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Stationary Light in Atomic Media

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Stationary Light in Atomic Media Stationary light in atomic media Jesse L. Everett1,2, Daniel B. Higginbottom1,3, Geoff T. Campbell1, Ping Koy Lam1, and Ben C. Buchler1 1Centre for Quantum Computation and Communication Technology, Research School of Physics and Engineering, Australian National University, Canberra, ACT 2601, Australia 2Light-Matter Interactions for Quantum Technologies Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa, 904 - 0495, Japan. 3Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 When ensembles of atoms interact with co- counterpropagating optical fields and can be tuned to herent light fields a great many interesting and manipulate the localization of light fields and atomic useful effects can be observed. In particular, excitations inside the ensemble. the group velocity of the coherent fields can be The original SL proposal [3] was followed in quick modified dramatically. Electromagnetically in- succession by a first experimental demonstration in duced transparency is perhaps the best known hot atomic vapor [6] and alternative SL schemes, some example, giving rise to very slow light. Care- with an alternative physical picture of the underlying ful tuning of the optical fields can also produce mechanism [7]. Although early demonstrations were stored light where a light field is mapped com- described in terms of standing wave modulated grat- pletely into a coherence of the atomic ensem- ings, this picture does not apply to hot atoms due to ble. In contrast to stored light, in which the thermal atomic motion. A subsequent multi-wave mix- optical field is extinguished, stationary light is ing formulation [7, 8] more comprehensively explains a bright field of light with a group velocity of the complex behaviours resulting from combinations zero. Stationary light has applications in sit- of counterpropagating optical fields. uations where it is important to maintain an The same all-optical tunable bandgap that lies be- optical field, such as attempts to engineer large hind SL effects has been considered as a flexible al- nonlinear interactions. In this paper we review ternative to fixed photonic crystals [9] with applica- the stationary light demonstrations published tions in quantum light storage and fast optical switch- to date and provide a unified theoretical frame- ing. Furthermore, because of the nonlinear behaviour work that describes the experimental observa- of atomic ensembles, reversibly trapping a SL field in tions. We also discuss possible applications of such a dynamic bandgap holds promise for enhancing stationary light with a particular focus on all- nonlinear photon-photon interactions. The use of SL optical phase gates for quantum information for this purpose is strongly motivated by the devel- technology. opment of photonic quantum information processing [10]. The size of a nonlinear phase shift scales with the product of the interaction strength and time. Con- 1 Introduction sequently, nonlinear interactions usually involve high intensity fields. Photonic qubit gates, however, require Several decades of research into coherent atom-light in- nonlinear interactions for fields down to the single- teractions have precipitated a multifarious menagerie photon level. SL therefore provides a path to this end by localizing optical fields in the atomic medium and arXiv:1905.00589v1 [quant-ph] 2 May 2019 of optical phenomena for storing and manipulating light fields inside atomic ensembles [1, 2]. In 2002 providing a longer time for a nonlinear phase shift to Andr´eand Lukin proposed that dynamically modu- be accumulated. lating the refractive index along the optical axis of Although slow-light schemes have been proposed for an ensemble could be used, not only to slow or store such quantum information applications, they feature light, but also to reversibly trap a light field within an inherent trade-off between interaction time and in- the atoms [3]. In contrast to prior methods of creating teraction strength, because the photonic component `stored light' [4], the optical component of this `sta- of the polariton is inversely proportional to the time tionary light' (SL) field remains considerable even as the probe spends inside the ensemble [11]. SL schemes the light's group velocity vanishes. Rather than map- allow greater flexibility in the configuration of the op- ping the optical field to an entirely atomic state, the tical field potentially enabling larger conditional phase non-zero light field is prevented from propagating by shifts at the single photon level [12] and photon-photon a dynamic optical bandgap, analogous to the static entanglement [13]. bandgap caused by the structure of photonic crystals These SL phase-gate schemes are in some ways or ordered atoms [5]. This bandgap is controlled by analogous to the well known cavity QED techniques 1 for atom-mediated photon-photon gates [14], with the with near-resonant control fields and subsequent re- photonic band gap trapping the light field in place of search has largely focused