# Majorana qubit readout by a point-contact detector under finite bias
voltages

Huizi Xie  School of Physics, Hangzhou Normal University, Hangzhou, Zhejiang
311121, China  Sirui Yu  School of Physics, Hangzhou Normal University,
Hangzhou, Zhejiang 311121, China  Hong Mao<EMAIL_ADDRESS>School of Physics,
Hangzhou Normal University, Hangzhou, Zhejiang 311121, China  Jinshuang Jin
<EMAIL_ADDRESS>School of Physics, Hangzhou Normal University, Hangzhou,
Zhejiang 311121, China

###### Abstract

In this work we revisit the problem of four Majorana-zero-modes (4-MZMs) qubit
readout by a point-contact (PC) detector. The logic states of the 4-MZMs qubit
are converted to different charge occupations of a tunnel-coupled quantum dot
(QD), which is further measured by a nearby PC detector. This type of
measurement in principle requires that the PC detector is to be biased at
small voltages, which should be comparable to the energy scales of the QD
level and its coupling to the qubit. Going beyond existing studies on this
problem, we present a unified treatment for both the steady-state current and
the current power spectrum, which allow us to infer the qubit state
information from either the QD occupation or the characteristic peak at the
oscillation frequency. Moreover, we carry out the peak-to-pedestal ratio of
each single characteristic peak (associated with each logic state of the
qubit) and the signal-to-noise ratio of the two peaks. From these two figures
of merit, we reveal the optimal bias voltage window for the PC detector at low
temperature limit.

##  I Introduction

Owing to the unique nature of non-Abelian statistics and spatial nonlocality,
Majorana zero modes (MZMs) have a potential application in topological quantum
computations Kit01131 ; Kit032 ; Nay081083 ; Lei11210502 ; Das1515001 ;
Aas16031016 ; Plu17012001 ; Kar17235305 . Largely based on the MZMs which may
be realized from the semiconductor/superconductor hybrid structures
Lut10077001 ; Ore10177002 , a variety of of Majorana qubits have been
proposed, such as the Majorana box qubits (MBQs) Plu17012001 , tetrons and
hexons Kar17235305 . For the Majorana topological quantum computation, the
qubit measurement is an important problem to be studied in the relatively
early stage. Indeed, for instance, for the readout of MBQs, a variety of
schemes have been proposed Plu17012001 ; Gri19235420 ; Smi20020313 ;
Qin19043027 ; Zho21010301 ; Mun20033254 ; Ste20033255 , such as measuring the
frequency shift of a double-dot system Plu17012001 , dispersive shift from
qubit-resonator interaction Smi20020313 , and interferometric conductance
Qin19043027 ; Zho21010301 . In particular, we are aware of the recent scheme
of four-MZMs qubit readout by a point-contact (PC) detector Plu17012001 ;
Kar17235305 , where the fermion parity of the qubit is converted to occupation
of a tunnel-coupled quantum dot (QD) Mun20033254 ; Ste20033255 . In Ref.
Mun20033254, , it was found that the scheme by means of measuring the steady-
state QD occupation can be effective only under low bias voltage. On the
contrary, in Ref. Ste20033255, , instead of QD-occupation dependent current
measurement under low bias voltage, a scheme based on measuring the power
spectrum of the PC detector current was analyzed, at a large-bias-voltage
limit.

In this work, we revisit the problem of 4-MZMs qubit readout analyzed in
Refs.Mun20033254 ; Ste20033255 . Going beyond their existing studies, we
present a unified treatment for both the steady-state current and the current
power spectrum under finite (small) bias voltages, which allow us to infer the
qubit state information from either the QD occupation or the characteristic
frequencies in the current spectrum. We will carry out useful figures of
merit, such as peak-to-pedestal ratio and signal-to-noise ratio, and reveal
optimal bias voltage window for the PC detector at low temperature limit.

The paper is organized as follows. In Sec. II, we first give the model
description and outline the basic measurement scenario. We then introduce the
quantum master equation approach to calculate the transport current through
the detector and the current power spectrum in a unified way. In Sec. III, we
carry out the analytical and numerical results of the detector current. We
also address the underlying physical mechanism of the parity-dependent current
related to the bias voltage. In Sec. IV, the analytical and numerical results
of the current power spectrum are illustrated. We further obtain the formula
of the signal-to-noise ratio (SNR) of the two parity-dependent Rabi peaks.
Especially, the optimal bias voltage window for the PC detector at low
temperature limit is provided. Finally, we give our summary in Sec. V.

##  II Methodology

###  II.1 Model description

Figure 1: (Color online) Schematic setup for the readout of a Majorana box
qubit (MBQ). The MBQ is coupled to a quantum dot (QD) which is measured by a
point-contact (PC) detector. The MBQ consists of two topological
superconducting wires, hosting four Majorana zero energy modes
$\gamma_{1},\gamma_{2},\gamma_{3}$ and $\gamma_{4}$. The coupling of two
Majoranas ($\gamma_{1}$, $\gamma_{2}$) and the QD leads to a parity-dependent
current $I_{\pm}$ through the PC detector.

We consider a Majorana box qubit (MBQ) coupled to a quantum dot (QD), and the
QD is in turn continuously measured by a voltage-biased point-contact (PC), as
sketched in Fig. 1. The MBQ is formed in a floating superconducting island,
consisting of two long topological superconducting nanowires Plu17012001 ;
Kar17235305 . They each host a Majorana zero-energy mode (MZM) $\gamma_{i}$ at
both ends. Each pair of MZMs are related to the regular fermion through the
transformation of $\gamma_{1}=(\hat{f}_{\mbox{\tiny
L}}+\hat{f}^{\dagger}_{\mbox{\tiny L}})$, $\gamma_{2}=-i(\hat{f}_{\mbox{\tiny
L}}-\hat{f}^{\dagger}_{\mbox{\tiny L}})$ for the left pair, and
$\gamma_{3}=(\hat{f}_{\mbox{\tiny R}}+\hat{f}^{\dagger}_{\mbox{\tiny R}})$,
$\gamma_{4}=-i(\hat{f}_{\mbox{\tiny R}}-\hat{f}^{\dagger}_{\mbox{\tiny R}})$
for the right pair, respectively. The corresponding occupation number
operators are $\hat{n}_{\mbox{\tiny L}}=\hat{f}^{\dagger}_{\mbox{\tiny
L}}\hat{f}_{\mbox{\tiny L}}$ and $\hat{n}_{\mbox{\tiny
R}}=\hat{f}^{\dagger}_{\mbox{\tiny R}}\hat{f}_{\mbox{\tiny R}}$. Here,
$\\{\gamma_{i},\gamma_{j}\\}=2\delta_{\ij}$ and $\gamma^{2}_{i}=1$. The qubit
states $|n_{\mbox{\tiny L}}n_{\mbox{\tiny R}}\rangle$ can then be encoded in
the even parity space $\\{|0_{\mbox{\tiny L}}0_{\mbox{\tiny
R}}\rangle,|1_{\mbox{\tiny L}}1_{\mbox{\tiny R}}\rangle\\}$ or odd parity
space $\\{|0_{\mbox{\tiny L}}1_{\mbox{\tiny R}}\rangle,|1_{\mbox{\tiny
L}}0_{\mbox{\tiny R}}\rangle\\}$.

