Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

74 Rashba spin-orbit coupling in quantum dots
The potential which is induced by a bulk acoustic phonon with (three-dimensional)
wave vector q is given by [139]
V ep (q) = λ q e iq·r( )
b q + b † −q , (3.51)
with the phonon operators b q and b † q, and the coupling parameter λ q depending
on the mechanism of electron-phonon interaction, e.g. deformation potential or
piezo-electric coupling. §
In the following, we are interested in phonon-induced transitions from the
upper to the lower eigenstate within a JCM subspace, representing excited and
ground state of the qubit under consideration, Ψ + → Ψ − , with Ψ ± given by
Eq. (3.29). Since the electron-phonon interaction does not depend on the spin,
only transition matrix elements which are diagonal in spin space contribute,
〈Ψ + |V ep (q)|Ψ − 〉 =cosθ + cosθ − 〈n + ,n − ,↑ |V ep (q)|n + ,n − ,↑〉
+ sinθ + sinθ − 〈n + ,n − + 1,↓ |V ep (q)|n + ,n − + 1,↓〉. (3.52)
In appendix B, matrix elements of the electron-phonon interaction are calculated
in the Fock–Darwin basis {|n + ,n − ,σ〉}, leading to
〈Ψ + |V ep (q)|Ψ − (
〉 = λ q L n+ |α
+
q | 2) e − 2(|α 1 + q | 2 +|α − q | 2 )
[
(
× cosθ + cosθ − L n− |α
−
q | 2) + sinθ + sinθ − L n− +1(
|α
−
q | 2)] , (3.53)
with the Laguerre polynomials L n and complex phonon wave vectors
α ± q := ±˜l(q y ± iq x )/2. For the ease of computation we concentrate on the lowest
qubit, n + = n − = 0. Generalisation to higher qubits is straightforward. With
L 0 (x) = 1 and L 1 (x) = 1 − x we have
〈Ψ + 0 |V ep(q)|Ψ − 0 〉 = λ q e − 1 2(|α + q | 2 +|α − q | 2 ) [ cosθ + cosθ − + sinθ + sinθ −
− |α − q | 2 sinθ + sinθ −
]. (3.54)
The q-independent summand on the right vanishes because of the orthogonality
of Ψ ± 0 , cosθ + cosθ − + sinθ + sinθ − = 〈Ψ + 0 |Ψ− 0
〉 = 0, (3.55)
leading to
〈Ψ + 0 |V ep(q)|Ψ − 0 〉 = −λ (˜lq ‖ ) 2
q sinθ + sinθ − e − 4(˜lq 1 ‖ ) 2 , (3.56)
4
§ An introduction to electron-phonon interaction is given in chapter 5.