The validity of the discrete nonlinear Schr\"odinger equation and related semi-classical transport equations is studied with the help of an exact calculation of the spin-boson system in the experimentally relevant context of the fluorescence depolarization of stick dimers. The consequences of an initial excitation corresponding to low-energy dynamics are investigated. A number of features are found including the sensitive role of the polarization angle of incident light in determining the regimes of validity. Our earlier finding that the semiclassical approximation improves in validity in a well defined ``massive oscillator'' limit involving small phonon frequency is borne out. In the opposite limit of large phonon frequency, the semiclassical approximation and the discrete nonlinear Schr\"odinger equation are found to coincide with each other but to differ significantly from the exact evolution. \textcopyright{} 1996 The American Physical Society.
The validity of the discrete nonlinear Schr"odinger equation and related semi-classical transport equations is studied with the help of an exact calculation of the spin-boson system in the experimentally relevant context of the fluorescence depolarization of stick dimers. The consequences of an initial excitation corresponding to low-energy dynamics are investigated. A number of features are found including the sensitive role of the polarization angle of incident light in determining the regimes of validity. Our earlier finding that the semiclassical approximation improves in validity in a well defined ``massive oscillator'' limit involving small phonon frequency is borne out. In the opposite limit of large phonon frequency, the semiclassical approximation and the discrete nonlinear Schr"odinger equation are found to coincide with each other but to differ significantly from the exact evolution. \textcopyright{} 1996 The American Physical Society.
The validity of the semiclassical approximation is studied for a system comprising one quasiparticle coupled to a boson degree of freedom. Using a two-site Holstein model as an example, it is shown that the semiclassical approximation becomes exact in a nontrivial adiabatic limit. Furthermore, in the model's polaron regime, there exists a hierarchy of time scales that rationalizes the quantum dynamics of the Holstein model. For the single-mode case considered, the discrete nonlinear Schr\"odinger equation is found to be valid only in a highly limited antiadiabatic regime.
We study a two-state quantum-mechanical system, equivalent to a 1/2-spin dipole, with frequency ${\mathrm{\ensuremath{\omega}}}_{0}$, linearly interacting with a set of quantum oscillators. The corresponding discrete nonlinear Schr\"odinger equation (DNSE) has been given an analytical solution by Kenkre and Campbell [Phys. Rev. B 34, 4959 (1986)]. We show that in the strong-coupling limit their analytical expression is recovered from our microscopic approach when the thermal and quantum fluctuations of the bath of oscillators are neglected. To take into account the influence of thermal and quantum fluctuations on the prediction of the DNSE in the strong-coupling limit, we adopt a straightforward procedure based on the direct evaluation of 〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)〉, under the assumption alone that the frequency ${\mathrm{\ensuremath{\omega}}}_{0}$ is very weak. A special initial condition is used, with 〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)〉=1 and the oscillator at equilibrium in the corresponding effective potential. It is shown that in the strong-coupling limit this expression coincides with the noninteracting-blip approximation of Leggett et al. [Rev. Mod. Phys. 59, 1 (1987)] and the result of the equivalent projection approach of Aslangul, Pottier, and Saint-James [J. Phys. (Paris) 46, 2031 (1985); 47, 1657 (1986)]. Then, this expression is used to study the special case where the 1/2 spin interacts with only one oscillator at zero temperature. It is shown that the fast oscillations predicted by Kenkre and Campbell in the strong-coupling limit are damped by a Gaussian-like relaxation process provoked by the quantum fluctuations of the oscillator, which eventually lead to the destruction of the trapped state. These collapses are followed by periodical revivals reminiscent of those observed in quantum optics.
Analytical expressions for the energy eigenstates of the boson-Hubbard (or quantum discrete-nonlinear Schr\"odinger) dimer are obtained by applying perturbation theory in the small-tunneling-amplitude limit. The results are relevant for a Bose-Einstein condensate trapped in a double-well potential. A detailed comparison with the numerical solutions arising from the direct diagonalization of the Hamiltonian (for even or odd numbers of bosons) is presented, and the limits of validity of the perturbative results are determined. For fixed values of the tunneling amplitude, the accuracy of the perturbative calculation varies with the total number of bosons in different ways, depending on the particular energy level. Results for the fluctuations of the boson-number difference in each eigenstate are also presented.
The semiclassical and quantum dynamics of two ultrastrongly coupled nonlinear resonators cannot be explained using the discrete nonlinear Schrödinger equation or the Bose-Hubbard model, respectively. Instead, a model beyond the rotating wave approximation must be studied. In the semiclassical limit this model is not integrable and becomes chaotic for a finite window of parameters. For the quantum dimer we find corresponding regions of stability and chaos. The more striking consequence for both semiclassical and quantum chaos is that the tunneling time between the sites becomes unpredictable. These results, including the transition to chaos, can be tested in experiments with superconducting microwave resonators.
Abstract Recent experimental progress in monitoring the dynamics of ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if time-dependent semiclassical initial value methodology, with propagators expressed as integrals over phase space and using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that application of the Herman–Kluk (HK) propagator conserves unitarity in the semiclassical limit, but a decay of the norm is seen for low particle numbers n . The frozen Gaussian approximation (FGA) (HK with unit prefactor) is explicitly shown to violate unitarity in the present system for non-vanishing interaction strength, even in the semiclassical limit. Furthermore, we show by evaluating the phase space integral in steepest descent approximation, that the HK propagator reproduces the exact spectrum correctly in the semiclassical limit ( ). An error is, however, incurred in next-to-next-to-leading order (small parameter ), as seen upon numerical evaluation of the integral and confirmed analytically by considering finite n corrections to the steepest descent calculations. The FGA, in contrast, is only accurate to lowest order, and an erroneous next-to-leading order term in the energy spectrum was found analytically. Finally, as an example application, we study the dynamics of wave packets by computing the time evolution of the Wigner function. While the often-used truncated Wigner approximation cannot capture any interferences present in the exact quantum mechanical solution (known analytically), we find that the HK approach, despite also using classical information only, reproduces the salient features of the exact solution correctly.
The nonlinear-oscillating system in action-angle variables is characterized by the dependence of frequency of oscillation $\ensuremath{\omega}(I)$ on action I. Periodic perturbation is capable of realizing in the system a stable nonlinear resonance at which the action I adapts to the resonance condition $\ensuremath{\omega}({I}_{0})\ensuremath{\simeq}\ensuremath{\omega}$, that is, ``sticking'' in the resonance frequency. For a particular physical problem there may be a case when $I\ensuremath{\gg}\ensuremath{\hbar}$ is the classical quantity, whereas its correction $\ensuremath{\Delta}I\ensuremath{\simeq}\ensuremath{\hbar}$ is the quantum quantity. Naturally, dynamics of $\ensuremath{\Delta}I$ is described by the quantum equation of motion. In particular, in the moderate nonlinearity approximation $\ensuremath{\varepsilon}\ensuremath{\ll}(d\ensuremath{\omega}/dI)(I/\ensuremath{\omega})\ensuremath{\ll}1/\ensuremath{\varepsilon}$, where $\ensuremath{\varepsilon}$ is the small parameter, the description of quantum state is reduced to the solution of the Mathieu-Schr\"odinger equation. The state formed as a result of sticking in resonance is an eigenstate of the operator $\ensuremath{\Delta}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{I}$ that does not commute with the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$. Expanding the eigenstate wave functions in Hamiltonian eigenfunctions, one can obtain a probability distribution of energy level population. Thus, an inverse level population for times lower than the relaxation time can be obtained.