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arXiv:2102.09450 · Physical Review A

Ideal pairing of the Stokes and anti-Stokes photons in the Raman process

Kishore Thapliyal, Jan Peřina

Published Mar 17, 2021• 10 citations
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Abstract

A quantum model of the Raman process with the independent Stokes and anti-Stokes nonlinear interactions is developed to study nonclassical correlations between the photons in the Stokes and anti-Stokes fields. The role of the laser pump amplitude, the ratio of the Stokes and anti-Stokes coupling constants, and the population and losses of the vibrational mode in forming the correlations is elucidated. The ${g}^{(2)}$ intensity cross-correlation function, noise-reduction factor, two-mode principal squeezing variance, logarithmic negativity, nonclassicality depth, steering parameter, and the Bell parameter are analyzed side by side to shed light to the correlations between the Stokes and anti-Stokes fields. Conditions for having the Stokes and anti-Stokes fields composed of only photon pairs, similarly as it occurs in twin beams in parametric down-conversion, are revealed. They allow for nonzero mean thermal phonon numbers.

DOI
10.1103/physreva.103.033708

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A quantum model of the Raman process with the independent Stokes and anti-Stokes nonlinear interactions is developed to study nonclassical correlations between the photons in the Stokes and anti-Stokes fields. The role of the laser pump amplitude, the ratio of the Stokes and anti-Stokes coupling constants, and the population and losses of the vibrational mode in forming the correlations is elucidated. The g(2){g}^{(2)}g(2) intensity cross-correlation function, noise-reduction factor, two-mode principal squeezing variance, logarithmic negativity, nonclassicality depth, steering parameter, and the Bell parameter are analyzed side by side to shed light to the correlations between the Stokes and anti-Stokes fields. Conditions for having the Stokes and anti-Stokes fields composed of only photon pairs, similarly as it occurs in twin beams in parametric down-conversion, are revealed. They allow for nonzero mean thermal phonon numbers.
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It is shown that the Stokes and anti-Stokes components, related by the conditions ω s + ω α = 2 ω L and k s + k α =2 k L , are correlated. When the temperature of the scattering material is high [ kT ≫ h ( ω L – ω s )] and/or stimulated scattering exists, this correlation assumes a classical form and is analogous to the Brown-Twiss intensity correlation. This intensity correlation is readily explained by modulation of the pump wave due to molecular vibrations so that the thermal fluctuations of energy of these vibrations are synchronously transferred into the s and α sidebands. At low temperatures and pump intensities (as well as outside Raman resonances) a photon correlation is found which is explained by the simultaneous production of pairs of s and α photons as a result of the spontaneous decay of two pump photons. The statistics of the scattering field in this case assumes a nonclassical form characterized by a relatively high ( m ≫1) probability of pair production compared with the product of the probabilities for the production of separate s and α photons (with the intensity correlation m≤1). An estimate of the rate of coincidences for the scattering of 1-W focused TEM 00 pump radiation in liquid nitrogen yielded ~6 sec –1 and m~10 3 . Analogs of the Kirchhoff equation were obtained for the inelastic scattering of light. These formulas, derived from the conditions of unitarity of the scattering process and equilibrium of matter, define the statistics of the scattered light in terms of the Raman gain and the parametric conversion coefficient.
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.
Statistical properties of optical fields in nonlinear couplers composed of two waveguides in which Raman or Brillouin processes (with classical pumping) are in operation and which are mutually connected through the mutual Stokes and/or anti-Stokes linear interactions are investigated within the framework of a generalized superposition of coherent fields and quantum noise. Heisenberg equations describing the couplers are solved both analytically under special conditions and numerically in general cases. Regimes for nonclassical properties of optical fields, such as sub-Poissonian photon-number statistics, negative reduced moments of integrated intensity and squeezing of quadrature fluctuations are discussed for the cases of single and compound fields. General results are compared with those from the short-length approximation.
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.
The light initially prepared in a coherent state runs the stimulated and spontaneous Raman processes in a second-order nonlinear medium. Using analytical expressions of field operators for the light modes, we determine the quantum statistical properties of the modes in the stimulated and spontaneous Raman processes. By a choice of phase, we obtain a boson-pair property correlating the pump and vibration modes and such a property correlating the pump and anti-Stokes modes. The Stokes and vibration modes exhibit the boson-pair property for all initial phase differences between the pump mode and the other ones. In the spontaneous Raman process only the Stokes and vibration modes show such a property.