**Part I – Abstract**

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

**I.2.3. Evaporation in the dipole trap**

The simulation of the evaporation process consists of the solution of two coupled differential equations for the number of atoms and for the energy. Each escaping particle is associated with a decrease in the total energy of the sample. On the other hand, loss of atoms reduces the density, and consequently, the phase space density. Since the objective of the evaporation process is to maximize the phase space density, the evaporation performance results on the relation between the phenomena that reduce the phase-space density, like the atom loss, and the ones that increase it, like the temperature decrease associated with the ejection of hot atoms.

Except from the variation in the atom number caused intentionally during the evaporation process, there are more phenomena on which the final atom number and temperature of the trap depends. These are principally losses related to inelastic collisions. Evaporation relies on elastic collisions in order to redistribute the temperature in the sample during the evaporation. During Fig. I.2.4: a) The loading time versus the waist of the dipole trap, b) The final number of atoms trapped in the dimple estimated as Nf,-Nloss. c) The loss fraction of atoms due to the two body (red line) and three body (blue line) inelastic collisions as well as due to the Majorana losses (black line). The two body losses are calculated by a factor of 100 in order to be able to show them in the same scale. d) The final phase space density. This calculation is done with the value of α fixed at ~3000 α0, which is a reasonable value for Cs.

In [Comp06] a similar simulation for α ~ 300 (case of Rb) is given.

elastic collisions, the internal energy of the particles does not change, and high elastic collision rates usually favors evaporation. Inelastic collisions on the other hand change the internal energy of the atoms. Usually, atoms are transferred to a lower energy states after the collision, while the excess energy is passed to the system in the form of kinetic energy, thus heating the sample. The lower energy state in which the atoms are transferred after an inelastic collision can be other atomic states like in the case of collision-assisted spin relaxation. It can also be a molecular state, as is the case in the three body recombination, where two out of the three colliding atoms form a molecule and the third escapes after having acquire the reaction's excess energy in the form of kinetic energy.

In this paragraph I try to describe the evaporation process in the dense dipole trap with the use of coupled differential equations for the number of atoms and energy. Each related phenomena is inserted in these equations with its corresponding rate, and contributes either in the increase or the decrease of the energy of the atomic sample. For the sake of clarity the equations are given first and then each of the term’s origin is discussed.

The equations for the evolution of the number of atoms and the energy are

*N*˙ =−[* _{ev}*

**

_{loss}_{3}]

*N*(2.22)

*E*˙ =−[ * _{loss}*

_{3}]

*E*−E

**

_{ev}**

_{ev}*E*

**

_{pot}**

_{pot}*N*

_{laser}*k*

_{B}*T*

**

_{recoil}_{3}2

3*k*_{b}*T**k*_{B}*T*_{h}*f** _{TBR}* (2.23)
Starting from Eq.2.22, we see that the number of atoms is decreased at a rate that is the sum of the
rates with which the number of atoms is decreased due to the evaporation Γ

*ev*, the three body recombination rate Γ

*3*and an additional loss rate Γ

*loss.*This final parameter includes both the losses due to collisions between the trapped atoms

*Г*

*in*

*=K*

*2*

*n*

*av*

*(where n*

*av*is the averaged density) and the background collision rate Γ

*bg*.

Equation 2.23, which describes the evolution of energy appears more complicated. Initially,
we see the term proportional to the sum of Γ*3 *and Γ*loss *which describes the energy drop due to the
decrease of the atom number. Another source of energy decrease is described by the term E*ev*, which
describes the energy decrease due to the intentionally removed atoms. The evaporation rate Γ*ev* is
related to the elastic collisions that thermalize the sample. The elastic collision rate is *Г**el **= n**0*σu*r *

where u*r**=4(k**B**T/πm)** ^{1/2}* is the average atomic velocity, and σ=8πa2/(1+k

*a*

^{2}

^{2}*) the energy dependent*cross section with a being the scattering length. The wavevector k = mu

