Topological Phase for Spin-Orbit Transformations on a Laser Beam

Abstract

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.PACS numbers: PACS: 03.65. Vf, 03.67.Mn, 07.60.Ly, 42.50.Dv The seminal work by S. Pancharatnam [1] introduced for the first time the notion of a geometric phase acquired by an optical beam passing through a cyclic sequence of polarization transformations. A quantum mechanical parallel for this phase was later provided by M. Berry [2]. Recently, the interest for geometric phases was renewed by their potential applications to quantum computation. The experimental demonstration of a conditional phase gate was recently provided both in nuclear magnetic ressonance [3] and trapped ions [4]. Another optical manifestation of geometric phase is the one acquired by cyclic spatial mode conversions of optical vortices. This kind of geometric phase was first proposed by van Enk [5] and recently found a beautiful demonstration by E. J. Galvez et al [6].The Hilbert space of a single qubit admits an useful geometric representation of pure states on the surface of a sphere. This is the Bloch sphere for spin 1/2 particles or the Poincaré sphere for polarization states of an optical beam. A Poincaré sphere representation can also be constructed for the first order subspace of the spatial mode structure of an optical beam [7]. Therefore, in the quantum domain, we can attribute two qubits to a single photon, one related to its polarization state and another one to its spatial structure. Geometrical phases of a cyclic evolution of the mentioned states can be beautifully interpreted in such representations as being related to the solid angle of a closed trajectory. However, in order to compute the total phase gained in a cyclic evolution, one should also consider the dynamical phase. When added to the geometrical phase, it leads to a total phase gain of π after a cyclic trajectory. This phase has been put into evidence for the first time using neutron interference [8]. The appearence of this π phase is due to the double connectedness of the three dimensional rotation group SO(3). However, in the neutron experience, only two dimensional rotations were used, and this topological property of SO(3) was not unambiguously put into evidence, as explained in details in [9,10].As discussed by P. Milman and R. Mosseri [9,11], when the quantum state of two qubits is considered, the mathematical structure of the Hilbert space becomes richer and the phase acquired through cyclic evolutions demands a more careful inspection. The naive sum of independent phases, one for each qubit, is applicable only for product states. In...

Topological phase structure of entangled qudits

“…The role of entanglement in the phase evolution of two-qubit systems was investigated in refs. [13,14], and the topological nature of the corresponding geometric phases was investigated both theoretical [15][16][17] and experimentally in the context of spin-orbit transformations on a paraxial laser beam [18] and in nuclear magnetic resonance [19]. In a recent work, we investigated the crucial role played by the dimension of the Hilbert space on the topological phases acquired by entangled qudits [20,21].…”

Orbital-angular-momentum mixing in type-II second-harmonic generation

“…In the quantum domain, qubits and qudits can be encoded on Laguerre-Gaussian (LG) or Hermite-Gaussian (HG) modes that combined with the photon polarization allows creation of entanglement between internal photonic degrees of freedom. Many works have been devoted to schemes for implementing and applying this spin-orbit coupling [2][3][4][5][6][7][8][9][10][11]. Beyond the intrinsic beauty of this subject, one may find interesting applications to quantum information tasks like optical communication [12,13], teleportation schemes [14][15][16][17][18][19][20], alignment free quantum cryptography [21][22][23], controlled gates for quantum computation [24,25], quantum simulations [26,27] and metrology [28][29][30].…”

Tripartite nonseparability in classical optics

“…One example are vector vortex beams, which are non-separable superpositions of transverse modes and polarization states of a laser beam [9][10][11]. This analogy was used to demonstrate the topological phase acquired by entangled states evolving under local unitary operations [12]. Recently, it has attracted a growing interest due both to the fundamental aspects involved, but also for potential applications to classical optical information processing [13][14][15][16][17][18][19][20].…”