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EIT Cavity of D = 500 and Ωc = 2Γ Within 2 Μs

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EIT Cavity of D = 500 and Ωc = 2Γ Within 2 Μs All-optical control of superradiance and matter waves using a dispersive cavity Shih-Wei Su,1 Zhen-Kai Lu,2 Nina Rohringer,3, 4, 5 Shih-Chuan Gou,1 and Wen-Te Liao3, 4, 5, 6, ∗ 1Department of Physics and Graduate Institute of Photonics, National Changhua University of Education, Changhua 50058 Taiwan 2Max Planck Institute for Quantum Optics, D-85748 Garching, Germany 3Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany 4Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany 5Center for Free-Electron Laser Science, 22761 Hamburg, Germany 6Department of Physics, National Central University, 32001 Taoyuan City, Taiwan (Dated: Version of October 1, 2018.) Cavity quantum electrodynamics (CQED) [1] plays an elegant role of studying strong coupling between light and matter. However, a non-mechanical, direct and dynamical control of the used mirrors is still unavailable. Here we theoretically investigate a novel type of dynamically controllable cavity composed of two atomic mir- rors. Based on the electromagnetically induced transparency (EIT) [2,3], the reflectance of atomic mirror is highly controllable through its dispersive properties by varying the intensity of applied coupling fields or the optical depth of atomic media. To demonstrate the uniqueness of the present cavity, we further show the pos- sibility of manipulating vacuum-induced diffraction of a binary Bose-Einstein condensate (BEC) [4,5] when loading it into a dispersive cavity and experiencing superradiant scatterings [6]. Our results may provide a novel all-optical element for atom optics [7] and shine new light on controlling light-matter interaction. For decades, CQED has served as an elegant model for a backward probe field with Rabi frequency Ω− [13], and vise demonstrating atom-light interaction and fundamental princi- versa. For showing the controllability and novelty of the dis- ples of quantum mechanics [1]. In conjunction with the state- persive cavity, we consider an excited quasi one-dimensional of-art experiments using ultracold atoms, CQED has been able binary BEC which transversely flies through it and emits pho- to facilitate the realization of quantum metrology [8], quan- tons [5]. As illustrated in Fig.1 c, the binary BEC consists tum teleportation [9] and quantum memory storage [10], etc., of atoms condensed in the ground state jgi and excited state experiments paving the way for the fulfillment of quantum jei. The photonic polarization and frequency of transition technology. By its very nature, cavity is an indispensable in- jgi ! jei match that of j1i ! j3i of two atomic mirrors such gredient to CQED, which reflects photons and has them in- that emitted photons with wavelength λ from BEC can be re- teract with atoms in a confined space. However, conventional flected by two side EIT media. The vacuum fluctuations [15] CQED systems consisting of reflecting mirrors limiting them- and the geometric shape of the BEC will speed up the decay selves from being networked at large scale. Here, we propose and result in the emission of superradiance (SR) along BEC’s a novel cavity system where mechanical mirrors are replaced long axis [6, 15, 16]. Therefore when the long axis of the by atomic ensembles whose dispersion can be controlled by, loaded BEC is parallel to the cavity axis, the SR will circulate e.g., EIT, and so we term it as dispersive cavity which of- within a dispersive cavity and become subsequently releasable fers unique controllability. Moreover, the separation between by switching off coupling fields. The coexistence of BEC and the atomic mirrors and their size can be much smaller than superradiant photons within a cavity enables the multiple scat- that of mechanical ones. Therefore, the cavity cooperativ- tering between them. This renders the generation of matter ity is enhanced due to smaller cavity volume and it becomes waves with high order harmonic of photon wave number pos- more feasible to reach strong coupling regime and scalable for sible, namely, the diffraction of matter waves [7]. Because more demanding systems. Both advantages render fabricating the effect is triggered by vacuum fluctuations, we term it as a more functional matter-wave interferometry or atomic cir- vacuum induced diffraction of matter waves. cuit possible. The dispersive cavity can also be integrated on We first investigate the controllability of the novel cavity atom chips [11] or used to study the cavity-mediated spin- and calculate the reflectance of an EIT mirror by numerically orbit coupling in the ultracold atoms [12], bringing both the solving the optical-Bloch equations [14, 17] controllability and the tiny size into full play. 1 h i As depicted in Fig.1 a, our dispersive cavity is com- @ ρˆ = Hˆ ; ρˆ + ρˆ ; (1) t i~ dec posed of two remote collections of three-level-Λ type atoms. ! 