Trapping a particle of a quantum walk on the line
  We observe that changing a phase at a single point in a discrete quantum walk results in a rather surprising localization effect. For certain values of this phase change the possibility of localization strongly depends on the internal coin-state of the walker. 
Antoni Wojcik, Tomasz Luczak, Pawel Kurzynski, Andrzej Grudka, Tomasz   Gdala and Malgorzata Bednarska-Bzdega
quant-ph