Raman scattering and vacuum fluctuation: An Einstein-coefficient-like equation for Raman cross sections

Since it was first predicted 100 years ago, Raman scattering has been a cornerstone of molecular spectroscopy with a widespread impact on science and technology. Nearly all theoretical frameworks have employed Raman cross sections (σRaman) to characterize and quantify molecular Raman response. The recently introduced absolute stimulated Raman scattering cross section (σSRS), on the other hand, provides an alternative way of interpreting molecular responses under two coherent laser sources. However, the theoretical connection between σRaman and σSRS remains unclear. Herein, we are inspired by Einstein’s A and B coefficients for spontaneous and stimulated emissions and derived an analogous equation [Eq. (16)] for Raman scattering from an approach along quantum electrodynamics. Equation (16) decomposes Raman cross sections into a contribution from the vacuum electromagnetic field and an underlying molecular response captured by stimulated Raman cross sections (in the unit of Göppert–Mayer). This theoretical relation is supported by recent experimental measurements on methanol as a model compound. Foremost, it provides a connection between experimentally defined σRaman and σSRS under certain approximations. In addition, it quantitatively shows that it is the weak vacuum field of the Stokes channel that makes Raman cross sections appear so small, corroborating the conventional Raman theory. Moreover, it suggests stimulated Raman cross sections to be a vacuum-decoupled intrinsic quantity for characterizing molecular response during Raman scattering. Remarkably, stimulated Raman cross sections turn out to be not weak when compared to two-photon absorption, narrowing the conventional gap of cross sections between spontaneous Raman and UV–vis absorption by more than 1010 folds.


INTRODUCTION
Raman scattering, an inelastic light scattering process by molecules, is a cornerstone of molecular spectroscopy with a wide impact on physical science, life science, and technology.4][5] Hence, 2023 marks the 100th anniversary of Raman scattering.
All Raman literature studies have employed Raman cross sections, σ Raman , as a measure of the molecular response for the interaction strength between light and molecule. 6The rate of energy (unit in J/s) scattered into the Stokes channel (considered to be a differential rate into solid angle dΩ), P Raman , is written as the product of the number of vibrational modes, n, differential Raman cross section (unit in cm 2 ) of the mode, dσ Raman , with the incident intensity [unit in J/(s⋅cm 2 )] of the pump beam, dP Raman = n ⋅ dσ Raman ⋅ Ipump, (1)   where σ Raman can be theoretically expressed with other spectroscopic quantities from quantum mechanics.Popular theoretical frameworks include Kramers-Heisenberg formulation derived from the second-order perturbation theory or third-order perturbation expansion of the density matrix operator. 7,8The effect of vacuum electromagnetic field is implicitly treated in most of these frameworks.
Very recently, a new framework was proposed to understand molecular response under stimulated Raman scattering (SRS), 9 a process closely related to the classic Raman scattering.7][18] To quantify the response of the emerging Ramanactive imaging probes, absolute stimulated Raman cross section, σ SRS , was recently introduced, carrying a unit of Göppert-Mayer (1 GM = 10 −50 cm 4 ⋅ s ⋅ photon −1 ). 9 A systematic comparison between the measured σ SRS and the reported two-photon absorption cross sections for real molecules has provided useful insights.While σ SRS was phenomenologically introduced to describe molecules under SRS excitation (i.e., strong pump and strong Stokes beams), we hypothesize that its significance can extend beyond SRS experiments, as it might provide the vacuum-field-independent molecular response for all Raman scattering processes.However, an important question needs to be addressed: is stimulated Raman cross section compatible with the traditional framework of Raman scattering [Eq.(1)]?If so, can σ Raman and σ SRS of the same molecular mode be related to each other?
Herein, we attempt to connect σ Raman and σ SRS without referring to the full quantum theory.The physical picture is to treat normal Raman as a stimulated Raman process driven by zeropoint energy of vacuum electromagnetic field as the Stokes beam, an accepted view in quantum electrodynamics (QED) theory. 19n doing so, we have derived an Einstein-coefficient-like relation, Eq. ( 16), which decomposes σ Raman into a contribution from the vacuum field and an underlying molecular response captured by σ SRS .This relation [Eq.(16)] is analogous to the Füchtbauer-Ladenburg equation describing fluorescence.This relation is interesting in several aspects.Foremost, it provides a connection between experimentally defined σ Raman and σ SRS under certain approximations.In addition, the agreement with experimental measurement supports that it is indeed plausible to think of spontaneous Raman as stimulated Raman driven by zero-point radiation of the vacuum.Moreover, this relation provides a quantitative perspective to reconcile the apparently feeble Raman cross sections.In short, it is the weak vacuum field, as calculated to be the equivalent of a propagating beam of microwatt power level, that makes Raman cross sections so small.1][22][23] When compared with other nonlinear optical processes, this molecule-intrinsic Raman response is not weak after all, narrowing the gap of conventional comparison of cross sections by over 10 10 folds.

