280
M. Sato
ˆ
H eff = ˆ
H mag −
B +
B
2
0
2ω
ˆ
S
z
tot + O
(ω)
−2
.
(11.25)
One finds that an effective Zeeman interaction −(B
2
0 /(2ω)) ˆ
S
z
tot emerges owing to
the applied laser. The Floquet Hamiltonian indicates that circularly polarized THz
laser can change the value of magnetization ˆ
S
z
tot . An image of this Floquet theory
is depicted in Fig. 11.3. It is well known that if we apply a circularly polarized highfrequency wave (such as visible light) to metallic systems with SO coupling, an
effective Zeeman interaction also appears through the combination between chargelight and SO couplings. This Floquet engineering is called inverse Faraday effect [53]
and it has been utilized in various studies of condensed-matter and applied physics [8].
Equation (11.25) shows that an inverse Faraday effect can also be generated even by
applying circularly polarized THz (low-frequency) laser through the direct spin-light
coupling.
However, we should note that if the spin Hamiltonian ˆ
H mag is SU(2)-symmetric
(i.e., spin-rotation symmetric) like the Heisenberg model
r,r
ˆ
S r · ˆ
S r , the static
(time averaged) magnetization ˆ
S
z
tot does not change even under the existence of the
laser-driven Zeeman term [24]. This is understood by mapping the driven system to
that on a rotating frame via the Unitary transform
ˆ
U(t) = exp(i ˆ
S
z
tot ωt).
(11.26)
This mapping has been often used in the study of magnetic resonance [6] and it
eliminates the time-periodic Zeeman coupling since the Unitary rotation has the
same frequency ω as the laser. After acting ˆ
U(t) to the Schrødinger equation from
the left side, the transformed Hamiltonian ˆ
H u = ˆ
U(t) ˆ
H (t) ˆ
U
†
(t) − i ˆ
U(t)(∂ t ˆ
U
†
(t)) is
given by
ˆ
H u = ˆ
H mag − B ˆ
S
z
+ B 0 ˆ
S
y
− ω ˆ
S
z
= ˆ
H mag − B u · ˆ
S,
(11.27)
where we have defined a new field B u = (0, −B 0 , B + ω). We note that the spin
Hamiltonian ˆ
H mag is invariant through the mapping due to the SU(2) symmetry. As
expected, the in-plane AC field is mapped to a static field B 0 along the S
y axis, while
an additional Zeeman term −ω ˆ
S
z emerges. From this static model (11.27), we see
that the spin along the new field B u is conserved and therefore the static magnetization
cannot grow even by applying any circularly polarized laser. For instance, a magnetic
anisotropy such as single-ion terms D z
r ( ˆ
S
z
r )
2 and Ising ones z
r,r
ˆ
S
z
r
ˆ
S
z
r is
necessary to break this conservation. Such anisotropies stem from SO coupling and
after all it means that even the THz inverse Faraday effect requires an interaction
connecting real and spin spaces like the usual high-frequency inverse Faraday effect.
In [24, 68], the detailed condition for generating a large ˆ
S
z
tot is discussed and its
value is numerically computed.
M. Sato
ˆ
H eff = ˆ
H mag −
B +
B
2
0
2ω
ˆ
S
z
tot + O
(ω)
−2
.
(11.25)
One finds that an effective Zeeman interaction −(B
2
0 /(2ω)) ˆ
S
z
tot emerges owing to
the applied laser. The Floquet Hamiltonian indicates that circularly polarized THz
laser can change the value of magnetization ˆ
S
z
tot . An image of this Floquet theory
is depicted in Fig. 11.3. It is well known that if we apply a circularly polarized highfrequency wave (such as visible light) to metallic systems with SO coupling, an
effective Zeeman interaction also appears through the combination between chargelight and SO couplings. This Floquet engineering is called inverse Faraday effect [53]
and it has been utilized in various studies of condensed-matter and applied physics [8].
Equation (11.25) shows that an inverse Faraday effect can also be generated even by
applying circularly polarized THz (low-frequency) laser through the direct spin-light
coupling.
However, we should note that if the spin Hamiltonian ˆ
H mag is SU(2)-symmetric
(i.e., spin-rotation symmetric) like the Heisenberg model
r,r
ˆ
S r · ˆ
S r , the static
(time averaged) magnetization ˆ
S
z
tot does not change even under the existence of the
laser-driven Zeeman term [24]. This is understood by mapping the driven system to
that on a rotating frame via the Unitary transform
ˆ
U(t) = exp(i ˆ
S
z
tot ωt).
(11.26)
This mapping has been often used in the study of magnetic resonance [6] and it
eliminates the time-periodic Zeeman coupling since the Unitary rotation has the
same frequency ω as the laser. After acting ˆ
U(t) to the Schrødinger equation from
the left side, the transformed Hamiltonian ˆ
H u = ˆ
U(t) ˆ
H (t) ˆ
U
†
(t) − i ˆ
U(t)(∂ t ˆ
U
†
(t)) is
given by
ˆ
H u = ˆ
H mag − B ˆ
S
z
+ B 0 ˆ
S
y
− ω ˆ
S
z
= ˆ
H mag − B u · ˆ
S,
(11.27)
where we have defined a new field B u = (0, −B 0 , B + ω). We note that the spin
Hamiltonian ˆ
H mag is invariant through the mapping due to the SU(2) symmetry. As
expected, the in-plane AC field is mapped to a static field B 0 along the S
y axis, while
an additional Zeeman term −ω ˆ
S
z emerges. From this static model (11.27), we see
that the spin along the new field B u is conserved and therefore the static magnetization
cannot grow even by applying any circularly polarized laser. For instance, a magnetic
anisotropy such as single-ion terms D z
r ( ˆ
S
z
r )
2 and Ising ones z
r,r
ˆ
S
z
r
ˆ
S
z
r is
necessary to break this conservation. Such anisotropies stem from SO coupling and
after all it means that even the THz inverse Faraday effect requires an interaction
connecting real and spin spaces like the usual high-frequency inverse Faraday effect.
In [24, 68], the detailed condition for generating a large ˆ
S
z
tot is discussed and its
value is numerically computed.