Merge pull request #3 from jvdoorn/development

Development

Author: jvdoorn

.latexmkrc

@@ -1,5 +1,11 @@
-$pdf_mode = 1; # tex -> pdf
 @default_files = ('thesis.tex');
 
-$lualatex = 'lualatex -shell-escape -latexoption=-file-line-error -f';
+$pdflatex = 'lualatex %O %S';
 $pdf_previewer = 'start zathura';
+
+# Use lualatex for all types of document processing
+$pdflatex = 'lualatex %O %S';
+$pdf_mode = 1;
+
+# Automatically clean up auxiliary files after compilation
+$clean_ext = 'synctex.gz acn acr alg aux bbl blg fdb_latexmk fls glg glo gls idx ilg ind log out toc';

.zed/settings.json

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+// Folder-specific settings
+//
+// For a full list of overridable settings, and general information on folder-specific settings,
+// see the documentation: https://zed.dev/docs/configuring-zed#settings-files
+{
+  "lsp": {
+    "ltex": {
+      "settings": {
+        "ltex": {
+          "language": "en-EN"
+        }
+      }
+    }
+  }
+}

.zed/tasks.json

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+\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:
+\begin{equation}
+    \ddot{x} - \gamma \dot{x} + \omega_0^2 x = D(t)
+    \tag{damped h.o.}
+\end{equation}
+where $x$ is the signal amplitude (we will use meters, but the results are independent of units), $\gamma$ is the damping rate in \unit{\per\second}, $\omega_0 = 2\pi f_0$ is the resonance frequency in \unit{\radian\per\second} and $D(t)$ is the driving force. We specifically consider the case where $0 < \gamma \leq 1$. For a more general treatment of Q-factors, including non-linear systems, see \citeauthor{wang_rigorous_2017}.
+
+\section{Q-factor definition}
+There are two common definitions of the Q-factor. The first is the \textit{bandwidth} definition, where the Q-factor is given by the ratio between the resonance frequency and the bandwidth of the resonance.
+\begin{equation}
+    Q_\text{B} = \frac{f_0}{\Delta f} = \frac{\omega_0}{\Delta \omega}
+    \tag{bandwidth definition}
+\end{equation}
+where $\Delta \omega$ is the full width at half maximum (FWHM) of the resonance peak. The second definition is the \textit{energy} definition, where the Q-factor is given by the ratio between the energy stored in the oscillator and the energy dissipation (per cycle):
+\begin{align}
+    Q_\text{E} &= 2\pi \frac{\text{stored energy}}{\text{dissipated energy per cycle}} \nonumber \\
+               &= 2\pi f_0 \frac{\text{stored energy}}{\text{dissipation}} = \omega_0 \frac{\text{stored energy}}{\text{dissipation}} \tag{energy definition}
+\end{align}
+
+\section{FWHM}
+The first method of determining the Q-factor is by looking at the FWHM of the resonance peak. There are two ways of doing so, either by looking at the FWHM in the power spectrum or the FWHM in the Fourier spectrum.
+
+\subsection{Power spectrum}
+In the power spectrum (PSD), the peak is described by a Lorentzian distribution:
+\begin{equation}
+    \mathcal{L}(f) \propto \frac{\frac{1}{2} \Gamma}{(\omega - \omega_0)^2 + \left(\frac{1}{2} \Gamma\right)^2}
+\end{equation}
+where $\Gamma$ is the FWHM of the peak. Coincidentally, the FWHM is equal to the damping rate $\gamma$. The Q-factor in terms of the FWHM in the power spectrum is then given by:
+\begin{equation}
+    Q_\text{B} = \frac{\omega_0}{\Gamma} = \frac{\omega_0}{\gamma}
+\end{equation}
+
+\subsection{Fourier spectrum}
+There is a caveat however. When looking at a peak in a `normal' Fourier spectrum, meaning you look at the amplitude of the signal, the magnitude of the peak is instead described by (this can be found in most textbooks on classical mechanics, such as Chapter~3.6 in \citeauthor{fowles_analytical_2005}):
+\begin{equation}
+    A(f) \propto \sqrt{\frac{1}{(\omega^2 - \omega_0^2)^2 + \gamma^2\omega^2}}
+\end{equation}
+The FWHM in this case is approximated by $\sqrt{2}\gamma$. This means that the Q-factor of a `normal' Fourier spectrum deviates from the Q-factor of a power spectrum by a factor of $\sqrt{2}$! Since the energy of a system is more fundamental than the amplitude, we choose to convert the FWHM of the Fourier spectrum to the FWHM of the power spectrum.
+
+\section{Decay time}
+Another way of measuring the Q-factor of a resonator is by looking at the decay time of the signal. There are two ways of doing so, either by looking at the decay in amplitude or the decay in energy. They are related to $\gamma$ as $\gamma = 1 / \tau_\text{energy} = 2 / \tau_\text{amplitude}$. Where after a time $\tau_\text{energy}$ ($\tau_\text{energy}$) the energy (amplitude) has decreased by a factor $1/e$. To derive an expression for the Q-factor in terms of the decay time, we use that the energy dissipation is given by $\dot{E} = -\frac{1}{\tau_\text{energy}}E$. Substituting this into the energy definition of the Q-factor gives:
+\begin{equation}
+    Q_\text{E} = \omega_0 \frac{E(t)}{\frac{1}{\tau_\text{energy}} E(t)} = \omega_0 \tau_\text{energy} = \frac{1}{2} \omega_0 \tau_\text{amplitude} = \frac{\omega_0}{\gamma}
+\end{equation}
+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.
+
+\begin{table}[h]
+    \centering
+    \begin{tabular}{ccc}
+        \toprule
+        \textbf{Observation} & $\bm{\gamma}$ & $\bm{Q}$ \\
+        \midrule
+        FWHM in PSD ($\Delta \omega$ or $\Delta f$) & $\Delta \omega$ & $\omega_0 / \Delta \omega = f_0 / \Delta f$ \\
+        FWHM in FS ($\Delta \omega$ or $\Delta f$) & $\Delta \omega / \sqrt{2}$ & $\sqrt{2} \omega_0 / \Delta \omega = \sqrt{2} f_0 / \Delta f$ \\
+        Energy decay ($\tau_\text{energy}$) & $1 / \tau_\text{energy}$ & $\omega_0 \tau_\text{energy} = 2\pi f_0 \tau_\text{energy}$ \\
+        Amplitude decay ($\tau_\text{amplitude}$) & $2 / \tau_\text{amplitude}$ & $\frac{1}{2} \omega_0 \tau_\text{energy} = \pi f_0 \tau_\text{amplitude}$ \\
+        \bottomrule
+    \end{tabular}
+\end{table}

