Feshbach Resonances in Ultracold Bose-Fermi
Mixturesby
Daniele Carlo Enrico Bortolotti
B.S., Western Michigan University, 2001
M.S., University of Colorado, 2004

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
2007

This thesis entitled:
Feshbach Resonances in Ultracold Bose-Fermi Mixtures
written by Daniele Carlo Enrico Bortolotti
has been approved for the Department of Physics

John L. Bohn

Murray Holland

Date

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

iii
Bortolotti, Daniele Carlo Enrico (Ph.D., Physics)
Feshbach Resonances in Ultracold Bose-Fermi Mixtures
Thesis directed by Prof. John L. Bohn

In the wake of successful experiments in Fermi condensates, experimental attention is broadening to include resonant interactions in degenerate BoseFermi mixtures. In this thesis we wish to study the equilibrium properties of thefermionic molecules that can be created in such a mixture.
To this end, we first discuss the two body properties of the system, andintroduce the model Hamiltonian we use to describe the resonant physics, highlighting its virtues, as well as its limitations. We then proceed by analyzing themean field solution of this model, by studying both the equilibrium problem, andthe non-equilibrium equations of motion, thus developing a powerful language todiscuss the system. We then highlight the limitations of the mean-field approach,and develop a numerically tractable generalized version of this theory, which isable to correctly describe the two-body properties of the system in the low-density
limit.
Finally, we study the properties of the system using this generalized meanfield theory, by first analyzing the two-body scattering matrix in the many-bodyenvironment, assessing its complex poles in order to understand the stability properties of the Feshbach molecules in the gas. Secondly we solve the equilibrium
equations self-consistently, to study the molecular populations and density distributions at equilibrium, as a function of external bias magnetic field.

iv

Acknowledgements

First and foremost a huge thanks to John Bohn, who was the greatest advisor
anyone can hope for. Aside from teaching me to be a physicist, he has allowed me
to pursue my research interests freely, but always keeping me honest about myresults. He has also given me a lot of trust and understanding, which helped me
work through of some personal challenges throughout the years.
I also wanted to thank Alexandr Avdeenkov, for teaching me most of what
I know about many-body physics, and for his patience with many of my trivial
questions. I would also like to thank Shai Ronen, Murray Holland and Victor
Gurarie for the substantial amount of time they spent helping me. A big thanks
to Massimo Inguscio and the LENS cold atoms groups, for letting me spend time
there, and for their help throughout these years.
On a more personal note, I would also like to acknowledge the moral supportof my good friends Stefano Tonzani and Josh Dunn, who made coffee at JILA amuch more pleasant experience, and for their physical insights and suggestions,and Ed Meyer, for helping me release some steam by means of the most terrifyingof Horror movies.
Finally I must thank my wife, Khouloud Manai, and my daughters Sara
and Nur, for inspiring me and motivating me, and for putting up with me in theharder times. My parents and my brother for their help, and for making me close
to home in spite of the many miles.
Support for this work came from the DOE and The Keck foundation.

v

Contents

Chapter
1 Introduction

1

2 Scattering Theory: The Starting Point5

2.1

Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Low Energies Collisions and Resonances . . . . . . . . . . . . . .

7

2.2.1

Effective Range Expansion . . . . . . . . . . . . . . . . . .

8

2.2.2

Resonances at Low energy . . . . . . . . . . . . . . . . . .

10Types of Resonances . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.3.1

Potential Resonances . . . . . . . . . . . . . . . . . . . . .

13

2.3.2

Shape resonances . . . . . . . . . . . . . . . . . . . . . . .

15

2.3.3

Feshbach Resonances . . . . . . . . . . . . . . . . . . . . .

15

Magnetic Feshbach Resonances in Alkali Gases . . . . . . . . . . .

16

2.4.1

1 ⊗ 1-Particle Physics . . . . . . . . . . . . . . . . . . . . .

19

2.4.2

The Interparticle Potential . . . . . . . . . . . . . . . . . .

21

2.3

2.4

3 Two-Body Physics: The Model

26

3.1

The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.2

Two Body Scattering Parameters . . . . . . . . . . . . . . . . . .

28

3.3

Solution of the Model in the Two-Body Limit . . . . . . . . . . .

30

3.3.1

34

Poles of the T-Matrix: The Two-body case . . . . . . . . .

vi
4 Mean-Field Theory: The Language
4.1

4.2

4.3

Equilibrium Approach . . . . . . . . . . . . . . . . . . . . . . . .

40

4.1.1

The Formalism . . . . . . . . . . . . . . . . . . . . . . . .

40

4.1.2

Analysis and Results . . . . . . . . . . . . . . . . . . . . .

45

Non Equilibrium Approach . . . . . . . . . . . . . . . . . . . . . .

53

4.2.1

The Formalism . . . . . . . . . . . . . . . . . . . . . . . .

53

4.2.2

The Equations of Motion . . . . . . . . . . . . . . . . . . .

55

4.2.3

Analysis and Results . . . . . . . . . . . . . . . . . . . . .

56

The Importance of Noncondensed Bosons . . . . . . . . . . . . . .

61

5 Beyond Mean Field
5.1

39

63

Scattering in the Medium . . . . . . . . . . . . . . . . . . . . . .

63

5.1.1

Poles of the T-Matrix: The Two-body Case Revisited . . .

64

5.1.2

Poles of the T-Matrix: The Many-body Case . . . . . . . .

65

5.1.3

Conditions for molecular stability . . . . . . . . . . . . . .

72

5.2

Many-Body Physics . . . . . . . . . . . . . . . . . . . . . . . . . .

77

5.3

Non-Equilibrium Mean-Field Theory Revisited . . . . . . . . . . .

83

6 Conclusion

88

Bibliography

89

Appendix
A Green Function Methods for Fermions94

A.1 Free Green functions . . . . . . . . . . . . . . . . . . . . . . . . .

94

A.2 Interacting Green functions . . . . . . . . . . . . . . . . . . . . .

95

vii
B Derivation of Mean-Field Equation of Motion

98

C Gaussian Integrals

101

D Reduced Units

103

viii

FiguresFigure2.1

Typical l = 0 cross-section versus momentum profile near an s-wave
resonance (solid line), and its unitarity limit (dashed line). . . . .

2.2

12

Conceptual representation of a potential resonance. A weakly attractive potential with small negative scattering length is represented in the top figure. As the depth of the potential is increased
it becomes more and more attractive, and the scattering length increases, as a resonant state approaches threshold. Eventually theresonance crosses threshold, and a bound state is created, as thescattering length remains large, but changes sign abruptly. As thepotential becomes deeper, the bound state becomes more deeply
bound, and the scattering length decreases, eventually becoming
negative once again. The whole process repeats as more and morebound states are created . . . . . . . . . . . . . . . . . . . . . . .

2.3

14

Pictorial representation of the three main types of resonances described in the text. . . . . . . . . . . . . . . . . . . . . . . . . . .

17

ix
2.4

a) Hyperfine levels as a function of external magnetic field of 40 K
in its electronic orbital groundstate. b) Same as a) but for 87 Rb. c)Thresholds for a 40 K−87 Rb collision, incoming in the |9/2−9/2i|11i
channel. Since m1 + m2 is conserved in the collision, only thestates plotted in black in the a) and b) channel participate. It is
informative to note the energy scale of the threshold energies, andcompare it with the depths of the singlet and triplet potentials in
fig 2.5.

2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22Singlet and triplet potential curves for the Rb-K collision. Note thepotential depth is of the order of 105 Kelvins, while the thresholdenergies are of the order of hundreds of milliKelvins or less. (seefig 2.4)

3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

Feynman diagrams representing the resonant collision of a fermion
and a boson. Solid lines represent fermions, dashed lines bosons,and double solid-dashed lines represent the effective composite fermions. 32

3.2

The top panel shows the scattering length versus magnetic field
for the 544.7 G resonance, present in the |9/2, −9/2i|1, 1i states of
the 40 K-87 Rb collision. The bottom panel shows the poles of themodel two-body T-matrix (eq. 3.14) parametrized for the same
resonance, as a function of magnetic field. Thick solid and dashedlines denote the real parts of relevant poles, representing bound andresonance states respectively. The thin dashed lines are the realparts of unphysical poles. The empty circles and squares, represent
the position of the resonance and the bound state, obtained via
a virtually exact close coupling calculation, and are presented toshow the level of accuracy of the model. . . . . . . . . . . . . . .

36

x
4.1

The thick solid lines represent the “renormalized” mean-field energy
levels λα,β (kf ) , while the thin dashed lines represent their barecounterparts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

46

Equilibrium chemical potentials (top panel) and populations (bottom panel) as a function of detuning for a non interacting gas with
rbf = .6. The solid lines represent fermions, dashed lines molecules,and dashed-dotted lines bosons. The dotted lines in the top panel
represent the bare molecular and fermionic internal energies, respectively ν and 0. The vertical lines labeled a) and b) are discussed in the text, and represent the detuning at which molecularformation begins and ends, respectively

4.3. . . . . . . . . . . . . .

48

Equilibrium chemical potentials (top panel) and populations (bottom panel) as a function of detuning for a non interacting gas with
rbf = 6. The solid lines represent fermions, dashed lines molecules,and dashed-dotted lines bosons. The dotted lines in the top panel
represent the bare molecular and fermionic internal energies, respectively ν and 0. . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

49

Mean-field equilibrium chemical potentials (top panel) and molecular fraction (bottom panel) as a function of detuning. In the top
panel, solid lines represent fermions, dashed lines molecules, anddashed-dotted lines bosons. In the bottom panel, the solid line
represents the full mean-field result, while the dashed-dotted line
represents results obtained using the approximations in describedin the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

xi
4.5

The top panel represents the time evolution of the population ofcondensed atoms after detuning is suddenly shifted from infinitely
positive to -5.1 K (−.1G magnetic detuning) around the 659.2G
resonance in 87 Rb −40 K described in table 3.1. The bottom panel
shows the absolute value of Fourier transform of said time evolution.
The main peak in this graph represents the computed value of thebinding energy, which we see is about 5.410−6 K. The system underconsideration is composed of fermionic densities of 1015 cm−3 , for aconstant density ratio of six bosons per fermion.

4.6. . . . . . . . .

59

Plot representing the poles of the scattering t-matrix for the 659.2G
resonance in 87 Rb-40 K described in table 3.1. The dotted line represents the “bare” molecular detuning as a function of field, as definedin the text. The solid black line is the correct binding energy ofthe molecular state, obtained by means of full close coupling calculations, while the gray solid lines are the eigenenergies obtained
from equations 4.18.a-4.18.d, for different atomic densities. From
top to bottom on the right the grey lines refer to fermionic densities of 1013 cm−3 ,1014 cm−3 , and 1015 cm−3 , for a constant density
ratio of five bosons per fermion. We note that for lower and lower
densities the calculated binding energy incorrectly approaches thebare detuning instead of the correct two-body binding energy. . .

5.1

60

Complex poles of the two-body T-matrix, as a function of detuning. Solid and dashed lines denote the real and imaginary parts ofphysically relevant poles, respectively. The dotted and dash-dottedlines are real and imaginary parts of unphysical poles, respectively.

66

xii
5.2

Complex poles of the many-body T-matrix, as a function of detuning. Solid and dashed lines denote the real and imaginary parts ofphysically relevant poles. In (a) the center-of-mass momentum of
the molecule is P = kf , i.e., equal to the Fermi momentum of theatomic gas. In (b), this momentum is P = 0.1kf . . . . . . . . . .

5.3

70

Contours of molecular population under various combinations ofmolecular center-of-mass momentum P and detuning ν. The uppermost contour identifies the detuning at which bound molecules
first appear for a given momentum P . Numbers indicate contours
with a equal molecule fraction. The dots represent the result obtained analytically for the critical detuning, Eq. (5.16). . . . . . .

5.4

74

The internal energy that a molecule of center-of-mass momentum
P/kf would require to overcome Pauli blocking and dissociate. The
shaded area represents numerical calculations, and the dots represent the analytical kinematic arguments in Eqs. (5.12,5.15). . . . .

5.5

76

Feynman diagrams included in the mean-field theory. Thin (thick)solid lines represent free (renormalized) fermions, thin double dashedsolid lines represent free molecules, and thick double dashed lines
represent renormalized molecules. The little lightning bolts represent condensed bosons, whereby the arrow indicates whether they
are taken from or released into the condensate.

. . . . . . . . . .

78

xiii
5.6

Feynman diagrams included in the generalized mean-field theory.
Like in the mean field case (fig 5.5), thin (thick) solid lines represent
free (renormalized) fermions, thin double dashed-solid lines represent free molecules, and thick double dashed lines represent renormalized molecules. The little lightning bolts represent condensed
bosons, whereby the arrow indicates whether they are taken fromor released into the condensate. The novelty here in the inclusionof the 2-body dressed molecules from fig. 3.1. . . . . . . . . . . .

5.7

80

The thick lines represent the poles of D GM F (kf ), and correspond tomolecular and fermionic energies. The dashed lines represent themolecular two-body poles, corresponding to the molecular binding
and resonant energies. Comparison with the top panel of fig 5.2
highlights the fact that, as in the mean-field theory case, the effect of the interaction with the condensate is to create an avoided
crossing between the atomic and molecular states. . . . . . . . . .

5.8

81

Equilibrium molecular population as a function of detuning for the
narrow 492.49G resonance. The solid line represents results obtained via the generalized mean-field theory presented in the text,
while the dashed-dotted line represents the mean-field results. . .

5.9

84

Equilibrium molecular population as a function of detuning for thewider 544.7G resonance. The solid line represents results obtained
via the generalized mean field theory presented in the text, while
the dashed-dotted line represents the mean field results.

. . . . .

85

xiv
5.10 Transition probability into molecular state via a magnetic-field ramp
across the 492.5G Feshbach resonance. The solid lines are obtained
via numerical solutions of equations 4.18.a-4.18.d, while the dashedlines represent the Landau-Zener equivalent. In the top panel thegas is composed of more bosons than fermions rbf = 6, while in thebottom panel the opposite is true rbf = .6 . . . . . . . . . . . . .

87

Chapter 1
Introduction

The use of magnetic Feshbach resonances to manipulate the interactions
in ultracold quantum gases has enabled the study of an astounding amount ofphysics.

Notable examples, include the study of the crossover between BCS(Bardeen-Cooper-Schrieffer [1]) and BEC (Bose-Einstein Condensate [2, 3]) superfluidity in ultracold Fermi gases [4, 5, 6], and the “Bose Nova” collapse in
Bose gases [7]. Recent experimental developments, [8, 9, 10, 11] have enabled
the creation of an ultracold mixture of bosons and fermions, with an interspecies
Feshbach resonance, opening the ground for what may very well be a new prolificsource of physical insight.
From the theoretical point of view, on the other hand, studies of BoseFermi mixtures to date have been mostly limited to non resonant physics, focusing
mainly on mean field effects in trapped systems [12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22], phases in optical lattices [23, 24, 25, 26, 27],or equilibrium studies ofhomogeneous gases, focusing mainly on phonon induced superfluidity or beyondmean-field effects [28, 29, 30, 31, 32, 33, 34]. Pioneering theoretical work on theresonant gas include [35], whereby a mean-field equilibrium study of the gas issupplemented with a beyond mean-field analysis of the bosonic depletion, and[36], where an equilibrium theory is developed using a separable-potentials model.
The aim of this thesis is to develop and solve a mean-field theory describ-2
ing an ultracold atomic Bose-Fermi mixture in the presence of an interspecies
Feshbach resonance. This goal appears innocuous enough at first sight, sincemean-field theories for resonant Bose-Bose [37, 38] and Fermi-Fermi [39, 40, 41]gases exist, and have studied extensively. In both of these theories, the mean-fieldapproximation consists of considering the bosonic Feshbach molecules as being
fully condensed, and this greatly simplifies the treatment, since the Hamiltonian
reduces to a standard Bogoliubov-like integrable form [42].
The fundamental difference between these examples and the Bose-Fermi
mixtures is that in the latter the Feshbach molecules are fermions, and therefore
their center of mass momentum must be included explicitly. The most obvious
mean-field approach consists in considering the atomic gas to be fully condensed.
However, as we will see in more detail in the body of the thesis, resonant molecules
are really composed of two bound atoms, which spend their time together vibratingaround their center of mass. It follows that outright omission of the bosonic fluctuations of the atoms, disallows the bosonic constituents to oscillate (i.e. fluctuate)
at all, and therefore this leads to an improper description of the the two-body
physics of the gas.
This thesis is organized as follows. In chapter 2 we lay the groundwork
by briefly introducing the main scattering theory concepts necessary to approach
the problem. We then proceed by describing the low-energy resonance physics
in general terms, and to introduce the main classes of resonances which exist.
An introduction to the physics of magnetic Feshbach resonances in alkali atoms
follows, where we outline in detail the nature of the interatomic interaction using
the language of ultracold scattering physicists.
In chapter 3 we introduce the field theory model used to study these resonances, describing in detail the paramerization used, and outlining the exact
solution of this model in the two-body limit. The chapter ends with a test of

3
this two-body theory, by comparing the binding and resonance energies predictedby the model and the virtually-exact analogues obtained from two-body close
coupling calculations.
Chapter 4 introduces the simplest mean-field many-body theory of the gas,
obtained by disregarding all bosonic fluctuations. The solution of the theory isoutlined, and its limitations highlighted. We recognize here that, in spite of theselimitations, mean-field theory gives us a language useful to deal with the problem,
which will persist even beyond the limits of applicability of the theory itself. The
chapter then proceeds to describe a non-equilibrium approach to the problem,
whereby the beyond mean-field HFB equation of motion are shown to reduce totheir mean-field equivalent, highlighting the importance of properly accounting
for the bosonic fluctuations.
Finally in chapter 5 we introduce our generalized mean-field theory, whichis, in short, similar to the mean-field theory described in chapter 4, but with thenotable improvement of using properly renormalized molecules as building blocks,
instead of their bare counterparts. This approach is not trivially described in
the Hamiltonian formalism, where substituting dressed molecules for free ones
would lead to double counting of diagrams. In this chapter we therefore shift
to the Green-function/path-integral language, where this double-counting can be
avoided quite easily. Finally we proceed to the numerical solution of this theory,and note that for narrow resonances the results are in line with their mean-field
equivalents. This encourages us to develop a simple theory to study the molecularformation via magnetic-field ramps, and, using an approach based on the LandauZener formalism [43, 44], we derive analytic expressions.
Throughout this thesis, we work with zero temperature gases, in the freespace thermodynamic limit. These are limitations which render the results obtained here hard to directly compare with experimental results. One of the main

4
possible future directions of this work should include solving the same problem ina trap, and generalization to higher temperatures.
This thesis, by its very nature, represents a synthesis of the work of thisauthor during the last few years. We therefore draw heavily from published work
which resulted from this effort, and in particular [45] and [46].

Chapter 2
Scattering Theory: The Starting Point

The rich physics of the system we intend to study is in large part due
to the exotic nature of the interactions between its various constituents. An
understanding of the two-particle dynamics is therefore paramount to any analysis
which may follow, and it is the aim of this chapter to introduce this physics, by
first providing a general outline of cold collision theory, and in particular resonant
scattering theory, and then describing how Feshbach resonances arise in ultracold
collisions between alkali atoms.
2.1

Fundamental Concepts
The essence of the two-body physics we are interested in is low energy reso-

nant scattering, in general, and magnetic-field Feshbach resonances in alkali gases
in particular. In the following we introduce some of the basic concepts and equations we will need to talk about these resonances; focusing on interactions between
distinguishible particles, interacting via a central potential. The conventions we
use here are those found in [47].
A prototypical scattering experiment consists of a beam of identical particles
incident upon a target, from which they scatter. The beam is assumed to be described as a plane wave of well defined momentum k and from a specific direction,
which we will choose as our ẑ axis. Furthermore the properties of the scattered6
particles are measured far enough away from the target (r → ∞) that interactions
with it no longer influence them. Under these circumstances the wavefunction of
the incident particles can be written, in center of mass coordinates,
#

"

1
eikr
ikz
ψ(r) =
.
e
+
f
(k,
θ)
(2π)3/2r(2.1)

The first term in this expression represents the incoming plane wave, and thesecond term represents the scattered spherical wave, where f (θ) is the amplitudeof the scattered wave, and is therefore known as “scattering amplitude”. The
scattering amplitude contains all the physical information about the scattering
process, and it can be used to calculate any observable. Also we should point out
that the wavefunction is normalized to unity incoming flux.
For central potentials, such as (for the most part) alkali atom collisions, it
is often convenient to separate the angular degrees of freedom by expanding thewave function in spherical harmonics, or “ partial waves”, whereby the scattering
amplitude can be written asf (k, θ) =X

(2l + 1)fl (k)Pl (cosθ),

(2.2)

l

where where Pl is a Legendre polynomial, and fl are the partial-wave amplitudes.
We then expand the incoming plane wave in spherical waves, and take the asymptotic form, obtaining
ikz

e

"

eikr − ei(kr−lπ)→ (2l + 1)Pl (cosθ)
2ikr
l
X

#

(2.3)

Inserting 2.2 and 2.3 into eq. 2.1, and expanding the incoming plane wave in
spherical waves, we obtain, after some simplifications,"#

1 X Pl (cosθ)
eikr e−i(kr−lπ)ψ(r) =
(1
+
2ikf
(k))
.
−
l
(2π)3/2 l
2ik
r
r

(2.4)

From this expression we see that the quantity 1 + 2ikfl (k, θ) determines thephysics of the scattering, such that if fl → 0 the outgoing wave is identical to7
the incoming wave, and no scattering occurred. Furthermore if we note that theincoming flux must equal the outgoing flux, then |1 + 2ikfl (k, θ)|2 = 1, and we cantherefore define a quantity δl (k), such that 1 + 2ikfl (k, θ) = e2iδl (k) , and insert intoeq. 2.4, the effect of the scattering event is therefore to generate a “phase-shift”
δl (k) in the outgoing wave which would vanish, had no scattering occurred.
The quantity
Sl = e2iδl (k) = 1 + 2ikfl (k)(2.5)

is so ubiquitous in scattering theory, that it has been dubbed the “scattering (S-)
matrix”. A close cousin of Sl is the T-matrix, defined as
T =πi(S − 1),µk(2.6)

Where µ is the reduced mass. This quantity will be of fundamental importancein the many-body treatment, and the definition in eq.2.6 will be much more convenient for that purpose which is the reason why we will adopt it.
An observable of particular interest in scattering theory is the scatteringcross section σ, defined as the number of particles scattered per unit time as afraction of the number of incident particles per unit area per unit time. From thediscussion above about the role of the scattering amplitude we can readily deduce
σ(k) =Z|f (k, θ)|dΩ =

X 4π
l

k

(2l + 1)sin2 (δl ) =2X

4π(2l + 1)|fl (k)|2 .

(2.7)

l

It is often useful to define partial cross sections σl so that
σ(k) =Xσl (k).

(2.8)

l

2.2

Low Energies Collisions and Resonances
The physics of low energy collisions is characterized by the fact that thegreat majority of relevant observables can be characterized universally with a8handful of parameters, by only knowing the features of the long-range physics.
This universality is quite remarkable in its own right, and has the useful side
effect of allowing one to explore the fundamental scattering properties of very
complex systems without the need for sophisticated potential surfaces.
If we could define a length-scale R0 separating short-range and long-range
physics, a collision would be defined as “low energy” if R0 << 1/k, where k is thecollision momentum [48], and 2π/k is the deBroglie wavelength. The use of theconditional here is due to the fact that in the great majority of physically relevant
systems, potentials generally asymptote to zero, so that a clear cut definition of
R0 is not available. However if the potentials approach zero fast fast enough (we’ll
see shortly what “enough” means), then R0 can be defined heuristically.
In the case of alkali atoms, R0 is generally identified with the van der Waals
length R0 = (µC6 /8h̄2 )1/4 [49], where C6 is the parameter defining the perturbative
long-range van der Waals potential −C6 /r 6 .
2.2.1

Effective Range Expansion

We will begin with the radial Schrödinger equation for the scattering of two
particles (setting h̄ = 1)
u′′l +!

l(l + 1)
ul = 0,k − 2mV −r22

where ul is the radial wavefunction, k =

√

(2.9)

2µE is the collision momentum, V is theinteraction potential, and l is the orbital angular momentum quantum number.
Since we here postulate that the energy is low, and 1/k >> R0 , we can saythat if r << R0 , then k 2 << 2mV , and in this region eq. 2.9 reads!

l(l + 1)u′′l + −2mV −ul = 0.
r2(2.10)

9
On the other hand if r >> 1/k then since 1/k >> R0 , and by definition of R0 ,V → 0, and eq. 2.9 reads!

l(l + 1)
ul = 0.
u′′l + k 2 −r2(2.11)In this region the solution is readily found to be
Al (cos(δl )jl (kr) + sin(δl )nl (kr)),(2.12)

where j and n are the regular and irregular spherical Bessel functions, δl , is thescattering phase shift, remarkably the same phase shift described above. Thiscan be verified by plugging eq. 2.5 into 2.4, and expanding the exponentials in
spherical Bessel functions. Finally Al is an unimportant (at least for the purposesof this discussion) flux normalization constant.
Near threshold the short-range solution does not depend significantly onenergy, leaving all of the energy dependence in the long-range region. In theintermediate R0 < r < 1/k region, it turns out that the solution is also energy
independent, provided that V approaches zero faster than 1/r 2l+3 . This can beverified by guessing the solution ul = c1 r l+1 + c2 r −l , and checking it by plugging
into eq.2.9. In this regime, r > R0 >> 1, therefore both V ul and k 2 ul are muchsmaller than the other terms, and eq.2.9 becomesu′′l −

l(l + 1)
ul = 0.
r2(2.13)

Since the solution is virtually energy independent for the range of r’s inlwhich V acts, then so is the logarithmic derivative Yl = 1/ul du. The scatteringdrphase shift is in turn found by matching Y with the logarithmic derivative of thelong-range solution (eq. 2.12) at some large distance r = R, obtainingcot(δl ) =kRn′l (kR) − Yl jl (kR).
kRjl′ (kR) − Yl nl (kR)

(2.14)

10
Finally, since the only energy dependence in eq. 2.14 is in k, (and not in
Yl ) it is a simple, though tedious, algebraic exercise, using the properties of thespherical Bessel functions, to show that as k → 0, k 2l+1 cot(δl ) → const., and that
only even powers of k appear in the Taylor expansion of this quantity around zero,
k 2l+1 cot(δl ) = Fl (k 2 ) ≈ −1
1+ rl k 2 + ·.
al 2

(2.15)In the case of s-wave collisions, (l = 0) the quantities a0 and r0 are known as“scattering length” and “effective range.”
As we saw in section 2.1, the scattering amplitude is a measure of thestrength of the scattering. A particularly useful form in which to write thisquantity is derived by solving eq. 2.5 for fl in terms of δl , thus obtaining, in
combination with equation 2.15
fl (k) =1.
k cot(δl ) − ik

(2.16)

From this expression we can readily writefl (k) =2.2.2

1
k −2l Fl (k 2 ) − ik≈1k −2l (−1/al + rl k 2 ) − ik.

(2.17)

Resonances at Low energy

It is instructive to write the approximate version of equation 2.17, in termsof the collision energy E = k 2 /2µ, where µ is the reduced mass. We then obtain
the expression
fl =1
Γk 2l+1,
k (E − Er ) − iΓk 2l+1

(2.18)

where we have defined Γ = 2r1l µ , and Er = al r1l µ . From equation 2.7, we then get
σl =(Γk 2l+1 )2
4π
(2l
+
1)
.
k2
(E − Er )2 + (Γk 2l+1 )2(2.19)

This expression is known as a Breit-Wigner profile, in honor of its discoverers.

11
The most remarkable feature of equation 2.19 if that as E → Er , the crosssection is enhanced, and reaches its maximum possible value. Also in this regime
the S-matrix reaches 1, and we say that the systems reaches “unitarity”. We callthis phenomenon a resonance. Furthermore, if the resonance energy Er is large
enough, and Γ is small enough that over the width of the resonance variations in
1/k 2 l are negligible, then we see that the resonance is Lorentzian in shape, withfull width at half maximum equal to 2Γ, see fig.2.1. We should also note thatthe enhancement of the cross-section to unitarity requires that the phase shiftδl = π/2.
To better understand the physics of a resonant state, we turn to equation
2.4, and imagine a scattering event at “negative energy.”(i.e. k → iκ). We thenget!

e−κr eκrψ ∝ A S(κ).
+
rr

(2.20)

Solutions to the Schrödinger equation in this regime should lead to bound states,with boundary conditions implying exponentially decaying wavefunctions. Therefore, the coefficient A must vanish, as the s-matrix S diverges, such that the
product SA is held constant, so that the normalization of the wavefunction isfinite. We see in this case that a pole in the s-matrix implies a bound state. At
a resonance, we saw above, the scattering amplitude has a (complex) pole, whichmeans that the s-matrix also has one. In this light we can think of poles in the
scattering amplitudes as either bound states, if real, or as resonances, if complex.
In somewhat qualitative terms, we may say that resonances indicate thepresence of the “shadow” of a bound states (known as a quasi-bound state),
whereby when the two particles scatter at this energy they spend an unusuallylarge time close together, in a bound-like state, but eventually come apart. The
amount of time they spend together is inversely related to the imaginary part of12

σ l =0

4π
k2

σ0

kr

k→

Figure 2.1: Typical l = 0 cross-section versus momentum profile near an s-wave
resonance (solid line), and its unitarity limit (dashed line).

13
the resonant energy, or, more precisely, to width of the resonance.
2.3

Types of Resonances
In the previous section we have introduced the concept of scattering reso-

nance by disregarding the internal degrees of freedom of the colliding objects, andthus focusing on the simplest possible example. Here we will briefly introduce
other types of resonances, including the Feshbach resonance, complementing theprevious discussion by focusing mainly on the phenomenological characteristics ofresonant scattering.
2.3.1

Potential Resonances

In section 2.2.2 we mentioned that resonance states are “shadows” of real
bound states. Here we would like to better qualify this statement, by introducing
the simple example of a “potential resonance”, graphically illustrated in fig. 2.2.
In this thought experiment we will consider an attractive interparticle potential,
and we analyze what the effect on the physics would be if we could tune its depth.
For a perturbative attractive potential, the scattering length is negative.
This can be understood by noting that the wavefunction in the absence of thispotential is sin(kr). Since the potential is negative, for small r the deBroglie
wavelength will be slightly smaller, and the wavefunction will thus oscillate slightly
faster, accumulating a small positive phase. Since for asc = − limk→0 tan(δ)/k ≈−limk→0 δ/k, then if δ > 0, asc < 0.
As the potential is made deeper, and thus less perturbative, the phase shift
increases, and the scattering length becomes more and more negative. The phaseshift eventually reaches π, adding a node to the wave function, and causing thescattering length to diverge through negative infinity. This added node is a signature of a new bound-state having appeared. Increasing the potential further

14

a

s

a

a

a

c

s

c

s

c

s

<

c

0









<



0

Figure 2.2: Conceptual representation of a potential resonance. A weakly attractive potential with small negative scattering length is represented in the top
figure. As the depth of the potential is increased it becomes more and more attractive, and the scattering length increases, as a resonant state approaches threshold.
Eventually the resonance crosses threshold, and a bound state is created, as thescattering length remains large, but changes sign abruptly. As the potential becomes deeper, the bound state becomes more deeply bound, and the scattering
length decreases, eventually becoming negative once again. The whole processrepeats as more and more bound states are created15
causes the scattering length to initially change sign abruptly, and subsequently
diminish from positive infinity, through zero, and again through negative infinity
as another bound-state is added to the potential.
2.3.2

Shape resonances

Another example of single-channel resonance, is the shape resonance. This
resonance is characterized by the presence of a potential barrier in the interparticle
potential. If we imagined the barrier to be infinitely high, we could easily convince
ourselves of the existence of a bound-state behind it. As the barrier becomes finite,tunneling will allow the particles to separate, though the presence of the barrier
will cause them to spend a long time close to each other, in correspondence toenergies around where the original bound-state was. This phenomenon causes anamplification of the scattering cross-section, causing a resonance.
Examples of shape resonances are ubiquitous in nonzero partial-wave scattering between alkali atoms, where at long range the repulsive centrifugal potentiall(l+1)/2µr 2 is stronger than the (negative) Van der Waals potential −C6 /r 6 , whileat short range the opposite is true.
2.3.3

Feshbach Resonances

Feshbach resonances are different from all other resonances described so far,in that they are an intrinsically multi-channel phenomenon. What this meansis that they are the consequence of the complicated interplay between different
internal states of the colliding particles.
Using qualitative reasoning similar to that used above, we can think of a twochannel scattering problem, for example a spinless particle (particle 1) colliding
with a two state particle (particle 2), where state |1i and state |2i are separated byenergy ∆, |2i being the higher energy one. There are therefore two distinct ways

16
in which the two object can interact, namely particle 1 with particle 2 in eitherof the two states. This implies that there are two distinct interaction potentials,
characterizing the two collisions. Let us assume that the potential for state |2i
collisions hold a bound state ǫ, with binding energy B < ∆.
Let us now introduce the concept of coupling: the possibility, that is, thata collision between the two particles could change the state of particle 2. In theabsence of coupling, a particle of type 1, and one of type 2 in state |2i could bebound in state ǫ for ever. If coupling is introduced, since B < ∆, then it would
be energetically favorable for particle 2 to change state, and for the two particles
to break-up, implying that ǫ would no longer be a proper bound-state.
From a collisional point of view, consider a particle of type 1 colliding witha particle of type 2 in state |1i, at an energy smaller that ∆. In this example,
it is possible that that during the scattering event particle 2 will visit state |2i
temporarily, though it is energetically impossible it will leave the collision in thisstate. If the collision energy is close to ∆ − B (the position of state ǫ relative tostate |1i threshold), then the time the two particles spend together before breaking
up becomes quite long, and we identify a resonance.
2.4

Magnetic Feshbach Resonances in Alkali Gases
In this section we will briefly introduce the physics of ultracold collisionof alkali atoms in a magnetic field, from the perspective of an atomic physicist.
Because of the scope of this thesis, we will not dwell substantially on this point,
but only give a general overview. A more accurate and detailed account is given
in [50].
Magnetic Feshbach resonances in alkali atoms, are the consequence of theinterplay between the magnetic/hyperfine degrees of freedom, dominant at longrange, and the interatomic interaction, which is diagonal in the electronic spin

17

P

R

o

t

e

s

e

n

o

t

n

S

a

e

R

s

e

e

l

c

e

h

o

p

n

h
F

a

n

a

R

i

a

n

n

e

h
a

o

c

b

s

s

e

a

c

n

c

e

Figure 2.3: Pictorial representation of the three main types of resonances describedin the text.

