Measures of space-time nonseparability of electromagnetic pulses

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Measures of space-time nonseparability of electromagnetic pulses

Yijie Shen, Apostolos Zdagkas, Nikitas Papasimakis, and Nikolay I. Zheludev

Phys. Rev. Research 3, 013236 – Published 12 March 2021

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 nontrivial wave-matter interactions of pulses with complex space-time nonseparable 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 nonseparable few-cycle pulse with links to toroidal and nonradiating (anapole) excitations in matter. Here, we propose a quantum-mechanics-inspired methodology for quantitatively characterizing space-time nonseparability in structured pulses. In analogy to the mathematics of nonseparability in quantum mechanics, we introduce the concept of space-spectrum nonseparable states to describe the space-time nonseparability of a classical electromagnetic pulse and apply the state tomography method to reconstruct the corresponding density matrix. Using the example of the FD pulse, we calculate the fidelity, concurrence, and entanglement of formation as their quantitative measures, and we demonstrate that such properties dug out from quantum mechanics can quantitatively characterize the spatiotemporal evolution of general structured pulses. Our results highlight the potential of space-time nonseparable pulses as information carriers and facilitate their deployment in information transfer and cryptography applications.

Received 27 July 2020
Revised 17 December 2020
Accepted 21 January 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.013236

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Entanglement measures Ultrashort pulses

Spacetime symmetries

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Yijie Shen ^1,*, Apostolos Zdagkas ¹, Nikitas Papasimakis¹, and Nikolay I. Zheludev^1,2

¹Optoelectronics Research Centre and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom
²Centre for Disruptive Photonic Technologies, School of Physical and Mathematical Sciences and The Photonics Institute, Nanyang Technological University, Singapore 637378, Singapore

