\section{Introduction}
Superconducting qubits coupled to a quantum mechanical
harmonic oscillator have reached the strong coupling region, where
the coupling strength is much larger than the decay rates of
the cavity and the qubit.
The circuit quantum electrodynamics (QED) scheme has been
applied to the superconducting charge qubit \cite{Blais,Blais07},
phase qubit \cite{Sillanpaa}, and flux qubit \cite{Abd,Yang,Niem,Fedorov}.
Recently, in Refs. \cite{Abd,Niem,Fedorov} the phase-biasing scheme was studied
to provide a strong coupling strength between the resonator and the flux qubit
by sharing the qubit loop with the resonator.
This galvanic coupling, however, has difficulty in
switching on and off the coupling between the qubit and the resonator,
which is essential for a scalable design.
In this study, by introducing a {\it current-biasing} scheme for the flux qubit,
we offer a new circuit QED architecture,
where the flux qubit is coupled with the {\it current mode of
the resonator}, to obtain a strong coupling between the flux qubit and the resonator
without the galvanic link.
In this scheme, the flux qubit is biased by the oscillating current mode of
the resonator.

The current-biased dc-SQUID qubit (phase qubit) \cite{Martinis,Berkley,Steffen}
is controlled by an electric field, which provides a fast operation.
On the other hand, the flux qubit \cite{Mooij,Niskanen}
is operated at the optimal point where the first-order phase fluctuations vanish.
The present current-biased flux qubit operation is also performed
at a point optimally biased with respect to both the bias current
and the external magnetic flux, which may provide a long coherence time.
An oscillating bias current induces a Rabi oscillation between
qubit states.
If the bias current operation of the flux qubit is to be performed,
the number of the Josephson junctions in the flux qubit loop should be
three (in general, an odd number), and the junctions should be arranged in an
asymmetrical way.

\section{Current-biased flux qubit}
Figure \ref{fig1}(a) shows the  current-biased flux qubit, where
the current flows through the three-Josephson-junction qubit loop.
In the circuit QED architecture with the flux qubit, which we will
discuss later, the bias current comes from the oscillating current
mode of the resonator. For the time being, we consider an externally applied
oscillating bias current. The Hamiltonian of this system can be
derived semiclassically by using the quantum Kirchhoff relation.

First of all, let's consider a superconducting loop without a Josephson junction,
where the usual fluxoid quantization condition reads  \cite{Tinkham}
\begin{eqnarray}
\label{Tc}
-\Phi_t+(m_c/q_c)\oint \vec{v}_c \cdot d\vec{l}=n\Phi_0,
\end{eqnarray}
with $\vec{v}_c$ being the average velocity of Cooper pairs, $q_c$ being $2e$, and $m_c$ being $2m_e$.
The total magnetic flux threading the loop $\Phi_t$ is the sum of the external and the induced flux
$\Phi_{\rm t}=\Phi_{\rm ext}+\Phi_{\rm ind}$ with  the superconducting unit flux quantum $\Phi_0=h/2e$.
Then, the condition of Eq. (\ref{Tc}) is written as
\begin{eqnarray}
k{\it l}=2\pi (n+ f_t),
\end{eqnarray}
where  $f_t\equiv \Phi_t/\Phi_0=f+f_{\rm ind}$ with $f=\Phi_{\rm ext}/\Phi_0$ and
$f_{\rm ind}=\Phi_{\rm ind}/\Phi_0$, $l$ is the circumference of the loop, and $k$ is the
wave vector of the Cooper pair wavefunction.

For the current-biased flux qubit in Fig. \ref{fig1}(a), the fluxoid quantization
condition in Eq. (\ref{Tc}) is changed due to the phase differences $\phi_i$'s
in the circumference of the qubit loop as follows:
\begin{eqnarray}
\label{fqc}
(k_1+k_2)\frac{l}{2}=2\pi(n+f_t)-\phi_1-\phi_2-\phi_3.
\end{eqnarray}
Usually, for flux qubits, the contribution $(k_1+k_2)l/2$
is negligible in Eq. (\ref{fqc}) which then can be reduced to  $2\pi(n+f_t)-\phi_1-\phi_2-\phi_3=0$.
Rigorously, however, we keep this term for the time being
to derive the Hamiltonian of our qubit system.

The induced flux $\Phi_{\rm ind}$ is given by $\Phi_{\rm ind}=L_s (I_1-I_2)/2$ with
the self inductance $L_s$, and the current $I_{1(2)}$ in Fig. \ref{fig1}(a) is
\begin{eqnarray}
\label{I}
I_{1(2)}=\mp(n_cAq_c/m_c)\hbar k_{1(2)},
\end{eqnarray}
with $n_c$ being the Cooper pair
density and $A$ the cross section of the superconducting loop. Then, by
using Eq. (\ref{I}) and $\Phi_0=h/q_c$,
$f_{\rm ind}=\Phi_{\rm ind}/\Phi_0$ can be represented as $f_{\rm ind}=
-(L_s/L_K)(l/2)(k_1+k_2)/2\pi$ with $L_K = m_cl/An_cq^2_c$ being the
kinetic inductance \cite{flux}.
With this relation the condition in Eq. (\ref{fqc}) becomes
\begin{eqnarray}
\label{fqc2}
\left(1+\frac{L_s}{L_K}\right)(k_1+k_2)\frac{l}{2}
=2\pi\left(n+f-\frac{1}{2\pi}\sum^3_{i=1}\phi_i\right).
\end{eqnarray}

The dynamics of the Josephson junction is described in the
capacitively-shunted model, where the current relation is given by
$I=-I_{\rm c}\sin\phi+C\dot{V}$ with the capacitance of junction
$C$, $\dot{V}=dV/dt$, and voltage $V$ across the junction. This
relation can be rewritten by using the Josephson voltage-phase
relation $V=-(\Phi_0/2\pi){\dot \phi}$ as
\begin{eqnarray}
\label{Jc}
-(n_cAq_c/m_c)\hbar k_i=-I_{{\rm
c}i}\sin\phi_{i}-C_{i}(\Phi_0/2\pi)\ddot{\phi}_{i},
\end{eqnarray}
with the critical current of Josephson junction $I_{{\rm c}i}=2\pi E_{Ji}/\Phi_0$ ($i=1,2,3$)
and the Josephson coupling energy $E_{Ji}$.
Then, from Eqs. (\ref{fqc2}) and (\ref{Jc}) with
$k_0=k_1-k_2$, we have the quantum Kirchhoff relation
\begin{eqnarray}
\label{motion}
\left(\frac{\Phi_0}{2\pi}\right)^2C_{i}\ddot{\phi}_{i}&=&
\frac{\Phi^2_0}{2(L_s+L_K)\pi}\left(n+f-\frac{1}{2\pi}\sum^3_{i=1}\phi_i\right)\nonumber\\
&&-E_{Ji}\sin\phi_{i}\mp\frac{\Phi_0I_0}{4\pi},
\end{eqnarray}
where $I_{0}=-(n_cAq_c/m_c)\hbar k_{0}$.
Here, for $i=1,3$,  the last term of Eq. (\ref{motion}) is $+\Phi_0I_0/4\pi$
whereas for $i=2$ the sign is reversed as $-\Phi_0I_0/4\pi$.
For usual flux qubits, $L_K/L_s \sim 0.01$ \cite{Mooij};
thus, hereafter $L_K$ is neglected for simplicity.

The equation of motion, Eq. (\ref{motion}), can be obtained
from the Lagrange equation $(d/dt)\partial {\cal L}/\partial \dot{\phi}_i-\partial {\cal L}/\partial \phi_i=0$
with the Lagrangian
\begin{eqnarray}
\label{Lag}
{\cal L}(\phi_i,\dot{\phi}_i)&=&\sum^3_{i=1}\frac12 C_i\left(\frac{\Phi_0}{2\pi}\right)^2\dot{\phi}^2_i
-U_{\rm eff}(\{\phi_i\}),\\
\label{Ueff}
U_{\rm eff}(\{\phi_i\})&=&\sum^3_{i=1}E_{Ji}(1-\cos\phi_i)+\frac{\Phi_0I_0}{4\pi}(\phi_1+\phi_3-\phi_2)\nonumber\\
&&+\frac{\Phi^2_0}{2L_s}\left(n+f-\frac{1}{2\pi}\sum^3_{i=1}\phi_i\right)^2,
\end{eqnarray}
where the first term in Eq. (\ref{Lag}) comes from the charging energy
of the qubit system, $E_C=
\sum^3_{i=1}Q^2_i/2C_i$ with $Q_i=C_i(\Phi_0/2\pi)\dot{\phi_i}$.
The second term of Eq. (\ref{Ueff}) has a finite value owing to
the asymmetry of the qubit loop, giving rise to the coupling between
the bias current and the flux qubit.

\begin{figure}[t]
\vspace{-0cm}
\includegraphics[width=0.45\textwidth]{fpqubit.eps}
\vspace{-0cm}
\caption{(Color online) (a) Current-biased flux qubit. Two current states of the
three-Josephson-junction loop interact with the bias currents.
$I_{0}$ is the bias current applied across the capacitance
$C_{\mu\!\rm w}$, and $k_i$'s are the wave vectors of the Cooper pair wave function.
(b) Plot for the effective potential $U_{\rm eff}(\{\phi_\pm\})$
with $f=0.5$ and $\phi_-$=0. The potential tilts due to a finite bias current
$I_0$. The solid (dashed) line corresponds to positive (negative) $I_0$.
}
\label{fig1}
\end{figure}

In the usual experiments for flux qubits,
$\Phi^2_0/2L_s \sim O(10^3E_J)$ is much larger than the other energy
scale in the Lagrangian of Eq. (\ref{Lag}). Hence, the last term in
Eq. (\ref{Ueff}) can be removed, leaving  the  constraint
$g(\phi_i)=\phi_1+\phi_2+\phi_3-2\pi(n+f)=0$,
which is  the familiar flux quantization condition.
The Lagrangian then has the effective potential
\begin{eqnarray}
U_{\rm eff}(\{\phi_i\})\!=\!\!\!\sum^3_{i=1}E_{Ji}(1\!-\!\cos\phi_i)
\!+\!\!\frac{\Phi_0I_0}{4\pi}(\phi_1\!+\!\phi_3\!-\!\phi_2)
\end{eqnarray}
with the constraint $g(\phi_i)=0$. The Lagrange
equation of motion with the above constraint
produces the Kirchhoff
equations in the qubit loop of Fig. \ref{fig1}(a): $I_0=I_1+I_2$ and $I_1=I_3$.

