Field time-dependent

Peskin U, Miller W H and Ediund A 1995 Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditioner J. Chem. Phys. 103 10 030... 

Choosing a non-zero value for uj corresponds to a time-dependent field with a frequency u, i.e. the ((r r)) propagator determines the frequency-dependent polarizability corresponding to an electric field described by the perturbation operator QW = r cos (cut). Propagator methods are therefore well suited for calculating dynamical properties, and by suitable choices for the P and Q operators, a whole variety of properties may be calculated. " ... 

C. J. F. Bottcher and P. Bordewijk, Theory of Polarization, Vol. II. Dielectrics in Time-Dependent Fields, Vol. 18, Elsevier, Amsterdam, 1978. 

E. O. Stejskal, J. E. Tanner 1965, (Spin diffusion measurements Spin echoes in the presence of a time-dependent field gradient),/. Chem. Phys. 42 (1), 288—292. 

Time Evolution in Time-Dependent Fields and Time-Independent Reactive-Scattering Calculations via an Efficient Fourier Grid Preconditioner. 

Thus, the perhaps unfamiliar constitutive relations (2.23)-(2.25) yield familiar results when the fields are time harmonic moreover, because of (2.26) and (2.27), physical meaning can now be attached to the phenomenological coefficients even for arbitrarily time-dependent fields. 

The theory has been generalized by us to finite temperatures and to qubits driven by an arbitrary time-dependent field, which may cause the failure of the rotating-wave approximation (RWA) [11]. It has also been extended to the analysis of multilevel systems, where quantum interference between the levels may either inhibit or accelerate the decay [19]. 

INTRODUCTION. A standard and universal description of various nonlinear spectroscopic techniques can be given in terms of the optical response functions (RFs) [1], These functions allow one to perturbatively calculate the nonlinear response of a material system to external time-dependent fields. Normally, one assumes that the Born-Oppenheimer approximation is adequate and it is sufficient to consider the ground and a certain excited electronic state of the system, which are coupled via the laser fields. One then can model the ground and excited state Hamiltonians via a collection of vibrational modes, which are usually assumed to be harmonic. The conventional damped oscillator is thus the standard model in this case [1]. 

Fig. 3. Time dependent field of the 3.9fs pulse reconstructed from the FROG trace. The solid curve is the field and the dotted curve is the envelope of the field. The central wavelength is 595nm corresponding to the oscillation period of 2 fs. The carrier envelope cannot be determined and arbitrary phase is used for the description.

Remark. We assumed that Y(t) is a Markov process. Usually, however, one is interested in materials in which a memory effect is present, because that provides more information about the microscopic magnetic moments and their interaction. In that case the above results are still formally correct, but the following qualification must be borne in mind. It is still true that p y0) is the distribution of Y at the time t0, at which the small field B is switched off. However, it is no longer true that this p(y0) uniquely specifies a subensemble and thereby the future of Y(t). It is now essential to know that the system has aged in the presence of B + AB, so that its density in phase space is canonical, not only with respect to Y, but also with respect to all other quantities that determine the future. Hence the formulas cannot be applied to time-dependent fields B(t) unless the variation is so slow that the system is able to maintain at all times the equilibrium distribution corresponding to the instantaneous B(t). 

Exercise. Rotational Brownian motion of a dipole in an external time-dependent field has been described by ... 

Equation 1.3 can be generalized further by considering a time-dependent field c(r, t) the instantaneous rate of change of c with velocity v(r) is then... 

Here a0 is the static polarizability where the perturbation is time-independent. For time-dependent fields (e.g. optical fields) the perturbing hamiltonian is H0> = — a(F,f)Fa(0. and the problem is treated in a similar way using time-dependent perturbation theory. 

Two interesting comments are in order first, the stabilization phenomenon, originally demonstrated within the framework of the Floquet fomalism, i.e. for constant amplitude fields, is also observed in numerical simulations for an explicitly time-dependent field. Regarding the role of... 

The states correspond to wave packet controlled in the far past and in the far future, respectively. Let us see what this means. In the absence of external time-dependent fields, the scattering component of the time-dependent wave function i/r(f) can be expanded in terms of either of the two sets of scattering states for example, those with incoming boundary conditions... 

K. Osaki, N. Bessho, T. Kojimoto, and M. Kurata, Flow birefringence of polymer solutions in time-dependent field, J. Rheol., 23,457 (1979). 

