#major Initial final version of thesis

Author: jvdoorn

.github/workflows/latex.yml

@@ -23,7 +23,7 @@ jobs:
           GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
           WITH_V: true
           RELEASE_BRANCHES: main
-          DEFAULT_BUMP: ${{ github.ref == 'refs/heads/main' && 'minor' || 'patch' }}
+          DEFAULT_BUMP: patch
           DRY_RUN: true # We want to manually create the release.
       - run: |
           echo "${{ steps.version.outputs.new_tag }}" > version

.zed/settings.json

@@ -7,9 +7,14 @@
     "ltex": {
       "settings": {
         "ltex": {
-          "language": "en-EN"
+          "language": "en"
         }
       }
     }
+  },
+  "languages": {
+    "LaTeX": {
+      "soft_wrap": "editor_width"
+    }
   }
 }

abstract.tex

@@ -1 +1 @@
-TODO: Abstract
+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.

chapters/appendices/q_factors.tex

@@ -1,12 +1,12 @@
 \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 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}.
+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 \textcite{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.
@@ -34,7 +34,7 @@ \subsection{Power spectrum}
 \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}):
+However, there is a caveat. 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 \textcite{fowles_analytical_2005}):
 \begin{equation}
     A(f) \propto \sqrt{\frac{1}{(\omega^2 - \omega_0^2)^2 + \gamma^2\omega^2}}
 \end{equation}

chapters/conclusion.tex

@@ -1,11 +1,11 @@
 \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.
+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 $\omega_{\gamma,\beta} \propto \sqrt{B_0}$.
 
-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 $\Omega$.
+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 teh \xmode and \ymode. This might shed light on the origin of the damping but further investigation is needed.
+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.
+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 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}.
+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}.