  Randomly moving active particles can be herded into directed motion by
asymmetric geometric structures. Although such a rectification process has been
extensively studied due to its fundamental, biological, and technological
relevance, a comprehensive understanding of active matter rectification based
on single particle dynamics remains elusive. Here, by combining experiments,
simulations, and theory, we study the directed transport and energetics of
swimming bacteria navigating through funnel-shaped obstacles -- a paradigmatic
model of rectification of living active matter. We develop a microscopic
parameter-free model for bacterial rectification, which quantitatively explains
experimental and numerical observations and predicts the optimal geometry for
the maximum rectification efficiency. Furthermore, we quantify the degree of
time irreversibility and measure the extractable work associated with bacterial
rectification. Our study provides quantitative solutions to long-standing
questions on bacterial rectification and establishes a generic relationship
between time irreversibility, particle fluxes, and extractable work, shedding
light on the energetics of non-equilibrium rectification processes in living
systems.
