（人造）原子—光场强相互作用系统的量子性质
|
摘要
最近,随着量子物理、凝聚态物理的发展,以及微加工、极低温、量子态测量等技术的飞速提高,量子体系与量子谐振子的作用越来越强,实验上已实现了超导磁通量子比特、量子点、超导电路QED系统,其耦合强度已经进入超强耦合区间。但这些强耦合系统的动力学仍然可以用Rabi模型来模拟。弱耦合作用下忽略Rabi模型非旋波项的JC模型已不再不适用。由于超强耦合,Rabi模型中的非旋波项对量子比特的相干性、纠缠的演化、光谱结构、拉比振荡、量子态的演化等产生了重要的影响,在实验上和理论分析中产生了一些新奇的物理现象。例如非对称的真空Rabi劈裂、光子阻塞、非经典态、超辐射跃迁等。只有考虑Rabi模型非旋波项的作用,才能真正地解释在强耦合作用下所出现的这些新奇的物理现象。因此,非旋波近似(NRWA)的Rabi模型的研究正是当前量子信息、量子光学和凝聚态物理的研究热点。
首先,我们研究了非旋波项对量子迹距离的影响。为了便于从物理上分析非旋波项对量子迹距离的影响,我们选取了两个特殊的初始态。这两个特殊的初始态是Rabi模型在旋波近似下且共振时的本征态。如果对玻色场求迹,这两个初始态对于量子比特而言是同一个态。通过对量子迹距离的分析,我们发现：在以两个特殊的初始态为条件下,量子比特的量子迹距离只与两个量子比特激发态布居差有关,而与两个量子比特的相干性无关。如果不考虑非旋波项的影响,这两个特殊初始态所对应的迹距离为零。但是,在考虑非旋波项的影响时,两个量子态的迹距离随时间振荡,并且随着耦合强度的增强,量子迹距离的振荡幅度和频率也增加。结果明显不同于旋波近似下的结果。
其次,我们讨论了在强耦合作用区间Rabi模型物理发射谱的性质。我们将旋波近似下量子比特的物理发射谱定义进行推广,使之适用于强耦合作用下非旋波近似的情况。结果发现：即使玻色场和量子比特初始态为基态|9,0>,多峰结构的真空Rabi劈裂谱出现了,并且谱峰不再对称；随着耦合强度和玻色场初始强度的增强,更多的本征态被激发,谱峰变得更加复杂。通过微扰可知：这些新的特征来源于非旋波项的影响。同时,在强耦合作用区间能级交错也同时被观察到。
接着,还讨论了非旋波近似下Rabi模型的Berry相位。主要是运用两个单位幺正变换来近似获得Rabi模型的本征能和本征态。为了说明本案解析方法的可靠性,同时给出了Rabi模型Berry相位的高精度数值解。通过比较发现：本案与高精度数值解在近共振时吻合得较好。为了进一步从物理上说明非旋波项对Berry相位的影响,采用了二阶微扰方法。通过微扰方法发现：本征态的基态所对应的Berry相位依赖于耦合强度的二次方和更高次方,而第一、第二激发态的berry相位依赖于耦合强度的零次、一次方和更高次方。
最后我们运用Beaudoin等人提出的、量子光学更为普适的方法-主方程方法,对强耦合作用区间腔模的输出谱进行了讨论。在弱探测场的情况下,发现：当弱探测场与基态和第一激发态共振时能够显现一个较强的峰,并且随着耦合作用的增强,峰的高度有所增加。当弱探测场与其它本征态共振或大失谐,出现的峰都较弱。这是因为在稳态的情况下,粒子布居绝大多数分布在基态,而探测场的强度较弱,对粒子数布居影响不大。当弱探测场与基态和第一激发态共振时,少部分的布居激发到第一激发态,因此能够显现一个较强的峰。
The strong interaction between a qubit and a quantum harmonic oscillator is now realized experimentally in different kinds of systems such as Josephson charge qubit, semiconducting dots and circuit QED, etc.. The dynamics of such strong coupling systems can be modeled by the Rabi model. However, the qubit is coupled to the oscillator so strongly that the well-known rotating wave approximation(RWA) is broken, and the counter-rotating wave terms must be considered. In such strong coupling regime, the qubit-oscillator interacting strength is com-parable to the oscillator frequency or the natural transition frequency of the qubit。Theoretical and experimental investigations have found novel phenomena, such as the asymmetry of vac-uum Rabi-splitting, photon blockade, nonclassical states generation, superradiance transition, and collapse and revival dynamics. Only when the effects of counter-rotating wave terms in the Rabi model are considered, these new phenomena can be satisfactorily explained in the strong coupling regime. Therefore, the researches of the effect of counter-rotating wave terms in the Rabi model have become one of the most hot points of quantum information, quantum optics and condensed matter physics.
Based on these, some quantum actions of Rabi model in strong coupling regime are inves-tigated in this dissertation. The main contents are as follows:
Firstly, the dynamic of the quantum trace distance for the states of the qubit strongly in-teracting with a bosonic field is studied. In order to gain physical insight into the effect of the counter-rotating wave terms on the dynamic of the qubit, two specific initial states which are the eigenstates of the qubit and field coupling system with the resonant interaction under the standard RWA are chosen. The reduced density matrices of the qubit in these two initial states are the same. Starting from these two initial states, we find that the time-evolution of the trace distance for the qubit states is only related to the difference between the populations in the qubit excited state but independent of the coherence of the qubit. If the RWA is applied under the resonant interaction, the dynamic of the trace distance for the qubit states remains zero. But the effect of the counter-rotating wave terms brings the deviation of the qubit states starting from the two initial states, which are exhibited by the time-dependence trace distance. The stronger the coupling between the qubit and the field is, the more obvious the deviation is.
