Brief introduction to Full Counting Statistics is given. The procedure of noise determination for nanostructures is given.

1. Introduction to Full Counting Statistics
Krzysztof Pomorski
Nagoya University
E-mail: kdvpomorski@gmail.com
June 1, 2016
Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 1 / 25

2. Overview
1 Motivation to use FCS
2 Definition of counting field
3 Landauer formula
4 Lesovik-Levitov formula
5 Quantum state of detector under measurement
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3. Motivation to use Full Counting Statistics (FCS)
There is significant difference between laws at macroscale and microscale.
In case of large current intensity and voltages above electron binding
energy the experimentalist observes the continuous electric current flow.
However if current flow has small rate in the structures of small
dimensions the quantum mechanic effects become visible. Then it will be
shown that quantum noise is very important source of knowledge about
the system. Therefore we can state Noise is information!!!(Landauer).
Figure: Dynamics of flow has stochastic character for small systems in short time
scales!!!
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4. One finds for the average transferred charge < Q > in a time period t0 at
zero temperature < Q >= e2VTt0
.
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5. Definition of counting field χ
The fundamental quantity of interest in quantum transport between A1
and A2 point is the probability distribution P(N)t0 in given observational
time t0.
exp(F(χ)) =< exp(iχN)P(N)t0 >
It is discrete Fourier transform of probability for integer number of
electrons. Here F(χ) is cumulant generating function and χ is counting
field and < . > is statistical average. In equivalent way we have
F(χ) = log[< exp(iχN)P(N)t0 >]
We can compute the statistical quantities by taking derivatives with
respect to counting field and going in limit of χ to zero.
In case of multiterminal system we represent N and χ as vectors.
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6. Computing various statistical moments
Figure: The distribution of the number of transmitted electrons N. The mean C1,
the variance C2, the skewness C3 and the kurtosis C4 characterize the peak
position, the width, the asymmetry and the sharpness of the distribution,
respectively
Ck = (i)
dk
dχk
F(χ)|χ→0 (1)
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7. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 7 / 25

8. Various cumulant generating functions and statistics
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9. Microscopic picture and scattering/transferring matrix
Considerations are easy to be conducted for the case of planar waves
ψq(x) = Aqexp(ikqx), where index q stands for left, central or right region.
There is continuity of wavefunction and its derivative ... in all regions.
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10. Landauer formula, G = 2e
2 n tn, T=0K
Relation between microscopic (transmission coefficients tn for different
channels) and macroscopic quantities (conductance G). Lesovik-Levitov
formula goes beyond first cumulant.
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11. Lesovik-Levitov formula: noise from scattering matrix
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12. Example 1:
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13. Example 2: Limit of low transmission ...
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14. Short introduction to Quantum Mechanics
In QM we have single particle Hamiltonian H = H0 + V and its
eigenstates |ψ(x) > (bra state that in particular case is |ψ >= ψ(x)) that
fulfill the relation H|ψ(x) >= E|ψ(x) > and E is energy and eigenvalue of
the system.
In case of Josephson junction we have
H = HL + HR + HT , (2)
where HL, HR are Hamiltonians of left and right physical system and HT is
tunneling Hamiltonian.
We have the quantum state to be of the form H|ψ >= E|ψ >. In our case
we assume
|ψ >=
ψL
ψR
, H =
HL ET
ET HR
,
where ψL = |ψL|exp(iφL), ψR = |ψR|exp(iφR).
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15. Concept of projector in QM
A = A1|1 >< 1| + A2|2 >< 2|
spectral decomposition of two eigenvalue operator A.
If
|ψ >=
ψ1
ψ2
,
then
< ψ| = ψ†
1 ψ†
2 ,
In such case we have
|ψ >< ψ| = ψ†
1 ψ†
2
ψ1
ψ2
=
ψ†
1ψ1 ψ†
2ψ1
ψ†
1ψ2 ψ†
2ψ2
,
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16. Concept of density matrix in Quantum Mechanics
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17. Two interacting quantum systems A and B
In non-interacting case we have (tensor product of two Hilbert spaces)
|ψ >= |ψA > ×|ψB > (3)
with
ˆH = ˆHA × ˆI + ˆI × ˆHB (4)
Thus matrix H has diagonal blocks.
In case of nonzero interaction between A and B matrix ˆH has non-zero
non-diagonal terms so
ˆH = ˆHA × ˆI + ˆI × ˆHB + HAB (5)
.
We can also introduce tensor product of two density matrices
ˆρA × ˆρB =
ρA,11 ρA,12
ρA,21 ρA,22
×
ρB,11 ρB,12
ρB,21 ρB,22
=
ρA,11 ˆρB ρA,12 ˆρB
ρA,21 ˆρB ρA,22 ˆρB
,
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18. Single spin as galvanometer
In case of two quantum systems B (spin as galvanometer) and A (current
flow) with no interaction the density matrix can be factorized as
ˆρ(t = 0) = ˆρe(0) × ˆρs(0) (6)
.
Then we can turn on interaction between the current flowing in conductor
and external spin (our galvanometer).
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19. Quantum mechanical definition of CMG
In order to provide a quantum mechanical definition of the CGF of
electrons we will follow the approach proposed by Levitov and Lesovik.
The key step is to include the measurement device in the description. As a
gedanken scheme a spin-1/2 system is used as a galvanometer for the
charge detection. This spin is placed near the conductor and coupled
magnetically to the electric current. Let the electron system be described
by the Hamiltonian H(q, p). We further assume that the spin-1/2
generates a vector potential A(r) of the form A(r) = (1/2)χ f (r).
Here the function f(r) smoothly interpolates between 0 and 1 in the
vicinity of the cross-section at which the current is measured, and χ is an
arbitrary coupling constant so far. It will be shown below that it plays a
role of counting field. If one further restricts the coupling of the current to
the z-component of the spin then the total Hamiltonian of the system
takes the form Hσ = H(q, p − Aσz).
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20. In semiclassical approximation when the variation of f (r) on the scale of
Fermi wave length λf is weak it is possible to linearize electron spectrum
at energies near to Fermi surface. Thus one arrives to the Hamiltonian
Hσ = H(q, p) + Hint (7)
where
Hint = −
1
e
σz
+∞
−∞
d3
rA(r)j(r) = −
χ
2e
σzIS (8)
Here j(r) is the current density and IS = d3rj(r) f (r) the total current
across a surface S. On the quasi-classical level last equation shows that a
spin linearly coupled to the measured current Is(t) will precess with the
rate proportional to the current. If the coupling is turned on at time t=0
and switched off at t0 the precession angle Θ = χ
t0
0 Is(t)dt/e of the spin
around the z-axis is proportional to the transferred charge through
conductor. In this way the spin 1/2 turns into analog galvanometer.
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21. Definition of trace and partial trace
Trace of matrix ˆA is defined as Tr( ˆA) = i Aii = A11 + A22 + ... + Ann
and and partial trace of tensor of matrices is defined as
TrA( ˆA × ˆB) = Tr( ˆA) × ˆB.
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22. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 22 / 25

