Most of my thesis is there, except for discussion and conclusion.

Author: jvdoorn

chapters/conclusion.tex

@@ -0,0 +1,16 @@
+\chapter{Conclusion}
+\label{chap:conclusion}
+In this thesis we have shown successful levitation of a \qty{12}{\micro\meter} sized \ce{NdFeB} particle 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 $x$/$y$ and $\gamma$/$\tilde\beta$ 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 $\beta$ mode. The dependence of $\omega_{x,y}$ on $i_1$ and $\Omega$ follows the expected relation from theory.
+
+At lower pressures the Q-factors increase untill we reach roughly \qty{1E-2}{\milli\bar} where the Q-factor tends to a constant value. We attribute this to Eddy current damping, though we do not know where it occurs exactly. Further more careful measurements of the Q-factors at low pressures are needed to determine the origin of the damping.
+
+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. We also suspect that the levitation height is very low, because of this we think that we do not need a hole in the top cover glass. In the future this will help us to use diamonds with NV centers to couple to the particle more easily.
+TODO: Readout or also sideband cooling
+- Many spins to cool
+- Many spins to change angular momentum
+- Single spin for stern gerlach experiment
+
+- smaller particle
+- elongated particle (Huellery, Gabriel etet ferromagnet diamond)
+- hoger B0 veld (met kern)
+- magnetisatie (Milan)

chapters/discussion.tex

@@ -1,2 +1,9 @@
+\chapter{Discussion}
+\label{chap:discussion}
 - Use of laser for readout and heating
 - why we fit the lorentzians individually
+- magnetization
+- niet meer willen zweven
+- zweefhoogte (no B2 dependence)
+- eddy current magnisme or other sources (noise)
+- difference between laser and video readout

chapters/introduction.tex

@@ -1 +1,4 @@
 \chapter{Introduction}
+The Hensen Lab aims to understand the interplay of quantum mechanics and gravity. Recently a experiment was proposed by \citeauthor{bose_spin_2017} to probe the quantum mechanical nature of gravity. The central idea is to entangle two particles through gravity which is only possible if gravity is a quantum entity. The idea is to levitate two small ($\approx \qty{1}{\micro\meter}$) particles and cool them to their ground state. The particles are dropped through a Stern-Gerlach interferometer and the entanglement is measured by the interference pattern.
+
+Currently, we as a group, are working towards the levitation of micrometer sized particles. Previous projects within our group worked on the levitation of a \qty{100}{\micro\meter} sized particle in a planar magnetic Paul Trap\cite{eli, mart}. This trap was realised on PCB and the goal of this project is to miniturize the trap. Miniturization is key to reach the quantum regime. Furthermore a on-chip trap enables easier integration with other components such as NV centers.

chapters/method.tex

@@ -39,7 +39,7 @@ \section{Sample fabrication}
         \centering
         \includegraphics{figures/sample/trap_geometry.pdf}
     \end{subfigure}%
-    \caption{An optical microscope image of the sample (\textbf{left}) and a schematic of the sample geometry (\textbf{right}). A couple features can be seen in the image 1) in the bottom left is a discoloration due to the epoxy 2) the dark streak in the bottom right is the edge of the cover glass 3) the dark discoloration is the \ce{Pt} layer of the glass. In the center of the trap you can clearly see the \ce{NdFeB} particle. Furthermore, you can see that their is a small misalignment between the hole in the \ce{Si} and hole in the cover glass. A definition of the symbols in the schemtic and their numerical value can be found in \autoref{tab:sample-dimensions}.}
+    \caption{An optical microscope image of the sample (\textbf{left}). A couple features can be seen in the image 1) in the bottom left is a discoloration due to the epoxy 2) the dark streak in the bottom right is the edge of the cover glass 3) the dark discoloration is the \ce{Pt} layer of the glass. In the center of the trap you can clearly see the \ce{NdFeB} particle. Furthermore, you can see that their is a small misalignment between the hole in the \ce{Si} and hole in the cover glass. On the \textbf{right} you see a schematic of the sample. A definition of the symbols and their numerical value can be found in \autoref{tab:sample-dimensions}.}
     \label{fig:sample-dimensions}
 \end{figure}
 
@@ -60,13 +60,14 @@ \section{Experimental setup}
 A microscope objective with a $10\times$ magnification and large working distance is used to provide a visual image of the sample. Additionally a laser (\qty{635}{\nm}, \qty{1}{\milli\watt}) is coupled in. The reflection of the laser is then imaged on a photodiode (ThorLabs PDA36A2). The photodiode is connected to a lock-in amplifier (Zurich Instruments MFLI) allowing for the detection of the motion of the particle. In addition to the photodiode, we can use the camera feed to analyse the motion of the particle. This is done using Python and OpenCV using the CSRT tracker. This tracker is quite robust at the cost of being computationally expensive. As such it cannot be used in real-time.
 
