Some more minor changes.

Author: jvdoorn

chapters/conclusion.tex

@@ -4,7 +4,7 @@ \chapter{Conclusion and outlook}
 
 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 $\omega_{x,y,z} \propto 1/\Omega$.
 
-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 to determine the relation between $\gamma$ and $i_1$.
+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.
 

chapters/discussion.tex

@@ -45,7 +45,7 @@ \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.
 
 \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. This is something that should be investigated further.
+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.
 
 \section*{Loss of levitation}
 A major source of confusion (and frustration) was the sudden loss of levitation. For long periods of time our particle was easily trapped without major issues. However, just before the Christmas holiday it became very difficult to trap. We had previously seen that heating can cause the particle to lose its magnetization, but didn't appear to be the case as it still responded to the magnetic field. An attempt was mode to remagnetize the particle which did not matter much. Using an optical microscope we noted a black `smut' in and around the trap. This was further confirmed using SEM measurements. This `smut' was not found on earlier images before we used the trap. See \autoref{fig:smut-optical-microscope}.

chapters/results.tex

@@ -44,7 +44,7 @@ \section{Measurements at low pressure}
 \label{sec:measurements-at-low-pressure}
 At \qty{1}{\milli\bar} the readout was performed using the Thorcam and analysed using our custom object tracking algorithm. The reason to no longer use laser readout is due to the heating of the particle. By equating the power of the laser to Stefan-Boltzmann law we find that, in equillibrium assuming no heat dissipation through air, the temperature of the particle could reach \qty{2500}{\kelvin}. Even if only a fraction of the laser power would reach the particle this would be enough to reach its Curie temperature of \qty{320}{\celsius}\cite{magnequench}. Lower laser powers leads to an insufficient signal-to-noise ratio. We will return to this in the discussion. For a more thorough analysis of heating refer to \cite{millen}.
 
-\begin{SCfigure}
+\begin{SCfigure}[][h]
     \centering
     \includegraphics{figures/data/xyz_mode_dependence_on_driving_frequency_spectrum.pdf}
     \caption{The dependence of $\omega_{x,y,z}$ on $\Omega$ at \qty{1}{\milli\bar}. Shown are two directions (horizontal and vertical), they live in the $xy$-plane but do not directly match to the \xmode or \ymode, a small rotation angle needs to be taken into account.}
@@ -53,7 +53,7 @@ \section{Measurements at low pressure}
 
 \autoref{fig:xyz-mode-dependence-1mbar} shows the dependence of $\omega_{x,y,z}$ at \qty{1}{\milli\bar}. We note that compared to the measurements at atmospheric pressures that the width of the peak is substantially smaller. Whilst this measurement shows a clear dependence of $\omega_{x,y,z}$ on $\Omega$, it is very difficult to see the \zmode. These measurements were obtained by continiously sweeping the driving frequency. Instead, we can do a `lock-in like' measurement as described in \autoref{chap:method}. An example of such a measurement is shown in \autoref{fig:xyz-mode-spectrum-1mbar}.
 
-\begin{SCfigure}
+\begin{SCfigure}[][h]
     \centering
     \includegraphics{figures/data/xyz_mode_spectrum.pdf}
     \caption{A PSD with a visible \xmode, \ymode and \zmode. The diagonal line is caused by the changing driving frequency. By looking at the magnitude along the diagonal line the eigenfrequencies and Q-factors can be }
@@ -95,13 +95,15 @@ \section{Q-factor dependence on pressure}
 \label{sec:q-factor-dependence-on-pressure}
 Using an external coil we are able to excite the \xmode  and \ymode seperately. Ringdown measurements are performed to determine the pressure dependence on the Q-factor. The Q-factor in this case is defined as $Q = \omega_{x,y} \cdot \tau / 2$ where $\tau$ is the decay time of the oscillation ($\sim \exp\left(t / \tau\right)$). We qualitatively observed a strong dependence of the Q-factor on the driving amplitude. When the driving amplitude is too high we observe non-linear time dependent behaviour. The results in \autoref{fig:q-factor-pressure-dependence} are obtained with a sufficiently low driving amplitude.
 
