Impurity in an interacting medium: a perturbative approach 

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Magnetic tuning of Feshbach resonances

As mentioned earlier, a magnetic field can be used to shift the energy of the open channel with respect to the closed channel. In alkali atoms, this is done by making use of the Zeeman shift in the triplet channel potential (when mS 6= 0) due to its non-zero total spin while the singlet channel undergoes no shift since its total spin is zero.
In Eq. (1.24) we arrived to an expression where the scattering length diverges when the resonance is matched. By writing Dm as the magnetic moment difference between the incoming open channel state and the closed uncoupled molecular state, we have e e0 = Dm(B B0).
By taking into account collisions in the open channel in the model we can write the scattering length in Eq. (1.24) around resonance showing explicitly its open-channel bare value without any coupling to the molecular channel (abg) with respect to the magnetic field change in the commonly used expression [124]: a(B) = abg D B m L2 ¯ 2 1 with DB = = h (1.27).
where B0 designates the position of the resonance and DB the width of the resonance, and abg the background scattering length corresponding to the limit a(B ! ¥), for instance in Fig.2.3 we recover the famous huge scattering length of the triplet channel when B ! ¥ for 6Li. Again, the width of the resonance is larger for small resonance ranges Re and closer magnetic moments between the two channels.
Also, from Eq. (1.27) we notice that when a = 0 we get DB = B B0 which marks the closest zero-crossing point for the scattering length around resonance (as in Fig.2.3). Which adds to the point that for a broad resonance (DB 1) the scattering length is modified within a large interval around resonance and for a narrow resonance the scattering length is modified in a very small interval around resonance. For instance in Fig.2.3 there is actually a resonance around 543 G but its width is around 200 mG which makes it invisible on such a scale used in the figure and practically useless.

Efimov trimers properties and domain

In general , for a given three-particle system, there are three inter-particle interac-tions. At least two of these interactions are required to be resonant for the Efimov effect to occur. This can be understood simply from the picture of mediated inter-action we mentioned earlier where in order for one particle to mediate an effective long-range interaction between two other particles, it must interact resonantly with these two particles. If it interacts resonantly with only one particle, then the media-tion to another particle is not possible.
Generally speaking, bosonic particles are favorable to the Efimov effect, whereas fermionic particles tend to prevent the Efimov effect, since their Pauli exclusion may overcome the Efimov attraction.
The lighter a particle is, the better it mediates interaction between other particles. Thus, mass-imbalanced systems tend to enhance the Efimov attraction, and enable the Efimov effect in fermionic systems.
It is useful to add a discussion on the domain of existence of the Efimov trimers. The spectrum in Eq. (1.34) does not have a lower limit, meaning that the energy is not bounded from below and the ground-state energy lies at ¥ giving smaller and smaller trimer sizes. This leads to the mentioned Thomas Collapse [129] encountered in two-body potentials with spatial dependence as 1=r2. However, if one accounts for the physical fact of finite range interactions and the limits of the zero-range ap-proximation it is possible to set upper and lower limits on the size of the trimer. In the case of large binding energies, the trimer size becomes comparable to the range of the two-body potential rvdw defined in Eq. (1.2), in this case the short-range approximation is no longer valid and the Efimov scenario breaks down. For low binding energies, the trimer size becomes larger than inter-particle distance n 1=3 and the surrounding atoms create additional interactions which smear the trimer state.
In practice the number of accessible trimers is small, if we take ln0 = rvdw and ln1 = n 1=3 where n is the particle density introduced in 1.1.1 we have ln1 =ln0 = 1=a(n1 n0) giving n = n0 n1 = ln(n 1=3=rvdw)=ln(a) ’ 2.
where a = exp(p=s0) ’ 22.7 is the recurrent factor encountered in Efimov physics (or its square for energy dependence).

Experimental evidence of Efimov physics

Efimov physics had stayed for decades a mere theoretical prediction until the first experimental evidence was discovered. We will only focus in this section on experi-ments in the field of cold atoms relevant to the scope of this manuscript.
Most atomic species have interactions that decay as a µ 1=r6 Van der Waals potential. Thus, to observe the Efimov trimer described earlier, neutral atoms appear to give an ideal system. As for resonant interactions, this can be achieved by means of the Fesh-bach resonances studied earlier. Shortly after these techniques were well-established, many groups turned to the old problem of Efimov.
Experimental evidence of the Efimov physics was first obtained with 133Cs atoms [138, 139] by studying the three-body losses in their Bose gas. The basic principle behind the measurement is that each time an Efimov trimer couples to a three-atom or to an atom-dimer threshold, the particle loss dramatically increases, and the cor-responding scattering rate coefficients provide well-suited observables to detect Efi-mov physics in experiments.
Although the experiment observed only one trimer state, its Efimovian nature was convincing since it was observed in regions where the dimer state is known to be unbound.
This experiment was later extended to other atomic species and the scaling factor a was tested experimentally through the observation of multiple resonances and the value of a was measured as well as the atom-dimer scattering value which deter-mines the value of a when the trimer state dissolves into the dimer state [140–142]. The binding energy was also directly measured via radio-frequency association, with a three-component Fermi gas of 6Li [143, 144].

