In this thesis, we explore the properties of photons generated by parametric down-conversion (PDC) and entangled in continuous variables of transverse momentum, with the aim of using this degree of freedom for developing a new method for generating en-tangled states in discrete and multidimensional spaces. Such states have a great potential for applications in fundamental problems of quantum mechanics as well as in quantum information protocols. Our study is based on the theory of angular spectrum transfer from the laser beam tothe two-photon state, proposed and demonstrated by Monken, Souto Ribeiro and P Ladua [Phys. Rev. A 57, 3123 (1998)]. Using it, we investigate theoretically the evolution of the quantum state of photon pairs when they propagate from the crystal and are transmitted through linear optical systems like lenses and apertures. The two-photon state is calculated when there is only one optical system in the path of each photon and we also discuss the generalization of the method employed in the calculation, for more complex experiments. With the two-photon state obtained, we propose a method for generating entangledstates of qubits (two-dimensional quantum systems) and qudits (D-dimensional quantum systems), using the transverse momentum of photons from PDC. The method is based on the selection of specific transverse spatial modes of each photon, by using an array of D-slits (D . 2) in the path of these photons. In this method we have the control of the dimension, D, of each qudit and also the degree of entanglement between them, by manipulating the transverse laser beam profile, which is due to the transfer of angular spectrum. In order to test the theory, we make an experiment for a specific case, which has a great interest in practical applications of entanglement: we demonstrate the generation of maximally entangled states of qudits with D = 4 and D = 8. The quantum character of the correlations is assured qualitatively by measuring two-photon interference patterns with conditional fringes. We also investigate the propagation of these entangled states of qudits through free space and with a simple experiment we show that they can be distributed for distant and spatially separated parts, keeping the correlations. Finally, we perform an experiment for quantifying entanglement in this type of state that we are producing. In this work, we restrict ourselves to the simplest case of two qubits. We prepare different states by manipulating the transverse laser beam profile and their degree of entanglement is measured in two ways: (i) through the degree ofconditionality of the two-photon interference patterns and (ii) through the measurement of marginal probability. We discuss the extension of the method (i) for two-qudit states.
In this thesis, we explore the properties of photons generated by parametric down-conversion (PDC) and entangled in continuous variables of transverse momentum, with the aim of using this degree of freedom for developing a new method for generating en-tangled states in discrete and multidimensional spaces. Such states have a great potential for applications in fundamental problems of quantum mechanics as well as in quantum information protocols. Our study is based on the theory of angular spectrum transfer from the laser beam tothe two-photon state, proposed and demonstrated by Monken, Souto Ribeiro and P Ladua [Phys. Rev. A 57, 3123 (1998)]. Using it, we investigate theoretically the evolution of the quantum state of photon pairs when they propagate from the crystal and are transmitted through linear optical systems like lenses and apertures. The two-photon state is calculated when there is only one optical system in the path of each photon and we also discuss the generalization of the method employed in the calculation, for more complex experiments. With the two-photon state obtained, we propose a method for generating entangledstates of qubits (two-dimensional quantum systems) and qudits (D-dimensional quantum systems), using the transverse momentum of photons from PDC. The method is based on the selection of specific transverse spatial modes of each photon, by using an array of D-slits (D . 2) in the path of these photons. In this method we have the control of the dimension, D, of each qudit and also the degree of entanglement between them, by manipulating the transverse laser beam profile, which is due to the transfer of angular spectrum. In order to test the theory, we make an experiment for a specific case, which has a great interest in practical applications of entanglement: we demonstrate the generation of maximally entangled states of qudits with D = 4 and D = 8. The quantum character of the correlations is assured qualitatively by measuring two-photon interference patterns with conditional fringes. We also investigate the propagation of these entangled states of qudits through free space and with a simple experiment we show that they can be distributed for distant and spatially separated parts, keeping the correlations. Finally, we perform an experiment for quantifying entanglement in this type of state that we are producing. In this work, we restrict ourselves to the simplest case of two qubits. We prepare different states by manipulating the transverse laser beam profile and their degree of entanglement is measured in two ways: (i) through the degree ofconditionality of the two-photon interference patterns and (ii) through the measurement of marginal probability. We discuss the extension of the method (i) for two-qudit states.
We review recent theoretical and experimental works where are proposed and demonstrated how to use photon pairs created by spontaneous parametric down-conversion to generate entangled states of D-dimensional quantum systems, or qudits. This is the first demonstration of high-dimensional entanglement based on the intrinsic transverse momentum entanglement of the type-II down-converted photons. The qudit space is defined by an aperture made up of an opaque screen with D slits (paths), placed in the arms of the twin photons. By manipulating the pump beam profile we can prepare different entangled states of these possible paths. We focus our attention on an important case for applications in quantum information: the maximally entangled states. Experimental results for qudits with D=4 and D=8 are shown and measuring a two-photon conditional interference, we also demonstrate the nonclassical character of the correlations.
