Resonant and Nonresonant Interactions in Cold
Quantum Gases
byJochen Wachter
Vordiplom, Universität Konstanz, Germany, 1998
M. S., University of Colorado at Boulder, 2000

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:
Resonant and Nonresonant Interactions in Cold Quantum Gases
written by Jochen Wachterhas been approved for the Department of Physics

Prof. Murray Holland

Prof. John L. Bohn

Date

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

iii
Wachter, Jochen (Ph. D.)
Resonant and Nonresonant Interactions in Cold Quantum Gases
Thesis directed by Prof. Murray Holland

In the first part of this thesis, we present a unified kinetic theory that describes thefinite-temperature, non-equilibrium dynamics of a Bose–Einstein condensed gas interacting witha thermal cloud in a trap. This theory includes binary interactions to second order in theinteraction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov
quasiparticles. The Hartree–Fock–Bogoliubov interactions include the pairing field and are
expressed as many-body T matrices to second order. The interactions thus include the correct
renormalized scattering physics. This renormalized theory is automatically gapless. Thus, theexcited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic
theory is a complete second-order theory that reduces to the Gross–Pitaevskii equation and thequantum Boltzmann equation in the respective limits and thus is capable of describing thesystem over a wide temperature range.
In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbachresonance. Experiments explore the crossover physics between the Bardeen–Cooper–Schrieffer(BCS) superfluidity of a two-spin Fermi gas, and the Bose–Einstein condensation (BEC) of
composite bosons. We consider correlations between a composite boson and a fermion pairand show that such correlations are the minimal ingredients needed in a many-body theory to
generate the correct boson-boson scattering length in the Bose–Einstein limit of the crossover.
We also use imaginary-time propagation to find zero-temperature ground states in the
BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-orderinteractions between bosons and fermions. In particular, we calculate the Hartree term acrossthe resonance. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numericallycalculate the full time dependence in ramps across the resonance in this regime and find dif-iv
ferent two-body and many-body time scales in the system. We calculate molecular conversion
efficiencies as a function of temperature and phase-space density, and find good agreement withresults from JILA potassium experiments.

v

Acknowledgements

I acknowledge financial support from the National Science Foundation, the GermanAmerican Fulbright commission, the German National Merit Foundation, the W. M. Keck
Foundation, the National Security Agency, and the American Physical Society. The cast of
Holland-group characters that have contributed to this thesis includes: Murray Holland, Jinx
Cooper, Marilu Chiofalo, Reinhold Walser, Servaas Kokkelmans, Chiara Menotti, Simon Gardiner, Satyan Bhongale, Josh Milstein, Rob Chiaramonte, Meret Krämer, Dominic Meiser, Jami
Kinnunen, Rajiv Bhat, Brian Seaman, Sarah McGee, Brandon Peden, Ron Pepino, Boris Nowak,
and Dave Tieri. There are yet more people to thank for maintaining my sanity.

Contents

1 Introduction

1

1.1 Theoretical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2 Separation of Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3 Markov Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 The Kadanoff–Kane Formulation of Kinetic Theory

11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2 Nonequilibrium Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.3 Kadanoff–Baym Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.4 Transformation to the Energy Basis . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.5 Conclusion

22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Gaplessness—T Matrices24

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.2 Single-Particle Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.2.1

Mean-Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.2.2

Equations for Normal Densities and Anomalous Fluctuations . . . . . . .

31

3.3 Gaplessness—T Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

vii
3.4 Off-Diagonal Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.5 Diagonal Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.6 Renormalized Self Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.7 Ladder Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4 Quasiparticle Kinetic Equations

41

4.1 Quasiparticle Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.2 Kinetic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.3 Orders of Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

4.4 Quasiparticle T Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.5 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

5 Resonance Theory for Fermions
5.1 Bose–Einstein Condensation and Superfluidity

53. . . . . . . . . . . . . . . . . . .

53

5.2 Description of a Superfluid in a Dilute Atomic Gas . . . . . . . . . . . . . . . . .

56

5.3 Breakdown of the Mean-Field Picture—Resonance Superfluids. . . . . . . . . .

58

5.4 Single-Channel versus Two-Channel Approaches . . . . . . . . . . . . . . . . . .

59

5.5 Poles of the Molecular Propagator . . . . . . . . . . . . . . . . . . . . . . . . . .

62

5.6 The Equivalent Single-Channel Theory . . . . . . . . . . . . . . . . . . . . . . . .

64

5.7 Connection with the Theory of Feshbach Resonances . . . . . . . . . . . . . . . .

66

5.8 The BCS–BEC Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

6 Imaginary-Time Propagation for Fermions
6.1 Imaginary-Time Methods for Single- and Two-Channel BCS Models . . . . . . .

69
69

6.1.1

Single-Channel BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . .

69

6.1.2

Imaginary-Time Propagation for Bosons . . . . . . . . . . . . . . . . . . .

70

viii
6.1.3

Imaginary-Time Propagation for Fermions . . . . . . . . . . . . . . . . . .

71

6.1.4

Imaginary-Time Algorithm for the Single-Channel Model . . . . . . . . .

73

6.1.5

Imaginary-Time Propagation for the Two-Channel Model . . . . . . . . .

73

6.2 A Mean-Field Description for the Crossover Problem . . . . . . . . . . . . . . . .

77

6.2.1

Boson Scattering Length . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

6.2.2

Beyond Pair Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

7 Zero-Temperature Correlation Effects the BCS–BEC Crossover83

7.1 Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

7.2 Pairs in the BCS–BEC Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

7.3 Mean Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

7.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

7.5 Adiabatic Elimination of Three-Operator Correlations . . . . . . . . . . . . . . .

91

7.6 Dressed Pair Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

7.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

8 Many-Body Dynamics of the BCS–BEC Crossover in the Normal Phase98

8.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9 Summary and Outlook

110

A Collisional Self Energies in the Single-Particle Energy Basis

113

A.1 Mean-Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
A.2 Normal and Anomalous Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 115
B Collisional Terms in the Quasiparticle Basis

120

ix
C Kinetic Equations for Fermions
C.1 Kinetic Equations

126. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

C.2 Transformation to the Position Basis . . . . . . . . . . . . . . . . . . . . . . . . . 129
C.3 Center-of-Mass Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
C.4 Wigner Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
C.5 Scattering Equations for the Crossover Problem . . . . . . . . . . . . . . . . . . . 133
D Fermionic Pairing

136

D.1 Pairs of Fermions as Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
D.2 Pairs in a Homogeneous System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
E Rate Equations in the Normal Phase141

E.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
E.2 Step (1): Cut Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
E.3 Step (2): Eliminate Four-Operator Cumulants . . . . . . . . . . . . . . . . . . . . 143
E.4 Step (3): Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
E.5 Equations of Motion to Order g 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
E.6 Rate Equations to Order g 2 and Effective Damping Rates . . . . . . . . . . . . . 147
Bibliography149

List of Figures

1.1 Superfluid transition in liquid 4 He. . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Vortex in a trapped 87 Rb BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3 A Feshbach resonance in 40 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4 Molecular BEC in 40 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.5 Separation of collision time scales. . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1 The first-order Hartree–Fock self-energy diagrams. . . . . . . . . . . . . . . . . .

17

2.2 The second-order, collisional self-energies in the gapless Beliaev approximation. .

18

3.1 Second-order terms in the Gross–Pitaevskii equation. . . . . . . . . . . . . . . . .

30

3.2 Second-order terms in the generalized Boltzmann equation. . . . . . . . . . . . .

33

4.1 Quasiparticle collision rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

5.1 Vortex schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

5.2 Vortex phase and density plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

5.3 Schematic potential plot of a Feshbach resonance. . . . . . . . . . . . . . . . . . .

59

5.4 Two-body T -matrix for 40 K as a function of scattering energy. . . . . . . . . . .

60

5.5 Poles of the molecular propagator in the complex plane. . . . . . . . . . . . . . .

63

5.6 Schematic comparison of BCS theory and the BCS–BEC crossover theory. . . . .

68

6.1 Normal and anomalous density for a single-channel model at kF a = −1. . . . . .

74

6.2 Normal and anomalous density for a two-channel model at kF a = −1. . . . . . .

76

6.3 Schematic illustrating the crossover between fermions and composite molecules. .

77

6.4 Schematic of the interaction between two dimers of paired fermions. . . . . . . .

78

6.5 Atom-molecule correlation function. . . . . . . . . . . . . . . . . . . . . . . . . .

79

xi
7.1 Released energy of a harmonically trapped gas as a function of the detuning. . .

83

7.2 Pair fractions of the total fermion number for 40 K and 6 Li. . . . . . . . . . . . .

86

7.3 Closed-channel fraction for the wide resonance in 6 Li. . . . . . . . . . . . . . . .

87

7.4 Schematic of the cumulant hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . .

89

7.5 Correlation functions in the BCS/BEC crossover. . . . . . . . . . . . . . . . . . .

94

7.6 Energy plots for the narrow resonance in 6 Li. . . . . . . . . . . . . . . . . . . . .

95

7.7 Self energy plots for 40 K.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

8.1 Schematic of the BBGKY cumulant hierarchy for the normal phase. . . . . . . .

99

8.2 Scattering rates for the crossover Hamiltonian. . . . . . . . . . . . . . . . . . . . 100
8.3 Normal-phase distribution functions. . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.4 Time evolution of the population fractions. . . . . . . . . . . . . . . . . . . . . . 104
8.5 Fits to equilibrium distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.6 Energy and entropy for BCS side ramp. . . . . . . . . . . . . . . . . . . . . . . . 106
8.7 Fitted detuning and temperatures for several ramp speeds.

. . . . . . . . . . . . 106

8.8 Fermion fraction and entropy for return ramps. . . . . . . . . . . . . . . . . . . . 107
8.9 Fitted detuning and temperatures for a ramp across the resonance. . . . . . . . . 108
8.10 Molecular conversion efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Chapter 1
Introduction

In 1924, S. N. Bose and A. Einstein predicted the phenomenon of Bose–Einstein condensation (BEC) [1, 2], where a macroscopic number of noninteracting bosons (particles withinteger total spin, for example 87 Rb atoms) collapse into a single quantum state. This behavior
is a consequence of quantum statistics, which modifies the physical properties of the gas at very
low temperatures.

The superfluid transition in liquid Helium was discovered in 1938 [4, 5]. Shortly thereafterin the same year, F. London suggested that the observed superfluid λ-transition (see Fig. 1.1)
was due to BEC [6]. However, even at zero temperature, only 10% of the Helium atoms actuallycondense into a single state, because Helium is a strongly interacting liquid, whereas Bose andFigure 1.1: Superfluid transition in liquid 4 He [3]. The specific heat as a function of temperatureshows the characteristic λ behavior at Tλ = 2.71 K. Note that the λ-shape is scale invariant over
six orders of magnitude in temperature, indicating that the superfluid transition is a secondorder phase transition.

2

Figure 1.2: Vortex in a trapped 87 Rb BEC [11]. Images (a) through (c) show nondestructive
absorption images (density plots) of the condensate. The images are 0.1 mm × 0.1 mm, which
is macroscopic. Image (a) shows a quantized-vortex state, (c) a non-rotating state, and (b) the
superposition of states (a) and (c). Due to the phase winding of the vortex in (a), the particles
interfere constructively/destructively on the left/right sides of cloud (b).

Einstein predicted complete condensation for the case of a non-interacting, ideal gas.
The observation of BEC in a dilute, weakly interacting quantum gas of atomic 87 Rb at
JILA [7] and shortly after in 23 Na at MIT [8] and 7 Li at Rice [9] thus opened a new field, in whichcorrections to the ideal-gas model of BEC could be calculated directly. The experimental systems
are usually contained in harmonic trapping potentials, which is a complication compared to thehomogeneous Helium case. Many interesting consequences of macroscopically occupied quantum
states have since been observed experimentally: interference fringes between condensates [10],quantized vortices (see Fig. 1.2 (a)), superposition of condensate wave-functions or matter waves(ibid. (b)), and more. These systems exhibit quantum phenomena on macroscopic length scales;
the plots in Fig. 1.2, for example, have a scale of a tenth of a millimeter.

These quantum phenomena can be understood as a consequence of U (1) symmetry breaking below the transition temperature [12]. The breaking of the gauge symmetry causes theexistence of a well-defined phase in the condensate. Only phase differences are physically observable; the absolute value of the phase can thus be changed without energy cost. There thus
exists a low-frequency phonon mode, the zero-energy Goldstone mode [13]. This means that thecollective-excitation spectrum is gapless and linear for small momenta. Another consequenceof the broken gauge symmetry is the existence of long-range order [14]. This means that in a

3

Figure 1.3: A Feshbach resonance in 40 K [17]. The scattering length as a function of magnetic
field shows the |9/2, −5/2i-|9/2, −9/2i Feshbach resonance.

condensate correlations between distant points exist. These correlations allow the low-energy
collective excitations required by the Goldstone theorem.
The new cooling and trapping techniques [15] that lead to the creation of BECs werethen also applied to fermions, which have half-integral total spin and at low temperatures obey
quantum statistics that differ from those of bosons. In fact, Fermi-Dirac statistics predicts
a maximum population of one particle per state (Pauli exclusion principle), such that at zerotemperature all available quantum states up to the Fermi energy are singly occupied. This Pauliblocking behavior was first observed in a dilute gas of 40 K at JILA in 1999 [16]. This experiment
was performed with an equal mixture of two different magnetic sublevels (|F = 9/2, m F = 7/2i
and |F = 9/2, mF = 9/2i) of the hyperfine ground state with total atomic spin F and magnetic
quantum number mF , because s-wave scattering and thus thermalization during evaporationfor one species is suppressed for due to the fermion’s anti-symmetry requirement.

An intriguing possibility for fermion systems is pairing of two fermions to form a bosonicmolecule. One inspiration for this idea is the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity [18]. It explains the resistance-less conduction below the transition temperature by
the formation of weakly bound Cooper pairs, which are made up of two electrons interacting via

4

Figure 1.4: Time-of-flight images of a molecular cloud in 40 K [21]. Left images are above the
BEC transition temperature (470, 000 molecules), right images below (200, 000 molecules). a)
Surface plots of the optical density. b) Cross-sections through above images (dots) with bimodal
fits (red lines).

lattice vibrations of the conductor. However, it turns out that the BCS transition temperatures
for typical non-resonant s-wave scattering in a cold quantum gas are a few orders of magnitude
below temperatures that can be reached in experiments.
Fortunately, the BCS transition temperature rises with increasing interaction strength,
so that resonantly enhanced interactions near a Feshbach resonance opened the possibility ofrealizing BCS states in dilute quantum gases [19]. A Feshbach resonance is a scattering resonancewith an energetically closed channel that strongly modifies the scattering in the open channel(see Chap. 5 for a more detailed introduction to Feshbach scattering). The relative position of
the open and closed channels can be shifted by the Zeeman effect with an external magnetic field,
which leads to the characteristic scattering behavior shown in Fig. 1.3. The Figure shows thatthe two-component Fermi gas is brought to the strongly interacting regime [20, 17]. Specifically,
the Feshbach resonance connects loosely bound Cooper pairs on the attractive BCS limit on theright side of Fig. 1.3 to tightly bound molecules with weak repulsive interactions of the BEC(left) side of the figure.

5
The next series of results from groups all over the world reported the condensation of
long-lived molecules on the BEC side of the resonance [22, 23, 24, 25]. These molecules are
immersed in the residual Fermi sea of unbound atoms, which stabilizes the molecules due toPauli blocking of final states for molecular decay. Figure 1.4 shows the resulting absorption
images for the JILA experiment. Finally, many groups have reported observing condensation
all the way through the BCS–BEC crossover [21, 26, 27, 28].

1.1

Theoretical Treatment
An exact many-body treatment of finite-temperature boson or fermion systems—even if

we only considered two-body interactions—would have to involve correlations between as many
particles as the system contains, because a tree of binary collisions would eventually entangle asingle atom with every other atom in the system.
Fortunately, these many-body correlations are typically suppressed in the case of dilute,
weakly interacting gases, because the duration of a collision τ is very small compared to thetime between collisions ∆τ , so that the atoms oscillate essentially interaction-free in the externalpotential between isolated collision events [29]. This separation of time scales allows the highorder correlations to decay between collision events and thus justifies a coarse-grained description
with a reduced set of Master variables that only contain correlation functions between two or
three particles, because for times long compared to the duration of a collision τ , the higherorder correlations can be expressed as functionals of these variables [30]. We calculate the time
evolution of these Master variables using the Markov approximation.

1.2

Separation of Time Scales

The diluteness of a gas is characterized by the following relation between its average
number density n and its two-particle interaction strength, as given by the s-wave scattering

6

∆τ

τ

Figure 1.5: Separation of collision time scales: The duration of a collision τ is small comparedto the time between subsequent collision ∆τ . The red circles illustrate the range of the binary interaction potential. Note that this range remains small even when the interactions areresonantly enhanced.

length as ,na3s  1.

(1.1)

This means that, on average, the volume available for each atom is large compared to its interaction volume. See Fig. 1.5 for an schematic illustration. In the case of a Feshbach resonance,
where the scattering length diverges, the diluteness criterion is given by the range of the twobody potential r0 as nr03  1. Equation (1.1) is equivalent to the following relation between the
duration of a collision τ and the time between collisions ∆τ
τ=

as
1

= ∆τ,
v
na2s v(1.2)

where v is the average velocity. This quasi-classical argument shows that the diluteness of thesystem causes a separation of time scales. Choosing realistic experimental parameters for a cloudat 100 nK, we get, for example, τ ≈ 5 µs and ∆τ ≈ 0.4 s. Since many collision events for eachparticle are needed to establish local or even global equilibrium, the times associated with thesestates are again orders of magnitude larger than ∆τ . Some BEC experiments are performed on
the time scale of ∆τ , which means that the observed system is far from equilibrium and should
be described by a non-equilibrium theory.
We can distinguish between three distinct stages of evolution of a dilute quantum-gas system:• The dynamical stage, where we follow the system on time scales shorter than the du-7ration of a collision t . τ . On this time scale, the system is mostly dependent on its
initial condition. In the case of an initially uncorrelated system, the time scale is too
short to build up high-order correlations and a reduced description is still possible. If,
however, the initial condition is highly correlated, the evolution in the dynamical stage
is non-Markovian and we need the full N -particle density matrix to describe the system.
• The kinetic stage of evolution is characterized by time scales on the order of the time
between collisions t ≈ ∆τ . Here, we can assume that the correlations decay because of
the long separation between successive collisions and the presence of large intermediatefluctuations. This allows us to make the Markov approximation and track the systemwith a reduced set of low-order Master variables. As pointed out above, this is thetime scale on which many quantum-gas experiments are performed. Thus the kinetic
description of these systems given in Chapters 2 to 4 is very useful.
• Finally, the hydrodynamic stage of evolution takes place on time scales large compared
the time between collisions t  ∆τ , that is, when the system has come into a stateof local equilibrium. Hydrodynamic equations for trapped Bose gases have been derived [31]. However, this state is more relevant in strongly interacting Bose systems,
such as superfluid 4 He, where the kinetic stage vanishes (τ ≈ ∆τ ).

1.3

Markov Approximation
The separation of time scales explained in the last Section allows us to make the far-reaching Markov approximation, which renders the problem of a kinetic theory for dilute,
trapped, Bose–Einstein condensed gases feasible. It consists of the principle of rapid attenuation of quantum correlations: Individual collision events can correlate and entangle multiple
particles. However, the separation of time scales leads to a decay of these correlations due tointermediate fluctuations.
Since correlations are the result of the past evolution of the system, we can interpret

8
the Markov approximation as saying that the future evolution is exclusively determined by thepresent state of the system. This is the defining property of a Markov process in the theory of
stochastic differential equations, and is exactly the same property that is used for deriving thequantum-optical Master equation [32].
Another important feature of this approximation is that it implies the key observable
Master variables that completely describe the system’s irreversible evolution: The decay ofhigher-order correlation and distribution functions suggest using only few-particle quantities ina reduced but macroscopically sufficient description of the system. On the other hand, thisdecay means that information is lost in approximating the reversible, microscopic many-bodytreatment including N -particle correlations by a reduced description. This information loss
is exactly the way in which irreversibility is usually introduced in statistical mechanics: Thegigantic loss of information in going from a completely microscopic description in terms of atom
coordinates to a macroscopic description in terms of thermodynamic quantities makes the latter
appear irreversible. (Looking at decoherence in quantum systems, we see that similarly tracing
over unobserved degrees of freedom, which involves loss of information, irreversibly replaces purestates by mixtures, which have to be described by density matrices.)
This principle of rapid attenuation of correlations was historically introduced by Bogoliubov [33, 29] to explain Boltzmann’s assumption of molecular chaos (Stoßzahlansatz) in thederivation of his collision equation. He furthermore used it as a boundary condition in solving
the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy of recursively coupled equations of motion for s-particle distribution functions, where each couples to the s + 1-particle
distribution.
In the Markov approximation, the reduced set of Master variables completely describes
the system, that is, its full N -particle density operator can be approximated by a density operator
that is a functional of these variables only and does not contain an explicit time dependence.
This is analogous to the Chapman-Enskog procedure used in classical statistical mechanics [29].
The reduced description is equivalent to coarse graining the quantum Liouville equation over

9a kinetic time interval. Coarse graining is performed in the derivation of the quantum-optical
Master equation to incorporate the weak-coupling approximation, which here corresponds to
the rapid decay of correlations.
Within the Chapman-Enskog approach [34], expectation values are performed with the
reduced density operator. Since the Markovian Master variables are our sole dynamic quantities
and we can change many of the degrees of freedom of this density operator without changing
these expectation values of the relevant operators, we can replace the reduced density operator
by a more convenient Gaussian reference distribution, which is an exponential of quadratic
combinations of the field operators. This reference distribution then enables one to expand the
expectation values of operator products, which appear in the Heisenberg equations of motion
for the relevant operators, using Wick’s theorem [35, 36, 37, 38].

1.4

Overview
This thesis has two parts. The first begins with Chap. 2, where we derive a quantum-

kinetic theory for atomic BECs using the Kadanoff–Kane Green-function formalism. In Chap. 3
we show explicitly that this theory has a gapless energy spectrum and discuss the scattering
properties to second order in the binary interaction. In Chap. 4 we express the full quantum
dynamics of the BEC in terms of quasiparticles interacting via a Boltzmann collision integral.
In the second part, we discuss dilute Fermi gases, in particular the crossover between BCS
superfluidity and BEC of composite molecules. In Chap. 5 we discuss one- and two-channel
models of Feshbach resonances. In Chap. 6 we introduce an imaginary-time method to findzero-temperature ground states for these strongly interacting fermion systems. In Chap. 7 weuse a cumulant expansion to find a zero-temperature many-body theory that correctly describes
the scattering physics of the BCS–BEC crossover. We also develop a cumulant expansion in
the normal phase above the transition temperature in Chap. 8 and show numerical results forthe full time dependence of thermal bosons and fermions across a resonance. We conclude in10Chap. 9 with the summary and outlook.

Chapter 2
The Kadanoff–Kane Formulation of Kinetic Theory [39]2.1

Introduction
Binary collisions are the essential mechanism for the formation of a Bose–Einstein con-densate in an atomic gas. Moreover, many aspects of the system’s dynamics require two-particle
collisions, for example, sound propagation, the damping of elementary excitations, and the verymechanism that leads to the quantum phase transition—evaporative cooling. However, the conventional Hartree–Fock–Bogoliubov approach to generalize the Gross–Pitaevskii equation for
dilute, trapped gases includes binary collisional interactions only as first-order energy shifts.
Second-order kinetic theories that include collisional redistributions of excited atoms offer amore complete microscopic description of the gaseous system.
Why is a simplified kinetic description possible, when the evolution of the Bose–Einsteincondensate might involve correlations between as many particles as the system contains? Would
not binary collisions eventually entangle the quantum state of each atom in the system with thatof every other atom? Fortunately, such complexity is not necessary to describe the measurable
properties of a dilute, weakly interacting gas, because the duration of a collision, τ , is very shortcompared to the essentially interaction-free oscillation in the external potential between isolated
collision events [29].
Because of this characteristic separation of time scales, correlations that arise during an
individual collision decay rapidly before the next collision takes place. This rapid decay, in turn,implies the possibility of a Markov approximation, which assumes that only the current config-

12
uration of the system determines its future evolution. Furthermore, this decay of correlations
allows us to parameterize the system’s state by a reduced set of master variables, because we areinterested in the system’s time evolution only on the kinetic time scale, that is, for times large
compared to the duration of a collision τ . This reduced description with a set of master variables
is possible, because for kinetic times the higher-order correlation functions can be expressed asfunctionals of these variables [30].
This set of master variables is common to both the kinetic theories that we discuss: In
the Kadanoff–Baym approach, abstract real-time Green’s functions parameterize the condensating gas, whereas in the Walser et al. case [40], single-time density matrices, which containthe physical density and coherences of thermal atoms, as well as the mean field, represent thesystem. The equivalence of these two approaches is a general principle in nonequilibrium statistical mechanics [41, 42]. However, it is not trivial to verify this fact in detail by explicitly
connecting the complementary microscopic equations. Strictly speaking, we find equivalence
after the Kadanoff–Baym theory has been restricted to single-time quantities using the Markov
approximation.
We present the formulation of the quantum kinetic theory of dilute Bose–Einstein condensed gases in terms of nonequilibrium, real-time Green’s functions and their Kadanoff–Baymequations of motion [43], which were generalized in Refs. [44, 45] to include the condensate.
By transforming these equations to the single-particle energy basis and taking the singletime limit of the two-time Green’s functions by means of the Markov approximation, we reproduce the equations of motion of the Walser et al. kinetic theory as presented in Ref. [40], thus
providing an independent confirmation of these equations. Following Imamović-Tomasović and
Griffin [46], we use the gapless Beliaev approximation for the self-energies in the Kadanoff–Baymequations, and thus prove the Walser et al. kinetic theory to be gapless as well.

13

2.2

Nonequilibrium Green’s Functions
We begin the introduction to the Kadanoff–Baym description of the dilute Bose gas

by defining its variables. Neglecting three-body interactions, the second-quantized many-body
Hamiltonian H describing the atoms is
H=

Z

dx

Z

dy



†

a (x) hx| H

(0)


1 †
†
|yi a(y) + a (x)a (y) Vbin (x − y) a(y)a(x) ,
2

(2.1)

where a† (x) is the bosonic creation operator and Vbin (x − y) the binary interaction potential.
The single-particle Hamiltonian
H (0) =

p2
+ Vext (x)
2m

(2.2)

contains the kinetic energy of a boson with mass m and the external potential Vext (x).
To represent the master variables in terms of nonequilibrium Green’s functions, we firstwrite the system’s degrees of freedom in terms of spinor operators [47] a(1)  and A† (1) =†A(1) = a (1) a(1) ,


†
a (1)

(2.3)

where we now follow Kadanoff–Baym and abbreviate (1) = (x1 , t1 ). The master variables arethen contained in the following two-time propagators:
h(1, 2) = −ihA(1)ihA† (2)i,g(1, 2) = −i T A(1)A† (2)

(2.4)
,

(2.5)

where h·i denotes the grand-canonical average and T {·} the time ordering operator, which sorts
its arguments in order of decreasing time. These two propagators are defined for real times byanalytic continuation of the finite-temperature propagators for imaginary time, following [43,Chap. 8]. We subtract the condensate propagator h from the full propagator g and thus define
the Green’s function for the fluctuationsg̃(1, 2) = g(1, 2) − h(1, 2).

(2.6)

g̃ < (1, 2) = g̃(1, 2) for t1 < t2

(2.7)

The two time orderings of g̃,14andg̃ > (1, 2) = g̃(1, 2) for t1 > t2 ,(2.8)

define the generalized two-time fluctuation-density matrices. This can be seen by explicitly
writing these two time orderings in terms of the fluctuating part ã(1) of the field operators,ã(1) = a(1) − ha(1)i = a(1) − α(1),(2.9)

as follows: f˜12g̃ < (1, 2) = 

m̃∗12

m̃12
(1 + f˜)∗12g̃ > (1, 2) = σz + g̃ < (1, 2),




,


(2.10)(2.11)

where we defined the two-time normal (f˜) and anomalous (m̃) averages of the fluctuations in
the position basis asf˜12 = ã† (2)ã(1)and m̃12 = ã(2)ã(1) .

(2.12)In the case t1 = t2 , the propagators in Eqs. (2.10) and (2.11) correspond to the dynamicalquantities in the kinetic equations for the fluctuations given in Eqs. (24) and (25) of Ref. [40];for t1 = t2 , the averages in Eq. (2.12) correspond to the density of thermal atoms around thecondensate and correlations between these atoms. We have thus represented the condensate (h)and its fluctuations (g̃ < ) and can now look for their corresponding evolution equations.

2.3

Kadanoff–Baym Equations
The equations of motion for the nonequilibrium Green’s functions h and g̃ < are the

Kadanoff–Baym equations; these equations are equivalent to the Dyson equation. In the second
part of this Section, we discuss the second-order Beliaev approximation for the self-energies that
we use. For the condensed part of the atom cloud, which is parameterized by the propagator h(1, 2) defined in Eq. (2.4), we can write the Kadanoff–Baym equations as [44]
Z ∞

−∞


d1̄ g0−1 (1, 1̄) − SHF (1, 1̄) h(1̄, 2) =

Z t1

−∞

d1̄

 >
S (1, 1̄) − S < (1, 1̄) h(1̄, 2)(2.13)

15
and
Z ∞

−∞


d1̄ h(1, 1̄) g0−1 (1̄, 2) − SHF (1̄, 2) = −

Z t2

−∞


d1̄ h(1, 1̄) S > (1̄, 2) − S < (1̄, 2) .

(2.14)

We write the corresponding equations for the fluctuations g̃ < (1, 2) and g̃ > (1, 2) [Eqs. (2.10) and(2.11)] around the condensate mean field as
Z ∞

−∞

=

>

d1̄ g0−1 (1, 1̄) − ΣHF (1, 1̄) g̃ < (1̄, 2)

Z t1

d1̄

−∞



>

<

(2.15)><

Σ (1, 1̄) − Σ (1, 1̄) g̃ (1̄, 2) −

Z t2

−∞

>

d1̄ Σ< (1, 1̄) g̃ > (1̄, 2) − g̃ < (1̄, 2)

and
Z ∞

−∞

=

>

d1̄ g̃ < (1, 1̄) g0−1 (1̄, 2) − ΣHF (1̄, 2)

Z t1

−∞

>

d1̄ g̃ > (1, 1̄) − g̃ < (1, 1̄) Σ< (1̄, 2) −(2.16)
Z t2

−∞

>

d1̄ g̃ < (1, 1̄) Σ> (1̄, 2) − Σ< (1̄, 2) .

In Eqs. (2.13) through (2.16), we use the definition of the matrix inverse of the interaction-free
propagator g0 ,on∇2d
+ 1 − Vext (1) + µ δ(1, 2),g0−1 (1, 2) = iσ z
dt1
2m

(2.17)

with the third Pauli matrix σ z = diag(11, −11) and an energy shift µ, which removes mean-fieldoscillations. We define the δ function by δ(1, 2) = δ(x1 − x2 )δ(t1 − t2 ) and integration d1̄ asintegration dt1̄ over time within the given time limits and dx1̄ over all space. The approximations
we choose for the Hartree–Fock self-energies for the condensate SHF and for the fluctuations ΣHF
as well as the second-order collisional self-energies S < and Σ< are discussed below.
Kadanoff and Baym derived these equations without including the condensate [43] andde Dominicis and Martin formulated a very general mathematical account [48]. The Green’s
function formalism traces back to Schwinger [49] and originally made use of the correspondence
between the partition function and the time evolution operator in imaginary time (eβH = eiHt
for t = −iβ). To get information about measurable quantities, the dynamic variables andequations of motion were extended to real times by analytic continuation (see [43, Chap. 8]
and [50, 51, 42] for more details).

16This nonequilibrium Green’s function description was developed 40 years ago to eventually explain the behavior of superfluid helium [52]. Since this description involves a weakcoupling approximation but helium atoms are strongly interacting, the results at that time weredisappointing and, for example, could not explain all predictions of the phenomenological Landau model. However, since the Green’s function description holds for a dilute, weakly interacting
gas, its application to Bose–Einstein condensation in this system is more appropriate.
To complete our exposition of the Kadanoff–Baym equations (2.13) through (2.16), we
have to choose the Hartree–Fock and collisional self-energies. We draw the Hartree–Fock selfenergy diagrams for both the condensate h and the thermal cloud g̃ < in Fig. 2.1 and write them,
respectively, asi2

Zd2̄ v(1, 2̄)Tr {g(2̄, 2̄)} δ(1, 2) + iv(1, 2)g̃(1, 2)(2.18)i
ΣHF (1, 2) =2

Zd2̄ v(1, 2̄)Tr {g(2̄, 2̄)} δ(1, 2) + iv(1, 2)g(1, 2),(2.19)SHF (1, 2) =andwith the local-time, binary interaction potential v(1, 2) = Vbin (x1 − x2 ) δ(t1 − t2 ) and thematrix trace Tr. When we evaluate the time-ordered propagator g at equal times, we follow theconvention T {a(1)a† (2)} = a† (2)a(1).

