Interacting Bose gas

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Table of contents

1 Interactions in cold atoms 
1.1 Interactions between two particles
1.1.1 Interaction potential
1.1.2 Low-energy scattering theory
1.1.3 Born’s approximation
1.1.4 Transition matrix
1.1.5 Regularization of the pseudopotential
1.2 Feshbach resonance
1.2.1 Two-channel model
1.2.2 Magnetic Feshbach resonances
1.2.3 Bound state near the resonance
1.2.4 Narrow and broad Feshbach resonances
1.3 Three-body problem
1.3.1 A two-channel model for the three-body problem
1.3.2 Properties of the Efimov trimers
1.3.3 The atom-dimer scattering length
2 Ultracold quantum gases: fermionic superfluidity and impurity problems 
2.1 Ideal quantum gases
2.2 Interacting Bose gas
2.3 Interacting Fermi gas : the BEC-BCS crossover
2.3.1 The BCS limit
2.3.2 Molecular BEC domain
2.3.3 Unitary Fermi gas
2.3.4 Fermi equation of state at zero-temperature
2.4 Tan’s contact for a two-component Fermi gas
2.4.1 Universality hypothesis
2.4.2 Momentum distribution and adiabatic sweep theorem
2.4.3 Short-range correlations in a many-body system
2.4.4 Two-body contact in the BEC-BCS crossover
2.4.5 Other measurements of the contact
2.5 The spin-polarized Fermi gas
2.5.1 Imbalanced ultracold Fermi gases
2.5.2 The N+1 body problem: the Fermi polaron
2.5.3 Fermi polaron to molecule transition
2.5.4 Repulsive branch
2.6 The Bose polaron
3 Producing a superfluid Bose-Fermi mixture 
3.1 Overview of the set-up
3.2 Lithium atoms
3.2.1 Atomic structure
3.2.2 Feshbach resonances
3.2.3 Stability of the mixture
3.3 Laser system
3.4 Loading the magnetic-optical trap
3.4.1 The Lithium source
3.4.2 Zeeman slower
3.4.3 Magneto-optical trap
3.5 Magnetic trapping
3.5.1 Optical pumping
3.5.2 Lower magnetic trap and transport
3.5.3 Ioffe-Pritchard trap
3.5.4 Doppler cooling
3.5.5 RF evaporation
3.6 The final hybrid magnetic-dipolar trap
3.6.1 The magnetic-dipolar trap
3.6.2 Loading of the trap
3.6.3 Mixture preparation
3.6.4 Final evaporation
3.7 Imaging
3.7.1 Absorption imaging
3.7.2 Double and Triple imaging
3.8 Analysis of the profiles
3.8.1 Degenerate bosons and thermometry
3.8.2 Superfluidity of the fermions
3.9 Calibrations
4 Lifetime of an impurity in a two-component Fermi gas 
4.1 Three-body recombination
4.1.1 General description
4.1.2 Expected scalings in our system
4.1.3 From the three-body loss rate to the two-body contact
4.2 Lifetime measurements
4.2.1 Investigating the nature of the losses
4.2.2 Losses on the BEC side
4.2.3 Losses at unitarity
4.3 Lifetime at finite temperatures
4.3.1 Previous results on the unitary contact
4.3.2 Lifetime measurements at finite temperature (preliminary)
4.3.3 Effect of the finite size of the impurity cloud
5 Counterflow of a dual Bose-Fermi superfluid 
5.1 Dipole mode excitations
5.1.1 Creating the counterflow
5.1.2 Uncoupled oscillations
5.2 Long-lived oscillations: probing the interactions between the impurities and the superfluid
5.2.1 Effect of the Fermi superfluid on the impurities oscillations
5.2.2 Frequency shift through the BEC-BCS crossover
5.3 Damping of the oscillations
5.3.1 Higher amplitudes: critical velocity
5.3.2 Higher temperatures: out of the superfluid phase
5.4 Conclusion
6 The 2N+1 body problem 
6.1 Perturbative expansion of the polaron energy
6.1.1 Theoretical framework
6.1.2 Asymptotic limit and structure factor
6.2 Regularization of the three-body scattering amplitude
6.2.1 T-matrix in Faddeev’s formalism
6.2.2 Diagrammatic representation of the solutions
6.2.3 Calculation of ti
6.2.4 Power counting
6.2.5 Calculation of the diverging term
6.2.6 Three-body contact interaction
6.3 Renormalization of the polaron energy
6.3.1 Expression of the polaron energy
6.3.2 The F function: Asymptotic expansions
6.3.3 Comparison with other theories
6.3.4 The atom-dimer scattering problem
6.3.5 Infinite-mass impurity
6.4 Consequences on the experiment: frequency shift corrections
6.4.1 BCS side
6.4.2 BEC side: corrections to the atom-dimer scattering length
6.4.3 Unitarity: interactions with the many-body background
A Determination of the fermionic peak density 
A.1 The inverse Abel transformation
A.2 Peak density of the unitary fermi gas
A.2.1 The EoS of the Unitary Fermi gas
A.2.2 Using the EoS to determine the peak density
A.3 A new method: using the curvature of the integrated profile
A.3.1 Principle of the method
A.3.2 Calibration at T=0
A.3.3 Measurements at finite temperature
B BCS Theory 
B.1 Elements of BCS Theory
B.2 Perturbative expansion of the polaron energy within BCS Theory
B.2.1 Mean-field compressibility
B.2.2 Perturbative calculation of the energy
B.2.3 The F function
C Phase diagram of an impurity immersed in a Fermi superfluid 
C.1 Polaron
C.2 Dimeron
C.3 Trimeron
C.4 Building the diagram
D Determination of the R3 and Cad constants 
D.1 Calculating R3
D.2 Atom-dimer scattering
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