Analysis of Side-band Inequivalence

Frequency shifts of red- and blue-scattered (Stokes/anti-Stokes) side-bands in quantum optomechanics are shown to be counter-intuitively inequal, resulting in an unexpected symmetry breaking. This difference is referred to as Side-band Inequivalenve (SI), which normally leans towards red, and being a nonlinear effect it depends on optical power or intracavity photon number. Also there exists a maximum attainable SI at an optimal operation point. The mathematical method employed here is a combination of operator algebra equipped with harmonic balance, which allows a clear understanding of the associated nonlinear process. This reveals the existence of three distinct operation regimes in terms of pump power, two of which have immeasurably small SI. Compelling evidence from various experiments sharing similar interaction Hamiltonians, including quantum optomechanics, ion/Paul traps, electrooptic modulation, Brillouin scattering, and Raman scattering unambiguously confirm existence of a previously unnoticed SI.

It should be noted that optically confined particles in side-band resolved regime can display remarkably strong SI. Given the fact that the nature of these experiments are markedly different than cavity quantum optomechanics, it is quite exciting indeed to observe the fact that SI does survive. A recent such experiment [S12], with the reported uncertainties taken into account, exhibits an SI as large asδ = 0.8%, if we assume that the zero calibration is sufficiently more accurate than 0.1 kHz. Otherwise, referencing with respect to the fitted central resonance givesδ = 0.43% ± 0.18%. In any case, it is positive as expected within a reasonable confidence interval, that is the red peak is definitely further away from the resonance as opposed to the blue. For this experiment, the raw measurement data was made available to the author, providing an accurate analaysis, as depicted in Fig.  S1. Optical confinement of a single-atom using tweezers [S13] also leads to measurement of single-atom sideband spectra,  Figure S1. Motional sidebands of an optically trapped ion in a tweezer before sideband cooling obtained from spectroscopic raw measurement data [S12].
which in absence of cooling can be measured to have a normalized SI ofδ = 1.04%. In a very recent landmark experiment carried out [S14] side-bands were generated as a result of controlled detuned acoustooptic modulation, where detuning frequency effectively replaces modulation frequency. The authors have made the measurement data available online [S15] from which the SI may be calculated with respect to the modulation frequency. The behavior of SI here is dependent on the parameters of modulator and not the optomechanical device, so we had to look for fitting parameters. The appropriate expression in the limit of very small Γ follows (17) in the main article. Assuming γ ≈ κ and g = g 0 √n , it takes the form δ (Ω) = 8g 2 κ 2 Ω 2 sgn[Ω] noticing that Ω represents the detuning frequency and can be positive or negative. Hence, the normalized SIδ may be considered as a function of Ω, too, and sgn(·) is the sign function. This expression (S1) very well fits the calculated behavior based on the analysis of raw measurement data [S15], giving a remarkably nice fit. However, it has to be pointed out again that the optimum fitting parameters (κ, g) = 2π(620, 90)kHz are irrelevant to the optomechanical system parameters.
When ι < 1, then the expression within the brackets is composed of an even real part and an odd imaginary part. Hence, the Fourier transform of this hyperbolic expression within the brackets is real-valued, which we here denoted by G (w). Obviously, the magnitude of Fourier transform, or the spectrum |G (w)| no longer needs to be even. Hence, the Fourier transform of ϒ(t), which is given by J (w) = F {ϒ(t)}(w) is simply the shifted transform G (w − θ 2 |∆|), and the amount of frequency shift is nothing but the SI. This fact is a direct result of the multiplying term exp(iθ 2 τ) in the analytical solution (S3), and thus the SI.
While, the algebraic hyperbolic form of (S3) with ι < 1 disallows analytical evaluation of the spectrum J (w), however, calculation of J (w) from (S3) with ι > 1, which leads to periodic nonlinear oscillatory solutions, becomes explicitly possible as shown in the following. To do so, we may proceed with the substitution ι = |(Ω + iΓ/2)/g| ≈ Ω/g where g = g 0 √ n is the enhanced optomechanical interaction rate, which in the weakly-coupled limit obviously satisfies ι >> 1. Hence, we have iϑ = 2θ 2 ι √ ι 2 − 1 ≈ 2θ 2 ι 2 . Now, we can rearrange (S3) as where the replacement iϑ τ = ϖt has been made and ϖ = Ω + δ Ω is the shifted mechanical frequency while taking the optical spring effect δ Ω into account. The expression within brackets in (S4) can be simplified using trigonometric identities as , and Ψ = 2Ξς . We here may notice that δ = θ 2 |∆| > 0 is nothing but the SI. The function f (t) can be rewritten as
All remains now is to solve iϑ τ = ϖt, which gives rise to the approximate solution This can be further written as since δ Ω ≈ −2g 2 ∆(Ω 2 − 1 4 κ 2 )/(Ω 2 + 1 4 κ 2 ) 2 . It is here seen that the correction arising from spring effect is O(g 4 ) while the SI is O(g 2 ). Hence, the normalized SIδ = δ /Ω can be finally approximated for the side-band resolved regime Ω >> κ as showing that the correction of optomechanical spring effect to the ansatz (1), ultimately yielding the expression for SI (14) had been safely ignored indeed. There is also a higher-order correction to the optomechanical spring effect δ Ω [S2] as a result of non-zero coherent phonon populationm, which results in an extra correction to (S8) by replacing g = g 0 √n within the brackets of (S9) with g ≈ g 0 √n +m + 1. But this leaves our derivations and conclusions regarding the SI unchanged. This solution (S5) is a bi-periodic and complex-valued product of two periodic functions f (t) and exp(iδt). Fourier transform of the periodic function Hence, it is straightforward to see that how its spectrum looks like. Defining the spectrum of f (t) as |F(w)| in the Fourier domain w simply is F(w) = ∑ ν | f ν |δ(w − νϖ), which consists of Dirac deltas at w = ±νϖ ≈ ±νΩ, ν ∈ N corresponding to the side-bands. A practical system obviously does not exactly follow the breather solution (S3) and hence side-bands all have finite non-zero linewidths. Ultimately, we have I(w) = |J (w)|.
Therefore the ultimate spectrum of the cavity within the approximation of breather solutions is I(w) ≈ |F(w − 1 2 δ )| + R(w), where R(w) is the reflection from cavity at central resonance w = 0. Since the reflected central resonance R(w) normally masks out the zeroth harmonic, therefore the side-bands appear to be positioned asymmetrically in frequency equal to the SI δ . Hence, I(w) may be written conveniently as where the change of variables z = exp(iϖt) has taken place, and the integration is taken counter-clockwise on the unit circle in the complex u−plane. The only contributing pole of h(z) is at z = iζ − ≈ −i/2ι 2 . Furthermore, u(·) is the unit-step function, which allows the second term to contribute only if ν ≥ 0. The function h(z) = (z − κ)/(z − iζ + )(z − iζ − ) was also defined in (S6). In (S10), the positive odd harmonics identically vanish, and f ν and f −ν respectively correspond to Stokes and anti-Stokes amplitudes. There are no odd-ordered Stokes components in the breather nonlinear oscillatory wave (S4), and also anti-Stokes components diminish in strength with their order −ν increasing as (ζ − ) −ν according to (S10).
The total power P = ∞ −∞ I(w)dw is now simply P ≈ ∑ ν | f ν |, and total harmonic distortion shall be given by the simple expression THD ≈ ∑ |ν|≥2 | f ν |/ ∑ |ν|≥1 | f ν |. The first-order mechanical side-bands correspond to f ±1 with sharp peaks located at ±ϖ − 1 2 δ ≈ ±Ω − 1 2 δ , confirming the initial ansatz (1) and speculation regarding the existence of SI. In summary, the breathing analytical solution (S3) actually highlights the existence of a non-zero SI, simply because of the multiplying term exp(iθ 2 τ) and its bi-periodic form, and furthermore SI has to be always towards red (Stokes) simply because θ 2 > 0 is always positive.
The ratios of coefficients f ν in (S10) can be estimated using binomial expansion of denominator in (S5) and the original form, resulting in approximate expressions for the ratios of side-band powers. For instance, the ratio of optical amplitude in the first-order side-bands with respect to the central resonance is roughly The accuracy of breathing solutions for second-and higher-order harmonics is insufficient to obtain a meaningful ratio such as (S11), nevertheless, it exhibits a positive and unmistakable SI towards red.

