Pumped spin-current and shot noise spectra in a single quantum dot
Bing Dong$^{1,2}$
H.L. Cui$^{1,3}$
and X.L. Lei$^{2}$
We exploit the pumped spin-current and current noise spectra under equilibrium condition %%@ in a single quantum dot connected to two normal leads, as an electrical scheme for %%@ detection of the electron spin resonance (ESR) and decoherence. We propose spin-resolved %%@ quantum rate equations with correlation functions in Laplace-space for the analytical %%@ derivation of the zero-frequency atuo- and cross-shot noise spectra of charge- and %%@ spin-current. Our results show that in the strong Coulomb blockade regime, ESR-induced %%@ spin flip generates a finite spin-current and the quantum partition noises in the %%@ absence of net charge transport. Moreover, spin shot noise is closely related to the %%@ magnetic Rabi frequency and decoherence and would be a sensitive tool to measure them.
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\begin{document}

\title{Pumped spin-current and shot noise spectra in a single quantum dot}

\author{Bing Dong$^{1,2}$, H.L. Cui$^{1,3}$, and X.L. Lei$^{2}$} 
\affiliation{$^{1}$Department of Physics and Engineering Physics, Stevens Institute of %%@
Technology, Hoboken, New Jersey 07030 \\
$^{2}$Department of Physics, Shanghai Jiaotong University,
1954 Huashan Road, Shanghai 200030, China \\
$^{3}$School of Optoelectronics Information Science and Technology, Yantai University, %%@
Yantai, Shandong, China}

\begin{abstract}

We exploit the pumped spin-current and current noise spectra under equilibrium condition %%@
in a single quantum dot connected to two normal leads, as an electrical scheme for %%@
detection of the electron spin resonance (ESR) and decoherence. We propose spin-resolved %%@
quantum rate equations with correlation functions in Laplace-space for the analytical %%@
derivation of the zero-frequency atuo- and cross-shot noise spectra of charge- and %%@
spin-current. Our results show that in the strong Coulomb blockade regime, ESR-induced %%@
spin flip generates a finite spin-current and the quantum partition noises in the %%@
absence of net charge transport. Moreover, spin shot noise is closely related to the %%@
magnetic Rabi frequency and decoherence and would be a sensitive tool to measure them. 

\end{abstract}

\pacs{72.70.+m, 73.23.Hk, 73.50.Td, 85.35.-p}

\maketitle

\textit{Introduction}---There have been extensive investigations focused on electron %%@
spin detection and measurement via charge transport in mesoscopic quantum dot (QD) %%@
system,\cite{Engel,Balatsky,Martin} which is motivated from the fact that easy %%@
preparation and manipulation of electron spins, as well as the remarkably long spin %%@
coherence time, provide the way for applications in spintronics and quantum information %%@
processing.\cite{Prinz} In these systems, transport is governed not only by the charge %%@
flow, but also by the spin dynamics. Recently, a pure spin current has been reported by %%@
direct optical injection without generation of a net charge current.\cite{Stevens} %%@
Theoretically, a spin source device has also been proposed to carry pure spin flow based %%@
on electron spin resonance (ESR) in a QD-lead system with sizable Zeeman %%@
splitting.\cite{Wang1,Zhang} Moreover, both auto- and cross-correlation noise spectra of %%@
spin-current have been studied for this QD-based spin battery.\cite{Wang2}   