Definitions of Raman cross sections and stimulated Raman cross sections
We begin by comparing the definitions of the two sets of cross sections.Equation (1) defines the commonly utilized Raman cross section σ Raman found in the literature.Note that σ Raman is defined in terms of energy flux due to historic reasons, rather than photon flux, a unit more commonly used in many other optical processes.We can rewrite Eq. ( 1) in terms of photon flux as where ̵ hωp and ̵ hω S are the energy of the pump and Stokes photons, respectively, dR Raman represents the generation rate of Stokes photons into solid angle dΩ, in the unit of photon ⋅ s −1 , and ϕpump is the photon flux of the pump beam, in the unit of photon/(s ⋅ cm 2 ).Note that dR Raman also describes the rate of vibrational transition of the target vibrational mode that accompanies the differential photon generation, as each Stokes photon generation is accompanied by a quantum transition to the vibrational excited state of the mode.
Inspired by the theory of two-photon absorption, a new framework has been recently introduced to define absolute stimulated Raman cross section, σ SRS , for a single molecule, 9 where R SRS is the rate of stimulated Raman gain or loss (in the unit of photon ⋅ s −1 ) and ϕpump and ϕ Stokes are the photon flux of the pump and Stokes beam, respectively.R SRS also describes the rate of vibrational transition of the target mode during SRS.However, this equation did not explicitly account for the frequency dependence of σ SRS .Herein, we start with a differential form of R SRS and redefine the frequency-dependent absolute SRS cross section σ SRS as where ωp and ω S are the angular frequency of the pump and Stokes photon, respectively.F(ωp) and F(ω S ) represent the spectral photon flux density of the pump and Stokes beam, in the unit of (photon ) is a function of ωp − ω S , centered around Ω 0 , the central frequency of the target vibrational mode.Equation (4) can be viewed to describe the SRS process for a particular solid angle dΩ, which is the case for most SRS experiments.
Integrating Eq. ( 4) over both ωp and ω S , we have the integrated form as In typical narrowband SRS experiments using picosecond excitation lasers, F(ωp) and F(ω S ) are narrowly distributed functions, and ω S ≈ ωp − Ω 0 , when tuning at vibrational resonance.Under this experimental condition, the integral in Eq. ( 5) can be carried out and simplified into the following form: where σ SRS (Ω 0 ) is the peak value of σ SRS (ωp − ω S ).Obviously, Eq. ( 6) recovers Eq. (3) introduced recently. 9In that study, σ SRS values determined experimentally are, hence, the peak values of the corresponding σ SRS (Ω 0 ).

