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arXiv:2110.03462 · Physical Review Applied

Characterizing and Tailoring Spatial Correlations in Multimode Parametric Down-Conversion

Vatshal Srivastav, Natalia Herrera Valencia, Saroch Leedumrongwatthanakun, Will McCutcheon, Mehul Malik

Published Nov 2, 2022• 16 citations
arXivHTMLPDF

Abstract

Photons entangled in their position-momentum degrees of freedom serve as an elegant manifestation of the Einstein-Podolsky-Rosen paradox, while also enhancing quantum technologies for communication, imaging, and computation. The multimode nature of photons generated in parametric down-conversion has inspired a generation of experiments on high-dimensional entanglement, ranging from complete quantum state teleportation to exotic multipartite entanglement. However, precise characterization of the underlying position-momentum state is notoriously difficult due to limitations in detector technology, resulting in a slow and inaccurate reconstruction riddled with noise. Furthermore, theoretical models for the generated two-photon state often forgo the importance of the measurement system, resulting in a discrepancy between theory and experiment. Here we formalize a description of the two-photon wave function in the spatial domain, referred to as the collected joint-transverse momentum amplitude (JTMA), which incorporates both the generation and measurement system involved. We go on to propose and demonstrate a practical and efficient method to accurately reconstruct the collected JTMA using a simple phase-step scan known as the $2D\ensuremath{\pi}$ measurement. Finally, we discuss how precise knowledge of the collected JTMA enables us to generate tailored high-dimensional entangled states that maximize discrete-variable entanglement measures such as entanglement of formation or entanglement dimensionality, and optimize critical experimental parameters such as photon heralding efficiency. By accurately and efficiently characterizing photonic position-momentum entanglement, our results unlock its full potential for discrete-variable quantum information science and lay the groundwork for future quantum technologies based on multimode entanglement.

DOI
10.1103/physrevapplied.18.054006

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Photons entangled in their position-momentum degrees of freedom serve as an elegant manifestation of the Einstein-Podolsky-Rosen paradox, while also enhancing quantum technologies for communication, imaging, and computation. The multimode nature of photons generated in parametric down-conversion has inspired a generation of experiments on high-dimensional entanglement, ranging from complete quantum state teleportation to exotic multipartite entanglement. However, precise characterization of the underlying position-momentum state is notoriously difficult due to limitations in detector technology, resulting in a slow and inaccurate reconstruction riddled with noise. Furthermore, theoretical models for the generated two-photon state often forgo the importance of the measurement system, resulting in a discrepancy between theory and experiment. Here we formalize a description of the two-photon wave function in the spatial domain, referred to as the collected joint-transverse momentum amplitude (JTMA), which incorporates both the generation and measurement system involved. We go on to propose and demonstrate a practical and efficient method to accurately reconstruct the collected JTMA using a simple phase-step scan known as the 2D\ensuremathπ2D\ensuremath{\pi}2D\ensuremathπ measurement. Finally, we discuss how precise knowledge of the collected JTMA enables us to generate tailored high-dimensional entangled states that maximize discrete-variable entanglement measures such as entanglement of formation or entanglement dimensionality, and optimize critical experimental parameters such as photon heralding efficiency. By accurately and efficiently characterizing photonic position-momentum entanglement, our results unlock its full potential for discrete-variable quantum information science and lay the groundwork for future quantum technologies based on multimode entanglement.
We report an eight-element, linear-array, single-photon detector that uses multiple fibers of differing lengths coupled to a single detector, the timing information from which reveals the position in which the photon was measured. Using two such arrays and two detectors we measure the correlations of photons produced by parametric downconversion, without recourse to mechanical scanning. Spatial light modulators acting as variable focal length lenses positioned between the downconversion crystal and the arrays allow us to switch between measurement of position, transverse momentum, or intermediate bases. We observe the product of the variances of the conditional probabilities for position and momentum to be more than an order of magnitude below the classical limit, realizing a full-field demonstration of the Einstein-Podolsky-Rosen paradox. Such, multistate measurement technologies allow access to the higher information content of the photon based upon spatial modes.
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Abstract The measurement of the position–momentum Einstein–Podolsky–Rosen (EPR) correlations of a two-photon state is important for many quantum information applications ranging from quantum key distribution to coincidence imaging. However, all the existing techniques for measuring the position–momentum EPR correlations involve coincidence detection and thus suffer from issues that result in less accurate measurements. In this article, we propose and demonstrate an experimental scheme that does not require coincidence detection for measuring the EPR correlations. Our technique works for two-photon states that are pure, irrespective of whether the state is separable or entangled. We theoretically show that if the pure two-photon state satisfies a certain set of conditions then the position–momentum EPR correlations can be obtained by doing the intensity measurements on only one of the photons. We experimentally demonstrate this technique for pure two-photon states produced by type-I spontaneous parametric down-conversion, and to the best of our knowledge, we report the most accurate experimental measurement of position–momentum EPR correlations.
Biphoton systems exhibiting entanglement in position-momentum variables, known as spatial entanglement, are among the most intriguing and well-studied phenomena in quantum optics. A notable subset of these are phase entangled states, where entanglement manifests purely through correlations in the spatial phase of the wavefunction. While the generation of such states from biphotons via spontaneous parametric down-conversion has been explored, their physical implications and applications remain under-investigated. In this work, we theoretically and experimentally examine a unique form of phase entanglement known as `pure' phase entanglement. This state exhibits the unusual feature that the position of one photon is correlated with the momentum of the other. Unlike typical spatially entangled states, it shows no direct correlation in position or momentum between the two photons, underscoring that all correlations arise purely from the spatial phase of the wavefunction. We delve deeper into the theory of this state and experimentally construct it from known phase-entangled states. To certify its properties, we propose a setup that performs a "one-particle momentum measurement" and explore the various tunable parameters. We also highlight potential applications of this state in quantum optics and imaging experiments.
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.
Structured optical fields have led to several ground-breaking techniques in classical imaging and microscopy. At the same time, in the quantum domain, position-momentum entangled photon fields have been shown to have several unique features that can lead to beyond-classical imaging and microscopy capabilities. Therefore, it is natural to expect that position-momentum entangled two-photon fields that are structured can push the boundaries of quantum imaging and microscopy even further beyond. Nonetheless, the existing experimental schemes are able to produce either structured two-photon fields without position-momentum entanglement, or position-momentum entangled two-photon fields without structured correlations. In this article, by manipulating the phase-matching condition of the spontaneous parametric down-conversion process, we report experimental generation of two-photon fields with various structures in their spatial correlations. We experimentally measure the minimum bound on the entanglement of formation and thereby verify the position-momentum entanglement of the structured two-photon field. We expect this work to have important implications for quantum technologies related to imaging and sensing.