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Physical Review A

Polarization state tomography of Stokes–anti-Stokes photon pairs from degenerate four-wave mixing in diamond

Tiago A. Freitas, Paula Machado, Riichiro Saito, Marcelo França Santos, Carlos Henrique Monken, Ado Jório

Published Nov 25, 2024• 2 citations
Publisher

Abstract

Degenerate four-wave mixing in diamond generates a quantum state ${\ensuremath{\rho}}_{\mathrm{SaS}}$ composed of Stokes (S) and anti-Stokes (aS) time-correlated photon pairs (SaS). Here, polarization state tomography of ${\ensuremath{\rho}}_{\mathrm{SaS}}$ is performed in four pertinent scattering configurations: Raman shift in resonance and out of resonance with the phonon energy, and incident light polarized in the direction parallel to or $\ensuremath{\pi}/4$ rotated from a diamond crystallographic axis. We reconstruct the density matrix, calculate concurrence and purity for the four configurations, and directly measure and analyze the diagonal terms of ${\ensuremath{\rho}}_{\mathrm{SaS}}$ in different bases. We establish ${\ensuremath{\rho}}_{\mathrm{SaS}}$ as an entangled linear combination of electronic four-wave mixing and SaS Raman process, mixed with uncorrelated S and aS photons, with the weight constants depending on the scattering conditions, which can be engineered for future applications.