Connecting spontaneous Raman with stimulated Raman scattering
A key task here is how to connect σ SRS (ωp − ω S ) with σ Raman .It has been proposed that spontaneous Raman scattering can be interpreted as arising from fluctuations in the vacuum zero-point energy. 24A parallel thought experiment could be as follows: if we were to reduce the Stokes laser power to zero, the rate of the scattering event should gravitate toward the limit of spontaneous Raman scattering rather than vanishing completely.This line of thought suggests that the effective Stokes flux shall derive from two distinct sources: the laser field from the Stokes beam and another equivalent field originating from the zero-point energy of the vacuum.Hence, we are motivated to generalize Eq. ( 5) as where Fvacuum(ω S ) is the effective spectral Stokes photon flux density that is generated from vacuum zero-point fluctuations.This treatment is in the same spirit as the photon occupation interpretation of SRS, where n Stokes + 1 was used for Stokes photon creation. 13quation ( 7) shall be to be able to describe spontaneous Raman scattering when F(ω S ) → 0 from the external Stokes beam.In this limit, it becomes which describes spontaneous Raman scattering from the perspective of SRS induced by vacuum field.An implicit assumption here is that quantum pathways probed in SRS are equally accessible through the spontaneous Raman, which is often true for ground-state SRS far from electronic resonance. 13In a typical spontaneous Raman experiment, a continuous-wave pump laser [i.e., narrowband excitation profile for F(ωp)] is used.Equation ( 8) then becomes As discussed above, Eq. ( 9) describes the rate of vibrational transition of the target mode from the perspective of SRS induced by vacuum field.This should recover Eq. ( 2) that also describes the same quantity from the perspective of σ Raman .Both are specifying a solid angle.Equating Eq. ( 2) with Eq. ( 9), we have Equation ( 10) provides a route of connecting spontaneous Raman and stimulated Raman cross sections, which are experimentally defined quantities by Eqs. ( 1) and ( 6), respectively, via the effective spectral photon flux density of vacuum, Fvacuum(ω S ), which will be determined in the following.

Contribution from vacuum fluctuation
Next, we seek an expression of Fvacuum(ω S ) suitable for Raman scattering.According to classical electromagnetism, the number of modes of the electromagnetic field in free space in volume V in the frequency range (ω, ω + dω) in the polar coordinates is given by dN = V 8π 3 ω 2 c 3 dωdΩ for each polarization. 25dΩ is an element of solid angle.Thus, the mode density (or photon density of states), ρ mode (ω), per unit frequency in volume V is given by In quantum electrodynamics (QED), the ground state of vacuum consists of one virtual photon in each of these modes.In other words, each field mode is mathematically equivalent to a harmonic oscillator in its lowest energy (zero-point energy), and in the case of electromagnetic fields, a virtual photon. 26Hence, ρ mode (ω) is equivalent to the (virtual) photon density, ρ photon (ω), of the vacuum ground state: ρ photon (ω) = ρ mode (ω).These are called virtual photons because they are non-radiative "photons" from zero-point vacuum fluctuation, which do not exist in the classical sense.Virtual photons are created in the vacuum out of nothing and then disappear again after an extremely short time.Although these transient virtual photons cannot be observed directly, they contribute measurably to the probabilities of observable events.For example, if these photons interact during their short existence with the electrons of an atom, the binding energies of the electrons shift ever so slightly (i.e., the Lamb shift).For another example, when two mirrors are placed facing each other in a vacuum, more virtual photons can exist around the outside of the mirrors than between them, generating a seemingly mysterious force (i.e., Casimir force) that pushes the mirrors together.Now, let us consider that the molecules under Raman excitation are within an area A in the time duration of τ.The effective volume for the virtual photon will be V = A cτ.The effective spectral photon flux density (after area and time normalization) for the vacuum, Fvacuum(ω S ), can be calculated by Fvacuum(ω S ) = ρ photon (ω) which represents the number of virtual photons, per unit frequency, crossing a unit area in a unit time.

