  Although coherent manipulation of electronic states can be achieved in
quantum dot (QD) devices by harnessing nanofabrication tools, it is often hard
to fathom the extent to which these nanoelectronic devices can behave quantum
mechanically. Witnessing their nonclassical nature would thus remain of
paramount importance in the emerging world of quantum technologies, since the
coherent dynamics of electronic states plays there a crucial role. Against this
backdrop, we resort to the general framework of Leggett-Garg inequalities (LGI)
as it allows for distinguishing the classical and quantum transport through
nanostructures by way of various two-time correlation functions. Using the
local charge detection at two different time, we investigate here theoretically
whether any quantum violation of the original LGI exists with varying device
configurations and parameters under both Markovian and non-Markovian dynamics.
Two-time correlators within LGI are derived in terms of the non-equilibrium
Green's functions (NEGFs) by exactly solving the quantum Langevin equations.
The present study of non-Markovian dynamics of quantum systems interacting with
reservoirs is significant for understanding the relaxation phenomenon in the
ultrafast transient regime to especially mimic what happens to high-speed
quantum devices. We can potentially capture the effect of finite reservoir
correlation time by accounting for level broadening at the electrodes along
with non-Markovian memory effects. Furthermore, the large bias restriction is
no longer imposed in our calculations so that we can safely consider a finite
bias between the electronic reservoirs. Our approach is likely to open up new
possibilities of witnessing the quantumness for other quantum many-body systems
as well that are driven out of the equilibrium.
