Light Shifts From Optical Pumping To Cavity QED

Light ShiftsFrom Optical Pumping to Cavity QEDClaude Cohen-TannoudjiFRISNO 12Ein Gedi , Israël24th February – 1st March 20131

Purpose of this lecture- Describe the perturbations experienced by an atom whenit interacts with an incident quasi resonant light beamPhysical interpretation of these effectsExperimental observation- Point out the connections which exist between theseeffects and the absorption and anomalous dispersionfor lightIndex of refraction- Show that these perturbations are also useful formanipulating atoms and light and for preparing newinteresting quantum statesAtom traps, atom mirrors, optical lattices“Schrödinger cats”, Non destructive detection of photons2

OUTLINE1 - Light shifts and absorption rates. Theory2 - Experimental observations3 - Perturbation of the field. Absorption and dispersion4 - Using light shifts for manipulating atoms and fields3

Approach followed in this lectureDressed atom approachTwo-level atom {g,e} coupled to a single mode of thequantum radiation field with frequency ωLI g, N 1 : Atom in g in thepresence of N 1 photonsI e, N : Atom in e in thepresence of N photonsħδħ ωLEe- Eg ħ ω0Eg,N 1 – Ee,N ħ δ δ ωL – ω0 Detuningħδe, N VAL g, N 1 ( / 2 ) Ω N 1ħδħ ωLΩ N 1 Ω 0 N 1Ω 0 : Rabi frequency in the manifold E1 { g,1 , e,0g, N 1EN 1e, Ng, Ne, N-1 ENg, N-1e, N-2 EN-1}We keep here the N-dependence of the Rabi frequency becausewe will also consider the case of an atom in a real cavity4

Reduced evolution within E (N)EN 1 : Manifold of 2 states { g,N 1 ; e,N }e,N is coupled not only to g, N 1 (by stimulated emission),but also to g,N,k (by spontaneous emission of a photon k).One can show that the reduced evolution of the system in thismanifold is govened by an effective Hamiltonien obtained byadding an imaginary term – i Γ / 2 to the energy of eΓ : Natural width of e, describing the radiative instability of eH eff%δΩ N 1 / 2' ' Ω / 2 i Γ / 2& N 1(**)In sections 1, 2, we consider an atom in free space and afield in a coherent state, so that we take all the ΩN equal to Ωfor all NIn sections 3.4, we consider an atom in a cavity and we keepthe N-dependence of ΩN5

Light shifts and absorption ratesSince Heff is not hermitian, its eigenvalues are not real.The eigenvalue which tends to ħ δ when Ω 0 can thus be written :h(δ δ g i γ g / 2 )where δg et γg are realδ g : Displacement of the state gγ g : Departure rate from the state g. Width of g The interaction with theincident wave displaces gδg is called « light shift » This interaction also« contaminates » the stablestate g by the unstable statee which gets a lifetime 1/γg.γg is the absorption rate ofa photong,N 1ħggħdg Ω / 2e,N6

Ω δ or ΓWeak intensity limitA perturbative calculation of δg et γg is then possible and gives:δg Ωδ2Γγg Ω4δ 2 Γ224δ 2 Γ2Variations with the light intensity ILΩ 2 is proportional to N , thus to I Lγ g and δ g are thus proportional to I LVariations with the detuning δγgδgδLimitδ Γγ g : Absorption curveδ g : Dispersion curveILIL ΓΓδg γ g 2 δg4δδ4δ7

Semiclassical interpretationThe atomic dipole is driven by the monochromatic fieldand has a component in phase with the field and acomponent in quadrature related to the field by adynamic polarizability α(ω).The component in quadrature with the field absorbsenergy. It varies with d as an absorption curve. It is thiscomponent which is responsible for the absorption rate.The component in phase with the field gives rise to apolarization energy. It varies with d as a dispersioncurve. It is this component which is responsible for thelight shift. This effect is analogous to the Stark effectdescribing the interaction of a static electric field with thestatic dipole that it induces. The light shift δg is oftencalled for that reason « dynamical Stark shift ».8

