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Register a new userDynamics of the Pionium with the Density Matrix Formalism
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L Afanasyev
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The evolution of pionium, the $pi^+ pi^-$ hydrogen-like atom, while passing through matter is solved within the density matrix formalism in the first Born approximation. We compare the influence on the pionium break-up probability between the standard probabilistic calculations and the more precise picture of the density matrix formalism accounting for interference effects. We focus our general result in the particular conditions of the DIRAC experiment at CERN.
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Chiral Perturbation Theory predicts the lifetime of pionium, a hydrogen-like $pi^+ pi^-$ atom, to better than 3% precision. The goal of the DIRAC experiment at CERN is to obtain and check this value experimentally by measuring the break-up probability of pionium in a target. In order to accurately measure the lifetime one needs to know the relationship between the break-up probability and lifetime to a 1% accuracy. We have obtained this dependence by modeling the evolution of pionic atoms in the target using Monte Carlo methods. The model relies on the computation of the pionium--target atom interaction cross sections. Three different sets of pionium--target cross sections with varying degrees of complexity were used: from the simplest first order Born approximation involving only the electrostatic interaction to a more advanced approach taking into account multi-photon exchanges and relativistic effects. We conclude that in order to obtain the pionium lifetime to 1% accuracy from the break-up probability, the pionium--target cross sections must be known with the same accuracy for the low excited bound states of the pionic atom. This result has been achieved, for low $Z$ targets, with the two most precise cross section sets. For large $Z$ targets only the set accounting for multiphoton exchange satisfies the condition.
In this work, we simulate the electron dynamics in molecular systems with the Time-Dependent Density Matrix Renormalization Group (TD-DMRG) algorithm. We leverage the generality of the so-called tangent-space TD-DMRG formulation and design a computational framework in which the dynamics is driven by the exact non-relativistic electronic Hamiltonian. We show that, by parametrizing the wave function as a matrix product state, we can accurately simulate the dynamics of systems including up to 20 electrons and 32 orbitals. We apply the TD-DMRG algorithm to three problems that are hardly targeted by time-independent methods: the calculation of molecular (hyper)polarizabilities, the simulation of electronic absorption spectra, and the study of ultrafast ionization dynamics.
We report the progress in the measurement of the pionium lifetime by the DIRAC Collaboration at CERN (PS212). Based on data collected in 2001-2003 on Ni targets we have achieved the precision of 11% in the measurement of the pionium lifetime, which corresponds to the measurement of S-wave pion-pion scattering lengths difference |a0-a2| with the accuracy of 6%.
We present a method to apply the well-known matrix product state (MPS) formalism to partially separable states in solid state systems. The computational effort of our method is equal to the effort of the standard density matrix renormalisation group (DMRG) algorithm. Consequently, it is applicable to all usually considered condensed matter systems where the DMRG algorithm is successful. We also show in exemplary cases, that polymerisation properties of ground states are closely connected to properties of partial separability, even if the ground state itself is not partially separable.
The density matrix formalism and the equation of motion approach are two semi-analytical methods that can be used to compute the non-equilibrium dynamics of correlated systems. While for a bilinear Hamiltonian both formalisms yield the exact result, for any non-bilinear Hamiltonian a truncation is necessary. Due to the fact that the commonly used truncation schemes differ for these two methods, the accuracy of the obtained results depends significantly on the chosen approach. In this paper, both formalisms are applied to the quantum Rabi model. This allows us to compare the approximate results and the exact dynamics of the system and enables us to discuss the accuracy of the approximations as well as the advantages and the disadvantages of both methods. It is shown to which extent the results fulfill physical requirements for the observables and which properties of the methods lead to unphysical results.
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