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A critical part of observing Rydberg dressing is obtaining a high Rabi frequency on the Rydberg transition. We measure the Rabi frequency of the coupling beam using Autler- Townes splitting, this is sometimes challenging for coupling beams of comparable size to the cloud.

We have measured Autler-Townes splitting in the broadband MOT using two coupling beam sizes and in the single frequency MOT using a single beam size.

Autler-Townes splitting in a broadband MOT

We observe Autler-Townes splitting using two-photon excitation with a resonant coupling beam and varying the probe beam frequency, shown in Figure 4.6. As we increase the coupling beam intensity we see the resonance feature split into two distinct features, the separation of which is the Rabi frequency that we couple to the Rydberg state with.

(a) (b) (c) x/µm z/ µ m

Figure 4.6: Autler-Townes splitting in a broadband MOT with a large (a) and a small (c) coupling beam with respect to the cloud. Focussing the coupling beam gives higher Rabi frequencies, illustrated through the increased splitting, but also blurs the features out due to the range of Rabi frequencies experienced by the cloud. (b) shows an averaged fluorescence image of the cloud, the red ring indicates the size of the large coupling beam and the blue ring indicates the size of the small coupling beam - both are 1/e2 radii.

In the broadband MOT we can see splitting of up to 400 kHz at full coupling beam power at the 5s37s 3_S

1 state. This is not sufficient for Rydberg dressing experiments,

as to obtain a dressed interaction of 20 kHz with this Rabi frequency we would need a maximum coupling beam detuning of 580 kHz, where we would not be in the ΩC  δC

regime.

size, expecting a rise in peak intensity of 52 _{and a rise in peak Rabi frequency of 5. The}

broadband MOT typically has a 1/e2 _{horizontal radius of 420 µm and vertical radius}

of 150 µm. By comparison, the coupling beam before focussing has a 1/e2 _horizontal

and vertical radius of 790 µm and 930 µm. The focussing lens is outside the vacuum chamber, so measuring the exact size of the coupling beam at the position of the atoms is challenging.

Initially, we attempted to see Autler-Townes splitting in the broadband MOT with the focussed coupling laser beam. This had some success, shown in Figure 4.6(c), as we observe a power dependent splitting, but the features are not as resolved as we would like. The peaks are more separated than those in Figure 4.6(a) but are also much more broad, with less dip in the centre than we would expect extrapolating from the previous data. This is due to averaging over a range of Rabi frequencies due to using an atom cloud that is larger than the coupling beam; to avoid this we switch to using a smaller, single frequency MOT.

Autler-Townes splitting in a single frequency MOT

To avoid this averaging effect we use a single frequency MOT that is much smaller (65 µm by 140 µm 1/e2 radii) than the expected coupling beam size, eliminating the averaging effect and allowing us to see well resolved Autler-Townes peaks. Figure 4.7 shows Autler- Townes spectra taken coupling to the 5s36d3_D

1mJ = −1 state, where we expect a larger

Rabi frequency due to the larger dipole matrix element [74].

The peaks are much more separated due to the higher intensity of the focussed coupling beam and stronger coupling of the 5s36d 3_D

1 state than the 5s37s 3S1 state, allowing

us to reach Rabi frequencies of 4 MHz. This is sufficient to off-resonantly Rydberg dress the MOT with - a 4 MHz Rabi frequency and a detuning of 12 MHz gives rise to a two- body dressed interaction strength of 18 kHz (Equation 2.21), comparable to the cloud temperature. The averaging effect observed in the broadband MOT is largely suppressed, allowing us to be confident that all of the atoms experience strong coupling to the Rydberg state.

