Download - Abstract
1
Laser noise and decoherence are generally viewed as deleterious in quantum control. Numerical simulations show that optimal fields can cooperate with laser noise and decoherence when seeking modest control yields, and it’s possible to find optimal fields to fight with them while seeking a high control yield. The theoretical foundations for the ability of a control field to cooperate with laser noise and decoherence are established. d
Abstract
2
The use of instantaneous and continuous observations(measurements) acting as controls is explored. Quantum observations can break dynamical symmetries, and a time-dependent observation can even transfer a state to another state. Suitably optimized observations could be powerful tools in the manipulation of quantum dynamics. d
Abstract: Continued
3
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
4
Laser Noise*: Model
Noise Model:
l ll l A l lA A 0 0
,
Objective Function
22
20200
00
,
1,
NNN
llTNllN
NNll
tEOtEOtE
AOtEOAJ
tEJAJ
* J.Chem.Phys 121, 9270 (2004)
Deterministic part
noise part
5
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
6
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
8
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
* Submitted to J.Chem.Phys
9
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
34
23
12
01
=0.0 fs-1
34
23
12
01
=0.01 fs-1
Frequency (rad fs-1)Frequency (rad fs-1)
34
23
12
01
=0.03 fs-1
23
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
22
2122
22
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
*In Progressobserved operator
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Cooperating or Fighting with Instantaneous Observations During Control
(a). Yield from control field with (O[E(t),Q]) or without (O[E(t)]) observation Q
(b). Fluence of control field optimized with (F) or without (F0) observation.
20 40 60 80 100
20
40
60
80
100
20
40
60
80
100
20 40 60 80 1000.00
0.02
0.04
0.06
0.08(a)
Target Yield (%)
Co
ntr
ol Y
ield
(%
)
O[E(t),Q1]
O[E(t)]
(b)
Target Yield (%)F
luen
ce
F F0
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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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Time-dependent Observations
The Quantum Anti-Zeno Effect A time-dependent observation can transfer a state 0 to a target state f, and may be a useful tool in the control of quantum dynamics. d
fttt
tttA
sincos 0
0.5 1.0 1.5 2.020
40
60
80
100
Yie
ld o
f O
bse
rvat
ion
%
: Strenght of Observation
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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