>For the second-order collisional self-energies Σ< we choose the gapless and energy- andnumber-conserving Beliaev approximation [46, 53, 54]. This means that, compared to Kaneand Kadanoff [44], we include the exchange terms, which they deliberately excluded to obtainthe simplest conserving approximation as proven in [55], and compared to Hohenberg and Martin [45], we include the terms containing no condensate contributions, which give rise to thequantum Boltzmann terms for the fluctuations.
We depict the resulting self-energy diagrams in Fig. 2.2 and represent them mathemati-

17







 












Figure 2.1: The first-order Hartree–Fock self-energy diagrams. The solid lines depict the noncondensate propagator g̃, the wiggly lines the condensate propagator h, and the dashed lines
the interaction potential v. The first two terms give the energy shifts due to both the mean
field Uf c and the normal fluctuations Uf˜. The third term in SHF gives rise to a factor of 2 for Uf˜and to Vm̃ . The fourth term which only appears in ΣHF causes the difference in the mean-field
shifts that are experienced by the condensate and the fluctuations, respectively.

cally as>S < (1, 2) = −

1
2

Z

d2̄

Z

d3̄ v(1, 2̄)v(2, 3̄)

 >
n <
o
>
g̃ < (1, 2)Tr g̃ > (3̄, 2̄)g̃ < (2̄, 3̄)><>+2g̃ < (1, 3̄)g̃ > (3̄, 2̄)g̃ < (2̄, 2)(2.20)

for the condensed part and>ZZ1d2̄ d3̄ v(1, 2̄)v(2, 3̄)(2.21)
2
 >
o
o
n <
n <
>
>
× g̃ < (1, 2)Tr g > (3̄, 2̄)g < (2̄, 3̄) − h(3̄, 2̄)h(2̄, 3̄) + h(1, 2)Tr g̃ > (3̄, 2̄)g̃ < (2̄, 3̄)
o
n <
o
n <
>
>
>
<
>
<
>
<
+2g̃ (1, 3̄) g (3̄, 2̄)g (2̄, 2) − h(3̄, 2̄)h(2̄, 2) + 2h(1, 3̄) g̃ (3̄, 2̄)g̃ (2̄, 2)

Σ< (1, 2) = −for the fluctuations.
Instead of using lines for the matrix-valued propagators g̃ and h as in Fig. 2.2, one canalso draw diagrams for the four elements of the matrix separately. The resulting diagrams forthe first-order and second-order Beliaev terms can be seen in Figs. 15 and 17 of Ref. [56], where
the interaction potential is replaced by a two-body T matrix.

18

$#

#

#


    "!
%






$#


$#

#

    "!
& 


%

%

%

%

%

%



$#

#

%




Figure 2.2: The second-order, collisional self-energies in the gapless Beliaev approximation. The
solid lines depict the noncondensate propagator g̃, the wiggly lines the condensate propagator h,and the dashed lines the binary interaction potential v. The second diagram of S corresponds
to the last four of Σ, when we replace each of the three fluctuation propagators by an open
condensate one.

2.4

Transformation to the Energy Basis
We now demonstrate the key steps that connect the kinetic theory presented in theprevious Section to the work of Walser et al. presented in [40]: We rewrite the Kadanoff–BaymEqs. (2.13) through (2.16) in the single-particle energy (SPE) basis and obtain the equationsof motion for the master variables—the measurable quantities in our reduced description of thesystem—in this basis, exactly as given in the Walser et al. paper.
First, we define our master variables in the SPE basis {|10 i}10 = {|10 i} 0 and determine1

the relation to their position basis counterparts, the Green’s functions given in Eqs. (2.4), (2.10),
and (2.11). The time-dependent, two-component mean-field state vector

is defined in terms of α = α10 |10 i =

 α 

χ=


∗
α

(2.22)

0
10 ha10 i |1 i and also contains the time-reversed meanPfield α∗ . The time-dependent, fluctuating annihilation and creation operators ã and ã† transform19
asand ã† (1) = h10 | 1iã†10 ,ã(1) = h1 | 10 iã10(2.23)

where |1i = |x1 i are the position eigen states. The fluctuating part of the master variables isthen contained in the single-time fluctuation-density matrix G̃< , which we define as


m̃
 f˜

,
G̃< = ∗∗m̃ (1 + f˜)(2.24)

with the normal fluctuation density f˜ = hã†20 ã10 i |10 i ⊗ h20 | and the anomalous average m̃ =hã20 ã10 i |10 i ⊗ |20 i in the SPE basis.
Second, we can recognize the fluctuation-density matrix G̃< as the single-time limit of
its position-basis counterpart g̃ < (1, 2) in Eq. (2.10). The mean-field state vector χ, on the other
hand, can be combined with its Hermitian conjugate into the matrix −iχχ† , which correspondsto h(1, 2) in Eq. (2.4). We thus define χ(1) = hA(1)i. This allows us to explicitly connect
the condensate mean-field state vectors χ(1) expressed in the position basis and χ(t 1 ) in theSPE basis as follows:




0



0
 h1 | 1 i α10 (t1 )  = T (1) χ(t1 ),χ(1) = 



0
∗
0
h1 | 1i
α10 (t1 )(2.25)

with a time-independent 2 × 2n transformation matrix T (1) = T (x1 ). Because of the completeness of the position basis, we can also writeχ(t1 ) =Z

dx1 T † (1)χ(1).

(2.26)For the fluctuation density, we obtain similarly
i g̃ < (1, 2) t1 =t2 = T (1)G̃< (t1 )T † (2)(2.27)

and−i G̃< (t1 ) =

Z

dx1

Z

dx2 T † (1)g̃ < (1, 2)T (2) t1 =t2 .

(2.28)

20
We can now use the transformation Eq. (2.27) to write the condensate’s Hartree–Fockself-energy SHF (1, 1̄) in Eq. (2.18) as




Vm̃
 †
 Uf c + 2Uf˜
 T (1̄)δ(t1 − t1̄ ).
T (1) 


†
†
†
Vm̃
Uf c + 2Uf˜

(2.29)We here use the definitions of [40], where energy shifts due to both the mean field and thenormal fluctuations are given by the matrices
0 0 0 0

Uf = 2 φ1 2 3 4 f30 20 |10 i ⊗ h40 | ,(2.30)

whereas the first-order anomalous coupling strength is given by
0 0 0 0Vm̃ = 2 φ1 2 3 4 m̃30 40 |10 i ⊗ |20 i .

(2.31)The symmetrized two-body interaction matrix elements φ are here defined by
0 0 0 0

φ1 2 3 4 =0 0 0 0
φ1u 2 3 4 =Z

0 0 0 00 0 0 00 0 0 0
1 10 20 30 40(φu+ φ1u 2 4 3 + φ2u 1 3 4 + φ2u 1 4 3 ),
4

dx1

Z

dx2 h10 | 1ih20 | 2i

(2.32)

Vbin (x1 − x2 )h1 | 30 ih2 | 40 i.
2(2.33)

Like the first-order Hartree–Fock self-energies, we can rewrite the second-order selfenergies in Eqs. (2.20) and (2.21) using the transformations in Eqs. (2.25) to (2.28) and (2.33).
In particular, we now have to transform two potential factors, which makes the computation
more complicated. Furthermore, the integrals over time to t1 and t2 in Eqs. (2.13) through (2.16)
modify one of the binary potentials according to Eq. (65) of [37] to an approximately energyconserving two-particle matrix element0 0 0 00 0 0 0φ1η 2 3 4 = φ1 2 3 41πδη (∆) + i Pη∆,

(2.34)

with an energy difference ∆ between the incoming and outgoing states of the collision event. Thisdefinition of the matrix elements φη introduces the Markov approximation into the Kadanoff–>

Baym equations. We obtain the second-order damping rates and energy shifts Υ < for the21>condensate, corresponding to S(1, 2) in Eq. (2.20), and Γ< for the fluctuations, correspondingto Σ(1, 2) in Eq. (2.21); these are the collision integrals defined in [40]. These second-order terms
appear in combinations Γ< G̃> −Γ> G̃< that contain, for example, the Boltzmann collision termsnoΓf˜f˜(1+f˜) (1 + f˜) − Γ(1+f˜)(1+f˜)f˜f˜ 0 0 =(2.35)

15

0 0 0 000 00 00 00

8φ1 2 3 4 φ1η 2 3 4n

˜ 400 20 (1 + f˜)300 50
f˜30 100 f˜40 200 (1 + f)

˜ 40 200 f˜400 20 f˜300 50− (1 + f˜)30 100 (1 + f)oand similar contributions involving the anomalous averages m̃ and m̃∗ . See Appendix A for a
detailed derivation of the collisional terms in the SPE basis.
We can now exactly reproduce the coupled equations for the condensed fraction as wellas the normal and anomalous fluctuations stated in Eqs. (10) and (26) of Ref. [40]. Considering
the first column of the matrix Eq. (2.13) for the condensate at t1 = t2 , we obtain the generalized
Gross–Pitaevskii equationdχ = (−i Π + Υ< − Υ> )χ,dtwith the symplectic first-order propagator


 ΠN
Π=

−Π∗A

ΠA
−Π∗N




.


(2.36)(2.37)

This propagator consists of the normal Hermitian HamiltonianΠN = H (0) + Uf c + 2Uf˜ − µ,(2.38)

which contains the usual single-particle Hamiltonian H (0) given in Eq. (2.2) and the mean-field
and fluctuation shifts Uf given in Eq. (2.30); furthermore, the symmetric anomalous couplingΠA = Vm̃

(2.39)

is defined in Eq. (2.31). The propagator Π contains the Hartree–Fock shifts, which are given in
Eq. (2.29), and originally were contained in SHF (1, 2) [see Eq. (2.18)].

22
To obtain the equation of motion for the fluctuations, we subtract Eq. (2.15) from (2.16)
and evaluate at t1 = t2 to obtaind <G̃ = −i ΣG̃< + Γ< G̃> − Γ> G̃< + H.c.
dt

(2.40)The reversible evolution of the fluctuations G̃< is governed by the Hartree–Fock–Bogoliubov
self-energy operator


 ΣN
Σ=

−Σ∗A

ΣA
−Σ∗N

which in turn consists of the Hermitian Hamiltonian




,


ΣN = H (0) + 2Uf c + 2Uf˜ − µ(2.41)(2.42)and the symmetric anomalous couplingΣA = V m .

(2.43)The propagator Σ corresponds to ΣHF (1, 2) in Eq. (2.19). Its mean-field shift is twice as large
as that of the condensate propagator Π, which is a well known property of first-order Hartree–
Fock–Bogoliubov theories. Further details of this transformation can be found in Appendix A.

2.5

Conclusion
We independently rederive the kinetic equations of Walser et al. from the Kadanoff–Baymnonequilibrium Green’s function formulation of kinetic theory, and recover identical factors inall second-order damping rates and energy shifts. This shows that for dilute, weakly interacting
gases the Kadanoff–Baym nonequilibrium, real-time Green’s function approach is microscopically equivalent to the density-matrix approach used by Walser et al. [40]. The latter approach
is more physical in two respects: First, its variables are measurable quantities: the mean field
and the density and coherences of thermal atoms. Second, the variables’ equations of motion
reduce to the Gross–Pitaevskii equation and the quantum Boltzmann equation in the low- andhigh-temperature limits, respectively.

23
Starting from the gapless Beliaev approximation for the collisional self-energy in the
Kadanoff–Baym equations, we furthermore learn that the full second-order kinetic theory ofWalser et al. is gapless itself [54, 46]. This shows that the gap that appears in the first-order
Hartree–Fock–Bogoliubov spectrum [57] is closed by the second-order energy shifts.
Furthermore, this work connects the kinetic theory of Walser et al. with work done by
M. Imamović-Tomasović et al. [46, 53, 58], because they start from the same Kadanoff–Baymequations.

Chapter 3
Gaplessness—T Matrices [59]3.1

Introduction
The extension of finite-temperature theories of Bose–Einstein condensation (BEC) to thereal-time domain has been a very active field of study. The goal is a unified description of a dilute,
atomic gas of bosons in a harmonic trap in terms of a condensate mean field interacting with athermal cloud. The success of the zero-temperature Gross–Pitaevskii (GP) theory in describing
BEC experiments spurred interest in effects not contained in this framework. Examples are
damping of collective excitations and condensate growth through collisional redistributions ofthermal atoms. This redistribution cannot be treated in theories that are still of first order inthe binary interaction, such as, for example, Hartree–Fock–Bogoliubov (HFB) theories.
We present a kinetic theory of second order in the interaction, which is formulated in termsof Bogoliubov quasiparticles and contains collisional terms beyond the HFB approximation. The
HFB interactions are expressed as many-body T matrices to second order in the binary potential
and thus include the correct renormalized scattering physics. This theory thus contains noultraviolet divergences and has a gapless energy spectrum. We extend the papers of Walser et
al. [37, 40], which are the basis of the present results, by clearly demonstrating these two essential
aspects of the theory.
We go beyond theoretical approaches that drop the anomalous pair matrix m̃ = hããi in
the Popov approximation [31, 60, 53, 61]. Monte Carlo simulations based on the semi-classical
Zaremba–Nikuni–Griffin theory [31, 62, 63, 64] show very good agreement for experimentally25
observed damping rates and response frequencies [65]. However, recent experiments [66] andtheir theoretical explanations [67, 68] have shown that the pairing field plays an important rolein Bose gases with resonance interactions.
Keeping the pair matrix m̃ in this theory would cause ultraviolet divergences if we replaced
the non-local interaction potential with a contact-potential, whose strength is given by s-wave
scattering length as . The vacuum part of the pairing field’s self interaction, for example, diverges,
because the delta-function potential contains unphysically high energy contributions. These
divergences can be resolved by writing the interactions in terms of scattering T matrices, which
subsume the divergent sums over intermediate scattering states and correctly reduce to thescattering length as in the zero-energy and -momentum limit [69, 70, 71, 72, 73, 74, 75, 76].
Using these scattering T matrices, we can consistently eliminate all divergences to second orderin the interaction.
We thus obtain a renormalized HFB operator, which has a gapless spectrum. The zeroenergy eigen space is spanned by the condensate, and excitations with non-vanishing energy arethus automatically orthogonal to the condensate. This is similar to the renormalized gapless
HFB equations proposed in Ref. [71]. Other approaches have to explicitly project the excitations
orthogonal to the condensate [77, 72, 78]. Using the condensate as the ground state of thisadiabatic basis also simplifies the representation of a condensate band in the Gardiner-Zoller
master equation formulation [79, 80].
The Monte Carlo simulations of Jackson and Zaremba [62, 63] show that considering
dynamic population-exchange between the condensate and the thermal cloud and within thethermal cloud leads to good agreement with the observed response spectra. We include thesekinetic effects, thus going beyond collisionless descriptions [81, 82].
Our presentation builds on the papers by Walser et al. [37, 40] and we begin by summarizing the kinetic equations derived in these papers in Sec. 3.2. In Sec. 3.3, we examine thefirst-order HFB self energy and rewrite it in terms of second-order T matrices, by including
second-order energy shifts and adiabatically eliminating the pairing field m̃. This shows that26
the theory is explicitly gapless and renormalized. We then make use of the gaplessness andwrite the kinetic equations in terms of Bogoliubov quasiparticles, which are orthogonal to thecondensate by construction. The results in Chap. 4 show that the complicated collision terms
presented in the Walser et al. papers can be simplified dramatically by a basis transformation.
Practical calculations of the quantum Boltzmann equation then require only diagonal elementsof the quasiparticle population matrix.

3.2

Single-Particle Kinetic Equations
We present the Walser et al. formulation of kinetic theory [40], which was originallyderived using a statistical operator approach [37]. In Chap. 2, we derived the kinetic theory fromthe Kadanoff–Baym [43, 44] theory of non-equilibrium Green functions. We used the gaplesssecond-order Beliaev approximation [53, 83] and showed equivalence to the work of Walser et al.
under the Markov approximation. Further, another independent derivation [84, 85] connects thisapproach to kinetic theories by Morgan and Proukakis. Rey et al. [86, 87, 88] use an effectiveaction formalism for calculations in optical lattices that also includes Beliaev effects and reduces
to the Kadanoff–Baym equations in the two-time limit.
Neglecting three-body and higher interactions, we can describe the weakly interacting,dilute gas by the following Hamiltonian
H = H (0)where H (0)10 20

0 0 0 0
10 20 †
a10 a20 + φ1 2 3 4 a†10 a†20 a30 a40 ,(3.1)= h10 | H (0) |20 i denotes the matrix elements of the interaction-free single-particle

Hamiltonian
H (0) =

p2
+ Vext (x)
2m

(3.2)

with external harmonic potential Vext . The bosonic creation operator a†10 creates a particle inthe state |10 i, where 10 stands for a complete set of quantum numbers, which label a constant,single-particle energy basis, such as, for example, harmonic oscillator states or eigen states of

27
the GP equation. We use the summation convention for these abbreviated indices and indicate
the single-particle basis with primes.
The two-particle matrix elements of the binary interaction potential Vbin (x1 , x2 ) aredefined by
0 0 0 0

φ1 2 3 4 =14Z

dx1 dx2 h10 | x1 ih20 | x2 iVbin |x1 − x2 |×nohx1 | 30 ihx2 | 40 i + hx1 | 40 ihx2 | 30 i .



(3.3)

These matrix elements are symmetric in the first and last two indices:0 0 0 00 0 0 00 0 0 0

φ1 2 3 4 = φ 2 1 3 4 = φ 1 2 4 3 .

(3.4)

To determine the measurable quantities we want to calculate in this theory, we first define
the condensate mean field α as the expectation value of the destruction operator
α = α10 |10 i = ha10 i |10 i .

(3.5)

The total density matrix f is defined by
f = ha†20 a10 i |10 i ⊗ h20 | .

(3.6)

Subtracting the condensate density matrix
f c = α ⊗ α∗ = α∗20 α10 |10 i ⊗ h20 | ,(3.7)

we obtain the density matrix of thermal atoms f˜ = f − f c . The anomalous average m is split
analogously
m = ha10 a20 i |10 i ⊗ |20 i = mc + m̃,(3.8)

in the condensate part mc = α ⊗ α and fluctuations m̃.
This set of variables contains all possible combinations of up to two field operators. The
reference distribution parameterized by these variables is thus Gaussian, and we can use Wick’s
theorem [36] to truncate the coupling to higher-order correlation functions, that is, expectation
values of more than two field operators. This approximation is valid because of the diluteness28of the condensed gas. In a dilute gas the duration of a collision event is short compared to theessentially free evolution between collisions, which allows higher-order correlations to dampen.
The full coarse-grained density operator includes higher-order non-Gaussian corrections, whichgive rise to collisional redistributions of populations.
The procedure followed by Walser et al. [37] is then to write the Heisenberg equationsof motion for the variables above and expand the expectation values using Wick’s theorem.
Walser et al. thus obtain the equations of motion given in the following two Sections [40].

3.2.1

Mean-Field Equations
Since the anomalous fluctuations m̃ couple the mean field α to its conjugate α∗ = α∗10 h10 |,it is convenient to write the generalized Gross–Pitaevskii (GP) equation in a two-by-two matrixformdχ = (−iΠ + Υ< − Υ> )χ,
dt

(3.9)

where the two-component state vector




 α 

χ=


∗
α

(3.10)

is defined in terms of α = α10 |10 i and also contains the conjugate mean field α∗ .
The generalized GP operator representing the reversible evolution of χ is defined as


 ΠN
Π=

−Π∗A

ΠA

−Π∗N


.


(3.11)

This symplectic operator consists of the Hermitian HamiltonianΠN = H (0) + Uf c + 2Uf˜ − µ,(3.12)

where µ removes rapid oscillations of the mean field, and the symmetric anomalous couplingΠA = Vm̃ .

(3.13)

29
The energy shifts due to both the mean field and the normal fluctuations are given by thematrices0 0 0 0

Uf = 2 φ1 2 3 4 f30 20 |10 i ⊗ h40 | ,(3.14)

whereas the first-order anomalous coupling-strength is given by
0 0 0 0Vm = 2 φ1 2 3 4 m30 40 |10 i ⊗ |20 i .

(3.15)The second-order irreversible evolution, consisting of damping rates and energy shifts, isgiven by the collision operatorΥ< = 


Υ<
N

Υ<
A

−Υ>∗
A

−Υ>∗
N






(3.16)and its time-reversed counterpart Υ> = −σ1 Υ<∗ σ1 , where σ1 is the first Pauli matrix exchanging
the positive- and negative-energy components of vectors. The matrix elements of the collision
operator are given by the in-rates
Υ<N= Γf˜f˜(1+f˜) + 2Γf˜m̃m̃∗ ,(3.17a)Υ<A= Γm̃m̃m̃∗ + 2Γf˜m̃(1+f˜) ,(3.17b)Υ>N= Γ(1+f˜)(1+f˜)f˜ + 2Γ(1+f˜)m̃m̃∗ ,(3.17c)Υ>A= Γm̃m̃m̃∗ + 2Γ(1+f˜)m̃f˜,(3.17d)and the out-rates

which are defined in terms of individual collisions Γ. These elementary collision processes aredefined explicitly as
0 0 0 000 00 00 00

0 0 0 000 00 00 00

0 0 0 000 00 00 00

0 0 0 000 00 00 00

Γf f f= 8φ1 2 3 4 φ1η 2 3 4 f30 100 f40 200 f400 20 |10 i ⊗ h300 | ,Γf mf= 8φ1 2 3 4 φ1η 2 3 4 f30 200 m40 300 f400 20 |10 i ⊗ |100 i ,Γf mm∗= 8φ1 2 3 4 φ1η 2 3 4 f30 100 m40 400 m∗200 20 |10 i ⊗ h300 | ,Γmmm∗= 8φ1 2 3 4 φ1η 2 3 4 m30 400 m40 300 m∗200 20 |10 i ⊗ |100 i .

The in-rates of the collision operator Υ are depicted in Fig. 3.1.

(3.18a)(3.18b)(3.18c)(3.18d)30'")( *,+ -.+ +' :( * + * + +

/

01

01

4657
231 1
081

291 1

01

4657
/
0 11

0 11

Figure 3.1: These diagrams depict the second-order terms Υ< in the GP equation (3.9). The
dashed potential lines correspond to the symmetrized binary potential φ in the single-particle
energy basis. The directed propagators represent the normal density f˜, the remaining ones theanomalous average m̃ and its conjugate. Note that all diagrams are topologically equivalent,and only propagators are exchanged.

In this theory, collisional interactions are considered to second order. The effect of higher
order terms, which lead to a finite duration of a collision, can be modeled by introducing
a parameter η, such that every second-order collision operator contains dispersive as well as
dissipative parts from the complex-valued matrix element
00 00 00 00

00 00 00 00

φ1η 2 3 4 = φ1 2 3 41,
η − i∆(3.19)

where the energy difference ∆ has to be smaller than the energy uncertainty η to get a sizable
contribution. The energy difference ∆ = −(0100 + 0200 ) + 0300 + 0400 is defined in terms of thesingle-particles eigen energies H (0) 010 = 010 010 . Note that the papers [37, 40] contain a sign
error in the definition of ∆. For small η we obtain1
η − i∆−→η→0πδη (∆) + iPη

1,∆(3.20)

where P indicates that the Cauchy principal value has to be taken upon integration. The
parameter η thus represents off-the-energy-shell propagation after a collision. Most off-theenergy-shell coherences decay during subsequent propagation, but, due to the finite time between
collisions ∆τ , energy cannot be conserved exactly, because η has to be larger than the collision
rate 1/∆τ .

31
3.2.2

Equations for Normal Densities and Anomalous Fluctuations
The equations of motion for the fluctuation densities f˜ and m̃ are coupled and canalso conveniently be written in terms of two-by-two matrices. To achieve this, we define thegeneralized single-time fluctuation-density matrix G̃< as


˜
m̃

 f
,
G̃< = ∗∗˜m̃ (1 + f )(3.21)

where f˜ = f˜10 20 |10 i ⊗ h20 | and m̃ = m̃10 20 |10 i ⊗ |20 i are the matrix representations of the mastervariables. In Ref. [39], we use the property that this density matrix is the single-time limit of
the corresponding time-ordered two-time Green’s function. This showed that the other timeordering is given by (1 + f˜)G̃> = 

m̃∗



m̃ 
 = σ1 G̃<∗ σ1 = σ3 + G̃< .
f˜∗(3.22)

Here, we use the third Pauli matrix σ3 = diag(11, −11). Note that our naming of the fluctuationdensity matrices G̃< and G̃> is consistent with the two-time formalism in Ref. [39], but differsfrom Ref. [40].
The generalized Boltzmann equation of motion for this fluctuation-density matrix can be
written asd <G̃ = −iΣG̃< + Γ< G̃> − Γ> G̃< + H.c.
dt

(3.23)

This equation has to be solved under the constraintsα∗ f˜ = 0 and α∗ m̃ = 0,(3.24)

which force the fluctuations to be orthogonal to the condensate.
Again, the equation of motion (3.23) has two parts: The reversible evolution is governedby the Hartree–Fock–Bogoliubov self-energy operator

 ΣN
Σ=

−Σ∗A

ΣA

−Σ∗N




,


(3.25)

32
which in turn consists of the Hermitian HamiltonianΣN = H (0) + 2Uf c + 2Uf˜ − µ(3.26)and the symmetric anomalous couplingΣA = Vmc + Vm̃ .

(3.27)The irreversible evolution introduced by second-order collisional contributions now consists of the collisional operatorΓ< = 


Γ<
N

Γ<
A

−Γ>∗
A

−Γ>∗
N




,


(3.28)and its time-reversed counterpart Γ> = −σ1 Γ<∗ σ1 . The diagonal components of the collision
operator are defined as
Γ<
N

Γ>
N

= Γ(f˜+f c )f˜(1+f˜) + Γf˜f c (1+f˜) + Γf˜f˜f c

+2 Γ(f˜+f c )m̃m̃∗ + Γf˜mc m̃∗ + Γf˜m̃mc∗ ,= Γ(1+f˜+f c )(1+f˜)f˜ + Γ(1+f˜)f c f˜ + Γ(1+f˜)(1+f˜)f c(3.29)(3.30)

+2 Γ(1+f˜+f c )m̃m̃∗ + Γ(1+f˜)mc m̃∗ + Γ(1+f˜)m̃mc∗ ,and the off-diagonal, anomalous components as
Γ<
A

Γ>
A

= Γ(m̃+mc )m̃m̃∗ + Γm̃mc m̃∗ + Γm̃m̃mc∗
+2 Γ(f˜+f c )m̃(1+f˜) + Γf˜mc (1+f˜) + Γf˜m̃f c ,= Γ(m̃+mc )m̃m̃∗ + Γm̃mc m̃∗ + Γm̃m̃mc∗(3.31)(3.32)

+2 Γ(1+f˜+f c )m̃f˜ + Γ(1+f˜)mc f˜ + Γ(1+f˜)m̃f c .

Both the diagonal and off-diagonal incoming rates are depicted diagrammatically in
Fig. 3.2. For every term that appears in the collisional terms of the generalized GP equation in Υ< [Fig. (3.1)], we here [Fig. (3.2)] have three additional terms, where in each of them,

33

;=>
< ?A@ BC@ @

H
D

E8F

H

H

H

H

H

H

H

H

H

H

G3F F

HJILKNM

HEF; P< ?A@Q?A@ @

H

EF

O

G FF

D

H

E FF
EF
HJILK

M

O

E FF
Figure 3.2: These diagrams correspond to the second-order terms Γ< in the generalized Boltzmann equation (3.23). The dashed lines depict the symmetrized binary potential φ in the singleparticle energy basis. The directed propagators represent the normal density f˜, the remaining
ones the anomalous average m̃ and its conjugate. The first column of diagrams is identical tothose depicted in Fig. 3.1. The remaining diagrams each have one of the three propagators
replaced with an open condensate line.

one of the three fluctuating contributions is replaced with the corresponding mean-field quantity.
This replacement rule can be seen in the Beliaev collisional self energies presented in Ref. [39]and is a consequence of the fact that the Boltzmann equation (3.23) can be generated from theGP equation (3.9) by functional differentiation [45].
When the collision operator Γ< is multiplied by G̃> as in Eq. (3.23), we get terms like˜ −Γ ˜˜Γf˜f˜(1+f˜) (1 + f)
(1+f )(1+f˜)f˜f,(3.33)

where the second part comes from the time-reversed contribution Γ> G̃< . The diagonal parts
are exactly the in and out terms of the quantum Boltzmann equation for the single-particle
distribution function f˜. The remaining second-order contributions couple to the anomalous
fluctuations m̃ and do not have an analogue in the quantum Boltzmann equation. In Chap. 4,
we rewrite the kinetic equations presented here in terms of Bogoliubov quasiparticles. Then all
collisional contributions take the form of Boltzmann terms.

34

3.3

Gaplessness—T Matrices
Our goal in this Section is to explicitly show that the kinetic equations (3.23) and (3.9) are

gapless. This should on one hand be obvious, because the previous Chapter showed them to beequivalent to the Kadanoff–Baym equations [43, 44] in the gapless Beliaev approximation [53, 54].
The first-order HFB self energy Σ, which appears in the kinetic equations, is, on the other hand,
known to exhibit a non-physical energy gap in the long-wavelength, homogeneous limit [57].
We resolve this discrepancy by including second-order collisional energy shifts P{Γ} into
the HFB operator and adiabatically eliminating the anomalous average m̃ in the first-orderanomalous potential Vm̃ (3.15). This upgrades the bare interaction potentials in the first-order
operators Σ and Π to the real parts of many-body T matrices. We then have a systematic way
to approximate the many-body T matrices by two-body T matrices, whose low-energy limit is
the s-wave scattering length. We can thus find a contact-scattering model without incurring
ultra-violet divergences.
The upgraded HFB self energy Σ, where all binary interactions are written as manybody T matrices, is explicitly gapless and thus obeys the Hugenholtz-Pines theorem [89]. The
self energy thus has zero-energy modes, which are completely specified by the value of thecondensate α. If we use the non-zero energy Bogoliubov modes of Σ as a basis for the thermal
excitations, the excitations will automatically be orthogonal to the condensate. We follow thisidea in Chap. 4.

3.4

Off-Diagonal Potentials
Here, we update the off-diagonal potentials V(mc +m̃) in the HFB self energy Σ by adia-

batically eliminating the pairing field m̃. We integrate the first-order equation of motion for the
anomalous average m̃d˜ ∗ − if˜ΣA ,m̃ = −iΣN m̃ − im̃ΣN − iΣA (1 + f)
dt

(3.34)

35
which is obtained by taking the m̃ component of the generalized Boltzmann equation (3.23)
and dropping the second-order terms, because we want to substitute the result for m̃ into theanomalous potential Vm̃ and only keep terms up to second order. In stationarity, that is, forvanishing time derivatives, we solve for m̃ in the dressed eigen basis of ΣN ,
ΣN 10 = (10 − µ) 10 ,(3.35)and obtain0 00
˜ 0 00 + f˜ 0 00 Σ200 20
Σ1A 2 (1 + f)
22
12
Am̃10 20 = P
2µ − (10 + 20 )(3.36)as an adiabatic solution. Adiabatic here means that this solution only includes time-variations
with characteristic times long compared to the duration of a collision. We use the Cauchy
principal value P to indicate omission of the divergent term in an energy integral or sum. This
divergent δ-function term gives rise to the imaginary part. We insert this result into the offdiagonal potential (3.15),
0 0φVm̃1 2 = 4P

˜ 00 00 φ300 200 30 40(1 + 2f)
4 2
mc30 40 ,2µ − (300 + 400 )10 20 300 400(3.37)

where we dropped the recursive Vm̃ term in the anomalous coupling ΣA in order to keep
Eq. (3.37) at second order. We discuss the recursive term in Sec. 3.7.
We then recognize that we can write the off-diagonal element of the HFB operator Σ A
as the real part of a many-body T matrix
ΣA= Vmc + Vm̃ = Tmc (2µ),(3.38)

which is defined byT

10 20 30 40() = 2φ

10 20 30 40

0 0 00 00
˜ 00 00 φ300 200 30 40
φ1 2 3 4 (1 + 2f)
4 2
+ 4P
.
 − (300 + 400 )(3.39)The energies 10 are dressed by the normal and mean-field shifts, but are not the full quasiparticle
energies, because they do not include the effect of the pairing field, which comes in at higher
order. Contractions of this T matrix with anomalous averages are defined by
0 0

0 0 0 012Tm() = T 1 2 3 4 ()m30 40 .

