WavePacket: A Matlab package for numerical quantum dynamics. III: Quantum-classical simulations and surface hopping trajectories

WavePacket is an open-source program package for numerical simulations in quantum dynamics. Building on the previous Part I [Comp. Phys. Comm. 213, 223-234 (2017)] and Part II [Comp. Phys. Comm. 228, 229-244 (2018)] which dealt with quantum dynamics of closed and open systems, respectively, the present Part III adds fully classical and mixed quantum-classical propagations to WavePacket. In those simulations classical phase-space densities are sampled by trajectories which follow (diabatic or adiabatic) potential energy surfaces. In the vicinity of (genuine or avoided) intersections of those surfaces trajectories may switch between surfaces. To model these transitions, two classes of stochastic algorithms have been implemented: (1) J. C. Tully's fewest switches surface hopping and (2) Landau-Zener based single switch surface hopping. The latter one offers the advantage of being based on adiabatic energy gaps only, thus not requiring non-adiabatic coupling information any more. The present work describes the MATLAB version of WavePacket 6.0.2 which is essentially an object-oriented rewrite of previous versions, allowing to perform fully classical, quantum-classical and quantum-mechanical simulations on an equal footing, i.e., for the same physical system described by the same WavePacket input. The software package is hosted and further developed at the Sourceforge platform, where also extensive Wiki-documentation as well as numerous worked-out demonstration examples with animated graphics are available

Multidimensional spectroscopy with a single broadband phase-shaped laser pulse

We calculate the frequency-dispersed nonlinear transmission signal of a phase-shaped visible pulse to fourth order in the field. Two phase profiles, a phase-step and phase-pulse, are considered. Two dimensional signals obtained by varying the detected frequency and phase parameters are presented for a three electronic band model system. We demonstrate how two-photon and stimulated Raman resonances can be manipulated by the phase profile and sign, and selected quantum pathways can be suppressed.Comment: 26 pages, 15 figure

A Simplified Approach to Optimally Controlled Quantum Dynamics

A new formalism for the optimal control of quantum mechanical physical observables is presented. This approach is based on an analogous classical control technique reported previously[J. Botina, H. Rabitz and N. Rahman, J. chem. Phys. Vol. 102, pag. 226 (1995)]. Quantum Lagrange multiplier functions are used to preserve a chosen subset of the observable dynamics of interest. As a result, a corresponding small set of Lagrange multipliers needs to be calculated and they are only a function of time. This is a considerable simplification over traditional quantum optimal control theory[S. shi and H. Rabitz, comp. Phys. Comm. Vol. 63, pag. 71 (1991)]. The success of the new approach is based on taking advantage of the multiplicity of solutions to virtually any problem of quantum control to meet a physical objective. A family of such simplified formulations is introduced and numerically tested. Results are presented for these algorithms and compared with previous reported work on a model problem for selective unimolecular reaction induced by an external optical electric field.Comment: Revtex, 29 pages (incl. figures

Time Ordering in Kicked Qubits

We examine time ordering effects in strongly, suddenly perturbed two-state quantum systems (kicked qubits) by comparing results with time ordering to results without time ordering. Simple analytic expressions are given for state occupation amplitudes and probabilities for singly and multiply kicked qubits. We investigate the limit of no time ordering, which can differ in different representations.Comment: 26 pages, 5 figure

Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field

We propose a new monotonically convergent algorithm which can enforce spectral constraints on the control field (and extends to arbitrary filters). The procedure differs from standard algorithms in that at each iteration the control field is taken as a linear combination of the control field (computed by the standard algorithm) and the filtered field. The parameter of the linear combination is chosen to respect the monotonic behavior of the algorithm and to be as close to the filtered field as possible. We test the efficiency of this method on molecular alignment. Using band-pass filters, we show how to select particular rotational transitions to reach high alignment efficiency. We also consider spectral constraints corresponding to experimental conditions using pulse shaping techniques. We determine an optimal solution that could be implemented experimentally with this technique.Comment: 16 pages, 4 figures. To appear in Physical Review

