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arXiv:1208.6509 · Optics Express

Entangled Bessel-Gaussian beams

Melanie G. McLaren, Megan Agnew, Jonathan Leach, Filippus Stefanus Roux, Miles J. Padgett, Robert W. Boyd, Andrew Forbes

Published Oct 1, 2012• 126 citations
arXivHTMLPDF

Abstract

Orbital angular momentum (OAM) entanglement is investigated in the Bessel-Gaussian (BG) basis. Having a readily adjustable radial scale, BG modes provide an alternative basis for OAM entanglement over Laguerre-Gaussian modes. We show that the OAM bandwidth in terms of BG modes can be increased by selection of particular radial wavevectors and leads to a flattening of the spectrum, which allows for higher dimensionality in the entangled state. We demonstrate entanglement in terms of BG modes by performing a Bell-type experiment and showing a violation of the Clauser-Horne-Shimony-Holt inequality for the ℓ = ±1 subspace. In addition, we use quantum state tomography to indicate higher-dimensional entanglement in terms of BG modes.

DOI
10.1364/oe.20.023589

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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.

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Physical Review A•Oct 29, 2018•22 citations
Orbital angular momentum (OAM) entanglement is investigated in the Bessel-Gaussian (BG) basis. Having a readily adjustable radial scale, BG modes provide an alternative basis for OAM entanglement over Laguerre-Gaussian modes. We show that the OAM bandwidth in terms of BG modes can be increased by selection of particular radial wavevectors and leads to a flattening of the spectrum, which allows for higher dimensionality in the entangled state. We demonstrate entanglement in terms of BG modes by performing a Bell-type experiment and showing a violation of the Clauser-Horne-Shimony-Holt inequality for the ℓ = ±1 subspace. In addition, we use quantum state tomography to indicate higher-dimensional entanglement in terms of BG modes.
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.
Bessel-Gaussian (BG) modes possess unique characteristics that have been exploited in the classical world and which may also offer advantages over other modes in the quantum regime. The easily adjustable radial scale of BG modes provides a more favourable basis of orbital angular momentum (OAM) entanglement over Laguerre- Gaussian (LG) modes, where the radial dependence is often ignored. We demonstrate high-dimensional entanglement with the BG modes and show a higher fidelity than the LG modes. We use the reconstruction property of BG modes to recover the degree of entanglement of our quantum state after encountering an obstruction. By moving the obstruction along the path of propagation of the entangled photon pairs, we quantitatively show a increase in the degree of entanglement as the obstruction was moved beyond that minimum distance.
The OAM or spiral bandwidth indicates the dimensionality of an entangled state that is produced by the spontaneous parametric down-conversion process. Normally this bandwidth is determined by modulating the signal and idler beams with helical phase functions with opposite azimuthal indices on the spatial light modulators in the signal and idler beams, respectively. We added an additional binary Bessel function to the helical phase, thereby specifying the radial dependence of the mode to be Bessel-Gaussian (BG) modes. This comes down to a post selection process, which is known to have the ability to increase entanglement. The result is a modification to the shape of the OAM spectrum, which leads to a higher dimensionality for the quantum states. We perform analytical calculations to show that the bandwidths obtained by measuring in the BG modal basis are larger than those for the LG modes. These theoretical predictions are confirmed by experimental measurements of the bandwidths for LG modes and for BG modes with different transverse scales.
The most common approach to create orbital angular momentum (OAM) carrying vortex beams is by modulating a Gaussian beam with an azimuthal phase profile. This results in a beam with many radial modes and hence reduced power in the desired mode. Here we show that it is possible to select the mode size of the Laguerre-Gaussian basis so that the power in the detected p = 0 mode is maximised. We outline the idea theoretically and confirm it experimentally with both classical and quantum tests, demonstrating a power increase by as much as 50x. The consequence is higher signal-to- noise in optical communication and larger available dimensions in quantum entanglement experiments when using OAM modes as a basis.
Twisted photons enable the definition of a Hilbert space beyond two dimensions by orbital angular momentum (OAM) eigenstates. Here we propose a feasible entanglement concentration experiment, to enhance the quality of high-dimensional entanglement shared by twisted photon pairs. Our approach is started from the full characterization of entangled spiral bandwidth, and is then based on the careful selection of the Laguerre–Gaussian (LG) modes with specific radial and azimuthal indices p and l. In particular, we demonstrate the possibility of high-dimensional entanglement concentration residing in the OAM subspace of up to 21 dimensions. By means of LabVIEW simulations with spatial light modulators, we show that the Shannon dimensionality could be employed to quantify the quality of the present concentration. Our scheme holds promise in quantum information applications defined in high-dimensional Hilbert space.
The Bell inequalities for azimuthal quantum number, i.e., orbital angular momentum, have been extensively reported to show quantum correlations between the spatial degree of freedom of entangled photon pairs. Here we investigate instead the Bell inequalities for radial quantum number. We demonstrate, both theoretically and experimentally, the violation of a suitable Clauser-Horne-Shimony-Holt Bell inequality for the radial index ppp of the Laguerre-Gaussian mode, thus manifesting the entanglement in the radial structure of two-photon wave functions.