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Progress of Theoretical Physics

Quantum Nonlinear Resonance

Hidehiko Tsukuma, Fumihiko Sakata, Toshio Marumori, K. Iwasawa, H. Itabashi, Yukio Hashimoto, Toshiaki Tanaka

Published Jun 1, 1994• 3 citations
Publisher

Abstract

To visualize complex structure of the excited states in the many-fermion system, a profile function of the eigenstate within the quantum phase space is defined by using the boson expansion theory. When the anharmonic interaction is sufficiently weak where a corresponding semi-quantum system described by the time-dependent Hartree-Fock (TDHF) theory shows isolated resonant structure, there exists a remarkable similarity between the Poincaré section map of the TDHF trajectories and the profile functions of the eigenstates. It is shown that these eigenstates are well understood by using a new concept of quantum nonlinear resonance. It is also shown that a pair of newly appeared states due to the quantum resonance is not associated with any dissolution in a number of the good quantum numbers.

DOI
10.1143/ptp.91.1135

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To visualize complex structure of the excited states in the many-fermion system, a profile function of the eigenstate within the quantum phase space is defined by using the boson expansion theory. When the anharmonic interaction is sufficiently weak where a corresponding semi-quantum system described by the time-dependent Hartree-Fock (TDHF) theory shows isolated resonant structure, there exists a remarkable similarity between the Poincaré section map of the TDHF trajectories and the profile functions of the eigenstates. It is shown that these eigenstates are well understood by using a new concept of quantum nonlinear resonance. It is also shown that a pair of newly appeared states due to the quantum resonance is not associated with any dissolution in a number of the good quantum numbers.
In order to try to open a new scope to explore the mutual dependence between the single-particle and collective modes of motion near to the level crossing region, a general method is developed to investigate the nonlinear resonant structure of the time-dependent Hartree-Fock (TDHF) manifold, without depending on the adiabatic assumption. By using the Lie canonical transformation method with the Deprit perturbation treatment, in this method, the maximal integrable-form representation of the TDHF manifold is introduced. This representation plays an essential role in exploring the nonlinear resonant structure of the TDHF manifold, which characterizes complex topology of the manifold. Aiming at relating the nonlinear resonance in the TDHF manifold with the dynamics between the single-particle and collective modes of motion near to the level crossing region, structure of the TDHF wave function in the nonlinear resonant region is investigated. It is clarified that an isolated nonlinear resonant region of the TDHF manifold is characterized by a local constant of motion (dynamical symmetry) and generates a new type of dynamical stable single-Slater-determinant states, which is topologically different from the TDHF states near the HF ground state, and cannot be reached by the conventional static Hartree-Fock method, constrained Hartree-Fock method, nor the adiabatic TDHF theories. One may expect that the appearance mechanism of the new dynamical stable single-Slater-determinant states gives us a new scope for understanding occurrence mechanism of a variety of collective sideband structure near to the level crossing region.
We develop a new general theory for large quantum fluctuations in a Fermion many-body system that cannot be described by fluctuations around the Hartree-Fock ground state but arises from resonance of different correlation structures. We start with an exact coherent state representation of a Fermion system on a unitary group. We show that the Hamiltonian in the coherent state representation has a close connection with the Hartree-Fock energy functional. From this, we can derive a new approximation called the resonating Hartree-Fock approximation in which a state is approximated by a superposition of non-orthogonal Slater determinants with different correlation structures. We derive the variation equations to determine a resonating Hartree-Fock wavefunction. We show that the resonance between degenerate broken symmetry Slater determinants may partially recover the symmetry. We discuss how to choose trial Slater determinants in a resonating Hartree-Fock wavefunction. We suggest that resonance of Slater determinats representing localized defects, such as solitons, polarons and breathers, produced in the long range order of the HF ground state may be the most important content of large quantum fluctuations in condensed matter systems.
To find out how quantum bifurcation phenomena manifest themselves in the space of states, various periodic trajectories accompanied with several sets of bifurcated trajectories are studied for a simple quantum model Hamiltonian under a time-dependent mean-field approximation. Making use of the quantum-classical correspondence, a method of classifying the eigenstates and of visualizing the global structures of the whole space of eigenstates is proposed. By means of the profile functions of the eigenstates, appropriate sets of eigenstates representing the quantum correspondents of the bifurcations are found. The idea of quantum bifurcations is supported by similar behavior of the energy vs period time ( E-T ) plots of the periodic trajectories and the eigenstates.
The applicability of the concept of “optimal” collective submanifold determined by the self-consistent collective-coordinate (SCC) method is investigated by using a simple model Hamiltonian. Long-time behavior of the SCC-method trajectories on the optimal collective submanifold is analyzed, by comparing the geometrical structure of the SCC-method trajectories with that of the time-dependent Hartree-Fock (TDHF) trajectories in terms of the Poincaré-mapping method. The concept of the optimal collective submanifold determined by the SCC method turns out to be applicable for a fairly large domain of the TDHF manifold where the separatrix is not included, in the sense that it can represent an approximate time-averaged trajectory of the complicated TDHF trajectories.