Storage and retrieval of (3 + 1)-dimensional weak-light bullets and vortices in a coherent atomic gas

OPEN SUBJECT AREAS: NONLINEAR OPTICS SOLITONS Storage and retrieval of (3 1 1) -dimensional weak-light bullets and vortices in a coherent atomic gas Zhiming Chen1, Zhengyang Bai1, Hui-jun Li2,1, Chao Hang1 & Guoxiang Huang1 Received 19 September 2014 Accepted 8 December 2014 Published 3 February 2015 Correspondence and requests for materials should be addressed to G.H. (gxhuang@phy. ecnu.edu.cn) 1 State Key Laboratory of Precision Spectroscopy and Department of Physics, East China Normal University, Shanghai 200062, China, 2Institute of Nonlinear Physics and Department of Physics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China. A robust light storage and retrieval (LSR) in high dimensions is highly desirable for light and quantum information processing. However, most schemes on LSR realized up to now encounter problems due to not only dissipation, but also dispersion and diffraction, which make LSR with a very low fidelity. Here we propose a scheme to achieve a robust storage and retrieval of weak nonlinear high-dimensional light pulses in a coherent atomic gas via electromagnetically induced transparency. We show that it is available to produce stable (3 1 1)-dimensional light bullets and vortices, which have very attractive physical property and are suitable to obtain a robust LSR in high dimensions. T he investigation of light storage and retrieval (LSR), a key technique for realizing optical quantum memory, has received much attention in recent years1–3. One of important techniques for LSR is electromagnetically induced transparency (EIT)4, a quantum interference effect typical occurring in a three-level atomic system interacting with a probe and a control laser fields. The origination of EIT is the existence of dark state, which makes not only the absorption (dissipation) of the probe field largely suppressed but also the LSR possible through an adiabatical manipulation of the control field. Up to now, nearly all studies on LSR have been carried out in various schemes working in linear regime5,6. Such schemes are simple but encounter the inevitable problem of pulse spreading due to the existence of dispersion, which may result in a serious distortion for retrieved pulse. Recently, the EIT-based LSR has been generalized to weak nonlinear regime, where the storage and retrieval of a (1 1 1)-dimensional [(1 1 1)D] (i.e., the first ‘1’ refers to one spatial dimension, and the second ‘1’ refers to time) soliton pulse is suggested7,8. However, because the (1 1 1)D soliton pulse is unstable in high dimensions due to the existence of diffraction, such scheme is still not realistic or quite limited. For practical applications of optical quantum memory, a challenged problem is to obtain a light pulse that is robust (i.e., with a high fidelity) during storage and retrieval in (3 1 1)D. Before proceeding, we note that in recent years there is much effort focused on high-dimensional optical solitons due to their rich nonlinear physics and important applications9,10. Although in recent works11–13 (3 1 1)D light bullets and vortices in coherent atomic systems have been studied, the possibility of their storage and retrieval is not explored yet to the best of our knowledge. Here we propose an EIT-based new scheme to realize a robust LSR for (3 1 1)D light pulses in a coherent atomic ensemble working in a free space. Based on Maxwell-Bloch equations governing the evolution of atoms and light field we derive a nonlinear equation controlling the motion of the envelope of a probe field. We show the possibility for obtaining (3 1 1)D light bullets (or called (3 1 1)D spatiotemporal optical solitons9,10) and vortices, which have ultraslow propagating velocity and extremely low generation power. We further show that these highdimensional light pulses can be stabilized by using the balance between dispersion, diffraction, nonlinearity, and by a far-detuned laser field. We demonstrate that these high-dimensional light pulses can be stored and retrieved very stably by switching off and on a control field. Results Model. We consider a cold, lifetime-broadened L-type three-level atomic gas interacting with a probe field (with pulse length t0, center angular frequency vp, and half Rabi frequency Vp) that drives the j1æ « j3æ transition, and a continuous-wave control field (with the center angular frequency vc and half Rabi frequency Vc) that drives j2æ « j3æ transition; see the inset of Fig. 1(a). SCIENTIFIC REPORTS | 5 : 8211 | DOI: 10.1038/srep08211 1 www.nature.com/scientificreports Figure 1 | Model and linear dispersion relation. (a) Possible experimental arrangement of beam geometry. The probe (with angular frequency vp and half Rabi frequency Vp) and continuous-wave control (with angular frequency vc and half Rabi frequency Vc) fields propagate nearly along z direction. The (orange) thick arrow denotes the Stark field (with angular frequency vs) used to stabilize (3 1 1)D light bullets and vortices. Cold atomic gas are represented by yellow dots. The inset shows the energy-level diagram and excitation scheme of the L-type three-level atoms. D2 and D3 are detunings, C13 (C23) is the decay rate from | 3æ to | 1æ ( | 3æ to | 2æ). The atoms are initially populated on the ground state | 1æ. (b) The linear dispersion relation K(v) of the probe field as a function of v. For simplicity, we assume the Xelectric field propagates along z direction with the form E~ e E eiðkl z{vl tÞ zc:c:, where l~p,c l l el ðE l Þ is the unit polarization vector (envelope). A fardetuned laser field (Stark field) used to stabilize (3 1 1)D light bullets and vortices (see below) is applied to the system [see Fig. 1(a)] with the form pffiffiffi EStark ðx,y,t Þ~es 2Es ðx,yÞcosðvs t Þ, where es, Es, and vs are the unit polarization vector, field amplitude, and angular frequency, respectively. Due to the existence of the Stark field, . an energy shift for the  2   level j jæ occurs, i.e., DEj,Stark ~{aj EStark t 2~{aj jEs ðx,yÞj2 2. Here aj is the scalar polarizability of the level j jæ, and h  i denotes the time average in one oscillating cycle. Under electric-dipole and rotating-wave approximations, the Hamiltonian of the system in the interaction picture reads X3   H^int ~{ hD’j j jih jj{ h Vp j3ih1jzVc j3ih2jzH:c: , with D’j j~1      ~Dj z aj ð2hÞ jEs j2 , Vp ~ ep :p13 E p  h, and Vc ~ðec :p23 ÞE c = h. Here D2 5 vp 2 vc 2 v21 and D3 5 vp 2 v31 are respectively the two- and one-photon detunings, pjl is the electric-dipole matrix element related to the levels j jæ and jlæ,  hvjl 5 Ej 2 El is the energy difference between the level jjæ and the level jlæ with Ej the eigenenergy of the level jjæ. The equation of motion for density matrix s in the interaction picture reads L zC Lt s~{  i ^ Hint ,s , h  SCIENTIFIC REPORTS | 5 : 8211 | DOI: 10.1038/srep08211 ð1Þ where s is a 3 3 3 density matrix, C is a 3 3 3 relaxation matrix denoting the spontaneous emission and dephasing. (...truncated)