11 Floquet Theory and Ultrafast Control of Magnetism
275
From these statements, we see that the present Floquet theory can correctly predict
the short-time behavior when we consider isolated periodically driven many-body
systems. For a long-time driving, the systems are generally heated and the Floquet
theory does not work well. However, I emphasize that the forms of low-order terms
such as ˆ
H
(0) , ˆ
H
(1) , and ˆ
H
(2) are very important to qualitatively understand which
kinds of quantities can be controlled by an AC field. A more quantitative computation
beyond Floquet Hamiltonian (e.g.., numerical analysis) is often necessary to more
accurately predict the values and time evolution of AC-field driven quantities, but
finding set of possible engineering observables is the most fundamental process to
propose a novel Floquet engineering.
11.3 Laser and Typical Excitations in Solids
Some parts of theoretical proposals for Floquet engineering have been experimentally
realized by using cold-atom systems [21, 22] in which one can prepare nearly isolated
(closed) quantum systems in a few (sub-)seconds. It means that Floquet engineering
in cold-atom systems has already grown to a certain degree, while that in materials
still offers various open issues. Therefore, now is the time to develop theories and
experiments for Floquet engineering in usual materials such as solids, liquids, and
gases. Use of usual materials has the advantage compared with cold atoms. For
instance, the lifetime of materials may be considered as infinity, and thereby one
can repeatedly perform Floquet-engineering process with single material. Floquet
engineering in materials can be done in a table-top manner by applying a suitable
AC field while that in cold atoms requires application of many sorts of tuned lasers
together with atom-trapping techniques.
As I mentioned in Introduction, I will review a few examples of the Floquet
engineering in solid crystals in the next section. A typical high-frequency field for
solids is electromagnetic wave or laser beams. Therefore, the information about
currently-available laser and excitations in solids is important. Below I discuss it
from a quantitative viewpoint.
Figure 11.2 depicts typical excitations (quasi particles) of solids in a wide range
of laser frequency (photon energy). Magnetic (electron spin) excitations in solids are
usually distributed from 1 GHz to 10 THz. Nuclear spin dynamics is much slower
than electron spin motion and nuclear magnetic resonance (NMR) [6] is usually done
in a megahertz (MHz = 10
6 Hz) range. Phonons (lattice vibration) and molecular
oscillations are located from 1 THz to infrared wave (∼ 100 THz). Electron charge
excitations are around the visible-light regime (0.1 − 10 petahertz (PHz = 10
15 Hz))
in both band and Mott insulators.
Electromagnetic waves and quasi-particle dynamics in MHz and GHz (lowfrequency) range have been widely used in electronics and spintronics [9]. In this
low-frequency regime, it is difficult to create intense coherent waves like laser beams
(although there exists maser technique), and instead various methods of generating
electromagnetic waves have been established. For instance, Gunn diode has been
275
From these statements, we see that the present Floquet theory can correctly predict
the short-time behavior when we consider isolated periodically driven many-body
systems. For a long-time driving, the systems are generally heated and the Floquet
theory does not work well. However, I emphasize that the forms of low-order terms
such as ˆ
H
(0) , ˆ
H
(1) , and ˆ
H
(2) are very important to qualitatively understand which
kinds of quantities can be controlled by an AC field. A more quantitative computation
beyond Floquet Hamiltonian (e.g.., numerical analysis) is often necessary to more
accurately predict the values and time evolution of AC-field driven quantities, but
finding set of possible engineering observables is the most fundamental process to
propose a novel Floquet engineering.
11.3 Laser and Typical Excitations in Solids
Some parts of theoretical proposals for Floquet engineering have been experimentally
realized by using cold-atom systems [21, 22] in which one can prepare nearly isolated
(closed) quantum systems in a few (sub-)seconds. It means that Floquet engineering
in cold-atom systems has already grown to a certain degree, while that in materials
still offers various open issues. Therefore, now is the time to develop theories and
experiments for Floquet engineering in usual materials such as solids, liquids, and
gases. Use of usual materials has the advantage compared with cold atoms. For
instance, the lifetime of materials may be considered as infinity, and thereby one
can repeatedly perform Floquet-engineering process with single material. Floquet
engineering in materials can be done in a table-top manner by applying a suitable
AC field while that in cold atoms requires application of many sorts of tuned lasers
together with atom-trapping techniques.
As I mentioned in Introduction, I will review a few examples of the Floquet
engineering in solid crystals in the next section. A typical high-frequency field for
solids is electromagnetic wave or laser beams. Therefore, the information about
currently-available laser and excitations in solids is important. Below I discuss it
from a quantitative viewpoint.
Figure 11.2 depicts typical excitations (quasi particles) of solids in a wide range
of laser frequency (photon energy). Magnetic (electron spin) excitations in solids are
usually distributed from 1 GHz to 10 THz. Nuclear spin dynamics is much slower
than electron spin motion and nuclear magnetic resonance (NMR) [6] is usually done
in a megahertz (MHz = 10
6 Hz) range. Phonons (lattice vibration) and molecular
oscillations are located from 1 THz to infrared wave (∼ 100 THz). Electron charge
excitations are around the visible-light regime (0.1 − 10 petahertz (PHz = 10
15 Hz))
in both band and Mott insulators.
Electromagnetic waves and quasi-particle dynamics in MHz and GHz (lowfrequency) range have been widely used in electronics and spintronics [9]. In this
low-frequency regime, it is difficult to create intense coherent waves like laser beams
(although there exists maser technique), and instead various methods of generating
electromagnetic waves have been established. For instance, Gunn diode has been