report.tex

\documentclass[12pt, twoside]{report}
	\usepackage[margin=1in]{geometry}
	\usepackage[hidelinks, linktoc=all, pagebackref]{hyperref}
	\usepackage{indentfirst}
    \usepackage{subcaption}
   	\usepackage[superscript,biblabel]{cite}
    \usepackage[font=small,labelfont=bf]{caption}
    \usepackage{abstract}
    \usepackage{mathtools}
    \usepackage[russian, english]{babel}
    \usepackage{mathrsfs}  
    \usepackage{fontawesome}
    \usepackage{cleveref}
    \usepackage{fancyhdr}
    \usepackage{enumitem}
    \setlength{\parskip}{0.1cm}   
	\setlength{\parindent}{0.7cm}
    \usepackage{graphicx} % Used to insert images
    \usepackage{adjustbox} % Used to constrain images to a maximum size 
   	\usepackage{color} % Allow colors to be defined
    \usepackage{geometry} % Used to adjust the document margins
    \usepackage{amsmath} % Equations
    \usepackage{amssymb} % Equations
    \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
    \usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document
    \usepackage{fancyvrb} % verbatim replacement that allows latex
    \usepackage{grffile} % extends the file name processing of package graphics 
                         % to support a larger range 
    % The hyperref package gives us a pdf with properly built
    % internal navigation ('pdf bookmarks' for the table of contents,
    % internal cross-reference links, web links for URLs, etc.)
    \usepackage{longtable} % longtable support required by pandoc >1.10
\usepackage{bbm}

\newcommand{\diff}{\,\mathrm{d}} 	
 \renewcommand{\figureautorefname}{Fig.}%
\renewcommand*\thesection{\arabic{section}}
\DeclarePairedDelimiter\bra{\langle}{\rvert}
\DeclarePairedDelimiter\ket{\lvert}{\rangle}
\DeclarePairedDelimiterX\braket[2]{\langle}{\rangle}{#1 \delimsize\vert #2}
\newcommand{\rbrkt}[1]{\left( #1 \right)}
\newcommand{\sbrkt}[1]{\left[ #1 \right]}

\renewcommand*{\backreftwosep}{ and~}
\renewcommand*{\backreflastsep}{ and~}
\renewcommand*{\backref}[1]{}
\renewcommand*{\backrefalt}[4]{%
\ifcase #1 %
\relax%
\or
(referenced on p. [#2])%
\else
(referenced on p. [#2])%
\fi
}

\numberwithin{equation}{section}
\setcounter{tocdepth}{3}

\renewcommand*\thesection{\arabic{chapter}.\arabic{section}}
\lhead[\rm\thepage]{\fancyplain{}{\nouppercase{\sl{\rightmark}}}}
\rhead[\fancyplain{}{\nouppercase{\sl{\leftmark}}}]{\rm\thepage}
\chead{}\lfoot{}\rfoot{}\cfoot{}
\pagestyle{fancy}

\title{Reverse engineering of the Xmon qubits}

\graphicspath{{Pictures/}, {MIPT NIR presentation/Pics/}}


\begin{document}

\selectlanguage{russian}
\begin{titlepage}
\par 
\vspace*{-2cm}
\begin{center}
Министерство образования и науки Российской Федерации\\
Федеральное государственное автономное образовательное учреждение высшего\\ 
профессионального образования \\
\glqq Московский физико-технический институт (государственный университет) \grqq \\
Факультет общей и прикладной физики\\
Кафедра физики и технологии наноструктур
\end{center}

\vspace*{0.2cm}
\begin{flushright}
На правах рукописи\\
УДК 539.12
\end{flushright}

\vfill

\begin{center}
Федоров Глеб Петрович

\vspace*{0.5cm}

{\large Проектирование и исследование высококогерентных сверхпроводниковых квантовых систем}

\vspace*{1cm}

{\bf Магистерская диссертация}

\vspace*{1cm}

\begin{tabular*}{0.8\textwidth}{l}
Направление подготовки 03.04.01 <<Прикладные математика и физика>>\\
Магистерская программа 010988 <<Физика микроволн и наноматериалов>>\\
\end{tabular*}

\vspace*{2cm}

\begin{tabular*}{0.8\textwidth}{lll}
Заведующий кафедрой & \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } &	/ Лебедев В.В. / \\
Научный руководитель & \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \hspace*{-4cm} \includegraphics[width=100pt]{ryazanov_sign}& / Рязанов В.В. / \\
Студент		    & \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \hspace*{-4cm} \includegraphics[width=100pt]{empty_sign} & / Федоров Г.П. / \\
\end{tabular*}

\vfill

Москва

2017 г. 
\end {center} 
\end{titlepage}


\selectlanguage{english}

\tableofcontents

\input{report_theory}


\input{report_experimental_methods}


\chapter{Experimental results}

\section{Proof-of-concept design}

\subsection{Geometry and parameters}

The first sample that was measured was created from a proof-of-concept design, which was aimed at testing general properties of the simplest Xmon-based cQED systems. The easiest way to gain insight about the structure of the sample is to look at \autoref{fig:first_chip_design_full} where some of the main parts of its design are shown. The chip is 8 mm long and 4 mm wide and consists basically of six isolated qubit-resonator systems. 

All of the resonators were $\lambda/4$ with frequencies designed to be $6, 6.5, 6.9,7,7.1,8$ GHz for devices I-VI, respectively. Their coplanar parameters are $7\,\mu$m for the central wire width and $4\,\mu$m for the gaps.

The qubits were designed to be identical, with $C_\Sigma \approx 80$ fF and $I_{C, \Sigma} = 60$ nA, giving the $\omega_{01}/2\pi \approx 7$ GHz at their flux sweet-spot ($\Phi_{ext}=0$) and anharmonicity of approximately $-230$ MHz.

Two test structures at the sides of the chip were also included to allow direct DC measurement of the SQUIDs created during the shadow evaporation.

\subsection{Purposes}

The main targets for the chip were:
\begin{enumerate}[label=(\alph*), leftmargin=1.5cm]
\itemsep0pt
 \item to check the measurement setup
 \item to check the calculations for the frequencies and the coupling strengths
 \item to roughly check the coherence of Xmons
 \item to estimate the reproducibility of the junctions 
 \item to observe the AC-Stark shift 
 \item to observe high-power multiphoton transitions and sidebands
\end{enumerate}

\begin{figure}[h!]
\centering
\includegraphics[width=0.9\textwidth]{chip_design_full}
\caption{\textbf{(a)} Large-scale image of the design used for the proof-of-concept sample. The chip (8x4 mm) consists of six $\lambda/4$ CPW resonators coupled capacitively to the feedline each with an Xmon qubit at the open end. \textbf{(b)} Zoomed area around one of the qubits, showing its cross-shaped capacitor and the ``claw'' coupler of its resonator. \textbf{(c)} Zoom around one of the qubits' SQUIDs. Pink areas denote the e-beam mask that was used for shadow evaporation.}
\label{fig:first_chip_design_full}
\end{figure}

\subsection{Fabrication}
The sample was fabricated by D. Gusenkova in the cleanroom facility at MIPT, in a two-step process. Firstly, electron beam lithography was used to form the shadow evaporation mask for the pads with junctions and Al was shadow evaporated in the Plassys unit ($\pm 7^\circ$). Secondly, photolithograpy was used to pattern larger structures like the feedline, resonators and Xmons' capacitors aligned with e-beam lithography, and finally the second layer of Al was deposited.	The order of lithographies was chosen this way to facilitate alignment.

\begin{figure}
\includegraphics[width=\textwidth]{chip_photo}
\caption{Micrograph of the chip implemented in silicon and aluminium. \textbf{(a)} Large-scale image of the design pattern on the sample. The chip (8$\times$4 mm on the 10$\times$10 substrate) consists of six $\lambda/4$ CPW resonators coupled to the feedline, each with an Xmon qubit at the open end. \textbf{(b)} Zoomed area around one of the cQED systems showing a meandering resonator with the ``claw'' coupler \cite{barends2013} on the open end and a cross-shaped capacitor of the qubit.}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{X_mon_JJ_SQUID}
\caption{\textbf{(a)} SEM micrograph of the SQUID of one of the test structures on the chip. One can see that photo- and electron lithography are aligned, however fine structures \autoref{fig:first_chip_design_full}~(c) were not resolved during the second step. \textbf{(b)} Enlarged view on one of the junctions (upper). Its area is approximately $120\times 330\approx 0.4\, \mu m^2$, in the design it's $0.3\,\mu m^2$.} 
\end{figure}

\subsection{Measurement setup}

The sample was measured at ISSP in laboratory of RQC. Cryogenic equipment was represented by BlueFors LD250 dilution refrigerator, with base temperature of 16 mK. The microwave equipment included R\&S ZNB 10 kHz-20 GHz vector network analyser,  Agilent E8257D 100 kHz - 40 GHz analog signal generator. The sample was flux biased using Keithley 6221 current source.

