Lujing Jiang(姜露靜), Kang Lan(藍(lán)康), Zhenyu Lin(林振宇), and Yanhui Zhang(張延惠),?

1School of Physics and Electronics,Shandong Normal University,Jinan 250014,China

2School of Physics,State Key Laboratory of Crystal Materials,Shandong University,Jinan 250100,China

Keywords: double quantum dots,fluctuating environment,electron transfer,noise

1. Introduction

With the rapid development of quantum information technology,solid-state quantum computing based on semiconductor quantum dot system has become a research focus in quantum measurement and quantum information processing.[1–9]In the practical quantum computation, the system inevitably interacts with the surrounding environments, which leads to the loss of the coherence of the quantum system.[10–14]Recently, it has become an urgent problem to study the decoherence process caused by the coupling of the quantum system and the external environment, and research on the non-Markovian dynamics of the open quantum system has attracted extensive attention.[15–26]Studying the process of electron transport[27–34]in an open quantum system is of great significance for understanding the dynamical evolution of the system,restraining the decoherence and improving the efficiency of quantum computation.

In recent years, driven by the potential application of quantum information science,the double quantum dot(DQD)system with its unique quantum properties and strong controllability has become an effective model for quantum measurement and quantum transport.[35–38]As a measuring instrument,the quantum point contact(QPC)is widely used to measure the position of electron in the DQD owing to its high sensitivity.[39–43]The disturbance of the external environment leads to many interesting phenomena in the dynamical evolution of open quantum systems.A great many theoretical works and experiments related to quantum dots have made breakthroughs,which inspires us to research the dynamical properties of the quantum systems.

The information about the probability distribution of n electrons tunneling to the collector electrode can be obtained from full counting statistics (FCS).[44–53]The first-order cumulant of the number of electrons transferred during transport represents the average current of the system,and the shot noise which represents the fluctuation of the system is indicated by the second-order cumulant. The intensity of shot noise is usually expressed numerically by the value of the Fano factor, F = 1, F < 1, and F > 1, which correspond to the distributions of Poissonian, sub-Poissonian, and super-Poissonian.[54–59]As indexes reflecting system dynamics in the short-time limit,single electron transfer probability and the average waiting time can effectively show the electron transfer properties.[60]

2. Theoretical framework

First,we consider a DQD,as shown in Fig.1.The Hamiltonian of the DQD can be described as[47]

Fig.1. A double quantum dot two-level system.

The dynamical evolution of the density matrix of the DQD satisfies

The reduced density matrix of the system is represented by the column vector, ρ(t) = [ρLL(t),ρRR(t),ρRL(t),ρLR(t)]?, and the evolution of the elements of the reduced density matrix can be written as

Fig.2. The DQD system coupled with a QPC. μl and μr denote the chemical potentials of the left and right reservoirs, and V =μl ?μr is the bias voltage in QPC.

In order to study the electron transfer properties in the DQD,QPC is introduced to measure the position of the electron in the DQD,as shown in Fig.2. The Hamiltonian of the system is written as[63]

By introducing a QPC,the dynamical evolution of the reduced density matrix of the system is given by[6]

Considering the influence of the detector,the reduced density matrix elements of the system are expressed as

where x is a statistical parameter. The quantum master equation for the generalized particle number resolution of the system is written as[61]

The operators L and LJrespectively describe the continuous evolution of the system and the quantum jump in electron transfer,and can be expressed as

We consider a DQD system measured by a QPC in fluctuating environment. First, we only consider the influence of noise in transverse direction on the system; the random fluctuation in Hamiltonian of the DQD system can be expressed as

where HSis the Hamiltonian of the DQD without environmental effect,and Hδ(t)is the fluctuation caused by the coupling between the DQD system and transverse noise,T(t)is the random fluctuation of the tunneling amplitude in the two-level system, and σzand σxdenote the Pauli matrices. According to the statistical properties of GWN, the noise in transverse direction can be represented by the second-order cumulant

where σTand τTrespectively denote the amplitude and damping coefficient of the transverse noise. Based on the properties of GWN,the exact dynamical evolution equation of the DQD system is given by a time-convolutionless master equation[58]

The first and second terms on the right respectively describe the unitary evolution of the DQD system and the decocoherence caused by QPC measurement, and the third term corresponds to the fluctuation of the transverse noise on the system.The super operator Lδ(t) corresponding to the fluctuation of the system caused by the transverse noise can be written as

With only the longitudinal noise taken into account, the Hamiltonian of the DQD system can be written as

