Fix many spelling mistakes.

Author: jvdoorn

.zed/settings.json

@@ -48,7 +48,7 @@ \section{Decay time}
 We now see that the definition of the Q-factor in terms of the decay time is equivalent to the definition in terms of the FWHM in the power spectrum.
 
 \section{Summary}
-In summary, calculating a Q-factor can be done in four ways: by looking at the decay time of the energy, the decay time of the amplitude, the FWHM in the power spectrum or the FWHM in the Fourier spectrum. The table below provides a overview on how to calculate the Q-factor based on your observations. The resulting Q-factor is the same for all methods, thus it can be used to convert between different observations.
+In summary, calculating a Q-factor can be done in four ways: by looking at the decay time of the energy, the decay time of the amplitude, the FWHM in the power spectrum or the FWHM in the Fourier spectrum. The table below provides an overview on how to calculate the Q-factor based on your observations. The resulting Q-factor is the same for all methods, thus it can be used to convert between different observations.
 
 \begin{table}[h]
     \centering

chapters/appendices/q_factors.tex

@@ -1,11 +1,11 @@
 \chapter{Conclusion and outlook}
 \label{chap:conclusion}
-In this thesis we have shown successful levitation of a \ce{NdFeB} particle with a \qty{12}{\micro\meter} diameter in a planar magnetic Paul trap. The trap was fabricated using a combination of nanofabrication techniques. Levitation was observed at atmospheric pressure all the way down to \qty{1E-4}{\milli\bar}. At atmospheric pressure we succesfully observed the $x$, $y$, $\gamma$ and $\beta$ modes. Due to the low Q-factors at atmospheric pressure it was not possible to tell the difference between the $x$ and $y$ mode or the $\gamma$ and $\beta$ modes. We furthermore observed the expected relations $\omega_{x,y} \propto i_1 \propto 1/\Omega$ and $\omega_{\gamma,\beta} \propto \sqrt{B_0}$.
+In this thesis we have shown successful levitation of a \ce{NdFeB} particle with a \qty{12}{\micro\meter} diameter in a planar magnetic Paul trap. The trap was fabricated using lithography and FIB milling. Levitation was observed at atmospheric pressure all the way down to \qty{1E-4}{\milli\bar}. At atmospheric pressure we successfully observed the $x$, $y$, $\gamma$ and $\beta$ modes. Due to the low Q-factors at atmospheric pressure it was not possible to tell the difference between the $x$ and $y$ mode or the $\gamma$ and $\beta$ modes. We furthermore observed the expected relations $\omega_{x,y} \propto i_1 \propto 1/\Omega$ and $\omega_{\gamma,\beta} \propto \sqrt{B_0}$.
 
-At lower pressures the Q-factors increase untill we reach roughly \qty{1E-2}{\milli\bar} where the Q-factors tend to a constant value. We think this is due to Eddy current damping, though we have been unable to reproduce it in simulations. More detailed measurements of the Q-factors at low pressures are needed to determine the origin of the damping. At low pressures we also observed the \zmode. Again at low pressures we saw the expected relation between $\omega_{x,y,z} \propto 1/\Omega$.
+At lower pressures the Q-factors increase until we reach roughly \qty{1E-2}{\milli\bar} where the Q-factors tend to a constant value. We think this is due to Eddy current damping, though we have been unable to reproduce it in simulations. More detailed measurements of the Q-factors at low pressures are needed to determine the origin of the damping. At low pressures we also observed the \zmode. Again at low pressures we saw the expected relation between $\omega_{x,y,z} \propto 1/\Omega$.
 
 The direct gaps in our knowledge are: the dependence of the Q-factor on pressure for the $z$, $\gamma$ and $\beta$-mode; the dependence of $\omega_z$ on $B_0$; the dependence of $z_0$ on $B_2'$; and the origin of the damping at low pressures. We note that the damping of the \zmode is statistically significantly higher than the damping of the \xmode and \ymode. This might shed light on the origin of the damping, but further investigation is needed. We suggest performing a time dependent simulation in COMSOL and properly model potential sources of damping such as eddy currents inside the particle or tracks.
 