(3.40)

36
The T matrix defined in Eq. (3.39) is a function of energy through its last two indices inthe sense that its argument  = 30 + 40 .

3.5

Diagonal Potentials
In this Section, we want to redefine the diagonal potentials Uf c and Uf˜ as the real partsof T matrices by using the second-order energy shifts P{Γ}. With P{Γ} we here denote theprincipal-value part of the collisional terms in Eqs. (3.28) and (3.16) according to Eq. (3.20).
We begin by considering the condensate potential.
−i 2Uf c + P 2Γf c f˜(1+f˜) + Γf˜f˜f c−2Γf c (1+f˜)f˜ − Γ(1+f˜)(1+f˜)f c= −i 2Uf c − P{Γ1(1+2f˜)f c }(3.41)We here assume real eigen functions for the single-particle energy basis and do not include3/(∆)
any Γ terms involving the anomalous average m̃, because they are at least of order V bin

2according to Eq. (3.36). The second-order terms that contain only normal fluctuations f˜ areused in Eq. (3.46) to rewrite the fluctuation potential Uf˜. The term in Eq. (3.41) can again be
written in terms of a many-body T matrix
Uf c +1P{Γ1(1+2f˜)f c } = Tf c ,2i(3.42)

which is given by
0 0 0 00 0 0 0T 1 2 3 4 () = 2φ1 2 3 4 + 4P

0 0 00 00
˜ 00 00 φ300 200 30 40
φ1 2 3 4 (1 + 2f)
4 2.
 − (300 + 200 )(3.43)The slight difference compared to Eq. (3.39) is resolved when we assume diagonal quasiparticle
populations P1̄2̄ = P1̄ δ1̄1̄ as will be justified in Chap. 4:
˜
f˜10 20 = U11̄0 P1̄ U21̄∗
0 = f 20 10 ,(3.44)

where U is the transformation matrix to the quasiparticle basis. Alternatively, we note that
the T matrix is essentially constant for energy differences up to the duration of a collision.

37
Contractions of the T matrix with normal averages are performed according to
0 0

0 0 0 0Tf1 4 = T 1 2 3 4 (30 + 40 )f30 20 .

(3.45)

We now consider the fluctuation potential Uf˜, again include only the truly second-orderenergy shifts, and obtain−i 2Uf˜ + P{Γf˜f˜(1+f˜) − Γ(1+f˜)(1+f˜)f˜}= −i 2Uf˜ − P{Γ1(1+f˜)f˜} = −i 2Tf0˜,(3.46)

where we get a different T matrix defined by
φ1 2 3 4 (1 + f˜)400 200 φ3 2 3 4,
 − (300 + 200 )0 0 00 00

0 0 0 00 0 0 0

T 01 2 3 4 () = 2φ1 2 3 4 + 4P

00 00 0 0(3.47)

˜ This difference
which does not have a factor of 2 in the intermediate-population term (1 + f).
is due to the fact that the mean field α is not bosonically enhanced. If we assume diagonal
population f˜10 20 = δ10 20 f˜10 10 , which is not a good approximation in this basis, we reproduce
the results of Ref. [90] for the GP equation to second order in the interaction potential. In
particular, the factor of 2 in their many-body T matrix in the term corresponding to Eq. (3.46)
gets canceled with a negative term from adiabatically eliminating their triple average.

3.6

Renormalized Self Energies
Using the T matrices defined in the previous Sections, we can now rewrite the generalized

GP operator Π given in Eq. (3.11) and the generalized Boltzmann operator Σ given in Eq. (3.25).
The Hamiltonian of the GP equation is now
Π0N = H (0) + Tf c + 2Tf0˜ − µ,(3.48)and the anomalous coupling Π0A vanishes, because of the identity
Vm̃ α∗ = {Tmc (2µ) − Vmc } α∗ = {Tf c (2µ) − Uf c } α.

(3.49)

This means that the coupling between α and α∗ vanishes and they are no longer independent
quantities, and that we can without loss of generality treat α as real.

38
When we write the Boltzmann operator in terms of T matrices, we can explicitly showthat the energy spectrum is gapless; this theory thus fulfills the Hugenholtz-Pines theorem [89].
The diagonal part of the operator Σ is now
Σ0N = H (0) + 2Tf c + 2Tf0˜ − µ,(3.50)

Σ0A = Tmc (2µ).

(3.51)and the anomalous coupling is

We now show that this renormalized Boltzmann operator Σ0 has a zero-energy eigen
vector, that is, its spectrum is gapless. We begin by writing the generalized GP equation forthe condensate ground stateno
H (0) + Tf c (2µ) + 2Tf0˜(2µ) α = µα.

(3.52)

This equation can be written in terms of the renormalized GP Hamiltonian (3.48) as ΠN α =0, where the energies are now measured relative to the adiabatic chemical potential µ. An
immediate consequence of Eq. (3.52) is that the quasiparticle ground statePα = √

α 
1 
,



2Nc
∗
−α

(3.53)

is a zero-energy eigen vector of the renormalized Σ0 , because of the identity
Tmc (2µ)α∗ = Tf c (2µ)α.

(3.54)The ground state Pα is normalized by the number of condensate atoms
Nc = α† α,(3.55)

as we find in Sec. 4.1.
These zero-energy eigen vectors of the HFB operator are proportional to the condensate
mean field α. All non-zero-energy eigen vectors WE1̄ 6=0 of Σ are thus automatically orthogonal
to the condensate, and we can use the complete set {α, WE1̄ 6=0 } as a basis to describe the39condensate interacting with thermal excitations [91]. Other approaches to finite-temperature
theories [72, 77] have to explicitly orthogonalize their bases using projection operators. We
discuss this new basis in Chap. 4.

3.7

Ladder Approximation
We can extend the second-order T matrices introduced in the previous Sections to include

ladder diagrams to all orders by using consistency arguments. We first note that by keeping therecursive Vm̃ term on the right side of Eq. (3.37), we can extend our definition of the off-diagonal
T matrix to
˜ 00 00 T 3 2 3 4 ()φ1 2 3 4 (1 + 2f)
4 2.
 − (300 + 400 )0 0 00 00

0 0 0 00 0 0 0T 1 2 3 4 () = 2φ1 2 3 4 + 4P

00 00 0 0(3.56)

This is the real part of the many-body T matrix in the ladder approximation.
In the previous Section, we showed that the HFB operator Σ is gapless with the T
matrices defined to second order. If we used the ladder T in Eq. (3.56) on the off-diagonal while
keeping the second-order T in Eq. (3.43) in the diagonal, we would find an energy gap of thirdorder, because the cancellation in Eq. (3.54) only works if the two T s are identical. However,the Hugenholtz-Pines theorem tells us that the full theory should again be gapless. We thus
conclude that we have to upgrade the T matrix on the diagonal of Σ to the ladder approximation
as well.
We would like to finish our discussion of the scattering matrices in terms of the singleparticle energy basis with two remarks. First, as the Liouville-space formulation [92] of densitymatrix evolution shows, the scattering should really be formulated in terms of Liouville-spacescattering T matrices [93], which can be expressed in terms of Hilbert-space T matrices as
T = T ⊗ 11 + 11 ⊗ T † + T ⊗ T † .

(3.57)

Since we only consider Hilbert-space T matrices, we thus would miss higher-order terms of thetype T ⊗ T † , even if we included the full many-body Hilbert-space scattering matrix. The
imaginary part of T gives rise to the inelastic rates in the kinetic equations in Chap. 4.

40Second, since the asymptotic states in this scattering problem are typically trapped
harmonic-oscillator states for dilute, trapped atomic gases, we are strictly speaking dealing
with bound-state R matrices [94] instead of T matrices.

Chapter 4
Quasiparticle Kinetic Equations [59]4.1

Quasiparticle Basis
We now want to write the kinetic equations (3.23) in the Bogoliubov quasiparticle basis.

In this basis, the complicated and non-linear evolution due to the HFB self energy Σ0 is replacedwith a simple commutator with the eigen energies and a slow basis rotation. As this theoryis gapless, the E 6= 0 quasiparticle states together with the condensate α form an orthogonalbasis, that is, the thermal fluctuations are by definition orthogonal to the condensate. Anothermotivation for transforming to a diagonal first-order Hamiltonian is that the reversible firstorder evolution leaves the quasiparticle populations constant. Thus, in the quasiparticle basis,
only the second-order collisional terms change the populations. A more detailed account of thetransformation to the quasiparticle basis can be found in Appendix B.
Since the quasiparticles consist of the eigen vectors of the self-energy operator Σ 0 , weconsider the operator’s 2n by 2n (n is the number of single-particle states considered) eigenvector matrix W defined by
Σ0 W = W E(4.1)

at each time with the diagonal quasiparticle eigen-energy matrix E, which is labeled with thequasiparticle indices 1̄ = E1̄ . The eigen-value equations (4.1) are the Bogoliubov-de-Gennes
equations. We decompose their solution W into two n by 2n matrices U and V ∗ ,  U  u+
=
W =

 
∗
∗
V
v+

u− 
,

∗
v−

(4.2)

42
∗
which are in turn split into n by n matrices for positive (u+ , v+
) and negative quasiparticle∗
energies (u− , v−). These quasiparticle eigen energies 1̄ are the column indices and the original

single-particle energies 10 the row indices.
Since the operator σ3 Σ0 is positive semi-definite, the eigen values E in Eq. (4.1) arereal and come in positive and negative pairs of equal magnitude [95]. We demonstrated in theprevious Section that there exists a zero-energy eigen vector Pα defined in Eq. (3.53). Thismeans that the operator Σ0 is defective and has two zero-energy eigen values. The eigen-value
equation (4.1), however, does not yield the second linearly independent zero-energy eigen vector.
Instead, the associated vector Qα is given by [95]
Σ0 Qα = −i

Pα
,
M

(4.3)

with a positive constant M . In our case, we find




α
1 
 .
Qα = −i √2Nc  ∗ 
α(4.4)

In order to find a complete set of basis states, we now define the quadrature components of P α
and Qα :
W +0

W −0

=

 

1
1 α ,√ (Pα + iQα ) = √ 
Nc  
2
0
 

0
1
1 
 .
= − √ (Pα − iQα ) = √Nc  ∗ 2
α(4.5a)(4.5b)

These linearly independent and normalizable vectors are not zero-energy eigen vectors, but theyspan the space corresponding to E = 0. If we now substitute these two states W +0 and W −0
into the zero-energy columns of W , we can normalize all eigen vectors using the symplectic norm
W † σ3 W = σ 3 .

(4.6)

The negative-energy states thus have negative norm, and the positive-energy states positive
norm. In particular, the zero-energy vector W +0 has positive norm and thus belongs to thepositive-energy part of W .

43
The completeness relation for the quasiparticle basis also has symplectic structure
W σ3 W † σ3 = 112n ,(4.7)

where 112n indicates the 2n-dimensional unit-matrix. The symplectic structure of the orthonormalization and completeness relations Eqs. (4.6) and (4.7), that is, the appearance of the matrix σ3 in these expressions, guarantees that the transformation to the quasiparticle basis is
canonical [95]. Canonical here means that the new quasiparticle operators obey the boson
commutation relations.
To further examine the structure of the quasiparticle states W , we consider the following
symmetry of the HFB operator Σ0Σ0 = −σ1 Σ0∗ σ1 ,(4.8)

which holds according to its definition in Eq. (3.25). This symmetry implies [40, IV.B.] thefollowing relation for the quasiparticle states W :


 u+
W =

∗
v+



u−  u+
=
 
∗
∗
v−
v+



v+ 
.

u∗+

(4.9)

This relation in Eq. (4.9) shows that the positive- and negative-energy eigen vectors of Σ0
are not independent but related by
W 1̄ = σ1 W −1̄∗ .

(4.10)

Inserting W from Eq. (4.9) into the full completeness relation Eq. (4.7), we find a completeness
relation for the independent elements of W alone:X11̄ 1̄†(u1̄+ u1̄†αα† ++ − v+ v+ ) = 11n .
Nc(4.11)

1̄>0This is the basis we want to use for the kinetic equations. The explicit split in a condensate mode α and the orthogonal fluctuation modes u1̄>0 and v 1̄>0 is similar to the formalismof Ref. [91]. While the condensate mode α evolves according to Eq. (3.9) with the updated
operator Π0 , we have to find an evolution equation for the quasiparticle populations.

44
We thus write the generalized single-time fluctuation-density matrix G̃< defined in Eq.
(3.21) in terms of quasiparticles as† UP UG̃< = W P W † = V ∗P U †UP V>

V ∗P V >




,


(4.12)

where P is the not necessarily diagonal 2n by 2n quasiparticle population matrix. The timereversed density matrix G̃> transforms according to its definition Eq. (3.22). From the same
equation, we can also deduce that the quasiparticle population matrix P is Hermitian and fulfills
a similar identity:
σ3 + P = σ 1 P ∗ σ1 = σ 1 P > σ1 .
This identity implies that P can be written as

p
P =

q∗

q
(1 + p)∗




,


(4.13)(4.14)

in a structure similar to G̃< in Eq. (3.21). Since the classical mean field α only undergoes
stimulated emission, the ground-state factors P+0+0 = P−0−0 do not contain enhancement
terms. Thus, the zero-energy components of σ3 in Eq. (4.13) and similar enhancement terms
associated with α actually have to be zero.
We also represent the condensate state vector (3.10) by a population matrix P c in thequasiparticle basisχχ† = W P c W † .

(4.15)In order to maintain orthogonality between the condensate and the thermal excitations,
we have to demand an analogue to the constraints α∗ (t)f˜ = 0 and α∗ (t)m̃ = 0 in the quasiparticle
basis. We here have to distinguish two cases.
In the first case, the system evolves slowly enough that the ground-state of the adiabatic
basis given in Eq. (4.11) is the true condensate state α(t) = α (real). We can then explicitly
give the condensate matrix as
P c = Nc e0 e†0 ,(4.16)

45
with a vector e1̄0 = δ1̄,±0 , such that all non-zero-energy components of Pc vanish. To implement
the orthogonality constraint, we then explicitly set the E = ±0 elements of the quasiparticle
matrix P to zero:
P01̄ = P1̄0 = 0,

for all 1̄.

(4.17)In the second case, the system evolves too fast for the quasiparticle basis to follow. Then,the condensate matrix contains components of non-zero energy. In this case, the more generalorthogonality constrainthα(t)| P = hα(t) | 1̄iP1̄2̄ h2̄| = 0(4.18)

has to be fulfilled. This constraint is particularly important when the condensate is coherently
excited in linear response. For adiabatic evolution, Eq. (4.18) reduces to Eq. (4.17). In the next
Section, we find the evolution equation for P .

4.2

Kinetic Equations
With these ingredients, we can now try to obtain an equation of motion for the occu-

pation number matrix P from the kinetic equation (3.23). We use Eq. (4.12) to substitute thefluctuation-density matrices G̃ and obtain
d
P
dt

n
o
dW
= −i[E, P ]− + W −1
P + H.c.
dtno+
W −1 Γ< W (σ3 + P ) − W −1 Γ> W P + H.c. .

(4.19)We are now left with the task of transforming the collisional contributions to the quasiparticle
basis. To this end, we define new two-particle matrix elements in the quasiparticle energy basis
by0 0 0 02̄∗ 3̄ 4̄
Φ1̄2̄3̄4̄ = φ1 2 3 4 V11̄∗
0 V2 0 U 3 0 U 4 0 .

(4.20)

These quasiparticle matrix elements have the same symmetries as the original ones given in
Eq. (3.4). Furthermore, they fulfillΦ∗1̄2̄3̄4̄ = Φ1̄2̄3̄4̄∗= Φ−3̄−4̄−1̄−2̄ .

(4.21)

46
The careful examination of symmetries in Appendix B shows that the collision operator
Eq. (3.28) reduces to terms containing the completely symmetric matrix elements
1 X π(1̄2̄3̄4̄)ψ 1̄2̄3̄4̄ = √Φ
,
3 π(4.22)

which are defined as a sum over all index permutations π.
The resulting kinetic equations for the quasiparticle populations P ared
P = −iEP + BW P + CP P + CαP + H.c.,dt(4.23)

with the non-adiabatic basis-rotation term
BW = W −1

dWdW= σ 3 W † σ3,dt
dt

(4.24)and the Boltzmann collision rates between fluctuations
CP P= CP<P (σ3 + P ) − CP>P P(4.25)= ΓP P (σ3 +P ) (σ3 + P ) − Γ(σ3 +P )(σ3 +P )P P
and between fluctuations and the condensate
CαP = 3ΓP c P (σ3 +P ) (σ3 + P ) − 3ΓP c (σ3 +P )P P.

(4.26)

These collision rates are defined in terms of the quasiparticle collision operator
σ3 Γ P P P =1 1̄2̄3̄4̄ 1̄0 2̄0 3̄0 4̄0P−1̄−1̄0 P−2̄−2̄0 P4̄0 4̄ |3̄i ⊗ h3̄0 | .
ψψη
2(4.27)In this operator, P can stand for any one of the three possibilities P , (σ3 + P ), or P c , as neededin Eqs. (4.25) and (4.26). The approximately energy conserving matrix element ψη is explicitly
given by0 0 0 00 0 0 0ψη1̄ 2̄ 3̄ 4̄ = ψ 1̄ 2̄ 3̄ 4̄ πδη (E1̄0 + E2̄0 + E3̄0 + E4̄0 ),(4.28)

where the quasiparticle energies can be positive or negative depending on their index. In an
on-shell scattering event, this delta function forces two of the indices to be positive and theremaining two to be negative. This has to be considered in interpreting the collision terms C P P

47and CαP , because there all the sums run over positive and negative indices. The principal-value
part in Eq. (3.20), which appears in the single-particle kinetic equations, is absorbed in the T
matrices in Sec. 3.3 and thus does not appear anymore in the quasiparticle equations.

RTLS TVU
W RS
X ZT Y[UFigure 4.1: The incoming collision rates for collisions between thermal atoms CP<P and between<
a thermal and a condensate atom CαP. The potential lines are now totally symmetric according
to Eq. (4.22). The propagator lines represent the quasiparticle propagator P , which contains
both the anomalous average m̃ and the normal density f˜. Because of the total symmetry of theinteraction line, the three distinct condensate diagrams on each row of Fig. 2.2 reduce to one
diagram with a weight of 3.

To complete the presentation of the quasiparticle kinetic equations, we write the generalized Gross–Pitaevskii equation (3.9) with the updated GP operator Π0 and the second-ordercollisional contributions expressed in terms of quasiparticle populations P and obtaindχ = −iΠ0 χ + W σ3 ΓP P (σ3 +P ) − Γ(σ3 +P )(σ3 +P )P W −1 χdt

(4.29)In Sec. 4.6, we write this equation for α(t) alone. This equation together with Eq. (4.23) gives acomplete description in terms of Bogoliubov quasiparticles of a condensate coupled to a thermal
cloud at finite temperatures.

4.3

Orders of Magnitude
We now want to discuss the orders of magnitude of several of the quantities of this theory.

This suggests some approximations to the full quasiparticle kinetic equations (4.23).
We first consider the basis transformation W . The completeness relation for the quasi-

48
particle basis Eq. (4.7) tells us that
W = O(1).

(4.30)For example, for high-energy eigen functions, the effect of the condensate becomes small, andthe quasiparticle transformation reduces tou −→ 1 and v −→ 0,

as E −→ ∞.

(4.31)

Now, considering the basis-rotation term defined in Eq. (4.24) we find
BW = W −1

 1 
O(1)
dW
,
=
= O(Γ) < O
dt
dt
∆τ

(4.32)

because the time-scale for population changes, which change the quasiparticle basis W , is limitedby the time between collisions dt > ∆τ . In equilibrium, the populations are constant due todetailed balance of the in and out rates. Thus, the net collision rate Γ = CP P + CαP + H.c.
gives a better estimate for population changes dt ≈ Γ−1 . This also confirms that BW = 0 in
equilibrium, since dW/dt = 0.
We now show that the stationary solutions P of the Boltzmann equation (4.23) areddiagonal. Considering the stationary solution dt
P1̄2̄ = 0 of Eq. (4.23) for an off-diagonal elementwith 1̄ 6= 2̄, we obtain
P1̄6=2̄ = i Γ BW P + CP P + CαP + H.c.
=O.
E2̄ − E1̄∆

(4.33)

This shows that the off-diagonal elements of the quasiparticle population are small compared tothe diagonal ones, which are of the order of the number of particles N , and vanish at equilibrium.

4.4

Quasiparticle T Matrices
The T matrix in Eq. (3.39) (or the ladder extension Eq. (3.56)) has been obtained ap-

propriately to second order. The second-order term with the factor of 1 in Eq. (3.39) is thedivergent part, which is renormalized in Eq. (3.56) when replaced by the T -matrix. The term
containing 2f˜ of Eq. (3.39) is a convergent many-body second-order term. It is thus reasonable

49
to replace the T of Eq. (3.56) with
0 0 0 00 0 0 01234
T 1 2 3 4 (E) = T2B(E + 2µ) + 8P

0 0 00 00
00 00 0 0
φ1 2 3 4 f˜400 200 φ3 2 3 4,
E − (E300 + E400 )(4.34)

where we replaced the single-particle energies in the denominator by quasiparticle energies,
because the difference is of higher order. We here use the following two-body T matrix
0 0 00 00

0 0 0 00 0 0 01234
T2B() = 2φ1 2 3 4 + 4P

00 00 0 0

φ1 2 3 4 φ3 4 3 4, − (0300 + 0400 )(4.35)

which is given in terms of single-particle energies 0 defined by
H (0) 010 =

 p2

2m

+ Vext (x)010 = 010 010 .

(4.36)

The collisional terms of the kinetic equations (4.23) correspond to the imaginary part ofa Liouville-space T -matrix (3.57). Examining the argument of the δη -function in Eq. (4.28),
which defines ψη , in comparison with the ladder T in Eq. (3.56) shows that (when the energies
are correlated with the appropriate elements of W, see Appendix B) two of the energies in the
denominator must be positive and two negative, which, together with the fact that P is diagonalclose to equilibrium, leads to terms in the kinetic equation of the form P P (1 + P )(1 + P ), etc.
The kinetic equation (4.23) then becomes the quantum Boltzmann equation for quasiparticle
populations. The equilibrium solution is therefore the expected Bose–Einstein distribution for
the quasiparticles, as the steady-state solution of Eqs. (4.23) and (4.29) shows [40, Sec. V]. Theinteraction matrix elements φ in Eq. (4.20) can also be upgraded to T s. These equations interms of T are now consistent with an impact approximation treatment (with elastic scattering
not contributing when T ⊗ T † terms are considered, cf Eq. (3.57)).

4.5

Conservation Laws
Because of the non-vanishing pairing field, the trace of the quasiparticle density matrix P

is in general not equal to the number of excited particles in the system. Hence, the numberof quasi-particles is not conserved. Our single-particle kinetic equations (3.9) and (3.23) do,

50
however, conserve the mean total number
˜ = Nc + Tr{f}˜ = const;
hN i = α† α + Tr{f}(4.37)dthis can be proven explicitly by inserting these kinetic equations into dt
hN i and cancelingterms. We thus adopt a self-consistent procedure for the quasiparticle kinetic equations Eqs.
(4.23) and (4.29), by requiringhN i = Nc + Tr{U P U † } = const.

(4.38)

This equation self-consistently constrains the number of condensate atoms Nc : in equilibrium
at temperature β −1 , the quasiparticle matrix P consists of q = 0 andp(E) =1eβ(E−µ) − 1(4.39)

according to Eq. (4.14) with the chemical potential µ given by the GP equation (3.52). As
temperature tends to zero, we obtain the usual corrections for the anomalous average andcondensate depletion [72]. Note that the number of excited atoms is not given by the trace of p,
but has to include the basis transformation U as discussed above. If we drop the basis-rotation
term BW in numerical simulations, we incur number-non-conservation on the order of Γ, while
away from equilibrium.
Since we use a Markovian collision integral with a damping function of finite width η inorder to include off-the-energy-shell propagation, this theory is not exactly energy conserving.
Markovian theories fail to track the decay of initial correlations and thus do not account for the
decay of the correlation energy [96]. In our case, with a self-consistent η, energy is conserved toorder η, which is consistent with the order of approximation. For a detailed discussion of thesememory effects and how they affect the conservation laws see [97].

4.6

Summary
We summarize the main results of Chapters 3 and 4. We formulate a kinetic theory interms of Bogoliubov quasiparticle modes W , which are defined by the eigen value equation for

51
the renormalized Hartree–Fock–Bogoliubov operator Σ0H (0) + 2Tf c + 2Tf0˜ − µ


−Σ0∗
A

T mc 
 W = W E.

−Σ0∗
N

(4.1)

The T matrices are defined to second order in Eqs. (3.39) and (3.47) and Eq. (3.56) gives anextension to the ladder approximation. The Gross–Pitaevskii equationno
H (0) + Tf c (2µ) + 2Tf0˜(2µ) α = µα(4.52)

for the ground state α is contained in Eq. (4.1), because the renormalized Σ0 is gapless. The
GP equation defines the value of the chemical potential µ. To find a complete basis W , we haveto find the second zero-energy mode and form quadrature components as discussed in Sec. 4.1.
We find the following Boltzmann equation for the thermal excitations in terms of Bogoliubov quasiparticles:
d
P
dt

n
o
dW
= −i[E, P ] + σ3 W † σ3
P + H.c.
dtno+ΓP P (σ3 +P ) (σ3 + P ) − Γ(σ3 +P )(σ3 +P )P P + H.c.

(4.23)no+ 3 ΓP c P (σ3 +P ) (σ3 + P ) − ΓP c (σ3 +P )P P + H.c. .

dThe basis-rotation term containing dt
W can be dropped for adiabatic evolution. However, this is

not the case if the system is driven, as in linear response calculations. The quasiparticle density
matrix P is diagonal close to equilibrium, and its elements obey an equilibrium Bose–Einstein
distribution as the second and third lines show. The collision terms containing the general
condensate matrix P c defined in Eq. (4.15) represent population exchange between the thermal
cloud and the condensate. They are balanced in the following equation for the condensate
d
α(t)
dt

=



o
H (0) + Tf c + 2Tf0˜ − µ α(t)(4.29)

+ U σ3 ΓP P (σ3 +P ) − Γ(σ3 +P )(σ3 +P )P σ3
× U † α(t) − V > α∗ (t) .

In general, α(t) can be different from the α used as the ground state of the adiabatic basis. A
non-adiabatic change in a driving force, for example, would cause α(t) to change quickly.

52
The coupled Eqs. (4.23) and (4.29) have to be solved under the orthogonality constraints(4.18) or (4.17) depending on whether the evolution is adiabatic or not. Furthermore, the total
particle number has to fulfill the self-consistent constraint Eq. (4.38).

4.7

Conclusions
We have extended the non-equilibrium kinetic theory of Walser et al. [37, 40] in two

important respects. First, we write the binary interactions as many-body T matrices to second
order in the interaction by subsuming ultra-violet divergent terms. This procedure removes
divergences caused by the anomalous average m̃ and contained in the second-order terms. We
can then replace the low-energy limit of the T matrix with the s-wave scattering length fornumerical calculations. We present a consistent treatment of these difficulties associated withthe anomalous average in a theory that includes second-order collisional terms. The updated
Hartree–Fock–Bogoliubov operator Σ0 is then gapless, which greatly facilitates a consistent treatment of the condensate coupled to thermal fluctuations.
The second extension of the Walser et al. theory makes use of the gapless HFB operator
by using its quasiparticle eigen modes to parameterize the thermal fluctuations, which are thenautomatically orthogonal to the condensate mean field. We find that this basis greatly simplifies
the second-order collision terms of the Walser et al. theory. Another important result reported
here is that, in equilibrium, the Bogoliubov modes diagonalize the quasiparticle population
matrix P . This means that, close to equilibrium, the second-order terms can be evaluated in
n4 operations, where n is the number of energy levels considered. This is a vast improvementcompared to n8 operations for a basis that does not diagonalize the population matrix, such as,
for example, the single-particle basis used by Walser et al.

Chapter 5
Resonance Theory for Fermions [98]In this Chapter, we formulate a many-body theory of a dilute gas focusing specifically on
the description of a scattering resonance. This situation is relevant to current experiments on
quantum gases in atomic physics involving Feshbach resonances. Utilizing a Feshbach resonance,experimentalists are able to probe the crossover physics between the Bardeen–Cooper–Schrieffer(BCS) superfluidity of a two-spin Fermi gas, and the Bose–Einstein condensation (BEC) of
composite bosons. The physical situation giving rise to a Feshbach resonance, a two-channel
system, is the principal focus of this Chapter. In particular, we highlight the criteria necessary
for a two-channel model to be reducible to a single-channel situation. When such reduction
is not possible, additional microscopic parameters which characterize the resonance itself are
found to play an essential role in the many-body problem.
We begin with an overview of the main ideas of superfluid quantum gases, and examine
the general problem of describing resonance interactions.

5.1

Bose–Einstein Condensation and Superfluidity
Bose–Einstein condensation, as the textbook problem was originally posed, occurs in a

non-interacting gas at equilibrium as a consequence of the bosonic statistics of the particles.
Perhaps the most striking aspect is the phenomenon that below a critical temperature a finite
fraction of all the particles are found in a single quantum state, the ground state of the system,
representing the Bose–Einstein condensate. This persists even when one considers the limit in54
which the single quantum ground state is a state of zero integral measure, for example in thecase of a continuous spectrum. Although Bose–Einstein condensation is remarkable in itself,
there is an even more subtle and profound aspect which is the connection with superfluidity.
The phenomenon of superfluidity requires that there be interactions between the particles.
Superfluidity occurs in a variety of physical systems bridging many fields of physics and a wide
range of energetic and spatial scales. Superfluidity can be defined in a number of ways, but
perhaps the most powerful is the connection of superfluidity with the presence of an orderparameter or macroscopic wave function which describes all the superfluid particles. As inany scalar field of complex numbers, a macroscopic wave function ψ(x) contains precisely two
degrees of freedom at each point in space, which may be interpreted in terms of the local
superfluid density n(x) and the local superfluid phase φ(x), that is, ψ(x) =pn(x) exp(iφ(x)).

Since the global phase of a wave function is not measurable and plays no physical role, it is only
the variation in space of the superfluid phase that is of physical significance.
To make this clear, we point out that for the simple situation of a de Broglie matterwave of a free particle of mass m with wave vector k, the wave function is simply a plane wave
proportional to exp(ik · x). In this case it is the gradient of the quantum mechanical phasemultiplied by ~/m which leads to the particle’s velocity, that is, ~k/m. We define the superfluid
velocity in analogous manner from the gradient in the phase of the macroscopic wave function
v=

~
∇φ.
m

(5.1)

While this may appear straightforward, such a construction has truly nontrivial consequences.
Any vector field formed from the gradient of a scalar field is irrotational, that is, ∇×v = 0. Thismeans that a superfluid in which the density is non-zero everywhere cannot contain circulation.
The natural question which immediately arises therefore is how can a superfluid possibly rotate?
The dilemma is naturally resolved by accounting for the fact that it is not required thatthe density be non-zero everywhere, and indeed a superfluid can rotate if there are lines ofzero superfluid density which penetrate the fluid. These are known as vortex lines and have

(a)

55

(b)

π/2

π

vortex
core

0

3π/2
Figure 5.1: (a) The circulation integral in a superfluid depends on the number of vortex lines
enclosed by the loop, in this case seven. (b) A characteristic phase pattern around a vortex core.

been imaged in a number of physical systems, including by direct methods in dilute atomic
gases [11, 99]. Interestingly, when there are many vortex lines, it is generally the case that the
motion of the vortex lines themselves mimics that of rigid body rotation of the system. In any
case, when one performs a circulation integral inside any superfluid, as illustrated in Fig. 5.1,
the circulation is quantizedIloop

v · dl = n

h
m

(5.2)

where n is an integer and is equal to the number of vortex lines enclosed by the loop.

Figure 5.2 shows an example of the density and phase profile of a vortex in a dilute atomic
gas trapped in a harmonic potential. For the state 1 the situation shown is a Bose–Einsteincondensate with no vortex cores and a flat phase profile. In 2 , a single quantized vortex isshown, and the characteristic 2π phase winding around the vortex line of zero density is evident.
The density drops to zero in the center of the vortex core since the superfluid velocity diverges in
the region. The characteristic size of this density dip—the vortex core—coincides with the regionin which the superfluid velocity exceeds the speed of sound. Compare also to the experimental
images in Fig. 1.2, where (c) corresponds to state 1 and (a) to state 2 above.

