Trapped Ultracold Atoms with Tunable
Interactions
byJavier von Stecher
B.A., Universidad de Buenos Aires, 2002

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
2008

iii
von Stecher, Javier (Ph.D., Physics)
Trapped Ultracold Atoms with Tunable Interactions
Thesis directed by Prof. Chris H. Greene

In this dissertation, we analyze both many- and few-body systems under external
confinement with tunable interactions. First, we develop a density-renormalization approach for describing two-component fermionic systems with short-range interactions.
This renormalized zero-range interaction eliminates the instabilities produced by a bareFermi pseudopotential and provides a simple description of the interactions from theweakly interacting BCS region up to unitarity.
In the second part of the thesis, we focus on few-body systems in the BCS-BEC
crossover. To obtain the solutions, we implement two different numerical techniques:a correlated-Gaussian-basis-set expansion and a fixed-node diffusion Monte Carlo technique. We also develop an innovative numerical technique for obtaining solutions to thefour-body problem in the hyperspherical representation.
Our solutions provide an accurate description of few-body trapped systems. The
analysis of two-, three-, and four-body systems, for instance, provides a few-body perspective on the BCS-BEC crossover problem. The analysis of the spectrum of such
systems allows us to visualize important pathways for molecule formation. We then use
the four-body solutions to extract key properties of the system such as the dimer-dimer
scattering length and the effective range.
We also explore the qualitative change of behavior in the BCS-BEC crossoverby analyzing the spectrum and structural properties. We investigate the dynamics ofthese few-body systems and analyze them using a Landau-Zener model. At unitarity, we
study the universal properties of few-body systems and verify the absence of many-bodybound states up to N=6.

iv
Finally, we present preliminary results on the four-boson system. We analyze
the structure of the spectrum and find a family of four-body states attached to thethree-body thresholds. These four-body states follow the universal scaling propertiesof the Efimov states. We explore the collisional implications of these four-body statesand find relations between the atom-dimer and dimer-dimer collisional properties. In
particular, we predict that these four-body states will produce resonances in the dimerdimer scattering length.

Para Leila y Faustino.

vi

Acknowledgements

First, I want to thank Chris Greene, for his perseverance and confidence in thisproject. I especially want to acknowledge his patience with me at the beginning of myresearch, when it seemed fruitless. During my research with him, I always felt I could
only absorb just a small fraction of his comprehensive knowledge of atomic physics.
His passion for physics combined with his rigorous research methodology make him an
inspirational figure for my work. Most importantly, I will always be grateful to Chrisfor constantly putting my career above all other things.
During my first years of research in Chris Greene group, I was lucky to share theoffice with Josh Dunn and Stefano Tonzani. They were the most helpful colleagues aperson could wish for. They were reliable sources of knowledge when I was taking myfirst steps in computer programming. I appreciate their generous team spirit and theway in which they welcomed me to the group.
I also want to thank the various collaborators who made this work possible. Seth
Rittenhouse and Jose D’Incao were always very generous in sharing their knowledgeof few-body physics and hyperspherical methods. I especially want to thank Doerte
Blume. I had the pleasure of collaborating with her during her visit to JILA in 2007.
Not only did she teach me a lot about Monte Carlo techniques and ultracold gases ingeneral, but she also showed me how to be more effective in my research. She was always
critical (in a positive way) of my research, and that helped me grow as a physicist.
I want to thank Julie Phillips for her careful and patient editing of this disserta-

vii
tion.
I thank my parents, brothers, and sister for their constant support and company.
Finally, and foremost, I want to thank my wife, Leila, for generously understanding mypassion for my work and for living every moment of my career as if it were hers. Noneof this would have been possible without her wise advice and love.
I would like to thank the National Science Foundation for funding my research
as a graduate student.

viiiContents

Chapter
1 Introduction

1

2 Theoretical introduction

7

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

The interaction potential . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.1

Scattering properties of ultracold gases . . . . . . . . . . . . . . .

8

2.2.2

Zero-range interactions . . . . . . . . . . . . . . . . . . . . . . . .

13

2.3

Spectrum of two particles in a trap . . . . . . . . . . . . . . . . . . . . .

14

2.4

BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.5

Hyperspherical representation . . . . . . . . . . . . . . . . . . . . . . . .

20

2.5.1

23

The noninteracting limit: hyperspherical harmonics. . . . . . .

3 Numerical Techniques

26

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.2

Fixed-node diffusion Monte Carlo approach . . . . . . . . . . . . . . . .

26

3.2.1

Imaginary time propagation and diffusion Monte Carlo . . . . . .

27

3.2.2

Trial Wave functions for a Trapped Two-Component Fermi Gas .

30

3.3

Correlated Gaussian Method . . . . . . . . . . . . . . . . . . . . . . . .

36

3.4

Correlated Gaussian Hyperspherical method . . . . . . . . . . . . . . . .

40

3.4.1

42

Expansion of the channel function in a basis set . . . . . . . . . .

ix
3.4.2

Unsymmetrized Matrix Elements evaluation of three particles . .

44

3.4.3

General considerations . . . . . . . . . . . . . . . . . . . . . . . .

49

4 Renormalized Mean-Field Methods

52

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.2

Perturbative behavior of a trapped two-component Fermi gas . . . . . .

54

4.2.1

The noninteracting two-component Fermi gas in a trap

. . . . .

54

4.2.2

Effective versus exact wave function . . . . . . . . . . . . . . . .

57

4.2.3

Perturbative behavior of two-component Fermi gas . . . . . . . .

59

4.3

Density renormalization idea . . . . . . . . . . . . . . . . . . . . . . . .

61

4.4

Renormalization procedure for two trapped particles . . . . . . . . . . .

65

4.4.1

Exact Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

4.4.2

Mean-field solution . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Application to many-particle systems . . . . . . . . . . . . . . . . . . . .

70

4.5.1

Variational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.5.2

Thomas-Fermi results . . . . . . . . . . . . . . . . . . . . . . . .

77

4.5.3

Hartree-Fock method . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.6

Comparison with other approaches . . . . . . . . . . . . . . . . . . . . .

83

4.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

4.5

5 Spectrum and Dynamics of few-body trapped systems

89

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.2

Spectrum in the BCS-BEC crossover . . . . . . . . . . . . . . . . . . . .

90

5.3

Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5.3.1
5.4

Landau-Zener analysis of few-body trapped systems. . . . . . . . . . . . 104
5.4.1

5.5

The Landau-Zener model and the P-matrix analysis . . . . . . . 101

Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Comparison with experimental results. . . . . . . . . . . . . . . . . . . . 112

x

5.6

5.5.1

Molecule fraction as a function of the speed . . . . . . . . . . . . 113

5.5.2

Atom-molecule coherence . . . . . . . . . . . . . . . . . . . . . . 115

5.5.3

Conversion efficiency as a function of temperature. . . . . . . . . 117

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6 BCS-BEC Crossover of few-body trapped Systems

121

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2

Energy crossover curves for N = 3 and N = 4 . . . . . . . . . . . . . . . 123

6.3

Extraction of dimer-dimer collisional properties . . . . . . . . . . . . . . 131

6.4

A two-channel model for the BEC-BCS crossover. . . . . . . . . . . . . . 139

6.5

Structural properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.6

6.5.1

Analysis of the validity of the numerical structural properties . . 144

6.5.2

Structural properties along the BEC-BCS crossover. . . . . . . 147

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7 Unitary Gas

155

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.2

Universal properties in trapped systems . . . . . . . . . . . . . . . . . . 158

7.3

Numerical verification of universal properties for trapped systems . . . . 164

7.4

Renormalization and unitarity . . . . . . . . . . . . . . . . . . . . . . . . 173

7.5

Monte Carlo calculations for trapped systems: β parameter and the gap

7.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8 Four Bosons with large scattering length

175

184

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.2

Efimov physics in a three-body system.

8.3

Analysis of the four-boson system with the correlated Gaussian method

8.4

Analysis of the four-boson system in the hyperspherical representation . 192. . . . . . . . . . . . . . . . . . 186
187

xi
8.5

Scattering properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

8.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9 Summary and Outlook

205

Bibliography

209

Appendix
A Symmetrization of the basis functions and evaluation of the matrix elements

222

B Evaluation of unsymmetrized basis functions

225

C Jacobi vectors and CG matrices

227

C.1 Mass-scaled Jacobi vectors for three particles . . . . . . . . . . . . . . . 228
C.2 Mass-scaled Jacobi vectors for four particles . . . . . . . . . . . . . . . . 229
D Selection of the basis set

231

E Controlling Linear dependence

233

F Stochastical variational method

234

G CGHS unsymmetrized matrix elements for four particles

236

xii

TablesTable
4.1

Angular momentum L and coefficients CN1 ,N2 for the ground state of
two-component Fermi gases in the weakly attractive regime. Here, we
consider N2 = N1 for even N and N1 = N2 + 1 for odd N . . . . . . . . .

62

4.2

Exact numerical values of ζ(x). . . . . . . . . . . . . . . . . . . . . . . .

71

6.1

Dimer-dimer scattering length, add , and dimer-dimer effective range, rdd ,obtained using (a) the CG spectrum and (b) the FN-DMC energies. The
reported uncertainties reflect the uncertainties due to the fitting procedure; the potential limitations of the FN-DMC method to accurately
describe the energetically lowest-lying gaslike state, e.g., are not included
here (see Sec. IIIB of Ref. [184]). . . . . . . . . . . . . . . . . . . . . . . 137

7.1

CG energies Eνn for N = 3 − 6 (the range r0 of the Gaussian potential
is 0.01aho for N = 3 and 4, and 0.05aho for N = 5 and 6). Finite
range effects are expected to be ≈ 0.02~ω and 0.1~ω for r0 = 0.01~ω and0.05~ω, respectively. Results from Ref. [19]. . . . . . . . . . . . . . . . . 167

7.2

Coefficients sν of the hyperradial potential curves Vsν (R), Eq. (7.18), for
the N = 4 system with L = 0 for various mass ratios κ. Results are from
Refs. [185]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.3

Experimental and theoretical predictions of β. . . . . . . . . . . . . . . . 174

xiii
8.1

Lowest three- and four-body energies at unitarity. All the energies arein units of ~2 /(mr02 ). The three-body energies (left column) where calculated with CG without a trapping potential. The N = 4 energies were
calculated with CG (center column) and CGHS in the adiabatic approximation (right column). The CG provides exact results for true boundstate (E13b , E23b , E33b , E14b , and E24b ) but it is only approximated for quasibound states (E34b and E44b ). The adiabatic approximation provides arigorous upper bound of the exact energies. Therefore, the most reliable
E34b and E44b are obtained with the CGHS in the adiabatic approximation. 198

xiv

FiguresFigure2.1

Behavior of the scattering length and the zero-energy wave function asfunctions of the potential depth. In panels (a) and (b), the red curve
corresponds to the potential interaction. The black curve is the zeroenergy u0 (r), and the dashed blue line represents the extrapolated longrange behavior of u0 (r) [Eq. 2.4].(c) Schematic behavior of the scattering
length as a function of the potential depth. The solid blue line is thescattering length, and the dashed green line is the binding energy of the
weakly bound dimer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Spectrum of the relative motion of two particles in a trap as a functionof the inverse scattering length aho /as .

4.1. . . . . . . . . . . . . . . . . .

16Orbital energies ²nl as a function of the quantum number l. The numberof particles in a closed shell Ncs is written next to each shell. . . . . . .

4.2

12

56

CN1 ,N2 coefficients divided by EN I as a function of N . Circles correspondto L = 0 ground states, squares to L = 1 ground states and triangles to
L = 2 ground states. A solid line connects the odd-N values, while adashed line connects the even-N values. Figure from Ref. [185]. . . . . .

4.3

60

Spectrum Erel as a function of as . The dashed blue line corresponds tothe energy branch selected for the renormalization procedure developedin this study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

xv
4.4

Effective scattering length ζ(k f as ) (circles) and its analytical approximation ζ0 (k f as ) (solid line). Figure from Ref. [183]. . . . . . . . . . . . . .

4.5

Fractional error in our analytical approximation to the numerical renormalization function, ζ0 (k f as ). Figure from Ref. [183]. . . . . . . . . . . .

4.6

70

71

Ratio of the total energy to the noninteracting energy for a sphericallytrapped two-component degenerate Fermi gas in the large N limit. The
circles correspond to HF calculations for 2280 particles using ζ(kf as ),while the solid line corresponds to either the variational solution, Eq. (4.33),
or the TF solution, Eq. (4.45) using the approximate renormalization
function ζ0 (kf as ); note that the curves are indistinguishable on the scaleof the figure. Figure from Ref. [183]. . . . . . . . . . . . . . . . . . . . .

4.7

73

(a) Variational energy obtained with the Fermi pseudopotential prediction. Black curve is the noninteracting energy. Blue curve is the prediction for kf0 as = 1.5. The red and green curves correspond to the negative
scattering length side with kf0 as = 0.5 and kf0 as = −1.5, respectively.
(b)Variational energy as a function of λ for the renormalized interaction,in units of the noninteracting total energy. The solid curve corresponds
to kf0 as = 0; the dashed curve corresponds to kf0 as = −∞; the dotteddashed curve corresponds to kf0 as = ∞. The minimum of the energyfunctional for kf0 as = −∞ occurs at λ = 0.844, which represents theratio between the cloud radius at unitarity and the noninteracting cloud
radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

76

Chemical potential in units of the noninteracting Fermi energy. The blacksolid line is the prediction obtained via the renormalization function. The
black circles represent the BCS prediction, and the blue dashed curve isthe prediction obtained in Ref. [43]. Figure from Ref. [183]. . . . . . . .

79

xvi
4.9

The speed of sound as a function of kf as shown in units of the Fermi
velocity vf = ~kf /m for a uniform two-component Fermi gas. Figure
from Ref. [183]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.10 Total energy of eight fermions in a trap shown in oscillator units as afunction of kf0 as . FN-DMC results are shown in open red circles, while thesolid blue line corresponds to HF results using the effective renormalized
scattering length. The dashed and dash-dotted curves correspond tosolutions using first-order or first- and second-order terms in an expansion
into powers of kf as . See the discussion of Eqs. (4.58, 4.59). Figure from
Ref. [183]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.11 E/EN I for an infinite homogeneous Fermi gas kf as in the mean-fieldapproach (solid line). The dashed curve corresponds to the local density
BCS solution, and the circles correspond to FN-DMC results obtained inRef. [7]. Figure from Ref. [183]. . . . . . . . . . . . . . . . . . . . . . . .
5.1

86

Spectrum of two particles with J = 0 in a trap obtained with CG method.
The black curve corresponds to the ground state that evolves in a dimer
configuration. Red curves correspond to excited states that evolve in a
two free-atom configuration. . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

91

Energy spectrum as a function of λ in the unitarity region for N = 4 with
J = 0. The thin solid black lines correspond to the adiabatic spectrum.
The wide black line with circles is the diabatic ground state labeled Ψ1 .
The blue curve with circles is the diabatic first-excited state labeled Ψ2 ,the wide red curve with circles is the diabatic state Ψ5 , and the wide
green curve with circles is the diabatic state Ψ13 . . . . . . . . . . . . . .

92

xvii
5.3

Energy spectrum of L = 0 three-particle states. The black line withcircles shows the diabatic ground state. The blue curves with circles
show the diabatic excited states, which represent a bound dimer plus an
extra particle. The red curves with circles show the diabatic states where
no bound dimer is formed. . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

94

Energy spectrum for four particles with J = 0 in the crossover region(lowest 20 diabatic states). The black curve corresponds to the groundstate. The blue curves are the states that go diabatically to excited
dimer-dimer configurations. The red curves correspond to states that go
diabatically to configurations of a dimer plus two free atoms, and thegreen curves correspond to states that go adiabatically to configurationsof four free atoms. The lowest green curve is the atomic ground state on
the BEC side of the resonance. Results from Ref. [182]. . . . . . . . . .

5.5

96

Hyperspherical potential curves in the BCS-BEC crossover for N = 4
with J = 0. (a) Potential curves in the BCS regime, as ∼ −0.3aho . (b)Potential curves in the BEC regime, as ∼ 0.3aho . . . . . . . . . . . . . .

5.6

96

Probabilities of the N = 4 system during a unidirectional ramp at constant speed from the BCS to the BEC side. The initial configuration is|ΨBCS
i. The black curve corresponds to pg (t). The blue curve corre1
sponds to pdd (t). The red curve corresponds to pd2a (t), and the green
curve corresponds to p4a (t). (a) Probabilities in a ramping at a speedof χ ≈ 13. (b) Probabilities in a ramping at a speed of χ ≈ 52. (c)Probabilities in a ramping at a speed of χ ≈ 128 . . . . . . . . . . . . . 100

5.7

Schematic description of the Landau-Zener model. Solid black and red
curves correspond to the nonadiabatic states ψ1 and ψ1 , respectively.
Dashed blue and green curves correspond to the adiabatic states ψ− andψ+ , respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xviii
5.8

The most important P matrices involved in a ramping that starts at the
BCS side with all the probability in the ground state. The black curve
represents P1−2 , the blue P1−5 , the red P2−5 , and the green P5−13 . . . . 107

5.9

Numerical results and Landau-Zener model for N = 2.(a) circles correspond to P12 obtained numerically. The solid line corresponds to the bestf it
fit with P12. (b) Transition probabilities as a function of the speed ξ.

Black symbols represents evolution in the dimer family (ground-state).
Red symbols represents evolution in the two-atom family (red energy
curves in Fig. 5.1). Black and red lines correspond to p1 and p2 obtained
from the Landau-Zener model. . . . . . . . . . . . . . . . . . . . . . . . 107
5.10 Probability of evolving in a given configuration as a function of the dimensionless speed parameter χ. The symbols correspond to the numerical evolution, while the curves correspond to the Landau-Zener theory.
Black curve and symbols correspond to the ground state. Blue curve andsymbols corresponds to the excited atom-dimer configuration, and red
curve and symbols corresponds to the three-atom configuration. . . . . . 109
5.11 (a) Energy in the BCS-BEC crossover of the important states in theLandau-Zener approximation. The black curve corresponds to |Ψ1 i which
represents the ground state configuration. The blue curve corresponds
to |Ψ2 i which represents the excited dimer-dimer configuration. The red
curve corresponds to |Ψ5 i which represents the configuration of a dimer
plus two atoms. The green curve corresponds to |Ψ13 i which represents
the four-atom configuration. (b) Probability of evolving in a given configuration as a function of the dimensionless speed parameter χ. The
symbols correspond to the numerical evolution, while the curves correspond to the Landau-Zener theory. The colors follow the same conventionof Figure (a). Results from Ref. [182]. . . . . . . . . . . . . . . . . . . . 111

xix
5.12 Molecular fraction as a function of χ. The dashed red curve is the N = 2
prediction, and the black solid line is the N = 4 prediction. . . . . . . . 114
5.13 (upper panel) Molecule fraction as a function of the delay ∆t in theramp scheme. (lower panel) Fourier transform of the upper panel figure.
The peaks of the spectrum correspond to the most important energy
transitions. Figure from Ref. [182].

. . . . . . . . . . . . . . . . . . . . 116

5.14 Molecule fraction as a function of T /TF . The black circles are the experimental results showing the maximum molecule fraction as a functionof the temperature [90]. The blue and curves are the theoretical predictions from two particles in a trap using TF2b and TFT F , respectively.
Experimental data courtesy of Cindy Regal. . . . . . . . . . . . . . . . . 117
6.1

(κ)

Energy crossover curve Λ4(2µ)

as a function of aho /as for (a) κ = 1 and(b) κ = 8. Solid lines are calculated by the CG approach, and circles
and crosses by the FN-DMC method using ψN =4 and ψP , respectively.
Figure from Ref. [184].
6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Normalized energy (E − E(1, 1) − 3~ω/2)/~ω for N = 3 as a functionof aho /as calculated by the CG approach (lines). E denotes the threebody energy for L = 0 (solid lines) and for L = 1 (dashed lines). (a)Equal-mass atoms [κ = 1, E = E(2, 1) = E(1, 2)]. (b) Two heavy atoms
and one light atom [κ = 4, E = E(2, 1)]. (c) Two light atoms and one
heavy atom [κ = 4, E = E(1, 2)]. The normalized energy crossover curve(κ)Λ3 , Eq. (6.3), coincides with the dashed and solid lines, respectively,
depending on whether the three-particle ground state has L = 1 or 0.
In the CG calculations, the range r0 of the two-body potential is fixed
at 0.01aho . For comparison, crosses and circles show selected FN-DMC
energies for L = 0 and L = 1, respectively. Figure from Ref. [185]. . . . 127

xx
6.3

Limiting behavior of the ground-state energy for N = 3 equal mass
fermions. (a) Energy correction ∆E = E(2, 1) − E(1, 1) − 3~ω/2 on
the BEC side. Circles show the CG results, while the solid line shows
the first order correction for aad ≈ 1.2as . (b) Energy E(2, 1) on the BCS
side. Circles show the CG results while the solid line shows the first-ordercorrection on the BCS side. Figure from Ref. [185]. . . . . . . . . . . . . 129

6.4

Excitation gap ∆(N ) for N = 3 as a function of aho /as calculated by
the CG approach for κ = 1 (solid line) and κ = 4 (dashed line). Circles
present the BEC limiting behavior 3~ω/2−E(1, 1)/2 which is independentof κ. The inset shows a blow-up of the region where ∆(3) is smallest;in this region, the dependence of ∆(3) on κ is most pronounced. The
dash-dotted line shows the limiting behavior for κ = 1 obtained by approximating the E(N ) in Eq. (6.6) by their perturbative values, Eq. (6.4).
Figure from Ref. [185].

6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Four-body energies of the three energetically lowest-lying dimer-dimer(M )states as a function of as /aho for κ = 8. Panel (a) shows the energeticallylowest lying energy level (i = 0), panel (b) the energetically second lowest(i = 1) and panel (c) the energetically third lowest state (i = 2). Circles
and crosses show our CG and FN-DMC results, respectively. Solid lines
show the zero-range model results. Figure from Ref. [184]. . . . . . . . . 134
6.6

Circles and crosses show add /as as a function of κ extracted from thefour-fermion CG and FN-DMC energies, respectively. For comparison,a solid line shows the results from Fig. 3 of Ref. [140]. Diamonds andsquares show rdd /as extracted from the four-fermion CG and FN-DMC
energies, respectively. Figure from Ref. [184]. . . . . . . . . . . . . . . . 136

xxi
6.7

Hyperspherical potential curves Uν for a scattering length as = 100r0 .
The dashed red curve is the dimer threshold and the dashed blue curveis the dimer-dimer threshold. Therefore, the blue curve is a dimer-dimer
potential curve and the red curves are dimer–two-atom potential curves.
Dashed-black curves are predictions from Ref. [56]. . . . . . . . . . . . . 138

6.8

Comparison of exact calculations with the two-channel model. Solid blacklines correspond to the noninteracting energy and the pair-state energy,
respectively. Blue dashed curves correspond to the two-channel model
solutions. Red and green lines with circles correspond to the exact CG
results for the ground state and the lowest atomic state in the BEC side,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.9

Pair-distribution functions P12 (r), multiplied by r2 , at unitarity for equalmass Fermi systems with (a) N = 3 (L = 1) and (b) N = 4 (L = 0) atoms
calculated by the CG method (solid lines) and by the FN-DMC method(circles). The agreement is excellent. Figure from Ref. [185].

. . . . . . 145

6.10 Pair-distribution functions P12 (r), multiplied by r2 , for equal-mass–twocomponent Fermi systems with N = 2 (dashed lines) and N = 4 (solidlines). The N = 2 pair-correlation function has an arbitrary norm selected to match the N = 4 pair correlation in the small r regime. . . . . 146
6.11 Pair-distribution functions P12 (r), multiplied by r2 , for equal-mass–twocomponent Fermi systems with N = 3 and L = 0 (dashed lines), N = 3
and L = 1 (dash-dotted lines), and N = 4 and L = 0 (solid lines)
obtained by the CG approach for three different scattering lengths, as :(a) as = −aho (BCS regime), (b) 1/as = 0 (unitarity), and (c) as = 0.1aho(BEC regime). The pair distribution function for N = 4 and as = 0.1aho[solid line in panel (c)] is shown in more detail in Fig. 6.12. Figure from
Ref. [185].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xxii
6.12 (a) Circles show the pair-distribution function, P12 (r), multiplied by r2 ,for as = 0.1aho (BEC regime) calculated by the CG approach for N = 4
and κ = 1 [note, this quantity is also shown by a solid line in Fig. 6.11(c)].
For comparison, the blue dash-dotted line shows P12 (r)r2 for two atoms
of mass m with the same scattering length but normalized to 1/2, thered dashed line shows P12 (r)r2 for two trapped bosonic molecules of mass2 m interacting through a repulsive effective potential with add = 0.6as ,and the green dotted line shows P12 (r)r2 for two trapped noninteracting
bosonic molecules of mass 2 m. Panel (b) shows a blow-up of the small r
region. Figure from Ref. [185]. . . . . . . . . . . . . . . . . . . . . . . . 149
6.13 Pair-distribution function P12 (r), multiplied by r2 , for two-component
Fermi gases with κ = 4 for different scattering lengths as : (a) as =−aho (BCS regime), (b) 1/as = 0 (unitarity), and (c) as = 0.1aho (BEC
regime). Dashed and dash-dotted lines show P12 (r)r2 for N = 3 (twoheavy particles) with L = 0 and 1, respectively. Circles and squares show
P12 (r)r2 for N = 3 (two light particles) with L = 0 and 1, respectively.
Solid lines show P12 (r)r2 for N = 4 with L = 0. Figure from Ref. [185].

150

6.14 One-body densities ρ1 (r) and ρ2 (r) for N = 4 and (a) κ = 1 and (b) κ = 4
for different scattering lengths as [for κ = 1, ρ1 (r) and ρ2 (r) coincide andonly ρ2 (r) is shown]: Circles and solid lines show ρ1 (r) and ρ2 (r) for
as = 0, squares and dashed lines show ρ1 (r) and ρ2 (r) for 1/as = 0,
and triangles and dash-dotted lines show ρ1 (r) and ρ2 (r) for as = 0.1aho .
Note, ρ2 (r) for κ = 4 and as = 0 [solid line in panel (b)] is multiplied bya factor of three to enhance the visibility. Figure from Ref. [185]. . . . . 151

xxiii6.15 Ground and excited dimer-dimer states for as = 0.1aho .(a) Pair-distributionfunction 4πr2 P1 1(r) for the lowest dimer-dimer states (b) Lowest twobody solutions, 4πr2 φn0 (r), of a noninteracting trapped system of bosonicparticles of mass 2 m.
7.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Dash-dotted lines show V (R) as a function of R/aho for (a) N = 4 with
ν = 0, (b) N = 4 with ν = 1, and (c) N = 6 and ν = 0. Circles
2 determined numerically andand solid lines show the corresponding Fνnrel .
analytically, respectively. Dashed horizontal lines show the energies EνnResults are from Refs. [185, 19]. . . . . . . . . . . . . . . . . . . . . . . . 168
7.2

Four-body energy spectrum for L = 0 at unitarity as a function of κ.
Circles correspond to the numerical results obtained by the CG approach.
Solid, dashed and dash-dotted lines show the energies Eν0 + 2n~ω forν = 0, 1 and 2, respectively (n = 0, 1, · · · ). Results are from Ref. [185]. . 170

7.3

Hyperspherical potential curves for the 4-fermion system multiplied by
2µR2 /~2 . The solid lines represent the predictions from analyzing thespectrum obtained with the CG method. The symbols correspond todirect evaluation of the potential curves with the CGHS method. Unpublished results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.4

Circles and crosses show the fixed-node diffusion Monte Carlo (FN-DMC)
energies E(N ) in units of ~ω as a function of N at unitarity. Circles correspond to even-N systems and crosses correspond to odd-N systems. Solidlines show a fit of the even FN-DMC energies to Eq. (7.34). Numericalresults are from Refs. [184, 19]. . . . . . . . . . . . . . . . . . . . . . . . 176

7.5

Excitation spectrum for N = 3, 4 and 5 at unitarity. Black circles correspond to L = 0 states and red circles correspond to L = 1 states. . . . . 178

xxiv
7.6

Circles show the excitation gap ∆(N ) determined from our FN-DMC
energies. A solid line shows ∆LDA (N ). For comparison, triangles show∆(N ) determined from the DFT energies [34]. Figure taken from Ref. [19].180

7.7

Hyperradial potential curves V (R) for equal-mass two-component Fermi
systems with (a) vanishing interactions and (b) infinitely strong interactions as a function of R. The hyperradial potential curves naturally
appear ordered as N increases: Solid lines correspond, from bottom totop, to N = 4 − 20 (N even), while dashed lines correspond, from bottom
to top, to N = 3 − 19 (N odd). Figure taken from Ref. [185]. . . . . . . 182

7.8

Normalized coefficients C̄N , Eq. (7.25) with EN I replaced by EN I,ET F , asa function of N ; values for even N are shown by circles and values for odd
N by crosses. The dash-dotted line shows the value ξ = 0.42 obtained
by FN-DMC calculations for the homogeneous system [7, 37], while adashed curve shows the value ξ = 0.508 obtained with a renormalization
procedure [183]. The inset shows the same quantities as a function of
1/N instead of N . Figure taken from Ref. [185].

8.1. . . . . . . . . . . . . 182

Schematic hyperradial potential curve for three identical bosons having
as < 0. The dashed red line corresponds to a quasi-bound Efimov state
above threshold. As the attraction increases, the state becomes bound.
The blue line corresponds to a bound Efimov state. . . . . . . . . . . . . 187

8.2

Spectrum of the four-boson system in a trap. The use of the function
F (x) = sgn(x) ln(1 + |x|) allows to visualize effects at different orders ofmagnitude in a single graph. Blue lines correspond to four-body states.
Black lines correspond to trimer+atom thresholds, and red lines correspond to dimer+atom+atom (upper) and dimer+dimer (lower) thresholds.190

xxv
8.3

Lowest pair-correlation functions of the four-boson system at unitarity.
The black solid line corresponds to the lowest four-body state; the dashed
red curve is the second four-body state, and the dash-dotted curve is thelowest 3+1 state. The lowest 3+1 state has a double-peak structure witha second peak around aho = 104 r0 that does not appear in the figure. . . 191

8.4

Rescaled pair-correlation functions of the 3rd and 4th four-body states.
The pair correlation for |Ψ3 i has not been rescaled, but the |Ψ4 i paircorrelation function is rescaled by eπ/s0 ≈ 22.7. . . . . . . . . . . . . . . 192

8.5

Potential curves in the scattering-length region near a1f ∼ −6r0 , where
the first Efimov state appears. (a) Potential curves at as > a1f [as ≈−5.31] . The three-body bound state has not appear yet. (b) Potential
curves at as & a1f [as ≈ −6.16]. For these scattering length values aquasi-bound state appears. (c) Potential curves at as < a1f [as ≈ −7.29].
For these scattering length values the attraction increases and the threebody state becomes bound. . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.6

Lowest potential curves when three-body states are at threshold. Black
curve correspond to the lowest potential curve at the scattering length
value for which the first Efimov state is at threshold. The red curveis the second potential curve at the scattering length value for which
the second potential Efimov state is at threshold. Green dashed lines
present the linear behavior of the curves (note that the potential curves
are multiplied by 2µR2 and that the x-axis is in logarithmic scale). The
linear behavior of the curves in logarithmic scale is evidence of Eq. 8.2.

196

xxvi
8.7

Lowest three rescaled potential curves Uν are shown as functions of the
four-body hyperradius. The black curve corresponds to the lowest curve.
The blue curve is the second potential curve and the red curve is thethird potential curve. The agreement between the rescaled second andthird potential curves is numerical evidence for the universal behavior of
the four-boson system, as is discussed. . . . . . . . . . . . . . . . . . . . 196

8.8

Potential curves at positive scattering-length values. (a) Potential curves
at as ∼ 37r0 . (b) Potential curves at as ∼ 21r0 . Solid blue line corresponds to the atom-trimer threshold. Dashed-red line corresponds to thedimer-dimer threshold, and dashed-green line corresponds to the dimer–
two-atom threshold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

8.9

Real and imaginary parts of the dimer-trimer scattering length. Left andright panels present the real and imaginary parts of the dimer-trimer
scattering length: ardd and aidd , respectively. The vertical dashed lines
correspond to atrim−dd and atet−dd . Figure courtesy of Jose P. D’Incao. . 203

8.10 Real and imaginary parts of the atom-trimer scattering length. Left andright panels present the real and imaginary parts of the atom-trimer
scattering length: arat and aiat , respectively. The vertical dashed lines
correspond to atrim−dd and atet−dd . Figure courtesy of Jose P. D’Incao. . 204
C.1 Mass-scaled Jacobi vector for three and four particles. . . . . . . . . . . 228

Chapter 1
Introduction

Superconductivity is a striking phenomenon discovered in 1911 by Heike Kamerlingh Onnes [132]. Superconducting materials at very low temperatures are characterized by zero electrical resistance and the expulsion of the magnetic field from theirinterior. Since the discovery of superconductivity, important efforts have been devoted
to the theoretical understanding the of mechanism that leads to it. It took more than
40 years until physicists arrived at a solid understanding of this mechanism. During the1950s, condensed matter physicists developed the phenomenological Ginzburg-Landau
theory [112] and the microscopic Bardeen-Cooper-Schrieffer (BCS) theory [10]. These
theories form the basis of our current theoretical understanding of superconductivity
and superfluidity. Superfluids are neutral gases with analogous properties to superconductors, i.e., superfluids are irrotational gases with zero viscosity.
The BCS theory in particular was a great triumph since it was able to explain
the microscopic quantum mechanical mechanism that leads to superconductivity. BCS
theory views superconductivity as a macroscopic quantum mechanical effect in which
the interacting particles feel an effective attraction that leads them to form pairs. Thesepairs, usually called Cooper pairs [47], would not bind in free space. They are boundby the influence of the medium, and their binding energy depends on the density of thesystem. This binding energy produces a gap in the single-particle excitation spectrum
that is the key for the understanding of many superconducting properties.

2
An interesting question arises if we consider what would happen if the attractionin the superconducting or superfluid system is changed. If the attraction increases
the interacting particles could be bound, possibly even in free space, and the system
would evolve into a system of diatomic molecules. This scenario was first considered byEagles [59] and Leggett [113] who realized that the BCS theory could be extended toany interaction strength; furthermore, it would predict a smooth transition between thesystem of particles and the system of molecules. These seminal studies were followedby Nozieres and Schmitt-Rink [129] and Sá de Melo, Randeria, and Engelbrecht [155]
who also considered temperature effects on these systems.
This transition from an atomic to a molecular system entails a qualitative change
in the system. The constituent particles of the superconducting system described by
BCS theory are fermions. However, when two fermions are bound, they form a bosonicmolecule. An ultracold system of bosons is described by Bose-Einstein condensation(BEC). Therefore, the transition between interacting particles described by BCS theory
and the system of molecular bosons described by the BEC theory is usually called the
BCS-BEC crossover. Fermions and bosons are characterized by completely different statistical behaviors. Therefore, the BCS-BEC crossover describes the transition between
two different statistical regimes. For that reason, the BCS-BEC crossover problem istheoretically both very appealing and very challenging.
For starters, there are no condensed-matter or nuclear systems in nature thatallow the experimental study of the BCS-BEC crossover. An ideal system that would
allow the experimental realization of the BCS-BEC crossover should be cold enough toreach quantum degeneracy and have interactions that can be externally controlled.
The experimental achievements of BEC [5, 30, 50] and Fermi degeneracy [53, 177]
in ultracold gases opened the door to the analysis of superfluidity in a controllable
manner. In particular, Fermi gases with weak attractive interactions should be describedby BCS theory and exhibit superfluidity. Since fermions of the same component repel

3each other because of Pauli repulsion, it is necessary to create and cool a two-component
Fermi gas to achieve attractive interactions, as it was done in Ref. [53].
In these first experiments, the interactions between fermions were not strong
enough to attain superfluidity. It was necessary to find a way to change the interactions of the system. It was soon realized that interactions in cold atoms could be
experimentally tuned using Fano-Feshbach resonances [69, 70, 74]. In the vicinity of a
Fano-Feshbach resonance, the scattering length, i.e., the parameter that characterizes
the interaction strength, depends strongly on external magnetic field. Therefore, by
making small changes in the external magnetic field, we can widely tune the interaction and even change its sign. Fano-Feshbach resonances were first utilized to tune theinteractions in a bosonic gas [96, 48], but strongly interacting regimes were also soon
obtained in fermionic systems using a Fano-Feshbach resonance [130, 24].
After all this experimental progress, the condensation of atom pairs in a twocomponent Fermi gas was observed. The first observations were in the BEC region [82,99, 194, 25], where the pairs formed real molecules. Next, pairs were also condensed
in the BCS region [144, 195]. After these encouraging results, experimentalist tried toverify the superfluid nature of the system. Measurement of collective excitations [12,104] and the pairing gap [44] were consistent with the superfluidity ansatz. However,those predictions can also be attributed to a normal state. The experimental verificationof superfluidity in a Fermi gas in the BCS-BEC crossover directly probed the irrotational
nature of the fluid by rotating the system and observing quantized vortices [193].
In another remarkable experimental accomplishment, cold atoms were loaded in
optical lattices [81], i.e., strong periodic potentials. These optical lattices provide new
ways to control the system such as changing the distance between the minima of theperiodic potential (usually called the lattice sites), or changing the dimensionality ofthe system, or controlling the number of atoms in each lattice site. Most experiments
focussed on the experimental realization of the Hubbard Hamiltonian [81]. However,4
optical lattices were also used to measure properties of a two-body trapped system [168].
The theoretical exploration of the BCS-BEC crossover has also significantly increased in the last few years. In contrast with dilute bosonic systems, for which theGross-Pitaevskii equation provides a highly successful many-body theory, fermionic systems in the BCS-BEC crossover lack a comprehensive many-body theory. Therefore, it
has been necessary to resort to numerical or approximate methods. For instance, an
extension of the BCS theory proposed by Eagles [59] and Leggett [113] is only qualitatively correct and has been consistently used as a starting point for most sophisticatedanalytical models. One of the most reliable descriptions of this system was obtained
through Monte-Carlo techniques [36, 7]. The Monte-Carlo results are generally used
to test theories or models as well as experimental results. In addition, there is an impressive amount of theoretical work on the BCS-BEC crossover problem that cannot
be covered in this introduction; however this work will be partially explored in thisdissertation.
Our approach to the BCS-BEC crossover problem is unique, since it tries to avoid
conventional many-body approaches and focusses on the few-body physics. In contrast
with many-body theories, which usually focus only on the ground state and have to rely
on approximations, few-body calculations can describe the complete spectrum without
resorting to approximations. From a few-body perspective, the solution of the fourfermion problem represents the next big challenge -a challenge we partially meet in thisdissertation. Although the three-body problem might be considered as largely “solved”
by existing numerical techniques, only specific and limited solutions to the four-bodysystem exist. Furthermore, the four-body problem is the smallest system that can
describe most of the physics of the BCS-BEC crossover, from a gas of attractive fermions
to a system of two interacting bound boson pairs. Here, we present such a solution.
However, we do not limit ourselves to the four-body system. We extend our
calculations to smaller and larger systems, in the process providing a comprehensive

5
analysis of few-body trapped systems. Our results are organized this thesis as follows:
Chapter 2 presents the theoretical elements that are particularly relevant to thisdissertation. The basic elements of ultracold scattering theory are introduced. The
concept of a zero-range pseudopotential is discussed and applied to the description of
two particles in a trap. A brief description of BCS theory is presented. Finally, thehyperspherical representation is presented.
Chapter 3 develops three different numerical techniques used throughout this dissertation. Quantum Monte Carlo techniques are described at the beginning of the chapter. Special attention is devoted to the qualitative understanding of the method and theproper selection of trial wave functions that describe fermionic systems in the BCS-BEC
crossover. Then, the correlated Gaussian (CG) method is discussed. The general ideas
and implementation of the CG method are presented here and many of the details areincluded in a set of Appendices. Finally, the implementation of the correlated-Gaussian
hyperspherical (CGHS) method is presented. This method combines the ideas of thestandard CG method with the hyperspherical method. This method was developed near
the end of my thesis research and the results have not yet been published.
Chapter 4 presents our initial attempts to describe two-component Fermi gases in
the BCS region. Analysis of the perturbative regime in an open-shell system is described.
Then, a density renormalization technique is presented. Finally, different mean-field
methods – namely Thomas-Fermi, variational-trial wavefunction, and Hartree-Fock withbare and renormalized interactions – are used to analyze the behavior of trapped Fermi
gases. The work presented in this chapter has been published in Refs. [148, 149, 183].
Chapter 5 analyzes the topology of the spectrum and dynamics of few-bodytrapped systems (N ≤ 4). The spectrum of these few-body systems is analyzed throughout the BCS-BEC crossover region. Then, the dynamics of these systems are probed bytuning the scattering length throughout the BCS-BEC crossover region. The numericalresults are analyzed with the Landau-Zener theory and compared with experimental

6
results. The results for the four-body system presented here have been published inRef. [182], and the rest of the results remain unpublished.
Chapter 6 analyzes more quantitatively the spectrum of few-body trapped systems. The CG results are compared with Quantum Monte Carlo predictions. Fromthese calculations, the dimer-dimer scattering length and effective range are obtained.
Finally, structural properties in the BCS-BEC crossover are analyzed. The studies
presented here have been published in Refs. [184, 185].
Chapter 7 focusses on the properties of trapped fermionic systems at unitarity.
The theoretical predictions of the properties at unitarity are reviewed. These propertiesare verified using different numerical methods. The numerical calculations are extended
to larger trapped systems using Quantum Monte Carlo methods, and the results areanalyzed using a local density approximation. The excitation gap and the universal
parameter β is extracted. This work has been published in Refs. [19, 185].
Finally, in the last chapter, Chapter 8, we shift gears and present preliminary results on the four-boson system. We analyze the spectrum as a function of the scattering
length. Then we consider the collisional aspects of the four-boson system: dimer-dimer
collisions and atom-trimer collisions. This work was performed in collaboration with J.
P. D’Incao and Chris H. Greene and has not been published yet.

Chapter 2Theoretical introduction

2.1

Introduction
In this chapter, we review important theoretical elements used in this dissertation.

First, we review important aspects of ultracold collisions, the definition of scattering
length, and the selection of the appropriate interaction potential for numerical calculations. Second, we analyze the solution of two particles in a trap interacting through azero-range pseudopotential. This solution represents an ideal case because of its simplicity and the way in which scattering-length dependence is introduced. We will use
the two-particle solution extensively throughout the thesis.
Third, we present mean-field BCS theory in Sec. 2.4. The BCS theory is the standard theory for describing the BCS-BEC crossover from a many-body viewpoint. For
that reason, it is important to consider it here and review its strengths and limitations.
Finally, we introduce the hyperspherical framework. The hyperspherical framework
becomes very useful once some intuition is built around it. A simple analysis of hyperspherical potential curves usually provides a qualitative understanding of the system.
Also, an exact solution can be obtained by solving coupled differential equations involving the hyperspherical potential curves. In additions, properties of strongly interacting
trapped system can be easily understood from the hyperspherical perspective. The
hypersherical representation will be used throughout this dissertation.

8

2.2

The interaction potential
In this section, we analyze the collisional properties of ultracold gases with s-wave

interactions. We focus on the single-channel model that is used throughout this thesis.

2.2.1

Scattering properties of ultracold gases

The scattering properties of ultracold alkali atoms can be understood considering
the simple case of two particles interacting with a central potential V (r). Since therelative coordinate Schrödinger equation has spherical symmetry, the wave function canbe written as
ψk (r) =uk (r)Ylm (Ω).
r(2.1)

Here, k is the momentum of the collision, and Ylm (Ω) are the well-known sphericalharmonics. Using Eq. 2.1, the radial Schrödinger equation that describes the collision
takes the form

·

¸
~2 d2
~2 k 2uk (r) = 0,−
+ Vef f (r) −2µ dr22µ(2.2)

where µ is the reduced mass, and energy of this collision is E = k 2 /2µ. The two particles
feel an l-dependent effective potential,
Vef f (r) = V (r) +~2 l(l + 1)
,
2µr2

(2.3)

where l is the relative angular momentum between the particles. The second term in
Eq. (2.3) is called the centrifugal barrier.
For each angular momentum l there is a scattering solution to Eq. (2.2), andall these solutions contribute to the scattering observables. However, at ultracold temperatures only the l = 0 contribution is important. We can neglect higher angularmomentum solutions because the centrifugal barrier introduces an effective repulsion
between the particles. The centrifugal barrier controls particle interaction at large distances is controlled if the long-range behavior of the bare interaction V (r) decays faster9than 1/r2 . At low energies (ultracold temperatures), particles colliding with l > 0 do
not feel the interaction of the bare potential V (r) because they cannot overcome thecentrifugal barrier. Therefore, particles colliding with l > 0 can be considered noninteracting, reducing significantly the complexity of the problem.
The ultracold collisional properties of the two-body system are characterized by
the long-range behavior of the radial-scattering solutions. The asymptotic solutions of
the radial equation take formuk (r)r→∞ ≈A(k)sin[k r + δ(k)],
k

(2.4)

where δ(k) is the phase shift that characterizes all the two-body scattering properties.
For example, the scattering cross section for two distinguishable particles is given by
σ(k) =

4π
sin2 [δ(k)].
k2

(2.5)

Therefore, it is important to understand the behavior of the phase shift in the ultracold
limit.
The phase shift is characterized by the interaction potential. For alkali atoms,
the long-range behavior of the interaction between is generally described by a van der
Waals force of the form V6 (r) = −C6 /r6 . This r−6 long-range behavior decays fastenough that it can be treated as a “short-range” potential. Short-range potentials havenice scattering properties at low energy. In particular, the s-wave scattering phase shift
δ(k) depends linearly in the collision momentum k as it goes to zero. This property
allows us to define the s-wave scattering length as
as = − lim

k→0tan[δ(k)]
.
k

(2.6)

All the scattering properties reduce to a simple expression involving the s-wave scattering length as , in the ultracold limit. For example, the scattering cross section [Eq. (2.5)]
for two distinguishable particles in the ultracold limit reduces to σ(0) = 4πa2s , which isexactly the classical prediction of the cross section for hard spheres with radius as .

10Since all scattering properties depend only on as , the shape of the real potentialof the interacting alkali atoms is irrelevant for describing ultracold scattering. Furthermore, different potentials with the same scattering length will then have identicalscattering properties. Consequently, the behavior of dilute ultracold atoms can be considered to be universal. Such universal behavior give us some flexibility in selecting a
potential to simplify the numerical calculations. In general, we use a potential of theform V (r) = V0 v(r). Here, v(r) is a purely attractive potential of a simple form likea Gaussian or a square well, and V0 is the potential depth that can be tuned to thedesired scattering-length value as . The potential v(r) has a range r0 , and its ultracold
scattering properties will behave universally if as À r0 .
There is also a simple interpretation of the scattering length in terms of the zeroenergy wave function u0 (r). In the zero-energy limit, the long range behavior of u0isAuA0 (r)r→∞ ≈ u0 (r) = A(0)(r − as ).

