Some final changes.

Author: jvdoorn

abstract.tex

@@ -1 +1 @@
-The twentieth century for physics was marked by the succesfull theories of quantum mechanics and the theory of general relativity. However, a unification of these two theories has not yet been achieved and is one of the biggest challenges in modern physics. To test the quantum nature of gravity, \citeauthor{bose_spin_2017} proposed an experiment to entangle two massive particles through gravity\cite{bose_spin_2017}. This thesis is a step towards this experiment and builds on our previous work \cite{janse_characterization_2024,eli,mart}. We succesfully demonstrate stable levitation of a \ce{NdFeB} particle with a diameter of $\qty{12}{\micro\meter}$ in an on-chip planar magnetic Paul trap. 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, but with very low Q-factors ($Q \approx 5$). At lower pressures the Q-factor increases ($Q \approx 3000$), and we are limited by a different source of damping. The on-chip design opens the possibility to integrate the trap with NV centers to groundstate cool the particle. We expect that we will be able to trap a $\qty{1}{\micro\meter}$ particle using the same method.
+The twentieth century for physics was marked by the successful theories of quantum mechanics and the theory of general relativity. However, a unification of these two theories has not yet been achieved and is one of the biggest challenges in modern physics. To test the quantum nature of gravity, \citeauthor{bose_spin_2017} proposed an experiment to entangle two massive particles through gravity\cite{bose_spin_2017}. This thesis is a step towards this experiment and builds on our previous work \cite{janse_characterization_2024,eli,mart}. We demonstrate stable levitation of a \ce{NdFeB} particle with a diameter of $\qty{12}{\micro\meter}$ in an on-chip planar magnetic Paul trap. Levitation was observed at atmospheric pressure all the way down to $\qty{1E-4}{\milli\bar}$. At atmospheric pressure we observed the $x$, $y$, $\gamma$ and $\beta$ modes, but with very low Q-factors ($Q \approx 5$). At lower pressures the Q-factor increases ($Q \approx 3000$), and becomes independent of the chamber pressure. The on-chip design opens the possibility to integrate the trap with NV centres to ground state cool the particle. Our results suggest that this method allows the trapping of a \qty{1}{\micro\meter} particle.

chapters/appendices/q_factors.tex

@@ -1,7 +1,7 @@
 \chapter{Definition and conversion of Q-factors}
 \label{app:q_factors}
 
-The goal of this appendix is ot provide a clear definition of the Q-factor and to derive (using simulations) how the Q-factor can be obtained. In particular, we will study a system described by the following differential equation:
+The goal of this appendix is to provide a clear definition of the Q-factor and to derive (using simulations) how the Q-factor can be obtained. In particular, we will study a system described by the following differential equation:
 \begin{equation}
     \ddot{x} - \gamma \dot{x} + \omega_0^2 x = D(t)
     \tag{damped h.o.}

chapters/conclusion.tex

@@ -1,6 +1,6 @@
 \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 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 $\omega_{\gamma,\beta} \propto \sqrt{B_0}$.
+In this thesis we have shown successful levitation of a \ce{NdFeB} particle with a \qty{12}{\micro\meter} diameter in an on-chip 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 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 $\omega_{x,y,z} \propto 1/\Omega$.
 

chapters/introduction.tex

@@ -2,7 +2,7 @@ \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 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 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.
+\citeauthor{bose_spin_2017} suggest entangling two particles through gravity\cite{bose_spin_2017}. If this is possible, it would be a direct proof that gravity is a quantum entity. This experiment requires two cat states. A cat state is a superposition of a macroscopic object. To achieve this we need to cool two objects of roughly \qty{1}{\micro\meter} to their ground state. The superpositions are created using a Stern-Gerlach interferometer. The particles are then allowed to interact through gravity. 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 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.
 

chapters/method.tex

@@ -52,7 +52,7 @@ \subsection{Nano fabrication}
     \label{fig:FIB}
 \end{figure}
 
-Using a micromanipulator a \ce{NdFeB} particle of \qty{12}{\um} is placed inside the \ce{Si} hole. The easiest way to do so is by sticking the particle to the bottom of the needle and then scraping the particle off on the sides of the \ce{Si} hole. The coverslip is placed on top of the \ce{Si} substrate. The holes are carefully aligned by moving the coverslip using a micromanipulator. The coverslip is then glued to the \ce{Si} substrate using an epoxy. The coverslip creates a enclosed environment for the particle to move in. This prevents the particle from escaping the trap. The particle is magnetized by putting the whole sample in a magnetic field of approximately \qty{1.3}{\tesla} for several minutes at room temperature and pressure. \autoref{fig:optical-microscope-image-sample} shows an optical microscope image of the sample.
+Using a micromanipulator a \ce{NdFeB} particle of \qty{12}{\um} is placed inside the \ce{Si} hole. The easiest way to do so is by sticking the particle to the bottom of the needle and then scraping the particle off on the sides of the \ce{Si} hole. The coverslip is placed on top of the \ce{Si} substrate. The holes are carefully aligned by moving the coverslip using a micromanipulator. The coverslip is then glued to the \ce{Si} substrate using an epoxy. The coverslip creates a enclosed environment for the particle to move in. This prevents the particle from escaping the trap. The particle is magnetized by putting the whole sample in a magnetic field of approximately \qty{1.4}{\tesla} for several minutes at room temperature and pressure. \autoref{fig:optical-microscope-image-sample} shows an optical microscope image of the sample.
 
