Optimizing H1 cavities for the generation of entangled photon pairs

We report on the theoretical investigation of photonic crystal cavities etched on a suspended membrane for the generation of polarization entangled photon pairs using the biexciton cascade in a single quantum dot. The implementation of spontaneous emission enhancement effect increases the entanglement visibility, while the concomitant preferential funneling of the emission in the cavity mode increases the collection of both entangled photons. We demonstrate and quantify that standard cavity designs present a polarization dependent emission diagram, detrimental to entanglement. The optimization of H1 cavities allows to obtain both high collection efficiencies and polarization independent emission, while keeping high Purcell factors necessary for high quality entangled photon sources. PACS numbers: 42.55.Tv, 03.67.Bg, 03.67.Mn,78.67.Hc Submitted to: New J. Phys. ha l-0 03 33 04 4, v er si on 1 22 O ct 2 00 8 Optimizing H1 cavities for the generation of entangled photon pairs 2 Entangled photon sources play an important role in quantum communication networks or quantum information processing [1, 2]. For the former, they are a necessary resource for the realization of quantum repeaters [3] based on quantum teleportation or quantum entanglement swapping. In the first demonstrations of such relays, parametric down conversion sources have been used for the generation of entangled photon pairs [4, 5, 6, 7]. Such non-linear sources of entanglement can combine narrow spectral bandwidths with a maximal generation rate [7, 8, 9]. Although these sources may be very useful and easy to implement, they always suffer from the Poissonian statistics of the emitted photons pairs leading to multipair emission and thus decreasing the fidelity of entanglement [10]. Being able to produce polarization entangled photon pairs on demand would be an important step towards robust quantum relays. Such sources can be obtained from the biexciton-exciton cascade emission of a single quantum dot [11], and first experimental demonstrations have been reported [12, 13]. Obtaining entangled photons pairs, however, from such quantum dots sources with both high fidelity and high collection efficiency remains a problem. Implementing Cavity Quantum ElectroDynamics effects by embedding a single quantum dot in a microcavity could not only improve the fidelity of the emitted pair [14] by taking advantage of the Purcell effect, but also by enhance the collection efficiency [15, 16]. One promising microcavity for such purpose is the single defect hole cavity in a triangular lattice of holes (H1) etched on a suspended membrane, due to its small mode volume and its polarization degeneracy. However, in a standard H1 cavity, the radiation pattern of the two fundamental degenerate modes do not overlap, leading to photon distinguishability and thus destroying entanglement. Theoretical calculations demonstrate that this radiation pattern can be strongly modified by changing, for instance, the position of the holes surrounding the defect [17]. This modification of the design is necessary to avoid distinct emission patterns. This emission pattern distinguishability is related to the mode overlap. In this paper, we report on the theoretical investigation of H1 photonic crystal cavities etched on slab membrane, in order to obtain both high collection efficiencies for both photons and a high overlap between the two fundamental energy-degenerate modes. The dependency of the Bell inequalities as a function of the mode overlap is derived. We also investigate the impact of the position of the quantum dot inside the cavity on entanglement visibility and collection efficiency. 1. Entangled state density matrix for non overlapping modes Polarization entangled photon pairs can only be obtained if and only if, even in principle, the polarization of the photon can not be determined by measuring another degree of freedom as for example the photon’s energy. In the same way, if the emission mode of one of the photons of the pair does not perfectly match the emission mode of the other photon, the non-maximal overlap between the two emission modes will reduce the fidelity of entanglement. Our analytical derivation of this non-maximal mode overlap ha l-0 03 33 04 4, v er si on 1 22 O ct 2 00 8 Optimizing H1 cavities for the generation of entangled photon pairs 3 effect is based on the density matrix of the photon pair emitted by the cascade emission from the biexcitonic level of a single quantum dot. Figure 1. Schematic description of the two-photon cascade in a typical quantum dot four-level system with an energy splitting 2h̄δω of the relay level, yielding two collinearly polarized photons (either H or V ). The eigenbasis of the dot involves four levels : |2〉 (biexcitonic level), |1H〉 and |1V 〉 (two excitonic levels with opposite angular momenta) and |0〉 (fundamental level). In this eigenbasis, the emitted photons are linearly polarised along the horizontal (H) or vertical (V) directions. The density matrix of the photon pair in this particular basis B = [H1H2, H1V2, V1H2, V1V2] where the subscript i = 1, 2 is related to the photon emitted by the biexcitonic level and excitonic level respectively, can be written in the form [14]: