Lecture 25. atomic vapor. One determines how the response of the medium to the probe wave is modified by the presence of the pump wave.

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1 Optical Wave Mixing in o-level Systems () Saturation Spectroscopy setup: strong pump + δ eak probe Lecture 5 atomic vapor δ + measure transmission of probe ave One determines ho the response of the medium to the probe ave is modified by the presence of the pump ave. () Multiave mixing strong pump + δ eak probe atomic vapor + δ δ b + δ a δ Forard 4-ave mixing (generation of the symmetric sideband at frequency ) δ

2 At lo intensities of the pump laser: (perturbation theory) absorption and dispersion experienced by the probe ave is somehat reduced by the presence of the pump ) I( δ ) I pump At high intensities of the pump laser: (perturbation theory is not enough) atomic energy levels are strongly modified => ne resonances. Solution of the Density Matrix Equations for a -level Atom in the Presence of Pump and Probe Fields Eɶ = E exp ( it ) + c. c. = ρ ρ bb aa complex dipole amplitude p = µ σ ab ba dɶ pɶ = p exp ( it ) + c. c. i pɺ = i p µ ba E ħ eq 4 ɺ = + Im ħ ( pe ) pɶ : expectation value

3 = ba E = E E exp( ) 0 + iδ t E E 0 Eɶ ( t) = E exp[ it ] + E exp [ i ( + δ ) t] + c. c. E Solution: exact for the field 0 and the loest order in the amplitude the steady-state solution is in the form: p = p + p exp( iδ t) + p exp( iδ t) 0 = + exp( iδ t) + exp( iδ t) 0 0 p0, 0 E0 p ± p 0 ± 0 - the solution hen only the pump field is present = ( ) = + cos( δ + φ ) t t 0 phase oscillations of the pump-probe frequency difference ( + ) ( eq ) = Ω E 3

4 if from (the result of the perturbation theory limit) D( δ ) resonances at ( p = 0 ) no resonance at ( + ) ( eq ) = Ω i i E E δ + δ + ħ i D( δ ) i i i i D( δ ) = δ + δ δ δ Ω + 0 = 0 µ ba Ω 0 δ = + χ (3) δ = δ = 0 δ = + in general case: D( δ ) δ δ Ω + Ω i + = δ + ( ) / Ω = Ω + detuned Rabi frequency 4

5 Real part vanishes for if then near δ = 0 δ = ± Ω + + Ω (resonances are separated) δ δ Ω D( δ ) = δ ( δ Ω ) i + δ = 0 D( δ ) δ = 0, ± Ω - resonances ( δ i ) Ω + Γ0 + Ω + Ω 0 Γ = / Γ0 hen Ω Γ0 hen Ω near δ = ±Ω D( δ ) Ω ( δ ± Ω ) ± iγ± [ ] ( ) ( ) Ω + Γ + Ω ± = + Ω Γ± hen Ω Γ + ± hen Ω 5

6 he positions of these resonances can be understood in terms of the energies of the dressed atomic states. We next consider the response of the atomic dipole at sideband frequencies p µ E = + ħ response at (probe field frequencies) ba 0 ( ) [ F( δ ) ] + δ + i ( δ + i )( δ + i ) F( δ ) = Ω ( i ) D( ) δ ( )( )( D( δ ) = δ + i δ + i + δ + i ) Ω ( δ + i ) F( δ ) dδ = 0 + δ st term is the result of the dc part of the population difference. nd term is result of population oscillations. Resonances from D( δ ) does not modify the integrated absorption; leads to spectral redistribution of probe-ave absorption. 6 δ = ±δ

7 he positions of these resonances can be understood in terms of the energies of the dressed atomic states. p at ( )( + ) ( + i )( δ + i ) D ( δ ) 4 µ ba 0 E0 E δ i δ i = 3 ħ Ω 0 p 3 ( (3) E χ ) Nonlinear Susceptibility and Coupled-Amplitude Equations P( + δ ) = Np () P( + δ ) = χ ( + δ ) E eff () N 0 eff ( ) µ ba i i δ χ + δ = δ + δ + Ω ε 0 D( δ ) ħ i effective linear susceptibility (depends on intensity of pump beam) effective 3 rd order susceptibility ( ) = = 3 χ [ δ = + ( + δ )] P δ Np E E χ ( δ ) = (3) eff 0 ( )( + ) ( i ) 4 (3) N0 µ ba δ i δ i eff 3 3 ε 0ħ ( δ + i ) D ( δ ) + 7

8 δ aking into account that the ave is generated in the medium. [ ] P( + δ ) = χ () ( + δ ) E + 3χ (3) + δ = + ( δ ) E E eff eff 0 [ ] P( δ ) = χ ( δ ) E + 3χ δ = + ( δ ) E E () (3) eff eff 0 Describe the propagation of these aves by means of coupled-amplitude equations: ( ) E = A exp ik z ± ± ± k = n ( ± δ ) / c ± ± () n = + 4π Re ± χeff ( ± δ ) sloly-varying-amplitude approximation da dz da dz = αa exp + κa ( i kz) = α A + κ A ( i kz) exp + δ () α± = π Im χeff ( + δ ) n c k = k k k δ κ = 6 πi χ ( ± δ ;,, ( δ )) (3) ± eff 0 n± c (can be stated exactly; the gain is particularly large hen is close to + ± 0 δ Ω A 8

9 We consider the solution for the case if angle is large no phase-matching equations decouple into da dt da = α A = α dt A Rayleigh resonance δ = +Ω δ = Ω 3 features: one is centered on the laser frequency and the other occur at the Rabi sidebands α can become negative! the gain feature at δ = 0 can be considered to be a form of stimulated Rayleigh scattering 9

10 + Ω Ω + + Ω + Ω P: 3-photon resonance (atom makes a transition from the loest dressed level to the highest dressed level by the simultaneous absorption of pump photons and the emission of a photon at the Rabi sideband Ω amplification of ave ith = Ω probe RL: Stimulated Rayleigh resonance: corresponds to a transition from the loer level of the loer doublet to the upper (loer level of the upper doublet). ransitions centered at AC: Usual absorptive resonance of the atom as modified by the ac=stark effect. For the sign of the detuning used in the diagram, the atomic absorption is shifted to higher frequencies. (can lead only to the absorption; at small Ω transfer to the usual resonance) 0