Abstract
We calculate Berry phases and entanglement of adiabatic states for a two spin-1/2 system described by the Heisenberg model with Dzyaloshinski-Moriya (DM) interaction; one of the spins is driven by a time-varing rotating magnetic field and the other is coupled with a static magnetic field. This static magnetic field can be used for controlling as well as vanishing the Berry phases and entanglement of the system state. Besides, we show that the Berry phase and entanglement are not always exact but useful to detect energy levels approach. Additionally, we find that a nontrivial two-spin unitary transformation, purely based on Berry phases, can be obtained by using two consecutive cycles with the opposite direction of the static magnetic field, opposite signs of the exchange constant as well as DM interaction, and a phase shift of the rotating magnetic field. This unitary transformation presents a two-qubit geometric phase gate.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Berry M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45 (1984)
Aharonov Y., Anandan J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596 (1987)
Samuel J., Bhandari B.: General setting for Berry’s phase. Phys. Rev. Lett. 60, 2339–2342 (1988)
Wilczek F., Zee A.: Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984)
Uhlmann A.: Parallel transport and quantum holonomy along density operators. Rep. Math. Phys. 24, 229–240 (1986)
Uhlmann A.: A gauge field governing parallel transport along mixed states. Lett. Math. Phys. 21, 229–236 (1991)
Sjöqvist E., Pati A.K., Ekert A., Anandan A., Ericsson E., Oi D.K.L., Vedral V.: Geometric phases for mixed states in interferometry. Phys. Rev. Lett. 85, 2845–2849 (2000)
Layton E., Huang Y.H., Chu S.I.: Cyclic quantum evolution and Aharonov-Anandan geometric phases in SU(2) spin-coherent states. Phys. Rev. A 41, 42–48 (1990)
Lai Y.Z., Liang J.Q., Muller-Kirsten H.J.W., Zhou J.G.: Time-dependent quantum systems and the invariant Hermitian operator. Phys. Rev. A 53, 3691–3693 (1996)
Yan F.L., Yang L.J., Li B.Z.: Invariant Hermitian operator and geometric phase for the Heisenberg spin system in a time-dependent magnetic field. Phys. Lett. A 259, 207–211 (1999)
Yang L.G., Yan F.L.: The area theorem of the Berry phase for the time-dependent externally driven system. Phys. Lett. A 265, 326–330 (2000)
Yang L.G., Yan F.L.: Berry phase for many-spin system with the uniaxial anisotropic exchange interaction in a time-dependent magnetic field. Phys. Lett. A 298, 73–77 (2002)
Zhu S.L., Wang Z.D., Zhang Y.D.: Nonadiabatic noncyclic geometric phase of a spin-1/2 particle subject to an arbitrary magnetic field. Phys. Rev. B 611, 1142–1148 (2000)
Sjöqvist E.: Geometric phase for entangled spin pairs. Phys. Rev. A 62, 022109 (2000)
Tong D.M., Sjöqvist E., Kwek L.C., Oh C.H., Ericsson M.: Relation between geometric phases of entangled bipartite systems and their subsystems. Phys. Rev. A 68, 022106 (2003)
Ge X.Y., Wadati M.: Geometric phase of entangled spin pairs in a magnetic field. Phys. Rev. A 72, 052101 (2005)
Yi X.X., Wang L.C., Zheng T.Y.: Berry phase in a composite system. Phys. Rev. Lett. 92, 150406 (2004)
Sun H.Y., Wang L.C., Yi X.X.: Berry phase in a bipartite system with general subsystem-subsystem couplings. Phys. Lett. A 370, 119–122 (2007)
Li X.: Interacting spin pairs in rotational magnetic fields and geometric phase. Phys. Lett. A 372, 4980–4984 (2008)
Oh S.: Geometric phases and entanglement of two qubits with XY type interaction. Phys. Lett. A 373, 644–647 (2009)
