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The Journal of Chemical Physics

Fundamental detectability of Raman scattering: A unified diagrammatic approach

Wei Min, Xin Gao

Published Mar 5, 2024• 17 citations

Abstract

Both spontaneous Raman scattering and stimulated Raman scattering (SRS) are cornerstones of modern photonics, spectroscopy, and imaging. However, a unified understanding of the ultimate detectability of Raman scattering is lacking, due to both historical and technical reasons. Starting from quantum electrodynamics, we formulate the fundamental detectability for both spontaneous Raman scattering and SRS. The key concept is recognizing spontaneous Raman scattering as stimulated Raman process driven by vacuum field fluctuation. A simple and unified expression, Eq. (17), is derived, which can be depicted on a two-dimensional phase-diagram-like graph with inherent symmetry. It turns out that the particle nature of light dictates the ultimate detectability of spontaneous Raman scattering, which can be represented by a line on this detectability diagram. Importantly, if provided with a reasonably strong Stokes photon flux, SRS can breach this fundamental limit and open uncharted territory of drastically accelerated measurement speed and much lower detection concentration relevant to biological imaging. Such new territory in the detectability diagram is otherwise forbidden by the spontaneous counterpart. Diagrammatical analysis explains the empirical observations, provides quantitative insights, and makes new predictions. Notably, recent experimental applications of SRS microscopy can almost entirely be captured by this diagram, further supporting the explanatory power of the theory. Thus, this unified diagrammatic approach outlines a framework to understand all Raman-based measurement and provides a theoretical explanation for the remarkable utility of the emerging SRS microscopy.

DOI: 10.1063/5.0197977

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A theory of stimulated Raman scattering (SRS) is presented which treats in a unified manner the buildup of SRS from spontaneous Raman scattering and the spatial propagation which leads to gain. Maxwell-Bloch equations describe the coupling between the Stokes field operator and the collective atomic operators, which are driven by a classical laser field under low-signal-gain conditions, where the atomic ground states and laser field remain undepleted. An analytic expression is derived for the forward-propagating Stokes field operator in a one-dimensional spatial approximation when the Fresnel number of the excited medium is unity. Steady-state and transient Stokes intensities are evaluated under low-gain (spontaneous) and high-gain (stimulated) conditions. The Stokes intensity is found to be exactly independent of a laser bandwidth arising solely from phase fluctuations. The power spectrum of steady-state SRS is evaluated, giving a gain-narrowed spectral profile when the laser is narrow band, and a spectrum identical to the laser spectrum when the laser is broad band. A strong analogy between SRS and two-level superfluorescence is found; in both cases the macroscopic, collective dipole moment is initiated by quantum fluctuations.

We present a theoretical framework for spontaneous Raman scattering that fundamentally bridges quantum-mechanical and nonlinear-optical approaches. By conceptualizing spontaneous Raman scattering as a stimulated Raman gain or loss event seeded by the quantum vacuum field, we rigorously derive the spontaneous Raman cross-section directly from the third-order nonlinear susceptibility. Crucially, this framework predicts the existence of a hitherto unrecognized phenomenon: "spontaneous Raman loss" (sRL), which acts as the vacuum-seeded counterpart to stimulated Raman loss, complementing traditional spontaneous Raman scattering (spontaneous Raman gain, sRG). Furthermore, we establish a rigorous connection to the traditional Kramers-Heisenberg-Dirac (KHD) theory, revealing that the spontaneous process is governed by interference before a detector between the signal field emitted from molecules and the vacuum field itself that stimulates the molecules. This insight uncovers a direct correspondence between the sRG susceptibility and the rotating/counter-rotating interference terms in the KHD formula. Ultimately, we extend the foundational KHD theory by incorporating previously unrecognized essential terms, achieving perfect analytical agreement between the quantum mechanical and nonlinear optical descriptions of Raman scattering.

Abstract Stimulated Raman scattering (SRS) is a powerful technique for imaging and spectroscopy, in particular in chemistry and biology. By introducing a full quantum description, we determine the ultimate quantum limits of precision and optimal strategies to approach them within the rigorous framework offered by quantum metrology. This approach reveals optimal uses of squeezed states, and the possibility of enhancing the resolution beyond recent experiments. Our work establishes a basis for approaching quantum precision limits in SRS and integrating nonlinear spectroscopy and imaging with emerging quantum technologies.

Quantum-enhanced stimulated Raman scattering (QE-SRS) microscopy sits at the intersection of biomedical imaging, laser microscopy, and quantum optics toward realizing sensitive molecular-vibrational imaging. This Perspective explores the current status of this emerging field, integrating viewpoints from these diverse disciplines to highlight its potential for advancing biological and medical imaging. We discuss key challenges and future directions in leveraging quantum light to push the sensitivity limits of SRS microscopy, paving the way for next-generation imaging modalities.

In our studies of soliton formation in stimulated Raman scattering (SRS), we have discovered that a fully quantum mechanical theory predicts large shot-to-shot fluctuations in the SRS power spectrum, which leads to soliton decay. This theory, which includes a Langevin operator to simulate collision-induced fluctuations and maintain operator consistency, can be used to generate the statistics of the macroscopic SRS output. We have also studied a simpler model which uses a random vacuum field as an input to the semiclassical SRS equations. The vacuum field was numerically generated by summing a set of electromagnetic field modes that spanned the collisional Raman linewidth. Each mode was separated in frequency by less than the resolution of the system and given a random phase for each shot. Despite the lack of Langevin terms, the shot-to-shot variations and two-frequency correlation function from the simple approach were essentially indistinguishable from the fully quantum mechanical theory. This suggests that a simulation of SRS from spontaneous emission does not always require Langevin terms in the quantum noise.

Manifestations of quantum fluctuations in stimulated Raman scattering ( SRS ) have been intensively studied lately1. Quantum noise present in spontaneous scattering stimulates fluctuations both in phase and in amplitude characteristics of output Stokes emission. At present statistical properties of amplitude fluctuations are studied in detail and it is shown that in linear transient regime of SRS without pump depletion statistical scatter of Stokes energy Es has negative exponential distribution2. In saturation regime of SRS the amplitude fluctuations are almost suppressed simultaneously with pump depletion. The phase fluctuations manifest both in linear and in nonlinear SRS and produce beam pointing fluctuations3, a change in Stokes emission spectra from pulse to pulse4 and cause soliton appearance in Rayleigh component of SRS5. Phase noise is observed in saturation regime due to relation between phases of Stokes and phonon waves which is preserved in processes of stationary and quasistationary SRS amplification. Spectrum phase fluctuations manifest as fine structure of SRS spectrum4, fluctuations of average Stokes frequency and Stokes linewidth fluctuations. The evolution of phase fluctuations in time is determined by the nature of spontaneous scattering and polarization of Raman media and is of great interest as one can follow its development from vacuum modes level to saturation regime. However up to now this problem has not been studied yet.