劉新建,李衛(wèi)東
(山西大學(xué)理論物理研究所,中國(guó) 太原 030006)
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弱聯(lián)接玻色愛(ài)因斯坦凝聚體中勢(shì)壘寬度對(duì)非線性耦合及其動(dòng)力學(xué)的影響
劉新建,李衛(wèi)東
(山西大學(xué)理論物理研究所,中國(guó) 太原 030006)
利用解析與數(shù)值方法,對(duì)處于對(duì)稱雙勢(shì)阱中的玻色愛(ài)因斯坦凝聚體中,勢(shì)壘寬度對(duì)系統(tǒng)非線性耦合及其動(dòng)力學(xué)的影響進(jìn)行了研究.研究發(fā)現(xiàn)當(dāng)勢(shì)壘寬度較大時(shí),系統(tǒng)的線性耦合強(qiáng)度可迅速減??;在勢(shì)壘寬度大于0.3且非線性強(qiáng)度較大時(shí),線性耦合強(qiáng)度遠(yuǎn)小于非線性耦合項(xiàng),此時(shí)玻色約瑟夫森結(jié)模型的動(dòng)力學(xué)特性由非線性耦合強(qiáng)度來(lái)決定.同時(shí)對(duì)勢(shì)壘寬度對(duì)BEC約瑟夫森振蕩的周期和發(fā)生宏觀量子自俘獲時(shí)的非線性臨界值進(jìn)行了詳細(xì)的研究.
對(duì)稱雙方勢(shì)阱;雙模近似; 玻色約瑟夫森結(jié); 宏觀量子自俘獲
自玻色愛(ài)因斯坦凝聚體(BECs)實(shí)現(xiàn)以來(lái),理論[1-9]和實(shí)驗(yàn)[10-11]都對(duì)其進(jìn)行了廣泛的研究.雙勢(shì)阱模型作為一個(gè)簡(jiǎn)單的物理模型,主要研究量子干涉效應(yīng)和約瑟夫森效應(yīng)的典型結(jié)構(gòu).早在1997年Smerzi等人就利用玻色約瑟夫森模型對(duì)雙勢(shì)阱中排斥相互作用原子的隧穿動(dòng)力學(xué)進(jìn)行了研究[1],并在理論上得出了一種非線性現(xiàn)象:宏觀量子自俘獲(MQST)[2-4].十年后在光晶格中人們觀察到了約瑟夫森振蕩[10],在雙勢(shì)阱中觀察到了約瑟夫森振蕩和宏觀量子自俘獲[11].這些工作都是基于用一束失諧的激光把一個(gè)磁諧振子勢(shì)從中一分為二,實(shí)現(xiàn)雙勢(shì)阱,這樣囚禁在其中的BECs就一分為二,適當(dāng)調(diào)節(jié)雙勢(shì)阱之間的勢(shì)壘高度就形成了兩團(tuán)弱耦合BECs.李衛(wèi)東等人指出在修正的玻色約瑟夫森結(jié)模型中不僅存在線性耦合項(xiàng)而且存在非線性耦合項(xiàng),并且發(fā)現(xiàn)在弱耦合,即滿足雙模近似的條件下,勢(shì)壘高度和寬度不發(fā)生變化的情況下隨著非線性增加非線性耦合項(xiàng)將達(dá)到甚至超過(guò)線性耦合項(xiàng),從而影響雙勢(shì)阱中BECs的動(dòng)力學(xué)性質(zhì)[7-8].我們知道在實(shí)驗(yàn)中可以通過(guò)調(diào)節(jié)Feshbach共振改變非線性強(qiáng)度[10-11].同樣可以改變激光脈沖的強(qiáng)度來(lái)調(diào)節(jié)勢(shì)壘的高度和寬度,從而影響兩勢(shì)阱的耦合強(qiáng)度.本文在對(duì)稱雙方勢(shì)阱這一模型中,利用文獻(xiàn)[8]的方法,研究了勢(shì)壘寬度對(duì)對(duì)稱雙方勢(shì)阱中BECs非線性耦合強(qiáng)度及其動(dòng)力學(xué)影響.
零溫下對(duì)稱雙方勢(shì)阱中弱相互作用的BECs的波函數(shù)滿足一維GP方程:
(1)
其中V(x)是雙勢(shì)阱函數(shù)形式為:
(2)
(3)
將雙模近似代入GP方程可推導(dǎo)出修正的非線性雙模近似動(dòng)力學(xué)方程:
(4)
其中R(..)表示括號(hào)中式子的實(shí)部.其中κ為線性耦合項(xiàng),為非線性耦合項(xiàng).
