PhD thesis in English

where a BG is the off-resonant scattering length, B ∞ is the resonance position and∆ B is the resonance width. Feshbach resonance is a very useful tool that allowsfine and versatile tuning of the interaction strength in the cold atom systems overa range of several orders of magnitude from highly repulsive to highly attractiveregime. A phenomenon is well known in atomic and nuclear physics as Feshbach-Fano resonance [26, 27]. Beside the original reference by Tiesinga et al. [28] whichanalyzed the benefits of using the resonance in the experiments with ultracold atoms,the underlying theory has been discussed in the review paper by Chin et al. [6] andin several textbooks [19, 29]. Here we give only a brief explanation, based on a factthat two atoms can interact via an energetically open or closed channel. These twochannels are coupled and include Zeeman terms. By tuning the external magneticfield it is possible to make a bound state of a closed channel resonant with theincome energy in the open channel. In that case the scattering length diverges, asgiven in Eq. (1.23) for B = B ∞ . The properties of Feshbach resonances are exploitedin many experiments and one of the topics in this thesis will explore some relatedrecent experimental results.To summarize the subsection, the achievement of the BEC regime took a lot ofeffort and many techniques had to be developed for this purpose. However, once thishas been achieved, the cold atomic systems have become the cleanest experimentalsetting for studying macroscopic quantum phenomena. All parameters of thesesystems are highly controllable and tunable: the geometry, temperature, densityand even the type and the strength of interaction. For this reason the field of ultracoldatoms is still in the process of strong expansion and cross-collaboration withother fields, and further new important insights are expected.1.2.3 Interacting bosons at low temperaturesAfter discussing physical characteristics of cold bosonic atoms, we are ready to writedown the Hamiltonian of the system:∫Ĥ =d⃗r(− ˆψ † (⃗r) 22M ∇2 ˆψ(⃗r) + V (⃗r) ˆψ† (⃗r) ˆψ(⃗r) + g )2 ˆψ † (⃗r) ˆψ † (⃗r) ˆψ(⃗r) ˆψ(⃗r) . (1.24)Here ˆψ † (⃗r) and ˆψ(⃗r) are bosonic field operators in the second quantized form, andsatisfy commutation relations [ ˆψ(⃗r), ˆψ(⃗r ′ )] = 0, [ ˆψ † (⃗r), ˆψ † (⃗r ′ )] = 0, [ ˆψ † (⃗r), ˆψ(⃗r ′ )] =δ(⃗r−⃗r ′ ). On the right hand side of Eq. (1.24) we have the kinetic energy term, poten-13