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Journal of Physics G Nuclear and Particle Physics

Quantum dissipation in cluster decay phenomena: I. Smoothly joined quadratic potentials

Şerban Mişicu

Published Aug 25, 2000• 1 citation
Publisher

Abstract

This paper is devoted to the study of quantum dissipation in nuclear cluster decay phenomena within the framework of the Lindblad approach to quantum open systems. Typical examples of nuclear decay phenomena, such as cluster radioacivity, spontaneous cold fission of actinide nuclei or quasi-fission of super-heavy nuclei can be described by considering the tunnelling of a metastable state across a piecewise quadratic potential. Two study cases are considered: (i) a harmonic well smoothly joined to an inverted parabola which simulates the barrier and (ii) the inverted parabola of case (i) is continued with a second harmonic well. The width and depth of the second harmonic oscillator well are varied over a wide range of values in order to encompass particular cases of tunnelling in the double-well potential. The evolution of the averages and covariances of the quantum sub-system is studied in both under- and over-damped regimes. For a Gaussian initial wavepacket we compute the tunnelling probability for different values of the friction coefficient and fixed values of the diffusion coefficients. The ansatz used for these coefficients corresponds to the case with temperature T = 0.

DOI
10.1088/0954-3899/26/10/301

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This paper is devoted to the study of quantum dissipation in nuclear cluster decay phenomena within the framework of the Lindblad approach to quantum open systems. Typical examples of nuclear decay phenomena, such as cluster radioacivity, spontaneous cold fission of actinide nuclei or quasi-fission of super-heavy nuclei can be described by considering the tunnelling of a metastable state across a piecewise quadratic potential. Two study cases are considered: (i) a harmonic well smoothly joined to an inverted parabola which simulates the barrier and (ii) the inverted parabola of case (i) is continued with a second harmonic well. The width and depth of the second harmonic oscillator well are varied over a wide range of values in order to encompass particular cases of tunnelling in the double-well potential. The evolution of the averages and covariances of the quantum sub-system is studied in both under- and over-damped regimes. For a Gaussian initial wavepacket we compute the tunnelling probability for different values of the friction coefficient and fixed values of the diffusion coefficients. The ansatz used for these coefficients corresponds to the case with temperature T = 0.
Quantum diffusion equations with time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytical formulas for calculating friction and diffusion coefficients in nuclear processes are obtained. The asymptotics of the transport coefficients and of the correlation functions are investigated. The problem of correlation decay in quantum dissipative systems is studied. A comparative analysis of diffusion coefficients for the harmonic and inverted oscillators is performed. The role of quantum statistical effects during passage through a parabolic potential barrier is investigated. Sets of diffusion coefficient assuring the purity of states at any time instant are found in cases of non-Markovian dynamics. The influence of different sets of transport coefficients on the rate of decay from a metastable state is studied in the framework of the master equation for reduced density matrices describing open quantum systems. The approach developed is applied to investigation of fission processes and the processes of projectile-nuclei capture by target nuclei for bombarding energies in the vicinity of the Coulomb barrier. The influence of dissipation and fluctuation on these processes is taken into account in a self-consistent way. The evaporation residue cross sections for asymmetric fusion reactions are calculated from the derived capture probabilities averaged over all orientations of the deformed projectile and target nuclei.
As a prelude to a discussion of tunneling in the presence of dissipation, we discuss the damped harmonic oscillator. We show that for a harmonic-oscillator potential, the Caldeira-Leggett Hamiltonian can be reduced to a sum of independent harmonic-oscillator Hamiltonians by a normal-mode transformation. We then discuss tunneling through a parabolic barrier and explain the apparently contradictory results in the literature. The decay of a metastable state is then discussed. We argue that much of the published work on tunneling in dissipative systems has assumed unphysical initial conditions. We calculate the tunneling out of a metastable state, on the assumption that the environment is initially at zero temperature, and find that quantum tunneling is enhanced by dissipation.
In the framework of Bohmian quantum mechanics supplemented with the Chetaev theorem on stable trajectories in dynamics in the presence of dissipative forces we have shown the possibility of the classical (without tunneling) universal description of radioactive decay of heavy nuclei, in which under certain conditions so called noise-induced transition is generated or, in other words, the stochastic channel of alpha decay, cluster radioactivity and spontaneous fission conditioned by the Kramers diffusion mechanism. Based on the ENSDF database we have found the parametrized solutions of the Kramers equation of Langevin type by Alexandrov dynamic auto-regularization method (FORTRAN program REGN-Dubna). These solutions describe with high-accuracy the dependence of the half-life (decay probability) of heavy radioactive nuclei on total kinetic energy of daughter decay products. The verification of inverse problem solution in the framework of the universal Kramers description of the alpha decay, cluster radioactivity and spontaneous fission, which was based on the newest experimental data of alpha-decay of even-even super heavy nuclei (Z=114, 116, 118) have shown the good coincidence of the experimental and theoretical half-life depend upon of alpha-decay energy.
Quantum diffusion equations with transport coefficients explicitly depending on time are derived from the generalized non-Markovian Langevin equations. The asymptotic behavior of the friction and diffusion coefficients is investigated in the case of the FC and RWA couplings between the collective and internal subsystems. An asymptotic expression is obtained for the propagator of the density matrix of the open quantum system with the general quadratic Hamiltonian, linearly coupled (in coordinate and momentum) to internal degrees of freedom. The effect of different sets of transport coefficients on the decoherence and decay rate of the metastable state is investigated using the master equation for the reduced density matrix of open quantum systems. The developed approach is used to study the capture of the projectile nucleus by the target nucleus at energies near the Coulomb barrier. Capture cross sections in asymmetric reactions are well described with allowance for the calculated capture probabilities. Particular cases where dissipation favors penetration through the potential barrier are found. The generalized Kramers formula for the quasi-stationary decay rate of the quantum metastable systems is analytically derived.
The elements of the quantum mechanical Markovian diffusion matrix leading to a Gibbs equilibrium state for a set of N coupled quantum harmonic oscillators are derived within Lindblad's axiomatic approach. Consequences of the fundamental constraints on the quantum friction coefficients are discussed. We derive the equations of motion for the expectation values and variances, and we solve them analytically. We apply our results to the description of the charge and mass asymmetry coordinates in heavy-ion collisions, and we investigate the effect of dissipation on tunneling in sub-barrier processes.
The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger, Heisenberg and Weyl-Wigner-Moyal representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that not all of these equations are satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions and a comparative study is made for the Glauber P representation, the antinormal ordering Q representation, and the Wigner W representation. The density matrix is represented via a generating function, which is obtained by solving a timedependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided. The solution of the master equation in the Weyl-Wigner-Moyal representation is of Gaussian type if the initial form of the Wigner function is taken to be a Gaussian corresponding (for example) to a coherent wavefunction. The damped harmonic oscillator is applied for the description of the charge equilibration mode observed in deep inelastic reactions. For a system consisting of two harmonic oscillators the time dependence of expectation values, Wigner function and Weyl operator, are obtained and discussed. In addition models for the damping of the angular momentum are studied. Using this theory to the quantum tunneling through the nuclear barrier, besides Gamow’s transitions with energy conservation, additional transitions with energy loss are found. The tunneling spectrum is obtained as a function of the barrier characteristics. When this theory is used to the resonant atom-field interaction, new optical equations describing the coupling through the environment of the atomic observables are obtained. With these equations, some characteristics of the laser radiation absorption spectrum and optical bistability are described.