The composite Hamiltonian of the device in Fig. 1 is $H_{\rm
tot}=H_{\mbox{\tiny S}}+H_{\mbox{\tiny B}}+H^{\prime}$. The system is modeled
by the low-energy Hamiltonian Mun20033254 ; Fle11090503 ,

$H_{\mbox{\tiny S}}=\varepsilon_{\rm
d}\hat{d}^{\dagger}\hat{d}+\sum_{i=1,2}\big{(}\lambda_{i}\gamma_{i}\hat{d}+{\rm
H.c.}\big{)},$ (1)

where we assume a pair of MZMs described by $\gamma_{1}$ and $\gamma_{2}$ and
a single electronic level $\varepsilon_{\rm d}$ in the QD with the energy
level of $\varepsilon_{\rm d}$. The second term in Eq. (1) describes the
tunnel coupling between the QD and two MZMs. Without loss of generality, the
tunneling coefficients $\lambda_{i}$ can be formally defined as,
$\lambda_{1}=|\lambda_{1}|e^{i\varphi_{1}}$ and
$\lambda_{2}=|\lambda_{2}|e^{i\varphi_{2}}$, and the gauge invariant phase
difference $\varphi=\varphi_{2}-\varphi_{1}$ can be tuned such as by varying
the magnetic flux penetrating the enclosed area of the interference loop (see
Fig. 1). The PC detector is described by the two electron reservoirs,
$H_{\mbox{\tiny B}}=\sum_{\alpha k}\varepsilon_{\alpha
k}\hat{c}^{\dagger}_{\alpha k}\hat{c}_{\alpha k}$ with $\alpha={\rm L,R}$. The
interaction Hamiltonian between the dot and the PC detector is given by
Gur9715215 ; Kor995737 ; Goa01235307

$\displaystyle H^{\prime}$ $\displaystyle=\sum_{k,q}({\cal T}+\chi n_{\rm
d})\hat{c}^{\dagger}_{{\mbox{\tiny L}}k}\hat{c}_{{\mbox{\tiny R}}q}+{\rm
H.c.}.$ (2)

It describes the electron tunneling through PC, with the tunnel coupling
amplitude that depends on electron occupation of the QD ($\hat{n}_{\rm
d}=\hat{d}^{\dagger}\hat{d}$). Here, we consider weak interaction with
$\chi\ll{\cal T}$. For the later convenience, we introduce the system-related
operator $\hat{Q}={\cal T}+\chi\hat{n}_{\rm d}=\hat{Q}^{\dagger}$ and the PC
bath operator $\hat{F}=\sum_{kq}\hat{c}^{\dagger}_{\mbox{\tiny
L}k}\hat{c}_{{\mbox{\tiny R}}q}$. The interaction Hamiltonian Eq. (2) is then
straightforwardly rewritten as
$H^{\prime}=\hat{Q}(\hat{F}+\hat{F}^{\dagger})$.

The measurement effects of PC reservoirs on the combined QD-MBQ system are
characterized by the interaction bath correlation functions Li04085315 ;
Li05066803 ,
$C^{(+)}(t)\equiv\langle\hat{F}^{\dagger}(t)\hat{F}(0)\rangle_{\mbox{\tiny
B}}$ and
$C^{(-)}(t)\equiv\langle\hat{F}(t)\hat{F}^{\dagger}(0)\rangle_{\mbox{\tiny
B}}$ with $\hat{F}(t)=e^{iH_{\mbox{\tiny B}}t}\hat{F}e^{-iH_{\mbox{\tiny
B}}t}=\sum_{kq}\hat{c}^{\dagger}_{\mbox{\tiny L}k}\hat{c}_{{\mbox{\tiny
R}}q}e^{i(\varepsilon_{\mbox{\tiny L}k}-\varepsilon_{\mbox{\tiny R}q})t}$.
Here, $\langle\cdots\rangle_{\mbox{\tiny B}}$ stands for the statistical
average over the bath (PC electron reservoirs) in thermal equilibrium. The
interaction bath spectrum is defined as the Fourier transform of the
reservoir-electron correlation function Xu02023807 ; Li04085315

$\displaystyle C^{(\pm)}(\omega)$ $\displaystyle=\int^{\infty}_{-\infty}{\rm
d}t\,e^{i\omega t}C^{(\pm)}(t)=\frac{\eta x}{1-e^{-\beta
x}}\Big{|}_{x=\omega\mp V},$ (3)

which satisfies the detailed-balance relation
$C^{(+)}(\omega)=e^{\beta(\omega-V)}C^{(-)}(-\omega)$ for arbitrary bias
voltage $V=\mu_{\rm L}-\mu_{\rm R}$. Here we introduced the parameter
$\eta=2\pi g_{\rm L}g_{\rm R}$ under the consideration of the energy-
independent density of states ($g_{\rm L}$ and $g_{\rm R}$ of the two
reservoirs). Throughout this work, we adopt units of $e=\hbar=1$ for the
electron charge and the Planck constant.

