Axial Parametric Feedback

Up to now, all feedback routines are only applied on the radial direction of the atomic motion away from the cavity axis. Here we will extend our scope and show a first implementation of axial feedback. Since the axial motion occurs on much faster timescales, i.e. two order of magnitude, only the parametric feedback routine offers the possibility to tackle this ambitious challenge. The low count-rates caused by the low transmission of the probe beam through our atom-cavity system in combination with the necessary fast response time require that the feedback decision is based on the timing of a single photon click. An overview of the settings employed to demonstrate the axial feedback is given in Table 5.3.

General parameters

Probe-cavity frequency detuning Δ ⁄2𝜋 0 MHz Atom-cavity frequency detuning Δ ⁄2𝜋 5 MHz

Probe power 𝑃 6 𝑀𝐻𝑧/ 𝜂 ⋅ 𝜂 ⋅ ℏω

Dipole trap power (base-value) 𝑃 950 𝑛𝑊 Dipole trap modulation depth Δ𝑃 0.36 ⋅ 𝑃 Feedback parameters (parametric feedback)

Frequency of parametric feedback oscillator f 500 kHz Integration time of feedback T 2/𝑓 4 µs

Phase advance scanned ϕ 0 360°

Internal FPGA parameters

DDS increment for local oscillator feedback_phase_ddsinc = 21474836

Cycles between feedback iteration feedback_phase_fiforate = 0

Integration steps of feedback feedback_phase_fifoshift = 400

Table 5.3: Overview of the settings used for the axial parametric feedback scan. The probe power is significantly increased compared to the previous radial scans.

Control of Atomic Motion 5.5 Feedback on a Single Atom

5.5.4.1. Implementation 

The high frequency of the axial oscillation in the few hundred kHz regime also requires a modulation of the same frequency which will be output by the parametric feedback routine.

This frequency value is way beyond the bandwidth of the PID controller, which lies well below 100kHz. Hence it is not possible to use the feedback output as set-point of the PID loop as it has been done for the radial feedback routine. In Figure 5.24 the required changes on the dipole trap AOM driver, when going from radial to axial feedback, are depicted. If high modulation frequencies are used as for axial feedback, the set-point for the dipole-trap stabilization PID is a fixed value output by the main program controlling the experimental run. The output of the PID is then multiplied with the oscillator generating the fixed AOM frequency via a mixer (Mini Circuits ZLW-3+). Now a second mixer of the same kind is used to modulate this signal with the frequency of the feedback routine output by an analog channel of the FPGA module.

In total this results in a fast modulation of the radio frequency driving the AOM and with that also of the dipole trap beam. The limited bandwidth of the PID, which is below the high modulation frequency, ensures that the PID controller itself does not get “distracted” by the modulation and consequently tries to compensate for it. Instead, it only “sees” a time-averaged value of the signal, which makes the modulation invisible.

5.5.4.2. Phase Dependency 

The fast timescale of the axial motion in combination with its low quality factor, as it can be seen in the number of peaks showing up in correlation measurements, necessitates to increase the information rate emitted from the cavity. In order to do so, the probe power impinging on the cavity is increased by a factor of 10 compared to the measurements of the radial oscillation, resulting in an empty cavity photon number of 1.1. It is important to keep in mind that this value is determined for the case where the cavity is on resonance with no atom inside. The presence of an atom significantly reduces the actual number of intracavity photons once the atom is

Figure 5.24: Setup of the PID and AOM mixer for slow and fast modulations as they are used in the radial and axial feedback, respectively. A PID is used to stabilize the intensity of the dipole trap laser via a photodiode (PD). In case of the radial feedback, the signal of the FPGA is directly fed into the PID controller. The output of the PID subsequently controls the amplitude of the modulation by passing through a mixer. In case of axial feedback, a fixed value controls the set-point of the PID. The output of the PID controls the amplitude of the modulation while passing through a first mixer. A second mixer is used to create an amplitude modulated signal controlled by the output of the FPGA.