23. Microscopic form of CGF F(χ)
One now can identify Z(χ) with the CGF
Z(χ) = exp(−F(χ)) (9)
and the spin density matrix ρs(t0) can be represented as a superposition of
the form
ˆρs(t0) =
N=+∞
N=−∞
Pt0 (N)RΘ=Nχ(ˆφ) (10)
where Pt0 (N) has meaning of the probability to observe the precession at
angle Θ = Nχ. For a classical spin a precession angle Θ = χ corresponds
to a current pulse carrying an elementary charge e =
t0
0 Is(t)dt. Using
the correspondence principle we conclude that the quantity Pt0(N) can be
interpreted as the probability of transfer the multiple charge Ne.
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24. Connection of CGF with Keldysh contour.
One can represent Z(χ) in the form of Keldysh partition function
Z(χ) = exp(−F(χ)) =< TK exp(−i
contourK
dtHχ(t)) > (11)
with counting field of different sign on upper and lower branch of contour
Hint(t) = 1
2e χ(t)Is.
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25. Literature
1. Full Counting Statistics of Interacting Electrons,D.Bagrets, Y.Utsumi, D.Golubev, G.Schon, Arxiv:0605263v1 (2006)
2. Lectures of Maciejko on non-equilibrium Green functions-www:stanford:edu= maciejko=nonequilibrium:pdf (2006)
3. Introduction to Keldysh techniques, A.Kamenev. (2006)
4. Quantum statistical mechanics, Kadanoff,Baym (1962). (2006)
5. Full counting statistics in electrical circuits, M.Kindermann, Nazarov, Arxiv:0303590v1 (2003). (2006)
6. Noise and the full counting statistics of a Coulomb blockaded by quantum dot, Thomas Phoenix (2010), PhD (2006) thesis,
University of Birmingham (2006)
7. Full counting statistics of electron tunneling between two superconductors, W.Belzig, PRL 87 (2001) (2006)
8. Belzig lectures on FCS (2006) (2006)
9. Quantum Noise and Quantum Optics in the Solid State, Workshop materials from Bad Honnef (2007) (2006)
10. Frequency-dependent shot noise in nanostructures, presentation by R.Aguado (2006)
11. Full counting statistics-An elementary derivation of Levitovs formula, I. Klich (2006)
12. Dc transport in superconducting point contacts: A full-counting-statistics view, J. C. Cuevas, PRB 70 (2004) (2006)
13. Full counting statistics in electric circuits, M.Kindermann, Arxiv:0303590v1 (2003)
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