[Paper Review] Decoherence of orbital angular momentum entanglement in a turbulent atmosphere

Filippus S. Roux|arXiv (Cornell University)|Sep 10, 2010

Orbital Angular Momentum in Optics3 citations

TL;DR

This paper develops an infinitesimal propagation model for orbital angular momentum (OAM) entangled photons in turbulent atmosphere using a Markovian master-equation approach in the paraxial limit, without simplifying turbulence assumptions. It shows that while OAM mode scattering occurs rapidly (at r₀ ≈ ω₀), entanglement decays slowly due to small coefficients in the decoherence term, implying mode scattering is a greater challenge than entanglement loss for free-space quantum communication.

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.

Motivation & Objective

• To model the spatial evolution of OAM-entangled photon states through atmospheric turbulence without simplifying turbulence assumptions.
• To analyze decoherence effects on OAM-based quantum states in realistic atmospheric conditions using a first-principles approach.
• To quantify the relative impact of mode scattering versus entanglement decay in free-space quantum communication.
• To provide a comprehensive formalism applicable to any spectral model of atmospheric turbulence and any OAM state.

Proposed method

• Uses an infinitesimal propagation approach to derive a differential equation for the density operator's evolution in distance, analogous to a master equation but with respect to propagation length.
• Expresses OAM modes as momentum-space wave functions via 2D Fourier transforms of Laguerre-Gaussian modes, enabling treatment of turbulence-induced phase distortions.
• Models refractive index fluctuations via a three-dimensional power spectral density Φ₀(k), with random complex spectral functions satisfying statistical independence (Markov approximation).
• Derives a perturbative transformation for the momentum-space wave function over dz, incorporating free-space propagation and turbulence-induced phase modulation via convolution with N(K,z).
• Expands the distorted wave function back into the orthogonal OAM basis to update density matrix elements, enabling tracking of entanglement evolution.
• Truncates the infinite system of differential equations to study a two-photon singlet Bell state in the OAM basis, yielding closed-form solutions for trace and concurrence.

Experimental results

Research questions

• RQ1 How does atmospheric turbulence affect the evolution of OAM-entangled photon states in free space?
• RQ2 What is the relative timescale of OAM mode scattering versus entanglement decoherence in turbulent conditions?
• RQ3 How do the parameters ω₀ (beam waist) and r₀ (Fried parameter) influence the survival of OAM entanglement?
• RQ4 To what extent does the outer scale of turbulence affect OAM entanglement decay?
• RQ5 Can a general, turbulence-agnostic formalism be derived for OAM-based quantum states in the paraxial regime?

Key findings

• The trace of the density matrix decays exponentially with propagation distance, with decay rate dependent on the ratio ω₀/r₀, indicating photon loss from the desired OAM modes.
• The concurrence of the OAM-entangled state decays slowly, with the decay governed by small coefficients B_q (e.g., 0.0303 for q=1), leading to longer entanglement persistence than previously estimated.
• For q=1,2,3, the concurrence remains significant even when ω₀/r₀ exceeds 1, indicating that higher-OAM states retain entanglement longer than lower-OAM states.
• The outer scale of turbulence does not affect the final decoherence rate, as its contribution cancels exactly in the final expression.
• Scattering into other OAM modes occurs at a scale where r₀ ≈ ω₀, consistent with prior work, but entanglement decay is at least an order of magnitude slower than scattering, making mode scrambling the dominant challenge.
• The model confirms that the primary obstacle in free-space OAM quantum communication is not entanglement decoherence per se, but the rapid scattering of photons into unintended OAM modes.