Quantum systems entanglement in continuous variables has been receiving increasing attention from the scientific community. The transverse spatial shape of photons (i.e. transverse position and momentum degree of freedom) is such a continuous variable systems, which due to its ease of control is prone to be used for testing some of the new physical predictions that continuous variables quantum systems can offer, as the improvement of the security of quantum key distribution cryptography and the efficiency enhancement of quantum communication protocols. Following this line, the mean goal of this Thesis has been the characterization and measurement of the spatial shape of photons. Another outstanding problem in Quantum Optics Information Technology (QOIT) is the coherent and reversible storage of photons states in matter, i.e. the necessity of a quantum memory. Theoretical and experimental results about this topic have been showed.
Below we summarize the more important contributions of this Thesis:
Demonstration about the non-validity of the OAM transformation rule FORr highly focused light beams: We have shown from a theoretical and experimental point of view that the OAM transformation rule is not valid for highly focused light beams, since Dove prisms inherently introduce astigmatism, and therefore further OAM changes. We have analyzed how light beams with a well defined value of the OAM per photon, after traversing the Dove prism, are transformed into a superposition of states with well defined OAM. The violation of the rule, turns out to be more important for highly focused light beams. We have provided a quantitative study of the properties of the Dove prism, and we have verified experimentally the validity of the theoretical results in a series of experiments with a commercially available Dove prism.
Presentation of a new method to measure the amplitude and phase of Laguerre-Gaussian like beams: A method to measure the amplitude an phase of Laguerre-Gaussian like beams have been presented, from a theoretical and experimental point of view. This method is based on a small variation of the spiral phase interferometry technique, which allows to avoid some technical problems that can be found in the reconstruction of finite sized beams and beams with phase singularities. A few examples of the use of our technique for the characterization of complex beams have been presented.
Description of the transverse spatial shape of bipartite entangled photons generated in Spontaneous Parametric Down Conversion (SPDC) using the Wigner function formulism: We have show some theoretical and experimental results regarding the spatial Wigner function of entangled paired photons generated in SPDC. Concretely, we have studied the remote preparation of one of the photons of the pair, for the photon being in a Gaussian beam and in a superposition of two modes. Also, we have analyzed and measured the case of the Wigner function of only one of the two photons of the pair, which shows that one photon of the pair is in a statistical mixture of modes. Finally, an experiment to completely measure the spatial state of the two photons was proposed, and also analytical expressions for the results one would expect in this case were given.
Investigation about the storage of spatial light polarization grating into the Zeeman ground states of cold cesium atoms: We have investigated theoretically and experimentally the dynamics of light grating stored in electromagnetically induced transparent (EIT) medium associated with a degenerate two-level system. The dependence of the stored light grating with the intensities of the incident and reading beams has been investigated. Bragg diffraction into the stored grating has been employed to probe the dynamics under different conditions showing reasonable results in agreement with the theoretical model developed. Moreover, we also have observed the coherent evolution of the stored grating in the presence of an applied magnetic field, which allows collapses and revivals of the stored grating, which is associated with the Larmor precession of the induced grating around the applied magnetic field.
We think that the results of this Thesis are an important step in order to explore the quantum capabilities of the transverse spatial shape of photons and can encourage going further in this line of research. Moreover, the investigation on the dynamics of light grating stored in an EIT medium is of considerable importance for a better understanding of the coherent memory for multidimensional state spaces. Thus, a further and more detailed investigation of the topics presented here should be of special interest for the scientific community.
Quantum systems entanglement in continuous variables has been receiving increasing attention from the scientific community. The transverse spatial shape of photons (i.e. transverse position and momentum degree of freedom) is such a continuous variable systems, which due to its ease of control is prone to be used for testing some of the new physical predictions that continuous variables quantum systems can offer, as the improvement of the security of quantum key distribution cryptography and the efficiency enhancement of quantum communication protocols. Following this line, the mean goal of this Thesis has been the characterization and measurement of the spatial shape of photons. Another outstanding problem in Quantum Optics Information Technology (QOIT) is the coherent and reversible storage of photons states in matter, i.e. the necessity of a quantum memory. Theoretical and experimental results about this topic have been showed.
Below we summarize the more important contributions of this Thesis:
Demonstration about the non-validity of the OAM transformation rule FORr highly focused light beams: We have shown from a theoretical and experimental point of view that the OAM transformation rule is not valid for highly focused light beams, since Dove prisms inherently introduce astigmatism, and therefore further OAM changes. We have analyzed how light beams with a well defined value of the OAM per photon, after traversing the Dove prism, are transformed into a superposition of states with well defined OAM. The violation of the rule, turns out to be more important for highly focused light beams. We have provided a quantitative study of the properties of the Dove prism, and we have verified experimentally the validity of the theoretical results in a series of experiments with a commercially available Dove prism.