(2.7)

Therefore, the asymptotic behavior of u0 (r), called uA
0 (r), is a straight line that crosses
zero at r = as . Figure 2.1 provides a qualitative understanding of the scattering length
by analyzing the zero-energy wave function. In Figs. 2.1 (a) and (b), the red curves are
the potential V0 (r), which is a Gaussian interaction potential in this case. The black
curves are the zero-energy solutions u0 (r), and the dashed blue lines are the extrapolated
asymptotic behavior uA
0 (r). Figure 2.1 (c) presents the behavior of the scattering length
as a function of the potential depth V0 .
Another important quantity for the qualitative understanding of the scattering
length is the phase accumulated in the interacting region. This accumulated phaseis related to the “curvature” of u0 (r) in that region. For a shallow potential depth,
the phase accumulated is very small since the potential is not strong enough to produce
important changes. Therefore, this potential produces a small curvature in u0 (r) leading

11
to a negative scattering length [see Fig. 2.1 (a)]. In contrast, for larger values of V0 ,the accumulated phase is large enough that u0 (r) turns around in the interacting region
and has node at r = as , a positive scattering length. Therefore, there is a criticalaccumulated phase, associated with a critical potential depth at which the scattering
length changes from positive to negative. That critical accumulated phase is π/2 andoccurs when the curvature of u0 (r) is just enough to produce asymptotic behavior
parallel to the coordinate r. In this particular case, the scattering length diverges, as
shown by the dashed vertical line in Fig. 2.1 (c).
The divergence of the scattering length is also associated with the existence of abound state right at the zero-energy threshold. Therefore, for deeper potentials, i.e.,
positive scattering lengths, the potential holds a weakly bound state. The properties ofthis weakly bound state only depend on as as long as as À r0 . For example, this weakly
bound state has a binding energy of Eb = 1/(2µa2s ) [schematically shown in Fig. 2.1 (c)]and a size of ∼ as /2.
However, realistic potentials differ significantly from the model potential presented above. A complete description of ultracold two-body collisions usually requires
the analysis of a multichannel system where the hyperfine splitting is incorporated. In
experiments with ultracold gases, the scattering length as is tuned using Fano-Feshbachresonances. A description of Fano-Feshbach resonances requires at least a two-channel
model. The incoming collision occurs in the lower channel while the upper channel isclosed. The resonance occurs at the energy at which the uncoupled closed channel has
a bound state. For ultracold systems, the collision energy is very close to threshold.
Therefore, Fano-Feshbach resonances occur when a bound state of the uncoupled closed
channel coincides with the open channel threshold. As in the single channel case, thebound state at threshold produces divergence of the scattering length. For positive
scattering lengths, the bound state is below threshold. This state is analogous to theweakly bound state of the one-channel model, i.e., its properties are only characterized

12

as < 0(a)as < 0rV0(b)as > 0as > 0

a0

as

(c)

Eb

r
V0

as

V0

Figure 2.1: Behavior of the scattering length and the zero-energy wave function asfunctions of the potential depth. In panels (a) and (b), the red curve corresponds tothe potential interaction. The black curve is the zero-energy u0 (r), and the dashed blueline represents the extrapolated long range behavior of u0 (r) [Eq. 2.4].(c) Schematic
behavior of the scattering length as a function of the potential depth. The solid blueline is the scattering length, and the dashed green line is the binding energy of the
weakly bound dimer.

13
by the scattering length.
Realistic multichannel calculations are important for predicting the interaction
properties of a particular pair of alkali atoms (see e.g. [106]), i.e., the scattering-length
dependence on the magnetic field, the position of resonances, the regions of universality,
etc. Once a large scattering-length region has been theoretically or experimentally identified for a particular pair of alkali atoms, we can use a simple single-channel potential
to model the interaction. Furthermore, we can even use a zero-range pseudopotential that explicitly introduces the scattering length dependence. The description of thezero-range pseudopotential is described in the next subsection.

2.2.2

Zero-range interactions

From the analysis of ultracold collisions with large scattering lengths presented
above, we can understand the zero-range pseudopotential used to mimic the interaction.
This pseudopotential is valid in the universal regime. The universal regime is achievedin a dilute system 1 when r0 ¿ as . Since this range is very small and does not enterin the theory, we can replace the realistic potential with a zero-range pseudopotential.
By introducing pseudopotentials that only depend on the scattering length, we makeexplicit the universal behavior of the interaction.
To obtain the pseudopotential, we consider the scattering wave function solution.
In the limit of r0 → 0, the asymptotic behavior of the scattering solution, Eq. (2.7), isvalid for all r values. In particular, consider a many-particle system. When two particles
get very close together while other particles remain far apart, then the behavior of the
many-body wave function is controlled by the scattering of the two particles given by
Eq. (2.7). This reasoning leads to the Bethe-Peierls contact condition [15],uA (rij )lim Ψ(r1 , ..., ri , ..., rj , ...) ∝ 0=Crij →0
rij
1

µ

1
1
−
rij
as

¶
.

(2.8)

Dilute system refers in this case to the condition r0 ¿ rmean , where rmean is the mean interparticle
distance.

14Here, rij is the distance between particles i and j, and C is a smooth function of theremaining coordinates. The boundary condition of Eq. (2.8) can be achieved through azero-range pseudopotential of the form [95]v(r) =

2π~2 as∂δ(r) r.
µ
∂r

(2.9)

This pseudopotential, or contact condition, tries to reproduce the long-range behavior of the scattering solution. To some extent, it represents an ideal interaction thathas the minimum ingredients for describing an ultracold system.
In the next section, we apply the zero-range pseudopotential to the system of two
trapped particles.

2.3

Spectrum of two particles in a trap
A system of two particles in a trap with tunable interactions provides a qualita-

tive understanding of trap few-body systems and is a useful tool for describing more
complicated systems.
Consider the system of two particles in an harmonic trap with masses m1 andm2 interacting in the s-wave channel. Both particles experience a trapping potential
characterized by the same frequency. Thus, when we select the appropriate center ofmass and relative vectors, i.e., RCM = (m1 r1 + m2 r2 )/M and r = r1 − r2 , the centerof-mass motion decouples from the system and is characterized by a reduced mass
µ = m1 m2 /M .
We also assume that the interaction potential V (r) is spherically symmetric andshort range. When the range of the interaction is much smaller than the two-bodypscattering length as and the characteristic trap length (aho = ~/2µω) that controls
the mean interparticle distance, then the interaction can be replaced by a zero-range
pseudopotential from Eq. (2.9).
The center-of-mass motion is described by a one-particle noninteracting system

15
of mass M = m1 + m2 , and the energies are ECM = (2n + l + 3/2)~ω, where n andl are radial and angular quantum numbers. The relative motion also has a sphericalsymmetry that allows us to consider each relative angular momentum lrel separately.
However, we are only interested in lrel = 0 solutions since the pseudopotential [Eq. (2.9)]
was designed to describe s-wave collisions. The remaining lrel > 0 solutions can beconsidered noninteracting.
Taking advantage of the spherical symmetry, we write the relative wave functionas
1 u(r)
Ψ(r) = √.
4π r(2.10)The radial Schrödinger equation for lrel = 0 collisions is given by
µ 2 2
¶
~ d
1 2 2
−
+ µω r + v(r) u(r) = Er u(r).
2µ dr2 2(2.11)

There are different techniques for solving Eq. (2.11). In the method outlined inRef. [35], the zero-range interaction is interpreted as a boundary condition given by1
u0 (0)
=− .
u(0)
as

(2.12)

Therefore, the system can be described by a noninteracting Hamiltonian, and all theinteraction effects are introduced as the boundary condition given by Eq. (2.12). The radial wave function u(r) can be then expanded in solutions of the noninteracting system,
i.e., harmonic-trapped solutions. Using properties of the Laguerre polynomials that
describe the harmonic trapped solutions, the imposed boundary condition, Eq. (2.12),
leads to a transcendental equation that relates the energies with the scattering length:¡ Er¢√ Γ − 2~ω+ 43a
¢ = ho .
2 ¡ Er1aΓ − 2~ω + 4(2.13)

Here, Γ is the gamma function. Note that the energies only depend on the ratio a0 /aho .
Figure 2.2 presents the spectrum predicted in Eq. (2.13) as a function of aho /as . Interestingly, we observe smooth behavior in the strongly interacting regime [−1 . aho /as . 1].

16
In the positive scattering length, we observe the molecule formation described by thelowest energy curve. The appearance of the molecule affects all the higher states connecting the n vibrational trapped state at as ∼ 0− [left extreme of Fig. 2.2] to the n − 1
vibrational trapped state at as ∼ 0+ [right extreme of Fig. 2.2].

8

Er /h̄ω

6
4
2
0
−2
−4
−6
−5

0

5

aho /aFigure 2.2: Spectrum of the relative motion of two particles in a trap as a function of
the inverse scattering length aho /as .

It is interesting to analyze the limiting behaviors of the lowest energy of thesystem. In the small and negative scattering length limit, the lowest energy isrµ¶
2 − 2 ln(2) as 22 as+
E0 /~ω = 3 ++ ... .
π aho
π
aho

(2.14)

The first term is the noninteracting energy with the center-of-mass contribution included. The second term can be obtained using the Fermi pseudopotential and is associated with a mean-field effect.
In the small and positive scattering-length region, the ground-state energy is
µ
¶
µ
¶
aho 2
7 as 2E0 /~ω = −+ 3/2 −.
(2.15)
as
8 aho
The first term is the molecule-binding energy, −~ωa2ho /a2s = −~2 /ma2 , and the second
is the zero-point energy of the molecule.
After analyzing the energies, we now consider the wavefunction. The zero-range
pseudopotential approximation also allow us to obtain analytical expressions for the17wave functions, e.g.,
µ
¶ µ
¶
r2
3 3 r2Er
Ψ(r) = A exp − 2 U −+ , ,.
2~ω 4 2 2a2ho
4aho(2.16)

Here, U is the confluent hypergeometric function, and A is a normalization constant.
A series expansion of Eq. (2.16) around r ≈ 0 shows that the wave function obeys theBethe-Peierls boundary condition where
Ã
¡ Er
¢!
¶
µ
+ 34
A
A
aho √ Γ − 2~ω
aho aho¢ + O(r) = √Ψ(r) ≈ √+ O(r).
− 2 ¡ Er
−
3/2
3/2
r
r
as
Γ − 2~ω + 14
2πa
2πa
ho

ho

(2.17)

Note that we have used Eq. (2.13) to obtain the scattering-length dependence in Eq. (2.17).
A second method for obtaining solutions of the two-particle trapped system isbased on quantum defect theory techniques [17]. Since the potential v(r) is only nonzero
at the origin, the solution u(r) of Eq. (2.11) should be a linear combination of the regularand irregular solutions of the noninteracting system. This linear combination at a given
energy is chosen so that the logarithmic derivative boundary condition, i.e., Eq.(2.12),
is obeyed. This procedure leads to a transcendental equation equivalent to Eq. (2.13)
and provides an interesting new insight into the problem.

2.4

BCS theory
In 1956, Cooper found that two opposite-spin fermions immersed in a Fermi sea

with arbitrarily weak attractive interactions would form a bound pair [47]. The large
degeneracy of states in the Fermi surface allows such formation. This idea was extended
to many fermion pairs by Bardeen, Cooper, and Schieffer when they developed what is
now known as the BCS theory [10, 11]. An important prediction of the BCS theory was
the existence of an energy gap (pairing gap) in the single particle excitation spectrum.
Cooper pairs and the pairing gap are key ingredients in our modern understanding ofsuperconductivity and superfluidity.

18
A particularly interesting problem arises when interactions in fermionic systems
can be tuned. In such cases, we can access the BCS-BEC crossover. Eagles [59] andLeggett [113] were the first to realize that BCS theory is capable of qualitatively describing both limiting behaviors and predicting an smooth transition between them. Here,
we outline the main results relevant to this dissertation.
To arrive at the BCS equations we assume that the wave function takes theform [113] ofΨ(r1 , ..., rN ) = A{ψ(r1,10 ), ψ(r2,20 ), ..., ψ(rN/2,N 0 /2 ).

(2.18)

Next, we consider a system of distinguishable fermions for which the spin part factors
out, and unprimed and primed indices refer to spin-up and spin-down particles, respectively. To obtain the BCS equations, we apply a variational approach using an extensionof Eq. (2.18) where the pair wave function ψ is the variational parameter [113]. The
extended version of Eq. (2.18) does not correspond to definite number of particles, andthe minimization process constrains the wave function to a mean number of particles.
Alternatively, the BCS equation can be obtained by solving the effective BCS Hamiltonian using a Bogoliubov transformation. Both methods lead to the same set of coupledequations. The first equation is usually called the gap equation and relates the gap∆ and the interaction strength (in this case characterized by the two-body scattering
length as ):

¶
1
1
−
.
(2.19)
2²k
2Ekp
Here, we define ²k = ~2 k 2 /2m and Ek =∆2 + (²k − µ)2 , where µ is the chemim=
4π~2 as

Z

dk
(2π)3

µ

cal potential. Note that a momentum renormalization has been carried out to obtain
Eq. (2.19). The second equation describes the one-component density ρ1 as
ρ1 =

1
2

Z

dk
(2π)3

µ
¶
²k − µ1−.
Ek(2.20)The density is written in general as ρ1 = kF3 /(6π 2 ), and the system is then characterized
by a single dimensionless parameter kF as . In Eqs. (2.19) and (2.20), the gap ∆ and the

19chemical potential µ are obtained by fixing the scattering length as and the density µ.
These coupled equations can be solved analytically [123], but the general solution is notparticularly enlightening. Here, we solve these equations numerically.
Several observables can be obtained once ∆ and µ are known. In particular, we
are interested in the energy of the system, which is given by
Z
E0 =

dk
(2π)3

µ µ
¶
¶
²k − µ∆2²k 1 −−.
Ek
2Ek(2.21)

The first term describes the kinetic contribution, and the second describes corrections
due to the interactions. The behavior of the energy in the BCS-BEC crossover can beeasily obtained numerically. The numerical results of E0 are presented in Fig. 4.11.
Furthermore, the limiting behaviors of the energy and the first-order corrections can beobtained analytically. For |kF as | ¿ 1 and as < 0, i.e., in the weakly interacting BCS
limit, the energy is
3
E0 /N = EF
5

µ
¶
40
1 + 4 eπ/kF as .
e

(2.22)

Here, the first term corresponds to the noninteracting energy where EF = kF2 /2m.
The second term is the first-order correction, which is exponentially small in kF as . In
chapter 4 , we will see that the Hartree term, neglected in this approximation, predicts
a first-order correction linear in kF as . Thus, the Hartree term is the leading correctionin the weakly interacting BCS regime.
In contrast, in the weakly interacting molecular BEC limit, i.e., for |kF as | ¿ 1
and as > 0, the energy is~2
3
E0 /N = −
+ EF
2
2ma
5

µ

¶
5kF a
+ ... .
9π

(2.23)

The first term is half the binding energy of the weakly bound molecules. The second
term describes a repulsive interaction between molecules with a dimer-dimer scattering
length of add = 2as . More accurate calculations, like the ones presented in Chapter 6 ,show that the interaction is indeed repulsive, but the dimer-dimer scattering length is20
add ≈ 0.6as . Therefore, both the BCS and BEC first-order corrections predicted by the
BCS model are incorrect. However, this simple model can still describe the transition
from fermionic atoms to bosonic molecules.
Now we shift gears and we introduce the hyperspherical representation which will
be use to describe both few- and many-body systems.

2.5

Hyperspherical representation
The hyperspherical formulation has been mainly developed to describe few-bodysystems [51, 52, 119, 116, 159, 42]. However, some properties of Bose and Fermi gases
have been successfully explained with this formulation within the K harmonic approximation [20, 149]. Therefore, the hyperspherical formulation is particularly suitable
for studying the transition from few- to many-body systems since it provides a unified
framework for treating this transition. The study of both few- and many-body systems
can be, in turn, greatly simplified using an adiabatic approximation. In this approximation, the description of a many-body system can be reduced to a simple one-dimensional
Schrödinger equation.
The hyperspherical formulation will be extensively used in this dissertation. In
Chapter 3, we present an efficient way to numerically implement the full hyperspherical formulation for few-body systems. In Chapters 5 and 6, we present preliminary
hyperspherical calculations that allow a qualitative interpretation of the spectrum inthe BCS-BEC crossover and the dimer-dimer collisional properties. In Chapter 7, weverify important properties of unitary trapped systems derived from the hypersphericalframework. Finally, in Chapter 8, we use the full solution in the hyperspherical framework to analyze the spectrum and collisional properties of the four-boson system. Here,
we outline the hyperspherical method.
The main objective of the hyperspherical method is to solve the time-independent
Schrödinger equation in a convenient and efficient way. To do this, we calculate eigenval-

21
ues and eigenfunctions of the fixed-hyperradius Hamiltonian, constructing an adiabatic
hyperspherical representation. These eigenvalues and eigenfunctions are then used toconstruct a set of one-dimensional coupled equations in the hyperradius R. The hyperradius is a smooth collective coordinate. In a system described by N coordinate vectors
r1 , . . . , rN , the hyperradius R is defined by
2

µR =

N
X

mi r2i .

(2.24)

i=1

Here, µ is the hyperradial reduced mass, and mi are the masses corresponding to theparticle i. The remaining coordinates are described by a set of hyperangles, generally
called Ω.
The dimension of this system is d = 3 N . The total wave function ψ is rescaled byR, Ψ = R(d−1)/2 ψ, so that the hyperradial equation resembles a coupled one-dimensional
Schrodinger equation. In the adiabatic representation, the wave function ΨE (R, Ω) is
expanded in terms of a complete orthonormal set of angular wave functions Φν andradial wave functions FνE , such thatΨE (R, Ω) =XFνE (R)Φν (R; Ω).

(2.25)

νThe adiabatic eigenfunctions, or channel functions Φν , depend parametrically on R and
are eigenfunctions of a 3 N − 1 partial differential equation:
µ 2 2
¶
~ Λ
(d − 1)(d − 3)~2+
+ V (R, Ω) Φν (R; Ω) = Uν (R)Φν (R; Ω).
2µR2
8µR2

(2.26)

Here, Λ is the grand angular momentum operator, which is related to the kinetic termby−

X ~2 ∇2
i

i

2mi

=−

~2 1 ∂ d−1 ∂
Λ2 ~2
R
+
.
2µ Rd−1 ∂R∂R 2µR2(2.27)The Uν (R) obtained in Eq. (2.26) are effective hyperradial potential curves of aset of one-dimensional differential equations:
¸
·
· 2 2
¸
~2 X
~ d
d
0
0
+ Uν (R) FνE (R) −2Pνν (R)+ Qνν (R) Fν 0 E (R) = EFνE (R).
−2µ dR22µ 0dR
ν(2.28)

22
These differential equations [Eq. (2.28)] are coupled through the Pνν 0 (R) and Qνν 0 (R)couplings defined as¯d¯|Φν 0 (R; Ω)i ¯ ,
dR
R
¯
d2
¯
Qνν 0 (R) = hΦν (R; Ω)| 2 |Φν 0 (R; Ω)i ¯ .
dRRPνν 0 (R) = hΦν (R; Ω)|(2.29)(2.30)

Since the basis set expansion of the wave function, Eq. (2.25), is complete in
the 3 N -dimensional space, Eqs. (2.26) and (2.28) reproduce exactly the original ddimensional Schrödinger equation. As in most numerical methods, the solutions areapproximated by truncating the Hilbert space. In this case, the Hilbert space is truncated by considering a finite number of channels in Eq. (2.28). This approximation
can be easily tested analyzing the convergence with respect to the number of channels
included in the calculation.
The utility of the hyperspherical representation relies on the assumption that thehyperradius R is a smooth variable. In such cases, only a few channels are relevant,and the couplings are small and have a smooth behavior. Furthermore, a fairly good
approximation to the solutions can be achieved by truncating the expansion in Eq. (2.25)
to a single term:
ΨE (R, Ω) = FνE (R)Φν (R; Ω).

(2.31)

This adiabatic hyperspherical approximation leads to an effective one-dimensional Schrödinger
equation,

¸
· 2 2
~ d
+ Wν (R) FνE (R) = EFνE (R),−2µ dR2(2.32)

where the effective potential isWν (R) = Uν (R) −~2Qνν (R).
2µ(2.33)

Here, the first term is the hyperradial potential curve, and the second term is the
kinetic contribution of the hyperradial dependence of the channel function, which isalways positive. If the potential curves are well spaced and do not have strong avoided

23crossings, then the adiabatic approximation can be very good. This approximation
comes from a truncation of the Hilbert space and, for that reason, follows the variational
principle. Any energy solution obtained with this method is an upper bound of the exact
energy solution.
Another useful approximation is obtained by neglecting the second term in Eq. (2.33),
i.e., replacing Wν (R) by Uν (R) in Eq. (2.32). This is usually called the hypersphericalBorn-Oppenheimer approximation. As in the standard Born-Oppenheimer approximation, the approximated energy represents a lower bound of the exact energy [46].
Next, we consider the noninteracting limit.

2.5.1

The noninteracting limit: hyperspherical harmonics

The eigenfunctions of an N -particle system acquire a simple form in the noninteracting limit. Since Λ is an operator that acts only on the Ω coordinates, the solutionsof Eq. (2.41), Φν , become independent of R. Therefore, any derivative of the channel
function with respect to R is zero, all the coupling terms vanish, and the adiabatic
approximation becomes exact. In other words, the kinetic operator [Eq. (3.47)] is separable in Ω and R and, for that reason, its solutions take the form of Eq.(2.31), with Φν
independent of R.
The solutions of Λ2 , Φλ (Ω) are usually labeled by λ and called hypersphericalharmonics. Here λ is the solution of[Λ2 − λ(λ + d − 2)]Φλ (Ω) = 0.

(2.34)For each symmetry and each number of particles, there is a λmin and particulardegeneracies corresponding to that problem. To obtain the λmin and degeneracies, we
can compare the energies of the trapped system in both the single particle and thehyperspherical representation. In the trapped system, the external potential is simplyP
V (R, Ω) = ω 2 /2 di=1 mi x2i = ω 2 µR2 /2. This potential commutes with the Λ2 operator

24
and does not introduce any coupling. Therefore, the channel functions of the free system
Φλ (Ω) also describe a trapped system. Then, the wave functions of the trapped systemtake the form
Ψλn (R, Ω) = Fλn (R)Φλ (Ω),(2.35)

where Fλn (R) obeys the effective one-dimensional Schrödinger equation,
µ

¶
−~2 ∂
+ Vef f (R) R(d−1)/2 F (R) = ER(d−1)/2 F (R).
2m ∂R2(2.36)

Here, the effective potential isVef f (R) =(d − 1)(d − 3) λ(λ + d − 2) µω 2 R2
+
+
.
8µR2
2µR2
2

(2.37)

Defining ` = λ + (d − 3)/2, the effective potential takes the known form
Vef f (R) =`(` + 1) µω 2 R2
,
+
2µR2
2

(2.38)

which is identical to the radial Schrödinger equation of a particle with angular momentum ` and mass µ in a spherical trap of frequency ω. Thus, the energies areµ¶dE`,n = (` + 2nR )~ω = λ + + 2nR ~ω.
2(2.39)

A useful method to obtain the λ values and their degeneracies compares thesolutions of Eq. (2.39) with those obtained from the single particle framework [159]. By
comparing both energy solutions, we obtain the relation
λ + 2nR =NJ
X(li + 2ni ).

(2.40)

i=1

Here, NJ is the number of Jacobi vectors. Considering all the possible options in
the right-hand-side solutions, we can infer the solutions in the left-hand side. The
degeneracy of λ can be obtained by combining all the possible li and ni on the righthand side of Eq. (2.40) allowed by the symmetrization rules of the wave function.
As an example, consider the system of three distinguishable particles. We remove
the center-of-mass coordinate and are left with two Jacobi coordinates. Thus, Eq. (2.40)

25
is simply
λ + 2nR = l1 + 2n1 + l2 + 2n2 .

(2.41)

We focus on the L = 0 states. The case λ = 0 is not degenerate and corresponds to
l1 = l2 = n1 = n2 = 0. The next case is λ = 2 (The λ = 1 can only be formed with
L > 0). The single particle representation predicts three cases ( l1 = l2 = n1 = 0 and
n2 = 1; l1 = l2 = n2 = 0 and n1 = 1; l1 = l2 = 1 and n1 = n2 = 0 ). One of thesecases corresponds, in the hyperspherical representation, to a hyperradial excitation of
the λ = 0 case. The other two are degenerate states of λ = 2. The eigenvalue of the Λ2
operator λ(λ + d − 2) = 12. This procedure can be continued to higher λ values.

Chapter 3
Numerical Techniques

3.1

Introduction
In this Chapter, we present the numerical methods used in this Thesis. We focus

initially on the Monte Carlo methods. In the fixed-node diffusion Monte Carlo section,
we pay special attention to the importance of physical intuition for the construction of
the guiding wave functions. In particular, we present the different trial wave functions
used to describe trapped Fermi gases in the BCS-BEC crossover. Then we present
a powerful technique to describe few-body trapped systems where the solutions are
expanded in correlated Gaussian (CG) basis set. Finally, we present an innovating
method which combines the hyperspherical technique with the CG method. This chapter
presents these numerical methods in a general way; and it is complemented with a setof Appendices: A, B C, D E, F, G.

3.2

Fixed-node diffusion Monte Carlo approach
In 1949, Metropolis and Ulam presented the basic idea of what they called The

Monte Carlo Method [126]. They proposed a general statistical approach to thestudy of differential equations. In particular, motivated by the work of Fermi, they
noted that the Schrödinger equation could be expressed as a diffusion equation andsimulated by a system of “particles” that move randomly and multiply themselves.
Since then, quantum Monte Carlo methods, i.e., applications of Monte Carlo27
techniques to solve the Schrödinger equation, have been extensively used and perfected.
There have been important developments such as the implementation of importantsampling [125, 101] and different ways [4, 146, 39, 111] to overcome the limitations of
the Monte Carlo methods to deal with wave functions that have nodes or, equivalently,
changes of sign. These limitations are usually called the “sign problem” and affect the
study of fermionic systems whose ground state wave functions have nodes.
In this section, we present the fixed-node diffusion Monte Carlo (FN-DMC)
method [4] that overcomes the “sign problem” by forcing the ground state solution
to have the same nodes as a suitably-chosen trial wave function. This trial wave function is crucial for a reliable estimation of the physical observables. Many of the detailsof the implementation of the FN-DMC method can be found extensively in the literature [84]. Here, we present the general ideas that describe the FN-DMC method. Then,
we discuss in detail the trial wave functions used to describe trapped two-component
Fermi gases.

3.2.1

Imaginary time propagation and diffusion Monte Carlo

We present the concepts of imaginary time propagation and its application to
the diffusion Monte Carlo method. Then, we discuss general aspects of the FN-DMC
method.
Consider the time-dependent Schrödinger equation for a Hamiltonian H,i~∂Φ(x, t)= (H − ET )Φ(x, t).
∂t

(3.1)

For convenience, we have introduced an energy shift ET . The quantity x = {x1 , ...xN }includes the coordinates of all the particles. The solutions of Eq. (3.1) for a timeindependent Hamiltonian take the form,
Φ(x, t) =Xk

Ck Ψk (x)e−i(Ek −ET )t .

(3.2)

28
Equation (3.1) can be extended to imaginary time, τ = it, and be written as−~∂Φ(x, τ )= (H − ET )Φ(x, τ ).
∂τ

(3.3)

The solutions of Eq. (3.3) are simply Φ(x, t) from Eq. (3.2) extended to imaginary time:
Φ(x, τ ) =∞X

Ck Ψk (x)e−(Ek −ET )τ .

(3.4)

k=0

After a propagation in imaginary time, the term with lowest Ek − ET will become themost prominent term in Eq. (3.5). The lowest energy is, of course, the ground-stateenergy E0 , and the dominant term would be proportional to the ground state Ψ0 . In
particular, we can set ET = E0 in Eq. (3.5) and obtain
Φ(x, τ ) = C0 Ψ0 (x) +∞X

Ck Ψk (x)e−(Ek −E0 )τ .

(3.5)

k=1

In this case, Φ(x, τ ) converges to a steady state proportional to Ψ0 (x).
The concept of imaginary time propagation and the derivation in Eqs. (3.1-3.5) arecompletely general and independent of the method used to make the propagation. The
diffusion Monte Carlo method takes advantage of these properties of the Schrodinger
equation in imaginary time and its similarities with the classical diffusion equation.
Consider Eq. (3.3) with the Hamiltonian now explicitly written as∂Φ(x, τ )~2 2=
∇ Φ(x, τ ) + (ET − V (x))Φ(x, τ ).
∂t
2m

(3.6)

If the term (ET −V (x)) is ignored, then Eq. (3.6) is the usual diffusion equation. Ignoring
the kinetic term leads to a first-order rate equation. Both the diffusion and rate processes
can be simulated separately by the Monte Carlo method, but there is no general exact
solution for Eq. (3.6). However, a propagation of Eq. (3.6) in a differential δτ , theleading order correction can be understood as a sum (or product) of the independent
diffusion and rate processes. In other words, the exact propagation operator factorizes
in a diffusion and rate term in the small time-step limit.

29
There is a close analogy between a diffusion Monte Carlo simulation and theexperimental study of the behavior of a gas. To theoretically obtain the temperatureand other global properties of a gas, we solve the corresponding diffusion equation.
Alternatively, we could release the real gas from an initial configuration and wait until itreaches equilibrium and experimentally measure the temperature and the other global
properties. The second approach is more like the diffusion Monte Carlo simulation.
However, instead of experimentally studying the ”gas,” it numerically simulates a gasthat obeys the diffusion equation [Eq. (3.6)]. The simulation starts from an initialconfiguration of an artificial gas, the diffusion Monte Carlo method propagates this gas
until it reaches equilibrium. The gas remains in equilibrium for some measuring time
during which its properties are computed. At equilibrium, the gas describes the groundstate of the system and any observable of the ground state can be obtained.
The constituent parts of this artificial gas are usually called “walkers”, and theydescribe a configuration of the N -particle system. For example, a Schrodinger equationof a 3D N -body system, a walker represents a given configuration of all the bodies, i.e.,x = {x1 , ...xN }, and live in a 3N dimensional space. The diffusion process is simulated
by producing a random displacement of the initial configuration, or walker, with aprobability dictated by the diffusion term in Eq. (3.6). The rate process is simulatedby allowing the walkers to die or to duplicate themselves.
Although solving Eq. (3.6) is in principle possible with the “pure” diffusion Monte
Carlo method, the fluctuations are generally very large. However, the fluctuations can begreatly reduced by the Monte Carlo technique of importance sampling [125, 101]. In thisprocedure, an initial guess ΨT , usually called the trial wave function or guiding function,
is used to bias the “walkers” to reproduce the distribution f (x, τ ) = Φ(x, τ )ΨT (x). An
equation similar to Eq. (3.6) describes the imaginary time evolution of f (x, τ ) (seedetailed description in Ref. [84]). If ΨT is a good approximation to the exact groundstate, then the fluctuations will be greatly reduced. Therefore, the construction of ΨT

30
is a very important step in FN-DMC calculations.
Another vital aspect of the importance sampling technique is that ΨT can be used
to overcome the sign problem. In general, the propagation in imaginary time does nottake into account the symmetry of the wave function. Thus, independently of the type ofparticles we are considering, such a propagation will converge to the same ground state,
the nodeless bosonic ground state. This is unsatisfying if we are interested in fermionicsystems. To remedy this difficulty, we can force the wave function to have the same
nodal structure of a guiding wave function, ΨT . This method is usually called FN-DMC.
To within statistical uncertainties, the FN-DMC algorithm provides an upper bound to
the exact ground-state energy, i.e., to the lowest-lying state with the same symmetry as
ΨT . If the nodal surface of ΨT coincides with that of the exact eigenfunction, then the
FN-DMC method produces the exact eigen energy of the system.
An important step in the FN-DMC calculation is the optimization of ΨT . The
variational Monte Carlo (VMC) method is used to test and optimize the trial wave
function. In the VMC method, the energy expectation value of ΨT is evaluated using
Monte Carlo integration techniques. The trial wave function is characterized by a setof parameters that are optimized by minimizing the energy of the trial wave function.
The optimized ΨT is then used as a guiding function in the FN-DMC process.
A detailed description on how the VMC and FN-DMC methods are implemented
will not be presented here but can be found in Ref. [84].

3.2.2

Trial Wave functions for a Trapped Two-Component Fermi Gas

The selection of the guiding wave function is a crucial step in the FN-DMC calculations. Physical intuition is used to select a suitable wave function. For homogeneous
two-component Fermi gases, there are a couple of guiding wave functions that have beenused to describe the gaslike ground state at different regimes of the crossover [7, 36].
The “Jastrow-Slater” wave function is used to describe a system in the weakly interact-

31
ing BCS limit (as < 0 and |kF as | ¿ 1) while the “paired” wave function describes theweakly interacting molecular BEC (as > 0 and |kF as | ¿ 1). Here, we extend the use ofsuch wave functions for trapped systems.
We begin with the Jastrow-Slater wave function,
ΨJ (r1 , r2 , ..., rN ) =Y

J(ri − ri0 )ΨN I (r1 , r2 , ..., rN )

(3.7)

ii0

Here, i and i0 correspond to different-spin fermions, and ΨN I is the noninteracting
ground state wave function. Alternatively, we can replace ΨN I by the noninteracting
wave function with an effective oscillator length that can be different from the baretrap length of the problem. This effective scattering length is used as a variationalparameter. For strongly attractive systems, we find that the optimal effective oscillator
length is smaller than the bare trap length. This reduction of the effective oscillator
length is a consequence of the squeezing of the gas produced by the attraction in thesystem.
The Jastrow term, i.e.,

Q

ii0 J(ri − ri0 ), describes the correlation between spin-

up and spin-down particles. The Jastrow term is constructed to be nodeless for thetrial wavefunction ΨJ . Therefore, the purpose of the Jastrow term in the FN-DMC
calculations is only to reduce the statistical fluctuations. In constrast, in the VMC
calculations, the energies depend on the form of the Jastrow correlations.
Since ΨN I is in most cases degenerate, and there are different ways to construct it.
In Eq. (3.7), the noninteracting wave function is a product of two Slater determinants,
one for each component,
ΨN I = A(φnx ny nz (r1 ), · · · , φnx ny nz (rN1 )) A(φnx ny nz (r10 ), · · · , φnx ny nz (rN20 )) =
¯
¯
¯
¯
¯
¯
¯
¯
¯ φ111 (r 0 )¯ φ111 (r1 )···φ111 (rN20 ) ¯¯···
φ111 (rN1 ) ¯¯
1
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
..
..
..
..
det ¯
¯ (3.8)¯ det ¯
.
.
.
.
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
0
0
φ
(r
)
)
·
·
·
φ
(r
φ
(r
)
·
·
·
φ
(r
)
¯ nx ny nz 1
¯ nx ny nz 1
nx ny nz N2 ¯
nx ny nz N1 ¯
The determinant form is particularly suitable for numerical evaluation. Furthermore,32
the evaluation of the derivatives can be greatly simplified using the inverse of the Slater
matrix (see e.g., Appendix B in Ref. [84]). Each Slater matrix is formed with the harmonic oscillator orbitals. In general, it is convenient to use harmonic oscillator orbitals
obtained in Cartesian coordinates, φnx ny nz . For closed shells, it is straightforward toselect the harmonic oscillator orbitals. For open shells, the selection of the orbitals
is more cumbersome since the ground state is degenerate. However, this degeneracy
can be broken by perturbation theory with the zero-range Fermi pseudo-potential as
discussed in Sect. 4.2. Thus, ideally we could use the ground state obtained from thisperturbation approach as ΨN I . However, such ΨN I is an expansion of the product ofSlater determinants and is very expensive to compute. Therefore we look for alternative
trial wave functions to describe some of the degenerate noninteracting systems.
Now we turn the discussion to a different type of trial wave function. The behaviorof the two-component Fermi gas on the BEC side is qualitatively different. For attractive
potentials with small, positive as , comparatively strongly-bound two-body dimers exist
and the system is expected to form a molecular Bose gas of dimers. Such a system is
not described even qualitatively correctly by the guiding function ΨJ , which assumes
that every spin-up fermion is “simultaneously” correlated with every spin-down fermion.
Therefore, we use instead a “paired” guiding function ΨP [36, 7],©
ªΨP (r1 , r01 , ..., rN ) = A Ψ2 (r1 , r10 )Ψ2 (r2 , r20 )...Ψ2 (rN/2 , rN/20 ) ,(3.9)

where A is the antisymmetrizer operator. The paired guiding function is much better
suited for describing a Fermi gas that behaves as a weakly interacting molecular Bosegas. In this case, ΨP correlates the first spin-up fermion with the first spin-down
fermion, the second spin-up fermion with the second spin-down fermion, and so on;then it antisymmetrizes this “paired state.” For its part, the guiding function ΨP is
expected to accurately describe the system when the size of the dimer pairs becomes
small compared to the oscillator lengths.

33
In Eq. (3.9), we have assumed that we have an even-N system where the numberof spin-up and spin-down particles is the same. The other case we are going to consider
is the odd-N system where N1 = N2 + 1. ΨP can be extended to treat odd-N systems
by simply adding a single particle orbital φnl [23, 36],no
ΨP (r1 , r01 , ..., rN ) = A Ψ2 (r1 , r10 )Ψ2 (r2 , r20 )...Ψ2 (rN1 −1 , rN20 ), φnl (rN1 ) .

(3.10)

We consider a number of different single particle orbitals φnl in spherical coordinates,and determine the optimal nl values by performing a series of FN-DMC calculations.
For the lowest n and l (the m quantum number is set to zero), the unnormalized orbitals
2

2

2

2

2

2

are φ00 (~r) = e−r /2b , φ01 (r) = e−r /2b z/aho , φ20 (r) = e−r /2b (1 − 2(r/aho )2 /3) and22

φ02 (r) = e−r /2b (3(z/aho )2 − (r/aho )2 ). The b parameter can be treated variationally.
If b = aho , the φnl agree with the noninteracting trapped orbitals.
Another important point that must be decided is the angular momentum of thetrial wave function ΨP . Since each pair function Ψ2 (r1 , r10 ) has vanishing relative orbital
angular momentum, the total angular momentum L of ΨP is 0 for even N and N1 = N2 .
For odd N , L of ψP is determined by the angular momentum of φnl , i.e., L = l.
Both paired wave functions, Eqs. (3.9,3.10), are constructed in from a two-bodywave function. The two-body wave function, Ψ2 , can be obtained from the trapped
two-particle system. Interestingly, the ground-state wave function of two particles in a
trap can be separated exactly into a Jastrow term and noninteracting orbitals, in whichΨ2 (r1 , r2 ) = ψ(r1 )ψ(r2 )J(r1 − r2 ),(3.11)

where ψ is the non-interacting ground state orbitals. We will show that this relation
is valid when the center-of-mass wave function and the noninteracting orbitals have a
Gaussian form, like in the trapped system. For this discussion, we will not consider
normalization factors or proportionality constants. If we define RCM = (1/2)(r1 + r2 )and r = (r1 − r2 ) we know that the pair wave function can be separated in the centerof-mass and relative-coordinate terms, Ψ2 (r1 , r2 ) = Ψ(RCM )φ(r). Also, we know that34
the center-of-mass wave function is unaffected by the two-body interaction and is
r2

r2

2

2

r
r
− 21 − 22
R2
2a
2a
4a2
4a2
ho e
ho e ho = ψ(r1 )ψ(r2 )e ho ,Ψ(RCM ) = exp(− CM
)
=
e
a2ho

where ψ(r1 ) = e

−

r2
2
2a2
ho

(3.12)

is the 1s (ground-state) orbital. Thus, the total wave function isΨ(RCM )φ(r) = ψ(r1 )ψ(r2 )e

r2
4a2
ho

φ(r) = ψ(r1 )ψ(r2 )J(r),(3.13)

φ(r).

(3.14)

where
J(r) = e

r2
4a2
ho

The evaluation of the relative coordinate wave function, φ(r), usually requires a numerical calculation.
These two-body solutions are used to build both the Jastrow and the paired wave
function. In the Jastrow wave function, the Jastrow term is expected to describe shortrange correlations. Therefore, the two-body solutions are matched outside the range ofthe interaction to a function that decays rapidly to a constant. For our calculations, we
use A exp(−c r) + B. The parameters A and B are selected so that the J(r) and J 0 (r)are continuous at the matching point rm . In general, both c and rm can be used as
variational parameters in a VMC calculation. In the weakly interacting BCS side, we
observe that the results are roughly insensitive to the values of c and rm if we set rm toa few times the range of the potential. In the noninteracting case, i.e., for as = 0, theguiding function ΨJ with J(r) = 1 coincides with the exact eigenfunction. For weakly
attractive Fermi systems, the attractive nature of the two-body potential introduces
correlations but does, to a good approximation, leave the nodal surface unchanged. Indeed, we find that the variational energy for ΨJ in this regime is nearly indistinguishable
from the FN-DMC energies, indicating that the Jastrow product over all pair functions
accounts properly for the two-body correlations of the system. Furthermore, it suggeststhat the Jastrow-Slater wave function is very close to the exact wave function in thisregime.

35
The exponentially growing term of the Jastrow function [Eq. (3.14)] compensates
the exponential decaying behavior of φ due to the trapping potential. Therefore, J(r)decays with a power law behavior at as < 0. As we will see in the following this type oflong-range behavior is similar to the Jastrow term obtained from the free system. Thus,
it is sensible to replace the Jastrow term obtained for the trapped system with the one
for the free system.
For the initial calculations for N = 8 in Ref. [183], we used a Gaussian interaction
and the numerical solutions of the trapped system to obtain the Jastrow function.
These simulations can be significantly sped up, if we use a square-well potential andthe analytical Jastrow function of the free system. In this case, the Jastrow function
J(r) is given by the free-space two-body solution [7]. For positive scattering lengths
as , J(r) coincides with the free-space–two-body bound-state solution. For negative as ,J(r) is the free-space scattering solution, calculated at the scattering energy Erel . For
N = 6, 8 at unitarity, we treat Erel as a variational parameter and find a reduction of
the energy of 1 or 2% for a finite Erel compared to Erel = 0. For larger N , we simply
use Erel = 0 which implies that outside the range of the interaction the Jastrow term
goes like J(r) ∝ (1/r − 1/as ).
Another description of the system is given by an alternative wave function ΨCB ,proposed by Chang and Bertsch [40]. This wavefunction combines the ideas of Eqs. (3.7,
3.9). For closed shell noninteracting systems, Eq. (3.15) can be written as a single
Slater determinant of paired functions. Finally, the guiding function ΨCB is constructedfollowing Eqs. (3) and (4) of Ref. [40]. At unitarity, for most N systems either ΨJ or
ΨP provide the lowest energy so we do not use ΨCB . However, for N = 11, we find that
ΨCB gives the lowest energy.
The calculations presented in Sec. 4.6 were done for closed shell (N = 8), andconsequently we used Eq. (3.15). As an example, we compare the energies at unitarity
obtained with the Jastrow-Slater, EJ , and the paired wave function EP . The results

36
are EJ = 12.63(2) and EP = 12.58(2). In this case we see that the energies are very
close.
For N=4, the constructed guiding wave function that describes the system on the
BCS side has been inspired by the picture for a homogeneous system, i.e.
¯
¯
¯
¯
0
0
¯
Y
0)
0) ¯
ψ(r
)ψ(r
)
cos(krψ(r)ψ(r)
cos(kr
1
1
11
12
1
2
¯
¯
Ψ(N =4) ==J(ri − ri0 )det ¯¯ (3.15)
¯
¯
0
0
0
ii
¯ ψ(r2 )ψ(r1 ) cos(kr210 ) ψ(r2 )ψ(r2 ) cos(kr220 ). ¯
The Jastrow term describes the short-range interactions. The determinant part describes a nodal surface similar to that of the noninteracting system if the parameter k
is chosen to be very small, i.e., k ∼ 10−5 /aho . This trial wave function can be thoughtof as a variant of the Chang and Bertsch guiding function. We empirically verified thatsuch a guiding function produces accurate results (see Sec. 6.2).
Finally, in the strongly-interacting regime, i.e., for |as | → ∞, it is not a priori
clear which of the two guiding functions provides a better description of the system.

3.3

Correlated Gaussian Method
Now we turn to an alternative method for solving the few-body Schrödinger equa-tion variationally which does not rely on Monte Carlo techniques. Different types ofGaussian basis functions have long been used in many different areas of physics. In
particular, the usage of Gaussian basis functions is one of the key elements of the success of ab initio calculations in quantum chemistry. The idea of using an explicitly
correlated Gaussian to solve quantum chemistry problems was introduced in 1960 byBoys [26] and Singer [157]. The combination of a Gaussian basis and the stochasticalvariational method SVM was first introduced by Kukulin and Krasnopol’sky [110] in
nuclear physics and was extensively used by Suzuki and Varga [179, 178, 180, 181].
These methods were also used to treat ultracold many-body Bose systems by Sorensen,
Fedorov and Jensen [162]. A detailed discussion of both the SVM and CG methods canbe found in Sorensen’s Master’s thesis [161] and, in particular, in Suzuki and Varga’s

37book [173]. In the following, we present the CG method and its application to few-bodytrapped systems.
Consider a set of coordinate vectors that describe the system {x1 , ..., xN }. In thismethod, the eigenstates are expanded in a set of basis functions,Ψ(x1 , · · · , xN ) =X

CA ΦA (x1 , · · · , xN ) =A

X

CA hx1 , · · · , xN |Ai .

(3.16)A

Here A is a matrix with a set of parameters that characterize the basis function. In
the second equality we have introduced a convenient ket notation. Solving the timeindependent Schrödinger equation in this basis set reduces the problem to a diagonalization of the Hamiltonian matrix:~ i = Ei OC~i
HC(3.17)~ i is a vector form with the coefficients CA
Here, Ei are the energies of the eigenstates, C
and H and O are matrices whose elements are HBA = hB|H|Ai and OBA = hB|Ai. For
a 3D system, the evaluation of these matrix elements involves 3N -dimensional integrations which are in general very expensive to compute. Therefore, the effectiveness ofthe basis set expansion method relies mainly on the appropriate selection of the basis
functions. As we will see, the CG basis functions permit a fast evaluation of overlap
and Hamiltonian matrix elements; they are flexible enough to correctly describe physicalstates.
To reduce the dimensionality of the problem we can take advantage of its symmetry properties. Since the interactions considered are spherically symmetric, the total
angular momentum, L, is a good quantum number. For simplicity, we will restrict
ourselves to L = 0 solutions. This restriction allow us to reduce the Hilbert space byintroducing restrictions to the basis functions. In particular, if the basis functions only
depend on the interparticle distances, then Eq. (3.16) can only describe states withzero angular momentum and positive parity (LP = 0+ ). Furthermore, we can consider

38that the center-of-mass motion decouples from the system. In such cases, the CG basis
functions take the form



N


X
2


Φ{αij } (x1 , · · · , xN ) = ψ0 (RCM )S exp −,αij rij /2

(3.18)

j>i=1

where S is a symmetrization operator and rij is the interparticle distance between
particles i and j. The set of parameters {αij } are related with the matrix elements of
A. Here, ψ0 is the ground state of the center-of-mass motion. For trapped systems,
2M

ψ0 takes the form, ψ0 (RCM ) = e−RCM /2aho . Because of its simple Gaussian form, theψ0 can be absorbed in the exponential factor. Thus, in a more general way, the basis
function can be written in terms of a matrix A that characterizes them,


¾
½
N


X
1
1
Aij xi .xj ) ,ΦA (x1 , x2 , ..., xN ) = S exp(− xT .A.x) = S exp(−
2
2(3.19)

j>i=1

where x = {x1 , x2 , ..., xN }, and A is a symmetric matrix. The matrix elements Aij
are directly related with the αij (see Appendix C). Because of the simplicity of thebasis functions, Eq. (3.18), the matrix elements of the Hamiltonian can be calculated
analytically.
The key idea that allows the analytical evaluation of the matrix elements is selecting the set of coordinates that simplifies the evaluations. For basis functions of the formof Eq. (3.19), the matrix elements are characterized by a matrix M in the exponential.
In such case, the matrix element integrand can be greatly simplified if we write it interms of the Jacobi vectors that diagonalize that matrix M . This change of coordinates
permits, in many cases, the analytical evaluation of the matrix elements. The explicit
evaluation of the matrix elements can be found in Appendices A and B.
There are two properties of the CG method that worth mentioning. First, theCG basis set is linearly-dependent and over-complete, so a systematic increase in thenumber of basis functions will converge to the exact eigenvalues [161]. Secondly, thebasis functions ΦA are squared integrable only if the matrix A is positive definite. We can

39further restrict the basis function by introducing a real widths dij such that αij = 1/d2ij .
With this transformation, we make sure that A is positive definite. Furthermore, thesewidths are proportional to the mean interparticle distances of the basis function. Thus,
it is easy to select them after considering the physical length scales relevant to the
problem. Even though we have restricted the Hilbert space with this transformation,
we have numerical evidence that that the results converge to the exact eigenvalues.
The linear dependence in the basis set causes problems in the numerical diagonalization of the Hamiltonian matrix equation 3.17. There are different ways to minimize
or eliminate such problems which are explained in the Appendix E.
Finally, we want to stress the importance of the appropriate selection of theinteraction potential. For the problems considered in this dissertation, the interactions
are expected to be characterized only by the scattering length, i.e., to be independentof the shape of the potential. For that reason, we can select a model potential that
permits rapid evaluation of the matrix elements. We found that a model potential witha Gaussian form,¶µ
r2
V0 (r) = −d exp − 2 ,2r0(3.20)

is particularly suitable for this basis set expansion. If the range r0 is much smaller
than the scattering length, then the interactions are effectively characterized only by
the scattering length. The scattering length is tuned by changing the strength of theinteraction potential, d, while the range, r0 , of the interaction potential remains unchanged. This is particularly convenient in this method since it implies that we only
need to evaluate the matrix elements once and we can use them to solve the Schrödinger
equation at any given potential strength ( or scattering length). Of course, this procedure will give accurate results only if the basis set is complete enough to describe thedifferent configurations that appear at different scattering lengths.
In general, a simple version of this method includes four basic steps: the gen-

40
eration of the basis set, the evaluation of the matrix elements, the elimination of thelinear dependence and the evaluation of the spectrum. The SVM, presented in Appendix F, combines the first three of these steps in a optimization procedure where thebasis functions are selected randomly.