Jones J.A., Verdral V., Ekert A., Castagnoli G.: Geometric quantum computation using nuclear magnetic resonance. Nature(London) 403, 869–871 (1999)
Duan L.M., Cirac J.I., Zoller P.: Geometric manipulation of trapped ions for quantum computation. Science 292, 1695–1697 (2001)
Zhu S.L., Wang Z.D.: Unconventional geometric quantum computation. Phys. Rev. Lett. 91, 187902 (2003)
Shao L.B., Wang Z.D., Xing D.Y.: Implementation of quantum gates based on geometric phases accumulated in the eigenstates of periodic invariant operators. Phys. Rev. A 75, 014301 (2007)
Wang Z.S., Wu C.F., Feng X.L., Kwek L.C., Lai C.H., Oh C.H., Vedral V.: Nonadiabatic geometric quantum computation. Phys. Rev. A 76, 044303 (2007)
Sjöqvist E.: A new phase in quantum computation. Physics 1, 35 (2008)
Macchiavello C., Palma G.M., Zeilinger A.: Quantum Computation and Quantum Information Theory. World Scientific, Singapore (2000)
Zhou Y., Zhang G.F.: Geometric phase of a bipartite system with DzyaloshinskiGMoriya interaction. Opt. Commun. 281, 5278–5281 (2008)
Zheng S.B., Guo G.C.: Efficient scheme for two-atom entanglement and quantum information processing in cavity QED. Phys. Rev. Lett. 85, 2392–2395 (2000)
Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Communication. Cambridge University Press, Cambridge (2000)
Bouwmeester D., Pan J.W.: Experimental quantum teleportation. Nature(London) 390, 575–579 (1997)
Deutsch D., Jozsa R.: Rapid solution of problems by quantum computation. Proc. R. Soc. London A 439, 553–558 (1992)
Loss D., DiVincenzo D.P.: Quantum computation with quantum dots. Phys. Rev. A 57, 120–126 (1998)
DiVincenzo D.P.: Two-bit gates are universal for quantum computation. Phys. Rev. A 51, 1015–1022 (1995)
Coish, W.A., Loss, D., http://arxiv.org/abs/cond-mat/0606550; Burkard, G., Loss, D., DiVincenzo, D.P.: Coupled quantum dots as quantum gates. Phys. Rev. B 59, 2070–2078 (1999)
Kheirandish F. et al.: Effect of spin-orbit interaction on entanglement of two-qubit Heisenberg XYZ systems in an inhomogeneous magnetic field. Phys. Rev. A 77, 042309 (2008)
DiVincenzo D.P., Bacon D., Kempe J., Burkard G., Whaley K.B.: Universal quantum computation with the exchange interaction. Nature 408, 339–342 (2000)
Bennett C.H., DiVincenzo D.P., Smolin J., Wootters W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996)
Dzyaloshinskii I.: A thermodynamic theory of weak ferromagnetism of antiferromagnetics. J. Phys. Chem. Solid 4, 241–255 (1958)
Moriya T.: New mechanism of anisotropic superexchange interaction. Phys. Rev. Lett. 4, 228–230 (1960)
Zhu S.-L.: Scaling of geometric phases close to the quantum phase transition in the XY spin chain. Phys. Rev. Lett. 96, 077206 (2006)
Oh S., Huang Z., Peskin U., Kais S.: Entanglement, Berry phases, and level crossings for the atomic Breit-Rabi Hamiltonian. Phys. Rev. A 78, 062106 (2008)
Ekert A., Ericsson M., Hayden P., Inamori H., Jones J.A., Oi D.K.L., Vedral L.: Geometric quantum computation. J. Mod. Opt. 47, 2501–2513 (2000)
Shi Y.: Geometric vs. dynamical gates in quantum computing implementations using Zeeman and Heisenberg Hamiltonians. Europhys. Lett. 83, 50002 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Amniat-Talab, M., Rangani Jahromi, H. On the entanglement and engineering phase gates without dynamical phases for a two-qubit system with Dzyaloshinski-Moriya interaction in magnetic field. Quantum Inf Process 12, 1185–1199 (2013). https://doi.org/10.1007/s11128-012-0463-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-012-0463-y