Title: Laser Cooling of Molecules: A Theory of Purity Increasing Transformations
1Laser Cooling of MoleculesA Theory of Purity
Increasing Transformations
David Tannor Alon Bartana Yuri Rozhdestvensky Shlo
mo Sklarz Navin Khaneja
2The Challenge Direct Laser Cooling of Molecules
ATOMS
MOLECULES
Why traditional laser cooling fails for molecules
33 Laser Cooling Schemes
I) DOPPLER COOLING TDhg/2KB 240mK II)
SISYPHUS COOLING TRh2k2/2MKB 2.5mK III)
VELOCITY SELECTIVE COHERENT POPULATION TRAPPING
(VSCPT) T0? nK
4- I) Atom Cooling Schemes
- Questions
- 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, Dahleb and Rabitz 1988
- Kosloff, Rice, Gaspard, Tersigni
- and Tannor 1989
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6Introduction to Optimal Control
equations of motion with control
(penalty) objective
optimal field
Tannor, Kosloff, Rice (1985-89) Rabitz et al.
(1988)
Iteration!
7Optimal Control of Cooling
dissipation!
optimal field
8Laser Cooling of MoleculesVibrations Rotations
- Optimal Control meets Laser Cooling
- Tannor, Bartana, Kosloff
9Rotaional Selective Coherent Population Trapping
--Projection onto 0gtlt0 --Largest eigenvalue of
r --Purity Tr(r2)
10What is Cooling?
Tr(r2) is a measure of coherence. The essence of
cooling is increasing coherence!
11PHASE SPACE PICTURE
12Bombshell Hamiltonian Manipulations
Cannot Increase Tr(r2)!
Control
(Ketterle Pritchard 1992)
Need Dissipation
BUT DISSIPATION (G) CANNOT BE CONTROLLED!
13- 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?
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d
14Interplay of control fields and spontaneous
emission
15Generalization to N-Level systems (simplified
assumptions)
- Definitions
- nt number of target states
- nr other participating states.
- ??? sum of spontaneous emission rates into
target.
- Assumptions
- Populations are evenly and non coherently
distributed between states.
nt Target states
Maximum Cooling Rate (3rd Law)
16N-Level systems Complete treatment (with
Navin Khaneja)
- 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
- Geometrical principals N. Khaneja et al Phys.
Rev. A, 63 (2001) 032308. - G-unitary group
- K-subgroup generated by the control Hamiltonians,
Kexp(HjLA). - 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.
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- The problem reduces to finding the fastest way to
get between cosets in G/K space
17Controllability Analysis
Definition A system is completely controllable
if any kinematically admissible target state can
be dynamically reached from the initial state.
- Hamiltonian
- Controllable if the Lie Algebra L generated by
iH0, iHj is of order N2 and thus spans the
entire space of unitary transformations, U(N).
V. Ramakrishna et al Phys. Rev. A, 51 960
(1995)
18THERMODYNAMICS
0th law of thermo
Definition of Cooling Tr(r2)
Tr(r2)0 for Hamiltonian manipulations
Optimal Control Theory
3rd law of thermo
2nd law of thermo
19Constant T (uncontrollable)
Carnot cycle
Constant S (controllable)
Spontaneous emission (uncontrollable)
Laser Cooling
Coherent Fields (controllable)
Thermalization, Collisions (uncontrollable)
Evaporative Cooling
Trap Lowering (controllable)
Universality of the interplay of controllable
uncontrollable in cooling
20Conclusions
- New frontier for optimal control
- Increasing Tr(r2) 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. - New strategy for cooling molecules. Experiments,
anyone? - Thermodynamic analysis of laser cooling 0th, 2nd
3rd law - Cooling and Lasing as complementary
ProcessesLasing as cooling light!
21Other applications of Optimal Control
- Bose Einstein Condensates
- Quantum Computing
22Flat Phase Loading of a Bose-Einstein Condensate
onto an Optical Lattice
23Krotov MethodOptimization of Non Linear Problems
LINEAR
NON-LINEAR
24Optimization of Non Linear Schrodinger
Eq.(NLSE)-Results.
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