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Physical Review A

Four-wave mixing in three-level systems: Interference and entanglement

Shengwang Du, Eun Oh, Jianming Wen, Morton H. Rubin

Published Jul 3, 2007• 53 citations
Publisher

Abstract

The interference of degenerate four-wave mixing (FWM) in three-level systems is examined in both the classical and quantum regimes with a backward configuration. In classical FWM, we find that the phase difference between two indistinguishable FWM transition paths can be varied by different driving laser parameters, and leads to interference in the amplitude and polarization of the generated conjugate field. In the paired-photon generation case, the interference in the nonlinearity disappears because of the time ordering in biphoton generation. However, because of the slow group velocity at the degenerate frequency and polarization, the biphoton-amplitude interference between two Feynman paths can erase the time-ordering information at the detectors. For small group delay, the biphoton correlation, determined by the third-order nonlinearity, shows antibunching and damped Rabi oscillations. For large group delay, where the biphoton bandwidth is determined by phase matching, we show that the biphoton interference leads to a bunching effect. The feasibility of generating polarization entanglement is also discussed.

DOI
10.1103/physreva.76.013803

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The interference of degenerate four-wave mixing (FWM) in three-level systems is examined in both the classical and quantum regimes with a backward configuration. In classical FWM, we find that the phase difference between two indistinguishable FWM transition paths can be varied by different driving laser parameters, and leads to interference in the amplitude and polarization of the generated conjugate field. In the paired-photon generation case, the interference in the nonlinearity disappears because of the time ordering in biphoton generation. However, because of the slow group velocity at the degenerate frequency and polarization, the biphoton-amplitude interference between two Feynman paths can erase the time-ordering information at the detectors. For small group delay, the biphoton correlation, determined by the third-order nonlinearity, shows antibunching and damped Rabi oscillations. For large group delay, where the biphoton bandwidth is determined by phase matching, we show that the biphoton interference leads to a bunching effect. The feasibility of generating polarization entanglement is also discussed.
We find that in a two-level system there are two kinds of four-wave mixing processes which destructively contribute to the third-order nonlinear susceptibility. In a paired-photon generation scheme by using a single retroreflected pump beam, these two processes occur with definite time orders. The biphoton temporal correlation, which is measured with a back-to-back geometry in a laser-cooled two-level atomic ensemble, shows a damped Rabi oscillation and photon antibunching. Without suppressing the background of Rayleigh scattering, the upper limit of Cauchy criteria is estimated.
We have theoretically studied the space-time entangled biphoton state generated from a two-level atomic system. In the photon counting measurement, the two-photon coincidence counting rate is a damped oscillation. The oscillation period is determined by the effective Rabi frequency and the damping rate is determined by the linewidth of the inhomogeneous-broadened ground state and the dipole dephasing rate. In an optical-pathway-balanced configuration, the two-photon temporal correlation shows an antibunching effect which corresponds to the interference between two types of nonlinear four-wave mixing processes occurring in such a two-level system. The visibility of the normalized second-order quantum coherence function ${g}^{(2)}(\ensuremath{\tau})$ increases along with the increase of the effective Rabi frequency, but has an upper limit at 45%. We find agreement between the theory and the experiments [P. Kolchin et al., Phys. Rev. Lett. 97, 113602 (2006); S. Du, J.-M. Wen, M. H. Rubin, and G. Y. Yin, ibid 98, 053601 (2007)].
We predict the possibility of the interference of narrow-band biphotons generated by spontaneous four-wave mixing with double electromagnetically induced transparency configuration in cold atoms. In an N-type four-level system, an auxiliary optical field Ω m can create double transparency windows for anti-Stokes photons. When the slow light effects in the double transparency windows are very strong, two four-wave mixing channels could exist due to the splitting of the phase matching condition. The biphoton generated from the two four-wave mixing channels can cause interference and shows Rabi oscillations in two-photon correlation. This interference mechanism will complement the understanding of interference at the two-photon level.
Polarization properties of pure four-wave mixing (FWM) and dressed-FWM processes in a two-level system and a cascade three-level atomic system are theoretically and experimentally investigated. The relative intensities and polarization characteristics of the FWM signals in different laser polarization configurations and different level systems are experimentally investigated and compared. Also, the results are theoretically explained by different transition paths combinations. In the dressed-FWM processes, we study the dependence of dressing effect on the incident field's polarization. The FWM signal generated by a linearly polarized pumping field is suppressed more by the dressing field than the one generated by a circularly polarized pumping field. However, an opposite effect was observed when the probe field's polarization is changed. The multidressing mechanisms are used to explain these effects. In addition, the interference and polarization dependence of the coexisting FWM signals in the same atomic system are discussed.
We analyze theoretically the quantum beat and the polarization interference phenomena observed as oscillations in pulsed four-wave mixing. We show that resolution of the signal with respect to the delay between the incoming pulses as well as the detection frequency provides valuable information on the nature of the oscillations. Most importantly, it is very simple to distinguish between polarization interference from two independent optical transitions and quantum beats from a three-level system. In case of polarization interference, the phase of the oscillations shifts \ensuremath{\pi} when the detected frequency passes any of the resonances. In case of quantum beat, the phase has little dependence on the detected frequency near the resonance.