-8
500

600
(f)

700
800
B (Gauss)

900

1000

Figure 5.4: (a)-(e) Examples of the hyperangular eigenvalues from each region of magnetic field are shown as a function of the hyperradius in atomic units. (f) The threes-wave scattering lengths are shown as a reference plotted versus the magnetic fieldstrength. The dotted lines, from right, show the magnetic field at which the hyperangular eigenvalues from (a)-(e) were found, B = 675, 695, 805, 820 and 845 gauss respectively.

90
As a final examination of this system, I extract the scaling of the low energy threebody recombination rate, i.e. the rate at which three particles collide and form a dimer
and a free particle. The lowest 3-body curve, the lowest potential that goes to the threefree-particle threshold, is the potential that dominates this process. Contributions fromhigher hyperradial potentials will be suppressed due to larger tunnelling barriers. One
limitation of the zero-range pseudo-potential is that it only admits at most one dimerof each type. The process of three-body recombination releases the binding energy ofthe dimer state as kinetic energy between the dimer and remaining particle. For the
purposes of this thesis I will concentrate on the three-body recombination processes that
result in trap loss processes, where the energy released in the recombination is enoughto eject the remaining fragments from a trap.
The rate coefficient for N particles transitioning from a hyperspherical potential
curve with hyperangular eigenvalue λ to a lower lying final state is given by [117, 26] d d−1 Γ
~k
2π
2 X
KN =
NS
(2L + 1) |Tf λ |2 ,(5.44)
d/2
2µ
k
2π
λ
where d is the total dimension of the system (in the case of three-body recombinationd = 6), Tf λ is the transition matrix element between an initial channel labeled by it’s√
free space hyperangular eigenvalue, λ, and the final channel, and k = 2µE/~ is thewave number of the asymptotic hyperradial wavefunction. The sum in this equation
runs over all the initial, asymptotic channels with total angular momentum L that
contribute to the scattering process. In Eq. 5.44, NS is the number of permutational
symmetries in the system. For three distinguishable particles, which is the case studied
here, NS = 1, but it can be different, for instance for N bosons, NS = N !. To find the
T matrix element, it is sufficient to examine the Wentzel–Kramers–Brillouin (WKB)
tunnelling probability to this small R region [112]:
 !
Z RT s 
2µ~2 1/4− E dR ,P (R0 ) = exp −2
U (R) +
~2
2µ R2
R0

(5.45)

91
where RT is the outer classical turning point. The extra repulsive hyperradial barrier
term, ~2 (1/4) /2µR2 , appears due to the Langer correction [118]. The T -matrix element
will be proportional to the WKB tunneling probability that the system will reach thepoint at which a transition is made. The total T matrix will depend on the detailed
nature of the real short range interactions, but the scaling behavior with the scattering
lengths will be determined by Eq. 5.45. In each region of magnetic field, there aredifferent length scale discrepancies and different numbers of bound states. As a result,
I will examine each region separately.

5.3.0.1

Region I (a(1) ∼ a(3)  a(2) )Figure 5.4(a) shows the behavior of the first few hyperangular eigenvalues inregion I. The first three eigenvalues correspond to dimer states, while the fourth corresponds to the lowest three-body potential and is the entrance channel that will control
three-body recombination. The lowest two dimer states are relatively deeply boundwith binding energies, ~2 /ma2 , on the order of 10−12 Hartree. This is close the sameenergy as the trap depth of a normal magneto-optical trap for experiments with 6 Li
[94, 95] meaning that recombination into these dimer channels releases enough energy
to cause trap losses. In the limit where R  a(3) , the three atoms are far enough apartto be in the noninteracting regime. This means that the hyperangular eigenfunction
limits to the lowest allowed three-body hyperspherical harmonic with its corresponding
eigenvalue, (ν + 2)2 → 4. In this limit the hyperspherical potential becomes~2 4 − 1/4
U R  a(3) =.
2µ R2(5.46)For very low energy scattering, the classical turning point in Eq. 5.45, is approximately
RT =

1
,
2k

(5.47)

using this potential. In fact, this will be the turning point for all of the three-bodyrecombination processes discussed in this chapter.

92
It is possible for recombination to occur directly between the lowest three-body
curve and the deep dimer channels, but this direct process is strongly suppressed dueto the large tunneling barrier in the three body-curve at small R. The favored path isthrough a transition to the weakly bound dimer channel, shown schematically in Fig.
5.5. The coupling between the lowest three-body channel and the weakly bound dimer

Figure 5.5: A schematic of the path for three-body recombination in region I is shown.

channel peaks at approximately a(2) , while the coupling peak between the weakly bounddimer channel and the two deep dimer channels occurs at approximately a(3) ∼ a(1) . In
the regime where a(1) ∼ a(3)  R  a(2) the three particles are so far apart that theycannot see the smaller scattering lengths a(1) and a(3) , but the third scattering length
is so large compared to the hyperradius that it might as well be infinite. This leads toa universal potential whose hyperangular eigenvalue can be found by solving Eq. 5.32
with a(1) = a(3) = 0 and a(2) → ∞, i.e
U (R) =

~2 p21 − 1/4,
2µ R2(5.48)

p1 = (ν + 2) = 1.
The behavior of each channel can be approximated by the universal behavior of

93
the hyperradial potential in each region. Under this assumption, using Eq. 5.45, thetransition probability is given by"!#rr
Z a(2)
Z RT


2
p
4
1
P a(3) ∝ exp −2
dR
−E+
−E
.
dR
R2
R2
a(3)
a(2)

(5.49)

2
If the scattering energy is very small, E  ~2 /m a(2) , then the energy dependence inthese integrals becomes negligible leaving,
2
P a(3) ∝ k 4 a(2) a(3) .

(5.50)

Inserting this in for the T -matrix element in Eq. 5.44 gives the scaling behavior of therecombination rate with the scattering lengths [112]:
2

K3 ∝ a(2) a(3) .

(5.51)

It was assumed here the final transition occurs at R ∼ a(3) leading to the scaling behavior
with a(3) , but the transition could just as easily have occurred at R ∼ a(1) . a(1) and a(3)are approximately equal here, and which one dominates the transition depends on the
short range behavior of the real two-body interaction. To extract the scaling behavior
with respect to a(1) , one can simply replace a(3) with a(1) in Eq. 5.51 as long as a(1)and a(3) are close.

5.3.0.2

Region II (a(1) ∼ a(3)  a(2) )The recombination in region II is simpler as there is no weakly bound state.
Again, I assume that the trap loss recombination is dominated by transitions to the tworemaining dimer states seen in Fig. 5.4(b). The lowest three-body potential has coupling
to these channels that peaks at R ∼ a(1) and R ∼ a(3) . For R  a(2) the hyperangulareigenvalue takes on the non-interacting value (ν + 2) → 2. For a(1) , a(3)  R  a(2)the universal hyperangular eigenvalue (ν + 2) = p1 = 1 is seen again [111, 112, 113].
Ignoring the transitional region between these two regimes the transition probability is

94
given by

"

P a(3) ∝ exp −2

Z |a(2) |

r
dR

a(3)

r !#
Z RT
p21
4
+
dR
.
2
R
R2
|a(2) |

(5.52)

Inserting this into Eq. 5.44 gives a recombination rate that has the same scaling behavior
as in region I [112]:(2) (3)K3 ∝ aa2.

(5.53)

Again it is assumed that the final transition occurs at R ∼ a(3) , but it could occur ata(1) as well. As in Region I, the scaling behavior with respect to a(1) can be found bysimply replacing a(3) with a(1) in Eq. 5.51 as long as a(1) and a(3) are close.

5.3.0.3

Region III ( a(2)  a(1) ∼ a(3) ) and Region IV ( a(2)  a(1) ∼ a(3) )In Region III, none of the dimers predicted by the zero-range model have enoughbinding energy to cause trap loss. While recombination can occur into these channels,
I will focus on the process of recombination to deeply bound states here. In reality, thedeep interaction potential between two Li atom in different spin states admits many
deeply bound dimer states, and a true hyperspherical description of the system would
have channels going to each possible dimer-atom threshold. The energy released inrecombining into these deep states is enough to kick the atoms out of any normal trap.
Because the deeply bound states are of the size of the range of the interaction, coupling
to the deeply bound hyperradial channels will peak at small hyperradius, R ∼ r0 , andthe rate can be found by studying the tunneling probability of reaching these states.
As with the recombination process in Region I, the most favorable pathway involves multiple steps. Starting from the lowest three-body channel, a transition is madeto either the first or second weakly bound dimer channel. Because a(1) and a(3) aresimilar in magnitude, the coupling to these channels peaks in the same region. If thetransition is made to the highest dimer channel, then another transition is made directly
to the second.

95This pathway is shown schematically in Fig. 5.6. An interesting thing occurs

Figure 5.6: A schematic of the potentials and the path for three-body recombinationin region III is shown. Again the grey regions indicate a transition from one universal
potential behavior to another. The green line represents the hyperradial potential fora deeply bound dimer state. The blue area is the short range region not described byzero-range interactions.

in the lowest weakly bound potential when a(2)  R  a(1) ∼ a(3) : the universal
potential becomes attractive. This region of attractive potential gives rise to a number ofphenomena. For instance, in the limit a(1) , a(3) → ∞, the universal attractive potential
supports an infinite number of geometrically spaced three-body bound states, giving
rise to the Efimov effect. In the process of three-body recombination to deeply-bounddimmer states, though, there is no tunnelling suppression in this channel, and thehyperradial wavefunction merely accumulates phase in this region. As a result the
WKB tunnelling probability is controlled by the transition at R ∼ a(1) , a(3) :
" Z
r #


RT
4
(1)
P a
∝ exp −2
dR
.
R2
a(1)
Inserting this into Eq. 5.44 gives the scaling of three-body recombination to deep dimer
states ashi4
K3 ∝ a(1) .

(5.54)

96
Again, it is assumed here that a(1) and a(3) are similar in magnitude. If this is not thecase, for instance if a(1)  a(3) , then a scaling behavior similar to that of Eq. 5.51 isrecovered:hi2K3 ∝ a(1) a(3) .

(5.55)In Region IV there is only a single weekly bound dimer state available, and trap
loss will occur through recombination to deeply bound dimers. The path here is similar
to that of Region III, where a transition happens from the lowest three-body channel
to the weakly bound dimer channel. From there the hyperradial wavefunction can go
to the small R region without further suppression. This process then yields the same
three-body recombination scaling behavior as Eq. 5.54 when a(1) ∼ a(3) . Whena(3)  a(1) , the scaling predicted by Eq. 5.55 is recovered.

5.3.0.4

Region V ( a(2)  a(3)  a(1) )In this regime the recombination process is entirely controlled by the lowest threebody channel, shown schematically in Fig. 5.7. The hyperradial potential has three
universal regimes. The first, when r0  R  a(2)  a(3)  a(1) , is identical tothat of three strongly interacting bosons. The hyperangular eigenvalue, (ν + 2)2 , is thefirst solution to Eq. 5.33 in the limit where R/a → 0, yielding the hyperradial potential,
U (R) =

~2 − (s0 )2 − 1/4,
2µR2(5.56)

s0 = 1.0062.
In the next regime, when r0  a(2)  R  a(3)  a(1) , the three particles are far
enough apart so as not to see the smallest scattering length. As a result the hyperangulareigenvalue is governed by Eq. 5.32 with the BBX symmetry of Table 5.1 imposed:
U (R) =

~2 − (s1 )2 − 1/4,
2µR2

s1 = 0.4137.

(5.57)

97

Figure 5.7: A schematic of the lowest hyperradial potential is shown with the pathfor three-body recombination to deeply bound states. The green line represents thehyperradial potential for a deeply bound dimer state. Grey areas indicate transition
regions from one universal behavior to another, and the blue region indicates the shortrange regime.

In the regime where r0  a(2)  a(3)  R  a(1) , there is only one scattering
length that is seen by the system, and the universal potential becomes that of Eq.
5.48. In the final regime, where the hyperradius is much larger than all of the scattering
lengths, the potential goes to the non-interacting behavior of a hyperspherical harmonic.
The transition to a deeply bound dimer state occurs at R ∼ r0 following the pathshown in Fig. 5.7. To get to this region, the wavefunction must first tunnel through abarrier, leading to suppression of the recombination rate. Once through the barrier, thewavefunction accumulates phase in the attractive potential regime. If enough phase canbe accumulated in this regime, then a three-body bound state (a so called Efimov state)
can be present leading to a resonance in the recombination rate. The final recombination
rate for this process is [26, 111, 112, 61]
K3 ∝ Asinh 2η,sin (φW KB ) + sinh2 η2(5.58)

where η is controlled by the short range properties of the system, φW KB is the WKBphase accumulated in the attractive regime from r0 to a(3) , and A is proportional to

98
the tunneling suppression through the barrier:hi2
A ∝ a(3) a(1) ,!
a(1)φW KB = s1 ln+ s0 lna(2)(5.59)
a(2)
r0

!
.

(5.60)

Figures 5.8 (a-c) show a log-log plots of the recombination rate as a function ofa(1) , a(2) and a(3) respectively for magnetic fields from 834.15 to1000 gauss with
η = 0.01. The short range length scale here is chosen to be approximately the vander Waals length of 6 Li, r0 = rd ≈ 60 atomic units. In Fig. 5.8(a), for smaller a(1) ,Figure 5.8: (a-c) The log of the three-body recombination rate from Eq. 5.58 for 6 Li
is shown for magnetic fields ranging from 834.15 to 1000 gauss is shown plotted versusa(1) , a(2) and a(3) respectively with η = 0.01 and the short range length scale chosen
to be approximately the van der Waals length, r0 = rd ≈ 30 a.u. The beginning of an2
Efimov resonance can be seen in (a) and (b) while the scaling law K3 ∝ a(3) can beseen in (c).

the scaling predicted in Eq. 5.59 cannot clearly be seen because of deviations due toan Efimov state beginning to appear. Though the Efimov state has not quite becomebound, its effects on the recombination rate can clearly be seen. In Fig. 5.8(b), there
is no scaling behavior with a(2) , though the effects of the Efimov state are present.
Figure 5.8(c) shows only slight deviations from the predicted scaling behavior in Eq.

995.59 as a function of a(3) , this is because a(3) takes on a large range of values in thisrange of magnetic fields, from −4000 a.u. to −∞, but does not play a role in the sizeof the phase accumulation regime.
With three overlapping resonances, 6 Li provides a rich hunting ground for the
study of three-body physics. Further, because it is a fermionic atom, three-body interactions involving only two of the three lowest components are strongly suppressed
meaning that the majority of the three-body physics is controlled by a system of three
distinguishable particles. While only the processes of three-body recombination thatlead to trap losses were studied in this section, there is still a rich and complex array of
behaviors not discussed that can be described using the model presented here.

5.4

Multi-channel interactions
In this section I develop a multi-channel generalization of the Green’s function

methods, in which each particle in the three-body system can be in one of two different
internal states. While the exact nature of the internal states is not required, for reference
one can think of them as Zeeman states of atoms in an ultracold gas. Allowing theatoms to be in different internal states leads to an interaction that is intrinsically multichannel in nature, where each channel corresponds to a different two-body combinationof internal states. The benefits of having a true multi-channel interaction are many fold.
For instance, instead of tuning a single channel interaction’s scattering length to modela broad s-wave Fano-Feshbach resonance, using a multi-channel interaction allows forthe real resonance to be mimicked. In the previous section a deeply bound two bodystate had to be put in by hand with assumptions about the coupling behavior to thisstate. With a multi-channel interaction a deeply bound state can be included directly inthe two body interaction. Another benefit is that the three-body system can be studied
at excited scattering thresholds with coupling to lower thresholds. This topic will be
the main focus of this section.

100
To begin, I will assume that each atom in the three-body system can be in oneof two internal states, |1i and |2i, and that these internal states have energy 0 andε respectively. To reduce the large parameter space involved here, I will assume thatthe particles are identical bosons. Two atoms can interact in different symmetric in√ternal state channels labeled by {|σi} = |11i , (|12i + |21i) / 2, |22i with asymptotic
threshold energies, Eσ = 0, ε, 2ε, respectively. The interaction between the two particles
is assumed to be a zero-range multi-channel extension of the pseudo-potential,v̄ (r) =4π~2 (3)∂Āδ (~r) r,
2µ2B∂r(5.61)

where µ2B = m/2 is the two body reduced mass. The symmetric matrix, A11 A12 A12 Ā =  A12 A22 A23  ,
A13 A23 A33(5.62)

is expressed in the |σi basis and can be considered as a multi-channel generalization of
the scattering length in Eq. 5.20. Figure 5.9 gives a schematic picture of this interaction.
When the channels are weakly coupled, the diagonal elements of Ā can be thought of as

Figure 5.9: A schematic picture of the multi-channel two body interaction is shown.

