Femtosecond wave-packet dynamics in cesium dimers studied through controlled stimulated emission Page: 3
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FEMTOSECOND WAVE-PACKET DYNAMICS IN CESIUM ...
calculating the absorption of the probe pulse in this four-level
system, vibrational levels in the ground and excited states are
rechosen and the calculations are repeated for another set of
vibrational levels. The total absorption of the probe pulse is
defined as a sum over all possible absorption (emission) paths
for such four-level configurations.
Because of the large relaxation time of cesium dimer
(- 1 ns), the equation of motion for the density matrix is [20]p = - [H, p].
h,(6)
Since the durations of the pump pulse and the probe pulse
(<100 fs) are much smaller than the time delay of the probe
pulse (> 1 ps), we can calculate the corresponding molecular
polarizations induced by them separately. Let us consider
one radiation field, Ei(t) = Ecos(vit + oi), where i p
for pump and i = pr for probe, and do the rotation-wave
approximation (RWA)Pab -- abe-ivit-o ,
Pac -- ace-ivt-i o
Pa'b -- abe-ivit-i ,
Pa'c = Pa'ce(7)
(8)
(9)
(10)We can write down the density matrix elements explicitly as
i PabE (
lab - -i(c0ab - Vi) ab _ (Paa
h 2i ac EcE
+ 52Pcb
h 2
Pac = -i (ac - Vi) ac
i &ab Ei
+ 2Pbc
h 2
Pa'b = -i(a'b - i)Pa'b
i &acE0
+ iPcb
h 2
Pa 'c = -i a'c - i) a'c
i Pa'bEO
+ 2Pbc
h 2i &a'bE
h 2 Paa' ,
h9 2
acE (Paa-
h 2
i P&alcE
S Paa',
h 2
i albE
h 2 Paa
i Pab Ei
i a Pa'a ,
h 2
h2 (Pa'a'
i & aac EO
h 2Pa'a,Pbb)
(11)
Pcc)
PHYSICAL REVIEW A 81, 053405 (2010)
Pbb =' 2 iab 2 Pa'b
Pa'b EO a)
2 Pba'
Pc ca Ea+ca Ea
C h\ 2 2 Pa
2 Pca ,
Pa'a Qa 2 E Pba' 2 Pca
ca' E 2 .
2 /'&ab Er
2 ba
(18)
PacE
2 Pca
(19)
Pba' E
2 Pa'b
(20)The general equation for the absorption of this field is
[24]
Q fPEdt
S(&abba + Paclca +Pa'bba' + a'cca' +c.c.)Ei(t)dt.
(21)
After RWA and using the approximation << vpr p, this
absorption equation will turn to
& ba Er ca E pr
Q = 2Im vpr Pab + 2 PacPba' Er 'ca Er
+ pab +a' s dt.
2 2(12)
Pbb)
Pcc)i pbaE E &ba' E
Pbc = iCcbPbc 2 Pac 2 a'c
h 2 h 2
i PacE i PacE
h 2 ba 2 ba,
baa = i a a Paa' -4 - a' - - Pea a
P aa' PabEO i bca' ac
Pab - Pac,
h 2 h 2S(ab E9 Pac E
Paa = y2 Pba + 2 Pca
2 ac ,&ba E
2 Pab
2(22)
Therefore, the density matrix elements can be found when
the pump pulse with frequency Vp and phase op comes in first
if we use the Eqs. (11) to (20). Then these results can be the
initial conditions for the calculation when the probe pulse with
(13) frequency Vpr and phase qopr comes in with the time delay At.
Validity of this assumption rests on the fact that the decay time
scale of Cs2 is too long (- 1 ns) compared to probe delay time
At. Matrix elements can be found with the Eqs. (11) to (20).
The absorption can also be found by the Eq. (22) with the time
(14) delay. We have to average over all possible phases because of
the noncollinear beam geometry. Pulses do not copropagate
in our experiments. Even if the phase of each pulse is stable,
it is still different in different parts of the sample. Therefore,
we should average our results of absorption with different
(15) phases in each pairs of pulses to find our final numerical
result.
Now let us consider how to find the analytical solution.
First, we consider one pump pulse coming at time t 0 with
the Gaussian pulse shape, Ep(t) Ep cos(vpt + qp), where
(16))2 0
Eo - 8pe 2ye . 2 Since Pbb Pbb Paa, Pcb, and paa* where
pb is the initial condition, Eq. (11) can be approximated
with(17)
i PabEp O
Pab ' -i(Clab - Vp)ab + Pbb
h 2(23)
053405-3
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Yuan, Luqi; Ariunbold, Gombojav O.; Murawski, Robert K.; Pestov, Dmitry; Wang, Xi; Patnaik, Anil K. et al. Femtosecond wave-packet dynamics in cesium dimers studied through controlled stimulated emission, article, May 12, 2010; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc103267/m1/3/?q=%22Ariunbold%2C%20Gombojav%20O.%22: accessed May 1, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.