Of course, it is never guaranteed that in any given quantum system a single macroscopic
wave function exists to define the superfluid. The condensate can, at least in principle, be

56
|2

2π

density

phase

|1

y

y

x

x

Figure 5.2: 1 The phase (top) and density (bottom) profile of a Bose–Einstein condensate in aharmonic trap in the ground state. 2 As for 1 but showing a condensate containing a single
quantized vortex [100].

fragmented over a few quantum states. Indeed, there are many situations in which a more sophisticated description is necessary. A few examples include strongly correlated gases where theinteraction effects are large or resonantly enhanced, gases where the ground state has a macroscopic degeneracy (a situation which occurs for harmonically confined gases at high rotation),and gases in the vicinity of critical points where the ground state symmetry changes.

5.2

Description of a Superfluid in a Dilute Atomic Gas
In a dilute atomic gas, typically composed of ground state alkali-metal atoms, the descrip-tion of interactions in the superfluid phase is typically straightforward and depends on very few
microscopic quantities. The primary physical reason for this is that the de Broglie wavelength
associated with the atoms in the temperature scale at which Bose–Einstein condensation occurs
is much larger than the characteristic range R of the interaction potential.
This characteristic range is determined by matching the spatial scale at which the kinetic
energy coincides with the potential energy for collisions of alkali-metal atoms. If we assume thethermal collision energy to be sufficiently low, and the atoms to be in the ground internal state,then only the long-range part of the van der Waals potential plays an important role. The vander Waals potential is given in general form by C6 /R6 where C6 is a constant coefficient which

57
encapsulates the induced polarizability. In this case, the matching is given by the spatial scale
R for which
C6
~2
= 6.
2
mR
R

(5.3)

In a dilute gas, the resulting scale R for alkali-metal atoms is generally much less than the typicalinter-particle spacing so that the simple picture of contact interactions is relevant. Furthermore,
at low temperature, that is, when the de Broglie wavelength is much larger than R, one can
expect that the low-energy scattering properties do not depend on the detailed structure of theinteraction potential at small internuclear separation.
A further simplification occurs when one considers the effects of quantized angular momentum. Collisions between atoms which have a non-zero value for the orbital angular-momentum quantum number l see a centrifugal barrier proportional to ~2 l(l + 1)/mr2 , which becomeslarge and repulsive at short internuclear separation r. At sufficiently low temperature, the presence of a centrifugal barrier prevents the atoms reaching small enough separation for the true
interatomic potential to have appreciable effect. In that case, collisions only occur in the l = 0
channel where the centrifugal barrier is absent. This channel is known as the s-wave channel.
Thus, in ultracold quantum gases, the interactions are generally parameterized by thes-wave scattering phase shift which may be expressed in length units as the s-wave scattering
length a, which then determines the zero-energy two-body scattering T -matrix, T = 4π~ 2 a/m.
A consequence is that the theory of superfluidity in dilute Bose–Einstein condensates, for themost part, has involved solutions of the nonlinear Schrödinger equation known as the Gross–
Pitaevskii equation
i~

dψ
=
dt



−


~2 2
∇ + V (x) + T |ψ|2 ψ.
2m

(5.4)

The full condensate evolution depends here on three energy contributions in the bracketed
expression on the right hand side of the equation: the kinetic energy, the potential energy V (x)of an externally applied potential, and the internal mean-field energy proportional to both thetwo-body T -matrix and condensate density |ψ|2 . The fact that interactions are parameterized

58
by this particularly concise form of a constant T -matrix is not the only simplification. The
many-body state is also taken to be completely factorisable into orbitals which depend only ona single coordinate; an approximation equivalent to completely dropping explicit two-particle
and higher correlations. In the next Section, we begin to consider the effects of a failure of thisapproximation.

5.3

Breakdown of the Mean-Field Picture—Resonance Superfluids
There are important and relevant situations in which this approach fails. Scattering res-onances can modify the qualitative character since it is possible to tune the two-body scattering
length through infinity by appropriate modification of the details of the potential. When the
scattering length is infinity, clearly the Gross–Pitaevskii equation cannot be applied as written.
The resonance can be of many types: a direct Feshbach resonance, a shape resonance, a potential resonance, or even a Feshbach resonance induced through photo-associative laser coupling.
Regardless of the detailed mechanism, the principles we now outline are almost universally applicable. The case of a Feshbach resonance is illustrated in Fig. 5.3. A closed-channel potential,
typically corresponding to a distinct spin configuration, can support bound states with energies in close proximity to the scattering threshold. The difference in the magnetic moments of
the open and closed channels allow the detuning ν̄ to be varied by application of an external
magnetic field.

When the bound state crosses threshold, the scattering length passes from positive infinity to negative infinity. In the vicinity of this point, the two-body T -matrix is not constantand becomes strongly dependent on the scattering energy. The T -matrix may even acquire asubstantial imaginary component as shown in Fig. 5.4. The assumption that the fields factorize
into single-particle orbitals is no longer valid and quantum correlations must be included as
an essential part of the description. For a single Feshbach resonance, as considered here, the59

Closed Channel

EnergyνOpen Channel

Internuclear Separation
Figure 5.3: A Feshbach resonance. A bound state of a closed potential is in close proximity(with detuning ν̄) to the scattering threshold (dashed line)

behavior of the scattering length as a function of magnetic field is universal and is shown in theinset of Fig. 5.4. Note that the separation of scales arguments that were discussed previously
apply here as well. In its minimal form only two parameters are required to characterize thisresonance: the matrix element between the open and closed channels g, and the detuning fromresonance ν̄. The behavior of the scattering length in this approximation is given by
T =

5.4

4π~2 a
g2
=− .
m
ν̄

(5.5)

Single-Channel versus Two-Channel Approaches
We now turn to the formulation of a many-body description of a dilute gas including

scattering resonances. In accomplishing this task, it is necessary to ensure that the microscopic
physics just explained is correctly incorporated. We begin by presenting two alternative starting
points and then proceed to establish their connection. We focus our attention solely on a systemof fermions, rather than considering directly bosons as described by the Gross–Pitaevskii theory.
This is a good starting point, since a dilute gas of bosons emerges indirectly when the interaction
properties are tuned, such that the fermions pair-up to form composite bosonic molecules.
Since fermions require an anti-symmetric wave function under exchange, s-wave interactions require at least two spin components, which we label as ↑ and ↓. We may then write a

60

0
−1
−2

0

−4

10

−5
0

a (103 a )T−Matrix (103 a )

−3

−6
−7
−8

−2

∆B (Gauss)
0
1

2

5
0
−5

−10
−0.2

−9

−1

−0.1

0
ν (mK)

0.1

0.2

0

−10
0

0.1

0.2

0.3
0.4
0.5
Scattering Energy (µK)

0.6

0.7

Figure 5.4: Real (solid line) and imaginary (dashed line) components of the two-body T -matrix
for collisions of the lowest two spin states of 40 K at a detuning of 20 F (F is a typical Fermi
energy for a dilute gas), shown in length dimensions, that is, T /(4π~2 /m) [101]. The scattering
length is the intercept at zero scattering energy which for this case is approximately −10000 a 0,
where a0 is the Bohr radius. The inset shows the scattering length as a function of detuning,with 20 F detuning indicated by the dashed-dot line.

Hamiltonian for the system, keeping pairwise or binary interactions of general form,
Hsingle =Xkσk c†kσ ckσ +

X

qkk0

Uk,k0 c†q/2+k↑ c†q/2−k↓ cq/2−k0 ↓ cq/2+k0 ↑ ,

(5.6)
(†)

where k = ~2 k 2 /2m is the free-fermion dispersion relation. The operators ckσ annihilate(create) fermions with momentum k and spin σ, and the momentum indices enforce momentum
conservation. The interaction potential U need not be the physical interaction potential forthe atoms being considered, because there is no unique potential that reproduces the physicaltwo-body T -matrix for the given atoms over the relevant energy scale. In other words, one
is free to choose U to be of a particularly simple and convenient form, a procedure known
as renormalization. One way of doing this is to impose the constraint that the potential be
independent of momentum and to cut all momentum sums in the theory off at a maximum
momentum, K. This leads to the following relationship between U and T :T =U,
1 − αU(5.7)

61
where α = mK/(2π 2 ~2 ).
There exists an alternative approach to the many-body formulation. This is based on constructing the many-body Hamiltonian by considering the microscopic formation and dissociationof molecules in the Feshbach resonance state [19, 102]
Hres =Xk a†kσ akσ +k,σ=↑,↓X  q
2

q



X 
gk b†q aq/2−k↓ aq/2+k↑ + H.c. ,+ ν b†q bq +

(5.8)

qk

where gk is the matrix element relating two free fermions in the open channel to the closedchannel bound state near threshold, and ν is the bare detuning of the bound state, which we(†)

relate to the physical detuning by renormalization below. The operators akσ annihilate (create)(†)

open-channel fermions with momentum k and spin σ, while bk

annihilate (create) closed-

channel bosons. See Appendix D for a discussion of when a pair of fermions can be treated as aboson. Said Appendix also discusses a more general Hamiltonian that includes a non-resonantbackground interaction for the fermions.
So we may pose the question: what is the connection between the single- and two-channel
formulations? The relationship can be well understood by considering the two-fermion relative
wave function. An eigen state of Eq. (5.8) can be constructed by considering the following linear
combination of basis statesΨk = χk + Ck φ,(5.9)

where χk = ha−k↓ ak↑ i and φ = hb0 i. The coefficient Ck will be determined later. The evolutionof Ψk is given by taking vacuum expectation values of the Heisenberg equations of motion
generated by the resonance Hamiltonian in Eq. (5.8):


X
dΨk
d
0
0
i~
= i~ (χk + Ck φ) = 2k χk + gk φ + Ck νφ +g k χk .
dt
dt
0(5.10)kThis can be rewritten to eliminate the explicit dependence on the open-channel fermions by
substituting χk = Ψk − Ck φ to give



X
X
dΨk
0
0
0
0
i~
= 2k Ψk + Ck
gk Ψk + gk + Ck ν − 2k −gk Ck φ,
dt
0
0
k

k

(5.11)

62
where the term in brackets contains the residual explicit dependence on the bosons in the
Feshbach resonance state. If this term in brackets vanishes, we obtain an effective single-channel
theory for the dressed eigen-state solution Ψk
i~

X
dΨk
g k0 Ψ k0 .
= 2k Ψk + Ck
dt
0

(5.12)k

The following choice for the coefficient Ck in Eq. (5.9) gives us the above single-channel solution
Ck = Pgk,
2k − E(5.13)

where P denotes the Cauchy Principal Value and E is defined by the solution of an integral
equation
E =ν −PXkgk2.
2k − E

(5.14)The nature of the solution depends on the presence or absence of a bound state indicated by
the sign of the renormalized detuning ν̄. This is defined asν̄ = ν −

X g2

k

k

2k

,

(5.15)and is physically related to the magnetic field shift from the Feshbach resonance [103]. The
case of ν̄ < 0 corresponds to the side of the resonance in which the scattering length is positive.
There, a bosonic dimer bound state exists and the solution of Eq. (5.14) coincides at small
detuning with the bound-state energy E = −~2 /ma2 [104, 105, 67]. For ν̄ > 0 there is no boundstate and the solution is E = ν̄.

5.5

Poles of the Molecular Propagator
Finding the correct physical solutions for the energy in Eq. (5.14) requires some care. If

we simply were to drop the principal value from Eq. (5.14), we would find the following equations

63
for a complex-frequency pole of the molecular propagator
ω=ν−

g2
2π 2

Z ∞
0

g2 m
= ν̄ − 2 2
2π ~= ν̄ −dk

k2~2 k 2 /m − ω + iδ

Z ∞

dk

Z

dz

0

g 2 m3/2
4π 2 ~3,

δ = 0+(5.16a)ω(5.16b)~2 k 2 /m − ω + iδω.
z 2 − ω + iδ

(5.16c)The subtle point here is that the solution one finds depends on the integration path in thecomplex plane. If we perform the integral, the poles arise as roots of the following quadratic
equationz2 ± ig 2 m3/2z − ν̄ ,4π ~3√

ω = z.

(5.17)

4
3
2
1

E

0
−1
−2
−3
−4
−2

−1

0

_
ν

1

2

Figure 5.5: Poles of the molecular propagator as a function of detuning (solid line) real part,(dashed line) imaginary part. The insets show the various integration contours which lead tothe solutions shown. The true bound state is the upper solid curve for ν̄ < 0.

Figure 5.5 shows the various possible solutions. Note the prediction of two bound-statesolutions on the positive side of the resonance (ν̄ > 0), and the shift of the real part of the64pole to higher detunings than ν̄ at large detuning. With the exception of the true bound state
at negative detuning (ν̄ < 0), these curves are misleading and do not correspond to physicalsolutions of the two-channel scattering problem.

5.6

The Equivalent Single-Channel Theory
Using the definition for E in Eq. (5.14), we may write the prefactor bracket of φ in

Eq. (5.11) asX g k g k0 
gk
(2k − E) + ν − 2k − P,
2k − E2k0 − E0(5.18)kand substituting Eq. (5.14), which defines E, this isi
gk h2k − E + ν − 2k − ν + E = 0,2k − E

(5.19)

as required. The evolution of Ψk is then given byi~

X g k g k0dΨk
= 2k Ψk + P
Ψ k0 .
dt2k − E0(5.20)kThis is nothing more than a time-dependent Schrödinger equation for an effective single-channel
problem. In other words, an effective single-channel theory has now been shown to be encapsulated by the resonance Hamiltonian theory. The interaction potential, defined in Eq. (5.6), canbe directly read off from Eq. (5.20)
Uk,k0 = Pg k g k0.
2k − E

(5.21)

What remains is to show that this potential generates the correct scattering length at
all detunings ν̄. To this end, we must obtain the two-body T -matrix by solving the LippmannSchwinger equation [106]Tk,k0 = Uk,k0 +X
q

Uk,q Tq,k0,
2k0 − 2q + iδ

δ → 0+ .

(5.22)

We wish to solve this in the limit of zero scattering energy k → 0 and constant gk → g.
Equation (5.22) can then be rewritten as a series by recursive substitutions
T =−

g2 X
g2
g2
+ P
+... .
E
E
2k00 (2k00 − E)00
k(5.23)

65
Now from the definition of E in Eq. (5.14)
E =ν −PXkg2
2k − E

(5.24a)=ν −P

X g 2 (2k − E + E)

= ν̄ − PXkk(5.24b)2k (2k − E)

g2E,2k (2k − E)

(5.24c)which leads toPν̄ − E
g2
=
.
2
(2
−
E)
E
k
k
00

X
k

(5.25)

Substituting this expression into Eq. (5.23) and continuing similarly for the rest of the terms,
we arrive at the geometric series
g2
T =−
E
=−
=−

E − ν̄
1++E



E − ν̄
E

g2
E(1 − ((1 − ν̄)/E))
g2
.
ν̄

2

+...

!

(5.26a)(5.26b)(5.26c)Equation (5.26) provides the correct behavior of the tuning of the scattering length aroundresonance, with the usual definition T = 4π~2 a/m, and confirms that the potential Uk,k0 leads
to the correct effective fermion interaction properties.
We have thus presented a detailed mathematical proof of the equivalence of the two initial
Hamiltonians for the single- and two-channel models describing the scattering of two fermionsin vacuum. One must extend this result to consider the equivalence in systems that containmore than two fermions. The structure of the mathematical proof can be continued along
the presented lines. The result for the important case of four fermions is that the equivalence
between the single and two-channel theories has been shown to hold, but requires that the widthof the Feshbach resonance be sufficiently broad [107].

66

5.7

Connection with the Theory of Feshbach Resonances
We can equivalently express the two-channel model in terms of the original language ofthe open P and closed Q channels as done by Feshbach [108, 109, 110]. In terms of these separate
Hilbert subspaces, the time-independent Schrödinger equation takes the following coupled form:
E |ΨP i = HPP |ΨP i + HPQ |ΦQ i ,(5.27a)E |ΦQ i = HQQ |ΦQ i + HQP |ΨP i .

(5.27b)

We may formally solve Eq. (5.27a),|ΨP i =

1E − HPP

HPQ |ΦQ i ,(5.28)and substitute the result into Eq. (5.27b) and obtain
E − HQQ − HQP

1
E − HPPHPQ!

|ΦQ i = 0.

(5.29)The effective interaction due to the coupled spaces is thereforeeff
HQQ= HQQ + HQP

1
E − HPP

HPQ .

(5.30)

This expression is, in fact, similar to Eq. (5.14); except that in this case it is in operator form,
whereas Eq. (5.14) is represented in a basis. The chosen basis involves a continuum k for the
P subspace, a single quantum resonance state φ for the Q subspace, and an explicit form forthe matrix elements of momentum dependent coupling. The mapping is thus φ HQQ φ = ν,k HPP k = 2k , and φ HQP k = k HPQ φ = gk .

5.8

The BCS–BEC Crossover
One of the first attempts to understand the crossover between the phenomena of BCS and

BEC was put forth by Eagles in a 1969 paper on pairing in superconducting semiconductors [111].
He proposed moving between these two limits by doping samples, in this case by decreasing
the carrier density in systems of SrTiO3 doped with Zr. In a 1980 paper by Leggett [112],

67
motivated by the early ideas of quasi-chemical equilibrium theory, he modeled the crossover atzero temperature by way of a variational wave function:
ψBCS =Yk(uk + vk a†k a†−k )|0i.

(5.31)

This wave function is simply the BCS wave function and assumes that at T = 0 all the fermionsform Cooper pairs. What Leggett was able to show was that he could smoothly interpolate
between conventional BCS theory and the occurrence of BEC.
In 1985, Nozières and Schmitt-Rink (NSR) extended this theory to finite temperatures, inorder to calculate the critical temperature TC [113]. NSR derived the conventional BCS gap andnumber equations, but introduced into the number equation the self-energy associated with theparticle-particle ladder diagram (or scattering T-matrix) to lowest order. This very influential
paper was built upon by many other groups and was transformed into a functional form byRanderia et al. [114].
A compelling motivation for understanding the crossover problem comes from the factthat many high-TC superconductors fall within the intermediate region between loosely bound
BCS pairs and a BEC of tightly bound pairs. In the copper oxides, for instance, the coherence
length of the Cooper pairs has been measured to be only a few times the lattice spacing. In
contrast, in conventional superconductors, the coherence lengths are usually much greater thanthe lattice spacings. An understanding of the crossover may be one of the keys to understanding
and manipulating high-TC materials.

Dilute quantum gases have already played a very important role in experimentally probing
the BCS–BEC crossover. This crossover is, in fact, a special case of a more general frameworkof resonance superfluids. In the broad resonance limit, which is generally the experimentally
relevant one, the system maps on to the BCS–BEC crossover problem originally introduced in thecontext of condensed matter systems. Figure 5.6 illustrates some of the important distinctionsof resonance superfluids. In particular, the pairing in a weakly coupled BCS superconductor

T > Tc
BCS
Superconductivity

(a)

(c)

kF
(b)

Resonance
Superfluidity

T < Tc

68

kF

kF
(d)

kF

Figure 5.6: Schematic comparison of BCS theory and the BCS–BEC crossover theory of resonance superfluidity. Resonance superfluidity describes closed-channel, tightly bound pairs offermions (green) in addition to the loosely bound BCS pairs (red/yellow). Below the transition
temperature TC , the closed-channel pairs condense and also mediate pairing of open-channel
fermions away from the Fermi sphere.

occurs primarily at the Fermi surface in momentum space and the superfluid appears out of thedegenerate Fermi sea at a critical temperature TC much less than the Fermi temperature. As
Fig. 5.6 illustrates, the physical situation for resonance superfluids can be quite different, withpairing throughout the Fermi surface, and molecular condensation of the composite bosons.
Furthermore the critical temperature for superfluidity in this case can be comparable to the
Fermi temperature. This is very important, since current experiments in dilute quantum gases
can, at the lowest, reach temperatures on the order of a tenth of the Fermi temperature, which
is far above the critical temperatures predicted by simple BCS theory in the region in which it
can be applied.

Chapter 6
Imaginary-Time Propagation for Fermions [98]6.1

Imaginary-Time Methods for Single- and Two-Channel
BCS Models
The method of steepest descents has been widely applied for finding condensate wave

functions in Boson systems. In this Section, we want to generalize this method and calculate
the single- and two-channel BCS solution for interacting fermions. Our imaginary-time approach
can be generalized to include beyond-BCS interactions.
The most important advantage of our imaginary-time method for fermions is that itgives direct access to zero-temperature ground states for fermion systems without diagonalizing
the BCS self-energy matrix. One could, for example, study topological excitations of the BCS
superfluid by imposing symmetry constraints. For example, imposing a 2π phase winding arounda central core results in a vortex state as depicted in Fig. 5.2.

6.1.1

Single-Channel BCS Theory
BCS theory is a single-channel theory for fermions, whose interactions are characterized

by the scattering length a as the single microscopic parameter [18]. The BCS Hamiltonian canbe diagonalized analytically by solving the following number and gap equations
n=

Z ∞

k2
dk 2
2π

Z ∞

k2
dk 2
4π

0

m
=
4π~2 a0

k − µ F1− p(k − µ)2 + ∆2

!

1
1
−p
k
(k − µF )2 + ∆2

and
!

,

(6.1a)(6.1b)

70
where k = ~2 k 2 /(2m) is the kinetic energy, µF the chemical potential, ∆ the superfluid gap
and n = n↑ + n↓ the total particle density. These equations are self-consistently solved for thechemical potential and the gap, and one obtains the Bogoliubov quasi-particle modes u k and vk ,which are given by1
u2k = 1 − vk2 =21− pk − µ F(k − µF )2 + ∆2

!

(6.2)

The normal and anomalous averages fk and mk at zero temperature are then given by
fk↑ = ha†k↑ ak↑ i = vk2

andmk = ha−k↓ ak↑ i = uk vk(6.3a)(6.3b)The normal average fk↑ is the density of spin-up atoms at momentum k, and the anomalous
average a pair-correlation function between atoms of opposite momentum and spin. We wantto find the solutions (6.3) for the averages by using imaginary-time propagation.

6.1.2

Imaginary-Time Propagation for Bosons
How does imaginary-time propagation work for bosons? The basic idea is to replace thetime variable t in the Gross–Pitaevskii (GP) equation for the condensate wave-function ψ,
dψ
=
i~
dt




~2 2
2
−
∇ + V + T |ψ| ψ = HGP ψ,
2m

(6.4)

with the imaginary time variable −it. The time-evolution under the GP equation (6.4) can be
written in terms of its eigen states φn , which are defined by HGP φn = En φn , with the eigen
energies En :


iEn t
ψ(t) =
cn exp −
~
n
X



φn ,

(6.5)

where the coefficients cn are defined by the expansion of the initial condition ψ(t = 0) =P

n cn φn . Propagating the GP equation in imaginary time changes the above time evolution toEn tψ(t) =
cn exp −
~
n
X



φn .

The unitary time-evolution in Eq. (6.5) has turned into an exponential decay.

(6.6)

71
The algorithm for the imaginary-time method is now to use the imaginary-time evolution over a time interval and renormalizing the resulting wave function after each step using anormalization condition or number equation, in this case,N=Z

d3 x|ψ(x)|2 .

(6.7)

This procedure converges on the lowest-energy ground state solution φ0
ψ(t)

−it

−→

φ0 ,

(6.8)

provided that the ground state is nondegenerate. Due to numerical errors, this even works if
the initial wave function ψ(t = 0) does not contain a contribution of the ground state, thatis if c0 = 0. The convergence can, however, be accelerated in practice by choosing ψ(t = 0)
appropriately.
Imaginary-time propagation can include symmetry, topological, or orthogonality constraints, and one can thus calculate topological condensate states or higher-excited states ofthe Hamiltonian. See Fig. 5.2 for an example of a vortex state calculated using imaginary-time
propagation. We now generalize this powerful approach to fermions.

6.1.3

Imaginary-Time Propagation for Fermions
In the case of bosons above, we learned how to propagate a wave-function equation in

imaginary time and thus find ground-state solutions. Time-dependent BCS theory, which is thesimplest single-channel theory for interacting fermions, has two equations for the normal f k andanomalous density mk ,dfk↑
= i2U = (p∗ mk ) ,
and
dt
dmk
= 2(k − µF )mk + U p (1 − fk↑ − fk↓ ) ,
i~
dt

i~

with the pairing field p =P(6.9a)(6.9b)

mk and the renormalized potential U , which is derived from theT -matrix, and where = indicates the imaginary part. The second equation for the pairingcorrelation can be evolved like the GP equation in Sec. 6.1.2. The first equation for the density,

72
however, is a density matrix equation, which does not evolve like a wave function. Density
matrices evolve under two time-evolution operators called tetradics with positive and negative
energies, such that the diagonal elements do not evolve at all. The conventional imaginary-timealgorithm would thus not change the initial particle distribution function.
We here propose a new solution to finding the evolution of the density matrix equationby using the Bloch–Messiah at zero temperature.

Bloch–Messiah Theorem
In this Section, we use the Bloch–Messiah theorem [115, 116, 117] to find a relation
between the density fk and the pairing correlation mk that we can use instead of the densitymatrix equation to determine the evolution of fk .
We motivate the theorem by first discussing the Cauchy–Schwartz inequality. The inequality holds for any inner-product space and can be written in the usual bra-ket notation ashα|αihβ|βi ≥ |hβ|αi|2 .

(6.10)

Choosing values for |αi and |βi, we can prove the following relationha†k↑ ak↑ iha−k↓ a†−k↓ i ≥ |ha−k↓ ak↑ i|2 .

(6.11)

At zero temperature, this relation becomes an identityfk↑ (1 − fk↓ ) = |m2k |,(6.12)

as one can see from the quasi-particle vacuum relations Eqs. (6.3) and the properties of the
Bogoliubov modes in Eq. (6.2). One can prove the identity in Eq. (6.12) at zero temperature forany set of evolution equations for which one can find a quasi-particle transformation. This is the
Bloch–Messiah theorem. It can be generalized [118, 119] and extended to finite temperature [120,121], and has also been discussed for Bosons [122, 123].
In order to be able to use Eq. (6.12) as planned, we have to assume spin symmetryfk↑ = fk↓ = fk .

(6.13)

73
On closer examination, we note that the solution for fk of Eq. (6.12) has two branches1
fk = + sgn(µF − k )2

r1
− |mk |2 .
4

(6.14)The sign function here picks the positive branch for energies below the chemical potential andthe negative branch for higher energies, as is the case for the BCS solutions given in Eqs. (6.3).

6.1.4

Imaginary-Time Algorithm for the Single-Channel Model
With these ingredients, we can now formulate the new algorithm for finding zero-tempera-ture ground states in interacting fermion systems.
1. Pick an initial pairing correlation mk and chemical potential µF .
2. Calculate the pairing field p =P

mk and the density fk according to Eq. (6.14).

3. Evolve the anomalous density mk for a time step dt in imaginary time using~dmk= −2(k − µF )mk − U p (1 − fk↑ − fk↓ ) .
dt(6.15)

4. Repeat (2) and (3) until convergence.
5. Adjust chemical potential µF in (1) until total density n =Pk,σ fkσ is correct.

We have verified numerically that this algorithm yields the BCS solution both for local and a
Gaussian, non-local potential. One can, in fact, show analytically that the BCS equations area solution to the imaginary-time equations. See Fig. 6.1 for the results for the local potential.
Can we generalize this algorithm to the two-channel case?

6.1.5

Imaginary-Time Propagation for the Two-Channel Model
The two-channel model with contact interactions is equivalent to a single-channel modelwith non-local interactions. However, models with contact interactions are much easier computationally. We further show in the following Sections how we can extend the two-channel contact

741

Normal density fkAnomalous density |mk|

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.5

1

1.5

Momentum k/kF

2

2.5

3

Figure 6.1: Normal (full line) and anomalous (dashed) density for a single-channel model at
kF a = −1. The paired fermions predominantly occupy states near the Fermi energy.

model to sufficiently high-order correlations to properly reproduce the composite boson-boson
scattering length. This extension is not feasible for the single-channel model with a contact
interaction.

Two-Channel Equations of Motion
We begin by deriving the equations of motion of the relevant mean fields for the twochannel model. The crossover Hamiltonian for a homogeneous system is, again, given byH=Xk,σ=↑,↓

k a†kσ akσ +X  q
q

2



X 
+ ν b†q bq +gk b†q aq/2−k↓ aq/2+k↑ + H.c. ,(6.16)

qk

where we now have composite-boson fields bq coupling to the fermions.
The minimal set of mean fields that we now have to derive equations of motion for isthe anomalous density mk and now also the condensate wave function φm = hb0 i. The normal
density is again given by the Bloch–Messiah result in Eq. (6.14). We ignore the lowest-orderthermal molecular mean fieldsb†q bq≡ hb†q bq i − |φm |2 δq0 andb−q bq≡ hb−q bq i − φ2m δq0 ,which neglects the quantum depletion of the molecular condensate. Note that there is no thermal
depletion, since we only consider zero-temperature ground states.
To derive the equations of motion for the relevant mean fields, we first write the Heisen-

75
berg equations of motion for the three individual operators
i~

i~

dak↑
dt

i~

X

g−q/2+k a†q−k↓ bq ,X

gq/2−k a†q−k↑ bq ,

q

dak↓
dt

= k ak↑ += k ak↓ −(6.17a)and(6.17b)

q



X
dbq
q
=
+ ν bq +gk aq/2−k↓ aq/2+k↑ .
dt
2(6.17c)k

We then take the average of Eq. (6.17c) to obtain the equation of motion for the condensate
wave function φm
i~

X
dφm
mk = νφm + gp.
= νφm + gdt

(6.18)

k

We similarly combine Eqs. (6.17a) and (6.17b) to obtain the equation of motion for the anomalous density mk
i~

X
dmk
bq a†q+k↑ ak↑ .
= 2k mk + gφm (1 − fk↑ − fk↓ ) − 2gdt
q(6.19)

This equation for the anomalous density mk couples to the three-operator cumulantba† a . The cumulant notation again indicates that the lower-order factorized averages havebeen subtracted out. We drop this cumulant for now, but in Sec. 6.2 we discuss its importancefor reproducing the correct molecule-molecule scattering length on the BEC side of the crossover.
With these two equations of motion (6.18) and (6.19), we can now update the algorithm
for the steepest-descent method.

Imaginary-Time Algorithm for the Two-Channel Model
The algorithm now has two coupled wave functions that need to be evolved.
1. Pick an initial pairing correlation mk , condensate wave function φm , and chemical potential µF .
2. Calculate the pairing field p =P

mk and the density fk according to Eq. (6.14).

3. Evolve the anomalous density mk and condensate wave function φm for a time step dt

761

Normal density fkAnomalous density |mk|

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.5

1

1.5

Momentum k/kF

2

2.5

3

Figure 6.2: Normal (full line) and anomalous (dashed) density for a two-channel model at
kF a = −1. This is calculated for the broad resonance in 6 Li at 834 G, with a bare molecular
fraction of 4 · 10−6 [124]. One can see the good agreement with the single-channel result in
Fig. 6.1 in the broad resonance limit.

in imaginary time usingdφm= −(ν − 2µF )φm − gp anddtdmk~= −2(k − µF )mk − gφm (1 − fk↑ − fk↓ ) .
dt(6.20a)~(6.20b)

4. Repeat (2) and (3) until convergence.
5. Adjust chemical potential µF in (1) until total density n =Pk,σ fkσ + 2|φm |2is correct.

In Fig. 6.2, we show results for this algorithm for a contact two-channel model. Theylook very similar to the ones we found in the single-channel case in Fig. 6.1. Is that what wewould expect? The superfluid gaps of both theories turn out to be the same
∆ = U p = gφm .

(6.21)

However, the number equations are slightly different. In the single-channel case discussed
in Sec. 6.1.3, only the dressed fermions are summed over
n1C =

X
kσ

fkσ ,

(6.22)

77

Figure 6.3: Schematic illustrating the crossover between fermions, whose interactions can be
described by BCS theory with scattering length a, and composite bosonic molecules, with interactions given by 0.6a, as a function of detuning ν̄, that is, magnetic field.

whereas, in the two-channel case, both the bare fermions and bare molecules contribute to thetotal fermion density
n2C =

X
kσ

fkσ + 2|φm |2 .

(6.23)

Each molecule contributes two fermions to the total density. This difference results in a correction to the chemical potential µF ; a small correction in the broad-resonance case.

6.2

A Mean-Field Description for the Crossover Problem
In this Section, we want to determine the minimal ingredients for a mean-field theory

that wants to correctly reproduce the molecule-molecule scattering between composite bosonson the BEC side of the resonance. Consider first a schematic picture of the crossover in Fig. 6.3.
The picture shows how the overlapping, loosely bound Cooper pairs on the right side of theresonance contract as the detuning is lowered and changes sign. On the BEC side, at negative
detuning, the pairs turn into tightly bound molecules, which are interacting with a moleculemolecule scattering length of approximately 0.6a [125], where a is the atom-atom scattering
length.

78

Figure 6.4: Schematic of the interaction between two dimers of paired fermions.