We  introduce a coordinate transformation such as
$\phi_+=(\phi_2+\phi_3)/2$ and $\phi_- = (\phi_2-\phi_3)/2$. In the
usual flux qubit experiment, the two Josephson junctions are nearly
identical; thus, we set $E_{J2}=E_{J3}=E_J$. Although one can
treat the general case numerically, this case gives a clear and intuitive
picture for our qubit system. In this case, the effective potential
is given by
\begin{eqnarray}
\label{UeffT}
U_{\rm eff}(\{\phi_\pm\})&=&-E_{J1}\cos(2\pi f-2\phi_+)
-2E_{J}\cos\phi_+\cos\phi_- \nonumber\\
&&+\frac{\Phi_0I_0}{2\pi}(\pi f-\phi_+ -\phi_-).
\end{eqnarray}
Without the last term representing the linear coupling of the phase
to the external bias current $I_0$, the effective potential of Eq. (\ref{UeffT}) can have
local minima only for $\cos\phi_-=\pm 1$, {\it i.e.}, $\phi_-= j\pi$,
with $j$ being an integer, where the qubit states are formed (See Fig. \ref{fig2}(a)).

Let the flux qubit be at the degeneracy point $f=0.5$.
When there is no bias current $I_0=0$, $\phi_+ \approx
\pi/3(-\pi/3)$ for the counterclockwise (clockwise) current state
$|\downarrow\rangle ~(|\uparrow\rangle)$ with $\phi_-= 0$,
and  the qubit energy levels corresponding to the local minima of $U_{\rm eff}(\{\phi_\pm\})$
are degenerate,  $E_{\downarrow}=E_{\uparrow}$.
In this case, the qubit is optimally biased
with respect to both the current $I_0$ and the  magnetic flux $f$.
A finite bias current tilts the effective potential as shown in Fig. \ref{fig2}(b),
where  the direction of energy level tilt depends on the sign of $I_0$.
Consequently, the effective potential of Eq. (\ref{UeffT}) is expressed as
$(E_{\downarrow}-\Phi_0I_0\alpha/2\pi)|\downarrow\rangle\langle\downarrow|
+ (E_{\uparrow}+\Phi_0I_0 \alpha/2\pi)|\uparrow\rangle\langle\uparrow|$ apart from the constant,
where
\begin{eqnarray}
\alpha \approx |\phi_+|\approx \pi/3.
\end{eqnarray}

\begin{figure}[t]
\vspace{0.5cm}
\includegraphics[width=0.4\textwidth]{pot.eps}
\vspace{0.cm}
\caption{(Color online) (a) Effective potential of the current-biased flux qubit
in the plane of $(\phi_+, \phi_-)$, where the effective potential of the local minima decreases as
$\phi_+$ or $\phi_-$ increases. Here, we set $I_0/I_{\rm c}$=0.05, $f=0.5$ and $E_{J1}/E_J$=0.8.
(b) Plot for $U_{\rm eff}(\{\phi_\pm\})$
along the dashed line in the upper panel.
}
\label{fig2}
\end{figure}

The transitions between the two states,
$|\downarrow\rangle$ and $|\uparrow\rangle$, are induced by the charging
energy given by the first term of the Lagrangian in Eq. (\ref{Lag}).
Using the tight-binding approximation, we can write the Hamiltonian for our qubit system as
\begin{eqnarray}
\label{H}
{\cal H} \!&=&\! E_{\downarrow}|\downarrow\rangle\langle\downarrow|\!+\!
E_{\uparrow}|\uparrow\rangle\langle\uparrow|
- t_q(|\downarrow\rangle\langle\uparrow|\!+\!|\uparrow\rangle\langle\downarrow|) \nonumber\\
&& -\frac{\Phi_0I_0}{2\pi}\alpha (|\downarrow\rangle\langle\downarrow| -
|\uparrow\rangle\langle\uparrow|),
\end{eqnarray}
where $t_q$ is the transition rate between
$|\downarrow\rangle$ and $|\uparrow\rangle$ states.

In order to operate the single qubit states,
we apply a oscillating current,
\begin{eqnarray}
I_0=-I_{\rm b}\sin\omega t.
\end{eqnarray}
As shown in Fig. \ref{fig1}(b), the effective potential vibrates with the frequency $\omega$,
which produces oscillating diagonal terms in Eq. (\ref{H}).
In transformed coordinates, these terms appear off-diagonal,
giving rise to a Rabi oscillation.
The qubit states, $|0\rg$ and $|1\rg$, are the symmetric and the antisymmetric superpositions
of $|\da\rg$ and $|\ua\rg$, such that
\begin{eqnarray}
|0\rg&=&(|\da\rg+|\ua\rg)/\sqrt{2},\nonumber\\
|1\rg&=&(|\da\rg-|\ua\rg)/\sqrt{2}.
\end{eqnarray}
In the basis of $\{|0\rangle, |1\rangle\}$, the Hamiltonian is represented as
\begin{eqnarray}
\tilde{\cal H}=\frac{\hbar\omega_0}{2}\sigma_z +g\sin\omega t
\sigma_x, \label{OneT}
\end{eqnarray}
where $\hbar \omega_0=2t_q$ is the qubit frequency, and $\sigma_{x,z}$ and $I$
are the Pauli  and the identity matrices, respectively.
The coupling strength $g$ between the oscillating current and the qubit is given by
\begin{eqnarray}
\label{g}
g\equiv \frac{\Phi_0I_{\rm b}}{2\pi}\alpha.
\end{eqnarray}
Near resonance, $\omega\approx \omega_0$, with
a weak coupling $g/\hbar \ll \omega_0$,
one can apply the rotating wave approximation. \cite{Jaynes}
Then, we can observe a Rabi oscillation between the two qubit states $|0\rangle$ and$|1\rangle$
with a Rabi frequency $\Omega_{\rm R}^0=g/\hbar$.

Although we do not present it explicitly, in general the number of Josephson junctions
can be any odd integer larger than one.
In this case, at the two sides of the qubit loop divided by the bias current line,
the numbers of Josephson junctions should be different from each other so that
the symmetry of the flux qubit loop is broken.
Then, the resulting coupling  between the current and the phase of junction
enables current-driven operation of the flux qubit.
For the dc-SQUID (2-junction) qubit, the bias current flows across the Josephson junctions,
but the symmetry of the dc-SQUID does  not allow  bias-current
operation of the qubit states.

\section{Circuit QED}
The present current-biasing scheme for the flux qubit is implemented in
the circuit QED architecture. In Fig. \ref{fig3}, the qubit is
coupled to the resonator by an ac current flowing through a
capacitance. The Lagrangian ${\cal L}_r$ of the transmission line
resonator consists of the charge density mode $ q(x,t)$ and the
current density mode $I(x,t)$:
\begin{eqnarray}
{\cal L}_r=\int^{\frac{L}{2}}_{-\frac{L}{2}}dx\left(\frac{l}{2}I^2(x,t)-\frac{1}{2c} q^2(x,t)\right),
\end{eqnarray}
where $l$ and $c$ represent the inductance and the capacitance
per unit length, and $L$ the length of the resonator.
Introducing the variable $\theta(x,t)\equiv \int^x_{-L/2}dx'q(x',t)$,
the Lagrangian becomes
\begin{eqnarray}
{\cal L}_r=\int^{L/2}_{-L/2}dx[(l/2)(\partial_t\theta)^2-(1/2c) (\partial_x \theta)^2],
\end{eqnarray}
where the voltage and the current  of the resonator are given by
$V(x,t)=\frac{1}{c}\frac{\partial\theta(x,t)}{\partial x}$ and
$I(x,t)=\frac{\partial\theta(x,t)}{\partial t}$, respectively.
For example, the voltage and the current of the resonator for the second harmonic mode can be represented in terms of
the boson creation and annihilation operators $a^\dagger(t)$ and $a(t)$ as
 \cite{Blais}
\begin{eqnarray}
\label{VI}
V(x,t)&=&\sqrt{\frac{\hbar\omega_r}{cL}}\cos\left(\frac{2\pi x}{L}\right)[a(t)+a^\dagger(t)],\nonumber\\
I(x,t)&=&-i\sqrt{\frac{\hbar\omega_r}{lL}}\sin\left(\frac{2\pi x}{L}\right)[a(t)-a^\dagger(t)],
\end{eqnarray}
where $\omega_r=2\pi v/L$ and $v=\sqrt{1/lc}$.
The current profile for the second harmonic mode is shown in Fig. \ref{fig3}.

\begin{figure}[t]
\vspace{0.5cm}
\includegraphics[width=0.48\textwidth]{cQED.eps}
\vspace{0.3cm}
\caption{(Color online) Schematic diagram  and equivalent lumped circuit representation
of a three-junction flux qubit coupled to a transmission line resonator.
The flux qubit interacts with the current mode of the resonator. The arrows
in the schematic diagram and equivalent circuit show the flow of oscillating current.
As shown in the equivalent circuit, the oscillating current flows between the resonator and the
qubit through the capacitance.
Here, $L$ is the length of the resonator, and $d$ is the width of qubit loop.}
\label{fig3}
\end{figure}