 \begin{figure}
     \centering

chapters/results.tex

@@ -60,17 +60,19 @@ \section{Measurements at low pressure}
     \label{fig:xyz-mode-spectrum-1mbar}
 \end{SCfigure}
 
-By performing this `lock-in like' measurement at multiple driving frequencies we can determine clear relations between $\omega_{x,y,z}$, the damping and the driving frequency. We do so by fitting a Lorentzian for each peak in the spectrum. The results are shown in \autoref{fig:xyz-mode-dependence-on-trapping-frequency-1mbar}. The fits are given by:
-\begin{align*}
-    \omega_x &\propto \frac{\qty{ 11.394193305319426 \pm 0.6168138640026458 }{\square\kilo\hertz}}{\Omega} \\ % + (\qty{ -130.8583313665632 \pm 35.52389434927459 }{\hertz}) \\
-    \omega_y &\propto \frac{\qty{ 8.61031330605224 \pm 0.3506111962572132 }{\square\kilo\hertz}}{\Omega} \\ % + (\qty{ -47.01583771770941 \pm 21.39358314176167 }{\hertz}) \\
-    \omega_z &\propto \frac{\qty{ 19.211603271753344 \pm 0.9219280210953733 }{\square\kilo\hertz}}{\Omega} % + (\qty{ -142.4727024226977 \pm 53.88080820095942 }{\hertz})
-\end{align*}
+By performing this `lock-in like' measurement at multiple driving frequencies we can determine clear relations between $\omega_{x,y,z}$, the damping and the driving frequency. We do so by fitting a Lorentzian for each peak in the spectrum. The results are shown in \autoref{fig:xyz-mode-dependence-on-trapping-frequency-1mbar}.
+
+% The fits are given by:
+% \begin{align*}
+%     \omega_x &\propto \frac{\qty{ 11.394193305319426 \pm 0.6168138640026458 }{\square\kilo\hertz}}{\Omega} \\ % + (\qty{ -130.8583313665632 \pm 35.52389434927459 }{\hertz}) \\
+%     \omega_y &\propto \frac{\qty{ 8.61031330605224 \pm 0.3506111962572132 }{\square\kilo\hertz}}{\Omega} \\ % + (\qty{ -47.01583771770941 \pm 21.39358314176167 }{\hertz}) \\
+%     \omega_z &\propto \frac{\qty{ 19.211603271753344 \pm 0.9219280210953733 }{\square\kilo\hertz}}{\Omega} % + (\qty{ -142.4727024226977 \pm 53.88080820095942 }{\hertz})
+% \end{align*}
 
 \begin{figure}
     \centering
     \includegraphics{figures/data/xyz_mode_dependence_on_driving_frequency.pdf}
-    \caption{The dependence of $\omega_{x,y,z}$ and $\gamma_{x,y,z}$ on $\Omega$ at \qty{1}{\milli\bar}. The $\gamma$ in this case is the width of the Lorentzian fitted on the PSD. The dashed lines are a fit. The error bars are left out in the first figure because they are too small to be visible. The errorbars in the bottom figure are mostly dependent on the standard deviation on $\gamma$.}
+    \caption{The dependence of $\omega_{x,y,z}$ and $\gamma_{x,y,z}$ on $\Omega$ at \qty{1}{\milli\bar}. The $\gamma$ in this case is the width of the Lorentzian fitted on the PSD. The dashed lines are a fit. The error bars are left out in the first figure because they are too small to be visible. The errorbars in the bottom figure are mostly dependent on the standard deviation on $\gamma$. The \zmode at $\Omega/2\pi = \qty{3}{\kilo\hertz}$ was determined in two separate measurements.}
     \label{fig:xyz-mode-dependence-on-trapping-frequency-1mbar}
 \end{figure}
 
@@ -93,7 +95,7 @@ \section{Measurements at low pressure}
 
 \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 separately. 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.
+Using an external coil we are able to excite the \xmode  and \ymode separately. Ringdown measurements are performed to determine the pressure dependence on the Q-factor. The Q-factor in this case is defined as $Q = \omega_{x,y,z} \tau / 2$ 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}[h]
     \centering
@@ -102,7 +104,7 @@ \section{Q-factor dependence on pressure}
     \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.
+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
@@ -115,7 +117,7 @@ \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}-layer 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