Optimal Control of Quantum Dynamics : A New Theoretical Approach

A New theoretical formalism for the optimal quantum control has been presented. The approach stems from the consideration of describing the time-dependent quantum system in terms of the real physical observables, viz., the probability density rho(x,t) and the quantum current j(x,t) which is well documented in the Bohm's hydrodynamical formulation of quantum mechanics. The approach has been applied for manipulating the vibrational motion of HBr in its ground electronic state under an external electric field.Comment: 4 figure

Interacting classical and quantum particles

We apply Hall and Reginatto's theory of interacting classical and quantum ensembles to harmonically coupled particles, with a view to understanding its experimental implications. This hybrid theory has no free parameters and makes distinctive predictions that should allow it to be experimentally distinguished from quantum mechanics. It also bears on the questions of quantum measurement and quantum gravity.Comment: 7 pages, 6 figure

Extracting quantum dynamics from genetic learning algorithms through principal control analysis

Genetic learning algorithms are widely used to control ultrafast optical pulse shapes for photo-induced quantum control of atoms and molecules. An unresolved issue is how to use the solutions found by these algorithms to learn about the system's quantum dynamics. We propose a simple method based on covariance analysis of the control space, which can reveal the degrees of freedom in the effective control Hamiltonian. We have applied this technique to stimulated Raman scattering in liquid methanol. A simple model of two-mode stimulated Raman scattering is consistent with the results.Comment: 4 pages, 5 figures. Presented at coherent control Ringberg conference 200

Unification of the conditional probability and semiclassical interpretations for the problem of time in quantum theory

We show that the time-dependent Schr\"odinger equation (TDSE) is the phenomenological dynamical law of evolution unraveled in the classical limit from a timeless formulation in terms of probability amplitudes conditioned by the values of suitably chosen internal clock variables, thereby unifying the conditional probability interpretation (CPI) and the semiclassical approach for the problem of time in quantum theory. Our formalism stems from an exact factorization of the Hamiltonian eigenfunction of the clock plus system composite, where the clock and system factors play the role of marginal and conditional probability amplitudes, respectively. Application of the Variation Principle leads to a pair of exact coupled pseudoeigenvalue equations for these amplitudes, whose solution requires an iterative self-consistent procedure. The equation for the conditional amplitude constitutes an effective "equation of motion" for the quantum state of the system with respect to the clock variables. These coupled equations also provide a convenient framework for treating the back-reaction of the system on the clock at various levels of approximation. At the lowest level, when the WKB approximation for the marginal amplitude is appropriate, in the classical limit of the clock variables the TDSE for the system emerges as a matter of course from the conditional equation. In this connection, we provide a discussion of the characteristics required by physical systems to serve as good clocks. This development is seen to be advantageous over the original CPI and semiclassical approach since it maintains the essence of the conventional formalism of quantum mechanics, admits a transparent interpretation, avoids the use of the Born-Oppenheimer approximation, and resolves various objections raised about them.Comment: 10 pages. Typographical errors correcte

How do wave packets spread? Time evolution on Ehrenfest time scales

We derive an extension of the standard time dependent WKB theory which can be applied to propagate coherent states and other strongly localised states for long times. It allows in particular to give a uniform description of the transformation from a localised coherent state to a delocalised Lagrangian state which takes place at the Ehrenfest time. The main new ingredient is a metaplectic operator which is used to modify the initial state in a way that standard time dependent WKB can then be applied for the propagation. We give a detailed analysis of the phase space geometry underlying this construction and use this to determine the range of validity of the new method. Several examples are used to illustrate and test the scheme and two applications are discussed: (i) For scattering of a wave packet on a barrier near the critical energy we can derive uniform approximations for the transition from reflection to transmission. (ii) A wave packet propagated along a hyperbolic trajectory becomes a Lagrangian state associated with the unstable manifold at the Ehrenfest time, this is illustrated with the kicked harmonic oscillator.Comment: 30 pages, 3 figure