# Modelling and Control of Quantum Measurement Induced Backaction in Double
Quantum Dots

Wei Cui, and Daoyi Dong This work was supported by the National Natural
Science Foundation of China under Grant 11404113, the Australian Research
Council’s Discovery projects funding scheme under Project DP130101658, and the
Guangzhou Key Laboratory of Brain Computer Interaction and Applications under
Grant 201509010006. W. Cui is with the School of Automation Science and
Engineering, South China University of Technology, Guangzhou 510641, China
(e-mail: aucuiwei@scut.edu.cn).D. Dong is with the School of Engineering and
Information Technology, University of New South Wales, Canberra, ACT 2600,
Australia (e-mail: daoyidong@gmail.com).

###### Abstract

Quantum measurements disturb the quantum system being measured, and this is
known as measurement-induced backaction. In this work, we consider a double
quantum dot monitored by a nearby quantum point contact where the measurement-
induced backaction plays an important role. Taking advantage of the quantum
master equation approach, we calculate the tunnelling current, and propose a
simple feedback-control law to realize and stabilize the tunnelling current.
Theoretical analysis and numerical simulations show that the feedback control
law can make the current quickly convergent to the desired value.

###### Index Terms:

Quantum control, quantum measurement, measurement-induced backaction, quantum
master equation.

##  I Introduction

Controlling quantum systems is an essential task in many applications
including quantum information, atomic physics and molecular chemistry [1, 2,
3, 4, 5, 6]. In contrast to classical systems, where measurements do not alter
the state of the system, quantum projective measurement [7, 8, 9] collapses a
quantum system into one of its eigenstates in a probabilistic manner: the
“measurement-induced backaction” (see, e.g., [10, 11, 12, 13] and references
therein). Recent progress in macroscopic quantum manipulation and quantum weak
measurements have made it possible to perform experiments in which individual
quantum systems can be monitored and feedback-controlled in real time [14, 15,
16, 17]. In this paper, we derive a non-Markovian master equation for a double
quantum dot (DQD) [18] monitored by a nearby quantum point contact (QPC) where
the measurement-induced backaction plays a significant role. In this master
equation, time-dependent coefficients can characterize non-Markovian effects.
Furthermore, we consider how to reduce the measurement-induced backaction and
improve the quantum measurement efficiency by using quantum control theory
[19, 20, 21, 22].

Several strategies [23, 24] have already been proposed to reduce quantum
measurement-induced backaction. For example, Wiseman [23] considered the
measurement-induced backaction based on quantum trajectory theory. In the case
of damping, the jump operator is not Hermitian, and so the measurements
permitted by damping are necessarily quantum-demolition measurements. He
proved that by feeding back the measurement result to control the system
dynamics, the quantum-demolition measurement can be turned into a quantum-
nondemolition measurement [23]. The scheme to reduce and suppress the
measurement-induced backaction is to obtain a feedback Hamiltonian based on
the measurements on the system. A crucial issue in this approach is the
necessity to solve a non-linear equation to find the feedback control law. The
authors in [24] used measurement and feedback control to attempt to correct
the state of the qubit. They demonstrated that projective measurements are not
optimal for this task, and that there exists a nonprojective measurement with
an optimal measurement strength which achieves the best trade-off between
gaining information about the system and disturbing it through measurement
backaction. Similar to [23], the feedback control is a unitary rotation whose
angle depends on the measurement result.

DQD has been considered as a promising candidate of solid state qubits for
quantum computation. This system can be monitored by coupling it to a nearby
biased quantum point contact. It is interesting to understand the backaction
disturbance resulting from such a detection approach. For a zero-bias DQD, the
effect of charge-detector-induced backaction was theoretically studied to
explain experimental observations of inelastic electron tunneling [25, 26].
Ref. [27] exposed that under certain conditions the QPC-induced backaction has
profound effects on the counting statistics, e.g., changing the shot noise
from being sub-Poissonian to super-Poissonian, and changing the skewness from
positive to negative. However, how to effectively reduce this measurement-
induced backaction is still an open question. Two possible ways have been
proposed to reduce the backaction effects by making use of this interference:
1) turn the DQD to an operating point where destructive interference
suppresses phonon absorption, and 2) manipulate the electron-phonon
interaction. There are various limitations on the applications of these
existing strategies [23]. First, it is difficult to estimate quantum states
and the measurement-induced backaction from incomplete measurement data.
Second, the system of interest is generally open, which means that the
dynamics of its environment will affect its evolution. Thus, an accurate
master equation is needed to characterize the whole measurement model.

In this paper, a non-Markovian master equation for a DQD monitored by a nearby
QPC is derived [28]. Based on an ensemble description of the system of
interest, we consider the quantum detection over many identical systems
simultaneously. Moreover, we develop a simple feedback control strategy based
on counting statistics to estimate and compensate the measurement-induced
backaction in the DQD.

This paper is organized as follows. In Section II we describe a model of a
double quantum dot measured by a nearby quantum point contact. Using the
quantum master equation, the tunnelling current is calculated in Section III.
In Section IV, we design a feedback controller to realize and stabilize a
target tunnelling current. Conclusions are given in Section V.

##  II Physical System

DQD has been considered as a promising candidate to prepare solid state qubits
for quantum computation [29, 30, 31]. This system can be monitored by a nearby
QPC and controlled by electronic signals or picosecond laser pulses [32, 33].
The measurement mechanism can be understood as follows. QPC measurements rely
on the Coulomb interaction between the DQD and the QPC. The current in the QPC
is sensitive to the charge state of the DQD. By measuring the current passing
through the QPC one can infer the quantum state of the DQD. However, the
measurement-induced backaction will collapse the quantum state. Theoretical
and experimental results demonstrated that the backaction of the QPC on the
DQD leads to a quantum limit of detection and decoherence [34].

We consider a DQD in the Coulomb-blockade regime with strong intradot and
interdot interaction (see Fig. 1(a)). The Hilbert space of the DQD system
spanned by three states $|0\rangle,|1\rangle$, and $|2\rangle$. Here, the two
basis states $|1\rangle\equiv|1,0\rangle$ and $|2\rangle\equiv|0,1\rangle$
describe, respectively, one additional electron in the left and right dot
above the empty state $|0\rangle\equiv|0,0\rangle$. Accordingly, the current
through the QPC switches between these three different values. The total
Hamiltonian of the DQD system is

$\displaystyle H_{0}=H_{{\rm DQD}}+H_{{\rm Leads}}+H_{{\rm T}},$ (1)

where the dynamics of the interdot DQD system is described by the Hamiltonian
($\hbar=1$),

$\displaystyle H_{{\rm
DQD}}=\epsilon_{1}d_{1}^{{\dagger}}d_{1}+\epsilon_{2}d_{2}^{{\dagger}}d_{2}+\Omega(d_{1}^{{\dagger}}d_{2}+d_{2}^{{\dagger}}d_{1}).$
(2)

Here, $\epsilon_{i}~{}(i=1,2)$ are the energies for a single electron state in
each dot, $d_{i}^{{\dagger}}$ is the transpose and conjugate of $d_{i}$,
$\Omega$ is the coupling strength between the two dots, and
$d_{i}~{}(d_{i}^{{\dagger}})$ represents the electron annihilation (creation)
operator in each dot. This system can be approximated as a two-level system,
and it is characterized by the energy offset,
$\epsilon=\epsilon_{2}-\epsilon_{1}$. The effective Hamiltonian can be written
as

$\displaystyle H_{{\rm DQD}}=\frac{\epsilon}{2}\sigma_{z}+\Omega\sigma_{x}$
(3)

where $\sigma_{z}=d_{2}^{{\dagger}}d_{2}-d_{1}^{{\dagger}}d_{1}$ and
$\sigma_{x}=d_{2}^{{\dagger}}d_{1}+d_{1}^{{\dagger}}d_{2}$ are the Pauli
matrices for the pseudospin bases of $|1\rangle$ and $|2\rangle$. By
diagonalizing the DQD Hamiltonian, we can rewrite it as

$\displaystyle H_{{\rm DQD}}=\omega_{0}(|e\rangle\langle e|-|g\rangle\langle
g|)/2=\omega_{0}\varrho_{z}/2,$ (4)

where $\varrho_{z}=|e\rangle\langle e|-|g\rangle\langle g|$, and the
eigenstates are

$|g\rangle=\alpha|1\rangle-\beta|2\rangle,$
$|e\rangle=\beta|1\rangle+\alpha|2\rangle,$

with $\alpha\equiv\cos(\theta/2),$ and $\beta\equiv\sin(\theta/2)$. Here
$\theta$ is given by $\tan\theta=2\Omega/\epsilon$. The coherent oscillations
of the system have a frequency $\omega_{0}=\sqrt{\epsilon^{2}+4|\Omega|^{2}}$.

The Hamiltonian of the two electrodes in the DQD is

$\displaystyle H_{{\rm
Leads}}=\sum_{k}\epsilon_{{l}k}c_{{l}k}^{{\dagger}}c_{{{l}k}}+\epsilon_{{r}k}c_{{r}k}^{{\dagger}}c_{{r}k},$
(5)

where $c_{{l}k}~{}(c_{{r}k})$ is the annihilation operator of an electron in
the left (right) lead with quantum numbers $k$, and
$\epsilon_{{l}k}~{}(\epsilon_{{r}k})$ is the energy of the annihilation
operator $c_{{l}k}~{}(c_{{r}k})$. The tunneling Hamiltonian of the DQD and the
two electrodes is

$\displaystyle H_{{\rm
T}}=\sum_{k}\bigg{(}\Omega_{{l}k}c_{{l}k}d_{1}^{{\dagger}}+\Omega_{{r}k}c_{{r}k}^{{\dagger}}\Lambda_{{r}}^{{\dagger}}d_{2}$
(6)
$\displaystyle+\Omega_{{l}k}^{\ast}c_{{l}k}^{{\dagger}}d_{1}+\Omega_{{r}k}c_{{r}k}\Lambda_{{r}}d_{2}^{{\dagger}}\bigg{)},$

which depends on the tunneling coupling strengths $\Omega_{{l}k}$ and
$\Omega_{{r}k}$. Here, $\Lambda_{{r}}^{{\dagger}}$ is the operator to count
the number of tunnelled electrons [25]. The behavior of the DQD tunnelling
process is schematically depicted in Fig. 1(b).