101
the scattering length in each channel while the off diagonal elements are the coupling.
In a more rigorous treatment, Ā is simply minus the scaled reactance matrix K̄0 frommulti-channel quantum defect theory [114, 119, 120]. The regularization factor serves toimpose the generalization of the Bethe-Peierls boundary condition. This multi-channelinteraction has been used in previous three-body studies as well [121].
To extend this interaction to a three-body system, I first use the Jacobi coordinates described in Eq. 5.21 and the hyperspherical coordinates described by Eqs. 5.26
and 5.27. The possible three-body internal states are given by {|Σi} = {|111i, |112i,|121i, |211i, |122i, |212i, |221i, |222i} with threshold energies EΣ = 0, ε, ε, ε, 2ε, 2ε,
2ε, and 3ε. The hyperangular piece of the three-body Hamiltonian with the center ofmass motion removed is given byHad =~2 Λ2 ¯
I + V̄ (R, Ω) + ĒT h ,2µ R2(5.63)

where I¯ is the identity matrix, V̄ (R, Ω) is a matrix representing the sum of all possible
two-body interactions, and Ēth is the threshold energies of each internal state. The
resulting Schrödinger equation is then solved by the Lippmann-Schwinger equation,~ (Ω) =ΦZ X
kwhere d =

p



 
 2µR2(k)0 ~(k)0(k)dΩ(k)0 ,dρΦΩv̄Ḡ Ω(k) , Ω(k)0
1~2µ/µ2B =

(5.64)

p
2/31/2 . The two-body interaction, v̄ (k) , is the zero-range

pseudo-potential from Eq. 5.61 written in the |Σi basis by considering the third particle

102
as a spectator. For example when k = 1 the two body interaction is given by, 4π~2∂ (1)(1)
v̄ (1) dρ1=Ā(1) (1) ρ1 ,
2µ2B
∂ρ1

A12 A12√
 A11 √
22
 A
 √12 A
0
22


2
 A
 √12
0
A22


2


0
0
 0

Ā(1) = AA A13 √23 √23

2
2


 0
0
0



 0
0
0



0
0
0

(5.65)

0
0
0

A13
A23
√
2
A23
√
2

0

0

0

0

0

0

A11

0

A12
√
2

A12
√
2

0

A33

0

0

0

A22

0

0

0

A22

0

A23
√
2

A23
√
2

A12
√
2
A12
√
2
A13

0



0 




0 



A13 

,

0 


A23 
√ 
2 

A23 
√ 
2 


A33

where Aij is defined in terms of the two-body interaction given by Eq. 5.62. The
somewhat odd form of this scattering length matrix has been chosen so that the two√internal states, |12i and |21i, have the same interactions. Extra factors of 1/ 2 in somematrix elements are included for symmetrization purposes.
In Eq. 5.64 Ḡ (Ω, Ω0 ) is the multi-channel generalization of the hyperangular
Greens function whose matrix elements are given byG Ω, Ω0ΣΣ0
= δΣΣ0 GνΣ Ω, Ω0 ,(5.66)

where GνΣ (Ω, Ω0 ) is the single channel Green’s function from Eq. 5.12 with the hyperangular eigenvalue modified to include the threshold energy offsets, i.e.
vΣ (νΣ + 4) = ν (ν + 2) −2µR2
[ET h ]ΣΣ .
~2

It is clear from this definition that Ḡ (Ω, Ω0 ) is the solution to the multi-channel Green’s
equation,



2µR2
¯
¯
¯
Λ I − ν (ν + 2) I −ĒT h I Ḡ Ω, Ω0 = δ Ω − Ω0 I,
2
~
2¯



103
where νΣ can be interpreted as the hyperangular eigenvalue with respect to the Σththree-body threshold. Hyperangular solutions to the multi-channel three-body problem
give the hyperradial Schrödinger equation
"
#
~2 d2
~2 (ν + 2)2 − 1/4−+
R5/2 F (R) = ER5/2 F (R) ,2µ dR2 2µR2where the nonadiabatic matrices P̄ and Q̄ have been omitted.
As in Sec. 5.2, each integral in Eq. 5.64 is performed in the Jacobi coordinates
∂ (k)
(k)
where ρ~1 = d (~ri − ~rj ). The regularization factor, (k) ρ1 , acting on the wavefunction
∂ρ1

(k)
(k)
~
at α → 0 gives a boundary condition vector,
Φ Ω
∂





(k) ~
(k)
~ (k) yLM ω (k) .
=
C
ρ
Φ
Ω
2
1
α(k) →0 ∂ρ(k)1lim(5.67)

where yLM (ω) is a normal spherical harmonic. The three boundary condition vectors
are related by assuming that any two particles in the same internal state are identical: C1 

 C2 




 C 
 3 




 C4 


(1)
~ =
C
,


 C5 




 C 
 6 




 C7 

C8~ (2) = P̄− C~ 1,
C~ (3) = P̄+ C~ 1,
Cwhere the matrices P+ and P̄− correspond to cyclic and anti-cyclic permutations of the104internal states:

 1


 0


 0



 0

P̄+ = 

 0


 0



 0


0


0 0 0 0 0 0 0 
0 0 1 0 0 0 0 
1 0 0 0 0 0 0 
0 1 0 0 0 0 0 
0 0 0 0 1 0 0 
0 0 0 0 0 1 0 
0 0 0 1 0 0 0 
0 0 0 0 0 0 1

P̄− = P̄+T .

(5.68)(5.69)~ proceeds in
Evaluating the integral in the LS equation for each component of Φ
the same way as in Sec. 5.2:fΣ− α(k)1 X(1) ~ (1)
(1)
y
ω
CΣ0
Ā
NνΣ L
ΦΣ (Ω) =
0
LM
2
ΣΣ
dR 0
(sin α)2 (cos α)2 W [f + , f − ]
Σ




fΣ− α(3)
1 X
(3) 
(2)
~ (1)
+
ω
P̄
NνΣ L
Ā
y
0
+
LM
0 Σ00 CΣ00
2
ΣΣ
2
2
Σ
+
−
dR 0 00(sin α) (cos α) W [f , f ]ΣΣ(5.70)



fΣ− α
1 X
(3) 
(3)
~ (1)
y
ω
P̄− Σ0 Σ00 C
Ā
NνΣ L
0
LM
2
ΣΣ
Σ00 ,
2
2
+
−
dR 0 00(sin α) (cos α) W [f , f ]

(3)

+

ΣΣ

where fΣ− (α) and (sin α)2 (cos α)2 W [f + , f − ] are given by Eqs. 5.17 and 5.18 respectively with ν taken to νΣ , l1 = 0, l2 = L, and d1 = d2 = 3. The pseudo-potential
treatment is that of enforces s-wave boundary conditions on each interacting pair of(k)particles, meaning that in Eq. 5.70 the spherical harmonics, yLM ω2 , in each termhold the entire three-body spatial angular momentum.
For the purposes of this thesis I consider only zero total angular momentum states,
L = 0, but it is clear how this can be generalized to arbitrary total angular momentum.
To find the hyperangular eigenvalue, ν, the boundary condition in Eq. 5.67 is applied

105
to both sides of Eq. 5.70 for k = 1. Using the behavior of the Jacobi vectors describedby Eq. 5.31 and a bit of algebra, the matrix equation becomes



1
1
1
(1)
(2)
(3)
¯
~ (1) = 0M̄ +M̄ P̄+ +M P̄− − I C
dR
dR
dR

where the 8 × 8 matrices M̄ (k) are given by A(k)k=1
ΣΣ0 (νΣ + 2) cot [(νΣ + 2) π/2] ,(k)MΣΣ0 =.
4 (k) sin [(νΣ + 2) π/6]


√
A
−
,
k
=
2,
3

0
3 ΣΣ cos [(νΣ + 2) π/2]

(5.71)(5.72)

Solving this matrix equation will give both the hyperangular eigenvalue, ν, and the~ (1) up to an arbitrary constant.
boundary condition vector C
There is one last issue that must be dealt with. In the description of the internal
states it was assumed that two interacting particles could have unsymmetrized internal
states. Correcting this is a simple matter of projecting out the antisymmetric linear
combination of internal states in each two particle interaction. Following this procedure
gives the final matrix equation describing the hyperangular eigenvalue for three bosons
with multi-channel two-body interactions,



1
1
1
(1)
(2)
(3)
¯
~ (1) = 0,
M P̄− − I S̄ T CS̄M̄ +M̄ P̄+ +
S
dR
dR
dR

(5.73)(1)~
where CS is a 6 dimensional vector of boundary conditions in the set of symmetrized√√states given by ΣS = |111i , (|112i + |121i) / 2, |211i , |122i , (|212i + |221i) / 2, |222iwith each interacting pair in one of the symmetrized internal states shown in Fig. 5.9.
Here S̄ is the matrix that projects the original |Σi basis onto this subset of three-bodyinternal states

106


 1


 0



 0

S̄ = 

 0


 0



0


0

0

0 0

0

0

1
√
2

1
√
2

0 0

0

0

0

0

1 0

0

0

0

0

0 1

0

0

0

0

0 0

1
√
2

1
√
2

0

0

0 0

0

0

0 


0 



0 

.

0 


0 



1

(5.74)

Solving the transcendental equation implied by Eq. 5.71 is by no means simple, as
the determinant of the resulting 6 × 6 determinantal equation has many roots with avery complex structure. Further, the complex system of coupled hyperradial differential
equations can have many very narrow avoided crossings and sharp features. As a result
I will limit the implementation in what follows to the prediction of qualitative featuresof the system.

5.4.1

Efimov States embedded in the continuum

The adiabatic hyperspherical method has proven especially useful in the description of Efimov physics. In Sec 5.3, it was seen that when three particles are interacting
resonantly, they can form a series of bound states called Efimov states. In the case of
three bosons, in the limit where a → ±∞, the lowest adiabatic hyperradial potential
takes on a universal 1/R2 attractive potential when the hyperradius is much greaterthan a short range length scale, r0 ,U (R) =

~2 −s20 − 1/4
,
2µ
R2

q
where s0 = − (ν + 2)2 = 1.00624 is the first solution to Eq. 5.33 in the limit R/a → 0.
This attractive dipole–like potential supports an infinite number of geometrically spacedbound states that accumulate at the zero-energy threshold [32, 60]. The position of thefirst bound state will depend on the short range length scale r0 , and each trimer after107that has energy
En = En−1 e−2π/s0 ≈En−1.
(22.7)2(5.75)

The first strong experimental evidence of Efimov states has recently been seen experimentally in the work of Kramer et. al. [62] in observations of trap losses causedby three-body recombination. Unfortunately, this experiment has only seen one resonant feature. To really show that the observation corresponds to an Efimov state, two
resonances must be observed with the characteristic geometric scaling factor between
scattering lengths, a2 ≈ e−2π/s0 a1 = 22.7a1 . Of course, increasing the scattering length
by more than an order of magnitude poses its own problems. For instance the threebody recombination rate scales with a4 [26, 111, 61], a fact that can easily be confirmed
using the WKB tunneling methods described in Sec. 5.3, severely limiting the lifetimeof the gas in the large scattering length limit. In this section I propose a method of
observing Efimov states utilizing the multi-channel nature of the two-body interactions.
To limit the 6 dimensional parameter space of the interaction scattering length
matrix Ā, in this study I investigate only a parameter regime where A11 = A33 , A22 =2A33 and A12 = A23 = A33 /4. The qualitative results found here are not changed bythis particular choice of parameters. The only requirement is that A33 must be positive.
For the initial study I will assume that the first and third two body channel in Fig.
5.9 are not directly coupled, i.e. A13 = 0. Effects of weak coupling between these twochannels will be discussed later. The main idea here is to tune the interactions to aresonance in an excited two-body threshold. This occurs when a quasistable two-bodybound state attached to the third two-body scattering threshold is degenerate with thesecond two-body threshold. This is achieved by tuning the asymptotic two-body energy
to ε = ~2 /mA233 . By choosing these parameters, I have also chosen a rough short range
length scale of r0 ∼ A33 .
Inserting these parameters into Eq. 5.73 and solving gives a somewhat intimidat-

108
ing set of potentials. Figure 5.10(a) shows the first 300 adiabatic hyperradial potentials
for these parameters in units of ε plotted as a function of R/A33 .In this figure the

Figure 5.10: (a) The first 300 adiabatic hyperradial potentials are shown for the multichannel three-boson problem in units of ε as a function of R/A33 . (b) The adiabatic
three-body potentials near the second three-body scattering threshold are shown. The
attractive Efimov diabat can be clearly seen approaching the second scattering threshold.

various possible asymptotic three-body threshold energies and the corresponding internal states in the ΣSinternal state basis where particle 1 is treated as a spectator

particle can clearly be seen. At first glance, it seems likely impossible to extract any
useful information from this figure, but upon close examination several features become
apparent.
First, there is a clear three-body scattering threshold at E = 0 with three bodypotential curves converging to this value. In fact there is a series of diabatic curves going
through a series of very sharp avoided crossing limiting to each three-body internal state

109
threshold, EΣ = 0, ε, 2ε, and 3ε. Looking below the zero energy threshold, two potentials
corresponding to states consisting of a dimer and a free particle can be seen. This isa clear difference between the multi-channel model for three identical bosons and thesingle channel model. In the single channel model of three identical bosons, there is
only a single length scale, the scattering length, allowing only at most a single two-bodybound state. In the multi-channel model there are multiple length scales, and multiple
bound states are allowed attached to different thresholds. In this case the deepest twobody state is attached to the |111i scattering threshold and corresponds to a |11i dimer
with binding energy Eb = (1.11) ε. This binding energy is about what one would expectfrom the scattering length in this channel:√ 
180 − 2a11 =
A33 = 0.92A33 .
194
With this scattering length, the zero range single channel model predicts a binding
energy of Eb ≈ ~2 /ma211 = (1.18) ε. The difference in these two energies is due to theextra short range physics added in the multi-channel interaction. The second, weakly
bound hyperradial potential has an asymptotic energy Eb = 0.11ε. This state comesfrom a |11i bound state attached to the second three body scattering threshold and is
not present in the zero-range single channel predictions.
Examining Fig. 5.10(a) further reveals a series of avoided crossings at E ≈ 0.6ε,√
these correspond to a quasi-bound two-body state with an internal state (|12i + |21i) / 2
attached to the second three body channel. Another series of avoided crossings is seen in
Fig. 5.10(a) at E ≈ 1.6ε corresponding to the same type of two-body quasi-bound state
attached to the third three-body threshold. Further discussion characterizing the features and potentials seen in this figure would take more time and space than is available,
and as such I will limit myself to one more interesting region.
The original purpose stated in this section was to analyze the three-body hyperradial potentials when a quasi-bound two-body state was made resonant with an excited

110
two-body scattering threshold. By tuning the threshold spacing, ε to ε = ~2 /mA233 ,this is exactly the case. With the chosen parameters, A11 = A33 , A22 = 2A33 and√A12 = A23 = A33 /4, the |σ = 2i = (|12i + |21i) / 2 threshold scattering length can befound:
Im
a22 = aRe
22 + a22 ,


aRe
22 =

(5.76)



1
A33 
30 +
16

1
r
1 − A33

mε
~2

+

1


,
mε

1 + A233 2
~

r

mε
1
~2
aIm
.
22 = −
16 1 + A2 mε
33 2
~
A233

With a resonance at the second two-body scattering threshold, it might be a good idea
to consider the behavior of the three-body potentials at the second three-body threshold. Figure 5.10(b) shows an enlarged view of the adiabatic hyperradial potential in
the region near this threshold. It becomes obvious that the adiabatic potentials here
go through a series of very narrow avoided crossings here forming a set of diabatic hy√
perradial potentials asymptotically approaching the |Σsymm = 2i = (|112i + |121i) / 2
threshold including an attractive potential. The attractive diabatic potential of interestis given by~2 (νΣ=2 + 2)2 − 1/4+ε2µR2~2 −s21 − 1/4
→
+ε
2µ
R2

Udiab (R) =(5.77)(5.78)

where νΣ=2 is the hyperangular eigenvalue with respect to the second threshold. Figure5.11 shows (νΣ=2 + 2)2 as a function of the hyperradius. In the limit R  A33 this
hyperangular eigenvalue approaches a constant, (νΣ=2 + 2)2 → −s21 = − (0.4137)2 wheres1 is the universal constant given in Eq.5.57 that describes three equal mass particles
with two resonant interactions [32, 122], the exact behavior one might would predict fora resonance in the |σ = 2i two body channel. With this universal attractive potential,

111
the Efimov diabat can support an infinite number of geometrically spaced states thatare bound with respect to the second three-body threshold:
En = ε − E0 e−2π/s1 ≈ ε − 2.5 × 10−7 E0 .

(5.79)

Because the diabat goes through some avoided crossings with finite width, it is coupled
to the three-body |111i continuum, making the Efimov states quasi-stable. Becauseof the multi-channel nature of the three-body interaction, these states are interpreted
as fully three-body Fano-Feshbach resonances in a three-body continuum [18, 123, 19].
Similar resonances have been predicted due to Efimov states in the context of particle
dimer scattering in nuclear halo systems [124].
0
1

-0.04

0.5

(ν2+2)

2

-0.06
-0.08

(ν2+2)

2

-0.02

-0.1

0
-0.5
-1

-0.12

-1.5
10000 10001 10002 10003 10004 10005
R/A33

-0.14
-0.16
-0.18
-0.2 1
10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

R/A33

Figure 5.11: The hyperangular eigenvalue corresponding to the Efimov diabat is shown
as a function of R/A33 . (inset) The hyperangular eigenvalue is shown for A13 = 0,0.005A33 and 0.01A33 (black, red and green respectively). As A13 increases the widthof the avoided crossings that create the Efimov diabat can be seen to increase.

5.4.2

Non-adiabatic matrices for the multi-channel problem

The states predicted in this section can only exist if the nonadiabatic diagonalcorrection falls off fast enough. Specifically, if Qdiab
nn (R) for the diabatic Efimov potential
dies off more slowly than s21 /R2 , the long range 1/R2 behavior that allows for the infinite

112
number of states will be cut off. If the diagonal correction dies of as 1/R2 then it canchange the spacing of the states.
The derivation of the non-adiabatic matrix elements Pmn is shown in Appendix
B the result of which is:iiih12µR (4π)2 h ~ (1) h(1) ~ (1)(1)(1) ~ (1)~C
(5.80)
Ē
Ā
C
−
C
·
Ē
Ā
C
·
Th
th
m
n
n
m
dR (εn − εm )2

3 (4π)2
1 ~ (1)  (1) ~ (1) 
1 ~ (1)  ~ (1) ~ (1)0 
1 ~ (1)0  (1) ~ (1) +−−C· Ā CC · Ā Cn0 +C · A Cn 0
(εn − εm )
dR n
dR2 m
dR m

Pmn = −

where εn = (νn + 2)2 − 4 and the prime indicates a derivative with respect to the hyper~ n (Ω), is normalizedradius. This equation holds if the hyperangular channel function, Φcorrectly. The normalization condition can be found through the boundary value vechi
~ n(1) , by first “pre-normalizing” with C~ n(1) = N C~ (1) where C~ (1) · C̄ (1) = 1, andtor, C0n0n0n
the constant N is defined by−1
3 (4π)2 h ~ (1)~ 10
C0n · ε0n − 4µRĒT h
ĀC(5.81)ndR~ (1)0 · ε0 − 4µRĒT h −1 ĀC~ (1) − 1 C~ (1) · ε0 − 4µRĒT h −1 ĀC~ (1) .
−C
n
n
0n
0n
0n
0n
R

1 = N2The second derivative non-adiabatic diagonal matrix elements, Qnn , are somewhat
more complex, and are given in Appendix B. The resulting diagonal correction matrix
element for the diabatic Efimov potential is shown in Fig. 5.12. It is clear that for large
R, the diagonal correction, Qnn , falls as 1/R3 , and that including this non-adiabatic
correction does not destroy the Efimov states.
The lifetime of the quasi-stable Efimov states will be partly controlled by theshort range coupling to the continuum and to deeply bound states. A WKB velocity
can be used to find the time it takes for the Efimov state to travel out to the far classicalturning point and back to the short range region,
Z Rt
µ
p
dR,
T =
2µ [−En − U (R)] + 1/4R2
r0(5.82)

where RT is the classical turning point at large R, r0 is the inner turning point controlled
by the short-range details of the interaction, En is the binding energy of the nth Efimov

113
-3

10

-Qnn(R)/2µε

-6

10

-9

10

-12

10

-15

10

4

10

5

6

10

10

7

10

8

10

R/A33

Figure 5.12: The diagonal correction, −Qnn /2µ is shown in dimensions of the two-bodythreshold separation ε plotted as a function of R. The red dashed line shows 1/R3
scaling while the blue dot-dashed line shows 1/R2 scaling.

state with respect to the second three-body threshold, and U (R) is the diabatic Efimov
potential from Eq. 5.78. If it is assumed that every time the Efimov state oscillates
through the short range region there is a fixed probability of transitioning to a deepstate, the lifetime of the state is found to be
Γn =

PSR
≈ Γn−1 e−2π/s1 ,2T(5.83)

where PSR is the short-range transition probability, T is given by Eq.5.82, and Γn is thewidth of the nth Efimov state. This means that if the width of the first Efimov state isnarrow enough, then each of these resonances will be isolated from the others.
The final mechanism that can limit the lifetime of the quasistable Efimov statesis the width of the avoided crossings at large R. Direct calculation of the P matrix
elements is numerically difficult as it requires a large number of adiabatic potentials,
calculated to high accuracy, to construct the diabatic Efimov potential and find the P̄matrix coupling at large R. A good estimate of the widths of these couplings is the
lifetime of the two-body resonant state. This means that the Efimov state can remain
as long as two-body interactions keep the system in the resonant two-body channel.

114
The parameter that controls the width of the large R avoided crossings is the coupling
A13 . The inset of Fig. 5.11 shows (νΣ=2 + 2)2 for large R for of A13 = 0 (black line),
0.005A33 (red circles), and 0.01A33 (green crosses).As A13 grows, the width of each√crossing increases. The |σ = 2i = (|12i + |21i) / 2 two-body scattering length can be
determined as a function of A13 :a22 = aRe + iaIm ,aRe =(5.84)

A333,
16A213


A33 32A213 + 2A13 A33 + A233
aIm =.
16A213The lifetime of the quasi-bound resonant state can be approximated by one over theimaginary part of the energy Eres ≈ ~2 /ma222 . For A13  A33 this yields
T ≈

~
mA633
≈
.
Im (Eres )~128A413(5.85)

This clearly demonstrates that for A13 → 0 the the large R avoided crossings in the
diabatic Efimov potential become level crossings, and for A13  A33 the crossings will
be very narrow, and the lifetime of the Efimov states will be mostly controlled by theshort range couplings.
The prediction of the Efimov state induced Fano-Feshbach resonances leads me
to suggest a new method of experimentally probing these states with the use of radio
frequency (RF) spectroscopy. In a cold atomic gas, the different internal states are
Zeeman states which are weakly coupled by an RF field. By probing an ultracold gasof bosons in the lowest internal state using an RF field, resonances in the loss rate fromthe gas should appear when the frequency of the field is at the energy of one of thequasistable Efimov states. Further, the atoms in the gas will spend the majority of
their time near the E = 0 threshold and in the |1i state. Because the scattering length
between atoms in the |1i state is not necessarily large, the gas will be stable with respect
to the a4 scaling of three-body recombination, allowing for long probing times. If it is115
assumed that the first Efimov state appears at E1 ≈ ~ × 2π (40 MHz) below the second
threshold, an energy that is not unreasonable for a three-body state of 85 RB atoms
that is approximately the van der Waals length rd ≈ 200 a.u. in size, Eq. 5.79 showsthat the second state will appear at approximately E2 = ~ × 2π (40 MHz) e−2π/s1 ≈~ × 2π (10 Hz). This binding energy is very small compared to the 2 GHz spacing of thehyperfine levels of 85 Rb, but still possibly within experimental resolution. The geometricscaling factor in the spacing of the Efimov states, e−2π/s1 , can be made considerably
better by creating a resonance at the |22i two-body scattering threshold and considering
Efimov states attached to the |222i identical boson threshold. Another tactic to improve
the geometric spacing is that of using more favorable mass ratios [111, 112, 113]. The
multi-channel treatment of these scenarios is a task beyond the scope of this thesis, butshould be accessible using the methods shown here.

Chapter 6

Four Fermions and Dimer-Dimer Scattering

The physics of strongly interacting fermions is of fundamental importance to many
areas of physics, including condensed matter physics, nuclear and particle physics, andastrophysics. Specifically the last few years have seen explosive growth, both experimentally [12, 13, 14, 15, 16, 17] and theoretically [46, 47, 48, 49], in the area of degenerate
Fermi gases of atoms. Combined with a Fano-Feshbach resonance that allows for tuningof the s-wave interactions, these systems exhibit a variety of interesting phenomena including the Bose-Einstein condensate (BEC) and the Bardeen-Cooper-Schreiffer (BCS)
crossover that smoothly links two fundamentally different types of superfluidity, BEC(a > 0) to BCS (a < 0), through a resonance. In these systems, few-body correlations
play a fundamental role in understanding the many-body behavior [23]. For instance,
a full solution of the four-fermion problem is required to understand the energy anddynamics of a BEC of bosonic molecules composed of two fermions each.
The Hamiltonian for the four-fermion problem is given by
H=

4
X
i=1

−

~2 2 X∇ +V (rij ) .
2m i

(6.1)

i>j

Generally speaking the resulting 12 dimensional Schrödinger equation is unmanageable.
In this chapter I incorporate a variety of existing theoretical methods together to first
reduce the dimensionality of this partial differential equation, and then to develop a basis
set that allows for manageable variational calculations of the scattering potentials. This

117
chapter is organized as follows: in Section 6.1, the various coordinate systems used insolving the four-fermion problem are discussed as well as various properties of eachsystem; in Section 6.2 the variational basis set used to describe the system is developed;
in Section 6.3 the resulting dimer-dimer wave function is analyzed; in Section 6.4 theenergy dimer-dimer scattering length is examined; and finally in Section 6.5 the processof dimer-dimer relaxation is discussed.

6.1

Coordinate systems
The use of different coordinate systems plays a significant role in the treatment of

few-body scattering problems. Depending on the symmetries, interactions, and fragmentation channels inherent in the problem, different coordinates may significantly affectthe ease with which the problem can be described. For example, in the four fermion
problem, the fermionic symmetry of the system can be used to significantly reduce thesize of the basis needed to describe the possible scattering processes. Describing thissymmetry in the wrong coordinate system, as I will show in this section, can create aconsiderable amount of difficulty. Here I detail the variety of coordinate systems thatare used to describe the four-fermion problem in the adiabatic hyperspherical frameworkand the necessary transformations to describe one set in terms of another. The coordinate systems used here are not only needed to describe correlations between particles,
they also allow the system to be reduced in dimensionality by removing the center ofmass motion and moving into a body fixed frame.

6.1.1

Jacobi coordinates

The first and most obvious symmetry in the Hamiltonian described by Eq. 6.1 isthat of translational symmetry. By describing the system in the center of mass frame,
the dimensionality of the system can be reduced from d = 12 to d = 9. This is donewith the use of Jacobi coordinates. In the interest of brevity, I constrain myself here

118
to only those coordinates directly concerned with the problem at hand. The Jacobi
coordinates relevant to the four-body problem may be broken into two sets, H-type andK-type, shown schematically in Fig. 6.1.

Figure 6.1: The two Jacobi coordinates configurations possible in the four body problem
are shown schematically.

H-type Jacobi coordinates are constructed by considering the separation vector
for two two-body subsystems, and the separation vector between the centers of mass of
those two subsystems, i.e.
ρ
~Hσ
=
1

r

µij
(~ri − ~rj ) ,
µ

r

µkl
(~rk − ~rl ) ,µrµij,kl mi~ri + mj ~rj
mk ~rk + ml~rl
Hσ
ρ
~3 =
−
,
µ
mi + mj
mk + ml

ρ
~Hσ
=
2

(6.2)

(m1~r1 + m2~r2 + m3~r3 + m4~r4 ),m1 + m2 + m3 + m4
mi mj
(mi + mj ) (ml + mk )µij =, µij,kl =.
mi + mj
m1 + m2 + m3 + m4

ρ
~cm =

Here the superscript σ enumerates the 24 different H-type coordinates that may beobtained through particle permutation, ρ~cm is the position of center of mass of the
four-body system, and µ is an arbitrary reduced mass for the four-body system. The
prefactors in each Jacobi vector, which are given in terms of the various reduced masses

119
in the problem, are chosen to give the so-called mass scaled Jacobi vectors. The kinetic
energy in these coordinates can be written as
4
X
~2

3
~2 2
~2 X 2
2
−
∇ =−
∇
−
∇ρj ,
2mi ri2M ρcm 2µi=1
j=1

where M is the total mass of the four particles. The reduced mass, µ, can be chosen to preserve the differential volume element for the full 3D problem, ensuring thatd3 ρσ1 d3 ρσ2 d3 ρσ3 d3 ρcm = d3 r1 d3 r2 d3 r3 d3 r4 :µ=m1 m2 m3 m4
m1 + m2 + m3 + m4

1/3.

Physically, the H-type coordinates are useful for describing correlations between two
particles, for example a two body bound state or a symmetry between two particles,or two separate two-body correlations. It is interesting to note that when two particles
are on top of each other, for instance in a bound dimer, the H-type coordinate system
reduces to a three body system, i.e. when ~ri = ~rj in Eq. 6.2:
ρ
~Hσ
= 0,
1
r
µkl
(~rk − ~rl ) ,
ρ
~Hσ
=
2
µ


r
µij,klmk ~rk + ml~rl
Hσ
ρ
~3 =
~ri −
.
µ
mk + ml
This observation will be useful later when constructing a basis to examine the four
fermion problem. This point is called a coalescence point and finding a way of accurately
describing these point is crucial for describing interactions between particles.
K-type Jacobi coordinates are constructed in an iterative way by first constructing
a three body coordinate set as in Eq. 5.21, and then taking the separation vector between

120
the fourth particle and the center of mass of the three particle sub-system, yieldingrµij(~ri − ~rj ) ,µrµij,k mi~ri + mj ~rj
Kσ
ρ
~2 =
− mk ~rk ,µmi + mjrµijk,l mi~ri + mj ~rj + mk ~rk
Kσ
ρ
~3 =
− ml~rl ,µmi + mj + mk
ρ
~Kσ
=
1

(6.3)

(m1~r1 + m2~r2 + m3~r3 + m4~r4 ),m1 + m2 + m3 + m4
(mi + mj ) mk
mi mj
, µij,k =µij =,mi + mjmi + mj + mk

ρ
~cm =

µijk,l =(mi + mj + mk ) ml.
m1 + m2 + m3 + m4

Again σ enumerates the 24 different K-type coordinates that result from particle permutations. Examining Fig. 6.1 shows that K-type Jacobi coordinate systems are usefulfor describing correlations between three particles within the four particle system. In
the four fermion system, there are no weakly bound trimer states meaning that K-type
Jacobi coordinates will not be used here, but the methods described in this report canbe easily generalized to include these type of states. Unless explicitly stated all Jacobi
coordinates from here on will be of the H-type, and for notational simplicity, I will dropthe H superscripts.

6.1.2

Coalescence points and symmetry

The proper description of coalescence points is crucial for describing two-bodyinteractions, but they are also important for describing points of symmetry. For instanceif two identical fermions are on top of one another it is known that the wave function
must vanish at this point owing to the anti-symmetry of fermionic wave functions. In
this thesis I will be concerned with four fermions in two different “spin” states. Away
from a p-wave resonance, the interactions between identical fermions can be neglected
for low energy collisions. This means that there are two types of coalescence points thatmust be described; two “symmetry” points, when two fermions of the same type are121
on top of each other, and four “interaction” points, places where two distinguishable
fermions interact via an s-wave potential.
It might be tempting at this point to choose a single Jacobi coordinate systemand then try to describe the interactions and symmetries in the same coordinates, but
this leads to problems. For instance if it is assumed that particles 1 and 3 are spin up
and particles 2 and 4 are spin down one might start with coordinates that are simple
to anti-symmetrize the system in:
s
ρ
~s1 =

41/3
(~r1 − ~r3 ) ,
2

s

41/3
(~r2 − ~r4 ) ,2p~r1 + ~r3 ~r2 + ~r4s
1/3
ρ~3 = 4
−
,
2
2
ρ
~s2 =

(6.4)

where it has been assumed that all of the particle masses are equal, m1 = m2 = m3 =m4 = m leaving µ = m/41/3 . The generalization to distinguishable fermions of different
masses is clear. I will refer to this Jacobi coordinate system as the symmetry coordinates
for fairly obvious reasons. If a permutation of two identical fermions is considered, forinstance 1 and 3, the transformation is simple:
P13 ρ~s1 = −~
ρs1 ,

(6.5)

P13 ρ
~s2 = ρ
~s2 ,

(6.6)

P13 ρ
~s3 = ρ
~s3 .

(6.7)

Similarly for the exchange of particle 2 and 4,
P24 ρ~s1 = ρ
~s1 ,

(6.8)

P24 ρ
~s2 = −~
ρs2 ,

(6.9)

P24 ρ
~s3 = ρ~s3 .

(6.10)The points where two identical fermions are on top of each other are also simply
described by taking either ρ~s1 → 0 or ρ
~s2 → 0. Everything seems to be progressing

122
nicely until an interaction between two distinguishable fermions, for instance 1 and 2,
is considered. This interaction occurs around the point ~r1 = ~r2 . In the symmetry
coordinates this means thatρ
~s1 = ρ
~s2 −

√

2~
ρs3 .

This equation describes a 6 dimensional sheet in the 9 dimensional space, something thatis not easy to describe directly in any basis set. To get around this problem I introduce
two more Jacobi coordinate systems that are useful for describing interactions,
s
41/3
ρ
~i1
=
(~r1 − ~r2 ) ,
1
2
s
41/3
ρ
~i1
(~r3 − ~r4 ) ,(6.11)2 =2p~r1 + ~r2 ~r3 + ~r4
i1
1/3
ρ~3 = 4
−
,
2
2
and
s
ρ
~i2
1 =

41/3
(~r1 − ~r4 ) ,
2

s

41/3
(~r3 − ~r2 ) ,2p~r1 + ~r4 ~r2 + ~r3
i2
1/3
ρ~3 = 4
−
.
2
2
ρ
~i2
2 =

(6.12)The superscript i1 and i2 in Eqs. 6.11 and 6.12 indicate that these Jacobi coordinates
are appropriate for interactions between distinguishable fermions. For instance, a coalescence point between particles 1 and 2 is described by ρ~i1
1 → 0. Another benefit ofthese coordinates is that they are well suited to describing a dimer wavefunction. If particles 2 and 3 are in a weakly bound molecule then the wavefunction for that molecule
is only a function of ρ
~i2
2 .
~i1
~i2
Using combinations of these three coordinate systems, ρ~sj , ρj and ρj , can describe
all of the possible two-body correlations of the fermionic system. This assumes that thesystem in question is that of four equal mass fermions in two internal states with swave interactions only. However, the method used is quite general. Coordinates can be123chosen to describe the possible three-body correlations that can arise due to things likethe Efimov effect [59, 60, 113]. The only requirement in constructing the appropriate
Jacobi coordinates for a given problem is a bit of physical intuition.

6.1.3

Kinematic rotations

Since I am going to be using different Jacobi systems to describe different types ofcorrelations, a method of transforming between different sets of coordinates is needed.
In the above section, equal mass particles are considered. This is because extension toarbitrary masses was fairly obvious. To describe the kinematic rotations I will keep themasses arbitrary and specify for equal masses later. It is convenient here to deal withtransforming all of the Jacobi coordinates at once. Thus the matrices whose columns
are made of the Jacobi vectors are used:%̄s = {~
ρs1 , ρ~s2 , ρ~s3 } i1 i1 i1%̄i1 = ρ~1 , ρ
~2 , ρ
~3(6.13)

 i2 i2 i2%̄i2 = ρ~1 , ρ
~2 , ρ
~3 .
The transformation that takes one coordinate system to another cannot stretch or shrink
the differential volume element, and thus it must be a unitary transformation. Further,
the transformation cannot mix the Cartesian components of the Jacobi vector, i.e. ρi1x
has no part of ρsy in it. This means that the transformation will be a unitary matrix
that acts from the right, e.g.
%̄i1 = %̄s Ūs→i1 .

(6.14)The matrices that perform these operations are called kinematic rotations [125,126, 127], and they will be put to extensive use in the calculations that follow. In truth,transformations between coordinates systems that do not require an inversion should
be considered, but the general principle still holds if improper rotations are included.

124
Note that all of the matrix elements must be real, so that the inverse transformation is
given merely by the transpose.
I employ a direct “brute force” method of finding these matrices where the systemof equations given in Eq. 6.2 are solved for ~r1 , ~r2 , ~r3 and ~r4 in a given Jacobi system.
These normal lab-fixed coordinates can then be inserted into the definition of the Jacobi
coordinates that I wish to describe. The kinematic rotation can then be extracted fromthe resulting relations. Following this procedure givesm3
m1 +m3

q

µ12
µ13



q

µ12
4
Ūs→i1 =  − m m+m
µ24
2
4


q
µ12
µ13,24

m3
m1 +m3

q

µ14
µ13



q

µ14
4
Ūs→i2 =  m m+m
 2 4 µ24
 q
µ14
µ13,241− m1m+m
3

m2
m2 +m4

q

q

q

µ34
µ13

q

µ34
µ24

q

µ24
µ12,34µ34µ13,24√ m1 m4 −m2 m3

1− m1m+m
3

q

µ23
µ13

q

2
− m2m+m
4
q

q

µ23
µ24

q
24
− µµ14,23µ23µ13,24µ13µ12,34q(m1 +m2 )(m3 +m4 )µ12,34µ13,24(6.15a)µ13µ12,34√ m1 m2 −m3 m4




,



q

(m2 +m3 )(m1 +m4 )µ14,24µ13,24




,



(6.15b)
TTŪi1→s = Ūs→i1 ; Ūi2→s = Ūs→i2 ,(6.15c)
TTŪi1→i2 = Ūi1→s Ūs→i2 = Ūs→i1 Ūs→i2 ; Ūi2→i1 = Ūi1→i2 .

(6.15d)The same method can be used to find the kinematic rotations to other Jacobi systems,
for instance to K-type coordinates.

6.1.4

Hyperangular coordinates

As with Jacobi coordinates, there is no unique way to construct the hyperangles
of a system. In this section I construct the hyperangular coordinates used in the fourfermion problem. The choice of hyperangular parameterization has physical meaning.
Different parameterizations can be used to describe different correlations within thesystem. Also, in the case of body-fixed coordinates, hyperspherical coordinates can be125
used to remove the Euler angles of solid rotation, reducing the dimensionality of thesystem.
The first hyperangular parameterization used here is in the form of Eq. 2.10 for
three 3D vectors. The hyperradius is defined in the same way as in Eq. 2.1,
2

R =

d
X

x2i ,

(6.16)

i=1

where each xi is a Cartesian component of one of the Jacobi vectors. The hyperspherical
trees that will be used for the four fermion problem will of the type in Fig. 2.4 inwhich N vectors are described by the spherical polar angles of each vector and a setof hyperangles correlating the lengths of each vector. Specifically, the hyperangles aredefined by the tree shown in Fig. 6.2 combined with the spherical polar angles of each
Jacobi vector. Following the rules described in Chapter 2 gives the hyperangles,

Figure 6.2: The hyperspherical tree used to parameterize the hyperangular coordinatesin the four-fermion problem is shown. See Chapter 2 or Ref. [5] for details.

|~
ρσl |
,
|~
ρσm |
q
2
ρ
~σl + |~
ρσm |2σ−1αlm,n = tan
.
|~
ρσn |
σ
αl,m
= tan−1(6.17)

126
Here the superscript σ = s, i1, i2 indicates which Jacobi system is being used, whilel, m, n indicate the three Jacobi vectors from that system. In principle there are six ofthese hyperangular coordinate systems that can be constructed from each set of Jacobi
vectors giving a total of 18 different hyperangular systems. Fortunately I will not need
all of these to tackle the four fermion problem. If one considers two particles, say 1
and 2, the tree defined by Fig. 6.2 with σ = i1, l = 2, m = 3 and n = 1 can be
decomposed into two subtrees. The right branch describes the hyperangular behaviorof the dimer alone, while the left branch describes the behavior of the remaining threebody system composed of a dimer and two free particles. This type of decomposition will
be important for evaluating kinetic energy matrix elements and defining basis functions.