However, Coulomb blockade (CB) effect on this QD spin battery needs investigation, since %%@
CB effect is an essential feature of transport through a QD. More importantly, the spin %%@
configuration of the confined electrons on QD is apparently affected by CB effect, which %%@
directly determine the efficiency of this device. As well, these previous studies also %%@
neglected the inevitable spin decoherence due to coupling of the single spin with %%@
environment. Therefore, it is the purpose of this paper to study this ERS-pumped %%@
spin-current and its fluctuations for a QD connected with two normal leads in the strong %%@
CB regime at zero temperature. The setup is schematically depicted in Fig.~1: The single %%@
electron levels in the dot are split by an external magnetic field $B$, %%@
$\epsilon_{\downarrow}-\epsilon_{\uparrow}=g_{z}\mu_{B}B$ ($=\Delta$ Zeeman energy), %%@
where $g_{z}$ is the effective electron g-factor in the $z$ direction and $\mu_{B}$ is %%@
the Bohr magneton. The gate voltage controls the chemical potentials $\mu$ of two leads %%@
located between the two split levels and no bias voltage is applied to the two leads. %%@
After a spin-up electron tunnels into the QD, an oscillating magnetic field $B_{\rm %%@
rf}(t)=(B_{\rm rf}\cos\Omega t,\, B_{\rm rf}\sin\Omega t$) applied perpendicularly to %%@
the constant field $B$ with the frequency $\Omega$ nearly equal to $\Delta$ can pump %%@
electron to the higher level where its spin is flipped, then the spin-down electron can %%@
tunnel out to the leads. Because the two currents with opposite directions have %%@
different spin orientations, spin-currents are established in both leads with equal %%@
values. If the Coulomb interaction in the QD is strong enough to prohibit the double %%@
occupation, no more electrons can enter the QD before the spin-down electron exits. As a %%@
result, the number of electrons exiting from the QD is equal to that of electrons %%@
entering the QD, namely, the charge currents exactly cancel out each other. Otherwise, %%@
in the case $\epsilon_{\uparrow}+U < \mu$ the ESR pumping will generate both %%@
spin-current and charge-current. Note that both leads have contributions to the %%@
spin-current of one lead, causing the correlation of spin-currents in different leads, %%@
which provides possibility of studying cross-correlation of spin currents without bias %%@
voltage.\cite{Sauret}            

\begin{figure}[t]
\includegraphics[height=5cm,width=6cm]{fig1}
\caption{Schematic view for the QD-based spin battery.}
\label{fig1}
\end{figure}

\textit{Theoretical formalism}---We write the Hamiltonian of the ESR-induced spin %%@
battery under consideration as:
\bn
H &=& \sum_{\eta, k, \sigma}\epsilon _{\eta k\sigma} 
c_{\eta k\sigma }^{\dagger }c_{\eta k\sigma }^{\pdag}+ \sum_{\sigma} \epsilon_{\sigma} 
c_{d \sigma }^{\dagger }c_{d \sigma }^{\pdag} + Un_{d \uparrow }n_{d \downarrow } \cr 
&& + \sum_{\eta, k, \sigma} (V_{\eta} c_{\eta k\sigma }^{\dagger }c_{d \sigma}^{\pdag} %%@
+{\rm {H.c.}}) + H_{\rm rf}(t),
\label{hamiltonian}
\en
where $c_{\eta k \sigma}^{\dagger}$ ($c_{\eta k \sigma }$) and $c_{d \sigma}^{\dagger}$ %%@
($c_{d \sigma}$) are the creation (annihilation) operators for electrons with momentum %%@
$k$, spin-$\sigma$ and energy $\epsilon_{\eta k \sigma}$ in the lead $\eta$ ($={\rm %%@
L,R}$) and for a spin-$\sigma$ electron on the QD, respectively. The third term %%@
describes the Coulomb interaction among electrons on the QD. $n_{d\sigma}=c_{d %%@
\sigma}^{\dagger} c_{d \sigma}^{\pdag}$ is the occupation operator in the QD. The fourth %%@
term represents the tunneling coupling between the QD and the reservoirs. The last term %%@
$H_{\rm rf}(t)$ describes the coupling between the spin states due to the rotating field %%@
$B_{\rm rf}(t)$ and can be written, in the rotating wave approximation (RWA), as
\bq
H_{\rm rf}(t)=R_{\rm rf} \big ( c_{d \uparrow}^{\dagger} %%@
c_{d\downarrow}^{\pdag}e^{i\Omega t} + c_{d\downarrow}^{\dagger} c_{d\uparrow}^{\pdag} %%@
e^{-i\Omega t} \big ),
\eq
with the ESR Rabi frequency $R_{\rm rf}=g_{\perp}\mu_{B}B_{\rm rf}/2$, with the g-factor %%@
$g_{\perp}$ and the amplitude of the rf field $B_{\rm rf}$.

