Abstract
Electromagnetic pulses are typically treated as space-time (or space-frequency) separable solutions of Maxwell's equations, where spatial and temporal (spectral) dependence can be treated separately. In contrast to this traditional viewpoint, recent advances in structured light and topological optics have highlighted the non-trivial wave-matter interactions of pulses with complex topology and space-time non-separable structure, as well as their potential for energy and information transfer. A characteristic example of such a pulse is the "Flying Doughnut" (FD), a space-time non-separable toroidal few-cycle pulse with links to toroidal and non-radiating (anapole) excitations in matter. Here, we propose a quantum-mechanics-inspired methodology for the characterization of space-time non-separability in structured pulses. In analogy to the non-separability of entangled quantum systems, we introduce the concept of space-spectrum entangled states to describe the space-time non-separability of classical electromagnetic pulses and develop a method to reconstruct the corresponding density matrix by state tomography. We apply our method to the FD pulse and obtain the corresponding fidelity, concurrence, and entanglement of formation. We demonstrate that such properties dug out from quantum mechanics quantitatively characterize the evolution of the general spatiotemporal structured pulse upon propagation.
Highlights
The generation, diagnostics, and applications of spatiotemporal electromagnetic pulses have attracted growing interest from diverse research communities, including metamaterials [1,2], communications [3], particle acceleration [4,5], laser machining [6,7,8], and nonlinear and topological photonics [9,10,11,12]
Based on the analogous mathematical description and physical origin, we introduce the new concept of a spacespectrum nonseparable state that allows us to quantitatively describe pulses with prescribed space-time nonseparability (STNS), such as the flying doughnut” (FD), i.e., |ψ =
If the target state is a pure state |ψ1, the density matrix is given by ρ1 = |ψ1 ψ1|, and the fidelity is simplified to F = Tr(ρ1ρ2) = ψ1|ρ2|ψ1 [42]
Summary
The generation, diagnostics, and applications of spatiotemporal electromagnetic pulses have attracted growing interest from diverse research communities, including metamaterials [1,2], communications [3], particle acceleration [4,5], laser machining [6,7,8], and nonlinear and topological photonics [9,10,11,12] Such pulses are treated as space-time (or, equivalently, space-frequency) separable solutions of Maxwell’s equations, which can be expressed as a product of a spatial mode and a temporal (or spectral) function, following the traditional separation of variables for solving partial differential equations. A measure of entanglement, the Schmidt number, was used to characterize nondiffracting properties in optical wave packets [61] These useful applications of quantum mechanics in classical optics have motivated the development of novel methods in (tele)communications [62,63,64], cryptography [65], and metrology [66,67,68]. The approach proposed here will lead to insights into light-matter interactions with ultrafast structured pulses and extend applications in spectroscopy, cryptography, and communications
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