S2 Ion/Paul Traps
The volume of existing literature on atomic and ion traps is truly vast, and we limit the study to a collection of selected works in this area. SI numbers are typically large and quite noticeable. Detection of motional side-bands around 5GHz in a linear Paul trap placed on an optical cavity [S18] gives rise tō δ = 1.32% ± 0.32% at anti-node andδ = 1.42% ± 0.24% at node. Another measurement [S19] clearly shows red and blue side-bands separately from which one may obtain the fairly accurate estimation ofδ = 0.021% ± 0.0021%. Doppler cooling on a microchip multi-segmented ion trap [S20] givesδ = 0.047% ± 0.0078%. Doublet features of side-bands are very much visible in the next research [S21], which was the primary assumption as displayed in (1) at the beginning of our analysis. One report considers a high-resolution measurement of motional side-bands of trapped Ca + ions [S22], which contains four very sharp resonances on either side. All of these resonances exhibit significant SI and the values arē δ = {1.8% ± 0.29%, 1.3% ± 0.21%, 0.50% ± 0.12%, 0.87 ± 0.11%%} sorted in terms of increasing shift frequencies, shown in Fig. S3. Planar and vertical modes in ion traps can also individually have SI, as the recent measurements [S23] may give the respective valuesδ = 0.096%, 0.18% within ±0.008%. Axial motional sidebands in another microfabricated ion trap design [S24] are measured from whichδ = 0.037% ± 0.0085% can be computed. Side-band Inequivalence (%) Figure S3. Side-band Inequivalences computed from digitization of microfabricated ion trap experiment [S22].
A very large SI also can be noticed in the side-bands of a calcium ion confined in a linear Paul trap [S25] asδ = 6.3% ± 0.18%. Doppler cooling does not seem to remove this effect, as the valueδ = 0.10% ± 0.0070% corresponds to cooled vibrational states of trapped atomic ion [S26]. Motional side-bands of two 40 Ca + ions before sympathetic ground-state cooling [S27] show a clear SI as large asδ = 0.35% ± 0.0030%. Also, an article [S28] which reports cooling of trapped 111 Cd + ions showsδ = 0.21% ± 0.0086%.
Finally, a very recent experimental study at Max-Planck Institute [S29] reports resolved-sideband cooling of an optical lattice with zoomed-out side-bands both before and after cooling takes place. The SI is so large that can be easily seen on the frequency scales equal toδ = 1.9%. It turns out that the authors had noticed this large difference and had tried to explain it using anharmonicity of the confining potential [S30] in weak limit. It has to be mentioned that if the cause of side-band S5/S21 Side-band Inequivalence (%) Figure S4. Side-band Inequivalences of an ion trap measured for the radial mode spectrum after Doppler cooling, computed from raw measurement data [S31].
inequivalence were anharmonicity, then the sign of expression for SI must follow the sign of anharmonic potential, that is δ ∝ g 0 . However, the analysis in the current study shows that δ ∝ g 2 0 holds to a good accuracy, regardless of the sign of anharmonic optomechanical interaction.
Another article, the measured radial mode spectrum of a nine-ion linear crystal has been reported [S31] and the raw measurement data were made available to the author. There are multiple resonances on either of the blue and red sides, which are measured both after Doppler cooling and after Doppler/EIT cooling. After EIT cooling, the red side-bands almost disappear, and it has been possible to find the SI only after Doppler cooling, shown here in Fig. S4. The behavior of SI, while clearly non-zero, does not conform to the expectation and that could possibly because of Doppler cooling which dislocates resonances a bit. Nevertheless, nearly all of the 15 resonances contain non-zero SI.
Shown in Fig. S5, the SI of the motional state of a trapped 171 Yb + ion placed in a state-dependent potential generated by a running optical lattice is computed from raw measurement data, made available to the author [S32]. Measurements here correspond to the absence of cooling of any type. Every point is furthermore the average of 100 traces, and that provides an accurate estimation of SI. The maximum normalized SI is aboutδ = 0.99.
Interestingly, in the limit of small optomechanical interaction rate g 0 and low optical power, one may use (18) to obtain the approximate proportionality dependencē where Ω and γ are respectively the mechanical/modulation frequency and optomechanical decay rate. This information can be used now to check whether SI decreases with frequency in accordance to (S12). This is shown in red in Fig. S5.
As the last remark of this section, not all reported measurements apparently show SI in the desired way [S33, S34]. What these works have in common is that they include measured side-bands before (and after) cooling, and while the measureable SI seems significant, however, it is in the opposite direction. Normally, it is convenient to see the central resonance to make sure zero-referencing is accurately done, and it has come to the attention of the author that in most cases and in particular for Raman and Brillouin scattering measurements, the accurate SI cannot be found without such referencing. Anyhow, these three articles remain as unanswered questions, which need further investigation.