3.4

Correlated Gaussian Hyperspherical method
Several techniques to solve few-body systems have been developed in the last

decades [68, 173, 121, 190, 119]. Among these method, the Correlated Gaussian (CG)technique, presented in the previous Section, has proven to be capable of describing
trapped few-body system with short-range interactions. Because of the simplicity ofthe matrix element calculation, the CG method provides an accurate description of theground and excited states up to N = 6 [19]. However, CG can only describe boundstates. For this reason, it is numerically convenient to treat trapped systems where all
the states are quantized. The CG cannot (without substantial modifications) describe
states above the continuum nor the rich behavior of atomic collisions such as dissociation
and recombination.
The hyperspherical representation, on the other hand, provides an appropriate
framework to treat the continuum. In the adiabatic hyperspherical representation, theHamiltonian is solved as a function of the hyperradius R, reducing the many-body
Schrödinger equation to a single variable form with a series of different couple effective potentials. The asymptotic behavior of the potentials and the channels describe
different dissociation or fragmentation pathways, providing a suitable framework foranalyzing collisional physics. However, the standard hyperspherical methods expand
the channel functions in splines or finite element basis functions [133, 192, 65, 172], andthe calculations become computationally very demanding for N > 3 systems.
Ideally, we would like to combine the fast matrix element evaluation of the CG
basis set with the capability of the hyperspherical framework to treat the continuum.

41Here, we explore how the CG basis set can be used within the hyperspherical framework. We call the use of CG basis function to expand the channel functions in thehyperspherical framework the CG hyperspherical method (CGHS).
In the hyperspherical framework, we need to evaluate matrix elements of the
Hamiltonian at fixed R. To evaluate such matrix elements, consider first how the matrix
element evaluation is carried out in the standard CG approach.
In the CG method, we select, for each matrix element evaluation, a set of coordinate vectors that simplifies the integration, i.e., the set of coordinate vectors that
diagonalize the basis matrix M which characterizes the matrix element. The flexibility
to choose the best set of coordinate vectors for each matrix element evaluation is key
to the success of the CG method.
This selection of the optimal set of coordinate vectors is formally applied by anorthogonal transformation from an initial set of vectors x = {x1 , ..., xN } to a final setof vectors y = {y1 , ..., yN }: T x = y, where T is the orthogonal transformation matrix.
The hyperspherical method is particularly suitable for such orthogonal transformations
because the hyperradius R is an invariant under them. Consider the hyperradius definedin terms of a set of mass scaled Jacobi vectors [51, 52, 172, 124], x = {x1 , ..., xN },µR2 = µX

xi 2 ,(3.21)i

If we applied an orthogonal transformation to a new set of vectors y, thenµR2 = µXi

xi 2 = µyT T T y = µ

X

yi 2

(3.22)iwhere we have used that T T T = I, and I is the identity. Therefore, in the hypersphericalframework we can also select the most convenient set of coordinate vectors for eachmatrix element evaluation. This will be the key to reducing the dimensionality of thematrix element integration. This transformation is the equivalent of saying that, foreach matrix element evaluation, we can select the set of hyperangles (Ω) that simplify
the matrix-element evaluation.

42
As an example of how the dimensionality of matrix-element integration is reduced,
consider a three dimensional N -particle system with the center of mass removed and
L = 0 solutions. We will show that this technique reduces a (3N − 7) numericalintegration 1 to a sum over the symmetrization permutation of (N − 3) numericalintegrations (see Sec. 3.4.2 and Appendix G). This result implies that for N = 3 thematrix element evaluation can be done analytically and that for N = 4, it requires asum of one-dimensional numerical integrations.
The next three subsections discuss the implementation of the CGHS. Many of
the techniques used in the standard CG method can be directly used in the CGHS. For
example, the selection and symmetrization of the basis function can be directly appliedin the CGHS. Also, the optimization through the VMS method can be used to optimize
the basis set at different values of the hyperradius R. Subsection 3.4.1 describes how thehyperangular Schrödinger equation (Eq. 2.26) can be solved using a basis set expansion.
Subsection 3.4.2 shows, as an example, how the unsymmetrized matrix elements can becalculated analytically for a three particle system (the calculation of the unsymmetrized
matrix elements for N = 4 is presented in the Appendix G). Finally, subsection 3.4.3
discuss in general how this method is implemented.

3.4.1

Expansion of the channel function in a basis set

In the hyperspherical method (see Sec. 2.5), channel functions are solutions of
the adiabatic Hamiltonian HA (R; Ω),
HA (R; Ω)Φν (R; Ω) = Uν (R)Φν (R; Ω).
1(3.23)The (3N-7) numerical integration results from the following reasoning: initially we have 3N numerical integration but 3 dimensions are removed by decoupling the center of mass motion, 3 dimension are
removed fixing the Euler angles and 1 dimension is removed fixing R.

43The eigenvalues of this equation are the hyperspherical potential curves Uν (R). The
adiabatic Hamiltonian has the form,
HA (R; Ω) =~2 Λ2(d − 1)(d − 3)~2+
+ V (R, Ω).
2µR2
8µR2

(3.24)

Here, d = 3NJ where NJ is the number of Jacobi vectors.
A standard way to solve Eq. (3.23) is to expand the channel functions in a basis,|Φµ (R; Ω)i =X

ciµ (R) |Bi (R; Ω)i .

(3.25)i

Here µ labels the channel function. The |Bi (R; Ω)i are the basis functions. With thisexpansion, Eq. (3.23) reduces to the eigenvalue equation
HA (R)~cµ = Uµ (R)O(R)~cµ .

(3.26)The vectors ~cµ = {c1µ , ..., cDµ }, where D is the dimension of the basis set. HA and O are
the Hamiltonian and overlap matrices whose matrix elements are given by¯¯
HA (R)ij = hBi |HA (R; Ω)|Bj i ¯ ,R¯¯O(R)ij = hBi |Bj i ¯ .
R(3.27)(3.28)

Once the hyperradial potential curves are calculated, we still need to evaluate the
non-adiabatic couplings between the channel functions. The P and Q couplings, definedin Sec. 2.5, aredΦµ (R)i ,dRddQνµ = hΦν (R)|
Φµ (R)i .
dRdRPνµ = hΦν (R)|

(3.29)(3.30)In the basis set expansion, the couplings P and Q can be calculated as matrix multiplication. Using the expansion in Eq. (3.25),
X dciµ (R)dBi (R)d|Φµ (R)i =, |Bi (R)i + ciµ (R) |
i.
dR
dR
dR
i

(3.31)

44
For simplicity, we have omitted the angular dependence. The P coupling takes the form
Pνµ =

X

cjν (R)ij

dciµ (R)dBi (R)hBj (R)|Bi (R)i + cjν (R)ciµ (R) hBj (R)|idR
dR

(3.32)

which can be written as simple matrix multiplication,Pνµ = ~cTν O(R)~c˙µ + ~cTν P(R)~cµ .

(3.33)

Here and in the following, ~c˙ is the derivative of ~c with respect to R and P(R) is definedin Eq. (3.36). The same procedure can be done for the Q couplings withQνµ =X dcjν (R) dciµ (R)
ij

dR

dR

cjν (R)hBj (R)|Bi (R)i +dcjν (R) idBi (R)cµ (R) hBj (R)|idRdR

dciµ (R) dBj (R)dBj (R) dBi (R)h|Bi (R)i + cjν (R)ciµ (R) h|i (3.34)
dR
dR
dR
dR

and can also be written as a matrix multiplication.
Qνµ = ~c˙Tν O(R)~c˙µ + ~c˙Tν P(R)~cµ + ~cTν P T (R)~c˙µ + ~cTν Q(R)~cµ .

(3.35)In Eqs. (3.33, 3.35) we have used the overlap matrix O and defined the matrices P and
Q whose matrix elements are
dBi (R)i,dR
dBj (R) dBi (R)Q(R)ij = h
|
i.
dR
dR
P(R)ij = hBj (R)|

(3.36)(3.37)The derivatives of the cjµ (R) coefficients that form the ~c˙µ are calculated numerically using the three point rule.

3.4.2

Unsymmetrized Matrix Elements evaluation of three particles

A basis function in terms of the interparticle distances isΨ(r12 , r13 , r23 ) = exp(−

2
2
2
r13
r23
r12
)
exp(−
)
exp(−
).
2d212
2d213
2d223

(3.38)

45
These basis functions do not depend on the center-of-mass coordinate and can only
reproduce the J P = 0+ spectrum. For equal mass systems, we can write Eq.(3.38) interms of the following Jacobi coordinates:1x1 = √ (r1 − r2 ),
2
r µ
¶
2
r1 + r2
x2 =
r3 −
.
3
2

(3.39)(3.40)The basis functions [Eq. (3.38)] can be written as|Ai = exp(−

xT .A.xx1 .x1 a11 + 2x1 .x2 a12 + x2 .x2 a22) = exp(−)
22(3.41)

where x ≡ {x1 , x2 } and A is a 2 by 2 symmetric matrix whose elements are a11 =√2/d212 +1/2(1/d213 +1/d223 ) , a12 = a21 = 3(1/d223 −1/d213 ), and a22 = 3/2(1/d213 +1/d223 ).
In Eq. (3.41), we can clearly see that the state |Ai depends only on the distances x1 ,and x2 plus the angle θ12 between them, cos θ12 = x1 .x2 /x1 x2 .
We want to obtain the matrix elements corresponding to these basis function at
fixed hyperradius R. We define the hyperradius to be R2 = x21 + x22 . The integrand of
the overlap matrix element is
B.A = exp(−

xT .(A + B).x).
2

(3.42)We change to the Jacobi basis set that diagonalizes A + B and we call β1 and β2 theeigenvalues and y ≡ {y1 , y2 } the eigenvectors. In this new coordinate basis, Eq. (3.42)
has a simple form,
B.A = exp(−

β1 y12 + β2 y22).
2(3.43)

We integrate over the angles of the vectors y1 and y2 and we fix the hyperradius, so
y1 = R cos θ and y2 = R sin θ. In this set of coordinates, the matrix element at fixed R
is
Z π/2
¯
β1 R2 cos2 θ + β2 R2 sin2 θ
¯
2
exp(−
) cos2 θ sin2 θdθ
hB|Ai ¯ = (4π)
2R0(3.44)

46
This integration has an analytical form,¶µ
β1 +β2 2¯¯
3 exp(− 4 R )2 β1 − β2hB|Ai ¯ = 8π
I1 R
R2 (β1 − β2 )4R(3.45)

To simplify the interaction matrix element evaluation, we can use a Gaussianmodel potential as we have done in the CG method. In this case, the interaction term canbe evaluated in the same way we have calculated the overlap term since the interactionr2

is also a Gaussian. Each pairwise interaction can be written as Vij = V0 exp(− 2dij2 ) =0

V0 exp(−xT .M (ij) .x/(2d20 )) (see Appendix C for the definition of M (ij) ). Therefore, tocalculate the interaction matrix element, we need to evaluateZdΩ exp(−hB|Vij |Ai = V0xT .(A + B + M (ij) /d20 ).x).
2(3.46)

This integration can be done following the same steps of the overlap matrix element.
Equation (3.45) can be used directly if we multiply it by V0 , and β1 and β2 are replacedby the eigenvalues of A + B + M (ij) /d20 . Note that for each pairwise interaction, thematrix M (ij) changes and requires a new evaluation of the eigenvalues.
The third term we need to evaluate is the hyperangular kinetic term at fixed R.
This kinetic term is proportional to the grand angular momentum operator Λ definedfor the N = 3 case as
X ~2 ∇2
Λ2 ~2~2 1 ∂ 5 ∂
i
+
R
.
=
−
2µR2
2µ
2µ R5 ∂R ∂R(3.47)i

The expression can be formally written asTΩ = TT − TR ,(3.48)

where
TΩ =

Λ2 ~2
,
2µR2

TT = −

X ~2 ∇2ii

2µ, and TR = −~2 1 ∂ 5 ∂R.
2µ R5 ∂R ∂R(3.49)

In typical calculations, TΩ is evaluated by directly applying the corresponding derivatives
in the hyperangles Ω. However, in this case, it is convenient to evaluate TT and TR
separately, and make use of (3.48).

47
The integrand of the total kinetic term TT takes the form
Ã 2
!
X ~2
xT .B.x
xT .A.x
B|TT |A = exp(−) −∇2i exp(−
).
2
2µ
2

(3.50)iFirst, we diagonalize A and use the eigenvectors and eigenvalues of A, obtaining,
B|TT |A = −¢xT .(A + B).x
~2 ¡−Tr[A] + xT .A2 .x exp(−).
2µ2(3.51)

Here Tr is the trace function. We can use Tr[A] = (α1 + α2 ), where α1 and α2 are theeigenvalues of A. Now we diagonalize A+B. We call T the matrix with the orthonormal
eigenstates in columns and β1 and β2 are the eigenvalues of A + B. We make a changeof coordinates to the basis set that diagonalizes A + B. We obtain
B|TT |A = −~2
β1 y12 + β2 y22(−3(α1 + α2 ) + y.G.y) exp(−),
2µ2(3.52)

where G = T T .A2 .T , and y1 and y2 are the vectors in the new eigen basis. The
integration over the angles of these vectors is trivial. After this integration, we fix thehyperradius and integrate over the hyperangle θ defined by y1 = R cos θ and y2 = R sin θ,Z¯o(4π)2 π/2 n¯hB|TT |Ai ¯ = −− 3(α1 + α2 ) + g11 R2 cos2 θ + g22 R2 sin2 θ2
R0
β1 R2 cos2 θ + β2 R2 sin2 θ) cos2 θ sin2 θdθ. (3.53)
exp(−
2This integration can be done analytically and the results expressed in terms of the Bessel
functions I1 and I0 :(β +β )R2½·¸¯
n
− 1 22π318e¯
2hB|TT |Ai ¯ = −−8(g11 − g22 )I0 (β1 − β2 )R +8(g11 − g22 )4(β1 − β2 )4
(β1 − β2 )R2
R
·
¸¾
¡
¢o
1
2
2
+(β1 − β2 ) −6(α1 + α2 ) + (g11 + g22 )RI1 (β1 − β2 )R. (3.54)
4

Now we will evaluate TR , the hyperradial kinetic term. It is written asµ¶~2
1 ∂ 2 5/2
15
TR = −R −.
2µ R5/2 ∂R2
4R2
Therefore, the integrand takes the form
µ
¶
µ T
¶
xT .B.x
1 ∂ 2 5/2
15
x .A.x
~2
)
R −
exp −
,
B|TR |A = − exp(−
2µ
2
4R2
2
R5/2 ∂R2(3.55)(3.56)

48
We use the property that xT .A.x = R2 FA (Ω) to evaluate the derivatives withrespect to R. This allows a simple application of the derivatives in Eq. (3.56), yielding· T¸¤x .(A + B).x
~2 £ TT
2
6x .A.x − (x .A.x) exp −.
(3.57)
B|TR |A =2µR2
2
We diagonalize A + B. Defining D = T T .A.T , we obtain
µ
¶
¤
~2 £
β1 y12 + β2 y22
2
B|TR |A =6y.D.y−(y.D.y)
exp
−
2µR2
2

(3.58)The terms y.D.y and (y.D.y)2 depend on the polar angles of the vectors. The integration
over the polar angles (Ω1 = {φ1 , θ1 } and Ω2 = {φ2 , θ2 }) of these terms is
Z

n
£
¤
£
¤o
6y.D.y − (y.D.y)2 dΩ1 dΩ2 = (4π)2 6d11 y12 +6d22 y22 − d211 y14 +(2d11 d22 +4d212 /3)y12 y22 +d222 y24 .
(3.59)

Now we carry out the integration over the hyperangle θ. Using y1 = R cos θ and y2 =R sin θ, we obtainZ¯(4π)2 ~2 π/2 n¯hB|TR |Ai ¯ =6d11 R2 cos2 θ + 6d22 R2 sin2 θ−
2µR2 0Rod211 R4 cos4 θ − (2d11 d22 + 4d212 /3)R4 cos2 θ sin2 θ − d222 R4 sin4 θ¶µβ1 R2 cos2 θ + β2 R2 sin2 θcos2 θ sin2 θdθ. (3.60)
exp −2This integration has the analytical form2

(β1 +β2 )R
¯
h
¡
4
~2 e−
π3 n
¯
hB|TR |Ai ¯ = −−8− 8d212 + (d11 − d22 ) 6(−β1 + β2 + d11 − d22 )+2
2µ 4R (β1 − β2 )
R
¸
·
h
¢i
8
1
2
2
(β1 − β2 )(d11 + d22 )R I0 (β1 − β2 )R +− 64d212 +4(β1 − β2 )R248(d11 − d22 )(−β1 + β2 + d11 − d22 ) + 2(β1 − β2 )(−3β1 + 3β2 + 4d11 − 4d22 )(d11 + d22 )R2
¸¾
i ·1
2
2 2
2
4
. (3.61)+(β1 − β2 ) (d11 + d22 )R I1 (β1 − β2 )R
4Combining Eqs. (3.54, 3.61), we obtain TΩ . The expression for TΩ can be simplified using the relation G = D2 to write G matrix elements of Eq. (3.54) in terms of theones of D.

49
A useful test to verify the functional form of the matrix elements is to integratethem with respect to R, with the corresponding volume element, and compare that
result with the CG matrix elements presented in Appendix B. Another important test
is to verify that TΩ is symmetric under the exchange of the basis functions A and B.
This is not a trivial test since neither TT nor TR are symmetric. Both tests can be
carried out numerically.

3.4.3

General considerations

Many of the procedures of the standard CG method can be easily extended to the
CGHS. The selection, symmetrization, and optimization of a basis follow the same ideasof the standard CG method (see Appendices A, C, D, F and E). However, the evaluationof the unsymmetrized matrix elements at fixed R is clearly different. Furthermore, thehyperangular Hamiltonian [Eq. 3.23] need to solved at different hyperradius R.
There are several properties that makes this method particularly efficient. For
the model potential used, the scattering length is tuned by varying the potential depthsof the two-body interaction. Therefore, as in the CG case, the matrix elements need
only be calculated once; then they can be used for a wide range of scattering lengths.
Of course, the basis set should be complete enough to describe the relevant potential
curves at all the desired scattering length values.
The selection of the basis function generally depends on R. To avoid numericalproblems, the mean hyperradius of each basis function hRiB should be of the same orderof the hyperradius R in which the matrix elements are evaluated. We can ensure that
hRiB ∼ R by selecting some (or all) the weights dij to be of the order of R.
We consider two different optimization procedures. The first possible optimization
procedure is the following: First, we select a few basis functions and optimized them todescribe the lowest hyperspherical harmonics. The widths of these basis functions arerescaled by R at each hyperradius so that they represent the hyperspherical harmonics

50
equally well. These basis functions are used at all R, while the remaining are optimized
at each R. Starting from small R (of the order of the range of the potential), we optimize
a set of basis functions. As R is increased, the basis set is increased and reoptimized.
At every R step, only a fraction of the basis set is optimized, and those basis functions
are selected randomly. After a several R-steps, the basis set is increased.
Instead of optimizing the basis set at each R, one can alternatively try to create
a complete basis set at large Rmax . In this case, the basis functions should be completeenough to describe the lowest channel functions with interparticle distances varying frominteraction range r0 up to the hyperradius Rmax . Such a basis set can be rescaled to any
R < Rmax and should efficiently describe the channel functions at that R. The rescaling
procedure is simply dij /R = dmax
ij /Rmax . This procedure avoids the optimization ateach R. Furthermore, the kinetic, overlap, and couplings matrix elements at R are
straightforwardly related with the ones at Rmax . So, the interaction potential is the
only matrix element that need to be recalculated at each R. This property can beunderstood by dimensional analysis. The kinetic, overlap and couplings matrix elements
only depend on R, so a rescaling of the widths is straightforward related to a rescalingof the matrix elements. In contrast, the interaction potential introduces a new lengthscale, so the matrix elements depends on both R and d0 , and the rescaling does not
work.
These two methods, the complete basis set or the small optimized basis set, canbe appropriate in different circumstances. If a large number of channels are needed,
probably the complete basis method is the best choice. But, if only a couple of particularchannels are needed, then the optimization might be more efficient.
The most convenient way we have found to optimize the basis function in thefour-boson and four-fermion problem is the following: First we select an hyperradius
Rm that is Rm ≈ 300 d0 where the basis function will be initially optimized. The basis
set is increased and optimized until the relevant potential curves are converged and,

51
in that sense, the basis is complete. This basis is then rescaled, as proposed in thesecond optimization method, to all R < Rm . For R > Rm , it is too expensive to have a“complete” basis set. For that reason, we use the first optimization method that allow
a reliable description of the lowest potential curves.
Note that for standard correlated Gaussian calculations, the matrices A and B
need to be positive definite. This condition restricts the Hilbert space to exponentiallydecaying functions. In the hyperspherical treatment, this is not necessary since thematrix elements can be always calculated at fixed R, even for exponentially growing
functions. This gives more flexibility in choosing the optimal basis functions.

Chapter 4
Renormalized Mean-Field Methods

4.1

Introduction
In this chapter, we analyze the behavior of a two-component Fermi gas in thenegative-scattering-length or BCS region. In recent years, the BCS-BEC crossover problem has become experimentally accessible [82, 99, 195, 104], enabling tests of different
many-body theories. The BCS theory has been successful in explaining superfluidity
in Fermi gases, but this theory is incomplete because it neglects the Hartree term of
the interaction, 4πaρ/m (ρ is the density of one spin component). Comparatively little
research has included the Hartree term in considering Fermi gases [76, 113, 33, 86, 54].
In the cases the Hartree term was included, it was primarily to study the perturbative
regime where ρa3s << 1. These studies usually analyze the “normal state” of the gas,and contrasts with the superfluid state that is study with the BCS theory. Quantum
Monte Carlo (QMC) simulations include (in a sense) both the Hartree term and pairing
physics, but a complete theory that contains both ingredients is still required. A recent
approach [54] started from the mean-field BCS theory and reproduced the Hartree termby including quantum fluctuations. Therefore, this theory succeeded in reproducing
both normal and superfluid contributions. However, its validity in the strongly interacting regime and the importance of higher order corrections are still unclear. This
chapter analyzes mean-field theories with zero-range interactions that reproduce thecorrect Hartree term in the perturbative limit.

53
The Hartree term can be easily incorporated using a mean field approach andeffective zero-range interactions. When the range of interaction is much smaller than
the interparticle distance, the potential can be replaced by a delta-function interaction. However, this replacement must be done with caution because the delta function
interaction is too singular to be exactly solvable, even in principle. In the weaklyinteracting limit, ρa3s << 1, the coupling parameter in the delta-function interaction
is proportional to the two-body scattering length as ; this interaction is known as the
Fermi pseudopotential [73]. Here, we analyze the first-order corrections introduced by
the Fermi pseudopotential. By including this approximation beyond first-order correction, mean-field theories have been applied to Fermi gases [33, 149]. However, the use ofthis approximation in strongly interacting regimes leads to an unphysical collapse of the
Fermi gas. To go beyond the weakly interacting regime using a zero-range pseudopotential, it is crucial to renormalize the coupling constant. In this Chapter, we introduce
a new and convenient way to achieve this renormalization.
The renormalization strategy presented here is based on a zero range interactionof the Fermi pseudopotential form but with as replaced by an effective scattering length
aef f . The aef f is then calculated by using the exact energies of few particles in a trap.
We compare many-particle predictions obtained by using the renormalized interaction
potential with diffusion Monte Carlo simulations and alternative mean-field calculations.
We find that our renormalization automatically gives the correct behavior in both thestrong and weak interaction limits, for both positive- and negative-scattering lengths.
This Chapter is organized as follows. In Section 4.2, we review the noninteracting
solutions of the trapped two-component Fermi gas and analyzes corrections introducedby the Fermi pseudopotential. Section 4.3 describes the general ideas of the density
renormalization. Section 4.4 develops the renormalization procedure and shows that asimple two-parameter analytical formula can be utilized as an excellent approximation
over the whole range from positive-to-negative–two-body scattering lengths. Section 4.5

54
applies the renormalization to many-particle–mean-field theory and presents some of its
predictions. Section 4.6 compares the results obtained using our renormalized meanfield theory with quantum Monte Carlo calculations and with perturbative mean-fieldcalculations. Finally, in Section 4.7, we present the conclusions.

4.2

Perturbative behavior of a trapped two-component Fermi gas
In this section, we analyze the weakly interacting Fermi gas in a harmonic trap.

First, we review the solutions of noninteracting Hamiltonian with an arbitrary numberof particles [N1 in component one and N2 in component two]. Then, we discuss theuse of the Fermi pseudopotential to analyze first-order corrections. Finally, we apply
degenerate perturbation theory to analyze open-shell systems.

4.2.1

The noninteracting two-component Fermi gas in a trap

Consider an N -particle system in a spherical trap. In the noninteracting limit, theHamiltonian of the N -particle system can be expanded in single-particle Hamiltonians,
e.g.,
H=

N
X

h(ri ),

(4.1)

i=1

where the single-particle Hamiltonians in spherical coordinates areh(r) = −~2 2 mω 2 r2
∇ +
.
2m
2

(4.2)

Here, ω is the frequency of the trap and m is the mass of the particles. The singleparticle Hamiltonian is spherically symmetric and describes a particle in a trap. The
single-particle Schrödinger equation ish(r)ψnlm = ²nl ψnlm ,(4.3)

where the eigenfunctions ψnlm can be written asψnlm (r) =unl (r)Ylm (Ω),
r

(4.4)

55
where Ylm (Ω) are the standard spherical harmonics, and Ω = {θ, φ} are the angularcoordinates. The radial orbitals are solutions of an effective one-dimensional Schrödinger
equation given bys
3/2

unl (r) = aho
where aho =2n!
r2exp(− 2 )(r/aho )l+1 Ll+1/2(r2 /a2ho ),nΓ(n + l + 3/2)
2aho

(4.5)

p
~/(mω) is the trap length. The single particle energies are²nl = (2n + l + 3/2)~ω,(4.6)

where n = 0, 1... is the number of radial nodes, and l = 0, 1, ... is the orbital angularmomentum.
Up to this point, we have omitted the spin dependence in the orbitals. The different components of the Fermi gas can be treated as distinguishable particles whichsimplifies the calculation. In this case, the spin part of the many-body wave functions
completely factors out. The spatial part of the many-body wave functions is now antisymmetric in each component but there is no symmetrization between particles ofcomponent one and two. Therefore, the spatial wavefunction of a system with N1 particles in component 1 and N2 particles in component 2, where N1 + N2 = N ; the spatial
wave function is given by
ΨkN I = A(φni li mi (r1 ), · · · , φn1 l1 m1 (rN1 )) A(φni li mi (r10 ), · · · , φnf lf mf (rN20 )) =
1 1 1
2 2 2
2 2 2
¯
¯
¯
¯
¯
¯
¯
¯
¯ φ i i i (r1 ) · · · φ i i i (rN ) ¯¯ φ i i i (r 0 ) · · · φ i i i (r 0 ) ¯1¯ n1 l1 m1¯¯ n2 l2 m2 1
n1 l1 m1
n2 l2 m2 N2 ¯
¯
¯
¯
¯
¯
¯
¯
¯
..
..
..
..
det ¯
¯ det ¯¯ . (4.7)
.
.
.
.
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯ φnf lf mf (r1 ) · · · φnf lf mf (rN1 ) ¯¯ φnf lf mf (r10 ) · · · φnf lf mf (rN20 ) ¯
1 1

1

1 1

1

2 2

2

2 2

2

Here, unprimed coordinates refer to component 1 and primed coordinates to component 2. The quantum numbers {ni1 l1i mi1 } and {nf1 l1f mf1 } correspond to the first andlast occupied orbitals of component 1. Similarly, the quantum numbers {ni2 l2i mi2 } and{nf2 l2f mf2 } correspond to the first and last occupied orbitals of component 2. These setsof orbitals characterize the wave function that is labeled by a general quantum number

56
k. A is the antisymmetrization operator that can be realized by constructing a Slater
determinant [second line of Eq. (4.7)]. The Slater determinant vanishes if an orbital isdoubly occupied. Thus, the wave function ΨiN I will be nonzero only if all the orbitals
in each Slater determinant are different. This is just a mathematical manifestation of
the Pauli exclusion principle.
Any ΨiN I constructed as in Eq. (4.7) is an eigenfunction of the Hamiltonian (4.1)
with energy
nf2 l2f mf2

nf1 l1f mf1

X

k
EN
I =

X

²nl +

²nl .

(4.8)

nlm=ni2 l2i mi2

nlm=ni1 l1i mi1

Thus, to construct the ground state, we just need to minimize Eq. (4.8). This is achievedby simply selecting the N1 and N2 orbitals with the lowest energy.
10²nl /h̄ω8

Ncs = 84
Ncs = 56

6

Ncs = 35
Ncs = 20

4

Ncs = 10
Ncs = 420

Ncs = 1

0

2

4

6

8

l
Figure 4.1: Orbital energies ²nl as a function of the quantum number l. The number ofparticles in a closed shell Ncs is written next to each shell.

As in Eq. (4.8), the ground-state noninteracting energy EN I can be evaluated asp
the sum of the noninteracting energies of polarized Fermi gases EN
I with N1 and N2
pp
particles, i.e., EN I (N ) = EN
I (N1 ) + EN I (N2 ). To select the lowest Ni orbitals, we firstconsider Fig. 4.1. Here, the energies of the orbitals as functions of the angular quantum
number l show a clear shell structure. In each shell, the energies of the orbitals are57(ns + 1/2)~ω and degeneracy of the orbitals is ns (ns + 1)/2. The shells are closed at
the magic numbers
Ncs =ns (ns + 1)(ns + 2)
.
6

(4.9)

The ground-state energy of the polarized system in a closed shell has a simple closedform
cs (n )
EN(ns − 1)ns (ns + 1)(ns + 2) 3N cs
I s
=
+
.
~ω
8
2

(4.10)

p
The ENI (Ni ) can be written in terms of the shell number ns , the energy of the closed
cs (n ), and the corresponding magic number N cs :shell subsystem EN
I s

µ
p
cs
EN
I (Ni ) = EN I (ns ) +¶3+ ns (N − Ncs )~ω.
2

(4.11)On the right hand side of Eq. (4.11), the first term is the contribution from the closedshell fermions while the second term is the contribution from the fermions in the open
shell. Here, the shell number ns represents the number of closed shells. It can be written
in terms of the number of particles Ni as·
ns = Intwhere¸
g[Ni ]
1+−1 ,g[Ni ]
3(4.12)s µ
g[Ni ] =3¶q2
3 27Ni − 3(243Ni − 1) ,(4.13)and Int[x] is the integer part of x.
For most N values, the ground-state solutions are degenerate. Only if N1 and N2
correspond to the magic Ncs is the ground state system not degenerate. Such closed-shell
systems are special cases in which all the orbitals couple to zero-angular momentum.
The degeneracy for open-shell systems is generally broken by interactions.

4.2.2

Effective versus exact wave function

Here, we discuss first order corrections to the noninteracting energy and the use ofzero-range potential instead of finite-range potential. Note that the zero-range potential58
presented here provides a qualitatively different treatment of correlations in the systemthan the zero-range potential presented in Sect.2.2.
Consider the two-component Fermi gas interacting through a short-range potential Vint . Consider each component as a distinguishable particle such that the noninteracting wave functions can be written as a product of two Slater determinants, i.e.,
Eq. (4.7). If the system is dilute, the interaction only depends on the scattering length
as . However, the first-order perturbation theory correction is
Z Z
E1 = hVint i =ρ1 (r1 )Vint (r1 − r2 )ρ2 (r2 )dr1 dr2 .

(4.14)

Here, ρ1 and ρ2 are the densities of components 1 and 2, respectively. Clearly, thiscorrection depends on the shape of the two-body potential Vint . To obtain the scattering
length as dependence we need to include many orders in perturbation theory. In otherwords, we have to be able to describe correlations of the order of the range of theinteraction r0 . Such a description would imply a large configuration interaction (CI)
expansion. This treatment is impossible in large systems.
An alternative option for describing the correction due to the scattering properties
of Vint can be obtained using a Jastrow term to describe the correlations:Ψ = J(r110 , · · · , r220 )ΨN I (r1 , · · · , rN ).

(4.15)The Jastrow-Slater wave function was discussed in the fixed-node diffusion Monte Carlo(FN-DMC) section in chapter 3. In the Jastrow term, all the pair correlations are
explicitly built from the two-body solutions and describe the correct scattering behavior. Thus, the energy correction depends mainly on a and not on the form of Vint .
However, it is very difficult to work with Eq. (4.15), and its exact implementation isusually restricted to Monte Carlo methods. There are other methods like lowest-orderconstrained variational [41, 100] that use Eq. (4.15) as a starting point but require someapproximations to obtain the energy of the system.

59
To avoid similarly complicated techniques that deal with real two-body potentials,
Fermi proposed a zero-range pseudopotential that explicitly depends on the scattering
length and not on the shape of the real potential. This pseudopotential is expressed as
4π~2 a0δ(r).
m

VF (r) =(4.16)

Here as is the two-body scattering length, and m is the mass of the particles. The
first-order correction of this potential isZ
4π~2 a0F
E1 = hVp i =mρ1 (r)ρ2 (r)dr.

(4.17)

This first-order correction depends on the scattering length and not on the shape of
the potential. In the dilute limit, both the Jastrow correlations or the CI expansion for
the realistic potential should lead to the same correction expressed in Eq. (4.17). With
this pseudopotential, we are able to reproduce the correct energy correction without including the corresponding correlations. Thus, in the Fermi pseudopotential philosophy,
the exact solution is replaced by a mean-field solution. In that way, the many-bodyproblem is greatly simplified. All the information about the correlations is lost but the
new problem reproduces mean-field properties.

4.2.3

Perturbative behavior of two-component Fermi gas

Here, we use the Fermi pseudopotential to analyze how the degeneracy in open
shells is broken in first-order perturbation theory. The Fermi pseudopotential [Eq. (4.16)]
can be used to describe the atomic ground state in weakly interacting regimes where|as | ¿ aho . To obtain the correct first-order correction, we use degenerate perturbation
theory. The wave function in a first order correction Ψ(1) is expanded in terms of all
the degenerate noninteracting states asΨ(1) =

X
k {E k =EN I }ck ΨkN I(4.18)

60
Here, the sum is over all the states ΨkN I whose energy corresponds to the groundstate noninteracting energy EN I . The coefficients ck are obtained by diagonalizing thedegenerate submatrix of the Hamiltonian in this basis. Different ΨkN I states are coupledwith each other through the Fermi pseudopotential.
We can use the fact that Eq. (4.18) resembles a CI expansion to evaluate thematrix elements. Since the noninteracting wave functions ΨkN I are formed with singleparticle orbitals, only states that differ by less than one orbital per component arecoupled. Furthermore, the interaction matrix elements can be easily expanded in matrix
elements of the occupied single-particle orbitals.
Since the potential and interactions are spherically symmetric, the total angularmomentum is a good quantum number. Therefore, we could could couple states ΨkN I
to a given total angular momentum and diagonalize the Hamiltonian in that reduced
representation. However, since we are only going to consider small N systems, we candiagonalize the “entire” Hamiltonian matrix and then analyze the angular momentumof the solutions.

CN1 ,N2 h̄ω/EN I

0.4

0.35

0.3

0.25

0.2

5

10

15

20

N

Figure 4.2: CN1 ,N2 coefficients divided by EN I as a function of N . Circles correspondto L = 0 ground states, squares to L = 1 ground states and triangles to L = 2 groundstates. A solid line connects the odd-N values, while a dashed line connects the even-N
values. Figure from Ref. [185].

61
Degenerate perturbation theory predicts an energy
E ≈ EN I + ~ω CN1 ,N2

as
,
aho

(4.19)

where CN1 ,N2 is a dimensionless quantity obtained through diagonalization. When bothN1 and N2 correspond to closed shells there is no degeneracy; then CN1 ,N2 can bestraightforwardly calculated to be [184]Z
CN1 ,N2 = 4πa3ho

I
I
ρN
r)ρN
r)d~r.
1 (~2 (~(4.20)I r ) is the density of a one-component non-interacting gas with N fermions ofHere, ρNii (~RI r )d~r = Ni . Alternatively, one can approximate themass mi , normalized so that ρNi (~I with Thomas-Fermi density profiles. This approximation should be quite accurateρNiin the large N limit but it cannot describe the splitting of the energies related to the
angular momentum of the states.
To obtain the CN1 ,N2 for open-shell systems, we apply first-order degenerate perturbation theory. This calculation additionally allows us to obtain the angular momentum quantum number L of the ground state. Figure 4.2 and Table 4.1 present theresults for N ≤ 20. The coefficients CN1 ,N2 increase monotonically with increasing N
and show a slight odd-even staggering. In general, the coefficients CN1 ,N2 for even N
are comparatively higher than those for odd N , implying a smaller energy for even N
than for odd N and suggesting that, even in the perturbative regime, the odd-evenoscillations are already present. We note that the CN1 ,N2 coefficients for even N shownin Fig. 4.2 clearly reflect the shell closure at N = 8.

4.3

Density renormalization idea
The use of the Fermi pseudopotential has been extended beyond first-order per-turbation theory [20]. This extension has to be done with caution because a full diagonalization of a Hamiltonian with an attractive delta-function interaction (in 3D) would

62
Table 4.1: Angular momentum L and coefficients CN1 ,N2 for the ground state of twocomponent Fermi gases in the weakly attractive regime. Here, we consider N2 = N1 foreven N and N1 = N2 + 1 for odd N .
N
2
3
4
5
6
7
8
9
10
11

L
0
1
0
1
0
1
0
0
0
0

CN1 ,N2
√2
2π
√3
2π
13
√
2 2π
15
√
2 2π
√11
2π
√12
2π
31
√
2 2π
145
√
8 2π

9.21052
10.1980

N
12
13
14
15
16
17
18
19
20

L
0
0
0
2
0
2
0
2
0

CN1 ,N2
12.2274
13.1651
15.2382
16.164218.2445
19.1735
21.2476
1779
√
32 2π
1945
√
32 2π

always lead to an infinitely bound state. This unphysical collapse can be avoided if thedescription of the system is restricted to a mean-field level of approximation.
For example, in Ref. [20], the Gross-Pitaevskii equation with Fermi-pseudopotential
interactions is solved to describe a BEC. The Gross-Pitaevskii equation emerges frommean-field theory, and, for that reason, the solutions are well behaved. When the scattering length becomes large enough in magnitude and negative, the system collapses.
However, this is not an artifact of the zero-range interaction, but a real physical effectthat has been experimentally verified [151].
A similar approach can be used to analyze fermionic systems. For two-component
Fermi gases, the use of the Fermi pseudopotential can also be extended beyond firstorder perturbation theory. For this system, we would use the Hartree-Fock (HF) levelof approximation, which produces well-behaved results for small as . However, for large
and negative scattering lengths, the HF solutions collapses. This collapse occurs in a
regime where the Fermi pseudopotential approximation is no longer justified. In thissection, we propose a renormalization method to extend the validity of a zero-range
pseudopotential to large scattering lengths.

63
For the remainder of the Chapter, we consider a system of fermions in a sphericallysymmetric harmonic oscillator trap at temperature T = 0. The Hamiltonian that we
adopt is¶ X
X µ ~24π~2 aef f
1
2
2 2
H=
−
∇i + mω ri +δ(ri − ri0 ).
2m
2
m
0
i

(4.21)

i<iThis Hamiltonian cannot be diagonalized exactly, since the delta-function interaction
is too singular and would produce divergent results [62]. Therefore, full diagonalizationof a Hamiltonian with delta-function interactions requires a momentum cut-off renormalization even in the weakly interacting limit. This type of renormalization has been
carried out, for example, in Refs.[107, 28, 114, 152, 98, 165, 156, 131, 108], to name afew such studies. However, such a renormalization is unnecessary at small or modest
scattering lengths when the system is studied at a mean-field level with zero-range interactions. The mean-field theories are well behaved in this limit. To go beyond the weaklyinteracting limit of mean-field theory, we propose a density-dependent renormalizationof the coupling parameter that is intended to apply even in the long-wavelength limit.
The idea is to extract aef f from the calculation of a few-body system that can besolved exactly. The premise is that the renormalized scattering length extracted fromfew-body systems will be close to the aef f that governs the large N limit. The level ofapproximation we adopt to solve the many-body Hamiltonian is the same one we use
to solve the corresponding Schrödinger equation for a few-body system. For example, if
we want to diagonalize this Hamiltonian in the random-phase approximation (RPA), wewould use RPA for the few-body system and obtain the renormalization by matchingexact and RPA observables.
However, it is important to carefully select the appropriate level of approximation. An explicitly correlated wave function or an extensive configuration interaction(CI) wave function can produce divergent results, and a momentum cutoff renormalization is necessary. Since we want to obtain and apply a density renormalization without

64
the necessity of introducing a momentum renormalization, we need to carefully selectthe level of approximation in the wave function. The HF approximation does not introduce explicit interparticle correlations, as the only correlations included are “exchange
correlations” from the Pauli exclusion principle. This makes the HF wave function asuitable approximation to adopt in our renormalization technique, since it does not
require a momentum renormalization.
Before proceeding with the extraction of aef f , it is important to analyze its functional dependence. Since we want to extract aef f valid in the large N limit, we consider
the homogenous system. Dimensional analysis suggests that, in an infinite, dilute anduniform Fermi gas the only parameter that characterizes the behavior of the system isthe dimensionless combination kf as of the Fermi momentum kf and as [113]. Throughout this chapter the Fermi momentum is defined as kf ≡ (6π 2 ρ)1/3 where the density isjust the one-spin component density. If we were applying the renormalized scattering
length aef f to an infinite uniform system, the only relevant parameter would be kf aef f .
This suggests that kf aef f has to be a function of kf as . So, we propose the following
functional dependence for the effective scattering length:
aef f ≡

ζ(kf as ).
kf(4.22)We see below that the renormalization function, ζ(kf as ), will have the desired behavior
in the limiting cases, becoming independent of kf as in the unitarity limit (|as | → ∞)and reproducing the relation aef f = as in the weakly interacting limit (kf as ¿ 1). The
unrenormalized Fermi pseudopotential can be obtained by setting the renormalization
function to ζ F (kf as ) = kf as . We consider that Eq.(4.22) holds even with the inclusion of a trapping potential. The renormalized scattering length aef f can be viewed
as accounting, to some extend, for the correlations neglected in the mean-field wave
functions.
The renormalization function is intimately related with the equation of state,

65
as we will show in using the T-F approximation. Therefore, sophisticated many-bodysolutions of the energy per particle in the BCS-BEC crossover can be readily used to
extract the renormalization function. For example, quantum Monte Carlo results from
Ref. [7] provide probably the most reliable solutions to extract ζ. However, it is not in
the spirit of the method presented here to use such a sophisticated method to extract
ζ, but, rather to use simple few-body solution to obtain an approximate ζ.

4.4

Renormalization procedure for two trapped particles
The simplest system that can be used to extract the renormalization function is

two particles in a trap. To obtain the renormalized scattering length we solve (4.21) for
two opposite-spin fermions in the HF approximation. The ground state energies in thisapproximation are matched with the exact energies of the system for different values ofthe bare two-body scattering length as . From this procedure we extract the functional
dependence of aef f on as . The spectrum of two opposite-spin fermions in a trap having a
specified scattering length as and zero-range interactions can be determined exactly [35,17, 22, 21]. This solution represents the ideal case in which the spectrum only depends
on as and the density (characterized by the trap length). Then, Eq. (4.21) can be solved
numerically for two particles using a HF wave function.
In choosing to extract aef f from a two-body system, we are implicitly assumingthat two-body correlations are the most important in the many-body system. This
assumption is reasonable for two-spin-component fermions with short-range interactions
because the probability of finding more than two fermions close enough to interact issignificantly reduced by Pauli repulsion.

4.4.1

Exact Energies

The zero-range pseudopotential proposed by Yang and Huang [95] and presented
in Sec. 2.2 has the necessary ingredients to represent the exact solutions. The boundary

66
conditions imposed by this pseudopotential produce the correct energies and correlationsof the system. The solutions of two particles in trap interacting through the YangHuang pseudopotential were presented in Sec. 2.3. Here, we select the appropriate
energy branch to generate the renormalization function.

6

Erel /h̄ω

4
2
0
−2
−5

0

5

as /ahoFigure 4.3: Spectrum Erel as a function of as . The dashed blue line corresponds to theenergy branch selected for the renormalization procedure developed in this study.

Figure 4.3 shows the spectrum of Erel [Eq. (2.13)] as a function of the scattering
length. Since we are interested in describing the ground state solution, we focus on thelowest energy curves. The lowest curve for a > 0 of the spectrum describes the formationof a molecule, where interparticle correlations are fundamental. One anticipates that a
HF wave function would be a terrible approximation for such a state in which the two
atoms are bound together to form a molecular eigenstate. However, because we do notconsider molecule formation in this work, we will consider instead the second branch inFig. 4.3 for the renormalization. The renormalization for the positive scattering length
will only be valid when the two-body potential does not support a bound state or weare interested in describing the atomic ground state with no molecule formation. Theenergies used for the renormalization are ECM = 3/2~ω and the energy branch where
1/2~ω < Erel < 5/2~ω. This branch of solutions is a smooth curve that gives the correct

67
noninteracting energy at as = 0.
4.4.2

Mean-field solution

Next we determine a renormalization function ζ(kf as ) that, applied to the systemof two particles in a trap and using the HF approximation, yields the exact results
obtained in the previous section.
To obtain the HF solution of two opposite-spin particles in a trap, we utilize a
product wave function having the same orbital for both particles. Thus, the two-bodyspatial wave function isΨ(r1 , r2 ) = ψ(r1 )ψ(r2 ),(4.23)and the spin part is antisymmetric. We introduce this trial wave function into the
Hamiltonian and obtain the energy functional
µ
¶
¶
Z µ
4π~2 ζ(kf as )~2 2 12 2
4E(ψ) =
2ψ(r) −
∇ + mω r ψ(r) +ψ(r) dr.
2m2
mkf(4.24)

Minimization of this energy functional determines the ground-state energy and wave
function. The minimization is done with respect to the orbital ψ(r) as in a standard
HF procedure [66]. But prior to carrying out this minimization of Eq. (4.24), we mustchoose how to evaluate kf . Since its formal definition is kf ≡ (6π 2 ρ)1/3 , this means thatkf depends at each r-value on ψ(r). For many-particle systems, we will use local density
approximation to evaluate kf . However, the application of a local density approximation
for a system of two particles does not seem physically correct. Thus, for two particles,
we consider the expectation value of k f to be the more appropriate quantity
Z
kf ≡

Z
kf (r)ψ(r)2 dr =(6π 2 ψ(r)2 )1/3 ψ(r)2 dr.