Figure 1: (a) Schematic diagram of a double quantum dot (DQD) measured by a
nearby quantum point contact. The s and d in the QPC denote the source and the
drain. (b) Energy-level diagrams for the three states of the DQD. The arrows
represent the directions of the electron transitions. The tunnelling rates
$\Gamma_{1,4}$ are defined in Eq. (III). The decay rates $\Delta_{1,2}$ are
defined in Eq. (III).

The DQD is measured by a low-transparency QPC, and the total Hamiltonian of
this composite system is

$\displaystyle H_{{\rm tot}}=H_{0}+H_{{\rm QPC}}+H_{{\rm int}},$ (7)

where the Hamiltonians of the QPC and the DQD-QPC interaction are

$\displaystyle H_{{\rm
QPC}}=\sum_{m}\epsilon_{{s}m}a_{{s}m}^{{\dagger}}a_{{s}m}+\sum_{n}\epsilon_{{d}n}a_{{d}n}^{{\dagger}}a_{{d}n},$
(8) $\displaystyle H_{{\rm
int}}=\sum_{m,n}(D-\chi_{1}d_{1}^{{\dagger}}d_{1}-\chi_{2}d_{2}^{{\dagger}}d_{2})(a_{{s}m}^{{\dagger}}a_{{d}n}+a_{{d}n}^{{\dagger}}a_{{s}m}).$
(9)

Here $a_{{s}m}~{}(a_{{d}n})$ is the electron annihilation operator for
electrons in the source (drain) of the QPC with momentum $m\ (n)$, and
$\epsilon_{{s}m}~{}(\epsilon_{{d}n})$ is the corresponding energy. When a QPC
is placed near a DQD, it can be used as an ideal detector to implement quantum
weak measurements on the DQD, where the tunnelling barrier of the QPC is
modulated by the charge in the nearby DQD. In Eq. (9), $D$ is the tunnelling
amplitude of an isolated QPC, $\chi_{1}~{}(\chi_{2})$ provides the variation
in the tunnelling amplitude when the extra electron stays at the first
(second) dot. We assume $\chi_{1}<\chi_{2}$, because the QPC is located closer
to the second dot. The current through the QPC is sensitive to the electron
location and switches between three values corresponding to the three charge
states of the DQD. Figure 1(b) depicts the electron tunnelling in different
directions. The current in the QPC allows to obtain quantum state information
on the DQD [35].

##  III Non-Markovian master equation

Quantum master equation approach [36] provides a theoretical tool to
understanding the dynamics of open quantum systems. In the high-bias case, the
behavior of the system density matrix can be described with a master equation
of the Lindblad form [25, 27, 34, 37]. In the interaction picture with respect
to

$H^{\prime}=H_{{\rm DQD}}+H_{{\rm QPC}},$

the quantum system is governed by

$\displaystyle\dot{\rho}_{s}(t)=-\int_{0}^{t}{\rm d}t^{\prime}{\rm Tr}_{{\rm
E}}\bigg{[}\big{[}\rho_{{s}}(t)\otimes\rho_{{\rm E}},H_{{\rm
I}}(t^{\prime})\big{]},H_{{\rm I}}(t)\bigg{]},$ (10)

where ${\rm Tr}_{{\rm E}}$ indicates the trace over the environmental degrees
of freedom, $\rho_{E}$ is the state of the environment and $\rho_{s}$ is the
system state. The commutator $[\cdot~{},~{}\cdot]$ is defined by $[A,B]=AB-
BA$. The DQD interacts with two environments: the QPC and two electrodes.
Thus, the interaction Hamiltonian is [25]

$\displaystyle H_{{\rm I}}(t)=A(t)Y(t)+H_{{\rm T}}(t).$ (11)

The first term of the right hand side of Eq. (11) is the interaction with the
QPC and

$\displaystyle A(t)$ $\displaystyle=$ $\displaystyle e^{iH_{\rm
DQD}t}\big{(}D-\chi_{1}d_{1}^{{\dagger}}d_{1}-\chi_{2}d_{2}^{{\dagger}}d_{2}\big{)}e^{-iH_{\rm
DQD}t}$ (12) $\displaystyle=$
$\displaystyle\sum_{j=1}^{3}P_{j}e^{-i\omega_{j}t}$ $\displaystyle Y(t)$
$\displaystyle=$
$\displaystyle\sum_{m,n}a_{{s}m}^{{\dagger}}a_{{d}n}e^{i(\epsilon_{{s}m}-\epsilon_{{d}n})t}+a_{{s}m}a_{{d}n}^{{\dagger}}e^{-i(\epsilon_{{s}m}-\epsilon_{{d}n})t}$

with

$\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle-\alpha\beta~{}\chi_{d}~{}\sigma_{+}$ $\displaystyle P_{2}$
$\displaystyle=$ $\displaystyle-\alpha\beta~{}\chi_{d}~{}\sigma_{-}$
$\displaystyle P_{3}$ $\displaystyle=$ $\displaystyle
D-\big{(}\chi_{1}|e\rangle\langle e|+\chi_{2}|g\rangle\langle
g|\big{)}+\alpha^{2}~{}\chi_{d}~{}\varrho_{z},$

and $\omega_{1}=-\omega_{2}=-\omega_{0},~{}\omega_{3}=0$,
$\chi_{d}=\chi_{1}-\chi_{2}$. Here $H_{\rm T}(t)$ describes the interaction
with the two leads in the Heisenberg picture,

$\displaystyle H_{{\rm
T}}(t)=\sum_{k}\bigg{[}\Omega_{{l}k}c_{{l}k}e^{-i\epsilon_{{l}k}t}\left(\alpha
a_{g}^{{\dagger}}e^{i\omega_{0}t}+\beta
a_{e}^{{\dagger}}e^{i\omega_{0}t}\right)$
$\displaystyle+\Omega_{{r}k}c_{{r}k}^{{\dagger}}e^{i\epsilon_{{r}k}t}\Lambda_{{r}}^{{\dagger}}\left(-\beta
a_{g}e^{-i\omega_{0}t}+\alpha a_{e}e^{-i\omega_{0}t}\right)$
$\displaystyle+\Omega_{{l}k}^{\ast}c_{{l}k}^{{\dagger}}e^{i\epsilon_{{l}k}t}\left(\alpha
a_{g}e^{-i\omega_{0}t}+\beta a_{e}e^{-i\omega_{0}t}\right)$
$\displaystyle+\Omega_{{r}k}^{\ast}c_{{r}k}e^{-i\epsilon_{{r}k}t}\Lambda_{{r}}\left(-\beta
a_{g}^{{\dagger}}e^{i\omega_{0}t}+\alpha
a_{e}^{{\dagger}}e^{i\omega_{0}t}\right)\bigg{]}$ (14)

where $a_{g}=|0\rangle\langle g|$ and $a_{e}=|0\rangle\langle e|$ are defined
by $d_{1}^{{\dagger}}=|1\rangle\langle 0|=\alpha a_{g}^{{\dagger}}+\beta
a_{e}^{{\dagger}}$. The above results can also be found in Ref. [25]. We will
use them to derive the dynamic model in this paper.

We first consider the interaction of the DQD with the QPC. Applying Eq. (12)
and Eq. (LABEL:Y) to the master equation (10), one obtains

$\displaystyle\dot{\rho_{s}}(t)=\int_{0}^{t}{\rm d}t^{\prime}\bigg{\\{}{\rm
Tr}_{{E}}\left[Y(t^{\prime})\rho_{E}Y(t)\right]\times$
$\displaystyle\bigg{[}A(t^{\prime})\rho_{s}(t)A(t)+A(t)\rho_{s}(t)A(t^{\prime})\bigg{]}-{\rm
Tr}_{{E}}\left[Y(t)Y(t^{\prime})\rho_{E}\right]$
$\displaystyle\bigg{[}A(t)A(t^{\prime})\rho_{s}(t)+\rho_{s}A(t^{\prime})A(t)(t)\bigg{]}\bigg{\\}},$
(15)

where

$\displaystyle A(t^{\prime})\rho_{s}A(t)$ $\displaystyle=$ $\displaystyle
P_{2}\rho_{s}P_{1}e^{i\omega_{0}(t-t^{\prime})}+P_{1}\rho_{s}P_{2}e^{i\omega_{0}(t^{\prime}-t)}$
(16)
$\displaystyle+P_{1}\rho_{s}P_{1}e^{i\omega_{0}(t+t^{\prime})}+P_{2}\rho_{s}P_{2}e^{-i\omega_{0}(t+t^{\prime})}$
$\displaystyle+P_{3}\rho_{s}P_{1}e^{i\omega_{0}t}+P_{3}\rho_{s}P_{2}e^{-i\omega_{0}t}$
$\displaystyle+P_{1}\rho_{s}P_{3}e^{i\omega_{0}t^{\prime}}+P_{2}\rho_{s}P_{3}e^{-i\omega_{0}t^{\prime}}$
$\displaystyle+P_{3}\rho_{s}P_{3}$

and

$\displaystyle A(t)A(t^{\prime})\rho_{s}$ $\displaystyle=$ $\displaystyle
P_{1}P_{2}\rho_{s}e^{i\omega_{0}(t-t^{\prime})}+P_{2}P_{1}\rho_{s}e^{i\omega_{0}(t^{\prime}-t)}$
(17)
$\displaystyle+P_{1}P_{1}\rho_{s}e^{i\omega_{0}(t+t^{\prime})}+P_{2}P_{2}\rho_{s}e^{-i\omega_{0}(t+t^{\prime})}$
$\displaystyle+P_{1}P_{3}\rho_{s}e^{i\omega_{0}t}+P_{2}P_{3}\rho_{s}e^{-i\omega_{0}t}$
$\displaystyle+P_{3}P_{1}\rho_{s}e^{i\omega_{0}t^{\prime}}+P_{3}P_{2}\rho_{s}e^{-i\omega_{0}t^{\prime}}$
$\displaystyle+P_{3}P_{3}\rho_{s}\bigg{.}.$

To simplify the notation, we ignore the subscript $s$ (for system, not
source). Neglecting the fast-oscillating terms, we obtain the master equation:

