Intellectual Merit: The central theme of this project is the memory of the initial state in quantum systems in between order and chaos. While this question is immensely complicated in classical systems, it turns out to be possible to identify a limited class of quantum systems for which relatively simple quantitative predictions can be made. This project is devoted to worked examples. The first of them is two atoms in a guide. The second consists of five two-dimensional strongly interacting atoms on a lattice. The third is a two-component mixture of atoms in two coupled traps. In all three cases, the group will be testing a simple formula that relates the values of observables after relaxation from an initial excited state to their initial values.
Broader impact: Interdisciplinary by nature, the project lies on the interface between Atomic,Many-Body, and Mathematical Physics, Nonlinear Science, and Complex Systems. It will provide analytical and numerical experience for undergraduate and graduate students at the University of Massachusetts Boston. The PI will also work on an undergraduate-oriented book [Quantum Mechanics by order of magnitude, World Scientific, c. 2010] devoted to qualitative methods, a set of skills used by many and taught nowhere.
At the first glance, a definite direction of the arrow of time contradicts the equivalence between the past and the future present in the underlying equations. The butterfly effect of classical mechanics---the exponential sensitivity of the outcomes to the tiny details of the past---that partially explains the past-future asymmetry, is absent in the quantum world. The purpose of this project is to address this inconsistency. What we found was that the classical instabilities are still implicitly present in the quantum systems, but hidden due to the energy-time quantum uncertainty. The energy of a quantum state is well defined, but the phase of the motion is completely uncertain. What we see is effectively a foggy superposition of many stages of system's evolution, where the fine details of the motion, e.g. exponential instabilities, are hidden from view. Most of the familiar concepts can be restored using a proper classical quantum dictionary, and our results provide this dictionary. Classical deviations of a long time behavior from thermal predictions gets translated as state-to-state variations of the quantum expectation values. Classical temporal fluctuations become quantum fluctuations. These findings provide a unified view on the nature of irreversibility. We study these concepts using several examples. They include: two atoms in a guide, quantum billiards, atoms in an optical lattice, Bose condensates in a two-dimensional magnetic trap. Systems with nontrivial symmetries---e.g. systems that are invariant under a change in units of measurement---traditionally play a role of a test bench in studies on irreversibility; our project followed this pattern heavily.
