Theoretical Analysis of the Resonance Four-Wave Mixing Amplitudes: a Fully Nondegenerate Case Alexander Kouzov Department of Physics, Saint-Petersburg State University, Ulyanovskaya str. 3, Peterhof, Saint-Petersburg 198504, Russia Email: alex@ak1197.spb.edu

Outline 1. Key characteristics of 4-wave mixing 2. Resonant 4WM: Pros & cons 3. Particular schemes to excite resonance 4WM: A/ 3 photons are absorbed, one is emitted (AAAE): sum frequency generation scheme B/ 2 photons are absorbed, 2 are emitted a/ resonance CARS-type schemes (AEAE) b/ unfolded scheme (AAEE) C/ 1 photons is absorbed, 3 are emitted (AEEE): difference frequency generation scheme 4. Decoupling procedure 5. Polarization dependence 6. Looking ahead

Basics of 4WM Theory 48 terms res TC-RFWM: CARS:

Fully Resonant 4WM: Pros and Cons High frequency and spatial resolution High sensitivity (pressures as low as 1 mbar) Nonintrusive tool Applies both to isotropic and anisotropic media TC-RFWM: unique tool to directly trace the state-to-state collisional transfer Flexible polarization setup allowing to study rotational anisotropy at highest state of art Radiation coherence Limitations Weakly absorbing media (low-pressure gases)

Theoretical techniques Double-sided Feynman diagrams S.Y. Yee et al, Phys. Rev. A. vol. 18, 1597 (1978). Line-space (Liouville) techniques D. Forster “Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions” (Benjamin, Reading, MA, 1975). Irreducible spherical tensor (IST) formalism D. A. Varshalovich, A. N.Moskalev, and V. K. Khersonskii, “Quantum Theory of Angular Momentum” (World Scientific, Singapore, 1988)

Sum frequency generation 3 p 2 4 n 1 i 4 time f i 3 p 2 n 1

Decoupling procedure (based on IST formalism) TC-RFWM (Zare’s group, Sandia Labs): S. Williams et al, J. Chem. Phys. 1994-1997

8 diagrams contribute at finite and CARS-like schemes 8 diagrams contribute at finite and resulting in: Conventional CARS (single Raman resonance) 3 4 2 1 n i Doubly resonant CARS schemes Triply resonant CARS scheme f f 1-2-3 p p 3 4 3 3 4 2-1-3 2-3-1 3-2-1 1 2 1 2 4 1 2 n n n i i i 4 diagrams: 1( )+3( ) - no cancellation at

Unfolded schemes (extended TC-RFWM) 4 f f 3 p 3 2 3 2 p n n f 1 4 1 2 p n 4 i i 1 4 4 f 4 3 p 2 f p p n f 2 n 3 1 2 1 n i 3 1

Difference frequency generation 2 i n 1 p f 4 1 3 2 n 3 n i 4 p 2 i 2 p 3 f 1

Difference frequency generation 4 4 4 f n p p n 2 2 2 p 2 p n 4 1 3 f n f 3 f 3 1 3 i 1 1 4 4 4 f 4 f p p p f f p 2 1 2 n 1 2 n 2 3 n n 3 1 i 1 3 3

Thanks for your attention ! Pros FR-FWM allows To study dipole-allowed transitions from/within unpopulated vibronic states Up and down frequency shifting in a wide spectral range Cons Sofisticated techniques (use of 3 lasers, finer frequency tunings) Looking ahead Generation of coherent MW; its indicatrix Extensions to time-domain Thanks for your attention !