18
degrees of freedom, and dominates the short-range physics.
The Hamiltonian can be schematically written as
H = T (R) + H M + H HF + V (R),

(2.21)

1 ∂where T (R) = − 2µ+ l(l+1)
is the kinetic term in the interparticle coordinate∂R
R2

R, V (R) is the interparticle potential, H M is the Zeeman Hamiltonian, and H HF
is the hyperfine Hamiltonian.
We should note that while V (R) → 0 as R → ∞, H M and H HF are very
nearly R-independent, and they persist at R → ∞, thus defining the thresholdsof the problem. From this follows that the basis we use to describe the problem
must diagonalize these two components of the Hamiltonian. Simultaneously, we
will see that this basis does not diagonalize V (R), and these off-diagonal elements
will provide the coupling which will lead to the existence of Feshbach resonances.
The operational procedure, which we will describe in more detail below, may
be outlined as follows:• basis |ai diagonalizes V (r)• basis |bi diagonalizes H HF .
• basis |b̃i diagonalizes H HF + H M .
We need to express everything in basis |b̃i, which are the physically relevant
scattering stares at R → ∞; to this end we take the following steps:(1) Express H HF + H M in basis |bi; this can be done analytically, and leads
to a non-diagonal matrix.
(2) Diagonalize numerically this matrix, to obtain the diagonal eigenvalue
matrix λb̃ , and the eigenvector transformation matrix Ub→b̃ = hb|b̃i. This
step is known as “field dressing”

19(3) Transform V (R) from the |ai basis representation (in which it is diagonal)
to the |bi basis; this can also be done analytically, and leads to a nondiagonal matrix(4) Transform V (R) from the |bi basis representation to the |b̃i basis representation, using Ub→b̃ .
At this point the whole Hamiltonian is expressed in the appropriate long-range
basis, and it is possible to run a close coupling scattering calculation.
In the following section we will discuss this procedure in more detail, fillingin the missing parts of this outline, and, in particular, the quantum numbers
associated with the |ai, |bi and |b̃i basis.
2.4.1

1 ⊗ 1-Particle Physics

We will begin our discussion by studying the R → ∞ Hamiltonian, todefine the operational long-range basis |b̃i. The long-range physics of the problem
consists, by definition, of two distant non-interacting atoms. This implies that themost physically relevant representation must be a direct product of single-atom
representations. This allows us to treat each atom individually, and then put thetwo pictures together. We therefore write the magnetic Hamiltonian for atoms 1
and 2 as
HaM = H Z + H HF = H1Z + H1HF + H2Z + H2HF

(2.22)

Assuming without loss of generality that the magnetic field B = B ẑ,a a
HaZ = −B [ge µe saz + µN gN
iz ]





HaHF = f~a2 − ~s2a − ~i2a C(2.23)In these expressions ~sa and ~ia are respectively the electron spin and nuclear spin
operators of atom a (which refers to either 1 or 2), and f~a = ~sa + ~ia is the total

20
spin, while saz , iaz and fza are the corresponding ẑ projections. Also µe and ge areathe electron magnetic moment and g-factor, while µaN and gN
are their nucleara, where ∆a is the hyperfine energy
counterparts for atom a. Finally C = Λa 2i∆a +1

splitting, and Λa = sign(µaN ). If Λ = 1, then the hyperfine structure is said to be“standard”, otherwise it is said to be “inverted.”
With these definitions, we can readily identify the basis which diagonalizes
HaHF , namely |fa ma i, and we can writehfa′ m′a |HaHF |fa ma i = Λa∆a
δf ,f ′ δm ,m′ .
2ia + 1 a a a a(2.24)The matrix HaZ in this basis, on the other hand, is not diagonal, and its matrix
elements readhfa′ m′a |HaZ |fa ma i = −BX

msa +mia =maams [ge µe msa + µN gN
mia ] ×hfa ma |sa msa ; ia mia ihfa′ m′a |sa msa ; ia mia i,(2.25)

where hf m|s ms ; i mi i is a Clebch-Gordon coefficient coupling s and i into f .
Diagonalizing the matrix hfa′ m′a |HaHF + HaZ |fa ma i generates a set of eigenvalues ǫa , and an eigenvector transformation matrix
Ufa ma →fg= hfa ma |fg
a ma i.
a ma(2.26)It should be noted that f˜ is not a strictly good quantum number in the dressedbasis, but given the small magnetic fields generally involved in the experiments(recall that f˜ → f in zero field), the f˜ state is usually still “recognizable”, andthe single-particle states are still referred to using f˜.
So far we have worked focusing on a single atom; since H Z +H HF is separable
into 1-atom components. Obtaining a two-atom version of the Hamiltonian isjust a matter of taking direct products, so that the basis |b̃i can be written as
g
g
g
|fg
a ma i|fa ma i (or equivalently|fa ma ; fa ma i , and the full transformation matrix

21
becomesg
Uf1 m1 f2 m2 →fg= Uf1 m1 →fg⊗ Uf2 m2 →fg= hf1 m1 |fg
1 m1 ihf2 m2 |f2 m2 i.
1 m1 fg
2 m2
1 m1
2 m2

(2.27)

A plot of the hyperfine levels of 40 K and 87 Rb appears in fig. 2.4, along witha plot of the thresholds for the collision in the |9/2 − 9/2i|11i state.
2.4.2

The Interparticle Potential

Alkali atoms have only a single valence electron, and we are only concernedwith their orbital ground state which is an s-shell. It follows that the only electronic degrees of freedom which influence the interatomic interaction is the spin of
this electron. In this regime, the interatomic potential V (R) is therefore diagonalin the basis |IMI ; SMS >, so thathS ′ MS′ ; I ′ MI′ |V (R)|SMS ; IMI i = δI,I ′ δMI ,MI′ δS,S ′ δMS ,MS′ VS (R)(2.28)~ = ~s1 + ~s2 , I~ = ~i1 + ~i2 and i1 , i2 are the nuclear spins of the atoms, while
where S
s1 = s2 = 1/2 are the electronic spins.
Although the details of how the actual potentials are derived are beyond
the scope of this thesis, and we refer the reader to [51] for more details, we would
like to point out that the potentials VS (R) are not in general entirely ab-initio,
as they generally require some tweaking via empirical parameterizations. Highprecision Feshbach spectroscopy experiments have been used heavily to determine
these parameterizations, with outstanding results, such as [4, 7, 8, 9]. Potential
curves for the Rb-K collision were obtained in [52], and are plotted in fig. 2.5.
Our goal is to integrate the above representation of V (R) with our previous
representations. To this end we first wish to represent V in the hyperfine basis

22

Energy (K)0.1|7/2 -7/2>|7/2 7/2>|9/2 9/2>

0

a)

-0.1

|9/2 -9/2>|2 2>

Energy (K)

0.2
0.1

|2 -2>

0-0.1|1 -1>

b)

-0.2

|1 1>

Energy (K)0.4|7/2 -7/2>|2 0>|7/2 -5/2>|2 -1>|7/2 -3/2>|2 -2>|9/2 -9/2>|2 -1>|9/2 -7/2>|2 0>|9/2 -5/2>|2 -1>|9/2 -3/2>|2 -2>

0.2|7/2 -5/2>|1 -1>|7/2 -7/2>|1 0>

0|9/2 -5/2>|1 -1>|9/2 -7/2>|1 0>|9/2 -9/2>|1 1> : Incoming

c)

-0.2
0

200

400
600
Magnetic Field (G)

800

1000

Figure 2.4: a) Hyperfine levels as a function of external magnetic field of 40 K in
its electronic orbital groundstate. b) Same as a) but for 87 Rb. c) Thresholds fora 40 K−87 Rb collision, incoming in the |9/2 − 9/2i|11i channel. Since m1 + m2 isconserved in the collision, only the states plotted in black in the a) and b) channel
participate. It is informative to note the energy scale of the threshold energies,
and compare it with the depths of the singlet and triplet potentials in fig 2.5.

23

V1 (R)V0(R)5Energy (K)

10

0

5

-10

0

5

10
Interparticle Separation (a.u.)

15

20

Figure 2.5: Singlet and triplet potential curves for the Rb-K collision. Note thepotential depth is of the order of 105 Kelvins, while the threshold energies are of
the order of hundreds of milliKelvins or less. (see fig 2.4)

24
with the following basis transformation:
hf1′ m′1 ; f2′ m′2 |V (R)|f1 m1 ; f2 m2 i =X

S,S ′ ,MS ,M ′S
I,I ′ ,MI ,M ′I

hf1′ m′1 f2′ m′2 |S ′ MS′ ; I ′ MI′ i ×

hS ′MS′ ; I ′MI′ |V (R)|SMS ; IMI ihSMS ; IMI |f1 m1 ; f2 m2 i =XSVS (R)hSMS ; IMI |f1 m1 ; f2 m2 ihf1′ m′1 f2′ m′2 |SMS ; IMI i,(2.29)

where in the first step we introduced two complete sets of the form |SMS ; IMI ihSMS ; IMI |,and in the second step we used equation 2.28.
To make the final expression in equation 2.29 useful, we need the matrix
elements hSMS ; IMI |f1 m1 ; f2 m2 i, which, it turns out, are well known [53]hSMS ; IMI |f1 m2 ; f2 m2 i =X
f,mhSMS ; IMI |f mihf1 m1 ; f2 m2 |f miq(2f1 + 1)(2f2 + 1)(2S + 1)(2I + 1) ×





s1





s1






 s

1

i1




f1 





i1 f1  .



(2.30)


i1 f1 Here hSMS ; IMI |f mi and hf1 m1 ; f2 m2 |f mi are Clebsch Gordon coefficients, andthe quantity in curly brackets is an object known as a “nine-J” symbol [53].
From this expression we note that m1 + m2 = m = MS + MI . Since from theequivalent “primed” expression we get that m′1 + m′2 = m = MS + MI , then wecan conclude that m′1 + m′2 = m1 + m2 . This makes perfect physical sense, given
the cylindrical symmetry of the problem around the field axis, which we also use
as quantization axis.
The final step is to transform the matrix elements in eq.2.29 in the dressedbasis. To this end we use the transformation matrices U derived in the previous
section, obtaining
g
′ ′
′ ′
g
g
hfg
1 m1 ; f2 m2 |V (R)|f1 m1 ; f2 m2 i =X

f1 f2 f ′ f ′1 2
m1 m2 m′ m′
1 2

g
hf1 m1 |fg
1 m1 ihf2 m2 |f2 m2 i ×

ghf1′ m′1 ; f2′ m′2 |V (R)|f1 m1 ; f2 m2 ihf1 m1 |fg
1 m1 ihf2 m2 |f2 m2 i.

(2.31)

25
Now that we managed to write the Hamiltonian matrix elements in a physically relevant basis, we need to add the kinetic energy part, whereby our basisgbecomes |fg
1 m1 ; f2 m2 i|LML i. Within our approximation L and ML are good quan-tum numbers.

Finally, the last step to take is symmetrization of the wavefunctions, butsince we are concerned here with collision between distinguishible particles, wewill not require it.
Where necessary, in throughout the thesis we have solved the coupledchannel equations for K-Rb collisions using the log-derivative propagator method
described in [54, 55].
Feshbach resonances are very common in multichannel problems with thislevel of complexity. What truly renders these resonances special is the motion of
the thresholds relative to one-another as a function of magnetic field, as depicted
in fig 2.4. This causes the resonances to shift in line with the thresholds, which,in many cases, provides a very powerful and direct mean to control the creationof molecuar dimers.

Chapter 3
Two-Body Physics: The Model

In order to describe the many-body properties of the system we wish to
employ the powerful tools offered by statistical field theory. To this end we needto rephrase the concepts from the last chapter in the language of field theory, andto develop a simplified model able to retain most of the relevant two-body physics.
In this chapter we present a reasonably well studied field theoretical modelof resonant collisions, which we tailor for our specific needs. The model is exactly
solvable in the two-body limit, and we show it can reproduce the equations derived in section 2.2.2. We then proceed by testing quantitatively the model bycomparing its predictions to the results of (virtually exact) close-coupling calculations.
3.1

The Hamiltonian
We are interested primarily in the effects of resonant behavior on the other-wise reasonably understood properties of the system. To this end we use a model
which, in the last few years, has become one of the standards in the literature,
and which was used to study the effect of resonant scattering in systems composed of bosons [38, 56, 57], and fermions [58, 59, 60, 61, 62, 63]. Because there isalready a significant literature explaining the details involved in the choice of theappropriate model Hamiltonian, we only outline the extent of the approximation

27
involved in such a choice.
An accurate approach to the problem would have to incorporate several
scattering channels, since the resonance in question is a consequence of the intertwined behavior of the complex internal structures of the atoms. In a field
theoretical sense, that would imply having to consider vector fields for the bosons
and fermions with as many components as there are spin states involved in theinteraction, and a non local interaction tensor of adequate size, to account for all
coupling between such components. Fortunately, if we assume that the resonancesin the system are sufficiently far from each other, such that it is possible to define a “background,” or away from resonance behavior, we can focus on only one
resonance at a time, which in turn makes it possible for an effective two channel
model to describe the resonance.
Furthermore, since the closed channel threshold is energetically unaccessibleat the temperatures of interest, we can “integrate out” the closed channel components of the fermion and boson field, in favor of a fermion field which we identify
as representative of the motion of one boson and one fermion, and which we dub
the “molecular field.” In the appropriate limit the molecular field identifies boundstates between fermions and bosons.
We emphasize that the molecular field is a theoretical artifice that alleviates
the need to treat relative motion of two atoms on the natural scale of the interaction (tens of Bohr radii). However, this model is appropriate for the study ofthe systems at hand, typically composed of 1012 40 K atoms per cubic centimeter,whereby the characteristic length scale associated with the many body system isof the order of the inverse Fermi wavenumber, (thousands of Bohr radii), implying an average interparticle distance, which is given by



9
4π

1/3

1
, of the samekF

order. Lastly, since the coupling terms in the Hamiltonian represent an effective
interaction, we can choose its functional form, and we do so by choosing to deal

28
with contact interactions, which simplify the calculations immensely.
The resulting Hamiltonian has the following form:

H = H0 + HI ,

(3.1)

where
H0 =

X

ǫFp â†p âp +

p

X

†
ǫB
p b̂p b̂p +pγ X † †+b̂ b̂ ′ b̂p b̂p′
2V p,p′,q p−q p +q

X

ǫM
p +ν

p

Vbg X † †â b̂ ′ âp b̂p′
V p,p′,q p−q p +q
g X †
+ √
(ĉq â−p+q/2 b̂p+q/2 + h.c.).
V q,p



ĉ†p ĉp

HI =

(3.2)

Here âp , b̂p , are the annihilator operators for, respectively, fermions and bosons, ĉp
is the annihilator operator for the molecular field [58, 62, 63]; γ = 4πab /mb is theinteraction term for bosons, where ab is the boson-boson scattering length; andVbg , ν, and g are parameters related to the Bose Fermi interaction, yet to be determined. Also we define single particle energies ǫα = p2 /2mα , where mα indicatesthe mass of bosons, fermions, or pairs, and V as the volume of a quantization box
with periodic boundary conditions.
3.2

Two Body Scattering Parameters
The first step is to find the values for Vbg , ν, g, in terms of measurable pa-

rameters. We will, for this purpose, calculate the 2-body T-matrix resulting fromthe Hamiltonian in eq. 3.2. Integrating the molecular field out of the real time
path integral, (see Appendix C for clarifications) leads to the following Bose-Fermi

29interaction Hamiltonian
HI2body =

1
V

!

X
g2
Vbg +
â†p b̂†−p âp b̂−p .
E−ν p

(3.3)

This expression is represented in center of mass coordinates, and E is the collisionenergy of the system. From the above equation we read trivially the zero energy
scattering amplitude in the saddle point approximation:
T = (Vbg −

g2
),
ν

(3.4)

which corresponds to the Born approximation 1 . We emphasize that this approximation is only valid at exactly zero energy, and it does not, therefore, describe
the correct binding energy as a function of detuning, and will be improved intreatment in Sec. 3.3. However, with this approach we obtain an adequate description of the behavior of scattering length as a function of detuning, whichallows us to relate the parameters of our theory to experimental observables via
the conventional parameterization [58, 56]!

