1. 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. NUS;
Erik Uppsala Univ.;
H. T. Cui, L. C. Wang, X. L. Dalian Univ. of Tech.

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

3. Why study geometric phase
Classical counterpart of Berry Phase;
it is connected to the intrinsic
curvature of the sphere.
Parallel transport

4. Parameter dependent system:
Adiabatic theorem:
Geometric phase:

5. Well defined for a closed path

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”
A-B effect, A-C effect, quantum Hall effect etc.

7. Geometric quantum computation
Dynamical evolution
Geometric phase
Geometric gates can be more robust against different sources of noise

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, (2003).
It is independent of systematic errors.

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?

10. Berry’s phase of entangled spin pair
E. Sjoqvist, PRA 62, (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
Geometric phase
The Hamiltonian which governs the evolution of the system is H=\omega_1\sigma_1^z+ \omega_2\sigma_2^z; I.e., the free Hamiltonian
Of the system, both driven by a time-dependent magnetic field.

11. Geometric phase in coupled bipartite systems
PRL 92,150406(2004),by X. X. Yi etal.
Trichloroethylene: 三氯乙烯
Schematic of trichloroethylene, a typical molecule used for QIP.

12. A rescaled coupling constant
Azimuthal angle
Berry phase corresponding
different eigenstates
A rescaled coupling constant

13. phase for the subsystems
The geometric phase of the bipartite system is a sum over the one-particle geometric
phase for the subsystems
Schmidt decomposition
For the composite system evolving adiabatically
The ‘one particle geometric phase’ of subsystem 1
Berry phase of the whole system

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, )
In most of the cases these definitions do not agree on account of different constraints imposed,
namely different generalizations of the parallel transport condition.

15. Via state purification
The key idea in Uhlmann’s approach to the mixed state geometric phase is to lift the system’s density operator acting on the Hilbert space to an extended Hilbert space by attaching an ancilla.
Then the geometric phase depends on the dynamics of the ancilla.
Mixed State Geometric Phases, Entangled Systems,
and Local Unitary Transformations Marie Ericsson, Arun K. Pati, Erik Sjöqvist, Johan Brännlund, and Daniel K. L. Oi Phys. Rev. Lett. 91, (2003)  

16. through an interferometric procedure
For unitary evolution (PRL 85,2845(2000))
This definition is based on interferometry
Mirror,
Beam-splitters,
Phase shift
Experiment:
J. Du et al.
Phys. Rev. Lett. 91, (2003)  
M. Ericsson etal.,  Phys. Rev. Lett. 94, (2005)   
The matrix represents an operator written in a basis spanned by two wave-packets, in external space.

17. By perturbative expansion
Initial state
B(t)
Probability to state is
(a) The field is (instantaneously) prepared at its initial value,
which is at angle theta to the z axis. At the same
time the spin is (instantaneously) placed in the above initial state
(b) we adiabatically rotate the magnetic field clockwise,
n times around a closed loop
(c) the spin is flipped and
(d) the field is rotated anticlockwise, the same times n .
(e) the spin is flipped again and (f) the spin state is measured.

18. Do we really need to define geometric phase for mixed states?
The Quantum jump approach
A system interacting with an enviroment the state evolves according to the master equation:
By dividing the time into small steps Δt, the evolution can be cast into this form:
where ( ) å + = D i W t r
“No-jump” operator
i-th jump operator
A.C., M. Franca Santos, I. Fuentes-Guiridi, V. Vedral, PRL 2003

19. If the system starts from a pure state Ψo
the evolution of the open system can be pictured as a “tree” of trajectories of pure states:
At each time step tn=nΔt, the effect of the environment can be regarded as the action of the jump operator Wi with probability pi:
The master equation evolution is recovered by averaging the trajectories according to their probabilities: pi(1)pi(2)…pi(n)

20. Geometric phase is well defined for any sequence of pure states
(Pancharatnam formula):
Using Pancharatnam formula we can easily calculate the geometric phase for each trajectory:
(The area enclosed inside the trajectory + geodesic joining final and initial states)

21. Using this quantum jump approach is possible to show that under specific kind of decoherence models, the geometric phase is robust.
Example: Decoherence for a 2 level system:
Suppose that the systems evolves under the Hamiltonian:
In case of no jump, after a time T=2π/ω
the geometric phase is:
In case of 1 jump, after the same time T=2π/ω
the geometric phase remain the same

22. 1 jump trajectory:

23. 1 jump trajectory:

24. 1 jump trajectory:

25. 1 jump trajectory:

26. 1 jump trajectory:

27. 1 jump trajectory:

28. 1 jump trajectory:

29. 1 jump trajectory:

30. 1 jump trajectory:

31. In case of spontaneous decay:
This result remain the same in case of a general number N of jumps, in case of dephasing
Then the geometric phase is independent of the number of jumps!!!
In case of spontaneous decay:
Bad news: For any jump the geometric phase is completely spoiled.
Good news: We can monitor spontaneous decay
In case of no jump:

32. For dephasing system [ X. X. Yi etal, New J. Phys. (2005) ;Phys. Rev
For dephasing system [ X. X. Yi etal, New J. Phys. (2005) ;Phys. Rev. A 73, (2006)]
Hamiltonian
Initial state
Final state
with   
Phys. Rev. Lett. 93,

34. For dissipative bipartite systems
Master equation
Berry’s phase without any quantum jump
Cyclic is defined by H(X)
Biorthogonal to each other

38. Geometric phase and QPTs
On the geometric phase of the many-body system itself
A. C. M. Carollo and J. K. Pachos, Phys. Rev. Lett. 95, ;
S. L. Zhu, Phys. Rev. Lett. 96, ;
On the geometric phase induced in auxiliary
particle by the Many-body sys.
X. X. Yi etal. ,Phys. Rev. A 75, (2007)

39. Geometric phase and QPTs
Case 1
satisfy
Case 2
satisfy

40. Geometric phase and QPTs
A spin-1/2 coupled to a XY spin chain

41. Geometric phase and QPTs
Geometric phase as a function of . The figure was plotted for N=1245 sites, g=0.1 and
\gamma= a 0.2, b 0.5, c 0.8, and d 1.0;

42. Conclusions
The inter-system coupling would wash out the Berry phase of the composite system
The Berry phases of the subsystem add up to the Berry phase of the composite system
The geometric phase change due to the coupling with the bath depends sharply on the initial state

43. Thanks

44. Papers worth reading Phys Rev A 73, 012107(2006), by P. Zanardi etal.
Quant-ph/ , by E. Sjoqvist etal.