###  II.2 Basic measurement scenario

Taking the even parity space ($\\{|0_{\mbox{\tiny L}}0_{\mbox{\tiny
R}}\rangle,|1_{\mbox{\tiny L}}1_{\mbox{\tiny R}}\rangle\\}$) as a typical
example, we give a brief description of the basic measurement protocol of the
MBQ Mun20033254 ; Ste20033255 . During the entire measurement, the total
parity of the superconducting island is assumed to be conserved. Therefore,
the readout of the MBQ states $|0_{\mbox{\tiny L}}0_{\mbox{\tiny R}}\rangle$
and $|1_{\mbox{\tiny L}}1_{\mbox{\tiny R}}\rangle$ can map to the readout of
the states $|0_{\mbox{\tiny L}}\rangle$ and $|1_{\mbox{\tiny L}}\rangle$,
respectively. Then we can denote $|0_{\mbox{\tiny L}}\rangle$
($\rightarrow|0_{\mbox{\tiny L}}0_{\mbox{\tiny R}}\rangle$) and
$|1_{\mbox{\tiny L}}\rangle$ ($\rightarrow|1_{\mbox{\tiny L}}1_{\mbox{\tiny
R}}\rangle$) as a logical qubit states. The readout of this logical qubit
states can be realized by the measurement of the combined fermion parity of
the dot and the left pair of MZMs ($\gamma_{1}$ and $\gamma_{2}$),

$p=(-1)^{n_{\rm d}+n_{\mbox{\tiny L}}},$ (4)

since this parity is conserved by the total composite Hamiltonian $H_{\rm
tot}$. For instance, the MBQ is assumed to be prepared in the initial state
e.g., $|\psi_{0}\rangle=a|0_{\mbox{\tiny L}}\rangle+b|1_{\mbox{\tiny
L}}\rangle$, with the complex coefficients $a$ and $b$ satisfying
$|a|^{2}+|b|^{2}=1$. The dot is supposed initially empty, $n_{\rm d}=0$. We
then have the initial state of the dot-MBQ,

$|\Psi_{0}\rangle=|0_{\rm d}\rangle\otimes(a|0_{\mbox{\tiny
L}}\rangle+b|1_{\mbox{\tiny L}}\rangle)=a|0_{\rm d}0_{\mbox{\tiny
L}}\rangle+b|0_{\rm d}1_{\mbox{\tiny L}}\rangle.$ (5)

Consequently, one knows that the MBQ state is $|0_{\mbox{\tiny L}}\rangle$
($|1_{\mbox{\tiny L}}\rangle$) if $p=+1$ ($p=-1$) has been measured.

Described by the dot-MZM interaction Hamiltonian of Eq. (1), there are
intrinsic coherent Rabi oscillations in the both subspaces of the odd parity
$\\{|0_{\rm d}1_{\mbox{\tiny L}}\rangle,|1_{\rm d}0_{\mbox{\tiny
L}}\rangle\\}$ (${p}=-1$) and even parity $\\{|0_{\rm d}0_{\mbox{\tiny
L}}\rangle,|1_{\rm d}1_{\mbox{\tiny L}}\rangle\\}$ (${p}=+1$). The former is
induced by the electron normal tunneling processes, and the latter arises from
the Andereev reflection processes. That is, the Hamiltonian Eq. (1) can be
written as,

$H_{\mbox{\tiny S}}={\left(\begin{array}[]{cc}H_{-}&0\\\
0&H_{+}\end{array}\right)},$ (6)

where the parity-dependent Hamiltonian in each subspace is

$H_{{\rm p}=\pm}={\left(\begin{array}[]{cc}0&\lambda_{1}-i{p}\lambda_{2}\\\
\lambda^{*}_{1}+i{p}\lambda^{*}_{2}&\varepsilon_{\rm d}\end{array}\right)}.$
(7)

Apparently, $H_{\rm p}$ indicates the intrinsic coherent Rabi oscillation. The
corresponding Rabi frequency ($\Delta_{\rm p}$) reads,

$\Delta_{\rm p}=\sqrt{\varepsilon^{2}_{\rm d}+4\lambda^{2}_{\rm p}(\varphi)},$
(8)

with

$\lambda^{2}_{\rm
p}(\varphi)\equiv|\lambda_{1}|^{2}+|\lambda_{2}|^{2}+2p|\lambda_{1}||\lambda_{2}|\sin\varphi.$

Note that the Rabi frequency $\Delta_{\rm p}$ is the energy difference between
the two eigenstates ($\\{|e_{\rm p}\rangle,|g_{\rm p}\rangle\\}$) of $H_{\rm
p}$, i.e., $\Delta_{\rm p}=\varepsilon_{e_{\rm p}}-\varepsilon_{g_{\rm p}}$,
with the excited energy $\varepsilon_{e_{\rm p}}=(\varepsilon_{\rm
d}+\Delta_{\rm p})/2$ and the ground energy $\varepsilon_{g_{\rm
p}}=(\varepsilon_{\rm d}-\Delta_{\rm p})/2$. The resulting Rabi frequency
expressed by Eq. (8) is a periodic function of $\varphi$ with $2\pi$
periodicity and parity dependent for $\varphi\neq k\pi$.

Thus, the MBQ can be in principle read out via any $\Delta_{\rm p}$-dependent
observable quantities with $\varphi\neq k\pi$, such as the dot occupation
number, the detector current, and the current power spectrum. It has been
demonstrated in Ref. Mun20033254  that the readout of MBQ from the dot
occupation number is effective only at low bias voltage. On the contrary, in
Ref. Ste20033255 , it has been analyzed that the information on the parity is
not carried by the average current as well as the dot occupation number, but
generically encoded in the power spectrum of the detector current at large
bias voltage limit. In the present work, we will revisit such measurement
problem at any bias voltage regime. We will explore the underlying physical
mechamism and reveal the optimal bias voltage window for the PC detector for
the readout of MBQ in the current power spectrum. All studies on the steady-
state dot occupation number, detector current and its power spectrum will be
uniformly based on the quantum master equation approach as introduced below.

