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把三能级开放系统的密度矩阵按照Gellmann矩阵展开,然后将展开系数和Bloch球中的方位角对应, 从而获得了Poincaré球内部点和复三维Hilbert空间的非单位矢量即波函数的映射.进一步建议用该非单位矢量来定义混合态的量子几何相位.结果显示该几何相位仅仅与复Hilbert投影空间的几何结构有关, 与开放系统具体的演化路径无关;并且该混合态的几何相位依赖于开放系统的反转粒子数,也是描述开放系统混合度的单值光滑曲线,这个结果意味着混合态的演化的确按照几何相位保持其运动记忆.此外,在纯态的限制下,Berry相位是本文定义的几何相位极限情况.By expanding the density matrix of the open system in terms of Gell-mann matrix in a three-level system, we parameterize coefficients of expansion by some azimuthal angles and find an identity mapping of the density matrices onto interior points of the unit Poincaré sphere. Thus, the relations between the points on the unit Poincaré sphere and wave functions are extended to connect the interior points in the sphere with the nonunit vector rays corresponding to an open system in complex Hilbert space. Thus,the geometric phases for the open system are proposed to be observed by the nonunit vector rays,where the geometric phase of the pure state is the limiting case of our definition. The results show that this geometric phase merely with duplicate three-dimensional Hilbert projection space geometry structure related, has nothing to do with the open system concrete evolution way; and it depends on population inversion and is a slippy and single-value curve of Bloch radius. Therefore, the mixed state of open system retains indeed a memory of its motion in the form of a geometric phase factor.
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Keywords:
- pure state /
- mixed state /
- three-level open system /
- geometric phase
[1] Aharonov Y, Bohm D 1959 Phys. Rev. 115 485
[2] Berry M V 1984 Proc. R. Soc.(London),Ser A 392 45
[3] Fonseca-Romero K M, Aguiar-Pinto A C, Thomaz M T 2002 Physica A 307 142
[4] Li C F, Guo G C 1996 Acta Phys. Sin. 45 897(in Chinese )[李春芳, 郭光灿 1996 45 897]
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[7] Zheng L M, Wang F Q, Liu S H 2009 Acta Phys. Sin. 58 2430(in Chinese )[郑力明, 王发强, 刘颂豪 2009 58 2430]
[8] Berr-Aryeh Y 2004 J. Opt. B:Quantum Semiclass. Opt. 6 R1
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[10] Falci C, Fazio R, Palma G M, Siewert J, Vedral V 2000 Nature 407 355
[11] Wang Z S,Wu C F, Feng X L, Kwek L C, Lai C H, Oh C H, Vedral V 2007 Phys. Rev. A 76 044303
[12] Carollo A, Fuentes-Guridi I, Franca Santos M, Vedral V 2003 Phys. Rev. Lett. 90 160402
[13] Fonseca Romero K M, Aguiar A C, Thomaz M T 2002 Physica A 307 142
[14] Nazir A, Spiller T P, Munro W J 2003 Phys. Rev. A 65 042303
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[17] Tong D M, Sjoqvist E, Kwek L C, Oh C H 2004 Phys. Rev. Lett. 93 080405
[18] Whitney R S, Makhlin Y, Shnirman A, Gefen Y 2005 Phys. Rev. Lett. 94 070407
[19] Carollo A, Palma G M , zinski A, Santos, Vedral V 2006 Phys. Rev. Lett. 96 150403
[20] Wang Z S, Kwek L C, Lai C H, Oh C H 2006 Europhys. Lett. 74 958
[21] Jiang Y Y, Ji Y H, Xu H L, Hu L Y, Wang Z S, Chen Z Q, Guo L P 2010 Phys. Rev. A 82 062108
[22] Wang Z S, Wu C F, Kwek L C, Lai C H, Oh C H 2007 Phys. Rev. A 75 024102
[23] Lindblad G 1976 Commun. Math. Phys. 48 119
[24] Wang Z S, Kwek L C, Lai C H, Oh C H 2005 The European Physical Journal D 33 285
[25] Wang Z S 2009 Int. J. Theor. Phys. 48 2353
[26] Yu Y X, Chen Z Q, Hu L Y, Tang H S, Wang Z S 2011 Int. J. Theor. Phys. 50 148
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[1] Aharonov Y, Bohm D 1959 Phys. Rev. 115 485
[2] Berry M V 1984 Proc. R. Soc.(London),Ser A 392 45
[3] Fonseca-Romero K M, Aguiar-Pinto A C, Thomaz M T 2002 Physica A 307 142
[4] Li C F, Guo G C 1996 Acta Phys. Sin. 45 897(in Chinese )[李春芳, 郭光灿 1996 45 897]
[5] Li B Z, Zhang D G , Wu J H, Yan F L 1997 Acta Phys. Sin. 46 227(in Chinese )[李伯臧, 张德刚, 吴建华, 阎凤利 1997 46 227]
[6] Li H Z 2004 Acta Phys. Sin. 53 1643(in Chinese )[李华钟 2004 53 1643]
[7] Zheng L M, Wang F Q, Liu S H 2009 Acta Phys. Sin. 58 2430(in Chinese )[郑力明, 王发强, 刘颂豪 2009 58 2430]
[8] Berr-Aryeh Y 2004 J. Opt. B:Quantum Semiclass. Opt. 6 R1
[9] Jones J A, Vedral V, Ekert A, Castagnoli G 1999 Nature 403 689
[10] Falci C, Fazio R, Palma G M, Siewert J, Vedral V 2000 Nature 407 355
[11] Wang Z S,Wu C F, Feng X L, Kwek L C, Lai C H, Oh C H, Vedral V 2007 Phys. Rev. A 76 044303
[12] Carollo A, Fuentes-Guridi I, Franca Santos M, Vedral V 2003 Phys. Rev. Lett. 90 160402
[13] Fonseca Romero K M, Aguiar A C, Thomaz M T 2002 Physica A 307 142
[14] Nazir A, Spiller T P, Munro W J 2003 Phys. Rev. A 65 042303
[15] Whitney R S, Gefen Y 2003 Phys. Rev. Lett. 90 190402
[16] Chiara G De, Palma M 2003 Phys. Rev. Lett. 91 090404
[17] Tong D M, Sjoqvist E, Kwek L C, Oh C H 2004 Phys. Rev. Lett. 93 080405
[18] Whitney R S, Makhlin Y, Shnirman A, Gefen Y 2005 Phys. Rev. Lett. 94 070407
[19] Carollo A, Palma G M , zinski A, Santos, Vedral V 2006 Phys. Rev. Lett. 96 150403
[20] Wang Z S, Kwek L C, Lai C H, Oh C H 2006 Europhys. Lett. 74 958
[21] Jiang Y Y, Ji Y H, Xu H L, Hu L Y, Wang Z S, Chen Z Q, Guo L P 2010 Phys. Rev. A 82 062108
[22] Wang Z S, Wu C F, Kwek L C, Lai C H, Oh C H 2007 Phys. Rev. A 75 024102
[23] Lindblad G 1976 Commun. Math. Phys. 48 119
[24] Wang Z S, Kwek L C, Lai C H, Oh C H 2005 The European Physical Journal D 33 285
[25] Wang Z S 2009 Int. J. Theor. Phys. 48 2353
[26] Yu Y X, Chen Z Q, Hu L Y, Tang H S, Wang Z S 2011 Int. J. Theor. Phys. 50 148
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