\section{Introduction}

In quantum information science, solid state approaches are attractive as they are easily scalable.
The coherent transport of information in such systems is usually coded into the degrees of freedom of the electron, either spin \cite{delft-esr} or charge \cite{hayashi,petta_10}.
The advantage of the spin degree of freedom lies in the fact that it is well protected from the electrostatic environment whereas the charge degree of freedom is easily measurable \cite{field}.
Over the last decade, a variety of interesting electronic devices have been proposed and tested, such as Fabry-Perot  \cite{camino_05,willet_pnas_09,halperin_11} or Mach-Zehnder interferometers in the quantum Hall regime \cite{heiblum_03, samuelsson_04,neder_07,preden_08}.
In particular, these quantum Hall systems are particularly appealing as they could be operated as flying qubits, where the quantum information can be manipulated during flight, due to the absence of backscattering.
The fact that now an individual electron charge can be controlled and manipulated in such systems, opens the possibility to perform electron quantum optics experiments at the single electron level \cite{feve_ses_07,hermelin_nature_11,mc-nail_nature_11,feve_science-ex_13,glattli_nature_13}.
In a recent experiment by Yamamoto et al. \cite{michi_nnano_12}, the charge degree of freedom has been exploited to gain full electrical control of a flying qubit.
The system is composed of an Aharonov-Bohm (AB) ring connected to two tunnel-coupled wires at each side, which can act as beam splitters for the ballistic electrons.
In this case, the system behaves like a Mach-Zehnder interferometer for ballistic electrons \cite{heiblum_03,preden_08}.
Compared to quantum Hall systems, this system is more easily scalable and no magnetic field is in principle necessary to operate the device.
It is made from a semiconductor heterostructure and electrostatic surface gates define the borders of the interferometer by locally depleting the two-dimensional electron gas while the two middle gates allow to tune the tunnel-coupling of the two wires at the entrance and exit of the AB ring (see Fig.\ \ref{fig:sample}) \cite{wieck_apl,d-coupler}.
In the original experiment, several interesting experimental features have been observed.
By applying a relatively small negative gate voltage on the right tunnel gate $V_{t}$ and connecting the left tunnel gate to ground, only the middle island will be depleted, but essentially no tunnel barrier will be formed in the tunnel-coupled wire region.
In this case the system behaves as a two-terminal device and the currents $I_{u}$ and $I_{d}$ are identical. When sweeping the gate voltage more negative, the tunnel barrier between the upper and lower channel can be tuned.
For a sufficiently negative voltage the \emph{in-phase} oscillations of $I_{u}$ and $I_{d}$ turn surprisingly into \emph{anti-phase} oscillations.
Other interesting observations have been made such as the possibility to control the partition between the two output currents by using the tunnel-coupling gates.
It has also been shown that this interferometer does not suffer from backscattered electrons which encircle the AB loop and hence allows to perform reliable phase shift measurements \cite{shintaro_13}.
In this article, we address all the experimental findings of ref.\cite{michi_nnano_12}  by means of a simplified theoretical model that can account for several features observed in the experiments. In order to capture the more subtle features we perform numerical simulations and confront them with the experiment. Our simulations show that the majority of the experimentally observed features can be well explained within the Landauer-B\"uttiker scattering formalism \cite{landauer_IBM,landauer_PM,buttiker_multi-channel,buttiker_4-terminal}.
\section{Summary of the main experimental features}
The flying qubit sample is tailored within a two-dimensional electron gas (2DEG) of density $n_s= 3.2 \cdot 10^{11} \mathrm{cm^{-2}}$ and a mobility $\mu=0.86 \cdot10^6 \mathrm{cm^2/Vs}$ made from a GaAs/AlGaAs heterostructure by metallic (Ti/Au) surface gates (see Fig.\ \ref{fig:sample}).
The 2DEG is situated 90 nm below the surface, hence the electrostatic potential applied to the surface gates leads to a smooth potential change in the 2DEG over roughly the same distance.
For the measurements, a bias voltage is applied to the lower left contact (the upper left contact is floating) and the current is measured simultaneously in the right upper and lower contact.
To observe Aharonov-Bohm oscillations, the magnetic field is swept over a field range of approximately $\pm$\,80 mT. Above this field one suffers from Shubnikov-de-Haas oscillations.
\begin{figure}[htbp]
	\includegraphics[width=7cm]{sample-grenoble.png}
	\caption{\label{fig:sample} Scanning electron microscope image of the flying qubit sample. The outer metallic gates (light grey) define the borders of the interferometer while the right tunnel gate (blue) allows for depletion of the centre AB island as well as for adjustment of the tunnel barrier of the split-wire. Changing the voltage of the side gate (red) allows to induce a phase shift in the lower branch (see text).  White squares represent the ohmic contacts.  }
\end{figure}
As the right tunnel gate voltage $V_T$ is switched on, the sample goes through a series of different regimes. Initially, the Aharonov-Bohm loop does not exist and one only observes universal conductance fluctuations.
As $V_T$ is swept more negative, the AB region gets more depleted, yet the gate does not affect much the tunnel-coupled region. We refer to this regime as the Strong Coupling Regime (SCR).
Upon further increasing negative $V_T$, one enters the regime of main interest:
the Weak Coupling Regime (WCR) where the 2DEG is also partly depleted underneath the tunnel gate. In this regime, there is a finite coupling between the up and down channels in the wire region. Finally, upon further increase of $V_T$, one enters a regime where the upper and lower channels are decoupled.
Below we list the different experimental features when either scanning the tunnel gate through the different regimes or by scanning the side gate voltage of $V_s$ (see Fig.\ \ref{fig:sample}) and which we attempt to reproduce with analytical as well as numerical approaches.
\begin{table}[h!]
  \begin{tabular}{ ll }
    \multicolumn{2}{l}{\ }\\
    $\bullet$ (P1a)\  & \ \textbf{Magnetic field sweep in SCR:} \\
                      & \ \ \ $I_{u}$ and $I_{d}$ show almost identical  in-phase\\
                      & \ \ \ oscillations with magnetic field \\
    $\bullet$ (P1b)\  & \ \textbf{Magnetic field sweep in WCR} \\
    							    & \ \ \ $I_{u}$ and $I_{d}$ show anti-phase oscillations \\
  	  					  	  & \ \ \ with magnetic field \\
    $\bullet$ (P2a)\  & \ \textbf{Side gate sweep in SCR:} \\
    					  			& \ \ \ AB oscillations show phase jumps for \\
  	  			  				& \ \ \ $I_{u}(B)$ and $I_{d}(B)$  \\
    $\bullet$ (P2b)\  & \ \textbf{Side gate sweep in WCR:}  \\
    			  					& \ \ \ AB oscillations show a smooth phase \\
  	  	  						& \ \ \ shift for $I_{u}$ and $I_{d}$  \\
    $\bullet$ \ (P3)\ & \ \textbf{Anti-phase oscillations in WCR:}\\
                      & \ \ \ $I_{u}$ and $I_{d}$ show oscillations with respect \\
                      & \ \ \ to the tunnel gate voltage $V_T$\\

  \end{tabular}
\end{table}

\section{A minimum (scattering) theory of the flying qubit.}

In this section we develop a minimum scattering approach which captures the main features of the experiment. In Fig.\ \ref{fig:cartoon} we show a sketch of the actual device used in Ref.\ \onlinecite{michi_nnano_12}.
Here, the two left contacts of the original device have been replaced by a single contact.
This is a simplification which has no effect on the experimentally observed results.
As we show below, the flying qubit can be manipulated even in such a three-terminal configuration \cite{wieck_apl,d-coupler}.
The difference from the four-terminal device only appears as a shift of the AB oscillation phase by $\pi$/2.

The device consists of several distinct regions: the injecting region on the left, the central Aharonov-Bohm region and the tunnel-coupled wire on the right which are modelled by their respective scattering matrices, $S_{inj}$, $S_{ab}$ and $S_{tw}$.
In this section, we assume for simplicity that a single channel, labeled \textbf{u}p and \textbf{d}own, is propagating inside each arm of the interferometer (this assumption will be relaxed in the numerics performed on the microscopic model).

\begin{figure}[htbp]
	\includegraphics[width=8.5cm]{cartoon}
	\caption{\label{fig:cartoon} Schematic of the system used for the modelling. We divide the sample into several regions: the injection region on the left, the central Aharonov-Bohm region  and the tunnel-coupled wire on the right. The tunnel-coupled wire can further be split into three regions I, II and III:  the two wires are coupled only in region II and decoupled upon entering in region I or III.  a), b) correspond to two possible backscattered electron trajectories that could, in principle, contribute to the reflection amplitude. }
\end{figure}

A general property of a 2DEG is the smoothness of the electrostatic potential generated by the surface gates felt by the electrons in the 2DEG as it is situated approximately 90 nm below the surface.
Here we assume that the scattering is mostly forward, which is valid in a specific configuration. Suppose the width of each wire is kept unchanged between the ring and coupled-wire region, the potential change $\Delta V$  of each wire at this transition region is simply defined by the tunnel coupling. For small $\Delta V$ (smooth potential change), the length scale of the tunnelling $\hbar \,v_x / \Delta V $ becomes much larger than that of the potential change at the transition region, which suppresses tunnel-backscattering into the other arm. Since the potential is also smooth with respect to the Fermi wave length, intra-wire backscattering is also suppressed.