###  II.3 Quantum master equation approach

It is well-known that the primary QD-MBQ system is described by the reduced
density operator $\rho(t)\equiv{\rm tr}_{\mbox{\tiny B}}[\rho_{\rm tot}(t)]$,
i.e., the partial trace of the total density operator $\rho_{\rm tot}$ over
the bath space (the electron reservoirs of PC detector). Regarding the
interaction Hamiltonian $H^{\prime}$ in Eq. (2) as a perturbation and adopting
the Born-Markovian approximation, one can easily get the master equation for
QD-MBQ system. It reads Yan982721 ; Yan002068 ; Xu02023807 ,

$\displaystyle\dot{\rho}(t)$ $\displaystyle=-i{\cal L}\rho(t)-{\cal
R}\rho(t),$ (9)

where the first term, ${\cal L}\bullet\equiv[H_{\mbox{\tiny S}},\bullet]$,
describes the coherent dynamics and the second term,

$\displaystyle{\cal R}\bullet\equiv\frac{1}{2}\big{[}Q,{\cal
Q}\bullet-\bullet{\cal Q}^{\dagger}\big{]},$ (10)

arises from the measurement back-action of the PC detector. The resolvant
operators are $Q={\cal T}+\chi n_{\rm d}$ and ${\cal Q}={\cal Q}^{(-)}+{\cal
Q}^{(+)}$, with ${\cal Q}^{(\pm)}\equiv C^{(\pm)}(-{\cal L})Q$. Here, the bath
correlation function $C^{(\pm)}(-{\cal L})$ is given by Eq. (3) and satisfies
the detailed-balance relation for arbitrary bias voltage. The Liouvillian
operator“${\cal L}$” in the correlation function $C^{(\pm)}(-{\cal L})$
contains the information of energy transfer between the detector and the
measured system. In the eigen-state basis $\\{|e_{\rm p}\rangle,|g_{\rm
p}\rangle\\}$, “${\cal L}$” would be replaced by “$\Delta_{\rm p}$” which
enters the correlation function, i.e., $C^{(\pm)}(\pm\Delta_{\rm p})$. All the
observable quantities measured by PC detector as demonstrated below are
closely related to $C^{(\pm)}(\pm\Delta_{\rm p})$, thus they carry the
information of the parity.

For the calculation of the current through PC detector and its power spectrum,
one can extend the master equation of Eq. (9) into the number-resolved
(conditional) master equation and/or its conjugate counting field formalism
Zwa01 ; Mak01357 . An alternative method is the introduction of the current-
related density operator based on the energy-dispersed dissipaton
decomposition Xu22064130 . In the framework of the Born-Markov approximation,
we can obtain the same formulas of the detector current and its power spectrum
based on the above two methods as follows. Explicitly, the average current
through PC detector is given by Li05066803 ; Xu22064130

$\displaystyle I(t)$ $\displaystyle={\rm tr}\big{[}{\cal
J}^{(-)}\rho(t)\big{]},$ (11)

and its power spectrum can be calculated via

$\displaystyle S(\omega)$ $\displaystyle\\!=\\!2{\rm
tr}\\!\big{[}{\mathcal{J}}^{(+)}\bar{\rho}\big{]}\\!+\\!4{\rm
tr}\\!\big{[}{\mathcal{J}}^{(-)}{\Pi}(\omega){\cal J}^{(-)}\bar{\rho}\big{]}.$
(12)

Here the superoperators are defined as

$\displaystyle{\mathcal{J}}^{(\pm)}\bullet$
$\displaystyle\equiv\frac{1}{2}\big{[}{\cal Q}^{(-)}\pm{\cal
Q}^{(+)}\big{]}\bullet Q+{\rm H.c.},$ (13a) $\displaystyle{\Pi}(\omega)$
$\displaystyle\equiv\frac{1}{i({\cal L}-\omega)+{\cal R}},$ (13b)

and the stationary state $\bar{\rho}\equiv\rho(t\rightarrow\infty)$ is
obtained by setting $\dot{\rho}(t)=0$ in Eq. (9).

We would like to note that in the following numerical calculations, all
energies are measured in (arbitrary) unit of $\lambda$. We set
$|\lambda_{1}|=\lambda$ and $|\lambda_{2}|=0.8\lambda$ for the consideration
of a certain degree of the coupling asymmetry between the QD and MZMs in
experiments. Moreover, we consider weak interaction between the PC detector
and the QD with ${\cal T}=\lambda$ and $\chi=0.2\lambda$ in Eq. (2). The
energy-independent density of states with $g_{\mbox{\tiny L}}=g_{\mbox{\tiny
R}}=\lambda$ and finite temperature $k_{\rm B}T=0.5\lambda$ are adopted
throughout the paper, unless otherwise stated. .

Figure 2:  (Color online) The parity-dependent steady-state current $I_{\rm
p}$ (in $e\lambda/\hbar$) at low bias voltage $V=\lambda$, (a) as a function
of flux $\varphi$ with $\varepsilon_{\rm d}=\lambda$ and (b) as a function of
$\varepsilon_{\rm d}$ with $\varphi=\pi/2$. (c) The signal-to-current ratio as
a function of $\varepsilon_{\rm d}$ with different temperature (in $\lambda$)
for $\varphi=\pi/2$ and $V=\lambda$. (d) The detector current $I_{\rm p}$ as a
function of the bias voltage for $\varphi=\pi/2$ and $\varepsilon_{\rm
d}=2\lambda$.

##  III The steady-state current

By replacing $\rho(t)$ with $\bar{\rho}$ in the expression of Eq. (11), the
steady-state current is obtained as

$\displaystyle I_{\rm p}$
$\displaystyle=g_{u}V-\frac{1}{2}(g_{u}-g_{v})V\Big{[}1+\frac{\Delta_{\rm
p}}{G^{(+)}(\Delta_{\rm p})}\Big{]}$
$\displaystyle\quad+g_{b}V\big{[}1-\frac{\Delta_{\rm
p}}{V}\frac{G^{(-)}(\Delta_{\rm p})}{G^{(+)}(\Delta_{\rm p})}\big{]},$ (14)

where, $g_{u}=\eta u^{2}$, $g_{v}=\eta v^{2}$, and $g_{b}=\eta|b|^{2}$, with
$u={\cal T}+\frac{\chi}{2}(1+\frac{\varepsilon_{\rm d}}{\Delta_{\rm p}})$,
$v={\cal T}+\frac{\chi}{2}(1-\frac{\varepsilon_{\rm d}}{\Delta_{\rm p}})$, and
$|b|^{2}=\frac{\chi^{2}\lambda^{2}_{\rm p}(\varphi)}{\Delta^{2}_{\rm p}}$.
Here, we introduced

