Fitting the experimental data

Since Eq.5.1 can treat a decrease of the number of transferred atoms during the loading process, we can try to fit our experimental results with it. The interest in such a fit is to see weather the time dependance of this decrease can indeed be correlated to the lifetime of the atomic reservoir in our experiments. Apart for the reservoir lifetime γ, the remaining parameters R0, γL and βL are irrelevant to the atom number decrease and can be seen as free parameters for the time being. In Fig. I.5.2.1 we see the fit of 5.1 in three different studies. A 89µm waist and 60W power crossed dipole trap is loaded a) by a magnetic trap, b) by a Dark-SPOT and c) by a C-MOT (data corresponding to figures 4.10a, 4.16.b and 4.19a). The values used for the parameters R0, γ, γL and β of Eq.5.1 in each case are shown in Table 5.1.

The parameter R0 represents the rate in which atoms are loaded in the dipole trap by the reservoir in this description and γL the density independent losses, while no density depended losses were considered. This fact does not change much in this analysis, since the losses are there only to balance the loading rate into a final atom number that corresponds to the maximum value experimentally obtained. The losses, as well as the loading rate R0, are irrelevant to the atom number decrease during the loading, which is described only by γand which this analysis aims to determine.

The inverse of γcorresponds to the lifetime of the atomic reservoir which could explain such a temporal evolution. We see that for the case of loading a dipole trap from a magnetic trap reservoir, the required lifetime is 3 ms, which is ~4 order of magnitudes less than the values experimentally measured (in the order of 20 s). This result shows that the finite lifetime of the atomic reservoir cannot explain the temporal evolution of the atom number versus the loading time of the magnetic trap. For the case of the Dark-SPOT shown in Fig.5.2.b the required lifetime is 150 ms. The lifetime of the Dark-SPOT has not been measured, but the lifetime of an ordinary MOT is ~15 s, and it is not expected to be very different, thus the 150 ms required by Eq.5.1 is still considered a very small value. Finally, for the case of loading from a C-MOT shown in Fig.5.2.c, the required lifetime is 2.5 s, a value that is less than ten times smaller than the lifetime of an ordinary MOT, and can be seen as more realistic.

Unfortunately, the lifetime of the C-MOT was never measured in the experiment, so the prediction of Fig.5.2.c cannot be verified. However, the value of 2.5 s does not lye far from the value observed for a regular MOT, while the lifetime of a compressed MOT, is expected to be reduced due to temperature increase. The relative success of Eq.5.1 in this case was expected, since the loading method involving the C-MOT is very close to the experiment reported in [Kupp00] and the fitting process yields only a rescaling of γ by a factor of 10. On the other hand, the values extracted for the parameter γ in the case where the dipole trap was loaded by a magnetic trap or a Dark-SPOT, suggest that a different explanation, other than the finite reservoir lifetime has to be proposed.

Fig. I.5.2.1: Fitting with 5.1 the evolution of the atom number loaded in the dipole trap by a) a magnetic trap, b) a Dark-SPOT and c) a C-MOT.

R0 (atoms/s) γ (s^-1) γL (s^-1)

a) Magnetic Trap 3.0 10⁸ 333,33 200

b) Dark – SPOT 2.2 10⁷ 6,66 100

c) C-MOT 1.3 10⁷ 0,4 50

TABLE 5.1. Values of the various parameters of Eq.5.1 used for the fits shown in Fig. I.5.2.1

The analysis presented in Chapter I2 and [Comp06] can be useful in the search of an explanation after inserting some modifications. As suggested in Fig.5.1, no type of losses can justify the atom number decrease during the loading, which can only be explained by a modification in the 'reservoir'. The modification of the number of atoms in the reservoir considered so far does not yield reasonable results for the case of a magnetic trap or a Dark-SPOT in the role of the reservoir, but the atoms number is not the only thing that can be modified. An additional parameter that could be modified during the loading process is the reservoir temperature. Indeed, this parameter is considered to be fixed in the analysis of Chapter I2, since there the reservoir is considered to have an infinite size and number of atoms with respect to the dipole trap. The temperature of the atoms in the reservoir is an important parameter in which the total number of atoms transferred from the reservoir to the dipole trap is strongly depended. In Fig. I.5.2.2 we see this dependance for the initial reservoir temperature ranging from 20 to 300 µK. The number of atoms transferred in the dipole trap is calculated by numerically integrating Eq.2.13 and taking into account the losses by substructing Eq.2.22 for a waist of 89 µm and a power of 60 W.

