266
M. Sato
THz or gigahertz (GHz) waves (10 GHz–1.0 THz) have been long investigated [6,
7] (GHz = 10
9 Hz). The control of magnetism with laser or electromagnetic wave [5,
8] has also gathered much attention as a large branch of spintronics [9].
Recently, several theoretical methods for non-equilibrium systems have also
developed, and they have gradually become widespread in broad fields of condensedmatter physics. Non-equilibrium Green’s function method [10–14], approaches based
on quantum master equation [15–19], and Floquet theory [20–23] are representative
of them. These techniques have a high potential to capture different aspects of laserdriven non-equilibrium dynamics. In fact, with such methods, I and collaborators
have theoretically explored/proposed several ways of controlling physical properties
of materials, especially, focusing on magnetic systems [24–32].
Among studies for laser-driven phenomena, the concept “Floquet engineering”
has been an important keyword and provided us various research directions. This
terminology stands for controlling physical (especially static) properties of target
systems by applying an AC field whose frequency (photon energy) is much higher
than the energy scale of the systems. Excitations or quasi particles of focused systems
cannot be directly coupled to such a high-frequency AC field, but it is known that static
or low-frequency properties of the systems can be changed through the nonlinear
effects of the AC field. This effect is theoretically formulated by Floquet theorem
and related techniques developed in recent years [20–23].
In this chapter, I would like to review the theoretical basis of Floquet engineering
and its application to simple, realistic magnetic systems. I will explain that basic
magnetic quantities such as magnetization, spin chirality, and spin current can be
controlled by application of intense THz laser or wave to magnets.
11.2 Floquet Engineering
This section is devoted to the explanation about the theoretical basis of Floquet
engineering [20–23]. As I mentioned, Floquet engineering means creating nonequilibrium states with desirable (static) physical properties by periodically driving
(i.e., by applying an external AC field to) a material. This concept stems from Floquet
theorem, and so I start from the explanation about the theorem. Then I will derive the
Floquet effective Hamiltonian through so-called high-frequency (Floquet-Magnus)
expansion. The effective Hamiltonian is the most important instrument in Floquet
engineering from its conceptual viewpoint, and it describes slow dynamics of the
driven system. Finally, I will state some remarks on the physical meaning of Floquet
Hamiltonian.
As one will see soon later, the Floquet theory based on Floquet theorem assumes
that (i) the external AC field is treated as a classical number (not operator) in the
Hamiltonian considered, and (ii) the driven system is decoupled to any environment,
i.e., we consider “isolated” quantum systems. The assumption (i) might be sometimes
justified. For instance, if the intensity of applied laser is large enough, its AC electric
and magnetic fields can be approximated by classical external fields. The condition
M. Sato
THz or gigahertz (GHz) waves (10 GHz–1.0 THz) have been long investigated [6,
7] (GHz = 10
9 Hz). The control of magnetism with laser or electromagnetic wave [5,
8] has also gathered much attention as a large branch of spintronics [9].
Recently, several theoretical methods for non-equilibrium systems have also
developed, and they have gradually become widespread in broad fields of condensedmatter physics. Non-equilibrium Green’s function method [10–14], approaches based
on quantum master equation [15–19], and Floquet theory [20–23] are representative
of them. These techniques have a high potential to capture different aspects of laserdriven non-equilibrium dynamics. In fact, with such methods, I and collaborators
have theoretically explored/proposed several ways of controlling physical properties
of materials, especially, focusing on magnetic systems [24–32].
Among studies for laser-driven phenomena, the concept “Floquet engineering”
has been an important keyword and provided us various research directions. This
terminology stands for controlling physical (especially static) properties of target
systems by applying an AC field whose frequency (photon energy) is much higher
than the energy scale of the systems. Excitations or quasi particles of focused systems
cannot be directly coupled to such a high-frequency AC field, but it is known that static
or low-frequency properties of the systems can be changed through the nonlinear
effects of the AC field. This effect is theoretically formulated by Floquet theorem
and related techniques developed in recent years [20–23].
In this chapter, I would like to review the theoretical basis of Floquet engineering
and its application to simple, realistic magnetic systems. I will explain that basic
magnetic quantities such as magnetization, spin chirality, and spin current can be
controlled by application of intense THz laser or wave to magnets.
11.2 Floquet Engineering
This section is devoted to the explanation about the theoretical basis of Floquet
engineering [20–23]. As I mentioned, Floquet engineering means creating nonequilibrium states with desirable (static) physical properties by periodically driving
(i.e., by applying an external AC field to) a material. This concept stems from Floquet
theorem, and so I start from the explanation about the theorem. Then I will derive the
Floquet effective Hamiltonian through so-called high-frequency (Floquet-Magnus)
expansion. The effective Hamiltonian is the most important instrument in Floquet
engineering from its conceptual viewpoint, and it describes slow dynamics of the
driven system. Finally, I will state some remarks on the physical meaning of Floquet
Hamiltonian.
As one will see soon later, the Floquet theory based on Floquet theorem assumes
that (i) the external AC field is treated as a classical number (not operator) in the
Hamiltonian considered, and (ii) the driven system is decoupled to any environment,
i.e., we consider “isolated” quantum systems. The assumption (i) might be sometimes
justified. For instance, if the intensity of applied laser is large enough, its AC electric
and magnetic fields can be approximated by classical external fields. The condition