(4.25)The minimization procedure leads to a Schrödinger-type equation, where ψ 2 (r) is the68
1-particle density:Ã
4π~2 ζ(k f as )~2 2 1−∇ + mω 2 r2 +ψ(r)2 +
2m
2
mk f
Ã
!
!
8π~2 (6π 2 )1/3 ρ ζ 0 (k f as )as ζ(k f as )
2/3
−
ψ(r)
ψ(r) = ²ψ(r), (4.26)
2
3m
kf
kf
where ² is a Lagrange multiplier that represents the chemical potential. The relation
between ² and the energy is not as straightforward as in the HF case, primarily becauseof the appearance of ζ 0 (kf a0 ). Note that ψ(r)2/3 is supposed to be evaluated on a branch
for which it is real and positive everywhere. Here and in the following, ζ 0 (x) ≡ dζ(x)/dx.
Equation (4.26) corresponds to the Gross-Pitaevskii equation for 2 particles witha renormalized scattering length aef f = ζ(k f as )/k f . After solving Eq. (4.26) we use
Eq. (4.24) to evaluate the energy. The basic idea is that, for any chosen bare two-bodyscattering length as , we need to find ζ(k f a0 ) such that the energy of the ground stateof Eq. (4.24) matches exactly E0 = ECM + Erel with ECM = 3~ω/2 and Erel obtained
from Eq. (2.13). From our numerical experience, the functional dependence of ζ on k f asappears to be uniquely defined by the set of equations Eqs. (2.13), (4.24), (4.25), and(4.26).
There are two self-consistent procedures involved in this calculation. To solve
Eq. (4.26), we follow the standard HF procedure, adopting noninteracting solutions
as the initial guess for the orbitals. Then, we iterate Eq. (4.26) until convergence isachieved. For this procedure, we need the functional forms of ζ(k f as ) and ζ 0 (k f as )over a range of k f as values since k f is changing in each iteration. This means that wecannot find the exact renormalization function ζ(k f as ) at any fixed value of as without
knowledge of the functional form of ζ(k f as ) at nearby values. To solve this problem, we
calculate ζ(k f as ) self-consistently over the entire range in k f as that is of interest. First,
we select a set of scattering length (as ) values that cover the entire range of interest.
For an initial trial ζ (0) (k f as ), we solve Eqs. (4.26) and (4.24) at each as , obtaining
the energy E, k f , the wave function and ². Then, to obtain a new ζ (1) (k f as ), we look

69
for the value of the scattering length e
a0 for which Eexact (ea0 ) = E, and we generate a
new renormalization function that satisfies ζ (1) (k f ea0 ) = ζ (0) (k f as ). The modificationof the renormalization function is evidently in the abscissa rather than in the ordinate.
This is a convenient way to approach this calculation. Once we have carried out thematching procedure with the whole set of scattering-length (as ) values, we generate thenext iteration for ζ (1) (k f as ) and its derivative by interpolation.
In the next iteration, ζ (0) (k f as ) is replaced by ζ (1) (k f as ), and we repeat theenergy-matching step for the whole set of as values. This procedure is repeated a
few times until it converges to give a single correct renormalization function ζ(k f as ).
Note that this iterative procedure determines a “numerically exact” renormalization
function ζ(k f as ). Because the iteration procedure is efficient, in five iterations we obtainnine digits of agreement between Eexact and E over the entire as range. However, it
is important to introduce a sensible initial trial renormalization function ζ (0) (k f as ).
Many trial ζ (0) (k f as ) functions, like ζ (0) (k f as ) = k f as , would produce collapse of thetwo-fermion wave function for large and negative as . To avoid this collapse, we propose
an initial trial ζ (0) (k f as ) which is close to the correct ζ(k f as ). We do this by choosing
a qualitatively correct functional form with a few free parameters, and we then find theset of parameters that best reproduce the exact two-body energies.
The final numerical results obtained for the renormalization function ζ(k f as ) areaccurately approximated by the monotonic functional form ζ0 (k f as ) = A+B arctan(C +Dk f as ), where A and B are chosen to have the corresponding maximum and minimum
values at as → ±∞, and C and D are chosen to obey ζ(k f as ) → k f as for k f as << 1.
The maximum value is ζ max = 2.182, and minimum value is ζ min = −1.392. These
values lead to A = 0.395 and B = −1.138. Getting the correct behavior for k f as << 1requires C ≡ arctan(−A/B) ≈ 0.362 and D ≡ −(1 + C 2 )/B ≈ −0.994. Thus there
are only two independent parameters A, and B specified at this level of approximation.
Figure 4.4 compares our numerical results for ζ(k f as ) with this arctangent approxima-

70
tion:
ζ0 (k f as ) = 0.395 − 1.138 arctan(0.362 − 0.994k f as ).

(4.27)

Figure 4.5 displays the fractional error in ζ0 (k f as ) defined as (ζ(k f as )−ζ0 (k f as ))/ζ(k f as )and shows a maximum error of approximately 5%.
2.5
2

ζ(k f as )

1.5
1
0.5
0
−0.5
−1
−1.5
−10

−5

0

5

10

k f asFigure 4.4: Effective scattering length ζ(k f as ) (circles) and its analytical approximation
ζ0 (k f as ) (solid line). Figure from Ref. [183].

Now that the renormalization function has been determined, other observables
can be tested for the two-particle system. Interestingly, there is a numerically exact
agreement between the the trapping-potential-energy expectation values measured withthe exact wave function and the mean-field-renormalized wave function.However, theone-particle density profiles calculated using the exact wave functions and the meanfield-renormalized wave function are only in qualitative agreement, i.e., for scattering
lengths of large magnitude where |k f as | >> 1.

4.5

Application to many-particle systems
This section presents different many-particle approximations for which the renor-

malized scattering length can be used. The renormalization procedure is designed to
be used in the HF approximation. However we will see that simpler approximations

71
Table 4.2: Exact numerical values of ζ(x).
x−∞-11.937-11.394-10.85-10.306-9.7626-9.2193-8.6758-8.1326-7.5895-7.0467-6.504-5.9617-5.4197-4.8782-4.3373-3.7973-3.2585-2.7216-2.1875ζ(x)-1.392-1.2818-1.2767-1.2712-1.2651-1.2584-1.251-1.2428-1.2336-1.2231-1.2113-1.1977-1.182-1.1635-1.1416-1.1153-1.0829-1.0423-0.99004-0.92043x-1.6582-1.3097-0.96693-0.63317-0.31351-0.014329
0.26087
0.51684
0.76339
1.00721.25071.61661.98292.34972.71663.08373.45093.81814.18534.5526ζ(x)-0.82404-0.73716-0.62355-0.47254-0.27173-0.014242
0.28630.591220.86101
1.07871.24741.43091.55851.65081.72011.77371.8164
1.8512
1.88
1.9042ζ(x)1.92481.9426
1.95811.97171.98381.99462.00422.01292.02082.028
2.03462.04062.02382.03982.04896
2.05362.06572.06992.07632.182

x4.91995.28725.65456.02196.38926.7566
7.12397.49137.85868.226
8.59338.96078.00568.91179.523
9.866910.89511.31111.998+∞(ζ − ζ0 )/ζ

0.04
0.02
0
−0.02
−0.04
−0.06
−10

−5

0

5

10

k f asFigure 4.5: Fractional error in our analytical approximation to the numerical renormalization function, ζ0 (k f as ). Figure from Ref. [183].

72
like TF or a variational trial wave function will yield equally effective results in thelarge N limit. The variational wave function we use is the noninteracting wave function
rescaled in the radial direction by a factor λ [as in Eq. (4.28)], where λ is the variationalparameter. For example, Figure 4.6 a comparison for the ground state energy of a twocomponent Fermi gas in an spherical trap in the large N limit. The result obtained withthe approximate ζ0 (kf as ) in conjunction either with a variational trial-wave function
or with the TF method are in good agreement with the full HF calculation with theexact ζ(kf as ). The difference between the results stems mainly from the replacementof the exact ζ(kf as ) with the approximate ζ0 (kf as ). If we use the exact ζ(kf as ) for all
the methods, the energies agree to at least 3 digits. In systems having a small numberof particles, the HF method is, of course, more reliable than the variational or the TF
method.

4.5.1

Variational

The simplest approximation [148] utilizes a trial wave function that is a simple
radial rescaling of the noninteracting wave function:
Ψλ (r1 , r2 , ..., rN ) =

1
λ3N/2

ΨN I (r1 /λ, r2 /λ, ..., rN /λ).

(4.28)The expectation value of the renormalized Hamiltonian (4.21) can be separated into
two terms, E(λ) = EHO (λ) + Eint (λ, as ), where¶
X µ ~21
2
2 2∇ + mω ri |Ψλ i ,EHO (λ) = hΨλ |
−
2m i
2
i

Eint (λ, as ) = hΨλ |X 4π~2 aef fi<i0mδ(ri − ri0 ) |Ψλ i .

(4.29)The energy of this trial wave function is calculated as a function of the variationalscale parameter λ for the renormalized Hamiltonian (4.21). The noninteracting wave
function is a Slater determinant formed with the occupied spin orbitals. The EHO
is simple to calculate, as it requires only a change of variables to determine the λ73dependence in Eq. (4.29) in conjunction with the known results of the noninteracting
ground state, i.e.,µEHO (λ) = EN I

1
λ2
+
2λ2
2

¶
.

(4.30)

When we apply the renormalization locally as a function of the density, the interactionenergy Eint can be written as
4π~2
Eint (λ, as ) =mZζ(kfλ (r)as ) 2
ρλ (r)dr.
kfλ (r)(4.31)

1.3

E/ENI

1.2
1.1
1
0.9
0.8
0.7
−10

−5

0
0
kf as

5

10

Figure 4.6: Ratio of the total energy to the noninteracting energy for a sphericallytrapped two-component degenerate Fermi gas in the large N limit. The circles correspond to HF calculations for 2280 particles using ζ(kf as ), while the solid line corresponds to either the variational solution, Eq. (4.33), or the TF solution, Eq. (4.45)
using the approximate renormalization function ζ0 (kf as ); note that the curves are indistinguishable on the scale of the figure. Figure from Ref. [183].

In this equation, ρ is the density of one spin component, and kfλ (r) ≡ (6π 2 ρλ (r))1/3 .
In the large N limit, the density of the noninteracting wave function can be replacedby the TF density of the noninteracting system [148]. The density corresponding to ourtrial wave function is a simple radial rescaling in which the density in the high-N limit

74
becomesρλ (r) =√6N
3π 2 a3ho λ3

³
´3/2
2
1 − 2a2 λ2r(3N )1/3

if r2 < Rc2 ,

ho


 0

(4.32), otherwise.

Here N is the total number of particles, and Rc =√2aho λ(3N )1/6 is the radius of

the Fermi gas. In the large N limit, the total energy can be expressed in units of thenoninteracting energy:
λ2
11
E/EN I = 2 +
+ 2F
2λ
2
λ
Here λ is the scaling parameter, kf0 =Ã

kf0 as!

λ.

(4.33)√
2(3N )1/6 /aho is the Fermi momentum of thenoninteracting system at the trap center, and F is44
F (γ) = 2
9π

Z 1

³ p
´
(1 − x2 )5/2 x2 ζ γ 1 − x2 dx.

(4.34)

0Equations (4.33)and (4.35) can be used to describe either the bare Fermi or the renormalized interaction. First, consider the bare Fermi pseudopotential prediction. To
use the Fermi pseudopotential within our renormalization approach, we replace ζ with
ζ F (y) = y. Then Eq. (4.35) can be easily solved, e.g.,
44
F (γ) = 2
9π

Z 1

0
³ p
´
4096 kf as.
(1 − x2 )5/2 x2 ζ F γ 1 − x2 dx =2835π 2 λ0(4.35)

Therefore, the energy of the system is given by
0

E/EN I =1
λ24096 kf as
+
+
.
2
2λ
2
2835π 2 λ3

(4.36)

Figure 4.7 (a) presents the energies [Eq. (4.36)] as functions of λ for some selected kf0 as .
We can clearly see that the interaction term, which goes as 1/λ3 , is the leading term forsmall λ. Therefore, for as < 0, the energy becomes infinitely negative. This is always
true independent of the scattering-length strength. However, for small and negative
kf0 as , there is a metastable state corresponding to a local minimum of E [cf. the red
curve in Fig. 4.7 (a)]. This metastable state is considered the physical gaslike state. For
some critical value of kf0 as ≈ −1.21 the local minimum disappears, which produces a

75collapse of the gaslike state (see, e.g., the green curve in Fig. 4.7 (a)). This reasoning
is similar to the collapse picture of a trapped bosonic system presented in Ref. [20].
However, this analysis is incorrect in our case since the Fermi pseudopotential is only
valid for |kf0 as | ¿ 1. Now, we see how this behavior changes with the introduction of
the renormalized interaction.
The evaluation of Eq. (4.35) for the renormalized ζ function obtained from thetwo-body problem must be calculated numerically unless further approximations are
made. The energy results obtained using Eq. (4.33) are shown in Fig. 4.6. We observethat the energies remain finite for all scattering-length values, and no collapse occurs.
In the unitarity limit, the behavior can be calculated exactly:Z
44 ζ min 1
5ζ min2 5/2 2.
F (γ → −∞) =
(1
−
x
)
x
dx
=
9π 2
9π
0

(4.37)

Therefore, at unitarity, the energy is:0

EU /EN I =1
λ2 5ζ min kf as
+
+
.
2λ2
2
9π λ2

(4.38)

To illustrate the behavior of the energies obtained with the renormalization, weshow three curves predicted by Eq. (4.33) in Fig. 4.7 (b). For the entire range ofinteractions, the energy of the system (the minimum of the curve) remains finite, ranging
from 0.713 EN I to 1.33 EN I . Thus our renormalization circumvents the collapse thatwould occur for the bare Fermi pseudopotential.
Interestingly, the variational method presented here is similar to the K-harmonic
hyperspherical method developed in Refs. [149, 150]. Furthermore, it was shown in
Ref. [150] that the matrix elements calculated with the variational and K-harmonic hyperspherical method are identical in the large N limit if we replace λ by the hyperradius
R. For that reason, Eqs. (4.36,4.38) are equivalent to Eq. (3.11) from Ref. [149] andEq. (33) from Ref. [150], respectively. Therefore, the energy curves in Figs. 4.7 (a) and(b) can be seen as hyperspherical potential curves. This relationship is an informative
new insight.

76

4

4

(a)

3

E/ENI

E/ENI

3

2

1

0
0

(b)

2

1

1

2

λ

3

0
0

1

λ

2

3

Figure 4.7: (a) Variational energy obtained with the Fermi pseudopotential prediction.
Black curve is the noninteracting energy. Blue curve is the prediction for kf0 as = 1.5.
The red and green curves correspond to the negative scattering length side with kf0 as =0.5 and kf0 as = −1.5, respectively. (b)Variational energy as a function of λ for the
renormalized interaction, in units of the noninteracting total energy. The solid curve
corresponds to kf0 as = 0; the dashed curve corresponds to kf0 as = −∞; the dotteddashed curve corresponds to kf0 as = ∞. The minimum of the energy functional for
kf0 as = −∞ occurs at λ = 0.844, which represents the ratio between the cloud radiusat unitarity and the noninteracting cloud radius.

77
4.5.2

Thomas-Fermi results

In this subsection we review the TF approximation using the renormalization
function. The TF approximation was previously used to study a two-component Fermi
gas with zero-range pseudopotentials [153], but in that study no renormalization wasconsidered. As in the previous subsection, all the results presented here can be applied
to the Fermi pseudopotential predictions by replacing the renormalized function byζ F (y) = y or aef f by as .
TF is a local density approximation in which, at each position r inside the trap,
the single particle orbitals are approximated by plane waves. Since we consider eachcomponent as a distinguishable particle, the wave function at each r is a product of
two Slater determinants, one for each component. The orbitals are characterized by
the vector momentum k. The orbitals are filled uniformly up to a level kf (r), whichis the same for spin-up and spin-down fermions. The value of kf (r) will depend on
the distance r from the trap center, and on the number of particles in the system. Fora uniform system, the value of kf is a constant that characterizes the density of thesystem.
To calculate the local energy, we need to sum over all the states at that position.
For example, the kinetic energy term K for the one-spin component isK=

1 X~2 X 2V ~2
hk|p2 |ki =k hk|ki =2m2m2m(2π)3k

kZk 2 d3 k =V ~2 5k .
20π 2 m f

(4.39)

Here V is the volume of integration that will disappear when we consider the local
energy. This volume is small in comparison with the external potential (in this casethe trap) characteristic length but is big enough to contain many particles. Thus, kf
and Vext can be considered constant during the integration. The calculation of the
expectation value of an external-trapping potential is then straightforward. We obtainhVext i =V
Vext kf3 ,6π 2(4.40)

78
which is just the volume times the density of particles of one component ρ↑ = ρ↓ =kf3 /(6π 2 ) times the external potential at that position. Now we proceed to evaluate theinteraction term,
hVint i =X

hkk0 |Vint |kk0 i =kk0

1 4πaef f ~2(2π)6m

Z kf Z kf Z Z0

δ(x − x0 )dx0 dxd3 k 0 d3 k0354πaef f ~2 kf (kf )3
4π~2 kf ζ(kf as )=V=Vm
6π 2 6π 2m(6π 2 )2

(4.41)The second to last equality is simply 4πaef f ~2 /m V ρ↑ ρ↓ , which is the Hartree term.
Combining the expectation value of the kinetic energy, the external potential
energy, and the interaction energy, we obtain the total energy. For the case of two
equally numerous spin components, the local energy (per unit volume) is
E(kf ) = E(kf )/V =

5
5
kf3
~2 kf
4π~2 kf ζ(kf as )
+
+
.
V
ext
2m 5π 23π 2m
(6π 2 )2

(4.42)In an infinite uniform system, where Vext = 0, the energy is
55

4π~2 kf ζ(kf as )
~2 kf
+
.
E(kf ) =2m 5π 2m
36π 4(4.43)

Equation (4.43) gives the equation of state for a uniform gas The ratio between the totalenergy and the noninteracting energy has a simple form that only depends on kf as :E(kf )/EN I (kf ) = 1 +10ζ(kf as ).
9π

(4.44)

Using Eq. (4.42), we construct an energy functional by integrating the local energy overall space.
µ

Z
E=

dr

kf (r)3 4π~2 kf (r)5 ζ(kf as )~2 kf (r)5
+
V
(r)
+
ext
2m 5π 2
3π 2
m
36π 4

¶
.

(4.45)

To find the ground state, we have to minimize the energy under the constraint that the
total number of particles is fixed. This constraint can be implemented by introducing
a Lagrange multiplier µ0 , which is usually called the chemical potential. Thus, the79
minimization of Eq. (4.45) for a fixed number of particles is reduced to the minimizationofZΛ ≡ E − µ0 N = E − µ0drkf (r)3,3π 2(4.46)

where the variational parameter is kf (r). The necessary, but not sufficient, condition
for kf (r) to minimize Λ is that∂Λ= 0.
∂kf (r)(4.47)

1
0.9

µ/Ef

0.8
0.7
0.6
0.5
0.4
−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

1/kf asFigure 4.8: Chemical potential in units of the noninteracting Fermi energy. The blacksolid line is the prediction obtained via the renormalization function. The black circles
represent the BCS prediction, and the blue dashed curve is the prediction obtained inRef. [43]. Figure from Ref. [183].

This condition leads to a relationship between the local chemical potential, definedas µ(r) ≡ µ0 − Vext (r), and the local Fermi momentum kf (r):µ(r) =~2 kf2 (r)2mµ¶kf (r)as 0
101+
ζ(kf (r)as ) +ζ (kf (r)as ) .
9π
18π 2

(4.48)The value of µ0 fixes the number of particles, and with this relationship, we can calculate
the density profile and the energy of the system. Figure 4.8 shows the chemical potential

80
dependence on kf as obtained with the renormalization function and other models. In
Fig. 4.6, we see the energy obtained using Eqs. (4.45, 4.48) in the large N limit.
In the unitarity limit [as → −∞], we obtainµ¶
10 min
µ=
.
1+
ζ
2m
9π
~2 kf2(4.49)At unitarity, when the scattering length is much larger than the interparticle distance,
the only relevant parameter is the density [85, 130]. Dimensional analysis suggests thatµ ∝ ρ2/3 ∝ kf2 . The expected relation between µ and kf is usually written asµ=~2 kf22m(1 + β).

(4.50)

From our calculations this relation appears naturally with a universal parameter β =10ζ min /9π = −0.492. A more detailed discussion about the predictions of renormalization and the properties of a unitary Fermi gas will be presented in chapter 7.
It is well established [91, 130, 193] that an ultracold two-component Fermi system
exhibits superfluidity. Even though our renormalization scheme does not explicitly
consider superfluidity, it reproduces a number of properties of the Fermi gas sensibly,
including the equation of state and the chemical potential. Consequently, these results
can be used in a hydrodynamic theory to extract information about dynamics of thesystem such as the speed of sound or normal modes of excitation. For example, thespeed of sound in a uniform two-component system is given by [85]
~ ∂
v =
m ∂ρ
2

µ
¶
2 ∂E/Nρ.
∂ρ

(4.51)

Using Eq. (4.44) we can thus evaluate the speed of sound, generating the results shownin Figure 4.9. The speed of sound results reproduce the expected limiting behaviors. Inp√the noninteracting limit v = vf / 3, while at unitarity v = vf (1 + β)/3 [87]. This is
one example of a nontrivial observable quantity for this system that can be predictedby the renormalization technique.

81
4.5.3

Hartree-Fock method

The HF method for a many-particle system is an extension of the two-particle
calculation done in Section II. Consider a wave function written as the product of two
Slater determinants, as in Eq. (4.7). The HF method looks for the best set of orbitals
that minimize the energy of the system. The expectation value of the energy of the trial
wave function for a Hamiltonian of Eq. (4.21) can be written as

µ
¶
Z
N/2
X
~2 2 12 2
E(ψ) =2
ψi (r) −∇ + mω r ψi (r)
2m
2
i

¶
4π~2 ζ(kf (r)as )2+
ρ(r) dr. (4.52)
kf (r)m

Here, the orbitals ψi (r) are the variational parameters, and Eq. (4.52) is the energyfunctional to minimize. The density ρ(r) is the one-component density.

v/vf

0.55

0.5

0.45

0.4
−10

−8

−6

−4

−2

0

1/kf asFigure 4.9: The speed of sound as a function of kf as shown in units of the Fermi velocity
vf = ~kf /m for a uniform two-component Fermi gas. Figure from Ref. [183].

In Eq. (4.52), we have considered restricted spin orbitals, which are constrained
to have the same spatial function for spin-up and spin-down orbitals. Also, we focus on
closed-shell systems, which have an even number N of particles, with all particles paired
such that n = N/2 spatial orbitals are doubly occupied. In this situation, the density is

82
expected to be spherically symmetric, and the problem becomes spherically symmetric.
Therefore, we can assume that the orbitals also have the spherical symmetry of Eq. 4.4,i.e.,ψnlm (r) =unl (r)Ylm (Ω)
r(4.53)

Here, the Ylm (Ω) are the spherical harmonics. The radial orbitals unl (r) are not thenoninteracting orbitals of Eq. (4.5) but are obtained from the minimization of the energy.
Using the orbitals of Eq. (4.53), the density and the local Fermi momentum take theformρ(r) =X (2l + 1) u2

nl

nl

4π

r2

,

kf (r) = (6π 2 ρ(r))1/3 .

(4.54)(4.55)The sum in Eq. (4.54) is over all the n and l of the occupied orbitals of a given shell.
Differentiating Eq. (4.52) over the orbitals, we find that the extreme condition is
fulfilled when the orbitals obey the following set of nonlinear equations:½~2 ∂ 2l(l + 1)~2 1−++ mω 2 r2
2m ∂r2
2mr2
2
"
Ã
!#)
kf (r) as ζ 0 (kf (r)as ) ζ 0 (kf (r)as )4π~2 ζ(kf (r)as )+ρ(r) +−unl (r) = ²nl unl (r).
mkf (r)6ρ(r)kf (r)
kf2 (r)(4.56)
Each of these equations can be seen as a Schrödinger equation of a particle immerse ina mean-field. The first line of the equation describes the kinetic and external potential
terms. The term in the second line is the interaction with the mean field. The firstterm in the square brackets is the standard mean-field term that is proportional to thedensity. The second term in the square brackets comes from the density dependence ofthe effective scattering length.
Replacing the renormalization function by the Fermi pseudopotential prediction,

83
zetaF (y) = y, we obtain the standard HF equation of a normal gas used in Ref. [33].
¾½4π~2 a~2 ∂ 2l(l + 1)~2 1
2 2−++ mω r +ρ(r) unl (r) = ²nl unl (r).
2m ∂r22mr2
2
m(4.57)The set of coupled equations for the orbitals, Eq. (4.56) or Eq. (4.57), are solved
self consistently. Starting from the noninteracting solution, kf (r) and ρ(r) are evaluatedand introduced on the left hand side of Eq. (4.56) to obtain a new set of orbitals. The
new set of orbitals are then used to obtain new estimates of kf (r) and ρ(r). We repeat
this iteration procedure until we reach convergence. During this procedure, we observethat the energy of system decreases verifying that equilibrium correspond to a minimumof energy.
Figure 4.6 shows results for the HF energy of 2280 particles. This approximation
is particulary useful for systems with small numbers of particles, in which the TF
approximation has limited applicability. In Sec. 4.6 below, this method is used toobtain the energies of eight fermions in a trap.

4.6

Comparison with other approaches
To compare the predictions based on our renormalized scattering length withother methods, we have carried out fixed-node diffusion Monte Carlo (FN-DMC) simulations for equal mixtures of different-spin fermions. Interactions are considered only
between different-spin fermions, which are treated here as distinguishable particles. Theinteraction potential is a purely attractive Gaussian, and its width d is chosen so thatρd3 ≈ 10−4 . The FN-DMC method was discussed in detail in Sec. 3.2. Since we focus
on the BCS region, the results presented here correspond to the Jastrow-Slater wave
function Eq. (3.7).
We have calculated FN-DMC energies for eight particles in the BCS side of thecrossover. In Fig. 4.10, these energies are compared with HF calculations the include
the first- and second-order corrections in the kf as expansion [95] and with full HF

84calculations that use the renormalized scattering length directly,
Eint /N =~2 kf2mµ¶
kf as 6(11 − 2 ln 2)
2
+
(k
a
)
+
...
.
f s
3π
105π 2(4.58)The idea of using this type of expansion to construct energy functionals has been applied
for bosons [16, 28]. The expansion [Eq. (4.58)] can be introduced locally in variational
treatments, yielding an energy functionalZ
E(ψ) =


2

N/2
X
i

µ
¶
~2 2 12 2
ψi (r) −∇ + mω r ψi (r)2m2¶
4π~2 a0
2 4/3
7/3
2
2 12(11 − 2 log(2))ρ(r) + as(6π ) ρ(r)dr, (4.59)+m105π 2where ρ = ρ↑ = ρ↓ . If we only consider the first term in Eq. (4.58), we obtain the Fermi
pseudopotential contribution.
18

E/h̄ω

17

16

15

14
−4

−3

−2

−1

0

kf0 asFigure 4.10: Total energy of eight fermions in a trap shown in oscillator units as afunction of kf0 as . FN-DMC results are shown in open red circles, while the solid blueline corresponds to HF results using the effective renormalized scattering length. The
dashed and dash-dotted curves correspond to solutions using first-order or first- andsecond-order terms in an expansion into powers of kf as . See the discussion of Eqs.
(4.58, 4.59). Figure from Ref. [183].

Previous authors [33, 153] have considered the Fermi pseudopotential approximation to study the weak interacting limit. This approximation corresponds to the85first term in the energy expansion (4.58). Applying Eq. (4.58) in the local density
approximation is a convenient way to introduce higher-order corrections to mean-field
theories. We obtain an expansion of the density-dependent renormalization function
using Eq. (4.58);
ζ(kf as ) = kf as +6(11 − 2 ln 2)(kf as )2 + ... .
35π(4.60)

Insertion of this result into Eq. (4.21), with the local density approximation and a
Slater determinant wave function, yields Eq. (4.59).
A power expansion of ζ(kf as ) obtained by the renormalization method should
agree with this expansion. The first term is reproduced exactly but the second one
is only in qualitative agreement. While the coefficient of the second-order expansion
in Eq. (4.60) is approximately 0.525, in the density renormalization from Sec. 4.4, thecoefficient is 0.422. This disagreement may be due to the level in approximation of thedensity renormalization procedure.
We find very good agreement between the mean-field results calculated using
the renormalized interaction developed in this paper and the FN-DMC (Fig. 4.10).
The variational methods including the perturbative corrections [Eq. 4.59] show good
agreement in the small kf as region, but deviate from the FN-DNC results when thecorrections to the expansion [Eq. (4.58)] become important.
It is also possible to compare our results with other quantum Monte Carlo calculations. For example, Astrakharchik and co-workers [7] have studied a homogenous
doubly degenerate Fermi gas using FN-DMC methods. In their calculations, they considered up to 60 particles. We compare the energy of this system as obtained using thedensity renormalization procedure in Eq. (4.44). A comparison between the two calculations and a local density BCS result (see, discussion in Sec. 2.4 and Refs. [113, 123])
is shown in Fig. (4.11).
In this section, we compared the predictions of the renormalized interaction with

86
FN-DMC , mean-field BCS, and other local-density predictions. We showed that therenormalization approach is in generally good agreement with the FN-DMC results andprovides a better description of the ground-state energy than the mean-field BCS results.
1

E/ENI

0.9
0.8
0.7
0.6
0.5
0.4

−6

−4

−2

0

1/kf asFigure 4.11: E/EN I for an infinite homogeneous Fermi gas kf as in the mean-field approach (solid line). The dashed curve corresponds to the local density BCS solution, andthe circles correspond to FN-DMC results obtained in Ref. [7]. Figure from Ref. [183].

4.7

Conclusions
In this chapter, we have presented different methods for analyzing the Hartree

term correction. Starting with the Fermi pseudopotential, we have analyzed the first order correction in the weakly interacting BCS limit. Then we proposed a renormalization
method to extend the use of zero-range interactions.
The density renormalization method applies the philosophy of many effective
theories. It is the goal of most many-body theoretical studies to derive predictive power
for numerous observables of interest, using simpler methods that bypass the actual
calculation of this ”true” ground-state wave function for the trapped atomic gas. At
the heart of many such treatments are the following two steps: (i) replacement of thetwo-body potential energy by a zero-range Fermi pseudopotential, followed by (ii) a87mean-field wave function ansatz and the computation of observables. The basic levelof description for a Bose gas incorporates no correlations whatsoever. For a Fermi
gas, correlations are generally treated at either the bare minimalist level of exchange
correlations alone, using Slater determinant wave function. A more sophisticated level
is often considered for a system of mutually attractive fermions, which are frequently
described with BCS-type correlations built into the description. One way of visualizing
the value of a Fermi-type zero-range pseudopotential adopted in most such theories is toremember that it has been specifically designed to give a meaningful interaction energy
for each pair of particles even when the wave function structure is too simplistic toincorporate any appreciable correlations.
The renormalization scheme presents an alternative implementation of this general philosophy. We developed a procedure for renormalizing the coefficient of a zerorange potential, based entirely on an analysis of the nonperturbative two-body systemsolved first with and then without wave-function correlations. When we applied thisprocedure to the many-body Fermi gas, we find agreement with the standard dilute gas
limit, an important prerequisite for any realistic theory. But in addition, our procedure
is able to treat higher densities ρ, including the regime |ρas 3 | >> 1. We studied anumber of observables that have been explored both experimentally and theoreticallyin the BCS-BEC crossover regime, and found good agreement using our renormalized
HF approach all the way to the unitarity limit, as → ∞. Perhaps surprisingly, thisgood agreement is achieved without incorporating explicit BCS-type correlations into
the many-body wave function. One result of this study is an approximate expression
for the renormalization function in closed analytical form that may prove to be useful inother studies of the two-component degenerate Fermi gas. Another interesting result isthat at unitarity, the chemical potential exhibits the expected density dependence characterized by the parameter β = −0.492, which, interestingly, is consistent with recent
experiments [134, 25, 105, 167].

88
Unfortunately, the current implementation of the renormalization cannot describe
the BEC side. An possible way to extend the renormalization method to the BEC side
is to treat the dimers as bosonic particles and describe the system with the GrossPitaevskii equation. The renormalized dimer-dimer interaction would be obtained bymatching the two-boson trapped system with the “exact” four fermions calculations
from Chapter 6. This renormalization on the BEC side would complement the renormalization presented here for the BCS side. However, the combination of the BCS and
BEC renormalization treatments would not predict a smooth transition in the crossover
region. In order to achieve such smooth transition, we should use a more flexible manybody wave function which can represent a Fermi gas in the weak interacting region anda gas of Bose molecules in the BEC region as it is achieved by the BCS wave function.
Such desired theory should also be able to include both Hartree and pairing corrections.

Chapter 5
Spectrum and Dynamics of few-body trapped systems

5.1

Introduction
In this chapter, we analyze the spectrum of fermionic few-body trapped systems

in the BCS-BEC crossover. In particular, we focus on systems with N = 2, 3, 4. We
will also analyze the dynamics of these systems when the scattering length is tuned in
the BCS-BEC crossover.
Few-body systems can provide an alternative approach to the BCS-BEC crossoverproblem. A four-body system, made up of two spin-up and two spin-down fermions,is the smallest system that can describe the BCS-BEC crossover. Since a four-bodysystem can describe the formation of two atom pairs or two bound molecules, we cananalyze its evolution from an interacting atomic Fermi gas to an interacting molecularbosonic system.
The BCS-BEC crossover problem is usually studied by many-body theories that
focus mainly on ground-state properties. In contrast, few-body calculations not only
describe the ground state, but also the excitation spectrum. Furthermore, these calculations do not require the standard approximations applied to many-body systems. Thus,
few-body systems allow exact numerical solutions that can be studied with many-bodytheories; they also allow us to test nontrivial behavior.
In this chapter, we first solve the time-independent Hamiltonian for different
scattering length values as . Then we use these solutions to analyze the dynamics of the

90system. The model Hamiltonian that describes a trapped two-component Fermi gas is
given by
¶ X
¶ NX
N1 µ
N2 µ
1 ,N2
X
~2 2 1~2 2 12 2
2 2
H=−∇ + mω ri +−∇ 0 + mω ri0 +V (rii0 ).
2m i
2
2m i
2
0
0
i

i

(5.1)

i,i

Here, unprimed indices label spin-up and spin-down particles. The interaction potential
V has a purely attractive Gaussian form (see Eq. 3.20). The relevant length scale ofp
the problem is the trap length aho = ~/mω. In Eq. (5.1), the center-of-mass motion
can be decoupled, reducing the dimensionality of the numerical calculations. All thesolutions presented here have the center-of-mass wave function in its ground state.
To obtain the spectrum, we use a correlated-Gaussian (CG) basis-set expansion [157, 180], which allows a simple evaluation of the matrix elements. A diabatization
procedure allows us to reduce the system to a tractable set of eigenfunctions. Then wesolve the time-dependent Schrödinger equation using the adiabatic representation. The
numerical results are then compared with a Landau-Zener model and with experimental
results. Finally, we analyze temperature effects in the two-body system and comparethem with experimental predictions for ultracold Fermi gases.

5.2

Spectrum in the BCS-BEC crossoverIn this section we use the CG method to obtain the spectrum of N = 2, 3, 4

systems. All solutions have a relative angular momentum of Lref = 0.
First, we consider the N = 2 system. Solutions for this system are very easy tofind. In chapter 2 we presented semianalytical solutions for a zero-range pseudopotential.
Using this model potential, the dynamics of the system have also been studied [22].
Here, we solve the N = 2 system with a short, but finite, two-body interaction using
the CG method. These solutions help us to test both the CG method used to obtain the
spectrum and the algorithm we use to solve the time-dependent Schrödinger equation.
Our results are compared with the N = 3, 4 systems, yielding a general description of

91
few-body trapped systems.

10

E/h̄ω

5
0
−5
−10
−15
−4

−2

0

2

4

aho /asFigure 5.1: Spectrum of two particles with J = 0 in a trap obtained with CG method.
The black curve corresponds to the ground state that evolves in a dimer configuration.
Red curves correspond to excited states that evolve in a two free-atom configuration.

Figure 5.1 presents the L = 0 spectrum obtained with a CG of two particles
in a trap. This spectrum resembles the zero-range spectrum presented in Fig. 2.2.
Both finite and zero-range results are nearly indistinguishable because the range of theinteracting potentials is much shorter than the scattering length as . In the system of
two particles in a trap, we see a clear topology when the scattering length is tuned acrossa resonance. In the weakly interacting regime (small and negative as ), the energies areapproximately separated by 2~ω. As as is tuned closer to the unitarity region, a setof avoided crossings appears that leads to the appearance of a two-body bound statein the as > 0 region. In free space, this state has a binding energy of ~2 /(ma2s ). The
presence of a trap alters this result, leading to a trap energy given by Eq. 2.13. In thelimit as ¿ aho , the two particles form tightly bound dimers of size ∼ as /2, and thetrap energies given by Eq. 2.13 converge to their free-space value of ~2 /(ma2s ). Anotherinteresting point is that the unbound spectrum on the weakly interacting BEC side
reproduces the spectrum of the weakly interacting BCS side.

92
We are interested in understanding how this spectrum changes when we consider more particles. Here, we analyze the L = 0 spectra for N = 3 and N = 4 in
the BCS-BEC crossover. To obtain the spectra, we use the CG method to solve thetime-independent Schrödinger equation for different values of as . Like most numericalmethods, this method provides an adiabatic spectrum, i.e., the energies of the spectrum
are labeled according to their energy values as as changes. The N = 3 and N = 4
spectra present a series of crossings or narrow avoided crossings when the scattering
length is tuned in the BCS-BEC crossover. For this reason, it is convenient to use a
representation where these narrow avoided crossings are treated diabatically, and thespectrum smoothly evolves from the BCS to the BEC side. The diabatic representation
is more relevant from the physical point of view since the diabatic states are usuallyassociated with good or “approximately good” symmetries of the problem.

12

E/h̄ω

10

8

6

4
−1

−0.5

0

0.5

1

1.5

aho /asFigure 5.2: Energy spectrum as a function of λ in the unitarity region for N = 4 with
J = 0. The thin solid black lines correspond to the adiabatic spectrum. The wide blackline with circles is the diabatic ground state labeled Ψ1 . The blue curve with circles isthe diabatic first-excited state labeled Ψ2 , the wide red curve with circles is the diabatic
state Ψ5 , and the wide green curve with circles is the diabatic state Ψ13 .

To illustrate the diabatization procedure, we consider the spectrum of the N = 4
system in the strongly interacting region as shown in Fig. 5.2. We see a series of crossings

93
and avoided crossings when the adiabatic parameter λ ≡ 1/as is tuned in the stronglyinteracting region. The avoided crossings can be roughly characterized by their width∆λ, which is the range where the two adiabatic energy curves are coupled. There are
narrow crossings where ∆λ ¿ 1/aho and there are wide crossings where ∆λ & 1/aho . To
obtain smooth energy values, we use variation of the diabatization procedure presented
in Ref. [88].
The objective of the diabatization algorithm is to make the one-to-one connection
between states and energies in consecutive points of the λ grid that maximize the sum of
the overlaps between connected states. The diabatization procedure starts from the BCS
side of the resonance and connects the states (and their energies) between consecutive
values of λ for which their overlap is maximum. When two initial energies connect tothe same final energy, a refinement of the diabatization procedure is applied.
Diabatization is controlled by the spacing between consecutive values of λ givenby ∆λg . If the width of the avoided crossing is smaller than ∆λg , then that crossing
is diabatized. But if the width of the avoided crossing is larger than ∆λg , then thatcrossing is not diabatized. Thus, ∆λg is selected so that narrow crossings are diabatized
and wide crossings remain adiabatic. For example, in Fig. 5.2 we see how this procedure
diabatizes the narrow crossings of Ψ13 of the N = 4 system. However, wide crossings
such as the one between Ψ5 and Ψ13 , are still adiabatic in this representation.
This structure of avoided crossings permits a global view of the manner in whichstates evolve from weakly interacting fermions at a < 0 to all the different configurationsof a Fermi gas at a > 0, i.e., molecular bosonic states, fermionic states, and molecularboson-Fermi mixtures. Furthermore, it allow us to visualize concretely the possible
pathways of the time-dependent sweep experiments. Here, we analyze the diabatic
spectrum for N = 3 and N = 4.
The diabatic spectrum of the N = 3 system is presented in Fig. 5.3. The systemof three particles in a trap presents a rich structure of avoided crossings related to the94appearance of a bound dimer state. There is a family of states with energies separated
by approximately 2~ω that are attached with the two-body threshold. These states
represent the ground state and excitations of a system formed with a dimer and afermion in an L = 0 state. There are other states corresponding to the red energy
curves in Fig. 5.3 that do not form dimer states. These three-atom states reproduce
the three-particle noninteracting spectrum in the BEC limit, i.e., as → 0+ . Finally, we
note that no trimer formation occurs.

10

E/h̄ω

0
−10
−20
−30
−40

−5

0

5

aho /asFigure 5.3: Energy spectrum of L = 0 three-particle states. The black line with circles
shows the diabatic ground state. The blue curves with circles show the diabatic excitedstates, which represent a bound dimer plus an extra particle. The red curves with circles
show the diabatic states where no bound dimer is formed.

We next consider the four-body diabatic spectrum presented in Fig. 5.4. The
structure of avoided crossings is more complicated because of the appearance of two
different thresholds for the dimer and dimer-dimer, respectively. We identify three different families of diabatic states in this spectrum. The dimer-dimer family, represented
by the black and blue curves, describes the ground and excited dimer-dimer states.
These states are approximately separated by 2~ω on the BEC side. The dimer–twoatom family, represented by the red energy curves, follows the dimer threshold. In the

95
BEC limit, the dimer–two-atom family reproduces the degeneracies of three distinguishable particles: a spin up, a spin down and a dimer. The third family describes four-atomsystems. In this family, none of the atoms form dimers, and the energy remains positivein the crossover region. In the BEC limit, the four-atom family reproduces the spectrumof the noninteracting four-body system.
The evolution of the N = 3 and N = 4 spectra in the BCS-BEC crossover canbe qualitatively understood by considering the important quantum numbers for thedescription of the dimer. In the N = 3 system, for each vibrational excitation of 2~ω,there is one state in the noninteracting limit that diabatically becomes a atom-dimer
state. This atom-dimer state corresponds qualitatively to a states where the relative
angular momentum of a spin-up–spin-down pair is zero [L↑↓rel = 0], and the relative
angular momentum between the pair and the remaining atom is also zero. The spinup–spin-down pair is in the lowest vibrational state 1s, that evolves in a weakly bounddimer. In the weakly interacting BCS limit where the degeneracy of the vibrational
states is broken, such pair-atom states correspond to the lowest states.
A similar effect occurs in the N=4 system. For each vibrational excitation of2~ω in the noninteracting limit, there is one state that diabatically becomes a dimerdimer state. These states correspond qualitatively to states where the relative angularmomentum of two spin-up–spin-down pairs is zero [L↑↓rel = 0], and the relative angularmomentum between the pairs is also zero. The spin-up–spin-down pairs are in the lowest
vibrational state. In the weakly interacting BCS limit, where the degeneracy of the thevibrational states is broken, pair-pair states correspond to the lowest states.
A direct and more concrete way to visualize the structure of the spectrum is to analyze the evolution of the adiabatic hyperspherical potential curves. Figure 5.5 presents
the N = 4 hyperspherical potential curves Uν (R) as functions of the hyperradius R obtained with the correlated Gaussian hyperspherical method (CGHS). Panel (a) presents
the potential curves in the BCS regime. These potential curves are clearly grouped

96

10

E/h̄ω

0
−10
−20
−30
−40

−6

−4

−2

0

2

4

6

aho /asFigure 5.4: Energy spectrum for four particles with J = 0 in the crossover region(lowest 20 diabatic states). The black curve corresponds to the ground state. The
blue curves are the states that go diabatically to excited dimer-dimer configurations.
The red curves correspond to states that go diabatically to configurations of a dimer
plus two free atoms, and the green curves correspond to states that go adiabatically toconfigurations of four free atoms. The lowest green curve is the atomic ground state on
the BEC side of the resonance. Results from Ref. [182].

Uν /h̄ω

30

(a)

30

25

20

20

10

15

(b)

0

10

−10

5
−20
0
0

2

4

R/aho

6

0

2

4

6

R/aho

Figure 5.5: Hyperspherical potential curves in the BCS-BEC crossover for N = 4 with
J = 0. (a) Potential curves in the BCS regime, as ∼ −0.3aho . (b) Potential curves in
the BEC regime, as ∼ 0.3aho .

97
in families. The potential curves belonging to the same family are degenerate in thenoninteracting limit. Thus, the weak interactions in the BCS break the degeneracies ofthe potential curves forming these families of potential curves. Panel (b) describes thesystem in the BEC regime. In this case, the description of the system is quite clear.
The lowest potential curve is more than twice as deep as the rest of the curves andis associated with the dimer-dimer threshold. The family of dimer-dimer states lives
mainly in the lowest potential curve. The rest of the potential curves are associatedwith the dimer–two-atom threshold. The dimer–two-atom states are mainly describedby this family of potential curves. There is a third family of potential curves, not shownin Fig. 5.5 (b), that describes four-atom states. This family of potential curves has adifferent large R asymptotic behavior.

5.3Time evolution
The diabatic representation can be used to ramp an initial configuration throughthe BCS-BEC crossover, mimicking experiments carried out at different laboratories
at JILA and Rice University. The initial configuration is propagated using the timedependent Schrödinger equation
i~

d |Ψi
= H[λ(t)] |Ψi .
dt

(5.2)

The time dependence of the Hamiltonian comes from the λ(t) term. In our work, we
focus on unidirectional ramps. Starting from the ground state on the BCS side, theparameter λ is ramped through the resonance to the BEC side at different speeds,
ν = dλdt . The relevant dimensionless speed quantity is ξ = aho ν/ω.
To propagate the initial configuration, we use the diabatic representation obtained
previously. First, we divide the BCS-BEC crossover range into sectors. Starting from the
BCS side at λ ≈ λBCS to the BEC side at λ ≈ λBEC , the BCS-BEC crossover is divided
in 40–80 sectors. At the middle of each sector, the time-independent Hamiltonian [Eq.

98(5.1)] is solved using the CG method. For the N = 3 and N = 4 systems, thousandsof CG basis functions are usually needed to describe the spectrum. We could use thisbasis set to solve Eq. (5.2) but this large basis would make the numerical propagation
very slow. Instead, we use the diabatic representation obtained at the middle of thesector to expand the time-dependent wave function through the sector j, i.e.,NdX|Ψ(t)i =cji (t) |Ψji i .

(5.3)

i

Here, |Ψji i is the diabatic basis i of sector j, and Nd is the number of diabatic statesconsidered. The time dependence only appears on the complex coefficients cji (t). Selecting only the lowest 20–100 diabatic states at that point, we reduce the size of the
Hamiltonian matrix in Eq. (5.2). Since the inverse scattering length λ changes very
little in each sector, the relevant diabatic states are well described by this reduced basis
throughout the sector.
The time-dependent Schrödinger equation, Eq. (5.2), is propagated from one extreme of the sector to the other using an adaptive step Runge-Kutta method. Whenλ changes from a sector to the following one, the wave function in the basis set is projected onto the new basis set. It is important to test that no information is lost in
this projection. To avoid this potential problem, the relevant states for the particularpropagation must be well described in each sector. For example, in most of the time
propagations considered here, the ground state at the BCS side, |ΨBCS
i, is propagated1
to the BEC side at different speeds. If the velocity is fast enough, the propagation isalmost a projection of the initial states into states at different λ values. In other words,
the propagation is so fast that the initial state does not have time to change, it is simply
projected in the different basis sets as a function of λ. To accurately describe this fast
propagation, the diabatic basis set in each sector should be able to accurately describe|ΨBCS
i. This state is well described in the BEC side by the lowest atomic state. How1
ever, in the strongly interacting unitarity region, there is no state that qualitatively

99
describes |ΨBCSi.
1
For that reason, we do observe in the numerical propagations that some probability is lost during these fast ramps. Also, we observe that most probability is lost in
the strongly interacting region. To avoid this problem, we introduce a new state in our
diabatic basis set at each sector. This new state is orthogonal to the diabatic basis set
and completes the description of the ground state in the BCS side. To obtain this newstate, we apply a Gram-Smith orthogonalization of the ground state on the BCS sidewith respect to the diabatic basis set:Ã|ΨjNd +1 i = N

|ΨBCS
i−
1

Nd
X

!
hΨji |ΨBCS
i |Ψji i
1

.

(5.4)

i

Here N is a normalization factor. The new state |ΨjNd +1 i ensures that the ground statein the BCS side is well described at any λ value.
To understand the time propagation of this system, we visualize how the probabilities evolve in the BCS-BEC crossover. At each point of the time propagation, theprobability of being in state i is given by pi (t) = |cji (t)|2 . Here j denotes the sector that
includes λ(t). We can obtain the probabilities of evolving in a given family by summing
the probabilities of all states belonging to the same family. For two particles, there are
two families, the dimer family, which only includes the lowest state, and the two-atomfamily, which includes the rest of the states. In this case, we define pd (t) = |cj1 (t)|2 , andP d j
2P2a (t) = N
i=2 |ci (t)| .
Similarly, for the N = 3 system we can define three families: the ground state,
the excited atom-dimer state, and the three-atom state. For each of these families we
can define a probability: pg (t), pad (t), p3a (t), respectively.
Finally, in the N = 4 system, we can define four families: the ground state,
the excited dimer-dimer states, the dimer–two-atom states, and the four-atom states.
These families are characterized by the probabilities pg (t), pdd (t), pd2a (t), and p4a (t),respectively.

