Photon polarisation entanglement from distant dipole sources

It is commonly believed that photon polarisation entanglement can only be obtained via pair creation within the same source or via postselective measurements on photons that overlapped within their coherence time inside a linear optics setup. In contrast to this, we show here that polarisation entanglement can also be produced by distant single photon sources in free space and without the photons ever having to meet, if the detection of a photon does not reveal its origin – the which way information. In the case of two sources, the entanglement arises under the condition of two emissions in certain spatial directions and leaves the dipoles in a maximally entangled state. PACS numbers: 03.67.-a, 42.50.Dv, 42.50.Lc Secure quantum cryptographic protocols [1] rely on the creation of entangled photon pairs or at least the presence of effective entanglement in the scheme [2]. In order to establish a shared secret key, the sender (Alice) produces a stream of photons that she sends to the receiver (Bob). Each photon should be prepared in a state known to Alice and encodes one random bit of information. The secret key is extracted from the outcomes of the measurements that Bob performs on the incoming photons. To prevent an eavesdropper from obtaining information about the key without being noticed, it is important that each bit is encoded in the state of only one photon [3]. Other applications for single photon states can be found in linear optics quantum computing [4]. Due to the variety of interesting applications, a lot of effort has been made in the last years to develop new and reliable photon sources. Each of them has its respective merits. Current single photon sources [5] include atom-cavity schemes [6] as well as quantum dots [7], NV color centres in a diamond [8, 9] and tunable photonic band gap structures [10]. It is commonly believed that entangled photon pairs can only be created within the same source as in atomic cascades [11], in parametric down conversion schemes [12] and in the biexciton emission of a single quantum dot in a cavity [13]. If the entanglement is not created within the same source, single photons can be brought together to overlap within their coherence time on a beamsplitter where a postselective entangling measurement has to be performed on the output ports [14]. In contrast to this, we show that polarisation entanglement can also be obtained when the photons are created by distant sources without the photons ever having to meet and without having to control their emission times precisely. As an example we analyse the photon emission from two dipole sources that might be realised in the form of trapped atoms, diamond NV color centres, quantum dots or by using single Photon polarisation entanglement from distant dipole sources 2 atoms doped onto a surface. An interaction between the sources is not required. Each source should possess a Λ-type three-level configuration with the two degenerate ground states |0〉 and |1〉, the exited state |2〉 and optical transitions corresponding to the two orthogonal polarisations “+” and “−”. Polarisation entanglement arises under the condition of the emission of two photons in different but carefully chosen directions independent from the initial state of the sources. To understand how the proposed scheme works, it is important to recall that a detector always observes an integer number of photons. At the same time, fluorescence from two distant dipole sources can produce an interference pattern on a far away screen, if the distance of the screen from the sources is much larger than the distance between the sources [15, 16, 17]. This wave-particle dualism implies that both sources contribute coherently to the creation of each photon. Consequently, the emission of one photon leaves a trace in the states of all its potential sources, depending on its polarisation and the direction of its wave vector [17], and can thus affect the state of the subsequently emitted photon. In the following, the detectors of Alice and Bob are placed such that all wave vector amplitudes contributing to the creation of a second photon with the same polarisation as the first one interfere destructively. In case of the collection of two photons (one by Alice and one by Bob) the shared pair has to be in a superposition of the state where Alice receives a photon with polarisation “+” and Bob a photon with polarisation “−” and the state where Alice receives a photon with polarisation “−” and Bob a photon with polarisation “+”. Both share a maximally entangled pair, if the amplitudes for these two states are of the same size. In summary, polarisation entanglement is obtained with the help of postselection and interference effects. Related mechanisms have been proposed in the past to create atom-atom entanglement [18]. The pair creation scheme proposed in this paper is feasible with present technology and might offer several advantages to quantum cryptography. In contrast to parametric down conversion [5], the setup guarantees antibunching between subsequent photon pairs since the creation of a new pair is not possible without reexcitation of both sources. Furthermore, the scheme is robust. The final photon state does not depend on the initial state of the sources in case of a successful collection. Another important advantage to note is that the scheme offers the possibility to generate multiphoton entanglement by incorporating more than two radiators in the setup [19]. Let us now discuss the creation of an entangled photon pair in detail. We describe the interaction of the dipole sources with the surrounding free radiation field by the Schrödinger equation. The annihilation operator for a photon with wave vector k, polarisation λ and polarisation vector‡ ǫ̂ k̂λ is akλ. The two dipole sources considered here are placed at the fixed positions r1 and r2 and should be identical in the sense that they have the same dipole moment D2j for the 2-j transition (j = 0, 1). The energy separation between the degenerate ground states and level 2 is h̄ω0 while ωk = kc and L is the quantisation volume of the free radiation field. Using this notation, the system Hamiltonian becomes within the rotating wave approximation and with respect to the interaction-free Hamiltonian