6.1.4.1

Hyperspherical harmonics

Using the recursive definition of the hyperangular momentum operator in Eq. 2.11
the total hyperangular momentum operator can be written in terms of the hyperangularcoordinates asσΛ2 =∆1 αlm,n+
σ∆1 αlm,n
=



1
−
2 σ
σ cos ασsin αlm,n sin αl,m
l,m

"

∂
σ
∂αl,m

#2
σ
σ
(6.18)
cos αl,msin αl,m

ˆl2
ˆl2
ˆl2
m
n
l
+
+
,
σ
σ
σ
σ
σ
cos2 αlm,nsin2 αm,ncos2 αm,nsin2 αlm,nsin2 αlm,n

∂
−1
1
∂
σ
σ
σ
,
sin αlm,nsin2 αlm,ncos αlm,nσσσσσ
sin αlm,n∂αlm,n∂αlm,nsin2 αlm,ncos αlm,nwhere ˆll , ˆlm and ˆln are the normal spatial angular momentum operators for each Jacobi
vector. This can also be written directly from Eq. 2.11 asΛ2σ= ∆1 αlm,n



+

Λ2l,m

ˆl2
n
,
σ
cos2 αlm,n#2+sin2 αlm,n"1∂Λ2l,m = −σσσ
sin αl,m cos αl,m ∂αl,mσ
σ
sin αl,mcos αl,m+(6.19)
ˆl2
ˆl2
m
l
+
,
σ
σ
cos2 αm,nsin2 αm,nwhere all of the hyperangular behavior above the second node in Fig.6.2 is described bya sub-hyperangular momentum, Λ2lm,n .

127Constructing the hyperspherical harmonics for the four-body system is accomplished following the procedure in Appendix A giving

λ

(4b)

+5/2,l +1nl,mNλ63l,m ln ,λ sinλl,m (αlm,n ) cosln (αlm,n ) P λ−λ(cos 2αlm,n )Y λλ l ,l ,l (Ω) =Nl33
l lm λl,m[ l,m l m n ](l,m −ln )/2(6.20)
λm +1(cos 2αl,m )
sinll (αl,m ) coslm (αl,m ) P llλ+1,l−l
× Nll ,ll,mm( l,m l −lm )/2

× yll ml (ωl ) ylm mm (ωm ) yln mn (ωn ) ,where Pγα,β (x) is a Jacobi polynomial of order γ, ylm (ω) is a normal spherical harmonicde is a normalization constant [5, 66]:
with spherical polar solid angle ω, and Nabc

"
de
Nabc
=

a+b+c+d+e−2
c−a−b
2
2


c+b−a+ec+a−b+dΓ22(2c + d + e − 2) ΓΓ

 #1/2!

.

In Eq. 6.20 the degeneracy quantum number µ has been replaced with an explicit tabulation of the hyperangular momentum quantum numbers, i.e. λµ → [λλl,m ll , lm , ln ]. The(4b)total four-body hyperspherical harmonics satisfy the eigenvalue equation Λ2 Y λλ l ,l ,l (Ω) =[ l,m l m n ](4b)

λ (λ + 7) Y λλ l ,l ,l (Ω). The sub-harmonics that are eigenfunctions of Λ2l,m can be[ l,m l m n ]found as well:
(3b)

m +1
(cos 2αl,m )
sinll (αl,m ) coslm (αl,m ) P llλ+1,l−l
Y λ l ,l (Ωl,m ) =Nl33
l ,lm λl,m
( l,m l −lm )/2[ l,m l m ](6.21)

× yll ml (ωl ) ylm mm (ωm ) .
Here the superscript, (3b), indicates that this eigenfunction behaves as a 3-body hyperspherical harmonic. For instance if a hyperspherical tree is used with Jacobi vectors
defined in the i1 interaction coordinate system and l = 1, m = 3, and n = 2, thisthree-body harmonic describes the free-space behavior of a dimer with two free parti(3b)cles. The three body harmonics obey the eigenvalue equation, Λ2l,m Y λ l ,l (Ωl,m ) =[ l,m l m ]

128
(3b)

λl,m (λl,m + 4) Y λ l ,l (Ωl,m ). The restrictions on the values of λ and λl,m are[ l,m l m ]λl,m = ll + lm + 2j,(6.22)

λ = λl,m + ln + 2k
= ll + lm + ln + 2j + 2k,
where j, k = 0, 1, 2, .... The quantum numbers ll , lm and ln are the spatial angularmomentum quantum numbers associated with each Jacobi vector, and each has a zprojection quantum number associated with it which I have suppressed in Eqs. 6.20
and 6.21.

6.1.4.2

Democratic coordinates

Using Delve’s coordinates greatly simplifies evaluating hyperangular momentum
matrix elements, but it still leaves the 8 dimensional space of hyperangles. I will only
be considering systems with total angular momentum L = 0. Therefore it is convenient
to move into a body-fixed coordinate system, as the final wavefunction for the fourbody problem will not depend on the Euler angles that produce a solid rotation of thesystem. Removing the Euler angle dependence is accomplished by transforming into
the so-called democratic, or body-fixed coordinates. Four-body democratic coordinates
are developed in several references (see Refs. [125, 126, 127]). In this work I use theparameterization of Aquilanti and Cavalli. For a detailed derivation of the coordinate
system see their work in Ref. [125].
At the heart of democratic coordinates is a rotation from a space fixed frame toa body fixed frame:%̄ = D̃ (α, β, γ) %̄bf

(6.23)

where %̄ is the matrix of Jacobi vectors defined in Eq. 6.13, %̄bf is the set of body fixed
Jacobi coordinates, and D (α, β, γ) is an Euler rotation matrix defined in the standard

129
way as








 cos α − sin α 0   cos β 0 sin β   cos γ − sin γ 0 








D̄ =  sin α cos α 0  


.
0
1
0
sin
γ
cos
γ
0








0
0
1
− sin β 0 cos β001(6.24)The “˜” in Eq. 6.23 indicates a transpose has been taken.
The body-fixed coordinates are defined in a system whose axes are defined by theprinciple moments of inertia, I1 , I2 and I3 . In this coordinate system the body-fixed
Jacobi coordinates are given by%̄bf = ΠD̄ (φ1 , φ2 , φ3 ) ,(6.25)

where D̄ is defined in the same way as in Eq. 6.24 with φ1 , φ2 and φ3 replacing α, β,and γ. Π is a 3 × 3 diagonal matrix whose diagonals are given by ξ1 , ξ2 and ξ3 which
are parameterized by the hyperradius and two hyperangles Θ1 and Θ2 :R
ξ1 = √ cos Θ1 ,3R p
ξ2 = √
3 sin2 Θ1 sin2 Θ2 + cos2 Θ1 ,3R p
ξ3 = √
3 sin2 Θ1 cos2 Θ2 + cos2 Θ1 .
3

(6.26)

To avoid double counting and to allow for different chiralities, Θ1 and Θ2 are restricted
to 0 ≤ Θ1 ≤ π and 0 ≤ Θ2 ≤ π/4. With this parameterization the moments of inertia
are given byI1
R2= ξ22 + ξ32 =2 + sin2 Θ1 ,
µ
3
2

R
I2
= ξ12 + ξ32 =3 sin2 Θ1 cos2 Θ2 + 2 cos2 Θ1 ,
µ
3

I3
R2
= ξ12 + ξ22 =3 sin2 Θ1 sin2 Θ2 + 2 cos2 Θ1 .
µ
3(6.27)The hyperradius in terms of the principle moments of inertia can be then written as
R2 = ξ12 + ξ22 + ξ32 = (I1 + I2 + I3 ) /2µ.

130With this parameterization, all 8 hyperangles have been defined. The first three
are the Euler angles {α, β, γ}, which are external degrees of freedom describing solid
rotations of the four body system. The two angles, Θ1 and Θ2 , defined in Eq. 6.26,
describe the overall x, y and z extent of the four-body system in the body fixed frame.
From Eq. 6.27, if Θ1 = 0, π, then the principle moments of inertia are all equal, i.e.
I1 = I2 = I3 , meaning that the four particles are arranged at the vertices of a regulartetrahedron. When Θ1 = π/2, Eq. 6.26 shows that the particles are in a planar
configuration. The remaining angles, {φ1 , φ2 , φ3 }, are kinematic rotations within thesystem, and coalescence points and operations like particle exchange are described inthese angles. Broadly speaking, the democratic angles Θ1 and Θ2 can be thought ofas correlating the overall x, y, and z spatial extent of the four-body system in thebody-fixed frame, while the kinematic angles φ1 , φ2 , and φ3 parameterize the internal
configuration of the particles.
Since transformations from one Jacobi set to another are merely rotation matrices(sometimes combined with an inversion), the democratic parameterization can always
be written in the same form for any given Jacobi coordinate system. For example, thesymmetry coordinates (Eq. 6.4) can be transformed into the second set interaction
coordinates (Eq. 6.12) using the kinematic rotation defined by Eq. 6.15b. If thedemocratic parameterization defined in Eqs. 6.23 and 6.25 is used, this transformationreads%̄i2 = D̄ (α, β, γ) Π̄D̃ (φ1 , φ2 , φ3 ) Ūs→i2 ,= D̄ (α, β, γ) Π̄D̃ φ01 , φ02 , φ03

where I have used the fact that the product of two rotations in 3D is itself a rotation.
From this it is clear that within a given type of Jacobi coordinate (H-type or K-type),
all coalescence points are equally well described. This is an important feature as it does
not appear in Delve’s type coordinates, which are strongly dependent on which Jacobi

131
tree is used to define them. For the purposes of this paper, to make symmetrization of
the wave function easier, we will always define the body fixed coordinates in terms of
the symmetry Jacobi system %̄s .
Putting this all together, the body-fixed Jacobi vectors in terms of the internal
hyperangles can be defined:Rρs1x = √ cos Θ1 (cos φ1 cos φ2 cos φ3 − sin φ1 sin φ3 ) ,3R p 2
ρs1y = √sin Θ1 sin2 Θ2 + cos2 Θ1 (sin φ1 cos φ2 cos φ3 + cos φ1 sin φ3 ) ,3p−R
sin2 Θ1 cos2 Θ2 + cos2 Θ1 sin φ2 cos φ3 ,(6.28)
ρs1z = √3−Rρs2x = √ cos Θ1 (cos φ1 cos φ2 sin φ3 + sin φ1 cos φ3 ) ,3−R p 2sin Θ1 sin2 Θ2 + cos2 Θ1 (sin φ1 cos φ2 sin φ3 − cos φ1 cos φ3 ) ,ρs2y = √3
R p 2
ρs2z = √sin Θ1 cos2 Θ2 + cos2 Θ1 sin φ2 sin φ3 ,3R
ρs3x = √ cos Θ1 cos φ1 sin φ2 ,3
R p 2
ρs3y = √sin Θ1 sin2 Θ2 + cos2 Θ1 sin φ1 sin φ2 ,3
R p 2
sin Θ1 cos2 Θ2 + cos2 Θ1 cos φ2 ,
ρs3z = √3π
0 ≤ Θ1 ≤ π; 0 ≤ Θ2 ≤ ,4
0 ≤ φ1 , φ2 , φ3 ≤ π.
The restriction on the range of the internal hyperangles is to avoid double counting
configurations and allows for configurations of different chirality .
By moving into democratic coordinates, the dimensionality of the four-body problem can be decreased from 9 to 6, but, as with many simplifications, there is a cost.

132This cost comes in the form of the differential volume element dΩ [125]:√dΩ = (dα sin βdβdγ) 

3 cos3 Θ2 sin3 Θ2 cos 2Θ2 sin9 Θ1
1/2
cos2 Θ2 + 3 sin2 Θ1 cos2 Θ2 cos2 Θ2 + 3 sin2 Θ1 sin2 Θ2(6.29)× dΘ1 dΘ2 dφ1 sin φ2 dφ2 dφ3 .
The first factor is purely from the Euler angle rotation and will always yield a factor of
8π 2 for functions that are independent of α, β and γ.
Another price that is paid using democratic coordinates comes in the form of thehyperangular momentum operator, Λ2 . In terms of the democratic hyperangles, Λ2 isquite complex and can be found in Ref. [125]:
2




I1
2
2
2 L1 + J1
2 (I2 − I3 )


I2
I3
2
2
2
2
+
2 L2 + J2 +2 L3 + J3
2 (I1 − I3 )
2 (I1 − I2 )h
i1/2
2 I12 − (I2 − I3 )2+
L1 J1(I2 − I3 )2hi1/2
2 I22 − (I1 − I3 )2
L2 J2+(I1 − I3 )2

h
i1/2


2 I32 − (I1 − I2 )2
+
L
J
,
3 3

(I1 − I2 )2Λ = − ∆ (Θ1 , Θ2 ) + 2µR

2where J~ is the total angular momentum operator, and


 2

∂
∂
2
∂
4
∂
7
∆ (Θ1 , Θ2 ) = 7
sin Θ1++ cot Θ1−1
∂Θ1 sin2 Θ1 ∂Θ22
∂Θ1
sin Θ1 ∂Θ1
sin2 2Θ2


4
1
∂
∂
2
1 + 3 cos2 2Θ2+
sin 4Θ2+ cot2 Θ12∂Θ2 3
sin Θ1 4 sin 4Θ2 ∂Θ2
sin2 2Θ2




1 ∂2
cot 2Θ2 ∂
1
∂
∂ ∂
×
+
−
+ cot Θ1 cot 2Θ2.
4 ∂Θ222∂Θ2
sin 4θ2 ∂Θ2∂Θ2 ∂Θ2
1

Terms in Ji2 are centrifugal contributions, terms in Li Ji are Coriolis contributions and133
terms in L2i are contributions from internal kinematic angular momentum with ∂ 
sin φ1
 sin φ1 cot φ2 cos φ1 − cos φ2  
∂φ1 



∂ 


sin
φ
1
~

.
L = i~  cos φ1 cot φ2 − sin φ1 −




∂φ
sin
φ
2 
2 

 ∂ 
1
0
0
∂φ3
Fortunately, the methods for evaluating matrix elements in what follows will not directly require this form of the hyperangular momentum, but it is included here forcompleteness.
The final element needed from the democratic coordinates is the inter-particle
spacing. The ability to define these will be necessary to describe pairwise interactions
and correlations. Using Eqs. 6.2, 6.14 and 6.25,
r

i
† s
µ h s
,
%̄bf Ūs→i1%̄bf Ūs→i1µ12
11rµ h s† s i|~r13 |2 =%̄ %̄,µ13 bf bf 11
r
† s
i
µ h s
|~r14 |2 =%̄bf Ūs→i2%̄bf Ūs→i2
,
µ14
11
r
† s
i
µ h s
2
%̄bf Ūs→i2%̄bf Ūs→i2,|~r23 | =µ23
22r
µ h s† s i
2|~r24 | =%̄ %̄,µ24 bf bf 22
r
i
† s
µ h s
2
,
%̄bf Ūs→i1|~r34 | =%̄bf Ūs→i1µ23
22|~r12 |2 =(6.30)(6.31)(6.32)(6.33)(6.34)(6.35)

where [ ]ij indicates the ijth element of a matrix. In this equation, only the body fixed
Jacobi coordinates from Eq. 6.25 are used. This is because the unitary Euler rotation
used to rotate into the body fixed frame is the same for all Jacobi coordinates cancelingout the {α, β, γ} dependence in the inter-particle spacings.
Figure 6.3 shows the surfaces in {φ1 , φ2 , φ3 } for constant rij in a planar configuration for Θ2 = π/4, π/6, and π/12 for equal mass particles. The φ1 coordinate axis
has been transformed to φ1 − πΘ (φ1 − π/2), where Θ (x) is the unit step function, toemphasize the symmetry of the surfaces. The red surfaces correspond to the interact-

134
ing particles in the four-fermion system (r12 , r14 , r23 and r34 ) while the blue surfaces
correspond to the identical fermions (r13 and r24 ). The identical particle surfaces surround a coalescence point that must be a Pauli exclusion node in the final four-bodywave function. The simple nature of these coalescence points makes clear the reason forchoosing to base the democratic coordinates on the symmetry Jacobi vectors. The red
surfaces will play an important role in the pairwise interaction as these surfaces outline
the valleys of the potential. As the system becomes more linear (Θ2 becomes smaller) it
can be seen that the surfaces become broader in the φ1 direction. In fact, when Θ2 = 0(in perfectly linear configurations) these surfaces become independent of φ1 .

Figure 6.3: Surfaces surrounding the coalescence points in the body-fixed democraticπππ
coordinates are shown for θ1 = π/2 and θ2 =(a), (b), and(c) respectively.
4
6
12Blue surfaces surround interaction coalescence points while red surfaces surround Pauli
exclusion nodes.

6.2

Variational Basis Elements
In the previous section, the 12 dimensional four-body problem was reduced to a 6

dimensional problem for total spatial angular momentum J = 0, by removing the centerof mass coordinate and rotating into a body-fixed frame. By treating the hyperradius

135
adiabatically, the remaining 5 dimensional hyperangular partial differential equation,


2
X
Λ

+
V (rij ) Ψ (R; Ω) = U (R) Ψ (R; Ω) ,(6.36)
2µR2i,jmust be solved to obtain the adiabatic channel functions and potentials used in the
adiabatic hyperspherical formulation, Eq. 5.1. Here V (r) is a short-range s-wave pairwise interaction between distinguishable fermions, and the sum runs over all possible
pairs of distinguishable fermions. This thesis will only consider a potential whose zeroenergy s-wave scattering length a is positive and large compared with the range r0 of theinteraction. Further, unless otherwise stated, I assume that the potential can support
only a single weakly bound dimer.
The 5 dimensional PDE given by Eq. 6.36 is still too much for most numericalmethods, such as B-splines and finite elements. In fact if 40 − 100 B-splines were used(a common number in three body calculations [26, 128, 112]) in each dimension, there
would be 108 − 109 basis functions resulting in 1011 − 1013 non-zero matrix elements ina banded matrix. The computational power required for such a calculation is currently
beyond reach. Therefore, to proceed a different strategy must be developed.
The strategy used here is not unknown [129]. It involves using a variational basisthat diagonalizes the adiabatic Hamiltonian (Eq. 6.36) asymptotically (R  a) and at
small distances (R  r0 ). It is thought that linear combinations of these basis elements
will provide a variationally accurate description of the wavefunction in the intermediate
regime. In the asymptotic regime, there are three scattering thresholds to consider; athreshold corresponding to weakly bound dimers at twice the dimer binding energy, athreshold consisting of a single weakly bound dimer and two free particles, and finally athreshold consisting of four free particles. In general, it would be necessary to consider
another set of thresholds with trimer states (for instance a set of Efimov states forbosons). In the case of fermions though, such considerations are not needed as there
are no weakly bound dimers in the a  r0 regime. I will proceed by developing the

136variational basis based on these threshold behaviors.

6.2.0.3

Dimer-dimer threshold

The asymptotic behavior of the four-body system must include a description of
two s-wave dimers separated by a large distance. To incorporate this behavior thevariational basis must include a basis function of the form,
Ψ2+2 (R, Ω) = φd (r12 ) φd (r34 ) ,(6.37)

where the subscript 2+2 indicates the dimer-dimer nature of this function, and thedimer wavefunction, φd , is given by the two-body Schrödinger equation~2 ∂ 2−+ V (r) rφd (r) = −Eb rφd (r) .
2µ2b ∂r2(6.38)

Here µ2b is the reduced mass of the two distinguishable fermions, and Eb ≈ ~2 /2µ2b a2
is the binding energy of the weakly bound dimer. At first glance the right hand sideof Eq. 6.37 depends only implicitly on the hyperradius and hyperangles. To makethis dependence explicit, Eqs. 6.30 and 6.35 are employed to extract r12 (R, Ω) andr34 (R, Ω). It can also be noted that the basis function, Eq. 6.37, does not respect thesymmetry of the identical fermions, i.e. P13 Φ2+2 6= −Φ2+2 . The anti-symmetrization of
the variational basis will be discussed later in this thesis.

