Application of spin echo technique to Bose-Einstein condensates: time evolution of the spins by simulating and computing the Gross-Pitaevskii equation

The principles of Nuclear Magnetic Resonance (NMR) can be understood thanks to the magnetic properties of the nuclei derived from the intrinsic spin and the orbital angular momentum, enabling a precise control on the dynamics of the nuclear spin system by means of radio-frequency (RF) pulses.
NMR has revolutionised the field of medical imaging, being one of the few techniques where no ionizing radiation is employed, besides ultrasound.
However, not exclusively medicine benefits from the outstanding applications of this physical phenomenon. Structural and chemical information of diverse atomic species can be likewise investigated through NMR. Furthermore, in the recent cutting-edge research in quantum computing, magnetic resonance is an alternative technique to create the first quantum computer. Moreover, the application of NMR to unique states of matter at very low temperatures explores the behaviour of the spins in different elements, revealing very interesting physical properties. In this thesis, the application of an important and precise pulse sequence, spin echo, to a specific type of Bose-Einstein condensate, a spin-1 spinor, is studied. The interest of this work focusses on analysing the spin evolution when only the short range interaction constants are taking into account. The simulations are computed applying numerical algorithms, by making use of GPELab, a MAT LAB toolbox developed to model non linear Schrodinger equations, the so called Gross-Pitaevskii equations. The techniques are explored more in detail, analysing the type of algorithm to be applied to compute the dynamics of the system. These results can be implemented in numerous experimental possibilities, and open up a new alternative to apply NMR in systems at very cold temperatures.