王 瓊, 冷 洋

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量子拉比模型中的對(duì)稱性與對(duì)稱性破缺研究

王 瓊*1, 冷 洋2

(1. 湖南文理學(xué)院 物理與電子科學(xué)學(xué)院, 湖南 常德, 415000; 2. 三一重工 智能研究院, 湖南 長(zhǎng)沙, 410100)

量子拉比模型(QRM)是描述物質(zhì)與光相互作用最簡(jiǎn)單的模型之一. 本文發(fā)現(xiàn)在存在對(duì)稱性的拉比模型中, 偏置的引入將破壞模型的對(duì)稱性, 系統(tǒng)出現(xiàn)標(biāo)度行為; 通過(guò)引入一個(gè)新的伊辛相互作用, 模型的對(duì)稱性將重新修復(fù). 發(fā)現(xiàn)這種對(duì)稱性與對(duì)稱性破缺規(guī)律性的結(jié)論在N態(tài)量子拉比模型同樣存在.

量子拉比模型; 對(duì)稱性; 對(duì)稱性破缺

量子拉比模型(QRM)[1]在20世紀(jì)30年代已被大家所了解,是量子力學(xué)最簡(jiǎn)單和最典型的模型之一. 它描述的是一個(gè)二能級(jí)原子與單模玻色場(chǎng)間的相互作用,即所謂的單模耗散兩能級(jí)系統(tǒng). 拉比模型在離子阱[2]、電路量子電動(dòng)力學(xué)[3—5]以及光子系統(tǒng)[6]等領(lǐng)域都有廣泛的應(yīng)用. 近些年來(lái), 人們?cè)趯ふ襋RM封閉形式的解析解做了許多工作[7—13]. 但相比于JC模型, 由于QRM中存在反旋波項(xiàng), 這樣使得QRM中量子激發(fā)數(shù)不再是一個(gè)守恒量, 但從對(duì)稱性角度考慮, 我們?nèi)匀豢梢詫RM當(dāng)成是一個(gè)可積分系統(tǒng)[10]. 由于這個(gè)原因, Casanova等發(fā)現(xiàn)在二態(tài)QRM中存在一個(gè)無(wú)限維不變量子子空間[14], 并被Albert等推廣到N態(tài)QRM[15]. 然而, 當(dāng)對(duì)QRM引入偏置場(chǎng)后, 情況將完全不一樣. 我們將引入偏置后的QRM稱為偏置拉比模型(BRM), 可以用以下的Hamiltonian來(lái)描述:

1 對(duì)稱性與對(duì)稱性破缺

在BRM引入一個(gè)新的自旋, 系統(tǒng)的Hamiltonian可以寫(xiě)為:

這里是本征能量,D為:

2 標(biāo)度行為

圖1 H2(虛線, η = 0)以及H2B (實(shí)線, η≠ 0)的四個(gè)最低本征能級(jí)隨拉比耦合強(qiáng)度λ的函數(shù), 參數(shù)取值.

由基態(tài)波函數(shù), 我們計(jì)算得到:

圖2 本證基態(tài)下的標(biāo)度行為. (a) 隨變化, 表現(xiàn)兩種不同的行為取決于與的差. (b)隨變化, 對(duì)變現(xiàn)為標(biāo)度不變性, 這里. 黑色圓點(diǎn)對(duì)應(yīng)參數(shù)取值, . 黑色圓點(diǎn)對(duì)應(yīng)參數(shù)取值, .

我們得出的對(duì)稱性與對(duì)稱性破缺規(guī)律性的結(jié)論在N態(tài)量子拉比模型同樣成立. 考慮一個(gè)N態(tài)系統(tǒng), 且第一個(gè)自旋與單模玻色場(chǎng)存在拉比相互作用, 同剩下N-1自旋存在伊辛相互作用, 系統(tǒng)的Hamilto- nian可以寫(xiě)為:

3 結(jié)論

我們研究了量子拉比模型中的對(duì)稱性與對(duì)稱性破缺問(wèn)題. 發(fā)現(xiàn)存在對(duì)稱性的拉比模型中, 偏置的引入將破壞模型的對(duì)稱性, 系統(tǒng)出現(xiàn)標(biāo)度行為; 通過(guò)引入一個(gè)新的伊辛相互作用, 模型的對(duì)稱性將重新修復(fù), 并且這種規(guī)律性的結(jié)論能推廣到N態(tài)情形.

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Parity symmetry and parity breaking in the quantum Rabi model

WANG Qiong1, LEN Yang2

(1. College of Physics and Electronics, Hunan University of Arts and Science, Changde 415000, China; 2. Institute of Intelligence Research, Sany Heavy Industry, Changsha 410100, China)

The well-known quantum Rabi model strictly describes the simplest interaction between matter and the quantum light. We explore the possibility to generate new parity symmetry in the quantum Rabi model after a bias is introduced. We consider a physically realistic method by involving an additional spin into the quantum Rabi model to couple with the original spin by an Ising interaction. The rule can be found that the parity symmetry is broken by introducing a bias and then restored by adding new degrees of freedom.

quantum Rabi model; parity symmetry; parity breaking

O 413.2

1672-6146(2014)02-0018-04

10.3969/j.issn.1672-6146.2014.02.004

通訊作者email: zuiguiyixiu@163.com.

2014-05-04

國(guó)家自然科學(xué)基金專項(xiàng)基金(11347142); 光電信息集成與光學(xué)制造技術(shù)湖南省重點(diǎn)實(shí)驗(yàn)室; 湖南文理學(xué)院重點(diǎn)學(xué)科建設(shè)項(xiàng)目(光學(xué)).

(責(zé)任校對(duì):譚長(zhǎng)貴)