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
Overview
•
New approach for measuring the absolute transition dipole moment μ of
molecular rovibronic transitions between the ground and the excited states by a 4level 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.
Extended  Excitation Scheme
1
2 g
1Π
g
20
L3
A 
3
|4>
X 1g+
3.6
3s+3s
1 +
3.4
3.2
X g
0
3.0
2
4
6
8
Internuclear distance (Å)
X 1g+
|1>
3.8
e(R) (a.u.)
8
4
L1
4.0
Coupling Laser L
12
4.2
1 
16
-1
A1u+
A1u+
|2>
3s+3p
1
3
L2: 21Πg(25,20)  A1u+(25,20)
L3: A1u+(25,20)— X 1g+(38,21)
Probe Laser L
24
Energy (cm )x10
L2
2
2
L1: A 1u+(25,20) X 1g+(1,19)
Pump Laser L
|3>
1
100
+
1
+
Na2 A g - X g Franck - Condon factors
0
0.01000
Autler-Townes split spectrum
0.02500
80
Intensity (Arbitrary Units)
413MHz
0.05000
0.1000
350mW
0.2000
0.3320
1
+
v ' (A u )
60
40
20
0
0
10
20
30
1
40
v'' (X g )
0
-0.5
0.0
0.5
Probe laser detuning (GHz)
50
+
______________________________
Annie Hanson, Peng Qi and Li Li
60
Experimental Setup and AT Splitting vs. Coupling Laser
Power
Experimental Setup
Experimental Data set 1
Experimental Data set 2
Linear fit
500
TiSa
L3
M
Verdi V10
AT Splitting,  (MHz)
M
M
PMT
Lock-in
Amplifier
400
300
200
Monochromator
P - Coupling laser (L3) power
Verdi V10
DCM
R6G
Sabre SBRC-DSW 25
L2
BS
M
L1
Lasers (699-29 or 899-29)
100
Sodium
Heatpipe
Oven
8
10
12
14
16
18
20
1/2
Square Root (P), mW
M
Mechanical modulator
413 MHz
303 MHz
Intensity
Intensity
Intensity
440 MHz
200 mW
350 mW
450 mW
465422.5
465423.0
465423.5
Probe laser detuning (GHz)
465422.5
465423.0
465423.5
Probe laser detuning (GHz)
465424.0
465422.5
465423.0
465423.5
Probe laser detuning (GHz)
465424.0
22
Simulation – Density Matrix Formalism
Excitation spectrum in the presence
of the coupling laser (Power 450mW)
OODR excitation spectrum
Intensity (arb. units)
1 = 28 MHz
2 = 52 MHz
Intensity (arb. units)
A1u+(25,20)— X 1g+(38,21)
1 = 28 MHz
2 = 52 MHz
0.0
0.0
-0.5
0.0
0.5
Probe laser detuning (GHz)
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 = 4.77 MHz;
Transit relaxation rate wt/2 = 0.38 MHz.
-1.0
-0.5
0.0
0.5
Probe laser detuning (GHz)
1.0
The Rabi frequency 3 of the coupling
field is used as fitting parameter 3 = 755 ( 10) MHz.
3 
  E3

 exp  5.65 D
  vJ  e R vJ 
ab initio  5.92D
E. Ahmed et. al., J. Chem. Phys. 124, 084308 (2006)
Electronic Transition Dipole Moment
 e R 
Calculating  e R  from the Experimentally Measured Dipole Moment Matrix
Elements

  vJ  e R vJ 
 e R     i R i
i 0
ith R-centroid
v' J '  e R  v" J "  v' J ' v" J "

