A sum rule for nonlinear optical susceptibilities

Abstract

It is explicitly shown, for optical processes arbitrarily comprising two-, three-or fourphoton interactions, that the sum over all matter states of any optical susceptibility is exactly zero. The result remains true even in frequency regions where damping is prominent. Using a quantum electrodynamical framework t o render the photonic nature of the fundamental interactions, the result emerges in the form of a traceless operator in Hilbert space. The generality of the sum rule and its significance as a thermodynamic limit are discussed, and the applicability to real systems is assessed. IntroductionIn the theory of linear and nonlinear optical response from atomic and molecular systems, a common expedient for calculational and interpretive simplicity is the adoption of a two-level approximation, in which it is assumed that any material of interest has only one significant electronic excited state in the relevant frequency range. It has been known for some time that this approximation leads to an excited state polarisability that is the exact negative of the ground state polarisability, and the same proves true for the ground and excited state first hyperpolarisability. In the bulk, similar relationships hold for the linear and second order optical susceptibility tensors, and the consequential possibility of observing novel harmonic resurgence phenomena in suitable media has been noted [1]. Recently, it has also been shown that the susceptibility relationships retain validity even when resonance damping is entertained [2,3], provided the correct choice is made for the sign of the damping corrections, as exacted by the principles of time-reversal invariance.In this paper we develop a more general theory by considering multi-level systems with an arbitrary number of excited states, also considering higher-order optical processes such as those involved in four-wave mixing. Equations are developed using a quantum electrodynamical framework that faithfully renders the photonic nature of the fundamental interactions involved. As a result it is explicitly shown, for two-, three-and four-photon interactions, that for any optical process the sum of the susceptibilities for all matter states is zero. The result remains true even in frequency regions where damping is prominent. Applications range from Rayleigh scattering, through harmonic generation to CARS, the ac Stark effect and phaseconjugate reflection. The generality of the sum rule and its physical significance are discussed, and the paper concludes with an assessment of the applicability to real systems. interactions is then evaluated from the probability amplitude;whereis the resolvent operator for the sub-system, the kets i and f signifying its initial and final states, respectively. A more detailed description of this formalism can be found elsewhere [6]. For an n-photon process the leading contribution in the perturbative development of the probability amplitude is the matrix element Here the first label corresponds to the material state and the second o...

Perturbation theory and the two-level approximation: A corollary and critique

“…An example is afforded by recent work that has identified a rule dictating that the sum over all states of optical susceptibilities of any order has to be zero [42]. The result emerges in the form of a traceless operator in Hilbert space, symbolized by P r r h j r j i ¼ 0, where r denotes a matter state and the operator is here to be identified with the operator form of the polarizabilityâ ij , hyperpolarizabilityb ijk or higher-order response tensor.…”