† Corresponding author. E-mail:
Project supported by the Natural Science Foundation of Shandong Province, China (Grant No. ZR2014AM030).
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.
The quantum state, after undergoing adiabatic and cyclic evolution, acquires a phase that contains a component which is only related to the geometry of the path traced by the system called the geometric phase.[1] The conception of geometric phase has been extended to diverse directions.[2–5] In the realistic implementations of quantum computation, it is necessary to consider the environmental effect on the evolution of the quantum systems, which is driven by the coupling of the system and the surrounding environments.[6–17] Moreover, the experiments about the geometric phase have proved that its intrinsic fault-tolerate feature is significant for implementing quantum information processing.[18–34] Thus, it is important to study the geometric phase in different environments for investigating quantum phase transition, quantum error correction, and so on.
With the rapid development of quantum information, a large amount of attention has been focused on exploiting the feasible programs so as to develop quantum computing. In these schemes, the double-quantum-dot (DQD) system is a promising candidate in quantum information processing on account of its long coherent time and powerful controllability.[35–39] Meanwhile, the quantum point contact (QPC) is widely used to measure the transport properties of electrons due to its high sensitivity. The extensive theoretical and experimental works associated with it have been done,[40–43] which has inspired us to study further the physical phenomenon in the quantum dot system.
The study of the geometric phase in different situations has made great progresses,[44–65] such as a two-level system coupled to a radiation field at zero temperature[66] and a spin-1/2 particle interacting with N independent spins.[67] There are also many achievements on the geometric phase of the quantum dot systems, including the influence of the temperature on the geometric phase in two coupled quantum dots[68] and the current method to obtain the geometric phase in the DQD system.[69,70] The geometric phase of the DQD system measured by a single-electron transistor (SET) has been reported, in which an electron inside the quantum dot of the SET would cause the fluctuations in the coupling strength between the quantum dots and the energy level in the nearest quantum dot.[71–75] However, there are still many valuable questions in the study of the geometric phase in quantum dot systems. For example, when we use the QPC as a detector, how the geometric phase of the DQD system evolves during a quasiperiod in the pure dephasing or dissipative environment and what is the physical mechanism of the QPC and the environmental effect on it? Solving these questions provides some reference for taking advantage of the DQD systems to construct geometric quantum logic gates.
In this paper, we employ the QPC as a detector to study the properties of the geometric phase in a DQD system and then analyze its effect on the geometric phase. We calculate the geometric phase of the DQD system coupled to a QPC in two kinds of environments according to the Bloch-type rate equations and the kinematic approach of the geometric phase given in mixed states under nonunitary evolution. It is shown that the coupling strength between the two quantum dots has an enhancement on the geometric phase. Moreover, there is a notable near-zero region in the geometric phase due to the coupling between the system and the QPC. Additionally, the influences of the pure dephasing environment and the dissipative environment on the geometric phase are analyzed. The geometric phase is reduced with the increase of the dephasing rate which is generated only by the phase damping of the system in the pure dephasing environment. In the dissipative environment, the geometric phase decreases as the relaxation rate increases, which results from both the energy dissipation and phase damping of the system.
We consider a DQD system measured by a QPC as shown in Fig.
![](cpb_29_3_030302/cpb_29_3_030302_ieqn1.gif)
![](cpb_29_3_030302/cpb_29_3_030302_ieqn2.gif)
![](cpb_29_3_030302/cpb_29_3_030302_ieqn3.gif)
![](cpb_29_3_030302/cpb_29_3_030302_ieqn4.gif)
![]() | Fig. 1. The model of the DQD system coupled with a QPC. The chemical potentials of the left and right reservoirs are μα, μβ and μα > μβ. |
The dynamical evolution of the reduced density matrix for the system satisfies the Liouville equation
Firstly, we consider that the DQD system is only coupled to the QPC, and the evolution of the reduced density matrix elements of the electron is written as[40]
![](cpb_29_3_030302/cpb_29_3_030302_ieqn5.gif)
![](cpb_29_3_030302/cpb_29_3_030302_ieqn6.gif)
![](cpb_29_3_030302/cpb_29_3_030302_ieqn7.gif)
![](cpb_29_3_030302/cpb_29_3_030302_ieqn8.gif)
![](cpb_29_3_030302/cpb_29_3_030302_ieqn9.gif)
Given that this DQD system is an open system which is affected by the QPC and the environments, the formulation of the geometric phase in the mixed states under nonunitary evolution is used[5]
![](cpb_29_3_030302/cpb_29_3_030302_ieqn10.gif)
![](cpb_29_3_030302/cpb_29_3_030302_ieqn11.gif)
The instantaneous eigenvalues of the reduced density matrix are given by
For simplicity, we consider the condition where the system is prepared in a pure initial state with
To associate with the geometric feature of the evolution, we can express ρ(t) in the Bloch sphere representation as
In this section, we will discuss the properties of the geometric phase of the DQD system when we use a QPC as a detector in the presence of two kinds of environments, namely, the pure dephasing and dissipative environments, and investigate numerically the influences of the detector and these two environments on the phase.
Firstly, we consider only the interaction between the DQD system and the pure dephasing environment. Then we use the rotation operation,
![](cpb_29_3_030302/cpb_29_3_030302_ieqn19.gif)
Figure
In order to better understand the relation between the geometric phase γg and the coupling strength Ω, we show the evolution path of the open system in the Bloch sphere representation for different Ω in Fig.
In Fig.
In the subsection, we introduce the dissipative environment into the DQD system coupled to a QPC. In the case of the DQD system and the dissipative environment, the reduced density matrix proceeded by the same rotation operation,
In this process,
Figure
![]() | Fig. 5. Geometric phase of the DQD system versus the coupling strength Ω between the two quantum dots in quasiperiod τ with the fixed relaxation rate Γr = 0.1ε for different decoherence rate Γd. |
As shown in Fig.
In Fig.
We study the geometric phase of an open DQD system coupled with a QPC in the pure dephasing and dissipative environments and focus on analyzing the influences of the detector and the two kinds of environments on it. It is demonstrated that the coupling strength between the two quantum dots has an enhanced impact on the geometric phase. Because the expansion of the width of the tunneling channel connecting the two quantum dots accelerates the oscillations of an electron between the quantum dots and increases the length of the evolution path in the Bloch sphere. Moreover, the geometric phase shows a near-zero region due to the quantum Zeno effect that stronger coupling between the system and the QPC leads to the electron to be frozen in one quantum dot and the solid angle enclosed by the evolution path is close to zero. In the pure dephasing environment, the geometric phase is decreased with the increase of the dephasing rate which originates from the phase damping of the system. In the dissipative environment, it is illustrated that the geometric phase is reduced as the relaxation rate grows which results from both the energy dissipation and phase damping of the system. The results provide a theoretical reference for using the geometric phase to construct quantum logical gates based on quantum dot systems. More researches on the geometric phase in the DQD system can be carried out, such as the relationship between the current and the geometric phase in different environments.
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