 R
i 0
i
i
R 
i
v' J ' R i v" J ' '
v' J ' v" J ' '
R-centroid Approximation
 0  e Rc  
Rc  R1 
vJ   e R  vJ 
vJ  vJ 
1
+
1
+
A u (10,20)
v' J ' R v" J ' '
X g (17,21)
v' J ' v" J ' '
R
 e R 
 ' r ' ' r dr
0
J. Tellinghuisen, The Franck-Condon Principle in Bound-Free
Transitions, Advances in Chemical Physics Vol. 60, 1985
2
3
4
R, Å
5
6
Electronic Transition Dipole Moment as function of R
A1u+
(v',J')
25,20
X1g+
(v'',J')
38,21
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
34, 20
44, 19
2.82
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
,Debye
11.0
5.65
10.5
10.0
e(R), (Debye)
9.5
9.0
8.5
8.0
7.5
7.0
AT based R-centroid approximation results
AT based results using  e R    0   1R   2 R 2
2
Intensity based results (J. Huennekens)  e R    0   1R   2 R
ab initio pseudo-potential method (S. Magnier)
ab initio relativistic configuration method (S. Kotochigova)
6.5
6.0
4
5
6
R, (Å)
7
8
Quadruple Resonance Spectroscopy
In collaboration with Peng Qi
Na2dimer
energy
levelslevels
Sodium
energy
35
1
2 g
|3>
1 +
4 g
|5>
30
L3
L2
25
L4
+
1
+
Na2 A g - X g Franck - Condon factors
0
0.01000
0.02500
1 +
20
|4>
0.05000
0.1000
|2>
0.2000
60
0.3320
+
15
A u
80
v ' (A u )
L4
10
L1
|5> 1 +
X g
5
0
1
-1
Energy (cm )x10
-3
1
100
40
20
|1>
0
0
4
6
8
10
Internuclear distance R (Å)
12
10
14
20
30
1
40
+
v'' (X g )
50
60
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
Intensity (Arbitrary units)
L2
L3
L4
|4>
|2>
L1
Laser
wavenumbers
, cm-1
FranckCondon
Factor
we,
m
|1>  |2>
X1Σg+ (1,21) A1Σu+ (22,20)
16874.14
0.0079
300
|1>  |2>
A1Σu+ (22,20)21Πg (19,20)
15305.00
0.0502
280
|3>  |4>
21Πg (19,20) A1Σu+ (23,20)
15204.10
0.1676
405
|4>  |5>
A1Σu+ (23,20)41Σg+ (14,21)
12545.28
0.1898
660
|1>
0
-0.4
-0.2
Detuning0.0
of L2,GHz
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 = 4.77 MHz;
Transit relaxation rate wt/2 = 0.38 MHz.
0.2
0.4
The transtion |3>--|4> is coherently driven, =228MHz
Only spontaneous decay from |3> to |4> present
Simulations
Density Matrix Formalism
3 ,3
W32
Density matrix equation of motion in the interaction picture:
d
i
  H I ,     
dt

5 ,5
W34
W54
W52
4 , 4
2 , 2
7*
The total Hamiltonian H for the system:
4
i
H I    j i  1 i  1 
i 1 j 1
W41
 4
 i  i  1 i  i i  1 
2 i 1
W21
6*
1 , 1  0
 represents all relaxation terms:
3 ,3
W32


n 1


ij     ij   Wi  ij   ( k   i )Wki  kk   1   ij  ij  ij
k 1


k i


W34
7*
4 , 4
2 , 2
k 
Ek

 nm
1
c
  Wnk  Wmk    nm
2 k
W45
nm
5 ,5
6*
W21
1 , 1  0
W41
Experimental Results – Autler-Townes Splitting
400
|3
Experimental Data
Linear fit
350
AT Splitting,  (MHz)
L3
L2
|4>
|2>
L4
|5
L1
300
250
200
150
100
P - Coupling laser (L4) power
|1
50
5
0
455799.2
455799.6
455800.0
15
20
1/2
Square Root (P), mW
-0.6
-0.4
-0.2
0.0
3, GHz
0.2
0.4
25
Coupling laser power = 700mW
Coupling laser power = 500mW
Coupling laser power = 100mW
455798.8
10
0.6
-0.6
-0.4
-0.2
0.0
3, GHz
0.2
0.4
0.6
Simulations-Density Matrix Formalism
Simulation of the experimental fluorescence spectra from level |4> with 4 as adjustable
parameter
Experimental spectrum P =450mW
Simulation =940MHz
Intensity (Arbitrary Units)
Transition
-0.4
-0.2
0.0
0.2
Franck-Condon
Factor
we,
m
|1>  |2>
X1Σg+ (1,21)  A1Σu+ (22,20)
16874.14
0.0079
300
|1>  |2>
A1Σu+ (22,20)21Πg (19,20)
15305.00
0.0502
405
|3>  |4>
21Πg (19,20) A1Σu+ (23,20)
15204.10
0.1676
278
|4>  |5>
A1Σu+ (23,20) X1Σg+ (36,19)
12533.75
0.2324
505
0.4
Detuning of L3,GHz
exp  4.87 D
Laser
wavenumbers
, cm-1
ab initio  5.05D
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 = 4.77 MHz;
Transit relaxation rate wt/2 = 0.38 MHz.
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.
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
Density Matrix Formalism
Density matrix equation of motion in the interaction picture:
d
i
  H I ,     
dt

