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The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting

Exprimental Demonstration of the Topological Phase for Entangled Qubits

C. E. R. Souza, J. A. O. Huguenin, P. Milman, Antonio Zelaquett Khoury

Published Jan 1, 2007• 0 citations
Publisher

Abstract

We investigate the topological phase associated with the SO(3) representation in terms of maximally entangled states. An experimental demonstration of this topological phase is provided for polarization and spatial mode transformations of a laser beam.
DOI
10.1364/cqo.2007.jwc16

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We investigate the topological phase associated with the double connectedness of the SO(3) representation in terms of maximally entangled states. An experimental demonstration is provided in the context of polarization and spatial mode transformations of a laser beam carrying orbital angular momentum. The topological phase is evidenced through interferometric measurements, and a quantitative relationship between the concurrence and the fringes visibility is derived. Both the quantum and the classical regimes were investigated.
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.
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.
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.
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.