On Small Beams with Large Topological Charge II: Photons, Electrons and Gravitational Waves

Beams of light with a large topological charge significantly change their spatial structure when they are focused strongly. Physically, it can be explained by an emerging electromagnetic field component in the direction of propagation, which is neglected in the simplified scalar wave picture in optics. Here we ask: Is this a specific photonic behavior, or can similar phenomena also be predicted for other species of particles? We show that the same modification of the spatial structure exists for relativistic electrons as well as for focused gravitational waves. However, this is for different physical reasons: For electrons, which are described by the Dirac equation, the spatial structure changes due to a Spin-Orbit coupling in the relativistic regime. In gravitational waves described with linearized general relativity, the curvature of space-time between the transverse and propagation direction leads to the modification of the spatial structure. Thus, this universal phenomenon exists for both massive and massless elementary particles with Spin 1/2, 1 and 2. It would be very interesting whether other types of particles such as composite systems (neutrons or C$_{60}$) or neutrinos show a similar behaviour and how this phenomenon can be explained in a unified physical way.Comment: 8 pages, 3 figure

Physical meaning of the radial index of Laguerre-Gauss beams

The Laguerre-Gauss modes are a class of fundamental and well-studied optical fields. These stable, shape-invariant photons - exhibiting circular-cylindrical symmetry - are familiar from laser optics, micro-mechanical manipulation, quantum optics, communication, and foundational studies in both classical optics and quantum physics. They are characterized, chiefly, by two modes numbers: the azimuthal index indicating the orbital angular momentum of the beam - which itself has spawned a burgeoning and vibrant sub-field - and the radial index, which up until recently, has largely been ignored. In this manuscript we develop a differential operator formalism for dealing with the radial modes in both the position and momentum representations, and - more importantly - give for the first time the meaning of this quantum number in terms of a well-defined physical parameter: the "intrinsic hyperbolic momentum charge".Comment: 12 pages, 4 figures, comments encourage

Quantifying high dimensional entanglement with two mutually unbiased bases

We derive a framework for quantifying entanglement in multipartite and high dimensional systems using only correlations in two unbiased bases. We furthermore develop such bounds in cases where the second basis is not characterized beyond being unbiased, thus enabling entanglement quantification with minimal assumptions. Furthermore, we show that it is feasible to experimentally implement our method with readily available equipment and even conservative estimates of physical parameters.Comment: 17 pages, 1 figur

Quantum gate description for induced coherence without induced emission and related phenomena

We introduce unitary quantum gates for photon pair creation in spontaneous parametric down-conversion nonlinear crystals (NLs) and for photon path alignment. These are the two key ingredients for the method of "induced coherence without induced emission" and many ensuing variations thereof. The difficulty in doing so stems from an apparent mixing of the mode picture (such as the polarization of photons) and the Fock picture (such as the existence of the photons). We illustrate utility of these gates by obtaining quantum circuits for the experimental setups of the frustrated generation of photon pairs, identification of a point-like object with undetected photons, and creation of a Bell state. We also introduce an effective nonunitary description for the action of NLs in experiments where all the NLs are pumped coherently. As an example, by using this simplifying picture, we show how NLs can be used to create superposition of given quantum states in a modular fashion.Comment: 4+3 page

Quantum Experiments and Graphs: Multiparty States as coherent superpositions of Perfect Matchings

We show a surprising link between experimental setups to realize high-dimensional multipartite quantum states and Graph Theory. In these setups, the paths of photons are identified such that the photon-source information is never created. We find that each of these setups corresponds to an undirected graph, and every undirected graph corresponds to an experimental setup. Every term in the emerging quantum superposition corresponds to a perfect matching in the graph. Calculating the final quantum state is in the complexity class #P-complete, thus cannot be done efficiently. To strengthen the link further, theorems from Graph Theory -- such as Hall's marriage problem -- are rephrased in the language of pair creation in quantum experiments. We show explicitly how this link allows to answer questions about quantum experiments (such as which classes of entangled states can be created) with graph theoretical methods, and potentially simulate properties of Graphs and Networks with quantum experiments (such as critical exponents and phase transitions).Comment: 6+5 pages, 4+7 figure