模糊層次分析法FAHP(Fuzzy Analytic Hierarchy Process)是在傳統(tǒng)的層次分析法基礎(chǔ)上,考慮到人們對(duì)復(fù)雜事物判斷的模糊性,引入模糊一致性矩陣的決策方法。模糊層次分析法很好地解決了AHP判斷矩陣的一致性問(wèn)題。本文在具體采用模糊層次分析法確定各個(gè)路由度量權(quán)重系數(shù)之前,首先作如下假設(shè):
(5)
定義兩個(gè)勢(shì)阱中的粒子布局?jǐn)?shù)差z(t)≡n1(t)-n2(t)和相位差φ(t)=θ2(t)-θ1(t),則非線性雙模近似的動(dòng)力學(xué)方程可寫(xiě)為:
(6)
(7)
表1 線性耦合強(qiáng)度與非線性耦合強(qiáng)度隨勢(shì)壘寬度和非線性強(qiáng)度的變化表
取勢(shì)壘高度V0=200, 通過(guò)計(jì)算表(1)發(fā)現(xiàn)當(dāng)勢(shì)壘寬度b≥0.30,即兩勢(shì)阱間的耦合強(qiáng)度較弱,同時(shí)非線性參數(shù)非線性較強(qiáng)時(shí),會(huì)出現(xiàn)κ?χ 即非線性耦合項(xiàng)占主導(dǎo)作用.此時(shí)方程(6)可以忽略κ簡(jiǎn)化為:
(8)
系統(tǒng)的總哈密頓量為:
(9)
此式比方程(6)更簡(jiǎn)潔.但此式必須在弱耦合和強(qiáng)非線性同時(shí)滿足的條件下才能滿足.對(duì)于雙勢(shì)阱模型所采用的雙模近似在耦合強(qiáng)度越弱的情況下,近似程度越高;同時(shí)非線性強(qiáng)度可以通過(guò)調(diào)節(jié)Feshbach共振改變.所以上式能很好地描述雙模近似模型下強(qiáng)相互作用的量子氣體.
2.1 約瑟夫森振蕩
利用(6)式取勢(shì)壘高度V0=200,勢(shì)壘寬度分別取b=0.15,0.30,0.45.粒子布局?jǐn)?shù)差初始值z(mì)(0)=0.01,初始相位差為φ(0)=0計(jì)算出其粒子布局?jǐn)?shù)差隨時(shí)間變化圖像,及其對(duì)應(yīng)的相位差圖像.如圖1可知,粒子數(shù)布局?jǐn)?shù)差平均值為零,所以BECs進(jìn)行約瑟夫森振蕩.勢(shì)壘寬度增加,勢(shì)阱間耦合強(qiáng)度減弱,其他條件相同的情況下,勢(shì)壘寬度越大,雙勢(shì)阱系統(tǒng)中BECs做約瑟夫森振蕩的周期越長(zhǎng);相位差的變化周期與粒子數(shù)布局差周期規(guī)律一致.
圖1 不同勢(shì)壘寬度和η=5,40時(shí),粒子布局?jǐn)?shù)差和相位差隨時(shí)間演化Fig.1 The time evolution of particle population imbalance and phase difference for various barrier width and η=5,40
2.2 宏觀量子自俘獲
同樣利用公式(6)取勢(shì)壘高度V0=200,初始相位差為φ(0)=0,且勢(shì)壘寬度分別取b=0.15,0.30,0.45.與小幅振蕩不同的是這里選取粒子布局?jǐn)?shù)差的初始值z(mì)(0)=0.1.計(jì)算出其粒子布局?jǐn)?shù)差隨時(shí)間變化圖像及其對(duì)應(yīng)的相位差圖像.由圖2可以看出不同的勢(shì)壘寬度在其他條件一致的情況下出現(xiàn)的振蕩形式不同;勢(shì)壘寬度越大的越容易出現(xiàn)粒子數(shù)布局?jǐn)?shù)差平均值為非零,即宏觀量子自俘獲,關(guān)于這一點(diǎn)文章后續(xù)還有說(shuō)明.從圖2還可以看出當(dāng)粒子布局?jǐn)?shù)差進(jìn)入宏觀量子自俘獲相時(shí),左右勢(shì)阱中波函數(shù)相位差變成了相位的持續(xù)增加.