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle\Gamma_{+}(t)(P_{2}\rho
P_{1}-P_{1}P_{2}\rho)+\Gamma_{-}(t)(P_{1}\rho P_{2}-P_{2}P_{1}\rho)$
$\displaystyle+\Gamma_{+}^{\prime}(t)(P_{1}\rho P_{2}-\rho
P_{2}P_{1})+\Gamma_{-}^{\prime}(t)(P_{2}\rho P_{1}-\rho P_{1}P_{2})$
$\displaystyle+\Lambda(t)(P_{3}\rho
P_{3}-P_{3}P_{3}\rho)+\Lambda^{\prime}(t)(P_{3}\rho P_{3}-\rho P_{3}P_{3}),$

where

$\displaystyle\Gamma_{\pm}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\langle Y(s)Y(0)\rangle\exp({\pm i\omega_{0}s}){\rm
d}s$ $\displaystyle\Gamma_{\pm}^{\prime}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\langle Y(0)Y(s)\rangle\exp({\pm i\omega_{0}s}){\rm
d}s$ $\displaystyle\Lambda(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\langle Y(s)Y(0)\rangle{\rm d}s$
$\displaystyle\Lambda^{\prime}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\langle Y(0)Y(s)\rangle{\rm d}s.$ (19)

Here, we define $t-t^{\prime}\equiv s$ and

${\rm Tr}_{{E}}\\{Y(t)Y(t^{\prime})\rho_{E}\\}\equiv\langle Y(s)Y(0)\rangle.$

Note that the two-time reservoir correlation function $\langle
Y(s)Y(0)\rangle$ is

$\displaystyle\langle Y(s)Y(0)\rangle$ $\displaystyle=$
$\displaystyle\sum_{mm^{\prime}nn^{\prime}}\langle
Y_{m,n}(s)Y_{m^{\prime},n^{\prime}}(0)\rangle\delta_{m,m^{\prime}}\delta_{n,n^{\prime}}$
(20) $\displaystyle=$
$\displaystyle\sum_{mn}N_{{s}m}(1-N_{{d}n})e^{i(\epsilon_{{s}m}-\epsilon_{{d}n})s}$
$\displaystyle+(1-N_{{s}m})N_{{d}n}e^{-i(\epsilon_{{s}m}-\epsilon_{{d}n})s}.$

The notation $\langle~{}\cdot\cdot\cdot~{}\rangle$ stands for the statistical
average over the source and the drain reservoirs of the QPC. They are assumed
to be in thermal equilibrium with the Fermi-Dirac distribution function

$\displaystyle
N_{{s}m}=\left\\{\exp\Big{[}\frac{(\epsilon_{{s}m}-\mu_{{s}})}{k_{B}T}\Big{]}+1\right\\}^{-1},$
(21) $\displaystyle
N_{dn}=\left\\{\exp\Big{[}\frac{(\epsilon_{dn}-\mu_{d})}{k_{B}T}\Big{]}+1\right\\}^{-1},$
(22)

where $\mu_{{s}}=\mu_{F}\pm eV_{{\rm QPC}}/2$ and $\mu_{d}=\mu_{F}\pm eV_{{\rm
QPC}}/2$ are the applied QPC bias voltages. Thus, the coefficients (19) become

$\displaystyle\Gamma_{\pm}(t)$ $\displaystyle=$ $\displaystyle\int_{0}^{t}{\rm
d}s\sum_{mn}\bigg{\\{}N_{sm}(1-N_{dn})\cos(\epsilon_{{s}m}-\epsilon_{{d}n}\pm\omega_{0})s$
$\displaystyle+(1-N_{sm})N_{dn}\cos(\epsilon_{{s}m}-\epsilon_{{d}n}\mp\omega_{0})s$
$\displaystyle+i\bigg{[}N_{sm}(1-N_{dn})\sin(\epsilon_{{s}m}-\epsilon_{{d}n}\pm\omega_{0})s$
$\displaystyle-(1-N_{sm})N_{dn}\sin(\epsilon_{{s}m}-\epsilon_{{d}n}\mp\omega_{0})s\bigg{]}\bigg{\\}},$
$\displaystyle\Gamma_{\pm}^{\prime}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}{\rm
d}s\sum_{mn}\bigg{\\{}N_{sm}(1-N_{dn})\cos(\epsilon_{{s}m}-\epsilon_{{d}n}\mp\omega_{0})s$
$\displaystyle+(1-N_{sm})N_{dn}\cos(\epsilon_{{s}m}-\epsilon_{{d}n}\pm\omega_{0})s$
$\displaystyle+i\bigg{[}-N_{sm}(1-N_{dn})\sin(\epsilon_{{s}m}-\epsilon_{{d}n}\mp\omega_{0})s$
$\displaystyle+(1-N_{sm})N_{dn}\sin(\epsilon_{{s}m}-\epsilon_{{d}n}\pm\omega_{0})s\bigg{]}\bigg{\\}},$
$\displaystyle\Lambda(t)$ $\displaystyle=$ $\displaystyle\int_{0}^{t}{\rm
d}s\sum_{mn}\bigg{\\{}i(N_{sm}-N_{dn})\sin(\epsilon_{{s}m}-\epsilon_{{d}n})s$
$\displaystyle+(N_{sm}+N_{dn}-2N_{sm}N_{dn})\cos(\epsilon_{{s}m}-\epsilon_{{d}n})s\bigg{\\}},$
$\displaystyle\Lambda^{\prime}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}{\rm
d}s\sum_{mn}\bigg{\\{}-i(N_{sm}-N_{dn})\sin(\epsilon_{{s}m}-\epsilon_{{d}n})s$
$\displaystyle+(N_{sm}+N_{dn}-2N_{sm}N_{dn})\cos(\epsilon_{{s}m}-\epsilon_{{d}n})s\bigg{\\}}.$

For the sake of simplicity, we define the new variables

$\displaystyle\omega_{m}\equiv\epsilon_{{s}m}-\mu_{{s}}$ (27)
$\displaystyle\omega_{n}\equiv\epsilon_{{d}n}-\mu_{{d}}.$ (28)

It is clear that

$\epsilon_{{s}m}-\epsilon_{{d}n}=\omega_{m}-\omega_{n}+eV_{{\rm QPC}}.$

Thus, the master equation (III) reduces to

$\displaystyle\frac{d}{dt}\rho$ $\displaystyle=$
$\displaystyle-i\bigg{[}\eta\Delta_{3}(t)\sigma_{z}+\Delta_{5}(t)P_{3}P_{3},\rho_{s}\bigg{]}+\Delta_{4}(t)\mathcal{D}[P_{3}]\rho$
(29)
$\displaystyle+\eta\bigg{[}\Delta_{1}(t)+\Delta_{2}(t)\bigg{]}\mathcal{D}[\sigma_{+}]\rho$
$\displaystyle+\eta\bigg{[}\Delta_{1}(t)-\Delta_{2}(t)\bigg{]}\mathcal{D}[\sigma_{-}]\rho.$

Now we consider the interaction of the DQD with the two leads. The Hamiltonian
of the two electrodes in the DQD is

$\displaystyle H_{{\rm
Leads}}=\sum_{k}\bigg{(}\epsilon_{{l}k}c_{{l}k}^{{\dagger}}c_{{{l}k}}+\epsilon_{{r}k}c_{{r}k}^{{\dagger}}c_{{r}k}\bigg{)},$
(30)

where $c_{{l}k}~{}(c_{{r}k})$ are the annihilation operators of an electron in
the left (right) lead with quantum numbers $k$. The tunnelling Hamiltonian of
the DQD and the two electrodes is

$\displaystyle H_{{\rm
T}}=\sum_{k}\bigg{(}\Omega_{{l}k}c_{{l}k}d_{1}^{{\dagger}}+\Omega_{{r}k}c_{{r}k}^{{\dagger}}\Lambda_{{r}}^{{\dagger}}d_{2}$
$\displaystyle+\Omega_{{l}k}c_{{l}k}^{{\dagger}}d_{1}+\Omega_{{r}k}c_{{r}k}\Lambda_{{r}}d_{2}^{{\dagger}}\bigg{)},$
(31)

which depends on the tunnelling coupling strengths $\Omega_{{l}k}$ and
$\Omega_{{r}k}$. Here, $\Lambda_{{r}}^{{\dagger}}$ is the operator used to
count the number of electrons that have tunnelled into the right lead. Also,
$H_{\rm T}(t)$ is the interaction with the two leads in the Heisenberg picture
as described in (14).

Similarly, using the interaction Hamiltonian $H_{T}(t)$ in the master equation
(10), we obtain the master equation for the system of DQD interacting with the
leads (For details, please refer to the Appendix)

$\displaystyle\frac{d}{dt}\rho=\Gamma_{1}\mathcal{D}\bigg{[}\alpha
a_{g}^{{\dagger}}+\beta
a_{e}^{{\dagger}}\bigg{]}\rho+\Gamma_{2}\mathcal{D}\bigg{[}\alpha a_{g}+\beta
a_{e}\bigg{]}\rho$
$\displaystyle+\Gamma_{3}\mathcal{D}\bigg{[}\alpha\Lambda_{r}a_{e}^{{\dagger}}+\beta\Lambda_{r}a_{g}^{{\dagger}}\bigg{]}\rho+\Gamma_{4}\mathcal{D}\bigg{[}\alpha\Lambda_{r}^{{\dagger}}a_{e}+\beta\Lambda_{r}^{{\dagger}}a_{g}\bigg{]}\rho.$

Now, converting the obtained equation into the Schrödinger picture, we have
the master equation as follows

$\displaystyle\frac{d}{dt}\rho$ $\displaystyle=$
$\displaystyle-i\bigg{[}H_{\rm
DQD}+\eta\Delta_{3}(t)\sigma_{z}+\Delta_{5}(t)P_{3}P_{3},\rho_{s}\bigg{]}$
(33)
$\displaystyle+\Delta_{4}(t)\mathcal{D}[P_{3}]\rho+\eta\bigg{[}\Delta_{1}(t)+\Delta_{2}(t)\bigg{]}\mathcal{D}[\sigma_{+}]\rho$
$\displaystyle+\eta\bigg{[}\Delta_{1}(t)-\Delta_{2}(t)\bigg{]}\mathcal{D}[\sigma_{-}]\rho+\Gamma_{1}\mathcal{D}\bigg{[}\alpha
a_{g}^{{\dagger}}+\beta a_{e}^{{\dagger}}\bigg{]}\rho$
$\displaystyle+\Gamma_{2}\mathcal{D}\bigg{[}\alpha a_{g}+\beta
a_{e}\bigg{]}\rho+\Gamma_{3}\mathcal{D}\bigg{[}\alpha\Lambda_{r}a_{e}^{{\dagger}}+\beta\Lambda_{r}a_{g}^{{\dagger}}\bigg{]}\rho$
$\displaystyle+\Gamma_{4}\mathcal{D}\bigg{[}\alpha\Lambda_{r}^{{\dagger}}a_{e}+\beta\Lambda_{r}^{{\dagger}}a_{g}\bigg{]}\rho,$

where $\eta=\alpha^{2}\beta^{2}\chi_{d}^{2}$, and the superoperator
$\mathcal{D}$ is defined as

