Optimal laser impulsion for controlling population
within the Ns = 1, Nr = 5 polyad of 12C2H2
L. Santos1, N. Iacobellis1, M. Herman1, D.S. Perry2, M. Desouter-Lecomte3, N. Vaeck1
1
Laboratoire de Chimie quantique et Photophysique, Université Libre de Bruxelles, Belgium
2 Department of Chemistry, The University of Akron, USA
3 Laboratoire de Chimie Physique, Université Paris-sud, Orsay, France
5
0
−5
0
5
10
15
Time (ps)
20
0
1
5
10
15
Time (ps)
20
25
0.5
|0 , 0 , 0 , 0 0 , 0 0 i
5
10
15
20
|Γ 1 i
|Γ 3 i
1
Probability
Optimal pulses and
evolutions of
populations
Laser control
0.5
0
0
5
10
15
Time (ps)
20
Spectroscopic
observation of
vinylidene
Global acetylene Hamiltonian and the polyad (Ns = 1,
30 years of extensive investigations of the 12C2H2 vibrationrotation spectrum under high-resolution spectroscopic conditions
have led to the construction of an effective Hamiltonian[1]. The
energy precision is typically between 10−4 and 10−3 cm−1. In
this work, we focus on the polyad Ns = 1, Nr = 5 of the acetylene.
Nr
ν1
3397.12
ν2
1981.80
ν3
3316.86
ν4
608.73
ν5
729.08
The polyad Ns = 1, Nr = 5 corresponds
to the following zero-order states: (0010000),
(010111−1), (0101111) according to the notation
l l
(v1v2v3v44 v55 )[2]. In the figure, the time evolution
of the zero-order states, the bright state (in red) and
the two dark states (in white), is shown as well as
the spectroscopic observation.
Spectro. obs.
ν
20
25
−5
0
5
10
15
Time (ps)
20
25
ν2
↑
4350
4400
4400
4450
0.04
→ ν3
ν1
↑
0.02
6000
ν2
↑
5000
→ν
3
4390
4000
→ν
2
4380
3000
4370
2000
→ ν1
4360
0
4300
4350
4400
Wavenumber (cm−1)
4450
4350
0
1000
5
10
15
Time (ps)
Pulse driven evolutions
Free evolutions
(a)
(a)
1
Probability
Transition frequencies from the groundstate
1
0.5
|0 , 0 , 0 , 0 0 , 0 0 i
5
10
15
20
10
15
Time (ps)
40
60
|Γ 2 i
0.5
5
0.5
0
|Γ 1 i
|Γ 3 i
20
80
100
120
νΓ2 = EΓ2 = 4382.7 cm−1
|0 , 1 , 0 , 1 1 , 1 1 i
(b)
1
|0 , 1 , 0 , 1 1 , 1 −1 i
0.5
0
40
60
80 100
Time (ps)
120
Eigenstates
Zero-order states
case(a)
case(b)
Outlook
- Choose an higher polyad which includes pure bending dark states (ν1,ν2,ν3 = 0). Those are believed to be
involved in the isomerization of acetylene to vinylidene[6]:
Functional: Fidelity[3]:
0
Z T
0
#
i
k
∂
λ(t) H + ψ (t) dt
h̄
∂t
• Amplitude of the guess pulse, A: 107V/m
• Penalty factor, α0: 50
• Shape of the initial guess pulse 0(t)
πt 2
0(t) = A sin
cos(νΓ1 t) + cos(νΓ2 t) + cos(νΓ3 t)
T
- Taking into account :
• Rotational coupling (l-resonances) between vibrational states.
• Coriolis coupling between vibrational states.
• Interactions with polayds that have similar total energies.
• Duration of the pulse, T: 24.2 ps
– Time step: 48.4 as
Requirements for the polyad:
• High vibrational bend excitation (ν4,ν5)
• Include a bright state
Evolution equations: Variational method
Resolution: Rabitz’s algorithm[4]
Representation: Dirac
Optimized parameters:
F = 0.9999 after 200 iterations
• Influence on the pulse intensity from states accessed by hot band transitions[7].
– Number of step: 500000
References
[1]
[2]
[4]
[5]
[6]
[7]
140
Pulse driven evolutions: We have reached the respective target populations in both cases. We see that the population of the ground state decreases
monotonously with oscillations of populations between the ground state and
the states Γ1 and Γ2.
Free evolutions: The field-free evolutions of 121 ps show that there are two
periods of oscillations. One about 3 ps and the other about 67.6 ps. We also
see that φ+ (case a) is a better target state in order to keep high population in
a zero-order dark state.
Specification of the optimal control
α(t) |(t)|2 dt − 2Re hψ(T ) | φi
|0 , 0 , 1 , 0 0 , 0 0 i
νΓ3 = EΓ3 = 4394.55 cm−1
E
|φ+i = 0, 1, 0, 11, 11 = −0.133994 |Γ1i+0.314427 |Γ2i+0.939777 |Γ3i
E
|φ−i = 0, 1, 0, 11, 1−1 = −0.641475 |Γ1i+0.69532 |Γ2i−0.324099 |Γ3i
F = |hψ(T )| φi|2 −
140
|0 , 0 , 0 , 0 0 , 0 0 i
νΓ1 = EΓ1 = 4370.36 cm−1
The hamiltonian matrix is diagonalized in order to obtain the eigenstates. The target
states of the control are defined as a superposition of the eigenstates and corresponds to
the two dark states of the polyad. The transitions from the ground state to each of the
eigenstates of the polyad, νΓ1 , νΓ2 and νΓ3 are the only transitions allowed.
"
25
Results : Evolutions
0
0
Z T
20
Pulses: The pulses have a duration of 25 ps and are rather simple.
Fourier Transforms: As expected, the frequencies of the guess field are present. In
both cases the higher intensity is for νΓ3 .
Gabor transform of field (case b): We see that the transition νΓ3 is always present
while the transition for νΓ1 and νΓ2 are respectively present in the middle and at the end
of the pulse.
1


a
b
EΓ1 0
0
E(0010000)









⇒
 0 EΓ2 0 
a
E(010111−1)
c






0
0 EΓ3
b
c
E(0101111)
Zero-order states
Eigenstates
0
4300
4410
(b)
(b)

ν1
↑
0.2
(b)
0
0
Diagonalization of the hamiltonian matrix
Target states
15
0
Eigenstates

10
5
I
Zero-order states
5
→ ν3
0.4
= 5*)
Polyad Ns = 1, Nr = 5
Time
−10
0
H C C H cm−1
*Ns = v1 + v2 + v3 and Nr = 5v1 + 3v2 + 5v3 + v4 + v5
(a)
0
|Γ 2 i
(b)
(a)
10
Intensity (arb. unit)
−5
0
0
0
Populate a dark
state of this polyad
25
(a)
Ground state
The polyad of
acetylene Ns = 1,
Nr = 5
25
Wavenumber (cm−1)
(b)
5
20
Probability
25
15
Gabor transform
(b)
Probability
20
10
Fourier transforms
FT(ν) (arb. unit)
15
5
Outlooks
FT(ν) (arb. unit)
10
Results
Probability
X
Dark state
5
−10
0
Pulses
(a)
Field (108 Vm−1)
−10
0
0
Field (108 Vm−1)
0
Field (108 Vm−1)
Bright state
(a)
10
10
Field (108 Vm−1)
Method
Field (108 Vm−1)
Goal
Probability
System
Results : Optimal pulses
Field (108 Vm−1)
Preview - Summary
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