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arXiv:1102.5166 · Physical Review A

Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence

Filippus Stefanus Roux

Published May 17, 2011• 80 citations
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Abstract

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.

DOI
10.1103/physreva.83.053822

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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 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.
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.
We study the evolution of an orbital angular momentum (OAM) entangled bipartite photonic state for the case where one of the photons propagates through Kolmogorov turbulence, using the concurrence as a measure of entanglement. Quantum state tomography was performed to reconstruct the two qubit density matrices for a range of scintillation strengths. Our results give the first direct experimental confirmation of the existing theories for decay of entanglement due to atmospheric turbulence. We also show how the modal scattering increases with increasing scintillation and we discuss the impact of the scale at which entanglement dissipates due to atmospheric turbulence on free-space quantum communication.