  We measure the noise correlations of two-dimensional (2D) Bose gases after
free expansion, which allows us to characterize the in-situ phase coherence
across the Berezinskii-Kosterlitz-Thouless (BKT) transition. The noise
correlation function features a characteristic spatial oscillatory behavior in
the superfluid phase, which gives direct access to the superfluid exponent.
This oscillatory behavior vanishes above the BKT critical point, as we
demonstrate for both single-layer and decoupled bilayer 2D Bose gases. Our work
establishes noise interferometry as a general tool to probe and identify
many-body states of bilayer quantum gases.
