\section{Coherent state in a QD pump under a strong magnetic field}
We show that in a 2D QD formed by time-dependent confinement potential, the evolution of a Gaussian state with spatial width $\simeq l_B$
follows the classical equation of motion of a coherent state
under the strong-magnetic-field conditions that the QD has a size much larger than the magnetic length $l_B$ and that the QD confinement potential $U_\textrm{QD}$ changes slowly in the length scale of $l_B$ and the time scale of $\omega_c^{-1}$.
We derive Eq.~(1) and the expression of the wave function overlap $|\ovl{\psi_c}{\psi}|^2$ mentioned in the main text.
In order to elucidate the behavior and the validity of the result, we discuss an example, the time evolution of the ground state in a time-dependent anisotropic harmonic QD confinement potential.

We consider the ground state of the Hamiltonian $H_\textrm{QD}$ at initial time $s=0$, and study its time evolution under $H_\textrm{QD}(s) = (\mb{p}- e\mb{r}\times B \hat{\mb{z}}/2)^2 / (2m_e^*)  + U_\textrm{QD} (\mb{r}, s)$. Since at low energy the confinement potential $U_\textrm{QD} (s=0)$ is well approximated as an anisotropic harmonic potential,
the ground state can be approximately expressed~\cite{Madhav-2} as a Gaussian form with certain harmonic frequencies $\omega_{0,x}$ and $\omega_{0,y}$,
\begin{equation}
\psi (\mb{r}, s=0) \approx \exp[ -\{\frac{x^2}{ l_B^2}\frac{\omega_{0,x}}{\omega_{0,x} +\omega_{0, y}}+\frac{y^2}{ l_B^2}\frac{\omega_{0,y}}{\omega_{0,x} +\omega_{0, y}}\}(1+ \frac{(\omega_{0,x} +\omega_{0, y})^2}{2\omega_c^2}) +i \frac{xy}{l_B^2} \frac{\omega_{0,x}-\omega_{0,y}}{\omega_{0,x} +\omega_{0, y}} (1+\frac{(\omega_{0,x} +\omega_{0, y})^2}{{ 4} \omega_c^2})], \label{init_state} \end{equation}
up to the normalization constant.
This approximation becomes more valid as $\omega_c$ becomes much larger than $\omega_{0,x}$ and $\omega_{0,y}$; when $\omega_c \gg \omega_{0,x}, \omega_{0,y}$,
$\psi (\mb{r}, s=0) \to \exp [ - \mb{r}^2 / (2 W) ]$ with $W \sim l_B$ and the overlap between the exact ground state and $\exp [ - \mb{r}^2 / (2 W) ]$ is approximately $1 - \bo (4 \omega_{0, x (y)}^4 / \omega_c^4)  $.

Next, we study the time evolution of the ground state $\psi (\mb{r}, s=0)$ in Eq.~\eqref{init_state} under $H_\textrm{QD}(s)$.
The evolution is decomposed into two parts, one from the kinetic Hamiltonian and the other from $U_\textrm{QD}(\mb{r}, s)$,
\begin{equation}\label{small_time_evol}
\psi(\mb{r'}, s + \delta s ) = \int d\mb{r} \, K_B(\mb{r'}, s + \delta s; \mb{r}, s) e^{-i {U_\textrm{QD}(\mb{r}, s)} \delta s +\bo[\delta s^2]}  \psi(\mb{r}, s)
\end{equation}
for infinitesimal $\delta s$. $K_B (\mb{r}', s; \mb{r},0) \equiv \langle \mb{r'}| e^{-i s (\mb{p}- e\mb{r}\times \mb{B}/2)^2 / (2m_e^*)} | \mb{r} \rangle$ is obtained~\cite{Papadopoulos1971-2} as
\begin{equation}
K_B (\mb{r}', s; \mb{r}, 0) = \left(\frac{m_e^*\omega_c}{4\pi i \sin \frac{\omega_c s}{2}}\right)^{3/2} \exp\left[\frac{i}{2 l_B^2} \left( \cot \frac{\omega_c s}{2}(\mb{r}-\mb{r'})^2 +2 (\mb{r'} \times \mb{r})_z \right) \right]. \nonumber
\end{equation}
To compute the time evolution further, (i) the infinitesimal $\delta s$ is considered so that the commutator between the kinetic Hamiltonian and $U_\textrm{QD}(\mb{r}, s)$ is neglected in Eq.~\eqref{small_time_evol}, and (ii) at each instant $s$, $U_\textrm{QD}(\mb{r}, s)$ is approximately expressed as
\begin{eqnarray}
U_\textrm{QD}(\mb{r}, s) \simeq U_\textrm{QD}(\mb{r}_0 , s) +\sum_{i=x,y} \frac{\partial}{\partial x_i} U_\textrm{QD}(\mb{r}, s)|_{\mb{r}_0} (\mb{r}-\mb{r}_0)_i +\sum_{i,j=x,y}\frac{\partial^2}{2\partial x_i \partial x_j} U_\textrm{QD}(\mb{r}, s)|_{\mb{r}_0} (\mb{r}-\mb{r}_0)_i (\mb{r}- \mb{r}_0)_j, \label{TaylorExp} \end{eqnarray}
where
$U_\textrm{QD}(\mb{r}_0 , s)$ is the potential at the mean position $\mb{r}_0$ of $\psi(\mb{r}, s)$ at time $s$. This Talyor expansion up to the second order is sufficient when $U_\textrm{QD}(\mb{r}, s) - U_\textrm{QD}(\mb{r}_0, s)$ is much smaller than $\hbar \omega_c$ for $|\mb{r}-\mb{r}_0| < l_B$;
when the total potential $U_\textrm{QD}(\mb{r}, s) + m_e^* \omega_c^2 \mb{r}^2 / 8$ from $U_\textrm{QD}$ and the magnetic confinement is Taylor expanded around $\mb{r}_0$,
the expansion up to the second order dominates over the higher-order terms. Note that the drift motion of the state evolution is described by the first two terms of Eq.~\eqref{TaylorExp}, while the anisotropic shape of the Gaussian form is determined by the last term of Eq.~\eqref{TaylorExp}.
To compute the time evolution of the state, we apply Eq.~\eqref{TaylorExp} to Eq.~\eqref{small_time_evol} and perform Gaussian integrals, assuming that $U_\textrm{QD}$ changes slowly in time $< \omega_c^{-1}$, namely $\partial U_\textrm{QD} / \partial s \ll \omega_c U_\textrm{QD}$.
The result shows that the time evolved state remains in a Gaussian form.
\begin{equation}\label{finite_time_evol}
\psi(\mb{r}, s) = N \exp\left[ (\mb{r}-\mb{r}_c(s))^{\intercal} { \mb{R}} (\mb{r}_c(s))^\dagger { \begin{bmatrix} A(s) & C(s)/2 \\ C(s)/2 & B(s) \end{bmatrix} } { \mb{R}} (\mb{r}_c(s)) (\mb{r}-\mb{r}_c(s)) /l_B^2 \right]
 e^{i \mb{p}_c(s) \cdot \mb{r} }
\end{equation}
where $N \equiv \sqrt[4]{4 \mr{Re}[A(s)] \mr{Re}[B(s)] -\mr{Re}[C(s)]^2}/(\sqrt{\pi}l_B)$.
The matrix $\mb{R}(\mb{r}_c(s)) \equiv \begin{bmatrix} \cos \phi(\mb{r}_c(s)) & \sin \phi(\mb{r}_c(s)) \\ -\sin \phi(\mb{r}_c(s)) & \cos \phi(\mb{r}_c(s)) \end{bmatrix}$ rotates the coordinate by  angle $\phi(\mb{r}_c(s)) \equiv \frac{1}{2} \tan^{-1}[ \partial_{xy}^2 U_\textrm{QD}(\mb{r}_c(s))/\{\partial_{yy}^2 U_\textrm{QD}(\mb{r}_c(s))- \partial_{xx}^2 U_\textrm{QD}(\mb{r}_c(s))\}]$ so that the anisotropic directions (the major and minor axes) of the Gaussian form align the  rotated axes $\mb{R}(\mb{r}_c(s)) \hat{\mb{x}}$ and $ \mb{R}(\mb{r}_c(s))  \hat{\mb{y}}$.
{
Remarkably, We notice that the mean position $\mb{r}_c(s)$ and mean momentum $\mb{p}_c(s)$ of the state at time $s$ are determined by the classical equation of motion (this is why we call $\psi$ a coherent state),
\begin{equation} \label{eom_rp}
\begin{aligned}{}
\frac{d \mb{r}_c}{ds} &= \frac{\partial H_\textrm{QD}}{\partial \mb{p}_c} = \frac{\mb{p}_c}{m_e^*} -\frac{\omega_c}{2} \hat{\mb{z}} \times \mb{r}_c,   \\
\frac{d \mb{p}_c}{ds} &= - \frac{\partial H_\textrm{QD}}{ \partial \mb{r}_c} =   -\frac{m_e^* \omega_c^2}{4} \mb{r}_c + \frac{\partial}{\partial \mb{r}} U_\textrm{QD}(\mb{r}, s)|_{\mb{r}_c} - \frac{\omega_c}{2}\hat{\mb{z}} \times \mb{p}_c. \\
\end{aligned}
\end{equation}
Therefore, the ground state at the initial time $s=0$ evolves in time, propagating along
the $\mb{E} \times \mb{B}$ drift determined by $\partial U_\textrm{QD}(\mb{r}, s)/ \partial \mb{r} $ and the magnetic field. The shape (the width and the anisotropy) of the Gaussian wave packet are determined by $A(s)$, $B(s)$ and $C(s)$, which are governed by the differential equations
\begin{equation} \label{eom_c}
\begin{aligned}{}
\frac{d A}{d s} &= \frac{i}{4}  (-1 + 4A^2) \omega_c + \frac{1}{2} (-C + i C^2/2) \omega_c -\frac{i \kappa_+^2(s)}{\omega_c},  \\
\frac{d B}{d s} &= \frac{i}{4}  (-1 + 4 B^2) \omega_c + \frac{1}{2} (C + i C^2/2) \omega_c -\frac{i \kappa_-^2(s)}{\omega_c},  \\
\frac{d C}{d s} &= \omega_c(A -B)  +i\omega_c (A +B)  C,
\end{aligned}
\end{equation}
where $\kappa_{\pm}(s) \equiv [ (\partial_{xx}^2 + \partial_{yy}^2) U_\textrm{QD}|_{\mb{r}_c}/2 \pm \{((\partial_{xx}^2- \partial_{yy}^2) U_\textrm{QD}|_{\mb{r}_c} )^2/4+ {( \partial_{xy}^2  U_\textrm{QD}|_{\mb{r}_c} )^2} \}^{1/2}]^{1/2}/\sqrt{m_e^*}$.
The state in Eq.~\eqref{init_state} provides the initial condition of the differential equations in Eq.~\eqref{eom_c}. The solution of Eq.~\eqref{eom_c} is
\begin{equation}
\begin{aligned}{} \label{sol}
A(s) &= -\frac{\kappa_{+}(s)}{\kappa_{+}(s)+\kappa_{-}(s)}  -\frac{ \kappa_+(s) (\kappa_+ (s) + \kappa_- (s))}{2\omega_c^2} + \bo(\frac{\kappa_\pm^4}{\omega_c^4}) + \bo(\frac{1}{\kappa_\pm \omega_c}\frac{d\kappa_{\pm}}{ds}), \\
B(s) &= -\frac{\kappa_{-}(s)}{\kappa_{+}(s)+\kappa_{-}(s)} -\frac{\kappa_-(s) (\kappa_+ (s) + \kappa_- (s))}{2\omega_c^2} + \bo(\frac{\kappa_\pm^4}{\omega_c^4}) + \bo(\frac{1}{\kappa_\pm \omega_c}\frac{d\kappa_{\pm}}{ds}), \\
C(s) &= i \frac{\kappa_{+}(s)-\kappa_{-}(s)}{\kappa_{+}(s)+\kappa_{-}(s)} +i \frac{(\kappa_{+}(s)-\kappa_{-}(s))(\kappa_+ (s) + \kappa_- (s))}{4 \omega_c^2}  + \bo(\frac{\kappa_\pm^4}{\omega_c^4}) + \bo(\frac{1}{\kappa_\pm \omega_c}\frac{d\kappa_{\pm}}{ds}).
\end{aligned}
\end{equation}
Note that the errors $\bo(\cdots)$ are small under the conditions of $U_\textrm{QD}$ mentioned above; the smoothness of $U_\textrm{QD}$ in the length scale $l_B$, $U_\textrm{QD}(\mb{r} -\mb{r}_0, s) -U_\textrm{QD}(\mb{r}_0 s) \ll \hbar \omega_c$ for $|\mb{r} -\mb{r}_0| <l_B $, implies $\kappa_\pm \ll \omega_c$, while the smoothness of $U_\textrm{QD}$ in the time scale $\omega_c^{-1}$, $\partial U_\textrm{QD}/\partial s \ll \omega_c U_\textrm{QD} $, implies $\partial \kappa_{\pm}/\partial s \ll \omega_c \kappa_\pm$.
Under the conditions, the second terms of Eqs. \eqref{sol} are also negligible as they become much smaller than the first terms. Then, the solution shows that the time evolved wave packet has an anisotropic Gaussian form of width of $l_B [1 - (\kappa_+-\kappa_-)/(\kappa_+ +\kappa_-)]^{-1/2}$ along the major axis and $l_B [1 +(\kappa_+-\kappa_-)/(\kappa_+ +\kappa_-)]^{-1/2}$ along the minor axis.
When $U_\textrm{QD}$ is an isotropic harmonic potential, the Gaussian form is also isotropic with $\kappa_+ = \kappa_-$. For general anisotropic QD confinements, the Gaussian form is anisotropic, but this effect is not significant in realistic QDs as discussed below. Thus, $\psi (\mb{r}, s)$ is well approximated as
\begin{equation}
\psi_c(\mb{r}, s) =\frac{1}{\sqrt{\pi l_B^2}} \exp [-\frac{(\mb{r}-\mb{r}_c (s))^2}{2 l_B^2}+i \mb{p}_c (s) \cdot \mb{r}]. \nonumber
\end{equation}