-\begin{figure}
+\begin{figure}[h]
     \centering
     \includegraphics{figures/data/q_factor_pressure_dependence.pdf}
     \caption{The pressure dependence of the Q-factor for the $x$-, $y$- and $z$-mode. The dashed lines are a prediction based on the theory from \autoref{sec:damping}. The experimental Q-factors are determined using the ringdown method. The \zmode only been observed at \qty{1}{\milli\bar} and extrapolated from the data in \autoref{fig:xyz-mode-dependence-on-trapping-frequency-1mbar}. The driving frequency is $\Omega/2\pi = \qty{1.8}{\kilo\hertz}$.}
     \label{fig:q-factor-pressure-dependence}
 \end{figure}
 
+An attempt was made to model the damping at low pressures. This was done in COMSOL where we used the estimated maximum velocity of the particle as a Lorentz factor in the magnetic field. We considered two cases: the particle induced a current in the tracks and the particle induced a current in a \ce{Ga} layer of \qty{1}{\micro\meter} thick around the trap. The reason to model the \ce{Ga}-layer is due to contamination during FIB milling. The results are shown in \autoref{tab:dissipation}. The dissipation of the \zmode is not included in this table because it has not been measured. Eddy current in the particle itself have not been taken into account.
+
 \begin{table}
     \centering
     \begin{tabularx}{\textwidth}{Xccc}
@@ -113,11 +115,11 @@ \section{Q-factor dependence on pressure}
         Ringdown measurements & \num{21.55 \pm 0.8} & \num{20.7 \pm 2.1} & --- \\
         \midrule
         Track dissipation sim. & \num{1.3878E-3} & \num{5.8018E-4} & \num{3.6026E-6} \\
-        \ce{Ga+} dissipation sim. & \num{0.595} & \num{0.393} & \num{0.272} \\
+        \ce{Ga}-layer dissipation sim. & \num{0.595} & \num{0.393} & \num{0.272} \\
         % Thin wire model & \num{1.02} & \num{0.631} & \num{3.26} \\
         % Induction dissipation sim. & ??? & ??? & ??? \\
         \bottomrule
     \end{tabularx}
-    \caption{The measured dissipation at low pressures ($<\qty{1E-3}{\milli\bar}$) for the \xmode and \ymode (data in \autoref{fig:q-factor-pressure-dependence}) compared to the simulated dissipation of various sources.}
+    \caption{The measured dissipation at low pressures ($<\qty{1E-3}{\milli\bar}$) for the \xmode and \ymode (data in \autoref{fig:q-factor-pressure-dependence}) compared to the simulated dissipation through induction and eddy currents in either the tracks and a thin \ce{Ga} layer.}
     \label{tab:dissipation}
 \end{table}

chapters/theory.tex

@@ -20,14 +20,14 @@ \subsection{Fields}
     \label{eq:homogeneous_field}
 \end{equation}
 
-The static field is given in \autoref{eq:saddle_point}. $B_1''$ is the curvature of $\vec{B_1}$ and $\Omega / 2\pi$ the frequency of the oscillation. The field is created using two coplanar loops of with a current in opposite directions. These loops are the main focus of this thesis. Given the radius of the inner (outer) loop $r_1$ ($r_2$) we can express the curvature as $B_1'' = -\frac{9}{16}\mu_0i_1/r_1^3$ where $i_1$ ($i_2$) is the current through the inner (outer) loop\cite{perdriat}. This assumes that $i_2/i_1 = -r_2/r_1 = -\xi$.
+The dynamic field is given in \autoref{eq:dynamic_field}. $B_1''$ is the curvature of $\vec{B_1}$ and $\Omega / 2\pi$ the frequency of the oscillation. The field is created using two coplanar loops of with a current in opposite directions. These loops are the main focus of this thesis. Given the radius of the inner (outer) loop $r_1$ ($r_2$) we can express the curvature as $B_1'' = -\frac{9}{16}\mu_0i_1/r_1^3$ where $i_1$ ($i_2$) is the current through the inner (outer) loop\cite{perdriat}. This assumes that $i_2/i_1 = -r_2/r_1 = -\xi$.
 \begin{equation}
     \vec{B_1}(\vec{r}, t) = \frac{B_1''}{2} \begin{pmatrix}
         -xz \\
         +yz \\
         z^2 - \frac{1}{2}\left(x^2 + y^2\right)
     \end{pmatrix} \cos(\Omega t)
-    \label{eq:saddle_point}
+    \label{eq:dynamic_field}
 \end{equation}
 
 Finally, we have the gradient field, given in \autoref{eq:gradient_field}. The required gradient can be expressed as $B_2' = mg/\mu$ where $m$ is the mass of the levitated object, $g$ the gravitational acceleration and $\mu$ the magnetic moment of the object. The gradient serves to offset the effect of gravity. As such the gradient is also oriented in the $z$-axis. The gradient field can be created by using a larger current in one of the Helmholtz coils.