^*y.shen@soton.ac.uk

Article Text

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Supplemental Material

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References

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Issue

Vol. 3, Iss. 1 — March - May 2021

Subject Areas

Optics

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Images

Figure 1
(a) Spatiotemporal structure of the FD pulse: The spatial isosurfaces of the electric field $E (t, r, z)$ of the TE FD at different times of $t = 0$ and $\pm q_{2} / (2 c)$ , at amplitude levels of $E = \pm 0.2$ . (a1) The $y − z$ cross section of the instantaneous electric field $E$ of the FD pulse at $x = 0$ for $t = 0$ and $\pm q_{2} / (2 c)$ . (a2) The $x − z$ cross section of the electric field intensity ${| E |}^{2}$ of the FD pulse at $x = 0$ , at $t = 0$ and $\pm q_{2} / (2 c)$ . (a3) The $x − y$ cross section of the electric field intensity of the FD pulse integrated over all times at the $z = 0$ plane $\int_{- \infty}^{\infty} {| E (t, r, 0) |}^{2} d t$ . The inset at top left shows the electromagnetic vector structure of the FD pulse at the focus ( $z = 0$ ). (b) Spectral structure of the FD pulse: The slices in the $r − λ$ domain show $| \tilde{E} {(λ, r, z) |}^{2}$ of the spectral components of the FD pulse at propagation distances $z$ . The color map in the $r − z$ plane shows a false-color image constructed by the distribution of monochromatic components at the corresponding radial position; the inset shows a similar false-color image in $x − y$ plane.
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Figure 2
(a) Profiles of radial distribution of normalized intensity $I (λ, r, z) = | \tilde{E} {(λ, r, z) |}^{2} / {max}_{(r, λ)} [| \tilde{E} (λ, r, z) |^{2}]$ of different spectral components of the FD pulse at the focus ( $z = 0$ ). Lines of different color represent monochromatic components of different wavelengths. (b) The color-coded traces of the radial positions where the intensity $I (λ, r, z)$ reaches its maximum for different wavelengths of the FD pulse. (c) Ratio $ξ (λ, z) = r_{λ} / r_{λ_{n}}$ , where $r_{λ}$ is the radial position of the peak of the intensity $I (λ, r, z)$ of the monochromatic component at wavelength $λ$ . Here, the radius $r_{λ_{n}}$ is used for normalization and corresponds to the position of peak intensity for a given wavelength $λ_{n}$ of the FD pulse. (d) Peak value of the intensity, $I_{m} (λ, z) = I (λ, r_{λ}, z)$ , for each monochromatic component of the FD pulse as a function of propagation distance $z$ . (e) The normalized total field $I_{0} (r, z) = \int I (λ, r, z) d λ / {max}_{r} [\int I (λ, r, z) d λ]$ plotted versus the value of $η (r, z) = r / r_{max} (z)$ at each propagation distance $z$ , where $r_{max} (z)$ is the radius at which $I_{0} (r, z)$ reaches its maximum. Note the isodiffraction property: $ξ (λ)$ , $| {\tilde{E}}_{m} {(λ) |}^{2}$ , and $I (η)$ do not depend on $z$ . The red and blue arrows demonstrate the positions of spectral and spatial states. Spectral states are represented by the trajectories of the electric field intensity maxima of the various monochromatic components, i.e., $r_{λ} (z)$ , and spatial states are represented by the trajectories of prescribed positions $r (z)$ fulfilling the prescribed radial ratios of $η_{i} = r / r_{max} (z)$ .
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Figure 3
(a) and (b) The transverse intensity patterns $I_{0} {(r) |}_{z = 0}$ of the FD pulse (a) and a wideband LG beam (see Note 1 of the Supplemental Material [72]) (b). (c) and (d) The propagation profiles of spectral ( $| λ_{i} 〉$ ) and spatial ( $| r_{i} 〉$ ) states of the FD beam (c) and wideband LG beam (d). The two insets in (d) show the spatial profiles of different wavelength components of the wideband LG beam at two different propagation distances, $z = 0$ (d1) and $z = 200 z_{0}$ (d2), respectively, where $z_{0}$ is $1 / 100$ of the Rayleigh length (averaged over all monochromatic components of the beam). (e) and (f) The $η − z$ map of spectral and spatial states of the FD beam (e) and wideband LG beam (f). Note that the LG beam experiences dramatic distortion upon propagation, whereas the FD pulse profile remains invariant owing to its isodiffracting nature. In this illustration, we have considered 20 different spectral and spatial states, $| λ_{i} 〉$ and $| η_{i} 〉$ ( $i = 1, 2, ..., 20$ ), where wavelength values are set as $λ_{i} = i q_{1}$ and radial ratios are selected as $η_{i} = r_{λ_{i}} / r_{max}$ .
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Figure 4
(a1) and (a2) Procedure for experimental determination of spectral states $| λ_{i} 〉$ ( $i = 1, 2, ..., 5$ ): (a1) The transverse intensity profiles corresponding to different monochromatic components can be obtained by capturing an image of the pulse after propagation through spectral filters at selected wavelengths $λ_{i}$ . (a2) The radial distribution of the intensity patterns presented in (a1). The radius $r_{λ_{i}}$ marks the position at which the intensity of the monochromatic component of wavelength $λ_{i}$ reaches its maximum value. (b1) and (b2) Procedure for experimental determination of spatial states $| η_{i} 〉$ ( $i = 1, 2, ..., 5$ ): (b1) The captured transverse profile of the total electric field intensity with the marked position of the recorded radius of $r = r_{max}$ ( $η = 1)$ at which the total intensity reaches its maximum. Based on the recorded position of $η = 1$ , the positions of various spatial ( $| η_{i} 〉$ ) states can be recorded through radially scaling the position of $η = 1$ by corresponding ratios of $η_{i}$ . (b2) Spatial states represented by trajectories in the $η − z$ plane. As the trajectories of the spatial states are parallel to the $z$ axis in the $η − z$ map, each position of the spatial state can be determined by the corresponding normalized radius $η_{i} = r / r_{max}$ . (c) Measurement matrix of state tomography of space-spectrum entangled states, where the inner products are noted as $〈 i | j 〉 = 〈 η_{i} | λ_{j} 〉$ ( $i, j = 1, 2, ..., 5$ ). The values of $η_{i}$ are selected so that the measurement matrix is diagonal for the ideal isodiffracting pulses. Here, the sample number is set to 5 for spectral and spatial states for illustration purposes.
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Figure 5
(a)–(c) The numerical results of quantum-analogous state tomography of the ideal FD pulse (a), the FD pulse with noise (b), and the wideband LG beam (c). The insets at the top right of (a)–(c) are the corresponding results of the intensity-normalized measurements. In (c), the tomography matrix is obtained by averaging multiple measurements at various propagation distances from $z = - 250$ to $z = 250$ with a step of 50 (unit: $q_{1}$ for FD and $z_{0}$ for LG). (d) The selected tomography matrices of the wideband LG beam at distances of $z = \pm 250, \pm 150, \pm 50$ (unit: $z_{0}$ ). (e)–(g) The reconstructed density matrices for the ideal FD pulse (e), the FD pulse with noise (f), and the wideband LG beam (g), with marked values of fidelity, concurrence, and EoF, respectively. (h) and (i) The tomography matrix (h) and density matrix (i) for a monochromatic beam. In all panels, red and blue indices, $i$ and $j$ , denote spatial $| η_{i} 〉$ and spectral states $| λ_{j} 〉$ . See the full data set of tomography and density matrices of the wideband LG beam at various propagation distances in Note 3 of the Supplemental Material [72].
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