Quantum tomography of the full hyperfine manifold of atomic spins via continuous measurement on an ensemble

Quantum state reconstruction based on weak continuous measurement has the advantage of being fast, accurate and almost non-perturbative. In this work we present a pedagogical review of the protocol proposed by Silberfarb et al (2005 Phys. Rev. Lett. 95 030402), whereby an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner so as to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time through a maximum-likelihood estimate. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sublevels of a large-spin alkali atom, (133)Cs. We detail two different protocols for control. Using magnetic interactions and a quadratic ac Stark shift, we can reconstruct a chosen hyperfine manifold F, e. g. the seven-dimensional F = 3 manifold in the electronic ground state of Cs. We review the procedure as implemented in experiments (Smith et al 2006 Phys. Rev. Lett. 97 180403). We extend the protocol to the more ambitious case of reconstruction of states in the full 16-dimensional electronic ground subspace (F = 3 circle plus F = 4), controlled by microwaves and radio-frequency (RF) magnetic fields. We give detailed derivations of all physical interactions, approximations, numerical methods and fitting procedures, tailored to the realistic experimental setting. For the case of light-shift and magnetic control, reconstruction fidelities of similar to 0.95 have been achieved, limited primarily by inhomogeneities in the light-shift. For the case of microwave/RF-control we simulate fidelity >0.97, limited primarily by signal-to-noise.