Microwave line was thermalized with 60 dB of attenuation, additional 20 dB of attenuation were introduced on a directional coupler which added the second tone from the $\mu$-wave source. After leaving the sample the signal passed through two isolators and a hybrid coupler, which was used before to measure two samples during single cooldown. Finally, the signal was amplified with 4-8 GHz LNF amplifier at 4 K and a with a room-temperature amplifier.

The sample holder that was used was designed for 10$\times$10 mm chips, so the bondwires had a relatively large length of 1 mm which of course deteriorated the overall transmission. Chip lay directly on the copper disk of the bottom part of the sample holder with no hole carved under it. Around the sample holder a superconducting coil was wound which has been connected to the current source mentioned above.

The magnetic shielding of the sample holder was achieved via a cryoperm shield. A superconducting shield was not installed in this run due to the lack of space inside the magnetic shield, which may have influenced the noise background.


\subsection{Characterization of the resonators}

As a first step of characterizing the sample a study of the resonances was performed. In \autoref{fig:first_resonators} the power transmission through the cryostat is shown. In can be seen that in overall the transmission level is at approximately $-35$ dB. As long as the amplifiers add 60 dB the directional coupler and the hybrid coupler subtract 23 dB, it can be inferred that the sample in the sampleholder itself has approximately -10 dB transmission. This should be improved with better impedance matching to reduce noise. Secondly, there is a clear 300 MHz-wide dip in the transmission around 6.2 GHz, which should also be eliminated.

\begin{figure}
\centering
\includegraphics[width=\textwidth]{xmon-first-try_general15x5}

\includegraphics[width=\textwidth]{q-factors-and-freqs_xmon_first_try}

\caption{\textbf{(Top)} General view of the resonances. All six resonances are visible, however shifted far down in frequency. This shift is due to a mistake in the design which didn't compensate for the ``claw'' couplers at the resonator ends. \textbf{(Bottom)} Various quality factors and frequencies depending on radiation power. Single photon limit is near -40 dBm on the VNA summed with 80 dB of attenuation, saturation limit near 5 dBm on the VNA. The shifts of frequency and the dips in the internal Q at around -15 dBm indicate presence of functional qubits in each cavity.}
\label{fig:first_resonators}
\end{figure}

All resonators are functional which can be seen from the presence of six sharp dips in transmission. The frequencies at which the dips occur are significantly lower than expected because there was a flaw in the macro code which draws the design. The frequency compensation routine made specifically to calculate the frequency shift\cite{Sank2014} caused by the ``claw'' coupler at the end of the resonator was not executed, so the length of the resonators became larger than it should have been. At higher frequencies the phase shift of the ``claw'' is larger so the frequency discord is larger there.

Below the results obtained using the \textit{circlefit}\cite{probst2015} fitting method are presented. Each peak from \autoref{fig:first_resonators}~(top) was enlarged and scanned with a fine resolution and averaged to reduce noise (more averages on low and less on high powers). Then for each power complex $S_{21}$ data for each scan area around a resonance was recorded.

After all of the data had been obtained, the fitting procedure has been applied for every scan at each power. The full fitting process is described in depth in the original publication\cite{probst2015}. In practice the whole algorithm is encapsulated in several function calls of the library called \textit{resonator tools} that the authors have kindly provided via \href{https://github.com/sebastianprobst/resonatortools}{GitHub}. Fitting results are summarized in \autoref{fig:first_resonators}~(bottom).

It is well-known\cite{wang2009} that for superconducting microwave resonators the internal quality factor experiences an increase in value when probed with higher power. This effect is believed to occur due to the presence of two-level defects or two-level systems (TLS) with a dipole moment in the areas of high electric fields which resonator creates. As long as TLSs have same frequency as the measured resonator and coupled strong enough, they will drain excitations from the resonator. However, TLSs can only accommodate only one photon at a time; thus, at high probe powers they saturate and do no more participate in resonator relaxation. Therefore, an increase of the internal Q-factor is observed when the resonator is driven with strong microwave fields.

Most importantly, in \autoref{fig:first_resonators}~(bottom) we can see a distinct dips in internal Q-factors and significant shifts in frequency. This is actually a very useful feature of the cQED systems since it allows to determine whether the qubit is alive in the resonator or not. This happens because at that power the qubits begin to break down because of the too large induced current. The dispersive shift disappears, shifting the resonator back into its bare position. However, in the experiment, the VNA scans the frequency around the resonance, and the qubits at first only break when the VNA's signal is exactly resonant with the cavity (because in this case the resonator's field amplitude is the largest). With following increase of power, the breakdown starts to happen earlier in the scan, and the resonator splits in two shifted versions of itself. By fitting these incorrect line shapes incorrectly, the fitting algorithm transitions from the dispersively shifted frequency to the bare cavity frequency, and the Q factor actually remains the same during the whole transition. 

\begin{figure}
\centering
\includegraphics[width=\textwidth]{first_resonators_on_flux}
\caption{Magnetic field influence on frequencies of four devices. Resonators III and IV didn't demonstrate any significant dependence on flux (not shown here).}
\label{fig:first_resonators_on_flux}
\end{figure} 

\subsection{Characterization of the full cQED systems}

From six systems on the chip only three were studied in detail partly due to lack of time and partly because the overall picture was more or less clear at that point. Firstly the behaviour of the resonators was investigated when the flux bias was being changed. This measurement revealed obvious periodic frequency oscillations in four devices from six, see \autoref{fig:first_resonators_on_flux}. The other two devices didn't show any noticeable periodic flux dependence because one of the qubits (III) had a broken SQUID and the other (IV) probably was much lower in frequency than its resonator.

Firstly system II was measured as long as it has shown the most prominent flux response of all. Then system VI was investigated and, finally, system I. For each of them a two-tone spectroscopy was performed and for II and VI a high-resolution single-tone one as well. Then the data was used to fit the parameters in the model \eqref{eq:hamiltonian} described in \autoref{sec:cQED}. Accordingly, the graphs that are shown below are composed of the experimental spectra overlayed with theoretical curves obtained from numerical solution of the corresponding eigenproblems.

\subsubsection{System II}

\paragraph{Model.} Below all the experimental data will be provided for system II along with theoretical fits of the spectral lines that were observed using the model \eqref{eq:hamiltonian}. Model parameters were fit based on all data on this system and are same for all figures in this section. Before turning to the comparison of the theoretical predictions and experimental results it would be useful to present the model parameters used for fitting which are summarized in \autoref{tab:first_II_params}.

\begin{table}[h]
\centering
\begin{tabular}{l|c}
Parameter & Value \\
\hline 
$C_\kappa$ & 0 fF \\
\hline
$C_g$ & 1.9 fF \\
\hline
$C_q$ & 95 fF \\
\hline
$E_C$ & 200 MHz
\end{tabular}~
\begin{tabular}{l|c}
Parameter & Value\\
\hline
$C_r$ & 444 fF \\
\hline
$L_r$ & 1.72 nH \\
\hline
$I_{C, \Sigma}$ & 69.5 nA \\
\hline
$E_{J, \Sigma}$ & 34.5 GHz
\end{tabular}
\caption{Values of the main parameters defining the spectrum of the model \eqref{eq:hamiltonian}.}
\label{tab:first_II_params}
\end{table}

From these values it's possible to calculate the coupling strength $g \approx 19.6$ MHz between the qubit and the resonator, bare qubit frequency of $\omega^{(0)}_{ge}/2\pi = 7.3$ GHz and bare resonator frequency $\omega^{(0)}_r = 5.758$ GHz. In the coupled system these values are shifted strongly, $\omega_{ge}/2\pi = 7.23$ GHz and $\omega_r = 5.746$ GHz. These shifts are so large (much greater than what would be predicted by the usual dispersive shifts) because of the large value of the Xmon capacitance. Usually cQED systems are treated in the limit of large $C_r \gg C_q, C_g, C_\kappa$ and in this case factors in the terms $\mathcal{\hat H}_q, \mathcal{\hat H}_r $ from \eqref{eq:hamiltonian} can be reduced to their bare uncoupled values. This means the only difference from the uncoupled case is the term $\mathcal{\hat H}_i$, which defines the shifts from the uncoupled frequencies, which are in this case by definition equal to the dispersive shifts. In the studied case $C_q \approx 0.25\, C_r$ and cannot be neglected; thus, the interacting systems are not the same as before coupling, and only the full model \eqref{eq:hamiltonian} taking account of all capacitances can be used to obtain correct results. Despite that after acquiring the new parameters for the interacting systems from \eqref{eq:hamiltonian} the dispersive shifts may be calculated as usual.