$\displaystyle G^{(+)}(\Delta_{\rm p})$
$\displaystyle\equiv\frac{C(\Delta_{\rm p})+C(-\Delta_{\rm p})}{2\eta},$ (15a)
$\displaystyle G^{(-)}(\Delta_{\rm p})$
$\displaystyle\equiv\frac{\bar{C}(\Delta_{\rm p})-\bar{C}(-\Delta_{\rm
p})}{2\eta},$ (15b)

with $C(\omega)\equiv C^{(-)}(\omega)+C^{(+)}(\omega)$ and
$\bar{C}(\omega)\equiv C^{(-)}(\omega)-C^{(+)}(\omega)$. Apparently, the
steady-state current $I_{\rm p}$ in Eq. (III) is $\Delta_{\rm p}$-dependent
through the bath correlation function $C^{(\pm)}(\pm\Delta_{\rm p})$ and its
periodic variation with the flux is numerically shown in Fig. 2 (a). The MBQ
can thus be read out from $I_{\rm p}$ for $\varphi\neq k\pi$. Especially, the
signal $\delta I=|I_{+}-I_{-}|$ is maximum at $\varphi=(2k+1)\frac{\pi}{2}$
with $k$ the integer.

Actually, the parity-dependent average current ($I_{\rm p}$) is quite similar
to that of the steady-state dot occupation number ($n_{\rm p}$). Via the
definition of $n_{\rm p}=\langle\hat{d}^{\dagger}\hat{d}\rangle_{\rm p}$, one
can straightforwardly recover the result of $n_{\rm
p}=\frac{1}{2}\big{[}1-\frac{\varepsilon_{\rm d}}{G^{(+)}(\Delta_{\rm
p})}\big{]}$ in Ref. Mun20033254, . The dot occupation number $n_{\rm p}$ is
parity-dependent via also the bath correlation function of
$G^{(+)}(\Delta_{\rm p})$ (expressed by Eq. (15a)), and thus can read out the
MBQ. Apparently, for $\varepsilon_{\rm d}=0$, the dot occupation number is
simplified to $n_{\rm p}=\frac{1}{2}$ and the steady-state current is also
parity-independent as displayed in Fig. 2 (b). The reason is that the coherent
dynamics between the two states in each subspace, e.g., $|00\rangle$
($|01\rangle$) and $|11\rangle$ ($|10\rangle$) for even (odd) parity, are the
equivalent Rabi resonances for $\varepsilon_{\rm d}=0$. Furthermore, we plot
the signal-to-current ratio, i.e., $\delta I/I=|I_{+}-I_{-}|/I$ with
$I=\frac{I_{+}+I_{-}}{2}$, as shown in Fig. 2 (c). It indicates that the read
out of MBQ from the detector current $I_{\rm p}$ should away from (but nearby)
the resonance of $\varepsilon_{\rm d}=0$ at low temperature regime with
$k_{B}T\ll\lambda$.

Figure 2 (d) displays the steady-state current as a function of the bias
voltage. Evidently, _in the large bias voltage regime_ , i.e.,
$V\gg\Delta_{\rm p}$, the current is parity-independent and thus cannot read
out the MBQ. The behind reason can be understood as follows. In line with the
expression of the current in Eq. (III), the involved correlation functions are
simplified to be energy-independent, i.e., $C^{(+)}(\pm\Delta_{\rm p})=0$ and
$C^{(-)}(\pm\Delta_{\rm p})=\eta(V\pm\Delta_{\rm p})\rightarrow\eta V$ as can
be inferred from Eq. (3) under large bias voltage limit. Physically, this is
because the energy emitted/absorpted by the system ($\Delta_{\rm p}$) is
negligible in comparison with the large bias voltage. That is, the average
current in Eq. (III) becomes

$\displaystyle I_{\rm p}\rightarrow I_{\rm c}$ $\displaystyle=g_{0}V+g_{1}V,$
(16)

where $g_{0}=\eta({\cal T}+\chi/2)^{2}$ and $g_{1}=\eta(\chi/2)^{2}$.
Apparently, the current in Eq. (16) under large bias voltage limit is
independent of the parity. In fact, it can be easily obtained as $I_{\rm
c}=(I_{1}+I_{0})/2$ with $I_{1}=\eta({\cal T}+\chi)^{2}V$ and $I_{0}=\eta{\cal
T}^{2}V$, which are the currents corresponding to the dot states $|1\rangle$
and $|0\rangle$, respectively. Consequently, we cannot read out the MBQ from
the detector current as well as the steady-state dot occupation number
($n_{\rm p}\rightarrow 1/2$) at large bias voltage regime. Similar result has
also been reported in Refs Mun20033254 ; Ste20033255 .

Therefore, as far as the steady-state current and the dot occupation number
are concerned, there have two conditions what should be satisfied for the
readout of the parity number experimentally. One is that the energy level of
the dot should be tuned away from the resonance ($\varepsilon_{\rm d}\neq 0$).
The other is that the applied bias voltage should be low with
$V\lesssim\Delta_{\rm p}$.

##  IV Power spectrum

As well-known, the current power spectrum carries the information of the
system beyond the average current. As we can infer from the formula Eq. (12),
the power spectrum characterizes not only the back action of the detector
described by the first term, but also the intrinsic coherent dynamics of the
measured system depicted by the second term. This leads to that the current
power spectrum is directly related to the parity-dependent Rabi frequency
$\Delta_{\rm p}$ and is not limited by both the dot energy level and bias
voltage. Namely, as long as $\delta\Delta\equiv|\Delta_{+}-\Delta_{-}|\neq 0$
with $\varphi\neq k\pi$, the MBQ can be read out from the current power
spectrum as we will demonstrated in detail below. Consequently, for clarity,
we will consider the resonance case with $\varepsilon_{\rm d}=0$ to get the
analytical result of power spectrum, and take $\varphi=\pi/8$ as a typical
example in the following numerical calculations.