∆B
2π
abg 1 −,
T (B) =
mbf
(B − B0 )(3.5)

where abg is the value of the scattering length far from resonance, ∆B is the width,
in magnetic field of the resonance, mbf is the reduced mass, and B0 is the field at
which the resonance is centered.
The identification of parameters between eqns, (3.4) and (3.5) proceeds asfollows: far from resonance, |B − B0 | >> ∆B , the interaction is defined by a2πabackground scattering length, via Vbg = mbfbg . To relate magnetic field dependent
quantity B − B0 to its energy dependent analog ν, requires defining a parameter δB = ∂ν/∂B, which may be thought of as a kind of magnetic moment forthe molecules. It is worth noting that ν does not represent the position of the1

This is akin to identifying the scattering amplitude f = asc in the Gross-Pitaevskii equation,
where the interaction term would be m2πbf asc

30
resonance nor the binding energy of the molecules, and that, in general δB isa field-dependent quantity, since the thresholds move quadratically with field,
because of nonlinear corrections to the Zeeman effect. For current purposes we
identify δB by its behavior far from resonance, where it is approximately constant.
Careful calculations of scattering properties using the model in eq. (3.2), however,leads to the correct Breit-Wigner behavior of the 2-body T-matrix, as we show insection 3.3
Finally we get the following identifications:
2πaVbg = mbfbg
g=q

Vbg δB ∆Bν = δB (B − B0 ).
(3.6)
3.3

Solution of the Model in the Two-Body Limit
In this section we will abandon the Hamiltonian formalism in terms of thepath integral formalism, which is better suited to the perturbative treatment wewish to present. This implies that we will be using the field operators φ, ψ, ξinstead of the second quantized operators b, a, c used so far [64].
The perturbative analysis begins by recasting the Hamiltonian in eq. (3.2)
in terms of a 2-body action, in center of mass coordinates :S[ψ, ψ † , φ, φ†, ξ, ξ †] = SB [φ, φ† ]+SF [ψ, ψ † ]+SM [ξ, ξ †]+SC [ψ, ψ † , φ, φ† , ξ, ξ †], (3.7)
where the field φ represents the bosons, ψ the fermions, and ξ the fermionicmolecules, and where†SB [φ, φ ] =

Z

dω X
†
(−h̄ω + ǫBp ) φω,p φω,p +
2π p

31
1
γ
2V
Z

Z

dω X †
φω,p−q φ†ω,p′ +q φω,p′ φω,p
2π p,p′ qdω X†(−h̄ω + ǫFp ) ψω,p
ψω,p
2π p
Z
dω X
†
(−h̄ω + ǫM
SM [ξ, ξ †] =p + ν) ξω,p ξω,p2π pZ
Vbg dω X ††††SC [ψ, ψ , φ, φ , ξ, ξ ] =ψω,p−q φ†ω,p′ +q ψω,p′ φω,pV
2π p,p′ q†

SF [ψ, ψ ] =+gZdω X †(ξ ψω,q−p φω,p + c.c),
2π pq ω,p(3.8)

where h̄ω is the frequency associated with the motion of the various fields.
The initial difficulty is that there are two distinct interaction pathways between bosons and fermions: a direct one of the form |φ|2 |ψ|2 , and an indirect
molecule-mediated one, of the form ξφ† ψ † , corresponding respectively to open andclosed channel events. In order to proceed, we will introduce an alternate effective
molecular field, which will exactly reproduce both these interaction pathways via
a single mediated interaction term. This is accomplished by first integrating out
the original molecular degree of freedom [64] to get :SC′ [ψ, ψ † , φ, φ†]

=

Z

!

g2
dω dω ′ X†′
(3.9)φω′ ,p′ ψE−ω′ ,−pφ†ω,p ψE−ω,−pVbg +2π 2π pp′E−ν

where E is the collision energy between the fermions and the bosons. After collecting the background and resonance terms of the interaction, we note that thisnew direct interaction contains both the resonant and background channels. Next
we undergo the inverse transformation to eliminate all direct boson-fermion interactions in favor of interactions mediated by new effective molecules, to obtain:′′SM[ξ, ξ † ]=−SC′′ [ψ, ψ † , φ, φ† , ξ, ξ †] =

Z

Z

dω X
2π

dω X
2π

p




−1

Vbg
1

+
2
p
2
g
w − 2(m +m ) − ν
b

′†
′
ξω,p
ξω,p

f

′†
g 2(ξE,0ψE−ω,−p φω,p + c.c).

p(3.10)

32
Here ξ ′ represents the new effective (i.e. primed) molecules. The first line of fig.
3.1 shows the diagrams describing the effective resonant collisions between bosons
and fermions. Here the solid lines refer to fermions, the dashed lines to bosons,and the double solid-dashed lines to effective molecules.

Figure 3.1: Feynman diagrams representing the resonant collision of a fermion
and a boson. Solid lines represent fermions, dashed lines bosons, and double
solid-dashed lines represent the effective composite fermions.

Since we are looking for poles of the S-matrix, we can disregard the trivial fermion
and boson propagators, and proceed, as outlined in fig. 3.1, to calculate therenormalized propagator for ξ ′ , denoted as D̂, represented there as a heavy broken
line. This object is proportional to the T-matrix of the system, and shares its
poles. Using the definition of the retarded molecular self energy Π̂ given in fig. 3.1,
and calling the free molecular propagator D̂ 0 (again for ξ ′ ), we get the following
Dyson series:
T = g D̂ 0 g + g D̂ 0 g Π̂ g D̂ 0 g + g D̂ 0 g Π̂ g D̂ 0g Π̂ g D̂ 0g + · · · = g D̂g(3.11)

where T is the T-matrix for the collision, and which has formal solution
T = g D̂g =g2.
(D̂ 0 )−1 − g 2 Π̂These quantities take the explicit form
0D̂ (E) =

1
Vbg
+
2
g
E−ν

!

(3.12)

33
Π̂(E)

= −i

Z

1
dω dp
2
p2
p
3
+
2π (2π) (h̄ω − 2m + i0 )(E − h̄ω − 2m
+ i0+ )
b

3/2
mbf √

≈ i√

2π

E+

f

mbf Λ
,
π2

(3.13)

where mbf is the boson-fermion reduced mass, and Λ is an ultraviolet momentum
cutoff needed to hide the unphysical nature of the contact interactions; we will
dwell more on that shortly. Finally inserting eq. 3.13 into eq. 3.12, we obtain thefollowing expression for the T-matrix:


−1

3/2

mbf √
1
mbf Λ √T (E) = 
+
i
E
+
2
g
π2
2π
Vbg + E−ν(3.14)To show that this expression correctly represents the low-energy two-body
T-matrix for resonant scattering, we will compare it to the s-wave (l = 0) versionof equation 2.18, remembering that T = 2π/mbf f
2π√
T (E) =
mbf 2mbf!

Γ√.
(E − Er ) − iΓ E

(3.15)The quantities Γ and Er can be extracted from experimental observables, throughaccurate two-body scattering calculations.
From the parameterization of the zero energy T-matrix in eq.(3.5), and the√
E → 0 limit of (3.15), we easily derive Γ = 2mbf abf δB ∆B. With these definitions we can relate equations (3.15) and (3.14), to find a regularization scheme
for the theory, by substituting the non observable parameters g, ν, and Vbg by
the Λ dependent (renormalized) quantities ḡ, ν̄, and V¯bg , such that the observable
T-matrix will not be itself Λ dependent
Following [58] we compare equations (3.14) and (3.15), in the limit E → 0,
where we have (once we include the definitions of the bare quantities)−1(g)2 
mbf Λ 
1
Vbg −
=
2 +
ḡ
¯
ν
π2
Vbg − ν̄.

(3.16)

Since we have one equation and three unknowns, we will have to consider somephysical circumstances, analyzing 3.16 in various limits. The first limit is far from34
resonance, where ν → ∞V¯bg = Vbg 

11−

mbf ΛVbg
π2(3.17)We are now left with the task of defining the resonant quantities, and wehave no more leeway to make physically motivated simplifications. The equations
which remain are ambiguous, which leaves us with a set of possibilities for the
choice of ḡ and ν̄. One way is to proceed as follows: insert eq. (3.17) into (3.16),
and solve for ν̄, to getν̄ = ḡ 2 (1 −ν
mbf ΛVbg mbf Λ)( 2 + 2 ).
2ππg(3.18)

From inspecting the above equation we can choose a definition of ḡ, which
will also imply one for ν̄, and we get (reporting also eq(3.17) for completeness)V¯bg = Vbg ḡ = g 

1
1−

mbf ΛVbg
π2

1
m

ΛV








1 − bfπ2 bg
mbf ΛVbg
ν̄ = ν + ḡg
π2(3.19)

Using these definitions of V¯bg , ḡ and ν̄, together with the policy of imposing Λ as
the upper limit of momentum integrals, will guarantee that observables will notdepend on the choice of Λ, as long as it is chosen to be bigger than momentum
scales relevant to experiment.
3.3.1

Poles of the T-Matrix: The Two-body caseAs described in section 2.2.2, bound states and resonances of the two-bodysystem are identified in the structure of poles of the T-matrix (eq.3.14). This is
illustrated in Figure 3.2, where real and imaginary parts of the poles’ energies
are plotted as a function of magnetic field. The resonance portrayed in the figure

35
is the 544.7G resonance present in the |9/2, −9/2i|1, 1i state of 40 K-87 Rb. For
B < 544.7G (corresponding to detunings ν < 0) the two-body system possessa true bound state, whose binding energy is denoted by the solid line. In thiscase, the pole of 3.15 occurs for real energies. This bound state vanishes as thedetuning goes to zero, where the resonance occurs.
For positive detunings, ν > 0, on the other hand, the poles are complex, andthe inverse of the imaginary part is proportional to the lifetime of the metastable
resonant state. In this regime, there is no longer a true bound state, but there
may be a scattering resonance, indicated in fig. 3.2 by a thick dashed line. This
resonance appears for magnetic fields B > 544.7 for this particular resonance,well before the disappearance of the bound state. This is highly dependent upon
the value of the background potential. We will see in chapter 5 that for Vbg = 0,
the resonance actually appears at positive detunings. In the case of 40 K-87 Rb,
Vbg < 0, implying that there is a weak potential resonance in the open channel
which interferes with the closed-channel resonance, and causes it to cross the axis
at negative detunings. For VBG > 0 ([65]) the positive background scattering
length is set by a bound state in the open channel, which does not affect theresonance states, but which interferes with the bound state at negative detunings.
An in depth analysis of this physics is highly interesting, but is beyond the scopeof this thesis.
The thin dashed lines in fig. 3.2 are physically meaningless solutions to the
Schrödinger equation, in which the amplitude in the resonant state would growexponentially in time, rather than decay. These poles do not therefore identify
any particular features in the energy-dependent cross section of the atoms, andwill not modify the physics of the system.
Finally fig. 3.2 contains data obtained from virtually exact close-coupling
calculations (see section 2.3.3), which show the extent of validity of the model.

36

3×10

4

a sc (a.u.)

544.7 G

0

-3×10

4

Energy (mK)

2

1

0

-1
530

540
Magnetic Field (G)550

Figure 3.2: The top panel shows the scattering length versus magnetic field forthe 544.7 G resonance, present in the |9/2, −9/2i|1, 1i states of the 40 K-87 Rb
collision. The bottom panel shows the poles of the model two-body T-matrix(eq. 3.14) parametrized for the same resonance, as a function of magnetic field.
Thick solid and dashed lines denote the real parts of relevant poles, representing
bound and resonance states respectively. The thin dashed lines are the real partsof unphysical poles. The empty circles and squares, represent the position of
the resonance and the bound state, obtained via a virtually exact close coupling
calculation, and are presented to show the level of accuracy of the model.

37
For the purposes at hand this agreement is sufficient.
It should be noted that the agreement is not as good for positive background scattering length systems, since the open-channel bound state determining
this scattering length is not adequately described by the model, which treats the
background physics as an essentially zero range interaction. This implies that therelation between the background scattering length and open channel bound-stateenergy is exactly Eb = 1/2µa2bg , while in the physical system this relation depends
on the details of the interaction potential. This problem has been addressed in
the literature [49], but no treatable field theory has yet been proposed.

38

B0 (G)∆B (G)

δB (K/G)abg (a.u.)492.49

0.134

544.7

3.13

3.624 × 10−5−176.5659.2

1.0

1.576 × 10−4

2.017 × 10−4

−176.5−176.5

Table 3.1: Parametrization of the three main Feshbach resonances used in thisthesis. All Three resonances are in the |9/2, −9/2i|1, 1i states of the 40 K −87 Rb
collision

Chapter 4
Mean-Field Theory: The Language

In this chapter we wish to introduce the many-body physics of the system,by first analyzing the mean-field approach. Because of the statistical propertiesof the system, we will see right away that mean-field theory does not recover thecorrect two-body physics in the low density limits. In spite of this substantial
weakness, however, the approach has several qualitative features which persist
even in the improved theory which we will later introduce. Furthermore, since themodel is exactly solvable, it will allow us to develop a language which will help us
to understand the problem in simpler terms, and to identify some small physicaleffects, which, when ignored, can greatly simplify the beyond mean-field approachpresented in the next chapter.
We will then develop the Hartree-Fock-Bogoliubov equations of motion, as afirst attempt to gain beyond mean-field insight. We will observe that the peculiar
physical structure of the problem is such that the HFB equations reduce to themean-field equations, and higher order correlations play an unusually importantrole in the physics of mixture.
This fact is of itself a novelty: the physics of other resonant systems, composed of either fermions or bosons, is known to be mainly driven by atom-atomcorrelations, (at least qualitatively). We will show that even some of the mostfundamental qualitative physical aspects of resonant Bose-Fermi mixtures require40
that at least three-body correlations be considered to be reasonably well understood, implying a peculiar complexity of the system that sets it apart from its
Bose-Bose and Fermi-Fermi counterparts.
4.1

Equilibrium Approach
The mean-field theory approach to the resonant Bose-Fermi mixture lends

itself very naturally to treatment in the Hamiltonian formalism, and in the next
section we will mainly focus on this particular approach. Nonetheless, to createa stronger link to the beyond-mean-field approach of chapter 5 we will also introduce the Green’s function formalism, and discuss the similarity between the two
approaches. All calculations in this and next Chapters assume a gas in free space,at zero temperature.
4.1.1

The Formalism

Starting with the Hamiltonian described by equation 3.1, we obtain themean-field Hamiltonian by substituting the boson annihilator b̂ by its expectation
value φ = hb̂i, a complex number. The number operator b̂†p b̂p therefore becomes|φ|2 = Nb , where Nb is the number of condensed bosons. The grand canonicalHamiltonian therefore becomes

H = Eb +XpǫFp − µf + Vbg nb â†p âp +√ X †ĉp âp + h.c. ,+ g nbX
pǫM
p + ν − µmĉ†p ĉp(4.1)

pwhere nb is the density of condensed bosons, Eb /V = γnb2 − µb nb is the energy
per unit volume of the (free) condensed bosons, a constant contribution to thetotal energy of the system, and µ(b,f,m) are the chemical potentials. These are
Lagrange multipliers that serve to keep the densities constant as we minimize the41energy to find the ground state. In the following we will drop the volume term,
absorbing it in the definition of the creator/annihilator operators, such that the
expected value of the number operator represents a density, instead of a number.
Before proceeding with the analysis of this Hamiltonian, we should introduce
the set of self-consistent equations we wish to solve. To this end we define thequantities n0b,(f ) , representing the total density of bosons (fermions) in the system,at detuning ν → ∞. At finite detunings some of these atoms will combine intomolecules, and the densities will be denoted as n(b,f,m) for bosons, fermions andmolecules respectively.
The system, therefore, is composed of six unknowns, namely three densities, and three chemical potentials, which require six equations to solve. Theseequations, which can be derived by number-conservation constraints and energy
minimization arguments, are:nf + nm − n0f = 0(4.2.a)nb + nm − n0b = 0(4.2.b)nf =(4.2.c)

dΩ
dµf
dΩ
nm =
dµm
dΩ
=0
dφ

(4.2.d)µb + µf = µm ,(4.2.f)(4.2.e)where Ω = hHi/V is the Gibbs free energy.
Equations 4.2.a and 4.2.b follow from the simple counting argument thatfor every molecule created, there is one less free boson and one less free fermion in
the gas. Equations 4.2.c, and 4.2.d are simply the Lagrange multiplier constraint
equations, equation 4.2.e follows from the mean-field approximation, whereby thebosonic field is simply a complex number, and minimization of the energy can42
therefore be done directly. Finally equation 4.2.f, known as the “law of massaction,” is just a condition on the multipliers that follows from the fact that to
make a molecule it takes one free atom of each kind.
The next step is to write down Ω for the system, by taking the expectation
value of the Hamiltonian in equation 4.1, obtainingΩ = Eb /V +

X
p



ǫFp − µf + Vbg nb ηf (p) +√ X+ 2g nbηmf (p),

X
p



ǫM
p + ν − µm ηm (p) (4.3)

pwhere ηf (p) = hâ†p âp i and ηm (p) = hâ†p ĉp i are the fermionic and molecular momentum distributions, ηmf (p) = hĉ†p âp i is an off-diagonal correlation term arisingfrom the interactions in the system, and the densities are given by nf,m,mf =R

dpη(p). Equations 4.2.a-4.2.f then read
2π 2 f,m,mf

nf + nm − n0f = 0(4.4.a)nb + nm − n0b = 0(4.4.b)nf,m,mf =(4.4.c)Zdp
ηf,m,mf (p)
2π 2

√
3/2
g nmf − µb nb + λnb = 0(4.4.d)µb + µf = µm(4.4.e)The remaining task is now to find expressions to calculate the expected values
ηf,m,mf (p). To this end we follow a Bogoliubov-like approach, similar to that described in [35]. The mean-field Hamiltonian is bilinear in all creation/annihilation
operators, which means that it can be diagonalized via a change of basis, whereby
introducing the operators
αp = Aα ap + Cα cp
βp = Aβ ap + Cβ cp ,

(4.5)

43
for some appropriately chosen coefficients Aα,β and Bα,β , the Hamiltonian willread
H ′ = E0 +Xλα (p)αp† αp +pX

λβ (p)βp† βp .

