
The electronhole system in semiconductors excited by a highintensity light field has been one of the main topics on excitedstate processes in condensed matter. This system is expected to exhibit macroscopic quantum phenomena. In highdensity regime, the electronhole system is regarded as a twocomponent Fermi liquid in which the cooperative pairing of electrons and holes arises in momentum space at low temperature, quite similar to the BCS state in superconductors. While, in lowdensity regime, the electron hole pairs are regarded as welldefined excitons which are the bound states in the real space. In a dense gas of excitons where the excitons are still regarded as bosons, their BoseEinstein condensatation is expected.
Superfluidity is intimately connected with the BoseEinstein condensation. The experiments to observe the superfluidity of excitons have been carried out, but there are no direct evidence. Since their charge is neutral, the current carried by excitons cannot be detected directly. In order to overcome such difficulty, we propose an exciton system in a typeII quantum well. To characterize the superfluidity, we should find phenomena in which the phase of a macroscopic wave function of the condensed state plays a crucial role. We showed that, in a certain condition, the spatial structure of the phase are ordered, which lead to the formation of vortex lattice. The observation of this behavior could be the crucial evidence of the superfluidity of exciton systems. 

Liquid helium go through the superfluid transition at a temperature below the lambdapoint. This causes the anomalous property of the caacit, thus this is a phase transition related to the BoseEinstein condensation.
Vortices in superfluid state have characteristic nature without in classical vortices. For example, the quantized vortices have a circulation of quanta, and their core size is the atomic size. As theoretical prediction of the quantized vortices by Onsager and Feynman in 1950's, the physics of quantized vortices yielded by the macroscopic quantum state in the Bose condensed system has been studied extensively as the topic of the origin of the quantum fluid. For example, when the superfluid velocity extends over a certain critical velocity, the persistent current decay. This fact would be caused by the superfluid turbulent state, thus the dynamics of the quantized vortex plays the important role to the stability of the persistent current (superfluidity). The calculation of three dimensional vortex dynamics was performed by Schwarz in 1980's. There are two point of view to describe the motion of quantized vortices. If the mean separation of vortices or the curvature is sufficiently larger than the radius of a vortex core, the vortex line approximation which neglect the structure of the vortex core is valid. In this case, the velocity field around the vortex line is represented by the potential field which obeys the BiotSavart's low. While, when we consider the microscopic phenomena such as the nucleation or the reconnection of quantized vortices, we use the GrossPitaevskii equation for the order parameter of BoseEinstein condensate. We have treated this problem in both point of view described above. more details 