 \section{Simulations}
-There are two assumptions in the theory: infinitely thin tracks; and perfectly symmetric loops. Because of this the ideal current ratio will be different in practice. A simulation in COMSOL is used to guide our choice of the current ratio. The simulation models the \ce{Si} wafer, the \ce{Au} tracks and a air box surrounding them. The simulation is performed in 3D using a stationary dc-analysis.
+There are two assumptions in the theory: infinitely thin tracks; and perfectly symmetric loops. Because of this the ideal current ratio will be different in practice. A simulation in COMSOL is used to guide our choice of the current ratio. The simulation is performed in 3D using a stationary dc-analysis.
 
 \subsection{Q-factor}
 \label{subsec:q-factor}
 In addition to this we estimate our effective Q-factor. A finite Q-factor is caused by damping. Below we discuss several sources of damping.
 
 \subsubsection{Gas damping}
+\label{subsubsec:gas-damping}
 Gas damping on levitated micrometer sized particles has been studied by \citeauthor{millen}. For pressures above \qty{1E-6}{\milli\bar} and $K_n \ll 1$ stachastic forces dominate the damping rate. The damping rate in this case is given by \autoref{eqn:gas-damping-rate}.
 
 \begin{equation}
@@ -84,7 +85,11 @@ \subsubsection{Gas damping}
 \end{SCfigure}
 
 \subsubsection{Inductive damping}
-TODO
+\label{subsubsec:inductive-damping}
+The enclosed flux in the loops will change as the particle moves. A changing flux induces a current in the loops which is a dissipative process. Using COMSOL we make an estimate of the dissipation by calculating the induced current in the loops.
+
+TODO: we still need to do this simulation, but I think it will be very similar to the Eddy current damping? Need to check what COMSOL does exactly.
 
 \subsubsection{Eddy current damping}
-TODO
+\label{subsubsec:eddy-current-damping}
+Eddy currents are small `whirlpools' of current that are induced by a changing magnetic field. Due to the resistance of the conductor this will lead to dissipation of energy. We consider the case that the moving particle creates a changinging magnetic field which induces the Eddy currents in either the tracks or the residual \ce{Ga} around the trap due to the FIB milling. To estimate the dissipation we perform a simulation in COMSOL where we use the estimated maximum velocity of the particle to include a Lorentz transformation in the magnetic field. The simulation is done in 3D using a stationary dc-analysis.

chapters/results.tex

@@ -2,15 +2,24 @@ \chapter{Results}
 \label{chap:results}
 The datasheet of our \ce{NdFeB} particles specifies that their residual magnetic field is \qtyrange{730}{760}{\milli\tesla}. Full saturation (\qty{>95}{\percent}) is achieved when a \qty{2}{\tesla} field is applied. Our method of magnetising the particle was limited to a \qty{1.4}{\tesla} field. As such, it is likely that we did not achieve the full saturation of the particle and that the residual magnetic field is lower than specified. Assuming a roughly linear relation between the two values, we estimate the residual magnetic field of our particles to be \qtyrange{510}{530}{\milli\tesla}. The density of the particles is given as \qty{7430}{\kilo\gram\per\cubic\meter}.
 
