PHYSICAL REVIEW A 85, 053837 (2012) Wave-vector mismatch effects in electromagnetically induced transparency in Y-type systems Azeem B. Mirza and Suneel Singh * School of Physics, University of Hyderabad, Hyderabad 500 046, India (Received 9 March 2012; published 29 May 2012) We study electromagnetically induced transparency (EIT) of a probe field in a four-level Y-type atomic system driven by two strong laser (coupling) fields. Both homogeneously (radiative) and Doppler broadened systems are considered. The effect of residual Doppler broadening on EIT is demonstrated for various wave-vector mismatches occurring when the frequency of coupling fields is equal (k c = k p ), higher (k c >k p ), or lesser (k c <k p ) than that of the probe field frequency. Contrary to usual belief it is found that for the k c >k p wave-vector mismatch case, the probe absorption profile displays very wide and almost complete single or double EIT windows whose width, depth, and location depend upon the wave-vector mismatch, Rabi frequencies, and atom field detuning of the coupling fields. Analytical results are also obtained to explain these interesting features. DOI: 10.1103/PhysRevA.85.053837 PACS number(s): 42.50.Gy, 42.50.Ct, 42.50.Hz I. INTRODUCTION Electromagnetically induced transparency (EIT) [1] is one of the many unusual and interesting phenomena produced by atomic coherence and interference effects that enable propagation (without significant attenuation) of light through an otherwise opaque atomic medium. EIT is of tremendous interest due to possibility of wide applications in optical switching via light velocity control [2], quantum informa- tion [3], and nonlinear optics [4]. Initial works on EIT primarily focused on one-photon transitions between states of opposite parity in simple three- level , V , and cascade (ladder) systems [1–4]. However, currently there is considerable interest in the study of EIT and its effect on nonlinear optical interactions in four-level systems of various configurations. Pertinent to the present work is a four-level Y-type system interacting with two strong laser (coupling) fields and a low-intensity probe field. Agarwal and Harshawardhan [5] predicted theoretically that a Y-type atomic system under one strong-field and two weak-field excitations can give rise to interesting effects such as inhibition and enhancement of the two-photon absorption. This effect was first experimentally observed by Gao et al. in sodium atoms [6] and subsequently electromagnetically induced two-photon transparency was experimentally ob- served in both 85 Rb and 87 Rb atomic vapors [7]. In a similar model, the resonance fluorescence [8] and vacuum-induced interference effects [9] were also investigated theoretically. Experimental observation of competing four-wave mixing (FWM) and six-wave mixing (SWM) processes [10] and two FWM processes [11] due to atomic coherence and interference in the four-level Y-type atomic systems were also reported. These experiments utilized two-photon Doppler-free configurations for propagation of the (pump, coupling, and probe) laser beams in an atomic Rb vapor cell. More recently a theoretical study of dynamical control of soliton formation and propagation [12] in a four-level Y-type homogeneously broadened system interacting with two strong laser fields and a low-intensity probe field was reported. * suneelsp@uohyd.ernet.in In atomic vapors the probe transparency (or absorption) characteristics are governed by the nature of the residual (two- photon) Doppler broadening originating from the thermal mo- tion of the atoms and mismatch k p − k c , of applied probe and coupling field wave vectors k p and k c , respectively [13–15]. It is the usual belief that a Doppler-free medium is essential for observing reduced probe absorption at much lower coupling (or pump) laser power. Hence most experiments tend to utilize a two-photon Doppler-free geometry where the coupling and probe beams are counterpropagating and have similar frequencies so that k p = k c and the residual Doppler width vanishes. Realizing perfect Doppler-free two-photon transition in real atomic systems is not feasible due to a rather large dissimilarity between the wavelengths (or wave vectors) of the (upper) transition driven by coupling field(s) and the lower transition connected by the probe field. Even the often- utilized nearly-Doppler-free two-photon process involving the 5 s 1/2 → 5 p 3/2 → 5 d 5/2 (or 5 d 3/2 ) transition in Rb [10,11] has a nonzero (k p <k c ) residual width of 1.6 MHz [14], which is much larger than the dephasing rate (0.4 MHz) of the two- photon transition and thus can significantly affect the probe transparency. On the other hand it was found that the k p <k c case is actually conducive to the observation of reduced probe absorption [13,15]. However, in the opposite k p >k c (fre- quency upconversion) case, where no transparency can occur for a single coupling field, Silva et al. [16] demonstrated that using a standing-wave configuration of counterpropagating coupling fields, it is possible to induce a large transparency in a three-level cascade system whenever the ratio of probe to drive field frequency is close to half integer values. It therefore would be of interest to assess the influence of the various broadening mechanisms and different regimes of wave-vector mismatches in the inhomogeneous broadening case (particularly when k p <k c ) on probe transparency (absorption) characteristics in a Y-type four-level system. In this work we study the transparency of a weak probe field in a Doppler broadened four-level Y-type system interacting with two strong laser (coupling) fields. To compare and con- trast, an analysis of a homogeneously (radiatively) broadened four-level Y-type system is also presented. The organization of the paper is as follows: Sec. II (which also contains several subsections) deals with density-matrix formulation. We solve for the steady-state density-matrix equations for 053837-1 1050-2947/2012/85(5)/053837(8) ©2012 American Physical Society