Driven Quantum Systems - Institut für Physik

264 Driven Quantum Systems
where |c 1 (t)| 2 + |c 2 (t)| 2 =1. With2¯hλ ≡−µE 0 and ϕ = π/2, yielding a pure cos(ωt)
perturbation, the Schrödinger equation has the form
⎛
i¯h d c ⎞
1(t)exp(i∆t/2¯h)
⎝
⎠
dt c 2 (t)exp(−i∆t/2¯h)
⎛
= ⎝
−∆/2
−2¯hλ cos ωt
−2¯hλ cos ωt ∆/2
⎞ ⎛
⎠ ⎝
c 1(t)exp(i∆t/2¯h)
c 2 (t)exp(−i∆t/2¯h)
⎞
⎠ . (5.83)
With ¯hω 0 ≡ ∆, (5.83) provides two coupled first-order equations for the amplitudes,
dc 1
dt =iλ( exp[i(ω − ω 0 )t]+exp[−i(ω + ω 0 )t] ) c 2 ,
dc 2
dt =iλ( exp[−i(ω − ω 0 )t]+expi(ω+ω 0 )t] ) c 1 . (5.84)
With an additional differentiation with respect to time, and substituting ċ 2 from the
second equation, we readily find that c 1 (t) obeys a linear second order ordinary differential
equation with time periodic (T =2π/ω) coefficients (Hill equation). Clearly,
such equations are generally not solvable in analytical closed form. Hence, although
the problem is simple, the job of finding an analytical solution presents a hard task! To
make progress, one usually invokes, at this stage, the so-called rotating-wave approximation
(RWA), assuming that ω is close to ω 0 (near resonance), and λ not very large.
Then the anti-rotating-wave term exp(i(ω + ω 0 )t) is rapidly varying, as compared to
the slowly varying rotating-wave term exp(−i(ω − ω 0 )t). Therefore it cannot transfer
much population from state |1〉 to state |2〉. Neglecting this anti-rotating contribution,
one has in terms of the detuning parameter δ ≡ ω − ω 0 ,
dc 1
dt =iλexp(iδt)c 2,
dc 2
dt =iλexp(−iδt)c 1. (5.85)
From (5.83) one finds for c 1 (t) a linear second-order differential equation with constant
coefficients — which can be solved readily for arbitrary initial conditions. For example,
setting c 1 (0) = 1, c 2 (0) = 0, one obtains
[ ( ) 1
c 1 (t) =exp(iδt) cos
2 Ωt − i δ ( ) ] 1
Ω sin 2 Ωt ,
c 2 (t) =exp(−iδt) 2iλ
Ω sin ( 1
2 Ωt )
, (5.86)
where Ω denotes the celebrated Rabi frequency
Ω= ( δ 2 +4λ 2) 1/2
. (5.87)