Laser Cooling of Molecules: A Theory of Purity Increasing Transformations

Transcription

1 Laser Cooling of Molecules: A Theory of Purity Increasing Transformations COHERENT CONTROL Shlomo Sklarz Navin Khaneja Alon Bartana Ronnie Kosloff LASER COOLING QUANTUM INFORMATION/ DECOHERENCE

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

3 3 Laser Cooling Schemes I) DOPPLER COOLING T D =hγ/k B 40µK II) SISYPHUS COOLING T R =h k /MK B.5µK Force III) VELOCITY SELECTIVE COHERENT POPULATION TRAPPING (VSCPT) T=0? nk Normalized velocity ν,σ+ a,p> ν,σ b -,p-hk> b +,p+hk> Energy Atomic Position

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 985 (Calculus of variations) Peirce, Dahleh and Rabitz 988 Kosloff, Rice, Gaspard, Tersigni and Tannor 989

5 Introduction to Optimal Control ih ψ t = H[ε(t)]ψ(t) J = lim t <ψ(t) P ψ(t) > equations of motion with control (penalty) objective ψ(0) χ(0) χ(t) ψ(t) ε(t) = i λ [<ψ b µ χ a > < χ b µ ψ a >] optimal field Iteration! Tannor, Kosloff, Rice (985-89) Rabitz et al. (988)

6 Optimal Control of Cooling ρ t = ih [ H[ ε(t)], ρ] + Γρ Bartana, Kosloff and Tannor, 993, 997, 00 Γρ = FρF + [F + Fρ + ρf + F] dissipation J = lim ρ(t)â T  0 0 ρ(0) Â(0) t  + = L (Â) ρ(t) Â(T) ε(t) = i λ Tr[ ˆ ρ c ˆ µ A ˆ e + ( ˆ ρ g ˆ µ ˆ µ ˆ ρ e ) A ˆ c ˆ µ ˆ ρ ˆ c B g ] optimal field

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

8 Rotational Selective Coherent Population Trapping Ĥ Ĥ = e/g = B j e/g Ĥ ε j e (t) * µ ˆ l(l + ) j j ε Ĥ j (t) µ ˆ g j --Projection onto 0><0 --Largest eigenvalue of ρ --Purity Tr(ρ )

9 What is Cooling? = n n P n < P n Tr(ρ ) = n n P n = P n Tr(ρ) is a measure of coherence. The essence of cooling is increasing coherence!

10 PHASE SPACE PICTURE ( ) A ˆ = Tr ρa ˆ ˆ ρ = Tr( ρ ) = Tr ( W ρ ) W = πh dpdqρ W (p,q)

11 Bombshell: Hamiltonian Manipulations Cannot Increase Tr(r)! ih [ H[ ε(t ρ] ρ & = )], Control Tr( ρ& ) = Tr( ρρ& ) Need Dissipation: H ρ & = [, ρ] + Γρ i h = ih Tr( ρ[h, ρ]) (Ketterle + Pritchard 99) Tr( ρ& ) = Γρ = Tr( ργρ) i = 0 {[Vρ,V i 0 + i Tr(ρ ) = ] + [V, ρv BUT DISSIPATION (Γ) CANNOT BE CONTROLLED! i + i ]}

12 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? Tr( ρ& γ + ) γ Tr( ρ ) 0,0 d 0,0 d..99 a ρ = c b d 0 a + d = bc ad bc = γcd 0 γ

13 Interplay of control fields and spontaneous emission γ Tr( ρ& ) Tr( ρ γ ) 0,0 Optimal cooling strategy d Strange but interesting form! 0,0 max Tr( ρ& ) d, γ d + λtr( ρ ) Physical significance of optimal strategy = T T Algorithm: optimal trajectory Tr( ρ& ) = Tr( ρ keep coherence off the off-diagonal. ) + d, γ Tr( ρ Tr( ρ ) ) Diff. eq. for Tr(ρ) vs t: 3rd law of thermodynamics! ~ d, ~ γ Tr( ρ& )

14 Purity Increasing Transformations: Bloch Sphere Representation Purity decreasing Tr(r.) Dissipative Tr(r ) Unitary ρ = ρ ρ ρ ρ. Purity increasin g Tr(r ) does depend on the control E(t) indirectly

15 Constant T (uncontrollable) Constant S (controllable) Carnot cycle Spontaneous emission (uncontrollable) Coherent Fields (controllable) Laser Cooling Thermalization, Collisions (uncontrollable) Trap Lowering (controllable) Evaporative Cooling Universality of the interplay of controllable + uncontrollable in cooling

16 Beyond two-level systems: Two simplifying assumptions Instantaneous unitary control U= e ih[e]t is infinitely fast compared with G Criterion: w ij ³g Complete unitary control Any U in SU(N) can be produced by e ih[e]t Lie algebra criterion: dim {H, H } LA =N - è Complete and Instantaneous Unitary Control

17 Representation of the problem in terms of spectral transformations r r L r L L=U+rU L L=U+rU

18 Modified Control problem I II Eqn. of Motion ρ = i [H,ρ] + Γρ h Control E(t) U(t) Objective Tr(ρ ), ρ

19 Hamilton-Jacobi-Bellman Theorem (Dynamical Programming) V(l,t) l t

20 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.

21 Greedy strategy for 3 level L system is optimal The Greedy strategy: Maximize dp/dt at each instant Maintain maximal population of the excited state Keep r Diagonal (Q={P} ) (No coherences) and Ordered (P=I) (Ordered Eigenvalues) Theorem: The greedy trajectorydiag(r)=l is optimal

22 time Greedy strategy for N+ level g ñ ñ g g 3 4 3ñ 4 ñ g n n ñ Greedy=. No coherences Q={P i }. Ordered Eigenvalues P i =I l (populations) system; Spectral evolution 4ñ g g ñ ñ g 3 3ñ

23 4 levels G=[0.05, 0.045, 0.000] l (populations) Investment Return 4ñ g g ñ ñ g 3 3ñ

24 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

25 THERMODYNAMICS Definition of Cooling Tr(ρ ) 0 th law of thermo Tr(ρ )=0 for Hamiltonian manipulations Optimal Control Theory nd law of thermo 3 rd law of thermo

26 Conclusions New frontier for optimal control Increasing Tr(ρ )= 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 0 th, nd + 3 rd law Cooling and Lasing as complementary Processes Lasing as cooling light! ρ33 LASING IWL LWI COOLING Re ρ Kocharovskaya + Khanin 988