Stokes–Anti-Stokes Coherence

• Stokes–anti-Stokes coherence is a phenomenon where phase-sensitive correlations link Stokes and anti-Stokes scattering via shared vibrational or acoustic excitations.
• It is characterized by measurable signatures such as intensity cross-correlations, Fano resonances, and quantum beat revivals in techniques like coherent Raman spectroscopy.
• Experimental observations show that pump power, temporal delays, and optical mode geometry critically influence non-classical correlations and photon pairing efficiency.

Stokes–anti-Stokes coherence denotes a family of phase-sensitive and correlation-based phenomena in which Stokes and anti-Stokes scattering channels are linked by a shared vibrational or acoustic excitation, by a common nonlinear polarization, or by coherent overlap of sideband pathways, so that the two channels cannot be treated as independent. In the literature, this linkage appears as phonon-mediated photon correlations in Raman scattering, collective vibrational superpositions selected by measurement geometry, phase-opposed interferograms in coherent Raman spectroscopy, complex Fano resonances in Brillouin scattering, coherent anti-Stokes generation in four-wave mixing, and drive- and linewidth-induced interference in dispersively coupled systems (Kasperczyk et al., 2015, Vento et al., 2021, Ideguchi et al., 2014, Lai et al., 2020, Chathanathil et al., 2022, Zhang et al., 29 Mar 2026).

1. Conceptual scope and physical meaning

At the most basic level, Stokes scattering creates or emits a vibrational quantum, whereas anti-Stokes scattering annihilates or absorbs one. In the Brillouin formulation of coherent acoustic phonons, Stokes scattering means that the probe photon emits a phonon into the acoustic wavepacket and loses energy, while anti-Stokes scattering means that the probe photon absorbs a phonon from the wavepacket and gains energy (Lai et al., 2020). In Raman formulations, the same asymmetry is expressed as a Stokes event that “writes” a phonon into the medium and an anti-Stokes event that “reads” it out (Kasperczyk et al., 2015).

The term coherence is used in more than one technical sense. In several Raman papers it denotes pair correlation mediated by a shared phonon memory, rather than first-order phase coherence between two classical optical fields (Kasperczyk et al., 2015, Guimarães et al., 2020). In Brillouin and coherent Raman spectroscopy, by contrast, the emphasis is on interference between amplitudes or on opposite-phase modulations produced by the same vibrational coherence (Lai et al., 2020, Ideguchi et al., 2014). This suggests that Stokes–anti-Stokes coherence is best regarded as an umbrella concept whose precise meaning depends on whether the relevant observable is a coincidence statistic, an interferometric signal, a Fano lineshape, or a driven four-wave-mixing response.

Setting	Link between Stokes and anti-Stokes	Principal observable
Raman in diamond membranes	the phonon excited by the S process is consumed in the aS process	$g_{\rm S,aS}^{(2)}(0)$ (Kasperczyk et al., 2015)
Coherent acoustic Brillouin scattering	reflected coherent acoustic phonon enables simultaneous Stokes and anti-Stokes access	complex Fano resonance (Lai et al., 2020)
Impulsive coherent Raman spectroscopy	same vibrational coherence produces opposite-phase Stokes and anti-Stokes modulations	differential interferogram (Ideguchi et al., 2014)
CARS	pump and Stokes prepare coherence that is read out as anti-Stokes light	directional anti-Stokes field (Chathanathil et al., 2022)
Collective liquid/fiber Raman	geometry and mode selection determine whether collective coherence is visible	oscillatory or suppressed $g^{(2)}$ (Vento et al., 2021, Panyukov et al., 14 May 2026)

A recurring conceptual correction is that ordinary thermal anti-Stokes scattering and correlated Stokes-induced anti-Stokes scattering are distinct regimes. Treating the anti-Stokes channel solely through Bose–Einstein phonon occupancy is therefore incomplete once the correlated SaS channel becomes appreciable (Parra-Murillo et al., 2015).

2. Phonon-mediated Raman correlations and pair formation

In diamond membranes, Stokes–anti-Stokes coherence is realized through a shared vibrational quantum: the phonon created by a Stokes scattering event can later be annihilated in an anti-Stokes event (Kasperczyk et al., 2015). The experiment used a 785 nm pulsed laser, a freestanding 50 $\mu$m diamond membrane, the diamond optical phonon at 1332 cm$^{-1}$, Stokes photons at 880 nm, and anti-Stokes photons at 710 nm. Because diamond has a large phonon energy, thermally generated anti-Stokes scattering is weak at room temperature, which helps the Stokes-induced anti-Stokes channel stand out (Kasperczyk et al., 2015).