\subsection{Generalities}
Let us first discuss the structure of the different scattering matrices for the three distinct regions. $S_{inj}$ is rather arbitrary. It is characterised by the amplitude for an electron injected from the left to be transmitted into the upper $a_u$ and lower $a_d$ channel (which, we assume, does not depend on the injecting channel).
The probability $|S_{bs}|^2$ for an electron to be backscattered into the injecting electrode is obtained from the current conservation law $|S_{bs}|^2+|a_u|^2+|a_d|^2=1$.
The Aharonov-Bohm region is translation invariant along $x$, and therefore, no backscattering occurs there and the upper/lower electrons are simply transmitted into their respective arms.
The down transmission amplitude picks up an Aharonov-Bohm phase $e^{i 2\pi \Phi/\Phi_0}$ with respect to the upper one where $\Phi_0=h/e$ is the flux quantum and $\Phi= S B$ is the magnetic flux through the Aharonov-Bohm region ($B$: magnetic field, $S$: effective surface of the AB ring).
Next we describe the tunnel-coupled wire region.
If the variation of the lateral confinement potential at the transition region between the AB ring and the coupled wire is smooth, the electrons can only be transmitted and ring encircling paths due to backscattering can be ignored. Once the electrons are scattered into the tunnel-coupled wire, the corresponding transmission matrix can be parametrised as
\begin{equation}
t_{tw}=
\left(\begin{array}{cc} t_{uu} & t_{ud} \\ t_{du} & t_{dd} \end{array} \right)
\end{equation}
with $t_{tw}t_{tw}^\dagger=1$ (current conservation).
Summing up all the probability amplitudes for the different paths taken by the electrons, we arrive at a Mach-Zehnder like expression
\begin{equation}
t_{u}=  t_{uu} a_u +  e^{i 2\pi \Phi/\Phi_0} t_{ud} a_d
\end{equation}
\begin{equation}
t_{d}=  t_{du} a_u +  e^{i 2\pi \Phi/\Phi_0} t_{dd} a_d
\end{equation}
where $t_{u/d}$ is the total amplitude for an electron injected from the left to be transmitted in the upper/lower right electrode. The corresponding currents are given by the Landauer formula\cite{landauer_IBM,landauer_PM},
\begin{equation}
\frac{dI_{u/d}}{dV_b}=\frac{2e^2}{h}  T_{u/d}.
\end{equation}
Here $T_{u/d}=|t_{u/d}|^2$ is the total transmission probability from the left to the right upper/lower electrode.

Before going further, let us discuss the conclusions that can be already drawn at this stage. First, properties (P1a) and (P2a) are rather natural: in the strong coupling limit, the system is essentially a {\it two-terminal}
Aharonov-Bohm device where forward scattering in the wire makes the upper and lower current homogeneous (P1a).
Onsager symmetry imposes that $G(B)=G(-B)$ which leads to phase rigidity \cite{buttiker_phase_rigidity_1,buttiker_phase_rigidity_2} as the phase of the conductance can only take multiples of $\pi$ at zero magnetic field, as observed in many experiments \cite{benoit-webb_prl86,benoit+hasselbach_prl_97}.
Current-conservation imposes that the injected current $I_{inj}$ = $I_{bs} + I_u + I_d$, hence property (P1b) simply translates into $I_{bs}$ being {\it independent} on $B$.
From the sample geometry we can also make some assumptions about the electron trajectories that dominantly contribute to the transport properties.
Figure \ref{fig:cartoon} shows two different backscattered trajectories that could potentially
contribute to the electronic transport. In particular, in order to observe an Aharonov-Bohm effect on $I_{bs}$, one needs both trajectories.
We have argued, however, that trajectory b) can be neglected due to the smoothness of the confining potential that prevents backscattering at the interface between the Aharonov-Bohm and tunnel wire region, hence only a) contributes to the reflection amplitude and $I_{bs}$ is essentially independent of the magnetic field (P1b).
Property (P2b) is also due to the absence of the backscattering trajectory b). It straightforwardly leads to a realization of the two-slit experiment once the electron is injected into the AB ring.

\subsection{Scattering matrix of the tunnel-coupled wire}

Let us now focus on the tunnelling region and compute the transmission matrix $t_{tw}$.
As seen from Fig.\ \ref{fig:cartoon}, at the entrance and exit of tunnel-coupled wire (region I and III), the tunnel barrier is infinitively high and, as a consequence, the two separated wires are fully decoupled, whereas in the central region (region II), the coupling is finite.
We suppose that the transition between the regions happens smoothly.

The eigenstate in region II for mode $\alpha$ takes the form,
\begin{equation}
\Psi_\alpha (x,y)=\Psi_\alpha (y) e^{ik_\alpha x}
\end{equation}
and we consider the situation where only two modes can propagate in the wire, hereafter labeled  the symmetric $\ket {S_{II}}$ and anti-symmetric $\ket {A_{II}}$ mode. Indeed, $\Psi_\alpha (y)$ corresponds to the solution of the 1D Schr\"odinger equation along the transverse direction $y$,
\begin{equation}
[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial y^2} +V(y)]\Psi_\alpha (y)=E \Psi_\alpha (y)
\label{s-equation}
\end{equation}
with $E= E_F -\frac{\hbar^2}{2m}k_\alpha^2$ and its two solutions in the absence of the tunnelling gate are respectively a symmetric and anti-symmetric function of $y$.
The actual wave function of these two eigenstates of the tunnel-coupled region are displayed in Fig.\ \ref{fig:S+AS}. The discussion of the evolution of these states when changing the tunnel barrier height is postponed to the next subsection where we treat this issue semi-analytically.

Upon going from region I to II, $\ket {A_{II}}$ is essentially unaffected (the weight of $\Psi_{A_{II}} (y)$
in this tunnelling region is very small so the gate hardly affects this mode) while $\ket{S_{I}}$ is smoothly transformed into $\ket {S_{II}}$ whose wave function is essentially $\Psi_{S_{I}} (y)=|\Psi_{A_{I}} (y)|$.
The two modes $\ket {S_I}$ and $\ket {A_I}$ are degenerate and can also be rewritten as combinations of the
modes that propagate in the upper ($\ket{\uparrow}$) or lower ($\ket{\downarrow}$) parts:
\begin{equation}
\ket{S_I}=\ket{\uparrow}+\ket{\downarrow} ; \ket{A_I}=\ket{\uparrow}-\ket{\downarrow}
\end{equation}
 The transmission matrix $t_{tw}$ can now be obtained by the following adiabatic argument: let us start with an electron in mode $\ket{\uparrow}=(\ket{S_I}+\ket{A_I})/2$. In the beginning of region II, the wave function has smoothly evolved into $(\ket{S_{II}}+\ket{A_{II}})/2$. Towards the end of region II, the wave function has picked up mode-dependent phases and reads $(e^{ik_S L}\ket{S_{II}}+e^{ik_A L}\ket{A_{II}})/2$ where $L$ is the total length of region II. Then, the wave function is smoothly transformed into $(e^{ik_S L}\ket{S_{III}}+e^{ik_A L}\ket{A_{III}})/2$ which can be re-expressed as $(e^{ik_S L} [\ket{\uparrow}+\ket{\downarrow}]+
e^{ik_A L} [\ket{\uparrow}-\ket{\downarrow}])/2$. We can directly read from this expression the amplitude to be transmitted in the up ($(e^{ik_S L}+e^{ik_A L})/2$) and down ($(e^{ik_S L}-e^{ik_A L})/2$) channel. Repeating the procedure for  spin down we arrive at,
\begin{equation}
t_{tw}=
\exp(i\frac{k_S+k_A}{2}L) \left(\begin{array}{cc} \cos(\frac{k_A-k_S}{2}L)& i\sin(\frac{k_S-k_A}{2}L) \\  i\sin(\frac{k_S-k_A}{2}L) & \cos(\frac{k_A-k_S}{2}L) \end{array} \right)
\end{equation}

Putting everything together, the Landauer formula finally provides
\begin{align}
\label{eq:simple}
\frac{dI_{u/d}}{dV_b} &=\frac{2e^2}{h}
\big[
\frac{|a_u|^2 + |a_d|^2}{2}
\pm \frac{|a_u|^2 - |a_d|^2}{2} \cos [(k_A-k_S)L]  \nonumber \\
& \pm |a_u a_d| \sin [(k_A-k_S)L] \cos \left( 2\pi \Phi/\Phi_0 +\phi \right) \big]
\end{align}

Equation \ref{eq:simple} now provides a general analytical description of the interferometer.
Changing the amplitudes $a_u$ and $a_d$ allows to control the symmetry of the injected wave function.
When injecting into one arm only, the last term of equation \ref{eq:simple}  cancels and the system reduces to a simple split-wire.
This simple analysis shows that the currents in the upper and lower branches oscillate anti-phase as a function of $\Delta k=k_A-k_S$. Varying $\Delta k$ is equivalent to changing the tunnel gate voltage and hence explains the experimentally observed oscillations with respect to $V_T$ [property (P3)]. In a similar way, for a given $\Delta k$ the two output currents have opposite sign and will also oscillate anti-phase as a function of magnetic field (P1b).