(4.6)

p

At this point we note that the Hamiltonian is just a separable sum of free-particle
Hamiltonians, where the free particles are fermions, with dispersion relations
λα,β (p). We can readily write down the distribution
ηα,β (p) = Θ(−λβ,α (p))(4.7)

, where Θ is the step function, and calculate the densities nα,β . The step function
could be replaced by the free Fermi distribution for non-zero temperatures, but
the mean-field assumption that all bosons are condensed would no longer hold.
Also Note that if these were ordinary free fermions with dispersion p2 /2m − µ,equation 4.7 would reduce to the standard zero-temperature Fermi distribution.
We will see below that λα,β (p) are dispersion relations of quasi-particles, which
are a mixture of atoms and molecules.
Below we show how these ideas, together with equation 4.2.a - 4.2.f, give us
the tools we require to calculate the observable atomic and molecular densities asa function of the chemical potentials.
To illustrate more explicitly the diagonalization procedure we define thevectorsA† = ( a†p c†p ),

(4.8)



B † = ( αp† βp† ),(4.9)

 ap 



A=


and

cp



 αp 



B=


βp

whereby the Hamiltonian can be written as A† ĤA, and B † Ĥ ′ B, where Ĥ = 
ǫFp − µf + Vbg nb
√
g nb



√
g nb


ǫM
p + ν − µm





 ,

(4.10)44and λα (p)Ĥ ′ = 00λβ (p)

Diagonalizing Ĥ, we get the two eigenvalues
λα,β (p) =.


(4.11)

hf (p) + hm (p) 1 q 2
4g nb + (hm (p) − hf (p))2 ,
±
1
2




(4.12)



where we have defined hf (p) = ǫFp − µf + Vbg nb , and hm (p) = ǫM
p + ν − µm ,and the unitary eigenvector matrix


 Aα

U =




Bα 

Aβ Bβ

.


(4.13)The transformation in eq. 4.5 can then be written as A = U † B, and its inverse
B = UA.
Our goal now is to write the densities ηm,f,mf (p) in terms of the known
densities ηα,β (p). In component notation, (where Ai = ap , etc.), we can writehA†i Al i = hBj† Uji (U † )lj Bk i = Ulj Uij∗ hBj† Bj i,(4.14)

where we have used the fact that since the Hamiltonian is diagonal in the B basis,then hBj† Bk i = hBj† Bj iδjk .
Using this formalism we get the relations:ηf (p) = |Aα |2 ηα (p) + |Bα |2 ηβ (p)(4.15)

ηf (p) = |Aβ |2 ηα (p) + |Bβ |2 ηβ (p)ηf m (p) = A∗α Aβ ηα (p) + Bα∗ Bβ ηβ (p).
Using these expressions in conjunction with eqs. 4.4.a-4.4.e will then allow
us to compute the equilibrium properties of the system.

45
4.1.2

Analysis and Results

For concreteness, in this section, as well as in some of the following ones, wewill consider a free gas of fermionic 40 K atoms, with a density of 8.2 × 1014 cm−3 ,corresponding to the Thomas-Fermi density of 106 such atoms in the center of a√
100Hz spherical trap. The coupling term g nb in equation 4.10 is the perturbative
expansion parameter for the problem, and since it has units of energy, it must becompared with the characteristic non-perturbed energy of the gas, which in thiscase is Ef . Also since in the perturbative expansions it always appear squared(see eq. 3.11), we can define the unitless small parameter for the system asǫSM = g 2 nb /Ef2 = g 2nf /Ef2 rbf , where rbf is the boson to fermion density ratio.
For the 492G resonance in table 3.1, we have ǫSM = 6.35 × 10−2 rbf , and since wewill be mostly concerned with values of rbf generally smaller than 10 (106 87 Rb
atoms correspond to a bosonic density in the center of the same 100Hz trap about
six times larger than the fermionic one i.e rbf = 6), the small parameter is of order10−1 , appropriate for perturbative treatment.
Figure 4.1 shows the quasi-particle energy levels λα,β (kf ), as a functionof detuning. On the leftmost and rightmost part of the graph we see that the
two lines asymptote to the detuning (mean field molecular energy) and 0 (atomic
internal energy). Looking in the region near the resonance we see that the two lines
avoid crossing each other, and from the form of the Hamiltonian in equation 4.10
we can easily deduce that the avoiding is due to the coupling term g 2 nb , which,for a specified resonance, implies that the size of the crossing is proportionallyrelated to the density of condensed bosons in the gas.
If we consider a gas with more fermions than bosons, we easily see that on
the left side of the plot, both states are populated, and there exist a well defined
atomic Fermi surface, as well as a well defined molecular Fermi surface. However,

46

10

8

6

2

(E − 2pm + µ)/µ

4

2

0

−2

−4

−6

−8

−10
−10

−8

−6

−4

−2

0

2

4

6

8

10

ν /µ

Figure 4.1: The thick solid lines represent the “renormalized” mean-field energy
levels λα,β (kf ) , while the thin dashed lines represent their bare counterparts.

47
if the density of bosons is much larger than the density of fermions, the crossing
may be so large, that the higher state is energetically inaccessible, and only thebottom state is ever populated, such that a single Fermi surface is ever populated.
This implies that the atoms on the large positive detuning side of the resonance“morph” into molecules as the detuning is shifted, to the negative size, and it is
no longer strictly appropriate to talk about atoms or molecules separately in thisregime.
To illustrate this difference, we start by an example problem, in which weturn off the interactions (i.e. g, Vbg , γ → 0). Using this simplified system, westudy an example of a gas with two Fermi surfaces, rbf = 0.6, and one with one
Fermi surface, with rbf = 6. Results appear respectively in figures 4.2 and 4.3,
where we plot the chemical potentials and populations as a function of detuning.

At zero temperature, the chemical potential in a Fermi gas, represents its
Fermi energy. Analyzing the top panel of figure 4.2, starting from the positive
detuning region, we see that the molecular and fermionic chemical potentials coincide, which is reasonable, since the chemical potential for a condensate vanishes atzero temperature. Plotted in the graphs are also the internal energies of fermionsand molecules, which for the noninteracting gas are equal to 0 and ν respectively. In the large positive detuning regime, the only Fermi surface present is
the fermionic one, since µf > 0 (and 0 is the internal energy of the fermionic
atoms), while µm < ν, which implies that it is energetically not favorable to create a molecule. As the detuning diminishes, and becomes smaller than 1 (region
marked “a” in the graph), it starts to “cut” into the energetically favorable regime,
so that creating low-momentum molecules becomes favorable to high momentum
atoms. This reduces the number of atoms present in the gas, and causes µf = µm
to diminish. This process continues until there are no bosons left (region marked

48

nb < nf2µ /E f

1
0
−1
b

a

−2
−3
−2

−1.5

−1

−0.5

0
ν /E f

0.5

1

1.5

2

1.5

2

0.8
b

a

N /N t ot

0.6

0.4

0.2

0
−2

−1.5

−1

−0.5

0
ν /E f0.5

1

Figure 4.2: Equilibrium chemical potentials (top panel) and populations (bottom
panel) as a function of detuning for a non interacting gas with rbf = .6. The solidlines represent fermions, dashed lines molecules, and dashed-dotted lines bosons.
The dotted lines in the top panel represent the bare molecular and fermionicinternal energies, respectively ν and 0. The vertical lines labeled a) and b) are
discussed in the text, and represent the detuning at which molecular formation
begins and ends, respectively

49nb > nf2µ /E f

1

0

−1

−2
−2

−1.5

−1

−0.5

0
ν /E f

0.5

1

1.5

2

−1.5

−1

−0.5

0
ν /E f

0.5

1

1.5

2

1

N /N t ot

0.8
0.6
0.4
0.2
0
−2

Figure 4.3: Equilibrium chemical potentials (top panel) and populations (bottom
panel) as a function of detuning for a non interacting gas with rbf = 6. The solidlines represent fermions, dashed lines molecules, and dashed-dotted lines bosons.
The dotted lines in the top panel represent the bare molecular and fermionicinternal energies, respectively ν and 0.

50“b” in the graph), and no more molecules can be formed. Since there are nobosons left in the gas, the bosonic chemical potential can become negative, andthis allows µf and µm to separate, so that two separate Fermi seas are created: amolecular one, from ν to µm , and an atomic one, from 0 to µf .
The case of only one Fermi surface, is represented in figure 4.3. Like theprevious case, no molecules are generated until the detuning cuts into the atomic
Fermi gas. Since there are enough bosons to turn all the fermions into molecules,
the atomic and molecular chemical potentials move in unison. When they crosszero, then no more fermions can exist in the gas, since µf < 0 (the atomic internalenergy), and µm > ν, (the molecular internal energy), making molecules the only
energetically accessible state.
Since, under standard experimental conditions, in the center of the trap
the bosonic density is larger than the fermionic one, we only analyze the one
Fermi-surface case in the mean-field approximation. Figure 4.4, shows the equilibrium chemical potentials for the system obtained via a self-consistent solutionof equations 4.16 and 4.4.a-4.4.e.
The first main difference which appears when comparing the graph in figs
4.4 with its non-interacting equivalent in fig 4.3, is the fact that µb > 0, causes a
split between the molecular and fermionic chemical potential. At first sight thismay suggest the existence of two Fermi surfaces, as it was the case in fig 4.3.
The reason why µb is nonzero, is a consequence of equation 4.4.d, whereby theboson-boson interaction energy generates a shift. In physical terms, this meansthat there is an energy cost in maintaining bosons unpaired, and therefore weneed to take this into account in the kinematic analysis, by realizing that whilein the noninteracting case pairing becomes energetically favorable when ν crossesµf , in the interacting case this happens when ν crosses µf + µb . Therefore theonly consequence of this interaction energy is to shift the molecular population

51

2µ /E f

1

0

−1

−2
−2

−1

0

1
ν /E f

2

3

4

−1

0

1
ν /E f2

3

4

1

N /N t ot

0.8
0.6
0.4
0.2
0
−2

Figure 4.4: Mean-field equilibrium chemical potentials (top panel) and molecular
fraction (bottom panel) as a function of detuning. In the top panel, solid lines
represent fermions, dashed lines molecules, and dashed-dotted lines bosons. In thebottom panel, the solid line represents the full mean-field result, while the dasheddotted line represents results obtained using the approximations in described in
the text.

52
curve up and to the left by exactly and amount µb (see bottom panel of fig 4.4).
Another substantial difference between the interacting and noninteracting
case, is that molecule creation takes place in a much narrower range of detunings
in the noninteracting case. This can be understood by the fact that the avoided
crossing makes for a gentler sloping internal molecular energy than the bare detuning. This will be emphasized even further when we will discuss the beyond
mean-field case, where the molecular binding energy slopes even more gently.
The last goal of this section is to introduce some approximations to facilitate the beyond mean-field approach to the problem. These approximations havebeen tested numerically, and they give corrections of the order of .1% or less incalculated molecular populations for all regimes of interest here.
• Vbg can be disregarded• As per the arguments above, the interaction γn3b is included by appropriately shifting the detuning, and the molecular potential.
• hηb f (p)i (analogous to the boson polarization operator in the Green function formalism) can also be disregarded, since its contribution to µb is
much smaller than that of γn3b .
A comparison plot of the full calculation and the approximate calculation of themolecular population as a function of detuning appear in the second panel of
figure 4.4. With exception of the last item, and some aspects of the first, weare unable to directly test the validity of these approximations in the generalized
beyond mean-field theory which we introduce in the next chapter. However, wehave reason to believe that this approximations will be valid also, since, as we willsee, the generalized mean field theory is, after all, a mean-field theory at heart.

53
4.2

Non Equilibrium Approach
Having discussed the equilibrium mean-field properties of the system we will

now develop the Hartree-Fock-Bogoliubov equations of motion, as a first attempt
to gain beyond-mean-field insight. We will then notice that the peculiar physicalstructure of the problem is such that the HFB equations reduce to the mean-field
equations, and higher order correlations play a peculiarly important role in thephysics of mixture.
4.2.1

The Formalism

We now move on to derive the Heisenberg equations of motion for the many
body system. The way this is done is to find equations of motion for correlation
functions, fs (x1 , ....xs ), which represent the probability of finding s particles at
positions x1 , ..., xs . As it turns out, the equation of motion for the correlation
function f1 will depend on the function f2 , which in turn will depend on f3 , andso on all the way to fN , where N is the total number of particles in the system.
This is known as a Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy[66]. In practice we will be concerned with momentum space correlation functions,but the idea is the same.
Given the large number of particles in the system, it is impossible to calculate equations of motion for all correlation functions, and we need to invoke
an approximation. In practice, correlation functions are often calculated only
up to two-body correlations, s = 2. This is justified under the assumption thatinteractions are suitably “weak.” Higher-order correlations are included in an
approximate way by considering, not the actual atomic constituents, but rather
combinations called quasiparticles. The quasiparticles are defined to be noninteracting, so that their higher-order correlation functions can be written in terms of

54
second order correlation functions [64].
Using this qualitative idea we proceed to develop a more formal understanding. In statistical field theory, given an operator O, and Hamiltonian H, we definenothe thermal average of O with respect to H as hOiH = 1/ZT r Oe−βH , wherenβ = (kB T )−1 is the inverse temperature, and Z = T r e−βHois the partition

function. In this framework, the 1-particle correlation function is defined as thethermal average of the number operator, with respect to the Hamiltonian of thesystem.
In the quasiparticle representation, we define the annihilation operator for
quasiparticles as a′ , reminding ourselves that it is a complicated function ofa, a† , b, b† , c, c†. In momentum space, the 1-particle correlation function in thisrepresentation, will then be ha′p1 a′p2 iHqp , where Hqp is the (noninteracting) quasiparticle Hamiltonian.
Now we introduce the real approximation, namely that the quasi-particles
can be written as linear combinations of all possible products of two operators(except for averages involving one fermionic and one bosonic operator, which areeasily shown to vanish). The procedure is then to find the Heisenberg equationsof motion for these pairs of operators, and then average over the quasi-particle
Hamiltonianih̄∂
hOiHqp = h[O, H]iHqp ,∂t(4.16)

which, being Gaussian, allows us to invoke Wick’s theorem to decompose all higher
order correlations in 1-particle correlations, thus truncating the BBGKY hierarchy.

55
4.2.2

The Equations of Motion

Before generating Heisenberg equations, we need to take a little care in thetreatment of the Bose field, to properly treat the condensed part. To this end
we perform the usual separation of mean field and fluctuations of the Bose field,
substituting b0 ( the zero-momentum component of the Bose gas) with a c-number
φ = hb0 iHqp , and identifying it with the condensate amplitude, while hbp6=0 iHqp = 0
are the fluctuations. We insert these definitions in the Hamiltonian in eq. (3.2),
then proceed to calculate commutators.
Since we wish to consider a homogeneous gas, the correlation functions
f1 (x, x′ ) can be written in terms of a relative coordinate y = x − x′ . Thus in
momentum space f1 (p) is the probability to find a particle with momentum p in
the gas, or in other words it is the momentum distribution of the system.
Having taken all appropriate commutators, and applied Wick’s theorem,(for more details on the procedure see [67], or Appendix B for the derivation ofa sample equation.), we obtain the following self consistent set of equations of
motion for the system:ih̄∂
φ = Vbg ρF φ + γ (2φρ̃B + ∆B φ∗ ) + gρ∗M F + γ|φ|2 φ (4.17.a)
∂t

h
i
∂
2
η˜B (p) = 2γ ℑm κB (p)(φ∗ + ∆∗B )
∂t
h
i
∂
2
ih̄ κB (p) = ǫB+2V
ρ+4γ(|φ|+ρ˜)
κB (p) +
bg
F
B
P
∂t

h̄

γ(2η˜B (p) + 1)(φ2 + ∆B )∂
ηF (p)∂t∂ih̄ κF (p)∂t∂h̄ ηM (p)∂t∂ih̄ κM (p)∂th̄= −2g ℑm(φηM F (p))=

h(4.17.b)(4.17.c)(4.17.d)iǫFp + 2Vbg (ρ˜B + |φ|2) κF (p)(4.17.e)= 2g ℑm(φηM F (p))(4.17.f)ǫM
P + ν κM (p)(4.17.g)

=

h

i

56
ih̄

ih̄

h
i
∂
2
ηM F (p) = ǫFp − ǫM
−
ν
+
V
(
ρ
˜
+
|φ|
)
ηM F (p) −
bg
B
p
∂t

∂
κM F (p) =∂thgφ∗ (ηF (p) − ηM (p))(4.17.h)i2ǫFp + ǫMp + ν + Vbg (ρ˜B + |φ| ) κM F (p) −g [φκF (p) + φ∗ κM (p)] ,(4.17.i)where η˜B (p) = hb†p6=0 bp6=0 iHqp is the momentum distribution of non-condensed
bosons, and ρ̃B =R

dp 2
p η˜B (p) is the density of non-condensed bosons; κB (p) =2π 2

hbp6=0 bp6=0 iHqp is the anomalous distribution of bosonic fluctuations, and ∆B =R

dp 2
p κB (p)2π 2the anomalous density. Similarly ηF,M (p) are the fermionic andmolecular distributions, ρM,F the densities, and κF,M (p), and ∆F,M the anomalous
molecular and fermionic distributions and densities. Finally ηM F (p) = hc†p ap iHqp
and κM F (p) = hcp ap iHqp are the normal and anomalous distribution for moleculefermion correlation, with the associated densities ρM F and ∆M F .
4.2.3

Analysis and Results

Equations (4.17.a-4.17.i) describe the complete self-consistent set of HFBequations for the resonant BF mixture. Inspection of these equations, however,allows us to simplify the set quite dramatically, without sacrificing almost any ofthe physics thereby contained. First, we notice that the evolution of the anomalous fermionic densities κM F (p), κF (p), and κM (p) is entirely decoupled from theevolution of all other quantities, and can therefore be considered separately. Thisimplies that, since we are mainly interested in the evolution of the normal densities, we can eliminate without approximation all the anomalous ones.
The next thing we notice is that the evolution of the normal and anomalous
bosonic averages is completely independent of the resonant interaction, and iscontrolled only by the background interactions between bosons and with fermions.
For typical background interaction strengths, and cold enough temperatures, it is57
well established that the role of noncondensed bosons is minor, and the system iswell described at the Gross-Pitaevskii level of approximation.
We can therefore write the following reduced set of equations:ih̄∂
φ = (Vbg ρF + γ|φ|2)φ + gρ∗M F∂t∂
ηF (p) = −2g ℑm(φηM F (p))∂t∂h̄ ηM (p) = 2g ℑm(φηM F (p))
∂t
i
h
∂
2
ih̄ ηM F (p) = ǫFp − ǫMηM F (p) −
−
ν
+
V
|φ|
bg
p
∂t
h̄

gφ∗ (ηF (p) − ηM (p)) .