$\displaystyle\mathcal{D}[O]{\rho}=O{\rho}O^{{\dagger}}-\frac{1}{2}(O^{{\dagger}}O{\rho}+{\rho}O^{{\dagger}}O).$
(34)

The time-dependent parameters in Eq. (33) are

$\displaystyle\Gamma_{1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{0}^{t}\\!\\!\sum_{k}\left|\Omega_{l,k}\right|^{2}\left(\\!\\!1-\tanh\frac{\epsilon_{l,k}-\mu_{l}}{2k_{B}T}\right)\cos(\epsilon_{l,k}-\omega_{0})s{\rm
d}s$ $\displaystyle\Gamma_{2}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{0}^{t}\\!\\!\sum_{k}\left|\Omega_{l,k}\right|^{2}\left(\\!\\!1+\tanh\frac{\epsilon_{l,k}-\mu_{l}}{2k_{B}T}\right)\cos(\epsilon_{l,k}-\omega_{0})s{\rm
d}s$ $\displaystyle\Gamma_{3}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{0}^{t}\\!\\!\sum_{k}\left|\Omega_{r,k}\right|^{2}\left(\\!\\!1-\tanh\frac{\epsilon_{r,k}-\mu_{r}}{2k_{B}T}\right)\cos(\epsilon_{r,k}-\omega_{0})s{\rm
d}s$ $\displaystyle\Gamma_{4}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{0}^{t}\\!\\!\sum_{k}\left|\Omega_{r,k}\right|^{2}\left(\\!\\!1+\tanh\frac{\epsilon_{r,k}-\mu_{r}}{2k_{B}T}\right)\cos(\epsilon_{r,k}-\omega_{0})s{\rm
d}s,$

and additional parameters are given by:

$\displaystyle{\Delta}_{1}(t)$ $\displaystyle=$
$\displaystyle\sum_{mn}\Lambda_{1}\int_{0}^{t}{\rm
d}s\cos(\omega_{m}-\omega_{n}+eV_{{\rm QPC}})s\cos\omega_{0}s,$
$\displaystyle{\Delta}_{2}(t)$ $\displaystyle=$
$\displaystyle\sum_{mn}\Lambda_{2}\int_{0}^{t}{\rm
d}s\sin(\omega_{m}-\omega_{n}+eV_{{\rm QPC}})s\sin\omega_{0}s,$
$\displaystyle{\Delta}_{3}(t)$ $\displaystyle=$
$\displaystyle\sum_{mn}\Lambda_{1}\int_{0}^{t}{\rm
d}s\cos(\omega_{m}-\omega_{n}+eV_{{\rm QPC}})s\sin\omega_{0}s,$
$\displaystyle{\Delta}_{4}(t)$ $\displaystyle=$ $\displaystyle
4\sum_{mn}\Lambda_{1}\int_{0}^{t}{\rm d}s\cos(\omega_{m}-\omega_{n}+eV_{{\rm
QPC}})s,$ $\displaystyle{\Delta}_{5}(t)$ $\displaystyle=$
$\displaystyle-2\sum_{mn}\Lambda_{2}\int_{0}^{t}{\rm
d}s\sin(\omega_{m}-\omega_{n}+eV_{{\rm QPC}})s.$ (36)

The coefficients $\Lambda_{1}$ and $\Lambda_{2}$ are the Fermi functions

$\displaystyle\Lambda_{1}(\omega_{m},\omega_{n})=1-\tanh\\!\\!\left(\frac{\omega_{m}}{2k_{B}T}\right)\tanh\\!\\!\left(\frac{\omega_{n}}{2k_{B}T}\right),$
$\displaystyle\Lambda_{2}(\omega_{m},\omega_{n})=\tanh\\!\\!\left(\frac{\omega_{m}}{2k_{B}T}\right)-\tanh\\!\\!\left(\frac{\omega_{n}}{2k_{B}T}\right).$
(37)

In the zero temperature limit ($k_{B}T=0$), the functions $\Lambda_{1}$ and
$\Lambda_{2}$ reduce to

$\displaystyle\Lambda_{1}=2\bigg{|}\Theta(\omega_{m})-\Theta(\omega_{n})\bigg{|},$
$\displaystyle\Lambda_{2}=2\bigg{[}\Theta(\omega_{m})-\Theta(\omega_{n})\bigg{]},$
(38)

where $\Theta$ is the Heaviside function [36]. Clearly, for different values
of $\omega_{m}$ and $\omega_{n}$, $\Lambda_{1}$ and $\Lambda_{2}$ reduce to
different constant values, as shown in Table I.

TABLE I. The Fermi functions $\Lambda_{1}$ and $\Lambda_{2}$.

$\omega_{m}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ | $\omega_{n}$ | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Lambda_{1}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ | $\Lambda_{2}$
---|---|---|---
$<0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ | $<0$ |  0 | 0
$>0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ | $>0$ |  0 | 0
$>0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ | $<0$ |  2 | 2
$<0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ | $>0$ |  2 | $-2$

The master equation obtained in this paper is non-Markovian, but the Born
approximation (i.e., second-order perturbation) is still used. The master
equation (33) is the result under such an approximation. This is the first
main result of this paper. The advantages of this master equation over
previous ones are: (i) it has the Lindblad form and the non-Markovian effects
are embodied in the time-dependent coefficients, (ii) the tunnelling and decay
effect can be clearly understood in the master equation, and (iii) it can be
controlled by manipulating the Hamiltonian.

These results are significant because two very different kinds of behaviours,
Markovian and non-Markovian, are embodied in these time-dependent coefficients
(III). In the zero-temperature limit, the Fermi functions (III) become $0$ or
$\pm 2$ in certain cases (see Table 1), and the coefficient
$\Delta_{1}(t)-\Delta_{2}(t)$ of the master equation (33) can be reduced to

$\displaystyle\frac{\Delta_{1}(t)-\Delta_{2}(t)}{2}=\sum_{\omega_{m}>0,~{}\omega_{n}<0}\frac{\sin(\omega_{m}-\omega_{n}+eV_{\rm
QPC}+\omega_{0})t}{\omega_{m}-\omega_{n}+eV_{\rm QPC}+\omega_{0})t}$
$\displaystyle+\sum_{\omega_{m}<0,~{}\omega_{n}>0}\frac{\sin(\omega_{m}-\omega_{n}+eV_{\rm
QPC}-\omega_{0})t}{\omega_{m}-\omega_{n}+eV_{\rm
QPC}-\omega_{0})t}.~{}~{}~{}~{}$ (39)

If the coefficient $\Delta_{1}(t)-\Delta_{2}(t)$ is positive, the information
is always lost during the time evolution of the DQD quantum system. In certain
case, the coefficient becomes negative within certain intervals of time, which
displays the non-Markovian behaviour. Obviously, in these time intervals, the
environment compensates some lost information to the quantum system of
interest. The corresponding non-Markovian properties of the negative
coefficient were shown in Fig. 1 of Ref. [38]. It is clear that the Markovian
dynamics is solely responsible for the long-time limit. We describe the
measurement-induced backaction by the tunneling current. We will show that the
current is directly determined by the coefficients in Section III.
Furthermore, we propose a simple feedback control law to stabilize the
current.

##  IV Measurement output and induced backaction

We now investigate electrons tunnelling in the DQD, and use a feedback control
method to stabilize a particular current and reduce the measurement-induced
backaction. We consider a zero bias across the DQD, and set
$\mu_{l}=\mu_{r}=0$. From equation (33) and the following relations [25, 27]

$\displaystyle\langle
n|\Lambda_{d}^{{\dagger}}\Lambda_{d}\rho|n\rangle=\rho^{(n)},$
$\displaystyle\langle
n|\Lambda_{d}\Lambda_{d}^{{\dagger}}\rho|n\rangle=\rho^{(n)},$
$\displaystyle\langle
n|\Lambda_{d}^{{\dagger}}\rho\Lambda_{d}|n\rangle=\rho^{(n-1)},$
$\displaystyle\langle
n|\Lambda_{d}\rho\Lambda_{d}^{{\dagger}}|n\rangle=\rho^{(n+1)},$ (40)

we obtain a $n$-resolved equation for each reduced density matrix elements:

$\displaystyle\dot{\rho}_{gg}^{(n)}$ $\displaystyle=$
$\displaystyle\eta\bigg{(}\Delta_{1}-\Delta_{2}\bigg{)}\rho_{gg}^{(n)}-\eta\bigg{(}\Delta_{1}+\Delta_{2}\bigg{)}\rho_{ee}^{(n)}$
$\displaystyle+\Gamma_{1}\alpha^{2}\rho_{00}^{(n)}-\Gamma_{4}\beta^{2}\rho_{gg}^{(n)},$
$\displaystyle\dot{\rho}_{ee}^{(n)}$ $\displaystyle=$
$\displaystyle\eta\bigg{(}\Delta_{1}+\Delta_{2}\bigg{)}\rho_{ee}^{(n)}-\eta\bigg{(}\Delta_{1}-\Delta_{2}\bigg{)}\rho_{gg}^{(n)}$
$\displaystyle+\Gamma_{1}\beta^{2}\rho_{00}^{(n)}-\Gamma_{4}\alpha^{2}\rho_{ee}^{(n)},$
$\displaystyle\dot{\rho}_{00}^{(n)}$ $\displaystyle=$
$\displaystyle-\Gamma_{1}\rho_{00}^{(n)}+\Gamma_{4}\bigg{(}\beta^{2}\rho_{gg}^{(n-1)}+\alpha^{2}\rho_{ee}^{(n-1)}\bigg{)},$
(41)

where $n$ is the number of electrons that have tunnelled through the DQD via
the right tunnelling barrier at time $t$ so that the density matrix elements
$\rho_{{ij}}=\sum_{n}\rho_{ij}^{(n)}~{}(i,j=0,g,e)$. An electron can tunnel
from the left lead into one of the eigenstates $|g\rangle$ and $|e\rangle$
with tunnelling rates $\Gamma_{1}\alpha^{2}$ and $\Gamma_{1}\beta^{2}$,
respectively. Later, it may tunnel out of the DQD to the right lead with the
rates $\Gamma_{4}\beta^{2}$ from the ground state or $\Gamma_{4}\alpha^{2}$
from the excited state.