An Einstein-coefficient-like equation for Raman cross sections
Plugging the vacuum contribution of Eq. ( 12) into Eq.( 10), we arrive at Thanks to the narrow bandwidth of Raman spectral response, Eq. ( 13) can be approximated as follows.Recall that σ SRS (ωp − ω S ) is a sharp function centered around Ω 0 .For example, in the case of methanol C-O mode, Ω 0 corresponds to the vibrational band centered at 1030 cm −1 , with a full-width-at-half-maximum (FWHM) of ∼20 cm −1 . 27For a typical spontaneous Raman scattering experiment, ωp is located at 3.543 × 10 15 rad ⋅ s −1 (equivalent to 532 nm wavelength) for a visible laser excitation, and ω S is mostly ranging from 3.344 to 3.354 × 10 15 rad ⋅ s −1 to cover the C-O mode.Within this range, ω S squared only changes about 1% and can, therefore, ARTICLE pubs.aip.org/aip/jcpbe approximated as a constant.Hence, we can take the ω S variable outside the integral, and Eq. ( 13) approximates to where we have used ω S = ωp − Ω 0 and the definition of differential Raman cross section dσ Raman /dΩ.We next introduce the lineshape function G(ω) for the frequency dependence of σ SRS around one vibrational mode.Since the absolute position of the peak is not important in the integral, the normalization requires ∫ G(ωp − ω S ) ⋅ dω S = 1.Then, we can rewrite Eq. ( 14) as Note that G(ω) has the same frequency dependence as σ SRS (ω). Hence, where we have used the normalization condition of G(ω).For simplicity, here, we assume the scattering is isotropic, and integrating dσ Raman /dΩ over all spatial angles and considering the two polarizations of light gives the total cross section, If we model the Raman lineshape function G(ω) by a nor- 2 , where Γ is the FWHM of the peak, the peak value G(Ω 0 ) = L(ν)| ν=ν 0 = 2 πΓ , then Eq. (16b) becomes which is a simple practical form of Eq. (16b).It is interesting to see how the peak width Γ plays a role here.Intuitively, Γ controls the spectral window of the vacuum photon modes that can contribute.
The factors in front of σ SRS (Ω 0 ) together display a unit of photon/(s ⋅ cm 2 ), suggesting a physical quantity of photon flux.Hence, we are prompted to group them into an effective photon flux arising from the vacuum field fluctuation, Equation ( 16) is a key result that relates normal Raman cross section with absolute SRS cross section, which are experimentally defined by Eqs. ( 1) and ( 6), respectively.It carries the same spirit of Einstein's A and B coefficients relating spontaneous emission rate with stimulated emission.
In summary, we have established a connection between spontaneous Raman and stimulated Raman cross sections without explicitly referring to full quantum mechanics.The general methodology adopted here is similar to Einstein's derivation of the A and B coefficients: we started from the experimental definition of two sets of cross sections in the form of transition rate, and we identified a logical link between the two formulas to determine their relationship.In Einstein's derivation, the connection is established by comparing to Planck's blackbody radiation law and microscopic reversibility, and the linkage we used here is how the vibrational transition rate of SRS should recover that of regular Raman when the external laser field vanishes.
It is constructive to compare our result to the Füchtbauer-Ladenburg equation, which was introduced long time ago for treating fluorescence in the literature of atomic physics, 28,29 1 where σem(ω) is the stimulated emission cross section and τ rad is the lifetime of the upper level.The Füchtbauer-Ladenburg equation, which is often regarded as a generalized form of Einstein's coefficients, is usually not evaluated in close forms.This is due to the relatively large bandwidth of electronic transition and the term ν 2 is no longer a constant in the integration. 28In contrast, Eq. (16b) gives a simple close form, thanks to the narrow bandwidth of Raman spectral response.In fact, if we replace the angular frequency ω with linear frequency ν, we shall transform Eq. ( 13) into the following form (Note S1): This form of Eq. ( 18) is almost identical to the Füchtbauer-Ladenburg equation, supporting the validity of our equation.It is interesting to note that a coherent process (such as stimulated emission or stimulated Raman) is connected to an incoherent process (such as fluorescence or spontaneous Raman), which can be viewed from the perspective of a single molecule.

