Driven Quantum Systems - Institut für Physik

268 Driven Quantum Systems
relations in (5.88), which in this case are exact. In particular, the transition probability
W 1→2 (t) =|〈2|Ψ(t)〉| 2 = |a 2 (t)| 2 = |b 2 (t)| 2 = |c 2 (t)| 2 obeys
( )
W 1→2 (t) = 4λ2 1
Ω 2 sin2 2 Ωt . (5.105)
At resonance, δ =0,ω=ω 0 , it assumes with Ω 2 =4λ 2 its maximal value. We also
note that the quasienergies are given by the — in this case exact — result in (5.96).
5.4.3 Quantum systems driven by circularly polarized fields
The fact that the time evolution of a TLS in a circularly polarized field can be factorized
in terms of a time-independent Hamiltonian in (5.103) is surprising. We note that this
factorization involves a rotation around the z-axis,
|a(t)〉 −→ |b(t)〉 =exp(−iS z ωt/¯h)|a(t)〉. (5.106)
This feature can be generalized to any higher-dimensional system, such as a magnetic
system or a general quantum system that can be brought into the structure which, in
a representation where J z is diagonal, is of the form
H(t) =H 0 (J 2 )+H 1 (J z )−4λ[J x cos ωt − J y sin ωt]. (5.107)
Here, H 0 (J 2 ) contains all interactions that are rotationally invariant (Coulomb interactions,
spin-spin and spin-orbit interactions). Setting R(t) ≡ exp(−iJ z ωt/¯h) and upon
observing that
R(t)J x R(t) −1 = J x cos ωt + J y sin ωt,
R(t)J y R(t) −1 = −J x sin ωt + J y cos ωt,
R(t)J z R(t) −1 = J z , (5.108)
one finds upon substituting (5.108) into (5.107)
H˜ (t) ≡ R(t)H(t)R −1 (t) =H 0 (J 2 )+H 1 (J z )−4λJ x . (5.109)
Hence, the transformed Hamiltonian becomes independent of time. With the propagator
obeying
∂
∂t K(t, t 0)=− ī h H(t)K(t, t 0)
= − ī h R−1 (t)H˜ R(t)K(t, t 0 ),
we find from
∂
∂t [R(t)K(t, t 0)R −1 (t 0 )] = − ī h Ĥ[R(t)K(t, t 0)R −1 (t 0 )] (5.110)