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Optics Letters

Generation of squeezing in two-photon three-level media

Sunghyuck An, Murray Sargent

Published Jun 1, 1988• 10 citations
Publisher

Abstract

We present a fully quantum-mechanical theory of nondegenerate four-wave mixing processes in three-level cascades with a two-photon pump. The results are applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong pump intensities and tunings within the Rabi sidebands. We find good broadband squeezing for low pump intensities and tunings outside a small region around central tuning. Both cases avoid regions of significant spontaneous emission.

DOI
10.1364/ol.13.000473

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Journal of the Optical Society of America B•Jan 1, 1991•10 citations
We present a fully quantum-mechanical theory of nondegenerate four-wave mixing processes in three-level cascades with a two-photon pump. The results are applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong pump intensities and tunings within the Rabi sidebands. We find good broadband squeezing for low pump intensities and tunings outside a small region around central tuning. Both cases avoid regions of significant spontaneous emission.
We present a fully quantum-mechanical theory of nondegenerate multiwave mixing processes in three-level cascades with a two-photon pump. The explicit formulas for the resonance fluorescence spectrum and the quantum combination-tone source term are derived. The theory is applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong-pump intensities and tunings within the Rabi sidebands. We find good broadband squeezing for low-pump intensities and tunings outside a small region around central tuning. Both cases avoid regions of significant spontaneous emission.
In this summary we extend the quantum theory of multiwave mixing by Holm and Sargent1,2 to a three-level cascade model with a two-photos pump and apply the results to the generation of squeezed states. The experimental observation of the suppresion of amplified spontaneous emission by the four-wave mixing process in this model has been reported by Malcuit, Gauthier, and Boyd.3 This experiment has been interpreted semiclassically by the same authors4 and quantum mechanically by Agarwal.5 Agarwal also has shown the generation of squeezed states in this model, but simplified his treatment by ignoring the spontaneous emission to and from the intermediate level. In contrast our treatment allows for arbitrary detunings of a classical pump and quantized sidemode fields, and includes a complete treatment of spontaneous emission.
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.
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.
This dissertation presents theories of multiwave mixing in two- and three-level media. The first part of the dissertation treats the semiclassical theories in two-level media. Chapter 2 gives the simple semiclassical theory of four-wave mixing when the two pump frequences differ by more than the reciprocal of the population-difference lifetime. This difference washes out the pump spatial holes as well as one of the two reflection gratings. We compare the results to the degenerate treatment of Abrams and Lind and find significant differences in the reflection coefficient spectra. Chapter 3 presents the semiclassical theory of multiwave in a squeezed vacuum characterized by unequal in-phase and in-quadrature dipole decay times. For a highly squeezed vacuum, we find sharp resonances in both probe absorption and reflection coefficients, which provide sensitive ways to measure the amount of squeezing in the vacuum. The second part of the dissertation treats the quantum theories in two- and three-level media. Chapter 4 develops the fourth-order quantum theory of multiwave mixing to describe the effects of sidemode saturation in two-level media. We derive explicit formulas for the fourth-order quantum coefficients and show that the fourth-order quantum theory reproduces the third-order semiclassical coefficient obtained by truncating a continued fraction. We apply the results to cavity problems and find significant differences in the sideband spectra given by the second- and fourth-order treatments, particularly as the sidemode approaches the laser threshold. The final chapter presents a quantum theory of multiwave mixing in three-level cascades with a two-photon pump. The explicit formulas for the resonance fluorescence spectrum and the quantum combination-tone source term are derived. The theory is applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong pump intensities and good broad-band squeezing for low pump intensities.
We apply our quantum theory of nondegenerate multiwave mixing [ Phys. Rev. A37, 2017 ( 1988)] to squeezed-state generation experiments with two-level atoms. Our main interest is to predict the amount of squeezing achievable with a Doppler-broadened two-level medium. We are particularly interested in the single-beam configuration, in which all four interacting beams are spatially degenerate. We analytically solve the coupled-mode quantum Langevin equations for nondegenerate four-wave mixing. The solutions are used to compute the amount of squeezing. In the computation the effects of pump–probe phase mismatch, collisions, Doppler broadening, and Gaussian-intensity variation are comprehensively taken into account for the first time to our knowledge. Simple rules of thumb as to where one can see squeezing in both degenerate- and nondegenerate-frequency cases are derived by examining the limit of a short medium. We then present the case of an infinitely long medium, in which maximum squeezing is achieved when there is no pump–probe phase mismatch. With the inclusion of pump–probe phase mismatch, however, the maximum amount of squeezing is obtained with a finite-length medium instead. This prompts us to investigate in detail the finite-length medium case. Our results show that the effects of Doppler broadening and Gaussian-intensity variation can be largely circumvented by detuning the pump frequency more than three Doppler half-widths from resonance and that good broadband squeezing can be achieved even with a Doppler-broadened medium that has a moderate amount of collision broadening. Under these circumstances it is found that the effect of pump self-focusing or defocusing will be the major factor that limits the amount of achievable squeezing. In particular, the spatially varying nonlinear refractive indices seen by the pump and the probe modes are quite different, which causes the former to become spatially mismatched with the latter in the region in which strong squeezing is otherwise expected.