趙 佳

(太原工業(yè)學(xué)院 理學(xué)系,山西 太原 030008)

自旋相干態(tài)法研究自旋軌道耦合BEC系統(tǒng)的基態(tài)性質(zhì)

趙 佳

(太原工業(yè)學(xué)院 理學(xué)系,山西 太原 030008)

文章采用自旋相干態(tài)的方法研究自旋軌道耦合玻色愛因斯坦凝聚(BEC)系統(tǒng)的基態(tài)性質(zhì),在自旋相干態(tài)表象下把系統(tǒng)的哈密頓量對角化, 由變分法求出系統(tǒng)的基態(tài)能譜、原子布居數(shù)和光子數(shù),從而得到系統(tǒng)的量子相變,與Holstein-Primakoff方法所得的結(jié)果一致．

自旋相干態(tài);自旋軌道耦合;能譜;原子布居數(shù);光子數(shù)

0 引言

自旋軌道耦合(SOC)描述了粒子的自旋和軌道自由度之間的相互作用．在凝聚態(tài)物理中,電子的自旋軌道耦合不僅產(chǎn)生了一些重要的量子現(xiàn)象,例如自旋和反?；魻栃?yīng)、拓樸絕緣體和拓樸超導(dǎo)等,在實現(xiàn)自旋電子學(xué)和拓?fù)淞孔佑嬎惴矫嬉舶l(fā)揮了重要的作用[1]．本文用自旋相干態(tài)方法研究自旋軌道耦合玻色愛因斯坦凝聚(BECs)系統(tǒng)的基態(tài)性質(zhì),主要思路:首先由玻色子算符取平均場近似后,得出等效贗自旋哈密頓算符,然后應(yīng)用自旋相干態(tài)變換將其對角化[2],最后將能量泛函對復(fù)參數(shù)求變分并取極小值,從而得出基態(tài)能量和原子布居數(shù)的精確解．

1 模型及哈密頓量

圖1 自旋軌道耦合實驗裝置示意圖

(1)

(2)

(3)

既然s波散射長度在實驗可以測量:c0=100.86aB和c2=-0.46aB[3],其中aB是玻爾半徑,那么可以得到g↑↑=g↑↓≈g↓↓,這說明囚禁BEC表現(xiàn)出原子間碰撞的強(qiáng)相互作用,而在這種強(qiáng)相互作用下,所有超冷原子都被限制在相同的基態(tài)上,每個原子的動量都完全一樣．引入兩個玻色算符

(4)

通過計算可以得到自旋軌道耦合驅(qū)動BEC系統(tǒng)的有效哈密頓量[4]

(5)

在圖1所示的系統(tǒng)中,玻色子模在y方向上與超冷原子間沒有相互作用,所以該系統(tǒng)的性質(zhì)可以用下面的單模Dicke模型哈密頓量來描述[5]

(6)

2 理論方法

在玻色場相干態(tài)表象中,計算方程(6)所對應(yīng)哈密頓算符的期待值

(7)

其中復(fù)數(shù)α=u+iυ是玻色子湮滅算符a的本征值,有a|α〉=α|α〉．

(8)

那么,基態(tài)能譜

(9)

在熱力學(xué)極限(N→∞)下,HP方法給出了同樣的結(jié)果[9,10],而自旋相干態(tài)法對任意原子數(shù)都成立．從圖2可以看出,基態(tài)能量一階導(dǎo)在臨界點附近連續(xù)

在北極規(guī)范下,激發(fā)的宏觀量子態(tài)能譜

(10)

原子布居數(shù)

(11)

圖2 基態(tài)能譜隨耦合參數(shù)λ2的變化曲線

圖3 宏觀量子態(tài)能譜隨耦合參數(shù)λ2的變化曲線

圖4 原子布居數(shù)隨耦合參數(shù)λ2的變化曲線

圖5 光子數(shù)隨耦合參數(shù)λ2的變化曲線

3 結(jié)果與討論

參數(shù)ω=Ω=1,則有相變臨界點λc=1．當(dāng)λ<λc時,系統(tǒng)處于正常相;當(dāng)λ>λc時,系統(tǒng)處于超輻射相．隨著耦合參數(shù)λ2的增大,系統(tǒng)從正常相變化到超輻射相,發(fā)生了量子相變．基態(tài)能譜在相變臨界點處的突然變化說明系統(tǒng)發(fā)生了量子相變,原子布居數(shù)和光子數(shù)在相變臨界點處從零開始急劇增加．

[1] LIAN Jinling,YU Lixian,LIANG J Q.Orbit-induced spin squeezing in a spin-orbit coupled Bose-Einstein condensate[J].Sci Rep,2013,3(1):3166-3166

[2] LAI Y Z,LIANG J Q.Time-dependent quantum systems and the invariant Hermitian operator[J].Phys Rev A,1996,53:3691-3693

[3] LIN Y J,GARCIA K J,SPIELMAN I B.Spin-orbit-coupled bose-einstein condensates[J].Nature,2011,471:83-86

[4] ZHANG Y,CHEN G, ZHANG C.Tunable spin-orbit coupling and quantumphase transition in a trapped Bose-Einstein condensate[J].Sci Rep,2013,3(1):1937-1937

[5] DICKE R H.Coherence in spontaneous radiation processes[J].Phys.Rev,1954,93:99-110

[6] LIAN Jinling,ZHANG Yuanwei,LIANG Jiuqing.Macroscopic quantum states and quantum phase transition in the dicke model[J].Chin Phys Lett,2012,29(1):060302-060302

[7] EMARY Clive, BRANDES Tobias.Quantum chaos triggered by precursors of a quantum phase transition: The Dicke model[J].Phys Rev Lett,2003,90:044101-044104

[8] EMARY Clive, BRANDES Tobias.Chaos and the quantum phase transition in the Dicke model[J].Phys Rev E,2003,67(1):066203-066203

[9] CASTAFLOS O,ACHAR E N, HIRSCH J G.No singularities in observables at the phase transition in the Dicke model[J].Phys Rev A,2011,83(1):05160-05160

[10] EMARY Clive, BRANDES Tobias.Phase transitions in generalized spin-boson (Dicke)models[J].Phys Rev A,2004,69(1):053804-053804

Studying the Ground-State Properties of the Spin-Orbit-Coupled BEC by Means of the Spin Coherent-State Transformation

ZHAO Jia

(Department of Science, Taiyuan Institute of Technology, Taiyuan 030008, China)

We study the ground-state characteristics of the spin-orbit-coupled BEC by means of the spin coherent-state transformation. In the picture of the spin coherent-state,the Hamiltonian of the spin-Orbit-coupled BEC can be diagonalized by the spin coherent-state transformation. Based on the variational method we are able to obtain the expression of the ground-state energy spectra, the atomic population and the photon number. Consequently, The quantum phase transition is detected by these physical quantities. the same results are exactly obtained by using the Holstein-Primakoff approach.

the spin-orbit-coupled; the spin coherent state transformation; the energy spectra;the atomic population;photon

2015-10-11

趙 佳(1983-),女,山西太原人,太原工業(yè)學(xué)院理學(xué)系助教,主要從事量子理論研究.

1672-2027(2015)04-0004-04

O175

A