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Title: Laser Cooling of Molecules: A Theory of Purity Increasing Transformations


1
Laser Cooling of MoleculesA Theory of Purity
Increasing Transformations
David Tannor Alon Bartana Yuri Rozhdestvensky Shlo
mo Sklarz Navin Khaneja
2
The Challenge Direct Laser Cooling of Molecules
ATOMS
MOLECULES
Why traditional laser cooling fails for molecules
3
3 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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6
Introduction to Optimal Control
equations of motion with control
(penalty) objective
optimal field
Tannor, Kosloff, Rice (1985-89) Rabitz et al.
(1988)
Iteration!
7
Optimal Control of Cooling
dissipation!
optimal field
8
Laser Cooling of MoleculesVibrations Rotations
  • Optimal Control meets Laser Cooling
  • Tannor, Bartana, Kosloff

9
Rotaional Selective Coherent Population Trapping
--Projection onto 0gtlt0 --Largest eigenvalue of
r --Purity Tr(r2)
10
What is Cooling?
Tr(r2) is a measure of coherence. The essence of
cooling is increasing coherence!
11
PHASE SPACE PICTURE
12
Bombshell 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?

1
d
14
Interplay of control fields and spontaneous
emission
15
Generalization 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)
16
N-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.

1
  • The problem reduces to finding the fastest way to
    get between cosets in G/K space

17
Controllability 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)

18
THERMODYNAMICS
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
19
Constant 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
20
Conclusions
  • 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!

21
Other applications of Optimal Control
  • Bose Einstein Condensates
  • Quantum Computing

22
Flat Phase Loading of a Bose-Einstein Condensate
onto an Optical Lattice
23
Krotov MethodOptimization of Non Linear Problems
LINEAR
NON-LINEAR
24
Optimization of Non Linear Schrodinger
Eq.(NLSE)-Results.
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