Setter, Ashley J.
(2020)
Cooling of a mesoscopic single-particle harmonic oscillator using real-time digital control systems.
*University of Southampton, Doctoral Thesis*, 171pp.

## Abstract

Optically levitated systems provide an ideal environment for isolating objects extremely well from the environment such that their behaviour can be more easily studied without external influences. Moreover, using optical detection, very high levels of sensitivity to that behaviour can be achieved, allowing precise detection of their motion and measurement of forces upon such a system.

In the experiment we describe in this thesis, one can construct an optical gradient force trap by simply directing a collimated light beam onto a parabolic mirror with a sufficiently high numerical aperture. The resulting focusing not only generates the optical trapping potential but also allows detection as a particle trapped at the focal spot of the mirror has the light it scatters onto the mirror collimated and sent back in the direction of the incident trapping light beam. The trap described here allows levitation of dielectric nanoparticles with radii from 10nm to 300nm at pressures from 10^{3} —10^{-8} mbar.

In this thesis I describe the experimental setup and the theory describing a particle levitated in this setup, detailing how, from the detected light intensity on a photodetector, we can extract the properties describing the physical system, including the frequency of motion, the damping upon the motion, the temperature associated with the motion, the radius of the trapped nanoparticle and the conversion factor which allows us to convert from the photo-current measured by the detector to the physical displacement of the particle.

Cooling of the translational motion to the ground state is a much targeted goal in the field of levitated optomechanics as it has applications in producing quantum states, performing matter-wave interferometry, testing collapse models and as a very precise force sensor. In this thesis I demonstrate and investigate a number of different cooling methods applied to the translational motion.

First, I describe how one can utilise an FIR (Finite Impulse Response) band-pass filter to isolate the signal due to the motion of a single translational degree of freedom and utilise this to perform active feedback cooling upon this degree of freedom. I demonstrate this by implementing an FIR filter on a FPGA (Field Programmable Gate Array) and utilising it in real-time to cool the motion to 2.5K ± 0.4 K.

Secondly I utilise a tracking algorithm called a Kalman Filter, I describe how this algorithm functions to track the state of objects from noisy partial measurements of the state and how this can be applied to track levitated nanoparticles. I then detail how this was implemented in an FPGA so that the Kalman Filter could be used in real-time to estimate the position of a levitated nanoparticle. We demonstrate cooling of the translation centre-of-mass motion along the optical axis by 3 orders of magnitude to a temperature of 162mK ± 15 mK.

Next I describe a classical model of the motion of a levitated nanoparticle and describe how this model can be simulated numerically to produce simulated time traces of the motion. I then demonstrate how this simulation can be used to reproduce several theoretical relations as well as reproduce experimental results and be utilised to simulate and test the performance of cooling algorithms without requiring experimental implementation. Next I use this simulation to model the effect of a Dung non-linearity in the trapping potential. I match the behaviour of the simulation when exposed to as a pulsed reduction of the laser power with the behaviour of the experimental system when exposed to the same stimulus and in doing so extract the Dung non-linearity in the optical trapping potential generated by the parabolic mirror, which takes a value of ξ = -0.100 m^{-2}. Finally I describe a quantum model of the motion of a levitated nanosphere, which takes into account the various decoherence sources in the experimental system, namely the interaction with the background gas, the laser photons and decoherence due to the measurement. Using this I investigate several different forms of active feedback cooling, including the "optimal" cooling as determined from the tools of control theory, which takes the form of a square wave, and the more traditional sinusoidal feedback signal. In addition I compare both linear cooling by applying a force directly to the nanoparticle and quadratic cooling by modulating the power of the trapping laser. I determine that cooling to mean phonon numbers of less than 10 with the current experimental setup is achievable utilising any of the feedback schemes described here when they are tuned appropriately.

**Cooling of a Mesoscopic Single-Particle Harmonic Oscillator using Real-time Digital Control Systems - Version of Record**

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