Phase factors in bases for solutions of time-dependent schrödinger equations

The time-dependent Schrödinger equation is formulated within a model space by means of a finite set of coupled, linear differential equations. The basis is spanned by a set of orthogonal and well-defined many body wave-functions, which are solutions of a model Hamiltonian in a “moving frame”. As a by-product one is able to separate approximatively collective potential, collective kinetic, and intrinsic excitation energy for arbitrary collective motion. For the two types of motion discussed in greater details (i.e. center of mass and quadrupole motion), the expressions for the collective kinetic energy approach their correct asymptotic values.     

Theory of non-adiabatic cranking and nuclear collective motion

The cranking model is extended to the case of a general non-adiabatic motion. The time-dependent many-body Schrödinger equation is solved, where the time dependence of the collective motion is determined by the classical Lagrange equations of motion. The Lagrangian is obtained from the expectation value of the energy. In the case of one collective degree of freedom the condition that the expectation value of the energy is constant in time is sufficient to determine the collective motion. An iteration procedure is applied, of which the zeroth order is shown to be the common cranking formula. In an alternative approach the energy conservation is expressed in differential form. This leads in the case of one collective degree of freedom to a set of coupled, non-linear first-order differential equations in time for the expansion coefficients of the many-body wave function and for the collective variable. As an illustrative example we solve the case of two coupled linear harmonic oscillators.     

Dipole oscillations in a Bose-Fermi mixture in the time-dependent approach

We study the dipole collective oscillations in a Bose-Fermi mixture, which are formulated with the time-dependent Gross-Pitaevskii equation and the Vlasov equation, using a dynamical time-dependent approach. We find a large difference in the behavior of fermion oscillations between the time-dependent approach and the usual approaches such as the random-phase approximation. The dipole oscillation of the Bose-Fermi mixture cannot be described with simple center-of-mass motions of the two gases.     

Laplace transform approach for the dynamics of N qubits coupled to a resonator

An approach to use the method of Laplace transform for the perturbative solution of the Schrödinger equation at any order of the perturbation for a system of N qubits coupled to a cavity with n photons is suggested. We investigate the dynamics of a system of N superconducting qubits coupled to a common resonator with time-dependent coupling. To account for the contribution of the dynamical Lamb effect to the probability of excitation of the qubit, we consider counter-rotating terms in the qubit-photon interaction Hamiltonian. As an example, we illustrate the method for the case of two qubits coupled to a common cavity. The perturbative solutions for the probability of excitation of the qubit show excellent agreement with the numerical calculations.     

Stratified quantization approach to dissipative quantum systems: Derivation of the Hamiltonian and kinetic equations for reduced density matrices

A technique for describing dissipative quantum systems that utilizes a fundamental Hamiltonian, which is composed of intrinsic operators of the system, is presented. The specific system considered is a capacitor (or free particle) that is coupled to a resistor (or dissipative medium). The microscopic mechanisms that lead to dissipation are represented by the standard Hamiltonian. Now dissipation is really a collective phenomenon of entities that comprise a macroscopic or mesoscopic object. Hence operators that describe the collective features of the dissipative medium are utilized to construct the Hamiltonian that represents the coupled resistor and capacitor. Quantization of the spatial gauge function is introduced. The magnetic energy part of the coupled Hamiltonian is written in terms of that quantized gauge function and the current density of the dissipative medium. A detailed derivation of the kinetic equation that represents the capacitor or free particle is presented. The partial spectral densities and functions related to spectral densities, which enter the kinetic equations as coefficients of commutators, are evaluated. Explicit expressions for the nonMarkoffian contribution in terms of products of spectral densities and related functions are given. The influence of all two-time correlation functions are considered. Also stated is a remainder term that is a product of partial spectral densities and which is due to higher order terms in the correlation density matrix. The Markoffian part of the kinetic equation is compared with the Master equation that is obtained using the standard generator in the axiomatic approach. A detailed derivation of the Master equation that represents the dissipative medium is also presented. The dynamical equation for the resistor depends on the spatial wavevector, and the influence of the free particle on the diagonal elements (in wavevector space) is stated.     

A quasi-linear response theory of statistical heavy-ion collisions: (I). Basic formulation : Transport equations and roles of Green functions

We develop a time-dependent theory of heavy-ion collisions which consistently treats the relative and the intrinsic motions by coupled equations derived from the many-body von Neumann equation. The structure of the equations determining the mean trajectory and the fluctuations of the relative motion is the same as that of the corresponding equations in the known linear response theory. The present theory differs, however, from the linear response theory, in that it presumes neither weak coupling between the relative motion and the intrinsic excitations, nor the canonical distribution function for the density operator of the intrinsic motions. We apply the theory especially to deep inelastic collisions, where the relative motion couples to intrinsic excitations through a stochastic hamiltonian. Based on the stochastic assumption, we study the properties of the Green functions that take into account the higher order effects of the coupling hamiltonian. We then discuss, in particular, the effects of the Green functions on the time evolution of the intrinsic state, which is described in terms of a coarse-grained master equation, the friction tensor and fluctuation dissipation theorems.     

Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians

For a large class of time-dependent non-Hermitian Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis–Riesenfeld invariants making use of the metric picture, which is an equivalent alternative to the Schrödinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian.     

One-body dynamics modified by collective fluctuations

We show how the fluctuating part of the residual coupling between collective and intrinsic motion of a dissipative heavy-ion collision induces correlations in either subspace. They lead in general to a transport equation for the collective motion, and to a new term in the equation for the one-body density which describes collisions with the collective fluctuations. The resulting redistribution of the single-particle occupation numbers ρ_α and the evolution of the fluctuations are coupled with each other due to the dependence of the transition rates in the master equation on the fluctuations, and of the transport coefficients on ρ_α. Considering the special case of a long contact phase, we find the fluctuations to be most effective, with respect to a randomization of ρ_α, within a certain critical region where they pass from stable to unstable behaviour. Estimates are made for the corresponding relaxation times employing a schematic model.     

Nucleus-nucleus collisions: Intrinsic motion with external correlations

A consistent treatment of the relative and intrinsic motion which goes beyond the mean-field approach allows to include the fluctuations of the time-dependent mean field for the intrinsic as well as for the relative motion. Starting with the v. Neumann equation for the total density matrix, we derive a modified equation for the intrinsic many-body density matrix. This equation is used to obtain the quantum kinetic equations for the one-body density matrix and the two-body correlator. In the time-dependent single-particle basis, the occupation numbers change in time due to a collision term originating from residual two-body interactions which account for equilibration, and due to the fluctuations of the external mean field. The relations to TDHF with collision term are discussed. Special attention is paid to the conditions for a diabatic evolution of the single-particle states and to finite size effects which play an important role to make two-body collisions operative in finite nuclei.     

Breathing oscillations of Bose-Fermi mixtures with Yb atoms in the prolate deformed traps

We study the breathing oscillations in bose-fermi mixing gases with Yb atoms in the prolate deformed trap, which are realized in Kyoto group. We choose the three combinations of the Yb isotopes, ¹⁷⁰Yb–¹⁷¹Yb, ¹⁷⁰Yb–¹⁷³Yb and ¹⁷⁴Yb–¹⁷³Yb, whose boson-fermion interactions are weakly repulsive, strongly attractive and strongly repulsive. Then we calculate the collective oscillations in the deformed trap using a dynamical time-dependent approach, which is formulated with the time-dependent Gross-Pitaevskii equation and the Vlasov equation.     

Variational principles for collective motion: Relation between invariance principle of the Schrödinger equation and the trace variational principle

The invariance principle of the Schrödinger equation provides a basis for theories of collective motion with the help of the time-dependent variational principle. It is formulated here with maximum generality, requiring only the motion of intrinsic state in the collective space. Special cases arise when the trial vector is generalized coherent state and when it is a uniform superposition of collective eigenstates. The latter example yields variational principles uncovered previously only within the framework of the equations of motion method.     

Formally exact quantum variational principles for collective motion based on the invariance principle of the Schrödinger equation

Utilizing the concept of invariant collective subspace of a many-body system (or invariance principle of the time-dependent Schrödinger equation), we derive a number of formally exact variational principles to characterize the subspace. Previous studies based on time-dependent or adiabatic time-dependent Hartree-Fock theory are, in principle, contained as approximations.     

Order parameter analysis of synchronization transitions on star networks

The collective behaviors of populations of coupled oscillators have attracted significant attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynamical mechanism of collective synchronizations, by adopting the star-topology of coupled oscillators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe–Strogatz transformation, Ott–Antonsen ansatz, and the ensemble order parameter approach. Different solutions of the order parameter equation correspond to the diverse collective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions in the star-network model are revealed by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis.     

The Exact Solution of Milburn Equation Without Diffusion Approximation for the Two-Model Raman Coupled Model

By making use of the dynamical algebraic approach, we study the two-mode Raman coupled model governed by the Milburn equation and find the exact solution of the Milburn equation without diffusion approximation. The exact solution is then used to discuss the influence of intrinsic decoherence on the revivals of atomic inversion, oscillation of the photon number distribution and squeezing of radiation field in the whole ranges of the decoherence parameter γ.     

Entanglement Evolution of Two-Coupled Spins in a Time-Dependent Rotating Magnetic Field

We study the dynamics evolution of a two-qubit Heisenberg XXX spin chain under a time-dependent rotating magnetic field. Based on the algebraic structure of the non-autonomous system, the exact solution of the Schrödinger equation is obtained by using the method of algebraic dynamics. Based on the time-dependent analytical solution, we further study the entanglement evolution between the two coupled spins for different initial states, and find that the entanglement is determined by the coefficients of the initial state and the coupling constant $J$ of the system.