Here, in order to anticipate the CB and the intrinsic spin relaxation, we utilize the %%@
quantum rate equations for the system density matrix elements: $\rho_{00}$ and %%@
$\rho_{\sigma \sigma}$ describe the occupation probability in the QD being, %%@
respectively, unoccupied and spin-$\sigma$ states, and the off-diagonal term %%@
$\rho_{\uparrow \downarrow (\downarrow \uparrow)}$ denotes coherent superposition of the %%@
two coupled spin states in the QD.\cite{Gurvitz} The doubly-occupied state is prohibited %%@
due to the infinite Coulomb interaction $U\rightarrow \infty$. Here we focus our %%@
interest on the nonadiabatic pumping where the photo-assisted resonance is achieved and %%@
neglect cotunneling processes.
For the purpose of evaluating the noise spectrum, we introduce the spin-resolved (in %%@
both terminals) density matrices $\rho_{ab}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})}(t)$, meaning that the QD is on the electronic state %%@
$|a\rangle$ ($a=b=0,\,\uparrow,\,\downarrow$) or on quantum superposition state ($a\neq %%@
b$) at time $t$ together with $m_{L\uparrow}$ ($m_{R\uparrow}$) spin-up electrons and %%@
$m_{L\downarrow}$ ($m_{R\downarrow}$) spin-down electrons in the left (right) lead. %%@
Obviously, $\rho_{ab}(t)=\sum_{m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}} \rho_{ab}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})}(t)$ and the quantum rate equations for these %%@
spin-resolved density matrices in the rotating frame with respect to $\Omega$ are
\begin{subequations}
\label{rateq}
\bn
\dot{\rho}_{00}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})} %%@
& = & \Gamma_{L\downarrow} \rho_{\downarrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} %%@
\atop m_{L\downarrow}-1,m_{R\downarrow}})} + \Gamma_{R\downarrow} \rho_{\downarrow %%@
\downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}-1})} %%@
\cr
&& - (\Gamma_{L\uparrow}+\Gamma_{R\uparrow}) \rho_{00}^{({m_{L\uparrow},m_{R\uparrow} %%@
\atop m_{L\downarrow},m_{R\downarrow}})}, \label{r0} \\
\dot{\rho}_{\uparrow \uparrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})} &=& \Gamma_{L \uparrow} %%@
\rho_{00}^{({m_{L\uparrow}+1,m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})} + %%@
\Gamma_{R\uparrow} \rho_{00}^{({m_{L\uparrow},m_{R\uparrow}+1 \atop %%@
m_{L\downarrow},m_{R\downarrow}})} \cr
&& + \frac{1}{2T_1} \rho_{\downarrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})} - \frac{1}{2T_1} \rho_{\uparrow %%@
\uparrow}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})} \cr
&& + i R_{\rm rf} (\rho_{\uparrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})} - \rho_{\downarrow %%@
\uparrow}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})}), %%@
\label{r1} \\
\dot{\rho}_{\downarrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})} &=& -(\Gamma_{L\downarrow} + \Gamma_{R \downarrow}) %%@
\rho_{\downarrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})} + \frac{1}{2T_1} \rho_{\uparrow %%@
\uparrow}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})} \cr 
&& \hspace{-2cm} - \frac{1}{2T_1} \rho_{\downarrow %%@
\downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})} + i %%@
R_{\rm rf} (\rho_{\uparrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})} - \rho_{\downarrow %%@
\uparrow}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})}), %%@
\label{r2} \\   
\dot{\rho}_{\uparrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})} &=& iR_{\rm rf} (\rho_{\uparrow %%@
\uparrow}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})} - %%@
\rho_{\downarrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})}) \cr
&& \hspace{-2.5cm} + i \delta_{\rm ESR} \rho_{\uparrow %%@
\downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})}
- [\frac{1}{2} ( \Gamma_{L\downarrow} + \Gamma_{R \downarrow} ) + \frac{1}{T_2}] %%@
\rho_{\uparrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})}, \label{r3} \en
\end{subequations}
and the normalization relation $\rho_{00}+ \sum_{\sigma} \rho_{\sigma \sigma}=1$. The %%@
equation of motion for $\rho_{\downarrow \uparrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})}$ can be obtained by implementing complex conjugate on %%@
Eq.~(\ref{r3}). In these equations, $\delta_{\rm ESR}=\Delta-\Omega$ is the ESR detuning %%@
and $\Gamma_{\eta \sigma}=2\pi \sum_{k} |V_{\eta}|^2\delta (\omega-\epsilon_{\eta k %%@
\sigma})$ denotes the strength of coupling between the QD and the lead $\eta$ involving %%@
spin $\sigma$. In wide band limit, these tunneling amplitudes are independent of energy %%@
and for simplicity we set $\Gamma_{\eta \uparrow}=\Gamma_{\eta \downarrow}=\Gamma/2$. %%@
Furthermore, we describe the coupling of single spin with the environment in a %%@
phenomenological way via introducing two time scales: the spin relaxation time $T_1$ of %%@
an excited spin state into the thermal equilibrium, and the spin decoherence time $T_2$ %%@
related to the loss of phase coherence of the spin superposition state. The measurement %%@
of $T_2$ in QDs is currently active topic because it is the limit time scale for %%@
coherent spin manipulation and thus quantum information processing. In typical GaAs QD, %%@
$\Gamma=8~\mu$eV, corresponding to a tunneling time $\tau\sim \Gamma^{-1}=0.5$~ns. To %%@
observe single- and multi-photon effects in tunneling (nonadiabatic regime), it must %%@
require the spin flipping time $T_{\Omega}<\tau$. If one takes $B=1~$T, the Zeeman %%@
energy is $\Delta=26~\mu$eV which determine the optimal driving frequency %%@
$\Omega=13\pi~$GHz and the corresponding $T_{\Omega}=0.15~$ns. These proposed %%@
spin-device parameters can be easily realized by present technology.     

