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The Journal of Chemical Physics

Nonperturbative susceptibility of a three-level system interacting with a monochromatic field

Theresa C. Kavanaugh, R. Silbey

Published May 1, 1992• 15 citations
Publisher

Abstract

We derive an exact, intensity-dependent expression for the susceptibility of a three-level system interacting with one monochromatic electric field. This expression, once expanded, is equivalent to the usual perturbative series χ=χ(1)+χ(3)E2+⋅⋅⋅ within the rotating wave approximation. We consider two types of resonant processes: one- and two-photon resonances corresponding to the intensity-dependent susceptibility χ(3)(−ω;ω,−ω,ω), and one-, two-, and three-photon resonances corresponding to χ(3)(−3ω;ω,ω,ω). As an example of current interest, we use a model three-level system that mimics the excited electronic states of typical nonlinear optical polymers and show that near resonance, successive terms in the perturbative series approach the same order of magnitude for experimentally realizable fields.

DOI
10.1063/1.462609

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We derive an exact, intensity-dependent expression for the susceptibility of a three-level system interacting with one monochromatic electric field. This expression, once expanded, is equivalent to the usual perturbative series χ=χ(1)+χ(3)E2+⋅⋅⋅ within the rotating wave approximation. We consider two types of resonant processes: one- and two-photon resonances corresponding to the intensity-dependent susceptibility χ(3)(−ω;ω,−ω,ω), and one-, two-, and three-photon resonances corresponding to χ(3)(−3ω;ω,ω,ω). As an example of current interest, we use a model three-level system that mimics the excited electronic states of typical nonlinear optical polymers and show that near resonance, successive terms in the perturbative series approach the same order of magnitude for experimentally realizable fields.
We investigate the properties of a homogeneously broadened three-level system interacting with two optical fields, Eω1(x,t) and Eω2(x,t), with central frequencies ω1 and ω2 whose amplitudes are assumed to be slowly varying in time compared with dephasing times of the optical transitions. We calculate Raman susceptibilities allowing each transition to be induced by both fields and without employing the rotating wave approximation (RWA); this enables correct calculation of the optical response of the medium when the fields are on-resonance or arbitrarily far off-resonance with the Raman transition. In some frequency regions the RWA causes substantial errors. A distinction is drawn between two different types of RWA: (a) a neglect of off-resonant components of the polarization for direct optical transitions, and (b) neglect of off-resonant components of off-diagonal density matrix elements of order Eω1(x,t)Eω2(x,t)* in the field amplitudes. The first RWA is known to become inaccurate when the pump frequency is significantly off-resonance with the pump transition. We show that the second RWA becomes inaccurate when ‖ω1−ω2‖ is small or comparable to the active Raman mode frequency (even when the field are close to resonance). Calculations based upon our expressions show that when the pump frequency is close to resonance with the optical transition from the ground state, a nonlinear absorption at the Stokes shifted frequency occurs. This absorption can significantly decrease the Raman gain. Also, we find that Rabi splittings in a three-level system can occur even when the pulse durations of the incident fields are short compared with the T1 times of the system. We solve analytically for the optical response of the system to two fields of arbitrary strength and frequency. We analyze this solution and its perturbative expansion in the field amplitudes. We use the Maxwell–Bloch equations developed here to propagate time-dependent pulses.
Perturbative corrections are derived for the rotating-wave approximation (RWA) for the single-photon and multiphoton resonance profiles due to the interaction between a two-level molecule, with nonzero permanent dipoles, and a sinusoidal time-dependent electric field. The derivation is carried out through the use of a Floquet secular equation that introduces the effects of the permanent dipole moments into the problem through the argument, d\ensuremath{\cdot}e^scrE/\ensuremath{\omega}, of Bessel functions contained in molecule-laser coupling parameters; here d=${\ensuremath{\mu}}_{22}$-${\ensuremath{\mu}}_{11}$, where ${\ensuremath{\mu}}_{\mathrm{ii}}$ is the permanent dipole of molecular state i, and scrE, e^, and \ensuremath{\omega} are the field strength, direction of polarization unit vector, and the circular frequency of the electromagnetic field (EMF). Expansions for the N-photon resonance profiles and the associated resonance frequencies and full widths at half maximum, in powers of the couplings between the transition and permanent dipole moments and the applied EMF, are obtained by expanding the Bessel functions in the perturbative results and are compared with those in the literature (available mostly for d=0 only). The perturbative results, and a series of exactly calculated two-level model absorption spectra, are used to discuss (1) the ability of the explicit perturbative corrections to the RWA to explain the differences between exact and RWA resonance profiles (for example, the positive or negative Bloch-Siegert shifts and dynamic backgrounds absent in the RWA), (2) the usefulness of the perturbative corrections in general, and (3) the effects of permanent dipoles on single-photon and multiphoton absorption spectra.
