Conclusion
2.2. Spin selective detection
173Yb:3P1(F = 7/2)state, we shift the frequency using an AOM cascade of two double passes, each shifts with400 MHz, and one single pass with a shift of200 MHz. Details about why this particular frequency was chosen can be found in the following section. We simultaneously use the last AOM in front of the fiber link for switching the beam. To keep the optical setup after the fiber as simple as possible we use fiber-polarization control to set the beam to a circular polarization (σ±) and avoid further polarizing optics. Since we can use almost all the light of the laser, we reach a power of about45 mWbehind the telescope despite the limited efficiency of the AOMs.
Different to the magnetic potentials that can usually be engineered for spin separation, the spatial extension of typical OSG beams is very limited. According to Eq. (2.1) the potential is directly connected to the intensity profile of the beam. For typical polarizabilities and available laser powers the waists range from several tens of µmto few hundreds of µm. These small foci are essential to reach sufficient forces at the slope of the Gaussian beam profiles [Sie86]. Due to the relative size between atomic cloud and OSG beam the spatial dependence on the dipole force can lead to strong deformation of the atomic cloud [Ste11]. For this reason ideally the waist is as large as possible. An upper limit is given by the available laser power, because the maximum force is given by
Fmaxw0 2
∝ I0 w0, and the Intensity scales with
I0 ∝ P0
w20,
with intensityI0, powerP0and waist w0. As a compromise we decided for a waist of100µm, which showed good separation for pulses on the order of100µsduration.
The telescope for producing the respective waist inside the science cell was put on top of a rigid aluminum post with an angle of incidence of11°. It was manually assembled using a lens tube. For collimation we used an achromatic lens14 with a focal length off = 15 mmand for focusing a singlet lens15withf = 750 mmto cover a large distance towards the atoms. Due to the big lever of the beam we use differential-µmscrews to position the focus slightly above the atomic cloud. In addition, the post can be translated along the optical axis.
2.2.2. Polarizability of the
3P
1state for the isotopes
171Yb and
173Yb
Essential for the separation of different mF states are the individual polarizabilities, because FmF ∝ αmF according to Eq. (2.1). Therefore, we calculated the actual values for 171Yb and 173Yb using atomic parameters from [Pan09]. The results can be found in Fig. 2.10.
Both plots show the behavior exemplary for σ+ light. Regarding the polarizability differ-ences in case of 173Yb, a situation slightly blue detuned with respect to the F = 7/2 tran-sition seems optimal. A balance has to be found between the smallest potential difference Udip(mF = −3/2)−Udip(mF = −5/2)and the overall scattering rate. In case of 171Yb, the situation is more relaxed, because of the overall large polarizability difference between both
14Thorlabs AC064-015 ML
15Thorlabs LA1978-A-ML
2.2. Spin selective detection 39
F = 7/2 F = 5/2 F = 3/2
1S0
a b A
OM
Vacuum glass cell
Fiber link
Atoms
Experiment control
OSG
z x
y
Polarization control
Focussed beam 45mW 173
Yb:
3P
11.5GHz 4.7GHz
1.4GHz
DL-SHG 556nm
c
OSG beam
Atomic cloud mF=5/2 -5/2
Figure 2.9.: OSG level scheme for173Yb and laser setup. (a)The illustration shows the level scheme relevant for OSG application with173Yb. Indicated is the laser frequency about1.4 GHzblue detuned with respect to the
3P1(F= 7/2)state. Level distances are not to scale.(b)Both images illustrate the geometry between OSG beam and atomic cloud. Depending on themFstate the beam focus can act repulsive or attractive. Sizes are not to scale.
(c)The graphic shows a simplified version of the OSG laser setup. We use a SHG system as source for556 nm light. The single AOM is representative for an AOM cascade of three units. We set the proper light polarization (σ±) using fiber-polarization control. The OSG-beam focus is slightly adjusted above the atomic cloud, has a waist of100µm×100µmand an incident angle of about11°.
mF states.