S3 Electrooptic Modulation
One of the pioneering works on optical fibers [S35, S36] reports the optical intensity spectra scanned with a Fabry-Perot analyzer of a single frequency laser modulated by an electrooptic modulator. It is straightforward to identify an SI as large as 0.85% ± 0.2%, due to a modulation frequency f m on a carrier with frequency ν 0 therein.
Similarly, one may find an SI as large as 2.20% ± 0.27% and 0.42% ± 0.14% for the depolarized spectrum of an LiNbO 3 cell [S37] held at 45 • and modulated at respectively at f 1 = 3GHz and f 1 = 8GHz.
It can be easily verified here that whether (S12) holds, as the normalized SI should roughly decrease with the second power of modulation frequency (or mechanical frequency where relevant). This actually happens to be the case since one S6/S21 Side-band Inequivalence (%) Figure S5. Side-band Inequivalences of an ion trap measured for the radial mode spectrum after Doppler cooling, computed from raw measurement data [S32]. The red fit is according to (S12) Checking the either side while taking care of error bounds gives 14. These numbers are obtained from graph digitization, while such an agreement is later observed to a much higher accuracy for the case of Raman scattering where direct raw measurement data has been available. The next reference [S38] is a thesis reporting the spectra of the Stokes/anti-Stokes signals at the output of a single-mode fiber which has been 50km long, modulated at 10.94GHz using an optical QPSK (Quadrature Phase-Shift Keying) modulator. The first three Stokes and anti-Stokes resonances are sharply measured and clearly visible, which is consistent with an unnormalized and positive SI of δ = (250 ± 25)MHz.
A recent article [S39] reports measurements of the optical spectrum of a phase modulated signal and the measurement data were made accessible to the author. The modulation provided up to seven side-bands on either side, which made the evaluation of SI possible. One would expect, likewise, that the largest SI would go to the lowest order, and that happens to be the case where an SI as large as 1.94% has been observed, which is shown in Fig. S6. There is yet another high resolution experiment on electrooptic modulation of light at telecommunications wavelength at 1545.91nm using a LiNbO 3 modulator with various modulation frequencies, where raw measurement data were made available [S40]. Shown in Fig. S7, the largest SI happens to occur for the lowest modulation frequency, and is very large up to 9.4%, indeed. Pressurized hydrogen as a result of molecular optical modulation [S41] has been shown to generate sidebands, due to ultrafast variation of molecule polarizability arising from coherent molecular motion. The author was also given access to the raw measurement data which was noticed to give rise to an SI ofδ = 0.047%.  Figure S7. Side-band Inequivalence of an electrooptically modulated light computed from raw measurement data [S40].
It has to be stressed out that not every source of nonlinearity (such as cross-Kerr) or multiplying element could exhibit SI. It necessarily must be an Electrooptic medium such as externally driven LiNbO 3 cell or equivalent, with interaction Hamiltonian of the desired type. It remains as a subject for next studies to investigate the excessive distortion caused by this type of modulation and its implications on microwave and optical communication technologies.

S4 Brillouin Scattering
In the contexts of Brillouin Scattering (BS) and Stimulated Brillouin Scattering (SBS), SI appears very clearly which is typically easy to measure due to sharp resonances, and it could take on large magnitudes up to a few percent. One study [S42] has clearly zoomed out the Stokes and anti-Stokes resonances at various power levels, which enables to examine the variation of SI with respect to optical power to a great precision. The measured SI for illuminating optical powers of P op = {12, 14, 20}mW is respectivelyδ = {0.059%, 0.059%, 0.035%} showing that an optimal operation point around (P max ,δ max ) = (13mW, 0.059%) has indeed been hit. Had the authors taken more measurement points above and below this power level, it would have been possible to reconstruct the accurate dependence. These SI numbers are accurate better within than an error margin of 0.017%. The next article [S43] reports a very high-resolution spectrum of pump together with the blue and red side-bands in an optical fiber due to Brillouin scattering. The corresponding wavelengths are also clearly marked, enabling one to easily calculate the SI after conversion of wavelengths to frequencies. Once done, the resulting SI isδ = 0.027% which leans towards the red side-band as it should be. Similarly in another report [S44], the SI can be found in spectra of the transmitted light through a single-mode optical fiber 3.0% ± 0.21%, which should be evaluated with respect to the dislocated central peak.