Let us derive the analytical result of the current power spectrum based on the
formula Eq. (12). For convenience, we separate the power spectrum into a
frequency-independent part $S^{\rm p}_{0}$ and a frequency-dependent part
$S^{\rm p}_{1}(\omega)$, i.e., $S_{\rm p}(\omega)=S^{\rm p}_{0}+S^{\rm
p}_{1}(\omega)$. Explicitly, we get

$\displaystyle S^{\rm p}_{0}$
$\displaystyle=2g_{0}V\coth\frac{V}{2T}+2g_{1}\Big{[}G^{(+)}_{\rm
p}-\frac{\Delta^{2}_{\rm p}}{G^{(+)}_{\rm p}}\Big{]},$ (17a) $\displaystyle
S^{\rm p}_{1}(\omega)$ $\displaystyle=\frac{F_{1}(\Delta_{\rm p})\Gamma_{\rm
p}\Delta^{2}_{\rm p}}{(\omega^{2}-\Delta^{2}_{\rm p})^{2}+\Gamma_{\rm
p}^{2}\omega^{2}}+\frac{F_{2}(\Delta_{\rm p})G^{(-)}_{\rm
p}}{\omega^{2}+\Gamma_{\rm p}^{2}},$ (17b)

where $G^{(\pm)}_{\rm p}\equiv G^{(\pm)}(\Delta_{\rm p})$, $F_{1}(\Delta_{\rm
p})=I^{2}_{\rm d}\Big{[}1\\!-\\!\frac{\Delta_{\rm p}}{2V}\frac{G^{(-)}_{\rm
p}}{G^{(+)}_{\rm p}}\Big{]}$ with $I_{\rm d}\\!=\\!I_{1}\\!-\\!I_{0}$,
$F_{2}(\Delta_{\rm p})=8g^{3}_{1}\big{[}2\Delta_{\rm p}V-\Delta^{2}_{\rm
p}\frac{G^{(-)}_{\rm p}}{G^{(+)}_{\rm p}}-G^{(-)}_{\rm p}G^{(+)}_{\rm
p}\big{]}$. In particular, $\Gamma_{\rm p}$ in Eq. (17b) is the dephasing rate
and it can be expressed as,

$\Gamma_{\rm p}=2g_{1}G^{(+)}_{\rm p}(\Delta_{\rm p}).$ (18)

Furthermore, by taking the large bias voltage limit ($V\gg\Delta_{\rm p}$),
the power spectrum in Eq. (17) becomes

$\displaystyle S^{\rm p}_{0}$ $\displaystyle\rightarrow 2I_{\rm p},$ (19a)
$\displaystyle S^{\rm p}_{1}(\omega)$
$\displaystyle\rightarrow\frac{I^{2}_{\rm d}\Gamma_{\rm d}\Delta^{2}_{\rm
p}}{(\omega^{2}-\Delta^{2}_{\rm p})^{2}+\Gamma_{\rm
d}^{2}\omega^{2}}+\frac{4g_{1}\Gamma_{\rm d}\Delta^{2}_{\rm
p}}{\omega^{2}+\Gamma_{\rm d}^{2}},$ (19b)

where the dephasing rate of Eq. (18) is simplified to $\Gamma_{\rm
d}=2g_{1}V=\eta V\chi^{2}/2$.

Figure 3:  (Color online) The finite-frequency current power spectrum for
$\varepsilon_{\rm d}=0$ and $\varphi=\pi/8$ with (a) at low bias voltage
$V=\lambda$ and (b) at large bias voltage $V=8\lambda$.

As we expected, the current power spectrum has more advantages over the
steady-state current and the dot occupation number. Since the power spectrum
given by Eq. (17) depicts the parity-dependent even for the resonance case
($\varepsilon_{\rm d}=0$) at any bias voltages, including large bias voltage
regime [see Eq. (19b)]. The MBQ can be read out via the locations of the
parity-dependent peaks around the parity-dependent Rabi frequencies, i.e.,
$\omega\simeq\pm\Delta_{\rm p}$. The peaks which come from the intrinsic
coherent Rabi oscillation of the QD-MBQ system is determined by $\Pi(\omega)$
in the second term of Eq. (12).

The corresponding numerical results are illustrated in Fig. 3 (a) at low
voltage ($V=\lambda$) and Fig. 3 (b) at large voltage ($V=8\lambda$),
respectively. We find that the raise of the measurement voltage leads to two
effects on the power spectrum. One is the increase in the height of the peaks
and the other is the enhancement of the peak width. The former is beneficial
for the measurement, while the latter is not. Therefore, it is necessary to
find the optimal bias voltage range of the PC detector for the qubit
measurement visibility.

To characterize the effect of the voltage on the peak width, we recast the
frequency-dependent power spectrum formula Eq. (17b) into the Lorentz form.
After some algorithm, we obtain

$\displaystyle S^{\rm p}_{1}(\omega)$ $\displaystyle\simeq\frac{4f(\Delta_{\rm
p})F_{1}(\Delta_{\rm p})\Delta_{\rm p}^{2}}{\Gamma_{\rm p}(4\Delta_{\rm
p}^{2}-\Gamma_{\rm p}^{2})}\sum_{\pm}\frac{\sigma^{2}_{\rm
p}}{(\omega\pm\widetilde{\Delta}_{\rm p})^{2}+\sigma^{2}_{\rm p}}$
$\displaystyle\quad+\frac{g(\\!\Delta_{\rm p}\\!)F_{1}(\\!\Delta_{\rm
p}\\!)\Gamma_{\rm p}}{\Delta_{\rm p}^{2}}\frac{\sigma^{2}_{0\rm
p}}{\omega^{2}\\!+\\!\sigma^{2}_{0\rm p}}\\!+\\!\frac{F_{2}(\Delta_{\rm
p})G^{(-)}_{\rm p}}{\omega^{2}\\!+\\!\Gamma_{\rm p}^{2}},$ (20)

where

$\displaystyle\widetilde{\Delta}_{\rm p}$ $\displaystyle=\sqrt{\Delta^{2}_{\rm
p}-\Gamma^{2}_{\rm p}/2}\approx\Delta_{\rm p},$ (21a)
$\displaystyle\sigma_{\rm p}$ $\displaystyle=\frac{\widetilde{\Delta}_{\rm
p}}{2}\Big{(}\sqrt{1+\Gamma_{\rm p}/\widetilde{\Delta}_{\rm
p}}-\sqrt{1-\Gamma_{\rm p}/\widetilde{\Delta}_{\rm
p}}\,\Big{)}\approx\frac{\Gamma_{\rm p}}{2},$ (21b) $\displaystyle\sigma_{0\rm
p}$ $\displaystyle=\sqrt{\widetilde{\Delta}^{2}_{\rm p}+\sqrt{\Delta^{4}_{\rm
p}+\widetilde{\Delta}^{4}_{\rm p}}}\approx\sqrt{1+\sqrt{2}}\Delta_{\rm p}.$
(21c)

Here, Eqs. (21) come from the consideration of the weak interaction between
the QD and the detector with $\Gamma_{\rm p}\ll\Delta_{\rm p}$. Since the
functions of $g(\Delta_{\rm p})$ and $f(\Delta_{\rm p})$ are quite complicated
and they are not explicitly shown.