For quantifying the efficiency of the separation we calculated the potential differences∆Udip between adjacentmF states and normalized them with the mean scattering rateΓ¯sc. The result can be found in Fig. 2.11a. It can be seen that this measure confirms the frequency range be-tween hyperfine statesF = 7/2andF = 5/2to be optimal. Similar values can be found to the far left (red detuning). However, these frequencies are technically hard to reach. As Fig. 2.10 already indicated, the critical part is the separation of the least affectedmF state. In this case these aremF =−5/2andmF =−3/2. Fig. 2.11 even suggests to increase the detuning a little to reach the maximum value of the blue solid line. All other pairs are better separated than these two.
Finally, Fig. 2.11b shows the scattering rateγsc along a trajectory for differentmF states. The calculation was carried out exemplary for an OSG duration of250µs, beam waist100µmand a power of 45 mW using a Runge-Kutta method for ordinary, differential equations [Sha97].
The initial position of the particles was set to thew/2. Here, the initial force is maximized by the largest gradient. The plot indicates that scattering is relatively constant during illumination.
The values for both lowestmF states actually increases a little, which is expected, because these states are attracted towards the beam focus. Integrating each graph results in the total amount of scattered photons:
m−5/2 : 0.03, m−3/2 : 0.04, m−1/2 : 0.06, m1/2 : 0.08, m3/2 : 0.11, m5/2 : 0.14. (2.6)
- 1 0 - 5 0 5 1 0 - 4
- 2
024
- 1 0 - 5 0 5 1 0
- 4 - 2
024
1 7 3
Y b
Udip/h (Hz)
D e t u n i n g ( G H z ) m F = - 5 / 2
m F = - 3 / 2 m F = - 1 / 2 m F = + 1 / 2 m F = + 3 / 2 m F = + 5 / 2
F = 7/2 F = 5/2 F = 3/2 F = 3/2 1 7 1Y b
Udip/h (Hz)
D e t u n i n g ( G H z ) m F = - 1 / 2
m F = + 1 / 2
F = 1/2
Figure 2.10.: Polarizability of the 3P1 state for 171Yb and 173Yb. Both plots show the dipole potential of differentmF states for an intensityI= 1 W m−2. The light is purely circular (σ+) polarized. The center (0 GHz) is set to the resonance of176Yb. Atomic properties for the calculation are taken from [Pan09]. The black dotted line indicates the frequency position of our OSG laser. Here, the polarizability differences show large values. For further discussion see main text.
These numbers indicate that up to a portion of14%atoms a subject to off-resonant scattering.
This can be observed in the following section.
2.2.3. OSG imaging
For determining the spin distribution we initially apply an OSG sequence before commencing actual measurements. An exemplary image of a six-component gas of 173Yb can be found in Fig. 2.12. In this case the OSG beam was switched on for250µsat a power of45 mWand the picture was taken after a time-of-flight (TOF) of13.5 ms.
The image shows very good separation for the higher mF states, while the lowest ones can barely be distinguished. This behavior is in agreement to the polarizabilities shown in Fig. 2.10.
The total separation between|mF|= 5/2is about650µm. This value is in very good agreement with the numerically determined trajectory.
Furthermore, scattered atoms are visible in between and below the different mF states. Re-capturing the numbers from 2.6 it is apparent that during illumination off-resonant photons are scattered. Based on the estimation, in case of mF = 5/2 about 14% of the atoms scatter a photon. The visible atoms next to the main peaks can be attributed to this process. For instance the cloud below the mF = 5/2 peak is located within a distance of 94µm. The direction of separation coincides with the optical axis of the OSG beam. The displacement can therefore be explained by the absorption of556 nmphotons. The velocity change due to one photon in case of Yb is 4.1 mm s−1 which is not sufficient to explain this distance by mere propagation dur-ing TOF (55µm). Instead, the scattering event happens during illumination, which additionally changes the actual trajectory in the potential of the OSG beam. The atoms in between the peaks can be likewise explained by scattering processes during illumination. Thus they are subject to differently applied forces. In addition, eachmF state needs to propagate through all other states