Full counting statistics (FCS) has been widely used to analyze this type of
phenomena [37, 40, 39]. The FCS method concentrates on the probability
distribution for the number of electron tunneling during a given period of
time. With the help of FCS, one can extract not only the average or variance,
but also higher order moments of electron correlations. This method has also
been experimentally verified in detecting the higher-order cumulant of
electrical noise [41, 42]. Complete information about the transport properties
is contained in the probability distribution

$P(n,t)={\rm
Tr}_{s}\\{\rho^{(n)}(t)\\}=\rho_{00}^{(n)}+\rho_{gg}^{(n)}+\rho_{ee}^{(n)}.$

We assume that the time period is much larger than the inverse current
frequency. This ensurers that $n\gg 1$, on average. The corresponding cumulant
generating function (CGF) $\mathcal{F}(\zeta,t)$ is [43, 35]

$\displaystyle\exp\left[{\mathcal{F}(\zeta,t)}\right]=\sum_{n}P(n,t)\exp(in\zeta),$
(42)

where the auxiliary variable $\zeta$ is usually called counting field. The
power series expansion of the CGF is:

$\displaystyle\mathcal{F}(\zeta,t)=\sum_{k}C_{k}\frac{i^{k}\zeta^{k}}{k!}.$
(43)

It can be proven [43] that every moment of $n$ is finite and
$\mathcal{F}(\zeta,t)$ is $C^{\infty}$, from which the $k$th order cumulant is
[35, 43]

$\displaystyle
C_{k}=\left.\frac{\partial^{k}\mathcal{F}(\zeta,t)}{\partial(i\zeta)^{k}}\right|_{\zeta\rightarrow
0}.$ (44)

Clearly,

$\displaystyle F(\zeta,t)$ $\displaystyle=$
$\displaystyle\ln\left(\sum_{n}P(n,t)e^{in\zeta}\right)$ $\displaystyle=$
$\displaystyle\ln\left(1+\sum_{k=1}^{\infty}\frac{(i\zeta)^{k}}{k!}\left\langle
n^{k}\right\rangle\right)$ $\displaystyle=$
$\displaystyle\nabla_{1}-\frac{\nabla_{2}}{2}+\frac{\nabla_{3}}{3}-\frac{\nabla_{4}}{4}+\cdots$

where

$\langle n^{k}\rangle=\sum_{n}n^{k}P(n,t).$

When

$\left|\sum_{n}P(n,t)\left(e^{in\zeta}-1\right)\right|<1,$

one can use the Taylor expansion to derive the cumulants

$\displaystyle\nabla_{1}$ $\displaystyle=$ $\displaystyle(i\zeta)\left\langle
n\right\rangle+\frac{(i\zeta)^{2}}{2!}\left\langle
n^{2}\right\rangle+\frac{(i\zeta)^{3}}{3!}\left\langle n^{3}\right\rangle+$
$\displaystyle\frac{(i\zeta)^{4}}{4!}\left\langle n^{4}\right\rangle+\cdots$
$\displaystyle\nabla_{2}$ $\displaystyle=$ $\displaystyle(i\zeta)^{2}\langle
n\rangle^{2}+(i\zeta)^{3}\langle n\rangle\langle n^{2}\rangle+(i\zeta)^{4}$
$\displaystyle\cdot\left(\frac{\langle n^{2}\rangle^{2}}{4}+\frac{2\langle
n\rangle\langle n^{3}\rangle}{3!}\right)+\cdots$ $\displaystyle\nabla_{3}$
$\displaystyle=$ $\displaystyle(i\zeta)^{3}\langle
n\rangle^{3}+(i\zeta)^{4}\frac{3\langle n\rangle^{2}\langle
n^{2}\rangle}{2}+\cdots$ $\displaystyle\nabla_{4}$ $\displaystyle=$
$\displaystyle(i\zeta)^{4}\langle n\rangle^{4}+\cdots$

Figure 2: The mean transport density $C_{1}=\langle n\rangle$, the Fano factor
$C_{2}/C_{1}$, the normalized skewness $C_{3}/C_{1}$, and the normalized
sharpness $C_{4}/C_{1}$ versus the detection time for various electron
tunneling rates. Here, the cyan-dash lines denote $\Gamma_{4}=0$, the blue-
dash-dot lines denote $\Gamma_{1}=0.1,\Gamma_{4}=0.1$, the purple-dot lines
denote $\Gamma_{1}=0.5,\Gamma_{4}=0.1$, and the red-solid lines denote
$\Gamma_{1}=0.1,\Gamma_{4}=0.5$.

From Eq. (44), the first four cumulants of the generating function can be
expressed as

$\displaystyle C_{1}(t)$ $\displaystyle=$ $\displaystyle\langle n\rangle$ (45)
$\displaystyle C_{2}(t)$ $\displaystyle=$ $\displaystyle\langle
n^{2}\rangle-\langle n\rangle^{2}$ $\displaystyle C_{3}(t)$ $\displaystyle=$
$\displaystyle\langle n^{3}\rangle-3\langle n\rangle\langle
n^{2}\rangle+2\langle n\rangle^{3}$ $\displaystyle C_{4}(t)$ $\displaystyle=$
$\displaystyle\langle n^{4}\rangle-3\langle n^{2}\rangle^{2}-4\langle
n\rangle\langle n^{3}\rangle+12\langle n\rangle^{2}\langle
n^{2}\rangle-6\langle n\rangle^{4}.$

They denote the mean, variance, asymmetry (“skewness”) and kurtosis
(“sharpness”), respectively. The mean transport density $C_{1}(t)$, the Fano
factor $C_{2}/C_{1}$, the normalized skewness $C_{3}/C_{1}$, and the
normalized sharpness $C_{4}/C_{1}$ are plotted in Fig. 2 for various electron
tunneling rates. Here, the cyan-dash lines denote $\Gamma_{4}=0$, the blue-
dash-dot lines denote $\Gamma_{1}=0.1,\Gamma_{4}=0.1$, the purple-dot lines
denote $\Gamma_{1}=0.5,\Gamma_{4}=0.1$, and the red-solid lines denote
$\Gamma_{1}=0.1,\Gamma_{4}=0.5$. The coefficients are set as
$\chi_{d}=0.5,\epsilon=108\ \mu eV,\Omega=32\ \mu eV$, $V_{{\rm QPC}}=400\ \mu
eV$, and the initial condition of Eq. (IV) is
$\rho_{gg}^{(n)}(0)=\delta_{n,0}$. From Fig. 2(a), we find that if the
tunneling rate $\Gamma_{4}=0$, the cumulants will keep zero. If we set
$\Gamma_{4}=0.1$ and change $\Gamma_{1}$ from $0.1$ to $0.5$, there are no
significant changes. However, when the tunneling rate of the drain is enhanced
to $0.5$, drastic changes occur to all the cumulants. The tunnelling current
can be extracted from the mean $C_{1}$ in Eq. (45),

$\displaystyle I(t)$ $\displaystyle=$ $\displaystyle e\frac{d}{dt}\langle
n\rangle=e\sum_{n}n\dot{P}_{n}$ (46) $\displaystyle=$ $\displaystyle
e\Gamma_{{4}}\sum_{n}n\left[\beta^{2}(\rho_{gg}^{(n-1)}-\rho_{gg}^{(n)})+\alpha^{2}(\rho_{ee}^{(n-1)}-\rho_{ee}^{(n)})\right]$
$\displaystyle=$ $\displaystyle
e\Gamma_{{4}}\left(\beta^{2}\rho_{gg}(t)+\alpha^{2}\rho_{ee}(t)\right).$

This series is cut-off with $\rho_{gg}^{(-1)}=0$ and $\rho_{ee}^{(-1)}=0$, and
normalization holds as

$\sum_{n}\rho_{gg}^{(n)}=\rho_{gg}$

and

$\sum_{n}\rho_{ee}^{(n)}=\rho_{ee}.$

It is clear that the current linearly depends on the electron tunneling rate
$\Gamma_{4}$ and the population of the quantum state.

Figure 3:  Stationary transport current $I_{s}$ as a function of the energy
detuning for different parameters. Fig. 3(a) demonstrates that by increasing
the coupling strength $\chi_{d}$, the current resonance becomes asymmetrically
broadened a bit and its maximum values decreases; Fig. 3(b) indicates that the
current is more significant for a larger tunneling rate $\Gamma_{4}$.

In the long-time limit, the stationary current $I_{s}=I(t\rightarrow\infty)$
reads

$\displaystyle I_{s}=\frac{e\Gamma_{{1}}\Gamma_{{4}}(2\beta^{2}\chi_{{\rm
d}}^{2}\Delta_{2}+\Gamma_{4})}{I_{1}+I_{2}},$ (47)

where

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\Gamma_{{1}}\Gamma_{{4}}\left(\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}\right)+\Gamma_{{4}}^{2}-\chi_{{\rm
d}}^{2}\Gamma_{{4}}\Delta_{2}(2\alpha^{2}-1),$ $\displaystyle I_{2}$
$\displaystyle=$ $\displaystyle\chi_{{\rm
d}}^{2}(\Gamma_{4}(\Delta_{1}+2\Delta_{2})+2\Gamma_{4}\Delta_{1}).$

It is clear that the stationary current is determined by the tunneling rates
[$\Gamma_{1}$ and $\Gamma_{4}$, defined in Eq. (III)], the measurement-induced
decay rates [$\Delta_{1}$ and $\Delta_{2}$, defined in Eq. (III)] and the
coupling strength $\chi_{d}^{2}$. In Fig. 3, we plot the stationary current as
a function of the energy detuning $\epsilon$ for various values of the
coupling strength $\chi_{d}^{2}$ and tunnelling rates $\Gamma_{4}$. In Fig.
3(a), by increasing the coupling strength $\chi_{d}$, the current resonance
becomes asymmetrically broadened a bit and its maximum value decreases. This
is because the measurement-induced backaction becomes stronger when $\chi_{d}$
is larger. From Fig. 3(b), we can also find that the current is more
significant for a larger tunnelling rate $\Gamma_{4}$. The reason is that the
larger the tunnelling rate the more electrons can tunnel. In this paper, we
investigate the backaction by considering the influence of the energy
detuning. It is worth noting that recently Ref. [27] demonstrated considerable
QPC-induced backaction by investigating the influence of the QPC parameter
$eV_{\rm QPC}$, which is embodied in the decay coefficients of the master
equation (33). The authors in [27] showed that when the QPC bias energy
$eV_{\rm QPC}$ goes beyond a constant value, the current increases with the
magnitude of the voltage applied across the QPC in a nearly linear manner.
Here, the energy detuning influence is embedded in the Hamiltonian in the
master equation (33). Within this particular structure, the quantum
Hamiltonian control method [44] may be used to stabilize a target quantum
state and optimize the resulting convergence speed. Below, we use this method
to stabilize the tunneling current.