100

1

(a)

0.5

P

0
1

(b)

0.5
0
1

(c)

0.5
0

−6

−4

−2

0

aho /as2

4

6

Figure 5.6: Probabilities of the N = 4 system during a unidirectional ramp at constant
speed from the BCS to the BEC side. The initial configuration is |ΨBCS
i. The black1
curve corresponds to pg (t). The blue curve corresponds to pdd (t). The red curve corresponds to pd2a (t), and the green curve corresponds to p4a (t). (a) Probabilities in a
ramping at a speed of χ ≈ 13. (b) Probabilities in a ramping at a speed of χ ≈ 52. (c)Probabilities in a ramping at a speed of χ ≈ 128.

101
Figure 5.6 presents examples of the numerical time evolution of a N = 4 system
during a unidirectional ramp at constant speed from the BCS to the BEC side. As
expected, the probability of staying in the ground state decreases with speed. The
probability of evolving in the four-atom configuration increases with speed, which isin agreement with the projection argument. The transfer of probability occurs mainlyin the strongly interacting regime, −2 . aho /as . 2. In this region, the diabatic
states are sometimes mixed, producing jumps in the probabilities. For example, aroundaho /as ≈ 1, the red and blue curves have a kink due to an avoided crossing between an
excited dimer-dimer state and a dimer–two-atom state.
We can also study the probabilities at the end of the time evolution as functionsof the speed ξ. Before analyzing these numerical results, however, we must consider thesimple Landau-Zener model that provides insights about our numerical calculations.

5.3.1

The Landau-Zener model and the P-matrix analysis

The Landau-Zener model provides a simple description of nonadiabatic transitions. We review this model and analyze the relationship between the P-matrix between
adiabatic states and the Landau-Zener parameter, as derived in Ref. [45].
We first consider a two-channel model as a function of an adiabatic parameter λthat is changed at constant speed, i.e., λ = νt. The Landau-Zener model assumes thefollowing Hamiltonian,


 
ψ
²
²
ψ
1 
12  
 1   1
H
.

=
ψ2
²12 ²2
ψ2
(5.5)

Following the Landau-Zener model, we set ²1 −²2 = αλ, and we consider ²12 independentof λ, i.e., d²12 /dt = 0. The time evolution of this model can be easily solved. We areparticularly interested in the nonadiabatic transition probability Tna . Tna describes theprobability of evolving to a final adiabatic state different from the initial adiabatic state
after the parameter λ is tuned through an avoided crossing. To obtain this probability,

102
the time-dependent Schrödinger equation is propagated starting at t = −∞ [with ψ(t) =ψ1 ] to t = +∞. The nonadiabatic probability is then given by Tna = | hψ(+∞)|ψ2 i |2 .
Landau and Zener solved this problem analytically and showed that
Tna (ν) = e−δ ,(5.6)

where δ = 2π²12 /(αν).

+

+
1

E

2

1

2
-

-

Figure 5.7: Schematic description of the Landau-Zener model. Solid black and red
curves correspond to the nonadiabatic states ψ1 and ψ1 , respectively. Dashed blue andgreen curves correspond to the adiabatic states ψ− and ψ+ , respectively.

Unfortunately, Eq. (5.6) is not very useful in its current form when it comesto analyzing numerical results. In particular, it requires a knowledge of nonadiabatic
quantities such as α and ²12 . In numerical calculations, in which the adiabatic spectrum
is obtained, these quantities cannot be directly calculated. To obtain the coupling ²12 ,we can calculate ∆, i.e., the difference between adiabatic energy curves at the closest
approach. In the Landau-Zener model ∆ = 2²12 . However, the quantity α is difficult toestimate unless you have a very clean avoided crossing (like the one in Fig. 5.7).
Nevertheless, Clark [45] showed how α can be obtained from an analysis of theadiabatic P-matrix coupling. The P-matrix between two adiabatic states |ψ+ i and |ψ− i

103
is defined as
P+− (λ) = hψ+ |

dψ−
i.
dλ

(5.7)

As we show below, the properties of P+− (λ) depend on ∆ and α. Therefore, by analyzing
the P-matrices, we can extract the Landau-Zener parameters.
To show the relation between the P+− (λ) and ∆ and α, we consider the timeindependent solutions of the Hamiltonian [Eq. (5.5)] to be a function of the adiabatic
parameter λ, whereH(λ)ψ± (λ) = ²± (λ)ψ± (λ).

(5.8)

µ
¶
q
´
p
1³
2
2
²1 + ²2 ± (²2 − ²2 ) + 4²12 =²1 + ²2 ± α2 λ2 + ∆2 .
2

(5.9)

The eigenvalues are given by
1
²± =
2

Here, we set ²1 − ²2 = αλ, and ∆ = 2²12 . The adiabatic eigenvectors can be now written
as











 ψ+   cos θ sin θ   ψ1 

=

.
ψ−
− sin θ cos θψ2(5.10)

Next, we consider the P-matrix for this system:
P+− (λ) = hψ+ |hψ+ | dHdψ−dλ |ψ− ii=.
dλ²+ − ²−(5.11)The second equality is a consequence of the Hellman-Feynman theorem, and we use it
to extract P+− (λ). Using Eqs. (5.5) and (5.10), we obtain
hψ+ |

dH
tan(θ)|ψ− i = −α cos(θ) sin(θ) = −α.
dλ
1 + tan2 (θ)(5.12)

Analyzing the eigenvectors and setting ²1 − ²2 = αλ and ∆ = 2²12 , we obtain thattan(θ) = −∆√.
αλ + α2 λ2 + ∆2

(5.13)

Finally, combining Eqs. (5.11-5.13), we obtain
LZ
P+−
(λ) =

1 ³α´
1
.
2
2 ∆ 1 + ( αλ∆)(5.14)

104
Thus, the P-matrix only depends on the ratio α/∆. Since the Landau-Zener parameter
depends on ∆2 /α, the P-matrix analysis is not enough to extract the Landau-Zener
parameter δ. For that reason, ∆ is extracted from the spectrum. Another interesting
point is that the area under the P-matrix in entirely independent of both α and ∆.

5.4

Landau-Zener analysis of few-body trapped systems.
The N = 2 trapped system can be treated with the two-channel Landau-Zenermodel. However, the N = 3 and N = 4 systems are more complicated. Their spectra have several avoided crossings that are potentially important for the unidirectional
ramps considered in this section. Thus, the dynamic of these systems cannot be described by a simple Landau-Zener transition. For that reason, we model the transition
probabilities pi of the N = 3 and N = 4 systems with a sequence of Landau-Zenertransitions. Analyzing the spectra and the numerical P-matrices, we can extract theLandau-Zener parameters δ. Since the P-matrices peak at different λ values, we can use
the positions of the peaks to understand the order in which the Landau-Zener transitions
occur.
We begin by extending the definitions presented in the Sect. 5.3.1 for describing
several Landau-Zener transitions. The Landau-Zener transition probability between the
adiabatic Ψi and Ψj is
Tij (ν) = e−δij = e

η

− ξij

.

(5.15)The Landau-Zener parameter δij characterizes the transition and has the form δij =π∆2ij
2αij ν , where ∆ = Ei − Ej

is evaluated in the transition region (at the peak of theP-matrix), and αij is obtained from P-matrix analysis. The transition probabilities Tij
can be conveniently described as functions of the dimensionless speed parameter ξ anddimensionless quantities ηij that describe the specific transitions.
Similarly, we extend the definitions of the P-matrix to several states. The P-

105
matrix between two adiabatic states Ψi and Ψj is
Pij (λ) = hΨi |hΨi | dHdΨjdλ |Ψj ii=,dλEi − Ej(5.16)

where λ is the adiabatic parameter, and Ei and Ej the energies of these states. To
numerically evaluate Pij (λ), we use
H = H0 + V0 (λ) Hint andH= V00 (λ) Hint .
dλ(5.17)In the CG basis set expansion, H0 and Hint , are matrices and V0 (λ) is the potential
depth of the interaction whose derivative is V00 (λ). Using Eq. (5.17), the evaluation ofa P-matrix takes the form
Pij (λ) =V00 (λ)hΨi (λ)|Hint |Ψj (λ)i .
Ei (λ) − Ej (λ)(5.18)In Eq. (5.18), we include all the explicit dependence on λ. Note that Hint does notdepend on λ and is only evaluated once before being used to describe the complete
crossover region. For the Gaussian interaction used for these systems, V0 (λ) is, to agood approximation, a linear function, and V00 (λ) is then constant. The energies Ei (λ)
and Ej (λ) are extracted from the spectra presented in Figs. 5.1, 5.3, and 5.4. The
term hΨi (λ)|Hint |Ψj (λ)i implies just a matrix multiplication of the eigenstates with thematrix Hint .
Figure 5.8 presents some of the P-matrices we obtained numerically. We observethat all these Pij have a Lorentzian-like form. The P1,2 has a smooth behavior but theother Pij are not very smooth. The series of crossings or narrow avoided crossings affectthe P-matrices because the diabatization procedure cannot avoid a possible mixing ofstates when the energies are close to becoming degenerate. Nevertheless, as we show
below, the analysis of these P-matrices is possible and provides good estimates of the
Landau-Zener parameters.
According to the Landau-Zener theory presented above, for each transition be-106
tween Ψi and Ψj has a Lorentzian form of
PijLZ (λ) =

αij
2∆ij

h
1+

1

i ,
αij (λ−λij ) 2∆ij(5.19)

where ∆ij and αij are the parameters that characterize the Landau-Zener parameter δij .
For Landau-Zener transitions, we can more rigorously define the width of the avoiding
crossing as ∆λij ≡ ∆ij /αij . The position of the PijLZ (λ) peaks are given by λij . In theLandau-Zener formula, the height and width of the Lorentzian are characterized by asingle parameter ∆ij /αij . However, because most of numerical calculations cannot be
described accurately by PijLZ (λ), a more flexible function is needed. To fit our numericalresults we propose
Pijf it (λ) =

aij
2

h
1+

1

i .
(λ−λij ) 2
bij(5.20)The height of the Lorenztian is given by aij , and the width is given by bij . In principle,
we can use either aij , bij , or a combination of both quantities to estimate ∆ij /αij . We
observe that the extraction of ∆ij /αij using the width of the Lorentzian, bij , usually
gives results closer to the numerical predictions. For the N = 2 case, the Landau-Zener
formula works well, and aij ≈ bij , but in the N = 4 case, the width and height of thefitted Lorentzian can differ up to a factor of 2.

5.4.1

Numerical results

We first analyze the dynamics of the N = 2 system. Starting at λBEC ≈ −7/aho ,we ramp the interaction at different speeds ν to the BEC side, where λBEC ≈ 7/aho .
The probability of remaining in the ground state (dimer state) is then measured. The
symbols in Fig. 5.9 (b) present the numerical results. The black circles correspond to
the probability of evolving in the dimer state (red energy curves in Fig. 5.1), and the redcircles correspond to the probability of evolving in two-atom states (red energy curves
in Fig. 5.1).

107

0.4
0.35

Pij /a−1
ho

0.3
0.25
0.2
0.15
0.1
0.05
0
−6

−4

−2

0

2

4

6

aho /asFigure 5.8: The most important P matrices involved in a ramping that starts at the
BCS side with all the probability in the ground state. The black curve represents P1−2 ,the blue P1−5 , the red P2−5 , and the green P5−13 .

1

(a)

(b)

0.3

0.8
0.6

0.2

P

Pij /a−1
ho

0.25

0.15

0.4

0.1
0.2

0.05
0

−5

0

aho /as

5

0
0

50

100

ξ

Figure 5.9: Numerical results and Landau-Zener model for N = 2.(a) circles correspondf it
to P12 obtained numerically. The solid line corresponds to the best fit with P12. (b)Transition probabilities as a function of the speed ξ. Black symbols represents evolution
in the dimer family (ground-state). Red symbols represents evolution in the two-atomfamily (red energy curves in Fig. 5.1). Black and red lines correspond to p1 and p2
obtained from the Landau-Zener model.

108
The simple form of the transition-probability numerical results and the spectrum
suggest that the N = 2 system can be modeled by a single Landau-Zener transition
between the ground state, labeled by 1, and the first-excited state, labeled by 2. We
analyze the numerical P12 and find it has a clear Lorentzian shape [see Fig. 5.9 (a)].
For this P12 , the difference between a12 and b12 is less than 15%. Still, the b12 parameter provides a better Landau-Zener parameter η12 . The Landau-Zener model predicts
probabilities p1 to remain in the ground state and p2 to evolve in the first excited state
as given by
p1 = (1 − T1,2 ),(5.21)

p2 = T1,2 .

(5.22)The Landau-Zener parameter corresponding to this transition is η12 ≈ 4.625. For large
speed parameters ξ, we note that the Landau-Zener prediction is slightly higher than
the numerical results. This discrepancy might be due to limitations in the numericalcalculation for predicting a full conversion efficiency. However, we note an excellent
agreement between the numerical results and the Landau-Zener prediction.
Things get more complicated as we turn the discussion to a N = 3 system.
Figure 5.10 presents the numerical results and the Landau-Zener prediction. In this case,
we cannot describe the transition probabilities with a single Landau-Zener transition;
instead we propose a sequence of Landau-Zener transitions. Even so, we want to restrict
our model to the simplest possible case. Since in the N = 3 system there are threefamilies of states, we consider only three states, one for each family, in describing thissystem in the Landau-Zener approximation. Analyzing the spectrum and the numericalP-matrices, we observe that the most important P-matrices describe transitions between
the lowest energy states of each family. These states correspond to the ground state(labeled as 1), the first-excited state (labeled as 2), and the state corresponding to thelowest red energy curve in Fig. 5.3. This last state is the second-excited state in the

109
BCS regime, and we label it as 3. Analyzing the P-matrices, we note that the firsttransition is 1 → 2, the second transition is 1 → 3, and finally the third transition is
2 → 3. This order in the transitions implies probabilities
p1 = (1 − T1,2 )(1 − T1,3 ),(5.23)

p2 = T1,2 (1 − T2,3 ),(5.24)

p3 = T1,2 T2,3 + T1,3 (1 − T1,2 ).

(5.25)

It is easy to test whether the total probability is conserved, i.e., p1 + p2 + p3 = 1.
1
0.8

P

0.6
0.4
0.2
0
0

10

20

30

40

ξ
Figure 5.10: Probability of evolving in a given configuration as a function of the dimensionless speed parameter χ. The symbols correspond to the numerical evolution, while
the curves correspond to the Landau-Zener theory. Black curve and symbols correspondto the ground state. Blue curve and symbols corresponds to the excited atom-dimer
configuration, and red curve and symbols corresponds to the three-atom configuration.

Analyzing the spectrum and the P-matrices, we obtain Landau-Zener transitionsof η12 ≈ 9.1, η13 ≈ 13.4, and η12 ≈ 2.5. The probability pi results from the LandauZener model are presented as solid lines in Fig. 5.10. From the figure we can see that theLandau-Zener model qualitatively describes the numerical results. However, we can also
clearly see the limitations of this model, including important deviations between it andthe numerical results. In particular at large ξ, the conversion efficiency for three-atom

110states according to the Landau-Zener model is much higher than the numerical results.
We carried out several tests of the numerical calculations that confirmed the resultspresented in Fig. 5.10. We observe that the P-matrices have large overlaps and theirpeaks are very close together. As a result, the approximation of a sequence of isolated
Landau-Zener transitions fails. A theory that takes into account multiple simultaneoustransitions will be needed. Such theory is beyond the scope of this dissertation.
Finally, we consider the N = 4 system. Figure 5.11 (b) displays our numericalresults. The black symbols correspond to the dimer-dimer ground state. The blue
symbols correspond to the excited dimer-dimer family, the red symbols to the dimer–
two-atom family, and the green symbols to the four-atom family. For slow ramps (smallξ), the probability of forming a condensate, i.e., remaining in the ground state, is large.
For intermediate ramps, the greatest probability is breaking one bond and ending up
with a dimer plus two particles. For fast ramps (large ξ), the probability of staying in
the atomic ground state on the BEC side is the most important. The probability of thesystem evolving in an excited dimer-dimer configuration remains small for all ramping
speeds.
To analyze these transitions within the Landau-Zener approximation, we label the
diabatic states according to their energies in the BCS regime. This labeling is arbitrary
since many of the states are almost degenerate. To select the potentially importantstates, we consider the possible pathways according to the P-matrix couplings. Startingfrom the ground state, we note that |Ψ1 i has important couplings with states |Ψ2 i
and |Ψ5 i. Here, |Ψ2 i is the first excited dimer-dimer state, i.e., the lowest state of theexcited dimer-dimer family. State |Ψ5 i is the first excited state of the dimer–two-atom
configuration. Since an important probability is transferred to states |Ψ2 i and |Ψ5 i, we
analyze the couplings of these states to follow the flow of probability. State |Ψ2 i has
an important coupling with |Ψ5 i, and state |Ψ2 i has an important coupling with |Ψ13 i.
Here, |Ψ13 i is the lowest state of the four-atom configuration, i.e., the atomic ground

111

15

1

(a)

0.8

10

0.6

P

E/h̄ω

(b)

5

0.4
0

−5

0.2

−5

0

aho /a0

5

0
0

50

100

ξ

Figure 5.11: (a) Energy in the BCS-BEC crossover of the important states in theLandau-Zener approximation. The black curve corresponds to |Ψ1 i which represents
the ground state configuration. The blue curve corresponds to |Ψ2 i which represents
the excited dimer-dimer configuration. The red curve corresponds to |Ψ5 i which represents the configuration of a dimer plus two atoms. The green curve corresponds to|Ψ13 i which represents the four-atom configuration. (b) Probability of evolving in agiven configuration as a function of the dimensionless speed parameter χ. The symbols
correspond to the numerical evolution, while the curves correspond to the Landau-Zenertheory. The colors follow the same convention of Figure (a). Results from Ref. [182].

112
state in the BEC side. Figure 5.11 (a) presents the energy curves of these four states.
Conveniently, each of these states represents a different configuration. For that reason,
this is the minimal set of states that can describe the numerical results. We could
include more states in our analysis but we restrict ourselves to the simplest possible
case.
Figure 5.8 presents the relevant numerical P-matrices. The order of the peaks
reveals the following sequence: the first transition is 1 → 2, then 2 → 5, then 1 → 5,
and finally 5 → 13. The Landau-Zener prediction for this sequence is
p1 = (1 − T1,2 )(1 − T1,5 ),p2 = T1,2 (1 − T2,5 ),p5 = ((1 − T1,2 )T1,5 + T2,5 T1,2 )(1 − T5,13 ),(5.26)

p13 = ((1 − T1,2 )T1,5 + T2,5 T1,2 )T5,13 .
Again, the sum of all these probabilities is, by construction, 1. The Landau-Zener
parameters obtained from the P-matrix analysis are η12 ≈ 5.4, η15 ≈ 6.6, η25 ≈ 2.1
and η5,14 ≈ 13.8. We note that the P-matrix P5−13 in Fig. 5.8 is only in qualitative
agreement with the Lorentzian form, so the uncertainty in η5,13 is large. However, thesequence of Landau-Zener transitions in the model shows very good agreement with thenumerical results even though many possible transitions were neglected. In comparison
with the N = 3 case, the sequence of transitions is better defined, suggesting a reason
for the better agreement with the numerical results.

5.5Comparison with experimental results.
In this section, we compare our few-body calculations with experimental results

obtained with ultracold Fermi gases. First, we compare the molecule fraction as afunction of the speed of the ramping. Then, we analyze atom-molecule coherence.
Finally, we study conversion efficiency as a function of temperature.

113
5.5.1

Molecule fraction as a function of the speed

To compare our results with Fermi gas experiments, we have to find a way to relate
few-body calculations with many-body calculations. We first consider a homogeneous
system where λ is tuned from λ = −∞ (BCS) to λ = +∞ (BEC). The only relevant
quantities of this system are the speed ν, the density ρ, and the mass of the particles m.
m dλTherefore, the dimensionless speed parameter that characterizes the ramp is χ = ~ρdt .

This is the relevant parameter for comparing homogeneous systems at different densities,
ramp speeds ν, or masses. We also consider χ to be the relevant parameter for comparing
our calculations to larger, experimentally accessible systems.
Here, we consider that it is the density of the system (and not the trap length asin ξ) that characterizes both few- and many-body systems. The trap only plays a rolein confining the gas to a given density. In other words, to relate a large Fermi gas witha few body calculation, we solve the few-body system in a trap tight enough so thatthe average density of this few-body system is the same as the average density of the
Fermi gas in the experiment. This strategy is similar to the frequency-scaling method
presented in Ref. [22].
This strategy allows us to write Eq. (5.27) in terms of the parameter χ, i.e.,κ

Tij (χ) = e

− χij

,

(5.27)

where κij are now dimensionless parameters. In the experiments carried out at JILA
and Rice, the Landau-Zener parameter for atom-molecule transitions is usually written¡ ¢−1[143]. If the dependence of as on the magnetic field can be approxas δexp = β dB
dt³´
wimated by as (B) = abg 1 + B−B0 , then β has been theoretically predicted [77] to be
β = cρwabg /m, where c is a proportionality constant, w is the width of the resonance,
and abg is the background scattering length.
The same results can be obtained from our dimensional analysis arguments. Usingas (B), we can write χ in terms of the experimental parameters. Then, the Landau-Zener

114
parameter κexp /χ = β¡ dB ¢−1dt, where β = cρwabg /m. Thus, by considering χ as therelevant dimensionless parameter, we obtain agreement in the functional dependenceof the Landau-Zener parameter with Ref. [77]. The dependence of δexp on ρ has been
experimentally verified [90]. Like in the experiments, we use the average density on the
weakly interacting BCS side to evaluate χ. Using noninteracting wave functions, we√√
obtain ρ2 = 2/( 2πaho )3 ≈ 0.127/a3ho and ρ4 = 29/(12( 2πaho )3 ) ≈ 0.153/a3ho .

molecule fraction

1
0.8
0.6
0.4
0.2
0
0

100

χ

200

300

Figure 5.12: Molecular fraction as a function of χ. The dashed red curve is the N = 2
prediction, and the black solid line is the N = 4 prediction.

The molecule fraction, i.e., the fraction of particles that become molecules after
the ramping procedure, is probably the most relevant quantity to compare with experiments. Only even-N systems can produce full conversion efficiency and, for thatreason, are considered for molecule-fraction analysis. In the N = 2 system, the molecule
fraction is just the probability of staying in the ground state. For N = 4 systems, themolecule fraction is defined as the probability of evolving in a dimer-dimer configuration plus one-half the probability of evolving in a dimer plus two-atom configuration.
In Fig. 5.12, we compare the two-body prediction with four-body predictions.
In the experiments, the molecule fraction is fitted with a Landau-Zener function,
fm (1 − e−κmol /χ ), where fm is is the conversion efficiency that depends on temperature. Whether a Landau-Zener function is the correct functional form to describe the115molecule-formation fraction in large systems remains a question that existing experiments have not resolved [136, 3, 189]. The Landau-Zener model presented in this workfor four particles does not predict a single Landau-Zener function but rather a combination of different Landau-Zener terms. However, the final molecule fraction predictedby this model and the numerical results for the molecule-formation fraction can be approximately fitted by the Landau-Zener function, leading to κmol4 ≈ 59 ± 6; this value
is higher than the two-body prediction of κmol2 ≈ 39. This result is consistent withthe experimental Landau-Zener parameter obtained of Ref. [145] for 40 K. The fit of theexperimental data to a Landau-Zener formula predicted a β = 20 ± 6 µs/G, which canbe expressed in terms of the dimensionless form. For the experimental conditions, thisresult gives a value of κexp ≈ 62 ± 15. Note that this experiment was carried out at
T /TF = 0.33 where a conversion efficiency of approximately fm = 0.6 was observed.
In contrast, our calculations are at T = 0 where the conversion efficiency is 100%. In
addition, the experiments carried out at Rice measured the Landau-Zener parameterof β = 1.3 ms/G for 6 Li [170]. Taking into account the experimental conditions andproperties of the 6 Li Feshbach resonance at B ≈ 543.8, we estimate a κexp ∼ 90. In
this second experiment, the conversion efficiency was of the order of fm = 0.5, andthe estimated temperature T ≈ 0.1 TF . Both experiments are in qualitative agreementwith the four-body predictions, suggesting that χ is the relevant quantity for comparing
few-body results with many-body results.
Up to this point, we have only considered unidirectional ramps. However, moresophisticated ramp schemes can be implemented. Next, we consider ramps scheme thatmanifest the atom-molecule coherence in the system.

5.5.2

Atom-molecule coherence

Another interesting phenomenon studied experimentally is atom-molecule coherence. For bosonic condensates, atom-molecule oscillations [58], or quantum beats [22],116
have previously been explored. Fermionic systems near a narrow Fano-Feshbach resonance have been also predicted to exhibit aatom-molecule coherence [6]. Here, wemol. frac.

analyze atom-molecule oscillations in the N = 4 system.

0.6
0.4
0.2
0
0

2

4

6

8

10

50

60

Amp. (a.u.)∆t (1/ω0 )
0.1
0.05
0
0

10

20

30

40

ω (ω0 )

Figure 5.13: (upper panel) Molecule fraction as a function of the delay ∆t in the ramp
scheme. (lower panel) Fourier transform of the upper panel figure. The peaks of thespectrum correspond to the most important energy transitions. Figure from Ref. [182].

To study atom-molecule coherence in a Fermi gas, we proposed the following
ramping scheme. Starting in the ground state on the BCS side, we ramp at medium
speed (χ ≈ 32) to the BEC side. We stop at a value λstill , we wait a time ∆t, andthen we go back with the same speed to the BCS side. Finally, we slowly ramp λ to
the BEC (χ ≈ 2) side, and we measure the molecule fraction. The molecule fraction as
a function of ∆t is shown in Fig. 5.13 (upper panel). This ramping scheme produces alarge molecule-fraction oscillation that varies from 0.2 to 0.8. To analyze the frequencies
involved in the molecule-fraction oscillations, we Fourier transform the molecule fraction,
as shown Fig. 5.13 (lower panel). The peaks in the Fourier transform can be easily
related with the most important transition at λstill . The peaks around ω ≈ 28 ω0correspond to transitions between configurations that have one extra or one fewer dimer.
For example, the highest peak is a transition between Ψ5 and Ψ13 , while the second

117highest is a transition between Ψ1 and Ψ5 . The peaks around ω ≈ 57ω0 correspond toa transition between Ψ1 and Ψ13 as well as Ψ2 and Ψ13 . In the real experiment, where
the atom-molecule transitions are much larger than the trap frequency, this multipeak
structure in the Fourier transform around the atom-molecule transition frequency will
disappear, yielding only a single peak.

5.5.3

Conversion efficiency as a function of temperature.

In the experiments with Fermi gases carried out at JILA, the maximum molecule
fraction was measured versus temperature [90]. To explore the dependence of themolecular-formation probability on the initial temperature, we have carried out a twobody calculation in the canonical ensemble.

molecule fraction

1.0

0.5

0.0
0.0

0.5

1.0

1.5

T/T

F

Figure 5.14: Molecule fraction as a function of T /TF . The black circles are the experimental results showing the maximum molecule fraction as a function of the temperature [90]. The blue and curves are the theoretical predictions from two particles in a
trap using TF2b and TFT F , respectively. Experimental data courtesy of Cindy Regal.

We assume there is complete adiabaticity during the ramping with regard to themolecular formation process. However, the ramps were not so slow as to allow thermal
equilibrium to be reached at each moment during the ramp. Therefore, we can deducethe molecule fraction by simply “counting” the states that would form molecules in the118initial thermal distribution. The resulting molecular fraction is extracted as a functionof initial temperature by starting from the initial thermal distribution and counting thestates that, in a completely adiabatic ramp, would form molecules. For this study, we
begin from thermal equilibrium in a weakly interacting gas on the BCS side. The twobody states under study are filled accordingly to Boltzmann factors, and the partition
function isZ(T ) =X − Eqe kB T .

(5.28)q

Here, kB is the Boltzmann constant, and Eq are the eigen energies of the N = 2 systemlabeled by the collective quantum number q. Separating in relative and center-of-mass
coordinates and solving the noninteracting system in spherical coordinates, the energyof the N = 2 system can be written as
EnCM ,lCM ,nr ,lr = ECM + Er = (2nCM + lCM + 2nr + lCM )~ω.

(5.29)

Only those atom pairs that are in the relative coordinate ground state will form molecules.
This pairs have nr = lr = 0. Both the partition function, Eq. (5.28), and the sum overthe pairs that form molecules can be solved analytically. These calculations lead to amolecule fraction given by³´3
F (T ) = e−3~ω/kB T e~ω/kB T − 1 .

(5.30)

To compare this two-body prediction with experimental results, we plot the results as afunction of T /TF . Since TF ultimately depends on the density, this technique involves
a density scaling.
Considering the highest occupied orbital energy, TF2b = EF /kB = 3~ω/(2kB ),where EF is the Fermi energy. However, the Fermi energy EF for small N systems
depends strongly on the shell effects. For that reason, we propose to use a smooth
semiclassical Fermi energy obtained from a Thomas-Fermi approximation, i.e., EFT F ≈1.211~ω. Therefore, TFT F ≈ 1.211~ω/(kB ). Figure 5.14 presents the comparison between

119
the experimental results and the N = 2 prediction. The red curve is obtained using
TF2b , and the blue curve is obtained using TFT F . Both predictions are in good agreementwith the numerical results.
The experimental results presented in Fig. 5.14 were successfully described by amany-body model proposed by Cornell [90] in 2005. This model is based on similararguments. As in our N = 2 calculation, the conversion efficiency only depends on theinitial conditions before the adiabatic ramp. According to Cornell’s model, two particlesof the gas would form a molecule depending on their “distance” in phase space δps . For
c , the particles would form a molecule, and for
distances δps smaller than some cutoff δps
c , the particles would not form a molecule. This simple idea was implemented
δps > δpsc as a fitting parameter to the experimental results.
using Monte Carlo methods and δpsExcellent agreement was obtained between the theory and the experiment. In contrast,
our theory does not have fitting parameters besides the uncertainty in defining the most
relevant EF .

5.6

Conclusions
In this chapter, we have presented an accurate description of few-body trappedsystems. First, we showed how the N = 2, N = 3, and N = 4 spectra evolve in the
BCS-BEC crossover. These results were used to propagate in time trapped atoms in
the ground state from the BCS region through the crossover, while investigating theimportant nonadiabatic transitions. Even though the spectra present a rich structureof avoided crossings, we have shown that a simple Landau-Zener model approximately
describes the dynamics of unidirectional ramps.
Finally, we compared our few-body results with experimental measurements. We
find reasonable agreement in molecule-fraction predictions as functions of the speed of
the ramping and temperature. We also investigated atom-molecule coherence using anovel ramping scheme. These results show that few-body calculations can successfully

120
describe an appreciable fraction of many-body experimental results.

Chapter 6
BCS-BEC Crossover of few-body trapped Systems

6.1

Introduction
In this chapter, we analyze the energetics and structural properties of fermionicfew-body trapped systems in the BCS-BEC crossover. In particular, we focus on systems
with N = 3, 4.
The systems considered here are relevant for experiments in optical lattices. Ina deep three-dimensional optical lattice, the tunneling probability between lattice sites
is negligible, and each lattice site can be considered an harmonic trap. Therefore, theoptical lattice becomes an ensemble of microscopic harmonic traps in which the fewbody properties can be probed.
Interestingly, in some cases few-body physics controls the many-body phenomena
in an optical lattice. A clear example is an optical lattice with N = 3 particles ineach site. The ground state changes its angular momentum from L = 1 for small andnegative scattering lengths to L = 0 for small and positive scattering lengths. Thischange produces a quantum-phase transition in the optical lattice as the scattering
length is tuned in the BCS-BEC crossover [103]. In addition, if the optical lattice is
not that deep and the tunneling probability is small but not negligible, then few-bodytrapped solutions can be used as starting point in perturbative treatments of many-bodytheories.
Solving the Schrdinger equation for more than a few fermions by first-principles

122
methods is, despite the increasing available computer power, still a challenging task.
Monte Carlo methods are one of the most suitable methods for such studies. Unfortunately, however, assessing the accuracy of the assumptions going into Monte Carlocalculations, such as the nodal surface employed in the FN-DMC approach, remains achallenge. Our calculations for N = 3, 4, which use the CG and FN-DMC approaches
in parallel, benchmark the strengths and limitations of the nodal surface employed in
the FN-DMC calculations and validates, to some extent, the FN-DMC for N > 4.
The calculations for trapped systems can also be used to extract properties ofthe free systems. If the range of the interactions and the scattering length are muchsmaller than the trapped size, then the trapping potential provides confinement without
significantly affecting the few-body physics. Consequently, the trapped results can be
related with those of the free systems. When the magnitude of scattering length issimilar to the trapping potential, then the confinement plays a major role affecting theenergies and the correlations.
The main objective of this chapter is to obtain and interpret solutions of few-bodytrapped systems. The model Hamiltonian that describes these systems is given by
H=

N1 µ
X
−~2
i=1

2m1

1
∇2i + m1 ω 2~ri2
2

¶
+

N2 µ
X
−~2
i0 =1

2m2

1
∇2i0 + m2 ω 2~ri20
2

¶
+

N1 X
N2
X

V (rii0 ). (6.1)

i=1 i0 =1

Here, unprimed indices label mass m1 and primed indices mass m2 fermions, and N is
assumed to be even, ω denotes angular trapping frequency, and ~ri the position vectorof the ith fermion. The interaction potential V has a purely attractive Gaussian form
(see Eq. 3.20). The mass ratio κ is defined by m1 /m2 , and throughout the analysis
we assume m1 ≥ m2 . For each specie or component, there is a trap length associatedpp(i)aho = ~/mi ω. We also define a trap length associated with the pair aho = ~/2µω,
(1)

(2)

where µ = m1 m2 /(m1 + m2 ). For equal mass systems, aho = aho = aho .
For N = 3 systems, there are two types of systems: the two heavy and one lightparticle system [N1 = 2 and N2 = 1] and the two light and one heavy particle system

123[N1 = 1 and N2 = 2]. For N = 4 systems, N1 = 2 and N2 = 2.
In the following section, we first analyze the spectrum of Eq. (6.1) as the scattering is tuned in the BCS-BEC crossover. Then, we compare the correlated Gaussian(CG) lowest energy results with both the fixed-node diffusion Monte Carlo (FN-DMC)
predictions and the theoretical predictions in the limiting behaviors. Next we study
the lowest state in the BEC side and extract collisional properties of these systems.
Then we show how a simple two-channel model can describe qualitatively the BCSBEC crossover. Finally, we analyze the structural properties of these few-body trappedsystems. Most of the work in this chapter was done in collaboration with D. Blume andpresented in Refs. [184, 185].

6.2

Energy crossover curves for N = 3 and N = 4
This section analyzes in detail the ground-state energy of N = 3 and N = 4

systems. The calculations are extended to unequal mass systems, and comparisons
between CG results and FN-DMC results are presented.
Consider initially the N = 4 system. From the ground-state energy, we construct(κ)

the energy crossover curve Λ4 , defined in Refs. [184, 185] as
(κ)

Λ4 =

E(2, 2) − 2E(1, 1)
.
2~ω

(6.2)

Here E(2, 2) is the ground-state energy of the four-particle system and, E(1, 1) is theground-state energy of the two-particle system. Both energies depend on the mass ratio
κ and the scattering length. The energy crossover curve is convenient for comparisonsbecause any effects of finite-range interactions on the two-body binding energy are
significantly reduced by the subtraction in Eq. (6.2). Therefore, even though both E(4)(κ)

and E(2) are not completely universal, Λ4(κ)

The energy crossover curve Λ4(2µ)

is universal.

is shown in Figure 6.2 for four fermions as afunction of aho /as calculated by the CG and FN-DMC approaches for (a) κ = 1 and

124
(2µ)

(b) κ = 8. Here, the oscillator length aho is defined in terms of the reduced massp(2µ)µ = m1 m2 /(m1 + m2 ), i.e., aho = ~/(2µω). The solid lines in Fig. 6.2 are obtainedusing E(4) calculated by the CG approach, while the circles and crosses are obtainedusing E(4) calculated by the FN-DMC approach using ΨN =4 and ΨP , respectively. The
ranges r0 of the two-body potentials used in Fig. 6.2 are much smaller than the oscillator(2µ)

lengths, i.e., r0 ≈ 0.01aho . From our CG energies for different r0 , we estimate that the(κ)ΛN curves shown in Fig. 6.2 deviate by at most 1% from the corresponding curves forzero-range interactions. For m1 = m2 , e.g., the energy at unitarity calculated by the CG(2µ)

approach for the Gaussian interaction potential is E = 5.027~ω for r0 = 0.01aho

and

(2µ)

E = 5.099~ω for r0 = 0.05aho . For comparison, the FN-DMC energy for the square(2µ)

well potential with r0 = 0.01aho is E = 5.069(9), which is in good agreement with theenergies calculated by the CG approach.
1

Λ4

(1)

(a)
0.5

0

Λ4

(8)

1

(b)

0.5

0

−6

−4

−2

0
2
(2µ)
aho /as

(κ)

4

6

(2µ)

Figure 6.1: Energy crossover curve Λ4 as a function of aho /as for (a) κ = 1 and (b)κ = 8. Solid lines are calculated by the CG approach, and circles and crosses by the
FN-DMC method using ψN =4 and ψP , respectively. Figure from Ref. [184].

As expected, the energy-crossover curve smoothly connects the limiting valuesof one on the BCS side and zero on the BEC side. Importantly, the lowest FN-DMC
energies and the CG energies agree well, implying that the functional forms of Ψ(N =4)

125
and ΨP are adequate. For equal masses [panel (a)], the FN-DMC energies at unitarity
calculated using the two different trial wave functions agree approximately. In contrast,
for unequal masses [panel (b)], the nodal surface of ψ(N =4) leads to a lower energy at
unitarity than that of ψP , and the crossing point between the energies calculated using
Ψ(N =4) and ΨP moves to the BEC side. The normal Ψ(N =4) describes the system in a
wider region which can be understood by realizing that the densities of the heavy andlight particles do not overlap fully, leading to a reduced pairing.
The good agreement between both numerical methods suggests two things. First,(κ)Λ4 is indeed universal. Second, numerical methods accurately described the BCS-BEC
crossover of the four-body system.
We continue our analysis by studying the crossover curve for N = 3 for different
mass ratios κ. Here, the behavior of odd N systems is rich and, in many cases, qualitatively different from that of even N systems. One characteristic of odd N systems is apossible change of the angular momentum of the ground state as the scattering length istuned through the BEC-BCS crossover region [103, 166, 185]. The ground state in the
BCS region can have nonzero angular momentum (see Sec. 4.2), but in the BEC limit,
the ground state has always L = 0. Therefore, to obtain the ground-state energy for
odd N systems, we need to consider L > 0. This is a problem since CG only describes
L = 0 states. This problem was overcome in Refs. [19, 185]) by introducing a spectator
particle that couples to L = 0 in the N particle system. This method allows one, in
principle, to obtain the spectrum of the N particle system with an arbitrary L. We will
explore this method in more detail in chapter 7.
As in the the N = 4 system, we can define an energy-crossover curve. The energy
crossover curve for the N = 3 systems takes the form
(κ)

Λ3 =

E(3) − E(1, 1) − 3~ω/2
,
~ω

(6.3)

where E(3) is the ground state energy of the N = 3 system and corresponds to E(2, 1) for126
two heavy and one light atoms and E(1, 2) for two light and one heavy atoms. For equal(κ)

masses E(3) = E(2, 1) = E(1, 2). The definition of Λ3 is sometimes convenient because
it depends on the ground state E(3) regardless of its angular momentum. However, tosee the energy crossings between states, it is more convenient to analyze the lowest states
for each angular momentum L. Figure 6.2 shows the three-particle energy E, with theenergy E(1, 1) + 3~ω/2 subtracted for L = 0 (solid lines) and L = 1 (dashed lines). The
upper panel shows results for κ = 1, and the two lower panels for κ = 4 [panels (b)and (c) consider the three-particle system with a spare heavy and a spare light particle,
respectively]. The ground state has L = 1 for aho /as → −∞ and L = 0 for aho /as → ∞.
These limiting behaviors are independent of κ and independent of whether the spareparticle is heavy or light. For equal masses, the change of symmetry occurs at as ≈ aho .
In contrast, for κ = 4 it occurs at as ≈ 0.3aho if the extra particle is a heavy atom
[panel (b)] and at as ≈ 3aho if the extra particle is a light atom [panel (c)]. The dashed(κ)

and solid lines shown in Fig. 6.2 coincide with the normalized crossover curve Λ3 ,
Eq. (6.3), in the region where the ground state of the three-particle system has L = 1(κ)

and 0, respectively. The normalized crossover curve Λ3changes from 1 in the weaklyinteracting molecular BEC regime to 0 in the weakly interacting BCS regime.
We find that the normalized L = 1 energy curve for two heavy atoms and one
light atom [Fig. 6.2(b)] depends on the range of the underlying two-body potential if thescattering length as is positive. For example, the normalized energy curve changes by asmuch as 20% if the r0 of the two-body potential changes from 0.01aho to 0.02aho . This
comparatively large dependence on r0 indicates that the properties of the system with
two heavy atoms and one light atom are not fully determined by the s-wave scattering
length for the ranges considered. In the r0 → 0 limit, the κ = 4 system is expected tobehave universally [140, 102]. We speculate that the comparatively strong dependenceof the normalized energy curve on the range for as > 0 is related to the fact that thethree-particle system supports weakly bound states for sufficiently large κ.

1273(E − E(1, 1) − h̄ω)/h̄ω
2

2

(a)

1
0
2

(b)

1
0
2

(c)

1
0

−5

0

5

aho /asFigure 6.2: Normalized energy (E − E(1, 1) − 3~ω/2)/~ω for N = 3 as a functionof aho /as calculated by the CG approach (lines). E denotes the three-body energy
for L = 0 (solid lines) and for L = 1 (dashed lines). (a) Equal-mass atoms [κ = 1,
E = E(2, 1) = E(1, 2)]. (b) Two heavy atoms and one light atom [κ = 4, E = E(2, 1)].
(c) Two light atoms and one heavy atom [κ = 4, E = E(1, 2)]. The normalized energy(κ)
crossover curve Λ3 , Eq. (6.3), coincides with the dashed and solid lines, respectively,
depending on whether the three-particle ground state has L = 1 or 0. In the CG
calculations, the range r0 of the two-body potential is fixed at 0.01aho . For comparison,crosses and circles show selected FN-DMC energies for L = 0 and L = 1, respectively.
Figure from Ref. [185].

128
For comparison, circles and crosses in Fig. 6.2 show selected three-particle energies
calculated by the FN-DMC method for L = 0 and L = 1, respectively. The good
agreement with the CG results (lines) indicates that the FN-DMC method can be used
to accurately describe different symmetry states.
Our CG energies for equal-mass systems interacting through short-range potentials presented in Fig. 6.2(a) can be compared with those of Kestner and Duan [103]obtained for zero-range interactions. Our L = 1 energy curve agrees with that of Kestner and Duan for all scattering lengths as considered. The L = 0 energy curve, however,
only agrees for as < 0. For as > 0, our results are noticeably lower than those of Kestnerand Duan. As shown below, our as > 0 results for L = 0 predict the correct atom-dimer
scattering length, suggesting that our energies should be very close to those for r0 = 0
and that the disagreement is not due to finite-range effects. We speculate that theresults of Kestner and Duan might not be fully converged for as > 0 although otherpossibilities cannot be excluded.
Now, we compare our numerical results with the theoretical predicted limiting
behaviors. As presented in Sec. 4.2, the first order corrections in the BCS- and BEClimiting behaviors can be obtained using the Fermi pseudopotential. In general, we findgood agreement between the numerical results and these theoretical predictions. Furthermore, the numerical results can be used to extract information about the collisional
aspect of these systems. As an example, consider the equal mass N = 3 system. On
the weakly interacting BCS side,
3 as.
E(3) ≈ EN I + ~ω √2π aho(6.4)

The first term corresponds to a zero order correction, the noninteracting energy. The
second term is a first-order correction corresponding to the Hartree term of the Fermi
pseudopotential.
In the weakly interacting molecular BEC side, the N = 3 system behaves as a

129
dimer and an atom interacting with a short-range potential characterized by a scattering
length aad . Therefore,3~ω
E(3) ≈ E(1, 1) +
+ ~ω
2

r

2 aad
.
π a(ad)(6.5)

ho

The first two terms correspond to the energy of the dimer and the atom, respectively,∆E/h̄ω

and the third term corresponds to the first-order correction proportional to aad .

0.2

(a)

0.1

E(2, 1)/h̄ω

0
0
5.5

0.05

0.1

0.15

0.2

−0.01

0

(b)

5.48
5.46
5.44
−0.05

−0.04

−0.03

−0.02

as /ahoFigure 6.3: Limiting behavior of the ground-state energy for N = 3 equal mass fermions.
(a) Energy correction ∆E = E(2, 1) − E(1, 1) − 3~ω/2 on the BEC side. Circles show
the CG results, while the solid line shows the first order correction for aad ≈ 1.2as . (b)Energy E(2, 1) on the BCS side. Circles show the CG results while the solid line shows
the first-order correction on the BCS side. Figure from Ref. [185].

Figures 6.3(a) and (b) present the BCS- and BEC-limiting behaviors for an equalmass system with N = 3. The perturbative expression, Eq. (6.5), on the BEC side
is expected to be applicable if r0 ¿ as ¿ aho ; thus, we choose a small r0 , i.e., r0 =0.005aho , in the CG calculations. The energy in this region is determined by the atomdimer scattering length aad . The CG energies change linearly with as , showing that
aad is proportional to as , i.e., aad = cad as . A simple linear fit to the CG results
predicts cad ≈ 1.21, in good agreement with previous studies [158, 138], which foundaad ≈ 1.2as . A solid line in Fig. 6.3(a) shows the resulting linear expression. (A more

130
sophisticated analysis accounting for the energy-dependence is presented presented in
the next section.) On the BCS side, the first-order correction varies also linearly with as .
Circles in Fig. 6.3(b) show the CG results, while the solid line shows the prediction from
Eq. (6.4). Good agreement is observed in both limiting behaviors. We perform a similar
analysis of the BCS- and BEC-limiting behaviors for the N = 4 system. Again, good
agreement is obtained between the numerical results and the theoretical predictions.
25

15

1.5

∆(3)/h̄ω∆(3)/h̄ω

20

10

1
0.5
0
−8

−6

−2

0

aho /as

5
0
−8

−4

−6

−4

−2

0

2

4

6

aho /asFigure 6.4: Excitation gap ∆(N ) for N = 3 as a function of aho /as calculated by the
CG approach for κ = 1 (solid line) and κ = 4 (dashed line). Circles present the BEC
limiting behavior 3~ω/2 − E(1, 1)/2 which is independent of κ. The inset shows a blowup of the region where ∆(3) is smallest; in this region, the dependence of ∆(3) on κ ismost pronounced. The dash-dotted line shows the limiting behavior for κ = 1 obtainedby approximating the E(N ) in Eq. (6.6) by their perturbative values, Eq. (6.4). Figure
from Ref. [185].

Finally, we turn the discussion to analyze the odd-even staggering in the N =2, 3, 4 systems. Our CG energies for N = 2, 3 and 4 can be readily combined todetermine the excitation gap,∆(3) =

E(2, 1) + E(1, 2) − E(2, 2) − E(1, 1)
.
2

(6.6)

Figure 6.4 shows the excitation gap ∆(3) as a function of aho /as for two different mass
ratios, i.e., κ = 1 and 4. In the weakly-interacting molecular BEC regime, the excitation
gap approaches 3~ω/2−E(1, 1)/2 (circles), independent of the mass ratio. In the weakly-

131interacting BCS regime, however, the excitation gap depends on the mass ratio (see inset
of Fig. 6.4). For equal masses, ∆(3) is very well described by the perturbative expression
for as . −0.5aho (dash-dotted line in the inset). Figure 6.4 shows that ∆(3) is smallerfor κ = 4 than for κ = 1. Intuitively, this might be expected since the radial densitiesof the two species do not fully overlap for unequal masses (recall, we consider the casewhere species one and two experience the same trapping frequency). Thus, the pairing
mechanism is expected to be less efficient in the unequal-mass system, especially on the
BCS side, than in the equal-mass system. The next chapter discusses the behavior of
the excitation gap at unitarity in more detail.
In this section, we showed that the CG and FN-DMC numerical results agree andbehave universally, describing a smooth crossover for the N = 4 system. For N = 3,
we showed that the ground state of the systems changes its angular momentum in
BCS-BEC crossover. Finally, we verified that the theoretical BCS- and BEC-limiting
behaviors are reproduced.