In order to elucidate the behavior and the validity of the result, we discuss the time evolution of the ground state in a time-dependent anisotropic harmonic QD, $U_\textrm{QD}(\mb{r}, s) = (\omega_{0,x}^2(s) x^2 +\omega_{0,y}^2(s) y^2)/(2 m_e^*) + F(s) x$.
Here, the force $F(s)$ describes a time-dependent shift of the center of the confinement in the $x$ direction. In this case, the error terms $\bo(\cdots)$ in Eq.~\eqref{sol} vanish and the time evolution $\psi(\mb{r}, s)$ is obtained from Eqs.~\eqref{finite_time_evol}, \eqref{eom_rp}, and \eqref{sol} as
\begin{equation}
\begin{aligned}{}
\psi (\mb{r}, s) &= \exp \Big[ -\Big\{\frac{(x-x_c(s))^2}{ l_B^2}\frac{\omega_{0,x}(s)}{\omega_{0,x}(s) +\omega_{0, y}(s)}+\frac{(y-y_c(s))^2}{ l_B^2}\frac{\omega_{0,y}(s)}{\omega_{0,x}(s) +\omega_{0, y}(s)}\Big\} \big(1+ \frac{(\omega_{0,x}(s) +\omega_{0, y}(s))^2}{2\omega_c^2} \big)  \\
&\quad\quad \quad +i \frac{(x-x_c(s)) (y-y_c(s))}{l_B^2} \frac{\omega_{0,x}(s)-\omega_{0,y}(s)}{\omega_{0,x}(s) +\omega_{0, y}(s)} \big(1+\frac{(\omega_{0,x}(s) +\omega_{0, y}(s))^2}{ 4 \omega_c^2 }\big)\Big] e^{i \mb{p}_c(s) \cdot \mb{r}}.
\end{aligned} \label{EXACT}
\end{equation}
We note that when $\omega_{0,x(y)}(s)$ is time independent, Eq.~\eqref{EXACT} is identical to the result analytically obtained~\cite{Madhav-2} by diagonalizing the Hamiltonian.
The overlap between $\psi$ and $\psi_c$ is obtained as
\begin{equation} \label{ovl}
|\ovl{\psi_c(s)}{\psi(s)}|^2 \approx 1 - \left(\frac{\omega_{0,x}(s) +\omega_{0,y}(s)}{2 \omega_c}\right)^4 -\frac{1}{2} \left(\frac{\omega_{0,x}(s)-\omega_{0,y}(s)}{\omega_{0,x}(s) +\omega_{0,y} (s)} \right)^2.
\end{equation}
The second term means that $\psi$ is well approximated by the Gaussian packet Eq.~\eqref{finite_time_evol} in a sufficiently strong magnetic field, and the last term shows that $\psi(s)$ is well described by the isotropic Gaussian packet $\psi_c$ when $U_\textrm{QD}$ is not too anisotropic.
For example when $|\omega_{0,x}(s)-\omega_{0,y}(s)|$ is $\sim$30\% of $(\omega_{0,x}(s)+\omega_{0,y}(s))/2$ (this is the value that we find in the numerical simulation of an realistic QD pump in Fig.~1), the second term of Eq.~\eqref{ovl} is less than $10^{-2}$.

}

\section{Time dependent scattering theory}

We here describe the scattering theory for the QD pump, which follows a Floquet theory~\cite{floquet-2},
and derive the emitted wave functions in Eqs.~(3) and (5).

In the regime of $v_U \tau_d \gtrsim \Delta$, the QD pump is simplified into a scattering model in Fig.~\ref{figs1}.
The coherent state propagates along a loop of coordinate $l$ which couples with the edge of the 2DEG outside the QD via the exit barrier located at $l=0$.
The loop represents the trajectory of the state at $s > s_\textrm{mx}$.
The state gains the potential energy $U(s)$ by $V_\textrm{G1}$.
$U(s)$ increases linearly and then stays at its maximum $U_\textrm{mx}$ at $s > s_\textrm{mx}$, following $V_\textrm{G1}(s)$ in Fig.~1.
The exit barrier is parameterized by scattering amplitudes $t_\mathcal{E}$, $r_\mathcal{E}$, $t'_\mathcal{E}$, and $r'_\mathcal{E}$ connecting plane waves of the QD ($\sim Y e^{i k l}$, $Y$ being the wave function in the transverse direction) and those of the 2DEG edge ($\sim Y e^{i k x}$) at the same energy $\mathcal{E}$.
The 2DEG edge states belong to the lowest Landau level, since the edge state of the higher levels are located farther from the exit barrier (by distance longer than $l_B$) hence the coupling from the QD to them is much weaker.
At time $s_0$, the coherent state has so low energy that $t_\mathcal{E} = 0$ within its energy window, and is located at $l=0^{-}$ for simplicity.

\begin{figure}[t]
\includegraphics[width=0.7\textwidth]{fig1_supp.pdf}
\caption{Scattering problem (a) before and (b) after the gauge transformation where the information of the time dependence of the potential $U$ is attached onto the scattering amplitudes.}
\label{figs1}
\end{figure}

We solve the scattering problem, using the gauge transformation where the dynamical phase by $U(s)$ is attached onto the scattering amplitudes, as in a Floquet theory~\cite{floquet-2}.
We consider a phase $\Lambda (l,s) = \Theta(l-0^{+}) \Theta(L^{-}-l) \int_{-\infty}^s U(s')ds' $,
where $L$ is the total length of the loop, $\Theta (x) = 1$ for $x > 0$, and $\Theta (x) = 0$ for $x < 0$; in this Supplementary Materials, we use the convention of $\hbar \equiv 1$. Then the potential $U$ and the vector potential $\mb{A}$ are gauge transformed as
\begin{equation}
\begin{aligned}{}
\Phi =U(t) \,\,\,\,\,\, &\rightarrow \,\,\,\,\,\, \Phi - \partial \Lambda/\partial s =0 \\
\mb{A} =0 \,\,\,\,\,\, &\rightarrow \,\,\,\,\,\, \mb{A} + \nabla \Lambda = [\delta(l-0^{+})-\delta(l-L^{-})] \int_{-\infty}^s U(s')ds'.
\end{aligned}
\end{equation}
After the transformation, the loop becomes time independent, and instead the coupling at $l=0$ (at $l=0^+$ and $l=L^-$) between the loop and the 2DEG edge becomes time dependent, carrying the information of $U(s)$.

Then, to apply the gauge transformation, we decompose the loop into three regions, $l \in [L^-, L]$, $l \in [ 0, 0^+]$, and $l \in [0^+, L^-]$,  and we assign state amplitudes $a_1$, $b_1$, and $c$, to the regions; cf. Fig.~\ref{figs1}(b). For example, a scattering state incoming from the 2DEG edge can be decomposed into an incident edge state of amplitude $a_{2, \mathcal{E}}$, an edge state with amplitude $b_{2, \mathcal{E}}$ outgoing from the coupling point, a loop state with amplitude $b_{1, \mathcal{E}}$ in $l \in [ 0, 0^+]$, a loop state with amplitude $a_{1, \mathcal{E}}$ in $l \in [L^-, L]$, and a loop state with amplitude $c_\mathcal{E}$ in $l \in [0^+, L^-]$. Here, $\mathcal{E}$ is the energy of the incident state. At $l=0^+$ and $l=L^-$, the information of the time-dependent $U(s)$ is attached to wave functions such that a wave function $\psi_{c, \mathcal{E}}(l,s)$ of energy $\mathcal{E}$ in $l \in [0^+, L^-]$ couples with $\psi_{b_1, \mathcal{E}}(s)$ in $l \in [0,0^+]$ and $\psi_{a_1, \mathcal{E}}(s)$ in $l \in [L^-,L]$ as
\begin{align}
\psi_{c, \mathcal{E}}(l=0^+, s) &= \psi_{b_1,\mathcal{E}} (s)  e^{i\phi(s)}, \label{cb1} \\
\psi_{a_1, \mathcal{E}} (s) &= \psi_{c, \mathcal{E}} (l=L^-, s) e^{-i\phi(s)}, \nonumber
\end{align}
where $\phi (s) = \int_{-\infty}^s U(s') ds'$. Since $\psi_{c, \mathcal{E}}(l = L^-,s) = \psi_{c, \mathcal{E}} (l = 0^+, s - \tau_d)$, one finds
\begin{equation}
\psi_{a_1, \mathcal{E}} (s) = e^{-i\phi(s) +i\phi(s-\tau_d)}  \psi_{b_1, \mathcal{E}} (s-\tau_d). \nonumber \end{equation}
From the Fourier transformation of this, the relation between the amplitudes $a_1$ and $b_1$ is found as
\begin{align}
a_{1,\mathcal{E}} (E') &= \int dE'' g(E'-E'') b_{1, \mathcal{E}}(E'')  e^{i E'' \tau_d} \label{a1}
\end{align}
where $g(E) \equiv \int dt ^{-i\phi(t)+i\phi(t-\tau_d)} e^{iE t}$.
And, $a_1$ and $a_2$ are related with $b_1$ and $b_2$ as
\begin{align}
b_{1, \mathcal{E}}(E') &=  \delta (E'-\mathcal{E}) t'_{\mathcal{E}} a_{2,\mathcal{E}} + e^{i\phi_\textrm{AB}} r_{E'} a_{1, \mathcal{E}}(E') \label{S1} \\
b_{2, \mathcal{E}} (E') &= \delta(E'-\mathcal{E})  r'_{\mathcal{E}} a_{2, \mathcal{E}} + t_{E'} a_{1 \mathcal{E}}(E').  \nonumber
\end{align}
Note that the Aharonov-Bohm phase $\phi_\textrm{AB} = 2\pi B \pi (l/2\pi)^2/(h/e) $ is attached to the reflection event in the loop.