Brillouin scattering at various angles relative to a SiOC:H/Si film has been measured and reported [S45], which shows peaks for longitudinal guided and bulk modes. Here, the SI for longitudinal bulk mode isδ (θ = 65 • ) = 0.69% andδ (θ = 25 • ) = 0.93% where θ is the illumination angle, which more or less preserves its frequency regardless of θ . The situation for guided modes is a lot more complex since it appears to be combined with the dispersion of modes like Raman scattering from Dirac materials (to be discussed next sub-section). Anyhow, one may observe approximately thatδ (θ = 65 • ) = −1.43% and δ (θ = 25 • ) = −5.01% hold for longitudinal guided modes.
Backward Brillouin scattering may also give rise to SI. This fact may be verified on the measurements carried out in an ultra-high Q resonator [S47] which exceptionally the Stokes/anti-Stokes pair could be seen visibly. For this case, a value of δ = −2.4% ± 0.6% can be estimated, where the negative sign results from backward configuration which is equivalent to the replacement Ω by −Ω.
A more recent research [S48] concerns Brillouin forward and backward scattering in various materials under two (s, p) orthogonal polarizations. For the case of microscope slab in forward scattering two resonances (L, T ) can be distinguished S8/S21 for each of (s, p) orthogonal polarizations, and the resulting numbers for SI mark significant deviations from zero. For p−polarization, we haveδ = {3.2%, 2.0%} corresponding to the two resonances while for s−polarization, we haveδ = {4.2%, 0.83%} all within ±0.14%. It is instructive to examine the measurements corresponding to the forward and backward configurations which are both available for the case of cover glass. Both of the two resonances (L, T ) are sharp enough in the forward configuration for both (s, p) polarizations, while in the backward configuration only one resonances can be clearly seen. The distinguishable resonance/polarization pairs here are (L; s) and (T ; p). In the forward configuration we haveδ (L; p) = 0.28%δ (T ; s) = 1.87% both within ±0.14%. Meanwhile, the SI for forward modes areδ (T ; p) = 1.87%, δ (L; s) = 0.83% within ±0.14% and for backward modes areδ (T ; p) = −0.37%,δ (L; s) = −0.67% within ±0.09%. Hence, it can be confirmed well that the SI for the backward Brillouin scattering can assume negative values, as we have verified for two very different types of measurements [S47, S48].
The comprehensive set of raw measurement data of hypersonic Brillouin scattering of surface acoustic waves in bulk transparent materials has been also made available to the author [S48]. These contain a total of 12 traces shown in Fig. S8, each exhibiting 1 to 3 major resonances. By isolation of these resonances individually, and taking care of zero-calibration according to the available center resonances in the data, the SI of each resonance has been calculated one by one. There are too many plots to put here, and instead only one plot containing all SI of all traces versus frequency of each resonance is shown in Fig. S9. This shows that even taking into consideration of uncertainties, the SI of each resonance could be as large as 4% in magnitude, a remarkably large deviation from zero.  Figure S8. Hypersonic Brillouin scattering measurements for various polarizations and excitation conditions from raw measurement data [S49].
Finally, the last article in this category [S49] reports Brillouin scattering from phononic structures where accurate locations of side-bands are clearly marked throughout all figures therein, and is perfectly consistent with the positiveness of SI. The SI observed at various illumination angles for the two fabricated structures is always positive and reaches a value of 3.6%. The author was granted access to the raw measurement data, from which the figures S10,S11 were generated by making Lorentzian fits accurately to the relevant peaks. Side-band Inequiavalence (%) All traces Figure S9. Collective SI points computed from raw measurement data of backward and forward Brillouin scattering of light for traces in Fig. S8, respectively with negativeδ < 0 and positiveδ > 0 SI for majority of points [S48].

S5 Raman Scattering
Very few researchers actually happen to have noticed the existence of an anomaly in frequency asymmetry, and clearly made a mention of it. But in majority of published works, it has gone unnoticed. Among the bulky archives of available works over the S10/S21  Figure S11. Side-band Inequivalences computed from raw measurement data [S49].
past decades, Raman scattering experiments provide the strongest SI of all. However, Raman scattering is also the most complex one in the broad family of optomechanical types of interactions, where even for the simplest case of perfectly defect-free and ideal crystals, an integration over various phonon polarizations and momenta should be carried out. Hence, any non-ideal effects such as point or line defects and internal stress in piezo-electric materials could significantly alter the Raman response. For that reason, it is believed that measurements carried out on ultra-pure epitaxial cubic crystals such as Silicon (Si) are among the most reliable sources to verify the behavior of SI.
