Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons.

Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, an observable that rapidly approaches a value proportional to the Zak phase during the free evolution of the system. Then we measure the Zak phase in a photonic quantum walk of twisted photons, by observing the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent timeframe and combine the two windings to characterize the full phase diagram of this Floquet system. Finally, we prove the robustness of the measure by introducing dynamical disorder in the system. This detection method is extremely general and readily applicable to all present one-dimensional platforms simulating static or Floquet chiral systems.

Statistical moments of quantum-walk dynamics reveal topological quantum transitions.

Many phenomena in solid-state physics can be understood in terms of their topological properties. Recently, controlled protocols of quantum walk (QW) are proving to be effective simulators of such phenomena. Here we report the realization of a photonic QW showing both the trivial and the non-trivial topologies associated with chiral symmetry in one-dimensional (1D) periodic systems. We find that the probability distribution moments of the walker position after many steps can be used as direct indicators of the topological quantum transition: while varying a control parameter that defines the system phase, these moments exhibit a slope discontinuity at the transition point. Numerical simulations strongly support the conjecture that these features are general of 1D topological systems. Extending this approach to higher dimensions, different topological classes, and other typologies of quantum phases may offer general instruments for investigating and experimentally detecting quantum transitions in such complex systems.

Quantum walks and wavepacket dynamics on a lattice with twisted photons.

The "quantum walk" has emerged recently as a paradigmatic process for the dynamic simulation of complex quantum systems, entanglement production and quantum computation. Hitherto, photonic implementations of quantum walks have mainly been based on multipath interferometric schemes in real space. We report the experimental realization of a discrete quantum walk taking place in the orbital angular momentum space of light, both for a single photon and for two simultaneous photons. In contrast to previous implementations, the whole process develops in a single light beam, with no need of interferometers; it requires optical resources scaling linearly with the number of steps; and it allows flexible control of input and output superposition states. Exploiting the latter property, we explored the system band structure in momentum space and the associated spin-orbit topological features by simulating the quantum dynamics of Gaussian wavepackets. Our demonstration introduces a novel versatile photonic platform for quantum simulations.

Directly measuring mean and variance of infinite-spectrum observables such as the photon orbital angular momentum.

The standard method for experimentally determining the probability distribution of an observable in quantum mechanics is the measurement of the observable spectrum. However, for infinite-dimensional degrees of freedom, this approach would require ideally infinite or, more realistically, a very large number of measurements. Here we consider an alternative method which can yield the mean and variance of an observable of an infinite-dimensional system by measuring only a two-dimensional pointer weakly coupled with the system. In our demonstrative implementation, we determine both the mean and the variance of the orbital angular momentum of a light beam without acquiring the entire spectrum, but measuring the Stokes parameters of the optical polarization (acting as pointer), after the beam has suffered a suitable spin-orbit weak interaction. This example can provide a paradigm for a new class of useful weak quantum measurements.

Observation of quantum recoherence of photons by spatial propagation.

Entanglement is at the heart of many unusual and counterintuitive features of quantum mechanics. Once two quantum subsystems have become entangled, it is no longer possible to ascribe an independent state to either; instead, the subsystems are completely described only as part of a greater, composite system. As a consequence of this, each entangled subsystem experiences a loss of coherence following entanglement. We refer to this decrease in coherence as decoherence. Decoherence leads inevitably to the leaking of information from each subsystem to the composite entangled system. Here, we demonstrate a process of decoherence reversal, whereby we recover information lost from the entanglement of the optical orbital angular momentum and radial profile degrees of freedom possessed by a photon pair. These results carry great potential significance, since quantum memories and quantum communication schemes depend on an experimenter's ability to retain the coherent properties of a particular quantum system.

Dynamics of laser-induced radial birefringence in silver-doped glasses.

Silver ion-exchanged glass exhibits nonlinear optical properties upon interacting with intense light beams. The thermal effect due to the nanoparticles' light-absorption induces radial stress, and consequently, a radial birefringence on the glass surface. The induced birefringence possesses a topological charge of 1 in the transverse plane of the glass, i.e., cylindrical symmetry. Therefore, when the glass is illuminated with a circularly polarized light beam, a portion of the incoming beam flips its polarization handedness, since the plate is birefringent, and gains an orbital angular momentum of ±2 in units of the Planck constant. This is referred to as optical spin-to-orbital angular momentum conversion, and can be understood by means of the Pancharatnam-Berry phase. Here, we design a pump-probe setup to study and observe the dynamics of optical angular momentum coupling in real time. We show that this effect can be permanent or reversible, depending on the power and interaction time of the pump beam. In particular, an intrinsic power-dependent birefringence hysteresis is observed on the sample after interaction with and the relaxation of the irradiated point.