-These values, together with the simulation of the magnetic field distribution in the trap, allows us to estimate the required driving frequency and the resulting eigen frequency.
+These values, together with the simulation of the magnetic field distribution in the trap, allows us to estimate the required driving frequency and the resulting eigen frequency. \autoref{fig:magnetic-field-curvature} shows the $z$-component of the ac magnetic field ($\vec{B_1}$) for a range of $\chi$ values as well as the curvature. Using \autoref{eq:eigenfrequencies} and \autoref{eq:frequency-constraint} we can then estimate the required driving frequency and the resulting eigen frequencies. These results are shown in \autoref{fig:eigen-frequency-xi-dependence}.
 
 \begin{figure}
     \centering
     \includegraphics{figures/data/magnetic_field_curvature.pdf}
-    \caption{the \textbf{top} figures from left to right show $z$-component of the magnetic field evaluated along a line parallel to the $x$-, $y$- and $z$-axis respectively through the origin. The gray zones indicate the boundaryies of the trap. The \textbf{bottom} figure shows the curvature of the magnetic field for these calculations evaluated in the extrema. The simulations were performed for $i_1=\qty{0.5}{\ampere}$ and $\chi$ between \numrange{-0.5}{-5}. The curves have been labelled with the corresponding value for $\chi$.}
+    \caption{the \textbf{top} figures from left to right show $z$-component of the magnetic field evaluated along a line parallel to the $x$-, $y$- and $z$-axis respectively through the origin. The gray zones indicate the boundaryies of the trap. The \textbf{bottom} figure shows the curvature of the magnetic field for these calculations evaluated in the extrema. The simulations were performed for $i_1=\qty{0.5}{\ampere}$ and $\chi$ between \numrange{0.5}{5}. The curves have been labelled with the corresponding value for $\chi$.}
     \label{fig:magnetic-field-curvature}
 \end{figure}
 
+\begin{figure}
+    \centering
+    \includegraphics{figures/data/eigen_frequency_xi_dependence.pdf}
+    \caption{Simulation results of the dependence of the eigen frequency on $\chi$. These results are based on the data presented in \autoref{fig:magnetic-field-curvature}. The \textbf{top} figure shows the minimal trapping frequency such that $q_z \leq 0.4$. In the \textbf{middle} the corresponding eigenfrequencies of the modes are shown for $\Omega / 2\pi = \qty{2.5}{\kilo\hertz}$. The \textbf{bottom} figure shows the deviation from the center of the \ymode. This is due the fact that the slit is in the direction of the \ymode. The dashed lines are a linear interpolation of the data and only serve as a guide to the eyes.}
+    \label{fig:eigen-frequency-xi-dependence}
+\end{figure}
+
+As an additional visual reference, \autoref{fig:magnetic-field-distribution} shows the magnitude of $\vec{B_1} \parallel \zhat$ in the trap for $i_1=\qty{0.5}{\ampere}$ and $\chi=2$. The important takeaway from this figure is that the magnetic field is stronger towards the edges of the trap (nearer to the tracks). TODO: just leave this figure out? Does not add much.
+
 \begin{figure}
     \centering
     \begin{subfigure}{0.5\textwidth}
@@ -23,12 +32,6 @@ \chapter{Results}
     \label{fig:magnetic-field-distribution}
 \end{figure}
 
-\begin{figure}
-    \centering
-    \includegraphics{figures/data/eigen_frequency_xi_dependence.pdf}
-    \caption{Simulation results of the dependence of the eigen frequency on $\chi$. These results are based on the data presented in \autoref{fig:magnetic-field-curvature}. The \textbf{top} figure shows the minimal trapping frequency such that $q_z \leq 0.4$. In the \textbf{middle} the corresponding eigenfrequencies of the modes are shown for $\Omega / 2\pi = \qty{2.5}{\kilo\hertz}$. The \textbf{bottom} figure shows the deviation from the center of the \ymode. This is due the fact that the slit is in the direction of the \ymode. The dashed lines are a linear interpolation of the data and only serve as a guide to the eyes.}
-\end{figure}
-
 \section{Measurements at atmospheric pressure}
 \label{sec:measurements-at-atmospheric-pressure}
 TODO: change this to only mention the x-mode? As we did not drive the y-mode separately.
@@ -51,7 +54,7 @@ \section{Measurements at atmospheric pressure}
 \begin{SCfigure}
     \centering
     \includegraphics{figures/data/rotational_mode_dependence_on_B0.pdf}
-    \caption{The dependence of $\omega_{\gamma,\tilde\beta}$ on $B_0$ at atmospheric pressure.}
+    \caption{The dependence of $\omega_{\gamma,\beta}$ on $B_0$ at atmospheric pressure.}
     \label{fig:rotational-mode-dependence-1bar}
 \end{SCfigure}
 