Secondly, we present a simple and straightforward analytical method to investigate the physical emission spectrum of the Rabi model without the RWA. we generalize the definition of the physical emission spectrum valid with the RWA in order to meet without the RWA with some modifications due to the physical consideration in the strong coupling regime. The multi-peak Rabi splitting, even the qubit initially in its ground state and the bosonic field initially in vacuum, are obviously observed in the strong coupling regime. These new features of physical emission spectrum originate from the effect of counter-rotating wave terms. Moreover, the intercrossing of energy level can also be observed in the strong coupling regime.
Finally, we present a simple and straightforward analytical method based on two unitary transformations to investigate the Berry phase of the Rabi model without the RWA. To compare the validity of our analysis explicitly, we have given the high-accuracy numerical results of the Berry phase. It is found that our results agree well with the high-accuracy numerical results near the resonance situations. The Berry phase of ground state is relevant to the second-order terms of the coupling strength, and the Berry phases of the first excited and second excited state are mainly relevant to the first-, second-, and third-order terms of the coupling strength when the effect of the CRTs is taken into account.
首先,我们研究了非旋波项对量子迹距离的影响。为了便于从物理上分析非旋波项对量子迹距离的影响,我们选取了两个特殊的初始态。这两个特殊的初始态是Rabi模型在旋波近似下且共振时的本征态。如果对玻色场求迹,这两个初始态对于量子比特而言是同一个态。通过对量子迹距离的分析,我们发现：在以两个特殊的初始态为条件下,量子比特的量子迹距离只与两个量子比特激发态布居差有关,而与两个量子比特的相干性无关。如果不考虑非旋波项的影响,这两个特殊初始态所对应的迹距离为零。但是,在考虑非旋波项的影响时,两个量子态的迹距离随时间振荡,并且随着耦合强度的增强,量子迹距离的振荡幅度和频率也增加。结果明显不同于旋波近似下的结果。
其次,我们讨论了在强耦合作用区间Rabi模型物理发射谱的性质。我们将旋波近似下量子比特的物理发射谱定义进行推广,使之适用于强耦合作用下非旋波近似的情况。结果发现：即使玻色场和量子比特初始态为基态|9,0>,多峰结构的真空Rabi劈裂谱出现了,并且谱峰不再对称；随着耦合强度和玻色场初始强度的增强,更多的本征态被激发,谱峰变得更加复杂。通过微扰可知：这些新的特征来源于非旋波项的影响。同时,在强耦合作用区间能级交错也同时被观察到。
接着,还讨论了非旋波近似下Rabi模型的Berry相位。主要是运用两个单位幺正变换来近似获得Rabi模型的本征能和本征态。为了说明本案解析方法的可靠性,同时给出了Rabi模型Berry相位的高精度数值解。通过比较发现：本案与高精度数值解在近共振时吻合得较好。为了进一步从物理上说明非旋波项对Berry相位的影响,采用了二阶微扰方法。通过微扰方法发现：本征态的基态所对应的Berry相位依赖于耦合强度的二次方和更高次方,而第一、第二激发态的berry相位依赖于耦合强度的零次、一次方和更高次方。
最后我们运用Beaudoin等人提出的、量子光学更为普适的方法-主方程方法,对强耦合作用区间腔模的输出谱进行了讨论。在弱探测场的情况下,发现：当弱探测场与基态和第一激发态共振时能够显现一个较强的峰,并且随着耦合作用的增强,峰的高度有所增加。当弱探测场与其它本征态共振或大失谐,出现的峰都较弱。这是因为在稳态的情况下,粒子布居绝大多数分布在基态,而探测场的强度较弱,对粒子数布居影响不大。当弱探测场与基态和第一激发态共振时,少部分的布居激发到第一激发态,因此能够显现一个较强的峰。
The strong interaction between a qubit and a quantum harmonic oscillator is now realized experimentally in different kinds of systems such as Josephson charge qubit, semiconducting dots and circuit QED, etc.. The dynamics of such strong coupling systems can be modeled by the Rabi model. However, the qubit is coupled to the oscillator so strongly that the well-known rotating wave approximation(RWA) is broken, and the counter-rotating wave terms must be considered. In such strong coupling regime, the qubit-oscillator interacting strength is com-parable to the oscillator frequency or the natural transition frequency of the qubit。Theoretical and experimental investigations have found novel phenomena, such as the asymmetry of vac-uum Rabi-splitting, photon blockade, nonclassical states generation, superradiance transition, and collapse and revival dynamics. Only when the effects of counter-rotating wave terms in the Rabi model are considered, these new phenomena can be satisfactorily explained in the strong coupling regime. Therefore, the researches of the effect of counter-rotating wave terms in the Rabi model have become one of the most hot points of quantum information, quantum optics and condensed matter physics.
Based on these, some quantum actions of Rabi model in strong coupling regime are inves-tigated in this dissertation. The main contents are as follows:
Firstly, the dynamic of the quantum trace distance for the states of the qubit strongly in-teracting with a bosonic field is studied. In order to gain physical insight into the effect of the counter-rotating wave terms on the dynamic of the qubit, two specific initial states which are the eigenstates of the qubit and field coupling system with the resonant interaction under the standard RWA are chosen. The reduced density matrices of the qubit in these two initial states are the same. Starting from these two initial states, we find that the time-evolution of the trace distance for the qubit states is only related to the difference between the populations in the qubit excited state but independent of the coherence of the qubit. If the RWA is applied under the resonant interaction, the dynamic of the trace distance for the qubit states remains zero. But the effect of the counter-rotating wave terms brings the deviation of the qubit states starting from the two initial states, which are exhibited by the time-dependence trace distance. The stronger the coupling between the qubit and the field is, the more obvious the deviation is.
Secondly, we present a simple and straightforward analytical method to investigate the physical emission spectrum of the Rabi model without the RWA. we generalize the definition of the physical emission spectrum valid with the RWA in order to meet without the RWA with some modifications due to the physical consideration in the strong coupling regime. The multi-peak Rabi splitting, even the qubit initially in its ground state and the bosonic field initially in vacuum, are obviously observed in the strong coupling regime. These new features of physical emission spectrum originate from the effect of counter-rotating wave terms. Moreover, the intercrossing of energy level can also be observed in the strong coupling regime.
