## Abstract

Since classical interference has become ubiquitous in our modern optical devices and experiments, it is reasonable to assume that its quantum counterparts could become as prevalent in the next wave of technologies. The first signs of this can already be seen in the plethora of potential quantum technological applications that heavily rely on non-classical multi-photon interference effects. One degree of freedom where multi-photon interference has not been extensively studied yet is the transverse-spatial degree of freedom. This lack of studies is surprising when considering the potential benefits this degree of freedom has demonstrated in many other technological applications. Hence, our work here aims to increase our understanding of quantum interference in transverse spatial modes. We do this by investigating two-photon interference effects in this degree of freedom and by exploring some of their applications in metrology.

In this thesis, we first define a theoretical framework with which one can calculate the quantum effects of structured paraxial light fields. We also describe in detail how one can use phase-only spatial light modulators to shape and measure the transverse structure of photons. This is done both for the case of a single mask transformation and so-called multi-plane light conversion schemes. The experimental work of this thesis builds upon these methods and the theoretical framework to manipulate and utilize the quantum states of paraxial photons.

In our first experiments, we observed two-photon interference in the transverse-spatial degree of freedom. Initially, this was done in a complex unitary device which performs a beamsplitter-like transformation solely in the transverse-spatial degree of freedom. Thus, this experiment performs the exact analogue of the famous Hong-Ou-Mandel experiment in the space of transverse spatial modes. The unitary device was a multi-plane light conversion system. Subsequently, we removed the unitary device and performed similar interference effects in a simplified setup which was easier to operate. This setup allowed us to produce high-quality two-photon N00N states between transverse spatial modes, in a single beam of light.

After demonstrating our capability of producing high-quality N00N states, we explored their applicability in different quantum metrological tasks. This was done theoretically with the help of a quantity called quantum Fisher information, and experimentally using the simplified quantum interference system. We applied the N00N states to superresolution experiments in rotations and longitudinal translations.

Lastly, we were also able to investigate, using our N00N state experiments, the Gouy phase of a photon number state. As this phase has not been previously investigated in such a context, it provided us with some new insights into previously investigated phenomena. For instance, this experiment provides a simple example of why the so-called effective de Broglie wavelength of photons is not always a good description of photon number state evolution. It also allowed us to reaffirm the predictive power of a specific physical interpretation of the Gouy phase.

In this thesis, we first define a theoretical framework with which one can calculate the quantum effects of structured paraxial light fields. We also describe in detail how one can use phase-only spatial light modulators to shape and measure the transverse structure of photons. This is done both for the case of a single mask transformation and so-called multi-plane light conversion schemes. The experimental work of this thesis builds upon these methods and the theoretical framework to manipulate and utilize the quantum states of paraxial photons.

In our first experiments, we observed two-photon interference in the transverse-spatial degree of freedom. Initially, this was done in a complex unitary device which performs a beamsplitter-like transformation solely in the transverse-spatial degree of freedom. Thus, this experiment performs the exact analogue of the famous Hong-Ou-Mandel experiment in the space of transverse spatial modes. The unitary device was a multi-plane light conversion system. Subsequently, we removed the unitary device and performed similar interference effects in a simplified setup which was easier to operate. This setup allowed us to produce high-quality two-photon N00N states between transverse spatial modes, in a single beam of light.

After demonstrating our capability of producing high-quality N00N states, we explored their applicability in different quantum metrological tasks. This was done theoretically with the help of a quantity called quantum Fisher information, and experimentally using the simplified quantum interference system. We applied the N00N states to superresolution experiments in rotations and longitudinal translations.

Lastly, we were also able to investigate, using our N00N state experiments, the Gouy phase of a photon number state. As this phase has not been previously investigated in such a context, it provided us with some new insights into previously investigated phenomena. For instance, this experiment provides a simple example of why the so-called effective de Broglie wavelength of photons is not always a good description of photon number state evolution. It also allowed us to reaffirm the predictive power of a specific physical interpretation of the Gouy phase.

Original language | English |
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Place of Publication | Tampere |

ISBN (Electronic) | 978-952-03-3013-2 |

Publication status | Published - 2023 |

Publication type | G5 Doctoral dissertation (articles) |

### Publication series

Name | Tampere University Dissertations - Tampereen yliopiston väitöskirjat |
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Volume | 844 |

ISSN (Print) | 2489-9860 |

ISSN (Electronic) | 2490-0028 |