Upper and lower bounds on the control field and the quality of achieved optimally controlled quantum molecular motion

A large class of problems in optimally controlled quantum or classical molecular dynamics has multiple solutions for the control field amplitude. A denumerably infinite number of solutions may exist depending on the structure of the design cost functional. This fact has been recently proved with the aid of perturbation theory by considering the electric field as the perturbating agent. In carrying out this analysis, an eigenvalue (i.e., a spectral parameter) appears which gives the degree of deviation of the control objective from its desired value. In this work, we develop a scheme to construct upper and lower bounds for the field amplitude and spectral parameter for each member of the denumerably infinite set of control solutions. The bounds can be tightened if desired. The analysis here is primarily restricted to the weak field regime, although the bounds for the strong field nonlinear case are also presented.     

Parameter estimation in exponentially fitted hybrid methods for second order differential problems

In this work we deal with exponentially fitted methods for the numerical solution of second order ordinary differential equations, whose solutions are known to show a prominent exponential behaviour depending on the value of an unknown parameter to be suitably determined. The knowledge of an estimation to the unknown parameter is needed in order to apply the numerical method, since its coefficients depend on the value of the parameter. We present a strategy for the practical estimation of the parameter, which is also tested on some selected problems.     

On the Structure of the Spectra for a Class of Combustion Waves

The stability of a premixed laminar flame supported by a general combustion reaction system is considered using the Evans function method. The spectrum of the linearised second-order differential operator is investigated in detail. The special structure of the differential equations due to an Arrhenius temperature dependence is exploited. It is shown that, for certain combustion systems, the limit of the Jacobian of the reaction terms as the travelling wave coordinate approaches the front and rear of the flame is a lower triangular matrix. For this type of system a simple geometrical method is shown for the study of the essential spectrum of the linearised operator, and for determining the domain of the Evans function. The results are applied to some representative combustion reactions.     

A probabilistic evolution approach trilogy, part 1: quantum expectation value evolutions, block triangularity and conicality, truncation approximants and their convergence

This is the first one of three companion papers focusing on the “probabilistic evolution approach (PEA)” which has been developed for the solution of the explicit ODE involving problems under certain consistent impositions. The main purpose here is the determination of the expectation value of a given operator in quantum mechanics by solving only ODEs, not directly using the wave function. To this end we first define a basis operator set over the Kronecker powers of an appropriately defined “system operator vector”. We assume that the target operator’s commutator with the system’s Hamiltonian can be expressed in terms of the above-mentioned basis operators. This assumption leads us to an infinite set of linear homogeneous ODEs over the expectation values of the basis operators. Its coefficient matrix is in block Hessenberg form when the target operator has no singularity, and beyond that, it may become block triangular when certain conditions over the system’s potential function are satisfied. The initial conditions are the basic determining agents giving the probabilistic nature to the solutions of the obtained infinite set of ODEs. They may or may not have fluctuations depending on the nature of the probability density. All these issues are investigated in a phenomenological and constructive theoretical manner in this paper. The remaining two papers are devoted to further details of PEA in quantum mechanics, and, the application of PEA to systems defined by Liouville equation.     

Making More Extensive Use of the Coupled-cluster Wave Function: from the Standard Energy Expression to the Energy Expectation Value

The standard coupled-cluster (CC) scheme with single and double excitations in the cluster operator (CCSD) includes only up to quadruple excitations in the equations. The CCSD exponential expansion generates, however, all possible excitations out of the reference function through products of the cluster operators. Clearly, in all standard approximate CC approaches only a part of the CC wave function is used in the equations. If the standard CCSD wave function is inserted into the energy expectation value expression then the complete CCSD wave function contributes to the energy. Such an energy expectation value expression can be presented as a sum of the standard CCSD energy formula plus correction terms. The correction terms provide an information about the quality of the total CC function. Contributions associated with the presence of higher than double excitations in the bra CCSD wave function supplement the CCSD energy obtained within the standard scheme. These contributions can be generated in a sequential way by considering intermediate excitation levels for the bra CCSD wave function in the expectation value expression before reaching the highest excitation level. In this way the importance of particular components differing in the standard and expectation value CCSD energies can be investigated. Some of the contributions can be recognized as close to or identical with the so-called renormalized noniterative corrections to the CC methods. We try to see to what an extent the nonstandard energy expressions, like the energy expectation value or the asymmetric energy formula, can be used to extend the applicability of the CCSD method illustrating our considerations with some numerical examples.Dedicated to Professor Jean-Paul Malrieu to honor his contribution to quantum chemistry and physics     

Pulse shaping for optimal control of molecular processes

In this paper, a new method is proposed to design optimized control fields with desired temporal and/or spectral properties. The method is based on penalizing the difference between an optimized field obtained from an iterative scheme and a reference field with desired temporal and/or spectral properties. Compared with the standard optimal control theory, the current method allows a simple, experimentally accessible field be found on the fly; while compared with parameter space searching optimization, the iterative nature of this method allows automatic exploration of the intrinsic mechanism of the population transfer. The method is illustrated by examing the optimal control of vibrational excitation of the Cl-O bond with both temporally and spectrally restricted pulses.     