The flux normalization on the x-axis was done using data from \autoref{fig:first_resonators_on_flux}~(II) knowing the fact that $\Phi_0$ should be the period of the pattern and that zero flux point is situated at the center of one of the lower branches.

Below in the description of the figures a bit different notation for the transition frequencies may be used than those which denote the theoretical curves on the legends of the graphs. This is due to the fact that in the presence of coupling the qubit spectrum $\omega_{ge}(\Phi_{ext})$ is shared between transitions $\omega_{01}(\Phi_{ext})$ and $\omega_{02}(\Phi_{ext})$ of the full Hamiltonian, so when we say $\omega_{ge}(\Phi_{ext})$ we imply $\omega_{01}(\Phi_{ext})$ and $\omega_{02}(\Phi_{ext})$ for the cases of $\Delta_\omega > 0 $ and $\Delta_\omega < 0$, respectively.

\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{first_II_anticrossing_fit}
\caption{Anticrossing spectrum of system II with fitting lines obtained by numeric calculation from the model\eqref{eq:hamiltonian} (z-axis normalized). It can be seen that lower branch has some artefacts near the anticrossings which are caused by multiphoton and sideband transitions, signifying that probe power was not at single-photon level. The x-axis is normalized to one flux quantum through the SQUID of the Xmon.}
\label{fig:first_2nd_res_anticrossing}
\end{figure} 

\paragraph{Anticrossings spectrum.} Firstly, a high-resolution scan of one of the anticrossings from \autoref{fig:first_resonators_on_flux}~(II) was obtained. The power level on the VNA was set to -40.0 dBm, which means around -120 dBm at the sample. It is presented in \autoref{fig:first_2nd_res_anticrossing}. It can be seen that a really good agreement between experiment and theory was attained. However there are some discrepancies that can be pointed out: a slight asymmetry of the left and right anticrossings and the reduced brightness of the main branch $\omega_{01}$ on the right. It can be seen clearly from the data that additional multi-photon transitions or sidebands are visible similar to what have been observed before\cite{bishop2009} which indicates that the power was higher than at the single-photon level. Theoretical curves for those transitions were not displayed because the picture becomes too crowded. It is not clear why this effect is stronger in the right anticrossing than in the left one. Further investigation is needed, because at that cooldown due to the setup limitations measurements at lower power were too time consuming because of the low transmission.

\begin{figure}[t]
\centering
\includegraphics[width=.49\textwidth]{first_II_2tone_fit}\includegraphics[width=.49\textwidth]{first_II_2tone_higher_power_fit}

\caption{The two-tone spectra of system II at different power levels on the $\mu$-wave source with fitting lines obtained from the model \eqref{eq:hamiltonian}. \textbf{(Left)} At -20 dBm. Various transitions are visible, most pronounced are the horizontal resonator $0n$ transition, the hyperbolic qubit $ge$ transition ($\omega_{01},\ \omega_{02}$) and lower branch of the hyperbolic two-photon $gf$ transition ($\omega_{03}/2$). Also, a sideband $\ket{1,g}\rightarrow \ket{0,f}$ is visible ($\omega_{15}$). \textbf{(Right)} At -10 dBm. More transitions are visible, new compared to \autoref{fig:first_II_2tone} are the upper branch of the two-photon $gf$ transition ($\omega_{05}/2$), the lower branch of the three-photon $gd$ transition ($\omega_{06}/3$) and also the lower branch of the sideband $\ket{1,g}\rightarrow \ket{0,f}$ ($\omega_{23}$).}
\label{fig:first_II_2tone}
\end{figure}


\paragraph{Two-tone spectroscopy.} Next measurement was a standard two-tone spectroscopy. The results of such measurement at different second tone powers are presented. Firstly, the lowest possible power (-20 dBm) scan was acquired, it is shown in \autoref{fig:first_II_2tone}. It was not possible to set a lower power with the step attenuator of the $\mu$-wave source, however for studied system that power was low enough to observe only single-photon processes without apparent multi-photon transitions and sidebands around the qubit's degeneracy point. However due to the fact that the second tone was introduced from the feedline through the resonator the effective driving power was increasing when the qubit-resonator detuning was decreasing; thus, some secondary transitions become visible near the anticrossing regions. For example, a sideband transition which uses one photon from a resonator and one photon from the incident microwave radiation ($\omega_{15}$) is vaguely visible above the resonator line and a two-photon transition $\omega_{gf}/2$ is clearly visible below it.

If the power of the second tone is increased, the probability of the transitions with lesser matrix elements also rises allowing to see these transitions more clearly. Such measurement yields a spectrum as in \autoref{fig:first_II_2tone}~(right). It was done at -10 dBm; thus, a power 10 times higher than in the previous case was sent at the sample. Now the two-photon hyperbolic line $\omega_{gf}$ under the main one $\omega_{ge}$ is clearly visible (the upper complementary branch $\omega_{05}/2$ of the previously visible transition $\omega_{03}/2$). The upper branch of the sideband $\ket{1,g}\rightarrow \ket{0,f}$ is visible also, see \autoref{fig:first_II_2tone_zoom_fit} for the scan around $\Phi_{ext}=0$. Also there are two lines below the resonator at the sides of the graph whose origin is not clear. One of them might be the lower branch of that sideband (fit with $\omega_{23}$ in \autoref{fig:first_II_2tone}~(right)), however it can be seen that the theoretical line is not very accurate there. The other line which is visible between $\omega_{23}$ and $\omega_{06}/3$ was not fitted because no appropriate transition was found. Surely, one should be able to find it, but this task looks hard enough to give it up.

	
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{first_II_2tone_zoom_fit}
\caption{Zoomed area around the main qubit transition $ge$, two-photon transition $gf$ and the sideband transition $\ket{1,g}\rightarrow \ket{0,f}$ with fitting lines. Second tone power was -10 dBm, transmission amplitude is displayed. Bright horizontal line at 7 GHz is one of the higher-frequency resonators.}
\label{fig:first_II_2tone_zoom_fit}
\end{figure}

Interestingly enough, as well as the transition corresponding to the system's resonator, some other narrow horizontal transitions are visible. Three of these lines are visible best of all around 6.1 GHz near the qubit line in \autoref{fig:first_II_2tone}~(left)~($|S^2_{21}|$) and near 7 GHz in the high-resolution area of \autoref{fig:first_II_2tone}~(right)~($|S^2_{21}|$). They correspond to the other resonators which lie higher in frequency. This effect was already observed before but was not thoroughly studied. At this moment it seems that the effect of changed transmission on the probe frequency when the other resonator is resonantly excited with large power (second tone power is 10$^4$-10$^5$ times higher than the probe power) is not due to the coupling of the resonators but due to non-linear effects or suppressed superconductivity in the Al film itself.

\paragraph{AC-Stark effect and multi-photon transitions.} These experiments are more advanced than the previous ones. The results that I've managed to obtain are presented in \autoref{fig:first_II_ac_mp}. The data from the previous experiments were already rather noisy, so the noise could be expected to become more evident with the more subtle experiments. The \autoref{fig:first_II_ac_mp}~(left) shows the result of the AC-Stark shift experiment. The data were obtained not in the sweet spot to increase contrast (see two-tone spectra, near the sweet spot the qubit line is barely visible). According to the plot, the single-photon limit is located near -45 dBm for this sample. The \autoref{fig:first_II_ac_mp}~(right) shows the multi-photon processes emerging from the high driving power (compare with \autoref{fig:transmon_transitions}). Interesting effect can be seen here: there are waves of color intensity in the area where the transmon is strongly driven. By detuning the qubit, I could bring the y-axis into the area near 6.15 GHz where 3 resonators are located. There this wave-ish behaviour was significantly increased and displayed direct connection with 3 visible resonator lines. This means that it is caused by the presence of spurious modes interacting resonantly with the qubit. However, this is not a desired behaviour and was not thoroughly analysed. For \autoref{fig:first_II_ac_mp}~(right) I have chosen the flux point near the sweet spot that was near only a single resonance.


\begin{figure}
\includegraphics[width=0.49\textwidth]{ac-stark}\includegraphics[width=0.49\textwidth]{anharm_probing}
\caption{More advanced experiments with system II \textbf{(Left)} AC-Stark shift. It is observed as expected when the resonator is populated with photons from the 1$^\text{st}$ tone. \textbf{(Right)} Emergence of the $n$-photon peaks from transmon transitions $\ket{E_0^q}\rightarrow\ket{E_n^q}$ as the driving power of the 2$^\text{nd}$ tone is increased. Interesting wave-ish effects due to interaction with spurious modes are noticeable, too.}
\label{fig:first_II_ac_mp}
\end{figure}