\subsection{Semi-analytical analysis}

In this subsection we would like to get some more physical insight into the experimental system by calculating the precise dependence of $\Delta k$ on the tunnel gate voltage $V_T$. This can be done by numerically solving the Schr\"odinger equation (Eq.\ \ref{s-equation}) of our system \cite{ref-quant}.
For this we first implement the electrostatic potential felt by the electrons which are at a depth of 90 nm below the surface gates by following the approach of Ref.\ \onlinecite{sukhorukov_apl_95}, where the electrostatic potential created by a polygon surface gate is calculated by solving the Laplace equation (screening effects by the electrons in the 2DEG are however not taken into account).
The obtained electrostatic potential profile of the split-wire along the y-direction for different tunnel gate voltages is shown in Fig.\ \ref{fig:wire-potential}.
It resembles qualitatively the one of the experimental situation of the data we present later on and has been used to realize the numerical simulation in section IV.
As can be seen from Fig.\ \ref{fig:wire-potential}  we explore in detail the crossover region between the SCR and the WCR regime.

\begin{figure}[htbp]
	\includegraphics[width=7.5cm]{potential.png}
	\caption{Electrostatic potential $V(y)$ created by the electrostatic gates defining the split-wire for different tunnel gate voltages. The horizontal lines correspond to the quantized energies of the symmetric and anti-symmetric state for each tunnel gate voltage.}
	\label{fig:wire-potential}
\end{figure}

Assuming an infinitely long tunnel-coupled wire we can then calculate numerically the actual wave function of the symmetric and anti-symmetric state as displayed in Fig.\ \ref{fig:S+AS}.
At zero tunnel gate voltage the weight of the symmetric state is pinned in the centre of the split-wire, whereas the anti-symmetric wave function has its weight within each tunnel-coupled wire. When increasing the tunnel barrier, the symmetric and the anti-symmetric wave functions are displaced differently. However, upon further increasing the tunnel barrier, the symmetric and anti-symmetric wave function become similar and finally degenerate when completely decoupled.

\begin{figure}[htbp]
	\includegraphics[width=8.5cm]{s_as.png}
	\caption{Normalized wave function probability of the symmetric (left) and anti-symmetric (right) state in the tunnel-coupled wire for different tunnel gate voltages.}
\label{fig:S+AS}
\end{figure}

This can easily be seen when plotting the corresponding eigenenergies of the symmetric and anti-symmetric state as illustrated in panel a) of Fig.\ \ref{fig:bonding+antibonding}.
At zero tunnel barrier height the energy difference is simply the energy separation of the two lowest energy states of the potential created by the approximately harmonic confinement of the outer electrostatic gates of the split-wire, whereas at large tunnel barrier height, the energy difference vanishes as the two minima of the split-wire potential are decoupled.
The absolute energy values of the two states move to higher energy as the confinement potential is stronger due to the strong influence of the tunnel barrier and eventually cross the Fermi energy (in our case 11.4 meV, see section IV).

\begin{figure}[htbp]
	\includegraphics[width=8cm]{bonding.png}
	\caption{ a) energy dependence of the symmetric (green) and anti-symmetric (blue) state as a function of tunnel gate voltage for $B=$ 0 T. b) tunnel gate voltage dependence of $\Delta k$ for different magnetic fields. c) magnetic field dependence of $\Delta k$ for different tunnel gate voltages. d) surface area increase with respect to the AB ring calculated using the results of c). }
	\label{fig:bonding+antibonding}
\end{figure}

Similarly, the dispersion relation of our system as well as the values for $\Delta k$ = $k_A-k_S$ as a function of tunnel gate voltage and magnetic field can be evaluated, which is detailed in the appendix.
At low energy, which is of interest here, we observe that the energy bands for the symmetric and anti-symmetric states are affected by the tunnel coupling as well as the magnetic field.
For zero magnetic field, $\Delta$k is decreasing when increasing the negative tunnel gate voltage (Fig.\ \ref{fig:bonding+antibonding}b). This is expected since the symmetric and anti-symmetric state become degenerate.
The contrary is observed when applying a magnetic field. The influence of the magnetic field is to displace the wave functions with respect to the centre of the tunnel-coupled wire (see appendix). As a consequence, $\Delta$k for a given tunnel gate voltage is increasing with magnetic field.
This can also be seen in the field dependence of $\Delta$k for fixed tunnel gate voltages (Fig.\ \ref{fig:bonding+antibonding}c). The stronger the tunnel barrier, the stronger is the increase in $\Delta$k. For a completely decoupled wire the slope of $\Delta$k with respect to magnetic field finally saturates.
As a consequence, the electrons will pick up an additional phase difference when traversing the tunnel-coupled wire. This will eventually lead to a change in oscillation period of the magneto conductance oscillations. We will come back to this point in Section IV.

Having numerically determined the values of $\Delta k$  for different tunnel barrier heights, we can then compute the current in the upper (lower) branch using equation \ref{eq:simple}.
The corresponding conductance versus $V_T$ trace is shown in Fig.\ref{fig:VT}.
We clearly see that the two output currents oscillate in anti-phase with respect to the tunnel gate voltage $V_T$ as observed in the experiment.
At zero tunnel gate voltage the two output currents are equal since the upper and lower channels are strongly coupled.
Increasing the negative tunnel gate voltage induces anti-phase oscillations until the tunnel gate completely separates  the two channels (P3).
This demonstrates that the tuning of the tunnel gate allows to reach a fully electrical control of the repartition of  the output currents  into the upper/lower branch.
When the two output currents are equal, the tunnel-coupled wire behaves like a perfect beam splitter.

\begin{figure}[htbp]
	\includegraphics[width=8cm]{G_vs_V_T.png}
	\caption{Conductance of the upper/lower channel as a function of tunnel gate voltage as given by Eq.(\ref{eq:simple}). The mapping between $V_T$ and $\Delta k$ was performed numerically.}
	\label{fig:VT}
\end{figure}

  \section{Microscopic theory: model and simulation}

In the preceding section we have been able to understand the underlying physics of the Aharonov-Bohm interferometer coupled to a tunnel-coupled wire by means of a simplified analytical model (complemented with a numerical calculation of the mapping between $V_T$ and $\Delta k=k_A-k_S$).
Even though the analytical model provides basic concepts for designing the flying qubit device, it relies on the assumption that encircling paths induced by backscattering are fully suppressed. In the following, we make use of a detailed microscopic model to confirm that we can indeed suppress the encircling paths for the weak tunnel coupling by correctly choosing the device configuration. We show that the main experimental features (P1)-(P3) are very well reproduced with the simulations.
Neglecting screening and Coulomb interactions, our potential calculations do not allow to discuss precise quantitative agreement between experiments and simulations. Interestingly however, we find that for a certain parameter range, both the conductance and the visibility of the oscillation can be tuned close to what is observed in the experiments.

\subsection{Microscopic model}

In the following numerical simulations, the sample is modelled by a simple single-band Schr\"odinger equation that includes the confining potential $V(x,y)$ due to the gate structure as well as an uniform magnetic field B.
\begin{equation}
\frac{1}{2m}[   i\hbar \vec\nabla - e\vec A(x,y)]^2 \psi(x,y) + V(x,y)\psi (x,y) = E\psi(x,y)
\label{eq:sql}
\end{equation}
For the actual simulations, the model is discretized on a square lattice with lattice constant $a$ and
we introduce the wave function $\psi_{n_x,n_y}\equiv \psi (n_x a,n_y a)$.  The discretized Schr\"odinger equation reads,
\begin{align}
   &-e^{-i\phi n_y}\psi_{n_x+1,n_y}- e^{+i\phi n_y}\psi_{n_x-1,n_y} -
\psi_{n_x,n_y+1}-\psi_{n_x,n_y-1} \nonumber \\
&+ V_{n_x,n_y} \psi_{n_x,n_y}= (E/\gamma+2) \psi_{n_x,n_y}
\end{align}
where $\gamma=\hbar^2/(2ma^2)$ and $\phi=eBa^2/\hbar$.
The numerical calculations of the differential conductances are performed with the Kwant code\cite{kwant_groth}. Kwant is a general purpose
library designed for quantum transport.
The calculations are done within the standard framework of the Landauer-B\"uttiker approach\cite{Blanter2000} which is
also equivalent, in this context, to the non-equilibrium Green's function formalism (NEGF) \cite{Wingreen-Meir_1992}.

The system that we used for the simulations is composed of approximately 800 $\times$ 100 lattice sites as shown in Fig.\ \ref{fig:sites}a.
We have taken a sufficiently large number of lattice sites such that no influence of the discretisation on the transport properties is observed and we can hence safely assume that we are in the continuum limit. The separation of the tunnel-coupled wire on the right side has been implemented by a smooth transition towards the ohmic contacts.  These contacts are mimicked by semi-infinite wires following the standard approach in NEGF.
In order to convert the tight-binding parameters into \emph{experimental} units \cite{waintal_KNIT},
we fix the Fermi energy of the system to 11.4 meV corresponding to an electron density of $n_s$= 3.2$\cdot$ $10^{11}$ cm$^{-2}$.
Using a lattice constant of  $a\,=\,5\,$nm we define gates and distances in real space units.

In order to provide a realistic electrostatic potential associated with the different electrostatic gates of the sample,
we follow the approach of Ref.\ \onlinecite{sukhorukov_apl_95} as briefly mentioned in section II.
The two-dimensional potential at a depth of 90 nm below the surface obtained with this approach is plotted in Fig.\ \ref{fig:sites}b.
For convenience, we also separate the right tunnel-coupled wire from the middle island, such that we can sweep its voltage independently.
This allows us to investigate the influence of the tunnel-coupled wire on the AB oscillation frequency without affecting the depleted AB area.
When performing the simulations, we adjust the gate voltages for the simulations by the following procedure:
First, we fix the energy to match the  Fermi energy and then sweep simultaneously all the outer gates to obtain the desired conductance similar to the one of the experiment. Afterwards, we sweep the desired parameter ($V_T, V_S$ or $B$) and record the two output currents.

\begin{figure}[htbp]
	\includegraphics[width=8cm]{lattice.png}
	\caption{Top: lattice site model of the sample (see Fig.1). For clarity we only display few lattice sites. In the actual sample the lattice grid was much finer e.g. 800 $\times$ 100 lattice sites. Bottom: electrostatic potential felt by the electrons in the 2DEG, about 90 nm below the surface. }
	\label{fig:sites}
\end{figure}

\subsection{Comparison between numerics and experiment}

In the following, we first address the issue of the magneto oscillations in the strong coupling regime (P1a \& P2a).
We apply a finite gate voltage to the centre island in order to form an Aharanov-Bohm ring
and then sweep the magnetic field as well as the side gate voltage $V_{S}$. The simulated data is confronted with the experimental data in Fig.\ \ref{fig:in-phase}.
For all simulations we set the total conductance (transmitted as well as backscattered signal) to approximately five channels, similar to the experimental conditions.
In this regime, we can safely assume that electron interactions can be efficiently screened and the Landauer-B\"uttiker approach is valid.
In this two-terminal setup, the upper and lower current oscillate in-phase and several phase jumps are observed when sweeping the side gate voltage $V_s$ as imposed by the phase rigidity.  One also remarks the symmetry with respect to magnetic field as imposed by the Onsager relations. Let us note that the values for the side gate voltage are much smaller for the simulated data to induce a phase jump.
This difference is simply due to electron screening, which is not taken into account in the simulations. One can evaluate a scaling factor of about 20-30 by comparing the pinch-off voltages of the individual gates between experiment and simulations.

