Entanglement is a fascinating concept in quantum physics and a precious resource for quantum communication and computing. It is usually created in particle pairs, frequently by means of a conserved quantity in pair creation processes. Very little is known about the classification and quantification of entanglement in more complex systems, therefore entanglement itself is a phenomenon that is studied intensively worldwide. There are numerous applications of entanglement in quantum communication, computation, metrology and fundamental studies. It enables quantum teleportation and quantum key distribution, provides improved resolution in measurements, and is intrinsically connected with the paradox of nonlocality.

It is not straightforward to create entangled states of more than two particles in a controlled way. One of the most successful approaches has been to start out with pairs of entangled photons produced in spontaneous parametric down-conversion. In this process a high frequency photon is split into two lower-frequency ones so that energy is conserved. The requirement that the process be coherent in the nonlinear optical crystal provides for various kinds of entanglement of the converted photons. More complex entangled states can then be built by interferometric techniques and projective measurements.

While parametric down-conversion in bulk crystals has been very successful it is also limited in many ways. Two of the problems are that the dispersion is fixed by the material and the efficiency is, in general, very low. In order to get higher efficiencies one must usually increase the interaction length, which in turn due to dispersion decreases the bandwidth. The interferometric techniques of creating multi-photon entangled states rely on the down-conversion of ultrafast pulses with a relatively large bandwidth and their creation rate goes with the square of the efficiency for two pairs and even higher powers for more than two.

Therefore it is highly desirable to have a medium with managed dispersion if one wants to get efficient photon pair creation from ultrafast pulses.

Photonic crystals are media with a periodic variation in their dielectric constant. The periodicity controls the propagation of electromagnetic waves resulting in a band structure for light. The band structure can have gaps, i.e. frequencies where no light can propagate.

Waveguides can be defined by surrounding a guiding region with suitable photonic crystal material. In the design of these waveguides we have many degrees of freedom to adjust the modes and their dispersion. Our purpose is to design a structure in which parametric down-conversion can happen with phase and group velocities matched in order to allow for efficient conversion without the dispersion that is detrimental to the quality of entanglement.

For this purpose we have been studying slab-type quasi-two-dimensional photonic crystals in AlGaAs. GaAs is a highly nonlinear optical material (~ 100pm/V = 50x BBO) used heavily for optoelectronic devices. The structures of interest are created by patterning an epitaxial layer of GaAs with an array of holes and then etching out an Al-rich layer underneath. The resulting membrane (”slab”) can confine light vertically by total internal reflection and in-plane via forbidden frequencies.

The simplest waveguide is defined by leaving away a row of holes. Waveguides like this have been produced up to 1cm in length [Sugimoto et al., Opt. Exp. 12, 1090 (2004)].

The defect induces states that are localized orthogonal to the line and can propagate along the line. The dispersion and the number of modes can be changed by changing the width of the waveguide and by other techniques.

Our goal is to identify suitable modes at frequencies w_dc and w_p = 2w_dc so that k_p = 2k_dc + G, i.e. phase-matching in the first Brillouin zone. Preliminary calculations (2D) show that such modes do exist. For practical realizations we may need to ensure single-mode behaviour, in order to avoid mode mixing introduced by fabrication irregularities and surfaces.

Eventually it would be desirable to have the two down-conversion photons in different spatial or polarization modes. Because the nonlinear interaction is local good spatial overlap of the various modes is necessary. Therefore the best solution would be to have a conversion process where the two converted waves have orthogonal (quasi-) polarization, one of them being parallel and one of them orthogonal to the pump wave. Then is is easy to create polarization entanglement by combining two of these sources on a polarizing beam-splitter.

Coupling into and out of the structure is a difficult task. Since we cannot efficiently couple from the top we need to create cleaved faces while keeping the membrane intact. Special patterning of the terminating hole rows can be used to improve the coupling efficiency.