Finally, we present a simple and straightforward analytical method based on two unitary transformations to investigate the Berry phase of the Rabi model without the RWA. To compare the validity of our analysis explicitly, we have given the high-accuracy numerical results of the Berry phase. It is found that our results agree well with the high-accuracy numerical results near the resonance situations. The Berry phase of ground state is relevant to the second-order terms of the coupling strength, and the Berry phases of the first excited and second excited state are mainly relevant to the first-, second-, and third-order terms of the coupling strength when the effect of the CRTs is taken into account.
引文
[1]Irish E K and Schwab K 2003 Phys. Rev. B 68 155311
[2]Schwab K C and Roukes M L 2005 Phys. Today 58 36
[3]Blais A, Huang R S, Wallraff A, Girvin S M and Schoelkopf R J 2004 Phys. Rev. A 69 062320
[4]Wallraff A, Schuster D I, Blais A, Frunzio L, Huang R S, Majer J, Kumar S, Girvin S M and Schoelkopf R J 2004 Nature (London) 431162
[5]Chiorescu I, Bertet P, Semba K, Nakamura Y, Harmans C J P M and Mooij J E 2004 Nature (London) 431159
[6]Johansson J, Saito S, Meno T, Nakano H, Ueda M, Semba K and Takayanagi H 2006 Phys. Rev. Lett.96 127006
[7]Hofheinz M, Wang H, Ansmann M, Bialczak R C, Lucero E, Neeley M, O'Connell A D, Sank D, Wenner J, Martinis J M and Cleland A N 2009 Nature (London) 459 546
[8]Anappara A A, DeLiberato S, Tredicucci A, Ciuti C, Biasiol G, Sorba L and Beltram F 2009 Phys. Rev. B 79 201303
[9]Todorov Y, Andrews A M, Colombelli R, DeLiberato S, Ciuti C, Klang P, Strasser G and Sirtori C 2010 Phys. Rev. Lett.105196402
[10]Huber R, Anappara A A, Gunter G, Sell A 2011 J. Appl. Phys.109 102418
[11]Anappara A A, Liberato S D, Tredicucci A, Ciuti C, Biasio G, Sorba L and Beltram F 2009 Phys. Rev. B 79 201303
[12]Todorov Y, Andrews A M, Colombelli R, De Liberato S, Ciuti C, Klang P, Strasser G and C Sirtori 2010 Phys. Rev. Lett.105 196402
[13]Hofheinz M, Wang H, Ansmann M, Bialczak R C, Lucero E, Neeley M, OConnell A D, Sank D, Wenner J, Martinis J M and Cleland A N 2009 Nature (London) 459 546
[14]J Bourassa, Gambetta J M, A Abdumalikov A, Astafiev J, Nakamura Y and Blais A 2009 Phys. Rev. A.80 032109
[15]Nataf P and Ciuti C 2011 Phys. Rev. Lett.107 190402
[16]Lizuain I, Casanova J, Garcia-Ripoll J J, Muga J G and Solano E 2010 Phys. Rev. A 81 062131
[17]LaHaye M D, Suh J, Echternach P M, Schwab K C and Roukes M L 2009 Nature (London) 459960
[18]Fedorov A, Feofanov A K, Macha P, Forn-Diaz P, Harmans C J P M and Mooij J E 2010 Phys. Rev. Lett.105 060503
[19]Gunter G, Anappara A A, Hees J, Sell A, Biasiol G, Sorba L, DeLiberato S, Ciuti C, Tredicucci A, Leitenstorfer A and Huber R 2009 Nature (London) 458178
[20]Oconnell A D et al.2010 Nature 464 697
[21]Groblacher S et al.2009 Nature Phys.5 485
[22]Van der Wal C H et al.2000 Science 290 773
[23]McKeever J, Boca A, Boozer A D, Buck J R and Kimble H J 2003 Nature 425 268
[24]Birnbaum K M, Boca A, Miller R, Boozer A D, Northup T E and Kimble H J 2005 Nature 43687
[25]Boozer A D, Boca A, Miller R, Northup T E and Kimble H J 2007 Phys. Rev. Lett.98 193601
[26]Wilk T, Webster S C, Kuhn A and Rempe G 2007 Science 317 488
[27]Weber B, Specht H P, Muller T, Bochmann J, Mucke M, Moehring D L and Rempe G 2009 Phys. Rev. Lett 102 030501
[28]Lettner M, Miicke M, Riedl S, Vo C, Hahn C, Baur S, Bochmann J, Ritter S, Diirr S and Rempe G 2011 Phys. Rev. Lett.106 210503
[29]Specht H P, Nolleke C, Manue A R, Uphoff L M, Figueroa E, Ritter S and Rempe G 2011 Nature 473 190
[30]Koch M, Sames C, Balbach M, Chibani H, Kubanek A, Murr K, Wilk T and Rempe G 2011 Phys. Rev. Lett.107 023601
[31]Rabi I I 1937 Phys. Rev.51652
[32]Jaynes E T and Cummings F W 1963 Proc. IEEE 5189
[33]shore B W and Knight P L 1993 J. Mod. Opt.40 1195
[34]Ciuti C, Bastard G and Carusotto 12005 Phys. Rev. B 72 115303