Optics. Observation of optical polarization Möbius strips.

Möbius strips are three-dimensional geometrical structures, fascinating for their peculiar property of being surfaces with only one "side"­or, more technically, being "nonorientable" surfaces. Despite being easily realized artificially, the spontaneous emergence of these structures in nature is exceedingly rare. Here, we generate Möbius strips of optical polarization by tightly focusing the light beam emerging from a q-plate, a liquid crystal device that modifies the polarization of light in a space-variant manner. Using a recently developed method for the three-dimensional nanotomography of optical vector fields, we fully reconstruct the light polarization structure in the focal region, confirming the appearance of Möbius polarization structures. The preparation of such structured light modes may be important for complex light beam engineering and optical micro- and nanofabrication.

Generation of a spin-polarized electron beam by multipole magnetic fields.

The propagation of an electron beam in the presence of transverse magnetic fields possessing integer topological charges is presented. The spin-magnetic interaction introduces a nonuniform spin precession of the electrons that gains a space-variant geometrical phase in the transverse plane proportional to the field's topological charge, whose handedness depends on the input electron's spin state. A combination of our proposed device with an electron orbital angular momentum sorter can be utilized as a spin-filter of electron beams in a mid-energy range. We examine these two different configurations of a partial spin-filter generator numerically. The results of this analysis could prove useful in the design of an improved electron microscope.

Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram.

A phase-only hologram applies a modal transformation to an optical transverse spatial mode via phase encoding and intensity masking. Accurate control of the optical field crucially depends on the method employed to encode the hologram. In this Letter, we present a method to encode the amplitude and the phase of an optical field into a phase-only hologram, which allows the exact control of spatial transverse modes. Any intensity masking method modulates the amplitude and alters the phase of the optical field. Our method consists in correcting for this unwanted phase alteration by modifying the phase encryption accordingly. We experimentally verify the accuracy of our method by applying it to the generation and detection of transverse spatial modes in mutually unbiased bases of dimension two and three.

Test of mutually unbiased bases for six-dimensional photonic quantum systems.

In quantum information, complementarity of quantum mechanical observables plays a key role. The eigenstates of two complementary observables form a pair of mutually unbiased bases (MUBs). More generally, a set of MUBs consists of bases that are all pairwise unbiased. Except for specific dimensions of the Hilbert space, the maximal sets of MUBs are unknown in general. Even for a dimension as low as six, the identification of a maximal set of MUBs remains an open problem, although there is strong numerical evidence that no more than three simultaneous MUBs do exist. Here, by exploiting a newly developed holographic technique, we implement and test different sets of three MUBs for a single photon six-dimensional quantum state (a "qusix"), encoded exploiting polarization and orbital angular momentum of photons. A close agreement is observed between theory and experiments. Our results can find applications in state tomography, quantitative wave-particle duality, quantum key distribution.

Generation and dynamics of optical beams with polarization singularities.

We present a convenient method to generate vector beams of light having polarization singularities on their axis, via partial spin-to-orbital angular momentum conversion in a suitably patterned liquid crystal cell. The resulting polarization patterns exhibit a C-point on the beam axis and an L-line loop around it, and may have different geometrical structures such as "lemon", "star", and "spiral". Our generation method allows us to control the radius of L-line loop around the central C-point. Moreover, we investigate the free-air propagation of these fields across a Rayleigh range.

Reconstructing the Poynting vector skew angle and wavefront of optical vortex beams via two-channel moiré deflectometery.

An approach based on the two-channel moiré deflectometry has been used to measure both wavefront and transverse component of the Poynting vector of an optical vortex beam. Generated vortex beam by the q-plate, an inhomogeneous liquid crystal cell, has been analyzed with such technique. The measured topological charge of generated beams are in an excellent agreement with theoretical prediction.

Time-division multiplexing of the orbital angular momentum of light.

