Model for surface induced atom loss

rms stability ∆zt,0 =µ0/2π q

(∆II/Bb,y)2+ (II∆Bb,y/Bb,y2 )2 = 3.3 nm. A larger con- tribution comes from the background magnetic noise in the lab of ∆B = 20 mG peak to peak, which in the worst case (when pointing exactly along our y-axis) would lead to ∆zt,0 = 7.4 nm rms. Thus, the estimated positioning reproducibility is at the level

expected from background magnetic field fluctuations. By implementing a magnetic shield and using a better current source for theBb,y field, position fluctuations could be brought down below 1 nm. However, at this level also the mechanical stability of the coils becomes relevant. E.g. if our Bb,y-coils drift in position by 1 µm along (x, y, z), the relative change in the field is ∆Bb,y/Bb,y = (3.1,0.1,9.3)×10−6 and the trap shifts by ∆zt,0 = (0.2,0.06,0.6) nm for our trap parameters. A temperature

change of only 1◦C gives rise to a differential position shift alongz between the atom chip and the coils by ∼ 2 µm due to the different thermal expansion of the Pyrex vacuum cell (αex = 3×10−6 /K) and the steel coil mounts (αex = 13×10−6 /K).

5.3. Model for surface induced atom loss

Here we discuss the model for the loss of atoms in the attractive surface potential Us of the undriven cantilever. A simple model describing such measurements was de- veloped by Linet al. [24] and summarized in chapter 2.4.2. When the magnetic trap is ramped to the surface, the trap depth is reduced to U0 by the surface potential.

The model assumes that this leads to a sudden loss of atoms with energy E > U0.

For a thermal cloud this is described by the truncation of the tail of the Boltzmann distribution, while for partially condensed clouds at temperature T > 0, only the residual thermal cloud coexisting with the condensate is affected. Furthermore, it includes 1D evaporation from the trap to account for collisional repopulation of the high energy states. In summary, the remaining fraction of atoms in the trap is given by

χ= (1−e−η)e−Γ(η)th_{, (5.15)}

whereη=U0/kBT is the ratio of the trap depth and the thermal energy, and Γ(η) is the 1D evaporation rate according to equation 2.80. Evaporation is important when th τel. In the measurement of figure 5.4, τel = 0.2−0.6 ms and th = 1 ms, and evaporation has only a small effect.

Finally, atoms can be lost from the trap by tunneling through the barrier. The loss rate can be estimated by calculating the transmission coefficient T(E, U) of the barrier in the WKB approximation according to equation 2.82. This determines the tunneling rate Λ = ωz/2πT(E, U) (Eq. 2.83). For our measurements, tunneling will contribute when Λ(E) ≈1/th. As Λ increases exponentially with E, the effect can be accounted for by setting the potential depth to the value where the tunneling rate equals 1/th, or Ueff =E(Λ(E) = 1/th).

0 0.1 0.2 0.3 0.4 0.5 0.85 0.9 0.95 1 barrier height U₀ [MHz] U eff /U 0 a) 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 d [µm] χ GPE (ii) (i,ii) data b)

Figure 5.6.: (a) Calculated relative reduction of the barrier heightUeff/U0 due to tunneling as a function ofU0 for Λ(Ueff) = 1 kHz. Red (blue) circles correspond to a trap with ωz/2π = 10 kHz (5 kHz). (b) Measured fraction of remaining atoms

compared to a numerical simulation of the Gross-Pitaevskii equation, to the surface loss model with modification (ii) and T =Tc, or both (i) and (ii) with T = 0.8Tc.

Parameters are ωz/2π = 10 kHz,th = 1 ms,Us=UCP−200C4/(z−zc)4.

Figure 5.6 (a) shows a numerical simulation of the relative trap depth reduction Ueff/U0 as a function of the barrier height U0 for th = 1 ms. The behaviour of the (absolute) barrier reduction ∆U =U0−Ueff can be approximated by a polynomial of

the form ∆U ∼=cU₀0.3, and for the traps used in the experiments we findc(5 kHz) = 5.6×10−3 _{andc(10 kHz) = 1.3×10}−2_{. The effect is of the order of a few percent and}

becomes important only for very shallow traps (U0 <100 kHz). Overall, tunneling

shifts surface loss curves by z ∼20−60 nm compared to the case where tunneling is neglected. It thus has only a small impact for the interpretation of atom loss at the surface.

5.3.1. Improvements of the model

Although this simple model already describes our data fairly well, several improve- ments and alternative approaches are possible:

(i) The model describes loss only from the thermal cloud. A more accurate description can be obtained by assuming a bimodal cloud forT < Tc and including also loss from the condensate. The condensate and thermal atom numbers are obtained by equation2.20or2.21. The thermal cloud is lost forU0 ≥µcas described above with a modifiedη = (U0−µc)/kBT. The condensate is lost forU0 < µc, where the number of remaining atoms Nr can be determined from µc[Nr] = U0, resulting

inχBEC = (U0/µc)5/2.

