Tim Ehret

An open quantum system theory for dynamics in disordered environments

The propagation of twisted photons carrying orbital angular momentum (OAM) through a turbulent atmosphere has been a subject of intense study, owing to the fact that the OAM basis offers the possibility to encode high-dimensional quantum systems (qudits), thus paving the way for higher channel capacities and enhanced security of free-space quantum communication compared to two-dimensional encoding. However, free-space propagation through the atmosphere is naturally accompanied by deleterious effects – namely, distortions of the twisted photon’s phase and intensity profile due to fluctuations of the refractive index, which induce losses and coupling (crosstalk) among different OAM modes. To compensate for these errors by means of adaptive optics (i.e., deformable mirrors reversing phase distortions [1]) or to identify turbulence-resilient singular modes [2], it is necessary to simulate the twisted photon’s propagation through the turbulent medium – a task that, given the stochastic nature of refractive-index fluctuations, can only be approached from an ensemble-average perspective.
Classically, the standard approach to this problem is the numerical solution of the stochastic parabolic equation that incorporates the refractive index as a stochastic quantity. Based on this, formidable numerical tools employing multiple phase screens have been developed [3], with the latter accounting for both diffraction and refraction. In this approach, a series of equidistantly placed, randomly generated screens (obeying the statistics dictated by the phase power spectrum) induces refraction, whereas diffraction occurs between screens.
Notwithstanding the success of this standard approach, a fully quantum treatment of the propagation of photons through turbulence is highly desirable. This problem is important not only from a fundamental perspective, but also because it enables us to take advantage of the open-quantum-systems toolbox. While a brief survey of the problem and the underlying challenges can be found in Ref. [4], it has remained essentially unsolved until now.
Revisiting the fully quantum approach in our work, we rigorously derive a quantum master equation for the ensemble-averaged density matrix of twisted photons that are phase-distorted by turbulence. This is obtained from a Hamiltonian constructed by quantizing the electromagnetic field in a non-homogeneous dielectric medium in a basis of Laguerre-Gaussian modes [4,5], together with a novel approach to stochastic-average dynamics put forward in Ref. [6]. Our master equation covers the general N-photon case and can be numerically solved for appropriately truncated Hilbert spaces, allowing for comparison to the state of the art in the research field. Interestingly, we were able to show the equivalence of our master equation to the one presented by Roux [7], which was derived in a fundamentally different way starting from the stochastic parabolic equation. Further, we elucidate how our master equation is related to other maps [8] and master equations in the literature devised for the same purpose. Ongoing efforts are dedicated to the efficient numerical implementation of our master equation and to the verification of analytical and numerical results by reproducing established features in the crosstalk matrix of OAM modes.

[1] Giacomo Sorelli et al., Entanglement protection of high-dimensional states by adaptive optics, In: New Journal of Physics 21.2 (2019), p. 023003. issn: 1367-2630. doi: 10.1088/1367-2630/ab006e
[2] David Bachmann et al., Highly Transmitting Modes of Light in Dynamic Atmospheric Turbulence, In: Phys. Rev. Lett. 130 (2023), p. 073801. doi: 10.1103/PhysRevLett.130.073801
[3] David Bachmann et al., Accurate Zernike-corrected phase screens for arbitrary power spectra, In: Optical Engineering 64.5 (2025), p. 058102. doi: 10.1117/1.OE.64.5.058102
[4] Giacomo Sorelli, Quantum state transfer in diffractive and refractive media, PhD thesis, Universität Freiburg, (2019), doi: 10.6094/UNIFR/150962
[5] David Bachmann et al., Highly Transmitting Modes of Light in Dynamic Atmospheric Turbulence, In: Phys. Rev. Lett. 130 (2023), p. 073801. doi: 10.1103/PhysRevLett.130.073801
[6] Li Yu and Daniel F. V. James, Average quantum dynamics of closed systems over stochastic Hamiltonians, In: Scientific Reports 15.1, (2025), p. 30984. issn: 2045-2322. doi: 10.1038/s41598- 025- 14825- z
[7] Filippus S. Roux, The Lindblad equation for the decay of entanglement due to atmospheric scintillation, Journal of Physics A: Mathematical and Theoretical 47.19, (2014), Doi: 10.1088/1751-8113/47/19/195302
[8] David Bachmann, Vyacheslav N Shatokhin, and Andreas Buchleitner, Universal entanglement decay of photonic orbital angular momentum qubit states in atmospheric turbulence: an analytical treatment, In: Journal of Physics A: Mathematical and Theoretical 52.40, (2019), p. 405303. issn: 1751-8121. doi: 10.1088/1751-8121/ab3f3c

Supervisor: Andreas Buchleitner