Recently, $T_1$ in a single QD was probed to be an order of microsecond via transport %%@
experiment,\cite{Fujisawa} which is notably longer than other time scales $T_1\gg %%@
T_2,\,\tau,\,T_{\Omega}$. Hence it is a good approximation to assume $T_1\rightarrow %%@
\infty$ in the following calculation. Consequently, one recovers the usual quantum rate %%@
equations for the reduced density matrix elements $\rho_{ab}(t)$ for a single QD with %%@
spin coupling: $\dot {\bm \rho}(t)=(\dot {\rho}_{00}, \dot {\rho}_{\uparrow \uparrow}, %%@
\dot {\rho}_{\downarrow \downarrow}, \dot {\rho}_{\uparrow \downarrow}, \dot %%@
{\rho}_{\downarrow \uparrow})={\cal M} {\bm \rho}(t)$ with\cite{Martin,Gurvitz}  
\bq
{\cal M}= \left (
\begin{array}{ccccc}
-\Gamma_{\uparrow} & 0 & \Gamma_{\downarrow} & 0 & 0 \\
\Gamma_{\uparrow} & 0 & 0 & iR_{\rm rf} & -iR_{\rm rf} \\
0 & 0 & -\Gamma_{\downarrow} & -iR_{\rm rf} & iR_{\rm rf} \\
0 & iR_{\rm rf} & -iR_{\rm rf} & - \Gamma_{\downarrow}V + i\delta_{\rm ESR} & 0 \\
0 & -iR_{\rm rf} & iR_{\rm rf} & 0 & - \Gamma_{\downarrow}V + i\delta_{\rm ESR}
\end{array}
\right),
\eq
and $V=\frac{1}{2} + \frac{1}{\Gamma_{\downarrow} T_2}$.

\textit{Spin-current}---The spin-related currents $I_{\eta \sigma}$ can be evaluated by %%@
the time change rate of spin-$\sigma$ electron number in the $\eta$ lead
\bn
I_{\eta \sigma} &=& e\dot {N}_{\eta \sigma} = e\frac{d}{dt} %%@
\sum_{m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}} m_{\eta \sigma} %%@
P(t) {\big |}_{t\rightarrow \infty}, \label{cur} \\
P(t) &=& \rho_{00}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})}(t) + \rho_{\uparrow %%@
\uparrow}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})}(t)
+ \rho_{\downarrow \downarrow}^{({m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}})}(t). \nonumber 
\en
In particular, using Eqs.~(\ref{rateq}) we find $I_{L\uparrow}=-e\Gamma_{L\uparrow} %%@
\rho_{00}$ and $I_{L\downarrow}= e \Gamma_{L \downarrow}\rho_{\downarrow \downarrow}$. %%@
The stationary solution of Eqs.~(\ref{rateq}) is:
\bn
\rho_{\uparrow \uparrow} &=& \frac{2R_{\rm rf}^2 V + \delta_{\rm ESR}^2 + \Gamma^2 %%@
V^2}{\Xi},\\  
\rho_{\downarrow \downarrow} &=& \frac{2R_{\rm rf}^2 V} {\Xi},\\
\rho_{\uparrow \downarrow} &=& \frac{R_{\rm rf}[i \Gamma V - \delta_{\rm ESR}]}{\Xi},
\en
with $\Xi=6R_{\rm rf}^2 V + \Gamma^2 V^2 + \delta_{\rm ESR}^2$. Obviously, the %%@
stationary charge-current is exactly zero $I_{\eta}^{c}=I_{\eta\uparrow}+ I_{\eta %%@
\downarrow}=0$, while the spin-current is 
\bq
I_{L}^{s}=I_{L \downarrow} - I_{L\uparrow}= e \frac{2R_{\rm rf}^2 \Gamma %%@
V}{\Xi},\,\,\,\,I_{R}^{s}=-I_{L}^{s}.
\eq
Explicitly, the spin-current is proportional to the excitation power, which is %%@
consistent with the prototype of this ESR-based spin battery. And it's amplitude %%@
exhibits a saturation behavior with increasing driving field as a consequence of the %%@
nonlinear photon-absorption of a single spin [Fig.~2(a)]. The saturated value is %%@
$e\Gamma/3$ independent on the decoherence time $T_2$ and detuning $\delta_{\rm ESR}$. %%@
Figure 2(b) shows that the detailed dependence of the spin-current on the driving %%@
frequency is determined by the spin decoherence time $T_2$. In the inset of Fig.~2(a), %%@
we show that nonzero charge-current is also pumped by the driving field unless the %%@
double occupation is prohibited owing to the strong charging effect $\Gamma'=0$.\cite{U} %%@
This verifies our statement in the introduction.     

