Measurements and calculations of the absolute carrier-envelope phase (CEP) effects in the photodissociation of the simplest molecule, H2+, with a 4.5-fs Ti:Sapphire laser pulse at intensities up to (4 +- 2)x10^14 Watt/cm^2 are presented. Localization
of the electron with respect to the two nuclei (during the dissociation process) is controlled via the CEP of the ultra-short laser pulses. In contrast to previous CEP-dependent experiments with neutral molecules, the dissociation of the molecular ions is not preceded by a photoionization process, which strongly influences the CEP dependence. Kinematically complete data is obtained by time- and position-resolved coincidence detection. The phase dependence is determined by a single-shot phase measurement correlated to the detection of the dissoziation fragments. The experimental results show quantitative agreement with ab inito 3D-TDSE calculations that include nuclear vibration and rotation.
