Complete Quantum Characterization of a Pulse echo in an Optical Lattice

Error correction is the major problem facing quantum information science. An ideal tool to characterize quantum errors and corrections is quantum process tomography \footnote {S. Myrskog et.al., {\it quant-ph/0312210}; M. W. Mitchell et. al. {\it PRL}, \textbf{91}, 120402(2003).}(QPT). In previous experiments with cold atoms in optical lattices decoherence has been observed in the center-of-mass motion. We introduce quantum mechanical methods to completely characterize the state of the system (density matrix) and the evolution process (superoperator). We apply these methods to a pulse-echo technique for correction of dephasing errors due to inhomogeneous broadening. Restoration of oscillations is achieved by applying a series of pulses consisting of lattice displacements. We achieve large echo amplitudes compared to amplitudes previously observed in optical lattices\footnote{ F. B. J. Buchkremer et. al., {\it PRL} \textbf{85}, 3121 (2000). }, and use QPT to compare different pulse sequences\footnote{Y.S. Weinstein et. al, {\it J. Chem. Phys. } \textbf{121(13)}, 6117 (2004). }. We study techniques for using QPT data to optimize error-correction protocols. We believe this approach will be essential for devising appropriate error- correction schemes for general quantum information systems with no a priori knowledge of the errors.