[35]Niemczyk T, et al.2010 Nature Phys.6 772
[36]Casanova J, Romero G, Lizuain I, Garcia-Ripoll J J and Solano E 2010 Phys. Rev. Lett. 105263603
[37]Irish E K 2007 Phys. Rev. Lett.99 173601
[38]Cao X,You J Q, Zheng H, Kofman A G and Nori F 2010 Phys. Rev. A 82 022119
[39]DeLiberato S, Ciuti C and Carusotto 12007 Phys. Rev. Lett.98 103602
[40]Zueco D, Reuther G M, Kohler S and Hanggi P 2009 Phys. Rev. A 80033846
[41]Niemczy K T 2010 Nature. Phys.6 772
[42]Johansson J R, Johansson G, Wilson C M and Nori F 2009 Phys. Rev. Lett.103 147003
[43]Johansson J R, Johansson G, Wilson C M and Nori F 2010 Phys. Rev. A 82 052509
[44]DeLiberato S, Gerace D, Carusotto I and Ciuti C 2009 Phys. Rev. A 80 053810
[45]Uchiyama C, Aihara M, Saeki M and Miyashita S 2009 Phys. Rev. E 80 021128
[46]Ciuti C and Carusotto 12006 Phys. Rev. A 74 033811
[47]Devoret M, Girvin S and Schoelkopf R 2007 Ann. Phys.16 767
[48]Braak D 2011 Phys. Rev. Lett.107100401
[49]Travenec I 2012 Phys. Rev. A 85 043805
[50]Bloch F and Siegert A 1940 Phys. Rev.57 522
[51]Shirley J H 1965 Phys. Rev. B 138 979
[52]Wei C,Holmstromk S A, Sellarsy M J, Mansony N B and Ficek Z 1997 J. Phys. B:At. Mol. Opt. Phys.30 2735
[53]Swain S 1973 J. Phys. A:Math. Gen.6 192
[54]Swain S 1973 J. Phys. A:Math. Gen.6 1919
[55]Zaheer K and Zubairy M S 1988 Phys. Rev. A 37 1628
[56]Fang M F and Zhou P 1995 J. Mod. Opt. 42 1199
[57]Phoenix S J D 1989 J. Mod. Opt.36 1163
[58]Bishop R F and Emary C 2001 J. Phys. A:Math. Gen.34 5635
[59]Werlang T, Dodonov A V, Duzzioni E I and Villas-Boas C J 2008 Phys. Rev. A 78 053805
[60]Durstt C, Sigmundt E, Reineke P and Scheuing A 1986 J. Phys. C 19 2701
[61]Tur E A 2000 Opt. Spectrosc.89 574
[62]Feranchuk I D, Komarov L I and Ulyanenkov A P 1996 J. Phys. A 29 4035
[63]Chen Q H, Zhang Y Y, Liu T and Wang K L 2008 Phys. Rev. A 78 051801
[64]Liu T, Wang K L and Feng M 2009 Europhys. Lett.86 54003
[65]Casanova J, Romero G, Lizuain I, Garci a-Ripoll J J and Solano E 2010 Phys. Rev. Lett. 105 263603
[66]Chen Q H, Liu T, Zhang Y Y and Wang K L 2011 Europhys. Lett.9614003
[67]Feng M, Zhu X W, Fang X M, Yan M and Shi L 1999 J. Phys. B:At. Mol. Opt. Phys.32 701
[68]Sornborger A T, Cleland A N and Geller M R 2004 Phys. Rev. A 70 052315
[69]Liu T, Wang K L and Feng M 2007 J. Phys. B:At. Mol. Opt. Phys.401967
[70]Beaudoin F, Gambetta J M and Blais A 2011 Phys. Rev. A 84 043832
[71]Irish E K, Gea-Banacloche J, Martin I and Schwab K C 2005 Phys. Rev. B 72 195410
[72]Hausinger J and Grifoni M 2011 Phys. Rev. A 83 030301
[73]Zheng H, Zhu S Y and Zubairy M S 2008 Phys. Rev. Lett.101200404
[74]Chen Q H, Yang Y, Liu T and Wang K L 2010 Phys. Rev. A 82 052306
[75]Cao X, You J Q, Zheng H and Nori F 2011 New J. Phys.13 073002
[76]Zhao C J and Zheng H 2010 Phys. Rev. A 82 043844
[77]Peng J S and Li G X. Introduction to Modern Quantum Optics (World Scientific, Singapore, 1998)
[78]Meystre P and Sargent M. Elements of Quantum Optics (Springer, New York,1999)
[79]Eberly J H and Wodkiewicz K 1977 J. Opt. Soc. Am.671252
[80]Berry M V 1984 Proc. R. Soc. A 392 45
[81]Abaronov and Bohm Y D 1959 Phys. Rev.115 485
[82]陈成明,张全1991物理学报40(3)345
[83]Simon B H 1983 Phys Rev Lett.512167
[84]Aharonov Y and Anandan J 1987 Phys. Rev. Lett.58 1593
[85]Shapere A and Wilczek F 1989 Geometric Phases in Physics (Singapore:World Scientific)
[86]Mead C A 1992 Rev. Mod. Phys.64 51
[87]Xiao D, Chang M C and Niu Q 2010 Rev. Mod. Phys.821959
[88]Zhu S L and Wang Z D 2002 Phys. Rev. Lett.89 097902
[89]Moler D, Madsen L and Mflmer K 2007 Phys. Rev. A 75 062302
[90]Liu J B, Li J H, Wei H, Xie X T and Li W B 2006 J. Phys. B:At. Mol. Opt. Phys.39 1199
[91]Joshi A and Xiao M 2006 Phys. Lett. A 359 390
[92]Yin S and Tong D M 2009 Phys. Rev. A 79 044303
[93]Jones J A, Vedral V, Ekert A and Castagnoli G 2000 Nature 403 869
[94]Filipp S, Klepp J, Hasegawa Y, Plonka-Spehr C, Schmidt U, Geltenbort P and Rauch H 2009 Phys. Rev. Lett.102 030404