In the circuit QED scheme with superconducting charge qubit,
the qubit interacts with the voltage mode $V(x,t)$. In contrast, the present current-biased flux
qubit is coupled to the temporal fluctuation of local charge density, $I(x,t)$,
corresponding to the bias current applied to the qubit.
In Fig. \ref{fig3} the resonator and the qubit are coupled by a capacitance
through the region $-d/2<x<d/2$, where the charge fluctuation ${\dot q}(x,t)$ in the resonator
produces the current flow into the qubit,
$I_0(t)=\int^{d/2}_{-d/2}{\dot q}(x,t)dx.$
As a result, following the current conservation condition in the resonator,
${\dot q}(x,t)=\partial I(x,t)/\partial x$, and  Eq. (\ref{VI}), the current flowing into
the qubit with width $d$ is given by
\begin{eqnarray}
\label{I0}
I_0(t)&=&I(d/2,t)-I(-d/2,t)\nonumber\\
&=&-2i\sqrt{\frac{\hbar\omega_r}{lL}}\sin\left(\frac{\pi d}{L}\right)[a(t)-a^\dagger(t)].
\end{eqnarray}
The interaction between the flux qubit and the bias current $I_0(t)$
of the resonator is given by the last term of Eq. (\ref{H}).
Then, from Eqs. (\ref{H}) and (\ref{I0}), the total Hamiltonian
at the degeneracy point ($E_\downarrow=E_\uparrow$) is given by
\begin{eqnarray}
 {\cal H}&=&\hbar\omega_r\left(a^\dagger
a+\frac12\right)-t_q(|\downarrow\rangle\langle\uparrow|\!+\!|\uparrow\rangle\langle\downarrow|)\nonumber\\
&&+i g(|\downarrow\rangle\langle\downarrow| -
|\uparrow\rangle\langle\uparrow|)(a-a^\dagger),
\end{eqnarray}
where the first term comes from the oscillating mode in the
resonator. The first two terms of Eq. (\ref{H}) disappear
at the degeneracy point.
In the basis of $|0\rangle$ and $|1\rangle$, the Hamiltonian is
written as
\begin{eqnarray}
\label{Hc} {\tilde {\cal H}}=\hbar\omega_r a^\dagger a
+\frac{\omega_a}{2}\sigma_z+i g\sigma_x(a-a^\dagger),
\end{eqnarray}
with $\hbar \omega_a=2t_q$.
Here, the single qubit gate is performed by a resonant
external driving microwave field \cite{Blais07}.

Since usually $d\ll L$, we have the expression of the coupling $g$ as follows:
\begin{equation}
\label{gc}
g \approx \alpha \Phi_0 \sqrt{\frac{\hbar\omega_r}{lL}}\left(\frac{d}{L}\right).
\end{equation}
Using the usual experimental values for the parameters \cite{Abd} such that
$d=3 ~\mu$m, $L$=5 mm, $lL$=2.5 nH, $\omega_r/2\pi=$15 GHz, and $\alpha=\pi/3$,
we estimate the coupling strength to be $g/h\sim 120$ MHz.
For inductive coupling between the qubit and the resonator,
the coupling strength, $g_{\rm IC}=\Phi I_c\sin\alpha$
with $\Phi$ being the magnetic flux threading the qubit due to
the resonator current, can also be estimated to be
\begin{eqnarray}
g_{\rm IC}\approx \frac{\mu_0 d^2}{2\pi R}\sqrt{\frac{\hbar\omega_r}{lL}}I_c\sin\alpha,
\end{eqnarray}
where $R$ is the mean distance  between the resonator and the qubit loop
and $I_c=2\pi E_J/\Phi_0$.
With the same parameter values, we see that $g_{\rm IC}$ is smaller than $g$
by an order of magnitude.
We obtain the inductive coupling strength $g_{\rm IC}/h\sim 12$ MHz with
$R=5~\mu$m and $E_J/h=200$ GHz.
Note that since the qubit is located at the nodal point of the current oscillation for the second harmonic mode,
the inductive coupling at this point is negligible.
The above value of inductive coupling has been obtained for the first harmonic mode.
We have also assumed that the capacitance density is nearly uniform along the resonator.
In reality, the capacitance around the center of the resonator can be much higher due to
the presence of the qubit loop, which can potentially  provide a much
stronger coupling $g$ in the present scheme.

\section{Decoherence property}

The qubit state of the present current-biased flux qubit is
driven in a different way from that of the flux-driven flux qubit;
thus, the decoherence property will also differ from each
other. According to a recent review for the phase qubit
\cite{QIP}, the dephasing times of the phase qubit and the
three-Josephson junction flux qubit are comparable with each
other. Since the only difference between the present
current-biased flux qubit and the usual current-biased dc-SQUID
qubit (phase qubit) is the number of Josephson junctions in the
qubit loop, the decoherence property of both qubits can be analyzed
in a similar way.

In a recent experiment \cite{Steffen2} for the phase qubit, the
capacitance of Josephson junction is $\sim$ 50fF while the
three-Josephson junction flux qubit has a typically smaller value of
capacitance of $\sim$ 3fF ~\cite{Wal} with a small area of the
Josephson junction. For both qubits, a large shunted capacitance
may reduce the decoherence from charge fluctuation. We employ the
typical parameters of the three-Josephson junction flux qubit for
the present current-biased qubit. Since both the capacitance and
the critical current of the Josephson junction scale as the area of
the junction, the critical current $I_c$ of junction is also much
smaller.

For the present current-biased flux qubit, the dephasing due to the
bias-current noise may cause  decoherence of the qubit state,
as in the phase qubit \cite{QIP}.  The current noise is related to
the $1/f$ charge noise with a spectral density $S^*_q(1 {\rm Hz})/f$.
The phase noise is given by $\langle\phi^2\rangle\approx
[S^*_q(1{\rm Hz})/C\Delta U] (2/3)\ln(1.778\omega_{10}t)$,
\cite{Nam} where $C$ is the capacitance of the Josephson junction,
$\Delta U$ is the barrier height of the potential, and
$\omega_{10}$ is the qubit level spacing. Although for the present
qubit,  the spectral density $S^*_q(1 {\rm Hz})$ and the
capacitance $C$ are small, these contributions cancel each other in
$\langle\phi^2\rangle$ because both the spectral density and the
capacitance scale as the area of the junction \cite{Nam}. Further,
the qubit level spacing $\omega_{01}$ is similar between the two
qubits. For the operation of the present qubit,
we need not tilt the potential; thus, the barrier hight of the
potential  $\Delta U$ remains large. Hence, we consider
the dephasing due to current noise in present qubit to still
be comparable to the flux-driven flux qubit.
On the other hand, the phase noise due to the critical-current fluctuation and
to the flux $1/f$ fluctuations is given by $\langle\phi^2\rangle\approx (S^*_I(1{\rm
Hz})L_J/\Delta U) \ln(0.401/f_m t)(\omega_{01}t)^2/6$  \cite{Nam},
with the Josephson inductance $L_J=\Phi_0/2\pi I_c\cos\phi$ and
the low-frequency cutoff $f_m$. For the phase noise,
the argument is similar because
both the spectral density $S^*_I(1{\rm Hz})$ and the critical
current $I_c$ scale as the area of the junction.

\section{Summary}
We propose a new circuit QED architecture for the three-Josephson-junction flux qubit,
where the flux qubit is coupled to the temporal charge density fluctuations of the
transmission line resonator.
The flux qubit is controlled by using a bias current.
When three Josephson junctions are arranged asymmetrically,
the energy levels of  two qubit states with different
chiralities couple to the external bias current.
Rabi oscillations can be induced by an ac current at an optimal point
with respect to both the bias current and the external magnetic flux.
Remarkably, the coupling between
the qubit and the resonator is strongly enhanced compared to conventional inductive coupling.

\begin{acknowledgments}
We wish to thank S. Girvin for useful discussions and valuable suggestions.
This work was supported by the Korea Research Foundation Grant
funded by the Korean Government (MOEHRD, Basic Research Promotion
Fund) through KRF-2008-313-C00243.
\end{acknowledgments}


\subsection{Type II Quasars} Type II quasars are the obscured counterparts
of the classical quasar population predicted by AGN unification models.  In
the optical, Type II quasar candidates are traditionally selected as
objects with narrow permitted emission lines and high ionization line
ratios. Zakamska et al (2003) were the first to compile large  samples of
Type II quasar candidates in the Sloan Digital Sky Survey.  Their objects
were selected to lie in the redshift range $0.3 < z < 0.83$ in order to
disfavour selection of low luminosity objects. The  main disadvantage of
applying such a redshift cut in SDSS is that the Type II quasars are
not selected from a magnitude-limited survey of galaxies, so their
demographics and contribution to black hole growth cannot be studied in
detail, nor can direct comparisons be made to other classes of AGN.

Zakamska et al's  Type II  selection was based on a cut in the [OIII]/H$\beta$ line ratio
as well as the presence of very high-ionization lines such as [NeV]. The
[OIII] line luminosities of the sample range from $3 \times 10^7 L_{\odot}$
to close to $10^{10} L_{\odot}$ and the [OIII]/[OII] line ratios lie in the
range 1-10, i.e. very similar to the most luminous objects in our sample.
In follow-up work, Zakamska et al (2004) found that 143 of these objects
had counterparts in the FIRST radio catalogue. They speculate that this may
represent an overestimate of the true fraction, because the SDSS targetted
many FIRST radio sources for spectroscopy.  Hubble Space Telescope
imaging of the host galaxies of a subset of 9 Type II quasars with [OIII]
line luminosities greater than $3 \times 10^8 L_{\odot}$ reveal  that 6 out
of the 9 are elliptical galaxies well-fit by de Vaucouleurs light profiles
and the other 3 have a minor disk component (Zakamska et al 2006).  Most recently, Liu et al
(2013a,b) have obtained IFU data for 11 of the  most luminous, radio-quiet
objects in their sample and show the the [OIII] emission is very extended
with a mean diameter of 28 kpc and is spherical in morphology.  The
majority of nebulae show blue-shifted excesses in their line profiles across
most of their extents, signifying gas outflows. These authors estimate a
median outflow velocity of 760 km/s, similar to or above the escape
velocities from the host galaxies.