For the measurement visibility analysis, we further define the signal by the
position difference between the two peaks of the even and odd parities,

$\displaystyle{\cal
S}\equiv\widetilde{\delta}{\Delta}=|\widetilde{\Delta}_{+}-\widetilde{\Delta}_{-}|\approx\delta{\Delta}.$
(22a) While the noise is defined by the sum of the peak widths of the two
parities, $\displaystyle{\cal
N}\equiv\sigma_{+}+\sigma_{-}\approx\frac{1}{2}(\Gamma_{+}+\Gamma_{-}).$ (22b)

Therefore, the so-called signal-to-noise ratio (SNR) of the two characteristic
peaks can be calculated via

SNR $\displaystyle=\frac{\cal S}{\cal
N}\approx\frac{2\delta\Delta}{\Gamma_{+}+\Gamma_{-}}.$ (23)

It indicates that the SNR which is proportional to $\delta\Delta$ is
determined by the intrinsic quantities involved in the system’s Hamiltonian of
Eq. (1), such as the coupling coefficients $\lambda_{i}$ and the flux
$\varphi$. Whereas, SNR is inversely proportional to the dephasing rates given
by Eq. (18) due to the back-action of the PC detector on the QD-MZM system.
Hence, SNR is affected by both the applied bias voltage and temperature of the
detector.

Figure 4:  (Color online) The SNR as a function of the bias voltage with
different temperature (in $\lambda$). The inset is the peak-to-pedestal ratio
as a function of the bias voltage at low temperature $k_{B}T=0.1\lambda$. The
other parameters are the same as in Fig. 3.

Figure 4 depicts that the SNR is reduced by increasing the measurement
voltage, and even less than 1 at large voltage limit. Furthermore, at low bias
voltage regime, the SNR is enhanced with the reduction in temperature. We now
focus on the low temperature limit with $k_{B}T\ll\lambda$. Then the SNR of
Eq. (23) is simplified to

SNR
$\displaystyle=\frac{4\delta\Delta}{\eta\chi^{2}(\Delta_{+}+\Delta_{-})},~{}{\rm
for}~{}~{}V<\Delta_{\rm p},$ (24a) SNR
$\displaystyle=\frac{2\delta\Delta}{\eta\chi^{2}V},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}V\geq\Delta_{\rm p}.$ (24b)

Seemingly, the measurement voltage should be low at $V<\Delta_{\rm p}$, since
the SNR determined by Eq. (24a) reaches a maximum saturation value which is
independent of the bias voltage. This feature can also be seen in Fig. 4 at
low temperature with $k_{B}T=0.1\lambda$.

However, from the perspective of measurement visibility, we have to consider
the height of the single peak in the experiments. That is, we have to consider
another type of “signal-to-noise” ratio which is also known as peak-to-
pedestal ratio of each single characteristic peak, i.e., $\gamma_{\rm p}\equiv
S^{\rm p}_{1}(\Delta_{\rm p})/S^{\rm p}_{0}(\Delta_{\rm p})$. In contrast with
the SNR feature discussed above, the peak-to-pedestal increases with the bias
voltage as shown in the inset of Fig. 4. It has the fundamental upper bound
limit of 4 at high-voltage regime and also decreases with the increase in the
temperature. This characteristic is consistent with the demonstrations in
previous works Kor01085312 ; Rus03075303 ; Li05066803 .

Therefore, considering the merits of SNR and peak-to-pedestal ratio, the
optimal condition of the measurement visibility should satisfy the relations
of both ${\it SNR}>1$ and $\gamma_{\rm p}\rightarrow 4$ at low temperature
limit. For $V<\Delta_{\rm p}$, the SNR given by Eq. (24a) can easily satisfy
${\it SNR}>1$, but $\gamma_{\rm p}<2$. So that the low bias voltage regime
with $V<\Delta_{\rm p}$ is not good for the MBQ measurement. For
$V\geq\Delta_{\rm p}$ in which $\gamma_{\rm p}\lesssim 4$, the condition of
${\it SNR}>1$ leads to $V<\frac{2\delta\Delta}{\eta\chi^{2}}$ as inferred by
Eq. (24b). Consequently, we get the optimal range of the measurement voltage
for the MBQ readout as,

$\displaystyle\Delta_{\rm p}\leq V<\frac{2\delta\Delta}{\eta\chi^{2}}.$ (25)

Then, based on Eq. (25), all the parameters involved in the experimental
measurement can be appropriately adopted. We believe it significantly
simplifies the current experimental demonstrations.

##  V Summary

In summary, we have thoroughly investigated the detector current and its power
spectrum at all voltages for the hybrid QD-MBQ system continuously measured by
the PC detector. The analytical and numerical results have been carried out
uniformly based on the Born-Markovian quantum master equation approach. It
maintains the detail-balance relation and is potentially applicable for any
bias voltage.

We have explored and demonstrated that the readout of the MBQ from the parity-
dependent detector current as well as the dot occupation number is only
suitable for low bias voltage situations. The underlying reason is that the
parity information contained in the current ($I_{\rm p}$) and the dot
occupation number ( $n_{\rm p}$) is via the energy-dependent bath correlation
function. It comes from the back-action of the detector and accounts for the
energy exchange (characterized by the parity-dependent Rabi frequency
$\Delta_{\rm p}$) between the measured system and detector. Nevertheless, the
energy ($\Delta_{\rm p}$) emitted/absorpted by the QD-MBQ system is negligible
in comparison with the large bias voltage. This leads to the energy-
independent bath correlation function. Thus, $I_{\rm p}$ (also $n_{\rm p}$)
are parity-independent and cannot readout MBQ at large bias voltage regime
($V\gg\Delta_{\rm p}$).