In crystalline solids, this normally reduces in effect to interaction with longitudinal optical phonons at van Hove singularities, which have nearly-identical frequencies and group velocities close to zero. For the exceptional case of Dirac materials such as graphite and graphene and metallic carbon nanotubes, the phonon dispersion together with the linear dispersion of photons can source another cause of frequency asymmetry, which normally goes in the opposite direction than that of SI. This effect however is optically linear, and does not scale with optical power.
Temperature contributions to Raman scattering are relatively significant, however, it mostly translates into enhanced optical spring effect which goes symmetrical in frequency shifts of red and blue side-bands. Since SI is all about the asymmetries, this contribution is automatically canceled out. Therefore, for graphite, the frequency asymmetry due to Anomalous Raman Phenomenon (ARP) is large enough to cause anti-Stokes shift exceed than that of Stokes resulting inδ < 0.
In gapped crystalline dielectrics such as Si, SI is strong enough to cause Stokes shift being larger than anti-Stokes, and this is particularly easily observable for the SI in characteristic Raman line 520cm −1 of Si.
In summary, there appears to be two competing effects which collectively contribute to frequency asymmetry in Raman scattering: 1. Side-band Inequivalence (SI): which is frequency independent but optically nonlinear, and 2. Anomalous Raman Phenomenon (ARP): which is independent of optical power but frequency dependent.
Normally, SI and ARP cause asymmetry biased towards respectively red and blue. With the exception of Dirac materials, only for which ARP has been known to exist as early as 1998 [S50], in the rest of materials, ARP could be simply neglected. This implies that effectively ARP takes over for Dirac materials, which for other crystalline solids such as Si, SI does.
The authors of the first article referring to the ARP boldly mentioned that ARP does not arise from the error of the instrument [S50]. That was the first major step towards realizing that Stokes and anti-Stokes pairs are not necessarily symmetrical in frequency. Initial explanations were based on the formation of doublets [S51, S52], before it was found out that it was actually the slight deviation from linear photon dispersion across the Dirac point, which could be held responsible for ARP [S53-S56].
The expression for ARP was given as δ = −E s (∂ ω s /∂ ε L ) with E s being the energy of Raman Stokes line, ω s being the frequency of Raman shift, ε L being energy of light photon, and the expression within the parentheses being the dispersion of Raman peak determined from electronic energy band structure. Clearly, ARP is independent of optical power and therefore is optically linear. However, it could play measureable role in Raman spectroscopy of graphene [S57-S59] and even corrections to Raman thermometry and determination of temperature, if it were to be obtained from Stokes/anti-Stokes difference [S57].
Raman spectrometry of crystalline solids contain some of the most interesting results. In a recent article, Raman scattering of the zincblende semiconductor ZnTe at various temperatures has been carried out [S60] and the raw measurement data were made available to the author. Interestingly, the longitudial (LO) and transverse (TA) optical phonons contribute differently to S11/S21 the Raman spectra and the second harmonic of 2LO can also be seen. Interestingly, for most of the temperature range, both of LO and 2LO modes follow the same behavior, within a factor of 2. This is in complete agreement with the formula for the SI of second-order side-bands in quantum optomechanics in the weakly nonlinear regime [S3], and thus another way to verify the higher-order operator algebra presented therein.  Figure S12. Lorentzian peaks for Stokes(red)/anti-Stokes(blue) pairs of Raman scattering measurements from ZnTe at various temperatures [S60].
In another study [S61], SI of three different materials, namely graphene on Si, MoS 2 on Si, and crystalline Si (with a 100nm native oxide) has been carried out at various temperatures. These are among the most extensive collection of measurements ever made available to the author in raw data format. Measurements were done using a 600gr/mm grating and for each trace both Stokes and anti-Stokes along with the central resonance were recorded at one shot. For those traces corresponding to the same material and the same temperature, trace-averages were made. Each trace exhibited a number of Raman lines which were extracted and after fitting Lorentzians and referencing to the frequency determined from the central Rayleigh peak, are shown in Fig. S14 at various temperatures. As it could be seen in Fig. S15, all SI values are positive and can be as large as 1% and decrease with Raman line frequency. Agreement to aδ ∝ 1/Ω 2 fit from (S12) is excellent.
