量子位元系統的非馬可夫動力行為的研究

Non-Markovian Dynamics of Quantum Bit Systems

簡崇欽(Chung-Chin Jian)

指導教授 ： 管希聖

國立臺灣大學/理學院/物理研究所/博士(2012年)

https://doi.org/10.6342/NTU.2012.00233

摘要

摘要
在本論文中，我們研究兩個量子位元(quantum bit)系統的非馬可夫
(non-Markovian)動力行為。第一個量子位元系統，是一單顆電子在兩個耦合量子點中做相干性地穿隧，同時被一個對電荷感應靈敏的量子點接觸(quantum point contact)偵測元件所量測。第二個是一個被具非馬可夫性質的環境所影響而做純粹退相位(pure-dephasing)行為的量子位元系統。
在不做一般在奈米結構元件中電子傳輸行為的理論研究所常採用的寬能帶極限（wideband limit）以及馬可夫近似的情況之下，我們將系統-熱庫作用強度當作微擾參數並用二階微擾的理論去推導耦合量子點系統的非馬可夫量子主方程式(quantum master equation)以及計算量子點接觸（可視為一個熱庫）中的傳輸電流。耦合量子點系統的約化密度矩陣(reduced density matrix)的非馬可夫性質來自於量子點接觸中電子的穿隧振幅和電極的狀態密度和能量有關的效應。我們用一個有限帶寬（或具截止頻率）的簡單羅倫茲譜密度(Lorentzian spectral density)來模擬量子點接觸中和能量有關的電子穿隧振幅與電極狀態密度，並藉此來闡明這個耦合量子點系統的非馬可夫動力行為。我們發現在所推導的主方程式和傳輸電流中和時間相關的耗散係數包含了來自於量子點接觸熱庫的時間關聯函數的實部和虛部的貢獻。若忽略來自於虛部的貢獻，我們可用所推導的主方程式取各種
不同的近似得到文獻中所對應的各種不同馬可夫形式的主方程式。然而，在某些參數範圍內，虛部的貢獻明顯地影響了帶電量子位元系統和傳輸電流的動力行為。我們也許可以藉由穩態電流的數值，觀察到虛部的效應。不過，量子點接觸中的瞬變電流在非馬可夫的參數的行為與寬能帶極限之下所對應的馬可夫瞬變電流之間有相當不一樣的動力行為。因此，瞬變電流的差異的觀測可做為耦合的量子點接觸-量子位元系統的非馬可夫行為的見證。此外，即使是在馬可夫的參數，
與時間相關的係數中的虛部效應也可以從電流噪聲頻譜(current noise spectrum)中觀測到。為了這個目的，我們在馬可夫範圍內，藉著運用量子回歸定理(quantum regression theorem)，去計算量子點接觸中電流-電流的時間關聯函數，然後再執行傅立葉轉換去得到電流噪聲頻譜。
在非馬可夫的情況下，量子回歸定理不能被應用來計算時間關聯函數，這是由於熱庫中的記憶效應。為了計算非馬可夫情況下的電流噪聲頻譜，我們必須超越並推廣量子回歸定理。因此我們在弱的系統-環境耦合極限之下，推導了一個在有限溫度的環境下可適用於計算系統算符的時間關聯函數的非馬可夫演化方程式。為了闡明我們所推導出來的演化方程式的正確性與用處，我們用兩種不同的方式去計算一個純退相位的自旋-波色子模型(spin-boson model)中的時間關聯
函數。一種是藉由直接的精確算符技巧，另一種則是藉由所推導的演化方程式。此兩種不同方式所得到結果的一致性清楚地說明了我們所推導的演化方程式可正確地將量子回歸定理推廣到非馬可夫的情況。我們相信我們所推導出來的演化方程式將可應用到許多不同的物理領域上。

關鍵字

量子位元 ； 非馬可夫 ； 量子點接觸 ； 純退相位 ； 有限帶寬 ； 量子回歸定理 ； 時間關聯函數

並列摘要

Abstract
In this thesis, we investigate non-Markovian dynamics of two quantum bit systems. The first system is a charge qubit consisting of an electron tunneling coherently between two coupled quantum dots (CQD's) subject to a measurement by a charge-sensitive detector of a quantum point contact (QPC). The second system is a qubit under the influence of a pure-dephasing non-Markovian environment.
Going beyond the wideband limit (WBL) and the Markovian approximation usually employed in the theoretical study for electron transport properties in nanostructure devices, we derive perturbatively the non-Markovian quantum master equation for the CQD's system and calculate the transport current through the QPC (considered as reservoirs) to second-order in the system-reservoir interaction strength. The non-Markovianity of the reduced density matrix of the CQD's system comes from the fact that electron tunneling amplitudes and also the electrodes' densities of states of
the QPC are in general energy-dependent. We model the energy-dependent tunneling amplitudes and electrodes' densities of states of the QPC through a simple model
of a Lorentzian spectral density with a finite bandwidth (or a cut-off frequency) to illustrate the non-Markovian dynamics of the CQD's qubit system. We find that the calculated time-dependent decay coefficients in the derived master equation and transport current involve the contributions from both the real and imaginary parts
of the QPC reservoir correlation functions. By neglecting the contributions from the imaginary parts, the derived master equation reduces, in appropriate limits, to various
Markovian versions of master equations in the literature. However, the contributions of imaginary parts significantly influence the dynamics of the charge qubit and thus
the transport current in certain parameter regime. One may be able to observe the effect of imaginary parts from the values of the steady-state current. Furthermore,
the behavior of the transient currents through the QPC in the non-Markovian parameter regime differs considerably from the WBL Markovian counterparts and thus may serve as a witness for the non-Markovian behavior in the QPC-qubit system. The effect of imaginary parts of the coefficients can also be found in the current noise spectrum even in the Markovian parameter regime. To this end, we calculate the
QPC current-current two-time correlation functions (CF's) in the Markovian regime by applying the quantum regression theorem (QRT) and then perform the Fourier transform to obtain the current noise spectrum.
The QRT cannot be applied to calculate the two-time CF's in the non-Markovian case due to the memory effect of the reservoirs. To find the non-Markovian current noise spectrum, we need to go beyond the QRT. We thus derive in the weak system-environment coupling limit an evolution equation capable of calculating the two-time CF's of system operators in non-Markovian finite-temperature environments.
We thus calculate the two-time CF's of a pure-dephasing spin-boson model by two different ways, one by the direct exact operator technique and the other by the derived
evolution equation. The agreement of the results between the two different approaches demonstrates clearly that the derived evolution equations generalize correctly the QRT to non-Markovian cases. It is believed that these evolution equations will have applications in many different branches of physics.

並列關鍵字

non-Markovian ； qubit ； quantum point contact ； pure-dephasing ； WBL ； the imaginary parts of the coefficients ； two-time correlation functions

參考文獻

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