Comparing theoretical prediction with experimental measurements
We now compare the prediction of Eq. (16c) with experimentally measured Raman cross sections.First, we modify Eq. (16c) into a more ready-to-use format for experimentalists.For example, ωp is replaced by wavelength λp, and Ω 0 and Γ are replaced by wavenumbers ṽ0 and ṽΓ .Then, we obtain (Note S2) Assuming λp and λS are in nm, ṽΓ is with a unit of cm −1 , and σ SRS is in GM, we then have numerically (Note S2) The Journal of Chemical Physics

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The methanol C-O ṽ0 = 1030 cm −1 mode is a commonly used standard for Raman scattering, and its cross section has been carefully measured and documented in the literature.In particular, σ SRS,C -O = 0.04 GM = 4 × 10 −52 cm 4 ⋅ s −1 photon −1 has been recently measured as the peak value of σ SRS (Ω 0 ) in a narrowband SRS experiment. 9The SRS pump excitation and Stokes scattering wavelengths are at λp = 960 nm and λS = 1064 nm, respectively.Γ is experimentally estimated to be corresponding to ∼20 cm −1 . 27Plugging these parameters, we obtain σ Raman,predicted = 7.5 × 10 −31 cm 2 . ( Meanwhile, the spontaneous Raman cross section has been experimentally determined by several reports to range between 5.1 × 10 −31 and 1.6 × 10 −30 cm 2 .We have summarized the results in Fig. 1. Results are converted into an excitation profile at 960 nm using the ωpω 3 S dependency, and differential cross sections are integrated to total cross sections using the following equation described in the literature: 18 σ R = 8π 3 where ρ = 0.21 is the depolarization ratio measured for methanol C-O stretching and ∂σ ∂Ω ∥ is the differential cross-section measured in both the parallel and perpendicular directions.Remarkably, the Raman cross section predicted from Eq. ( 21) closely aligns with the experimental results (Fig. 1).Taking the inhomogeneous broadening effect into consideration, we have applied the same method to the Gaussian profile, and the result is still within the range of the reported values (Note S3).Note that these two sets of experiments were performed in rather different instruments and the cross sections were calculated using other parameters (such as concentration, focal volume, laser power, collection efficiency, and slit width) with varying degrees of uncertainty.Hence, this agreement is considered to be satisfactory under current measurement efforts.This result could be further improved by rigorous calibration of SRS instrument errors coming from detector response, optical alignment, and

DISCUSSIONS AND CONCLUSIONS
Raman scattering as a vacuum fluctuation-induced effect Contemporary physicists, when asked to give a physical explanation for the occurrence of spontaneous emission, generally invoke the vacuum electromagnetic field. 26This view was first introduced by Weisskopf in 1935 35 and was mentioned later on multiple occasions. 36,37It was subsequently adopted in Raman spectroscopy by interpreting spontaneous Raman scattering as arising from fluctuations in the vacuum zero-point energy. 24However, a quantitative relation between σ Raman and σ SRS did not exist in the literature, to the best of our knowledge.This is likely because σ SRS has not been explicitly defined and measured for molecule systems until very recently.Now, one can have a more comprehensive list of effects that are the physical consequences of the fluctuating vacuum field.These effects include phenomena such as spontaneous emission, spontaneous Raman, Casimir force, van der Waals forces, as well as the Lamb shift.To some extent, this further proves the significance of vacuum in the understanding of nature and possibly opens the door for connecting Raman and quantum-vacuum-related studies, such as quantum-entangled Raman 38 and relativistic Raman scattering.
Why do Raman cross sections appear so small?Raman scattering has been acknowledged in textbooks and reviews for decades as an extremely weak process. 39The values of σ Raman (10 −30 to 10 −28 cm 2 for small chemical bonds) turn out to be many orders of magnitude smaller in comparison to other molecular spectroscopies, such as UV-vis absorption cross sections (10 −16 to 10 −15 cm 2 ) or infrared cross sections (10 −19 to 10 −17 cm 2 ) of similar bonds.Even with the boost of the electronic resonance effect in resonance Raman spectroscopy (σ Raman can increase up to 10 −24 to 10 −23 cm 2 ), the gap is still as wide as many orders of magnitude. 12,40,41espite the shared unit of cm 2 for cross section, linear absorption and Raman scattering are two very different processes.Both UV-vis and infrared cross sections describe absorption processes, and all absorptions are stimulated transitions by nature.However, Raman scattering naturally includes the spontaneous emission process.The participation of vacuum in the spontaneous emission process underlies the differences.Fortunately, Eq. (16d) suggests that σ Raman is not an intrinsic response of the molecule but rather a special case of SRS where the vacuum field instead of an external laser beam is providing the Stokes photon flux.The vacuum-induced contribution is being absorbed into σ Raman .Quantitatively, this contribution can be treated as an effective vacuum photon flux, as explicitly shown in Eq. (16d).We can easily estimate ϕvacuum from Eq. (16d) using the above-calculated values for methanol C-O bond,

