Quantum speed limit of the double quantum dot in pure dephasing environment under measurement

Zhenyu Lin(林振宇)， Tian Liu(劉天)， Zongliang Li(李宗良)， Yanhui Zhang(張延惠)，?， and Kang Lan(藍康)

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: quantum speed limits，the DQD system，the pure dephasing environment，quantum measurement

1. Introduction

The quantum speed limit (QSL) has been a hot research field with the development of quantum computation and quantum information technology.[1–10]The QSL is the minimum evolution time between the initial and distinguishable states，which is usually applied to prolong the life-time of coherence and shorten the evolving time.[11–16]Originally，Mandelstam–Tamm(MT)type and Margolus–Levitin(ML)type of the QSL bound have been proposed under the unitary dynamics. People have summarized the definiting expression of the QSL time which has described the speed of system evolving to orthogonal state by combining the two types of bound.[17–19]In recent years， people have paid much attention on the QSL in open systems. The QSL bounds of the ML and the MT type have been derived by using geometric approach，which is based on the bures angle between states of system， and those bounds are generalized to generic time-dependent dynamics of open quantum system.[20]Furthermore， a QSL bound which contains the energy fluctuation term has been proposed and it is applicable to general open system.[21]

The QSL bound determines the upper limit of evolving speed. It also represents the maximal degree which the system can be accelerated， and it is significant to improve the efficiency of quantum computing.[22–24]Recently，lots of works have been done for different way to accelerate the evolution and sharp the QSL bound.[25–30]It proves that the non-Markovian effects can speed up quantum evolution and therefore lead to a smaller QSL time in dissipative environment.[20]How to shorten the minimum evolving time under the pure dephasing environment is the important field of research. For a single quantum dot system which is coupled to purely-dephasing bosonic reservoir，the potential acceleration of the quantum evolution could be induced by the periodic dynamical decoupling pulses.[31]Moreover，the double quantum dot (DQD) system with its strong controllability has become an useful model for developing the unit devices of quantum computing.[32，33]So the variation of the QSL bound of DQD system in the pure dephasing environment is an attractive object of study.

In this paper， we explore the quantum evolution of the DQD system under the pure dephasing environment and analyze the effect of the dephasing rate(Γφ)，the decoherence rate(Γd)， the energy displacement (ε) and the coupling strength(Ω)between qubits to the QSL bound. The ratio of the QSL time (τqsl) to the driving time (τ) represent the CPS of the DQD system. Firstly， the CPS reduce to steady level along the driving time. The altitude of the stable value is risen by enhancing theΓd， and it reveals that increasing the frequency of measurement can weaken the CPS to some extent in the DQD system. By contrast，the increase ofΓφis more efficient on decreasing the CPS， which is according to that the strong coupling with the pure dephasing environment would accelerate the diminish of coherence. Secondly， the enlargement of the range of the QSL bound owing to the increase ofΓdis narrowed byΓφ. However， the expansion of the scope of the QSL bound according to the increase ofΓφis also reduced byΓd. Thirdly， the CPS would be raised with the increasing of the energy displacement，while it decreases for enhancing the coupling strength between two quantum dots. It is interesting that there has an inflection point， and the reducing effect of the CPS from theΩwould be dominant if theΩis greater than the point， otherwise the increasing influence of the CPS from theεwould suppress the impact of the coupling strength.Our results and analysis can not only reveal the accelerating mechanism of evolution in the DQD system，but also provide reference for the fabrication of quantum computing devices.

The paper is organized as follows. In Section 2，we make a brief presentation with the measurement of the DQD system，and extend the QSL bound of ML type to a general two-level system. In Section 3， we apply the equation of QSL time to the evolution of the DQD system under the dephasing environment and analyze the influence to the QSL time for different factors. In Section 4，we summarize the conclusion drawn from the present study.

2. Theoretical method

2.1. The QSL bound of ML type

In order to quantify the distinguishability between an initial stateρ(0) and a final stateρ(t)， the trace distance has been employed to measure the distance between two states. Then combining with the von-Neumann trace inequality one can derive the QSL bound of ML type[38]

Actually， this equation not only relates the initial state(ρ(0)) and the target state (ρ(t)) closely， but also intuitively shows the ratio relationship of the QSL time(τqsl)to the driving time(τ). It is convenient for us to observe the changes of QSL in the evolution of a specific system.

