Title: Gain engineering and topological atom laser in synthetic dimensions

Abstract: In the recent rapid progress of quantum technology, controlling quantum states has become an important subject of study. Of particular interest is the control of open quantum systems, where the system of interest couples to the environment in an essential way. One formalism to describe open systems is the non-Hermitian quantum mechanics. Photonics systems have been a major platform to study non-Hermitian quantum mechanics due to its flexibility in engineering gain and loss. Ultracold atomic gases have also used to study non-Hermitian quantum mechanics. However, unlike in photonics, gain is not easily controllable in ultracold atomic gases, and exploration of non-Hermitian physics has been limited to control of losses. In this paper, we report engineering of effective gain through evaporative cooling of judiciously chosen initial thermal atoms. We observe resulting formation of Bose-Einstein condensation (BEC) in excited eigenstates of a synthetic lattice. We realize formation of BEC in a topological edge state of the Su-Schrieffer-Heeger lattice in the synthetic hyperfine lattice, which can be regarded as atomic laser oscillations at a topological edge mode, i.e. topological atom laser. Gain-loss engineering in ultracold atoms opens a novel prospect to explore open many-body quantum systems.