Eur. Phys. J. D 18, 247–250 (2002) DOI: 10.1140/epjd/e20020029 T HE EUROPEAN P HYSICAL JOURNAL D c  EDP Sciences Societ` a Italiana di Fisica Springer-Verlag 2002 Testing Bell inequalities in photonic crystals D.G. Angelakis 1, 2 , a and P.L. Knight 1 1 Quantum Optics and Laser Science Group, Blackett Laboratory, Imperial College, London SW7 2BW, UK 2 St Catharines’s College, Cambridge CB2 1RL, UK Received 31 July 2001 and Received in final form 30 November 2001 Abstract. We show how entangled atomic pairs can be prepared in order to test the Bell inequalities. The scheme is based on the interaction of the atoms with a highly localized field mode within a photonic crystal. The potential of using optically separated transitions and the stability of the entangled state to spontaneous emission could lead to the closure of the communication and the detection loopholes appearing in experiments so far. The robustness of the scheme against detector inefficiencies, the spread in the atomic velocities and the fact that the entangled pairs are not generated simultaneously is also studied. PACS. 03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell’s inequalities, GHZ states, etc.) – 42.50.-p Quantum optics – 42.70.Qs Photonic bandgap materials During the past twenty years, a number of interesting ex- periments for testing the Bell inequalities [1,2] have been proposed and carried out. In most, the entangled particles are photons, [3,5,8]. Violation of the inequalities occurs, supporting the quantum mechanical description of Nature against local realistic theories. However in all these pho- ton experiments, low detection efficiency combined with low analyzing speed of the results prevented the complete closure of the detection and/or the communication loop- hole. Experiments [8] which partially or even completely close one of the loopholes exist, e.g. the experiment by Weihs et al. [9] where the communication loophole was closed. However no photon experiment closing both loop- holes has been reported so far. In addition to photon based experiments there have also been proposals using atoms, based on the direct ma- nipulation of the atomic degrees of freedom, by interac- tion with the quantized field of the micromaser [10–13] or by photo-dissociation of dimers [4,15] and more recently by laser manipulating ions trapped inside a cavity [16]. Although in these elegant experiments the detection effi- ciency was higher than the photon based ones, problems such as as sequential detection (for the micromaser based experiments) and fragility of the Rydberg atoms at room temperature did not allow the complete closure of all the loopholes. We propose to create entangled atomic states using a photonic crystal (or photonic band gap material-PBG) for a Bell inequalities test. The potential of using opti- cally separated transitions which can be detected very ef- ficiently, the stability against background radiation, and the inhibition of spontaneous emission inside the crystal a e-mail: da241@hermes.cam.ac.uk can lead to the closure of all loopholes appearing in pre- vious experiments. In our scheme the entanglement originates from the in- teraction of two atoms with a resonant defect mode inside a photonic crystal. Photonic crystals are highly porous three dimensional periodic materials of high refractive in- dex with pore periodicity on the length scale of the rel- evant wavelength of light. They exclude electromagnetic modes over a continuous range of frequencies [17,18,20,25] (a photonic band gap). By introducing voids that are larger than the rest of the array, strongly localized sin- gle modes of light can be engineered within the otherwise optically empty PBG [19]. Our system consists of two two level atoms, the first of which is initially prepared in the upper of two opti- cally separated states, denoted by |e 1 〉 and the second in the lower one |g 2 〉 [10,11]. The two atoms propagate sequentially in orthogonal directions through the defect region of the crystal (see Fig. 1). The defect mode is ini- tially prepared in the vacuum state |0〉 and it is on res- onance with the atomic transition |e i 〉→|g i 〉 (Fig. 2). Although our scheme has some similarities with those pro- posed by [10,11] and implemented by [13], the potential of entangling and manipulating conventional, rather that Rydberg atoms in a spontaneous emission free environ- ment (the photonic crystal) opens the way to a new class of loophole free Bell experiments. The dynamics of a two level atom passing through a point defect [21–23], under the dipole and rotating wave approximations are described by the Jaynes-Cummings Hamiltonian [7,21]: H(r)= ω a 2 σ z + ω d a † a + G(r)(aσ + + a † σ − ) , (1)