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Understanding the non-Markovian dynamics of dissipative processes induced by memory effects of the environment is a fundamental subject of open quantum systems. Because of the complexity of open quantum systems, e.g., the multiple energy scales involving that of the system, the environment, and their mutual coupling, it is generally a challenging task to characterize the relationship among the parameters of the system dynamics and the reservoir spectra. For the two-level spontaneous emission model within structured environments, it was shown in a recent literature (Opt. Lett. 38, 3650) that a functional relation could be established between the asymptotically non-decaying population and the spectral density of the reservoir as the system undergoes a long-time evolution. It hence renders a distinct perspective to look into the character of long-lived quantum coherence in the corresponding non-Markovian process. This article is devoted to further investigate the phenomena of limit cycle oscillations possibly occurring in such non-Markovian dissipative systems in a long-time evolution. For a two-level system subjected to an environment with Ohmic class spectra, due to the presence of a unique bound-state mode of the system, the evolution trajectory of the given initial states will converge to a limit cycle in the Bloch space. The dependence of the radius and the location of the limit cycle on the spectral density function of the reservoir are manifested by virtue of the described functional relation. For the model subjected to a photonic crystal environment with multiple bands, our studies reveal that, owing to the presence of two or more bound states, the evolution trajectory of the system will converge to a toric curve of a paraboloid in the Bloch space and the phenomena of periodic or quasi-periodic oscillations could exhibit. While the equation of the parabolic curve is fully determined by the initial values of the state vector in the Bloch space, our results reveal that the scope of the evolution trajectory inside the toric curve is related to the spectral density of the reservoir and their quantified relation is distinctly characterized. Finally, the asymptotic dynamics of the correlations of a two-qubit system is discussed when it is subjected locally to the non-Markovian dissipative process.
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Keywords:
- open quantum system /
- non-Markovian dissipation /
- limit cycle /
- bound state
[1] Breuer H P, Petruccione F 2002 The Theory of Open Quantum Systems (London: Oxford University Press) pp460-498
[2] John S, Wang J 1990 Phys. Rev. Lett. 64 2418
[3] John S, Wang J 1991 Phys. Rev. B 43 12772
[4] John S, Quang T 1994 Phys. Rev. A 50 1764
[5] Kofman A G, Kurizki G, Sherman B 1994 J. Mod. Opt. 41 353
[6] Kofman A G, Kurizki G 1996 Phys. Rev. A 54 R3750
[7] Chen S, Xie S Y, Yang Y P, Chen H 2003 Acta Phys. Sin. 52 853 (in Chinese) [陈三, 谢双媛, 羊亚平, 陈鸿 2003 物理学报 52 853]
[8] Lodahl P, Driel A F van, Nikolaev I S, Irman A, Overgaag K, Vanmaekelbergh D, Vos W L 2004 Nature 430 654
[9] Xu X, Yamada T, Ueda R, Otomo A 2008 Opt. Lett. 33 1768
[10] Hoeppe U, Wolff C, Kchenmeister J, Niegemann J, Drescher M, Benner H, Busch K 2012 Phys. Rev. Lett. 108 043603
[11] Bellomo B, Franco R Lo, Maniscalco S, Compagno G 2008 Phys. Rev. A 78 060302
[12] Breuer H P, Laine E M, Piilo J 2009 Phys. Rev. Lett. 103 210401
[13] Rivas á, Huelga S F, Plenio M B 2010 Phys. Rev. Lett. 105 050403
[14] Tong Q J, An J H, Luo H G, Oh C H 2010 Phys. Rev. A 81 052330
[15] Zhang P, You B, Cen L X 2013 Opt. Lett. 38 3650
[16] Zhang P, You B, Cen L X 2014 Chin. Sci. Bull. 59 3841
[17] Perko L 2001 Differential Equations and Dynamical Systems (New York: Springer-Verlag) pp315-540
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[1] Breuer H P, Petruccione F 2002 The Theory of Open Quantum Systems (London: Oxford University Press) pp460-498
[2] John S, Wang J 1990 Phys. Rev. Lett. 64 2418
[3] John S, Wang J 1991 Phys. Rev. B 43 12772
[4] John S, Quang T 1994 Phys. Rev. A 50 1764
[5] Kofman A G, Kurizki G, Sherman B 1994 J. Mod. Opt. 41 353
[6] Kofman A G, Kurizki G 1996 Phys. Rev. A 54 R3750
[7] Chen S, Xie S Y, Yang Y P, Chen H 2003 Acta Phys. Sin. 52 853 (in Chinese) [陈三, 谢双媛, 羊亚平, 陈鸿 2003 物理学报 52 853]
[8] Lodahl P, Driel A F van, Nikolaev I S, Irman A, Overgaag K, Vanmaekelbergh D, Vos W L 2004 Nature 430 654
[9] Xu X, Yamada T, Ueda R, Otomo A 2008 Opt. Lett. 33 1768
[10] Hoeppe U, Wolff C, Kchenmeister J, Niegemann J, Drescher M, Benner H, Busch K 2012 Phys. Rev. Lett. 108 043603
[11] Bellomo B, Franco R Lo, Maniscalco S, Compagno G 2008 Phys. Rev. A 78 060302
[12] Breuer H P, Laine E M, Piilo J 2009 Phys. Rev. Lett. 103 210401
[13] Rivas á, Huelga S F, Plenio M B 2010 Phys. Rev. Lett. 105 050403
[14] Tong Q J, An J H, Luo H G, Oh C H 2010 Phys. Rev. A 81 052330
[15] Zhang P, You B, Cen L X 2013 Opt. Lett. 38 3650
[16] Zhang P, You B, Cen L X 2014 Chin. Sci. Bull. 59 3841
[17] Perko L 2001 Differential Equations and Dynamical Systems (New York: Springer-Verlag) pp315-540
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