大连理工大学：《物理学》课程教学资源（PPT讲稿）Geometric Phase in composite systems

Geometric Phase in composite systems .X.yⅰ Department of Physics, Dalian University of Technology With C. H. Oh, L. C Kwek, D. M. Tong e NUS Erik Sjoqvist e Uppsala Univ H. T. Cui, L. C. Wang, X L Huang e Dalian Univ of tech

Geometric Phase in composite systems 衣学喜 X. X. Yi Department of Physics, Dalian University of Technology With C. H. Oh, L. C. Kwek, D. M. Tong @ NUS; Erik Sjoqvist @ Uppsala Univ.; H. T. Cui, L. C. Wang, X. L. Huang @ Dalian Univ. of Tech

Outline Why study geometric phase Geometric phase in bipartite systems Geometric phase in open systems Geometric phase in dissipative systems Geometric phase in dephasing systems Geometric phase and QPTs Conclusion

2 Outline • Why study geometric phase • Geometric phase in bipartite systems • Geometric phase in open systems – Geometric phase in dissipative systems – Geometric phase in dephasing systems – Geometric phase and QPTs • Conclusion

Why study geometric phase Classical counterpart of berry Phase it is connected to the intrinsic curvature of the sphere Parallel transport 4 http://www.mi.infmit/manini/berryphase.html

3 Why study geometric phase Classical counterpart of Berry Phase; it is connected to the intrinsic curvature of the sphere. Parallel transport http://www.mi.infm.it/manini/berryphase.html

Parameter dependent system: H(n) 41(2,|w(2)}1() Adiabatic theorem: ()=v(()ee Geometric phase d (vn in y

4  , |     n n ( ) ( )  ( ) ( ( )) ( ) 0 / | | t n n i dt i t n t t e e     −  −   =  0 t n n n d i         =   Geometric phase: Adiabatic theorem: Parameter dependent system: H ( )    n ( ) 

Well defined for a closed path ,=乎aww

5 n n n C     d i   =   Well defined for a closed path x  y C

Why study geometric phase? It is an interesting phenomenon of Quantum mechanics, which can be observed in many physical systems It has interesting properties that can be exploited to increase the robustness of Quantum Computation Geometric Quantum Computation

6 Why study geometric phase? • It is an interesting phenomenon of Quantum mechanics, which can be observed in many physical systems... • It has interesting properties that can be exploited to increase the robustness of Quantum Computation: “Geometric Quantum Computation

Geometric quantum computation 100)->00)01〉-01) 10)〉10),1)-1) U =eXpRE(tdt Dynamical evolution Geometric phase Geometric gates can be more robust against different sources of noise

7 exp[ ( ) ]  U = −i E t dt i Dynamical evolution Geometric phase Geometric gates can be more robust against different sources of noise | 00 | 00 ,| 01 | 01 , |10 |10 ,|11 |11 ,  →   →   →   → −   g Geometric quantum computation

Why geometric phase robust? Geometric phase is robust against classical fluctuation of the phase(of the first order)see for example G. D. Chiara. G. M. Palma, Phys. Rev Let.91,090404(2003) It is independent of systematic errors

8 Why geometric phase robust? • Geometric phase is robust against classical fluctuation of the phase (of the first order) see for example: G. D. Chiara, G. M. Palma, Phys. Rev. Lett. 91, 090404 (2003). ◼It is independent of systematic errors

Geometric phase in bipartite systems Almost all systems in Qip are composite If entanglement change Berry's phase what role the inter-subsystem couplings may play

9 Geometric phase in bipartite systems • Almost all systems in QIP are composite. • If entanglement change Berry’s phase. • What role the inter-subsystem couplings may play?

Berry' s phase of entangled spin pair E. Sjoqvist, Pra 62, 022109(2000 without inter-subsystem coupling) Separable pair, total BP=sum over individual BP For entangled pair, with one particle driven by the external magnetic field Initial state|H()≥=c0s个+sia geometric phase O. =T(cos a cos 8-1) B 6=arc cos[cos a cos g]

10 Berry’s phase of entangled spin pair E. Sjoqvist, PRA 62, 022109(2000)(without inter-subsystem coupling) • Separable pair, total BP=sum over individual BP. • For entangled pair, with one particle driven by the external magnetic field | (0) cos | sin | 2 2   •Initial state   =  +  (cos cos 1)     g = − •Geometric phase    B  S  '    = arc cos[cos cos ]