A high dimensional model representation based numerical method for solving ordinary differential equations

A new numerical method for solving ordinary differential equations by using High Dimensional Model Representation (HDMR) has been developed in this work. Higher order ordinary differential equations can be reduced to a set of first order ODEs. Although HDMR is generally used for multivariate functions, univariate functions are taken into account throughout the work because of the ODEs’ natures. Not the numerical solution but its image under an appropriately chosen linear ordinary differential operator is expressed as a linear combination of the positive deviation powers of independent variable from its initial value. The linear combination of these image functions are expected to form a basis set under consideration. The unknown constants in the linear combination are found by maximizing the constancy measurer formed in terms of the HDMR components after they are evaluated. Results are compared with well-known step size based numerical methods. A semi qualitative error analysis of the proposed method is also established.     

Optimal control theory for a target state distributed in time: optimizing the probe-pulse signal of a pump-probe-scheme

Optimal control theory (OCT) is formulated for the case of a two-color pump-probe experiment. The approach allows to calculate the pump-pulse shape in such a way that the probe-pulse absorption signal is maximized. Since the latter quantity is given by the time-averaged expectation value of a time dependent operator (the probe-pulse field-strength times the dipole operator) a version of OCT has to be used where the target state is distributed in time. The method is applied to a molecular three-level system with the pump-pulse driving the transition from the electronic ground state into the first-excited electronic state and the probe-pulse connecting the first-excited state with a higher lying electronic state. Depending on the probe-pulse duration, the vibrational wave packet becomes localized or at least highly concentrated in the Franck-Condon window for the transition into the higher-excited state. The dependence on the probe-pulse duration and on the delay time between the optimized pump-pulse and the probe-pulse is discussed in detail. The whole study demonstrates the feasibility of laser pulse induced temporal wave packet localization and the use of spectroscopic quantities as target states in experiments on femtosecond laser pulse control.     

Local topology at limited resource induced suboptimal traps on the quantum control landscape

In a quantum optimal control experiment a system is driven towards a target observable value with a tailored external field. The underlying quantum control landscape, defined by the observable as a function of the control variables, lacks suboptimal extrema upon satisfaction of certain physical assumptions. This favorable topology implies that upon climbing the landscape to seek an optimal control field, a steepest ascent algorithm should not halt prematurely at suboptimal critical points, or traps. One of the important aforementioned assumptions is that no limitations are imposed on the control resources. Constraints on the control restricts access to certain regions of the landscape, potentially preventing optimal performance through convergence to limited resource induced suboptimal traps. This work develops mathematical tools to explore the local landscape structure around suboptimal critical points. The landscape structure may be favorably altered by systematically relaxing the control resources. In this fashion, isolated suboptimal critical points may be transformed into extensive level sets and then to saddle points permitting further landscape ascent. Time-independent kinematic controls are employed as stand-ins for traditional dynamic controls to allow for performing a simpler constrained resource landscape analysis. The kinematic controls can be directly transferred to their dynamic counterparts at any juncture of the kinematic analysis. The numerical simulations employ a family of landscape exploration algorithms while imposing constraints on the kinematic controls. Particular algorithms are introduced to meet the goals of either climbing the landscape or seeking specific changes in the topology of the landscape by relaxing the control resources.     

Study of simulation method of time evolution of atomic and molecular systems by quantum electrodynamics

We discuss a method to follow step‐by‐step time evolution of atomic and molecular systems based on quantum electrodynamics. Our strategy includes expanding the electron field operator by localized wavepackets to define creation and annihilation operators and following the time evolution using the equations of motion of the field operator in the Heisenberg picture. We first derive a time evolution equation for the excitation operator, the product of two creation or annihilation operators, which is necessary for constructing operators of physical quantities such as the electronic charge density operator. We, then, describe our approximation methods to obtain time differential equations of the electronic density matrix, which is defined as the expectation value of the excitation operator. By solving the equations numerically, we show “electron‐positron oscillations,” the fluctuations originated from virtual electron‐positron pair creations and annihilations, appear in the charge density of a hydrogen atom and molecule. We also show that the period of the electron‐positron oscillations becomes shorter by including the self‐energy process, in which the electron emits a photon and then absorbs it again, and it can be interpreted as the increase in the electron mass due to the self‐energy. © 2014 Wiley Periodicals, Inc.     

Birefringence and dielectric relaxation decay transients and anisotropic rotational diffusion of macromolecules in a strong external electric field

The decay transient response of polar and polarizable rigid bodies (macromolecules) diluted in a nonpolar solvent after a sudden switch-off of a strong external dc field is evaluated in the context of the anisotropic noninertial rotational diffusion model. On solving the differential-recurrence equations for the statistical moments (expectation values of Wigner's D functions), the decay transients of the birefringence and dielectric relaxation are obtained. The solutions (valid for arbitrary strengths of an external electric field) are given in a closed form suitable for comparison with experiment and Brownian dynamics simulations.     

