Geometric phase for degenerate states of spin-1 and spin-1/2 pair

• Authors: Guo-Qiang Zhu
• Languages of publication: EN

Abstracts

EN

The geometric phase of a bi-particle model is discussed. One can drive the system to evolve by applying an external magnetic field, thereby controlling the geometric phase. The model has degenerate lowest-energy eigenvectors. The initial state is assumed to be the linear superposition or mixture of the eigenvectors. The relationship between the geometric phase and the structures of the initial state is considered, and the results are extended to a more general model.

Keywords

EN

geometric phase; dynamical evolution

Discipline

• 03.65.Vf: Phases: geometric; dynamic or topological
• 03.65.Yz: Decoherence; open systems; quantum statistical methods(see also 03.67.Pp in quantum information; for decoherence in Bose-Einstein condensates, see 03.75.Gg)

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2007
• Volume: 5
• Issue: 4
• Pages: 463-470

Dates

published

1 - 12 - 2007

online

1 - 12 - 2007

Contributors

author

Guo-Qiang Zhu

• Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, 310027, P.R. China

References

• [1] M.V. Berry “Quantal Phase Factors Accompanying Adiabatic Changes”, Proc. R. Soc. London A, Vol. 392, (1984), pp. 45–57. http://dx.doi.org/10.1098/rspa.1984.0023[Crossref]
• [2] Y. Aharonov and J. Anandan “Phase change during a cyclic quantum evolution”, Phys. Rev. Lett., Vol. 58, (1987), pp. 1593–1596. http://dx.doi.org/10.1103/PhysRevLett.58.1593[Crossref]
• [3] J. Samuel and R. Bhandari “General Setting for Berry’s Phase”, Phys. Rev. Lett., Vol. 60, (1988), pp. 2339–2342. http://dx.doi.org/10.1103/PhysRevLett.60.2339[Crossref]
• [4] A. Uhlmann “Parallel transport and quantum holonomy along density operators”, Rep. Math. Phys. Vol. 24, (1986), pp. 229–240; “A gauge field governing parallel transport along mixed states”, Lett. Math. Phys. Vol. 21, (1991), pp. 229–236. http://dx.doi.org/10.1016/0034-4877(86)90055-8[Crossref]
• [5] E. Sjöqvist, A.K. Pati, A. Ekert, J.S. Anandan, M. Ericsson, D.K.L. Oi and V. Vedral “Geometric Phases for Mixed States in Interferometry”, Phys. Rev. Lett., Vol. 85, (2000), pp. 2845–2848. http://dx.doi.org/10.1103/PhysRevLett.85.2845[Crossref]
• [6] J. Du, P. Zhou, M. Shi, L.C. Kwek, J.-W. Pan, C.H. Oh and A. Ekert “Observation of Geometric Phases for Mixed States using NMR Interferometry”, Phys. Rev. Lett., Vol. 91, (2003), pp. 100403–100406. http://dx.doi.org/10.1103/PhysRevLett.91.100403[Crossref]
• [7] D.M. Tong, E. Sjöqvist, L.C. Kwek and C.H. Oh “Kinematic Approach to the Mixed State Geometric Phase in Nonunitary Evolution”, Phys. Rev. Lett., Vol. 93, (2004), pp. 080405–080408. [Crossref]
• [8] D.M. Tong, E. Sj"öqvist, S. Filipp, L.C. Kwek and C.H. Oh “Kinematic appreach to off-diagonal geometric phases of nondegenerate and degenerate mixed states”, Phys. Rev. A, Vol. 71, (2005), pp. 032106–032111. http://dx.doi.org/10.1103/PhysRevA.71.032106[Crossref]
• [9] X. X. Yi, L.C. Wang and T.Y. Zheng “Berry Phase in a Composite System”, Phys. Rev. Lett., Vol. 92, (2004), pp. 150406–150409. http://dx.doi.org/10.1103/PhysRevLett.92.150406[Crossref]
