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arXiv:1607.05259 · Optics Letters

Mode analysis of higher-order transverse-mode correlation beams in a turbulent atmosphere

Hakob Avetisyan, Carlos Henrique Monken

Published Dec 20, 2016• 6 citations

Abstract

Due to the transfer of the angular spectrum of the pump beam to the two-photon state in spontaneous parametric downconversion, the generated twin photons are entangled in their spatial degrees of freedom. This spatial entanglement can be observed through correlation measurements in any set of modes in which one may choose to perform measurements. Choosing, e.g., a Hermite-Gaussian (HG) set of spatial modes as a basis, one can observe correlations present in their spatial degrees of freedom. In addition, these modes can be used as alphabets for quantum communication. For global quantum communication purposes, we derive an analytic expression for two-photon detection probability in terms of HG modes, taking into account the effects of the turbulent atmosphere. Our result is more general as it accounts for the propagation of both signal and idler photons through the atmosphere, as opposed to other works considering one photon's propagation in vacuum. We show that while the restrictions on both the parity and order of the downconverted HG fields no longer hold, due to the crosstalk between modes when propagating in the atmosphere, the crosstalk is not uniform: there are more robust modes that tend to keep the photons in them. These modes can be employed in order to increase the fidelity of quantum communication.

DOI: 10.1364/ol.42.000101

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Due to the transfer of the angular spectrum of the pump beam to the two-photon state in spontaneous parametric downconversion, the generated twin photons are entangled in their spatial degrees of freedom. This spatial entanglement can be observed through correlation measurements in any set of modes in which one may choose to perform measurements. Choosing, e.g., a Hermite-Gaussian (HG) set of spatial modes as a basis, one can observe correlations present in their spatial degrees of freedom. In addition, these modes can be used as alphabets for quantum communication. For global quantum communication purposes, we derive an analytic expression for two-photon detection probability in terms of HG modes, taking into account the effects of the turbulent atmosphere. Our result is more general as it accounts for the propagation of both signal and idler photons through the atmosphere, as opposed to other works considering one photon's propagation in vacuum. We show that while the restrictions on both the parity and order of the downconverted HG fields no longer hold, due to the crosstalk between modes when propagating in the atmosphere, the crosstalk is not uniform: there are more robust modes that tend to keep the photons in them. These modes can be employed in order to increase the fidelity of quantum communication.

Higher order correlation beams, that is, two-photon beams obtained from the process of spontaneous parametric down-conversion pumped by Hermite-Gauss or Laguerre-Gauss beams of any order, can be used to encode information in many modes, opening the possibility of quantum communication with large alphabets. In this paper we calculate, analytically, the fourth-order correlation function for the Hermite-Gauss and Laguerre-Gauss coherent and partially coherent correlation beams propagating through a strong turbulent medium. We show that fourth-order correlation functions for correlation beams have, under certain conditions, expressions similar to those of intensities of classical beams and are degraded by turbulence in a similar way as the classical beams. Our results can be useful in establishing limits for the use of two-photon beams in quantum communications with larger alphabets under atmospheric turbulence.

Turbulence presents challenges for free-space light-based protocols. In this work, we investigated the effect of turbulence on the correlations of twin photons prepared in Spontaneous Parametric Down Conversion (SPDC) by a pump beam with different transverse modes. We demonstrate that the strategy of coordinate inversion for mitigating turbulence-induced distortions on transverse coincidence profiles is effective for twin photons generated by HG/LG modes. The conservation of orbital angular momentum in SPDC enables us to encode information in the transverse degrees of freedom of the down-converted photon pairs. This information can have its correlation protected by the coordinate inversion technique against weak turbulence effects in free-space links.

We extend the theory of detection probabilities of single-photon communication in turbulent atmosphere to the entangled case and derive analytical expressions for detection probabilities. We show that the rotational field correlation in the entangled case cannot be expressed as a straightforward extension from the disentangled case. By considering the orbital angular momentum (OAM) entangled Laguerre-Gaussian (LG) beams for illustration, we explore the effect of non-Kolmogorov atmospheric turbulence on the probabilities of both the biphoton detection and the single-photon detection. The entanglement of OAM LG beams contributes positively to the faithful detection probability. The change of maximal crosstalk probability in the detection process is also explored, which displays the non-linear dependence on the crosstalk quantum number.

We analyse the effect of atmospheric Kolmogorov turbulence on entangled orbital angular momentum states generated by parametric down-conversion. We calculate joint and signal photon detection probabilities and obtain numerically their dependence on the mode-width-to-Fried-parameter ratio. We demonstrate that entangled photons are less robust to the effects of Kolmogorov turbulence compared to single photons. In contrast, signal photons are more robust than single photons in the lowest-order mode. We also obtain numerically a scaling relation between the value of the mode-width-to-Fried-parameter ratio for which the joint detection probability is a maximum and the momentum mismatch between signal and idler photons after propagation through the medium.

It is known that the entanglement present in the transverse spatial properties of photon pairs generated by spontaneous parametric down-conversion depends explicitly on the characteristics of the pump laser beam. Here we show that, for pump beams described by a Hermite-Gaussian mode, the entanglement depends on the beam width as well as the order of the beam. The amount of entanglement generally increases with the order of the beam and is less dependent on the beam width for higher orders. We propose a method to measure the spatial entanglement directly, independent of the dimension of the subsystems.

Two photonic qubits, initially maximally entangled in OAM modes of Hermite-Gaussian HG vortex beam, propagating through non-Kolmogorov turbulence are investigated numerically. The influences of turbulence parameter (i.e., the generalized exponent, the outer scale of turbulence, and the inner scale of turbulence) and OAM values on the entanglement evolution of two photon are investigated and compared with those of the Laguerre-Gaussian (LG) beam. The results indicate that the OAM entanglement to be more robust in turbulence for the higher OAM values, the higher generalized exponent. Moreover, the influence of the turbulence inner scale and outer scale on OAM entanglement can be ignored. The OAM entanglement of HG vortex beam is better than the OAM entanglement of LG beam for resisting the entanglement decoherence of atmosphere turbulence under the same conditions.