We analyze the recurrence probability (Pólya number) for d-dimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a by-product we find a simple criterion for localization of quantum walks. In contrast with classical walks, where the Pólya number is characteristic for the given dimension, the recurrence probability of a quantum walk depends in general on the topology of the walk, choice of the coin and the initial state. This allows us to change the character of the quantum walk from recurrent to transient by altering the initial state.

We propose the use of a balanced 2N-port as a technique to measure the pure quantum state of a single-mode light field. In our scheme the coincidence signals of simple, realistic photodetectors are recorded at the output of the 2N-port. We show that applying different arrangements both the modulus and the phase of the coefficients in a finite superposition state can be measured. In particular, the photon statistics can be so measured with currently available devices.

Like classical fluids, quantum gases may suffer from hydrodynamic instabilities. Our paper develops a quantum version of the classical stability analysis in fluids, the Bogoliubov theory of elementary excitations in unstable Bose-Einstein condensates. In unstable condensates the excitation modes have complex frequencies.We derive the normalization conditions for unstable modes such that they can serve in a mode decomposition of the noncondensed component. Furthermore, we develop approximative techniques to determine the spectrum and the mode functions. Finally, we apply our theory to sonic horizons-sonic black and white holes. For sonic white holes the spectrum of unstable modes turns out to be intrinsically discrete, whereas black holes may be stable.