To test the validity of the 3D MC simulation we compare several predictions with
experimental data from a^{87}Rb MOT of which the properties were studied in great
detail[146, 154, 155]. We are primarily interested in the comparison of the MC results
with the experiment. The main goal is to verify the MC estimate of the reduction in
the loading rate due to geometrical reasons.

For the simulation we use the parameters from table 2.2. The repump laser is tuned
*to the F= 1 → F = 2 resonance and its intensity is typically around 1.4 mW/cm*^{2}.

The data in table 2.2 are preliminary and are from the Madison group at UBC, Vancouver[146]. More details about their setup and the used methods can be found

*2.4 Comparison of 3D simulation with experimental observations* 31

**Table 2.2: The parameters used for the Vancouver**^{87}*Rb MOT.*

*x, y axis* *z axis*

Aperture diameter (mm) 9 10

1/e^{2}intensity diameter (mm) 7.4 8.4
Magnetic ﬁeld gradient (Gauss/cm) 14 28

** experiment (m/s)**
**v****e**

**0** **2** **4** **6** **8** **10** **12** **14** **16** **18** **20** **22**

** MC (m/s)****e****v**

**0**
**2**
**4**
**6**
**8**
**10**
**12**
**14**
**16**
**18**
**20**
**22**

**Figure 2.6: The prediction for experimental measured values for the escape velocity for a**^{87}*Rb*
*MOT and the predictions from the 3D Monte Carlo (MC) simulation (see text).*

**Table 2.3: The atom number for a given trap depth and the corresponding MOT parameters for the**

87*Rb Vancouver setup. The trap depth is measured with the Photo Association technique (see text).*

Trap depth Detuning Peak intensity Atom number

(K) (MHz) (mW/cm^{2}) (10^{7})

2.2 -12 84 7.7

2.0 -12 21 9.5

1.8 -12 16 10

1.1 -10 6 3.0

0.90 -8 6 3.2

0.64 -5 6 1.4

**8****)**

**Figure 2.7: Comparisons between experimental values, results of the 3D Monte Carlo (MC)***simulation and the prediction by the Reif model. The points are obtained with different laser*
*detunings and**/or laser intensity (see table 2.3). The correlation between the number of trapped*
*atoms observed experimentally and the prediction of the MC model (a). The relation between*
*the escape velocity and the capture velocity in the plane (for a Zeeman slowed beam) (b). The*
*relation between the capture velocity (equation 2.16) and the capture velocity in the plane (c).*

*The number of trapped atoms predicted, using the value of v*_{c}*from (b) and the number of trapped*
*atoms predicted by the MC model (d).*

*2.4 Comparison of 3D simulation with experimental observations* 33

in[154, 155]. To measure the trap depth of the MOT, they used the Photo Association
(PA) technique, pioneered by Walker’s group[156]. They determined the number of
*trapped atoms with a method developed by Chen et al.*[175], which is supposed to
be more robust than the standard ﬂuorescence and absorption method to determine
the atom number^{8}. We calculate the number of atoms by combining the result from
the MC simulation with the result of the loss rate model (appendix A),

*N*= 1

4*〈v〉6π(d/2)*^{2} *εη*

*〈σv〉* . (2.24)

where *〈v〉 is the average velocity of the atoms which end up trapped, assuming a*
thermal gas. In the MC simulation the atoms are uniformly released from a circle
with diameter*d (the laser beam diameter) andε is the fraction of the Boltzmann*
distribution, which is captured by the MOT. Finally, we take into account*η = 0.28,*
the isotopic abundance of^{87}Rb in the source used.

In ﬁgure 2.6 we compare the predictions of the model with experimental values from the Madison group, found using Photo Association (PA). The mean value for the simulation is the velocity for which 50% of the atoms are lost, the error is the velocity range for which 10% and 90% of the atoms escape from the trap. The ﬁt of the scaling factor gives 1.1± 0.05. The model is thus in good agreement with the data.

In ﬁgure 2.7 the measured atom numbers for different MOT parameters are com-pared with the results of the 3D MC simulation and the Reif model prediction (equation 2.18). Figure 2.7a shows the number of trapped atoms observed experimentally and the prediction of the MC model. Only the data point for a trap depth of 1.1 K deviates signiﬁcantly.

We conclude that the MC simulation overestimates the number of trapped atoms with a factor 2.2. This mismatch is either due to an (combination of an) overestimate of the loading rate in the MC simulation, or an underestimate of the collision rate in the loss rate model. From ﬁgure 2.3 we already concluded that the collision rate might be underestimated by about a factor of 1.4. This would leave a disagreement in the observed and the calculated atom number of a factor 1.6, which is very reasonable considering the simplicity of the MC simulation and the difﬁculty in obtaining the experimental observables.

In ﬁgure 2.7b we show the relation between the escape velocity and the capture
velocity. The capture velocity is determined by simulating a Zeeman slowed beam,
with no initial beam size, entering the MOT beams under 45^{◦}. The axial direction
of the MOT is perpendicular to the direction of the atomic beam. This is a usual
conﬁguration for a MOT loaded from a Zeeman slower. The ﬁtted linear function has
an offset of 3 m/s and a slope of 0.72. So from the MC simulation the relationship
*between the capture and escape velocity is v*_{c}*= 1.4v*e. This is in very good agreement
with the experimental value of 1.29(0.12) [155].

8The atom number found in this way was a factor of about 1.4 higher than estimated by measuring the ﬂuorescence of the atoms.

We compare in ﬁgure 2.7c the mean speed (equation 2.16) with the capture velocity, determined via the MC for a Zeeman slowed beam. The data are ﬁtted with a linear function, the offset is 3 m/s and the slope is 0.70. The slope would be 1 if all atoms below the capture velocity would be captured.

Finally, in ﬁgure 2.7d the number of trapped atoms predicted, using the value of
*v*_{c} from ﬁgure 2.7b and the number of trapped atoms predicted by the MC model
are compared. The data is ﬁtted with a scaling factor. The conclusion is that the Reif
model overestimates the number of trapped atoms with a factor 2.7. This factor is
independent of the laser intensity and detuning. The loading rate for the Reif model
is thus

*R*_{mod}= 1

2.7*R .* (2.25)

Note that our 1D estimate of a factor 1/2 (equation 2.20) is close to this value.