[95]Liu J B, Li J H, Wei H, Xie X T and Li W B 2006 J. Phys. B:At. Mol. Opt. Phys.39 1199
[96]Fuentes-Guridi I, Carollo A, Bose S and Vedral V 2002 Phys. Rev. Lett.89 220404
[97]Carollo A, Franfca Santos M and Vedral V 2003 Phys. Rev. A 67 063804
[98]Liu Y, Wei L F, Jia W Z and Liang J Q 2010 Phys. Rev. A 82 045801
[99]Leek P J, Fink J M, Blais A, Bianchetti R, Goppl M, Gambetta J M, Schuster D I, Frunzio L, Schoelkopf R J and Wallraff A 2007 Science 318 1889
[100]Chibotaru L F and Ungur L 2012 Phys. Rev. Lett.109 246403
[101]周世勋,陈灏著.量子力学教程(第2版)北京：高等教育出版社,2011
[102]Law C K, Zhu S Y and Zubairy M S 1995 Phys. Rev. A 52 4095
[103]Carmechael H J. An Open Systems Approach to Quantum Optics (Springer, Berlin,1993)
[104]Carmechael H J. Statistical Methods in Quantum Optics (Springer, Berlin,2002)
[105]Dajka J and Luczka J 2010 Phys. Rev. A 82 012341
[106]Dajka J, Luczka J and Hanggi P 2011 Phys. Rev. A 84 032120
[107]Ban M, Kitajima S and Shibata F 2011 Phys. Rev. A 84 042115
[108]Gan C J and Zheng H 2010 Eur. Phys. J. D 59 473
[109]Nataf P and Ciuti C 2010 Phys. Rev. Lett.104 023601
[110]Ashhab S and Nori F 2010 Phys. Rev. A 81042311
[111]Oliveira F A M, Kim M S and Knight P L,1990 Phys. Rev. A 412645
[112]Agarwal G S and Tara K,1991 Phys. Rev. A 43 492
[113]Smirne A, Breuer H P, Piilo J and Vacchini B 2010 Phys. Rev. A 82 062114
[114]Smirne A, Brivio D, Cialdi S, Vacchini B and Paris M G A 2011 Phys. Rev. A 84 032112
[115]Dakic B, Vedral V and Brukner C 2010 Phys. Rev. Lett.105 190502
[116]Ballester D, Romero G, Garcfa-Ripoll J J, Deppe F and Solano E 2012 Phys. Rev. X 2
[117]Zheng S B 2005 Phys. Rev. A 71062335
[118]Romero G, Ballester D, Wang Y M, Scarani V and Solano E 2012 Phys. Rev. Lett.108 120501
[119]Peng J S and Li G X 1992 Phys. Rev. A 45 3289
[120]He S, Wang C, Chen Q H, Ren X Z, Liu T and Wang K L 2012 Phys. Rev. A 86 033837
[121]Li G X, Luo M and Ficek Z 2009 Phys. Rev. A 79 053847
[122]Li G X 2000 Opt. Commun.184 397
[123]Peng J S, Li G X, Zhou P and Swain S 2000 Phys. Rev. A 61063807
[124]Zaheer K and Zubairy M S 1989 Phys. Rev. A 39 2000
[125]Lappas D G, M V and Eberly J H 1993 Phys. Rev. A 47 1327
[126]Reithmaier J P, Sek G, et al.2004 Nature 432 197
[127]Yoshie T, Scherer A, Hendrickson J, et al.2004 Nature 432 200
[128]Savenko I G, Kibis O V and Shelykh I A 2012 Phys. Rev. A 85 053818
[129]Ridolfo A, Leib M, Savasta S and Hartmann M J 2012 Phys. Rev. Lett.109 193602
[130]Falci G, Fazio R, Palma G M, Siewert J and Vedral V 2000 Nature 407 355
[131]Zhu S L, Wang Z D and Zanardi P 2005 Phys. Rev. Lett.94 100502
[132]Liang S D and Huang G Y 2013 Phys. Rev. A 87 012118
[133]Hsieh C Y, Rene A and Hawrylak P 2012 Phys. Rev. A 86 115312
[134]Horsley S A R and Babiker M 2007 J. Phys. B:At. Mol. Opt. Phys.40 2003
[135]Mironova P V, Efremov M A and Schleich W P 2013 Phys. Rev. A 87 013627
[136]Larson J 2012 Phys. Rev. Lett.108 033601
[137]Larson J 2012 arXiv:1208.1891v1
[138]Fuentes-Guridi I, Carollo A, Bose S and Vedral V 2002 Phys. Rev. Lett.89 220404
[139]Liu J B, Li J H, Wei H, Xie X T and Li W B 2006 J. Phys. B:At. Mol. Opt. Phys.391199
[140]Zheng S B 2012 Phys. Rev. A 85 022128
[141]Feofanov A K, Macha P, Forn-Diaz P, Harmans C J P M and Mooij J E 2010 Phys. Rev. Lett.105 060503
[142]Wu Y and Yang X X 2007 Phys. Rev. Lett.98 013601
[143]Chen G, Li J Q and Liang J Q 2006 Phys. Rev. A 74 054101
[144]Liu T, Feng M and Wang K L 2011 Phys. Rev. A 84 062109
[145]Zheng H and Takada Y 2011 Phys. Rev. A 84 043819
[146]Deng W W and Li G X 2013 J. Phys. B:At. Mol. Opt. Phys.46 035505
[147]Deng W W and Li G X 2013 Eur. Phys. J. D 67 91
[148]Scalari G, et al.2012 Science 3351323
[149]Portolan S, Di Stefano O, Savasta S, Rossi F and Girlanda R 2008 Phys. Rev. B 77 035433
[150]Ridolfo A, Di Stefano O, Portolan S and Savasta S 2010 J. Phys.:Conf. Ser.210 012025
[151]Ridolfo A, Stelitano S, Patanne S, Savasta S and Girlanda R 2010 Phys. Rev. B 81075313
