Measuring the quantum state of a Bose-Einstein condensate

Abstract

We describe an atomic interferometer based on Raman transitions between two hyperfine states. We show how atom count probabilities at the output of the interferometer can be used to reconstruct the joint multiparticle state of a two-mode Bose-Einstein condensate. Two data collection schemes are compared, via numerical solution for the density matrix from the probability data and analytical estimates of the errors are made. In the first scheme, only the phase of the interferometer is varied. We prove that state reconstruction is impossible for the case of a balanced beam splitter; otherwise, the problem is ill-conditioned. In the second scheme, both the phase and transmission parameters are varied, the $Q$ function can be measured directly, and the reconstruction of the state is well conditioned.