The feasible generation of entangled photon states by using linear optical elements

We present a feasible scheme to produce a polarization-entangled photon states 1 √ 2 (|H > |V > +|V > |H >) in a controllable way. This scheme requires single-photon sources, linear optical elements and photon detectors. It generates the entanglement of spatially separated photons. The interaction takes place in the photon detectors. We also show that the same idea can be used to produce the entangled N -photon state 1 √ 2 (|0, N > +|N, 0 >). PACS number:03.65.Ud,42.50.-p, The generation of entangled quantum states plays a prominent role in quantum optics. An experimental realization in this context can be achieved with trapped ions [1], Cavity QED [2] or Bose-Einstein condenses [3]. Currently, experiments with polarization entangled two photons has opened a whole field of research. Such polarization entanglement has been used to test the Bell-inequality [4] and to implement quantum information protocols like quantum teleportation [5], quantum dense coding [6] and quantum cryptography [7]. More recently, the experimental generation of GHZ-states of three or four photons were reported [8]. In practice, these polarization entangled photon states have only been produced randomly, since there is no way of demonstrating that polarization entanglement was generated without measuring and destroying the outgoing state [9]. Some quantum protocols like error correction have been designed for maximally entangled quantum states without random entanglement [10]. Thus, a photon source is needed, which produces a maximally polarization entanglement of outgoing photons. Remarkably, efficient quantum computation with linear optics has been put forward [11]. Such scheme can be used directly to generate a polarization entanglement quantum state. It is suggested to arrange an array of beam splitters in order to implement a basic non-deterministic gate [11]. More recently, a feasible linear optical scheme [12] was proposed to produce polarization entanglement with the help of single-photon quantum non-demolition measurement based on atom-cavity system [13]. On the other hand, entangled N -photon states of the form Ψ = 1 √ 2 (|0, N > +|N, 0 >) (1) are of potential interest in respect to phase sensitivity in two mode interferometer. They should allow a measurement at the Heisenberg uncertainty limit [14]. Recently it is also shown that such states allow sub-diffraction limited lithography [15]. In the case of N = 2 the entangled N -photon state (1) can be generated easily by using linear optical elements. For higher values of N , a scheme was proposed by using nonlinear media [16]. It was supposed that the generation of such quantum states can become possible by a linear optical scheme. In this paper, we present the realizable scheme to generate the two-photon polarization entanglement with a single linear optical element and a single-photon source. We also show that the same idea can be used to generate the entangled N -photon state (1). At first we consider protocols to produce two-photon polarization entanglement. Figure 1 shows the required experimental set up. Two photons were emitted by two independent single-photon sources. The initial state of the two-photon system is |χ1 > |χ2 >, with |χi >= |H >i; i ∈ {1; 2}. The abbreviation H(V ) denotes the horizontal(vertical) linear polarization. We consider two polarized photons passing the symmetric beam splitter BS Φ1 = 1 √ 2 (|2H >1 |0 >2 −|0 >1 |2H >2) . (2) The two output modes of BS pass through two polarization rotators with rotation angle calibrated to cos θ1 = √ 1 3 . If the action of the polarization rotator on the output modes corresponds to a transformation of angle x ( aH → (cosx)aH+(sin x)aV ; aV → (cosx)aV − (sin x)aH), the quantum state becomes Φ2 = √ 2 3 [0.5|2H >1′ |0 >2′ +|HV >1′ |0 >2′ +|2V >1′ |0 >2′ −0.5|0 >1′ |2H >2′ −|0 >1′ |HV >2′ −|0 >1′ |2V >2′ ] . (3) The two output modes pass two polarization beam splitters: PBS1 and PBS2. Since the H(V ) polarization photon is transmitted(reflected) by the PBS, the state evolves into Φ3 = √ 2 3 [0.5|2H >3 +|H >3 |V >4 +|2V >4 −0.5|2H >5 −|H >5 |V >6 −|2V >6] . (4) The spatially separated polarized photon modes 4 and 6 are the input of the symmetric beam splitters BS1 and BS2. The second input ports of the beam splitters are assumed to be single-photon states produced by single-photon sources. After passing the symmetric beam splitters BS1 and BS2, the quantum state of the auxiliary mode is measured. The outcome of this BS-transformation is accepted only if the measurement of the auxiliary mode gives the same number of photons like the ancilla state was initially prepared: 1 photon. Thus, the quantum state (4) is transformed to the quantum state Φ4 = 1 2 [|2H >3 −|2V >4′ −|2H >5 +|2V >6′ ] . (5) After PBS3, PBS4, we obtain the Φ5 = 1 2 [|2H >7 −|2V >7 −|2H >8 +|2V >8] . (6) The two output modes 7 and 8 pass through two polarization rotators with an angle of rotation: θ2 = π 4 . The quantum state becomes Φ6 = 1 √ 2 [|HV >7′ −|HV >8′] . (7) Finally, mode 7′ and mode 8′ are incident on a polarization beam splitter. The twophoton polarization entangled quantum state 1 √ 2 (|H > |V > −|V > |H >) is obtained. It was shown, that two-photon polarization entangled quantum states can be generated with the probability of success 1/16 by using a non-deterministic gate [11]. Our method will in principle provide a slightly smaller probability of success: 1/18. But compared to the scheme presented in [11], which is used to generate entangled twophoton polarization entangled state, the experimental set up is simple. Instead of four detectors our scheme needs only 2 photon detectors. It can be also shown, that this concept can be used to generate the maximally entangled N-photon state (1). For simplify, we consider the entangled 2N -photon state 1 √ 2 (|0, 2N > +|2N, 0 >). The same scheme can be used to generate the entangled 2N +1-photon state 1 √ 2 (|0, 2N +1 > +|2N +1, 0 >). Figure 2 shows the experimental set up schematically. N photons in mode 1 and N photons in mode 2 are the input of the symmetric beam splitter (BS). The output of the BS is in the form: