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

7

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.

10

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

11

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

12

Observation-assisted Control*

o Instantaneous Observations

o Continuous Observations

k

kkjk

kj ,

tAAttEHitt

,,,0

*In Progressobserved operator

13

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

14

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

15

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

16

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