Bose-Einstein condensation in interacting gases
While the processus of Bose-Einstein condensation was described above for noninteracting par-ticles, Bose-Einstein condensation is also possible in the presence of atomic interactions. The role of interactions generally becomes more important with increasing atomic density. While interactions can be neglected in a thermal gas with low atomic density, they cannot be neglected in a BEC. In a BEC all atoms populate the same quantum state, which leads to a high atomic density in the trap center2. In this case interactions visibly modify the equilibrium properties of the condensates.
The scattering length
Interactions between two particles are generally described by a scattering potential. If this potential is radially symmetric, the scattering process can be treated with the partial wave expansion . For collisions of low energy, already the first term of this expansion, the s-wave scattering potential, describes the scattering process accurately. This potential is characterized by a single parameter, the s-wave scattering length a.
The Thomas-Fermi approximation
For many properties of BECs it is useful to examine the possible approximations of the GP-equation. To do so, one has to look at the diﬀerent parts of the total energy of the system. It can be distinguished in three parts: the kinetic, the potential and the interaction energy:E = Ec + Ep + Ei. From equation (1.14), one can estimate the scale of these contributions:Ec ∝ = ~ωho ho mR2 R2 where R is the geometric mean of the radii of the condensate, and aho = ~ the harmonic mωho oscillator length. The assumption Ei ≫ Ec is valid if (aho/R)(N0a/aho) ≫ 1. The dimensionless parameter N0a/aho gives a measure of the strength of the interactions, and for typical parameters of our 3D condensates, N0 ∼ 105 and aho ∼ 1µm, one gets N0a/aho ∼ 500. Typical radii of the cloud are of the order of R ∼ 10 µm, and for these parameters it is thus possible to neglect Ec. This regime is called the Thomas-Fermi (TF) regime.
Measuring the radii of a BEC is thus a method to determine its atom number. This method is used in our experiments, when other methods fail due to small optical densities of the cloud (see chapter 2). It is however not of high accuracy, as an error in the radius measurement is considerably amplified in the calculation of the atom number, due to the R5-dependence.
The above equations rely on the TF-approximation and are valid for all experiments with 3D condensates described in this thesis. Part II of this thesis is however dedicated to the description of quasi-2D BECs. In this case the axial trapping frequency in z-direction is larger than µ/~. The axial profile of the cloud is then well approximated by a gaussian, as only the ground state of the motion along z is populated (see chapter 3).
The hydrodynamic approach
The dynamic properties of BECs can in principle be derived from the time-dependent GP-equation (equation (1.12)). It exists however a more intuitive approach, which directly uses the velocity field as a variable. This hydrodynamic approach is presented in the following.
The condensate wave function is fully described by the two macroscopic parameters n(r, t) and ϕ(r, t), the density and the phase of the condensates: Ψ(r, t) = eiϕ(r,t).
Irrotational flow and hydrodynamic equations
According to equation (1.20) the motion of the condensate corresponds to a potential flow; the velocity field being the gradient of the scalar velocity potential ~ϕ/m. If ϕ is not singular, the motion of the condensate must be irrotational, that is ∇ × v = m~ ∇ × ∇ϕ = 0.
The possible motions of a condensate are thus more restricted than those of a classical fluid. The only rotational motion is in fact induced by quantized vortices, as introduced in part III of this thesis.
To obtain the hydrodynamic equations of a BEC one inserts the condensate wave function in the time-dependent GP-equation (equation (1.12)), which leads to a system of coupled equations for n and v.
Dynamics of a condensate
The time-dependent behavior of a condensed cloud is an important source of information about its physical nature. One can for example understand the phenomenon of superfluidity in a BEC from the structure of its excitation spectrum.
This section contains a brief presentation of the methods to derive the excitation spectrum. The most complete approach follows the theory of Bogoliubov. This approach allows in principle to derive the excitations spectrum under arbitrary conditions. In most cases it does however not lead to analytic calculations.
In a second section an especially simple case is thus presented in more detail: the case of a trapped condensate at zero temperature. This section is based on the hydrodynamic treatment of a condensate, as presented in section 1.3. Within this approach some of the most studied modes of BECs are derived: the collective excitations of low energy (see e.g. [67–69] for theory, and [70, 71] for experiments). These low-lying surface modes are also of some relevance in the experimental parts of this thesis.
The Bogoliubov approach
For the derivation of the Gross-Pitaevskii equation (equation (1.12)), the ansatz N0φ0 (1.25) Ψ = Ψ + δΨwithΨ had been introduced, where Ψ is the condensate wave function, and δΨ the condensate depletion. The condensate depletion was then neglected, which is in general a reasonable approximation for Bose-Einstein condensates in atomic gases.
For a more complete description of Bose-Einstein condensates at temperatures T = 0 the condensate depletion must however be taken in to account. The expansion of the Heisenberg equation must thus be taken out to a superior order. This approach was developed by Bogoli-ubov  and is given in great detail in  and , and here only its general approach is briefly sketched out.
Inserting the ansatz (1.25) in the Heisenberg equation leads to a system of coupled linear ˆ ˆ † equations for δΨ and δΨ . The field operator can be then be rewritten using the eigenmodes of the system: Under typical experimental conditions the temperature is of the order of kBT ∼ µ ∼ 10 ~ω⊥, and a large number of Bogoliubov excitations ωk is populated. For some selected trap geometries the spectrum of the eigenmodes can be calculated analytically, and the solutions are presented in the following.
