## Quantum Entanglement of Complex Structures of PhotonsThis thesis casts new light on quantum entanglement of photons with complex spatial patterns due to direct coincidence imaging. It demonstrates novel methods to generate, investigate, and verify entanglement of complex spatial structures. Quantum theory is one of the most successful and astonishing physical theories. It made possible various technical devices like lasers or mobile phones and, at the same time, it completely changed our understanding of the world. Interestingly, such counterintuitive features like entanglement are an important building block for future quantum technologies. In photonic experiments, the transverse spatial degree of freedom offers great potential to explore fascinating phenomena of single photons and quantum entanglement. It was possible to verify the entanglement of two photons with very high quanta of orbital angular momentum, a property of photons connected to their spatial structure and theoretically unbounded. In addition, modern imaging technology was used to visualize the effect of entanglement even in real-time and to show a surprising property: photons with complex spatial patterns can be both entangled and not entangled in polarization depending on their transverse spatial position. |

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### Contents

1 Preamble | 1 |

2 Introduction and Theoretical Background | 5 |

3 Entanglement of High Angular Momenta | 35 |

4 Coincidence Imaging of Spatial Mode Entanglement | 61 |

5 Entanglement of Complex Photon Polarization Patterns in Vector Beams | 77 |

6 Conclusion and Outlook | 93 |

Appendix ATheoretical Formalism of the SlitWheelMeasurement | 96 |

Curriculum Vitae | 101 |

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### Common terms and phrases

angle angular position Anton Zeilinger bi-photon Bloch sphere classical bound coincidence counts coincidence images coincidence-imaging coordinates crystal degree of freedom described detection efficiency ellipticity entangled photons Entanglement of Complex entanglement of high entanglement pattern entanglement witness experiment experimental HG vector high OAM hybrid-entangled ICCD camera Ince-Gauss modes intensity structure interferometer investigated ization Krenn LG modes LG vector photons light field linear polarization mask mode family mode structure modulation non-classicality novel OAM entanglement OAM values Optics Express orbital angular momentum paraxial paraxial approximation partner photon phase pattern phase shift photon number photon pairs Physical Review Letters pixels Poincaré beams Poincaré photons Poincaré sphere polarization entanglement polarization pattern possible Quantum Entanglement quantum information quantum mechanical quantum numbers qubit Ramelow regions rotation second photon slit-wheel slits Springer Structures of Photons superposition theoretical thesis transfer setup transferred photon transverse spatial modes trigger photon trigger polarizations vector beams Zeilinger