6.2.1

Boson Scattering Length

To find an expression for the dimer-dimer scattering length add , which is the effective interaction of the composite bosonic molecules, Petrov et al. [125] start with a four-body
Schrödinger equation in the set of coordinates defined in Fig. 6.4,

1mE
− ∇2r 1 + ∇2r2 + ∇2R + 2 Ψ =2~!
X  r1 + r2
m
±R
Ψ,
U
− 2 U (r 1 ) + U (r2 ) +~2±(6.24)

where U (r) is the two-body potential in real space. This equation is simplified by assuming a
pseudopotential boundary condition
r →0

1
Ψ(r1 , r 2 , R) −→f (r2 , R)



1
1
−
r1
a



,

(6.25)

which is valid, because the effective range of the interatomic potential U is small compared tothe scattering length a. The factor multiplying f (r 2 , R) on the right-hand side of Eq. (6.25) isan expansion of the bound-state wave function exp(−r1 /a)/r1 near threshold. This boundary
condition Eq. (6.25) implies that we do not need the full four-body wave function Ψ, which is sixdimensional in a homogeneous system, to describe the dimer-dimer scattering correctly. Instead,
it suffices to solve for the reduced wave function f (r 2 , R), which has only three independent
dimensions in a homogeneous system. This simplification allowed the authors of [125] to solve
the scattering equation (6.24) and find the dimer-dimer scattering length as add ≈ 0.6a, a resultthat has been supported experimentally [126, 127, 28].

What is the physical meaning of the wave function f (r 2 , R)? The schematic in Fig. 6.5
depicts f (r2 , R) as an atom-molecule correlation function between a tightly bound dimer and79
two loosely bound fermions. As we have seen in Sec. 6.1, this correlation function is not partof BCS theory or the lowest-order mean-field picture of the crossover we discussed in Sec. 6.1.5.
We did, however, see in that Section how to extend the equations of motion: Equation (6.19)
couples to a three-operator correlation function that is of the same vector structure as f (r 2 , R).
To extend the set of equations in Sec. 6.1.5, we would have to derive an equation of motion for
the new correlation functionba† a . This correlation in turn couples to other three-operator

correlation functions. We assume that we can drop all couplings to still higher-order correlations
and solve the coupled three-operator equations. This yields a theory that includes the Hartree
self-energy shift in the crossover and yields the observed dimer-dimer scattering length within
our approximations as discussed above.
We may thus anticipate that one should be able to combine the quantity f (r 2 , R) withthe mean-field description of the crossover that we began to present in Sec. 6.1.5 to get a more
complete picture of the crossover as presented in Table 6.1.

6.2.2

Beyond Pair Correlations
In the last Section, we learned that we need to include four-particle correlation functions

in order find the correct value for the molecule-molecule interactions on the BEC side. Here,we want to revisit the single- and two-channel models discussed in the context of BCS theory
in Sec. 6.1 and see how they can be extended to include these beyond-pair correlations.

Figure 6.5: This atom-molecule correlation function is the minimum ingredient needed to recover
the boson-boson scattering length for the composite molecules as 0.6a. The schematic on theright shows the dimensionality of f (r 2 , R) in momentum space.

80
Table 6.1: A more complete picture of the crossover
φm = hb0 i

BEC: Interactions mediated by fermions⇑b−q aq/2−k↓ aq/2+k↑Crossover⇓mk = ha−k↓ ak↑ i

BCS: Interactions mediated by bosons

Four-Particle Correlations in the Single-Channel Model
The Hamiltonian of the single-channel model isH=Xk a†kσ akσ +

kσ

X

qkk0

Uk−k0 a†q/2+k↑ a†q/2−k↓ aq/2−k0 ↓ aq/2+k0 ↑ .

(6.26)The minimum necessary mean-field to include the required four-particle correlations in thismodel isa−q/2−k↓ a−q/2+k↑ aq/2−k0 ↓ aq/2+k0 ↑ .

(6.27)

With this Hamiltonian, it is impossible to contract a fermion pair into a boson directly, so wehave to treat all four particles explicitly. The four-particle correlation above is a function of
three vectors, and thus has six degrees of freedom in a homogeneous system, which is numericallyvery difficult.

Four-Particle Correlations in the Two-Channel Model
Let us now see whether the two-channel model has an advantage in describing the necessary four-particle correlations. The Hamiltonian for this model isH=Xk,σ=↑,↓

k a†kσ akσ +X  qq2X+ ν b†q bq + gb†q aq/2−k↓ aq/2+k↑ + H.c. ,(6.28)

qk

which shows that this model contains composite molecules explicitly. The minimum correlation
function to include four-particle interactions is now
b−q aq/2−k↓ aq/2+k↑ ,(6.29)

81
which is a function of only two momentum vectors. The dimensionality of this correlation
function is thus only three in a homogeneous system, which is directly accessible in numericalcalculations.
The two-channel model thus gives naturally a minimal description that is at this levelof approximation consistent with the vacuum scattering properties of four-particle scattering
discussed in Sec. 6.2.1.

6.3

Summary
Atomic physics has provided a wealth of information on a variety of aspects of super-

fluidity, both in bosonic and fermionic systems. We have presented the foundation concepts ofsuperfluids, and discussed the vortices which support rotation in superfluid systems. We haveshown how the separation of scales, both in energy and in physical space, lead to a simplified
parametrization of the interaction effects in dilute quantum gases. Of particular interest has
been Feshbach resonances, which allow the collision effects to be resonantly enhanced.
We have shown that for two fermions in vacuum, one is able to prove the equivalence
between the two-channel approach which arises naturally in the description of Feshbach resonances, and the single-channel approach which is a typical starting point for condensed mattertheories. We should emphasize here that in the case of a many-body system, the single-channel
and two-channel theories do not generally coincide if simple contact potentials are chosen.
The description of Feshbach resonances in dilute atomic gases has required the development of a many-body theory able to describe strong correlations and specifically the point of
infinite scattering length. Careful consideration must therefore be made of the breakdown of
simple mean-field approaches which contain the scattering length explicitly. We have shown how
one may include the two-channel Feshbach formulation in the many-body Hamiltonian. This
problem is relevant to the theoretical description of many current experimental efforts, exploring
the formation and dissociation of molecules around Feshbach resonances, and the crossover from82
fermionic to bosonic superfluidity.

Chapter 7
Zero-Temperature Correlation Effects the BCS–BEC CrossoverIn this Chapter, we formulate a many-body mean-field theory of the BCS–BEC crossoverproblem. In particular, we want to extend the imaginary-time method presented in Chap. 6 to go
beyond pair correlations. Figure 7.1 shows the deviation between an experiment in the Jin group
at JILA [129] and a theory that is calculated to include only pair correlations [128]. Including
higher-order correlations, beginning with the Hartree term, might remedy this discrepancy.
We first need a suitable description of correlations and introduce the notion of a cumulant
6
5

3

rel

E /E0

kin

4

2
1
0
−1

−0.5

0

0.5

1

−1/(k(0)a)
F

Figure 7.1: Released energy of a harmonically trapped gas as a function of the detuning−1/(kF0 a(0)) for a ramp rate of 2µs/G (blue line) [128]. The red circles are the experimental results from Ref. [129]. The blue theory line is calculated with pair correlations only. The
lower, green line is the corresponding result solving the two-body problem associated with themolecular state. The energy is normalized to the kinetic energy of the non-interacting gas0
Ekin= 30F /8.

84mean field, which is a many-particle correlation function. The order of a cumulant is given by
the number of particles—more precisely, the number of operators— involved. The equationsof motion for cumulants form a hierarchy, because the evolution of a cumulant of order n isdependent on lower-order cumulants and only the cumulant of order n + 1. In a dilute system,
this Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy can be cut, because higherorder correlations rapidly dampen between subsequent two-body collisions [33, 29]. This cutof the hierarchy gives a closed set of coupled equations, which can be simplified further by
adiabatically eliminating the remaining highest-order cumulants using a secular approximation.
We show how this elimination renormalizes the detuning as previously discussed in Chap. 5.
This renormalization removes the cutoff dependence of the bare detuning and relates it to theapplied magnetic field in the experiments.
We present two examples for this method of cumulant expansion using the two-channel
BCS–BEC crossover Hamiltonian Eq. (5.8):
1. Condensed bosonic molecules and fermions interacting at zero temperature. In this case,
we use the imaginary-time methods discussed in the preceding Chap. 6.
2. Thermal bosons and fermions interacting in the normal phase, that is, above the criticaltemperature, as discussed in the following Chap. 8. We show how the cumulant expansion gives rise to rate equations similar to those found by Williams et al. [130, 131].
We begin by defining the type of correlation function we use to derive the BBGKY
hierarchy.

7.1

Cumulants
A cumulant expansion is a systematic way of classifying correlation functions of differ-ent orders [132, 133, 134, 74]. The order of a cumulant is given by the number of operators
involved, that is, the number of operators in the correlation function. Cumulants are essentially
expectation values of operators, where the lower-order contributions, which can be obtained by

85
factorization, are removed. We have already encountered several of these correlation functions
for composite bosons bq and fermions ak↑↓ in a homogeneous system:
Bare molecular wave function φm = b0 .

(7.1)

Fermion density fk↑ = a†k↑ ak↑ .

(7.2)

Pairing correlation mk = a−k↓ ak↑ .

(7.3)

These lowest-order correlation functions are just given by the thermal average, because they do
not factorize. The single-operator averages for fermions vanish, since macroscopic occupationof a single state is not possible for fermions, that is, they do not Bose–Einstein condense. We
here use a symmetry-breaking approach as discussed in [135, 136], where it is shown that thisapproach can be rigorously justified for Bose–Einstein condensed systems. Alternative numberconserving approaches [137, 138, 139, 78] yield equivalent results, but may differ in the formal
details.
A simple example of an actual cumulant is the density of thermal molecules:b†q bq = hb†q bq i − |φm |2 δq0 ,(7.4)

where δ is the Kronecker delta function. Here, the average hb†q bq i contains the factorizable
component |φm |2 , which gets subtracted out in the case of zero momentum to define the twoboson cumulant b†q bq . The pairing function of the molecules is analogously given by:
b−q bq = hb−q bq i − φ2m δq0 .

(7.5)

A more complicated three-operator correlation between a boson and a pair of fermions isbq a†q+k↑ ak↑ = hbq a†q+k↑ ak↑ i − φm fk δq0 .

(7.6)

The following four-fermion average contains two lower-order contributions, which have to be
removed to define the cumulanta†q/2+k↑ a†q/2−k↓ aq/2−k0 ↓ aq/2+k0 ↑ = ha†q/2+k↑ a†q/2−k↓ aq/2−k0 ↓ aq/2+k0 ↑ i− m∗k mk0 δq0 − fk0 ↑ f−k↓ δq0 .

(7.7)
(7.8)

40
1

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1
−3

−2

−1

0

1

Detuning −1/(kF as)

2

3

Paired fraction M+Z
Closed−channel fraction Z

0.9

0.8

0
−4

Li resonance at 834 G

1Paired fraction M+Z
Closed−channel fraction Z

0.9

86

6

K resonance at 224 G

4

0
−4

−3

−2

−1

0

1

Detuning −1/(kF as)

2

3

4

Figure 7.2: Pair fractions of the total fermion number for the resonance at 224.2 G in 40 K (left)
and the wide resonance at 843 G in 6 Li (right).

To apply the cumulant method, we discuss the extension of the imaginary-time propagation
method for finding zero-temperature ground states presented in Chap. 6 to include higher-ordercorrelations. We begin by discussing some results we obtained for our two-channel model.

7.2

Pairs in the BCS–BEC Crossover

In a two-channel model, pairing takes place both in the open and closed scattering channels. However, the closed- (Z = 2|φm |2 ) and open-channel (M = 2

2
k |mk | ) contributions inP

Fig. 7.2 are just projections of the physical molecular bound state onto the Feshbach subspaces.
In the case of 6 Li on the right side of the Figure, it seems that the closed-channel contribution isvery small and thus insignificant. Remember, however, that with our model Hamiltonian (5.8)
the fermions do not interact directly. The pairing in the open channel is thus only due to thesmall closed-channel contribution. Furthermore, the closed-channel part of the 6 Li has actually
been observed, as can be seen in Fig. 7.3. The experiment measures the photoexcitation rateof the singlet closed-channel bound state to a free state [124]. The open-channel pairs are in a
triplet state, and thus do not get excited. The agreement of our zero-temperature simulation
with the experimental points is very good on the BEC side of the resonance (left), which indicates that neglecting thermal molecules is a good approximation. On the BCS side (right),

87

6

Li resonance at 834G

−3

Closed channel fraction Z

10

−4

10

−5

10

−6

10

−7

10

Rice experiment (T<TC=.22TF)
Simulation (T=0,ρ=1012/cm3)
Simulation (T=.11TF)
Simulation (T=.22TF)
−2

−1

0

1

Detuning −1/(k a )F2sFigure 7.3: Closed-channel fraction for the wide resonance in 6 Li [124].

finite-temperature effects are more important for reducing the closed-channel fraction, as our
finite-temperature results show. Finite temperature effects can thus account for the discrepancy
between zero-temperature theory and experiment. Note in particular that the finite-temperature
theory lines coincide with the zero-temperature results on the BEC side. We use a generalizationof the imaginary-time procedure using an equilibrium distribution for the excited quasiparticle
states. The self-energy operator defining the quasiparticle states can include the Hartree andBose mean-field corrections discussed below, and we can derive a quantum-Boltzmann equation
for the quasiparticles, along the lines of Chap. 4.

7.3

Mean Fields
The model we consider for the imaginary-time algorithm is a homogeneous system atzero temperature with symmetrically populated spins up and down. There is no pairing withineach spin state, because s-wave interactions are suppressed. We thus use the following set of

88elementary mean-field correlation functions:
Bare molecular wave function

φ m = b0 ,(7.9)

Fermion density fk↑ = a†k↑ ak↑ = fk↓(spin symmetry),

Pairing correlation mk = a−k↓ ak↑ .

(7.10)(7.11)We also neglect the following correlation functions:
Cross-level magnetizationa†k↑ ak↓ = 0,

Pairing for equal spinsa−kσ akσ = 0(7.12)(no s-wave pairing for fermions),b†q bq = b−q bq = 0

Thermal molecules(zero temperature).

(7.13)(7.14)The number of particles is fixed by the following density equation
ntot =Xkσ

fkσ + 2|φm |2 ,(7.15)

where the momentum sums are spherically symmetric and evaluated as follows
Xk1
F (k) = 22πZ

k 2 F (k) dk.

(7.16)The next Section shows how these sets couple to higher-order cumulants, and how we obtainclosed equations for the above mean fields by adiabatic elimination.

7.4

Equations of Motion

In this Section, we show how to obtain closed equations for the mean fields listed inSec. 7.3 by adiabatically eliminating three- and four-operator cumulants. Figure 7.4 schematically shows the BBGKY hierarchy of correlation functions, which we cut by dropping thecoupling to five-operator correlation functions. This amounts to a kinetic approximation, which
is valid in a dilute system, where the inter-particle distance n−1/3 is small compared to the rangeof the two-body potential r0 ,nr03  1,

(7.17)

89
Cut

1)
2)

Bloch-MessiahgFigure 7.4: Schematic of the cumulant hierarchy. We cut the coupling to five-operator cumulants
and adiabatically eliminate the steady-state solutions at levels 1) and 2) to obtain a closed
equation for the pair correlation mk .

and many-particle correlations can thus decay between collisions. The first elimination step 1)of the four-operator cumulants introduces the renormalization of the detuning ν̄. The second
elimination 2) of the resulting three-operator equations yields an updated equation for the pairfunction mk including the Hartree and higher-order interaction terms.
We begin by repeating the coupled equations for the molecular wave function φm (6.18)
and the pair function mk (6.19), which we have derived from the Heisenberg equations (6.17)
previously:
i~

X
dφm
= νφm + g
mkdt

(7.18)

k

i~

X
dmk
bq a†q+k↑ ak↑= 2k mk + gφm (1 − fk↑ − fk↓ ) − 2gdt
q(7.19)The normal density is again calculated from the Bloch-Messiah theorem [140, 141]:fk↑ (1 − fk↓ ) = |mk |2 .
The pair correlation mk couples to the three-operator cumulant(7.20)
bq a†q+k↑ ak↑ , whose

90
equation of motion,i~d bq a†q+k↑ ak↑dt= ν − 2q/2+k + 2k bq a†q+k↑ ak↑ − g fq+k↑ mk
+g

X
k0

aq/2−k0 ↓ aq/2+k0 ↑ a†q+k↑ ak↑(7.21)− gφ∗m bq a−k−q↓ ak↑ + gφm bq a†q+k↑ a†−k↓ ,is also derived from the Heisenberg equations (6.17). The equation of motion for the threeoperator correlation again couples up to the next order. We have, however, already dropped
the coupling to the four-operator correlationbb† aa , because this correlation function does

not contribute to the four-fermion scattering problem. In writing the following equation for theother required four-operator correlation function we cut the BBGKY hierarchy and drop the
coupling to five-operator correlations:
i~

d
†
a
a
0 a
0 a
dt q/2−k ↓ q/2+k ↑ q+k↑ k↑= (−2q/2+k + 2k + 2k0 ) aq/2−k0 ↓ aq/2+k0 ↑ a†q+k↑ ak↑+ g bq a†q+k↑ ak↑= 2µ aq/2−k0 ↓ aq/2+k0 ↑ a†q+k↑ ak↑ .

(7.22)(7.23)(7.24)(7.25)We also neglect all many-body terms proportional to the normal density and pair function andonly keep two-body terms, because the two-body terms dominate in the high-momentum limitthat is important for the renormalization. We set the time-derivative on the left-hand side equal
to a global phase given by the steady-state energy 2µ, which is determined by the number ofcreation and destruction operators. Each fermion creation operator evolves with −µ and each
destruction operator with µ, which results in the 2µ given above. The boson operators each
evolve with 2µ, respectively. The steady-state energy 2µ is constrained by the total number.
If we included correlations to all orders and were in full thermodynamic equilibrium, µ would
be the chemical potential of the fermions. Since we cut the correlation hierarchy, we need to
explicitly examine the change in energy as a particle is added to find the chemical potential [142].
This steady-state approximation allows us to adiabatically eliminate the four-operator

91
correlation as
aq/2−k0 ↓ aq/2+k0 ↑ a†q+k↑ ak↑ = −

g
a
b a†
,
∆E q q+k↑ k↑(7.26)

where the energy denominator is given by∆E = −2µ − 2q/2+k + 2k + 2k0 .

(7.27)

We can substitute this result (7.26) into the full equation for the three-operator correlation (7.21)
and find that the new term enters the kinetic energy.
i~d bq a†q+k↑ ak↑
dt



X g2
= ν −P− 2q/2+k + 2k bq a†q+k↑ ak↑∆E0k(7.28)− g fq+k↑ mk − gφ∗m bq a−k−q↓ ak↑ + gφm bq a†q+k↑ a†−k↓ .
We here only keep the real principal-part contribution according to the relation
11=P− iπδ(∆E).
∆E + i∆E(7.29)In Appendix E, we demonstrate in the normal phase how the imaginary delta-function terms
give rise to the collisional quantum-Boltzmann rates, which determine equilibrium. This allows
us to define a diagonal quasiparticle representation for fermions and composite bosons, which
can be used to prove the Bloch-Messiah theorem Eq. (7.20).
The new term in Eq. (7.28) renormalizes the bare detuning ν to the physical valueν̄ = ν − P

X g2
k0

2k0

= ν − g2

4π
P
8π 3

Z K
0

mk 02 0mK
dk = ν − 2 2 g 2 ,~2 k 022π ~(7.30)as discussed in Sec. 5.4.
Adiabatically eliminating the coupling to the four-operator cumulant thus updates thebare detuning ν to the renormalized physical detuning ν̄. This happens analogously in theequations for all three-operator correlation functions.

7.5

Adiabatic Elimination of Three-Operator Correlations
We now continue by further eliminating the three-operator correlations, in order to obtainclosed equations for the pairing function and molecular mean field. At first, we only keep terms to92order g 2 to illustrate the lowest-order corrections. We start by finding the steady-state solutionof Eq. (7.28),
E1 bq a†q+k↑ ak↑ = −g fq+k↑ mk +g2 ∗g2
φm mq+k mk +φm fq+k↑ fk↓ ,E2
E3(7.31)

with the following energy denominators
E1 = 2µ − ν̄ + 2q/2+k − 2k ,(7.32a)E2 = 4µ − ν̄ − q − 2q/2+k , and(7.32b)E3 = 2q/2+k − ν̄.

(7.32c)Substituting Eq. (7.31) into Eq. (7.19) for the pairing field we obtain the following set of equations
i~

X
dφm
= (ν − 2µ)φm + g
mk ,dt(7.33)ki~dmk= 2(k + Uk + Vk − µ)mkdt
X
+ gφm 1 − fk↑ − fk↓ − 2 Pqg2
fq+k↑ fk↓ ,E1 E3(7.34)

together with the Bloch-Messiah relation (7.20). The lowest-order correction terms are theHartree term,
Uk = g 2 P

X fq+kqE1,(7.35)

and a Bose term, which is the lowest contribution to the molecular self-energy on the BEC sideof the resonance,
Vk = −g 3 P

X φ∗ mq+k
m

q

E1 E2

.

(7.36)In the following, we neglect the last term in Eq. (7.34), because it is a higher-order correction
to the many-body Pauli-blocking factor (1 − fk↑ − fk↓ ). We now consider the resulting equationof motion for the dressed paired state to find how the Bose term (7.36) enters the molecular
self-energy.

93

7.6

Dressed Pair Correlations
We are looking for an eigenvalue solution for Eqs. (7.33) and (7.34) at energy E. Thissolution describes a dressed pair. We replace the steady-state energy 2µ with the eigen energy E
and try the following Ansatz for a pair wave functionXχ m = N φm + P
k


g
mk ,
E − 2k(7.37)

with eigen energy
E =ν+PX g 2 1 − fk↑ − fk↓
E − 2kk



,(7.38)

and a normalization constant N which is chosen so that the norm of χm matches the density ofpairs|χm |2 = np =Xk|mk |2 + |φm |2 .

(7.39)The density of unpaired fermions can then be defined as nf = ntot − 2np . The eigen energy E
only coincides with the steady-state energy 2µ if all higher-order correlations are included and
we are in full thermodynamic equilibrium.
In the two-body case, the Pauli-blocking factor in the expression for the energy in
Eq. (7.38) vanishes and we have
E2B = ν + P

X
k

g2
,
E − 2k(7.40)

for the two-body dressed energy. Equation (7.40) corresponds to Eqs. (7) and (8) in Ref. [143].
In that paper, Fano discusses the interaction of a discrete state with a continuum of states andfinds the numerator of Eq. (7.40) to be the absolute value squared of a matrix element of the
Hamiltonian. This is the case for the two-body formula above. The full many-body case in
Eq. (7.38), however, has a Pauli-blocking factor in the numerator, which can become negativeand can thus not be written as the square of a matrix element. This means, that the many-body
dressed state above is not an eigenstate of a two-body Hamiltonian. The dressed pairs canthus not be interpreted as two-body molecules, because of their inherent many-body nature.
A generalization of BCS quasiparticles along the lines of Chap. 4 for the boson case is a more1

=0 (resonance)=−4 (BEC)

0.45
k

0.8

Anomalous average m

k

0.5

F

0.9

Distribution function f−1/(k a)=4 (BCS)

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

94
−1/(k a)=4 (BCS)F=0 (resonance)=−4 (BEC)

0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05

0.5

1

Momentum k/k

1.5

0
0

2

0.5

F

1

Momentum k/kF

1.5

2

Figure 7.5: Correlation functions in the BCS/BEC crossover. We plot the normal densityfk (left) and pair correlation function mk (right) as a function of momentum for different
detunings −1/(kF a). The blue line is in the BCS regime, the green line on resonance, and the
red line in the BEC limit.

appropriate picture for the pairs. See also App. D for a discussion of a density matrix for thepaired fermions.
Using Eqs. (7.33) and (7.34) for closed and open channel pairs, we obtain the following
equation for the dressed pair correlation,i~

X Uk + V kdχm= Eχm + 2g P
mk .
dt
E − 2k(7.41)

k

7.7

Numerical Results

We use the imaginary-time algorithm discussed in the previous Chapter and now use
Eqs. (7.33) and (7.34), which include the Hartree and Bose correlation corrections to the selfenergy. We iterate the equations and calculate new self-energy corrections at each step, toobtain a self-consistent result. The final change in the distribution functions is below 10−4 ,indicating a good level of convergence. In Fig. 7.5 we show the resulting correlation functions
across the BCS/BEC crossover. In the BCS limit (blue line), the fermions obey a sharp FermiDirac distribution, and pairing is limited to close to the Fermi momentum. With decreasing
detuning, pairing spreads through the Fermi sphere, as can also be seen in Fig. 7.2.

0.2

1

Un
0.6 Un/2
E

0.15

0.9

Hartree

0.1

E

n/n

E

+E

Bose

Hartree

0.05

OC

E/N−EBound/2 [3/5 EF]

Self Energies [EFermi]

95

BCS pairs (open channel)

n/nOC

Bose

0
−0.05
−0.1
−0.15
−0.2
−15

0.8
0.7

EBCS
µ

BCS

Esc
µsc

0.6
0.5
0.4
0.3
0.2
0.1

−10

−5

0

5

Detuning −1/(k a)

10

15

0
−15

−10

−5

0

5

Detuning −1/(kF a)

F

10

15

Figure 7.6: Energy plot for the narrow resonance in 6 Li at 543 G as a function of detuning −1/(kF a). On the left we plot the Hartree and Bose contributions to the self energy,
together with the dashed asymptotic values for the BCS side (positive detuning) in blue andthe BEC side (negative detuning) in green. On the right we plot the steady-state energy 2µ
and energy per particle, after subtracting the binding energy on the BEC side, which is theasymptotic value for these quantities. The plot on the right is in units of the ideal-gas energy
per particle 3/5EF.

In Fig. 7.6 we show energy plots for the narrow resonance at 543 G in 6 Li. On the left,
we plot the Hartree- and Bose-contributions to the self energy as a function of detuning. The
asymptotic value for the self-energy corrections on the BCS side (blue dashed) is given byUn =4π~2 an,m(7.42)

where n is the density of fermions, m is their mass, and a the s-wave scattering length. On
the BEC side, the expected dimer-dimer scattering length is add = 0.6a [125], which reduces
the asymptotic value of the self-energy correction (green dashes). We divide the Bose termby the open-channel fraction nOC /n to match the asymptotic behavior. The numerical resultsinterpolate between the two limits. On the right, we plot the energy per particle minus theasymptotic value on the BEC side in units of the ideal-gas value of 3/5E F . Including theself-energy corrections lowers the total energy.
To calculate the energy per particle, we evaluate the expectation value of the Hamiltonian (5.8). We first consider the BCS level, where we drop correlations beyond pairs:
EBCS = Hres =Xk,σk fkσ + ν|φm |2 + 2gXk<(φ∗m mk ).

(7.43)

Dressed pairs

0.4

0.1
0
−0.1
−0.2

Bose

0.5
0
−0.5
−1

−0.3
−0.4

Hartree

E

]

Bose

Fermi

0.2

Un
E

1

HartreeE

Self Energies [E

Fermi

]

0.3

Self Energies [E1.5Un.6 Un/2
E

96

BCS pairs (open channel)

−3

−2

−1

0

1

Detuning −1/(k a)

2

−1.5

3

−3

F

−2

−1

0

1

Detuning −1/(k a)

2

3

F

Figure 7.7: Self energy plots for 40 K as a function of detuning −1/(kF a), together with the dashedasymptotic values for the BCS side (positive detuning) in blue and the BEC side (negative
detuning) in green. The plot on the right shows the self energies for the open-channel BCS
pairs, on the left we show dressed pairs, which are a superposition of open- and closed-channel
contributions

The self-consistent energy Esc , on the other hand, includes a three-operator contribution andthe energy is calculated from the self-consistent averages fk and mk .
Esc = EBCS + 2g

Xqk

< bq a†q/2−k↓ a†q/2+k↑ .

(7.44)The right-hand side is given by the following adiabatic, principal value solution for the threeoperator correlation:2q/2+k − ν̄bq a†q/2+k↑ a†q/2−k↓ = g fq+k↑ fk↓ + gφ∗m

bq a†k+q↓ ak↓ + bq a†−k↑ a−k−q↑(7.45)In Fig. 7.7 we show the self-energy corrections for the resonance at 224 G in 40 K. In thiswider resonance, we find that the Hartree term dominates across the resonance. On the right,
we plot the corrections for the open-channel BCS pairs, and find the same asymptotic behavior
on both sides of the resonance. Only considering dressed pairs (left) recovers the expected
dimer-dimer scattering behavior on the BEC side.

97

7.8

Summary
We extend the imaginary-time algorithm developed in Chap. 6 by using a cumulant

expansion to include higher-order correlation effects. In particular, we include the Hartree
term, and the lowest order contribution to the molecular self-energy on the BEC side and show
numerically that we obtain results consistent with the observed dimer-dimer scattering for theeffective bosons. This means that our many-body mean-field theory includes the four-fermion
correlations necessary to properly describe the effective bosons. It is straightforward to extend
the cumulant method we use to include correlations of higher order than considered here.

Chapter 8
Many-Body Dynamics of the BCS–BEC Crossoverin the Normal Phase

In this Chapter, we consider fermions coupled to bosonic composite molecules above thecritical temperature for the BEC or BCS transition in a homogeneous system. We use the
BCS–BEC crossover Hamiltonian (5.8), which models a Feshbach resonance with a two-channel
model , and use the cumulant expansion discussed in the previous Chapter to derive equationsof motion for the thermal densities of composite bosons,nq = b†q bq ,(8.1)

fk = fk↑ = ha†k↑ ak↑ i = fk↓ ,(8.2)

and fermions,in a spin-symmetric system. We assume that the temperature is high compared to the transition temperature so that we can neglect all symmetry-broken terms, such as the fermion paircorrelation mk and the molecular mean field φm . The number equation for this model is thus
ntot =Xk,σfkσ + 2

X

nq .

(8.3)

q

Figure 8.1 shows schematically the correlation hierarchy we find for this model. We can cut thehierarchy at the following levels:
Cut A: If we cut here, we decouple the fermions and bosons and thus cut out the physics weare interested in.

99

Cut C

Cut B
Cut A

Figure 8.1: Schematic of the BBGKY cumulant hierarchy for the normal phase. The levels of
this hierarchy are determined by the number of operators involved. In the dilute-gas systems
we are interested in, a separation of time scales (the time between collisions is much larger thanthe duration of a collision) allows us to cut this hierarchy by dropping the correlation functions
at a certain level.

Cut B: This yields a theory that describes the crossover problem for coupled, thermal bosons
and fermions. However, we numerically find that the resulting equations are not positive
definite. Furthermore, the renormalization of the detuning, Eq. (5.15), borrows terms
from the next level of the hierarchy and this cut can thus not be performed cleanly.
Cut C: This is consistent with the renormalization used and yields a positive-definite theory.
However, the four-operator cumulants are functions of three momenta and have too
many degrees of freedom for a full numerical treatment.
In the following discussion, we derive the required equations of motion and show how adiabatic
elimination of the four-operator cumulants leads to a closed set of three coupled equations, whosetime dependence we numerically simulate. Simulations are performed for the narrow resonancein 6 Li, where the open and closed channels are coupled weakly, that is, the coupling constant g in
the crossover Hamiltonian Eq. (5.8) is of order one in units of the Fermi energy and the systemdensity. The experimentally explored wide resonances in 6 Li and 40 K have coupling constants
that are two orders of magnitude larger, which makes the full time dependence very unstable.
The zero-temperature method discussed in the last Chapter does not have this limitation.
Numerical results show that, above the Feshbach resonance, fermion and boson distributions stay in thermal equilibrium, even as the detuning is changed as a function of time. In
Appendix E we derive effective coupled Boltzmann-type rate equations for the fermions and100a) Fermion 1

b)

c)
g

g
Boson

g

g

∗

g∗

Fermion 2

Figure 8.2: Scattering rates for the crossover Hamiltonian Eq. (5.8). a) Interaction node forthe crossover Hamiltonian. This is the diagram for the first-order conversion of fermions tobosons, which is energetically suppressed on the BEC side of the resonance. b) Effective fermionfermion interaction through intermediate boson state. c) Second-order fermion-boson interaction
through particle exchange.

bosons similar to those found by Williams et al. [130, 144, 131], which explicitly show the collisional contributions depicted in Fig. 8.2. Each of these contributions contains energy-conserving
delta functions between the incoming and outgoing lines. Rate b) thus allows off-shell intermediate boson states, which can not be represented by combining two collisions of type a), because
they each conserve energy.

8.1

Equations of Motion
We list below the full set of normal-phase equations up to cut C illustrated in Fig. 8.1.