\subsubsection{System VI}

\paragraph{Model.} Below all the experimental data will be provided for system VI along with theoretical fits of the spectral lines that were observed using the model \eqref{eq:hamiltonian}. Model parameters were fit based on all data on this system and are same for all figures in this section. Before turning to the comparison of the theoretical predictions and experimental results it would be useful to present the model parameters used for fitting which are summarized in \autoref{tab:first_VI_params}.

\begin{table}[h]
\centering
\begin{tabular}{l|c}
Parameter & Value \\
\hline 
$C_\kappa$ & 0 fF \\
\hline
$C_g$ & 1.8 fF \\
\hline
$C_q$ & 95 fF \\
\hline
$E_C$ & 200 MHz
\end{tabular}~
\begin{tabular}{l|c}
Parameter & Value\\
\hline
$C_r$ & 367 fF \\
\hline
$L_r$ & 1.41 nH \\
\hline
$I_{C, \Sigma}$ & 63.4 nA \\
\hline
$E_{J, \Sigma}$ & 31.5 GHz
\end{tabular}
\caption{Values of the main parameters defining the spectrum of the model \eqref{eq:hamiltonian}.}
\label{tab:first_VI_params}
\end{table}

\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{first_VI_anticrossing_fit} \includegraphics[width=.49\textwidth]{first_VI_2tone_fit}

\caption{\textbf{(Left)} Anticrossing spectrum for the 6$^\text{th}$ system (z-axis normalized) with fitting lines obtained from the model \eqref{eq:hamiltonian}. \textbf{(Right)} Two-tone spectrum of the 6$^\text{th}$ system (z-axis normalized) at -20 dBm on the $\mu$-wave source with fitting lines obtained from the model \eqref{eq:hamiltonian}. As long as the qubit is very close to the resonator in frequency the effective driving power on it is very high, and thus a lot of sideband and multiphoton transitions are visible.}
\label{fig:first_VI_spectroscopy}
\end{figure}

\paragraph{Anticrossing spectrum.} The anticrossing spectrum of the sixth system is presented in \autoref{fig:first_VI_spectroscopy}~(left). It can be directly seen that the qubit is very close in frequency to its resonator; thus, the resonator line is bent slightly and the qubit line is looking ordinary. Using the theoretical model \eqref{eq:hamiltonian} these two lines were fit, and there's a good agreement between theory and data. It can be seen that at the degeneracy point, where the qubit is closest in frequency to the resonator, the resonator line becomes dimmer; it indicates that the qubit is less coherent than the resonator, reducing its Q-factor when approaching resonant interaction. 



\paragraph{Two-tone spectroscopy.} The results of the two-tone spectroscopy are presented in \autoref{fig:first_VI_spectroscopy} (right). It was performed at the lowest possible power of -20 dBm on the $\mu$-wave source, yet, due to a very small detuning of the qubit at the degeneracy point, at this power already the multiphoton transitions have emerged. The lines were more or less accurately fitted, most interesting one was a very vague upside-down sideband in the upper part.

\subsection{Conclusion} 

The sample has lived up to its purposes. It was the first working cQED design made in Russia, and it was enough to demonstrate the full range of the spectroscopic experiments that can be done with single cQED system. However, a lot of noise and obviously low coherence still has to be overcome in the future.

\newpage

\section{On-chip control lines testing design}

\subsection{Geometry and parameters}

The second design that was developed was intended to test the properties of the control lines, i.e. flux bias lines and microwave driving lines, implemented on the chip. The design is presented in \autoref{fig:second_design_full} where some of its key features are highlighted. It's 10x5 mm.

\begin{figure}[h!]
\centering
\includegraphics[width=0.7\textwidth]{chip_design2}
\includegraphics[width=0.7\textwidth]{chip_design2_zoom}
\caption{\textbf{(a)} Large-scale image of the lines testing design. The chip (8x4 mm) consists of eight $\lambda/4$ CPW resonators coupled capacitively to the feedline, six low-Q$_e$ with Xmon qubits at the open end and two high-Q$_e$ with no qubits. \textbf{(b)} Zoomed area around one of the qubits, showing the microwave antenna configuration. \textbf{(c)} Zoomed area around another qubit, showing the flux bias line. \textbf{(d)} Zoom around one of the qubits' SQUIDs. Pink areas denote the part of the mask which produces the SQUID and should be patterned with higher resolution.}
\label{fig:second_design_full}
\end{figure}

Firstly, differently from the previous design, this design has more resonators, additional two (TI, TII, see \autoref{fig:second_design_full}(a)) were inserted at the ends of the main resonator-qubit array (I-VI). These resonators have high Q$_e$ from their geometry to yield accurate results for possibly high internal Q-factors.

Secondly, the frequencies of the devices were changed. The qubits still are all calculated to have the same frequency, but it's now 6 GHz, with $C_\Sigma \approx 80$ fF and $I_{C, \Sigma} = 40$ nA. The size of each junction in the qubit's SQUID is 100$\times$200 nm$^2$. The resonators had frequencies of 7, 7.1, 7.2, 7.3, 7.4, 7.5 GHz for the cQED systems I-VI, and the bare resonators TI and TII had 8 and 8.25 GHz, respectively.

Finally, as the main change made to the design, some coplanar control lines were introduced. For the upper qubits they are microwave antennas, i.e. open-ended coplanar line pieces, to induce transitions while for the lower qubits they are flux-bias lines to change the energy level structure of the devices. The lines were separated in such a way to make the design more fault-tolerant.

Four test structures at the sides of the chip were also included to allow direct DC measurement of the SQUIDs created during the shadow evaporation.


\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{xmon_al_bmstu_1_in_pcb}
\caption{Bonded Xmon Al BMSTU 1 in the PCB of the sample holder, as seen through the microscope of the bonding machine.}
\label{fig:first_tight_fit}
\end{figure}

\subsection{Purposes}


Purposes for the design:
\begin{enumerate}[label=(\alph*), leftmargin=1.5cm]
\itemsep0pt
\item to test Q of resonators without qubits (test resonators added)

\item to test new PCBs

\item to test the on-chip excitation and bias lines

\item to test again the qubits' basic parameters

\item to test the semi-sawing of the substrate before fabrication with following cracking

\end{enumerate}


\subsection{Fabrication}

Before actual fabrication a 15$\times$15 substrate of high-resistivity Si ($> 6$ kOhm/cm) was pre-sawed at MISIS on the back side approximately 150 $\mu$m deep along the borders of two 10$\times$5 chips to allow the substrate to be cracked around the chips precisely. This was done to avoid sawing after the fabrication was done for convenience reasons. The e-beam mask for the chip was prepared in the cleanroom facility of Bauman Moscow State Technical University. The e-beam lithography included both the junctions and the large structures; then Al (25 and 45 nm) was shadow-evaporated at ISSP, so everything on the chip was two-layer. Finally, the residual metallization and resist residuals were removed via lift-off.

The twin chip was lost during the separation of the chips because the process was not yet mastered. But nevertheless, the remaining sample fitted in the PCB cutout really well (see \autoref{fig:first_tight_fit}), and thus the method will be further improved.

\begin{figure}[h!]
\centering
\includegraphics[width=0.9\textwidth]{xmon_al_bmstu_1_general}

\includegraphics[width=0.9\textwidth]{xmon_al_bmstu_1(2)_general}

\includegraphics[width=0.9\textwidth]{xmon_al_bmstu_1(3)_general}


\caption{\textbf{(Top)} General view of the resonances. All six resonances are visible, however shifted down in frequency. This shift more or less consistent throughout the devices, and thus it's most probably caused by effective $\epsilon_{Si}$ different from the one used in the calculation. \textbf{(Middle)} General view of the resonances (2$^\text{nd}$ cooldown). Transmission now lower as a 5 dB attenuator was replaced by a 10 dB one. Seven resonances are visible, as the second resonator presumably coupled to something. The shapes of the other peaks also changed. \textbf{(Bottom)} General view of the resonances (3$^\text{d}$ cooldown).  Transmission is 20 dB higher as the directional coupler was altered. Again, six resonances are visible, though some spurious dip is present between devices III and IV, and the shapes changed again.}
\label{fig:second_resonators_general}
\end{figure}


\begin{figure}[h!]
\centering

\includegraphics[width=\textwidth]{q-factors-and-freqs_xmon_al_bmstu_1}

\caption{Various quality factors and frequencies depending on radiation power. Some resonators show interesting dip in internal Q near -25 dBm and significant change in frequency, implying they are coupled to functioning qubits saturation of which causes the effect. Other resonators do not show such features.}
\label{fig:second_q_factors}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth]{q-factors-and-freqs_xmon_al_bmstu_1(2)}
\caption{Same figure for the 2$^\text{nd}$ cooldown. Significant deviations are present in comparison with the first cooldown, both to the greater and to the lower Q-factors on different devices.}
\label{fig:second_q_factors(2)}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth]{q-factors-and-freqs_xmon_al_bmstu_1(3)}
\caption{Same figure for the 3$^\text{d}$ cooldown. The Q-factors degraded completely both for the cQED resonators and the test resonators.}
\label{fig:second_q_factors(3)}
\end{figure}

\subsection{Characterization of the resonators}

The resonators on the sample were measured several times (after each new cooldown) as they seemed to degrade. Indeed, looking at the Q-factors over the cooldowns reveals that the resonators got worse and worse, even if nothing at all was done with the chip.

