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Coherent and collective phenomena in quantum transport 8levitov/8513/lec1.pdf · Coherent and...
Transcript of Coherent and collective phenomena in quantum transport 8levitov/8513/lec1.pdf · Coherent and...
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Coherent and collective phenomena in quantum transport
8.513● Lectures T, TR 1-2:30 ● Homework handed out in class, collected two
weeks later – except holidays and HW#1● Term paper: choose topic by the end of October,
submit before or in the last lecture● Instructor: Leonid (aka Leo) Levitov, office 6C-345,
email: [email protected]● Course webpage: http://www.mit.edu/~levitov/8513
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Quantum transport; Lecture I
● Localization vs. diffusion● Anderson model, localization transition;● Quantum-coherent effects in diffusive
conductors: weak localization, negative magnetoresistance, Aharonov-Bohm effect (2 and 0)
● Transport as a scattering problem: conductance = transmission; Landauer formula
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from lectures by Boris Altshuler
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Quantum effects in diffusive transport
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Weak localizationfrom lectures by Ben Simons
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Quantum diffusion
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Suppression of conductance
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Long-range quantum coherence
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Scattering approach
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from lectures by Markus Buttiker
Landauer formula
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Electrochemical potential -eU
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Discuss again later
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Dissipation for elastic scattering?!
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Dissipation for elastic scattering?! Energy is lost to the reservoirs.
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General properties of S-matrix
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Time reversal: Hamiltonian is invariant when all momenta and B field are reversed
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General properties of the S-matrix
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UnitarityCurrent conservation
Example 1: single channel, a 2x2 unitary matrix
Example 2: many channels, an NxN unitary matrix,eigenvalues of the form Sj = exp(2ij)
j scattering phases
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Causality in EM and in QMCause-effect relation, linear response
Dielectric function analytic in the upper half-plane
b/c of exponentially decreasing
in-stateout-state
at at
S() could have singularities at the zeros of A(), but this is impossible b/c S is a property of the scattering potential, independent of the in-state
As a function of energy, S-matrix is analytic is the upper complex half-plane
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?