The dispersion interaction between an atom and a metal surface was first calculated by Lennard-Jones in 1932, who considered the metal as a perfect conductor for static and time-dependent fields, using a point dipole for the molecule [44], Although these results overestimate the dispersion energy, the correct l/d3 dependence was recovered (d is the metal-molecule distance). Later studies [45 17] extended the work of Lennard-Jones to dielectrics with a frequency-dependent dielectric constant [48] (real metals may be approximated in this way) and took into account electromagnetic retardation effects. Limiting ourselves to small molecule-metal distances, the dispersion interaction of a molecule characterized by a frequency-dependent isotropic polarizability a embedded in a dielectric medium with permittivity esol (note that no cavity is built around the molecule) reads ... 

The extension of density functional theory (DFT) to the dynamical description of atomic and molecular systems offers an efficient theoretical and computational tool for chemistry and molecular spectroscopy, namely, time-dependent DFT (TDDFT) [7-11]. This tool allows us to simulate the time evolution of electronic systems, so that changes in molecular structure and bonding over time due to applied time-dependent fields can be investigated. Its response variant TDDF(R)T is used to calculate frequency-dependent molecular response properties, such as polarizabilities and hyperpolarizabilities [12-17]. Furthermore, TDDFRT overcomes the well-known difficulties in applying DFT to excited states [18], in the sense that the most important characteristics of excited states, the excitation energies and oscillator strengths, are calculated with TDDFRT [17, 19-26]. 

Similarly the quantities which are functions describing the response of the system to external time dependent fields can be modeled by the MD approach. 

Initially, at f < 0, the isolated system is described by the Hamiltonian H0 that does not depend on time. Thus, any expectation value averaged with the density matrix p0 is time-independent. At t > 0, the system is disturbed by an external time-dependent field. Then, the evolution of the expectation value of operator O can be presented as follows ... 

For simplicity, the QD and SM indices in the e-ph constants have been omitted in Eq. (118) however, the frequencies and e-ph constants are obviously different in the both subsystems. In the proposed description, we assume that equilibrium Green s functions of the semiconductor and the quantum dot are known. However, to find QD equilibrium Green s function in a time-dependent field is not an easy task because it is not even clear whether Dyson equations for SM and QD Keldysh functions exist for different types of fermions interacting with each other. This problem is rather complicated even for molecular wires [54], Thus, we expect this problem to be even more complicated for solar cell systems where the interaction with light makes the problem essentially time dependent. In this section, we prove that Dyson equations for nonequilibrium Green s functions do exist. In our description, we adopt a graduate approach to the problem introducing different approximations step by step. As the first and the easiest step, we consider only uncorrelated electrons. 

Bottcher CJF and Bordewijk P, "Theory of Electric Polarisation", Vol. 2, "Dielectrics in Time-dependent Fields", Elsevier, Amsterdam, 2nd Ed, 1973. 

Refs. [i] Bottcher CJF (1973) Theory of electric polarization Dielectrics in static fields, vol. 1. (1978) Dielectrics in time-dependent fields, vol. 2. Elsevier, Amsterdam [ii] Tide DR (2003) Dipole Moments. In Tide DR (ed) CRC handbook of chemistry and physics, 84til edn. CRC Press, Boca Raton, pp 9-42 - 9-51 [Hi] Miller TM (2003) Atomic and molecular polarizabilities. In Tide DR (ed) CRC handbook of chemistry and physics, 84th edn. CRC Press, Boca Raton, pp 10-163 -10-177 [ivJFred-erikse HPR (2003) Polarizabilities of atoms and ions in solids. In LideDR (ed) CRC handbook of chemistry and physics, 84til edn. CRC Press, Boca Raton, pp 12-17 -12-18... 

This chapter concentrates on the results of DS study of the structure, dynamics, and macroscopic behavior of complex materials. First, we present an introduction to the basic concepts of dielectric polarization in static and time-dependent fields, before the dielectric spectroscopy technique itself is reviewed for both frequency and time domains. This part has three sections, namely, broadband dielectric spectroscopy, time-domain dielectric spectroscopy, and a section where different aspects of data treatment and fitting routines are discussed in detail. Then, some examples of dielectric responses observed in various disordered materials are presented. Finally, we will consider the experimental evidence of non-Debye dielectric responses in several complex disordered systems such as microemulsions, porous glasses, porous silicon, H-bonding liquids, aqueous solutions of polymers, and composite materials. 

If the molecule in its ground state is subjected to a coupling with a time-dependent field with the interaction hamiltonian ... 

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