Simulated Annealing

In Figure 5.3 we overlay a modified resonance curve on the data that accounts for the AC Stark shift. This section will use the dressed state picture and the MOT resonance curves from Chapter 2 to calculate these resonance curves. To do this we will consider when the MOT beams are resonant with the dressed state |e0i rather than the bare state |ei.

Position/µm P osition / µ m

Figure 5.4: A composite image from the data in Figure 5.3, made using a range of MOT beam detunings to visualise the effect of the coupling beam.

from the bare state |ei, and the position dependence of this shift.

In Chapter 2 we demonstrated that the AC Stark shift of the dressed state |e0i compared to the bare state |ei is given by δAC = −δ₂C ±

√

δ2 C+Ω2C

2 , where δC is the coupling beam

detuning and ΩC is the coupling beam Rabi frequency. The coupling beam intensity is

Gaussian, I(x, z) = I0exp

 −2(x−x0)2 ω2 x − 2(z−z0)2 ω2 z 

, where I0is the peak intensity, ωx,zis the

1/e2 radius in the horizontal and vertical direction, and x0 and z0 describes the coupling

beam position. The Rabi frequency is proportional to the square root of intensity, so Ω2 C(x, z) = Ω2C(x0, z0) exp  −2(x−x0)2 ω2 x − 2(z−z0)2 ω2 z  .

From this we obtain an expression for the AC Stark shift of the dressed state |e0i that reflects the intensity profile of the coupling beam, which can be simplified when ΩC  δC:

δAC = − δC 2 ± r δ2 C + Ω2Cexp  −2(x−x0)2 ω2 x − 2(z−z0)2 ω2 z  2 ≈ Ω2 C 4δC exp  −2(x − x0) 2 ω2 x − 2(z − z0) 2 ω2 z  . (5.1) This is simply introducing a position dependent term to the AC Stark shift to reflect the coupling beam intensity profile. The AC Stark shift is shown in Figure 5.5(b) for the parameters used in Figure 5.3.

This treatment assumes that the Rabi frequency position dependence is purely dependent on the coupling beam intensity. In practice, the angle between the magnetic field and the coupling beam polarisation will influence what transitions are coupled and the effective Rabi frequency, but at this point we consider only Regime I MOTs, where the magnetic field is primarily vertical. This matches the coupling beam polarisation axis and the configuration under which the Rabi frequency was measured. In Appendix F we consider

the effect of magnetic field direction variation.

Having considered the AC Stark shift that the dressed state experiences we now consider when the MOT beams are resonant with the dressed state. The atoms will be trapped where the MOT beams are resonant.

In Chapter 2.5 we did this by equating the MOT beam detuning from the bare state δMOT to the Zeeman shift mJgJµBdB_dz · z, shown in Figure 2.8. Now we are interested in

the dressed state |e0i so we replace the MOT beam detuning from the bare state δMOT

with the MOT beam detuning from the dressed state δMOT− δAC to obtain:

δMOT− δAC = mJgJµB

dB

dz · z , (5.2)

The MOT beam detuning δMOT is constant, the Zeeman shift and AC Stark shift are

shown in Figure 5.5(a-b). We can numerically solve this to obtain a curve on the x − z plane where the MOT beams are resonant with the dressed state, shown in Figure 5.5(c). Included in this calculation is an offset on the 689 nm frequency due to an offset on the laser lock point, which has been estimated based from the MOT shape. Parameters used in this calculation were chosed to match experimental parameters and are: ΩC = 2.4 MHz,

δC = 6 MHz, ωx = 120 µm, ωz = 160 µm, x0 = 60 µm, z0 = 250 µm, dB/dz = 7.2 G/cm,

and a 689 nm laser lockpoint offset of -40 kHz.

In Figure 5.3(d-h) we see good agreement between the resonance curve and the cloud shape, subject to the vertical offset due to the cloud forming above the resonance curve. The largest limitation is the two-dimensional nature of this model. The imaging axis is at 30◦ to the coupling beam propagation axis, so the position offset will not be constant across the imaged plane. As a result, the position of the dimple will be dependent on the depth of the plane being imaged, resulting in a blurring of the dimple across the image. The larger the MOT, the stronger this effect will be; this is particularly apparent in Figure 5.5(g).

Whilst the agreement between the model and the data is good, there is a position offset between the resonance curve and the cloud. We expect the cloud to be positioned slightly above the resonance curve, where the scattering force is equal to gravity, rather than where scattering is strongest [115]. This is observed in Figure 2.8 in the undressed MOT; by calculating when the scattering force is equal to gravity we identify the true cloud position. We extend this treatment to the dressed MOT in Figure 5.6 for a range of coupling beam positions and detunings.

P

osition

/

µ

m

Position/µm Position/µm Position/µm

Figure 5.5: Modelling the AC Stark shift due to the coupling beam. (a) and (b) show the Zeeman shift and the AC Stark shift of the 5s5p 3_P

1 mJ = −1 state through a

plane perpendicular to the coupling beam through the quadrupole centre. (c) shows the combined effect; overlaid are contour plots for the detunings used in in the data. These contours are compared to data in Figure 5.3.

The data in Figure 5.6 show four coupling beam detunings and three MOT positions, labelled high, medium and low. Moving the cloud with the shim coils is equivalent to moving the coupling beam and the images are corrected for the cloud movement to illustrate the effect of coupling beam position.

Having accounted for the relative strength of scattering and gravity we see excellent agree- ment between data and the model. We can see the dimple becoming more pronounced both as the coupling laser frequency nears resonance, and as the coupling laser alignment on the cloud is improved.

Two ellipses at which the scattering force equals gravity are observed, an inner curve and an outer curve. Atoms outside these curves are lost, atoms inside the outer curve will be pushed to the inner curve, where a stable equilibrium between gravity and scattering is reached. This is particularly apparent for the −6 MHz detuned case - at high alignment, the lower curve encompasses the entire position of the undressed cloud, consequently all of the atoms remain trapped. At the lowest alignment, the outer resonance curve is shifted above the position of part of the undressed cloud. Atoms that are below this curve when dressing is initiated are not trapped and are observed falling under gravity. This model therefore provides insight both into the cloud shape and the recapture fraction.

Position/µm Position/µm Position/µm

Figure 5.6: Rydberg dressed MOTs for four coupling beam detunings and three coupling beam positions. The dashed black line indicates the modelled position of the coupling beam and the solid blue lines indicate positions where the scattering force magnitude is equal to gravity, which occurs above and below the position of strongest scattering. Taken with δMOT = −400 kHz, PMOT= 90 µW and 5 ms of dressing with ΩC = 2.4 MHz.

The modelled coupling beam includes a position offset from the quadrupole centre of x0 = 100 µm and z0 = 25 µm (high), 100 µm (medium) and 175 µm (low).

This technique is only suited to Regime I and III MOTs as it neglects the direction of scattering. It also does not account for different MOT beam power ratios in the vertical and horizontal MOT beams, and considers only a single x − z plane. We use an imaging axis that is at 30◦angle from the coupling beam axis. Given these limitations, the success of this model is very satisfactory.