##  V Stabilization of the tunneling current

In this section we specifically study how to stabilize a particular tunneling
current. This is similar to tracking control [45] in classical control theory.
Feedback is an essential concept in control theory, for its prominent
capability in dealing with various uncertainties. In general, quantum semi-
classical feedback control consists of applying a conditional Hamiltonian to a
quantum system to obtain desired results, i.e., engineering the system
evolution in the form

$\displaystyle-i\sum_{j}u_{j}(t)[H_{j},~{}\rho].$ (48)

The control law $u_{j}(t)$ is a function of the measurement output

$\displaystyle u_{j}(t)=f(t,I(t)).$ (49)

Direct quantum feedback with no time delay is possible if the control is
sufficiently strong and fast. In order to obtain a particular tunnelling
current, here we design a feedback control law to the double quantum dot. From
Eq. (3) there are two adjustable parameters that are experimentally
accessible,i.e., the energy level $\epsilon$ and the coupling $\Omega$ in the
system Hamiltonian. We assume that the pulses are ideal and act as an
instantaneous unitary operation on the system, so that no coupling to the
surroundings needs to be considered during this process. The corresponding
system Hamiltonian becomes

$\displaystyle H_{{\rm
DQD}}^{\prime}=\frac{\epsilon+u_{1}(t)}{2}\sigma_{z}+\Big{[}\Omega+u_{2}(t)\Big{]}\sigma_{x}.$
(50)

Explicitly, $u_{1}(t)$ can be achieved by applying the time-dependent gate
voltages on the two dots to vary the level difference between them, and
$u_{2}(t)$ can be achieved by applying a time-dependent gate voltage between
the two dots to vary the interdot barrier. Based on the above analysis, we
propose a simple feedback control law to realize and stabilize a particular
tunnelling current $I_{0}$

$\displaystyle u_{i}(t)={\rm
sgn}\big{[}I_{0}-I(t)\big{]}\eta_{i}\exp\Big{(}-\frac{1}{k|I_{0}-I(t)|}\Big{)},$
(51)

where $\eta_{i}>0$ is the control amplitude, $k>0$ is an adaptive factor,
$I_{0}$ is the target electron current, and ${\rm sgn}$ is the sign function,

${\rm sgn}(x)=\left\\{\begin{array}[]{rcl}\\!\\!\\!&1&~{}~{}~{}{\rm
if}~{}~{}~{}x>0,\\\ \\!\\!\\!&0&~{}~{}~{}{\rm if}~{}~{}~{}x=0,\\\
\\!\\!\\!&-1&~{}~{}~{}{\rm if}~{}~{}~{}x<0.\\\ \end{array}\right.$ (52)

Thus, the tunneling current $I(t)$ in Eq. (51) determines whether to increase
or decrease the tunnelling coupling and the energy level (speed up or slow
down the electron tunnelling). It is clear that the controller (51) has the
characteristics

Figure 4:  Current difference, $I_{0}-I(t)$ as a function of $\omega_{0}t$,
for various control amplitudes. The blue dot-dashed line corresponds to
$\eta_{1}=\eta_{2}=1$, the red dotted line corresponds to
$\eta_{1}=0,~{}\eta_{2}=2$, and the purple solid line for
$\eta_{1}=\eta_{2}=2$. Note that the feedback control law can quickly drive
the current $I(t)$ to the expect one $I_{0}$.

$\left\\{\begin{array}[]{rcl}\\!\\!\\!\\!&&\lim_{I(t)\rightarrow I_{0}}\
u_{i}(t)\ =\ 0,\\\ \\!\\!\\!\\!&&u_{i}(t)\to-\eta_{i},\ \ \ \ \ \textrm{if}\ \
I(t)\gg I_{0},\\\ \\!\\!\\!\\!&&u_{i}(t)\to+\eta_{i},\ \ \ \ \ \textrm{if}\ \
I(t)\ll I_{0}.\end{array}\right.$ (53)

Equation (53) demonstrates that if the current $I(t)$ is smaller than the
target one, a stronger coupling strength is applied, which will speed up the
electron transport. If the current $I(t)$ is larger than the target one, a
weaker coupling strength is applied, which will slow down the electron
tunnelling. This algorithm (feedback control process) will end when $I(t)\to
I_{0}$.

In Fig. 4 we show the numerical results for the current difference
$I_{0}-I(t)$ as a function of $\omega_{0}t$ for various control amplitudes.
Here we set $I_{0}=0.1$ and the factor $k=5\times 10^{4}$. Note that all the
parameters are dimensionless. It is clear that with the feedback control law,
the current $I(t)$ quickly convergences to the target value $I_{0}$. It is
also demonstrated in Fig. 5 that for various control amplitudes $\eta_{i}$,
the behaviours of the feedback controller are quite different. Usually, the
larger the amplitude, the more oscillation behaviour. The reason is that the
sign function determines whether to increase or decrease the tunnelling
coupling and the energy level, the amplitude indicates how large the control
law is and the factor indicates how quick the current $I(t)$ convergences to
the target value. We consider that the feedback control (30) acts on an
ensemble and the measurement is implemented over many identical systems
simultaneously. Thanks to the current technology, the measurement and feedback
control of a charge qubit in a DQD can be implemented in the experiment; e.g.,
see [46].

##  VI Conclusion

We considered the non-Markovian dynamics of open quantum systems and a time-
dependent master equation for a DQD interacting with the QPC was derived. We
further illustrated several basic features of quantum non-Markovianity by
analyzing the first four cumulants. Moreover, we propose a control algorithm
to realize and stabilize a particular tunnelling current from its surrounding
environment. The strategy can be summarized as follows: we calculate the
cumulants of full counting statistics by the master equation, and then deduce
the tunnelling current. Based on the tunnelling current, a Hamiltonian
feedback control law was proposed to realize and stabilize a particular
measurement current. We considered a specific physical set-up of a double
quantum dot measured by a nearby quantum point contact. Theoretical analysis
and numerical simulations show that the control algorithm can make the current
quickly convergent to the target value, and thus enhance the performance of
the quantum measurement.

## Appendix: Master equation for DQD interacting with the leads

Using the interaction Hamiltonian $H_{T}(t)$ in the master equation (10), we
obtain the master equation for the system of DQD interacting with the leads as
follows [28]:

$\displaystyle\dot{\rho}=\alpha^{2}{\rm I}+\beta^{2}{\rm II}+\beta^{2}{\rm
III}+\alpha^{2}{\rm IV},$ (54)

where

$\displaystyle{\rm I}$ $\displaystyle=$
$\displaystyle\Gamma_{l,-}\big{(}a_{g}^{{\dagger}}\rho
a_{g}-a_{g}a_{g}^{{\dagger}}\rho\big{)}+\Gamma^{\prime}_{l,-}\big{(}a_{g}\rho
a_{g}^{{\dagger}}-\rho a_{g}^{{\dagger}}a_{g}\big{)}$
$\displaystyle+\Gamma_{l,+}\big{(}a_{g}\rho
a_{g}^{{\dagger}}-a_{g}^{{\dagger}}a_{g}\rho\big{)}+\Gamma^{\prime}_{l,+}\big{(}a_{g}^{{\dagger}}\rho
a_{g}-\rho a_{g}a_{g}^{{\dagger}}\big{)}$ $\displaystyle=$
$\displaystyle-i\left[\frac{\Gamma_{l,-}-\Gamma_{l,+}^{\prime}}{2i}a_{g}a_{g}^{{\dagger}}-\frac{\Gamma_{l,+}-\Gamma_{l,-}^{\prime}}{2i}a_{g}^{{\dagger}}a_{g},\rho\right]$
$\displaystyle+(\Gamma_{l,-}+\Gamma^{\prime}_{l,+})\mathcal{D}[a_{g}^{{\dagger}}]{\rho}+(\Gamma_{l,+}+\Gamma^{\prime}_{l,-})\mathcal{D}[a_{g}]{\rho},$

$\displaystyle{\rm II}$ $\displaystyle=$
$\displaystyle\Gamma_{l,-}\big{(}a_{e}^{{\dagger}}\rho
a_{e}-a_{e}a_{e}^{{\dagger}}\rho\big{)}+\Gamma^{\prime}_{l,-}\big{(}a_{e}\rho
a_{e}^{{\dagger}}-\rho a_{e}^{{\dagger}}a_{e}\big{)}$
$\displaystyle+\Gamma_{l,+}\big{(}a_{e}\rho
a_{e}^{{\dagger}}-a_{e}^{{\dagger}}e_{g}\rho\big{)}+\Gamma^{\prime}_{l,+}\big{(}a_{e}^{{\dagger}}\rho
a_{e}-\rho a_{e}a_{e}^{{\dagger}}\big{)}$ $\displaystyle=$
$\displaystyle-i\left[\frac{\Gamma_{l,-}-\Gamma_{l,+}^{\prime}}{2i}a_{e}a_{e}^{{\dagger}}-\frac{\Gamma_{l,+}-\Gamma_{l,-}^{\prime}}{2i}a_{e}^{{\dagger}}a_{e},\rho\right]$
$\displaystyle+(\Gamma_{l,-}+\Gamma^{\prime}_{l,+})\mathcal{D}[a_{e}^{{\dagger}}]{\rho}+(\Gamma_{l,+}+\Gamma^{\prime}_{l,-})\mathcal{D}[a_{e}]{\rho},$