Next, we derive the Fabry-Perot type scattering state resulting from the incident state.
Combining Eqs.~\eqref{a1} and \eqref{S1}, we find the recursive equations for $b_1$ and $b_2$,
$b_{1, \mathcal{\mathcal{E}}}(E') = \delta (E'-\mathcal{E}) t'_{\mathcal{E}} a_{2,\mathcal{E}} + e^{i\phi_\textrm{AB}} r_{E'} \int dE'' g(E'-E'') e^{i E'' \tau_d}  b_{1, \mathcal{E}}(E'')$,
$ b_{2,\mathcal{E}}(E')=   \delta(E'-\mathcal{E}) r'_\mathcal{E} a_{2, \mathcal{E}} +t_{E'} \int dE'' g(E'-E'') e^{iE'' \tau_d} b_{1, \mathcal{E}}(E'')$.
Their Fourier transformations are
\begin{align}
b_{1, \mathcal{E}}(s) &=  t'_\mathcal{E} e^{-i \mathcal{E} s} a_{2, \mathcal{E}} + e^{i\phi_\textrm{AB}} \int ds' r(s') e^{-i \{\phi(s-s') -\phi(s-s'-\tau_d) \}} b_{1, \mathcal{E}}(s-s'-\tau_d ) \label{b1e22} \\
b_{2, \mathcal{E}}(s) &=   r'_\mathcal{E} e^{-i \mathcal{E} s} a_{2, \mathcal{E}} + \int ds' t(s') e^{-i \{\phi(s-s') -\phi(s-s'-\tau_d) \}} b_{1, \mathcal{E}}(s-s'-\tau_d ). \nonumber
\end{align}
$r(s) \equiv \int dE r_E e^{-i Es}$ and $t(s) \equiv \int dE t_E e^{-i Es}$ are the Fourier transforms of $r_E$ and $t_E$, and it can be approximated as a peak structure with width $1/\Delta$ at $s=0$.
The integral in Eq.~\eqref{b1e22} is further evaluated by using the peak structure and under
the condition of $\ddot{U} / (2\dot{U}) \ll \Delta$,
$\int ds' r(s') e^{-i \{\phi(s-s')-\phi(s-s'-\tau_d)\}} e^{i \mathcal{E} s'} = \int ds' r(s') \exp[-i \{ \phi(s)-\phi(s-\tau_d) - (U(s) -U(s-\tau_d))s' +\frac{\dot{U}(s)-\dot{U}(s-\tau_d)}{2} (s')^2 + \bo(s'^3) \} ]  e^{i \mathcal{E} s'}\approx r_{\mathcal{E} +U(s)-U(s-\tau_d)} e^{-i \{\phi(s)-\phi(s-\tau_d)\}} $; in the first equality, we use the Taylor expansion of $\phi(s-s')$ and $\phi(s-s'-\tau_d)$ at $s'=0$, considering the peak structure of $r(s')$ at $s'=0$; in the second equality, we ignore the quadratic term, applying the condition of $\ddot{U} / (2\dot{U}) \ll \Delta$. Then, $b_{1, \mathcal{E}}$ in Eq.~\eqref{b1e22} is iteratively solved as
\begin{equation} \label{b1}
b_{1, \mathcal{E}}(s) = t'_\mathcal{E} e^{-i\mathcal{E} s} a_{2, \mathcal{E}} +\sum_{M=1}^{\infty} e^{-i\phi(s)+i\phi(s-M\tau_d)} e^{i M \mathcal{E}\tau_d} e^{iM \phi_\textrm{AB}} \left[ \Pi_{m'=1}^M  r_{\mathcal{E}+ U(s-(M-m'-1)\tau_d) -U(s-M\tau_d)} \right] t'_\mathcal{E} e^{-i\mathcal{E} s} a_{2, \mathcal{E}}
\end{equation}
Note that the condition of $\ddot{U} / (2\dot{U}) \ll \Delta$ is satisfied in usual experiments, because the smallest time scale for variation of $U$ is limited by the band width of signal generator. Namely, the smallest time scale is $\sim 50$ ps for 10 GHz bandwidth, and then $\ddot{U} / (2\dot{U}) \sim 0.02$ meV, while $\Delta \sim 0.5$ meV (see the main text).
Plugging Eq.~\eqref{b1} into Eqs.~\eqref{cb1} and \eqref{b1e22}, we obtain the Fabry-Perot type expression of $c$ and $b_2$ in time domain,
\begin{align}
&c_\mathcal{E} ( s) = e^{i\phi(s)}  t'_\mathcal{E} e^{-i\mathcal{E} s} a_{2, \mathcal{E}} +\sum_{M=1}^{\infty} e^{i\phi(s-M\tau_d)} e^{i M \mathcal{E}\tau_d} e^{iM \phi_\textrm{AB}} \left[ \Pi_{m'=1}^M  r_{\mathcal{E}+ U(s-(M-m'-1)\tau_d) -U(s-M\tau_d)}   \right]  t'_\mathcal{E} e^{-i\mathcal{E} s} a_{2, \mathcal{E}} \label{ct}  \\
&b_{2, \mathcal{E}}(s) =  r'_{\mathcal{E}} e^{-i\mathcal{E} s}a_{2, \mathcal{E}}  \nonumber \\
&+ \sum_{M=1}^{\infty} t_{\mathcal{E}+ U(s)-U(s-M\tau_d) }  e^{-i\phi(s)+i\phi(s-M\tau_d)}
  e^{iM \mathcal{E} \tau_d} e^{i (M-1) \phi_\textrm{AB}} [ \Pi_{m'=1}^{M-1}   r_{\mathcal{E}+ U(s-(M-m'-1)\tau_d) -U(s-M\tau_d)}  ] t'_\mathcal{E} e^{-i\mathcal{E} s} a_{2, \mathcal{E}}  \label{b2t}
\end{align}
Here each term of index $M$ in Eq.~\eqref{b2t} describes the process that the incident electron with amplitude $a_{2,\mathcal{E}}$ enters the loop at time $s-M\tau_d$, circles the loop $M$ times, and then escapes from the loop at time $s$.  In the $M=1$ term we use $\prod_{m'=1}^{0}\equiv 1$ instead of 0, for brevity.

Next, we determine the incident amplitudes $a_{2,\mathcal{E}}$
with which the resulting scattering state is identical to the coherent state $\psi_\textrm{coh} =(\sqrt{\pi/2} v\tau_p)^{-1/2} \exp[-l^2/(v\tau_p)^2] \exp[i\epsilon_0 l /v ]$ at time $s_0$,
$\int d \mathcal{E} \psi_{c, \mathcal{E}}(l, s_0) = \psi_\textrm{coh}(l)$. We compute $a_{2,\mathcal{E}}$, using  $\psi_{c, \mathcal{E}}(l, s_0) = c_{\mathcal{E}}(s_0-l/v)$ and Eq.~\eqref{ct}, and choosing the time dependence of $U(s)$ at $s \le s_0$ as $U(s\le s_0) = U(s_0)$ to simplify the calculation (note that the emitted wave function of $b_{2, \mathcal{E}}(s)$ at $s_\textrm{rd} \ge s_0$ does not rely on this specific choice):
\begin{eqnarray}
a_{2,\mathcal{E}} & \propto & e^{-i\phi(s_0)} e^{i \mathcal{E} s_0 }   e^{-(\mathcal{E}-U(s_0)-\epsilon_0)^2 \tau_p^2/4}  \frac{(t'_\mathcal{E})^*}{1-r_{\mathcal{E}} e^{-i(\mathcal{E}-U(s_0))\tau_d -i\phi_\textrm{AB}}} \label{coeff} \\
& = & e^{-i\phi(s_0)} e^{i \mathcal{E} s_0 }   e^{-(\mathcal{E}-U(s_0)-\epsilon_0)^2 \tau_p^2/4} \sum_n \frac{(t'_\mathcal{E})^*}{|t_\mathcal{E}|^2/2 - i (\mathcal{E} - \mathcal{E}_n) \tau_d },
\nonumber \end{eqnarray}
where $\mathcal{E}_n = 2 \pi \hbar n / \tau_d + U(s_0) -\phi_\textrm{AB}\hbar/\tau_d$ is the resonance energy of the loop at $s_0$.
In the second equality, the expression is decomposed into the resonance states, since $r_\mathcal{E} \to 1$ at $s_0$.

Finally, we compute the emitted state of $\psi(x,s) = \int d\mathcal{E}  b_{2, \mathcal{E}}(s_\mathrm{rd})$, where the $s_\mathrm{rd} \equiv s - x/v_\mathrm{ed}$, and
obtain Eq.~(3)
\begin{equation} \label{gew}
\begin{aligned}{}
\psi(x,s) &\propto \sum_{n,m=1}^{\infty}
 e^{-(\mathcal{E}_n-\epsilon_0)^2 \tau_p^2/4}
 t_{\mathcal{E}_n+U(s_\mathrm{rd}) -U(s_0) } e^{i m\phi_\textrm{AB}} \left[ \Pi_{m'=0}^{m-1} r_{\mathcal{E}_n+U(s_\mathrm{rd}-(m-m') \tau_d) -U(s_0) } \right] \\
& \quad \quad \quad \times e^{-i (\mathcal{E}_n -U(s_0)) (s_\mathrm{rd}-s_0)} e^{im(\mathcal{E}_n-U(s_0))\tau_d  } e^{-i \phi(s_\mathrm{rd}) + i \phi (s_0)} \zeta_m (s_\mathrm{rd}),
\end{aligned}
\end{equation}
where  $\zeta_{m}(s_\mathrm{rd}) = 1$  for $s_\mathrm{rd} \in [s_0+ m \tau_d, s_0 + (m+1) \tau_d )$ and $\zeta_m (s_\mathrm{rd}) = 0$ otherwise. Here, we performed the integral $\int d\mathcal{E}   b_{2, \mathcal{E}}(s_\mathrm{rd})$, based on the fact that the Lorentzian function in the integrand becomes the delta function when $r_\mathcal{E} \to 1$.
Note that Eq.~\eqref{gew} does not include the emission process with no circulation, since we choose $s_0$ such that the energy $\epsilon_0$ of the coherent state is much lower than the exit barrier height and hence $t_{\epsilon_0} \ll 1$.

Next, we derive Eq.~(5) when $\tau_\mr{tun} \gg \tau_p$. This limit is achieved when $|t_{\epsilon_0 + U_\textrm{mx}}|$ is small.
Then the emission occurs during the time of $U(s) = U_\textrm{mx}$, as $|t_\mathcal{E}|$ is more smaller at $s < s_\textrm{mx}$.
This allows us to write the product term in Eq.~\eqref{gew} as an exponential decaying function in $s_\mathrm{rd}$ as
$\Pi_{m'= 1}^{m} r_{\mathcal{E}_n+U(s_\mathrm{rd}-  m')\tau_d) -U(s_0)} \approx (r_{\mathcal{E}_n+U(s_\mathrm{mx}) -U(s_0)})^{\lfloor (s-s_\mathrm{mx})/\tau_d \rfloor}
\approx \exp[\, (\ln r_{\mathcal{E}_n+U(s_\mathrm{mx}) -U(s_0)}) \,(s_\mathrm{rd}-s_\mathrm{mx} )/\tau_d \, ]
 \approx    \exp[\,- (s_\mathrm{rd}-s_\mathrm{mx})  |t_{\mathcal{E}_n+U(s_\mathrm{mx}) -U(s_0)}|^2 /(2\tau_d) \,]$, where $\lfloor x \rfloor$ means the integer obtained by flooring x; in the first approximation, the reflection amplitudes before $s_\textrm{mx}$ are replaced by 1 and in the other approximations, it is used that $r_{\mathcal{E}_n+U(s_\textrm{mx}) -U(s_0)}$ is close to 1. And, the phase gain by $U$ reduces to $e^{-i\phi (s_\mathrm{rd})+i\phi(s_0)} \propto e^{-i U(s_\mathrm{mx}) s_\mathrm{rd}  }$
Then, the Eq. \eqref{gew} is expressed as
\begin{equation}\label{ew_res}
\psi(x,s) \propto
\sum_{n=1}^{\infty} e^{-(\mathcal{E}_n-\epsilon_0)^2 \tau_p^2/4}   t_{\mathcal{E}_n  +U(s_\mathrm{mx})-U(s_0)} e^{-\frac{ |t_{\mathcal{E}_n+U(s_\mathrm{mx})-U(s_0)}|^2}{2\tau_d} (s_\mathrm{rd}-s_\mathrm{mx} )}  e^{-i (\mathcal{E}_n+U(s_\mathrm{mx})-U(s_0)) s_\mathrm{rd}} \Theta(s_\mathrm{rd}-s_\mathrm{mx}),
\end{equation}
where $\Theta(x)$ is 1 for $x>0$ and 0 otherwise.
Each $n$ term of this equation describes the emission of the $n$-th resonance state whose life time is $\tau_d/|t_{\mathcal{E}_n +U(s_\mathrm{mx})-U(s_0)}|^2$.
The equation~(5) is derived from Eq.~\eqref{ew_res} using the fact that $t_{\mathcal{E}_n +U(s_\mathrm{mx})-U(s_0)}$ is a rapidly increasing function of $n$ in the limit of $\tau_{\mr{tun}} \gg \tau_p$.
The detailed steps for the derivation are as follows.
We first express the transmission amplitude as $t_{\mathcal{E}_n+U(s_\mathrm{mx})-U(s_0)} = t_{\epsilon_0+U(s_\mathrm{mx})-U(s_0)} \exp[\, \tau_\mr{tun} (\mathcal{E}_n -\epsilon_0) \,]$, using the definition of $\tau_\textrm{tun}$.
Then the factor depending on $t_{\mathcal{E}_n+U(s_\mathrm{mx})-U(s_0)}$ in Eq.~\eqref{ew_res},
$t_{\mathcal{E}_n+U(s_\mathrm{mx})-U(s_0)}
  \exp[\, - |t_{\mathcal{E}_n+U(s_\mathrm{mx})-U(s_0)}|^2 (s_\mathrm{rd}-s_\mathrm{mx})/(2\tau_d) \, ]$,
has a non-vanishing value only for the resonance levels $\mathcal{E}_n$
$\in [\epsilon_0  +
( \ln \frac{2\tau_d}{|t_{\epsilon_0+U(s_\mathrm{mx})-U(s_0)}|^2 (s_\mathrm{rd}-s_\mathrm{mx})} )/\tau_\mr{tun}
 - 1/\tau_{\mr{tun}}, \epsilon_0  +
( \ln \frac{2\tau_d}{|t_{\epsilon_0+U(s_\mathrm{mx})-U(s_0)}|^2 (s_\mathrm{rd}-s_\mathrm{mx})} )/\tau_\mr{tun}
 + 1/\tau_{\mr{tun}}]$.
This means that at retarded time $s_\textrm{rd}$, the emitted wave function is determined only by the resonance levels whose lifetime is similar to $s_\textrm{rd}-s_\textrm{mx}$.
When $\tau_p \ll \tau_{\mr{tun}}$, $t_{\mathcal{E}_n+U(s_\mathrm{mx})-U(s_0)}
  \exp[\, - |t_{\mathcal{E}_n+U(s_\mathrm{mx})-U(s_0)}|^2 (s_\mathrm{rd}-s_\mathrm{mx})/(2\tau_d) \, ]$ has a sharp peak of width $1/\tau_\textrm{tun} \ll 1/\tau_p$, thus the gaussian factor in Eq. \eqref{ew_res} can be approximated as
$e^{-(\mathcal{E}_n-\epsilon_0)^2 \tau_p^2/4}
\approx \exp[\, -\frac{\tau_p^2}{4\tau_\mr{tun}^2}(\ln \frac{|t_{\epsilon_0+U(s_\mathrm{mx})-U(s_0)}|^2 (s_\mathrm{rd}-s_\mathrm{mx})}{2\tau_d})^2 \,] $.
Finally, using the identities,
$\int d\mathcal{E}\,  e^{\mathcal{E}/2} e^{-a e^\mathcal{E}} e^{i \mathcal{E} s} = a^{- 1/2 -i s} \Gamma[ 1/2+ i s] $,
$\Gamma \left[ 1/2+ i s \right] \approx \sqrt{\pi \mr{sech} (\pi s )} \approx \sqrt{\pi/2} \mr{sech}(s/2)$, and
$ \int ds\, \mr{sech}(as) e^{-i \omega s}= \pi \mr{sech} (\frac{\pi}{2a}\omega)/a $,
we obtain Eq.~(5),
\begin{align} \label{ll}
\psi(x, s)   \propto &
\sqrt{\frac{2\tau_d}{(s_\mathrm{rd}-s_\mathrm{mx})|t_{\epsilon_0}|^2}}
e^{-   \left( \frac{\tau_p}{\tau_\mr{tun}} \ln \frac{|t_{\epsilon_0}|^2 (s_\mathrm{rd}-s_\mathrm{mx})}{\tau_d}\right)^2} \nonumber \\
&  \times \sum_{n=1}^{\infty} e^{-i (\mathcal{E}_{n} +U(s_\textrm{mx})-U(s_0)) (s_\mathrm{rd}-s_\mathrm{mx})} \mr{sech} \left(\pi \ln  \frac{(s_\mathrm{rd}-s_\mathrm{mx}) |t_{\mathcal{E}_{n} +U(s_\mathrm{mx})-U(s_0)}|^2}{2\tau_d} \right)  \Theta(s_\mathrm{rd}-s_\mathrm{mx}).
\end{align}


\section{Introduction}

The outburst of V445 Pup was discovered on 30 December 2000 by Kanatsu
\citep{kan00}.  The outburst shows unique properties such as
absence of hydrogen, unusually strong carbon emission lines as well as
strong emission lines of Na, Fe, Ti, Cr, Si, Mg etc. \citep{ash03,iij08}.
The spectral features resemble those of classical slow novae except
absence of hydrogen and strong emission lines of carbon \citep{iij08}.
The development of the light curve is very slow (3.3 mag decline
in 7.7 months) with a small outburst amplitude of 6 mag.
From these features, this object has been
suggested to be the first example of helium novae \citep{ash03,kat03}.

