Laser Noise, Decoherence &Observations in the Optimal Control of Quantum Dynamics
双 丰Department of Chemistry, Princeton University
Frontiers of Bond-Selective Chemistry
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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,
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Control of Quantum Dynamics
)(0 tEHH
Hamiltonian:
Control Field
lll
f tAT
ttE cos2
exp)(2
Objective Function
l
lT AOtEOtEJ 22
Closed Loop Feedback Control
Genetic Algorithm
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Laser Noise: Model
Noise Model:
Objective Function
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20200
00
,
1,
NNN
llTNllN
NNll
tEOtEOtE
AOtEOAJ
tEJAJ
Deterministic part
noise part
ll llAll AA 00 ,
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Cooperating with Laser Noise
0.01 0.03 0.05 0.07 0.09
0.0
0.5
1.0
1.5
2.0
2.5
noise alone
optimal field alone
optimal field with noise
Yie
ld %
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
l
lAtEO 2
Control Yield from perturbation theory
Averaged over the noise distribution
NllllAlllNl
lNl
xAdxxPxAA
AtEO
220202
2__
)(
Minimize the objective function,
Const220 Nll xA
symmetric noise distribution function
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Fighting with Laser Noise
.Tr)(
,
,
2
2
2
ttRtRtR
ttR
ttR
dc
kkkd
jkkjc
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
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Decoherence: Model
Decoherence described by the Lindblad Equation
nllnlnl
nnnllll ttt
tttEHitt
''ln''
0
2
1
,
Objective Function:
OTtEO
AOtEO
f
llT
Tr,
,tEJ 02
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Cooperating with Decoherence
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
Po
wer
Sp
ectr
um
Po
wer
Sp
ectr
um
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01
=0.0 fs-1
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12
01
=0.01 fs-1
Frequency (rad fs-1)Frequency (rad fs-1)
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12
01
=0.03 fs-1
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12
01
=0.05 fs-1
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
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2122
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jTjjjjj
jkkj
AOFFAAP
AAPJ
•Minimize objective function:
Const212 jjjj FFA
Independent of Aj and j
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Fighting with Decoherence
Decoherence is deleterious for achieving a high target value, but a good yield is still possible.
0.00 0.01 0.02 0.03 0.04 0.050
20
40
60
80
100
Yie
ld f
rom
op
tim
al fi
eld
s (
%)
: Strength of decoherece
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Observation-assisted Control
o Instantaneous Observations
o Continuous Observations
k
kkjk
kj ,
tAAttEHitt
,,,0
observed operator
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Cooperating or Fighting with Instantaneous Observations During Control
(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
20 40 60 80 100
0
20
40
60
80
100
0
20
40
60
80
100
20 40 60 80 100
0.0
0.1
0.2
0.3
0.4
O[E(t),]
O[E(t),]
(a)
Expected Yield (%)
Yie
ld (
%)
F0
F
(b)
Expected Yield (%)
Flu
ence
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Cooperating or Fighting with Instantaneous Observations During Control
1 3 5 7 90
20
40
60
80
O[PN]
O[0,PN]
Po
pu
lati
on
(%
)
N0
O[E(t),PN]
Yield from a series of instantaneous observations with or without optimal control field.
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Optimized Continuous Observations to Break Dynamical Symmetry
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
To control an uncontrollable system. Goal: 01
a: Operator observed between times T1 and T2 with strength Pk indicates population at level k;
b: Yield in state 1 from optimizing the control field E(t), T1, T2 and
2
1
0
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Optimized Continuous Observations to Break Dynamical Symmetry
0 50 100 150 200
0
20
40
60
80
100
0 50 100 150 200
0
20
40
60
80
100
P2
P1 T
2
Po
pu
lati
on
(%)
Time(fs)
T1
P0
P1
P2
P0
T2T
1
(b)
Po
pu
lati
on
(%)
Time(fs)
(a)
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Observation assisted optimal Control
0.00 0.05 0.10 0.15 0.200
20
40
60
80
100
P1'
P3
Yie
ld (%
): 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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2
1'
1
0
(c)
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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) 严以京( HKUST)
Herschel Rabitz ( Princeton) Mark Dykman ( MSU)
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Thanks, Family