In Figure 15, we present a compilation of images of the mid-IR excess AGN
in our sample with the highest [OIII] luminosities ($> 10^8 L_{\odot}$). As
can be seen, the majority also have elliptical morphologies and centrally
concentrated light profiles.  The galaxy in the bottom row with a strange
red handle-liked protuberance is the famous ``teacup'' AGN, the nearest
known radio-quiet type II quasar with a redshift z=0.08056, first identified
by Reyes et al (2008). In recent work, Harrison
et al (2015) have studied the ionized gas kinematics in this object and
find evidence for an outflow with velocity 740 km/s. We thus believe it is
likely that there is a close correspondence between the Type II quasar
population and the brightest objects in our mid-IR excess sample.
We have also checked whether there is a significant population of
optically-selected AGN with [OIII] luminosities  greater than ($> 10^8 L_{\odot}$)
that are not included in our mid IR-excess sample. We find only 25 out of 128 such
objects, indicating that the mid-IR and type II quasar
selection techniques yield essentially the same set of objects at the very
highest [OIII] luminosities.

\begin{figure*}
\includegraphics[width=135mm]{radiocentral_mostlum.eps}
\caption{A compilation of SDSS cut-out images of the mid-IR excess AGN
in our sample with the highest [OIII] luminosities ($> 10^8 L_{\odot}$).
\label{models}}
\end{figure*}

What about the less luminous mid-IR excess AGN? In Figure 16, we
present a compilation of objects with D$_n$(4000) in the range 1.0-1.2,
indicative of current bursts of star formation. As shown in Figures 9 and
10, these galaxies have more moderate [OIII] luminosities in the range
$10^7-10^8 L_{\odot}$.  As can be seen, there are many more interacting
pairs and triples in this sample than in the high-luminosity sample.  If
the galaxies in Figures 15 and 16 constitute different phases of the same
merger-induced black hole fuelling events, the host galaxies shown in
Figure 16 could be said to be  an earlier stage of the merging process.

\begin{figure*}
\includegraphics[width=135mm]{radiocentral_moststarf.eps}
\caption{A compilation of SDSS cut-out images of the mid-IR excess AGN
in our sample with D$_n$(4000) in the range 1.0-1.2, indicative
of current bursts of star formation.
\label{models}}
\end{figure*}

\subsection {AGN selected only by a mid-IR colour criterion}
As discussed in Paper I, the selection of AGN using WISE photometry has generally
been carried out using colour cuts designed to avoid the main locus of star-forming
galaxies (e.g. Stern et al 2012). We showed in this paper that the W1-W2 colours
of a significant fraction of such objects remain very red out to large ($>$ 5 kpc)
radii, suggesting that a significant fraction of the mid-IR emission may arise
from an extended distribution of dust in the galaxy heated by collisions with
electrons from  surrounding hot halo gas, rather from a central, parsec-scale
torus. We include an additional radio-loud criterion
to increase the likelihood that the sample contains a high proportion of
galactic nuclei with black holes that are currently accreting.
In other words, our  goal is to maximize the purity of the
AGN sample with the data at hand.

The danger with the procedure adopted in Paper I with respect to past
mid-IR selection procedures, is that such an AGN sample is not complete.
This may occur, for example, if the emission comes from a jet that is variable over timescales
that are short compared to the lifetime of the torus.
Figure 17 compares some of the key properties of the full mid-IR excess sample
and the sample with the additional cut on radio luminosity.
Mid-IR excess galaxies are  identified as outliers in SFR/$M_*$ versus D$_n$(4000)
space for galaxies with $\log$ SFR/$M_* > -11$ and in H$\delta_A$ versus D$_n$(4000) space
for galaxies with lower
specific star formation rates. We bin up the two planes in
intervals of 0.15 in D$_n$(4000), 0.25 dex in $\log$ SFR/$M_*$ and 0.1 in H$\delta_A$
and calculate the distribution of W1-W2 colours in
each bin. Outliers or mid-IR excess galaxies are defined to have colours
that lie above the upper
95th percentile point of the distribution.
In Figure 17, we plot galaxy properties as a function of the quantity
$\Delta$(W1-W2), the difference between the measured W1-W2 colour of the galaxy and the
colour that delineates the upper 95th percentile cut.

In the top left panel, we  plot the fraction of galaxies as a function of
$\Delta$(W1-W2) for the mid-IR excess sample without the radio-loud cut (black
histogram) compared to the fiducial sample (black
triangles). As can be seen, the sample with the radio loud cut includes a
much more pronounced tail of objects with large $\Delta$(W1-W2), i.e. with
mid-IR colours that are very far from the stellar locus. This
supports our claim that the radio selection is increasing the purity of the
AGN sample.

\begin{figure}
\includegraphics[width=90mm]{threshprope.ps}
\caption{ The top left panel shows the fraction of the sample
in bins of $\Delta (W1-W2)$, where $\Delta (W1-W2)$  is the difference between the W1-W2 colour and the
colour that delineates the 95th percentile cut.
The result for the mid-IR excess sample without any constraint from
the radio is shown as a black histograms, while black triangles show
the result for the fiducial sample.
In the next three panels,  stellar mass, [OIII] line luminosity and
ionization parameter
are plotted as a function of $\Delta (W1-W2)$.
Each plotted point
corresponds to a bin containing a fixed number (200) galaxies, so
the noise due to Poisson sampling of the underlying distribution
remains constant in each diagram. Black line/triangles  show the median
of the distribution, red lines/triangles  the upper 75th percentile of the
distribution and blue lines/triangles  the lower 25th percentile. Once again,
lines denote the full mid-IR excess sample, while triangles denote the
fiducial sample.
\label{models}}
\end{figure}

In the next three panels, trends in stellar mass of the host
galaxy, [OIII] line luminosity and ionization parameter [OIII]/[OII] are
shown as a function of $\Delta$(W1-W2) for the two samples. As can be seen,
[OIII] luminosity and
ionization parameter  increase for systems with larger W1-W2 colours in a similar
manner for both classes of object.
This implies  that the main effect of the radio
selection is to boost the fraction of galaxies with the most extreme W1-W2
colours, which are also the most optically luminous systems with the
highest ionization parameters.  More detailed studies using spatially
resolved data will be necessary to figure out  the physical nature
of the central radio emission (jet or central starburst) and whether
or not the radio source is influencing the structure of the dust emission
in the central regions of the galaxy.

\section {Summary} We now summarize the main results of our analysis.  We
have  studied the narrow emission line properties and stellar populations
of a sample of 1385 radio-detected, mid-IR excess AGN in order to
understand the physical conditions in the interstellar medium of these
objects. We compare these systems with a control sample of 50,000 AGN
selected by their optical emission line ratios that do not have a
significant mid-IR excess. Our main conclusions
are the following: \begin{itemize} \item The mid-IR excess AGN populate the
high ionization branches of the [OIII]/H$\beta$ versus
[OI]/H$\alpha$/[SII]/H$\alpha$ BPT diagrams, whereas the control sample AGN
cluster near the star-forming locus and have lower ionization parameters on
average.  \item The mid-IR excess AGN have [OIII] luminosities that are an
order of magnitude large on average than the control sample AGN.  \item The
mid-IR excess AGN have higher electron densities, but similar metal
abundances to the control sample.  \item The H$\delta_A$ versus D$_n$(4000)
diagrams show that a much larger fraction of the host galaxies of mid-IR
excess AGN have experienced recent bursts of star formation. These recent
starburst galaxies have lower stellar metallicities and higher Mg/Fe
ratios.  \item The number densities of mid-IR excess AGN are a 1000 times
smaller than those of control sample AGN at low [OIII] luminosities ($\sim
10^{6} L_{\odot}$, but at the very highest [OIII] luminosities probed by
our sample ($\sim 10^{9} L_{\odot}$), mid-IR excess AGN become more
populous by a factor of 10.  \item  Mid-IR excess AGN contribute about half
the total present-day black hole growth in galaxies with  stellar
masses larger than $10^{11} M_{\odot}$, whereas control sample AGN are
currently the dominant contributor in lower
mass systems.  \end {itemize}

It is well known that the AGN population evolves strongly to higher
luminosities at higher redshifts, and it is likely that AGN similar to the
mid-IR excess population studied in this paper become much more populous.
We note that more than 95\% of all AGN in the parent sample with
[OIII] luminosities greater than $10^{8}$ L$_{\odot}$ are included in
the mid-IR excess/radio sample studied in this paper, suggesting that
the two selection techniques converge at the highest luminosities.

The future usefulness of our low redshift sample will lie in spatially
resolved spectroscopic follow-up studies of various kinds, as in the
Harrison et al study of the teacup AGN. These studies are required in order
to understand how accretion onto the central supermassive black hole is
occurring, the physical origin and location of the very high ionization gas
in these systems, and the impact of the energetic processes occurring near
the black hole on the interstellar medium of the host galaxy. The
establishment of a technique that selects a {\em population} of AGN seen at
different phases along a starburst cycle is also interesting for more
detailed follow-up programs. Although `AGN feedback' in the form of
extended outflowing gas is now established in a variety of  AGN
sub-populations such as the most luminous Type II quasars and radio
galaxies, understanding the global ubiquity, energetics  and duty cycle of
the feedback  will require more carefully controlled statistical
approaches.

Finally, our sample is an interesting one for understanding
the relation between AGN activity, galaxy-galaxy interactions,
mergers between black holes, and the origin of powerful AGN driven outflows of gas.
It is rather interesting that although the number densities of IR excess
and control sample AGN are very different, the integrated [OIII] emissivity
in both classes of objects peaks at a stellar mass of $\sim 10^{10.5} M_{\odot}$.
We note that this was shown for the AGN population as a whole in Heckman \& Kauffmann
(2004, see their Figure 4). This value  ($10^{10.5} M_{\odot}$)
corresponds closely to the transition mass where the galaxy population switches
over from a blue, star-forming, disk-dominated population to a red, passive, bulge-dominated
one. Energetic feedback from AGN has been hypothesized to cause this transition,
but considerable uncertainty remains as to how this occurs in practice. Some models
assume that a feedback mode associated with radio galaxies accreting hot gas
at the centers of massive dark matter halos is responsible
for this transition (Croton et al 2006), while others invoke quasar-driven feedback
triggered by galaxy-galaxy mergers (Hopkins et al 2006). Identification of
a population of very luminous AGN clearly associated with galaxy-galaxy mergers
is the first step to answering this question empirically.

