Eigenvector continuation and strongly interacting cold atom systems

Abstract

New experimental techniques have made it possible to trap and cool atoms to extremely low temperatures, which have initiated a new research field—the study of ultracold atoms. In this thesis, we consider one-dimensional many-body quantum systems of ultracold atoms with a zero-range interaction. The atoms are trapped in a tightly confining harmonic oscillator potential and the resulting Hamiltonian depends linearly on the tunable interaction strength. The aim of this study is to describe the evolution of the energy spectrum as a function of the interaction strength by exploiting eigenvector continuation—a recently developed method that has been proposed as a very efficient and accurate emulator for solving quantum many-body problems. Emulation is achieved by utilizing a limited set of calculated eigenvectors as training data to construct subspace-projected Hamiltonian matrices that can be used to approximately extrapolate the smooth eigenvector trajectory in the full Hilbert space. In this study, we show that the method of eigenvector continuation is very efficient and provides accurate predictions of the energy spectrum for both attractive and repulsive interaction strengths with just a few training vectors. The method performs extremely well for the two-particle system, where the training vectors correspond to wave functions in a relative coordinate basis, but also for the many-particle system with a basis constructed from singleparticle states. In addition, we perform an extensive analysis regarding the performance of the emulator and its dependence on the choice of training data. As a result, we propose an algorithm for finding the optimal set of training data and measure the emulator performance using K-fold cross validation.