(4.18.a)(4.18.b)(4.18.c)(4.18.d)

Together with the prospect of simulating time dependent experiments, sucha set of equations allow us to calculate many characteristics of the system, which
we could use to understand further physics or, more importantly at this stage, totest the theory against our knowledge of the system in various limits.
A relevant quantity we can calculate to this end is the binding energy ofthe molecules. This can be done by an instantaneous jump of the detuning fromlarge and positive values, where we know the equilibrium distributions very well,
to some other arbitrary value. The system thus perturbed oscillates at a specificcharacteristic frequency, which is identified as the (unique) pole of the HFB many
body T-matrix of the system. For negative detunings, as shown below, this polecorresponds to the binding energy of the molecules, dressed by the interactions inthe system.
Figure 4.5 shows a representative example of time evolution of the condensate population (number conservation guarantees that all three populations
oscillate with the same frequency) under the conditions described above. In thisparticular example, at time t=0 the detuning is suddenly shifted a magnetic field

58
detuning of approximatively 0.1G. The response of the population shows an envelope function, indicated by the gray shaped area, that arises from nonlinearities
in the equations of motion. The inset shows that under this envelope is a welldefined sinusoidal oscillation.
The nearly monochromatic character of the response is made clearer by
Fourier transforming the time-dependent population. The Fourier transform shownin the second panel of Fig 4.5 is strongly peaked at 5.4×10−6 K. Similarly, the position of the peak in the frequency spectrum, for different final detunings, should
map the molecular binding energy as a function of magnetic field.
Figure 4.6 shows the results obtained by this method. This plot represents
the binding energy of the molecules, dressed by the interactions in the system.
This dressing is expected to be weaker for smaller densities of atoms and molecules.
In this limit, we should thus recover the two-body molecular binding energy,
obtained via coupling calculations (solid line in fig 4.6), and which, as describedin chapter 3 can be well approximated by our model.
Instead we see that the pole behavior approaches the bare detuning (dashedline in fig 4.6), indicating that the renormalization of the binding energy obtainedat the presented level of approximation is inadequate to correctly include thetwo-body physics. This behavior is in sharp contrast to the Bose-Bose resonantinteraction, where the correct binding energy is preserved at the HFB level [38].
This is also true for the Fermi-Fermi case [68].
This discrepancy is due to the fact that the creation of molecules requires theformation of correlations between bosons and fermions, which, as shown in the following, cannot exist if the density matrix is assumed to be Gaussian. Specificallywhat is required is a more careful consideration of the noncondensed bosons.

59

Population

Population

0.84
0.828

0.824

1.0

0.83

1.1

1.05

time (ms)

0.82

Intensity (arb. units)

0

2.5

5
time (ms)

10

7.5

1
-1

10

-2

10

-3

10

-4

10

-8

10

-7

10

-6

10
Energy (K)

-5

10

-4

10

Figure 4.5: The top panel represents the time evolution of the population ofcondensed atoms after detuning is suddenly shifted from infinitely positive to 5.1 K (−.1G magnetic detuning) around the 659.2G resonance in 87 Rb −40 K
described in table 3.1. The bottom panel shows the absolute value of Fouriertransform of said time evolution. The main peak in this graph represents thecomputed value of the binding energy, which we see is about 5.410−6 K. The
system under consideration is composed of fermionic densities of 1015 cm−3 , for aconstant density ratio of six bosons per fermion.

60

Figure 4.6: Plot representing the poles of the scattering t-matrix for the 659.2G
resonance in 87 Rb-40 K described in table 3.1. The dotted line represents the “bare”
molecular detuning as a function of field, as defined in the text. The solid blackline is the correct binding energy of the molecular state, obtained by means of fullclose coupling calculations, while the gray solid lines are the eigenenergies obtained
from equations 4.18.a-4.18.d, for different atomic densities. From top to bottomon the right the grey lines refer to fermionic densities of 1013 cm−3 ,1014 cm−3 , and1015 cm−3 , for a constant density ratio of five bosons per fermion. We note thatfor lower and lower densities the calculated binding energy incorrectly approaches
the bare detuning instead of the correct two-body binding energy.

61
4.3

The Importance of Noncondensed Bosons
The reason for the failure of the HFB theory is not immediately clear fromthe theory itself. To bring out the inadequacy of this theory in the dilute limit,
in Chapter 5, we recast the problem in an alternative perturbative form that can
reproduce the correct behavior in the two-body limit. This path integral approach
will also lay bare the role of noncondensed bosons.
What we will see in the upcoming analysis may be qualitatively understoodin the following simple terms. A molecule in the gas can decay into a pair of “virtual” (i.e. non energy conserving) atoms, which can then meet again and reformthe molecule. These events modify the behavior of the molecule, and an appropriate treatment of these virtual excitations is therefore necessary to correctly
include the two-body properties of the molecules in the many-body theory. In
particular, the molecules can decay forming a virtual non condensed boson, andthe contribution of this set of events to the physics of the molecules turns out tobe very important. An appropriate theory would therefore consider the couplingof the molecules to non condensed bosons explicitly, which implies that one has toinclude in the equations of motion three-point averages, such as hc†q a−p+q/2 b(p+q/2) i.
Since the HFB theory disregards three-point averages, it only contains moleculeatom-atom couplings of the form hc†q aq iφ0 , where molecules can only decay forming
a condensed boson.
It is straightforward to see that the HFB theory treats 3-body correlation
functions differently depending on the quantum statistics of the constituents. Fora Bose-Bose mixture, the correlation function is approximated (schematically) byhbbm† i ≈ hm† ihb(−q)b(q)i + 2hbihbm† i,(4.19)

where b is a boson and m is a molecule. The first term of the right of this expression
allows explicitly for virtual bosonic pairs of arbitrary momentum, provided that62
the molecular field hmi accounts for most of the molecules, which is assumed tobe the case. Similarly, in a mixture of distinct fermions, the correlation function
reads
hf1 f2 mi ≈ hmihf1 (q)f2 (−q)i,(4.20)and the same argument applies, since the molecules are bosons.
For the Bose-Fermi mixture, on the other hand, the correlation function
would be approximated by
hbf mi ≈ hbihf mi + hf ihbmi + hmihbf i.

(4.21)The required virtual atom-atom pairs would arise from the third term on theright-hand-side of this expression. However, these molecules are fermions, whichhave no mean field, hmi = 0. The only surviving term is then the first one, which
accounts only for condensed bosons, and somehow correlates the fermionic atoms
to the fermionic molecules. This is only an indirect way to get the bosons andfermions correlated.

Chapter 5
Beyond Mean Field

In chapter 4 we reached the conclusion that the mean-field approach to theresonant Bose-Fermi system does not properly account for the correct two-body
physics of the system. In this chapter we wish to improve on this, by introducing
a generalization to mean-field theory, via an appropriate renormalization of themolecular propagator, which is able to reproduce the correct two-body physics inthe low-density limit. To accomplish this, we will have to abandon the Hamiltonian treatment of the previous chapter, in favor of a perturbative approach basedon the Green’s function formalism, much as was done for two bodies in chapter 3.
We begin this treatment, by first describing the effect of the many-body
medium on the collisional properties of the constituents. We will find that thekinematic implications of Pauli blocking play an important role in the physics ofthe stability of the molecules. We then proceed by introducing a generalized meanfield theory, which we solve self-consistently to calculate equilibrium moleculardensities and distribution in the gas, as a function of detuning.
5.1

Scattering in the Medium
The behavior of the atom pairs in a resonant BF mixture depends subtly

on the momentum of the pairs. Roughly speaking, as discussed in chapters 2 and3 for a single, free pair of atoms, a true molecular bound state exists on one side64of the resonance, denoted as the “negative detuning” side. On the other, “positive detuning,” side, the pair is not rigorously bound but may exhibit resonant
scattering. (see fig. 3.2). For free molecules this demarcation at zero detuning
between bound and resonant states is clearly independent of the pair’s center-ofmass momentum. In the many-body environment, however, this situation changes
dramatically. We will show that slowly moving molecules can be stable against
decay even on the positive detuning side of resonance. The reverse is also true:
pairs that are moving fast enough will become unstable and exhibit only resonances on the negative detuning side, even though their two-body analog would
be completely stable. This unusual behavior is connected to the fact that thepairs are themselves fermions, and must obey the correct Fermi statistics. This
is of course different from the case of either a boson-boson or fermion-fermion
resonance, where the pairs are always bosons.
In this section we will consider only one aspect of BF mixture near FR,
namely how the stability of a composite fermionic molecule will be affected by
the many-body medium. To do this we will assess the poles of the many-body
T-matrix of a molecular pair propagating in the many-body medium. All the
calculations in this section refer to the 659.2G resonance in 87 Rb −40 K describedin table 3.1
5.1.1

Poles of the T-Matrix: The Two-body Case Revisited

The mean-field analysis in section 4.1.2 led us to conclude that the role ofVbg close to resonance is negligible with regards to the many-body properties ofthe system. In this section we present once more the analysis of section 3.3.1, andmodify it to include this approximation, to establish a closer connection to the
many-body analysis which follows. In this light, as Vbg → 0, after the appropriate

65
renormalization, equation 3.14 becomes
T2B (E) = −

2πh̄
1
q
,
mbf −a−1 + r0 mbf E − i 2mbf E(5.1)

where r0 = −2π/m2bf g 2 is the effective range of the interaction resulting from theunderlying model. The main difference in the pole structure, as anticipated in
section 3.3.1, is that the resonant state crosses threshold for the positive detuningν = µ < g 4m3bf /4π 2 , where we have introduced the chemical potential µ to provide
a direct link to the many-body analysis which will follow.
At the same detuning, there emerges a positive imaginary part of the poleenergy (dashed line), which denotes the energy width of the resonance. Interestingly, for detunings 0 < ν/µ < g 4m3bf /4π 2 , the poles of (5.1) are purely imaginary,and the imaginary part is negative. These poles stand for physically meaninglesssolutions to the Schrödinger equation, in which the amplitude in the resonant
state would grow exponentially in time, rather than decay. These poles do nottherefore identify any particular features in the energy-dependent cross section of
the atoms. There is a characteristic detuning scale on which these events occur.
3/2 √This scale is given by the width parameter γ 2 = g 2Mbf / 2π [60], also indicatedin the figure.
5.1.2

Poles of the T-Matrix: The Many-body Case

In a many-body environment, the T -matrix and its poles depend on thecenter-of-mass momentum. The importance of taking into account this dependence was demonstrated in [69] for the BCS-BEC crossover at T 6= 0 when noncondensed fermion pairs lead to pseudogap effects above Tc and non BCS- behavior
below. The influence of the many- body medium on a Feshbach resonance between
two fermions was demonstrated in [70] for a composite boson as well. In the BF
mixture the composite object is a fermion, so its momentum dependence cannot

66

40
2

20

2

(E-P /2m+ )/

/2

0

-20
-40

-20

0

20

40

/

Figure 5.1: Complex poles of the two-body T-matrix, as a function of detuning.
Solid and dashed lines denote the real and imaginary parts of physically relevant
poles, respectively. The dotted and dash-dotted lines are real and imaginary partsof unphysical poles, respectively.

67
be omitted even for T = 0. Thus the T -matrix near the FR in a BF medium must
be considered as a function of densities of both the Bose and Fermi subsystems
as well as the center-of-mass motion of a given BF pair. We have found that
T-matrix poles and residues are quite sensitive to all these ingredients.
The T -matrix T̂ of the system in the many-body medium is defined, similarly
to the two-body version case discussed in chapter 3 by the Lippmann-Shwinger
equation:T̂ = ĝ D̂ gˆ† = ĝ D̂ 0 gˆ† + ĝ D̂ 0 gˆ† GˆB GˆF T̂ ,(5.2)

ˆ is the bowhere D̂ is the renormalized molecular Green function, and GB/F
son/fermion renormalized Green function. These quantities are defined in termsof their many-body non-interacting counterpartsD̂ 0 =
Ĝ0 F =1ω − ξ M (p) + iη sign(ξ(p))

1ω − ξ F (p) + iη sign(ξ(p))
1,Ĝ0 B =Bω − ξ (p) + iη

(5.3)

where ξ M,F,B (p) = ǫM,F,B− µM,F,B , and the role of η is described in depth inpappendix A), by the self-consistent set of equationsD̂ = D̂ 0 + D̂ 0 Π̂D̂GˆB = Gˆ0B + Gˆ0B ĝ GˆF D̂ gˆ† GˆB
GˆF = Gˆ0F + Gˆ0F ĝ GˆB D̂ gˆ† GˆF
where Π̂ is the molecular self energy. All quantities are functions of the energy
E and center-of-mass momentum P of the molecules. The complete solution tothese equations is beyond current computational capabilities, except perhaps byMonte Carlo methods.
We therefore make a few simplifying assumptions, namely we account for
propagation of the atomic fermions and bosons using their free Green functions

68
only (i.e., setting GˆF ≈ Gˆ0F and GˆB ≈ Gˆ0B ). An important consequence of thesechoices is that the many-body T -matrix is approximated by its ladder series, whichmeans that Π̂ ≈ g Gˆ0B Gˆ0F g † analogously to that introduced chapter 3, but obtained
integrating the free Green functions in eq. 5.3. This standard approximation has
the property of being exact in the two-body limit, where Eq. 5.1 satisfies 5.2.
This implies that the two-body physics is accounted for exactly in the many-bodyproblem.
Another appealing characteristic of this approach is that the molecular selfenergy Π can be calculated exactly [71], leading to the following expression
2

2

2



mb k 2



m2 P

g
g
f
− mbfb − mPb D
Π(E, P ) = πg 2 mbf Λ − 4π
2 mbf kf − 8π 2
P


√


√
(kf +P mbf /mb )2 −D(kf + D)2 −(P mbf /mb )2g2
ln (kf −P mbf /mb )2 −D + 4π2 mbf D ln (k −√D)2 −(P m /m )2 ,f

where D =

bf

b

(5.4)

q

2mbf (E − P 2 /2(mf + mb ) + µ), and Λ is the ultraviolet cutoff in-troduced in chapter 3, which in the Vbg → 0 approximation used here can be2

regularized by a shift in the detuning ν → ν − πg 2 mbf Λ.
Further approximations include considering a homogeneous system where
the density of the fermionic subsystem is much larger than that of the bosonic
one, whereby the avoided crossing between atomic and molecular state introducedin chapter 4 becomes approximately a real crossing. This condition is introduced
for the purposes of this section, to allow us to isolate the effects of Pauli blocking
on the collision physics, and will be relaxed later, when we study the equilibrium
properties of the system.
The set of equations we obtain with these approximations is therefore
T (E) = g 2D(E),(5.5)

where D(E) is the pair propagator defined by:
D(E) = D 0 (E) + D 0 (E)Π(E)D(E),

(5.6)

69
Π(E) = g 2G0B (E)G0F (E),which lead to
T (E, P ) =g2,
P2− ν + µ − Π(E, P )E − 2m(5.7)

where m = mf + mb is the molecular mass.
In order to understand the stability of a BF molecule, we study the structures of the poles and residues of the T-matrix in Eq. (5.7). To determine numerically the poles, we consider this equation as a system of two nonlinear equations forreal and imaginary parts of the internal energy, E − P 2 /2(mf + mb ) + µ. The resulting nonlinear system of equations always has some unphysical solutions which
can be rejected by the following analysis: i) the residues for physical solutions,which represent the distribution of population in that state (see appendix A)
must be less than unity; ii) for the imaginary solutions the relative momentum,D=q

2mbf (E − P 2 /2m + µ) in Eq. (5.4), should dwell on the lower half of thecomplex momentum plane, to ensure that resonances decay with time [72]; iii)
the sum rule (5.11, see below) should be fulfilled including both the discrete andcontinuum parts.
As an example, we have calculated the poles of (5.7) for a dual species
gas with fermion density 1013 cm−3 and boson density 1014 cm−3 . The results
are shown in Figure 5.2 for two representative center-of-mass momenta P of themolecules. (Note the different scale from Figure 5.1). Unphysical poles are notshown in this figure. These results are cast as a kind of “binding energy,” bysubtracting the center-of-mass kinetic energy, and adding the chemical potentialµ. They can therefore be compared directly to the two-body results in Fig. 5.1.
The structure of these poles is quite different near resonance, although westress that far from resonance (ν ≫ γ 2 /2) they return to the two-body values. A
main difference from the two-body case is that now there may exist physical poles

70(a) P = k

f

2

0

2

(E-P /2m+ )/

4

-2

-4
-20

-10

0

10

20

10

20

/

(b) P = 0.1k

f

2

0

2

(E-P /2m+ )/

4

-2

-4
-20

-10

0

/

Figure 5.2: Complex poles of the many-body T-matrix, as a function of detuning.
Solid and dashed lines denote the real and imaginary parts of physically relevant
poles. In (a) the center-of-mass momentum of the molecule is P = kf , i.e., equal
to the Fermi momentum of the atomic gas. In (b), this momentum is P = 0.1kf .

