Classical and quantum parametric excitations of Bose-Einstein condensates - Finke, Andreas (2017)
In this thesis, we study the parametric excitations of a homogeneous Bose-Einstein condensate(BEC) at finite temperature obtained from modulating the interaction strength. In Bogoliubov theory, a result connecting the real part of the diagonal Bogoliubov coefficient and instability, i.e. exponential growth of excitations, is obtained for general periodic modulations. Consequently, an analytic solution for an arbitrary square wave modulation is derived that can be studied to understand similar drivings like the sinusoidal case. This adds to previous works studying only the case for small driving amplitudes. In the recent experimental and theoretical literature well-established nonclassicality criteria from the field of quantum optics have been directly applied to the case of excitations in matter-waves like those studied here. Among these are violations of Cauchy-Schwarz inequalities, Glauber-Sudarshan P-nonclassicality, sub-Poissonian number difference squeezing (also known as the two-mode variance) and the criterion of nonseparability. We review the strong connection of these criteria and their meaning in quantum optics, and point out differences in the interpretation between light and matter waves. By proposing a classical theory of a thermal BEC that gives the same predictions as the quantum theory in the high temperature limit, and using it as a reference, we can isolate real quantum effects like spontaneous emission caused by quantum noise. We conclude that to date in experiments relevant for this scenario nonclassical effects have not conclusively been observed and conjecture that additional, noncommuting, observables have to be measured to this end. Moreover this has important implications for proposed analog gravity models where the observation of nonclassical effects is a major goal. In the second part of the thesis, we present simulational methods with focus on the Truncated Wigner Approximation (TWA). We take a path integral approach inspired by [Polkovnikov, 2010] and derive a set of classical equations from the semi-classical limit that is more general than TWA. We add some proofs to the literature and identify problems in this approach. We also present a highly optimized software package as an extension of previous work [Jain, 2007]. It can carry out the relevant simulations and we use it to confirm the theoretical predictions of the first part.