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We study theoretically the geometric quantum speed limits (QSLs) of open quantum systems under Markovian dynamical evolution. Three types of QSL time bounds are introduced based on the geometric inequality associated with the dynamical evolution from an initial state to a final state. By illustrating three types of QSL bounds at the cases of presence or absence of system driving, we demonstrate that the unitary part, dominated by system Hamiltonian, supplies the internal motivation for a Markovian evolution which deviates from its geodesic. Specifically, in the case of unsaturated QSL bounds, the parameters of the system Hamiltonian serve as the eigen-frequency of the oscillations of geodesic distance in the time domain and, on the other hand, drive a further evolution of an open quantum system in a given time period due to its significant contribution in dynamical speedup. We present physical pictures of both saturated and unsaturated QSLs of Markovian dynamics by means of the dynamical evolution trajectories in the Bloch sphere which demonstrates the significant role of system Hamiltonian even in the case of initial mixed states. It is further indicated that whether the QSL bound is saturated is ruled by the coummutator between the Hamiltonian and reduced density matrix of the quantum system. Our study provides a detailed description of QSL times and reveals the effects of system Hamiltonian on the unsaturation of QSL bounds under Markovian evolution.

We study theoretically the geometric phase of a double-quantum-dot (DQD) system measured by a quantum point contact (QPC) in the pure dephasing and dissipative environments, respectively. The results show that in these two environments, the coupling strength between the quantum dots has an enhanced impact on the geometric phase during a quasiperiod. This is due to the fact that the expansion of the width of the tunneling channel connecting the two quantum dots accelerates the oscillations of the electron between the quantum dots and makes the length of the evolution path longer. In addition, there is a notable near-zero region in the geometric phase because the stronger coupling between the system and the QPC freezes the electron in one quantum dot and the solid angle enclosed by the evolution path is approximately zero, which is associated with the quantum Zeno effect. For the pure dephasing environment, the geometric phase is suppressed as the dephasing rate increases which is caused only by the phase damping of the system. In the dissipative environment, the geometric phase is reduced with the increase of the relaxation rate which results from both the energy dissipation and phase damping of the system. Our results are helpful for using the geometric phase to construct the fault-tolerant quantum devices based on quantum dot systems in quantum information.

We theoretically study the electron transfer properties of a double quantum dot system in dissipative and pure dephasing environments based on a quantum dot contact detector. Theoretical results show that in the dissipative environment, the decoherence caused by the detector would increase the stable value of the average current and Fano factor as functions of time. Meanwhile, we find the existence of the quantum Zeno effect during the process of dynamical evolution. In the case of symmetric DQD, the relaxation caused by the dissipative environment would decrease the amplitude of the average current with time evolution and increase the value of the Fano factor in the long time limit. In the case of asymmetric DQD, the relaxation reduces the peak value of Fano factor over time. In the pure dephasing environment, we find that the frequent measurement would hinder the switch between different current channels during the cotunneling process. This results in a high value of Fano factor. In the case of symmetric DQD, increasing the pure dephasing rate would improve the value of Fano factor. In the case of asymmetric DQD, the dynamical evolution with time is not sensitive to the pure dephasing rate. In addition, it is indicated that the transfer probability of electron in the detector is only affected by the coupling between QPC and DQD. The environments have no effect on the transfer of a single electron in the detector. Our theoretical results provide theoretical references for experimental researchers to study the electron transport properties.

Using the innovative method of the additional Bloch vector, the electron transfer properties of a double quantum dot (DQD) system measured by a quantum point contact (QPC) in a fluctuating environment are investigated. The results show that the environmental noises in transverse and longitudinal directions play different roles in the dynamical evolution of the open quantum systems. Considering the DQD with symmetric energy level, the Fano factor exhibits a slight peak with the increase of transverse noise amplitude σ T, which provides a basis for distinguishing dynamical phenomena caused by different directional fluctuation noises in symmetric DQD structures by studying the detector output. In the case of asymmetric DQD, the dependence of a detector current involving the level displacement is distinct when increasing the transverse noise damping coefficient τ T and the longitudinal noise damping coefficient τε respectively. Meanwhile, the transverse noise damping coefficient τ T could significantly reduce the Fano factor and enhance the stability of the quantum system compared with the longitudinal one. The Fano factors with stable values as the enhancement of noise amplitudes show different external influences from the detector measurement, and provide a numerical reference for adjusting the noise amplitudes in both transverse and longitudinal directions appropriately in a microscopic experimental process to offset the decoherence effect caused by the measurements. Finally, the research of average waiting time provides unique insights to the development of single electron transfer theory in the short-time limit.