6.3

Extraction of dimer-dimer collisional properties
The four-body problem of two-component Fermi system in the BEC side was firstconsidered by Petrov et al. 2004 [139]. Their solution showed that the system of bosonicdimers is stable and that the interaction between dimers is characterized by an effective
scattering length of ad d = 0.6as . This value was later derived using diagrammatic
techniques [115, 31]. Here, we extract the dimer-dimer scattering length and its firstenergy correction, the effective range from the four-body spectrum. These calculations
are extended to two fermionic species with unequal mass ratio.
In the BEC limit, the lowest N = 3 states behave as a dimer and an atom.
Similarly, the lowest N = 4 states describe different vibrational states of a dimer-dimer
configuration. Therefore, both the N = 3 and N = 4 systems can be treated effectively
as two-particle systems. A comparison between the two particle solutions with the

132
N = 3 or N = 4 solutions allows us to extract information on the effective atom-dimer
or dimer-dimer interactions.
The two-particle system is much simpler to solve than the corresponding N = 3
or N = 4 systems. However, we do not know the form of the dimer-dimer or atomdimer effective potentials. A simple way to deal with this problem is to approximate
the effective potentials with zero-range pseudopotentials. Since the size of the dimers
are of the order of a, we expect that the range of the effective potentials should also
be of order of a. Thus, we consider including effective range effects using an energy
dependent scattering length. We recognize that the inclusion of the energy dependenceof the scattering length notably extends the validity regime of the zero-range pseudopotential when applied to the description of the scattering of two atoms with finite-range
potentials under external confinement [17, 21].
Therefore, we opt to use the zero-range pseudopotential two-particles solutions
found in Sec. 2.2 to describe these effective two-body systems. In the zero-range solutions, we include the energy dependence of the scattering length by using the effective
range expansion−11
1≈ − + k 2 ref f .
aErel
af
2

(6.7)

Here, aErel is the energy-dependent scattering length parameterized by the (zero-energy)
scattering length af and the effective range ref f . The momentum k is associated withthe relative kinetic energy of the dimer. Thus, k 2 /2µ = Erel where Erel = E3b − E2b for
N = 3 and Erel = E4b −2E2b for N = 4. For N = 4, relative mass µ is µdd = M/2, where
M is the mass of the bosonic molecules, M = m1 + m2 . For N = 3 we only consider
the equal mass case [m1 = m2 ], thus relative mass µ is µad = M m/(M + m) = 2m/3,
where M = 2m.
Using the effective range expansion [Eq. (6.7)], the regularized zero-range potential V (r) [95] takes the form V (r) = g(E)δ(~r)(∂/∂r)r. The scattering strength g is133parameterized by the scattering length a0 and the effective range ref f , i.e.,·¸
2π~2 afµErel ref f af −1
g(E) =
1−
.
µ
~2

(6.8)

As presented in Sec. 2.3, the L = 0 spectrum of the two particle system is then given
by

³
´
Erel
(µ)
+ 34
√ Γ − 2~ωa
´ = ho .
2 ³
aErel
rel
Γ − E2~ω
+ 14

(6.9)

Equation (6.9) is a transcendental equation that can be easily solved numerically. The
solutions of Eq. (6.9) are obtained as functions of the af and ref f parameters andfitted to the numerical results. The calculation can be carried out at different valuesof the two-body scattering length as and, in this way, one can obtain a more reliable
estimation of af and ref f . To compare calculations at different values of as , we have tofirst understand the dependence of af and ref f on as .
To relate af and ref f with the two-body scattering length as we can use dimensional analysis considerations. If the particles interact through short-range interactions
characterized by a scattering length as , both the size of the dimers and the atom-atominteraction only depend on as . For this reason, we expect that the effective dimer-atomor dimer-dimer interactions only depends on as and not on the details of the short-range
interactions. This implies that aad , add , rad , and rdd should be proportional to the twobody scattering length as . Therefore, we propose aad = cad as , add = cdd as , rad = dad as ,and rdd = ddd as . The parameters cad and dad are obtained by fitting the zero-range
two-particle solution to the atom-dimer states of the N = 3 system. Similarly, theparameters cdd and ddd are obtained by fitting the zero-range–two-particle solution to
the dimer-dimer states of the N = 4 system.
First, we analyze the equal-mass atom-dimer system. Considering the three lowest
energy levels on the BEC side, we obtain aad ≈ 1.18(1)as and rad ≈ 0.08(1)as . The
atom-dimer scattering length agrees with that of Refs. [158, 138, 154]. As discussed
earlier [141], the atom-dimer system can be characterized by a soft-core repulsion with

134

(Ei (4) − 2E(2))/h̄ω̄

5

(c)

4.5
4
3

(b)

2.5
2
1

(a)

0.5
0
0

0.2

0.4

0.6

0.8

(M )
as /ahoFigure 6.5: Four-body energies of the three energetically lowest-lying dimer-dimer states(M )as a function of as /aho for κ = 8. Panel (a) shows the energetically lowest lying
energy level (i = 0), panel (b) the energetically second lowest (i = 1) and panel (c)the energetically third lowest state (i = 2). Circles and crosses show our CG and FNDMC results, respectively. Solid lines show the zero-range model results. Figure from
Ref. [184].

135
range of the order of as ; our calculations support this general picture but predict a
range about 10 times smaller than as .
Second, we analyze the dimer-dimer states of the N = 4 systems. To illustrate
the fitting procedure, the circles in Figs. 6.5(a)–(c) show three energetically lowestlying dimer-dimer energy levels, referred to as Ei (4), where i = 0–2 with the center-ofmass energy and the dimer-binding energy subtracted. The results of Figs. 6.5(a)–(c)correspond to κ = 8 and where obtained by the CG approach Solid lines show theenergy levels obtained by fitting these four-body energies to the two-boson energies
obtained using the energy-dependent zero-range pseudopotential. We find that inclusionof the effective range rdd extends the validity regime over which the four-fermion system
can be described by the two-boson model and additionally allows for a more reliable
determination of add . Figure 6.5 illustrates that the two-boson spectrum reproduces
the dimer-dimer states of the four-fermion spectrum well over a fairly large range ofatom-atom scattering lengths, as .
For comparison, crosses in Fig. 6.5(a) show the corresponding FN-DMC energies
for the energetically lowest-lying dimer-dimer state. In the application of Blume’s FNDMC, we do not attempt to construct a guiding function that would allow for thedetermination of excited dimer-dimer states. We find that the FN-DMC energies areslightly larger than the CG energies and that the deviation increases with increasing as .
Presumably, this increasing deviation can be attributed to the functional form of thenodal surface used in the FN-DMC calculations, which should be best in the very deep
BEC regime. The increasing deviation between the FN-DMC and CG energies withincreasing as explains why the effective range predicted by the analysis of the FN-DMC
energies is somewhat larger than that predicted by the analysis of the CG approach (seediscussion of Fig. 6.6 below).
Formation of trimers or tetramers can affect this dimer-dimer analysis. If weuse short-range Gaussian two-body potentials that support no two-body–s-wave-bound136
state for negative as and one two-body–s-wave-bound state for positive as , the fourbody energy that enters the calculation of the energy crossover curves shown in Fig. 6.2
is the true ground state of the system, i.e., no energetically lower-lying bound trimer
or tetramer states with LP = 0+ symmetry exist. For larger mass ratios, bound trimer
states exist. The mass ratio at which these nonuniversal trimer states appear depends
on the range r0 of the two-body potential employed. In the regime where three-bodybound states exist, the four-body spectrum calculated by the CG approach also contains
universal states that are separated by approximately 2~ω; they can be best describedas two weakly interacting composite bosons. For fixed as (as > 0), the energy of these“dimer-dimer states” changes smoothly as a function of κ even in the regime wherebound trimer states appear. In the following, we use these dimer-dimer states to extract
the dimer-dimer scattering length as a function of κ up to κ = 20.
Our analysis of the dimer-dimer states is carried out for selected values of themass ratio κ. Table 6.1 summarizes the results. Circles and crosses in Fig. 6.6 show therdd /as , add /as

resulting dimer-dimer scattering lengths, add , extracted from the energies calculated by

1

0.5

0
0

5

10

κ

15

20

Figure 6.6: Circles and crosses show add /as as a function of κ extracted from the fourfermion CG and FN-DMC energies, respectively. For comparison, a solid line shows theresults from Fig. 3 of Ref. [140]. Diamonds and squares show rdd /as extracted from thefour-fermion CG and FN-DMC energies, respectively. Figure from Ref. [184].

the CG and the FN-DMC approach, respectively, as a function of κ. For all mass ratios

137
Table 6.1: Dimer-dimer scattering length, add , and dimer-dimer effective range, rdd ,obtained using (a) the CG spectrum and (b) the FN-DMC energies. The reported
uncertainties reflect the uncertainties due to the fitting procedure; the potential limitations of the FN-DMC method to accurately describe the energetically lowest-lying
gaslike state, e.g., are not included here (see Sec. IIIB of Ref. [184]).
κ
1
4
8
12
16
20

add /as (a)0.608(2)0.77(1)0.96(1)1.10(1)1.20(1)1.27(2)

add /as (b)0.64(1)0.79(1)0.98(1)1.08(2)
1.21(3)
1.26(5)rdd /as (a)0.13(2)0.15(1)0.28(1)0.39(2)0.55(2)0.68(2)rdd /as (b)0.12(4)0.23(1)0.38(2)0.55(2)0.60(5)0.74(5)considered in Fig. 6.6, we include up to three dimer-dimer energy levels in our analysisof the CG results and only the lowest dimer-dimer level in our analysis of the FN-DMC
results. Our dimer-dimer scattering lengths agree well with those calculated by Petrov
et al. within a zero-range framework [140] (solid line in Fig. 6.6). The calculations
by Petrov et al., performed for the free and not the trapped four-fermion system,
terminate at κ ≈ 13.6, beyond which a three-body parameter is needed to solve thefour-body equations within the applied framework. Our calculations show the existenceof deeply-bound “plunging” states, that consist of a trimer plus a free atom. Thesestates signal a qualitative change of the energy spectrum, in agreement with Petrov etal. [140]. At the same time, our calculations for finite-range potentials predict that add
continues to increase smoothly when the mass ratio κ exceeds 13.6. This can possibly
be explained by the fact that the presence of the external confining potential may “washout” some of the features present in the free-space system. As already mentioned, thestudy of the stability of the four-fermion system, consisting of two dimers, with large
mass ratios is beyond the scope of this work. Nevertheless, a recent theoretical study
found good agreement with our predictions beyond 13.6 [122].
We now turn to an analysis of the effective range results. Diamonds and squares
in Fig. 6.6 show the effective range, rdd , extracted from our CG and FN-DMC energies,

138
respectively. We estimate the uncertainty of rdd obtained from the CG approach to beabout 10%; this uncertainty is quite a bit larger for that extracted from the FN-DMC
energies. Figure 6.6 shows that the ratio rdd /add increases from about 0.2 for κ = 1
to about 0.5 for κ = 20. While earlier work already suggested that the dimer-dimer
potential may be best characterized as a broad soft-core potential [140], implying a
non-negligible value for the effective range of rdd , our work makes the first quantitative
predictions for rdd as a function of κ. The large value of rdd suggests that effective-range
corrections may need to be considered in analyzing the physics of molecular Fermi gases.
54

V (R)/Eb

3
2
1
0
−1
−2
−3
0

200

400

600

800

1000

1200

R/r0
Figure 6.7: Hyperspherical potential curves Uν for a scattering length as = 100r0 .
The dashed red curve is the dimer threshold and the dashed blue curve is the dimerdimer threshold. Therefore, the blue curve is a dimer-dimer potential curve and the red
curves are dimer–two-atom potential curves. Dashed-black curves are predictions from
Ref. [56].

The development of four-body hyperspherical methods allows an analysis of thefull energy dependence of the dimer-dimer scattering length. Figure 6.7 presents thefour-fermion potential curves obtained with correlated Gaussian hyperspherical (CGHS).
The lowest curve represents the dimer-dimer channel and the rest represent different
dimer–two-atom channels. Standard multichannel scattering techniques, like the Rmatrix method, can be applied to solve the hyperspherical coupled differential equa-

139
tions. This analysis was performed in a recent study by D’Incao et. al. [56] whereenergy dependence of the dimer-dimer scattering length for equal mass systems was
obtained. Black dashed curves in Figure 6.7 represent the potential curves of Ref. [56].
As we can see, the CGHS method presented in this dissertation predicts very similarpotential curves. The dimer-dimer potential curves obtained with the different methods
are almost indistinguishable. For dimer–two-atom potential curves, the CGHS predicts
lower potential curves suggesting that the CGHS calculation is slightly better. At large
R, the asymptotic behavior of both methods agree. This is very encouraging since inD’Incao et. al. method, the asymptotic behavior of the channels is correct by construction but in the CGHS implies an important nontrivial test. Preliminary calculations
with the CGHS potential curves predict a similar energy dependence of the dimer-dimer
scattering length. Therefore, we consider that the CGHS opens the possibility for accurately analyzing four-body scattering events, as we will see in the four-boson problem
presented in chapter 8.
To conclude, we have shown that few-body trapped systems can be used to extract
information on the collisional properties of free systems. Atom-dimer and dimer-dimer
scattering lengths can be extracted by analyzing the trapped few-body spectrum fordifferent two-body scattering length values. Furthermore, energy-dependent corrections
to aad and add can also be obtained with this method.

6.4

A two-channel model for the BEC-BCS crossover.
We observed in Sec. 3.2, that the BCS- and BEC-limiting behaviors can be de-scribed by completely different wave functions. The BCS-limiting behavior is describedby a Jastrow-Slater type of wave function, while the BEC behavior is described by apaired wave function. These type of wave functions where discussed from a technicalpoint of view in the FN-DMC section of chapter 3 to construct the appropriate trial wave
function. Here, we use these two trial wave functions to propose a simple interpretation

140of the BCS-BEC crossover problem.
Since the BCS and BEC regimes are described by qualitatively different wave
functions, it is natural to suspect that the BCS-BEC crossover can be understood as
an avoided crossing between these two limiting cases. This interpretation suggests a
crossover from a normal to a superfluid state. The paired wave function is consistent
with superfluidity because of its long-range correlations, while the Jastrow-Slater wave
function only contains short-range correlations that are characteristic of a normal state.
Therefore, the crossover would imply an avoided crossing between a normal and a superfluid state.
Here, we propose a two-channel model to qualitatively describe the BCS-BEC
crossover. In this model, the trial wave function ansatz is a linear combination of the
Jastrow-Slater wave function and the paired wave function. We apply it to the fourfermion system and compare our model to the full CG results.
To implement this model, we have to construct the paired and Jastrow-Slater
states and evaluate the matrix elements in this reduced two-state Hilbert space. Next,
we solve the Schrödinger equation for this two-channel system. The paired wave function
for four particles takes the form,ΨP (r1 , r2 , r10 , r20 ) = A{ψ2p (r1 , r10 ), ψ2p (r2 , r20 )}= Np (ψ2p (r1 , r10 )ψ2p (r2 r20 ) − ψ2p (r1 , r20 )ψ2p (r2 , r10 )) . (6.10)
Here Np is a normalization constant, and ψ2p is the two-body solution of the trappedsystem. The Jastrow-Slater wave function isΨJ (r1 , r2 , r10 , r20 ) =Y

J(rii0 )ΨN I (r1 , r2 , r10 , r20 ).

(6.11)

i,i0

The correlations in the Jastrow-Slater term are usually obtained by using the smallr region of the zero-energy solution matched to a fast-decaying function (for example
with an exponential decay). In the negative scattering length, the Jastrow-Slater term

141
does not have nodes, but in the positive scattering length region, each Jastrow-Slater
correlation has a node that describes the zero-energy correlations of the excited atomic
state, but not the weakly bound molecular ground state. Thus, the total wave function
describes the atomic ground state in both the BCS- and BEC-limiting cases.
We use the two-body solutions of two particles in a trap to construct ψ2p . These
solutions can be separated in center-of-mass and relative coordinates, RCM and r, as
22

ψ2p (r1 , r2 ) = e−2RCM /aho φ(r).

(6.12)The evaluation the matrix elements requires the evaluation of multidimensional
integrals. Thus, it is convenient to use a correlated-Gaussian expansion that allows ananalytical computation of the matrix elements. To obtain the paired wave function,
we first solve the two-body Schrödinger equation in the relative coordinate using CG,
then we use the ground-state–two-body solution to expand Eq. (6.10) in a CG basis
functions.
To construct the Jastrow-Slater wave function, we use a different approach. In
practical terms, it is not very convenient to construct the Jastrow-Slater wave function
from the two-body solution and the noninteracting wave function. This solution would
produce an unnecessarily large CG basis-function expansion. Instead, it is easier to solve
the exact Hamiltonian in the perturbative regime where we know that the exact solution
has the form of Eq. (6.11) and then use that solution in the two-channel model. Here, it
is assumed that the Jastrow term does not depend strongly on a and that we can use asolution at one a value for a wide region of the crossover. This assumption is consideredvalid in the BCS and BEC regimes but not in the strongly interacting region where
an extra node appears in the correlations. Thus, Jastrow terms in the BCS and BEC
side are significantly different because the appearance of that extra node. Therefore, we
use the perturbative solution on the BCS side to describe the BCS-unitarity region and
we use the peturbative solution on the BEC side to describe the BEC-unitarity region.

142
This approximation seems to be sensible for our qualitative two-channel model.
Figure 6.8 present the results of the two-channel model. The black solid lines
correspond to the noninteracting energy [E = 8~ω] and the energy of the paired wave
function. The blue dashed lines correspond to the solutions of the two-channel model.
The red lines with circles corresponds to the exact ground state obtained with the CG
method, and green lines with circles corresponds to the state that diabatically becomesthe “atomic ground state” in the BEC. First, we note that the paired state represents adiabatic state that crosses the trapped states in the BCS regime and correctly describes
the dimer-dimer states in the BEC regime. The lowest dashed curve, which is the groundstate of the two-channel model, follows closely the true ground state. As expected, thedeviations of the two-channel model solution to the exact solution are maximal in thestrongly interacting unitarity regime. Another important result is that the excited
solution of the two-channel model describes the atomic ground state in the BEC side.

20

E (h̄ω)

10
0
−10
−20
−30
−40

−5

0

5

aho /a0
Figure 6.8: Comparison of exact calculations with the two-channel model. Solid blacklines correspond to the noninteracting energy and the pair-state energy, respectively.
Blue dashed curves correspond to the two-channel model solutions. Red and green lines
with circles correspond to the exact CG results for the ground state and the lowest
atomic state in the BEC side, respectively.

Finally, we have shown that the two-channel model qualitatively describes the
BCS-BEC crossover providing, an alternative interpretation to the crossover physics.

143

6.5

Structural properties
The analysis of the spectrum in the BCS-BEC crossover can be complemented

with the analysis of the wave functions and their structural properties. Here, we obtain
the one-body densities and pair-distribution functions of two-component Fermi systems
in the crossover regime for different κ. We determine the averaged radial densities,Rρi (r), normalized so that 4π ρi (r)r2 dr = Ni ; 4πr2 ρi (r)/Ni tells one the probability offinding a particle with mass mi at a distance r from the center of the trap. If N1 = N2
and m1 = m2 , the radial one-body densities ρ1 (r) and ρ2 (r) coincide. If m1 and m2
or N1 and N2 differ, however, the radial one-body densities ρ1 (r) and ρ2 (r) are, in
general, different. We also determine the averaged radial pair distribution functionsR
Pij (r), normalized so that 4π Pij (r)r2 dr = 1; 4πr2 Pij (r) tells one the probability offinding a particle of mass mi and a particle of mass mj at a distance r from each other.
For notational simplicity, we refer to the radial one-body densities as one-body densities
and to the radial-pair distribution functions as pair-distribution functions in from nowon.
To compute these structural properties we use CG. Since the CG method only
calculates L = 0 states, all the structural properties extracted from CG are sphericallysymmetric. When N + 1 systems are studied to obtain solutions for the N systemwith L > 0, the N -particle system and the spectator particle are coupled to L = 0.
Therefore, even though the structural properties of the N -system are not sphericallysymmetric, the structural properties obtained from the N + 1 calculation are sphericallysymmetric. We can recover the corresponding angular dependence by multiplying the
CG result by the corresponding spherical harmonic. However, for plotting purposes, it
is convenient to present the spherically symmetric CG results, which is equivalent toaveraging over the angular degrees of freedom.

144
To evaluate the structural properties, we calculateZ24πr F (r) = hΨn |δ(x − r)|Ψn i =dr1 ...drN δ(x − r)|Ψn (r1 , r10 , ..., rN1 , rN2 )|2 . (6.13)

Here, F (r) is a generic structural property, i.e., the density profiles ρ1 or ρ2 , the interspecies pair-correlation function P12 , or the intraspecies pair-correlation functions P11
or P22 . x is the length of the coordinate vector that describes the structural property.
For ρ1 and ρ2 , x = r1 and x = r10 , respectively. For P12 , x is the interparticle distance
between opposite-spin or different species, x = r110 . For P11 and P22 , x is the same-spinor same species interparticle distance, with x = r12 and x = r10 20 , respectively. To
evaluate 4πr2 F (r), we expand Ψn in the CG basis set. The integral in Eq. (6.14) can
be carried out analytically for the CG basis functions (see Appendix G).

6.5.1

Analysis of the validity of the numerical structural properties

The first question that we want to answer is how reliable are our structuralproperties. To answer this question, we first compare structural properties obtainedby CG and FN-DMC methods. Then, we analyze how well these structural properties
reproduce the expected behavior from a zero-range pseudopotential.
Figures 6.9 (a) and (b) compare the pair distribution functions P12 (r) for thethree-particle system with L = 1 and the four-particle system with L = 0, respectively, calculated by the CG and the FN-DMC methods. The pair correlation functions
P12 (r) at unitarity allow the most stringent comparisons between the CG and FN-DMC
predictions. The correlations between opposite spins or different species describe a nontrivial behavior controlled by the properties of the short-range interactions. Therefore,
these correlations depend strongly on the scattering length as , and the shape and rangeof the interaction r0 . Since we use different short-range potentials in CG and FNDMC, the good agreement in Figs. 6.9(a) and (b) suggest that these correlations aremainly described by the scattering length. Another important point is that we make145
the comparison at unitarity in the strongly interacting regime where the correlations
are nontrivial.
The agreement between the pair-distribution functions calculated by the CG
method (solid lines) and the FN-DMC method (circles) is very good, supporting both
numerical results. This is particularly important for the FN-DMC results since it suggests that the construction of the nodal surface of the trial wave function ψT is close to
the exact one. Furthermore, the good agreement suggests that the technique used to
extract the FN-DMC results, i.e., the evaluation of the mixed estimator, provides structural properties very close exact ones estimator (see discussion on the mixed estimatorin Chapter 3). This agreement is very encouraging. In particular, it suggest that the
N > 4 structural properties obtained in Ref. [185] with FN-DMC are reliable.

4πr2 P12 (r)/a−1
ho

(a)
0.4
0.2
0

(b)
0.4
0.2
0
0

1

2

3

4

5

r/aho
Figure 6.9: Pair-distribution functions P12 (r), multiplied by r2 , at unitarity for equalmass Fermi systems with (a) N = 3 (L = 1) and (b) N = 4 (L = 0) atoms calculatedby the CG method (solid lines) and by the FN-DMC method (circles). The agreement
is excellent. Figure from Ref. [185].

The second question we want to answer is how these numerical results compare to
the zero-range pseudopotential predictions. The comparison presented in Figures. 6.9(a)and (b) made the first step in this direction by showing that the numerical solutions

146
are, to a good approximation, independent of the model potential used. To further
investigate the universal behavior of the structural properties we analyze zero-range
predictions. For zero-range pseudopotential, the opposite-spin pair-correlation function
obeys a boundary condition [118, 76][rP12 (r)]0r=02=− .
[rP12 (r)]r=0
as(6.14)

This boundary condition is a consequence of the Bethe-Peierls boundary condition.
Equation (6.14) can be derived by taking into account that the wave functions obey
the boundary condition [r12 Ψ(r12 )]0r12 =0 /[r12 Ψ(r12 )]r12 =0 = −1/as . The factor of 2 in
Eq. (6.14) can be understood by noticing that the pair correlation is proportional to thesquare of the wave function, i.e., P12 (r) ∝ Ψ(r12 )2 . A direct consequence of Eq. (6.14)
is that at unitarity, i.e., when |a| = ∞, r2 P12 (r) has zero slope as r → 0. This condition

4πr2 P12 (r)/a−1
ho

is qualitatively fulfilled in the numerical results presented in Figs. 6.9 (a) and (b).

1
0.8
0.6
0.4
0.2
0
0

1

2

3

4

r/aho
Figure 6.10: Pair-distribution functions P12 (r), multiplied by r2 , for equal-mass–twocomponent Fermi systems with N = 2 (dashed lines) and N = 4 (solid lines). The
N = 2 pair-correlation function has an arbitrary norm selected to match the N = 4
pair correlation in the small r regime.

The numerical verification of Eq. (6.14) is very difficult. For systems with finite
range interactions, Eq. (6.14) is valid in a narrow regime of r values. For as < 0,
Eq. (6.14) is valid when r is much larger than the range of the potential and much

147
smaller than the mean interparticle distance, i.e., the r0 ¿ r ¿ aho regime. For as > 0,
Eq. (6.14) is valid when r is much larger than the range of the potential and muchsmaller than the size of the dimer (given by a) and the mean interparticle distance, i.e.,
the r0 ¿ r ¿ min[as , aho ] regime. This regime is almost nonexistent for our numericalcalculations, which consider only the ah o/r0 = 100 case.
An alternative procedure is to compare the numerical results with the two-bodyanalytical results of the zero-range pseudopotential presented in Sec. 2.3. This comparison is more convenient because the two-body analytical results include corrections to
Eq. (6.14) for trapping effects and finite r. Figure 6.10 presents the N = 2 and N = 4
pair correlation r2 P12 (r). The calculations are done on the BEC side [as = aho ], where
both the N = 2 and N = 4 pair correlations in the small r region are controlled bymolecule formation (see discussion below). The good agreement between the N = 2
and N = 4 pair correlations in the small r region suggests that the N = 4 numericalresults correctly describe the zero-range pseudopotential limit.

6.5.2

Structural properties along the BEC-BCS crossover

Here, we analyze how structural properties change in the BCS-BEC crossover.
We focus in particular on the structural properties of the ground state for N = 4 andthe lowest N = 3 state with L = 0, 1.
Figure 6.11 shows the pair distribution function P12 (r) for N = 3 (dash anddash-dotted lines correspond to L = 0 and 1, respectively) and N = 4 (solid lines) along
the crossover for equal mass systems [κ = 1]. Panel (a) shows results for as = −aho ,panel (b) for 1/as = 0 and panel (c) for as = 0.1aho . Interestingly, the pair distributionfunctions for N = 3 and N = 4 show a similar overall behavior. In the BCS regime[Fig. 6.11(a)], the quantity P12 (r)r2 shows a minimum at small r (for very small r,
P12 (r)r2 goes smoothly, but steeply, to zero; this rapid change of P12 (r)r2 is hardly
visible on the scale shown in Fig. 6.11). At unitarity [Fig. 6.11(b)], P12 (r)r2 shows a

148
maximum at small r and a second peak at larger r. In the BEC regime [Fig. 6.11(c)],the two-peak structure is noticeably more pronounced. The peak at small r indicatesthe formation of weakly bound dimers (one dimer for N = 3 and two dimers for N = 4),
while the peak between 1 aho and 2 aho is related to the presence of larger atom-atom
distances set approximately by the atom-dimer distance for the three-body system andthe dimer-dimer distance for the four-body system. This interpretation suggests thatthe three-particle system has one small and one large interspecies distance, and the fourparticle system has two small and two large interspecies distances. Indeed, integrating
P12 (r) for N = 3 and N = 4 from 0 to the r value at which P12 (r)r2 exhibits theminimum, we find that the likelihood of being at small distances (forming a molecule)and being at large distances is the same. Finally, we note that in the BCS-BEC crossoverthe slope of r2 P12 (r) at small r changes from positive to zero to negative as predicted
by Eq. (6.14).
We now analyze the pair distribution function P12 (r) for N = 4 more quantitatively. Dash-dotted lines in Figs. 6.12 (a) and (b) show the pair-distribution function, P12 (r), multiplied by r2 , for two trapped atoms with as = 0.1aho (normalized
to 1/2). This dimer curve is essentially indistinguishable from the small r part of
the four-particle pair distribution function (circles). To describe the large r part of
the four-particle pair-distribution function, we consider two bosonic molecules of mass
2m, which interact through an effective repulsive potential with dimer-dimer scattering
length add ≈ 0.6as [140, 184]. The dashed line in Fig. 6.12 (a) shows the pair-distributionfunction for this system under external confinement. This dashed curve is essentiallyindistinguishable from the large r part of the pair-distribution function for the fourparticle system. For comparison, a dotted line shows the pair-distribution function for
two noninteracting trapped bosons of mass 2 m. Figure 6.12 indicates that the effective
repulsive interaction between the two dimers is crucial for accurately reproducing thestructural properties of the four-body system. Our analysis shows that the entire pair

149
(a)

4πr2 P12 (r)/a−1
ho

0.5

0

(b)

0.5

0

(c)

0.5

0
0

1

2

3

4

r/aho
Figure 6.11: Pair-distribution functions P12 (r), multiplied by r2 , for equal-mass–twocomponent Fermi systems with N = 3 and L = 0 (dashed lines), N = 3 and L = 1(dash-dotted lines), and N = 4 and L = 0 (solid lines) obtained by the CG approachfor three different scattering lengths, as : (a) as = −aho (BCS regime), (b) 1/as = 0(unitarity), and (c) as = 0.1aho (BEC regime). The pair distribution function for N = 4
and as = 0.1aho [solid line in panel (c)] is shown in more detail in Fig. 6.12. Figure
from Ref. [185].

(a)4πr2 P12 (r)/a−1
ho

0.4

0.2

0
0

0.5

1

1.5

2

2.5

3

(b)

8
6
4
2
0
0

0.05

0.1

0.15

0.2

r/aho
Figure 6.12: (a) Circles show the pair-distribution function, P12 (r), multiplied by r2 , for
as = 0.1aho (BEC regime) calculated by the CG approach for N = 4 and κ = 1 [note,
this quantity is also shown by a solid line in Fig. 6.11(c)]. For comparison, the blue dashdotted line shows P12 (r)r2 for two atoms of mass m with the same scattering length but
normalized to 1/2, the red dashed line shows P12 (r)r2 for two trapped bosonic moleculesof mass 2 m interacting through a repulsive effective potential with add = 0.6as , andthe green dotted line shows P12 (r)r2 for two trapped noninteracting bosonic moleculesof mass 2 m. Panel (b) shows a blow-up of the small r region. Figure from Ref. [185].

150
distribution function, P12 (r), of the four-body system in the weaklyinteracting molecular
BEC regime can be described quantitatively in terms of a “dimer picture.”
The pair-correlation function P12 (r) also reflects the ground-state symmetry inversion of the N = 3 system (see Sec. 6.2). In the BCS regime and at unitarity [Figs. 6.11 (a)and (b)], P12 (r) shows less structure for L = 1 than for L = 0. In the weakly-interacting
molecular BEC regime [Fig. 6.11(c)], the pair-distribution function for L = 0 nearly coincides with that for L = 1 at small r, but is more compact than that for L = 1 at larger.

(a)

4πr2 P12 (r)/a−1
ho

0.5

0

(b)

0.5

0

(c)

0.5

0
0

1

2

3

4

r/aho
Figure 6.13: Pair-distribution function P12 (r), multiplied by r2 , for two-component
Fermi gases with κ = 4 for different scattering lengths as : (a) as = −aho (BCS regime),(b) 1/as = 0 (unitarity), and (c) as = 0.1aho (BEC regime). Dashed and dash-dottedlines show P12 (r)r2 for N = 3 (two heavy particles) with L = 0 and 1, respectively.
Circles and squares show P12 (r)r2 for N = 3 (two light particles) with L = 0 and 1,
respectively. Solid lines show P12 (r)r2 for N = 4 with L = 0. Figure from Ref. [185].

Next, we analyze how the behaviors of the pair distribution functions P12 (r) for
N = 3 and 4 change along the crossover if the mass ratio is changed from κ = 1 to 4.
Figure 6.13 shows the pair distribution functions for κ = 4. For N = 3, we consider
three-particle systems with either a spare light particle or a spare heavy particle. The
pair distributions for the three-particle system with two light particles and one heavy

151
particle are notably broader than those for the three-particle system with one light
(1)

(2)

particle and two heavy particles. We attribute this behavior to the fact that aho > aho .
Besides this, a comparison of the pair-distribution functions shown in Fig. 6.13 for κ = 4
and those shown in Fig. 6.11 for κ = 1 reveals that the overall behavior of the P12 (r) issimilar.

ρi (r)/a−3
ho

2

(a)

1.5
1
0.5

ρi (r)/a−3
ho

0

(b)

3
2
1
0
0

0.5

1

1.5

2

2.5

r/aho
Figure 6.14: One-body densities ρ1 (r) and ρ2 (r) for N = 4 and (a) κ = 1 and (b) κ = 4
for different scattering lengths as [for κ = 1, ρ1 (r) and ρ2 (r) coincide and only ρ2 (r)is shown]: Circles and solid lines show ρ1 (r) and ρ2 (r) for as = 0, squares and dashedlines show ρ1 (r) and ρ2 (r) for 1/as = 0, and triangles and dash-dotted lines show ρ1 (r)and ρ2 (r) for as = 0.1aho . Note, ρ2 (r) for κ = 4 and as = 0 [solid line in panel (b)] is
multiplied by a factor of three to enhance the visibility. Figure from Ref. [185].

The analysis of the pair correlation function is complemented with the analysis of
the one-body densities. Figures 6.14(a) and (b) show the one-body densities for κ = 1
and 4, respectively. In the noninteracting limit [the solid lines show ρ1 (r) and the circles
(1)

(2)

show ρ2 (r)], the sizes of ρ1 (r) and ρ2 (r) are determined by aho and aho , respectively.
As is evident in Fig. 6.14, the density of the light particles extends to larger r thanthe density of the heavy particles. The density mismatch for κ = 4 between the two
one-body densities decreases as as is tuned through the strongly-interacting regime to
the weakly-interacting molecular BEC side. In the weakly-interacting molecular BEC

152
regime, two molecules consisting each of a heavy and a light particle form. In thisregime, the size of the system is determined by the molecular trap length and thedensities ρ1 (r) and ρ2 (r) [triangles and dash-dotted line in Fig. 6.14(b)] nearly coincide.
Furthermore, the densities are to a very good approximation described by the one-bodydensity for two bosonic molecules of mass m1 + m2 interacting through an effective
repulsive interaction characterized by the dimer-dimer scattering length (add ≈ 0.77asfor κ = 4 [140, 184]).

(a)

0.7
0.6
0.5
0.4
0.3
0.2

0.7
0.6
0.5
0.4
0.3
0.2
0.1

0.1
0
0

(b)

0.8

4πr2 φ2n0 (r)/a−1ho

4πr2 P11 (r)/a−1
ho

0.8

1

2

r/aho

3

4

0
0

1

2

3

4

r/aho

Figure 6.15: Ground and excited dimer-dimer states for as = 0.1aho .(a) Pair-distributionfunction 4πr2 P1 1(r) for the lowest dimer-dimer states (b) Lowest two-body solutions,
4πr2 φn0 (r), of a noninteracting trapped system of bosonic particles of mass 2 m.

On the weakly interacting molecular BEC side, where the size of the molecules
is much smaller than the trap length, aho , P11 of the dimer-dimer states describes, toa good approximation, the molecule-molecule pair-correlation function. To understandwhy, consider the molecule-molecule pair-correlation function. If we treat the dimers
as point particles, the molecule-molecule pair correlation function would measure thedensity probability as a function of the distance between the center of mass of two
different dimers. But since the dimers are very small in comparison with the dimer-dimer
distance, measuring the dimer-dimer distance from the center of masses or from each153of the spin up particles yields almost identical results. For this reason, on the weaklyinteracting molecular BEC side, P11 describes the molecule-molecule pair-correlation
function.
As an example, consider the P11 (r) for as = 0.1aho . Figure 6.15(a) shows P11 (r)of the ground and excited dimer-dimer states. For comparison, Fig. 6.15(b) presents
the square of the relative-coordinate wave functions φnl of two noninteracting particles
of mass M = 2 m in a trap. The pair-correlation function of the N = 2 system issimply the square of relative-coordinate wave functions. Since the dimer-dimer statesof the N = 4 system describe vibrational states with dimer-dimer angular momentum
l = 0, Fig. 6.15(b) presents the l = 0 two-body solutions. The number of nodes in theP11 (r) and φ2nl correspond to the number of vibrational excitations. The good agreement
between panels (a) and (b) in Fig. 6.15 confirms that the dimer-dimer P11 (r) in the BEC
side describes the molecule-molecule pair-distribution function.

6.6

Conclusions
In this chapter, we have presented an accurate description of few-body trappedsystems. First, we showed how the N = 3 and N = 4 spectrum evolves in the BCSBEC crossover. Then, we analyzed more quantitatively the ground state energies andcompared them both theoretical predictions and FN-DMC numerical results.
We also showed that the spectrum in the BEC regime reveals atom-dimer anddimer-dimer collisional properties such as scattering lengths and effective-range corrections. We have also presented calculations of the hyperspherical potential curves for
N = 4. The analysis of the hyperspherical potential curves for free systems allows a
direct measurement of the collisional properties. For this reason, the development the
CGHS for four-body systems opens the possibility to deeper understanding of dimerdimer collisional properties.
Finally, we analyzed the structural properties of N = 3 and N = 4 systems. We

154showed how the qualitative change in the behavior of the systems in the BCS-BEC
crossover is reflected in the structural properties.

Chapter 7
Unitary Gas

7.1

Introduction
In many-body systems, the strongly-interacting region is characterized by kF |a| &1. In this region, the scattering length becomes larger than the interparticle distance,
which, in turn, is much larger than the range of the two-body potential. Thus, the gas
is both dilute and strongly interacting. This unusual situation is difficult to study physically since there is no small parameter for perturbation analysis. Even understanding
the stability of such systems is challenging. Bosonic systems in such conditions are
known to collapse, and fermionic systems, even though they are expected to be morestable because of Pauli repulsion, are not guaranteed to be stable. Sophisticated numerical simulations and experimental observations are required to show that fermionicsystems are indeed stable.
Similarly for few-body trapped systems, the strongly-interacting region is alsocharacterized by kF |a| & 1, which is equivalent to |a| & aho . In the few-body case,difficulties similar to those of the many-body problem arise. There is no small parameterfor perturbative treatments and there exist semianalytical solutions only for some casesof N ≤ 3 [35, 187]. For N > 3, computationally demanding numerical techniques, likethe ones presented in this chapter, have been used to analyze such systems.
A particular point in the strongly-interacting region, where the s-wave interaction
strength reaches its maximal value, is usually called unitarity. Therefore, this regime is156characterized by a universal behavior of the scattering amplitude:
f0 (k) =

i
.
k

(7.1)

Unitarity is alternatively characterized by a divergent s-wave scattering length, |a| = ∞.
In this situation, if the range of the interaction is much smaller than the typical interparticle distance and the scattering length is divergent, then there is no relevant length scalethat characterizes the interaction. This situation is similar to the noninteracting limit,
where the absence of interactions implies, of course, the absence of a length scale that
describes the interaction. The absence of a length scale that describes the interaction
allows us to extract the functional form of different quantities via dimensional analysis.
Furthermore, it allows us to relate quantities at unitarity to those in the noninteracting
limit.
For example, in the homogenous system at unitarity, the only relevant lengthscale is the interparticle distance of the noninteracting system, or, equivalently, kF−1 .
Therefore, the energy and the chemical potential of the system at unitarity should be
proportional to those of the noninteracting system. In general, this relation is writtenasµU = (1 + β)µN I ,(7.2)

EU = (1 + β)EN I .

(7.3)

Here, the subscripts U and N I refer to the unitary and noninteracting limits, respectively, and E is the energy per unit volume. The relation between these unitary andnoninteracting quantities is characterized by the parameter β. This parameter is universal, i.e., it is the same for any equally populated two-component Fermi gas andis independent of the nature of the interaction. An equivalent parameter, defined as
ξ = 1 + β, has been extensively used in the literature. The same proportionality constant, (1 + β), relates the noninteracting and unitarity pressures of the system. Also,157
other quantities are related by powers of this constant. For example, the speed of sound(c) and the size of a trapped cloud R(s) are related as [75, 87]
cU =
(s)

p
1 + βcN I ,(s)

RU = (1 + β)1/4 RN I .

(7.4)
(7.5)

Because of these relationships, much effort has been expended to obtain the parameterβ. The value of β has been calculated by, among other methods, numerical simulations
using quantum Monte Carlo techniques [7]. Section presents a table with many different
theoretical and experimental values for β. Other properties of unitary homogenous
systems have been determined using dimensional analysis arguments [89].
The same dimensional analysis arguments used in homogeneous systems can beextended to few- and many- body systems using the hyperspherical framework. Dimensional analysis becomes particularly interesting in the hyperspherical framework of afree system, where the hyperradius is the only coordinate with units of length. Since
the potential curves have units of energy and the only length scale is given by R, thenit follows that VU (R) ∝ 1/R2 . This is equivalent of saying that VU (R) ∝ VN I (R), since
the noninteracting potential curves have the form 1/R2 (see Sec. 2.5). The inclusion ofa trapping potential discretizes the spectrum and introduces similar scaling properties
which influence the topology of the spectrum. We will analyze such universal propertiesof the trapped system in Sec. 7.2, where we will also explore their consequences.
However, not every system at unitarity obeys these universal properties. There
are some cases where the short-range physics needs to be included to obtain converged
results. A typical example is the three boson system. In this case, the lowest potential
curve has an attractive 1/R2 form. This type of dipole interaction is too divergent as
R → 0. Therefore, the zero-range model predicts an unphysical Thomas collapse [176],
i.e., there is no lower bound to the ground state, and there is an infinite number ofbound states with divergent binding energies. The introduction of more-realistic short-158
range physics is crucial for remedying this unphysical situation, because realistic shortrange physics introduces a lower bound to the series of weakly bound states known
as Efimov states [60, 61]. Thus, a new length scale that describes the short-range
physics is necessary for describing the system. Consequently, the properties obtainedby dimensional analysis are no longer valid for the all the solutions, but, they remain
valid for some of the solutions.
The universal behavior of a system and its stability are closely related. The
collapse of a trapped system is a many-body effect. Nevertheless, the existence orabsence of weakly bound few-body systems clearly influence the stability of a system.
For example, a clusterization process, i.e., formation of continuously enlarging boundsystems, would also lead to a collapse. On the other hand, if, there is no weakly boundstate for N < Nc but there exists a weakly bound Nc -body state, then it is requiredthat Nc + 1 particles get close together to form a Nc -body state. Therefore, the larger
Nc is the more unlikely the clusterization process will occur 1 .
The need for a short-range physics length scale, the appearance of an attractive
1/R2 potential, and the formation of weakly bound states appear to be related. Verification of these scaling properties can tell us if a system behaves universally or if theformation of weakly bound molecules occurs. As we said before, formation of weakly
bound molecules has consequences for the stability of the system.

7.2

Universal properties in trapped systems
In this section, we review the universal properties of trapped systems at unitarity.

These properties, and the derivation presented in this Section, were obtained by Wernerand Castin [38, 186].
As presented in the Introduction, short range interactions with large scattering
1

In this discussion we are omitting the possible formation of tightly bound molecules which are also
unlikely.

159
lengths can, in some cases, be replaced by a zero-range pseudopotential as in Eq. (2.9).
The pseudopotential is equivalent to the Bethe-Peierls boundary condition [15] that, at
unitarity, takes the formlim Ψ ≈

rij →0F+ O(rij ).
rij

(7.6)

Here rij is any of the distances between interacting particles, and F is a smooth functionof all coordinates except the interparticle distance rij . This unitary condition implies
that when two particles get very close together, the wave function should behave as thescattering solution at unitarity, i.e., Ψ ∝ 1/rij .
Except for these boundary conditions in the vicinity of coalescent points, i.e.,
rij is much smaller than the typical interparticle spacing, the wavefunction obeys thenoninteracting Schrodinger equation H0 Ψ = EΨ. The Hamiltonian is
H0 =

X
i

−

~2 2 1∇ + mi ω 2 r2i .
2mi i
2

(7.7)

This Hamiltonian is identical to that of an ideal gas. The effects of interactions areincluded in the above boundary conditions, Eq. (7.6), which introduces a great mathematical difficulty to the problem.
To analyze this problem we use hyperspherical coordinates. We define the hyperspherical vector R = {r1 , ..., rN }. R describes all the coordinates of the system and canbe divided into the hyperradius R = |R| and a set of angles encapsulated in a single
coordinate called Ω. The contact conditions at unitarity do not have a length scale associated with them, since a → ∞. Thus, they become invariant under the rescaling of thespatial coordinates. This behavior implies that for any wave function ψ that obeys thecontact conditions, Eq. (7.6), we can define a rescaled wave function ψλ (R) = ψ(R/λ)
that also obeys the contact conditions. Here λ > 0 is the scaling factor.
In free space, this scale invariance implies that if ψ is an eigenstate of energy
E, then there is an eigenstate of energy E/λ2 for any λ. This relationship implies theabsence of bound states in free space which otherwise would be an unphysical continuum

160
of states. In the trapped system, the appearance of a length scale, the trap length aho ,breaks down the scale invariance properties of the eigenstates. However, as we will seein the following, a similar scale invariance analysis can be applied to trapped systems
to extract unique universal properties.
To analyze the properties of a trapped system, consider the evolution of an initial stationary state Ψ0 when the frequency of the external potential becomes timedependent, i.e., ω = ω(t). Castin proposed the following ansatz for the time-dependent
wavefunction,
2 0Ψ(R, t) = N (t)eimRj λ (t)/2~λ(t) Ψ0 (R/λ(t))(7.8)

where the scaling factor λ(t) and the normalization N (t) need to be determined. To
verify to the validity of Castin’s ansatz, we first note that Eq. (7.8) obeys the boundary
condition, Eq.(7.6). Eq. (7.8) also obeys the time-dependent Schrödinger equation governed by the Hamiltonian of Eq. (7.7) provided that the time evolution of λ is described
by
λ00 =

ω 2 (0)− ω 2 (0)λ.
λ3

(7.9)

In analyzing the perturbative time-dependent behavior of a stationary state ofenergy E, Castin showed that there exist states with energies E ±2~ω [38]. Furthermore,in this time-dependent analysis, the operators responsible for ±2~ω are obtained, andcan be associated with raising and lowering operators [186]:
H
mωR2
3N
+ R∂R +
−
,
2
~ω
~
3N
H
mωR2
L− = −− R∂R +
−
.
2
~ω
~
L+ =

(7.10)(7.11)

The commutations relations of H, L+ and L− have the structure of the SO(2,1) Lie161
algebra;[H, L+ ] = 2~ωL+ ,(7.12)[H, L− ] = −2~ωL− ,(7.13)[L+ , L− ] = −4H/~ω.

(7.14)

From this point on, all the machinery for raising and lowering operators can be applied.
By successively applying the lowering operator to any eigenstate, we obtain a new
eigenstate with arbitrary negative energy. To avoid this unphysical scenario, we haveto impose the condition that for some eigenstate ψ0 ,L− |ψ0 i = 0.

(7.15)

Without a loss of generality, we assume that the energy of ψ0 takes the formEν,0 = (sν + 3/2)~ω.

