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Physical Review A

Spontaneous parametric down-conversion in a three-level system

Jianming Wen, Shengwang Du, Morton H. Rubin

Published Jul 20, 2007• 33 citations
Publisher

Abstract

We have studied the space-time entangled Stokes-anti-Stokes photons generated from a three-level system. In the presence of counterpropagating pump and control lasers, paired Stokes and anti-Stokes fields are spontaneously emitted into opposite directions. The two-photon wave packet is generally a convolution of the phase matching and the third-order nonlinear susceptibility and the feature of the joint temporal correlation is determined by either of them. When the phase matching plays a major role in determining the spectral width of biphotons, the two-photon interference exhibits as conventional type-II spontaneous parametric down-conversion (SPDC). However, if the spectral width of paired photons is determined by the third-order nonlinear susceptibility, two types of four-wave mixings (FWMs) destructively interfere and contribute to the two-photon coincidences. In such a case, the coincidence counting rate appears as a damped Rabi oscillation and exhibits the photon antibunching. The oscillation frequency is equal to the effective Rabi frequency. We have also discussed polarization-entanglement generation in a three-state case. It is found that if the phase matching is crucial in determining the properties of paired Stokes and anti-Stokes photons, the generated state is the same as the degenerate type-II SPDC. However, if the spectral width is mainly controlled by the resonant linewidths in the third-order nonlinearity, one can select one FWM to produce polarization-entangled biphotons, but paired photons from the other FWM process is nonpolarization entangled.

DOI
10.1103/physreva.76.013825

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We have studied the space-time entangled Stokes-anti-Stokes photons generated from a three-level system. In the presence of counterpropagating pump and control lasers, paired Stokes and anti-Stokes fields are spontaneously emitted into opposite directions. The two-photon wave packet is generally a convolution of the phase matching and the third-order nonlinear susceptibility and the feature of the joint temporal correlation is determined by either of them. When the phase matching plays a major role in determining the spectral width of biphotons, the two-photon interference exhibits as conventional type-II spontaneous parametric down-conversion (SPDC). However, if the spectral width of paired photons is determined by the third-order nonlinear susceptibility, two types of four-wave mixings (FWMs) destructively interfere and contribute to the two-photon coincidences. In such a case, the coincidence counting rate appears as a damped Rabi oscillation and exhibits the photon antibunching. The oscillation frequency is equal to the effective Rabi frequency. We have also discussed polarization-entanglement generation in a three-state case. It is found that if the phase matching is crucial in determining the properties of paired Stokes and anti-Stokes photons, the generated state is the same as the degenerate type-II SPDC. However, if the spectral width is mainly controlled by the resonant linewidths in the third-order nonlinearity, one can select one FWM to produce polarization-entangled biphotons, but paired photons from the other FWM process is nonpolarization entangled.
We have theoretically analyzed nonclassical paired photons generated spontaneously from a four-level inverted-Y atomic system. We discuss the feasibility of biphoton generation due to different pumping arrangements. Two types of correlated photon pair emissions have been carefully examined: dressed cascade and dressed double-$\ensuremath{\Lambda}$ emissions. In the dressed cascade two-photon emission, the coincidence counting rate may exhibit a damped Rabi oscillation, in contrast with a simple exponential decay feature in the literatures. In the dressed double-$\ensuremath{\Lambda}$ configuration, we show that not only is the temporal correlation of entangled Stokes--anti-Stokes photons in agreement with discussions presented by Wen et al. [Phys. Rev. A 76, 013825 (2007)], but also the oscillation period may be manipulated by altering either the control Rabi frequency or the dressing Rabi frequency or both. All the observed damped Rabi oscillations in the two-photon coincidences result from the destructive interference among the possible four-wave mixing processes occurring in the atom-field interaction system. This feature of the two-photon amplitude is governed by the convolution between the phase matching and third-order nonlinear susceptibility. The methodology adopted here can be applied to other atomic-level configurations and provides a useful tool to study the spectroscopy of the system. The generated narrow-band photon pairs may have potential applications in long-distance quantum communication and quantum information processing.
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)].
Spontaneous parametric down-conversion (SPDC) is the most often used source of entangled photon states in quantum optics. Recently, it has been suggested in theory that the reduction of polarization entanglement between the photon pair generated via the type-II SPDC process pumped by a ultrafast laser is due to the asymmetry in the joint spectrum function, which is the indicative of the frequency entanglement between the two photons. In this paper, we report the measurement of the joint spectrum function (i.e., direct observation of the degree of frequency entanglement) of the two-photon state of type-II SPDC in both cw- and ultrafast-laser pumping conditions. We observed that, in agreement with the theory, the degree of frequency entanglement (the shape of the joint spectrum function) is indeed different for cw- and ultrafast-pumped type-II SPDC. Our results are of importance in "engineering" the two-photon state for specific quantum applications, such as quantum cryptography.
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.
Summary form only given. Pairs of entangled photons display a complex quantum state that cannot be represented through a product of single-particle states. The spontaneous parametric down-conversion (SPDC) process, that takes place when a laser pulse pumps a second-order nonlinear crystal has become recently a convenient source of such nonclassical states of light in quantum optics experiments. Nonstationary effects are mainly observed when the pump pulse temporal duration is significantly shorter than the characteristic interaction time in the nonlinear crystal. We introduce a nonstationary theory of SPDC to describe the results of fourth-order quantum-interference experiments when no spectral filtering is applied to the down-converted fields. Experiments involving nonlinear SPDC crystals with three different lengths were performed. The ultraviolet laser pulses were obtained from a Ti:sapphire femtosecond laser with a second-harmonic generator. These pulses were used to pump nonlinear crystals to initiate the process of nonstationary spdc and generate orthogonally-polarized photon pairs. These pairs then went into a polarization interferometer, and the coincidence counts of the detectors at the two output ports of the interferometer were recorded.
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.