Lecture 1. Physics of light forces and laser cooling

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1 Lecture 1 Physics of light forces and laser cooling David Guéry-Odelin Laboratoire Collisions Agrégats Réactivité Université Paul Sabatier (Toulouse, France) Summer school "Basics on Quantum Control, August 2008

2 Outline Illustrative plot of various phenomena along a scale of temperature plotted against Lecture 1 the de Broglie Wavelength ( ) Lecture 2 ( )

3 Goal of this lecture: understand how to use light to manipulate the external degrees of freedom of atoms, with an application to laser cooling 1. Classical picture Outline 2. Elementary processes: Spontaneous and stimulated emissions 3. Forces on an atom at rest 4. The radiation pressure force and the dipole force 5. Doppler cooling 6. The magneto-optical trap 7. Sisyphus cooling

4 Simple Classical Picture (1) Atom =nucleus + electron The electron is harmonically bounded to the nucleus and subjected to the classical radiation field of the laser

5 Simple Classical Picture (2) Driven oscillations of the electron obey the Newton equation: Dissipative term according to Larmor well known formula for the power radiated by an accelerated charge: Γ = ω e πε ω mc e 3

6 Simple Classical Picture (3) Physical interpretation = interaction between the laser field and the the atomic dipole that it induces p = ( ex ) =αe we deduce the atomic polarizability α = e 2 1 ω ω + ωγ 2 2 m 0 i ω with the on resonance damping rate

7 Simple Classical Picture (4) Conservative component (Dispersive shape) Dissipative component (Lorentzian shape) Im( α) Re( α) ω ω 0

8 Semiclassical approach in quantum mechanics In a semiclassical approach, the atomic polarizability can be calculated by considerinq a two-level atom as a two level quantum system interacting with a classical radiation field. One finds that, when saturation effects can be neglected, the semiclassical calculation yields exactly the same result as the classical calculation with only one modification: The damping rate can no longer be calculated from Larmor s Formula, but it is determined by the dipole matrix element: Γ= ω 3πε 0 3 c 3 edg ˆ 2

9 Two-level atom and elementary processes e e hν hν g g Emission The atom goes from a state e to a lower state g by emitting a photon h ν Absorption The atom goes from g to e by absorbing a photon h ν In his attempts to derive Planck s law for blackbody radiation from an analysis of the energy exchanges between a 2-level atom and a radiation in thermal equilibrium, Einstein was led in 1917 to introduce 2 types of emission 1-9

10 Spontaneous emission of a photon (dissipative) An atom does not remain indefinitely in the excited state e. After a finite time τ R, it falls down to the ground state g by spontaneously emitting a photon in all possible directions. τ R : Radiative lifetime of e, on the order of 10-8 s e g Stimulated emission of a photon (conservative) h ν e g A photon with energy h ν = E e -E g, impinging on an atom in the excited state e stimulates (or induces) this atom to emit a photon exactly identical to the impinging photon (same energy, same direction of propagation, same polarization)

11 Light forces: momentum exchange The absorption and emission are accompanied with a momentum exchange. Example with emission The recoil velocity of the atom, assumed initially at rest, is 1 Example: = 6 mm.s for rubidium atoms v R v R k = m

12 How should we proceed to describe quantum mechanically the force exerted by quasi-resonnant light on atoms without any restriction on saturation? 1 internal degrees of freedom (g and e) 2 external degrees of freedom ( ˆR and ˆP )

13 The systems in interaction

14 The hamiltonian of the problem

15 The timescales of the problem

16 The broadline condition

17 Concept of mean force and Heisenberg inequalities To define the concept of mean force F(x) in a given point x, we need to atomic wave packets sufficiently well localized in position and velocity: Δ x 1/ k k Δv Γ Δx. Δv Γ/ k 2 L L L This localization condition is compatible with Heisenberg relation Δ x. Δv h / M only if : Γ 2 / kl h/ M One recovers the condition : hγ Erec or in other terms

18 The force acting on an atom

19 The average force

20 The forces on an atom at rest

21 The radiation pressure force

22 Slowing down an atomic beam ω L ω 0 OVEN 300 m/s Solenoid The atom is progressively out of resonance because of Doppler effect ω L +kv μβ(z) Zeeman slower 300 m/s = x 6 mm/s The atoms are stopped After a one meter interaction With the laser A stopped sodium beam W. Phillips, 1985

23 The dipole force at low intensity

24 Physical interpretation #1 of the dipole force Plane wave, no dipole force absorption - stimulated emission occurs but with no consequence on the atom motion I 0 Dipole force is exerted on the atom Many wave vectors are involved