71
for any detuning across the resonance, depending on momentum. The appearanceof poles depends, however, on the center-of-mass momentum P of two atoms. If
P ≫ 2kf , the two-body physics will not be influenced much and the poles structure
will be as it is in Fig. 5.1. We have verified this numerically, but have not shown
it in this thesis.
Figure 5.2(a) shows the case where the center-of-mass momentum of theatom pair is equal to the Fermi momentum of the atomic gas, P = kf . The
criterion of molecular stability is set by the vanishing of the imaginary part of thepole. Figure 5.2(a) shows that the imaginary part remains zero until ν/µ ∼ −5,
after which the pair becomes unstable. Thus, for detunings −5 ≤ ν/µ ≤ 0, someof the molecules that would have been stable become de-stabilized in the manybody environment. They may still be rather long-lived, however. In the limitof very large momenta, P ≫ 2kf , we find that the pole structure returns to thetwo-body value. This makes sense, since a very rapidly-moving molecule does notinteract strongly with the gas at all. In particular, for P ≥ 2kf , we find thata “gap” reappears, in which no physical poles exist for some range of positive
detuning. In the P → ∞ limit, this gap returns to its two-body value, γ 2 /2.
Figure 5.2(b) shows an alternative case in which the molecular momentum
is much smaller than the atomic Fermi momentum, P = 0.1kf . In this case, theimaginary part of the pole only differs from zero at positive detunings, ν/µ > 8.
Therefore, quite the opposite to the P = kf case, here the molecules that would
have been unstable are stabilized by the presence of the many-body environment.
Roughly, this is due to Pauli blocking of the fermionic atoms into which themolecules would dissociate. If such an atom already occupies the state into which
the molecule would drop its atom, then the process is forbidden. We discuss
this further below. For any molecule with P ≤ 2kf , there exist physical poles
at all detunings. Thus positive-detuning molecules are always present in the BF

72
mixture.
5.1.3

Conditions for molecular stability

To make a more global picture of whether molecules are stable or not for agiven detuning, we can consider the spectral function for atom pairs. In general,
when the molecular propagator possesses an imaginary part, this part alludes
to the decay rate of the pair due to interactions with the rest of the gas, inthe same sense that the oscillator strength of an atom alludes to its decay rateby spontaneous emission. Following a standard approach (see appendix A), thespectral function is defined as1
ρ(P, E) = − ImD(P, E).
π

(5.8)

In the case of a true, bound molecular state, the spectral function reduces to adelta function at the energy of the state E0 :
ρ(P, E) = 2πZ(P )δ(E − E0 ),(5.9)

where the coefficient is given by the “spectral weight” function
Z(P ) =

1.
1 − dRe(Π(P, E))/dE|E0(5.10)

Just as for an oscillator strength, the discrete and continuum parts of the spectral
density must satisfy a sum rule:Z(P ) +Z

dEρ(E, P ) = 1(5.11)

for each momentum P . We have explicitly verified the sum rule in each casewe computed, as a test of the numerical procedure, and to distinguish between
physical and unphysical poles of the T-matrix.
As mentioned above, the spectral weight Z is associated with a specific pole
E0 of the T-matrix, an energy level of the system, and represents its population.

73
Therefore, if Z vanishes, so does the probability of finding stable molecules in thegas. To this end, Figure 5.1.3 plots contours of the function Z(P ) as a function ofdetuning ν and center-of-mass momentum P of the atom pairs. These calculations
are performed for fermion density nf = 1013 cm−3 . The contour of Z = 0 thus
represents the borderline between conditions where molecules exist and are stable(to the left of this line) and where they are unstable to decay (to the right of thisline, in the white region of the graph).
Figure 5.1.3 thus shows that: i) molecules are still stable for a continuumof positive detuning when P is small; ii) molecules that would have been stable atnegative detuning may not be such stable at intermediate momenta P (although at
small negative detuning they may possess small widths); and iii) in the limit P →∞, the borderline between stable and unstable again returns to zero detuning.
Thus far, these are exact results, at least within the simplifying approximations made above. Once this is done, the effective dissociation energy of themolecules within the medium is determined. The relation between the molecule’s
total energy at dissociation and the molecule’s momentum is then easily determined from kinematics, plus simple considerations on the Pauli blocking of theatomic fermions. For example, consider the case where the molecule’s kinetic
energy is greater than twice the atomic Fermi energy, i.e., P 2 /2(mf + mb ) ≥2 × kf2 /2m. At the same time, the molecule is assumed to be exactly at its dissociation threshold, so that it could live equally well as a molecule or as two free
atoms. Upon dissociating, each atom would carry away half the energy, so inparticular the fermionic atom is at the top of the Fermi sea, and this dissociation is not prevented by the Pauli exclusion principle. The total energy of themolecule at its dissociation threshold is then determined simply by the molecular
kinetic energy, and no contribution is required from the molecular binding energy.
Thus, if B represents the internal energy of the pairs relative to threshold, bound

74

7
0.25
6
0.2

P/k

f

5

0.15

4

3
0.1
2
0.05
1

−10

−8

−6

−4

−2

ν/µ

0

2

4

6

0

Figure 5.3: Contours of molecular population under various combinations ofmolecular center-of-mass momentum P and detuning ν. The uppermost contour identifies the detuning at which bound molecules first appear for a given
momentum P . Numbers indicate contours with a equal molecule fraction. The
dots represent the result obtained analytically for the critical detuning, Eq. (5.16).

75
molecules are possible whenB≤0v
uu 2(mb + mf )
P ≥t
k .

for

mf

(5.12)f

Alternatively, suppose the molecules have less than twice the atomic Fermi
energy, P 2 /2(mf + mb ) ≤ 2 × kf2 /2m. Now it is no longer guaranteed thatthe molecules can automatically decay in the many-body environment, since thefermion’s kinetic energy may lie below the Fermi level of the atomic gas. In sucha case, the molecule can sustain a positive internal energy without dissociating,
simply due to Pauli blocking. To decide how high this binding energy can be, we
examine the conservation of energy and momentum in the dissociation process:P = pf + pb
p2b2

(5.13)
p2f

P
+B =
+
.
2(2m)
2mb 2mf(5.14)

Here pf , pb , and P are the momenta of the atomic fermions, atomic bosons,and molecules, respectively. To ensure that the atomic fermion emerges with the
maximum possible kinetic energy, we consider the case where P and pf pointin the same direction. To ensure that pf > kf , where kf is the atomic Fermi
momentum, along with (5.14), implies that molecules are stable whenB≤

(mf P − (mb + mf )kf )2mb mf (mb + mf )

2

for

v
u
u 2(mb + mf )
P ≤t
k .

mf

f

(5.15)

Figure 5.4 shows the internal energy of the molecules evaluated at the stability boundary, as described above, as a function of center-of-mass momentum.
The solid line in this figure is determined numerically from the Z = 0 contourof Fig. 5.1.3. Subtracting the kinetic energy contribution and chemical potentialfrom the pole of the T-matrix evaluated on the contour, we obtain the molecularinternal energy. Also shown, as dots, are the kinematic estimates (5.12,5.15).
Analytical expressions for the detuning at the boundary as a function ofcenter-of-mass momentum, plotted as dots in Fig. 5.1.3, are readily obtainable

76

2

1.5

B/µ

1

Z=0

0.5

0

−0.5

Z>0
−1

−1.5
0

1

2

3

4

5

6

7

P / kfFigure 5.4: The internal energy that a molecule of center-of-mass momentum
P/kf would require to overcome Pauli blocking and dissociate. The shaded area
represents numerical calculations, and the dots represent the analytical kinematicarguments in Eqs. (5.12,5.15).

77
analytically, with similar accuracy by inspecting the denominator of Eq. (5.7).
The total energy of the molecule, measured from the chemical potential, is, infact, given by the pole of (5.7). In general this energy can be written as E0 =P2−µ+B, where B is a complicated function of all the parameters. However,2(mf +mb )2

since Etot is a pole of (5.7), then E0 = 2(mfP+mb ) − µ + ν + Π(E0 , P ), so!

P2− µ + B, P .
νcrit = B − Π2(mf + mb )(5.16)

Plugging the stability boundary value of B from Eqs. (5.12,5.15) into this formula
leads to an analytic, albeit complicated, expression for the critical detuning as afunction of cente-of-mass momentum.
5.2

Many-Body Physics
Having described the effect of the many-body medium on the resonant colli-sion physics, we now proceed to an equilibrium analysis of the system. In chapter
4, we introduced the mean-field theory for the gas; here we will introduce a generalized version of that same theory, which will properly take into account for the
intricacies of the resonant two-body physics.
We begin this discussion by introducing the mean-field theory approach in
the language of Green-Functions, within the limits of the three approximations
we tested in the previous chapter, namely• Vbg → 0
• γn3b → 0• hηb f (p)i → 0 .
The self-consistent Dyson equations that describe this system are:
F
0
2
0
MF
GM
F (E, P ) = GF (E, P ) + g nb D (E, P ) GF (E, P )D M F (E, P ) = D 0 (E, P ) + g 2nb G0F (E, P ) D M F (E, P ).

(5.17)

78
The processes which these equations describe, depicted diagrammatically in fig
5.5, are simply the facts that a free fermion may encounter a condensed bosons on
its path and associate with it creating a molecule, or that a free molecule may split
into a fermion and a condensed boson. The self-consistency ensures that theseprocesses may be repeated coherently an infinite number of times. The omissionof the bosonic renormalization equation φM F = n0b + g 2 nb G0F (E, P ) D 0 (E, P ),whereby a boson sitting in the condensate may pick-up a fermion to create amolecule, is a restatement of the third of the approximations described above.

Figure 5.5: Feynman diagrams included in the mean-field theory. Thin (thick)solid lines represent free (renormalized) fermions, thin double dashed-solid lines
represent free molecules, and thick double dashed lines represent renormalized
molecules. The little lightning bolts represent condensed bosons, whereby thearrow indicates whether they are taken from or released into the condensate.

Solutions to these equations take the formF
GM
F (E, P ) =D M F (E, P ) =1
G0F (E, P )−1 − g 2 nb D 0 (E, P )1

D 0 (E, P )−1 − g 2 nb G0 (E, P ).

(5.18)

Plugging in the definitions of G0F (E, P ) and D 0 (E, P ) from eq 5.3 and solving
for the poles, we notice that we get exactly the mean-field eigenvalues λα,β (p)from chapter 3, appropriately modified to include the three approximations abovementioned.
Another interesting property of this theory, is that it is perfectly symmetric79with respect to interchange of GF and D, which implies that both renormalized
Green functions have the same poles, and the same residues, which means that
we can, for example, study the properties of the fermions by only looking at themolecules. This is not completely striking, since, given that the condensed bosons
are relatively inert, every molecules corresponds exactly to a missing fermion, andvice-versa.
The simple discussion above, in terms of the fundamental processes includedin the theory, highlight numerous limitations, but one is particularly striking,namely that we only allow molecules to decay into a free fermion and a condensed boson, disregarding the possibility that the bosonic byproduct may benoncondensed. The fundamental mean-field assumption is that the gas is at zerotemperature, and therefore the noncondensed population should be negligible atequilibrium. Furthermore, if a molecule is made by a zero-momentum boson anda fermion from the Fermi sea, dissociating into a noncondensed boson implies thatthe outgoing fermion would have momentum lower than the Fermi momentum,
and will need to be “fit” into quite a tightly packed Fermi sea, an event which
Pauli blocking makes quite unlikely.
In this light, we should therefore feel relatively confident about the fact that
if a molecule does indeed decay yielding a non-condensed boson, it should be inorder to recapture it right-away, in a virtual process such as that described in fig.
3.1. It is only convenient that these events are exactly the kind of events which
will correctly renormalize the binding energy of the molecules, leading to a theory
which will reproduce the exact two-body resonant physics.
The Dyson equation describing this theory are:FF
(E, P )
(E, P ) = G0F (E, P ) + g 2nb D(E, P ) GGM
GGMFF

D GM F (E, P ) = D(E, P ) + g 2 nb G0F (E, P ) D GM F (E, P ).

(5.19)

80
where D is the renormalized molecular propagator from equation 5.7. A diagrammatic representation of this theory appears in fig. 5.6, and the solution to theseequations are:F
GGM
(E, P ) =F

D GM F (E, P ) =1
G0F (E, P )−1 − g 2nb D(E, P )1

D(E, P )−1 − g 2 nb G0 (E, P ).

(5.20)

Figure 5.6: Feynman diagrams included in the generalized mean-field theory. Likein the mean field case (fig 5.5), thin (thick) solid lines represent free (renormalized)
fermions, thin double dashed-solid lines represent free molecules, and thick double
dashed lines represent renormalized molecules. The little lightning bolts represent
condensed bosons, whereby the arrow indicates whether they are taken from orreleased into the condensate. The novelty here in the inclusion of the 2-body
dressed molecules from fig. 3.1.
These equations preserve the symmetrical nature of the mean-field theory
described above, and also the two avoiding states structure discussed in the previous chapter. This is demonstrated in figure 5.7, where we reproduce the P = kf
pole of D from fig 5.1, and the corresponding poles for the generalized mean-field
theory. As in figure 4.1 we note the splitting in two energy levels, avoiding eachother around ν = 0. The fundamental difference in this case is that the molecular

81

15

10

p = kf2p(E − 2m+ µ)/µ

5

0

−5

−10

−15

−40

−30

−20

−10

0

10

20

30

40

ν /µ

Figure 5.7: The thick lines represent the poles of D GM F (kf ), and correspondto molecular and fermionic energies. The dashed lines represent the molecular
two-body poles, corresponding to the molecular binding and resonant energies.
Comparison with the top panel of fig 5.2 highlights the fact that, as in the meanfield theory case, the effect of the interaction with the condensate is to create an
avoided crossing between the atomic and molecular states.

82
curve does not asymptote to the bare detuning, but rather to the correct molecularbinding and resonance energies.
Studying the equilibrium properties of the system is now a matter solving
the self-consistent set of equations 4.4.a-4.4.e, remembering to disregard ηmf andλ. To do this we first need to extract the distributions ηf,m from the GreenF
functions D GM F and GGM; to avoid taking a distracting detour here, we referF

the reader to appendix A.
As in the previous section, we will consider a mixture composed free gas offermionic 40 K atoms, with a density of 8.2 × 1014 cm−3 , and a gas of condensed
87

Rb bosons with density 4.9 × 1015 (corresponding to the respective Thomas-

Fermi densities of 106 atoms of either species in the center of a 100Hz sphericaltrap).
As mentioned in Chapter 4, the generalized mean-field theory presented
here is very similar, in spirit, to the simple mean-field theory presented there.
The real difference between the two, is simply a shift in the molecular internalenergy structure, in line with the two-body results of Chapter 3. This means thatthe difference between the two approaches will be quantitiatve only, and that theinterpretation of the results can be done in the same exact way as we did in section
4.1.2.
Figure 5.8 shows the equilibrium molecular population as a function of detuning, for the 492.5G resonance, where, for the conditions mentioned above, themean-field parameter ǫSM = g 2 nb /Ef2 ≈ .38 is indeed perturbative. In this regime,
the agreement between mean-field and generalized mean-field is quite good, unlike
the example in fig. 5.9, where the mean-field parameter is ǫSM = 38.7. The difference in this latter example is due to the fact that the actual molecular bound-stateenergy is actually higher than the bare detuning, by an amount proportional toǫSM . Using the chemical potential arguments analogous to the ones used in sec.

