Generalized Kramers-Heisenberg expressions for stimulated Raman scattering and two-photon absorption

The frequency-domain pump-probe signal of a material system interacting with two quantum modes of the radiation field is recast in terms of products of scattering amplitudes T matrix elements rather than the third-order susceptibility Im 3 . The resulting expression offers a more intuitive physical picture for the optical process compared with the semiclassical approach which treats the radiation field as classical. It can be derived and interpreted using closed-time-path-loop diagrams which represent the joint state of the matter and the field for each contribution to the signal. The signal has two components representing stimulated Raman scattering 12 and two-photon absorption 1+2 two-photon resonances. Both are expressed as nonequi- librium steady-state photon and matter fluxes, as is common in the description of dissipative processes in open quantum systems. Pump probe spectroscopy is the simplest nonlinear optical technique. The system is subjected to two laser beams: the pump beam with wave vector k1 and frequency 1 and the probe beam k 2 , 2 . The signal is defined as the change in the transmitted probe beam intensity caused by the pump beam. Apart from single-photon resonances obtained by tun- ing 1 and 2 which can also be observed in linear spec- troscopy, the technique also shows new two-photon reso- nances. Stimulated Raman scattering SRS spectroscopy 1-4 focuses on the 12 resonances, whereas the two- photon absorption TPA focuses on 1+2 5. Both tech- niques have been extensively employed since the early days of nonlinear optics. Recent applications include imaging and microscopy 6,7 and femtosecond techniques 8-11. Nonlinear optical spectroscopy is commonly formulated within the semiclassical SC approach whereby a quantum material interacts with classical fields. This convenient theo- retical framework has been very successful in the theories of the laser 12 and in describing processes where quantum effects of the fields are negligible 1,13,14. Most of the current understanding and design of such experiments is based on this language. In an n+1-wave mixing process a material sample is exposed to n+1 optical fields. Energy exchange between the modes, mediated by the matter-field interaction is observed by monitoring one of these modes, labeled as the signal mode. In spectroscopic applications we extract information about the sample by tuning the fields on- or near-resonance with the material optical transitions. Two detection schemes are commonly used. In heterodyne detec- tion one measures the electric field itself both amplitude and phase; in homodyne detection the field intensity is measured 14. Semiclassically, an n+1 wave mixing experiment is de- scribed as a two-step process. First, the n incoming waves induce an nth order polarization in the material, which is calculated microscopically by solving the quantum Liouville equation. In a second step, this nonlinear polarization acts as a source in the macroscopic Maxwell equations to generate the signal field. In macroscopic samples the signal field wave vector is given by a linear combination of the incoming pulse wave vectors 1,13-15 and is detected interferrometrically using an external reference field in the same direction, called the "local oscillator" LO. This somewhat awkward nomen- clature originated in radio wave technology where the re- ceiver is indeed local. The difference in intensity of the LO with and without the signal field defines the heterodyne de- tected signal: