shot noise in mesoscopic conductors - Low Temperature Laboratory

18 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
Table 2
Output probabilities for one particle incident in each input arm (from Ref. [8])
Probability Classical Bosons Fermions
P(2, 0) R¹ R¹(1#J) R¹(1!J)
P(1, 1) R#¹ R#¹!2R¹J R#¹#2R¹J
P(0, 2) R¹ R¹(1#J) R¹(1!J)
particles went which way at the scatterer and there is thus no interference. The outcome is classical:
P(1,1)"R for an initial state with a spin up in arm 1 and a spin down in arm 2. For the same
state we have P(1,1)"¹.
Now let us assume that there is no way of detecting the spin state of the outgoing particles. For
a given initial state we have P(1, 1)" P(1,1) where and are spin variables. If we consider
all possible incident states with equal probability, we "nd P(1, 1)"¹#R#¹RJ i.e. a result
with an interference contribution which is only half as large as given in Eq. (26). For further
discussion, see Appendix B.
The scattering experiments considered above assume that we can produce one or two particle
states either in a single mode or by exciting many modes. Below we will show that thermal sources,
the electron reservoirs which are of the main interest here, cannot be described in this way: In
the discussion given here the ground state is the vacuum (a state without carriers) whereas in an
electrical conductor the ground state is a many-electron state.
2.3. The scattering approach
The idea of the scattering approach (also referred to as Landauer approach) is to relate transport
properties of the system (in particular, current #uctuations) to its scattering properties, which are
assumed to be known from a quantum-mechanical calculation. In its traditional form the method
applies to non-interacting systems in the stationary regime. The system may be either at equilibrium
or in a non-equilibrium state; this information is introduced through the distribution functions of
the contacts of the sample. To be clear, we consider "rst a two-probe geometry and particles
obeying Fermi statistics (having in mind electrons in mesoscopic systems). Eventually, the generalization
to many probes and Bose statistics is given; extensions to interacting problems are
discussed at the end of this Section. In the derivation we essentially follow Ref. [9].
2.3.1. Two-terminal case; current operator
We consider a mesoscopic sample connected to two reservoirs (terminals, probes), to be referred
to as `lefta (L) and `righta (R). It is assumed that the reservoirs are so large that they can be
To avoid a possible misunderstanding, we stress that the long-range Coulomb interaction needs to be taken into
account when one tries to apply the scattering approach for the description of systems in time-dependent external "elds,
or "nite-frequency #uctuation spectra in stationary "elds. On the other hand, for the description of zero-frequency
#uctuation spectra in stationary xelds, a consistent theory can be given without including Coulomb e!ects, even though
the #uctuations themselves are, of course, time-dependent and random.