11 Floquet Theory and Ultrafast Control of Magnetism
271
11.2.3 Floquet-Magnus Expansion and Floquet Hamiltonian
It is known that one can compute an effective static Hamiltonian (called Floquet
Hamiltonian) [37, 38] acting on the original space of the Schrødinger equation (11.1)
from the eigenvalue equation (11.11) or (11.12). The approximation is based on the
power series expansion of 1/ω. There are several sorts of the expansions and they
are related with each other [39]. In general, they are called Floquet-Magnus (highfrequency) expansion.
Here, I explain one of the expansion methods which is based on degenerate perturbation theory, by making use of the matrix representation (11.12) and Fig. 11.1. Since
the bra-ket notation is generally useful for the perturbation calculation, we re-write
wave functions n and (t) as | n and |(t), respectively, in this subsection.
We consider the case where the photon energy ω is sufficiently larger than other
energy scales (i.e., all the eigenvalues of { ˆ
H n }). In this condition, the “distance” ω
between neighboring subspaces is large enough and therefore the averaged Hamiltonian ˆ
H 0 and Fourier components ˆ
H ±n (n = 0) connecting different subspaces
may be viewed as the perturbation with respect to the the diagonal photon energies
diag(· · · , 2ω, ω, , 0, −ω, −2ω, · · · ).
Below, I will construct the effective Hamiltonian for a subspace with a fixed photon number. Hereafter, we call the subspace with photon number n as nth subspace,
and ignore the unperturbed photon energies nω since they are just constants. In fact,
owing to periodicity of the quasi energy , we arrive at the same effective Hamiltonian
in each subspace with photon number n, expect for the constant nω. We define the
orthonormal basis {| −n j } ( j = 1, 2, · · · , d) for nth subspace, where d is the dimension of the subspace. Let us introduce the photon-number state |n that is convenient
for the perturbation calculation. With it, the projection operator ˆ
P −n to the nth subspace may be expressed as ˆ
P −n =
d
j=1 | −n j j −n | = |nn| ×
d
j=1 | j j |.
The basis | j is common to all the subspaces and in that sense, two projection
operators ˆ
P −n and ˆ
P −m (n = m) are equal to each other, ˆ
P −n ∼ ˆ
P −m , except for their
photon states. Similarly, the averaged Hamiltonian ˆ
H 0 in the diagonal part of (11.12)
should be represented as ˆ
H 0 |nn|, and ˆ
H m (m = 0) connecting nth and (n + m)th
subspaces as ˆ
H m |n + mn|. One can perform the perturbation calculation utilizing
these instruments.
The first-order Hamiltonian is equivalent to putting the perturbation term between
the projection operators. Therefore, we obtain
ˆ
H
(1)
≡ ˆ
P −n ˆ
H 0 ˆ
P −n = ˆ
H 0 .
(11.13)
Fourier components ˆ
H n =0 all disappear due to the sandwich of ˆ
P −n . We also note that
d
j=1 | j j | is unity when we focus on a subspace with a fixed photon number.
Following the standard formula of the second-order perturbation, we write down the
second-order term as
271
11.2.3 Floquet-Magnus Expansion and Floquet Hamiltonian
It is known that one can compute an effective static Hamiltonian (called Floquet
Hamiltonian) [37, 38] acting on the original space of the Schrødinger equation (11.1)
from the eigenvalue equation (11.11) or (11.12). The approximation is based on the
power series expansion of 1/ω. There are several sorts of the expansions and they
are related with each other [39]. In general, they are called Floquet-Magnus (highfrequency) expansion.
Here, I explain one of the expansion methods which is based on degenerate perturbation theory, by making use of the matrix representation (11.12) and Fig. 11.1. Since
the bra-ket notation is generally useful for the perturbation calculation, we re-write
wave functions n and (t) as | n and |(t), respectively, in this subsection.
We consider the case where the photon energy ω is sufficiently larger than other
energy scales (i.e., all the eigenvalues of { ˆ
H n }). In this condition, the “distance” ω
between neighboring subspaces is large enough and therefore the averaged Hamiltonian ˆ
H 0 and Fourier components ˆ
H ±n (n = 0) connecting different subspaces
may be viewed as the perturbation with respect to the the diagonal photon energies
diag(· · · , 2ω, ω, , 0, −ω, −2ω, · · · ).
Below, I will construct the effective Hamiltonian for a subspace with a fixed photon number. Hereafter, we call the subspace with photon number n as nth subspace,
and ignore the unperturbed photon energies nω since they are just constants. In fact,
owing to periodicity of the quasi energy , we arrive at the same effective Hamiltonian
in each subspace with photon number n, expect for the constant nω. We define the
orthonormal basis {| −n j } ( j = 1, 2, · · · , d) for nth subspace, where d is the dimension of the subspace. Let us introduce the photon-number state |n that is convenient
for the perturbation calculation. With it, the projection operator ˆ
P −n to the nth subspace may be expressed as ˆ
P −n =
d
j=1 | −n j j −n | = |nn| ×
d
j=1 | j j |.
The basis | j is common to all the subspaces and in that sense, two projection
operators ˆ
P −n and ˆ
P −m (n = m) are equal to each other, ˆ
P −n ∼ ˆ
P −m , except for their
photon states. Similarly, the averaged Hamiltonian ˆ
H 0 in the diagonal part of (11.12)
should be represented as ˆ
H 0 |nn|, and ˆ
H m (m = 0) connecting nth and (n + m)th
subspaces as ˆ
H m |n + mn|. One can perform the perturbation calculation utilizing
these instruments.
The first-order Hamiltonian is equivalent to putting the perturbation term between
the projection operators. Therefore, we obtain
ˆ
H
(1)
≡ ˆ
P −n ˆ
H 0 ˆ
P −n = ˆ
H 0 .
(11.13)
Fourier components ˆ
H n =0 all disappear due to the sandwich of ˆ
P −n . We also note that
d
j=1 | j j | is unity when we focus on a subspace with a fixed photon number.
Following the standard formula of the second-order perturbation, we write down the
second-order term as