In our earlier work \citep{kat03} we presented a theoretical light curve
model with the assumption of blackbody emission from the photosphere,
which resulted in the best fit model of a very massive white dwarf (WD)
($M_{\rm WD} \gtrsim 1.3 ~M_{\odot}$) and a relatively short distance
of $d \lesssim 1$ kpc. However, \citet{iij08} recently showed that
there are absorption lines of Na I D1/2 at the velocities of 16.0 and
73.5 km~s$^{-1}$, suggesting that V445 Pup is located in or beyond
the Orion arm and its distance is as large as $3.5 -  6.5$ kpc.
\citet{wou08} also suggested a large distance of $4.9$ kpc. Moreover,
color indexes during the outburst are consistent with those of
free-free emission.  With these new observational aspects we have revised
the light curve model of V445 Pup.

Helium novae were theoretically predicted by \citet{kat89} as a
nova outburst caused by a helium shell flash on a white dwarf.
Such helium novae have long been a theoretical object
until the first helium nova V445 Pup was discovered in 2000.
\citet{kat89} assumed two types of helium accretion: one is
helium accretion from a helium star companion, and the other is
hydrogen-rich matter accretion with a high accretion rate and,
beneath the steady hydrogen burning shell, ash helium accumulates
on the WD.  The latter case is divided into two kinds of systems
depending on whether hydrogen shell burning is steady (stable) or not.
To summarize, helium accretion onto a WD occurs in three types
of systems: (1) helium accretion from a helium star companion;
(2) steady hydrogen accretion with a rate high enough to keep
steady hydrogen shell burning such as supersoft X-ray sources, e.g.,
RX J0513.9-6951 \citep[e.g.,][]{hac03kb, mcg06}
and CAL 83 \citep[e.g.,][]{sch06},
(3) hydrogen accretion with a relatively low rate resulting in
a recurrent nova such as RS Oph \citep[e.g.,][]{hac07kl}
and U Sco \citep[e.g.,][]{hkkm00}.
In the present paper, we regard that V445 Pup is Case (1), because
no hydrogen lines were detected \citep{iij08}.

In helium novae, mass loss owing to optically thick wind is relatively
weak compared to hydrogen novae, and a large part of the accreted
helium burns into carbon and oxygen and accumulates on the WD
\citep{kat99,kat04}.
Our previous model indicated that V445 Pup contains
a massive WD ($M_{\rm WD} \gtrsim 1.3 ~M_{\odot}$) and the WD mass is
increasing through helium shell flashes.
Therefore, V445 Pup is
a strong candidate of Type Ia supernova progenitors.
Although there are two known evolutional paths of the single
degenerate scenario toward Type Ia supernova \citep[e.g.,][]{hkn99},
helium accreting WDs such as V445 Pup do not belong to either of them.
Therefore, it may indicate the third path to Type Ia supernovae.
If various binary parameters of V445 Pup are determined,
they provide us important clues to binary evolutions to Type Ia supernovae.

In the next section we introduce our multi-band photometric
observations, which indicates that free-free emission dominates
in optical and near infrared.  In \S 3 we present
our free-free emission dominated light curve model and its
application to V445 Pup.  Then we discuss brightness
in quiescence in \S 4.
Discussion and conclusions follow in \S 5 and \S 6, respectively.

\section{Observation}

Shortly after the discovery of V445 Pup outburst, one of us (S.K.)
started multi-band photometry with a 25.4 cm telescope
[focal length $=1600$~mm, CCD camera $=$ Apogee AP-7 (SITE SIA502AB
of $512 \times 512$ pixels)].  $V$ and $R_{\rm c}$ magnitudes are
obtained from January 4 to May 6, 2001, and $I_{\rm c}$ from
January 4, 2001 to January 15, 2007, with the comparison star,
TYC 6543-2917-1 ($V=8.74$, $B-V=0.304$).  All of our data can be taken
from the data archive of Variable Star Observers League in Japan
(VSOLJ)\footnote{http://vsolj.cetus-net.org/}.

\begin{figure}
\epsscale{1.15}
\plotone{f1.color.epsi}
\caption{
$V$ and $I_{\rm c}$ light curves of V445 Pup (large open circles).
The $I_{\rm c}$ data are shifted up by one magnitude in order to
separate it from the $V$ light curve.  Open triangles denote
$V$ magnitudes (IAUC No. 7552,
7553, 7557, 7559, 7569, 7620).  Small open circles and downward arrows
are taken from AAVSO to show very early and later phases of the outburst.
Large downward arrows show an upper limit observation of
$I_{\rm c} > 16.2$ (this work), $V > 20$ and $I > 19.5$ \citep{hen01}.
Straight lines indicate the average decline rates of
$1.9 ~{\rm mag} / 130 $~days $= 0.0146 ~{\rm mag}~{\rm day}^{-1}$ and
$3.3 ~{\rm mag} / 230$ days $= 0.0143 ~{\rm mag}~{\rm day}^{-1}$
for  $I_{\rm c}$ and $V$, respectively.
}
\label{lightobs.only}
\end{figure}

Figure \ref{lightobs.only} shows our $V$ and $I_{\rm c}$ magnitudes
as well as other observations taken from literature.
These two light curves show very slow evolution
($\sim 0.014$ mag day$^{-1}$) followed by an oscillatory behavior
in the $V$ magnitude before it quickly darkened by dust blackout
on about JD 2452100, i.e., 7.5 months after the discovery
\citep{hen01, wou02, ash03}.
Here, we assume the decline rate of the light curves as shown
in Figure \ref{lightobs.only} ignoring the later oscillatory phase
(JD 2452040 -- 2452100) because we assume steady-state in the
nova envelope as mentioned below and, as a result, our theoretical
light curve cannot treat unsteady oscillations.

\begin{figure}
\epsscale{1.15}
\plotone{f2.epsi}
\caption{
Color indexes of $V-I_{\rm c}$ and $V-R_{\rm c}$ as well as $I_{\rm c}$
and $V$.  Open circles: this work. Open triangles: Gilmore
(IAUC 7559, 7569) and Gilmore \& Kilmartin (IAUC 7620).
The horizontal dashed lines indicate the mean values of
$V-R_{\rm c}=0.5$ and $V-I_{\rm c}=1.12$.
}
\label{VmIc}
\end{figure}

Figure \ref{VmIc} shows evolution of color indexes $V-I_{\rm c}$ and
$V-R_{\rm c}$ as well as $V$ and $I_{\rm c}$ themselves.
We can see that both of $V-I_{\rm c}$ and  $V-R_{\rm c}$ are
almost constant with time, i.e., $\approx 1.12$ and $\approx 0.5$,
respectively, during our observation.  This means that the three
light curves of $V$, $I_{\rm c}$, and  $R_{\rm c}$ are almost parallel
and each light curve shape is independent of wavelength.  These
are the characteristic properties of free-free emission dominated
light curves as explained below.

The flux of optically thin free-free emission is inversely proportional to the
square of wavelength, i.e., $F_\lambda \propto \lambda^{-2}$.
This spectrum shape is unchanged with time although the electron
temperature rises during nova outburst because the emission
coefficient depends only very slightly on the electron temperature
\citep[e.g.,][]{bro70}.
Therefore, the color indexes of free-free emission dominated
light curves are unchanged with time.  On the other hand,
the color indexes change with time if blackbody emission dominates
because the photospheric temperature rises with time in nova outbursts.
Figures  \ref{lightobs.only} and \ref{VmIc} indicate
that the emission from V445 Pup is dominated by free-free emission
rather than by blackbody emission.

\begin{figure}
\epsscale{1.15}
\plotone{f3.epsi}
\caption{
Color indexes against wavelength.  Our observational color indexes
of V445 Pup, $V-R_{\rm c}$ and $V-I_{\rm c}$, are plotted
for individual data (dots) and their mean values of 0.5 and 1.12
(open squares).  The color indexes of V1500 Cyg are added
for $y$ magnitude instead of $V$ (filled triangles)
\citep[taken from][]{hac07kc}.  Arrows indicate the effective
wavelength of six bands of $V$, $R$, $I$, $J$, $H$, and $K$.
Solid curves denote color indexes of free-free
emission for $E(B-V)=0$ and 0.5.  }
\label{magnitude_calibration}
\end{figure}

The color index of free-free emission is calculated from
$\lambda F_\lambda \propto \lambda^{-1}$.
When the reddening is known, the reddened color is obtained as
\begin{equation}
m_V - m_\lambda = (M_V - M_\lambda)_0 + c_\lambda E(B-V),
\label{color_reddening_relation}
\end{equation}
where $(M_V - M_\lambda)_0$ is the intrinsic color and
$c_\lambda$ is the reddening coefficient
(these values are tabulated in Table 4 in \citet{hac07kc}
for five colors of $V-R$, $V-I$, $V-J$, $V-H$, and $V-K$).
Figure \ref{magnitude_calibration} shows the color indexes relative
to $V$ for two cases of reddening, i.e., $E(B-V)=0$ and 0.5.

For comparison, we have added four relative colors of V1500 Cyg
\citep{hac06} because its continuum flux is known to be that of
free-free emission \citep{gal76} except for the first few days
after the optical peak.  Here we use Str\"omgren $y$ magnitude
instead of $V$, because the intermediate-wide $y$ band
is almost emission-line free.
Three ($y-I$, $y-H$, and $y-K$) of four color indexes are consistent
with those of color indexes with $E(B-V)=0.5$ \citep{fer77, hac06}.
The $J$ band is strongly contaminated with strong emission lines
of \ion{He}{1} \citep[e.g.][]{bla76, kol76, she76}.
This is the reason why the $y-J$ color index deviates from
that of free-free emission \citep*[see][for more details]{hac07kc}.

Our observed color indexes are shown in Figure \ref{magnitude_calibration}.
The open squares are centered at the mean values of $V-R_{\rm c}=0.5$ and
$V-I_{\rm c}=1.12$, whereas dots represent individual observations.
In V445 Pup, no strong emission lines dominate the continuum
in $V$, $R$, and $I$ bands \citep{kam02, iij08}.  Therefore we conclude
that the color indexes of V445 Pup are consistent with those of
free-free emission with the reddening of $E(B-V)=0.51$ \citep{iij08}.

\section{Modeling of Nova Light Curves}
\label{model_nova_outburst}

\subsection{Basic Model}
After a thermonuclear runaway sets in on the surface of a WD,
the envelope expands to a giant size and the optical luminosity
reaches its maximum.  Optically thick winds occur and
the envelope reaches a steady state.  Using the same method
and numerical techniques as in \citet{kat94h,kat03},
we have followed evolution of a nova by connecting steady state
solutions along the sequence of decreasing envelope mass.
We have solved the equations of motion, continuity, radiative diffusion,
and conservation of energy, from the bottom of the helium envelope
through the photosphere.
The winds are accelerated deep inside the photosphere.
Updated OPAL opacities are used \citep{igl96}.
As one of the boundary conditions for our numeral code, we assume that
photons are emitted at the photosphere as a blackbody with a photospheric
temperature, $T_{\rm ph}$. \citet{kat03} calculated the visual magnitude
$M_V$ on the basis of blackbody emission and constructed the light curves.
In the present work, we calculate free-free emission dominated
light curves.  The envelope structure, wind mass loss rate,
photospheric temperature, and photospheric radius of the WD envelope
are essentially the same as those in the previous model.