Ultra-cold Fermi gases

The first theoretical study of fermions goes back to Paul Drude’s model for the con-ductivity in solids [145]. Drude considered metals to be composed of heavy positively charged particles and light electrons1. Then he modeled the motion of these particles using a purely classical treatment and using this simple approach, it became possible to explain the basic properties of metals like conductivity and resistance.
However, the observation at the beginning of the 20th century of superconductivity in metals changed this view. It was Heike Kamerlingh Onnes, then a professor at the University of Leiden, who, after having succeeded in liquefying 4He earlier, used his liquid 4He to cool down mercury when he observed the remarkable effect of the resistivity dropping to non-measureable values at low enough temperatures2.
The condensation of electrons was an utterly strange observation, since for fermions, they cannot be in one and the same quantum state and an obvious scenario for this to happen might be the formation of tightly bound pairs of electrons that can behave as bosons and condense. However, at the time no known interaction could possibly overcome the Coulomb repulsion between electron pairs. It was not until four decades later that L. Cooper realized that fermions interacting via an arbitrarily weak attractive interaction mediated by the crystal lattice vibra-tions (phonons) on top of a filled Fermi sea can form a bound pair [146], and by a result could be responsible for superconductivity.
Soon afterwards, Bardeen, Cooper and Schrieffer (BCS) developed a full theory of superconductivity starting from a new, stable variational ground state in which pair formation was included in a self-consistent way [147]3. Later, Popov [148], Leggett [149] and Eagles [150] realized that the BCS formalism and its variational ansatz provides also a description of a Bose-Einstein condensate of a dilute gas of tightly bound pairs.
In this section, we will describe the rich physics involved in the low temperature interacting Fermi gas, with its asymptotic regimes, the molecular Bose-Einstein con-densate, the Barden Cooper Schriefer superfluid, and the unitary Fermi gas. To begin, we will provide some results for non-interacting gases.

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Interacting Fermi gas and the BEC-BCS crossover

In the case of a gas of fermions composed of two equally populated spin states, in-teractions arise and become more important and the physics described earlier needs to be modified due to large scale coherence which leads to many-body phenomena arising.
Unlike Bose gases which suffer from huge three-body losses in the limit of strong interactions due to three-body recombination, Fermi gases do not have this problem since Pauli exclusion principle blocks these combinations which leads to gases with life times longer than the times needed to run an experimental sequence in cold atoms experiments.
In the case of strongly attractive interaction kF a 1, the ground state of the system should be a BEC of tightly bound molecular pairs. When the binding energy largely exceeds the Fermi energy, the fermionic nature of the gas becomes irrelevant since paired fermions have different momenta and therefore no Pauli blocking happens.
In the other case of weak attractive interactions kF a 1, there is no bound state for two isolated fermions, but Cooper pairs form in the medium for fermions close to the Fermi sphere. The ground state of the system is a condensate of Cooper pairs as described by BCS theory. In contrast to the case of molecular condensate, the binding energy of these pairs is much smaller than the Fermi energy and thus the Pauli principle plays a major role.
In between, it crosses the unitary limit where the scattering length diverges and the properties of the gas become scale invariant. It was then realized by Leggett [151] building upon the work of Popov [148] that the transition from the BCS to the BEC regime is a smooth crossover. In light of what was explained this sounds a bit odd since the two-body physics shows a threshold behavior at the unitary limit, below which there is no bound state for two particles. However, in the presence of the Fermi sea, the transition is manifested simply by a crossover from a regime of tightly bound pairs to a regime where these pairs are of much larger size than the interparticle spacing.