2.2. The quantum measurement in DQD system

The DQD system have important research value for realizing the measurement of QSL in experiment. We show the DQD system in Fig.1，and the Hamilton of the system is written as

represent the Hamiltonians of the DQD， the quantum point-contact (QPC) detector and its interaction term，respectively.[39–41]

Fig.1. A point-contact detector monitoring the electron position in the double quantum dots，Ua and Ub denote the chemical potentials in the left and right reservoirs. EL and ER represent the energy level of electrons in two quantum dots. If the electron stay in right(left)dot，the tunnelling probability of the electrons in QPC detector will be P(P′).

The right-side of Eq.(8)consists of two parts. The first term represents the unitary evolution of the system， and the decoherence which produced by the coupling between the system and the environment is the second term.

For the influence of the detector in the whole system， it can be described in the following density-matrix elements:

3. The QSL time in pure dephasing environment

In this section， we systematically investigate the evolution of system in the pure dephasing environment. In order to facilitate the calculation， the DQD system has been transformed to a parallel two-level system by employing the following method. Bringing the first term of Eq.(7)into the Liouville equation is the primary step and the corresponding matrixL0of the derivation of density matrix with respect to time is shown as

It is worth noting that the dephasing rate will cause the vanishment of off-diagonal matrix elements in long-time limit.In the presence of the interaction between the QPC detector and the system， it is necessary to reverse the diagonalization by using the transfer matrix selected before. The corresponding evolution matrix reads as follows:

Also we combine the matrix elements with Eq. (5) to derive the QSL time.

After taking an elementary operation to Eq.(5)，the ratio of the QSL time to the driving time has derived. With the consideration of above details，the value of the ratioτqsl/τreflects the potential capacity for the quantum dynamical speedup. Ifτqsl/τequal to one，it represents that the system could not be speeded up， because the speed of evolution reaches the maximum in this condition. In addition， the greater theτqsl/τis，the weaker the CPS could be.

Figure 2 shows the variation ofτqsl/τwith respect to the driving timeτin the DQD system. The coupling strength between two qubits isΩ= 1 and the energy displacement isε=5(in units ofΩ). There obviously displays that the value ofτqsl/τdecreases to be steady with the increase of the driving time. The coherence of system diminishes continuously along the driving time which implies that the loss of coherence would promote the system evolving to a stable state. Just like the evolutionary trend ofτqsl/τwhich is shown in the dissipative environment，[28]the oscillation reduction behavior ofτqsl/τappears when the coupling between system and dephasing environment(Γφ)is chosen as zero in Figs.2(a)and 2(b).

During the period ofτfrom 0 to 3， we notice that the decreasing process ofτqsl/τis delayed with the increase ofΓd(4≤Γd≤6) in Fig. 2(c). It means that the evolution of system from the initial state to the stable state is hindered by enhancing theΓdin small time. Actually，it is because that the frequently measurement localizes the electron in the DQD system.In the region of 0≤τ ≤3，it is not difficult to find that the evolution from the initial state to the stable state is prolonged with the increase ofΓφ(4≤Γφ ≤6) in Fig. 2(d). It can be interpreted that the increasing rate of the CPS is impeded by theΓφ. In other words， the interaction with the pure dephasing environment has stimulative effect on the oscillation of the electron during the evolution.[45]The enhancement of the CPS represents that the oscillation of the electron is slowed down and the probability of the electron’s tunnelling in the DQD system would be reduced.

The two figures withΓφ=0 in Fig. 2(a) andΓφ=5 in Fig. 2(c) show thatτqsl/τchanges as the function of the decoherence rate(Γd). It is interesting that the decreasing range ofτqsl/τalongτturns to be narrow with the increase ofΓφby comparing the two graphs. It represents thatΓφdetermines the range of the CPS，and the range will be tighter whenΓφis larger.The CPS is weakened obviously with the increase ofΓφafter the system arriving at the target state. In other words，the coupling between the system and the pure dephasing environment can accelerate the state’s evolution of the DQD system.

It is obvious that the value ofτqsl/τtends to be zero alongτforΓφ=0 withΓd=0 in Fig.2(b). On the other hand， the stable value ofτqsl/τforΓφ=0 is near 0.4 withΓd=5 in Fig. 2(d). Comparing two pictures， the stable value ofτqsl/τhas been increased for differentΓd. It is easy to find thatΓdwould shrink the reducing range ofτqsl/τ. It means that the coupling between the system and the detector can improve the speed of the DQD system evolving to target state to a certain extent.

