Laser Cooling of Molecules: A Theory of Purity Increasing Transformations COHERENT CONTROL LASER COOLING QUANTUM INFORMATION/ DECOHERENCE Shlomo Sklarz Navin Khaneja Alon Bartana Ronnie Kosloff
Dec 20, 2015
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
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
‘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
•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