\begin{figure}[t]
\includegraphics[height=5cm,width=8cm]{fig2}
\caption{The spin-current vs (a) strength and (b) detuning of the driving field for %%@
different spin decoherence time. Inset in (a): Overview of the charge- and spin-current %%@
with the Coulomb correlation $\Gamma'$. The calculated strength of the oscillation %%@
magnetic field corresponds to $0\sim 1~$T.}
\label{fig2}
\end{figure}

\textit{Spin shot noise}---Recently nonequilibrium quantum shot noise is another current %%@
active subject in mesoscopic physics, because the current correlation (CC) is inherently %%@
related to the quantization of electron charge and thus give unique information about %%@
electronic correlation, which cannot be obtained by probing the conductance %%@
only.\cite{Blanter} For instance, the cross-correlation for charge-current (the %%@
correlation between different terminals) is negative for a normal single-electron %%@
transistor, meaning anti-bunching of the wavepacket. Interestingly, much studies have %%@
been devoted to find a system where the cross-correlation changes its %%@
sign.\cite{Samuelsson} Here we would like to address that the spin-resolved CC is more %%@
useful to describe electron correlation, because the electronic wavepacket with opposite %%@
spins is uninfluenced by the Pauli exclusion principle and only reflects unambiguous %%@
information about the interaction. For the purpose of evaluating the charge and spin CC %%@
from the quantum rate equations (\ref{rateq}), we extend the MacDonald's formula for the %%@
spin-resolved situation:\cite{MacDonald}
\bn
S_{\eta \eta'}^{\sigma \sigma'}(\omega) &=& 2\omega e^2 \int_{0}^{\infty} dt \sin %%@
(\omega t) \frac{d}{dt}  {\Big \{} \sum_{m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}} m_{\eta \sigma} m_{\eta' \sigma'} \cr
&& \times P(t) - t^2 I_{\eta \sigma} I_{\eta' \sigma'}/e^2 {\Big \}}.
\en
Specially, the zero-frequency shot noise spectrum is
\bq
S_{\eta \eta'}^{\sigma \sigma'}(0) = 2 e^2 \frac{d}{dt}  {\Big \{} %%@
\sum_{m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}} m_{\eta \sigma} %%@
m_{\eta' \sigma'} P(t) - \frac{t^2}{e^2} I_{\eta \sigma} I_{\eta' \sigma'} {\Big %%@
\}}{\big |}_{t\rightarrow \infty}. \label{sn}
\eq
The auto- and cross-noise spectra of the charge-current and spin-current can be obtained %%@
from these correlations $S_{\eta \eta'}^{\sigma \sigma'}$: $S_{\eta \eta'}^{c/s}=S_{\eta %%@
\eta'}^{\uparrow \uparrow}+ S_{\eta \eta'}^{\downarrow \downarrow} \pm S_{\eta %%@
\eta'}^{\uparrow \downarrow} \pm S_{\eta \eta'}^{\downarrow \uparrow}$.