However, for the parity-dependent current power spectrum $S_{\rm p}(\omega)$,
the MQB can be readout from the parity-dependent Rabi oscillation peak
signals, in principle, at all bias voltage. This is because the current power
spectrum characterizes not only the back action of the detector described by
the bath correlation, but also the intrinsic coherent dynamics of the measured
system depicted by its intrinsic parity-dependent Hamiltonian. Importantly,
for the measurement visibility analysis, we carry out the signal-to-noise
ratio of the two parity-dependent characteristic peaks and the peak-to-
pedestal ratio of each single peak. From these two figures of merit, we obtain
the optimal bias voltage range of the PC detector for the MBQ readout at low
temperature limit. The present results can enlighten the current experimental
demonstrations.

###### Acknowledgements.

We acknowledge helpful discussions with Prof. Xin-Qi Li and Prof. YiJing Yan.
The support from the Natural Science Foundation of China (Grant No. 12175052)
is acknowledged.

## References

  * (1) A. Y. Kitaev,  “Unpaired Majorana fermions in quantum wires,” Physics-Uspekhi 44, 131 (2001).
  * (2) A. Kitaev,  “Fault-tolerant quantum computation by anyons,” Annals of Physics 303, 2 (2003).
  * (3) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma,  “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
  * (4) M. Leijnse and K. Flensberg,  “Quantum Information Transfer between Topological and Spin Qubit Systems,” Phys. Rev. Lett. 107, 210502 (2011).
  * (5) S. D. Sarma, M. Freedman, and C. Nayak,  “Majorana zero modes and topological quantum computation,” npj Quantum Inf 1, 15001 (2015).
  * (6) D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flensberg, and J. Alicea,  “Milestones Toward Majorana-Based Quantum Computing,” Phys. Rev. X 6, 031016 (2016).
  * (7) S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg,  “Majorana box qubits,” New Journal of Physics 19, 012001 (2017).
  * (8) T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman,  “Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes,” Phys. Rev. B 95, 235305 (2017).
  * (9) R. M. Lutchyn, J. D. Sau, and S. Das Sarma,  “Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
  * (10) Y. Oreg, G. Refael, and F. von Oppen,  “Helical Liquids and Majorana Bound States in Quantum Wires,” Phys. Rev. Lett. 105, 177002 (2010).
  * (11) A. L. Grimsmo and T. B. Smith,  “Majorana qubit readout using longitudinal qubit-resonator interaction,” Phys. Rev. B 99, 235420 (2019).
  * (12) T. B. Smith, M. C. Cassidy, D. J. Reilly, S. D. Bartlett, and A. L. Grimsmo,  “Dispersive Readout of Majorana Qubits,” PRX Quantum 1, 020313 (2020).
  * (13) L. Qin, X.-Q. Li, A. Shnirman, and G. Schön,  “Transport signatures of a Majorana qubit and read-out-induced dephasing,” New Journal of Physics 21, 043027 (2019).
  * (14) K. Zhou, C. Zhang, L. Qin, and X.-Q. Li,  “Double-dot interferometer for quantum measurement of Majorana qubits and stabilizers*,” Chinese Physics B 30, 010301 (2021).
  * (15) M. I. K. Munk, J. Schulenborg, R. Egger, and K. Flensberg,  “Parity-to-charge conversion in Majorana qubit readout,” Phys. Rev. Res. 2, 033254 (2020).
  * (16) J. F. Steiner and F. von Oppen,  “Readout of Majorana qubits,” Phys. Rev. Res. 2, 033255 (2020).
  * (17) K. Flensberg,  “Non-Abelian Operations on Majorana Fermions via Single-Charge Control,” Phys. Rev. Lett. 106, 090503 (2011).
  * (18) S. A. Gurvitz,  “Measurements with a noninvasive detector and dephasing mechanism,” Phys. Rev. B 56, 15215 (1997).
  * (19) A. N. Korotkov,  “Continuous quantum measurement of a double dot,” Phys. Rev. B 60, 5737 (1999).
  * (20) H. S. Goan and G. J. Milburn,  “Dynamics of a mesoscopic charge quantum bit under continuous quantum measurement,” Phys. Rev. B 64, 235307 (2001).
  * (21) X. Q. Li, W. K. Zhang, P. Cui, J. S. Shao, Z. S. Ma, and Y. J. Yan,  “Quantum measurement of solid-state qubit: A unified quantum master equation approach,” Phys. Rev. B 69, 085315 (2004).
  * (22) X. Q. Li, P. Cui, and Y. J. Yan,  “Spontaneous relaxation of a charge qubit under electrical measurement,” Phys. Rev. Lett. 94, 066803 (2005).
  * (23) R. X. Xu, Y. J. Yan, and X. Q. Li,  “Cavity quantum electrodynamics in the presence of energy relaxation and pure dephasing: A unified quantum master-equation approach,” Phys. Rev. A 65, 023807 (2002).
  * (24) Y. J. Yan,  “Quantum Fokker-Planck theory in a non-Gaussian-Markovian medium,” Phys. Rev. A 58, 2721 (1998).
  * (25) Y. J. Yan, F. Shuang, R. X. Xu, J. X. Cheng, X. Q. Li, C. Yang, and H. Y. Zhang,  “Unified approach to the Bloch-Redfield theory and quantum Fokker-Planck equations,” J. Chem. Phys. 113, 2068 (2000).
  * (26) R. Zwanzig,  Nonequilibrium Statistical Mechanics,  Oxford University Press, New York, 2001.
  * (27) Y. Makhlin, G. Schön, and A. Shnirman,  “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357 (2001).
  * (28) Y. Xu, J. Jin, S. Wang, and Y. Yan,  “Memory-effect-preserving quantum master equation approach to noise spectrum of transport current,” Phys. Rev. E 106, 064130 (2022).
  * (29) A. N. Korotkov,  “Output spectrum of a detector measuring quantum oscillations,” Phys. Rev. B 63, 085312 (2001).
  * (30) R. Ruskov and A. N. Korotkov,  “Spectrum of qubit oscillations from generalized Bloch equations,” Phys. Rev. B 67, 075303 (2003).