The feasible generation of entangled spin - 1 state using linear optical element

We firstly present a feasible scheme to produce entangled spin-1 photon states from Fock state. The scheme requires only linear optical element and photon detector. The probability of success of method is relatively low. We further give a scheme to show that, if two maximally entangled photon state is prepared in advance, the probability of success can be improved. PACS number:03.65.Ud,42.50.Dv,03.67.-a The generation of entangled state occupies a central position in quantum optics. Popular candidate for experimental investigation in this context include trapped ions[1], Cavity QED[2] and Bose-Einstein condenses[3]. Many scheme has been proposed for the purpose of generating entanglement between atoms[4]. In fact, the GHZ state of many particles have been controllably produced in the trapped ion and atom cavity system[5, 2]. Currently, experiments with photon entanglement has opened a whole field of research. Such photon entanglement has been used to test bell inequality[6] and implement quantum information protocols like quantum teleportation[7], quantum dense coding[8] and quantum cryptography[9]. More recently, experimental GHZ state of three or four photons has also been reported[10]. Remarkably, efficient quantum computation with linear optics has been put forward[13]. Such scheme can directly be used to generate the photon polarization entanglement. In this proposals, a sequence of beam splitter is arranged carefully to implement basic non deterministic gate. More recently, a feasible linear optical scheme is[14] proposed to produce photon polarization entanglement with the help of single-photon quantum nondemolition measurement based on atom-cavity system[15]. In recent paper[16], we propose a scheme to generate entangled N photon state of the form 1 √ 2 (|0, N > +|N, 0 >) via linear optical element. Recently, there are increasing interest in the study of entangled state of spin-s objects (S > 1/2), which , apart from its fundamental interest[17, 18], are of clear interest for application in quantum information such as quantum cryptography[19] due to higher dimensional Hilbert space associated to these states. Experimental violation of a spin-1 bell inequality has been reported by using polarization entangled four photon state of pulsed parameter down conversion, which is formally equivalent to two maximally entangled spin-1 particle[20]. In practice, such polarization entangled photon state have only been produced randomly, since we have no way of telling that polarization entanglement was produced without measuring(and hence destroying) the outgoing state. In this paper, we propose a scheme to generate entangled spin-1 photon states of the form |Ψ >= 1 √ 2 (|2, 0, 2, 0 > −|1, 1, 1, 1 > +|0, 2, 0, 2 >) (1) using linear optical element. Consider the experiment shown schematically in Fig.1. A pair of photon in mode 1 and mode 2, are incident on a symmetric Beam Splitter BS1. The initial state of the system is |1 >1 |1 >2, Here |m >i denotes the Fock state of the ith mode. After the Beam Splitter, the state become |Ψ1 >= 1 2 (a†2 − b†2)|0 > |0 > (2) Let mode a pass through the Beam Splitter BS2. The second input port of Beam Splitter BS2 is assumed to be vacuum state. The auxiliary photons are measured and the outcome is accepted only when no photon is found. Thus the (unnormalized)state is projected into |Ψ2 >= 1 2 (cos θa†2 − b†2)|0 > |0 > (3) where cos θ are transimittance of beam splitter which is later determined. Let mode b pass through the Beam Splitter BS3, whose transformation is given by