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Physical Review A

Alternative nonlinear-optical expansion and diagrammatic approach:Nth-order dephasing-induced effects

Rick P. Trebino

Published Sep 1, 1988• 23 citations
Publisher

Abstract

We show that dephasing-induced nonlinear-optical effects whose amplitudes are proportional to trinomials of dephasing rates, ${\ensuremath{\Gamma}}_{\mathrm{ij}}$+${\ensuremath{\Gamma}}_{\mathrm{jk}}$-${\ensuremath{\Gamma}}_{\mathrm{ik}}$, exist in all nonlinear orders of perturbation theory. To do this, we derive a new form for the perturbation expansion of the susceptibility, equivalent to the currently used expansion, but which writes each order as a sum of (1) terms due to single-sided diagrams, and (2) correction terms, all of which are proportional to trinomials of dephasing rates. The proof of this result utilizes a diagrammatic approach involving a new type of double-sided diagram for the latter terms. These diagrams immediately yield the terms proportional to trinomials of dephasing rates in all orders without the need for algebraic manipulation. This approach also reveals some general properties of the Nth-order susceptibility.

DOI
10.1103/physreva.38.2921

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We show that dephasing-induced nonlinear-optical effects whose amplitudes are proportional to trinomials of dephasing rates, \ensuremathΓij{\ensuremath{\Gamma}}_{\mathrm{ij}}\ensuremathΓij​+\ensuremathΓjk{\ensuremath{\Gamma}}_{\mathrm{jk}}\ensuremathΓjk​-\ensuremathΓik{\ensuremath{\Gamma}}_{\mathrm{ik}}\ensuremathΓik​, exist in all nonlinear orders of perturbation theory. To do this, we derive a new form for the perturbation expansion of the susceptibility, equivalent to the currently used expansion, but which writes each order as a sum of (1) terms due to single-sided diagrams, and (2) correction terms, all of which are proportional to trinomials of dephasing rates. The proof of this result utilizes a diagrammatic approach involving a new type of double-sided diagram for the latter terms. These diagrams immediately yield the terms proportional to trinomials of dephasing rates in all orders without the need for algebraic manipulation. This approach also reveals some general properties of the Nth-order susceptibility.
We study perturbation expansions for quantum systems which are nonlinear in the sense that they may contain derivative couplings. The characteristic feature of these systems is the occurrence of products of noncommuting operators in the interaction Hamiltonian. In order to perform perturbation expansions in the Dyson-Wick manner it is in general necessary to order these operators according to some ordering scheme. We determine the most general scheme of this sort which allows a Dyson-Wick expansion with pair contractions. It involves a set $\ensuremath{\lambda}$ of three dimensionless ordering parameters which are related to the three aaa priori undetermined equal-time contractions. Different $\ensuremath{\lambda}$-ordering schemes are related via an "equal-time" or "static" Wick theorem. In order to be able to contract operators which may be taken either at equal or at different times we introduce a time-ordering operation ${T}_{\ensuremath{\lambda}}$ which extends Dyson's time-ordering operation to equal times in a fashion dependent on the $\ensuremath{\lambda}$ ordering. We arrive at a "dynamic" Wick theorem which enables us to decompose the ${T}_{\ensuremath{\lambda}}$ product of a set of operators taken at arbitrary times into a sum of ${\ensuremath{\lambda}}^{\ensuremath{'}}$ products, where the most important case is that in which ${\ensuremath{\lambda}}^{\ensuremath{'}}$ represents normal ordering. Contrary to the usual Wick theorem ours works also in the case when two or more times coincide. Applying this theorem to the scattering operator expressed as a ${T}_{\ensuremath{\lambda}}$ product we obtain a class of Dyson-Wick expansions for nonlinear systems. Different expansions correspond to different choices of the ordering $\ensuremath{\lambda}$ and lead to different Feynman rules. The resulting diagrammatic technique for nonlinear systems is seen to have the following special features: (i) The interaction vertices include terms which arise from "quantum corrections," which are dependent on the ordering \ensuremath{\lambda} chosen for the noncommuting factors in the interaction Hamiltonian, and (ii) the equal-time contractions which occur in the closed-loop diagrams must be assigned corresponding $\ensuremath{\lambda}$-dependent values. In this way individual diagrams are $\ensuremath{\lambda}$ dependent. Nevertheless their sum is $\ensuremath{\lambda}$ independent, confirming the fact that the complete scattering operator is independent of the particular ordering chosen. The ordering multiplicity is also reflected in functional versions of the Dyson-Wick expansion. In the path-integral version this multiplicity lurks in the different possible discretization prescriptions. We discuss the relevance of our considerations to the question of the invariance of a quantum theory with respect to nonlinear point transformations. For the special case of a reducible system we present explicit calculations which show how this invariance is maintained in perturbation theory, for any ordering scheme belonging to the $\ensuremath{\lambda}$ class. In any particular scheme the invariance is ensured by a more or less intricate cancellation mechanism between the individual diagrams.