5.3 Model for surface induced atom loss 101

evaporation law is no longer valid and leads to unphysically large Γ. We correct this by introducing a cutoff at the cross dimensional mixing rate [267], which we implement by setting Γ−1 =τel 1 f(η) exp(−η) + 2.7 . (5.16)

However, this is only a simple patch and does not account for the qualitatively different situation at small η.

(iii) The repulsive interaction between the condensate and the thermal cloud pushes the latter out of the trap center (see equations 2.46 and figure 2.6). This leads to a broadening of the loss curves. This effect could be included by an effective potential Uth(r) = 1 2m(ω 2 xx 2_+ω2 yy 2 _+ω2 zz 2_)−µ c (5.17) for the thermal cloud. However, this no longer permits to treat the sudden loss in the manner discussed above.

(iv) In chapter 5.8.2 we use a numerical 1D simulation of the Gross-Pitaevskii equation to study the dynamics of a BEC in a modulated trap. In the context here, we can use it to simulate the loss in the static surface potential.

(v) Technical heating, three-body collisional loss, and cooling due to evaporation of the atoms are not included in the model. These effects have a strong dependence on the trap frequency and in part also on the atom number.

Figure 5.6 (b) shows a comparison of two of the modifications of the model to- gether with a numerical simulation for a pure BEC (iv) and experimental data. The measurements often display a characteristic kink, which indicates a partially con- densed cloud with different loss behaviour of the thermal cloud and the condensate fraction. The data can be reproduced by using the improvements (i) and (ii).

When applying the above improvements (i) - (iv), we find that the resulting change in the calibration of the atom-cantilever distance d is ±80 nm, which is within our error bar on d. Furthermore, we point out that the data can also be analyzed without a detailed model for atom loss by simply exploiting that χ= 0 corresponds to the values of the atom-cantilever distance d where the trap has vanished (which is well described by the condition U0 < ~ωz/2). This analysis depends only on the knowledge of the trapping potential U, and again yields similar results as the model described above for the short th of the measurements in Figure 5.4, where evaporation does not play an important role.

5.3.2. Heating rate analysis with the surface loss model

In chapters 2.5.3and 4.4.2we have discussed limitations due to technical heating in the trap. A difficulty is that temperature and heating rate measurements in TOF

1 1,5 2 0 0.2 0.4 0.6 0.8 1

atom surface distance d [µm]

χ 2.1× T_c ∆ RF=70kHz 1.5× T_c ∆_RF=45kHz 0.8× T_c ∆ RF=20kHz a) 0 5 10 15 100 101 102 103 trap frequency [kHz] heating rate [ µK/s] b)

Figure 5.7.: (a) Surface loss measurements for different cloud temperatures. The temperature is set by changing the final value of the radio frequency for rf- evaporation. For the data we quote the detuning of the RF stop-frequency ∆RF

from the trap bottom νRF,0 = 4.950 MHz. The fitted temperatures are slightly higher but in reasonable agreement with the temperature following from the RF stop-frequency (T = (2.0,1.3,0.6)×Tc). Such a reference measurement serves as

”calibration” of the thermometer. (b) Heating rates extracted from the surface loss model applied to several measurements with given initial temperature and varying holding time. The solid line is the heating rate calculated with Eq. 2.99 for the measured current noise level.

show large uncertainty for small clouds. In an alternative approach we can use the fit results of the surface loss model to obtain the temperature of the cloud. Even though the accuracy of the evaluated temperature might be worse due to systematic errors of the model, the reproducibility of the determination is very good. When applied to measurements with varying interaction time th, the temperature as a function of time and thereby a heating rate can be extracted. In Figure 5.7 (a), surface loss measurements are shown where different cloud temperatures have been set by changing the final value of the radio frequency during rf-evaporation. The measurement illustrates, how well temperature differences can be discerned. From repetitive measurements with fixed parameters we find a variation of the resulting temperature ∆T = ±0.1 Tc. Note that the only free parameters of the fit are the position of the surface (which is the same for all fits in the figure) and the temperature.

To extract heating rates, we perform measurements for fixed initial temperature with varying holding time at the surface for several trap frequencies. The results are summarized in Figure 5.7 (b) together with a prediction of trap heating. The observed heating rate dependence follows a ω4

z scaling as expected from technical noise induced heating (Eq. 2.97, 2.99), and can be quantitatively explained by the current noise of the source that drives the magnetic coils for the Bb,y offset field (FUG 15A 20V, see chapter 4.4.2).