First, we have the normal densities of fermions and composite bosons,
i~

o
nX
dfk↑
bq a†k↑ a†q−k↓ ,= 2ig =
dt
q

i~

nX
o
dnq
= −2ig =bq a†q/2+k↑ a†q/2−k↓ ,dtk(8.4a)(8.4b)

101
where = indicates the imaginary part. These couple to the three-operator correlation function,i~d bq a†q/2+k↑ a†q/2−k↓dt= (ν − 2k ) bq a†q/2+k↑ a†q/2−k↓+gXaq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓−gXbq b†q0 aq0 −q/2+k↑ a†q/2+k↑ + bq b†q0 aq0 −q/2−k↓ a†q/2−k↓

k0

q0



+ g fq/2+k↑ fq/2−k↓ (1 + nq ) − g nq (1 − fq/2+k↑ )(1 − fq/2−k↓ ),(8.4c)
which in turn depends on the following two four-operator correlation functions,i~d bq b†q0 aq0 −q/2+k,σ a†q/2+k,σdt= 2 (q0 −q)/2+k − k
± g fq/2+k,σ + nqbq b†q0 aq0 −q/2+k,σ a†q/2+k,σb†q0 aq/2−k,−σ aq0 −q/2+k,σ∓ g fq0 −q/2+k,σ + nq0



(8.4d)

bq a†q/2+k,σ a†q/2−k,−σ ,where the signs on the right-hand side correspond to the two possible spin directions, and,i~d aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓dt= 2 (k0 − k ) aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓+ g 1 − fq/2+k0 ↑ − fq/2−k0 ↓− g 1 − fq/2+k↑ − fq/2−k↓bq a†q/2+k↑ a†q/2−k↓b†q aq/2−k0 ↓ aq/2+k0 ↑ .
(8.4e)

These equations of motion are derived from the Heisenberg equations (6.17). Equations (8.4) are exact in the normal phase, apart from the highest-order ones, Eqs. (8.4d) and(8.4e), where we dropped the couplings to five-operator cumulants to cut the BBGKY hierarchyof correlation functions.
The three-operator correlation (8.4c) is a three-dimensional quantity in a homogeneous
system. This is the maximum number of degrees of freedom that can be treated numerically. We
thus eliminate the four-operator cumulants Eqs. (8.4d) and (8.4e) by adiabatically solving forthe four-operator cumulants and substituting into the equation for the time-dependent threeoperator correlation bq a†q/2+k↑ a†q/2−k↓ . This yields the equations we simulate numerically.

102
In adiabatically solving the four-operator equations, we pick the imaginary delta-function
contribution according to the following relation for energy denominators
11=P+ iπδ(∆E),∆E − i∆E(8.5)

for small  > 0. This choice gives us the collisional rates we are interested in, as can beseen in App. E. The resulting quantum-Boltzmann rates are depicted in Fig. 8.2: the couplings a) and c) are between fermions and bosons, b) is an effective fermion-fermion interaction.
Only rate a) allows for particle exchange between bosons and fermions. The off-shell, principal
value contributions renormalize the energy-denominators, but we neglect this correction of theintermediate-state energies. We find for the mixed four-operator cumulant:bq b†q0 aq0 −q/2+k,σ a†q/2+k,σ = −iπ δ 2(q0 −q)/2+k − 2k gh
× ± fq/2+k,σ + nq b†q0 aq/2−k,−σ aq0 −q/2+k,σ∓ fq0 −q/2+k,σ + nq0bq a†q/2+k,σ a†q/2−k,−σWe analogously consider Eq. (8.4e) and find the following adiabatic expression:

i

(8.6)
.

aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓ = −iπ δ(2k0 − 2k ) g×h

1 − fq/2+k0 ↑ − fq/2−k0 ↓− 1 − fq/2+k↑ − fq/2−k↓bq a†q/2+k↑ a†q/2−k↓b†q aq/2−k0 ↓ aq/2+k0 ↑

i

.

(8.7)

Substituting these expressions (8.6) and (8.7) on the right-hand side of the three-operator
Eq. (8.4c) gives a closed set of cumulant equations. We are thus including four-operator correlations without explicitly keeping the four-operator functions as dynamical quantities. This
procedure is necessary, because we do not assume a Gaussian reference distribution, which would
lead to a straight-forward way of cutting the BBGKY hierarchy depicted in Fig. 8.1, because
correlation functions beyond pairs would factorize with a Gaussian reference distribution.

103

Figure 8.3: Normal-phase distribution function for a) fermions and b) bosons during a ramp
from detuning ν̄ = 2EFermi close to resonance at Tinitial = 0.33TFermi as a function of time andmomentum. The time grid has 2000 points and is measured in units of tFermi ≈ 10 µs. The
momentum grids have a) 200 and b) 60 points.

8.2

Numerical Results
We here show results for the narrow resonance in 6 Li at 543 G [145, 146] for various

detuning ramps. The initial distributions are equilibrium Bose–Einstein and Fermi–Dirac distributions at a given temperature Tinitial , where the fermion chemical potential µF is found byroot finding using a number constraint Eq. (8.3) and the chemical potential for the bosons isgiven by µB = 2µF . The time evolution is then calculated using a Runge-Kutta algorithm [147].

Figure 8.3 shows the fermion and boson distribution functions as a function of momentum
and time during a ramp from an initial detuning of ν̄ = 2EFermi to 0.2EFermi just above theresonance. The ramp takes place in the first half of the time axis and is followed by equilibration
at constant detuning. Some of the fermions get converted to composite bosons during the ramp:
Fig. 8.4 shows on the left how the fermion fraction drops from an initial value of 85% to a final
value of just below 30%. The plot on the right shows the resulting initial and final distributionsof bosons and fermions. We can fit these distributions with equilibrium Bose–Einstein andFermi–Dirac distributions.
From fits to the distributions in Fig. 8.3 we can extract the detuning ν̄, the temperatures

1

0.8

Fermions (initial)
Fermions (final)
Bosons (initial)
Bosons(final)

1.2

Particle distributions

Total fermion number

0.9

104

1.4

Fermion fraction
Boson fraction
Total

0.7
0.6
0.5
0.4
0.3

1
0.8
0.6
0.4
0.2

0.2
0.1
0

10

20

Time [t

30

]

40

50

0
0

0.5

Fermi

1

1.5

2

Momentum [kFermi]

2.5

3Figure 8.4: Time evolution of the population fractions during the ramp (left plot). Most of theunpaired fermions are converted to tightly bound molecules (this is for the narrow-resonancelimit). The plots on the right show the initial and final distributions of the particles. These are
cuts through the plots in Fig. 8.3.

of the fermion and boson distributions, and the chemical potential µF of the fermions. Figure 8.5
shows on the left the detuning as a function of time. The blue, solid line is the applied magnetic
field ramp, and the dashed, green line is the detuning obtained from fitting the distributions.
The fitted detuning quickly follows the ramp and briefly oscillates to the applied value after
the ramp. On the right side of Fig. 8.5 we plot the temperatures of the two distributions as
a function of time; the bosons and fermions are driven out of thermal equilibrium during theramp, and reequilibrate at constant detuning after the ramp. The system also heats during theramp and ends up at a final temperature of Tfinal = 0.47TFermi.

Figure 8.6 shows the energy and entropy as a function of time during the ramp. Theenergy is given by the expectation value of the crossover Hamiltonian Eq. (5.8)
Etot = Hres =Xk,σk fkσ +X  qq2

X+ ν nq + 2g
< bq a†q/2−k↓ a†q/2+k↑ .

(8.8)

qk

To calculate the entropy, we use the following formula [29, Chap. 2.2.3] for the combinatorial
entropy of the two distributions and neglect the correlation contribution in this case
Stot = −

Xh
k,σ

i Xh
i
fkσ ln(fkσ ) + (1 − fkσ ) ln(1 − fkσ ) −nq ln(nq ) + (1 + nq ) ln(1 + nq ) . (8.9)
qFermions
Bosons

0.48
Fermi

1.5

Fitted temperature [EDetuning [E

Fermi

]

]

2

105

0.5

Fitted equil. distributions
Applied magnetic field

1

0.5

0.46
0.44
0.42
0.4
0.38
0.36
0.34

0
0

10

20

30

Time [tFermi]

40

50

0.32
0

10

20

30

Time [tFermi]

40

50

Figure 8.5: We fit the distributions shown in Fig. 8.3 to equilibrium Bose and Fermi distributions with the fermion chemical potential, the boson detuning ν̄ and the two temperatures
as parameters. The plot on the left shows the fitted detuning and the applied detuning, thatis, magnetic field as a function of time. The plot on the right shows the temperatures of the
distributions.

Neglecting the correlation contribution to the entropy is not a good approximation close to theresonance, where the ramp in the previous plot ends. We will now consider a case, where westay a Fermi energy away from the resonance.

Figure 8.7 shows ramps from a detuning of 2EFermi to 1EFermi and back up for threedifferent, constant ramp speeds, now at a higher initial temperature of Tinitial = 0.8 TFermi. On
the left, we plot the applied and fitted detunings as a function of time on a logarithmic scale.
By the middle ramp, the fitted detuning already tracks the applied magnetic field very well.
The plot on the right of Fig. 8.7 shows on the other hand that only the very slowest ramp
brings the temperatures back to the initial value at the end of the ramp. This illustrates thatthe reversibility of the ramp is determined by a much longer many-body adiabaticity time scalethan the transfer of populations, which is determined by a two-body adiabaticity time scale.
The same difference in time scales appears in Fig. 8.8. The transfer of fermions during the ramp
saturates quickly, whereas it takes longer for the entropy to return to its initial value.

Fermions
BosonsCorrelations
Total energy

0.8

106

2
1.8

Fermions
BosonsTotal entropy

1.6

0.6

1.4

B

Entropy [k ]Energy per particle [E Fermi]

1

0.4
0.2
0

1
0.8
0.6

−0.2
−0.4
0

1.2

0.4
10

20

30

40

Time [tFermi]

0.2
0

50

10

20

30

40

Time [tFermi]

50

Figure 8.6: We plot the energy (left) and entropy (right) of the system as a function of time.
The green, dashed line is the boson part, the red, dash-dotted line the fermion part, and theblue, full line the total number. In the energy plot, the black dotted line is the correlation energy
due to the three-operator cumulant ba† a† , which is included in the total energy.

The discussion of rate equations in App. E finds that the collision processes depicted
in Fig. 8.2 are the ones contained in this theory. The second-order processes b) and c) in
Fig. 8.2, which maintain local equilibrium in each of the Fermi and Bose distributions, are ofsecond order in the interaction g and are thus much slower than the first-order interaction a),which directly couples a pair of fermions to a composite boson and thus leads to global thermal
equilibrium. The first-order rate, however, is energetically suppressed for negative detuning,
because the bound state of the composite boson lies below the open-channel threshold of thefree fermions. Figure 8.9 for a ramp across the resonance illustrates this point: The fitted
3

0.87

2.8

Fitted distributions
Applied magnetic field

2.4
2.2
2
1.8
1.6
1.4

0.85

BosonsFermions

0.84
0.83
0.82
0.81
0.8

1.2
1 0
10

0.86

Fitted temperature [EFermi]

Detuning [EFermi]

2.6

1

10

Time [tFermi]

2100.79 010110Time [tFermi]

210Figure 8.7: Fitted detuning and temperatures for return ramps. Shown are results for threedifferent ramp speeds as a function of time.

0.8

107

2.78
2.77
2.76

0.7

2.75

B

Entropy [k ]Fermion fraction

0.75

0.65

2.74
2.73
2.72
2.71

0.6

2.7

0.55 010110Time [t

]

2

10

Fermi

2.69 010110Time [t

]

2

10

Fermi

Figure 8.8: Fermion fraction (left) and entropy (right) as a function of time for return ramps atdifferent speeds.

detuning follows the applied magnetic field until just below the resonance. At this point thefirst-order coupling between the fermions and bosons becomes energetically suppressed, andglobal equilibrium is not maintained any more as the ramp continues into the BEC regime. The
plot on the right shows the much slower relaxation under the second-order rates that does notbring the temperatures of the two distributions into equilibrium any more after the ramp has
finished. Three-body collisions, which we have neglected in this approach, become important inthis regime and ultimately lead to thermalization.

We next consider the temperature dependence of the same ramp across the resonance fromdetuning ν̄ = 2EFermi to −2EFermi. Figure 8.10 shows on the left the initial and final boson
fraction as a function of temperature. The lowest temperature points are outside the rangeof validity of this normal-phase theory, because they are below the Bose-Einstein transition
temperature TBEC = 0.35TFermi for the experimental density and the mass of 6 Li. Accordingly,
the initial boson population goes to zero and the final fraction does not head for the analyticalzero-temperature result of full conversion given by Landau-Zener theory [149]. We have alsoplotted an experimental result for the narrow Lithium resonance [22], which deviates significantly
from our results. Our simulation is, however, in much better quantitative agreement with resultsin 40 K experiments at JILA [148]. This is demonstrated in the graph on the right of Fig. 8.10,

2.5
2

]

Fermi

1
0.5
0
−0.5
−1
−1.5

0.48
0.46
0.44
0.42
0.4
0.38
0.36
0.34

−2
−2.5
0

Fermions
Bosons0.5

Fitted temperature [EDetuning [EFermi]

1.5

108

0.52

Fitted equil. distributions
Applied magnetic field

10

20

30

40

Time [t

50

]

60

70

80

0.32
0

10

Fermi

20

30

40

50

Time [tFermi]

60

70

80

Figure 8.9: Fitted detuning (left) and temperatures (right) for a ramp across the resonance fromdetuning ν̄ = 2EFermi to −2EFermi.

where we plot the same data as a function of peak phase-space density ρ, which is given by thethermal de-Broglie wave length λth and the peak density npeak at the center of the trap as
ρ = λ3th npeak .

(8.10)The plot on the right of Fig. 8.10 shows that the Rice result still deviates, but the measurements
at JILA [25, 148] agree very well with our quantum simulation. We also plot two theory lines
due to Williams et al. [131]. The authors of the latter paper find the following transcendental
equation for the molecular production efficiency χ as a function of phase-space density ρ for a
two-component Fermi gas
2χ + ln χ − ln(1 − χ) = ln ρ

(8.11)

They derive this relation using a classical-gas approximation and entropy conservation to relate
the initial and final values of the phase-space density. We plot the numerical solution of thisrelation Eq. (8.11) as the green dashed line on the right side of Fig. 8.10. Williams et al. [131]also find an analytical prediction including quantum-statistical corrections, which we plot as thegreen full line and agrees with the 2005 JILA data better than the classical prediction, but notquite as well as our fully time-dependent quantum simulation.

1

0.8

Molecular conversion efficiency χ

0.9

Boson fraction

1

Simulation final
Simulation initial
Rice experiment final

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.2

0.4

0.6

Temperature [TFermi]

0.8

1

0.9
0.8
0.7

109
Quantum simulation
Analytical theory
Classical theory
Rice experiment
JILA 2003
JILA 2005

0.6
0.5
0.4
0.3
0.2
0.1
0

−1

10

0

10

3

1

10

Phase−space density ρ=λthnpeak

Figure 8.10: Molecular conversion efficiency as a function of temperature (left) and peak phasespace density (right) for a ramp across the resonance from detuning ν̄ = 2E Fermi to −2EFermi.
We also include experimental data from Rice [22] and JILA [25, 148] and theory results due toWilliams et al. [131].

8.3

Summary
We discuss how thermal fermions and bosons in the normal phase couple in a mean-

field theory of the BCS/BEC crossover near a Feshbach resonance. The cumulant method is a
powerful tool for deriving a hierarchy of coupled equations of motion for distribution functions
and higher-order correlations. We show how to consistently truncate this hierarchy and obtaina set of equations that we simulate numerically. The full time dependence of the crossoverproblem in the case of the narrow resonance at 543 G in 6 Li shows rapid local and global thermal
equilibration of the distribution functions. The first-order rates that lead to global equilibrium
are energetically suppressed below the resonance. We also find two adiabaticity time scales as
the ramp speed is varied. We further examine the temperature dependence of the molecularconversion efficiency and find good agreement with experiments in 40 K.

Chapter 9
Summary and Outlook

In the first part of this thesis, we discussed a kinetic theory for a dilute quantum gas ofbosons. In dilute gases, particles propagate freely for a long time between successive collision
events, which leads to an attenuation of high-order correlations. Because of this attenuation,
tracking a few, low-order correlation functions or Master variables is sufficient to describe thebehavior of the system. This simplification allows us to develop microscopic theories, unlike thestrongly interacting case of liquid Helium.
We used the Kadanoff–Baym nonequilibrium Green’s function formalism [43, 44] to derive a kinetic theory for Bose–Einstein condensed bosons. The resulting equations reproduced
the quantum-Boltzmann limit at high temperature and the Gross–Pitaevskii equation at low
temperature. We also recovered the results of Walser et al. [37], who used a statistical-operator
approach to derive the same kinetic equations. Next we examined the interaction diagrams
present in this theory and identified many-body scattering T matrices to the level of approximation used. This allowed us to explicitly demonstrate that the excitation spectrum is gapless,as required for condensed bosons. We also diagonalized the renormalized self energies and used
the resulting quasiparticle basis to write a diagonal Boltzmann equation for the dressed states.
This first part of the thesis is all theoretical, and calculating finite-temperature excitations
and damping of a BEC would make comparison to experiments such as [65] and other theories
possible. The work of Morgan et al. [75, 76] seems particularly interesting , since they use anumber-conserving approach, which is different from the symmetry-breaking language we use.
In the second part of this thesis, we considered the BCS–BEC crossover near a Feshbach

111
resonance. Resonantly enhanced interactions between a pair of free fermions cause the fermionsto cross over into a tightly bound, composite molecule. These effectively bosonic molecules
can then condense. The bosons do not behave quite as expected. For example, the effective
boson-boson scattering length is just 60% of the initial fermion-fermion interaction strength.
We first discussed one- and two-channel models of the BCS–BEC crossover and the Feshbach scattering theory that gives rise to the crossover. We then introduced an imaginary-time
technique for fermions that finds zero-temperature ground states across the resonance. We used
the Bloch-Messiah theorem, which relates the density and the pair function of fermions, to find asteepest-descent algorithm that works even for density-matrix evolution. We use this algorithm
to find generalized BCS states.
We also examined the minimum ingredients for a mean-field theory that reproduces theobserved boson-boson scattering length on the BEC side of the crossover and found that a threepoint correlation function between a pair of free fermions and a composite boson needs to betreated. We introduced the concept of a cumulant, in order to include this required three-point
correlation function. We then applied the imaginary-time method to the resulting equations of
motion and numerically showed the observed dimer-dimer scattering behavior on the BEC side.
Next we used the cumulant expansion in a different regime and considered the coupling
between thermal molecules and fermions in the normal phase above the transition temperatureby numerically solving the time-dependent equations of motion including the three-point correlation function. We showed that the equations in the normal phase contain the Boltzmann rates
for fermions and bosons and that the interconversion rate becomes energetically suppressed on
the BEC side of the crossover. We showed numerical results in the normal phase for the narrow
resonance at 543 G in 6 Li and discuss different time scales for many-body and two-body relaxation. We also compare molecular conversion efficiencies from our simulations to results in JILA
potassium experiments and find good agreement.
To extend the work in the last two Chapters, one could imagine using the zero-temperature
results of the imaginary-time method as an initial condition for a more general time-dependent112
code that includes symmetry-broken mean fields. We have in fact worked with a program like
that, but before we developed the imaginary-time method, and finding stable solutions proved
difficult without good initial conditions. Another possible direction would be to calculate moremeasurable quantities, such as the superfluid gap and the universal β parameter, which describes
the self-energy shift on resonance.

Appendix A
Collisional Self Energies in the Single-Particle Energy BasisIn this Appendix we give details omitted in Chap. 2, where the Kadanoff–Kane kinetic
equations are written in the single-particle energy basis. The basis transformation is discussed
in Section 2.4 and the steps for the collisional self energy are exactly the same as that of thefirst-order Hartree–Fock self energies discussed in Chap. 2. However, since the collisional selfenergies are quadratic in the interaction potential, this Section is more involved [150].

A.1

Mean-Field Equations
We start by writing down the analytical form of the collisional self energy given in

Eq. (2.20) as
S < (1, 2) = −

1
2

Z

d2̄

Z



d3̄ v(1, 2̄)v(2, 3̄)nog̃ < (1, 2)Tr g̃ > (3̄, 2̄)g̃ < (2̄, 3̄)<><+2g̃ (1, 3̄)g̃ (3̄, 2̄)g̃ (2̄, 2) .

(A.1)

This is the gapless Beliaev approximation for the second-order contributions as pointed out in
Chap. 2. Using the transformation Eq. (2.27) for the first propagator g̃ < and their definitions
Eqs. (2.7) and (2.8) for the remaining ones, we obtain for the self energy1
S (1, 2) = −
2
<

Z

d2̄



Z

"

n
o
˜) f˜ + m̃ m̃∗
T
2i
(1
+
f
d3̄ v(1, 2̄)v(2, 3̄) T130 G̃<
0
00
00
31
1 2
3̄2̄ 2̄3̄
3̄2̄ 2̄3̄

˜ f˜ + m̃ m̃∗ (1 + f)3̄2̄ 2̄2
3̄2̄ 2̄2<
+G̃30 20 T20 3̄ 2i m̃∗3̄2̄ f˜2̄2 + f˜2̄3̄ m̃∗2̄2

#˜˜(1 + f )3̄2̄ m̃2̄2 + m̃3̄2̄ (1 + f )22̄ 
 .

∗
˜
˜
m̃3̄2̄ m̃2̄2 + f2̄3̄ (1 + f )22̄

(A.2)

114
We now write all normal and anomalous fluctuation densities in the energy basis by using theirrespective transformation laws:
S < (1, 2) = −i

−i

Z

d2̄


Z

Z

d2̄

Z

d3̄ h20 | 2̄iv(1, 2̄)h2̄ | 40 i h200 | 3̄iv(2, 3̄)h3̄ | 400 iT130 G̃<
30 100on
˜ 00 0 f˜ 0 00 + m̃ 00 0 m̃∗0 00 T 00
× (1 + f)
4 2 424 4
22
1 2d3̄ h20 | 2̄iv(1, 2̄)h2̄ | 40 i v(2, 3̄)T130 G̃<30 200 T200 3̄ T3̄40 (1 + f˜)400 20 f˜40 100 + m̃400 40 m̃∗20 100

×

˜ 00 0 m̃ 0 00 + m̃ 00 0 (1 + f)
˜ 00 0 (1 + f)
4 2
41
4 4
1 2
 T 00 . (A.3)
 1 2∗m̃400 20 m̃40 100 + f˜40 400 (1 + f˜)100 20m̃∗400 20 f˜40 100 + f˜40 400 m̃∗20 100

Finally, using Eq. (2.33) to write the interaction potential in terms of its symmetrized energy
matrix-elements, we obtain the condensate self energy in the energy basis projected back onto
the position basis as0

h1 | 1 i
0
00 00 00 00 
0 0 0 0
S < (1, 2) = −4i φ1t1 2 3 4 φ1t2 2 3 4 00h3 | 1i"˜om̃30 300
 f30 100n
 (1 + f˜)400 20 f˜40 200 + m̃400 40 m̃∗20 200
× m̃∗10 100 (1 + f˜)300 10



#
∗
˜
˜
˜
˜
2(1 + f )400 20 m̃40 300m̃30 400 (1 + f )400 20 f40 100 + m̃400 40 m̃20 100
 f30 200


+



∗
∗
∗
˜
˜
˜
˜
m̃200 20 m̃40 300 + f40 200 (1 + f)300 20
2m̃200 20 f40 100m̃10 200 (1 + f)400 10 h300 | 2i
×

0

0

00

h2 | 1 i


,


(A.4)

0 0 0 0where the time subscript of the two-particle matrix elements φt11 2 3 4 means that the field operators that are contracted with that matrix element have to be evaluated at t 1 . We can collectsome terms using the symmetries of the two-particle matrix element φ given in Eq. (3.4) to115
obtain0

h1 | 1 i0
0 0 0 000 00 00 00 
S < (1, 2) = −8i φ1t1 2 3 4 φ1t2 2 3 4 00h3 | 1i"˜om̃30 300
n f30 100
˜ 00 0 f˜ 0 00 + m̃ 00 0 m̃∗0 00
 (1 + f)
× 
4 2 424 422
˜ 00 0m̃∗10 100 (1 + f)
3 1
 


#
˜
˜
0
(1 + f)400 20 m̃40 300   h300 | 2im̃30 400
 f30 200
 

+
 


∗
∗
˜
˜
0
0
m̃ 00 0 f 0 00m̃ 0 00 (1 + f ) 00 0
12

2 2

4 1



0
h2 | 100 i

41

Now using the definition of the non-conserving collision operators .(A.5)


0 0 0 000 00 00 00

0 0 0 000 00 00 00

0 0 0 000 00 00 00(A.6c)

0 0 0 000 00 00 00(A.6d)

Γ0f f f = 8φ1t1 2 3 4 φ1t2 2 3 4 f30 100 f40 200 f400 20 |10 i h300 | ,(A.6a)

Γ0f mf = 8φ1t1 2 3 4 φ1t2 2 3 4 f30 100 m40 300 f400 20 |10 i |200 i ,(A.6b)

Γ0f mm∗ = 8φ1t1 2 3 4 φ1t2 2 3 4 f30 100 m40 300 m∗200 20 |10 i h400 | , andΓ0mmm∗ = 8φ1t1 2 3 4 φ1t2 2 3 4 m30 400 m40 300 m∗200 20 |10 i |100 i ,we get the collisional self energy for the condensate in the position basis expressed in terms of
the energy-basis collision operators




0
0
h12| Γm̃ h1| Γ0f˜f˜(1+f˜) + 2Γ0f˜m̃m̃∗ |2im̃m̃∗ + 2Γf˜m̃(1+f˜).
S(1, 2)< = −i 




0†
0†
0∗
0∗
h2| Γ(1+f˜)(1+f˜)f˜ + 2Γ(1+f˜)m̃m̃∗ |1iΓm̃
m̃m̃∗ + 2Γ(1+f˜)m̃f˜ |12i





(A.7)

The other time ordering S > (1, 2) can be obtained by exchanging f˜ for (1 + f˜) and vice versa.

A.2

Normal and Anomalous Fluctuations
Similar to the condensate case, the collisional self energy for the fluctuations given in

Eq. (2.21) is transformed:
ZZ1
Σ< (1, 2) = −d2̄ d3̄ v(1, 2̄)v(2, 3̄)
2

n
o
n
o
× g̃ < (1, 2)Tr g > (3̄, 2̄)g < (2̄, 3̄) − h(3̄, 2̄)h(2̄, 3̄) + h(1, 2)Tr g̃ > (3̄, 2̄)g̃ < (2̄, 3̄)no+2g̃ <(1, 3̄) g > (3̄, 2̄)g < (2̄, 2) − h(3̄, 2̄)h(2̄, 2) + 2h(1, 3̄)g̃ > (3̄, 2̄)g̃ < (2̄, 2) .

(A.8)

116
We now substitute the first propagator in each term by using Eq. (2.27) and the analogous expression for h(1, 2). For the remaining propagators we substitute the definitions Eqs. (2.7) and (2.8)
and obtain1Σ< (1, 2) = −
2
"

Z

d2̄

Z

d3̄ v(1, 2̄)v(2, 3̄) T130

n
o
∗
c c
c
c∗
× G̃<
T
2i
(1
+
f
)
f
+
m
m
−
f
f
−
m
m
00
0
00
1 2
31
3̄2̄ 2̄3̄
3̄2̄ 2̄3̄
3̄2̄ 2̄3̄
3̄2̄ 2̄3̄on
˜ f˜ + m̃ m̃∗+H30 100 T100 2 2i (1 + f)3̄2̄ 2̄3̄
3̄2̄ 2̄3̄ (1 + f )3̄2̄ f2̄2 + m3̄2̄ m∗2̄2

+G̃<
30 20 T20 3̄ 2i m∗3̄2̄ f2̄2 + f2̄3̄ m∗2̄2

c+ mc3̄2̄ mc∗
 f3̄c2̄ f2̄2
2̄2


−G̃<
30 20 T20 3̄ 2i 



c
c
c∗
mc∗
3̄2̄ f2̄2 + f2̄3̄ m2̄2

(1 + f )3̄2̄ m2̄2 + m3̄2̄ (1 + f )22̄ ∗m3̄2̄ m2̄2 + f2̄3̄ (1 + f )22̄

f3̄c2̄ mc2̄2 + mc3̄2̄ f2c2̄ 


c∗ c
c c
m3̄2̄ m2̄2 + f2̄3̄ f22̄˜ f˜ + m̃ m̃∗ (1 + f)3̄2̄ 2̄2
3̄2̄ 2̄2+H30 20 T20 3̄ 2i m̃∗3̄2̄ f˜2̄2 + f˜2̄3̄ m̃∗2̄2

#˜(1 + f˜)3̄2̄ m̃2̄2 + m̃3̄2̄ (1 + f)
22̄ 
 .

m̃∗3̄2̄ m̃2̄2 + f˜2̄3̄ (1 + f˜)22̄

(A.9)

Now writing all normal and anomalous fluctuation densities in the energy basis, we obtainZZ
Σ< (1, 2) = −i d2̄ d3̄ h20 | 2̄iv(1, 2̄)h2̄ | 40 i h200 | 3̄iv(2, 3̄)h3̄ | 400 i T130

n
o
∗
c
c
c
c∗
× G̃<
(1
+
f
)
f
+
m
m
−
f
f
−
m
m
00
0
0
00
00
0
0
00
00
0
0
00
00
0
0
00
0
00
4
2
4
2
4
4
2
2
4
2
4
2
4
4
2
2
31
o
n
∗
˜
˜
0
00
+H3 1 (1 + f )400 20 f40 200 + m̃400 40 m̃20 200 T100 2ZZ−i d2̄ d3̄ h20 | 2̄iv(1, 2̄)h2̄ | 40 i v(2, 3̄) T130"∗
 (1 + f )400 20 f40 100 +m400 40 m20 100 (1 + f )400 20 m40 100 +m400 40 (1 + f )100 20 


× G̃<
30 200 T200 3̄ T3̄400 m∗400 20 m40 100 +f40 400 (1 + f )100 20
m∗400 20 f40 100 +f40 400 m∗20 100


f4c00 20 mc40 100 +mc400 40 f1c00 20 
 f4c00 20 f4c0 100 +mc400 40 mc∗
20 100


T
−G̃<
T
0
00
00
00
32
2 3̄ 3̄4 

c
c
c∗
c
c
c
mc∗
mc∗
400 20 f40 100 +f40 400 m20 100
400 20 m40 100 +f40 400 f100 20


#
∗
˜
˜
˜
˜
 (1 + f )400 20 f40 100 + m̃400 40 m̃20 100 (1 + f)400 20 m̃40 100 + m̃400 40 (1 + f )100 20 
+H30 200 T200 3̄ T3̄400 m̃∗400 20 m̃40 100 + f˜40 400 (1 + f˜)100 20m̃∗400 20 f˜40 100 + f˜40 400 m̃∗20 100

×T100 2 .