@@ -76,22 +79,22 @@ \section{Measurements at low pressure}
 \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 &= \frac{\qty{ 11.394193305319426 \pm 0.6168138640026458 }{\square\kilo\hertz}}{\Omega} + (\qty{ -130.8583313665632 \pm 35.52389434927459 }{\hertz}) \\
-    \omega_y &= \frac{\qty{ 8.61031330605224 \pm 0.3506111962572132 }{\square\kilo\hertz}}{\Omega} + (\qty{ -47.01583771770941 \pm 21.39358314176167 }{\hertz}) \\
-    \omega_z &= \frac{\qty{ 19.211603271753344 \pm 0.9219280210953733 }{\square\kilo\hertz}}{\Omega} + (\qty{ -142.4727024226977 \pm 53.88080820095942 }{\hertz})
-\end{align}
+\begin{align*}
+    \omega_x &= \frac{\qty{ 11.394193305319426 \pm 0.6168138640026458 }{\square\kilo\hertz}}{\Omega} \\ % + (\qty{ -130.8583313665632 \pm 35.52389434927459 }{\hertz}) \\
+    \omega_y &= \frac{\qty{ 8.61031330605224 \pm 0.3506111962572132 }{\square\kilo\hertz}}{\Omega} \\ % + (\qty{ -47.01583771770941 \pm 21.39358314176167 }{\hertz}) \\
+    \omega_z &= \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.}
+    \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 $\gamma$.}
     \label{fig:xyz-mode-dependence-on-trapping-frequency-1mbar}
 \end{figure}
 
-Visually it appears that $\gamma$ is constant as a function of $\Omega$. The question is whether $\gamma_z$ differs significantly from $\gamma_{x,y}$. To test this we perform a $T$-test on the mean values, the results can be found in \autoref{tab:gamma-t-test}. From this we can conclude that $\gamma_x = \gamma_y$ and $\gamma_z \gg \gamma_{x,y}$ at \qty{1}{\milli\bar}.
+Visually it appears that $\gamma$ is constant as a function of $\Omega$. The question is whether $\gamma_z$ differs significantly from $\gamma_{x,y}$. To test this we perform a $T$-test on the mean values, the results can be found in \autoref{tab:gamma-t-test}. The T-test suggests that $\gamma_x = \gamma_y$ and $\gamma_z \gg \gamma_{x,y}$ at \qty{1}{\milli\bar}.
 
-\begin{SCtable}
+\begin{SCtable}[50]
     \centering
     \begin{tabular}{cc}
         \toprule
@@ -100,15 +103,12 @@ \section{Measurements at low pressure}
         $\gamma_x = \gamma_y$ & \textcolor{x_axis_color}{$0.897 \gg 0.05$} \\
         $\gamma_y = \gamma_z$ & \textcolor{y_axis_color}{$0.005 \ll 0.05$} \\
         $\gamma_x = \gamma_z$ & \textcolor{y_axis_color}{$0.005 \ll 0.05$} \\
-        % $\gamma_x, \gamma_y \leq \gamma_z$ & \num{0.003} \\
         \bottomrule
     \end{tabular}
     \caption{The results of the $T$-test on the mean values of $\gamma_{x,y,z}$ from the data in \autoref{fig:xyz-mode-dependence-on-trapping-frequency-1mbar}.}
     \label{tab:gamma-t-test}
 \end{SCtable}
 
-
-
 \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.
@@ -119,3 +119,22 @@ \section{Q-factor dependence on pressure}
     \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{subsec:q-factor}. The experimental Q-factors are determined using the ringdown method. The \zmode has not been observed experimentally. The \xmode and \ymode were driven sinusoidally at their respective resonance frequencies.}
     \label{fig:q-factor-pressure-dependence}
 \end{figure}
+
+\begin{table}
+    \centering
+    \begin{tabularx}{\textwidth}{Xccc}
+        \toprule
+        Source & \multicolumn{3}{c}{Dissipation (\unit{\atto\watt})} \\
+        \cmidrule(r){2-4}
+        & \xmode & \ymode & \zmode \\
+        \midrule
+        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} \\
+        % 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.}
+\end{table}