In order to evaluate these correlations, we introduce the following generating %%@
functions:
\bq
G_{\eta \sigma}^{ab}(t) = \sum_{m_{L\uparrow},m_{R\uparrow} \atop %%@
m_{L\downarrow},m_{R\downarrow}} m_{\eta \sigma} %%@
\rho_{ab}^{({m_{L\uparrow},m_{R\uparrow} \atop m_{L\downarrow},m_{R\downarrow}})}(t).
\eq
With the help of Eqs.~(\ref{rateq}), all noise spectra are therefore relevant with these %%@
auxiliary functions and $\rho_{ab}(t)$ as:
\bn
\frac{S_{\eta \eta'}^{\sigma \sigma'}(0)}{2e^2} &=& {\big \{} \frac{1}{2} %%@
\Gamma_{\sigma} \delta_{\sigma \sigma'} [\delta_{\eta \eta'} %%@
(\rho_{00}(t)\delta_{\sigma\uparrow} + \rho_{\downarrow \downarrow}(t) \delta_{\sigma %%@
\downarrow}) \cr
&& \hspace{-1cm} + \sum_{\eta''} (G_{\eta'' \sigma}^{\downarrow\downarrow}(t) %%@
\delta_{\sigma \downarrow} - G_{\eta'' \sigma}^{00}(t) \delta_{\sigma \uparrow})]+ %%@
\delta_{\sigma \bar{\sigma'}}[\frac{1}{2} \Gamma_{\downarrow} G_{\eta %%@
\uparrow}^{\downarrow \downarrow}(t) \cr
&& \hspace{-1cm} - \frac{1}{2} \Gamma_{\uparrow} G_{\eta' \downarrow}^{00}(t) ]  
- \frac{2t}{e^2} I_{\eta \sigma }I_{\eta' \sigma'} {\big \}} {\big |}_{t\rightarrow %%@
\infty}.
\en
On the other hand, using Eqs.(\ref{rateq}), the equations of motion for ${\bf G}_{\eta %%@
\sigma}(t)$ is explicitly obtained in matrix form: $\dot {\bf G}_{\eta \sigma}(t)= {\cal %%@
M} {\bf G}_{\eta \sigma}(t) + {\cal G}_{\eta\sigma} {\bm \rho}(t)$ with
\bq
{\cal G}_{\eta\sigma}= \left (
\begin{array}{ccccc}
0 & 0 & \Gamma_{\eta \downarrow}\delta_{\sigma \downarrow} & 0 & 0 \\
-\Gamma_{\eta \uparrow}\delta_{\sigma \uparrow} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}
\right).
\eq
Applying Laplace transform to these equations yields
\bq
{\bf G}_{\eta \sigma}(s) = (s {\bf I}-{\cal M})^{-1} {\cal G}_{\eta\sigma} {\bm %%@
\rho}(s),
\eq
where ${\bm \rho}(s)$ is readily obtained by performing Laplace transform on its %%@
equations of motion and the normalization relation. Due to inherent long-time stability %%@
of the physics system under investigation, all real parts of nonzero poles of ${\bm %%@
\rho}(s)$ and ${\bf G}(s)$ are negative definite. Consequently, the large-$t$ behavior %%@
of the auxiliary functions is entirely determined by the divergent terms of the partial %%@
fraction expansions of ${\bm \rho}(s)$ and ${\bf G}_{\eta \sigma}(s)$ at $s\rightarrow %%@
0$. 

Resolving both ${\bf G}_{\eta \sigma}(s)$ and ${\bm \rho}(s)$ into the partial fraction %%@
expansion forms and performing inverse Laplace transform, we can eventually derive the %%@
analytical expressions for these large-$t$ asymptotic correlations.
Surprisingly, we find the zero-frequency charge shot noises being proportional to the %%@
spin-current $S_{LL}^{c} = -S_{LR}^{c}= e^2 |I_{L}^{s}|$, which manifests that pumping %%@
processes do generate shot noise even in the absence of net charge-current. This feature %%@
is underlying analog of the quantum partition noise of photo-excited electron-hole pairs %%@
demonstrated in the recent experiment.\cite{Reydellet} Spin-up and spin-down electrons %%@
in this spin battery are dissociated by tunneling events in the presence of pumping %%@
field but without bias: spin-up electrons inject into the QD, but spin-down electrons %%@
flow off in an opposite direction. As expected, the cross-correlation of the charge %%@
noise is always negative definite, showing anti-bunching statistics, but it has the same %%@
magnitude with the auto-correlation of the charge-current originated from the %%@
conservation law of charge.\cite{Wang2}

\begin{figure}[t]
\includegraphics[height=5cm,width=8cm]{fig3}
\caption{The zero-frequency auto- and cross-correlations of spin-current vs (a) strength %%@
and (b) detuning of driving field.}
\label{fig3}
\end{figure}

