Ultracold quantum gases: fermionic superfluidity and impurity problems 

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Interacting Bose gas

We consider a gas of Nb spinless bosons, with an interaction characterized by the scattering length abb, that can be changed by taking advantage of a Feshbach resonance.
In order to describe the weakly interacting Bose gas, we make an assumption that we will call the universality hypothesis: we assume that the properties of a many-body system with short-range interactions do not depend on the details of the interaction potential. This hypothesis can be justified saying that the wavelength associated to the particles is much bigger than the range of the potential, but it does not always work: as we saw before in Chapter 1, close to unitarity, non-universal three-body physics emerges for strongly interacting bosons.
Within the universality hypothesis, we can use the model of a contact interaction in the mean-field approximation, and the Hamiltonian of the system without any external potential reads bH = Z d3r 􀀀 ~2 2mb b yr2 b + gbb 2 b y b y b b .

Interacting Fermi gas : the BEC-BCS crossover

In this section, we will study the behaviour of the balanced two-component Fermi gas at low temperature. We consider then a Fermi gas constituted of spin 1/2 fermions, each species being referred to either  » or # fermions, with a balanced distribution N » = N#. We write aff the scattering length characterizing the interaction between a  » and a # fermion.
To describe the properties of the system, we again assume we are within the universality hypothesis. Due to Pauli blocking, three-body physics are suppressed and this hypothesis is valid even for strong interactions. At T = 0, only two length scales are available: the scattering length aff which encompasses the two-body interactions and 1=kF , the inverse of the Fermi wavevector, defined with the Fermi energy EF = ~2k2F =2m, and related to the Fermi density in a homogeneous system through kF = (32nf )1=3 (for a two-component Fermi gas). We see with this equation that 1=kF is directly proportional to the interparticle distance n􀀀1=3 of the universal dimensionless quantity5 1=kF aff .
At T = 0, the properties of the two-component Fermi gas depend on the interactions. From the BCS limit 1=kF aff ! 􀀀1 to the BEC limit 1=kF aff ! +1, while crossing the unitary limit 1=kF aff = 0, the properties of the Fermi gas vary considerably. The transition between the BCS and the BEC regimes for a Fermi gas in the vicinity of a Feshbach resonance is not a phase transition but a smooth crossover, known as the BEC-BCS crossover, represented in Fig. 2.1. The BEC-BCS crossover was first proposed by Leggett [46], Nozi`eres and Schmitt-Rink [47] and confirmed by various theoretical approaches [48] and experiments [123, 124]. In the whole BEC-BCS crossover, the twocomponent Fermi gas is superfluid and is characterized by the pairing of fermions of opposite spins, but the nature of this pairing changes drastically with the interactions. In this section, we will detail the universal properties of Fermi gases in the BEC-BCS crossover.

Tan’s contact for a two-component Fermi gas

We saw in the last section that in the strongly interacting regime, where many-body effects have to be taken into account, we can measure experimentally the equation of state in the BEC-BCS crossover for a Fermi gas but finding a theory without approximations to describe these systems is very difficult.
In this section, we introduce a new parameter, Tan’s contact C2 first introduced by Shina Tan in 2008 [144, 145], a fundamental thermodynamic quantity that appears in a set of exact universal relations connecting thermodynamic observables to various other microscopic or macroscopic quantities that would seem otherwise unrelated. Those relations actually hold for any temperature, number of atoms, trap geometry or interaction strength, in particular they hold in the strongly interacting regime. Detailed review on the contact can be found in [146, 147], we will present in this section its most relevant properties.

Other measurements of the contact

We already presented a first method to measure the contact, with the disadvantage of not being able to recover precisely the contact at unitarity (as we have a “bump” at unitarity). Another method consists in using RF-spectroscopy where interacting atoms are transferred to a non-interacting state, with a transition rate directly linked to the contact [152, 153] 􀀀(!) / C2 !3=2 : (2.50).
This method was used in several experiments [154, 155] and also compared to the results obtained by looking to the tail of the momentum distribution [156].
Other methods include photoassociation of atoms to form deeply-bound molecules [54, 157] or the use of the structure factors measured by Bragg spectroscopy [158, 159]. In Fig. 2.4, we compare several measurements of the contact in the crossover performed using those different techniques. It shows that all measurements are in agreement and validates all the different relations involving the Contact that would seem otherwise disconnected.

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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
List of publications 

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