The competition between the two factors is discussed via Fig. 3(a)， where the driving time is equal to one. Then the value ofτqsl/τcorresponds to the degree of sharp of QSL bound under this condition.As is shown in Fig.3(a)，the scope of the change ofτqsl/τis from 0.2 to 0.8 under the condition ofΓd=2. While the rising range ofτqsl/τis from 0.25 to 0.65 when it meetsΓφ=2. The sharpening effect of the QSL bound is induced by increasing the decoherence rate，and it would be hindered byΓφ. Similarly，Γdcould weaken the sharpening effect generated by enhancing the dephasing rate.The competition betweenΓφandΓddetermines the velocity of the evolution. It is worth noting thatΓφhas more obvious tightening effect on the QSL bound compared withΓd. Based on this competitive effect， the value ofτqsl/τcan reach the maximum via modulating the relative strength betweenΓφandΓd. In other words， it would achieve the most efficient acceleration of evolution in the DQD system under the appropriate dephasing environment and conditions of measurement.

The variation ofτqsl/τunder the condition ofτ=1 is plotted in Fig.3(b). The value ofτqsl/τis decrease from 0.98 to 0.70 with the increase ofεforΩ=8. The figure shows that the CPS is enhanced monotonously with increasing the energy displacement for a fixed coupling strength. It can be ascribed to the property ofεthat the greater the energy displacement is，the more difficult the transfer of electron could be.In addition，there has an inflection point which meetsΩ=0.74εin the change ofτqsl/τ. Forε=5，the value ofτqsl/τbegins to rise with the increase ofΩfor the conditionΩ ＞0.74ε. The seducement of the CPS is interpreted that broadening the width of the tunneling channel connecting the qubits could increase the tunnelling probability of electron. Furthermore， the decreasing effect of the CPS fromΩis stronger than the increasing influence fromεunder the case ofΩ＞0.74ε， but the increasing tendency of the CPS is suppressed by the action of theεwhenΩ ≤0.74ε. It is the reason why the value ofτqsl/τdecreases rapidly from 0.83 to 0.68 in the region 0≤Ω ≤0.74ε.

Fig.2. The ratio of the QSL time to the driving time τ for the DQD system in the dephasing environment. The dephasing rate is chosen as Γφ =0 in (a) and Γφ =5Ω in (c). The decoherence rate is setting as Γd =0 in (b) and Γd =5Ω in (d). The energy displacement ε =5 and the coupling strength parameter is Ω =1.

Fig. 3. (a) The τqsl/τ vs. the dephasing rate Γφ and the decoherence rate Γd. The coupling strength between two quantum dots is Ω =1 and the energy displacement is ε =5(in units of Ω). The driving time is τ =1. (b)τqsl/τ vs. the energy displacement ε and the coupling strength Ω. The dephasing rate Γφ =5 and the decoherence rate Γd=5. The driving time is τ =1.

4. Conclusion

The QSL of the DQD system has been investigated by adopting the detecting of the quantum point contact (QPC)in the pure dephasing environment. The CPS would be decreased by enhancingΓdto some extent in the DQD system.And the process in which the system evolves to the stable state would be delayed for frequently measurement. Furthermore，the improvement of the CPS has been prolonged because of the enhancement ofΓφ， namely， the pure dephasing environment would vary the oscillation of electron. The degree of sharpen the QSL bound is enhanced with increasingΓφ， andΓdwould weaken the sharpening effect and vice versa. The competition between the dephasing rate and the decoherence rate determines the degree of the evolution in the DQD system.The increase of energy displacement can raise the CPS because the strongεcould hinder the transfer of electron. However，the tunneling probability of electron would be increased by strengthening the coupling between two qubits， that is the reason for augmenting the CPS alongΩ. It is interesting that there has an inflection point， and the increasing effect of the CPS fromεwould be dominant ifΩis less than the point，otherwise the effect fromεwould restrain the impact ofΩ. More physical quantities can be connected with the QSL bound in the DQD system，such as the current of detector，the geometric phase of the system，etc.

Acknowledgment

Project supported by the National Natural Science Foundation of China(Grant No.11974217).