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Quantum correlations in the Stokes and anti-Stokes Raman scattering: the photonic counterparts of Cooper pairs

Filomeno Soares de Aguiar Júnior

Published Nov 25, 2020• 0 citations
OpenAlex

Abstract

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.

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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.
In this work, the Stokes-anti-Stokes¿ time-correlation phenomena was studied. The timecorrelated scattering process occurs when the same vibrational quantum from a material participates in the Stokes and anti-Stokes photon generation process in the inelastic scattering of light. We demonstrate that the SaS phenomenon occurs in different bands of the Raman spectra, from different transparent liquid materials. The observation of time-correlated SaS pairs in energy bands out of the resonance region indicates the possibility of energy exchange between photons via virtual phonons, which allows drawing a parallel with the Cooper pair formation phenomenon for electrons, from the BCS superconductivity theory. In this dissertation, we discuss the fundamentals of the theory and the instrumentation related to the concepts about the discussed phenomena, as well as demonstrate that the rate of SaS pair formation exhibits a linear relation with the supertransparency gap, concept brought from the BCS superconductivity theory.
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.
In this study, the correlated Stokes--anti-Stokes Raman scattering, denominated as the SaS process, is investigated in a decane sample, generalizing for a liquid the results observed in crystalline diamond. The production of SaS photon pairs is observed in both real and virtual processes by measuring nonclassical values of the second-order cross-correlation function gSaS2(0){g}_{\text{SaS}}^{2}(0)gSaS2​(0). The correlated Raman spectrum is observed only for Raman shift values below approximately $2900\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$, which is equivalent to the highest vibrational mode of the decane molecule. Furthermore, despite the fact that decane is a liquid, the polarization and spatial profile of the SaS process in decane presents the same properties as those observed in diamond.
In the correlated Stokes--anti-Stokes scattering (SAS) an incident photon interacts with a Raman-active material, creating a Stokes photon and exciting a quantum vibrational mode in the medium, which is posteriorly annihilated on contact with a second incident photon, producing in turn an anti-Stokes photon. This can be accomplished by real and virtual processes. In the real process the quantum mode shared between the Stokes and anti-Stokes events is a real particle, whereas in virtual processes the pair formation is mediated by the exchange of virtual particles. Here, we introduce a Hamiltonian to describe the pair production in SAS scattering, for both types of process, when stimulated by two orthogonally polarized laser pulses in a pump-and-probe configuration. We also model the effect of the natural decay of the vibration created in the Stokes event and compute the probability of producing SAS pairs. Additionally, we follow the dynamics of the vibration by considering the Stokes and anti-Stokes fields as external reservoirs, obtaining thus a master equation for the reduced density matrix for the vibrational population. Finally, we compare our theoretical results to recently published experimental data.
The photon wave function formalism provides an alternative description of some quantum optical phenomena in a more intuitive way. We use this formalism to describe the Stokesanti-Stokes (SaS) correlated scattering process present in Raman scattering. In this process, two photons from a laser beam are inelastically scattered by a phonon created by the first photon (Stokes processes) and annihilated by the second photon (anti-Stokes process), producing a pair of Stokes–anti-Sokes (SaS) photons. . We arrive at an expression for the two-photon wavefunction, in the time and frequency domains, of the scattered SaS photon pair, which is in agreement with a number of experimental results. It was shown, from the phase matching analysis, that the scattering pattern is compatible with the experiments. It was seen that, in the application for Pump-and-probe experiments, it was possible to generate the correct curves for the decay of the population of phonons in the sample. Finally, it was verified that, through the analysis of the joint spectrum of the pairs, information was obtained about the entanglement in frequency of the scattered photons. Thus, it is concluded that, from the results, the photon wave function formalism provides an intuitive and robust way to calculate the SaS scattering properties.
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.