shot noise in mesoscopic conductors - Low Temperature Laboratory

14 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
by a unitary transformation which is just the scattering matrix,
bK
bK "s a( . (12)
a(
Similarly, the creation operators a( and bK are related by the adjoint of the scattering matrix s.
Note that the mirror generates quantum mechanical superpositions of the input states. The
coe$cients of these superpositions are determined by the elements of the scattering matrix.
For bosons, the a( obey the commutation relations
[a( , a( ]" . (13)
Since the scattering matrix is unitary, the bK obey the same commutation relations. In contrast, for
fermions, the a( and bK obey anti-commutation relations
a( , a( " . (14)
The di!erent commutation relations for fermions and bosons assure that multi-particle states
re#ect the underlying symmetry of the wave function.
The occupation numbers of the incident and transmitted states are found as n( "a( a( and
n( "bK bK ,
n(
n( " R ¹
¹ R n(
, (15)
n(
where we introduced transmission and re#ection probabilities, ¹"t and R"r.
2.2.1. Single, independent particle scattering
Before treating two-particle states it is useful to consider brie#y a series of scattering experiments
in each of which only one particle is incident on the mirror. Let us suppose that a particle is incident
in arm 1. Since we know that in each scattering experiment there is one incident particle, the
average occupation number in the incident arm is thus n "1. The #uctuations away from the
average occupation number n "n !n vanish identically (not only on the average). In
particular, we have (n )"0. Particles are transmitted into arm 4 with probability ¹, and thus
the mean occupation number in the transmitted beam is n "¹. Similarly, particles are re#ected
into arm 3 with probability R, and the average occupation number in our series of experiments is
thus n "R. Consider now the correlation n n between the occupation numbers in arms
3 and 4. Since each particle is either re#ected or transmitted, it means that in this product one of the
occupation numbers is always 1 and one is zero in each experiment. Consequently, the correlation
of the occupation numbers vanishes, n n "0. Using this result, we obtain for the #uctuations
of the occupation numbers n "n !n in the re#ected beam and n "n !n in the
transmitted beam,
(n )"(n )"!n n "¹R . (16)
The #uctuations in the occupation numbers in the transmitted and re#ected beams and their
correlation are a consequence of the fact that a carrier can "nally only be either re#ected or
transmitted. These #uctuations are known as partition noise. The partition noise vanishes for