Spatial Raman solitons in far-off resonant atomic systems

Since the original suggestions of self-trapping for laser beams 1–3 , there has been a growing interest in generation of spatial optical solitons due to their rich nonlinear dynamics, and their potential technological applications 4 . Over the last few decades, spatial optical solitons have been predicted and demonstrated in a variety of physical systems utilizing different nonlinear optical processes 5–9 . The formation of spatial optical solitons require a balance between the lensing effect of the medium and diffraction. In three spatial dimensions, it is well known that such a balance may be unstable, leading to collapse and filamentation of an optical wave 3 . An exciting practical application of spatial optical solitons is optical information processing. Collision processes between spatial solitons can be used to efficiently perform alloptical gates between laser beams. For such practical applications, there are several key challenges that have to be met: 1 The optical solitons for the studied physical system must be sufficiently stable to allow for well-defined collision properties, 2 the required optical power for each soliton must be low enough to keep the power requirement for a possible logic device within reasonable limits. Recently, spatial Raman solitons formed in molecular systems were predicted 10,11 . In this Rapid Communication, we extend this suggestion and analyze the formation, propagation, and collision properties of spatial Raman solitons formed in far-off resonant atomic systems. Noting Fig. 1, the key idea is to drive a Raman transition with two laser beams whose frequency difference is slightly detuned from the frequency of the Raman resonance 12–15 . For sufficiently intense laser beams, almost half of the atomic population can be adiabatically transferred from ground Raman state a to excited Raman state b . For this case, the magnitude of the coherence of the Raman transition off-diagonal density matrix element approaches its maximum value, ab 1/2. The adiabatically prepared atomic coherence significantly modifies the refractive indices of the driving laser beams. Depending on the sign of the Raman detuning, , the refractive indices of the driving laser beams are either enhanced 0 or reduced 0 . This modification of the refractive index can cause self-focusing or selfdefocusing of the driving lasers. As a result, under appropriate conditions, bright 0 or dark 0 twofrequency solitons are formed. Operating near maximum coherence, ab 1/2, assures the stability of these solitons against perturbations and allows well-defined collision properties. As we will show below, spatial Raman solitons in atomic systems decisively meet the required criteria discussed in the previous paragraph. Before proceeding we note that there is extensive literature on self-trapping and pattern formation of laser beams in atomic vapor cells utilizing one-photon resonances 16 . There has also been substantial work on electromagnetically induced focusing in a variety of near-resonance, twofrequency systems, including three level ladder, , and V systems 17–22 . We begin by developing the formalism for a model atomic system, with two Raman states a and b and an arbitrary number of excited states i , interacting with two driving lasers termed the pump and the Stokes . We follow closely the formalism of Harris and colleagues 12–14 . Noting Fig. 1, we consider threedimensional propagation of the driving lasers with electric field envelopes Ep x ,y ,z , t and Es x ,y ,z , t such that the total field is Ê x ,y ,z , t =Re Ep x ,y ,z , t exp j pt−kpz +Es x ,y ,z , t exp j st−ksz where kp= p /c and ks= s /c. The two photon detuning from the Raman resonance is defined as = b− a − p− s . When the two laser beams have the same polarization, new frequencies separated by the Raman transition higher-order Stokes and anti-Stokes sidebands will be generated as described in Refs. 13,14 . Throughout this paper, for simplicity, we will ignore these