In recent decades, Parametric Down Conversion (PDC) has been the main method of production of entangled photonic quantum states, most often bipartite. Inaddition, the discretization (withinacertainrange) of transversal momentum of the photon pair generated by PDC is the main tool for producing entangled qudits. Such discretization, in conjunction with different methods of experimental manipulation of photon beamsproduced in PDC, constitutes the various methods of engineering of slit states, which are well established only to states with dimensionality of the kind n×n. Inthismasters thesis, a study is presented of each of those methods of production applied to states of the form 2×2. As the central point of the work, is also presented their extension to case 2×3, aiming the generalization to any dimensionality of the form NxM. These twoparty discrete quantum states with different dimensions in each party is called Hybrid Dimension Quantum State.
We report an experiment to generate and propagate two entangled spatial qudits, [Formula: see text]-dimensional quantum systems, using spontaneous parametric down-conversion. The manipulation, via pump beam, of the transverse spatial correlation between the photon pairs is explored. Inserting apertures with [Formula: see text]-slits in the arms of the down-converted photons, we associate the qudit space with the [Formula: see text] possible paths followed by each photon. Experimental results for qudits with [Formula: see text] and 8 are shown. We demonstrate that the generated states cannot be classically correlated. We also show the propagation of entangled states of spatial qudits. Their free-space distribution is performed at the laboratory scale and the propagated states maintain a high fidelity with their original form.
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.
Using angular position–orbital angular momentum entangled photons, we propose an experiment to generate maximally entangled states in D-dimensional quantum systems, the so called qudits, by exploiting correlations of parametric down-converted photons. Angular diffraction masks containing N angular slits in the arms of each twin photon define a qudit space of dimension N2, spanned by the alternative pathways of the photons. Numerical results for N angular slits with N = 2, 4, 5, 10 are reported. We discuss relevant experimental parameters for an experimental implementation of the proposed scheme using Spatial Light Modulators (SLMs), and twin-photons produced by Spontaneous Parametric Down Conversion (SPDC). The entanglement of the qudit state can be quantified in terms of the Concurrence, which can be expressed in terms of the visibility of the interference fringes, or by using Entanglement Witnesses. These results provide an additional means for preparing entangled quantum states in high-dimensions, a fundamental resource for quantum simulation and quantum information protocols.
During the past thirty years the field of Quantum Information has beenvery successful in proposing protocols and algorithms that involve two-level quantum systems known as qubits. It has been shown that cryptographic keys can be distributed securely using quantum resources and that quantum algorithms can solve tasks such as the factoring of large numbers with an exponential speedup over classical algorithms. Most implementations, however, are restricted to proof-of-concept experiments using few qubits, due to the difficulty of scaling the physical systems utilized. Quantum systems with more than two levels, called qudits, present themselves as an alternative way of obtaining a space with large dimensionality without recurring to a large number of qubits. This reduces the number of particles necessary for implementing protocols in a scale that is relevant for practical applications. Additionally, pairs of entangled qudits are more robust agains noise than their qubit counterparts and are also more adequate for experiments that test the non-local and contextual behavior of nature, as predicted by its description using Quantum Mechanics. In this thesis we present the construction and characterization of a source of states of four qudits. We used a pulsed laser as pump for type-II Spontaneous Parametric Down Conversion (SPDC) to occur on a PPKTP, a KTiOPO4 (KTP) crystal with a periodic ferroelectric domain inversion, a technique called periodic poling. In type-II SPDC a pair of photons, also called biphoton, is generated in which one has horizontal polarization while the other hasvertical polarization. Due to the use of a pulsed pump beam and the highdown-conversion efficiency of our PPKTP, the probability that a single pump pulse will originate two pairs of photons is high. In order to generate qudits we used the multiple-slit implementation, meaning that a multiple slit was placed between the PPKTP and the detectors, encoding discrete qudit states onto the continuous transverse mode of the biphoton. Using double, triple and quadruple slits we produced, respectively, qubit, qutrit and ququart states. A system of lenses placed between the PPKTP and the slits manipulated the amplitude of the transverse biphoton state impinging on the slits and, thus, the quantum correlations between qudits after the slits. By projecting either the near or far field of the crystal onto the slits, the lenses changed the characteristics of the biphoton amplitude function and allowed us to obtain either states with very high entanglement, in the case of the near field, or very low entanglement, in the case of the far field. A lens in the f-f configuration between the plane of the slits and the plane of the detectors projected the far field of the slits, so we could observe conditional interference patterns. We detected the photons in coincidence asa function of the position of two detectors These were used to generate maps of coincidences between two photons with different polarization and for two photons with the same polarization. The first case contains contributions from photons of a single pair and from photons from two different pairs, while the second case contains contributions from different pairs only. In order to isolate the single pair contribution on different-polarization coincidences, we removed the background component from photons of different pairs, given by the same-polarization coincidences. Besides serving to isolate the singlepair signal, same-polarization coincidences confirm the existence two-pair events and give us information about the purity of the reduced state of a single photon, which is related to the degree of entanglement in a biphoton. We conclude that manipulating the biphoton transverse amplitude is a promising technique for generating qudit states while controlling their entanglement and correlation properties, especially if more exotic transverse amplitudes are explored, such as Hermite-Gaussian pump beams or amplitudes engineered with a spatial light modulator. The use of high-efficiency SPDC sources allows for states of more than one biphoton, engineered using transverse amplitude manipulation, to be employed in many different protocolsin Quantum Information, serving as sources of heralded pure or mixedstates, or playing an important role in tests of the non-local and contextual nature of physical systems, as predicted by Quantum Mechanics.