1. Experimental Mapping of the Absolute Value of the Transition Dipole Moment Function μe(R) of the Na2 A1Σu+ - X1Σg+ Transition
E. Ahmed1, B. Beser1, P. Qi1, S. Kotochigova1, A. M. Lyyra1 and
J. Huennekens2,
1Physics Department, Temple University
2Physics Department, Lehigh University

2. Overview
New approach for measuring the absolute transition dipole moment μ of molecular rovibronic transitions between the ground and the excited states by a 4- level extended Λ scheme.
Using the R-centroid method, we determine the electronic transition dipole moment μe(R) as function of the internuclear distance R.
To extend the range of accessible transitions beyond the ones available with the extended Λ scheme we have demonstrated a new 4-laser excitation scheme.

3. Extended  Excitation Scheme
X 1g+
|1>
|2>
|3> L1 L2 L3
|4>
A1u+
2 1Πg
L1: A 1u+(25,20) X 1g+(1,19)
L2: 21Πg(25,20)  A1u+(25,20)
L3: A1u+(25,20)— X 1g+(38,21)
Autler-Townes split spectrum
______________________________
Annie Hanson, Peng Qi and Li Li

4. Experimental Setup and AT Splitting vs. Coupling Laser Power
TiSa M
Verdi V10 M M
Lock-in
Amplifier
PMT
Monochromator
Verdi V10 L2 BS
Sodium
Heatpipe Oven
DCM L1 M
R6G M
Sabre SBRC-DSW 25
Lasers ( or )
Mechanical modulator

5. Simulation – Density Matrix Formalism
Excitation spectrum in the presence
of the coupling laser (Power 450mW)
OODR excitation spectrum
A1u+(25,20)— X 1g+(38,21)
1 = 28 MHz
2 = 52 MHz
1 = 28 MHz 2 = 52 MHz
The Rabi frequency 3 of the coupling
field is used as fitting parameter 3 = 755 ( 10) MHz.
Parameters:
Lifetime A 1u+ 2 = 12.5 ns, 21g 3 = 18.3 ns;
branching ratios W32/W3 = 0.076, W21/W2 = 0.001, W24/W2 = 0.16;
Doppler width 1.15 GHz;
Collisional dephasing rates ij/2 = MHz;
Transit relaxation rate wt/2 = 0.38 MHz.
E. Ahmed et. al., J. Chem. Phys. 124, (2006)

6. Electronic Transition Dipole Moment
Calculating from the Experimentally Measured Dipole Moment Matrix Elements
ith R-centroid
R-centroid Approximation
J. Tellinghuisen, The Franck-Condon Principle in Bound-Free Transitions, Advances in Chemical Physics Vol. 60, 1985

7. Electronic Transition Dipole Moment as function of R
A1u+ (v',J')
X1g+
(v'',J')
,Debye
25,20
38,21
5.65
28, 20
42, 21
2.02
28, 20
40, 21
3.40
28, 20
43, 21
3.33
28, 20
41, 21
4.45
33, 20
43, 21
3.20
33, 20
43, 19
3.26
33, 20
46, 19
2.96
33, 20
48, 21
3.38
33, 20
51, 21
2.10
34, 20
44, 21
2.66
AT based results using
34, 20
44, 19
2.82
Intensity based results (J. Huennekens)
34, 20
48, 21
1.97
35, 20
45, 19
1.82
10, 20
20, 21
2.58
10, 20
17, 21
2.01
10, 20
23, 21
4.00
8, 20
20, 21
3.10
14, 20
27, 19
3.18

