Recoil limit

k_B T_r=(ħk_L )^2/ma
last photon which is emitted from the cooled atom in an excited state

Cooling below Recoil limit

 Recoil limit: polarisation gradient cooling
 atoms with p=0 should not be excited anymore (dark state)
 achieved when we use VSCPT cooling

Polarisation gradient cooling

 polarization gradient by two counter prop. lasers with lin┴lin configuation
 „dressed states“
 when atoms reach a position where potential energy large they are pumped into another state for which the energy is close to minimum
 atoms against potential, where pumped, fall in a minimum and then begin again

Rabi oscillations

Resonant Rabi freq. Ω_0=((d_10 ) ⃗ε ⃗E_0)/ħ
C_0 (t)^2=cos^2〖(Ω_0/2t);C_1 (t)^2=sin^2〖(Ω_0/2t)〗 〗
oscillation of population between ground and excited state (periodic alternation of stim. absorption and emission)
generalized Rabifreq. : Ω=√(δ^2+Ω_0^2 )

Dipole/rotating wave approximation

E ⃗(r ⃗,t)≈E ⃗(t)≈ε ⃗E_o cos(ω_L t);for a_0≪λ_L
neglect rapidly oscillating terms (ω_L+ω_10) (nonresonant, to fast for field(atom) to react)

two level atom

atom with a closed two level transition (does not exist/sketch)
Ψ(r ⃗,t)=C_0 (t) e^(iω_0 t) ├ 0⟩+C_1 (t) e^(iω_1 t) ├ 1⟩

electric dipole

Interaction energy: V(t)=(d_el ) ⃗E ⃗(t)
d ⃗_el=∫▒〖ρ ⃗_el (r) r ⃗ d^3 r〗 ; ρ_el=eψ(r ⃗ )^2
Eigenfunction of atomic Hamiltonian(12S1/2) →del=0
superposition state:
d ⃗_el^ind ∞E ⃗(t)
d ⃗_ij=〈id ⃗j〉=e ∫▒〖u_1^* (r ⃗ ) r ⃗ u_0 (r ⃗ ) d^3 r〗

classical electro magn. field

E ⃗(r ⃗,t)=ε ⃗E_o cos(ω_L tk ⃗_L r ⃗ )
k_L=2π/λ_L =ω_L/c
classical Laser field, Maxwell eq. wave eq.

Detuning

δ=ω_Lω_10; deviation from resonance → higher Rabifreq.
δ>0 (blue detuning) repelled in trap
δ<0 (red detuning) attracted in trap

Density matrix pure/mixed states

ρ ̂=∑_(i=0)^n▒〖p_n ├ Ψ_n ⟩⟨Ψ_n ┤ 〗 off diagonal elements (coherences phase relation)
Pure states: Superposition→1/√2(0>+1>)
Mixed states: Mixture source

Spontaneous emission (decay rate)

decay of an excited state into all possible modes (WignerWeißkopf theory)
destroys coherences
τ_1=1/γ_10 (lifetime)

Optical Bloch Equations

 time evolution oft he (components of) density matrix
 steady state (long times)
 →Population,inversion
 Louiville equation

Inversion

ω=(ρ_11 ) ̂(ρ_00 ) ̂ ; Population exited state – Population ground state
ω=1/(1+δ)
not possible to invert the population in a 2level atom (ρ_11 ) ̂→1/2 as maximum in steady state

Saturation parameter; Saturation intensity

saturation parameter: S=(S_0 γ_10^2)/(γ_10^2+4δ^2 );
res. S_0=(2Ω_0^2)/(γ_10^2 )
S_0=I/I_s (measures how strong laser field compared to atomic quantities)

Photon scattering rate

 the rate at which an atom absorbs a photon and reemitts it
 γ_phase: a Lorentzian curve as function of detuning
 FWHM is γ10 for low intensities
