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Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields

Complex energy eigenstates in quantum decay models

T. K. Bailey, William C. Schieve

Published Sep 1, 1978• 36 citations
Publisher
DOI
10.1007/bf02778054

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Applying the doublet representation we analyze the solutions of a Hamiltonian system which has eigenstates with complex eigenvalues. The example of the Friedrichs model allows us to show how the appearance of solutions with non-Hilbert initial conditions is linked to the energy degeneration of the Hamiltonian spectrum. We discuss the difficulties of giving a physical meaning to the growing or decaying non-Hilbert solutions. We also suggest a way to circumvent the problem of the anomalous probabilities related to both complex energy eigenvalues and degeneration of the spectrum.
We study the complex quantum dynamics of a system of many interacting atoms in an elongated anharmonic trap. The system is initially in a Bose–Einstein condensed state, well described by Thomas–Fermi profile in the elongated direction and the ground state in the transverse directions. After a sudden quench to a coherent superposition of the ground and lowest energy transverse modes, quantum dynamics starts. We describe this process employing a three-mode many-body model. The experimental realization of this system displays decaying oscillations of the atomic density distribution. While a mean-field description predicts perpetual oscillations of the atomic density distribution, our quantum many-body model exhibits a decay of the oscillations for sufficiently strong atomic interactions. We associate this decay with the fragmentation of the condensate during the evolution. The decay and fragmentation are also linked with the approach of the many-body model to the chaotic regime. The approach to chaos lifts degeneracies and increases the complexity of the eigenstates, enabling the relaxation to equilibrium and the onset of thermalization. We verify that the damping time and quantum signatures of chaos show similar dependences on the interaction strength and on the number of atoms.
The dynamics of a wave in an open quantum system can be described by the eigenstates of the non-Hermitian Hamiltonian, provided that it is not at an exceptional point. Naively, one would expect that the decay of the energy eigenstates directly determines the decay of the total wave intensity. We show, however, that this is not the case for systems with localized losses. Instead, the nonorthogonality of the energy eigenstates plays an important role. It can to a large extent compensate for the initial decay of the eigenstates leading to a transient power-law decay of the wave intensity with a potentially large exponent.