The flux of free-free emission from the optically thin region outside
the photosphere dominates in the continuum flux of optical and
near infrared wavelength region and is approximated by
\begin{equation}
F_\nu \propto \int N_e N_i d V
\propto \int_{R_{\rm ph}}^\infty {\dot M_{\rm wind}^2
\over {v_{\rm wind}^2 r^4}} r^2 dr
\propto {\dot M_{\rm wind}^2 \over {v_{\rm ph}^2 R_{\rm ph}}},
\label{free-free-wind}
\end{equation}
where $F_\nu$ is the flux at the frequency $\nu$, $N_e$ and $N_i$
are the number densities of electrons and ions, respectively,
$V$ is the volume, $R_{\rm ph}$ is the photospheric radius,
$\dot M_{\rm wind}$ is the wind mass loss rate, and $v_{\rm ph}$ is
the photospheric velocity \citep{hac06, hac07kc}.

The proportionality constant in equation (\ref{free-free-wind}) cannot be
determined a priori because we do not calculate radiative transfer
outside the photosphere: These proportionality constant are determined
by fitting with observational data \citep{hac06, hac07k, hac07kc}.

\subsection{Free-free Light Curve and WD Mass}
We have calculated free-free emission dominated light curves of V445 Pup
for WD masses of 1.2, 1.3, 1.33, 1.35, 1.37, and $1.377 ~M_{\odot}$.
The last one is the upper limit of the mass-accreting WDs \citep{nom84}.
The adopted WD radius and chemical composition are listed in Table
\ref{composition}.  Here we assume that chemical composition is constant
throughout the envelope.

\begin{deluxetable}{llll}
\tabletypesize{\scriptsize}
\tablecaption{Model Parameters\tablenotemark{a}
 \label{composition}}
\tablewidth{0pt}
\tablehead{
\colhead{$M_{\rm WD}$} & \colhead{$\log R_{\rm WD}$} & \colhead{$Y$} &
\colhead{$X_{\rm C+O}$} \cr
\colhead{($M_{\odot}$)} & \colhead{($R_{\odot}$)} & \colhead{}
 & \colhead{}
}
\startdata
1.2 & -2.193  & 0.68 & 0.3 \cr
1.3 & -2.33   & 0.48 & 0.5 \cr
1.33 & -2.417 & 0.38 & 0.6 \cr
1.35 & -2.468 & 0.38 & 0.6  \cr
1.37 & -2.535 & 0.58 & 0.4  \cr
1.37 & -2.535 & 0.38 & 0.6  \cr
1.37 & -2.535 & 0.18 & 0.8  \cr
1.377 & -2.56 & 0.38 & 0.6
\enddata
\tablenotetext{a}{The heavy element content is assumed to be $Z=0.02$
that includes carbon and oxygen by the solar ratio.
}
\end{deluxetable}

\begin{figure}
\epsscale{1.15}
\plotone{f4.epsi}
\caption{
Free-free emission dominated light curves for various WD masses.
The chemical composition is assumed to be $Y=0.38$,
$X_{\rm C+O}= 0.6$, and $Z=0.02$
throughout the envelope.  The WD mass is attached to each curve.
The dashed and dotted curves denote $M_{\rm WD}=1.37  ~M_\sun$ model
but with different chemical composition of $X_{\rm C+O}= 0.4$ and 0.8,
respectively.  Small open circles denote the starting points of
light curve fitting in Fig. \ref{lightcurve_fitting}:
upper and lower points of $M_{\rm WD}= 1.37 ~M_\sun$ (solid
line) correspond to Models 4 and 3
and those of 1.377 $M_\odot$ correspond to Models 8 and 7, respectively.
A short solid line indicates the decline rate of observed $V$ data in Fig.
\ref{lightobs.only} (3.3 mag / 230 days).
}
\label{lightcurve}
\end{figure}

Figure \ref{lightcurve} shows the calculated free-free light curves.
More massive WDs show faster decline.  This is mainly because
the more massive WDs have the less massive envelope and the smaller
mass envelope is quickly taken off by the wind.
The figure also shows two additional models of $M_{\rm WD}= 1.37
~M_\sun$ with different compositions of $X_{\rm C+O} =0.4$ and 0.8.
These models show almost similar decline rates to that of
$M_{\rm WD}= 1.37 ~M_\sun$ with $X_{\rm C+O}=0.6$.

The starting point of our model light curve depends on the initial
envelope mass.  When an initial envelope mass is given,
our nova model is located somewhere on the light curve.
For a more massive ignition mass, it starts from an upper point of
the light curve.  Then the nova moves rightward with the decreasing
envelope mass due to wind mass loss and nuclear burning.

The development of helium shell flashes is much slower than that of
hydrogen shell flashes mainly because of much larger envelope masses
of helium shell flashes \citep{kat94h}.
The short straight solid line in Figure \ref{lightcurve} represents
the decline rate of $V$ light curve in Figure \ref{lightobs.only}.
The length of the line indicates the observed period which is relatively
short because of dust blackout 230 days after the optical maximum.
We cannot compare the entire evolution period of the light curve
with the observational data so that fitting leaves some ambiguity
in choosing the best fit model.
Even though, we may conclude that WDs with masses of
$M_{\rm WD} \lesssim 1.33 ~M_\sun$ are very unlikely
because their light curves are too slow to be comparable with
the observation.

\begin{deluxetable*}{lllllllllr}
\tabletypesize{\scriptsize}
\tablecaption{Parameters of Fitted Model
 \label{fitting_parameter}}
\tablewidth{0pt}
\tablehead{
\colhead{model } & \colhead{$M_{\rm WD}$} & \colhead{$Y$}
& \colhead{$X_{\rm C+O}$} &
\colhead{$\Delta M_{\rm He,ig}$\tablenotemark{a}}
& \colhead{$\Delta M_{\rm ej}$\tablenotemark{b}}
& \colhead{$\eta_{\rm He}$\tablenotemark{c}}& \colhead{Distance}
& \colhead{$\dot M_{\rm He}$\tablenotemark{d}} & \colhead{$\tau_{\rm rec}$\tablenotemark{d}} \cr
\colhead{No. } & \colhead{($M_\sun$)} & \colhead{} & \colhead{}
&\colhead{($M_{\odot}$)} &\colhead{($M_{\odot}$)} & \colhead{}
& \colhead{(kpc)} & \colhead{($M_\sun$~yr$^{-1}$)}
& \colhead{(yr)}
}
\startdata
1& 1.35 & 0.38  & 0.6 &3.3E-4 &1.8E-4 & 0.45 & 6.6 & 6.E-8 & 5000 \cr
2& 1.37 & 0.58  & 0.4 & 1.6E-4 &9.2E-5 & 0.42  &5.0  &7.E-8  &2000 \cr
3& 1.37 & 0.38  & 0.6 &1.9E-4 &8.7E-5 & 0.53 & 4.8 & 6.E-8 & 3000 \cr
4& 1.37 & 0.38  & 0.6 &2.1E-4 &1.1E-4 & 0.49 & 5.5 & 5.E-8 & 4000 \cr
5& 1.37 & 0.18  & 0.8 &3.5E-4 &1.2E-4 &0.64 & 6.0 & 4.E-8  & 9000\cr
6& 1.37 & 0.18  & 0.8 &3.8E-4 &1.5E-4 &0.61 & 6.9 &4.E-8 &10000 \cr
7& 1.377 & 0.38 & 0.6 &1.8E-4 &7.4E-5 & 0.60 & 4.8 & 5.E-8 & 4000\cr
8& 1.377 & 0.38 & 0.6 &2.2E-4 &9.6E-5 & 0.56 & 5.2 & 4.E-8 & 5000 \cr
\enddata
\tablenotetext{a}{Envelope mass at ignition is assumed to be equal to the
mass at the optical peak, which we assume as JD 2451872
(28 days before the discovery).}
\tablenotetext{b}{Total mass ejected by the wind}
\tablenotetext{c}{Mass accumulation efficiency}
\tablenotetext{d}{estimated from Fig. \ref{dMenvMacc}}
\end{deluxetable*}

Figure \ref{lightcurve_fitting} represents the light curve fitting
with observational data.  Here, the model light curves of $I_{\rm c}$
are identical to those of $V$ but are lifted up by 1.12 mag,
which is the mean value of $V-I_{\rm c}$ obtained in Figure \ref{VmIc}.
It should be noted that, in this figure, these light curves are further
lifted up by 1.0 mag because of $I_c - 1$.

\begin{figure}
\epsscale{1.15}
\plotone{f5.epsi}
\caption{
Our model light curves are fitted with the observation.
Dotted line: Model 1. Thin dashed line: Model 2. Thin solid line: Model 3.
Thick solid line: Model 4.  Dash-three dotted line: Model 5.
Thin dash-dotted line: Model 6. Dashed line: Model 7.
Dash-dotted line: Model 8.
Our model $I_{\rm c}$ light curves are
identical with those of $V$ but are lifted up by 1.12 mag
(and further shifted up by 1 mag for $I_{\rm c} - 1$).
The ordinate on the right vertical axis represents
the absolute magnitude of Model 4.
For the other models, the model light curves are shifted down
by 0.4 mag (Model 1), up by 0.2 mag (Model 2) up by 0.3 mag (Models
3 and 7), down by 0.2 mag (Model 5), down by 0.5 mag (Model 6),
up by 0.1 mag (Model 8).
 The observational data are same as those
in Fig. \ref{lightobs.only}.
}
\label{lightcurve_fitting}
\end{figure}

These light curves are a part of the model light curves
in Figure \ref{lightcurve}.
In the early phase of the outburst the light curve declines almost
linearly so that fitting is not unique.  We can fit any part of
our model light curve if its decline rate is the same as
$\approx 0.014-0.015 ~{\rm mag}~{\rm day}^{-1}$, for example,
either top or middle part of the same light curve of $1.377 ~M_\sun$
WD in Figure \ref{lightcurve}.  In such a case we show two
possible extreme cases, i.e., the earliest starting point and
the latest one by open circles, as shown in Figure \ref{lightcurve}.
These model parameters are summarized in Table \ref{fitting_parameter},
where we distinguish the model by model number.

Thus we have selected several light curves which show a reasonable
agreement with the observation.
However, we cannot choose the best fit model among them because
the observed period is too short to discriminate a particular
light curve from others because the difference among them
appears only in a late stage.

For relatively less massive WDs of $M_{\rm WD} \lesssim 1.33 ~M_{\odot}$,
we cannot have reasonable fits with the observation for any part of
the model light curves.  Among relatively more massive WDs
of $M_{\rm WD} \gtrsim 1.35 ~M_{\odot}$ in Table \ref{fitting_parameter},
Model 1 shows slightly slower decline.  Therefore, the
$1.35 ~M_{\odot}$ WD may be the lowest end for the WD mass.
Thus we may conclude that V445 Pup contains a very massive
WD of $M_{\rm WD} \gtrsim 1.35 ~M_{\odot}$.

\subsection{Mass Accumulation Efficiency}

During helium nova outbursts, a part of the helium envelope is blown
off in winds, while the rest accumulates on the WD.  We define the mass
accumulation efficiency, $\eta_{\rm He}$, as the ratio of the envelope
mass that remains on the WD after the helium nova outburst to the helium
envelope mass at ignition, $\Delta M_{\rm He,ig}$ \citep{kat04}.

The mass accumulation efficiency is estimated as follows.
We have calculated the mass lost by winds during the outburst,
$\Delta M_{\rm ej}$.  Note that $\Delta M_{\rm ej}$ is the calculated
total ejecta mass which is ejected during the wind phase,
and not the mass ejected during the observing period which may be
shorter than the wind phase.

The ignition mass is approximated by the envelope mass at the optical
peak,  i.e., $\Delta M_{\rm He,ig} \approx \Delta M_{\rm He,peak}$.
Since the exact time of the optical peak is unknown, we assume
that the optical peak is reached on JD 2451872, i.e., the earliest
prediscovery observation in the brightest stage reported to IAU Circular
No. 7553, 28 days earlier than the discovery.
The envelope mass at the optical peak is estimated from our
wind solution and is listed as $\Delta M_{\rm He,ig}$
in Table \ref{fitting_parameter}.
The resultant accumulation efficiency,
\begin{equation}
\eta_{\rm He} \equiv {{\Delta M_{\rm He,peak}-\Delta M_{\rm ej}}
 \over {\Delta M_{\rm He,peak}}},
\end{equation}
is also listed in Table \ref{fitting_parameter}.
The efficiencies are as high as $\sim 50$\%.
The WD of V445 Pup is already very massive
($ M_{\rm WD} \gtrsim 1.35 ~M_{\odot}$) and its mass has increased
through helium nova outbursts.  Therefore, V445 Pup is a strong
candidate of Type Ia supernova progenitors.

\section{Quiescent Phase}
Before the 2000 outburst, there was a 14.5 mag star
at the position of V445 Pup (taken from the archive of
VSNET\footnote{http://vsnet.kusastro.kyoto-u.ac.jp/vsnet/}),
but no bright star has been observed since the dust blackout.
We may regard that 14.5 mag is the preoutburst magnitude of the binary.
There are two possible explanations for this quiescent phase luminosity,
one is the accretion disk luminosity and the other is the luminosity of
bright companion star. In the following subsections, we discuss how
these possible sources contribute to the quiescent luminosity.