The spectrum of small energy modes with ~2k 2/(2m) ≪ µ corresponds to a phonon spec-trum linear in k, while the high energy modes (~2k2/(2m) ≫ µ) have an energy quadratic in k and correspond to free particles.
• Harmonic trap: For a condensate in a harmonic trap the modes are quantized in energy. For modes with a wavelength much smaller than the size of the cloud, the spectrum is similar to that of the homogenous case. For modes with a wavelength that is similar to the size of the condensate, some normal modes corresponding to shape oscillations of the condensate can be identified . A description of these modes within the hydrodynamic approach is given in section 1.4.2.
Collective modes: the hydrodynamic approach
To investigate elementary excitations within the hydrodynamic approach, one considers small deviations from the equilibrium density of the gas: n = neq +δn, where neq is the equilibrium den-sity. This ansatz is inserted in the hydrodynamic equations in the TF-limit (equations (1.23)), which are then linearized by treating δn as a small quantity.
The mode ω+ corresponds to the l = m = 0 monopole mode, which is of some importance in our experiments. It is radially symmetric, and corresponds to a periodic increase and decrease of the cloud’s diameter (figure 1.2(a)). For a 2D gas in a harmonic potential the breathing mode has a frequency of exactly 2ω⊥, and for the case of hard core interactions it was shown by Pitaevskii and Rosch  that this solution is “universal”, i.e. not dependent on other properties of the trapped gas.
The low-lying eigenmodes of atomic BECs have been studied thoroughly in harmonic traps with various geometries (see e.g. [70, 71]). As their properties are thus well known, they are used for several purposes in the experiments: to measure the frequencies of the magnetic or optic traps (usually through the frequency of the monopole or dipole mode), or to set the condensate into rotation (by resonantly exciting the quadrupole mode). In this thesis only in one case an eigenmode is explicitly studied: chapter 10 describes an experimental study of the monopole-oscillation of a fast rotating gas in a quadratic+quartic potential.
Coherence properties of Bose-Einstein condensates
For the experiments described in this manuscript especially two notions play an important role: phase coherence and superfluidity. While superfluidity manifests itself in the excitation spectrum and the rotational properties of the BEC, phase coherence properties are especially well illustrated in interference experiments. Such experiments constitute the main experimental tool of part II of this manuscript. In the following the notion of phase coherence is briefly introduced.
The coherence length
The Gross-Pitaevskii equation is a non-linear diﬀerential equation for a classical field, the con-densate wave function. This has some analogy to a classical light field, governed by the Maxwell-equations. Indeed a Bose-Einstein condensate without interactions can be understood as the matter equivalent to a laser. This equivalence inspired various experiments with BECs, which all have their analog in optics: interference experiments [3, 35, 73], bosonic amplification [74, 75] and the construction of continuous atom lasers [76–78]. The notion of coherence thereby is associated with the variations of the phase ϕ of the condensate wave function and is crucial to the understanding of interference.
Table of contents :
I Bose-Einstein condensates: Properties and realization
1 Bose-Einstein condensation in harmonic traps
1.1 The BEC phase transition in ideal gases
1.2 Bose-Einstein condensation in interacting gases
1.3 The hydrodynamic approach
1.4 Dynamics of a condensate
1.5 Coherence properties of Bose-Einstein condensates
2 Experimental realization of a BEC
2.1 Experimental cycle: brief overview
2.2 The magnetic trap
2.3 Evaporative cooling
2.4 The imaging system
II 2-dimensional condensates
3 2-dimensional Bose gases
3.1 Ideal Bose gases in 2D
3.2 Interacting gases: Condensates and quasicondensates
3.3 Phase-fluctuations and vortex-antivortex pairs: The KT-transition
4 Experimental realization of 2D condensates
4.1 The optical lattice
4.2 Selecting sites from the lattice
4.3 Experimental results
5 Detection of phase defects in 2D condensates
5.1 Interferometric detection of phase defects
5.2 Interpretation of the interference pattern
5.3 Axial imaging: Possibility for vortex detection in 2D clouds
5.4 Probability of thermal vortex configurations
6 Interference of 30 independent condensates
6.1 Experimental routine
6.2 Experimental results
6.3 Theoretical discussion of the results
6.4 Number squeezing and Mott-Insulator-transition
III Fast rotating condensates in quartic potentials
7 Introduction to rotating condensates and the quantum Hall effect
7.1 Rotating condensates: vortices and vortex lattices
7.2 Rotating Bosons and the quantum Hall effect
7.3 Bosonic systems in the LLL
8 Fast rotation of a BEC in a quartic potential
8.1 Rotating bosons in anharmonic traps
8.2 Fast rotation of an ultra-cold Bose gas
9 Condensation temperature in a quadratic+quartic potential
9.1 Condensation temperature of a rotating gas in a harmonic trap
9.2 Results for the combined quadratic + quartic potential
9.3 Numerical solution for the quartic potential
9.4 Relevant results for the experimental situation
10 Monopole in fast rotating BECs