\begin{figure}[htbp]
	\includegraphics[width=8cm]{in-phase.png}
	\caption{Magneto conductance oscillations in the strong coupling regime after subtraction of a smooth background. Left: simulations, right: experimental data. Top: magneto conductance oscillations for small tunnel gate voltage $V_T$. The blue (green) curve corresponds to the current in the upper (lower) contact. Bottom: 2D colour plot of the magneto conductance oscillations of the total transmitted current (upper - lower) as a function of side gate voltage $V_s$. Phase jumps are clearly observed in the simulated as well as experimental data. }
	\label{fig:in-phase}
\end{figure}

The more interesting regime, however, is the weak coupling regime when a finite tunnel coupling is induced by the right tunnel-coupled wire.
In this case one observes the appearance of anti-phase oscillations when imposing a finite gate voltage on the tunnel gate as shown in Fig.\ \ref{fig:anti-phase}.
The simulated data (left panel) reproduces nicely the experimentally observed  anti-phase oscillations (right panel).
From the simulations we find that anti-phase oscillations appear even by imposing only a very small tunnel barrier.
For the present simulations $V_T$ has been set to -8.3 mV where we are basically in a single wire regime with a very flat bottom for the electrostatic potential.
We associate the appearance of the anti-phase oscillations to a situation where the potential change at the transition region between the AB ring and the tunnel-coupled wire is such that the symmetric modes of the AB ring can be smoothly coupled to the symmetric and anti-symmetric modes within the tunnel-coupled wire which finally leads to anti-phase oscillations. It should hence be possible to induce anti-phase oscillation in a single wire when shaping carefully the potential landscape.
In such a "peculiar" single wire regime, one is also able to induce a smooth shift of the AB oscillations when sweeping the side gate voltage $V_S$.
This is shown in the bottom panels of Fig.\ \ref{fig:anti-phase}. Features (P1b \& P2b) are hence nicely reproduced by the simulations.
The absolute amplitude of the conductance oscillations is very sensitive to the side gate voltage and can vary between (0.01 - 0.1) $\times$ $2e^2/h$ for the investigated gate voltage scan.
Let us note that the smoothness of the phase shift is sensitive to symmetry of the gate voltages applied to the individual gates.
For instance, if we induce an asymmetry of two equivalent gates the phase shift becomes more irregular and the anti-phase oscillations are not perfectly anti-phase any more.
This is also observed in the experiments.

\begin{figure}[htbp]
	\includegraphics[width=8cm]{anti-phase.png}
	\caption{Magneto conductance oscillations in the crossover region between the SCR and WCR after subtraction of a smooth background: left simulations, right experimental data. Top: magneto conductance oscillations for $V_T$ = -8.3 mV. The blue (green) curve corresponds to the current in the upper (lower) contact. Bottom: 2D colour plot of the magneto conductance oscillations of the total transmitted current  (upper - lower) as a function of side gate voltage. For both data sets one observes a smooth phase shift of the AB oscillations as a function of side gate voltage $V_{s}$.}
	\label{fig:anti-phase}
\end{figure}

The most interesting observation of the experiment is certainly the conductance oscillations with respect to the tunnel gate voltage in the WCR.
This allows to partition the output current into the upper/lower channels and hence tune the tunnel-coupled wire into a beam splitter regime.
In this case the left tunnel gate is fully depleted to inject the current only into the lower branch of the AB interferometer.
In Fig.\ \ref{fig:V_T-oscillation} we show the simulated as well as the experimental data.
While at very small tunnel gate voltage (SCR) the two output currents are basically equal, we observe anti-phase oscillations for both data sets when approaching the WCR. For strongly negative gate voltage the tunnel barrier splits the tunnel-coupled wire into two independent wires.
Again the correspondence between experiment and simulation is quite remarkable. Let us note, however, that in the experiment for a gate voltage regime below the 2D pinch-off ($V_T \approx$ -0.3 to -0.5 V) the oscillations are suppressed.
This is most probably due to density alignment of one of the subbands caused by electron-electron interactions \cite{energy,michi_nnano_12}, which sets the corresponding channel into off-resonance. Such influences of the Coulomb interaction is not taken into account in our model.

\begin{figure}[htbp]
	\includegraphics[width=8cm]{VToscillation.png}
	\caption{Conductance as a function of tunnel gate voltage after subtraction of a smooth background: left simulations, right experimental data. The blue (green) curve corresponds to the current in the upper (lower) contact. }
	\label{fig:V_T-oscillation}
\end{figure}

Analysing the magneto conductance oscillations as a function of tunnel gate voltage $V_T$, we observe in the numerical simulations a change in the magneto conductance oscillation frequency when passing from the SCR to the WCR as displayed in Fig.\ \ref{fig:surface}. For the SCR ($V_T$ = 0V) we observe in-phase oscillations as expected and the oscillation period corresponds simply to the surface area enclosed by the AB loop.
When increasing the tunnel barrier height (decrease of tunnel gate voltage) one clearly observes an increase of the number of periods for a given magnetic field scan.
This implies that the effective AB surface area increases. We emphasize this by taking the fast Fourier transform (FFT) of the magneto oscillations and by plotting the FFT peak position as a function of the tunnel gate voltage (bottom panel).
One clearly sees an increase of the Fourier peak as a function of tunnel gate voltage as indicated by the dashed line in Fig.\ \ref{fig:surface}.

\begin{figure}[htbp]
	\includegraphics[width=8cm]{surface.png}
	\caption{Simulations of the evolution of the AB conductance oscillations for different values of the tunnel gate voltage $V_T$. A smooth background has been subtracted from the data. Bottom right panel:  frequency  of the magneto oscillations (obtained by FFT of the other panels) as a function of $V_T$.}
	\label{fig:surface}
\end{figure}

To explain this observation, the electron has to pick up an AB phase over a significant distance in the tunnel-coupled wire. We associate this effect to the presence of the magnetic field which displaces the wave function with respect to the center of the tunnel-coupled wire due to the Lorentz force.
As a consequence, the electrons will acquire an additional Aharonov-Bohm phase which is proportional to $\frac{\hbar}{e} \frac{\partial \Delta k }{ \partial B} L$, in agreement with the semi analytical results of Fig.\ \ref{fig:bonding+antibonding} (bottom panel).
This can be interpreted as a surface area increase and explains the observed change in the magneto oscillation period in the simulations when going from the SCR to the WCR.
Note however, that the Lorentz force makes the symmetric and anti-symmetric states more localized in either of the two wires (see appendix) and induces an imbalance of the coupling of these states to the upper and lower wire. As a consequence, the visibility decreases with increasing surface area. In addition, increasing the length of the tunnel-coupled wire enhances this surface area increase almost linearly. These effects could readily be tested with the experimental set-up of ref. \cite{michi_nnano_12}.
Naturally, it would also be interesting to implement electron interactions into the numerical simulations \cite{sukhorukov_09} to allow for better quantitative agreement between theory and experiment as well as the possibility to study other effects such as decoherence \cite{lindelof_01,niimi_09,niimi_10,preden_08,sukhorukov_09}.

 \section{\bf Conclusion}

We have presented a minimum scattering theory as well as realistic simulations of an Aharonov-Bohm interferometer connected to two tunnel-coupled wires, a solid state implementation of a flying qubit.
While our simplified model could account for most experimental observations by assuming suppression of backscattered induced loop trajectories, our numerical simulations of the actual experimental system with realistic parameters can reproduce the majority of the experimentally observed features as well as suppression of  multiple loops in the AB ring. These simulations are important in particular for the understanding of rather subtle, geometry related aspects.
The good agreement between experiment and theory shows, that the physics of the flying qubit is well described within the Landauer-B\"uttiker scattering formalism. In addition to the interpretation of the experiments, the sort of simulations performed with Kwant could be a useful tool for quantum device design and signal optimisation.

 \section{\bf acknowledgements}

C.B. would like to thank M. B\"uttiker, L. Glazman, T. Jonckheere, T. Kato, T. Martin, C. Texier and  R. Withney for useful discussions.
T.B. acknowledges financial support from the Nanoscience Foundation Grenoble. S. Takada acknowledges support from JSPS Research Fellowships for Young Scientists. M.Y. acknowledges financial support by Grant-in-Aid for Young Scientists A (no. 23684019). S. Tarucha acknowledges financial support by MEXT KAKENHI "Quantum Cybernetics", MEXT project for Developing Innovation Systems, and JST Strategic International Cooperative Program. X.W. acknowledges financial support from the ERC grant MesoQMC. C.B. acknowledges partial financial support from the French National Agency (ANR) in the frame of its program BLANC FLYELEC project no anr-12-BS10-001 and from CNRS (DREI) - JSPS (PRC0677).

 \section{\bf appendix}

{\bf Dispersion relation: }For each set of experimental parameters, e.g. tunnel gate voltage and magnetic field, we calculate the dispersion relations for the symmetric and anti-symmetric state as shown in Fig. \ref{fig:appendix01}  for  $V_T = 0\,$V and $B =0\,$T. From this we can extract the wave vectors in propagation direction $\hat x$ for each mode at the Fermi energy E$_F$ and hence $\Delta k$. By taking $k_F^2=k_y^2+k_x^2$ we can also compute the eigenenergies of these two states due to confinement.

\begin{figure}[htbp!]
	\includegraphics[width=8cm]{appendix01.png}
	\caption{Dispersion relations for the symmetric (anti-symmetric) state in blue (red). The wave vectors in propagation direction have been extracted at the Fermi energy as indicated by the dashed lines.}
	\label{fig:appendix01}
\end{figure}

{\bf Magnetic field dependence:} At zero magnetic field, the symmetric and anti-symmetric state are degenerate at high tunnel gate voltage.
However, when a magnetic field is applied, the situation becomes rather subtle.
To understand the underlying physics, we consider the two-dimensional Schr\"odinger equation (\ref{eq:sql}).