25 Physical interpretation #1 of the dipole force If the laser wave is a superposition of several plane waves r (,,,.k ω = ω i = 123,i )i L the atom can absorb one photon in the wave i and emit, in a stimulated way, one photon in another wave i j No energy is absorbed from the laser wave in such a cycle since hω = hω = hω r r i j L But since hk hk the atomic momentum changes by an j amount,i r r h k k ( ) i j The reactive force is thus a force due to a «redistribution» of photons between the various plane waves forming the laser wave. This is why it is called also «redistribution force» ( i j j i) The sense of the redistribution or depends of the relative phase between the 2 waves i and j at the atom position. Redistribution is a coherent process

26 Physical interpretation #2 of the dipole force

27 Prospect 2 : Dipole beam Transport of a packet of cold atoms Transport of atoms Yb-fiber laser: 100 W λ L = 1070 nm Rubidium atoms ωl ω0 λ 0 = 780 nm

28 Prospect 2 : Dipole beam Transport of a packet of cold atoms Transport BEC of atoms

29 Prospect 2 : Dipole beam Transport of a packet of cold atoms Transport of atoms Non adiabatic transport: A. Couvert, T. Kawalec, G. Reinaudi and D. Guéry-Odelin, Europhys. Lett. 83, (2008).

30 Physical interpretation #3 of the dipole force: dressed atom approach

31 Dipole trap gallery

32 C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, Phys. Rev. Lett. 71, 3083 (1993) Atomic mirror with blue detuned evanescent light probe Photo-detector Blue evanescent wave at the surface of the prism

33 Radiation pressure force Plane wave r,e r k ω L L x F = F = k L 2 Γ s( δ) F x ( δ ) δ = ω ω A L s( δ) 2 = Ω1 /2 2 2 δ +Γ / 4 33

34 Atom moving in a plane wave. Doppler effect The atomic center of mass moves in a plane wave. r r r r r = + v t = v t (we take = )R The velocity v is considered ω ω k r as fixed. Laser wave r. v Doppler effect L L L r r (, ) r r r r r cos. cos ( r E.v) L R t = el E L ωlt kl R = el EL ωl kl t Optical Bloch equations keep the same form as for an atom at rest with: F v()x ω A ω L k L v r v r r,k ω e r x L L Couterpropagating atom and laser r r k r L = k L e r x k v = k v.l L Resonant excitation when: r r ω ω ω L kl. v = L + klv = A ω < ω L A We suppose here : (red detuning) 34

35 1975 Force exerted by the +k L wave Principle of Doppler cooling ( for a red detuning : δ = ω ω < 0 ) One supposes that one can add independently the radiation pressure forces of the 2 waves. F v()x L Total force A r k ω r,l L,k ω e r L L x ω A ω L ω L ω A k L v Force exerted by the -k L wave 35

36 Principle of Doppler cooling klγ F = F = ( s+ ( v) s ( v) ) αv 2 Friction coefficient s with 2 /2 2 δ + Γ /L 2 2hks when δ 2 L s 1 α = 2hk s α = = Γ Order of magnitude of the velocity damping time ± 2 Ω1 /2 ( v) = 2 2 ( δ k. v) +Γ /4 δ Γ L ( 4) ///τ = M α = M k s τ = M k if s = Damp Damp 2 2 2h 10 h 0 1 L.L For Rb atoms and s=0.1, one finds τ Damp 100 μs Very short damping time Velocity capture range For δ=-γ/2, the variation with k L v of the total force shows that the velocity interval v capt over which the force is appreciable is given by:

37 Physical interpretation Quasi-resonant, red-detuned counter propagating beams ω L ω A Lab frame (LF) ωl V ωl «V»frame (VF) ω kv L L ω + kv L L = ω A VF VF ωa ωa LF ω L V ω V v R + kv= ω + 2kV A L L L

38 Energy balance and entropy considerations Energy balance The energy of the reemitted photon is on average larger than the energy of the absorbed one. The energy of the radiation field increases while the energy of the atoms decreases. Qualitative considerations about the entropy Cooling of atoms results in a decrease of the entropy of the atoms. But photons are absorbed from the laser beam, which is a low entropy system and transformed into fluorescence photons emitted in all possible directions. The fluorescence field is a disordered system with a high entropy. The entropy of the radiation field thus increases while the entropy of the atoms decreases. 38

39 3D version S. Chu et al. PRL 55, 48 (1985).

