Gamow vectors formalism applied to the Loschmidt echo
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Gamow vectors formalism applied to the Loschmidt echo S. Fortin1,a , M. Gadella2 , F. Holik3 , M. Losada4 1 CONICET - Universidad de Buenos Aires, Buenos Aires, Argentina 2 Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, 47011 Valladolid,
Spain
3 Instituto de Física La Plata, UNLP, CONICET, Facultad de Ciencias Exactas, La Plata, Argentina 4 CONICET - FAMAF, Universidad Nacional de Córdoba, Córdoba, Argentina
Received: 10 May 2020 / Accepted: 5 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Gamow vectors have been developed in order to give a mathematical description for quantum decay phenomena. Mainly, they have been applied to radioactive phenomena, scattering and to some decoherence models. They play a crucial role in the description of quantum irreversible processes and in the formulation of time asymmetry in quantum mechanics. In this paper, we use this formalism to describe a well-known phenomenon of irreversibility: the Loschmidt echo. The standard approach considers that the irreversibility of this phenomenon is the result of an additional term in the backward Hamiltonian. Here, we use the non-Hermitian formalism, where the time evolution is non-unitary. Additionally, we compare the characteristic decay times of this phenomenon with the decoherence ones. We conclude that the Loschmidt echo and the decoherence can be considered as two aspects of the same phenomenon and that there is a mathematical relationship between their corresponding characteristic times.
1 Introduction Gamow vectors have been introduced in the context of quantum unstable states, also called quantum resonances. Initially, they were used in nuclear physics to describe radioactive decay [1]. As is well known, experiments have shown that quantum resonances decay exponentially—at least for most observable times. Nevertheless, there are deviations of the exponential decay law [2] for very short (Zeno effect [3]) as well as for very long times (Khalfin effect [4]), which are difficult to be experimentally observed [5,6]. Noise effects may also contribute to slight deviations of the exponential law for intermediate times [2]. In any case, this exponential law gives a good approximation, so that resonance states can be very well approximated by vector states (or wave functions) decaying exponentially with time. These are the Gamow vectors or Gamow functions. This means that we have either to generalize time evolution so as to allow for non-unitary time evolutions or to extend the Hilbert space to a larger space containing the Gamow vectors. The exponential decay of Gamow vectors is accomplished if they are defined as eigenvectors (with complex eigenvalues) of a total Hamiltonian H = H0 + V , where V is a potential responsible for the decay [7]. Since Hamiltonians are self-adjoint, this is not possible in the
a e-mail: [email protected] (corresponding author)
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