\subsection{Accretion Disk}
In some nova systems, an accretion disk mainly contributes to the brightness
in their quiescent phase.  If the preoutburst luminosity of V445 Pup
comes from an accretion disk, its absolute magnitude is approximated by
\begin{eqnarray}
M_V {\rm (obs)}&=& -9.48 -{5\over 3}\log\left({M_{\rm WD}\over M_\sun}
{{\dot M_{\rm acc}}\over {M_\sun ~{\rm yr}^{-1}}} \right) \cr
 & & -{5\over 2} \log(2\cos i),
\label{accretion-disk-Mv}
\end{eqnarray}
where $M_{\rm WD}$ is the WD mass, $\dot M_{\rm acc}$ the mass accretion
rate, and $i$ the inclination angle \citep[equation (A6) in][]{web87}.
Assuming that $M_{\rm WD}= 1.37 ~M_{\odot}$ and $i=80$\arcdeg
\citep{wou08}, we have $M_V =1.4$, 2.3, and 3.1 for the accretion rates
of $1 \times 10^{-6}$, $3 \times 10^{-7}$,
and $1 \times 10^{-7} ~M_{\odot}$~yr$^{-1}$, respectively.

The apparent magnitude of the disk is calculated from
\begin{equation}
m_V-M_V = A_V + 5 \log D_{10},
\label{distance-modulus}
\end{equation}
where $D_{10}$ is the distance divided by 10 pc.
With the absorption of $A_V = 1.6$ and the distance of 3 kpc,
the apparent magnitude is estimated to be $m_V= 15.4$, 16.3 and 17.1,
for the above accretion rates,
respectively.  For the distance of 6.5 kpc,
we obtain brightnesses of $m_V = 17.1$, 18.0, and 18.8,
respectively. All of these values are much fainter than 14.5 mag.
Therefore, it is very unlikely that an accretion disk mainly
contributes to the preoutburst luminosity.

\begin{figure}
\epsscale{1.15}
\plotone{f6.color.epsi}
\caption{
Evolutional tracks of helium stars with masses of $M_{\rm He}= 0.6$,
0.7, 0.8, 0.9, 1.0 (thick line), 1.2, 1.4, 1.6, 1.8, 2.0 (thick line),
2.5, and $3.0 ~M_\sun$ in the H-R diagram.
The 0.6 and $0.7~M_\sun$ stars evolve
blueward and do not become red giants.  We stopped calculation when
carbon ignites at the center for 2.5 and $3.0 ~M_\sun$ stars.
Open circles and squares denote stars with a 14.5 mag
brightness at the distance of 6.5 kpc and 4.9 kpc, respectively, for
$A_V=1.6$.
}
\label{hestarevolution}
\end{figure}

\subsection{Helium Star Companion}
Another possible source of the preoutburst brightness is
a helium star companion.  Figure \ref{hestarevolution} shows
evolutional tracks of helium stars with masses between 0.6 and
$3.0 ~M_\sun$ from the helium main-sequence to the red giant stage
in the H-R diagram.  We use OPAL opacities and an initial
chemical composition of $X=0.0$, $Y=0.98$ and $Z=0.02$.
The numerical method and input physics are the same as those
in \citet{sai95}.

As shown in Figure \ref{hestarevolution},
low mass helium stars do not evolve to a red giant.
In our new calculation, the $0.8 ~M_{\odot}$ helium
star evolves to a red giant, but the 0.6 and $0.7 ~M_{\odot}$ helium
stars evolve toward blueward.  \citet{pac71} showed that  stars of
$M_{\rm He} \gtrsim 1.0 ~M_{\odot}$ evolve to a red giant while 0.5,
0.7, and $0.85 ~M_{\odot}$ do not.  Our calculations are essentially the
same as those of \citet{pac71} and the difference is attributed
mainly to the difference between the adopted opacities.

Figure \ref{hestarevolution} also shows locations of stars
whose apparent magnitudes are $m_V= 14.5$ for $A_V= 1.6$.
Here, the distance is assumed to be 4.9 (open squares) or 6.5 kpc
(open circles).  For example, a star of (temperature, luminosity)
$= (\log T_{\rm ph} ~({\rm K})$,  $\log L_{\rm ph}/L_\sun) =$
(4.2, 2.81), (4.4, 3.25), and (4.6, 3.75) could be observed as a 14.5 mag
star for 6.5 kpc and $(\log T_{\rm ph} ~(K)$,
$\log L_{\rm ph}/L_\sun) =$ (4.2, 2.57), (4.4, 3.0),
and (4.6, 3.51) for 4.9 kpc.  Therefore, the observed 14.5 mag is
consistent with the luminosities and temperatures of
slightly evolved helium stars of $M_{\rm He} \gtrsim 0.8 ~M_\sun$
if the companion is a blue star of $\log T_{\rm ph} \gtrsim 4.5$.

On the other hand, if the companion is redder than $\log T_{\rm ph}
\lesssim 4.4$, there is no corresponding evolution track
in the low luminosity region of the H-R diagram as shown in Figure
\ref{hestarevolution}.  In such a case the preoutburst luminosity
cannot be attributed to a helium star companion.
Kato (2001)\footnote{http://www.kusastro.kyoto-u.ac.jp/vsnet/Mail/alert5000/msg00493.html}
suggested that preoutburst color of V445 Pup was
much bluer than that of symbiotic stars (which have a red giant companion)
but rather close to that of cataclysmic variables (main-sequence or
little bit evolved companion). This argument is consistent with
our estimate of relatively high surface temperature
$\log T_{\rm ph} \gtrsim 4.5$ (see Fig. \ref{hestarevolution}).

\section{Discussions}

\subsection{Distance}
As explained in \S\S 3.1 and 3.2, we do not solve energy transfer outside
the photosphere and, thus, we cannot determine the proportionality
constant in equation (\ref{free-free-wind}).  Therefore, we cannot
determine the distance to V445 Pup directly from the comparison of
theoretical absolute magnitude with the observational apparent magnitude.
Instead, we can approximately estimate the distance by assuming
that the free-free flux is larger than the blackbody flux during
the observing period.  This assumption gives a lower limit to
the distance.  We expect that the distance estimated thus is
close to the real value.  The 8th column of Table \ref{fitting_parameter}
lists the distance estimated for each model.  These distances are
consistent with the observational estimates of $3.5 \lesssim d \lesssim 6.5$ kpc
\citep{iij08} and $\sim 4.9$ kpc \citep{wou08}.

\subsection{Helium Ignition Mass and Recurrence Period}

\begin{figure}
\epsscale{1.15}
\plotone{f7.color.epsi}
\caption{
The helium ignition mass, $\Delta M_{\rm He, ig}$, of helium-accreting
WDs is plotted against the helium accretion rate, $\dot M_{\rm He}$.
The WD mass is attached to each curve. Straight dashed lines indicates
the recurrence period. The open square indicates
the ignition mass of each model in Table \ref{fitting_parameter}.
}
\label{dMenvMacc}
\end{figure}

We have calculated evolution of C+O white dwarfs accreting helium
at various rates until the ignition of a helium shell flash.
Chemical compositions assumed are $X_{\rm C}=0.48$ and
$X_{\rm O}=0.50$ for the WD core, and $X=0.0$ and  $Y=0.98$ for
the accreted envelope.
The initial model adopted for a given core mass and accretion rate is
a steady-state model in which the heating due to the accretion is balanced
with the radiative energy flow.
Adopting such an initial model is justified because the WD has accreted
matter from the companion for a long time and experienced many
shell flashes.

As the helium accretion proceeds, the temperature at the bottom of
the envelope gradually increases.
When the temperature becomes high enough the
triple-alpha reaction causes a shell flash.
The mass of the helium envelope at the ignition depends on the
WD mass and the accretion rate as shown in Figure~\ref{dMenvMacc}.
The envelope mass required to ignite a shell flash tends
to be smaller for a higher accretion rate and for a larger WD mass.
Helium flashes are weaker for higher accretion rates and lower WD masses.
In particular they are very weak for accretion rates higher than
$\sim 10^{-6} M_{\odot}$ yr$^{-1}$.

The envelope mass at the ignition depends also on the core temperature.
If the core temperature
of the initial WD is lower than that of the steady state model
employed for our calculations, the envelope mass at the ignition
would be larger.

The recurrence period,  $\Delta M_{\rm He,ig}/{\dot M}_{\rm He}$, is
also plotted in Figure~\ref{dMenvMacc} by dashed lines.
The recurrence period corresponding to the ignition mass of our
fitted models are listed in Table \ref{fitting_parameter}.

\subsection{Mass Transfer from Helium Star Companion}

We have estimated the mean accretion rate ${\dot M}_{\rm He}$
of the WD as in Table \ref{fitting_parameter}.
The companion star feeds its mass to the WD via the Roche lobe overflow
or by winds depending on whether it fills the Roche lobe or not.
Here we examine if this accretion rate is comparable to the mass
transfer rates of the Roche lobe overflow from a helium star
companion.  We followed stellar evolutions of helium stars from
the main sequence assuming that the star always fills its effective
Roche lobe of which radius is simply assumed to be $1.5 ~R_\sun$.
The resultant mass loss rates are shown in Figure \ref{Hestar.dmdt}
for stars with initial masses of  1.2, 1.0 and $0.8 ~M_\sun$.
These results are hardly affected even if we change the Roche lobe
radius.

The mass loss rate decreases with the companion mass almost
independently of the initial value. Except for
the short initial and final stages, these rates are
as large as $\dot M_{\rm He} \sim 10^{-6} ~M_\sun$~yr$^{-1}$
and much larger than the mass accretion rate estimated from our
fitted models in Table \ref{fitting_parameter}.
Since almost all of the mass lost by the companion accretes
onto the WD in the case of Roche lobe overflow,
the accretion may result in very weak shell flashes
for such high rates as $10^{-6} M_\sun$~yr$^{-1}$.
A strong shell flash like V445 Pup is realized only when
the final stages of mass-transfer in which the mass transfer rate
quickly drops with time. This rare case may correspond to a
final stage of binary evolution after the WD had grown up in mass to
near the Chandrasekhar limit from a less massive one and the companion
had lost a large amount of mass via Roche lobe overflow.
Such information will be useful for modeling a new path of
binary evolution scenario that will lead to a Type Ia supernova.

\begin{figure}
\epsscale{1.15}
\plotone{f8.epsi}
\caption{
Mass loss rate of a helium star that fills its Roche lobe of the radius
$1.5 ~R_\sun$. The initial stellar mass is 1.2, 1.0 (dotted line), and
$0.8 ~M_\sun$. Time goes on from left to right. See the text for more detail.
}
\label{Hestar.dmdt}
\end{figure}

\subsection{Comparison with Other Observation}
The mass of the dust shell can be estimated from infrared continuum flux
with the assumption that the emission originated from warm dust.
With infrared 10 $\mu$m spectrum, \citet{lyn04b} estimated the dust
mass to be $2 \times 10^{-6} M_{\odot}$ for a distance of 3 kpc.
This value would be increased if we adopt a larger distance
instead of 3 kpc, but still consistent with our estimated
ejecta mass $\Delta M_{\rm ej}\sim (0.7-1.8) \times 10^{-4}
M_{\odot}$ in Table \ref{fitting_parameter},
because the dust mass is a small part of the ejected mass.

\citet{lyn04a} reported the absence of He II or coronal lines on
their near IR spectra and suggested that the ionizing source was
not hot enough on 2004 January 14 and 16 (1146 days after
the optical maximum).   \citet{lyn05} also reported,
from their near IR observation on 2005 November 16
that the object had faded and the thermal dust emission
had virtually disappeared. They suggested that the dust had
cooled significantly until that date (1818 days).  This suggestion
may constrain the WD mass of V445 Pup because
Figure \ref{lightcurve} shows the outburst lasts more than
1800 days for less massive WDs ($ \lesssim 1.33 ~M_\sun$).
If the WD had cooled down until the above date we may exclude
less massive WD models ($ \lesssim 1.33 ~M_\sun$) because
these WDs evolve slowly and their hot surfaces emit high energy photons
at least until 1800 days after the optical peak.

We have shown that the WD mass of V445 Pup is increasing with time.
When the WD continues to grow up to the Chandrasekhar limit,
central carbon ignition triggers a Type Ia supernova explosion
if the WD consists of carbon and oxygen. We regards the star as
a CO WD instead of an O-Ne-Mg WD because no indication of neon
were observed in the nebula phase spectrum \citep{wou05}. Therefore,
we consider that V445 Pup is a strong candidate of Type Ia supernova.

When a binary consisting of a massive WD and a helium star
like V445 Pup becomes a Type Ia supernova, its spectrum may more or less
show a sign of helium.  The search for such helium associated with
a Type Ia supernova has been reported for a dozen Type Ia supernovae
\citep{mar03, mat05}, but all of them are negative detection.
It may be difficult to find such a system because this type of
Type Ia supernovae are very rare.

The binary seems to be still deeply obscured by the optically thick
dust shell even several years after the outburst. This blackout period
is much longer than that of the classical novae such as OS And ($\sim 20$
days), V705 Cas ($\sim 100$ days), and DQ Her ($\sim 100$ days).
As the ejected mass estimated in Table \ref{fitting_parameter} is
not much different from those of classical novae, the difference of
the blackout period may be attributed to a large amount of dust in
the extremely carbon-rich ejecta of a helium nova.
Moreover, observed low ejection velocities of $\sim 500$ km~s$^{-1}$
\citep{iij08} is not much larger than the escape velocity of
the binary with a relatively massive total mass [e.g., $1.35 ~M_\sun
+ (1 - 2)  ~M_\sun$].  Both of these effects lengthen the dust
blackout period.  When the dust obscuration will be cleared
in the future the period of blackout provides useful information on
the dust shell.

