Laser cooling (temperature measurement)

Temperature of a lot of cold atoms
expansion (from trap or molasses)
image after time Δt
measurement of position spread Δx
T∝(Δx/Δt)^2

Magnetic trap

class: E=μ_m B cosϑ
quant: E=gμ_B m_F B
for an atom in state mF a trap is formed if E(mF) has a local minimum
gmF>0 local minimum low field seekers
gmF<0 local maximum high field seekers
the trap centre is always where B has local minimum

Majorana losses

 due to nonadiabatic transitions
 spin flips for small or vanishing BFields
 if magnetic field is finite like in a JoffePritchard trap, Majorana a losses are supressed

Optical lattices

 1D: the electrical field of a onedim. standing wave formed by two counter prop. lasers
 to avoid interference:
polarisations orthogonal
frequency detuning
 one can have Bragg scattering on lattices

Trap geometries

 simplest optical trap: Gaussian beam (red detuned δ<0) potential can be approximated as harmonic
 weak confinement along laser beam axis
 crossed dipole trap: overlap of two laser beams
 Advantage: strong confinement in all directions
 but: small volume

Magnetooptical trap(MOT)

combination of optical (laserfield) trap and magnetic field
F=0 →F‘=1 transition
F=β ̃vkz (damped harmonic oscillator)
010 ms overdamped
capture velocity ≈50 m/s

Optical traps

trapping due to optical potential
U_opt (r ⃗ )≈(ħδ_0 γ_10^2)/8δ
δ>0 atoms are repelled
δ<0 atoms are attracted
(due to radiative force)

MOT: Temperature vs. density regime

temperature limited:
overdamped oscillator
1/2 k_B T=1/2 mv_rms^2=1/2 kz_rms^2
z_rms=√((k_B T)/κ)
density limited
if the atom number is increased atoms start to interact →repulsion
max. ≈10^11 atoms/cm³→ Volume increases with number of atoms

Evaporative cooling

o fast particles escape
o „rest“ thermalizes through collisions
o Abkühlung
o E/N↓→T↓
 theoretical:
o infinitly slow evaporation
o But: in experiment are losses

Evaporation in a magnetic trap

 transfer all energetic atoms to a magnetic sublevel, that is not trapped
 RF Knife
 inharmonic trap: density may increase while atom number shrinks
 run away evaporation: nσv (n=NT3/2→∞T1/2)
 RF induced: manipulate potential such, that fast atoms evaporate

Feshbach resonances

 tune the scattering length by applying an external field (magnetic)

What is a BEC?

quantum mechanical phase transition from thermal gas at T≠0
nλ_dB^3=2.612;
caused by quantum statistics not interactions
p ⃗=0⟩ has macroscopic occipation
can be described as matter wave
Atomic oven → Zeeman slower → MOT →optical molasses → magnetic trap → evaporative cooling → BEC
n(ε_p )=1/(e^β(ε_pμ) 1) ;μ≤min(εp) for n≥0

Critical temperature macroscopically occupation

below Tc the ground state is occupied with N0 atoms in the ground state
T_c=(2πħ^2)/(mk_B (2.612)^(2/3) ) p^(2/3)
N_0=N(1(T/T_c )^(3/2)

BEC in a harmonic trap

 Δx ist Breite des harmonischen Potentials

GrossPitiaevskiEquation

iħ ∂/∂t Φ(r ⃗,t)=((ħ^2 ∇^2)/2m+V_ext (r ⃗ )+gΦ(r ⃗ )^2
„Mean field“: eff.Potential =trap potential +U meanfield Φ(r ⃗,t)=√(ρ(r ⃗ ) ) e^iΦ(r ⃗,t) valid for ρa³ <<1 (a is scattering length)
Interaction energy of BEC much larger than kinetic