The vector potential can be expressed within the Landau-gauge
\begin{equation}
\vec{A}=-By\hat{e_x},
\end{equation}
which leads after separation of variables to
\begin{equation}
\begin{split}
[\hbar^2k_x^2+2e\hbar Byk_x+e^2B^2y^2-\hbar^2\Delta+V(y)]\Psi(y)\\
=E\Psi(y)2m.
\end{split}
\end{equation}

We can now identify three spatially dependent terms: V(y) denotes the electrostatic potential in the tunnel-coupled wire created by the surface gates. The quadratic term increases the parabolic confinement symmetrically, whereas the second term induces a tilt in the potential landscape which is linear in y, B and $k_x$. This leads to a energy difference of the symmetric and anti-symmetric state and hence to a finite $\Delta k$ as depicted in Fig. \ref{fig:appendix02}.

\begin{figure}[htbp!]
	\includegraphics[width=8cm]{appendix02.png}
	\caption{Effective confinement potential under magnetic field of 60 mT. Blue (red) solid line corresponds to the symmetric (anti-symmetric) state. The dashed lines show their respective eigenenergies. Green solid line shows the confinement potential for $B = 0\ {\rm T}$ for comparison. The blue (red) dotted curves correspond to the symmetric (anti-symmetric) wave function under magnetic field.}
	\label{fig:appendix02}
\end{figure}

\newpage


\section{Introduction}

\label{intro}

One of the most widely used fermion discretizations is
that of Wilson fermions. The increase of computational power
and the improvement of algorithms has allowed for
simulations in the deep chiral regime.
To extrapolate the lattice data to the continuum limit
it is crucial to understand the lattice discretization effects
as well as the interplay between finite lattice spacing
and chiral symmetry breaking.
This problem was attacked in numerous ways.
There has been a large ongoing effort to determine the
Low Energy Constants (LECs) of Wilson chiral pertubation theory
\cite{SS}
both from an analytical point  of view
\cite{Damgaard:2010cz,Akemann:2010zp,NS,Kieburg,Kieburg:2011uf,Kieburg:2012fw,Kieburg:2013xta}
as well as numerically
\cite{Baron,Michael,757044,DWW,BBS,DHS,Herdoiza:2013sla}
in order to elucidate the new phases that one encounters at finite
values of the lattice spacing. Those discretization artifacts have
also been studied for QCD-like theories like two-color QCD and QCD with
adjoint fermions \cite{Kieburg:2013tca}.

In our analysis we incorporate the effect of the three
LECs at order $a^2$. As a starting point we employ Wilson random matrix theory (RMT) for the non-Hermitian Wilson Dirac
operator originally proposed in \cite{Damgaard:2010cz}.
We refer the reader to \cite{KimLat12} for a review of
the recent developments of RMT for the Wilson Dirac operator. This
proceeding is a summary of the main results of
Ref.~\cite{Kieburg:2013xta}. Readers who are interested in the
derivation of those results are referred to this work and related
 mathematical developments
published in \cite{Kieburg}.

The contents of this proceeding is as follows.
In Sec.~\ref{effects} we briefly introduce the RMT and describe
the effect of $W_6$ and $W_7$ on the spectrum of the
Wilson Dirac operator by switching off $W_8$.
In Sec.~\ref{densities} we summarize the effects of all three LECs
on the level densities and discuss the limits of large and small
lattice spacing. Thereby we provide comparisons of the analytical
results with Monte-Carlo simulations of Wilson RMT.
Moreover we propose some observables accessible by
lattice simulations to extract the LECs.

\section{The effects of $W_6$ and $W_7$}\label{effects}

The introduction of the Wilson term in the lattice
discretization of the Dirac operator explicitly breaks chiral symmetry.
As a result the low energy effective theory for QCD will be
affected and consequently new terms have to be incorporated
in order to describe this effect.
 Denoting the quark masses by $m$, the spacetime volume by $V$
and the chiral condensate by $\Sigma$, the effective partition function
for $N_{\rm f}$ fermionic flavors reads
\begin{eqnarray}
 Z^\nu_{N_{\rm f}}(m)&=&\int\limits_{{\rm U\,}(N_{\rm f})}d\mu(U)\exp\left[\frac{\Sigma V}{2}{\rm tr\,} m(U+U^{-1})-a^2 VW_6{\rm tr\,}^2( U+U^{-1})\right]\nonumber\\
 &&\hspace*{-0.5cm}\times\exp\left[-a^2VW_7{\rm tr\,}^2 (U-U^{-1})-a^2 VW_8{\rm tr\,} (U^2+U^{-2})\right]{\det}^{\nu}U.\label{ZnuNf}
\end{eqnarray}
Hereby we adopt the conventions of \cite{Damgaard:2010cz} for the
LECs $W_{6/7/8}$.
The partition function~\eqref{ZnuNf} is obtained in the $\epsilon$-regime of chiral perturbation theory (chPT) which is the limit of large volume, $V\to\infty$, where $\hat{m}=mV\Sigma$, $\hat{a}_{6/7}^2=-a^2VW_{6/7}$ and $\hat{a}_8^2=a^2VW_{8}$ are fixed\footnote{For $\hat{a}_{6/7/8}$ we employ the sign convention of \cite{Kieburg:2013xta}.}. In this limit there is an exact equivalence between the partition functions of chiral perturbation theory and chiral RMT \cite{Shuryak:1992pi,Verbaarschot:1994qf,Osborn:1998qb}.
The corresponding RMT is
\begin{eqnarray}\label{model}
 D_{\rm W\,}=\left(\begin{array}{cc} a A & W \\ -W^\dagger & a B \end{array}\right)+m_6\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}+\lambda_7\gamma_5,
\end{eqnarray}
where the random matrix $W$ generates the $a\to 0$ limit of Eq.~\eqref{ZnuNf}, the Hermitian matrices $A$ and $B$ generate the term proportional to $W_8$ and the two scalar random variables $m_6$ and $\lambda_7$ generate the terms proportional to $W_6$ and $W_7$, respectively.
All random variables are Gaussian distributed. In the microscopic limit,
where  the matrix dimension $n\to\infty$
while fixing $\widehat{m}_6=2nm_6$, $\widehat{\lambda}_7=2n\lambda_7$ and $\widehat{a}_8^2=na^2/2$ the random matrix partition function reduces to the
chiral partition function~(\ref{ZnuNf}).

First, let us focus on the effect of $W_6$ and $W_7$ and set $W_8=0$.
In this case the massive Dirac operator reads
\begin{eqnarray}
D_{\rm W\,}= D_{\rm W\,}|_{a=0} +(m +m_6)\leavevmode\hbox{\small1\kern-3.8pt\normalsize1} +\lambda_7 \gamma_5,\label{massDirac}
\end{eqnarray}
with eigenvalues given by
\begin{eqnarray}
z_\pm = m+m_6\pm \imath\sqrt{\lambda_{\rm W\,}^2-\lambda_7^2},
\label{ev67}
\end{eqnarray}
where we denote an eigenvalue of the continuum Dirac operator $D_{\rm W\,}|_{a=0}$ by $\imath\lambda_{\rm W\,}$.
In Fig.~\ref{fig1} the different effects of $W_{6/7}$ are shown
schematically.

\begin{figure}[h!]
 \includegraphics[width=\textwidth,height=5cm]{effect_a7_a6.pdf}
 \caption{\label{fig1}Schematic plots of the effects of $W_6$ (left plot) and of $W_7$ (right plot) on the microscopic spectrum. The former zero modes (green crosses) are broadened by a Gaussian while the complex conjugate pairs (red crosses) are broadened parallel to the real axis by $W_6$ and are pushed into the real axis by $W_7$.
}
\end{figure}

The LEC $W_6$ broadens the microscopic spectrum
parallel to the real axis by a Gaussian with a width proportional to $\widehat{a}_6$,
while the shape of the spectrum projected to the real axis is not changed at all.  The reason is the additive nature of $\widehat{m}_6$ to the eigenvalues~\eqref{ev67} resulting in a convolution. The effect of $W_7$ is more drastic since the purely imaginary eigenvalues invade the real axis through the origin.
Only the former zero modes are broadened by a Gaussian
with width proportional to $\widehat{a}_7$.

