Lifetime and Polarization for Real and Virtual Correlated Stokes-anti-Stokes Raman Scattering in Diamond

The production of correlated Stokes (S) and anti-Stokes (aS) photons (SaS process) mediated by real or virtual phonon exchange has been reported in many transparent materials. In this work, we investigate the polarization and time correlations of SaS photon pairs produced in a diamond sample. We demonstrate that both S and aS photons have mainly the same polarization of the excitation laser. We also perform a pump-and-probe experiment to measure the decay rate of the SaS pair production, evidencing the fundamental diference between the real and virtual (phonon exchange) processes. In real processes, the rate of SaS pair production is governed by the phonon lifetime of $(2.8 \pm 0.3)$ ps, while virtual processes only take place within the time width of the pump laser pulses of approximately 0.2 ps. We explain the diference between real and virtual SaS processes by a phenomenological model, based on probabilities of phonon creation and decay.


I. INTRODUCTION
In correlated Raman scattering, the same phonon participates in both Stokes (S) and anti-Stokes (aS) frequency conversions, characterizing the SaS process 1,2 . The phonon created in the Stokes process is annihilated by the anti-Stokes one, generating a time-correlated photon pair. The phonon generated in a resonant S scattering has a lifetime τ p , which is typically of the order of a few picoseconds, and the correlated photon pair is created only if the aS scattering happens within a delay time not much longer than τ P 3 . This so-called real SaS scattering has been studied in several materials [4][5][6] , explored as the implementation of a Raman quantum memory for light in diamond 3,7,8 and gases [9][10][11] , and used to measure the lifetime of one-phonon Fock state, of the order to 3.9-ps 12,13 .
In recent studies, it was shown that the formation of SaS correlated photon pairs can also occur mediated by the exchange of virtual phonons, which is referred to as virtual SaS processes 14 . The photon pair produced by a virtual process is analogous to the electronic Cooper pair in superconductivity 15 , and this analogy has been explored in diamond samples 16,17 . The virtual SaS emerges as a source of correlated photon pairs in a wider range of energies, different from the real SaS that is restricted to E L ± E P , where E L is the excitation laser energy and E P is the phonon energy.
While the real SaS process has a characteristic time scale dictated by τ P , in the virtual SaS process the exchange of virtual phonons is expected to be nearly instantaneous, limited by the inverse bandwidth of the excitation pulse. In this work, we study the production rate of photonic Cooper pairs as a function of the time delay between the S and aS scattering in the SaS process to elucidate this fundamental difference between the real and virtual phonon exchange processes. As it will be discussed, photon polarization has also to be studied to enable performing the lifetime measurements.  For the lifetime measurements, the 633 nm pulse train is divided in two by a 50/50 beam-splitter(BS), both with polarization horizontal (H) with respect to the optical table.
One of these pulsed beams passes through a delay line and has its polarization changed to vertical (V) by a half-wave plate (HWP). The two pulse trains are then recombined into a single beam by a polarization beam splitter (PBS) and directed to an inverted microscope, where the sample is located. The combined beam crosses neutral density filters (ND) used to reduce the power, and a band-pass filter (BP L ) used as a laser line filter for 633 nm. The evidence of occurrence of correlated SaS processes is the presence of a distinguished peak at the central bin (∆τ = 0 ± 96 ps) of the histogram (Figure 1(b)) with total counting number more than twice as high as the set of the other peaks. The coincidence counts corresponding to S and aS photons scattered by the same phonon will fall within this central peak 2-9 , referred to as ∆τ = 0 hereafter.

A. Polarization
To study the polarization dependence of the SaS process, we chose to excite the sample with H polarized light pulses. The polarization of the S and aS photons are then investigated using two linear polarizers (P S and P aS ), placed in front of each APD (see Fig.1(a)).
Considering the representation (L P ol. ; S P ol. , aS P ol. ) for laser, Stokes and anti-Stokes po- We investigated the polarization in the SaS emission for both the real process, when the Stokes and anti-Stokes energies correspond to the first-order Raman peak from diamond at ±1332 cm −1 (Fig.2, right panels), and for the virtual process, placing the S and aS band pass filters at ±900 cm −1 (Fig. 2, left panels). We observe that the SaS emission occurs majorly  Fig. 2 (d,h) show the measured second-order cross-correlation function g SaS (∆τ = 0), which is g Analyzing the real SaS (red circles in Fig. 3), the intensity I Real SaS starts to increase when the pulses start to overlap, reaches a maximum for a delay time δτ ≈ 0.5 ps, and decreases for larger delays. The constant non-zero I Real SaS ≈ 0.2 value is related to SaS processes from each pulse individually and amounts to the much less probable but still existing cases when δτ , creating an aS photon is given by where B = Aπσ 2 L e σ 2 L /(2τ 2 P ) /2, with A depending on the Raman cross-sections, and erf is the error function. The SaS intensity as a function of delay time between laser pulses is given by where I SP SaS is a constant that does not depend on δτ and represents the intensity of the SaS processes generated in a single pulse (stars in Fig. 3). In Fig. 3, the normalized intensity of real SaS (red circles) is fitted by Eq. 2 (solid red line), with τ P = (2.8 ± 0.3) ps and I SP SaS = 0.22 ± 0.02 as fitting parameters, for a laser pulse width of 0.40 ps. Qualitatively, the process is maximized when there is enough time for the first pulse to create the phonon (hence the delay) and not enough time for this phonon to decay.
Considering now the virtual SaS (blue circles in Figure 3), the process is expected to happen within the overlap time of the H and V laser pulses. By analyzing the blue data, we observe that, indeed, the intensity of the virtual SaS is better fitted by the convolution of two Gaussians with time width of 0.40 ps FWHM (same time width of the laser pulse).
When we increase or decrease δτ , reducing the overlap of the pulses, the intensity of the SaS process decreases (solid blue line in Fig. 5).

IV. CONCLSION
In summary, based on the strong polarization correlation between the excitation laser and the scattered SaS photon pairs, we have elucidated a fundamental difference between real and virtual SaS pair production processes in a diamond sample. By means of a pumpand-probe experiment with cross-polarized and time-delayed laser pulses, we showed that the production rate of real SaS pairs decreases with the decay of the phonon population generated by the Stokes process. We measure an SaS time correlation profile compatible with a phonon population lifetime τ P = (2.8 ± 0.3) ps in diamond 3, 19 . Quite differently from the real process, the virtual SaS pair production occurs just as long as the pump and probe laser pulses overlap, indicating that it is faster than the duration of a single pulse, which is 0.40 ps FWHM in our experiment. Assuming that the virtual SaS pair production is a coherent process, the time correlation between S and aS photons in this regime is probably limited by the bandwidth of the detection system. Further investigation is needed to clarify this point. Furthermore, it is important to expand these findings to other materials/media for testing the generality of these results. Similar polarization behavior has been observed for water 6 . However, the lifetime experiment in liquids is more challenging because of the smaller phonon lifetimes [REF] and the smaller SaS production rate 16 .