[152]A Ridolfo, M Leib, S Savasta and M J Hartmann 2013 Phys. Scr. T153 014053
[153]Walls D F and Milburn G J 1994 Quantum Optics (Cambridge:Cambridge University Press)
[154]zhou P and Swain S 1997 Phys. Rev. A 56 3011
[155]Tan H T and Li G X 2010 Phys. Rev. A 82 032322
[156]Gardiner C W and Zoller P 2000 Quantum Noise (Berlin:Springer)
[157]Milonni P W, James D F V and Fearn H 1995 Phys. Rev. A 52 1525
[2]Schwab K C and Roukes M L 2005 Phys. Today 58 36
[3]Blais A, Huang R S, Wallraff A, Girvin S M and Schoelkopf R J 2004 Phys. Rev. A 69 062320
[4]Wallraff A, Schuster D I, Blais A, Frunzio L, Huang R S, Majer J, Kumar S, Girvin S M and Schoelkopf R J 2004 Nature (London) 431162
[5]Chiorescu I, Bertet P, Semba K, Nakamura Y, Harmans C J P M and Mooij J E 2004 Nature (London) 431159
[6]Johansson J, Saito S, Meno T, Nakano H, Ueda M, Semba K and Takayanagi H 2006 Phys. Rev. Lett.96 127006
[7]Hofheinz M, Wang H, Ansmann M, Bialczak R C, Lucero E, Neeley M, O'Connell A D, Sank D, Wenner J, Martinis J M and Cleland A N 2009 Nature (London) 459 546
[8]Anappara A A, DeLiberato S, Tredicucci A, Ciuti C, Biasiol G, Sorba L and Beltram F 2009 Phys. Rev. B 79 201303
[9]Todorov Y, Andrews A M, Colombelli R, DeLiberato S, Ciuti C, Klang P, Strasser G and Sirtori C 2010 Phys. Rev. Lett.105196402
[10]Huber R, Anappara A A, Gunter G, Sell A 2011 J. Appl. Phys.109 102418
[11]Anappara A A, Liberato S D, Tredicucci A, Ciuti C, Biasio G, Sorba L and Beltram F 2009 Phys. Rev. B 79 201303
[12]Todorov Y, Andrews A M, Colombelli R, De Liberato S, Ciuti C, Klang P, Strasser G and C Sirtori 2010 Phys. Rev. Lett.105 196402
[13]Hofheinz M, Wang H, Ansmann M, Bialczak R C, Lucero E, Neeley M, OConnell A D, Sank D, Wenner J, Martinis J M and Cleland A N 2009 Nature (London) 459 546
[14]J Bourassa, Gambetta J M, A Abdumalikov A, Astafiev J, Nakamura Y and Blais A 2009 Phys. Rev. A.80 032109
[15]Nataf P and Ciuti C 2011 Phys. Rev. Lett.107 190402
[16]Lizuain I, Casanova J, Garcia-Ripoll J J, Muga J G and Solano E 2010 Phys. Rev. A 81 062131
[17]LaHaye M D, Suh J, Echternach P M, Schwab K C and Roukes M L 2009 Nature (London) 459960
[18]Fedorov A, Feofanov A K, Macha P, Forn-Diaz P, Harmans C J P M and Mooij J E 2010 Phys. Rev. Lett.105 060503
[19]Gunter G, Anappara A A, Hees J, Sell A, Biasiol G, Sorba L, DeLiberato S, Ciuti C, Tredicucci A, Leitenstorfer A and Huber R 2009 Nature (London) 458178
[20]Oconnell A D et al.2010 Nature 464 697
[21]Groblacher S et al.2009 Nature Phys.5 485
[22]Van der Wal C H et al.2000 Science 290 773
[23]McKeever J, Boca A, Boozer A D, Buck J R and Kimble H J 2003 Nature 425 268
[24]Birnbaum K M, Boca A, Miller R, Boozer A D, Northup T E and Kimble H J 2005 Nature 43687
[25]Boozer A D, Boca A, Miller R, Northup T E and Kimble H J 2007 Phys. Rev. Lett.98 193601
[26]Wilk T, Webster S C, Kuhn A and Rempe G 2007 Science 317 488
[27]Weber B, Specht H P, Muller T, Bochmann J, Mucke M, Moehring D L and Rempe G 2009 Phys. Rev. Lett 102 030501
[28]Lettner M, Miicke M, Riedl S, Vo C, Hahn C, Baur S, Bochmann J, Ritter S, Diirr S and Rempe G 2011 Phys. Rev. Lett.106 210503
[29]Specht H P, Nolleke C, Manue A R, Uphoff L M, Figueroa E, Ritter S and Rempe G 2011 Nature 473 190
[30]Koch M, Sames C, Balbach M, Chibani H, Kubanek A, Murr K, Wilk T and Rempe G 2011 Phys. Rev. Lett.107 023601
[31]Rabi I I 1937 Phys. Rev.51652
[32]Jaynes E T and Cummings F W 1963 Proc. IEEE 5189
[33]shore B W and Knight P L 1993 J. Mod. Opt.40 1195
[34]Ciuti C, Bastard G and Carusotto 12005 Phys. Rev. B 72 115303
[35]Niemczyk T, et al.2010 Nature Phys.6 772
[36]Casanova J, Romero G, Lizuain I, Garcia-Ripoll J J and Solano E 2010 Phys. Rev. Lett. 105263603
[37]Irish E K 2007 Phys. Rev. Lett.99 173601
[38]Cao X,You J Q, Zheng H, Kofman A G and Nori F 2010 Phys. Rev. A 82 022119
[39]DeLiberato S, Ciuti C and Carusotto 12007 Phys. Rev. Lett.98 103602
[40]Zueco D, Reuther G M, Kohler S and Hanggi P 2009 Phys. Rev. A 80033846