Presentation of a new method to measure the amplitude and phase of Laguerre-Gaussian like beams: A method to measure the amplitude an phase of Laguerre-Gaussian like beams have been presented, from a theoretical and experimental point of view. This method is based on a small variation of the spiral phase interferometry technique, which allows to avoid some technical problems that can be found in the reconstruction of finite sized beams and beams with phase singularities. A few examples of the use of our technique for the characterization of complex beams have been presented.
Description of the transverse spatial shape of bipartite entangled photons generated in Spontaneous Parametric Down Conversion (SPDC) using the Wigner function formulism: We have show some theoretical and experimental results regarding the spatial Wigner function of entangled paired photons generated in SPDC. Concretely, we have studied the remote preparation of one of the photons of the pair, for the photon being in a Gaussian beam and in a superposition of two modes. Also, we have analyzed and measured the case of the Wigner function of only one of the two photons of the pair, which shows that one photon of the pair is in a statistical mixture of modes. Finally, an experiment to completely measure the spatial state of the two photons was proposed, and also analytical expressions for the results one would expect in this case were given.
Investigation about the storage of spatial light polarization grating into the Zeeman ground states of cold cesium atoms: We have investigated theoretically and experimentally the dynamics of light grating stored in electromagnetically induced transparent (EIT) medium associated with a degenerate two-level system. The dependence of the stored light grating with the intensities of the incident and reading beams has been investigated. Bragg diffraction into the stored grating has been employed to probe the dynamics under different conditions showing reasonable results in agreement with the theoretical model developed. Moreover, we also have observed the coherent evolution of the stored grating in the presence of an applied magnetic field, which allows collapses and revivals of the stored grating, which is associated with the Larmor precession of the induced grating around the applied magnetic field.
We think that the results of this Thesis are an important step in order to explore the quantum capabilities of the transverse spatial shape of photons and can encourage going further in this line of research. Moreover, the investigation on the dynamics of light grating stored in an EIT medium is of considerable importance for a better understanding of the coherent memory for multidimensional state spaces. Thus, a further and more detailed investigation of the topics presented here should be of special interest for the scientific community.
Multi-particle entangled states are becoming more and more interesting for different applications in quantum information, such as quantum multi-particle teleportation, entangled based cryptography and quantum communication schemes based on multi-particle states, but also for fundamental reasons such as testing local realism by means of the Greenberger-Horne-Zeilinger states (GHZ) arguments . The advantage of multiparticle entanglement versus the most common two particle one lies in the fact that the greater the number of particles involved in the entangled state, the more clearly the quantum effects are exhibited. Our final goal is to exploit the multi-photon entangled states for quantum metrology. By the interaction of those states with nanostructures (mostly nanoholes, but in a future also nanospheres), we expect to reach sub-wavelength sensitivity phase measurements beating the Standard Quantum Limit as well as to detect plasmonic effects. In this thesis we present our work consisting on designing and building an experimental setup for the generation of states of 4-photon entangled in the transverse linear momentum. In the first part we explain some basics in the form of a technical introduction. We then adapt a general theoretical model to our particular case, explaining the physics of the process of generation of entangled states of photons. We proceed explaining the details of our setup, as well as the procedure of measurement. We then present the results of our tests in a "standard" 2-photon regime to show the correct functioning of the setup itself. Some more sophisticated measurements using structured light in the form of a first-order Hermite-Gaussian mode has been performed to generate entanglement in higher spatial modes are also presented. We conclude with theoretical and experimental details regarding the 4-photon experiments feasible with our setup. Our source is based on Spontaneous Parametric Down-Conversion (SPDC). In the last two decades it has been extensively studied for the generation of two-photon entangled states, since Kwiat et al. published in 1995 the new scheme based on polarization entangled photons. Most of the studies regarding entangled states involving more than two photons are indeed based on quantum correlations in the polarization or in the path of the down converted beams. The great advantage of entangling a spatial degree of freedom (as we do) is related to the dimensionality of the Hilbert space in which the state is defined. In the particular case of linear momentum, which is a continuous variable, we are dealing with infinite dimensions. Since in each dimension of the state one quanta of information can be encoded, the advantage of having multidimension entanglement for the field of quantum communication and computing is clear. The specs of our setup include a periodically-poled KTP crystals offering high conversion efficiency and cut for type-0 SPDC. The photons generated are well known to be inherently entangled in different degrees of freedom such as polarization, energy-time, linear momentum, and orbital angular momentum. Since each single SPDC event generates one photon pair, in order to reach four photon we entangle two consecutive SPDC events per each state. Furthermore, it is possible to manipulate (to enhance) the dimensionality of the entanglement by properly tailoring the pump spatial profile. The key point for the generation of four-photon entangled states is that, under some conditions, two consecutive pairs generated via SPDC cannot be described as two independent pairs but necessarily as a four-photon state showing genuine entanglement. This can happen when using a pulsed laser to generate the entangled photons; in particular, in the regime in which the coherence time of the down-converted photons τc is (much) longer than the temporal length of the pumping pulse Δt. In this condition the emission of a second pair is stimulated by the presence of the first one. That effect can be ultimately seen as a constructive multi-particle interference effect, the deep origin of which has to be sought in the boson statistic of photons.
