In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the three-body system, (2) the solution of the Schr\"odinger equation in two steps: determination of adiabatic states for a fixed hyper-radius and then solution of a set of coupled hyper-radial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer $^{4}\mathrm{He}_{3}$ , isotopomers of the $\mathrm{H}_{3}{}^{+}$ ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the three-body system, (2) the solution of the Schr\"odinger equation in two steps: determination of adiabatic states for a fixed hyper-radius and then solution of a set of coupled hyper-radial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer $^{4}\mathrm{He}_{3}$, isotopomers of the $\mathrm{H}_{3}{}^{+}$ ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.

Physical Review AJan 10, 200638 citations

Simple Quantum Systems

Philipp O. J. Scherer

Springer eBooksJan 1, 20100 citations

Statistical reconstruction of the quantum states of three-level optical systems

Yu. I. Bogdanov, Leonid A. Krivitsky, S. P. Kulik

The procedure of measurement followed by the reconstruction of the quantum state of a three-level optical system is implemented for a frequency-and spatially degenerate two-photon field. The method of statistical estimation of the quantum state from a solution to the likelihood equation and the analysis of the statistical properties of the obtained estimators is developed. Using the root method of estimating quantum states, the initial two-photon (qutrit) wave function is reconstructed from the measured fourth-order field moments.The procedure of measurement followed by the reconstruction of the quantum state of a three-level optical system is implemented for a frequency-and spatially degenerate two-photon field. The method of statistical estimation of the quantum state from a solution to the likelihood equation and the analysis of the statistical properties of the obtained estimators is developed. Using the root method of estimating quantum states, the initial two-photon (qutrit) wave function is reconstructed from the measured fourth-order field moments.

Journal of Experimental and Theoretical Physics LettersSep 1, 200321 citations

Decoherence in a driven three-level system

A. R. P. Rau, Weichang Zhao

Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. Master equations such as the Lindblad equation preserve the trace of this matrix. Viewing them as first-order time-dependent operator equations for the elements of the density matrix, a unitary integration procedure can be adapted to solve for these elements. A simple model for decoherence preserves the Hermiticity of the density matrix. A single, classical Riccati equation is the only one requiring numerical handling to obtain a full solution of the quantum evolution. The procedure is general, valid for any number of levels, but is illustrated here for a three-level system with two driving fields. For various choices of the initial state, we study the evolution of the system as a function of the amplitudes, relative frequencies, and phases of the driven fields and of the strength of the decoherence. The monotonic growth of the entropy is followed as the system evolves from a pure to a mixed state. An example is provided by the $n=3$ states of the hydrogen atom in a time-dependent electric field, such degenerate manifolds affording an analytical solution.Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. Master equations such as the Lindblad equation preserve the trace of this matrix. Viewing them as first-order time-dependent operator equations for the elements of the density matrix, a unitary integration procedure can be adapted to solve for these elements. A simple model for decoherence preserves the Hermiticity of the density matrix. A single, classical Riccati equation is the only one requiring numerical handling to obtain a full solution of the quantum evolution. The procedure is general, valid for any number of levels, but is illustrated here for a three-level system with two driving fields. For various choices of the initial state, we study the evolution of the system as a function of the amplitudes, relative frequencies, and phases of the driven fields and of the strength of the decoherence. The monotonic growth of the entropy is followed as the system evolves from a pure to a mixed state. An example is provided by the $n=3$ states of the hydrogen atom in a time-dependent electric field, such degenerate manifolds affording an analytical solution.

arXiv:quant-ph/0307039Physical Review ANov 4, 200321 citations

Simple Quantum Systems

Elsevier eBooksJan 1, 19710 citations

^{4}\mathrm{He}_{3}

, isotopomers of the

\mathrm{H}_{3}{}^{+}

ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.

n=3

states of the hydrogen atom in a time-dependent electric field, such degenerate manifolds affording an analytical solution.