(7.16)

Here, ν is a general quantum number that labels the state and its energy. In addition,
Eq. (7.15) can be easily solved in hyperspherical coordinates. We obtain,2

2~ = Rsν +3(N −1)/2 e−R /2aho Φν (Ω).
ψν,0 (R)(7.17)The state ψ0 is the lowest of a ladder of eigenstates |ψq i = Lm
+ |ψ0 i with energies
Eν,m = (sν + 3/2 + 2m)~ω,(7.18)

where m is a positive integer. The coefficients sν dictate the beginning of the ladder,and to obtain them, we have to solve the complete many-body Schrodinger equation.
Even though we do not know where these ladder of state begin, we showed that thestructure of the spectrum consists of families of eigenstates, labeled by ν, with energies
separated by 2~ω.
The functional form of the unnormalized states obtained by successive applications of L+ to the lowest state are~ = Rsν +3(N −1)/2 e−R2 /2a2ho Lsν +1/2 (R2 /a2 )Φν (Ω).
ψν,m (R)mho(7.19)

162
These eigentates are separable in the hyperspherical coordinates (R, Ω). In the hyperspherical framework, the separability of the eigenstates implies that the adiabatic
approximation is exact [119]. Furthermore, the channel functions Φν do not depend on
the hyperradius. Since the L+ only acts on the collective coordinate R, the 2~ω spacing
is associated with collective or breathing mode excitations.
From the functional form of the eigenstates [Eq. (7.19)], we can infer the effective
hyperradial potentials. We define the hyperradius wave function Fν,m with the correct~ = R1−3N/2 Fν,m (R)Φν (Ω). The hyperradial wave functions havenormalization, ψν,m (R)the same functional form as the noninteracting system, so we expect that the functional
form of the hyperradial potentials should also agree. This agreement can be verified byinserting the wavefunction solutions of Eq. (7.19) into the hyperspherical Schrodinger
equation [Eq. (2.32)]. This procedure leads,µ¶~2 ∂ 2mω 2 R2 ~2 sν (sν + 1)+−+
Fν,m = Eν,m Fν,m .
2m ∂R222mR2(7.20)The effective potential curves include a trapping term and a term proportional to 1/R2that includes the kinetic energy and the effects of the two-body interactions. By taking
the limit ω → 0 in Eqs. (7.19,7.20) we can verify that the 1/R2 term remains unchangedin the absence of a trap. In this limit, the trapping potential disappears and all thetrapping states from Eq. (7.19) tend to the zero energy eigenstate of the free system.
We can now easily show that these state obey a virial theorem [175, 186]. Consider
2 P
2the expectation value of the external potential Vtrap = mωi ri over an eigenstate,2hψν,m |Vtrap |ψν,m i =

X
mω 2
mω 2
ri2 |ψν,m i =hψν,m |
hψν,m |R2 |ψν,m i .
22(7.21)iEquation 7.21 can be evaluated analytically in hyperspherical coordinates. The integration over the hyperangles is trivial since the eigenstates are separable in this coordinates
and Vtrap does not depend on Ω. We are then left with a one dimensional integration
over R, using the properties of the Laguerre polinomials we obtain the virial theorem

163
relationhψν,m |Vtrap |ψν,m i =

Eν,m
.
2

(7.22)

Since all the eigenstates obey the virial theorem, a thermal distribution would also obey
the same virial theorem.
All the properties presented above are valid for both few- and many-body systems.
For large N systems, it is convenient to introduce variables rescaled by the noninter0
acting quantities. If one defines x = R0 /RN
I and ²νn = Eνn /EN I , the hyperradial

Schrödinger equation can be rewritten asµ−¶sν (sν + 1) 1 2
1 d2++ x F̄νn (x)2µef f dx2
2µef f x22= ²νn F̄νn (x),(7.23)

2 /(~ω)2 . R
where µef f = EN
N I in the above definition of x denotes the rms radiusIof the non interacting system. The virial theorem [Eq. 7.22] implies that RN I can be
expressed in terms of the energy EN I of the non-interacting two-component Fermi gas,
r
r
p
~
EN I
2
.
(7.24)RN I = hR iN I =Mω~ωThe dimensionless coefficient CN is defined byCN =s0 (s0 + 1)~2 ω 2
s0 (s0 + 1)
=
2
µef f
EN
I

(7.25)and it characterizes the ground state of the system at unitarity. The scaled hyperradiusx and the scaled energies ²νn remain finite in the large N limit. For example, in thelimit N → ∞,
²00 ≈

p

C∞ or E00 ≈p

C∞ EN I .

(7.26)

Here, we have used that CN converges to a finite value C∞ in the large N limit. Thus, thescaled quantities are particularly well suited to discussing the large N limit. For small
systems, in contrast, some properties of the system can be highlighted more naturallyusing the unscaled hyperradius R.

164
Finally, all the properties explained in this section remain valid if the center-ofmass motion is removed and the particles have different masses. Both for a systemwith the center of mass decoupled and for an unequal mass system, there exists a setof Jacobi vectors which obey the boundary condition of Eq. (7.6) and transform the
Hamiltonian to that of the form Eq. (7.7), reproducing the problem considered above.
The coefficients sν describe both the trapped and free systems and are related tothe universal parameter ξ of the homogeneous system [19, 174]. Thus, the hypersphericalframework connects few- and many-body quantities and allows one to bridge the gap
between atomic and condensed matter physics.

7.3

Numerical verification of universal properties for trapped systems
It cannot be known a priori in which cases the theory presented in the previous

section is valid. The theory can only say that, if the system behaves universally, thenit has the mentioned properties. To verify that a particular system is universal we needto solve the complete problem, i.e., we have to obtain solutions of the spectrum andwavefunctions and test whether the universal properties are fulfilled.
For example, a two-component Fermi gas is expected to behave universally. Pauli
repulsion is considered responsible for stabilizing the system against collapse and avoiding clustering. On the other hand, bosonic systems with attractive interactions are
known to collapse and form cluster systems. Therefore, it is clear that bosonic systems
are not universal. However, there can be part of the bosonic spectrum that follows these
universal properties.
The stability of two-component Fermi gases is supported by experimental results
and numerical simulations. The most relevant numerical results are obtained by FNDMC, which shows that the energy of a homogenous Fermi gas at unitarity is positive[36, 7]. However, these results do not constitute proof that the two-component Fermi

165
gas is universal. Furthermore, FN-DMC calculations have important limitations due tothe nodal constraint, so we cannot know for sure that the E = (1 + β)EN I is the true
ground state (for a zero-range pseudopotential) or just the lowest “gas-like” state. In
addition, unphysical purely attractive potentials, used in these FN-DMC calculations,produce many-body bound states for large N systems. It is not completely clear what
would happen with these many-body bound states if the potentials would get replacedby realistic potential or by zero-range pseudopotentials.
In this section, we analyze the few-body two-component Fermi gas at unitarity
by solving the many-body Schrodinger equation with the CG method. The universal
properties described in the previous section are valid with and without the inclusion of
the center-of-mass motion [186]. In the numerical calculations, we removed the centerof-mass motion to reduce the computational demands. The center-of-mass motion is
described by a simple one-particle harmonic oscillator wave function and can be trivially
included after the numerical calculations.
For even systems, we consider N1 = N2 = N/2, while for odd systems we consider
N1 = N2 +1. As we noted in the Introduction, the perturvative BEC and BCS behaviorsof the even N systems predict an L = 0 ground state while for odd N systems L canbe different from zero. This is consistent with the pairing arguments that suggest thatan even N system should have all the pairs couple to L = 0. Therefore, we expect thateven at unitarity the even-N systems should have L = 0. However, to analyze odd N
systems we have to consider L > 0 solutions. The CG method as implemented above
only describes L = 0 states [182].
To obtain L > 0 states, we introduce a spectator particle. Specifically, we solvean (N + 1)-atom system. This spectator particle does not interact with the rest of theparticles but couples to them to form an L = 0 state. Through this introduction of an
extra particle to an N particle system, the wave function will now describe “solutions of
the N -particle”+ “solutions of the extra particle” coupled to zero angular momentum.

166
Even though the extra particle does not interact at all with the N -particle system, theconstraints of the wave function force it to be coupled with the N -particle system tozero angular momentum. Thus, if the extra particle has angular momentum L, thenthe N -particle system has also angular momentum L so they couple to zero angularmomentum. To know the angular momentum of the N system, we measure the energyof the spectator particle, which should agree with the energy of one particle in a trap.
From the energy of the spectator particle, we can deduce the angular momentum of
the system. For example, consider that we want to extract the spectrum of the N particle system with L = 1 from the L = 0 spectrum of the N + 1-particle system.
In such case, we know that the L = 1 ground-state energy of the extra particle in a
trap is 5/2~ω. Therefore, by measuring the energy of the extra particle, we can know
the angular momentum of the N -particle system, and we can predict its energy, via
EN = EN +1 − 5/2~ω. Table 7.1 summarizes a few selected results for N = 3 − 6.
A property of the even-N systems that is important for numerical calculations
is the symmetry under the exchange of spin-up and spin-down particles. In the equalmass case, the different spin components are treated as distinguishable, particles andthere is only interaction between spin-up and spin-down particles. Therefore, both the
Hamiltonian and the symmetry of the wave function remain unchanged if we replace
all the spin-up particles with spin-down particles and vice versa. This exchange operation reveals a symmetry of the problem that we call “spin exchange” symmetry. The
eigenfunctions have to be even or odd under this symmetry transformation. The groundstate of an even-N system is expected to be even under spin exchange, since the “oddspin” wave function is forced to have an extra node. In the N = 4 system, we test
this property. First, we calculate the complete L = 0 spectrum, and then we calculate
the even-spin spectrum imposing the even-spin symmetry in the wave function. The
even-spin spectrum accurate reproduces the ground state and part of the excitation
spectrum. For the N = 6 calculations presented in Table 7.1, we use this property to167
enforce the even-spin symmetry to the wave function. This significantly reduces theHilbert space and allows an accurate description of several states.
Table 7.1: CG energies Eνn for N = 3 − 6 (the range r0 of the Gaussian potential
is 0.01aho for N = 3 and 4, and 0.05aho for N = 5 and 6). Finite range effects are
expected to be ≈ 0.02~ω and 0.1~ω for r0 = 0.01~ω and 0.05~ω, respectively. Resultsfrom Ref. [19].
N
3
3
4
4
5
5
6

Lrel
0
1
0
0
0
1
0

E00 /(~ω)
4.682
4.275
5.028
7.11
8.03
7.53
8.48

E01 /(~ω)
6.685
6.276
7.032

E02 /(~ω)
8.688
8.279
9.039

10.04

12.06

10.50

12.52

E10 /(~ω)
7.637
6.868
7.464
7.19
8.83
9.13
10.44

E20 /(~ω)
9.628
8.229
8.051
7.74
10.38
11.00

The 2~ω spacing reported in Table 7.1 strongly suggests the validity of the universal properties. However, as the energy increases, there is a large quasi-degeneracy ofstates and the states separated by 2~ω are hard to identify. The difficulty to identify the2~ω spacing occurs even more if we consider the numerical uncertainty of the energies
and the finite range effects. Therefore, it would be convenient to compute an independent observable that would also reveal the universal properties and complement the
2 (R).
spectrum analysis. Hence, we analyze the behavior of the hyperradial density Fν,n

The numbers of nodes tell us the number of 2~ω excitations and the analytical formsof the hyperradial densities are known, see Eq. (7.19). The analytical solutions canbe directly compared with the numerical results. The combination of the analysis of
the spectrum with the verification of the functional forms of the hyperradial densities
provides strong numerical evidence of the universal nature of the spectrum.
To obtain the hyperradial densities, we integrate the square of the wave function
over all the hyperangles. This multidimensional integration would be computationallydifficult using standard integration techniques. Accordingly, we use two unconventional
numerical techniques that work remarkably well for this specific problem. We use the168
first technique for N ≤ 4 and take advantage of the properties found in correlated
Gaussian hyperspherical method (CGHS). As explained in Sec. 3.4, the overlap matrix
element of two basis functions can be reduced to an analytical expression for N = 3 or toa one-dimensional numerical integration for N = 4. Thus, by using the basis expansionof the wave function we can significantly reduce the computational demands of theintegration. As the number of particles N increases, the dimension of the numericalintegration goes as N − 3, so this method becomes very computationally demanding for2
Vsν (R)/h̄ω or Fνn(R)/a−1ho

N > 4.
10

(b)

12

8

11

6

10

4
0

2

9

4

(a)

6

(c)

8
7

4
6
2
0

2

R/aho

4

0

2

4

6

R/aho

Figure 7.1: Dash-dotted lines show V (R) as a function of R/aho for (a) N = 4 with
ν = 0, (b) N = 4 with ν = 1, and (c) N = 6 and ν = 0. Circles and solid lines show
2 determined numerically and analytically, respectively. Dashedthe corresponding Fνnrel . Results are from Refs. [185, 19].
horizontal lines show the energies Eνn

The second method uses Monte Carlo techniques to carry out the integration. The
universal wavefunctions [see Eq. (7.19)] are separable in hyperradius and hyperangles.
Therefore, if the numerical solutions are of the form of Eq. (7.19), we could simply
fix the hyperangles Ω to some value, and evaluate the square of the wavefunction as afunction of the hyperradius. That procedure should give us a function proportional to
2 (R). However, our numerical solutions are not exactly separable in hyperspherical
Fν,n

169
coordinates because of the finite range of the interaction. Also, numerical limitations
can affect the separability in hyperspherical coordinates. Therefore, instead of selecting
a single set of hyperangles and extracting the square of the wave function, we canrepeat this process for many different sets of hyperangles and average over them. The
different sets of hyperangles are selected randomly, and this process is equivalent to aMonte Carlo integration. Since the numerical solutions are approximately separable in
hyperspherical coordinates, we do not need a large number of randomly selected setsof hyperangles to converge the solution. In other words, the standard deviation of thenumerical integration goes to zero as the separability becomes exact.
Figure 7.1 presents numerical results for N = 4 and N = 6 hyperradial densities
and compares them to the analytical solutions. The analytical solutions are obtainedby extracting the sν values from the numerical energies shown in Table 7.1. The N = 4
solutions were obtained with the basis set expansion technique while the N = 6 method
is obtained with the Monte Carlo technique. In both cases, good agreement is obtained
between the numerical and the analytical solutions.
The analysis of the universal properties can be extended to unequal mass systems.
The universal properties predicted in Sec. 7.2 are also valid for universal systems of two
fermionic species with arbitrary mass ratio so long as all the particles feel the same
trapping frequency. Here, we analyze the spectrum of four trapped fermions, two foreach fermionic species, characterized by a mass ratio κ. In Fig. 7.2, the spectrum
is analyzed as a function of the mass ratio. The circles correspond to the numerical
spectrum. We can clearly identify families of solutions separated by 2~ω. To identify
these families, we generate a series of solid lines that correspond to E00 + 2n~ω, a seriesof dashed lines that correspond to E10 + 2n~ω, and a series of dash-dotted lines that
correspond to E20 + 2n~ω. The energies (circles) that agree with the lines correspondto the same family. The energies Eν0 determine the coefficients sν of the hyperradialpotential curves Vsν (R). Table 7.2 summarizes the three smallest coefficients for various

170
κ.

10

E/h̄ω

9
8
7
6
5
2

4

6

κ

8

10

12

Figure 7.2: Four-body energy spectrum for L = 0 at unitarity as a function of κ.
Circles correspond to the numerical results obtained by the CG approach. Solid, dashedand dash-dotted lines show the energies Eν0 + 2n~ω for ν = 0, 1 and 2, respectively(n = 0, 1, · · · ). Results are from Ref. [185].

From the numerical results presented in this section, we have indirectly verified
the form of the hyperspherical potential curves. The 2~ω spacing clearly suggests theuniversality of the potential curves and the 1/R2 free-space hyperspherical potential.
The development of the CGHS method allows us to carry out a hyperspherical calculation for the four-fermion problem and to directly verify the form of the hypersphericalpotentials. Also, it allows us to analyze deviations from the zero-range solutions due tofinite-range effects.
The 20 lowest four-body potential curves 2µR2 Uν (R)/~2 for the equal-mass system are presented in Figure 7.3. We can identify three regimes in these potential curves.
The region R . r0 is controlled by the kinetic energy. The kinetic energy effects are more
important than the interaction energy and the potential curves are well approximatedby the non-interacting potential curves. In other words, 2µR2 Uν (R)/~2 ≈ Λ2 + 12 andthe eigenchannels are well approximated by the hyperspherical harmonics (see Sec. 2.5).
For that reason, there is a large degeneracy in the R . r0 region which corresponds to171

Table 7.2: Coefficients sν of the hyperradial potential curves Vsν (R), Eq. (7.18), for the
N = 4 system with L = 0 for various mass ratios κ. Results are from Refs. [185].
κ
1
2
3
4
5
6
7

s0
2.03
2.09
2.18
2.27
2.34
2.40
2.43

s1
4.46
4.41
4.27
4.15
4.04
3.95
3.88

s2
5.05
4.88
4.90
4.98
5.06
5.15
5.22

κ
8
9
10
11
12
13

s0
2.45
2.45
2.42
2.37
2.29
2.17

s1
3.81
3.74
3.68
3.62
3.57
3.51

s2
5.29
5.35
5.39
5.39
5.30
5.18

2µR2 Uν (R)/h̄2

120
100
80
60
40
20
0

0

10

1

10

2

10

R/r0
Figure 7.3: Hyperspherical potential curves for the 4-fermion system multiplied by
2µR2 /~2 . The solid lines represent the predictions from analyzing the spectrum obtainedwith the CG method. The symbols correspond to direct evaluation of the potential
curves with the CGHS method. Unpublished results.

172
the degeneracy of the Λ2 operator. Furthermore, the potential curves are, to a good
approximation, proportional to 1/R2 . The second region is r0 . R . 20r0 . In thisregion both the kinetic and the interaction terms are important and finite range effects
are important. In the third region, R & 20r0 , the potential curves recover their universal behavior. The potential curves are, again, approximately proportional to 1/R2 . As
R/r0 increases, finite range effects tend to zero and we obtain the zero-range potential
curves at unitarity. Therefore, in this region, the eigenvalues of 2µR2 Uν (R)/~2 are approximately sν (sν + 1). Thus we can compare these results with the ones deduced fromtrapped calculations for r0 /aho = 0.01 presented in Table 7.2. The red, blue and green
solid lines correspond to s0 (s0 + 1), s1 (s1 + 1) and s2 (s2 + 1) respectively. There is good
agreement between the predictions from the trapped system and the direct computationof the potential curves.
To quantify this last statement, we analyze the value of s0 . Several groups [182,40, 184, 2, 166] have tried to benchmark the four-body value of E00 , which is straightforwardly related with s0 . The calculations from Ref. [2] use zero-range interactions
explicitly.and they report a value of E00 ≈ (5.045 ± 0.003)~ω. To extract the s0 value
in the zero-range limit, we carry out two different calculations. First, we study the
E00 energy obtained with CG as a function of the range of the two-body interaction
and then we extrapolate to zero-range limit. This method was previously applied forthe three-body system and the numerical results agreed with the analytical predictions
up to 7 digits [185]. The same procedure applied to the four-body system, leads tos0 ≈ 2.0096. The second calculation analyzes the long-range behavior of the potential
curves. To eliminate finite-range effects, we extrapolate the potential curve U0 (R) to
R/r0 → ∞. In this limit, U0 (R) is characterized by a value s0 ≈ 2.0092. These twodifferent methods provide a value of s0 which agrees in four digits. These values areslightly lower than s0 ≈ 2.045 ± 0.003 predicted in Ref. [2]. Therefore, we consider thatthe uncertainty in Ref. [2] is underestimated.

173

7.4

Renormalization and unitarity
The dimensional arguments of the unitary system have been taken into accountin functional form of the effective scattering length. The effective scattering length isassumed to take the following functional form [183],
kf aef f = ζ(kf as ).

(7.27)

If the renormalization ζ function remains finite in the as → ∞ limit, the interaction at
unitarity is only characterized by the Fermi momentum,
aef f = ζ min /kf .

(7.28)In the Thomas-Fermi approximation, the renormalization technique leads toµ¶
10 min
µU =
1+
ζ
,
2m
9π
µ
¶
10 minEU /EN I = 1 +ζ.
9π~2 kf2(7.29)(7.30)

Here, ζ min is the asymptotic behavior for as → ∞. In this renormalization approach,the β parameter is intimately related with the value of the renormalization at unitarity. Comparing Eq. (7.2) to Eq. 7.29 we obtain β = 10ζ min /9π. The renormalization
procedure described in Sec. 4.4 leads to a β ≈ −0.492. Table 7.3 shows different experimental and theoretical values of β. Experimental groups, like Thomas and Huletgroups, measure the ratio of the size of the Fermi cloud at unitarity, RU , and in thenoninteracting limit, RN I [134, 105]. In the Thomas-Fermi approximation, the ratio is
RU /RN I = (1 + β)1/4 [75]. This relation was obtained previously using dimensional
arguments [75] and appears naturally using the renormalization approach.
The renormalization applied to the hyperspherical method has predicting power
also.

Rittenhouse and Greene [150] showed that a renormalization of the form ofEq. (7.27) applied in the K-harmonic approximation leads to a hyperspherical potential

174
Table 7.3: Experimental and theoretical predictions of β.

Experiments
Experiments
QMC
Padé asymptotes
Green’s function
BCS∗ ——² Expansion†
Other methods−0.54(5)13−0.68+0.13−0.10−0.58(1)6-0.6748−0.5451 0−0.4112 ∗−0.314β−0.64(15)25−0.54+0.05−0.12−0.56(1)7−0.4329−0.59911−0.52513 †−0.56415−0.49(4)4−0.49216

1 Ref. [134] 2 Ref. [25] 3 Ref. [13] 4 Ref. [105] 5 Ref.[167] 6 Ref. [7] 7 Ref. [41] 8 Refs. [9, 85]
9 Ref. [9] 10 Ref. [137] 11 Ref. [92] 12 This is a well known result, see for example Ref. [137]
13 Ref. [128] 14 Ref. [32] 15 Ref. [43] 16 Result obtained using the renormalization functionof this thesis (and from Ref. [183])

at unitarity of the form,
V (R) =

~2 (1 + β),2M R2

(7.31)

where β = 10ζ min /9π. Here, the hyperradius can be understood as a measure of the
global size of the cloud and breathing mode oscillations can be understood as small
oscillations at the bottom of this potential curve. Therefore, oscillations in R are collective oscillations. For a spherically symmetric trap, the functional form of the potential[Eq. (7.31)] predicts ω0 = 2~ω, in agreement with the universal properties presented inSec. 7.2 and several other many-body predictions.
The renormalization function is applied at the mean-field level of approximation,
which does not take into account pair correlations. The pair correlations are essential for the description of the superfluid nature of the system. For that reason, thisrenormalization approach cannot predict whether the system is superfluid or not; itcannot describe superfluid observables. However, it does predict the correct relation
between chemical potential µ and the density ρ that control the collective excitationsin the hydrodynamic theory [49, 93]. These collective excitations have been measured
experimentally and are in good agreement with theory.

175

7.5

Monte Carlo calculations for trapped systems: β parameterand the gap
For systems with a large number of particles, the calculation of the complete

spectrum using a method such as the CG becomes computationally prohibitive. In
contrast, quantum Monte Carlo simulations can be extended to larger systems andallow a reliable determination of the β parameter and the excitation gap for a twocomponent Fermi gas. For example, the first calculations, carried out in 2003, considereda homogenous system and used both variational and fixed-node diffusion Monte Carlotechniques (VMC and FN-DMC). The universal parameter was determined to be β =−0.56(1) [36]. These calculations were followed in 2004 by the authors of Ref. [7] inwhich the BCS-BEC crossover was considered; a similar parameter was obtained, with
β = −0.58(1). Another important parameter calculated with FN-DMC is the excitation
gap. The excitation gap ∆ of the homogenous system was calculated by Carlson andReddy [37] to the ∆ = 0.84(5)EF G , where EF G denotes the energy per particle ofnoninteracting homogeneous Fermi gas.
Initial calculations of the energy of trapped two-component Fermi gases werereported in Ref.[183]. These calculations were significantly improved and extended
to larger systems by Chang and Bertsch [40] and D. Blume who performed the FNDMC calculations in Refs. [184, 19]. I participated in the implementation FN-DMC
simulations for different trial wave functions and in the analysis of the results. Using
my own implementation of FN-DMC method and considering more flexible trial wave
functions, I was able to reproduce some of D. Blume results. In some cases, more
flexible trial wave functions predict lower energies than those of Ref. [184, 19]. However,
the improvement was usually less than 2%, very small considering the computational
demands of my calculations. For detailed discussion of Monte Carlo methods in generaland the trial wave functions considered in the trapped two-component Fermi gas, please176
see Sec. 3.2. Here, we analyze the numerical results.
Figure 7.5 presents the total energy at unitarity as a function of the number ofparticles N . We can clearly see an odd-even behavior in the energies as a function of N
that is related, as we will see later, to the excitation gap.

70
60

E/h̄ω

50
40
30
20
10
0
0

5

10

15

20

25

30

N
Figure 7.4: Circles and crosses show the fixed-node diffusion Monte Carlo (FN-DMC)
energies E(N ) in units of ~ω as a function of N at unitarity. Circles correspond toeven-N systems and crosses correspond to odd-N systems. Solid lines show a fit of theeven FN-DMC energies to Eq. (7.34). Numerical results are from Refs. [184, 19].

For sufficiently large N , the energy of the trapped system at unitarity can becalculated using the homogeneous result of Eq. (7.2) in a local density approximation(LDA). As with Eq. (7.2), we obtain an equation that relates the noninteracting energy
(T )

(T )

of the trapped system EN I with the energy at unitarity of the trapped system EU :(T )EU (N ) =p
(T )1 + βEN I (N ).

(7.32)

Equation (7.32) allows us to extract β from the energy of trapped system at unitarity.
(T )The noninteracting energies EN I present a shell structure. However, the energies atunitarity up to N = 30 show little shell structure and can be described to a very good
approximation by a smooth function. Therefore, for Eq. (7.32) to actually work in the(T )small N (N < 30) region, we have to approximate EN I by a smooth function. Such

177
an approximation can be accomplished by applying the extended Thomas-Fermi (ETF)(T )model [29] to EN I so that
(3N )4/3(T )EN I,ET F ≈ ~ω4Ã(3N )−2/3
1+
+ ···
2

!
.

(7.33)

Using ETF in Eq. (7.32), we obtain
(T )

EU

≈

p
(T )
1 + βEN I,ET F .

(7.34)

Fitting our energies to Eq. (7.34) and treating β as a variational parameter, we find
β = −0.535. Our β extracted from the trapped system is about 10% larger than that
determined for the bulk system, i.e., ξ1 = 0.58(1) [7, 37], suggesting that one has to go to
somewhat larger trapped systems to extrapolate to the bulk β with high accuracy within
the LDA. Although the β obtained from the fit to the energies of the trapped systemis larger than the corresponding bulk value, it is worthwhile to note that the simple
functional form given in Eq. (7.34) provides an excellent description of the energies of
the trapped system even for small N -particle systems.
The Monte Carlo calculations for odd and even systems can be combined to obtain
the excitation gap ∆(N ) for the trapped unitary Fermi gas as a function of N . The
definition of the excitation gap for the homogeneous system [36], which equals half theenergy it takes to break a pair, can be readily extended to the trapped system [40, 34]
where∆(N ) = E(N ) − [E(N − 1) + E(N + 1)]/2.

(7.35)The excitation gap ∆(N ) obtained from our FN-DMC energies is shown in Fig. 7.6 (circles). ∆(N ) increases from ≈ 0.75~ω for N = 3 to ≈ 1.5~ω for N = 29. For comparison,
we show the excitation gap ∆(N ) calculated using the DFT energies (triangles) obtained
recently by Bulgac [34]. The agreement between Bulgac’s ∆(N ) and our results is quite
good.

178
Alternatively, the excitation gap can be deduced from the excitation spectrum.
According with pairing theories, the ground state of even N systems has all the particles
paired up and it is separated of the excited states by the energy necessary to break
the pair. In contrast, odd N system have one unpaired particle and the spectrum
does not reveal the excitation gap. In nuclear systems, the excitation gap is observedby comparing the excitation spectrum from odd and even nucleon systems (see e.g.
Ref. [147]). In two-component trapped systems, the gap for small N -system is of thesame order of the trapping excitation energy ~ω. Therefore, it is more difficult to identify
the effects of the excitation gap in the excitation spectrum. Despite these difficulties,
we analyze the behavior of the excitation spectrum for N = 3, 4 and 5.
10
9

E/h̄ω

8
7
6
5
4
3
2.5

3

3.5

4

4.5

5

5.5

N
Figure 7.5: Excitation spectrum for N = 3, 4 and 5 at unitarity. Black circles correspondto L = 0 states and red circles correspond to L = 1 states.

Figure 7.5 presents the L = 0 and L = 1 excitation spectrum for N = 3, 4, 5.
The states corresponding to the same hyperspherical potential curve are separated by
2~ω. Clearly, these excitations are not associated with the excitation gap. To see
the effects of the excitation gap, we consider the energy difference between the lowest
L = 0 and L = 1 states for odd and even N systems. Here, we assume that the

179change of angular momentum ensures the breaking of the pair in the even N system.
Considering Table 7.1, we see that the N = 4 has an energy difference much larger thanthe N = 3 and N = 5 systems. To estimate the excitation gap we consider that the
N =4 − E N =4 | has two contributions: the energy for breaking theenergy difference |EL=0
L=1

pair and the energy for the change of symmetry. In this picture, the odd systems have
only the contribution from the change of symmetry. Therefore, we can estimate thegap, i.e., half the energy for breaking the pair, as∆(4) =12µ¶
1 N =3
N =4
N =4
N =3
N =5
N =5
|EL=0
− EL=1| − [|EL=0− EL=1| + |EL=0− EL=1|] .
2(7.36)The second term correspond to the energy due to the change of symmetry for N = 4
which is estimated averaging over the N = 3 and N = 5 values. Using the energies
from Table 7.1, we estimate ∆(4) ≈ 0.81~ω which is close to ∆(3) ≈ ∆(5) ≈ 0.75~ω
obtained with Eq. (7.35). This qualitative agreement support the assumption that the
odd-even behavior is a consequence of pairing physics. Unfortunately, the analysis of
the spectrum can only be done for N ≤ 5. For larger systems, only the ground state is
known and we need to use Eq. (7.35). Next, we consider the dependence of the gap on
N and its large N limit behavior.
To analyze the large N limit behavior of the excitation gap, we use LDA andobtainp∆LDA (N ) = 3αhom (3N )1/3 ~ω/(8 1 + β).

(7.37)The universal parameter α describes the even-odd oscillations and αhom equals 0.85
for the bulk system [37]. Using α as a fitting parameter we find αtr = 0.60 for thetrapped system. The ∆LDA (N ) with the ξtr and αtr parameters, shown in Fig. 7.6 by asolid line, provides a good description of our FN-DMC results. It is the fact that αtr isnoticeably smaller than αhom which suggests that the extra particle is not distributed
uniformly throughout the cloud; rather it has a greater probability of residing at thesurface of the cloud where the gap is smaller. Indeed, our density profiles for N & 11

180
(not shown) indicate that the extra particle sits near the surface of the cloud [185]. The
unpaired particle location is verified by comparing the density profiles of the spin-up
and spin-down particles. In general, the LDA is expected to describe the center regionof the trap well, but to fail in the surface region, raising questions about the validityof our LDA treatment [Eq. (7.37)]. Recently, Son reached similar conclusions [160]. In
his analysis, the gap increases with N 1/9 as N → ∞. A fit of our results for ∆(N ) forN ≥ 9 shows consistency with the N 1/9 dependence but does not conclusively confirm
it.

∆(N )/h̄ω

2
1.5
1
0.5
0
0

10

20

30

N
Figure 7.6: Circles show the excitation gap ∆(N ) determined from our FN-DMC energies. A solid line shows ∆LDA (N ). For comparison, triangles show ∆(N ) determinedfrom the DFT energies [34]. Figure taken from Ref. [19].

The FN-DMC energies provide interesting insights on the evolution from few- to
many-body systems. Such evolution can be studied in the hyperspherical framework.
We can use the FN-DMC energies to extract the lowest hyperspherical potential curves
that describe these systems. The ground-state energies E(N ) determine the coefficientss0 [see Eq. (7.16)] of the hyperradial potential Vs0 (R) [see Eq. (7.20)]. Figures 7.7 (a)and (b) show the lowest hyperradial potential curves V (R) [V (R) = Vs0 (R) + Vtrap (R),where Vtrap (R) = 21 µN ω 2 R2 and µN = m] for N = 3−20 in the noninteracting limit andat unitarity, respectively. The small R behavior of V (R) is dominated by Vs0 (R), while

181
the large R behavior of V (R) is dominated by Vtrap (R). Comparison of Figs. 7.7 (a)and (b) shows that the attractive interactions lead to a lowering of the potential curves
at unitarity compared to those of noninteracting system. Furthermore, the V (R) at
unitarity appear to be “staggered,” i.e., odd-even oscillations are visible, reflecting thefinite excitation gap at unitarity. In the noninteracting limit, in contrast, the excitation
gap [Eq. 7.35] is zero and no odd-even staggering of the hyperradial potential curves isvisible.
Another important aspect of these potential curves is their relation to the parameter β. As was pointed out previously by Tan [174] and was shown using therenormalized hyperspherical approach [150], the small R behavior of the wavefunction
and the hyperspherical potential curves in the large N limit are intrinsically related tothe universal parameter β. To analyze the large N limit, we use the rescaled coefficientsC̄N presented at the end of Section 7.2. Figure 7.5 shows the normalized coefficientsC̄N . As in our analysis of the energies EU , we find it useful to smooth the energies
EN I ). Therefore C̄N stands for CN from Eq. (7.25) with EN I replaced by EN I,ET F , asa function of N . The coefficients C̄N oscillate between two smooth curves, a curve foreven N (circles) and a curve for odd N (crosses), reflecting the odd-even staggering. As
N increases, the difference between the two curves decreases. In the large N -limit, thevalue of C̄N for two-component Fermi gases at unitarity should approach the universal
parameter ξ = 1 + β [19]. This relationship can be shown by comparing the groundstate energy obtained within the hyperspherical framework, Eq. (7.26), to the LDA
prediction, or by applying renormalized zero-range interactions within the hyperspherical framework [150]. The dash-dotted and dashed lines in Fig. 7.5 show the ξ value
obtained by FN-DMC calculations for the homogeneous system (ξ = 0.42) [7, 37] andthe ξ value obtained with a renormalization procedure (ξ = 0.508) [183], respectively.
For comparison, our energies for the trapped system predict ξtr = 0.467. The circles inFig. 7.5 approach this value. The fact that ξtr is larger than the corresponding valueV (R)/h̄ω

182

(a)

60
40
20

V (R)/h̄ω

0

(b)

60
40
20
0
0

2

4

6

8

10

R/aho
Figure 7.7: Hyperradial potential curves V (R) for equal-mass two-component Fermi
systems with (a) vanishing interactions and (b) infinitely strong interactions as a function of R. The hyperradial potential curves naturally appear ordered as N increases:
Solid lines correspond, from bottom to top, to N = 4 − 20 (N even), while dashed lines
correspond, from bottom to top, to N = 3 − 19 (N odd). Figure taken from Ref. [185].

0.55
0.5

0.4

0.55
0.5

0.35

C̄N

C̄N

0.45

0.45

0.3

0.4

0.25

0.35
0

0.05

0.1

0.15

1/N
0.2
0

5

10

15

20

25

30

N
Figure 7.8: Normalized coefficients C̄N , Eq. (7.25) with EN I replaced by EN I,ET F ,as a function of N ; values for even N are shown by circles and values for odd N
by crosses. The dash-dotted line shows the value ξ = 0.42 obtained by FN-DMC
calculations for the homogeneous system [7, 37], while a dashed curve shows the valueξ = 0.508 obtained with a renormalization procedure [183]. The inset shows the same
quantities as a function of 1/N instead of N . Figure taken from Ref. [185].

183
of the bulk system, i.e., ξ = 0.42, might be attributable to the fact that comparatively
small system sizes (N ≤ 30) are considered in our analysis. If finite size effects are responsible for such disagreement, we would expect the circles in the main part of Fig. 7.5
to turn around at larger N values approaching to the homogeneous prediction ξ = 0.42
(dashed-dotted curve). Nevertheless, we cannot rule out that the nodal surface entering
our FN-DMC calculations might not be optimal. Even though the trial wave functions
for the trapped system are analogous to the wave functions of the homogenous case [7],trapping effects and the shell structure might affect differently the trapped calculations
and more sophisticated trial wave functions might be required.

7.6

Conclusions
The verification of the universal properties of few-body systems has importantconsequences in the understanding of the unitary regime. In the two-component Fermi
gas, we show quantitatively that the system behaves universally up to N = 6. This result
implies that there are no weakly bound states for N ≤ 6 which explains, from a few-bodyperspective, the stability of two-component Fermi gases. The FN-DMC calculations do
not exclude the presence of negative energy states; but, we do not find evidence of suchstates. The stability of the FN-DMC calculations reveals, to some extent, the stability ofthe system [18]. If the system is unstable, the walkers evolve into cluster configurations
with very large and negative energies which produces an instability of the simulation.
This effect does not occur in the two-component Fermi gas. The FN-DMC simulations
are remarkably stable, suggesting that the state described is also quite stable.

Chapter 8
Four Bosons with large scattering length

8.1

Introduction
The behavior of few-body bosonic systems with finite range interaction potentials

and large scattering lengths is quite different from fermionic systems. A remarkableprediction by Efimov [60] in 1970 stated that a system of three bosonic particles withresonant interactions, but without a two-body bound state, has an infinite family of
weakly bound three-body states. The existence of weakly bound Efimov states implies
a completely different topology of the spectrum as compared to fermionic systems. In
bosonic systems, for example, a “three-body parameter” is required to describe theseweakly bound states. Therefore, two parameters are needed to describe the system:
the scattering length as and the three-body parameter. When the scattering length
diverges, the three-body bound states obey interesting scaling properties. The energies
and sizes of two consecutive states are related to each other by a scaling factor that is
independent of the nature of the interaction.
Fundamental questions arise when another particle is added to such a system.
Does the four-boson system have an equivalent to the three-body Efimov effect? Does
the system requires a new parameter, i.e., a “four-body” parameter? Unfortunately, only
limited or approximate solutions to the four-boson problem exist [1, 163, 142, 83] becauseof the complexity of such systems and these questions have not yet been resolved.
Here, we present solutions to the four-boson system obtained by using the correlated-

185
Gaussian (CG) and correlated-Gaussian-hyperspherical (CGHS) methods. In contrast
with previous calculations, these methods essentially do not rely on any approximation;
thus limited only by numerical convergence.
We show that when a three-body Efimov state is at the zero-energy threshold,there is only a finite number of four-body states. From this perspective, we conclude
that there is no four-body Efimov effect. However, when two-body interactions areresonant, i.e., when the two-body bound state is at threshold, there is a family of fourbody bound states that are intercalated in energy with the family of three-body Efimov
states and this four-body family follows the same scaling properties. These results are
consistent with the prediction from Ref. [83] that no four-body parameter is needed.
Other important phenomena concern the effects of Efimov physics on the collisional properties of four-boson systems. We know that Efimov physics significantly
affects the collisional aspects of identical bosons with tunable interactions producing
striking features in the three-body recombination [63, 27]. These features were experimentally observed for the first time in 2006 in an ultracold gas of cesium atoms [67, 109].
Here, we use our solutions to the four-boson problem to predict the consequences of Efimov physics in dimer-dimer collisions. We show that for particular scattering length
values, an atom-trimer mixture can be obtained with high efficiency by colliding dimers.
We have also been able to study atom-trimers collisions which will help deduce the lifetime of the atom-trimer mixture.
In this chapter, we first review the three-body Efimov effect from the hyperspherical point of view. This solution will help us understand the four-boson system. Next, we
use the CG method to extract information about the spectrum and structural propertiesof the four-boson system. We then apply the CGHS method and extract the four-body
hyperspherical potential curves. We use these results to confirm the topology of thespectrum and study dimer-dimer and atom-trimer collisions. The results presented inthis chapter are preliminary, and the research on the four-boson system is still underway.

186

8.2

Efimov physics in a three-body system.
In this section we analyze the Efimov effect from the hyperspherical perspec-tive. This perspective reduces the description of the Efimov effect to an analysis of thefunctional form of the relevant hyperspherical potential curve.
In the hyperspherical representation, the Efimov effect is revealed in the form of
the adiabatic hyperspherical potential curve W0 (R). In the region where r0 ¿ R ¿ |as |,the lowest potential curve behaves likes2 − 1/4,
W0 (R) ≈ − 0
2µR2

(8.1)

where s0 = 1.0062378 [55, 127]. This attractive 1/R2 potential resembles that of acharge-dipolar interaction [80, 79, 78, 65, 71, 191]. At unitarity, when the scattering
length diverges, the 1/R2 behavior of the potential extends all the way up to R → ∞.
This long-range behavior is known to have an infinite series of bound states [61]. This
infinite series of states becomes finite on both sides of the unitarity point, where the
1/R2 behavior is only valid up to approximately |as |. Beyond that point, the potential
becomes repulsive [for as < 0] or converges quickly to the dimer-binding energy [for as >0]. The bound states whose sizes are much greater than r0 are mainly controlled by 1/R2
behavior and are therefore considered universal, i.e., independent of the nature of theinteraction. Note that this universality is slightly different from the universal behaviorof fermions discussed previously. Here, two parameters are necessary to characterize theEfimov states: the scattering length as and a three-body parameter. The three-bodyparameter is related to the range of the two-body potential (not to be confused withthe two-body effective range), and it controls the size and energy of the “first” Efimov
state.
The form of the potential curve at unitarity is responsible of certain scaling properties. The energy ratio between consecutive states is given by En /En−1 = e−2πs0 ≈
1/515, while the ratio of the mean sizes rn /rn−1 = eπs0 ≈ 22.7. These scaling laws only

187
depend on s0 which is a universal quantity. The value of s0 only depends on the natureof the particles and their corresponding symmetry. The value of s0 = 1.0062378 is onlyUν (R) mr02 /h̄2valid for three identical bosons.

R/r0
Figure 8.1: Schematic hyperradial potential curve for three identical bosons having
as < 0. The dashed red line corresponds to a quasi-bound Efimov state above threshold.
As the attraction increases, the state becomes bound. The blue line corresponds to abound Efimov state.

An interesting phenomenon, called “Efimov-like shape resonance” [63], occurs in
the negative scattering length region when an Efimov state is about to appear. The
attractive 1/R2 is valid up to R ∼ |as |. Beyond that point the interaction diminishes,
and the effective potential is controlled mainly by the kinetic energy which produces
a repulsive 1/R2 . This change of behavior produces a potential curve of the form in
Fig. 8.1. In the outer region, there is a potential barrier. This potential barrier appears
ar R ∼ as and controls the size of the trimer. Therefore, even at threshold, the size ofthe trimer remains finite. As the attraction increases, this state becomes bound.

8.3

Analysis of the four-boson system with the correlated Gaussian
methodIn this section, we present the analysis of the four-boson spectrum using the CG

method. We also analyze the pair-correlation functions and extract information on the188structure of the four-body states.
The study of the four-boson systems is significantly more challenging than thefour-fermion system. The scaling laws of the three-boson subsystem should appear insome form in the four-boson system. Thus, to describe a few Efimov states, we have tobe able to accurately represent states whose sizes varies by couple order of magnitudes.
For example, to produce three bound Efimov states in a trap, we have to make weakenough so that the trap length at least 103 r0 , where r0 is the range of the two-bodyinteraction. Even for such a weak trapping potential, the size of the 4th Efimov state is
too big to fit in the trap.
To analyze the spectrum, we introduce an external trapping. We are interested
in understanding not only the bound states in the four-boson system but also the quasibound states above the continuum described by three-body Efimov states and the extra
particle. Thus, the inclusion of a trapping potential is convenient for the numericalanalysis with CG since it allows us to discretize the continuum. However, numericallimitations do not allow us to converge the complete spectrum. For example, for a trap
size of 104 r0 , the binding energy of the lowest Efimov state is of the order of 107 ~ω.
The 3+1 states formed by an Efimov state and an extra particle can be treated as twoparticles in trap; they are spaced in energy by approximately 2~ω. So, between thelowest three-body threshold and the zero of energy there would be about 107 states. Of
course, we cannot describe accurately all these states. As the numbers of nodes increases,
the wave function becomes more difficult to describe. Numerically, we observe that only
a few of these 3 + 1 states at each three-body threshold are accurately described while
the rest remain unconverged. Intercalated with these unconverged states, there arequasi-bound four-body states. In general, these quasi-bound four-body states are easier
to converge than the 3+1 states.
To extract the quasi-bound four-body states from the complete spectrum, we
have to be able to distinguish them from the 3+1 states. To do that, we first analyze

189
the convergence of the spectrum as an optimization procedure is applied. In a standard
optimization procedure, the states converge according to their energy. The lowest states
converge faster than the highly excited states. However, in trapped four-body systems,
the quasi-bound four-body states converge much faster than any 3+1 state of the sameenergy. As we increase and optimize the basis, we can observe how unconverged 3+1
states plunge down, crossing the quasi bound-body states that remain remarkably stable.
By analyzing the convergence of the spectrum, we can distinguish unconverged statesfrom stable states that are good candidates for describing quasi-bound four-body states.
Next, we analyze the pair-correlation function of the states. The 3+1 pair correlation function presents two peaks. The first peak describes the trimer, while the second
peak describes the trimer-extra particle correlation. Unconverged states produce unphysical pair correlation functions, while stable quasi-bound states have smooth andwell-behaved pair correlation functions.
The combination of these analyses allows us to identify the quasi-bound four-bodystates. Of course, this semiempirical method does not guarantee the extraction of all
the quasi-bound four-body states. Unconverged four-body states cannot be detected bythis method.
Figure 8.2 presents the four-boson energy spectrum as a function of the inverse
scattering length. Here, we introduce a function F (x) = sgn(x) ln(1 + |x|) that allows
us to deal with the logarithmic nature of the Efimov scaling. The blue lines are thefour-body states; the black lines correspond to three-body thresholds; the red lines
correspond to dimer and dimer-dimer thresholds.
The first thing we notice is that there are two four-body states below the lowest
Efimov threshold across the entire range of scattering length values considered. At
unitarity, the deeper 4-body state has an energy E14b ≈ 5.88E13b , while the second has
an energy E24b ≈ 1.10E13b . These ratios were also calculated by Platter et al., who
obtained 5 and 1.01, respectively [83]. Note that even though the our values differ

190
quantitatively from those of Platter et al., the structure of the bound four-body system
is similar, with one ”tight” bound state and another state very close to the three-bodythreshold.
0

F (E/h̄ω)

−5
−10
−15
−20
−5

0

5

F (aho /as )Figure 8.2: Spectrum of the four-boson system in a trap. The use of the function
F (x) = sgn(x) ln(1 + |x|) allows to visualize effects at different orders of magnitude ina single graph. Blue lines correspond to four-body states. Black lines correspond totrimer+atom thresholds, and red lines correspond to dimer+atom+atom (upper) anddimer+dimer (lower) thresholds.

To understand the form of these states, we analyze the pair-correlation function.
The pair correlation function for the lowest four-body and 3+1 states are presented inFig. 8.3. The sizes of these three- and four-body states are of similar order to the rangeof the interaction; thus they are mainly nonuniversal. The mean interparticle distancein the lowest four-body state is 1.76r0 clearly showing that this has a size too small to
be a universal state. The lowest three-body state has a mean interparticle distance of
2.55r0 .
The second four-body state has an energy very close to the first three-body state.
Analyzing the pair correlation functions of these three- and four-body states we observethat the four-body state can be understood as a three-body Efimov state with the 4th
particle weakly bound to it. The four-body pair correlation function presents a peak

191
at around 2 r0 with a fairly long tail that extends out to approximately 10 r0 . The
small-r region of the four-body–pair-correlation function describes the 3-body Efimov
state, while the long tail describes the correlation between the three-body state and the
4th particle. The mean distance between the three-body state and the 4tth particle is∼ 8 r0 .

4πr2 P (r)/r0−1

0.5
0.4
0.3
0.2
0.1
0
0

5

10

15

20

r/r0
Figure 8.3: Lowest pair-correlation functions of the four-boson system at unitarity. The
black solid line corresponds to the lowest four-body state; the dashed red curve is thesecond four-body state, and the dash-dotted curve is the lowest 3+1 state. The lowest
3+1 state has a double-peak structure with a second peak around aho = 104 r0 that does
not appear in the figure.