$\displaystyle{\rm III}$ $\displaystyle=$
$\displaystyle\Gamma_{r,-}\big{(}\Lambda_{r}a_{g}^{{\dagger}}\rho\Lambda_{r}^{{\dagger}}a_{g}-\Lambda_{r}^{{\dagger}}a_{g}\Lambda_{r}a_{g}^{{\dagger}}\rho\big{)}$
$\displaystyle+\Gamma^{\prime}_{r,-}\big{(}\Lambda_{r}^{{\dagger}}a_{g}\rho\Lambda_{r}a_{g}^{{\dagger}}-\rho\Lambda_{r}a_{g}^{{\dagger}}\Lambda_{r}^{{\dagger}}a_{g}\big{)}$
$\displaystyle+\Gamma_{r,+}\big{(}\Lambda_{r}^{{\dagger}}a_{g}\rho\Lambda_{r}a_{g}^{{\dagger}}-\Lambda_{r}a_{g}^{{\dagger}}\Lambda_{r}^{{\dagger}}a_{g}\rho\big{)}$
$\displaystyle+\Gamma^{\prime}_{r,+}\big{(}\Lambda_{r}a_{g}^{{\dagger}}\rho\Lambda_{r}^{{\dagger}}a_{g}-\rho\Lambda_{r}^{{\dagger}}a_{g}\Lambda_{r}a_{g}^{{\dagger}}\big{)}$
$\displaystyle=$
$\displaystyle(\Gamma_{r,-}+\Gamma^{\prime}_{r,+})\mathcal{D}[\Lambda_{r}a_{g}^{{\dagger}}]{\rho}+(\Gamma_{r,+}+\Gamma^{\prime}_{r,-})\mathcal{D}[\Lambda_{r}^{{\dagger}}a_{g}]{\rho}$
$\displaystyle-i\left[\frac{\Gamma_{r,-}-\Gamma_{r,+}^{\prime}}{2i}\Lambda_{r}^{{\dagger}}a_{g}\Lambda_{r}a_{g}^{{\dagger}},~{}\rho\right]$
$\displaystyle-i\left[-\frac{\Gamma_{r,+}-\Gamma_{r,-}^{\prime}}{2i}\Lambda_{r}a_{g}^{{\dagger}}\Lambda_{r}^{{\dagger}}a_{g},~{}\rho\right],$

$\displaystyle{\rm IV}$ $\displaystyle=$
$\displaystyle\Gamma_{r,-}\big{(}\Lambda_{r}a_{e}^{{\dagger}}\rho\Lambda_{r}^{{\dagger}}a_{e}-\Lambda_{r}^{{\dagger}}a_{e}\Lambda_{r}a_{e}^{{\dagger}}\rho\big{)}$
$\displaystyle+\Gamma^{\prime}_{r,-}\big{(}\Lambda_{r}^{{\dagger}}a_{e}\rho\Lambda_{r}a_{e}^{{\dagger}}-\rho\Lambda_{r}a_{e}^{{\dagger}}\Lambda_{r}^{{\dagger}}a_{e}\big{)}$
$\displaystyle+\Gamma_{r,+}\big{(}\Lambda_{r}^{{\dagger}}a_{e}\rho\Lambda_{r}a_{e}^{{\dagger}}-\Lambda_{r}a_{e}^{{\dagger}}\Lambda_{r}^{{\dagger}}a_{e}\rho\big{)}$
$\displaystyle+\Gamma^{\prime}_{r,+}\big{(}\Lambda_{r}a_{e}^{{\dagger}}\rho\Lambda_{r}^{{\dagger}}a_{e}-\rho\Lambda_{r}^{{\dagger}}a_{e}\Lambda_{r}a_{e}^{{\dagger}}\big{)}$
$\displaystyle=$
$\displaystyle(\Gamma_{r,-}+\Gamma^{\prime}_{r,+})\mathcal{D}[\Lambda_{r}a_{e}^{{\dagger}}]{\rho}+(\Gamma_{r,+}+\Gamma^{\prime}_{r,-})\mathcal{D}[\Lambda_{r}^{{\dagger}}a_{e}]{\rho}.$
$\displaystyle-i\left[\frac{\Gamma_{r,-}-\Gamma_{r,+}^{\prime}}{2i}\Lambda_{r}^{{\dagger}}a_{e}\Lambda_{r}a_{e}^{{\dagger}},~{}\rho\right]$
$\displaystyle-i\left[-\frac{\Gamma_{r,+}-\Gamma_{r,-}^{\prime}}{2i}\Lambda_{r}a_{e}^{{\dagger}}\Lambda_{r}^{{\dagger}}a_{e},\rho\right].$

The coefficients are

$\displaystyle\Gamma_{l,-}=\int_{0}^{t}\sum_{k}\left|\Omega_{l,k}\right|^{2}\left\langle
c_{l,k}^{{\dagger}}c_{l,k}\right\rangle e^{i(\epsilon_{l,k}-\omega_{0})s}{\rm
d}s,$
$\displaystyle\Gamma_{l,+}=\int_{0}^{t}\sum_{k}\left|\Omega_{l,k}\right|^{2}\left\langle
c_{l,k}c_{l(r),k}^{{\dagger}}\right\rangle
e^{-i(\epsilon_{l,k}-\omega_{0})s}{\rm d}s,$
$\displaystyle\Gamma_{l,-}^{\prime}=\int_{0}^{t}\sum_{k}\left|\Omega_{l,k}\right|^{2}\left\langle
c_{l,k}c_{l,k}^{{\dagger}}\right\rangle e^{i(\epsilon_{l,k}-\omega_{0})s}{\rm
d}s,$
$\displaystyle\Gamma_{l,+}^{\prime}=\int_{0}^{t}\sum_{k}\left|\Omega_{l,k}\right|^{2}\left\langle
c_{l,k}^{{\dagger}}c_{l,k}\right\rangle e^{-i(\epsilon_{l,k}-\omega_{0})s}{\rm
d}s,$
$\displaystyle\Gamma_{r,-}=\int_{0}^{t}\sum_{k}\left|\Omega_{r,k}\right|^{2}\left\langle
c_{r,k}^{{\dagger}}c_{r,k}\right\rangle e^{i(\epsilon_{r,k}-\omega_{0})s}{\rm
d}s,$
$\displaystyle\Gamma_{r,+}=\int_{0}^{t}\sum_{k}\left|\Omega_{r,k}\right|^{2}\left\langle
c_{r,k}c_{r,k}^{{\dagger}}\right\rangle e^{-i(\epsilon_{r,k}-\omega_{0})s}{\rm
d}s,$
$\displaystyle\Gamma_{r,-}^{\prime}=\int_{0}^{t}\sum_{k}\left|\Omega_{r,k}\right|^{2}\left\langle
c_{r,k}c_{r,k}^{{\dagger}}\right\rangle e^{i(\epsilon_{r,k}-\omega_{0})s}{\rm
d}s,$
$\displaystyle\Gamma_{r,+}^{\prime}=\int_{0}^{t}\sum_{k}\left|\Omega_{r,k}\right|^{2}\left\langle
c_{r,k}^{{\dagger}}c_{r,k}\right\rangle e^{-i(\epsilon_{r,k}-\omega_{0})s}{\rm
d}s,$

with the Fermi-Dirac distribution function

$\displaystyle\left\langle c_{l,k}^{{\dagger}}c_{l,k}\right\rangle$
$\displaystyle=$
$\displaystyle\left[{e^{\left(\epsilon_{l,k}-\mu_{l}\right)/k_{B}T}+1}\right]^{-1}$
(59) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1-\tanh\frac{\epsilon_{l,k}-\mu_{l}}{2k_{B}T}\right]$

and

$\displaystyle\left\langle c_{r,k}^{{\dagger}}c_{r,k}\right\rangle$
$\displaystyle=$
$\displaystyle\left[{e^{\left(\epsilon_{r,k}-\mu_{r}\right)/k_{B}T}+1}\right]^{-1}$
(60) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1-\tanh\frac{\epsilon_{r,k}-\mu_{r}}{2k_{B}T}\right].$

Thus, the master equation (54) reduces to

$\displaystyle\frac{d}{dt}\rho=\Gamma_{1}\mathcal{D}\bigg{[}\alpha
a_{g}^{{\dagger}}+\beta
a_{e}^{{\dagger}}\bigg{]}\rho+\Gamma_{2}\mathcal{D}\bigg{[}\alpha a_{g}+\beta
a_{e}\bigg{]}\rho$
$\displaystyle+\Gamma_{3}\mathcal{D}\bigg{[}\alpha\Lambda_{r}a_{e}^{{\dagger}}+\beta\Lambda_{r}a_{g}^{{\dagger}}\bigg{]}\rho+\Gamma_{4}\mathcal{D}\bigg{[}\alpha\Lambda_{r}^{{\dagger}}a_{e}+\beta\Lambda_{r}^{{\dagger}}a_{g}\bigg{]}\rho,$

where

$\displaystyle\Gamma_{1}=\Gamma_{l,-}+\Gamma_{l,+}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\sum_{k}\left|\Omega_{l,k}\right|^{2}\left(1-\tanh\frac{\epsilon_{l,k}-\mu_{l}}{2k_{B}T}\right)\cos(\epsilon_{l,k}-\omega_{0})s{\rm
d}s$ $\displaystyle\Gamma_{2}=\Gamma_{l,+}+\Gamma_{l,-}^{\prime}$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\sum_{k}\left|\Omega_{l,k}\right|^{2}\left(1+\tanh\frac{\epsilon_{l,k}-\mu_{l}}{2k_{B}T}\right)\cos(\epsilon_{l,k}-\omega_{0})s{\rm
d}s$ $\displaystyle\Gamma_{3}=\Gamma_{r,-}+\Gamma_{r,+}^{\prime}$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\sum_{k}\left|\Omega_{r,k}\right|^{2}\left(1-\tanh\frac{\epsilon_{r,k}-\mu_{r}}{2k_{B}T}\right)\cos(\epsilon_{r,k}-\omega_{0})s{\rm
d}s$ $\displaystyle\Gamma_{4}=\Gamma_{r,+}+\Gamma_{r,-}^{\prime}$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\sum_{k}\left|\Omega_{r,k}\right|^{2}\left(1+\tanh\frac{\epsilon_{r,k}-\mu_{r}}{2k_{B}T}\right)\cos(\epsilon_{r,k}-\omega_{0})s{\rm
d}s.$

## Acknowledgment

The authors would like to thank Prof. Franco Nori and Prof. Jianqiang You for
helpful comments.