We have formulated expressions providing the number of terms (Y-G diagrams)1contributing to an n-th order susceptibility of a non-linear optical process. Obtaining an accurate count of these terms is important for diverse reasons: a) To accurately specify the number of terms needed to evaluate the nonlinear optical susceptibility either analytically or numerically, b) To help develop sequences of diagrammatic contributions which can be summed to provide analytical results for the strong interaction limit. An example is the two-level system for semiclassical and quantized fields, c) To treat two or more processes (terms) which could give rise to interference, resulting, for example, in extra resonances due to a lack of complete coherence, d) To generalize results from quantum electrodynamics to situations for which multiple photon interactions and zero point fluctuations associated with non-vacuum states need to be considered. We shall discuss the number of possible diagrams in terms of three factors, which we call a permutation factor relating to the number of possible frequencies that can be assigned to each interaction vertex, a commutation factor relating to interaction vertex assignment on the two sides of the diagram, and a state factor relating to the number of diagrams generated from an enumeration of specific material state assignments between interaction vertices. We will discuss the application of these results to several situations of interest including second harmonic generation, degenerate four-wave mixing, and resonant CARS (coherent anti-Stokes Raman scattering) in a four-level system. The relevance of the present results to experimental results which have been obtained will be discussed along with future directions which need to be addressed including the summability of diagrams under specific approximations, the calculation of expectation values, and the nonperturbative representation of processes.
We have investigated the optical response of superradiant atoms, which undergoes three different damping mechanisms: radiative dissipation (γrγ_rγr​), dephasing (γdγ_dγd​), and nonradiative dissipation (γnγ_nγn​). Whereas the roles of γdγ_dγd​ and γnγ_nγn​ are equivalent in the linear susceptibility, the third-order nonlinear susceptibility drastically depends on the ratio of γdγ_dγd​ and γnγ_nγn​: When γd≪γnγ_d \ll γ_nγd​≪γn​, the third-order susceptibility is essentially that of a single atom. Contrarily, in the opposite case of γd≫γnγ_d \gg γ_nγd​≫γn​, the third-order susceptibility suffers the size-enhancement effect and becomes proportional to the system size.
Diagrammatic methods for the calculation of the nonlinear response of a system that is dressed by resonant external fields are developed. The double-sided diagrams used earlier by Prior [IEEE J. Quantum Electron. QE-20, 37 (1984)] for the calculation of nonlinear susceptibilities in the presence of weak fields are generalized. Additional types of double-sided diagrams are given that incorporate explicitly (a) the relaxation-induced population transfers among the dressed levels and (b) the effects of the nonvanishing of the diagonal elements of the dipole operator in the dressed-state basis. The rules for the diagrams are worked out from the analytical structure of the nonlinear response. Applications of the diagrammatic techniques to several intense-field phenomena are discussed.
Abstract The book so far is almost entirely concerned with linear optical processes, where the response of a material to incident light is described by its linear susceptibility and the rates of optical processes are proportional to the incident intensity. However, examples of nonlinear behaviour occur in the saturation of atomic transitions, discussed in §1.7, and in the power-broadened susceptibility, derived in eqn (2.8.6). Atomic saturation is also an important component in the laser theory of§§t.10 and 7.3. The expressions derived in these examples contain the electromagnetic energy density or photon number in denominators whose expansions include terms proportional to all positive powers of the optical energy. The higherorder, or nonlinear, terms are normally much smaller than the linear term for conventional light sources but their influence is often significant in experiments that use laser sources. The present chapter treats the effects of the nonlinear terms in modifying the properties of incident light beams.
We apply a diagrammatic density matrix perturbation technique to analyze third order nonlinear optical processes. The diagrammatic approach is shown to yield a significant simplification in the computation of both steady state and transient response while including the effects of damping and linewidth. The topology of the diagrams is stressed in identifying and interpreting their associated perturbation terms as well as predicting frequency resonance and transient behavior. We demonstrate the practical utility of the technique by presenting a detailed analysis and discussion of the processes of stimulated Raman scattering, hot luminescence, and pressure induced resonances in four wave mixing.