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Optics Letters

Effects of dynamic Stark shifts on two-photon side-mode instabilities

Shlomo Ovadia, Sami T. Hendow, Murray Sargent

Published Oct 1, 1985• 13 citations
Publisher

Abstract

We have extended our two-photon two-level instability model to include the effects of level Stark shifts and find that the shifts tend to expand the regions of side-mode gain that allow single-wavelength and multiwavelength instabilities to occur. In addition, we find that the instabilities are strongly affected by conjugate scattering off the pump-induced two-photon coherence.

DOI
10.1364/ol.10.000505

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We have extended our two-photon two-level instability model to include the effects of level Stark shifts and find that the shifts tend to expand the regions of side-mode gain that allow single-wavelength and multiwavelength instabilities to occur. In addition, we find that the instabilities are strongly affected by conjugate scattering off the pump-induced two-photon coherence.
We recently1 calculated the amount of squeezing predicted by our quantum theory of multiwave mixing for a one-photon two-level model. Here we use the corresponding theory for the two-photon two-level model,2 which allows for nonsaturating nondegenerate pump and probe waves and includes dynamic Stark shifts. We simplify the more complicated general model using a fourthorder perturbation limit and find very simple expressions that show even for very weak pumps with no Stark shifts or atomic detuning significant squeezing can be achieved. Much of this squeezing is due to scattering of the conjugate field off the pump-induced two-photon coherence, which does not arise in the one-photon case. In general, Stark shifts induce large regions of significant gain, resulting in instabilities. We also show results for high intensity pumps that require larger atomic detunings to obtain a large amount of squeezing. The low intensities required in the two-photon case should make squeezing easier to observe than in the one-photon case.
In a recent paper,1 we considered the stability of single-mode operation in two-photon lasers and two-photon optical bistability.2 The models assumed that two photons of equal or nearly equal frequencies span the atomic two-photon transition, with no resonant intermediate levels. The paper neglects possible Stark shifts of the two levels due to virtual transitions to nonresonant levels and back. For one-photon processes, these virtual transitions can be neglected because they have one nonresonant denominator. They cannot in general be neglected for the two-photon case, however, since all transition probabilities have at least one nonresonant denominator. In this paper we give formulas and illustrations for the two-photon two-level absorption and coupling coefficients including Stark shifts. For vanishing shifts, the coefficients have symmetric spectra that look very similar to their one-photon counterparts, but for nonzero shifts, they have asymmetric spectra with exaggerated resonances. These resonances appear to increase the likelihood of instabilities for certain combinations of pump detuning and Stark shifts.
We apply our quantum theory of multiwave mixing for the two-photon two-level model to the generation of squeezed states of light. Because of the two-photon transition, a larger number of interaction potentials are present compared to one-photon models, leading to effects absent in the one-photon case. The two most significant of these are the appearance of dynamic Stark shifts and conjugate scattering off the pump induced two-photon coherence. We show that this can lead to almost perfect squeezing, even for low pump intensities and low detunings from the two-photon resonance.
Multiphoton transitions are of great interest in laser spectroscopy because they permit the study of new physical phenomena and because of their potential applications. Malcuit et al.1 studied two-photon transitions in atomic sodium and showed that for appropriate phase-matching four-wave mixing can overcome the usual spontaneous decay processes. We have recently developed a quantum theory2 to treat such multiwave mixing processes in two-photon media. Because of the greater complexity of two-photon transitions, many effects arise that are absent in the one-photon case, some of which have important consequences in the generation of squeezed states. For the two-photon two-level model, the field modes have frequencies approximately one-half the frequency difference between the levels. We find that our coefficients differ from the corresponding one-photon theory in that dynamic Stark shifts have a major effect and that the pump-induced coherence between the two levels has a new term in the conjugate coupling terms, which leads to improved squeezing.
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
We extend the semiclassical two-photon two-level model to include upper-level ionization and upper-level decay via an intermediate level. We examine the effects of these relaxation mechanisms on multiwave interactions, finding the generalized one- and two-wave two-photon absorption coefficients and the three-wave coupling coefficient. These mechanisms cause asymmetries to appear in the absorption spectra in addition to those caused by Stark shifts. The Stark-shift term becomes more complicated and fails to vanish even when it does in the simpler model. For recombination rates and ionization decay rates comparable to the coherence decay rate, coherent dips fill in.