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

2.2 Spectral properties, various limits 31
with eigenstates
|n,↑↓〉 = |n〉 ↑↓ |↑↓〉, |n〉 ↑↓ = 1 √
n!
(A † ↑↓) n
|0〉↑↓ . (2.25)
Thus, at the level of this extended longitudinal-SO approximation, the simultaneous
effect of weak SO coupling and magnetic field leads to a spin-dependent shift
of the wavefunction. In comparison with the zero-B result Eq. (2.19), however, the
spectral properties are only affected by a constant shift of the ground state energy
by −ξ 2 2 /4Ω, as one might have expected from the analogy with the electric field.
Remark: Equation (2.25) gives good insight into the approximate properties of
eigenfunctions and energies in the low-coupling limit. However, in the context of
a numerical diagonalisation of the full Hamiltonian H 0 +H mix , the basis {|n,↑↓〉}
is of little practical use. This is because the orbital wavefunctions |n〉 ↑↓ correspond
to differently displaced oscillators for opposite spin, hence not fulfilling
orthogonality, ↑↓ 〈n|m〉 ↓↑ ≠ δ nm .
2.2.4 High-field limit
For the zero magnetic field case, the coupling of adjacent subbands is symmetric,
see Eq. (2.17). With increasing field this coupling becomes more and more asymmetric
due to ξ 2 and ξ 3 in Eqs. (2.20) and (2.21). This amounts to the convergence
of the Hamiltonian against the Jaynes–Cummings model (see appendix A) in the
high field limit: By introducing σ ± = (σ x ± iσ y )/2, the Hamiltonian (2.5) can be
rewritten as
H(k)
ω 0
= Ω
(
a † k a k + 1 2
)
+ δ 2 σ z + 1 2 ξ 1kl 0 σ x + 1 (kl 0 ) 2
2 Ω 2
+ 1 l
[ 0
γ r
(a k σ + + a †
2 l
k σ −
SO
) (
+ γ c a k σ − + a † k σ +
)]
,
(2.26)
where (l 0 /l SO )γ r,c = ξ 2 ∓ iξ 3 . In quantum optics, terms preceded by γ r(c) are
called (counter)-rotating. This distinction becomes apparent when transforming
into the interaction picture and comparing the time evolution of these terms. The
first two terms on the right hand side of Eq. (2.26), H 0 := Ω(a † k a k +1/2)+δσ z /2,
correspond to the free evolution of the boson field of subbands and the spin. Thus,
in the interaction picture with respect to H 0 , boson and spin operators evolve as
â = ae −iΩτ , â † = a † e iΩτ , ˆσ ± = σ ± e ±iδτ , τ := ω 0 t. (2.27)
Close to resonance Ω ≈ δ in Eq. (2.26), counter-rotating terms are rapidly oscillating
in time as exp(2iΩτ) whereas rotating terms become almost stationary. In