  Tunneling of atoms between two trapped Bose-condensed gases at finite
temperatures is explored using a many-body linear response tunneling formalism
similar to that used in superconductors. To lowest order, the tunneling
currents can be expressed quite generally in terms of the single-particle
Green's functions of the isolated Bose gases. A coherent first-order tunneling
Josephson current between two atomic Bose-condensates is found, in addition to
coherent and dissipative contributions from second-order
condensate-noncondensate and noncondensate-noncondensate tunneling. Our work is
a generalization of Meier and Zwerger, who recently treated tunneling between
uniform atomic Bose gases. We apply our formalism to the analysis of an
out-coupling experiment induced by light wave fields, using a simple
Bogoliubov-Popov quasiparticle approximation for the trapped Bose gas. For
tunneling into the vacuum, we recover the results of Japha, Choi, Burnett and
Band, who recently pointed out the usefulness of studying the spectrum of
out-coupled atoms. In particular, we show that the small tunneling current of
noncondensate atoms from a trapped Bose gas has a broad spectrum of energies,
with a characteristic structure associated with the Bogoliubov quasiparticle
u^2 and v^2 amplitudes.
