Driven Quantum Systems - Institut für Physik

266 Driven Quantum Systems
for the description of optical properties such as the dipole moment. We have for the
expectation
µ(t) =tr{ϱ TLS σ x } =tr{ϱ T σ z }, (5.92)
where ϱ ... is the density matrix in the corresponding representation. Note that a static
asymmetry energy can be included if the field assumes a static component, i.e. λ sin(ωt+
φ) → λ sin(ωt + φ)+λ 0 .
Explicit results for the time-periodic Schrödinger equation require numerical methods,
cf. Sect. 5.5, one must solve for the quasienergies ɛ αn and the Floquet modes
Ψ αn (x, t). Without proof we state here some results that are very important in discussing
driven tunneling in the deep quantum regime. For example, Shirley [11] already
showed that in the high-frequency regime ∆ ≪ max[ω, (λω) 1/2 ] the quasienergies obey
the difference
ɛ 2,−1 − ɛ 1,1 =¯hω 0 J 0 (4λ/ω), (5.93)
where J 0 denotes the zeroth order Bessel function of the first kind. The sum of the two
quasienergies obeys the rigorous relation [11]
ɛ 2n + ɛ 1k = E 1 + E 2 =0 (mod¯hω). (5.94)
For weak fields, one can evaluate the quasienergies by use of the stationary perturbation
theory in the composite Hilbert space R⊗T. In this way one finds:
(i) Exact crossings at the parity forbidden transitions where ω 0 =2nω, n =1,2,...,
so that
ɛ 2,1 =0 (mod¯hω). (5.95)
(ii) At resonance ω = ω 0 :
ɛ 2,1 = ± 1 2¯hω 0
(
1+ 2λ √
)
1−λ
ω 2 /4ω0
2
0
(mod ¯hω). (5.96)
(iii) Near resonance, one finds from the RWA approximation readily the result
ɛ 2,1 = ± 1 (
2¯hω 1+ Ω )
(mod ¯hω), (5.97)
ω
where Ω denotes the Rabi frequency in (5.87). Correcting this result for counterrotating
terms, an improved result, up to order O(λ 6 ), reads [27]
ɛ 2,1 = ± 1 (
2¯hω 1+ ¯Ω )
(mod ¯hω), (5.98)
ω
with the effective Rabi frequency ¯Ω
¯Ω 2 = δ 2 + 8ω 0λ 2
(ω + ω 0 ) − 8ω 0λ 4
(ω + ω 0 ) 3 . (5.99)