**Part I – Abstract**

**I.2. OUR APPROACH TO THE BEC**

**I.2.1. Basic parameters and geometry of the dipole trap**

Even though dipole traps are realized with relatively simple configurations of powerful cw lasers, their characteristics and behavior during evaporation can vary significantly, depending on the wavelength and the chosen geometry. In our experiment, the dipole trap is created with an IPG 100 Ytterbium fiber laser (maximum output power 100 W), but the geometrical characteristics that are needed for the evaporation are achieved with application of magnetic field gradients as well.

An electric field interacts with an atom due to the atoms electric moment, permanent or induced by the field. This can be true for static and for oscillating electric fields as the oscillating field of a laser beam. In this case, the potential of this interaction is

*U** _{dip}*=±〈

*p E*〉

2 (2.1)

where p is the electric dipole moment, E the electric field (where the angled brackets stand for
*average). The sign depends on the detuning of the laser light; if the light is detuned at a frequency *

lower than the nearest transition (red detuned), then the atom minimizes its energy in the area where the electric field is maximized, and the interaction is attractive. In the opposite case of blue detuning the interaction is repulsive. The situation reminds the behavior of atoms in magnetic fields, where atoms either increase (low field seekers) either decrease (high field seekers) their energies with magnetic fields, only that here the creation of a local field maximum is allowed and easily realized.

Most of the dipole traps are realized with red detuned lasers, that is they are based on attractive interactions. The electric dipole moment p is proportional to the electric field E

*p**A*=*a**E* (2.2)

and a(ω) is the atomic polarizability, and can be calculated in the context of the classical Lorentz model [Grim00]. The result for the dipole potential is

*U** _{dip}*

*x , y , z*=−3

*c*

^{2}2

^{3}

_{0}−

_{0}*I**x , y , z* (2.3)

where is Γ relaxation rate, ω*0* is the atomic transition frequency, ω the laser frequency and I the
laser intensity. With x we label the direction in which the beam propagates; as the beam diverges
the intensity depends on this dimension. In the case were the dipole trap is realized with a laser that
is tuned close to an atomic resonance, as is the case for the traps that are used for Raman Sideband
Cooling (see section I.4.4), the rotating wave approximation can be applied and the second part of
the sum in the parenthesis of Eq. 2.3 can be omitted. For the case of the Far Off Resonance Traps
(FORT) like the one used here, Eq. 2.3 is still valid, only that ω*0* is replaced by ω*eff,* which is a
'weighted' average of the transitions that are mainly excited by the far off resonance laser field.

Additionally, Γ is substituted by Γ*eff*, the 'weighted' relaxation rate [Web03b].

As far as the frequency of the dipole trap is very far detuned with respect to the atomic transitions, the main difference between the use of a CO2 laser (λ = 10.6 µm) that has been used in the previous BEC experiment [Web03b] and an Ytterbium fiber laser, is the different beam Fig. I.2.1: The potential of a crossed dipole trap of 50 µm realized either with a) an Ytterbium fiber laser or b) by a CO2 laser. Note that in the second case the ‘arms’ (non-crossed region) of the trap are less profound.

convergence, which can lead to different evaporation efficiency for the two traps. The dipole trap is usually realized in minimum waist region of a focalized beam, and when crossed dipole traps are discussed, the beams are crossed in this minimum waist region. Convergence becomes important for the evaporation, since it is essential that the dipole potential drops fast outside of the crossed region (arms). If this is not the case, then as the potential depth of the trap is lowered in order for the hottest atoms to be ejected from the trap, some of them are guided back to the trap by the potential that is created by the ‘arms’, and thus heat the sample. In Fig. I.2.1 we compare two dipole traps of the same potential depth and dimensions, one created with a CO2 and one created with an Ytterbium fiber laser. We see the ‘arms’ of the trap for the case Ytterbium fiber laser are much more profound with respect to the crossed region, than the ones for the case of a CO2 laser. This difference is due to the dependence of the laser waist evolution on the wavelength as this is given by the Gaussian formula

*w**z*=*w*_{0}

###

^{1}

^{z}^{z}

^{R}^{}

^{2}

^{}

^{where}

^{z}

^{R}^{=}

^{}

^{}

^{w}^{0}

^{2}the Raleigh range (2.4)

In our experiment, the slow beam divergence is a disadvantage for the evaporation process, due to the possibility of the ejected atoms to be guided back to the trap region by the 'arms'. This problem is confronted with the use of magnetic field gradients. Initially the atoms are found in a low field seeking state, where a quadrupole magnetic field can be used in order to trap the atoms and create the reservoir. After the end of the loading process the atoms are polarized in a high field seeking state, such that the quadrupole field alone would expel them from its center. Thus, by having placed the crossed dipole trap in the center of a quadrupole magnetic field we create a potential which has a local minimum in the crossed region but is repulsive out of it. The combined potentials are shown in Fig. I.2.2.

A quadrupole magnetic potential is given by

*U** _{magn}*=

*m*

_{F}*g*

**

_{F}

_{B}*B*(2.5)

where g*F* is the Landé factor which for the F = 3 ground cesium state is calculated to be equal to -
-¼ [Steck02]; µ*B* is Bohr’s magneton, m*F* the spin projection and B the magnetic field. The magnetic
field variation in the center of a quadrupole trap varies linearly, thus the potential becomes

*U** _{magn}*r=−3

4* _{B}*∇

*B r*(2.6)

where with ∇*B* we symbolize the magnetic field gradient. The total potential of the dipole trap is
Fig. I.2.2: The trapping potential (black) which is a combination of the dipole (red) and the magnetic (blue)
potential for atoms in the F = 3, mF = 3 high field seeking state. Effects of gravity are not visible in this scale.

*U** _{tot}*

*x , y , z*=−U

**

_{dip}*z*e

where with w*x*, w*y* we note the minimum waist for the x and y directions respectively. The geometry
of the trap that results the combination of the dipole and the magnetic potential is given in Fig. I.2.2.

The laser intensity is 100 Watt and the laser is focalized in 100 µm, while the quadrupole field gradient is 1 mT/cm.