1 Chiral Coupling to Magnetodipolar Radiation
19
The equation of motions of the Kittel modes in the nanowire and film spin waves
with momentum q in the coupled system read
d ˆ
m L
dt
= −iω K ˆ
m L −
q
ig
∗
q e
iq R 1 ˆ
α q −
κ
2
ˆ
m L −
√
κ ˆ
N L ,
d ˆ
m R
dt
= −iω K ˆ
m R −
q
ig
∗
q e
iq R 2 ˆ
α q −
κ
2
ˆ
m R −
√ κ ˆ
N R ,
d ˆ
α q
dt
= −iω q ˆ
α q − ig q e
−iq R 1 ˆ
m L − ig q e
−iq R 2 ˆ
m R −
κ q
2
ˆ
α q −
√ κ q ˆ
N q ,
(1.42)
where κ is caused by the same Gilbert damping in both nanowires, and ˆ
N L and ˆ
N R
represent the thermal noise in the left and right nanowires, with ˆ
N
†
η (t) ˆ
N η (t
) =
n η δ(t − t
)δ ηη . Here, η ∈ {L , R} and n η = 1/
exp
˜
ω K /(k B T η )
− 1
and T R is
also the film temperature. Integrating out the spin-wave modes in the film, we obtain
equations for dissipatively coupled [60, 61] nanowires. In frequency space,
−i(ω − ω K ) +
κ
2
+
1 + 2
2
ˆ
m L (ω) + 2 e
iq ∗ |R 2 −R 1 |
ˆ
m R (ω)
(1.43)
=
q
ig
∗
q e
iq R 1 √
κ q G q (ω) ˆ
N q (ω) −
√
κ ˆ
N L (ω),
−i(ω − ω K ) +
κ
2
+
1 + 2
2
ˆ
m R (ω) + 1 e
iq ∗ |R 2 −R 1 |
ˆ
m L (ω)
=
q
ig
∗
q e
iq R 2 √
κ q G q (ω) ˆ
N q (ω) −
√
κ ˆ
N R (ω),
(1.44)
where 1 = |g q ∗ |
2
/v q ∗ and 2 = |g −q ∗ |
2
/v q ∗ are assumed constant (for the Kittel
mode). Here, q ∗ is the positive root of ω q ∗ = ˜
ω K .
For perfectly chiral coupling with 2 = 0 the solutions of (1.44) read
ˆ
m L (ω) =
q ig
∗
q e
iq R 1 √
κ q G q (ω) ˆ
N q (ω) −
√
κ ˆ
N L (ω)
−i(ω − ω K ) +
κ
2
+
1
2
,
ˆ
m R (ω) =
q ig
∗
q e
iq R 2 √
κ q G q (ω) ˆ
N q (ω) −
√
κ ˆ
N R (ω) − 1 e
q ∗ (R 2 −R 1 )
ˆ
m L (ω)
−i(ω − ω K ) +
κ
2
+
1
2
.
(1.45)
With ˆ
m L ,R (t) =
e
−iωt
ˆ
m L ,R (ω)dω/(2π), the Kittel modes are occupied according
to