6.2.0.4

Dimer-atom-atom threshold

Another fragmentation possibility that must be incorporated into the asymptotic
behavior of the four fermion system is that of an s-wave dimer with two free particles.
The description of this configuration at fixed hyperradius is somewhat more difficult
than the dimer-dimer configuration. Fortunately, the hyperspherical tree described by
Eq. 6.17 can be used with interaction Jacobi coordinates, e.g. σ = i1, with l = 2, m = 3
and n = 1. If we assume that particles 1 and 2 are bound in a dimer, then the system

137
may be thought of as three distinguishable particles consisting of two distinguishable
fermions and a dimer. The hyperangles, Ω3B , of this three-body sub-system are thendescribed by the three-body sub-tree seen in Fig. 6.2. Furthermore, the three freeparticle behavior is merely given by a three-body hyperspherical harmonic, Eq. 6.21.
Assembling all of this together, a set of basis functions can be constructed of the form,Ψλ3B µ3B (R, Ω) = φd (r12 ) Φλ3B µ3B Ω123B ,(6.39)

where λ3B is the three body hyperangular momentum of the system and φd is, again,
defined by Eq. 6.38. Here the superscript 12 in Ω12
3B indicates that the third particle in
the three body subsystem is a dimer made from particles 1 and 2. Further, for notational
simplicity, µ3B has been used to denote the set of quantum numbers, {l2 , l3 , m2 , m3 },which enumerate the degenerate states for each λ3B .
So far the basis function defined by Eq. 6.39 can easily be written in Delve’s
coordinates. Unfortunately, this function is not written in a way that uses the total
angular momentum L of the system as a good quantum number. To get around this wemust couple the angular momenta corresponding to the interaction Jacobi coordinates i1
to total angular momentum L = 0. The angular momentum of the dimer is by definition
zero and all that remains is to restrict the angular momentum of the three-body subsystem to zero. This can be achieved by recognizing that the angular momentum of theindividual Jacobi vectors are good quantum numbers in the hyperspherical harmonics
defined by Eq. 6.21, meaning that we may proceed by normal Clebsch-Gordan angularmomentum coupling, i.e.
λ3B l1 l2
Ψ2+1+1(R, Ω) = φd (r12 )

l2
X

l3
X


hl2 m2 l3 m3 |00i Φλ3B µ3B Ω123B ,(6.40)

m2 =−l2 m3 =−l3where hl2 m2 l3 m3 |LM i is a Clebsch-Gordan coefficient, and l2 (l3 ) is the angular momentum quantum number associated with ρ
~i1
ρi1
2 (~
3 ) from the interaction Jacobi coordinates
defined in Eqs. 6.11. Now with the total angular momentum set to L = 0, there can be138
no Euler angle dependence in the total wavefunction. The Delve’s coordinates may thenbe defined for this system using Eq. 6.17. The Delve’s hyperangles can then be written
in terms of the democratic coordinates without including the Euler angle dependence.

6.2.0.5

Four free particles

The final asymptotic threshold that must be considered is that of four free particles. Using Delve’s coordinates, this behavior is already described by the four-body
hyperspherical harmonics given by Eq. 6.20, i.e.
λ(4b)+5/2,l +1nl,m(cos 2αlm,n )Nλ63l,m ln sinλl,m (αlm,n ) cosln (αlm,n ) P λ−λΦλµ (Ω) =Nl33
l lm λl,m(l,m −ln )/2

λm +1(cos 2αl,m )
sinll (αl,m ) coslm (αl,m ) P llλ+1,l−l
× Nll ,ll,mm( l,m l −lm )/2

× Yll ml (ωl ) Ylm mm (ωm ) Yln mn (ωn ) ,where µ has again been used to denote the set of quantum numbers {λ12 , l1 , l2 , l3 , m1 , m2 , m3 }that enumerate the degenerate states for each λ. Here li is the spatial angular momentum quantum number associated with the Jacobi vector ρ~σi with z-projection mi ,and λ12 is the sub-hyperangular momentum quantum number associated with the subhyperangular tree in Fig. 6.2.
The choice of quantum numbers here does not give the total angular momentumof the four particle system as a good quantum number. To accomplish this, the three
angular momenta of the Jacobi vectors must be coupled to a total momentum L = 0.
This gives a variational basis element of the form
λλ12 l1 l2 l3
Ψ1+1+1+1
(Ω) =

L12
X

l3
X

l2
X

l1
X

hL12 M12 l3 m3 |00i

(6.41)

M12 =−L12 m3 =−l3 m2 =−l2 m1 =−l1(4b)× hl1 m1 l2 m2 |L12 M12 i Φλµ (Ω) .
Now that the total angular momentum is set to L = 0 the same procedure used for3B l1 l2the Ψλ2+1+1
basis functions may be employed. However, this time the hyperangular139parameterization is defined using the symmetry Jacobi coordinates in Eqs. 6.4. As
there is no dependence on the Euler angles, the Jacobi coordinates may then be definedin the body fixed frame given by Eqs. 6.28.

6.2.1

Symmetrization

The definition of the basis functions developed in the previous subsection do notinclude the fermionic symmetry of the four particle system in question. Until this point,
only the symmetry and the first interaction coordinate systems have been employed.
Imposing the S2 × S2 symmetry of two sets of two identical fermions will now require
us to incorporate the extra interaction coordinate Jacobi set given in Eq. 6.12. First
we define the projection operator,

P̄ =



1 ¯
I − P̄13 I¯ − P̄24 ,
4(6.42)

where I¯ is the identity operator, and P̄ij is the operator that permutes the coordinatesof particles i and j. This operator will project any wavefunctions onto the space of
wavefunction that are antisymmetric under exchange of identical fermions. Since I am
treating the fermion species as distinguishable, permutations of members of different
species is ignored. Using this projection operator on the dimer-dimer basis wave function
yields(symm)

Ψ2+2(R, Ω) = P̄ Ψ2+2 (R, Ω) =1
(φd (r12 ) φd (r34 ) − φd (r14 ) φd (r23 )) ,2

(6.43)

where the inter-particle spacings r14 and r23 given by Eqs. 6.32 and 6.33.
Imposing the antisymmetric constraints on the dimer plus two free particle basis

140
functions in Eq. 6.40 yields(symm)λ3b l2 l3

Ψ2+1+1

3b l2 l3
(R, Ω)(R, Ω) = P̄ Ψλ2+1+1

l3
X

l2
X
1
= φd (r12 )4

hl2 m2 l3 m3 |00i Yλ3B µ3B Ω12
3B(6.44)

m2 =−l2 m3 =−l3l2
X1
− φd (r23 )
4

l3
X

l2
X
1
− φd (r14 )
4

l3
X

l2
X
1
+ φd (r34 )4l3
X

hl2 m2 l3 m3 |00i Yλ3B µ3B Ω23
3B

hl2 m2 l3 m3 |00i Yλ3B µ3B Ω143Bm2 =−l2 m3 =−l3m2 =−l2 m3 =−l3hl2 m2 l3 m3 |00i Yλ3B µ3B Ω34
3B ,m2 =−l2 m3 =−l3where Ωij
3B is the set of three-body hyperangles associated with particles i and j ina dimer and the remaining two particles free. The democratic parameterizations forthe inter-particle distances from Eqs. 6.30-6.35 can be used in the dimer wavefunction
directly. By using the symmetry coordinates, the hyperangles of the four-body system
can be divided into a dimer subsystem and a three body subsystem where the third
particle is the dimer itself. Using the three-body hyperangles in the three-body harmonicin each term in Eq. 6.44 combined with the kinematic rotations from Eqs. 6.15aand 6.15b, the three body harmonics are then fully described in the hyperangles from(symm)λ3b l2 l3

symmetry Jacobi coordinates. Since Ψ2+1+1

has been constrained to zero total

spatial angular momentum, L = 0, the body-fixed parameterization of the Jacobi vectors
can be inserted directly without worrying about the Euler angles α, β and γ.
The final set of basis functions that must be symmetrized with respect to identical fermion exchange are the four-body hyperspherical harmonics. Permutation of theidentical fermions is accomplished in the symmetry coordinates using Eqs. 6.5-6.10.
Using these permutations gives
l1 λλ12 l1 l2 l3
12 l1 l2 l3
P13 Ψλλ1+1+1+1 (Ω) = (−1) Ψ1+1+1+1 (Ω) ,l2
12 l1 l2 l3
P12 Ψλλ1+1+1+1 (Ω) = (−1) ,141which clearly indicates that the four free particle basis functions are anti-symmetrized
by choosing l1 and l2 to be odd.
Another symmetry in this system is that of inversion where all Jacobi coordinates
are sent to their negatives,
ρ~σj → −~
ρσj ,where σ = s, i1, i2 and j = 1, 2, 3. Following the definitions of the Jacobi coordinates
12 l1 l2 l3
positive inversion symmetry in the 1+1+1+1 basis functions, Ψλλ1+1+1+1 (Ω), is imposed(symm)λ3b l2 l3

by choosing λ to be even. The 2 + 1 + 1 basis functions,Ψ2+1+1(R, Ω), mustalready have positive inversion symmetry since φd (r) is an s-wave dimer wavefunction
and l2 = l3 for zero total spatial angular momentum, L = 0. The dimer-dimer basis(symm)

function, Ψ2+2

(R, Ω), is already symmetric under inversion and does not need further

restrictions placed on it.
The final symmetry to be imposed is not quite as obvious as the symmetries
discussed so far. By performing a “spin-flip” operation in which the distinguishable
species of fermions are exchanged, i.e. P̄12 P̄34 , the Hamiltonian in Eq. 6.1 remains thesame. This operation is identical to inverting the two dimers in the dimer-dimer basis(symm)

function. One can see that Ψ2+2

is unchanged under this operation. Because I will

limit myself to dimer-dimer collisions in this chapter, I will only be concerned with basis
functions that have this symmetry. This symmetry is imposed on both the 2 + 1 + 1(symm)λ3b l2 l3(Ψ2+1+1

λλ12 l1 l2 l3) and the 1 + 1 + 1 + 1 (Ψ1+1+1+1) basis functions by the condition

l3 = evenλλ12 l1 l2 l3
for the Ψ1+1+1+1
basis functions.

6.3

Four-fermion potentials and the dimer-dimer wavefunction
Calculating the hyperradial potentials and channel functions using the variational

basis is conceptually simple. All that is required are matrix elements of the hyperangular142
part of the full Hamiltonian,Had =~2 Λ2 X+
V (rij ) ,
2µ R2
i,j

where the sum runs over all interacting pairs of distinguishable fermions. In the previous
sections, the specific two-body interaction was kept general. All that was required wasa weakly bound dimer state and a positive scattering length much larger than the rangeof the interaction. At this point I choose the so called Posch-Teller potential,
V (r) = −U0,cosh (r/r0 )2(6.45)

where r0 is the range of the interaction. Unless otherwise stated U0 is tuned so that
V (r) gives the appropriate scattering length with only a single bound state. Thispotential is chosen because the bound state wavefunctions and binding energies are
known analytically [130], but any two-body interaction can be used with interpolated
bound state wavefunctions and numerically obtained energies.
Using the variational basis results in a generalized eigenvalue problem,
H̄ (R) ~xν (R) = Uν (R) S̄ (R) ~xν (R)(6.46)

where Uν (R) is the νth adiabatic hyperradial potential, and ~xν is the channel function
expansion in the variational basis. The matrix elements of H̄ are given by matrix
elements of the adiabatic Hamiltonian at fixed hyperradius,H̄nm = hΨn |Had | Ψm i .
Because the variational basis is not orthogonal, an overlap matrix, S̄, appears in thismatrix equation. While the method employed here is conceptually simple, the actual
calculation of the matrix elements is numerically difficult. For instance the valleys in thePhyperangular potential surface, i,j V (rij ), become localized in the hyperangular spacefor large hyperradius. Further, examination of Fig. 6.3 shows that these coalescence
points have a complex structure in the five dimensional body-fixed hyperangular space.

143
To accurately calculate the matrix elements in Eq. 6.46 numerically, a large numberof integration points must be placed within these valleys. Calculation of these matrix
elements is in fact the time limiting step in this treatment with each interaction matrix
element taking approximately 0.5 cpu hours on a modern computer.
Calculation of the non-adiabatic matrices, P̄ and Q̄, is done directly throughnumerical differentiation of the channel functions,∂Pνν 0 = Φν (R; Ω) |
Φν 0 (R; Ω)
∂R


∂2
Qνν 0 = Φν (R; Ω) | 2 Φν 0 (R; Ω)∂R
where Φν (R; Ω) is the νth hyperangular channel function that results from solving Eq.
6.46.
With all of these complications in mind the adiabatic potential can be foundapproximately. Figure 6.4 shows the full set of hyperradial potentials including the
diagonal non-adiabatic correction (solid curves) calculated using 8 variational basis elements, one 2 + 2 element, four 2 + 1 + 1 elements, and three 1 + 1 + 1 + 1 elements.
Also shown are the expectation values of the basis elements themselves (dashed curves).
All calculations here are for a = 100. It is clear that the lowest potential converges
very quickly with respect to the number of variational basis elements used. The lowest
potentials converge well when only a few variational basis elements are included, while
the higher potentials are somewhat suspect. The adiabatic potentials become universal
for large scattering lengths. In other words, the potentials look the same when scaledby the scattering length and the binding energy. Comparison with a correlated Gaussian method of calculating these potentials shows excellent agreement in the lowest
dimer-dimer potential with reasonable agreement in the lowest few dimer-atom-atom
potentials [8].
In Fig. 6.4 the lowest hyperradial adiabatic potential approaches twice the binding
energy of a single dimer indicating that this channel corresponds to the dimer-dimer

144

Figure 6.4: The hyperradial potentials are shown (solid lines) calculated for a = 100
as a function of R/a. Also shown are the expectation values of the variational basis
elements used (dashed curves).

potential. Examining this potential further, one can see that at hyperradii less than the
scattering length, R < a, the dimer-dimer potential becomes strongly repulsive. Thiscan almost be thought of as hard wall scattering giving a dimer-dimer scattering length,
add , similar in magnitude to the two-body scattering length,
add ∼ 3.5a.

(6.47)

Higher potential curves are seen to approach the dimer binding energy, indicating thatthese potentials correspond to a dimer with two free particles in the large R limit. As
the scattering length becomes much larger than the range of potentials, the potential
becomes universal in the range of r0  R  a:U (R) →

~2 p20 − 1/4,
2µ R2(6.48)

where p0 = 2.55. This universal potential was extracted by von Stecher, Greene, andBlume [131, 58] by examining the behavior of the ground state energy of four fermionsin a trap in the unitarity limit.
2 / {2µ [(U (R) − U (R))]}, beFigure 6.5 shows the coupling strengths, ~2 Pnmmntween the dimer-dimer potential and the lowest three dimer-atom-atom adiabatic potentials for a two-body scattering length of a = 100r0 . In each case the coupling

145
strength peaks strongly near the short range region, R ∼ r0 , and near the scattering
length, R ∼ a, and then falls off quickly in the large R limit. This behavior indicatesthat recombination from a state consisting of a deeply bound dimer and two free particles to the dimer-dimer state occurs mainly at hyperradii near a. Looking at Fig. 6.5
one might think that a recombination path that occurs at small R, R ∼ r0 , will also
contribute, but the strong repulsion between R ∼ r0 and R ∼ a in the dimer-atom-atom
potentials, shown in Fig. 6.4, suppresses this pathway.
0.005
0.00450.004
0.00350.003
0.00250.002
0.00150.001
0.0005
0
0

2000

4000

6000

8000

10000

Figure 6.5: The coupling strength between the dimer-dimer potential and the lowest
dimer-atom-atom potential is shown as a function of R/a. The blue dashed line shows
the position of the coupling peak at R/a ≈ 3.5.

Figure 6.6 shows an isosurface of the hyperangular probability density in theconfigurational angles {φ1 , φ2 , φ3 } after integrating out Θ1 and Θ2 at a fixed hyperradiusof R = 0.41a. The φ1 axis has been modified here by shifting the region π/2 ≤ φ1 ≤ π
to emphasize the symmetry of the system. Each cobra-like surface corresponds to apeak in the four-body probability density. By examining Fig. 6.3, it is clear that thespine of each cobra corresponds to an interaction coalescence point. For each choice of{φ1 , φ2 , φ3 }, the maximum of the probability density in Θ1 and Θ2 is given in a planar

146
geometry, Θ1 = π/2. The coloring of each cobra indicates the value of Θ2 at which the
maximum occurs. Darker colors indicate a more linear geometry, i.e. Θ2 is closer to 0.
Figure 6.7 shows the same plot for the 2 + 2 basis function only. Comparing Figs. 6.6
and 6.7 it is clear that the added variational basis elements are critical for describing thefull dimer-dimer channel function for hyperradii less than the scattering length. Figure
6.8 shows the cobra isosurface of Fig 6.6 in a slightly more picturesque setting. No extra
physical insight is gained from showing this, but it does look very nice, though somehave suggested a palm tree would be a good addition [132].

Figure 6.6: An isosurface of the dimer-dimer probability density is shown. The surfaces
are found by integrating the total probability over θ1 and θ2 and plotting with respect
to the remaining democratic angles (φ1 , φ2 , φ3 ). The peak probability always occurs inplanar symmetry, θ1 = π/2. The coloring (light to dark) indicate the value of θ2 at the
peak.

147

Figure 6.7: The same as Fig. 6.6, but only using the 2 + 2 basis function. The dashedgray lines are purely for perspective.

6.4

Dimer-Dimer ScatteringWith the hyperradial potentials and non-adiabatic couplings in hand, low energy

dimer-dimer scattering properties can be examined. The zero-energy dimer-dimer scattering length in the large two-body scattering length limit was first calculated by Petrov
et. al [23] and found to beadd (0) = 0.60 (2) a,(6.49)

where the number in the parentheses indicates ±0.02, the 4% error stated in this work.
This prediction has been confirmed using several different theoretical approaches [133,131, 58].
Using the adiabatic potentials shown in Fig. 6.4 and the resulting non-adiabatic

148

Figure 6.8: The isosurfaces shown here are the same as in Fig. 6.6, but in a pleasing
environment.

couplings, the energy dependent dimer-dimer scattering length defined byadd (Ecol ) =− tan δdd
kdd

(6.50)

can be calculated. Here Ecol is the collision energy of the two dimers with respect to thedimer-dimer threshold, and δdd is the s-wave dimer-dimer phase shift. An interesting
thing occurs when the collision energy becomes greater than the dimer binding energy.
At this point the two dimers collide with enough energy to dissociate one of them. When
this happens, the four fermion system leaves in an excited channel causing a loss of flux
from the dimer-dimer channel. This process is parameterized by the imaginary part ofthe dimer-dimer scattering length which will become non-zero when Ecol > Eb .
Figures 6.9 and 6.10 respectively show the real and imaginary parts of the dimerdimer scattering length calculated with different numbers of adiabatic channels plotted149
as a function of Ecol in units of the dimer binding energy. Also shown in Fig. 6.9 is thedimer-dimer scattering length calculated from the variational potential that results fromusing a single variational basis element. It is important to note that the single adiabatic
channel calculation and the single basis element calculation are not the same. In theformer, the single potential used is the lowest potential resulting from a calculation using
multiple basis elements while the latter is the result of using only the 2 + 2 variational
basis element. Not surprisingly, the scattering length at collision energies found near
the binding energy depends strongly on the number of channels used. With just a single
channel in use, there is no decay pathway available for the system. As more channels
are included the system has more pathways available to fragment into, modifying thehigh energy behavior.
2

Re(add)/a

1.5

1

0.5

0

-0.5
0.1

1

10

100

1000

Ecol/Eb

Figure 6.9: The real part of the energy dependent dimer-dimer scattering length is shownin dimensions of a plotted versus the collision energy in units of the binding energy. The
calculation if done with one, two, three, four, and five adiabatic channels (blue, black,
red, green, and purple curves respectively) from the 8 basis element calculation. The
red dashed line shows add = 0.6a, the prediction of Ref. [6].