 When measuring using a laser at low pressures we observed a loss of magnetization. A future project will work on interferometric readout, which allows us to use a lower laser intensity. This will enable us to fill most of our knowledge gaps about the parameter dependences. In addition to this we are also looking to increase the remnant magnetization of the particle and to reach a higher $\vec{B_0}$ field by adding a core to the Helmholtz coils. In addition to this we are also looking to trap a smaller (\qty{1}{\micro\meter} diameter) particle as a step towards the quantum regime. We expect that we will be able to trap this particle using the same approach as we used for the \qty{12}{\micro\meter} particle.
 
-Even more long term we are looking to use NV centers. If we replace the cover glass with a diamond we can use the NV centers as a readout. Due to the movement of the particle the emission of the NV centers will split in two bands due to the Zeeman splitting of the $\ket{+1}$ and $\ket{-1}$ states of the NV centers. An additional use of the NV centers is to cool the particle using sideband cooling, similar to the work of \textcite{delord_spin-cooling_2020}. A key step in this case are high eigenfrequencies and good Q-factors in order to reach a sideband resolved regime. The idea is to use the rotational modes (which have an order of magnitude of \qty{1}{\kilo\hertz}) to cool the particle. These rotational modes can be `boosted' by using an elongated particle\cite{huillery_spin-mechanics_2020}.
+Even more long term we are looking to use NV centres. If we replace the cover glass with a diamond we can use the NV centres as a readout. Due to the movement of the particle the emission of the NV centres will split in two bands due to the Zeeman splitting of the $\ket{+1}$ and $\ket{-1}$ states of the NV centres. An additional use of the NV centres is to cool the particle using sideband cooling, similar to the work of \textcite{delord_spin-cooling_2020}. A key step in this case are high eigenfrequencies and good Q-factors in order to reach a sideband resolved regime. The idea is to use the rotational modes (which have an order of magnitude of \qty{1}{\kilo\hertz}) to cool the particle. These rotational modes can be `boosted' by using an elongated particle\cite{huillery_spin-mechanics_2020}.

chapters/conclusion.tex

@@ -3,10 +3,12 @@ \chapter{Discussion}
 In this chapter we will discuss the results presented in \autoref{chap:results}. We will also discuss the limitations of the current setup and possible improvements.
 
 \section*{Laser heating}
-As mentioned in \autoref{chap:results}, the particle lost its magnetization when irradiated with the laser at low pressures ($<\qty{1}{\milli\bar}$). Decreasing the laser intensity also ment a smaller SNR. Due to this tradeoff a decision was made to only use camera measurements at low pressure. A comparision between the two methods will follow. Besides the total loss of magnetization we are also not sure how well the particle retains its magnetization over time. In addition to this we are also not sure what the resulting magnetization is since we do not reach the saturation field. A future project will focus on designing a device that can fully saturate the particle.
+As mentioned in \autoref{chap:results}, the particle lost its magnetization when irradiated with the laser at low pressures ($<\qty{1}{\milli\bar}$). Decreasing the laser intensity also resulted in a smaller SNR. Due to this tradeoff a decision was made to only use camera measurements at low pressure. A comparison between the two methods will follow. Besides the total loss of magnetization we are also not sure how well the particle retains its magnetization over time. In addition to this we are also not sure what the resulting magnetization is since we do not reach the saturation field. A future project will focus on designing a device that can fully saturate the particle.
 