\section{The eigenvalue densities and their dependence on $W_{6/7/8}$}\label{densities}

Due to $\gamma_5$-Hermiticity the eigenvalues of the
 Wilson Dirac operator come in complex conjugate pairs or are real. A Dirac operator with fixed index $\nu$ has $\nu$
generic real eigenvalues which correspond to the zero modes in
the continuum limit. In addition,  other real
modes resulting from complex conjugate pairs entering the real axis pairwise.
 Moreover, all real modes have non-zero chirality while the
chirality of the complex modes vanishes.
Thus, one can define three different eigenvalue densities,
namely the one of the complex eigenvalues, $\rho_{\rm c}$,
and the one of all real modes, $\rho_{\rm real}=\rho_{\rm right}+\rho_{\rm left}$,
 splitting into the density of the right handed real modes
($\langle\psi|\gamma_5|\psi\rangle>0$), $\rho_{\rm right}$,
and the one of the left handed real modes ($\langle\psi|\gamma_5|\psi\rangle<0$), $\rho_{\rm left}$. Another important distribution is the distribution of the chirality over the real eigenvalues  \cite{Akemann:2010zp}
\begin{equation}\label{chidef}
\rho^\nu_{\rm \chi}(\widehat{\lambda})
 \equiv  {\sum}_{\widehat{\lambda}_k \in {\mathbb R}}
\delta(\widehat \lambda -\widehat \lambda_k)  {\rm sign }\langle k| \gamma_5 |k \rangle,
\end{equation}
which is equal to the imaginary part of the resolvent
 \begin{eqnarray}
\label{chicond}
\rho^\nu_{\rm \chi}(\widehat{\lambda}) =\lim_{V\to\infty}\frac{1}{\pi}{\rm Im}
\left[G^\nu(\widehat{\lambda})\equiv \left\langle{\rm tr\,}\frac{1}{V\Sigma D_{\rm W\,}+\widehat{\lambda}\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}-\imath\epsilon\gamma_5}\right\rangle\right].
\end{eqnarray}
 With the help of this distribution one can define the density of additional real modes $\rho_{\rm add}=\rho_{\rm real}+\rho_\chi$. The splitting of the level density of real modes into $\rho_{\rm add}$ and $\rho_\chi$ has proven quite convenient \cite{Kieburg:2011uf,Kieburg:2013xta} as will be seen in the ensuing discussion. This discussion summarizes the extensive analytical study performed in \cite{Kieburg:2013xta}.

\begin{figure}[h!]
 \includegraphics[width=\textwidth,height=6cm]{Nadd.pdf}
 \caption{\label{fig3} Log-log plots of the additional real modes versus $\widehat{a}_8$ for $\nu=0$ (left plot) and $\nu=2$ (right plot).
The analytical results (solid curves) are compared to Monte-Carlo
simulations of RMT (symbols).
Notice that a non-zero value of $\widehat{a}_7$ yields a saturation at small $\widehat{a}_8$.}
\end{figure}

First we discuss the additional real modes. The average number of additional real modes,
\begin{equation}\label{Nadd-def}
N_{\rm add}=\int_{-\infty}^\infty\rho_{\rm add}(\widehat{x})d\widehat{x},
\end{equation}
has shown to be a powerful quantity for measuring the
strength of the lattice artifacts. Since we integrate
over the real axis $N_{\rm add}$ is independent of $W_6$.
At small lattice spacing $\widehat{a}_{7/8}\approx0.1$
the average number of additional real modes behaves as $\widehat{ a}^{2\nu+2}$
while it scales linearly with ${\widehat a}$ and becomes independent of $\nu$ in the limit of large lattice spacing, see Fig.~\ref{fig3}. Thus, at small lattice spacing almost all additional real modes come from the sector with index $\nu=0$
given by
\begin{eqnarray}
  N_{\rm add}^{\nu=0}&\overset{{\widehat{a}}\ll1}{=}&2Va^2(W_8-2W_7).\label{addrealsmall}
\end{eqnarray}
The distribution $\rho_{\rm add}$ lives on the scale $\widehat a$ at small lattice spacing and on the scale $\widehat a^2$ at large lattice spacing. The latter limit is close to the mean field limit where
the support of the real part of the eigenvalues scales with ${\widehat a}^2$.

\begin{figure}[h!]
 \includegraphics[width=7.5cm,height=6cm]{rhocp-new.pdf}\hfill\includegraphics[width=7.5cm,height=6cm]{rhochi.pdf}
 \caption{\label{fig4} Analytical results for the projected level density of complex eigenvalues onto the imaginary axis, $\rho_{\rm cp}$, (left)
 and for the distribution of the chirality over the real modes, $\rho_\chi$, (right) compared
to Monte-Carlo simulations of Wilson RMT for $\nu=1$.
Notice that both distributions are symmetric around the origin
in the quenched theory and we plotted only the positive imaginary and real axis, respectively. The black curve in the left plot is the continuum result and shows good agreement with $\rho_{\rm cp}$ for small
lattice spacing while the deviation is quite substantial for large lattice spacing.}
\end{figure}

The behavior of the projected density of complex eigenvalues,
\begin{equation}\label{rhocp-def}
 \rho_{\rm cp}(\widehat{y})=\int_{-\infty}^\infty\rho_{\rm c}(\widehat{x}+\imath \widehat{y})d\widehat{x},
\end{equation}
when varying $W_7$ and $W_8$ can be seen in Fig.~\ref{fig4}. Also this quantity is $W_6$ independent since the integration cancels the convolution
with the random variable $\widehat{m}_6$ and thus increases the statistics of the numerical simulation. The LEC $W_8$ smoothens
 the distribution such that the oscillations completely disappear
when increasing this LEC. In contrast to this effect, $W_7$
dampens the height of $\rho_{\rm cp}$ near the real axis since
the complex eigenvalue pairs are pushed into the real axis while
the oscillations seem to be more persistent with regard to this
LEC. Luckily the distribution $\rho_{\rm cp}$ smoothly
converges to the continuum result in the limit of small lattice spacing
and gives a hint that it is still a good quantity for extracting
the chiral condensate via the Banks-Casher relation,
\begin{eqnarray}\label{Banks-Casher}
\Delta\overset{{\widehat{a}}\ll1}{=}\frac \pi{\Sigma V},
\end{eqnarray}
where $\Delta$ is the average level spacing of the imaginary part of the eigenvalues several eigenvalue spacings from the origin. Furthermore,
 the section of $\rho_{\rm c}$ parallel to the real axis is a Gaussian of width
\begin{eqnarray} \label{width}
 \frac{\sigma^2}{\Delta^2}&\overset{{\widehat{a}}\ll1}{=}&\frac 4{\pi^2}a^2 {V(W_8-2W_6)}
\end{eqnarray}
at small lattice spacing and becomes box-shaped on the scale $\widehat{a}_8^2=a^2VW_8$ in the limit of large lattice spacing as well as in the mean field limit.

Finally, we consider the distribution of the chirality over the real
eigenvalues, $\rho_\chi$. This distribution is always normalized
to  the index of the Dirac operator, i.e.
$\int \rho_\chi(\widehat{x})d\widehat{x}=\nu$.
At small lattice spacing the effects of $W_6$ and $W_7$ are almost the same and yield a Gaussian broadening on the scale $\widehat{a}$. In particular the variance of this distribution at fixed index $\nu$ takes a simple form, i.e.
\begin{eqnarray}\label{chismall}
\frac{\langle x^2\rangle_{\rho_\chi}}{\Delta^2}
&\overset{{\widehat{a}}\ll1}{=}&\frac 8 {\pi^2}Va^2(\nu W_8-W_6-W_7),\ \nu>0,
\end{eqnarray}
which is convenient for measuring the LECs.
Moreover $\rho_\chi$ dominates the level density of the
real eigenvalues at small lattice spacing, i.e.
$\rho_\chi{\approx}\rho_{\rm real}$ (for $\widehat{a}\ll1$), since its height is of order
 ${\widehat a}^{-1}$ in contrast to the height of $\rho_{\rm add}$ which is of order ${\widehat a}^{2\nu+1}$.
 This behavior is
 reversed in the limit of large lattice spacing.
Then the height of $\rho_\chi$ is of order ${\widehat  a}^{-2}$ with
support of order ${\widehat a}^2$ while the height of
$\rho_{\rm add}$ is of order ${\widehat a}^{-1}$.
Hence one has
$\rho_{\rm add}= \rho_{\rm real}+ \rho_{\chi} {\approx}\rho_{\rm real}$ (for $\widehat{a}\gg1$).

\section{Conclusions}\label{conc}

By utilizing powerful RMT techniques, see \cite{Kieburg,Kieburg:2011uf,Kieburg:2013xta},  we analytically calculated the spectral densities of the real and complex eigenvalues of the non-Hermitian Wilson Dirac operator $D_{\rm W\,}$ and summarized the most important results here. All the results were presented for the quenched theory but will be generalized to dynamical flavors in forthcoming publications.

We studied the explicit effects of all three LECs on the spectrum of the Wilson Dirac operator. Thereby we derived simple relations between the LECs, $\Sigma$ and $W_{6/7/8}$, and some measurable quantities like the average level spacing of the projected eigenvalues onto the imaginary axis and the average number of additional real modes, see Eqs.~\eqref{addrealsmall} and (\ref{Banks-Casher}-\ref{chismall}). Those relations apply at small lattice spacing $|a^2VW_{6/7/8}|\leq0.1$. In this regime these quantities may serve for fixing the LECs in lattice simulations.
In particular the linear relations between $N_{\rm add}^{\nu=0}$, $\sigma$ and $\langle x^2\rangle_{\rho_\chi}$ are linearly
dependent requiring the consistency relation
\begin{eqnarray}
\frac{\langle  x^2 \rangle_{\rho_\chi}^{\nu = 1}}{\Delta^2} =
\frac{\sigma^2}{\Delta^2} + \frac 2{\pi^2} N^{\nu=0}_{\rm add}
\end{eqnarray}
valid in the limit of small $a^2VW_{6/7/8}$. Such consistency relations as well as the relations ~(\ref{addrealsmall}) and~(\ref{Banks-Casher}-\ref{chismall}) will certainly improve the accuracy of the LECs fitted in \cite{DWW,DHS}.

Moreover we also considered the limit of large lattice spacing which is
closely related to the mean field limit. The real part of the complex eigenvalues of $D_{\rm W\,}$ as well as the real eigenvalues themselves have a support
 in the interval $[-8 a^2VW_8,8 a^2VW_8]$ agreeing with the discussion about the complex eigenvalues in \cite{Kieburg:2012fw}.
 Although the impact of $W_6$ and $W_7$ on the density of the complex
eigenvalues is negligible in the mean field limit,
they have a strong impact on the distribution of the real eigenvalues.
In particular, in the mean field limit of the quenched theory,
a non-zero $W_7$ generates a square root singularity
at the boundary of the support of $\rho_{\rm add}$ while the density is uniform for  $W_7=0$.
 Nevertheless we expect that the phase diagram of the quenched as well as
 the unquenched theory will not really change by this effect
which will be checked  in a forthcoming publication.

\section*{Acknowledgement}

MK acknowledges partial financial support by the Alexander-von-Humboldt Foundation. JV and SZ acknowledge support by U.S. DOE Grant No. DE-FG-88ER40388. We thank Gernot Akemann, Poul Damgaard, Urs Heller and Kim Splittorff for fruitful discussions.