What is more surprising is the low energy behavior seen in Fig. 6.9. For a single
variational basis element, the dimer-dimer zero energy scattering length is found to be
add = 0.72a, which is already fairly close to the result of Ref. [23], add (0) = 0.6a. A

150-Im(add)/a

2

1

0
0.1

1

10

100

1000

Ecol/Eb

Figure 6.10: The imaginary part of the energy dependent dimer-dimer scattering lengthis shown in dimensions of a plotted versus the collision energy in units of the binding
energy. The calculation if done with one, two, three, four, and five adiabatic channels (blue, black, red, green, and purple curves respectively) from the 8 basis element
calculation.

single channel calculation using the dimer-dimer potential and channel function that
results from using 5 basis elements improves considerably on this yielding add (0) =0.64a, indicating that including correlations describing two free particles at hyperradii
less than a gives a significant contribution to the physics of dimer-dimer scattering.
It is somewhat unexpected that the single channel calculation is only 8% off of thepredicted value. As the scattering energy approaches zero, the higher fragmentation
channels become strongly closed but still apparently play a small role in the dimerdimer scattering process. By including progressively more channels in the scattering
calculation the zero-energy dimer-dimer scattering can be extracted for large two bodyscattering length:add (0) = 0.605 (5) a.

(6.51)

This result is in agreement with the results of Ref. [131, 58] which found the zero-energy
dimer-dimer scattering length to similar accuracy using different methods.

151
6.4.1

Energy dependence

By examining the low energy behavior of the energy dependent dimer-dimer scattering length, the effective range can be extracted. The two dimers “see” each otherwhen their wavefunctions are overlapping, i.e. when the hyperradius is approximately
equal to the scattering length, R ∼ a. If one thinks of the effective range of an interaction as proportional to the size of the interaction region, then one would expect theeffective range for dimer-dimer scattering to be proportional to the scattering length.
By fitting the low energy scattering phase shift to the effective range expansion,k cot δdd = −11+ rdd k 2 ,add (0) 2(6.52)

this intuitive behavior is born out giving an effective range,
rdd = 0.13a,(6.53)

where a is the two-body scattering length. Figure 6.11 shows both the real and imaginary parts of the energy dependent dimer-dimer scattering length as a function ofcollision energy in units of the binding energy compared to the effective range expansion, Eq. 6.52. This clearly shows that, while the low energy behavior of dimer-dimer
scattering is well described by the effective range expansion, it is only accurate over asmall range of collision energies. In fact, for collision energies larger than the binding
energy, add (Ecol ) actually turns over and decreases as dimer breakup channels become
open. Further, when the collision energy is equal to the dimer binding energy, Ecol = Eb ,the dimer-dimer scattering length becomes complex, with an imaginary part that parameterizes inelastic processes. These results indicate that both the real and imaginary
dimer-dimer scattering lengths are universal functions of the collision energy, i.e. insensitive to the short range nature of the two-body interaction, for scattering lengths muchlarger than the two-body interaction length scale, r0 . Because very few basis functions
were used in these calculations, the results at higher energies, Eb  Ecol  ~2 /mr02 , are152
not well converged, though their qualitative nature is expected to persist. Well above
the dissociation threshold, oscillations in the real part of the dimer-dimer scattering
length can be seen. These oscillations are caused by interference between different scattering paths. As more basis functions are included and the high energy results converge,
the large number of available pathways will generally cancel the oscillatory behavior,
but the decrease in the real part of the dimer-dimer scattering length should persist.

Figure 6.11: The real (red) and imaginary (green) parts of the energy dependent dimerdimer scattering length is shown in dimensions of a plotted versus the collision energy
in units of the binding energy. Also shown is the energy dependent scattering length
using the effective range expansion. Figure from Ref. [7].

An interesting aspect of these results is in the two-body scattering length dependence of the dimer-dimer scattering length at finite collision energy. In the large
scattering length limit, the dimer binding energy becomes Eb ≈ ~2 /ma2 , so that as

153
the scattering length increases the binding energy decreases. At the critical scattering
length,ac = √~,
mEcol

the collision energy is the same as the binding energy. As a result, Fig. 6.11 shows that
one expects the real part of the dimer-dimer scattering length to turn over and remain
finite for all two-body scattering lengths. This behavior is demonstrated in Fig. 6.12
which shows the real part of the dimer-dimer scattering length at several fixed collisionenergies compared to the zero-energy result, add (0) = 0.6a. The scattering length were
fixed by setting the range of the interaction to be approximately the Van der Waals
length of 40 K, r0 ≈ 100 a.u. Another aspect of the finite collision energy behavior isthat at large scattering length, the dimer-atom-atom channels become open, and dimer
dissociation is allowed. This leads to the idea that near unitarity the Fermi gas might
be thought of as a coherent mixture of atoms and weakly bound dimers.
5

10

5

Re(add) (a.u.)

10

5

10

5

10

0
3

10

4

10
a (a. u.)10

5Figure 6.12: The real part of the energy dependent dimer-dimer scattering length isshown plotted as a function of the two-body scattering length in atomic units is shown
calculated for several collision energies; Ecol /kb =250nK,100nK, 25nK, 10nK, 2.5nK,
1nK, 10−1 nK, and 10−2 nK. Also shown is the zero energy prediction (black dashedcurve). Figure from Ref. [7].

154

6.5

Dimer-dimer relaxation
A significant loss process in an ultracold gas of bosonic dimers is that of dimer-

dimer relaxation in which two dimers collide, and in the process one of the dimers
relaxes to a deeply bound two-body state. The excess energy is released as kinetic
energy and is then enough to eject the remaining fragments from the trap. This process
was studied by Petrov, Salomon and Shlyapnikov [23, 6] by assuming that this process
is controlled by the probability of three particles being in proximity to one another.
With this assumption and the further assumption that the fourth particle is far away
and plays no role in the scattering process, they predict that the relaxation rate is
dd ∝ a−2.55 .
suppressed at large two-body scattering lengths with a scaling law, Vrel

In this section I introduce a new method for finding the dimer-dimer relaxation
rate based directly on Fermi’s golden rule. The key observation in this section is thatthe final allowed states in the hyperspherical appear as an infinite set of hypersphericalpotentials corresponding to a deeply bound dimer with two free atoms. The transition
rate to a single one of these states can be described by Fermi’s golden rule, i.e.
Tpλ ∝ |hΨdd (R; Ω) |V (R, Ω)| Ψλ (R, Ω)i|2 .

(6.54)

Here Ψλ is the final outgoing state, Ψdd is the dimer-dimer wavefunction, and V (R, Ω)
is the sum of the two-body interactions. The evaluation of this matrix element and thesum over the final states is shown in Appendix D. The final result of this analysis isgiven as an integral over the hyperradius,
dd
Vrel
∝

Z

PW KB (R) F (R)ρ (R) dR
Rκ (R)(6.55)

where PW KB (R) is the WKB probability of the dimer-dimer wavefunction at hyperradius R, κ (R) is the WKB wavenumber given bys 
2µ~2 1/4
κ (R) =
Vdd (R) +− Ecol .
~22µ R2(6.56)

155
In Eq. 6.55 ρ (R) is the nearly constant density of final states, and F (R) is the probability of finding three particles in near proximity to one another in the dimer-dimer
wavefunction at hyperradius R, given by,
F (R) = hΦdd (R; Ω) |f (R, Ω)| Φdd (R; Ω)i .

(6.57)

Here Φdd is the hyperangular dimer-dimer channel function, and f (R, Ω) is a proximity
function that only peaks when three particles are within approximately the range of thetwo-body interaction of one another.
Equation 6.55 makes physical sense upon closer examination. It says that the rate
at which a dimer relaxes to a deeper state is determined, with some extra factors, by theprobability that three particles are close enough together that two of them can fall intoa deeply bound state and release the extra binding energy to the third particle. Figure
6.13 shows the integrand from Eq. 6.55 for several scattering lengths as a function of thehyperradius in units of the scattering length. This quantity can be interpreted as being
proportional to the transition rate per unit hyperradius, i.e. the probability that thetransition will occur between R and R + dR. The overall transition rate is determinedby the nature of the interaction at short range and is not predictable using this method.
By examining the relaxation rate as a function of scattering length, a scaling law canbe extracted at each fixed hyperradius.
Figure 6.14 shows the relaxation rate per unit hyperradius for several fixed valuesof R/a as a function of the scattering length, a. The large a behavior in each caseappears to follow a scaling law, but the scaling law changes with R/a. This behavior
indicates that, contrary to the prediction of Ref. [6], when the integral in Eq. 6.55
is evaluated, the relaxation rate will not be determined by a mere scaling law. By
integrating over different hyperradial regions, contribution to the transition rate fromdifferent processes can be extracted. For instance, if integral in Eq. 6.55 is performed
only over small hyperradii, R . 5r0 , the result is the transition rate due to processesTp (arbitrary units)

156

0.5

1
R/a

1.5

2

Figure 6.13: The integrand from Eq. 6.55 is shown for a = 50r0 , 64r0 , 80r0 , and 100r0
as a function of R in units of a.

in which all four particles are in close proximity. If the integral is evaluated over larger
hyperradii, R > 10r0 , the result is the rate due to three-body processes only.

Figure 6.14: The integrand from Eq. 6.55 is shown for various values of R/a as afunction of a in units of r0 .

Figure 6.15 shows the the relaxation rate as a function of the scattering length
in atomic units as a red solid line. In this result the range of the interaction is set
to the van der Walls length of 40 K, r0 ≈ 100 a.u. Also shown in Fig 6.15 are the157

Figure 6.15: The relaxation rate is shown as a function of scattering length (see text).
The red curve represents all possible relaxation pathways. The blue dot-dashed curve
represents only the contribution to relaxation for processes occurring at R > 10r0 . The
green dashed line shows the contributions for prosses occurring at R < 5r0 . Figure from
Ref. [7]contribution to this relaxation rate due to four-body processes (dashed blue curve), andthe contribution to this rate due to three-body processes (dotted green curve). Also
dd ∝ a−4.02 , for transitions that occur at small
shown is the expected scaling law, Vrel
dd ∝ a−2.55 ,hyperradius, R = 5r0 . Figure 6.15 also shows the predicted scaling law, Vrel

from Ref. [6], and an intermediate scaling law that can be extracted. In this regime,
because the hyperradius is small, the probability of three particles being in proximity isnear unity, meaning that the transition probability per unit hyperradius is determinedby the probability that the system can tunnel through the repulsive potential seen in
Fig. 6.4 at R . a. The universal repulsive potential in this regime [8, 7],
U (R) =

~2 p20 − 1/4,
2µ R2p0 = 2.55,(6.58)(6.59)

leads to a scaling law for transitions in the small R regime that behaves asddVrel∝ a1−2p0 = a−4.20 .

(6.60)

158
Figure 6.15 also shows the experimentally determined relaxation rates from Ref. [134].
Both the scaling law predicted in Ref. [6] of a−2.55 and the prediction using Eq. 6.55
fit the experimental data in the regime for 1000 a.u.. a . 4000 a.u.. The experimental
data for a > 3000 a.u. are in the regime where the average dimer separation distanceis less than the dimer size where the dimer-dimer scattering picture discussed here nolonger applies.

Chapter 7

Summary

In this dissertation I have analyzed the effects of few-body physics on many-bodysystems, and the dynamics that result from fundamental few-body systems. Within
the framework of hyperspherical coordinates, complex many dimensional systems canbe described by a set of effective coupled 1D Schrödinger equations, allowing all of
the intuition and understanding of standard 1D Schrödinger quantum mechanics to bebrought to bear on the problem. While obtaining the effective hyperradial potentials
is often a significant challenge, once these potentials have been obtained, a variety ofcomplex dynamics can be analyzed such as low energy excitation frequencies in trapped
many-body systems and inelastic scattering in few-body systems.
In the first chapters of this thesis I generalized known few-body methods to a
many-body system through the use of a variational ansatz wavefunction. While thevariational wavefunction used does not allow for complex correlations, it does provide
an intuitive picture for the many-fermion system in the form of an effective potential. This approach also provides an initial link between standard methods in few-bodysystems and the complex world of many-body physics. By tuning the few-body parameters through a Fano-Feshbach resonance the behavior of the degenerate Fermi gas was
explored throughout the resonance.
By applying the K-harmonic approximation in Chapter 3, a simple effective potential was extracted which gave the ground state energy and rms size of the gas in good

160
agreement with those predicted from the Hartree-Fock method. In the case of the barezero-range Fermi pseudo-potential, the effective potential predicted an unphysical collapse on the negative side of a resonance in the strongly interacting regime. This collapse
can be avoided by incorporating a density dependent interaction. Using this interaction
the ground state energy is predicted with excellent agreement with the Hartree-Fockprediction, both of which yield the universal constant β = −0.49. This value of β islarger than the variational prediction from quantum Mote Carlo studies, β = −0.58, but
lower than the β value predicted from a normal degenerate Fermi gas, β = −0.44. Thismethod does not describe the behavior of the gas on the positive side of the resonance,as neither the interaction or the variational wave function are capable of describing a
BEC of dimers. By further extending the K-harmonic method to incorporate multiple
internal states, a dynamic instability is predicted in a degenerate Fermi gas with four
components. The three-component gas was predicted to be barely stable at unitarity
using a density dependent interaction, and by modifying the interaction to fit the twocomponent gas, the three-component system is predicted to be unstable. These results
do not include the possibility of a ground state consisting of trimer states. Further
examination, both theoretically and experimentally, is needed to clarify whether thethree-component gas will collapse in a manner similar to the Bosenova first, or boil off
through inelastic three-body recombination. This question is increasingly pertinent as
recent experiments with the three lowest hyperfine states of 6 Li are near the predicted
density required for collapse.
In the case of an anisotropic trap, the K-harmonic method is difficult to apply.
By generalizing the hyperspherical picture to that of the hyper-vectorial method in
Chapter 4, the anisotropic nature of the trap is easily incorporated. The ground stateenergy and rms longitudinal and transverse size of the gas are in perfect agreement with
the K-harmonic prediction in an anisotropic trap. However, the low energy excitation
frequencies disagree strongly with both previous theoretical predications and with ex-161
perimental observations. This is possibly due to the somewhat unphysical assumptionof the distribution of oscillator quanta in the variational wavefunction used. It is hopedthat this work can provide a starting point for later studies in this system.
In the second half of this dissertation, several fundamental few-body systems were
explored in the context of the adiabatic hyperspherical method. This method lends itself
well to scattering dynamics by incorporating the various fragmentation channels thatcan result in few-body systems. By including the non-adiabatic coupling, the qualitative
behavior of inelastic scattering was easily found. In Chapter 5 I introduced a novel formof the free-space hyperangular Green’s function. When used in a Lippmann-Schwinger
equation based approach, this Green’s function was shown to yield the same results forthe three-body hyperradial potential curves that are found using conceptually different
theoretical approaches, such as Faddeev decompositions. The results of this study werethen used to predict the scaling behavior of trap-loss recombination events across thethree resonances in the three lowest hyperfine states of 6 Li. The resulting scaling at
higher magnetic fields (834 to1000 gauss) are in agreement with other theoretical predictions, but further experimental investigation is required to explore this system.

The Green’s function method was then applied to the complex systems of threebosons with multi-channel two-body interactions leading to the prediction of a new classof Efimov states. While other theoretical treatments have examined this multi-channel
system, this is the first prediction of three-body Fano-Feshbach resonances embedded in
the three-body continuum due to the presence of a quasistable Efimov state. There areno current experiments looking for these states, but current radio frequency spectroscopy
techniques could possibly be used to probe them. Using this method, multiple Efimov
states might be seen experimentally for the first time.
Finally, in Chapter 6 I explored the four-fermion problem with large positive
scattering lengths. While no specific new method was developed to describe this complex

162
system, a variety of existing ideas were incorporated in new ways to extract a set ofvariationally accurate hyperradial potentials. This method gives the zero-energy dimerdimer scattering length to high precision in good agreement with previous predictions.
At higher energies, the energy dependent dimer-dimer scattering length is in agreementwith previous effective range expansions but deviates from this prediction at collisionenergies well below the dimer binding energy. Further, it was shown that at any finite
collision energy, the dimer-dimer scattering length will never diverge.
By incorporating the full multi-threshold behavior of the system, inelastic dimer
breakup scattering was also incorporated in the form of a complex energy dependent
dimer-dimer scattering length. By developing a new Fermi’s golden rule based method,
the dimer-dimer relaxation rate was also predicted, disagreeing the previous predictions,dd ∝ a−2.55 , that ignored the effects of the presence of a fourth particle in the relaxation
Vrel

process. The predictions given here agree with the experimentally measured behavior
in the relevant regime, and explain the agreement of experiment with the previously
predicted scaling law through the effects of short-range corrections.

Bibliography[1] S. T. Rittenhouse, M. J. Cavagnero, J. von Stecher, and C. H. Greene,
Phys. Rev. A 74, 053624 (2006).
[2] S. T. Rittenhouse and C. H. Greene, J. Phys. B 41, 205302 (2008).
[3] J. von Stecher and C. H. Greene, Phys. Rev. A 75, 022716 (2007).
[4] M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen, S. Jochim, C. Chin,
J. H. Denschlag, R. Grimm, A. Simoni, E. Tiesinga, et al., Phys. Rev. Lett.
94, 103201 (2005).
[5] Y. F. Smirnov and K. V. Shitikova, Sov. J. Part. Nucl. 8, 44 (1977).
[6] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. A 71, 012708(2005).
[7] J. P. D’Incao, S. T. Rittenhouse, N. P. Mehta, and C. H. Greene, Phys.
Rev. A 79, 030501 (2009).
[8] J. V. Stecher, Trapped Ultracold Atoms with Tunable Interactions, PhD thesis,
University of Colorado, 2008.
[9] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, andE. A. Cornell, Science 269, 198 (1995).
[10] C. Bradley, C. Sackett, J. Tollett, and R. Hulet, Phys. Rev. Lett. 75,
1687 (1995).
[11] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. Van Druten, D. S.
Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).
[12] B. DeMarco and D. S. Jin, Science 285, 1703 (1999).
[13] C. Regal, M. Greiner, and D. Jin, Phys. Rev. Lett. 92, 40403 (2004).
[14] J. Kinast, S. Hemmer, M. Gehm, A. Turlapov, and J. Thomas, Phys. Rev.
Lett. 92, 150402 (2004).
[15] M. Zwierlein, C. Stan, C. Schunck, S. Raupach, A. Kerman, and W. Ketterle, Phys. Rev. Lett. 92, 120403 (2004).

