11 Floquet Theory and Ultrafast Control of Magnetism
273
If the phase factor of the right hand side can be expanded with respect to power of
1/ω, we obtain the series expansion formula of ˆ
H eff . It is known that if the time
evolution operator is assumed to be decomposed in the following form, one can
compute the expanded form of the effective Hamiltonian:
ˆ
U (t 2 , t 1 ) = exp(−i ˆ
G(t 2 )) exp
− i ˆ
H (t 2 − t 1 )
exp(i ˆ
G(t 1 )).
(11.19)
Here, ˆ
G(t) is a periodic operator ˆ
G(t) = ˆ
G(t + T ), and the exponential exp(±i ˆ
G(t))
describes the high-frequency fluctuation in one cycle T . ˆ
G(t) is called kick operator
or micro-motion operator. On the other hand, ˆ
H is the time-independent operator
describing the slow dynamics longer than the period T . Generally, ˆ
H depends on
the initial time t 1 , but the t 1 dependence is eliminated under the condition that ˆ
G(t)
satisfies
T
0 dt ˆ
G(t) = 0. In this case, the high-frequency expansion of ˆ
H is shown
to be equal to that of ˆ
H eff in (11.16). Similarly, the kick operator can be expanded
as ˆ
G(t) =
n=1 G
(n) . The lower-order terms are given by
i ˆ
G
(1)
(t) = −
∞
m=−∞
(m =0)
ˆ
H m
mω
e
−imωt
,
(11.20)
i ˆ
G
(2)
(t) =
∞
m=−∞
(m =0)
[ ˆ
H m , ˆ
H 0 ]
m 2 (ω) 2 e
−imωt
+
∞
m=−∞
(m =0)
∞
n=−∞
(n =0,m)
[ ˆ
H n , ˆ
H m−n ]
2mn(ω) 2 e
−imωt
. (11.21)
11.2.4 Physical Meaning of Floquet Hamiltonian
In the previous subsection, I explained that the static Floquet Hamiltonian ˆ
H eff of
(11.16) is computed through the Floquet-Magnum high-frequency expansion. What
can we understand from ˆ
H eff ? One might expect that thermal equilibrium or ground
states of the “Hamiltonian” ˆ
H eff is realized by continuously applying an AC field.
However, such a naive expectation is not correct. Below, I will comment on a few
important results related with the Floquet Hamiltonian ˆ
H eff , focusing mainly on
many-body systems.
[Short time behavior] When one applies the expansion formulas of ˆ
H eff and ˆ
G(t)
to a periodically-driven system, the Floquet-Magnus expansion should be terminated at a certain order in a practical sense. If we focus on a small finite-size
system, the expansion is often well-defined. On the other hand, it is known that if
we consider a wide class of locally-interacting many-body systems, their FloquetMagnus expansion is usually of an asymptotic-expansion type like the perturbation
expansion of quantum field theories. Let us define a truncated Floquet Hamiltonian
as ˆ
H q ≡
q
k=0
ˆ
H
(k) . The following statements about ˆ
H q have been theoretically
shown for locally interacting many-body systems [40, 41]. (i) There is an optimal
273
If the phase factor of the right hand side can be expanded with respect to power of
1/ω, we obtain the series expansion formula of ˆ
H eff . It is known that if the time
evolution operator is assumed to be decomposed in the following form, one can
compute the expanded form of the effective Hamiltonian:
ˆ
U (t 2 , t 1 ) = exp(−i ˆ
G(t 2 )) exp
− i ˆ
H (t 2 − t 1 )
exp(i ˆ
G(t 1 )).
(11.19)
Here, ˆ
G(t) is a periodic operator ˆ
G(t) = ˆ
G(t + T ), and the exponential exp(±i ˆ
G(t))
describes the high-frequency fluctuation in one cycle T . ˆ
G(t) is called kick operator
or micro-motion operator. On the other hand, ˆ
H is the time-independent operator
describing the slow dynamics longer than the period T . Generally, ˆ
H depends on
the initial time t 1 , but the t 1 dependence is eliminated under the condition that ˆ
G(t)
satisfies
T
0 dt ˆ
G(t) = 0. In this case, the high-frequency expansion of ˆ
H is shown
to be equal to that of ˆ
H eff in (11.16). Similarly, the kick operator can be expanded
as ˆ
G(t) =
n=1 G
(n) . The lower-order terms are given by
i ˆ
G
(1)
(t) = −
∞
m=−∞
(m =0)
ˆ
H m
mω
e
−imωt
,
(11.20)
i ˆ
G
(2)
(t) =
∞
m=−∞
(m =0)
[ ˆ
H m , ˆ
H 0 ]
m 2 (ω) 2 e
−imωt
+
∞
m=−∞
(m =0)
∞
n=−∞
(n =0,m)
[ ˆ
H n , ˆ
H m−n ]
2mn(ω) 2 e
−imωt
. (11.21)
11.2.4 Physical Meaning of Floquet Hamiltonian
In the previous subsection, I explained that the static Floquet Hamiltonian ˆ
H eff of
(11.16) is computed through the Floquet-Magnum high-frequency expansion. What
can we understand from ˆ
H eff ? One might expect that thermal equilibrium or ground
states of the “Hamiltonian” ˆ
H eff is realized by continuously applying an AC field.
However, such a naive expectation is not correct. Below, I will comment on a few
important results related with the Floquet Hamiltonian ˆ
H eff , focusing mainly on
many-body systems.
[Short time behavior] When one applies the expansion formulas of ˆ
H eff and ˆ
G(t)
to a periodically-driven system, the Floquet-Magnus expansion should be terminated at a certain order in a practical sense. If we focus on a small finite-size
system, the expansion is often well-defined. On the other hand, it is known that if
we consider a wide class of locally-interacting many-body systems, their FloquetMagnus expansion is usually of an asymptotic-expansion type like the perturbation
expansion of quantum field theories. Let us define a truncated Floquet Hamiltonian
as ˆ
H q ≡
q
k=0
ˆ
H
(k) . The following statements about ˆ
H q have been theoretically
shown for locally interacting many-body systems [40, 41]. (i) There is an optimal