\section{Introduction}
Many pieces of evidence support the existence of dark matter in the universe. One of the most attracting candidates is weakly interacting massive particles (WIMPs). Three methods can be used to search for WIMPs: direct, indirect and collider experiments. Direct detection aims to catch the signal of nuclear recoil by the WIMPs. Many direct dark matter search experiments have been performed in the world.
Among the variations of direct detection experiments, directional dark matter searches are said to provide a ''smoking gun'' evidence of the galactic dark matter\ \cite{TANIMORI2004241}.
Several directional dark matter search experiments utilizing low-pressure gaseous time projection chambers (TPCs), such as DRIFT\ \cite{BATTAT201765}, MIMAC\ \cite{Santos:2013oua}, and NEWAGE\ \cite{Nakamura:PTEP2015}, are being performed.
As is the case for other direct dark matter search experiments, background-reduction is one of the most important R\&D items for the gaseous TPC detectors. Large volume detectors are also required because the expected event rate is very low. However, electron diffusion during drift is large compared to its track length in the conventional TPCs, in which the electrons are drifted by the electric field; thus constructing a large TPC (e.g., 1\ $\mathrm{m^{3}}$ scale) is very challenging.

A negative-ion TPC (NITPC) is a gaseous TPC with negative-ion gas, or gas with a large electron affinity gas. This type of TPC, in which negative-ions are drifted instead of the electrons, has been studied for the high position resolution applications because the ion diffusion is smaller than that of the electrons\ \cite{MARTOFF2000355}.
The DRIFT group pioneered the development of NITPCs for direct dark matter search and succeeded in operating a larger volume TPC.
The ''minority carrier,'' which makes the z-fiducialization possible\ \cite{Battat:2014van}\ \cite{PHAN2015}.
After the nuclear recoil and ionizations of gas molecules, several species of negative-ion, called main and minority charges, are created in the NITPCs. These ion species have different drift velocities; hence, the absolute z-position of the incident position can be known from the arrival times. This feature enables detectors to cut radioactive decay events from the drift and readout planes.
First NITPC with $\mathrm{CS_2}$ and a few percent $\mathrm{O_{2}}$ mixture was demonstrated \cite{BATTAT201765}. $\mathrm{SF_6}$ was also recently identified as a good negative-ion gas candidate for NITPC \cite{PHAN2015}. $\mathrm{SF_6}$ is a safer, non-flammable and non-toxic gas compared to the  $\mathrm{CS_{2} + O_{2}}$ mixture.

NITPC is a relatively new technology; thus, no method has yet been established to simulate micro pattern gaseous detectors (MPGDs) in negative-ion gas with Garfield++ \cite{Garfield:url}, which is a MPGD simulation toolkit.
The NITPC simulation method and their results together with the related experimental results are described herein to establish the MPGD simulation method for negative-ion gas.

\section{Measurements}
\subsection{Measurement Setup}
Several fundamental properties of the MPGD performance in negative-ion gas were measured as a comparison target of the simulation.
Fig.\ref{fig:exp_setup} schematically shows the setup used to evaluate the performance of GEMs\ \cite{SAULI1997531} in $\mathrm{SF_6}$. Two or three GEMs (100\ $\mathrm{\mu m}$ thickness, liquid crystal polymer, made by Scienergy) were used. The difference between these setups was only the number of GEMs. The drift mesh, made of stainless steel, was set on the top, and GEM1 (three-GEM case) or GEM2 (double-GEM case) was located 10\ mm below the drift mesh. The transfer gap, which is the distance between the GEMs, was 3.5\ mm, while the induction gap, which is the distance between GEM3 and the readout plane, was 2\ mm. One-dimensional 24\ strips with a pitch of 400\ $\mu$m were used as the readout electrodes. The signals from these strips were grouped to one channel, decoupled with a 1\ nF capacitor, and read by the charge sensitive amplifier, Cremat CR-110 (Gain:\ 1.4\ V/pC, $\tau$\ =\ 140\ $\mu$s). The waveforms were obtained using a USB oscilloscope UDS-5206S. Pure $\mathrm{SF_6}$ gas with various low pressures of 60--120\ Torr were used. The gas gains were measured with different applied voltages and pressures.
\begin{figure}[]
    \centering
    \includegraphics[width=8cm]{exp_setup_schemEN.pdf}
    \caption{Gain measurement setup}
    \label{fig:exp_setup}
\end{figure}
\subsection{Measurement result}
Fig.\ref{fig:GEM_gain} shows the measurement results.
Gas gains of up to 6,000 were obtained with $\mathrm{SF_6}$ at 100\ Torr in the double-GEM measurement (Fig.\ref{fig:doubleGEM_gain}).
Gas gains of up to 10,000 for 60 to 120\ Torr were obtained in the triple GEM measurement (Fig.\ref{fig:tripleGEM_gain}).
Figs.\ref{fig:dep_results}a--c show the gas gain dependence on the drift, the transfer and the induction, respectively.
No dependence on the drift electric field was observed, as shown in Fig.\ref{fig:drift_dep_result}.
Some dependence on the transfer and induction electric fields was observed as shown in Fig.\ref{fig:transfer_dep_result} and Fig.\ref{fig:induction_dep}. In both cases, the gains increased at the lower electric field region and reached plateaus at the higher electric field region. In addition, the induction-dependence measurement result showed a slight gain rise after the plateau, indicating that some electron multiplication may occur at the induction gap.

The gas gains relevant to the single GEM were calculated from the double and triple-GEM results by assuming that the total gain was a product of two or three GEM gains.
Fig.\ref{fig:gain_per_single} shows the result. The single GEM gains in the case of double and three GEM were on the same gain curve.
This result indicates that there is no significant charge loss in the transfer region between the GEMs.
\begin{figure}
    \begin{minipage}[b]{0.48\linewidth}
        \centering
        \includegraphics[keepaspectratio, scale=0.4]{double_gain.pdf}
        \subcaption{Double GEM gas gain at 100\ Torr pure $\mathrm{SF_{6}}$}
        \label{fig:doubleGEM_gain}
    \end{minipage}
    \begin{minipage}[b]{0.45\linewidth}
        \centering
        \includegraphics[keepaspectratio, scale=0.4]{triple_gasgain.pdf}
        \subcaption{Triple GEM gas gain in pure $\mathrm{SF_{6}}$}
        \label{fig:tripleGEM_gain}
    \end{minipage}
    \caption{Double and Triple GEM gas gain curve}
    \label{fig:GEM_gain}
\end{figure}

\begin{figure}
    \begin{minipage}[b]{0.48\linewidth}
        \centering
        \includegraphics[keepaspectratio, scale=0.4]{drift_gain.pdf}

        \subcaption{Gas gain dependence on $E_{drift}$. Measurement of the conditions are $\mathrm{SF_{6}}$ at 100\ Torr,$E_{transfer}=2.9\ \mathrm{kV/cm}$ and $E_{induction}=2.5\ \mathrm{kV/cm}$}
        \label{fig:drift_dep_result}
    \end{minipage}
    \begin{minipage}[b]{0.48\hsize}
        \centering
        \includegraphics[keepaspectratio, scale=0.4]{transfer_gain.pdf}

        \subcaption{Gas gain dependence on $E_{transfer}$. Measurement of the conditions are $\mathrm{SF_{6}}$ at 100\ Torr,$E_{drift}=1.0\ \mathrm{kV/cm}$ and $E_{induction}=2.5\ \mathrm{kV/cm}$}
        \label{fig:transfer_dep_result}
    \end{minipage}\\
    \centering
    \begin{minipage}[b]{0.48\linewidth}
        \includegraphics[keepaspectratio, scale=0.4]{induction_gain.pdf}
        \subcaption{Gas gain dependence on $E_{induction}$. Measurement of the conditions are $\mathrm{SF_{6}}$ at 100\ Torr,$E_{drift}=1.0\ \mathrm{kV/cm}$ and $E_{transfer}=2.9\ \mathrm{kV/cm}$}%

        \label{fig:induction_dep}
    \end{minipage}

    \caption{Gas gain dependence on the electric fields in $\mathrm{SF_{6}}$ at 100\ Torr}
    \label{fig:dep_results}
\end{figure}

\begin{figure}[]
    \centering
    \includegraphics[width=.6\textwidth]{gain_per_single2.pdf}

    \caption{Gas gain per single GEM at 100\ Torr $\mathrm{SF_{6}}$}
    \label{fig:gain_per_single}
\end{figure}

\section{MPGD simulation}
The MPGD simulation study for the negative-ion gas is indispensable for the optimization of the design and operation.
Garfield++\ \cite{Garfield:url} and Magboltz\ \cite{Magboltz:url} version 9.01 were used herein.
Garfield++ is a simulation toolkit for the gaseous detector study written in C++. It imports detector geometry and electric field from the external finite element method software's output and uses the Monte Carlo microscopic and macroscopic methods for the charge propagation.
Magboltz calculates the transport parameters of the electrons in given gas, electric field, and magnetic field from the cross-section data. These parameters were passed to Garfied++ and used for the gas avalanche calculation.

\subsection{Simulation method}
The cross-section data for the reactions between the electron and the negative-ion gas, such as $\mathrm{SF_6}$, were implemented in Magboltz. However no code was implemented for the negative-ion transportation and the electron detachment.
The negative-ion transportation process and two electron detachment models were introduced herein and implemented in the Garfield++ customization code.
The whole process is presented as follows.
\begin{enumerate}
    \item Negative-ions drift in the electric field and reach the MPGD region.
    \item The electron is detached from the negative-ion gas molecule by the high electric field\ (electron detachment).
    \item An avalanche starts from the detached electron.
    \item The electron-ion pairs create the signal.
\end{enumerate}

As a simulation setup, a single 100\ $\mathrm{\mu m}$ thick GEM, which has 140\ $\mathrm{\mu}$m holes pitch and 70\ $\mathrm{\mu}$m hole diameter, was used. The gas was pure $\mathrm{SF_{6}}$\ 100\ Torr and only the main charge $\mathrm{SF_{6}^{-}}$ was considered in the drift process\ (i).
The electron detachment cross-section of $\mathrm{SF_{6}^{-}}$ was used for the electron detachment process\ (ii) and two models, namely the cross-section and threshold models, were introduced and used to simulate the detachment process. The electron~/~$\mathrm{SF_{6}}$ cross-section which Magboltz has was used for the avalanche process\ (iii).