Before the second cooldown the chip was moved out of the PCB to solder additional SMP connectors on it, and then moved back in. No mechanical damage or contamination was inflicted to it. However, we can see dramatic changes in quality factors of the resonators, some got better and some got worse for unknown reasons (see \autoref{fig:second_q_factors(2)}). Devices II and III got so bad it was impossible to fit them as their widths compare to the S$_{12}$ general roughness features (see \autoref{fig:second_resonators_general} (Top) vs (Middle)).

After that before the third cooldown nothing at all was done to the chip. After the cooldown and before the Q-factor measurement there was an accidental excessive current through the surrounding coil. Though, it didn't change the general appearance of the resonances which was again found different at this cooldown (see \autoref{fig:second_resonators_general} (Bottom)). At this run the Q-factors became unacceptably low (see \autoref{fig:second_q_factors(3)}) at all powers. 

From the Q-factor plots it is possible to see that systems III and V do not have working qubits in them. Most possibly, they got destroyed in the sample separation process.


\subsection{Characterization of the full cQED systems}

The two systems were possible to study in this sample, I and VI. Others either did not indicate presence of the functional qubits or experience strong flux hopping while tuned via the superconducting coil and don't have the flux bias lines attached to work around this problem. Fortunately, system I has the qubit with the flux bias line and system VI has the microwave antenna, so both these on-chip devices were successfully tested.



\subsubsection{System I}

\paragraph{Spectroscopy.} The system I was biased by the superconducting loop right on the chip. This allows to make wide scans without affecting negatively the resonator with the width limited only by the current source and the heating. The periodic spectrum of the anticrossings for system I is displayed in \autoref{fig:second_I_anti}. The pattern is shifted noticeably to the left due to some residual field around the sample.

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{second-I-anti}
\caption{Periodic anticrossing picture for the system I (at high power). The current on the current source spans 80 mA and is actually comparable at the endpoints to it's maximum possible output of 105 mA.}
\label{fig:second_I_anti}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{second-I-spec-lines}

\includegraphics[width=.95\textwidth]{ge-linewidth}
\caption{Spectroscopy for Xmon Al BMSTU 1, I. \textbf{(Left)} Qubit spectrum at 0 dBm on the output of the 2$^{\text{nd}}$ tone generator. The additional lines below the main one are most probably the $ef$ transition at $\omega_{ge}-E_C$ and $fd$ transition at $\omega_{ge}-2E_C$. \textbf{(Right)} Dependence of the linewidth on the driving power. \textbf{(Bottom)} high-resolution scan of the two-tone peak at -20 dBm.}
\label{fig:second_I_spec_lines}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=0.49\textwidth]{second_I_rabi}\includegraphics[width=0.49\textwidth]{second_I_relax} 
\caption{Time-resolved measurement results for Xmon Al BMSTU 1, I. Blue lines are the experimental data, orange lines are least-squares fits. \textbf{(Left)} Rabi oscillations. \textbf{(Right)} Relaxation. Note that the y-axis values are not exactly the same as in the Rabi oscillations. The amplitude of decay is larger than the amplitude of the Rabi. This is probably caused by a frequency shift of the qubit between the experiments.}
\label{fig:second_I_td}
\end{figure}


The experiment to determine the qubit spectrum and the linewidth at the degeneracy point was also done on the system. The resulting graphs are presented in \autoref{fig:second_I_spec_lines}. On the right plot the spectrum of the qubit is visible; it consists of three lines corresponding to the transitions $ge,\ ef$ and $fd$ and visible due to the elevated temperature near the sample, possibly due to the flux line. These lines are not multi-photon processes, since is was established in another experiment with higher power that they appear in between the lines visible now. 

On the left plot the spectral line width is recorded over the incident power of the drive. It's possible to estimate the linewidth of the bare qubit from the phase 2-tone data (see Section \ref{sec:2tone}). The line experiences significant broadening with increased power; though, at the minimal possible power of -20 dBm it's width is around 0.5 MHz, which allows to set a lower bound of 300 ns on the qubit $T_1$ (assuming that there's no pure dephasing due to the flux fluctuations at the sweet spot and to the excessive population of the resonator as $\delta\omega = \frac{1}{T_1}+\frac{2}{T_\phi}$). If we suppose that pure dephasing and relaxation rates are approximately equal, then we find that $T_1\approx T_\phi \approx 1\ \mu$s.


\paragraph{Time-resolved measurements.} Almost 6 months later, I used this chip to test my first time-domain setup at MIPT. Despite some noticeable problems and noise, I have managed to observe Rabi oscillations, relaxation and Rabi chevrons experiments. The results are presented in \autoref{fig:second_I_td} for the former two; Rabi chevrons are too noisy to show in a work like this. As can bee seen, pretty high $T_1=1.7\ \mu$s was established from the fit compared to the lower bound I have found before. Maybe the driving power was still too high that time. Maybe the pure dephasing that was neglected has played a role in this because the Rabi oscillations that should ideally have a decay constant of $\frac{3}{4}\gamma$ and thus larger $T_R = \frac{4}{3} T_1$ in the absence of dephasing, and we observe only $T_R=1.3\ \mu$s=$\frac{3}{4} T_1$. Thus, pure dephasing was definitely present and could increase the linewidth. Maybe the flux line which was not connected during the time resolved experiments was bringing additional flux noise to the qubit during spectroscopy.


\subsubsection{System VI}

System VI had an on-chip microwave antenna nearby, but had no flux bias line (see \autoref{fig:second_design_full}), and thus had to be biased by the external coil. With the studied sample this method of biasing was very inconvenient, as even small magnetic field from the coil influenced the resonators tremendously, inducing continuous and discontinuous frequency shifts in them, presumably due to flux creep and hopping. Fortunately, is was still possible to tune the qubit of the VI$^{\text{th}}$ system enough to observe it's spectrum and to test the microwave antenna.

\begin{figure}[h!]
\centering
\includegraphics[height=0.25\textheight]{second_VI_anti}\includegraphics[height=0.25\textheight]{second_VI_spec_antenna}

\caption{\textbf{(Left)} Anticrossing of the VI$^{\text{th}}$ cQED system. It can be seen that the picture is not symmetric and  the lines are bent downwards at the ends, as the resonator frequency change was caused not only by the qubit, but also directly by the magnetic field. \textbf{(Right)} Two-tone spectroscopy of the VI$^{\text{th}}$ system using the microwave antenna mounted on the chip.}
\label{fig:second_VI_spectra}
\end{figure}

The anticrossing measured for this system is shown in \autoref{fig:second_VI_spectra}~(left). It was hard to measure, though, as the flux hopping was interfering with the tuning of the qubit. The resonator frequency change due to the magnetic field has bent the picture, so it would be impossible to fit it within the model described before. Nevertheless, the spectrum of this qubit was gathered with 2-tone spectroscopy using the microwave antenna and with a second tone driving the resonator. The results are presented in \autoref{fig:second_VI_spectra}~(right) for the measurement via an antenna and with an additional tone in the feedline driving the resonators at the qubit frequency.

\subsection{Conclusion} 

This chip has also fulfilled the expectations concerning all the targets it was aimed at. Most pleasant change for me was the reduced noise which allowed to perform much cleaner experiments. The on-chip control lines were functional; however, I didn't check for crosstalk, mostly because I only had 2 functional devices on the chip. 

The qubit frequencies were too high compared to the expected values, meaning there's still a need to calibrate oxidation process, since the capacitive parameters look accurate, and thus the $I_C$ is the only parameter that could be wrong.

Flux jumps were very disappointing and persisted in all working systems except for the I and VI, which were more stable, through all the cooldowns. The behaviour of these jumps were so reproducible in different cooldowns, that I started to suspect they were inherent to the sample itself. To test this, I have moved forward to the next design and the new sample.