\section*{Acknowledgments}
I thank Patricia Sanchez-Blazquez and Tim Heckman for helpful discussions and Mazda Adli
for his support.


\section{Introduction}
\label{section:introduction}

\enlargethispage{0.5 cm}

Radioactive decays in which photons with energies above $Q=2039$~keV
are emitted are expected to be a significant source of background for
the GERmanium Detector Array, {\sc GERDA}~\cite{proposal}. {\sc GERDA}
is an experiment which is currently being constructed and has as aim
the search for the neutrinoless double beta-decay ($0\nu\beta\beta$)
of the germanium isotope $^{76}$Ge. \\

Methods to distinguish between electrons and multiply scattered
photons using the time structure of the germanium detector response,
or pulse shape, are presented in this paper. The pulse shape depends
on the location and the spatial distribution over which energy is
deposited inside the detector in a single event. Photons in the
MeV-energy region will predominantly Compton-scatter and deposit
energy at locations separated by centimeters. These events are
referred to as {\it multi-site events}. In contrast, electrons in the
same energy region have a range of the order of a millimeter. Events
of this kind are referred to as {\it single-site events}. \\

Pulse shape analysis methods have been developed for nuclear
experiments such as {\sc AGATA}~\cite{agata} and {\sc
GRETA}~\cite{greta} as well as for double beta-decay
experiments~\cite{Hellmig:2000xp,HM,IGEX,Aalseth:2000hy,Elliott:2005at}. In
the context of the latter these techniques are now extended to
segmented detectors. In this study the focus is on the pulse shape
analysis after the application of a single segment requirement as
presented in~\cite{Abt:2007rg}. The performance of the pulse shape
analysis with and without segment information is compared based on
data taken with an 18-fold segmented {\sc GERDA} prototype
detector. \\

The experimental setup and the collected data sets are described in
Section~\ref{section:setup}. The accompanying Monte Carlo simulation
is introduced in Section~\ref{section:simulation}. A parameter
accessible in simulations which is a measure of the volume over which
energy is deposited inside the detector is defined. A definition of
single-site and multi-site events is derived from the Monte Carlo data
sets based on this parameter.  The fraction of single-site and
multi-site events in the data sets is estimated. Three analysis
methods are presented in Section~\ref{section:methods} and these
methods are applied to the data sets taken with the prototype
detector. The results are summarized in
Section~\ref{section:results}. Conclusions are drawn in
Section~\ref{section:conclusions}.

\section{Experimental setup and data sets}
\label{section:setup}

\subsection{Experimental setup and data taking}

The segmented germanium detector under study is the first segmented
{\sc GERDA} prototype detector. The true coaxial cylindrical crystal
has a height of 70~mm, an outer diameter of 70~mm and a central bore
with a diameter of 10~mm. It is 18-fold segmented with a 6-fold
segmentation in the azimuthal angle $\phi$ and a 3-fold segmentation
in the height $z$. It was operated in a conventional test
cryostat. Signals from the core and the segment electrodes were
amplified and subsequently digitized using a 14-bit ADC with a
sampling rate of 75~MHz. The energy and the pulse shapes of the core
and the 18~segment electrodes were recorded for each event. The pulse
shape data consists of 300 13.3~ns samples of the integrated charge
amplitude. The onset of the signal was delayed by one~$\mu$s. The
(full width at half maximum) energy resolution of the core electrode
was 2.6~keV at energies around 1.3~MeV, the energy resolutions of the
segment electrodes ranged from 2.4~keV to 4.8~keV with an average
segment energy resolution of 3.3~keV. Details of the experimental
setup and the detector performance can be found
in~\cite{Abt:2007rf}. \\

A 100~kBq $^{228}$Th source was placed at $z=0$~cm and $r=17.5$~cm
with respect to the detector center ($z=0$~cm, $r=0$~cm) facing
towards the center of a segment, $S$, located in the middle row. Two
data sets were taken with different trigger conditions labeled
$TR_{C}$ and $TR_{S}$. The former trigger condition requires the core
electrode to show an energy above 1~MeV. The collected data set is
referred to as {\it core data set} and contains $127\,000$~events. The
latter trigger condition requires segment~$S$ to show an energy above
1~MeV. The collected data set is referred to as {\it segment data set}
and contains $420\,000$~events. As an example,
Figure~\ref{fig:example} shows a pulse shape measured with the core
(left) and with the segment~$S$ electrode (right) for an event in the
segment data set. The core-energy spectra will be shown in
Section~\ref{subsection:application}. \\

\begin{figure}[ht!]
\center
\mbox{\epsfig{file=example_7_new.eps,width=0.9\textwidth}}
\caption{Pulse shape measured with the core (left) and with the
segment~$S$ electrodes (right) for an event in the segment data
set. The energy of $1758$~keV seen in the core is completely contained
in segment~$S$. The starting time is chosen arbitrarily in this
example. The amplitude is in arbitrary units but the scale is the same
for both pulse shapes. The pulse shapes are dominated by different
charge carrier types.
\label{fig:example}}
\end{figure}

\pagebreak

\subsection{Event selection}
\label{subsection:selection}

A pre-selection applied to the segment data set collects events with
energy deposited only in one segment. It requires the energy measured
in segment~$S$ to be the same as the energy measured in the core
within $\pm5$~keV, according to about $\pm 4~\sigma$ given the energy
resolution. In total, $150\,396$~events fulfill the pre-selection
criterion. \\

Four data samples each are selected from the core and segment data
sets. The data samples are defined by the energy measured in the core
and are labeled:
\begin{itemize}
\item $DEP$: The sample contains events with a core energy in the region
of \mbox{$(1593\pm5)$~keV}. These events are associated with the
double escape peak of the $2615$~keV $^{208}$Tl photon. The photon
produces electron-positron pairs of which the positron subsequently
annihilates. Both 511~keV annihilation photons escape the
detector. The energy is predominantly deposited on a millimeter-scale;
i.e., locally.
\item $\Gamma_{1}$: The sample contains events with a core energy in the
region of \linebreak \mbox{$(1620\pm5)$~keV}. These events are
associated with photons of this energy produced in the decay of
$^{212}$Bi. The photons mostly scatter multiple times before their
energy is fully deposited inside the detector.
\item $\Gamma_{2}$: The sample contains events with a core energy in the
region of \linebreak \mbox{$(2615\pm5)$~keV}. These events are
associated with photons of this energy produced in the decay of
$^{208}$Tl. The photons mostly scatter multiple times before their
energy is fully deposited inside the detector.
\item $ROI$: The sample contains events with a core energy in the region
of interest, $(2039\pm50)$~keV. These events are predominantly
associated with Compton-scattered photons from $^{208}$Tl.
\end{itemize}

The requirements of the trigger, pre-selection and event selection are
listed in Table~\ref{table:datasets}. Also the number of events in the
corresponding data samples are shown. The amount of background in each
data sample, as estimated from taking spectra without the $^{228}$Th
source present, was found to be less than 1\%. \\

\begin{table}[ht!]
\caption{Requirements of the trigger, pre-selection and event selection,
and the number of events in the corresponding data samples. $E_{C}$
and $E_{S}$ are the energies seen in the core and in segment~$S$,
respectively.
\label{table:datasets}}
\center
\begin{tabular}{llr}
\\
\hline
Cut                      & Condition                                                          & Events \\
\hline
Trigger ($TR_{C}$)       & $E_{C} > 1$~MeV                                                 & $127\,000$ \\
Pre-selection            & -                                                               & $127\,000$ \\
Selection ($DEP$)        & $\left| E_{C} - 1593\mathrm{~keV} \right| < \phantom{0}5$~keV & $1673$ \\
Selection ($\Gamma_{1}$) & $\left| E_{C} - 1620\mathrm{~keV} \right| < \phantom{0}5$~keV & $1965$ \\
Selection ($\Gamma_{2}$) & $\left| E_{C} - 2615\mathrm{~keV} \right| < \phantom{0}5$~keV & $22\,924$ \\
Selection ($ROI$)        & $\left| E_{C} - 2039\mathrm{~keV} \right| < 50$~keV           & $6\,431$ \\
\hline
Trigger ($TR_{S}$)       & $E_{S} > 1$~MeV                                                 & $420\,000$ \\
Pre-selection            & $\left| E_{C} - E_{S} \right| < 5$~keV                          & $150\,396$ \\
Selection ($DEP$)        & $\left| E_{C} - 1593\mathrm{~keV} \right| < \phantom{0}5$~keV & $3492$ \\
Selection ($\Gamma_{1}$) & $\left| E_{C} - 1620\mathrm{~keV} \right| < \phantom{0}5$~keV & $1972$ \\
Selection ($\Gamma_{2}$) & $\left| E_{C} - 2615\mathrm{~keV} \right| < \phantom{0}5$~keV & $19\,243$ \\
Selection ($ROI$)        & $\left| E_{C} - 2039\mathrm{~keV} \right| < 50$~keV           & $7707$ \\
\hline
\end{tabular}
\end{table}

\section{Monte Carlo simulation}
\label{section:simulation}

The GEANT4~\cite{geant4} based MaGe~\cite{MaGe} framework was used to
simulate the prototype detector setup (for details and a validation of
this particular simulation see~\cite{Abt:2007rg}). A Monte Carlo study
was performed to estimate the spatial distribution over which energy
is deposited in the detector for events in the different data
samples. A $^{228}$Th source was simulated. The trigger, pre-selection
and event selection requirements discussed in the previous section
were applied to the Monte Carlo data. The data sets are referred to as
{\it core} and {\it segment Monte Carlo data sets}.

