  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.
