2 Mechanism of Population Inversion in N 2
+
29
where
H βv K m αvK m (t) =
∞
0
π
0
2π
0
ψ
∗
βv K m (r)H (t)ψ αvK m (r)dr sin θ dθ dφ.
(2.18)
Under an intense laser pulse, the total Hamiltonian H of the system interacting with
the laser field becomes
H (t) = −
2
2μ
∇
2
+ V + H 1 (t) = H 0 + H 1 (t),
(2.19)
where μ is the reduced mass of N 2
+ , V is the interatomic potential energy, and H 1
stands for the interaction with the laser field. Therefore, H βv K m αvK m (t) becomes
H βv K m αvK m (t) = (H 0 ) βv K m αvK m + (H 1 ) βv K m αvK m (t),
(2.20)
where
(H 0 ) βv K m αvK m =
ψ βv K m
H 0 |ψ αvK m
= ε αvK m δ αβ δ v v δ K K δ m m ,
(2.21)
and
(H 1 ) βv αv (t) =
ψ βv K m
H 1 (t) |ψ αvK m
=
ψ βv K m
D αβ F
αβ
θ |ψ αvK m E(t).
(2.22)
In (2.22), representing the coupling between the rotational levels via the transition
dipole moment D αβ , E(t) stands for the laser field strength at time t defined as
E(t) = E 0 f (t) cos(ωt),
(2.23)
where f (t) is an envelope function, ω is the frequency of the laser pulse. F
αβ
θ is the
angular factor representing the polar-angle dependent coupling between the X, A
and B states defined as
F
αβ
θ = cos(θ )(δ Bα δ Xβ + δ X α δ Bβ ) + sin(θ )(δ Aα δ Xβ + δ X α δ Aβ ).
(2.24)
The envelope function is defined as
f (t) = e
−t
2 /2σ
2
0 ,
(2.25)
where σ is chosen according to the intensity half-width of the laser pulse adopted from
the experimental conditions and the simulation is conducted from t = 0, resulting in
a half laser pulse with a sudden turn-on behavior.