chapters/appendices/q_factors.tex

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+\chapter{Conclusion and outlook}
+\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 $x$, $y$, $\gamma$ 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. 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. At low pressures we also observed the \zmode. Again at low pressures we saw the expected relation between $\omega_{x,y,z}$ on $\Omega$.
+
+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 teh \xmode and \ymode. This might shed light on the origin of the damping but further investigation is needed.
+
+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.
+
+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 experience Zeeman splitting. 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 Q-factors in order to reach a sideband resolved regime. The idea is to use the rotational modes (which have a 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

@@ -1,3 +1,9 @@
 \chapter*{Conventions}
 \label{chap:conventions}
-In this thesis we regularly have to project 3D structures onto a 2D plane. To clarify the x-, y- and z-directions they have been color coded. This is done consistently throughout the thesis and matches with the colors assigned to the axes by COMSOL. The x-direction is colored in \textcolor{x_axis_color}{red}, the y-direction in \textcolor{y_axis_color}{green} and the z-direction in \textcolor{blue}{blue}. Additionally an attempt has been made to optimize figures for colorblind readers based on Paul Tol's color schemes\cite{paul_tol}.
+In this thesis we regularly have to project 3D structures onto a 2D plane. To clarify the x-, y- and z-directions, they have been color coded. This is done consistently throughout the thesis and matches with the colors assigned to the axes by COMSOL. The x-direction is colored in \textcolor{x_axis_color}{red}, the y-direction in \textcolor{y_axis_color}{green} and the z-direction in \textcolor{blue}{blue}. Additionally an attempt has been made to optimize figures for colorblind readers based on Paul Tol's color schemes\cite{paul_tol}.
+
+Furthermore, when a damping rate ($\gamma$) or Q-factor is given, we use the following definition:
+\begin{equation*}
+    Q = \omega_0 \frac{\text{energy stored}}{\text{dissipation}} = \frac{\omega_0}{\gamma}
+\end{equation*}
+where $\omega_0$ is the resonance frequency. More information on the definition and conversion of Q-factors can be found in Appendix~\ref{app:q_factors}.

