shot noise in mesoscopic conductors - Low Temperature Laboratory

16 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
for bosons, whereas for fermions the two probabilities vanish P(2, 0)"P(0, 2)"0. For classical
partition of carriers the probability to "nd the two particles in the same detector is R¹, which is
just one-half of the probability for bosons.
The average occupation numbers are n "n "1, since we have now two particles in
branch 3 with probability P(2, 0) and one particle with probability P(1, 1). Consequently, the
correlations of the #uctuations in the occupation numbers n( "n( !n are given by
n( n( "!4R¹ , (22)
for bosons and by n( n( "0 for fermions. For bosons the correlation is negative due to the
enhanced probability that both photons end up in the same output branch. For fermions there are
no #uctuations in the occupation number and the correlation function thus vanishes.
2.2.3. Two-particle scattering: Wave packet overlap
The discussion given above implicitly assumes that both `particlesa or `wavesa arrive simultaneously
at the mirror and `seea each other. Clearly, if the two particles arrive at the mirror with
a time-delay which is large enough such that there is no overlap, the outcome of the experiments
described above is entirely di!erent. If we have only a sequence of individual photons or electrons
arriving at the mirror we have for the expectation values of the occupation numbers
n "n "R#¹"1, and the correlation of the occupation number n n "0 vanishes.
Consequently, the correlation of the #uctuations of the occupation number is n n "!1.
Without any special sources at hand it is impossible to time the carriers such that they arrive
simultaneously at the mirror, and we should consider all possibilities.
To do this, we must consider the states at the input in more detail. Let us assume that a state
in input arm i can be written with the help of plane waves (k, x )"exp(!ikx ) with x the
coordinate along arm i normalized such that it grows as we move away from the arm toward the
source. Similarly, let y be the coordinates along the output arms such that y vanishes at the mirror
and grows as we move away from the splitter. A plane wave (k, x )"exp(!ikx ) incident from
arm 1 thus leads to a re#ected wave in output arm 3 given by (k, y )"r exp(!iky ) and to
a transmitted wave (k, y )"t exp(!iky ) in output arm 4. We call such a state a `scattering
statea. It can be regarded as the limit of a wave packet with a spatial width that tends towards
in"nity and an energy width that tends to zero. To build up a particle that is localized in space at
a given time we now invoke superpositions of such scattering states. Thus, let the incident particle
in arm i be described by (x , t)"dk (k) exp(!ikx ) exp(!iE(k)t/), where (k) is a function
such that
dk (k)"1 , (23)
and E(k) is the energy of the carriers as a function of the wave vector k. In second quantization the
incident states are written with the help of the operators
AK (x , t)" dk (k ) (k , x )a( (k ) exp(!iE(k )t/) , (24)