You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
{{ message }}
This repository was archived by the owner on Feb 24, 2025. It is now read-only.
Copy file name to clipboardExpand all lines: chapters/appendices/q_factors.tex
+2-2Lines changed: 2 additions & 2 deletions
Original file line number
Diff line number
Diff line change
@@ -6,7 +6,7 @@ \chapter{Definition and conversion of Q-factors}
6
6
\ddot{x} - \gamma\dot{x} + \omega_0^2 x = D(t)
7
7
\tag{damped h.o.}
8
8
\end{equation}
9
-
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 $0 < \gamma\leq1$. For a more general treatment of Q-factors, including non-linear systems, see \citeauthor{wang_rigorous_2017}.
9
+
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 $0 < \gamma\leq1$. For a more general treatment of Q-factors, including non-linear systems, see \textcite{wang_rigorous_2017}.
10
10
11
11
\section{Q-factor definition}
12
12
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}
34
34
\end{equation}
35
35
36
36
\subsection{Fourier spectrum}
37
-
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}):
37
+
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 \textcite{fowles_analytical_2005}):
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.
3
+
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 $\omega_{x,y}\proptoi_1\propto1/\Omega$and $\omega_{\gamma,\beta} \propto\sqrt{B_0}$.
4
4
5
-
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 $\Omega$.
5
+
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 $\omega_{x,y,z}\propto1/\Omega$.
6
6
7
-
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.
7
+
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 to use a time dependent simulation in COMSOL and to determine the relation between $\gamma$ and $i_1$.
8
8
9
9
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.
10
10
11
-
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}.
11
+
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 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}.
Copy file name to clipboardExpand all lines: chapters/discussion.tex
+44-37Lines changed: 44 additions & 37 deletions
Original file line number
Diff line number
Diff line change
@@ -1,5 +1,49 @@
1
1
\chapter{Discussion}
2
2
\label{chap:discussion}
3
+
In this chapter we will discuss the results presented in \autoref{chap:results}. We will also discuss the limitations of the current setup and possible improvements.
4
+
5
+
\section*{Laser heating}
6
+
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.
7
+
8
+
\section*{Damping at low pressures}
9
+
We observed a limit in the Q-factor for pressures below \qty{1E-2}{\milli\bar}. Attempts to model this have been unsuccessful and do not match the experimental results (see \autoref{tab:dissipation}. Proper modelling of the dissipation should involve a time dependent simulation in COMSOL instead of using a Lorentz term. The main reason for this is that a displacement of the particle brings it closer to its surrounding meaning any effects due to the magnetization of the particle will be stronger. Another source of damping could be eddy currents inside of the particle itself. In addition to this we should consider additional sources of dampign, such as noise from the electronics or anisotropy (differences in magnetization) inside the particle\cite{millen}. Additionally a recent paper also discusses eddy currents in more detail and how they would behave in superconductors as well\cite{fuwa_stable_2023}. If the damping is caused by eddy currents inside of the particle, then we should expect a relation between $\gamma$ and $i_1$.
10
+
11
+
\section*{Laser v.s. camera readout}
12
+
The disadvantages and advantages of the camera and laser readout are summarized in \autoref{tab:laser-vs-camera-readout}.
\item SNR is dependent on the laser intensity, background light and electronic noise;
31
+
\item No immediate visual feedback about the behaviour of the particle;
32
+
\item High laser intensity can cause the particle to loose its magnetization;
33
+
\end{itemize} \\
34
+
\bottomrule
35
+
\end{tabularx}
36
+
\caption{Advantages and disadvantages of camera and laser readout.}
37
+
\label{tab:laser-vs-camera-readout}
38
+
\end{table}
39
+
40
+
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 may also allow for sideband cooling.
41
+
42
+
\section*{Lorentzian fits}
43
+
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.
44
+
45
+
\section*{Trapping at low pressures}
46
+
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.
3
47
4
48
\section*{Loss of levitation}
5
49
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}.
@@ -17,40 +61,3 @@ \section*{Loss of levitation}
17
61
18
62
\section*{Levitation height}
19
63
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.
20
-
21
-
\section*{Loss of magnetization}
22
-
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.
23
-
24
-
\section*{Damping at low pressures}
25
-
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.
\item SNR is dependent on the laser intensity, background light and electronic noise;
44
-
\item No immediate visual feedback about the behaviour of the particle;
45
-
\item High laser intensity can cause the particle to loose its magnetization;
46
-
\end{itemize} \\
47
-
\bottomrule
48
-
\end{tabularx}
49
-
50
-
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.
51
-
52
-
\section*{Lorentzian fits}
53
-
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.
54
-
55
-
\section*{Trapping at low pressures}
56
-
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.
0 commit comments