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

Appendix B
Fock–Darwin representation of
electron-phonon interaction
For representing the electron-phonon interaction Eq. (3.51),
V ep (q) = λ q e iq·r( )
b q + b † −q ,
in the Fock–Darwin basis of the quantum dot, we rewrite the coordinate operators
in terms of creation and annihilation operators of the ω ± fields [see Eq. (3.14)],
x = ˜l (
)
a − + a † − + a + + a † + , y = ˜l (
)
a − − a † − − a + + a † + . (B.1)
2
2i
It is convenient to introduce complex phonon wave vectors
and define displacement operators
α + q = ˜l
2 (q y + iq x ), α − q = − ˜l
2 (q y − iq x ), (B.2)
D ± (α) = e αa† ± −α∗ a ±
,
(B.3)
leading to
( )
V ep (q) = λ q e iqzz D + (α + q )D − (α − q ) b q + b † −q . (B.4)
Thus, the matrix elements for spontaneous emission of a phonon in the Fock–
Darwin basis {|n + ,n − ,σ〉} factorise to
〈n ′ +,n ′ −,σ ′ |V ep (q)|n + ,n − ,σ〉 = δ σσ ′λ q 〈n ′ +|D + (α + q )|n + 〉〈n ′ −|D − (α − q )|n − 〉. (B.5)
115