Driven Quantum Systems - Institut für Physik

270 Driven Quantum Systems
5.5 Numerical approaches to periodically driven
quantum systems
Except for the special cases discussed in Sect. 5.4, exactly solvable quantum systems
with explicitly time-dependent interaction potentials are extremely rare. As demonstrated
with (5.83), this is true already for the periodically driven two-level-system in
a linearly polarized monochromatic field [11] for which no exact closed form solution
can be found. Thus, we generally have to invoke numerical procedures.
5.5.1 Method of Floquet matrix
Since the Hamiltonian H(x, t) and the Floquet modes are time-periodic, we can expand
the Floquet solutions into the Fourier vectors |n〉, n =0,±1,±2,..., such that 〈t|n〉 =
exp(inωt),
Φ α (x, t) =
∞∑
n=−∞
c n α (x)exp(inωt). (5.116)
The functions c n α(x) can be expanded in terms of a complete orthonormal set {ϕ k (x),
k =1,...,∞}, yielding in terms of the unperturbed eigenfunctions of H 0 (x),
Φ α (x, t) =
∞∑
∞∑
k=1 n=−∞
c n α,kϕ k (x)exp(inωt), (5.117)
with c n α,k = 〈ϕ k|c n α 〉. Hence, in terms of the kets |ϕ k〉, 〈x|ϕ k 〉 = ϕ k (x), the Floquet
equation (5.14) reads
∞∑
∞∑
k=1 n=−∞
Hc n α,k |ϕ k〉 exp(inωt) =
∞∑
∞∑
k=1 n=−∞
ɛ α c n α,k |ϕ k〉 exp(inωt). (5.118)
Setting 〈ϕ k |〈m| ≡〈ϕ k m|and multiplying (5.118) with 〈ϕ j m| exp(−imωt) fromtheleft,
yields after a time-average over one period of driving, the system of equations
∞∑ ∞∑
〈〈ϕ j m|H|ϕ k n〉〉c n α,k = ɛ αc m α,j . (5.119)
n=−∞ k=1
Here we used the scalar-product notation in (5.19). With the definition
∫ T
H m−n = 1 dtH(t)exp[−i(m − n)ωt], (5.120)
T 0
one finds the Floquet-matrix representation for (5.119),
∞∑ ∞∑
〈〈ϕ j m|H F |ϕ k n〉〉c n α,k = ɛ α c m α,j, (5.121)
k=1 n=−∞
with the Floquet matrix defined by
〈〈ϕ j m|H F |ϕ k n〉〉 ≡ 〈ϕ j |H m−n |ϕ k 〉 + n¯hωδ n,m δ j,k . (5.122)