The master equation of particle number resolution describing electron transfer of the QPC with the effect of transverse and longitudinal noises is[58]

In order to describe the influences of transverse noise and longitudinal noise on the electron transfer properties of the system clearly, we introduce the method of additional Bloch vector to represent the cumulants in the transport process;the traditional Bloch vectors are[51]

and the additional vector is

The cumulative moment generation function of the electron can be obtained by deriving the additional Bloch vector with respect to x[52]

3. Results and discussion

3.1. Electron transfer affected by transverse noise

Figure 3 shows the evolutions of average current I and the Fano factor F with different decoherence rates Γdand transverse noise parameters σT,τTfor symmetric and asymmetric DQDs in the long-time limit. We find that the average current always presents an enhancement behavior with the increase of Γdinduced by the QPC in Fig.3(a). The increase of the output current is due to the electron transfer in DQD being inhibited,which leads to a relatively low QPC barrier and makes the electrons pass through the barrier more easily in the detector. We can see the interesting phenomenon in Fig.3(b)where the Fano factor increases rapidly with Γdand reaches a very high peak value in the case of the asymmetric DQD because of the slow switching between different current channels connecting with the left and right reservoirs of the detector.[57,58]Meanwhile, the Fano factor in the symmetric case is much smaller than that in the asymmetric case. Whether the DQD is symmetric or not, the Fano factor always presents a super-Poissonian distribution due to the cotunneling effect caused by the tunneling of electrons between different energy levels in the detector.[57,62]

Fig.3. The evolution of(a)the average current I and(b)the Fano factor F with decoherence rate Γd at different level displacement ε0 in the long-time limit,with parameters σT=0.3T0 and τT=T0. The evolution of(c)the average current I and(d)the Fano factor F with the transverse noise amplitude σT at different level displacement ε0 in the long-time limit, with parameters τT =T0 and Γd =0.01T0. The evolution of (e) the average current I and (f) the Fano factor F with the transverse noise damping coefficient τT at different level displacement ε0 in the long-time limit, with parameters σT=0.3T0 and Γd=0.01T0 .

Figure 4 shows the influence of the transverse noise on the probability of single electron transfer P0as a function of the time t for different damping coefficient τTand amplitude σTin the short-time limit. As shown in Fig.4(a),the increase of σTreduces the time of P0reaching stability, meaning that σTpromotes the transfer of a single electron in the detector.Different from the strong dependence of σT, there are no obvious distinctions in the probability of single electron transfer P0at different τTin Fig.4(b). It is not difficult to see that the influence of σTon single electron transfer probability P0is greater than that of τT. Therefore,it is the best way to acquire the noise properties by the sensitivity of the single electron transfer probability P0to σT.

Fig.4. Asymmetric DQD . (a) The probability of single electron transfer P0 versus time for different amplitude σT,other parameters are chosen as Γd =0.01T0,τT =T0. (b)The probability of single electron transfer P0 versus time for different damping coefficient τT with Γd =0.01T0 and σT=0.3T0.

Fig.5. (a) The average waiting time versus amplitude σT at different level displacement ε0,other parameters are chosen as Γd=0.01T0,τT=T0.(b)The average waiting timeversus damping coefficient τT at different level displacement ε0 with Γd=0.01T0 and σT=0.3T0.

3.2. Electron transfer affected by longitudinal noise

Fig.6. The evolution of (a) the average current I and (b) the Fano factor F with decoherence rate at different level displacement ε0 in the long-time limit, with parameters σε =0.3T0 and τε =T0. The evolution of (c) the average current I and(d)the Fano factor F with the longitudinal noise amplitude at different level displacement ε0 in the long-time limit,with parameters τε =T0 and Γd =0.01T0. The evolution of (e) the average current I and (f) the Fano factor F with the longitudinal noise damping coefficient at different level displacement ε0 in the long-time limit, with parameters σε =0.3T0 and Γd=0.01T0.

Fig.7. Asymmetric DQD. (a)The probability of transferring single electron P0 versus time at different amplitude σε, other parameters are chosen as Γd =0.01T0, τε =T0. (b) The probability of transferring single electron P0 versus time at different damping coefficient τT with Γd=0.01T0 and σε =0.3T0.

Fig.8. (a) The average waiting time versus amplitude σε at different level displacement ε0,other parameters are chosen as Γd=0.01T0,τε =T0.(b)The average waiting timeversus damping coefficient τε at different level displacement ε0 with Γd=0.01T0 and σε =0.3T0.

4. Conclusion