8. In collaboration with Peng Qi
Quadruple Resonance Spectroscopy
In collaboration with Peng Qi
Na2 energy levels

9. Experimental Results and Simulations – Stimulated Emission
Quadruple resonance single channel fluorescence spectra from level |5>. Comparison
between coherently driven and spontaneous decay only |3>|4> transition.
|3
|5
Transition
Laser wavenumbers, cm-1
Franck-Condon
Factor
we, m L2 L3 L4
|1>  |2>
X1Σg+ (1,21) A1Σu+ (22,20)
0.0079
300
|4>
|1>  |2>
A1Σu+ (22,20)21Πg (19,20)
0.0502
280
|2>
|3>  |4>
21Πg (19,20) A1Σu+ (23,20)
0.1676
405
|4>  |5>
A1Σu+ (23,20)41Σg+ (14,21)
0.1898
660 L1
|1>
Parameters for the simulation:
Lifetime A 1u+ 2 = 12.5 ns, 21g 3 = 18.3 ns, 41g+ 4 = 12.2 ns,
Rabi frequencies 1=56MHz, 2=104MHz, 3=228MHz
Doppler width 1.15 GHz;
Collisional dephasing rates ij/2 = MHz;
Transit relaxation rate wt/2 = 0.38 MHz.

10. The total Hamiltonian H for the system:
Density Matrix Formalism
Density matrix equation of motion in the interaction picture:
W32
W34
W54
W52
The total Hamiltonian H for the system:
W41
W21
represents all relaxation terms:
W32
W34
W45
W21
W41

11. Experimental Results – Autler-Townes Splitting
|3 L3 L2
|4>
|2> L4
|5 L1
|1

12. Simulations-Density Matrix Formalism
Simulation of the experimental fluorescence spectra from level |4> with 4 as adjustable parameter
505
0.2324
A1Σu+ (23,20) X1Σg+ (36,19)
|4>  |5>
278
0.1676
21Πg (19,20) A1Σu+ (23,20)
|3>  |4>
405
0.0502
A1Σu+ (22,20)21Πg (19,20)
|1>  |2>
300
0.0079
X1Σg+ (1,21)  A1Σu+ (22,20)
we, m
Franck-Condon
Factor
Laser wavenumbers, cm-1
Transition
Parameters for the simulation:
Lifetime A 1u+ 2 = 12.5 ns, 21g 3 = 18.3 ns;
Rabi frequencies 1=58MHz, 2=91MHz, 3=185MHz
Doppler width 1.15 GHz;
Collisional dephasing rates ij/2 = MHz;
Transit relaxation rate wt/2 = 0.38 MHz.

13. Conclusion
Using the extended Λ scheme we have measured the absolute value of the transition dipole moment between A1Σu+ and X1Σg+ states of Na2 for a number of rovibrational transitions.
Using the R-centroid method, we have investigated the internuclear distance R dependence of e(R).
To extend the range of accessible transitions, we have demonstrated a new 4-laser excitation scheme . To predict and simulate the experimental spectra, a theoretical model based on the density matrix formalism was developed.

14. Acknowledgments Prof. L. Li, Tsinghua University
Prof. R. W. Field, MIT
Prof. S. Magnier, Rennes, France
Prof. R. Le Roy, University of Waterloo (Level program)
Annie Hansson
Teodora Kirova
Jianmei Bai
Omer Salihoglu
Bill Stevenson
Ed Kaczanowicz

15. The total Hamiltonian H for the system:
Density Matrix Formalism
Density matrix equation of motion in the interaction picture:
The total Hamiltonian H for the system:
represents all relaxation terms:

16. Density Matrix Equations (scheme A)
where:
Each equation involving the time derivative of the off diagonal matrix elements on the left side has a complex conjugate equation.
The set of equations are solved in the limit of steady state approximation, along with a condition for conservation of the population.

17. Diagrams of the Excitation and Decay Processes
Excitation scheme A
Excitation scheme B
1,1
W32
W21
W34
W41
W52
W54
4,4
3,3
2,2
W32
W21
W34
W41
W45
3,3
4,4
2,2
1,1
*Levels |6> and |7> represents all other ro-vibrational levels of the ground and first excited electronic
states, respectively. They are not coherently coupled to the system.

18. Using razor blade technique one can measure
Measuring the amplitude E of the electric field
For a Gaussian beam we have:
- beam waist, the radius at which the intensity
drops 1/e2 from the maximum value of I0
Using razor blade technique one can measure
where C is: d

21. -Transition dipole moment matrix element
Transition Dipole Moment Measurements Using the Autler-Townes (AT) effect
The AT splitting arises from the
two dressed States
|2>
Laser field, E
|1>
-Transition dipole moment matrix element
Laser field, E
|1>
|2>
Probe Laser
Laser field, E
|1>
|2>
Probe Laser d