## References

  * [1] H. M. Wiseman and G. J. Milburn, _Quantum Measurement and Control_ Cambridge, Cambridge University Press, 2010.
  * [2] D. Dong and I. R. Petersen, “Quantum control theory and applications: A survey,” IET Control Theory $\&$ Applications, vol. 4, no. 12, pp. 2651-2671, 2010.
  * [3] D. Dong, M. A. Mabrok, I. R. Petersen, B. Qi, C. Chen, and H. Rabitz,“Sampling-based learning control for quantum systems with uncertainties,” _IEEE Transactions on Control Systems Technology_ , vol. 23, no. 6, pp. 2155-2166, 2015.
  * [4] T. M. Zhang, R.-B. Wu, F. H. Zhang, T. J. Tarn, and G. L. Long, “Minimum-time selective control of homonuclear spins,” _IEEE Transactions on Control Systems Technology_ , vol. 23, no. 5, pp. 2018-2025, 2015.
  * [5] J. Zhang, L. Greenman, X. Deng, K.B. Whaley, “Robust control pulses design for electron shuttling in solid state devices,” _IEEE Transactions on Control Systems Technology_ , Vol.22, No.6, pp.2354-2359, 2014.
  * [6] W. Zhang, D. Dong, and I. R. Petersen,“Adaptive target scheme for learning control of quantum systems,” _IEEE Transactions on Control Systems Technology_ , in press, DOI: 10.1109/TCST.2017.2715007, 2017.
  * [7] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton: Princeton University, 1955.
  * [8] A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Physics Reports, vol. 520, no.2, pp. 43–134, 2012.
  * [9] K. Jacobs, Quantum Measurements Theory and Its Application, Cambridge: Cambridge University, 2014.
  * [10] J. Dressel and F. Nori, “Certainty in Heisenberg’s uncertainty principle: Revisiting definitions for estimation errors and disturbance,” Physical Review A, vol. 89, p. 022106, 2014.
  * [11] M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, “Quantum back-action of an individual variable-strength measurement,” Science, vol. 339, no.6116, pp.178-181, 2013.
  * [12] W. Cui, N. Lambert, Y. Ota, X. Y. Lü, Z. L. Xiang, J. Q. You, and F. Nori, “Confidence and backaction in the quantum filter equation,” Physical Review A, vol. 86, no. 5, p. 052320, 2012.
  * [13] M. Kulkarni, O. Cotlet, and H. E. Türeci, “Cavity-coupled double-quantum dot at finite bias: Analogy with lasers and beyond,” Physical Review B, vol. 90, p. 125402, 2014.
  * [14] C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J. M. Raimond, and S. Haroche, “Real-time quantum feedback prepares and stabilizes photon number states,” Nature (London), vol. 477, pp. 73-77, 2011.
  * [15] H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Argo, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science vol. 337, no.6101, pp. 1514-1517, 2012.
  * [16] I. Iida, M. Yukawa M, H. Yonezawa, N. Yamamoto, and A. Furusawa, “Experimental demonstration of coherent feedback control on optical field squeezing,” IEEE Transactions on Automatic Control, vol. 57, pp. 2045-2050, 2012.
  * [17] G. F. Zhang and M. R. James, “Direct and indirect couplings in coherent feedback control of linear quantum systems,” IEEE Transactions on Automatic Control, vol. 56, pp. 1535-1550, 2011.
  * [18] X. Mi, J. V. Cady, D. M. Zajac, J. Stehlik, L. F. Edge, and J. R. Petta ,“Circuit Quantum Electrodynamics Architecture for Gate-Defined Quantum Dots in Silicon,” Applied Physics Letters vol. 110, p. 043502, 2017.
  * [19] D. Dong and Ian R. Petersen, “Sliding mode control of two-level quantum systems,” Automatica, vol. 48, no.5, pp. 725–735, 2012.
  * [20] M. R. James and J. E. Gough, “Quantum dissipative systems and feedback control design by interconnection,” IEEE Transactions on Automatic Control, vol. 55, pp. 1806–1821, no. 8, 2010.
  * [21] B. Qi, “A two-step strategy for stabilizing control of quantum systems with uncertainties,” Automatica, vol. 49, no. 3, pp. 834–839, 2013.
  * [22] N. Yamamoto and L. Bouten, “Quantum risk-sensitive estimation and robustness,” IEEE Transactions on Automatic Control, vol. 54, pp.92–107, 2009.
  * [23] H. M. Wiseman, “Using feedback to eliminate back-action in quantum measurements,” Physical Review A, vol. 51, vol.3, p. 2459, 1995.
  * [24] A. M. Brańczyk, P. E. M. F. Mendonça, A. Gilchrist, A. C. Doherty, and S. D. Bartlett, “Unambiguous discrimination of special sets of multipartite states using local measurements and classical communication,” Physical Review A vol. 75, no. 1, p. 012329, 2007.
  * [25] S. H. Ouyang, C. H. Lam, and J. Q. You, “Backaction of a charge detector on a double quantum dot,” Physical Review B, vol. 81, p. 075301, 2010.
  * [26] Z. Z. Li, S. H. Ouyang, C. H. Lam, and J. Q. You, “Probing the quantum behavior of a nanomechanical resonator coupled to a double quantum dot,” Physical Review B, vol. 85, p. 235420, 2012.
  * [27] Z. Z. Li, C. H. Lam, T. Yu, and J. Q. You, “Detector-induced backaction on the counting statistics of a double quantum dot,” Scientific Reports, vol. 3, p. 3026, 2013.
  * [28] W. Cui, “Dynamics of a double quantum dot charge qubit measured by a quantum point contact,” Proceedings of the 34th Chinese Control Conference, pp. 8281-8286, Hangzhou, China, 28th-30th July 2015.
  * [29] B. B. Chen, B. C. Wang, G. Cao, H. O. Li, M. Xiao, G. C. Guo, H. W. Jiang, X. Hu, and G. P. Guo, “Spin blockade and coherent dynamics of high-spin states in a three-electron double quantum dot,” Physical Review B, vol. 95, p. 035408, 2017.
  * [30] A. Stockklauser, P. Scarlino, J. Koski, S. Gasparinetti, C. K. Andersen, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, “Strong coupling cavity QED with gate-defined double quantum dots enabled by a high impedance resonator,” Physical Review X, vol. 7, p. 011030, 2017.
  * [31] M. J. A. Schuetz, G. Giedke, L. M. K. Vandersypen, and J. I. Cirac, “High-fidelity hot gates for generic spin-resonator systems,” Physical Review A, vol. 95, p. 052335, 2017.
  * [32] T. Brandes, “Feedback control of quantum transport,” Physical Review Letters vol. 105, p. 060602, 2010.
  * [33] G. G. Gillett, R. B. Dalton, B. P. Lanyon, M. P. Almeida, M. Barbieri, G. J. Pryde, J. L. O’Brien, K. J. Resch, S. D. Bartlett, and A. G. White, “Experimental feedback control of quantum systems using weak measurements,” Physical Review Letters, vol. 104, p. 080503, 2010.
  * [34] X. Q. Li, P. Cui, and Y. J. Yan, “Spontaneous relaxation of a charge qubit under electrical measurement,” Physical Review Letters vol. 94, p. 066803, 2005.
  * [35] C. Pöltl, C. Emary, and T. Brandes, “Feedback stabilization of pure states in quantum transport,” Physical Review B, vol. 84, p. 085302, 2011.
  * [36] H. P. Breuer and F. P. Petruccione, The Theory of Open Quantum Systems, Oxford: Oxford University Press, 2002.
  * [37] S. A. Gurvitz and Y. S. Prager, “Microscopic derivation of rate equations for quantum transport,” Physical Review B, vol. 53, p. 15932, 1996.
  * [38] W. Cui, Z. R. Xi, and Y. Pan, “Controlled population transfer for quantum computing in non-Markovian noise environment,” Proceedings of the IEEE Conference on Decision and Control and Chinese Control Conference, pp. 2504-2509, Shanghai, China, 15th-18th December 2009.
  * [39] G. Schaller, G. Kiesslich, and T. Brandes, “Transport statistics of interacting double dot systems: Coherent and non-Markovian effects,” Physical Review B, vol. 80, p. 245107, 2009.
  * [40] A. J. Keller, J. S. Lim, D. Sánchez, R. López, S. Amasha, J. A. Katine, H. Shtrikman, and D. Goldhaber-Gordon, “Cotunneling drag effect in Coulomb-coupled quantum dots, ” Physical Review Letters, vol. 117, p. 066602, 2016.
  * [41] N. Ubbelohde, C. Fricke, C. Flindt, F. Hohls, and R. J. Haug, “Measurement of finite-frequency current statistics in a single-electron transistor,” Nature Communications, vol. 3, p. 612, 2012.
  * [42] V. F. Maisi, D. Kambly, C. Flindt, and P. Pekola, “Full Counting Statistics of Andreev Tunneling,” Physical Review Letters vol. 112, p. 036801, 2014.
  * [43] A. Braggio, K. König, and R. Fazio, “Full counting statistics in strongly interacting systems: non-Markovian effects,” Physical Review Letters, vol. 96, p. 026805, 2006.
  * [44] F. Ticozzi, R. Lucchese, P. Cappellaro, and L. Viola, “ Hamiltonian control of quantum dynamical semigroups: stabilization and convergence speed,” IEEE Transactions on Automatic Control, vol. 57, pp. 1931-1944, 2012.
  * [45] S. Xue, M. R. Hush, and I. R. Petersen, “Feedback tracking control of non-Markovian quantum systems,” IEEE Transactions on Control Systems Technology, vol. 25 , no. 5, pp. 1552-1563, 2017.
  * [46] G. Cao, H. O. Li, L. Wang, C. Zhou, M. Xiao, G. C. Guo, H. W. Jiang, and G. P. Guo, “Ultrafast universal quantum control of a quantum-dot charge qubit using Landau-Zener-Stuckelberg interference,” Nature Communications, vol. 4, p. 1401, 2012.