However, the situation is very different for the zero-frequency spin shot noise, as %%@
shown in Fig.~3. The auto-correlation of the spin-current is surely positive definite, %%@
while the cross-correlation is either positive or negative depending on a number of %%@
parameters: the pumping amplitude $R_{\rm rf}$, the detuning $\delta_{\rm ESR}$, and the %%@
decoherence $T_2$. For small decoherence rates, the spin cross-correlation experiences %%@
the change of sign with increasing excitation $R_{\rm rf}$ at resonant pumping %%@
$\delta_{\rm ESR}=0$. Interestingly, this cross-correlation is always positive (1) at %%@
large excitation $R_{\rm rf}> \Gamma$ regardless of decoherence $T_2$ [Fig.~3(a)] or (2) %%@
if apart away from quantum resonance [Fig.~3(b)]. The significant role of decoherence is %%@
to evidently reduce these noise spectra and even to eliminate the change of sign in the %%@
spin cross-correlation. It is addressed that spin shot noise is more sensitive to the %%@
spin decoherence than the spin-current and the charge shot noise. This could provide a %%@
way to measure the spin decoherence time $T_2$.

In summary, we have presented a generic method for analytical calculation of the atuo- %%@
and cross-shot noise spectra of charge- and spin-current in an ESR-based single QD %%@
system. We found that in the strong CB regime, indeed ESR pumping generates a finite %%@
spin-current and quantum partition noises in the absence of net charge transport. And %%@
the spin shot noises display complicated behaviors depending on the pumping parameters %%@
and the spin decoherence time. The measurement of spin-current and shot noise provides a %%@
scheme for the Rabi oscillation and spin decoherence detections in the QD system.                 

BD and HLC are supported by the DURINT Program administered by the US Army Research %%@
Office. XLL is supported by Major Projects of National Natural Science Foundation of %%@
China, the Special Founds for Major State Basic Research Project and the Shanghai %%@
Municipal Commission of Science and Technology.


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D.M. Treger, Science \textbf{294}, 1488 (2001).

\bibitem{Stevens}{M.J. Stevens, A.L. Smirl, R.D.R. Bhat, A. Najmaie, J.E. Sipe, and H.M. %%@
van Driel, Phys. Rev. Lett. \textbf{90}, 136603 (2003); J. H\"ubner, W.W. R\"uhle, M. %%@
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\textbf{90}, 216601 (2003).}

\bibitem{Wang1}{B.G. Wang, J. Wang, and H. Guo, Phys. Rev. B {\bf 67}, 92408 (2003).}

\bibitem{Zhang}{P. Zhang, Q.K. Xue, and X.C. Xie, Phys. Rev. Lett. {\bf 91}, 196602 %%@
(2003).}

\bibitem{Wang2}{B.G. Wang, J. Wang, and H. Guo, cond-mat/0305066 (2003).}

\bibitem{Sauret}{O. Sauret and D. Feinberg, Phys. Rev. Lett. {\bf 92}, 106601 (2004).}

\bibitem{Gurvitz}{S.A. Gurvitz and Ya.S. Prager, Phys. Rev. B {\bf 53}, 15932 (1996); B. %%@
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\bibitem{Fujisawa}{T. Fujisawa, Y. Tokura, and Y. Hirayama, Phys. Rev. B {\bf 63} %%@
81304(R) (2001).}

\bibitem{U}{Extension of zero-temperature spin-resolved quantum rate equations and %%@
current formula are easily achieved for considering the doubly occupied state. See %%@
Ref.~\onlinecite{Gurvitz}. Charging effect is modeled by tunneling rate $\Gamma'$.}

\bibitem{Blanter}{Ya.M. Blanter and M. B\"uttiker, Phys. Rep. {\bf 336}, 1 (2000).}

\bibitem{Samuelsson}{P. Samuelsson and M. B\"uttiker, Phys. Rev. Lett. {\bf 89}, 46601 %%@
(2002); F. Taddei and R. Fazio, Phys. Rev. B {\bf 65}, 134522 (2002); D. S\'anchez, R. %%@
L\'opez, P. Samuelsson, and M. B\"uttiker, Phys. Rev. B {\bf 68}, 214501 (2003).}

\bibitem{MacDonald}{D.K.C. MacDonald, Rep. Prog. Phys. {\bf 12}, 56 (1948); L.Y. Chen %%@
and C.S. Ting, Phys. Rev. B {\bf 46}, 4714 (1992).}

