  Thermodynamic geometry provides a physically transparent framework to
describe thermodynamic processes in meso- and micro-scale systems that are
driven by slow variations of external control parameters. Focusing on periodic
driving for thermal machines, we extend this framework to ideal quantum gases.
To this end, we show that the standard approach of equilibrium physics, where a
grand-canonical ensemble is used to model a canonical one by fixing the mean
particle number through the chemical potential, can be extended to the slow
driving regime in a thermodynamically consistent way. As a key application of
our theory, we use a Lindblad-type quantum master equation to work out a
dynamical model of a quantum many-body engine using a harmonically trapped Bose
gas. Our results provide a geometric picture of the BEC-induced power
enhancement that was previously predicted for this type of engine on the basis
of an endoreversible model [New J. Phys. 24, 025001 (2022)]. Using an earlier
derived universal trade-off relation between power and efficiency as a
benchmark, we further show that the Bose-gas engine can deliver significantly
more power at given efficiency than an equally large collection of single-body
engines. Our work paves the way for a more general thermodynamic framework that
makes it possible to systematically assess the impact of quantum many-body
effects on the performance of thermal machines.
