  Working within the Nonequilibrium Green's Function (NEGF) formalism, a
formula for the two-time current correlation function is derived for the case
of transport through a nanojunction in response to an arbitrary time-dependent
bias. The one-particle Hamiltonian and the Wide Band Limit Approximation (WBLA)
are assumed, enabling us to extract all necessary Green's functions and self
energies for the system, extending the analytic work presented previously
[Ridley et al. Phys. Rev. B (2015)]. We show that our new expression for the
two-time correlation function generalises the B\"uttiker theory of shot and
thermal noise on the current through a nanojunction to the time-dependent bias
case including the transient regime following the switch-on. Transient terms in
the correlation function arise from an initial state that does not assume (as
is usually done) that the system is initially uncoupled, i.e. our approach is
partition-free. We show that when the bias loses its time-dependence, the long
time-limit of the current correlation function depends on the time difference
only, as in this case an ideal steady state is reached. This enables derivation
of known results for the single frequency power spectrum and for the zero
frequency limit of this power spectrum. In addition, we present a technique
which for the first time facilitates fast calculations of the transient quantum
noise, valid for arbitrary temperature, time and voltage scales. We apply this
to the quantum dot and molecular wire systems for both DC and AC biases, and
find a novel signature of the traversal time for electrons crossing the wire in
the time-dependent cross-lead current correlations.

Mesoscale and Nanoscale Physics