24
Y. Zhang et al.
where the time-dependent Hamiltonian H is
H (t) =
ε 1
−μE(t)
−μE(t)
ε 2
,
(2.2)
where ε i is the eigenenergy of the i-th level, μ is the dipole moment between the two
levels, and E(t) is the laser field defined as
E(t) = E 0 g(τ ) cos(τ ),
(2.3)
where E 0 is the peak field strength and g(τ ) is the envelope function of the sudden
turn-on pulse,
g(τ ) =
0
ifτ < 0,
e
−
τ 2
σ 2
if τ ≥ 0,
(2.4)
where σ determines the pulse width and τ = ωt with ω being the laser frequency.
In order to examine the sudden turn-on behavior, we modify the pulse shape using
an envelope function g k (τ ) given with a steepness parameter k as
g k (τ ) =
⎧
⎨
⎩
e
−
τ 2
σ 2 −
τ 2
k 2
if τ < 0,
e
−
τ 2
σ 2
if τ ≥ 0.
(2.5)
When k 1, the pulse becomes close to a sudden turn-on pulse, and when k 1,
the pulse becomes a full Gaussian pulse.
The final population in the excited state at the two-photon resonance after the laser
pulse vanishes is plotted as a function of the coupling strength parameter a = −
μE 0
2ω
,
where ω represents the one photon energy in laser pulse.
As shown in Fig. 2.2, when the pulse loses the sudden turn-on behavior at k = 10,
population inversion requires a larger value of a ≈ 0.89, which is larger than a = 0.6
required in the case of a sudden turn-on pulse at k = 0.01.
2.1.4 Population Inversion Achieved by Sudden Turn-On
Pulse in N 2
+
In [6, 15], using a sudden turn-on intense laser pulse, a numerical simulation of
population transfer in N 2
+ generated by intense laser pulse was performed and the
population inversion was shown to be achieved between the X
2
g
+ state and the
B
2
u
+ state of N 2
+ with the same laser parameters as in their experiment. Later in
[33], by quasi-stationary Floquet theory with a sudden turn-on pulse, we investigated
the mechanism of the population inversion achieved between the X
2
g
+ state and
the B
2
u
+ state of N 2
+ through electronic and vibrational excitations induced by