Quantum-Mechanical Theory of Linear and Nonlinear Susceptibility

Robert W. Boyd

Robert W. Boyd

Stefan Scheel, Dirk‐Gunnar Welsch

We present a consistent quantum theory of the electromagnetic field in nonlinearly responding causal media, with special emphasis on Chi(2) media. Starting from QED in linearly responding causal media, we develop a method to construct the cubic Hamiltonian expressed in terms of the complex nonlinear susceptibility in a quantum mechanically consistent way. In particular, we show that the method yields the nonlinear noise polarization, which together with the linear one is responsible for intrinsic quantum decoherence.

A. R. Kolovskii

Hervé Leblond

The use of the matrix density formalism, and of a multiscale expansion, allows the derivation of a macroscopic nonlinear evolution equation for a short light pulse (a nonlinear Schrödinger equation), directly from the microscopic quantum mechanics equations. We consider the simple case of a monochromatic plane wave, interacting with independent two-level atoms, to show that such a computation is possible. For their linear part, the results agree with that of the linear dispersion theory, but the obtained nonlinear coefficient differs appreciably from that derived from the computation of the so-called nonlinear susceptibilities, except in one simple particular situation.

D. Leporini

The nonlinear response to external disturbances of a system of interest scrA (the ``atom'') in contact with a thermal bath is studied. The central issue is the relationship between the nonlinear susceptibility and the equilibrium relaxation spectra of observables of scrA. Paralleling the results of the linear-response theory, the nth-order nonlinear susceptibility is factorized by quantum-mechanical arguments into two terms pertaining to the disturbances and the spin-bath system, respectively. The role of slow atom states, i.e., states relaxing on times T longer than the bath correlation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$, is pointed out, by recurring to recent findings on their general features [Phys. Rev. A 46, 6222 (1992)]. In the frame of a stochastic picture of the bath and a multilevel scheme of the system scrA, the conditions under which the nonlinear susceptibility provides the spectrum of the slow atom states are clarified. Illustrations are drawn from magnetic resonance to demonstrate that techniques based on the nonlinear response of a spin system to multiple continuous waves compete favorably with pulsed techniques to provide information on longitudinal spin relaxation.