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arXiv:1712.03355 · Physical Review A

Angular Schmidt spectrum of entangled photons: Derivation of an exact formula and experimental characterization for noncollinear phase matching

Girish Ashok Kulkarni, Lavanya Taneja, Shaurya Aarav, Anand K. Jha

Published Jun 22, 2018• 18 citations
arXivHTMLPDF

Abstract

We derive an exact computationally-efficient formula for the angular Schmidt spectrum of orbital angular momentum (OAM) entangled states produced by parametric down-conversion (PDC). Our formula yields the true spectrum and does not suffer from convergence issues arising due to infinite summations, as has been the case with previously derived formulas. We use this formula to experimentally characterize the angular Schmidt spectrum of entangled photons produced by PDC with noncollinear phase matching. We report measurements of very broad angular Schmidt spectra, corresponding to the angular Schmidt numbers up to 229. Our work could have important implications for OAM-based quantum information applications.

DOI
10.1103/physreva.97.063846

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•Dec 1, 2016•1 citation
We derive an exact computationally-efficient formula for the angular Schmidt spectrum of orbital angular momentum (OAM) entangled states produced by parametric down-conversion (PDC). Our formula yields the true spectrum and does not suffer from convergence issues arising due to infinite summations, as has been the case with previously derived formulas. We use this formula to experimentally characterize the angular Schmidt spectrum of entangled photons produced by PDC with noncollinear phase matching. We report measurements of very broad angular Schmidt spectra, corresponding to the angular Schmidt numbers up to 229. Our work could have important implications for OAM-based quantum information applications.
The existing methods for measuring the orbital-angular-momentum (OAM) spectrum suffer from issues such as poor efficiency, strict interferometric stability requirements, and too much loss. Furthermore, most techniques inevitably discard part of the field and measure only a post-selected portion of the true spectrum. Here, we propose and demonstrate an interferometric technique for measuring the true OAM spectrum of optical fields in a single-shot manner. Our technique directly encodes the OAM-spectrum information in the azimuthal intensity profile of the output interferogram. In the absence of noise, the spectrum can be fully decoded using a single acquisition of the output interferogram, and, in the presence of noise, acquisition of two suitable interferograms is sufficient for the purpose. As an important application of our technique, we demonstrate measurements of the angular Schmidt spectrum of the entangled photons produced by parametric down-conversion and report a broad spectrum with the angular Schmidt number 82.1.
The generation of paired photons entangled in orbital angular momentum (OAM), provides a new degree of freedom that is increasingly being used as a source of quantum states. Elucidation of the OAM spectrum of OAM spectrum of the generated photons is of paramount importance, since quantum information applications require the ability to generate arbitrary entangled states with the appropriate OAM correlations. Here we discuss the OAM spectrum of the photon generated via SPDC in two complementary scenarios. On one hand, we should consider the whole geometry of the nonlinear process, taking into account azimuthal variations of the nonlinear coefficient or the phase matching conditions. On the other hand, all relevant experiments reported to date detect only a small section of the full down-conversion cone. In this scenario, the measured OAM correlations depend of the emission angle of the photons and the strength of the Poynting vector walk-off. We will present experiments in SPDC where this dependence is clearly revealed.
We implement an interferometric method to measure the orbital angular momentum (OAM) spectrum of photon pairs generated by spontaneous parametric down-conversion. In contrast with previous experiments, which were all limited by the modal capacity of the detection system, our method operates on the entire down-conversion cone and reveals the complete distribution of the generated OAM. In this geometry, new features can be studied. We show that the phase-matching conditions can be used as a tool to enhance the azimuthal Schmidt number and to flatten the spectral profile, allowing the efficient production of high-quality multidimensional entangled states.
The orbital angular momentum (OAM) of photons provides a high-dimensional resource for quantum information protocols. The dimensionality of OAM-entangled states generated via spontaneous parametric down-conversion (SPDC) is quantified by the angular Schmidt spectrum. Here, we experimentally investigate the dependence of the angular Schmidt spectrum on the thickness of the nonlinear crystal. Contrary to previous studies reporting a monotonic decrease in the Schmidt number with increasing crystal thickness, we report the first experimental observation of a nonmonotonic behavior, as we demonstrate an increase in the Schmidt number beyond a certain crystal thickness. We attribute this to the spatial walk-off effect in the anisotropic nonlinear crystal and explain it using a theoretical model that is devoid of standard phase-matching approximations. These findings can have important implications for high-dimensional entangled state generation.
The full Schmidt decomposition of spatiotemporally entangled states generated from spontaneous parametric down-conversion (SPDC) has not been carried out until now due to the immense computational complexity arising from the large dimensionalities of the states. In this Letter, we utilize the rotational symmetry of the states to reduce the complexity by at least four orders of magnitude and carry out the decomposition to reveal the precise forms of the spatiotemporal Schmidt modes and the Schmidt spectrum spanning over 10^4 modes. We show that the Schmidt modes have a phase profile with a transverse spatial vortex structure that endows them with orbital angular momentum at all frequencies. In the high-gain regime, these Schmidt modes broaden and the Schmidt spectrum narrows with increasing pump strength. Our work can spur novel applications at the intersection of quantum imaging and spectroscopy that utilize entangled states produced from SPDC.
The existing methods for measuring the orbital-angular-momentum (OAM) spectrum suffer from issues such as poor efficiency, strict interferometric stability requirements, and too much loss. Furthermore, most techniques inevitably discard part of the field and measure only a post-selected portion of the true spectrum. Here, we propose and demonstrate an interferometric technique for measuring the true OAM spectrum of optical fields in a single-shot manner. Our technique directly encodes the OAM-spectrum information in the azimuthal intensity profile of the output interferogram. In the absence of noise, the spectrum can be fully decoded using a single acquisition of the output interferogram, and, in the presence of noise, acquisition of two suitable interferograms is sufficient for the purpose. As an important application of our technique, we demonstrate measurements of the angular Schmidt spectrum of the entangled photons produced by parametric down-conversion and report a broad spectrum with the angular Schmidt number 82.1.