In conclusion, a coherent system of one or more weakly coupled quantum dots with Coulomb interaction can be well described using Green's functions of a probabilistic nature. In a double dot, the modes of the individual dots are coupled together. The resulting level repulsion can lead to a complex form of the Ohmic conductance and the current. Regions of negative differential conductance are likely to occur. The Ohmic conductance through two dots with multiple levels is dominated by the matching energy levels.
The maximum conductance through a double dot 
is reached at low
temperatures in the
symmetric case of aligned levels 
, identical coupling to the reservoirs
and 
.
The peaks that occur in the current as the gate potential of one of the dots is varied while keeping the voltage bias across the dots fixed have a Lorentzian line shape with a width at least as large as the combined broadening of the individual dot levels.
Figure 1: Current and associated differential conductance through a dot
with a spin-split level for the various approximation schemes
(
, 
, 
,
, 
)
Figure 2: Ohmic conductance through a dot with a spin-split level for the
various approximation schemes (, ,
).
Figure 3: Occupation of a dot with a spin-split level for the
various approximation schemes (, ,
). The two sets of curves refer to the occupation
of the 
and 
states.
Figure 4: Ohmic conductance through two dots each with a level for both spins,
using various coupling constants 
(
, , 
,
, 
, 
).
In the occupation plots the solid lines correspond to the states
of spin , the dotted lines to the states of spin .
Figure 5: Current through two dots each with a level for both spins,
using various coupling constants
(
, , , ,
, , ).
In the occupation plots the solid lines correspond to the states
of spin , the dotted lines to the states of spin .
Figure 6: Ohmic conductance through two dots with multiple levels
in the limit of small coupling 
(
, 
, 
, 
,
). The empty and filled circles indicate the
positions at which the average occupation increases by one
for dot 1 and 2 respectively.