A measure for the spatial distribution over which energy is
distributed inside the detector is the radius $R_{90}$. This is
defined as the radius inside which 90\% of the energy in a single
event is deposited; for a detailed discussion
see~\cite{segmentation}. Figure~\ref{fig:R90} shows the distribution
of $R_{90}$ for the $DEP$, $\Gamma_{1}$, $\Gamma_{2}$ and $ROI$
samples for the core (left) and segment (right) Monte Carlo data
sets. All distributions are normalized to unity. The $R_{90}$
distributions range from 0.1~mm ($\log_{10}(R_{90})=-1$) up to 7~cm
($\log_{10}(R_{90})=1.8$). The $DEP$ samples are dominated by events
with $R_{90}$ in a region from 0.1~mm to 1~mm. A long tail towards
larger radii is visible and mostly due to events in the underlying
Compton-shoulder of $^{208}$Tl and events in which electrons undergo
hard bremsstrahlung processes. The $R_{90}$ distributions for the
$\Gamma_{1}$ and $ROI$ samples have two prominent regions each, one at
radii from 0.3~mm to 1~mm and a second from 3~mm to 6~cm.  The latter
one is due to multiply scattered photons whereas the former is due to
photons with higher energy which only scatter once and then leave the
detector. The $R_{90}$ distributions for the $\Gamma_{2}$ samples
range from 0.3~mm to about 7~cm with a maximum at around 2~cm for the
core Monte Carlo data sample and at around 1~cm for the segment Monte
Carlo data sample. The sample is dominated by events in which photons
scatter multiple times. No peak at small $R_{90}$ is visible. \\

It is expected that the single segment requirement rejects events with
large values of $R_{90}$. Indeed, the distributions of $R_{90}$ in the
segment Monte Carlo data samples are suppressed in the region above
1~cm. The peaks between 0.1~mm and 1~mm in the $DEP$, $\Gamma_{1}$ and
$ROI$ samples are more pronounced in this case. \\

\begin{figure}[ht!]
\center
\begin{tabular}{cc}
\begin{minipage}[ht!]{0.45\textwidth}
\mbox{\epsfig{file=R90_core.eps,width=\textwidth}}
\end{minipage}
&
\begin{minipage}[ht!]{0.45\textwidth}
\mbox{\epsfig{file=R90_segment.eps,width=\textwidth}}
\end{minipage} \\
\end{tabular}
\caption{Normalized distributions of $R_{90}$ for the $DEP$,
$\Gamma_{1}$, $\Gamma_{2}$ and $ROI$ samples for the Monte Carlo core
(left) and segment data sets (right). Single-site events (SSE) and
multi-site events (MSE) are defined by requiring $R_{90} < 2$~mm and
$R_{90} > 2$~mm (dashed line) as discussed in the text.
\label{fig:R90}}
\end{figure}

Single-site and multi-site events are defined by requiring
$R_{90}<\overline{R}$ and $R_{90}>\overline{R}$, respectively, where
$\overline{R}$ is a chosen parameter value. The distributions of
$R_{90}$ for the $DEP$ samples suggest $\overline{R}=2$~mm
($\log_{10}(\overline{R})=0.3$). Also, due to the sampling rate of
75~MHz and the average drift velocity of charge carriers
($\mathcal{O}(10^{8})$~mm/s) energy deposits closer than about 2~mm
cannot be resolved. The fractions of single-site events in the Monte
Carlo data samples are thus defined and summarized in
Table~\ref{table:fraction}. Also listed are the corresponding
systematic uncertainties of the fractions which are derived by varying
the parameter $\overline{R}$ by $\pm1$~mm.

\begin{table}[ht!]
\caption{Fractions of single-site events in the Monte Carlo data samples.
The errors are derived by varying the parameter $\overline{R}$ by
$\pm1$~mm.
\label{table:fraction}}
\center
\begin{tabular}{lcccc}
\\
\hline
Monte Carlo data samples & $DEP$ & $\Gamma_{1}$ & $\Gamma_{2}$ & $ROI$ \\
& ($1593$~keV) & ($1620$~keV) & ($2615$~keV) & (2039~keV) \\
\hline
Core samples    & $(77.9^{+1.6}_{-3.4})$\% & $(30.5^{+4.0}_{-3.6})$\% & $(12.2^{+\phantom{0}6.0}_{-\phantom{0}7.6})$\%  & $(52.4^{+3.8}_{-7.6})$\% \\
Segment samples & $(89.0^{+1.1}_{-3.0})$\% & $(55.0^{+5.0}_{-4.4})$\%  & $(30.0^{+10.0}_{-16.8})$\%                      & $(77.6^{+3.4}_{-6.7})$\% \\
\hline
\end{tabular}
\end{table}

The Monte Carlo data samples are not purely composed of single-site or
multi-site events. The $DEP$ samples are dominated by single-site
events, the $\Gamma_{1}$ and $\Gamma_{2}$ have large fractions of
multi-site events. Events in the $DEP$ samples are referred to as {\it
electron-like} while events in the $\Gamma_{1}$ and $\Gamma_{2}$
samples are referred to as {\it photon-like} in the following. Note,
that these two labels do not describe an intrinsic property of an
event (such as the range of energy deposition), but they are used to
emphasize the different probabilities of the event being single-site
or multi-site.

\section{Analysis methods}
\label{section:methods}

Three analysis methods were tested. Half of the $DEP$ and $\Gamma_{1}$
data samples were used to train the methods. The other half of the
samples, together with the $\Gamma_{2}$ and $ROI$ samples, were used
to test the analysis methods. The $DEP$ and $\Gamma_{1}$ samples were
selected for training in order to avoid biases due to the difference
in energy of events in the two samples. For the same reason the
maximum of each pulse shape was normalized to unity for each event. \\

The analyses were applied to the core and segment data samples in
order to study the effect of pulse shape analysis before and after the
application of a single segment requirement. In the former case, only
the core pulse shape was used. In the latter case, the core pulse
shape was used and, optionally, the segment~$S$ pulse shape in
addition.

\subsection{Likelihood discriminant method}

Four quantities are calculated for each pulse shape. These quantities
provided separation power in previous
studies~\cite{Aalseth:2000hy,Elliott:2005at}. Interpolation algorithms
were applied to the pulse shapes to obtain continuous
distributions. Figure~\ref{fig:quantities} shows an ideal pulse and
the quantities calculated are indicated. All quantities are given
subscripts $C$ and $S$ for the core and segment pulse shapes,
respectively.

\enlargethispage{0.5 cm}

\begin{itemize}
\item Risetime $\tau_{10-30}$, defined as the difference between the
times the integrated charge amplitude has reached 10\% and 30\% of its
maximal amplitude;
\item risetime $\tau_{10-90}$, defined as the  difference between the
times the integrated charge amplitude has reached 10\% and 90\% of its
maximal amplitude;
\item left-right asymmetry $\zeta$, defined as the asymmetry of the
area below the left and the right half of the current pulse, $A_{l}$
and $A_{r}$, measured from the maximum\footnote{The definition differs
from the one given in~\cite{Aalseth:2000hy,Elliott:2005at}.}, $\zeta =
\frac{A_{l}-A_{r}}{A_{l}+A_{r}}$;
\item current pulse width $\delta$, defined as the full width at half maximum
of the current pulse.
\end{itemize}

\begin{figure}[ht!]
\center
\mbox{\epsfig{file=quantities.eps,width=0.45\textwidth}}
\caption{Ideal pulse shape: the integrated charge (thick line) and the
current (thin line). Indicated are the quantities $\tau_{10-30}$,
$\tau_{10-90}$, $\delta$, $A_{l}$ and $A_{r}$ (see text).
\label{fig:quantities}}
\end{figure}

The variables are histogrammed for both training samples and their
integrals are normalized to unity.  As an example,
Figure~\ref{fig:quantities_distributions} shows the normalized
distributions of the four quantities calculated from the core pulse
shape in the two segment data samples. The average risetime of pulses
in the $DEP$ sample is larger than that of pulses in the $\Gamma_{1}$
sample~\footnote{This behavior was also found in a simple calculation
of pulse shapes assuming a perfect crystal and not taking into account
any effects from the electronics.}. The relative frequencies are used
to define discriminants, given that the event is electron-like ($DEP$
sample) or photon-like ($\Gamma_{1}$ sample). The respective overall
discriminants, $p_{e^{-}}$ and $p_{\gamma}$, are calculated by
multiplying the individual discriminants:

\begin{eqnarray}
p_{e^{-}}^{k} & = & p(\tau_{\mathrm{10-30},k}|e^{-}) \cdot p(\tau_{\mathrm{10-90},k}|e^{-}) \cdot
                    p(\zeta_{k}|e^{-}) \cdot p(\delta_{k}|e^{-}) \ , \\
&& \nonumber \\
p_{\gamma}^{k} & = & p(\tau_{\mathrm{10-30},k}|\gamma) \cdot p(\tau_{\mathrm{10-90},k}|\gamma) \cdot
                     p(\zeta_{k}|\gamma) \cdot p(\delta_{k}|\gamma) \ ,
\end{eqnarray}
with $k=C$~or~$S$ for the core and segment pulses, respectively. Note
that no correlations among these quantities are taken into account.