We have considered the excitation spectra of three-photon processes arising from the interaction of a three-level atom, in the cascade configuration, with two strong electromagnetic fields whose initially populated modes ω a and ω b are near resonance with the atomic transition frequencies ω21 and ω32, respectively. In the limit of high photon densities, expressions for the Green's functions are derived in a closed form describing third order nonlinear processes in the region of frequencies ω = ω a + 2ω b and ω = ω b + 2ω a , respectively. Numerical calculations are then performed and the computed spectra are presented graphically for different values of the Rabi frequencies Ω a and Ω b , and detunings ζ a and ζ b , respectively. The excitation spectrum near ω = ω a + 2ω b is calculated as a function of the parameters η a = Ω a /γ0, η b = Ω b /γ0, ζ a = (ω21 - ω a )/γ0 and ζ b = (ω32 - ω b )/γ0 where γ0 is the spontaneous transition probability. With no detuning the spectral intensities vary slowly with respect to variations of the Rabi frequencies and are shifted by amounts governed by the field strengths η a and η b . In the presence of detunings, asymmetries, characteristics of such off-resonance situations, arise enhancing certain spectral peaks while diminishing the intensity of others.
A theoretical analysis is made of the nonlinear (in respect of the radiation intensity) rotation of the plane of polarization occuring in an atomic medium subjected to a static magnetic field near one-photon (two-level atoms) and two-photon (three-level atoms) resonances. A study is also made of the influence of a strong monochromatic pump wave on the propagation of a weak nonmonochromatic wave. The wave functions and quasienergies are calculated for a three-level system in static magnetic and monochromatic electromagnetic fields near a two-photon resonance. It is shown that the Verdet constant and the refractive index of such a system depend on the magnitude and relative sign of the frequency detuning from resonance. Proximity to a two-photon resonance destroys the polarizability of a medium for either sign of the detuning.
The rate of absorption and emission of radiation is calculated for two laser fields interacting with a homogeneously broadened, three-level molecular system. The time-dependent Schrödinger equation is used in the density matrix formalism and the laser fields are treated classically. The laser fields, at frequencies ωpand ωs, are off resonance by δpand δs, respectively, such that|\delta_{p}|, |\delta_{s}| \ll \omega_{p}, \omega_{s}. Analytic expressions are derived for the imaginary part of the electric susceptibility,\chi"(\omega_{p})and\chi"(\omega_{s}). The present work differs from previous calculations in that the laser fields may have arbitrary values of field strength and each field may be on or off resonance. A discussion is presented of the properties of the solutions for\chi"(\omega_{p})and\chi"(\omega_{s}), for both strong and weak laser fields, based on computer evaluation of the general analytic expressions. It is shown that the solutions may be divided into single-photon and two-photon, or Raman, contributions. Conditions for optimizingdI(\omega_{s})/dzand the photon conversion efficiency are derived. These results may be applied to analyzing the pulsed, optically pumped submillimeter laser and to studying two-photon absorption in a gas.
A description of the second–order nonlinear optical susceptibility of semiconductor materials is considered. The Lie algebra su(3) arises from a 3–level quantum model for each state in the Brillouin zone, which is obtained by truncation of the true infinite–dimensional basis. The accuracy of several approximations inherent in this approach is analysed, in particular the electric dipole and minimal replacement Hamiltonians which specify the electrodynainical interaction in a finite–dimensional quantum system. The three–level model is extended by adding relaxation processes and utilising the rotating wave approximation to account for resonant nonlinear effects. Analytical forms for the transition rates associated with two–photon absorption and electromagnetically induced transparency are derived.