Similar measurements were done for MoS 2 on Si samples, and the SI values at various temperatures are shown for four individual Raman lines in Fig. S16. In general, a temperature variation from room temperature to 150 • C does not seem to alter the SI values appreciably.
Among these sets of measurements [S61], the most interesting is the last set done for Si, which exhibits only two resonances at 303/cm and 520/cm, and the experiment was redone and traces recorded a few times to later provide for averaging. The SI values are shown in Fig. S17 for all available traces in two forms of versus temperature and versus optical power. In after averaging over all traces at identical temperatures, as we learned from the previous two sets of traces. This will make it possible to estimate a linear fit as y = mx going through origin, where (x i , y i ) samples are available. This is so since in the weakly coupled limit, SI should zero at zero optical power, behaving roughly as δ ≈ g 2 0n /Ω [S3]. Then, the least-squares slope is simply obtained as m = ∑(xiyi)/ ∑ x 2 i . The ratio of obtained slopes for the two Raman lines is to be S13/S21  Figure S15. Side-band Inequivalences computed from raw measurement data processed in Fig. S14: SI for various Raman lines at different temperatures (left); SI for various Raman lines temperature-averaged [S61]. Note that SI values for each trace are renormalized to their maximum, to allow a simple and visible fit to (S12) which is in remarkable agreement.  Figure S16. Variation of four identified Raman lines from MoS 2 on Si substrate with respect to substrate temperature [S61]. compared with the inverse ratio of squared frequencies and the results agree very well. Doing this will give m 1 m 2 = 0.33, (S13) where subscripts 2 and 1 respectively correspond to the 303/cm and 520/cm lines. Careful inspection of Raman scattering on two-dimensional Transition Metal Di-Chalcogenides (TMDCs) [S62, S63] exhibits significant SI for all materials at all resonances, which deserves a thorough and in-depth study.
A very recent article [S64] has reported Raman scattering study of two pervoskites CsPbBr 3 and MAPbBr 3 at various temperatures. Pervoskites are known to be very complex materials with peculiar optoelectronic and chemical properties, which are not fully understood, yet exhibit numerous Raman lines. The raw measurement data were made available to the author from which SI values of identified Raman peaks were extracted. After analysis and averaging over temperatures, the overall SI values have been found and plotted in Fig. S19. Calculated SI numbers are typically large in magnitude up to 3% and do not follow a well understood pattern. Also, Raman measurements on NbSe 2 [S65] for the 11-layer and bulk samples respectively show SI values of 0.46% and 4.56%, the latter being remarkably large.
In a valuable book [S66] which also contains the demonstration version of OPUS [S67] software for plotting Raman spectra, a database of measurement is supplied out of which traces of four materials contain both Stokes and anti-Stokes side-bands. Typically, these reference measurements are averaged over hundreds of Raman scattering to remove background noise and improve the visibility of resonances. To a great extent, these measurements seem reliable and furthermore, contain the central Rayleigh resonance which allows accurate zero centering. These four mentioned substances are Ceramide, heavy Ethanol, Stearic Acid, and Sulphur. With the exception of first resonance of Stearic Acid, all these exhibit remarkably strong and positive SI, as shown in Si Raman Spectra (line 303/cm)

Side-band Inequiavalence (%)
Pump Power (%) Figure S17. Variation of SI for two characteristic lines of Si versus temperature (left in black) and versus optical power (right in red) for available traces [S61].  Figure S18. Variation of SI versus optical power for the two major lines at 303/cm and 520/cm [S61].
The Coherent company has developed a new technology for precision remote Raman analysis of various materials, many examples of which are publicized in their newest brochure [S68]. It is very interesting to take a look at the calculated SI values from their available data. All these very different substances, (1) Carbamazepine (a common prescription for epilepsy and other S15/S21  Figure S19. Variation of SI after temperature averaging of identified Raman lines for two pervoskites CsPbBr 3 and MAPbBr 3 [S64] from raw measurement data. neurologic disorders), (2) Carbamazepine hydrate, (3) Crystallized Caffeine (the extract and essential ingredient of coffee), and (4) Sulphur in three different configurations (α: crystalline; β : amorphous; λ : liquid) exhibit large SI numbers up for many Raman lines to a few percent. In all four studied cases, SI is positive and significantly larger than zero.