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