Impurity in a two-component Fermi gas

Experiments on dual superfluids raised many questions regarding the behavior of an impurity immersed in a superfluid of spin 1=2 fermions [111, 225, 226]. In these experiments, the polaron is weakly coupled to the background superfluid and the interaction could be accurately modeled within mean-field approximation.
Further theoretical works explored the strong coupling regime between the impu-rity and the background fermions using mean-field theory to describe the fermionic superfluid [114, 117, 130]. They highlighted the role of Efimov physics in the phase diagram of the system and as a consequence some results were plagued by unphys-ical ultraviolet divergences.
Indeed, in the Fermi polaron case, no three-body effects are possible since Pauli blocking forbids interactions between spin polarized fermions. On the contrary, the Bose polaron is subject to Efimov effects [132] and three-body interactions play an essential role in the strongly interacting regime. The nature of the transition between the polaron and the trimer state is intrigu-ing since it depends heavily on the background interactions. Indeed, it was shown in [130] using a mean-field approach to describe the superfluid, that superfluid ex-citations provide a strong coupling between the polaron and trimer state making for a smooth avoided crossing between the two branches. However, for a background Fermi sea in the normal state, it was shown in [118] that this transition takes the form of a sharp first order transition.
The presence of the two-component Fermi sea allows access to Efimov effects even for a fermionic impurity and a trimer bound state becomes accessible in the phase diagram of the problem along with the dimer state present in the strongly interacting regime. A phase diagram for the problem without the presence of the Fermi sea is shown in Fig.1.5. It was obtained (See supplemental material in [117]) by searching values where the mean field energies of the three sectors are equal.
Moreover, the presence of the Fermi sea allows us to study the transition between a polaronic state in the presence of two component fermionic superfluid and a Bose polaron when the fermion-fermion interaction is increased.
The study of a polaron immersed in a two-component Fermi sea will be a central part of this manuscript. We will see how the calculation of the energy of the polaron in an interacting superfluid presents divergent terms which are reminiscent from the three-body physics which will be the main focus of Chapter 4.
The problem mentioned earlier regarding an impurity immersed in a non-interacting Fermi sea is going to be detailed in Chapter 5 where we will use a variational ansatz to explore the phase diagram.

Table of contents :

1 Ultracold Fermi gases: From few to many 
1.1 Two-body problem
1.1.1 Universal dynamics and scaling
1.1.2 Scattering theory
1.1.3 Feshbach resonances
1.2 Three-body problem
1.2.1 Two-channels model for the three identical bosons
1.2.2 Efimov trimers properties and domain
1.2.3 Experimental evidence of Efimov physics
1.3 Ultra-cold Fermi gases
1.3.1 Non-interacting Fermi gas
1.3.2 Interacting Fermi gas and the BEC-BCS crossover
1.4 Impurity physics
1.4.1 Bose polaron
1.4.2 Fermi polaron
1.4.3 Impurity in a two-component Fermi gas
I The Lithium Experiment 
2 A new generation Lithium machine 
2.1 Overview of the setup
2.2 The 6Li atom
2.2.1 Level structure
2.2.2 Feshbach resonances of 6Li
2.3 Vacuum setup
2.4 671 nm Laser setup
2.5 Absorption imaging
2.6 Magneto-optical trap
2.6.1 Atomic beam
2.6.2 Zeeman Slower
2.6.3 Magneto-optical trap (MOT) and compressed MOT
2.7 Optical molasses
2.7.1 D2 molasses
2.7.2 D1 gray molasses and sub-Doppler cooling
2.8 Optical dipole traps
2.8.1 Optical transport
2.8.2 Cross dipole trap
2.9 Evaporative cooling
2.9.1 Working principle
2.9.2 State populations
2.9.3 Magnetic fields in the science cell
2.9.4 Cooling to degeneracy
3 From superfluidity to single atom imaging 
3.1 Thermodynamics of ultracold Fermi gases
3.1.1 Equation of state
3.2 Quantitative analysis of density distributions
3.2.1 Non-interacting Fermi gas in a harmonic trap
3.2.2 Unitary Fermi gas
3.2.3 Implementation and results
3.3 Searching fermionic superfluidity
3.3.1 Spin imbalanced systems
3.4 Single-atom imaging of fermions
3.5 Prospects of the 6Li machine
II Impurity immersed in a two-component Fermi sea 
4 Impurity in an interacting medium: a perturbative approach 
4.1 Perturbative expansion of the impurity energy
4.1.1 Preliminary calculation
4.1.2 Asymptotic behavior
4.2 Green’s function for an interacting system
4.2.1 Green’s function: Definition
4.2.2 Time evolution operator
4.2.3 Adiabatic activation
4.2.4 Vacuum polarisation
4.3 Perturbative expansion using Green’s function formalism
4.3.1 The impurity’s Green function
4.3.2 Expectation value of the density-density correlation function .
4.3.3 Ladder approximation
5 Impurity immersed in a double non-interacting Fermi sea 
5.1 Variational ansatz of the full problem
5.2 Polaron sector
5.3 Efimov sector
5.3.1 Trimer in vacuum
5.3.2 Cooper-like trimer
5.3.3 General case
5.4 Polaron-trimeron coupling
5.5 Dimer energy
A Calculation of the first contribution to the diagrammatic expansion 
B Ladder diagram terms 
C Derivation of polaron-trimeron coupled equations 
D Numercial solution of Skornyakov-Ter-Martirosyan’s equation 
E Cooper-like trimer for different values of kFRe 


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