11 Floquet Theory and Ultrafast Control of Magnetism
267
(ii) is too ideal, especially, if we consider usual materials such as solids, liquids, and
gases. However, as I will explain below, the ideal conditions (i) and (ii) enable us to
reveal a clear, simple picture of Floquet engineering. The engineering in dissipative
driven systems [14, 15, 19] is a front line of the non-equilibrium physics.
11.2.1 Floquet Theorem
Floquet theorem is an old mathematical result for a class of linear differential equations with a time periodic term, but in the field of physics, it has been recognized
as a theorem for Schrødinger equation (i.e., equation of motion) for periodicallydriven quantum systems. Following this convention, I will prove Floquet theorem
for quantum systems in this subsection. Hereafter, I will often use the unit of = 1.
The statement of Floquet theorem is as follows. We start from the time-dependent
Schrødinger equation for a periodically driven quantum system
i
∂
∂t
(t) = ˆ
H (t))(t).
(11.1)
Here, we assume that the Hamiltonian ˆ
H (t) is time periodic, ˆ
H (t + T ) = ˆ
H (t),
where T = 2π/ω is the period and ω is the (angular) frequency. If we focus on a
system driven by laser or electromagnetic wave, ω is the laser frequency. For this
driven quantum system, the theorem shows that the solution of (11.1) is given by
(t) = exp(−it))(t),
(11.2)
where the “wave function” (t) is a periodic one satisfying (t + T ) = (t) and
the real number is called Floquet quasi energy. Namely, Floquet theorem states that
the solution of Schrødinger equation for a periodically driven system is given by the
product of a plane wave e
−it (i.e., solution of vacuum) and a periodic function (t).
In this sense, Floquet theorem can be viewed as the time version of Bloch theorem
(See, e.g., [33, 34]) for spatially-periodic quantum systems.
Let us prove the above statement. We define the one-cycle time-evolution operator
as
ˆ
U (t + T, t) ≡ T
exp
−
i
t+T
t
dτ ˆ
H (τ )
,
(11.3)
where the symbol T denotes time-ordered product. From the periodicity of the
Hamiltonian, ˆ
U (t + T, t) = ˆ
U (t + (n + 1)T, t + nT ) for n ∈ Z . If ˆ
U (t + T, t) acts
on i∂ t (t) = ˆ
H (t))(t) from the left in both sides, we obtain i∂ t (t + T ) =
ˆ
H (t + T ))(t + T ) = ˆ
H (t))(t + T ). That is, (t) and (t + T ) both satisfy the
same Schrødinger equation. Therefore, for a solution (t), there always exists
another solution (t + T ) which is proportional to (t): (t + T ) = c T (t))(t).
267
(ii) is too ideal, especially, if we consider usual materials such as solids, liquids, and
gases. However, as I will explain below, the ideal conditions (i) and (ii) enable us to
reveal a clear, simple picture of Floquet engineering. The engineering in dissipative
driven systems [14, 15, 19] is a front line of the non-equilibrium physics.
11.2.1 Floquet Theorem
Floquet theorem is an old mathematical result for a class of linear differential equations with a time periodic term, but in the field of physics, it has been recognized
as a theorem for Schrødinger equation (i.e., equation of motion) for periodicallydriven quantum systems. Following this convention, I will prove Floquet theorem
for quantum systems in this subsection. Hereafter, I will often use the unit of = 1.
The statement of Floquet theorem is as follows. We start from the time-dependent
Schrødinger equation for a periodically driven quantum system
i
∂
∂t
(t) = ˆ
H (t))(t).
(11.1)
Here, we assume that the Hamiltonian ˆ
H (t) is time periodic, ˆ
H (t + T ) = ˆ
H (t),
where T = 2π/ω is the period and ω is the (angular) frequency. If we focus on a
system driven by laser or electromagnetic wave, ω is the laser frequency. For this
driven quantum system, the theorem shows that the solution of (11.1) is given by
(t) = exp(−it))(t),
(11.2)
where the “wave function” (t) is a periodic one satisfying (t + T ) = (t) and
the real number is called Floquet quasi energy. Namely, Floquet theorem states that
the solution of Schrødinger equation for a periodically driven system is given by the
product of a plane wave e
−it (i.e., solution of vacuum) and a periodic function (t).
In this sense, Floquet theorem can be viewed as the time version of Bloch theorem
(See, e.g., [33, 34]) for spatially-periodic quantum systems.
Let us prove the above statement. We define the one-cycle time-evolution operator
as
ˆ
U (t + T, t) ≡ T
exp
−
i
t+T
t
dτ ˆ
H (τ )
,
(11.3)
where the symbol T denotes time-ordered product. From the periodicity of the
Hamiltonian, ˆ
U (t + T, t) = ˆ
U (t + (n + 1)T, t + nT ) for n ∈ Z . If ˆ
U (t + T, t) acts
on i∂ t (t) = ˆ
H (t))(t) from the left in both sides, we obtain i∂ t (t + T ) =
ˆ
H (t + T ))(t + T ) = ˆ
H (t))(t + T ). That is, (t) and (t + T ) both satisfy the
same Schrødinger equation. Therefore, for a solution (t), there always exists
another solution (t + T ) which is proportional to (t): (t + T ) = c T (t))(t).