The rest of the four-body states are more weakly bound and behave universally.
Since they are above the lowest three-body threshold, these four-body states are quasibound. There is one quasi-bound four-body state below each Efimov threshold. Thesestates are universal and follow the same scaling law of the three-body Efimov physics;
they are characterized by a scaling factor of eπ/s0 ≈ 22.7. The binding energy of thisseries of four-body states (Ei4b ) are proportional to the three-body states (Ei3b ) and4b ≈ 4.4E 3b for i > 1 at unitarity.
follows the relation Ei+1i

We have numerically shown that pair-correlation functions follow the same Efimovscaling law. Figure 8.4 compares the rescaled pair-correlation functions of the 3rd and192
4th four-body states at unitarity. Within the accuracy of the CG calculations, we notethat pair-correlation functions of four-body states follow the same scaling law as the

4πr2 P (r)/(r0−1 e−π(n−3)s0 )three-body pair-correlation functions of the Efimov states.
0.025
0.02
0.015
0.01
0.005
0
0

50

r/(r0 e100π(n−3)s0

150)

Figure 8.4: Rescaled pair-correlation functions of the 3rd and 4th four-body states. The
pair correlation for |Ψ3 i has not been rescaled, but the |Ψ4 i pair-correlation function isrescaled by eπ/s0 ≈ 22.7.

8.4

Analysis of the four-boson system in the hyperspherical representation
In this section, we analyze the four-boson system within the hyperspherical rep-resentation. Here, we consider an untrapped system. Using the correlated-Gaussian
hyperspherical method (CGHS), we obtain channel functions and hyperspherical potential curves.
Figure 8.6 presents the hyperspherical potential curves in the as region where thefirst three-body Efimov states appear. It is useful to consider the situation in the threebody system (depicted in Fig. 8.1) in order to understand the qualitative behavior of
the four-body potential curves. Figure 8.6 (a) corresponds to the case where the threebody system does not support a bound or quasi-bound state. Figure 8.6 (b) corresponds

193
to the case where the three-body system supports a quasi-bound state (schematicallydescribed by the dashed red line in Fig. 8.1). This quasi-bound state produces a series of
avoided crossings in the four-body potential curves that describe an effective potential
curve converging to the quasi-bound threshold. Finally, Fig. 8.6 (c) describes the casewhere the bound state lies below zero-energy threshold1 (schematically described by
the solid blue line in Fig. 8.1). The series of avoided crossings disappears, and the lowest
potential curve is now seen to converge to the three-body bound-state threshold.
−4

Uν (R) mr02 /h̄2

0.1

(a)

10

x 10

(b)

0.08

(c)

0.08

0.06

0.06

5

0.04

0.04

0.02

0.02

0

0
−0.02

0.1

0
50

R/r0

100 −5200 400 600

R/r0

−0.02

50

100

R/r0

Figure 8.5: Potential curves in the scattering-length region near a1f ∼ −6r0 , where thefirst Efimov state appears. (a) Potential curves at as > a1f [as ≈ −5.31] . The threebody bound state has not appear yet. (b) Potential curves at as & a1f [as ≈ −6.16].
For these scattering length values a quasi-bound state appears. (c) Potential curves at
as < a1f [as ≈ −7.29]. For these scattering length values the attraction increases andthe three-body state becomes bound.

In this region of scattering lengths, the lowest four-body potential curve supportsa bound state. In fact, as the magnitude of scattering length is being increased, twobound states are formed before the first three-body state appears. This finding is inagreement with the results obtained with the CG method (see Fig. 8.2). Therefore,
when the three-body state is right at threshold, the number of four-body states is finite.
In this sense, there is no “four-body” Efimov effect.
1

In the energy scale used here, the zero is selected at the four-body breakup threshold

194
The absence of a four-body Efimov effect can be understood by considering thequalitative differences between the three- and four-body systems. The three-body Efimov effect occurs when the weakly-bound dimer is right at threshold. The size of the
dimers is very large, with their wave functions decaying as 1/r, where r is the interparticle distance. These long-range correlations are responsible in part for the long-range
1/R2 behavior of the lowest hyperspherical potential curve that produces the Efimov
effect.
In contrast, in the four-body system, the size of the three-body state remains finite
at threshold. The potential barrier schematically presented in Fig. 8.1 constrains thestate to have a finite size. Therefore, for R much greater than the size of the trimer, thechannel function describes an atom-trimer two-body system that decays much fasterthan 1/R2 . This situation is qualitatively different than the three-body case in thatno attractive 1/R2 potential occurs. However, in an intermediate region, we observe
numerically that the potential takes the form
W0 (R) ≈A ln(R/as ) + B
.
2µR2

(8.2)

Here, A is a positive constant, and B is a negative constant such that W0 (R) is negativein that region. This behavior is observed for ranges from R ≈ 6r0 to R ≈ 36r0 for
the three-body first Efimov state and over the range from R ≈ 230r0 to R ≈ 1000r0
for the second Efimov state. In both cases, the long-range behavior changes after thisintermediate region, converging fast to the asymptotic threshold. Equation (8.2) is only
valid when the trimer energy is very close to threshold. Once the three-body Efimov
state becomes more tightly bound, this intermediate behavior [Eq. (8.2)] is no longer
valid.
As the magnitude of as increases, a second three-body bound state appears. In
this as region, the energy of the first three-body bound state is quite deep. Therefore,
the lowest potential curve has a deep threshold in comparison with the rest of the195potential curves. The remaining potential curves behave qualitatively as in Fig. 8.6.
The behavior of the second potential curve in this as region is qualitatively describedby the lowest potential curve of Fig. 8.6.
As in the lowest potential curve, the second potential curve supports two boundstates when the second three-body bound state is at the zero-energy threshold. The
second of the four-body bound states belonging to the second potential curve was not
identified in the CG calculation because of the limitations of the method. This second
bound state (included in Fig. 8.2) appears before the second Efimov state and becomesvery shallow as the scattering length approaches unitarity. In the adiabatic approximation this state disappears. However, a coupled-channel calculation suggests that thestate may reappear if these couplings are included. The binding energy of such a state
would be less than 1% lower than the three-body threshold. The existence of this statevery close to the trimer threshold should greatly influence the atom-trimer scattering
length.
The second potential curve in the as region is quantitatively different from thelowest potential curve of Fig. 8.6. The lowest potential curve of Fig. 8.6 is more attractive, and the bound states are comparatively deeper than the levels in of the second
potential curve. Furthermore, the second state in the second potential curve seems todisappear as as is tuned closer to unitarity while the second state continues to exist
at unitarity and in the ranges where 1/as > 0. These differences can be understoodby considering finite attractive interaction effects. The lowest potential curve has aminimum around R ≈ 2r0 where the effects of the two-body attractive interaction are
still visible. The second potential curve has a minimum around R ≈ 16r0 , and it is not
significantly affected by the two-body interaction. For this reason, we expect the secondand higher potential curves to be universal in the R À r0 . To verify this statement, we
now analyze the potential curves right at unitarity.
Figure 8.7 presents the lowest three potential curves rescaled by the length and196

2µR2 Wν (R) mr02 /h̄2

0

−2

−4

−6

−8 0
10

1

2

10

3

10

10

R/r0
Figure 8.6: Lowest potential curves when three-body states are at threshold. Black
curve correspond to the lowest potential curve at the scattering length value for which
the first Efimov state is at threshold. The red curve is the second potential curve at
the scattering length value for which the second potential Efimov state is at threshold.
Green dashed lines present the linear behavior of the curves (note that the potential
curves are multiplied by 2µR2 and that the x-axis is in logarithmic scale). The linear
behavior of the curves in logarithmic scale is evidence of Eq. (8.2).

Un (R)/e−2nπ/s0

0
−0.5
−1
−1.5
−2
−2.5
0

2

4

(R/r0 )/e

6

8

nπ/s0

Figure 8.7: Lowest three rescaled potential curves Uν are shown as functions of the
four-body hyperradius. The black curve corresponds to the lowest curve. The blue
curve is the second potential curve and the red curve is the third potential curve. The
agreement between the rescaled second and third potential curves is numerical evidence
for the universal behavior of the four-boson system, as is discussed.

197
energy scaling factors eπ/s0 and e−2π/s0 , respectively. We can clearly see that the lowest
potential curve (black symbols) is different from the second and third potential curves.
However, the second and third rescaled potential curves (blue and red symbols) agree
very well. Therefore, they are considered universal. This agreement strongly suggestthat the three-body scaling factor appears in the four body problem as an scaling factorof the potential curves. This universal behavior of the four-body system is consistent
with the speculations of Platter et. al [83]. Our results are also consistent with Platter
et. al prediction that no four-body parameter is needed to renormalize the four-bodysystem.
−3

4

x 10

(a)

Uν (R) mr02 /h̄2

2

0.01

(b)

0.005

0
0

−2

−0.005

−4
−6

−0.01

−8
−0.015

−10
−12
0

100

R/r0

200

−0.02
0

100

200

R/r0

Figure 8.8: Potential curves at positive scattering-length values. (a) Potential curves
at as ∼ 37r0 . (b) Potential curves at as ∼ 21r0 . Solid blue line corresponds to theatom-trimer threshold. Dashed-red line corresponds to the dimer-dimer threshold, anddashed-green line corresponds to the dimer–two-atom threshold.

Using the adiabatic hyperspherical approximation [Eq. (2.32)], we obtain thefour-body energies. Since each potential curve supports one bound state, there is afamily of quasi-bound four-body states which follow the same scaling relations of the
4b /E 4b =potential curves. Therefore, the energies of the four-body bound state obey En−1ne2π/s0 , which is the same relation between the energies of the three-body Efimov states.

198
Table 8.1: Lowest three- and four-body energies at unitarity. All the energies are in unitsof ~2 /(mr02 ). The three-body energies (left column) where calculated with CG without
a trapping potential. The N = 4 energies were calculated with CG (center column) and
CGHS in the adiabatic approximation (right column). The CG provides exact results
for true bound state (E13b , E23b , E33b , E14b , and E24b ) but it is only approximated for quasibound states (E34b and E44b ). The adiabatic approximation provides a rigorous upperbound of the exact energies. Therefore, the most reliable E34b and E44b are obtained with
the CGHS in the adiabatic approximation.
N = 3 (CG)E13b = −0.11922
3b
E2 = −0.2257 10−3
E33b = −0.438 10−6

N = 4 (CG)E14b = −0.7014
E24b = −0.1311
E34b = −0.99 10−3
E44b = −0.17 10−5

N = 4 (CGHS)
E14b = −0.69998
E24b = −0.1285
E34b = −0.1035 10−2
E44b = −0.205 10−5

However, the three-body system hyperspherical scenario is quite different since all thestates belong to the same hyperspherical potential curve. A comparison of the energies
4b ≈ 4.6E 3b . This relation isof the three- and four-body systems leads to the relation Ei+1islightly different than the one obtained with CG. Since both calculations are variational
and the adiabatic calculation predicts lower energies, we consider this second relation
more accurate. Table 8.1 summarizes CG and CGHS results.
Finally, we consider the positive scattering length region. In Fig. 8.2, we observed how different two- and three-body thresholds crossed each other as the scattering
length was tuned. Figure 8.8 shows the potential curves for two different scatteringlength values. The blue line corresponds to the energy of the second Efimov trimer(trimer+atom threshold), the dashed-red line corresponds to the dimer-dimer threshold, and the dashed-green line corresponds to the dimer threshold. There is a also deeper
potential curve that converges to the deep atom-trimer threshold that is not shown in
the figure. Figure 8.8 (a) presents the case where the trimer threshold is below the
dimer-dimer threshold. As the scattering length is decreased, the dimer-dimer threshold and the atom-trimer threshold cross each other. Figure 8.8 (b) shows the case where
the dimer-dimer threshold lies energetically below the trimer+atom threshold. In both

199
figures, a four-body bound state is contained in the attractive region of the lowest potential curve. If we continue decreasing the scattering length, the dimer+atom+atom andtrimer thresholds approach each other, and the atom-trimer potential curve smoothly
becomes a dimer+atom+atom potential curves.
The crossing of thresholds and formation or destruction of a four-body boundstate have important consequences in the collisional properties of the four-boson system.
These consequences will be explored in the next section.

8.5

Scattering properties
From the potential curves obtained in the previous section, we can analyze thescattering properties of the four-boson system. In particular, we focus on the dimerdimer and atom-trimer collisions.
Sharp features in ultracold collisional observables are expected when a bound or
quasi-bound state crosses a fragmentation threshold or when two different fragmentationthresholds coincide. By analyzing the two-, three- and four-body spectrum (Fig. 8.2),
we can deduce important scattering-length values in which sharp features would occur
in collisional observables. There is one value of the two-body scattering length where
the dimer energy equals one of the trimer energies. At this scattering-length value, we
expect a pole in the atom-dimer scattering length. Similarly, when a four-body-stateenergy equals twice the dimer energy, we expect a pole in the dimer-dimer zero energy
scattering length. Another sharp feature in the dimer-dimer scattering length should
occur when the trimer energy equals twice the dimer energy and the thresholds of theatom-trimer and dimer-dimer potential curves coincide.
We define atrim−d as the two-body scattering-length value where the trimer anddimer energies are equal. We also define atrim−dd as the two-body scattering-length value
where the trimer energy equals twice the dimer energy. Similarly, we define atet−dd as thetwo-body scattering-length value where the tetramer (four-body) energy equals twice

200
the dimer energy. These three scattering lengths depend on the three-body parameterthat controls three- and four-body state energies. Using the semi-analytical solutions
from the three-body system [27], it can be shown that the ratio atrim−d /atrim−dd isindependent of the three-body parameter. Similarly, since the four-body state has thesame universal properties of the three-body system, the ratios atrim−dd /atet−dd andatet−dd /atrim−d should also be independent of the three-body parameter.
These ratios can be used to identify these sharp features experimentally. If theposition of one of the scattering length values is known, then the other two can be
deduced. An observation of these three sharp features in the atom-dimer and dimerdimer collisions that obey the ratios predicted for atrim−d , atrim−dd ,and atet−dd would
be experimental verification that these features are a consequence of three and fourbody states. Such experimental verification of Efimov physics would be much simpler
than directly verifying the eπ/s0 ≈ 22.7 scaling law, which would require control of thescattering length over a couple orders of magnitude.
From an experimental point of view, it is also important to find atet−dd . In thevicinity of atet−dd , there is a pole in the dimer-dimer scattering length that can beused to tune dimer-dimer interactions. These bosonic dimers can have both positive
and negative scattering lengths. Therefore, the wide tunability of the dimer-dimer
interaction opens the door to studying new phenomena of bosonic dimers with attractive
interactions.
To obtain atrim−d , atrim−dd and atet−dd , we analyze the bound states associatedwith the third Efimov state. We select these three- and four-body states because they
are much larger than the range of the interaction and are expected to behave universallyat all scattering length values.
Our analysis of the two-, three- and four-body spectrum determines the following

201
values:atrim−dd ≈ 720r0 ,(8.3)

atrim−d ≈ 105r0 ,(8.4)

atet−dd ≈ 254r0 .

(8.5)

These scattering lengths lead to the ratios
atrim−dd≈ 2.84,atet−ddatet−dd≈ 2.41.
atrim−d(8.6)

To obtain atet−dd , we calculate the four-body energy using the hyperspherical adiabatic
approximation. These energies are more accurate than the ones obtained by the CG(see Sec. 8.3).
The atrim−dd and atrim−d only depend on two- and three-body physics. Therefore,
we can use all the machinery that has been built to obtain an independent verification of
the atrim−dd and atrim−d values. In particular, we use the semi-analytical formulas from
Ref. [27]. From the three-body energy of our third Efimov state at unitarity we extract
the three-body parameter that characterizes the system. With this single parameter,
we calculate atrim−dd and atrim−d to be
atrim−dd ≈ 720r0 ,(8.7)

atrim−d ≈ 107r0 .

(8.8)

We find very nice agreement, with corrections of less than 2% in atrim−d . This results
show that our numerical calculations behave universally to a good approximation. Also,
they give us confidence in our numerical procedure.
These scattering length ratios can be used to identify the sharp features but we
do not know the shape of these features. To directly observe the sharp features in the
dimer-dimer scattering length, we carry out scattering calculations using the hyperspherical representation and the R-matrix method [8] (these calculations were carried202
out by Jose P. D’Incao). The third Efimov state is quite large and scattering calculations would be very difficult. Instead we analyze the two-body scattering-length region
where the second Efimov and its four-body state crosses the dimer-dimer threshold.
These states are affected by finite range corrections. For example, the trimer energy
never crosses the dimer threshold 2 , and the ratio atrim−dd /atet−dd ≈ 3.47 instead of
2.84. Nevertheless, we consider that the behavior of the dimer-dimer scattering length
qualitatively describes universal behavior.
Figure 8.5 shows the real and imaginary part of the dimer-dimer scattering length
add as a function of the two-body scattering length. The real part of the scattering
length describes elastic scattering, while the imaginary part describes losses. The units
are selected such that r0 = 100(a.u.), which is close to the Cs van der Waals length. The
red lines and symbols correspond to the numerical results. The dashed vertical lines
correspond to atrim−dd and atet−dd . At atet−dd , we observe a pole in the real-part of
scattering length, and at atrim−dd we observe a discontinuity in the real part. For as >atrim−dd , the dimer-dimer scattering length converges to approximately add ≈ 1.3as .
However, it is not clear if this behavior is universal.
The imaginary part of the scattering length [right panel of Fig. 8.5] represents
inelastic scattering processes. The peak around atrim−dd ∼ 2900 a.u. describes theconversion of two dimers into trimer and an atom. The second peak around 800 a.u.
represents important losses close to the atet−dd .
Finally, we focus on atom-trimer collisions. Figure 8.10 presents the real andimaginary part of the atom-trimer scattering length. The main feature is the pole in
the real part of the scattering length at as ∼ 6r0 . The pole is due to the appearanceof a four-body state in the atom-trimer channel. This four-body state occurs at small
scattering-length values, and it is not universal. Also, a sharp feature occurs at atrim−dd2

According to the atrim−dd value and the universal ratios [Eq. 8.6], atrim−d should be approximately
4r0 . Clearly, scattering lengths around this value are not large and the behavior of the system is notuniversal.

203

50

20000

40

18000
16000

30

14000
20
12000
10
10000
0
8000
-10
6000
-20

4000

-30

2000

-40

0
0

1000

2000

3000

4000

5000

6000

0

1000

2000

3000

4000

5000

6000

Figure 8.9: Real and imaginary parts of the dimer-trimer scattering length. Left andright panels present the real and imaginary parts of the dimer-trimer scattering length:
ardd and aidd , respectively. The vertical dashed lines correspond to atrim−dd and atet−dd .
Figure courtesy of Jose P. D’Incao.

204
where an atom and a trimer can decay into two dimers. An understanding of this feature
will require a deeper analysis of the atom-trimer collisional properties.
300

300000200

2500001002000000150000-100

100000

-200

50000

-300

0
0

1000

2000

3000

4000

5000

6000

0

1000

2000

3000

4000

5000

6000

Figure 8.10: Real and imaginary parts of the atom-trimer scattering length. Left andright panels present the real and imaginary parts of the atom-trimer scattering length:arat and aiat , respectively. The vertical dashed lines correspond to atrim−dd and atet−dd .
Figure courtesy of Jose P. D’Incao.

8.6

Conclusions
We have presented a preliminary analysis of the four-boson problem using the CG

method and the hyperspherical representation. The system at unitarity is described bya set of potential curves that are related via the Efimov scaling factor. Each of thesepotential curves contains a four-body state that follows the same scaling laws of thethree-body system.
At positive scattering lengths, Efimov physics can be probed with dimer-dimer
collisions. The four-body states produce poles in the scattering length that can be used
to tune dimer-dimer interactions. Also, for scattering lengths slightly above atrim−dd ,dimer-dimer collisions can be used to obtain a trimer-atom mixture without significant
losses.

Chapter 9
Summary and Outlook

The first objective of this dissertation was to develop a renormalized interactionthat would eliminate the artificial instabilities that produce the Fermi pseudopotential
in mean-field theories. The density renormalization obtained from two-particle solutions
succeeds in this objective. It provides a simple model that both reproduces the Hartree
term in the weakly attractive Fermi gas system and the expected functional form in thestrongly interacting unitarity regime. This model has proven to be useful in HartreeFock, Thomas-Fermi, and hyperspherical calculations. However, the description of thesystem is limited. Pair correlations are not included, and the excitation gap is not
reproduced. Also, the energy of the system at unitarity is slightly high in comparison
with more sophisticated calculations such as fixed-node diffusion Monte Carlo (FNDMC). In addition, the model presented here cannot be extended to Bose-Binsteincondensation (BEC) because it does not incorporate the necessary pair correlations to
describe molecule formation. Incorporating pair correlations would be the logical next
step but this would require a completely new approach, one closer to standard BCS
theory.
The second and primary objective of this dissertation was to obtain an appropriate
solution to the four-body fermionic system in the BCS-BEC crossover. To this end, wedeveloped and implemented a variant of the correlated-Gaussian (CG) method. The
solutions obtained with CG accurately describe four trapped fermions. These solutions,

206
extended to N = 2 and 3, are described in chapter 5. They were first used to analyze
the behavior of the spectrum as the interaction was tuned in the BCS-BEC crossover.
Later, we solved the dynamics of few-body systems as the scattering length was tuned
to produce different types of ramping schemes.
A nice extension to the calculations presented in chapter 5 would be the generalization to arbitrary angular momentum. Even though few-body solutions with L = 1
were obtained in chapters 6 and 7, the method used to obtain these solutions is limited and would not be appropriate for calculating the complete spectrum for arbitrary
angular momentum. There are ways to extend the CG method to treat systems with
arbitrary angular momentum [173]. Such extensions of the CG method could be usefulin obtaining few-body solutions with arbitrary angular momentum.
The calculation of the spectrum and dynamics of a few-body system with different
angular momenta would allow the study of temperature effects. At a finite temperature, states with any angular momentum can be populated. Therefore, to describe thedynamics of systems at any temperature, we need to consider all angular momentum
cases.
In chapters 6 and 7, we quantitatively analyzed the spectrum and the structuralproperties of such a system. We extracted the dimer-dimer scattering length and effective range for different mass ratios. We verified universal properties of unitarity systems.
We also showed that no weakly bound state exists for two-component Fermi systems
with L ≤ 6. These calculations were extended to larger systems using FN-DMC, whichconnected few- and many-body perspectives. All solutions obtained with CG and FNDMC were restricted to describing bound systems. This restriction allowed us to extract
collisional properties only in an indirect way, as was done in chapter 6. The correlated
Gaussian hyperspherical (CGHS) method introduced a more suitable way of analyzing
collisional properties.
The development of the CGHS method was a major numerical accomplishment

207of this dissertation. This method opens up the possibility of studying the continuum
and collisional properties of four-body systems using hyperspherical coordinates. Furthermore, it provides results more accurate than the standard CG method. We present
examples of four-body fermionic potential curves obtained with CGHS in chapters 5, 6,
and 7. In chapter 8, we also solve the four-boson system using CGHS. In addition toits great success, this method can be improved and extended.
The creation of the CGHS basis set can be improved upon by using the CG
solutions of subsystems. These CG solutions can be selected to construct a particularchannel function. For example, if we want to describe an atom-trimer channel, we canobtain the trimer state with CG and use it to construct the basis functions of the atomtrimer form. With a few of those solutions, we might be able to accurately describe
the atom-trimer channel. Similarly, the dimer-atom-atom and dimer-dimer channels
can be generated using the CG two-body solutions. These solutions would have several
advantages. By construction, they would describe correctly the asymptotic behavior of
the channels. And, we can introduce the channels in a controllable way, identifying theimportance of each channel and obtaining a deeper understanding of the system.
Another major improvement would be the extension of CGHS method to describe
L > 0 solutions. We might be able to use the ideas applied to standard CG to extend
CGHS calculations to L > 0.
In chapter 8, we used both CG and CGHS methods to solve the four-boson system with tunable interactions. We showed that potential curves and energies follow
Efimov-scaling relationships. We found that three- and four-body energies at unitarity also follow a scaling relationship. We then analyzed the effect of these four-bodystates in atom-dimer and dimer-dimer collisional properties. Finally, we showed that
abrupt changes in the atom-dimer and dimer-dimer scattering properties occur at twobody scattering-length values whose ratios are independent of the three-body parameter. These scattering ratios can be useful in experimentally identifying poles in the208
dimer-dimer scattering length.
The four-boson scaling relationships suggest that the system behaves universally.
To verify universality, we could introduce a different short-range potential. If the system’s behavior follows the same scaling relationships, then we would have demonstrated
universality in the four-boson system.
Other studies would complement the results from chapter 8. The analysis of
the collisional properties for as < 0 has not been carried out yet. We expect thatsharp features in the four-body recombination would appear at two-body scatteringlength values where the four-body states are at zero-energy threshold. The ratio ofthese scattering-length values would be independent of the three-body parameter. In
addition, we could analyze atom-trimer collisions at unitarity to better understand the
expected lifetime of atom-trimer mixtures.
Finally, it would be interesting to analyze collisional properties of other four-bodysystems. In particular, we could use the CGHS method to study a Bose-Fermi mixture.
In this system, Efimov states can be formed with two bosons and one fermion. Therefore,
we expect that Efimov physics would have important consequences in a Bose-Fermi
mixture. Boson-trimer and fermion-trimer collisions could be analyzed by considering
three bosons and a fermion or a two-boson-two-fermion system. Also, fermionic dimers
could be formed with one boson and one fermion. The analysis of dimer-dimer collisions
would require the extension of CGHS method to treat L = 1 systems.

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Appendix A
Symmetrization of the basis functions and evaluation of the matrix
elements

The CG basis functions take the form¾½
1 T
ΦA (x1 , x2 , ..., xN ) = S exp(− x .A.x) .
2

(A.1)

The symmetrization operator S can be expanded in a set of simple particle permutations,|S(A)i =Np
X

sgn(Pi ) |Pi (A)i .

(A.2)

i=1

Here, Np is the number of permutations that characterize the symmetry S. Each ofthese permutations, Pi , has a sign associated, sgn(Pi ), and is a given rearrangement of
the spatial coordinates
Pi (ΦA (x1 , ..., xN ) = ΦA (xPi (1) , ..., xPi (N ) )(A.3)

The label i characterizes the rearrangement. This rearrangement of the spatial coordinates is equivalent to a rearrangement of the interparticle widths {dij } (or the {αij }),Pk ({dij }) = {dPk (ij) }.

(A.4)

Therefore, permutation operations can be easily applied and become transformations of
the matrix A.
In general the evaluation of the symmetrized matrix elements of an operator O
is,hS(A)|O|S(B)i =

Np Np
X
X
i=1 i0 =1

sgn(Pi )sgn(Pi0 ) hPi0 (A)|O|Pi (A)i ,

(A.5)

223
which implies an Np2 evaluation of unsymmetrized matrix elements. Fortunately, if
S(O) = O then,hS(A)|O|S(B)i = Np hA|O|S(B)i = Np hS(A)|O|Bi .

(A.6)

This property significantly reduces the numerical demands since the left hand side ofEq.(A.6) implies Np2 permutations, while the right-hand side only implies Np permutations. All operators of the Hamiltonian are invariant under the S, operator and theirmatrix elements obey Eq.(A.6).
To obtain density profiles and pair-correlation functions, we use the delta function
operator. A single delta function operator in a given coordinate is not invariant under
this transformation; for this reason, the computational evaluation is more expensive.
Alternatively, we can create a similar operator as a sum of delta functions. If the sumof delta functions reflects the symmetry of the problem, the this new operator would beinvariant under S.
The permutation operator clearly depends on the problem under consideration.
In this work, we consider bosons and two-component fermions. For identical bosons andfermions,S=

Np
X

αi Pi ,

(A.7)

i=1

where Np = N ! and αi = 1 for bosons and αi = (−1)p ; p = 0, 1 is the parity of theoperator Pi . For two-component systems (boson-boson, fermion-fermion, or a BoseFermi mixture),S=

Np1 Np2XX

αi1 αi2 Pi1 Pi2 ,(A.8)

i1 =1 i2 =1

where Np1 = N1 !, Np2 = N2 !, and N1 and N2 are the number of particles in component
1 and 2, respectively.
The symmetrization operation, if it involves a permutation with a negative sign,
can significantly reduce the accuracy of matrix elements. In certain cases, the unsymmetrized matrix elements can be almost identical. Because of the negative sign224
of the permutation, the symmetrized matrix elements can become a subtraction ofvery similar numbers.

Therefore, accuracy is reduced.

These basis functions areusually unphysical, so it is convenient to eliminate them. To do this, we evaluate| hS(A)|S(A)i |/ max(| hPi (A)|Pi (A)i |). If this is a small number, then the accuracy ofthe matrix elements is reduced. So, in general, we introduce a tolerance of the order of
10−3 to determine whether to keep or discard the basis functions.

Appendix B
Evaluation of unsymmetrized basis functions

For convenience, we introduce the following simplify notation,
1|Ai = exp(− xT .A.x).
2

(B.1)

As a simple example, consider the overlap matrix element
Z
hA|Bi =

1
dx1 ..dxN exp(− xT .(A + B).x).
2

(B.2)

Since the matrix A + B is real and symmetric, there exists a set of eigenvectors y ={y1 , ..., yN } with eigenvalues {β1 , ..., βN } that diagonalize the matrix. In this set ofcoordinates, Eq. (B.2) takes the simple form,NZ ∞hA|Bi = (4π)

02dy1 y12 e−β1 y1 /2 ...

Z ∞
0

µ
2 /2
2 −βN yNdyN yNe=(2π)Ndet(A + B)

¶3/2
. (B.3)Here, we used the product β1 .β2 ...βN = det(A + B). These basics steps can be followed
to evaluate the remaining matrix elements.
To evaluate the kinetic energy, we use the following property,
hA| −

~2 2
~2
∇xi |Bi =h∇xi A|∇xi Bi .
2m
2m

(B.4)

This property can be simply proven by applying an integration by parts. Also, it
simplifies the matrix element evaluation and provides an expression which is symmetricin A and B. Then, the matrix element takes the form,
N

hA| −

~2 X 2~2∇xi |Bi =3 Tr((A + B)−1 A.B) hA|Bi
2m
2m
i

(B.5)

226
Another important matrix element, which is similar to Eq.( B.5), is
hA|xT Cx|Bi =~23 Tr((A + B)−1 C) hA|Bi .
2m

(B.6)

Here, C is an arbitrary matrix. This matrix element is used to calculate the trapping
potential energy. In such case, C = mω 2 I/2, where I is the identity matrix.
Finally, we calculate the matrix element for a two-body central force:Z
hA|V (ri − rj )|Bi =d3 rV (r) hA|δ(bTij x − r)|Bi = Gcij [V ] hA|Bi ,(B.7)

where ri − rj = bTij x, c1ij = bTij (A + B)−1 bij , and Gc [V ] is the Gaussian transform of thepotential
Gc [V ] =³ c ´3/2 Z
2π2d3 rV (r)e−cr /2 .

These matrix elements are enough to describe few-body systems.

(B.8)

Appendix C
Jacobi vectors and CG matrices

In this Appendix, we present the construction of the matrices that characterize thebasis functions in terms of the widths dij . In the following r = {r1 , ..., rN } correspondto Cartesian coordinates, while ρ = {ρ1 , ..., ρN −1 } correspond to mass-scaled Jacobi
coordinates. First, consider the basis function with the center of mass included
X (r1 − r2 )2 = exp(− 1 rT .A.r).
|Ai = Ψ0 (RCM ) exp −
2
2d2ij

(C.1)

j≥i

In the equal-mass case for N particles, it is more convenient to simply use Cartesian
coordinates. The ground-state–center-of-mass wave function of particles in a harmonic2

2

trap takes, conveniently, a Gaussian form Ψ0 (RCM ) = e−N RCM /2aho . Thus, Ψ0 (RCM )T

can be written as Ψ0 (RCM ) = e−r .M

CM .r/2, where M CM is the center-of-mass matrix

CM = 1/(N a2 ) for all k and l. Then, for each interparticle
whose matrix elements are Mklho
2 = rT .M (ij) .r. The matrix elements
distance rij , there exists a matrix M (ij) so that rij(ij)

of the M (ij) matrices are Mii

(ij)

= Mjj

(ij)

= 1, Mij

(ij)

= Mji

= −1; the rest are zero,

yielding
A = M CM +X 1d2
j≥i ijM (ij) .

(C.2)

In some cases it is important to include the center-of-mass motion. For example,
this allows one to extract single-particle observables such as density profiles.
If the center of mass is not included, then Eq. (C.2) can be written as,
A=

X 1
d2
j>i ijM (ij) .

(C.3)

228
In the next subsections we present the mass-scaled Jacobi vectors (see Fig. C.1)
and the corresponding form of the matrices M (ij) .

Figure C.1: Mass-scaled Jacobi vector for three and four particles.

C.1

Mass-scaled Jacobi vectors for three particles
A three-particle system can be described by the set of vectors [52]
ρ1 = (r1 − r2 )/d,¶µm1 r1 + m2 r2,
ρ2 = d r3 −m1 + m2
m1 r1 + m2 r2 + m3 r3
RCM =,m1 + m2 + m3

(C.4)
(C.5)
(C.6)

where
µ

¶1/2
m1 m2 m3,
m1 + m2 + m3
µ
¶1/2
r
m3
m1 + m2d=.
µ m1 + m2 + m3
µ=

(C.7)
(C.8)

In Eq. (C.4) ρ1 and ρ2 are mass-scaled Jacobi vectors and RCM is the center-of-mass
vector.
All the interparticle vectors can be written in terms of the Jacobi vectors ρ1 andρ2 :r2 − r1 = d ρ1 ,dm2
1
ρ1 + ρ2 ,m1 + m2ddm2
1
r3 − r2 = −ρ1 + ρ2 .
m1 + m2dr3 − r1 =(C.9)(C.10)(C.11)

229
As is presented in the next subsection, all the M (ij) matrices can be obtained
from Eqs. (C.9,C.10,C.11).

C.2

Mass-scaled Jacobi vectors for four particles
To present the mass-scaled Jacobi vectors for four particles we introduce thefollowing definitions,µ¶1/3
m1 m2 m3 m4µ=,
m1 + m2 + m3 + m4
m1 m2
µ1 =
,
m1 + m2
m3 m4
µ2 =
,
m3 + m4(m1 + m2 )(m3 + m4 )µ3 =,m1 + m2 + m3 + m4pppd1 = µ1 /µ, d2 = µ2 /µ, d3 = µ3 /µ.

(C.12)(C.13)(C.14)(C.15)(C.16)

Here, the value of µ is just selected by convention. Actually, µ can be left as an arbitrary
mass factor.
This is usually called the H tree for the form of the Jacobi vectors,
ρ1 = d1 (r1 − r2 )µ

ρ2 = d2 (r3 − r4 )m1 r1 + m2 r2 m3 r3 + m4 r4−
m1 + m2
m3 + m4
m1 r1 + m2 r2 + m3 r3 + m4 r4
RCM =m1 + m2 + m3 + m4

ρ 3 = d3

(C.17)(C.18)¶(C.19)(C.20)

Here, ρ1 , ρ2 and ρ3 are the mass-scaled Jacobi vectors and RCM is the center-of-mass
vector
The interparticle distances can be written in terms of transformations of the three

230
Jacobi vectors:
r1 − r2 = ρ1 /d1 ,
µ
¶
µ2 d3
µ3
µ1 d3
r1 − r3 =
ρ1 −
ρ2 ,
ρ3 +
µ
m1 d 1
m3 d2
¶
r µ
µ3
µ1 d3
µ2 d3
ρ3 +
ρ +
ρ ,
r1 − r4 =µm1 d1 1 m4 d2 2
¶
r µ
µ2 d3
µ1 d3
µ3
ρ +
ρ ,
r2 − r3 =ρ3 −µm2 d1 1 m3 d2 2
¶
r µ
µ1 d3
µ2 d3
µ3
r2 − r4 =
ρ3 −
ρ +
ρ ,
µ
m2 d1 1 m4 d2 2

(C.21)

r3 − r4 = ρ2 /d2 .

(C.26)

r(C.22)(C.23)(C.24)(C.25)For both the N = 3 and N = 4 systems, the interparticle distances can be written
in terms of the Jacobi vectors
ri − rj =X (ij)
ck ρk .

(C.27)k

Now we can write the matrices M (ij) in these Jacobi vectors that describe an interpar(ij)ticle distance. The matrix elements of these matrices are simply Mkl(ij) (ij)= ck cl.

Appendix D
Selection of the basis set

There are different strategies for selecting a basis set. If the numbers of dimensionsof the system we are studying is not that large, then we can try to generate a large
basis set that is complete enough to describe several eigenstates at different interaction
strengths.
The Gaussian widths dij are selected randomly and cover a range of values fromd0 to the trap length aho . Specifically, the dij are selected randomly using a Gaussian
distribution of range 1 and then scaled to three different distances: d0 , an intermediate√
distance d0 aho , and aho . These three distances are fixed once the interparticle potential
range d0 is fixed.
The basis set selection depends on the correlation we want to describe. So, theselection process changes depending whether the particles are bosons or fermions. For
fermions, when there is no trimer formation, basis functions with more than two particles
close together are not important.
For example, the algorithm for the selection of the basis functions for a twocomponent four-fermion system divides the basis into three parts: the first subbasis
generates dij , which are all of the order of aho ; they are useful for describing weakly√interacting states. The second subbasis generates two dij of the order of d0 or d0 aho
and the rest of the order of aho ; they are useful to describe dimer-dimer states. The√
third subbasis has one dij of the order of d0 or d0 aho and the rest of the order of aho .

232
They are useful to describe dimer–two-free-atom states.

Appendix E
Controlling Linear dependence

The CG basis set is over complete. This, and the fact that the basis functions
are chosen semi-randomly can introduce linear dependence problems in the basis set.
There are several ways to control or eliminate these linear dependence of the basis set.
The first way is to simply eliminate the linear dependence. After wselecting thebasis set and evaluating the matrix elements, we reduce the linear dependent basis set
to a smaller basis set without linear dependence.
To do this, we first diagonalize the overlap matrix and then eliminate the eigenstates with negative or low eigenvalues. The remaining eigenstates form an orthonormal
basis set. Finally, we transform the Hamiltonian to the new orthonormal basis set.
The threshold for the elimination can be selected automatically taking into account the lowest eigenvalue. If the lowest eigenvalue O1 is small and positive, thetolerance can be selected as, for example, 103 O1 . If O1 is negative and the magnitudeis large, then the basis set has a lot of linear dependence, and it is more convenient tochange the initial basis set.

Appendix F
Stochastical variational method

The SVM has been developed in the context of nuclear physics to solve few-bodyproblems [178, 179, 180]. It allows a systematically improvement of the basis set. A
detailed discussion of the implementation of the SVM will not be presented here but
can be found in Refs. [161, 173]. In the following, we present the main concepts of the
SVM.
The SVM is based on the variational nature of the spectrum obtained by a basis
set expansion. Consider a basis set of size D with eigenvalues {²1 , ..., ²D }, if we add a newbasis function then the new eigenvalues {λ1 , ..., λD+1 } obey λ1 ≤ ²1 ≤ λ2 ...²D ≤ λD+1 .
Here, we assume that both set of eigenvalues are arranged in increasing order. Thus, byadding a new basis, all the D eigenvalues should decrease or remain the same. Therefore,
the lower the new eigenvalues are the better the improvement of the basis set. Thus we
can test the utility of the added basis function by considering the improvement in the
eigenvalues.
In most cases, we are not interested in improving the complete spectrum. To
select which states or energies we want to improve, we can construct an appropriate
minimization function. This function would depend only on the energies we want toimprove and is minimized by the SVM.
In order to optimize the basis set, the SVM utilizes a trial an error procedure.
Starting from an initial basis set of size D, several basis functions are selected stochas-

235
tically and added, one at a time, to the basis set. For each D + 1 basis set, the new
eigenvalues are evaluated. The basis function that produces the best improvement of theselected energies is kept while the remaining basis functions are discarded. The initialbasis function is then increased by one and the trial an error procedure is repeated.
If this procedure is continued indefinitely the size of the basis set has become large
and the calculations become forbiddingly slow. Therefore, it is convenient to increase
the basis up to a reasonable size and then continue the optimization process without
increasing the basis size. This optimization can be carried out by a refinement process.
Instead of adding a new basis function, we test the importance of the basis functions of
the basis set. The trial and error procedure is then applied to each of the functions of
the basis set.
For the SVM procedure to be efficient, the evaluation of both the matrix elements
and the eigenvalues need to be fast and accurate. It is particularly important to obtainvery accurate matrix elements because the improvement due to a single basis function
is usually very small and can only be evaluated reliably if the matrix element are very
accurate. The matrix element evaluation in the CG and CGHS is both fast and accurate
making these methods particularly suitable for SVM optimization.
Also, the evaluation of the eigenvalues can be significantly speeded up in thetrial and error procedure. The basis functions are added or replaced one by one whichallowing us to reduce the evaluation of the eigenvalues to a root finding procedure. This
root finding procedure is much faster than any diagonalization procedure.
The SVM automatically takes care of the selection of the basis function. Also, it
tries to avoid linear dependence in the basis set by constraining the normalized overlap√
between any two basis function, i.e., O12 / O11 O22 , to be below some tolerance Omax .
The tolerance Omax is usually selected between 0.95 and 0.99. For example, the size ofthe basis set of N = 3 and 4 can be increased up to 700 and 8000 respectively without
introducing significant linear dependence.

Appendix G
CGHS unsymmetrized matrix elements for four particles

The same strategy for evaluating the matrix elements in the N = 3 case can beextended to the N = 4 system. The N = 4 system can be described by three Jacobi
vectors, x ≡ {x1 , x2 , x3 }, once the center-of-mass motion is decoupled.
To evaluate the overlap matrix element, we change to the coordinate basis set
that diagonalizes A + B. We call β1 , β2 and β3 the eigenvalues and y ≡ {y1 , y2 , y3 } are
the eigenvectors. In this new coordinate basis set the overlap integrand takes the formµ¶
β1 y12 + β2 y22 + β3 y32
B.A = exp −
.
2

(G.1)

We integrate over the polar angles of y1 , y2 , y3 , obtaining
µ
¶
Z
¯
¯
β1 y12 + β2 y22 + β3 y32
(4π)3
¯
¯
2 2 2
hB|Ai ¯ =
exp
−
y
y
y
dy
dy
dy
¯ .
1
2
3
1 2 3
R8
2
R
R

(G.2)

To integrate over the remaining hyperangles, we fix the hyperradius such that y1 =R sin θ cos φ, y2 = R sin θ sin(φ) and y3 = R cos θ; yieldingµ¶Z¯R2 (β1 sin2 θ cos2 φ + β2 sin2 θ sin2 φ + β3 cos2 θ)
¯
3
exp −
hB|Ai ¯ = (4π)
2Rsin5 θ cos2 θ cos2 φ sin2 φdθdφ. (G.3)
The integration over one of the angles can be carried out analytically. Introducing a237
variable dummy y, the overap matrix element takes the form¯¯
hB|Ai ¯ =R(4π)3 π22R (β1 − β2 )µ¶R2exp − [(β1 + β2 )(1 − y 2 ) + 2β3 y 2 ]
4
0
·
¸
2
2 (β1 − β2 )(1 − y )I1 R
y 2 (1 − y 2 )dy. (G.4)
4

Z 1

Next we calculate the angular kinetic energy. To do this, we calculate the total
kinetic energy and the hyperradial kinetic energy. These two matrix elements are notsymmetric, but the angular kinetic energy, i.e., the total kinetic energy minus the hyperradial kinetic energy, is symmetric. To obtain an explicitly symmetric operator, wesymmetrize the operation hB|TΩ |Ai |R = (hB|TT − TR |Ai |R + hA|TT − TR |Bi |R )/2 andobtain¶µZ¯¯β1 y12 + β2 y22 + β3 y32
(4π)3
¯
¯
2 2 2
T
AF
(y
,
y
,
y
)y
y
y
dy
dy
dy
hB|TΩ |Ai ¯ =
exp
−
¯ ,
1
2
3
1
2
3
1
2
3
8
R
2
R
R
(G.5)
where
1
T AF (y1 , y2 , y3 ) =
2

( 3 ·
X
i=1

µ
¶ ¸
dβi
2
−3βi + βi − 2(A.B)ii + 2 yi2
R

!2
Ã 3

X βi y 2
(~
y
.A.~
y
)(~
y
.B.~
y
)
i
+
−
. (G.6)

R2
R2
i=1

It is easy to show that (A.B)ii =P3

j=1 aij bij since A and B are symmetric matrices.

Here the bar sign indicates the integration over the angular degrees of freedom of y1 , y2 ,and y3 . We then divide the total result by (4π)3 . Making these integrations analytically
we obtain(~y .A.~y )(~y .B.~y ) =

3
X
i=1

aii bii yi4 +

¶
3 µ
X
4
aii bjj + bii ajj + aij bij yi2 yj2 .
3

(G.7)

i>j

Using spherical coordinates for y1 , y2 , and y3 . We observe that one of the integrations in
Eq. (G.5) can be done analytically. To simplify the notation, we introduce the following

238
variables,
1
xx = R2 (β1 − β2 )(−1 + y 2 ),
4

(G.8)

c1 = −9(β1 − β2 )2 − 12(−4a12 b12 + 3(a11 − a22 )(b11 − b22 )),(G.9)

c2 = −6a13 b13 + 6a23 b23 ,(G.10)

c3 = (3β12 − 3β22 − 6β1 β3 + 6β2 β3 − 6a11 b11 + 6a33 b11+8a13 b13 + 6a22 b22 − 6a33 b22 − 8a23 b23 + 6a11 b33 − 6a22 b33 ),(G.11)

c4 = (3β22 − 6β2 β3 ),(G.12)

c5 = (3β12 − 6β1 β3 ),(G.13)

c6 = 3(2a12 b12 + a13 b13 + a23 b23 ),(G.14)

c7 = 6a12 b12 ,(G.15)

c8 = 3(a13 b13 + a23 b23 ),(G.16)

c9 = (3β32 − 3a11 b11 + 3a33 b11 + a13 b13 − 3a22 b22 +3a33 b22 + a23 b23 + 3(a11 + a22 − 2a33 )b33 ).

(G.17)

The final expression for TΩ is
"
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¢#
Z 1
xx (β1 + β2 )(1 − y 2 ) − 2β3 y 2512π 4
1TΩ = −exp −3(β1 − β2 )5 R8 0 (−1 + y 2 )3(β1 − β2 )(−1 + y 2 )nhi
xx2 y 2 2(−1 + y 2 ) c1 (1 − y 2 ) + 4xx(c2 + c3 y 2 ) Ic [xx]
h
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¢
¡
¢
+ xx c4 (−1 + y 2 ) 3 − (9 − 4xx)y 2 + c5 (−1 + y 2 ) − 3 + (9 + 4xx)y 2o¡¢i+8xx c6 − c7 y 2 + c8 y 4 − c9 (y 2 − y 4 ) I1 (xx) dy, (G.18)
where we definedIc [xx] =xx
I0 [xx] − I1 [xx].
2(G.19)

We still need to evaluate the matrix elements corresponding to the couplings¯d¯|Ai ¯ ,PBA (R) = hB|dRR¯dB dA ¯QBA (R) = h|i¯ .
dR dR R

(G.20)(G.21)

239
Using the standard tricks, we obtainZ(4π)3
β1 y12 + β2 y22 + β3 y32
PBA (R) = − 9exp(−)~y .D.~y y12 y22 y32 dy1 dy2 dy3(G.22)R2Z
β1 y12 + β2 y22 + β3 y32
(4π)3
exp(−
)((~y .K.~y )(~y .D.~y ))y12 y22 y32 dy1 dy2 dy3
QBA (R) =R10
2(G.23)
where D = T T .A.T and K = T T .B.T and~y .D.~y = d11 y12 + d22 y22 + d33 y32 ,
¶
3
3 µ
X
X
4
4
(~y .K.~y )(~y .D.~y ) =kii dii yi +kii djj + dii kjj + dij kij yi2 yj2 .
3i=1(G.24)(G.25)

i>j

Equations (G.22, G.23) can be written in spherical coordinates, and one of the integrations can be done analytically leading to equations similar to Eq. (G.18).