Quantum Observations in Optimal Control of Quantum Dynamics

Feng Shuang Herschel Rabitz

Department of Chemistry, Princeton University

ICGTMP 26th, June, 2006, NY

2

Overview

• Introduction: • Optimal Control of Quantum Dynamics• Quantum Observations

• Optimal Observations:

• w/o Control Field

• With Control Field

3

Control: Coherence + Decoherence

Coherence:

Decoherence:

Laser Noise: Cooperate and Fight (1)

Dissipation: Cooperate and Fight (2)

Observations: A tool to assist control (3)

1. F.Shuang & H.Rabitz, J.Chem.Phys, 121, 9270 (2004)

2. F.Shuang & H.Rabitz, J.Chem.Phys, 124, 204115(2006)

3. F.Shuang et al, In progress

0 ,t i H E t t tt

Control Field ( )E t

t

4

Optimal Control of Quantum Dynamics

0 ( )H H E t

Hamiltonian:

Control Field

2

2( ) exp / 2 cos2f

l ll

TE t t A t

Objective Function

l

lT AOtEOtEJ 22

Closed-Loop Feedback Control: Herschel Rabitz

Genetic Algorithm

5

Quantum Observations

Instantaneous Observations: Von Neuman• General Operator A:

• Projection Operator P

Continuous Observations: Feynman & Mensky Master Equations

,

i i ii

kj k j kk k kk j k

A a a a

a a a a

' , , , with P P P

, , ,i H t A t A t

6

Quantum Zeno and Anti-Zeno effect

• Quantum Zeno Effect (QZE)– Repetitive observations prohibits evolution of

quantum system

• Quantum Anti-Zeno Effect (QAZE)– Time-dependent observation induces state change of

quantum system

7

Optimal Observations w/o Control Field

• Two-Level: Initial state and Final state, Projection Operators

• Adiabatic Limit: 100% Population Transfer (1)

Number of Instantaneous Observation, N Strength of Continuous Observations:

• When N and are finite, What’s the best? (1). A.P.Balachandran & S.M.Roy, PRL, 84, 4019(2000)

0 0 0 1 1

, k k k

H

P P t t t

8

Optimal Instantaneous Observations

1 1, ,

cos 0 sin 12 2

k

k k k k k

k k k

ik kk

P P

P

e

11 cos 100%

2 1ON

kY

N

N Observations. Interaction Picture

After Optimization:

Yield of N Observations: (QAZE)

1 12 23 1,

11 cos cos

2cos cos cos sin sin

N N N N

mn m n m n m n

Y C C C

C

9

Optimal Continuous Observations

• Weak Observation:

• Strong Observation:

• no analytical solution for general • linear form: (t)= Bopt+Aopt t

, ,

, cos 0 sin 12 2

i t

t P t P t t

t tP t t t t e

2 4 6 8 10

Tf

0.2

0.4

0.6

0.8

AOptTf

2 4 6 8 10

Tf

0.1

0.2

0.3

0.4

0.5

BOpt

1( ) 0 1

2 2

(0) 0 , ( ) 1ff

t t

tt T

T

2 4 6 8 10

Tf

20

40

60

80

YOpt

10

Optimal Observations with Control Field

• N-Level system

• Control Field:

• Two Models:

– Cooperate & Fight– Symmetry-breaking

0

N

vl

H v v

2

2( ) exp / 2 cos2f

l l ll

TE t t A t

11

Optimal Control Field with Observations Model 1

• Five-level system: Population 0 4• Control field is fighting with observations

of dipole, energy, population at Tm=Tf/2

Operator Value of observation Yield with observation

0.66 94.03%

H0 3.94 85.17%

P0 0.0037 95.77%

P1 0.021 93.71%

P2 0.055 92.98%

P3 0.0010 97.27%

P4 0.0032 95.68%

4

3

2

1

0

12

Optimal Observations with Control Field: Model 1

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[0,]

O[E(t),]O[E(t),]

(a)

Expected Yield OT(%)

Yie

ld (

%)

F0

F

(b)

Expected Yield OT(%)

Flu

ence

Cooperating with the observation of dipole

2

2

Objective functional : ,

Fluence

T

ll

J E t O E t O F

F A

13

Optimal Observations with Control Field: Model 2

2

1

0

0

1 0 0

0 2 0

0 0 3

0 1 0

1 0 1

0 1 0

H

High symmetry system:

Only 50% population is possible from 0 to 1

Control Field Used: cosE t A t

14

Optimal Control Field with Observations: Model 2

• Instantaneous observation: Partial Symmetry Breaking

P O[E(t),P] O[E(t),0] F

- 49.99% 49.99% 0.0031

P0 66.90% 46.04% 0.76

P1 49.99% 50.00% 0.96

P2 66.66% 46.37% 0.49

15

Optimal Observations with Control Field: Model 2

• Continuous observation: Symmetry Breaking, QZE

• Optimize: A, T1,T2,Gama

0 0 1 2

1 2

, , , , ,

,

0, otherwise

i H E t t P P P P P P

T t Tt

P=P0 P=P2

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)

16

Conclusions• 1. Control field can fight and cooperate with

observations

• 2. Observation can assist optimal control

• 3. Quantum Zeno and Anti-Zeno effects are key

Question: How to implement the observations in experiments ?

17

Acknowledgements

• Herschel Rabitz

• Alex Pechen & Tak-san Ho

• Mianlai Zhou

• Other colleagues

• Funding: NSF, DARPA, ARO-MURI