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 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)
and the initial state of our two-particle scattering experiment is thus AK (x , t) AK (x , t)0. We are again interested in determining the probabilities that two particles appear in an output branch or that one particle appears in each output branch, P(2, 0)"dk dk n( (k )n( (k ), P(1, 1)"dk dk n( (k )n( (k ), and P(0, 2)"dk dk n( (k )n( (k ). Let us again consider P(1, 1). Its evaluation proceeds in much the same way as in the case of pure scattering states. We re-write P(1, 1) in terms of the absolute square of an amplitude, P(1, 1)" dk dk 0bK (k )bK (k ) . (25) We then write the a( operators in the AK in terms of the output operators bK . Instead of Eq. (20), we obtain P(1, 1)"¹#R$2¹RJ , (26) where J" dk H(k) (k) exp(ik(x !x )) (27) Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 17 is the overlap integral of the two particles. For the case of complete overlap J"1 we obtain Eq. (20). For the case that we have no overlap we obtain the classical result P(1, 1)"¹#R which is independent of whether a boson or fermion is incident on the scatterer. In the general case, the overlap depends on the form of the wave packet. If two Gaussian wave packets of spatial width and central velocity v are timed to arrive at time and at the scatterer, the overlap integral is J"exp[!v( ! )/2] . (28) A signi"cant overlap occurs only during the time /v. For wave packets separated in time by more than this time interval the Pauli principle is not e!ective. Complete overlap occurs in two simple cases. We can assume that the two wave packets are identical and are timed to arrive exactly at the same instant at the scatterer. Another case, in which we have complete overlap, is in the basis of scattering states. In this case (k)"(k!k ) for a scattering state with wave vector k and consequently for the two particles with k and k we have J" .The"rst option of timed wave packets seems arti"cial for the thermal sources which we want to describe. Thus in the following we will work with scattering states. The probabilities P(2, 0), P(1, 1) and P(0, 2) for these scattering experiments are shown in Table 2. The considerations which lead to these results now should be extended to take the polarization of photons or spin of electrons into account. 2.2.4. Two-particle scattering: spin Consider the case of fermions and let us investigate a sequence of experiments in each of which we have an equal probability of having electrons with spin up or down in an incident state. Assuming that the scattering matrix is independent of the spin state of the electrons, the results discussed above describe the two cases when both spins point in the same direction (to be denoted as P(1,1) and P(1,1)). Thus what remains is to consider the case in which one incident particle has spin up and one incident particle has spin down. If the detection is also spin sensitive, the probability which we determine is P(1,1). But in such an experiment we can tell which of the two
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Physically, this corresponds to ele
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Our starting point is Eq. (51), whi
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introducing the transmission coe$ci
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investigated experimentally. A self
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and the non-equilibrium resistance
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the sample. These electrons are des
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with D ,¹ (2!¹ ). Performing t
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For completeness, we mention that i
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We note here, however, that there i
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The noise power (201) is strongly v
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Fig. 32. Experimental results of Ku
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where Ya.M. Blanter, M. Bu( ttiker
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where the quantity is expressed th
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6.3. Interaction ewects Ya.M. Blant
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one "rst goes from the non-interact
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non-thermalized electrons, the high
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Fig. 34. Geometry of the chaotic ca
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which describes strictly ballistic
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approximately e/C. Between the peak
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Boltzmann}Langevin approach (see Se
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thermal to shot noise. It is, howev
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Note added in proof Ya.M. Blanter,
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and Lee et al. [340] obtain in this
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shot noise in mesoscopic conductors - Low Temperature Laboratory

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