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# Introduction to Full Counting Statistics

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

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### Introduction to Full Counting Statistics

1. 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. 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 Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 2 / 25
3. 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!!! Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 3 / 25
4. 4. One ﬁnds for the average transferred charge < Q > in a time period t0 at zero temperature < Q >= e2VTt0 . Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 4 / 25
5. 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. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 5 / 25
6. 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) Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 6 / 25
7. 7. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 7 / 25
8. 8. Various cumulant generating functions and statistics Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 8 / 25
9. 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. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 9 / 25
10. 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. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 10 / 25
11. 11. Lesovik-Levitov formula: noise from scattering matrix Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 11 / 25
12. 12. Example 1: Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 12 / 25
13. 13. Example 2: Limit of low transmission ... Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 13 / 25
14. 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). Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 14 / 25
15. 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 , Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 15 / 25
16. 16. Concept of density matrix in Quantum Mechanics Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 16 / 25
17. 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 , Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 17 / 25
18. 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). Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 18 / 25
19. 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). Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 19 / 25
20. 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. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 20 / 25
21. 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. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 21 / 25
22. 22. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 22 / 25
23. 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. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 23 / 25
24. 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. Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 24 / 25
25. 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) Krzysztof Pomorski (NU) Introduction to FCS June 1, 2016 25 / 25