(9121) Scattering phase shift for quantum dot with one lead

Consider a single level quantum dot that is attached to a lead. The energy and the decay rate of the level are \( E_0 \) and \( \Gamma_0 \). An electron that is emitted from the dot with energy \( E = E_0 \) will have velocity \( v_E \) in the lead. The length of the lead is \( L \).

(1) Write the resolvent \( G(E) \) of the quantum dot.

(2) Assuming that the matrix element for dot-lead transition is \( V \), write a relation between \( V \) and \( \Gamma_0 \).

(3) Find the \( S \) matrix of the dot using the T-matrix formalism.

(4) What is the scattering phase shift \( \delta_0 \).

(5) What is the scattering delay time \( \tau_0 \).

Guidance: In item 3 use the relation between the \( S \) matrix and the \( T \) matrix. This assumes energy-shell normalization of the free waves. For a finite lead-segment of length \( L \) the free waves are sine functions. Determine the normalization that is implied by the following conventions:
\( \langle n|n_0\rangle = \delta_{n,n_0} \)
\( \langle k|k_0 \rangle = 2\pi\delta(k-k_0) \)
\( \langle E |E' \rangle = 2\pi \delta(E-E') \)
Formally \( \delta_{n,n_0} \mapsto \delta(n-n_0) \) and use the relation between \( dn \) and \( dk \) and \( dE \) in order to relate the delta functions of Dirac.