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...

We report a simple quantum-key-distribution experiment in which Alice and Bob do not need to share a common polarization direction in order to send information. Logical qubits are encoded into nonseparable states of polarization and first-order transverse spatial modes of the same photon.

It is possible to prepare classical optical beams which cannot be characterized by a tensor product of vectors describing each of their degrees of freedom. Here we report the experimental creation of such a non-separable, tripartite GHZ-like state of path, polarization and transverse modes of a classical laser beam. We use a Mach-Zehnder interferometer with an additional mirror and other optical elements to perform measurements that violate Mermin's inequality. This demonstration of a classical optical analogue of tripartite entanglement paves the path to novel optical applications inspired by multipartite quantum information protocols. PACS numbers:A composite quantum system is said to be entangled when it is not fully described by the state of its components [1]. Besides indicating a departure from classical physics, entangled states represent an important resource for a number of quantum information protocols [2]. In classical optics, the mode structure associated with different degrees of freedom of the wave field can also be described by complex vector spaces. As examples, an arbitrary polarization can be written as a complex superposition of circularly polarized beams, and the spatial configuration of a paraxial beam can be decomposed in terms of Laguerre-Gaussian beams. These degrees of freedom can be represented on two independent Poincaré spheres [3], in complete analogy with the Bloch sphere used to represent qubit states [2]. Intriguingly, also in classical optics there are field configurations which cannot be described as a tensor product of definite modes of each individual degree of freedom of the system [4]. These non-separable structures display a classical analogue of quantum entanglement [5][6][7][8]. 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]. Nonseparable structures have also proved their utility in the quantum optical domain [22][23][24][25][26][27][28][29][30][31][32][33]. Analogously to its quantum counterpart, classical entanglement has been characterized via the violation of Bell-like inequalities [34][35][36].Composite quantum systems may have more than two parts. For tripartite systems, Mermin [37] simplified an earlier argument by Greenberger, Horne and Zeilinger * Corresponding author.[38], to show that any local hidden-variable theory for tripartite systems must satisfy (1) where Z, Y, Z represent the Pauli operators. This inequality is violated by the so-called GHZ-Mermin state:for which ZZZ = +1 and ZXX = XZX = XXZ = −1, resulting in M = 4, the maximum algebraic violation of Mermin's inequality (1). In [39], Spreeuw proposed a scheme in which t...

We propose and demonstrate experimentally a single lens design for an astigmatic mode converter that transforms the transverse mode of paraxial optical beams. As an application, we implement a controlled-not gate based on a Michelson interferometer in which the photon polarization is the control bit and the first order transverse mode is the target. As a further application, we also build a transverse mode parity sorter which can be useful for quantum information processing as a measurement device for the transverse mode qubit.

We demonstrate second harmonic generation performed with optical vortices with different topological charges imprinted on orthogonal polarizations. Besides the intuitive charge doubling, we implement arbitrary topological charge addition on the second harmonic field using polarization as an auxiliary parameter.Besides their intrinsic beauty, optical beams carrying orbital angular momentum (OAM) have proved to be a powerful tool for encoding and processing quantum information. First order Laguerre-Gaussian and Hermite-Gaussian (HG) modes carry the mathematical structure of a qubit [1] and allow for a two-qubit encoding when combined with polarization of a single photon. The interplay between the two degrees of freedom leads to interesting applications, including topological phases [2,3], quantum cryptography [4][5][6], Bell inequalities [7-9], quantum logic gates [10][11][12], and quantum teleportation [13][14][15]. Polarization controlled spatial correlations between entangled photon pairs were first demonstrated in [16,17]. Nowadays, a

We investigate the transverse mode structure of the down-converted beams generated by a type-II optical parametric oscillator (OPO) driven by a structured pump. Our analysis focus on the selection rules imposed by the spatial overlap between the transverse modes of the three fields involved in the non-linear interaction. These rules imply a hierarchy of oscillation thresholds that determine the possible transverse modes generated by the OPO, as remarkably confirmed with experimental results.

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The topological phase acquired by vector vortex optical beams is investigated. Under local unitary operations on their polarization and transverse degrees of freedom, the vector vortices can only acquire discrete geometric phase values, 0 or π, associated with closed paths belonging to different homotopy classes on the SO(3) manifold. These discrete values are demonstrated through interferometric measurements, and the spin-orbit mode separability is associated to the visibility of the interference patterns. The local unitary operations performed on the vector vortices involved both polarization and transverse mode transformations with birefringent wave plates and astigmatic mode converters. The experimental results agree with our theoretical simulations and generalize our previous results obtained with polarization transformations only.