164[16] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. Denschlag, and R. Grimm, Phys. Rev. Lett. 92, 120401 (2004).
[17] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann, L. Tarruell, S. Kokkelmans, and C. Salomon, Phys. Rev. Lett.
93, 50401 (2004).
[18] U. Fano, Nuovo Cimento 12, 154 (1935).
[19] H. Feshbach, Ann. of Phys. 5, 357 (1958).
[20] A. J. Leggett, J. Phys. C (Paris) 41, 7 (1980).
[21] M. Drechsler and W. Zwerger, Ann. Phys. (Germany) 504 (1992).
[22] M. Holland, S. Kokkelmans, M. L. Chiofalo, and R. Walser, Phys. Rev.
Lett. 87, 120406 (2001).
[23] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. Lett. 93,
90404 (2004).
[24] S. Y. Chang, V. R. Pandharipande, J. Carlson, and K. E. Schmidt, Phys.
Rev. A 70, 43602 (2004).
[25] H. Heiselberg, Phys. Rev. A 63, 43606 (2001).
[26] B. D. Esry, C. H. Greene, and J. P. Burke, Phys. Rev. Lett. 83, 1751 (1999).
[27] E. Nielsen and J. H. Macek, Phys. Rev. Lett. 83, 1566 (1999).
[28] P. F. Bedaque, E. Braaten, and H. W. Hammer, Phys. Rev. Lett. 85, 908(2000).
[29] N. P. Mehta, S. T. Rittenhouse, J. P. D’Incao, and C. H. Greene, Phys.
Rev. A 78, 020701 (2008).
[30] L. M. Delves, Nucl. Phys 9, 391 (1958).
[31] L. M. Delves, Nuc. Phys. 20, 275 (1960).
[32] V. Efimov, Sov. J. Nucl. Phys 12, 589 (1971).
[33] M. Fabre de la Ripelle and J. Navarro, Ann. Phys. 123, 185 (1978).
[34] J. Macek, J. Phys. B 1, 831 (1968).
[35] U. Fano, Phys. Today 29, 32 (1976).
[36] U. Fano, Rep. Prog. Phys 46, 97 (1983).
[37] C. W. Clark and C. H. Greene, Physical Review A 21, 1786 (1980).
[38] M. Cavagnero, Phys. Rev. A 33, 2877 (1986).
[39] M. Cavagnero, Phys. Rev. A 36, 523 (1987).

165[40] V. Kokoouline and C. H. Greene, Phys. Rev. A 68, 012703 (2003).
[41] A. Igarashi and N. Toshima, Phys. Rev. A 50, 232 (1994).
[42] J. Macek, M. Cavagnero, K. Jerjian, and U. Fano, Phys. Rev. A 35, 3940
(1987).
[43] J. L. Bohn and U. Fano, Phys. Rev. A 53, 4014 (1996).
[44] P. Nozieres and S. Schmitt-Rink, J. Low-Temp. Phys. 59, 195 (1985).
[45] G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Phys.
Rev. Lett. 93, 200404 (2004).
[46] A. Bulgac, J. E. Drut, and P. Magierski, Phys. Rev. Lett. 96, 090404 (2006).
[47] A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. Lett. 93, 100404 (2004).
[48] A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. Lett. 93, 100404 (2004).
[49] S. Tan and K. Levin, Phys. Rev. A 74, 43606 (2006).
[50] S. Tan, Annals of Physics 323, 2952 (2008).
[51] S. Tan, Annals of Physics 323, 2971 (2008).
[52] S. Tan, Annals of Physics 323, 2987 (2008).
[53] S. Zhang and A. J. Leggett, Phys. Rev. A 79, 023601 (2009).
[54] G. Baym, C. J. Pethick, Z. Yu, and M. W. Zwierlein, Phys. Rev. Lett. 99,
190407 (2007).
[55] E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008).
[56] C. De Melo, M. Randeria, and J. R. Engelbrecht, Phys. Rev. Lett. 71,
3202 (1985).
[57] P. Pieri and G. C. Strinati, Phys. Rev. B 61, 15370 (2000).
[58] J. Von Stecher, C. H. Greene, and D. Blume, Phys. Rev. A 77, 43619(2008).
[59] V. N. Efimov, Sov. J Nucl. Phys. 12, 589 (1970).
[60] V. N. Efimov, Nucl. Phys. A 210, 157 (1973).
[61] E. Braaten and H. W. Hammer, Phys. Rev. A 70 (2004).
[62] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H. Naegerl, et al., Nature
440, 315 (2006).
[63] J. P. D’Incao, C. H. Greene, and B. D. Esry, J. Phys. B 42, 044016 (2009).

166[64] S. Knoop, F. Ferlaino, M. Mark, M. Berninger, H. Schöbel, H. Nägerl,
and R. Grimm, Nature Physics 5, 227 (2009).
[65] J. V. Stecher, J. P. D’incao, and C. H. Greene, eprint arXiv:0810.3876(2009).
[66] J. Avery, Hyperspherical Harmonics: Applications in Quantum Theory, Kluwer
Academic Publishers, Norwell, MA, 1989.
[67] J. L. Bohn, B. D. Esry, and C. H. Greene, Phys. Rev. A 58, 584 (1998).
[68] Y. E. Kim and A. Zubarev, J. Phys. B 33, 55 (2000).
[69] R. Zhang and C. Deng, Phys. Rev. A 47, 71 (1993).
[70] N. K. Timofeyuk, Phys. Rev. C 65, 0406306 (2002).
[71] N. K. Timofeyuk, Phys. Rev. C 69, 034336 (2004).
[72] M. De La Ripelle and J. Navarro, Ann. Phys. 123, 185 (1979).
[73] M. Fabre de la Ripelle, S. Sofianos, and R. Adam, Ann. of Phys. 316, 107(2005).
[74] P. Pieri, L. Pisani, and G. Strinati, Phys. Rev. B 72, 12506 (2005).
[75] D. Kushibe, M. Mutou, T. Morishita, S. Watanabe, and M. Matsuzawa,Phys. Rev. A 70, 63617 (2004).
[76] D. Z. Goodson, M. López-Cabrera, D. R. Herschbach, and J. D. Morgan III, J. Chem. Phys. 97, 8481 (1992).
[77] J. G. Loeser, J. H. Summerfield, A. L. Tan, and Z. Zheng, J. of Chem.
Phys. 100, 5036 (1994).
[78] M. Dunn and D. K. Watson, Ann. Phys. 251, 266 (1996).
[79] E. Fermi, Ric. Sci. 7, 13 (1936).
[80] M. Houbiers, R. Ferwerda, H. Stoof, W. McAlexander, C. Sackett,and R. Hulet, Phys. Rev. A 56, 4864 (1997).
[81] G. Bruun and K. Burnett, Phys. Rev. A 58, 2427 (1998).
[82] R. Roth and H. Feldmeier, Phys. Rev. A 64, 43603 (2001).
[83] R. D. Cowan, The Theory of Atomic Structure and Spectra, University ofCalifornia Press, Los Angeles, CA, 1981.
[84] M. Brack and R. K. Bhaduri, Semiclassical Physics, Addison-Wesley, Reading,MA, 1997.
[85] B. D. Esry and C. H. Greene, Phys. Rev. A 60, 1451 (1999).
[86] L. Vichi and S. Stringari, Phys. Rev. A 60, 4734 (1999).

167[87] B. M. Fregoso and G. Baym, Phys. Rev. A 73, 043616 (2006).
[88] D. Blume, S. T. Rittenhouse, J. V. Stecher, and C. H. Greene, Phys.
Rev. A 77, 033627 (2008).
[89] T. C. Germann, D. R. Herschbach, M. Dunn, and D. K. Watson, Phys.
Rev. Lett. 74, 658 (1995).
[90] S. Tan, eprint cond-mat/0412764 (2004).
[91] K. O’Hara, S. Hemmer, M. Gehm, S. Granade, and J. Thomas, Science
298, 2179 (2002).
[92] S. Giorgini, L. P. Pitaevskii, and S. Stringari, preprint arXiv:0706.3360(2007).
[93] F. Werner and Y. Castin, Phys. Rev. A 74, 53604 (2006).
[94] T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz, and S. Jochim,
Phys. Rev. Lett. 101, 203202 (2008).
[95] J. Huckans, J. Williams, E. Hazlett, R. Stites, and K. O’Hara, Arxiv
preprint arXiv:0810.3288 (2008).
[96] C. A. Regal and D. S. Jin, Phys. Rev. Lett. 90, 230404 (2003).
[97] M. Taglieber, A. C. Voigt, F. Henkel, S. Fray, T. W. Hänsch, and
K. Dieckmann, Phys. Rev. A 73, 11402 (2006).
[98] E. Shuryak, Phys. Rev. A 54, 3151 (1996).
[99] H. Stoof, J. Stat. Phys. 87, 1353 (1997).
[100] E. Donley, N. Claussen, S. Cornish, J. Roberts, E. Cornell, and C. Wieman, Nature 412, 295 (2001).
[101] S. Stringari, Europhys. Lett. 65, 749 (2004).
[102] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, Phys. Rev. Lett. 92, 203201 (2004).
[103] Y. Zhou, C. D. Lin, and J. Shertzer, J. Phys. B 26, 3937 (1993).
[104] C. D. Lin, Phys. Rep. 257, 1 (1995).
[105] V. Kokoouline and C. H. Greene, Phys. Rev. A 68, 012703 (2003).
[106] U. Fano, Phys. Rev. A 24, 2402 (1981).
[107] M. Fabre de la Ripelle, Few-Body Systems 14, 1 (1993).
[108] R. Szmytkowski, J. Math. Phys. 47, 063506 (2006).
[109] J. Jackson and R. Fox, Am. J. Phys. 67, 841 (1999).

168[110] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, With
Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, NY,
1965.
[111] E. Nielsen, D. V. Fedorov, A. S. Jensen, and E. Garrido, Phys. Rep. 347,
373 (2001).
[112] J. P. DIncao and B. D. Esry, Phys. Rev. Lett. 94, 213201 (2005).
[113] E. Braaten and H.-W. Hammer, Phes. Rep. 428, 259 (2006).
[114] M. Ross and G. Shaw, Ann. Phys. 13, 147 (1961).
[115] V. Efimov, Physics Letters B 33, 563 (1970).
[116] O. I. Kartavtsev and A. V. Malykh, J. Phys. B 40, 1429 (2007).
[117] N. P. Mehta, S. T. Rittenhouse, J. von Stecher, J. P. D’Incao, andC. H. Greene, eprint arXiv:0903.4145 (2009).
[118] R. Langer, Phys. Rev. 51, 669 (1937).
[119] C. H. Greene, A. R. P. Rau, and U. Fano, Phys. Rev. A 26, 2441 (1982).
[120] J. P. Burke, C. H. Greene, and J. L. Bohn, Phys. Rev. Lett. 81, 3355 (1998).
[121] J. H. Macek, Few-Body Systems 31, 241 (2002).
[122] A. Bulgac and V. Efimov, Sov. J. Nucl. Phys. 22, 153 (1976).
[123] U. Fano, Phys. Rev. 124, 1866 (1961).
[124] I. Mazumdar, A. R. P. Rau, and V. S. Bhasin, Phys. Rev. Lett. 97, 062503(2006).
[125] V. Aquilanti and S. Cavalli,
Transactions 93, 801 (1997).

Journal of the Chemical Society, Faraday[126] A. Kuppermann, J. Phys. Chem. A 101, 6368 (1997).
[127] R. Littlejohn, K. Mitchell, and V. Aquilanti, Phys. Chem. Chem. Phys.
1, 1259 (1999).
[128] H. Suno, B. D. Esry, C. H. Greene, and J. P. Burke Jr, Phys. Rev. A 65,
42725 (2002).
[129] H. Sadeghpour, Very Highly-Excited States of Two-Electron Systems, PhD
thesis, Louisiana State University., 1990.
[130] L. Landau and E. Lifshitz, Quantum Mechanics: Non-Relativistic Theory,
Butterworth-Heinemann, 1981.
[131] J. Von Stecher, C. H. Greene, and D. Blume, Phys. Rev. A 76, 53613(2007).

169[132] Private communication with J. L. Bohn.
[133] J. Levinsen and V. Gurarie, Phys. Rev. A 73, 53607 (2006).
[134] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 083201 (2004).
[135] O. I. Kartavtsev and A. V. Malykh, Phys. Rev. A 74, 042506 (2006).
[136] M. V. Berry and K. E. Mount, Rep. Prog. Phys. 35, 315 (1972).

Appendix A

Consructing Hyperspherical Harmonics

In this appendix I describe the construction of hyperspherical harmonics. The
construction method is taken from the work of Smirnov and Shitikova [5], but it isdetailed here for completeness. In Sec. 2.2 the it was shown that each nod in a hyperspherical tree has an associated hyperangular momentum quantum number. In thehyperspherical trees there are four types of node; a node with two free branches, twotypes of nod with one free branch, and a nod with no free branches. Theses nodes areillustrated in Fig. A.1. The construction of hyperspherical harmonics follows from eachof these types.

Figure A.1: The four possible hyperangular node types are shown.

A.0.0.1

Two free branches

A node with two free branches is fairly straight forward as there are no subhyperangular momenta that need to be worried about. In fat a nod with two freebranches represents a normal polar coordinate parametrization. The hyperangular vol-171
ume element for a tree with this node pick up a factor of dφ. A hyperspherical harmonic
for a tree with this node pick up a factor of1
Ym (φ) = √ eimφ ,
2π

(A.1)

with hyperangular momentum quantum number m = 0, ±1, ±2, ...

A.0.0.2

One free branch

To simplify the description I will begin by discussing a node of the type shownin Fig. A.1(b) where the right hand branch is free. A node of this type has a subhyperangular momentum quantum number, λ1 , associated with it. The hyperangularvolume element picks up a factor of sind1 −1 βdβ. A hyperspherical harmonic for a treewith this node pick up a factor of|λ |+(d −3)/2,|λ1 |+(d1 −3)/2d1
1
1
Yλλ1 (β) = Nλ|λsin|λ1 | βPλ−|λ1|
1|
(cos β) ,

(A.2)
1/2

 Γ (λ + |λ1 | + d1 − 2) (2λ + d1 − 2) (λ − |λ1 |)! 
d1


 

Nλ|λ
=
|


1
(d
−
1)
(d
−
1)
1
1
2|λ
|+d
−2
2 1 1 πΓ λ +
Γ λ+
2
2

,

where λ1 = λ, λ − 1, λ − 2, ...1, 0, and Pnγ,δ is a Jacobi polynomial.
To find the hyperspherical harmonic for a node of type Fig A.1(c) all that must
be done is to make the transformation β → π/2 − β.

A.0.0.3

No free branches

A node of the type shown in Fig. A.1(d) has two sub-hyperangular momentum
quantum numbers associated with it, λ1 and λ2 associated with sub-spaces of dimension
d1 and d2 respectively. The hyperangular momentum volume element picks up a factorof sind1 −1 α cosd2 −1 αdα. A hyperspherical harmonic for a tree with this node pick up a

172
factor of
λ +d /2−1,λ +d /2−1

d1 d2
1
1
2
2
sinλ1 α cosλ2 αP(λ−λ(cos 2α) ,(A.3)
Yλλ1 λ2 = Nλλ
1 λ2
1 −λ2 )/2





1/2
λ1 + λ2 + λ + d1 + d2 − 2
λ − λ1 − λ2
(2λ
+
d
+
d
−
1)
Γ
!
1
2


2
2
d1 d2

 ,




=
Nλλ


1 λ2
λ + λ1 − λ2 + d1
λ + λ2 − λ1 + d2
ΓΓ22

with the restriction
λ − λ1 − λ2 = even number.

Appendix B

Non-Adiabatic Corrections for the Three-Body Problem

In this Appendix, I give the P̄ matrix and the diagonal correction Qnn matrices
discussed in Chapter 5 for the multi-channel three-boson problem. The derivation issimilar to that of Kartatsev and Malykh [135, 116].I begin by considering the free space
Schrödinger equation the hyperangular channel function,~ n = 0,
Λ2 − λ2n + 2µR2 ε̄ Φ

(B.1)

~ n is the nth channel function whose components are the hyperangular behavior
where Φin each internal three-body state, ε̄ is the internal state energy offset matrix, and fornotational simplicity I have written the hyperangular eigenvalue as (νn + 2)2 = λ2n .
Next consider the matrix element between the same components of the mth and nthchannel functions.
hφmΣ | Λ2 − λ2n + 2µR2 εΣ |φnΣ i = 0.
To find the P matrix I begin by taking a hyperradial derivative of the hyperangular
Schrödinger equation away from the interaction points. Reversing the indices yields twoequations that can be combined giving
hφmΣ |φnΣ i =φmΣ Λ2 φnΣ − φnΣ Λ2 φmΣ(λ2n − λ2m )Using the definition of Λ2 and applying the generalized Bethe-Peirles boundary condition,~ n (Ω) =lim Φr(k) →0Ā(k)1 − (k)
r

!
~ k,
C
n

(B.2)

174
gives a normalization equation(4π)2hφmΣ |φnΣ i = −i
1 P P h k(k)k − C σk A(k) C k
C
A
C
0
0
0
0
n
mΣ Σ Σ nΣΣ Σ mΣ
dR k Σ
.
22(λn − λm )(B.3)

To find the P̄ matrix, I consider the derivative of Eq. B.1,


∂  2
Λ − λ2n + 2µR2 εΣ |φnΣ i = 0,
∂R

(B.4)

Considering the matrix elements yields two equations,0−λ20
n + 4µRεΣ hφmΣ |φnΣ i + φmΣ
φ0nΣΛ2 − λ2n + 2µR2 εΣ

φ0nΣ = 0,Λ2 − λ2m + 2µR2 εΣ |φmΣ i = 0.