[41]Niemczy K T 2010 Nature. Phys.6 772
[42]Johansson J R, Johansson G, Wilson C M and Nori F 2009 Phys. Rev. Lett.103 147003
[43]Johansson J R, Johansson G, Wilson C M and Nori F 2010 Phys. Rev. A 82 052509
[44]DeLiberato S, Gerace D, Carusotto I and Ciuti C 2009 Phys. Rev. A 80 053810
[45]Uchiyama C, Aihara M, Saeki M and Miyashita S 2009 Phys. Rev. E 80 021128
[46]Ciuti C and Carusotto 12006 Phys. Rev. A 74 033811
[47]Devoret M, Girvin S and Schoelkopf R 2007 Ann. Phys.16 767
[48]Braak D 2011 Phys. Rev. Lett.107100401
[49]Travenec I 2012 Phys. Rev. A 85 043805
[50]Bloch F and Siegert A 1940 Phys. Rev.57 522
[51]Shirley J H 1965 Phys. Rev. B 138 979
[52]Wei C,Holmstromk S A, Sellarsy M J, Mansony N B and Ficek Z 1997 J. Phys. B:At. Mol. Opt. Phys.30 2735
[53]Swain S 1973 J. Phys. A:Math. Gen.6 192
[54]Swain S 1973 J. Phys. A:Math. Gen.6 1919
[55]Zaheer K and Zubairy M S 1988 Phys. Rev. A 37 1628
[56]Fang M F and Zhou P 1995 J. Mod. Opt. 42 1199
[57]Phoenix S J D 1989 J. Mod. Opt.36 1163
[58]Bishop R F and Emary C 2001 J. Phys. A:Math. Gen.34 5635
[59]Werlang T, Dodonov A V, Duzzioni E I and Villas-Boas C J 2008 Phys. Rev. A 78 053805
[60]Durstt C, Sigmundt E, Reineke P and Scheuing A 1986 J. Phys. C 19 2701
[61]Tur E A 2000 Opt. Spectrosc.89 574
[62]Feranchuk I D, Komarov L I and Ulyanenkov A P 1996 J. Phys. A 29 4035
[63]Chen Q H, Zhang Y Y, Liu T and Wang K L 2008 Phys. Rev. A 78 051801
[64]Liu T, Wang K L and Feng M 2009 Europhys. Lett.86 54003
[65]Casanova J, Romero G, Lizuain I, Garci a-Ripoll J J and Solano E 2010 Phys. Rev. Lett. 105 263603
[66]Chen Q H, Liu T, Zhang Y Y and Wang K L 2011 Europhys. Lett.9614003
[67]Feng M, Zhu X W, Fang X M, Yan M and Shi L 1999 J. Phys. B:At. Mol. Opt. Phys.32 701
[68]Sornborger A T, Cleland A N and Geller M R 2004 Phys. Rev. A 70 052315
[69]Liu T, Wang K L and Feng M 2007 J. Phys. B:At. Mol. Opt. Phys.401967
[70]Beaudoin F, Gambetta J M and Blais A 2011 Phys. Rev. A 84 043832
[71]Irish E K, Gea-Banacloche J, Martin I and Schwab K C 2005 Phys. Rev. B 72 195410
[72]Hausinger J and Grifoni M 2011 Phys. Rev. A 83 030301
[73]Zheng H, Zhu S Y and Zubairy M S 2008 Phys. Rev. Lett.101200404
[74]Chen Q H, Yang Y, Liu T and Wang K L 2010 Phys. Rev. A 82 052306
[75]Cao X, You J Q, Zheng H and Nori F 2011 New J. Phys.13 073002
[76]Zhao C J and Zheng H 2010 Phys. Rev. A 82 043844
[77]Peng J S and Li G X. Introduction to Modern Quantum Optics (World Scientific, Singapore, 1998)
[78]Meystre P and Sargent M. Elements of Quantum Optics (Springer, New York,1999)
[79]Eberly J H and Wodkiewicz K 1977 J. Opt. Soc. Am.671252
[80]Berry M V 1984 Proc. R. Soc. A 392 45
[81]Abaronov and Bohm Y D 1959 Phys. Rev.115 485
[82]陈成明,张全1991物理学报40(3)345
[83]Simon B H 1983 Phys Rev Lett.512167
[84]Aharonov Y and Anandan J 1987 Phys. Rev. Lett.58 1593
[85]Shapere A and Wilczek F 1989 Geometric Phases in Physics (Singapore:World Scientific)
[86]Mead C A 1992 Rev. Mod. Phys.64 51
[87]Xiao D, Chang M C and Niu Q 2010 Rev. Mod. Phys.821959
[88]Zhu S L and Wang Z D 2002 Phys. Rev. Lett.89 097902
[89]Moler D, Madsen L and Mflmer K 2007 Phys. Rev. A 75 062302
[90]Liu J B, Li J H, Wei H, Xie X T and Li W B 2006 J. Phys. B:At. Mol. Opt. Phys.39 1199
[91]Joshi A and Xiao M 2006 Phys. Lett. A 359 390
[92]Yin S and Tong D M 2009 Phys. Rev. A 79 044303
[93]Jones J A, Vedral V, Ekert A and Castagnoli G 2000 Nature 403 869
[94]Filipp S, Klepp J, Hasegawa Y, Plonka-Spehr C, Schmidt U, Geltenbort P and Rauch H 2009 Phys. Rev. Lett.102 030404
[95]Liu J B, Li J H, Wei H, Xie X T and Li W B 2006 J. Phys. B:At. Mol. Opt. Phys.39 1199
[96]Fuentes-Guridi I, Carollo A, Bose S and Vedral V 2002 Phys. Rev. Lett.89 220404
[97]Carollo A, Franfca Santos M and Vedral V 2003 Phys. Rev. A 67 063804
[98]Liu Y, Wei L F, Jia W Z and Liang J Q 2010 Phys. Rev. A 82 045801
[99]Leek P J, Fink J M, Blais A, Bianchetti R, Goppl M, Gambetta J M, Schuster D I, Frunzio L, Schoelkopf R J and Wallraff A 2007 Science 318 1889