5.5 Feedback on a Single Atom Control of Atomic Motion

strongly coupled; this way we are close to saturating the atom but still not in the saturation regime. However, the high probe powers cause significant heating and hence reduce the average storage time when no feedback is applied to a value as low as 7.1 ms. The exposure time is set to two oscillation periods of the expected modulation, i.e. 4 µs. This increase reduces the bandwidth of the feedback algorithm on one hand, but, on the other hand permits to increase the signal-to-noise ration by a factor of two. The trap depth without modulation is set to a value of 840 µK. In order to increase the effect of the feedback on the atom, the modulation depth of the dipole trap is set to a value of 36% of its mean value. The modulation is applied continuously with a fixed amplitude. The modulation is neither turned off nor reduced in times where no or only a weak oscillation is detected as it would be possible by the feedback routine. This permits to exclude additional effects stemming from pure changes in the amplitude of the modulation and hence permits to attribute the emerging pattern as being caused by a change in the phase.

For the scan a feedback frequency of 500 kHz was chosen, as this is on the high-frequency side of the peak in the correlation plot (cf. Figure 5.5) and hence closer to the harmonic oscillation frequency. An overview of the parameters is given in Table 5.3.

Applying feedback at the calculated harmonic oscillation frequency of the atom around 1 MHz, with which the atom oscillates for small excursions, would in principle enhance the efficiency of the feedback algorithm. However, only a very little modulation is visible at this harmonic oscillation frequency in the transmitted signal due to the little change in the coupling constant for small excursions, rendering this frequency choice impractical. As for the radial oscillation, the phase advance is scanned and the average storage time is measured. For each data point more than 100 individual atoms have been captured. The result is plotted in Figure 5.25. The

-180 -135 -90 -45 0 45 90 135 180

5 6 7 8 9 10

heating

average storage time (ms)

phase advance



without feedback

cooling

7.1

Figure 5.25: Influence of the phase advance on the average storage time for axial feedback. The average storage times for different phase advance settings is plotted. As a reference the average storage time with the same setting, however, with feedback disabled, is plotted as dotted dark cyan line. A dark green line shows a sinusoidal fit and serves as guide to the eye.

Control of Atomic Motion 5.6 Conclusion

average storage time varies around the value measured without any feedback and with no modulation applied. Around a phase advance of 45 to 90 degrees the storage time increases which indicates cooling of the atom. At opposite phase values the inverse behavior occurs, which clearly indicates heating of the atom. A sine curve is fitted and plotted as a guide to the eye. The values for the phase advance in this plot can only be seen as relative values since small delays in the signal path already lead to an additional phase offset. The overall increase in storage time is not as pronounced as for the radial case. This is caused by the very weak detection efficiency of the axial oscillation, due to the very limited amount of photons. Here the decision is solely based on the absence or the presence of a single photon at a certain point in time. Yet, this measurement shows that parametric feedback in principle permits an extension even to the axial oscillation.

5.6. Conclusion

In this chapter the motion of an atom trapped inside a high-finesse optical oscillator is studied in detail. Besides correlation methods, parametric heating is employed to determine the harmonic oscillation frequency. Unlike the first method the latter one yields values which agree well with the computed ones. In addition to the detection of the atomic motion, fast electronic feedback has been introduced as a method to effectively cool this motion. The very low number of photons emitted by our atom-cavity system poses a limit to the information, which can be gained about the atomic motion. Different strategies have been employed for the radial oscillation. While the so-called bang-bang method does not require any pre-knowledge about the system and relies on the evaluation of a change in the emitted photon flux, the radial parametric feedback relies on the input of the frequency at which the atom moves and outputs a modulation. This pre-knowledge about the system permits to have integration times which are on the same order or larger than one oscillation period. This significantly improves the quality of prediction and thus leads to an increase of the storage time by almost a factor three.

The possibility to have longer integration times with respect to the atomic oscillation additionally enables an extension of the parametric feedback strategy to the axial direction. This feedback, however, requires higher probe powers substantially increasing the heating rate of the atom in order to have a visible effect. While this renders it impractical for day-to-day use in our experiment, it is a powerful demonstration of the possibilities of fast electronic feedback.

Compared to experiments relying on transverse beams to trap and cool atoms, the feedback cooling strategy demonstrated in this thesis substantially benefits from its easy and flexible way to be implemented. As it only depends on information derived from the transmitted photons and does not require any transverse optical access, it is thus applicable also for systems incorporating short cavities or systems with limited optical access, consisting e.g. of optical fibers [144] or microtoroids [145]. Further advances in the processing algorithm of the feedback strategy to incorporate real-time estimation of the quantum state can even permit to extend this method into the quantum domain, rendering it possible to stabilize the quantum state of a trapped particle.

Atomic Antiresonance and Parametric Feedback in a Strongly Coupled Atom-Cavity Quantum System