Here the primes indicate an R derivative. Combining these two equations and solving
for hφmΣ |φ0nΣ i yields
φmΣ |φ0nΣ


2
0
2 0
−λ20
n + 4µRεΣ hφmΣ |φnΣ i + hφmΣ | Λ |φnΣ i − hφnΣ | Λ |φmΣ i=.
(λ2n − λ2m )The hφmσ | Λ2 |φ0nσ i − hφ0nσ | Λ2 |φmσ i term can be solved for by considering the boundary
condition again. Summing over the internal states yields the equation for the P̄ matrix
elements
D

2 h
E
h
i
h
ii
k
(k) ~ kk
~ m |Φ
~ 0n = − 12µR (4π)~m
~ nk · ε̄Ā(k) C
~m
Φ
C
·
ε̄
Ā
C
−
C
n
dR (λ2n − λ2m )2
X
(4π)
(k)
k0
k
−
CnΣ
AΣ0 Σ0 CmΣ022dR (λn − λm )
0

(B.5)

Σ,Σ ,k

−

(4π)

2

X

dR2 (λ2n − λ2m )Σ,Σ0 ,k

(k)

k
k
Cmσ
AΣ0 Σ0 Cnσ
0

X
(4π)2
(k)
k
k0
+
Cmσ
AΣ0 Σ0 Cnσ0dR (λ2n − λ2m )0Σ,Σ ,kThe diagonal part of the Q̄ matrix can be found through a similar, but much uglier

175
derivation which yields
D






E
2 ~0
1
1 ~ 100
1
2
00
0
0
~
~
~
~
Φn |Φn = 3 (4π)
C ·ĀCn +C ·ĀCn
3dR2 n (−λ20
3dR n
(−λ20
n + 4µRε̄)n + 4µRε̄)
(B.6)
"

#




−λ200
1
7 ~
1 ~ 10n + 4µε̄
00
~ 00
~
C ·
ĀCn +Cn ·−2 ĀCn3dR n(−λn20 + 4µRε̄)
6dR
(−λ20
+
4µRε̄)n#"
200 + 4µε̄−λ
1 ~ 00
1
7
n
00
~n −
~ ·
~
−
C ·
ĀC
C
2 ĀCn3dR2 n (−λ206dR n (−λ20n + 4µRε̄)n + 4µRε̄)
#
"



−λ200
2 ~
7 ~
λ2000
n + 4µε̄
n
~
~
+
Ā
C
Ā
C
+
C
·
C
·
n
n
n
n
2
2
3dR3
3dR2
(−λ20
(−λ20
n + 4µRε̄)n + 4µRε̄)
"
#
2


−λ200
7 ~
λ2000
2 ~
n + 4µε̄
n
0
~
~
+
Cn ·
3 ĀCn − 3dR Cn ·2 ĀCn
6dR2
(−λ20
(−λ20
n + 4µRε̄)n + 4µRε̄)
#
#
"
"


200 + 4µε̄ 2
−λ200
+
4µε̄
−λ
7 ~
7
n
n
~0
~ n0
~
−
ĀC
Cn ·
2 ĀCn − 6dR Cn ·20 + 4µRε̄)3
3dR2
(−λ20
+
4µRε̄)
(−λ
n
n




2 ~0
1
λ2000
n
~n + 2 C
~0 ·
~n
−
C
·
Ā
C
Ā
C
2
3dR3 n (−λ203dR n (−λ20n + 4µRε̄)n + 4µRε̄)
"
#)
2
−λ200
7 ~0
n + 4µε̄
~n
C ·
ĀC
+
6dR n (−λn20 + 4µRε̄)3Appendix C

N -Body Recombination Rate

In this appendix, I derive the N -body recombination rate given in Eq. 5.44 [117].
In Chapter 5 this formula was applied to three-body recombination, but it is applicable
to any loss process involving N free initial particles in a gas phase. The starting pointof the derivation is to assume a hyperspherically symmetric potential in d dimensions.
This can be thought of in terms of uncoupled λ-wave channels in direct analogy tospherically symmetric potentials in three dimensions. At very large hyperradius thesolutions to the Schrödinger equation can be written as~ ~ΨI = eik·R + f k̂, k̂ 0eikR
R(d−1)/2,(C.1)~ is a position vector in d dimensions. Alternatively the large R solution can bewhere R
written in terms of a hyperspherical harmonic expansion with a phase shift,
ΨII =

X

 h
i
Aλµ Yλµ R̂ jλd (kR) cos δλ − ndλ (kR) sin δλ ,

(C.2)

λµ

where jλd and ndλ are hyperspherical Bessel functions of the first and second kind [66],Γ (α) Jα+λ (kR),(d − 4)!! (kR)αdα = − 1,
2

jλd (kR) =177
where Jγ is a normal Bessel function. Here R̂ is used to denote the hyperangles of the~ It will be useful to have this written in the large kR limit,
vector R.

r cos kR − 2α + 2λ − 1 π
α−122kR→∞ Γ (α) 2
jλd (kR) −→.
α+1/2(d − 4)!!
π
(kR)

(C.3)

The d-dimensional plane wave can also be expanded in terms of hypersphericalBessel functions and harmonics [66] as~ ~eik·R = (d − 2)!!

  
2π d/2 X λ d∗i jλ (kR) Yλµk̂ Yλµ R̂ .
Γ (d/2)(C.4)

λ

Comparing Eqs. C.3 and C.4, the incoming planewave part of ΨII can be extracted andthe expansion coefficients Aλµ can be found. This can then be inserted into Eq. C.2 tofind the scattering amplitude:

f k̂, k̂

0




=

2π
k

(d−1)/2 X

λ −i(2α+2λ+1)/4

i e

∗
Yλµ

 
 

2iδλ
0
e
−1 .
k̂ Yλµ k̂

(C.5)

λµ



At this point, the scattering amplitude f k̂, k̂ 0 can be immediately generalized toinclude coupling between hyperradial channels, i.e.


f k̂, k̂

0




=

2π
k

(d−1)/2 X

 
 

∗
iλ e−i(2α+2λ+1)/4 Yλµk̂ Yλ0 µ0 k̂ 0 Sλµ,λ0 µ0 − δλλ0 δµµ0 .

λµ,λ0 µ0

(C.6)
From this the total elastic cross section can be extracted, given by
Z
σ=


2
f k̂, k̂ 0
dk̂.

(C.7)

This cross section is the for the incoming states with hyperspherical momentum directionk̂ 0 summed over all outgoing hyperspherical momentum directions k̂. For the crosssection in a gas phase, all of the initial momenta must be averaged over. Using theorthonormality of hyperspherical harmonics, this procedure yields
 
1 2π d−1 X
2Sλµ,λ0 µ0 − δλλ0 δµµ0 ,Ωd k
λµ,λ0 µ/Z
2π d/2Ωd = dΩ =.
Γ (d/2)σ dist =

(C.8)

178
This can immediately be interpreted as the generalized average cross section that takes
an initial state i ≡ λµ to a final state f ≡ λ0 µ0 , i.e.
dist
σi→f
=



2π
k

d−1

Γ (d/2)|Sf i − δf i |2 .
d/2
2π

(C.9)

This expression must only be summed over the initial and final states of interest. For example, when the scattering potential is purely hyperradial all µ substates are degenerate
and the sum over initial states yields a factor ofM (d, λ) =(2λ + d − 2) Γ (λ + d − 2),Γ (λ + 1) Γ (d − 1)which is exactly the degeneracy of hyperspherical harmonics for fixed λ [66]. The hyperspherically symmetric version of Eq. C.9 has been found using the Born approximation
by Fabre de la Ripelle [107]. Equation C.9 describes the behavior of distinguishable
particles in the initial state.
To specify indistinguishable particles, all possible identical configurations mustbe summed over in the scattering amplitude and only unique states included in the sum
over final states,
σ indist =

Z

dk̂
Np

Z


2
dk̂ 0
Np f k̂, k̂ 0
= Np σ dist ,Ωd(C.10)

where Np is the number of permutation symmetries in the system. For instance, forN identical particles, Np = N !, or for two species of identicle particles, Np = N↑ !N↓ !.
Using this gives the final scattering cross section for total angular momentum J andparity Π ,ΠσfJi = Np2πk(d−1)2Γ (d/2) XΠ(2J + 1) SfJi − δf i ,
d/2
2π
i

(C.11)

where I have explicitly included the angular momentum degeneracy. To turn this intoa rate coefficient, the cross section must be multiplied by the “velocity”, v = ~k/µd ,resulting in
Kd =~k J Π
σµd f i(C.12)

179
where µd is the d dimensional reduced mass of the system, and k =√2µd E/~. For N

particles in three dimensions with the center of mass motion removed, d = 3 (N − 1)Q
1/(N −1)
PN
Nyielding Eq.5.44.
and µd =
m
/
m
i
i
i=1
i=1

C.1

Scaling laws for N identical bosons
In this section, I apply the result of Eq. C.12 to a system of N identical bosons at

ultracold temperatures. Ultracold here means that only the lowest channel in the sum
in Eq. C.11 contributes to the inelastic rate coefficient. In general the lowest asymptotic
channel for N bosons has λ = 0 and J Π = 0+ . I assume that the process in question isan inelastic rate to a deeply bound channel yielding
~k
KN =
N!
2µN



2π
k

(3N −4)Γ (3 (N − 1) /2) 0+ 2
S0f .
2π 3(N −1)/2(C.13)For a thermal gas of identical bosons, the density, n, of the gas follows the equation,
N

max
X
∂n
=−
LN nN ,
∂t

(C.14)

N =2

where Nmax is the highest order recombination process that is allowed, and LN is the
N -body recombination rate scaled by the number of particles lost in the process andincluding symmetry factors.
From Eq. 2.19, the large R behavior of the lowest hyperradial potential has theform of an effective centrifugal barrier,~2 lef f (lef f + 1)
,
2µ
R2
d 3
lef f = λ + − .
2 2

U (R) =The energy normalized hyperradial wavefunction for this entrance channel at large R
for low energy behaves as F (R) ∝ k lef f +1/2 . If it is assumed that this entrance channel
dominates the behavior of the S matrix and that the only relevant length scale in the+0 ∝ (ka)lef f +1/2 , the recombination
problem is the s-wave scattering length, a, i.e. S0f

180
rate from Eq. C.13 is (3N −4)~k2πΓ (3 (N − 1) /2)
KN ∝N!
(ka)3N −5,2µNk2π 3(N −3)/2~
Γ (3 (N − 3) /2)=
N ! (2π)(3N −4) a3N −5.
2µN
2π 3(N −1)/2(C.15)

It is interesting to note that this indicates that the inelastic N -body recombination
rate for bosons is always a constant at threshold. For N = 3, this gives
K3 ∝192π 2 ~ 4
a ,2µ3(C.16)

which is the well know a4 scaling law for three boson recombination [111]. For N = 4
this gives
K4 ∝

C.2

20160π 4 7
a .
2µ4(C.17)

Four-bosons: beyond scaling laws
In this section I examine the four boson problem using the work of von Stecher[8, 65]. Specifically I will be concerned with the process of four-body recombination toan Efimov state at negative scattering lengths, i.e. K4B+B+B+B→B3 +B . This process
can be important in identifying Efimov states [65], creating Efimov states, or in the highdensity regime of an ultracold gas of bosons.
Several examples of the four-boson potentials at negative scattering length are
shown schematically in Figs. C.1(a-d) [8, 65]. These potentials are from the region
where at least one Efimov states is bound. For a full analysis of the properties of thesepotentials the reader is pointed towards Ref. [8]. The important aspect here is thateach potential corresponding to an Efimov state and a free particle supports two fourbody bound states. Further, as the scattering length becomes more negative, eventuallyanother Efimov state is formed. The potential corresponding to the new Efimov state is
the same as the previous potential scaled by the characteristic Efimov scaling, i.e. the181length scale is multiplied by a factor of 22.7 and the energy scale is divided by a factorof (22.7)2 ≈ 515.

Figure C.1: (a)-(d) Schematic of the four-boson potentials from Ref. [8] are shown, As
the scattering length becomes more negative (a-d inorder) an inner well appears and ispulled further down until a second Efimov state is bound.

I will model this process as a single channel scattering problem with a complex
scattering phase shift to simulate loss from the lowest four-free-boson potential. With
this assumption the S matrix element corresponding to the transition to the Efimov
state is given in the low energy limit by|SB+B+B+B→B3 +B |2 = 1 − |SB+B+B+B→B+B+B+B |2 ,= 4 Im δ,(C.18)

where δ is the single channel complex phase shift. Figure C.2 shows a schematic pathof this process where the transition is assumed to occur at the peak in the couplingstrength between the two potentials. It is important to note that when a new Efimov
state is formed, I assume that all recombination proceeds out the new channel, and therecombination out of the old channel is turned off. Assuming a WKB wavefunction and182
using the WKB connection formulas [136] the imaginary part of the phase shift is
4 Im δ =sinh 2ηe−2γ ,sin (φ − π/2) + sinh2 η2where φ is the WKB phase accumulated in an inner well such as the one seen in Fig
C.2, γ is the WKB tunnelling for the entire process, and η parameterizes the transition
probability to the Efimov channel. In this treatment η is kept as a free parameter, but
in truth it is fixed by the exact nature of the four-boson system. This connection will
not be discussed further here.

Figure C.2: A schematic of the four-body recombination process into an Efimov stateplus a free particle.

Inserting this into Eq. C.13 gives
K4 = A

20160π 4sinh 2ηe−2γ,
2µ4 sin2 (φ − π/2) + sinh2 η k 4(C.19)

where A is an overall scaling that is not determined by this model. This rate coefficientassumes that the recombination process is entirely controlled by the incoming channel.
Including the behavior due to the outgoing channel is a subject for later work. The
apparent k 4 behavior of this rate coefficient will be cancelled by the strong tunnelling
suppression in the asymptotic regime from the effective centrifugal barrier. Figure C.3

183shows the resulting four-body recombination rate coefficient. Here it is assumed thatthe transition occurs at the point where the coupling strength peaks. Several things
are readily apparent in this result. First the rate has an overall scaling of a7 predictedby the simple dimensional argument in Eq. C.17. Second, there is a cusp in the rate
at exactly the point where a new Efimov state becomes bound, a = aEf i ≈ −104r0 .
It is unclear if this cusp behavior is physical, due to the approximations used in thistreatment. Finally, two strong resonances can be seen. These correspond to four-bodystates appearing at the scattering threshold at a = 0.43aEf i and a = 0.9aEf i exactly as
predicted in Ref. [8]. This gives another handle on identifying Efimov states, a task thatcan be quite difficult. If such four-body resonances can be associated with a three-bodyresonance it gives strong evidence that the three-body resonance is in fact due to an
Efimov state [65].

184

Figure C.3: The four-body recombination rate is shown plotted as a function of |a| inunits of the range of the two-body interaction.

Appendix D

Dimer-Dimer Relaxation Rates

In this appendix I present the derivation of the dimer-dimer relaxation rate usedin Section 6.5. This process occurs when the two dimers collide causing at least oneof the dimers to relax to a deeply bound state. The difference of the binding energies
is then releases as kinetic energy. This process can be pictured in the hyperspherical
picture as an infinite series of very closely spaced crossings between the dimer-dimer
channel and channels consisting of a deeply bound dimer and two free particles. This
near continuum of crossings is shown schematically in Fig. D.1(a).
Using Fermi’s golden rule between the initial dimer-dimer state and the final
states gives
dd
∝
Vrel

X

|hΨdd (R; Ω) |V (R, Ω)| Ψλ (R, Ω)i|2 ,

(D.1)

λ

where Ψdd (R; Ω) is the dimer-dimer wavefunction, V (R, Ω) is the interaction potentialand Ψk is the λth deeply bound dimer state. I now will assume that the dimer-dimer
wavefunction is approximated byΨdd (R; Ω) ≈ Fdd (R) Φdd (R; Ω) ,(D.2)

where Φdd (R; Ω) is the dimer-dimer hyperangular channel function and Fdd (R) is thehyperradial wavefunction resulting from the single channel approximation. I further
assume that the outgoing deeply bound dimer wavefunction can be written as
Ψk (R, Ω) ≈ ψ (r12 ) θλ (~r34 , ~r12,34 )

(D.3)

186

Figure D.1: (a) A schematic of the channels involved involved inthe dimer-dimer relazaxtion process is shown. The dimer-dimer potential (red curve) goes through an inifinite
number of crossing with deeply bound states (green dashed curves). (b) The hyperradial
behavior of the outgoing wavefunction is shown.

where ψ (r12 ) is the wavefunction for an s-wave deeply bound dimer and θλ (~r34 , ~r12,34 )is the free space behavior of the resulting three particle system.
Examining one of the terms from the sum in Eq. D.1 with a single two-bodyinteraction givesdd(λ)
Vrel ∝dd(k)where VrelZ

2

Fdd (R) Φdd (R; Ω) V23 (r23 ) ψ (r12 ) θλ (~r34 , ~r12,34 ) dRdΩ ,(D.4)

is the contribution to the relaxation rate by the λth term in Eq. D.1. The

first thing to notice is in this is that the factor V23 (r23 ) ψ (r12 ) is non-zero only whenparticles 1, 2, and 3 are in close proximity, and when particles 1, 2, and 3 are in close

187
proximity the remaining degrees of freedom are simplified as well,~r34 ≈ C~r12,34 .

(D.5)

This means that the wavefunction θ (~r34 , ~r12,34 ) can be rewritten asθλ (~r34 , ~r12,34 ) ≈ Gλ (R) fλ (Ω) ,(D.6)

where Gλ and fλ are the hyperradial and hyperangular behavior associated with theλth outgoing channel. A further simplification can be made by realizing that fλ (Ω)
must be independent of Ω when particles 1, 2, and 3 are in close proximity because
the total wavefunction must have zero spatial angular momentum. Rewriting D.4 withthese simplifications yields
dd(λ)
Vrel ∝Z

2Z
Fdd (R) Gk (R)Φdd (R; Ω) V23 (r23 ) ψ (r12 ) dΩdR .

(D.7)

The hyperangular integral is approximated by the probability that three-particles areclose to each other in the dimer-dimer channel function.
And example of Gλ is shown in Fig. D.1(b). Away from the classical turning
point Gλ oscillates very rapidly. This fast oscillation will generally cancel out meaning
the main contribution to the hyperradial integral is from the region near the classicalturning point Rλ . Putting this all together yields
dd
Vrel
∝

X |Fdd (Rλ )|2λRλF (Rλ ) ,(D.8)

where F (Rλ ) is the probability that three particles are in close proximity in the dimerdimer channel function. The final step in this derivation is to turn the sum over λ into
an integral over Rλ ,
dd
Vrel
∝

Z
ρ (Rλ )

|Fdd (Rλ )|2
F (Rλ ) dRλ
Rλ

(D.9)

where ρ (Rλ ) is the, nearly constant, density of states. This is possible due to thenear-continuum nature of the outgoing states. By inserting the WKB approximation

188
wavefunction for Fdd (R), the result of Eq. 6.55 is obtained, i.e.
dd
Vrel
∝

Z
ρ (R)

PW KB (Rλ )F (R) dR.
κ (R) R

(D.10)

189

Appendix E

Related Publications• “Dimer-dimer Collisions at Finite Energies in Two-Component Fermi Gases,”J. P. D’Incao, S. T. Rittenhouse, N. P. Mehta and C. H. Greene, Phys. Rev. A
79, 030501 (2009).
• “Efimov States Embedded in the Three-Body Continuum,” N. P. Mehta, S. T. Rittenhouse, J. P. D’Incao and C. H. Greene, Phys. Rev. A 72, 020701 (2008).
• “The degenerate Ferm gas with density-dependent interactions in the large N
limite under the K-Harmonic approximation,” S. T. Rittenhouse and C. H. Greene,
J. Phys. B 41, 205302 (2008).
• “Stability of inhomogeneuos multicomponent Fermi gases,” D. Blume, S. T. Rittenhouse, J. von Stecher and C. H. Greene, Phys. Rev A 77, 033627 (2008).
• “Hyperspherical description of the degenerate Fermi gas: s-wave interactions,”
S. T. Rittenhouse, M. J. Cavagnero, J. von Stecher and C. H. Greene, Phys.
Rev A 75, 053624 (2006).
• “A hyperspherical variational approach tothe N-body problem,” S. T. Rittenhouse, M. J. Cavagnero, J. von Stecher and C. H. Greene, Few Body Systems
38, 85-90 (2005).