Likelihood discriminants (LHD) are constructed from $p_{e^{-}}$ and
$p_{\gamma}$ for each individual event:

\begin{eqnarray}
D^{C}   & = & \frac{p_{e^{-}}^{C}}{p_{e^{-}}^{C} + p_{\gamma}^{C}} \ , \\
D^{C+S} & = & \frac{p_{e^{-}}^{C} \cdot p_{e^{-}}^{S}}{p_{e^{-}}^{C} \cdot p_{e^{-}}^{S}+ p_{\gamma}^{C} \cdot p_{\gamma}^{S}} \ ,
\end{eqnarray}
\noindent
where $D^{C}$ uses information from the core electrode only and
$D^{C+S}$ uses information from the core and segment electrodes. $D$
varies between~0 and~1 by construction. $D$ peaks at~1 for
electron-like events; for photon-like events $D$ peaks at 0. Events
are identified as electron-like for $D>\overline{D}$ and as
photon-like for $D<\overline{D}$, where $\overline{D}$ is a chosen
parameter.

\begin{figure}[ht!]
\center
\begin{tabular}{cc}
\begin{minipage}[ht!]{0.45\textwidth}
\mbox{\epsfig{file=quantities_risetime1030.eps,width=\textwidth}}
\end{minipage}
&
\begin{minipage}[ht!]{0.45\textwidth}
\mbox{\epsfig{file=quantities_risetime1090.eps,width=\textwidth}}
\end{minipage} \\
\begin{minipage}[ht!]{0.45\textwidth}
\mbox{\epsfig{file=quantities_zeta.eps,width=\textwidth}}
\end{minipage}
&
\begin{minipage}[ht!]{0.45\textwidth}
\mbox{\epsfig{file=quantities_delta.eps,width=\textwidth}}
\end{minipage}
\end{tabular}
\caption{Quantities calculated from the core pulseshapes in the $DEP$
(open histogram) and $\Gamma_{1}$ (hatched histogram) segment data
samples. Top left: risetime $\tau_{10-30}$, top right: risetime
$\tau_{10-90}$, bottom left: left-right asymmetry $\zeta$, bottom
right: current pulse width $\delta$.
\label{fig:quantities_distributions}}
\end{figure}

\subsection{Library method}

The training $DEP$ samples are interpreted as libraries of
electron-like reference pulses. An average $\chi^{2}$ with respect to
all reference pulses is calculated for each pulse shape in the test
samples. For the $k$th reference pulse and the $l$th pulse shape under
study the average $\chi^{2}$ is defined as
\begin{equation}
\chi_{k,l}^{2} = \frac{1}{N}\sum_{i = 1}^{N} \frac{(x_{k, i} - x_{l,i})^{2}}{\sigma^{2}} \ ,
\end{equation}
where $N$ is the number of bins of the pulse shapes and
$x_{k,i}$ and $x_{l,i}$ are the pulse heights in bin $i$ of
the $k$th reference pulse and the $l$th pulse under
study. $\sigma^{2}$ is defined as
\begin{equation}
\sigma^{2} = \sigma_{\mathrm{k}}^{2} + \sigma_{\mathrm{l}}^{2},
\end{equation}
where $\sigma_{\mathrm{k}}$ and $\sigma_{\mathrm{l}}$ are
the noise amplitudes of the reference pulse shape and the pulse shape
under study. The noise amplitude is the RMS of the baseline measured
during the one $\mu$s before the onset of the pulse. \\

The minimum $\chi^{2}$ is selected with respect to the reference
pulses and denoted
$\chi^{2}_\mathrm{min}=\chi^{2}_{k_{\mathrm{min}},l}$ for each pulse
shape in the test sample. Ideally, the minimum $\chi^{2}$ for
electron-like events should be smaller than that of photon-like
events. Events are identified as electron-like for
$\chi^{2}_{\mathrm{min}} <
\overline{\chi^{2}}$ and as photon-like for $\chi^{2}_{\mathrm{min}} >
\overline{\chi^{2}}$, where $\overline{\chi^{2}}$ is a chosen
parameter.

\subsection{Neural network method}

Artificial neural networks (ANNs) are used to separate electron-like
from photon-like events. Input neurons are fed with samples of the
normalized pulse shape, starting from the time when the amplitude has
reached 10\%. 40 consecutive samples per pulse shape are used. The ANN
consists of 40 input neurons, 40 hidden neurons and one output neuron
for the core data samples. An additional 40 input neurons are used
optionally for the segment data samples. \\

The ANNs are trained by feeding them with pulse shapes from the two
training samples and simultaneously providing the information which of
the samples each pulse belongs to (0:~$DEP$ sample, 1:~$\Gamma_{1}$
sample). The ANNs adjust the internal neurons iteratively using the
Broyden, Fletcher, Goldfarb, Shanno (BFGS) learning
method~\cite{BFGS}. Each ANN is trained in about
$1000$~iterations. The output quantity, $NN$, is on average larger
for photon-like events than for electron-like events. Events are
identified as electron-like for $NN < \overline{NN}$ and as
photon-like for $NN > \overline{NN}$, where $\overline{NN}$ is a
chosen parameter.

\pagebreak

\section{Results}
\label{section:results}

The three analysis methods are applied to the data samples defined in
Section~\ref{subsection:selection}. The likelihood discriminant and
neural network analysis are performed on the segment data samples (a)
with information from the core electrode only and (b) with information
from the core and the segment~$S$ electrode. As an example,
Figure~\ref{fig:PSA_output} shows the output distributions for the two
segment training data samples $DEP$ and $\Gamma_{1}$ for the
likelihood method (left), the library method (middle) and the neural
network (right). The segment pulse shapes have not been taken into
account for these examples.

\begin{figure}[ht!]
\center
\begin{tabular}{ccc}
\begin{minipage}[ht!]{0.30\textwidth}
\mbox{\epsfig{file=lhd.eps,width=\textwidth}}
\end{minipage}
&
\begin{minipage}[ht!]{0.30\textwidth}
\mbox{\epsfig{file=chi2.eps,width=\textwidth}}
\end{minipage}
&
\begin{minipage}[ht!]{0.30\textwidth}
\mbox{\epsfig{file=NN_1.eps,width=\textwidth}}
\end{minipage} \\
\end{tabular}
\caption{Output distributions for the two segment training data
samples $DEP$ (open histograms) and $\Gamma_{1}$ (hatched histograms)
for the likelihood method (left), the library method (middle) and the
neural network (right). The segment pulse shapes were not taken into
account in these examples.
\label{fig:PSA_output}}
\end{figure}

The results of the analysis are interpreted in the following. First,
it is shown that the electron-like and photon-like event samples can
be distinguished. In a second step, the results are interpreted to
distinguish between single-site and multi-site events. The estimate of
the power of such a distinction requires the knowledge of the fraction
of single-site and multi-site events in the data samples. That
information is taken from the Monte Carlo simulation presented in
Section~\ref{section:simulation} based on the parameter $R_{90}$.

\subsection{Distinction between electron-like and photon-like event samples}
\label{subsection:electronlike}

\enlargethispage{0.5 cm}

The power to distinguish between electron-like and photon-like event
samples is estimated. The events in the $DEP$ sample are assumed to
give the same output in the analyses as events from neutrinoless
double beta-decay. The cut values are chosen to keep 90\% of the
events in the $DEP$ training samples for the three analysis methods
and thus a high detection efficiency. The fraction of events in each
test data sample identified as electron-like are summarized in
Table~\ref{table:fractions}. The uncertainties are estimated from the
deviation from 90\% of the fraction of events identified as
electron-like in the $DEP$ test data samples and found to be about
2\%. Note that no deviation is found in case of the library method
since the $DEP$ training data sample is used as a reference library.

\begin{table}[ht!]
\center
\caption{Fraction of events in the test data samples identified as
electron-like for the three analyses. The uncertainties are estimated
to be about 2\%.
\label{table:fractions}}
\center
\begin{tabular}{lcccc}
\\
\hline
Data samples                    & $DEP$             & $\Gamma_{1}$    & $\Gamma_{2}$   & $ROI$ \\
                                & ($1593$~keV)  & ($1620$~keV)  & ($2615$~keV) & (2039~keV) \\
\hline
Likelihood method & & & & \\
\hline
Core samples                    & 89.3\% & 76.5\% & 75.4\% & 83.4\% \\
Segm. samples, core only        & 89.3\% & 67.1\% & 64.1\% & 84.8\% \\
Segm. samples, core \& segm. & 88.0\% & 66.7\% & 61.1\% & 83.4\% \\
\hline
Library method & & & & \\
\hline
Core samples                    & 90.0\% & 86.9\% & 85.8\% & 86.7\% \\
Segm. samples, core only        & 90.0\% & 68.4\% & 56.4\% & 83.1\% \\
\hline
Neural network method & & & & \\
\hline
Core samples                    & 90.4\% & 65.8\% & 63.2\% & 79.9\% \\
Segm. samples, core only        & 89.3\% & 54.1\% & 44.3\% & 80.8\% \\
Segm. samples, core \& segm. & 89.3\% & 56.1\% & 49.9\% & 79.6\% \\
\hline
\end{tabular}
\end{table}

The fraction of events identified as electron-like is significantly
lower than 90\% in the $\Gamma_{1}$, $\Gamma_{2}$ and $ROI$
samples. The fraction in the $\Gamma_{1}$ sample is found to be larger
than that in the $\Gamma_{2}$ sample with each method. This is
expected, as the mean free path of photons increases with the photon
energy. \\

The fraction of events identified as electron-like in the $\Gamma_{1}$
and $\Gamma_{2}$ segment data samples (using the core pulse shape
only) is found to be lower than that in the core data samples with all
three methods. The additional usage of the segment pulse shape in the
analyses reduces the fraction by maximally 3\%; in case of the neural
network it even increases the fraction by maximally 5\%. This
demonstrates that the additional information is highly correlated with
the existing information and only marginally contributes to the
analysis. \\

The neural network shows the best performance. This is expected, since
the ANN uses the largest fraction of information and also takes
correlations between input variables into account. \\

\subsection{Selection of single-site events and discrimination against multi-site events}

As demonstrated in Table~\ref{table:fraction}, neither the $DEP$ nor
the $\Gamma_{1}$, $\Gamma_{2}$ and $ROI$ samples are solely composed
of single-site or multi-site events. The probability to correctly
identify single-site and multi-site events as such, $\epsilon$ and
$\eta$, can be deduced from the fraction of single-site and multi-site
events in each sample (estimated from Monte Carlo) and the output of
the analyses, $D$, $\chi^{2}_\mathrm{min}$, $NN$:

\begin{eqnarray}
\epsilon & = & \frac{ N_{id}^{SSE}/N_{true}^{MSE} - M_{id}^{SSE}/M_{true}^{MSE} }{ N_{true}^{SSE}/N_{true}^{MSE} - M_{true}^{SSE}/M_{true}^{MSE} } \ , \label{eqn:epsilon} \\
&& \nonumber \\
\eta     & = & \frac{ N_{id}^{MSE}/N_{true}^{SSE} - M_{id}^{MSE}/M_{true}^{SSE} }{ N_{true}^{MSE}/N_{true}^{SSE} - M_{true}^{MSE}/M_{true}^{SSE} } \ , \label{eqn:eta}
\end{eqnarray}

