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Physical Review A

Efficient parametric amplification in double-Λ systems without maximal two-photon coherence

Hagay Shpaisman, A. D. Wilson‐Gordon, Harry Friedmann

Published Dec 1, 2004• 23 citations
Publisher

Abstract

We study four-wave mixing in a double-$\ensuremath{\Lambda}$ system, for cw laser beams whose transverse intensity profiles (TIP's) are initially Gaussian, by solving the Maxwell-Bloch equations numerically. For systems where coherent population trapping (CPT) is initially absent, we show that efficient frequency conversion, without focusing or ring formation, can occur even at distances which are much shorter than those required to establish CPT. We also show, for certain configurations, that blue-detuned beams become focused on propagation so that very high frequency conversion, even exceeding 100% at the center of the profile, can occur. This focusing is, however, accompanied by ring formation. We show that focusing, without ring formation, can occur for identical blue-detuned beams when the initial relative phase is $\ensuremath{\pi}$ so that CPT cannot be established on propagation. The behavior of the TIP's of the beams on propagation is explained by considering the effective linear and third-order contributions to the off-diagonal density-matrix elements.

DOI
10.1103/physreva.70.063814

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We study four-wave mixing in a double-\ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ system, for cw laser beams whose transverse intensity profiles (TIP's) are initially Gaussian, by solving the Maxwell-Bloch equations numerically. For systems where coherent population trapping (CPT) is initially absent, we show that efficient frequency conversion, without focusing or ring formation, can occur even at distances which are much shorter than those required to establish CPT. We also show, for certain configurations, that blue-detuned beams become focused on propagation so that very high frequency conversion, even exceeding 100% at the center of the profile, can occur. This focusing is, however, accompanied by ring formation. We show that focusing, without ring formation, can occur for identical blue-detuned beams when the initial relative phase is \ensuremathπ\ensuremath{\pi}\ensuremathπ so that CPT cannot be established on propagation. The behavior of the TIP's of the beams on propagation is explained by considering the effective linear and third-order contributions to the off-diagonal density-matrix elements.
Near the threshold for electromagnetically induced transparency (EIT) or coherent population trapping (CPT), two-photon-resonance-enhanced self-focusing of a $\ensuremath{\Lambda}$ system can be exploited to induce spatial confinement in a second, diffracting $\ensuremath{\Lambda}$ system. The diffracting $\ensuremath{\Lambda}$ system is characterized by parameters below the EIT or CPT threshold, and the two $\ensuremath{\Lambda}$ systems must be coupled to form a closed-loop double-$\ensuremath{\Lambda}$ system. The waveguiding effect is shown to be strongly phase dependent, indicating that it derives from the phase-dependent effective third-order susceptibility rather than the phase-independent effective first-order susceptibility, as is the case in previously studied systems. We also show that when the second $\ensuremath{\Lambda}$ system initially involves only a single laser beam, the loop is completed by the efficient generation of radiation at the four-wave-mixing frequency, within a propagation distance much shorter than the diffraction length. Both the applied and generated fields exhibit electromagnetically induced waveguiding.
We investigate a four-state system interacting with long and short laser pulses in a weak probe beam approximation. We show that when all lasers are tuned to the exact unperturbed resonances, part of the four-wave mixing (FWM) field is strongly absorbed. The part that is not absorbed has the exact intensity required to destructively interfere with the excitation pathway involved in producing the FWM state. We show that with this three-photon destructive interference, the conversion efficiency can still be as high as 25%. Contrary to common belief, our calculation shows that this process, where an ideal one-photon electromagnetically induced transparency is established, is not most suitable for high-efficiency conversion. With appropriate phase matching and propagation distance, and when the three-photon destructive interference does not occur, we show that the photon flux conversion efficiency is independent of probe intensity and can be close to 100%. In addition, we show clearly that the conversion efficiency is not determined by the maximum atomic coherence between two lower excited states, as commonly believed. It is the combination of phase matching and constructive interference involving the two terms arising in producing the mixing wave that is the key element for the optimized FWM generation. Indeed, in this scheme no appreciable excited state is produced, so that the atomic coherence between states |0〉 and |2〉 is always very small.
We show that nonadiabatic, resonant amplitude- and phase-modulated pulses can be frequency converted with greater efficiency than adiabatic resonant pulses in a coherently prepared $\ensuremath{\Lambda}$ system. Indeed, conversion efficiencies close to unity, similar to those achieved using highly detuned pulses, can been obtained by using highly nonadiabatic resonant pulses. Moreover, by solving the Maxwell-Bloch equations using Fourier transforms, we derive analytical expressions for the probe and the generated four-wave mixing (FWM) pulses as a function of time and propagation distance. From these expressions, which are valid for either adiabatic or nonadiabatic pulses, we derive the result that starting from a nonadiabatic probe pulse, an asymptotically matched probe-FWM pulse pair with the same shape as the initial probe pulse is obtained. In addition, we show that, starting with a nonadiabatic matched pulse pair or a pair of matched pulse trains, we obtain propagation of these pulses without either deformation or losses.
The atomic coherence effect can be used in enhancing the efficiencies of nonlinear optical processes. Particularly, nondegenerate four-wave mixing (NDFWM) in multilevel systems has recently gained attention in connection with the advantage of using coherent population trapping (CPT) or electromagnetically induced transparency (EIT) to enhance the efficiency of generated signals. Generally speaking, linear absorption competes with the nonlinear signal generation at the exact one-photon resonance condition. However, the linear absorption can, in principle, be reduced or even eliminated by preparing the system in the CPT or EIT state. We report experimental demonstrations of NDFWM in several multi-level configurations in rubidium atomic vapor. In one of the double-/spl Lambda/ systems an unusual behavior of signal output power as a function of pump power, i.e. a maximum in the generated signal power is reached as the pump power increases, is experimentally observed. By comparing with several related /spl Lambda/ and double-/spl Lambda/ systems, we have concluded that CPT is the mechanism for this interesting effect.
We study the field shift of coherent population trapping (CPT) resonance excited by a bichromatic field in an open Λ system with account for the Gaussian profile of the laser radiation intensity. Two methods for error signal formation are considered: the harmonic frequency modulation and the step phase modulation (phase jumps). It is shown that the spatial inhomogeneity of the light beam leads to an essentially nonlinear dependence of the error signal shift on the laser radiation intensity. We propose an approach for the linearization of this dependence, which is important for the development of methods for suppressing the field shift in atomic clocks based on CPT resonances.