 \section*{Damping at low pressures}
-We observed a limit in the Q-factor for pressures below \qty{1E-2}{\milli\bar}. Attempts to model this have been unsuccessful and do not match the experimental results (see \autoref{tab:dissipation}). Proper modelling of the dissipation should involve a time dependent simulation in COMSOL instead of using a Lorentz term. The main reason for this is that a displacement of the particle brings it closer to its surrounding meaning any effects due to the magnetization of the particle will be stronger. Another source of damping could be eddy currents inside the particle itself. In addition to this we should consider additional sources of damping, such as noise from the electronics or anisotropy (differences in magnetization) inside the particle\cite{millen}. Recent works also discuss eddy currents in more detail, including how they would behave in superconductors\cite{fuwa_stable_2023,gutierrez_latorre_chip_2023}.
+We observed a limit in the Q-factor for pressures below \qty{1E-2}{\milli\bar}. Attempts to model this have been unsuccessful and do not match the experimental results (see \autoref{tab:dissipation}). Proper modelling of the dissipation should involve a time dependent simulation in COMSOL instead of using a Lorentz term. The main reason for this is that a displacement of the particle brings it closer to its surrounding as such any effects due to the magnetization of the particle will be stronger. Another source of damping could be eddy currents inside the particle itself. In addition to this we should consider additional sources of damping, such as noise from the electronics or anisotropy (differences in magnetization) inside the particle\cite{millen}. Recent works also discuss eddy currents in more detail, including how they would behave in superconductors\cite{fuwa_stable_2023,gutierrez_latorre_chip_2023}.
+
+Another explanation for the damping is that the pressure inside the trap is not in equilibrium with the pressure in the chamber. This would result in a higher pressure inside the trap compared to the vacuum chamber. As a result a higher damping rate than expected is observed. One way to test this hypothesis would be to see if the damping rate changes over time when the chamber pressure is kept at a low value.
 
 A statistical test (see \autoref{tab:gamma-t-test}) suggests that the damping of the \zmode is significantly higher than the damping of the \xmode and \ymode. This might shed light on the origin of the damping, but it may also be caused by non-linear effects in the trap. Due to the difficulty of observing the \zmode, we needed quite strong driving forces which can cause non-linear effects. Improvements in the detection of the particle position will allow us to measure the \zmode at lower driving forces.
 
@@ -39,10 +41,10 @@ \section*{Laser v.s. camera readout}
     \label{tab:laser-vs-camera-readout}
 \end{table}
 
-In the future laser readout is key to properly study the $\gamma$ and $\beta$ modes at low pressures. The idea is to move to an interferometric setup to measure the position of the particle. This has a higher SNR if done correctly. In addition to this we are also investigating the use of NV centers in diamond to measure the position of the particle. An additional advantage of NV centers is that they may also allow for sideband cooling.
+In the future laser readout is key to properly study the $\gamma$ and $\beta$ modes at low pressures. The idea is to move to an interferometric setup to measure the position of the particle. This has a higher SNR if done correctly. In addition to this we are also investigating the use of NV centres in diamond to measure the position of the particle. An additional advantage of NV centres is that they may also allow for sideband cooling.
 
 \section*{Lorentzian fits}
-To obtain the data in \autoref{fig:xyz-mode-dependence-on-trapping-frequency-1mbar} we fitted the peaks in our data with a Lorentzian. This was done to obtain the Q-factor of the peaks. It is however better to fit all peaks at once instead of individually. The reason we did not do so is that the spectra were not very clean. An example of this is the fact that there was crosstalk between the horizontal and vertical spectra. A more careful analysis could properly rotate the spectra to avoid this crosstalk. This might make it easier to do a single fit per spectra instead of 2 seperate fits.
+To obtain the data in \autoref{fig:xyz-mode-dependence-on-trapping-frequency-1mbar} we fitted the peaks in our data with a Lorentzian. This was done to obtain the Q-factor of the peaks. It is however better to fit all peaks at once instead of individually. The reason we did not do so is that the spectra were not very clean. An example of this is the fact that there was crosstalk between the horizontal and vertical spectra. A more careful analysis could properly rotate the spectra to avoid this crosstalk. This might make it easier to do a single fit per spectra instead of 2 separate fits.
 
 \section*{Trapping at low pressures}
 Qualitatively we found it to be very hard to trap the particle at low pressures ($<\qty{1}{\milli\bar}$) when starting from an untrapped state. It is likely that more damping is needed to dissipate the energy of the particle or active feedback to trap the particle. Potentially we can artificially introduce damping by adding white-noise to a nearby coil. White-noise increases the damping rate\cite{millen}. This is something that should be investigated further.

chapters/discussion.tex

@@ -1,11 +1,11 @@
 \chapter{Introduction}
 \label{chap:introduction}
-The twentieth century brought us two major cornerstones in physics: quantum mechanics and general relativity. Whilst quantum mechanics successfully describes physics at the smallest scales and general relativity describes physics at the largest scales, a unified theory of quantum gravity is still missing. To test wether gravity is a quantum entity (the existence of gravitons) an experiment was proposed by \textcite{bose_spin_2017}.
+The twentieth century brought us two major cornerstones in physics: quantum mechanics and general relativity. Whilst quantum mechanics successfully describes physics at the smallest scales and general relativity describes physics at the largest scales, a unified theory of quantum gravity is still missing. To test whether gravity is a quantum entity (the existence of gravitons) an experiment was proposed by \textcite{bose_spin_2017}.
 