1 arXiv:1508.04878v1 [cond-mat.quant-gas] 20 Aug 2015 Based on EIT [13, 14], as illustrated in Fig.1 b and d, each ± ± @t ± @z Ω = iηEIT ρ31; (2) atomic medium constitutes a mirror when interacting with two c counter-propagating coupling fields with Rabi frequencies Ω± c whereρ ˆ is the density matrix for the state vector P3 B jii driving the j2i $ j3i transition. Under the action of four-wave i=1 i of the three-level atom in the Fig.1 a, b and d. Also,ρ ˆ mixing [14], an incident probe field with Rabi frequency Ω+ dec describes the spontaneous decay of the excited state j3i char- driving the transition j3i $ j1i will be reflected and generate acterized by rate Γ. Hˆ is the Hamiltonian for the interaction between atomic mirrors and counter-propagating fields (see opt opt Methods). Moreover, ηEIT = Γd =2L with d and L being ∗ [email protected] the optical depth and the length of the medium, respectively. 2 a e f g b 3 c e d 3 Ω− Ω+ Ω− Ω+ C C Ω− Ω− Ω+ Ω+ C C Ω+ Ω− 2 2 1 g 1 FIG. 1. Schematic plot of a dispersive cavity. a, a binary BEC sandwiched by two three-level EIT media, where the former is initially in the state jei and the latters are in state j1i. The red arrows represent two counter-propagating coupling fields driving the transition j2i ! j3i as shown in b and d. The quantum fluctuations of binary BEC spontaneously generate the forward and backward SR fields Ω± (yellow) as shown in c, which connect the EIT transition from j1i ! j3i. The composition of three-level atoms and coupling fields acts as a dispersive cavity, which reflects resonant yellow photons. The controllable reflectance of a dispersive mirror and the enhancement of cavity finesse are shown in e and f, respectively. g, the coherently trapped superradiant field [jΩ+(t; z)j2 + jΩ−(t; z)j2]=Γ2 is shown where EIT mirrors are located in the regions jzj > 0:2 mm. The resultant dispersive cavity has only one reflecting band dynamics of the generated SR fields is described by whose width is associated to the EIT transparency window ! p 1 2 ± (±1) ∆! = Ω = Γ dopt which is highly tunable by either vary- @t ± @z Ω = iηBECσ ; (4) EIT c c eg ing the optical depth or the coupling fields. As shown in Fig.1 opt (±1) e, at various detuning and a fixed optical depth d = 500, the where σeg is the coherence of the BEC associated with the ± ±ik z reflectance approaches unity when the fields Ω are resonant plane-wave factor e p (see Methods for details) and ηBEC = 2 and the width can be controlled by varying the intensity of 3ΓNBECλ =4πA with A being the transverse cross section of the coupling fields. To gain more insight into the dispersive the condensate. In the following, we consider the EIT mirror π cavity, we invoke the effective cavity finesse f (∆p) = opt 1−R(∆p) of optical depth d = 500 and coupling field of Ωc = 2Γ.A 7 where R(∆p) is the reflectance as a function of the detuning quasi one-dimentional BEC of Li at 2S1=2 (jgi) state can be ± ∆p of fields Ω [18]. The cavity finesse is plotted in Fig.1 f, prepared in a single beam dipole trap. An additional TEM01- which demonstrates that the cavity finesse can be controlled type laser mode beam is applied to cross the atomic ensemble by tuning the EIT transparency window [2,3]. [22]. The spatial intensity distribution of the cross beam al- lows to trap low density thermal clouds aside as the EIT mir- We are ready to show the cavity effect on the BEC dynamics 7 by numerically integrating equations (1) and (2) together with rors while leaving the central BEC undisturbed. The Li BEC the coupled Gross-Pitaevskii equation which reads [19, 20] can be coherently transferred to the excited 3P3=2 (jei) state by ∼ (see methods) a π pulse. The narrow linewidth of 3P3=2 state, Γ 760 kHz [23] offers a reasonable time scale for detection. Moreover, dopt = 1000 has already been experimentally achieved in cold ! g ! ! atom systems [24, 25]. As an example, with N = 3 × 105 g Hsp + Hint Hatom−light g BEC i~@t = ∗ e iΓ ; (3) and A = (π × 32) µm2, Fig.1 g depicts the intensity of the e Hatom−light Hsp + Hint − 2 e confined superradiance jΩ+(t; z)j2 + jΩ−(t; z)j2 in a dispersive cavity. where g;e are the wave functions of the binary BEC, Hsp the The three steps of vacuum induced diffraction of matter g;e single-particle Hamiltonian, Hint the atom-atom interaction, waves is shown in Fig.2 a, namely, from top to bottom, cre- Hatom−light the atom-light interaction (see Methods), and Γ is ating BEC in state jei, loading BEC into a dispersive cavity the spontaneous decay rate of the excited level jei introduced and releasing BEC.
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