In analogy with Bell's inequality for two-qubit quantum states, we propose an inequality criterion for the nonseparability of the spin-orbit degrees of freedom of a laser beam. A definition of separable and nonseparable spin-orbit modes is used in consonance with the one presented in Phys. Rev. Lett. 99, 160401 (2007). As the usual Bell's inequality can be violated for entangled two-qubit quantum states, we show both theoretically and experimentally that the proposed spin-orbit inequality criterion can be violated for nonseparable modes. The inequality is discussed in both the classical and quantum domains.

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.

We present a study of orbital angular momentum transfer from pump to down-converted beams in a type-II Optical Parametric Oscillator. Cavity and anisotropy effects are investigated and demostrated to play a central role in the transverse mode dynamics. While the idler beam can oscillate in a Laguerre-Gauss mode, the crystal birefringence induces an astigmatic effect in the signal beam that prevents the resonance of such mode.

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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 an all-optical setup, which couples different degrees of freedom of a single photon, to investigate entanglement generation by a common environment. The two qubits are represented by the photon polarization and Hermite-Gauss transverse modes, while the environment corresponds to the photon path. For an initially two-qubit separable state, the increase of entanglement is analyzed, as the probability of an environment-induced transition ranges from zero to one. An entanglement witness that is invariant throughout the evolution of the system yields a direct measurement of the concurrence of the two-qubit state.Comment: 11 pages, 3 figure

We report on an experiment demonstrating the conservation of the orbital angular momentum in stimulated down-conversion. It has been demonstrated that the orbital angular momentum is not transferred to the individual beams of the spontaneous down-conversion. It is also known that it is conserved when twin photons are taken individually. We observe the conservation law for an individual beam of the down-conversion through cavity-free stimulated emission. The cavity-free stimulated parametric down-conversion was first studied by Mandel and co-workers ͓1,2͔ and more recently it has been explored by other authors ͓3-5͔. One important aspect of this process is its connection with the spontaneous parametric down-conversion, where entangled states for two photons can be easily prepared. Signals obtained in stimulated down-conversion are much larger than those obtained in the spontaneous process and carry information about the details of the parametric interaction, such as phase matching conditions. This information is preserved thanks to the stimulation without optical cavities, where the optical mode properties are determined mainly by the cavity configuration. Therefore, stimulated down-conversion is a useful tool for understanding entanglement properties of the twin photons from the parametric down-conversion. We have recently demonstrated the transfer of coherence and images from the pump and auxiliary lasers to the stimulated downconversion field ͓5͔, in direct connection with the analogous process in the context of the quantum correlations observed in coincidence measurements ͓6͔.The possibility of preparing entangled photons in different degrees of freedom has also become a subject of interest. In particular, the orbital angular momentum ͑OAM͒ of the light has been studied in the context of classical ͓7͔ and quantum optics ͓8͔. Conservation of OAM in the up-conversion process ͓9,10͔, optical pumping of cold atoms ͓11͔, and quantum entanglement ͓8͔ have been observed experimentally for this degree of freedom. However, in the spontaneous parametric down-conversion process, the OAM is not transferred from the pump to each individual signal or idler beam ͓12͔. This is a consequence of the fact that signal and idler beams are incoherent when considered individually ͓13͔.In this work, we observe experimentally the manifestation of the conservation law for the OAM in the stimulated downconversion process, for the idler beam. In this case, besides the pump, a second auxiliary laser is aligned with one of the down-conversion modes, inducing emission. Conservation of the topological charge can be written as m p ϭm s ϩm i , where p, s, and i stands for pump, signal, and idler, respectively.Light beams with OAM can be described by LaguerreGauss LG l,m modes, where l and m are radial and azimuthal mode numbers. Under the paraxial approximation the angular momentum of a light beam can be separated into orbital and spin contributions ͓14͔ and the OAM is given by mប per photon ͓7͔. Therefore, the conservation of topological cha...