We also observe a central feature that occurs in a similar position to the unsplit feature. Close inspection suggests that this is due to weakly confined atoms that are not trapped at the quadrupole field centre. We are using a high MOT beam power of 500 µW and a MOT beam detuning of -120 kHz to reach such a small tightly confined MOT that

Position/µm

Figure 4.7: Autler-Townes splitting in a retrapped single frequency MOT. (a) Ion signals for high (blue, 50 mW) and low (orange, 200 µW, rescaled for comparison) coupling beam powers. At high coupling power Autler-Townes splitting is observed, as well as a third central peak caused by trap leakage illustrated in the optical depth images (b) and (c). The same cloud is shown with a different colorbar, allowing atoms trapped far from the quadrupole field centre to be observed.

can be retrapped, under these conditions atoms are observed leaking from the cloud. We believe that these atoms are being retrapped by leaked broadband red MOT light. Far from the focus of the coupling beam, they can still be excited to Rydberg states but will not be Autler-Townes split. The rate of atoms leaking from the trap is slightly density dependent, making this appear to be an effect of Rydberg atom interactions, but it is observed well below the blockade density for this state.

Coupling beam profile from Autler-Townes splitting

We can move the MOT position by changing the current in the external shim coils, mov- ing the cloud through the coupling beam and taking Autler-Townes spectra at different positions. This allows us to measure the beam size and ensure accurate alignment of the atoms in the coupling beam. Figure 4.8 shows the ion signal as a function of both cloud vertical position and probe beam detuning. We can clearly see splitting, as well as a position dependent shift, which is caused by the changing magnetic field due to the shim coils. The splitting is poorly resolved as the cloud was a similar size to the coupling beam width.

Position shift/µm P osition shift / µ m

Figure 4.8: Moving the MOT and scanning the probe beam detuning for a high power coupling beam, we observe position dependent Autler-Townes splitting on the left (colour indicates the ion number over the peak ion number). This is heavily blurred due to the comparable cloud size and coupling beam size, but by plotting the separation of the two peaks (shown on the right) we can fit the width of the coupling beam. Blue (orange) rings indicate Autler-Townes splitting as a function of vertical (horizontal) position.

as 160 µm and 120 µm in the vertical and horizontal direction. These numbers are comparable to those estimated from the increase in Rabi frequency due to focussing the coupling beam of 160 µm and 130 µm.

In future, an optical dipole trap will be implemented; this will allow the quantisation field to be on for long enough for eddy currents to decay, allowing consecutive excitations. Dipole traps can be switched off and on rapidly without the need for a 400 µs MOT coil switching time, allowing for easy recapture. It will also allow for a smaller trap size, reducing the averaging effect. The techniques described are sufficient for estimating Rabi frequency for our purposes; a particular strength of these techniques is that any observation of blurring of Autler-Townes splitting due to cloud size provides information on the distribution of Rabi frequencies that the atoms experience.

Rabi frequencies

Rabi frequency scales with the square root of beam intensity. From the data presented we have calculated the coupling strength as a function of coupling beam intensity to be Ω37s/ √ I = 190±30 kHz/pW/cm2 _{for the 5s5p} 3_P 1 mJ = −1 ↔ 5s37s 3S1 mJ = −1 transition and Ω36d/ √ I = 290±30 kHz/pW/cm2 _{for the 5s5p} 3_P 1 mJ = −1 ↔

coupling beam width measurements and, for the s-state, the position averaging of the relative MOT and coupling beam size. These values are sufficient for Rydberg dressing experiments as shown in Chapter 2.6.

A thorough comparison between these measurements and predicted values is not the fo- cus of this thesis, but we do compare our measured Rabi frequency to the predicted Rabi frequency achievable coupling to the 5s37s 3_S

1 state. Using the oscillator strengths set

out in [74] we expect a total coupling strength of 920 kHz/pW/cm2 _{on the 5s5p}3_P

1 ↔

5s37s 3S1 transition, giving a predicted coupling strength of 150 kHz/pW/cm2 on the

5s5p 3_P

1 mJ = −1 ↔ 5s37s 3S1 mJ = −1 transition. Given the paucity and lim-

ited accuracy of the spectroscopic data that [74] is based on, and the accuracy of our measurements, we consider a 21% difference between theory and experiment to be very reasonable.