Perturbation of the excited state eIn the weak intensity limit, the same perturbativecalculation as above shows that the excited state e isshifted by an amountδe δ gand that its width G changes:Γ γe Γ γ gThe light shifts of g and e are thus equal in absolutevalue but have opposite signsThe unstable state e is contaminated by the stablestate g and its radiative instability decreases.9

Case of a degenerate ground stateFor example, the angular momentum Jg of g is differentfrom 0 and there are several Zeeman sublevels Mg in g.The light shifts generally depend on the polarization ofthe light and vary from one Zeeman sublevel to another.Simple example of a transition 1/2 1/2 (case of 199Hg)Me -1/2σ-Mg -1/2Me 1/2σ Mg 1/2A σ- excitation onlydisplaces the sublevelMg 1/2A σ excitation onlydisplaces the sublevelMg -1/2For more complicated transitions, see:J. Dupont-Roc, C.C-T, Phys. Rev. A5, 968 (1972)10

High intensity limitWith laser sources, light shifts can be much larger, in the GHzrange. The Rabi frequency Ω can be much larger than Γ.The eigenstates of Heff in E(N) are no longer close to theunperturbed states g,N 1 and e,N. For example, if Ω » Γ, theyare equal to :1( N ) 1/ 2 !" g, N 1 e, N # ( )2( N ) (1/ 2 ) !" g, N 1 e, N # with eigenvalues : hΩN 1 / 2 i hΓ / 4At resonance (δ 0), the real parts of the 2 eigenvalues of Heffare equal as long as Ω Γ. When Ω Γ, these 2 real parts aredifferent. One can no longer describe the behavior of thesystem in terms of light shifts, but rather in terms of doubletsof dressed sates.These doublets are at the origin of effects like the Mollow tripletor the Autler Townes doublet11

OUTLINE1 - Light shifts and absorption rates. Theory2 - Experimental observations3 - Perturbation of the field. Absorption and dispersion4 - Using light shifts for manipulating atoms and fields12

First experimental studiesOptical pumping of 199Hg on the transition 1/2 1/2 with a dischargelamp (no laser sources at that time!) filled with the isotope 204Hg whichresonantly excites the transition F 1/2 F 1/2 .Optical detection of the magnetic resonance line between the2 Zeeman sublevels Mg 1/2 of the ground state g of 199Hg. Verynarrow line because relaxation times in g are very long.One adds a second perturbing light beam with a source filled withanother isotope (201Hg) in order to have a non resonant excitation(δ 0). This second beam is fitered with a cell filled with 204Hg inorder to eliminate all resonant frequencies from the second beam.The intensity of the second light beam, coming also from a dischargelamp is weak, so that one can apply the results derived above for theweak intensity limit. The light is not monochromatic, but, in a secondorder perturbation treatment, one can add independently the shiftsproduced by the various frequency components13

Differential light shiftMe -1/2σ-Me 1/2Mg -1/2σ σ-Mg -1/2Mg 1/2σ Mg 1/2Depending whether the second beam has a σ or σpolarization, it displaces only the sublevel Mg -1/2 or Mg 1/2, changing in this way the Zeeman splitting between these2 sublevels (the Zeeman is vey small compared to thedetuning d of the light beam).The magnetic resonance curve in g, optically detected with thefirst beam thus undergoes a light shift whose sign depends onthe polarization of the second perturbing beam.Very small light shifts can be detected if they are not too smallcompared to the width of the magnetic resonance curve in g14

Experimental observationFirst beamPumping beam σ AtomsSecond perturbing beamσ or σ- polarizedC. Cohen-Tannoudji,C.R.Acad.Sci. 252, 394 (1961)Possibility to detect very small light shifts on very narrowmagnetic resonance curves (relaxation times are very long in g)15

OUTLINE1 - Light shifts and absorption rates. Theory2 - Experimental observations3 - Perturbation of the field. Absorption and dispersion4 - Using light shifts for manipulating atoms and fields16

