Driven Quantum Systems - Institut für Physik

262 Driven Quantum Systems
yielding
{
i¯h ˙Ψ(y, t) = i¯h ˙ζ ∂
∂y − ¯h2 ∂ 2
2m ∂y + 1 }
2 2 mω2 0 (y + ζ)2 − (y + ζ)S(t) Ψ(y, t). (5.68)
Performing the unitary transformation
with ζ(t) obeying the classical equation of motion,
Ψ(y,t) =exp{−im ˙ζy/¯h}φ(y, t), (5.69)
m¨ζ + mω0 2 ζ = S(t), (5.70)
the term linear in y vanishes to yield
{−¯h2
i¯h ˙φ(y,t) ∂ 2
=
2m∂y + 1 }
2 2 mω2 0y 2 + L(ζ, ˙ζ,t) φ(y, t). (5.71)
Here, L(ζ, ˙ζ,t) is the classical Lagrangian of a driven oscillator,
L = 1 2 m ˙ζ 2 − 1 2 mω2 0ζ 2 + ζS(t). (5.72)
Another unitary transformation
{ ∫ t
φ(y,t) =exp −i dt ′ L(ζ, ˙ζ,t
}
′ ) χ(y, t) (5.73)
0
reduces the starting equation to the well-known Schrödinger equation of a stationary
harmonic oscillator,
{−¯h2 ∂ 2
i¯h ˙χ(y,t) =
2m∂y + 1 }
2 2 mω2 0y 2 χ(y, t). (5.74)
In terms of the eigenvalues E n = ¯hω 0 (n +1/2), and the well-known harmonic
eigenfunctions ϕ n , being proportional to the Hermite functions, the solutions of (5.66)
are of the form
{ [ ī
Ψ n (x, t) =ϕ n (x−ζ(t)) exp m
h
˙ζ(t)(x
∫ t ]}
− ζ(t)) − E n t + dt ′ L . (5.75)
0
The set {ϕ n (x)} forms a complete set in R; thus any general solution Ψ(x, t) canbe
expanded in terms of the solutions in (5.75). Next we consider the restriction to a
periodic monochromatic drive
A periodic solution ζ φ of (5.70) reads
S(t) =Ssin(ωt + φ), ω ≠ ω 0 . (5.76)
mζ φ (t) =Ssin(ωt + φ)/(ω 2 0 − ω2 ). (5.77)