\bibitem{Reydellet}{L.H. Reydellet, P. Roche, D.C. Glattli, B. Etienne, and Y. Jin, %%@
Phys. Rev. Lett. {\bf 90}, 176803 (2003).}

\end{thebibliography}


\end{document}

cond-mat0404064.extracted_bib
\begin{thebibliography}{99}

\bibitem{Engel}{H.A. Engel and D. Loss, Phys. Rev. Lett. {\bf 86}, 4648 (2001); Phys. %%@
Rev. B {\bf 65}, 195321 (2002).}

\bibitem{Balatsky}{A.V. Balatsky and I. Martin, cond-mat/0112407; D. Mozyrsky, L. %%@
Fedichkin, S.A. Gurvitz, and G.P. Berman, Phys. Rev. B {\bf 66}, 161313 (2002); J.X. Zhu %%@
and A.V. Balatsky, Phys. Rev. Lett. {\bf 89}, 286802 (2002).}

\bibitem{Martin}{I. Martin, D. Mozyrsky, and H.W. Jiang, Phys. Rev. Lett. {\bf 90}, %%@
18301 (2003).}

\bibitem {Prinz}G.A. Prinz, Science \textbf{282}, 1660 (1998); S.A. Wolf, D.D. %%@
Awschalom, R.A. Buhrman, J.M. Daughton, S.V. Molnar, M.L. Roukes, A.Y. Chtchelkanova, %%@
D.M. Treger, Science \textbf{294}, 1488 (2001).

\bibitem{Stevens}{M.J. Stevens, A.L. Smirl, R.D.R. Bhat, A. Najmaie, J.E. Sipe, and H.M. %%@
van Driel, Phys. Rev. Lett. \textbf{90}, 136603 (2003); J. H\"ubner, W.W. R\"uhle, M. %%@
Klude, D. Hommel, R.D.R. Bhat, J.E. Sipe, and H.M. van Driel, Phys. Rev. Lett. %%@
\textbf{90}, 216601 (2003).}

\bibitem{Wang1}{B.G. Wang, J. Wang, and H. Guo, Phys. Rev. B {\bf 67}, 92408 (2003).}

\bibitem{Zhang}{P. Zhang, Q.K. Xue, and X.C. Xie, Phys. Rev. Lett. {\bf 91}, 196602 %%@
(2003).}

\bibitem{Wang2}{B.G. Wang, J. Wang, and H. Guo, cond-mat/0305066 (2003).}

\bibitem{Sauret}{O. Sauret and D. Feinberg, Phys. Rev. Lett. {\bf 92}, 106601 (2004).}

\bibitem{Gurvitz}{S.A. Gurvitz and Ya.S. Prager, Phys. Rev. B {\bf 53}, 15932 (1996); B. %%@
Dong, H.L. Cui, and X.L. Lei, Phys. Rev. B {\bf 69} 35324 (2004).}

\bibitem{Fujisawa}{T. Fujisawa, Y. Tokura, and Y. Hirayama, Phys. Rev. B {\bf 63} %%@
81304(R) (2001).}

\bibitem{U}{Extension of zero-temperature spin-resolved quantum rate equations and %%@
current formula are easily achieved for considering the doubly occupied state. See %%@
Ref.~\onlinecite{Gurvitz}. Charging effect is modeled by tunneling rate $\Gamma'$.}

\bibitem{Blanter}{Ya.M. Blanter and M. B\"uttiker, Phys. Rep. {\bf 336}, 1 (2000).}

\bibitem{Samuelsson}{P. Samuelsson and M. B\"uttiker, Phys. Rev. Lett. {\bf 89}, 46601 %%@
(2002); F. Taddei and R. Fazio, Phys. Rev. B {\bf 65}, 134522 (2002); D. S\'anchez, R. %%@
L\'opez, P. Samuelsson, and M. B\"uttiker, Phys. Rev. B {\bf 68}, 214501 (2003).}

\bibitem{MacDonald}{D.K.C. MacDonald, Rep. Prog. Phys. {\bf 12}, 56 (1948); L.Y. Chen %%@
and C.S. Ting, Phys. Rev. B {\bf 46}, 4714 (1992).}

\bibitem{Reydellet}{L.H. Reydellet, P. Roche, D.C. Glattli, B. Etienne, and Y. Jin, %%@
Phys. Rev. Lett. {\bf 90}, 176803 (2003).}

\end{thebibliography}