Analog kara delikler ve bose einstein yoğunlaşmasında tek bir girdaptan süper-ışıma ile enerji çıkarımı

Analog models of gravity, in this case, the acoustic black holes, are based on the mathematical correspondence between the propagation of sound waves in a certain media and the propagation of fields in curved spacetime. Since the direct observation of certain phenomena arising from curved space-time, such as the Hawking radiation and the superradiance, is highly impractical, analog models provide an accessible and controllable way to investigate them in the laboratory. Bose-Einstein condensate provides a convenient system for the implementation of this analogy. Indeed, the propagation of acoustic perturbations in the velocity potential of a Bose-Einstein condensate (BEC) behaves like minimally coupled massless scalar fields in a curved (2+1) dimensional Lorentzian space-time, where their propagation is governed by the Klein-Gordon wave equation. For linearized perturbations, this geometric picture can still apply in the presence of a single vortex state of the BEC. Based on this setting, this thesis work investigates the amplified scattering of axisymmetric perturbations from a vortex, as a manifestation of the acoustic superradiance. We first employ the widely used constant background density under the hydrodynamic approximation and conduct a comparative analysis of the time-domain and asymptotic frequency-domain analysis of acoustic superradiance as a function of the rotational speed of the vortex and the frequency of the impinging fluctuations. This shows that the solutions in both domains are in good agreement near the characteristic rotational speed of the single quantized vortex in its lowest energy state. However, for larger rotational speeds time-domain calculations, especially near the event horizon becomes unreliable, shown by the constraint equations introduced by the excision technique. While the simulations in frequency domain do not suffer from the numerical instabilities due to the coordinate transformations, allowing to increase the rotational velocity of the fluid. In addition, the formulation predicts an upper bound of the reflection coefficient against the rotational velocity of the vortex. The second part of the thesis examines the validity of the constant background density approximation by calculating a self-consistent density profile through the Gross-Pitaevskii equation. The resulting radial density profile around the vortex implies a radially varying speed of sound, which modifies the entire propagation dynamics as well as the loci of the event horizon and the ergosphere. The main conclusions of this part are that the self-consistent density profile remedies the overshoot of the superradiance dynamics temporally and that the spectral profile of the superradiance differs significantly in the vortex-scaled low frequency regime between the constant density and self-consistent density formulations. In this thesis, the superradiance phenomena for a single vortex in Bose Einstein condensate under the hydrodynamic approximation is analyzed, showing that the self-consistent density profile of the condensate improves on the constant background density approximation, particularly at the low frequency regime of acoustic perturbations. And the qualitative corrections in the transient behavior of the scattered wave's energy within the ergosphere justifies the radially varying density. The analysis could be extended for external confining potentials which may be more attainable in terms of the experimental aspects of the study. In reality, setting up a BEC vortex with a drain in (2+1)-dimension may be a difficult task to achieve, acoustic black holes in BEC are already observed for a 1D-flow within a steplike potential combined with a harmonic potential.