(A.10)

117
In the next step, we use Eq. (2.33) to express the interaction potential in terms of its symmetrized
matrix elements in the energy basis and obtain
0

h1 | 1 i0
0 0 0 000 00 00 00 Σ(1, 2)< = −4i φ1t1 2 3 4 φ1t2 2 3 4 00h3 | 1i"om̃30 300
 f˜30 100
n (1 + f ) 00 0 f 0 00 + m 00 0 m∗0 00 − f c00 0 f c0 00 − mc 00 0 mc∗0 00
× 
4
2
4
2
4
4
2
2
4
2
4
2
4
4
2
2


m̃∗10 100 (1 + f˜)300 10


c
c
o
 f30 100 m30 300  n
˜ 00 0 f˜ 0 00 + m̃ 00 0 m̃∗0 00
 (1 + f)
+
4
2
4
2
4
4
2
2


mc∗
f3c00 1010 100
∗2(1 + f )400 20 m40 300m̃30 400 f˜30 200
 (1 + f )400 20 f40 100 + m400 40 m20 100


+



∗
∗
∗
2m20 200 f40 100m20 200 m40 300 + f40 200 (1 + f )300 20
m̃10 200 (1 + f˜)400 10



c
c
c
c∗
c
c
m̃30 400
2f400 20 m40 300
 f˜30 200
 f400 20 f40 100 + m400 40 m20 100



−



∗
c∗
c
c∗
c
c
c
m̃10 200 (1 + f˜)400 10
2m200 20 f40 100
m200 20 m40 300 + f40 200 f300 20



#
∗
c
c
˜ 00 0 f˜ 0 00 + m̃ 00 0 m̃ 0 00
2(1 + f˜)400 20 m̃40 300
 f30 200 m30 400  (1 + f)

21
4 4
4 2 41


+



∗
∗
c∗
c
˜ 00 02m̃200 20 f˜40 100m̃200 20 m̃40 300 + f˜40 200 (1 + f)
m10 200 f400 10
3 2 h300 | 2i
×

0

0

h2 | 100 i.
(A.11)

118
We collect some terms making use of the symmetries of the two-particle matrix element φ asgiven in Eq. (3.4)
h1 | 10 i0
0 0 0 000 00 00 00 Σ(1, 2)< = −8i φ1t1 2 3 4 φ1t2 2 3 4 0h30 | 1i"om̃30 300
 f˜30 100
n (1 + f )400 20 f40 200 + m400 40 m∗20 200 − f4c00 20 f4c0 200 − mc400 40 mc∗
× 
0 200
2


˜ 00 0m̃∗10 100 (1 + f)
3 1
 f3c0 100+mc∗
10 100 f˜30 200+m̃∗10 200
 f3c0 200+mc∗
10 200omc30 300  n (1 + f˜) 00 0 f˜ 0 00 + m̃ 00 0 m̃∗0 00
4
2
4
2
4
4
2
2

f3c00 10m̃30 400(1 + f˜)400 10






0

c
m∗20 200 f40 100 − mc∗
200 20 f40 100

0mc30 400  m̃∗200 20 f˜40 100
f4c00 10

(1 + f )400 20 m40 300 − f4c00 20 mc40 300






0

 
#
00
0
(1 + f˜)400 20 m̃40 300   h3 | 2i
 , (A.12)
 

 
00
0
h2 | 1 i0

and express the collisional self energy in terms of the collision operators Γ0 :




0
0
0
0
h12| Γf m̃(1+f ) + Γm̃mm∗
 h1| Γf˜f (1+f ) + Γf˜mm∗ |2iΣ(1, 2)< = −i 




0†
0†
0∗
0∗
Γ(1+f )m̃f + Γm̃mm∗ |12i h2| Γ(1+f˜)(1+f )f + Γ(1+f˜)mm∗ |1i




0
0
0
0
h12| Γf c m̃f c + Γm̃mc mc∗
 h1| Γf˜f c f c + Γf˜mc mc∗ |2i

+i 




0†
0†
0∗
0∗
Γf c m̃f c + Γm̃mc mc∗ |12i h2| Γ(1+f˜)f c f c + Γ(1+f˜)mc mc∗ |1i




0
0
0
0
 h1| Γf c f˜(1+f˜) + Γf c m̃m̃∗ |2i h12| Γf˜mc (1+f˜) + Γmc m̃m̃∗ (A.13)
−i 




0†
0†
0∗
0∗
Γ(1+f˜)mc f˜ + Γmc m̃m̃∗ |12i h2| Γf c (1+f˜)f˜ + Γf c m̃m̃∗ |1i




0
0
0
0
0
0
h12| Γf˜m(1+f ) − Γf˜mc f c + Γf c m̃(1+f˜) h1| Γf m̃m∗ − Γf c m̃mc∗ + Γf˜mc m̃∗ |2i
.
−i 




0†
0†
0†
0∗
0∗
0∗
Γ(1+f˜)mf − Γ(1+f˜)mc f c + Γf c m̃f˜ |12i h2| Γ(1+f )m̃m∗ − Γf c m̃mc∗ + Γ(1+f˜)mc m̃∗ |1i

119
Using f = f c + f˜ and m = mc + m̃, we can rewrite this equation to obtain the final result forthe collisional self energyi Σ(1, 2)< =(A.14)
00h12| Γ0mm̃m̃∗ +Γm̃m
c m̃∗ +Γm̃m̃mc∗
 h1| Γ0f f˜(1+f˜) +Γ0f˜f c (1+f˜) +Γ0f˜f˜f c |2i







0†
0†
0†
0∗
0∗
0∗
Γmm̃m̃∗ +Γm̃mc m̃∗ +Γm̃m̃mc∗ |12i h2| Γ(1+f )(1+f˜)f˜+Γ(1+f˜)f c f˜+Γ(1+f˜)(1+f˜)f c |1i




0
0
0
0
0
0
h12| Γf m̃(1+f˜) +Γf˜mc (1+f˜) +Γf˜m̃f c
h1| Γf m̃m̃∗ +Γf˜mc m̃∗ +Γf˜m̃mc∗ |2i


.
+2




0†
0†
0†
0∗
0∗
0∗
Γ(1+f )m̃f˜+Γ(1+f˜)mc f˜+Γ(1+f˜)m̃f c |12i h2| Γ(1+f )m̃m̃∗ +Γ(1+f˜)mc m̃∗ +Γ(1+f˜)m̃mc∗ |1i

˜
As in the condensate case, the self energy Σ> (1, 2) can be obtained by exchanging f˜ and (1 + f)
and vice versa.

Appendix B
Collisional Terms in the Quasiparticle BasisThis Appendix shows in detail how the symmetries of the quasiparticle basis allow us torewrite the second-order collisional terms as shown in Sec. 4.2 [150].
We first transform only the thermal terms of the collision operator, which do not containcondensate contributions and are denoted by Γ̃< . We begin by writing down their definition interms of individual collision processesΓm̃m̃m̃∗ + 2Γf˜m̃(1+f˜) Γf˜f˜(1+f˜) + 2Γf˜m̃m̃∗σ z Γ̃< = 


†
†
∗
∗
Γm̃m̃m̃∗ + 2Γ(1+f˜)m̃f˜ Γ(1+f˜)(1+f˜)f˜ + 2Γ(1+f˜)m̃m̃∗



(B.1)

Γf˜m̃(1+f˜)  Γf˜m̃m̃∗

.
≡ Γ̃<
1 +

†
∗
Γ(1+f˜)m̃f˜ Γ(1+f˜)m̃m̃∗(B.2)

For the moment, we drop the factors of two and substitute the definitions of the individual
collisional operators Γm̃30 300 |10 i ⊗ |100 i f˜30 100 |10 i ⊗ h300 |
10 20 30 40 100 200 300 400 

Γ̃<
=
8
φ
φ
1
η


m̃∗10 100 h30 | ⊗ h300 | (1 + f˜)300 10 h30 | ⊗ |100 ino˜ 00 0 f˜ 0 00 + m̃ 0 00 m̃∗0 00 .
× (1 + f)
4 2 42
44
22

(B.3)

Then we replace the expectation values of the normal and anomalous densities by writing
Eq. (4.12) in 2 by 2 components, m̃10 20
  u1̄10 P1̄2̄ u2̄∗ f˜10 20
20
<
=

G̃10 20 =  m̃∗10 20 (1 + f˜)20 10v11̄∗0 P1̄2̄ u2̄∗20

u1̄10 P1̄2̄ v22̄0
v11̄∗0 P1̄2̄ v22̄0




.


(B.4)

121
and substituting in Eq. (B.3):


1̄0 ∗
100



3̄0
300

|10 i ⊗ h300 | u3̄30 P3̄3̄0 v |10 i ⊗ |100 i  u3̄30 P3̄1̄0 u
10 20 30 40 100 200 300 400 Γ̃<φη
1 = 8φ00v11̄∗0 P1̄1̄0 u1̄100∗ h30 | ⊗ h300 | v11̄∗0 P1̄3̄0 v33̄00 h30 | ⊗ |100 i
n
o
0
0
0
0
× v22̄∗0 P2̄4̄0 v44̄00 u4̄40 P4̄2̄0 u2̄200∗ + u4̄40 P4̄4̄0 v44̄00 v22̄∗0 P2̄2̄0 u2̄200∗ .

(B.5)

−1̄
−1̄∗
1̄
1̄∗
Finally, we use Eq. (4.9) to substitute v11̄0 and u1̄∗10 according to v10 = u10 and u10 = v10 ,reverse the corresponding summations, and reintroduce the weighting factors to obtain for thefull collision operator
0 0 0 000 00 00 0000

σ z Γ̃< = 8 φ1 2 3 4 φ1η 2 3 4 v22̄00∗ u4̄400 P3̄−1̄0 P2̄−4̄0 P4̄−2̄0
 2̄∗ 4̄
 2̄∗ 4̄3̄ 1̄0 ∗4̄∗ 2̄
000
3̄ 1̄04̄∗ 2̄
000
 u30 v100 v20 u40 + 2v20 u40 |1 i ⊗ h3 | u30 u300 2v20 u40 + v20 u40 |1 i ⊗ |1 i  . (B.6)× 2̄∗ 4̄
 2̄∗ 4̄3̄∗ 1̄0 ∗4̄∗ 2̄
0
00
3̄∗ 1̄04̄∗ 2̄
000
v10 v100 2v20 u40 + v20 u40 h3 | ⊗ h3 | v10 u300 v20 u40 + 2v20 u40 h3 | ⊗ |1 i

To transform the collision operator Γ̃< to the quasiparticle basis {1̄}, we multiply it from theleft by W † and from the right by W to obtainW † σ z Γ̃< W = |5̄i ⊗ h5̄0 |0 0 0 0

0u5̄∗50 h5 | ⊗ 11
00 00 00 000

v55̄0 |50 i ⊗ 11


0







5̄0
500



 u 11 ⊗ |500 i 
σ z Γ< 

 0
00
5̄ ∗
v500 11 ⊗ h5 |(B.7)

= 8 φ1 2 3 4 φ1η 2 3 4 v22̄00∗ u4̄400 P3̄−1̄0 P2̄−4̄0 P4̄−2̄0 |−1̄i ⊗ h3̄0 |

h
0
0 
0
0 
× u3̄30 v11̄00∗ v22̄∗0 u4̄40 + 2v24̄∗0 u2̄40 v11̄∗0 u3̄300 + u3̄30 u1̄300 2v22̄∗0 u4̄40 + v24̄∗0 u2̄40 v11̄∗0 v13̄00∗

i
0
0 
0
0 
+v13̄∗0 v11̄00∗ 2v22̄∗0 u4̄40 + v24̄∗0 u2̄40 u1̄30 u3̄300 + v13̄∗0 u1̄300 v22̄∗0 u4̄40 + 2v24̄∗0 u2̄40 u1̄30 v13̄00∗ . (B.8)

We now define new two-particle matrix elements in the quasiparticle energy basis by
0 0 0 02̄3̄4̄
≡ φ1(η)2 3 4 v11̄∗0 v22̄∗0 u3̄30 u4̄40 ,Φ1̄(η)(B.9)

which have the same symmetries as the original matrix elements φ1234 [cf. Eq. (3.4)]:
1̄3̄4̄
2̄4̄3̄
1̄2̄3̄4̄= Φ2̄(η)= Φ1̄(η).
Φ(η)(B.10)

We now substitute these new matrix elements, defined in Eq. (B.9), into the collision operator

122
Eq. (B.8) and obtainW † σ z Γ̃< W = 8 P3̄−1̄0 P4̄−2̄0 P2̄−4̄0 |−1̄i ⊗ h3̄0 |h
0 0 0 00 0 0 00 0 0 00 0 0 0× Φ1̄2̄3̄4̄ Φ1̄η 2̄ 3̄ 4̄ + 2Φ1̄4̄3̄2̄ Φ1̄η 2̄ 3̄ 4̄ + 2Φ1̄2̄3̄4̄ Φ3̄η 2̄ 1̄ 4̄ + Φ1̄4̄3̄2̄ Φη3̄ 2̄ 1̄ 4̄

i
0 0 0 00 0 0 00 0 0 00 0 0 0+2Φ3̄2̄1̄4̄ Φ1̄η 2̄ 3̄ 4̄ + Φ3̄4̄1̄2̄ Φ1̄η 2̄ 3̄ 4̄ + Φ3̄2̄1̄4̄ Φ3̄η 2̄ 1̄ 4̄ + 2Φ3̄4̄1̄2̄ Φη3̄ 2̄ 1̄ 4̄ .(B.11)

The definition of the two-particle matrix elements Φ in Eq. (B.9) implies the following identityΦ∗1̄2̄3̄4̄ ≡ Φ1̄2̄3̄4̄

∗

0 0 0 00 0 0 01̄ −2̄ −3̄∗ −4̄∗
4̄∗
3412
u−
= Φ−3̄−4̄−1̄−2̄ ,= φ3 4 1 2 v11̄0 v22̄0 u3̄∗
3 0 u4 0 = φ
1 0 u2 0 v 3 0 v 4 0(B.12)

which we use to substitute the 1̄, 2̄, 3̄, and 4̄ indices and obtain, after reordering some terms,
W † σ z Γ̃< W = 8 P−1̄−1̄0 P−2̄−2̄0 P−4̄−4̄0 |3̄i ⊗ h3̄0 |no 0 0 0 0
× Φ∗1̄2̄3̄4̄ + 2Φ∗1̄4̄3̄2̄ + 2Φ∗3̄2̄1̄4̄ + Φ∗3̄4̄1̄2̄ Φ1̄η 2̄ 3̄ 4̄

no 0 0 0 0
+ 2Φ∗1̄2̄3̄4̄ + Φ∗1̄4̄3̄2̄ + Φ∗3̄2̄1̄4̄ + 2Φ∗3̄4̄1̄2̄ Φ3̄η 2̄ 1̄ 4̄

≡

1 1̄ 2̄ 3̄ 4̄
0
Ψ0 0 0 0P
0P
0P
0 |3̄i ⊗ h3̄ | ,2 1̄ 2̄ 3̄ 4̄ −1̄−1̄ −2̄−2̄ −4̄−4̄(B.13)(B.14)

where we have defined the combined matrix element Ψ, which is symmetrized under pairwise
exchange of 1, 2, and 4 in Eq. (B.13). We use this symmetry in the second line to obtaino 0 0 0 0n
1 1̄ 2̄ 3̄ 4̄
Ψ1̄0 2̄0 3̄0 4̄0 = Φ∗1̄2̄3̄4̄ + 2Φ∗1̄4̄3̄2̄ + 2Φ∗3̄2̄1̄4̄ + Φ∗3̄4̄1̄2̄ Φ1̄η 2̄ 3̄ 4̄
16
o 0 0 0 0n
+ 2Φ∗1̄4̄3̄2̄ + Φ∗1̄2̄3̄4̄ + Φ∗3̄4̄1̄2̄ + 2Φ∗3̄2̄1̄4̄ Φη3̄ 4̄ 1̄ 2̄

(B.15)

non 0 0 0 0o
0 0 0 0= Φ∗1̄2̄3̄4̄ + 2Φ∗1̄4̄3̄2̄ + 2Φ∗3̄2̄1̄4̄ + Φ∗3̄4̄1̄2̄ Φ1̄η 2̄ 3̄ 4̄ + Φ3̄η 4̄ 1̄ 2̄ .

(B.16)

We now exchange 1̄(0) and 2̄(0) in one half of the terms with a factor of two and make use of the1̄2̄3̄4̄
given in Eq. (B.10) to rewrite Ψ assymmetries of Φ(η)

no1 1̄ 2̄ 3̄ 4̄
Ψ1̄0 2̄0 3̄0 4̄0 = Φ∗1̄2̄3̄4̄ + Φ∗1̄4̄3̄2̄ + Φ∗2̄4̄3̄1̄ + Φ∗3̄2̄1̄4̄ + Φ∗3̄1̄2̄4̄ + Φ∗3̄4̄1̄2̄
16
on 0 0 0 0
0 0 0 0Φη1̄ 2̄ 3̄ 4̄ + Φ3̄η 4̄ 1̄ 2̄
=nΦ∗1̄2̄3̄4̄ + Φ∗1̄3̄2̄4̄ + Φ∗1̄4̄2̄3̄ + Φ∗2̄3̄1̄4̄ + Φ∗2̄4̄1̄3̄ + Φ∗3̄4̄1̄2̄
no
0 0 0 00 0 0 0Φη1̄ 2̄ 3̄ 4̄ + Φ3̄η 4̄ 1̄ 2̄ .

(B.17)o(B.18)

123
The first factor can be written in terms of a sum over all permutations Π of the indices 1̄, 2̄, 3̄,and 4̄. Again exchanging 1̄(0) with 4̄(0) and 2̄(0) with 4̄(0) , each in one third of the second factor,
the latter factor can also be written as a sum over index permutations
Ψ1̄1̄0 2̄2̄0 3̄3̄0 4̄4̄0 =

n1 X
4

Π

n

=

Φ∗Π(1̄2̄3̄4̄)o 16
30 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

Φ1̄η 2̄ 3̄ 4̄ + Φ4̄η 2̄ 3̄ 1̄ + Φ1̄η 4̄ 3̄ 2̄ + Φη3̄ 4̄ 1̄ 2̄ + Φ3̄η 1̄ 4̄ 2̄ + Φη3̄ 2̄ 1̄ 4̄

1 n X ∗Π(1̄2̄3̄4̄) on X Π(1̄0 2̄0 3̄0 4̄0 ) o
Φ
Φη
.
3
Π

o

(B.19)(B.20)

Π

Using the definition of the energy-conserving two-particle matrix element Φη in Eq. (B.9)
and Eq. (65) in [37] for the corresponding element in the single-particle energy basis, we canwrite Φη as
0 0 0 00 0 0 0000Φ1̄η 2̄ 3̄ 4̄ = φ1η 2 3 4 v11̄0 ∗ v22̄0 ∗ u3̄30 u4̄40 =

Z t

0

0

dt0 eητ φ1234 K11 † (τ )K22 † (τ )K33 (τ )K44 (τ )v11̄0 ∗ v22̄0 ∗ u3̄30 u4̄40 ,
0

0

0

0

0

−∞

(B.21)
where τ = t0 − t. The time-evolution operator K is given by the time-ordered exponential
R0

K(τ ) = T ei τ ds H

(0)

(s)

.

(B.22)For Eq. (B.21), we can approximate K(τ ) by replacing H (0) (s) by its value H (0) (0) = H (0)at τ = 0 because the energy uncertainty η exponentially dampens earlier contributions to obtain
K(τ ) ≈ e−iτ H(0).

(B.23)

Substituting this result back into the definition of the energy-conserving two-particle matrix
element in Eq. (B.21), we obtain0 0 0 0

Φ1̄η 2̄ 3̄ 4̄ ≈Z t

dt0 eητ φ1234 (eiτ H

−∞
0

0

0

= φ1234 v11̄ ∗ v22̄ ∗ u3̄3 u4̄4
0 0 0 0= Φ1̄ 2̄ 3̄ 4̄

1
η − i∆

Z t

(0)

−∞

−→

η→0

0

0

)11 v11̄0 ∗ (eiτ H

(0)

0

0

)22 v22̄0 ∗ (e−iτ H

(0)

0

0

)33 u3̄30 (e−iτ H

(0)

0

0

)44 u4̄40

dt0 eητ e−iτ (1̄0 +2̄0 +3̄0 +4̄0 )(B.24)

0 0 0 01,Φ1̄ 2̄ 3̄ 4̄ πδη (∆) + iPη
∆(B.25)

with the energy difference ∆ ≡ 1̄0 + 2̄0 + 3̄0 + 4̄0 . We can now rewrite the combined matrix
element Ψ from Eq. (B.20) as

0 0 0 01,Ψ1̄1̄0 2̄2̄0 3̄3̄0 4̄4̄0 = ψ ∗1̄2̄3̄4̄ ψ 1̄ 2̄ 3̄ 4̄ πδη (∆) + iPη
∆(B.26)

124
where we again defined a new, totally symmetric, matrix elemento
1 X Π(1̄2̄3̄4̄)
4 n
ψ 1̄2̄3̄4̄ ≡ √Φ= √ Φ1̄2̄3̄4̄ + Φ1̄3̄2̄4̄ + Φ1̄4̄2̄3̄ + Φ2̄3̄1̄4̄ + Φ2̄4̄1̄3̄ + Φ3̄4̄1̄2̄ .
3 Π
3(B.27)

To obtain the collision operator Γ< , which includes processes involving the condensate,from Γ̃< , we define extended two-particle matrix elements, modifying Eq. (B.9)∗

0 0 0 0Φα 2̄3̄4̄ ≡ φ1 2 3 4 α∗10 v22̄∗0 u3̄30 u4̄40(B.28)

0 0 0 0(B.29)

Φ1̄2̄α4̄ ≡ φ1 2 3 4 v11̄∗0 v22̄∗0 α30 u4̄40 ,and generalized matrix elements, modifying Eq. (B.27)
o
∗
∗
4 n ∗
ψ χ2̄3̄4̄ = √ Φα 2̄3̄4̄ + Φα 3̄2̄4̄ + Φα 4̄2̄3̄ + Φ2̄3̄α4̄ + Φ2̄4̄α3̄ + Φ3̄4̄α2̄ .
3(B.30)

This new generalized matrix element is also totally symmetric, in particularψ χ2̄3̄4̄ = ψ 2̄χ3̄4̄ = ψ 4̄2̄3̄χ .

(B.31)

0 0 01 Ψχχ2̄2̄0 3̄3̄0 4̄4̄0 ≡ ψ ∗χ2̄3̄4̄ ψ χ2̄ 3̄ 4̄ πδη (∆1 ) + iPη= Ψ2̄2̄0 χχ3̄3̄0 4̄4̄0 = Ψ4̄4̄0 2̄2̄0 3̄3̄0 χχ .
∆1

(B.32)As a consequence,

The energy-conserving matrix elements that contain condensate contributions can be written
analogous to Eq. (B.25) as
00 0 0 0
0 0 0 01 
Φ1η 2̄ 3̄ 4̄ α∗100 = Φ1 2̄ 3̄ 4̄ α∗10 πδη (∆1 ) + iPη
with ∆1 ≡ µ + 2̄0 + 3̄0 + 4̄0 and (B.33)∆1

0 0 0 00
0 0 0 01 Φη1̄ 2̄ 3̄ 4 α400 = Φ1̄ 2̄ 3̄ 4 α40 πδη (∆2 ) + iPη
with ∆2 ≡ µ + 1̄0 + 2̄0 + 3̄0 .
(B.34)∆2
We can now write the full collision operator Γ< asW † σ z Γ< W =n1|3̄i ⊗ h3̄0 | Ψ1̄1̄0 2̄2̄0 3̄3̄0 4̄4̄0 P−1̄−1̄0 P−2̄−2̄0 P−4̄−4̄02o+ Ψχχ2̄2̄0 3̄3̄0 4̄4̄0 P−2̄−2̄0 P−4̄−4̄0 + Ψ1̄1̄0 χχ3̄3̄0 4̄4̄0 P−1̄−1̄0 P−4̄−4̄0 + Ψ1̄1̄0 2̄2̄0 3̄3̄0 χχ P−1̄−1̄0 P−2̄−2̄0 .

(B.35)

Relabeling some indices, we obtainW † σ z Γ< W =n1|3̄i ⊗ h3̄0 | Ψ1̄1̄0 2̄2̄0 3̄3̄0 4̄4̄0 P−1̄−1̄0 P−2̄−2̄0 P−4̄−4̄02o+ Ψχχ2̄2̄0 3̄3̄0 4̄4̄0 + Ψ2̄2̄0 χχ3̄3̄0 4̄4̄0 + Ψ4̄4̄0 2̄2̄0 3̄3̄0 χχ P−2̄−2̄0 P−4̄−4̄0 .

(B.36)

125
We can now use the symmetry of the generalized matrix elements, as given in Eq. (B.32) toobtainW † σ z Γ< W =o
1 n 1̄ 2̄ 3̄ 4̄
Ψ1̄0 2̄0 3̄0 4̄0 P−1̄−1̄0 P−2̄−2̄0 P−4̄−4̄0 + 3 Ψχχ2̄2̄0 3̄3̄0 4̄4̄0 P−2̄−2̄0 P−4̄−4̄0 |3̄i ⊗ h3̄0 | . (B.37)
2

This is the final result for the in-rate of the second-order collisional contributions, which is much
simpler than the single-particle representation of these contributions given in Eq. (3.28).
The time-reversed collision operator for the fluctuations Γ̃> describes the collisions outof the distribution G̃< and is given by Γ(1+f˜)(1+f˜)f˜ + 2Γ(1+f˜)m̃m̃∗σ z Γ̃> = ∗Γm̃+ 2Γ∗f˜m̃(1+f˜)
m̃m̃∗



Γm̃m̃m̃∗ + 2Γ(1+f˜)m̃f˜ .
††Γf˜f˜(1+f˜) + 2Γf˜m̃m̃∗(B.38)

It can be transformed in the same way as shown above for the in-rates to obtainW † σ z Γ̃> W = 8 P2̄0 −4̄ P4̄0 −2̄ P1̄0 −3̄ |−1̄i ⊗ h3̄0 |
0 0 0 00 0 0 00 0 0 00 0 0 0× Φ1̄2̄3̄4̄ Φη1̄ 2̄ 3̄ 4̄ + 2Φ1̄2̄3̄4̄ Φη3̄ 2̄ 1̄ 4̄ + 2Φ3̄2̄1̄4̄ Φ1̄η 2̄ 3̄ 4̄ + Φ3̄2̄1̄4̄ Φη3̄ 2̄ 1̄ 4̄
0 0 0 00 0 0 00 0 0 00 0 0 0+2Φ1̄4̄3̄2̄ Φη1̄ 2̄ 3̄ 4̄ + Φ1̄4̄3̄2̄ Φ3̄η 2̄ 1̄ 4̄ + Φ3̄4̄1̄2̄ Φ1̄η 2̄ 3̄ 4̄ + 2Φ3̄4̄1̄2̄ Φ3̄η 2̄ 1̄ 4̄

= 8 P2̄0 2̄ P4̄0 4̄ P1̄0 1̄ |3̄i ⊗ h3̄0 |
0 0 0 00 0 0 00 0 0 00 0 0 0× Φ∗1̄2̄3̄4̄ Φ1̄η 2̄ 3̄ 4̄ + 2Φ∗1̄2̄3̄4̄ Φ3̄η 2̄ 1̄ 4̄ + 2Φ∗3̄2̄1̄4̄ Φ1̄η 2̄ 3̄ 4̄ + Φ∗3̄2̄1̄4̄ Φη3̄ 2̄ 1̄ 4̄+2Φ
=∗1̄4̄3̄2̄

0 0 0 00 0 0 00 0 0 00 0 0 0
Φ1̄η 2̄ 3̄ 4̄ + Φ∗1̄4̄3̄2̄ Φ3̄η 2̄ 1̄ 4̄ + Φ∗3̄4̄1̄2̄ Φ1̄η 2̄ 3̄ 4̄ + 2Φ∗3̄4̄1̄2̄ Φη3̄ 2̄ 1̄ 4̄

1
P 0 P 0 P 0 |3̄i ⊗ h3̄0 | Ψ1̄1̄0 2̄2̄0 3̄3̄0 4̄4̄0 ,2 1̄ 1̄ 2̄ 2̄ 4̄ 4̄

which is similar in form to Eq. (B.14).

(B.39)

(B.40)(B.41)

Appendix C
Kinetic Equations for Fermions

In this Appendix, we use the formalism of the Boson kinetic equations presented in
Chapters 2 through 4 to derive equations for fermions in a homogeneous system, similar to thoseused in the later Chapters 5 through 8. This illustrates how the two main parts of this thesis
are connected. The main steps are writing the kinetic equations in the position basis, writing
them in center-of-mass coordinates, and then Fourier-transforming the relative coordinate toa Wigner representation, which reduces to the usual distribution function in a homogeneous
system. We only demonstrate these transformations for the first-order terms; however, it isstraightforward to use the same steps for the higher-order collision terms and reproduce parts of
the rate equations we derived using the cumulant formalism in Chap. 7. We begin by considering
the kinetic equations for fermions in the single-particle energy basis as introduced in Chap. 2.

C.1

Kinetic Equations
We consider the following Hamiltonian
H = H (0)where H (0)10 20

0 0 0 0
10 20 †
a10 a20 + φ1 2 3 4 a†10 a†20 a30 a40 ,(C.1)= h10 | H (0) |20 i denotes the matrix elements of the interaction-free single-particle

Hamiltonian
H (0) =

p2
+ Vext (x),
2m

(C.2)

with external harmonic potential Vext . The fermionic creation operator a†10 creates a particle
in the state |10 i, where 10 stands for a complete set of quantum numbers, which label a con-

127
stant, single-particle energy basis {ϕ10 }10 , such as, for example, harmonic oscillator states. The
fermionic field operator in the position basis ψ(x) can be written in terms of the single-particle
energy basis
ψ(x) =

X10

ϕ10 (x)a10 = hx | 10 ia10 .

(C.3)

We use the summation convention for these abbreviated indices and indicate the single-particle
basis with primes.
We now define the distribution functions and potentials in the single-particle energy basis.
The Hermitian normal density matrix f is defined by
f = ha†20 a10 i |10 i ⊗ h20 |(C.4)

as in the boson case. The anomalous average m is defined as
m = ha20 a10 i |10 i ⊗ |20 i .

(C.5)

It is anti-symmetric for fermions: m = −m> . The anti-symmetrized two-particle matrix elements of the binary, spin-dependent interaction potential Vbin (x1 , x2 ) are defined byφ

10 20 30 40

1
= (A)
2

Z

dx1 dx2 h10 | x1 ih20 | x2 iVbin (x1 − x2 )hx1 | 30 ihx2 | 40 i.

(C.6)

These matrix elements are explicitly anti-symmetrized (A) such that:0 0 0 00 0 0 00 0 0 0

φ1 2 3 4 = −φ2 1 3 4 = −φ1 2 4 3 .

(C.7)

We anti-symmetrize these matrix elements, because the symmetric parts do not contribute to
the Hamiltonian in Eq. (C.1) due to the Fermi anti-commutation of the operators. The energy
shift due to the normal fluctuations is given by the Hermitian matrix
0 0 0 0

Uf = 2 φ1 2 3 4 f30 20 |10 i ⊗ h40 | ,(C.8)

whereas the first-order anomalous coupling-strength is given by the anti-symmetric matrix
0 0 0 0Vm = 2 φ1 2 3 4 m30 40 |10 i ⊗ |20 i .

(C.9)

128
To define the generalized density matrices, we start with the spinor
 
a

A=
 ,
†
a

(C.10)

where a = a10 |10 i and a† = a†10 h10 | are vectors of operators. Then, the single-time limit of one
time-ordering [43, 39] can be written as(1 − f ) −m.
G> = hAA† i = 


∗
∗
m
f

(C.11)The other time ordering gives the usual many-body density matrix fG< = −hσ1 AA† σ1 i> = − 

−m∗

m

(1 − f )∗.
(C.12)The generalized Boltzmann equation of motion for this fluctuation-density matrix can be
written asd <G = −iΣG< + Γ< G> − Γ> G< + H.c.
dt

(C.13)The reversible evolution above is governed by the Hartree–Fock–Bogoliubov self-energy operator


 ΣN
Σ=

−Σ∗A

ΣA 
,

−Σ∗N

(C.14)

which in turn consists of the Hermitian HamiltonianΣN = H (0) + 2Uf − µ(C.15)and the anti-symmetric anomalous couplingΣA = Vm̃ .

(C.16)In the following, we neglect the irreversible second-order evolution given by the collision operators Γ.
Considering the components of Eq. (C.13), we obtain the following first-order equations

129
for f and m alone:df = ΣN f − f ΣN − ΣA m∗ + mΣ∗Adtd∗i m = ΣN m + mΣ>
N + ΣA (1 − f ) − f ΣA
dt
i

C.2

(C.17)(C.18)

Transformation to the Position Basis
We explicitly identify the energy and spin parts of the single-particle energy basis by

writing |10 i = |10 , s10 i. We can then write the transformation to the position basis ash1 | 10 i = ψ10 (x1 ).

(C.19)

This wave function is used to transform the energy part of the matrices defined above to theposition basis. The spin labels remain after this transformation. The completeness relations forthe transformation are
h1 | 10 ih10 | 2i = δ(x1 − x2 ),Zanddx1 h10 | 1ih1 | 20 i = δ10 ,20 χ∗ (s10 )χ(s20 ),(C.20)(C.21)

where χ is a spin function and we use the summation convention for the energy basis.
We now write the distribution functions and energy shifts in the position basis. The
normal average f transforms as a matrix
fs1 s2 (x1 , x2 ) = h1 | 10 if10 20 h20 | 2i, andZ
f10 20 = dx1 dx2 h10 | 1ifs1 s2 (x1 , x2 )h2 | 20 i,(C.22)(C.23)

whereas the anomalous average m transforms as a tensor according to
ms1 s2 (x1 , x2 ) = h1 | 10 ih2 | 20 im10 20 , andZ
m10 20 = dx1 dx2 h10 | 1ih20 | 2ims1 s2 (x1 , x2 ).