\newpage

\section{Flux bias design}

\subsection{Design and parameters}

The design parameters of the next chip (called Xmon Al BMSTU S444 2) were exactly the same as for the previous one. The only difference was that I have changed all microwave antennas to flux bias lines to be able to surpass the flux jumps which occur when the coil is used to tune the qubits. However, in the end I didn't use them.

\subsection{Purposes}

This chip was aimed primarily at investigating in more detail the effects that have been observed in the previous devices. However, it also served for development of a time-domain setup at MIPT. So the full list of targets for the chip is as follows:
\begin{itemize}
\item to test flux jumps for similar behaviour as was observed in the previous sample
\item to test resonator Q-factors
\item to test qubit coherence
\item to test the improved attenuation configuration
\item to perform time-domain experiments
\end{itemize}

\subsection{Fabrication}

The fabrication process was exactly the same as for the previous chip.

\subsection{Characterization of the resonators}

\begin{figure}
\includegraphics[width=.95\textwidth]{Xmon Al BMSTU S444 2}

~

\includegraphics[width=\textwidth]{q-factors-and-freqs_xmon al bmstu s444 2}
\caption{Analysis of the resonators on the chip. \textbf{(Top)} General scan of the resonances. It can be seen that the frequency error is not very large and is consistent between all resonators, and with the previous chip results. Second resonator is missing. \textbf{(Bottom)} Quality factors and frequencies. It can be seen that the internal quality of the resonators is quite low, both for the loaded and test resonators. All of the working ones indicate the presence of the qubits, though.}
\label{fig:third_resonators}
\end{figure}

\begin{figure}
\includegraphics[width=\textwidth]{third_anticrossings}
\caption{Anticrossing scans for all the resonators. The scans are either as wide in current as possible without having a flux jump (I, IV and V), or have at least one flux jump in them (II and VI) just for illustration. As can be seen, system I can't be tuned into the sweet spot although the others more or less could possible be. I have extensively studied the system VI which is the most stable among all.}
\label{fig:third_anticrossings}
\end{figure}

The standard data is presented in \autoref{fig:third_resonators}. As usual, a general scan and a power scan of the quality factors are presented. The frequencies of the resonators are all shifted down approximately 6\%, and this is quite the same case as in the previous experiment with the control lines chip. This means that substrate $\epsilon$ should be corrected in the future to get exact frequency match with theory.

The Q-factors are very low compared to the previous sample in the first run, may be because this sample was waiting for loading in the fridge for two weeks in room conditions. Possibly, rather thin two-layer films may display such behaviour. However, in the long run we will switch from this approach to single-layer films, so I did not want to check this particular hypothesis.

From the behaviour of the frequencies and Q-factors, we can see that all the qubits are working fine.


\subsection{Characterization of the full cQED systems}


The standard spectroscopic experiments were conducted with all the resonators to reveal magnetic field dependence, shown in \autoref{fig:third_anticrossings}. The sample was tuned by a coil, the flux lines were not used in that run. All resonators have shown the expected anticrossing behaviour; however, rather violent flux jumps were present, too. As can be seen, I couldn't observe even a single period of the flux dependence in any of the systems. Moreover, the jumps had not been very susceptible to the absolute value of the magnetic field, i.e., they always happened nearly in the same point relatively to the anticrossing pattern. This meant that if I didn't see the qubit sweet spot in the range around zero current without jumps, then I couldn't tune the qubit into it at all even when going far away to large currents.

With this in mind, I decided to check first the system VI, and below there will be an extensive investigation of its parameters.

\subsection{System VI: spectroscopy}


\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{third-VI-2tone} \quad \includegraphics[width=.45\textwidth]{third-VI-2tone_lp}
\caption{Two-tone spectroscopy of the system VI. \textbf{(Left)} High-power scan, $|S_{21}|$. Sidebands and multi-photon processes are clearly visible. \textbf{(Right)} Low-power scan, $\angle S_{21}$. Transition $ef$ and a sideband are still visible besides the main line.}
\label{fig:third-VI-2tone}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.95\textwidth]{third-VI-2tone_powerscan}
\caption{Power scan of the transmon transitions near the sweet spot. Note how the colour of the phase plot is changing from red for the $ge$ transition to blue for the two-photon $gf/2$ and other higher-order lines. This is caused by a different dispersive shift for those transitions and the shape of the resonator phase response.}
\label{fig:third-VI-2tone-powerscan}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=.45\textwidth]{third-VI-linewidth} \quad 
\includegraphics[width=.45\textwidth]{third-VI-ac-stark}
\caption{\textbf{(Left)} Power scan to determine the linewidth of the qubit. Oscillations are due to the magnetic field fluctuations during the scan which was very long (12 hours). The full linewidth at half maximum is around 0.1 MHz for the lowest power, which gives 1.6 $\mu$s for the $T_1$ (without pure dephasing). \textbf{(Right)} AC-Stark shift of the qubit. Single-photon limit is near -30 dBm.}
\label{fig:third-VI-2tone-advanced}
\end{figure}

The results showing the spectral properties of system VI are presented in \autoref{fig:third-VI-2tone}, \autoref{fig:third-VI-2tone-powerscan} and \autoref{fig:third-VI-2tone-advanced}. 

Two-tone spectroscopy has discovered the values for the anharmonicity (around 250 MHz) and the $ge$ transition frequency (7.9 GHz). Near the sweet spot the qubit is now detuned only 900 MHz from the resonator, which has ensured a larger dispersive shift there than in the previous experiments. A large dispersive shift means high visibility of the spectral lines and reduces relative noise, so this was a very nice feature. Looking at the \autoref{fig:third-VI-2tone} one may be able to resolve $gf/2$ and $ef$ transitions separately. This is an important thing to do, as long as confusing these two lines will lead to an incorrect calculation of the aharmonicity.

The next experiment was to watch the emergence of the multiphoton peaks with increasing drive power of the second tone. The results cab be seen in \autoref{fig:third-VI-2tone-powerscan}, amplitude and phase. One interesting feature is the colours of the phase plot. While in the amplitude plot all spectral lines are of the same red color, the phase plot reveals more detail as some lines are blue. This can be attributed to the convolution of the movement of the resonator with its phase response shape. As for multiphoton processes the steady state is different from the one we obtain after a $ge$ transition, the resonator shift is different. The phase is changing non-monotonically during a shift in contrast to the amplitude which can only grow if the initial probing frequency is in resonance. Another feature is the split peak at 7.65 GHz. It was predicted actually by the \autoref{tab:tr_transitions}; the frequency $\omega_{ge}-E_C$ can be obtained from two different transitions. In reality, the frequencies of these two transitions are different due to the approximate nature of the equation \eqref{eq:tr_levels}, so we actually see a doublet there.

Finally, a linewidth scan was done and an AC-Stark experiment was preformed (see \autoref{fig:third-VI-2tone-advanced}). The linewidth scan was very long, approximately 12 hours, and revealed slow oscillations of the magnetic field that tuned the qubit frequency during the scan. These oscillations are clearly visible in \autoref{fig:third-VI-2tone-advanced}~(left). The period is approximately 1.5 hours. Despite this, it was still possible to estimate the linewidth and the corresponding relaxation rate of the qubit: they were around $2\pi\ 0.1$ MHz, so the qubit's $T_1$ should have been around $1.5\ \mu$s presuming the absence of pure dephasing. 

AC-Stark experiment in \autoref{fig:third-VI-2tone-advanced}~(right) has shown the single photon level around -30 dBm on the VNA. It was performed with using the strategy that reduces the averaging with the increasing power. Note how clear is the picture now when the dispersive shift is so large. Both the exponential shift of the qubit frequency and the broadening of the spectral line are visible.


\subsection{System VI: time-resolved results}

This system was the first system with which I could perform all the basic time-domain measurements. The setup I used is exactly the one depicted in \autoref{fig:td_setups}~(b).

