Time evolution external field

Abstract. This article reviews from both theoretical and numerical aspects three non-equivalent complete second-order formulations of quantum dissipation theory, in which both the reduced dynamics and the initial canonical thermal equilibrium are properly treated in the weak system-bath coupling limit. Two of these formulations are rather familiar as the time-local and the memory-kernel prescriptions, while another which can be termed as correlated driving-dissipation equations of motion will be shown to have the combined merits of the two conventional formulations. By exploiting the exact solutions to the driven Brownian oscillator system, we demonstrate that the time-local and correlated driving-dissipation equations of motion formulations are usually better than their memory-kernel counterparts, in terms of their applicability to a broad range of system-bath coupling, non-Markovian, and temperature parameters. Numerical algorithms are detailed for an efficient evaluation of both the reduced canonical thermal equilibrium state and the non-Markovian evolution at any temperature, in the presence of arbitrary time-dependent external fields. 

So long as the field is on, these populations continue to change however, once the external field is turned off, these populations remain constant (discounting relaxation processes, which will be introduced below). Yet the amplitudes in the states i and i / do continue to change with time, due to the accumulation of time-dependent phase factors during the field-free evolution. We can obtain a convenient separation of the time-dependent and the time-mdependent quantities by defining a density matrix, p. For the case of the wavefiinction ), p is given as the outer product of v i) with itself. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

Fig. 7.17 Time evolution of the nuclear forward scattering for metallic Ni foil. All measurements except for the upper curve were performed with external magnetic field B = 4 T. The solid lines show the fit. The arrows emphasize stretching of the dynamical beat structure by the applied magnetic field. The data at times below 14.6 ns had to be rescaled (from [34])...

Mozumder (1969b) pointed out that in the presence of freshly created charges due to ionization, the dielectric relaxes faster—with the longitudinal relaxation time tl, rather than with the usual Debye relaxation time T applicable for weak external fields. The evolution of the medium dielectric constant is then given by... 

Consider the extreme case where H is diagonal in the base set Qjk(q,Q). Accordingly, in absence of external fields, no time evolution is to be expected except for changes in time-phases. But, by hypothesis, we took H to be the generator of time displacement in Hilbert space. Such a situation is not useful in molecular physics because one search after a Hamiltonian that is able to generate the time evolution. This suggests the idea that the time generator H of interest contains two classes of operators H = + V. The Hamiltonian Ho is assumed to... 

We assume the system under consideration to be a single domain. Then the orientational state of the system can be specified by the order parameter tensor S defined by Eq. (63), The time evolution of S is governed by the kinetic equation, Eq. (64), along with Eqs. (62) and (65). This kinetic equation tells us that the orientational state in the rodlike polymer system in an external flow field is determined by the term F related to the mean-field potential Vscf and by the term G arising from the external flow field. These two terms control the orientation state in a complex manner as explained below. 

Coherent excitation of quantum systems by external fields is a versatile and powerful tool for application in quantum control. In particular, adiabatic evolution has been widely used to produce population transfer between discrete quantum states. Eor two states the control is by means of a varying detuning (a chirp), while for three states the change is induced, for example, by a pair of pulses, offset in time, that implement stimulated Raman adiabatic passage (STIRAP) [1-3]. STIRAP produces complete population transfer between the two end states 11) and 3) of a chain linked by two fields. In the adiabatic limit, the process places no temporary population in the middle state 2), even though the two driving fields - pump and Stokes-may be on exact resonance with their respective transitions, 1) 2)and... 

The modulus square of the amplitudes c (f) are the populations of the two states,/i (0 = c (0P, that is, the probabilities to find the electron in state n) after time t. The time evolution of the electron in the external laser field is governed by the TDSE... 

In order to relate the dressed state population dynamics to the more intuitive semiclassical picture of a laser-driven charge oscillation, we analyze the induced dipole moment n) t) and the interaction energy V)(0 of the dipole in the external field. To this end, we insert the solution of the TDSE (6.27) into the expansion of the wavefunction Eq. (6.24) and determine the time evolution of the charge density distribution p r, t) = -e r, f)P in space. Erom the density we calculate the expectation value of the dipole operator... 

A more complicated behavior of the system (3) is manifested if the time-dependent driving field and damping are taken into account. Let us assume that the driving amplitude has the form /1 (x) =/o(l + sin (Hr)), meaning that the external pump amplitude is modulated with the frequency around /0. Moreover,/) = 0 and Ai = 2 = 0. It is obvious that if we now examine Eq. (3), the situation in the phase space changes sharply. In our system there are two competitive oscillations. The first belongs to the multiperiodic evolution mentioned in Section n.D, and the second is generated by the modulated external pump field. Consequently, we observe a rich variety of nonlinear oscillations in the SHG process. 

All of the analyses described above are used in a predictive mode. That is, given the molecular Hamiltonian, the sources of the external fields, the constraints, and the disturbances, the focus has been on designing an optimal control field for a particular quantum dynamical transformation. Given the imperfections in our knowledge and the unavoidable external disturbances, it is desirable to devise a control scheme that has feedback that can be used to correct the evolution of the system in real time. A schematic outline of the feedback scheme starts with a proposed control field, applies that field to the molecular system that is to be controlled, measures the success of the application, and then uses the difference between the achieved and desired final state to design a change that improves the control field. Two issues must be addressed. First, does a feedback mechanism of the type suggested exist Second, which features of the overall control process are most efficiently subject to feedback control ... 

We note that the separation into the three types of transitions (7), (2), and (2) is somewhat artificial. In fact, molecular collisions and transitions due to external fields are special examples of prepared states. Time evolution of a system described by a time-independent Hamiltonian does occur in general, unless the initial state of the system is described by a ket which is an eigenket to the complete Hamiltonian. 

