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arXiv:quant-ph/9710038 · Europhysics Letters (EPL)

Quantum state reconstruction of a Bose-Einstein condensate

S Mancini, P Tombesi

Published Nov 15, 1997• 16 citations
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Abstract

We propose a tomographic scheme to reconstruct the quantum state of a Bose-Einstein condensate, exploiting the radiation field as a probe and considering the atomic internal degrees of freedom. The density matrix in the number state basis can be directly retrieved from the atom counting probabilities.
DOI
10.1209/epl/i1997-00470-4

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We propose a scheme to reconstruct the state of a two-mode Bose-Einstein condensate, with a given total number of atoms, using an atom interferometer that requires beam splitter, phase shift, and nonideal atom counting operations. The density matrix in the number-state basis can be computed directly from the probabilities of different counts for various phase shifts between the original modes, unless the beam splitter is exactly balanced. Simulated noisy data from a two-mode coherent state are produced and the state is reconstructed, for 49 atoms. The error can be estimated from the singular values of the transformation matrix between state and probability data.
We propose to use quantum tomography to characterize the state of a perturbed Bose-Einstein condensate. We assume knowledge of the number of particles in the zero-wave-number mode and of density distributions in space at different times, and we treat the condensate in the Bogoliubov approximation. For states that can be treated with the Gross-Pitaevskii equation, we find that the reconstructed density operator gives excellent predictions of the second moments of the atomic creation and annihilation operators, including the one-body density matrix. Additional inclusion of the momentum distribution at one point of time enables somewhat reliable predictions to be made for the second moments for mixed states, making it possible to distinguish between coherent and thermal perturbations of the condensate. Finally, we find that with the observation of the zero-wave-number mode anomalous second moment $⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{0}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{0}+{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{0}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{0}^{\ifmmode\dagger\else\textdagger\fi{}}⟩$, the reconstructed density operator gives reliable predictions of the second moments of locally amplitude squeezed states.