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arXiv:1807.11568 · Physical Review A

Golden ratio entanglement in hexagonally poled nonlinear crystals

A. Gatti, E. Brambilla, Katia Gallo, Ottavia Jedrkiewicz

Published Nov 19, 2018• 9 citations
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Abstract

This work analyzes the quantum state of twin photons and twin beams generated by parametric down-conversion in a hexagonally poled photonic crystal, characterized by the simultaneous presence of two nonlinear processes sustained by two vectors of the reciprocal lattice. In those special points of the fluorescence spectrum where the two processes coexist, we show that a tripartite entangled state is realized, equivalent to a single parametric process followed by a beam splitter. By proper angle tuning, a peculiar resonance condition is reached, with a transition to a four-mode entanglement, dominated by the golden ratio of the segment $\ensuremath{\phi}=(1+\sqrt{5})/2$. A maximal coherence between the two nonlinear processes is established here, as the overall process is shown to be equivalent to two independent parametric processes followed by a beam splitter. We offer an interpretation of the occurrence of the golden ratio in this system based on an analogy between the evolution of the light modes and the Fibonacci sequence.

DOI
10.1103/physreva.98.053827

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This work analyzes the quantum state of twin photons and twin beams generated by parametric down-conversion in a hexagonally poled photonic crystal, characterized by the simultaneous presence of two nonlinear processes sustained by two vectors of the reciprocal lattice. In those special points of the fluorescence spectrum where the two processes coexist, we show that a tripartite entangled state is realized, equivalent to a single parametric process followed by a beam splitter. By proper angle tuning, a peculiar resonance condition is reached, with a transition to a four-mode entanglement, dominated by the golden ratio of the segment \ensuremathϕ=(1+5)/2\ensuremath{\phi}=(1+\sqrt{5})/2\ensuremathϕ=(1+5​)/2. A maximal coherence between the two nonlinear processes is established here, as the overall process is shown to be equivalent to two independent parametric processes followed by a beam splitter. We offer an interpretation of the occurrence of the golden ratio in this system based on an analogy between the evolution of the light modes and the Fibonacci sequence.
We study the quantum state of twin photons generated by parametric down-conversion in a two-dimensional nonlinear photonic crystal, driven by a dual pump beam. The interplay between the grating of the ${\ensuremath{\chi}}^{(2)}$ nonlinear response and the transverse spatial modulation of the pump opens a rich scenario of possibilities for controlling the quantum state. We focus on a spatial resonance condition between the pump and the nonlinear grating, where a quadripartite entangled state is generated in proper light modes. The device is shown to implement the mixing of a pair of independent two-mode squeezed states onto four separate spatial modes, without the need of any external beam-splitter. Remarkably, both the squeezing parameters and the mixing coefficients can be reconfigured by modulating the relative phase and intensity of the two pumps, enabling thus the generation of a variety of quantum states in a single device.
We review recent experimental and theoretical progress in the study of optical parametric generation and cascaded frequency up conversion in two-dimensional purely nonlinear lattices, relying on acoherent coupling of multiple resonances in the quadratic lattice. Since their original proposal (1), two dimensional (2D) nonlinear photonic crystals in periodically poled materials have been the subject of a number of experimental and theoretical investigations, exploring the enhanced possibilities to engineer the nonlinear dynamics of quadratic interactions afiorded in such structures (2,3). Most of the studies have concerned frequency up-conversion, until the recent demonstration of twin-beam optical parametric generation (TB-OPG) in 2D hexagonally poled crystals (4,5). TB-OPG experiments in hexagonally poled MgO:LiTaO3 lattices demonstrated enhanced para- metric gains as well as a rich spectral and angular response underpinned bythe coherent coupling of concurrent nonlinear processes throughshared signal or idlermodes. They provided evidence forrich interaction scenarios, generating multiple outputs in the infrared as well in the visible and encompassing also self-phase modulation and cascaded upconversion processes arising from the interplay of multiple lattice resonances.A good agreement was found with a simple model in- volving the cross-seeding of two non-collinear OPG processes via a common signal (idler), while the development of advanced ad-hoc simulation tools enabled to fully account for the complex nonlinear dynamics in the 2D lattices and faithfully reproduce the rich spectral-angular response seen in the experiments (6).
We study parametric interactions in a new type of nonlinear photonic structure, which is realized in the volume between a pair of nonlinear crystals. In this kind of structure, which we call binary, multiple nonlinear optical processes can be implemented simultaneously, owing to multiple phase-matching conditions, fulfilled separately in the constituent crystals. The coupling between the nonlinear processes by means of modes sharing similar frequency is attained by the spatially broadband nature of the parametric fields. We investigate the spatial properties of the fields generated in the binary structure constructed from periodically poled crystals for the two examples: (1) single parametric downconversion and (2) coupled parametric downconversion and upconversion processes. The efficacy of the fields’ generation in these examples is analyzed through comparison with the cases of a traditional single periodically poled crystal and an aperiodic photonic structure, respectively. For this, we have devised a method that allowed us to solve the field equations numerically without resorting to statistically modeling the input vacuum states. It has been shown that the relative shift between the periodic crystal lattices has a crucial effect on the generated spatial field spectrum and overall efficiency. In addition, the influence of the intercrystal distance on these characteristics has been studied. Therefore, our study suggests that one can construct optical elements with sophisticated nonlinear properties from simpler elements without significant sacrifice of efficacy.
We experimentally investigate the parametric down-conversion process in a nonlinear bulk crystal, driven by two non-collinear pump modes. The experiment shows the emergence of bright hot-spots in modes shared by the two pumps, similar to the phenomenology recently observed in 2D nonlinear photonic crystals. By exploiting the spatial walk-off between the two extraordinary pump modes, we have been able to recreate a peculiar resonance condition, reported by a local enhancement of the parametric gain, which corresponds to a transition from a three-mode to a four-mode coupling. From a quantum point of view, this opens the way to the generation of multimode entangled states of light, such as tripartite or quadripartite states, in simple bulk nonlinear sources.