1. 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

2. The Challenge: Direct Laser Cooling of Molecules Why traditional laser cooling fails for molecules ATOMS MOLECULES

3. 3 Laser Cooling Schemes I) DOPPLER COOLING T D =h  /2K B 240  K II) SISYPHUS COOLING T R =h 2 k 2 /2MK B 2.5  K III) VELOCITY SELECTIVE COHERENT POPULATION TRAPPING (VSCPT) T=0? nK |a,p> |b +,p+hk> |b -,p-hk>     Normalized velocity Force Atomic Position Energy

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, Dahleh and Rabitz 1988 Kosloff, Rice, Gaspard, Tersigni and Tannor 1989

5. Introduction to Optimal Control equations of motion with control (penalty) objective optimal field Iteration! Tannor, Kosloff, Rice (1985-89) Rabitz et al. (1988)

6. Optimal Control of Cooling optimal field dissipation Bartana, Kosloff and Tannor, 1993, 1997, 2001

7. Laser Cooling of Molecules: Vibrations + Rotations Optimal Control meets Laser Cooling Spontaneous Emission Stimulated Emission Absorption VIBRATIONS ROTATIONS

8. Rotational Selective Coherent Population Trapping --Projection onto |0><0| --Largest eigenvalue of  --Purity Tr(  2 )

9. What is Cooling? Tr(  2 ) is a measure of coherence. The essence of cooling is increasing coherence!

10. PHASE SPACE PICTURE

11. Bombshell: Hamiltonian Manipulations Cannot Increase Tr(  2 )! Control (Ketterle + Pritchard 1992) Need Dissipation: BUT DISSIPATION (  ) CANNOT BE CONTROLLED!

12. Bombshell: Hamiltonian Manipulations Cannot Increase Tr(  2 )! Control (Ketterle + Pritchard 1992) Need Dissipation: BUT DISSIPATION (  ) 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? 0,0  d 1 1 d 1  1 + + + - - - 1010.1.99.3.7

14. Interplay of control fields and spontaneous emission 0,0  d 1 1  1 + + + - - - 1 d 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!

15. Tr(  2 ) does depend on the control E(t) indirectly Purity Increasing Transformations: Bloch Sphere Representation Tr (   ) Tr (   ) Dissipative Tr(   ) Tr(   ) Unitary Purity increasing Purity decreasing.

16. 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

17. Beyond two-level systems: Two simplifying assumptions Instantaneous unitary control –U= e iH[E]t is infinitely fast compared with  –Criterion:  ij  Complete unitary control –Any U in SU(N) can be produced by e iH[E]t –Lie algebra criterion: dim {H, H 1 …} LA =N 2 -1  Complete and Instantaneous Unitary Control

18. Representation of the problem in terms of spectral transformations      =U +   U   =U +   U  

19. III Eqn. of Motion U(t)E(t) Control Objective Modified Control problem

20. ‘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

21. THERMODYNAMICS Definition of Cooling Tr(  2 ) Tr(  2 )=0 for Hamiltonian manipulations Optimal Control Theory 0 th law of thermo 3 rd law of thermo 2 nd law of thermo

22. 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 0 th, 2 nd + 3 rd law Cooling and Lasing as complementary Processes Lasing as cooling light! LASING COOLING LWI IWL   Re   Kocharovskaya + Khanin 1988

23. Thermodynamics of light-matter interactions Erez Boukobza 22 11 00

24. 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({H j } 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 H 0. [1] G U V KV KU The problem reduces to finding the fastest way to get between cosets in G/K space

25. 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 4 6 5 6 4 5 6 t V(,t)

26. 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.

27. ‘Greedy’ strategy for N+1 level system;     nn   nn  Spectral evolution Greedy= 1. No coherences  ={P i } 2.Ordered Eigenvalues P i =I        time  populations) Spectral evolution

28. 4 levels  =[0.05, 0.045, 0.0001] time  populations)        Investment Return

29. 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 systems Coherences are required for optimality