\noindent
where $N_{id}^{SSE}$ and $N_{id}^{MSE}$ are the number of events in
the $DEP$ sample identified as single-site and multi-site events,
respectively. The numbers depend on the cut value chosen for each
analysis. $N_{true}^{SSE}$ and $N_{true}^{MSE}$ are the true number of
single-site and multi-site events in the same sample and are estimated
from the Monte Carlo simulation discussed in
Section~\ref{section:simulation}. $M_{id}^{SSE}$ and $M_{id}^{MSE}$
are the number of events in the $\Gamma_{1}$ sample identified as
single-site and multi-site events, respectively. $M_{true}^{SSE}$ and
$M_{true}^{MSE}$ are the true number of single-site and multi-site
events in the same sample. The probabilities $\epsilon$ and $\eta$ are
assumed to be the same for all samples. This assumption is reasonable
for the $DEP$ and $\Gamma_{1}$ samples as the average energies are
very close. \\

The cut values for the three analysis methods are chosen to maximize
the figure of merit, the identification efficiency
$\sqrt{\epsilon\cdot\eta}$. Note, that these cut values differ from
those used in Section~\ref{subsection:electronlike}. The probabilities
obtained from the data samples using Equations~\ref{eqn:epsilon}
and~\ref{eqn:eta} are listed in Table~\ref{table:efficiencies}.

\begin{table}[ht!]
\caption{Probabilities $\epsilon$ and $\eta$ obtained for all three
analysis methods. The errors are introduced by the choice of
$\overline{R}$ determining the fraction of single-site and multi-site
events.
\label{table:efficiencies}}
\center
\begin{tabular}{lccc}
\\
\hline
Analysis                        & $\epsilon$     & $\eta$          & $\sqrt{\epsilon\cdot\eta}$ \\
\hline
Likelihood method & & & \\
\hline
Core samples                    & ($74.8^{+1.8}_{-0.3}$)\% & ($\phantom{0}84.7^{+\phantom{0}3.4}_{-\phantom{0}2.4}$)\%  & ($79.6^{+1.4}_{-0.2}$)\% \\
Segm. samples, core only        & ($84.3^{+1.8}_{-0.2}$)\% & ($\phantom{0}97.7^{+10.4}_{-\phantom{0}5.9}$)\%  & ($90.8^{+4.8}_{-1.9}$)\% \\
Segm. samples, core \& segm. & ($83.9^{+1.7}_{-0.1}$)\% & ($\phantom{0}94.0^{+\phantom{0}9.9}_{-\phantom{0}5.6}$)\%  & ($88.8^{+4.6}_{-1.8}$)\% \\
\hline
Library method & & & \\
\hline
Core samples                    & ($68.7^{+\phantom{0}0.8}_{-\phantom{0}0.1}$)\%  & ($56.1^{+\phantom{0}1.4}_{-\phantom{0}1.0}$)\%  & ($62.1^{+0.7}_{-0.2}$)\% \\
Segm. samples, core only        & ($90.9^{+\phantom{0}0.1}_{-13.4}$)\% & ($80.4^{+10.1}_{-\phantom{0}9.1}$)\%  & ($85.6^{+4.8}_{-1.7}$)\% \\
\hline
Neural network method & & & \\
\hline
Core samples                    & ($85.6^{+2.4}_{-0.4}$)\% & ($\phantom{0}91.0^{+\phantom{0}4.3}_{-\phantom{0}0.3}$)\%  & ($\phantom{0}88.3^{+1.9}_{-0.3}$)\% \\
Segm. samples, core only        & ($96.4^{+2.5}_{-0.2}$)\% & ($121.6^{+15.0}_{-\phantom{0}8.5}$)\%  & ($108.3^{+6.6}_{-2.5}$)\% \\
Segm. samples, core \& segm. & ($90.6^{+2.3}_{-0.2}$)\% & ($115.4^{+13.4}_{-\phantom{0}7.7}$)\%  & ($102.3^{+5.9}_{-2.2}$)\% \\
\hline
\end{tabular}
\end{table}

The likelihood and library methods work better on events with only one
segment hit. The additional usage of the segment pulse shape in the
likelihood method does not improve the analysis results. \\

The analysis of the neural network output yields probabilities larger
than one for the segment data samples. The calculation of $\epsilon$
and $\eta$ depends on the real fraction of single-site and multi-site
events and is therefore model dependent. The current model assumes the
fraction of single-site and multi-site events to be completely
reflected by the parameter $R_{90}$. The validity of the assumed model
is limited and the extraction of the probabilities $\epsilon$ and
$\eta$ carries systematic uncertainties. The results should be taken
with care. The efficiencies do not exceed unity for the chosen cut
parameter for the core data samples. Figure~\ref{fig:PSA_efficiency}
shows $\epsilon$ and $\eta$ together with the identification
efficiency as a function of the neural network cut parameter for the
core data samples.

\begin{figure}[ht!]
\center
\epsfig{file=eta_epsilon_1000it.eps,width=0.5\textwidth}
\caption{Probabilities to correctly identify single-site, $\epsilon$,
and multi-site events, $\eta$, and the efficiency,
$\sqrt{\epsilon\cdot\eta}$, for the neural network analysis of the
core data samples. Probabilities above one are caused by uncertainties
in the extraction process.
\label{fig:PSA_efficiency}}
\end{figure}

\subsection{Application to the $^{228}$Th data set}
\label{subsection:application}

Figure~\ref{fig:spectrum} (left) shows the energy spectrum resulting
from a $^{228}$Th source in the region from 1.3~MeV to 2.7~MeV as seen
by the core electrode. The black line corresponds to all events with
only segment~$S$ hit, the gray line represents events with only
segment~$S$ hit and pulse shape analysis, using the ANN, applied. Only
the pulse shape of the core was used and the cut parameter was chosen
to keep 90\% of the events in the $DEP$ training data sample. \\

The gray spectrum is suppressed with respect to the black
spectrum. The suppression ranges up to a factor of about two at the
photon peaks. The suppression is weak in the double escape
peak. Figure~\ref{fig:spectrum} (right) shows a close-up of the
spectrum in the region from 1560~keV to 1650~keV. The application of
the pulse shape analysis removes photon induced events (1620~keV
photon line from the decay of $^{212}$Bi) but keeps most of the
electron induced events (double escape peak of the $2\ 615$~keV
$^{208}$Tl photon at $1593$~keV). Pulse shape analysis is thus
suitable to confirm the signal process. \\

\begin{figure}[ht!]
\center
\begin{tabular}{cc}
\begin{minipage}[ht!]{0.45\textwidth}
\mbox{\epsfig{file=spectrum_1000it.eps,width=\textwidth}}
\end{minipage}
&
\begin{minipage}[ht!]{0.45\textwidth}
\mbox{\epsfig{file=spectrum_closeup_1000it.eps,width=\textwidth}}
\end{minipage} \\
\end{tabular}
\caption{Spectrum of a $^{228}$Th source as seen by the core
electrode. The black line corresponds to all events with only
segment~$S$ hit, the gray line represents events with only segment~$S$
hit and pulse shape analysis, using the ANN, applied. Only the pulse
shape of the core was used and the cut parameter was chosen to keep
90\% of the $DEP$ events. Left: Spectrum from 1.3~MeV to
2.7~MeV. Right: Close-up of the region from 1560~keV to 1650~keV. For
a discussion see text.
\label{fig:spectrum}}
\end{figure}

\pagebreak

\section{Conclusions and outlook}
\label{section:conclusions}

Three methods using pulse shapes were introduced to distinguish
electrons from multiply scattered photons. They were applied on data
collected with a {\sc GERDA} prototype detector.  Single-site
dominated samples were distinguished from multi-site dominated
samples. The probability to correctly identify single-site and
multi-site events was estimated based on Monte Carlo calculations. \\

All three methods were trained with double escape events and events
from a nearby photon peak. The former events are expected to be
similar to the expected $0\nu\beta\beta$-events. \\

The methods are based on information from the core electrode and may
include information from the segment electrode or not. The power to
identify photon induced events does not increase with the
straightforward inclusion of the pulse shape of the segment. \\

The performance of the three methods is slightly worse than what was
reported in~\cite{Elliott:2005at}. A reason for this is the purity of
the samples. Also, the spatial distribution of energy deposited inside
the detector is not homogeneous in the $DEP$ sample. Methods to select
cleaner and more homogeneous training samples are currently being
tested. \\

The artificial neural network shows a better performance than both the
likelihood discriminant and the library method. Photon peaks remaining
after a single segment cut are suppressed by a factor of about two at
energies around 1.5~MeV. At the same time 90\% of the events in the
single-site dominated sample are kept. This demonstrates that the
association of a particular peak with the signal process can be
substantiated by this kind of analysis. \\

The calculation of the efficiency to correctly identify single-site
and multi-site events is limited by the assumed model based on the
$R_{90}$ parameter. Further studies are required; in particular, a
simulation of the development of pulse shapes is important and is
currently under development. Studies using additional information from
neighboring segments to distinguish single-site from multi-site events
are also planned. In addition, an improved experimental setup is
planned. \\

The rejection of events in the $1620$~keV peak using segment
anti-coincidences as presented in~\cite{Abt:2007rg} is about a factor
of two better than the sole application of pulse shape analysis as
presented in this paper. Nevertheless, the application of pulse shape
analysis after a single segment cut can further reject events in this
peak by an additional factor of about two.

\section{Acknowledgements}

The authors would like to thank A.~Bettini, P.~Grabmayr, L.~Pandola
and B.~Schwingenheuer for their helpful comments and suggestions. The
authors would also like to thank the {\sc GERDA} and {\sc Majorana}
Monte~Carlo groups for their fruitful collaboration and cooperation on
the {\sc MaGe} project.

\addcontentsline{toc}{section}{Bibliography}