Geometric phase in coupled bipartite systems PRL 92,150406(2004),by X.X. yi etal H=ao, B()+J(o o2+hcD 13C 130 B(0=Bo(Sin 8 cos o, sin Asind, cos 0) Schematic of trichloroethylene a typical molecule used for QIP ·a2+hc
11 Geometric phase in coupled bipartite systems 1 1 2 1 ( ) ( . .) 2 H B t J h c + + = + + PRL 92,150406(2004),by X. X. Yi etal. Schematic of trichloroethylene, a typical molecule used for QIP. 0 B t B ( ) (sin cos ,sin sin ,cos ) = 1 2 . . z z + h c
0 Azimuthal angle yi Berry phase corresponding different eigenstates 915 00 00 8=2J/abo a rescaled coupling constant
12 0 g J B = 2 / A rescaled coupling constant Azimuthal angle i Berry phase corresponding different eigenstates
The geometric phase of the bipartite system is a sum over the one-particle geometric phase for the subsystems Schmidt decomposition Y()=VP, IE, ()@le, (0)2 For the composite system evolving adiabatically The one particle geometric phase ∑+2/2 of subsystem 1 (1)|e1(m)d Berry phase of the whole system iEOITE()dt
13 1 2 | ( ) | ( ) | ( ) j j j j Schmidt decomposition = t p E t e t For the composite system evolving adiabatically j j j j 1, 2, j j = + p p Berry phase of the whole system 0 ( ) | ( ) T j j i E t E t dt t 0 ( ) | ( ) T j j i e t e t dt t The ‘one particle geometric phase’ of subsystem 1 The geometric phase of the bipartite system is a sum over the one-particle geometric phase for the subsystems
Geometric phase in open systems We need a description of geometric phase for mixed tate There have been various proposals for example via state purification (A. UhImann, Lett. Math. Phys. 21, 229(1991) through an interferometric procedure: E. Sjoqvist et al PRL 200 -or by pertur ative expansion (Gamliel, D. and Freed, J H. PRA 39, 3238(1989) R. Whitney, and Y. Gefen, Phys. Rev. Lett. 90, 190402)
14 Geometric phase in open systems • We need a description of geometric phase for mixed state! There have been various proposals, for example: • -via state purification: (A. Uhlmann, Lett. Math. Phys. 21,229 (1991)) • -through an interferometric procedure: (E. Sjoqvist et al. PRL 2000), • -or by perturbative expansion: (Gamliel, D. and Freed, J. H. PRA 39, 3238(1989); R. Whitney, and Y. Gefen, Phys. Rev. Lett. 90, 190402 )