\section{Conclusions}
Our main results are summarized as follows:

1. We  have reproduced $V$ and $I_{\rm c}$ light curves
using free-free emission dominated light curves calculated on the
basis of the optically thick wind theory.

2. From the light curve analysis we have estimated the WD mass
to be as massive as $M_{\rm WD} \gtrsim 1.35~M_\sun$.

3. Our light curve models are now consistent with a longer distance
of $3.5 \lesssim d \lesssim 6.5$ kpc \citep{iij08} and 4.9 kpc \citep{wou08}.

4. We have estimated the ejecta mass as the mass lost by optically
thick winds, i.e., $\Delta M_{\rm wind} \sim 10^{-4} M_\odot$.
This amounts to about a half of the accreted helium matter so that
the accumulation efficiency reaches $\sim 50$\%.

5. The white dwarf is already very close to the Chandrasekhar mass,
i.e., $M_{\rm WD} \gtrsim 1.35~M_\sun$, and the WD mass had increased
after the helium nova.  Therefore, V445 Pup is a strong candidate
of Type Ia supernova progenitors.

6. We emphasize importance of observations after the dense
dust shell will disappear, especially observations of the color and
magnitude, orbital period, and inclination angle of the orbit.
These are important to specify companion nature.

\acknowledgments
M.K. and I.H. are grateful to people at the Astronomical Observatory of
Padova and at the Department of Astronomy of the University of Padova, Italy,
for their warm hospitality.  Especially we thank Takashi Iijima for
fruitful discussion on V445 Pup, which stimulated us to start this work.
The authors thank Masaomi Tanaka for information on the helium detection
in Type Ia supernovae. We also thank the anonymous referee for
useful comments to improve the manuscript.  Thanks are also due to
the American Association of Variable Star Observers (AAVSO)
for the photometric data of V445 Pup and Taichi Kato for introducing us
the discussion in VSNET.  This research has been supported
in part by the Grants-in-Aid for Scientific Research (16540211, 20540227)
from the Japan Society for the Promotion of Science.


\section{Introduction}

Inspired by Shannon’s classic information theory \cite{shannon1948mathematical}, Weaver and Shannon proposed a more general definition of a communication system involving three different levels of problems, namely, (i) transmission of bits (the technical problem); (ii) semantic exchange of transmitted bits (the semantic problem); and (iii) effect of semantic information exchange (the effectiveness problem). The first level of communication, which is the transmission of bits, has been well studied and realized in conventional communication systems based on Shannon’s bit-oriented technical framework.
However, with the massive deployment of emerging devices, including Extended Reality (XR) and Unmanned Aerial Vehicles (UAVs), diverse tasks with stringent requirements pose critical challenges to traditional bit-oriented communications, which are already approaching the Shannon physical capacity limit. This imposes the Sixth Generation (6G) network towards a communication paradigm shift to semantic level and effectiveness level by exploiting the context of data and its importance to the task.

Initial works on ``semantic communications'' have  mainly focused on identifying the content of the traditional text and speech \cite{luo2022, shi2021semantic}, and the information freshness, i.e., AoI \cite{NowAoI} as a semantic metric that captures the timeliness of the information. However, these cannot capture the data importance sufficiently of achieving a specific task. In \cite{kountouris2021semantics}, a joint design of information generation, transmission, and reconstruction was proposed. Although the authors explored the benefits of including the effectiveness level in \cite{strinati20216g,popovski2020semantic}, an explicit and systematic communication framework incorporating both semantic level and effectiveness level has not been proposed yet. There is an urgent need for a unified communication framework aiming at task-oriented performances  for diverse data types.

  %

Motivated by this, in this paper, we propose a generic task-oriented and semantics-aware (TOSA) communication framework, which jointly considers the semantic level information about the data context, and effectiveness level performance metric that determines data importance, for different tasks with various data types. The main contributions of this paper are:
\begin{enumerate}

    \item We first present unique characteristics of the traditional text, speech, image, video data types, and emerging $360^\circ$ video, sensor, haptic, and machine learning models. For each data type, we summarize the semantic information definition and extraction methods in Section~II.
    \item We then propose a generic TOSA communication framework for typical time critical and non-critical tasks, where semantic level and effectiveness level are jointly considered. Specifically, by exploiting the unique characteristics of different tasks, we present TOSA information, their extraction and recovery methods, and effectiveness level performance metrics to guarantee the task requirements in Section~III.
    \item To demonstrate the effectiveness of our proposed TOSA communication framework, we present the TOSA solution tailored for interactive Virtual Reality (VR) data
    with the aim to maximize the long-term quality of experience (QoE) within the VR interaction latency constraints and analyze the results in Section IV.
    \item Finally, we conclude the paper in Section V.
\end{enumerate}

	\section{Semantic Information Extraction}

In this section, we focus on analyzing the characteristics of all data types, and summarizing the semantic information definition with corresponding extraction methods as shown in Table~\ref{feature_summarization}.

\begin{table}[!h]
	\caption{Semantic information extraction of different data types}
	\begin{center}
		\begin{tabular}{|c|c|c|}
			\hline
			\textbf{Data Type}  &Semantic Information &\makecell[c]{Semantic Information\\ Extraction Method}\\
			\hline
			Text &Embedding&BERT \\
			\hline
			Speech &Embedding&BERT \\
                \hline
                Image &Edge, Corner, Blob, Ridge&SIFT, CNN \\
			\hline
			Video &Temporal Correlation&\makecell[c]{CNN} \\
                \hline
                \makecell[c]{$360^\circ$ Video }&FoV&\makecell[c]{Biological Information\\ Compression}\\
                \hline
   			Haptic Data &JND&Web's Law\\
                \hline
                \makecell[c]{Sensor and\\ Control Data}&Freshness& AoI\\

			\hline
		\end{tabular}
		\label{feature_summarization}
	\end{center}
\end{table}

	\subsection{{{Speech and Text}}}

For a one-dimensional speech signal, the speech-to-text conversion can be first performed by speech recognition.
 With the extracted text information,
various approaches developed by Natural Language Processing (NLP) community can be applied to extract embedding as typical semantic information, which represents the words, phrases, or text as a low-dimensional vector. The most famous embedding extraction method is Bidirectional Encoder Representations from Transformers (BERT) proposed by Google, which can be pre-trained and fine-tuned via one additional output layer for different text tasks.
However, during the speech-to-text conversion process, the timbre and emotion conveyed in the speech may lose.

	\subsection{{Image and Video}}
	The image is a two-dimensional data type, where the image geometric structures, including edges, corners, blobs, and ridges, can be identified as typical semantic information. Although various traditional signal processing methods have been developed to extract image geometric structures such as SIFT (Scale-Invariant Feature Transform), the convolutional Neural Network (CNN) has shown stronger capability to extract complex geometric structures with its matrix kernel.

    Video is a typical three-dimensional data type as the combination of two-dimensional images with an extra time dimension. Therefore, the temporal correlation between adjacent frames can be identified as important semantic information, where the static background can be ignored during transmission. To extract temporal correlation information from video, different CNN structures have been utilized.

	\subsection{{$360^\circ$ Video}}
The $360^\circ$ rendered video is a new data type in emerging XR applications. The most important semantic information is identified as human field-of-view (FoV), which occupies around one-third of the $360^\circ$ video and only has the highest resolution requirement at the center \cite{Liu_vr}. In this case, biological information, such as retinal foveation and ballistic saccadic eye movements  can be leveraged for semantic information extraction. Therefore, biological information compression methods have been utilized to extract the semantic information, where retinal foveation and ballistic saccadic eye movements are jointly considered to optimize the semantic information extraction process.

 \subsection{{Haptic Data}}
	Haptic data consists of two submodalities, which are tactile information and kinesthetic \cite{Antonakoglou2018}. For tactile information,
 five major dimensions can be identified, which are friction, hardness perception, warmth conductivity, macroscopic roughness, and microscopic roughness.
 Kinesthetic information refers to the position/orientation of human body parts and external forces/torques applied to them.
	To reduce the redundant raw haptic data, Just Noticeable
Difference (JND) is identified as valuable semantic information to filter the haptic signal that cannot be perceived by the human, where the Weber's law serves as an important semantic information extraction criterion.

\subsection{Sensor and Control data}

Sensors are usually deployed to monitor the physical characteristics of the environment (e.g., temperature, humidity, or traffic) in a geographical area. The acquisition of data is transformed to status updates that are transmitted through a network to the destination nodes. Then these data are processed in order to extract useful information, such as control commands or remote source reconstruction, that can be further utilized in the prediction of the evolution of the initial source. The accuracy of the reconstructed data either in control commands or to predict the evolution is directly related to the relevance or the semantic value of the data measurements. Thus, one important aspect is the generation of traffic and how it can be affected in order to filter only the most important samples so the redundant or less useful data will be eliminated to reduce potential congestion inside the network.

The AoI has also a critical role in dynamic control systems, since it was shown that non-linear AoI and VoI are paradigm shifts and they can improve the performance of such systems. Furthermore, we have seen in early studies that the semantics of information (beyond timeliness) can provide further gains, by reducing the amount of information that is generated and transmitted without degrading the performance.

 \begin{figure*}[!h]
	\centerline{\includegraphics[scale=0.7]{framework_2.pdf}}
	\caption{TOSA communication framework for different tasks with diverse data types.}
	\label{TOSA_framework_fig}
\end{figure*}

 \subsection{{Machine Learning Model}}
 With the massive deployment of machine learning algorithms, machine learning-related model has been regarded as another important data type.

 \begin{itemize}
     \item \textbf{Federated Learning (FL) Model:} The FL framework has been considered as a promising approach to preserve data privacy, where each participating device uploads the model gradients or model weights to the server and receives the global model from the server.
     \item \textbf{Split Learning (SL) Model:} Due to the limited computation capability of devices and heavy computation burden, SL has been proposed to split the neural network model between the server and devices, where the device executes the model up to the cut layer and sends the smashed data to the central server to execute the remaining layers. Then the gradient of the smashed data is transmitted back from the server to update the local model.
 \end{itemize}

However, it is noted that explicit semantic information definition for machine learning model has not been proposed yet.

\begin{table*}[!h]
	\caption{TOSA Communication Framework Summarization of different tasks}
	\begin{center}
		\begin{tabular}{|c|c|c|c|c|c|}
			\hline
			\textbf{Data Type} & Task & Communication Entity& Recovery& Latency Type & \makecell[c]{Effectiveness Level\\ Perfromance Metrics}\\
			\hline
			\multirow{1}{*}{\makecell[l]{Speech}}&\makecell[c]{Speech Recognition} &  Human-Machine &Yes/No&Non-critical &\makecell[l]{$\bullet$F-measure\\$\bullet$Accuracy\\$\bullet$BLEU\\$\bullet$Perplexity}  \\
			\hline
			\multirow{5}{*}{\makecell[l]{Image}}& \makecell[c]{Face Detection} & Machine-Machine &No&Non-critical &  \makecell[l]{$\bullet$IoU\\$\bullet$mAP\\$\bullet$F-measure\\$\bullet$MAE} \\
                \cline{2-5}
                & \makecell[c]{Road Segmentation} & Machine-Machine&Yes & Critical &\makecell[l]{$\bullet$IoU\\$\bullet$Pixel Accuracy\\$\bullet$MPA\\$\bullet$Latency}\\
			\hline
			\multirow{5}{*}{\makecell[c]{$360^\circ$ Video}} & \makecell[c]{Display in AR}& Machine-Human& Yes& Non-Critical &\makecell[l]{$\bullet$PSNR\\$\bullet$SSIM\\$\bullet$Alignment Accuracy}  \\

                \cline{2-5}
                 & \makecell[c]{Display in VR} & Machine-Human& No& Non-Critical & \makecell[l]{$\bullet$PSNR\\$\bullet$SSIM\\$\bullet$Timing Accuracy\\$\bullet$Position Accuracy}\\
                \hline
                \makecell[c]{Haptic Data} & Grasping and Manipulation & Machine-Human&No & Critical& \makecell[l]{$\bullet$SNR\\$\bullet$SSIM}\\
			\hline
               Sensor& Networked control systems & Machine-Machine& No& Critical& \makecell[l]{$\bullet$LGQ}\\
			\hline
			\multirow{5}{*}{\makecell[c]{ML\\Model}}& Federated Learning & Machine-Machine &\rule[0pt]{1cm}{0.1em}& Critical/Non-Critical& \makecell[l]{$\bullet$Latency\\$\bullet$Reliability\\ $\bullet$Convergence Speed\\$\bullet$Accuracy} \\
                \cline{2-6}
                & Split Learning & Machine-Machine &\rule[0pt]{1cm}{0.1em}& Critical/Non-Critical & \makecell[l]{$\bullet$Latency\\ $\bullet$Reliability \\$\bullet$Convergence Speed \\$\bullet$Accuracy}\\

			\hline
		\end{tabular}
		\label{Tasks_summarization}
	\end{center}
\end{table*}

\section{TOSA Communication Framework}
	\label{different_task}
	In this section, we propose a generic TOSA communication framework incorporating both semantic level and effectiveness level, for typical time critical and non-critical tasks as shown in Fig.~\ref{TOSA_framework_fig}, where the TOSA information, its extraction and recovery methods,  and effectiveness level performance metrics are presented in detail.

