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

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: Adiabatic theorem: Geometric phase:

Well defined for a closed path

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.

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

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?

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 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.

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.

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

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

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

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, 090405 (2003)  

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, 100403 (2003)   M. Ericsson etal.,  Phys. Rev. Lett. 94, 050401 (2005)    The matrix represents an operator written in a basis spanned by two wave-packets, in external space.

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.

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

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)

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)

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

1 jump trajectory:

1 jump trajectory:

1 jump trajectory:

1 jump trajectory:

1 jump trajectory:

1 jump trajectory:

1 jump trajectory:

1 jump trajectory:

1 jump trajectory:

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:

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, 052103(2006)] Hamiltonian Initial state Final state with    Phys. Rev. Lett. 93, 080405

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

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, 157203 2005; S. L. Zhu, Phys. Rev. Lett. 96, 077206 2006; On the geometric phase induced in auxiliary particle by the Many-body sys. X. X. Yi etal. ,Phys. Rev. A 75, 032103(2007)

Geometric phase and QPTs Case 1 satisfy Case 2 satisfy

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

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;

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

Thanks

Papers worth reading Phys Rev A 73, 012107(2006), by P. Zanardi etal. Quant-ph/0608237, by E. Sjoqvist etal.