\subsubsection{Dispersive shift measurements}


Before turning to the 2D plots representing the read-out state of the qubit in dependence on pulse sequence paramters, I would like to demonstrate how actually this readout looks in reality and what actually happens to the resonance when we excite the qubit. To visualize the dynamical movement of the resonator during, e.g., the Rabi oscillations, it is necessary to perform several identical time-resolved experiments each time changing the readout frequency. This was done for the Rabi oscillations experiment, and the result can be seen in \autoref{fig:third-VI-disp_rabi}. 

\begin{figure}[h!]
\includegraphics[width=.5\textwidth]{third-VI-disp_rabi}\includegraphics[width=.5\textwidth]{third-VI-disp_rabi_long}
\caption{Resonator behaviour during the Rabi oscillations, $|S_{21}|$. Readout duration was 1 $\mu$s. Horizontal cross-sections of this graph will yield standard sinusoidal curves corresponding to the oscillations of level population. Note how the dark area caused by the resonator asymmetry is not moving.}
\label{fig:third-VI-disp_rabi}
\end{figure}

This plot does not fully reveal the big picture, as long as there is a definite dependence of the readout duration that I haven't investigated. There also is an interesting unexplained feature that the dark area which is actually the resonator's asymmetry (can be seen for resonator VI in \autoref{fig:third_resonators}~(top)) does not show any significant movement during the oscillations. It seems like only the internal part of the resonator is moving. May be this effect is dependent on the readout duration -- subject for future investigation.


\subsubsection{Basic experiments}

\begin{figure}
(a)\includegraphics[width=0.45\textwidth]{VI-rabi-real}
(b)\includegraphics[width=0.45\textwidth]{VI-decay-real}

(c)\includegraphics[width=0.45\textwidth]{VI-ramsey-real}
(d)\includegraphics[width=0.45\textwidth]{VI-hahn-echo-real}
\caption{Result of the basic time-domain experiments performed with system IV. Circles are experimental data, solid lines are theoretical fits. \textbf{(a)} Rabi oscillations. Rabi frequency of 6.1 MHz was obtained, meaning that $(\pi)^x$-pulse had $\approx$80 ns duration. \textbf{(b)} Relaxation. $T_1$ of 1.6 $\mu$s was calculated from the exponential fit. \textbf{(c)} Ramsey oscillations. Free induction decay time $T_2^*$ was 1.4 $\mu$s. Note the 5.00 MHz oscillation frequency. \textbf{(d)} Hahn echo (see Appendix A for details). Improved decay time $T_{2E}$ = 2.2 $\mu$s means that there was low-frequency correlated noise present in the system.}
\label{fig:td_basic}
\end{figure}

The results of the basic experiments (Rabi oscillations, relaxation, Ramsey oscillations and Hahn echo) are presented in \autoref{fig:td_basic}. First, the Rabi oscillations experiment was performed. The frequency was at first roughly chosen based on the spectroscopic data. Then, with estimated $(\pi)^x$ pulse the relaxation experiment was done (it is not sensitive to the detuning). Then, with estimated $(\pi/2)^x$ duration, the Ramsey experiment was done. The radiation frequency was chosen to be $\Delta \omega$=5 MHz detuned from the suggested qubit frequency; next, from the fit the real frequency $\Omega_{Ramsey}$ of the Ramsey oscillations was extracted and compared to this value. As long as it was different, the suggested qubit frequency was adjusted by the difference $\Delta \omega -\Omega_{Ramsey}$ until the extracted real qubit-radiation detuning became 5.00 MHz, which means less than 10 kHz error in qubit frequency estimation.

With exactly known frequency, the Rabi oscillations experiment was repeated to get more accurate $(\pi/2)^x$ pulses. With those pulses all the pictures in \autoref{fig:td_basic} were obtained.

Additionally, 2D scans were made, shown in \autoref{fig:third-VI-chevrons-fringes}. For such experiments, it is very advantageous to tune the sequences first and then tune the frequency in a double loop, because loading the sequences into the AWGs is much slower than changing the qubit LO frequency. One can see how the Ramsey oscillations slowly turn into the exponential decay when the drive becomes resonant. The pattern that is clearly seen has the shape of $1/|\omega-\omega_{ge}|$, as the period of the oscillations is inversely proportional to the detuning.

\begin{figure}
\includegraphics[width=.5\textwidth]{third-VI-chevrons.pdf}
\includegraphics[width=.5\textwidth]{third-VI-fringes.pdf}
\caption{Rabi chevrons (left) and Ramsey fringes, $\mathfrak{Re}[S_{21}]$. Standard experiments showing the influence of the detuning on the sequences. For the fringes, $(\pi/2)^x$-pulses were calibrated with Rabi oscillations on resonance.}
\label{fig:third-VI-chevrons-fringes}
\end{figure}

\subsubsection{APE}

\begin{figure}
\centering
\includegraphics[width=.75\textwidth]{APE}

\vspace{0.5cm}
\includegraphics[width=\textwidth]{APE_res}
\caption{\textbf{(Top)} The APE sequence. A standard Ramsey experiment with added Ramsey angle is modified by adding $n$ pairs of in ideal case mutually annihilating pulses ($I'$-pulses). In reality these pulses are introducing and amplifying phase error from a single $(\pi/2)^{\pm x}$-rotation. \textbf{(Bottom)} Result for the Gaussian and rectangular pulse windows. As can be seen from the right plots, Gaussian modulation for some reason becomes worse in a non-linear fashion, which is not expected. At low number of pulses it is better then the rectangular modulation, but after 8 pseudo-identity pulses it starts to lose. Taking into account that the Gaussian pulses are actually longer than the rectangular ones, this should be further investigated.}
\label{fig:APE}
\end{figure}

All the pulse sequences that I've used in the previous experiments were rectangular. These are not the best pulses to use with transmons, as long as they have wide spectrum which can at low durations excite the transitions out of the computational basis. This problem was already mentioned in the theoretical section. Gaussian pulses are believed to better suited for accurate manipulation of the transmons. To compare experimentally the two types of pulses, I have implemented a so-called Amplified Phase Error sequence\cite{Lucero2010}. The schematic of the sequence is presented in \autoref{fig:APE}~(top). It is based on an angular Ramsey sequence which is the same as the simple Ramsey sequence but with the second $(\pi/2)^\theta$-rotation made not around $x$-axis but around an axis $\theta$ degrees from the $x$-axis. Between the Ramsey pulses the so-called \textit{pseudo-identity} pulses are integrated. The number of such pulses determines the order of the sequence. The pseudo-identity pulse $I'$ is just simply a $(\pi/2)^x, (\pi/2)^{-x}$ operation which rotates the qubit state  $\pi/2$ up and back again around the $x$-axis. Ideally, such operation would just cancel itself. However, a non-ideal operation will amplify its phase error.\cite{Lucero2010} This error is then detected by the final Ramsey pulse by making it effectively around $\theta + \epsilon$-degrees-rotated axis.

In  \autoref{fig:APE}~(top) the result of such sequence implemented on the qubit VI. As can be seen, the shorter rectangular pulses are still winning against the longer Gaussian pulses. This was not expected. The Gaussian sequence tends to have a non-linearly growing error, which is also not expected. The padding of the pulses (i.e., the waiting periods between them) seem to play a role in errors, too. In the MIPT lab, though, we don't yet have the equipment capable of resolving such short modulated pulses to have a look at their possible overlay with each other. This situation will be further investigated.

However, with rectangular pulses the error is not very large (approximately 1.5 degrees per gate at 10$\times I'$). Moreover, this experiment has shown our ability to control qubits during the long sequences and to control the qubit's phase. Finally, APE is a great instrument on the way to pulse optimization, and will help to improve the Gaussian and DRAG/HD pulses in the future.

\subsection{Conclusion}

The chip analysed in this section has turned out to be extremely fruitful. Due to the large dispersive shift it had in the system VI, it made possible not only clean spectroscopic experiments, but also advanced time-resolved investigation of itself. Rather high coherence times allowed performing interesting and useful experiments such as the Hahn echo and the APE. 

Flux hopping is still an issue, but now we more and more attribute it to the lack of superconducting magnetic shielding of the sample holders and large diameters of the Cryoperm shields and not to the samples.

