  Due to strong interlayer correlations, the bilayer quantum Hall system is a
single coherent system as a whole rather than a weakly-coupled set of two
independent systems, which makes conventional tunnelling theories inapplicable.
In this paper, we develop a theory of interlayer tunnelling in coherent exciton
condensates of bilayer quantum Hall systems at total filling factor $\nu_T=1$.
One of the most important consequences of our theory is that the zero-bias
interlayer tunnelling conductance peak is strongly enhanced, but fundamentally
finite even at zero temperature. We explicitly compute the height of the
conductance peak as a function of interlayer distance, which is compared with
experiment. It is emphasized that the interlayer distance dependence of the
conductance peak is one of the key properties distinguishing between the
spontaneous coherence due to many-body effects of the Coulomb interaction and
the induced coherence due to the single-particle tunnelling gap. It is also
emphasized that, though the strongly enhanced tunnelling conductance originates
from the interlayer phase coherence, it is not the usual Josephson effect. We
propose an experimental setup for the true Josephson effect in couterflowing
current measurements for a coupled set of two bilayer quantum Hall systems,
which is a more precise analogy with the real Josephson effect in
superconductivity.
