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

52 Rashba spin-orbit coupling in quantum dots
with parabolic confinement of energy ω 0 [108],
H 0 = (p + e c A)2
2m
+ m 2 ω2 0(x 2 + y 2 ), (3.2)
where m is the effective mass of the electron. Applying a perpendicular magnetic
field in the symmetric gauge, in second quantised notation we have
H 0 = ˜ω(a † xa x + a † ya y + 1) + ω c
2i (a ya † x − a x a † y), (3.3)
with ω c ≡ eB/mc and ˜ω 2 ≡ ω 2 0 + ω2 c/4. Introduction of a ± = 2 −1/2 (a x ∓ ia y )
decouples the system into eigenmodes of frequency ω ± = ˜ω ± ω c /2.
We now include the Rashba interaction of Eq. (3.1), for which the coupling
strength α is related to the spin precession length l SO ≡ 2 /2mα. With magnetic
length l B ≡ √ /mω c , we have
H SO = α˜l
[
]
γ + (a + σ + + a † +σ − ) − γ − (a − σ − + a † −σ + ) , (3.4)
with coefficients γ ± ≡ 1 ± 1 2(˜l/l B
) 2 and ˜l ≡ √ /m˜ω.
Adding the Zeeman term, in which we take g to be negative as in InGaAs,
performing a unitary rotation of the spin such that σ z → −σ z and σ ± → −σ ∓ , and
rescaling energies by ω 0 we arrive at the Hamiltonian
H = ω + a † +a + +ω − a † −a − + 1 2 E zσ z
+ l2 [
]
0
γ − (a − σ + + a † −σ − ) − γ + (a + σ − + a † +σ + ) , (3.5)
2˜l l SO
where l 0 = √ /mω 0 is the confinement length of the dot and E z =|g|m/2m e (l B /l 0 ) 2
is the Zeeman energy with m e the bare mass of the electron.
This single-particle picture is motivated by the good agreement between Fock-
Darwin theory and experiment in the non-SO case [108], and by studies which
have shown that many-body effects in QDs play only a small role at the magnetic
fields we consider here [101, 102, 109].
We now derive an approximate form of this Hamiltonian by borrowing the
observation from quantum optics that the terms preceded by γ + in Eq. (3.5) are
counter-rotating, and thus negligible under the rotating-wave approximation [33]
when the SO coupling is small compared to the confinement. This decouples the
ω + mode from the rest of the system, giving H = ω + n + + H JC where
H JC (α) = ω − a † −a − + 1 2 E zσ z + λ(a − σ + + a † −σ − ), (3.6)