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arXiv:1009.1956

Decoherence of orbital angular momentum entanglement in a turbulent atmosphere

Filippus Stefanus Roux

Published Sep 10, 2010• 3 citations
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Abstract

The evolution of an entangled photon state propagating through a turbulent atmosphere is formulated in terms of a set of coupled first order differential equations, by using an infinitesimal propagation approach. The orbital angular momentum (OAM) basis is used to described the density matrix of the state. Although the analysis is done in the paraxial limit for a monochromatic optical field, the formalism is comprehensive in the sense that it does not require any assumptions about the strength of the turbulence and it can incorporate any spectral model for the turbulence. As a comparative example the case of entangled qubit OAM biphoton states is considered.

DOI
10.48550/arxiv.1009.1956

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The evolution of an entangled photon state propagating through a turbulent atmosphere is formulated in terms of a set of coupled first order differential equations, by using an infinitesimal propagation approach. The orbital angular momentum (OAM) basis is used to described the density matrix of the state. Although the analysis is done in the paraxial limit for a monochromatic optical field, the formalism is comprehensive in the sense that it does not require any assumptions about the strength of the turbulence and it can incorporate any spectral model for the turbulence. As a comparative example the case of entangled qubit OAM biphoton states is considered.
Orbital angular momentum (OAM) entangled bi-photons are a resource for the higher dimensional implementation of quantum cryptography, which allows secure communication over various channels. In the case where free-space is used as communication channel the initial OAM entangled bi-photon loses some or even all of its entanglement because of the scintillation that it experiences while propagating through the turbulence in the atmosphere. This decoherence of OAM entanglement has so far only been studied for the case of weak turbulence. Unfortunately, it is the more challenging strong turbulence scenario that is relevant for the practical implementation of free-space quantum communication through the atmosphere. Using an approach that differs from previous approaches, we derive a master equation for the evolution of an OAM entangled bi-photon during propagation through turbulence. However, in our approach the equation contains a derivative with respect to the propagation distance instead of time. The principle is to consider the propagation over an infinitesimal distance of OAM basis states through a random medium. This approach allows one to include, not only the effect of turbulence of arbitrary strength, but also the effect of the inner and outer scale of the turbulence, as represented by the Tartarskii and von Karman spectra. The resulting expression can predict the rates of decoherence for arbitrary initial OAM entangled states and can be used to calculate the concurrence, which measures the amount of entanglement, as a function of propagation distance for different initial entangled OAM states.
We derive a first-order differential equation for the decoherence of an orbital angular momentum entangled biphoton state propagating through a turbulent atmosphere. The derivation is based on the distortion that orbital angular momentum states experience due to propagation through a thin sheet of turbulent atmosphere. This distortion is treated as an infinitesimal transformation leading to a first-order differential equation, which we call an infinitesimal propagation equation. The equation is applied to a simple qubit case to show how the entanglement decays.
The orbital angular momentum (OAM) state of light can potentially be used to implement higher dimensional entangled systems for quantum communication. Unfortunately, optical fibers in use today support only modes with zero OAM values. Free-space quantum communication is an alternative to traditional way of communicating through optical fibers. However the refractive index fluctuation of the atmosphere gives rise to random phase aberrations on a propagating optical beam. To transmit quantum information successfully through a free-space optical channel, one needs to understand how atmospheric turbulence influences quantum entanglement. Here, we present a numerical study of the evolution of quantum entanglement between a pair of qubits. The qubits consist of photons entangled in the OAM basis. The photons propagate in a turbulent atmosphere modeled by a series of consecutive phase screens based on the Kolmogorov theory of turbulence. Maximally entangled initial states are considered, and the concurrence is used as a measure of entanglement. We show how the evolution of entanglement is influenced by various parameters such as the beam waist, the strength of the turbulence and the wavelength of the beam. We restricted our analysis to the OAM values l = ±1 and we compared our results to previous work.
Considering the evolution of the quantum state of a pair of entangled photons in a turbulent atmosphere and using the quadratic approximation to the Kolmogorov model of turbulence, we provide an analytical solution for the dynamical equation (an infinitesimal propagation equation) that describes this evolution in the plane-wave basis. As such, this solution fully incorporates the effect of multiple scattering, caused by the medium. After being converted into a discrete orbital angular momentum (OAM) basis, this solution retains the effect of coupling between different OAM modes on the twin-photon state for arbitrary propagation distances and arbitrary turbulence strengths. We define a minimal set of parameters that determines the entanglement evolution in the regime of strong scintillation. Furthermore, we show that in the limit of weak scintillation, our solutions reduce to those obtained from the single-phase-screen model.
The turbulence-induced decay of orbital-angular-momentum (OAM) entanglement between two photons is investigated numerically and experimentally. To compare our results with previous work, we simulate the turbulent atmosphere with a single phase screen based on the Kolmogorov theory of turbulence. We consider two different scenarios: in the first only one of the two photons propagates through turbulence, and in the second both photons propagate through uncorrelated turbulence. Comparing the entanglement evolution for different OAM values, we found the entanglement to be more robust in turbulence for higher OAM values. We derive an empirical formula for the distance scale at which entanglement decays in terms of the scale parameters and the OAM value.
Free-space optical communication based on orbital angular momentum (OAM) offers advantages such as high security, large information capacity, and high-speed transmission. However, turbulence can induce random perturbations to the wavefront, thereby affecting communication performance. Hence, accurately measuring turbulence intensity is crucial. We delve into the quantum Fisher information (QFI) of the spatial coherence length, a pivotal parameter in atmospheric turbulence. Within the framework of free-space optical channels, we conduct a comprehensive analysis of the QFI in models encompassing both single and biphoton systems. As photons propagate through free space, the QFI regarding turbulence parameter initially increases and then decreases, due to the coupling and decoherence between the photons and turbulence. Increasing the number of OAM modes can significantly enhance the QFI regarding turbulence parameters. We considered situations closer to practical preparation, our results indicate that the impact of entanglement on estimation precision depends on the purity of the prepared entangled state. Moreover, we discuss and compare the variations in the QFI of various parameters upon considering backward scattering (BS). We believe our work is the first theoretical attempt towards the exploration of estimating the turbulence parameter via QFI.