The standard diagnostic is the zero-delay intensity cross-correlation

$$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$

In the Stokes-induced anti-Stokes regime, the paper reports $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$, while the coincidence rate is mostly quadratic in pump power and requires a cubic term at high power (Kasperczyk et al., 2015). The quadratic term is consistent with a two-step process in which one interaction creates the phonon and another reads it out, whereas the cubic correction signals departure from the simple single-phonon picture through stimulated Raman scattering, coherent Raman scattering, multiple phonon-photon swapping cycles, or depletion and recycling effects (Kasperczyk et al., 2015).

Related Hamiltonian descriptions formalize the same mechanism as coupled pump, phonon, Stokes, and anti-Stokes modes. In one effective model,

$$\hat H=\hbar \omega_0 \hat a^{\dagger}\hat a + \hbar \nu \hat c^{\dagger}\hat c + \hbar \omega_{S} \hat b_S^{\dagger}\hat b_S + \hbar \omega_{aS} \hat b_{aS}^{\dagger}\hat b_{aS} + \hbar \lambda_S (\hat a \hat c^{\dagger}\hat b_{S}^{\dagger} + h.c.) + \hbar \lambda_{aS} (\hat a \hat c \hat b_{aS}^{\dagger} + h.c.),$$

with phonon dissipation included through a Lindblad master equation (Parra-Murillo et al., 2015). In this framework, the anti-Stokes intensity crosses over from $I_{aS}\propto P_L$ at low power, where thermal phonons dominate, to $I_{aS}\propto P_L^2$ when the correlated SaS channel dominates; Stokes intensity remains essentially linear in $P_L$ (Parra-Murillo et al., 2015). The measured ratio $g^{(2)}$0 therefore acquires a power-dependent SaS correction and is no longer determined by the Bose–Einstein factor alone (Parra-Murillo et al., 2015).

Pump–probe theory further distinguishes real SAS coherence, in which a real phonon survives from the Stokes event to the delayed anti-Stokes event, from virtual SAS coherence, in which the pair is generated through exchange of a virtual phonon and is governed mainly by pulse overlap (Diaz et al., 2020). In the resonant real-phonon regime, coincidence probability is broad in delay and follows the phonon lifetime; in the off-resonant virtual regime, the coincidence peak is centered at zero delay and is insensitive to phonon decay. The model was compared with experiment in diamond and reproduced a phonon lifetime of about $g^{(2)}$1 ps (Diaz et al., 2020).

A more stringent notion of pairing is developed in a quantum Raman model with independent Stokes and anti-Stokes nonlinear interactions (Thapliyal et al., 2021). The control parameter

$g^{(2)}$2

separates an exponential regime $g^{(2)}$3 from an oscillatory regime $g^{(2)}$4. In the oscillatory regime, special pump amplitudes

$g^{(2)}$5

yield ideal paired states in which the mean Stokes and anti-Stokes photon numbers coincide, the vibrational mode returns to vacuum, and the noise-reduction factor reaches $g^{(2)}$6 (Thapliyal et al., 2021). The same paper emphasizes that perfect pairing and maximal entanglement are not identical criteria.

3. Collective vibrational coherence and the role of geometry

In liquid CS$g^{(2)}$7, time-resolved Stokes–anti-Stokes coincidences show that spontaneous Raman scattering need not leave the molecular ensemble in an incoherent statistical mixture (Vento et al., 2021). Using a $g^{(2)}$8 fs, 80 MHz write pulse followed by a delayed read pulse, and collecting Stokes and anti-Stokes photons into a single-mode fiber in transmission, the experiment found oscillatory revivals over several picoseconds in $g^{(2)}$9 (Vento et al., 2021). A model based on a random single-molecule excitation predicts only multi-exponential decay and fails to reproduce these oscillations.

The data are instead reproduced by a heralded collective vibrational state. For ideal single-mode post-selection, the state after Stokes detection is

$\mu$0

For CS$\mu$1, four vibrational sub-ensembles with frequencies $\mu$2 and relative weights $\mu$3 define the collective state

$\mu$4

The anti-Stokes probability is proportional to

$\mu$5

so the observed revivals are quantum beats arising from interference among collectively excited sub-ensembles (Vento et al., 2021).