(C.24)(C.25)The matrix elements of the potential with explicit spin indices areφ

10 20 30 40

1
=
4

Z

n
o
dx1 dx2 h10 | 1ih20 | 2iVs1 s2 s3 s4 (x1 − x2 ) h1 | 30 ih2 | 40 i + h1 | 40 ih2 | 30 i ,(C.26)

130
where the energy labels under the integral are symmetric and the spin part of the binary potential
is anti-symmetricVs1 s2 s3 s4 (r) = −Vs2 s1 s3 s4 (r) = −Vs1 s2 s4 s3 (r),(C.27)in order to fulfill Eq. (C.7). The energy shifts in the position basis are
0 0

Ws1 s4 (x1 , x2 ) = h1 | 10 iUf1 4 h40 | 2i = Vs1 s2 s3 s4 (x1 − x2 )fs3 s2 (x1 , x2 ),

and

0 0

∆s1 s2 (x1 , x2 ) = h1 | 10 ih2 | 20 iVm1 2 = Vs1 s2 s3 s4 (x1 − x2 )ms3 s4 (x1 , x2 ).

(C.28)(C.29)

We now have all the elements of Eqs. (C.17) and (C.18) for the normal and anomalous
average and can write these equations of motion in the position basis:idfs s (x1 , x2 ) = [H (0) (x1 ) − H (0) (x2 )]fs1 s2 (x1 , x2 )dt 1 2Z+ dy [Ws1 s3 (x1 , y)fs3 s2 (y, x2 ) − fs1 s3 (x1 , y)Ws3 s2 (y, x2 )]−iZdy [∆s1 s3 (x1 , y)m∗s3 s2 (y, x2 ) − ms1 s3 (x1 , y)∆∗s3 s2 (y, x2 )],d
ms s (x1 , x2 ) = [H (0) (x1 ) + H (0) (x2 )]ms1 s2 (x1 , x2 )dt 1 2Z+ dy [Ws1 s3 (x1 , y)ms3 s2 (y, x2 ) + ms1 s3 (x1 , y)Ws3 s2 (y, x2 )]+Z(C.30)(C.31)dy [∆s1 s3 (x1 , y)(1 − f )∗s3 s2 (y, x2 ) − fs1 s3 (x1 , y)∆s3 s2 (y, x2 )],with the position-basis single-particle Hamiltonian
H (0) (x) =

C.3

2
∇x
+ Vext (x).
2m

(C.32)

Center-of-Mass Coordinates
We now transform to center-of-mass position coordinates. These coordinates are given

x2 and r = x − x . We find for the distribution functions:by R = x1 +
1
2
2

r
r
fs1 s2 (R, r) = fs1 s2 R + , R −
,
2
2

r
r
ms1 s2 (R, r) = ms1 s2 R + , R −.
2
2(C.33)(C.34)

131
The energy shifts are now
Ws1 s2 (R, r) = Vs1 s3 s4 s2 (r)fs4 s3 (R, r)(C.35)∆s1 s2 (R, r) = Vs1 s2 s3 s4 (r)ms3 s4 (R, r),(C.36)and the single-particle Hamiltonian can be written as a center-of-mass and relative contribution,
because we assume a harmonic trapping potential
H (0) (x1 ) + H (0) (x2 ) = H (0) (r) + H (0) (R),(C.37)

where the mass m in H (0) (x) is the appropriate reduced mass for the center of mass (2m) andrelative (m/2) Hamiltonian.
Since the range of the two-body potential r0 is small compared to the size of the system,
the distribution functions are weakly dependent on the center-of-mass coordinate, and we canmake the following approximations
fs1 s2 (R + r 0 , r) ≈ fs1 s2 (R, r) for any |r 0 | ≤ r0 ,(C.38)

ms1 s2 (R + r 0 , r) ≈ ms1 s2 (R, r) for any |r 0 | ≤ r0 .

(C.39)The equations of motion (C.30) and (C.31) can now be written in terms of center-of-mass
and relative coordinates:irdr
fs s (R, r) = [H (0) (R + ) − H (0) (R − )]fs1 s2 (R, r)
dt 1 2
2
2
Z

r 
r
+ dx [Ws1 s3 R + , x fs3 s2 (R, r − x) − fs1 s3 (R, r − x)Ws3 s2 R − , x ]
2
2
Z
r 
r 
− dy [∆s1 s3 R + , x m∗s3 s2 (R, r − x) − ms1 s3 (R, r − x)∆∗s3 s2 R − , x ],2
2
(C.40)

id
ms s (R, r) = [H (0) (R) + H (0) (r)]ms1 s2 (R, r) + Vs1 s2 s3 s4 (r)ms3 s4 (R, r)
dt 1 2Zr r + dx [Ws1 s3 R + , x ms3 s2 (R, r − x) + ms1 s3 (R, r − x)Ws3 s2 R − , x ]
2
2
Z
r 
r 
− dx [∆s1 s3 R + , x fs∗3 s2 (R, r − x) + fs1 s3 (R, r − x)∆s3 s2 R − , x ].
2
2
(C.41)

132

C.4

Wigner Representation
In this Section we perform a Fourier transform of the relative coordinate in the distribu-

tion function and obtain a Wigner representation with a spatial center-of-mass and momentum
relative coordinate. We start by writing the normal and anomalous averages in the Wigner
representation by Fourier transforming Eqs. (C.33) and (C.34)
Z1
dr e−ikr fs1 s2 (R, r),
fs1 s2 (R, k) =
(2π)3
Z
1
dr e−ikr ms1 s2 (R, r).
ms1 s2 (R, k) =(2π)3(C.42)(C.43)

These distribution functions are independent of the center-of-mass coordinate in a homogeneous
system, and correspond to the fk and mk used in Chap. 6. The potential transforms to
Vs1 s2 s3 s4 (k) =

1
(2π)3

Z

dr e−ikr Vs1 s2 s3 s4 (r),(C.44)and the relative part of the single-particle Hamiltonian is
H (0) (k) =k2
1+2m (2π)3Z

dr e−ikr Vext (r).

(C.45)The energy shifts are now
Ws1 s2 (R, k) =∆s1 s2 (R, k) =Z

Z

dk0 Vs1 s3 s4 s2 (k − k0 )fs4 s3 (R, k0 ),(C.46)

dk0 Vs1 s2 s3 s4 (k − k0 )ms3 s4 (R, k0 ).

(C.47)

We now obtain the equations of motion in the Wigner representation by Fourier transforming Eqs. (C.40) and (C.41):
Z
n
o
0

d
(0)
(0)
i fs1 s2 (R, k) = − H (R) + H (k) , fs1 s2 (R, k)+ dr dk0 e−i(k−k )r
dt
PB
h


r
r
(C.48)
× Ws1 s3 R + , k0 fs3 s2 (R, k0 ) − fs1 s3 (R, k0 )Ws3 s2 R − , k0
2
2
i
r 
r
− ∆s1 s3 R + , k0 m∗s3 s2 (R, k0 ) + ms1 s3 (R, k0 )∆∗s3 s2 R − , k0 ,22

where the Poisson-bracket term {, }PB is an approximation to the full single-particle term given

133
in [151, App. D].
iZ0d
ms1 s2 (R, k) = [H (0) (R) + H (0) (k)]ms1 s2 (R, k) + ∆s1 s2 (R, k) + dr dk0 e−i(k−k )r
dt
h

r
r 
× Ws1 s3 R + , k0 ms3 s2 (R, k0 ) + ms1 s3 (R, k0 )Ws3 s2 R − , k0
2
2

i
r
r
− ∆s1 s3 R + , k0 fs∗3 s2 (R, k0 ) − fs1 s3 (R, k0 )∆s3 s2 R − , k0 .
2
2

(C.49)

Dropping the r-dependence of the center-of-mass coordinates, these equations simplify to:
i

n
o

d
fs1 s2 (R, k) = − H (0) (R) + H (0) (k) , fs1 s2 (R, k)dtPB+ (2π)3 Ws1 s3 (R, k)fs3 s2 (R, k) − fs1 s3 (R, k)Ws3 s2 (R, k)(C.50)

− ∆s1 s3 (R, k)m∗s3 s2 (R, k) + ms1 s3 (R, k)∆∗s3 s2 (R, k) ,idms s (R, k) = [H (0) (R) + H (0) (k)]ms1 s2 (R, k) + ∆s1 s2 (R, k)
dt 1 2+ (2π)3 Ws1 s3 (R, k)ms3 s2 (R, k) + ms1 s3 (R, k)Ws3 s2 (R, k)(C.51)

− ∆s1 s3 (R, k)fs∗3 s2 (R, k) − fs1 s3 (R, k)∆s3 s2 (R, k) .

These are the first-order equations of motion in the Wigner representation. In a homogeneous
system, one can specify spin states as in the following Section to compare these equations toEqs. (72) to (75) in Ref. [103] and to the equations in [152].

C.5

Scattering Equations for the Crossover Problem
We now specify a set of spin states to model the two-channel crossover Hamiltonian

Eq. (5.8) discussed in Chap. 5 and show the resulting two-body scattering equations. We use
the following spin states: 1, 2 are hyperfine ground state (open channel), 3, 4 are hyperfine
excited (closed channel), electron up and down in each case. We assume spin symmetry in the
open channel, vanishing magnetizations and no populations in the closed channel,
f = f11 = f22 ,f12 = f34 = f33 = f44 = 0.

(C.52)(C.53)

134
No s-wave pairing for fermions in the same state means that the pair correlations vanish in thesame state,
m11 = m22 = m33 = m44 = 0.

(C.54)

Open channel pairs are given by the pair correlation function between spin up and down:m = m12 = −m21 .

(C.55)In the closed channel, we only consider bound pairs (see App. D for a discussion of when thesepairs can be treated as bosons):
φ0 = m34 .

(C.56)We define the following potential including some symmetries:
UF (r) = V1221 (r) = V2112 (r) = −V1212 (r),(C.57)UB (r) = V3443 (r) = V4334 (r) = −V3434 (r),(C.58)

g(r) = V1234 (r) = V3412 .

(C.59)

The combinations that can not be deduced from this list vanish, for example, V1111 = 0, because
there is no s-wave scattering of identical fermions.
To find the two-body scattering equations, we consider the center-of-mass equation of
motion (C.41) for the pair matrix m in the absence of normal fermions (f = 0):idms s (R, r) = [H (0) (R) + H (0) (r)]ms1 s2 (R, r) + Vs1 s2 s3 s4 (r)ms3 s4 (R, r).
dt 1 2(C.60)

We want to solve the eigen-value problem for the Hamiltonian on the right-hand side of
the above equation. As an Ansatz, we split off the spin-independent center-of-mass componentΨCM (R),
ms1 s2 (R, r) = ΨCM (R)ψs1 s2 (r),(C.61)and obtain the following two eigen-value equations:
H (0) (R)ΨCM (R) = EΨCM (R) and(C.62)

H (0) (r)ψs1 s2 (r) + Vs1 s2 s3 s4 (r)ψs3 s4 (r) = ψs1 s2 (r).

(C.63)

135
The second one can be split up into spin states using the definitions in Eqs. (C.55), (C.56),(C.57), and (C.59):[H (0) (r) + 2UF (r)]m(r) + 2g(r)φ0 (r) = m(r)(C.64)

νφ0 (r) + 2g(r)m(r) = φ0 (r),(C.65)

where ν is the eigen value to the bare closed-channel bound state φ0 given by[H (0) (r) + 2UB (r)]φ0 (r) = νφ0 (r).

(C.66)

Equations (C.64) and (C.65) are the two-body scattering equations for the open and closed
channels.

Appendix D
Fermionic Pairing

In this Appendix, we consider how a pair of fermions can be treated as an effective,
composite boson. We also examine the density-matrix evolution equations for pairs that aresuperpositions of bosons and fermions, and show their excitation spectrum to be gapless. We
here use a Hamiltonian that includes a non-resonant fermion interaction.

D.1

Pairs of Fermions as Bosons
First, we examine the commutation relation for a pair of fermions to find out when it

can be treated as a boson. The two-channel Hamiltonian (5.8) makes this approximation. The
fermionic field operator aσ (x) obeys the commutation relationhaσ (x), a†σ0 (x0 )i+= δ(x − x0 )δσσ0(D.1)

We define the field operator b(R) of a pair of these fermions byb(R) =

Z

dr φ(r)a↓ (R +rr)a↑ (R − ),
2
2

(D.2)

where φ(r) is the normalized pair wave function as a function of the relative coordinate r.
Examining the commutation relation for the pairs, we findb(R), b† (R0 ) − = δ(R − R0 )

 
X Z
r 
r
dr φ(r)φ∗ r + 2(R0 − R) a†σ 2R0 − R +−aσ R +
,
2
2
σ=↓,↑

(D.3)

137
and



b(R), b† (R0 ) − = δ(R − R0 )
 r


 r
X Z
−
+ R0 − R aσ
− (R0 − R) ,dr φ(r − 2R0 )φ∗ r + 2R a†σ
2
2

(D.4)

σ=↓,↑

where we used that the pair wave function φ is symmetric. If the second term on the righthand side of this equation can be neglected, we obtain a bosonic commutation relation for thepairs, that is, the pairs can be described as effective bosons. In what regime is this a good
approximation?
If we assume that the integral in Eq. (D.4) is maximal for R = R0 , we can write



X Z


r
,
dr |φ(r)|2 nσ R +b(R), b† (R0 ) − = δ(R − R0 ) 1 −
2

(D.5)

σ=↓,↑

where nσ is the number operator for spin σ. The pair wave function φ vanishes outside a relative
distance of r ≈ lpair , which is determined by the size of the pair. The condition for treating
pairs as bosons can then be written in terms of the number density of fermions nσ


D.2


b(R), b† (R0 ) − = δ(R − R0 )if

3nσ lpair
 1.

(D.6)

Pairs in a Homogeneous System
We now consider pairs of fermions and composite bosons in a homogeneous system forthe following HamiltonianH=X  qX + ν b†q bq +gk b†q aq/2−k↓ aq/2+k↑ + H.c.

Xk a†kσ akσ +X

Uk−k0 a†q+k a†q−k aq−k0 aq+k0

qk,σ=↑,↓

+

2

(D.7)

qk

(D.8)

qkk0

which is similar to the Bose-Fermi Hamiltonian discussed in Eq. (5.8), but now includes a nonresonant fermion background interaction Uk . The composite bosons and fermions obey the usual
commutation relations[bq , b†q0 ]− = δqq0

and [ak↑ , a†k0 ↑ ]+ = δkk0 .

(D.9)

138
The bosons and fermions commute, that is, the bound state does not contain continuum components.
[ak↑ , bq ]− = 0(D.10)The long-range fermion pairs in the open channel are defined by
pqk = a q −k↓ a q +k↑ .
2(D.11)

2

These pairs almost obey a bosonic commutation relationh[pqk , p†q0 k0 ]− i = δqq0 δkk0 NF (q, k),(D.12)

with the Pauli-blocking factor
NF (q, k) = 1 − GNq
− k − GN
+k ,
2
2

q

(D.13)

which represents Pauli blocking, and fermion populations
GN (k) = ha†k↑ ak↑ i = ha†k↓ ak↓ i.

(D.14)

We now define a density matrix for the fluctuations around the mean fields hbi and hpi.
We use the pair spinorA(q, k) =



bq

b†q

p†qk

pqk

>

(D.15)

to define the density matrix
Gab (q, k, q 0 , k0 ) = hA†b (q 0 , k0 )Aa (q, k)i.

(D.16)

This density matrix contains the following correlation functions
GM (q) = hb†q bq i,(D.17)GC (q, k) = hb†q pq(k− q ) i = hb†q aq−k↓ ak↑ i, and(D.18)

2

GP (q, k, k0 ) = hp†qk pqk0 i = ha†q +k↑ a†q −k↓ a q
2

2

0
2 −k ↓aq

0
2 +k ↑

i.

(D.19)

The functions GM and GP are the densities of closed- and open-channel molecules, respectively,
and GC describes a correlation function between the two kinds of pairs. Writing this density

139
matrix G in Eq. D.16 explicitly in terms of its components we obtain


q
0
∗
0
GC (q, 2 + k )0 GM (q)




q


0
0
1 + GM (q)0
GC (q, 2 + k )


0
.

G(q, k, k ) = G (q, q + k)
0
0
G
(q,
k
,
k)
0

 C
P
2




q
0
∗
0
δkk0 NF (q, k) + GP (q, k, k )0
GC (q, 2 + k)

(D.20)We here assume that b and p are quasiparticle operators in the boson (top right) and fermionpair (bottom right) parts of the density matrix respectively, that is, the pair (or four-point)correlation function hbbi and hppi vanish. The equation of motion for G for evolution under the
Hamiltonian (D.8) can now be written as (assuming GN = const.):i~dG(q, k, k0 ) X Σ(q, k, k00 )G(q, k00 , k0 ) − G(q, k, k00 )Σ† (q, k00 , k0 ) ,=dt
00(D.21)

kwith the self energyδkk0 (q + ν)



0

Σ(q, k, k0 ) =  N (q, k)g
 F
k


0

0

g k0

0

−δkk0 (q + ν)

0

−gk0
0

0

U (q, k, k )0−NF (q, k)gk

0

−U (q, k, k0 )an effective interaction

U (q, k, k0 ) = Uk−k0 NF (q, k)+ δkk0  q

2 +k

+  q −k + U0
2

Xh
q0

GN



and the free-particle dispersion relation
q =

















i
q
q
q 0 − + k + GN q 0 − − k 22~2 q 2.
2m(D.22)(D.23)(D.24)The self energy matrix Σ is gapless, because we have zero-energy eigen state P  hb0 i   φm 

 


 

 hb† i   φ∗ 
 0  m,
=P (k) =  −hp i −G (k)



A
0k 

 


 

†
∗
−GA (k)−hp0k i(D.25)

140
which consists of the pair mean fields. It is a non-trivial zero-energy eigen state,XΣ(0, k, k00 )P (k00 ) = 0,

(D.26)dφm
dGA (k)
=
= 0,
dt
dt

(D.27)k00

in equilibrium, that is, when

where the equations of motion of the mean fields are given by
i~

X
dφm
= νφm +
gk GA (k), anddt(D.28)ki~X
dGA (k)= NF (0, k)gk φm +U (0, k, k0 )GA (k0 ).
dt
0k(D.29)

Appendix E
Rate Equations in the Normal Phase

In this Appendix, we show how to derive effective kinetic rate equations from the cumulant expansion method discussed in Chap. 8. Step (2) discussed below is essentially thesame adiabatic elimination we used in Chap. 8 to close the three-operator equations we simulate
numerically.

E.1

Equations of Motion
We list below the full set of normal-phase equations up to cut C illustrated in Fig. 8.1.

First, we have the normal densities of fermions and composite bosons,
i~

nX
o
dfk↑
= 2ig =
bq a†k↑ a†q−k↓ ,
dt
q

i~

o
nX
dnq
bq a†q/2+k↑ a†q/2−k↓ ,= −2ig =dt(E.1a)(E.1b)kwhere = indicates the imaginary part. These couple to the three-operator correlation function,i~d bq a†q/2+k↑ a†q/2−k↓dt= (ν − 2k ) bq a†q/2+k↑ a†q/2−k↓+gXaq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓Xbq b†q0 aq0 −q/2+k↑ a†q/2+k↑ + bq b†q0 aq0 −q/2−k↓ a†q/2−k↓

k0

−g

q0



+ g fq/2+k↑ fq/2−k↓ (1 + nq ) − g nq (1 − fq/2+k↑ )(1 − fq/2−k↓ ),(E.1c)

142
which in turn depends on the following two four-operator correlation functions,i~d bq b†q0 aq0 −q/2+k,σ a†q/2+k,σdt= 2 (q0 −q)/2+k − k
± g fq/2+k,σ + nqbq b†q0 aq0 −q/2+k,σ a†q/2+k,σb†q0 aq/2−k,−σ aq0 −q/2+k,σ∓ g fq0 −q/2+k,σ + nq0(E.1d)

bq a†q/2+k,σ a†q/2−k,−σ ,where the signs on the right-hand side correspond to the two possible spin directions, and,i~d aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓dt= 2 (k0 − k ) aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓+ g 1 − fq/2+k0 ↑ − fq/2−k0 ↓− g 1 − fq/2+k↑ − fq/2−k↓bq a†q/2+k↑ a†q/2−k↓b†q aq/2−k0 ↓ aq/2+k0 ↑ .
(E.1e)

These equations are the same as listed in Chap. 8. In order to obtain rate equations forthe bosons and fermions we now use the following three steps:
1. Substitute the appropriate lowest-order adiabatic solution bq a†q/2+k↑ a†q/2−k↓ A of Eq.
(E.1c) into the equations for the four-operator cumulants Eqs. (E.1d) and (E.1e). This
cuts the recursive coupling of the three- and four-operator cumulants.
2. Adiabatically solve for the four-operator cumulants and substitute into the equation
for the time-dependent three-operator correlation bq a†q/2+k↑ a†q/2−k↓ . This yields theequations we simulate numerically.
3. To make sure that we choose the right expressions for the adiabatic eliminations in thetwo steps above, we further eliminate the three-operator cumulant altogether and areleft with coupled rate equations for the fermions and bosons. The Boltzmann collision
terms can be interpreted by inspection, which validates the previous two steps.

E.2

Step (1): Cut Recursion
We first cut the recursion between the three- and four-operator equations by adiabaticallysolving Eq. (E.1c) without any four-operator correlations and using this solution on the right-

143
hand sides of Eqs. (E.1d) and (E.1e). We here pick the real principal value according to thefollowing relation for energy denominators
11=P+ iπδ(∆E),∆E − i∆E(E.2)

for small  > 0. This choice gives us the collisional rates we are interested in in step (3).
The imaginary delta-function contribution renormalizes the energy-denominators, but we here
neglect this correction of the intermediate-state energies, since we expect it to be small.
The adiabatic principal-value solution of Eq. (E.1c) is
bq a†q/2+k↑ a†q/2−k↓ A = P−g 
fq/2+k↑ fq/2−k↓ − nq 1 − fq/2+k↑ − fq/2−k↓ ,ν − 2k(E.3)

which has the population factors of the collisional rate illustrated in Fig. 8.2 a):bq a†q/2+k↑ a†q/2−k↓ A = P




−g 
fq/2+k↑ fq/2−k↓ 1 + nq − nq 1 − fq/2+k↑ 1 − fq/2−k↓ ,ν − 2k(E.4)

This is the direct interconversion of a pair of fermions to a bare composite boson.

E.3

Step (2): Eliminate Four-Operator Cumulants
In this step, we adiabatically solve the four-operator equations (E.1d) and (E.1e) and ob-

tain the second-order collisional interactions depicted in Fig. 8.2 b) and c), an effective fermionfermion interaction through a virtual composite boson, and an interaction between a fermion
and a composite boson by exchanging a fermion. We use the adiabatic solution from step (1)on the right-hand side, and keep only the energy-conserving delta-function contributions of theenergy denominator, because the off-shell parts are energetically suppressed.
bq b†q0 aq0 −q/2+k,σ a†q/2+k,σ A = −iπ δ 2(q0 −q)/2+k − 2k gh
× ± fq/2+k,σ + nq b†q0 aq/2−k,−σ aq0 −q/2+k,σ A∓ fq0 −q/2+k,σ + nq0ibq a†q/2+k,σ a†q/2−k,−σ A .

(E.5)

144
Substituting the adiabatic three-operator solution (E.4) gives the fermion-boson rate in Fig. 8.2c):bq b†q0 aq0 −q/2+k,σ a†q/2+k,σ A = −iπδ 2(q0 −q)/2+k − 2k P

g2
ν − 2kh× − fq0 −q/2+k,σ nq (1 − fq/2+k,σ )(1 + nq0 )+ fq/2+k,σ nq0 (1 − fq0 −q/2+k,σ )(1 + nq )

(E.6)

i

These on-shell fermion-boson collisional rates redistribute particles, but obey number conservation, that is, they obey the following detailed-balance condition:
XX
q

q0

bq b†q0 aq0 −k,σ a†q−k,σ

A

= 0.

(E.7)

We analogously consider Eq. (E.1e) and find the rates depicted in Fig. 8.2 b):
aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓ A = −iπ δ(2k0 − 2k ) Pg2
ν − 2kh× fq/2+k0 ↑ fq/2−k0 ↓ (1 − fq/2+k↑ )(1 − fq/2−k↓ )i− fq/2+k↑ fq/2−k↓ (1 − fq/2+k0 ↑ )(1 − fq/2−k0 ↓ ) ,(E.8)

with the detailed-balance relation
XX
k

k0

aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓ A = 0.

(E.9)

These collisional processes enter into the rate equations for the fermion and boson populations.

E.4

Step (3): Rate Equations
Substituting the results of steps (1) and (2) back into the equations for the fermion andboson densities (E.1a) and (E.1b), we obtain the following coupled rate equations:
i~

hX
i
X
X
dfk↑
g2
= −2
aq/2−k0 ↓ aq/2+k0 ↑ a†k↑ a†q−k↓ A −P
bq b†q0 aq0 −q+k↑ a†k↑ Adt
ν − 2k−q/2
0
0
q
q

k

− 2πg 2Xqδ ν − 2k−q/2

 hfk↑ fq−k↓ (1 + nq ) − nq (1 − fk↑ )(1 − fq−k↓ )i(E.10a)

145

i~

hX
i
X
X
dfk↓
g2
P
= −2
aq/2−k0 ↓ aq/2+k0 ↑ a†q−k↑ a†k↓ A −bq b†q0 aq0 −q+k↓ a†k↓ Adt
ν − 2k−q/2
0
0
q
q

k

h
i
X 
− 2πg 2δ ν − 2k−q/2 fq−k↑ fk↓ (1 + nq ) − nq (1 − fq−k↑ )(1 − fk↓ )q(E.10b)i~i
X
g2 X hdnq= −2bq b†q0 aq0 −q/2+k↑ a†q/2+k↑ A + bq b†q0 aq0 −q/2−k↓ a†q/2−k↓ APdtν − 2k 0qk+ 2πg 2Xkhi
δ(ν − 2k ) fq/2+k↑ fq/2−k↓ (1 + nq ) − nq (1 − fq/2+k↑ )(1 − fq/2−k↓ )(E.10c)These rate equations contain the scattering processes depicted in Fig. 8.2. In particular, thesecond line in each equation contains the quantum-Boltzmann rates for the direct interconversion
of a pair of fermions and a composite boson as depicted in Fig. 8.2 a). The energy-conserving
delta function in these fermion-boson rates causes them to vanish for negative detuning, because
the closed-channel bound state lies below the continuum of the free fermions.

E.5

Equations of Motion to Order g 2
In this Section, we further simplify the equations of motion in Sec. E.1 by only keeping

the vacuum contributions from the four-operator level and introducing damping rates Γ 1 andΓ2 that model the neglected many-body and higher-order terms.
i~d bq a†q/2+k↑ a†q/2−k↓dt= (ν − 2k − iΓ1 ) bq a†q/2+k↑ a†q/2−k↓
+g

X
k0

aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓+ g fq/2+k↑ fq/2−k↓ (1 + nq ) − g nq (1 − fq/2+k↑ )(1 − fq/2−k↓ ),(E.11)

146i~d aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓= (2k0 − 2k − iΓ2 ) aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓dt+ gNF (q, k0 ) bq a†q/2+k↑ a†q/2−k↓ − gNF (q, k) b†q aq/2−k0 ↓ aq/2+k0 ↑(E.12)
with the fermion population factor
NF (q, k) = 1 − fq/2+k↑ − fq/2−k↓ .

(E.13)The damping rates Γ1 and Γ2 also model part of the effect of the couplings to five-operator
averages of the type b† a† aaa . We have neglected these couplings to close our equations andcut the BBGKY hierarchy.
We now eliminate the four-operator average aaa† a† using an initial condition ofaaa† a† (t = 0) = 0
aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓ =−g
NF (q, k0 ) bq a†q/2+k↑ a†q/2−k↓ − NF (q, k) b†q aq/2−k0 ↓ aq/2+k0 ↑ .
2k0 − 2k − iΓ2
In the limit k0  k we only keep the first vacuum term
−g
b a†
a†
2k0 − iΓ2 q q/2+k↑ q/2−k↓igΓ2 
g− 2= −bq a†q/2+k↑ a†q/2−k↓ ,2k0
4k0 + Γ22

aq/2−k0 ↓ aq/2+k0 ↑ a†q/2+k↑ a†q/2−k↓ =(E.14)(E.15)

since all the correlation functions go to zero for large momentum arguments.

Substituting the expression for the four-fermion cumulant Eq. (E.15) into the equation of
motion for the fermion-boson correlation function Eq. (E.1c), we obtain the following reduced
set of equations of motion
nX
o
dfk↑
= 2g =
b−q0 a†k↑ a†−q0 −k↓dt
0(E.16a)

q

nX
o
dnq
= −2g =
bq a†q/2+k0 ↑ a†q/2−k0 ↓dt0(E.16b)ki~d bq a†q/2+k↑ a†q/2−k↓dt= (ν̄ − 2k − iΓeff ) bq a†q/2+k↑ a†q/2−k↓+ gfq/2+k↑ fq/2−k↓ (1 + nq ) − gnq 1 − fq/2+k↑

1 − fq/2−k↓(E.16c)147
with the renormalized detuningν̄ = ν −

0
kX
=K

k0

K
g2
= ν − 2 g 2 = ν − αg 2
0
2k
2π

(E.17)

and an effective damping rate
Γeff = Γ1 + g 2

X
k0

Γ2
.
2
4k0 + Γ22(E.18)In the next Section, we try to find an estimate for the damping rate Γeff by looking at the rateequations for fermion and boson populations.

E.6

Rate Equations to Order g 2 and Effective Damping Rates
In this Section, we eliminate the Bose–Fermi correlation functionba† a†

to obtain acoupled set of rate equations for the populations of fermions and molecules. Adiabatically solving
for the imaginary part of Eq. (E.16c), we obtain:=nbq a†q/2+k↑ a†q/2−k↓

o

=


1
Γba (q, k) − Γab (q, k) ,2g

(E.19)

with fermion → boson and boson → fermion collision ratesΓab (q, k) = 2g 2Γeff
fq/2+k↑ fq/2−k↓ (1 + nq ) ,(ν̄ − k 2 )2 + Γ2eff

Γba (q, k) = 2g 2



Γeff
nq 1 − fq/2+k↑ 1 − fq/2−k↓ .
222(ν̄ − k ) + Γeff

(E.20a)(E.20b)

Substituting the resulting Eq. (E.19) into the population equations (E.16a) and (E.16b),
we obtain the rate equations
X
dfk↑
q0
q0 
=
Γba (q 0 , k − ) − Γab (q 0 , k − ) ,
dt
2
2
0

(E.21a)

q


X
dnq
=
Γab (q, k0 ) − Γba (q, k0 ) .
dt
0(E.21b)k

One easily sees that these equations obey conservation of the total number
N=

X
k,σ

fk,σ + 2Xqnq(E.22)

148
We now consider the limit Γeff → 0, where the Lorentzian in the rates in Eqs. (E.20)
reduces to a Dirac delta function πδ(ν̄ − 2k)
i
h
X 
dfk↑
δ ν̄ − 2k−q0 /2 nq0 (1 − fk↑ ) (1 − fq0 −k↓ ) − fk↑ fq0 −k↓ (1 + nq0 ) ,= 2πg 2dt
0(E.23a)

q

h
X

i
dnq
= 2πg 2
δ(ν̄ − 2k0 ) fq/2+k0 ↑ fq/2−k0 ↓ (1 + nq ) − nq 1 − fq/2+k0 ↑ 1 − fq/2−k0 ↓ .
dt
0k(E.23b)

We can next evaluate the remaining momentum integrals in the rate equations above
using the continuum limit of the momentum sums
2π
X= 3
8π
0

X
k

Z 1Z K−1

k 02 X dk 0 dx0(E.24)

0

For the molecule equation, we can solve the dk 0 -integral and obtain
√ Z

i
dx0
dnq
g 2 ν̄ 1 h=
fq/2+k0 ↑ fq/2−k0 ↓ (1 + nq ) − nq 1 − fq/2+k0 ↑ 1 − fq/2−k0 ↓√dt2π
k0 = ν̄ 2−1(E.25)In the fermion equation, it is easier to solve the dx0 integral first, and we obtain
πg 2
dfk↑
=
dt
k

Z K
0

i
h

q 0 nq0 (1 − fk↑ ) 1 − fk0 ↓ − fk↑ fk0 ↓ (1 + nq0 )dq 0
q /2−k2 +2ν̄ 2π 2

√ 02
0

k =

(E.26)

We can now estimate the effective decay rate Γeff for the Bose–Fermi correlation in Eq. (E.16c)asΓeff =

1
Γb + 2Γf
2(E.27)

where the boson and fermion decay rates are taken from the rate equations above:
√ Z

 dx0
g 2 ν̄ 1 
1 − fq/2+√ν̄↑ 1 − fq/2−√ν̄↑
2π −1
2
i
2 Z K hdq 0πg
q 0 fk0 ↑ (1 + nq0 ) 0 √ 02
Γf =k 0
k = q /2−k2 +2ν̄ 2π 2Γb =(E.28)(E.29)

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