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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23

12

01

=0.0 fs-1

34

23

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

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

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′

3

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