Spontaneous parametric down-conversion (SPDC) is one of the most important sources of entangled photons in quantum optics and underpins a wide range of applications in quantum communication, quantum imaging, quantum sensing, and photonic quantum information processing. In this nonlinear optical process, a pump photon is converted into a pair of lower-energy photons that exhibit strong correlations in multiple degrees of freedom. While the generation of photon pairs is governed by energy and momentum conservation, the resulting biphoton state possesses a rich spatial structure whose properties are determined by the interplay among the pump field, the nonlinear medium, and subsequent propagation. This thesis investigates the spatial properties of entangled photon pairs generated through SPDC, focusing on the evolution of their correlations in the near field and on the behavior of orbital-angular-momentum (OAM) entangled states under tight focusing conditions. A common theme throughout this work is the examination of physical effects that emerge when the full biphoton wavefunction is retained rather than reduced through the simplifying approximations often employed in theoretical treatments. The principal contribution of this thesis concerns the near-field propagation of SPDC photon pairs. The spatial correlations of SPDC-generated photons are commonly analyzed within the thin-crystal approximation, where the biphoton wavefunction is assumed to factorize into independent functions of the sum and difference coordinates. We demonstrate that birefringence-induced transverse walk-off breaks this factorization and introduces an intrinsic coupling between these coordinates. More importantly, this coupling survives free-space propagation and remains observable even for crystals that nominally satisfy the thin-crystal condition. As a result, the joint spatial correlations develop a characteristic tapering near the crystal image plane that cannot be captured by conventional factorized models. Numerical simulations and experimental measurements are found to be in excellent agreement, revealing a previously overlooked feature of SPDC and providing a more complete description of photon-pair generation and propagation in birefringent nonlinear media. The second part of the thesis explores the propagation of OAM-entangled photon pairs beyond the paraxial regime. While OAM correlations are typically studied under weak-focusing conditions, many emerging quantum technologies require structured quantum states to be manipulated in highly confined optical systems. We show that tight focusing modifies the OAM correlation spectrum, leading to a broadening of the accessible mode distribution. This broadening exhibits distinct transverse and longitudinal characteristics and arises from the emergence of nonparaxial field components absent in conventional scalar descriptions. These results provide new insight into the behavior of structured quantum light in strongly focused optical fields and establish a framework for understanding OAM entanglement beyond the paraxial approximation. Together, the results presented in this thesis reveal previously unexplored aspects of the structure and propagation of entangled photon pairs. By moving beyond widely used approximations, this work uncovers physical effects that influence the generation, evolution, and measurement of spatially entangled states, with implications for quantum imaging, high-dimensional quantum communication, and the broader development of structured-light-based quantum technologies.
