A Phase Space Fluctuation Method for Quantum Dynamics

Kazuma Nagao

Esma Meral, Metin Demіralp

A new method called “Fluctuation Expansion” has been recently proposed for solving quantum dynamical problems . This method is succesfully applied to the dynamics of expectation values such that not the wave function itself but the expectation values of the position and momentum operator are taken into consideration and then the expectation value of certain entities depending on position and momentum operators are expressed via fluctuation expansion. This brings an infinite number of new unknown, fundamental and associate fluctuation terms. The derivatives of expectation values of the position and momentum operators and fluctuation operators are expressed in terms of these terms again. The resulting infinite set of differential equations and accompanying initial conditions are truncated at certain fluctuation order and then tried to be solved. Here we take a one dimensional time independent Hamiltonian and consider the zeroth and first order truncations. Equations obtained in this way are investigated in the sense of phase space which is defined by the expextation values of the position and momentum operator and the fluctuations entering the truncated equations. We find certain subspaces of the phase space such that there, certain expressions remain constant . This helps us to reduce the dimension of the problem.

Susmita Kar, Ramon Carbó‐Dorca, Pratim Kumar Chattaraj

Quantum localization is observed in the Fermi accelerator model with monochromatic fluctuation using the Ehrenfest description of quantum dynamics as well as through the quantum theory of motion (QTM). The effects of variations in time period and amplitude of an oscillating wall for a particle in a box, on energy and quantum phase space structures are analyzed. The effects of fluctuation decay with an increase in distance from the fluctuating wall. It is also observed that small amplitude as well as high frequency fluctuation provides enhanced quantum localization. The “quantum phase space structure” in the Ehrenfest description is constructed by following time-dependent Fourier grid Hamiltonian method while that for the grid points are formed from their trajectories obtained from QTM. © 2015 Wiley Periodicals, Inc.

Vladimir S. Filinov

Bulent Yilmaz, Denis Lacroix A, Resul Curebal

In the stochastic mean-field (SMF) approach, an ensemble of initial values for a selected set of one-body observables is formed by stochastic sampling from a phase-space distribution that reproduces the initial quantum fluctuations. Independent mean-field evolutions are performed with each set of initial values followed by averaging over the resulting ensemble. This approach has been recently shown to be rather versatile and accurate in describing the correlated dynamics beyond the independent particle picture. In the original formulation of SMF, it was proposed to use a Gaussian assumption for the phase-space distribution. This assumption turns out to be rather effective when the dynamics of an initially uncorrelated state is considered, which was the case in all applications of this approach up to now. Using the Lipkin-Meshkov-Glick (LMG) model, we show that such an assumption might not be adequate if the quantum system under interest is initially correlated and presents configuration mixing between several Slater determinants. In this case, a more realistic description of the initial phase-space is necessary. We show that the SMF approach can be advantageously combined with standard methods to describe phase-space in quantum mechanics. As an illustration, the Husimi distribution function is used here to obtain a realistic representation of the phase-space of a quantum many-body system. This method greatly improves the description of initially correlated fermionic many-body states. In the LMG model, while the Gaussian approximation failed to describe these systems in all interaction strength range, the novel approach gives a perfect agreement with the exact evolution in the weak coupling regime and significantly improves the description of correlated systems in the strong coupling regime.

Anatoli Polkovnikov