The total Hamiltonian H for the system:


H I    k  k lk  lk k     j k k 
k 1  2
j

n 1
 represents all relaxation terms:


n 1


ij     ij   Wi  ij   ( k   i )Wki  kk   1   ij  ij  ij
k 1


k i


k 
Ek

 nm 
1
c
Wnk  Wmk    nm

2 k
nm
Density Matrix Equations (scheme A)
d11
 i1 12   21   W21  22  W41 44
dt
d 33
 i 2  23   32   W3 33  i 3  34   43 
dt
d 22
 i1 12   21   W2  22  i 2  23   32   W32  33  W52  55
dt
d 44
 i 3  34   43   W34 33  i 4  45   54   W4 44  W54 55
dt
d 55
 i 4  45   54   W5 55
dt
d12
 d1112  i1  22  i1 11  i 2 13
dt
d34
 i 2  24  i3 33  d 3334  i 4 35  i3 44
dt
d13
 i 2 12  d12 13  i3 14  i1 23
dt
d35
 i 2  25  i 4 34  d 34 35  i3 45
dt
d14
 i3 13  d1314  i 4 15  i1 24
dt
d 45
 i3 35  i 4 44  d 44  45  i 4 55
dt
d15
 i 4 14  d14 15  i1 25
dt
d 23
 i113  i 2  22  d 22  23  i 3  24  i 2 33
dt
d 24
 i114  i3  23  d 23 24  i 4  25  i 2 34
dt
d 25
 i115  i 4  24  d 24  25  i 2 35
dt
k
where:
dlk  i   j   lk 1
l , k  1,..,4
lk
j l
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.
N  11   22  33   44  55  66  77
Diagrams of the Excitation and Decay Processes
Excitation scheme A
Excitation scheme B
3 ,3
W32
W34
4 , 4
7 *
7 *
4 , 4
2 , 2
W45
1,1
1,1
1 , 1  0
4,4
W41
W21
3,3
2 , 2
2,2
W54
W32
4,4
3,3
2,2
W52
3 ,3
5 ,5
W34
5 ,5
6 *
6 *
W21
W41
1 , 1  0
*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.
Measuring the amplitude E of the electric field
1.0
For a Gaussian beam we have:
0.8
w2  z 
0.6
I/I0
I r , z   I 0  z e

r2
0.4
w( z ) - beam waist, the radius at which the intensity
0.2
w( z )
drops 1/e2 from the maximum value of I0
0.0
-800
Using razor blade technique one can measure w( z )
w z 
-400
0
400
800
r,m
1
d 75%  d 25% 
2C
where C is: erf C   , C  0.47
1
2
E0 
2 Ptot
2

c 0
w2
d
0.05< FCF < 0.07
0.07< FCF < 0.1
0.1< FCF
Pseudopotential calculations
ab initio data
11.0
10.5
µe(R), Debye
10.0
9.5
9.0
8.5
0.0
2
4
6
R, Å
8
10
1
+
1
+
A u (35,20)
X g (45,21)
2
3
4
R, Å
5
6
Transition Dipole Moment Measurements Using the AutlerTownes (AT) effect
 AT  
 AT
Laser field, E
|2>

 AT
|1>
The AT splitting arises from the
two dressed States
 1, n
E
 2, n  1 
2


-Transition dipole moment matrix element
|2>
Laser field, E
|1>
Probe Laser Laser field, E
|2>
Probe Laser
 AT
|1>