Questions on the Structure of Perfect Matchings inspired by Quantum Physics

We state a number of related questions on the structure of perfect matchings. Those questions are inspired by and directly connected to Quantum Physics. In particular, they concern the constructability of general quantum states using modern photonic technology. For that we introduce a new concept, denoted as inherited vertex coloring. It is a vertex coloring for every perfect matching. The colors are inherited from the color of the incident edge for each perfect matching. First, we formulate the concepts and questions in pure graph-theoretical language, and finally we explain the physical context of every mathematical object that we use. Importantly, every progress towards answering these questions can directly be translated into new understanding in quantum physics.Comment: 10 pages, 4 figures, 6 questions (added suggestions from peer-review

On Small Beams with Large Topological Charge

Light beams can carry a discrete, in principle unbounded amount of angular momentum. Examples of such beams, the Laguerre-Gauss modes, are frequently expressed as solutions of the paraxial wave equation. There, they are eigenstates of the orbital angular momentum (OAM) operator. The paraxial solutions predict that beams with large OAM could be used to resolve arbitrarily small distances - a dubious situation. Here we show how to solve that situation by calculating the properties of beams free from the paraxial approximation. We find the surprising result that indeed one can resolve smaller distances with larger OAM, although with decreased visibility. If the visibility is kept constant (for instance at the Rayleigh criterion, the limit where two points are reasonably distinguishable), larger OAM does not provide an advantage. The drop in visibility is due to a field in the direction of propagation, which is neglected within the paraxial limit.Comment: 6 pages, 2 figures; + supplementary informatio

Explaining strategic firm responsiveness to institutional processes in the evolution of corporate governance systems : the reform of director remuneration reporting in Germany

Due to economic and social globalization processes, the boundaries of national systems of corporate governance have become more permeable for the transfer of ideas and practices from other institutional contexts. I derive hypotheses from a multitheoretical framework to explain strategic firm responsiveness to national level pressures for corporate governance reform. This framework integrates institutional, resource dependence, social network, upper echelon, and organizational learning perspectives and portrays corporate governance reform as institutional change. I test hypotheses derived from this framework in the context of the issuance of the German corporate governance code. The code provision of interest recommends that German firms listed on the Frankfurt stock exchange publish a comprehensive director remuneration report for their management and supervisory boards, a practice that is arguably at odds with the traditional regulative, normative, and cognitive-cultural institutional pillars of the German corporate governance system. A unique longitudinal dataset of 189 stock exchange listed firms is used to explain strategic firm responsiveness to the issuance of this institutionally contested provision. In this context, this dissertation is the first study that (partly) operationalizes Oliver\u27s (1991) continuum of strategic responses to institutional processes. The findings reveal that in contrast to arguments advanced by financial economists and legal scholars, economic market forces do not significantly drive firms\u27 responsiveness to corporate governance reform pressures. Instead, firm ownership type and power, labor representatives, management characteristics, and different intra- and interorganizational learning processes are significant predictors of strategic firm responsiveness to national level corporate governance reform pressures. The findings generally provide support for the developed theoretical framework and help corporate governance research to expand beyond the traditional legal and financial economics perspective

Quantum Experiments and Graphs III: High-Dimensional and Multi-Particle Entanglement

Quantum entanglement plays an important role in quantum information processes, such as quantum computation and quantum communication. Experiments in laboratories are unquestionably crucial to increase our understanding of quantum systems and inspire new insights into future applications. However, there are no general recipes for the creation of arbitrary quantum states with many particles entangled in high dimensions. Here, we exploit a recent connection between quantum experiments and graph theory and answer this question for a plethora of classes of entangled states. We find experimental setups for Greenberger-Horne-Zeilinger states, W states, general Dicke states, and asymmetrically high-dimensional multipartite entangled states. This result sheds light on the producibility of arbitrary quantum states using photonic technology with probabilistic pair sources and allows us to understand the underlying technological and fundamental properties of entanglement.Comment: 7 pages, 7 figures; Appendix 3 pages, 5 figure

Gouy Phase Radial Mode Sorter for Light: Concepts and Experiments

We present an in principle lossless sorter for radial modes of light, using accumulated Gouy phases. The experimental setups have been found by a computer algorithm, and can be intuitively understood in a geometric way. Together with the ability to sort angular-momentum modes, we now have access to the complete 2-dimensional transverse plane of light. The device can readily be used in multiplexing classical information. On a quantum level, it is an analog of the Stern-Gerlach experiment -- significant for the discussion of fundamental concepts in quantum physics. As such, it can be applied in high-dimensional and multi-photonic quantum experiments.Comment: main text: 7 pages, 5 figures. Supplementary Information: 5 pages, 4 figure