Light pulse in Λ -type cold atomic gases Ran Wei, 1 Bo Zhao, 2, 3, ∗ Youjin Deng, 1, † Shuai Chen, 1 Zeng-Bing Chen, 1 and Jian-Wei Pan 1 $^{1}$Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China $^{2}$Institute for Theoretical physics, University of Innsbruck, A-6020 Innsbruck, Austria $^{3}$Institute for Quantum Optics and Quantum Information of the Austrian Academy of Science, A-6020 Innsbruck, Austria We investigate the behavior of the light pulse in Λ-type cold atomic gases with two counter- propagating control lights with equal strength by directly simulating the dynamic equations and exploring the dispersion relation. Our analysis shows that, depending on the length L 0 of the stored wave packet and the decay rate γ of ground-spin coherence, the recreated light can behave differently. For long L 0 and/or large γ , a stationary light pulse is produced, while two propagating light pulses appear for short L 0 and/or small γ . In the γ → 0 limit, the light always splits into two propagating pulses for sufficiently long time. This scenario agrees with a recent experiment [Y.-W. Lin, et al., Phys. Rev. Lett. 102 , 213601(2009)] where two propagating light pulses are generated in laser-cooled cold atomic ensembles. PACS numbers: 32.80.Qk 42.50.Gy Quantum information transfer between light and atomic ensembles has attracted much attention recently. In particular, electromagnetically induced transparency (EIT) [1, 2], a robust technique that renders a reso- nant opaque medium transparent by means of destruc- tive quantum interference, has been exploited to realize the storage and retrieval of light pulses in atomic ensem- bles [3–6]. In the storage process, a weak probe light pulse carrying quantum information and a strong cou- pling light are applied to an optically thick atomic en- semble. The probe light is then gradually converted into a ground-spin coherence as the coupling light is adiabat- ically switched off, and, as a result, the quantum infor- mation is stored in the atomic ensemble. The reading process is almost reverse to the storage: the control light is adiabatically turned on, and accordingly a new light pulse is created and propagates out of the atomic ensem- ble. In this way, the quantum information can be stored and read out without loss in principle. If two counter- propagating control light pulses with equal strength are adiabatically switched on [7], the retrieved light pulse will not propagate out of the atomic ensemble. Instead, it stops in the media and forms a stationary light pulse, as experimentally demonstrated by Bajcsy et al. [8]. On this basis, together with cold-atom techniques, many ap- plications are proposed, which includes simulation of the dynamics of massive Schr¨ odinger particles [9], of Dirac particles [10], and of the strong correlated Bose system confined in a hollow core fiber etc. [11]. However, thermal fluctuations are strongly suppressed in cold atoms, and thus higher-order terms decay much slower and should be considered. The light generation in cold atomic ensembles has been theoretically studied beyond the secular approximation by Hansen et al. [13]. They came to the conclusion that, when the decay rate of the ground-spin/optical coherence is zero, the gener- ated light is a pure stationary light pulse-i.e., a station- ary light without photon loss. Nikoghosyan et al. [14] took into account the relaxation of the upper state and demonstrated that, under the slow-light condition, the generated light is a stationary light pulse but with some photon loss. Nevertheless, a recent experiment demon- strated [15] that, in laser-cooled cold atomic ensembles, the retrieved light pulse is not stationary but splits into two propagating wave packets. The authors gave a sim- ple model which only involves the zeroth-order and first- order coefficients. Since there is no obvious reason why the cutoff should take place at the first-order term, our original minor motivation was to deal with the dynamic equations to a higher order. It turns out that, in the zero-decay limit and under adiabatic approximation, the dynamic equations can be analytically treated to any or- der as one wishes (see Appendix for details). Indeed, to the first order, one obtains two counter-propagating light pulses, consistent with Ref. [15]. However, as higher- order terms gradually come in, the relative group ve- locity of the two light pulses decreases and vanishes as √$_{2}$$_{ℓ}$$_{+ 1}$$_{/ℓ}$$_{, where}$ ℓ is the highest order of term in the cal- culation. It seems to confirm the results in Refs. [13, 14]. Given this discrepancy and the potential important ap- plications of the stationary light pulse in cold atoms, a careful and systematic study seems desirable. The dynamic equations describing the behavior of the recreated light in Λ-type atoms contain infinite-order terms. In conventional theoretical treatment, secular approximation is used and only the zeroth-order coef- ficient of the ground-spin/optical coherence is kept [12]. In hot atomic systems like in Ref. [8], such a treatment is reasonable, since the higher-order terms decay very fast due to the random atomic motions and collisions. In this paper, we directly simulate the dynamic equa- tions (given in Sec. II) to avoid further approximation. For a given set of parameters, a series of simulations is performed with cutoff at different order ℓ and the re- sult are extrapolated to the infinite-order limit ( ℓ → ∞ ). These results are presented in Sec. III. Section IV pro- arXiv:1001.2359v3 [quant-ph] 1 Mar 2010 I. INTRODUCTION