The same experiment showed that the coherence is not an intrinsic fixed property of the liquid alone. Visibility of the Stokes–anti-Stokes oscillations decreases when the collection geometry is changed from single-mode to few-mode to multimode fiber, because the single-mode geometry erases which-molecule information more effectively (Vento et al., 2021). This dependence on optical mode selection is reinforced by a later model for weakly guiding optical fiber, which shows that orthogonality of fiber modes makes different modal amplitudes uncorrelated in the standard detection scheme (Panyukov et al., 14 May 2026).

In that fiber model, the non-classical part of the measured cross-correlation scales as

$\mu$6

so increasing the number of guided modes suppresses both cross-correlations and autocorrelations toward the classical value $\mu$7 (Panyukov et al., 14 May 2026). At the same time, the normalized correlation does not vanish merely because the sample is macroscopic: numerator and denominator both scale as $\mu$8, so the explicit dependence on molecule number cancels (Panyukov et al., 14 May 2026). This combination of results makes spatial mode structure a central control parameter for whether collective Stokes–anti-Stokes coherence is revealed or diluted.

4. Interference, Fano structure, and coherent Raman readout

A distinct manifestation of Stokes–anti-Stokes coherence occurs in ultrafast Brillouin scattering from coherent acoustic phonons. For a coherent acoustic phonon traveling in a film, the Brillouin frequency is

$\mu$9

with $^{-1}$0 the refractive index, $^{-1}$1 the longitudinal sound speed, and $^{-1}$2 the probe wavelength (Lai et al., 2020). The crucial mechanism is reflection of the coherent acoustic phonon at the free surface, which abruptly changes phonon momentum and switches the phase-matching relation from one scattering channel to the other. During this transition, the reflected phonon can re-enter the optical penetration depth while preserving coherence, so the probe can experience both Stokes and anti-Stokes scattering simultaneously (Lai et al., 2020).

After Fourier transformation, the scattering cross-section contains two resonances centered at $^{-1}$3 and $^{-1}$4, corresponding respectively to anti-Stokes and Stokes scattering, and the amplitudes add coherently (Lai et al., 2020). Rewriting the result in Fano form gives a complex asymmetry parameter

$^{-1}$5

whose real part reflects resonance overlap and whose imaginary part encodes losses associated with reflection (Lai et al., 2020). The paper demonstrated this on a 110 nm tungsten film on Si with a tunable Ti:sapphire pump-probe setup at 820, 860, and 900 nm, observing acoustic echoes at about 42 ps and 84 ps and Fano-like dips in the Fourier-transformed spectra. Tracking the trajectory of $^{-1}$6 in the complex plane allows separation of partial reflection or energy loss from phase randomization or coherence loss at rough interfaces, providing a non-destructive probe of surface and buried interface quality (Lai et al., 2020).

In impulsive coherent Raman spectroscopy, the same vibrational coherence produces Stokes and anti-Stokes signals with opposite phase (Ideguchi et al., 2014). A sequence of excitation–probe pulse pairs with linearly increasing delay is generated by a scanning Michelson interferometer, and the delay-dependent modulation constitutes a time-domain interferogram. The anti-Stokes radiation is isolated with a shortpass filter, the Stokes radiation with a longpass filter, and the two signals are sent to a balanced differential detector (Ideguchi et al., 2014). Because the modulations are phase-inverted, subtraction enhances the coherent Raman contribution and suppresses common-mode noise. In the reported hexafluorobenzene spectra, the SNR for the $^{-1}$7 line is approximately 770 in the Stokes channel, 470 in the anti-Stokes channel, and 1520 in the differential channel; the measured spectral span is about $^{-1}$8, with about $^{-1}$9 resolution (Ideguchi et al., 2014).

Coherent anti-Stokes Raman scattering extends this phase-sensitive picture into four-wave mixing. In a chirped-pulse control scheme, pump and Stokes pulses prepare the vibrational coherence $$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$0, and a probe converts that coherence into anti-Stokes radiation at

$$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$1

After adiabatic elimination, the dynamics reduce to a “super-effective” two-level model in which maximum coherence corresponds to $$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$2, achieved when populations are balanced (Chathanathil et al., 2022). The required chirp relation is

$$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$3

and the pump chirp is reversed at the pulse center to preserve the maximal-coherence state rather than continue adiabatic transfer (Chathanathil et al., 2022). This establishes an explicitly coherence-engineered route to stronger and more selective anti-Stokes generation.

