用量子軌跡法模擬量子點系統之條件式計數統計

Conditional Counting Statistics of Quantum Dot Systems with Quantum Trajectory Method

葉宗鋼(Tsung-Kang Yeah)

指導教授 ： 管希聖

國立臺灣大學/理學院/物理研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.01334

摘要

Sukhorukov 等人用量子尖端接觸(Quantum Point Contact)作了單量子點元件條件式計數統計(Conditional Counting Statistics)的量測，並在理論上推導出相對應的解析解，而在這裡「條件式」指得是一個元件的計數統計量是以另一個元件的電流大小為條件。儘管他們的解析解和實驗結果在他們考慮的參數下是一致的，但他們的理論方法存在若干限制，因此我們又用了另外兩種方法去模擬條件式計數統計，他們分別稱作修正的生成方程法(Modified Generating Function Method)和量子軌跡法(Quantum Trajectory Method)。
　　我們在此考慮兩種系統，分別是一個接有源極(Source)與汲極(Drain)的量子點被量子尖端接觸測量，但用了和他們不一樣的參數；其二則是接有源極與汲極的兩個串聯的量子點，而其中一個量子點和量子尖端接觸耦合。我用非條件式主方程式(Unconditional Master Equation)推導量子擴散條件式主方程式(Quantum Diffusion Conditional Master Equation)，藉著這個方程式，我可以用數值方法模擬量子尖端接觸電流可能的量測的結果，以及量子點上電子數目的期望值，然後使用模擬出的量子尖端接觸的電流和量子點上的電子數，我們可以得到條件式計數統計的結果，這就是所謂的量子軌跡法。
　　我用量子軌跡法模擬這些系統的第一、二階量子點元件電流的條件式電流累積量(Cumulant)，並和我團隊內另一個成員以修正的生成方程法做的結果做比對，我們發現兩者相當吻合，顯示出兩種方法都能有效地模擬條件式計數統計。另一方面，量子軌跡法可以模擬量子點系統的動力學和在實驗上測量到的量子尖端接觸電流，因此它可以提供我們關於量子點被量子尖端接觸測量之系統的傳輸特性更多的洞見。

關鍵字

量子尖端接觸 ； 條件式記數統計 ； 量子軌跡法 ； 量子點

並列摘要

Sukhorukov et al. used a quantum point contact(QPC) to measure the conditional counting statistics of the quantum dot(QD) device, and they derived corresponding analytical results of conditional counting statistics theoretically. The conditional here means the counting statistics of one device is conditional on the current of the other device. Although their analytical results are in agreement with their experimental results for the parameters they used in their experiments, their theoretical method has some limitation or disadvantages. Thus we apply two other method to simulate the conditional counting statistics, namely the modifed generating function method and the quantum trajectory method.
We consider two systems here. One is a QD connected with a source lead and a drain lead is measured by a QPC the same system considered by Sukhorukov et al. but with different experimental parameters. The other is two quantum dots in series, one dot connected to a source lead and the other dot connected to a drain lead. One of the dots is coupled to a QPC. We derive the quantum diffusion conditional master equations (QDCME) with the unconditional master equations. Moreover,we can simulate the possible measurement results of the QPC and the expectation value of the electron number in the QD numerically with the QDCMEs. Then using the simulated QPC current and electron number on the QD, we can obtain the results of the conditional counting statistics with the QDCMEs. The above description is the so called quantum-trajectory method.
We simulate the firrst and second current cumulants of the QD device with quantum trajectory method, and compare our results with the results calculated by the modifed genertating function method performed by one of my group member. We find that the two results are in good agreement with each other. It indicates both methods can simulate the conditional counting statistics properly. In addition, the quantum-trajectory method can mimic the QD system dynamics and the measured QPC current in a single realization of the experiment, thus provide more insight into the transport properties of the nanostructure QD-QPC system.

並列關鍵字

Quantum Point Contact ； Conditional Counting Statistics ； Quantum Trajectory Method ； Quantum Dot

參考文獻

[26] Scott L. Miller and Donald Childers/ Probability and Random Process with Applications to Signal Processing and Communications 2nd edition/ Chp 10/ Elsevier(2012)

[5] Eugene V. Sukhorukov, Andrew N. Jordan, Simmon Gustavsson, Renaud

[4] Toshimasa Fujisawa, Toshiaki Hayashi, Ritsuya Tomita and Yoshiro Hirayama/

[1] Hermann Grabert, Michel H. Devoret/ Single Charge Tunneling Coulomb blockade Phenomena in Nanostructures/ Plenum Press(1992)

Science Vol.312, p.1634 (2006)