By following the same logic as the one used in Fig. I.5.2.1, we can try to fit our experimental data with this loading efficiency evolution. The idea is to see what kind of increase in the reservoir temperature can justify the atoms loss during the loading observed in each case. In fitting attempts shown in Fig. I.5.2.3, the dependence on the efficiency on the reservoir temperature, as shown in Fig. I.5.2.2 is plotted on top of the experimental data shown in Fig. I.4.2. The temperature scale is the only parameter used in the fit, and it is adjusted so that the number of atoms predicted for optimum reservoir temperature to coincide to the number of atoms detected for maximum tload.

We see that this simple operation provides with good fitting in cases (a) and (b) where the loading is done via a magnetic trap and a Dark-SPOT respectively, while this is not the case for (c) where the dipole trap is loaded by a C-MOT. In the case of the magnetic trap loading shown in Fig.

Fig. I.5.2.2: Evolution of the number of atoms loaded in the dipole trap versus the reservoir temperature, as predicted in by the analysis of the paragraph I.2.b (Eqs.2.13 and 2.22)

I.5.2.3(a), a temperature increase in the reservoir that could justify the atom number evolution is

~300 µK in ~40 ms, ie 7,5 mK/s. The required temperature increase for the case of a Dark-SPOT reservoir, shown in Fig. I.5.2.3(b) is in the order of 100 µK, during the period of 120 ms ie 0,83 mK/s. Finally, for the case of loading from a C-MOT, shown in Fig. I.5.2.3(c), the required temperature increase is ~40µK during a period of 800 ms ie 0,05 mK/s..

We see that the required reservoir heating is much higher in the case of the loading from a magnetic trap. If the hypothesis is made, that the loading decrease is due to heating inserted in the reservoir, and if this heating source is assumed to be more or less the same for all loading cases, it is reasonable to expect that it would affect the magnetic trap more than the Dark-SPOT or the C-MOT. The magnetic trap is not only small compared to the other traps, with the exception of the Dark-SPOT, but is also a conservative trap, and no cooling mechanism is there to balance the heating effect. While in both the Dark-SPOT and C-MOT cases such a mechanism is present. In the case of the Dark-SPOT, the atoms collide frequently with atoms in the border of the area where both the trapping and the repumping lasers are applied (bright MOT area), or even cross to that area, with a result for them to be in a thermal equilibrium with the laser cooled atoms. Finally, in the case of the C-MOT, all lasers are always present, and the laser cooling is maintained, perhaps slightly modified due to the dipole laser induced Stark shifts in the energy of the Cs levels.

If the temperature increase in the reservoir is a hypothesis that can explain the temporal evolution of the atom number during the loading process, the source of this heating has to be identified. Unfortunately, an experimental study of such the possibility of heating the atoms in the reservoir during the loading process has not been made. A possible heating source is the trapping laser. The Cs atoms scatter photons of the dipole laser at 1070 nm, in a rate which is given by [Web03b]. transition defined by the weighted average of both D lines. Thus

_eff=₁

In the center of a dipole trap realized by a 60 W crossed in a waist of 89 µm the photon scattering rate is 0,11 s^-1. Each scattering event is associated to heating due to the photon recoil of

~200 nK, which means that photon scattering can insert in the system 1,78 µK in one second. The heating associated with photon scattering form the dipole trap can be even bigger, if we consider that photons are scattered by the atoms not only in the crossed region but in all the length of the dipole laser beams that lies in the ~0.5 mm magnetic trap. But, in any case, the heating associated to absorption of the dipole laser radiation seems to be too small to justify the reservoir temperature increase that could justify the loading performance in each case.