We present an optical setup for generating a sequence of light pulses in which the orbital angular momentum (OAM) degree of freedom is correlated with the temporal one. The setup is based on a single q plate within a ring optical resonator. By this approach, we demonstrate the generation of a train of pulses carrying increasing values of OAM, or, alternatively, of a controlled temporal sequence of pulses having prescribed OAM superposition states. Finally, we exhibit an "OAM-to-time conversion" apparatus that divides different input OAM states into different time bins. The latter application provides a simple approach to digital spiral spectroscopy of pulsed light.

Radial coherent and intelligent states of paraxial wave equation.

Ladder operators for the radial index of the paraxial optical modes in the cylindrical coordinates are calculated. The operators obey the su(1,1) algebra commutation relations. Based on this Lie algebra, we found that coherent modes constructed as eigenstates of the destruction operator or resulting from the action of the displacement operator on the fundamental mode are different. Some properties of these two kinds of radial coherent modes are studied in detail.

Polarization pattern of vector vortex beams generated by q-plates with different topological charges.

We describe the polarization topology of the vector beams emerging from a patterned birefringent liquid crystal plate with a topological charge q at its center (q-plate). The polarization topological structures for different q-plates and different input polarization states have been studied experimentally by measuring the Stokes parameters point-by-point in the beam transverse plane. Furthermore, we used a tuned q=1/2-plate to generate cylindrical vector beams with radial or azimuthal polarizations, with the possibility of switching dynamically between these two cases by simply changing the linear polarization of the input beam.

Spin-to-orbital angular momentum conversion and spin-polarization filtering in electron beams.

We propose the design of a space-variant Wien filter for electron beams that induces a spin half-turn and converts the corresponding spin angular momentum variation into orbital angular momentum of the beam itself by exploiting a geometrical phase arising in the spin manipulation. When applied to a spatially coherent input spin-polarized electron beam, such a device can generate an electron vortex beam, carrying orbital angular momentum. When applied to an unpolarized input beam, the proposed device, in combination with a suitable diffraction element, can act as a very effective spin-polarization filter. The same approach can also be applied to neutron or atom beams.

Tunable liquid crystal q-plates with arbitrary topological charge.

Using a photoalignment technique with a sulphonic azo-dye as the surfactant aligning material, we fabricated electrically tunable liquid crystal q-plates with topological charge 0.5, 1.5 and 3 for generating optical vortex beams with definite orbital angular momentum (OAM) 1,3 and 6 per photon (in units of ¯h), respectively. We carried out several tests on our q-plates, including OAM tomography, finding excellent performances. These devices can have useful applications in general and quantum optics.

Efficient generation and control of different-order orbital angular momentum states for communication links.

We present an optical scheme to encode and decode 2 bits of information into different orbital angular momentum (OAM) states of a paraxial optical beam. Our device generates the four light angular momentum states of order ±2 and ±4 by spin-to-orbital angular momentum conversion in a triangular optical loop arrangement. The switching among the four OAM states is obtained by changing the polarization state of the circulating beam by two quarter-wave plates, and the 2 bit information is transferred to the beam OAM exploiting a single q plate. The polarization of the exit beam is left free for an additional 1 bit of information. The switching among the different OAM states can be as fast as a few nanoseconds, if suitable electro-optical cells are used. This may be particularly useful in communication systems based on light OAM.

Quantum information transfer from spin to orbital angular momentum of photons.

The optical "spin-orbit" coupling occurring in a suitably patterned nonuniform birefringent plate known as a "q plate" allows entangling the polarization of a single photon with its orbital angular momentum (OAM). This process, in turn, can be exploited for building a bidirectional "spin-OAM interface," capable of transposing the quantum information from the spin to the OAM degree of freedom of photons and vice versa. Here, we experimentally demonstrate this process by single-photon quantum tomographic analysis. Moreover, we show that two-photon quantum correlations such as those resulting from coalescence interference can be successfully transferred into the OAM degree of freedom.

Light propagation in a birefringent plate with topological charge.

We calculated the Fresnel paraxial propagator in a birefringent plate having topological charge q at its center, named "q-plate." We studied the change of the beam transverse profile when it traverses the plate. An analytical closed form of the beam profile propagating in the q-plate can be found for many important specific input beam profiles. We paid particular attention to the plate having a topological unit charge and found that if small losses due to reflection, absorption, and scattering are neglected, the plate can convert the photon spin into orbital angular momentum with up to 100% efficiency provided the thickness of the plate is less than the Rayleigh range of the incident beam.