Quantum state and mode profile tomography by the overlap

Any measurement scheme involving interference of quantum states of the electromagnetic field necessarily mixes information about the spatiotemporal structure of these fields and quantum states in the recorded data. Weshow that in this case, a trade-off is possible between extracting information about the quantum states and the structure of the underlying fields, with the modal overlap being either a goal or a convenient tool of the reconstruction. Weshow that varying quantum states in a controlled way allows one to infer temporal profiles of modes. Vice versa, for the known quantum state of the probe and controlled variable overlap, one can infer the quantum state of the signal. We demonstrate this trade-off. by performing an experiment using the simplest on-off detection in an unbalanced weak homodyning scheme. For the single-mode case, we demonstrate experimentally inference of the overlap and a few-photon signal state. Moreover, we show theoretically that the same single-detector scheme is sufficient even for arbitrary multi-mode fields.