	\subsection{One-hop Task}
We consider one-hop tasks with a single link transmission in this section, where each communication entity can be either human or machine as summarized in Table~\ref{Tasks_summarization}.

	\subsubsection{{Speech Recognition}} In a speech recognition task, the human speech needs to be transmitted to the server, and the speech recognition task can be further divided into conversation-type task (e.g., human inquiry) and command-type task (e.g., smart home control) depending on the speech content. The conversation-type task focuses on understanding the intent, language, and sentiment to provide human with free-flow conversations. The command-type task focuses on parsing the specific command over the transmitted speech and then controlling the target device/robot.

 In the conversation-type task, the TOSA information can be  keywords and emotions. The device can obtain the TOSA information by transforming the speech signal into text and extracting keywords and emotions via BERT. Then the server recovers the text via transformer decoder. In the command-type task, the TOSA information can be the binary command, the device can directly parse the speech signal and obtain the binary command signal for transmission, where no recovery is needed at the receiver side. The effectiveness level performance metrics include F-measure, accuracy, bilingual evaluation understudy (BLEU), and perplexity, where user satisfaction should also be considered.

	\subsubsection{{Face Detection and Road Segmentation}}

Face detection and road segmentation are two emerging image processing tasks \cite{Zhao2019, Minaee2022},  where the captured images are required to be transmitted to the central server for processing. However, the road segmentation task in autonomous driving applications imposes  stringent latency and reliability requirements due to road safety issues. This is because the vehicles need to instantaneously react to the rapidly changing  environment.

For the time non-critical face detection task, TOSA information can be the face feature that is extracted via CNN. After being transmitted to the central server, the regions with CNN features (R-CNN) can be applied to perform face detection. For the time critical road segmentation task, one possible solution is to identify the region of interest (ROI) features, i.e., road, as the TOSA information, and crop the images via region proposal algorithms. Then, the central server can perform image segmentation via mask R-CNN. Both tasks can be evaluated via effectiveness level performance metrics, including Intersection over Union (IoU), mean average precision (mAP), F-measure, and mean absolute error (MAE). It is noted that road segmentation in the autonomous driving application can be evaluated via pixel accuracy, and mean pixel accuracy (MPA). However, the trade-off between accuracy and latency remains to be an important challenge to solve.

		\subsubsection{{Display in Extended Reality}}
Based on the Milgram \& Kishino’s Reality–Virtuality Continuum, the XR can be  classified as Augmented Reality (AR), Mixed Reality (MR), and VR, where MR is defined as a superset of AR. Therefore, we focus on the AR and VR display tasks in the following.

In the AR display task, the central server transmits the rendered 3D model of a specific virtual object to the user. It is noted that the virtual object identification and its pose information related to the real world is the key to achieving alignment between virtual and physical objects. Therefore, the virtual object identification and pose information can be extracted as TOSA information to reduce the data size. Then, by sharing the same 3D model library, the receiver can locally reconstruct the 3D virtual object model based on the received TOSA information. To evaluate the 3D model transmission, Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity (SSIM) can be adopted as effectiveness level performance metrics. However, how to quantify the alignment accuracy among the virtual objects and physical objects as a performance metric remains to solve.

	In the VR display task, the central server transmits virtual $360^\circ$ video streaming to the user. To avoid the transmission of the whole $360^\circ$ video, the central server can predict the eye movements of the user and extract the corresponding FoV as TOSA information. Apart from the PSNR and SSIM mentioned in AR, timing accuracy and position accuracy are also important effectiveness level performance metrics to avoid cybersickness including: 1) initial delay: time difference between the start of head motion and that of the corresponding feedback; 2) settling delay: time difference between the stop of head motion and that of the corresponding feedback; 3) precision: angular positioning consistency between physical movement and visual feedback in terms of degrees; and 4) sensitivity: capability of inertial sensors to perceive subtle motions and subsequently provide feedback to users.

  \subsubsection{{Grasping and Manipulation}}
  Haptic communication has been incorporated by industries to perform grasping and manipulation for efficient manufacturing and profitable
production rates, where the robot transmits the haptic data to the manipulator. The shape and weight of the objects to be held are measured using cutaneous feedback derived from the fingertip contact pressure and kinesthetic feedback of finger positions, which should be transmitted within stringent latency requirement to guarantee industrial operation safety.

Due to the difficulty in supporting massive haptic data with stringent latency requirement, JND can be identified as an important TOSA information to ignore the haptic signal that cannot be perceived by the manipulator. Two effectiveness level performance metrics including SNR and SSIM have been verified to be applicable to vibrotactile quality assessment.

\subsubsection{Control}
In networked control systems (NCS), typically, multiple sensors measure the system state of their control processes and transmit the generated data over a resource-limited shared wireless network. These data usually are en-queued and then transmitted over unreliable channels that cause excessive delays resulting in outdated or even obsolete for decision-making based on less reliable information. Therefore, data freshness and importance are extracted as TOSA information via AoI and value of information (VoI) to guarantee the timing requirement, respectively \cite{TimingProcIEEE}. A typical effectiveness level performance metric that is used to minimize is the Linear Quadratic Gaussian (LQG) cost function, and usually the lower the value of the LQG function the higher the quality of control (QoC).

\subsubsection{{Machine Learning}}
In the following, we focus on task-oriented communications for two distributed ML models, which are FL and SL.

\paragraph{Federated Learning} For the time non-critical tasks, such as NLP and image classification, the goal of FL is to guarantee a high learning accuracy without latency constraints. Traditional loss functions for NLP and image classification, such as mean square error (MSE), MAE, and cross-entropy, can be directly used as effectiveness level performance metrics. However, for time critical tasks, such as object recognition in self-driving cars, the goal of FL is to balance the trade-off between learning accuracy, communication latency, and computation latency. The effectiveness level performance metrics are loss functions with latency constraints. Time critical tasks bring
 communication challenges, and communication-efficient FL should be designed to decrease the model size to satisfy latency constraints via federated dropout, federated pruning, and model compression.

Federated dropout is a simple way to prevent the learning
model from overfitting through randomly dropping neurons
and is only used during the training phase, which decreases
communication and computation latencies and slightly improves learning accuracy. However, during the testing phase, the extracted task-oriented information is the whole learning model and transmitted between the server and devices.  The extracted task-oriented information is the model with non-dropped weights. Meanwhile, federated dropout does not need model recovery.

Unlike federated dropout only temporarily removing neurons, federated pruning permanently removes neurons in either or both training and testing phases. The extracted task-oriented information in federated pruning is the pruned model. The decision of which parameters to remove is made by considering the importance of each parameter.  The pruning ratio should be carefully designed to guarantee learning accuracy and extra computation latency is required to calculate the importance of parameters. Thus, how to design federated pruning methods with low computation complexity needs to be investigated. In addition, federated pruning does not need model recovery.

Model-compression schemes, such as sparsification and quantization decrease the model size. The extracted task-oriented information is the sparse or quantized model. However, these methods slightly decrease the convergence rate and achieve a modest accuracy (about 85$\%$). Thus, how to design a model compression algorithm with high learning accuracy still needs to be investigated. In addition, compressed FL model needs to be recovered at the receiver.

\paragraph{Split Learning} In SL, the smashed data and its gradient associated with the cut layer are the extracted task-oriented information and transmitted between the server and devices, where no model recovery is required. When multiple devices exist in SL, all devices interact with the edge server in a sequential manner, resulting in high training latency. For time non-critical tasks, such as NLP and image classification, the goal of SL is to achieve high learning accuracy without latency constraints and the effectiveness level performance metrics are the same as that of the FL. However, for time critical tasks, the SL cannot guarantee the requirement of low latency because of its sequential training pattern.

To adapt the SL to time critical tasks, such as real-time object tracking, splitfed learning (SFL) \cite{thapa2022splitfed} and hybrid split and federated learning (HSFL) \cite{Xiaolan} are proposed, where they combine the primary advantages of FL and SL. The effectiveness level performance metrics of the SFL and HSFL are learning accuracy and training latency. However, SFL and HSFL assume that the model is split at the same cut layer and the server-side model is trained in a synchronous mode. Splitting at the same cut layer leads to asynchronization of device-side model training and smashed data transmission. Thus, how to select an optimal split point and deal with the asynchronization of SL remain important challenges to solve. Also, different split points can result in different smashed data. Thus, how to merge these smashed data in the server-side model should be considered.

 \subsection{Chain Task}
In this section, we analyze more complicated but practical chain tasks including XR-aided teleoperation and chain of control, where multiple entities cooperate through communication links to execute the task.

 \subsubsection{XR-aided Teleoperation}
XR-aided teleoperation aims to integrate 3D virtual objects/environment into remote robotic control, which can provide the manipulator with immersive experience and high operation efficiency \cite{xr-aided}. To implement a closed-loop XR-aided teleoperation system, the wireless network is required to support mixed types of data traffic, which includes control and command (C\&C) transmission, haptic information feedback transmission, and rendered $360^{\circ}$ video feedback transmission. Stringent communication requirements have been proposed to support XR-aided teleoperation use case, where over 50 $\mathrm{Mb/s}$ bandwidth is needed to support video transmission, and reliability over $1-10^{-6}$ within millisecond latency is required to support haptic and C\&C transmission.

  As XR-aided teleoperation task relies on both parallel and consecutive communication links, how to guarantee the cooperation among these communication links to execute the task is of vital importance. Specifically, the parallel visual and haptic feedback transmissions should be aligned with each other when arriving at the manipulator, and consecutive C\&C and feedback transmissions should be within the motion-to-photon delay constraint. Either violation of alignment in parallel links or latency constraint in consecutive links will lead to BIP and cybersickness. Therefore, both parallel alignment and consecutive latency should be quantified into effectiveness level performance metrics to guarantee the success of XR-aided teleoperation. Moreover, due to the motion-to-photon delay, the control error between the expected trajectory and actual trajectory will accumulate along with the time, which may lead to task failure. Hence, how to alleviate the accumulated error remains an important challenge to solve.

 \subsubsection{Chain of control}
In the scenario of a swarm of (autonomous) robots where they need to perform a collaborative task (or a set of tasks) within a deadline over a wireless network, an effective communication protocol that takes into account the peculiarities of such a scenario is needed. Otherwise, the generated and transmitted data will be of very high volume that eventually will face congestion in the network causing large delays and the operated control mechanisms will not be synced causing inefficient or even dangerous operation. Consider the simple case of two robots, let's say Robot A and Robot B that are communicating through a wireless network and they are not collocated. Robot A controls remotely Robot B such that to execute a task and the outcome of that operation will be fed to Robot A for performing a second operation to send the outcome back to Robot B. All this must happen within a strict deadline. The amount of information that is generated, transmitted, processed, and sent back can be very large with the traditional information agnostic approach. On the other hand, if we take into account the semantics of information and the purpose of communication, we change the whole information chain, from its generation point until its utilization. Therefore, defining TOSA metrics for the control loop and communication between a swarm of (autonomous) robots is crucial and it will significantly reduce the amount of information leading to a more efficient operation.

\section{Case Study}
We validate the effectiveness of ML algorithms in optimizing time-critical TOSA communication under one typical case, namely, MEC-enabled and reconfigurable intelligent surface (RIS)-assisted terahertz (THz) VR network.
\begin{figure}[!h]
    \centering
    \includegraphics[width=3.5 in]{latency_qoe.pdf}
    \caption{Average QoE and VR interaction latency of the MEC-enabled THz VR network of each time slot via CDRL with viewpoint prediction, where the VR interaction latency constraint is 20 ms.}
    \label{basic_modules}
\end{figure}
The goal of wireless VR networks is to guarantee high QoE under VR interaction latency constraints. Note that the QoE guarantees the seamless, continuous, smooth, and uninterrupted experience of each VR user. However, traditional wireless VR networks transmit whole $360^\circ$ videos, which leads to low QoE and high VR interaction latency. In the simulation, we extract the FoV of VR users as TOSA information to decrease the transmission data size. To extract the FoV, viewpoints of VR users are predicted via recurrent neural network (RNN) algorithms. Based on the predicted viewpoints, the corresponding FoVs of VR video frames can be rendered and transmitted in advance. Thus, the MEC does not need to render and transmit the whole $360^\circ$ VR video frames, which can substantially decrease the VR interaction latency and improve the QoE of VR users.
Fig. 2 plots the average QoE and VR interaction latency of each time slot via constrained deep reinforcement learning  with the viewpoint prediction via RNN. It is observed that the proposed ML algorithms maximize the long-term QoE of VR users under the VR interaction latency threshold.

	\section{Conclusion}
 In this article, we propose a generic task-oriented and semantics-aware (TOSA) communication framework incorporating both semantic  and effectiveness levels  for various tasks with diverse data types.
  We first identify the the unique characteristics of all existing and  new data types in 6G networks and summarize the semantic information with its extraction methods. To achieve task-oriented communications for various data types, we then present the corresponding TOSA information, their TOSA information extraction and recovery methods, and  effectiveness level performance metrics
for both time critical and non-critical tasks.
   Importantly, our results demonstrate that our proposed TOSA communication framework can be tailored for VR data  to maximize the long-term QoE within VR interaction latency constraints. The paradigm shift from conventional Shannon’s bit-oriented communication design towards  the TOSA communication design
   will flourish new  research on task-driven,  context and importance-aware  data transmission in 6G networks.

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