40 Does an optical molasses deserve its name? Consider an atom embedded in a 3D optical molasses. How this point distribution expand? According to the equation of motion, the atom "loose" the memory of its initial velocity after the damping time τ = M / α The spatial diffusion can be described by a random walk with a step size vrms τ given by the product of the rms velocity by the damping time 2 2 The randow walk model gives r = 2 where N is the number of steps, after a time T, N= T / τ N Mk T = = = = τ 2 2 B r 2 DT x with Dx vrmsτ 2 Numerical application: r 0.5 cm, requires, for Cs atoms, a time diffusion T = 1 second! D α 2

41 The Doppler temperature radiation pressure cooling force Each scattering event represent two random walk steps for ex.: Rubidium T min = 140 μk

42 The Doppler temperature min /B kt = hγ 2 The corresponding value of the velocity dispersion is given by: 2 ( ) h ( ) Δ v Γ Δ v hγ /M M min /Tmin ( ) ( ) min capt ////n fact, Δv Δv hγ M Γ k hγ E The first estimations of were mixing up Δv and Δv Γ ( ) ( ) ( ) ( ) 1 min capt First correct calculation of the limits of Doppler cooling: Moscow group : min L /L V. S. Letokhov, V. G. Minogin and B. D. Pavlik, Cooling and trapping of atoms and molecules by a resonant laser field, Opt. Commun. 19, 72 (1976). rec k

43 Doppler theory versus experiment T Temperatures much lower than T doppler have been reported, explanation relies on the sublevel structure of the ground state

44 Light shifts and polarization T

45 Sisyphus cooling Polarization Gradient Degenerate Ground State Interplay between dispersive and dissipative effect Claude Cohen-Tannoudji, Review of Modern Physics 70, 707 (1998)

46 Confrontation with experiment T Europhys. Lett 12, 683 (1990)

47 Use of optical molasses to load a dipole trap Superimpose the dipole beam on the optical molasses First experiment: S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, Experimental Observation of Optically Trapped Atoms, Phys. Rev. Lett. 57, 314 (1986). 500 sodium atoms were trapped in the laser beam

Cooling and trapping of atoms 1997 Physics

48 Cooling and trapping of atoms 1997 Physics Nobel prize W. Phillips, S. Chu and C. Cohen-Tannoudji Zeeman Slower Molasses First realization of the magneto-optical trap Sub-Doppler cooling mechanism "for development of methods to cool and trap atoms with laser light"

49 Radiation pressure trap: magneto-optical trap (MOT) Idea proposed by Jean Dalibard 1987 b' = 10 Gauss / cm I = a few mw per arm Coils in AntiHelmholtz configuration

50 Force exerted on an atom in a MOT Damping Doppler effect Trapping Zeeman effect

51 The limit of small number of atoms

52 The MOT in the limit of large atom numbers (1)

53 The MOT in the limit of large atom numbers (2)

Use of a MOT to load a dipole trap 5 10 7

54 Use of a MOT to load a dipole trap rubidium atoms are trapped in the laser beam

55 Can one cool without a dissipative force? (1)

56 Can one cool without a dissipative force? (2)

57 Laser cooling: the proposals [2.1] T. Hänsch and A. Schawlow, Cooling of gases by laser radiation, Opt. Commun. 13, 68 (1975). [2.2] D. Wineland and H. Dehmelt, Proposed Δν<ν laser fluorescence spectroscopy on TI + mono-ion oscillator III (side band cooling), Bull. Am. Phys. Soc. 20, 637 (1975). Theory of Doppler cooling [2.3] V. S. Letokhov, V. G. Minogin and B. D. Pavlik, Cooling and trapping of atoms and molecules by a resonant laser field, Opt. Commun. 19, 72 (1976). [2.4] D. Wineland and W. Itano, Laser cooling of atoms, Phys. Rev. A 20, 1521 (1979). [2.5] J. P. Gordon and A. Ashkin, Motion of atoms in a radiation field, Phys. Rev. A 21, 1606 (1980). [2.6] Y. Castin, H. Wallis and J. Dalibard, Limit of Doppler cooling, J. Opt. Soc. Am. B 6, 2046 (1989). Optical molasses [2.7] S. Chu, L. Hollberg, J. Bjorkholm, A. Cable, and A. Ashkin, Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure, Phys. Rev. Lett. 55, 48 (1985). [2.8] V. S. Letokhov and V. G. Minogin, Laser radiation pressure on free atoms, Phys. Rep. 73, 1 (1981). [2.9.] T. W. Hodapp, C. Gerz, C. Furtlehner, C. I. Westbrook, W. D. Phillips and J. Dalibard, Threedimensional spatial diffusion in optical molasses, Appl. Phys. B 60, 135 (1995). 57

58 Nobel Lectures [2.17] Steven Chu, Nobel Lecture: The manipulation of neutral particles, Rev. Mod. Phys. 70, 685 (1998). [2.18] Claude N. Cohen-Tannoudji, Nobel Lecture: Manipulating atoms with photons Rev. Mod. Phys. 70, 707 (1998) [2.19] William D. Phillips, Nobel Lecture: Laser cooling and trapping of neutral atoms, Rev. Mod. Phys. 70, 721 (1998). 58

Laser cooling and trapping

Laser cooling and trapping William D. Phillips wdp@umd.edu Physics 623 14 April 2016 Why Cool and Trap Atoms? Original motivation and most practical current application: ATOMIC CLOCKS Current scientific