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We can further gain insight by converting this value to an equivalent free-space laser that can produce such a level of photon flux.We imagine that this level of photon flux was to be produced by focusing a propagating real-space laser down to a diffraction-limited focal spot under a microscope.Assuming a λ = 532 nm CW laser beam with a diameter d = 600 nm, we would get an effective laser power of In a sense, this value can be regarded as the total zero-point energy (off by a factor of 2) of the virtual vacuum photon that drives the Raman scattering of methanol C-O bond in a microscope experimental setting.In other words, one can think of Raman scattering as being jointly excited by an external pump CW laser and a CW Stokes laser of a few microwatts (provided by the vacuum fluctuation) in a two-photon third-order nonlinear manner.In comparison, modern microscopy experiments (such as two-photon excited absorption) typically employ tens of milliwatts of laser power and 100 fs short pulses to induce stimulated absorption.One can estimate the corresponding photon flux during the pulse to be orders of magnitude (∼10 7 ) higher than the value in Eq. ( 23).Thus, the feeble vacuum-field-induced photon flux is the underlying mechanism of the apparently weak Raman response of σ Raman .
Intrinsically strong molecular response of stimulated Raman cross sections Contrary to common belief, the effect of vacuum might not be constant in all circumstances.Indeed, in cavity QED, 42 the rate of spontaneous emission could be controlled depending on the boundary conditions of the surrounding vacuum field.The possible enhancement or inhibition of the spontaneous emission rate is known as the Purcell effect. 43Similar effects have been observed in Raman scattering, although several decades later.In 1993, Cairo et al. put a Raman medium (C 6 H 6 ) in a cavity and observed that, as spontaneous emission, it is possible to enhance or inhibit spontaneous Raman scattering for a specific Raman line, just by spectral tuning of the cavity. 20][23] Having decoupled the contribution from the vacuum field that depends on the boundary conditions of the surrounding environment, Eq. ( 16) suggests that σ SRS should be a vacuumfield-independent, truly molecular intrinsic quantity for describing Raman response when compared to σ Raman .This view shall hold for both stimulated Raman processes and spontaneous Raman processes.σ SRS can be either converted from regular Raman cross sections theoretically or directly determined by SRS spectroscopy FIG. 2. Intrinsically strong stimulated Raman cross section: a case study.Normal Raman scattering has always been considered an extremely weak process due to the 13-15 orders of magnitude of difference in cross sections when compared to other processes such as linear absorption.However, after decoupling the effect of vacuum field, the molecular-intrinsic SRS cross section is much closer in value to the counterparts of two-photon absorption (TPA), with only three orders of magnitude difference.Shown in the figure are σ Raman and σ SRS for methanol C-O stretching mode at 1030 cm −1 , and the peak UV-vis absorption cross section σ UV-vis and two-photon absorption cross section σ TPA for Rhodamine 6G (Table S1).In essence, this tells us (σ UV-vis /σ Raman )/(σ TPA /σ SRS ) = 10 11 for this case.
The Journal of Chemical Physics ARTICLE pubs.aip.org/aip/jcpexperiments (as recently done for many Raman-active probes).An important consequence is that σ SRS carries a unit of Göppert-Mayer (1 GM = 10 −50 cm 4 ⋅ s photon −1 ), the same unit used in the twophoton absorption.This allows meaningful comparison in this new axis.
Figure 2 shows the comparison between the methanol C-O bond and Rhodamine 6G as a case study.Rhodamine dye is chosen here as it is a well-documented model compound.σ Raman of methanol C-O bond is 10 14 times smaller than the UV-vis absorption cross section of Rhodamine 6G, which is essentially the reason for criticizing Raman being an extremely weak process in many textbooks.However, the vacuum-decoupled σ SRS is only 10 3 times away from the two-photon absorption cross section σ TPA for Rhodamine 6G.Hence, this example highlights that molecule-intrinsic Raman response is not weak after all-the new perspective has narrowed the gap of conventional comparison of cross sections by over 10 10 folds.In fact, previous simultaneous measurements on SRS, TPA, and other pump-probe processes on the same molecules (such as melanin) have hinted that the SRS response has an appreciable magnitude. 44Recent invention of stimulated Raman photothermal microscopy also implies strong intrinsic response. 45Only after being driven by the weak vacuum field [Eq.( 23)], the resulting σ Raman becomes a small quantity.

