Laser Noise, Decoherence &Observations
in the Optimal Control of Quantum Dynamics
双
丰
Department of Chemistry, Princeton University
Frontiers of Bond-Selective Chemistry
2
3
4
Rabitz Group in Princeton
• Effect of environments on control of Quantum
Dynamics: Fighting & Cooperating
• Exploring Photonic Reagent Quantum Control
Landscape: no local sub-optimal
• Controlling Quantum Dynamics Regardless of the
Laser Beam Profile and Molecular Orientation
• Revealing Mechanisms of Laser-Controlled
Dynamics
• Experiment: SHG, C3H6,
5
Control of Quantum Dynamics
Hamiltonian:
H  H 0  E (t )
Control Field
  Tf
E (t )  exp   t 
2
 



2

 Al cost  l 
 l
Objective Function
J E t   OE t   OT    Al
2
2
l
Closed Loop Feedback Control
Genetic Algorithm
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Laser Noise: Model
Noise Model:
Deterministic part
Al  Al  Al ,l  l  l
0
0
Objective Function
noise part


J Al ,  l  J N 1   N E t 
0

0

J N Al ,  l  O E t  N  OT    Al
0
0
2
0 2
l
 N E t  
O 2 E t 
 O E t  N
2
N
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Cooperating with Laser Noise
2.5
optimal field with noise
Yield %
2.0
1.5
optimal field alone
1.0
noise alone
0.5
0.0
0.01
0.03
0.05
0.07
0.09
Noise Level A
The control yield under various noise conditions
with the low yield target of OT=2.25%. There is notable
cooperation between the noise and the field especially over
the amplitude noise range 0.06≤ΓA≤0.08.
d
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Laser Noise: Foundation of Cooperation
Control Yield from perturbation theory
OE t    Al2
l
Averaged over the noise distribution
OE t    A
__
l
2
l N
A
symmetric noise distribution function
2
l N
  A  xl Pl ( xl )dxl  A
0
l
2
0 2
l
A
 xl2
N
Minimize the objective function,
0 2
l
A

2
l
x
N
 Const
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Fighting with Laser Noise
Rc t     kj t  ,
2
k j
Rd t     kk t  ,
2
k
R(t )  Rc t   Rd t   Tr  2 t .
Time dependent dynamics driven by the optimal control field
with a large amount of phase noise. Plots (a1) and (a2) show the
dynamics when the system is driven by a control field with noise
while plots (b1) and (b2) show the dynamics of the system driven by
the same field but without noise. The associated state populations are
shown in plots (a2) and (b2).
d
10
Decoherence: Model
Decoherence described by the Lindblad Equation

 t   iH 0  E t ,  t    t 
t
1
 t ll '   ll '  ln  nn t    nl  nl '  ll ' t 
2 n
n
Objective Function:
JEt   OE t ,    OT    Al
2
OE t ,    Tr  T f O 
0
l
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Power Spectrum
Cooperating with Decoherence
=0.0 fs
34
0.0
Power Spectrum
-1
23
1.0
=0.03 fs
0.0
=0.01 fs
34
0.5
1.5
-1
23
34
12
01
2.0 0.0
23
0.5
01
1.0
=0.05 fs
12 01
12
-1
1.5
2.0
-1
23
12
01
0.5
1.0
1.5 -1 2.0 0.0
0.5
1.0
1.5-1 2.0
Frequency (rad fs )
Frequency (rad fs )
Power spectra of the control fields aiming at a low yield
of OT=5.0%. γ indicates the strength of decoherence. The
control field intensity generally decreases with the increasing
decoherence strength reflecting cooperative effects.
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Decoherence: Foundation of Cooperation
•When both the control field and decoherence are weak, the
objective cost function can be written in terms of the
contributions from each specific control field intensity Aj²
 
J  P Aj    Ak
2
 
2
k j
P Aj  A F   j F2 j  OT   Aj
2
2
2
j 1j
2
Independent of Aj and j
•Minimize objective function:
A2j F1 j   j F2 j  Const
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Fighting with Decoherence
Yield from optimal fields (%)
100
80
60
40
20
0
0.00
0.01
0.02
0.03
0.04
: Strength of decoherece
0.05
Decoherence is deleterious for achieving a high
target value, but a good yield is still possible.
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Observation-assisted Control
o Instantaneous Observations
    kj      kk
k, j
o
k
Continuous Observations

 t   iH 0  E t ,  t    A, A,  t 
t
observed operator
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Cooperating or Fighting with Instantaneous
Observations During Control
0.4
100
(a)
80
80
60
60
O[E(t),]
40
40
O[E(t),]
20
(b)
0.3
Fluence
Yield (%)
100
F
0.2
0.1
20
0
0
20
40
60
80
Expected Yield (%)
100
F0
0.0
20
40
60
80
100
Expected Yield (%)
(a). Yield from control field with (O[E(t),u]) or without (O[E(t)])
observation of dipole
(b). Fluence of control field optimized with (F) or without (F0)
observation of dipole
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Cooperating or Fighting with Instantaneous
Observations During Control
80
Population (%)
O[E(t),PN]
60
O[PN]
40
O[0,PN]
20
0
0
1
3
5
7
9
N
Yield from a series of instantaneous observations
with or without optimal control field.
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Optimized Continuous Observations
to Break Dynamical Symmetry
To control an uncontrollable system. Goal: 01
2
1
0
Qa
O[E(t),Q]b
T1
T2
No
49.9704%
\
\
P0
94.668%
131
200
P1
49.9661%
46
48
P2
98.4296%
129
193
Operator observed between times T1 and T2 with strength 1:
Pk indicates population at level k;
a:
b:
Yield in state 1 from optimizing the control field E(t), T1, T2 and .
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Optimized Continuous Observations
to Break Dynamical Symmetry
(a)
100
80
P0
P2
60
40
P1
T2
T1
20
Population(%)
Population(%)
100
P0
80
P1
60
40
20
T2
T1
P2
0
0
0
(b)
50
100
Time(fs)
150
200
0
50
100
150
200
Time(fs)
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Observation assisted optimal Control
(c) 3
100
2
1'
1
Yield (%)
80
60
40
20
0
0.00
0
P3
P1'
0.05
0.10
0.15
0.20
: Observation Strength
The control yield of desired state (P₃) and undesired state (P1’) under
different strength (κ) of continuous observations on level 1′
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Conclusions
 In the case of low target yields, the control field
can cooperate with laser noise, decoherence and
observations while minimizing the control fluence.
 In the case of high target yields, the control field
can fight with laser noise, decoherence and
observations while attaining good quality results
 An optimized observation can be a powerful tool
the in the control of quantum dynamics
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Where is Future of Modeling?
• Fighting with Noise, Decoherence. 100%
yield is expected Quantum Computation
• Simulate Controlled Real Chemical
Reaction: Systems investigated are too
simple.
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Thanks
朱清时（USTC）
Herschel Rabitz（Princeton）
严以京（HKUST）
Mark Dykman（MSU）23
Thanks, Family
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