Laser Cooling of Molecules:

A Theory of Purity Increasing Transformations

COHERENT CONTROL

LASER COOLING

QUANTUM INFORMATION/ DECOHERENCE

Shlomo SklarzNavin KhanejaAlon BartanaRonnie Kosloff

The Challenge: Direct Laser Cooling of Molecules

Why traditional laser cooling fails for molecules

ATOMS

MOLECULES

3 Laser Cooling SchemesI) DOPPLER COOLING

TD=h/2KB

240K

II) SISYPHUS COOLINGTR=h2k2/2MKB

2.5K

III) VELOCITY SELECTIVE COHERENT POPULATION TRAPPING (VSCPT)T=0?nK

|a,p>

|b+,p+hk>|b-,p-hk>

Normalized velocity

Forc

e

Atomic Position

En

erg

y

I) Atom Cooling SchemesQuestions:• Each new scheme seems to come out of the

blue. Is there a systematic approach?• Can the efficiency be improved?• Where is the thermodynamics?

II) Optimal Control Theory.Tannor and Rice 1985 (Calculus of variations)Peirce, Dahleh and Rabitz 1988Kosloff, Rice, Gaspard, Tersigni

and Tannor 1989

Introduction to Optimal Control

it

H[(t)](t)

J limt

(T) | P |(T)

(t) i

[b | | a b | |a ]

equations of motionwith control

(penalty) objective

optimal field

Iteration! Tannor, Kosloff, Rice (1985-89)Rabitz et al. (1988)

)(T)0(

)0( )(T

)],t([Hi

1

t

A)T(limJT

(t) i

Tr[ ˆ c ˆ A e ( ˆ g ˆ ˆ e) ˆ A c ˆ c ˆ B g ]

Optimal Control of Cooling

optimal field

dissipation

|00|A

)0()A(L

t

A

)T()0(A )T(A

Bartana, Kosloff and Tannor, 1993, 1997,

2001

]}V,V[]V,V{[2

1iii

ii

Laser Cooling of Molecules:Vibrations + Rotations

Optimal Control meets Laser Cooling

Spontaneous Emission

Stimulated Emission

Absorption

VIBRATIONS ROTATIONS

Rotational Selective Coherent Population Trapping

1)l(l

g/eg/e

gj

j*

j

jjje

BH

Hˆ)t(

ˆ)t(H

H

--Projection onto |0><0|--Largest eigenvalue of --Purity Tr(2)

What is Cooling?

Pnn

1

Pnn

21

Pnn

21

Pnn

1

Tr(2) is a measure of coherence. The essence of cooling is increasing coherence!

Tr ( 2) 1

PHASE SPACE PICTURE

ˆ A Tr ˆ A ˆ Tr 2 TrW W

2 1

2dpdq W

2 (p,q)

Tr ( 2) 1

)],t([Hi1

0]),H[(Tr)(Tr2)(Tr i22

Bombshell: Hamiltonian Manipulations Cannot Increase Tr(2)!

Control

(Ketterle + Pritchard 1992)

],i

H[

]}V,V[]V,V{[

2

1iii

ii

Need Dissipation:

0)(Tr2)(Tr 2

BUT DISSIPATION () CANNOT BE CONTROLLED!

Tr ( 2) 1

)],t([Hi1

0]),H[(Tr)(Tr2)(Tr i22

Bombshell: Hamiltonian Manipulations Cannot Increase Tr(2)!

Control

(Ketterle + Pritchard 1992)

],i

H[

]}V,V[]V,V{[

2

1iii

ii

Need Dissipation:

0)(Tr2)(Tr 2

BUT DISSIPATION () CANNOT BE CONTROLLED!

Questions:

• How can cooling be affected by external fields?

• What are the general rules for when spontaneous emission leads to heating and when to cooling?

0,0

d

1

1

)(Tr 2

d

1

0,0

1

)(Tr 2

+

++

-

-

-

10

.1.99

.3

.7

dc

ba

10

1

adbc

adbc

da

)ρΓρ(Tr2

Interplay of control fields and spontaneous emission

0,0

d

1

1

)(Tr 2

0,0

1

)(Tr 2

+++

-

-

-

1

d

)(Tr)(Trmax 22

,d

12 T2

1

T

1

)(Tr~,d~

)(Tr,d 22

1)(Tr2)(Tr21)(Tr 222

•Optimal cooling strategy Strange but interesting form!

•Physical significance of optimal strategy keep coherence off the off-diagonal.

•Algorithm: optimal trajectory

Diff. eq. for Tr() vs t: 3rd law of thermodynamics!

• Tr(2) does depend on the control E(t) indirectly

Purity Increasing Transformations:Bloch Sphere Representation

TrTr(()) Dissipative

Tr(Tr()) Unitary

Purity increasing

Purity decreasing

11 12

21 22

.

Universality of the interplay of controllable + uncontrollable in cooling

Constant T (uncontrollable)

Constant S (controllable) Carnot cycle

Spontaneous emission (uncontrollable)

Coherent Fields (controllable)

Laser Cooling

Thermalization, Collisions (uncontrollable)

Trap Lowering (controllable)Evaporative Cooling

Beyond two-level systems:Two simplifying assumptions

• Instantaneous unitary control– U= eiH[E]t is infinitely fast compared with – Criterion: ij

• Complete unitary control– Any U in SU(N) can be produced by eiH[E]t

– Lie algebra criterion: dim {H, H1…}LA=N2-1

Complete and Instantaneous Unitary Control

Representation of the problem in terms of spectral transformations

=U+U

=U+U

I II

Eqn. of Motion

Control E(t) U(t)

Objective

Modified Control problem

],[Hi

Tr(2),

‘Greedy’ strategy for

3 level system is optimal• The ‘Greedy’ strategy:

– Maximize dP/dt at each instant– Maintain maximal population

of the excited state– Keep

• Diagonal (={P} ) (No coherences)

• and Ordered (P=I)(Ordered Eigenvalues)

• Theorem:The greedy trajectory-diag()= is optimal

THERMODYNAMICS

Definition of CoolingTr(2)

Tr(2)=0for Hamiltonian manipulations

Optimal ControlTheory

0th law of thermo

3rd law of thermo2nd law of thermo

Conclusions• New frontier for optimal control

• Increasing Tr(2)= increasing coherence is relevant to more than laser cooling!

• It may be profitable to reexamine existing laser cooling schemes in light of purity increase. There is the potential for great improvement in rate/efficiency by exploiting all spontaneous emission.

• Potentially new strategies for cooling molecules

• Thermodynamic analysis of laser cooling 0th, 2nd + 3rd law

• Cooling and Lasing as complementary ProcessesLasing as cooling light!

LASING

COOLINGLWI

IWL

Re

Kocharovskaya + Khanin 1988

Thermodynamics of light-matter interactions

Erez Boukobza

2

1

0

0ss

H

H

Q

0ss

C

C

Q

0ssP

•G- “Liouville group”

•K- subgroup generated by the control Hamiltonians, assumed to be the whole unitary group U(N).

•Hamiltonian Motion is fast and governed by the controls

•Purity changing Motion is slow and determined by dissipation

N-Level systems: Complete treatment (with Navin Khaneja)

Geometrical principals [N. Khaneja et al Phys. Rev. A, 63 (2001) 032308].

• G-unitary group

• K-subgroup generated by the control Hamiltonians, K=exp({Hj}LA).

• G/K quotient space where each point represents some coset KU.

• Motion within a coset is fast and governed by the controls

• Motion between cosets is slow and determined by H0.

[1]

G

U

VKV

KU

•The problem reduces to finding the fastest way to get between cosets in G/K space

Hamilton-Jacobi-Bellman Theorem (Dynamical Programming)

1

6

5

2

4

3

4

3

3

6

5

6

5

4

5

5

4

6

6

5

4

46

5

6

4

5

6

t

V(,t)

Hamilton-Jacobi-Bellman Theorem

• Guaranteed to give GLOBAL maximum.• Capable of giving analytic optimal solutions.• Very Computationally expensive.• A possible method of solution:

guess optimal strategy and prove that HJB equations are satisfied.

‘Greedy’ strategy for N+1 level system;

n

n

Spectral evolution

Greedy=1. No coherences

={Pi}2. Ordered

EigenvaluesPi=I

time

p

opul

atio

ns)

Spectral evolution

4 levels =[0.05, 0.045, 0.0001]

time

p

opul

atio

ns)

Investment Return

Summary• The ‘Greedy’ cooling strategy is optimal for the

three-level L system

• ‘Investment & Return’ strategies rather than ‘Greedy’ are optimal for N>3 level systemsCoherences are required for optimality