\chapter{Summary}

During two years of work on my Master's degree I've done a tremendous amount of theoretical and experimental work. Here I would like to list things that were done in the chronological order just to recall everything again and not to forget:

\begin{itemize}

\item 2015 Summer - Internship at Karlsruhe Institute of Technology.
\subitem Learned how to simulate EM-structures, learned about the developments on the qubit coherence, dielectric loss,  materials etc. Learned how to create designs using macro programming. 
\subitem Simulated microwave coplanar resonators. Implemented a design with them. 
\subitem Simulated numerically a desired cQED system with a transmon qubit. Implemented it in a design based on the most advanced planar transmons named Xmons. 
\subitem Simulated 1/f noise and its influence on the dephasing of a qubit.
\item 2015 Autumn
\subitem Participated in building the Artificial Quantum System laboratory at MIPT. Learned how to operate BlueFors fridges.
\subitem First Nb resonators on oxidized Si from my design fabricated at MISIS. Measured at 4 K.
\subitem Investigation of the sample with 20 qubits in one resonator at MISIS. Later, based on these experiments a publication was made in JETP Letters, but not included here in this thesis.
\item 2016-2017 Winter
\subitem I became unofficially assigned by A. Ustinov as as responsible person for the lab of Russian Quantum Center at Institute of Solid State Physics.
\subitem First Nb resonators were measured there and then fitted with \textit{circlefit}, never have been done before in Russia.
\subitem Improvement of the setup at ISSP. Using a hybrid coupler and a 2-channel switch to improve the speed of experiments.
\subitem Wrote the code for more advanced measurements, improved the library left from my Bachelor works.
\item 2016 Spring
\subitem First aluminium resonators fabricated by O. Bolgar and J. Zotova measured at ISSP.
\subitem Visited the cleanroom at MISIS, learned from V. Chichkov about the fabrication and gave him information on the methods used to obtain high quality factors. Together we started systematic work on the improving the Q-factors.
\subitem Mastered the circular saw as MISIS for wafer cutting. It was used afterwards all the time, a very useful skill.
\subitem Finally, the qubits from my design were fabricated by D. Gusenkova (the proof-of-concept design).
\subitem Developed design for new PCBs which we had a strong shortage of, ordered them from a Russian commercial manufacturer named Rezonit.
\subitem Chosen and ordered microwave cables and connectors for our labs.
\item 2016 Summer
\subitem Measured the first working cQED systems in Russia on the proof-of-concept design. Measured parameters were in an excellent match with the theoretically calculated and predicted ones. More advanced spectroscopic experiments that I've ever done were conducted with this chip.
\subitem Participated in beginning of the collaboration with a superb cleanroom facility of Bauman Moscow State Technical university. Discussed the future prospects and fabrication method with the director of the facility I. Rodionov. Visited their cleanroom, learned about the capabilities of the equipment.
\subitem Supplied them with designs of cQED systems and bare resonators for e-beam lithography and mask forming.
\subitem Improved the cQED designs and the resonator designs. 
\subitem Still investigated the quality factors of Al and Nb resonators made at MIPT and MISIS, correspondingly.
\subitem Started working on the project of the Russian Foundation for Promising Developments on building a two-qubit quantum processor demonstrator.
\item  2016 Autumn
\subitem Continuation of the work on studying the microwave resonators. Some advances were achieved with both Nb and Al structures, however Nb was superior due to the dry-etching process and substrate cleaning, internal Q-factors at single-photon level of 4$\cdot 10^{5}$ were reached. First resonators from BMSTU were measured, too.
\subitem Further improvement of the designs; increased the number of resonators on a chip to improve statistical accuracy.
\subitem First qubit sample from the BMSTU was fabricated from a new design with on-chip control lines and measured in my lab at ISSP. The control lines were successfully tested there, too. 
\item  2016-2017 Winter
\subitem Participated in the preparation and 	delivery of the control experiment on superconducting resonators, done simulations and created specific designs to fulfil the committee requirements.
\subitem Invaded the Artificial Quantum Systems lab, improved the setup and wiring, brought new sample holders created at ISSP, developed separate PCBs for the old English sample holders they had. 
\subitem Began to build the time-domain setup there using the equipment I didn't have at ISSP at that moment. Wrote the code for fast and robust microwave mixer calibration for SSBSC excitation and readout.
\item 2017 Spring
\subitem Finished the time-domain setup for the dispersive readout at MIPT. Tested the fully-heterodyne setup, changed to the semi-heterodyne setup.
\subitem Measured two samples one old and one new spectroscopically and in time domain. The results were presented here, and are great.
\subitem Wrote a universal object-oriented library  in Python for qubit measurements to exploit inheritance and polymorphism in order to avoid redundancy and to fully capture the experimental logic in the code. Implemented a universal class logic for data storage that should be extendible to any time of experimental data in our domain.
\end{itemize}

Most of my work materials are available online \href{https://github.com/vdrhtc}{here \footnotesize{\faExternalLink}} on GitHub. Some memories, of course, got lost in time like tears... in rain. However, now I hope I won't forget the most important events and achievements of this wonderful period of my life.

And of course I would like to thank the people I have been working with. If not for such a wonderful community that gathered in our domain here, I think I would have already been working alone in the basement of some lab in Europe.

First of all I want to thank our supervisors Alexey Ustinov, Oleg Astafiev and Valeriy Ryazanov for providing again and again such amazing opportunities for all of us to work here in Russia on the subject that we love.

From the MISIS team I would like to thank Kirill Shulga, Ilia Besedin and Nikolay Abramov for the constant flow of new ideas and explanations for the things I wouldn't otherwise be able to understand... and of course for those ridiculous lunch meetings, saw room crowding and the too-loud-laughs (sorry Dasha, thank you very much for the first working qubit sample!). Here also goes a special thank you to Vladimir Chichkov: without your samples we would just have nothing to do! I also appreciate the thoroughness of your approach and admire the results that you have achieved with the fabrication at MISIS. Looking forward to what in the world you would do with the new Plassys! (when it is finally here, of course)

From the MIPT team I would like to thank Alexey Dmitriev for the nice talks, help with the equipment, with the experiments and with the code! And surely I will not forget to thank you for the rubber band madness, mayhem and destruction before going to eat some wok. Fundamental \textit{\textbf{COHERENT}} qubits forever! Next thanks goes to Julia Zotova and Oleksi Bolgar for the resonator samples. Finally, I would like to thank Alexander Korenkov for pointing out my mistakes and making me accept them. This can be hard sometimes, but definitely necessary; I think our collaboration is very useful for our community on the whole.

From the RQC team, first of all, I would like to thank my great and terrifying boss Ivan Khrapach for the fabrication insights, and help, and design advices, and in general such a friendly attitude concerning the questions, suggestions or problems that I had. Thank you for the samples, too! I would also like to thank Vladimir Milchakov for the help with the experiments, protecting the lab from intruders while I was gone and a very nice general vibe of travelling, distant places and crazy adventures.

Finally, from the KIT team I would like to thank Alexander Bilmes for incredibly thoughtful help with all the organizational questions, very fruitful scientific discussions and for the nice activities you have invited me to! I also would like to thank Jürgen Lisenfeld for the excursions in the lab, and for your very atmospheric BBQ party you have invited me to during my last stay, and for the bike, too!




\appendix
\renewcommand*\thesection{\Alph{chapter}.\arabic{section}}

\input{report_pure_dephasing}

\input{report_efficient_num_method}


\chapter{Design of the PCBs}

\begin{figure}[h!]
\includegraphics[width=\textwidth]{pcb_full}
\caption{Most advanced version of the PCB design for different ship sizes and shapes.}
\end{figure}

PCBs were generated by a macro code I wrote in the LayoutEditor EDA software. For a standard PCB we use, four layers are needed: top, bottom, board and via. Top and bottom are the simplest layers; they depict where the substrate should be and should not be metallized. They can be drawn using Gerber polygons. The via layer may also be drawn with polygons. Via holes are denoted as circles in a separate layer. The board layer should be drawn as a path and should go right at the edge of the substrate. It specifies where the openings will be in the PCB for screws. If the metallization covers the hole, its edges will be metallized, too.


\chapter{Investigation of the quality factors of the CPW resonators}

This subject I think deserves an additional thesis for itself, so I only will limit myself to putting the final result of it here in the end of this document. What you see in \autoref{fig:q_distributuions} is a box plot showing the distributions of the internal Q-factors at low and high powers, and the external Q-factors gathered from all the resonators from all the samples made of Al and Nb that I've ever analysed. Additionally, I had some NbN chips, but they are not included there. 

Each of the samples was prepared, installed in the sample holder, bonded, cooled down and measured (mostly this was done in ISSP by me). Then the data was fitted as usually and the Q-factors were plotted. Next, badly fitted or just bad resonances were thoroughly filtered out to obtain the resulting graph.

As you can see, the Nb samples are more numerous that their Al brothers, an this is in part the reason why they are so much better: the more work you do, the better the result. Besides, the Nb samples just have a superior process which includes deposition on a thoroughly cleaned and annealed Si substrate. Al samples cannot achieve the same quality of the interface because they all are created in a liftoff process where the Al is evaporated though an organic mask.

In external Q-factors plot it's possible to see how the design has changed for the Nb resonators from 6000 $Q_e$ to $10^5\ Q_e$. This was done to keep up with growing internal Qs to maintain the same fitting accuracy as before.

An important note is that among the Al samples some of the chips contained not bare resonators, but resonators with qubits.

\begin{figure}
\includegraphics[width=\textwidth]{Q_distributions.pdf}
\caption{Box plot of the Q-factor distributions in all the resonator samples studied by me during my Master's work.}
\label{fig:q_distributuions}
\end{figure}


\bibliographystyle{ugost2008}
\bibliography{report.bib}
\end{document}