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arXiv:quant-ph/0202080 · Physical Review A

Reconstruction of motional states of neutral atoms via maximum entropy principle

G. Drobný, Vladimír Bužek

Published May 6, 2002• 16 citations
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Abstract

We present a scheme for a reconstruction of states of quantum systems from incomplete tomographiclike data. The proposed scheme is based on the Jaynes principle of maximum entropy. We apply our algorithm for a reconstruction of motional quantum states of neutral atoms. As an example we analyze the experimental data obtained by Salomon and co-workers and we reconstruct Wigner functions of motional quantum states of Cs atoms trapped in an optical lattice.

DOI
10.1103/physreva.65.053410

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We present a scheme for a reconstruction of states of quantum systems from incomplete tomographiclike data. The proposed scheme is based on the Jaynes principle of maximum entropy. We apply our algorithm for a reconstruction of motional quantum states of neutral atoms. As an example we analyze the experimental data obtained by Salomon and co-workers and we reconstruct Wigner functions of motional quantum states of Cs atoms trapped in an optical lattice.
We apply the Jaynes principle of maximum entropy [Phys. Rev. 106, 620 (1957); 108, 171 (1957)] for a reconstruction of Wigner functions of quantum-mechanical states of light on different observation levels. We study how quantum interference between components of superpositions of coherent states, which is responsible for the appearance of nonclassical effects, can be detected on different observation levels. We analyze in detail the reconstruction of Wigner functions of squeezed states on different observation levels in the case of nonunit detection efficiency modeled as a decay of the state under consideration into a zero-temperature reservoir. \textcopyright{} 1996 The American Physical Society.
Using tomographic reconstruction we determine the complete internuclear quantum state, represented by the Wigner function, of a dissociating I2 molecule based on femtosecond time resolved position and momentum distributions of the atomic fragments. The experimental data are recorded by timed ionization of the photofragments with an intense 20 fs laser pulse. Our reconstruction method, which relies on Jaynes's maximum entropy principle, will also be applicable to time resolved position or momentum data obtained with other experimental techniques.
Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.