83
4.1.2, this can be shown to hinder molecular formation.
5.3

Non-Equilibrium Mean-Field Theory Revisited
In section 4.2, we outlined the limitations of the non-equilibrium theory, by

claiming that to obtain the correct two-body physics in the low-density limit it
would be necessary to include three point and possibly higher correlations. While
this fact is indeed true, we showed in the previous section that for experimentallyreasonable parameters, there exists a regime where mean-field theory accurately
describes the equilibrium properties.
In and of itself, this argument does not necessarily guarantee that the nonequilibrium approach will also be accurate, however the results do encourage us
to push forward, though the results should be taken with an appropriate amountof caution.
In the following we wish to study molecular formation via a time dependent ramp of the magnetic field across the resonance. To this end we use two
approaches: the first consists of propagating equations 4.18.a-4.18.d, ramping thedetuning linearly in time from a large positive value to a large negative one, andplotting the final molecular population as a function of detuning ramping rate R.
The second approach consists in noticing that if ν(t) is a linear function of time,then the mean-field Hamiltonian (eq. 4.10) is ideally suited to a Landau-Zenertreatment, whereby the final molecular population as a function of detuning canbe readily written as−R

nm /min(nb , nf ) = 1 − e τ(5.21)

Here nm /mim(nb , nf ) is the fraction of possible molecules formed, and R = 1/ ∂B∂tis the inverse ramp rate, and the exponential time constant is given by τ = gh̄δ2 nBb =mbf, where abg is the background scattering length, mbf the reduced mass, hhabg nb ∆B

84

1
0.9
0.8
0.7

N m /N f

0.6
0.5
0.4
0.3
0.2
0.1
0
−10

−5

0

5

10

15

ν /E fFigure 5.8: Equilibrium molecular population as a function of detuning for the
narrow 492.49G resonance. The solid line represents results obtained via thegeneralized mean-field theory presented in the text, while the dashed-dotted line
represents the mean-field results.

85

1
0.9
0.8
0.7

N m /N f

0.6
0.5
0.4
0.3
0.2
0.1
0
−100

−50

0

50

100

ν /E fFigure 5.9: Equilibrium molecular population as a function of detuning for thewider 544.7G resonance. The solid line represents results obtained via the generalized mean field theory presented in the text, while the dashed-dotted line
represents the mean field results.

86
is Plank’s constant, and ∆B is the magnetic field width of the resonance.
Remarkably, the τ does not depend on the fermionic density. This is an artifact due to fact that in mean-field theory the momentum states of the fermionic gas
are uncoupled, except via the depletion of the condensate. Since in the LandauZener approach the depletion is assumed small, it follows that the various fermionicmomentum states are considered independent, and that the probability of transition of the gas is equal to the probability of transition of each individual momentum state. This approximation is only valid for narrow resonances, such as the492.5G resonance in table 3.1.
The two approaches described above agree virtually exactly, and the results for an example set of parameters is shown in fig 5.10. We note from thebottom panel of the figure, that the Landau-Zener model agrees with the numerical simulations even when nb < nf . This is quite surprising, since the width ofthe crossing is proportional to density of leftover bosons, and we expect that thisnumber will change substantially as the bosonic population is depleted via the formation of molecules. This type of time dependent crossing should not be properly
described by the Landau-Zener formula. However, monitoring the time evolutionof the molecular population as a function of time shows that the majority of thetransfer takes place quite abruptly somewhat after crossing the zero-detuning region, whereby the change in bosonic density does not modify the energy levels
substantially.

87

n m /mi n (n b , n f )

1
0.8
0.6
0.4

rbf = 6

0.2
0

0

10

20

30
40
R am p R ate (µ s /G)

50

60

500

600

n m /mi n (n b , n f )

1
0.8
0.6
0.4

rbf = .6
0.2
0

0

100

200

300
400
R am p R ate (µ s /G)

Figure 5.10: Transition probability into molecular state via a magnetic-field ramp
across the 492.5G Feshbach resonance. The solid lines are obtained via numericalsolutions of equations 4.18.a-4.18.d, while the dashed lines represent the LandauZener equivalent. In the top panel the gas is composed of more bosons thanfermions rbf = 6, while in the bottom panel the opposite is true rbf = .6

Chapter 6
Conclusion

In this thesis, we developed and solved a generalized mean-field theory describing an ultracold atomic Bose-Fermi mixture in the presence of an interspecies
Feshbach resonance. The “generalized” attribute refers to the fact that we had
to modify the simple mean-field treatment obtained by neglecting bosonic fluctuations, to include the correct two-body physics.
We also attempted to give a reasonably self-consistent account of the twobody physics involved, and attempted to bridge the often obscure gap between it
and the many-body field-theoretical treatment.
The theory presented, as any mean-field theory, presents undeniable limitations. Nevertheless, any useful many-body treatment must start from a wellconceived mean-field theory, which we provide herein.
In light of these remarks, necessary future directions of this work may include
the generalization to finite temperature, and the inclusion of a trap, initially ina local-density approximation. These advances would be essential to check for
empirical confirmation (or lack there of) of the theory.

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Appendix A
Green Function Methods for Fermions

In this Appendix we briefly introduce some of the Green function techniquesthat we found useful in our calculations.
A.1

Free Green functions
We start from the Green function for a gas of free fermions, which is given,in the frequency-momentum representation by
G0 (w, q) =1,
ω − ξ(p) + iη sign(ξ(p))(1.1)

where ξ(p) = p2 /2m − µ. The momentum distribution, at equilibrium, is givenbyn(p) = −i lim+
η→0Z

dω iωη 0e G (w, q),
2π

(1.2)

Here the limit comes from the equilibrium condition; the frequency, in the Greenfunction definition, is the fourier space equivalent of time, whereby the real time
green function represents the evolution of the system from from time t to t’,
and the observables obtained this way, represent expected values of the kindhψ(t)|O|ψ(t′ )i. However, since we want equilibrium conditions, we need to takethe limit t → t′ , which is non trivial, since G0 is defined by a green function
equation of the form L G0 (t − t′ ) ∝ δ(t − t′ ), where L is some linear operator,
and which higlights a peculiar behaviour in the limit we desire. However, since we95know on physical grounds that observables, such as the momentum distribution,must be defined and well behaved at equilibrium, then by first taking the expectation value integral, and then the limit, we can circumvent the problem. In eq.
′

1.2 this implies we cannot quite get rid of the fourier transform exponent eiω(t−t )until after the ω integral.
To perform the integral in 1.2, we exploit the fourier exponent, by notingthat since η is positive, eiωη → 0 as ω → +i∞, so that the integral is identical toa contour integral over the path defined by the real ω axis, closed in the uppercomplex ω plain by an infinite radius semicircle, which, as we have just seen, gives
no contribution to the integral. We can now integrate using the residue theorem.
We note that the integrand in 1.2 has a simple pole at ω = ξ(p)−iη sign(ξ(p)).
Thus, if ξ(p) > 0, then the pole is in the lower complex plane, and the integral
vanishes, and if ξ(p) > 0, then the pole is in the upper complex plane, with residue
1. Using the residue theorem, and summarizing these results we finally getn(p) = Θ(−ξ(p)),(1.3)

which we recognize as the zero temperature fermi distribution.
A.2

Interacting Green functions
According to Dyson’s equation, the green function for an interacting system

has the form
G(w, q) =1,
ω − ξ(p) − Σ(ω, p)(1.4)

where Σ(ω, p) is an arbitrarily complicated function summarizing all the interactions in the system, which is known as self energy.
The prescription to find Σ is quite straightforward, and it consists of adding
all amputated connected feynman diagrams for the system. The fact that, in general, the number of such diagrams is infinite, makes this task virtually impossible.

96
Nonetheless, eq. 1.4 is very powerful, since it allows one to include the effect of
infinite subsets of the total number of diagrams in the system, by only having to
explicitly calculate a few representative ones.
An alternative standard approach, leads to the exact result (NOTE: Abrikosov[83] measures energy from µ, here we measure from 0, which is more standard)G(ω, p) =

Z ∞
0

dω

′

"

#

B(ω, p)A(ω, p),+ω − ω ′ + iη ω + ω ′ − iη

(1.5)

Where A and B are, again, arbitrary complicated functions, though they are
known to be finite.
To understand A and B more closely, we need to introduce the following
well known identity:
1
1= P ∓ iπδ(x),ν→0 x ± iν
x
lim

(1.6)

where P is a Cauchy principal value, which represents the contribution due to a
discontinuity in a Riemann sheet (branch cut), and the delta function represents
the contribution due to the pole.
Applying 1.6 to 1.5 we get
Re G(ω, p) = P
Im G(ω, p) =

Z ∞
0

dω

′

"

B(ω, p)A(ω, p)+ω − ω ′ + iη ω + ω ′ − iη
 −πA(ω, p) πB(−ω, p)if ω > 0

#

(1.7)

(1.8)

if ω < 0Finally, eq. 1.2 represents a fundamental property of green functions, and it
can be generalized to interacting systems simply substituting G0 with G. Applying
it to eq. 1.5, and performing the ω integral first, we getn(p) =Z ∞0′dω B(ω, p) =

Z 0

−∞

dω

−1
Im G(ω, p).
π

(1.9)

Introducing the function ρ(ω, p) = −2ImG(ω, p), generally called spectral function, the above equation can be written asn(p) =

Zdωρ(ω, p)Θ(−ω).
2π(1.10)

97
An important property of the spectral function is that for all p,Zdωρ(ω, p) = 1.
2π(1.11)

This can be understood as a sum rule in the following sense: if we wish to calculate
the number of holes in the system, we would take the η → 0− limit in equation 1.2.
R

The distribution would then have been nholes (p) = 1 − n(p) = 0∞ dω ′A(ω ′ , p), soR

that 1 = 0∞ dω ′ [A(ω ′ , p) + B(ω ′, p)] =

R dω
2π

ρ(ω, p).

Using eq.1.4, toghether with the definition of ρ, we can writeρ(ω, p) =−2ImΣ(ω, p).
[ω − ξ(p) − ReΣ(ω, p)]2 + [ImΣ(ω, p)]2

(1.12)

Furthermore, if Σ were to be real, or if, eqivalently, the pole of the green function
were to be real, for some momentum p, then taking the limit ImΣ → 0 of 1.4,
and using eq,1.6, we getρ(ω, p) = 2πδ(ω − ξ(p) − ReΣ(ω, p)),(1.13)

which can be simplified, using the properties of the delta function, toρ(ω, p) = 2πZ(p)δ(ω − ω0 (p)),(1.14)

where Z, known as spectral weight is given byZ(p) =

1∂ReΣ(ω, p)
1 − ∂ω

,(1.15)

w=ω0 (p)and ω0 (p) is the pole of the green function, defined byω0 (p) − ξ(p) − Σ(ω0 (p), p) = 0.

(1.16)The momentum distribution in this case is thus given byn(p) = Z(p)Z

δ(ω − ω0 (p))Θ(−ω) = Z(p)Θ(−ω0 (p)).

(1.17)

Appendix B
Derivation of Mean-Field Equation of Motion

In this appendix we will present a sample derivation of one of the equationsof motion, namely that for hδ † δi.
Starting with the Hamiltonian in coordinate space
Z

H=

Z

Z

dx ψ † (x)T F (x)ψ(x) +dx φ† (x)T B (x)φ(x) +dx ξ † (x)T M (x)ξ(x) +Z

1γ dx |φ(x)|4 +
2 Z

Vbg
g

Z

dx |φ(x)|2 |ψ(x)|2 +dx (ξ † (x)φ(x)φ(x) + c.c)
(2.1)

, where T α (x) is the kinetic energy of molecules, bosons or fermions.
We then write the bosonic field in terms of its average and fluctuations
around it φ(x) = φ0 (x) + δ(x), where φ0 is a complex number. Inserting thisexpression in the Hamiltonian, we get the following
H= E0 +Zγ

Zdx ψ † (x)(T F (x) + Vbg |φ0 (x)|2 )ψ(x) +dx δ † (x)T B (x)δ(x) +ZZdx ξ † (x)T M (x)ξ(x) +dx (4|φ0 (x)|2 |δ(x)|2 + φ∗0 (x)2 δ(x)δ(x) +99
φ0 (x)2 δ † (x)δ † (x)) +Z

dx φ∗0 (x)δ(x) + φ0 (x)δ † (x) ×γ

Zdx φ∗0 (x)δ † (x)δ(x)δ(x) + c.c +Z

dx ξ(x)† (φ0 (x) + δ(x)) φ(x) + c.c ,









γ
|φ0 (x)|2 + Vbg |ψ(x)|2 +
2



Z

γ
dx δ † (x)δ † (x)δ(x)δ(x) +2 Z
Vbg dx |δ(x)|2 |ψ(x)|2 +
g



h

i

(2.2)
where E0 is a constant which depends on φ0 , and its relevant for its motion, butdoes not contribute to that of δ † δ.
The next step is to calculate the commutator [δ † (z)δ(z ′ ), H], and to take its
average, thereby obtainingh[δ † (z)δ(z ′ ), H]i =

T B (z ′ ) − T B (z) hδ † (z)δ(z ′ )i +hγ 2|φ0(z ′ )|2 hδ † (z)δ(z ′ )i + φ20 (z)hδ † (z)δ † (z ′ )i−i2|φ0(z)|2 hδ † (z)δ(z ′ )i − φ∗0 2 (z ′ )hδ(z)δ(z ′ )i +

φ0 (z ′ ) γhδ † (z ′ )i|φ0(z ′ )|2 + Vbg
φ∗0 (z)γZ2γhδ(z)i|φ0 (z)| + Vbg
h

Z

Z



dx hδ † (z ′ )ψ † (x)ψ(x)i −†dx hδ(z)ψ (z)ψ(z)i +dx φ∗0 (x)(hδ † (z)δ(z ′ )δ † (x)δ(x)δ(x)i−hδ † (x)δ(x)δ(x)δ † (z)δ(z ′ )i)+φ0 (x)(hδ † (z)δ(z ′ )δ † (x)δ † (x)δ(x)ii−hδ † (x)δ † (x)δ(x)δ † (z)δ(z ′ )i) +γ hδ † (z)δ † (z ′ )δ(z ′ )δ(z ′ )i − hδ † (z)δ † (z)δ(z)δ(z ′ )i +Vbg hδ † (z)δ(z ′ )ψ † (z ′ )ψ(z ′ )i − hδ † (z)δ(z ′ )ψ † (z)ψ(z)i +

100




g hξ(z ′ )ψ † (z ′ )δ † (z)i − hξ † (z)ψ(z)δ(z ′ )i .

(2.3)

The next step is to apply Wick’s theorem to correlation functions of threeor more operators. This implies that all correlation functions of odd order will
vanish.We then geth[δ † (z)δ(z ′ ), H]i =

T B (z ′ ) − T B (z) hδ † (z)δ(z ′ )i +h2

γ 2|φ0(z ′ )| hδ † (z)δ(z ′ )i + φ20 (z)hδ † (z)δ † (z ′ )i−i2|φ0(z)|2 hδ † (z)δ(z ′ )i − φ∗0 2 (z ′ )hδ(z)δ(z ′ )i +γ hδ † (z)δ † (z ′ )ihδ(z)δ(z ′ )i + 2hδ † (z ′ )δ(z ′ )ihδ † (z)δ(z ′ )i−2hδ(z)δ † (z)ihδ † (z)δ(z ′ )i − hδ † (z)δ † (z)ihδ(z)δ(z ′ )iVbg hδ † (z)δ(z ′ )ihφ† (z ′ )φ(z ′ )i − hδ † (z)δ(z ′ )ihφ† (z)φ(z)i +(2.4)
In free space, φ0 becomes a constant, and all two point correlations, which
are functions of z,z’, become functions of z-z’, so that in momentum space they
become functions of a single momentum.

Appendix C
Gaussian Integrals

When we use the words “integrate out” in the text, we are referring toperforming one of the following integrals.
• BosonsZ

D[φ(x), φ† (x)]eT r[ −φ (x) G†−1 (x,x′ ) φ(x′ ) +J † (x) φ(x) + J(x) φ† (x)

]=

2πi
†
′
eT r[ J (x) G(x,x ) J(x) ]−1′[det G (x, x )]1

= e−T r[ ln G

−1 (x,x′ )[det G−1 (x, x′ )]]• FermionsZD[ψ(x), ψ † (x)]eT r[ −ψ (x) G†hi−1 (x,y) ψ(y) +J † (x) ψ(x) + J(x) ψ † (x)det G−1 (x, y) eT r[ J (x) G(x,y) J(x) ]

h

†

i

det G−1 (x, y) = eT r[ ln G

−1 (x,y)

]

]=

102
In the above expressions, φ refers to a bosonic field, while φ refers to afermionic (Grassman) field. Details of how these formulas are derived are very
well explained in chapter 1 of [64].

Appendix D
Reduced Units

In this section we wish to introduce a set of natural units which we founduseful to approach the problem. In particular we chose as fundamental energy
unit the Fermi energy Ef of the atomic Fermi gas at ν → ∞. As a length scale wechose the inverse of the Fermi wavenumber kf , and as time we chose the “Fermi
period” τf = h/Ef . In terms of these units the various quantities of interest canbe converted from standard units to reduce units by multiplying by the respective
factors in the table below.

Physical Quantity

Multiplicative Factor

mass

Ef /kf2

density

1/kf3

4-point coupling (Vbg , γ)

kf /Ef3-point coupling (g)kf3 /Ef

3/2

For magnetic field we define the unit via the product δB B = ν, which hasunits of energy.