Optimal control of classical molecular dynamics: A perturbation formulation and the existence of multiple solutions

This paper considers the prospect for there being multiple solutions to the control of classically modelled molecular dynamical systems. The research presented here follows up on a parallel study based on quantum mechanics. For polyatomic molecules it is generally expected that a classical mechanical model will be adequate and necessary as a means for designing optical fields for molecular control. The prospect for multiple control field solutions existing in this domain is important to establish in terms of ultimate laboratory realization of molecular control. A general formulation of the multiplicity problem is considered and the existence of a denumerably infinite number of solutions for the control field amplitude is shown to be the case under certain mild limitations on the physical variables.     

Complete dynamics of a two-level quantum system in interaction with a radiation field

Using the general theory of systems of linear differential equations with periodic coefficients, we derive a complete set of solutions of the equations of motion of a two-level quantum system in interaction with a classical radiation field. The structure of the solutions is investigated and a method of obtaining approximate solutions is discussed. These solutions are compared quantitatively with those in the rotating-wave approximation in dependence of field amplitude, detuning from resonance, time of interaction between two-level system and field, and initial state of the two-level system. In an appendix it is shown that the semiclassical dynamics of a harmonically driven two-level system may be derived from an associated fully quantum-mechanical motion by an asymptotic limit.     

Optimal control of multiple two-photon transitions

Optimal strategies to maximize the two-photon transition amplitude have been well studied. However, for a system with multiple intermediate states linking the same initial and final states, the question that how to achieve an optimal population transfer remains nontrivial. In this work, we propose a systematic block scheme to maximize the transition amplitude by explicitly considering the interferences among different transition pathways. The scheme can probably provide a quasi-optimal solution, even considering the uncertainties and noises in experiments. Cases with infinitesimal and finite spectral resolution are both investigated. A special example with all first transition frequencies being larger than half of the two-photon transition frequencies is employed to demonstrate our scheme. The analysis provides valuable insights on how to manipulate the interferences in control of quantum systems.     

Effects of Lead on Temporal Response Properties of Retinal Ganglion Cells in Developing Rats

Neonatal rats have taken in lead, during the period from their parturition to their weaning, from the milk of dams fed with water containing 0.2% lead acetate solutions. The alterations in the temporal response properties of retinal ganglion cells in adult rats (90 days) following the lead exposure at their developing stage have been studied. The results of this investigation demonstrate that the lead exposure in neonatal rats causes decreases in the optimal temporal frequency, bandwidth at half amplitude, temporal resolution and response phase of the retinal ganglion cells in adult rats. Compared with the sustained cells, the transient cells have a much greater alteration in temporal response properties.     

Gauging away the electron–electron interaction by the local phase of complex wave functions in two-electron systems

The electron–electron interaction is eliminated in the expectation values of the electronic Hamiltonian for two-electron systems. The part of the Hamiltonian referring to the repulsive interaction is gauged away by the local phase of the complex wave functions, much like a gauge field transformation, thereby leading to a one-electron Hamiltonian. Despite the appearance of complex wave functions, the expectation values of the total momentum operator vanish and Löwdin’s criterion holds for the stationary states.     

Feedback control system simulator for the control of biological cells in microfluidic cross slots and integrated microfluidic systems

Control systems for lab on chip devices require careful characterisation and design for optimal performance. Traditionally, this involves either extremely computationally expensive simulations or lengthy iteration of laboratory experiments, prototype design, and manufacture. In this paper, an efficient control simulation technique, valid for typical microchannels, Computed Interpolated Flow Hydrodynamics (CIFH), is described that is over 500 times faster than conventional time integration techniques. CIFH is a hybrid approach, utilising a combination of pre-computed flows and hydrodynamic equations and allows the efficient simulation of dynamic control systems for the transport of cells through micro-fluidic devices. The speed-ups achieved by using pre-computed CFD solutions mapped to an n-dimensional control parameter space, significantly accelerate the evaluation and improvement of control strategies and chip design. Here, control strategies for a naturally unstable device geometry, the microfluidic cross-slot, have been simulated and optimal parameters have been found for proposed devices capable of trapping and sorting cells.     

The interference in the decay of two ½ spins in a molecular medium,studied by the Nakajima–Zwanzig technique

This paper studies the decay, due to the spin-lattice coupling, of two ½ spins with slightly different Zeeman energies when the lattice is thermally excited. The analysis is based on obtaining, by means of the Nakajima–Zwanzig projection operator technique, an equation for the evolution of the reduced density operator of the spin system which manifests the influence of one spin on the relaxation process of the other. The zero-order solutions obtained for the evolution of the expectation values of the spin dynamics operators are essentially equivalent to the Bloch equations; higher order solutions are also obtained.