A phenomenological framework for connecting different photonic processes
We shall borrow the concept of reaction order from chemical kinetics to organize common optical processes into three categories (Fig. 3), each with a different unit.Zeroth-order processes have a unit of s −1 , representing the rate that we can directly measure in experiments.First-order photonics have a unit of cm 2 as is used in linear absorption, stimulated emission, and spontaneous Raman scattering.Second-order processes with a unit of GM are less common, the most popular of which are SRS and two-photon absorption.
These different orders of reactions can be related through the multiplication of proper molecular cross sections by a photon flux or several photon fluxes, which can derive either from vacuum (wavy lines) or from external laser beams (straight lines).Take Raman scattering for example, if we multiply the second-order σ SRS by the effective vacuum photon flux ϕvacuum, we shall get σ Raman , as shown in Eq. (16b).On the other hand, if we multiply σ SRS by the photon flux from an external Stokes laser beam ϕ Stokes , we shall get the apparent stimulated Raman cross section σapparent, defined as σapparent = σ SRS ⋅ ϕ Stokes previously. 9Further multiplication of the first-order σ Raman or σapparent by the pump photon flux ϕpump will lead to the zeroth-order rate in the unit of s −1 , which can be experimentally measured.Similarly, if we multiply the stimulated emission cross section σse of Rhodamine B by ϕvacuum, we would obtain the rate of transition in fluorescent emissions, usually expressed in the inverse form of lifetime (∼3 ns).A product of σse and ϕ laser would result in stimulated emission rate, a parameter commonly used in predicting stimulated emission depletion (STED) efficiency.Twophoton absorption (TPA) transitions are different in the sense that a direct transition from the second order to the zeroth order was achieved at once by multiplication σ TPA with the squared value of ϕ laser .Note that direct multiplication of cross section and photon flux would result in peak transition rate (i.e., rate within the laser pulse), and the average rate can be calculated by multiplying the peak rate by the duty cycle of the laser (Note S4).These results are organized in Fig. 3.
Based on this figure, it can be concluded that only those data points that are not connected to a higher order quantity represent the molecule or bond's true intrinsic property.Thus, they should only be compared to quantities located on the same axis.In this sense, we should only compare SRS with TPA and compare absorption with stimulated emission.Despite the shared unit of cm 2 , σ Raman and σ abs lie on very different grounds in terms of photonic nature because σ Raman contains the contribution from the vacuum field, whereas σ abs does not.This framework can also be expanded to even higher order processes such as three-photon absorption cross sections, which should be a third-order process.

SUPPLEMENTARY MATERIAL
See the supplementary material for the details of cross sections of Rhodamine 6G, calculation in linear frequency, calculation of predicted experimental Raman cross sections, calculation of cross sections under Gaussian profile, and calculation of cross sections and rates.

FIG. 3 .
FIG. 3. Connecting different photonic processes via cross sections of varying orders.Data points for two representative bonds/molecules (methanol C-O and Rhodamine B) and five photonic processes [including spontaneous Raman scattering, stimulated Raman scattering, two-photon absorption, fluorescence (i.e., spontaneous emission) and stimulated emission] are illustrated.These processes shall be connected with photon flux, either from vacuum (wavy lines connected to Raman scattering and fluorescence) or from external laser beams (straight lines).Only data points without a connection to a higher order quantity represent the intrinsic property of the molecule/bond.The peak zeroth rate and average zeroth rate can be connected via the duty cycle of the laser.Detailed calculations are shown in Note S4.