chapters/conventions.tex

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+\chapter{Discussion}
+\label{chap:discussion}
+
+\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 loose 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}.
+
+\begin{SCfigure}[50]
+    \centering
+    \includegraphics[width=0.4\textwidth]{figures/sample/dirt_optical_microscope.jpeg}
+    \caption{An optical microscope image showing the `smut' around the trap. This `smut' is also present inside of the trap.}
+    \label{fig:smut-optical-microscope}
+\end{SCfigure}
+
+The current hypothesis is that this `smut' limits the free movement of the particle. This could be the case if the particle has a very low levitation height. The origin of the `smut' is likely due to oil from the roughing pump. Due to the heat of the Helmholtz coils (roughly \qty{60}{\celsius}) we think the oil evaporated or burnt and was deposited around the trap. The oil hypothesis is strengthened by the fact that we found an oil like substance on the glass of the vacuum chamber. At the time of writing the group is working on creating a new sample.
+
+Due to the loss of levitation we were also unable to further study the dependence of the \zmode on other parameters or its Q-factor at low pressures.
+
+\section*{Levitation height}
+Changing the focus of the objective used to image the particle lets us estimate the levitation height of the particle. Based on this we noticed that the particle appears to levitate very close to the bottom of the trap. Furthermore, by changing the gradient field ($\vec{B_2}$) we did not notice any change. We think that the trap stiffness in $z$ is relatively much larger than the PCB version of the trap. A low levitation height also strengthens the hypothesis that the `smut' hindered the movement of the particle.
+
+\section*{Loss of magnetization}
+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.
+
+\section*{Damping at low pressures}
+We observed a limit in the Q-factor for pressures below \qty{1E-2}{\milli\bar}. The origin of this is likely Eddy currents. Our attempts to model this however have been unsuccessful, as can be seen in \autoref{tab:dissipation}. Another possibility is that the Eddy currents exist inside of the particle, something which we did not model. Furthermore, other sources of damping should be considered such as noise from the electronics (or other Brownian noise sources) or anisotropy in the particle\cite{millen}. The latter case might be influenced by the static field $\vec{B_0}$, it could be interesting to study the dependence of the dissipation on the static field.
+
+\section*{Laser v.s. camera readout}
+\begin{tabularx}{\textwidth}{XX}
+    \toprule
+    Camera readout & Laser readout \\
+    \midrule
+    \begin{itemize}[left=0pt,topsep=0pt,label=\textcolor{green}{\texttt{+}}]
+        \item Immediate visual feedback about the behaviour of the particle;
+        \item SNR only dependent on the contrast of the particle (which is really good with a flat field correction);
+    \end{itemize} \begin{itemize}[left=0pt,topsep=0pt,label=\textcolor{red}{\texttt{-}}]
+        \item Limited sample rate (roughly \qty{400}{\fps});
+        \item No `real' lock-in measurements possible;
+        \item Analysis is not directly possible;
+    \end{itemize} & \begin{itemize}[left=0pt,topsep=0pt,label=\textcolor{green}{\texttt{+}}]
+        \item High sample rate that is only limited by the gain bandwidth product of the photodiode;
+        \item `Real' lock-in measurements possible by connecting the photodiode to a lock-in amplifier;
+    \end{itemize} \begin{itemize}[left=0pt,topsep=0pt,label=\textcolor{red}{\texttt{-}}]
+        \item SNR is dependent on the laser intensity, background light and electronic noise;
+        \item No immediate visual feedback about the behaviour of the particle;
+        \item High laser intensity can cause the particle to loose its magnetization;
+    \end{itemize} \\
+    \bottomrule
+\end{tabularx}
+
+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 will 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 because there was 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 ($<\qty{1}{\milli\bar}$). 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 we did not investigate further.

chapters/discussion.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.