[100]Chibotaru L F and Ungur L 2012 Phys. Rev. Lett.109 246403
[101]周世勋,陈灏著.量子力学教程(第2版)北京：高等教育出版社,2011
[102]Law C K, Zhu S Y and Zubairy M S 1995 Phys. Rev. A 52 4095
[103]Carmechael H J. An Open Systems Approach to Quantum Optics (Springer, Berlin,1993)
[104]Carmechael H J. Statistical Methods in Quantum Optics (Springer, Berlin,2002)
[105]Dajka J and Luczka J 2010 Phys. Rev. A 82 012341
[106]Dajka J, Luczka J and Hanggi P 2011 Phys. Rev. A 84 032120
[107]Ban M, Kitajima S and Shibata F 2011 Phys. Rev. A 84 042115
[108]Gan C J and Zheng H 2010 Eur. Phys. J. D 59 473
[109]Nataf P and Ciuti C 2010 Phys. Rev. Lett.104 023601
[110]Ashhab S and Nori F 2010 Phys. Rev. A 81042311
[111]Oliveira F A M, Kim M S and Knight P L,1990 Phys. Rev. A 412645
[112]Agarwal G S and Tara K,1991 Phys. Rev. A 43 492
[113]Smirne A, Breuer H P, Piilo J and Vacchini B 2010 Phys. Rev. A 82 062114
[114]Smirne A, Brivio D, Cialdi S, Vacchini B and Paris M G A 2011 Phys. Rev. A 84 032112
[115]Dakic B, Vedral V and Brukner C 2010 Phys. Rev. Lett.105 190502
[116]Ballester D, Romero G, Garcfa-Ripoll J J, Deppe F and Solano E 2012 Phys. Rev. X 2
[117]Zheng S B 2005 Phys. Rev. A 71062335
[118]Romero G, Ballester D, Wang Y M, Scarani V and Solano E 2012 Phys. Rev. Lett.108 120501
[119]Peng J S and Li G X 1992 Phys. Rev. A 45 3289
[120]He S, Wang C, Chen Q H, Ren X Z, Liu T and Wang K L 2012 Phys. Rev. A 86 033837
[121]Li G X, Luo M and Ficek Z 2009 Phys. Rev. A 79 053847
[122]Li G X 2000 Opt. Commun.184 397
[123]Peng J S, Li G X, Zhou P and Swain S 2000 Phys. Rev. A 61063807
[124]Zaheer K and Zubairy M S 1989 Phys. Rev. A 39 2000
[125]Lappas D G, M V and Eberly J H 1993 Phys. Rev. A 47 1327
[126]Reithmaier J P, Sek G, et al.2004 Nature 432 197
[127]Yoshie T, Scherer A, Hendrickson J, et al.2004 Nature 432 200
[128]Savenko I G, Kibis O V and Shelykh I A 2012 Phys. Rev. A 85 053818
[129]Ridolfo A, Leib M, Savasta S and Hartmann M J 2012 Phys. Rev. Lett.109 193602
[130]Falci G, Fazio R, Palma G M, Siewert J and Vedral V 2000 Nature 407 355
[131]Zhu S L, Wang Z D and Zanardi P 2005 Phys. Rev. Lett.94 100502
[132]Liang S D and Huang G Y 2013 Phys. Rev. A 87 012118
[133]Hsieh C Y, Rene A and Hawrylak P 2012 Phys. Rev. A 86 115312
[134]Horsley S A R and Babiker M 2007 J. Phys. B:At. Mol. Opt. Phys.40 2003
[135]Mironova P V, Efremov M A and Schleich W P 2013 Phys. Rev. A 87 013627
[136]Larson J 2012 Phys. Rev. Lett.108 033601
[137]Larson J 2012 arXiv:1208.1891v1
[138]Fuentes-Guridi I, Carollo A, Bose S and Vedral V 2002 Phys. Rev. Lett.89 220404
[139]Liu J B, Li J H, Wei H, Xie X T and Li W B 2006 J. Phys. B:At. Mol. Opt. Phys.391199
[140]Zheng S B 2012 Phys. Rev. A 85 022128
[141]Feofanov A K, Macha P, Forn-Diaz P, Harmans C J P M and Mooij J E 2010 Phys. Rev. Lett.105 060503
[142]Wu Y and Yang X X 2007 Phys. Rev. Lett.98 013601
[143]Chen G, Li J Q and Liang J Q 2006 Phys. Rev. A 74 054101
[144]Liu T, Feng M and Wang K L 2011 Phys. Rev. A 84 062109
[145]Zheng H and Takada Y 2011 Phys. Rev. A 84 043819
[146]Deng W W and Li G X 2013 J. Phys. B:At. Mol. Opt. Phys.46 035505
[147]Deng W W and Li G X 2013 Eur. Phys. J. D 67 91
[148]Scalari G, et al.2012 Science 3351323
[149]Portolan S, Di Stefano O, Savasta S, Rossi F and Girlanda R 2008 Phys. Rev. B 77 035433
[150]Ridolfo A, Di Stefano O, Portolan S and Savasta S 2010 J. Phys.:Conf. Ser.210 012025
[151]Ridolfo A, Stelitano S, Patanne S, Savasta S and Girlanda R 2010 Phys. Rev. B 81075313
[152]A Ridolfo, M Leib, S Savasta and M J Hartmann 2013 Phys. Scr. T153 014053
[153]Walls D F and Milburn G J 1994 Quantum Optics (Cambridge:Cambridge University Press)
[154]zhou P and Swain S 1997 Phys. Rev. A 56 3011
[155]Tan H T and Li G X 2010 Phys. Rev. A 82 032322
[156]Gardiner C W and Zoller P 2000 Quantum Noise (Berlin:Springer)
[157]Milonni P W, James D F V and Fearn H 1995 Phys. Rev. A 52 1525