Time-of-arrival distributions from position-momentum and energy-time joint measurements

The position-momentum quasidistribution obtained from an Arthurs and Kelly joint measurement model is used to obtain indirectly an “operational” time-of-arrival (TOA) distribution following a quantization procedure proposed by Kochański and Wódkiewicz [Phys. Rev. A 60, 2689 (1999)]. This TOA distribution is not time covariant. The procedure is generalized by using other phase-space quasidistributions, and sufficient conditions are provided for time covariance that limit the possible phase-space quasidistributions essentially to the Wigner function, which, however, provides a nonpositive TOA quasidistribution. These problems are remedied with a different quantization approach which, on the other hand, does not guarantee normalization. Finally, an Arthurs and Kelly measurement model for TOA and energy is worked out, valid also for arbitrary conjugate variables when one of the variables is bounded from below. The marginal TOA distribution so obtained, a distorted version of Kijowski’s distribution, is time covariant, positive, and normalized.