Figure 6. Time evolution of ionic structure arid electronic key observables for Nag"1" in a laser field with intensity I = 2.2 1011 W/cm2 and frequency = 2.3 eV. Upper The z-positions of the 9 ions with multiplicity (arrangement in rings according to CAPS) as indicated. Middle Number of escaped electrons Nssc. Lower Energy Eext absorbed by the cluster from the external laser field. From [35].

The wavepacket /(t), on the other hand, is constructed in a completely different way. In view of (4.4), the initial state multiplied by the transition dipole function is instantaneously promoted to the excited electronic state. It can be regarded as the state created by an infinitely short light pulse. This picture is essentially classical (Franck principle) the electronic excitation induced by the external field does not change the coordinate and the momentum distributions of the parent molecule. As a consequence of the instantaneous excitation process, the wavepacket /(t) contains the stationary wavefunctions for all energies Ef, weighted by the amplitudes t(Ef,n) [see Equations (4.3) and (4.5)]. When the wavepacket attains the excited state, it immediately begins to move under the influence of the intramolecular forces. The time dependence of the excitation of the molecule due to the external perturbation and the evolution of the nuclear wavepacket /(t) on the excited-state PES must not be confused (Rama Krishna and Coalson 1988 Williams and Imre 1988a,b)... 

This section considers the theoretical background for calculating the molecular properties of a quantum mechanical subsystem exposed to a structured environment and interacting with an externally applied electromagnetic field. The time evolution of the expectation value of any operator A is determined using Ehrenfest s equation ... 

Recent years have also witnessed exciting developments in the active control of unimolecular reactions [30,31]. Reactants can be prepared and their evolution interfered with on very short time scales, and coherent hght sources can be used to imprint information on molecular systems so as to produce more or less of specified products. Because a well-controlled unimolecular reaction is highly nonstatistical and presents an excellent example in which any statistical theory of the reaction dynamics would terribly fail, it is instmctive to comment on how to view the vast control possibihties, on the one hand, and various statistical theories of reaction rate, on the other hand. Note first that a controlled unimolecular reaction, most often subject to one or more external fields and manipulated within a very short time scale, undergoes nonequilibrium processes and is therefore not expected to be describable by any unimolecular reaction rate theory that assumes the existence of an equilibrium distribution of the internal energy of the molecule. Second, strong deviations Ifom statistical behavior in an uncontrolled unimolecular reaction can imply the existence of order in chaos and thus more possibilities for inexpensive active control of product formation. Third, most control scenarios rely on quantum interference effects that are neglected in classical reaction rate theory. Clearly, then, studies of controlled reaction dynamics and studies of statistical reaction rate theory complement each other. 

We will in this section consider the mathematical structure for computational procedures when calculating molecular properties of a quantum mechanical subsystem coupled to a classical subsystem. Molecular properties of the quantum subsystem are obtained when considering the interactions between the externally applied time-dependent electromagnetic field and the molecular subsystem in contact with a structured environment such as an aerosol particle. Therefore, we need to study the time evolution of the expectation value of an operator A and we express that as... 

To end this section and the review, we mention briefly the first results from the simulation on laboratory-frame cross-correlation of the type (v(f)J (0)). Here v is the molecular center-of-mass linear velocity and J is the molecular angular momentum in the usual laboratory frame of reference. For chiral molecules the center-of-mass linear velocity v seems to be correlated directly in the laboratory frame with the molecule s own angular momentum J at different points r in the time evolution of the molectilar ensemble. This is true in both the presence and absence of an external electric field. These results illustrate the first direct observation of elements of (v(r)J (0)) in the laboratory frame of reference. The racemic modification of physical and molecular dynamical properties depends, therefore, on the theorem (v(r)J (0)) 0 in both static and moving frames of reference. An external electric field enhances considerably the magnitude of the cross-correlations. 

The dynamics of evolution become somewhat more complicated if the external field is not strictly periodic but rises and falls during a finite switch-on and switch-off time respectively, as will realistically be the case for an atom... 

At equilibrium, the box matrix h evolves to respond to the internal stresses with respect to an externally set pressure in the NPT ensemble (Eqs. [84]) in both the isotropic and flexible cell cases. The application of the external field contributes an additional term to its evolution. It is important to apply appropriate boundary conditions that are consistent with the nature of the dynamics. The contribution of the field to the time evolution of the simulation box can be represented as follows ... 

Here we have followed the notation of Ref. 12 and also noted that the time evolution operator will also act on the metric determinant. As described earlier, a change of variables is performed from F F,. Henceforth, the change of variables will be implicit and the subscript dropped for simplicity. If we assume that the external field and metric determinant have no explicit time dependence, we can take the derivative of both sides with respect to time to obtain ... 

Figure 2b Time evolution of chemical potential (/x,) when an atom is subjected to external electric fields (GS, ground state ES, excited state) el (—) monochromatic pulse e2( ) trichromatic pulse. Maximum amplitudes e0 = 10 6,0.01, 100 ...

The treatment in terms of induced current is in the mainstream of modem development of the time-dependent density functional theory (TDDFT). Moreover, the current density formalism has been proposed [4] as a variant of TDDFT. The evolution of current density presents properly the response of electrons on an external field. In general words, such a strong basis is promising for a theoretical treatment of many aspects of ion interactions with atoms, molecules and solids. 

The electric field in the film is calculated from the Poisson equation along with the constraint placed on the integral of the field by the fixed applied voltage. For each of a series of time steps, the carrier fluxes are calculated from the electric field and carrier concentrations. The time evolution of the system is calculated using fourth order Runge-Kutta integration of the carrier flux equations. Finite differences are used for spatial functions. The external current is given by the sum of injection and displacement currents. 

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