on this approach. More re- an optical resonator. What distinguishes SL photon- cently, Raman SL schemes with far-detuned control photon gate proposals is the wide degree of tunabil- fields have been introduced. What follows in this sec- ity: the spatial distribution of stationary atomic coher- tion is a short summary of SL work to date in each of ences and optical fields can be separately configured to these approaches. implement a wide range of potential interactions. Although SL can, in principle, be generated in any The purpose of this review is to consolidate the new sufficiently large ensemble of emitters, current demon- body of work on SL and to provide a unified model strations have been restricted to atomic systems by of SL phenomena under various conditions given the the difficulty of constructing optically deep and suffi- experimental evidence now available. We will also con- ciently coherent ensembles of ‘artificial atoms' such as sider the prospects and limitations of SL as a tool for quantum dots and diamond colour centres. In particu- quantum information applications in light of these re- lar, most demonstrations have been done in vapours of sults. rubidium or cesium atoms, which are dense ensembles with hydrogen-like spectra. SL has yet to be demon- 1.1 Structure strated in other optically deep atomic systems, such as rare-Earth ion crystals. Throughout this review we'll We have divided the following review into three sec- refer to the ensemble mediating SL as `atoms', but it's tions. We begin in Section 2 with the literature con- worth bearing in mind that SL could be generated in cerning schemes for generating SL fields. Our inten- any optically dense ensemble of coherent emitters. tion in reviewing this work is to convey a sense of the We'll see in the sections below that SL generation history of the field, with an emphasis on experimen- depends sensitively on the mobility of atoms within tal results and how they shaped our evolving picture of the ensemble, with qualitatively different behaviour in SL. We divide these results by generation scheme. The hot (≈ 400 K), cold (≈ mK), and ultracold (≈ µK) or bulk of results to date concern EIT-based SL, which stationary atoms. Hot, mobile atoms are typically the we cover in Section 2.1. The more recent Raman-based more complicated platform for quantum optics. Their schemes are covered in Section 2.2. In the interests of velocity with respect to optical beams Doppler broad- brevity, we will at first introduce only the bare mini- ens transitions and local coherences and excitations mum theory required to provide perspective for these are carried with the atoms as they move ballistically results. or diffusively through the ensemble. In this case, how- In Section 3 we provide a mathematical basis for the ever, motion has the effect of simplifying SL genera- physics in this review and introduce a comprehensive tion. Ultracold and stationary atoms can maintain a theoretical framework for SL in atomic ensembles. Our richer variety of coherences (higher-order coherences, goal is to give the reader sufficient tools to explain see Section 2.1.3) and have the more complicated dy- and model the effects discussed in Section 2 as well namics. as proposals which have yet to be implemented. We derive equations of motion for the optical fields and atomic coherences in a secular level scheme, i.e. one in which only the interactions of copropagating fields are 2.1 EIT-based stationary light considered. We use these to describe EIT and Raman SL. We return to a non-secular scheme to discuss how Electromagnetically induced transparency is a prop- to calculate the effect of higher-order coherences on the erty of three-level atoms interacting with two (usually SL. Finally, we discuss phase-matching requirements copropagating) optical fields. In the most common and transverse propagation. configuration, known as the Λ configuration and shown Section 4 concerns the proposed uses of SL, in par- in Figure 1(a), a weak probe field couples the atomic ticular for mediating gates in photonic quantum in- ground state j1i to an excited state j3i and a bright formation systems. We review early proposals for en- control beam couples the excited state to a meta-stable hancing nonlinear interactions, discuss the no-go theo- state j2i. The bright control field, with Rabi frequency rems these proposals inspired, and finally review more Ω, opens a narrow transparency window for the probe, ^ recent proposals that should overcome the obstacles E, at a resonant frequency that would otherwise be ab- raised by the no-go theorems. sorbed. When the two fields are equally detuned from the excited state, they are said to be in two-photon resonance and drive the ground-metastable state co- herence. When the two-photon resonance condition is 2 Generation of stationary light exactly met, the probe and control field excitations in- terfere such that the excited state is not driven at all, The SL schemes proposed and implemented to date and the atomic medium is rendered transparent for can be divided into two broad categories delineated by the probe field.
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