DOI
10.1103/physreva.110.053714

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•Nov 25, 2020•0 citations
Degenerate four-wave mixing in diamond generates a quantum state \ensuremathρSaS{\ensuremath{\rho}}_{\mathrm{SaS}}\ensuremathρSaS​ composed of Stokes (S) and anti-Stokes (aS) time-correlated photon pairs (SaS). Here, polarization state tomography of \ensuremathρSaS{\ensuremath{\rho}}_{\mathrm{SaS}}\ensuremathρSaS​ is performed in four pertinent scattering configurations: Raman shift in resonance and out of resonance with the phonon energy, and incident light polarized in the direction parallel to or \ensuremathπ/4\ensuremath{\pi}/4\ensuremathπ/4 rotated from a diamond crystallographic axis. We reconstruct the density matrix, calculate concurrence and purity for the four configurations, and directly measure and analyze the diagonal terms of \ensuremathρSaS{\ensuremath{\rho}}_{\mathrm{SaS}}\ensuremathρSaS​ in different bases. We establish \ensuremathρSaS{\ensuremath{\rho}}_{\mathrm{SaS}}\ensuremathρSaS​ as an entangled linear combination of electronic four-wave mixing and SaS Raman process, mixed with uncorrelated S and aS photons, with the weight constants depending on the scattering conditions, which can be engineered for future applications.
The last decades have been marked by intense research on the evolution of computer and communication systems. Within this context, quantum computing has been one of the areas that generated large interest and effort from researchers due, to the immense potential offered. Quantum information research has been through numerous evolutionary stages, and photonic transport is one of the most promising methods for information processing. Within the context of the use of photons for information transportation, Raman spectroscopy, which consists of an optical spectroscopy technique based on the inelastic scatter of light, where the scattered photons carry information from the vibration structure of the medium, is presented as an exciting way for coding and transportation of quantum information. In this work, we explore the phenomenon of correlated Stokes- anti-Stokes (SaS) Raman scattering, analyzing the quantum character of the phenomenon. The implementation of the experimental apparatus was performed for the measurement of the correlation of the photons generated in this phenomenon in diamond, confirming its quantum character and optimizing the phenomenon for a complete analysis of the process in order to confirm that besides being correlated, the pair of SaS photons would also be entanglement. Initially, we studied the dependence on phenomenon efficiency with different lasers (continuous, ps, and fs pulsed, and different wavelengths) and the dependence on the correlation function with the Raman shift. These studies have shown that the best configuration for the study regarding the possible entanglement of the process would be using the fs laser in 785nm. In the following steps, state tomography measurements were performed to completely characterize the polarization state of the SaS pair. During this study, it was possible to observe the influence of the crystallographic orientation of the diamond on the SAS polarization state, which directly influences the generation of entanglement. Finally, Bell’s inequality measurements of the CHSH type were performed to confirm the entanglement in polarization of the SaS state.
Quantum information research has been through numerous evolutionary stages, and photonic transport is one of the most promising methods for information processing. Within the context of the use of photons for information transportation, Raman spectroscopy, which consists of an optical spectroscopy technique based on the inelastic scatter of light, where the scattered photons carry information from the vibration structure of the medium, is presented as an exciting way for coding and transportation of quantum information. Herein, the phenomenon of correlated Stokes‐anti‐Stokes (SaS) Raman scattering is explored, analyzing the quantum character of the phenomenon. A state tomography is performed to fully characterize the SaS pair's polarization state. During this study, it is possible to observe the influence of the crystallographic orientation of the diamond on the SaS polarization state, which directly influences the entanglement generation. A Clauser, Horne, Shimony, and Holt‐type Bell's inequality measurement is conducted to confirm the entanglement in the polarization of the SaS state.
The present dissertation entails a detailed investigation into the asymmetry in the formation of correlated Stokes and anti-Stokes photon pairs before and after the resonance region in a diamond sample. The analysis undertaken in this study is based on the dependence on polarization and frequency, combined with two distinct crystallographic orientations of the sample, aiming to explore the physical characteristics of this asymmetry. Initially, the physical and structural properties of diamond are presented. Subsequently, the physical phenomenon under study is addressed, starting from uncorrelated Raman scattering to correlated Raman scattering. Following this, the experimental apparatus used to measure the desired physical quantities is described. The results obtained are then discussed. Finally, the conclusion section provides a concise and general analysis of the experimental results, which indicate that correlated Stokes and anti-Stokes Raman scattering constitutes a specific type of four-wave mixing. This four-wave mixing can be divided into a resonant part and a non-resonant part, providing an adequate explanation for the asymmetry issue.
The correlation between the Stokes and anti-Stokes components of Raman scattering has assumed an important role in the field of quantum information. When the Raman shifts match the energy of a phonon in the material, in the real-SaS, one-phonon Fock state are generated. When such resonance is not achieved, the energy exchange is mediated by virtual phonons, generating photonic Cooper pairs (PCPs). In this work, investigations of polarization correlations and scattering time dependences elucidate the fundamental difference between the real and the virtual phenomena.
The correlated Raman scattering, referred to here as the SaS process, occurs with the annihilation, within a material, of two photons of an excitation laser and the creation of a quantum correlated Stokes (S) and anti-Soktes (aS) photon pair where the phonon created in the S process is annihilated in the aS scattering. We found that, in addition to real SaS process where the energy exchange coincides with the energy of a vibrational state of the material, there is also the virtual SaS process, whose the exchange of energy and momentum between the photons of the laser, generating the S-aS photon pair, is mediated by a virtual vibrational excitation, setting up a photon-photon interaction similar to the electron-electron attraction responsible for the creation of Cooper pairs of the BCS theory of superconductivity. We observed the SaS process in transparent media in general, by measuring non-classical values of the normalized second-order cross-correlation function of the Stokes and anti-Stokes fields, 2(0). In diamond, we characterize the intensity of the virtual SaS scattering ( ) as a function of the virtual phonon energy, proving that the SaS process occurs only when the photon-photon interaction is attractive, as in the BCS theory for the Cooper pairs. We show that photons S and aS from the SaS process are scattered predominantly in the same polarization as the excitation laser of the sample and, with this, we implemented the pump-probe technique to study the lifetime of the real phonon (obtained in the diamond order 2.8 ps ) and the lifetime of the virtual excitation, which proved to be inferior to the temporal resolution of the measurement’s, characterizing one of the main differences between the two processes. We also observed that, due to the relations of energy and momentum conservation, have the same spatial intensity profile of the non-interacting laser that crosses the sample, different from the behavior observed for uncorrelated Raman scattering which spreads photons in all directions. We observed that the efficiency of the SaS scattering is proportional to the Raman scattering cross-section when we compare the process in different hydrocarbons and we generalize the observations made in the diamond to the sample of liquid decane. In water, the values of 2(0), predicted by the theoretical description of the process, corroborate the experimental data, validating the BCS theory for virtual SaS scattering, setting the pair of Stokes and anti-Stokes photons as counterparts of the Cooper pairs.