Orbital angular momentum (OAM) is one of the most recently discovered properties of light, and it is only in the past decade its quantum properties have been the subject of experimental investigations and have found applications. Unlike polarization, which is only bidimensional, orbital angular momentum provides, with relative ease, unprecedented access to a theoretically unbounded discrete state space. The process of spontaneous parametric down-conversion has long been used as a source of two-photon states that can be entangled in several degrees of freedom, including OAM. In this thesis, the properties of the natural OAM spectrum associated with the entangled states produced by parametric down-conversion were investigated. Chapters 2 and 3 describe the production and detection of tunable high-dimensional OAM entanglement in a down-conversion system. By tuning the phase-matching conditions and improving the detection stage, a substantial increase in the half-width of the OAM correlation spectrum was observed. The conjugate variable of OAM, angular position, was also considered when examining high-dimensional states entangled in OAM. In order to efficiently determine their dimension, high-dimensional entangled states were probed by implementing a technique based on phase masks composed of multiple angular sectors, as opposed to narrow single-sector analysers. Presented in chapter 4, this technique allows the measurements of tight angular correlations while maintaining high optical throughput. The states so produced were then used for a number of applications centred around the concept of mutually unbiased bases. One can define sets of mutually unbiased bases for arbitrary subspaces of the OAM state space. Two bases are mutually unbiased if the measurement of a state in one basis provides no information about the state as described in the other basis. Complete measurements in mutually unbiased bases of high-dimensional OAM spaces are presented in chapter 5. Measurements in sets of mutually unbiased bases are integral to quantum science and can be used in a variety of protocols that fully exploit the large size of the OAM state space; we describe their use in efficient quantum state tomography, quantum key distribution and entanglement detection. Caution is however necessary when dealing with state spaces embedded in higher-dimensional spaces, such as that provided by OAM. Experimental tests of Bell-type inequalities allow us to rule out local hidden variable theories in the description of quantum correlations. Correlations inconsistent with the states observed, or even with quantum mechanics, known as super-quantum correlations, have however been recorded previously in experiments that fail to comply with the fair-sampling conditions. Chapter 6 describes an experiment that uses a particular choice of transverse spatial modes for which super-quantum correlations persist even if the detection is made perfectly efficient. The sets of modes carrying OAM allow a complete description of the transverse field. The ability to control and combine additional degrees of freedom provides the possibility for richer varieties of entanglement and can make quantum protocols more powerful and versatile. One such property of light, associated with transverse modes possessing radial nodes in the field distribution, can be accessed within the same type of experimental apparatus used for OAM. In chapter 7, the radial degree of freedom is explored, together with OAM, in the context of Hong-Ou-Mandel interference.
We present an experimental method to measure the transverse spatial quantum state of an optical field in coordinate space at the single-photon level. The continuous-variable measurements are made with a photon-counting, parity-inverting Sagnac interferometer based on all-reflecting optics. The technique provides a large numerical aperture without distorting the shape of the wave front, does not introduce astigmatism, and allows for characterization of fully or partially coherent optical fields at the single-photon level. Measurements of the transverse spatial Wigner functions for highly attenuated coherent beams are presented and compared to theoretical predictions.
We present an experimental method to measure the transverse spatial quantum state of an optical field in coordinate space at the single-photon level. The continuous-variable measurements are made with a photon-counting, parity-inverting Sagnac interferometer based on all-reflecting optics. The technique provides a large numerical aperture without distorting the shape of the wavefront, does not introduce astigmatism, and allows for characterization of fully or partially coherent optical fields at the single-photon level. Measurements of the transverse spatial Wigner functions for highly attenuated coherent beams are presented and compared with theoretical predictions.
ENGLISH ABSTRACT: High-dimensional quantum entanglement offers an increase in information capacity per photon; a highly desirable property for quantum information processes such as quantum communication, computation and teleportation. As the orbital angular momentum (OAM) modes of light span an infinite-dimensional Hilbert space, they have become frontrunners in achieving entanglement in higher dimensions. In light of this, we investigate the potential of OAM entanglement of photons by controlling the parameters in both the generation and measurement systems. We show the experimental procedures and apparatus involved in generating and measuring entangled photons in two-dimensions. We verify important quantum tests such as the Einstein, Podolsky and Rosen (EPR) paradox using OAM and angle correlations, as well as a violation of a Bell-type inequality. By performing a full state tomography, we characterise our quantum state and show we have a pure, highly entangled quantum state. We demonstrate that this method can be extended to higher dimensions. The experimental techniques used to generate and measure OAM entanglement place an upper bound on the number of accessible OAM modes. As such, we investigate new methods in which to increase the spiral bandwidth of our generated quantum state. We alter the shape of the pump beam in spontaneous parametric down-conversion and demonstrate an effect on both OAM and angle correlations. We also made changes to the measurement scheme by projecting the photon pairs into the Bessel-Gaussian (BG) basis and demonstrate entanglement in this basis. We show that this method allows the measured spiral bandwidth to be optimised by simply varying the continuous radial parameter of the BG modes. We demonstrate that BG modes can be entangled in higher dimensions compared with the commonly used helical modes by calculating and comparing the linear entropy and fidelity for both modes. We also show that quantum entanglement can be accurately simulated using classical light using back-projection, which allows the study of projective measurements and predicts the strength of the coincidence correlations in an entanglement experiment. Finally, we make use of each of the techniques to demonstrate the effect of a perturbation on OAM entanglement measured in the BG basis. We investigate the self-healing property of BG beams and show that the classical property is translated to the quantum regime. By calculating the concurrence, we see that measured entanglement recovers after encountering an obstruction.