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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]
[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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[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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