Atom in a cavityTo study the perturbation of the field, we suppose now thatthe atom is put in a real cavity and we keep the Ndependence of the Rabi frequency.There are actually experiments of this type in a researchfield called « Cavity Quantum Electrodynamics »The field is supposed to be initially in a coherent state.One introduces in the cavity an atom whose frequency ω0 isclose to the frequency ω of the fieldHow are the amplitude and the phase of the field perturbedby the atom-field interaction?How do these effects vary with the detuning δ ω-ω0?17

N dependence of the perturbationWe keep now the N-dependent Rabi frequencyappearing in the effective Hamiltonian Heff governingthe reduced evolution in the manifold EN 1In the perturbative limit, which is realized when N issufficiently small, the eigenstates of Heff are veryclose to g,N 1 and e,N, and the state g,N 1 undergoesa shift δg and a broadening γg, which depend on Nand which will be denoted here δN et γN.δ N ( N 1) Ω20δ4δ 2 Γ2Γγ N ( N 1) Ω4δ 2 Γ220The state e,N undergoes a shift -δN18

Splitting between dressed states 3δ0E3E2g,3g,2E1g,1E0g,0 2δ0we,2-3δ0we,1-2δ0e,0-δ 0 δ0Eg , N 1 Eg , N h (ω δ 0 )Ee, N Ee, N 1 h (ω δ 0 )19

Frequency shift of the fielddue to the presence of the atomThe separation between the states g,N 1 (or e,N) displacedby light correspond to a new frequency of the fieldAtom in state gω ω δ0Atom in state eω ω - δ0The field-atom interaction thus displaces the frequencyof the intracavity field by opposite amounts dependingwhether the atom is in g or in e.If this interaction lasts for a time T, the oscillation ofthe field accumulates a phase shift Φ with respect tothe free oscillation in the absence of interactionAtom in state gAtom in state eΦ δ 0TΦ - δ0T20

Analogy with anomalous dispersionA light beam passing through a medium of length Lundergoes a phase shift proportional to L and to thereal part of the index of refraction.Near a resonance of the atoms of the medium, thereal part of the index of refraction varies like adispersion curve as a function of the frequency of thefield. This is what is called « anomalous dispersion ».The effect studied here for an intracavity field is ofthe same nature. It involves a single atom. The phaseshift increases with time and not in space. It varieslike a dispersion curve as a function of the fieldfrequency.ω ω02φ Ω0 T24 (ω ω 0 ) Γ 221

Damping of the fieldAtom initially in g field in a coherent state α α N α 2 /2ψ (0) g α eg, NN!N 0In the perturbative limit, each state g,N evolves with an energy:E N ω δ i γ / 2 N (ω δ i γ / 2)NNN00After a time t, the initial state becomes: ψ (t ) αNe2 α /2exp i N (ω δ 0 i γ 0 / 2 ) t g , NN! g α exp i (ω δ 0 i γ 0 / 2 ) t The field is still in a coherent state, evolving at the frequency ω δ0,and has an amplitude damped with a rate γ0/2, where:Γγ 0 Ω0224 (ω ω 0 ) Γ 2N 0This damping rate varies with the detuning as an absorption curve.Effect analogous to the one described by the imaginary part of theindex of refraction.22

OUTLINE1 - Light shifts and absorption rates. Theory2 - Experimental observations3 - Perturbation of the field. Absorption and dispersion4 - Using light shifts for manipulating atoms and fields23

Light shifts: a tool formanipulating atoms and fieldsLight shits are a perturbation for high resolution spectroscopysince they change the atomic frequencies that one tries tomeasure. An extrapolation to zero light intensity of the positionof the resonances has to be done for extracting theunperturbed frequency.Light shifts are however useful for creating potential wells,potential barriers for neutral atoms, spatially periodic arrays ofpotential wells whose parameters can be fully controlledThe depth of these wells, the height of these barriers aresmall. But they are large enough for trapping and reflectingthe ultracold atoms which can now be obtained by lasercooling methodsThe field in a cavity can be also manipulated by sendingatoms in this cavity24