-The experiment proposed by \citeauthor{bose_spin_2017} suggests entangling two particles through gravity. If this is possible, it would be a direct proof that gravity is a quantum entity. To realise this experiment, we need two small ($\approx \qty{1}{\micro\meter}$) but massive particles isolated from their environment and cooled to their ground state. An electronic spin is attached to the particles and a $\pi/2$-pulse is applied to the spins. This creates a superposition of spins. The particles are then dropped through a Stern-Gerlach interferometer which turns the superposition of spins into a superposition of positions. The two particles are then allowed to interact through gravity. This interaction entangles the particles. The outcome of the experiment is measured by the interference of the particles.
+The experiment proposed by \citeauthor{bose_spin_2017} suggests entangling two particles through gravity. If this is possible, it would be a direct proof that gravity is a quantum entity. To realize this experiment, we need two small ($\approx \qty{1}{\micro\meter}$) but massive particles isolated from their environment and cooled to their ground state. An electronic spin is attached to the particles and a $\pi/2$-pulse is applied to the spins. This creates a superposition of spins. The particles are then dropped through a Stern-Gerlach interferometer which turns the superposition of spins into a superposition of positions. The two particles are then allowed to interact through gravity. This interaction entangles the particles. The outcome of the experiment is measured by the interference of the particles.
 
-In this thesis we focus on the levitation of micrometer sized particles using a planar magnetic Paul trap. A magnetic Paul trap uses ac magnetic fields to obtain stable levitation. This builds on previous work within our group where a \qty{250}{\micro\meter} cubic \ce{NdFeB} magnet was levitated in a planar magnetic Paul trap realised on a PCB\cite{eli, mart}. In order to work towards the quantum regime, the system needs to be miniaturized.
+In this thesis we focus on the levitation of micrometer sized particles using a planar magnetic Paul trap. A magnetic Paul trap uses an ac magnetic field to obtain stable levitation. This builds on previous work within our group where a \qty{250}{\micro\meter} cubic \ce{NdFeB} magnet was levitated in a planar magnetic Paul trap realized on a PCB\cite{eli, mart}. In order to work towards the quantum regime, the system needs to be miniaturized.
 
 Two other well known levitation techniques are optical (eigenfrequencies between \qtyrange{10}{300}{\kilo\hertz}) and electrical traps (eigenfrequencies between \qtyrange{1}{10}{\kilo\hertz})\cite{levitodynamics}. Compared to these techniques, magnetic traps have the disadvantage of having a lower eigenfrequency (\qtyrange{1}{10}{\kilo\hertz}), often require cryogenic temperatures and no on-chip integration exists\cite{levitodynamics}. The advantage however is that magnetic traps have the potential to levitate much larger and heavier particles. Levitating heavier particles creates a strong gravitational interaction which is beneficial for the experiment.
 
-The cryogenic neccecity originates from the fact that many magnetic traps use the Meissner effect to levitate a particle. An alternative to this however is to use non-static magnetic fields which then also satisfies Earnshaw's theorem. Previous work has shown this to be possible\cite{perdriat,eli,mart}. The goal of this project is to create an on-chip variant of the magnetic Paul trap. Besides the advantage of an on-chip design, this will also increase the eigenfrequency of the trap\cite{perdriat}.
+The cryogenic necessity originates from the fact that many magnetic traps use the Meissner effect to levitate a particle. An alternative to this however is to use non-static magnetic fields which then also satisfies Earnshaw's theorem. Previous work has shown this to be possible\cite{perdriat,eli,mart}. The goal of this project is to create an on-chip variant of the magnetic Paul trap. Besides the advantage of an on-chip design, this will also increase the eigenfrequency of the trap\cite{perdriat}.