Additionally, the loss mechanisms described in Chapter I2, such as the background collisions, the inelastic two and three body losses, are associated with some kind of heating. In the case of inelastic two body collisions, one of the two colliding atoms has its spin changed, and inserts into the system heating equal to the energetic difference between the initial and the final spin state. In the case of three body recombination, two out of the three colliding atoms form a molecule, and the heating inserted to the system equals the energetic difference between the two atoms bound

and unbound state. These heating mechanisms are described in Chapter I2, in order for the temperature of the transferred atoms in the dipole trap to be calculated.

I.5.3. Conclusion

The aim of the analysis presented in this chapter was to present the efforts to understand the evolution of the atom number during the loading process. This is done with the use of a phenomenological theory that assumes modifications of the reservoir characteristics. After showing that the temporal evolution of the atom number during the loading process cannot be explained by losses such as those described in Chapter I2, we consider the analysis presented in [Kupp00], which was initially developed in order to fit the loading of ⁸⁷Rb atoms from a C-MOT to a dipole trap.

This simple analysis, successfully treats the similarly observed decrease of the atom number during the loading, since it takes into consideration the finite lifetime of the reservoir.

This analysis is used in order to fit our similar data, in 3 different cases of loading from a magnetic trap, a Dark-SPOT and a C-MOT. If we consider, that it is the reservoir finite lifetime that causes the atom number decrease observed during the loading, the required lifetime of the magnetic trap is found to be ~3ms, while is is 150 ms for the case of the Dark-SPOT and 2.5 s for the case of the C-MOT. The fact that the required lifetimes are much smaller than the ones expected for the case of the magnetic trap and the Dark-SPOT, means that such a consideration is not reasonable for these cases. However, the relatively small discrepancy between the required and the expected lifetime of the C-MOT, makes this explanation possible for this case of loading.

Since the idea that modification in the reservoir is necessary in order to explain the observed atom number decrease during the loading, I continue by considering modification of the reservoir temperature. The number of atoms loaded in the dipole trap, results from the equilibrium of the loading rate and the various losses. Since the loading rate is strongly depended from the reservoir temperature, we see that such a variation could perhaps explain the remaining cases of loading form a magnetic trap and a Dark-SPOT. The hypothesis of heating in the atomic reservoir during the loading, manages to fit these loading cases fairly well, and the required heating is fairly reasonable.

It is true, that my calculations on the heating due to the absorption of photons from the dipole trap shows that this mechanism alone, cannot explain the required heating. However, the accidental absorption of photons (coming either from the dipole trap, either from the trapping lasers of the MOT), leads to secondary effects, as depolarization, which can contribute to the reduction of the loading efficiency. Finally, parametric heating¹ can always be considered as the source of heating in

1 Meaning the heating due to vibrations on the optical table, which was a very old, marble table, in our experiments.

Fig. I.5.2.3: Fitting on the evolution of the atom number versus the loading time with the loading efficiency dependence curve on the reservoir temperature, as this is predicted in Eqs.2.13 and 2.22 and shown in Fig.

I.5.2.2.

the reservoir. The fact that the required heating in is in the case of the magnetic trap reservoir much larger than in the case of the Dark-SPOT, is easily explained; the magnetic trap is a conservative trap, so if it is heated by any reason it stays hot. On the other hand, the temperature of the Dark-SPOT (and all other non conservative traps), is the result of an equilibrium between the (laser) cooling and the heating mechanisms.

However, this modification on the reservoir characteristic during the loading process is only indicated through our studies, and not proved. However, despite the fact that some of the numbers resulting from the fitting procedure presented in Chapter I5, are not reasonable, the idea of a modification of the reservoir during the loading process, is the only one that can explain the loading dynamics. Perhaps the modification has not been successfully identified. An important limiting factor for this search, is the fact that this idea was considered long after the experiment was stopped, thus, no experimental studies could support my considerations.