\subsection{cross-section model}
The electron detachment probability in the cross-section model was calculated using the cross-section of the relevant chemical processes. Two possible detachment processes, namely direct(Eq.(\ref{eq:sf6_direct_reaction1}) or (\ref{eq:sf6_direct_reaction2})) and indirect one (Eq.(\ref{eq:sf6_indirect_reaction1})+(\ref{eq:sf6_indirect_reaction3})~or~(\ref{eq:sf6_indirect_reaction2})+(\ref{eq:sf6_indirect_reaction3})), were considered as possible processes.
The cross-sections of these reactions were reported in \cite{YWANG}.
The cross-section of Eq.(\ref{eq:sf6_direct_reaction1}) and (\ref{eq:sf6_direct_reaction2}) increased at 100\ eV (Fig.\ref{fig:detach_direct}) while that of Eq.(\ref{eq:sf6_indirect_reaction3})
increased at 10\ eV. This is ten times lower energy than what Eq.(\ref{eq:sf6_direct_reaction1}),\ (\ref{eq:sf6_direct_reaction2}) showed at 100\ eV.

Fig.\ref{fig:detach_indirect} indicates that the Eq.(\ref{eq:sf6_indirect_reaction1}) and (\ref{eq:sf6_indirect_reaction2}) reactions have larger cross-sections below 10\ eV compared to Eq.(\ref{eq:sf6_direct_reaction1}) and (\ref{eq:sf6_direct_reaction2}). In addition, $\mathrm{SF_{6}^{-}}$ has larger mass than $\mathrm{F^{-}}$ so $\mathrm{SF_{6}^{-}}$ is difficult to be accelerated and have sufficient energy for the reactions. %
Therefore the detachment rate can be assumed to be determined by Eq.(\ref{eq:sf6_indirect_reaction3}), such that the cross-section of Eq.(\ref{eq:sf6_indirect_reaction3}) can be used for the detachment calculation.

\begin{figure}
    \centering
    \begin{minipage}[b]{0.48\linewidth}
        \centering
        \includegraphics[keepaspectratio, scale=0.5]{detach_cross_direct.pdf}
        \subcaption{Direct detachment, Eq.(\ref{eq:sf6_direct_reaction1})(green square), Eq..(\ref{eq:sf6_direct_reaction2})(blue triangle) and Eq.(\ref{eq:sf6_indirect_reaction3})(red circle)}
        \label{fig:detach_direct}
    \end{minipage}
    \begin{minipage}[b]{0.48\linewidth}
        \centering
        \includegraphics[keepaspectratio, scale=0.5]{detach_cross_indirect.pdf}
        \subcaption{Fluorine ion ($\mathrm{F^{-}}$) creation reaction, Eq.(\ref{eq:sf6_indirect_reaction1})(green triangle) and Eq.(\ref{eq:sf6_indirect_reaction2})(blue triangle)}
        \label{fig:detach_indirect}
    \end{minipage}
    \caption{Direct electron detachment and fluorine ion ($\mathrm{F^{-}}$) creation reactions \cite{YWANG}}
    \label{fig:detachment_reaction_crosssection}
\end{figure}

\begin{eqnarray}
    \label{eq:sf6_direct_reaction1}
    SF_{6}^{-} + SF_{6} & \rightarrow & e^{-} + SF_{6} + SF_{6} \\
    \label{eq:sf6_direct_reaction2}
    SF_{5}^{-} + SF_{6} & \rightarrow & e^{-} + SF_{5} + SF_{6}
\end{eqnarray}
\begin{eqnarray}
    \label{eq:sf6_indirect_reaction1}
    SF_{6}^{-} + SF_{6} & \rightarrow & F^{-} + SF_{5} +SF_{6} \\
    \label{eq:sf6_indirect_reaction2}
    SF_{5}^{-} + SF_{6} & \rightarrow & F^{-} + SF_{4} +SF_{6} \\
    \label{eq:sf6_indirect_reaction3}
    F^{-} + SF_{6} & \rightarrow & e^{-} + F +SF_{6}
\end{eqnarray}
In the Eq.(\ref{eq:sf6_indirect_reaction1}) reaction, the $\mathrm{SF_{6}^{-}}$ energy can be calculated using the $\mathrm{SF_{6}^{-}}$ mobility in $\mathrm{SF_{6}}$\ \cite{Benhenni2012} and the fluorine ion $\mathrm{F^{-}}$ is expected to have energy up to approximately 2\ eV by a simple kinematics calculation.
The $\mathrm{F^{-}/SF_{6}}$ collision reaction has a cross-section of $\mathrm{\sigma=10^{-15}-10^{-14}\ cm^2}$\ \cite{Benhenni2012} in $\mathrm{SF_{6}}$ 100\ Torr at the corresponding energy range, and the number of density is $\mathrm{n = 3.5\times 10^{18}\ cm^{-3}}$. Thus the mean-free-path of this reaction can be calculated to be in an order of $\mathrm{10^{-6}\ m}$.
The energy of the $\mathrm{F^{-}}$ ion would obtain from the electric field
can be calculated along its path.
The energy obtained in the GEM hole, where typical electric field is  approximately 50\ kV/cm and above, was 10\ eV if the $\mathrm{F^{-}/SF_{6}}$ collision mean-free-path $\lambda$ is an order of $\mathrm{10^{-6}\ m}$, and this energy is close to Eq.(\ref{eq:sf6_indirect_reaction3}) reaction's threshold.
Given the $\mathrm{F^{-}}$ energy, the mean-free-path of the electron detachment reaction through the indirect reaction, $\mathrm{\lambda}$, was thus known based on the cross-section shown in Fig.\ref{fig:detach_direct}\ \cite{YWANG}.

The probability ($p$) of the electron detachment from $\mathrm{F^{-}}$ within 1\ $\mathrm{\mu m}$, the same as the step used in this simulation, was calculated as Eq.(\ref{eq:detach_prob}
\begin{eqnarray}
    p = 1 - \exp(- 1 \mu m / \lambda)
    \label{eq:detach_prob}
\end{eqnarray}
This detachment probabilities are shown with the blue triangles in Fig.\ref{fig:detach_prob}. The interpolated probability values were used in the simulation.

\subsection{Threshold model}
The other model, the threshold model, is a more phenomenological one. In this model, the negative-ion releases the electron when the electric field is more than a threshold value. In this simulation, the negative-ion releases an electron in the high electric field larger than 45\ kV/cm, where the avalanches are first observed in the corresponding experiments. This threshold value was known from the electric field map of the GEM geometry when the gas avalanche starts in the experiments. The red line in Fig.\ref{fig:detach_prob} shows the probability function.

\subsection{Simulation result}
\begin{figure}[]
    \centering

    \includegraphics[width=10cm]{deta_prob_comp2.pdf}
    \caption{Detachment probability dependence on the electric field. The blue marker denotes the cross-section model, while red line denotes the threshold model probability function.}
    \label{fig:detach_prob}
\end{figure}
\begin{figure}[]
    \centering
    \includegraphics[width=12cm]{final_gaincurve.pdf}

    \caption{Gas gain curve (pure\ $\mathrm{SF_6}$\ 100\ Torr). The cross mark denotes the measurement result and red square denotes threshold model, and blue triangle denotes the cross-section model.}
    \label{fig:final_result}
\end{figure}
The gas avalanche simulations were performed with the detachment models described above. Fig.\ref{fig:final_result} shows the simulation results. The blue triangles and the red squares show the results of the cross-section model and the threshold models, respectively.
The result of the measurements and its error are also depicted as the asterisk and the light blue region, respectively. It should be noted, as a primary outcome of this study, the absolute gas gain was reproduced by the simulation within factor 2 difference. This result can be claimed as a great success of this first study.
The main difference between the cross-section model and threshold models was the slope of the gain curve. In other words, the cross-section model had a larger gain than threshold model at low electric field but did not at higher electric field. These results can be explained by the detachment probability difference, shown in Fig.\ref{fig:detach_prob}, such that at low electric field cross-section model had a larger detachment probability and, on the contrary, at higher electric field threshold model had.
A comparison of the experimental and simulation results indicated that the threshold model produced the experimental result more than the cross-section model in terms of the slope of the gain curve.
Since the cross-section model is a more first principle method, tuning the cross-section method would be a important step to generalize the outcome of this study.

Studies for other MPGDs, such as MicroMegas and $\mathrm{\mu}$-PIC in negative-ion gas are also important future works.
It is also important to compare the gas gain dependences on the transfer and induction electric field. Studies on other parameters, such as the energy  resolution, signal formation, and signal collection, would help to validate this method. An optimization work are also planned for dark matter search experiment.

\section{Conclusion}
In this work, the first MPGD simulation study with $\mathrm{SF_{6}}$, one of negative-ion gases, was achieved with Garfield++ and Magboltz. Two detachment models, cross-section and threshold model, were introduced as the electron detachment process. In both models, the absolute gas gain was reproduced by the simulation within factor 2 difference. In the terms of the slope of the gain curve, the threshold model reproduced the experiments result more than cross-section one.
\section*{Acknowledgement}
This work was supported by KAKENHI Grants-in-Aids (15K21747, 26104005, 16H02189 and 19H05806), Grant-in-Aid for JSPS Fellows (17J03537 and 19J20376).
\section*{References}

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