• [10] A. Hamma: “Berry Phases and Quantum Phase Transitions”, Preprint: arXiv:quantph/0602091.
• [11] A.C.M. Carollo and J.K. Pachos “Geometric Phases and Criticality in Spin-Chain Systems”, Phys. Rev. Lett., Vol. 95, (2005), pp. 157203–157206. http://dx.doi.org/10.1103/PhysRevLett.95.157203[Crossref]
• [12] S.-L. Zhu: “Scaling of Geometric Phases Close to the Quantum Phase Transition in the XY Spin Chain”, Phys. Rev. Lett., Vol. 96, (2006), pp. 077206–077209. http://dx.doi.org/10.1103/PhysRevLett.96.077206[Crossref]
• [13] Z. Tang and D. Finkelstein “Geometric Phase of Polarized Hydrogenlike Atoms in an External Magnetic Field”, Phys. Rev. Lett., Vol. 74, (1995), pp. 3134–3137. http://dx.doi.org/10.1103/PhysRevLett.74.3134[Crossref]
• [14] X.X. Yi and E. Sjöqvist “Effect of intersubsystem coupling on the geometric phase in a bipartite system”, Phys. Rev. A, Vol. 70, (2004), pp. 042104–042108. http://dx.doi.org/10.1103/PhysRevA.70.042104[Crossref]
• [15] E. Sjöqvist, X.X. Yi and J. Åberg “Adiabatic geometric phases in hydrogenlike atoms”, Phys. Rev. A, Vol. 72, (2005), pp. 054101–054104. http://dx.doi.org/10.1103/PhysRevA.72.054101[Crossref]
• [16] C.-T. Xu, M.-M. He and G. Chen “Berry phase of coupled two arbitrary spins in a time-varying magnetic field”, Chinese Physics, Vol. 15, (2006), pp. 912–914. http://dx.doi.org/10.1088/1009-1963/15/5/006[Crossref]
• [17] M.-L. Liang, S.-L. Shu and B. Yuan “Aharonov-Anandan phases for spin-spin coupling in a rotating magnetic field”, Physica Scripta, Vol. 75, (2007), pp. 138–141. http://dx.doi.org/10.1088/0031-8949/75/2/003[WoS][Crossref]
• [18] L. Xing “A new concept of geometric phase in parameter space: coupling as a parameter”, J. Phys. A, Vol. 39, (2006), pp. 9547–9555. http://dx.doi.org/10.1088/0305-4470/39/30/010[Crossref]
• [19] X.X. Yi, L.C. Wang and W. Wang “Geometric phase in dephasing systems”, Phys. Rev. A, Vol. 71, (2005), pp. 044101–044104. http://dx.doi.org/10.1103/PhysRevA.71.044101[Crossref]
• [20] J. Pachos, P. Zanardi and M. Razetti “Non-Abelian Berry connections for quantum computation”, Phys. Rev. A, Vol. 61, (2000), pp. 010305–010308. http://dx.doi.org/10.1103/PhysRevA.61.010305[Crossref]
• [21] G. Falci, R. Fazio, G.M. Palma, J. Siewert, and V. Vedral “Detection of geometric phases in superconducting nanocircuits”, Nature, Vol. 407, (2000), pp. 355–358. [Crossref]
• [22] L.M. Duan, J.I. Cirac and P. Zoller “Geometric Manipulation of Trapped Ions for Quantum Computation”, Science, Vol. 292, (2001), pp. 1695–1697. http://dx.doi.org/10.1126/science.1058835[Crossref]
• [23] R. Bhandari “Singularities of the Mixed State Phase”, Phys. Rev. Lett, Vol. 89, (2002), pp. 268901–268901; J.S. Anandan, E. Sjöqvist, A.K. Pati, A. Ekert, M. Ericsson, D.K.L. Oi and V. Vedral: “Reply: Singularities of the Mixed State Phase”, Phys. Rev. Lett, Vol. 89, (2002), pp. 268902-268902. http://dx.doi.org/10.1103/PhysRevLett.89.268901
• [24] S. Filipp and E. Sjöqvist “Off-Diagonal Geometric Phase for Mixed States”, Phys. Rev. Lett., Vol. 90, (2003), pp. 050403–050406. http://dx.doi.org/10.1103/PhysRevLett.90.050403[Crossref]

Identifiers

DOI

10.2478/s11534-007-0026-5