*r*

*/ћ is modified as k*

*ev*

*2*

*= k*

^{2 }*ηπ/12 in order to take into account the energy dependent scattering in the potential. The evaporation*rate is related to the elastic collision rate trhough [Perr98]

* _{ev}*=

_{el}*e*

^{−}

###

2 [−52]*e*^{a}* ^{g}*−1

*a** _{g}* −1 (2.24)

where α*g* is the gravitational field written in temperature scale, equal to mgr*U**/k*BT, with r*U* being the
distance from earth's center. In the actual calculation, the ratio Γ*ev* is multiplied with the factor
*f(p**coll**), which is an empirical smoothing function which corrects this rate to take into account the *
effects of the hydrodynamical regime.

The three body recombination leads to losses at a rate Г*3** = L**3**n*^{2}*3** ^{-3/2 }*with n being here the
density. The three body recombination rate L

*3*is [Web03c]

*L*_{3}≈225 ℏ
*m*

*a*^{4}

10.1*k a*^{4} (2.25)

The next two terms in Eq.2.23 describe the energy variation due to the trapping potential's
decrease, and the heating rate due to photon absorption by the ultra-cold atoms. The potential
during the evaporation process decreases the energy of the sample by E*pot** =(3/2)ηk**B**T at a rate Г *

*=dU/dt. Photon absorption is taken into account in the rate Γ**laser*. The photon absorption rate is the
imaginary part of the polarizability a(ω) and is equal to

* _{laser}*=3

The final term in Eq. 2.23 describes the heating associate to three body recombination. An
important aspect of the three body recombination is that it leads to “anti-evaporation”, the loss of
the coldest atoms in the trap. Thus the average of many of these effects in an atomic sample leads to
heating which equal to 2δk*B**T/3 per event. Three body recombination leads to the creation of *
molecules in highly excited rovibrational states. Since these molecules remain in the sample they
can relax to a lower lying state after colliding with an atom, releasing their excess energy into the
sample and causing a heating equal to k*B**T**h** . In order to take into account the fact that such a process *
will not heat the sample if k*B**T**h**> ηk**B**T, but will lead to the expulsion of the atom, we multiply this *
factor by a smoothing function f*TBR**=1-f(**pcoll**)[1-f(k**B**T**h**/η k**B**T)]. *

With equations 2.22 and 2.23 we can simulate the evaporation process for a variety of
conditions, as long as the hydrodynamic regime is avoided. One of the most important parameters
to be defined is the value of the scattering length a since most of the collision losses depend on it. In
the simulation shown in Fig. I.2.5, the value of the scattering length is considered to be a ~ 2900a*0*,
a value which is close to reality for Cs. Other parameters are the initial number of atoms ~ 10^{8},
initial temperature 300 µK, laser waist 100 µm and initial laser power 100 Watts. The power is
decreased with the criteria to bring the value of η form the initial value of 9 to an asymptotically
reached final value of 6.

As noticed before, the evaporation in dipole traps is realized by lowering the trapping potential by reducing the trapping laser’s power. This has as an effect that the potential’s shape is modified during the process and become less confining, a fact that reduces the efficiency of the Fig. I.2.5: Simulation results for an evaporation process with constant waist and with η varying from 9 to 6.

a) The ratio of number of atoms and temperature with respect to their initial values, b) the phase space density evolution, c) the ratio of three body recombination rate on the elastic collision rate and the associated heating and d) the possibility for collisions on the hydrodynamic regime.

evaporation. As the potential becomes less confining, the frequency of the oscillation of the particles in the trap is lowered, reducing the number of collision and making thermalisation slower.

If this effect could be compensated, and the frequency of the trap could be kept constant, the evaporation could be more efficient. In Fig. I.2.6 we simulate a similar situation where the trap’s frequency is kept constant during evaporation. Experimentally, a similar situation can be realized since a piezoelectric stage gives us the ability to modify the trapping laser’s waist in real time (zoom).