5. Quantum formalisms, observables, and entanglement structure

Several complementary formalisms describe Stokes–anti-Stokes coherence. Effective Hamiltonian approaches treat pump, Stokes, anti-Stokes, and vibrational modes quantum mechanically and add dissipation through Lindblad terms (Parra-Murillo et al., 2015, Diaz et al., 2020). More recent work derives the correlated contribution directly from a fully quantized Raman polarization using Heisenberg perturbation theory (Corrêa et al., 24 Jul 2025). In that treatment, ordinary Raman scattering appears at zeroth order and depends on phonon occupation, while the correlated SaS contribution appears already in first order as a four-wave-mixing term and is independent of the initial phonon occupation (Corrêa et al., 24 Jul 2025). The resulting third-order susceptibility has the same functional form as the classical stimulated Raman susceptibility, but now with a microscopic quantum derivation appropriate to spontaneous correlated pair generation (Corrêa et al., 24 Jul 2025).

The observable used most widely is the second-order cross-correlation $$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$4, but the literature also uses the noise-reduction factor, two-mode principal squeezing variance, logarithmic negativity, non-classicality depth, steering parameter, and Bell parameter (Thapliyal et al., 2021). This broader set of diagnostics matters because large $$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$5 and ideal one-to-one pairing are not equivalent. The Raman twin-beam analysis identifies ideal pairing through conditions such as $$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$6, equal Stokes and anti-Stokes photon numbers, and a joint state containing only equal photon-number terms (Thapliyal et al., 2021).

A two-photon wave-function treatment provides a spatial-temporal picture of correlated SaS emission (Guimarães et al., 2020). In the stationary regime, the scattered pair amplitude has the form

$$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$7

with a phase-matching constraint and a correlation envelope set by the phonon decay rate $$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$8 (Guimarães et al., 2020). In this formulation, coherence is encoded in a two-photon amplitude showing temporal correlation, spatial or angular correlation, and polarization correlation inherited from the pump.

Near Raman resonance in diamond, Stokes–anti-Stokes coherence can also produce polarization entanglement through a coherent superposition of a purely electronic four-wave-mixing pathway and a phonon-mediated Raman pathway (Freitas et al., 2023). The two-photon state is written as

$$g_{\rm S,aS}^{(2)}(0)=\frac{P({\rm S,aS})}{P({\rm S})\,P({\rm aS})} =\frac{P({\rm S|aS})}{P({\rm S})}.$$9

For $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$0, the measured CHSH parameter is $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$1 at $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$2, $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$3 at $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$4, and $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$5 at $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$6; at $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$7, no CHSH violation is observed in the reported spectral windows (Freitas et al., 2023). The paper also reports $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$8 for the reconstructed two-photon state near resonance, indicating substantial mixedness (Freitas et al., 2023).

6. Platforms, control protocols, and conceptual boundaries

The practical implications of Stokes–anti-Stokes coherence are diverse. In ultrafast acoustics, the complex Fano parameter extracted from reflected coherent acoustic phonons can distinguish interfacial energy loss from coherence loss, offering a mechanism for non-destructive testing of interface quality and a possible link to interfacial thermal transport characterization (Lai et al., 2020). In multiplex coherent anti-Stokes Raman scattering, spatial division multiplexing in a few-mode microstructured fiber uses LP01 to generate a broadband continuum and LP11 to preserve a residual narrowband 1064 nm pump, thereby providing a self-referenced coherent source without an external delay line (Mansuryan et al., 2023). In the reported paraffin measurements, the linewidth of the $g_{\rm S,aS}^{(2)}(0)\propto 1/P_L$9 mode changes from 18 cm$$\hat H=\hbar \omega_0 \hat a^{\dagger}\hat a + \hbar \nu \hat c^{\dagger}\hat c + \hbar \omega_{S} \hat b_S^{\dagger}\hat b_S + \hbar \omega_{aS} \hat b_{aS}^{\dagger}\hat b_{aS} + \hbar \lambda_S (\hat a \hat c^{\dagger}\hat b_{S}^{\dagger} + h.c.) + \hbar \lambda_{aS} (\hat a \hat c \hat b_{aS}^{\dagger} + h.c.),$$0 FWHM in the standard setup with delay line to 28 cm$$\hat H=\hbar \omega_0 \hat a^{\dagger}\hat a + \hbar \nu \hat c^{\dagger}\hat c + \hbar \omega_{S} \hat b_S^{\dagger}\hat b_S + \hbar \omega_{aS} \hat b_{aS}^{\dagger}\hat b_{aS} + \hbar \lambda_S (\hat a \hat c^{\dagger}\hat b_{S}^{\dagger} + h.c.) + \hbar \lambda_{aS} (\hat a \hat c \hat b_{aS}^{\dagger} + h.c.),$$1 FWHM in the self-referenced setup, and a 20% amplification of the residual pump yields about a 20% increase in the M-CARS signal (Mansuryan et al., 2023).

