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arXiv:0705.3566 · Physical Review A

Experimental observation of a topological phase in the maximally entangled state of a pair of qubits

Jiangfeng Du, Jing Zhu, Mingjun Shi, Xinhua Peng, Dieter Suter

Published Oct 22, 2007• 42 citations
arXivHTMLPDF

Abstract

Quantum mechanical phase factors can be related to dynamical effects or to the geometrical properties of a trajectory in a given space---either parameter space or Hilbert space. Here, we experimentally investigate a quantum mechanical phase factor that reflects the topology of the SO(3) group: since rotations by $\ensuremath{\pi}$ around antiparallel axes are identical, this space is doubly connected. Using a pair of nuclear spins in a maximally entangled state, we subject one of the spins to a cyclic evolution. If the corresponding trajectory in SO(3) can be smoothly deformed to a point, the quantum state at the end of the trajectory is identical to the initial state. For all other trajectories the quantum state changes sign.

DOI
10.1103/physreva.76.042121

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Quantum mechanical phase factors can be related to dynamical effects or to the geometrical properties of a trajectory in a given space---either parameter space or Hilbert space. Here, we experimentally investigate a quantum mechanical phase factor that reflects the topology of the SO(3) group: since rotations by \ensuremathπ\ensuremath{\pi}\ensuremathπ around antiparallel axes are identical, this space is doubly connected. Using a pair of nuclear spins in a maximally entangled state, we subject one of the spins to a cyclic evolution. If the corresponding trajectory in SO(3) can be smoothly deformed to a point, the quantum state at the end of the trajectory is identical to the initial state. For all other trajectories the quantum state changes sign.
Entangled states play a crucial role in quantum physics, ranging from fundamental aspects to quantum information processing. We show here that entangled two-qubit states can also be used to characterize unambiguously the subtlety of the SO(3) rotation group topology. The well known two distinct families of path in this group are put in one-to-one correspondence with cyclic evolutions of these entangled states, resulting in a pi phase difference. We propose a simple quantum optics interference experiment to demonstrate this topological phase shift.
A representation of the SO(3)\mathrm{SO}(3)SO(3) group is mapped into a maximally entangled two qubit state according to the literature. To show the evolution of the entangled state, a model is set up on an maximally entangled electron pair, two electrons of which pass independently through a rotating magnetic field. It is found that the evolution path of the entangled state in the SO(3)\mathrm{SO}(3)SO(3) sphere breaks an odd or even number of times, corresponding to the double connectedness of the SO(3)\mathrm{SO}(3)SO(3) group. An odd number of breaks leads to an additional $\ensuremath{\pi}$ phase to the entangled state, but an even number of breaks does not. A scheme to trace the evolution of the entangled state is proposed by means of entangled photon pairs and Kerr medium, allowing observation of the additional $\ensuremath{\pi}$ phase.
We make a geometric study of the phases acquired by a general, pure bipartite two-level system after a cyclic unitary evolution. The geometric representation of the two particle Hilbert space makes use of Hopf fibrations. It allows for a simple description of the dynamics of the entangled state's phase during the whole evolution. The global phase after a cyclic evolution is always an entire multiple of $\ensuremath{\pi}$ for all bipartite states, a result that does not depend on the degree of entanglement. There are three different types of phases combining themselves so as to result in the $n\ensuremath{\pi}$ global phase. They can be identified as dynamical, geometrical, and topological. Each one of them can be easily identified using the presented geometric description. The interplay between them depends on the initial state and on its trajectory, and the results obtained are shown to be in connection to those on mixed state phases.
We discuss the representation of the SO(3)SO(3)SO(3) group by two-qubit maximally entangled states (MES). We analyze the correspondence between SO(3)SO(3)SO(3) and the set of two-qubit MES which are experimentally realizable. As a result, we offer a new interpretation of some recently proposed experiments based on MES. Employing the tools of quantum optics we treat in terms of two-qubit MES some classical experiments in neutron interferometry, which showed the πππ-phase accrued by a spin-1/21/21/2 particle precessing in a magnetic field. By so doing, we can analyze the extent to which the recently proposed experiments - and future ones of the same sort - would involve essentially new physical aspects as compared with those performed in the past. We argue that the proposed experiments do extend the possibilities for displaying the double connectedness of SO(3)SO(3)SO(3), although for that to be the case it results necessary to map elements of SU(2)SU(2)SU(2) onto physical operations acting on two-level systems.
We discuss the appearance of fractional topological phases on cyclic evolutions of entangled qudits. The original result reported by Oxman and Khoury [Phys. Rev. Lett. 106, 240503 (2011)] is detailed and extended to qudits of different dimensions. The topological nature of the phase evolution and its restriction to fractional values are related to both the structure of the projective space of states and entanglement. For maximally entangled states of qudits with the same Hilbert-space dimension, the fractional geometric phases are the only ones attainable under local SU(ddd) operations, an effect that can be experimentally observed through conditional interference.