圖2 不同勢(shì)壘寬度和η=5,40時(shí),粒子布局?jǐn)?shù)差和相位差隨時(shí)間演化Fig.2 The time evolution of particle population imbalance and phase difference for various barrier width and η=5,40
2.3 弱耦合強(qiáng)相互作用下的動(dòng)力學(xué)
當(dāng)勢(shì)壘的寬度較寬且非線性較強(qiáng)時(shí),對(duì)稱雙勢(shì)阱中BECs的隧穿動(dòng)力學(xué)可以用(8)式描述,選取勢(shì)壘寬度b=0.30,非線性強(qiáng)度為η=80,來(lái)比較保留線性耦合項(xiàng)和忽略線性耦合項(xiàng)對(duì)其動(dòng)力學(xué)的影響,驗(yàn)證簡(jiǎn)化式(8)的合理性.從圖3可以看出在上述條件下忽略線性耦合項(xiàng)對(duì)BECs的動(dòng)力學(xué)影響較小.因此可知(8)式能較好地刻畫(huà)強(qiáng)相互作用下BECs的動(dòng)力學(xué)行為.
圖3 φ(0)=0,z(0)分別為0.05,0.10時(shí)保留線性耦合項(xiàng)和忽略線性耦合項(xiàng)時(shí),粒子布局?jǐn)?shù)差和相位差隨時(shí)間演化對(duì)比Fig.3 The contrast time evolution images of particle population imbalance and phase difference between keeping the linear coupling term and ignoring it when φ(0)=0, z(0)=0.05 and 0.10
2.4 勢(shì)壘寬度對(duì)宏觀量子自俘獲的非線性臨界值的影響
宏觀量子自俘獲現(xiàn)象是玻色約瑟夫森結(jié)模型中的一種新奇的量子現(xiàn)象.對(duì)于給定的模型中當(dāng)初始相位差φ(0)=0,對(duì)應(yīng)的初始粒子布局?jǐn)?shù)差臨界值為:
(10)
圖4 不同勢(shì)壘寬度下初始粒子數(shù)差臨界值隨非線性強(qiáng)度變化圖像Fig.4 The threshold of initial population imbalance varies with the change of nonlinear intensity under different barrier widths
因此改變勢(shì)壘寬度和非線性強(qiáng)度都會(huì)改變臨界值z(mì)(0).由圖4可知,隨著勢(shì)壘寬度越大,在相同的粒子數(shù)差初始情況下,動(dòng)力學(xué)從約瑟夫森振動(dòng)向宏觀量子自俘獲轉(zhuǎn)變的非線性值越小.同時(shí)發(fā)現(xiàn)勢(shì)壘較寬的模型出現(xiàn)自俘獲的區(qū)間較大.
在對(duì)稱雙方勢(shì)阱中,利用修正的約瑟夫森模型,系統(tǒng)地研究了勢(shì)壘寬度對(duì)線性耦合項(xiàng)及其動(dòng)力學(xué)的影響.發(fā)現(xiàn)在b≥0.30弱耦合模型,強(qiáng)相互作用下,非線性耦合項(xiàng)將變成主要耦合項(xiàng),修正的約瑟夫森雙模動(dòng)力學(xué)模型(6)式可以簡(jiǎn)化為(8)式.同時(shí)發(fā)現(xiàn)在相同非線性強(qiáng)度下,勢(shì)壘越寬,約瑟夫森振蕩的周期越大,宏觀量子自俘獲的初始臨界值越?。ㄟ^(guò)計(jì)算發(fā)現(xiàn)在弱耦合強(qiáng)相互作用下簡(jiǎn)化的約瑟夫森模型能夠很好地描述雙方勢(shì)阱中BECs的動(dòng)力學(xué).
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(編輯 陳笑梅)
The Effect of the Barrier Width on the Nonlinear Coupling and Dynamics of Weakly Coupling Bose-Einstein Condensates
LIUXin-jian*,LIWei-dong
(Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China)
By using analytical and numerical methods, the effects of barrier width on the strength of nonlinear coupling and the dynamics of Bose-Einstein Condensates with symmetrical double square well were investigated. The results show that the linear coupling could be very small once the barrier width is large. When the width is larger than 0.3 and nonlinear parameters larger, the linear coupling can be safely neglected. In this case, the dynamics character of Bose-Jsoephson model is determined only by nonlinear coupling strength. Furthermore, the effects of this width on the period of the nonlinear Josephson oscillation and the nonlinear threshold value in macroscopic quantum self-trapping were presented in details.
symmetric double square well potential; two mode approximation; Bose-Josephson junction; macroscopic quantum self-trapping
2014-02-24
國(guó)家自然科學(xué)基金資助項(xiàng)目(11374197;11074155;10934004)
O469
A
1000-2537(2014)03-0053-05
*通訊作者,E-mail:15364839097@163.com