In dispersively coupled systems beyond the resolved-sideband limit, recent theory defines Stokes–anti-Stokes coherence as interference between sideband pathways activated simultaneously by classical driving and finite linewidth (Zhang et al., 29 Mar 2026). The asymmetry is quantified by

$$\hat H=\hbar \omega_0 \hat a^{\dagger}\hat a + \hbar \nu \hat c^{\dagger}\hat c + \hbar \omega_{S} \hat b_S^{\dagger}\hat b_S + \hbar \omega_{aS} \hat b_{aS}^{\dagger}\hat b_{aS} + \hbar \lambda_S (\hat a \hat c^{\dagger}\hat b_{S}^{\dagger} + h.c.) + \hbar \lambda_{aS} (\hat a \hat c \hat b_{aS}^{\dagger} + h.c.),$$2

with $$\hat H=\hbar \omega_0 \hat a^{\dagger}\hat a + \hbar \nu \hat c^{\dagger}\hat c + \hbar \omega_{S} \hat b_S^{\dagger}\hat b_S + \hbar \omega_{aS} \hat b_{aS}^{\dagger}\hat b_{aS} + \hbar \lambda_S (\hat a \hat c^{\dagger}\hat b_{S}^{\dagger} + h.c.) + \hbar \lambda_{aS} (\hat a \hat c \hat b_{aS}^{\dagger} + h.c.),$$3 corresponding to complete destructive interference of one channel (Zhang et al., 29 Mar 2026). The same framework predicts constructive interference and signal amplification, and extends to arrays with gain scaling as $$\hat H=\hbar \omega_0 \hat a^{\dagger}\hat a + \hbar \nu \hat c^{\dagger}\hat c + \hbar \omega_{S} \hat b_S^{\dagger}\hat b_S + \hbar \omega_{aS} \hat b_{aS}^{\dagger}\hat b_{aS} + \hbar \lambda_S (\hat a \hat c^{\dagger}\hat b_{S}^{\dagger} + h.c.) + \hbar \lambda_{aS} (\hat a \hat c \hat b_{aS}^{\dagger} + h.c.),$$4 or $$\hat H=\hbar \omega_0 \hat a^{\dagger}\hat a + \hbar \nu \hat c^{\dagger}\hat c + \hbar \omega_{S} \hat b_S^{\dagger}\hat b_S + \hbar \omega_{aS} \hat b_{aS}^{\dagger}\hat b_{aS} + \hbar \lambda_S (\hat a \hat c^{\dagger}\hat b_{S}^{\dagger} + h.c.) + \hbar \lambda_{aS} (\hat a \hat c \hat b_{aS}^{\dagger} + h.c.),$$5, depending on the driving protocol (Zhang et al., 29 Mar 2026). A related but distinct control paradigm uses zero-photon detection in optomechanics: null detection on the anti-Stokes channel enhances cooling beyond unconditional laser cooling, and, above a threshold detection efficiency, null detection on the Stokes channel can even overcome parametric-amplification heating and produce conditional cooling (Clarke et al., 2024).

Several conceptual boundaries are explicit in the literature. First, coherence does not always mean the same thing: in some papers it means coincidence-based pair correlation, in others phase-sensitive interference, collective post-selected superposition, or measurement-conditioned dynamics (Kasperczyk et al., 2015, Lai et al., 2020, Vento et al., 2021, Clarke et al., 2024). Second, standard thermometric use of the anti-Stokes/Stokes intensity ratio can become inaccurate when correlated SaS processes contribute appreciably (Parra-Murillo et al., 2015). Third, not every system with Stokes and anti-Stokes channels analyzes their mutual interference. In the domain-wall-string problem, Stokes and anti-Stokes scattering are treated as separate excitation and de-excitation processes of a shape mode, with no explicit coherent superposition or cross-term between them (Guo et al., 2024).

Taken together, these results show that Stokes–anti-Stokes coherence is not a single narrowly defined effect but a recurring structure of correlated light–matter scattering. Depending on platform and observable, it can manifest as shared-phonon memory, collective vibrational quantum beats, Fano asymmetry, anti-Stokes readout of a prepared coherence, entangled photon-pair generation, linewidth-induced interference, or geometry-dependent suppression of non-classical correlations (Kasperczyk et al., 2015, Vento et al., 2021, Lai et al., 2020, Chathanathil et al., 2022, Freitas et al., 2023, Panyukov et al., 14 May 2026).