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arXiv:quant-ph/0106089 · Journal of the Optical Society of America A

Toward a tomographic picture of a Bose–Einstein condensate

Stefano Mancini, Mauro Fortunato, Paolo Tombesi, Giacomo Mauro D’Ariano

Published Dec 1, 2000• 7 citations
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Abstract

The possibilities of applying tomographic techniques to a Bose-Einstein condensate to reconstruct its ground state are investigated by means of numerical simulations. Two situations for which the density-matrix elements can be retrieved from atom counting probabilities are considered. The methods presented here allow one to distinguish among various possible quantum states.

DOI
10.1364/josaa.17.002529

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The possibilities of applying tomographic techniques to a Bose-Einstein condensate to reconstruct its ground state are investigated by means of numerical simulations. Two situations for which the density-matrix elements can be retrieved from atom counting probabilities are considered. The methods presented here allow one to distinguish among various possible quantum states.
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.
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.