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arXiv:quant-ph/9703014 · Physical Review Letters

Reconstruction of the Joint State of a Two-Mode Bose-Einstein Condensate

Eric L. Bolda, Sze Meng Tan, Daniel F. Walls

Published Dec 15, 1997• 16 citations
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Abstract

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.

DOI
10.1103/physrevlett.79.4719

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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 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 QQQ function can be measured directly, and the reconstruction of the state is well conditioned.
Cold-atom interferometers with optical splitting and recombination use off-resonant laser beams to split a cloud of Bose-Einstein condensate (BEC) into two clouds that travel along different paths and are then recombined again using optical beams. After the recombination, the BEC in general populates both the cloud at rest and the moving clouds. Measuring a relative number of atoms in each of these clouds yields information about the relative phase shift accumulated by the atoms in the two moving clouds during the interferometric cycle. We derive the expression for the probability of finding any given number of atoms in each of the clouds, discuss features of the probability density distribution, analyze its dependence on the relative accumulated phase shift as a function of the strength of the interatomic interactions, and compare our results with experiment.
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.