Consequences of Time Reversal for Scattering Ma Sample Clauses

Consequences of Time Reversal for Scattering Ma trices The presence of time reversal has implications also for the symmetry of the scattering matrix. The exact way this symmetry is reflected in the scattering matrix depends on the basis chosen. We will here discuss a couple of cases. Symmetry of S Σ We consider a conventional two terminal scattering setup with NL(R) modes in the left (right) lead. We will label all incoming states on the left (right) with |n⟩ (|m⟩). The outgoing modes will then be |Tn⟩ (|Tm⟩). A general scattering state |ϕ⟩ will then have the following form in the left lead NL n n |ϕ⟩ = (cin,L |n⟩ + cout,L |Tn⟩), (1.66) n=1 and similar for the right lead (with L → R and n → m). The scattering matrix connects the vectors of coefficients cin to cout cout,R S xxx,X t r' xxx,X cout,L! = cin,L! = r t' ! cin,L! (1.67) If we have time reversal symmetry then Σ NL n n T |ϕ⟩ = [(cin,L)∗ |Tn⟩ − (cout,L)∗ |n⟩), (1.68) n=1 is also a scattering state with the same energy. That means that ! = (cin,L)∗ (cin,R)∗ S −(cout,L)∗ ! . −(cout,R)∗ (1.69) ! ! Multiplying from the left with S†, using unitarity of S and complex con- jugating cout,L = cout,R T cin,L −S xxx,X .
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