Algoritmos gen´eticos

In Figure 5.6 the cloud is larger than the coupling beam, so only a small fraction of the cloud experiences strong coupling to the Rydberg state. If most atoms experience a low coupling Rabi frequency we will not observe Rydberg dressed interactions.

To address this we must use smaller MOT beam detunings to reduce the cloud size. The breakdown of the MOT model for small clouds therefore poses a problem. In this section we present dressed MOT data using a smaller MOT beam detuning to reduce the size of the cloud to less than that of the coupling beam. We also develop the resonance curves considered above to better account for the small MOT size.

Dressing a smaller MOT

Figure 5.8 shows a single frequency MOT of δMOT = −140 kHz, PMOT = 30 µW split

in a power ratio of 3:1:1 in the vertical and two horizontal directions dressed for 10 ms. The cloud is similar in size to the coupling beam in the horizontal direction, and is substantially smaller in the vertical direction. We also move to dressing the MOT with the 5s36d3_D

1 Rydberg state, allowing us to Rydberg dress with a higher Rabi frequency

of ΩC/2π = 3.7 MHz, for a range of δC. Overlaid is a modified resonance curve that is

Position/µm Position/µm Position/µm Position/µm Figure 5.8: Single frequency MOTs dressed for 10 ms with a Rabi frequency of 3.7 MHz. Pixels in the images for δC = −6 MHz and δC = −14 MHz have been binned in 2 by 2

cells to improve the signal-to-noise ratio. The dashed black line indicates the coupling beam position, the solid blue lines indicate the modelled gravity-matched scattering force reflecting the angle between coupling beam polarisation and local magnetic field.

The smaller MOT beam detuning clearly results in a cloud that is smaller than the coupling beam. However, the cloud still experiences poor coupling to the Rydberg state. When the coupling beam is blue-detuned, atoms sag to the bottom of the coupling beam. When the coupling beam is red-detuned for small detunings, the dimple created by the coupling beam is larger than the cloud and the MOT is destroyed. We will consider techniques to rectify these issues, but first we consider the resonance curves that describe the smaller dressed MOT.

We had success reproducing the dressed MOT cloud shape using a simple resonance model given by Equations 5.1 and 5.2. We want to repeat this for the smaller MOT, but for smaller clouds, the approximation that all atoms experience a similar magnetic field breaks down - the magnetic field direction is no longer primarily vertical. We previously calculated the AC Stark shift in Equation 5.1 by assuming a Rabi frequency proportional to the square root of intensity, with no consideration of driven transitions, in this section we consider what transitions the coupling light can drive.

netic field, the vertical coupling beam polarisation and the quantisation axis align, and the coupling light drives π transitions. All of the coupling light can couple to the 5snl 3_l

1 mJ = −1 state.

When atoms are in the wings of a Regime II cloud and experience a horizontal mag- netic field, the coupling light can be considered as a combination of left-hand and right- hand circularly (LHC+RHC) polarised light that may drive σ± transitions. From the 5s5p 3P1 mJ = −1 intermediate state only transitions to the 5snl 3l1 mJ = 0 state can

then be driven, so only one component of the LHC + RHC polarised light may drive the transition. Consequently the coupling is weaker as less of the light couples the states. Treating the the coupling light as a combination of parallel and perpendicular polarisation to the quadrupole magnetic field, and treating the coupling to the two Rydberg states as independent, the AC Stark shift can be approximated as:

δAC = Ω2_Cexp  −2x2 ω2 x − 2z2 ω2 z  1 − r2 r2_+4z2/2  4δC , (5.3)

where ΩC  δC and r2 = x2+ y2 is the horizontal position from the quadrupole centre.

The vertical position from the quadrupole centre is z, so the term r2

/r2_+4z2 reflects the

component of the coupling light that is polarised orthogonal to the magnetic quadrupole field. The derivation of this expression is given in Appendix F.

In the two-dimensional case, we set y = 0 so that r = x. The exponential term accounts for the coupling beam intensity profile as in Equation 5.1, the term in the square bracket accounts for the weaker coupling strength when the coupling beam polarisation and the quantisation axis are not aligned.

This treatment considers only the shift of the 5s5p 3_P

1 mJ = −1 level, and assumes the

sublevels are defined only by the local magnetic field. This approach breaks down in some regimes, either where a second quantisation field is present, as will be considered in Section 5.5, or where the coupling light drives π and σ+ _{transitions with similar}

strengths. Appendix F demonstrates a six-level Hamiltonian solution (considering the three magnetic sublevels of the 5s5p 3_P

1 state and the Rydberg state) and considers

multiple planes of resonance, which more accurately replicate the cloud shape, but which requires numerical solution. We will therefore typically use Equation 5.3 and Equation 5.2 to reproduce the cloud shape.

is equal in magnitude to gravity, over the data. As in Chapter 2.5, we use a model MOT beam power ratio of 1:1:1 in the vertical and horizontal directions, compared to experimental parameters of 3:1:1, as it is simpler to calculate the scattering strength for a constant power in all beams.

This modified resonance condition shows good agreement to the experimental data taken with a smaller MOT, shown in Figure 5.8, although the agreement is not as good as in Regime I MOTs (Figure 5.6). This is partly because of the limited validity of this approach for small magnetic fields, as explained above. We expect these limitations to predominantly affect the fit close to the quadrupole field zero. There are several other factors.

Firstly, as the atoms can interact simultaneously with several MOT beams rather than a single MOT beam being dominant this model is not particularly effective even in the undressed regime. This is particularly true when the coupling beam is red-detuned, resulting in a smaller cloud.

Secondly, the resonance condition only shows when the Zeeman shift and AC Stark shift match the MOT beam detuning, but if the resonance ring does not encompass the quadrupole field zero, the scattering force will not be from all six MOT beams, so there will not be a restoring force in every direction. Consequently, the trap may not be sealed. As explained previously, we are using a two-dimensional model to compare to a three- dimensional system. The effects of this are exacerbated by the 30◦ angle between the coupling beam axis and the imaging axis. Whilst using a smaller MOT reduces the effect of the imaging axis angle, the added dependence on r, which varies through the imaged axis, reduces the effectiveness of the two-dimensional model. In Appendix F multiple planes are considered, yielding better agreement between the observed and predicted cloud shape, but with less intuitive insight.

Appropriate coupling beam parameters

As shown above, we can couple the MOT to a Rydberg state and for large clouds we reproduce the MOT outline through a simple AC Stark shifted resonance model. Smaller MOTs pose a greater challenge to reproduce the shape outline, but we see reasonable success modifying this simple model by considering the coupling beam polarisation and good agreement using a six-level Hamiltonian and considering multiple resonance curves at different imaging depths.

An optimum Rydberg dressed MOT regime involves atoms being confined to the coupling beam region; as is apparent from Figure 5.8 this is challenging whether red- or blue- detuned. We consider several techniques for keeping atoms in the coupling beam in Section 5.4; before we do this we must identify appropriate coupling beam parameters. To obtain strong two-body interaction strengths (given by Ω4_/8δ3

C) we want large cou-

pling beam Rabi frequencies and detunings. For achievable Rabi frequencies of 4 MHz coupling to the 5s36d 3D1 state and a detuning of +12 MHz we expect a two-body

dressed interaction strength of 19 kHz from Equation 2.21. This is comparable to the cloud temperature and will be used for the next stage of the experiment.