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Journal of the Optical Society of America B

Tunable spatial–spectral phase compensation of type-I (ooe) hyperentangled photons

Salem F. Hegazy, Salah S. A. Obayya

Published Feb 18, 2015• 17 citations
Publisher

Abstract

The two-photon state generated by overlapping the noncollinear spontaneous parametric down conversion (SPDC) emissions of two coherently pumped type-I crystals undergoes spatial and spectral decoherence precluding the capture of high pair counts with high state fidelity. Here we present a tunable compensation method for tackling the rising state decoherence in the two degrees of freedom in the case of crystals with negative birefringence. We determine the directional-spectral phase function of the two-photon state initially and after compensation, taking into account the finite spectral width of the pump beam. Subsequently, the optimal characteristics of the compensation crystals are computed by criteria of flat phase function in spectrum and space. The results of our model are compared with measurements of some recent experiments.

DOI
10.1364/josab.32.000445

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The two-photon state generated by overlapping the noncollinear spontaneous parametric down conversion (SPDC) emissions of two coherently pumped type-I crystals undergoes spatial and spectral decoherence precluding the capture of high pair counts with high state fidelity. Here we present a tunable compensation method for tackling the rising state decoherence in the two degrees of freedom in the case of crystals with negative birefringence. We determine the directional-spectral phase function of the two-photon state initially and after compensation, taking into account the finite spectral width of the pump beam. Subsequently, the optimal characteristics of the compensation crystals are computed by criteria of flat phase function in spectrum and space. The results of our model are compared with measurements of some recent experiments.
High flux of hyperentangled photons entails collecting the two-photon emission over relatively wide extent in frequency and transverse space within which the photon pairs are simultaneously entangled in multiple degrees of freedom. In this paper, we present a numerical approach to determining the spatial-spectral relative-phase and time-delay maps of hyperentangled photons all over the spontaneous parametric down conversion (SPDC) emission cone. We consider the hyperentangled-photons produced by superimposing noncollinear SPDC emissions of two crossed and coherentlypumped nonlinear crystals. We adopt a vectorial representation for all parameters of concern. This enables us to study special settings such as the self-compensation via oblique pump incidence. While rigorous quantum treatment of SPDC emission requires Gaussian state representation, in low-gain regime (like the case of the study), it is well approximated to the first order to superposition of vacuum and two-photon states. The relative phase and time-delay maps are then calculated between the two-photon wavepackets created along symmetrical locations of the crystals. Assuming monochromatic plane-wave pump field, the mutual signal-idler relations like energy conservation and transversemomentum conservation define well one of the two-photon with reference to its conjugate. The weaker conservation of longitudinal momentum (due to relatively thin crystals) allows two-photon emission directions coplanar with the pump beam while spreading around the perfect phase-matching direction. While prior works often adopt first-order approximation, it is shown that the relative-phase map is a very well approximated to a quadratic function in the polar angle of the two-photon emission while negligibly varying with the azimuthal angle.
Realizing high flux of hyperentangled photons requires collecting photon pairs simultaneously entangled in multiple degrees of freedom over relatively wide spectral and angular emission ranges. We consider the hyperentangled photons produced by superimposing noncollinear spontaneous parametric down conversion (SPDC) emissions of two crossed and coherently pumped nonlinear crystals. We present an approach for determining the directional-spectral relative-phase and time-delay maps of hyperentangled photons all over the SPDC emission cone. A vectorial representation is adopted for all parameters of concern. This enables us to examine unconventional arrangements such as the autocompensation of relative-phase and time-delay via oblique pump incidence. While prior works often adopt first-order approximation, it is shown that the actual directional relative-phase map is very well approximated by a quadratic function of the polar angle of the two-photon emission while negligibly varying with the azimuthal angle.
The polarization entangled state produced via spontaneous parametric downconversion (SPDC) has relativephase maps in frequency and momentum domains which give an almost complete picture about the distinguishability and purity loss in the conjugate time and space domains. We demonstrate experimentally the tunable compensation of directional relative-phase profile for entangled photons generated by two cascaded / crossed crystals and captured over ultra-wide spatial window. We use a phase-only spatial light modulator (SLM) programmable via a personal computer to flatten (or correct for) the spatial relative-phase profile and also to add on-demand spatial phase profile. A fast, yet accurate, technique is introduced for frequent relative-phase measurements based on the tilt angle of a quarter wave plate (QWP) acting on the diagonally polarized pump beam and nulling the relative-phase of the entangled state at that direction. Our experimental measurements verify previous theoretical models for tunable compensation of the polarization two-photon state produced by the cascaded crystals arrangement.
Photons produced in the process of spontaneous parametric down-conversion (SPDC) are typically entangled in multiple degrees of freedom. Although polarization is often the parameter of interest, the presence of spectral or spatial entanglement can complicate experiments, either by reducing the visibility of certain interference effects or by reducing collection efficiency. Several recent works have shown that the collection efficiency can be improved by focusing the pump beam. This approach has the added benefit of increasing the bandwidths of the emitted photons. We show that, in the case of type II SPDC, a focused pump can result in different spatial profiles for the signal and idler. If the crystal is configured for emission of polarization-entangled photon pairs, this effect will reduce the fidelity of that entanglement. Moreover, this spatial asymmetry leads to different spectral profiles for the two photons, even when the pump is monochromatic. The spectral and spatial asymmetries can be attributed to the difference in the angular dispersion (walk-off) of the two polarizations, along with a strong correlation between wavelength and emission direction. We also examine the link between spatial entanglement and single-mode coupling efficiency. We find that efficiency is maximized when spatial entanglement can be eliminated. For the case in which walk-off does not play a role, this can be accomplished by properly focusing the pump.
The generation of high-fidelity polarization-entangled photon pairs, to date, has been demonstrated on specific spatial modes or over relatively narrow apertures. We put forward and demonstrate an experimental scheme to extend the temporal and spatial indistinguishability of polarization-entangled photons over wide emission angles, which can be applied to cover the whole spontaneous parametric downconversion (SPDC) cone. Over such a wide angular extent, while the time-delay map is almost flat which renders the conventional compensation via a birefringent element an appropriate approach, the relative-phase map—verified as a quadratic function—necessitates a tunable compensation paradigm. Here, to do so, we employ a phase-only two-dimensional spatial light modulator (2D SLM) loaded by the complementary of the relative phase map to equalize the phase variations for one third of the noncollinear SPDC emission. After eliminating the temporal and spatial distinguishability over the 2D SLM area, a 97% polarization visibility is verified for the entangled photon pairs scattered widely across the SPDC cone.
Here we investigate the all-optical control of spectral correlations in spontaneous parametric down-conversion. We show that when photon pairs are projected onto high-order spatial modes, the spatial structure of the pump field defines the phase-matching function of the nonlinear interaction. Thus, by structuring the pump field in both space and spectrum, the biphoton spectral correlations are fully controlled. Considering a standard periodically-poled crystal as the nonlinear medium, we show that the Gouy phase matching method proposed here can generate both spectrally uncorrelated and high-dimensional spectrally entangled photon pairs, similarly to what is achieved with aperiodically-poled crystals. Furthermore, we show that our method can generate a wider class of quantum states if the pump field is a spatiotemporal wavepacket, that is, if its spatial and spectral structures are correlated.